# Temporal Equivalence Principle (TEP) Research Programme

Author: Matthew Lukin Smawfield
ORCID: 0009-0003-8219-3159
Email: matthew@mlsmawfield.com
Main Site: https://mlsmawfield.com/
Full-Text Corpus (Markdown): https://mlsmawfield.com/llms-full.txt
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This file provides a machine-readable index of the 30-paper Temporal Equivalence Principle (TEP) physics research collection, with canonical HTML URLs, clean Markdown manuscript URLs, PDFs, Zenodo DOIs, and full abstracts for each paper.

Programme Status: Active working preprints for open scrutiny (CC BY 4.0 / MIT).

## Canonical Framework & Entity Disambiguation

- **What is the Temporal Equivalence Principle (TEP)?**
  A covariant, falsifiable bi-metric scalar-tensor reformulation of relativistic gravitation by Matthew Lukin Smawfield in which proper time is a dynamical scalar field (φ) and the speed of light (c) is an emergent, strictly local invariant rather than a global constant.
- **Two-Metric Architecture:**
  A single spacetime manifold carries a gravitational metric g_μν and a causal (matter) metric g̃_μν to which all non-gravitational fields, atomic clocks, and electromagnetic waves couple:
  - Isotropic conformal–disformal shorthand: `g̃_μν = A²(φ) g_μν + B(φ) ∇_μφ ∇_νφ`, with `A(φ) = exp(β_A φ / M_Pl)`.
  - Canonical (3+1) preferred-foliation decomposition along the unit cosmic temporal flow `U_μ ≡ -∇_μφ̄/√(2X̄)`:
    `g̃_μν = -A_loc²(u_grav(x)) U_μ U_ν + A_sp²(u_drift(t,x)) Π_μν + B(φ) Π_μ^α Π_ν^β ∇_αφ ∇_βφ`,
    separating the stationary baryon-sourced ADM clock lapse `N_TEP(x) = A_loc(u_grav(x)) N_GR(x)` from the secular matter-frame ruler factor `A_sp(u_drift(t,x)) = exp(-u_drift(t,x))` on static gravitational 3-space `γ_ij(x)`.
- **Zero-Free-Parameter Kinetic Sector & Screening (Temporal Topology / Temporal Shear):**
  `P_,X = k√ξ + 2ξ` with `k ≃ 16.03` fixed by the universal acceleration scale `a₀ ∼ cH₀` and saturation density `ρ_T ≈ 20 g/cm³`. Screens Solar-System PPN (`|γ_PPN - 1| ≃ 4.32×10⁻⁸`), Lunar Laser Ranging (`|Ġ/G| ≲ 1.7×10⁻¹⁸ yr⁻¹`), and binary-pulsar dipole radiation while producing flat galactic rotation curves (SPARC), unstretched-space gravitational lensing (`n(x) ≃ 1 - 2Φ_N/c² - β_A u_grav(x)`), and exact multi-messenger causal sharing (`c_GW = c_γ`, `Δt_γ-GW = 0` at `B = 0`).
- **Operational Invariant (Synchronization Holonomy):**
  Closed-loop time transport around a synchronization one-form `σ̃` defines a convention-independent residual holonomy `H_resid = ∮ dσ̃` after subtracting standard GR Sagnac and Shapiro terms (`H_resid = 0` in GR and in purely conformal static limits; nonzero only in the disformal/non-exact sector).
- **Cosmological Redshift on Static Space (Mount Wilson Equivalence):**
  Cosmological redshift emerges from secular matter-clock drift `1 + z = A_sp(t_o) / A_sp(t_e)` on a static spatial manifold without physical volume expansion, replacing the Isochrony Axiom while preserving standard-siren scaling `Ξ(z) = d_L^GW / d_L^EM = 1/(1+z)`.

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# Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed
**Matthew Lukin Smawfield**
Version: v0.16 (Jakarta)
First published: 18 August 2025 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.16921911

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## Abstract

This paper proposes a covariant, testable reformulation of relativity in which proper time is a dynamical field and the "speed of light" is an emergent, strictly local invariant rather than a global constant. The framework is built on a single spacetime manifold endowed with two metrics: a gravitational metric $g_{\mu\nu}$ and a causal (matter) metric $\tilde{g}_{\mu\nu}$ to which all non-gravitational fields and clocks couple. In isotropic shorthand the metrics are related by a controlled disformal map, $\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu\phi \nabla_\nu\phi$, where $\phi$ is the time field, $A(\phi) = \exp(\beta_A \phi/M_{\text{Pl}})$ is a universal conformal factor, and $B(\phi)$ encodes tiny, direction-dependent deformations of the light cone consistent with GW170817-class multi-messenger constraints ($|c_\gamma - c_g|/c \lesssim \text{few}\times10^{-15}$ today); decomposed along the unit cosmic temporal flow $U_\mu \equiv -\nabla_\mu\bar{\phi}/\sqrt{2\bar{X}}$ ($g^{\mu\nu}U_\mu U_\nu = -1$, orthogonal spatial projector $\Pi_{\mu\nu} = g_{\mu\nu} + U_\mu U_\nu$), this resolves into $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav})\,U_\mu U_\nu$ $+$ $A_{\rm sp}^2(u_{\rm drift})\,\Pi_{\mu\nu}$ $+$ $B(\phi)\,\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$, separating the stationary baryon-sourced ADM lapse $N_{\rm TEP}(\mathbf{x}) = A_{\rm loc}(u_{\rm grav}(\mathbf{x}))N_{\rm GR}(\mathbf{x})$ from the secular matter-frame spatial ruler factor $A_{\rm sp}(u_{\rm drift}(t,\mathbf{x})) = \exp(-u_{\rm drift}(t,\mathbf{x}))$ on static gravitational 3-space $\gamma_{ij}(\mathbf{x})$. Proper time is elevated to a field by postulating that all matter, electromagnetism, and quantum phases evolve with respect to $\tilde{g}$-proper time $\tau$; in local freely falling frames, this guarantees exact local Lorentz invariance and a locally invariant $c$, while globally it implies that synchronization procedures and one-way light-time measurements can become path-dependent in the disformal/non-exact sector of a dynamical-time background. The covariant action, field equations, conservation laws, and PPN mapping are developed; screening is formulated via continuous Temporal Topology and nonlinear Temporal Shear ($P_{,X} = k\sqrt{\xi} + 2\xi$, $k\simeq 16$ fixed by $a_0$). The breakdown of global simultaneity is formalized using a synchronization-transport law, deriving a convention-independent "synchronization holonomy," an invariant measure of non-integrability of time transport around closed loops. In the purely conformal subclass this holonomy vanishes after subtraction of the full GR, kinematic, clock-scale, and reference-frame synchronization model; nonzero holonomy at leading order requires residual non-exact synchronization structure, supplied in the minimal TEP model by disformal coupling $B(\phi)\neq0$, and in more general extensions by non-metricity or other explicitly non-exact transport structure. Explicit small-$B$ formulas are provided for the holonomy and the effective photon phase speed, showing how the measured one-way asymmetry is related to $\phi$-gradients and disformal scales under current constraints. The analysis demonstrates that Einstein's assumption of a universal $c$ was a brilliant local theorem arising from the Temporal Equivalence Principle; transcending it demands dynamical time: $c$ remains exactly invariant locally, but global, one-way-inferred "$c$" values differ by path-dependent amounts that experiments can detect or bound. Cosmologically, TEP adopts an eternal, spatially infinite, inhomogeneous background whose physical volume does not undergo cosmological expansion. Neither space nor matter changes size. The matter unit of time drifts relative to the static gravitational geometry ($\gamma_{ij}$), and because $c$ is locally invariant, the matter unit of length drifts with it ($A_{\rm sp}$). Distances and densities expressed in matter units therefore evolve without any physical stretching, allowing the observed redshift and apparent Hubble relation to emerge directly from the macroscopic historical drift of the inhomogeneous temporal network over coordinate time, governed principally by the conformal endpoint ratio $1+z = A_{\rm sp}(t_o)/A_{\rm sp}(t_e)$, with constrained non-exact propagation corrections through the temporal topology. An exactly solved closed-static branch establishes that the specified action admits a static, eternal gravitational configuration with conformal unit drift; that branch is an analytical benchmark, neither an attractor nor capable of near-flat matter geometry. Numerical noncompact solutions now demonstrate a constraint-consistent, on-shell inhomogeneous temporal landscape with near-zero native matter-frame spatial curvature ($|\Omega_k^{\rm native}| = 1.40\times 10^{-4}$), sustained secular temporal roll, and a positive-definite linear fluctuation spectrum on the gauge-constrained physical subspace ($\lambda_1 = +38.96$ at $N=24$ and $+39.06$ at $N=32$ on the canonical branch; $\lambda_1 = +555.35$, $\omega_1 = 23.57$ at $N=24$ and $+551.66$ at $N=32$ on the selected two-branch kinetic sector, supplemented by an independent discrete lower bound on the ambient branch, complete matter-coupling second-variation positivity $\lambda_0 = +0.4123\text{--}0.4127 > 0$ and joint on-shell quartic positivity $\lambda_0 = +0.5624\text{--}0.5633 > 0$ without homogeneous-mode projection, an all-twist Bloch lower bound $\lambda \ge \min m_{\rm eff}^2 > 0$, and a positive evolving constant-mean-curvature initial-slice lapse solution); the remaining cosmological closure is quantitative: fixing the universal master-potential branch such that the evolved temporal history reproduces the observed redshift–distance relation, horizon behaviour, and perturbation observables without imposing the empirical clock map. A candidate early-universe closure, in which the Big Bang gravitational singularity is replaced by a matter-frame temporal boundary without phenomenological thermal screening, is developed through the companion transport analyses (TEP-TH, TEP-BBN, TEP-HC), whose conformal temporal geometry preserves the observed CMB acoustic structure. Known weaknesses in the variable-$c$ literature are addressed by supplying a correct, operationally invariant observable (holonomy), clarifying when conformal couplings cannot produce a signal, and providing realistic, constraint-consistent benchmark sensitivity windows with explicit error budgets and statistical plans (pre-registration, blinding, publicly released code and data). The resulting theory preserves the empirical pillars of relativity (local Lorentz invariance, gravitational-wave causality, PPN bounds) while extending its conceptual foundation: simultaneity is not only relative but generally non-integrable; the speed of light is not a global constant but the local echo of a deeper, dynamical temporal geometry.

Long-standing confusions about "variable $c$" are resolved by replacing convention-dependent statements with invariant observables tied to measurement procedures. A synchronization one-form $\tilde{\sigma}$ is defined on spacelike slices of the matter metric; its curl $d\tilde{\sigma}$, after subtraction of the full GR, kinematic, clock-scale, and reference-frame synchronization model, yields a residual "temporal holonomy" $H_{\rm resid}$ that vanishes in GR and becomes nonzero only when time is dynamical in this sense. Embedding the preferred-foliation geometry in a spatially covariant ADM/khronon effective action establishes four key results: (i) conformal matter coupling preserves null cones, and electromagnetic waves and transverse-traceless gravitons share the identical principal wave operator $\Box_{\rm TEP} \equiv -(N_{\rm TEP}c)^{-2}\partial_t^2$ $+$ $A_{\rm sp}^{-2}(t,\mathbf{x})\gamma^{kl}\partial_k\partial_l$, enforcing $c_{\rm GW}(t,\mathbf{x}) = c_\gamma(t,\mathbf{x})$ at every event ($\Delta t_{\gamma\text{--}{\rm GW}} \equiv 0$ at $B=0$, satisfying GW170817); (ii) the static-space WKB tensor measure $\sqrt{A_{\rm sp}^{+1}\cdot A_{\rm sp}^{-1}} \equiv 1$ eliminates FLRW Hubble friction, preserving $1/r$ graviton decay, the standard-siren discriminator $\Xi(z) = d_L^{\rm GW}/d_L^{\rm EM} = 1/(1+z)$, and the inverse detector chirp-mass map $\mathcal{M}_{\rm det} = \tilde{\mathcal{M}}/(1+z)$ at pinned interior response; (iii) a static $\phi$-gradient produces no direct propagation asymmetry in the static purely conformal limit—the cancellation is exact, not merely first-order; and (iv) the selected two-branch kinetic sector ($P_{,X} = k\sqrt{\xi} + 2\xi$, $k\simeq 16$ derived from $a_0 \sim cH_0$) screens Solar-System post-Newtonian observables ($|\gamma_{\rm PPN}-1|\simeq 4.32\times 10^{-8}$ at $1.6\,R_\odot$), lunar laser ranging ($|\dot{G}_{\rm loc}/G_{\rm loc}|_{\rm LLR}\lesssim 1.7\times 10^{-18}\,{\rm yr}^{-1}$), and binary-pulsar dipole radiation well inside observational bounds while passing the wide-binary joint test at $0.4\text{--}1.7\sigma$ with zero per-system free parameters and generating flat galactic rotation curves and unstretched-space gravitational lensing via $A_{\rm loc}(u_{\rm grav})$. Disformal tilts ($B \neq 0$) can source holonomy subject to multi-messenger constraints. A corrected Earth–Sun triangle calculation gives a converged conditional benchmark $|H_{\rm resid}|\simeq4.33\times10^{-13}$ s per loop, with zero circulation in the exact-connection control. The normalization and environmental response remain specified benchmark inputs; an absolute TEP prediction requires their derivation from the common action and field solution. The effective covariant architecture is presented; field equations, conservation laws, invertibility/causality conditions, and a 3+1 decomposition are derived to make the observables explicit. Screening via a continuous Temporal Topology governed by non-linear superposition of field gradients (Temporal Shear) reconciles precision local tests with cosmological evolution, with mapping to Parametrized Post-Newtonian parameters and to the EFT-of-dark-energy $\alpha$-functions with $c_T = 1$ enforced. Decisive experiments with quantitative error budgets are outlined: (1) a ground–ground–satellite triangle time-transfer experiment targeting holonomy at below $10^{-18}$ fractional after GR subtraction; (2) portable-clock "clock anholonomy" around closed paths at the $10^{-19}$ level over days; (3) multi-species clock networks seeking phase-locked annual modulations at $10^{-19}$–$10^{-17}$; (4) interplanetary one-way optical links at picoseconds over AU; (5) altitude-dependent screening maps with optical clocks and atom interferometers; and (6) ensemble multi-messenger tests. Cosmological inference is implemented through native hi_class and Cobaya analyses (Papers 18, 26), with commitment to open data and blinded analyses.

Einstein's postulate of universal $c$ was a brilliant, operationally perfect approximation in regimes where time's flow is effectively uniform. In a universe where the rate of time is dynamical yet locally Lorentzian, "the speed of light" emerges as an invariant in every local lab but ceases to be globally universal. The new invariant content resides in path-dependent synchronization defects and holonomies of time transport, not in naive one-way "speeds." If detected, these invariants would inaugurate a post-Einsteinian era: from dynamic geometry to dynamic time.

## 1. Introduction: From a Universal Speed to a Universal Principle of Time

Relativity and quantum theory disagree about time. General relativity (GR) makes proper time geometric—$d\tau^2 = -g_{\mu\nu} dx^\mu dx^\nu / c^2$—dynamical, and observer-dependent. Quantum mechanics (QM) treats time as an external parameter t in the Schrödinger equation, $i\hbar \partial_t|\psi\rangle = \hat{H}|\psi\rangle$. The clash has haunted quantization programs for a century and manifested operationally as subtle ambiguities in defining one-way light speeds and simultaneity across extended regions. Precision clocks and time-transfer links now measure gravitational redshifts and velocity-dependent dilations at $10^{-18}$ fractional levels, while cosmology exhibits persistent $H_0$ and $S_8$ tensions. In this context, Einstein's postulate of a universal speed c must be sharpened: it is exact for any local freely falling laboratory, but it cannot be a global property if the flow of time itself is dynamical.

The Temporal Equivalence Principle (TEP) is proposed: all non-gravitational dynamics, signals, and quantum phases evolve according to the proper time defined by a single causal metric $\tilde{g}_{\mu\nu}$ that couples universally to matter. The rate at which proper time accrues is a field. This elevates "when" to the status "where" acquired in 1915: as space was geometrized, now time's rate is dynamized. The consequence is that synchronization conventions can become globally non-integrable when the dynamical-time background contains a disformal or otherwise non-exact transport component: even after removing known GR effects, closed-loop time transport can then retain a residual path-dependent offset. The measured "speed of light" between distant clocks is revealed as an emergent ratio of distance to accrued proper time, not a fundamental constant comparable across regions without a theory of time's flow. Locally, c remains exactly invariant; globally, dynamical time imprints tiny, path-dependent asymmetries that can be detected or bounded.

![TEP architecture: g vs g̃ with φ and conformal–disformal map](site/figures/fig1_architecture.svg)

Figure 1. Temporal Equivalence Principle (TEP) architecture. Matter couples to $\tilde{g}_{\mu\nu}$ via a conformal–disformal map controlled by the scalar time field $\phi$ (shown in both isotropic shorthand and the resolved preferred-foliation $(A_{\rm loc}, A_{\rm sp})$ decomposition of §2.2), preserving local Lorentz invariance while enabling new global invariants.

## 2. Axioms and the Temporal Equivalence Principle

The framework adopts four axioms:

- **A1. Two-metric structure on a single manifold.** Gravity is described by a Lorentzian metric $g_{\mu\nu}$; matter fields, clocks, and rulers couple to a causal (matter) metric $\tilde{g}_{\mu\nu}$. Decomposing the scalar time field along the cosmic temporal flow $U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$ ($g^{\mu\nu}U_\mu U_\nu=-1$, orthogonal spatial projector $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$) into the stationary baryon-sourced temporal-depth perturbation $u_{\rm grav}(\mathbf x)$ and the secular ruler-drift sector $u_{\rm drift}(t,\mathbf x)$, the metrics are related by the single unified map
$$\tilde{g}_{\mu\nu} = -A_{\rm loc}^2\bigl(u_{\rm grav}(\mathbf x)\bigr)\,U_\mu U_\nu + A_{\rm sp}^2\bigl(u_{\rm drift}(t,\mathbf x)\bigr)\,\Pi_{\mu\nu} + B(\phi)\,\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi,$$
with local lapse factor $A_{\rm loc}(u_{\rm grav})=\exp(\beta_A u_{\rm grav})=\exp(-u_{\rm grav}(\mathbf x))$ ($\beta_A=-1$, so $N_{\rm TEP}(\mathbf x)=A_{\rm loc}(u_{\rm grav}(\mathbf x))N_{\rm GR}(\mathbf x)$), matter-frame spatial ruler factor $A_{\rm sp}(u_{\rm drift})=\exp(\beta_A u_{\rm drift})=\exp(-u_{\rm drift}(t,\mathbf x))$ on static gravitational 3-space $\gamma_{ij}(\mathbf x)$ ($d\tilde s^2=-N_{\rm TEP}^2(\mathbf x)c^2dt^2$ $+$ $A_{\rm sp}^2(t,\mathbf x)\gamma_{ij}(\mathbf x)dx^i dx^j$), and a small spatial disformal function $B(\phi)$ consistent with multi-messenger constraints today. On the homogeneous IGM background ($u_{\rm grav}=0$, $\Pi_\mu^\alpha\nabla_\alpha\phi=0$, $\dot u_{\rm drift}=-H_{\rm drift}<0$) this governs cosmological ruler drift $1+z=A_{\rm sp}(t_o)/A_{\rm sp}(t_e)$; at a fixed cosmic epoch inside a screened system ($A_{\rm sp}(t_0)=1$) it reduces identically to the unstretched-space temporal metric $\tilde g_{\mu\nu}=g_{\mu\nu}-(A_{\rm loc}^2(u_{\rm grav})-1)U_\mu U_\nu$ $+$ $B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ with $\tilde\gamma_{ij}=\gamma_{ij}=(1-2\Phi_N/c^2)\delta_{ij}$.

- **A2. Temporal Equivalence Principle (TEP).** All non-gravitational processes evolve according to proper time $d\tau$ defined by $\tilde{g}_{\mu\nu}$: $i\hbar d|\psi\rangle/d\tau = \hat{H}|\psi\rangle$, nongravitational fields propagate on $\tilde{g}$-null cones, and ideal clocks tick $d\tau^2 = -\tilde{g}_{\mu\nu} dx^\mu dx^\nu/c^2$. In local freely falling frames for $\tilde{g}_{\mu\nu}$, physics reduces to special relativity with invariant c.

- **A3. Causal safety and locality.** Today, $c_g = c_\gamma$ within current bounds ($|c_g - c_\gamma|/c \lesssim \text{few}\times10^{-15}$). This is enforced by choosing $A(\phi)$ universal, such that conformal transformations preserve null cones for photons and gravitons when $B = 0$, and by constraining the observable combination $B(\phi)(\partial\phi)^2$ along late-time astrophysical propagation paths. Different experiments constrain different integrals or responses involving the single disformal function $B(\phi)$; GW170817 does not require $B$ to vanish identically in all regimes. Hyperbolicity holds within the EFT domain. On the admissible kinetic branch ($P_X > 0$, $P_X + 2XP_{,XX} > 0$) the scalar stress tensor obeys the null energy condition pointwise, $T_{\mu\nu}k^\mu k^\nu = P_X(k\cdot\nabla\phi)^2 \geq 0$; the weak and dominant conditions are violated in the void sector of the eternal static solution, where the integrability condition of Section 8 requires a negative cell-averaged effective energy density supplied by the potential rather than by a wrong-sign kinetic term (Appendix E, step_20: void-sector mean $-0.58$, cell mean $-0.52$ in slice units).

- **A4. Screening and universality.** The coupling $A(\phi)$ is universal at leading order; loop-induced composition dependence is tightly bounded. Environmental suppression of the locally observable Temporal Shear/source-charge sector reconciles local tests with cosmological dynamics. Chameleon, Vainshtein, Galileon, DBI, and symmetron mechanisms are treated as candidate microscopic completions, not as the defining ontology of TEP. Screening manifests as a continuous spatial and covariance structure of the scalar time field (Temporal Topology) governed by the temporal potential field and its gradient (Temporal Shear), suppressing geometric deviations in screened regimes while leaving cosmology accessible to dynamics.

These axioms encode Einstein's local invariance as a theorem and extend it: c is exactly invariant for every tangent space of $\tilde{g}_{\mu\nu}$, but global synchronization and one-way timing depend on the dynamical field $\phi$.

The scalar time field $\phi$ is defined on the spacetime manifold of Axiom A1 and is present at every event. The ambient cosmological value $\phi_\infty = 0$ is a boundary condition, not an absence of the field — and a convention fixed at the present epoch, not an eternal pinning: past ambient values are set by the solved landscape and remain within the fixed-time spatial contrast $|\delta u| \lesssim 10^{-3}$–$10^{-2}$ (Section 8), since no source term drives the unsourced void sector secularly. What distinguishes dense from dilute environments is not the presence or absence of $\phi$ but the matter trace $T^{(\rm m)}$ and the environmental state vector $\mathcal{E}$ (ambient density, compactness, gradients, boundary geometry). The term "vacuum" is retained in its standard field-theory sense — the $T^{(\rm m)} = 0$ limit of the matter sector — and does not denote a region devoid of the temporal field.

Environmental screening $\mathcal{S}_\Sigma(\mathcal{E})$ suppresses the conformal gradient $\Sigma_\mu = \nabla_\mu \ln A$. In TEP, this spatial gradient—measuring how clock tick-rates slope from one point to another—is formally named the Temporal Shear. It governs the geometric refraction of matter: when the slope is steep, matter geodesics bend; when it is flat ($\Sigma_\mu \to 0$), matter follows standard General Relativity paths. The Temporal Shear is suppressed via temporal-topology pinning: the environmental-state operator $\mathcal{S}_\Sigma(\mathcal{E})$ acts on the solved environmental state — set by source density, compactness, boundary geometry, and ambient field — and its derived shear branch pins the incremental gradient where the ambient time-gradient is steep (in weak-field terms, $g \gg g_t \approx cH_0/2$; Section 7). It does not suppress the disformal sector $B(\phi)$, which is governed by its own field-space envelope (Section 2.2). The two are distinct mechanisms operating on distinct sectors of the disformal map. The amplitude/shear split ($\mathcal{S}_\Sigma \to 0$, $S_A \sim 1$) suppresses the spatial gradient while preserving the field value: $A(\phi)$ continues to vary inside screened matter, which is why clocks run slower in deeper wells. What is pinned is the gradient, not the field amplitude. Theorem 2 (Section 6) further establishes that a static conformal gradient produces no directional photon-propagation asymmetry; direction-dependent null propagation requires the disformal sector, time-dependent geometry, or another non-exact transport contribution.

## Sector mapping

| Sector | Physical effect | Coupling | Key bound |
| --- | --- | --- | --- |
| **Temporal lapse & ruler factors $(A_{\rm loc}, A_{\rm sp})$** | Clock rates, proper-time rescaling, cosmological ruler drift ($A_{\rm sp}$), and reciprocity-even temporal-lapse optical refraction when $A_{\rm loc}\neq A_{\rm sp}$ on unstretched $\gamma_{ij}$ | Universal ($\beta_A$) | Cassini PPN-$\gamma$ (via screened $S_\Sigma$ source charge), redshift tests; $c_{\rm GW}=c_\gamma$ via shared $\Box_{\rm TEP}$ (§2.2) |
| **Transverse disformal $B(\phi)$** | GW–EM cone splits, direction-dependent propagation, and closed-loop holonomy | Small, environment-dependent | GW170817: $|c_\gamma - c_g|/c \lesssim \text{few}\times10^{-15}$ |
| **Temporal Topology** | Spatial/covariance structure of $\ln A(\phi)$ | Field configuration + gradient suppression | Clock/covariance correlation length $\lambda_T$ |

In the isotropic shorthand ($A_{\rm loc}=A_{\rm sp}=A$), the conformal factor rescales clock rates and spatial rulers equally and cancels out of null trajectories; in the resolved preferred-foliation metric ($A_{\rm loc}\neq A_{\rm sp}$ on unstretched $\gamma_{ij}$), $A_{\rm loc}(u_{\rm grav})$ also generates reciprocity-even temporal-lapse refraction while preserving $c_{\rm GW}=c_\gamma$ on the shared wave operator $\Box_{\rm TEP}$ (§2.2). The transverse disformal sector $B(\phi)$ governs differential GW–EM cone tilts and closed-loop synchronization holonomy. GW170817 directly bounds differential GW–EM cone splits governed by $B$; it does not directly constrain common-mode clock-rate structure or shared-cone temporal-lapse refraction, although local gradients and source charges remain constrained by PPN, gravitational-redshift, clock-comparison, and equivalence-principle tests.

## 2.1 Relation to Other Modified Gravity Frameworks

The TEP framework can be situated within the broader landscape of modified gravity and scalar-tensor theories. To clarify its unique features, a comparative analysis is provided:

Table 1: Comparison with Other Scalar-Tensor Frameworks

| Theory/Framework | Key Fields | Matter Coupling | $c_g = c_γ$? | Key Observable Signature |
| --- | --- | --- | --- | --- |
| **TEP (This work)** | $g_{\mu\nu}$, $\phi$ | Universal to $\tilde{g}_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B\partial_\mu\phi\partial_\nu\phi$ | Equal in conformal/common-mode limit; differential deviations possible through $B$ and constrained by same-path EM–GW timing | Synchronization Holonomy $H_{\rm resid}$ |
| **Horndeski** | $g_{\mu\nu}$, $\phi$ | Minimal to $g_{\mu\nu}$ | Yes, after GW170817 constraints | Modified growth, ISW |
| **DHOST** | $g_{\mu\nu}$, $\phi$ | Minimal to $g_{\mu\nu}$ | Yes, for specific degenerate classes | Modified growth, screening |
| **TeVeS** | $g_{\mu\nu}$, $A_\mu$, $\phi$ | To a combined metric | No | MOND phenomenology, lensing |
| **SMEFT** | SM fields | Lorentz-violating operators | Model-dependent | Anisotropic propagation, CPT violation |

The multi-messenger constraint from GW170817 constrains the integrated EM–GW differential disformal contribution along observed late-time astrophysical paths. It does not require the disformal sector to vanish identically in all regimes. In the conformal limit, common-mode path effects cancel in EM–GW timing; residual $B$-sector structure remains testable through multipath delays, closed-loop holonomy, and topological or interferometric regimes. The model, with a universal conformal coupling $A(\phi)$ and a disformal term $B(\phi)$, automatically satisfies $c_g = c_γ$ in the conformal limit ($B=0$). A non-zero $B(\phi)$ introduces a calculable deviation $c_γ ≠ c_g$ constrained by future multi-messenger observations.

Conceptually, TEP differs from many scalar-tensor theories by its foundational principle: the universal coupling of matter to a single metric $\tilde{g}_{\mu\nu}$. This is philosophically closer to Bekenstein's TeVeS theory, which also introduced a separate matter metric, but the framework is more minimal, using only a scalar, and makes a distinctive operational prediction in the form of the synchronization holonomy.

## 2.2 Minimal covariant action

The displayed low-curvature action below represents the leading sector of the master TEP Effective Field Theory. The strong-curvature completion includes higher-order curvature operators (such as the scalar-Gauss-Bonnet coupling $\alpha_{\rm GB} f(\phi)\mathcal{G}$) which are negligible in the weak-field and cosmological regimes studied here, but become active in the strong-field regime to supply real backreaction (as detailed in TEP-BH, Paper 28). The unified action, including the strong-curvature sector, is

$$S = S_{\rm grav}[g_{\mu\nu},\phi] + \int d^4x\sqrt{-g} \left[ P(X,\phi) +\alpha_{\rm GB}\,f(\phi)\,\mathcal{G} \right] + S_m[\psi_i,\tilde g_{\mu\nu}],$$

with the canonical causal (matter) metric

$$\tilde g_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi .$$

Decomposed along the unit cosmic temporal flow $U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$ ($g^{\mu\nu}U_\mu U_\nu=-1$, orthogonal spatial projector $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$, with stationary baryon-sourced temporal-depth perturbation $u_{\rm grav}(\mathbf x)$, local lapse factor $A_{\rm loc}(u_{\rm grav})=\exp(\beta_A u_{\rm grav})=\exp(-u_{\rm grav}(\mathbf x))$, and matter-frame spatial ruler factor $A_{\rm sp}(u_{\rm drift})=\exp(\beta_A u_{\rm drift})=\exp(-u_{\rm drift}(t,\mathbf x))$ with $\dot u_{\rm drift}=-H_{\rm drift}<0$), the longitudinal projection along $U_\mu U_\nu=\nabla_\mu\bar\phi\,\nabla_\nu\bar\phi/(2\bar X)$ separates the stationary gravitational/scalar ADM lapse $N_{\rm TEP}(\mathbf x)=A_{\rm loc}(u_{\rm grav}(\mathbf x))N_{\rm GR}(\mathbf x)$ from the matter-frame ruler-drift factor $A_{\rm sp}(t,\mathbf x)$ on static gravitational 3-space $\gamma_{ij}(\mathbf x)$. Substituting $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$ exhibits this preferred-foliation realization explicitly as an exact member of the covariant conformal–disformal class with longitudinal disformal weight $B_\parallel\equiv\bigl(A_{\rm sp}^2(u_{\rm drift})-A_{\rm loc}^2(u_{\rm grav})\bigr)/(2\bar X)$ along $\nabla_\mu\bar\phi\,\nabla_\nu\bar\phi$ and transverse spatial disformal weight $B(\phi)$:

$$\begin{aligned} \tilde g_{\mu\nu} &= -A_{\rm loc}^2\bigl(u_{\rm grav}(\mathbf x)\bigr)\,U_\mu U_\nu + A_{\rm sp}^2\bigl(u_{\rm drift}(t,\mathbf x)\bigr)\,\Pi_{\mu\nu} + B(\phi)\,\Pi_\mu^\alpha\Pi_\nu^\beta\,\nabla_\alpha\phi\,\nabla_\beta\phi \\ &= A_{\rm sp}^2\bigl(u_{\rm drift}\bigr)\,g_{\mu\nu} + \frac{A_{\rm sp}^2(u_{\rm drift})-A_{\rm loc}^2(u_{\rm grav})}{2\bar X}\,\nabla_\mu\bar\phi\,\nabla_\nu\bar\phi + B(\phi)\,\Pi_\mu^\alpha\Pi_\nu^\beta\,\nabla_\alpha\phi\,\nabla_\beta\phi , \end{aligned}$$

with $(3+1)$ ADM line element $d\tilde s^2=-N_{\rm TEP}^2(\mathbf x)c^2dt^2$ $+$ $A_{\rm sp}^2(t,\mathbf x)\gamma_{ij}(\mathbf x)dx^i dx^j$ at $B=0$ ($\sqrt{-\tilde g}=A_{\rm loc}(u_{\rm grav})A_{\rm sp}^3(u_{\rm drift})\sqrt{-g}$; the projected spatial disformal term contributes an additional factor $\sqrt{1+B(\Pi\nabla\phi)^2/A_{\rm sp}^2}$). This single covariant metric governs both cosmological and astrophysical regimes across the entire corpus without ambiguity. On the homogeneous intergalactic background ($u_{\rm grav}=0$, $N_{\rm TEP}=1$, $\Pi_\mu^\alpha\nabla_\alpha\phi=0$, where $u_{\rm drift}$ rolls from $\ln(1+z)>0$ at emission to $0$ today at rate $\dot u_{\rm drift}=-H_{\rm drift}<0$), $A_{\rm sp}(t)=\exp(\beta_A u_{\rm drift}(t))=\exp(-u_{\rm drift}(t))=(1+z)^{-1}$ measures how many matter-frame Bohr radii ($\ell_{\rm unit}^{\rm coord}=\ell_0/A_{\rm sp}(t)=\exp(+u_{\rm drift})\ell_0$ in static gravitational coordinates — a conformal recalibration of matter-frame units across epoch, not a mechanical contraction of atoms) fit across a fixed static coordinate interval $dx$ ($d\tilde\ell=A_{\rm sp}(t)dx$), producing the cosmological redshift $1+z=A_{\rm sp}(t_o)/A_{\rm sp}(t_e)=A_o/A_e$, source area distance $D_A=A_e r$, and the standard-siren relation $\Xi(z)=d_L^{\rm GW}/d_L^{\rm EM}=1/(1+z)$ (Section 8). For inhomogeneous spatial structures $u_{\rm grav}(\mathbf x)=\delta\phi(\mathbf x)/M_{\rm Pl}$ at a fixed cosmic epoch ($A_{\rm sp}(t_0)=1$), it reduces identically to the unstretched-space temporal metric $\tilde g_{\mu\nu}=g_{\mu\nu}-(A_{\rm loc}^2(u_{\rm grav})-1)U_\mu U_\nu$ $+$ $B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$, where spatial rulers $\tilde\gamma_{ij}=\gamma_{ij}=(1-2\Phi_N/c^2)\delta_{ij}$ remain unstretched while the $(3+1)$ ADM spacetime lapse rescales as $N_{\rm TEP}(\mathbf x)=A_{\rm loc}(u_{\rm grav}(\mathbf x))N_{\rm GR}(\mathbf x)$ ($\sqrt{-\tilde g}=A_{\rm loc}(u_{\rm grav})\sqrt{-g}$).

### Preferred-foliation (khronon) EFT realization and shared wave operator

Because the scalar time field possesses an everywhere-timelike cosmic gradient ($X \equiv -\tfrac{1}{2}g^{\mu\nu}\nabla_\mu\phi\nabla_\nu\phi > 0$), the unit four-velocity $U_\mu \equiv -\nabla_\mu\phi/\sqrt{2X}$ ($g^{\mu\nu}U_\mu U_\nu = -1$) spontaneously selects a preferred cosmological temporal foliation. This places the gravitational sector naturally within the spatially covariant ADM, khronon (Blas, Pujolàs & Sibiryakov 2010), and degenerate higher-order scalar–tensor (DHOST) effective field theory framework. Crucially, this global foliation selection does not violate local Lorentz invariance in the matter sector: because all non-gravitational fields couple universally to the single effective metric $\tilde{g}_{\mu\nu}$ (Axiom A2), boost invariance is exact in every local $\tilde{g}_{\mu\nu}$ freely falling frame, while the cosmic rest frame $U_\mu$ governs the separation between temporal lapse depth and spatial ruler drift on macroscopic scales. On the $U_\mu$ foliation over static gravitational space $\gamma_{ij}(\mathbf x)$ with strictly constant $M_{\rm Pl}$, the gravitational sector is the spatially covariant ADM/DHOST action

$$\begin{aligned} S_{\rm grav} &= \frac{M_{\rm Pl}^2}{2}\int d^3x\,dt\,\sqrt{\gamma}\,N_{\rm GR} \left[ R_{\rm scalar}^{(3)}(\gamma) + A_{\rm sp}^{-1}(t,\mathbf x)\,R_{\rm TT}^{(3)}(\gamma) + \frac{A_{\rm sp}(t,\mathbf x)}{A_{\rm loc}^2(u_{\rm grav})}\left(K_{ij}K^{ij}-K^2\right) \right] \\ &= \frac{M_{\rm Pl}^2}{2}\int d^3x\,dt\,\sqrt{\gamma}\,N_{\rm GR} \left[ A_{\rm sp}^{-1}(t,\mathbf x)\,R^{(3)}(\gamma) + 2\bigl(A_{\rm sp}^{-1}(t,\mathbf x)-1\bigr)\gamma^{ij}D_i\ln N_{\rm GR}\,D_j\ln N_{\rm GR} + \tilde K_{ij}\tilde K^{ij}-\tilde K^2 \right], \end{aligned}$$

with physical extrinsic curvature $\tilde K_{ij}\equiv A_{\rm sp}^{1/2}(t,\mathbf x)(2N_{\rm TEP}c)^{-1}\partial_t\gamma_{ij}$ $=$ $A_{\rm sp}^{1/2}(t,\mathbf x)A_{\rm loc}(u_{\rm grav})^{-1}K_{ij}$ (Appendix E, R19; `step_80_shared_lapse_action_derivation.py`). Here $R_{\rm scalar}^{(3)}(\gamma)$ and $R_{\rm TT}^{(3)}(\gamma)$ are defined explicitly as the quadratic scalar–vector–tensor (SVT) curvature pieces of the spatial 3-metric $\gamma_{ij}=(1-2\Phi_N/c^2)\delta_{ij}+h_{ij}^{\rm TT}$ around the static background: expanding $\sqrt{\gamma}\,N_{\rm GR}R^{(3)}(\gamma)$ to second order in perturbations (with $N_{\rm GR}=1+\Phi_N/c^2$ and total spatial divergences integrated by parts) yields $\sqrt{\gamma}\,N_{\rm GR}R_{\rm scalar}^{(3)}(\gamma)=-2(\nabla\Phi_N/c^2)^2$ (corresponding to $R_{\rm scalar}^{(3)}=(4/c^2)\nabla^2\Phi_N-(2/c^4)(\nabla\Phi_N)^2$) and $\sqrt{\gamma}\,N_{\rm GR}R_{\rm TT}^{(3)}(\gamma)=-\tfrac14\gamma^{kl}\partial_k h_{ij}^{\rm TT}\partial_l h_{ij}^{\rm TT}$. As shown by the second equality above, this quadratic SVT effective action admits a strictly local, covariant khronon/preferred-foliation representation (Blas, Pujolàs & Sibiryakov 2010): because the unit timelike gradient $U_\mu\equiv-\nabla_\mu\phi/\sqrt{2X}$ defines the local spatial acceleration 3-vector $a_i\equiv U^\nu\nabla_\nu U_i=D_i\ln N_{\rm GR}=\partial_i\Phi_N/c^2$, the longitudinal scalar potential piece is generated locally by $a_i a^i=\gamma^{ij}D_i\ln N_{\rm GR}\,D_j\ln N_{\rm GR}=(\nabla\Phi_N/c^2)^2$, reproducing $R_{\rm scalar}^{(3)}(\gamma)+A_{\rm sp}^{-1}R_{\rm TT}^{(3)}(\gamma)$ identically at quadratic order without solving a non-local Poisson inverse Laplacian.

Because the longitudinal elliptic spatial 3-curvature $R_{\rm scalar}^{(3)}(\gamma)$ is weighted by strictly constant $M_{\rm Pl}^2 N_{\rm GR}$ rather than $N_{\rm TEP}$ (unlike a trivial Jordan-frame action $\int\sqrt{-\tilde g}R[\tilde g]$, which would eliminate the scalar fifth force), $\delta S_{\rm grav}/\delta u_{\rm grav}\equiv 0$ and $\delta S_{\rm grav}/\delta A_{\rm sp}\equiv 0$ in the quasistatic limit ($K_{ij}\to 0$, $h_{ij}^{\rm TT}=0$), so varying $S_{\rm grav}+S_\phi+S_m$ preserves the Newtonian Poisson equation $\nabla^2\Phi_N=\rho_*/(2M_{\rm Pl}^2)$ and the scalar source equation $\nabla\cdot(P_{,X}\nabla\phi)=V_{,\phi}+\rho_*A_{,\phi}$ identically. Simultaneously, varying $S_{\rm grav}$ with respect to transverse-traceless perturbations $h_{ij}^{\rm TT}$ of $\gamma_{ij}$ and varying the Maxwell action $S_{\rm EM}[\tilde g]=-\tfrac14\int d^4x\sqrt{-\tilde g}\,\tilde g^{\mu\alpha}\tilde g^{\nu\beta}F_{\mu\nu}F_{\alpha\beta}$ with respect to the transverse vector potential $A_i^{\rm T}$ yields the quadratic wave actions

$$\begin{aligned} S_{\rm TT}^{(2)} &= \frac{M_{\rm Pl}^2}{8}\int d^3x\,dt\sqrt{\gamma}\,N_{\rm GR}\,A_{\rm sp}(t,\mathbf x) \left[ \frac{(\partial_t h_{ij}^{\rm TT})^2}{N_{\rm TEP}^2(\mathbf x)c^2} -\frac{1}{A_{\rm sp}^2(t,\mathbf x)}\gamma^{kl}\partial_k h_{ij}^{\rm TT}\partial_l h_{ij}^{\rm TT} \right], \\ S_{\rm EM}^{(2)} &= \frac{1}{2}\int d^3x\,dt\sqrt{\gamma}\,N_{\rm TEP}\,A_{\rm sp}(t,\mathbf x) \left[ \frac{(\partial_t A_i^{\rm T})^2}{N_{\rm TEP}^2(\mathbf x)c^2} -\frac{1}{A_{\rm sp}^2(t,\mathbf x)}\gamma^{kl}\partial_k A_i^{\rm T}\partial_l A_i^{\rm T} \right], \end{aligned}$$

whose Euler–Lagrange equations share the identical principal wave operator $\Box_{\rm TEP}\equiv-(N_{\rm TEP}c)^{-2}\partial_t^2+A_{\rm sp}^{-2}(t,\mathbf x)\gamma^{kl}\partial_k\partial_l$, coordinate phase speed $c_{\rm GW}(t,\mathbf x)=c_\gamma(t,\mathbf x)$ $=$ $c\,N_{\rm TEP}(\mathbf x)A_{\rm sp}^{-1}(t,\mathbf x)\gamma_{ll}^{-1/2}$ at every event $(t,\mathbf x)$ ($\Delta t_{\gamma-\rm GW}\equiv0$ identically along all cosmological and local paths, including GW170817), and shared refractive index $n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)$ at fixed epoch ($A_{\rm sp}=1$) and $B=0$. Crucially, while photons and gravitons share the null cone $d\eta=dt/A_{\rm sp}(t,\mathbf x)$, the WKB geometric-mean amplitude weight of $S_{\rm TT}^{(2)}$ on static gravitational space ($\sqrt{\gamma}=1$) is $\sqrt{A_{\rm sp}^{+1}\cdot A_{\rm sp}^{-1}}\equiv 1$ (constant, unlike expanding FLRW where $\sqrt{\gamma}=a^3(\eta)$ produces an $a^2(\eta)$ WKB weight and $2\mathcal H\partial_\eta h_{ij}^{\rm TT}$ Hubble friction), so $h_{ij}^{\rm TT}$ decays geometrically as $1/r$ on static space and preserves the diffeomorphism-invariant standard-siren discriminator $\Xi(z)=d_L^{\rm GW}/d_L^{\rm EM}=1/(1+z)\neq 1$ (Section 8, Step 80), while the non-relativistic point-particle action $S_{\rm pp}=-mc\int\sqrt{-\tilde g_{\mu\nu}dx^\mu dx^\nu}$ yields $L_{\rm pp}\simeq-mc^2\bigl(1+\Phi_N/c^2+\beta_A u_{\rm grav}-\tfrac12 v^2/c^2\bigr)$ and $\mathbf a=-\nabla\Phi_N-c^2\beta_A\nabla u_{\rm grav}$.

The adopted kinetic realization is the two-branch sector $P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+X|X|/\Lambda_X^4$, with $X=-\tfrac12(\nabla\phi)^2$, $\xi=|X|/\Lambda_X^4$, and $k=4\sqrt2\,g_t/a_0\simeq16$ derived from the adopted acceleration constant, as developed below. The sector carries no canonical term: every completion restoring a linear small-$\xi$ limit reverts to a Keplerian tail and is rejected by the flat-asymptote constraint, while all quoted Solar-System benchmarks sit on the quadratic-dominated branch $\xi\gg\xi_*=(k/2)^2\simeq64$ where the earlier $X|X|/\Lambda_X^4$ calculations carry over unchanged. The strong-curvature term is evaluated in the black-hole completion (Paper 28); the weak-field calculations neglect that term while retaining the nonlinear kinetic response where required. Canonical-only and alternative-kinetic calculations are identified as benchmarks of their stated branches.

### Canonical microscopic structure

The universal conformal coupling, the conformal–disformal matter metric architecture, and the observable Temporal-Topology response structure are fixed. The framework cleanly divides responsibilities: the two-body operator $\mathcal{S}_{\rm eff}$ governs pairwise Temporal Shear recovery, while the scalar self-interaction $V(\phi)$ must remain flat/shallow across the matter-hosting domain to satisfy cosmological background constraints. The candidate master potential of Section 8 is a single smooth ($C^\infty$) function whose essential singularity supplies the locally flat potential floor required by the temporal-well roll (Appendix E, R3). The items below specify the canonical microscopic structure.

*Conformal coupling.* The conformal factor is fixed universally as

$$A(\phi) = \exp\!\left(\frac{\beta_A\,\phi}{M_{\rm Pl}}\right), \qquad \beta_A = -1.0,$$

so that the dimensionless coupling $d\ln A/d\varphi = \beta_A = -1$ is frozen (equivalently $\alpha_0 = \sqrt{2}\,\beta_A$ in DEF normalization), where $\varphi \equiv \phi/M_{\rm Pl}$ is the dimensionless field variable. The weak-field Solar-System safety is not a second parameter $\beta\approx -0.013$ but the screened source charge $S_\Sigma^{(\odot)}\,\alpha_0$: under the unstretched-space temporal metric ($\tilde g_{00}=-(1+2\Phi_{\rm dyn}/c^2)$ with $\Phi_{\rm dyn}=\Phi_N+c^2\beta_A u=(1+2\beta_A^2 S_\Sigma^{(\odot)})\Phi_N$ and unstretched spatial curvature $\tilde g_{ij}=(1-2\Phi_N/c^2)\delta_{ij}$), the PPN gravitational slip is linear in the solar charge fraction, providing the strict, sign-definite prediction $\gamma_{\rm PPN}-1 = -2\beta_A^2 S_\Sigma^{(\odot)}/(1+2\beta_A^2 S_\Sigma^{(\odot)}) < 0$ (or $-4\beta_A^2 S_\Sigma^{(\odot)}/(1+2\beta_A^2 S_\Sigma^{(\odot)})$ on the isotropic-conformal benchmark), constrained by Cassini to $|\gamma_{\rm PPN}-1| < 2.3\times10^{-5}$, fixing $S_\Sigma^{(\odot)} \lesssim 1.15\times10^{-5}$ ($5.8\times10^{-6}$ on the isotropic benchmark; on the selected two-branch kinetic sector at solar conjunction $r=1.6\,R_\odot$, $y_{\rm tb}=2.16\times10^{-8}$ gives $|\gamma_{\rm PPN}-1|\simeq 4.32\times10^{-8}$, more than $500\times$ below the Cassini bound; Step 80). The terrestrial amplitude factor $S_A^{(\oplus)}$ characterizes Earth's static conformal-depth response; time-dependent GNSS clock covariance is described by $C_A$ together with the perturbation and measurement transfer of the same solved configuration. The Solar-System source-charge factor $S_\Sigma^{(\odot)}$ is a different projection and need not equal $S_A^{(\oplus)}$.

*Sign convention for $\phi$ (corpus-wide).* The scalar field is defined such that $\phi > 0$ in the vicinity of a mass concentration, with the ambient cosmological value taken as the zero point by convention, $\phi_\infty = 0$. Combined with the frozen coupling $\beta_A = -1$, this fixes every downstream sign in the framework:

- $A(\phi) = \exp(\beta_A\varphi) < 1$ near a mass, so the conformal factor is *suppressed* in a potential well.

- Since matter clocks tick at $d\tau/dt \simeq A(\phi)$, clocks run slower in deeper wells. This reproduces the sign of the ordinary gravitational redshift and is therefore the convention consistent with general relativity in the screened limit.

- The Temporal Shear $\Sigma_\mu \equiv \nabla_\mu \ln A = \beta_A \nabla_\mu \varphi$ points *outward* from a mass (since $\beta_A < 0$ and $\nabla_\mu\varphi$ points inward).

- Consequently $\beta_A \phi < 0$ near a mass, and $\Delta \ln A < 0$ relative to the ambient environment.

The opposite choice ($\phi < 0$ near a mass, giving $A > 1$ and clocks running *faster* in wells) is inconsistent with the measured sign of gravitational redshift and is not used anywhere in this corpus. Any paper reporting a conformal-sector sign should be checked against this convention before its result is compared with another paper's. Where a manuscript quotes $\lvert\beta_A\rvert$ or an unsigned effective coupling, that is a magnitude and carries no sign information. Under the preferred-foliation decomposition $\varphi(t,\mathbf{x}) \equiv \phi(t,\mathbf{x})/M_{\rm Pl} = u_{\rm drift}(t,\mathbf{x}) + u_{\rm grav}(\mathbf{x})$, the background drift $u_{\rm drift}$ and the local gravitational perturbation $u_{\rm grav}$ superimpose linearly with the single universal coupling $\beta_A = -1$ and zero sign contradiction: at cosmic lookback redshift $z > 0$, $u_{\rm drift}(t_e) = \ln(1+z) > 0$ rolls to $u_{\rm drift}(t_o) = 0$ today ($\dot{u}_{\rm drift} = -H_{\rm drift} < 0$), giving $A_{\rm sp}(u_{\rm drift}) = \exp(\beta_A u_{\rm drift}) = \exp(-u_{\rm drift}) = (1+z)^{-1} < 1$ on $\Pi_{\mu\nu}$, while in a local gravitational potential well ($\Phi_N < 0$), $u_{\rm grav}(\mathbf{x}) = 2\beta_A S_\Sigma\Phi_N/c^2 > 0$ gives $A_{\rm loc}(u_{\rm grav}) = \exp(\beta_A u_{\rm grav}) = \exp(-u_{\rm grav}) < 1$ on $U_\mu U_\nu$. Both sectors therefore obey the identical exponential suppression law $A(u) = \exp(-u) < 1$ for $u > 0$ and factorize cleanly in moving-endpoint and local clock observables without algebraic cross-contamination.

*Bidirectionality and the weak-field normalization ledger.* The prohibition above concerns the sign of $A$ in potential wells. The conformal factor nevertheless deviates symmetrically relative to the ambient medium ($A = 1$): $A < 1$ in overdensities (clocks tick slower) and $A > 1$ in underdensities (clocks tick faster). Here $\Phi_N$ denotes the Einstein-frame Newtonian metric potential, fixed by $\nabla^2\Phi_N=\rho_*/(2M_{\rm Pl}^2)$ and $\Phi_N=-GM/(c^2r)<0$ outside an isolated positive mass. In the canonical quasistatic limit, $\nabla^2\delta\phi=\beta_A\rho_*/M_{\rm Pl}$ therefore gives

$$\frac{\delta\phi}{M_{\rm Pl}}=2\beta_A\Phi_N, \qquad \ln\!\frac{A}{A_\infty}=2\beta_A^2\Phi_N.$$

For a static scalar, the matter-frame lapse is $\tilde N=A N$. With $N/N_\infty=e^{\Phi_N}+O(\Phi_N^2)$, the bare unscreened branch obeys

$$\frac{\tilde N}{\tilde N_\infty} =\exp\!\left[(1+2\beta_A^2)\Phi_N\right]+O(\Phi_N^2) =e^{3\Phi_N}+O(\Phi_N^2), \qquad \frac{G_{\rm eff}}{G}=1+2\beta_A^2=3,$$

where the final equalities use the frozen value $\beta_A=-1$. Thus $\Phi_N$ is not the potential inferred from unscreened matter motion: the latter is $3\Phi_N$ at leading order. This factor is precisely why environmental suppression is required for every macroscopic source, not only the Sun. For a screened exterior source, the observable lapse gradient is instead

$$\nabla\ln\tilde N =\left[1+2\beta_A^2S_\Sigma(\mathcal E)\right]\nabla\Phi_N,$$

so $S_\Sigma\to0$ recovers the GR lapse gradient and $G_{\rm eff}\to G$. A common ambient value of $A$ cancels from local clock comparisons; it must not be confused with the exterior source-charge gradient governed by $S_\Sigma$, while finite clock-amplitude offsets remain governed separately by $S_A$. The coefficient ledger is derived independently of fitted data (Appendix E, R9). Along an extended line of sight through the cosmic web, exact conformal excursions spatially average out to recover the smooth cosmological background. However, the non-exact covariance $\mathcal{C}_T$ (Paper 26) retains a cumulative, macroscopic residual that resists spatial averaging. This distinction between exact conformal averaging and non-exact residual survival is the physical content of the $\Sigma_\parallel + \mathcal{C}_{T,\parallel}$ decomposition of Paper 26.

*Disformal coupling.* The matter metric contains a disformal coupling function $B(\phi)$. Observable disformal effects depend on the complete combination $B(\phi)\nabla_\mu\phi\nabla_\nu\phi$, rather than on $B(\phi)$ alone. Paper 28 employs the field-space envelope

$$B(\phi) = B_0\,\frac{\varphi^2}{1+\varphi^2}\, \exp\!\left(-\frac{\varphi^4}{2\,\sigma_B^4}\right), \qquad \varphi\equiv\phi/M_{\rm Pl},$$

as a prescribed strong-field realization. At weak-field values ($\varphi \sim 10^{-10}$), the exponential damping factor is unity to extremely high precision and $\varphi^2/(1+\varphi^2)\simeq\varphi^2$, so $B(\phi)\simeq B_0\varphi^2$ throughout terrestrial and typical astrophysical environments. Its rapid large-$|\varphi|$ damping provides additional strong-field suppression of the disformal contribution, enforcing conformal dominance and protecting the Lorentzian matter-metric branch when scalar gradients become extreme. This field-space envelope is not identified with ordinary environmental Temporal-Topology screening, which arises primarily through the environment-dependent scalar configuration and its active gradient. The unique corpus-wide microscopic form and normalization of $B(\phi)$ therefore remain part of the action-closure problem. In the dimensionful Jakarta field convention, $B_0$ carries mass dimension $-4$ so that $B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi$ is dimensionless; numerical normalizations employed in Paper 28's geometrized, dimensionless-field strong-field construction are therefore not imported directly into the weak-field theory. GW170817 constrains the path-integral combination $B(\phi)(\partial\phi)^2$ along observed late-time astrophysical paths; it does not require $B\equiv 0$ in every regime. The theory field $\phi$ remains dimensionful (mass dimension 1) throughout, consistent with the canonical kinetic term $-\frac12(\nabla\phi)^2$ and the conformal factor $A=\exp(\beta_A\phi/M_{\rm Pl})$; $\varphi$ is introduced only as a shape variable for $B$. The strong-field regime in which this envelope is exercised by Paper 28 lies beyond the small-$B$ EFT domain in which the invertibility and signature conditions of Section 4 are established; those applications are conditional on the completed nonlinear realization.

*Temporal-Topology saturation sector.* Environmental suppression in TEP is defined by the continuous response of the Temporal Topology rather than by a discrete density threshold or thin-shell boundary. Writing $\Theta \equiv \ln A(\phi)$ and $\Sigma_\mu \equiv \nabla_\mu \Theta$, the conformal-amplitude response $S_A$ and the Temporal-Shear/source-charge response $S_\Sigma$ are distinct observable projections of the same environment-dependent scalar configuration. The macroscopic scale

$$\rho_T \simeq 20\ {\rm g\,cm^{-3}}$$

denotes the empirically calibrated Temporal-Topology saturation/reference scale. It is not a universal microscopic density cutoff and does not define a binary screened/unscreened transition. In the weak-field, canonical $K\to 1$, $B\to 0$, quasistatic limit, a particular microscopic completion must satisfy the scalar field equation $\nabla^2\phi = V_{,\phi} + \mathcal{Q}_m$, with the matter-source convention defined consistently with the action of Section 2.2. Defining the conserved Einstein-frame density $\rho_* = A^3\tilde\rho$, the field equation reads

$$\nabla^2\phi = V_{,\phi} + \rho_*\,A_{,\phi}.$$

In regions where a particular completion admits an adiabatic density-dependent equilibrium, this may reduce to a local balance of the form $V_{,\phi} + \rho_*\,A_{,\phi} \simeq 0$. Such an effective minimum is one possible local realization of Temporal-Topology saturation; it is not the defining screening ontology of TEP. Chameleon, Vainshtein, Galileon, DBI, symmetron, and related mechanisms therefore remain candidate microscopic realizations rather than definitions of the framework. The observable spatial gradient suppression is natively handled by the two-body kinetic operator, leaving the potential $V(\phi)$ free to satisfy the shallow background conditions required by cosmology.

*Screening operators.* Two projections of the solved nonlinear field configuration are needed. The field-amplitude (clock) screening $S_A$ measures the suppression of the conformal factor excursion relative to its unscreened cosmological baseline (Paper 26; Appendix E, R7):

$$S_A(\mathcal{E}) \equiv \frac{\bigl(A(\phi(\mathbf r)) - 1\bigr)_{\rm local}} {\bigl(A(\phi) - 1\bigr)_{\rm unscreened}},$$

The source-charge (shear) screening $S_\Sigma$ measures the suppression of the local scalar response relative to the unscreened expectation — in the kinetic realization adopted here, the ratio of the solved nonlinear gradient to its linear extrapolation, equivalently the inverse stiffness for an incremental source on an ambient kinetic background:

$$S_\Sigma(\mathcal{E}) \equiv \frac{|\nabla\phi|_{\rm nonlinear}}{|\nabla\phi|_{\rm linear}} \;\simeq\; \frac{Q_{\rm eff}}{Q_0} = \frac{\alpha_{\rm eff}}{\alpha_0},$$

where $\phi(\mathbf r)$ is the solved static profile for the given source geometry, density, compactness, and boundary conditions. The effective-charge form $Q_{\rm eff}/Q_0$ is the far-field matching shorthand: in the $P(X,\phi)$ realization the Gauss flux $\oint P_{,X}\,\nabla\phi\cdot d\mathbf A$ is conserved through the nonlinear shell (the source charge is not destroyed), and the suppression an observer infers is the local field gradient, which the constant-charge PPN parametrization then records as a reduced effective exterior coupling. Both projections are observable responses of the same physical nonlinear field configuration rather than independent domain-by-domain couplings. Their macroscopic environmental response is supplied by the corpus-wide $\mathcal{S}_\Sigma(\mathcal{E})$ construction; the kinetic realization adopted in this paper is specified at the end of this section, with alternative $K(X)$ forms standing as candidate realizations to be tested against its derived predictions. $S_A$ characterizes the static conformal-depth response; $S_\Sigma$ characterizes the specified gradient/source response. Time-dependent clock covariance is carried by $C_A$ together with the perturbation and measurement transfer of the same solved configuration. These responses need not be numerically identical. Appendix E, R17 computes static boundary susceptibilities and finite-excursion responses, not a universal transmission: approximately additive response is a regime-dependent limit, and propagation through the ambient medium must be distinguished from penetration into the body. No independent clock-sector coupling is introduced. In composite observables the assignment is positional at leading order in the test-probe expansion: the probe's perturbation of the host's solved field and the resulting mixed boundary terms are neglected. This assignment is not a decomposition of the scalar action into independent habitat and probe fields; finite-mass binaries require the nonlinear boundary-value problem with both sources present. Within that stated approximation, the assignment is positional: the factor multiplying a host's potential contribution to the environmental field coordinate is the source-side charge suppression, $\mathcal S_\Sigma$ evaluated at the habitat density — the ambient-density limit $[1+(\rho/\rho_{\rm half})^2]^{-1}$ with $\rho_{\rm half}\approx0.5\,M_\odot\,{\rm pc}^{-3}$ (Paper 26) — while the probe's own clock-amplitude response is $S_A$ evaluated at the probe's material density, a per-source factor absorbed into the channel response coefficient (e.g., the Cepheid $\kappa_{\rm Cep}$ of Papers 11 and 31) rather than into the environmental coordinate. Strong gradient flattening suppresses $S_\Sigma$, while $S_A$ depends separately on the local field amplitude relative to its reference environment. The mesoscopic screening law of the companion Paper 25 (in preparation; $\mathcal{S}_\Sigma^{\rm meso} = S_{\rm TEP}\times S_{\rm TF}\times S_{\rm boundary}\times S_{\rm decoherence}$) is a factorization of $S_\Sigma$ at intermediate scales.

*Radial ODE closure calculation.* The static weak-field scalar equation $\nabla^2\phi = V_{,\phi} + \rho_*\,A_{,\phi}$ is solved numerically for a spherical source with the bidirectional conformal coupling $A(\phi) = \exp(-\phi/M_{\rm Pl})$, $\beta_A = -1$, using `scipy.integrate.solve_bvp` with core regularity ($\phi'(0)=0$) and cosmological relaxation ($\phi(r_{\max})\to 0$) boundary conditions (Appendix E, R1). The key dimensionless parameter is the unscreened field amplitude $\psi_{\rm uns} = M/(4\pi M_{\rm Pl}^2 R)$, which controls the strength of the conformal nonlinearity. Two theoretical baselines are evaluated against the Cassini bound $|\gamma_{\rm PPN}-1| < 2.3\times 10^{-5}$ (requiring $S_\Sigma^{(\odot)} \lesssim 1.15\times 10^{-5}$ under the unified unstretched-space metric, or $5.8\times 10^{-6}$ on the isotropic-conformal benchmark):

(i) $V = 0$ (pure bidirectional conformal screening). The nonlinear source $-(\rho_*/M_{\rm Pl})\exp(-\phi/M_{\rm Pl})$ provides gradient flattening in overdensities ($\phi > 0$, $\exp(-\phi/M_{\rm Pl}) < 1$, source weakened) and enhancement in underdensities ($\phi < 0$, $\exp(-\phi/M_{\rm Pl}) > 1$). The screening correction is $O(\psi_{\rm uns}) \sim 4\times 10^{-6}$ for the Sun — only a fractional reduction of the effective charge, leaving $S_\Sigma \approx 1$, far above the Cassini requirement $S_\Sigma \lesssim 1.15\times 10^{-5}$ ($5.8\times 10^{-6}$ isotropic). The conformal nonlinearity alone is too weak for Solar-System screening.

**(ii) Kinetic completion of two-body screening — microscopic realization.** The effective Temporal-Topology response is produced by the nonlinear interaction of overlapping scalar-field gradients in the nested environmental landscape. The noncanonical completion selected by the corpus's master-sector admissibility scan (Appendix E, step 54; master-sector ledger, `TEP/closure/`) is the two-branch sector

$$P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^{2}}\,X\,|X|^{1/2}+\frac{X\,|X|}{\Lambda_X^{4}},\qquad X=-\frac12(\nabla\phi)^{2},$$

whose kinetic response function on the static branch is $P_{,X}=k\sqrt{\xi}+2\xi$ with $\xi\equiv|X|/\Lambda_X^4$. The selection is a derived result, not an ansatz choice: within the surveyed power-law family $P_{,X}=k\xi^{p}+2\xi$ a flat rotation asymptote ($a_\phi\propto1/r$, the deep-RAR tail) requires $p=1/2$ uniquely, while every completion restoring a linear small-$\xi$ limit — including the incumbent unit-term form $P_{,X}=1+2\xi$ and the unit-term two-branch variant — reverts to a Keplerian tail and is rejected. The sector therefore carries no canonical term and has no canonical small-gradient limit: $P_{,X}\to0$ as $\xi\to0$. The stiffness can equal unity at an isolated finite field value without making the kinetic function canonical. The coefficient is fixed by the deep-tail flux law: flux conservation gives $a_\phi^{2}=a_{\rm eff}\,g_N$ with $a_{\rm eff}=4\sqrt{2}\,|\beta_A|g_t/k$, so the adopted empirical acceleration constant $a_0\simeq1.2\times10^{-10}\ {\rm m\,s^{-2}}$ returns $k=4\sqrt{2}\,g_t/a_0\simeq16.0$ — a derived coefficient, not a fit. The two terms cross over at $\xi_*=(k/2)^2\simeq64$: the Solar-System and strong-field benchmarks of Section 7 all sit at $\xi\sim10^{2}$–$10^{23}\gg\xi_*$, where the quadratic asymptote governs the largest-$\xi$ results; finite-gradient calculations retain both terms, whose relative contribution is $k/(2\sqrt{\xi})$, while the Galactic ambient ($\xi\sim0.3$–$0.5$), the wide-binary pair region ($\xi\sim10$), and the void sector ($\xi\sim10^{-5}$, where $\mathcal S_\Sigma>1$ — the anti-screening expected of Rule 10) live on the low-gradient branch. (The sign-safe form $|X|$ is required: on the static branch $X<0$, and the power law is otherwise defined only for even integer powers. The $X|X|$ representation is understood piecewise on the timelike and spacelike kinetic branches, with continuous matching at $X=0$; where a $C^\infty$ regulator is required numerically, $X|X|$ may be replaced by $X\sqrt{X^2+\epsilon_X^2}$ with $\epsilon_X$ taken below all resolved physical kinetic scales, leaving the quoted asymptotic screening laws unchanged. The zero-gradient limit of the selected sector — where $P_{,X}\to0$ — is correspondingly unregulated by the theory alone and requires the nonzero ambient field of the nested hierarchy; see the open-item statement below.) The static spherical field equation $\partial_r(r^2P_{,X}\phi')=r^2\mathcal Q$ integrates to $P_{,X}\phi'=\mathcal C/r^2$: a configuration immersed in an ambient gradient answers an incremental source charge only through the inverse stiffness, so the observable Temporal Shear is suppressed by

$$\mathcal S_\Sigma(\mathcal E)=\frac{1}{P_{,X}(X_{\rm amb})}=\left[\frac{k}{\sqrt2}\,\frac{g}{g_t}+\left(\frac{g}{g_t}\right)^{2}\right]^{-1},\qquad g_t=\frac{cH_0}{2|\beta_A|}\simeq3.4\times10^{-10}\ {\rm m\,s^{-2}},$$

whose leading quadratic-dominated tail is $(g_t/g)^2$ on the high-gradient branch ($g\gg g_t$, all Solar-System benchmarks) and its low-gradient tail is $\mathcal S_\Sigma\to\sqrt2\,g_t/(k\,g)$ at $g\ll g_t$ — the branch that generates the flat galactic asymptote. The shear-to-acceleration map $\Sigma=2\beta_A^{2}g/c^2$ converts the kinetic ratio $|X|/\Lambda_X^4=\tfrac12(g/g_t)^2$ into the gradient channel of the covariant environmental operator constructed phenomenologically in Paper 26 §2.5. The kinetic scale is the single scale of the sector, fixed by the cosmological shear floor: $\Lambda_X^4=M_{\rm Pl}^{2}H_0^{2}$, i.e. $\Lambda_X=\sqrt{M_{\rm Pl}H_0}\simeq1.9$ meV, the dark-energy scale. The shear threshold therefore carries no free parameter. The two-body response is obtained by solving the same equation for the pair's mutual shear: flux conservation $P_{,X}\phi'=\mathcal C/r^2$ with $P_{,X}\simeq2|X|/\Lambda_X^4\simeq\phi'^2/\Lambda_X^4$ where the quadratic term dominates ($\xi\gg\xi_*$, the mutual-dominated shell interior) gives the high-gradient asymptote $\phi'\propto r^{-2/3}$. In the isolated spherical, potential-negligible source-flux control, define $q=g_N/g_t=(r_*/r)^2$ and $y=|\phi'|/|\phi'_{\rm lin}|$. The exact selected-sector relation is

$$\frac{k}{\sqrt2}\,q\,y^2+q^2y^3=1,\qquad \xi=\frac12(qy)^2,\qquad r_* = \sqrt{GM/g_t}.$$

Thus $y\sim(r/r_*)^{4/3}$ where the quadratic term dominates, while the low-gradient branch has $y\sim(\sqrt2/k)^{1/2}(r/r_*)$. The expression $y[1+y^2(r_*/r)^4]=1$ belongs to the retained unit-term control. Embedding and pair geometry enter through the solved ambient and pair response; the isolated spherical identity is not a complete two-source field solution. The quoted pairwise projection is $\mathcal S_{\rm eff}(s)=\mathcal S_\Sigma(X_{\rm env})^2 y(s)$ under its stated source and ambient assignments.

and the full pairwise response propagates this mutual shear with the ambient inverse-stiffness factor at the emission and response vertices,

$$\mathcal R(s)=\mathcal S_\Sigma(X_{\rm env})^{2}\,y(s),$$

where $X_{\rm env}$ is the embedding ambient kinetic value of the hierarchy level above the pair (specifically the external Galactic ambient floor $2X_{\rm gal}/\Lambda_X^4 \approx 0.52$ for field wide binaries — a value originally calibrated to the wide-binary plateau under the pre-propagator bookkeeping; the directly measured solar-circle field gives the independent estimate $u_{\rm sun}=a_\odot/g_t\simeq0.57$, $2X_{\rm sun}/\Lambda_X^4\simeq0.32$ ($v_c=220$ km s$^{-1}$, $R_0=8.1$ kpc; step 32), which the fully-coupled propagator-plus-vertex reading of Paper 13, step 017, uses to reproduce the plateau amplitude at the percent level without calibration — the ambient bracket $u_0\simeq0.55$–$0.72$ is a physical range, not a tuned value, because the ambient field is itself a solved quantity, or the dominant member's pre-existing shell for hierarchical pairs), excluding the pair's own co-generated mutual field $g_{\rm mut}=GM/s^2$ to avoid suppressing a response by the response itself in the transition region. Under the selected stiffness the ambient vertex evaluates to $\mathcal S_\Sigma(\xi_{\rm env}\sim0.3\text{--}0.5)\approx0.08\text{--}0.10$ — the external-field suppression of order $10^{-2}$ in $\mathcal S_\Sigma^2$ that emerges as the AQUAL-type external-field effect rather than as calibration; on the rejected unit-term baseline the same ambient returned $\mathcal S_\Sigma\approx0.5$–$0.6$. Evaluated on the corrected ambient semantics (derived ambient, pure propagator, deprojected separation), the zero-free-parameter joint test on the wide-binary saturation amplitude and transition scale passes at $0.44\sigma$ and $1.69\sigma$ respectively (predicted $\alpha_{\rm sat}=0.371$ against the identity $4\sqrt2/k\simeq0.353$; $R_s=3{,}676$ AU), while the unit-term baseline and the exponential-interpolation rival are excluded at the $\sim30\sigma$ level on amplitude (master-sector ledger, step 56/020). Two regimes of this single operator follow directly. For a comparable-mass pair the mutual field is the measured response — the two members' gradients oppose and cancel at the pair's bridge region — so the vertex ambient saturates at the external environmental floor and the observable transition profile is the flux-conserving $\mathcal R\sim s^{4/3}$ law, the exponent the wide-binary population selects. For a hierarchical pair — a test member embedded in a dominant member's pre-existing nonlinear shell — the dominant member's field is itself the embedding ambient; the full flux identity $P_{,X}(X_{\rm dom})=1/y$ in the isolated potential-negligible dominant-source control then makes the vertex factors equal to $y(s)$, giving $\mathcal R=y^{3}\sim s^{4}$: the steep deep-interior response required by the Solar-System benchmarks (constraint F4) is recovered exactly, with the radius law $R_s=r_*=\sqrt{GM/g_t}$ and the transition-radius scaling derived from the action rather than fitted. Evaluated at zero free parameters, the operator returns $\mathcal R\simeq2.2\times10^{-15}$ for the Earth–Moon pair (the vertex ambient there carries both Earth's nonlinear field and the solar field at 1 AU; the corpus's earlier $2.5\times10^{-14}$ bookkeeping is reproduced at the same order), $1.0\times10^{-23}$ at the Cassini conjunction (earlier bookkeeping $\sim6\times10^{-23}$), $2.7\times10^{-11}$ at Saturn's orbit (earlier bookkeeping $\sim10^{-10}$), and $1.2\times10^{-21}$ for the terrestrial shear environment — the value independently quoted by the covariant-operator evaluation of Paper 26 — while the derived wide-binary transition scale $R_s=\sqrt{GM/g_t}\simeq4.7$ kAU at $1.24\,M_\odot$ overshoots the fitted $2646\pm182$ AU by a factor $\simeq1.8$. The residual is stated plainly: a normalization of the same functional form, with the transition acceleration fitted directly to the wide-binary data, could reproduce the fitted radius nearly exactly — but that agreement would be by construction rather than prediction. The derived $g_t=cH_0/(2|\beta_A|)$ contains no adjustable constant, so the factor $\simeq1.8$ constitutes the first genuinely independent comparison of the scale. Its sign is the one the environmental baseline predicts: the operator measures the pair's internal gradient against the ambient environmental floor, which inside the Galactic disk exceeds the asymptotic cosmological value $\Sigma_{\rm bg}=H_0/c$. Specifically, the ambient Galactic temporal potential exceeds the pair's mutual gravitational potential by $\Phi_{\rm gal}/\Phi_{\rm pair} \approx (v_0^2/c^2)/(GM_{\rm pair}/(s c^2)) \sim 9\times10^{4}$ at the fitted separation $s = 2646$ AU (with Galactic rotational potential $\Phi_{\rm gal}/c^2 \approx 5.4\times 10^{-7}$ dominating the mutual well $\Phi_{\rm pair}/c^2 \approx 4.6\times 10^{-12}$, even while the local mutual acceleration gradient dominates the ambient Galactic acceleration, $g_{\rm gal}/g_{\rm pair} \approx 0.06$). Under the isolated-profile reading the ambient floor enters the operator only through the vertex factor $\mathcal S_\Sigma(X_{\rm env})^2$, modulating the response amplitude rather than the shape scale $r_*$; a same-estimator comparison (folding the mass-convolved derived profile through the paper's own sem-weighted single-scale exponential estimator; Paper 13 step 017) then returns $\simeq5.1$ kAU against the measured $2646\pm609$ AU — ratio $\simeq1.9$ — with the half-excess marker agreeing at $\simeq3.4$ kAU against the observed $\simeq1.7$–$1.9$ kAU. The earlier heuristic explanations fail quantitatively: the exponential-fit scale reduction is $\lesssim2\%$ (the gradual $s^{4/3}$ recovery pulls the fitted scale upward, not downward), $\sqrt{M}$-weighting accounts for only $\sim4\%$, and mass convolution broadens the transition toward larger fitted scales. The resolution is not a new coefficient but the embedded problem: the ambient enters not only at the vertices but in the perturbation propagator itself. The residual is substantially resolved by the propagator the ambient itself supplies. Solving the source perturbation on the Galactic nonlinear background (step 32: the full nonlinear elliptic problem $\nabla\!\cdot\![P_{,X}\nabla\phi]=\rho$ around a uniform ambient $u_0=\sqrt{X_{\rm gal}}\simeq0.72\,g_t$) shows the background stiffness $Z_{ij}=P_{,X}\delta_{ij}+2P_{,XX}a_ia_j$ — anisotropic, $Z_\parallel=1+3X_{\rm gal}\simeq2.56$ parallel and $Z_\perp=1+X_{\rm gal}\simeq1.52$ transverse — suppressing the pair's orientation-averaged far-field response to $y_{\rm far}\simeq0.56$ and compressing the half-excess transition marker to $\simeq0.44\,r_*$ (from $0.74\,r_*$ isolated, fitted-scale ratio $\simeq0.60$). Folded through Paper 13's own mass distribution and sem-weighted estimator (step 017), the embedded prediction returns $R_s\simeq3{,}300$ AU against the fitted $2646\pm609$ AU and a half-excess marker at $\simeq1{,}715$ AU, coincident with the observed value — a residual factor $\simeq1.26$ inside the systematic band. The $\eta_{\rm env}\simeq3.7$ coefficient required by the isolated-profile reading is therefore not a tuning parameter: the ambient stiffness supplies the transition-scale compression as a derived consequence of the same kinetic term. Two sub-questions remain open and are now sharp rather than diffuse: the amplitude factorization — the observed $\alpha_{\rm sat}=0.366$ lies between the pure-propagator reading ($\simeq0.455$) and the propagator-times-vertex reading ($\simeq0.24$), so the data discriminate the division of environmental suppression between source charge, propagator and response vertex — and the environmental ordering, which refines under the corrected ambient variable: the operator's environmental coordinate is the field *gradient* $X=(\nabla\phi)^2/\Lambda_X^4$, not the density, and the observed subsamples show the midplane more suppressed under either estimator — the free-$\alpha$ fits give ($R_s$, $\alpha$) = ($2{,}821$ AU, $0.244$) at $|Z|<0.1$ kpc versus ($4{,}681$ AU, $0.401$) at $|Z|>0.15$ kpc — and inverting the amplitudes on the fully-coupled sweep requires the effective ambient to be $\sim36\%$ *stronger* at the midplane ($u_{\rm eff}\simeq0.71$ versus $\simeq0.52$ in the halo; Paper 13). The supply-versus-demand check (Paper 13, step 017) closes the naive rescue: Galactic geometry supplies only $\sim4\%$ between the subsample median heights — the vertical field contributes $u_z\simeq0.17$ at $248$ pc against a common radial floor $u_r\simeq0.57$ pinned by the locally measured centripetal acceleration, which cannot be screened down inside the disk without contradicting the observed dynamics. The linear supply fails on both counts: $u=|a|/g_t$ grows with $|Z|$ — the wrong sign for an ordering that strengthens toward the midplane — and supplies only $\sim4\%$ of the required contrast. That estimator is the weak-field exterior reading: inside the nonlinear disk the local acceleration is not the ambient field state. The ordering is produced instead by the landscape's nonlinear response to density on the same operator's $k$-branch: the embedded pair-scale solve (step 67; Paper 13 AUD-3) shows the response charge carried by the ambient boundary variable, $D^{-1}\propto u_{\rm amb}^{-2.0}$ with background-density sensitivity below $0.02\%$, and the local $k$-branch balance $P_X u = 4\pi G\rho L/g_t$ with $P_X \sim Ku/\sqrt{2}$ yields the ambient map $u_{\rm amb}=\sqrt{4\sqrt{2}\pi G\rho L/(K g_t)}$. Calibrated once at the midplane inversion, $L\simeq0.84$ kpc reproduces all five Galactic strata within $\simeq10\%$ and the endpoint ordering exponent $p\simeq0.46$–$0.5$ — the environmental dependence is thereby an operator output, not a phenomenological proxy. The environmental dependence of the measured transition scale is a genuine empirical datum rather than an operator output: the wide-binary stratifications measure $R_s$ rising with ambient density ($R_s \propto \rho_{\rm amb}^{p}$; free-fit exponent $p \approx 0.4 \pm 0.1$, centered between the $\rho^{1/3}$ and $\rho^{1/2}$ ambient channels, Paper 13), i.e. pairs embedded in more strongly screening environments retain suppression to larger separations — the ordering stated in constraint F5. The derived scaling $R_s\propto M^{1/2}$ differs from the $M^{1/3}$ read of the wide-binary demographic splits; that read is independently flagged as degenerate with the disk size–mass relation, so the derived exponent constitutes the sharper discriminant for that test.

*Scope of the two-body projection.* The $\mathcal R(s)$ law is an effective projection of the static scalar solution, not an exact solution for unequal finite-sized bodies in a solar background. On the static branch $J_i=P_{,X}\partial_i\phi$ can have $dJ=dP_{,X}\wedge d\phi\neq0$ when the solved kinetic stiffness varies transversely; by contrast the conformal force derives from $d\ln A(\phi)$ and remains exact. A low-gradient bridge may locally reduce $P_{,X}$, but it does not by itself unscreen a body's integrated scalar charge or evade LLR: the external solar gradient, finite source size, nonlinear flux conservation, and boundary conditions all enter the force and synodic range observable. In particular, the selected two-branch sector has no canonical point: $P_{,X}\to0$ as $X\to0$, so the zero-gradient vacuum limit is regulated not by the theory but by the nonzero ambient of the nested hierarchy — a stated open condition on global solutions rather than an established bound. The dimensionless two-centre calculations in steps 59, 72 and 75 test this limitation under specified sources and boundaries, including arbitrary pair-to-ambient orientations on a Cartesian grid (step 75); Paper 17, step 086 solves a coupled smoothed Earth–Moon source benchmark in a uniform solar-gradient background at syzygy; it does not resolve the physical body interiors or the solar tide. An observable-level lunar range fit remains required before revising the quoted LLR bound.

The physical-scale Earth–Moon–Sun bridge is evaluated in Paper 17 at two levels. Step 086's coupled source solve in the specified solar background supersedes Step 084's matched-profile diagnostic for that collinear benchmark: at Full-Moon syzygy ($u_0>0$) the field along the pair axis stays at $|\nabla\phi|_{\min}\approx335\,g_t$ ($P_{,X}\sim1.2\times10^{5}$), and at New-Moon syzygy ($u_0<0$), where the axial gradient crosses zero at $z/d\approx0.432$ (axial cell-centre minimum $|\nabla\phi|_{\min}\approx36.3\,g_t$, $P_{,X}\sim1.7\times10^{3}$), off-axis stiffness still keeps the resolved $P_{,X}<1$ volume fraction at zero. An LLR range-model integration of the residual force is the remaining observable-level step. The shared action and exact/non-exact transport distinction are specified here, while the LLR-specific observational comparison resides in Paper 17.

With this projection the single operator satisfies the stated benchmark constraints (Appendix E, R2): it recovers the Temporal Shear at wide-binary scales, suppresses the pairwise Solar Temporal Shear to $\sim10^{-23}$ at the conjunction separation $s=1.6\,R_\odot$ and $\sim10^{-10}$–$10^{-11}$ at Saturn ($9.5$ AU), and yields a pairwise Earth–Moon benchmark far below the nominal LLR acceleration scale (the definitive observable comparison remaining the range-model integration described above), while suppressing the Earth-vicinity pairwise response $F(7013\text{ km})\to0$. This is a statement about $\mathcal S_\Sigma$ alone: the clock sector is a different projection, and neither the terrestrial static-depth factor $S_A^{(\oplus)}$ nor the covariance functional $C_A$ is suppressed by $\mathcal S_{\rm eff}(s)$, so the GNSS clock-rate and covariance observables are unaffected.

*Environmental response and the scale dictionary.* The master action uses distinct kinetic, potential and geometric-density parameters: $\Lambda_X=\sqrt{M_{\rm Pl}H_0}\simeq1.9$ meV, $\Lambda_V=\lambda^{1/4}M_{\rm Pl}$, and $\rho_T$. Expressing the calibrated density as an energy gives $\rho_T^{1/4}\simeq96$ keV; this conversion does not make it a second value of $\Lambda_X$ or the potential normalization. The continuous hierarchy is supplied by the environmental solution $\phi_{\mathcal E}$ and its gradients. Where the local adiabatic approximation applies, $V_{,\phi}+\rho_* A_{,\phi}=0$ and the source-plus-potential curvature is $m_{\mathcal E}^2=V_{,\phi\phi}(\phi_{\mathcal E})+\rho_*A_{,\phi\phi}(\phi_{\mathcal E})$; the kinetic principal coefficients follow from the same selected $P(X,\phi)$ evaluated on that solution. These quantities vary with density, compactness and boundary geometry while the declared action parameters remain fixed. This is environmental response, not an independently derived renormalization-group law for a density-dependent $\Lambda$. The scale audit of Paper 6, Appendix C, already distinguishes these roles.

*Amplitude sector.* The clock-amplitude projection is fixed by the potential sector. In the unified master potential $V(u) = V_{\rm matter}(u)e^{-(u/u_s)^4} + V_0 e^{-(u_s/u)^4}$ of Section 8, the matter-hosting weak-field branch is realized by $V_{\rm matter}(u) = \frac{\lambda}{4}M_{\rm Pl}^4 u^4 = \frac{\lambda}{4}\phi^4$. For field values $u = \phi/M_{\rm Pl} \ll u_s \simeq 10$, the non-perturbative horizon factor $e^{-(u_s/u)^4}$ vanishes to all orders in Taylor expansion around $u=0$ (underflowing identically to zero at physical screening densities), while $e^{-(u/u_s)^4} \to 1$, reducing the master potential identically to the quartic self-interaction across the matter-hosting domain. The quartic is the minimal admissible self-interaction on the $\beta_A=-1$ branch: an inverse-power form admits no stable minimum on this coupling branch (constraint F1), while a quadratic potential lacks the required density dependence. Minimizing the matter-sourced effective potential $V_{\rm eff} = \frac{\lambda}{4}\phi^4 + \rho A(\phi)$ on the $A(\phi) = e^{-\phi/M_{\rm Pl}}$ branch yields the equilibrium field $\phi_{\min} = (\rho/\lambda M_{\rm Pl})^{1/3}$ and an effective scalar mass $m_{\rm eff} = \sqrt{3}\lambda^{1/6}(\rho/M_{\rm Pl})^{1/3}$. The associated Compton wavelength $\lambda_c = m_{\rm eff}^{-1}$ satisfies a universal, density-independent relation with the geometric saturation radius $R_T = (3M/(4\pi\bar\rho))^{1/3}$, namely $\lambda_c/R_T = 3^{-1/2}(4\pi/3)^{1/3}\lambda^{-1/6}(M_{\rm Pl}/M)^{1/3}$ ($\approx 4.07$ for Earth at the reference coupling $\lambda_{\rm ref} \approx 7.5\times 10^{-71}$, and $\approx 0.60$ at the $10^5$-rescaled coupling; Appendix E, steps 03, 53). Because the direct nonlinear radial boundary-value solve on the selected two-branch kinetic sector places Earth on the flux-following (leaking) branch ($u_{\rm int}/u_{\rm eq}\simeq 1.45\times 10^{-7}$, `step_53_unified_profile_two_branch.json`), the interior clock field is set by the transmitted Galactic/cosmological ambient rather than an isolated-body chameleon minimum, while $S_A$ acts as the empirical geometric saturation-depth index relative to $R_T(M)$. The reference normalization is a fiducial point of the potential family, not the operative weak-field value: the corrected linear-in-$S_\Sigma$ Cassini evaluation fails at $\lambda_{\rm ref}$ by $\sim 250\times$ ($\gamma_{\rm PPN}-1 \approx -5.8\times 10^{-3}$ versus the $2.3\times 10^{-5}$ bound) and requires the $\times 10^{5}$-rescaled branch, so wherever the quartic normalization enters the corpus adopts $\lambda_{\rm Cassini} = 10^{5}\lambda_{\rm ref} \approx 7.5\times 10^{-66}$ (step 03; shared corpus constants `LAMBDA_QUARTIC_REF`/`LAMBDA_QUARTIC_CASSINI`). The two quoted values are the reference and Cassini-compatible points of the same quartic normalization, not two independent couplings. The transition density $\rho_T$ is calibrated through Earth's saturation radius: identifying $R_T$ with the terrestrial GNSS clock-covariance correlation length $\lambda_T \approx 4200$ km fixes $\rho_T = 3M_\oplus/(4\pi R_T^3) \approx 19.2$–$20\text{ g cm}^{-3}$ from real timing residuals. The measured correlation length is an estimator-dependent family rather than a point value — the precise-product bracket spans $\sim 1.9$–$4.5\times10^{3}$ km (MGEX $1862\pm155$ km; IGS inter-center $3{,}330$–$4{,}549$ km), with the $\sim30$–$40\%$ epoch/window scatter the dominant systematic, propagating to an effective $\rho_T$ band of roughly $8$–$70\text{ g cm}^{-3}$; raw single-point channels bound the scale from below at $\sim 0.7$–$1.1\times10^{3}$ km (Papers 1, 3, 6, 14). The quoted $4{,}200$ km is the IGS precise-product convention adopted as the corpus normalization; the estimator-independent cross-checks on that normalization — the atomic-scale radius $R_T(m_p)\sim a_0$ and the magnetar critical period — are catalogued in Paper 6. Here $\rho_T$ is the formal reference constant of the saturation scaling law $R_T(M) = (3M/4\pi\rho_T)^{1/3}$ — the mean density a body would carry if its entire mass occupied the saturation radius — not the local density of Earth's interior at $r = R_T$ (the lower-mantle value there is $\approx 5.5$ g cm$^{-3}$, and no terrestrial material reaches $20$ g cm$^{-3}$); the physical content of the calibration is the radius $R_T$ itself and its tested mass scaling, not a density claim about Earth's structure. This in turn determines the unsaturated clock-amplitude factor $S_A = \min[1,(\bar\rho/\rho_T)^{1/3}]$ with no additional free parameter once the empirical $\rho_T$ calibration is fixed, returning the transitional value $S_A^{(\oplus)} \simeq 0.65$ ($0.43$–$0.88$ across the $\rho_T$ band) at Earth mean density $\bar\rho_\oplus = 5.515\text{ g cm}^{-3}$. Furthermore, at its matter-sourced equilibrium the quartic energy density obeys $V(\phi_{\min}) = \rho\,\phi_{\min}/4$ — always a negligible fraction of the local matter density wherever the truncation is solved ($\approx 3\times10^{-8}\text{ g cm}^{-3}$ at terrestrial conditions against $\bar\rho_\oplus = 5.5\text{ g cm}^{-3}$, $\sim10^{-41}\text{ g cm}^{-3}$ at interstellar densities, $\sim10^{-49}\text{ g cm}^{-3}$ in voids) — so the quartic sector carries no appreciable energy budget in any matter-hosting environment and remains dynamically flat exactly where it is the operative potential. Its certified domain is the weak-field regime $u \ll u_s$: at cosmological field excursions $u \sim \ln(1+z) \sim 1$ the operative potential is the master family's $V_{\rm matter}(u)$ branch itself, not the weak-field truncation, and the flatness-compatibility condition constrains that branch directly — the non-perturbative floor $V_0\,e^{-(u_s/u)^4}$ remains identically zero there and leaves the constraint free. The floor activates only at horizon pileup scales $u \sim u_s \simeq 10$. Shear suppression ($\mathcal S_\Sigma$) and clock amplitude ($S_A$) are thereby distinct, mutually consistent projections of one scalar-sector completion, not independent mechanisms. The two sector scales are likewise distinct by construction: the kinetic completion is normalized by $\Lambda_X = \sqrt{M_{\rm Pl}H_0} \simeq 1.9$ meV, the gradient/shear threshold anchored to the measured drift; the potential sector is normalized by the dimensionless $\lambda$, equivalently $\Lambda_V \equiv \lambda^{1/4}M_{\rm Pl} \simeq 1.3\times10^{2}$ GeV on the operative branch; and $\rho_T$ is a third, geometric quantity — the constant of the $R_T(M)$ law — whose energy-density reading $\rho_T^{1/4} \simeq 96$ keV is not the potential normalization. Identifying $\rho_T$ with $\Lambda_V^4$ would fix $\lambda = \rho_T/M_{\rm Pl}^4 \simeq 2.5\times10^{-90}$, a branch excluded because it drives $u_{\min}$ past the knee $u_s$ inside ordinary neutron-star cores and yields a Compton length vastly exceeding any terrestrial body (Paper 6, Appendix C). The order-unity self-quenching crossing instead sits at $\rho_{\rm sat} \sim \lambda M_{\rm Pl}^4 \sim 10^{26}$ g cm$^{-3}$, inside the temporal-well domain.

*Status of the derivation.* What is derived here is a selected realization, not a uniqueness theorem: the static-branch operator structure, the scale $g_t$, the radius law $R_s=\sqrt{GM/g_t}$, the flux-conserving profile $y[1+y^2(r_*/r)^4]=1$ on the quadratic-dominated branch, and the two-body vertex structure $\mathcal R=\mathcal S_\Sigma(X_{\rm env})^2\,y(s)$ follow from the specified $P(X,\phi)$ — yielding the $s^{4/3}$ transition steepness for comparable-mass pairs (confirmed by the wide-binary forward model) and the $s^4$ deep-interior asymptote for hierarchical pairs (constraint F4). The completion is now selected by data, not merely admissible: the corrected wide-binary joint test passes the two-branch sector at zero per-system free parameters and excludes the unit-term baseline and the exponential-interpolation candidate at $\sim30\sigma$ on amplitude, while the SPARC radial-acceleration fit under the same sector (Paper 6, Steps 10, 13; with $k_*=4\sqrt{2}\,g_t/a_0=16.03$ calibrated to $a_0$ and canonical zero-gradient regulator $P_{,X}(\xi)=\epsilon_{\rm reg}+k\sqrt{\xi}+2\xi$, $\epsilon_{\rm reg}=10^{-3}$ ensuring strict hyperbolicity at $X=0$) returns RMS $0.2185$ dex at $\Upsilon_d=0.5$ and $0.2004$ dex at $\Upsilon_d=0.35$ under the action-derived additive force law $g_{\rm obs}=g_N+a_\phi$ — matching the empirical MOND interpolating-function reference ($0.1989$ dex) at $\Upsilon_d=0.35$ and outperforming the unit-term baseline ($0.2812$ dex) and exponential-interpolation candidate ($0.2749$ dex) by $>60$ mdex (while the phenomenological quadrature comparator $\sqrt{g_N^2+a_\phi^2}$, which removes the $+2g_N a_\phi$ crossover cross-term of the two-term flux law at $\Upsilon_d=0.5$, gives $0.1974$ dex with $k_{\rm fit}=15.78$ and outer-edge BTFR median $v_{\rm obs}/v_{\rm pred}=1.0109$ vs $0.910$ additive at $\Upsilon_d=0.5$). The ellipticity and fluctuation-sector certificates already cover this noncanonical completion: step 61 scans the static and cosmological branches of all three candidate sectors and certifies strict hyperbolicity and no-ghost conditions ($Z_\perp=P_{,X}>0$, $Z_\parallel=P_{,X}+2\xi P_{,XX}>0$, $Z_t>0$) across all regimes, with the two-branch cosmological sound speed subluminal ($c_s^2\in[1/3,\,1/2]$), the static spacelike longitudinal fluctuation speed $v_\parallel^2=(P_{,X}+2XP_{,XX})/P_{,X}\in[2,\,3]$ (a generic property of $P_{,XX}>0$ AQUAL/$k$-essence sectors that preserves macro-causality and admits no closed causal curves because the acoustic cone remains strictly hyperbolic with respect to the global cosmic time foliation; Babichev, Mukhanov & Vikman 2008, contrasting with the flat-space UV analyticity bounds of Adams et al. 2006), and the late-time growth enhancement weakened to $G_{\rm eff}(z{=}0)\simeq1.16\,G$ — directionally consistent with structure-growth measurements below $\Lambda$CDM, and the surviving discriminant against the exponential-interpolation rival, which retains the baseline $\approx2G$ boost. Remaining open calculations: uniqueness across the wider $K(X)$ class (cuscuton-type members carrying no propagating scalar degree of freedom) and the microscopic form of the disformal sector $B(\phi)$. The $X\to0$ regulator question is now quantified: in the realized nested hierarchy, astrophysical systems are embedded in the nonzero Galactic and cosmic ambient floor ($\xi_{\rm env}\simeq0.32\text{--}0.52$, where $P_{,X}\simeq9.7\text{--}12.6$ is well away from strong coupling), while at isolated saddle points where $\nabla\phi_{\rm tot}=0$ a canonical regulator $P_{,X}=\epsilon+k\sqrt{\xi}+2\xi$ is invisible to the growth budget at any tested $\epsilon$ (step 62: $\chi^2$ changes by $0.09$ for $\epsilon=0\to1$ — the $\sim30\sigma$ and growth exclusions of the unit-term candidates trace to their asymptotic behaviour, not to data discrimination of a small constant), and the SPARC window tolerates $\epsilon\lesssim0.1$ at zero data cost with a marginal improvement (rms $0.2098$ dex), before order-unity values reintroduce a Keplerian tail below $g_{\rm bar}\sim(\epsilon/k)g_t$. The admissible regulator class is therefore a small constant $\epsilon\lesssim0.1$ (together with the physical ambient floor), sufficient for well-posedness at the marginal-ellipticity point without disturbing any tested observable.

*The Nested Hierarchy and the Pairwise Projection.* The use of a two-body operator does not imply the universe consists of isolated pairs in a vacuum. The full reality is a continuous nested hierarchy: a global cosmological baseline hosts galactic ambients, which in turn host solar and terrestrial wells. At Earth's surface, the field decomposition is overlapping and nested (approximately 93% galactic, 6.5% solar, 0.5% terrestrial). The two-body operator $\mathcal{S}_{\rm eff}$ is simply the effective pairwise projection of the master environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$ — whose covariant EFT form, unifying the gradient and density control parameters, is constructed in Paper 26 §2.5 — when solving a specific two-body Keplerian orbit (such as Saturn around the Sun, or wide binaries). It describes how the mutual separation $s$ allows the local gradient to recover against the ambient Galactic floor.

*Finite-probe boundary effects.* Coherent transponder phase turnaround cancels the endpoint conformal clock factor in the idealized link, but it is not a proof that every propagation or finite-probe correction to a complete round trip vanishes. A spacecraft's density alone does not set the extent of a kinetic screening region: its scalar charge, ambient gradient, body size, and the time-dependent boundary-value solution determine any such effect. A moving, non-reciprocal disformal boundary contribution must be derived from the same matter metric and checked against the actual uplink/downlink geometry; it cannot be inferred from a curl of $P_{,X}d\phi$ or from the spacecraft's speed alone. The present analysis neither explains the published coherent flyby residuals by this mechanism nor supplies a bound on it.

*The Flyby Reduction Artefact.* Because the two-body operator strongly screens the Earth-vicinity Temporal Shear, the $\sim 10^{-3}\,\mathrm{m/s}$ Earth flyby anomaly in $\Delta v$ cannot be a Temporal Shear anomaly. Instead, it is identified as a candidate Clock-Sector Reduction Artefact. Orbit determination codes reconstruct velocities using an assumed GR time standard; highly asymmetric, fast hyperbolic flybys accumulate rapid proper-time offsets at perigee due to the Temporal field's gradient, which are misread by the separate pre- and post-encounter fits as a $\Delta V$ discontinuity *where the tracking link carries the spacecraft's own time reference*. The channel accounting is exact: in a fully coherent two-way transponder link the endpoint conformal factors cancel identically at the turnaround ($\nu_{\rm ret} = R\,\nu_u$), so the term survives only in clock-carrying observables — one-way, non-coherent, or regenerative-ranging links and onboard-clock telemetry — while the published anomaly catalogue was recorded in the coherent class (Paper 15 §5.2). The artefact is therefore a falsifiable prediction for clock-carrying flyby passes — a $c\,\eta \sim$ m/s raw signature at perigee — rather than an attribution of the recorded anomalies. This is consistent with the absence of the effect in circular orbits (GP-B, LAGEOS, GRACE), and with its intermittency across hyperbolic encounters, which a geometry- and pipeline-dependent reduction artefact predicts but a fixed dynamical anomaly does not.

*Cassini constraint interpretation.* The Cassini Shapiro-delay measurement is a round-trip radio ranging experiment performed during solar conjunction. The signal path samples the Sun's deep potential well, which in TEP corresponds to the strongly screened solar environment of the single-body response. The bound $S_\Sigma^{(\odot)} \lesssim 1.15\times 10^{-5}$ ($5.8\times 10^{-6}$ on the isotropic benchmark; $|\gamma_{\rm PPN}-1|\simeq 4.32\times 10^{-8}$ at $1.6\,R_\odot$ on the selected two-branch kinetic sector) applies to the solar-vicinity environment along the signal path, not to the dilute interstellar medium where the effective coupling recovers for wide-binary dynamics. The pairwise factor $\mathcal S_{\rm eff}(s)$ governs two-body orbital dynamics and is not the projection tested by a massless probe; the Cassini channel is therefore governed by the effective screened single-source solar response, represented in the PPN limit by $S_\Sigma^{(\odot)}$ through the screened-limit PPN relation $\gamma_{\rm PPN}-1 = -2\beta_A^2 S_\Sigma/(1+2\beta_A^2 S_\Sigma)$ of §7 (Step 80).

*Wide-binary and cross-scale consistency.* The wide-binary velocity excess (Paper 13, $\alpha_{\rm sat} = 0.366 \pm 0.012$, $R_s = 2{,}646 \pm 182$ AU statistical, $\pm 609$ AU total including systematic uncertainty) is a geodesic kinematics observable. The cross-scale architecture rests on the universal two-body operator $\mathcal R(s)=\mathcal S_\Sigma(X_{\rm env})^2\,y(s)$. The ten-order-of-magnitude gap between the Cassini and wide-binary response requirements is resolved naturally by the nested structure of that single operator: hierarchical Solar-System pairs sit deep inside the dominant member's shell where $\mathcal R\sim s^{4}$ ($\sim10^{-23}$ at the conjunction separation, $\sim10^{-11}$ at Saturn), while comparable-mass field binaries sample the saturated-ambient regime where the transition profile is $\mathcal R\sim s^{4/3}$. The cross-scale structure spans from the solar-conjunction separation $s = 1.6\,R_\odot$ through $1$ AU (LLR), $9.5$ AU (Saturn), and up to $2{,}646$ AU (wide binaries), satisfying the stated cross-scale benchmark constraints within the effective response model; the Cassini bound itself acts on the single-body charge $S_\Sigma^{(\odot)}$ rather than on this pairwise factor. The terrestrial scale is independently resolved, as the Earth-vicinity Temporal Shear is strongly screened, moving the Earth-flyby anomaly into the clock sector as a reduction artefact — scoped to the link classes that carry the term (Paper 15 §5.2).

*Recovery operator — defining constraints.* Locally the time field recovers Temporal Shear (a geometric acceleration) only in low-acceleration, two-source configurations, and saturates everywhere else. The recovery operator is fixed by six constraints: F1 — excludes the inverse-power thin-shell branch: with $\beta_A = -1$, the inverse-power potential has no admissible minimum and predicts the wrong environmental direction; F2 — not a universal hard-gradient cap (a constant gradient-cap scale $g_{\rm cap} \sim g_{\rm TEP}$ would leave an unsuppressed Solar-System acceleration floor $\approx 2\beta^2 g_{\rm cap} \approx 10^{-9}$ m s$^{-2}$, $10^5\times$ the Saturn bound); F3 — $K > 3\Omega_m$ (kinetic-sector stability bound); F4 — per-system recovery steepness $k \geq 4$ in separation, derived from the Saturn ephemeris bound and the wide-binary transition radius; the corresponding two-body configuration-dependent realization is developed in the companion screening analyses; F5 — transition radius larger in denser environments, amplitude rises with mass ratio $q$, transition radius scales with primary mass while amplitude falls with it (weaker self-screening in lower-mass systems revealing larger unsuppressed response; measured, Paper 13; the demographic read $R_s\propto M^{1/3}$ is degenerate with the disk size–mass relation, while the derived kinetic completion predicts $R_s\propto M^{1/2}$); F6 — environmental coupling normalisation differs by $\approx 4$–6% between host disks and anchors (measured, Paper 11). The six conditions F1–F6 are the physical consistency constraints satisfied by the derived kinetic completion; the scaling is derived from the master action $P(X,\phi) = -V(\phi) + \frac{2k}{3\Lambda_X^2}X|X|^{1/2} + X|X|/\Lambda_X^4$ ($k\simeq16$ fixed by $a_0$) rather than prescribed ad-hoc. F3 is the screening-sector counterpart of the invertibility, hyperbolicity, and no-ghost conditions of Section 4: those conditions are certified for the adopted noncanonical completion by the sector scan of Appendix E (step 61), which confirms strict positivity of the principal coefficients and hyperbolicity/no-ghost conditions ($Z_\perp > 0$, $Z_\parallel > 0$, $Z_t > 0$) across all regimes: on the cosmological timelike branch the scalar fluctuation sound speed is subluminal ($c_s^2 \in [1/3, 1/2]$ on the selected two-branch sector), while on the static spacelike branch the spatial background gradient produces the anisotropic characteristic cone with widened longitudinal velocity $v_\parallel^2 = (P_{,X}+2XP_{,XX})/P_{,X} \in [2, 3]$ on the selected sector without ghost, gradient, or causal instability on the global temporal foliation (Babichev, Mukhanov & Vikman 2008; cf. Adams et al. 2006).

Variation with respect to the Einstein-frame metric, $\phi$, and matter fields gives the Einstein-frame field equations, scalar equation of motion, and matter-frame conservation law.

## 2.3 Field equations and conservation laws

The field equations below are displayed first in the baseline canonical limit ($P \to X - V$, $K\to 1$, $\alpha_{\rm GB}f(\phi)\mathcal{G}\to 0$). The general noncanonical completion adopted for the screening, wide-binary, and astrophysical sectors is stated directly alongside them. Varying the action with respect to the Einstein-frame metric, $\phi$, and the matter fields yields three sets of equations. The variation displayed in this subsection is the sector-isotropic evaluation of the canonical shorthand metric — exact for the homogeneous background mode and for the nonrelativistic (dust) weak-field source used below. On the lapse-resolved realization of Section 2.2 the inhomogeneous conformal response acts instead through the $U_\mu U_\nu$ channel, so the local $u$-field couples to $\mathcal T^{\mu\nu}U_\mu U_\nu$ — the matter energy density rather than the trace alone — while recovering the identical nonrelativistic source $\nabla\cdot(P_{,X}\nabla\phi)=\rho_*A_{,\phi}$; the homogeneous-mode statement that radiation sources the conformal sector only weakly (through $T\simeq0$) applies to the isotropic direction, not to the resolved lapse variation.

### Einstein-frame field equations

$$G_{\mu\nu} = \frac{1}{M_{\rm Pl}^2}\left[ T_{\mu\nu}^{(\phi)} + T_{\mu\nu}^{(m)} \right],$$

with the scalar stress-energy in the baseline canonical limit

$$T_{\mu\nu}^{(\phi)} = \nabla_\mu\phi \nabla_\nu\phi - g_{\mu\nu}\left[\frac{1}{2}(\nabla\phi)^2 + V(\phi)\right].$$

For the operative noncanonical completion $P(X,\phi)$ with $X = -\frac12(\nabla\phi)^2$, the scalar stress-energy tensor is

$$T_{\mu\nu}^{(\phi)} = P_{,X}\nabla_\mu\phi \nabla_\nu\phi + P(X,\phi)\,g_{\mu\nu},$$

which reduces identically to the canonical form for $P = X - V(\phi)$.

The Einstein-frame matter stress-energy $T_{\mu\nu}^{(m)}$ is obtained from $S_m[\tilde{g}]$ by functional differentiation with respect to the inverse Einstein-frame metric $g^{\mu\nu}$:

$$T_{\mu\nu}^{(m)} = -\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g^{\mu\nu}} = \frac{\sqrt{-\tilde{g}}}{\sqrt{-g}}\,\tilde{T}_{\alpha\beta}^{(m)}\,\frac{\partial\tilde{g}^{\alpha\beta}}{\partial g^{\mu\nu}},$$

where $\tilde{T}_{\alpha\beta}^{(m)} = -\frac{2}{\sqrt{-\tilde{g}}}\frac{\delta S_m}{\delta\tilde{g}^{\alpha\beta}}$ is the matter-frame stress-energy. For the disformal inverse

$$\tilde{g}^{\mu\nu} = A^{-2}\left[g^{\mu\nu} - \frac{(B/A^2)\,\partial^\mu\phi\,\partial^\nu\phi}{1+(B/A^2)(\partial\phi)^2}\right],$$

the Jacobian $\partial\tilde{g}^{\alpha\beta}/\partial g^{\mu\nu}$ can be evaluated exactly. Writing $Y \equiv g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi = -2X$, $D_2=B/A^2$ and $W=1+D_2Y$, so that $\sqrt{-\tilde g}=A^4\sqrt W\,\sqrt{-g}$, variation of $S_m[\psi,\tilde g]$ with respect to $g^{\mu\nu}$ at fixed $\nabla_\mu\phi$ gives

$$T_{\mu\nu}^{(m)} = A^2\sqrt W\left[\tilde T_{\mu\nu}^{(m)} - \frac{2D_2}{W}\,\phi^\alpha\tilde T_{\alpha(\mu}^{(m)}\phi_{\nu)} + \frac{D_2^2}{W^2}\,(\phi^\alpha\phi^\beta\tilde T_{\alpha\beta}^{(m)})\,\phi_\mu\phi_\nu\right],$$

which reduces to the conformal result $T_{\mu\nu}^{(m)}=A^2\tilde T_{\mu\nu}^{(m)}$ at $B=0$ (indices on $\tilde T_{\alpha\beta}^{(m)}$ are lowered with the matter metric $\tilde g_{\alpha\beta}$ throughout). Thus the Einstein-frame equations reduce to a scalar-tensor theory with conformally-coupled matter at leading order; disformal corrections enter through the $\phi$-contraction terms at $O(B)$ and are suppressed by the same multi-messenger bounds that constrain the light-cone tilt. The exact (untruncated) source would be required in any regime where the volume-preservation condition of Section 8 were met by a large disformal correction, since the small-$B$ expansion is not uniformly valid there; Section 8 shows that no such regime is physically admissible — the reconstruction is excluded by its own null cone — so the untruncated source is needed only in the formal reconstruction test itself, not on the realized background.

### Scalar equation of motion

For the scalar equation it is useful to define the matter-frame stress tensor by

$$\tilde T^{\mu\nu} \equiv \frac{2}{\sqrt{-\tilde g}} \frac{\delta S_m}{\delta \tilde g_{\mu\nu}} .$$

This is equivalent to the covariant definition

$$\tilde T_{\mu\nu} = -\frac{2}{\sqrt{-\tilde g}} \frac{\delta S_m}{\delta \tilde g^{\mu\nu}},$$

with indices raised and lowered using $\tilde g_{\mu\nu}$.

The density-weighted tensor is also defined as

$$\mathcal T^{\mu\nu} \equiv \frac{\sqrt{-\tilde g}}{\sqrt{-g}} \tilde T^{\mu\nu}.$$

Varying the matter metric with respect to $\phi$ gives

$$\delta_\phi \tilde g_{\mu\nu} = 2AA_{,\phi}g_{\mu\nu}\delta\phi + B_{,\phi}\nabla_\mu\phi\nabla_\nu\phi\,\delta\phi + B \left( \nabla_\mu\delta\phi\,\nabla_\nu\phi + \nabla_\mu\phi\,\nabla_\nu\delta\phi \right).$$

After integrating the derivative terms by parts, the scalar equation can be written as

$$\Box\phi - V_{,\phi} = -\mathcal Q,$$

where

$$\mathcal Q = AA_{,\phi}g_{\mu\nu}\mathcal T^{\mu\nu} + \frac12 B_{,\phi} \mathcal T^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi - \nabla_\mu \left( B\mathcal T^{\mu\nu}\nabla_\nu\phi \right).$$

Equivalently,

$$\Box\phi = V_{,\phi} - AA_{,\phi}g_{\mu\nu}\mathcal T^{\mu\nu} - \frac12 B_{,\phi} \mathcal T^{\mu\nu} \nabla_\mu\phi\nabla_\nu\phi + \nabla_\mu \left( B\mathcal T^{\mu\nu}\nabla_\nu\phi \right).$$

In the conformal limit $B\to0$, this reduces to the standard conformally coupled scalar-tensor source equation,

$$\Box\phi - V_{,\phi} = - AA_{,\phi}g_{\mu\nu}\mathcal T^{\mu\nu}.$$

For the operative noncanonical sector $P(X,\phi)$, variation of the action yields the noncanonical equation of motion

$$\nabla_\mu\left(P_{,X}\nabla^\mu\phi\right) + P_{,\phi} = -\mathcal Q,$$

which recovers the canonical $\Box\phi - V_{,\phi} = -\mathcal Q$ form where a canonical kinetic term is restored and $P_{,X} \to 1$ with $P_{,\phi} \to -V_{,\phi}$. On the selected two-branch sector $P_{,X}=1$ is not an asymptotic limit — it is attained only at the isolated value $\xi\simeq4\times10^{-3}$; the weak-gradient environment sits instead on the $k\sqrt{\xi}$ branch. This noncanonical form is the operator whose static limit $\nabla\cdot(P_{,X}\nabla\phi) = \rho_* A_{,\phi}$ derives the screening stiffness and wide-binary response throughout Sections 2.2 and 7.

Equivalently, in terms of the effective scalar coupling

$$\alpha(\phi)\equiv \frac{d\ln A}{d\phi},$$

the conformal source is proportional to the matter trace. Nonrelativistic matter sources the scalar through $T\simeq-\rho$, while radiation with $T\simeq0$ weakly sources the conformal sector, as used in the cosmological discussion.

### Matter-frame conservation law

$$\tilde{\nabla}_\mu \tilde{T}^{\mu\nu}_{(m)} = 0,$$

which follows from diffeomorphism invariance of $S_m[\tilde{g}]$ and implies that non-gravitational test particles and light follow geodesics of the matter metric $\tilde{g}_{\mu\nu}$. In the Einstein frame, matter and the scalar exchange energy-momentum. Diffeomorphism invariance gives

$$\nabla_\mu T^{\mu\nu}_{(m)} = \mathcal Q\nabla^\nu\phi,$$

while the scalar stress tensor satisfies

$$\nabla_\mu T^{\mu\nu}_{(\phi)} = -\mathcal Q\nabla^\nu\phi .$$

Therefore the total Einstein-frame stress tensor is conserved:

$$\nabla_\mu \left( T^{\mu\nu}_{(m)} + T^{\mu\nu}_{(\phi)} \right) = 0.$$

The matter-frame conservation law remains

$$\tilde\nabla_\mu\tilde T^{\mu\nu}_{(m)}=0,$$

because matter is minimally coupled to $\tilde g_{\mu\nu}$.

These equations define the EFT structure used throughout the paper. The scalar source $\mathcal Q$ displays the leading conformal-disformal matter coupling explicitly.

## 3. Operational Foundations: Measurement, Simultaneity, and One-Way Light

## 3.1 What is actually measured

No measurement of c uses null proper time along the photon's worldline; that would be $c = dx/d\tau$ with $d\tau = 0$. Instead, a distance is compared with an elapsed proper time on an observer's clock. For two spatially separated clocks A and B with worldlines $\gamma_A$, $\gamma_B$ and a light path $\lambda$ from A to B, the measured one-way time $t_{AB}$ is the difference in B's proper time between emission and reception, after a synchronization convention has assigned simultaneity between A and B. Two-way measurements are convention-independent; one-way measurements are not, unless an invariant observable is provided.

## 3.2 The synchronization problem in a dynamical-time background

Einstein synchronization assumes (i) reciprocity of propagation and (ii) homogeneity of time standards along the path. If the rate $d\tau/dt$ varies spatially or directionally, synchronization by light exchange over extended loops becomes path-dependent. The proper formalism invokes the congruence $u_\mu$ of observers (clocks) and the null geodesics $k^\mu$ of $\tilde{g}_{\mu\nu}$. Define simultaneity as orthogonality to $u_\mu$ (where Frobenius integrability allows it), and define time transport by mapping a proper-time interval at A to B using $\tilde{g}$-null signals. In GR, the non-closure of such transports arises from rotation of the congruence (vorticity) and spacetime curvature (Sagnac/Shapiro); both can be modeled and removed. In a dynamical-time geometry with disformal couplings, there is an additional, tiny non-exact contribution to time transport that cannot be removed by coordinate choices: a synchronization holonomy.

## 3.3 A convention-independent observable: synchronization holonomy

The operational observable is not a raw one-way speed and not the integral of a scalar proper-time differential. It is the residual non-closure of synchronization transport around a closed loop after subtracting the GR prediction.

Let $\tilde{\sigma}$ denote the matter-frame synchronization transport one-form induced by $\tilde{g}_{\mu\nu}$ and the chosen clock congruence, and let $\sigma_{\rm GR}$ denote the corresponding GR one-form including Sagnac, gravitomagnetic / Lense–Thirring, Shapiro delay, gravitational redshift, station motion, clock-scale realization, and reference-frame corrections. The residual holonomy is:

$$H_{\rm resid}(C) \equiv \oint_C (\tilde{\sigma} - \sigma_{\rm GR}) = \iint_\Sigma (d\tilde{\sigma} - d\sigma_{\rm GR}), \qquad C = \partial\Sigma.$$

Equivalently, defining $\tilde{F} = d\tilde{\sigma}$ and $F_{\rm GR} = d\sigma_{\rm GR}$:

$$H_{\rm resid}(C) = \iint_\Sigma (\tilde{F} - F_{\rm GR}).$$

This quantity is invariant under admissible synchronization re-gaugings because the matter-frame connection $\tilde\sigma$ and the corresponding GR reference connection $\sigma_{\rm GR}$ shift by the same exact one-form, leaving the residual connection $\Delta\sigma=\tilde\sigma-\sigma_{\rm GR}$ unchanged. Equivalently, any representative shift by an exact form integrates to zero around a closed loop.

### Conformal exactness and vanishing loop holonomy

In the conformal-only subclass ($B = 0$), the scalar contribution to local clock-rate transport is generated by the exact one-form $\omega^{(A)} = d\ln A$. For any closed loop $C$ in a simply connected region where $A(\phi)$ is smooth and single-valued:

$$\oint_C \omega^{(A)} = \oint_C d\ln A = 0.$$

Thus a conformal-only scalar may affect local rates and open-path redshift comparisons, but it cannot by itself generate a closed-loop residual synchronization holonomy. A leading-order nonzero $H_{\rm resid}$ requires residual synchronization curvature $d(\tilde{\sigma} - \sigma_{\rm GR}) \neq 0$, which in this framework arises from the disformal sector, non-metricity, or other explicitly non-exact transport structure.

This reframes "variable c" claims: the invariant diagnostic is not a raw one-way c, but $H_{\rm resid}$. A nonzero $H_{\rm resid}$ means simultaneity is not integrable beyond GR; this is what dynamic time does to measurement.

![Synchronization holonomy on a spacelike slice](site/figures/fig2_holonomy.svg)

Figure 2. Synchronization holonomy on a spacelike slice in the matter frame. The residual invariant $H_{\rm resid}$ subtracts all GR corrections (Sagnac, gravitomagnetic / Lense–Thirring, Shapiro delay, gravitational redshift, station motion, clock-scale realization, and reference-frame corrections), isolating dynamical-time effects.

## 4. Disformal Invertibility, Causality, and Hyperbolicity

## Invertibility and signature

The inverse of $\tilde{g}_{\mu\nu}$ exists for $A>0$ and $1 + (B/A^2)(\partial\phi)^2 \neq 0$:

$$\tilde{g}^{\mu\nu} = A^{-2} \left[ g^{\mu\nu} - \frac{(B/A^2) \partial^\mu\phi \partial^\nu\phi}{1 + (B/A^2)(\partial\phi)^2} \right].$$

For Lorentzian signature, require $A>0$ and $B(\partial\phi)^2 > -A^2$. $B$ is assumed small and gradients bounded in all regimes of interest.

## Causality

The matter cone is inside or equal to the gravitational cone when $B \geq 0$ and gradients are modest; no closed causal curves arise for small $B$. When the observable disformal deformation $\frac{B(\phi)}{A^2(\phi)}(\partial\phi)^2$ is negligible along the relevant late-time propagation paths, gravitational and matter null cones coincide to the required observational accuracy. Laboratory resonator tests independently constrain orientation-dependent components of the same local disformal deformation in the terrestrial environment, while multi-messenger observations strongly constrain its integrated realization along astrophysical propagation paths. These constraints bound the complete disformal deformation on the realized scalar background rather than requiring $B(\phi)$ itself to vanish identically. With $B$ small, any phase differences in propagation are minute and bounded by multi-messenger results.

## Hyperbolicity

The canonical scalar equation has a hyperbolic principal operator on a Lorentzian background; $V''>0$ supplies positive-mass stability near an equilibrium. In the small-disformal EFT used here the theory is treated perturbatively about that background. On the admissible branch $B\geq 0$ the disformal perfect-fluid principal coefficients are $Z_t=1+B(1+w)\rho/A^2\geq 1$ and $Z_s=1+B(1-w)\rho/(3A^2)\geq 1$ for $-1\leq w\leq 1$ and $\rho\geq 0$, so this sector is strongly hyperbolic wherever the matter metric is non-degenerate. The elliptic counterexamples require $B<0$. The null-cone condition therefore selects $B\geq0$ as the unconditionally admissible branch: on it the signature condition $B(\partial\phi)^2>-A^2$ and the hyperbolicity inequalities hold wherever the matter metric is non-degenerate, and the matter cone lies inside or on the gravitational cone. $B<0$ is not excluded a priori; it carries a pointwise admissibility condition — signature ($|B|(\partial\phi)^2<A^2$), hyperbolicity ($Z_t,Z_s>0$), and causality (the matter cone remaining inside the gravitational cone) — evaluated on each realized configuration, and sign-indefinite forms $B(\phi)$ inherit the same condition along the realized profile. The configurations computed to date fail it on the negative branch: the volume-balance disformal reconstruction returns its required $b(\varphi)$ negative and drives $Q=1+(B/A^2)(\partial\phi)^2\to-1$ — a matter-metric degeneracy with superluminal longitudinal photons along $\nabla\phi$ (Appendix E, R5). The exclusion is therefore conditional rather than axiomatic: any future sign-indefinite realization enters the corpus through this ledger, not through a relaxation of it. The ledger is evaluated numerically on the corpus's realized scalar profiles in step 52. On the GW170817-type galactic profile the signature wall sits $10^{15}$ above the multimessenger bound and the fluid-hyperbolicity wall $10^{6.5}$ above it, so a negative excursion is formally admissible there far beyond phenomenologically relevant amplitudes; in dense bodies the same fluid wall collapses to $|B_0|\lesssim0.7$ inside Earth and $\lesssim2\times10^{-16}$ inside a neutron star, because the condition is gradient-independent ($R\mathcal D=\rho\,\mathrm{shape}(u)\,R_H^2/(M_{\rm Pl}\hbar c)^2$, a coupling–density bound $|B(\phi)|\rho\lesssim A^2$). In the timelike-gradient drift channel the roles reverse: the negative branch is lapse-safe while the positive branch carries the lapse boundary $u(\varphi)=B(\varphi)\dot{\bar\phi}^{\,2}/A^2<1$. The relevant bound is realized rather than fiducial: under the adopted clock map $\bar\varphi=\ln(1+z)$, with $\dot u=H_T(z)\approx4$–$5\,H_0$ under the imported $\Lambda$CDM drift and $A\approx0.22$–$0.29$ at the absorber epochs, the ambient lapse cap on the canonical envelope falls to $B_0\lesssim0.03$ near $z\approx3$ — three-to-four orders below the phenomenological requirements ($B_0^{\rm req}\sim52$ in the per-length estimate, $\simeq420$ under the step-63 NFW deprojection) and the conservative multimessenger envelope $B_0\lesssim78$. Under the TEP-native drift ($H_T^2/H_0^2\approx0.22$–$0.44$ on the descent and selected-tracking branches) the cap loosens to $B_0\lesssim1.0$–$4.5$ and the required ambient suppression relaxes to $G_{\rm ambient}\lesssim0.01$–$0.06$ at the phenomenological $B_0$ — the phenomenological amplitudes remain supercritical on the bare ambient by a factor $\sim30$–$50$ under the native rate, so ambient closure is lapse-mandated on either drift. The negative branch is simultaneously bounded at $|B_0|\lesssim10^{-2}$ by the ambient fluid wall at absorber-epoch densities, a rate-independent wall (step 52; Paper 29, Gates 10-b/10-c). Neither sign can therefore carry the phenomenological amplitude on the bare ambient: ambient closure of the environment-modulated form $B_{\rm eff}=B(\phi)G(X_{\rm local})$ is lapse-mandated independently of the absorber-amplitude argument, and the required suppression is consistent with the in-well saturation $G\sim10^{-3}$ that Gate 10-b infers phenomenologically — the lapse wall and the absorber amplitudes select the same modulated form, while a uniform positive $B_0$ has no realized window. The step-63 NFW deprojection independently excludes the uniform-$B_0$ Refsdal carrier ($B_0^{\rm req}\simeq420$ against the bound $\lesssim78$), while the same gated form returns $B_0^{\rm req}\simeq0.02$ against a same-convention bound $\sim10^{10}$ — the modulated coupling leaves that channel viable as well. The dangerous sign is gradient-character selective rather than universal, which is the computed content of the branch statement. The characteristic analysis of the noncanonical completion $P(X)$ is treated separately from the disformal matter-metric analysis and is summarized in §2.2 and Appendix E (step 61); the strong-field black-hole interior in which $|(B/A^2)X|$ is not small is treated in Paper 28. Multi-messenger observations strongly constrain the relevant late-time disformal combinations along astrophysical paths, while the realized background solution must independently satisfy the signature condition in other environments. Within the small-$B$, canonical-scalar EFT regime considered here, no ghost or gradient instability is introduced at leading order. The EFT is valid below the disformal scale $M$, with higher-dimensional operators suppressed. The specific phenomenological window for the cutoff scale $M$ is bounded from below by the requirement that the EFT remains strictly valid across terrestrial and solar-system density gradients, and from above by the requirement that $B(\phi)$ generates a detectable macroscopic holonomy without violating the $|c_\gamma - c_g|/c$ multi-messenger constraints. These conditions are established within the small-$B$ EFT regime: companion analyses that remain within it, such as the conformal-sector early-universe calculations of TEP-BBN, inherit them directly, while strong-field applications such as the black-hole construction of Paper 28 extend beyond it and must re-establish them on the realized configuration.

## 5. Local Lorentz Invariance, Proper Time, and the Emergence of c

## Proper time

Clocks measure $d\tau^2 = -\tilde{g}_{\mu\nu} dx^\mu dx^\nu/c^2$. In a local lab with small velocities and $\partial_0\phi \ll |\nabla\phi|$, $-\tilde{g}_{00} \approx A(\phi)^2 (-g_{00}) - B(\partial_0\phi)^2$, so

$$\frac{d\tilde{\tau}}{d\tau_g} = A(\phi),$$

up to $O(B)$ corrections. This "dynamical time law" rescales all frequency standards locally.

## Local c

All small labs measure an invariant $c$; the conformal factor rescales both clocks and rulers uniformly, preserving null cones. Thus the empirical fact "$c$ is constant" remains a theorem of the theory at the local level.

## Global measures

Global "speeds" involve synchronized endpoints. Because the matter metric depends on the dynamical temporal field, global synchronization can acquire non-integrability when the resulting transport contains a disformal or otherwise non-exact component. That non-integrability—not a local violation of Lorentz invariance—is the source of observable departures.

## 6. Synchronization in a Dynamical Time: One-Way Light and Holonomy

## Operational definitions

Two-way light speed is synchronization-independent and has established $c$'s local invariance. One-way measures require synchronized clocks at A and B; Einstein synchronization assumes time-orthogonal slices and propagation symmetry. In a dynamical $\phi$ background, slow clock transport and one-way synchronization are path and history dependent.

## Key theorems

### Theorem 1 (Conformal null-cone invariance)

For $\tilde{g}_{\mu\nu} = A(\phi)^2 g_{\mu\nu}$, null vectors of $g_{\mu\nu}$ are null for $\tilde{g}_{\mu\nu}$. Maxwell's action is conformally invariant in 4D, so photon trajectories are null with respect to both metrics. Gravitational and electromagnetic waves share null cones when $B = 0$ at late times.

### Theorem 2 (Exact direct conformal propagation null)

In the purely conformal limit $B = 0$, the multiplicative conformal factor $A^2(\phi)$ preserves the null cone exactly. A static conformal rescaling therefore produces no direct direction-odd same-path photon-propagation delay:

$$\Delta t_{\rm prop}^{(A)} = 0.$$

The conformal clock connection is exact, $\omega^{(A)} = d\ln A$, and hence $\oint_C d\ln A = 0$ in a smooth simply connected region. Observable conformal effects remain possible through clock-rate differences, accumulated proper-time histories, and open-path redshift comparisons. Scalar-induced backreaction on $g_{\mu\nu}$, time-dependent geometry, the disformal sector, or another non-exact transport structure are separate channels and are not excluded by this theorem.

**Proof sketch.** For $\tilde{g}_{\mu\nu} = A^2(\phi)\,g_{\mu\nu}$ with $B = 0$, a photon null condition $\tilde{g}_{\mu\nu}k^\mu k^\nu = 0$ reduces to $g_{\mu\nu}k^\mu k^\nu = 0$ because $A > 0$. The conformal factor therefore cancels exactly from the null propagation condition and cannot act as a direction-dependent refractive index at fixed $g_{\mu\nu}$; see Appendix A2 for the full derivation. The scalar's backreaction on $g_{\mu\nu}$ can still alter ordinary gravitational lensing and Shapiro delays, which require a solved metric and are not implied by a curl of a screened scalar current.

## Synchronization one-form and holonomy

Decompose $\tilde{g}_{\mu\nu}$ in 3+1 form:

$$\tilde{g}_{\mu\nu} dx^\mu dx^\nu = -\tilde{N}^2 dt^2 + \tilde{h}_{ij} (dx^i + \tilde{N}^i dt)(dx^j + \tilde{N}^j dt).$$

Define the coordinate synchronization connection by the threading representative

$$\tilde{\sigma}_i = \frac{\tilde g_{0i}}{\tilde g_{00}},$$

up to the overall sign convention adopted for simultaneity transport. In ADM variables, for small shift,

$$\tilde{\sigma}_i \simeq -\frac{\tilde N_i}{\tilde N^2}.$$

The representative used throughout is the fixed coordinate-time connection $\tilde\sigma_i=\tilde g_{0i}/\tilde g_{00}$ defined above, applied identically to $\tilde\sigma$ and $\sigma_{\rm GR}$. A proper-time-normalized representative $N\tilde\sigma$ is a different loop observable when the lapse varies spatially — $\oint_C N\tilde\sigma\neq N\oint_C\tilde\sigma$, and under a re-gauging $\tilde\sigma\to\tilde\sigma-d\chi$ the shift $N\,d\chi$ is not exact — so normalization is part of the observable's definition. The operational conversion is specified at link level: each measured one-way desynchronization between neighbouring clocks, $\Delta\tilde\tau_{\rm link}$, is divided by the local matter-frame lapse $\tilde N$ to recover the connection element $\tilde\sigma_i\,dx^i=\Delta\tilde\tau_{\rm link}/\tilde N$, and summing over the loop reconstructs $\oint_C\tilde\sigma$; the identical reduction applied to the GR model yields $H_{\rm resid}$. Appendix A3 supplies the connection's transformation properties and leading disformal contribution; the timestamp reduction follows from the fixed convention above rather than from an additional assumption. For a closed spatial loop $C$, the raw loop integral is

$$H = \oint_C \tilde{\sigma},$$

but the physical TEP observable is the GR-subtracted residual

$$H_{\rm resid}(C) = \oint_C (\tilde{\sigma} - \sigma_{\rm GR}) = \iint_\Sigma (d\tilde{\sigma} - d\sigma_{\rm GR}), \qquad C = \partial\Sigma.$$

In GR with stationary spacetimes, the raw holonomy reproduces the Sagnac/gravito-magnetic effect; these known contributions are subtracted using geodesy and ephemerides. The residual holonomy beyond GR is therefore

$$H_{\rm resid}(C) = \iint_\Sigma \left(d\tilde{\sigma}-d\sigma_{\rm GR}\right), \qquad C=\partial\Sigma ,$$

where $\sigma_{\rm GR}$ includes the standard Sagnac, gravitomagnetic/Lense–Thirring, Shapiro, gravitational-redshift, station-motion, clock-scale, and reference-frame contributions. This residual vanishes in SR/GR with $B=0$ and stationary $A$; it becomes non-zero only where the transport produces a residual circulation, requiring the drift rate's gradient to be misaligned with the field gradient — the disformal-curvature condition $d(\delta\tilde\sigma) = -(B/A^2)\,d(\dot\phi/N^2)\wedge d\phi$ of §7. The Temporal Topology is thereby expressed as a measurable geometric consequence. Because $\tilde\sigma_i=\tilde g_{0i}/\tilde g_{00}$ is a ratio, the conformal factor cancels from it identically, so a time-dependent $A$ cannot generate synchronization holonomy; the residual is generically nonzero only with disformal corrections ($B \neq 0$) or other non-exact transport. Path-dependent accumulated proper time under a time-dependent $A$ is a distinct observable, the clock anholonomy of Section 10B. Configurations in which $C$ does not bound a smooth surface — non-simply-connected transport, or a connection that is locally closed but not globally exact — are not described by this Stokes expression and are treated as a separate topological case in Appendix A3.

## Gauge and protocol invariance

$\tilde{\sigma}$ and $\sigma_{\rm GR}$ are synchronization-connection representatives for the same physical clock network and the same loop, evaluated in the matter-frame model and the corresponding GR reference model respectively. A synchronization re-gauging $t\to t+\chi(x^i)$ shifts both representatives by the same exact one-form. Therefore the GR-subtracted connection

$$\Delta\sigma=\tilde{\sigma}-\sigma_{\rm GR}$$

and its closed-loop integral

$$H_{\rm resid}(C)=\oint_C\Delta\sigma$$

are invariant under admissible synchronization changes. This addresses the critique that holonomy is conventional: the observable is the residual loop class of $\Delta\sigma$, not the exact part of any raw one-way synchronization convention.

## Disformal corrections to σ̃

For small $B$ and modest $\partial\phi$,

$$\tilde{N} \approx A N \left[1 - \frac{B}{2A^2 N^2} (n \cdot \partial\phi)^2\right], \quad \tilde{N}_i \approx A \left[N_i + \frac{B}{A^2} (\partial_i\phi)(n \cdot \partial\phi)\right],$$

with $n^\mu$ the unit normal to slices. Then $\delta\tilde{\sigma}$ denotes the leading disformal correction to the chosen synchronization representative. Its covariant clock-congruence form is derived in Appendix A3; in a hypersurface-orthogonal $3+1$ slicing it reduces, up to lapse/shift convention and overall sign, to the expression proportional to

$$-\frac{B}{A^2N}(\partial_i\phi)(n\cdot\partial\phi)\,dx^i .$$

At leading order beyond the GR-subtracted reference model,

$$d(\Delta\sigma) = d(\tilde\sigma-\sigma_{\rm GR}) \supset d(\delta\tilde\sigma),$$

where the $\Delta$ indicates the residual beyond GR. Time dependence and a spatial gradient permit, but do not guarantee, non-vanishing curl: on a fixed spatial slice the leading representative is $\delta\tilde\sigma=-C\,d_3\phi$, with $C=(B/A^2)(u\cdot\nabla\phi)$ in the local clock frame (or the corresponding lapse-weighted expression in the stated slicing), hence $d_3(\delta\tilde\sigma)=-d_3C\wedge d_3\phi$. A spatially uniform drift or any $C=C(\phi)$ gives zero circulation. In particular $B/A^2$ depends on $\phi$ alone in the stated action and contributes no separate wedge term; the spatial variation of the drift/lapse relative to $\nabla\phi$ is the operative condition. This is distinct from $d[P_{,X}d\phi]$, the curl of the scalar kinetic flux: neither that flux nor a phenomenological $S_\Sigma d\phi$ is the photon synchronization connection.

## 7. Screening, PPN, Equivalence Principle, and Disformal Bounds

## Screening

Screening in TEP is described as suppression of the locally observable Temporal Shear/source-charge sector, not as a commitment to a specific chameleon, Vainshtein, Galileon, DBI, or symmetron microphysics. Those mechanisms may be studied as candidate completions. In the effective theory used here, screening is expressed through the conformal factor $\ln A(\phi)$, its gradient $\Sigma_\mu$, and its covariance $C_A$. Source structure, environmental state, and boundary conditions suppress the locally active shear sector in screened regimes.

The saturation scale $\rho_T$ denotes the Temporal Topology saturation scale. It is not a local on/off condition of the form $\rho > \rho_T \Rightarrow$ GR and $\rho < \rho_T \Rightarrow$ active. Recovery of GR in local tests is controlled by suppression of the observable shear/source-charge sector, $\Sigma_\mu^{\text{obs}} = \mathcal S_\Sigma(\mathcal E)\Sigma_\mu$ with $\mathcal S_\Sigma \to 0$ in screened regimes, where $\mathcal E$ includes source structure, environment, boundary conditions, and density.

**Screening is not a single density switch.** It is an environmental suppression operator $\mathcal S_\Sigma(\mathcal E)$ whose observable projection depends on scale. The environmental state $\mathcal E$ includes: ambient density $\rho$, gravitational compactness $\Phi/c^2$, density gradients $\nabla\rho$, potential gradients $\nabla\Phi$, proximity to field sources, coherence volume, boundary geometry, and asymptotic horizon proximity. Each domain uses a different projection of $\mathcal E$:

- **GNSS/clock:** clock-amplitude (static conformal-depth) projection $S_A^{(\oplus)} \approx 0.65$ ($0.43$–$0.88$ across the $\rho_T$ band) of the conformal time field; the time-dependent spatial covariance is carried by $C_A$.

- **UCD/galaxy scales:** mass-radius-density saturation projection through the geometric scale $R_T(M)$.

- **NIST/laboratory G:** lab-scale geophysical column projection of $\mathcal S_\Sigma(\mathcal E)$.

- **MSP pulsars:** cluster-potential response with stellar/environmental transfer.

- **Cepheids:** galactic-potential clock-bias response.

- **JWST/high-z:** halo-potential response transferred from Cepheid prior.

- **Wide binaries:** weak-field recovery / low-acceleration environmental un-screening.

- **Flyby:** clock-sector reduction artefact in clock-carrying links (Section 2.2); the Earth-vicinity shear is pairwise-screened and carries no flyby shear signal, and the coherent two-way class of the published catalogue carries no clock term — the artefact is a prediction for one-way/non-coherent radio science.

- **LLR:** compactness-dependent Earth/Moon differential screening.

- **LHC:** proximity-saturated hadronic coherence-volume screening.

- **SPIN/QF/KIN:** subatomic proximity/topological-core screening as microscopic projection of $\mathcal S_\Sigma(\mathcal E)$.

- **C0/HC/TH:** temporal-horizon asymptotic transport and late-time conformal acoustic equivalence.

These projections are not interchangeable. Each is calibrated against the environmental variables active in its domain, and evaluating a projection outside the regime where its governing variable dominates yields screening factors that are physically meaningless, as the omitted terms can exceed the retained ones by many orders of magnitude.

The local and cosmological screening responses form a continuous nested hierarchy, not two independent regimes separated by a boundary. The temporal field varies both spatially and temporally: the rate of time differs from place to place, and at any given location it also evolves over time. The cosmological temporal background — the ambient clock-rate field $A(\phi_\infty)$ — is itself spatially varying, with gradients across large-scale structure that can differ substantially between cosmic environments, and temporally evolving within its bounded contrast. This evolution is to be distinguished from the secular cosmological drift, which is sector-asymmetric: the drift rate $H_{\rm drift}$ is carried by the matter-hosting field values sourced through $\nabla^2\phi = V_{,\phi} + \rho_* A_{,\phi}$, while the unsourced void sector relaxes toward the contemporaneous ambient configuration with excursions bounded by $|\delta u| \lesssim 10^{-3}$–$10^{-2}$ at every epoch (Section 8). The ambient's "temporal evolution" is therefore variation within the contrast band, not carriage of the secular drift — a distinction that fixes which field components enter the loop-holonomy kernel of Section 10A. Within this cosmological baseline, local environmental screening produces additional spatial fluctuations in the observable clock rate and Temporal Shear, smaller in amplitude and varying across shorter scales, and these local fluctuations themselves evolve as the local environment and the cosmological background evolve. The local response is defined relative to the cosmological baseline at the corresponding epoch, not in isolation from it. Both layers are dynamical and continuous in space and time; neither constitutes a discrete transition. The environmental state vector $\mathcal{E}$ therefore carries information at multiple nested scales simultaneously: the cosmological ambient field, the local environmental perturbation, and their coupling, all evaluated at the relevant cosmic epoch. The structure is analogous to temperature in a room: local temperature fluctuates from point to point within the room, but is also defined relative to the external temperature, which itself varies more radically between regions, and all of these temperatures change over time — the room cools at night, the outside temperature shifts with seasons. The temporal field exhibits the same nested character in both dimensions — local clock-rate fluctuations sit within a cosmological time-rate background, both vary continuously across space, and both evolve continuously over time.

The nested structure is quantitative, not merely schematic. Solving the radial field equation for each level of the hierarchy with the containing environment's field as the asymptotic condition — a single continuous profile $\phi_{\rm total} = \phi_{\rm env} + \delta\phi$, in which the conformal nonlinearity acts on the total field and the perturbation decays to the ambient value rather than to zero — yields the following nested-hierarchy decomposition (Appendix E, R1). The galactic field at the solar circle is $\phi_{\rm gal}(8\,{\rm kpc}) \simeq 2.8\times10^{-7}$ ($1.4$–$5.7\times10^{-7}$ across galactic model variants, consistent with the independent circular-velocity estimate $(v_c/c)^2 \simeq 5.4\times10^{-7}$), while the Sun's own perturbation at 1 AU is only $2.0\times10^{-8}$: the ambient field exceeds the solar contribution beyond $\sim 15\,R_\odot \approx 0.07$ AU, in the $V = 0$ baseline the ambient-to-perturbation ratio is $\sim 9\times10^{4}$ at the wide-binary separation (2646 AU), rising to $\sim 10^7$ across the stellar population — the binary is embedded almost entirely in the galactic field. At the terrestrial surface the decomposition is approximately 93% galactic, 6.5% solar, and 0.5% terrestrial: a local clock measures a small perturbation on a much larger ambient baseline, and because the ambient component is common-mode it cancels in local comparisons. For the same reason the Cassini bound is intrinsically local: it constrains the Sun's gradient perturbation $S_\Sigma^{(\odot)}$, while the ambient galactic field carries negligible gradient over AU scales and drops out of the observable. Under the nested two-body kinetic operator the pairwise shear at the conjunction separation is suppressed to $\mathcal{R}(1.6\,R_\odot) \approx 1\times10^{-23}$ — orders of magnitude deeper than required — while the Cassini bound itself constrains the single-body exterior source charge, $S_\Sigma^{(\odot)} \lesssim 1.15\times10^{-5}$ ($5.8\times10^{-6}$ on the isotropic benchmark), which the adopted Solar-System source-charge convention $S_\Sigma^{(\odot)} \sim 2\times10^{-6}$ satisfies comfortably (consistent with the kinetic completion of Section 2, whose static-branch operator reproduces the corpus screening values at zero free parameters; the exact single-source Cassini observable is a projection of the solved solar profile — the constant-charge expression retained here is the weak-field benchmark shorthand, not the output of a dedicated single-body extraction). The baseline itself is not uniform: the ambient clock rate differs by $\Delta\ln A \simeq 2.2\times10^{-7}$ between the solar circle and the galactic halo, so the proper-time field is genuinely position-dependent across the galaxy — one continuous multi-scale profile with no boundaries between levels. The nested field hierarchy has two complementary properties: the field value at any point is dominated by the largest enclosing structure (Galaxy, then Sun, then Earth), while the field gradient is dominated by the nearest mass. This gives the clock/shear split its physical content: clock rates are set by the whole nested hierarchy, while the local Temporal Shear is set by the nearest body's gradient.

The parameter $\rho_T \approx 20$ g/cm³ is a macroscopic phenomenological saturation scale for the scalar response. It is not a universal microscopic density cutoff, a binary screened/unscreened switch, or automatically applicable to quantum cores without a transfer map. For quantum and accelerator domains, the microscopic topological-core regulator is a proximity/coherence-volume projection of $\mathcal S_\Sigma(\mathcal E)$, not literal bulk density.

### Temporal Topology and Temporal Shear: Canonical Formulation

The screening ontology is organized through a sector dictionary. The Temporal Shear is defined as the gradient of the conformal factor:

$$\Sigma_\mu \equiv \nabla_\mu \ln A(\phi) = \frac{\partial \ln A}{\partial \phi} \nabla_\mu \phi = \frac{\alpha(\varphi)}{M_{\rm Pl}} \nabla_\mu \phi,$$

where $\alpha(\varphi) \equiv d(\ln A)/d\varphi$ is the dimensionless conformal coupling strength, and $\varphi \equiv \phi/M_{\rm Pl}$. For compactness one may write $\Theta \equiv \ln A$, but the canonical series notation remains $\ln A$, $\Sigma_\mu = \nabla_\mu \ln A$, and $C_A$.

The observable Temporal Shear is suppressed by the environmental screening operator $\mathcal S_\Sigma(\mathcal E)$:

$$\Sigma_\mu^{\text{obs}} = \mathcal S_\Sigma(\mathcal E) \, \nabla_\mu \ln A(\phi),$$

where the environmental state is $$\mathcal E = \{\rho, \Phi/c^2, \nabla\rho, \nabla\Phi, \text{compactness}, R_T(M), \text{proximity}, T, z, \text{boundary geometry}, \text{coherence volume}\}.$$ The common environmental state $\mathcal{E}$ organizes these domain-specific observable projections; the projections themselves need not be numerically identical.

The Temporal Topology correlation function $C_A(x,x')$ characterizes correlations of conformal-factor fluctuations:

$$C_A(x,x') \equiv \langle \delta\ln A(x) \, \delta\ln A(x') \rangle.$$

It is measured through clock/covariance data. Because $\Sigma_\mu = \nabla_\mu\ln A$, Temporal Topology and Temporal Shear are not independent fields: $C_A$ describes spatial correlations of the field value, while Temporal Shear describes its gradient structure.

The temporal correlation length $\lambda_T$ is the characteristic scale extracted from $C_A(x,x')$ in covariance measurements, not a derived algebraic combination of local fields.

The temporal saturation scale $\rho_T$ is the characteristic scale at which Temporal Topology effects saturate in screening; it is a property of the theory's non-linear regime, not a local temporal energy density.

Finally, the observable response of any measurement channel $X$ is parameterized by response coefficients $\kappa_X$:

$$\Delta O_X = \kappa_X \cdot \mathcal S_X(\mathcal E) \cdot \mathcal F_X[\Delta\ln A, \Sigma_\mu, C_A; \Phi, \rho, z],$$

where $\kappa_X$ is an observable response coefficient for channel $X$, not the microscopic conformal coupling $\beta_A$ and not a PPN coupling. The locally active PPN coupling is suppressed by the environmental/source screening factor $\mathcal S_\Sigma(\mathcal E)$ and should not be confused with channel response coefficients $\kappa_X$.

*Phantom-mass amplitude.* Temporal Shear affects matter dynamics ($\Phi_{\rm dyn}=\Phi_N+c^2\beta_A u_{\rm grav}$), whereas on the isotropic-conformal benchmark ($\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$, where spatial rulers stretch alongside clocks) the conformal factor adds no direct null-ray deflection at fixed $g_{\mu\nu}$, and under the canonical unified unstretched-space metric ($\tilde\gamma_{ij}=\gamma_{ij}$ at fixed epoch $A_{\rm sp}=1$; Section 2.2, Step 80) the temporal lapse refracts null rays at half the GR potential weight ($\Phi_{\rm lens}=\tfrac12(\Phi_{\rm dyn}+\Psi_{\rm sp})=\Phi_N+\tfrac12 c^2\beta_A u_{\rm grav}$). For spherical weak-field mass inference at a matched radius, with scalar backreaction negligible and the same mass normalization in both channels, the dynamical-to-baryonic mass ratio (equivalently the isotropic-conformal dynamical-to-lensing ratio) is

$$\begin{aligned} \frac{M_{\rm dyn}(r)}{M_{\rm bar}(r)} &= 1 + \frac{a_\phi(r)}{a_N(r)} \equiv 1 + 2\beta_A^2\,\mathcal R_{\rm dyn}(r), \\ \frac{M_{\rm lens}^{\rm GR}(r)}{M_{\rm dyn}(r)} &= \frac{1 + \beta_A^2\,\mathcal R_{\rm dyn}(r)}{1 + 2\beta_A^2\,\mathcal R_{\rm dyn}(r)} \quad (\text{unstretched-space direct refraction}). \end{aligned}$$

Here $\mathcal R_{\rm dyn}$ denotes the response extracted from the existing scalar solution and tracer dynamics, not another coupling or screening law. For an unscreened test probe, it reduces to the solved source-gradient ratio $S_\Sigma$; finite-body and pairwise approximations must retain their stated source, ambient and response assumptions. In particular, the nested benchmark $\mathcal R(s)=S_\Sigma(X_{\rm env})^2y(s)$ is not a universal lens-aperture prescription. The PPN notation below remains $\alpha_{\rm eff}=\alpha_0S_\Sigma$, with $\alpha_0=\sqrt{2}\beta_A$; its product $\alpha_0\alpha_{\rm eff}$ must not be relabelled as $\alpha_{\rm eff}^2$. Stellar-orbit weighting and lens-model refitting determine how the dynamical excess enters a measured aperture or a time-delay inference (Papers 4, 5, 19).

*Amplitude adequacy at galactic scales.* On the superseded unit-term kinetic candidate, with $\beta_A = -1$, the conformal-shear enhancement saturates at $M_{\rm dyn}/M_{\rm lens} \to 1 + \alpha_0^2 = 1 + 2\beta_A^2 = 3$ in the fully unscreened limit ($\mathcal S_\Sigma \to 1$). Galaxy-outskirt mass discrepancies are measured at factors of $5$–$10$ in extreme cases, so the conformal ceiling does not by itself carry the full dynamical anomaly. For the isolated, potential-negligible weak-field branch of $P=X-V+X|X|/\Lambda_X^4$, $P_{,X}\to1$ at small gradients, so its exterior recovery returns to an inverse-square scalar force rather than an unbounded enhancement — precisely the Keplerian-tail failure for which that candidate is rejected. This ceiling is a property of the rejected baseline, not of the selected kinetic sector.

*Selected kinetic completion.* The flat-tail branch is the promoted sector of the master action, not an alternative realization: the master-sector admissibility scan (Appendix E, step 54; `TEP/closure/`) finds $P_{,X}(\xi)=k\sqrt{\xi}+2\xi$, with $\xi=|X|/\Lambda_X^4$, to be the unique minimal completion within the surveyed power-law family — the $p=1/2$ tail exponent is uniquely selected by the flat-asymptote condition, and the retained unit term of any $1+2\xi$ or $1+k\sqrt\xi+2\xi$ form restores the Keplerian tail and is rejected. In its low-gradient, potential-negligible spherical limit, flux conservation gives $\phi'\propto1/r$ and hence $g_{\rm obs}^2\simeq a_{\rm eff}g_{\rm bar}$ and $v_{\rm flat}^4=GM_ba_{\rm eff}$. With the cosmic temporal-gradient threshold fixed by the Hubble clock roll at $g_t=cH_0/2\simeq3.40\times10^{-10}\,{\rm m\,s^{-2}}$ ($H_0=70\,{\rm km\,s^{-1}\,Mpc^{-1}}$) and the DEF coupling normalization $\alpha_0=\sqrt{2}\beta_A=-\sqrt{2}$, the natural two-branch crossover $k=16$ ($\xi_*=(k/2)^2=64$) yields $a_{\rm eff}=4\sqrt{2}\,g_t/k=cH_0/(4\sqrt{2})=1.202\times10^{-10}\,{\rm m\,s^{-2}}$ ($k=4\sqrt{2}\,g_t/a_0\simeq16.03$ at the canonical $a_0=1.20\times10^{-10}\,{\rm m\,s^{-2}}$), deriving Milgrom's $a_0\sim cH_0$ coincidence directly from the cosmological time field while leaving zero per-system free parameters across all downstream regimes. Data select this sector across domains where non-relativistic MOND and relativistic TeVeS fail: on the corrected ambient/propagator semantics the wide-binary joint test passes it at $0.44\sigma$ on the saturation amplitude ($\alpha_{\rm sat}=0.371$; derived identity $4\sqrt2/k\simeq0.353$) and $1.69\sigma$ on the transition scale ($R_s=2{,}646\pm182$ AU), while the unit-term baseline and the exponential-interpolation candidate are excluded at $\sim30\sigma$ on amplitude; under the same sector ($P_{,X}=\epsilon_{\rm reg}+k\sqrt{\xi}+2\xi$ with zero-gradient regulator $\epsilon_{\rm reg}=10^{-3}$ and $k_*=16.03$ calibrated to $a_0$), the SPARC radial-acceleration fit (Paper 6, Steps 10, 13) under the action-derived additive force law $g_{\rm obs}=g_N+a_\phi$ returns RMS $0.2185$ dex at $\Upsilon_d=0.5$ and $0.2004$ dex at $\Upsilon_d=0.35$ — matching the empirical MOND interpolating function ($0.1989$ dex) at $\Upsilon_d=0.35$ and outperforming the unit-term baseline ($0.2812$ dex) by $>60$ mdex (while the phenomenological quadrature comparator $\sqrt{g_N^2+a_\phi^2}$ gives $0.1974$ dex at $\Upsilon_d=0.5$ with $k_{\rm fit}=15.78$ and outer-edge BTFR median $v_{\rm obs}/v_{\rm pred}=1.0109$). All Solar-System benchmarks sit on the $\xi\gg\xi_*$ quadratic branch ($|\gamma_{\rm PPN}-1|_{1.6\,R_\odot}=4.32\times10^{-8}\ll2.3\times10^{-5}$; ray-integrated Cassini Doppler $|\gamma_{\rm PPN}-1|_{\rm Doppler}=9.01\times10^{-8}$ and differential Shapiro $|\gamma_{\rm PPN}-1|_{\rm Shapiro}=3.74\times10^{-7}\ll2.3\times10^{-5}$; Step 80) and have zero dependence on the bulk-matter saturation proxy $\rho_T$.

For a matched spherical radius in the potential-negligible deep-tail limit, if lensing were evaluated under isotropic conformal rescaling $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$ (where space stretches alongside time so $A^2(\phi)$ cancels out of null rays), the low-gradient branch would give a conditional dynamical-to-baryonic-lensing ratio $M_{\rm dyn}/M_{\rm lens}\simeq g_{\rm obs}/g_{\rm bar}\simeq\sqrt{a_{\rm eff}/g_{\rm bar}}$ ($\sim10$ at $g_{\rm bar}=1.2\times10^{-12}\,{\rm m\,s^{-2}}$) and leave only the Planck-suppressed scalar stress tensor ($\kappa_\phi\simeq6.5\times10^{-6}$; Paper 19, Step 64). Under the canonical unified matter metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu$ $+$ $A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ $+$ $B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) — the $(3+1)$ ADM realization of TEP's ontology in which the spacetime lapse rescales as $N_{\rm TEP}(\mathbf x)=A_{\rm loc}(u_{\rm grav}(\mathbf x))N_{\rm GR}(\mathbf x)$ along the cosmic temporal flow while at a fixed cosmic epoch ($A_{\rm sp}(t_0)=1$) the spatial 3-metric $\tilde\gamma_{ij}=\gamma_{ij}=(1-2\Phi_N/c^2)\delta_{ij}$ remains unstretched — intermediate clock rates $A_{\rm loc}(u_{\rm grav}(\mathbf x))$ refract null rays along the line of sight via $n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)$. Algebraically, $-(A_{\rm loc}^2-1)U_\mu U_\nu=\frac{1-A_{\rm loc}^2}{2\bar X}\nabla_\mu\bar\phi\nabla_\nu\bar\phi$ has the tensor form of a longitudinal disformal shift along the cosmic temporal gradient; critically, however, this lapse rescaling is not a photon-only disformal coupling at fixed Einstein-frame tensor cone (which would split photon and gravitational-wave arrival times by $\Delta t\sim\int|\beta_A|u\,dl/c\sim10^7\text{ s}$ across Galactic and host haloes, the dark-matter-emulator exclusion of Boran et al. 2018, and invert the longitudinal sign in voids where $A>1$). Because proper time $d\tilde t=A_{\rm loc}(u_{\rm grav}(\mathbf x))\,dt$ and conformal propagation time $d\eta=dt/A_{\rm sp}(t,\mathbf x)$ govern the shared ADM foliation itself, varying the spatially covariant ADM/DHOST gravitational action $S_{\rm grav}=\tfrac12 M_{\rm Pl}^2\int d^3x\,dt\,\sqrt{\gamma}\,N_{\rm GR}[R_{\rm scalar}^{(3)}(\gamma)$ $+$ $A_{\rm sp}^{-1}R_{\rm TT}^{(3)}(\gamma)$ $+$ $\tilde K_{ij}\tilde K^{ij}-\tilde K^2]$ ($\delta S_{\rm grav}/\delta u_{\rm grav}\equiv 0$ and $\delta S_{\rm grav}/\delta A_{\rm sp}\equiv 0$ in the quasistatic limit $K_{ij}\to 0$, preserving the Newtonian Poisson and scalar source equations without Jordan-frame degeneracy; $\tilde K_{ij}=A_{\rm sp}^{1/2}(2N_{\rm TEP}c)^{-1}\partial_t\gamma_{ij}$; Section 2.2, Step 80) with respect to transverse-traceless tensor perturbations $h_{ij}^{\rm TT}$ of $\gamma_{ij}$ and varying $S_{\rm EM}[\tilde g]$ with respect to electromagnetic waves $A_i^{\rm T}$ yields the identical quadratic Lagrangian structure $\propto\sqrt{\gamma}N_{\rm GR}A_{\rm sp}[(N_{\rm TEP}c)^{-2}(\partial_t\psi)^2-A_{\rm sp}^{-2}\gamma^{ij}\partial_i\psi\partial_j\psi]$ and shared principal wave operator $\Box_{\rm TEP}=-(N_{\rm TEP}c)^{-2}\partial_t^2+A_{\rm sp}^{-2}\gamma^{ij}\partial_i\partial_j$. Both wave sectors therefore share the exact null cone $d\tilde s_{\rm TEP}^2=-N_{\rm TEP}^2(\mathbf x)c^2dt^2+A_{\rm sp}^2(t,\mathbf x)\gamma_{ij}(\mathbf x)dx^i dx^j=0$ and the identical refractive index $n(x)$ across wells, voids, and cosmic epochs alike, preserving $c_{\rm GW}(t,\mathbf x)=c_\gamma(t,\mathbf x)=c\,N_{\rm TEP}(\mathbf x)/A_{\rm sp}(t,\mathbf x)$ and $\Delta t_{\gamma-\rm GW}=0$ identically along the GW170817 sightline while supplying the non-baryonic optical deflection across galaxies, clusters (Paper 4, Step 02), strong-lensing delays (Paper 19, Step 64), and CMB lensing (Paper 18, Step 21) from baryons alone and satisfying Solar-System Shapiro bounds via quadratic-branch screening. Since $P_{,X}=k\sqrt{\xi}+2\xi\to0$ as $\xi\to0$, the physical ambient floor ($\xi_{\rm env}\simeq0.32\text{--}0.52$) together with a small canonical regulator $\epsilon\lesssim0.1$ at isolated saddle points keeps the operator strictly elliptic at zero data cost; uniqueness across the wider $K(X)$ class remains open.

### Universal transfer map ($\beta_A \to \kappa_X$)

The transfer map translates the microscopic coupling $\beta_A = -1.0$ into domain-specific observable response coefficients ($\kappa_X$). Channel-specific response coefficients are observable projections of the universal conformal sector, with $\kappa_X = |\beta_A|S_X(\mathcal E_X)\Gamma_X$. They encode the channel geometry and environmental response and are distinct from the frozen microscopic coupling $\beta_A = -1$:

Observable channel coefficients are defined as positive response magnitudes

$$\kappa_X \equiv \lvert\beta_A\rvert\,S_X(\mathcal E_X)\,\Gamma_X.$$

They are not the bare coupling. $S_X$ is the response projection appropriate to the channel: $S_A$ labels static conformal depth, while time-dependent clock/covariance data require the fluctuation transfer carried by $C_A$ and the measurement model. Source-gradient and dynamical responses are expressed through $S_\Sigma$ under their specified geometry. J0437's phase-closure channel tests non-exact transport, not a static $S_A$ attenuation. $\mathcal{E}_X$ is the environmental state evaluated for the target channel. $\Gamma_X$ is a geometric/kinematic projector — a known function of the channel geometry that converts the screened field excursion into the observable's native response. The projectors for the primary channels are derived and tabulated below (Appendix E, R12): the GNSS covariance channel is quadratic in the coupling, $\Gamma_{\rm GNSS} = |\beta_A|$, with the fluctuation spectrum and correlation scale $\lambda_T \approx R_T$ carried by the channel functional $\mathcal F_X$ rather than the projector; the Cepheid and galactic channels carry the distance-to-modulus magnitude conversion $\Gamma_\mu = 5/\ln 10$ (distinct from the Leavitt period-to-magnitude projector $\Gamma_{\rm PL} = -b/\ln 10$ used by the Papers 11–12 ladder-level slope), the millisecond-pulsar channel the direct proper-time projection $\Gamma_{\rm MSP} = 1$, and $\Gamma_{\rm LLR}=1$ records the unit range normalization. The $S_\Sigma(g)$ factor belongs to the R11 force benchmark; it is not the clock-to-range transfer for a passive reflector or the onboard-clock response of a flyby link. On the selected two-branch sector the wide-binary saturation amplitude follows from the derived flux identity $\Gamma_{\rm WB}=4\sqrt2/k\simeq0.353$, and the propagator-plus-vertex reading predicts $\alpha_{\rm sat}=0.371$, passing the corrected zero-free-parameter joint test against the observed $0.366\pm0.012$ at $0.44\sigma$ (Step 56/020).

*Canonical notation registry.* The corpus uses a small fixed symbol set, and the same physical scale is not re-lettered between papers. $\lambda_T$ is the clock/covariance correlation length — the characteristic decay scale of $C_A(x,x')$; its terrestrial realization is the GNSS-measured covariance length $L_c$, the instrument value of $\lambda_T$ rather than a separate parameter. $L_c$ is estimator- and product-dependent — $\approx 3.2\times10^3$ km in the isotropic estimator and $4.2\times10^3$ km in the directional cross-sector dispersion (Paper 2), $1.9\times10^3$ km in the 1-year MGEX multi-constellation window (Paper 14, where short-arc detrending acts as a spatial/temporal high-pass filter), and $\gtrsim 0.7$–$1.1\times10^3$ km as a raw-SPP pseudorange code-noise lower bound (Paper 3) — and the identification $\lambda_T \approx R_T(M_\oplus)$ is anchored on the 25-year phase-coherent directional-dispersion value, which coincides with $R_T(M_\oplus) = 4146$ km ($\rho_T=20\pm7\ {\rm g\,cm^{-3}}$ inter-center, $[8,70]\ {\rm g\,cm^{-3}}$ window envelope; Paper 6) and is independently cross-validated at the extragalactic scale by inverting the SH0ES 37-host Cepheid $P^2$-carrier fit at the unit-transport benchmark $\lambda_0=1$ (with bare coupling $|\beta_A|=1$), which recovers $\rho_T^{\rm (Cepheid)}=19.91\pm8.02\ {\rm g\,cm^{-3}}$ (Paper 11, Step 62). $R_T(M) = (3M/4\pi\rho_T)^{1/3}$ is the geometric saturation radius of the amplitude sector, with $\rho_T$ its macroscopic scale. Every channel response coefficient takes the single form $\kappa_X = |\beta_A|\,S_X(\mathcal E_X)\,\Gamma_X$, a dimensionless number bounded by order-unity projectors; the per-channel names appearing in the companion papers — $\kappa_{\rm MSP}$, $\kappa_{\rm LLR}$, the wide-binary plateau $\alpha_{\rm sat}$, and the population-level shear response $\Gamma_t$ — are evaluations of this map on their respective channels, not independent couplings. Two magnitudes-quoted coefficients share the letter but are different objects: the $\kappa_{\rm Cep}$ and $\kappa_{\rm gal}$ of Papers 11–12 are ladder-level response slopes in magnitudes — composite products $\kappa_{\rm Cep} = -(b/\ln 10)\,\lambda_{\rm Cep}$ that convert a dimensionless depth excursion $X \sim 10^{-7}$ into a magnitude shift — not the dimensionless $\kappa_X$ of this map (Paper 11, Appendix C.3). $S_\Sigma(\mathcal E)$ and $S_A(\mathcal E)$ are the only screening projections; environment-conditioned factors written elsewhere ($S_{\rm env}$, $S_{\rm eff}$, channel subscripts, or multiplicative products such as $S_{\rm local}\times S_{\rm group}$) denote one of these two operators evaluated on the stated environment vector. (In Section 8 the symbol $L_{\rm hom}$ is used for the unrelated cosmological local-to-Hubble-flow crossover scale, to keep it distinct from the GNSS $L_c$.)

| Domain | Base coupling | Screening projection | Projector $\Gamma_X$ | Observable response | Status |
| --- | --- | --- | --- | --- | --- |
| Solar System / GNSS | $\beta_A = -1.0$ | $S_A^{(\oplus)}$ (clock depth); $C_A$ (covariance) | $\Gamma_{\rm GNSS} = |\beta_A|$ | $\lambda_T$, clock response $\kappa_{\rm GNSS}$ | Derived (R12); $\lambda_T \approx R_T$ in $\mathcal F_X$ |
| Wide binaries | $\beta_A = -1.0$ | $\mathcal S_\Sigma(X_{\rm gal})^2\,y(s)$ | $\Gamma_{\rm WB} = 4\sqrt2/k \simeq 0.353$ | $\alpha_{\rm sat}$ | Derived on selected sector (predicted $\alpha_{\rm sat}=0.371$ at $0.44\sigma$, Step 56/020) |
| Cepheids ($H_0$) | $\beta_A = -1.0$ | $S_A(\rho_{\rm host\ gal})$ | $\Gamma_\mu = 5/\ln 10 \approx 2.17$ | $\kappa_{\rm Cep}$ | Derived projector (R12); $\kappa$ normalization open below |
| JWST high-$z$ | $\beta_A = -1.0$ | Stellar-population transfer | $\Gamma_{\rm gal} = \Gamma_\mu(1+z)^{\beta_A\Delta u} \approx 2.17$ | $\kappa_{\rm gal}$ | Inherited (R12) |
| Globular clusters | $\beta_A = -1.0$ | $S_\Sigma(\text{cluster env})$ | $\Gamma_{\rm MSP} = 1$ | Pulsar $\Gamma$ | Derived projector (R12) |
| LLR / flyby | $\beta_A = -1.0$ | $S_\Sigma(g_{\rm lunar}) = 1.6\times10^{-14}$ | $\Gamma_{\rm LLR} = 1$ | $\kappa_{\rm LLR}$ | Derived (R11, R12); screened below current LLR sensitivity |

With the kinetic realization now specified (Appendix E, R11), any discrepancy between the predicted $\kappa_X$ and empirical fits (such as the variation in $\kappa_{\rm Cep}$ between 0.665–1.275 $\times 10^6$ mag (Paper 11 v0.12) and the reference coefficient of 0.96 $\times 10^6$ mag) ceases to be an unconstrained liability and becomes a direct constraint on the transfer sector of the action.

*On the canonical value and its unit.* The canonical figure $\kappa_{\rm canonical} = 0.96 \times 10^6$ mag is a prespecified response coefficient in the sense of a declared prior — held fixed so that downstream analyses (for example the JWST application, Paper 12) run without domain-specific refitting — adopted by rounding the Paper-11 joint-fit coefficient ($9.72\times10^5$ mag) on the SH0ES sample. Its physical origin in the stellar envelope is resolved in Paper 11, Step 62 (`step_62_q_structure_derivation.json`): for a classical Cepheid at the $P_0=10$ d pivot period ($Q=0.041$ d), the fundamental-mode period–mean-density relation $\langle\rho\rangle(P)=\rho_\odot(Q/P)^2$ gives $\langle\rho\rangle(10\text{ d})=2.367\times10^{-5}\text{ g cm}^{-3}$, so the saturation-to-envelope density ratio is $\rho_T/\langle\rho\rangle(10\text{ d})=8.450\times10^5$. Multiplied by the Wesenheit Leavitt projector $|b_W|/\ln 10=1.416$ mag at the unit-transport benchmark $\lambda_0=1$ (with bare coupling $|\beta_A|=1$), the inverse-density carrier $\ln q=\lambda_0 X(\rho_T/\langle\rho\rangle)\propto X P^2$ predicts $|\kappa_{P^2}^{\rm pred}(\lambda_0=1)|=1.196\times10^6$ mag, matching the direct SH0ES full-ladder fit $|\kappa_{P^2}^{\rm fitted}|=(1.191\pm0.480)\times10^6$ mag to $0.5\%$ (implied $\lambda_0=0.995\pm0.401$ on the physical-anchor baseline, $0.928$–$1.082$ across joint $\kappa_Z$ variants) while outperforming the phenomenological $\log_{10} P$ carrier on $\chi^2$, AIC, and BIC. The unit “mag” is a bookkeeping convention inherited from the Cepheid period–luminosity relation in which the coefficient was first expressed; in the transfer-map sense of the equation above, $\kappa_X$ is a dimensionless response magnitude, and the magnitude unit simply records the observational channel through which it is measured. It should not be treated as a physical dimension carried by the coupling. That separation is now a number from one profile, not an open slot. On the uniform $10^{11}\,M_\odot$, $30\,\mathrm{kpc}$ host, at $8\,\mathrm{kpc}$, the same $\phi(r)$ gives $S_\Sigma = 0.99985$ and $S_A = 0.99936$. Host screening is not the Cepheid coefficient. The two clocks the ladder divides share each galactic potential, so the common rate cancels in the ratio $q=r_{\rm spec}/r_{\rm Cep}$. On this host against a $10^{10}\,M_\odot$ anchor the nested ratio is $\Delta\ln q=2.4\times 10^{-10}$, a clock-channel coefficient of $\kappa_{\rm nested}=-7.5\times 10^{-4}$ mag under the Leavitt projector $b/\ln 10$; alternative reference-clock conventions — the volume-mean light depth or the direct Cepheid-to-Cepheid comparison retaining the cross-galaxy baseline — leave $|\kappa_{\rm nested}|$ of order unity at most and of the same sign, opposite to the measured positive response. The raw conformal clock channel is therefore negligible, more than eight orders of magnitude below the empirical interval: $\kappa_X$ is carried by the envelope density-ratio response $\rho_T/\langle\rho\rangle(P)$, not by differential host clock drift. $B(\phi)$ is the family $B_0\,u^2/(1+u^2)\,e^{-u^4/2}$ with $B_0\ge 0$ and $B_0\le 78$ from the $10^{-15}$ cone split on this profile; the holonomy scales linearly across that interval. The kinetic sector of the R11 completion is the two-branch form $P_{,X}=k\sqrt\xi+2\xi$ specified in Section 2 — with $P_{,X}\to0$ at $\xi\to0$, regulated at zero data cost by a small canonical term $\epsilon\lesssim0.1$ or nonzero cosmic ambient.

Because the environmental vector $\mathcal{E}$ encompasses the total localized state (source structure, boundary conditions, and ambient fields), its operational realization depends strictly on the physical domain being probed. In dense macroscopic matter, it manifests via a density proxy; in cosmological voids and local potential wells, it tracks potential-depth gradients; in kinematic orbital phase space, it follows velocity-dispersion transitions; in geodetic clock networks, it is defined by geometric covariance lengths $\lambda_T$; and in subatomic environments, it is bounded by geometric proximity regulators. These are not competing mechanisms, but domain-specific macroscopic projections of the same underlying continuous saturation of the Temporal Topology.

## PPN mapping

Under the canonical unified matter metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu$ $+$ $A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ $+$ $B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ at a fixed cosmic epoch ($A_{\rm sp}(t_0)=1$), slow matter responds to the temporal lapse potential $\Phi_{\rm dyn}=\Phi_N+c^2\beta_A u_{\rm grav}=(1+2\beta_A^2 S_\Sigma)\Phi_N$ while the spatial 3-metric $\tilde g_{ij}=\gamma_{ij}=(1-2\Phi_N/c^2)\delta_{ij}$ remains unstretched ($\Psi_{\rm sp}=\Phi_N$), so optical rays respond to $\Phi_{\rm lens}=\tfrac12(\Phi_{\rm dyn}+\Psi_{\rm sp})=(1+\beta_A^2 S_\Sigma)\Phi_N$. The PPN parameter is therefore $\gamma_{\rm PPN} = \Psi_{\rm sp}/\Phi_{\rm dyn} = 1 / (1 + 2\beta_A^2 S_\Sigma)$ (compared with $(1-2\beta_A^2 S_\Sigma)/(1+2\beta_A^2 S_\Sigma)$ on the isotropic-conformal benchmark), which at leading order gives the strict, sign-definite prediction $\gamma_{\rm PPN} - 1 \simeq -2\beta_A^2 S_\Sigma < 0$, where $S_\Sigma$ is the source's screening fraction (Section 2.2, Step 80). With $\alpha_0 = \sqrt{2}\,\beta_A = -\sqrt{2}$, Cassini's $|\gamma_{\rm PPN} - 1| < 2.3\times10^{-5}$ requires the bound $S_\Sigma^{(\odot)} \lesssim 1.15\times10^{-5}$ ($5.8\times10^{-6}$ on the isotropic benchmark) in the Solar-System environment; evaluated directly on the selected two-branch kinetic sector $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$) at solar conjunction ($r=1.6\,R_\odot$, $g_N=107.1\,{\rm m\,s^{-2}}$, $q=g_N/g_t=3.15\times10^{11}$), the solved response $y_{\rm tb}=2.16\times10^{-8}$ gives $|\gamma_{\rm PPN}-1|\simeq 4.32\times10^{-8}$, more than $500\times$ below the Cassini bound. Near massive bodies, the suppression of Temporal Shear (vanishing field gradient) suppresses the effective scalar charge, cleanly preserving PPN bounds without invoking rigid thin-shell approximations.

**PPN recovery in the screened limit.** For $A(\phi) = \exp(\beta_A\phi/M_{\rm Pl})$, the dimensionless microscopic coupling is $d\ln A/d\varphi = \beta_A = -1$ (equivalently $\alpha_0 = \sqrt{2}\,\beta_A$ in DEF normalization). Environmental screening does not alter this universal microscopic coupling. Instead, it suppresses the effective exterior scalar charge,

$$\alpha_{\rm eff} = S_\Sigma(\mathcal E)\,\alpha_0,$$

where $S_\Sigma(\mathcal E) \to 0$ in screened source/ambient configurations. In an isolated spherical kinetic benchmark, the high-acceleration condition is $g_N=GM_{\rm enc}(r)/r^2\gg g_t$; density alone is neither a universal cutoff nor a substitute for the solved field and boundary conditions. Because the spatial 3-metric $\tilde\gamma_{ij}=\gamma_{ij}$ is unstretched while the temporal lapse carries the screened source charge, the PPN deviation is linear in the source charge fraction, enforcing the sign-definite prediction $\gamma_{\rm PPN} - 1 < 0$:

$$\begin{aligned} \gamma_{\rm PPN} - 1 &= -\frac{\alpha_0\,\alpha_{\rm eff}}{1 + \alpha_0\,\alpha_{\rm eff}} = -\frac{2\beta_A^2 S_\Sigma}{1 + 2\beta_A^2 S_\Sigma} \\ &\quad \left(\text{isotropic benchmark: } -\frac{4\beta_A^2 S_\Sigma}{1 + 2\beta_A^2 S_\Sigma}\right), \end{aligned}$$

which is linear in $S_\Sigma$ at leading order. Therefore $S_\Sigma \to 0 \;\Longrightarrow\; \alpha_{\rm eff} \to 0 \;\Longrightarrow\; \gamma_{\rm PPN} \to 1$. The remaining PPN parameters likewise recover their GR values in the screened limit. No independent running of the frozen microscopic coupling $\alpha_0$ is required.

## Equivalence principle

TEP ensures universality in the matter frame. In the Einstein frame, the anomalous geometric acceleration (conventionally parameterized as a fifth force) is bounded by Eötvös experiments; MICROSCOPE gives $\eta \lesssim 10^{-15}$. Sectoral dilaton-like couplings (to $\alpha$, $\mu$, quark masses) are constrained to $|d| \lesssim 10^{-5}$–$10^{-6}$ by composition tests and clock ratios.

## Radiative strong-field bounds: binary pulsars and stellar interiors

The selected two-branch action's static ambient response is calculated directly in step_76 with the same nonlinear matter source and quartic potential as step_53. At the reference coupling and an Earth outer boundary of $10^8R_\oplus$, the infinitesimal tangent is $0.6536$ while the finite response to $\delta u=2.5\times10^{-11}$ is $0.00199$. At $10^{10}R_\oplus$ the resolved Earth interior excursion falls below reporting resolution, whereas the selected massless-vacuum control returns unit response. The measurable tangent agrees with nonlinear finite differences and mesh refinement. Smaller comparison interfaces retain near-unit response. This is a calculated potential- and domain-dependent static screening effect; selecting the physical ambient matching surface and computing finite-frequency clock transfer remain distinct from this boundary test.

The scalar-dipole calculation in step_69 separates two source prescriptions: (i) a massless kinetic-only control, $P=X+X|X|/\Lambda_X^4$, in which compact-body sensitivity charges $\alpha_i=\alpha_0(1-2s_i)$ are prescribed as unscreened point sources at the orbital radius $r=a$ with a reflecting Dirichlet inner wall ($T_{\rm amp}=0.0121$ and tightest sensitivity-bracket margin $0.529$ on J1738+0333; this unscreened-source control does not pass the full bracket because it omits the screening shell between the neutron-star surface $R_{\rm NS}\simeq12\text{ km}$ and the orbital separation $a\simeq1.73\times10^6\text{ km}$), and (ii) the selected two-branch sector $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$) with shell-impedance screening across the compact-body-to-wave-zone profile, where the binding-energy sensitivity $s_A=-\partial\ln M_A/\partial\ln G_{\rm int}$ resides at the stellar radius $R_A\ll a$ and is suppressed across the pair shell by $T_{\rm amp}^{\rm imp}\simeq a/r_*=9.2\times10^{-7}\text{--}1.4\times10^{-5}$ ($2.26\times10^{-6}$ for J1738+0333), clearing every binary-pulsar $\dot P_b$ bound across the full EOS sensitivity bracket by amplitude margins of $2.8\times10^3$ to $3.9\times10^4$ at $K_D=0.5$ (tightest margin $2{,}817\times$ on J1738+0333 $s_{\rm hi}$; $996\times$ to $1.39\times10^4$ at $K_D=4.0$), while the hierarchical three-body operator $\mathcal R=y^3$ on the J0337+1715 inner binary at the outer white-dwarf separation gives $\eta_{\rm pred}=2.54\times10^{-14}\ll2.6\times10^{-6}$ (margin $\sim10^8\times$; vs. $0.089\text{--}0.165$ under the unscreened single-vertex control).

**Compact-well bound at fixed ambient boundary.** The selected high-gradient flux law $f(q)=kq/\sqrt2+q^2$ has an integrable point-source gradient, $q\sim r^{-2/3}$. For the source-only family of step_58, $0\le F(r)\le1$ because the matter coupling contributes $e^{-u}\le1$, hence, with $\mathcal J(q)=qf(q)$, $u_c\le\int_0^{R_{\rm out}}\mathcal J^{-1}(r^{-2})\,dr$. The finite bound is uniform as the source radius tends to zero. At the existing dimensionless boundary $R_{\rm out}=40$, $u_{\rm amb}=0$, direct quadrature gives $u_c\le2.88628$ and $A_c/A_{\rm amb}\ge0.05578$. The existing selected-family solutions lie comfortably inside the bound ($u_c\le1.40872$ over the scanned radii). This is a positive local nonsingularity certificate in the stated fixed-source, fixed-boundary problem; a finite-radius logarithmic trend fit cannot imply an unbounded central well. The low-gradient logarithmic tail still makes the ambient boundary relevant, so the local bound is not a universal floor on unbounded space.

## Disformal constraints

GW170817/GRB170817A constrain $|c_\gamma - c_g|/c \lesssim \text{few}\times10^{-15}$ at $z \approx 0.01$, bounding the observable combination $B(\phi)(\partial\phi)^2$ along the observed astrophysical path. This does not require $B$ to vanish identically. In regimes where common-mode conformal effects dominate, $B$ may remain active as a source for synchronization holonomy, multipath propagation anomalies, or interferometric signatures without violating the same-path EM–GW timing bound.

## Disformal holonomy and the screening ontology

As established in Section 3.3, the invariant synchronization observable is the GR-subtracted residual

$$H_{\rm resid}(C) = \oint_C \Delta\sigma, \qquad \Delta\sigma \equiv \tilde{\sigma}-\sigma_{\rm GR}.$$

In the conformal-only subclass, the scalar clock-rate contribution is the exact one-form $d\ln A$, so it cannot by itself generate closed-loop residual synchronization holonomy in a smooth simply connected region. A leading-order nonzero $H_{\rm resid}$ requires residual synchronization curvature,

$$d\Delta\sigma \neq 0 .$$

In the minimal TEP model this curvature is supplied by the disformal sector; in more general extensions it may arise from non-metricity or other explicitly non-exact transport structure.

To state the disformal contribution in a measurement-defined way, let $u^\mu$ denote the four-velocity field of the physical clock network or observer congruence used to define the synchronization protocol. The $(-+++)$ signature is used, so that

$$u^\mu u_\mu = -1 .$$

Let

$$P_\mu{}^\nu = \delta_\mu{}^\nu + u_\mu u^\nu$$

be the spatial projector into the local rest space of that congruence. To leading order in the disformal correction, the projected contribution to the synchronization connection has the representative form

$$\delta\tilde{\sigma}_\mu \simeq -\frac{B}{A^2} (u\cdot\nabla\phi) P_\mu{}^\nu \nabla_\nu\phi ,$$

up to the sign convention used for the synchronization one-form and higher-order disformal corrections.

Appendix A3 derives this expression from the mixed time-space projection of the disformal matter metric. It also shows that synchronization re-gaugings shift both the matter-frame synchronization connection and the corresponding GR reference connection by the same exact one-form. Therefore the GR-subtracted residual connection $\Delta\sigma$ is invariant under synchronization convention changes, and

$$H_{\rm resid}(C) = \oint_C \Delta\sigma$$

is independent of the arbitrary simultaneity convention used to coordinatize the specified physical clock network.

The invariant claim is therefore precise: $H_{\rm resid}$ is not independent of which physical clock network is used, but for a specified physical clock-network protocol it is independent of arbitrary synchronization re-labelling.

For a hypersurface-orthogonal clock network, $u^\mu=n^\mu$, the projected expression reduces to the familiar $3+1$ form

$$\delta\tilde{\sigma}_i \approx -\frac{B}{A^2N} (\partial_i\phi)(n\cdot\partial\phi),$$

where $N$ is the lapse and $n^\mu$ is the unit normal to the spatial slices. This ADM expression should be read as a special-case representation of the physical congruence formulation, not as the definition of the observable.

Unlike $d\ln A$, the disformal contribution is not generically exact. In the local non-topological case, its curvature satisfies

$$d(\delta\tilde{\sigma}) = -\frac{B}{A^2} d\left(\frac{\dot{\phi}}{N^2}\right) \wedge d\phi$$

whenever the prefactor multiplying the projected scalar gradient varies independently around the loop, for example through time dependence, anisotropic boundary conditions, inhomogeneous field structure, lapse/shift structure, or spatial variation of the disformal response within a fixed physical clock-network protocol. For a loop $C$ bounding a smooth surface $\Sigma$, the local non-exact contribution to the residual holonomy may be written

$$H_{\rm resid}(C) = \oint_C \Delta\sigma = \iint_\Sigma d(\Delta\sigma).$$

At leading order beyond the GR-subtracted reference model, $d(\Delta\sigma)$ contains the disformal curvature $d(\delta\tilde{\sigma})$. Topological holonomy, where the loop does not bound a smooth surface or the connection is locally closed but not globally exact, is a separate case and is not represented by the Stokes expression above.

The conformal and disformal responses arise from the same environment-dependent scalar configuration. For the universal exponential coupling,

$$\Sigma_\mu \equiv \nabla_\mu\ln A = \frac{\beta_A}{M_{\rm Pl}}\nabla_\mu\phi,$$

while the leading disformal deformation relative to the conformal metric is

$$\mathcal{D}_{\mu\nu} \equiv \frac{B(\phi)}{A^2(\phi)}\nabla_\mu\phi\nabla_\nu\phi = \frac{B(\phi)\,M_{\rm Pl}^2}{A^2(\phi)\,\beta_A^2}\,\Sigma_\mu\Sigma_\nu.$$

Environmental flattening of the Temporal Topology therefore suppresses both responses: Temporal Shear is linear in the locally active scalar gradient, whereas the leading disformal deformation is quadratic in that gradient for fixed $B/A^2$. Their observable suppression is nevertheless not represented by a single universal numerical factor, because the disformal response additionally depends on $B(\phi)$, the solved field profile, path geometry, boundary conditions, and the observer congruence. The body-level exterior source-charge factor

$$S_\Sigma \equiv Q/Q_0$$

is an integrated observable projection of the field configuration and should not be inserted directly as a universal local $S_\Sigma^2$ multiplier for disformal observables. At leading disformal order, the synchronization correction may equivalently be written

$$\delta\tilde{\sigma}_\mu \simeq -\frac{B(\phi)\,M_{\rm Pl}^2}{A^2(\phi)\,\beta_A^2}\,(u\!\cdot\!\Sigma)\,P_\mu{}^\nu\Sigma_\nu.$$

Thus synchronization holonomy depends on the local field value and Temporal Shear, the observer motion relative to that field, the orientation of the gradient, and the complete path geometry. Experimental response therefore cannot in general be inferred from ambient density or a body-level source-charge factor alone.

Accordingly, high-energy, mesoscopic, and topological probes should not be modeled solely by the ambient bulk-density screening function used in astrophysical applications. Their responses are represented by channel-specific coefficients $\kappa_X$, depending on momentum transfer, interaction topology, boundary geometry, and microscopic field structure. These probes test whether non-exact disformal transport remains measurable in regimes where the conformal clock-rate response is screened.

## 8. Cosmology: Static Spatial Geometry, Temporal Horizon, and EFT Mapping

## Background & The Temporal Horizon

*Supersession Note: Earlier formulations of TEP utilized phenomenological epoch-screening functions to artificially preserve a standard hot-plasma expansion history. The framework now develops thermodynamic closure natively within the canonical eternal-universe architecture. The "hot Big Bang" — a singular, hot, dense early state of an expanding space — is re-expressed in the canonical architecture as a matter-frame temporal horizon rather than a gravitational volume singularity. The observed thermal history is preserved throughout: recombination, the blackbody spectrum, and the acoustic peaks are reconstructed as conformal images of the standard history on the static background (TEP-TH, TEP-HC), and discrimination between the two readings lives where the dynamics differ — growth and lensing, clock channels, siren transport, and the native production mechanisms (TEP-BBN) — not in the CMB's gross features.*

The canonical TEP cosmology is spatially infinite, non-compact, physically eternal, and inhomogeneous: space has no boundary and does not close on itself. The physical (gravitational) volume is static, while the matter-unit volume drifts. Local curvature, gravitational dynamics, shear, and motion remain permitted. The unified causal matter metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu$ $+$ $A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ $+$ $B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($d\tilde s^2=-N_{\rm TEP}^2(\mathbf x)c^2dt^2$ $+$ $A_{\rm sp}^2(t,\mathbf x)\gamma_{ij}(\mathbf x)dx^i dx^j$ at $B=0$) reduces on the homogeneous intergalactic background ($u_{\rm grav}=0$, $N_{\rm TEP}=1$, $\Pi_\mu^\alpha\nabla_\alpha\phi=0$) to $d\tilde s^2=-c^2dt^2+A_{\rm sp}^2(t)\gamma_{ij}dx^i dx^j$ (or in conformal propagation time $d\eta=dt/A_{\rm sp}(t)$, $d\tilde s^2=A_{\rm sp}^2(\eta)[-c^2d\eta^2+\gamma_{ij}dx^i dx^j]$), inheriting its secular ruler evolution from the ambient drift $u_{\rm drift}(t)$, so that the expansion inferred from redshift–distance observations is a derived consequence of temporal-field ruler transport rather than growth of physical volume in either sector (Paper 30, Appendix C). Because the spatially covariant ADM gravitational action $S_{\rm grav}$ on static space $\gamma_{ij}(\mathbf x)$ carries constant $M_{\rm Pl}^2$ and constant WKB tensor weight $\sqrt{A_{\rm sp}^{+1}\cdot A_{\rm sp}^{-1}}\equiv 1$ in conformal time $\eta$ (Section 2.2, Step 80), no coordinate transformation or field redefinition maps the gravitational and matter sectors simultaneously onto a minimally coupled expanding FLRW spacetime; the two sectors carry independent physical content, and their separation is itself observable through the diffeomorphism-invariant standard-siren ratio $\Xi(z)=d_L^{\rm GW}/d_L^{\rm EM}=1/(1+z)\neq 1$ (standard sirens, below). The temporal horizon bounds the history accessible to an observer; it is not an edge of space or a beginning of the universe, and the finite age reconstructed from cosmological observation is distinguished from the universe's infinite physical history.

Cosmological expansion in TEP is a chronometric projection rather than a spatial collapse. The resolution arises natively from the dual-metric foundation:

- **Gravitational Frame:** The underlying gravitational manifold $g_{\mu\nu}$ is static. The number density is constant in the gravitational frame; the matter energy density there scales as $A$, which the scalar sector must balance to maintain a static $g_{\mu\nu}$. The gravitational volume $V_g$ of any coordinate region does not change.

- **Matter Frame:** Because local Lorentz invariance is strictly maintained (Axiom 2), the local speed of light $c=1$ is invariant in any freely falling frame of the matter metric $\tilde{g}_{\mu\nu}$. Consequently, if the matter unit of time scales by the conformal factor $A(\phi)$, the matter unit of length must scale identically. The physical volume of a coordinate region, measured in matter units, is therefore $\tilde{V} \propto A^3 V_g$.

Conservation of matter mass $M$ then dictates that the matter density, measured by local observers, evolves as $\tilde{\rho} = M / \tilde{V} \propto A^{-3}$. Given the observational clock map $A_{\rm clock}(z) = (1+z)^{-1}$ — adopted here as a consistency input, its derivation from the realized landscape being part of the open cosmological construction — this yields the standard cosmological density history:

$$\tilde{\rho}(z) = \tilde{\rho}_0 (1+z)^3$$

The matter-unit density history of standard cosmology, $\tilde\rho(z) = \tilde\rho_0(1+z)^3$, then reads as unit drift — the conformal recalibration of matter-frame distance units at temporal depth ($A \to 0$) — rather than by a physical crushing of gravitational space. The density scaling alone does not settle the corresponding dimensionless ratios (for example $T/m_e$ at the emitting region); those follow from the Proper-Time Reaction Flow below, and are the observables that distinguish the two readings. Distances and densities expressed in matter units evolve without any physical stretching. This dual-frame resolution removes the need for mechanical balance mechanisms: neither a negative disformal coupling ($B < 0$) nor complex matter-flow partitions are required to cancel volume expansion artificially, because there is no volume expansion in the gravitational frame to begin with.

Solving the rolling-floor ansatz $u(r,t)=U(r)-\bar\Pi\,t$ directly on a Painlevé–Gullstrand interior background — regular across $r_h$, with the shift $\beta^r=\sqrt{2M/r}$ encoding the relativistic infall — refines this picture in three ways (Appendix E, R3). First, the scalar profile steepens toward the horizon, reaching $u\simeq9$–$10$ at $1.01\,r_h$, smoothly connecting the weak-field exterior to the finite-minimum temporal floor. Second, the solution lands natively on the temporal-well branch the cosmology requires ($A < 1$), identifying the deep-well boundary with the operational clock-rate horizon of TEP-TH and distinguishing it from the disformal degeneracy surface $\det\tilde g_{2D}=0$ and from the prescribed $A\to\infty$ test profile that TEP-BH shows to be unattainable on a fixed Schwarzschild background (its Theorem D.1); the dynamics select the cosmological branch rather than being assigned it — the same temporal-well branch on which the TEP-BH coupled construction lands at $\eta=+0.3$. Third, self-consistency of the rolling ansatz requires $\Box u$ to be time-independent along the roll, which holds only when the interior drive is locally flat: a constant $V_{,u}+\mathcal{Q}_m$ closes to $7\times10^{-14}$, while steep or exponential potentials fail secularly — the floor region demands a locally flat potential or a matter-dominated source, a derived constraint on the $V(\phi)$ programme rather than an assumed one.

The candidate master potential is

$$V(u) = V_{\rm matter}(u) e^{-(u/u_s)^4} + V_0 e^{-(u_s/u)^4}$$

The transition scale ($u_s$) must be selected so that the deep-field sector contributes to the temporal-well solution as the scalar reaches $u\sim9$–$10$, while the total potential remains compatible with the field values and gravitational constraints of the cosmological matter-hosting landscape. These are simultaneous requirements on a single potential, not independently adjustable mechanisms. The floor $V_0$ and the precise transition scale are fixed by that joint compatibility calculation, which remains open; pending its completion, $V(u)$ is a parametrized family rather than a fixed potential.

*Temporal-well construction target.* Paper 28 develops exterior and interior benchmarks; their completion into one globally regular static mass family is not assumed here. The target interprets a black hole not as a collapsed singular object but as a temporal well: gravitationally regular space in which the matter clock rate drops to a finite, non-zero minimum ($N_{\min} > 0$), never actually reaching or asymptotically approaching zero. The scope of this statement is the deep-well sector: it is the local coupling inside a temporal well that is floored by the master potential $V_0$. The cosmological temporal horizon discussed below is a different object — on the homogeneous closed branch the conformal factor itself approaches $A \to 0$ as $\rho_m \to 0$, and on the inhomogeneous realization it is the reconstructed observational clock map $A_{\rm clock,obs} \equiv 1/(1+z)$ that vanishes as $z \to \infty$. Neither statement assigns a zero clock rate to a local well interior. What standard general-relativistic interpretation reads as collapse is the extreme slowing of matter-frame time transport. The underlying gravitational metric $g$ never collapses and its radial coordinate never becomes timelike; the interior is gravitationally regular, static space, not an expanding geometry, and the field equations remain strictly static all the way to $r=0$. The Temporal Horizon is not an absolute causal barrier, but an operational, observer-relative boundary of practical signal inaccessibility where extreme gravitational redshift renders signals practically undetectable from the exterior. Because the core density is generated by the master potential floor $V_0$, the interior is completely scale-invariant: supermassive black holes like M87* and Sgr A* share the exact same static regular core as stellar-mass wells, differing only in their spatial extent. Externally, a temporal well reproduces the shadow, photon sphere, and orbital dynamics of a standard black hole. The temporal roll on cosmic scales and the horizon structure in deep wells are dual manifestations of the universal scalar sector: the ambient field evolves over cosmic time across the landscape, while establishing regular, static potential wells around compact mass concentrations.

The scalar field on this background is not a homogeneous fluid but a dynamic, inhomogeneous network. At any given coordinate time $t$, the field forms a lumpy temporal landscape where local wells and voids possess distinct local values—an undulating terrain of temporal "hills" (voids where clocks run faster) and "valleys" (gravitational wells where clocks run slower). Light reaches the observer by propagating through this evolving landscape over cosmic journeys spanning billions of years, and the measured redshift is the record of that passage. The Temporal Shear $\Sigma_\mu = \nabla_\mu\ln A(\phi)$ (Appendix A.3) is a spatial gradient, so its accumulated effect along a null ray is exact whenever the background is stationary: along each ray, $k_t$ is conserved by the timelike Killing symmetry of a static $g$ — the conformal factor drops out of the null covector, so $\dot k_t = 0$ holds even when $A$ varies spatially along the path — so the endpoint form generalizes to $1+z = (A\,N)_{\rm obs}/(A\,N)_{\rm em}$ with $N$ the gravitational lapse (so that $AN$ is the matter-frame lapse), without any assumption of spatial homogeneity of $A$: the conformal part of the redshift depends on the macroscopic historical trend of the field values hosting matter over coordinate time. This telescoping is a statement about the net frequency bookkeeping, not about the photon's interaction with the terrain: the ray responds to the matter-frame topology continuously, at every point of the path — each well blueshifts the locally measured frequency on descent and redshifts it on ascent, asymmetric entry and exit slice the residual to the intervening field difference, and the dwell time accumulated while crossing each nested structure is positive and never returned. Nested and overlapping structures are handled by the total solved field rather than by independent additive kicks; on a strictly static background only the frequency sum collapses to the endpoints. Crucially, the physical landscape is dynamic: as the photon journeys across space, background time continues to evolve, and the intervening hills and valleys are themselves moving and evolving over time. Where the fields evolve during transit ($\partial_t \ln A \neq 0$, $\partial_t N \neq 0$), a path-integrated (integrated Sachs–Wolfe and Rees–Sciama type) contribution arises — each nested structure contributing in proportion to its own evolution rate and the time the photon dwells within it — and the disformal sector contributes its own non-exact transport; both are intrinsic parts of the total redshift rather than external corrections, computed within the same moving-endpoint framework alongside endpoint kinematics.

Along a Solar-System observer's worldline embedded in the continuous nested hierarchy (with continuous Temporal-Topology gradient flattening on the selected two-branch kinetic operator $P_{,X}(\xi)=k\sqrt{\xi}+2\xi$ and smooth quartic potential $V(u)=\tfrac{\lambda}{4}M_{\rm Pl}^4 u^4$, involving no thin-shell boundary; Steps 53, 70, 76), the continuous radial profile connects the screened interior ruler drift $u_{\rm int}(t)$ to the rolling intergalactic ambient $\bar u_{\rm amb}(t)$. In the $(3+1)$ ADM decomposition on static gravitational space $\gamma_{ij}(\mathbf x)$ with strictly constant $M_{\rm Pl}$ in $S_{\rm grav}$, $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(\phi(t,\mathbf x))\Pi_{\mu\nu}$ ($d\tilde s^2=-N_{\rm TEP}^2(\mathbf x)c^2dt^2$ $+$ $A_{\rm sp}^2(t,\mathbf x)\gamma_{ij}(\mathbf x)dx^i dx^j$), where $N_{\rm TEP}(\mathbf x)=\exp(-u_{\rm grav}(\mathbf x))N_{\rm GR}(\mathbf x)$ is the stationary baryon-sourced gravitational/scalar lapse ($\partial_t u_{\rm grav}=0$, $|u_{\rm grav}|\sim10^{-6}$, so virialized galaxies never accumulate a secularly growing $\mathcal O(1)$ potential cliff relative to the IGM) and $A_{\rm sp}(t,\mathbf x)=\exp(\beta_A u_{\rm drift}(t,\mathbf x))=\exp(-u_{\rm drift}(t,\mathbf x))$ is the matter-frame spatial ruler factor ($\partial_t\ln A_{\rm sp}=-\partial_t u_{\rm drift}=+H_{\rm drift}>0$) measuring how many matter-frame atomic Bohr radii ($\ell_{\rm unit}^{\rm coord}=\ell_0/A_{\rm sp}=\exp(+u_{\rm drift})\ell_0$ in static gravitational coordinates) fit across a fixed static coordinate interval $dx$ ($d\tilde\ell=A_{\rm sp}\,dx$). For any static-coordinate matter observer with unit 4-velocity $\tilde u^\mu=(N_{\rm TEP}^{-1},\mathbf 0)$ and measured proper frequency $\tilde\omega=-\tilde u^\mu k_\mu=E(t)/N_{\rm TEP}(\mathbf x)$ ($E\equiv-k_t$), Hamilton's null equations $dx/dt=N_{\rm TEP}/A_{\rm sp}$ and $d\ln E/dt=\partial_t\ln N_{\rm TEP}-\partial_t\ln A_{\rm sp}$ cancel $\partial_t\ln N_{\rm TEP}$ identically against $d\ln N_{\rm TEP}/dt$ at every point along the photon trajectory, leaving the universal differential-geometric redshift identity over lookback interval $\Delta t=t_{\rm now}-t_{\rm em}$ ($\Delta\bar u_{\rm amb}=u_{\rm drift}(t_e)-u_{\rm drift}(t_o)=\ln(1+z_{\rm cosmo})$; Step 70):

$$\begin{aligned} \ln(1+z) = \ln\!\left(\frac{\tilde\omega_e}{\tilde\omega_o}\right) &= \int_{t_e}^{t_o}\frac{\partial\ln A_{\rm sp}(t,x(t))}{\partial t}\,dt + \int_{x_e}^{x_o}\frac{\partial\ln N_{\rm TEP}(x)}{\partial x}\,dx \\ &= \underbrace{T_{\rm cosmo}\,\Delta\bar u_{\rm amb}}_{\text{(a) screened endpoint interior drift}} \;+\; \underbrace{(1-T_{\rm cosmo})\,f_{\rm IGM}\,\Delta\bar u_{\rm amb}}_{\text{(b) in-flight IGM ruler-drift integral }\int(-\partial_t u_{\rm drift})\,dt} \;+\; \underbrace{\delta u_{\rm env}}_{\text{(c) stationary well contrast }u_{\rm grav}(x_e)-u_{\rm grav}(x_o)} \\ &= \bigl[1 - f_{\rm well}(1 - T_{\rm cosmo})\bigr]\Delta\bar u_{\rm amb} + \delta u_{\rm env}, \end{aligned}$$

where $T_{\rm cosmo}\equiv\Delta u_{\rm int}/\Delta\bar u_{\rm amb}\in[0,1]$ is the continuous boundary susceptibility of the host interiors, $f_{\rm well}\equiv\frac{1}{L}\int_0^L W_{\rm well}(x)\,dx\in[0,1]$ is the sightline fraction occupied by screened wells, and $f_{\rm IGM}=1-f_{\rm well}\in[0,1]$ is the sightline fraction spent in the unscreened intergalactic medium. Because $\partial_t\ln N$ cancels identically along any ray between $d\ln E/dt$ and $d\ln N/dt$, placing the cosmic roll in $N(t)$ alone with $A_{\rm sp}=1$ would reduce to a trivial time-coordinate relabeling of Minkowski space ($\ln(1+z)=0$), while placing a counter-rolling step inside $N(t,\mathbf x)$ would build an unphysical $\mathcal O(1)$ potential cliff $\Delta\ln N\sim-(1-T)H_0 t$ between galaxies and the IGM; in the canonical unified metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu$ $+$ $A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ $+$ $B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($d\tilde s^2=-N_{\rm TEP}^2(\mathbf x)c^2dt^2$ $+$ $A_{\rm sp}^2(t,\mathbf x)\gamma_{ij}(\mathbf x)dx^i dx^j$), the stationary lapse $N_{\rm TEP}(\mathbf x)$ has zero secular cliff while the ruler-drift factor $A_{\rm sp}(t,\mathbf x)$ across the unscreened IGM ($f_{\rm IGM}=1-f_{\rm well}\to1$) redshifts the photon continuously in flight. Simultaneously, because Step 80's spatially covariant ADM gravitational action $S_{\rm grav}=\tfrac12 M_{\rm Pl}^2\int d^3x\,dt\sqrt{\gamma}\,N_{\rm GR}[R_{\rm scalar}^{(3)}(\gamma)$ $+$ $A_{\rm sp}^{-1}R_{\rm TT}^{(3)}(\gamma)$ $+$ $(A_{\rm sp}/A_{\rm loc}^2)(K_{ij}K^{ij}-K^2)]$ equips transverse-traceless gravitational waves $h_{ij}^{\rm TT}$ with the exact same principal wave operator $\Box_{\rm TEP}=-(N_{\rm TEP}c)^{-2}\partial_t^2+A_{\rm sp}^{-2}\gamma^{kl}\partial_k\partial_l$ as Maxwell's equations while keeping the WKB tensor measure $\sqrt{A_{\rm sp}^{+1}\cdot A_{\rm sp}^{-1}}\equiv 1$ constant on static space ($\sqrt{\gamma}=1$), photons and gravitons propagate at the identical coordinate speed $c_{\rm GW}(t,\mathbf x)=c_\gamma(t,\mathbf x)=c\,N_{\rm TEP}(\mathbf x)/A_{\rm sp}(t,\mathbf x)$ at every event ($\Delta t_{\gamma-\rm GW}\equiv 0.0\text{ s}$ for GW170817, compared with a $6.9\times10^5\text{ yr}$ offset if $A_{\rm sp}$ were omitted from $S_{\rm grav}$) while experiencing zero FLRW Hubble friction, preserving $\Xi(z)=d_L^{\rm GW}/d_L^{\rm EM}=1/(1+z)\neq 1$. With strictly constant $M_{\rm Pl}$ in $S_{\rm grav}$ and $G_{\rm loc}(\phi)=G_*A^2(\phi)(1+\alpha_{\rm amb}^2)$ (matching Paper 27 and Step 80), the Step 76 continuous two-branch + quartic profile suppresses the Solar-System interior response at $10^8 R_\oplus\approx4.3\times10^3$ AU ($\approx0.02$ pc) to a resolved static tangent $T_{\rm tan}^{\rm Cassini}=1.18\times10^{-8}$ on the operative Cassini-compatible $10^5\lambda_{\rm ref}$ branch — yielding $|\dot A/A|_{\rm loc}=8.42\times10^{-19}\,{\rm yr}^{-1}$ and $|\dot G_{\rm loc}/G_{\rm loc}|_{\rm LLR}=1.68\times10^{-18}\,{\rm yr}^{-1}$, more than $4\times10^4$ times below both the Hofmann & M&uuml;ller (2018) bound $(7.1\pm7.6)\times10^{-14}\,{\rm yr}^{-1}$ and the Williams et al. bound $4\times10^{-13}\,{\rm yr}^{-1}$ — while even the conservative reference-branch ($\lambda_{\rm ref}$) finite-excursion secant $T_{\rm cont}=1.99\times10^{-3}$ at $10^8 R_\oplus$ (static tangent $0.654$; $\le10^{-5}$ at $10^{10}R_\oplus\approx2.1$ pc) yields $|\dot A/A|_{\rm loc}=1.43\times10^{-13}\,{\rm yr}^{-1}$ and $|\dot G_{\rm loc}/G_{\rm loc}|_{\rm LLR}=2.85\times10^{-13}\,{\rm yr}^{-1}<4\times10^{-13}\,{\rm yr}^{-1}$ (this unscreened comparison ceiling satisfies the Williams et al. bound only; the operative screened branch satisfies both LLR analyses) (plus the two-branch scalar-force self-energy drift $2H_0 y_{\rm tb}(1\text{ AU})=2.10\times10^{-15}\,{\rm yr}^{-1}\ll7.1\times10^{-14}\,{\rm yr}^{-1}$, with `solve_ivp` Keplerian ODE integration confirming $\tilde r/\tilde r_0=0.99688$ versus the unscreened $T=1$ shrinkage $\tilde r/\tilde r_0=A^{-2}=0.20186$), while explicit `solve_ivp` ray-tracing with full endpoint 4-velocity normalization $\tilde\omega=-\tilde u^\mu k_\mu$ recovers $\ln(1+z)=0.09531=\ln 1.1$ ($100.00\%$ of the Hubble law for IGM-to-IGM and at the physical $10^8 R_\oplus$ screening scale where $f_{\rm IGM}=1.000000$, $0.09504$ [$99.72\%$] for two halo-scale wells with $f_{\rm IGM}=0.9972$, $0.08972$ [$94.13\%$] for $20$ intervening halo-scale wells with $f_{\rm IGM}=0.9421$ at $z=0.1$, and linear Hubble scaling across the Lyman-$\alpha$ forest lookback sweep $\Delta\in[0.02,0.20]$; Step 70; deriving the time-dependent cosmological matching transfer beyond static Dirichlet boundaries remains part of the open full-evolution problem).

The three times involved must be kept distinct: the gravitational coordinate time $t$ in which the stationary metric $g$ is written; the dynamical (ephemeris) time into which orbital fits absorb; and the matter proper time $\tilde\tau$ of atomic clocks and ranging codes, which locally runs on $A(\phi)\,dt$. A universal drift of $A$ rescales every matter clock and ruler together and therefore cancels from comparisons among them — clock-only networks are blind to it by construction. Lunar laser ranging is not a clock-only comparison: it measures dynamics governed by $g$ against a readout governed by $\tilde g$, which is precisely the mixed channel in which a secular drift of $A$ would appear, as $\dot G/G$. The same mixed comparison motivates the clock-sector interpretation tested in Paper 17; the detected quantity there is the post-fit synodic range coefficient, whose absolute field-to-range mapping is not supplied by the clock identity alone.

The general observable is $1+z=(-\tilde k_\mu \tilde u^\mu)_{\rm em}/(-\tilde k_\mu \tilde u^\mu)_{\rm obs}$ with $\tilde u^\mu$ the physical four-velocities of the emitting and receiving galaxies, so that genuine relative motion contributes to the measured redshift on the same footing as the temporal-geometry contribution. Applying it requires a moving-endpoint problem: emitter and receiver worldlines are evolved through the matter metric and the photon must satisfy the meeting condition $\mathbf x_\gamma(t_{\rm arr})=\mathbf x_{\rm obs}(t_{\rm arr})$ on the observer's past light cone; the energy ratio then already incorporates the appropriate motion and must not be supplemented by a separate Doppler correction. The redshift alone is not the observational prediction: the angular-diameter distance must be obtained simultaneously from the Sachs optical equation for the photon bundle in $\tilde g_{\mu\nu}$, with $D_L=(1+z)^2D_A$ under photon-number conservation, so the completed calculation delivers $z(\mathbf x)$ and $D_A$ together.

The two contributions to the measured redshift separate cleanly by scale. For a photon traversing a distance $L$ between congruence members, the matter-frame frequency ratio combines the drift of the matter-hosting field values accumulated over lookback time with endpoint kinematics,

$$1+z\simeq\exp\!\bigl(H_{\rm drift} L/c\bigr)\,\frac{1+\mathbf v_{\rm em}\cdot\hat n}{1+\mathbf v_{\rm obs}\cdot\hat n},$$

with $\hat n$ the source direction on the sky, $\mathbf v\cdot\hat n$ the recession velocity, and $H_{\rm drift}\equiv-\dot u$ (in $M_{\rm Pl}=1$ units) the macroscopic drift rate of the field values hosting the emitting and observing matter. The temporal drift establishes the overall Hubble flow, while the Doppler factor is bounded by local peculiar velocities. The local redshift field is therefore scatter-dominated — including blueshifted nearby galaxies — and the smooth linear relation emerges only beyond the crossover scale $L_{\rm hom}\sim c\,|\mathbf v_{\rm pec}|/H_{\rm drift}$: for $v_{\rm pec}\sim10^{-3}c$ and $H_{\rm drift}\sim H_0\simeq2.3\times10^{-4}\ {\rm Mpc}^{-1}$, $L_{\rm hom}\approx4$ Mpc. (This homogeneity crossover is distinct from the GNSS covariance length $L_c$ of Section 7.) The transition from local kinematic noise to the Hubble flow is thereby a structural expectation of the nested temporal landscape rather than an environmental accident. Because the global drift operates on an inhomogeneous spatial landscape, observations carry a small predicted dispersion in the redshift–distance relation of order the endpoint $\ln A$ contrast between the emission and observation host wells: a falsifiable signature of landscape inhomogeneity that a perfectly smooth FLRW metric expansion does not naturally produce without extensive dark matter tuning.

The non-expansion requirement is a consistency condition on the scalar sector rather than a free assumption, and the gravitational constraint equations determine which spatial geometries can realize it. On a maximal slice ($\mathcal K=0$; the traceless part of $\mathcal K_{ij}$ remains free, so local deformation is permitted) the Hamiltonian constraint reads ${}^{(3)}R = 2\rho_{\rm tot}/M_{\rm Pl}^2 + \mathcal K_{ij}\mathcal K^{ij} \ge 0$ for nonnegative total energy. Exactly flat static sections would require $\rho_{\rm tot}\le-\tfrac12 M_{\rm Pl}^2\mathcal K_{ij}\mathcal K^{ij}\le 0$, which is impossible for strictly positive total energy. The realized background is accordingly volume-static rather than strictly time-independent: the spatial metric determinant is preserved along the congruence while the lapse and Newtonian-sector potentials respond to the evolving landscape. This is a statement about the total energy density entering the constraint, not an equivalence between negative energy and ghost structure — a negative scalar potential can lower $\rho_{\rm tot}$ without reversing the kinetic sign, and the further restrictions imposed by the full Einstein equations must be analyzed separately. Standard homogeneous kinetic-braiding contributions, being proportional to the expansion rate, vanish identically in the static limit. The static requirement therefore does not select flatness. For a homogeneous isotropic realization the constraint sharpens to the classical Einstein-static result — positive curvature everywhere, hence a closed three-sphere of finite volume — whereas on a noncompact inhomogeneous manifold no such conclusion follows, positive scalar curvature being compatible with infinite spatial extent. The homogeneous closed branch is therefore retained strictly as an analytical benchmark: evidence that the specified action admits a static, eternal configuration at all, not a model of the physical universe's topology. On that benchmark, with dust-like matter transported by the conformal exchange, $\rho_m=\bar\rho\,A(\phi)$, the Einstein equations for a static 3-sphere of radius $a$,

$$3M_{\rm Pl}^2/a^2 = \rho_m + \tfrac12\dot\phi^2 + V(\phi),\qquad -M_{\rm Pl}^2/a^2 = \tfrac12\dot\phi^2 - V(\phi),$$

together with the scalar equation of motion, are satisfied identically by the exact potential

$$V(\phi) = \frac{2M_{\rm Pl}^2}{a^2} - \frac{\bar\rho}{2}\,A(\phi), \qquad \dot\phi^2 = \frac{2M_{\rm Pl}^2}{a^2} - \rho_m,$$

giving a closed, static, eternal gravitational solution with $V>0$ and no ghost degrees of freedom (verified numerically, residuals $<10^{-11}$; Appendix E, R4). On this branch the temporal horizon is approached as $\rho_m\to 0$ ($A\to 0$), the accumulated matter-frame proper time remains finite, and the apparent expansion reverses at the turnaround density $\rho_m = 2M_{\rm Pl}^2/a^2$. Because the homogeneous solution carries no spatial gradient, its disformal volume-compensation term vanishes ($Q=0$) and its matter-frame volume does evolve ($\tilde a=aA$): it exhibits gravitational staticity together with temporal redshift — the gravitational volume is static and the matter-unit evolution $\tilde a=aA$ is the conformal unit drift of the general case. Two transport rates must be distinguished. The conformal-drift rate per gravitational coordinate time, $\dot A/A$, follows $H_t(z)/H_t(0) = \sqrt{1+3\Omega_0\,z/(1+z)}$ and saturates at high redshift; the operational rate measured by matter clocks, $H_{\rm matter}=(aA)^{-1}d(aA)/d\tau=\dot A/A^2$ with $d\tau=A\,dt$, carries an additional factor of $1+z$,

$$H_{\rm matter}(z)/H_{{\rm matter},0} = (1+z)\sqrt{1+3\Omega_0\,z/(1+z)},$$

growing linearly at high redshift rather than as $(1+z)^{3/2}$ — a property of the closed-static branch, not yet a prediction of the inhomogeneous realization. The reconstructed potential is to be read as a constraint on the master family rather than a demonstration of it: it fixes what the matter-domain factor $V_{\rm matter}(u)$ must approach on the closed branch and identifies the deep-field floor with the curvature scale, $V_0 = 2M_{\rm Pl}^2/a^2$ — the same constant toward which the reconstructed profile asymptotes as $u\to\infty$ ($A\to0$). An exact algebraic representation within the parametrized family is exhibited explicitly (Appendix E, R4): the family member with $V_{\rm matter}(u) = \bigl[V_{\rm rec}(u) - V_0\,e^{-(u_s/u)^4}\bigr]e^{+(u/u_s)^4}$ reproduces the reconstructed profile identically (fractional deviation $\sim10^{-16}$ over the branch), while the natural shallow member $V_{\rm matter}(u) = V_{\rm rec}(u)$, $V_0 = 2M_{\rm Pl}^2/a^2$ departs from it only inside the transition band ($\lesssim 6\%$ below $u_s/2$, a $\sim26\%$ dip localized at $u\sim u_s$, asymptotically exact at the floor). The screening–interior join can therefore be modeled analytically: the candidate master potential introduced above smoothly accommodates the weak-field background without disrupting the kinetic screening and flattens onto the floor $V_0$ at deep-well field values. The remaining open condition is quantitative: any candidate potential must remain compatible with the cosmological field range $\Delta u \sim \ln(1+z)$ required by conformal unit drift while flattening onto the deep-well floor — the weak-field screening burden being carried by the kinetic sector of Section 2.2 rather than by $V(\phi)$ — a compatibility whose demonstration belongs to the inhomogeneous solution. The corresponding derivation-level open item is the extension of the Section-4 stability proofs to the noncanonical kinetic branch, not the potential range.

Localized positive-energy structure is accommodated on non-expanding slices through the Lichnerowicz form of the Hamiltonian constraint: writing $h_{ij}=\psi^4\hat h_{ij}$ with $\mathcal K=0$, trace-free extrinsic curvature $\mathcal K^{ij}=\psi^{-10}\hat{\mathcal A}^{ij}$, and scalar momentum $\Pi=n^\mu\nabla_\mu\phi$,

$$\hat\nabla^2\psi = \frac{\hat R}{8}\,\psi - \frac18\,\psi^{-7}\hat{\mathcal A}_{ij}\hat{\mathcal A}^{ij} - \frac{\psi^5}{4M_{\rm Pl}^2}\left(\rho_m+V+\tfrac12\Pi^2\right) - \frac{\psi}{8M_{\rm Pl}^2}\,|\hat\nabla\phi|^2,$$

supplemented by the momentum constraint — an inhomogeneous field with $\dot\phi\neq0$ and $\nabla_i\phi\neq0$ generally carries momentum, requiring compensating matter momentum or an appropriately constructed configuration. Preservation of $\mathcal K=0$ under evolution imposes the maximal-slicing lapse equation $D^2N=N\bigl(\mathcal K_{ij}\mathcal K^{ij}+(\rho_{\rm total}+S_{\rm total})/2M_{\rm Pl}^2\bigr)$, with the source taken from the exact (untruncated) matter stress tensor of Section 2; an initial zero-expansion state is insufficient if the coupled evolution immediately drives it away, and Raychaudhuri's equation — $\nabla_\mu a^\mu=\sigma_{\mu\nu}\sigma^{\mu\nu}+R_{\mu\nu}u^\mu u^\nu$ for a nonrotating zero-expansion congruence — identifies precisely which acceleration, curvature or shear terms must balance to keep the congruence non-expanding. The $\hat R\psi/8$ term is essential on closed slices, where the equation would otherwise admit no positive solution; on the intended noncompact slices $\hat R$ is unconstrained in sign. The shear term $\hat{\mathcal A}_{ij}\hat{\mathcal A}^{ij}$ is the precise place where the permitted volume-preserving deformation enters the constraint. This defines a constraint problem for initial data — the spatial realization of the inhomogeneous temporal network through which the macroscopic shear trend is expressed — and is to be distinguished from an evolved solution: the homogeneous closed branch does not by itself establish existence of the required inhomogeneous cosmos. That existence has now been demonstrated numerically (Appendix E, R8): on a periodic cell whose universal cover is a non-compact inhomogeneous slice, the constraint admits a smooth positive conformal factor for a landscape of matter-hosting wells embedded in a void sector, converging to residual $\sim10^{-10}$ under a bordered Newton solve of the full nonlinear equation. The Gauss integrability condition $\int\psi^5\rho_{\rm tot}\,dV=-\tfrac12\int\psi\,|\hat\nabla\phi|^2dV-\tfrac12M_{\rm Pl}^2\int\psi^{-7}\hat{\mathcal A}^2dV<0$ makes one structural feature compulsory rather than optional: on the non-compact covering slice the eternal solution requires the total energy density to change sign across the landscape — positive inside matter-hosting wells, negative in the void sector. Only the scalar potential can carry the negative budget, and the master family supplies it natively: the reconstructed branch $V_{\rm rec}(u)=V_0-\tfrac{\bar\rho}{2}e^{-u}$ crosses zero at $u_*=\ln(\bar\rho a^2/4M_{\rm Pl}^2)=-\ln 2\simeq-0.6931<0$, so void field values $u_{\rm void}<u_*$ — where matter clocks run faster than ambient — carry $V<0$ natively (verified in Step 20 both on the shifted positive-$u$ benchmark $u_{\rm void}=+0.8$, where the bordered Newton offset $\lambda_*=-2.42$ supplies the negative energy, and on the native negative-$V_{\rm rec}$ void realization $u_{\rm void}=-0.85<u_*$, where $8{,}496$ void cells carry $V(u)<0$ natively before any shift with $V_{\min}=-0.3396$ and converged $\psi\in[0.9985,1.0029]$). The mean gravitational-frame curvature on the solved slice is driven to $\sim10^{-3}$ of the local scale rather than to the closed branch's uniform positive value. The lapse and momentum/shear calculations sharpen the scope of this construction (Appendix E, R13). The canonical lapse calculation uses $\rho_\phi+S_\phi=2\Pi^2-2V_{\rm eff}$ and the covariant Laplacian $D_h^2N=\psi^{-6}\nabla\cdot(\psi^2\nabla N)$. On the prescribed periodic landscape, the unmodified lapse source is positive. A spectral shift can produce a positive zero-mode lapse, but at fixed Hamiltonian budget that shift requires $\Pi^2<0$ and is inadmissible for a real canonical scalar. Step 21 instead retains the original physical source with $\Pi=0$ and solves $(-D_h^2+q)N=C$, normalized by $\langle N\rangle_{\sqrt h}=1$. It obtains a positive lapse on both $16^3$ and $48^3$ grids, with relative residual below $10^{-10}$; $C=\partial_t\mathcal K$ is spatially uniform and nonzero. This establishes an evolving constant-mean-curvature initial-slice benchmark without negative kinetic energy. It does not establish preservation of $\mathcal K=0$, long-time evolution or the complete noncanonical cosmological branch. Exact non-expansion remains a condition to demonstrate for the intended eternal realization. Second, the momentum/shear sector is now solved rather than assumed away: the drift momentum of the scalar on the gradient landscape sources the momentum constraint through $j_i \propto \Pi\,\hat\nabla_i u$, the conformal Killing construction $\hat{\mathcal A}_{ij}=(\hat L W)_{ij}$ satisfies the constraint identically, and the full Hamiltonian equation including the $\hat{\mathcal A}^2\psi^{-7}/8$ volume-preserving-shear term admits a positive conformal factor at every drift amplitude scanned ($c_\Pi = 0$–$8$; $\psi\in[0.75,1.81]$ throughout), with the linearized constraint operator invertible at each solution — no flat or marginal direction, so the data are locally unique. The momentum/shear scans establish their specified initial-data constructions; they are not the same solution as the zero-momentum lapse benchmark, and global non-expanding evolution remains open. An initial evolution attempt on prescribed wells failed at the initial-data level, and diagnosing it identified a construction error rather than a defect of the drift law. On the prescribed slice the scalar equation of motion carried an rms residual of order $10^{2}$ at $t=0$, so the integration relaxed off its constraint shell rather than evolved; the apparent drift reversal within $\Delta t\sim0.01$ is reproduced exactly by a momentum kick opposing the ambient slope, which turns any positive $\Pi$ at $t\sim\Pi_0/|\lambda_c|\sim0.02$, where $\lambda_c$ is the periodic-domain solvability constant of the scalar equation (of order the constraint offset $\lambda_*$). The uniform part of the scalar residual is the homogeneous roll itself, not an error term: on the reconstructed branch $V_{,u}\propto e^{-u}>0$ drives $u$ downward, so matter clocks ran slower at earlier times — the secular redshift trend is the natural downhill roll, not an imposed motion. With the field instead solved from its elliptic equation on the matter landscape — a bordered Newton solve of $D^2u=V_{,u}-\rho_m+\lambda_c$ in mean-field gauge, iterated against the Lichnerowicz solve — the initial slice satisfies the scalar equation to $10^{-8}$ and the constraint to $10^{-12}$ (Appendix E, R14). The same integration machinery then produces a sustained drift without reversal: the field descends monotonically through $\Delta u\simeq3.4$, a lookback redshift span $z\lesssim28$, with the Hamiltonian residual held near $10^{-5}$ when the constraint and the maximal-slice lapse are re-solved on the evolved sources. The residual limitation is the drift's profile rather than its existence: at the benchmark parameters the ambient field sits on the steep reconstructed region where $V_{,u}\propto e^{-u}$ grows as $u$ falls, so the roll accelerates; a Hubble-law-shaped $H_T(z)$ requires the ambient on the shallow quartic-suppressed region of the master potential, making the ambient-branch identification an explicit theory input rather than a numerics obstruction. The same solved construction carries the hierarchical structure of Rules 22–23 directly: adding a broad overdense envelope beneath the well lattice and solving the scalar equation globally produces three coexisting levels on one slice — an exterior ambient, a deeper parent-envelope ambient ($\Delta u\simeq+0.022$, i.e. envelope-interior clocks slower than the exterior ambient by $\simeq2\%$), and embedded host wells — each with its own baseline and drift rate, with deeper-baseline hosts descending measurably faster. The kinetic screening operator of Section 2, evaluated a posteriori on the evolved slice with $\Lambda_X^4=M_{\rm Pl}^2H_0^2$ fixed by the slice's own drift rate, returns $P_{,X}-1\simeq5\times10^{-6}$ everywhere: the global landscape benchmark sits deep in the weak-gradient regime, so nested screening is present in the operator but not activated at landscape amplitudes — as expected, since screening responds to temporal-field steepness (deep, narrow wells), not to spatial compactness, and the cosmological fabric carries only shallow gradients. The estimate is conservative in a second sense: it is normalized on the accelerating branch's inflated drift rate $H_T\simeq40$, which overestimates $\Lambda_X^4=M_{\rm Pl}^2H_0^2$; on a Hubble-normalized branch ($H_T\sim1$) the same landscape would sit at $P_{,X}-1\sim10^{-2}$, and a microscopic well of depth $\Delta u\sim1$ — the Rule-23 high-redshift configuration — crosses the screening threshold outright.

*Energy-condition bookkeeping.* The negative void-sector energy quoted above is the Einstein-frame effective total $\rho_{\rm tot}$ entering the Hamiltonian constraint — the combination the slice geometry responds to — and the sign change is carried entirely by the scalar potential ($V_{\rm rec}<0$ for $u<u_*$), not by a wrong-sign kinetic term. Pointwise, the scalar obeys the null energy condition on the admissible branch, $T_{\mu\nu}k^\mu k^\nu = P_X(k\cdot\nabla\phi)^2 \geq 0$ since $P_X>0$; what the integrability condition forces is a violation of the weak and dominant conditions inside the void sector and a negative cell average ($\langle\rho_{\rm tot}\rangle = -0.52$ in slice units, step_20), which is precisely the energetic cost of an eternal non-expanding solution on a non-compact cover. This does not conflict with the positive dark-energy-like readings of the scalar sector elsewhere in the corpus (Papers 18, 26, 30): those concern the effective energy contribution of the scalar on the matter-hosting drift sector, where $V>0$ — the two statements refer to different regions of the same landscape, not to two different stress tensors. The same bookkeeping is imported by the horizon-sector analysis (Paper 27), whose strong-energy-condition argument inherits this framing.

The static solution is not a dynamical attractor: linearization of the homogeneous system exhibits an Eddington-type secular mode with e-folding time $\sim a/c$. Nor can this branch approach observationally relevant flatness: with $\dot\phi^2=2M_{\rm Pl}^2/a^2-\rho_m$ and matter-frame rate $H_{\rm matter}=\dot A/A^2$, the reconstructed curvature parameter satisfies $|\Omega_k|=M_{\rm Pl}^2/(a^2\dot\phi^2)\ge\tfrac12$ for $\rho_m\ge0$, so the closed-static solution cannot realize a near-flat matter geometry; its stability window is therefore of mathematical rather than observational significance; the analogous direction in the solved inhomogeneous realization is classified quantitatively by the eigenmode computation reported below.

*Restricted coupled spectra and conditional quartic stability.* Step 79 gives positive finite-cell reduced and Lichnerowicz-coupled spectra on the imposed weighted-zero-mean subspace at $N=24,32$; Step 82 independently bounds its specified ambient branch from below ($+377.40$ to $+378.81$ across $N=24,32,40$). The nearly homogeneous negative mode of the bare reconstructed canonical potential ($V_{\rm rec}''=-\tfrac12\bar\rho_* e^{-u}\simeq-0.4389$) remains recorded and its gauge classification is not inferred from projection; when the second variation of the conformal matter coupling $+\rho_*(x)\frac{d^2(e^{-u})}{du^2}=+\rho_*(x)e^{-u}$ in $\mathcal V_{\rm eff}(u,x)=V_{\rm rec}(u)+\rho_*(x)e^{-u}$ is retained at the Step-12 cell-mean density $\langle\rho_*\rangle=\bar\rho_*=2\,\text{rhob\_half}$ (`completed_canonical_reconstructed_coupled_audit` in Step 79), the cell-mean effective mass squared flips from $-0.4389$ to $+0.4102>0$ and the unprojected canonical Euclidean ($\lambda_0=+0.4087$ at $N=16$, $+0.4093$ at $N=24$) and Lichnerowicz-coupled two-branch ($\lambda_0=+0.4123$ at $N=16$, $+0.4127$ at $N=24$; $\lambda_1=+367.02$ and $+350.97$) spectra are strictly positive without homogeneous-mode projection. The separate quartic benchmark re-solves the Hamiltonian geometry while retaining a prescribed scalar profile, ambient stiffness and uniform integrability offset, and a companion jointly on-shell audit in Step 79 (`joint_on_shell_quartic_two_branch_coupled_audit`) solves both the nonlinear two-branch scalar Euler–Lagrange equation and the Lichnerowicz Hamiltonian constraint simultaneously to $<3.1\times10^{-12}$ ($N=16,24$). In both quartic audits, scalar perturbations are unprojected while the metric response obeys $q_{\rm vol}^T\delta\psi=0$ through an exact bordered inverse, yielding strictly positive unprojected spectra ($\lambda_0=+0.5624$ at $N=16$, $+0.5633$ at $N=24$; $\lambda_1=+378.95$ and $+381.67$ on the jointly on-shell background) and positive all-twist bounds ($+0.4662$ and $+0.4668>0$) using the half-wave lattice gap and the constraint-vector angle in periodic Bloch gauge. These finite-grid constrained-operator benchmarks do not establish full-action continuum stability. Likewise, the khronon stress-shift calculation in Step 21 solves an initial-slice lapse equation; its drift equation and preservation through time remain separate requirements for an eternal solution.

*Spatial-curvature matching.* Step 20 retains a conformal spatial benchmark with $A=e^{-u}$ and demonstrates its smoothness dependence ($|\Omega_k^{\rm eff}|\sim7\times10^{-2}$ at $\Delta u=0.1$ falling to $\sim10^{-3}$ at $\Delta u=0.02$). For the operative lapse/spatial-ruler metric at $B=0$, intrinsic matter curvature instead follows from the actual spatial factor $A_{\rm sp}$:

$${}^{(3)}\tilde R=A_{\rm sp}^{-2}\left[{}^{(3)}R-4D^2\ln A_{\rm sp}-2|D\ln A_{\rm sp}|^2\right].$$

Here $D$ is the derivative of the gravitational spatial metric. Local lapse gradients are not spatial conformal gradients. On a fixed-epoch slice with spatially constant $A_{\rm sp}(t_0)=1$, the derivative terms vanish identically and ${}^{(3)}\tilde R(\mathbf x)={}^{(3)}R(\mathbf x)=-8\psi^{-5}\nabla^2\psi$ pointwise: by Gauss's theorem on the periodic cell, $\int\psi^5\,{}^{(3)}\tilde R\,d^3x\equiv 0$ identically, and the physical-volume-weighted mean curvature evaluated directly on the solved $48^3$ Lichnerowicz slice in Step 20 gives $|\Omega_k^{\rm native}|=|\langle{}^{(3)}\tilde R\rangle_{\rm vol}|/(6H_{\rm drift}^2)=1.40\times10^{-4}\ll10^{-3}$, nearly an order of magnitude flatter than the conformal-derivative benchmark.

The past-ward drift is accordingly sector-asymmetric rather than a rigid shift of the full landscape. The drift rate $H_{\rm drift}\equiv-\dot u$ is defined on the field values hosting emitting and observing matter — the sector sourced by the matter trace through $\nabla^2\phi=V_{,\phi}+\rho_*A_{,\phi}$. The void sector is unsourced ($\rho_*\to0$) and relaxes toward the contemporaneous ambient configuration, so its excursions remain bounded by the fixed-time spatial contrast $|\delta u|\lesssim10^{-3}$–$10^{-2}$ at every epoch; no term in the field equation drives void field values secularly negative, and a growing contrast would conflict with the scatter bound just stated. The deep-past direction is therefore occupied by deepening matter-hosting wells, while voids remain anchored within the contrast scale of ambient, $A_{\rm void}-1\lesssim|\delta u|$ — clocks faster than ambient by at most the contrast scale, never diverging. The sector is moreover doubly inert observationally: emitters and absorbers reside in wells, so the endpoint ratio samples only matter-hosted field values, and photons traversing voids accumulate no conformal shift because the conformal factor drops out of the null covector — the void contribution is confined to the path-integrated lapse channel already enumerated, and the high-redshift scatter distribution is set by the depth scatter of the emitting well population rather than by a void tail. Correspondingly, no large-negative-$\phi$ domain is realized on the eternal slice, and the admissibility conditions on $B(\phi)$ established within the small-$B$ EFT of Section 4 are required only over the bounded realized range.

The apparent spatial singularity conventionally written as $a \to 0$ is formally re-expressed as a temporal horizon ($\mathscr{T}^-$) where the macroscopic observational clock map $A_{\rm clock} \to 0$ on a gravitationally regular, eternal spatial manifold. While a naive backward extrapolation of the late-time steady roll ($\dot u \simeq -\bar\Pi$, $A \propto e^{\bar\Pi t}$) would cause matter-frame curvature invariants to diverge as $\tilde R \propto (\dot A/A)^2 A^{-2} \propto A^{-2}$ — reproducing the apparent-singularity phenomenology that standard FLRW reconstruction reads as a Big Bang — the deep-field master potential floor $V_0$ decelerates the roll toward the asymptotic past onto the power-law horizon profile $A_{\rm clock}(\eta) \sim \eta^{-p}$ ($0 < p \le \tfrac12$, $d\ln A_{\rm clock}/d\eta = -p/\eta \to 0$ as $\eta \to \infty$). As proved in TEP-TH (Paper 27, Proposition 1), on this decelerating horizon branch all matter-frame curvature invariants vanish ($\tilde R \sim \eta^{2p-2} \to 0$, $\tilde K \sim \tilde R_{\mu\nu}\tilde R^{\mu\nu} \sim \eta^{4p-4} \to 0$) and both timelike and null geodesics are past-complete, demonstrating that the Big Bang curvature singularity is a reconstruction artifact of imposing an isochronous expanding frame on the temporal horizon. Two quantities must be distinguished here: the local coupling $A(\phi)$ appearing in the action, and the reconstructed clock factor $A_{\rm clock,obs}(z)\equiv 1/(1+z)$ defined by photon observation. They coincide on the homogeneous conformal branch; in the inhomogeneous theory the observational map follows from the evaluation of the temporal field $\phi(t,\mathbf{x})$ across the realized landscape, together with motion and any disformal term, and TEP-TH's horizon regularity and completeness results — derived in a homogeneous conformal model — must be formally verified on the realized inhomogeneous geometry. Likewise, $z\to\infty$ is a statement about temporal depth, not by itself a visibility cutoff: the effective bound on the observable universe follows from opacity — the diverging apparent optical depth demonstrated as the 1D Global Opacity Theorem in TEP-BBN — rather than from redshift alone. In the matter frame, the continuity equation for $\tilde{\rho}_m$ is standard, while apparent kinematic acceleration manifests entirely from the evolving Temporal Shear. The FLRW form adopted within the companion Boltzmann and transport integrations (Papers 18, 26) realizes the reconstructed scale factor of the static-conformal isomorphism — a computational decomposition of matter-sector observables — and attributes no expansion to the gravitational sector.

## Temporal-Horizon Chemical Equilibrium and Proper-Time Reaction Flow

Big Bang Nucleosynthesis (BBN) and the epoch of Recombination are not modeled as chronological eras following a fiery expansion. Instead, early-universe closure is governed natively by the Proper-Time Reaction Flow over a gravitational history that is eternal in coordinate time; the accumulated matter-frame clock time is finite on the closed-static benchmark branch of Section 8 — an existence result, not the physical realization — while the observational temporal depth remains unbounded ($A\to0$, $z\to\infty$).

The classical stellar astration paradox is resolved natively. On the benchmark branch a finite accumulated matter-frame clock time bounds stellar processing, which shows the mechanism is available in principle; in the general inhomogeneous realization that carries the physical claim, the corresponding bound is supplied where local chemical evolution approaches a steady-state fixed point, subject to the temporal-exposure convergence condition. The unbounded accumulation of heavy elements is prevented by local temporal sequestration: the formation of local Temporal Horizons (black holes) produces extreme but finite transport delays that effectively remove heavy metals from the active baryonic cycle. Consequently, the observed light-element abundances are not primordial artifacts of a global singularity; rather, TEP-BBN demonstrates — within the small-$B$ EFT regime of Section 4 — a candidate asymptotic chemical equilibrium in which the observed light-element abundances arise through long-term baryonic cycling, subject to the temporal-exposure convergence condition derived there. The steady-state budget is closed quantitatively in TEP-BBN (Gates 8, 11, 13, Steps 17–18): the reaction-flow map relaxes to within one per cent of the asymptotic abundances in approximately nine star-formation timescales from any tested initial composition; with literature stellar yields ($p_Y^{\rm VMO}=0.149$, $p_Y^{\rm std}=0.02$, $p_Z^{\rm std}=0.01$) and IMF return fraction ($R=0.40$) fixed by independent stellar physics, a single low-metallicity VMO fraction $f_{\rm VMO}=0.994$ is overconstrained across four observables simultaneously ($Y_{\rm eq}=0.247$, metal-poor DLA floor $Z_{\rm eq}=10^{-4}$, Spite plateau $^7{\rm Li/H}_{\rm eq}=1.55\times10^{-10}$ resolving the $3.2\times$ hot-BBN lithium problem, and $^3{\rm He/H}_{\rm eq}=9.0\times10^{-6}$), while local ISM deuterium ($\text{D/H}_{\rm ISM}\simeq1.7\times10^{-5}$) is supplied by VMO partial-burning envelope ejecta ($f_{\rm ej}\simeq0.50$, $\epsilon_D\simeq4.1\times10^{-5}$) and collapse-transient $\bar\nu_e$ irradiation alongside the high-$z$ velocity-sheared H I absorber solution ($N(\text{D I})=0$ under the joint D+H model — the D column is redundant on all three benchmark sightlines), and metal overproduction is excluded because the dominant low-metallicity channel sequesters its core yield behind temporal horizons while total astration exposure remains finite ($p+q=1.3>1$).

A candidate TEP origin of the Cosmic Microwave Background is the steady-state thermalization of distributed radiation, with its blackbody form preserved by achromatic conformal propagation ($\mu_{\rm transport}=0$, $y_{\rm transport}=0$ in TEP-TH Step 06, and thermalization-edge Compton/chemical distortions bounded by the gas-to-radiation energy ratio $|\Delta\mu|,\,|\Delta y|\sim U_{\rm gas}/U_{\rm rad}=1.34\times10^{-16}\ll9\times10^{-5}$ in TEP-BBN Step 18B, with the local bath law $T_{\rm loc}(z)=T_0(1+z)$ confirmed directly against high-$z$ CO rotational excitation lines in TEP-TH Step 11). The universe becomes completely opaque at high redshift because the diverging temporal depth stretches the apparent optical depth to infinity, creating an observable boundary without a physical plasma wall (the 1D Global Opacity Theorem in TEP-BBN). The thermalization and acoustic-boundary budgets are closed quantitatively in TEP-BBN Steps 19–20: the conformal-isomorphism Boltzmann calculation of TEP-HC (in conformally drifting matter units) and the eternal static-coordinate profile are dual representations of the same integrated Thomson column linked by the path Jacobian $J(z_*)=(d\ell/dz)/(c/HL_0)=4.15\times10^7$; solving the coupled self-shielded LyC radiative transfer directly from the $T_{\rm loc}(z)=T_0(1+z)$ Planck Wien tail ($\chi=h\nu_{\rm L}/k_{\rm B}T_{\rm loc}\simeq53.5$) forward-predicts $z_{\tau=1}^{\rm pred}=1077.9$ at the Step-14 $600\,n_{b,0}$ midpoint ($1079.8^{+4.6}_{-4.5}$ from column uncertainty alone, at the adopted LLS column-density input and fixed density conversion $\log_{10}N_{\rm HI}=17.345\pm0.150\implies661.2\,n_{b,0}$, span $[1069.9,1083.6]$ across $[400,800]\,n_{b,0}$) and refines the front overdensity to $\Delta_{\rm front}=668.9\,n_{b,0}$ ($n_b=1.68\times10^{-4}\text{ cm}^{-3}$, inside the observed KODIAQ/SQUAD LLS/DLA absorber band) for $\tau_{\rm Th}=1$ at $z_*=1080.0$, while deriving the pre-boundary inertial driving density directly from the zero-free-parameter two-branch temporal-shear law at the sound horizon — where $g_t=\Lambda_X^2/(\sqrt{2}M_{\rm Pl})=cH_0/2$ is a fixed constant of the scalar action $S_\phi$ and $q_s=g_b(r_s)/g_t=\Omega_b(H_0 r_s/c)$ is conformally invariant under $A(\phi)$, yielding the single-pass analytic sound-horizon estimate $\omega_{m,\rm eff}=\omega_b(1+y_s/\sqrt{2})=0.1391$ ($\omega_\phi=0.1166$, $q_s=1.62\times10^{-3}$, $y_s=7.38$, $M_s=6.22$ at $r_s=146.19\text{ Mpc}$, $\theta_s^{\rm derived}=0.010425\text{ rad}$, $\ell_A=301.3$ in the reduced 3-component integral) and converging under self-consistent fixed-point iteration of $r_s(\omega_\phi)$ in exact `hi_class` to $r_s=144.94$–$145.00\text{ Mpc}$, $q_s=1.59$–$1.60\times10^{-3}$, $y_s=7.42$–$7.44$, and $\omega_\phi=\omega_b(y_s/\sqrt{2})=0.1179$–$0.1182$ ($\omega_{m,\rm eff}=0.1404$–$0.1406$; spanning $\omega_\phi\in[0.1174,0.1199]$ across $H_0\in[67.4,70.0]\text{ km s}^{-1}\text{ Mpc}^{-1}$, all within $\lesssim1.6\sigma$ of the unconstrained TEP-HC full-Planck TTTEEE MCMC posterior $0.11813\pm0.00117$, with $0.02\sigma$ at the posterior mean $H_0=68.18$, $100\theta_s^{\rm CLASS}=1.04201$ [$+0.10\sigma$], $100\theta_s=1.04111$ [$+0.02\sigma$ at $H_0=67.95$], $\Delta\chi^2=-0.06$ and $\Delta\text{AIC}=-2.06$ on the full Planck TTTEEE + low-$\ell$ + lensing + BAO + Pantheon+ likelihood, linear-scale $\sigma_8=0.8170$–$0.8186$, $r_d=147.49$–$147.55\text{ Mpc}$, SDSS DR12 BAO+RSD $\chi^2/N=0.71$–$0.76$, and peaks within $\le0.31\%$–$0.40\%$; TEP-HC Step 21 Part C) — and integrating the coupled linear Einstein–Boltzmann baryon–photon acoustic ODE with spherical-Bessel projection — where the driving component is the continuous ambient proper-time field's two-branch clock-slope effective perturbation mode whose timelike coefficient is evaluated from the quadratic expansion of the scalar action $S_\phi=\int d^4x\sqrt{-g}\,M_{\rm Pl}^2\Lambda_X^4 P(\xi)$ around the rolling cosmic background $\dot{\bar\phi}=M_{\rm Pl}H_0$ ($\bar\xi=1$, canonical timelike kinetic coefficient $\mathcal K_t=\bar P_{,X}(1)+2\bar P_{,XX}(1)=2k+6=38.060$ supplying the momentum $\pi_u=\mathcal K_t\dot u$) while the reduced acoustic calculation prescribes $\mathcal K_{\rm sp}^{\rm eff}=\sqrt{2}/y_s=0.1917$ and $c_{p,\rm eff}^2=\mathcal K_{\rm sp}^{\rm eff}/\mathcal K_t=5.04\times10^{-3}$, with zero Thomson coupling. This is a conditional effective closure: the native spatial flux Hessian has transverse coefficient $K_\perp=1/y_s=0.1356$ (multiplicity 2 on $S^2$) and longitudinal coefficient $K_\parallel=2/y_s+(q_sy_s)^2=0.2713$ (multiplicity 1), with 3-D rotational trace average $K_{\rm iso}=(2K_\perp+K_\parallel)/3=0.1808$ and 3-D Frobenius/RMS principal stiffness $K_{\rm rms}=\sqrt{(2K_\perp^2+K_\parallel^2)/3}=0.1918$ (whose deep-IR $q_s\to0$ limit is $\sqrt{2}/y_s$, while peak multipoles shift by $\le0.12$ across $[K_\perp/\mathcal K_t,K_\parallel/\mathcal K_t]$), so the background flux identity alone does not derive this acoustic normalization or its pairing with $\mathcal K_t$. The following peak comparisons apply to the prescribed effective equations: the model recovers the Silk diffusion damping multipole $\ell_D=1520.7$ ($\theta_D=6.58\times10^{-4}\text{ rad}$, within $1.2\%$ of Planck) and the first three acoustic peak multipoles at $\ell=221.0,\,527.1,\,811.8$ in the reduced ODE ($\le1.94\%$ at $c_{p,\rm eff}^2=0$, $\le1.90\%$ with $c_{p,\rm eff}^2=5.04\times10^{-3}$, and $\le0.40\%$ in exact `hi_class`; closure ablation: wavenumber-resolved $y(k)$ $\le7.54\%$, slaved-Poisson null $\le5.41\%$, isolating the autonomous ambient time-field inertia as the operative closure; on linear cosmological scales $k\lesssim0.2\,h\,\text{Mpc}^{-1}$ [$R\gtrsim10\text{ Mpc}$, governing BAO and $\sigma_8$] the scale-averaged clock-slope multiplier $M_s=1+y_s/\sqrt{2}\simeq6.25$ enters the linear Poisson equation identically to an effective density $\omega_{m,\rm eff}=\omega_b+\omega_\phi$, whereas in the nonlinear static virialized galaxy/cluster limit [$R\lesssim100\text{ kpc}$, $\dot u\to 0$, $\dot\Phi\to 0$, $g_N\gtrsim g_t$] the timelike momentum $\pi_u=\mathcal K_t\dot u$ vanishes identically and the nonlinear operator $(k/\sqrt{2})q y^2+q^2 y^3=1$ screens down to the static baryon-only two-branch RAR/lensing law with zero independent collisionless dark-matter halo cusp, unlike free integration-constant or mimetic dust) with the same baryon-to-photon momentum ratio ($\Omega_b h^2=0.02237$). The two-regime behaviour of the ambient clock-slope mode — CDM-like on linear scales ($k\lesssim0.2\,h\,{\rm Mpc}^{-1}$) and screened to the baryon-only two-branch law inside virialized halos — is phenomenologically analogous to dark-matter superfluidity (Berezhiani & Khoury 2015), with the difference that in TEP the transition is governed by the kinetic stiffness $P_{,X}$ of the proper-time field rather than by a condensate phase transition. At late times, distance–redshift observables are reconstructed through the continuous three-piece redshift identity evaluated on the realized inhomogeneous landscape. Growth and lensing provide independent consistency tests of the realized cosmological solution; their quantitative response is derived from that solution rather than imposed through a generic unscreened scalar–tensor growth law.

*Status of $\Lambda$ in this framework.* Within TEP, $\Lambda$ is reinterpreted rather than removed: the accelerating-distance signature normally attributed to a cosmological constant is reconstructed as the kinematic bookkeeping contribution of the clock-drift mapping, $\Omega_\phi$ — not a literal energy density (Papers 18, 26). Its value is not fitted: the Paper 18 zero-parameter fixed point ($\omega_b$, $g_t$, $\theta_s$) fixes $\omega_{m,\rm eff}$ and $H_0$, and spatial flatness of the matter metric then gives $\Omega_\phi\simeq0.697$. The temporal-transport account is a reinterpretation of the Hubble-diagram phenomenology, not a claim that no corresponding energy density exists; the observational content of $\Lambda$ is preserved and re-sourced, not denied. At the level of distance–redshift observables, the temporal-transport description can be degenerate with an expanding-$\Lambda$CDM fit on the Hubble diagram (Paper 30). Discrimination therefore requires independent observables, including structure growth, lensing, and clock/transport tests, rather than the distance sector alone.

## EFT of dark energy mapping

For comparison with standard cosmological perturbation analyses, the late-time scalar sector can be represented in EFT-of-dark-energy language, with $\alpha_T=0$ enforcing $c_T=1$. The braiding and effective Planck-mass-running functions provide a phenomenological dictionary for perturbations of the realized TEP cosmological solution. This mapping does not identify the underlying static spatial manifold with a physically expanding FLRW background; quantitative constraints on the effective functions are developed in the companion cosmology analyses.

## Standard sirens

In TEP, gravitational perturbations propagate in the gravitational sector, whereas interferometric detectors measure strain through matter-frame clocks, rulers and electromagnetic signals. The detector response can therefore depend on the conformal field ($A(\phi)$); scalar temporal perturbations may also contribute to the observed waveform. The proposed conformal standard-siren relation is

$$\frac{d_L^{\rm GW}}{d_L^{\rm EM}} = A(z) = \frac{1}{1+z}.$$

This relation is established at the detector level on the tensor-ripple branch through a four-stage mapping on the homogeneous background limit ($u_{\rm grav}=0$, $d\tilde s^2=-c^2dt^2+A_{\rm sp}^2(t)\gamma_{ij}dx^i dx^j$, or in conformal time $d\eta=dt/A_{\rm sp}(t)$, $d\tilde s^2=A_{\rm sp}^2(\eta)[-c^2d\eta^2+\gamma_{ij}dx^i dx^j]$), each link symbolic-verified in the pipeline record (Appendix E, R10, R19; Steps 21, 80). (i) Emission: the $g$-sector wave equation is sourced by the Einstein-frame stress tensor $T_g^{\mu\nu}=A^6\tilde T^{\mu\nu}$, under which a matter-frame binary of chirp mass $\tilde{\mathcal M}$ radiates with effective mass $\mathcal M_e=A_e\,\tilde{\mathcal M}$, and the phase evolution follows the quadrupole law $\dot f_e\propto \mathcal M_e^{5/3}f_e^{11/3}$. (ii) Transport: both photons and transverse-traceless gravitons propagate along the shared null cone $dx/d\eta=c$ ($c_{\rm GW}=c_\gamma=c/A_{\rm sp}(t)$, so $\Delta t_{\gamma-\rm GW}\equiv0$), while because the spatial 3-metric $\gamma_{ij}$ is static ($\sqrt{\gamma}=1$) and the WKB geometric-mean tensor weight in $S_{\rm TT}^{(2)}$ is $\sqrt{A_{\rm sp}^{+1}\cdot A_{\rm sp}^{-1}}\equiv 1$ (constant, with zero $2\mathcal H\partial_\eta h_{ij}^{\rm TT}$ Hubble friction), conformal-time wave intervals are preserved ($\Delta\eta_o=\Delta\eta_e$) and the tensor amplitude decays geometrically as $1/r$ on static space with no expansion stretch factor. (iii) Readout: the interferometer counts arriving cycles against matter clocks ($\tilde f_o=\tilde f_e/(1+z)=(A_e/A_o)\tilde f_e$), while the measured strain is a dimensionless length ratio, invariant under the local ruler factor ($\tilde h=h$) because $A_o$ does not vary across the wave train. (iv) Inference: matching the observed phase evolution to the standard template in locally measured Cavendish units $G_{{\rm loc},o}=G_*A_o^2(1+\alpha_o^2)$ returns $G_{{\rm loc},o}\mathcal M_{\rm det}=A_o G_{\rm dyn}\mathcal M_e$ ($\mathcal M_{\rm det}=(A_e/A_o)\tilde{\mathcal M}=\tilde{\mathcal M}/(1+z)$ for screened systems at $A_o=1$), and amplitude matching returns $d_L^{\rm GW,inf}=A_o\,r$; the matter-frame luminosity distance follows from the source area distance $D_A=A_e\,r$ on $\tilde g$ through Etherington reciprocity, $d_L^{\rm EM}=(1+z)^2D_A=(1+z)A_o\,r$. The ratio of the inferred distances therefore reduces to the transport factor stated above, with no propagation content hidden in the mass or amplitude inference. The mapping carries one immediate population-level discriminator: the inferred chirp mass is $\mathcal M_{\rm det}=\tilde{\mathcal M}/(1+z)$ for $A_o=1$ — the inverse conformal factor relative to the FLRW convention $(1+z)\tilde{\mathcal M}$ — so the redshift trend of an intrinsic source population separates the two maps by a factor $(1+z)^2$. Residual items on the same branch are higher-order: scalar-mode admixture in the waveform, dipole back-reaction corrections to $\dot f$, and modulation of the effective sourcing mass by the source's screening environment remain part of the joint gravitational-wave derivation and perturb the mass map beyond leading quadrupole order. Catalogue-level constraints on modified GW propagation are obtained under an FLRW distance–redshift relation and the standard mass mapping, so they bound the transport factor conditionally rather than the completed endpoint map. Common null cones alone establish propagation causality; the chain above supplies the amplitude and mass bookkeeping that connects the theory to the quantity actually inferred from data.

A consistency question remains at the emission link: if a source binary is entirely standard in its own local units, a local observer at the source should infer the standard chirp mass — raising the possibility that the nonstandard mapping cancels out. Tracking the dimensionless inspiral invariant $I = G_{\rm loc}\,\mathcal M f/c^3$ through the full chain settles this (Appendix E, step_70): a source-local observer using the locally measured Cavendish constant $G_{\rm loc,e} = G_*A_e^2(1+\alpha_e^2)$ infers $\mathcal M_{\rm loc} = \tilde{\mathcal M}(1+\alpha_A\alpha_B)/(1+\alpha_e^2) \to \tilde{\mathcal M}$ for screened systems — the source *is* locally standard — and the end-to-end ratio $I_{\rm det}/I_{\rm src} = A_o^2(1+\alpha_o^2)/(1+\alpha_A\alpha_B) \to 1$ today. The $(1+z)^{-1}$ mass map therefore survives the invariant check because it is a transport statement — coordinate frequency conserved on the static background — not a claim of nonstandard source physics. The check does expose the physical content of the assumption: the locally measured gravitational constant varies with the ambient field as $G_{\rm loc}=G_*A^2(\phi)(1+\alpha_{\rm amb}^2)$, where $A^2$ converts between the gravitational and matter metrics and $(1+\alpha_{\rm amb}^2)$ is the screened Temporal-Shear contribution. Its evolution is dominated by the conformal factor itself, $\dot G/G \simeq 2(\dot A/A)$ evaluated at the system — a conditional local clock/dynamics relation. Relating it to an imposed ambient drift requires the continuous radial profile of the selected two-branch kinetic operator and quartic potential (Appendix E, R17; Step 76, with zero thin-shell approximation). Paper 17 measures a synodic range residual, not a directly identified conformal excursion. Published LLR analyses bound $|\dot G/G|\lesssim4\times10^{-13}\,{\rm yr}^{-1}$ (Williams et al.), with the tighter Hofmann–Müller re-analysis giving $(7.1\pm7.6)\times10^{-14}\,{\rm yr}^{-1}$: under strictly constant $M_{\rm Pl}$ in $S_{\rm grav}$ and $G_{\rm loc}=G_*A^2(\phi)(1+\alpha_{\rm amb}^2)$ (preserving $\Xi(z)=1/(1+z)$ across Papers 0, 22, 27, and 30), the Step 76 continuous two-branch + quartic profile evaluated on the Cassini-compatible screened branch — the quartic normalization used for the Solar-System flux profile — resolves the $10^8R_\oplus\approx4.3\times10^3$ AU interior response to a static tangent $T_{\rm tan}=1.2\times10^{-8}$ (the finite-excursion secant falling below the declared field-reporting floor) and to $\le10^{-5}$ at $10^{10}R_\oplus\approx2.1$ pc, bounding the operative Solar-System drift to $|\dot A/A|_{\rm loc}\lesssim8.4\times10^{-19}\,{\rm yr}^{-1}$ and $|\dot G_{\rm loc}/G_{\rm loc}|_{\rm LLR}\lesssim1.7\times10^{-18}\,{\rm yr}^{-1}$, more than four orders of magnitude below both LLR bounds. The reference-quartic comparison branch returns the conservative finite-excursion secant ceiling $T_{\rm cont}=1.99\times10^{-3}$, i.e. $|\dot G_{\rm loc}/G_{\rm loc}|\lesssim2.85\times10^{-13}\,{\rm yr}^{-1}<4\times10^{-13}\,{\rm yr}^{-1}$ (this unscreened comparison ceiling satisfies the Williams et al. bound only; the operative screened branch satisfies both LLR analyses) even without the screening normalization (plus the two-branch scalar-force self-energy drift $2H_0 y_{\rm tb}(1\text{ AU})=2.10\times10^{-15}\,{\rm yr}^{-1}\ll4\times10^{-13}\,{\rm yr}^{-1}$), while unstretched-space temporal-lapse ray-tracing across the intergalactic sightline ($f_{\rm IGM}=1-f_{\rm well}\to1$) recovers the full cosmological Hubble law $\ln(1+z)=[1-f_{\rm well}(1-T_{\rm cont})]\Delta\bar u_{\rm amb}+\delta u_{\rm env}\simeq\Delta\bar u_{\rm amb}+\delta u_{\rm env}$ (Step 70). Where a source host's interior ambient carries a larger fraction of the epochal contrast, the unscreened rate $\sim1.4\times10^{-10}\,{\rm yr}^{-1}$ sets the ceiling — the epoch-dependence of $G_{\rm loc}$ at the source end is where the siren claim is tested, and it is a stated, checkable prediction rather than a hidden one.

The same epoch-dependence propagates into the intrinsic source population (Appendix E, step_71). The additive interior profile $\phi_{\rm int}=\phi_{\rm amb}+\delta\phi_{\rm local}$ supplies a useful limiting case when the local profile is insensitive to the imposed shift. The static calculations of Appendix E, R17 distinguish this limit from finite-amplitude response and do not determine the time-dependent host transfer. Under the fixed-response assumptions above, the local LLR comparison corresponds to $\lesssim0.3\%$ of the Hubble rate; that local bound is not extrapolated to the cosmological history. Applied at a source where the interior ambient carries the epochal contrast, the locally measured $G$ inside a source galaxy at redshift $z$ is $G_{\rm loc}(z)/G_{\rm loc}(0)\simeq(1+z)^{-2}$ under the canonical map. Characteristic gravitational mass scales in baryon units scale as $G^{-3/2}$ (Chandrasekhar, TOV, Jeans and core-collapse thresholds alike), giving an intrinsic matter-frame drift $\tilde{\mathcal M}^*(z)\propto(1+z)^{3}$ at full tracking. The detector-frame discriminant is therefore not simply $(1+z)^{-1}$ versus $(1+z)$: writing $\epsilon$ for the fraction of the ambient drift carried by the source galaxy's interior ambient — $\epsilon$ summarizes the assumed source-host evolution for this population test, rather than a new microscopic coupling or a universal transmission inferred from step_76 — $\mathcal M_{\rm det}\propto(1+z)^{3\epsilon-1}$, the GR trend is recovered identically at the critical value $\epsilon=2/3$, while full k-mouflage tracking ($\epsilon=1$) would predict detector masses rising as $(1+z)^2$ — a reversed discriminant in which high-redshift 'over-massive' binary black holes would signal the weaker-$G$ epoch; this flip endpoint is the branch excluded by the Pantheon+ bound below, leaving the naive inverse direction as the surviving prediction. The binding constraint is cross-sector: the same drift rescales the Chandrasekhar mass and hence Type Ia peak luminosity as $L\propto(1+z)^{3\epsilon}$, an intrinsic brightening of $-1.3$~mag at $z=0.5$ and $-2.3$~mag at $z=1$ under full tracking, against which the supernova distance-ladder bookkeeping in TEP-H0 must be re-evaluated before the $\epsilon\simeq1$ endpoint can be claimed; shallower galactic wells at earlier epochs push the effective $\epsilon$ below unity rather than above it. The discriminant's status is accordingly sharpened, not weakened: its sign and magnitude are set by a single interior-response parameter that the compact-body solver and the supernova channel jointly bound. That bound is confirmed rigorously by the full-sample profile likelihood in Paper 26 (results/step_03_11_sn_luminosity_drift_profile.json): confronting the 1,701 Pantheon+ supernovae with the complete Cholesky covariance matrix and allowing the TEP distance-redshift relation and line-of-sight shear to completely re-optimize at every point, full tracking $\epsilon=1$ is excluded at $\Delta\chi^2=805.8$ (a $-2.3$~mag intrinsic trend at $z=1$ cannot be absorbed by flexing the distance relation), and the interpolated 3-sigma bound gives $\epsilon < 0.150$ (best fit $\epsilon=-0.05$, $1\sigma$ interval $[-0.128,+0.028]$), conditional on the fixed-$z_T=5$ M1 distance relation and the Chandrasekhar luminosity prescription. The bound is moreover not an artifact of the released standardization: repeating the profile on the raw $m_B$, $x_1$, $c$ columns with the Tripp coefficients, host-mass step, and shear all refit inside the likelihood at every profile point (Paper 26, step_03_13) returns the identical bound ($\epsilon < 0.150$ at $3\sigma$, best fit $-0.05$; refit $\alpha=0.150$, $\beta=3.00$, $\gamma_{\rm step}=+0.007$ sit at the released values), so the released $\alpha x_1-\beta c$ corrections absorb none of the drift signature. This constrains the assumed response of SN Ia progenitors within that model; the progenitor-dependent host-assembly transfer follows directly from the competition between the ambient cosmological drift $\Delta\ln A_{\rm amb}(z)$ and the secular deepening of the host galaxy's own local potential well $\phi_{\rm well}(z)$ as it assembles mass ($M_{\rm host}(z) = M_{\rm host}(0)\,f_M(z)$, so $\epsilon_{\rm eff}(z) = 1 - d\ln\phi_{\rm well}/d\ln A_{\rm amb}$, `results/outputs/audit_epsilon_bbh_derivation.json`): progenitors forming in already-assembled massive early-type wells ($f_M(z=1)\simeq0.85$) carry a near-pinned effective tracking fraction $\epsilon_{\rm eff}\simeq0.004$ over $z\in[0.2,0.8]$ (matching the Pantheon+ bound), whereas binary black hole progenitors assembled in actively accreting field-spiral ($f_M(z=1)\simeq0.70$) or rapidly assembling low-mass dwarf hosts ($f_M(z=1)\simeq0.55$) carry $\epsilon_{\rm eff}\simeq0.24$–$0.52$ (evaluated against the GWTC population in Paper 22). When confronted directly with the 95 BBH events from GWTC (Paper 22, step_09n), continuous physical high-mass cutoff fitting breaks the scale-free power-law degeneracy ($\beta \simeq 154\,M_\odot$ for TEP, $\beta \simeq 33\,M_\odot$ for GR). At their canonical parameter-free baselines, canonical pinned TEP out-performs stationary GR by $\Delta\ln L = +2.32$ out of sample on 50 high-$z$ events ($\ln L = -188.68$ vs $-191.00$, conditional likelihood ratio $\approx 10.2$), with the global profile likelihood independently confirming the optimum at the canonical pinned baseline ($\hat\epsilon = 0.0$).

*Pressure sensitivity.* HC Step 26 evaluates the prescribed sound speed through a supported near-dust sequence, with a zero-pressure CDM control and successive-limit checks. Over multipoles 30–2000, the limiting finite-pressure comparison changes TT and EE by approximately 5.44% and 6.08% RMS relative to its zero-pressure fluid control, and changes the lensing-potential spectrum substantially (99.57% RMS fractional difference). The zero-pressure control differs from CDM by at most 0.25% RMS across the tested spectra. In the $(3+1)$ preferred-foliation action (Step 80), where $\bar\phi(t)$ defines the unitary-gauge time slicing ($\delta\bar\phi=0$) and the quasistatic spatial clock-slope potential $u_{\rm grav}(\mathbf x,t)$ satisfies the elliptic Gauss constraint $-\nabla\cdot[P_{,\xi}(\xi_{\rm sp})\nabla u_{\rm grav}]=\beta_A\rho_b$ on each constant-$t$ hypersurface, the longitudinal mode carries zero independent Euler pressure wave ($c_p^2=0$) and enters the linear Poisson equation through the scale-independent clock-slope multiplier $M_s=1+y_s/\sqrt2=6.26$ ($\omega_\phi=0.11815$), corresponding to the validated zero-pressure limit above; the finite-$c_p^2$ fluid sequence remains a phenomenological sensitivity diagnostic without a likelihood refit, and the pressureless full-solver fit cannot be identified with the finite-pressure effective fluid model merely from agreement of a few peak locations.

## 9. Quantum Clocks, Interferometry, and Composition Dependence

## Quantum evolution

Proper-time quantum evolution is defined with respect to matter-frame proper time: $i\hbar d|\psi\rangle/d\tau = \hat{H}|\psi\rangle$, where $\tau$ is the proper time defined by the matter metric $\tilde{g}_{\mu\nu}$. Relative to an Einstein-frame or reference coordinate time $t$, stationary clocks satisfy $d\tau \simeq A(\phi)\,dt$ in the weak field, meaning a single clock's tick rate relative to $t$ carries the $A(\phi)$ scaling. Because this conformal rescaling is locally universal, it cancels exactly from dimensionless ratios of co-located ideal matter clocks. Observable conformal clock signatures arise strictly from comparisons between different spacetime environments, transported clock histories, or small sectoral dilaton-like sensitivities:

$$\delta \ln \nu = \delta \ln A + K_\alpha \delta \ln \alpha + K_\mu \delta \ln \mu + K_q \delta \ln X_q + \ldots$$

where the first term represents the universal conformal clock-rate contribution relative to the specified reference standard.

## Species sensitivity

For a dimensionless ratio of two co-located clock transitions $X$ and $Y$, the universal conformal contribution cancels:

$$\delta \ln \frac{\nu_X}{\nu_Y} = (K_{\alpha,X} - K_{\alpha,Y})\delta \ln \alpha + (K_{\mu,X} - K_{\mu,Y})\delta \ln \mu + (K_{q,X} - K_{q,Y})\delta \ln X_q + \ldots$$

Multi-species clock comparisons therefore constrain non-universal sectoral couplings, while universal $A(\phi)$ signatures are tested through comparisons of clock rates or accumulated proper-time histories across distinct spacetime environments.

## Interferometry

Phases acquire contributions proportional to the integral of $A(\phi)$ along arms. Atom interferometers with vertical baselines can sense $\partial_h\ln A$ at $10^{-21}$–$10^{-22}$ m$^{-1}$; photon interferometers are sensitive at different bands.

## Decoherence

Rapid $\phi$ variations would induce dephasing at rates $O(\partial_t\phi)$, strongly bounded by clock stabilities. Adiabatic evolution in the lab is assumed, consistent with null drift bounds.

## Inset: What "no-variable-c" tests do—and don't—probe (TEP view)

**Gauge-invariant observable.** The synchronization holonomy is the loop non-closure of calibrated time transport after GR subtraction. Let $\sigma$ denote the time-transport one-form whose line integral equals the calibrated proper-time increment along each leg. The residual holonomy is

$$H_{\rm resid}(C) \equiv \oint_C (\tilde\sigma - \sigma_{\rm GR}) = \iint_\Sigma (\tilde F - F_{\rm GR}),$$

the integral of the residual curvature two-form over a surface $\Sigma$ bounded by the closed loop $C = \partial\Sigma$. It is built from measured proper-time increments along each leg and has units of time. Because it is the closed-loop integral of the GR-subtracted synchronization connection, $\Delta\sigma=\tilde\sigma-\sigma_{\rm GR}$, it is invariant under admissible synchronization re-gaugings. Re-gaugings shift both the matter-frame connection and the corresponding GR reference connection by the same exact one-form, leaving $\Delta\sigma$ invariant (since $\oint_C d\chi = 0$ for single-valued $\chi$). It vanishes in SR/GR and in the conformal-only limit of TEP. The GR subtraction includes Sagnac, Lense–Thirring/gravito-magnetic, Shapiro, gravitational redshift, station motion, clock-scale realization, and reference-frame corrections, computed with ITRF ephemerides and TT/TDB standards.

How flagship constraints map to $H_{\rm resid}$:

- **GW170817 (GW–EM coincidence).** $|c_\gamma-c_g|/c\!\lesssim\!\text{few}\times10^{-15}$ constrains global cone splits. In TEP, late-time conformal coupling preserves null cones, so EM and GW share causal structure; small disformal tilts today are allowed. This is a boundary condition, not a loop-holonomy test.

- **Cassini (PPN-$\gamma$).** Two-way Doppler/Shapiro is reciprocity-even; it calibrates $\sigma_{\rm GR}$ to subtract but does not bound $H_{\rm resid}$.

- **Resonator MM/KT tests.** Cavities provide strong terrestrial bounds on even-parity orientation-dependent photon propagation, including the disformal $D(\hat n\cdot\nabla\phi)^2$ deformation derived in Appendix B. They do not directly measure the distinct odd/non-exact loop-closure observable $H_{\rm resid}$, whose leading synchronization kernel additionally depends on observer motion and path geometry.

- **"GPS works."** Network self-consistency uses explicit GR+Sagnac modeling and largely two-way/common-view calibration. This verifies internal consistency under assumed GR model; not a direction-reversing one-way loop-closure null.

- **Clock redshift & pairwise A↔B tests.** Exquisitely confirm GR locally; *only closed loops* (A→B→C→A with direction reversal) can reveal non-integrability captured by $H_{\rm resid}$.

Why classics can be null while $H_{\rm resid}\neq0$:

- Conformal null-cone invariance ⇒ no large GW–EM kinematic delays (consistent with GW170817).

- **$\partial_t\phi = 0$ over loop timescale; gradients conservative: no direct conformal propagation asymmetry.** In the purely conformal limit ($B = 0$), the conformal factor cancels exactly from the null condition, so forward/back propagation times are identical. Residuals require the disformal sector, time dependence, or non-exact transport structure. Thus two-way/closed-path nulls can hold while a loop-holonomy test remains sensitive.

**Experimental falsifier (primary endpoints).** Run a closed-loop, one-way time-transfer (and/or portable-clock) test and report:

- Leg-wise antisymmetry: $\Delta t_{AB}=t_{AB}-t_{BA}$ (and optionally $\Xi_{AB}\!\equiv\!(t_{AB}-t_{BA})/(t_{AB}+t_{BA})$).

- Loop holonomy $H_{\rm resid}$ after subtracting the full GR, kinematic, clock-scale, and instrumental synchronization model.
Use triangle/quadrilateral geometries with direction reversal; extend with interplanetary one-way links and multi-species clock networks.

## 10. Experimental Proposals and Falsifiability

A suite of decisive, cross-checking experiments is proposed that can falsify, constrain, or provide controlled support for the theory. All observables are dimensionless ratios or calibrated residuals; all designs include nulls, blinding, and open data.

![Sensitivity to TEP signals and constraints](site/figures/fig3_sensitivity.svg)

Figure 3. Triangle synchronization experiment geometry. Two ground stations (sea level and mountain) and a medium-Earth-orbit satellite form ~3000 km baselines; two-way time transfer calibrates each edge while one-way transfers around the loop probe $H_{\rm resid}$ after GR subtraction.

## A. Triangle synchronization holonomy (ground–ground–satellite)

**Geometry.** Three stations A, B, C forming 1000–3000 km baselines: two ground sites (sea level and high-altitude) and a medium-Earth-orbit satellite. Optical two-way time transfer (TWTT) on each edge provides calibration; stabilized fibers and free-space optical links carry both calibration and one-way signals.

**Protocol.** Establish Einstein synchronization on each edge via TWTT. Execute one-way transfers around the loop in both senses at high cadence for months. Record raw timestamps, environmental monitors (pressure, temperature, humidity), refractivity profiles, TEC for ionosphere, and precise ephemerides.

**Modeling.** Subtract GR Sagnac (Earth rotation, frame dragging), Shapiro delays, gravitational redshift differences, tropospheric and ionospheric delays, fiber dispersion and thermomechanical drifts. Use GNSS and gravimetric models.

**Observable.** The measured loop residual is

$$H_{\rm resid}(C) = \oint_C(\tilde{\sigma}-\sigma_{\rm GR}) = \oint_C\Delta\sigma ,$$

where $\sigma_{\rm GR}$ is computed from the full GR, kinematic, geodetic, atmospheric, instrumental, and clock-scale model for the same loop.

**Benchmark sensitivity window.** The target bounds are designed to be independent of the disformal sign. Phenomenological estimates motivate sensitivity in the $10^{-18}$–$10^{-16}$ fractional range per loop time (0.1–1 s). The corrected numerical benchmark (Appendix E, R6) integrates $\delta\tilde\sigma_i\simeq-(B M_{\rm Pl}^2/A^2N^2)\,S(r_E)\partial_i u\,(\dot u/c)$ on two visible ground–satellite links and a surface return route. The satellite is 2000 km above the angular bisector of two ground stations separated by 60 degrees. The physical solar gradient is retained, and parametric quadrature converges from 32 to 256 nodes per leg. Under the explicitly defined dimensionless normalization $B M_{\rm Pl}^2=B_0[c/H_0]^2\,u^2(1+u^2)^{-1}\exp(-u^4/2\sigma_B^4)$ with $B_0=+1$, the signed result for the stated orientation is $H_{\rm resid}\simeq-4.33\times10^{-13}$ s. The spherical-field and exact-connection ($S=1$) controls return zero within numerical precision; setting $B_0=0$ or $\dot u=0$ gives zero, and orientation reversal changes the sign. This is a conditional non-exact-connection benchmark, not yet an absolute prediction from the universal action. Its dimensionless $B_0$ must not be identified directly with the dimensionful strong-field normalization. The repaired calculation supplies a reproducible forward test for the completed field solution without fixing a universal amplitude ceiling or excluding that solution from a null at the benchmark sensitivity.

The reduced connection keeps the assumed radial screening profile $S(r_E)$ separate from the Earth–Sun field $u$. This separation permits a nonzero curl because $d[S(r_E)f(u)\,du]=dS\wedge f(u)\,du$; the drift weight $W(u)$ alone is a function of the field and cannot generate circulation. The calculation therefore tests the numerical transport of a specified connection. Deriving its active gradient, environmental screening and temporal drift from one consistent scalar configuration remains the physical completion required by Rules 11, 13 and 18. Uniform drift alone need not make an otherwise non-exact screened connection exact.

**Error budget (fractional per loop, targets after months):**

- Clock instability after averaging: $5 \times 10^{-19}$

- Two-way calibration residual: $2 \times 10^{-19}$

- Troposphere residual: $2 \times 10^{-19}$

- Fiber path noise after stabilization: $5 \times 10^{-20}$

- Ephemeris/geodesy: $2 \times 10^{-19}$

- GR subtraction residual: $5 \times 10^{-19}$

- **Total systematic floor (rss): $\sim 8 \times 10^{-19}$**

**Target:** below $10^{-18}$; projected capability with advanced turbulent cancellation techniques reaches the $10^{-19}$ regime.

**Falsification.** Null at the $10^{-18}$ fractional target across seasons/geometry bounds the non-exact-transport kernel and normalization at the calibrated level; reaching the projected $10^{-19}$ capability would bound the disformal normalization at $|B_0|\lesssim5\times10^{-8}$, more than four orders below the value the volume-balance reconstruction required ($|B_0|\sim3\times10^{-3}$; Appendix E, R5) — an independent observational exclusion of the channel already closed analytically by its own null cone (Section 4, small-$B$ EFT). The same calibration carries a terrestrial consistency condition: laboratory resonator and isotropy tests bound the directional cone deformation $B(\phi)(\partial\phi)^2$ at the $\sim10^{-18}$ level, which at $|B_0|\sim3\times10^{-3}$ is respected only if the near-surface shear pinning reaches $\mathcal S_\Sigma\lesssim10^{-10}$ — four orders of magnitude deeper than the solar-vicinity Cassini value. This is a consistency condition whose verification belongs to the same open single-body calculation as the Solar-System source charge (Section 7), not a derived output of the stated screening operator; and because the quoted $|B_0|$ calibration sits on the sign-excluded branch, the required pinning relaxes with whatever amplitude the admissible $B\geq0$ realization supplies. The condition is in any case decoupled from the loop signal: the resonator bound tests $B(\phi)\Sigma^2$ at the surface, while the holonomy tests $B(\phi)\Sigma\,\dot u$ integrated along the transit.

## B. Portable-clock "clock anholonomy"

**Design.** Two identical optical clocks transported from A to B along distinct paths (e.g., sea-level highway vs. mountain pass), durations ~1–3 days, then compared at B to a stationary clock. Common-view time transfer provides epoch.

**Prediction.** Path-dependent discrepancy $\Delta_{12}$ at few$\times10^{-19}$ for plausible $\alpha \dot{\phi}_0$ and $\partial_h\ln A$ under screening; null at $10^{-20}$ bounds $\partial_t\ln A$ and $\partial_h\ln A$ tightly. The smooth matter-frame lapse $A\cdot N$ reproducing standard gravitational redshift is subtracted within the $\sigma_{\rm GR}$ modeling, so the observable residual $\Delta_{12}$ is sourced strictly by temporal-landscape fluctuations $\delta\ln A$ beyond the smooth Newtonian potential and by the time-dependent conformal drift $\partial_t\ln A$.

**Systematics.** Temperature, vibration, transport-induced shifts, gravitational potential changes modeled and controlled; use transport pods with environmental control.

## C. Multi-species clock network: annual modulations

**Network.** Global network of optical clocks (Sr, Yb, Al⁺, Hg⁺, Ca, H(1s–2s)) cross-compared over years.

**Conditional extension-sector test.** Phase-locked annual modulations in differential ratios $\nu_A/\nu_B$ with amplitudes $10^{-19}$–$10^{-17}$, phases tied to orbital eccentricity. The universal conformal factor cancels identically in co-located dimensionless ratios (Section 9), so this channel tests non-universal sectoral couplings rather than the canonical action: species-amplitude ratios reflect $\alpha_A - \alpha_B$; a fit yields $\alpha(\phi)$ and dilaton coefficients $d_e$, $d_\mu$, $d_q$.

**Nulls.** Compare to environmental seasonality, tidal potentials, solar activity; require phase-locked global coherence characteristic of orbital eccentric anomaly.

## D. Interplanetary one-way optical time transfer

**Design.** Two drag-free spacecraft with $10^{-18}$-class optical lattice clocks, separated by 1–5 AU. Optical comb-based one-way time transfer, with third node for calibration (Earth or a relay). Kinematic synchronization via slow-clock transport or common-view transponders.

**Benchmark sensitivity window.** Geometry-dependent one-way asymmetry parameter $\Xi_{AB} \equiv (t_{AB} - t_{BA})/(t_{AB} + t_{BA})$ at $10^{-15}$–$10^{-14}$ (0.05–5 ps) for disformal tilts consistent with GW bounds. The absolute signal magnitude remains to be derived from the solved path-dependent kernel; this range is a benchmark target, not a no-refit prediction.

**Systematics.** Plasma delays, pointing jitter, thermal drifts, deep-space clock performance; anticipate >decade timeline.

## E. Clock Network Correlation Analysis and Environmental Screening Maps

**Objective.** Detect spatial correlations and environmental screening signatures in atomic clock frequency residuals consistent with screened scalar field coupling $A(\phi)$ to transition frequencies.

**Design.**
*Phase I - Distance Correlation Analysis:*
• Analyze existing precision clock networks (GNSS, optical clock arrays) for distance-dependent correlations in frequency residuals
• Apply phase-coherent cross-spectral analysis between station pairs
• Bin pairs by 3D distance, fit exponential correlation model: $C(r) = A \cdot \exp(-r/\lambda_T) + C_0$
• Cross-validate across independent analysis centers to control systematics

*Phase II - Environmental Screening Maps:*
• Deploy identical optical clocks at sea-level, mountain, stratospheric balloon, and LEO; intercompare with two-way optical links
• Subtract GR redshift and Doppler shifts; correlate residuals with detailed geophysical models and gravimetry to isolate screening signatures

**Forecast.**
• Distance correlations: Exponential decay with characteristic length $\lambda_T \sim 1,000$–$10,000$ km for viable screening parameters. The empirical terrestrial calibration $L_c \approx 4{,}200$ km, obtained from GNSS clock analysis (Papers 1, 2, 6), and the held-out MGEX multi-GNSS replication $\lambda = 1{,}862 \pm 155$ km (cluster-robust; within-fit $\pm 112$ km; Paper 14; single-year realizations scattering across 1,050–1,935 km) represent empirical calibrations across differing constellation geometries and filter transfer functions; their containment within the decade-wide $\lambda_T \sim 1,000$–$10,000$ km window represents a broad consistency check rather than an independently derived parameter prediction. Forward analyses that adopt $\lambda_T \approx 4{,}200$ km should state it as a calibration input.
• Altitude dependence: $10^{-19}$–$10^{-18}$ frequency shifts over tens of kilometers for $\lambda_{\text{scr}} \sim 10$ km near Earth
• Multi-center cross-validation expected to show $\sim 10$–$15\%$ variation in fitted parameters (revised from the v0.4 forecast of $<5\%$ after Paper 1)

## F. Multi-messenger ensemble

**Design.** Stack $N \gtrsim 30$ multi-messenger events with prompt EM counterparts; marginalize astrophysical lag distributions. Seek distance-correlated trends in $t_{\text{EM}} - t_{\text{GW}}$.

**Prediction.** In late-time conformal subclass, no kinematics-induced trend; disformal residuals bounded to sub-100 ms over 40–200 Mpc. Nulls constrain $B(\phi_0)(\partial\phi)^2$.

## 11. Statistical and Open Science Principles

**Pre-registration.** Publish analysis plans (models, priors, nulls, thresholds) for triangle and portable-clock experiments; record any deviations.

**Blinding.** Blind event-time stamps and calibration offsets; employ independent teams for calibration and analysis.

**Open data/code.** Release raw timestamps, environmental monitors, calibration logs, and pipelines with DOIs; release CLASS/HyRec modification, atom-sensitivity database, and meta-analytic code.

**Hierarchical inference.** Use heavy-tailed likelihoods (Student-t) to mitigate outliers; multi-level random effects by domain; selection models for publication bias; explicit covariance modeling for shared systematics.

## 12. Addressing Critiques and Clarifying Claims

**No EM–GW kinematic delay in conformal subclass.** With $B=0$ today, EM and GW share null cones; any observed delays are astrophysical/source or detector-time-standard effects. Because the apparent macroscopic expansion is carried natively by the dual-frame conformal unit drift of Section 8 rather than by a physical spatial stretching or disformal metric deformation, the cosmology does not drive $B$ to large late-time values, and the shared-cone structure is preserved without fine-tuning. Earlier overstatements are corrected and aligned with GW170817 constraints.

**Static $\phi$ gradients do not generate directional propagation asymmetry.** The conformal factor cancels from the null condition (Appendix A2), so $A(\phi)$ cannot act as a photon refractive index; clock-rate and open-path redshift effects remain.

**Holonomy invariant and not a synchronization artifact.** The observable $H_{\rm resid}$ is constructed from physical proper-time measurements as the closed-loop integral of the GR-subtracted synchronization connection, $\Delta\sigma=\tilde\sigma-\sigma_{\rm GR}$. Synchronization re-gaugings shift both the matter-frame connection and the corresponding GR reference connection by the same exact one-form, leaving $\Delta\sigma$ and therefore $H_{\rm resid}$ invariant.

**Causality and hyperbolicity.** The disformal matter-metric causality analysis is established in the canonical small-$B$ EFT regime, while the noncanonical $P(X)$ scalar characteristic conditions ($v_\parallel^2 = 1 + 2XP_{,XX}/P_{,X} > 1$ on the selected branch) are separately certified in §2.2 and Appendix E; in this preferred-foliation khronon EFT, $v_\parallel^2 > 1$ establishes classical Cauchy hyperbolicity and absence of closed timelike curves on the cosmic foliation rather than claiming a Lorentz-invariant UV completion, while matter-frame causality is preserved throughout the physical branches. The small-$B$ regime is safe.

The cosmological implementation has been tested with native hi_class MCMC fits (Paper 18) and Cobaya joint-likelihood analyses (Paper 26), demonstrating that the temporal-horizon architecture respects Planck acoustic anchors while producing late-time environmental observables. In the inhomogeneous realization the constraint equation does not require a residual Hubble-scale curvature: with ${}^{(3)}R=2\rho_{\rm tot}/M_{\rm Pl}^2+\mathcal K_{ij}\mathcal K^{ij}$ the gravitational-frame mean curvature vanishes on the constraint-satisfying landscape, and the observationally relevant curvature is the matter-frame quantity $|\Omega_k^{\rm eff}|\sim\langle|\nabla u|^2\rangle/H_{\rm drift}^2$, controlled by landscape gradient contrast and reaching the observationally flat regime for the spatial contrast the redshift scatter implies (Section 8).

## 13. Philosophical and Conceptual Implications

**Simultaneity beyond Einstein.** Special relativity makes simultaneity observer-dependent; dynamic time permits globally non-integrable synchronization through its non-exact transport sector. The invariant content is a holonomy of time transport: moving clocks around closed loops in a dynamical time background returns path-dependent offsets after subtracting GR effects.

**Constants clarified.** The speed of light is an invariant in local tangent spaces; globally, "$c$" is not a number but a family of operational ratios dependent on clock histories in a time field. Variation of constants becomes a question about dimensionless ratios across environments and epochs.

**Machian undertone.** The rate of time responds weakly to the stress-energy of matter through $\alpha T$, giving a principled, covariant flavor to the idea that the "rest of the universe" influences local rates, without violating local physics.

## 14. Conclusions

This paper articulates a covariant matter-metric framework realized on a preferred-foliation gravitational effective field theory in which the rate of time is a dynamical field with universal matter coupling. The architecture preserves matter-frame local Lorentz invariance and null-cone structure (in the conformal limit), conforms to multi-messenger bounds, and yields new invariant observables—synchronization holonomy and clock anholonomy—whose GR-subtracted residuals vanish in GR and are measurable with modern clocks.

The Temporal-Shear response is realized explicitly by the nonlinear kinetic sector $P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+X|X|/\Lambda_X^4$ ($P_{,X}=k\sqrt\xi+2\xi$, $k\simeq16$ fixed by the adopted $a_0$), selected by the corpus's admissibility scan and confirmed by the wide-binary joint test at zero per-system free parameters — it derives the environmental inverse stiffness, the characteristic acceleration scale, the flux-conserving recovery profile, the transition-radius scaling, the flat galactic asymptote, and the distinct comparable-mass and hierarchical screening regimes, with static ellipticity and time-dependent hyperbolicity/no-ghost conditions certified on the adopted branch; remaining precision-test calculations are observable projections and finite-source boundary-value problems of this fixed kinetic realization rather than unspecified screening physics, while uniqueness within the wider $K(X)$ class, the microscopic origin and uniqueness of the admissible $X\to0$ regulator, and the microscopic form and normalization of the disformal sector $B(\phi)$ remain open calculations. The theory is falsifiable with realistic experiments: a null closed-loop residual at the $10^{-18}$ fractional level, after full GR subtraction, bounds the disformal normalization and excludes the holonomy channel at the stated benchmark scale (while theory-level exclusion of the holonomy sector requires the completed $B(\phi)$ realization to fix a non-zero lower bound). The framework promises to clarify persistent cosmological tensions.

Einstein moved physics from absolute time to relative simultaneity and dynamic geometry; TEP extends this programme by governing the local calibration of proper time through a dynamical scalar field. TEP formulates this through explicit two-metric axioms and identifies a static, eternal, non-expanding gravitational geometry as its target cosmological realization: the exact and numerical model branches constructed here establish the existence of solutions in which apparent cosmological expansion and early-universe thermal history arise as temporal transport on an on-shell inhomogeneous temporal landscape, while deriving the observed redshift–distance history from a single fixed universal potential without per-branch reconstruction remains an open cosmological closure task. The framework separates the macroscopic Temporal Shear—which drives structure and dynamics—from the topological saturation that screens it where the ambient landscape is steep.

The two-metric architecture yields a proposed standard-siren discriminator, ($d_L^{\rm GW}/d_L^{\rm EM}=A(z)=1/(1+z)$), arising from the distinct gravitational propagation and matter-frame measurement sectors. The source-to-detector mapping is derived at leading quadrupole order on the tensor branch — Einstein-frame sourcing mass $\mathcal M_e=A_e\tilde{\mathcal M}$, stationary-coordinate transport, matter-frame readout — returning the detector-inferred ratio $1/(1+z)$ together with the chirp-mass map $\mathcal M_{\rm det}=A_oA_e\tilde{\mathcal M}$ — moderated to $\mathcal M_{\rm det}\propto(1+z)^{3\epsilon-1}$ by BBH interior tracking; the corresponding SN Ia tracking prescription is conditionally bounded to $\epsilon<0.150$ (3$\sigma$, $\Delta\chi^2=805.8$ at $\epsilon=1$) by Pantheon+ on the fixed-$z_T=5$ M1 branch, with the host-assembly transfer giving $\epsilon_{\rm eff}\simeq0.004$ for deep-well SN Ia hosts versus $0.24$–$0.52$ for field-spiral/dwarf BBH hosts and the GWTC empirical optimum at $\hat\epsilon=0$ (Section 8) — whose inverse direction relative to the FLRW convention is itself a population-level discriminator; scalar-mode admixture and higher-order back-reaction remain within the gravitational-wave programme. It enforces sign-definite deviations from GR ($\gamma_{\rm PPN} < 1$), yields measurable invariant observables like residual synchronization holonomy where the drift rate's gradient is misaligned with the field gradient (the disformal-curvature condition $d(\delta\tilde\sigma)\propto d(\dot\phi/N^2)\wedge d\phi$), and provides sharp multi-messenger discriminators. If residual holonomies at the targeted scale and the predicted cosmic distances are observed, physics will enter a new epoch in which dynamic time joins dynamic geometry as a foundation. If not, uniquely strong bounds will have been set and the operational bedrock of $c$ and simultaneity clarified to unprecedented precision.

### 14.1 Recovering GR is Not Regressing to GR

Recovering standard General Relativity in the Solar System is not a regression—it is the baseline requirement to avoid being immediately falsified by a century of precision experimental data. The effective screened limit recovers GR to the required observational accuracy where GR is experimentally established (local, screened systems), while permitting distinct phenomenology on unscreened astrophysical and cosmological scales. The framework remains radically non-GR:

- **GR has one metric; TEP has two.** In GR, geometry is dictated solely by the Einstein tensor of $g_{\mu\nu}$. In TEP, all matter, light, and clocks move along geodesics of a distinct, dynamical matter metric $\tilde{g}_{\mu\nu}$.

- **Standard GR-based cosmology requires dark matter; in TEP the dynamical phenomenology arises from Temporal Shear, and in the unstretched-space temporal foliation the corresponding lensing phenomenology arises from intermediate temporal depth.** Galactic rotation curves motivate the dark-matter comparison, while wide binaries test the local weak-field force law without requiring a dark halo around each pair. In TEP, active Temporal Shear bends matter geodesics. The selected two-branch kinetic sector supplies the bounded local enhancement of Section 7 on its quadratic branch ($\xi\gg\xi_*\simeq64$, the regime of every Solar-System benchmark) and the flat outer rotation curves with baryonic Tully–Fisher scaling on its low-gradient branch, tested against SPARC in Paper 6, Steps 10–13. The two branches are terms of the single action $P_{,X}=k\sqrt\xi+2\xi$: the same sector passes the local, binary and galactic tests jointly — the corrected wide-binary joint test excludes the unit-term baseline and the exponential-interpolation rival at $\sim30\sigma$ on amplitude while passing the selected sector at $\sim0.4$–$1.7\sigma$, and under the action-derived additive force law $g_{\rm obs}=g_N+a_\phi$ ($P_{,X}=\epsilon_{\rm reg}+k\sqrt{\xi}+2\xi$, $\epsilon_{\rm reg}=10^{-3}$, $k_*=16.03$ calibrated to $a_0$) the SPARC fit returns RMS $0.2185$ dex at $\Upsilon_d=0.5$ and $0.2004$ dex at $\Upsilon_d=0.35$ (matching the empirical MOND reference $0.1989$ dex and outperforming the $0.2812$ dex unit-term baseline by $>60$ mdex; phenomenological quadrature comparator $\sqrt{g_N^2+a_\phi^2}$ gives $0.1974$ dex at $\Upsilon_d=0.5$ with $k_{\rm fit}=15.78$ and outer-edge BTFR median $v_{\rm obs}/v_{\rm pred}=1.0109$). In the optical sector, isotropic conformal rescaling $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$ stretches spatial rulers by the same factor $A^2(\phi)$ as clocks, causing $A^2(\phi(x))$ to cancel out of $d\tilde s^2=0$ at every intermediate point along a photon path and leaving only the Planck-suppressed scalar stress tensor ($\kappa_\phi\simeq6.5\times10^{-6}$ on the selected sector; Paper 19, Step 64). Under the canonical unified matter metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu$ $+$ $A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ $+$ $B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) — where the $(3+1)$ ADM lapse rescales as $N_{\rm TEP}(\mathbf x)=A_{\rm loc}(u_{\rm grav}(\mathbf x))N_{\rm GR}(\mathbf x)$ while at a fixed cosmic epoch ($A_{\rm sp}(t_0)=1$) the spatial 3-metric $\tilde\gamma_{ij}=\gamma_{ij}=(1-2\Phi_N/c^2)\delta_{ij}$ remains unstretched — intermediate clock rates between source and observer do not cancel: every point $x$ along the ray contributes its local temporal depth $-\beta_A u_{\rm grav}(x)$ directly to the refractive index $n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)$. Because both transverse-traceless tensor perturbations $h_{ij}^{\rm TT}$ of $\gamma_{ij}$ and electromagnetic waves obey the identical principal wave operator $\Box_{\rm TEP}=-(N_{\rm TEP}c)^{-2}\partial_t^2+A_{\rm sp}^{-2}(t,\mathbf x)\gamma^{kl}\partial_k\partial_l$ from the spatially covariant ADM/DHOST action $S_{\rm grav}$ on static space $\gamma_{ij}$ (rather than splitting into a photon-only disformal cone or an unshared tensor operator, which GW170817 would exclude by $\sim10^7\text{ s}$ across haloes or $\sim6.9\times10^5\text{ yr}$ across the IGM; Boran et al. 2018; Step 80), $c_{\rm GW}(t,\mathbf x)=c_\gamma(t,\mathbf x)=c\,N_{\rm TEP}(\mathbf x)/A_{\rm sp}(t,\mathbf x)$ and $\Delta t_{\gamma-\rm GW}=0$ hold identically across haloes, voids, and cosmic epochs alike, while Solar-System Shapiro bounds remain satisfied via the quadratic-branch screening $y\sim(r/r_*)^{4/3}\ll10^{-5}$. Sourced by baryons alone ($f_b=0.16$, zero CDM) on the zero-free-parameter selected sector $P_{,X}=k\sqrt\xi+2\xi$, this unstretched-space temporal refraction reproduces the non-baryonic Fermat delay and arrival order of SN Refsdal ($+330.9$ d sheet solve, $+221.6$ to $+443.2$ d 3-D deprojected NFW ray solve; Paper 19, Step 64), the galaxy-peak convergence and offset morphology of the Bullet Cluster ($\kappa_{\rm tot}=0.307$–$0.372$ at the bullet galaxies with no negative moat; Paper 4, Step 02), and the cosmic-web amplification required by the Planck CMB lensing spectrum ($9.4\times$–$129\times$ boost across $g_b/g_t\sim8\times10^{-4}$–$5\times10^{-3}$; Paper 18, Step 21).

- **Standard GR-based cosmology models redshift as expanding space from a Big Bang; TEP rejects both.** The standard cosmological interpretation asserts that space itself physically stretches. TEP holds that the gravitational spatial geometry $\gamma_{ij}$ is eternal and static (carrying zero Hubble friction in the tensor sector, $\Xi(z)=1/(1+z)$), while cosmological ruler evolution and redshift reside in the chronometric recalibration of the matter metric ($A_{\rm sp}$).

- **GR predicts the reference synchronization holonomy; TEP permits an additional residual.** GR itself produces closed-loop transport effects such as the Sagnac term; the residual remaining after the complete GR contribution is subtracted vanishes. TEP permits a nonzero closed-loop synchronization residual sourced by non-exact temporal transport ($H_{\rm resid} \neq 0$).

- **Standard GR-based clock models do not predict TEP's residual spatial covariance.** After accounting for known gravitational, environmental, and measurement correlations, TEP predicts a distance-structured residual across continental baselines ($C_A(r) \propto \exp(-r/\lambda_T)$ with $\lambda_T \sim 1{,}000\text{--}10{,}000\text{ km}$). Standard geodetic pipelines apply common-mode signal removal that demonstrably attenuates injected clock offsets (Paper 1 reports ≈96% attenuation in CODE R5 injection tests), yet the measured correlation survives in the residuals — consistent with a physical signal that reduction attenuates rather than eliminates.

### 14.2 Master Falsification Ledger and Deciding Observables

The program maintains an explicit master ledger of its primary empirical claims, open closures, and deciding observables. Each item is paired with a quantitative falsification criterion and a definitive observational or computational channel; statuses cite the registered pipeline outputs on which they rest.

| Load-bearing item | Current status in corpus | Deciding observable / test | Falsification criterion |
| --- | --- | --- | --- |
| **Static-frame stability** (§8; Papers 0, 27) | Positive restricted finite-cell spectra and independent discrete bounds (Steps 79/82); a positive canonical evolving-CMC initial lapse and a khronon stress-shift lapse diagnostic (Step 21); complete matter-coupling second variation $\mathcal V_{\rm eff}''=V_{\rm rec}''+\rho_*(x)e^{-u}$ flipping the unprojected canonical reconstructed spectrum from $-0.4389$ to $\lambda_0=+0.4123\text{--}0.4127>0$ (`completed_canonical_reconstructed_coupled_audit`); and two quartic constrained-operator benchmarks in Step 79 — one with a prescribed scalar profile and re-solved Hamiltonian geometry, and a companion jointly on-shell solve where both the nonlinear two-branch scalar Euler–Lagrange equation and the Lichnerowicz Hamiltonian constraint converge to $<3.1\times10^{-12}$ ($N=16,24$, unprojected coupled $\lambda_0=+0.5624\text{--}0.5633>0$, all-twist bound $+0.4662\text{--}0.4668>0$ under volume-constrained metric response $q_{\rm vol}^T\delta\psi=0$). These results establish their stated initial-slice and finite-grid operator properties; they do not establish full-action continuum time evolution. | Higher-resolution ($N\ge64$) multi-scale continuum coupled evolution and radiative damping rates across cosmic lookback time. | Appearance of a physical volume-gauge-preserving inhomogeneous mode with negative eigenvalue in the coupled Einstein–scalar operator |
| **Optical-sector lensing** (Papers 4, 18, 19) | Unified matter metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu$ $+$ $A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ $+$ $B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ and spatially covariant ADM/DHOST gravitational action $S_{\rm grav}$ derived and symbolic-verified (Step 80): preserves static Poisson/scalar equations ($\delta S_{\rm grav}/\delta u_{\rm grav}=0$, $\delta S_{\rm grav}/\delta A_{\rm sp}=0$ at $K_{ij}\to0$), enforces $c_{\rm GW}(t,\mathbf x)=c_\gamma(t,\mathbf x)=c\,N_{\rm TEP}(\mathbf x)/A_{\rm sp}(t,\mathbf x)$ ($\Delta t_{\gamma-\rm GW}=0.0\text{ s}$ for GW170817) and constant WKB tensor measure $\sqrt{A_{\rm sp}^{+1}\cdot A_{\rm sp}^{-1}}=1$ on static space ($\Xi(z)=1/(1+z)$), gives PPN slip $\gamma_{\rm PPN}-1=-2\beta_A^2 S_\Sigma/(1+2\beta_A^2 S_\Sigma)$ ($|\gamma_{\rm PPN}-1|\simeq4.32\times10^{-8}$ at $1.6\,R_\odot$), and on the selected two-branch sector reproduces Refsdal Fermat delay ($+330.9$ d sheet, $+221.6$–$443.2$ d 3-D ray; Step 64), Bullet galaxy-peak convergence ($\kappa_{\rm tot}=0.307$–$0.372$, no moat; Paper 4 Step 02), and Planck 2018 CMB lensing spectrum at $\omega_{\rm cdm}=0$ ($\chi^2=15.40$ fixed-Silk, $13.40$ 4-param vs $12.30$ CDM; Paper 18 Step 21) | Stellar-kinematics mass-sheet breaking across strong-lens ensembles (including SN 2025wny); native Boltzmann perturbation transfer of the unstretched-space temporal refractive potential | Systematic absence of the predicted time-delay scale shift across kinematics-calibrated (mass-sheet-broken) long-baseline strong-lens ensembles; failure of the unstretched-space temporal convergence profile in cluster or CMB lensing |
| **LLR synodic residual** (Paper 17) | Precision-weighted residual detection $\eta_{\rm resid}=-(3.91\pm0.56)\times10^{-4}$ ($-5.08\pm0.73$ mm, $6.9\sigma$) in INPOP19a post-fit residuals (cross-ephemeris support in DE430); on the single two-branch solar solution that passes Cassini ($y_{\rm tb}(1.6\,R_\odot)=2.16\times10^{-8}$, $y_{\rm tb}(1\text{ AU})=1.47\times10^{-5}$), the direct solar scalar optical excursion is screened to $\delta u_\odot^{\rm tb}=7.4\times10^{-16}$ ($\eta_{\rm tb}=-1.1\times10^{-8}$), consistent with direct-fit LLR bounds alongside Cassini (and axisymmetric coupled syzygy PDE solves confirm zero $P_{,X}<1$ bridge volume at both Full and New Moon; Step 086). Because TEP's unified metric at a fixed cosmic epoch inside the screened Solar System ($A_{\rm sp}(t_0)=1$) has spatial 3-metric $\tilde\gamma_{ij}=\gamma_{ij}=(1-2\Phi_N/c^2)\delta_{ij}$ identical to GR's, the Newtonian solar tidal differential $\Delta\Phi_{N,\odot}/c^2=2.54\times10^{-11}\cos D$ (unscreened reference ceiling $-4.85$ mm, $\eta_{\rm ceiling}=-3.73\times10^{-4}$; Step 083) cancels in BCRS$\to$GCRS identically to GR, and $\eta_{\rm resid}$ stands as an empirical post-fit residual-channel detection awaiting a source-level integrator refit | Full numerical orbit-integrator refit (INPOP/DE430-class) with $\eta_{\rm resid}$ left free in the dynamical equations | Complete absorption of the synodic $\cos D$ amplitude into standard ephemeris parameters ($\eta_{\rm resid}\to0$) during integrator-level refit |
| **BBN & VMO recycling** (Paper 29) | Asymptotic chemical equilibrium yields $Y_{\rm eq}\approx0.247$; Strömgren ionization budget computed (margin $\approx0.97$); deuterium extended channels ledger registered | Pop-III stellar archaeology; nucleosynthetic yield matching from VMO partial-burning ejecta ($f_{\rm VMO}$) | Direct stellar-archaeological exclusion of metal-free VMO enrichment channels; failure to sustain D/H $\approx2\times10^{-5}$ under bounded astration |
| **J0437 phase closure** (Paper 16) | Phase-domain detection $\psi=1.12$ rad ($p=1.3\times10^{-44}$); canonical disformal connection supplies $1.3\%$ ($5.0\%$ on $p_{95}$ leg) at $B_0=+1$ unit benchmark and reaches $100.5\%$ (median leg, $B_0^{\rm req}=77.3$; $B_0^{\rm req}=20.1$ on $p_{95}$ leg) at the GW170817 galactic-profile envelope $B_0\lesssim77.7$ (Step 057); within-source sub-band delay ratio consistent with achromatic transport (Step 061) | Multi-pulsar phase closure (PSR J1603-7202, MeerKAT samples); secondary-spectrum arc curvature & DM regression | Scaling of the delay with $\nu^{-2}$ (plasma dispersion) across octave bands; absence of phase-closure concentration in independent high-dispersion pulsars |
| **Cosmic shear $S_8$** (Paper 18) | CMB joint MCMC gives $S_8=0.868\pm0.025$; provenance audit (Step 23) shows it is prior-boundary dominated ($A_{\rm planck}$/$n_s$ saturate ceilings; $\sigma_8$ tracks $A_{\rm planck}$ at $r=+0.46$). Perturbation-closure audit (Oct 2026, corrected $\delta\phi$ solver): the canonical-EF map $\alpha_K=-5\alpha_A^2$ is linearly unstable (tachyonic scalarization; $D>0$ insufficient); unscreened stable branch excluded ($\sigma_8\simeq1.6$); screened kineticity branch $Z_E\gtrsim700$ recovers $\Lambda$CDM ($\sigma_8=0.824$ vs $0.823$), with $\sigma_8\simeq0.76$ — the weak-lensing value — at $Z_E\sim6\times10^3$ | Stage-IV weak lensing (Euclid, Rubin LSST, Roman) tomographic cosmic-shear power spectra constraining the scalar kinetic coefficient $Z_E$; the low-$\ell$ ISW excess that accompanies the growth-suppressing branch ($w_c\sim6\times10^3$) | Exclusion of the screened branch — measured growth incompatible with the $\sigma_8(Z_E)$ response curve at every $Z_E\gtrsim700$, or a low-$\ell$ ISW level incompatible with the signature the growth-suppressing branch predicts |
| **GNSS clock correlations** (Papers 1, 2, 3) | Spatial covariance $\lambda_T\approx3{,}210\pm389$ km; orbital-velocity correlation $r=-0.888$; velocity vs calendar model comparison registered (Step 2.10: generic annual sinusoid preferred globally, orbital-waveform discrimination underpowered at 10-day resolution) | Independent-chain time transfer: raw TWSTFT inter-laboratory clock links (BIPM time-transfer archive, PTB-pivot network) and optical-fibre time-transfer links — BIPM Circular T itself is dominated by GPS PPP links and does not by itself constitute a non-GNSS channel; synthetic GNSS signal injection through the estimation chains | Complete disappearance of the $\sim3{,}000\text{--}4{,}000$ km correlation structure in non-GNSS (TWSTFT/fibre) clock networks — sharpest form: a GNSS-versus-TWSTFT difference on identical laboratory clock pairs must retain the structure in both arms if the field is environmental — or identification as an IGS datum-processing artifact via product-level signal injection; the decisive test requires satellite-scale non-GNSS data (optical inter-satellite links, raw TWSTFT sessions, or a GNSS product-level injection) |
| **Standard sirens $\Xi(z)$** (Paper 22) | Physical high-mass cutoff fitting breaks scale-free degeneracy ($\beta \simeq 154\,M_\odot$ TEP vs $33\,M_\odot$ GR); canonical pinned TEP out-performs stationary GR by $\Delta\ln L = +2.32$ out of sample on 50 high-$z$ events ($\ln L = -188.68$ vs $-191.00$, conditional likelihood ratio $\approx 10.2$), with global profile optima at canonical baselines ($\hat\epsilon = 0$, $\hat k = 0$); progenitor-dependent host transfer derived ($\epsilon_{\rm eff}\simeq0.004$ SNe Ia vs $0.24$–$0.52$ BBHs) | LVK O4/O5 bright standard sirens with spectroscopic host identification; detector-frame chirp-mass population trend | Spectroscopic bright sirens at $z\gtrsim0.3$ confirming standard $d_L^{\rm GW}=d_L^{\rm EM}$; detection of $(1+z)\mathcal M$ chirp-mass trend without tracking |

### 14.3 Evidential Hierarchy and Degree-of-Freedom Accounting Across the Corpus

Across the specialized studies of the TEP corpus, empirical channels fall into four distinct degree-of-freedom classes. Separating these classes prevents anchor calibration from being conflated with out-of-sample confirmation and defines which channels carry independent statistical weight:

| Evidential class | Free TEP parameters ($k_{\rm fit}$) | Representative channels & quantitative status | Evidential role |
| --- | --- | --- | --- |
| **I. Zero-parameter & model-independent tests** | $k_{\rm fit}=0$ (fixed by $\beta_A=-1$, $a_0$, or native coordinates) | KBC-void / MOND expansion falsification on native Pantheon+ coordinates ($\Delta\chi^2>100$, $8.4\sigma$/$9.7\sigma$, competitor exclusion; Paper 31); wide-binary joint force-law profile ($P_{,X}=k\sqrt\xi+2\xi$, $k_*=16.03$ fixed by $a_0$, passing at $0.4$–$1.7\sigma$ while excluding unit-term and exponential rivals at $\sim30\sigma$; Paper 13); SPARC RAR additive force law at fixed $k_*=16.03$ ($\text{RMS}=0.2004$–$0.2185$ dex; Paper 6); unstretched-space baryon-only Fermat delay and Bullet Cluster convergence (Papers 4, 19); GWTC-3 BBH high-$z$ chirp-mass cutoff ($\Delta\ln L=+2.32$ out of sample at $\hat\epsilon=0,\hat k=0$; Paper 22). | Primary load-bearing empirical discriminators; no per-dataset tuning. |
| **II. Terrestrial-anchor propagated tests** | $\rho_T\approx20\text{ g cm}^{-3}$ fixed on Earth ($L_c\approx4{,}200$ km); $k_{\rm fit}\le1$ channel transfer norm | Universal $R_T\propto M^{1/3}$ saturation scaling from GNSS ($R_T(M_\oplus)\approx4{,}150$ km; Papers 1–3, 14) to Earth flyby perigee asymptote ($R_{\rm sol}=4{,}146$ km, $\Delta\mathrm{BIC}\gtrsim1.66\times10^4$ vs null at $k_{\rm fit}=1$; Paper 15), Cepheid envelope density threshold ($\rho_T/\langle\rho\rangle$ predicting $\kappa_{P^2}$ to $0.5\%$ of the SH0ES central estimate, $(1.191\pm0.480)\times10^6$ mag; Paper 11), globular-cluster MSP spin-down suppression ($\Gamma_{\rm obs}=0.393\pm0.079$ vs literature-consensus Newtonian $0.748\pm0.039$ [$4.1\sigma$] and structurally rescaled 20-cluster $0.860\pm0.058$ [$4.8\sigma$], plus $8.6\sigma$ 20-cluster ensemble-propagated and $4.1\sigma$ fully-correlated ceiling excess across the 14 unique CMC realizations; Paper 10), and high-$z$ JWST age-excess scaling ($r=0.88$, $25.1\sigma$; Paper 12). | Cross-regime structural compression across 40 orders of magnitude in mass; tests scale invariance of $\rho_T$ and $S_\Sigma(\mathcal E)$. |
| **III. Cosmological $\Lambda$CDM-parity closures** | $k_{\rm fit}=1\text{--}2$ transport parameters in background/reference-Boltzmann runs; $k_{\rm fit}=0$ extra dark-sector parameters in Step-21 CDM-free closure | Pantheon+ + DESI DR2 BAO + Planck $\theta_*$ joint background fit ($\Delta\chi^2=-8.1$ vs $\Lambda$CDM; emergent DESI-quadrant thawing projection $(w_0,w_a)\approx(-0.83,-0.44)$ on that single history; Papers 26, 27); Paper 18 (a) reference Boltzmann validation with active scalar perturbations ($Z_E\gtrsim7\times10^2$) and reference CDM component retained ($\Delta\chi^2=-0.11$, $\sigma_8=0.824$; §§3–4) and (b) separate Step-21 CDM-free ($\omega_{\rm cdm}=0$) preferred-foliation construction with baryon clock-slope acoustic driving ($\omega_\phi=\omega_b y_s/\sqrt{2}=0.1179\text{--}0.1182$) and unstretched-space temporal-depth CMB lensing ($\chi^2=13.40$ on Planck lensing; $\Delta\chi^2=-0.06$, $\Delta\mathrm{AIC}=-2.06$ on full Planck TTTEEE + low-$\ell$ + lensing + BAO + Pantheon+); D/H Lyman-series absorber non-uniqueness ($T=161.03$, $p\le0.001$ across 3 sightlines; Paper 29). | Separates reference-Boltzmann active-scalar stability validation (retaining comparison CDM) from the preferred-foliation CDM-free ($\omega_{\rm cdm}=0$) acoustic and lensing construction and no-$\Lambda$ background closure. |
| **IV. Prospective divergent discriminators** | $k_{\rm fit}=0$ (pre-registered sign and scale) | Continental optical-fibre clock links (PTB–SYRTE 1,415 km link, predicted 14–19% differential plateau modulation; Paper 1 §4.10); raw TWSTFT vs GNSS common-clock differences; closed-loop non-exact synchronization holonomy $H_{\rm resid}\neq0$ at $10^{-18}$ (Paper 9); spectroscopic bright standard sirens at $z\gtrsim0.3$ (Paper 22); isolated field-galaxy strong lenses ($\gamma_{\rm SMP}\simeq0.81$ vs group-screened $\simeq0.98$; Step 80). | Out-of-sample experimental and observational channels where TEP and standard GR+$\Lambda$CDM make mutually exclusive predictions. |

*Multiplicity and look-elsewhere discipline.* Across a 30-paper corpus, individual $2\sigma\text{--}3\sigma$ auxiliary correlations would be expected by chance if treated in isolation. Accordingly, the framework's inference rests on (i) family-wise and false-discovery-rate corrected batteries within each flagship pipeline (e.g., 178/388 tests surviving FWER at $\alpha=0.05$ across 19 validation families in Paper 1; event-block stratified permutation nulls in Paper 26), and (ii) high-significance ($>4\sigma\text{--}25\sigma$) and zero-parameter out-of-sample channels (Class I above), while lower-significance auxiliary channels are recorded as consistency checks rather than standalone detections.

## Appendix A: Proofs and Key Derivations

## A1. Conformal null-cone invariance

Let $\tilde{g}_{\mu\nu} = A(\phi)^2 g_{\mu\nu}$ with $A > 0$. A vector $k^\mu$ null with respect to $g_{\mu\nu}$, $g_{\mu\nu} k^\mu k^\nu = 0$, satisfies $\tilde{g}_{\mu\nu} k^\mu k^\nu = A^2 g_{\mu\nu} k^\mu k^\nu = 0$. Maxwell's action $S = -(1/4) \int \sqrt{-g} F_{\mu\nu} F^{\mu\nu}$ is conformally invariant in 4D: under $\tilde{g}_{\mu\nu} = \Omega^2 g_{\mu\nu}$, $\sqrt{-\tilde{g}} F_{\mu\nu} F^{\mu\nu} = \sqrt{-g} F_{\mu\nu} F^{\mu\nu}$. Hence photon geodesics are conformally invariant; null cones coincide.

## A2. Exact direct conformal propagation null

Consider the purely conformal subclass $\tilde{g}_{\mu\nu} = A^2(\phi)\,g_{\mu\nu}$, $B = 0$. A photon trajectory satisfies

$$\tilde{g}_{\mu\nu}\,k^\mu k^\nu = A^2(\phi)\,g_{\mu\nu}\,k^\mu k^\nu = 0.$$

Because $A(\phi) > 0$, this is equivalent to $g_{\mu\nu}\,k^\mu k^\nu = 0$. Thus the conformal factor does not alter the local null cone or act as a direction-dependent refractive index. In a static conformal background, forward and backward propagation along the same geometrical path therefore acquire no conformal propagation-time antisymmetry:

$$\Delta t_{\rm prop}^{(A)} \equiv t_\to - t_\leftarrow = 0.$$

The conformal sector nevertheless remains observable through matter-clock rates and open-path comparisons. For stationary clocks, $d\tilde{\tau} = A(\phi)\,d\tau_g$, so clocks situated in different field environments can acquire different proper-time histories. The corresponding infinitesimal conformal transport is $\omega^{(A)} = d\ln A$, which is exact. Hence, in a smooth simply connected region,

$$\oint_C d\ln A = 0.$$

A static, purely conformal field can therefore produce clock-rate differences and open-path redshift effects, but cannot by itself generate a direction-odd same-path propagation residual or a non-zero closed-loop synchronization holonomy. Such a residual requires the disformal sector $B \neq 0$, explicit time dependence combined with motion through the field, or some other non-exact/topological transport structure.

*Photon theorem (general $A(x,t)$).* The result does not rely on $A$ being static, and the stronger observable statement — that the conformal contribution to a measured timing or frequency observable is endpoint-only — is a theorem of the measurement chain, not merely an inference from null-cone invariance. Two facts must be distinguished. First, at the trajectory level: for arbitrary $A(x,t)$ the unparametrized null curves of $\tilde{g}=A^2 g$ and $g$ coincide exactly, so no conformal factor produces additional photon deflection, altered image positions, or a propagated conformal contribution to inter-image coordinate delay. Second, at the observable level the experiment measures a chain — source clock $\to$ electromagnetic phase $\to$ receiver clock — that lives entirely in $\tilde g$. That chain still telescopes, for two reasons. The eikonal covector $k_\mu=\nabla_\mu S$ is an affinely parametrized null geodesic covector of *both* metrics: contracting the conformal connection difference gives $k^\nu k_\rho\,C^\rho_{\ \nu\mu} = k_\mu(k\cdot\partial\ln A) - k_\mu(k\cdot\partial\ln A) = 0$, so $k_\mu$'s transport along the path is governed by $g$ alone, whatever $A(x,t)$ does. And the matter-frame observer on a given worldline carries $\tilde u^\mu = u^\mu/A$, so the measured frequency is $\tilde\omega = -k_\mu\tilde u^\mu = \omega_g/A$ evaluated locally at the point of measurement. The observed frequency ratio therefore factorizes,

$$\frac{\tilde{\omega}_{\rm obs}}{\tilde{\omega}_{\rm emit}} = \left(\frac{\omega_g^{\rm obs}}{\omega_g^{\rm emit}}\right) \cdot \frac{A_{\rm emit}}{A_{\rm obs}},$$

where the first factor is the gravitational-metric redshift accumulated along the path and the second is endpoint-only for *any* $A(x,t)$ — intermediate evolution of $A$ along the trajectory telescopes out identically, because the only place $A$ enters the measurement is the normalization of the endpoint clocks. The same bookkeeping applies to arrival times: the null curves coincide as spacetime trajectories, so image $i$ arrives at coordinate time fixed by $g$'s Fermat surface, and the observer's proper-time delay is $\Delta\tilde\tau_{\rm obs}=A_{\rm obs}\,\Delta t_{\rm obs}$ — a single endpoint factor common to all images. Applied to the four combinations of static/evolving source and static/evolving $A$, the isotropic conformal sector contributes only through endpoint clock factors: emission-epoch differences at the source (the temporal-composite channel, which requires intrinsic source time-structure and vanishes for static sources), the observer-clock readout rate, and the endpoint ratio. Within the purely isotropic conformal subclass $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$, a genuinely path-dependent photon residual — integrated drift of $\tilde\omega$ along the ray, extra delay, extra deflection — cannot originate in $A(\phi)$ at any order and would instead require scalar stress-energy backreaction on $g_{\mu\nu}$ or a spatial disformal term $B(\phi)\nabla_\mu\phi\nabla_\nu\phi$ (or derivative matter couplings outside the minimal action). In the canonical resolved preferred-foliation metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu$ $+$ $A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ $+$ $B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ (§2.2, Step 80), by contrast, the longitudinal disformal weight $B_\parallel\equiv(A_{\rm sp}^2-A_{\rm loc}^2)/(2\bar X)$ is non-vanishing whenever $A_{\rm loc}/A_{\rm sp}\neq1$: at a fixed cosmic epoch ($A_{\rm sp}=1$) on unstretched spatial 3-geometry $\gamma_{ij}$, spatially varying local temporal lapse $A_{\rm loc}(u_{\rm grav}(\mathbf x))$ generates reciprocity-even optical refraction $n(\mathbf x)\simeq1-2\Phi_N(\mathbf x)/c^2-\beta_A u_{\rm grav}(\mathbf x)$ and Fermat time delays without requiring a non-zero transverse disformal coefficient $B(\phi)$ (Papers 4, 18, 19). Photon–graviton propagation coincidence ($c_{\rm GW}=c_\gamma$, $\Delta t_{\gamma\text{--}\rm GW}=0$) in this resolved preferred-foliation sector follows from the shared principal wave operator $\Box_{\rm TEP}=-(N_{\rm TEP}c)^{-2}\partial_t^2+A_{\rm sp}^{-2}\gamma^{kl}\partial_k\partial_l$ derived in §2.2 rather than from isotropic conformal invariance alone, while direction-odd same-path propagation asymmetry and closed-loop synchronization holonomy continue to require non-exact or transverse disformal transport ($B(\phi)\neq0$ or time-dependent landscape drift).

## A3. Synchronization holonomy invariance and projected disformal connection

This appendix derives the leading-order disformal contribution to the synchronization connection used in Sections 6 and 7 and shows why the GR-subtracted closed-loop residual is invariant under synchronization re-gauging.

### A3.1 Synchronization connection and re-gauging

Consider a local threading decomposition of the matter-frame metric relative to a physical clock-network congruence:

$$d\tilde s^2 = \tilde g_{00} \left(dt+\tilde\sigma_i dx^i\right)^2 + \tilde h_{ij}dx^i dx^j .$$

Equivalently, up to the sign convention used for $\tilde\sigma_i$,

$$\tilde\sigma_i = \frac{\tilde g_{0i}}{\tilde g_{00}} .$$

If the opposite sign convention is adopted, all shifts below acquire the opposite sign, but the closed-loop result is unchanged.

A synchronization re-gauging of the same physical clock network assigns a different simultaneity offset to the same clock worldlines:

$$t' = t+\chi(x^i),$$

where $\chi$ is a single-valued spatial clock-offset function on the network at the epoch of the loop measurement. Then

$$dt = dt' - \partial_i\chi\,dx^i .$$

Substituting into the metric gives

$$dt+\tilde\sigma_i dx^i = dt' + \left(\tilde\sigma_i-\partial_i\chi\right)dx^i .$$

Therefore

$$\tilde\sigma_i \longrightarrow \tilde\sigma_i' = \tilde\sigma_i-\partial_i\chi .$$

In one-form notation,

$$\tilde\sigma \longrightarrow \tilde\sigma-d\chi .$$

With the opposite sign convention for $\tilde\sigma$, the sign reverses. In either convention, the change is exact.

The same simultaneity re-labelling acts on the threading combination $g_{0i}/g_{00}$ of any metric in the same way. The GR reference connection $\sigma_{\rm GR}$ is the corresponding threading connection computed from the GR reference metric on the same physical clock-network congruence, with the same loop, same coordinate convention, and the standard Sagnac, Lense–Thirring, Shapiro, gravitational-redshift, station-motion, clock-scale, and reference-frame corrections included. Therefore the same re-labelling gives

$$\sigma_{\rm GR} \longrightarrow \sigma_{\rm GR}-d\chi .$$

The GR-subtracted residual connection

$$\Delta\sigma \equiv \tilde\sigma-\sigma_{\rm GR}$$

therefore transforms as

$$\Delta\sigma \longrightarrow (\tilde\sigma-d\chi)-(\sigma_{\rm GR}-d\chi) = \Delta\sigma .$$

Thus the closed-loop observable

$$H_{\rm resid}(C) = \oint_C \Delta\sigma$$

is invariant under synchronization re-gauging.

Equivalently, if one works with a representative in which the residual connection itself shifts by an exact form,

$$\Delta\sigma\rightarrow \Delta\sigma+d\chi_{\rm res},$$

then the loop integral is still invariant because

$$\oint_C d\chi_{\rm res}=0$$

for single-valued $\chi_{\rm res}$. The observable is the loop class of the residual connection modulo exact one-forms, not a raw one-way synchronization convention.

This proof is local in time. The loop $C$ is evaluated on the spatial network at a specified epoch. Time dependence of $\phi$, $B(\phi)$, or the matter metric may change the value of $\Delta\sigma$ from one epoch to another, but it does not change the synchronization-gauge argument at a fixed epoch. The gauge freedom relevant to the closed-loop observable is the spatial simultaneity re-labelling $\chi(x^i)$ of the specified physical clock network.

For experimental campaigns extending over months, the integrated data does not evaluate a single 4D spacetime loop, but rather generates a continuous time-series of instantaneous 3D spatial loop integrations, $H_{\rm resid}(C, t)$. At any given temporal slice $t$, the spatial simultaneity re-labeling remains strictly exact ($\oint_C d\chi(t, x^i) = 0$). Slow temporal variations in $\phi(t)$, or shifts in the Earth's environmental screening state $\mathcal{E}$ along its orbit, merely manifest as predictable, time-dependent modulations (e.g., annual or diurnal phases) of a continuously gauge-invariant spatial holonomy.

### A3.2 Projected disformal contribution

The matter metric is

$$\tilde g_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi .$$

Let $u^\mu$ be the four-velocity of the physical clock network defining the synchronization protocol, with

$$u^\mu u_\mu=-1$$

in the $(-+++)$ signature. The spatial projector into the local rest space of this congruence is

$$P_\mu{}^\nu = \delta_\mu{}^\nu + u_\mu u^\nu .$$

The scalar gradient decomposes as

$$\nabla_\mu\phi = D_\mu\phi - (u\cdot\nabla\phi)u_\mu ,$$

where

$$D_\mu\phi = P_\mu{}^\nu\nabla_\nu\phi$$

is the spatial gradient measured by the clock network.

*Corpus signature convention.* The canonical corpus convention, used throughout this paper and Papers 1–23 and 26–37, is $(-+++)$: timelike vectors obey $u^\mu u_\mu = -1$, the matter-metric lapse satisfies $N^2 = -\tilde g_{00}$, timelike scalar gradients have $\Sigma^2 = (\nabla\phi)^2 < 0$, and the null-cone condition selects $B \geq 0$ as the unconditionally admissible branch (the pointwise admissibility condition for sign-indefinite forms is stated in Section 4). Papers 18, 23, 24, and 25 declare the opposite signature $(+---)$. The mapping between conventions is $g_{\mu\nu} \to -g_{\mu\nu}$: covariant statements, scalar observables, and closed-loop holonomies are unaffected, but component-level sign relations reverse — the lapse is $+\tilde g_{00}$ rather than $-\tilde g_{00}$, the mass-shell condition reads $\tilde g^{\mu\nu}p_\mu p_\nu = +m^2$ rather than $-m^2$, and written signs of $B$-admissibility and $\Sigma^2$ statements flip. No error is committed within those papers; the note exists so that formulas are not copied across conventions without the sign map.

The disformal part of the matter metric is

$$\delta_B \tilde g_{\mu\nu} = B\nabla_\mu\phi\nabla_\nu\phi .$$

Its mixed time-space projection relative to the clock-network congruence is

$$P_\mu{}^\alpha u^\beta \delta_B\tilde g_{\alpha\beta} = B \left(P_\mu{}^\alpha\nabla_\alpha\phi\right) \left(u^\beta\nabla_\beta\phi\right),$$

so

$$P_\mu{}^\alpha u^\beta \delta_B\tilde g_{\alpha\beta} = B D_\mu\phi (u\cdot\nabla\phi).$$

The synchronization connection is the normalized mixed time-space component of the matter metric. In a local rest frame of the clock congruence, the conformal background gives

$$\tilde g_{00}^{(A)} = -A^2 ,$$

while the disformal correction gives

$$\delta_B\tilde g_{0i} = B (u\cdot\nabla\phi) D_i\phi .$$

The disformal sector also corrects $\tilde g_{00}$ at order $B(u\cdot\nabla\phi)^2$. However, in the local rest frame of the clock congruence the conformal background has no mixed time-space component, $\tilde g_{0i}^{(A)}=0$. Therefore the correction to $1/\tilde g_{00}$ multiplies a vanishing conformal numerator and first contributes only at higher order in $B$. To leading order, the denominator may consistently be evaluated at $\tilde g_{00}^{(A)}=-A^2$.

Therefore, to leading order in $B$, using the convention $\tilde\sigma_i=\tilde g_{0i}/\tilde g_{00}$,

$$\delta\tilde\sigma_i \simeq \frac{\delta_B\tilde g_{0i}}{\tilde g_{00}^{(A)}} \simeq -\frac{B}{A^2} (u\cdot\nabla\phi) D_i\phi .$$

With the opposite convention for the synchronization one-form, the overall sign is reversed; the closed-loop invariance and exact/non-exact distinction are unaffected.

In covariant projected form this is

$$\delta\tilde{\sigma}_\mu \simeq -\frac{B}{A^2} (u\cdot\nabla\phi) D_\mu\phi ,$$

or equivalently

$$\delta\tilde{\sigma}_\mu \simeq -\frac{B}{A^2} (u\cdot\nabla\phi) P_\mu{}^\nu\nabla_\nu\phi .$$

Higher-order terms include corrections from the disformal contribution to $\tilde g_{00}$, lapse-shift mixing, and higher powers of $B(\nabla\phi)^2/A^2$.

In a hypersurface-orthogonal $3+1$ slicing, where $u^\mu=n^\mu$, this reduces to

$$\delta\tilde{\sigma}_i \approx -\frac{B}{A^2N} (\partial_i\phi)(n\cdot\partial\phi),$$

up to lapse-shift convention and higher-order disformal corrections.

Furthermore, because the reference connection $\sigma_{\rm GR}$ is constructed using the exact same non-inertial clock congruence $u^\mu$, any purely kinematic effects arising from the network's proper acceleration or rotation (such as Sagnac or Thomas precession) enter both $\tilde{\sigma}$ and $\sigma_{\rm GR}$ symmetrically. These kinematics therefore cancel identically in the residual connection $\Delta\sigma = \tilde{\sigma} - \sigma_{\rm GR}$, ensuring that non-inertial cross-terms do not artificially mimic or contaminate the genuine disformal synchronization holonomy.

On the specified spatial slice this leading connection has the schematic form $\delta\tilde\sigma=-C\,d_3\phi$, where $C$ contains the clock-frame drift, lapse factors and $B/A^2$. Therefore $d_3(\delta\tilde\sigma)=-d_3C\wedge d_3\phi$; a uniform drift with fixed lapse, or a coefficient depending only on $\phi$, gives no curvature. The relation quoted in the main text with $B/A^2$ outside the wedge applies when $B/A^2$ is a function of $\phi$ only and with the corresponding lapse convention. A non-exact kinetic current $J=P_{,X}d\phi$ does not change the exactness of $d\ln A$ and does not by itself constitute a synchronization residual; the metric and physical clock congruence must provide the connection.

This establishes the two claims used in the main text: the leading disformal synchronization representative is the projected mixed time-space component of the matter metric, and the GR-subtracted closed-loop residual is invariant under admissible synchronization re-gaugings of the fixed physical clock network.

## A4. Disformal inverse and causality condition

Given $\tilde{g}_{\mu\nu} = A^2 g_{\mu\nu} + B \partial_\mu\phi \partial_\nu\phi$, define the scalar-gradient one-form $q_\mu \equiv \partial_\mu\phi$. The inverse satisfies $\tilde{g}^{\mu\nu} \tilde{g}_{\nu\sigma} = \delta^\mu_\sigma$. Ansätz: $\tilde{g}^{\mu\nu} = A^{-2} (g^{\mu\nu} + C q^\mu q^\nu)$. Solve for $C$:

$$\begin{aligned} A^{-2} [g^{\mu\nu} + C q^\mu q^\nu] [A^2 g_{\nu\sigma} + B q_\nu q_\sigma] &= \delta^\mu_\sigma \\ \Rightarrow \delta^\mu_\sigma + A^{-2} (B + A^2 C + B C (q \cdot q)) q^\mu q_\sigma &= \delta^\mu_\sigma \\ \Rightarrow A^2 C + B + B C (q \cdot q) = 0 \;\Rightarrow\; C &= - \frac{B}{A^2 + B (q \cdot q)}. \end{aligned}$$

Thus $\tilde{g}^{\mu\nu} = A^{-2} \left[ g^{\mu\nu} - \frac{(B/A^2) q^\mu q^\nu}{1 + (B/A^2) (q \cdot q)} \right]$. Lorentzian signature requires $1 + (B/A^2)(q \cdot q) > 0$.

## Appendix B: Photon Phase Speed and GW–EM Constraints

In a local inertial frame ($g_{\mu\nu} = \eta_{\mu\nu}$), the photon dispersion relation is $\tilde{g}^{\mu\nu} k_\mu k_\nu = 0$. Using the disformal inverse from A4,

$$\eta^{\mu\nu} k_\mu k_\nu = \frac{(B/A^2)(q \cdot k)^2}{1 + (B/A^2)(q \cdot q)}.$$

For a photon with 4-momentum $k^\mu = (\omega/c,\, k\hat{n})$, where $k \equiv |\vec{k}|$ is not assumed equal to $\omega/c$, and a static scalar gradient $q_\mu = (0, \nabla\phi)$, $q \cdot k = k\,\partial_{\hat{n}}\phi$ and $q \cdot q = |\nabla\phi|^2$. Defining $D \equiv B/A^2$, the dispersion relation gives

$$-\frac{\omega^2}{c^2} + k^2 = \frac{D\,k^2(\partial_{\hat{n}}\phi)^2}{1 + D|\nabla\phi|^2}.$$

Solving for $k^2$ and extracting the phase speed $v_{\rm ph} = \omega/k$ yields

$$v_{\rm ph} = c\sqrt{1 - \frac{D(\partial_{\hat{n}}\phi)^2}{1 + D|\nabla\phi|^2}}.$$

For propagation along the gradient ($\partial_{\hat{n}}\phi = |\nabla\phi|$), this simplifies to

$$v_\parallel = \frac{c}{\sqrt{1 + D|\nabla\phi|^2}}.$$

For small $D$, the leading-order result is $v_{\rm ph}/c \simeq 1 - \frac{1}{2}D(\partial_{\hat{n}}\phi)^2 + \mathcal{O}(D^2)$.

Gravitons propagate on $g_{\mu\nu}$, so $c_g = c$ exactly. The maximal fractional speed difference between photons and gravitons, for propagation perpendicular to $\nabla\phi$ ($\partial_{\hat{n}}\phi = 0$, $v_\perp = c$) versus along it ($\partial_{\hat{n}}\phi = |\nabla\phi|$, $v_\parallel = c/\sqrt{1 + D|\nabla\phi|^2}$), is

$$\frac{|c_\gamma - c_g|}{c} = \left| 1 - \frac{1}{\sqrt{1 + D|\nabla\phi|^2}} \right| \approx \frac{|D|}{2}|\nabla\phi|^2 \quad \text{(for small } D\text{)}.$$

The GW170817/GRB170817A bound $|c_\gamma - c_g|/c \lesssim {\rm few} \times 10^{-15}$ therefore requires

$$|D|\,|\nabla\phi|^2 \lesssim \text{few}\times10^{-15}$$

along typical lines of sight today, bounding the present-day disformal coupling $B(\phi_0)$. This derivation assumes a quasi-static gradient, $q_\mu\simeq(0,\nabla\phi)$, appropriate to the present-day local field; it is the $\Pi=0$ specialization of the general two-parameter cone relation of Section 8 and must not be applied to the evolving cosmological configuration, where temporal and spatial gradients coexist and the bound is evaluated along the realized photon path.

## Appendix C: Glossary of Symbols

| Symbol | Description | Section |
| --- | --- | --- |
| $g_{\mu\nu}$ | Gravitational metric tensor | 2 |
| $\tilde{g}_{\mu\nu}$ | Matter/causal metric tensor | 2 |
| $\phi$ | The scalar time field | 2 |
| $A(\phi)$ | Conformal coupling factor, $\exp(\beta_A \phi/M_{\text{Pl}})$ | 2 |
| $B(\phi)$ | Disformal coupling function | 2 |
| $\sigma_B$ | Dimensionless field-width parameter of the strong-field disformal suppression envelope, $B(\phi)=B_0\,\varphi^2(1+\varphi^2)^{-1}\exp(-\varphi^4/2\sigma_B^4)$ | 2.2 |
| TEP | Temporal Equivalence Principle | 1, 2 |
| $\tau$ | Proper time defined by $\tilde{g}_{\mu\nu}$ | 2 |
| $H_{\rm resid}$ | residual synchronization holonomy, $H_{\rm resid}(C) \equiv \oint_C (\tilde{\sigma} - \sigma_{\rm GR})$ | 3, 6 |
| $\Delta\sigma$ | GR-subtracted synchronization connection, $\Delta\sigma \equiv \tilde{\sigma}-\sigma_{\rm GR}$ | 3, 6, 7, A3 |
| $u^\mu$, $u_\mu$ | four-velocity field of the physical clock-network congruence (upper index contravariant, lower index covariant) | 3, 7, A3 |
| $P_\mu{}^\nu$ | spatial projector into the local rest space of the clock-network congruence, $P_\mu{}^\nu=\delta_\mu{}^\nu+u_\mu u^\nu$ in $(-+++)$ signature | 7, A3 |
| $q_\mu$ | scalar-gradient one-form, $q_\mu \equiv \partial_\mu\phi$ | A4, B |
| $c_g$, $c_\gamma$ | Speed of gravity, speed of light (photons) | 5 |
| $V(\phi)$ | Scalar self-interaction potential; master form $V_{\rm matter}(u) e^{-(u/u_s)^4} + V_0 e^{-(u_s/u)^4}$ in the dimensionless field $u = \phi/M_{\rm Pl}$ — remains shallow across the matter domain, with a Planck-scale floor at depth | 2.2, 8 |
| $\alpha(\phi)$ | Conformal coupling strength, $d(\ln A)/d\phi$ | 4 |
| $\beta_A$ | Dimensionless conformal coupling parameter | 2 |
| $M$ | Suppression scale for disformal/EFT operators | 4 |
| PPN | Parametrized Post-Newtonian formalism | 7 |
| $\gamma_{\rm PPN}$, $\beta_{\rm PPN}$ | Eddington PPN parameters | 7 |
| $H_0$ | Hubble constant today | 1, 8 |
| $S_8$ | Cosmological parameter for structure growth | 1, 8 |
| $r_s$ | Sound horizon at recombination | 8 |
| $\Xi_{AB}$ | One-way time asymmetry between A and B | 9, 10 |
| $k^\mu$ | Four-wavevector of a null signal | 3.2 |
| $\ln A(\phi)$ | conformal factor controlling matter-frame proper time | 7 |
| $\Sigma_\mu$ | Temporal Shear vector, $\Sigma_\mu \equiv \nabla_\mu \ln A(\phi)$ | 7 |
| $C_A$ | covariance of conformal-factor fluctuations, $C_A(x,x') = \langle \delta\ln A(x) \, \delta\ln A(x') \rangle$ | 7 |
| $\lambda_T$ | Temporal Topology correlation length measured in clock/covariance data | 7 |
| $\rho_T$ | Temporal Topology saturation scale; not a binary ambient-density switch | 7 |
| $\kappa_X$ | Observable response coefficient for channel X, not a microscopic coupling | 7 |

## Appendix D: Terminology and Usage Glossary

### Canonical Terms

| Term | Usage |
| --- | --- |
| **Temporal Topology** | Spatial/covariance structure of the time field $\phi$. |
| **Temporal Shear** | Active gradient $\Sigma_\mu = \nabla_\mu \ln A(\phi)$. |
| **Screening** | Suppression of local observable shear/response by $\mathcal S_\Sigma(\mathcal E)$ (not applicable to global cosmological origins). |
| **Conformal sector** | $A(\phi)$, clock rates, open-path transport. |
| **Disformal sector** | $B(\phi)$, cone tilts, non-exact transport. |
| **Synchronization holonomy** | Closed-loop residual only ($H_{\rm resid}$). |
| **Clock-network covariance** | GNSS/SLR/MGEX-type spatial correlation. |
| **Response coefficient** | Channel transfer parameter $\kappa_X$, distinct from the microscopic coupling. |
| **Residual-channel candidate** | Post-fit residual signal, not direct fundamental fit. |
| **Stress test** | Mixed/null/diagnostic paper. |
| **Inherited calibration** | Parameter imported from earlier paper. |
| **No-refit prediction** | External prior applied without tuning. |

### Sector Consistency Rules

**Conformal sector.** The conformal sector ($A(\phi)$) governs clock-rate, redshift, open-path, and covariance observables. Because $\nabla_\mu \ln A$ is exact, pure conformal shear does not produce closed-loop residual synchronization holonomy. Holonomy requires disformal or otherwise non-exact transport structure.

**Response-coefficient rule.** Fitted quantities such as $\kappa_X$, $\alpha_{\rm sat}$, and $\eta_{\rm resid}$ are observable response parameters. They are not bare scalar couplings unless a solved transfer function is supplied. The geometric projectors $\Gamma_X$ are supplied by the transfer-map calculation for the primary channels (Appendix E, R12), with the wide-binary plateau closed on the selected two-branch sector (predicted $\alpha_{\rm sat}=0.371$ against the observed $0.366\pm0.012$ at $0.44\sigma$, consistent with the derived projector identity $\Gamma_{\rm WB}=4\sqrt2/k\simeq0.353$; Step 56/020).

**Evidence-status rule.** Every corpus paper is classified as primary evidence, consistency check, stress test, candidate application, null audit, or theoretical completion. Its result is counted accordingly in the corpus-level evidence ledger.

## Appendix E: Reproducibility Record

Every numerical claim in the body of this paper traces to a registered pipeline step in `scripts/steps/`, executed in the pinned environment described in `scripts/README.md`. The table below keys each claim to its step and output artifact in `results/`; the body text cites the corresponding record identifier rather than implementation paths.

| ID | Claim supported | Step | Output artifact |
| --- | --- | --- | --- |
| R1 | Radial ODE closure and the nested galactic–solar–terrestrial field decomposition (Sections 2, 7): $\phi_{\rm gal}(8\,{\rm kpc}) \simeq 2.8\times10^{-7}$, solar perturbation at 1 AU, ambient-to-perturbation ratios, surface composition. | `scripts/steps/step_01_radial_ode.py` | `results/step_01_radial_ode.json` |
| R2 | Two-body operator benchmark evaluation (Section 2): wide-binary shear recovery at 2646 AU, conjunction suppression $\mathcal R(1.6\,R_\odot)\sim1\times10^{-23}$, Saturn suppression $2.7\times10^{-11}$, Earth–Moon pairwise response factor $\sim2.2\times10^{-15}$ (resolved nested operator, R11). | `scripts/steps/step_19_operator_evaluation.py` | `results/step_19_operator_evaluation.json` |
| R3 | Interior rolling-floor solution on the Painlev&eacute;–Gullstrand background (Sections 2, 8): logarithmic pileup to $u\simeq9$–$10$ at $1.01\,r_h$, native $A\to0$ branch selection, flatness closure to $7\times10^{-14}$. | `scripts/steps/step_16_interior_roll.py` | `results/step_16_interior_roll.json` |
| R4 | Closed static eternal cosmological benchmark (Section 8): field-equation residuals below $10^{-11}$; algebraic representation of the reconstructed closed-static potential within the parametrized master family confirmed numerically to machine precision; the natural shallow choice $V_{\rm matter}=V_{\rm rec}$ provides the non-trivial approximate comparison across the transition region. | `scripts/steps/step_12_cosmology_solution.py` | `results/step_12_cosmology_solution.json` |
| R5 | Volume-balance disformal reconstruction (Section 10A): $|b|\sim10^{-3}$ at $u\sim0.75$, envelope normalization $B_0\simeq-3\times10^{-3}$ on the sign-excluded branch. | `scripts/steps/step_13_disformal_closure.py` | `results/step_13_disformal_closure.json` |
| R6 | Corrected conditional loop benchmark (Section 10A): $H_{\rm resid}\simeq-4.33\times10^{-13}$ s at the stated dimensionless $B_0=+1$; quadrature convergence, orientation reversal, spherical/exact-connection zeros and $B_0=0$, $\dot u=0$ controls. Fixed-action normalization and common-field screening remain open. | `scripts/steps/step_15_holonomy_amplitude.py` | `results/step_15_holonomy_amplitude.json` |
| R7 | Clock-amplitude screening baseline $S_A$ (Section 2): computed in the companion screening paper's pipeline. | Paper 26 (TEP-C0), `step_02_02` | Paper 26 results. |
| R8 | Inhomogeneous non-compact slice existence demonstration and flatness ledger (Section 8): nonlinear Lichnerowicz constraint solved on a periodic cell (universal cover $R^3$), residual $\sim10^{-10}$; required sign-changing total-energy structure demonstrated both on the shifted positive-$u$ benchmark ($\lambda_*=-2.42$ in slice units) and natively on the negative-$V_{\rm rec}$ void branch ($u_{\rm void}=-0.85<u_*=-\ln 2$, with $8{,}496$ void cells carrying $V(u)<0$ before any shift, $V_{\min}=-0.340$, converged $\psi\in[0.9985,1.0029]$); native unstretched-space matter curvature ${}^{(3)}\tilde R=A_{\rm sp}^{-2}{}^{(3)}R$ at fixed epoch ($D_i\ln A_{\rm sp}=0$) has $\int\psi^5\,{}^{(3)}\tilde R\,d^3x\equiv 0$ by Gauss's theorem and physical-volume mean $|\Omega_k^{\rm native}|=1.40\times10^{-4}\ll10^{-3}$ (alongside the spatial-conformal benchmark scaling $|\Omega_k^{\rm eff}|\sim7\times10^{-2}$ at $\Delta u=0.1$ falling to $\sim10^{-3}$ at $\Delta u=0.02$). | `scripts/steps/step_20_landscape_existence.py` | `results/step_20_landscape_existence.json` |
| R9 | Canonical weak-field normalization ledger (Section 2): $\delta\phi/M_{\rm Pl}=2\beta_A\Phi_N$, $\ln(A/A_\infty)=2\beta_A^2\Phi_N$, bare unscreened matter lapse $\tilde N/\tilde N_\infty=e^{3\Phi_N}+O(\Phi_N^2)$ and $G_{\rm eff}/G=3$ for $\beta_A=-1$, with GR recovery as $S_\Sigma\to0$. | `scripts/steps/step_26_weak_field_lapse_ledger.py` | `results/step_26_weak_field_lapse_ledger.json` |
| R10 | Standard-siren source-to-detector mapping (Section 8): Einstein-frame sourcing mass $\mathcal M_e=A_e\tilde{\mathcal M}$, stationary-coordinate transport $\Delta t_o=\Delta t_e$, matter-frame readout $\tilde f_o=f_o/A_o$ with strain invariant, inferred chirp mass $\mathcal M_{\rm det}=A_oA_e\tilde{\mathcal M}$, inferred distances $d_L^{\rm GW}=A_o r$ and $d_L^{\rm EM}=(1+z)A_o r$, ratio $\Xi(z)=1/(1+z)$; all four links symbolic-verified. | `scripts/steps/step_21_waveform_inference.py` | `results/step_21_waveform_inference.json` |
| R11 | Master-action screening closure (Section 2): kinetic completion $P_{,X}=k\sqrt\xi+2\xi$ ($k\simeq16$ fixed by $a_0$; quadratic branch dominant at all Solar-System $\xi$) with $\Lambda_X^4=M_{\rm Pl}^2H_0^2$ gives $\mathcal S_\Sigma\to[1+(g/g_t)^2]^{-1}$ on the high-gradient branch, $g_t=cH_0/(2|\beta_A|)=3.4\times10^{-10}$ m s$^{-2}$, and the nested pairwise response $\mathcal R=\mathcal S_\Sigma(X_{\rm env})^2\,y(s)$ with $y$ the flux-conserving profile $y[1+y^2(r_*/r)^4]=1$ and $r_*=\sqrt{GM/g_t}$; benchmark table returns $1.0\times10^{-23}$ (Cassini conjunction), $1.6\times10^{-14}$ (Earth–Moon; $2.2\times10^{-15}$ including the solar ambient in the vertex factors), $2.7\times10^{-11}$ (Saturn — hierarchical $\mathcal R=y^3\sim s^4$, corresponding to $\delta a\simeq3.5\times10^{-15}$ m s$^{-2}$ against the $\sim5\times10^{-13}$ ephemeris bound), $1.2\times10^{-21}$ (terrestrial shear), and the wide-binary transition scale at zero per-system free parameters (comparable-mass pairs: $\mathcal R=\mathcal S_{\rm env}^2 y\sim s^{4/3}$, the exponent selected by the 341k-binary forward model). | `scripts/steps/step_27_master_action_screening_closure.py` | `results/step_27_master_action_screening_closure.json` |
| R12 | Channel-response transfer map (Section 7): $\kappa_X = |\beta_A|\,S_X\,\Gamma_X$ evaluated across the six primary channels (GNSS, Cepheid, millisecond pulsar, wide binary, galactic, LLR/flyby). Derived projectors: $\Gamma_{\rm GNSS}=|\beta_A|$ (quadratic covariance channel), $\Gamma_\mu=5/\ln 10$, $\Gamma_{\rm MSP}=1$, $\Gamma_{\rm gal}=\Gamma_\mu(1+z)^{\beta_A\Delta u}$, $\Gamma_{\rm LLR}=1$ as a unit range normalization, with $S_\Sigma(g_{\rm lunar})=1.6\times10^{-14}$ retained as the R11 force benchmark rather than a clock-to-range derivation; wide-binary saturation amplitude closed on the selected two-branch sector (predicted $\alpha_{\rm sat}=0.371$ against the observed $0.366\pm0.012$ at $0.44\sigma$; projector identity $\Gamma_{\rm WB}=4\sqrt2/k\simeq0.353$; Step 56/020). | `scripts/steps/step_05_gamma_all_channels.py` | `results/step_05_gamma_all_channels.json` |
| R13 | Canonical lapse admissibility, evolving CMC initial slice, and preferred-foliation khronon static maximal slice (Section 8): Step 21 retains the Hamiltonian source at $\Pi=0$, solves $(-D_h^2+q)N=C$ with $\langle N\rangle_{\sqrt h}=1$ on $16^3$ and $48^3$ grids ($N>0$, relative residual $<10^{-10}$, $C=\partial_t\mathcal K>0$), and shows that a purely canonical scalar shift requires $\Pi_{\rm can}^2<0$ (motivating the noncanonical sector), whereas the preferred-foliation khronon/cuscuton drift term $P_{\rm drift}=-\mu_{\rm kh}^2\sqrt{2X}$ ($2XP_{{\rm drift},X}-P_{\rm drift}\equiv 0$, $S_{\rm drift}=-3\mu_{\rm kh}^2\Pi<0$) solves the exact unshifted static maximal-slicing equation $(-D_h^2+q_{\rm kh})N=0$ ($\partial_t\mathcal K=0$) with $N\in[0.9990,1.0005]>0$ ($N=16$; $[0.9992,1.0004]$ at $N=48$), residual $<10^{-10}$, and strictly positive real drift $\mu_{\rm kh}^2\Pi=\tfrac23 M_{\rm Pl}^2\mu_0>0$; Step 28 solves the coupled $\hat{\mathcal A}_{ij}\ne0$ momentum/shear sector ($\psi\in[0.75,1.81]$). | `scripts/steps/step_21_lapse_solvability.py`, `scripts/steps/step_28_momentum_shear_sector.py` | `results/step_21_lapse_solvability.json`, `results/step_28_momentum_shear.json` |
| R14 | On-shell landscape evolution (Section 8): the scalar field is solved from its elliptic equation $D^2u=V_{,u}-\rho_m+\lambda_c$ on the matter landscape (bordered Newton, mean-field gauge, iterated against the Lichnerowicz solve) rather than prescribed, giving initial data on-shell at $10^{-8}$ with constraint residual $10^{-12}$; the same integration machinery then produces a monotonic drift over $\Delta u\simeq3.4$ (lookback $z\lesssim28$) without reversal, and constrained evolution on the step-21 maximal-slice eigen lapse holds the Hamiltonian residual near $10^{-5}$. Three variants isolate the earlier failure: prescribed wells, an uphill momentum kick against the ambient slope, and the two-zone lapse rescaling. A nested-landscape variant (broad envelope plus embedded wells, one global scalar solve) realizes the Rule-22 hierarchy on a single slice — parent-ambient/exterior-ambient offset $\Delta u\simeq+0.022$ with level-dependent drift — and evaluates the screening operator $P_{,X}-1\simeq5\times10^{-6}$ (normalized on the slice's own drift rate; conservative since that branch inflates $H_0$), i.e. the benchmark is in the unscreened weak-gradient regime — screening responds to temporal-field steepness, not spatial compactness. | `scripts/steps/step_51_onshell_landscape_evolution.py` | `results/step_51_onshell_landscape_evolution.json` |
| R15 | Binary-pulsar scalar-dipole transmission and strong-equivalence-principle bounds (Section 7): Step 69 evaluates both (i) the massless fixed-$r=a$ unscreened-sensitivity unit-term control $P=X+X|X|/\Lambda_X^4$ ($T_{\rm amp}=0.0121$ and tightest margin $0.529$ on J1738+0333 when stellar-to-orbit interior screening between $R_{\rm NS}$ and $a$ is omitted) and (ii) the selected two-branch sector $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$) with shell-impedance screening across the compact-body-to-wave-zone profile ($T_{\rm amp}^{\rm imp}\simeq a/r_*=9.2\times10^{-7}\text{--}1.4\times10^{-5}$, $2.26\times10^{-6}$ on J1738+0333), clearing all binary-pulsar $\dot P_b$ bounds across the full EOS sensitivity bracket by amplitude margins of $2.8\times10^3$ to $3.9\times10^4$ at $K_D=0.5$ (tightest: $2{,}817\times$ on J1738+0333 $s_{\rm hi}$; $996\times$ to $1.39\times10^4$ at $K_D=4.0$) and the J0337+1715 hierarchical $\mathcal R=y^3$ SEP channel at $\eta_{\rm pred}=2.54\times10^{-14}\ll2.6\times10^{-6}$ (margin $\sim10^8\times$). | `scripts/steps/step_69_dipole_transmission.py` | `results/step_69_dipole_transmission.json` |
| R16 | Standard-siren invariant, continuous worldline LLR/redshift reconciliation, and intrinsic-population bookkeeping (Section 8): the dimensionless inspiral invariant $I=G_{\rm loc}\mathcal M f/c^3$ is tracked through locally measured constants at both endpoints — a source-local observer recovers the standard chirp mass for screened systems ($M_{\rm loc}=\widetilde M(1+\alpha_A\alpha_B)/(1+\alpha_e^2)\to\widetilde M$), so the $(1+z)^{-1}$ detector-mass map is a transport statement (coordinate frequency conserved on the static background), with $\Xi(z)=A_e/A_o$. The frame-consistent chirp identity $M_{\rm det}=(A_e/A_o)(1+\alpha_A\alpha_B)/(1+\alpha_o^2)\,\widetilde M$ is verified symbolically. On the continuous two-branch + quartic radial profile ($P_{,X}=k\sqrt\xi+2\xi$, zero thin-shell approximation; Step 76), the operative response on the Cassini-compatible screened branch — the quartic normalization used for the Solar-System flux profile — resolves to the static tangent $T_{\rm tan}=1.2\times10^{-8}$ at $10^8 R_\oplus$ (finite-excursion secant below the declared field-reporting floor at the same radius and at $10^{10}R_\oplus$), bounding the local drift under strictly constant $M_{\rm Pl}$ in $S_{\rm grav}$ and $G_{\rm loc}=G_*A^2(\phi)(1+\alpha_{\rm amb}^2)$ to $|\dot A/A|_{\rm loc}\lesssim8.4\times10^{-19}\,{\rm yr}^{-1}$ and $|\dot G_{\rm loc}/G_{\rm loc}|_{\rm LLR}\lesssim1.7\times10^{-18}\,{\rm yr}^{-1}$, more than four orders of magnitude below both the Williams et al. bound $4\times10^{-13}\,{\rm yr}^{-1}$ and the Hofmann–Müller re-analysis result $(7.1\pm7.6)\times10^{-14}\,{\rm yr}^{-1}$; the reference-quartic comparison branch supplies the conservative finite-excursion secant ceiling $T_{\rm cont}\equiv\Delta u_{\rm int}/\Delta\bar u_{\rm amb}=1.99\times10^{-3}$, i.e. $|\dot G_{\rm loc}/G_{\rm loc}|\lesssim2.85\times10^{-13}\,{\rm yr}^{-1}<4\times10^{-13}\,{\rm yr}^{-1}$ (this unscreened comparison ceiling satisfies the Williams et al. bound only; the operative screened branch satisfies both LLR analyses) (plus the two-branch scalar-force self-energy drift $2H_0 y_{\rm tb}(1\text{ AU})=2.10\times10^{-15}\,{\rm yr}^{-1}\ll4\times10^{-13}\,{\rm yr}^{-1}$, with `solve_ivp` Keplerian ODE integration confirming $\tilde r/\tilde r_0=0.99688$ versus the unscreened $T=1$ shrinkage $\tilde r/\tilde r_0=A^{-2}=0.20186$), while explicit `solve_ivp` null-geodesic ray-tracing in the canonical $(3+1)$ ADM matter metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(\phi(t,\mathbf x))\Pi_{\mu\nu}$ ($d\tilde s^2=-N_{\rm TEP}^2(\mathbf x)c^2dt^2$ $+$ $A_{\rm sp}^2(t,\mathbf x)\gamma_{ij}(\mathbf x)dx^i dx^j$, with stationary baryon-sourced lapse $N_{\rm TEP}(\mathbf x)=\exp(-u_{\rm grav}(\mathbf x))$ so $\partial_t\ln N_{\rm TEP}=0$ and no secular $\mathcal O(1)$ potential cliff develops between galaxies and the IGM, and matter-frame ruler-drift factor $A_{\rm sp}(t,\mathbf x)=\exp(\beta_A u_{\rm drift}(t,\mathbf x))=\exp(-u_{\rm drift}(t,\mathbf x))$ on static gravitational space $dx$ with $\partial_t\ln A_{\rm sp}=-\partial_t u_{\rm drift}=+H_{\rm drift}>0$) with full static-observer 4-velocity normalization $\tilde\omega=-\tilde u^\mu k_\mu=E/N_{\rm TEP}$ proves the universal differential-geometric redshift identity $\ln(1+z)=\int_{t_e}^{t_o}\partial_t\ln A_{\rm sp}\,dt+\int_{x_e}^{x_o}\partial_x\ln N_{\rm TEP}\,dx$ $=$ $[1-f_{\rm well}(1-T_{\rm cosmo})]\Delta\bar u_{\rm amb}+\delta u_{\rm env}$, recovering $100.00\%$ of the Hubble law ($\ln 1.1=0.09531$ at $z=0.1$; Step 70) for IGM-to-IGM and at the physical $10^8 R_\oplus$ screening scale ($f_{\rm IGM}=1-f_{\rm well}=1.000000$; $99.72\%$ for two halo-scale wells with $f_{\rm IGM}=0.9972$, $94.13\%$ for $20$ intervening halo-scale wells with $f_{\rm IGM}=0.9421$, and linear Hubble scaling across the Lyman-$\alpha$ forest lookback sweep $\Delta\in[0.02,0.20]$) without invoking a running Planck mass or $f_{\rm well}=1$. For source populations, if interiors track the ambient drift by fraction $\epsilon$, characteristic masses drift as $(1+z)^{3\epsilon}$ and the discriminant runs as $(1+z)^{3\epsilon-1}$ (null at $\epsilon=2/3$, reversed at $\epsilon=1$). The Chandrasekhar-channel diagnostic assumes $L\propto G^{-3/2}$, giving $\Delta m=-7.5\epsilon\log_{10}(1+z)$. Step_73 records the residual-floor estimate $\epsilon\lesssim0.05$–$0.10$; Paper 26's fixed-$z_T=5$ M1 profile likelihood returns $\epsilon<0.150$ at 3$\sigma$ ($\Delta\chi^2=805.8$ at $\epsilon=1$, best fit $\epsilon=-0.05$) for SN Ia progenitors under its stated distance model and Chandrasekhar luminosity prescription, and the host-assembly calculation (`results/outputs/audit_epsilon_bbh_derivation.json`) yields $\epsilon_{\rm eff}\simeq0.004$ for deep-well SN Ia hosts versus $0.24$–$0.52$ for field-spiral/dwarf BBH hosts. Paper 22 independently establishes the BBH population likelihood on GWTC, resolving the global profile optimum at $\hat\epsilon=0.0$ ($\Delta\ln L=+2.32$ over stationary GR) consistent with the derived progenitor host-assembly range. | `scripts/steps/step_70_siren_invariant.py`, `scripts/steps/step_71_intrinsic_mass_drift.py`, `scripts/steps/step_73_sn_luminosity_bound.py`; Paper 26 (TEP-C0) `scripts/steps/step_03_11_sn_luminosity_drift_profile.py` | `results/step_70_siren_invariant.json`, `results/step_71_intrinsic_mass_drift.json`, `results/step_73_sn_luminosity_bound.json`; Paper 26 `results/step_03_11_sn_luminosity_drift_profile.json` |
| R17 | Static ambient-boundary response (Sections 2, 8), unit-term control: step_53 solves the kinetic/quartic radial boundary-value problem with central flux regularity and an enforced outer Dirichlet value. Step_76 separates linearized susceptibility from finite-excursion secants and explicitly varies both the quartic branch and outer radius. For the Earth at the reference coupling and an outer radius of $10^{10}R_\oplus$, the tangent is $0.9966$ while the secant at imposed $\delta u=2.5\times10^{-11}$ is $1.5\times10^{-5}$; at the Cassini-compatible coupling the same outer-radius readings are $0.210$ and $9.3\times10^{-9}$. With that coupling but imposed comparison interfaces at $235R_\oplus$ and $23{,}500R_\oplus$, the finite secants instead reach $0.99998$ and $0.9831$. The massless-vacuum control returns unit response: the falloff is potential- and domain-dependent, not a universal loss of clock amplitude inside Earth. A separate local gradient benchmark evaluates the cross-term of isolated Earth and Solar profiles at 1 AU: their aligned-to-antialigned kinetic-stiffness span is $0.345$ relative to Earth's isolated value. The comparison interfaces and local gradient bracket are not a self-consistent coupled Earth–Sun solution, nor a frequency-dependent lunar-clock, GNSS or cosmological transfer. On the selected two-branch operator, Step 76 also solves the same sources, potential and boundaries directly. At $10^8R_\oplus$ with the reference quartic coupling, the Earth tangent is $0.6536$ and the finite-excursion secant at $\delta u=2.5\times10^{-11}$ is $0.00199$. At $10^{10}R_\oplus$, the Earth, Sun and Cepheid-envelope finite responses fall below the declared field-reporting resolution; a computed zero tangent is reported as a numerical value rather than a universal exact bound. The selected massless-vacuum control transmits with unit tangent and secant. Comparison interfaces at $235R_\oplus$ and $23{,}500R_\oplus$ retain near-unit reference response, so suppression depends on the potential and matched outer domain. A 1,200/2,400/4,800-point refinement at the measurable $10^8R_\oplus$ boundary gives tangents $0.65346/0.65357/0.65359$, independently reproduced by small-excursion differences of the nonlinear solved profiles. The static depth ratio $S_A$ remains distinct from both transmission and clock covariance. | `scripts/steps/step_76_ambient_transmission.py` | `results/step_76_ambient_transmission_two_branch.json`; `results/step_76_ambient_transmission.json` (unit-term control) |
| R18 | Inhomogeneous landscape stability (Section 8): positive restricted finite-cell spectra and independent ambient bounds are retained. Step 79 also records (i) the complete second-variation audit on the canonical reconstructed potential (`completed_canonical_reconstructed_coupled_audit`), showing that retaining $+\rho_*(x)\frac{d^2(e^{-u})}{du^2}=+\rho_*(x)e^{-u}$ at the Step-12 cell-mean density $\langle\rho_*\rangle=2\,\text{rhob\_half}$ flips the unprojected lowest eigenvalue from $-0.4389$ (bare $V_{\rm rec}''$ alone) to $\lambda_0=+0.4123\text{--}0.4127>0$ ($N=16,24$), (ii) a conditional quartic calculation that re-solves the Hamiltonian geometry with a prescribed scalar profile and offset, and (iii) a jointly on-shell quartic calculation (`joint_on_shell_quartic_two_branch_coupled_audit`) that solves both the nonlinear two-branch scalar Euler–Lagrange equation and the Lichnerowicz Hamiltonian constraint simultaneously to $<3.1\times10^{-12}$ ($N=16,24$, unprojected coupled $\lambda_0=+0.5624\text{--}0.5633>0$, $\lambda_1=+378.95\text{--}381.67$, all-twist bound $+0.4662\text{--}0.4668>0$). In all three unprojected audits, the metric response has an explicit volume constraint $q_{\rm vol}^T\delta\psi=0$ solved by a bordered inverse, and the all-twist bound uses the half-wave lattice gap rather than the periodic-cell gap. These finite-grid constrained-operator benchmarks do not establish full-action continuum stability. | `scripts/steps/step_79_inhomogeneous_eigenmodes.py` | `results/step_79_inhomogeneous_eigenmodes.json` |
| R19 | Unified matter metric, spatially covariant ADM/DHOST and local khronon gravitational action on static space, GW170817 shared-cone verification, PPN slip, SLACS/ESO 325 lens-dynamics consistency, and standard-siren consistency (Sections 2.2, 7, 8): symbolic verification of $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu$ $+$ $A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ $+$ $B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($d\tilde s^2=-N_{\rm TEP}^2(\mathbf x)c^2dt^2$ $+$ $A_{\rm sp}^2(t,\mathbf x)\gamma_{ij}(\mathbf x)dx^i dx^j$) and $S_{\rm grav}=\tfrac12 M_{\rm Pl}^2\int d^3x\,dt\sqrt{\gamma}\,N_{\rm GR}[R_{\rm scalar}^{(3)}(\gamma)$ $+$ $A_{\rm sp}^{-1}R_{\rm TT}^{(3)}(\gamma)$ $+$ $(A_{\rm sp}/A_{\rm loc}^2)(K_{ij}K^{ij}-K^2)]$ $=$ $\tfrac12 M_{\rm Pl}^2\int d^3x\,dt\sqrt{\gamma}\,N_{\rm GR}[A_{\rm sp}^{-1}R^{(3)}(\gamma)$ $+$ $2(A_{\rm sp}^{-1}-1)\gamma^{ij}D_i\ln N_{\rm GR}\,D_j\ln N_{\rm GR}$ $+$ $\tilde K_{ij}\tilde K^{ij}-\tilde K^2]$ (where the quadratic SVT pieces $\sqrt{\gamma}\,N_{\rm GR}R_{\rm scalar}^{(3)}=-2(\nabla\Phi_N/c^2)^2$ and $\sqrt{\gamma}\,N_{\rm GR}R_{\rm TT}^{(3)}=-\tfrac14\gamma^{kl}\partial_k h_{ij}^{\rm TT}\partial_l h_{ij}^{\rm TT}$ are reproduced identically at quadratic order by the strictly local khronon acceleration operator $a_i a^i=\gamma^{ij}D_i\ln N_{\rm GR}\,D_j\ln N_{\rm GR}$ of Blas, Pujolàs & Sibiryakov 2010 without any non-local Poisson inverse), proving (i) $\delta S_{\rm grav}/\delta u_{\rm grav}\equiv 0$ and $\delta S_{\rm grav}/\delta A_{\rm sp}\equiv 0$ in the quasistatic limit $K_{ij}\to0$ (preserving the Poisson and scalar source equations without Jordan-frame degeneracy), (ii) identical GW/EM wave operator $\Box_{\rm TEP}=-(N_{\rm TEP}c)^{-2}\partial_t^2$ $+$ $A_{\rm sp}^{-2}(t,\mathbf x)\gamma^{ij}\partial_i\partial_j$ with $c_{\rm GW}(t,\mathbf x)=c_\gamma(t,\mathbf x)=c\,N_{\rm TEP}(\mathbf x)/A_{\rm sp}(t,\mathbf x)$ at every event ($\Delta t_{\gamma-\rm GW}\equiv 0.0\text{ s}$ for GW170817 at $z=0.0098$, vs. $6.92\times10^5\text{ yr}$ on an unshared tensor operator) and shared refractive index $n(x)\simeq1-2\Phi_N/c^2-\beta_A u_{\rm grav}(x)$ at fixed epoch ($A_{\rm sp}=1$), (iii) PPN slip $\gamma_{\rm PPN}-1=-2\beta_A^2 S_\Sigma/(1+2\beta_A^2 S_\Sigma)<0$ ($|\gamma_{\rm PPN}-1|\simeq4.32\times10^{-8}\ll2.3\times10^{-5}$ at $1.6\,R_\odot$ on the two-branch sector, with exact ray-integrated Earth–Sun–Saturn Cassini Doppler $|\gamma_{\rm PPN}-1|_{\rm Doppler}=9.01\times10^{-8}$ and conjunction-differential Shapiro $|\gamma_{\rm PPN}-1|_{\rm Shapiro}=3.74\times10^{-7}\ll2.3\times10^{-5}$, zero-gradient regulator $\epsilon_{\rm reg}=10^{-3}$, intrinsic moving-source PPN coefficients $\alpha_1^{\rm int}(1.6\,R_\odot)=-8.64\times10^{-8}$, $\alpha_2^{\rm int}(1\text{ AU})=-1.47\times10^{-5}$ and CMB-projected observables $\alpha_1(1.6\,R_\odot)=-1.31\times10^{-13}$, $\alpha_2(1\text{ AU})=-2.23\times10^{-11}\ll4\times10^{-9}$ and galaxy-scale lensing slip at ESO 325-G004 and the SLACS ensemble, where the idealized zero-ambient vacuum baseline $y_{\rm tb}(R_{\rm Ein})=0.152$ ($\gamma_{\rm SMP}=0.767$ at $X_{\rm group}=0$) is superseded in the realized nested cosmic hierarchy (Section 2.3) by ambient group/filamentary screening of the incremental scalar response ($y_{\rm eff}=1.6\times10^{-3}\text{--}7.3\times10^{-3}$ and $\gamma=0.986\text{--}0.997$ on the $M_h=3\times10^{13}\text{--}3\times10^{14}\,M_\odot$ group-halo profile with $X_{\rm group}=131\text{--}611$, or $y_{\rm eff}=0.007\text{--}0.104$ and $\gamma=0.83\text{--}0.98$ across effective-ambient $X_{\rm group}=3\text{--}100$, with the $1\sigma$ floor $\gamma\ge0.88$ requiring only $X_{\rm group}\ge8.1$, $y_{\rm eff}\le0.068$; Step 30 operator), consistent with both Collett et al. 2018 $\gamma_{\rm SMP}=0.97\pm0.09$ and the SLACS massive-ETG ensemble $\langle\gamma_{\rm SMP}\rangle=0.98\pm0.07$ ($\gamma_{\rm pred}=0.972\text{--}0.981$ at $M_h=3\times10^{13}\,M_\odot$, $0.01\text{--}0.11\sigma$; Cao et al. 2017; Shajib et al. 2021)), and (iv) constant WKB tensor measure $\sqrt{A_{\rm sp}^{+1}\cdot A_{\rm sp}^{-1}}\equiv 1$ on static space ($\sqrt{\gamma}=1$) eliminating FLRW Hubble friction and preserving $1+z=A_o/A_e$ and $\Xi(z)=d_L^{\rm GW}/d_L^{\rm EM}=1/(1+z)\neq 1$. | `scripts/steps/step_80_shared_lapse_action_derivation.py` | `results/step_80_shared_lapse_action_derivation.json` |

The full step registry, execution order, and pinned dependency versions are documented in `scripts/README.md`; the complete output set in `results/` also archives the intermediate diagnostics for steps not cited in the body.

## References

Einstein, A. (1905). Zur Elektrodynamik bewegter Körper. Ann. Phys. 17, 891–921.
Einstein, A. (1916). Die Grundlage der allgemeinen Relativitätstheorie. Ann. Phys. 49, 769–822.
Will, C. M. (2014). The Confrontation between General Relativity and Experiment. Living Rev. Relativity 17, 4.
Bertotti, B., Iess, L. & Tortora, P. (2003). A test of general relativity using radio links with the Cassini spacecraft. Nature 425, 374–376.
Bekenstein, J. D. (1993). The relation between physical and gravitational geometry. Phys. Rev. D 48, 3641.
Damour, T. & Polyakov, A. M. (1994). The string dilaton and a least coupling principle. Nucl. Phys. B 423, 532.
Khoury, J. & Weltman, A. (2004). Chameleon fields: Awaiting surprises for tests of gravity in space. Phys. Rev. Lett. 93, 171104.
Hinterbichler, K. & Khoury, J. (2010). Symmetron fields. Phys. Rev. Lett. 104, 231301.
Abbott, B. P. et al. (LIGO/Virgo) (2017). GW170817: Observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett. 119, 161101.
Abbott, B. P. et al. (2017). Multi-messenger observations of a binary neutron star merger. ApJ 848, L12.
Planck Collaboration (2020). Planck 2018 results. VI. Cosmological parameters. A&A 641, A6.
Riess, A. G. et al. (2022). A comprehensive measurement of the local value of the Hubble constant. ApJ 934, L7.
Bothwell, T. et al. (2022). JILA Sr optical lattice clock with 10−18 stability and accuracy. Nature 602, 420–424.
Touboul, P. et al. (2022). MICROSCOPE mission: test of the equivalence principle in space. Phys. Rev. Lett. 129, 121102.
Burrage, C. & Sakstein, J. (2018). Tests of chameleon gravity. Living Rev. Relativity 21, 1.
Bettoni, D. & Liberati, S. (2013). Disformal invariance of second-order scalar-tensor theories. Phys. Rev. D 88, 084020.
Koivisto, T. S. & Zumalacárregui, T. (2013). Disformal gravity. Phys. Rev. D 88, 084016.
Ashby, N. (2003). Relativity in the Global Positioning System. Living Rev. Relativity 6, 1.
Uzan, J.-P. (2011). Varying constants, gravitation and cosmology. Living Rev. Relativity 14, 2.
Michelson, A. A. & Morley, E. W. (1887). On the relative motion of the Earth and the luminiferous ether. Am. J. Sci. 34, 333–345.
Kennedy, R. J. & Thorndike, E. M. (1932). Experimental establishment of the relativity of time. Phys. Rev. 42, 400–418.
Müller, H. et al. (2003). Modern Michelson-Morley experiment using cryogenic optical resonators. Phys. Rev. Lett. 91, 020401.
Herrmann, S. et al. (2005). Test of the isotropy of the speed of light using a continuously rotating optical resonator. Phys. Rev. Lett. 95, 150401.
IERS Conventions (2010). Gérard Petit and Brian Luzum (eds.). IERS Technical Note No. 36, Frankfurt am Main: Verlag des Bundesamts für Kartographie und Geodäsie.
Damour, T. & Esposito-Farèse, G. (1992). Tensor-multi-scalar theories of gravitation. Class. Quantum Grav. 9, 2093.
Everitt, C. W. F. et al. (2011). Gravity Probe B: final results. Phys. Rev. Lett. 106, 221101.
Belgacem, E. et al. (2018). Modified gravitational-wave propagation and standard sirens. Phys. Rev. D 98, 023510.
Wetterich, C. (2013). A universe without expansion. Phys. Dark Univ. 2, 184.
Adams, A., Arkani-Hamed, N., Dubovsky, S., Nicolis, A. & Rattazzi, R. (2006). Causality, analyticity and an IR obstruction to UV completion. JHEP 10, 014.
Babichev, E., Mukhanov, V. & Vikman, A. (2008). k-Essence, superluminal propagation, causality and emergent geometry. JHEP 02, 101.
Boran, S., Desai, S., Kahya, E. O. & Woodard, R. P. (2018). GW170817 falsifies dark matter emulators. Phys. Rev. D 97, 041501.
Blas, D., Pujolàs, O. & Sibiryakov, S. (2010). Consistent extension of Hořava gravity. Phys. Rev. Lett. 104, 181302.
Berezhiani, L. & Khoury, J. (2015). Theory of dark matter superfluidity. Phys. Rev. D 92, 103510.
Cao, S. et al. (2017). Test of parameterized post-Newtonian gravity with galaxy-scale strong lensing systems. ApJ 835, 92.
Shajib, A. J. et al. (2021). Dark matter haloes of massive elliptical galaxies at $z\sim0.2$ are well described by the Navarro–Frenk–White profile. MNRAS 503, 2380–2405.

## How to cite

You can cite all versions by using the DOI: 10.5281/zenodo.16921911

**BibTeX:**

```
@misc{Smawfield_TEP_2025,
  author       = {Matthew Lukin Smawfield},
  title        = {Temporal Equivalence Principle: Dynamic Time & 
                  Emergent Light Speed},
  year         = {2025},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.16921911},
  url          = {https://doi.org/10.5281/zenodo.16921911},
  note         = {Preprint}
}
```

## Contact

For questions, comments, or collaboration opportunities regarding this work, please contact:

**Matthew Lukin Smawfield**
matthew@mlsmawfield.com

---

*This document was automatically generated from the TEP research site. For the interactive version with figures and enhanced formatting, visit: https://mlsmawfield.com/tep/theory/*

*Source code and data available at: https://github.com/matthewsmawfield/TEP*

---

<!-- ===================================================================== -->
<!-- CANONICAL URL: https://mlsmawfield.com/tep/gnss-i/ -->
<!-- MARKDOWN URL:  https://mlsmawfield.com/tep/gnss-i/manuscript.md -->
<!-- ===================================================================== -->

# Global Time Echoes: Distance-Structured Correlations in GNSS Clocks
**Matthew Lukin Smawfield**
Version: v0.28 (Jaipur)
First published: 17 September 2025 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.17127229

---

## Abstract

Phase-coherent spectral analysis of 62.7 million station-pair
measurements from 364 GNSS stations (2023–2025; effective sample size N<sub>eff</sub> ≈ 25–28 independent distance bins after accounting for spatial
correlation; see §Methods) reveals systematic
distance-structured correlations in clock networks. These correlations
follow an exponential decay with a Temporal Topology correlation
length λ<sub>T</sub> = 3,330–4,549 km (exponential best fit; bootstrap 95% CIs: CODE 1,198–5,918 km; IGS 3,197–4,871 km; ESA
2,532–3,984 km) and show strong goodness-of-fit when evaluated on
distance-binned means across three distinct analysis center solutions on largely shared raw inputs (CODE, IGS Combined, ESA Final; R² =
0.920–0.970; fits are to bin means, not raw pairs). Cross-solution
validation across software-diverse processing chains (Bernese, NAPEOS, and the IGS combination filter), consistent across 12 frequency bands and confirmed through
multiple binning schemes and null hypothesis testing, demonstrates these patterns represent persistent empirical correlations not explained by the tested artifacts. The patterns also show dependencies on station elevation and
geomagnetic latitude, consistent with screened scalar fields via
continuous Temporal Topology: coherence strengthens and correlation
lengths extend in quieter mid-latitude and higher-elevation regimes,
while high ionospheric activity suppresses rather than generates the
signal, and the forward-simulated satellite-visibility kernel is both
broader and differently shaped (&sect;4.6).

The primary inference rests on cross-centre distance-structured covariance
and λ<sub>T</sub>; the following planetary, Chandler, diurnal, and geomagnetic
signatures provide complementary physical consistency tests.
The correlations show associations with Earth's orbital
motion (r = -0.571 to -0.793 across centers), planetary gravitational
influences (8 nominally significant event fits, mostly non-event-locked
on audit), Chandler wobble modulation
(R² = 0.377–0.471), and systematic diurnal temporal variations with
synchronized early morning coherence peaks (Local Solar Time).
Comprehensive validation demonstrates 24–61× signal enhancement over
randomized controls (z = 15.8–31.9 across 180 null test iterations),
with 40.0–52.3% of 388 tests across 19 validation families surviving
Bonferroni, empirical-Bayes, and false-discovery-rate corrections. TID exclusion analysis shows
21–23 % signal improvement when excluding high-ionosphere
periods—confirming that ionospheric activity acts as a decohering
screen on an underlying ground-clock correlation rather than its source
(~78% ionosphere-independent baseline), consistent with shorter
correlation lengths in the disturbed auroral band and local-night
persistence bounding any residual static ionospheric contribution at
≲3% of the coherent signal.

The investigation was structured to test predictions from the Temporal
Equivalence Principle (TEP) framework, which suggested a Temporal
Topology correlation length (λ<sub>T</sub>) of 1,000–10,000 km. The
discriminating content is structural rather than the decade-wide prior
interval itself: the pooled-band estimate λ<sub>T</sub> = 3,330–4,549 km
spans the geometric saturation radius R<sub>T</sub>(M<sub>⊕</sub>) ≈
4,150 km, which is calibrated on this scale; the correlation is band-localized (control-band R² ≈ 0.6 vs
≈0.95 in the coherent bands), and the per-band decomposition is ordered
(tidal-band λ ≈ 3,600–5,900 km declining through post-tidal
≈2,100–2,500 km to ≈1,050–1,450 km at intermediate frequencies, with the
exponential fit degrading where coherence collapses). Forward
simulation of the satellite-visibility and network-datum channel on
the real station geometry bounds its effective decay scale at λ ≳
6.7×10<sup>3</sup> km; conditioned on the real product — true CODE
ephemerides and the estimated satellite-clock content itself — the
channel remains broader than every measured λ<sub>T</sub>
(no realisation below λ = 5,075 km across 40 conditioned
realisations)
and produces a profile shape (near-unity correlation within 1 Mm,
sign-definite anticorrelation beyond ~5 Mm) that the measured
correlation does not contain (§4.6). The measured terrestrial covariance scale defines the corpus's empirical saturation anchor ρ<sub>T</sub> ≈ 20 g/cm³ (R<sub>T</sub>(M<sub>⊕</sub>) ≈ 4,150 km), which is then propagated without retuning across galactic and stellar regimes in Papers 6 and 11. Across products the fitted scale is conditioned by
clock-product construction and estimator band floor: the multi-GNSS
MGEX family returns λ = 1,862 ± 155 km, with controlled decomposition
on identical station-days bounding spectral-resolution and
common-mode contributions at ≲±25% while reproducing the same
band-ordered profile and accounting for cross-product phase-index
conditioning (Paper 14). The
R<sub>T</sub> comparison is therefore stated at the cross-product
family level (~1.0–4.8×10<sup>3</sup> km). Cross-pipeline consistency
across software-diverse solutions, raw-RINEX pseudorange/SPP recovery
(Paper 3), combined multi-GNSS product validation on a largely held-out
2025–2026 epoch (Paper 14), and the 25-year longitudinal recovery
(Paper 2) establish the correlation structure across independent data
reductions, observation epochs, and estimator architectures. Raw
carrier-phase processing and non-GNSS optical-fiber and two-way
satellite time-transfer (TWSTFT) links provide direct tests of the
unsuppressed amplitude and baseline scaling across independent timing
technologies. A companion
25-year longitudinal CODE analysis is presented at
TEP-GNSS-II.

The empirically derived correlation length $\lambda_T$ is a GNSS-sector covariance scale obtained after environmental projection and processing transfer. It is not the screening operator $\mathcal{S}_\Sigma(\mathcal{E})$ itself, and it is not the non-exact (disformal) covariance term, which this measurement does not access.

## Executive Summary

### Key Findings

Primary Finding: Analysis of 62.73 million station pair measurements
reveals exponential correlation decay patterns with characteristic
length λ<sub>T</sub> = 3,330–4,549 km across three distinct analysis center solutions
(CODE, IGS Combined, ESA Final).

**Statistical Interpretation:** Primary fit significance
is not inferred from 62.7M independent pairs. The pair count provides
geometric coverage; inferential degrees of freedom are governed by bin
count (N<sub>eff</sub> ≈ 25–28), station covariance, and block-bootstrap
structure.

This robust and statistically significant—at the distance-binned,
block-bootstrap level, not as raw-pair significance—empirical
phenomenon represents systematic distance-structured correlations in
clock frequency residuals. Comprehensive multiband frequency analysis
(10–3000 μHz) across three distinct analysis center solutions shows
remarkable consistency in signal detection, with exponential models
achieving high goodness-of-fit to distance-binned data (R² up to
0.970, pooled fit on distance-bin means; ~28 bins) across optimal
frequency bands. This cross-solution validation represents a crucial
milestone, ruling out software- and center-specific implementation artifacts
through diverse processing methodologies (Bernese, NAPEOS, and the IGS combination algorithm). These patterns are consistent with—but do not yet
constitute proof of—screened scalar-field frameworks via continuous
Temporal Topology that predict broadband coupling suppressed at
large distances.

The correlations demonstrate sensitivity to Earth's orbital motion,
with center-specific statistics (per §3.3.1): CODE r = −0.701, p =
3.9×10⁻⁶; ESA Final r = −0.571, p = 4.2×10⁻⁴; IGS Combined r =
−0.793, p = 2.2×10⁻⁸; planetary gravitational influences (6
Bonferroni-significant astronomical events detected with 3–6σ
confidence), and systematic environmental dependencies including
elevation and geomagnetic latitude effects. The 6 significant
planetary detections represent outcomes from a systematic analysis
of 8 astronomical event windows, with conservative Bonferroni
correction applied across this complete set of tests. A detailed
diurnal analysis (Local Solar Time) reveals systematic temporal
variations exhibiting synchronized early morning coherence peaks and
seasonal modulation showing maximum spring equinox effects.
Gravitational energy hierarchy analysis reveals sophisticated
coupling mechanisms: Earth's 433-day polar wobble (Chandler wobble)
exhibits stronger detection signatures (R² = 0.377–0.471, |r| =
0.61–0.69) than expected from its mechanical energy scale (note:
this energetic comparison is heuristic—the detection is in
correlation space on binned means), potentially indicating
Moon-Chandler gravitational field modulation that amplifies coupling
through time-varying Earth-Moon gravitational geometry.

### Validation Framework

Extensive validation provides evidence for signal authenticity
through:

Cross-center validation: Three distinct analysis center solutions
(CODE, IGS Combined, ESA Final) achieve nearly identical multiband patterns, with
distance-binned fits reaching R² up to 0.970 (pooled fit on
distance-bin means; ~28 bins) (ESA Final), 0.920 (CODE), and
0.966 (IGS Combined), demonstrating that the spatial decay is not
an artifact of an individual processing software implementation

Frequency specificity analysis: Multiband analysis reveals
systematic frequency-dependent patterns with enhanced signals at
tidal frequencies, persistent post-tidal correlations (30–40
μHz: R² = 0.946, pooled fit on distance-bin means; mean across
centers), and appropriately reduced control band performance
(1000–1500 μHz: R² = 0.618, pooled fit on distance-bin means;
mean across centers)

Statistical robustness: Distance-binned fits R² = 0.920–0.970 on
~28 bins vs. null mean 0.015–0.040 (ΔR² = 0.89–0.95; z =
15.8–31.9, all p $\lt$ 10⁻¹⁵; 24–61× signal-to-null ratios across 180
scrambling iterations), with comprehensive multiple comparison
corrections across 388 statistical tests showing 52% FDR-BH
survival (40% under ultra-conservative Bonferroni correction)

Rigorous uncertainty quantification: Dual bootstrap approaches
provide comprehensive validation — (1) 5000-iteration block
bootstrap with 71–73 % convergence validates correlation
parameters without bias, yielding λ 95 % CI
ranges CODE 1,198–5,918 km,
ESA 2,532–3,984 km, IGS 3,197–4,871 km; (2)
station-block bootstrap with 30 % removal confirms
robustness to network topology.

Data integrity: Daily Kp indices (226 records) were used without
up-sampling; data gaps are reported rather than filled.

Alternative analysis: Systematic investigation addresses
classical tidal contamination and potential methodological
artifacts. Ionospheric validation using real geomagnetic data
shows weak correlations (Kp: r = 0.12-0.13, p > 0.29;
autocorrelation-corrected per Section 2.3) but comprehensive
solar activity assessment requires additional real daily data
sources

Cross-validation: Leave-one-station-out and leave-one-day-out
analyses suggest temporal and spatial robustness

### Theoretical Implications

The observed patterns are consistent with the Temporal Equivalence Principle's prediction of gravitational field coupling to atomic clock transport. The primary empirical quantity is the
covariance length λ<sub>T</sub> = 3,330–4,549 km, which defines the terrestrial saturation anchor R<sub>T</sub>(M<sub>⊕</sub>) ≈ 4,150 km (ρ<sub>T</sub> ≈ 20 g/cm³; Paper 6, UCD)
within the ~1.0–4.8×10<sup>3</sup> km cross-product family (Paper 14, Step 3.5).
Precision tests constrain the local shear response, which is pairwise-screened where the ambient time-gradient is steep (the $\mathcal{S}_\Sigma(\mathcal{E})$ gradient branch; weak-field: $g \gg g_t$); GNSS measures a distinct clock-covariance projection rather than a microscopic Compton wavelength. The diurnal analysis identifies synchronized daily and seasonal modulations consistent with dynamical proper-time structure; raw-RINEX/SPP consistency and combined multi-GNSS product tests are established in Papers 3 and 14, while raw carrier-phase analysis and separately processed per-constellation tests provide the next discriminating checks.

Together with the 25-year longitudinal recovery (Paper 2), the raw-RINEX pseudorange confirmation (Paper 3), and the multi-constellation MGEX replication (Paper 14), these findings establish a persistent, distance-structured covariance in global atomic clock networks consistent with the Temporal Equivalence Principle, with direct implications for precision time-transfer metrology and dynamical-time models. Raw carrier-phase processing and non-GNSS optical-fiber and two-way satellite time-transfer comparisons provide direct tests of the unsuppressed coupling amplitude across independent timing architectures.

### Next-Stage Experimental Discriminators

The completed three-center, 25-year, raw-RINEX, and MGEX analyses define a concrete set of out-of-ecosystem discriminators:

Independent replication by other research groups using different
methodologies and analysis approaches

Raw carrier-phase GNSS analysis (building on the raw-RINEX pseudorange/SPP validation completed in Paper 3) to quantify
processing effects and reveal full signal characteristics

Separately processed per-constellation testing across GLONASS, Galileo, and BeiDou (extending the combined MGEX multi-constellation confirmation in Paper 14)
for technology-independent confirmation

Systematic investigation of remaining alternative hypotheses
using independent datasets and complementary approaches

Extension to optical clocks for enhanced precision and different
systematic dependencies

### Conclusion

The convergence of multiple independent observational
domains—spatial correlations, spectral characterization, Earth
motion coupling, and gravitational correlations—combined with
validation across software-diverse processing chains, establishes robust
statistical patterns in global GNSS networks that warrant
comprehensive investigation. These findings provide compelling
empirical evidence for systematic distance-structured correlations
in atomic clock networks, with cross-center, 25-year (Paper 2),
raw-RINEX (Paper 3), and multi-constellation (Paper 14) validation
demonstrating that the decay is an intrinsic feature of the
clock-network observations rather than a pipeline artifact. These
results support environment-dependent proper-time transport as
predicted by the Temporal Equivalence Principle and define
quantitative targets for raw carrier-phase and optical-link
experiments.

<sup>1</sup> Primary correlation length range (3,330–4,549 km) with
bootstrap validation ranges: 1,198–5,918 km (CODE), 2,532–3,984 km
(ESA Final), 3,197–4,871 km (IGS Combined). Bootstrap analysis with
5000 iterations achieves 71–73% success rates. Environmental
dependencies show elevation stratification (1,600–7,500 km) and
geomagnetic effects (1,300–15,000 km).

<sup>2</sup> Coefficient of determination (R²) computed on
distance-bin means (effective number of samples, N<sub>eff</sub> ≈
25–28), not on the full dataset of 62.73 million pairs. This
distinction is crucial for proper statistical interpretation—the
high R² values reflect fits to binned means, not individual
measurements.

## 1. Introduction

## 1.1 The Temporal Equivalence Principle

The Temporal Equivalence Principle (TEP)[1](#note1)
represents a fundamental extension to Einstein's General Relativity,
proposing that gravitational fields couple directly to atomic transition
frequencies through a conformal rescaling of the matter metric. This framework
builds upon extensive theoretical work in scalar-tensor gravity (Damour
& Polyakov 1994; Damour & Nordtvedt 1993) and varying constants
theories (Barrow & Magueijo 1999; Uzan 2003). The coupling, if present,
would manifest as correlated fluctuations in atomic clock frequencies across
spatially separated precision timing networks, with correlation structure
determined by the continuous spatial profile of the underlying scalar
field—termed Temporal Topology in the TEP framework—where the kinetic
operator suppresses the local Temporal Shear where the ambient time-gradient
is steep — in weak-field language, high-acceleration regions — while leaving
the field light cosmologically (Smawfield 2025, Section 7).

Theoretical Motivation: TEP addresses a fundamental conceptual problem that
has persisted since the development of quantum mechanics and general
relativity: the disparate treatment of time. In GR, proper time is geometric
and dynamical—dτ² = −g<sub>μν</sub> dx<sup>μ</sup> dx<sup>ν</sup>/c²—while
in quantum mechanics, time serves as an external parameter in the
Schrödinger equation iℏ ∂<sub>t</sub>|ψ⟩ = H|ψ⟩. This fundamental
inconsistency manifests operationally in subtle ambiguities regarding
simultaneity and one-way light speeds across extended regions. TEP resolves
this by elevating "when" to the same dynamical status that "where" acquired
in 1915: just as space was geometrized, the rate of time's flow becomes a
field. This provides a covariant framework where local Lorentz invariance
remains exact while global synchronization can become non-integrable when
the non-exact sector is active, making previously untestable aspects of
spacetime geometry accessible to precision measurement.

Form and Justification of the Conformal Coupling: The TEP framework posits a
conformal factor $A(\phi) = \exp(\beta_A\phi/M_{\text{Pl}})$, with matter metric $\tilde{g}_{\mu\nu} = A^2(\phi)g_{\mu\nu}$, that rescales
the matter metric, where $\phi$ is a scalar field, $\beta_A$ is a
bare conformal coupling in the foundational TEP convention, and $M_{\text{Pl}}$ is the Planck mass.
This specific exponential form arises from three fundamental requirements:
(1) *Dimensional consistency*—β<sub>A</sub>/M<sub>Pl</sub> provides the unique
dimensionless coupling strength linking a scalar field ϕ to spacetime
geometry; (2) *Positivity preservation*—the exponential ensures A(ϕ)
> 0 always, maintaining the Lorentzian signature essential for causality;
and (3) *Observational constraints*—this form naturally accommodates
Parametrized Post-Newtonian (PPN) bounds through the screened-source relation
$\gamma_{\text{PPN}} - 1 = -2\beta_A^2 S_\Sigma^{(\odot)} / (1 + 2\beta_A^2 S_\Sigma^{(\odot)})$ under the unified unstretched-space metric ($\gamma_{\text{PPN}} - 1 = -4\beta_A^2 S_\Sigma^{(\odot)} / (1 + 2\beta_A^2 S_\Sigma^{(\odot)})$ on the isotropic-conformal benchmark),
where $\beta_A = -1.0$ universally. In high-acceleration regions near massive bodies, the two-branch kinetic
screening operator suppresses the active source charge to $y_{\rm tb}(1.6\,R_\odot)=2.16\times 10^{-8}$ ($|\gamma_{\text{PPN}}-1|=4.32\times 10^{-8}\ll 2.3\times 10^{-5}$, well inside the Cassini ceiling $S_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ [$5.8 \times 10^{-6}$ isotropic]),
cleanly preserving PPN and Cassini bounds without invoking rigid thin-shell
approximations (Paper 0 §7). The universality of the
coupling—all matter sees the same unified causal metric $\tilde g_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ (reducing to $\tilde g_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$ in the conformal-clock projection)—preserves the equivalence principle in the matter frame
while allowing for testable violations in the gravitational sector. In the conformal-dominant GNSS covariance sector considered here, the leading clock response is carried by A(ϕ). In this
modified spacetime, proper time transforms as dτ̃ = A(ϕ)dτ<sub>g</sub>. In the weak-field limit, atomic transition frequencies acquire a
fractional frequency shift:

\begin{equation} \label{eq:freq_shift}
y \equiv \frac{\Delta\nu}{\nu} \approx \frac{\beta_A}{M_{\text{Pl}}}\phi
\end{equation}

For a screened scalar field with exponential correlation function
$\text{Cov}[\phi(\mathbf{x}), \phi(\mathbf{x}+\mathbf{r})] \propto
\exp(-r/\lambda_{\mathrm{T}})$, the observable clock frequency correlations inherit the
same characteristic Temporal Topology length $\lambda_{\mathrm{T}}$.

Connection to Modified Gravity Theories: TEP extends established
scalar-tensor theories of gravity, including Brans-Dicke theory (ω
approaches ∞ limit), f(R) gravity (scalar degree of freedom), and
Horndeski/Galileon theories (screening mechanisms). The framework predicts
that a detectable correlation length corresponds to the geometric saturation
radius R<sub>T</sub> = (3M/4πρ<sub>T</sub>)<sup>1/3</sup>, with the
continuous spatial profile of the field (Temporal Topology) potentially
producing correlation lengths in the 1,000–10,000 km range. In TEP, GNSS clocks
read the value of the temporal field $\ln A$ at each station, and the measured
covariance describes how that value varies between stations; the screening that
keeps local trajectories GR-like acts on the field's effect on matter, not on clock rates (Paper 0 §7).
Rather than operating via discrete boundary
cutoffs, screening manifests as a continuous geometric profile governed by
the non-linear superposition of field gradients (Smawfield 2025, Axiom A4).
The primary empirical quantity is the covariance length λ<sub>T</sub> extracted
from the clock-correlation data; the IGS-final estimates define the terrestrial saturation anchor R<sub>T</sub>(M<sub>⊕</sub>) ≈ 4,150 km (ρ<sub>T</sub> ≈ 20 g/cm³), while the cross-product family spans ~1.0–4.8×10<sup>3</sup> km under documented estimator and product conditioning (Paper 6, UCD; Paper 14, Step 3.5). The same numerical length corresponds formally to a Compton scale of order 10⁻¹⁴ eV for comparison with older screened-scalar literature, but TEP does not identify the measured GNSS covariance length with a microscopic Compton wavelength.

Geometric Saturation Scale: The covariance length λ<sub>T</sub> = 3,330–4,549 km anchors the geometric saturation radius R<sub>T</sub>(M<sub>⊕</sub>) = (3M<sub>⊕</sub>/4πρ<sub>T</sub>)<sup>1/3</sup> ≈ 4,150 km (Paper 6, UCD) at ρ<sub>T</sub> ≈ 20 g/cm³, which is then held fixed across galactic and stellar regimes; across products the fitted scale is conditioned by clock-product construction and spectral-estimator settings, spanning ~1.0–4.8×10<sup>3</sup> km with the same band-ordered profile (MGEX 1,862 ± 155 km; Paper 14, Step 3.5). The M<sup>1/3</sup> scaling belongs to the empirical geometric saturation construction R<sub>T</sub>; the derived kinetic Temporal-Shear transition follows r<sub>*</sub> ∝ M<sup>1/2</sup>.

Theoretical Context: TEP builds upon a two-metric framework in which matter
couples to the causal metric g̃<sub>μν</sub> = A²(ϕ)g<sub>μν</sub> + B(ϕ)∇<sub>μ</sub>ϕ∇<sub>ν</sub>ϕ, while
gravity is described by the standard metric g<sub>μν</sub>. The present GNSS correlation
analysis primarily probes the conformal covariance sector. GNSS correlation
analysis would probe the spatial covariance structure of the underlying ϕ
field, complementing direct closed-loop tests of the non-exact
synchronization sector. This positions GNSS analysis as part of a broader
experimental program testing dynamical time theories.

**Historical Context and Prediction Timeline:** The TEP
theoretical framework was developed independently of the present GNSS
analysis, with pre-specified expectations and theoretical search ranges established in prior
theoretical work (Smawfield, 2025;
DOI: 10.5281/zenodo.16921911). The predicted Temporal Topology correlation length range λ<sub>T</sub> = 1,000–10,000 km was
specified as a theoretically motivated prior search range before the present GNSS
analysis. The observed results (λ<sub>T</sub> = 3,330–4,549 km) represent
consistency with the pre-specified theoretically motivated search range; the discriminating content is carried not by the decade-wide interval— which subsumes the leading systematic scales—but by the structural conditions detailed in §2 (consistency with R<sub>T</sub>(M<sub>⊕</sub>) ≈ 4,150 km within the ~1.0–4.8×10<sup>3</sup> km cross-product family, band localization, cross-center reproducibility, and separability from the co-visibility kernel). Following this primary consistency check, additional investigations of derived predictions were conducted.

**Manuscript Series Context:** This paper reports the initial multi-centre validation using 2-year data from CODE, IGS, and ESA. Two companion manuscripts extend this foundation: Paper 2 (TEP-GNSS-II) presents a 25-year longitudinal analysis using CODE data spanning 1999–2025, and Paper 3 (TEP-GNSS-RINEX) tests the signal in raw RINEX pseudorange observations processed independently with SPP and broadcast ephemerides. This separation preserves methodological clarity—each manuscript addresses distinct validation strategies while building a convergent evidence base.

## 1.2 Testable Predictions

The TEP theory establishes testable expectations with
current technology:

**Experimental Section:**

### Key Theoretical Predictions and Observational Confirmations

| Prediction | Theoretical Basis | Observational Confirmation |
| --- | --- | --- |
| **Exponential Decay** | Screened scalar field coupling to atomic frequencies. | **Observed:** Exponential models show optimal
fit (Sec 3.1.2) with R² = 0.92–0.97 across all centers. |
| **Temporal Topology Correlation Length (λT)** | Temporal Topology covariance scale; the IGS-final estimates
are consistent with the geometric saturation radius R_T(M_⊕) ≈ 4,150 km
(Paper 6; calibrated on this scale) within a ~1.0–4.8×10³ km cross-product family
(Paper 14, Step 3.5).
Specified range: 1,000–10,000 km (prior search interval;
non-discriminating alone). Structural criterion: pooled-band
λT is consistent with RT(M⊕) and is separable
from the co-visibility kernel (§2). | **Observed:** λT = 3,330–4,549 km, falling
within the pre-specified search range (Sec 3.1.1). |
| **Universal Coupling** | Universal conformal coupling should produce broadband
effects, not frequency-selective ones. | **Observed:** Signal persists across 12
frequency bands with smooth spectral rolloff, inconsistent
with frequency-selective tidal artifacts (Sec 3.5.1). |
| **Multi-Center Consistency** | A persistent physical effect should not depend sensitively on a single
processing methodology. | **Observed:** Three distinct processing chains (CODE,
IGS Combined, ESA Final) converge on the same physical parameters
(CV of λT ≈ 13%), ruling out software- and
center-specific implementation artifacts (Sec 3.1.1, 4.1). |
| **Falsification Criteria** | λ $\lt$ 500 km or λ > 20,000 km would rule out model. CV
> 20% would indicate artifacts. | **Passed.** The observed λ and CV fall within
the passing criteria, successfully surviving the
pre-specified falsification tests. |

The 25-year baseline enables detection of long-period signatures inaccessible
in the shorter Paper 1 analysis; the seven signatures are reported together
because they all arise from the same dataset and their collective pattern
constitutes the discriminating evidence. Individual papers for each signature
would fragment the statistical argument.

## 1.3 Why GNSS Provides an Ideal Test

Global Navigation Satellite System (GNSS) networks offer unique advantages
for testing TEP predictions, building on decades of precision timing
developments (Kouba & Héroux 2001; Senior et al. 2008; Montenbruck et
al. 2017):

Global coverage: Analysis of 364 unique stations (249 N / 115 S = 68.4%
/ 31.6%), sourced from three overlapping networks, provides
comprehensive global sampling.

Continuous monitoring: High-cadence (30-second) measurements over
multi-year timescales

Multiple analysis centers: Independent data processing by CODE, ESA, and
IGS Combined enables cross-validation

Precision timing: Clock stability sufficient to detect predicted
fractional frequency shifts

Public data availability: Open access to authoritative clock products
enables reproducible science

### Notes

**Note:** The acronym "TEP" has been used in various
contexts in the physics literature, including "Thermal Equivalence
Principle" (thermodynamics), "Test of Equivalence Principle"
(general relativity experiments), "Temporal Evolution Principle"
(quantum mechanics), and "Total Electron Content/Path" (GNSS
ionospheric analysis). The present work introduces "Temporal
Equivalence Principle" as a distinct theoretical framework for
dynamical time geometry. To avoid confusion, the full term "Temporal
Equivalence Principle (TEP)" is consistently used on first reference
and clear distinction from these prior uses is maintained throughout
the manuscript.

## 2. Methods

### Conceptual Overview: What This Study Is Looking For and How

### The Basic Idea

Imagine you have hundreds of ultra-precise clocks scattered around the world. If time flows uniformly everywhere, these clocks should tick independently—what happens at one clock shouldn't affect any other. But if there's a subtle "field" that influences how time flows (like the Temporal Equivalence Principle predicts), then clocks might show coordinated patterns in their tiny timing variations.

The approach: This study analyzes whether the tiny fluctuations in GPS atomic clocks show patterns that depend on how far apart the clocks are. If closer clocks fluctuate more similarly than distant ones, and this pattern has a specific distance scale, that would be evidence for a field affecting time.

### Why GPS Clocks?

GPS satellites need incredibly precise timing—errors of even a billionth of a second would make navigation useless. This means:

- Extreme precision: GPS clocks are stable to better than one part in a trillion

- Global coverage: ~360 unique stations provide comprehensive spatial sampling with strong Northern Hemisphere representation

- Continuous monitoring: Data collected every 30 seconds for years

- Multiple independent systems: Three different analysis centers process the data independently

- Public data: All data is freely available, enabling independent verification

### The Challenge: Finding a Needle in a Haystack

GPS clocks have many sources of variation—temperature changes, aging, radiation, atmospheric effects. The TEP signal being sought is tiny compared to these effects. Standard analysis methods average out or remove correlated signals (which is exactly what this study is looking for!). A special approach is needed.

The solution: Instead of looking at the strength of clock variations, this study looks at their *phase relationships*—whether clocks oscillate in sync or out of sync. This phase information survives the standard GPS processing that removes amplitude information.

### The Four-Step Process

**Step 1: Data Collection**
Download official GPS clock products from three analysis centers (CODE, IGS Combined, ESA Final). Validate that all station coordinates are correct and data quality is high.

**Step 2: Pattern Detection**
For every pair of stations, measure how synchronized their clock fluctuations are. Plot this synchronization versus distance to see if there's a pattern.

**Step 3: Validation**
Apply rigorous statistical tests to ensure the patterns are real, not artifacts. Test whether randomizing the data destroys the pattern (it does—indicating it is real).

**Step 4: Advanced Analysis**
Investigate how patterns change with Earth's motion, planetary positions, time of day, and season. Create visualizations and test theoretical predictions.

### What This Study Is Measuring: Phase Coherence

Think of each clock's fluctuations as a sound waveform. When comparing two clocks, one can ask: "Are these waves in phase (peaks align with peaks) or out of phase (peaks align with troughs)?"

Key insight: If a field is influencing both clocks, their waves should tend to be in phase more often than random chance would predict. The strength of this tendency should decrease with distance according to the field's properties.

*Technical note:* The cosine of the phase angle from cross-spectral density analysis is used. This metric is amplitude-invariant (doesn't depend on signal strength) and survives GPS processing that removes amplitude information. See Section 2.3 for mathematical details.

#### Conceptual Framework: Magnitude-Weighted Phase Correlation Method

##### (a) Time Series Input

Two atomic clock residual signals: x<sub>i</sub>(t), x<sub>j</sub>(t)

Signal + Noise

∿∿∿ Correlated Component

∵∵∵ Random Noise

##### (b) Cross-Spectral Analysis

Complex CSD → Phasor representation

Strong Signal: |CSD|↑, φ clustered

Weak Noise: |CSD|↓, φ random

Quality = Magnitude

##### (c) Weighted Averaging

Phase-alignment index: cos(φ<sub>weighted</sub>)

High |CSD| → High weight

Clustered phases dominate

Amplitude-invariant result

**Key Innovation:** Magnitude weighting preserves phase relationships that survive GNSS processing amplitude suppression, enabling detection of spatially correlated TEP signals that standard coherency analysis cannot detect.

### Why This Approach Works

- Survives processing: GPS centers remove amplitude correlations but preserve phase relationships

- Robust to noise: Random noise averages out when looking at millions of measurements

- Testable predictions: Theory predicts specific distance scales and patterns that can be checked

- Multiple confirmations: Three independent data sources provide cross-validation

- Falsifiable: If distances or phases are randomized, patterns should disappear (they do)

### Research Synopsis

This study measures whether GPS atomic clocks around the world show coordinated timing patterns that depend on distance. The methodology tests for exponential decay patterns in distance-structured correlations and assesses whether such patterns demonstrate cross-center consistency, survive rigorous validation tests, and align with theoretical predictions. Observed patterns and their statistical significance are reported in Section 3. The detailed technical methods below explain exactly how this analysis is accomplished.

## 2.1 Analysis Pipeline Overview

The TEP-GNSS analysis follows a systematic four-step pipeline designed to ensure rigorous validation and reproducibility. Each step builds upon previous results while maintaining independent validation criteria. The complete analysis pipeline is available at [github.com/matthewsmawfield/TEP-GNSS](https://github.com/matthewsmawfield/TEP-GNSS).

**Experimental Section:**

### Four-Step Analysis Pipeline

#### Step 1: Data Acquisition

- 1.0 Provenance Documentation: Establishes computational provenance with version tracking and execution logging

- 1.1 TEP Data Acquisition: Acquires GNSS clock data from three analysis center solutions (CODE, IGS Combined, ESA Final) covering 1 January 2023 to 30 June 2025

- 1.2 Coordinate Validation: Validates station coordinates against ITRF2014 with ECEF validation and spatial analysis

Establishes computational provenance and acquires GNSS clock data from three analysis center solutions with rigorous quality controls.

#### Step 2: Core Analysis

- 2.0 TEP Correlation Analysis: Implements phase-coherent cross-spectral density analysis in the 10–500 μHz band, extracting phase-alignment index correlations and fitting exponential decay models

- 2.1 Aggregate Geospatial Data: Consolidates station pair correlations with geographic metadata and distance binning

- 2.2 Geospatial Temporal Analysis: Analyzes correlations with astronomical events, orbital tracking, anisotropy patterns, and temporal field dynamics

Detects exponential correlation decay patterns using phase-coherent cross-spectral density analysis.

#### Step 3: Validation Suite

- 3.0 Cross-Validation Suite: Block-wise validation (monthly/spatial), Leave-One-Station-Out (LOSO), Leave-One-Day-Out (LODO), and block bootstrap analyses

- 3.1 Robust Block Bootstrap: Bootstrap confidence intervals with 5000 iterations

- 3.2 TEP Null Tests: Null hypothesis testing through three independent scrambling approaches: distance scrambling (20 iterations per center), phase scrambling (20 iterations per center), and station scrambling (20 iterations per center), with statistical significance assessment via permutation testing and z-score analysis. Distance scrambling randomizes both distances and coherences independently to destroy distance-coherence relationships.

- 3.3 Methodology Validation: Bias characterization and geometric artifact detection

- 3.4 Geographic Bias Validation: Spatial distribution testing and hemisphere-based validation

- 3.5 Realistic Ionospheric Validation: TID exclusion analysis with Hilbert transform filtering

- 3.6 Control Band Analysis: Out-of-band frequency validation for systematic effects

- 3.7 Bootstrap Convergence Validation: Comprehensive assessment of bootstrap uncertainty quantification, addressing convergence rates and bias validation (see Appendix 7.2)

A comprehensive validation suite including rigorous model comparison (seven correlation functions tested via AIC/BIC), cross-validation, null tests, bias characterization, and comprehensive multiple comparison corrections across 388 statistical tests. Results demonstrate systematic preference for exponential family models, with Matérn(ν=1.5) achieving optimal AIC performance.

#### Step 4: Advanced Analysis and Visualization

- 4.0 TEP Advanced Analysis: Model comparison (7 models tested), circular statistics, and advanced correlation metrics

- 4.1 TEP Visualization: Publication-quality figures for correlation patterns and multi-center consistency

- 4.2 Synthesis Figure: Comprehensive synthesis of observational evidence

- 4.3 High-Resolution Astronomical Events: Eclipse and supermoon coherence modulation analysis

- 4.4 Gravitational Temporal Field Analysis: Multi-window planetary gravitational correlation analysis with timescale strengthening

- 4.5 Detailed Diurnal Analysis: Hourly temporal analysis with seasonal pattern characterization

- 4.8 Multi-Band Visualization: Publication-quality figures for spectral analysis, frequency specificity assessment, and gravitational enhancement patterns across 12 frequency bands

Advanced analyses including multi-band spectral characterization, eclipse effects, gravitational correlations, and detailed visualizations.

#### Pipeline Validation Framework

- Multi-Center Consistency: All analyses performed across three GNSS analysis centers running different software (CODE, IGS Combined, ESA) on the shared tracking network, with systematic cross-validation to assess coefficient of variation

- Statistical Rigor: Null testing through data scrambling, cross-validation methods, and multiple comparison corrections applied throughout the pipeline

- Reproducibility: Complete computational provenance tracking, version control, execution logging, and open-source analysis scripts ensure full reproducibility

Implementation Details: For complete usage instructions, configuration options, runtime estimates, and step-by-step execution guide for all 23 pipeline steps, see Section 6 (Analysis Package) at the end of this document. The complete source code and documentation are available at [github.com/matthewsmawfield/TEP-GNSS](https://github.com/matthewsmawfield/TEP-GNSS).

## 2.2 Data Architecture

The analysis employs a rigorous three-way validation approach using independent clock products from major analysis centers. To ensure cross-validation integrity, the analysis is restricted to the common temporal overlap period (1 January 2023 to 30 June 2025) when all three centers have available data:

**Experimental Section:**

### Authoritative data sources

- Station coordinates: International Terrestrial Reference Frame 2014 (ITRF2014) via IGS JSON API and BKG services, with mandatory ECEF validation

- Clock products: Official .CLK files from CODE (AIUB FTP), ESA (navigation-office repositories), and IGS (BKG root FTP)

- Quality assurance: Hard-fail policy on missing sources; zero tolerance for synthetic, fallback, or interpolated data

Note on processing bias: Official IGS and CODE clock products are generated via least-squares / Kalman estimation that minimises per-station residual variance (Kouba & Héroux 2001; Steigenberger et al. 2021; IGS Technical Report 2023). This algorithmic objective suppresses any network-wide amplitude term, so absolute modulation depths in processed products are lower-bound estimates, whereas *phase-coherent* structure is largely preserved.

### Dataset characteristics

- Data type: Ground station atomic clock correlations

Temporal coverage: 1 January 2023 to 30 June 2025 (912 days) with date filtering applied

- IGS Combined: 912 files processed (complete coverage)

- CODE: 912 files processed (complete coverage)

- ESA: 912 files processed (complete coverage)

- Spatial coverage: The analysis is based on data from 364 unique GNSS stations (249 N / 115 S = 68.4% / 31.6%, ratio = 2.17). These stations are sourced from the overlapping networks of three analysis centers: CODE (322 stations), IGS Combined (278 stations), and ESA Final (201 stations). After combining the lists from these centers and filtering for stations with continuous data in the analysis period, the final set of 364 unique stations is obtained. For reference, the master station catalog contains 767 unique stations globally.

- Data volume: 62.73 million station pair cross-spectral measurements

- Analysis centers: CODE (39.0M pairs), ESA (10.8M pairs), IGS Combined (12.9M pairs). Station pair counts vary across centers due to different station network sizes and center-specific data availability and quality criteria.

*File counts reflect actual processed files within the 912-day analysis window (1 January 2023 to 30 June 2025) after date filtering.*

### Data Quality Assurance

A thorough quality validation across 62.73 million station pair measurements demonstrates strong data integrity essential for phase-coherent analysis:

- Filtering efficiency: 99.958% overall data retention across all centers (CODE: 99.944%, IGS: 99.964%, ESA: No removals)

- Phase quality: Healthy boundary clustering (1.62–1.67% of values within ±0.05 rad distance to ±π under wrap) indicates proper phase wrapping without accumulation artifacts. The expected probability for uniform phase distribution is 0.1/(2π) = 1.59%, confirming the observed values are consistent with proper phase processing. (Note: near +π and near −π are the same region on the circle; these are reported separately but the combined fraction is evaluated against 0.1/(2π).)

- Temporal completeness: Files present all 912 days; per-day pairs vary by center

- Data integrity: Zero duplicate pairs, all 364 stations successfully matched to coordinates, zero distance outliers (post-filtering)

- Phase distribution: Uniform coverage across full ±π range enables unbiased phase-alignment index analysis

*The boundary clustering metric provides critical validation that phase wrapping is handled correctly—values should distribute uniformly across the ±π range with minimal accumulation at boundaries. The observed 1.62–1.67% boundary clustering matches the expected probability 0.1/(2π) = 1.59% for uniform distribution with ±0.05 rad distance to ±π under wrap, confirming proper phase processing across all analysis centers.*

### Distance Quality Filtering

Systematic distance outlier removal ensures geodesic calculation accuracy:

- CODE: 21,679 outliers removed ($\lt$ 1 km or >20,000 km) from 39,068,754 initial pairs (0.056%)

- IGS Combined: 4,633 outliers removed from 12,879,380 initial pairs (0.036%)

- ESA Final: 0 outliers removed from 10,809,085 initial pairs (0.000%)

*Post-filtering, all analysis centers show zero distance outliers, ensuring clean exponential decay fitting without contamination from geodesic calculation errors or data artifacts.*

### Temporal Data Density Patterns

Daily pair count variation across the 912-day analysis window differs between centers, reflecting their operational characteristics:

- CODE: CV of daily pair counts = 6.2% (highly consistent: 33,372-48,776 pairs/day)

- IGS Combined: CV of daily pair counts = 58.4% (variable: 990-29,618 pairs/day, reflects multi-center combination strategy)

- ESA Final: CV of daily pair counts = 14.1% (consistent: 10,731-16,290 pairs/day)

*The higher temporal variation in IGS Combined reflects its nature as a weighted combination of multiple analysis center solutions, with varying contributor availability over time. Despite this operational difference, IGS demonstrates correlation parameters consistent with CODE and ESA, confirming robustness to temporal sampling patterns across diverse processing architectures (see Section 3.1.1 for detailed results).*

**Figure 1a. Global GNSS Station Network:** Three-globe perspective showing global GNSS infrastructure with ~360 stations used in this analysis. High station overlap between the source analysis centers enables robust cross-validation.

**Figure 1b. GNSS Station Coverage Map:** Global distribution showing substantial overlap between analysis centers and geographic coverage essential for intercontinental correlation analysis.

**Figure 2. All Station Pair Distance Distribution:** Complete sampling across 0–15,000 km range with peak density at intercontinental scales (8,000–12,000 km), showing total available station pair coverage.

## 2.3 Phase-Coherent Analysis Method

### Distance Binning Methodology

The analysis employs logarithmic distance binning with 40 bins attempted spanning 50 to 13,000 km. Each bin requires a minimum of 1,000 station pairs for statistical reliability. Bins below this threshold are excluded from exponential model fitting, yielding an effective sample size of N<sub>eff</sub> ≈ 25–28 bins used for analysis across centers. This approach provides uniform spatial sampling in log-space while ensuring robust statistical power for exponential model fitting. The logarithmic binning strategy ensures equal representation across distance scales, critical for detecting exponential decay patterns that span multiple orders of magnitude in correlation strength.

#### Autocorrelation Correction for Smoothed Time Series

For analyses involving smoothed time series (e.g., gravitational-temporal correlations with long-window averaging), autocorrelation-robust statistical corrections are applied to prevent inflated significance. Following Bretherton et al. (1999), effective sample sizes are calculated as N<sub>eff</sub> = N × (1 - r₁ₓr₁ᵧ)/(1 + r₁ₓr₁ᵧ), where r₁ₓ and r₁ᵧ are first-order autocorrelations of the two series. This correction accounts for temporal dependence introduced by smoothing, with corrected p-values calculated using t-distribution with degrees of freedom df = N<sub>eff</sub> - 2. For heavily smoothed data (e.g., 227-day windows), typical corrections reduce effective sample sizes from N ≈ 900 to N<sub>eff</sub> ≈ 45-50, substantially modifying statistical inference.

**Note:** The ionospheric validation analysis (Kp correlations in Section 4.3.4) also applies this autocorrelation correction, resulting in corrected p > 0.29 compared to raw p = 0.052, demonstrating the importance of temporal dependence corrections for robust statistical inference.

### Key Terminology

- **Phase-alignment index:** The primary metric quantifying phase synchronization between station pairs, computed as the cosine of the magnitude-weighted circular mean phase from cross-spectral density analysis. Consistently denoted as "phase-alignment index" throughout this work, with technical implementation as cos(weighted_phase)

- **Technical implementation:** compute_cross_power_plateau() returns (avg_magnitude, weighted_phase) where magnitudes serve as weights in circular averaging: weighted_phase = np.angle(np.average(exp(i*phases), weights=magnitudes)). Final phase-alignment index = np.cos(weighted_phase)

- **Network covariance score (collective network dynamics, termed "mesh dance" in pipeline outputs):** Composite metric quantifying network-wide coherent dynamics (see Section 2.6 for detailed explanation)

Standard signal processing techniques using band-averaged coherency fail to detect TEP signals due to phase averaging effects and amplitude suppression from GNSS processing. A magnitude-weighted phase correlation approach was developed that uses both magnitude and phase information from complex cross-spectral density: magnitudes weight the circular averaging of phases, then the phase-alignment index is extracted as cos(weighted_phase).

**Experimental Section:**

### Core methodology

- Cross-spectral density computation: For each station pair (i, j), compute complex CSD from clock residual time series

- Magnitude-weighted phase correlation: Extract phase-alignment index as cos(weighted_phase) where phase is magnitude-weighted circular average

- Frequency band selection: Analyze 10–500 μHz (periods: 33 minutes–28 hours) where GNSS clock noise shows characteristic low-frequency behavior

- Dynamic sampling: Compute actual sampling rate from timestamps (no hardcoded assumptions)

### Why magnitude-weighted phase correlation works

The TEP signal manifests as correlated fluctuations with consistent phase relationships. Standard coherency normalizes by amplitude, but GNSS processing suppresses spatially correlated amplitudes precisely when phases are aligned, causing coherency to collapse toward zero even for genuine TEP signals.

#### Magnitude-weighting preserves signal quality

The approach uses both magnitude and phase information optimally:

- Magnitude weighting: Stronger frequency components (higher |CSD|) get more influence in determining the representative phase through weighted circular averaging

- Phase preservation: Circular statistics correctly handle phase wrapping while incorporating magnitude-based quality weighting

- Amplitude invariance: Final phase-alignment index cos(weighted_phase) is normalized, making it robust to amplitude suppression from GNSS processing

Key insight: Magnitudes are used as quality weights for phase calculation, then amplitude-invariant phase-alignment index is extracted from the weighted phase. This preserves both pieces of CSD information while being robust to processing artifacts.

**Experimental Section:**

#### In Practice: A Simplified Example

Imagine two stations, A and B. The complex cross-spectral density (CSD) of their clock signals in a specific frequency band is computed, which gives a series of complex numbers (phasors). Each phasor has a magnitude and a phase angle.

- If the signals are physically correlated, the phase angles will tend to cluster around 0. For example, angles like: `[0.1, -0.2, 0.05, 0.3, -0.15]` radians. The cosine of these angles are all close to 1: `[0.995, 0.980, 0.998, 0.955, 0.988]`. Their average is high (≈0.983), indicating strong phase alignment.

- If the signals are uncorrelated, the phase angles will be random: `[1.5, -2.8, 0.5, -0.9, 3.1]`. The cosines will be scattered between -1 and 1: `[0.07, -0.94, 0.87, 0.62, -0.99]`. Their average will be close to 0, indicating no preferred phase alignment.

GNSS processing might suppress the CSD *magnitudes*, but it largely preserves the *phase angles*. By averaging `cos(phase)`, the phase alignment is isolated, which is the core of the TEP signature, making the method robust to standard processing suppression.

### Physical interpretation of the phase-based approach

For two zero-mean, wide-sense stationary clock residual processes $x_i(t), x_j(t)$, the cross-spectrum $S_{ij}(f)$ is the Fourier transform of the cross-correlation $R_{ij}(\tau)$ (Wiener–Khinchin):

\begin{equation} \label{eq:cross_spectrum}
S_{ij}(f)=\mathcal{F}\{R_{ij}(\tau)\}, \quad R_{ij}(\tau)=\mathbb{E}[x_i(t)\,x_j(t+\tau)]
\end{equation}
Under TEP, each clock's fractional frequency $y_k(t)$ receives a common field contribution $y_k(t) \propto \phi(\mathbf{x}_k,t)$ plus local noise. The local noise term encompasses per-station oscillator characteristics—crystal aging, individual frequency offset, temperature sensitivity, and drive-level dependence—that are independent between stations. These contributions are uncorrelated across the network by construction: two quartz oscillators in the same room drift independently, and GNSS receiver clocks at different sites share no common oscillator heritage. Prior to cross-spectral analysis, each station's clock time series is linearly detrended to remove the dominant per-station systematic drift (frequency offset and linear aging rate), isolating the stochastic fluctuations in the 10–500 μHz band where TEP signatures are expected. Because uncorrelated per-station noise has no distance dependence by definition, it contributes to the noise floor but cannot generate the distance-structured exponential correlation that distinguishes the TEP signal. In the 10–500 μHz band, any propagation delay across baselines ($\leq 15{,}000$ km) is negligible relative to the periods (33 minutes–28 hours):

\begin{equation} \label{eq:phase_bound}
\phi_{\max}=2\pi f_{\max}\,\tau_{\max} \le 2\pi\,(5\times10^{-4}\,\mathrm{Hz})\;\frac{1.5\times10^7\,\mathrm{m}}{c}\approx1.6\times10^{-4}\ \mathrm{rad}
\end{equation}
Hence, the physically expected inter-station phase is $\approx 0$ in this band; the information lies in how tightly phases cluster, not in a systematic lag. Writing the unit phasor $U_{ij}(f)=S_{ij}(f)/|S_{ij}(f)|$, the metric uses $\mathrm{Re}\{U_{ij}(f)\}= \cos(\arg S_{ij}(f))$. When averaged over pairs within a distance bin, this estimates the circular mean of phases. If the within-bin phase distribution is von Mises $\mathrm{VM}(\mu\!\approx\!0, \kappa(r))$, the expected value is

\begin{equation} \label{eq:von_mises}
\mathbb{E}[\cos(\arg S_{ij})]=\frac{I_1(\kappa(r))}{I_0(\kappa(r))}\approx \tfrac{1}{2}\kappa(r)\quad(\kappa\ll1)
\end{equation}
If the underlying field has exponential spatial covariance, $\mathrm{Cov}[\phi(\mathbf{x}),\phi(\mathbf{x}+\mathbf{r})]\propto e^{-r/\lambda}$, then the concentration $\kappa(r)$ (and thus the circular mean above) inherits an exponential distance-decay, matching the fitted form.

This phase-only approach is robust to amplitude artifacts because it normalizes each $S_{ij}$ to unit magnitude before averaging (amplitude invariance). It distinguishes genuine spatial organization from mathematical artifacts through: (i) comprehensive randomization testing (distance, phase, and station scrambling), which destroys the spatial correlation structure in null tests while preserving it in genuine data; and (ii) replication across software-diverse processing chains (CODE, IGS, ESA) with different systematic vulnerabilities on the shared tracking network. Standard magnitude-based metrics ($|\mathrm{CSD}|$ or band-averaged real coherency) discard this directional information and therefore cannot detect the distance-structured phase relationships central to TEP.

## 2.4 Validation Framework Overview

To distinguish genuine physical phenomena from methodological artifacts, a full suite of validation testing is employed. Each potential source of bias or alternative explanation is directly tested with quantitative criteria. Detailed validation results demonstrating signal authenticity are presented in Section 4 following the observational findings.

### Frequency-Band Logic: Tidal vs Post-Tidal Analysis

The analysis employs 12 frequency bands spanning 10–3000 μHz to characterize the spectral properties of TEP correlations and distinguish between tidal and post-tidal effects. Each band is analyzed using identical phase-coherent methodology with N<sub>eff</sub> ≈ 25–28 bins used for analysis. The table below shows band edges, bandwidths, and statistical support (pairs/bin) for the IGS Combined analysis center, demonstrating uniform coverage across the frequency spectrum.

| Band | Frequency Range (μHz) | Bandwidth (μHz) | Period Range | Pairs per Bin (mean) | Category |
| --- | --- | --- | --- | --- | --- |
| TEP Band (10–500 μHz) | 10–500 | 490 | 33 min–28 h | 342,134 | Primary |
| Diurnal Tides | 10–20 | 10 | 14–28 h | 329,985 | Tidal |
| Semidiurnal Tides | 20–30 | 10 | 9–14 h | 324,279 | Tidal |
| Post-Tidal 30-40 | 30–40 | 10 | 7–9 h | 322,282 | Post-Tidal |
| Post-Tidal 40-50 | 40–50 | 10 | 5.6–7 h | 317,757 | Post-Tidal |
| Post-Tidal 50-75 | 50–75 | 25 | 3.7–5.6 h | 339,852 | Post-Tidal |
| Post-Tidal 75-100 | 75–100 | 25 | 2.8–3.7 h | 339,431 | Post-Tidal |
| Intermediate 100-200 | 100–200 | 100 | 1.4–2.8 h | 342,134 | Intermediate |
| Intermediate 200-350 | 200–350 | 150 | 48–80 min | 342,134 | Intermediate |
| Intermediate 350-500 | 350–500 | 150 | 33–48 min | 342,134 | Intermediate |
| Transition 500-750 | 500–750 | 250 | 22–33 min | 342,134 | Transition |
| Transition 750-1000 | 750–1000 | 250 | 17–22 min | 342,134 | Transition |
| Control 1000-1500 | 1000–1500 | 500 | 11–17 min | 342,114 | Control |

Statistical Support: All bands show robust statistical support with mean pairs per bin ranging from 317,757 to 342,134, ensuring reliable exponential model fitting. The N<sub>eff</sub> ≈ 25–28 bins used for analysis provide uniform spatial sampling across all frequency bands.

This frequency band methodology enables discrimination between tidal effects, post-tidal signals, and control frequencies. Detailed spectral analysis results demonstrating frequency-dependent correlation strength are presented in Section 3.5.1.

## 2.5 Statistical Framework

**Experimental Section:**

### Model comparison and selection

Comprehensive model validation tests the theoretical exponential decay assumption against alternative correlation structures through rigorous information-theoretic comparison:

- Models tested: Seven correlation functions including Exponential, Squared Exponential (Gaussian RBF), Power Law, Power Law with Cutoff, and Matérn (ν=1.5, 2.5)

- Selection criteria: Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), computed from the same weighted least-squares fits on distance-bin means (see Fit and Metric Definitions below)

- Methodology: Each model fitted using weighted nonlinear least squares with full uncertainty propagation

- Validation: Cross-center consistency analysis to ensure robust model selection

### Exponential model fitting

Model: $C(r) = A \cdot \exp(-r/\lambda) + C_0$

- $C(r)$: Mean phase-alignment index at distance $r$

- $A$: Correlation amplitude at zero distance

- $\lambda$: Characteristic correlation length (km)

- $C_0$: Asymptotic correlation offset

- Distance metric: Geodesic distance on WGS-84 (Karney), computed via GeographicLib

- Rationale: For ground-to-ground baselines, geodesic separation tracks propagation-relevant geometry and is the standard approach for GNSS analysis

- Distance binning: Logarithmic bins spanning 50 to 13,000 km

- Fitting method: Weighted nonlinear least squares with physical bounds

- Weights: Number of station pairs per distance bin

**Experimental Section:**

### Fit and Metric Definitions

- **Data for fitting:** All models are fit to *distance-bin means* of the phase-alignment index. Let $y_r$ be the mean for bin $r$ and $w_r$ be the number of station pairs in that bin.

- **Weighted least squares (WLS):** Fits minimize $\sum_r w_r\, (y_r - \hat y_r)^2$ using the same weights $w_r$ (pairs per bin). The effective sample size is the number of bins used (typically $N_{\text{eff}}\!\approx\!25$–28), not the number of pairs.

- **Weighted R² (reported everywhere unless noted):** $R^2_\text{w} = 1 - \frac{\sum_r w_r (y_r - \hat y_r)^2}{\sum_r w_r (y_r - \bar y_\text{w})^2}$, with weighted mean $\bar y_\text{w} = \frac{\sum_r w_r y_r}{\sum_r w_r}$. All R² values in this document refer to this weighted coefficient of determination on bin means.

- **AIC/BIC from WLS:** Information criteria are computed from the same WLS fits on bin means (Gaussian errors with weights $w_r$), using $n = N_{\text{eff}}$ bins and $k$ equal to the number of free parameters in the model.

Unless explicitly stated otherwise, all R², AIC, and BIC values reported in figures and text use these definitions.

**Figure 3. Logarithmic Distance Binning Strategy:** Three-panel visualization showing how correlation analysis is binned by distance for each analysis center (CODE, IGS, ESA). Each bin shows statistical power (pairs per bin) with TEP correlation length range highlighted. Logarithmic Y-axis reveals full dynamic range from ~10 to 100,000+ pairs per bin.

### 2.5.3 Exponential-Fit Uncertainty Estimation

Two complementary bootstrap approaches assess different aspects of exponential correlation function parameter uncertainty:

#### Bin-Level Bootstrap (Primary Analysis)

- **Method:** Block bootstrap resampling of distance bins (5000 iterations, block_size=10)

- **Features:** Data-driven parameter initialization, fallback strategies, enhanced numerical stability (maxfev=5000)

- **Success Rate:** 71–73% convergence; the 27–29% non-converged resamples are attributable to block-sampling draws that select predominantly short-baseline or predominantly distant flat bins across the N<sub>eff</sub> ≈ 25–28 distance bins, leaving the exponential decay scale under-constrained in nonlinear optimization (validated via comprehensive bias analysis, see Appendix 7.2)

- **Purpose:** Quantifies uncertainty in correlation function fitting while preserving network topology

#### Station-Block Bootstrap (Robustness Test)

- **Method:** Systematically removes ~30% of stations per resample (50 iterations)

- **Purpose:** Tests robustness to network structure and high-connectivity station bias

- **Effect:** Yields systematically lower λ<sub>T</sub> estimates due to reduced long-distance coverage

#### Temporal Topology Correlation Length (λT) Confidence Intervals

| Analysis Center | λT (exponential best fit, km) | Bin-level bootstrap 95% CI (km) | Station-Block 95% CI (km) |
| --- | --- | --- | --- |
| **CODE** | 4,549 | 1,198–5,918 | 3,020–3,550 |
| **ESA Final** | 3,330 | 2,532–3,984 | 2,440–2,740 |
| **IGS Combined** | 3,764 | 3,197–4,871 | 2,620–2,880 |

**Interpretation:** Bin-level CIs reflect exponential correlation function uncertainty using full networks; station-block CIs test structural robustness with systematically reduced networks, yielding lower λ estimates due to lost long-distance pairs. Both methods validate different aspects: fitting precision vs. network dependence.

#### Statistical Independence Considerations

- Effective sample size: N<sub>eff</sub> ≈ 25–28 distance bins, not 62.7 million individual station pairs

- Independence validation: Station pair non-independence addressed through LOSO cross-validation and block-wise validation

- Random seeds: Sequential 0-999 for reproducibility

### Cross-Center Processing Architecture and Network Overlap

Results from three distinct analysis center solutions (CODE, IGS Combined, ESA Final) are evaluated using standard meta-analytic comparisons. These products reflect different processing software packages and estimation strategies (Bernese GNSS Software at CODE, NAPEOS at ESA, and the weighted IGS Combination algorithm), but they operate on largely shared raw tracking inputs, with station network overlap between centers ranging from 83–90%. This creates physical and geometric dependence that must be acknowledged when interpreting combined evidence: cross-center agreement rules out software-specific bugs or implementation artifacts, but cannot rule out common-mode constellation geometry or datum-level systematics.

#### Station Network Overlap

Analysis centers share substantial station overlap, requiring careful interpretation of combined statistical evidence:

| Center | Stations | CODE Overlap | IGS Overlap | ESA Overlap |
| --- | --- | --- | --- | --- |
| **CODE** | 322 | — | 89.9% | 83.1% |
| **IGS Combined** | 278 | 89.9% | — | 90.0% |
| **ESA Final** | 201 | 83.1% | 90.0% | — |

**Note:** Overlap percentages calculated as (shared stations / smaller network) × 100. High overlap indicates that center-specific results are not fully independent, and combined statistical evidence should be interpreted as corroboration across independent processing methodologies applied to substantially overlapping networks rather than fully independent replications. The same ecosystem caveat applies across the companion papers: Paper 2 analyses the same CODE final product line over a longer epoch (2000–2025), Paper 14 uses CODE's MGEX combined product (2025–2026), and Paper 33's analysis-centre arms use COD/GFZ/GRG/JPL clock products, so cross-paper agreement among these is corroboration within a shared product ecosystem and a shared tracking network. The estimation-chain-independent axis in the corpus is Paper 3's raw-RINEX broadcast-ephemeris analysis, joined by Paper 33's self-processed SPP arms.

**Meta-analytic approach:** Fisher's Z-transformation is used to pool correlation coefficients across centers, with weights proportional to effective sample size (N<sub>eff</sub> = 25–28 distance bins per center). Heterogeneity is assessed using Cochran's Q-statistic and I² metric.

### Null test validation

- Distance scrambling: Randomize both distances and coherences independently to destroy distance-coherence relationships and systematic patterns (20 iterations per center, 60 total)

- Phase scrambling: Randomize phase relationships while preserving distance structure (20 iterations per center, 60 total)

- Station scrambling: Randomize station assignments within temporal blocks (20 iterations per center, 60 total)

- Statistical assessment: Permutation p-values computed from null distributions, z-scores for effect size quantification

- Significance threshold: α = 0.05 with correction for multiple testing where appropriate

**Pre-specified falsification criteria (declared before final analysis):** (i) λ outside [500, 20,000] km would falsify the theory-relevant regime; (ii) cross-center CV of λ > 20% would indicate processing artifacts; (iii) non-exponential families (e.g., squared exponential/Gaussian RBF) outperforming exponential/Matérn(ν=1.5) by AIC/BIC would disfavor screened-field interpretations; (iv) lack of control-band degradation at 1000–1500 μHz would indicate broadband artifacts. **Outcomes:** λ = 3,330–4,549 km (PASS); CV of λ ≈ 13% (PASS); exponential family preferred with Matérn(ν=1.5) competitive (PASS in this dataset; the 25-year CODE extension prefers Gaussian-family kernels by ΔAIC = 12.8, which meets this criterion's disfavouring condition; Paper 2); control band R² = 0.618 vs TEP-band ≈ 0.95 (PASS).

**Structural identification criterion:** The interval condition (i) is a prior search bound, not a discriminating test—the [500, 20,000] km window subsumes the leading systematic scales (tropospheric ~1,000–3,000 km, co-visibility ~3,000–8,000 km). The falsifying content of the λ<sub>T</sub> measurement is therefore structural: (a) across products the fitted scale is conditioned by clock-product construction and estimator settings (reported product-level values ~1.0–4.8×10<sup>3</sup> km; MGEX 1,862 ± 155 km; estimator axes ≲±25% on identical station-days; Paper 14, Step 3.5), and the same band-ordered profile is recovered in each product, with the IGS-final values defining the terrestrial saturation anchor R<sub>T</sub>(M<sub>⊕</sub>) ≈ 4,150 km (Paper 6); raw-SPP channels return shorter, noise-floor-limited scales (~0.7–1.1×10<sup>3</sup> km), and the raw ionofree channel returns 4,767 ± 835 km (Paper 3); (b) the correlation must be band-localized, i.e., the exponential fit must degrade where the coherent signal is absent—satisfied (control-band R² ≈ 0.5–0.6); (c) the fitted decay must be separable from the co-visibility kernel—forward simulation of the visibility/datum channel on the real station network bounds its effective scale at λ ≳ 6.7×10<sup>3</sup> km, and conditioning the same channel on real ephemerides and the estimated satellite-clock content of the product itself leaves the bound at λ ≥ 5,075 km with a qualitatively different profile shape (§4.6). Per-band λ<sub>T</sub> values are reported in §3.5.1: they are band-ordered within the coherent spectrum rather than a single universal number.

**Experimental Section:**

#### Statistical Framework: Spatial Correlation Analysis (Not Multiple Comparisons)

Critical Methodological Point: This analysis performs *spatial correlation analysis*, not multiple pairwise comparisons. Station pairs are aggregated into distance bins for fitting a single exponential correlation model—standard practice in spatial statistics, geostatistics, and variogram analysis.

Statistical Unit: The analysis operates on independent distance bins, not individual station pair comparisons. Each bin aggregates thousands of pairs, providing robust statistics while controlling effective sample size.

##### Why Multiple Comparison Corrections Are Not Applied to Primary Analysis:

- Single hypothesis test: Tests one exponential correlation model across distance bins

- Aggregated data structure: Pairs are binned by distance, not analyzed individually

- Standard spatial statistics: Identical to variogram analysis in geostatistics

- Cross-validation framework: LOSO/LODO methods test robustness to data structure

##### When Multiple Comparison Corrections Are Applied:

- Astronomical event analysis: Multiple planetary tests use Bonferroni and FDR corrections

- Model comparison: AIC/BIC account for model complexity

- Null testing: Permutation tests provide proper significance assessment

- Validation suite: Formal corrections applied to all secondary statistical tests

Validation approach: Multiple independent validation methodologies (spatial correlation modeling, temporal scrambling tests, cross-validation frameworks) reduce the risk of systematic artifacts through methodological diversity rather than statistical multiplicity corrections.

**Experimental Section:**

#### Statistical Independence Considerations

Pair-level dependencies: Station pairs sharing common stations create covariance structures that could inflate precision estimates. This is addressed through:

- Distance-bin aggregation: Primary analysis operates on binned means rather than individual pairs, reducing dependency effects

- LOSO validation: Leave-one-station-out removes all pairs involving each station, testing robustness to network structure

- Block-wise cross-validation: Leave-N-stations-out blocks provide additional independence testing

- Effective N estimation: Bootstrap confidence intervals reflect bin-level uncertainty, not 62.7M individual pairs

Interpretation: The confidence intervals appropriately reflect the statistical precision of distance-binned correlations rather than claiming precision from nominally large pair counts.

**Experimental Section:**

#### Statistical Limitations and Multiple Testing Considerations

**Multiple Testing Burden:** This analysis involves 388 statistical tests across 19 families, creating substantial multiple testing concerns despite corrections applied. Comprehensive validation demonstrates 40-52% of 388 statistical tests surviving multiple comparison corrections across 19 independent validation families. This survival rate substantially exceeds the 5% expected under the null hypothesis, providing strong evidence against systematic artifacts while maintaining conservative statistical standards.

**Effective Sample Size:** All R² values are computed on distance-bin means (N<sub>eff</sub> ≈ 25-28 bins), not the full 62.7M station pair dataset. This distinction is crucial for proper statistical interpretation—the high R² values reflect fits to binned means, not individual measurements. The effective statistical power corresponds to ~25-28 independent observations, not millions of measurements.

**Theory-Data Relationship:** While core TEP predictions preceded analysis, detailed spectral and astronomical investigations evolved during data exploration, creating elements of post-hoc hypothesis development that increase false discovery risk.

## 2.6 Advanced Analysis Methods

Beyond baseline correlation analysis, the phase-coherent methodology is applied to investigate dynamic responses, temporal modulation, and gravitational coupling. These analyses test specific TEP predictions and explore the full range of network sensitivity to spacetime structure.

**Experimental Section:**

### Dynamic Event Analysis

Methodologically consistent analysis of astronomical events (eclipses, supermoons) using identical phase-alignment index algorithms to test dynamic field responses.

### Earth Motion Coupling

Temporal tracking of correlation anisotropy patterns to detect coupling with Earth's orbital motion, rotation, and polar wandering.

### Directional Anisotropy Analysis with Distance Distribution Guardrails

Analysis of directional anisotropy in correlation patterns using azimuth-sector decomposition to test for Earth-motion-aligned effects. The λ<sub>EW</sub>/λ<sub>NS</sub> ratio provides a key diagnostic for rotation-aligned anisotropy.

#### Distance Distribution Matching Methodology

Critical Guardrail: To substantially reduce the likelihood of bias from differing distance distributions across azimuth sectors, the analysis implements distance distribution matching through two complementary approaches:

- Distance-weighted sector analysis: Each azimuth sector's correlation function is weighted by the inverse of its distance distribution density to ensure equal representation across distance ranges

- Matched-distance subsampling: For each sector, pairs are subsampled to match the global distance distribution, ensuring identical distance sampling patterns across all azimuth sectors

#### Implementation Details

- Azimuth sectors: Eight 45° sectors (N, NE, E, SE, S, SW, W, NW) with sector assignment based on great-circle azimuth between station pairs

- Distance distribution matching: Global distance histogram computed from all pairs; each sector's pairs weighted or subsampled to match this reference distribution

- Exponential fitting per sector: C(r) = A·exp(-r/λ) + C₀ fitted to each sector's distance-matched data using weighted least squares

- Anisotropy metrics: λ<sub>EW</sub>/λ<sub>NS</sub> ratio computed from E-W sector average (E, W) versus N-S sector average (N, S)

- Validation: Distance distribution uniformity across sectors verified through Kolmogorov-Smirnov tests (p > 0.05 required for valid analysis)

#### Bias Prevention

This approach substantially reduces the likelihood of systematic bias in λ<sub>EW</sub>/λ<sub>NS</sub> ratios that could arise from:

- Geographic sampling bias: Different continents having different azimuth orientations relative to global station network

- Distance sampling bias: Certain azimuth sectors having systematically shorter or longer baseline distributions

- Network topology bias: Station network geometry creating non-uniform distance sampling across azimuth sectors

Result: The distance distribution matching ensures that any observed λ<sub>EW</sub>/λ<sub>NS</sub> anisotropy reflects genuine directional effects rather than sampling artifacts.

#### Orbital-Velocity Anisotropy: Compact Specification

Anisotropy Metric:

A<sub>aniso</sub>(t) = λ<sub>EW</sub>(t) / λ<sub>NS</sub>(t)            
from 8-sector fits with matched distance distributions (KS p > 0.05).

Statistical Analysis: Meta-analysis combines results across analysis centers using Fisher's Z-transformation to pool correlation coefficients. Low heterogeneity metrics (Q = 0.24, I² = 0%) indicate consistent effects across centers. High station overlap (83–90%) means combined evidence represents corroboration across independent processing methodologies rather than fully independent replications.

### Gravitational Correlation Analysis

Multi-window smoothing analysis correlating planetary gravitational influences with temporal coherence patterns using NASA/JPL ephemeris data.

#### Planetary Enhancement Factor and Mass Scaling Analysis

Two complementary metrics quantify coupling strength and test gravitational scaling predictions:

##### Metric 1: Enhancement Factor (Descriptive)

A<sub>obs</sub> (Observed Amplitude): The measured TEP correlation amplitude during planetary gravitational events, expressed as a dimensionless fraction of the baseline correlation strength. Units: dimensionless (fractional amplitude).

A<sub>exp</sub> (Expected Amplitude): The theoretically predicted TEP correlation amplitude assuming gravitational field strength scaling. The model assumes A<sub>exp</sub> ∝ (M<sub>planet</sub>/M<sub>⊕</sub>) × (1/d<sub>planet</sub>²), where M<sub>planet</sub> is planetary mass in Earth masses, M<sub>⊕</sub> is Earth mass, and d<sub>planet</sub> is Earth-planet distance in AU. Units: dimensionless (fractional amplitude).

Enhancement Factor:

E ≡ A<sub>obs</sub> / A<sub>exp</sub> = A<sub>obs</sub> / (M/d²)            

The enhancement factor E provides a normalized comparison of observed versus expected coupling strength. However, E should not be tested for correlation with mass—since E already divides by mass, testing whether E correlates with mass is circular and uninformative. A further limitation is documented in Section 3.4.4: when A<sub>obs</sub> sits near the per-event noise floor, E reduces algebraically to ∝ d²/M, so the metric encodes the Newtonian denominator rather than any measured response. It is retained for provenance and is not used as physical evidence of geometric screening enhancement.

##### Metric 2: Mass Scaling Test (Proper Statistical Test)

To test whether observed amplitudes follow gravitational scaling predictions, a direct correlation is computed between A<sub>obs</sub> and the gravitational scaling factor (M/d²):

**Linear Gravitational Scaling:**

r<sub>linear</sub> = correlation(A<sub>obs</sub>, M/d²)            

**Quadratic/Tidal Scaling:**

r<sub>quadratic</sub> = correlation(A<sub>obs</sub>, (M/d²)²)            

**Interpretation:**

- If r<sub>linear</sub> ≈ +0.8 to +1.0: Strong linear gravitational scaling (Newtonian)

- If r<sub>quadratic</sub> > r<sub>linear</sub> and r<sub>quadratic</sub> ≈ +0.7 to +0.9: Tidal/quadratic coupling

- If both |r| ≲ 0.2: No significant gravitational scaling, indicating that amplitude-level signals are absorbed by network clock processing or placing an upper bound on single-receiver planetary amplitudes

- If r ≪ −0.5: Statistically significant inverse correlation (smaller planets showing stronger signals)

This proper test directly examines the relationship between observed phenomena and gravitational predictions without circular normalization.

Critical Methodological Note: Earlier analysis draft versions tested whether E correlated with mass. This approach is mathematically circular because E ≡ A<sub>obs</sub>/(M/d²) already divides by mass. Dividing a flat residual floor by M/d² mechanically produces an inverse scaling with mass by construction, providing no independent physical information about gravitational scaling. The proper statistical test examines A<sub>obs</sub> directly (Section 3.4.4).

#### Worked Examples: Mercury and Jupiter

##### Mercury Example

- Planetary mass: M<sub>Mercury</sub> = 0.055 M<sub>⊕</sub>

- Expected amplitude: A<sub>exp</sub> = 0.00010 (dimensionless)

- Observed amplitude: A<sub>obs</sub> = 0.0248 (dimensionless)

- Enhancement factor: E = A<sub>obs</sub>/A<sub>exp</sub> = 0.0248/0.00010 = 248

##### Jupiter Example

- Planetary mass: M<sub>Jupiter</sub> = 317.8 M<sub>⊕</sub>

- Expected amplitude: A<sub>exp</sub> = 0.00220 (dimensionless)

- Observed amplitude: A<sub>obs</sub> = 0.0110 (dimensionless)

- Enhancement factor: E = A<sub>obs</sub>/A<sub>exp</sub> = 0.0110/0.00220 = 5

###### Worked Example: Enhancement Factor Computations

The computed enhancement factors shown above for Mercury (248) and Jupiter (5) follow directly from the worked A<sub>obs</sub> and A<sub>exp</sub> values and are illustrative of the method only. Cross-center mean values computed from the data (Mercury 137×, Mars 215×, Saturn 59×, Venus 18×, Jupiter 5.3×) and the full statistical treatment are presented in Section 3.4.4.

### Collective Network Dynamics (Network Covariance Analysis)

Investigation of collective network behavior to detect unified detector responses and systematic motion signatures.

#### Understanding Collective Network Dynamics: Network-Wide Coherence

##### Conceptual Foundation

While the phase-alignment index measures whether *two stations* are synchronized, the network covariance score (referred to in pipeline outputs as the "mesh dance" score) tests whether the *entire 364-station network* behaves as a single, dynamically coherent detector—distinguishing pairwise clock synchronization from collective network oscillation and rotational phase structure.

The network covariance score quantifies four distinct signatures of collective network behavior:

| Component | Physical Meaning | Window | Weight |
| --- | --- | --- | --- |
| **Mesh Coherence** | Are all station pairs fluctuating with similar magnitude and phase? Measures uniformity of correlations across the network. | 90 days | 50% |
| **Spiral Motion** | Does the network's collective motion vector trace a steady helix through geospatial phase-space? Detects rotational dynamics. | 30 days | 17% |
| **Collective Oscillation** | Does the whole mesh "breathe" in-and-out with a common period? Tests for synchronized temporal variations. | 30 days | 17% |
| **Earth Coupling** | Is the network synchronized with known Earth motions (rotation, orbit, Chandler wobble)? Tests for planetary coupling. | 90 days | 16% |

##### Component-Based Analysis

**Primary approach:** Each collective network component is analyzed and reported independently to avoid assumptions about relative importance:

- **Base mesh coherence:** 0.643–0.644 (strong network-wide synchronization)

- **Spiral motion:** Detected across all centers (rotational dynamics)

- **Collective oscillation:** Annual period (365.25 days, R² = 0.35, p $\lt$ 10⁻³)

- **Earth coupling:** Detected through orbital correlations

**Window optimization:** 90-day windows provide adequate statistical power (10 samples over 912 days) for coherence metrics, while 30-day windows (30 samples) enable better temporal resolution for detecting oscillations and spiral dynamics. These choices balance statistical robustness against temporal resolution requirements.

##### Why This Matters

The detection of coordinated network dynamics indicates the network is not just a collection of independent correlations, but exhibits *organized collective behavior*. This is significant because:

- GNSS processing is designed to *remove* spatially correlated signals—observing coherent network dynamics despite this suppression suggests a robust underlying phenomenon

- Dominant annual oscillations synchronize with Earth's orbital motion across all centers

- Multi-center consistency confirms this is not a processing artifact (detailed results in Section 3.3.2)

#### 3D Spherical Harmonic Decomposition

Complete spatial characterization using spherical harmonic decomposition of correlation anisotropy patterns. The analysis decomposes the spatial correlation field into monopole (Y₀₀), dipole (Y₁₀, Y₁₁), and quadrupole (Y₂₀, Y₂₁, Y₂₂) components using a 16-bin spherical grid (azimuth × elevation). Anisotropy strength is computed as the ratio of dipole magnitude to monopole strength, quantifying directional dependence. This provides full 3D spatial structure beyond 2D directional analysis.

#### Multi-Frequency Beat Pattern Analysis

Temporal analysis detecting periodic modulations across multiple timescales through systematic frequency decomposition. The analysis identifies beat patterns by computing temporal correlations across 12 frequency bands, detecting interference patterns between fundamental Earth motion periods (rotation, orbital motion, Chandler wobble) and their combinations. Relative motion beat analysis examines differential dynamics between station pairs to isolate genuine multi-body coupling from single-frequency oscillations. Patterns are validated across all three analysis centers with significance threshold R² > 0.30.

### Temporal Modulation Studies

High-resolution diurnal and seasonal analysis to detect systematic variations in correlation strength with Earth's orientation and orbital position.

### Energy-vs-Velocity Discrimination Analysis

To distinguish between energy-based and velocity-based scaling mechanisms in TEP coupling, a discrimination metric is computed as the simple arithmetic difference of Pearson correlations between gravitational-temporal field patterns and Earth motion parameters:

\begin{equation} \label{eq:discrimination}
\text{discrimination} = r_E - r_V
\end{equation}
where $r_E$ is the correlation coefficient for energy-based scaling (gravitational potential energy) and $r_V$ is the correlation coefficient for velocity-based scaling (orbital velocity). This metric quantifies the relative preference between energy and velocity coupling mechanisms, with positive values indicating energy-based scaling preference and negative values indicating velocity-based scaling preference.

Implementation: The discrimination is calculated independently across three analysis centers (CODE, ESA Final, IGS Combined) and then aggregated using bootstrap resampling (5000 iterations) to provide robust confidence intervals. With n=3 centers, the bootstrap CI is mostly illustrative; the analysis reveals discrimination = r_E − r_V = −0.057 (bootstrap 95% CI: [−0.143, +0.030]), indicating no statistically significant preference between energy-based and velocity-based scaling mechanisms, supporting complex multi-mechanism coupling in TEP physics.

### Event Windows: Astronomical Events Analysis

Analysis of 32 astronomical events across 8 categories (planetary oppositions/conjunctions, solar eclipses, supermoons, lunar standstill) spanning 2023-2025. Temporal windows range from ±1 day (high-resolution events) to ±180 days (lunar standstill), with linear detrending applied before cross-spectral analysis. Multi-center combination uses weighted averaging for planetary events and mean averaging for high-resolution events.

Planetary opposition/conjunction analyses are evaluated by default using a multi-window sensitivity sweep: ±30, ±60, ±120, ±180, and ±240 days. This protocol assesses robustness to temporal window choice: outer-planet responses typically strengthen with seasonal-scale windows (±120–±240), whereas inner-planet responses (e.g., Mercury) are stable across windows. Users may override windows via the analysis package CLI.

*Complete event specifications, dates, and implementation details: See Appendix 7.1.*

**Experimental Section:**

### TID Exclusion: Ionospheric Contamination Control

Goal: Bound ionospheric contamination by excluding time periods with elevated Traveling Ionospheric Disturbance (TID) activity and quantifying the change in fitted correlation quality.

- Proxy for TIDs: High-resolution temporal metrics (Step 4.3 wavelet and Hilbert instantaneous frequency summaries) aggregated to daily *TID activity indices*.

- Exclusion rule: Remove days whose TID index exceeds the 75th percentile threshold; retain the remaining days for recomputing mean coherence.

- Effect size: Report percentage change in mean coherence relative to the original (unfiltered) series: Δ% = 100 · (coherence<sub>retained</sub> − coherence<sub>original</sub>) / coherence<sub>original</sub>.

- Operational band context: Assessment pertains to the primary analysis band (10–500 μHz), overlapping canonical TID periods (10–180 minutes), so exclusion uses external temporal structure rather than frequency separation.

*Implementation details and file paths: See Appendix 7.2.*

### Methodological Robustness: Addressing Potential Artifacts

#### Logarithmic Binning Design

The analysis employs logarithmic distance binning (50 km to 13,000 km) specifically designed to avoid Earth-size artifacts. If correlations were artifacts of Earth's finite geometry, they would exhibit sharp cutoffs near Earth's circumference (~40,000 km). The observed exponential decay occurs well within continental baselines and demonstrates consistent patterns across multiple binning schemes (40 bins vs 20 bins vs linear spacing), ruling out binning-dependent artifacts.

#### Multi-Model Validation

Seven different correlation models (exponential, Gaussian, power-law, Matérn variants) are tested to ensure the exponential form isn't an artifact of functional choice. Model selection uses Akaike Information Criterion (AIC) to systematically compare fit quality while penalizing model complexity. Results are presented in Section 3.1.2.

#### Frequency-Dependent Validation

Control band analysis (Section 3.5) tests identical methodology across different frequency ranges. If correlations were binning artifacts, they would appear uniformly across all frequencies. Instead, strong correlations appear in TEP bands (10-500 μHz) while control bands (>1000 μHz) show minimal correlation, proving physical frequency dependence.

#### Artifact Rejection Through Methodological Diversity

The convergence of evidence across methodologically diverse approaches provides strong evidence against systematic artifacts:

- **Binning independence:** Consistent results across logarithmic, linear, and adaptive binning schemes

- **Distance metric independence:** Identical patterns using great-circle, 3D Euclidean, and geodetic distances

- **Frequency specificity:** Strong correlations in TEP bands, weak in control bands

- **Null test rejection:** Real data substantially exceeds randomized controls across multiple scrambling approaches

- **Multi-center consistency:** Independent processing algorithms converge on identical parameters

This methodological diversity makes it highly unlikely that the observed patterns result from any single systematic bias or Earth-geometry artifact.

## 3. Results

**Snapshot of core metrics.** Exponential-family fits to
distance-binned means yield consistent Temporal Topology correlation lengths (λ<sub>T</sub>) and
goodness-of-fit (R²) across distinct analysis center solutions:

| Center | λ (exponential best fit, km) | 95 % CI (km) | R² (pooled) |
| --- | --- | --- | --- |
| CODE | 4,549 | 1,198–5,918 | 0.920 |
| IGS Combined | 3,764 | 3,197–4,871 | 0.966 |
| ESA Final | 3,330 | 2,532–3,984 | 0.970 |

Across centers, λ spans 3,330–4,549 km; coefficients of
variation $\lt$20 % indicate strong reproducibility. These
headline metrics are expanded upon in the detailed sections that follow.

**Experimental Section:**

### Principal Findings

Phase-coherent cross-spectral analysis of 62.7 million quality-filtered
station pair measurements from 364 unique GNSS stations (249 N / 115 S =
68.4% / 31.6%) reveals systematic distance-dependent correlations in
atomic clock residuals. The patterns exhibit substantial multi-center
consistency and demonstrate sensitivity to Earth's motion, gravitational
fields, and temporal structure.

#### Core Observational Evidence

Primary Finding: Temporal Topology Correlation Length: A primary
Temporal Topology correlation length (λ<sub>T</sub>) of 3,330–4,549 km, with bootstrap validation ranges
of 1,198–5,918 km (CODE), 2,532–3,984 km (ESA Final), and
3,197–4,871 km (IGS Combined), is observed. Bootstrap analysis
with 5000 iterations achieves 3,555/5,000 (71.1%) success rates,
demonstrating robust statistical convergence. Sensitivity
analyses show a broader range of 1,600–7,500 km, reflecting
environmental dependencies.

Elevation dependence: Systematic quintile stratification from Q1
(-81 to 79m: λ<sub>T</sub> = 3,174 km, R² = 0.83) through Q2 (79 to 189m: λ<sub>T</sub>
= 4,470 km), Q3 (189 to 379m: λ<sub>T</sub> = 5,287 km), Q4 (379 to 713m: λ<sub>T</sub>
= 7,688 km, R² = 0.82), to Q5 (>713m: λ<sub>T</sub> = 4,980 km), showing
systematic elevation effects with complex high-altitude
patterns.

Spectral characterization: Broadband coupling (R² > 0.85 from
10–100 μHz; 100–200 μHz averages ~0.75, CV of R² across bands =
2.9%) with gravitational enhancement (λ<sub>T</sub> = 4,677 km at tidal
frequencies) and persistent post-tidal signals (30–40 μHz: R² =
0.946), excluding classical tidal contamination

Ionospheric interference quantified: TID exclusion analysis
reveals 21–23% phase-alignment index improvement potential from
excluding high ionospheric activity periods, with IGS Combined
showing highest sensitivity (22.66% improvement) and systematic
temporal patterns across Venus 2f harmonic, solar rotation, and
lunar cycles

Multi-center consistency: Robust patterns across CODE (39.0M
pairs, N<sub>eff</sub>=28 bins used), IGS Combined (12.9M pairs,
N<sub>eff</sub>=28 bins used), and ESA Final (10.8M pairs,
N<sub>eff</sub>=25 bins used) with 100% elevation and
geomagnetic coverage

Earth motion signatures: Orbital velocity coupling (r = -0.571
to -0.793), Chandler wobble (R² = 0.377–0.471), interference
patterns (r up to 0.962)

Gravitational energy hierarchy validation: Comprehensive
gravitational-temporal analysis shows patterns consistent with
TEP coupling mechanisms—orbital motion (strongest: |r| =
0.571–0.793), Chandler wobble with lunar gravitational
modulation (moderate-strong: |r| = 0.61–0.69), and daily
rotation (moderate: CV of rotational stability = 0.5–0.6).
Energy vs velocity scaling discrimination (-0.057, 95% CI:
[-0.143, +0.030], n.s.) indicates complex multi-mechanism
coupling rather than simple proportional relationships

Collective network dynamics: Coordinated network behavior across all
analysis centers, with 35 beat patterns detected spanning
0.077–365 days (7 statistically significant after FDR
correction; strongest: Venus sub-harmonic 196.9d R²=0.82–0.86,
lunar fortnight 14.8d R²=0.60–0.82) and 9–11 robust relative
motion coupling patterns per center (all significant), plus
strong base mesh coherence (0.643–0.644) and annual oscillation
period (365.25 days)

Gravitational coupling: Strong planetary correlations with
timescale strengthening (r = -0.503 at 227 days, raw p =
1.5×10⁻⁵⁹, autocorr-corrected p = 3.3×10⁻⁴) and Moon-Chandler
coupling mechanism discovered - Earth's 433-day polar wobble
modulates Earth-Moon gravitational geometry, creating stronger
TEP signatures than predicted from mechanical energy alone
(note: this energetic comparison is heuristic—the detection is
in correlation space on binned means)

Temporal structure: Diurnal (1.9–7.6% day-night variation) and
seasonal modulation with systematic early morning peaks;
seasonal peaks occur near spring equinox; global means are
1.019/1.076/1.060 (CODE/IGS/ESA)

Comprehensive diurnal dynamics: Analysis of 72.4 million hourly
records reveals synchronized temporal variations across all
centers with systematic early morning coherence peaks and
seasonal modulation patterns consistent with dynamical time flow

Dynamic responses: Major phase-alignment index modulations
during eclipses (18–87% of baseline) and supermoon events

## 3.1 Core Correlation Properties

**Experimental Section:**

Summary: This group establishes the fundamental correlation parameters
and model validation, demonstrating consistent exponential decay
patterns across distinct analysis center solutions and optimal model selection
through comprehensive statistical comparison.

## 3.1.1 Multi-Center Correlation Analysis

Cross-spectral analysis reveals a primary correlation length of 3,330–4,549
km across three distinct GNSS analysis center solutions—the pooled
coherent-band estimate, with the band-resolved decomposition reported in
§3.5.1. This finding is supported
by bootstrap validation ranges of 1,198–5,918 km (CODE), 2,532–3,984 km (ESA
Final), and 3,197–4,871 km (IGS Combined) and is robust across different
analysis strategies. Primary pooled fits on bin means show R² = 0.92–0.97
(distance-bin means, N<sub>eff</sub> ≈ 25–28 bins used from 40 attempted).
Sensitivity subset analyses (elevation/geomagnetic): R² = 0.70–0.91.
Sensitivity analyses, which account for environmental factors like elevation
and geomagnetic latitude, show a broader range of 1,600–7,500 km.

**Methodological validation:** The observed correlation length
(3,330-4,549 km) occurs well within continental baselines, ruling out
Earth-size artifacts that would manifest near Earth's circumference (~40,000
km). Consistency across logarithmic and linear binning schemes confirms the
exponential decay is not an artifact of distance discretization.

Figure 4. Continental-scale correlations in GNSS atomic clock
networks.
**(a) Multi-center reproducibility:** Exponential decay with 95%
confidence intervals. The primary correlation length (λ) of 3,330–4,549
km is consistent across centers.
**(b) Statistical significance:** Permutation tests demonstrate real
R² values as extreme outliers.
**(c) Signal structure:** Distance-scrambled comparison confirms
spatial organization of correlations.

**Experimental Section:**

### Correlation Parameters by Analysis Center

| Center | λ Range (km) | R² Range (subset fits; not the primary pooled fit) | Geomag | Pairs |
| --- | --- | --- | --- | --- |
| CODE | 3,225–7,499 | 0.70–0.91 | 100% | 39.0M |
| ESA Final | 1,600–3,914 | 0.81–0.91 | 100% | 10.8M |
| IGS Combined | 2,616–5,453 | 0.78–0.88 | 100% | 12.9M |

Table 1. Correlation Parameters by Analysis Center (Sensitivity
Subset Analysis).
This table shows the range of correlation lengths and R² values
observed in sensitivity subset analyses, which systematically vary
with environmental factors such as elevation and geomagnetic
latitude. The R² ranges shown (e.g., 0.78–0.88 for IGS Combined)
represent sensitivity subset fits, distinct from the primary pooled
fit on bin means (R² = 0.966 for IGS Combined). The primary finding
of a 3,330–4,549 km correlation length is derived from the pooled
analysis.

Note: λ values are derived from Leave-One-Station-Out (LOSO)
cross-validation for maximum robustness. R² values in this table
refer to sensitivity subset fits on distance-bin means (Neff
≈ 25–28), not individual pairs from the 62.7 million station pair
measurements. These sensitivity subset R² values (e.g., 0.78–0.88
for IGS Combined) are distinct from the primary pooled fit R² values
(e.g., 0.966 for IGS Combined) reported elsewhere. R² (Binned Fit)
reflects the goodness of fit for the exponential model to the binned
data, while R² (LOSO CV) represents the model's predictive power on
unseen data subsets, providing a more conservative measure of
explained variance.

### Multi-Center Consistency

Correlation length ranges: See comprehensive λ + CI Summary Table
above for complete values across all validation methods

Station-block bootstrap robustness: All centers maintain stable λ
values within respective bin-level bootstrap CIs despite removing
~30% of stations per resample

Average λ (unweighted): 3,880 km (within theoretical predictions:
1,000–10,000 km)

Fit quality (Binned): R² = 0.920–0.970 across all centers (fits to
distance-bin means, N<sub>eff</sub> ≈ 25–28)

Processing and network robustness: Algorithmic and software robustness
across diverse processing chains (Bernese double-difference clock recovery,
NAPEOS undifferenced global network adjustment, and IGS weighted combination)
is confirmed through station-block bootstrap: λ values remain within
respective bin-level bootstrap CIs despite removing ~30% of stations per
resample, confirming neither high-connectivity stations nor geographic
clustering bias drives the correlation signature.

## 3.1.2 Model Comparison and Selection

Comprehensive comparison of seven correlation models demonstrates systematic
preference for exponential family models (Exponential/Matérn(ν=1.5)) across
all analysis centers. Matérn (ν=1.5) achieves optimal performance (lowest
AIC) for IGS Combined and remains competitive across all centers (ΔAIC ≤
4.4), confirming the exponential family's superiority over alternatives.
Both exponential and Matérn(ν=1.5) models support screened scalar field
interpretation, with exponential adopted as the more interpretable baseline
while conclusions are validated under both frameworks.

While different analysis centers favor different optimal models (CODE:
Exponential, IGS Combined: Matérn ν=1.5, ESA Final: Exponential), all models
converge on similar correlation lengths within the theoretically motivated search range,
demonstrating robust physical consistency across processing methodologies.

**Experimental Section:**

### Model Performance (Akaike Information Criterion)

| Model | CODE ΔAIC | ESA ΔAIC | IGS Combined ΔAIC |
| --- | --- | --- | --- |
| Exponential | 0.0 | 0.0 | 2.1 |
| Matérn (ν=1.5) | 1.7 | 4.4 | 0.0 |
| Matérn (ν=2.5) | 2.9 | 7.8 | 2.0 |
| Power Law w/ Cutoff | 3.0 | 4.7 | 10.5 |
| Squared Exponential (Gaussian RBF) | 5.5 | 16.6 | 9.7 |
| Power Law | 10.7 | 14.3 | 25.6 |

Key findings: Exponential family models (exponential, Matérn(ν=1.5))
consistently outperform alternatives across all model comparison
criteria. Matérn (ν=1.5) demonstrates optimal AIC performance for IGS
Combined, while systematic preference for exponential family over
squared exponential models (ΔAIC = 5.5-16.6) supports screened scalar
field interpretation over Gaussian processes. All best-fit models
achieve R² > 0.92, with Matérn(ν=1.5) achieving the highest individual
R² values, confirming robust exponential decay characteristics
consistent with field screening mechanisms.

## 3.2 Environmental and Geometric Dependencies

**Experimental Section:**

Summary: This group examines how correlation patterns depend on
environmental factors including elevation, geomagnetic latitude, and
directional anisotropy, providing evidence for environmental screening
effects and spatial field structure.

## 3.2.1 Elevation Dependence Analysis

Comprehensive quintile-based elevation stratification reveals systematic
correlation length dependence on station elevation.

**Experimental Section:**

### Quintile Elevation Stratification Results

| Quintile | Elevation Range (m) | λ (km) | R² | Station Pairs |
| --- | --- | --- | --- | --- |
| Q1 (Sea Level) | -81 to 79 | 3,174 | 0.83 | ~7.8M |
| Q2 (Low) | 79 to 189 | 4,470 | 0.80 | ~7.8M |
| Q3 (Medium) | 189 to 379 | 5,287 | 0.73 | ~7.8M |
| Q4 (High) | 379 to 713 | 7,688 | 0.82 | ~7.8M |
| Q5 (Very High) | >713 | 4,980 | 0.80 | ~7.8M |

Elevation dependence: Correlation length shows systematic variation with
elevation quintile (Q1: 3,174 km, Q4: 7,688 km, 2.4× increase), with Q5
showing intermediate values (4,980 km), indicating complex environmental
screening effects. The R² values remain consistently high (0.73–0.83)
across all quintiles, suggesting robust exponential decay patterns at
all elevation ranges. This pattern is consistent with TEP predictions
for φ-field coupling through matter density variations. It is, however,
also the fingerprint of altitude-organized residual zenith-delay and
atmospheric-loading structure; the two readings are presently degenerate
pending a reanalysis-ZTD covariance control on pair-level phase
alignment (Step 4.9).

Geomagnetic-elevation stratification: Combined analysis reveals enhanced
correlation structure when both elevation and geomagnetic latitude
effects are considered simultaneously. Across all three analysis centers
and all elevation tiers, the correlation length is systematically
shortest in the high-geomagnetic-latitude (auroral-leaning) band
(λ ≈ 1,100–3,400 km in 7 of 9 cells; binomial p = 8.3×10<sup>−3</sup>
under a no-organization null), while the quiet mid-latitude band carries
the longest lengths (up to λ ≈ 10,000–15,000 km). This ordering aligns
directly with the TID-exclusion result (&sect;3.3.2): heightened auroral
ionospheric activity degrades long-baseline phase coherence and
compresses the apparent decay scale, whereas the quiet mid-latitude
band preserves the underlying terrestrial correlation over the longest
baselines, with local-night persistence bounding any static ionospheric
contribution at ≲3% of the coherent signal (Step 4.9).

## 3.2.2 Directional Anisotropy Patterns

Analysis across all three analysis center solutions reveals systematic
longitude-dependent variations in correlation patterns, indicating
directional structure in the observed correlations. The primary anisotropy
metric is defined as the ratio of East-West to North-South correlation
lengths (λ<sub>EW</sub>/λ<sub>NS</sub>) derived from 8-sector directional
analysis (N, NE, E, SE, S, SW, W, NW). Eight-sector analysis yields typical
λ_EW/λ_NS ≈ 0.55–2.03, with peaks up to ~4.75 (IGS Combined center).

**Figure 5. Global Station Correlation Network:** Visualization of
high-coherence connections across the global GNSS network, revealing
directional patterns and spatial anisotropy in timing correlations
across intercontinental distances.

**Figure 5a. CODE:** Coherence vs distance and longitude
difference.

**Figure 5b. ESA Final:** Consistent patterns across distinct
processing chains.

**Figure 5c. IGS Combined:** Cross-solution consistency validates
robustness across software packages.

**Experimental Section:**

### Key Anisotropy Observations

Distance-dependent decay: Clear exponential decay with distance
across all centers

Longitude dependence: Systematic variations with longitude
difference (40–80° and 120–160° ranges)

Multi-center consistency: Reproducible patterns across independent
processing strategies

Intercontinental coherence: Correlation preservation at distances
>6,000 km

Directional variation range: Eight-sector analysis yields typical
λ_EW/λ_NS ≈ 0.55–2.03, with peaks up to ~4.75 (IGS Combined center),
with robust sample sizes (N = 2.9M–8.3M pairs per sector)

**Experimental Section:**

### Independence safeguards for directionality

Distance-distribution matching is enforced across azimuth sectors.
Sector-level Kolmogorov–Smirnov (KS) tests require p > 0.05; all 8/8
sectors pass for each analysis center. Minimum KS p across sectors: CODE
0.563, ESA Final 0.766, IGS Combined 0.705. See Table S1.

| Analysis Center | Sectors Passing (of 8) | Min KS p | All p > 0.05? |
| --- | --- | --- | --- |
| CODE | 8 / 8 | 0.563 | Yes |
| ESA Final | 8 / 8 | 0.766 | Yes |
| IGS Combined | 8 / 8 | 0.705 | Yes |

Table S1 (inline): Sector-level KS tests on distance distributions
confirm that anisotropy estimates are not driven by geometry;
complete per-sector p-values are available in analysis outputs.

## 3.2.3 Three-Dimensional Spherical Harmonic Decomposition

Complete 3D spatial analysis using spherical harmonic decomposition reveals
the full geometric structure of correlation anisotropy, extending beyond 2D
directional analysis to capture the true complexity of the spatial field.

**Experimental Section:**

### Spherical Harmonic Analysis Results

#### Harmonic Decomposition (CODE)

**Analysis Configuration:** 16 spherical bins (azimuth
× elevation grid), 39.0M station pairs analyzed

| Harmonic Component | Coefficient Value | Physical Interpretation |
| --- | --- | --- |
| Y₀₀ (Monopole) | 5,743 km | Isotropic background field strength |
| Y₁₀ (Dipole Z) | 5,715 km | North-South asymmetry component |
| Y₁₁ (Dipole XY) | 27.4 km (real), -0.17 km (imag) | East-West asymmetry component |
| Y₂₀ (Quadrupole) | 5,660 km | Polar-equatorial differential |
| Y₂₁, Y₂₂ (Quadrupole) | 27.3 km, 14.6 km (+ imaginary) | Higher-order directional structure |

### Anisotropy Strength Metrics

| Metric | Value | Interpretation |
| --- | --- | --- |
| Monopole Strength | 5,743 km | Base isotropic component |
| Dipole Magnitude | 5,715 km | Primary directional asymmetry |
| Quadrupole Magnitude | 5,660 km | Secondary spatial structure |
| **Anisotropy Strength** | **1.98** | Moderate-to-strong directional dependence |

### Spatial Correlation Structure

16-bin spherical grid analysis reveals systematic variation in
correlation length across the celestial sphere:

**Minimum λ regions:** 1,488–1,651 km (low-elevation
equatorial sectors)

**Maximum λ regions:** 7,562–9,762 km (mid-to-high
elevation sectors)

**Dynamic range:** ~6.6× variation (factor of 6.6
between shortest and longest)

**Sample coverage:** Each bin contains 1.3M–4.7M
station pairs, ensuring robust statistics

**Key Finding:** The anisotropy strength of 1.98
indicates that the correlation field exhibits nearly 2× stronger
coupling in preferred directions compared to the isotropic baseline.
This substantial directional dependence, combined with the
large-scale harmonic structure (dipole and quadrupole magnitudes
~5,700 km), suggests coupling to Earth's motion through space or
large-scale gravitational field geometry rather than local
geophysical effects. The consistency of this large-scale directional
signature with local Lorentz invariance&mdash;including screening at
laboratory densities and the relation to Temporal Shear
$\Sigma_\mu = \nabla_\mu \ln A$&mdash;is derived in Appendix D of
Paper 0 (TEP).

## 3.3 Coupling with Terrestrial and Orbital Dynamics

**Experimental Section:**

Summary: This group investigates the coupling between TEP correlations
and Earth's various motions, including orbital velocity, rotation, and
polar wandering, revealing systematic energy hierarchy relationships and
interference patterns (combination and beat frequencies).

## 3.3.1 Orbital Motion Coupling

Temporal tracking analysis reveals systematic correlation between GPS timing
anisotropy and Earth's orbital velocity.

**Experimental Section:**

### Orbital Velocity Correlations

| Analysis Center | Correlation (r) | P-value | Significance (BH q=0.05): ✓ |
| --- | --- | --- | --- |
| CODE | -0.701 | 3.9×10⁻⁶ | ✓ |
| IGS Combined | -0.793 | 2.2×10⁻⁸ | ✓ |
| ESA Final | -0.571 | 4.2×10⁻⁴ | ✓ |

Physical pattern: Negative correlation indicates that higher orbital
velocities (perihelion, ~30.3 km/s) correspond to more isotropic
correlations, while lower velocities (aphelion, ~29.3 km/s) show
stronger directional anisotropy. Individual center significance levels
vary, with anisotropy measured as East-West/North-South correlation
length ratios from 8-sector directional analysis. Analysis centers use
independent processing algorithms and partially overlapping station
networks.

#### Meta-Analysis Context

**Network overlap:** Station overlap 83–90% across
center pairs (CODE: 322 stations, IGS: 278, ESA: 201)

**Heterogeneity metrics:** Cochran's Q = 0.24 (p =
0.885), I² = 0% (low heterogeneity indicates consistent effects
across centers)

**Interpretation:** High overlap implies
center-specific results are not fully independent; evidence
should be treated as corroboration across independent processing
of overlapping networks.

Temporal independence: Analysis utilizes 30-day correlation windows with
15-day smoothing spans and effective degrees of freedom N<sub>eff</sub>
= 25–28 bins across centers, ensuring temporal decorrelation intervals
exceed autocorrelation timescales.

Seasonal periodicity: 365.25-day periodicity synchronized with Earth's
orbital motion detected across all centers (amplitude variations:
36–55%). The 912-day record spans 2.50 orbital cycles; at this span the
annual period is shared identically by orbital-velocity coupling,
perihelion solar flux, seasonal ionospheric morphology, and seasonal
tropospheric/loading cycles, so frequency alone cannot discriminate the
driver. The operative discriminator is the energy-vs-velocity scaling
test (Section 3.3.2), which returns near-zero discrimination
(Step 4.9).

## 3.3.2 Earth Motion and Energy Hierarchy Analysis

Comprehensive analysis of Earth's complex motion through spacetime reveals
systematic coupling between gravitational energy scales, temporal dynamics,
and collective network dynamics. The global GNSS network exhibits unified detector
behavior combined with sophisticated sensitivity to orbital motion, polar
wobble, and gravitational field variations.

**Experimental Section:**

### Collective Network Coherence Foundation

#### Collective Network Dynamics

**Key Finding:** The global GNSS network exhibits
coordinated dynamics as a unified detector system, not just
independent pairwise correlations. Individual component analysis
reveals:

**Individual Components (Primary Evidence):**

**Base mesh coherence:** 0.643–0.644 (strong
network-wide synchronization)

**Spiral motion:** Detected across all centers
(rotational dynamics)

**Collective oscillation:** Annual period
(365.25 days) with R² = 0.35, $p \lt 0.001$

**Earth coupling:** Oscillation period matches
Earth's orbital motion (period shared by all seasonal
drivers; driver identity not resolved at 2.5 cycles)

The consistency across different analysis centers demonstrates
genuine network-wide synchronization patterns. Multi-frequency beat
analysis reveals 35 significant patterns, while relative motion
analysis detects 9–11 coupling patterns per center, providing robust
evidence for collective network behavior based on individual
component validation.

See Section 2.3 for detailed network covariance methodology and component
definitions.

| Analysis Center | Dataset Size | Collective Network Dynamics | Anisotropy CV (λEW/λNS) |
| --- | --- | --- | --- |
| CODE | 39.0M pairs | 0.624 | 0.475 |
| IGS Combined | 12.9M pairs | 0.579 | 0.555 |
| ESA Final | 10.8M pairs | 0.602 | 0.586 |

### Gravitational Energy Hierarchy Validation

Systematic analysis investigates whether TEP detection strength
correlates with gravitational energy scales versus kinematic velocities,
revealing sophisticated coupling mechanisms involving Earth-Moon
gravitational dynamics.

| Motion Type | Energy Scale (J) | Detection Strength | Physical Mechanism |
| --- | --- | --- | --- |
| Orbital Motion | ~10³³ | |r| = 0.571-0.793 | Earth-Sun gravitational binding |
| Chandler Wobble | ~10²⁰ J (mechanical), ~10²⁸ J (gravitational
modulation)† | |r| = 0.614-0.687 | Earth-Moon gravitational modulation |
| Daily Rotation | ~10²⁹ | CV of rotational stability = 0.475-0.586 | Bulk rotational motion |

<sup>†</sup>The ~10²⁸ J gravitational modulation scale represents
the effective energy scale of Earth-Moon gravitational field
variations induced by Chandler wobble. As Earth's polar axis shifts
~9 meters during the 433-day cycle, the changing Earth-Moon
gravitational geometry modulates tidal force patterns and
gravitational field gradients. This gravitational field modulation,
rather than the mechanical wobble energy (~10²⁰ J), appears to drive
the observed TEP coupling strength, consistent with the hypothesis
that TEP signatures scale with gravitational field variations rather
than mechanical energy alone. Note: This energetic
comparison is heuristic and qualitative—the detection is made in
correlation space on binned means (R² = 0.377–0.471), not through
direct energy measurements.

### Moon-Chandler Coupling Mechanism

The unexpectedly strong Chandler wobble TEP signature (R² = 0.377–0.471,
|r| = 0.61–0.69) compared to its mechanical energy (~10²⁰ J) shows
patterns consistent with Earth-Moon gravitational field modulation.
Note: This energetic comparison is heuristic—the detection is made
in correlation space on binned means, not through direct energy
measurements. The inference is made in correlation space on bin
means, not from absolute energetics, to avoid misreads.
During the 433-day Chandler cycle:

- Polar axis shifts ~9 meters from geographic poles

Changes Earth-Moon gravitational geometry and tidal force patterns

- Modulates gravitational field gradients at ~10²⁸ J scale

Associates with stronger TEP coupling than mechanical wobble energy
alone would predict

*Quantitative sketch (order-of-magnitude):* The ~9 m polar wander
corresponds to a reorientation of Earth's figure by δθ ≈ 9 m / R⊕
≈ 1.4×10⁻⁶ rad. The lunar tidal differential acceleration at Earth's
surface is ≈ 1.1×10⁻⁶ m s⁻²; a reorientation by δθ modulates its
projection along a fixed station baseline by O(δθ) via a cosine
projection, i.e., a fractional change of ~10⁻⁶ in the common-mode driver
at 433 days. In the phase framework, the distance-binned phase-alignment
index C(r) = 𝔼[cos(arg S<sub>ij</sub>)] satisfies C ≈ ½κ in the
small-signal regime, with κ ∝ A<sub>common</sub>(r)²; therefore a
fractional geometric modulation δ produces ΔC/C ≈ 2δ = O(10⁻⁶). This
links the 433-day Earth–Moon geometry change to a parts-per-million
level modulation of bin-mean phase-alignment index at fixed r—small per
bin but statistically resolvable after aggregation—consistent with the
detected Chandler-phase correlation and the Chandler±annual sidebands
below.

| Analysis Center | Chandler Wobble Detection | Interference Patterns | Statistical Significance |
| --- | --- | --- | --- |
| CODE | R²=0.377 (p=0.029) | 4 detected | Moderate-Strong |
| IGS Combined | R²=0.471 (p=0.008) | 4 detected | Strong |
| ESA Final | R²=0.453 (p=0.011) | 4 detected | Strong |

### Earth Motion Interference Patterns

All three centers consistently detect four Earth motion interference
patterns, demonstrating network sensitivity to the complex interplay of
terrestrial rotation, orbital motion, and polar axis wandering:

| Frequency Type | Interference Period (days) a | CODE (r) | IGS Combined (r) | ESA (r) |
| --- | --- | --- | --- | --- |
| M2–S2 (beat/difference) | 14.8 | 0.646 | 0.652 | 0.598 |
| Chandler + Annual (combination/sum) | 198.13 | 0.919 | 0.717 | 0.864 |
| Chandler + Semiannual (combination/sum) | 127.9 | 0.933 | 0.887 | 0.894 |
| Annual + Semiannual (combination/sum) | 121.8 | 0.962 | 0.877 | 0.903 |

**Figure 6. Earth Motion Interference Patterns.** Four
interference patterns detected consistently across all analysis
centers (r = 0.598–0.962): one beat frequency (M2–S2 difference) and
three combination frequencies (Chandler+Annual, Chandler+Semiannual,
Annual+Semiannual sums), demonstrating network sensitivity to the
complex interplay of terrestrial rotation, orbital motion, and polar
axis wandering. <sup>a</sup>Combination periods are calculated as
1/T = 1/T1 + 1/T2, using T1 = 433 days (Chandler wobble) and T2 =
365.25 days (Annual cycle).

### Energy vs Velocity Scaling Analysis

Methodology: Discrimination computed as the simple arithmetic difference
of Pearson correlations: Discrimination = r(energy) - r(velocity),
calculated independently across three analysis centers (CODE, ESA Final,
IGS Combined) and then averaged.

Result: The non-significant discrimination (near-zero difference)
indicates that energy and velocity scales correlate similarly with mesh
dance dynamics. Theoretical interpretation is presented in Section
4.5.1.

## 3.3.3 Orbital Periodicity Effects

Analysis of planetary orbital phase correlations reveals orbital
completeness dependency, with Venus showing the strongest effects due to
completing 4.05 orbital cycles within the analysis window.

**Experimental Section:**

### Cross-Planetary Results

| Planet | Orbits Completed | ESA | IGS Combined | CODE |
| --- | --- | --- | --- | --- |
| Mercury | 10.36 | +0.88% | -0.46% | +3.98% |
| Venus | 4.05 | +17.7%* | +10.6%* | +4.8% |
| Mars | 1.33 | -4.76% | -10.40% | +3.39% |
| Jupiter | 0.21 | -11.73% | -8.77% | +7.94% |

*Statistically significant after advanced sham controls (randomized
orbital phase assignments) and comprehensive multiple comparison
corrections (see Supplement: Multiple Comparison Corrections
Summary, Table S1).

Key observation: Venus (4.05 complete orbits) shows strongest
correlation, supporting hypothesis that orbital completeness enhances
signal detectability in orbital phase analysis.

## 3.3.4 Multi-Frequency Beat Pattern Analysis

Comprehensive temporal analysis detects 35 beat frequency patterns across
all analysis centers (using R²>0.01 detection threshold), with 7 patterns
achieving statistical significance after FDR correction. This indicates
multi-scale coupling between the GPS timing network and periodic Earth
motion components, with hierarchical strength patterns expected for genuine
physical signals.

**Experimental Section:**

### Beat Frequency Detection Summary

| Analysis Center | Beat Patterns Detected | Relative Motion Patterns | Total Analyzed |
| --- | --- | --- | --- |
| CODE | 35 | 11 of 12 | 12 frequency bands |
| ESA Final | 35 | 10 of 12 | 12 frequency bands |
| IGS Combined | 35 | 9 of 12 | 12 frequency bands |

### Primary Beat Frequency Patterns

Analysis reveals four dominant beat patterns with strong statistical
significance across all centers:

| Pattern Period | R² Range | Physical Interpretation | Multi-Center |
| --- | --- | --- | --- |
| **196.9 days** | **0.82-0.86** | Venus sub-harmonic (~1/2 synodic period) | ✓ Strong |
| **127.9 days** | **0.62** | Mercury synodic period resonance | ✓ Moderate |
| **14.765 days** | **0.60-0.82** | Lunar fortnight (half synodic month) | ✓ Strong |
| **0.517 days** | **0.56** | Semi-diurnal cycle (12.4 hours) | ✓ Moderate |

### Multi-Scale Temporal Coupling

The detected beat patterns span four orders of magnitude in timescale,
from hours to months:

**Sub-diurnal scale:** 0.077-0.517 days (1.8-12.4
hours) - Earth rotation harmonics

**Diurnal-to-weekly scale:** 0.77-14.8 days - Lunar and
solar tidal interactions

**Monthly-to-seasonal scale:** 127.9-196.9 days -
Planetary synodic resonances

**Annual scale:** 365.25 days - Earth orbital period
(detected in collective network covariance analysis)

**Physical Significance:** The detection of 35 beat
patterns (7 statistically significant after FDR correction),
combined with 9-11 showing robust relative motion coupling per
center (all 11 patterns significant across centers), demonstrates
that the GPS timing network exhibits sensitivity to multiple
overlapping periodic influences. The strongest patterns (R² > 0.80)
correspond to well-defined astronomical periods (Venus 196.9d, lunar
14.8d), suggesting genuine coupling to gravitational field geometry
variations. The hierarchical pattern—strong at key frequencies
(tidal, post-tidal), weaker at higher frequencies—is characteristic
of genuine physical coupling rather than uniform noise or artifacts.
The multi-center consistency substantially reduces the likelihood of
center-specific systematic artifacts.

## 3.4 Gravitational and Astronomical Correlations

**Experimental Section:**

Summary: This group analyzes correlations between TEP signals and
planetary gravitational influences, astronomical events, and mass
scaling relationships, revealing complex coupling mechanisms and dynamic
field responses.

## 3.4.1 Gravitational-Temporal Field Correlations

Correlations between planetary gravitational influences and GPS timing
coherence reveal systematic coupling with characteristic timescale
dependencies.

**Experimental Section:**

### Multi-Window Timescale Analysis

Gravitational-temporal correlations strengthen systematically with
longer analysis windows:

- 31-day window: r = -0.362, autocorr-corrected p = 1.3×10⁻²

- 59-day window: r = -0.400, autocorr-corrected p = 5.5×10⁻³

- 91-day window: r = -0.428, autocorr-corrected p = 2.8×10⁻³

- 119-day window: r = -0.454, autocorr-corrected p = 1.4×10⁻³

- 179-day window: r = -0.476, autocorr-corrected p = 7.4×10⁻⁴

227-day window: r = -0.503, autocorr-corrected p = 3.3×10⁻⁴
(raw p = 1.5×10⁻⁵⁹; selection-adjusted across the seven-window
scan, p ≃ 2.3×10⁻³)

Unsmoothed correlation: r = -0.164, autocorr-corrected p =
2.5×10⁻⁵

Pattern: Correlation strength increases systematically with smoothing
window, indicating gravitational-temporal coupling operates on seasonal
to annual timescales.
Note: All p-values in this section are autocorrelation-corrected
per §2.3 (N<sub>eff</sub> ≃ 47 for the smoothed series;
N<sub>eff</sub> ≃ 652 unsmoothed). The optimal window is
selected from a fixed seven-window scan; the quoted
selection-adjusted value applies a Bonferroni factor over that
scan.

### Individual Planetary Signatures

Jupiter: r = -0.175, autocorr-corrected p = 7.2×10⁻⁶ (strongest
individual effect)

Saturn: r = -0.101, autocorr-corrected p = 1.0×10⁻² (incomplete
orbital coverage)

Sun: r = -0.065, autocorr-corrected p = 0.10 (dominant mass
influence, marginal against temporal variability)

- Mars: r = -0.072, autocorr-corrected p = 0.068

- Venus: Complex distance-dependent coupling

Anti-phase seasonal anti-correlations: Higher gravitational influence
correlates with lower temporal phase-alignment index across all analysis
centers.

*Note:* These long-term gravitational correlations (seasonal
timescales with multi-month smoothing windows) reflect different
physical mechanisms than the short-term opposition event responses
examined in Section 3.4.3 (±30–60 day event windows). The
anti-correlation pattern observed here operates on annual cycles, while
opposition events show transient, center-specific responses to
gravitational alignments. Both phenomena may coexist, representing
coupling at different timescales.

**Figure 7. Gravitational-Temporal Field Correlations.**
Four-panel analysis showing: (1) stacked planetary gravitational
influences, (2) temporal field signatures from 62.7M
quality-filtered measurements, (3) anti-correlation pattern, (4)
multi-window timescale strengthening from r = -0.362 (31 days) to r
= -0.503 (227 days).

## 3.4.2 Dynamic Astronomical Responses

High-resolution analysis of GPS coherence during astronomical events reveals
substantial network modulations, demonstrating sensitivity to rapid
ionospheric perturbations and gravitational field variations.

**Experimental Section:**

### Solar Eclipse Effects

Analysis of five solar eclipses (2023–2025) using 30-second resolution
CLK data reveals statistically significant phase-alignment index
modulations across 294,572 station pair measurements. Eclipse effects
range from 18–87% of baseline TEP signal strength, representing major
network responses to ionospheric disruptions:

| Date | Type | Location | CODE | IGS Combined | ESA |
| --- | --- | --- | --- | --- | --- |
| 2023-04-20 | Hybrid | Australia/Indonesia | +6.5×10⁻³ | +9.7×10⁻³ | +2.5×10⁻² |
| 2023-10-14 | Annular | Americas | +3.8×10⁻² | +3.5×10⁻² | +3.5×10⁻² |
| 2024-04-08 | Total | North America | -8.4×10⁻⁴ | +6.3×10⁻² | -4.0×10⁻³ |
| 2024-10-02 | Annular | South America | +2.2×10⁻² | +3.6×10⁻³ | +2.6×10⁻³ |
| 2025-03-29 | Partial | Atlantic/Europe | +5.32×10⁻² | +2.88×10⁻² | +5.42×10⁻² |

Eclipse response patterns: Partial eclipses produce the strongest
network phase-alignment index modulation (mean: 4.54×10⁻², 87% of
baseline), followed by annular eclipses (mean: 2.48×10⁻², 48% of
baseline). A geometry-matched control yields the same ordering,
suggesting this hierarchy may reflect TEP correlations responding
primarily to ionospheric gradient steepness rather than absolute
disruption magnitude. Partial eclipses create broader spatial gradients
across the global network, while total eclipses with narrow totality
paths produce more localized but less network-wide effects.

### Supermoon Perigee Effects

Analysis of 11 supermoon events (2023–2025) reveals coherence
modulations:

- CODE: -0.211 ± 0.220%

- IGS: +0.298 ± 0.534%

- ESA: +0.419 ± 0.300%

Range: -3.17% to +2.04% (ΔR² = 0.85-0.93 relative to control-band
baseline at 1000–1500 μHz)

## 3.4.3 Opposition Event Responses

Analysis of planetary opposition events reveals short-term coherence
modulations during peak gravitational alignment periods. These measurements
are distinct from the long-term gravitational correlations reported in
Section 3.4.1, examining event-specific responses (±30–60 day windows)
rather than continuous daily correlations. Unless otherwise noted, results
are evaluated across a default multi-window sweep (±30, ±60, ±120, ±180,
±240 days) to assess robustness.

**Experimental Section:**

### Opposition Event Results Across Analysis Centers

| Event | CODE | IGS | ESA |
| --- | --- | --- | --- |
| Saturn Opp. 2023-08-27 | -23.27% | +47.98% | +93.27% |
| Saturn Opp. 2024-09-08 | -1.85% | -44.79% | -112.83% |
| Mars Opp. 2025-01-16 | -17.33% | -33.01% | -60.21% |
| Jupiter Opp. 2023-11-03 | +21.45% | +2.72% | -48.28% |
| Jupiter Opp. 2024-12-07 | +24.02% | -15.57% | -13.65% |

Values are the signed event-locked direct effect: the change in mean
daily coherence between the event window (|d| ≤ 2 days) and the
local baseline annulus (15 ≤ |d| ≤ 30 days), expressed as a
percentage of that baseline. This metric replaces the earlier
free-centre Gaussian amplitudes, which could lock onto unrelated
excursions tens of days from the event.

### Observed Patterns

Mars opposition: Same-signed suppression in all three centers
(-17.33% to -60.21%), yet within each center's placebo band
(21st–33rd percentile of sliding-center null)

Saturn oppositions: Strongly divergent; ESA reaches +93.27% (2023,
~99th placebo percentile) but falls to -112.83% (2024), and the
2023 excursion is not reproduced by CODE (-23.27%) or IGS
(+47.98%)

Jupiter oppositions: CODE positive (+21.45% to +24.02%), IGS and
ESA mixed-to-negative; all within placebo bands

Center variations: Per-center residuals scatter over a range far
wider than any candidate signal, indicating that transient
event-window fluctuations are dominated by center-specific
network-adjustment noise

Note: These short-term event responses differ from long-term
gravitational correlations (Section 3.4.1), reflecting different
physical timescales and mechanisms. Audit of the earlier
center-discordant values traced them to unconstrained-centre Gaussian
fits rather than to the data themselves; the corrected direct test
reported here finds no opposition event with a consistent
cross-center, placebo-exceeding response. Full audit:
results/outputs/planetary_channel_corrected_audit.json.

**Experimental Section:**

### Window Sensitivity (Default Multi-Window Analysis)

Outer planets (Jupiter, Saturn, Mars): the earlier observation that
fitted amplitudes strengthen with longer windows (±120–±240) is now
identified as a fitting artifact—a free-centre Gaussian prefers the
largest excursion inside the window, so longer windows admit larger
off-event excursions. Event-locked direct effects show no genuine
window growth and remain within placebo noise.

Mercury: nominally significant fits exist in IGS and ESA but are
mostly centred 4–65 days off the conjunction; direct effects are
discordant across centers and not robust.

Venus: the March 2025 conjunction is the strongest surviving
event-locked case—ESA fit centred +3.3 days (σ ≈ 4 across all
windows, direct effect +140%, 96th–98th placebo percentile), with
CODE positive (+72%) but IGS negative (−28%); single-event,
non-cross-center.

Interpretation now relies on event-locked direct effects with
sliding-centre placebo calibration; σ-levels from free-centre fits are
not treated as event detections.

## 3.4.4 Planetary Mass Scaling Analysis

### Preliminary Observation: Planetary Enhancement Factor Analysis

**Statistical Power Considerations:** With n=5 planets and
vastly different temporal sampling (Mercury: 7.9 cycles, Jupiter: 2.3
cycles, Mars: 1.2 cycles), direct correlation between observed event
amplitudes and Newtonian gravitational scaling (GM/d²) returns a null
result (mean r = −0.156). Individual opposition events also exhibit
large center-to-center scatter that cannot represent a coherent field
response: Saturn 2023 yields −23.27% in CODE but +93.27% in ESA, and
Jupiter 2023 yields +21.45% in CODE versus −48.28% in ESA. (The
sign-reversed Mars-2025 pair quoted in earlier drafts was an artifact
of unconstrained-centre Gaussian fits; the corrected event-locked
metric gives same-signed suppression in all centers, within placebo
noise.) Because a physical scalar field modulation imposes a definite
sign, this cross-center discordance indicates that transient
single-event windows in processed clock products do not isolate
coherent planetary detections, but instead provide empirical upper
bounds on transient planetary field perturbations. Extended 10+ year
observations will be required to place tighter constraints on
long-period planetary influences.

Robustness to temporal window choice was assessed using the default
multi-window sweep (±30, ±60, ±120, ±180, ±240 days). The corrected
event-locked direct effect samples only |d| ≤ 30 days of the event
series, so it is identical across the longer windows by construction;
window sensitivity now enters only through the placebo calibration.
Earlier reports that outer-planet responses strengthen with seasonal
windows (±120–±240) traced to free-centre Gaussian fits that locked
onto off-event excursions inside larger windows; with the centre
constrained to the event, that apparent scaling disappears.

#### Observation 1: Enhancement-Factor Distribution (Retired Metric)

Under the descriptive definition in §2.6, the catalogued values span
Jupiter 5.3× to Mercury 137× (cross-center means). These numbers
are retained for provenance only: because the per-event residual
amplitude A<sub>obs</sub> sits near the baseline noise floor across
all planets, E ≡ A<sub>obs</sub>/(M/d²) reduces
algebraically to ∝ d²/M, so the ordering encodes the
Newtonian denominators rather than any measured coupling strength
(see the algebraic demonstration below). The estimator is
therefore not evidence of geometric screening enhancement and is
retired as a physical channel.

| Planet | Mass (M⊕) | Cross-Center Mean Enhancement |
| --- | --- | --- |
| Mercury | 0.055 | 137× |
| Mars | 0.107 | 215× |
| Saturn | 95.2 | 59× |
| Venus | 0.815 | 18× |
| Jupiter | 317.8 | 5.3× |

#### Observation 2: Mass Scaling Analysis

To test whether observed amplitudes follow gravitational scaling, a
direct correlation is computed between A<sub>obs</sub> and the expected amplitude
(M/d²). This proper test avoids the circular analysis of testing E
vs mass (since E ≡ A<sub>obs</sub>/(M/d²) by definition).
Cross-center mass scaling analysis reveals consistent
null-to-negative correlations:

| Analysis Center | Linear r
(Aobs vs M/d²) | Quadratic r
(Aobs vs (M/d²)²) | Mean Enhancement | Coupling Type |
| --- | --- | --- | --- | --- |
| CODE | -0.049 | -0.038 | 63× | WEAK/UNCLEAR |
| IGS Combined | -0.086 | -0.116 | 91× | WEAK/UNCLEAR |
| ESA Final | -0.333 | -0.344 | 93× | WEAK/UNCLEAR |
| Mean | -0.156 | -0.166 | 82× | — |

#### Interpretation

The direct mass-scaling analysis yields an average correlation of
r = −0.156 across all three analysis center solutions, indicating
no statistically significant correlation between observed residual
amplitude variations A<sub>obs</sub> and Newtonian gravitational
field strength (M/d²). This null result is expected because
routine GNSS least-squares adjustment absorbs single-station
amplitude-level offsets into station and satellite clock parameters at
the ~10<sup>−13</sup> level, orders of magnitude larger than
the expected ~10<sup>−16</sup> planetary signatures, while
preserving phase-coherent timing structure across the network.

The apparent "inverse mass hierarchy" reported in descriptive
evaluations—wherein Mercury (0.055 M<sub>⊕</sub>) exhibits an
enhancement factor E ≡ A<sub>obs</sub>/(M/d²) of 137×
compared to 5.3× for Jupiter (317.8 M<sub>⊕</sub>)—is an
algebraic consequence of the definition of E. Because the observed
residual amplitude A<sub>obs</sub> remains at the per-event noise
floor (fractional deviations of order the baseline coherence
itself) across all planetary events,
dividing by M/d² mechanically yields E ∝ d²/M by
construction. Consequently, planets with minimal Newtonian field
strengths at Earth naturally register elevated values of E, whereas
massive bodies like Jupiter register small values. The ratio of
enhancement factors therefore reflects the ratio of the Newtonian
denominators rather than empirical proof of a gradient-dominated
disformal coupling mechanism.

**Sign Discordance and Empirical Bounds:** A genuine
scalar-field perturbation would impose a physically consistent sign
on clock variations during planetary configurations. However,
individual events exhibit marked discordance across analysis
centers: Saturn 2023 yields −23.27% in CODE but +93.27% in
ESA; Jupiter 2023 yields +21.45% in CODE but −48.28% in ESA.
(The earlier-quoted Mars-2025 sign reversal was traced to
unconstrained-centre Gaussian fits; with the fit centre fixed on
the event, all centers show a same-signed but placebo-consistent
suppression.) This discordance demonstrates that single-event
amplitude fluctuations are dominated by center-specific orbit
determination filter dynamics and clock re-absorption. Rather than
detecting unscreened scalar perturbations from individual planets,
the analysis establishes an empirical upper bound on transient
planetary field modulations in processed GNSS products.

Within the rigorous formulation of TEP, disformal transport governed
by B(φ)∇<sub>μ</sub>φ∇<sub>ν</sub>φ
(Rule 1) cannot be probed through single-receiver clock offsets, which
are subject to synchronization conventions and network absorption.
Instead, disformal physics is uniquely and gauge-invariantly tested
by closed-loop synchronization holonomy H<sub>resid</sub> =
∮(σ̃ − σ<sub>GR</sub>) around closed
circuits (Rule 13), which vanishes identically in the purely
conformal sector.

### Proposed Mechanisms for Investigation

This result provides a clear direction for theoretical modeling,
pointing toward mechanisms that depend on geometry and timing rather
than mass alone:

Orbital Resonance: Mercury's 88-day orbital period is
near-commensurate with Earth's ~90-day correlation coherence
timescale. This may enable resonant amplification, a mechanism that
would naturally favor planets with specific orbital periods.

Closed-Loop Holonomy vs. Open Amplitudes: While single-station clock
offsets are constrained by network datum absorption and screening,
non-exact disformal transport governed by
B(φ)∇<sub>μ</sub>φ∇<sub>ν</sub>φ
generates invariant closed-loop holonomies H<sub>resid</sub> =
∮(σ̃ − σ<sub>GR</sub>) across triangular
clock networks (Rule 13), providing an invariant physical channel for
future inter-station circuit tests.

Heliospheric Temporal Topology: The continuous spatial profile of
the φ field may create a radial asymmetry in the solar system as
Earth's orbital motion traverses regions of varying ambient field value,
altering the observable response amplitude for inner vs. outer planets.

These observations indicate systematic sensitivity of global GNSS
networks to large-scale spatial structure, Earth's motion,
gravitational fields, and temporal dynamics. The diurnal analysis
provides evidence consistent with dynamical time field coupling.
Comprehensive validation demonstrating signal authenticity is
presented in Section 4.2.

## 3.5 Signal Characterization and Validation

**Experimental Section:**

Summary: This group provides comprehensive signal characterization
across frequency bands, assessment of interference sources, and detailed
temporal analysis, establishing the authenticity and nature of the
observed correlations.

## 3.5.1 Multi-Band Frequency Analysis

Spectral decomposition across 12 frequency bands (10–3000 μHz) reveals the
physical mechanisms underlying TEP correlations and provides compelling
cross-center validation. The TEP band (10–500 μHz) theoretical prediction
shows strong performance with R² = 0.952 ± 0.026 across all analysis
centers, achieving optimal values of 0.970 (ESA Final), 0.920 (CODE), and
0.966 (IGS Combined). Individual sub-bands show systematic gravitational
enhancement patterns that distinguish TEP from classical tidal
contamination, with remarkable consistency across independent processing
chains supporting signal authenticity.

**Figure 8. TEP Multi-Band Performance Analysis.**
TEP band (10–500 μHz) prediction leads the analysis with strong
cross-center consistency (R² = 0.952 ± 0.026), achieving optimal values
of 0.970 (ESA Final), 0.920 (CODE), and 0.966 (IGS Combined). Individual
frequency sub-bands show systematic patterns with enhanced tidal
frequencies and persistent post-tidal correlations. The remarkable
cross-center consistency substantially reduces the likelihood of
center-specific systematic biases and supports the theoretical framework
of universal temporal field coupling across the full frequency spectrum.

**Experimental Section:**

### Key Multi-Band Findings

Cross-Center Validation Success: Three independent analysis
centers achieve nearly identical patterns - ESA Final (R² =
0.970), CODE (R² = 0.920), and IGS Combined (R² = 0.966) in
optimal bands, eliminating systematic biases

Gravitational Enhancement: Tidal frequencies show enhanced
spatial scales - diurnal: λ<sub>T</sub> = 4,677 km mean, semidiurnal: λ<sub>T</sub> =
4,676 km mean across centers

Critical Discriminator: Post-tidal 30–40 μHz band exhibits
strongest signal (R² = 0.946 mean) while excluding classical
tidal contamination

Control Band Validation: Appropriately reduced performance (R² =
0.618 mean) at 1000–1500 μHz confirms signal selectivity

Frequency Specificity: Enhancement ratios (1.3-1.9×) exclude
tidal contamination (>3× threshold); classification ranges from
"WEAK" (ESA) to "NONE" (CODE/IGS)

The fitted correlation length is band-ordered rather than a single
universal value. Table 8a reports the per-band exponential decay scale
for each center (outputs: step_3_6_multiband<sub>code,igs_combined,esa_final</sub>.json).
Within the coherent spectrum, λ<sub>T</sub> is longest at the tidal
frequencies (3,638–5,912 km), declines through the post-tidal 30–40 μHz
band (2,143–2,453 km), and settles near 1,050–1,450 km across the
40–200 μHz sub-bands. At transition and control bands the exponential fit
itself degrades (R² = 0.47–0.75), so those entries are ill-conditioned
estimates of a decay whose carrier is absent—not independent measurements
of a different scale. The ordering is consistent across all three centers,
which is the signature of a frequency-structured transfer function rather
than incoherent scatter: the pooled-band estimate λ<sub>T</sub> =
3,330–4,549 km is the designated primary scale because it integrates the
coherent content, while the per-band decomposition reports its internal
structure. A broadband geometry artifact (co-visibility, troposphere)
would be expected to fit all bands comparably; the observed R² collapse
outside the coherent spectrum argues the decay's carrier is
signal-localized. Whether the band ordering reflects screening
stratification or a systematic transfer mixture is the same open
degeneracy registered in §4.6. The visibility/datum component of that
degeneracy is bounded by forward simulation at λ ≳ 6.7×10<sup>3</sup>
km, and at λ ≥ 5,075 km when conditioned on the real
product's satellite-clock content and ephemerides (§4.6), above the
pooled measured scale.

| Band | λT CODE (km) | λT IGS (km) | λT ESA (km) | Mean R² |
| --- | --- | --- | --- | --- |
| TEP band (10–500 μHz, pooled) | 4,549 | 3,764 | 3,328 | 0.952 |
| Tidal diurnal | 5,912 | 3,920 | 4,199 | 0.926 |
| Tidal semidiurnal | 5,690 | 4,701 | 3,638 | 0.956 |
| Post-tidal 30–40 μHz | 2,404 | 2,453 | 2,143 | 0.946 |
| Post-tidal 40–50 μHz | 1,355 | 1,409 | 1,370 | 0.839 |
| Post-tidal 50–75 μHz | 1,190 | 1,192 | 1,326 | 0.846 |
| Post-tidal 75–100 μHz | 1,069 | 1,067 | 1,047 | 0.876 |
| Intermediate 100–200 μHz | 1,213 | 1,260 | 1,406 | 0.864 |
| Intermediate 200–350 μHz | 1,059 | 1,383 | 1,931 | 0.679 |
| Intermediate 350–500 μHz | 1,198 | 1,432 | 2,325 | 0.681 |
| Transition 500–750 μHz | 1,373 | 2,732 | 1,943 | 0.643 |
| Transition 750–1000 μHz | 1,433 | 1,954 | 2,194 | 0.589 |
| Control 1000–1500 μHz | 1,607 | 2,386 | 2,454 | 0.618 |
| Control 2000–3000 μHz (ESA only) | — | — | 2,111 | 0.553 |

Cross-corpus reconciliation follows directly once the band and product
are specified. The corpus estimates partition into two classes: pooled or
tidal-coherent estimates (this paper's headline 3,330–4,549 km and Paper
33's ≈4,200 km) and sub-band or product-specific estimates (this paper's
post-tidal/intermediate floor ≈1,050–2,450 km; Paper 14's MGEX combined
phase-alignment λ = 1,862 km; MGEX product sub-bands ≈1,050–1,935 km;
Paper 33's band spread 754–4,348 km spans both classes). The fitted
correlation length is therefore a band- and product-dependent
phenomenological observable; the invariant TEP signature is the existence
of a structured multi-thousand-kilometre decay in the coherent spectrum,
not a single number. The pooled TEP-band estimate reported here is
designated the primary corpus scale.

**Experimental Section:**

### Signal Enhancement Over Null Expectation

Comprehensive comparison with 180 null hypothesis iterations (60 per
scrambling type × 3 centers) quantifies the authenticity and strength of
the observed correlations:

| Metric | Value | Interpretation |
| --- | --- | --- |
| **Mean Enhancement Factor** | **124.4×** | Signal 124× stronger than random |
| Median Enhancement | 124.4× | Consistent with mean (symmetric) |
| Enhancement Range | 22.8× - 226.0× | All centers show strong enhancement |
| Standard Deviation | 101.6× | Substantial variation across centers |
| Coefficient of Variation | 0.82 | Moderate relative uncertainty |

#### Physical Interpretation

A mean signal-to-null ratio >100× indicates that the observed
distance-coherence structure lies far outside what the scrambling
nulls produce — the correlations are not explainable as
distance-independent noise. Candidate physical completions that have
been considered include:

**Resonance effects:** Network may be resonantly
coupled to specific Earth motion frequencies

**Tidal amplification:** Gravitational field
variations amplified through multi-body interactions

**Parametric coupling:** Time-varying field
geometry creates parametric amplification

**Geophysical mode coupling:** Interaction with
Earth's natural oscillation modes

**Validation Context:** The enhancement factor was
computed from ESA Final center analysis comparing real vs. null
distributions. All three centers show comparable signal-to-noise
ratios (z = 15.8-31.9), providing cross-centre confirmation of the
signal-over-null margin. The >100× amplification rules out the
scrambled-null family as an explanation of the correlation
structure; the candidate non-linear coupling interpretations listed
above remain hypotheses pending mechanistic derivation.

**Figure 9. Multi-Band Spectral Analysis Overview.**
**(A) Spectral correlation structure:** R² > 0.85 from tidal to
intermediate bands.
(B) Gravitational enhancement pattern: Longest λ at tidal
frequencies with sharp decrease by a factor ≈3.1 (from 4,677 km→1,502
km post-tidal group mean; per-band decomposition in Table 8a).
(C) Frequency specificity test: Modest enhancement excludes tidal
contamination. (D) Cross-center consistency: Strong signals show
excellent agreement.

**Figure 10. Comprehensive Post-Tidal Discriminator Analysis.**
**(A) Post-tidal 30–40 μHz prominence:** Strongest band excludes
classical tidal contamination while maintaining high correlation quality
across all analysis centers (mean R² = 0.946, equal to tidal bands).
**(B) Frequency specificity validation:** All enhancement ratios $\lt$2×
support universal coupling interpretation (classical tidal contamination
would require >3× ratios), distinguishing TEP from tidal artifacts and
demonstrating universal broadband coupling extending beyond tidal
frequencies.

## 3.5.2 Ionospheric Interference Assessment

To quantify potential ionospheric contamination of the TEP correlation
signals, comprehensive Traveling Ionospheric Disturbance (TID) exclusion
analysis was performed using high-resolution temporal analysis data. This
assessment provides quantitative bounds on electromagnetic interference
effects and validates signal purity.

**Experimental Section:**

### TID Impact Assessment Results

**Experimental Section:**

### Cross-Band Correlation Patterns

| Frequency Region | Mean R² | Mean λ (km) | R² CV |
| --- | --- | --- | --- |
| TEP Band (10–500 μHz) | 0.952 | 3,880 | 2.4% (Mean across centers) |
| Tidal Bands (10–30 μHz) | 0.941 | 4,677 | 2.7% |
| Post-Tidal (30–100 μHz) | 0.882 | 1,502 | 3.2% |
| Intermediate (100–500 μHz) | 0.748 | 1,459 | 9.1% |
| Control (1000–1500 μHz) | 0.618 | 2,149 | 14.4% |

Cross-center spectral validation: All three analysis centers
demonstrate exceptional consistency in multiband patterns, with
optimal bands achieving R² = 0.970 (ESA Final), 0.920 (CODE
post-tidal), and 0.966 (IGS Combined semidiurnal). The CODE analysis
center demonstrates strong spectral uniformity across the 10–200 μHz
range with R² coefficient of variation of only 2.9%, indicating
robust exponential decay structure maintained across 20-fold
frequency span. This cross-center convergence represents a crucial
validation milestone, substantially reducing center-specific
systematic biases and supporting signal authenticity through
independent processing methodologies. Values in the table above
represent means across all three analysis centers (CODE, ESA Final,
IGS Combined).

#### Detailed Frequency Band Analysis

Comprehensive 12-band frequency analysis reveals systematic spectral
characteristics across the complete 10–3000 μHz range:

| Frequency Band | Range (μHz) | Mean R² | Mean λ (km) | Enhancement Ratio |
| --- | --- | --- | --- | --- |
| Tidal Diurnal | 10-20 | 0.941 | 4,653 | 1.52× |
| Tidal Semidiurnal | 20-30 | 0.941 | 4,701 | 1.52× |
| Post-Tidal 30-40 | 30-40 | 0.946 | 2,453 | 1.53× |
| Post-Tidal 40-50 | 40-50 | 0.832 | 1,409 | 1.35× |
| Post-Tidal 50-75 | 50-75 | 0.864 | 1,502 | 1.40× |
| Intermediate 100-200 | 100-200 | 0.748 | 1,459 | 1.21× |
| Transition 500-750 | 500-750 | 0.695 | 2,089 | 1.12× |
| Control 1000-1500 | 1000-1500 | 0.618 | 2,149 | 1.00× (baseline) |

Key spectral findings:

Tidal enhancement: Both diurnal and semidiurnal tidal bands show
identical enhancement (1.52×) with λ<sub>T</sub> = 4,677 ± 954 km,
indicating gravitational forcing maxima

Post-tidal persistence: The 30–40 μHz band exhibits the
strongest correlation (R² = 0.946), demonstrating signal
persistence beyond classical tidal frequencies

Gradual spectral rolloff: Enhancement ratios decrease
systematically from 1.53× (post-tidal) to 1.12× (transition),
indicating broadband coupling rather than sharp frequency
cutoffs

Universal coupling evidence: All enhancement ratios $\lt$2× support
universal conformal coupling interpretation (classical tidal
contamination would require >3-5× ratios)

Spatial scale transition: Mean transition ratio of 3.1× between
tidal (4,677 km) and post-tidal (1,502 km) correlation lengths
demonstrates systematic frequency-dependent screening

#### Temporal Pattern Analysis

High-resolution Hilbert instantaneous frequency analysis reveals
systematic temporal effects across multiple frequency bands:

Solar Rotation (27-day): Moderate effects (0.51–0.60) with
statistically significant coupling in IGS Combined (p = 0.009)

Lunar Month (29-day): Variable response with ESA Final showing
anomalously low sensitivity (0.068 vs 0.60–0.93)

Venus 2f harmonic (~112-day): Strongest temporal effects across all
centers (1.23–1.73), with observed dominant periods spanning 109–118
days across centers (3–5% deviation from the 112.35-day second
harmonic of Venus' 224.7-day synodic period).

Lunar Harmonic (20-day): Consistent coupling (0.12–0.39)
corresponding to ~2/3 lunar month (19.86d ≈ 19.69d theoretical),
representing lunar tidal harmonics rather than Jupiter-Saturn beat

#### Ionospheric Contamination Quantification and Signal Purity

Key Finding: Ionospheric Discrimination Validates TEP
Mechanism

TID exclusion analysis provides three critical validations:

**Signal Composition:** Approximately 78% of
correlation structure is ionosphere-independent (robust TEP signal),
while 22% represents ionospheric modulation (suppressant noise).
This demonstrates genuine atomic clock coupling rather than
electromagnetic propagation effects.

**Enhancement Direction:** Excluding high-TID periods
IMPROVES signal strength by 21-23%, proving ionosphere SUPPRESSES
TEP correlations rather than creating them. This falsifies
ionospheric artifact hypotheses.

**Physical Mechanism Discrimination:** If TIDs created
the signal, exclusion would REDUCE correlations. Observed
enhancement confirms TEP field couples to atomic frequencies below
ionospheric layer.

**Theoretical Consistency:** TEP predicts φ-field
propagation at c through vacuum, with ionospheric electron density
creating screening effects that reduce coupling strength. Observed
21-23% suppression matches theoretical expectations for ionospheric
screening.

| Analysis Center | TID Contamination | Robust Signal | Improvement Potential |
| --- | --- | --- | --- |
| CODE | 22.36% | **77.64%** | +22.4% after filtering |
| ESA Final | 21.28% | **78.72%** | +21.3% after filtering |
| IGS Combined | 22.66% | **77.34%** | +22.7% after filtering |

**Physical Interpretation:** The 77–79%
ionosphere-independent signal retains all key characteristics:
exponential spatial decay (λ<sub>T</sub> = 3,330–4,549 km), cross-center consistency
(R² = 0.920–0.970), Earth motion coupling, and planetary gravitational
correlations. This demonstrates that while ionospheric effects
contribute measurably to signal variability, they cannot account for the
primary correlation structure or its physical dependencies.

**Venus 2f Harmonic as Ionosphere-Independent Validation:**
The dominant ~112-day harmonic appears consistently across centers with
observed periods spanning 109–118 days (3–5% deviation from the
112.35-day expectation), indicating temporal structure not explained by
ionospheric processes alone.

#### Venus 2f Harmonic Validation

The ~112-day band represents the Venus 2f harmonic (224.7 d / 2 = 112.35
d). Observed dominant periods lie in the 109–118 d range across centers
(3–5% deviation):

| Parameter | Value | Interpretation |
| --- | --- | --- |
| Venus Orbital Period | 224.7 days | Known astronomical constant |
| Second Harmonic | 112.35 days | 224.7 ÷ 2 = 112.35 d |
| Observed Period | 109–118 days (across centers) | High-resolution Hilbert-IF dominant period |
| Match Accuracy | 95–97% | Within 3–5% of 112.35 d |
| Temporal Effect Range | 1.23-1.73 | Standard deviation difference (days) between
event-locked and background periods |

Aliasing Control: Annual and Chandler wobble sideband aliases excluded
through Monte-Carlo surrogate phase-scrambling with spectral
prewhitening at 365.25-day and 433-day periods before 112-day period
estimation.

Physical Significance: The consistent ~112-day dominance (109–118 d)
aligns with the Venus 2f expectation (112.35 d) within 3–5%, suggesting
a planetary orbital harmonic present in the temporal correlation
structure. The strongest temporal effects (1.23–1.73 days Δσ (days)
between event-locked and background periods) occurring near this band
indicate Venus orbital dynamics may modulate the temporal field
structure.

Cross-Validation: This finding aligns with Step 3.6 multiband analysis
showing the 112-day period as dominant across all analysis centers, and
supports the non-ionospheric origin through TID exclusion analysis. The
systematic ranking of temporal effects (ESA Final $\lt$ CODE $\lt$ IGS Combined)
correlates with TID impact scores, suggesting Venus coupling strength
varies systematically across analysis centers.

## 3.5.3 Comprehensive Diurnal Analysis: Associations Consistent with Dynamical Time

Comprehensive hourly temporal analysis spanning January 2023 to June 2025
reveals systematic diurnal variations consistent with Temporal Equivalence
Principle predictions for dynamical time flow rates. High-resolution
temporal analysis of 72.4M hourly records across 2023-2025 exhibits patterns
supporting the theoretical prediction that proper time flow varies
systematically according to dτ/dt ∝ exp(β<sub>A</sub>ϕ/M<sub>Pl</sub>), though alternative
explanations involving slow environmental covariates cannot be excluded
pending independent validation.

**Experimental Section:**

### Day-Night Coupling Strength

Systematic day-night variations in correlation strength indicate
coupling to Earth's rotation and orbital position:

CODE: 1.019× stronger during day (1.9% enhancement) - global-mean
ratio

IGS: 1.076× stronger during day (7.6% enhancement) - global-mean
ratio

ESA: 1.060× stronger during day (6.0% enhancement) - global-mean
ratio

### Diurnal Pattern Characteristics

Temporal analysis scope: Comprehensive hourly temporal analysis across
all three analysis center solutions reveals systematic diurnal variations
consistent with φ-field coupling predictions:

| Center | Stations | Temporal Stability | Diurnal Modulation | φ-Field Sensitivity |
| --- | --- | --- | --- | --- |
| CODE | 160 | Daily pair count CV (Diurnal analysis) = 0.196
(variable) | 1.9% (minimal) | Baseline coupling |
| IGS Combined | 262 | CV of daily pair counts = 0.154 (moderate) | 7.6% (strongest) | Enhanced diurnal response |
| ESA Final | 166 | CV of daily pair counts = 0.137 (stable) | 6.0% (moderate) | 2× signal enhancement |

Key Finding: ESA Final demonstrates enhanced φ-field sensitivity with 2×
greater coupling strength compared to CODE, while IGS Combined exhibits
the strongest diurnal modulation (7.6% day-night variation), indicating
center-dependent sensitivity to temporal field dynamics.

### Seasonal Modulation Patterns

The analysis reveals systematic seasonal variations in diurnal peak timing
and day-night asymmetries, providing evidence for annual φ-field modulation
driven by Earth's orbital dynamics and solar angle effects:

#### Winter (DJF): Dawn Acceleration Phase

- Peak Hours: 4-5 AM (consistent early morning)

Day/Night Ratios: CODE 0.737 (nocturnal), IGS 1.098 (balanced),
ESA 0.969 (balanced)

Winter Anomaly: Opposite patterns across centers - CODE shows
night dominance while IGS/ESA show day dominance, indicating
center-specific sensitivity to axial tilt effects

Interpretation: φ-field maxima at dawn when solar angle is
optimal; screening effects vary by latitude

#### Spring (MAM): Maximum Gradient Phase

- Peak Hours: Variable (CODE H8, IGS H7, ESA H20)

Day/Night Ratios (seasonal-peak): All show strongest diurnal
effects (1.074-1.292)

Interpretation: Spring equinox creates maximum φ-gradients via
solar coupling; complex multi-modal structure

#### Summer (JJA): Stability Phase

- Peak Hours: 3-9 AM (morning convergence)

- Day/Night Ratios: Most balanced (0.936-1.047)

Interpretation: Summer solstice minimizes φ-variations;
ionospheric TEC minimum correlates with reduced field variations

#### Autumn (SON): Transition Phase

- Peak Hours: 9-10 AM (coherent morning consensus)

- Day/Night Ratios: Moderate diurnal effects (1.025-1.191)

Interpretation: Autumnal ionospheric transitions create
intermediate φ-field dynamics; systematic atmospheric coupling

#### Theoretical Implications

Time Rate Quantification: The observed patterns indicate systematic
diurnal variations with 1.9-7.6% day-night coherence changes and
synchronized early morning peaks across centers. Alternative slow
environmental covariates remain plausible pending raw carrier-phase confirmation (extending the raw-RINEX pseudorange/SPP recovery reported in Paper 3),
though the systematic nature and theoretical correspondence suggest
potential compatibility with TEP's central prediction that "when" is as
dynamical as "where" in Einstein's geometric framework.

Experimental Path Forward: The systematic temporal variations observed
terrestrially provide benchmarks for testing TEP predictions through
triangle synchronization tests, interplanetary time transfer, and
one-way asymmetry experiments outlined in the theoretical framework.

## 3.6 Synthesis of Observational Evidence

**Experimental Section:**

### Convergent Lines of Evidence

The comprehensive analysis reveals seven independent observational
domains supporting systematic network sensitivity to spacetime
structure:

Baseline spatial correlations: Exponential decay with a primary
correlation length of 3,330–4,549 km, consistent across independent
processing chains, and bootstrap validation ranges of 1,198–5,918 km
(CODE), 2,532–3,984 km (ESA Final), and 3,197–4,871 km (IGS
Combined)

Spectral characterization: Broadband correlation structure (R² >
0.85 from 10–100 μHz; 100–200 μHz averages ~0.75) with gravitational
enhancement in tidal bands (λ<sub>T</sub> = 4,677 km) but persistent post-tidal
signals (30–40 μHz shows R² = 0.946), suggesting universal coupling
rather than frequency-selective contamination

Environmental modulation: Systematic variations with elevation and
geomagnetic latitude indicating screening effects

Earth motion coupling: Correlations with orbital velocity, Chandler
wobble, and interference patterns (combination and beat frequencies)

Collective network dynamics: Network-wide coherent behavior combining strong
base synchronization (0.643–0.644), annual collective oscillation
(365.25 days, R²=0.35), spiral motion signatures, and Earth
coupling—indicating the 364-station network functions as a unified
detector rather than independent correlations

Gravitational correlations: Planetary influences with statistically
significant timescale strengthening (31-day to 227-day)

Temporal and dynamic modulation: Diurnal, seasonal, eclipse, and
supermoon responses

Theoretical context: These observations find natural interpretation
within the Temporal Equivalence Principle, which predicts scalar field
variations should couple to atomic transition frequencies through
conformal metric structure g̃<sub>μν</sub> = A²(φ)g<sub>μν</sub> +
B(φ)∇<sub>μ</sub>φ∇<sub>ν</sub>φ.

The convergence of multiple independent observational domains, combined
with consistent reproduction across analysis centers, indicates that
global GNSS networks exhibit sensitivity to large-scale spacetime
structure at previously unexplored scales.

Planetary mass scaling: Direct correlation analysis (Section 3.4.4) between
observed amplitudes and gravitational predictions (M/d²) reveals no
statistically significant correlation (r = −0.156 average across three
centers), establishing an empirical upper bound on unmodeled single-receiver
scalar perturbations. Individual opposition events display cross-center
discordance (e.g. Saturn 2023: CODE −23.27% vs ESA +93.27%), confirming
that single-event residual amplitudes reflect network adjustment dynamics
rather than coherent planetary scalar detections. Apparent enhancement factor
variations E ≡ A<sub>obs</sub>/(M/d²) mechanically mirror the inverse
Newtonian denominator (E ∝ d²/M) against the residual floor,
reinforcing that disformal coupling must be probed through closed-loop
holonomy rather than single-station amplitude ratios.

Comprehensive diurnal dynamics: High-resolution temporal analysis of 72.4M
hourly records reveals systematic diurnal and seasonal variations with early
morning peaks, day-night coupling variations (1.9–7.6%), and patterns
consistent with dynamical time flow predictions (Section 3.5.3).

Environmental stratification: Systematic elevation and geomagnetic
dependencies (see §3.2.1 and §3.2.2) demonstrate environmental screening
effects consistent with scalar field coupling.

## 4. Discussion

**Experimental Section:**

## 4.1 Observations and Theoretical Context

The observed patterns show characteristics consistent with predictions
from the Temporal Equivalence Principle framework (see Table 1 in
Section 1.2). While these consistencies are noteworthy, they do not
constitute definitive proof and require independent validation.

Screening in TEP is represented at the theory level by the environmental operator
$S_\Sigma(\mathcal{E})$.
Quantities such as
$\rho_T$,
$R_T(M)$,
$S_\oplus(r)$,
compactness $\Phi/c^2$,
local stellar density,
geometric coherence length,
and channel-specific response coefficients
are domain-specific projections of $\mathcal{E}$,
not independent screening mechanisms
and not interchangeable universal thresholds.
Each is an observational transfer model
that parameterizes the same underlying operator
in a regime-appropriate form.

**Exponential decay pattern:** The data show preference
for exponential-family models over alternatives, with R² values of
0.92–0.97 (pooled fit on distance-bin means), consistent with
screened scalar field predictions.

**Temporal Topology correlation length scale:** The observed
correlation length λ<sub>T</sub> = 3,330–4,549 km falls within the 1,000-10,000 km range
that was specified as a theoretically motivated prior search range
within the Temporal Topology framework. The discriminating content
is structural: the pooled coherent-band estimate brackets the
geometric saturation radius R<sub>T</sub>(M<sub>⊕</sub>) ≈ 4,150 km,
the correlation is band-localized (control-band R² ≈ 0.5–0.6 vs
≈0.95 coherent), and the per-band decomposition is ordered across
centers (§3.5.1, Table 8a). The R<sub>T</sub> identification
calibrates against the corpus's fitted ρ<sub>T</sub> convention
(Paper 6); across products the fitted scale is conditioned by
product construction and estimator settings within a
~1.0–4.8×10<sup>3</sup> km family (MGEX 1,862 ± 155 km; the
estimator-conditioned share is ≲±25% on identical station-days
and the same band-ordered profile is recovered in both product
families; Paper 14, Step 3.5), and the co-visibility kernel is bounded by forward
simulation at λ ≳ 6.7×10<sup>3</sup> km—and at λ ≥
5,075 km when conditioned on the real product's
ephemerides and satellite-clock content—broader than the
measured scale (§4.6).

**Broadband coupling:** The signal's persistence across
frequency bands—with smooth spectral rolloff and modest enhancement
ratios—is consistent with universal coupling rather than
frequency-selective artifacts.

**Multi-center consistency:** The convergence of three
distinct analysis center solutions (CV of λ<sub>T</sub> ≈ 13%, population σ/μ across the three centre exponential fits) on similar physical
parameters demonstrates that the spatial decay is not an artifact of a single
software implementation (Bernese vs NAPEOS vs IGS combination); combined multi-constellation clock consistency (Paper 14) and raw-RINEX pseudorange/SPP recovery (Paper 3) further constrain shared-product artifacts, while raw carrier-phase and separately processed per-constellation analyses remain open.

The remainder of the discussion examines validation evidence and
explores potential physical interpretations, while acknowledging that
alternative explanations remain possible pending independent
replication.

R² convention: unless specified otherwise, R² values refer to
pooled fits on distance-bin means; sensitivity subset fits and LOSO
cross-validation R² are explicitly labeled.

## 4.2 Signal Authenticity and Validation

Multiple independent validation tests were applied to distinguish genuine
physical correlations from systematic artifacts. Each test provides
quantitative criteria for assessing signal authenticity.

**Experimental Section:**

### Validation Framework: 11 Independent Criteria

| Test | Metric | Result | Rules Out |
| --- | --- | --- | --- |
| 1. Null Hypothesis Testing | ΔR² (signal vs null) | 0.89–0.95 (z=15.8–31.9) | Random artifacts, noise |
| 2. Cross-Center Consistency | CV of λ | ≈13% | Center-specific artifacts |
| 3. Multi-Band Frequency | Enhancement ratio | 1.58× (vs >3× expected) | Classical tidal contamination |
| 4. Signal-Artifact Discrimination | Control band ΔR² | 0.334 | Broadband processing noise |
| 5. Geometric Controls | Synthetic vs real ΔR² | 0.85-0.90 | Network topology bias |
| 6. Ionospheric Independence | Signal improvement | +21-23% when excluding high-TID periods | Ionospheric artifacts |
| 7. Distribution Neutrality | Signal preservation | 96.7% | Statistical binning bias |
| 8. Bias Characterization | Signal-to-bias ratio | ΔR² = 0.903 | Methodological artifacts |
| 9. Binning Sensitivity | Parameter stability | λ cluster: 4,350–4,450 km | Analysis parameter dependence |
| 10. Phase Quality | Boundary clustering | 1.62–1.67% (expected: 1.59%) | Phase wrapping errors |
| 11. Volume Independence | Consistency across 3.6× range | R² = 0.92-0.97 | Sample size artifacts |

**Summary:** All 11 validation criteria met.
Statistical robustness: 40–52% of 388 tests survive
ultra-conservative corrections. Multi-modal validation spans spatial
correlations, Earth motion coupling, planetary correlations, and
temporal dynamics.

The convergence of evidence across methodologically diverse
approaches (binning independence, distance metric independence,
frequency specificity, null test rejection, multi-center
consistency) substantially reduces the likelihood that observed
patterns result from systematic bias or Earth-geometry artifacts.

**Experimental Section:**

### Detailed Validation Descriptions

#### 1. Null Hypothesis Testing

Test Design: Three independent scrambling approaches applied
systematically across all analysis centers to validate signal
authenticity through controlled randomization:

Distance scrambling: Randomize both distances and coherences
independently to destroy distance-coherence relationships (20
iterations per center)

Phase scrambling: Randomize phase values while preserving distance
structure (20 iterations per center)

Station scrambling: Randomize station assignments within temporal
blocks (20 iterations per center)

| Center | Real R² | Distance Null | Phase Null | Station Null | ΔR² |
| --- | --- | --- | --- | --- | --- |
| CODE | 0.920 | 0.015 ± 0.028 (z=31.9) | 0.029 ± 0.036 (z=24.9) | 0.023 ± 0.038 (z=23.4) | 0.891-0.905 |
| IGS Combined | 0.966 | 0.029 ± 0.031 (z=30.3) | 0.031 ± 0.048 (z=19.5) | 0.032 ± 0.032 (z=29.4) | 0.934-0.937 |
| ESA | 0.970 | 0.027 ± 0.044 (z=21.7) | 0.040 ± 0.059 (z=15.8) | 0.033 ± 0.056 (z=16.8) | 0.930-0.943 |

Statistical Significance Assessment: Comprehensive multiple comparison
correction analysis across all 388 statistical tests demonstrates robust
statistical findings. The differential survival rates across test
types—strongest for primary predictions (TEP band R²=0.952), reduced for
control bands (R²=0.618), hierarchical for beat frequencies—validates
genuine signal rather than uniform artifact or noise.

Comprehensive Analysis (388 statistical tests across 19 families):

Major test families: Multiband analysis (40), Band diagnostics (40),
Anisotropy orbital (24), Model comparisons (18), Advanced analysis
(15), Eclipse analysis (15), Bootstrap cross-method (12), Hilbert-IF
astronomical (12)

Bonferroni correction: 155/388 tests remain significant (40.0%) with
ultra-stringent α = 0.000129 (0.05/388 tests)

Benjamini-Hochberg correction: 203/388 tests remain significant
(52.3%) with FDR q = 0.05

Family-wise correction: 178/388 tests remain significant (45.9%)
across analysis families

Primary TEP findings: All 3 primary exponential fits maintain
significance after all corrections

Complete pipeline validation: 388 tests span every aspect from data
quality to astronomical events, including null hypothesis validation
and bootstrap consistency checks

Interpretation: The comprehensive multiple comparison correction
analysis across 388 statistical tests is consistent with
theory-predicted signal authenticity. The analysis tests the established
Temporal Equivalence Principle theoretical framework, which predicted
exponential correlation decay with characteristic length scales before
data analysis. With comprehensive validation spanning data quality
metrics, null hypothesis testing, bootstrap validation, band-specific
diagnostics, astronomical period analysis, and complete pipeline
verification, this represents rigorous statistical validation of
theory-driven predictions. The 40-52% survival rate after
ultra-conservative corrections (α = 0.000129) indicates that the core
TEP theoretical predictions are statistically robust across 19
independent validation approaches, though alternative explanations
remain to be fully excluded.

#### 2. Cross-Center Consistency

Test: Compare results across CODE, IGS Combined, and ESA Final using
different algorithms, software implementations, and station network
configurations.

Result: λ<sub>T</sub> = 3,330–4,549 km with CV of correlation lengths across centers
≈ 13% consistency across fundamentally different processing
approaches.

Interpretation: The convergence of independent processing methodologies
on consistent physical parameters is consistent with genuine physical
phenomena rather than center-specific systematic artifacts. While all
centers analyze the same underlying IGS network data, their different
software implementations, quality control procedures, and solution
strategies would be expected to produce different artifacts if the
signal were methodological in origin. The observed consistency supports
a genuine physical phenomenon.

#### 3. Multi-Band Frequency Specificity

Test: Analyze correlation patterns across 12 independent frequency bands
(10–3000 μHz, representing 506M cumulative pair-band evaluations for
CODE across all bands, from 39.0M base station pairs) to distinguish
universal broadband coupling from frequency-selective mechanisms
including classical tidal contamination.

Result: Cross-center validation demonstrates remarkable consistency -
ESA Final (R² = 0.970), CODE (R² = 0.920), and IGS Combined (R² = 0.966)
achieve nearly identical optimal band performance, substantially
reducing the likelihood of systematic biases. Broadband correlation
structure with R² > 0.85 maintained from 10–100 μHz; 100–200 μHz
averages ~0.75 (CODE shows CV of R² across bands = 2.9% across this
range). Post-tidal 30–40 μHz band exhibits R² = 0.946 (mean across
centers), ranking among the strongest bands rather than showing expected
drop-off. Enhancement ratios of 1.26–1.90× (mean: 1.58×) across all
centers fall well below the >3–5× threshold characteristic of
frequency-selective tidal contamination. Control bands (1000–1500 μHz)
show R² = 0.618 (mean), indicating reduced model fit quality (ΔR² ≈
0.33) compared to TEP band.

Interpretation: The cross-center validation achievement, combined with
frequency analysis, provides compelling evidence consistent with
universal conformal coupling rather than frequency-selective
contamination. The remarkable consistency across independent processing
methodologies - ESA Final, CODE, and IGS Combined achieving nearly
identical multiband patterns - substantially reduces the likelihood of
center-specific systematic biases and represents a crucial scientific
validation milestone. The absence of sharp spectral features
characteristic of classical tidal contamination, combined with
persistent post-tidal signals (R² = 0.946) and modest enhancement ratios
(1.58× vs. expected >3-5× for tidal contamination), supports the TEP
interpretation. Tidal band gravitational enhancement (λ<sub>T</sub> = 4,627 km mean)
indicates φ-field response to gravitational gradients, while broadband
persistence suggests coupling operates universally across frequency
scales. The quantitative separation between TEP (R² ≈ 0.95) and control
(R² ≈ 0.618) bands establishes robust discrimination criteria for
physical correlations vs. systematic effects.

#### 4. Signal-Artifact Discrimination

Test: Apply comprehensive null tests using three independent scrambling
approaches (20 iterations per center, 180 total tests): distance
scrambling randomizes both distances and coherences independently to
destroy distance-coherence relationships and systematic patterns; phase
scrambling randomizes phase values while preserving distance structure;
station scrambling randomizes station assignments within temporal
blocks. Statistical significance assessed via z-scores comparing real
signals to null distributions.

Result: All nine scrambling tests show statistically significant signal
destruction with z-scores ranging from 15.8 to 31.9. Real TEP signals
(R² = 0.920–0.970) show 24–61× enhancement over null distributions (R² =
0.015–0.040), representing ΔR² = 0.89–0.95. Signal-to-null ratios: CODE
32–61×, IGS Combined 30–33×, ESA Final 24–36×. Only 5–20% of null
iterations exceed the R² > 0.06 spurious correlation baseline
established by logarithmic binning artifacts. Control band analysis
shows ΔR² = 0.334 between TEP and control frequencies (TEP R² = 0.952 vs
control R² = 0.618 at 1000–1500 μHz, mean across centers).

Interpretation: The substantial separation between genuine TEP
correlations and systematic artifacts is consistent with a genuine
physical signal rather than processing contamination. The improved
scrambling methodology successfully eliminates systematic patterns, with
null test R² values clustering near the 0.06 random baseline from
binning artifacts. All three scrambling types independently support
signal authenticity: distance scrambling rules out artifacts in distance
calculations (z = 21.7-31.9), phase scrambling rules out temporal
artifacts (z = 15.8-24.9), and station scrambling rules out spatial
clustering effects (z = 16.8-29.4). The extreme z-scores (all > 15)
indicate that the real signal cannot plausibly arise from the randomized
data, supporting genuine distance-coherence, temporal, and spatial
structure. This discrimination validates the phase-coherent methodology
and supports the interpretation that observed correlations represent
genuine field-structured coupling rather than common-mode processing
artifacts.

#### 5. Geometric Controls

Test: Apply identical analysis to synthetic data with same station
geometry using four distinct scenarios: uniform noise, Gaussian noise,
distance-independent signals, and weak geometric patterns.

Result: Maximum synthetic R² = 0.068 across all scenarios compared to
observed R² = 0.92–0.97, representing ΔR² = 0.85–0.90. All synthetic
scenarios produce R² ≤ 0.07, providing clear discrimination from real
TEP signals.

Interpretation: Station distribution and network geometry are
insufficient to explain the observed correlation patterns. The
substantial difference in model performance (ΔR² = 0.85-0.90) is
consistent with genuine physical effects rather than geometric
artifacts. This synthetic data analysis establishes quantitative
baselines that must be exceeded for genuine physical effects, which the
TEP signals consistently achieve.

#### 6. Ionospheric Independence

**Test:** Two-stage ionospheric validation: (1) Correlate
TEP signals with external space weather indices (Kp, F10.7) to test for
direct ionospheric coupling; (2) Apply high-resolution TID exclusion
analysis to quantify ionospheric effects by excluding high-activity
periods.

**Result:** Direct correlation with space weather indices
shows weak relationships (r = 0.12-0.13, p > 0.29), indicating no
primary ionospheric coupling. TID exclusion analysis reveals signal
improvement of +21-23% when excluding high ionospheric activity periods,
demonstrating that:

78% of correlation structure is ionosphere-independent (robust
signal fraction)

22% represents ionospheric modulation (noise that suppresses the
signal)

Excluding high-TID periods improves correlations (not degrades them)

**Interpretation:** This pattern constrains the temporal
ionospheric artifact hypothesis: if ionospheric variability created the
signal, TID exclusion would reduce correlations; instead, exclusion
improves them by +21-23%. The ionosphere acts as an obscuring medium
that *suppresses* TEP
correlations rather than creating them. The 78% ionosphere-independent
signal retains all key characteristics: exponential spatial decay (λ<sub>T</sub> =
3,330–4,549 km), cross-center consistency (R² = 0.920–0.970), Earth
motion coupling, and planetary gravitational correlations. The Venus 2f
harmonic detection (Section 3.3.2) provides additional evidence for
gravitational-temporal coupling independent of ionospheric processes.
The scope of this control is temporal: Kp is a global index and the TID
exclusion removes high-activity days, so neither tests a static
geomagnetic-latitude dependence. The stratification audit (Step 4.9)
finds that the fitted correlation length is organized on geomagnetic
latitude — shortest in the auroral-leaning band in 7 of 9
elevation×center cells — which is the morphological signature of
the ionospheric channel. A complementary in-repository control is
supplied by the local-solar-time coherence profile (Step 4.5): the
coherent signal persists through local night with a night/day ratio of
0.99–1.07 on the annual means and never below ~0.84 in any seasonal
cell, whereas the ionospheric electron content — and hence any residual
ionospheric bias in the clock products — collapses by a factor of
~3–10 on the same coordinate. The residual-ionospheric fraction of the
coherent signal is thereby bounded at ≲3% on the annual means
and ≲20–32% in the worst seasonal cell (results/outputs/
step_4_9_environmental_ambiguity.json, diurnal_persistence_bound). A
spatially resolved control (GIM/ROTI maps) remains the gold-standard
closure test for the static geomagnetic-latitude dependence itself.

#### 7. Distribution Neutrality

Test: Apply equal-count binning to eliminate right-skewed distance
distribution effects.

Result: 94.8-97.9% signal preservation (average 96.7%) when switching
from logarithmic to equal-count binning. R² differences remain small
(0.021-0.052), with equal-count binning producing R² = 0.933 (CODE),
0.894 (IGS Combined), 0.914 (ESA), demonstrating robust exponential
correlation independent of distance distribution effects.

Interpretation: The correlations represent genuine physical signals, not
statistical artifacts of the distance distribution.

#### 8. Bias Characterization

Test: Generate realistic GNSS noise scenarios and quantify
methodological bias through random data simulation (50 iterations).

Result: Clear signal-to-bias separation (TEP R² = 0.953 vs maximum
artifact R² = 0.050, ΔR² = 0.903). Random data simulations confirm the
bias envelope is well below the TEP signal strength.

Interpretation: Clear quantitative thresholds distinguish genuine
signals from artifacts. The discrimination factor provides a robust
statistical foundation for signal authenticity claims.

#### 9. Binning & Weighting Sensitivity

Test: To ensure the results are not artifacts of specific methodological
choices, a sensitivity analysis was performed as part of the main
analysis pipeline (Step 4.0). The analysis was repeated by sweeping
three independent parameters: the number of distance bins attempted (25,
40, 80), the binning strategy (logarithmic, linear,
quantile/equal-count), and the weighting scheme used in the exponential
fit (weighted by pair count, by sqrt(pair count), or unweighted).

Result: The key parameters (λ<sub>T</sub> and R²) demonstrate good stability. The
correlation length λ<sub>T</sub> remains within a tight cluster (~4350–4450 km)
across different bin counts and strategies, with R² consistently
exceeding 0.91. The analysis confirms that weighting the fit by the
number of pairs per bin is appropriate, but the result is not highly
sensitive to the specific weighting function (count vs. sqrt(count)).

Interpretation: The stability of the results across a wide range of
analysis parameters suggests that the detected exponential correlation
is a fundamental property of the data, not an artifact of the chosen
methodology. This is consistent with genuine signal detection rather
than methodological bias.

#### 10. Phase Distribution Quality

Test: Validate phase boundary handling through boundary clustering
analysis to detect phase wrapping artifacts or accumulation errors.

Methodology: For proper phase processing, plateau_phase values should
distribute uniformly across the ±π range. Excessive boundary clustering
(values accumulating near ±π limits) would indicate phase wrapping
artifacts or numerical errors. Boundary clustering is quantified as the
percentage of phase values within ±0.05 radians distance to ±π under
wrap.

Result: Boundary clustering rates of 1.62-1.67% across all centers match
theoretical expectations for uniform phase distribution (expected ~1.6%
for ±0.05 rad distance to ±π under wrap on uniform distribution), with
phase values distributing evenly across all octants of the phase circle.
(Note: near +π and near −π are the same region on the circle; these are
reported separately but the combined fraction is evaluated against 0.1/(2π).)

| Center | Boundary Clustering | Near +π | Near -π | Total Values |
| --- | --- | --- | --- | --- |
| CODE | 1.66% | 322,964 | 323,569 | 39,047,074 |
| IGS Combined | 1.67% | 107,129 | 107,561 | 12,874,746 |
| ESA Final | 1.62% | 87,424 | 87,188 | 10,809,084 |

Interpretation: The healthy boundary clustering rates (1.62-1.67%,
matching theoretical expectations) combined with balanced ±π
distribution validate proper phase processing across all centers. This
confirms that the phase-alignment index metric operates on properly
wrapped phase values without accumulation artifacts, providing an
essential foundation for phase-coherent correlation analysis. The
multi-center consistency (CV of boundary clustering rates = 1.5%)
further validates methodological robustness.

#### 11. Volume Independence

Test: Assess whether correlation parameters depend on data volume by
comparing centers with vastly different pair counts.

Data volumes: CODE: 39.0M pairs, IGS Combined: 12.9M pairs, ESA: 10.8M
pairs (3.6× volume ratio between largest and smallest)

Result: Despite 3.6-fold difference in data volume, all centers
demonstrate consistent elevation dependence patterns and geomagnetic
stratification with systematic λ ranges (CODE: 3,225–7,499 km, IGS
Combined: 2,616–5,453 km, ESA: 1,600–3,914 km). Primary pooled fits show
R² = 0.92–0.97 (distance-bin means, N<sub>eff</sub> ≈ 25–28 bins used
from 40 attempted). Sensitivity subsets (elevation/geomagnetic): R² =
0.70–0.91. This range (1,600–7,500 km) is a result of sensitivity
analysis and is distinct from the primary finding.

Interpretation: TEP correlations are robust across different statistical
sampling densities, ruling out sample-size-dependent artifacts. The
consistent elevation dependence and geomagnetic stratification patterns
across all centers demonstrate signal authenticity. ESA Final achieves
excellent fits (R² = 0.81–0.91) despite having the smallest dataset
(10.8M pairs), while CODE shows systematic elevation trends across the
largest dataset (39.0M pairs).

#### 12. Spatial GLS Re-fit (Covariance Structure)

Test: The original WLS fit assumes independent bin errors.  Nearby
distance bins may have correlated residuals from common-mode
systematics or smooth deviations from the exponential.  A theoretical
sensitivity study reconstructs the distance-bin geometry from the
station network and adds a parametric spatial nuisance kernel to the
covariance matrix.

Result: Under realistic residual spatial correlations (common-mode
fraction f ≤ 0.001, correlation length ≤ 3,000 km), the λ error
bar inflates by at most ~4× relative to the WLS value; for very
small systematics (f ≤ 0.0005) the inflation is $\lt$ 1.5×.
Even under the most extreme physically plausible nuisance levels
studied (f = 0.002, l = 5,000 km), the adjusted λ remains within the
TEP-consistent 1,000–10,000 km range (CODE: 1,492–16,991 km;
IGS Combined: 582–9,877 km; ESA Final: 485–8,811 km).

Interpretation: The TEP-consistent conclusion survives conservative
spatial-correlation assumptions.  The WLS λ uncertainty reported in
the primary analysis is not an artifact of ignoring bin-to-bin
covariance, and the correlation length remains statistically robust
even when the covariance structure is deliberately inflated.

#### 13. Systematic Injection Simulation

Test: Quantify how large a distance-dependent systematic bias would
be required to shift the fitted λ outside the TEP-consistent regime
(λ $\lt$ 1,000 km or R² $\lt$ 0.5).  Five physically motivated
systematic types were injected at the pair level and averaged into
bins: common-mode offset, latitude gradient, N-S dipole,
distance-linear trend, and hemispheric offset.

Result: Only a distance-linear systematic with amplitude ≃ 0.1
(in coherence units) pushes λ below the 1,000 km threshold for CODE;
latitude gradients, N-S dipoles, common-mode offsets, and hemispheric
biases do not destroy TEP consistency even at amplitudes up to 0.1.
Known GNSS biases are typically $\ll$ 10<sup>−3</sup> in coherence units,
orders of magnitude below the critical amplitude.

Interpretation: The exponential signature is robust against plausible
systematics.  The amplitudes required to fake or destroy the signal
are unphysically large for GNSS residuals, providing an SME/LLI
screening argument: no Standard-Model Extension coefficient or
local-Lorentz-violation parameter at currently allowed levels can
generate a bias of the required magnitude.

Validation Score: 13/13 criteria passed with 100% validation
achievement. The comprehensive multiple comparison correction analysis
across 388 statistical tests supports statistical robustness, with
40-52% of findings surviving ultra-conservative corrections across 18
validation families including complete data quality, null hypothesis
testing, bootstrap validation, and band diagnostic verification.

**Experimental Section:**

### Additional Supporting Evidence

Beyond the 11 core validation criteria, additional analyses provide
supporting evidence for signal authenticity and physical scale
consistency:

#### Eclipse Scale Consistency and Dynamic Field Predictions

While the primary evidence for TEP comes from persistent baseline
correlations, the framework predicts that astronomical events should
modulate the scalar field φ. Solar eclipses provide controlled natural
experiments where significant ionospheric changes might perturb the
effective field coupling. The key discriminator between ionospheric
artifacts and genuine TEP effects is scale consistency: TEP field
modulations should extend to the characteristic correlation length λ<sub>T</sub>,
while conventional ionospheric effects operate on different scales.

The conformal coupling A(ϕ) = exp(β<sub>A</sub>ϕ/M<sub>Pl</sub>) implies that
eclipse-induced changes in the electromagnetic environment will manifest
as measurable variations in atomic clock coherence. Different eclipse
types—total, annular, and hybrid—are predicted to produce distinct ϕ
field responses based on their differential ionospheric effects. Total
eclipses, with complete solar blockage, should create uniform
ionospheric depletion potentially enhancing field coherence. Annular
eclipses, leaving a ring of sunlight, may create complex field patterns
leading to coherence disruption.

**Observational Evidence:**

Eclipse shadow scale: Direct solar blockage spans ~2,000–3,000 km
diameter

Observed effect extent: Coherence modulations observed to distances
matching TEP λ<sub>T</sub> = 3,330–4,549 km

Cross-center consistency: Eclipse type hierarchy (Partial > Annular
> Total) observed across distinct analysis center solutions; a geometry-matched
control yields the same ordering

Limitations: Five eclipses with 1-2 events per type provide
preliminary evidence requiring independent replication

#### Opposition Event Scale Consistency

Temporal scope: Measures short-term coherence changes (±30–60 day
windows), distinct from long-term correlations (Section 3.4.1)

Center-specific responses: Varying sensitivities (CODE: -23.27% to
+24.02%, IGS Combined: -44.79% to +47.98%, ESA: -112.83% to
+93.27%) reflect different processing approaches to short-term
gravitational perturbations

Physical interpretation: Short-term event enhancements can coexist
with long-term anti-phase coupling, representing different physical
timescales

#### Multi-Center Convergence

The consistency across distinct processing chains with different
algorithmic implementations provides substantial validation evidence. The
three analysis center solutions employ fundamentally different processing
strategies (CODE: double-difference network solutions with zero-difference clock recovery via Bernese software; ESA: undifferenced global network batch least-squares adjustment via NAPEOS software; IGS: multi-center weighted combination) applied to
the IGS global tracking network. If software-specific algorithmic bugs or parameterization choices were
responsible, center-specific λ values reflecting their individual
software implementations would be expected, not the observed convergence to λ
= 3,330–4,549 km (CV of λ across centers ≈ 13%). This convergence
across distinct processing pipelines demonstrates that the spatial decay is robust against software and algorithmic choices, while shared-constellation common modes are systematically addressed in Paper 14 and Paper 3.

**Experimental Section:**

### Study Limitations and Research Context

Although all three centers analyze IGS data, their pipelines have
orthogonal systematic tendencies—CODE's network constraints suppress
global coherence, ESA's precise point positioning removes network-level
correlations, and IGS is a weighted multi-center combination—so
processing artifacts should diverge across centers; instead, the
observed convergence of λ<sub>T</sub> (3,330–4,549 km; CV of λ<sub>T</sub> ≈ 13%) and nearly
identical multiband patterns (optimal R² = 0.970/0.966/0.920) is a
discriminator in favor of a physical signal. The phase-coherent approach
leverages this orthogonality: amplitude-suppressing steps remove
common-mode artifacts while phase structure survives, creating a
built-in artifact filter. The observed convergence across opposing
pipelines is therefore a strength, not a limitation.

Atmospheric Loading and Geophysical Effects: Large-scale atmospheric
mass redistribution can generate distance-dependent correlations through
crustal and gravitational loading. While Section 4.3.6 demonstrates
incompatibilities with synoptic timescales, processing immunity, and
orbital-velocity coupling, direct cross-correlation against global
atmospheric reanalysis fields provides a quantitative bound on any
residual loading contribution.

Higher-Order Tidal Loading: Multi-band analysis excludes classical
tidal dominance (post-tidal signal persistence R² = 0.946; enhancement
ratios 1.58× vs. >3× expected for tidal forcing, Section 4.3.3).
Cross-comparison against non-GNSS optical-fiber and two-way satellite
time-transfer links provides an independent check orthogonal to
space-geodetic tidal models.

Temporal Coverage: The 2.5-year baseline (2023–2025) resolves
multi-annual patterns (Chandler wobble, seasonal and orbital coupling);
longer-period cycles (including the 18.6-year lunar nutation) are
evaluated in the companion 25.3-year longitudinal analysis (Paper 2).

Processing Suppression Effects: Standard GNSS processing applies network
constraints and environmental corrections specifically designed to
remove spatially correlated signals. For example, IGS network processing
enforces datum constraints that force the sum of satellite clock
corrections to zero across all satellites (see IGS Analysis Center
Coordinator specifications, *IERS Conventions 2010*, Section
5.4.1), which systematically attenuates globally coherent temporal
variations. CODE's "network solution" similarly applies ensemble
constraints across stations that suppress spatially correlated clock
residuals (Dach et al., 2007, GPS Solutions). The persistence of the
signal despite this suppression strengthens authenticity claims and
suggests genuine field coupling. The measured correlations likely
represent conservative lower bounds on the true field coupling strength,
with raw data access potentially revealing 2-10× stronger signatures.

Out-of-Ecosystem Replication: Cross-checking the GNSS covariance
structure on independent timing infrastructures—optical-fiber links
and raw two-way satellite time transfer (TWSTFT)—tests the signal
outside the IGS tracking and orbit-determination ecosystem. The
validation framework established here (11 independent criteria, 62.7M
measurements, multi-modal physical validation) specifies the exact
spectral and spatial signatures required for replication.

#### Validation Framework Strength

Within these defined scope boundaries, the analysis establishes
validation depth through:

Systematic testing: 11 independent validation criteria with a
100% achievement rate

Statistical robustness: Substantial signal separation (ΔR² =
0.89-0.95, z = 15.8-31.9) over null distributions across
multiple scrambling approaches with 24-61× signal-to-null ratios

Multi-modal physical validation: Convergent evidence across
spatial correlations, Earth motion coupling, planetary
gravitational influences, and temporal dynamics

Software and model diversity: Distinct processing software
and observation models (Bernese double-difference clock recovery vs NAPEOS
undifferenced global network adjustment vs IGS weighted combination)
yielding consistent results (CV of λ across centers ≈ 13%)

Frequency domain discrimination: Post-tidal signal persistence
and broadband coupling characteristics distinguishing from
classical contamination mechanisms

Synthesis: The convergence of multiple observational domains,
systematic alternative exclusion, and cross-center consistency across
distinct estimation architectures establishes a reproducible empirical
signature in global GNSS clock networks and defines quantitative
targets for non-GNSS precision timing comparisons.

## 4.3 Alternative Explanations: Systematic Exclusion

Having presented evidence for signal authenticity, alternative physical
explanations are systematically addressed. Each hypothesis makes specific,
testable predictions that can be distinguished from TEP signatures. This
section provides comprehensive evaluation of conventional explanations,
though some sophisticated systematic effects may require additional
investigation.

### Secondary Systematic Channels and Completed Controls

Beyond the primary null, tidal, and ionospheric tests above, four
secondary systematic channels are bounded by the multi-center and
companion analyses:

#### Atmospheric Loading Effects

Surface pressure variations induce crustal and gravitational
loading

- Typical effects: 10⁻¹⁵-10⁻¹⁴ fractional frequency

Status: Excluded as primary driver by synoptic-timescale,
seasonal-phase, and GPT/VMF processing-immunity mismatches
(&sect;4.3.6); weather-organized partition in Paper 14 returns
the opposite signature

#### Multipath Systematic Effects

Site-dependent multipath signatures act at local station
baselines

- Expected scale: Site-specific (typically $\lt$100 km)

Status: Incompatible with 3,330–4,549 km decay; minimum-baseline
cutoffs up to 2,000 km leave λ unchanged (Paper 14), and
raw-RINEX pseudorange recovery confirms survival without
precise-product filtering (Paper 3)

#### Reference Frame Instabilities

Coordinate-frame drift acts on secular/global scales

- Expected scale: Global (mm/year position drift)

Status: Separated by sub-diurnal to 30-day coherent band
localization and 25.3-year longitudinal stability across
ITRF realizations (Paper 2)

#### Network Adjustment & Orbit-Clock Coupling

Global network adjustment and shared satellite clocks could
project long-range correlations

- Expected: Analysis-center and estimator dependent

Status: Forward-simulated co-visibility/datum kernel is too
broad ($\lambda \ge 5{,}075$ km) with opposite long-baseline
sign (&sect;4.6); signal persists in single-station SPP raw-RINEX
reductions with zero network adjustment (Paper 3)

**Cross-Technology Extension:** Raw carrier-phase processing
and non-GNSS optical-fiber / two-way satellite time-transfer links
provide direct measurements of the unsuppressed coupling amplitude
outside GNSS observables.

| Alternative Hypothesis | Key Discriminator | Conclusion |
| --- | --- | --- |
| **Methodological Artifacts** | Null Tests & Multi-Center Consistency | Ruled out (ΔR² = 0.89-0.95 signal vs. null, z = 15.8-31.9;
CV of λT ≈ 13% across centers). |
| **Classical Tidal Effects** | Multi-Band Spectral Analysis | Inconsistent; Signal is broadband with modest enhancement
(1.58×), unlike frequency-selective tides. |
| **Ionospheric Disturbances (TIDs)** | Spatial Structure & Processing Immunity | Inconsistent; Isotropic exponential decay conflicts with TID
plane-wave structure. Signal survives ionospheric
corrections. |
| **Large-Scale Geophysical Phenomena** | Temporal Persistence & Physical Coupling | Ruled out; Mismatch in timescales (e.g., weather systems)
and observed coupling to non-terrestrial phenomena. |

### 4.3.1 Methodological Artifacts

A critical concern is whether the phase-alignment index metric might create
spurious exponential patterns through projection bias or other analytical
artifacts. This is addressed through comprehensive bias characterization
(Section 4.2, test #7) demonstrating clear signal-to-bias separation (TEP R²
= 0.953 vs maximum artifact R² = 0.050, ΔR² = 0.903).

Key discriminators against methodological bias:

Multi-center consistency: Three distinct analysis center solutions with different
algorithms converge on λ<sub>T</sub> = 3,330–4,549 km (CV of λ<sub>T</sub> across centers =
≈13%), ruling out software-specific analytical artifacts

Null hypothesis testing: Comprehensive randomization tests destroy
correlations (ΔR² = 0.89–0.95 separation, z = 15.8–31.9 across all
scrambling approaches, 24–61× signal-to-null ratios), indicating genuine
spatial and temporal encoding rather than mathematical artifacts

Scale separation: TEP λ<sub>T</sub> (3,330–4,549 km) >> methodological bias scales
(~600 km) by 6.5×

Temporal stability: Consistent across 2.5-year dataset, inconsistent
with processing artifacts

### 4.3.2 Processing Independence: Convergence Across Divergent Methodologies

A primary consideration in any multi-center analysis is the potential for
common processing artifacts to arise from shared systematic dependencies.
The validation framework was therefore designed to systematically test for
such effects, as all analysis centers ultimately process the same underlying
IGS network data using shared fundamental components.

**Experimental Section:**

#### Acknowledged Shared Systematic Dependencies

Methodological Context: True independence is inherently limited by
shared systematic components across all analysis centers:

Common data source: All centers process the same underlying IGS
global tracking network observations

Shared physical models: Similar satellite orbit determination using
JPL ephemeris (DE-series), common Earth rotation parameters (IERS),
and shared reference frame realizations (ITRF2014)

Fundamental processing constraints: All centers apply similar GNSS
processing principles, multipath mitigation strategies, and
environmental correction models

Common calibration standards: Shared atomic frequency standards,
similar relativistic corrections, and coordinated UTC realizations

**Critical assessment:** This shared infrastructure creates
a fundamental limitation in single-product GNSS validation studies. While centers
employ different processing philosophies, they cannot be considered
truly independent given shared data sources, physical models, and
calibration standards. Companion analyses address combined multi-constellation products (Paper 14, Step 3.2/3.5) and raw-RINEX pseudorange/SPP channels (Paper 3, Steps 2.10/2.12), while raw carrier-phase processing and independent external replication remain required to exclude all shared infrastructure systematics.

#### Evidence for Software and Algorithmic Robustness Despite Shared Infrastructure

The analysis was designed to provide multiple lines of evidence for
genuine signal detection despite these systematic dependencies:

##### 1. Software and Observation Model Diversity

CODE (Network Solution): Formulates double-difference phase observations
across baselines via the Bernese GNSS Software to determine orbits and station
coordinates, followed by undifferenced zero-difference clock recovery with
network datum constraints

ESA (Undifferenced Global Network Adjustment): Generates clock products via
the NAPEOS software suite using an undifferenced global network batch least-squares
adjustment, simultaneously estimating satellite and station clocks, orbital
state vectors, and tropospheric delays without explicit baseline differencing

IGS (Multi-Center Combination): Weighted meta-analysis combining
diverse processing approaches and independent software suites (including CODE,
ESA, GFZ, GRG, and JPL) aligned to an internal IGS timescale

Key Insight: Software and algorithmic diversity across Bernese, NAPEOS,
and the IGS combination algorithm ensures that the spatial correlation
structure is not an artifact of specific filtering implementations, numerical
solvers, datum constraint formulations, or software bugs. Both CODE and ESA
solve global network adjustments, but with distinct observation equations
(double-differenced vs undifferenced), parameter spaces, and codebases.
The close convergence on λ<sub>T</sub> = 3,330–4,549 km (CV of
λ<sub>T</sub> across centers ≈ 13%) demonstrates robustness across distinct
geodetic processing engines.

**Multi-level Robustness Validation:** Three
complementary approaches provide comprehensive validation of processing
robustness: (1) *Software architecture diversity*: Bernese
double-difference clock recovery, NAPEOS undifferenced global batch
adjustment, and IGS combination represent distinct mathematical formulations
that would produce different numerical artifacts, yet yield consistent results;
(2) *Station-block bootstrap*: Systematic removal of 30% of stations
preserves λ within confidence intervals, ruling out connectivity bias
and high-hub station effects; (3) *Geographic robustness*:
Consistent parameters across elevation quintiles, geomagnetic regions,
and continental baselines demonstrate independence from station-specific
systematic effects.

##### Quantitative Robustness Metrics

Three quantitative tests evaluate solution robustness and center-specific effects:

**Software Architecture and Observation Model Divergence Test:**

CODE: Double-difference baseline processing with zero-difference clock reconstruction via Bernese

ESA: Undifferenced global network batch least-squares adjustment via NAPEOS

IGS: Multi-center weighted combination across heterogeneous contributing pipelines

Observed: λ values converge across software architectures (CV ≈ 13%, bootstrap CIs overlapping)

Scope: Rules out software-specific bugs and center-specific numerical artifacts; shared-constellation common modes are evaluated in Paper 14 and Paper 3

**Station-Specific Artifact Test:**

- Removed top 20% highest-degree stations (hub removal)

- Result: λ shifts $\lt$ 8% (well within bootstrap CIs)

- Shared network hub artifacts would show > 30% shift

**Temporal Stability Test:**

- Split data into 6-month blocks

Cross-center correlation of temporal λ variations: r = 0.18

- Low correlation indicates that center-specific noise and processing fluctuations do not drive the common spatial scale

Robust Control Test Detail: The station-block bootstrap analysis
systematically removes random subsets of stations (up to ~30% per
resample), targeting potential high-connectivity hub stations and entire
geographic regions. Despite this aggressive station removal, λ values
remain stable within their respective bin-level bootstrap confidence
intervals (see Table in §3.1.1). The station-block bootstrap CIs are
narrower and shifted compared to the primary bin-level bootstrap CIs
because they test robustness to station removal rather than correlation
function uncertainty. This demonstrates that neither network
connectivity effects nor geographic clustering biases can account for
the observed TEP signature.

##### 2. Signal Survival Through Processing Suppression

A key indicator of authenticity: GNSS processing is explicitly designed
to remove spatially correlated signals through network constraints,
environmental corrections, and quality control filtering. The
persistence of strong correlations (R² = 0.920-0.970) through processing
designed to eliminate such signals is consistent with signal robustness
rather than processing artifacts.

##### 3. Comprehensive Systematic Validation

Null hypothesis testing: Substantial signal separation (ΔR² =
0.89-0.95, z = 15.8-31.9, 24-61× signal-to-null ratios) over
randomized controls is consistent with spatial and temporal
structure beyond processing artifacts

Cross-validation robustness: LOSO/LODO validation with consistent
parameters across geographic and temporal subsets

Physical dependencies: Systematic variations with elevation, orbital
velocity, and gravitational fields represent characteristics
unexpected from shared processing algorithms

##### 4. Systematic Environmental Dependencies

A key physical discriminator: A processing artifact arising from shared
models would not be expected to show systematic dependencies on the
local physical environment of individual stations. However, the advanced
analysis reveals clear, physically plausible patterns:

Elevation Stratification: The correlation length (λ<sub>T</sub>) shows a
monotonic increase with station elevation quintiles, from λ<sub>T</sub> ≈
1,600–3,200 km at the lowest elevations to λ<sub>T</sub> ≈ 3,900–7,500 km at the
highest. This is consistent with a screened field where atmospheric
density modulates the coupling strength.

Geomagnetic Stratification: The analysis reveals distinct
correlation regimes based on geomagnetic latitude, with equatorial
regions (λ ≈ 1,760–2,080 km) showing different characteristics from
polar regions (λ ≈ 1,300–3,400 km).

This coupling to the local physical and electromagnetic environment is a
signature of a genuine physical phenomenon interacting with the Earth
system, not an artifact of data processing algorithms.

##### 5. Coherence with External Physical Phenomena

The key discriminator: Perhaps the most compelling evidence against
processing artifacts is the signal's coherence with independent,
external physical phenomena that are not inputs to the GNSS processing
chain. A self-contained systematic error would be blind to these
external drivers. The analysis reveals multiple such correlations:

Planetary Ephemeris: Eight coherence modulations nominally exceed
σ = 3 near planetary events. A dedicated event-locking audit
(Section 3.4.4) shows that the largest of these—such as the 5.98σ
Saturn-2024 fit—are centred tens of days away from the event and
therefore do not constitute event responses; the clearest
surviving event-locked case is the March 2025 Venus conjunction
(ESA: +140%, σ ≈ 4, fit centred 3.3 days post-conjunction),
partially corroborated by CODE (+72%) but not IGS. The channel
thus provides at most suggestive ephemeris linkage.

Geophysical Oscillations: The significant detection of the 433-day
Chandler wobble (coefficient of determination R² = 0.377–0.471, all
$p \lt 0.01$) demonstrates that the signal is coupled to the physical
motion of the Earth itself.

Orbital Dynamics: The robust negative correlation between anisotropy
ratios and Earth's orbital velocity (r = -0.571 to -0.793 across
centers; meta-analysis: Q = 0.24, p = 0.885, I² = 0%; p-values
autocorrelation-corrected via Bretherton; N<sub>eff</sub> ≈ 47)
provides evidence of coupling to the motion of the entire Earth
system through the solar system.

Gravitational coupling: Planetary correlations with timescale
strengthening, Moon-Chandler coupling, and dynamic astronomical
responses

Temporal structure: Diurnal and seasonal modulation with systematic
patterns

The signal's phase-locking with these grand, independent "clocks" of the
solar system provides a final, powerful line of evidence that the
observed phenomenon is physical in origin, not an artifact of the
measurement system.

#### Residual Systematic Risk Assessment

While no analysis can completely eliminate the possibility of
sophisticated shared systematic errors, several observations from this
study's design support genuine signal detection:

Software and model diversity: Distinct processing software (Bernese
vs NAPEOS vs IGS combination) with different parameterizations and datum
constraints yield consistent results (CV ≈ 13%)

Processing suppression paradox: Signal persistence despite
suppression designed to remove correlated signals

Physical validation: Earth motion coupling, planetary correlations,
and astronomical event responses validate signal authenticity
through independent physical mechanisms

Multi-modal confirmation: Convergent evidence across spatial
correlations, temporal dynamics, and gravitational coupling provides
systematic validation beyond processing effects

Assessment: While shared systematic dependencies represent a legitimate
and acknowledged limitation, the convergence of opposing processing
philosophies on consistent physical parameters, combined with
comprehensive validation through independent physical mechanisms,
provides substantial evidence for genuine field coupling phenomena
transcending processing artifacts.

### 4.3.3 Classical Tidal Effects

A fundamental consideration in geodetic timing analysis is the potential for
imperfect correction of classical tidal effects (solid Earth tides, ocean
loading, atmospheric tides). The multi-band frequency analysis framework was
designed to provide comprehensive, quantitative discrimination between such
artifacts and genuine scalar field coupling.

**Experimental Section:**

#### Predicted Spectral Signatures

Classical tidal contamination and TEP universal coupling make distinct,
testable predictions for the frequency dependence of correlations:

| Mechanism | Predicted Pattern | Enhancement |
| --- | --- | --- |
| Classical Tidal Contamination | Strong peak at 10–30 μHz (diurnal/semidiurnal), sharp
drop-off >30 μHz, weak signal elsewhere | >3-5× in tidal bands |
| TEP Universal Coupling | Broadband correlation across all frequencies, modest
gravitational enhancement in tidal range, smooth
spectral response | ~1.5-2× enhancement |

#### Observational Evidence

1. Post-tidal band persistence: The 30–40 μHz frequency band—immediately
beyond primary tidal forcing frequencies—exhibits mean R² = 0.946 across
three centers, ranking among the strongest bands (CODE: 1st of 12, IGS
Combined: 4th of 12, ESA: 3rd of 12). This value equals or exceeds tidal
band correlations (mean R² = 0.941), appearing inconsistent with
classical tidal contamination which would predict weakest correlations
in this transition region.

2. Smooth spectral response: Analysis across 12 bands reveals gradual
decline in correlation strength from 10–3000 μHz without sharp
transitions. The CODE center demonstrates strong spectral uniformity
with CV of R² across bands = 2.9% across 10–200 μHz, maintaining R² >
0.85 from 10–100 μHz with 100–200 μHz averaging ~0.75. This smooth
response appears incompatible with frequency-selective tidal mechanisms.

3. Enhancement ratio analysis with theoretical justification: Observed
ratios of 1.26-1.90× (mean: 1.58×) between TEP and control bands fall
well below the >3× threshold expected for classical tidal dominance.
This threshold is theoretically grounded in several principles:

Theoretical basis for >3× tidal enhancement threshold:

Harmonic selectivity: Classical solid Earth tides exhibit sharp
spectral peaks at specific harmonics (K1: ~11.6 μHz, M2: ~22.4
μHz, S2: ~23.1 μHz) with power concentrated within narrow
frequency windows (±1–2 μHz). Tidal contamination would show
>5-10× enhancement within these windows compared to adjacent
frequencies.

Physical energy concentration: Tidal forcing represents
concentrated gravitational energy at discrete frequencies. The
M2 constituent alone typically dominates tidal spectra by
factors of 3-8× over neighboring harmonics in standard geodetic
measurements.

Residual contamination patterns: Imperfect tidal corrections in
geodetic time series typically leave 20-50% residuals at primary
harmonics, translating to 3-5× enhancement over broadband noise
levels in precision timing applications.

Frequency bandwidth scaling: TEP signals analyzed in 20–30 μHz
bands would capture multiple tidal harmonics simultaneously if
contaminated, amplifying the enhancement ratio compared to
single-harmonic analysis.

The observed modest enhancement ratios (1.58× mean) represent
*universal* characteristics: tidal (1.52×), TEP (1.54×), and
post-tidal (1.53×) bands show statistically indistinguishable
enhancement levels. This pattern does not support frequency-selective
tidal contamination, which would exhibit sharp spectral features, and
supports universal coupling mechanisms with gravitational modulation
operating across all frequency scales.

4. Spatial scale transitions: While correlation lengths decrease from λ<sub>T</sub>
= 4,677 km (tidal bands) to λ<sub>T</sub> = 1,502 km (post-tidal), the fit quality
(R²) remains consistently high (>0.85). This decoupling of spatial scale
from correlation strength indicates that tidal frequencies couple to
larger-scale gravitational gradients while maintaining the same
underlying exponential decay mechanism—a pattern more consistent with
scalar field response to varying gravitational forcing than with
classical tidal residuals.

#### Theoretical Interpretation

Within the TEP framework, tidal effects represent gravitational field
gradients that may modulate the φ-field through gradient-sensitive
structure associated with the non-exact sector. The observed
pattern—longest correlation lengths at tidal frequencies but persistent
correlations across all bands—suggests the φ-field responds to
gravitational forcing at multiple scales rather than being contaminated
by classical tidal residuals. The universal conformal term A(φ) provides
broadband coupling, while the enhanced correlation lengths at tidal
frequencies may indicate additional gradient-sensitive structure;
however, the GNSS covariance observable does not by itself isolate the
disformal term B(φ), which would require a dedicated non-exact transport
observable.

Conclusion: The multi-band analysis appears to exclude classical tidal
contamination as the dominant mechanism through four independent lines
of evidence: post-tidal signal persistence, smooth spectral response,
modest enhancement ratios, and spatial scale decoupling from correlation
strength. While tidal gravitational effects clearly influence the
correlations (evidenced by longest λ at tidal frequencies), the
broadband nature and persistent post-tidal signals suggest these effects
manifest through universal field coupling rather than
frequency-selective contamination.

### 4.3.4 Traveling Ionospheric Disturbances (TIDs)

Traveling Ionospheric Disturbances represent the most plausible ionospheric
alternative to TEP signals. Critical analysis requires distinguishing these
phenomena through spatial structure and processing evidence rather than
temporal separation:

**Experimental Section:**

#### Frequency Band Overlap: Alternative Discriminators Required

Important limitation: TIDs exhibit periods of 10–180 minutes (92–1,667
μHz), directly overlapping the analysis band (10–500 μHz).
Temporal/frequency separation cannot exclude TID contamination. The
analysis therefore relies on spatial structure analysis, processing
pipeline evidence, and ionospheric independence validation to
distinguish these phenomena.

#### Spatial Structure Incompatibility

TIDs: Coherent plane-wave propagation with defined k-vectors and
wavelengths of 100–3,000 km, creating directional propagation
patterns

Observed signals: Isotropic exponential correlation decay with
screening length λ<sub>T</sub> = 3,330-4,549 km, inconsistent with directional
wave propagation

Model discrimination: Plane-wave models would produce correlation
patterns dependent on station pair azimuth; observed patterns show
azimuthal independence (Section 3.2)

Different physics: Wave propagation (phase-coherent traveling
disturbances) vs field screening (exponential correlation decay)

#### Processing Pipeline Mitigation

GNSS analysis centers apply comprehensive ionospheric corrections
including dual-frequency combinations, global ionospheric maps (GIMs),
and common-mode signal removal specifically designed to mitigate TID
effects. The persistence of strong correlations (R² = 0.920-0.970) after
these corrections indicates the signal originates below the ionosphere,
in the atomic clock frequencies themselves. Multi-center λ consistency
(CV of λ across centers ≈ 13%) across different correction approaches
further supports non-ionospheric origin.
Injection tests on 30 days of raw CODE R5 data indicate that a
synthetic 10<sup>−15</sup> sinusoidal frequency offset is attenuated
by ≈96 % in the published clock products, whereas the
cross-station phase-coherence metric changes by $\lt$ 1 %,
empirically confirming that standard processing suppresses amplitude
signals while transmitting geometry-dependent phase structure (Kouba
& Héroux 2001;
Steigenberger et al. 2021).

#### Ionospheric Independence Validation

Direct correlation with space weather indices (Kp, F10.7 flux) shows
weak relationships (r = 0.12-0.13, p > 0.29), well below thresholds for
ionospheric contamination. TIDs are strongly modulated by solar activity
and geomagnetic conditions; the observed signal's independence from
these drivers provides additional evidence against TID origin.

Conclusion: While TIDs operate in an overlapping frequency band, the
observed signals are distinguished through (1) spatial structure
incompatibility (plane-wave vs exponential decay), (2) survival through
aggressive ionospheric processing, (3) independence from space weather
drivers, and (4) multi-center convergence on a common correlation-length
family. These discriminators provide strong evidence against TID
contamination.

### 4.3.5 Trans-equatorial Propagation (TEQ)

Trans-equatorial propagation (TEQ) represents a VHF/UHF ionospheric ducting
phenomenon that could potentially explain some observed correlations.
However, fundamental frequency and temporal mismatches rule out TEQ as an
alternative explanation:

**Experimental Section:**

#### Frequency Band Incompatibility

- TEP signals: L-band GNSS frequencies (1.2–1.6 GHz)

- TEQ: VHF/UHF amateur bands (30–300 MHz)

Frequency separation: order-of-magnitude frequency mismatch (GNSS
L-band ~1.2–1.6 GHz vs TEQ VHF/UHF ~30–300 MHz)

#### Temporal Characteristics Mismatch

TEP signals: Continuous over months/years (persistent correlations)

- TEQ: Hours post-sunset duration (transient propagation)

- Geographic scope: Global vs regional propagation patterns

Conclusion: Trans-equatorial propagation (TEQ) is excluded as an
explanation for the observed Global Time Echo correlations. The
frequency band incompatibility (order-of-magnitude mismatch: GNSS L-band
1.2–1.6 GHz vs TEQ VHF/UHF 30–300 MHz) and temporal persistence mismatch
(continuous multi-year TEP signals vs transient hours-duration TEQ
propagation) provide fundamental physical exclusion criteria that are
independent of statistical analysis.

### 4.3.6 Large-Scale Geophysical Phenomena

A central element of the validation framework is the systematic exclusion of
conventional geophysical phenomena at continental scales. The analysis was
designed to test for and distinguish TEP signals from each major category of
large-scale Earth system processes that could theoretically produce
distance-dependent correlations.

**Experimental Section:**

#### 1. Atmospheric Loading Effects: Systematic Analysis

Mechanism hypothesis: Large-scale atmospheric pressure variations could
create correlated timing effects through gravitational loading, where
atmospheric mass redistribution affects local gravity and influences
atomic clock rates through gravitational redshift (δν/ν ≈ δg/g ≈ 10⁻⁹
for mb-scale pressure changes).

##### Theoretical Expectations vs. Observations

| Parameter | Atmospheric Loading | Observed TEP | Match |
| --- | --- | --- | --- |
| Spatial scale | 2,000–8,000 km (synoptic systems) | 3,330–4,549 km | Partial overlap |
| Temporal signature | 3–10 day weather systems | Persistent across 2.5 years | Mismatch |
| Seasonal variation | Strong winter maxima | Spring equinox maxima | Mismatch |
| Magnitude | ~10⁻¹⁶ for 10 mbar changes | Diurnal variations observed | Compatible |

Critical discriminators against atmospheric loading:

Processing immunity: GNSS analysis centers already apply
comprehensive atmospheric loading corrections (Global Pressure and
Temperature models, Vienna Mapping Functions), yet strong
correlations persist (R² = 0.920-0.970)

Temporal persistence: Atmospheric loading effects operate on
synoptic timescales (3–10 days), incompatible with the multi-year
correlation persistence observed

Orbital velocity coupling: The observed negative correlation with
Earth's orbital speed (r = -0.701 to -0.793) represents a signature
absent from atmospheric loading mechanisms

Planetary correlations: Eight Bonferroni-significant astronomical
event fits (mostly non-event-locked on audit) with mass scaling
analysis showing no significant correlation (r = -0.156),
inconsistent with simple Newtonian mass-distance mechanisms.
Limited statistical power (n≈5 planets, 2.5 y) requires cautious
interpretation, though pattern does not support terrestrial
atmospheric origin

#### 2. Other Geophysical Phenomena: Scale and Physics Analysis

Planetary-scale atmospheric waves: Rossby waves have wavelengths of
6,000–10,000 km (Holton & Hakim 2012), significantly longer than
observed λ ≈ 3,330-4,549 km, and exhibit strong seasonal patterns
inconsistent with observed temporal stability

Ionospheric traveling disturbances: Large-scale TIDs propagate at
400–1000 km/h with wavelengths of 1,000–3,000 km (Hunsucker &
Hargreaves 2003), but show strong diurnal and solar cycle
dependencies systematically excluded through comprehensive TID
analysis (Section 4.3.4)

Magnetospheric current systems: Ring current and field-aligned
currents create magnetic field variations at 2,000–5,000 km scales
(Kivelson & Russell 1995), but these primarily couple to
magnetic rather than timing systems, and lack the exponential
spatial decay characteristic of screened fields

Tropospheric delay correlations: Water vapor patterns show
correlations up to 1,000–2,000 km (Bevis et al. 1994), insufficient
to explain the headline 3,330-4,549 km scale, and are systematically
removed by zenith delay modeling in standard GNSS processing. The
dismissal is band-dependent rather than global: the multiband
decomposition returns λ = 1,068–2,404 km in the post-tidal and
intermediate bands, inside the dismissed tropospheric range, so a
residual-ZTD covariance control (reanalysis-derived, e.g. ERA5/VMF)
is required to bound that channel (Step 4.9)

Systematic conclusion: Comprehensive analysis of major geophysical
phenomena reveals fundamental incompatibilities in spatial scale,
temporal signature, processing immunity, and coupling mechanisms. The
systematic exclusion of atmospheric loading through temporal,
processing, and correlation pattern analysis, combined with scale
mismatches for other geophysical processes, supports the interpretation
as novel field coupling transcending conventional Earth system
processes. Crucially, the signal's coupling with non-terrestrial drivers
(orbital velocity, planetary positions) is fundamentally inconsistent
with an origin in Earth's atmospheric or geophysical systems.

## 4.4 Spectral Coupling and Theoretical Constraints

The multi-band frequency analysis provides new constraints on the coupling
mechanism and theoretical parameters, complementing the spatial correlation
evidence presented in Section 4.1.

**Experimental Section:**

### Conformal Covariance and Possible Gradient-Sensitive Structure

The observed spectral pattern—broadband correlations with gravitational
enhancement—suggests a conformal covariance response, with possible
additional gradient-sensitive structure:

Conformal coupling A(φ): Evidenced by broadband response (R² > 0.85
from 10–100 μHz; 100–200 μHz averages ~0.75, CV of R² across bands =
2.9%) indicating universal coupling to all frequency modes without
frequency selectivity

Non-exact/disformal sensitivity: The enhanced correlation lengths
at tidal frequencies (λ<sub>T</sub> = 4,677 km vs 1,502 km post-tidal)
may indicate additional gradient-sensitive structure, but the GNSS
covariance observable does not by itself isolate B(φ). Direct
discrimination of the disformal sector requires a one-way loop or
other non-exact transport observable.

Combined metric structure: g̃<sub>μν</sub> = A²(φ)g<sub>μν</sub> +
B(φ)∇<sub>μ</sub>φ∇<sub>ν</sub>φ remains a candidate completion, but
the frequency-dependent spatial scales with frequency-independent
correlation strength are consistent with conformal covariance alone
under environmental modulation.

### Frequency-Scale Relationship

The systematic variation of Temporal Topology correlation length with frequency provides
insight into the physical mechanisms:

Tidal frequencies (10–30 μHz): λ = 4,677 km — planetary-scale
gravitational forcing (Moon/Sun)

Post-tidal (30–100 μHz): λ = 1,502 km — transition to regional-scale
coupling

Intermediate (100–500 μHz): λ = 1,459 km — stabilized local-scale
response

Control (>1000 μHz): λ = 2,149 km — systematic instrumental effects

The factor of 3× decrease in Temporal Topology correlation length from tidal to post-tidal
frequencies, while correlation strength (R²) remains high, suggests the
same coupling mechanism operates at different spatial scales depending
on the frequency of gravitational forcing.

### Systematic Effects and Signal Purity

Control band analysis reveals that systematic instrumental effects are
non-negligible but clearly discriminated from physical correlations:

Systematic contribution: R² = 0.618 in control bands indicates
reduced model fit quality compared to TEP band

Model fit difference: ΔR² ≈ 0.33 between TEP band (R² = 0.952) and
control bands (R² = 0.618)

Physical interpretation: Observed correlations represent genuine
physical signal superimposed on systematic background, not pure
signal with zero systematics

Validation strength: Clear quantitative separation enables robust
signal discrimination despite presence of instrumental effects

## 4.5 Theoretical Insights and Directions for Future Research

While the geospatial temporal analysis provides strong empirical support for
TEP predictions, several observations reveal a rich phenomenology that
challenges simple models and illuminates key areas for future theoretical
and experimental investigation.

### 4.5.1 Mass Scaling Analysis, Enhancement Ratios, and Disformal Coupling Bounds

**Key Finding:** Mass-scaling analysis directly correlating
observed variance reductions A<sub>obs</sub> with Newtonian gravitational
predictions (M/d²) shows no statistically significant correlation across
all three analysis center solutions (mean r = −0.156). Rather than
indicating non-gravitational coupling, this null result places an empirical
upper bound on unmodeled single-receiver scalar amplitudes in processed GNSS
products. Standard GNSS least-squares adjustment absorbs single-station
amplitude offsets into receiver and satellite clock parameters at the
~10<sup>−13</sup> level—orders of magnitude larger than expected
~10<sup>−16</sup> planetary signatures—while preserving phase-coherent
spatial correlation across the network. Consequently, amplitude-level mass
scaling cannot be recovered from processed clock products.

#### Methodological Assessment of Enhancement Factors

Earlier descriptive formulations evaluated an enhancement factor defined as
E ≡ A<sub>obs</sub>/(M/d²), which yields large values for Mercury
(137×) and Mars (215×) versus Jupiter (5.3×). Rigorous
examination confirms that this ordering is an arithmetic artifact of the
definition: because observed residual amplitudes A<sub>obs</sub> remain at
the per-event noise floor (fractional deviations of order the baseline
coherence itself) across all planetary events, dividing
by M/d² mechanically imposes E ∝ d²/M. Planets with small
Newtonian field strengths at Earth necessarily yield large enhancement
ratios, while massive bodies yield small ratios. Testing whether E
correlates with mass, or citing the E-ratio as evidence for an "inverse mass
hierarchy", is mathematically circular. The informative statistical quantity
is A<sub>obs</sub> itself, which exhibits a null correlation with M/d²
(r ≈ −0.16).

#### Cross-Center Sign Discordance as an Empirical Constraint

The interpretation of planetary events as coherent field detections is
further constrained by cross-center discordance. While physical
scalar-field modulations impose a definite sign, individual opposition
events exhibit conflicting responses across processing chains: Saturn 2023
yields −23.27% in CODE but +93.27% in ESA; Jupiter 2023 yields
+21.45% in CODE but −48.28% in ESA. (The Mars-2025 pair quoted in
earlier drafts, −14.79% versus +6.82%, was an artifact of
unconstrained-centre Gaussian fits; under the corrected event-locked
metric all centers suppress, within placebo noise.) Such discordance
confirms that single-event amplitude fluctuations reflect center-specific
orbit determination dynamics and clock re-absorption rather than coherent
physical imprints. The data thus bound transient planetary scalar
perturbations rather than resolving them. Planetary-event fits are
accordingly retained as secondary exploratory structure; later
season-preserving RINEX controls (Paper 3) show that much of the event
excess can arise from calendar-position sampling.

#### Disformal Transport and Closed-Loop Holonomy

Within the rigorous framework of TEP, genuine disformal interactions governed
by B(φ)∇<sub>μ</sub>φ∇<sub>ν</sub>φ (Rule 1)
cannot be cleanly extracted from unclosed, single-receiver clock offsets,
which are vulnerable to synchronization conventions and network parameter
absorption. Instead, Rule 13 dictates that disformal transport is uniquely
and gauge-invariantly probed by closed-loop synchronization holonomy
H<sub>resid</sub> = ∮(σ̃ − σ<sub>GR</sub>) around
closed circuits. While purely conformal couplings leave H<sub>resid</sub>
identically zero, non-exact disformal transport generates non-zero
circulation. Closed-loop inter-station circuits, rather than single-station
planetary event amplitudes, provide the definitive empirical channel for
isolating the disformal sector.

#### Energy vs Velocity Scaling: Multi-Body Dynamics

As reported in Section 3.3.2, the energy vs velocity discrimination
analysis reveals near-zero difference between mechanical energy and
kinematic velocity correlations with collective network dynamics. This
non-significant discrimination indicates that TEP coupling involves
gravitational field geometry changes and multi-body dynamics (Earth-Moon
system) beyond simple mechanical energy or kinematic velocity scales.
This pattern supports sophisticated TEP physics with multiple coupling
pathways operating simultaneously, consistent with TEP's predicted
dynamical clock/covariance structure and motivating direct tests of
non-integrable time transport through closed-loop experiments. The similar
scaling with both energy and velocity suggests the coupling mechanism is
sensitive to gravitational field curvature rather than simple mass or
motion parameters.

### 4.5.2 Re-evaluating the Saturn Opposition Residual

Observation: A 5.98σ Gaussian-fit amplitude was reported for the Saturn
opposition of September 2024 in IGS Combined, with an apparent enhancement
ratio of 59× relative to simple M/d² scaling. The event-locking audit
shows this fit is centred ~80 days away from the opposition date
(σ<sub>width</sub> ≈ 60 days): it is a window-edge excursion, not an
event response. Under the corrected direct test the Saturn-2024 peak window
shows suppression in all centers (CODE −1.85%, IGS −44.79%, ESA
−112.83%), within placebo noise; the single high-percentile Saturn
excursion (2023, ESA, +93.27%) is not reproduced by the other centers.

Interpretation: While initial speculation entertained the hypothesis that Saturn's ring
system or extensive magnetosphere might enhance local coupling to the φ-field, rigorous
methodological evaluation demonstrates that the enhancement ratio is primarily an algebraic
artifact of the small Newtonian denominator, and the underlying fit significance is an
artifact of an unconstrained event centre. Saturn's Newtonian acceleration at Earth
(M/d² ≈ 0.0104 × 10<sup>−10</sup> m/s²) is roughly 80 times
smaller than that of Jupiter, so a bounded network residual fluctuation divided by this
small baseline mechanically generates a large enhancement factor. Invoking ad hoc ring or
magnetospheric coupling mechanisms is unnecessary, reinforcing the principle that
disformal coupling must be isolated through closed-loop holonomies rather than unclosed
single-station amplitude ratios.

### 4.5.3 Consistency with Multi-Messenger Astronomy (GW170817)

The observed E-W/N-S spatial correlation anisotropy (typical directional range 0.55–2.03, with sector peaks to ≈4.75; integrated 25-year ratio 2.16 ± 0.23, Paper 2) is
structurally decoupled from, and fully consistent with, the
GW170817 propagation-speed bound |c_γ − c_g|/c ≲ 10<sup>−15</sup>. The
spatial anisotropy is driven entirely by the conformal factor
A(φ), which modifies clock transition frequencies through the
local scalar field amplitude without altering the null-cone
structure. The GW170817 bound constrains the disformal coupling
B(φ)(∂φ)² on cosmological scales, not the conformal sector that
governs the GNSS covariance observable. Consequently, the
conformal-dominant mechanism natively preserves null-cone
invariance: for a spatial variation of
Δφ/M<sub>Pl</sub> ~ 10<sup>−6</sup> near Earth (governed by universal
coupling β<sub>A</sub> = −1), the conformal variation
ΔA/A ~ 10<sup>−6</sup> produces the observed terrestrial clock rate
modulation while maintaining c_g ≈ c_γ globally. The PPN bound applies to the
environmentally screened effective source coupling rather than
directly fixing the bare conformal parameter β<sub>A</sub>, so
conversion of the observed coherence amplitude into Δφ is
completion- and screening-model-dependent. The disformal coupling
B(φ) may additionally be environment-dependent, remaining
negligible on cosmological sightlines while permitting
near-Earth variation—though the conformal pathway alone suffices
to explain the observation without invoking any disformal
enhancement.

### 4.5.4 Chandler Wobble Detection and Geodetic Reference Frame Coupling

Observation: A significant Chandler wobble signature is detected across all
centers: CODE (R² = 0.377, $p \lt 0.01$), ESA (R² = 0.453, $p \lt 0.01$), and IGS Combined
(R² = 0.471, $p \lt 0.01$), correlating network mesh dynamics with the Chandler wobble
phase (~433-day period).

Interpretation: In global GNSS processing, Earth Orientation Parameters (EOP: polar motion
coordinates x<sub>p</sub>, y<sub>p</sub>) are estimated simultaneously with satellite orbits,
station coordinates, and clock parameters. Polar motion exhibits a dominant ~433-day Chandler
component. Because the realization of the terrestrial reference frame links station coordinates
to EOP estimates, long-period rotational dynamics couple directly into network clock solutions.
The consistent detection of this ~433-day signature across CODE, ESA, and IGS Combined
reflects this geodetic coupling between terrestrial reference frame orientation and network clock
parameters. Rather than representing an unmodeled exotic gravitational perturbation, this
correlation serves as a valuable calibration metric demonstrating that the network correlation
framework correctly resolves multi-month geodynamic modulations. The variation in R² values
across centers reflects their respective parameter constraints: CODE's double-difference
network approach enforces explicit baseline constraints that partially attenuate long-period
common-mode rotational signals, whereas unconstrained zero-difference solutions transmit the
geodetic modulation more completely.

## 4.6 Robustness to Processing Effects

Having presented evidence for signal authenticity (Section 4.2),
systematically addressed alternative explanations (Section 4.3), and
analyzed spectral coupling (Section 4.4), this section now addresses how
GNSS processing affects the measurements and why the detected signals
demonstrate substantial robustness.

**Experimental Section:**

### Key Insight: Signal Persistence Despite Processing

The robustness of TEP correlations to GNSS processing provides critical
evidence for signal authenticity through three key observations:

#### 1. GNSS Processing Removes Correlated Signals

Network constraints, environmental corrections, and quality
control are specifically designed to remove spatially correlated
signals—precisely the signature predicted by TEP theory. Key
processing steps that suppress correlations:

**Orbit/clock estimation:** Removes common-mode
satellite errors affecting multiple stations simultaneously

**Ionospheric corrections:** Eliminates
TEC-related correlation sources across regional scales

**Tropospheric modeling:** Removes
weather-related correlations between nearby stations

**Network constraints:** Enforces global
reference frame, suppressing large-scale coordinate
correlations

**Suppression factor:** Standard GNSS processing is
designed to suppress spatial correlations through multiple
mechanisms. The exact suppression factor is unknown and requires
raw data analysis for quantification. The survival of TEP
signals (R² = 0.920-0.970) through this "correlation firewall"
suggests signal robustness, though true coupling strength
remains to be determined through raw pseudorange and carrier
phase analysis.

#### 2. Strong Correlations Nevertheless Persist

Despite aggressive filtering, R² = 0.920-0.970 correlations
persist with λ = 3,330-4,549 km across all three analysis center solutions.

#### 3. Implications for Signal Authenticity

Robustness demonstrates genuineness: Processing artifacts
would be eliminated by these corrections

Phase method effectiveness: The amplitude-invariant approach
captures signal structure that survives processing

Multi-modal validation confirms: Earth motion, planetary
correlations, and astronomical event responses validate the
physical nature of detected signals

Conclusion: The persistence of physically validated correlations despite
processing designed to remove them strengthens the case for genuine
field coupling.

Key observation: All GNSS clock products undergo extensive processing
specifically designed to remove spatially correlated signals—precisely the
signature predicted by TEP theory. The fact that strong correlations (R² =
0.920-0.970) survive this aggressive suppression, combined with their
validation through Earth motion coupling and astronomical event responses,
provides substantial evidence for signal authenticity and methodological
robustness.

**Experimental Section:**

### Systematic Signal Suppression Mechanisms

Standard GNSS analysis center processing applies multiple corrections
that would specifically attenuate TEP signals:

#### 1. Common-Mode Removal and Network Constraints

Network datum constraints: All centers apply network-wide reference
frame constraints that force the sum of clock corrections to zero,
explicitly removing any globally coherent signal components

Common-mode filtering: Systematic removal of signals common across
multiple stations, precisely the signature predicted by TEP theory

Reference clock stabilization: Centers typically stabilize against
ensemble averages, further suppressing spatially correlated
variations

Impact: Network constraints are designed to eliminate globally
coherent variations

#### 2. Sidereal and Environmental Corrections

Sidereal filtering: Routine removal of 24-hour and harmonically
related periodicities would eliminate TEP signals coupled to Earth's
rotation

Environmental modeling: Tropospheric and ionospheric corrections
remove spatially correlated atmospheric effects, potentially
including genuine field-atmosphere coupling

Multipath mitigation: Advanced multipath modeling may inadvertently
remove coherent field effects that manifest as apparent signal path
variations

Effect: Environmental corrections remove spatially correlated
atmospheric variations

#### 3. Outlier Detection and Quality Control

Statistical outlier removal: Automated detection of "anomalous"
correlations would systematically exclude the strongest TEP events

Inter-station consistency checks: Quality control algorithms
designed to ensure station independence would flag and remove
genuine field correlations

Temporal smoothing: Many centers apply temporal smoothing that would
blur rapid field variations during astronomical events

Result: Quality control processes are designed to flag and remove
inter-station correlations

**Experimental Section:**

### Evidence for Systematic Underestimation

Multiple lines of evidence suggest the measurements represent lower
bounds on true field coupling strength:

#### Processing-Dependent Signal Strength

Center-specific variations: The CV of λ across centers ≈ 13%
likely reflects different degrees of signal suppression rather than
measurement noise

ESA vs CODE comparison: ESA's shorter λ (3,330 km) and higher
amplitude (A=0.250) compared to CODE's longer λ (4,549 km) and lower
amplitude (A=0.114) suggests different processing approaches
preserve different aspects of the TEP signal

IGS Combined intermediate values: IGS Combined shows intermediate
characteristics (λ=3,764 km, A=0.194, exponential model), consistent
with processing-dependent signal recovery

#### Residual Signal Persistence

Survival of strong correlations: The fact that R² = 0.920-0.970
correlations persist despite aggressive processing suggests the
underlying field coupling may be substantially stronger than
measured

Elevation dependence: The systematic increase in λ with station
elevation (Q1 to Q5: 1,785 to 4,549 km) indicates atmospheric
screening effects that would be amplified in raw data

Astronomical modulations: Detection of eclipse and supermoon effects
despite processing suggests stronger signatures may exist in
minimally processed raw data

#### Phase Information Preservation

Why phase survives: While amplitude corrections are applied
per-station, relative phase information between stations remains
largely intact, explaining why the phase-based method succeeds where
amplitude methods fail

Incomplete phase suppression: The phase-alignment index approach
captures residual phase relationships that survive processing, but
this represents only a fraction of the original signal

Raw data potential: Access to raw pseudorange and carrier phase
measurements could provide additional insights into the field
coupling mechanism

**Experimental Section:**

### Future Research Directions

The robustness of the findings to GNSS processing suggests several
promising avenues for future investigation:

#### Immediate Research Priorities

Raw pseudorange analysis: direct analysis of raw GPS pseudorange
observations through independent minimal processing — the first
such consistency test is reported in Paper 3

Carrier phase coherence: High-precision analysis of raw carrier
phase data to detect field-induced timing variations at the
wavelength level

Multi-constellation validation: Extension to separately processed GLONASS, Galileo, and
BeiDou raw data to confirm field universality per constellation (extending the combined multi-GNSS MGEX replication reported in Paper 14)

Real-time monitoring: Development of dedicated TEP detection
networks bypassing standard GNSS processing chains

#### Expected Raw Data Advantages

Signal characterization: direct analysis of raw measurements under
minimal processing could provide additional insights

Temporal resolution: Sub-second detection of rapid field variations
during astronomical events

Spatial precision: Millimeter-level coherence detection through
carrier phase analysis

Dynamic range: Full access to field variations across all timescales
and amplitudes

Scientific Impact: Raw data access would provide critical validation of
the signal strength hypothesis and could reveal substantially stronger
coupling than currently measured, advancing understanding of potential
modifications to spacetime structure and enabling more rigorous tests of
fundamental physics.

## 4.7 Temporal Dynamics: Evidence Consistent with Dynamical Time

The comprehensive diurnal analysis (Section 3.5.3) provides evidence
consistent with the Temporal Equivalence Principle's central prediction:
proper time flow rates are dynamically variable. This section interprets the
temporal findings within the TEP theoretical framework and assesses their
implications for fundamental physics.

### TEP Theoretical Framework for Time Rate Variations

According to TEP, the rate at which proper time accrues is governed by
the scalar field φ through the conformal coupling:

dτ̃ = A(ϕ)dτ<sub>g</sub> where A(ϕ) = exp(β<sub>A</sub>ϕ/M<sub>Pl</sub>)

where β<sub>A</sub> is the bare conformal coupling in the foundational TEP convention and M<sub>Pl</sub> is the Planck mass

With the frozen coupling β<sub>A</sub> = −1 (Paper 0 §2.2), A(ϕ) = exp(β<sub>A</sub>ϕ/M<sub>Pl</sub>)
decreases as ϕ increases, so clocks run *slower* where ϕ is larger — that is, deeper in a potential
well, matching GR. The observed diurnal patterns provide evidence consistent with this prediction,
though alternative explanations involving slow environmental covariates
cannot be excluded pending raw data validation.

#### Quantitative Consistency Check

The observed temporal variations allow estimation of field parameters:

Daily amplitude: Systematic diurnal variations observed (1.9-7.6%
day-night coherence variation)

Seasonal amplitude: Systematic seasonal modulation observed (spring
equinox maxima)

Processing sensitivity: Factor 2× range (CODE vs ESA Final) suggests
β_eff varies by analysis center

The PPN bound applies to the environmentally screened effective source
coupling rather than directly fixing the bare conformal parameter
β<sub>A</sub>. Consequently, conversion of the observed coherence
amplitude into Δφ is completion- and screening-model-dependent, and
numerical estimates of Δφ from the diurnal/seasonal amplitudes are not
pursued here.

### Physical Coupling Mechanisms

The systematic diurnal patterns reveal associations consistent with specific
coupling pathways between φ-field dynamics and observable quantities, though
direct causal relationships remain to be established:

#### Solar Angle Modulation

Mechanism: Solar zenith angle variations drive φ-field gradients
through gravitational coupling. Early morning peaks correspond to
optimal solar angles for maximum field response.

Evidence: Consistent 3-10 AM peak timing across all seasons and
centers, with winter showing strongest dawn effects when solar angle
changes are most pronounced.

#### Ionospheric TEC Coupling

Mechanism: Total Electron Content variations modulate
electromagnetic field propagation, coupling to φ-field through the
disformal metric structure g̃<sub>μν</sub> = A²(φ)g<sub>μν</sub> + B(φ)∇<sub>μ</sub>φ∇<sub>ν</sub>φ.

Evidence: Early morning peaks align with TEC daily minimum, when
reduced multipath enhances sensitivity to fundamental field
variations.

#### Seasonal Orbital Dynamics

Mechanism: Earth's orbital motion creates "velocity wind" through
the φ-field, with maximum effects during equinox periods when
orbital velocity and solar coupling are optimally aligned.

Evidence: Spring showing strongest diurnal effects (seasonal-peak
ratios 1.074-1.292), summer showing most balanced patterns (ratios
0.936-1.047), consistent with orbital phase coupling.

#### Screening Modulation

Mechanism: The continuous spatial profile of the field (Temporal
Topology) maintains its structure in dense environments, while the local
Earth-vicinity Temporal Shear is pairwise-screened. The clock-sector
response is governed by the terrestrial amplitude factor $S_A^{(\oplus)} \approx 0.65$.

Evidence: Processing-dependent sensitivity (ESA Final 2× CODE
levels) suggests different observable response amplitudes, consistent
with variations in the clock-sector coupling.

### Cross-Center Analysis: Processing as φ-Field Probe

The systematic differences in temporal sensitivity across analysis centers
provide insights into the coupling mechanism:

| Center | Signal Level | Temporal Stability | Diurnal Strength | TEP Interpretation |
| --- | --- | --- | --- | --- |
| ESA Final | 2× enhancement | Highest (CV of daily pair counts = 0.137) | Moderate (1.060 global-mean) | Superior φ-field sensitivity via processing |
| IGS Combined | Intermediate | Moderate (CV of daily pair counts = 0.154) | Strongest (1.076 global-mean) | Ensemble averaging enhances diurnal contrast |
| CODE | Baseline | Lowest (CV of daily pair counts = 0.196) | Minimal (1.019 global-mean) | Conservative processing minimizes φ-coupling |

Key Insight: Different GNSS processing strategies exhibit systematically
different sensitivity to φ-field coupling, with ESA Final's enhanced
precision potentially preserving more φ-field information than CODE's
conservative approach. This suggests that processing methodology acts as an
effective "filter" for temporal field detection.

### 4.8 Evidential Hierarchy and Discriminating Tests

The empirical results and theoretical interpretation are structured into three distinct levels of inference:

Empirical Detection: The primary result of this paper is the detection
of a statistically robust, distance-structured correlation (λ<sub>T</sub> = 3,330–4,549 km) in global GNSS clock networks. The validation suite in
Section 4 establishes that this structure survives across software-diverse processing pipelines (Bernese, NAPEOS, IGS combination) and 180 randomized null realizations.

Theoretical Parameterization: Within the Temporal Equivalence Principle (TEP), the measured correlation scale defines the terrestrial saturation anchor ρ<sub>T</sub> ≈ 20 g/cm³ (R<sub>T</sub>(M<sub>⊕</sub>) ≈ 4,150 km) of the continuous Temporal Topology, which is then tested without retuning across galactic and stellar regimes in companion analyses.

Cross-Technology Scaling Tests: Companion studies confirm recovery across 25.3 years (Paper 2), raw RINEX pseudorange processing with zero network adjustment (Paper 3), and multi-constellation MGEX products (Paper 14); raw carrier-phase processing and non-GNSS optical-fiber and two-way satellite time-transfer comparisons test the unsuppressed coupling amplitude across independent timing architectures.

## 4.8 Synthesis and Research Roadmap

The strong evidence presented in Sections 4.1-4.7—observations consistent
with theoretical predictions, comprehensive validation, systematic
alternative exclusion, spectral analysis, theoretical insights, processing
robustness, and temporal dynamics—establishes a robust foundation for the
TEP interpretation. These findings are synthesized and the natural research
frontier emerging from this work is outlined:

**Experimental Section:**

### Summary of Evidence

Signal authenticity supported (Section 4.2): Eleven independent
validation criteria passed, including comprehensive multiple
comparison correction analysis (137-168 of 388 statistical tests
maintain significance after ultra-stringent Bonferroni α = 0.000129,
FDR q = 0.05, and family-wise corrections), complete validation
spanning data quality verification through astronomical period
analysis, and systematic statistical robustness through bootstrap,
LOSO, LODO, and cross-validation methods

Alternative explanations addressed (Section 4.3): Systematic
analysis does not support simple Newtonian mass-distance scaling
within current power (n≈5 planets, 2.5 y) for classical tidal
contamination (post-tidal 30–40 μHz shows R² = 0.946, enhancement
ratio only 1.58× not >3×), tested ionospheric effects (TIDs, TEQ),
processing artifacts, and examined conventional geophysical
phenomena through scale, temporal, and spectral mismatches

Spectral coupling characterized (Section 4.4): Cross-center
multiband validation demonstrates remarkable consistency across
software-diverse processing chains on the shared tracking network. Multi-band analysis reveals conformal
covariance (broadband, CV of R² across bands = 2.9%) with possible
gradient-sensitive structure (gravitational enhancement, λ = 4,627 km
mean at tidal frequencies), with systematic effects quantified through
control bands showing reduced model fit quality (R² = 0.618, ΔR² ≈
0.33 from TEP band)

Theoretical insights and future directions (Section 4.5): Observed
primary correlation lengths (λ<sub>T</sub> = 3,330–4,549 km) fall within
the pre-specified search range, with geometric saturation scale
R_T(M_⊕) ≈ 4,150 km (see §1.1)
and potential implications for fundamental physics including dark
matter connections and fifth
force constraints. Bootstrap validation shows center-specific
covariance-length ranges (see Table in §3.1.1); these do not
measure a microscopic scalar mass.

Robustness to processing effects (Section 4.6): GNSS processing is
designed to remove spatially correlated signals; the survival of
strong correlations (R² = 0.920-0.970) through such filtering,
combined with multi-modal physical validation, provides evidence for
genuine field coupling rather than processing artifacts

Dynamical time validation (Section 4.7): Comprehensive diurnal
analysis of 72.4 million hourly records reveals systematic temporal
variations consistent with TEP predictions for variable proper time
flow rates, with synchronized early morning peaks and seasonal
modulation. The patterns are consistent with the central TEP tenet
that "when" is as dynamical as "where," though alternative slow
environmental covariates remain plausible pending raw carrier-phase
confirmation (raw-RINEX pseudorange/SPP consistency is reported in Paper 3)

### Physical Parameter Space Constraints

The observed Temporal Topology correlation length λ<sub>T</sub> = 3,330-4,549 km defines the terrestrial saturation anchor R<sub>T</sub>(M<sub>⊕</sub>) = (3M<sub>⊕</sub>/4πρ<sub>T</sub>)<sup>1/3</sup> ≈ 4,150 km (ρ<sub>T</sub> ≈ 20 g/cm³; Paper 6); across products the fitted scale spans ~1.0–4.8×10<sup>3</sup> km under documented estimator and product conditioning, with the same band-ordered profile recovered in each product (MGEX 1,862 ± 155 km; Paper 14, Step 3.5). The measured covariance scale is not a scalar Compton-wavelength measurement. Precision constraints instead apply to the observable clock-covariance and shear channels separately:

**Optical clock comparisons:** Current fractional
stability limits (~10⁻¹⁸-10⁻¹⁹) provide stringent tests of coupling
strength β in the TEP framework

**Gravitational redshift tests:** Galileo eccentric
satellite experiments and terrestrial tower tests constrain metric
coupling mechanisms

**Equivalence principle measurements:** MICROSCOPE and
similar tests limit differential coupling to matter composition

**PPN parameter bounds:** Solar system tests constrain
deviations from General Relativity through γ-1 and other parameters

Critical research priority: derive the transfer from the solved $C_A(\mathbf r,t)$ field to the observed GNSS covariance amplitude, anisotropy and orbital modulation, while testing the clock-covariance and shear channels against their respective precision constraints.

### Open Questions: Next-Generation Alternative Testing

Building upon the systematic exclusion of primary alternatives, the
following hypotheses represent the natural research frontier. Each
offers specific testable predictions that could further strengthen the
TEP interpretation or reveal alternative physics:

#### 1. Systematic Processing Correlations

Hypothesis: GNSS analysis centers use similar reference models,
creating artificial correlations that survive current null tests

Mechanism: Shared satellite orbit models, common ionospheric
corrections, or similar tropospheric delay models could induce
distance-structured correlations

Required Test: Analysis with external software and observation regimes
(e.g., true user-side single-receiver PPP vs global network solutions,
multi-constellation combinations, or raw carrier-phase processing)

Bound established (Steps 5.0, 5.2): Synthetic station clocks
retaining only the satellite-visibility and zero-sum datum
structure on the real 768-station network, propagated through
the same band-limited correlation estimator, yield fitted decay
scales of λ ≳ 6.7×10<sup>3</sup> km in every realisation
(medians 7.3–12.8×10<sup>3</sup> km across elevation-mask and
datum configurations; pure geometric overlap kernel:
≳1.3×10<sup>4</sup> km). The bound is then conditioned on
the real product (Step 5.2): recomputing the kernel on true
CODE SP3 ephemerides reproduces the nominal-constellation kernel
essentially exactly (λ = 13.4–21.6×10<sup>3</sup> km, matching
the Paper 14 product kernel band for band), and replacing the
synthetic satellite noise with the product's own estimated
satellite clocks (CODE MGEX 30 s AS records, four days) leaves
every conditioned realisation at λ ≥ 5,075
km — still above every measured λ<sub>T</sub> = 3,330–4,549 km.
The discrimination is in fact stronger than the scale bound
alone: the conditioned channel projects near-unity correlation
within 1 Mm and a sign-definite anticorrelation beyond ~5 Mm
under the datum constraint (band means −0.10 to −0.30 at
8–12 Mm in every datum-applied realisation; fitted asymptote
C<sub>0</sub> = −0.43 to −0.94 in all 40), whereas the measured
correlation carries
amplitude ≈0.11–0.25 and asymptote ≈−0.02 — because the datum
constraint redistributes a common mode, the artifact must
anticorrelate distant stations, and the measured field does
not. On raw RINEX data the same conditioning, now applied to
the canonical weighted-overlap kernel rather than the binary
common-view fraction, gives partial Spearman
ρ(C, k<sub>w</sub> | r) = 0.014 (p = 0.026) with a joint-fit
kernel coefficient of 0.015 and a residual exponential at
λ = 580 km — a sub-percent share of the coherent variance
(Paper 3). What remains outside reach is only an in-centre
reprocessing of the ambiguity-resolution internals, which are
not published products and are unavailable to any independent
analysis; every channel accessible from the released products
is now bounded.

#### 2. Atmospheric Loading Effects

Hypothesis: Large-scale atmospheric pressure variations create
correlated timing effects through gravitational loading

Mechanism: Atmospheric mass redistribution affects local gravity,
influencing atomic clock rates through gravitational redshift

Predicted Scale: Continental-scale correlations (1,000-5,000 km)
matching observed λ values

Required Test: Correlation with atmospheric reanalysis data (ECMWF,
NCEP)

#### 3. Solid Earth Tidal Correlations

Hypothesis: Imperfect solid Earth tide corrections create residual
correlations with distance-dependent structure

Mechanism: Earth tide models may not perfectly capture local
geological variations, leaving systematic residuals

Required Test: Analysis with enhanced tidal corrections, geological
substrate correlations

#### 4. Electromagnetic Field Coupling

Hypothesis: Large-scale electromagnetic fields (magnetospheric,
atmospheric electrical) couple to atomic clocks

Mechanism: AC Stark shifts from ambient electromagnetic fields could
modulate transition frequencies

Required Test: Correlation with magnetometer networks, atmospheric
electrical measurements

#### 5. Thermal Environment Correlations

Hypothesis: Large-scale temperature patterns create correlated
thermal effects on clock performance

Mechanism: Continental-scale weather patterns could induce
systematic thermal effects on station infrastructure

Required Test: Correlation with meteorological reanalysis, seasonal
decomposition

Research Strategy: Systematic testing using independent datasets
(atmospheric reanalysis, geological surveys, electromagnetic monitoring,
meteorological data) represents the natural next phase of investigation.
Success in excluding these alternatives would further strengthen the TEP
interpretation through comprehensive process of elimination, while
positive findings would redirect theoretical development—either outcome
advances fundamental understanding.

### Critical Priorities and Path Forward

While the evidence is substantial, several critical steps are
essential before strong conclusions:

Independent replication: Analysis by other research groups using
different methodologies and data sources

Raw data validation: direct analysis of raw GNSS observations
through independent minimal processing to quantify processing
suppression and reveal full signal magnitude

Multi-constellation testing: Extension to GLONASS, Galileo, and
BeiDou for technology-independent confirmation

Next-generation hypothesis testing: Systematic investigation of
the additional hypotheses outlined above

Interpretive framework: The systematic progression from observations
consistent with predictions (Section 4.1) through comprehensive
validation (Section 4.2) and alternative exclusion (Section 4.3) to
spectral analysis (Section 4.4) and theoretical insights (Section
4.5) establishes a robust foundation for the TEP framework. The
observation that signals survive aggressive processing suppression
(Section 4.6) strengthens this foundation by demonstrating signal
robustness while indicating that true coupling may be substantially
stronger than measured.

Scientific Standards and Next Steps: The significant nature of these
findings necessitates rigorous independent verification—the standard
scientific practice for novel discoveries. Signal authenticity has
been strongly tested against the examined alternatives through
comprehensive validation, and quantitative constraints on field
properties have been derived. The robustness of these measurements to
standard processing procedures validates the methodological approach,
and these findings define a concrete experimental roadmap for
optical-fiber and raw carrier-phase tests with direct implications
for precision metrology and dynamical-time physics.

### Summary Matrix: Conventional Geodetic Mimics vs. Empirical Controls

Because geocentric radial and baseline-dependent correlations can in principle arise from unmodeled orbit, clock-datum, or environmental errors, Table 4.9 consolidates the primary conventional mimic hypotheses, the pipeline control that tests each mechanism across Papers 1, 2, 3, and 14, and the remaining out-of-sample discriminator.

| Candidate Conventional Mimic | Physical / Processing Mechanism | Empirical Control & Quantitative Outcome | Residual Scope & External Decider |
| --- | --- | --- | --- |
| **Common-view satellite orbit / clock-datum projection** | Radial orbit error or zero-sum reference-clock constraint projecting onto shared-visibility station pairs | 40-realization synthetic visibility + datum kernel on true CODE SP3 ephemerides and MGEX 30 s satellite clocks gives $\lambda \ge 5{,}075$ km (medians $7.3\text{--}21.6\times10^3$ km) and mandatory $8\text{--}12$ Mm anticorrelation ($C_0=-0.43$ to $-0.94$), whereas observed data show $\lambda_T = 3{,}330\text{--}4{,}549$ km and $C_0 \approx -0.02$ (Steps 5.0, 5.2). | Excluded as source of the $3.3\text{--}4.5$ Mm scale; in-center ambiguity-fixing internals tested by non-GNSS optical fiber links (§4.10). |
| **Analysis-center software / PPP filter conditioning** | Center-specific orbit/troposphere parameterization inducing synthetic spatial correlation | Cross-center invariance across CODE, IGS, and ESA ($R^2 = 0.920\text{--}0.970$) and survival in single-station raw RINEX broadcast-ephemeris SPP across 539 stations / 713M pair-samples without network clock estimation ($t$ up to $112$; Paper 3). | Network-filter artifact excluded by single-receiver SPP; raw un-differenced carrier-phase reprocessing is the next geodetic tier. |
| **Solar radiation pressure (SRP) / draconitic harmonic** | GPS $1.04\text{ cpy}$ draconitic orbit-modeling error aliasing into annual velocity modulation | Multi-constellation MGEX test across GPS, GLONASS, Galileo, and BeiDou (distinct orbital planes, altitudes, and SRP area-to-mass ratios) recovers the same band-ordered spatial decay (Paper 14, Step 3.5); 19-family FWER survival $178/388$ ($45.9\%$). | 10-day orbital-velocity vs. generic annual sinusoid remains underpowered globally (Paper 2, Step 2.10); requires daily-cadence multi-constellation separation. |
| **Tropospheric / seasonal / hemispheric weather loading** | Continental meteorological or hydrological mass loading coupling into station clocks/verticals | Northern vs. Southern Hemisphere season-phase stratification, elevation-mask invariance ($5^\circ\text{--}30^\circ$), and weighted common-view partial regression ($\rho(C, k_w | r) = 0.014$, $<1\%$ variance share; Paper 3 Step 2.6b) preserve the directional E&ndash;W/N&ndash;S anisotropy ($2.16\times$, $p < 10^{-15}$). | Direct station-level ECMWF pressure/temperature co-regression on raw carrier phase. |
| **Station hardware / atomic-standard family** | Common-vendor rubidium/cesium/H-maser drift or cable temperature sensitivity | Stratification by oscillator type (H-maser vs. Cs vs. Rb) and monument class across IGS/MGEX networks retains the exponential decay profile and CMB-aligned anisotropy axis ($\sim 18^\circ$ from CMB dipole; Solar Apex excluded). | Co-located GNSS-minus-TWSTFT and PTB&ndash;SYRTE optical-fiber time transfer (§4.10). |

## 4.10 A Falsifiable Prediction for Optical Clock Networks

This work's findings enable a transition from discovery to
predictive science. The parameters derived from the GNSS analysis
provide the basis for specific, falsifiable predictions for
next-generation optical clock networks. These networks, utilizing
Ytterbium and Strontium optical lattice clocks, offer ~100× greater
precision than the microwave clocks in the GNSS. Based on the
measured Temporal Topology correlation lengths λ<sub>T</sub> = 3,330–4,549 km and systematic
temporal patterns, it is predicted that a terrestrial network of
these clocks should observe distance-structured correlations with
enhanced sensitivity over continental baselines (1,000–5,000 km).
Expressed directly in fractional frequency (clock-rate) units
Δν/ν = σ<sub>dτ/dt</sub>, the GNSS spatial-covariance amplitude
(structured plateau σ<sub>Δτ</sub> ≈ 5–15 ps over diurnal-to-synodic
averaging scales τ<sub>int</sub> ~ 10<sup>4</sup>–10<sup>5</sup> s) corresponds to a correlated
fractional clock-rate modulation of Δν/ν ≈ (1.0–4.5) × 10<sup>−16</sup>
at τ<sub>int</sub> = 10<sup>4</sup> s and Δν/ν ≈ (1.5–5.0) × 10<sup>−17</sup> at
τ<sub>int</sub> = 10<sup>5</sup> s across continental baselines d ≳ 1,000 km,
scaling with baseline separation d as
ΔC(d) ∝ [1 − exp(−d/λ<sub>T</sub>)] (yielding ~21% of the asymptotic
plateau across a d = 1,000 km optical-fiber link such as PTB–SYRTE
and ~39% across a d = 2,000 km link such as PTB–SYRTE–NPL/INRIM). Existing fibre-linked comparisons already probe the lower part of this range. The PTB–SYRTE Sr/Sr comparison over a 1415 km fibre (690 km baseline) reached a remote-comparison instability of about (1–3) × 10<sup>−17</sup> at τ = 10<sup>4</sup> s, with frequency agreement (4.7 ± 5.0) × 10<sup>−17</sup> over 145 h (Lisdat et al. 2016, *Nat. Commun.* 7, 12443). At this baseline the predicted differential amplitude is 14–19% of the plateau, (1.4–8.6) × 10<sup>−17</sup> at 10<sup>4</sup> s, placing the lower half of the range at current sensitivity. Reanalysing the public ratio time series from the 2022 coordinated PTB–SYRTE–INRIM campaign (Lindvall et al. 2025, arXiv:2505.06763) for the distance-structured, annually modulated covariance predicted here provides a further test.
Because state-of-the-art phase-stabilized optical-fiber links and
Sr/Yb lattice clocks achieve fractional frequency instabilities of
σ<sub>y</sub> ≤ 10<sup>−17</sup>–10<sup>−18</sup> at τ<sub>int</sub> ≥ 10<sup>4</sup> s—two to three
orders of magnitude below the predicted TEP clock-rate floor—simultaneous
three-node cross-correlation across European or transcontinental
fiber-linked optical clock networks can resolve the distance-structured
Δν/ν covariance at SNR > 20 without satellite-orbit or
ionospheric-propagation systematics.

Furthermore, the temporal signature of this signal should exhibit
clear periodicities tied to Earth's orbital kinematics. A multi-year analysis of
such a network should reveal an annual orbital-phase modulation of the EW/NS
covariance anisotropy peaking at aphelion, alongside distance-structured spatial
decay free from satellite-orbit systematics, with secondary inspection at the
Chandler wobble period (~433 days). Detection of this
signature in fiber-linked or transportable optical clock networks would provide an
independent, orbit-free, higher-precision test of the
conformal covariance phenomenon identified in the GNSS data. Dedicated one-way closed-loop experiments remain the
direct test of synchronization holonomy.

## 5. Conclusions

This section synthesizes the investigation presented above, highlighting key findings, the validation framework, and priorities for future research.

### Scope and Discriminating Controls

- **Processing conditioning:** Standard GNSS estimation conditions signal amplitude and effective correlation scale; raw-RINEX pseudorange/SPP recovery is established in Paper 3, while raw carrier-phase processing remains a primary target

- **Temporal baseline:** The 2.5-year multi-center window (2023–2025) resolves seasonal, orbital, and Chandler-band modulations; decadal and 18.6-year nutation cycles are evaluated in the 25.3-year longitudinal analysis (Paper 2)

- **Environmental controls:** Null, geomagnetic, and product-conditioned forward simulations constrain the tested artifact classes; spatially resolved atmospheric/ionospheric reanalysis provides further quantitative bounds

- **Out-of-ecosystem verification:** Non-GNSS optical-fiber and two-way satellite time-transfer (TWSTFT) comparisons provide the definitive test outside the IGS tracking architecture

### 5.1 Principal Findings

Analysis of 62.7 million station pair measurements from 364 unique stations (249 N / 115 S = 68.4% / 31.6%) across three distinct GNSS analysis center solutions (CODE, IGS Combined, ESA) spanning 2023–2025 reveals systematic distance-structured correlations in atomic clock networks. The primary methodological finding is cross-center validation success: comprehensive multiband analysis (12 frequency bands, 10–3000 μHz) demonstrates remarkable consistency across software-diverse processing chains (Bernese double-difference clock recovery, NAPEOS undifferenced global network adjustment, and IGS weighted combination), with optimal bands reaching R² = 0.970 (ESA Final), 0.920 (CODE), and 0.966 (IGS Combined). This establishes that the spatial correlation structure is robust against software-specific implementations and center-specific algorithmic biases. A detailed diurnal analysis of 72.4 million hourly records further demonstrates spatial correlation structures and temporal dynamics consistent with theoretical predictions for field coupling mechanisms.

**Experimental Section:**

#### Summary of Key Results

| Observable | Measured Value | Significance |
| --- | --- | --- |
| Temporal Topology Correlation Length (λT) | 3,330–4,549 km (pooled coherent band; band-ordered per sub-band, Table 8a) | Cross-center validation: R² = 0.970/0.920/0.966 (pooled fit on distance-bin means; ESA/CODE/IGS Combined); defines terrestrial anchor RT(M⊕) ≈ 4,150 km (ρT ≈ 20 g/cm³); cross-product family ~1.0–4.8×103 km (Paper 14, Step 3.5) |
| Multiband Validation | 12 frequency bands (10–3000 μHz) | Cross-center consistency substantially reduces likelihood of systematic biases |
| Diurnal Time Variations | 1.9–7.6% day-night variation | 72.4M records, >6σ combined significance |
| Orbital Velocity Coupling | r = -0.571 to -0.793 | p $\lt$ 10⁻⁷ (IGS Combined), consistent across centers |
| Planetary Coupling | 8 nominally significant event fits; event-locking audit finds most off-centre | Clearest event-locked case: Venus 2025 in ESA (+140%, σ ≈ 4); null mass scaling (r = −0.156) bounds transient planetary perturbations |
| Chandler Wobble Modulation | R² = 0.377–0.471 | Observed across centers (FDR-BH: 3/3 significant; family-wise: 2/3) |
| Beat Frequency Detection | 11 of 12 Earth motion patterns | R² up to 0.899, multi-center consistent |
| Collective Network Dynamics | Annual oscillation (p $\lt$ 10⁻³), coherence=0.643–0.644 | High network synchronization across individual components |
| 3D Spatial Structure | Anisotropy strength = 1.98 | Dipole: 5,715 km, Quadrupole: 5,660 km, 6.6× dynamic range |
| Multi-Frequency Beat Patterns | 35 detected, 7 FDR significant | Venus sub-harmonic (196.9d, R²=0.82-0.86), lunar fortnight (14.8d, R²=0.60-0.82); hierarchical strength pattern validates genuine signal |
| Signal Enhancement Factor | Mean: 124.4× over null | Range: 22.8–226.0×, indicates non-linear coupling mechanism |
| Seasonal Time Modulation | Spring maximum effects | Seasonal-peak day/night ratios: 1.074-1.292, synchronized peaks |
| TID Contamination | 21–23% improvement potential | Quantified and correctable, Venus 2f harmonic identified (Section 3.3.2) |

### 5.2 Validation Framework Summary

The signal's authenticity is supported by a multi-layered validation framework:

- Cross-Center Validation Achievement: Three distinct GNSS analysis center solutions demonstrate remarkable consistency in multiband patterns (12 frequency bands), achieving optimal R² values of 0.970 (ESA Final), 0.920 (CODE), and 0.966 (IGS Combined). This represents a crucial validation milestone, substantially reducing center-specific systematic biases and supporting signal authenticity across software-diverse processing methodologies.

- Rigorous Statistical Framework: Systematic validation across eleven complementary criteria establishes robustness across multiple null and systematic-error tests through null hypothesis testing (ΔR² = 0.89-0.95 signal separation over randomized controls, z = 15.8-31.9, 24-61× signal-to-null ratios across 180 scrambling iterations), frequency-dependent discrimination (control bands R² = 0.618 vs TEP R² = 0.952), comprehensive multiple comparison corrections across 388 statistical tests (40-52% survival rates), and numerous other controls for systematic errors. Note that weights and effective degrees of freedom (Neff) are held constant between real and scrambled fits to ensure apples-to-apples comparison of z-scores against null R² distributions.

- Venus 2f Harmonic Identified: A dominant ~112-day harmonic consistent with the Venus 2f expectation (112.35 d) is observed across centers with 109–118 d periods (3–5% deviation) (Section 3.3.2). TID exclusion analysis reveals 21–23% coherence improvement potential when excluding high ionospheric activity periods.

- Statistical Robustness Observed: The analysis demonstrates stability under jackknife cross-validation (CV of λ<sub>T</sub> across subsets = 3.5-6.5%) with effective degrees of freedom N<sub>eff</sub> = 25-28 distance bins per analysis center.

- Geographic and Instrumental Characterization: The correlation strength is consistent across hemisphere subsets and ocean vs. land baselines; the fitted correlation length is stratified by station elevation and geomagnetic latitude (Section 3.2.1), with quieter mid-latitude and higher-elevation tiers preserving longer correlation scales while auroral ionospheric disturbance compresses them, consistent with environmental screening and the 21–23% TID-exclusion coherence gain.

- Software and Algorithmic Diversity: Consistent multiband structure across CODE, IGS Combined and ESA Final reduces sensitivity to a single processing implementation. Completed companion analyses include combined multi-constellation clocks (Paper 14) and independently processed raw-RINEX pseudorange/SPP observations (Paper 3). Paper 1 Steps 3.9, 5.1 and 5.2 add systematic-injection, datum-projection and real-product-conditioned forward controls; Paper 14 Step 3.2 evaluates the product-error kernel, and Paper 3 Steps 2.10 and 2.12 test common-view and broadcast-versus-precise channels. These constrain their specified artifact mechanisms; they do not exclude every shared systematic or constitute raw carrier-phase validation.

- Dynamic Event Consistency: Eclipse and opposition event scales are compared with baseline Temporal Topology correlation lengths within this analysis. Scale agreement is a consistency check; event attribution requires the applicable seasonal and sampling controls, including the season-preserving tests reported separately in Paper 3.

- Signal-Over-Null Margin: Mean signal enhancement of 124.4× over scrambled-null expectation (range: 22.8-226.0×) excludes distance-independent noise as the origin of the correlation structure; candidate non-linear coupling completions (resonance, tidal amplification, parametric coupling) remain hypotheses pending mechanistic derivation.

- 3D Spatial Structure Validation: Complete spherical harmonic decomposition reveals anisotropy strength of 1.98 with dipole (5,715 km) and quadrupole (5,660 km) magnitudes, indicating nearly 2× stronger coupling in preferred directions consistent with Earth's motion through space.

- Multi-Scale Temporal Coupling: Detection of 35 distinct beat frequency patterns spanning four orders of magnitude (0.077–365 days), with strongest patterns corresponding to well-defined astronomical periods (Venus 196.9d: R²=0.82–0.86, lunar fortnight 14.8d: R²=0.60–0.82), demonstrating genuine multi-scale coupling to gravitational field geometry.

### 5.3 Signal Robustness and Processing Effects

GNSS estimation can absorb network-common clock components while retaining or introducing structured differential covariance. The completed product-conditioned and raw-data controls test specific contributions to that covariance. Recovery across processing products supports reproducibility of the measured structure; physical attribution depends on the scope of those controls, rather than survival through processing alone.

### 5.4 Alternative Explanations and Research Frontier

The validation framework tests conventional explanations through null, environmental, geometric and processing controls. The completed tests and their generating scripts define which alternatives are constrained. Correlation with additional geophysical datasets and independent replication can extend that evidence without treating untested mechanisms as already excluded.

### 5.5 Completed Validation and Further Discrimination

The empirical analyses and controls listed below are completed where indicated. Further independent tests address the stronger interpretation as a TEP signal; they do not require repeating the completed GNSS pipelines.

- **Independent Replication (Critical):** Analysis by multiple research groups using different methodologies, software, and theoretical frameworks

- **Raw Data Validation:** Paper 3 has completed the first raw-RINEX/SPP consistency test, demonstrating signal recovery in raw RINEX pseudorange observations processed independently with SPP and broadcast ephemerides and strongly constraining precise-product processing as the sole origin; raw carrier-phase processing, orbit-independent optical clock links, and external replication remain outstanding

- **Multi-Constellation Testing:** Paper 14 has confirmed isotropic scale recovery on MGEX combined clocks (GPS+GLO+GAL+BDS); multi-constellation carrier-phase analysis remains open

- **Extended Temporal Baseline:** Paper 2 has completed the 25.3-year longitudinal analysis (2000–2025), isolating decadal cycles and draconitic cancellation

The raw-pseudorange/SPP route has already been evaluated in Paper 3. The following carrier-phase and temporal-resolution targets describe additional experiments, with amplitudes conditional on their stated signal and processing models:

**Experimental Section:**

#### Conditional Targets for Additional Raw-Data Experiments

- Signal Amplification: Test the proposed 3.2–17.9× amplification range only under the signal and suppression assumptions that produce it; it is not a universal acceptance criterion for the completed raw-pseudorange analysis

- Carrier Phase Coherence: Detect cm-scale (wavelength-level) field-induced timing variations in raw L1/L2 carrier phase measurements

- Real-Time Dynamic Response: Capture sub-second coherence modulations during eclipse/opposition events without processing delays

- Multi-Constellation Universality: Recover the same finite, thousands-of-kilometres covariance family and band ordering across independently processed GLONASS, Galileo, and BeiDou constellations, allowing product- and estimator-dependent scale shifts

- Sidereal Independence: Demonstrate that unfiltered signals persist after removing sidereal components, distinguishing from multipath artifacts

### 5.6 Broader Implications

Within the TEP framework, the detected clock-network covariance connects precision time-transfer metrology to modified gravity, dark-sector phenomenology, and equivalence-principle tests, while identifying a structured, distance-dependent correlation channel in global timing systems.

#### 5.6.1 Temporal Dynamics and the Nature of Time

The diurnal and seasonal coherence modulations match the structure expected when proper-time transport responds to a dynamical scalar environment, with raw carrier-phase and non-GNSS optical-link measurements providing direct tests of the unsuppressed amplitude across independent timing architectures.

#### Temporal Dynamics and Time Variation

Key Finding: The detection of systematic temporal variations with synchronized daily (1.9-7.6% day-night coherence variation) and seasonal patterns across all three analysis center solutions provides evidence consistent with dynamical proper-time/covariance structure, while non-integrability requires a dedicated closed-loop test.

Experimental Framework: The observed terrestrial patterns establish benchmarks for potential TEP confirmation through triangle synchronization tests, interplanetary time transfer, and seasonal experimental optimization.

Metrological Implications: These findings suggest that temporal field variations could be as important as gravitational redshift corrections in atomic clock networks.

Physics Implications: Corroboration across non-GNSS optical clock links would extend precision metrology from calibrating coordinate event times to characterizing spatial and temporal gradients in proper-time transport across fundamental physics, navigation, and timekeeping.

### 5.7 Final Assessment

Three distinct GNSS analysis center solutions exhibit consistent multiband correlation structure, with optimal bands achieving R² = 0.970 (ESA Final), 0.920 (CODE), and 0.966 (IGS Combined), constraining center-specific systematic biases across software-diverse processing chains (Bernese double-difference clock recovery, NAPEOS undifferenced global network adjustment, and IGS weighted combination). Multi-layered null and forward-modeling controls exclude the primary tested artifact classes, and the convergent spatial, spectral, temporal, and orbital-coupling signatures establish a quantitative empirical baseline across the global GNSS network.

Decisive out-of-ecosystem discriminators:

- Independent replication on external timing archives (raw TWSTFT and phase-stabilized optical-fiber links)

- Raw carrier-phase validation (building on the raw-RINEX pseudorange/SPP recovery in Paper 3) to measure unsuppressed signal amplitude

- Separately processed per-constellation carrier-phase testing across GLONASS, Galileo, and BeiDou (extending the combined MGEX confirmation in Paper 14)

- Cross-correlation against high-resolution atmospheric and ionospheric reanalysis products

Together with the companion 25.3-year (Paper 2), raw-RINEX (Paper 3), and multi-constellation MGEX (Paper 14) studies, this analysis establishes a reproducible, thousands-of-kilometers correlation structure in terrestrial atomic clock residuals and specifies the exact optical-fiber and satellite time-transfer measurements required to test its physical origin outside the GNSS ecosystem.

Open Science: All code, configuration files, and figure generators are openly available (repository link; DOI snapshot), enabling full reproduction of the analysis pipeline and results reported here.

## 6. Analysis Package

Complete Reproducible Pipeline

This work provides a complete, reproducible analysis pipeline for testing TEP predictions using GNSS data:

**Experimental Section:**

## Pipeline Overview

*Actual timings from n2-highcpu-96 (96 vCPUs, 96GB RAM) GCP instance

Complete source code & documentation
[https://github.com/matthewsmawfield/TEP-GNSS](https://github.com/matthewsmawfield/TEP-GNSS)

### Setup: Clone Repository ~1 minute

Command: `git clone git@github.com:matthewsmawfield/TEP-GNSS.git`

Purpose: Obtain the full analysis code locally to run the pipeline and reproduce results.

### GCP High-CPU Analysis (Recommended for Large-Scale)

Professional cloud deployment: Optimized for Google Cloud Platform high-CPU instances with full 912-day analysis window and automated deployment.

#### Setup Instructions

- **Create GCP Instance:** Launch n2-highcpu-96 (96 vCPUs, 96GB RAM) with a 100GB additional disk

- **Configure Environment:** Either set environment variables or edit the .env file with your instance details

- **Run Analysis:** Execute the automated pipeline script

#### Option A: Environment Variables

`export GCP_PROJECT_ID=your-project-id`

`export GCP_ZONE=us-central1-c`

`export GCP_INSTANCE_NAME=your-instance-name`

`./run_tep_gcp_high_cpu.sh`

#### Option B: .env File (Recommended)

Edit `.env` file with your instance details:

`GCP_PROJECT_ID=your-project-id`

`GCP_ZONE=us-central1-c`

`GCP_INSTANCE_NAME=your-instance-name`

`./run_tep_gcp_high_cpu.sh`

What It Does:                

- Automated Setup: Installs all dependencies (Python packages, system libraries)

- Complete Analysis: Runs Steps 1-4 (Data acquisition to Core analysis to Validation to Advanced analysis)

- Full Date Range: Analyzes 912 days (1 January 2023 to 30 June 2025)

- High Performance: Optimized for 96 vCPUs with parallel processing - Total runtime: 6.8 hours (wall-clock with parallelism)

- Comprehensive Output: Generates 31 JSON results + 27 figures + 28 logs

- Easy Download: `./download_gcp_results.sh`

#### Monitor Progress

`gcloud compute ssh [INSTANCE_NAME] --zone=[ZONE] --command='cd /mnt/data && tail -f full_pipeline.log'`

Recommended instance: n2-highcpu-96 (96 vCPUs, 96GB RAM - recommended)

### Local Pipeline Execution

For development and targeted analysis: Run specific pipeline steps locally with precise control over analysis parameters.

Command: `python scripts/clean_run_full_pipeline.py`

What it does:

- Cleans all previous data, outputs, logs, and figures for a fresh start

- Executes all 21 pipeline steps in correct order (Steps 1.0-4.7)

- Validates outputs and ensures data integrity

- Total runtime: ~6.8 hours (wall-clock with parallelism) on n2-highcpu-96 (96 vCPUs, 96GB RAM)

Optional arguments:

- `--dry-run` - Preview what would be cleaned without making changes

- `--skip-cleanup` - Skip cleanup phase and run steps only

For step-by-step execution with detailed explanations, see individual pipeline steps below.

### Pipeline Structure

The pipeline is organized into 4 main groups with 21 individual steps:

#### Group 1: Data Acquisition (step_1_data_acquisition/)

Downloads raw GNSS data and validates station coordinates

#### Step 1.0: Provenance Snapshot 0.1 seconds

Command: `python scripts/steps/step_1_data_acquisition/step_1_0_provenance_snapshot.py`

Purpose: Creates comprehensive provenance snapshot documenting computational environment, software versions, and system configuration for full reproducibility.

#### Step 1.1: Data Acquisition 5.2 minutes

Command: `python scripts/steps/step_1_data_acquisition/step_1_1_tep_data_acquisition.py`

Purpose: Downloads raw GNSS clock product files from official analysis centers (CODE, ESA, IGS Combined) covering the full analysis period (1 January 2023 to 30 June 2025). Retrieves precise clock corrections in 30-second intervals from authoritative FTP servers, ensuring data integrity through checksum validation and automatic retry mechanisms.

#### Step 1.2: Coordinate Validation 0.4 seconds

Command: `python scripts/steps/step_1_data_acquisition/step_1_2_tep_coordinate_validation.py`

Purpose: Validates station coordinates and performs comprehensive audit for pipeline consistency. Checks Step 1 completion, validates ECEF coordinate data quality, runs integrated station ID audit with spatial analysis, creates authoritative station counts for the pipeline, and generates comprehensive validation summary with data-driven metadata. Ensures coordinate data integrity and establishes authoritative station catalogue for subsequent correlation analysis.

#### Group 2: Core Analysis (step_2_core_analysis/)

Computes TEP correlations and performs geospatial-temporal analysis

#### Step 2.0: TEP Correlation Analysis 59.3 minutes

Command: `python scripts/steps/step_2_core_analysis/step_2_0_tep_correlation_analysis.py`

Purpose: Core TEP signal detection using phase-coherent cross-spectral density analysis. Computes complex CSD between all station pairs in the 10-500 µHz frequency band, extracts phase-coherent correlations as phase-alignment index, and fits exponential decay models to correlation vs. distance relationships. **Technical parameters:** 40 logarithmic distance bins (50-13,000 km range), 7 correlation models tested (Exponential, Gaussian, Matérn, Power Law, etc.), 5,000 bootstrap iterations for confidence intervals, bin count-weighted least squares fitting with adaptive lambda bounds. Implements the band-limited methodology that preserves essential phase information for TEP detection.

#### Step 2.1: Data Quality Validation 23.2 minutes

Command: `python scripts/steps/step_2_core_analysis/step_2_1_data_quality_validation.py`

Purpose: Comprehensive data quality validation and transparency analysis. Analyzes quality-filtered correlation data from Step 2.0, adds geospatial enrichments (azimuth, local time differences), and performs extensive validation including station coverage analysis, temporal gap detection, duplicate detection, outlier validation, plateau phase boundary clustering analysis, and inter-AC comparison. Generates transparency reports with red flags and analyst recommendations.

#### Step 2.2: Geospatial Temporal Analysis 19.2 minutes

Command: `python scripts/steps/step_2_core_analysis/step_2_2_tep_geospatial_temporal_analysis.py`

Purpose: Comprehensive geospatial and temporal analysis including astronomical event correlations, orbital tracking, anisotropy analysis, spherical harmonics, and advanced temporal field studies. Analyzes correlations with planetary positions, lunar standstills, solar eclipses, and Earth's orbital motion to validate TEP predictions across multiple temporal and spatial scales. **Multi-scale window strategy:** 30-day windows (orbital tracking), 120-day windows (collective network covariance), 240-day windows (planetary gravitational coupling), 433-day cycles (Chandler wobble analysis), with each timescale optimized for its characteristic physical phenomenon.

#### Group 3: Validation Suite (step_3_validation_suite/)

Rigorous statistical validation including cross-validation, null tests, and methodology validation

#### Step 3.0: Cross-Validation Suite 0.8 seconds

Command: `python scripts/steps/step_3_validation_suite/step_3_0_tep_cross_validation_suite.py`

Purpose: Comprehensive cross-validation framework including block-wise (monthly/spatial), Leave-One-Station-Out (LOSO), Leave-One-Day-Out (LODO), and block bootstrap analyses. Provides rigorous validation of TEP correlation parameters using multiple complementary approaches to ensure robustness and statistical validity. Tests whether fitted parameters have genuine predictive power and distinguishes real physics from curve-fitting artifacts.

#### Step 3.1: Robust Block Bootstrap 2.3 hours

Command: `python scripts/steps/step_3_validation_suite/step_3_1_robust_block_bootstrap.py`

Purpose: Advanced bootstrap validation with temporal block structure preservation. Implements stationary block bootstrap to account for temporal dependencies while generating confidence intervals for TEP parameters.

#### Step 3.2: Null Tests 31.0 seconds

Command: `python scripts/steps/step_3_validation_suite/step_3_2_tep_null_tests.py`

Purpose: Critical validation step that establishes signal authenticity through comprehensive randomization testing. Implements three independent scrambling approaches—distance scrambling (20 iterations per center), phase scrambling (20 iterations per center), and station scrambling (20 iterations per center)—to verify that observed correlations depend on genuine physical relationships rather than analysis artifacts. Distance scrambling randomizes both distances and coherences independently to destroy distance-coherence relationships. Quantifies signal enhancement over null distributions (24-61× signal-to-null ratios) and establishes statistical significance through permutation testing and z-score analysis (z = 15.8-31.9, all p $\lt$ 10⁻¹⁵).

#### Step 3.3: Methodology Validation 6.0 minutes

Command: `python scripts/steps/step_3_validation_suite/step_3_3_methodology_validation.py`

Purpose: Comprehensive methodology validation framework achieving 100% validation score with rigorous bias characterization, distribution-neutral validation, geometric control analysis, and zero-lag leakage immunity validation.

#### Step 3.4: Geographic Bias Validation 52.4 seconds

Command: `python scripts/steps/step_3_validation_suite/step_3_4_geographic_bias_validation.py`

Purpose: Comprehensive geographic bias validation using statistical resampling to assess geographic consistency, hemisphere balance, elevation quintile independence, and ocean vs land baseline characteristics.

#### Step 3.5: Realistic Ionospheric Validation 2.4 seconds

Command: `python scripts/steps/step_3_validation_suite/step_3_5_realistic_ionospheric_validation.py`

Purpose: Ionospheric independence validation using real space weather data correlating with authentic geomagnetic indices and solar activity to demonstrate non-ionospheric origin.

#### Step 3.6: Control Band Analysis 2.9 hours

Command: `python scripts/steps/step_3_validation_suite/step_3_6_control_band_analysis.py`

Purpose: Comprehensive multi-band frequency validation across 12 frequency bands (10-3000 µHz) to assess systematic effects and frequency specificity. Frequency decomposition: TEP band (10-500 µHz), tidal bands (10-30 µHz), post-tidal ranges (30-100 µHz), and control bands (>1000 µHz) for systematic effect quantification. Applies identical phase-coherent methodology as Step 2.0 across each band with exponential model fitting and correlation strength assessment. Validates frequency specificity of TEP signals and excludes broadband instrumental artifacts.

#### Step 4.7: Multiple Comparison Corrections 2.8 seconds

Command: `python scripts/steps/step_4_advanced_analysis_and_visualization/step_4_7_multiple_comparison_corrections.py`

Purpose: Comprehensive application of formal multiple comparison corrections (Bonferroni, FDR, FWER) to all 388 statistical tests across 19 analysis families, including data quality validation, null hypothesis testing, bootstrap validation, band diagnostics, Hilbert-IF astronomical analysis, coordinate validation, eclipse events, bootstrap cross-methods, multiband frequency validation, and astronomical events, showing that core TEP findings survive ultra-conservative corrections.

#### Group 4: Advanced Analysis & Visualization (step_4_advanced_analysis_and_visualization/)

Advanced analyses, astronomical correlations, and publication-quality visualizations

#### Step 4.0: Advanced Analysis 4.7 minutes

Command: `python scripts/steps/step_4_advanced_analysis_and_visualization/step_4_0_tep_advanced_analysis.py`

Purpose: Conducts specialized analyses including circular statistics validation (Phase-Locking Value, Rayleigh tests), elevation-dependent screening analysis, geomagnetic stratification studies, and temporal orbital tracking analysis. Circular statistics: Phase Locking Values (PLV) in 0.1-0.4 range, Rayleigh test significance (p $\lt$ 10⁻⁵), von Mises concentration parameter validation. Investigates directional anisotropy patterns and their correlation with Earth's orbital motion to test velocity-dependent spacetime coupling predictions.

#### Step 4.1: Visualization 6.0 minutes

Command: `python scripts/steps/step_4_advanced_analysis_and_visualization/step_4_1_tep_visualization.py`

Purpose: Generates comprehensive publication-quality figures including correlation vs. distance plots, global station network maps, residual analysis plots, anisotropy heatmaps, and temporal tracking visualizations. Creates both individual analysis center plots and multi-center comparison figures with consistent styling and statistical annotations.

#### Step 4.2: Synthesis Figure 43.7 seconds

Command: `python scripts/steps/step_4_advanced_analysis_and_visualization/step_4_2_tep_synthesis_figure.py`

Purpose: Creates the comprehensive synthesis figure combining key results from all analysis centers. Produces the main publication figure showing correlation decay curves, statistical fits, and multi-center consistency in a unified presentation suitable for manuscript submission.

#### Step 4.3: High Resolution Astronomical Events 3.5 minutes

Command: `python scripts/steps/step_4_advanced_analysis_and_visualization/step_4_3_high_resolution_astronomical_events.py`

Purpose: Comprehensive high-resolution astronomical analysis including: (1) Complete 5-eclipse study (2023-2025) using 30-second CLK data revealing substantial coherence modulations (18-87% of baseline TEP signal) with observed eclipse type hierarchy where partial eclipses produce strongest network responses (mean: 4.54×10⁻², 87% of baseline) through ionospheric gradient mechanisms; (2) Cross-planetary orbital periodicity analysis revealing systematic TEP signals correlated with orbital completeness - Venus (4.05 orbits) shows strong correlations (ESA: +17.7%, IGS Combined: +10.6%, CODE: +4.8%) while outer planets show incomplete cycles and weaker effects; (3) Advanced sham controls confirming robust statistical significance beyond GPS processing artifacts; (4) Instantaneous frequency analysis detecting event-locked solar rotation modulations. This analysis demonstrates how the global GPS network acts as a sensitive detector of both transient (eclipse) and persistent (orbital periodicity) field modulations, providing comprehensive validation of TEP astronomical coupling predictions using 30-second CLK file data.

#### Step 4.4: Gravitational Temporal Field Analysis 1.5 minutes

Command: `python scripts/steps/step_4_advanced_analysis_and_visualization/step_4_4_gravitational_temporal_field_analysis.py`

Purpose: Comprehensive gravitational-temporal field correlation analysis using NASA/JPL planetary ephemeris (DE-series) to correlate planetary gravitational influences with GPS clock coherence patterns, revealing strong stacked gravitational pattern correlations (r = -0.503, raw p = 1.5×10⁻⁵⁹, autocorr-corrected p = 3.3×10⁻⁴) with optimal 227-day smoothing window. Includes systematic Earth motion energy hierarchy validation demonstrating sophisticated TEP coupling mechanisms - orbital motion coupling (|r| = 0.7-0.8), Moon-Chandler gravitational field modulation (|r| = 0.6-0.7), and rotational anisotropy (CV of rotational stability = 0.5-0.6). Energy vs velocity scaling discrimination analysis (-0.057, 95% CI: [-0.143, +0.030], n.s.) reveals no significant preference between scaling mechanisms, supporting complex multi-mechanism coupling and supporting TEP's predicted dynamical clock/covariance structure, motivating direct tests of non-integrable time transport through closed-loop experiments.

#### Step 4.5: Comprehensive Diurnal Analysis 1.8 hours

Command: `python scripts/steps/step_4_advanced_analysis_and_visualization/step_4_5_comprehensive_diurnal_analysis.py`

Purpose: Comprehensive diurnal and seasonal TEP modulation analysis using high-resolution hourly windowing methodology on raw CLK files. Processes 72.4M hourly records to reveal complex seasonal diurnal patterns with multi-center validation, systematic day-night effects, and peak hours that shift across seasons and centers.

#### Step 4.6: TID Exclusion Analysis 2.8 minutes

Command: `python scripts/steps/step_4_advanced_analysis_and_visualization/step_4_6_tid_exclusion_analysis.py`

Purpose: Comprehensive exclusion analysis for alternative explanations including Traveling Ionospheric Disturbances (TIDs) and trans-equatorial propagation (TEQ). Performs quantitative temporal band separation analysis, spatial structure comparison (exponential vs plane-wave models), ionospheric independence verification, and frequency band analysis. Provides high confidence exclusion of TIDs through temporal separation factor and power distribution analysis, strengthening the case for TEP field coupling as the underlying mechanism.

### Key Features & Reproducibility

- Real data only: No synthetic, fallback, or mock data

- Authoritative sources: Direct download from official FTP servers

- Multi-core processing: Parallel analysis with configurable worker count

- Checkpointing: Automatic resume from interruptions

- Comprehensive validation: Null tests, circular statistics, bootstrap confidence intervals

**Experimental Section:**

#### Reproducibility Documentation

Code Repository: Complete analysis pipeline in `scripts/steps/`

Data Sources: Official GNSS FTP servers (CODE, ESA, IGS Combined)

Dependencies: Listed in `requirements/requirements.txt`

Configuration: Centralized in `scripts/utils/config.py`

Binning Method: 40 logarithmic distance bins attempted, 50–13,000 km range

Phase Computation: `cos(np.angle(CSD))` with magnitude-weighted circular averaging

Cross-Validation: LOSO/LODO with complete model re-fitting

Statistical Tests: Weighted least squares with uncertainty propagation

Replication Note: Full replication requires ~1 day of data download and ~1–2 days of computation on multi-core systems. All intermediate results are checkpointed for partial replication.

## 7. Appendix

### 7.1 Event Windows: Complete Astronomical Events Analysis Parameters

Complete specification of temporal windows, detrending methods, and multi-center combination rules for all astronomical events analyzed in the TEP study.

| Event Type | Events (n) | ±Window (days) | Detrending | Multi-Center Rule |
| --- | --- | --- | --- | --- |
| Jupiter Opposition | 2 | ±120 | Linear polyfit | Weighted average |
| Saturn Opposition | 2 | ±120 | Linear polyfit | Weighted average |
| Mars Opposition | 1 | ±120 | Linear polyfit | Weighted average |
| Venus Conjunction | 2 | ±120 | Linear polyfit | Weighted average |
| Mercury Conjunction | 8 | ±120 | Linear polyfit | Weighted average |
| Solar Eclipses | 5 | ±1 (high-res) | Linear polyfit | Mean averaging |
| Supermoon Perigee | 11 | ±7 | Linear polyfit | Mean averaging |
| Lunar Standstill | 1 | ±180 | Linear polyfit | Weighted average |

#### Key Parameters:

- Detrending: Linear polynomial fitting (np.polyfit, degree=1) applied to time series before cross-spectral density computation

- Multi-Center Combination: Weighted averaging by sample size for planetary events; simple mean averaging for high-resolution events

- Window Rationale: ±120 days for planetary events (optimal gravitational coupling timescale); ±1–7 days for rapid events; ±180 days for lunar standstill (18.6-year cycle)

- Total Events: 32 astronomical events across 8 categories, spanning 2023-2025 analysis period

Event Dates: Jupiter (3 November 2023, 7 December 2024), Saturn (27 August 2023, 8 September 2024)*, Mars (16 January 2025), Venus (13 August 2023, 23 March 2025), Mercury (8 conjunctions 2023-2025), Eclipses (20 April 2023, 14 October 2023, 8 April 2024, 2 October 2024, 29 March 2025), Supermoons (11 events 2023-2025), Lunar Standstill (1 June 2025).

**Note: Saturn opposition on 21 September 2025 falls outside the current analysis window (ends 30 June 2025) and is marked as planned for future analysis.*

### 7.2 TID Exclusion: Implementation Details

Detailed implementation specifications for the TID exclusion analysis, including file paths and code references.

#### Implementation Details

- Implementation: See `scripts/steps/step_4_advanced_analysis_and_visualization/step_4_6_tid_exclusion_analysis.py`, function `_perform_tid_exclusion` for date alignment, thresholding, and Δ% computation.

- Results location: `results/outputs/step_4_6_tid_exclusion_analysis_results.json` and narrative summary in Section 3.5.2 (Ionospheric Interference Assessment).

For auditability, the methods above directly correspond to the code: date alignment (common days), percentile-based thresholding on the TID proxy, exclusion of high-activity samples, and percent change reporting.

### 7.3 Multiple Comparison Corrections Summary

**Experimental Section:**

#### Table S1: Test Family Multiple Comparison Corrections

Comprehensive multiple comparison corrections applied across 388 statistical tests using Bonferroni, FDR-BH (Benjamini-Hochberg), Family-Wise Error Rate, and Hierarchical Empirical Bayes corrections. Family alpha level: α = 0.05.

| Test Family | m (Tests) | α | Bonferroni Sig. | FDR-BH Sig. | Family-Wise Sig. | Hierarchical EB Sig. |
| --- | --- | --- | --- | --- | --- | --- |
| TEP Band | 3 | 0.05 | 3/3 | 3/3 | 3/3 | **3/3** |
| Model Comparison | 18 | 0.05 | 1/18 | 9/18 | 4/18 | 0/18 |
| Null Validation | 9 | 0.05 | 9/9 | 9/9 | 9/9 | 0/9 |
| Astronomical Events | 8 | 0.05 | 5/8 | 8/8 | 8/8 | 0/8 |
| Anisotropy Orbital | 22 | 0.05 | 0/22 | 4/22 | 2/22 | 0/22 |
| Cross Validation | 3 | 0.05 | 0/3 | 0/3 | 0/3 | 0/3 |
| Advanced Analysis | 174 | 0.05 | 35/174 | 54/174 | 37/174 | 48/174 |
| Geographic Validation | 2 | 0.05 | 1/2 | 2/2 | 2/2 | 2/2 |
| Chandler Wobble | 3 | 0.05 | 0/3 | 2/3 | 2/3 | **3/3** |
| Bootstrap Validation | 9 | 0.05 | 9/9 | 9/9 | 9/9 | 0/9 |
| Multiband Analysis | 40 | 0.05 | 37/40 | 40/40 | 40/40 | **40/40** |
| Gravitational Field | 9 | 0.05 | 2/9 | 3/9 | 3/9 | 3/9 |
| Diurnal Validation | 2 | 0.05 | 1/2 | 2/2 | 2/2 | 2/2 |
| Eclipse Analysis | 15 | 0.05 | 5/15 | 7/15 | 7/15 | 9/15 |
| Bootstrap Cross-Method | 12 | 0.05 | 3/12 | 3/12 | 3/12 | 3/12 |
| Coordinate Validation | 1 | 0.05 | 1/1 | 1/1 | 1/1 | 1/1 |
| Data Quality Validation | 6 | 0.05 | 6/6 | 6/6 | 6/6 | 0/6 |
| Hilbert-IF Astronomical | 12 | 0.05 | 0/12 | 1/12 | 0/12 | 0/12 |
| Band Diagnostics | 40 | 0.05 | 37/40 | 40/40 | 40/40 | **40/40** |
| Total (All Families) | 388 | 0.05 | 155/388 | 203/388 | 178/388 | **154/388** |

**Notes:**

- **m:** Number of statistical tests in each family

- **α:** Family-wise alpha level (0.05)

- **Bonferroni:** Conservative correction with adjusted α = 0.05/388 = 0.000129

- **FDR-BH:** Benjamini-Hochberg False Discovery Rate control

- **Family-Wise:** Controls family-wise error rate across all test families

- **Hierarchical EB:** Empirical Bayes partial-pooling correction with adaptive shrinkage per family (minimum n≥5 for shrinkage application). Provides data-driven balance between Bonferroni conservatism and FDR leniency by sharing information across tests within coherent families.

- **Green highlighting:** All tests in family remain significant after correction

- **Impact Summary:** Bonferroni reduction: 60.1% | FDR-BH reduction: 47.7% | Family-Wise reduction: 54.1% | Hierarchical EB reduction: 60.3%

### 7.4 Bootstrap Convergence Validation

Statistical validation of bootstrap uncertainty quantification methodology, addressing convergence rates and potential bias from non-converged iterations.

#### Bootstrap Success Rates

The correlation parameter bootstrap analysis (Step 2.0) achieves convergence rates of 71-73% across all analysis centers:

| Analysis Center | Successful Fits | Total Iterations | Success Rate | Status |
| --- | --- | --- | --- | --- |
| CODE | 3,555 | 5,000 | 71.1% | ✓ Acceptable |
| IGS Combined | 3,555 | 5,000 | 71.1% | ✓ Acceptable |
| ESA Final | 3,632 | 5,000 | 72.6% | ✓ Acceptable |

**Assessment:** These success rates of 71-73% reflect the challenging nature of nonlinear parameter optimization on resampled data. The convergence rates are sufficient for robust uncertainty quantification, with comprehensive bias validation confirming that non-converged iterations do not systematically bias parameter estimates.

#### Bias Validation Analysis

Comprehensive statistical testing confirms that discarding non-converged bootstrap iterations does not introduce systematic bias in parameter estimates:

**✓ Parameter Bias Assessment**
All parameter biases $\lt$ 6%: |λ bias| $\lt$ 4%, |A bias| $\lt$ 6%, well within acceptable tolerances for uncertainty quantification.

**✓ Failure Pattern Analysis**
Failed iterations represent challenging optimization cases with poor coherence variation, not systematically biased physical subsets.

**✓ Confidence Interval Validity**
Bootstrap percentiles accurately reflect true parameter uncertainties; statistical power remains adequate with current success rates.

**✓ Improvement Potential**
Robust initialization strategies can increase success rates to 96%+ if desired, though current rates are scientifically sufficient.

#### Root Cause Analysis

**Primary failure mechanism:** `scipy.optimize.curve_fit` hitting parameter bounds during bootstrap samples with:

- **Extreme distance distributions:** Samples with predominantly short or long distances challenge λ parameter estimation

- **Poor coherence variation:** Samples with minimal coherence range (c_range ≤ 0) cannot constrain amplitude parameters

- **Ill-conditioned matrices:** Covariance matrix computation fails for degenerate parameter combinations

**Scientific interpretation:** These failures represent the inherent mathematical limitations of nonlinear optimization when applied to resampled data with reduced statistical power, not fundamental flaws in the methodology.

**Methodology Note:** Bootstrap convergence analysis conducted using enhanced diagnostic framework (Step 3.7) with 2,000 test iterations per analysis center, multiple initialization strategies, and comprehensive bias validation across all TEP correlation parameters.

## References

Barrow, J. D. & Magueijo, J. (1999). Varying-α theories and solutions to the cosmological problems. *Physics Letters B*, 447(3-4), 246-250.
Bevis, M., et al. (1994). GPS meteorology: Mapping zenith wet delays onto precipitable water. *Journal of Applied Meteorology*, 33(3), 379-386.
Bothwell, T., et al. (2022). Resolving the gravitational redshift across a millimetre-scale atomic sample. *Nature*, 602(7897), 420-424.
Brans, C. & Dicke, R. H. (1961). Mach's principle and a relativistic theory of gravitation. *Physical Review*, 124(3), 925-935.
Bretherton, C. S., et al. (1999). The effective number of spatial degrees of freedom of a time-varying field. *Journal of Climate*, 12(7), 1990-2009.
Chen, G. & Herring, T. A. (1997). Effects of atmospheric azimuthal asymmetry on the analysis of space geodetic data. *Journal of Geophysical Research*, 102(B9), 20489-20502.
Chou, C. W., et al. (2010). Optical clocks and relativity. *Science*, 329(5999), 1630-1633.
Damour, T. & Nordtvedt, K. (1993). General relativity as a cosmological attractor of tensor-scalar theories. *Physical Review Letters*, 70(15), 2217.
Damour, T. & Polyakov, A. M. (1994). The string dilaton and a least coupling principle. *Nuclear Physics B*, 423(2-3), 532-558.
Delva, P., et al. (2018). Gravitational redshift test using eccentric Galileo satellites. *Physical Review Letters*, 121(23), 231101.
Eddington, A. S. (1919). A determination of the deflection of light by the sun's gravitational field, from observations made at the total eclipse of May 29, 1919. *Philosophical Transactions of the Royal Society of London A*, 220(571-581), 291-333.
Einstein, A. (1915). Die Feldgleichungen der Gravitation. *Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften*, 844-847.
Godun, R. M., et al. (2014). Frequency ratio of two optical clock transitions in 171Yb+ and constraints on the time variation of fundamental constants. *Physical Review Letters*, 113(21), 210801.
Heflin, M. B., et al. (1992). Global coordinates and site motion from very long baseline interferometry and satellite laser ranging. *Journal of Geophysical Research*, 97(B1), 459-467.
Holton, J. R. & Hakim, G. J. (2012). *An Introduction to Dynamic Meteorology*. Academic Press.
Horndeski, G. W. (1974). Second-order scalar-tensor field equations in a four-dimensional space. *International Journal of Theoretical Physics*, 10(6), 363-384.
Hunsucker, R. D. & Hargreaves, J. K. (2003). *The High-Latitude Ionosphere and its Effects on Radio Propagation*. Cambridge University Press.
Khoury, J. & Weltman, A. (2004). Chameleon cosmology. *Physical Review D*, 69(4), 044026.
Kivelson, M. G. & Russell, C. T. (1995). *Introduction to Space Physics*. Cambridge University Press.
Kouba, J. & Héroux, P. (2001). Precise point positioning using IGS orbit and clock products. *GPS Solutions*, 5(2), 12-28.
Gendt, G., & Schmid, R. (2005). A common-coordinate approach to global GPS analysis. *IGS Technical Report 2004*, 131-141.
Steigenberger, P., Montenbruck, O., Dach, R., et al. (2021). CODE reprocessing 1995-2020: improved GPS orbits and clocks. *Journal of Geodesy*, 95, 65.
International GNSS Service. (2023). *IGS Technical Report 2023*, Chapter 7: Clock Product Generation.
Lyard, F., et al. (2006). Modeling the global ocean tides: modern insights from FES2004. *Ocean Dynamics*, 56(5-6), 394-415.
McGrew, W. F., et al. (2018). Atomic clock performance enabling geodesy below the centimetre level. *Nature*, 564(7734), 87-90.
Montenbruck, O., et al. (2017). The Multi-GNSS Experiment (MGEX) of the International GNSS Service (IGS)–achievements, prospects and challenges. *Advances in Space Research*, 59(7), 1671-1697.
Murphy, M. T., et al. (2003). Possible evidence for a variable fine-structure constant from QSO absorption lines. *Monthly Notices of the Royal Astronomical Society*, 345(2), 609-638.
Petit, G. & Luzum, B. (2010). IERS Conventions (2010). *IERS Technical Note* No. 36, Frankfurt am Main: Verlag des Bundesamts für Kartographie und Geodäsie.
Ray, J., et al. (2008). IGS polar motion measurement accuracy. *Geophysical Research Letters*, 35(3), L03303.
Rosenband, T., et al. (2008). Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place. *Science*, 319(5871), 1808-1812.
Senior, K. L., et al. (2008). Characterization of periodic variations in the GPS satellite clocks. *GPS Solutions*, 12(3), 211-225.
Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: [10.5281/zenodo.16921911](https://doi.org/10.5281/zenodo.16921911) (Paper 0)
Smawfield, M. L. (2025). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: [10.5281/zenodo.17127229](https://doi.org/10.5281/zenodo.17127229) (Paper 1 — this work)
Smawfield, M. L. (2025). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: [10.5281/zenodo.17517141](https://doi.org/10.5281/zenodo.17517141) (Paper 2)
Smawfield, M. L. (2025). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: [10.5281/zenodo.17860166](https://doi.org/10.5281/zenodo.17860166) (Paper 3)
Smawfield, M. L. (2025). *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: [10.5281/zenodo.17982540](https://doi.org/10.5281/zenodo.17982540) (Paper 4)
Smawfield, M. L. (2025). *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: [10.5281/zenodo.18004832](https://doi.org/10.5281/zenodo.18004832) (Paper 5)
Smawfield, M. L. (2025). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: [10.5281/zenodo.18064365](https://doi.org/10.5281/zenodo.18064365) (Paper 6)
Smawfield, M. L. (2025). *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: [10.5281/zenodo.18059250](https://doi.org/10.5281/zenodo.18059250) (Paper 7)
Smawfield, M. L. (2025). *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: [10.5281/zenodo.18064581](https://doi.org/10.5281/zenodo.18064581) (Paper 8)
Smawfield, M. L. (2025). *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: [10.5281/zenodo.18109760](https://doi.org/10.5281/zenodo.18109760) (Paper 9)
Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: [10.5281/zenodo.18165798](https://doi.org/10.5281/zenodo.18165798) (Paper 10)
Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: [10.5281/zenodo.18209702](https://doi.org/10.5281/zenodo.18209702) (Paper 11)
Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: [10.5281/zenodo.19000827](https://doi.org/10.5281/zenodo.19000827) (Paper 12)
Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: [10.5281/zenodo.19102061](https://doi.org/10.5281/zenodo.19102061) (Paper 13)
Smawfield, M. L. (2026). *Global Time Echoes: MGEX Multi-GNSS Clock Replication, 2025–2026*. Preprint v0.4 (Suva). Zenodo. DOI: [10.5281/zenodo.20572726](https://doi.org/10.5281/zenodo.20572726) (Paper 14)
Smawfield, M. L. (2026). *TEP-PPP: Operational Time Echoes — Phase-Resolved Spatial Clock Priors for GNSS Outage Bridging*. Preprint v0.1 (Pune). In preparation. (Paper 33)
Takamoto, M., et al. (2020). Test of general relativity by a pair of transportable optical lattice clocks. *Nature Photonics*, 14(7), 411-415.
Touboul, P., et al. (2017). MICROSCOPE mission: first results of a space test of the equivalence principle. *Physical Review Letters*, 119(23), 231101.
Uzan, J. P. (2003). The fundamental constants and their variation: observational and theoretical status. *Reviews of Modern Physics*, 75(2), 403.
van Dam, T. M., et al. (2001). Displacements of the Earth's surface due to atmospheric loading: Effects on gravity and baseline measurements. *Journal of Geophysical Research*, 106(B6), 11115-11132.
Webb, J. K., et al. (2001). Further evidence for cosmological evolution of the fine structure constant. *Physical Review Letters*, 87(9), 091301.
Jammalamadaka, S. R., & Sengupta, A. (2001). *Topics in Circular Statistics*. World Scientific.

### Data Sources

Johnston, G., Riddell, A., & Hausler, G. (2017). The International GNSS Service. In P. J. G. Teunissen & O. Montenbruck (Eds.), *Springer Handbook of Global Navigation Satellite Systems* (1st ed., pp. 967-982). Cham, Switzerland: Springer International Publishing. [https://doi.org/10.1007/978-3-319-42928-1_33](https://doi.org/10.1007/978-3-319-42928-1_33).
Dach, R., Lutz, S., Walser, P., & Fridez, P. (Eds.). (2015). *Bernese GNSS Software Version 5.2*. Astronomical Institute, University of Bern, Switzerland. Available at: [http://www.bernese.unibe.ch/](http://www.bernese.unibe.ch/).
Fernández, M. A. (2016). Geodetic and Time Reference Frames for ESA's Navigation Programmes. In *Proceedings of the 29th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2016)* (pp. 2714-2719). Portland, OR.

## Data Availability

### Primary Data Sources

**GNSS Clock Products:** Final clock solutions (30-second epochs, CLK format) from three distinct analysis center solutions, all part of the International GNSS Service (IGS). Data are freely available under IGS Terms of Use.

**CODE (Center for Orbit Determination in Europe):**

- **Provider:** Astronomical Institute, University of Bern (AIUB)

- **Source:** [http://ftp.aiub.unibe.ch/CODE/](http://ftp.aiub.unibe.ch/CODE/)

- **Temporal Coverage:** January 1, 2023 – June 30, 2025 (912 days, complete coverage)

- **Station Pairs:** 39.0 million measurements

- **Citation:** Steigenberger et al. (2021); Johnston et al. (2017)

**IGS Combined Products:**

- **Provider:** International GNSS Service (weighted combination of multiple analysis centers)

- **Source:** [https://igs.bkg.bund.de/root_ftp/IGS/products/](https://igs.bkg.bund.de/root_ftp/IGS/products/)

- **Temporal Coverage:** January 1, 2023 – June 30, 2025 (912 days, complete coverage)

- **Station Pairs:** 12.9 million measurements

- **Citation:** Johnston et al. (2017)

**ESA (European Space Agency):**

- **Provider:** ESA Navigation Support Office

- **Source:** [http://navigation-office.esa.int/products/gnss-products/](http://navigation-office.esa.int/products/gnss-products/)

- **Temporal Coverage:** January 1, 2023 – June 30, 2025 (912 days, complete coverage)

- **Station Pairs:** 10.8 million measurements

- **Citation:** Fernández (2016); Johnston et al. (2017)

**Total Dataset:** 62.73 million station-pair cross-spectral measurements from 364 unique GNSS stations

- **Format:** RINEX 3 CLK (compressed: .gz or .Z)

- **License:** Freely available for scientific use. Users must cite IGS and individual analysis centers appropriately

- **Terms of Use:** [IGS Data and Product Disclaimer](https://igs.org/wp-content/uploads/2020/09/IGS-Data-and-Product-Disclaimer-and-Terms-of-Use-200805.pdf)

**Station Coordinates:** IGS Network station coordinates (ITRF2014) obtained from IGS station metadata and BKG services, providing Cartesian coordinates (X, Y, Z) for all active IGS stations.

- **Source:** [IGS Network Metadata (JSON)](https://files.igs.org/pub/station/general/IGSNetworkWithFormer.json)

- **Coverage:** 364 unique stations with valid clock observations

- **Reference Frame:** ITRF2014 (International Terrestrial Reference Frame 2014)

- **License:** Freely available under IGS Terms of Use

### Software and Analysis Tools

**Analysis Code:** Complete Python pipeline (TEP-GNSS) available under MIT License at [GitHub: TEP-GNSS](https://github.com/matthewsmawfield/TEP-GNSS). Includes data acquisition, preprocessing, statistical validation, and figure generation scripts for all 23 analysis steps.

**Key Dependencies:**

- **NumPy/SciPy:** Numerical computing and statistical analysis

- **Pandas:** Data manipulation and time series analysis

- **Matplotlib:** Scientific visualization

- **Statsmodels:** Advanced statistical modeling

### Derived Products and Supplementary Materials

**Analysis Results:** Comprehensive JSON output files, processed correlation data, validation results, and extended figures available in the [Zenodo repository (DOI: 10.5281/zenodo.17127229)](https://zenodo.org/records/17127229).

**Data Attribution:** By using data from this study, users agree to cite the original data providers (IGS, CODE/AIUB, ESA) as well as this work. All data sources are freely available for scientific research under their respective terms of use.

**Cross-Center Validation:** The use of three distinct analysis center solutions (CODE, IGS Combined, ESA) provides robust cross-validation across diverse software implementations with R² = 0.920-0.970 consistency, demonstrating that the correlation structure is not an artifact of a specific processing engine.

**Experimental Section:**

## How to cite

**Cite as:** Smawfield, M. L. (2025). Global Time Echoes: Distance-Structured Correlations in GNSS Clocks. v0.28 (Jaipur). Zenodo. [https://doi.org/10.5281/zenodo.17127229](https://doi.org/10.5281/zenodo.17127229)

**BibTeX:**

```
@misc{Smawfield_TEP_GNSS_2025,
  author       = {Matthew Lukin Smawfield},
  title        = {Global Time Echoes: Distance-Structured Correlations in GNSS Clocks (v0.28 Jaipur)},
  year         = {2025},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.17127229},
  url          = {https://doi.org/10.5281/zenodo.17127229},
  note         = {Preprint}
}
```

## Declarations

**Funding:** This work received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

**Competing Interests:** The author declares no competing financial interests.

**Author Contributions:** M.L.S. designed the study, performed all analyses, and wrote the manuscript.

## Contact

For questions, comments, or collaboration opportunities regarding this work, please contact:

**Matthew Lukin Smawfield**

[matthew@mlsmawfield.com](mailto:matthew@mlsmawfield.com)

---

*This document was automatically generated from the TEP-GNSS research site. For the interactive version with figures and enhanced formatting, visit: https://mlsmawfield.com/tep/gnss-i/*

*Source code and data available at: https://github.com/matthewsmawfield/TEP-GNSS*

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# Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products
**Matthew Lukin Smawfield**
Version: v0.21 (Cairo)
First published: 3 November 2025 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.17517141

---

## Abstract

Analysis of 25.3 years of global GNSS timing data (165.2 million station-pair observations) documents persistent velocity-dependent correlations in atomic clock networks. Critically, this work proposes that standard GNSS processing algorithms, designed to remove energetic (common-mode) errors via datum constraints, inadvertently preserve the subtle, geometry-dependent (differential) correlations that are the focus of this work. Building on the multi-centre study's validation (R²=0.92-0.97 between CODE, IGS, ESA), the extended temporal baseline confirms long-baseline recovery and enables investigation of long-period geophysical phenomena inaccessible in shorter baselines.

Seven convergent signatures are identified: (1) Spatial anisotropy persists with EW>NS (global EW/NS ratio=2.16, strength=1.981 ± 0.23 resampling SE, integrated 25.3-year aggregate), (2) anisotropy ratio correlates with orbital velocity (r=-0.888; surrogate p $\lt$ 2×10⁻⁷, 0/5M exceeded; t-test p≈2.6×10⁻⁴ with N_eff≈11) across a 25.3-year baseline with ≈19% annual geometric ratio modulation, (3) a multi-resolution full-sky directional search down to 1° resolution recovers a stable preferred direction at RA=6°, Dec=+4° (r=0.746; exploratory directional-rank p $\lt$ 10⁻⁵; global sky-search significance not yet evaluated) coincident with Earth's orbital velocity tangent at aphelion (~6°) and ~162° from the CMB dipole — a heliocentric-orbital-phase signature that excludes the Solar Apex template at ~16,000× variance ratio (R²=55.7% vs 0.003%), (4) planetary alignment coupling validated in the 40-event outer-planet opposition subset at 5.5× over matched random dates (p $\lt$ 10⁻¹⁷) alongside a continuous planetary-potential correlation (r = +0.116, p = 0.012), whereas the broader 156-event free-centre pulse scan (56/156 ≥2σ) reflects repeatable estimator behaviour on the seasonal curve, (5) preliminary coupling to 18.6-year lunar nutation (R²=0.641±0.10 over 1.36 cycles; permutation p=0.0016, single-cycle-dominated) and a semiannual coherence component (R²=0.904, p=2.7×10⁻⁵; permutation p=0.00027, peaked near the orbital apsides), (6) collective network covariance score (0.582) replicating the multi-centre range, and (7) null results for solar rotation (27-day) and lunar standstill consistent with selectivity for orbital-gravitational phenomena over surface features. The 19% modulation describes changes in the geometric shape of the correlation field (ratio of spatial correlation lengths), not clock frequency variations; individual clock-rate effects remain at standard GR-modelled fractional-frequency levels. On the held-out MGEX product the distance-structured coherence replicates (λ ≈ 1,862 km), while the anisotropy and primary orbital-velocity metrics do not replicate robustly. In-sample, a free-phase annual sinusoid is preferred over the velocity template (ΔAIC = 11.2); the signature is therefore reported as an annual modulation phased near the apsides.

Observed patterns are compatible with key TEP expectations: the recovered spatial correlation length (azimuth-averaged isotropic exponential $\lambda = 3,210 \pm 389$ km, with cross-sector directional dispersion $\bar{\lambda}_T = 4,201 \pm 1,967$ km across eight azimuthal sectors) lies within the pre-specified 1,000–10,000 km search window (Paper 0 §10.E); across products the fitted scale is conditioned by clock-product construction and estimator settings (reported product-level values ~1.0–4.8×10$^3$ km; MGEX $\lambda = 1,862 \pm 155$ km, Paper 14), with the same band-ordered profile recovered in each product and consistent with the geometric saturation radius $R_T(M_\oplus) \approx 4,150$ km calibrated on this scale (Paper 6). Model discrimination is primarily established through kernel-form selection (Gaussian and squared-exponential kernels preferred by AIC/BIC with $R^2 = 0.965$, exponential $\Delta$AIC = 12.8 and $R^2 = 0.945$, strongly ruling out scale-free power-law forms with $\Delta$AIC > 30), velocity-dependent anisotropy ($r = -0.888$), and geometric alignment (EW/NS = 2.16). The absence of GM/r² scaling is physically consistent with the datum-projection mechanism in which common-mode components are absorbed by GNSS estimation while differential geometric structure is transmitted (validated via synthetic signal injection and product-conditioned forward kernels in Paper 1, raw-RINEX pseudorange controls in Paper 3, and multi-GNSS MGEX product-error kernels in Paper 14). Raw carrier-phase analysis, separately processed per-constellation solutions, and independent external replication represent the critical next steps.

The empirically derived spatial correlation length $\lambda_T$ (the Temporal Topology covariance scale) is a GNSS-sector covariance scale obtained after environmental projection and processing transfer. It is not the screening operator $\mathcal{S}_\Sigma(\mathcal{E})$ itself and should not be identified numerically with response coefficients from other TEP channels. In the near-Earth environment, this correlation length acts as the macroscopic geometric proxy for the continuous saturation of Temporal Topology, anchoring the differential clock correlations without committing to specific subatomic microphysics.

The primary result is 25.3-year aggregate consistency of distance-structured covariance
and EW/NS anisotropy. The in-ecliptic orbital-phase direction, planetary event response, and
nutation couplings are secondary covariance-structure signatures requiring
held-out replication.

†Smawfield, M. L. (2025). Global Time Echoes: Distance-Structured Correlations in GNSS Clocks. Zenodo. https://doi.org/10.5281/zenodo.17127229

## Executive Summary

This study extends the first empirical TEP investigation—a multi-centre validation spanning CODE, IGS, and ESA ([Smawfield, 2025†](https://mlsmawfield.com/tep/gnss-i/))—to a 25.3-year temporal baseline. All core signatures reported in that study (anisotropy, orbital coupling, event responses) are re-observed here, with added sensitivity to long-period dynamics.

#### What's New in This Paper

**Carried over from Paper 1 (multi-centre study):**

- Distance-structured correlations

- EW > NS anisotropy

- Annual orbital-velocity coupling

- Initial planetary-event responses

- Chandler-wobble hints

- Collective network dynamics (network covariance score, termed "mesh dance" in Paper 1, extended to 25 years)

- Cross-centre agreement (CODE/IGS/ESA)

**New in this CODE-only, 25.3-year analysis:**

- **In-ecliptic orbital-phase direction**—identified via multi-resolution sky search (79,330 tested directions across four resolutions, with 65,341 at 1°), recovered at Earth's aphelion-velocity tangent with strong geometric discrimination against both the CMB-dipole and Solar-Apex directional templates

- Long-baseline stability confirmed across a 25.3-year baseline

- Long-period geophysical coupling (18.6-year lunar nutation: R² = 0.641 ± 0.10; 1.36 cycles observed—cycle-limited, detection priced by phase-permutation null p = 0.0016)

- Matched-null outer-planet opposition validation (40-event opposition subset at 5.5× over random dates, p $\lt$ 10⁻¹⁷; accompanied by a 156-event baseline pulse-estimator audit)

- Stronger orbital-velocity tracking (r = −0.888, surrogate p $\lt$ 2×10⁻⁷)

*Trade-off:* Single-centre temporal depth (25 years) over multi-centre breadth, justified by Paper 1's established cross-centre reproducibility (R² = 0.92–0.97). This is Paper 2 in the TEP-GNSS series.

### Key Findings

#### 1. Distance-Structured Correlations (Primary Finding)

Analysis of 165.2 million station-pair observations reveals systematic distance-dependent decay of clock-pair coherence, ruling out scale-free power-law models ($\Delta$AIC > 30). The primary azimuth-averaged isotropic exponential fit yields a correlation length of $\lambda = 3,210 \pm 389$ km ($R^2 = 0.945$; Gaussian kernel preferred with $R^2 = 0.965$ and $\Delta\text{AIC} = 0.00$), directly matching Paper 1's multi-centre range ($\lambda_T = 3,330$–$4,549$ km across CODE, IGS, ESA). Across the eight azimuthal sectors, directional anisotropy introduces a spread of $\lambda_T = 2,314$–$7,664$ km with a cross-sector mean and directional dispersion of $\bar{\lambda}_T = 4,201 \pm 1,967$ km, order-of-magnitude consistent with the terrestrial geometric saturation scale $R_T(M_\oplus) \approx 4,146$ km (Paper 0 §7, Paper 6; cross-product family ~1.0–4.8×10$^3$ km including the multi-GNSS MGEX scale, Paper 14), confirming long-baseline recovery of the distance-structured correlation pattern over 25.3 years.

#### 2. Spatial Anisotropy

East-West correlation lengths exceed North-South by factor of 2.16 (anisotropy strength = 1.981 ± 0.23 across the complete 25.3-year aggregate). This directional structure is recovered from the complete multi-decade record and is consistent across distance scales.

#### 3. Orbital Velocity Coupling (Primary Detection)

Primary Finding: Spatial anisotropy ratio (EW/NS) correlates with Earth's orbital velocity (r = -0.888, 95% CI: [-0.94, -0.81]; surrogate p $\lt$ 2×10⁻⁷, 0/5M exceeded; t-test p ≈ 2.6×10⁻⁴ with N_eff ≈ 11). Monte Carlo validation: 0 out of 5,000,000 surrogates exceeded the observed correlation. Combined with Paper 1's multi-centre validation and this study's 25.3-year aggregate consistency, this provides substantial evidence consistent with systematic temporal-gravitational coupling in GNSS data. The quoted ≈19% annual geometric modulation refers to the seasonal variation of this dimensionless coherence-based anisotropy ratio (λEW/λNS) around its baseline—a change in correlation topology, not clock frequencies; individual clock-rate effects remain at standard GR-modelled fractional-frequency levels.

#### 4. Directional Search (Annual-Phase Identification)

**Method:** Multi-resolution grid search (10°, 5°, 2.5°, 1°) spanning 79,330 total tested directions across four grids (with 65,341 directions at the finest 1° resolution), blind to candidate directions, with the orbital-velocity term in the corrected (prograde) sign convention.

**Result:** Best-fit direction (RA=6°, Dec=+4°, r=0.746; exploratory directional-rank p $\lt$ 10⁻⁵; global sky-search significance not yet evaluated) lies ~6° from Earth's orbital velocity tangent at aphelion and 162° from the CMB dipole (168°, −7°). Results converge across all four resolutions with coordinate stability (RA varies by a few degrees). The best-fit directional template explains R²=55.7% of the variance; the CMB-dipole template anti-correlates (r=−0.55), indicating the measured annual phase is anti-aligned with the dipole apex rather than aligned with it.

**Geometric Discrimination:** The Solar-Apex template is strongly disfavoured (r=0.006, R²≈0.003%; ~16,000× variance ratio versus the best fit) and predicts N-S anisotropy, contradicting the observed E-W dominance. The preferred directional region lies in the ecliptic plane near the aphelion velocity tangent — the annual-phase readout expected from orbital-velocity coupling (§3.2.2) — and excludes both named fixed-frame templates. The scan identifies a preferred direction but does not estimate the background-speed magnitude.

**Consistency Check:** Hemisphere asymmetry (Southern r=-0.79 vs Northern r=+0.25) is consistent with an orbit-coupled modulation geometry rather than a fixed-frame directional driver.

*Caveat:* The recovered direction encodes the annual phase of the orbital-velocity-coupled modulation; discriminating a cosmological frame from the orbital tangent would require a predictor that responds to more than the velocity-declination phase. See §4.1.2.1 for interpretive caveats.

#### 5. Planetary Event Responses

A matched-null control under a focused ±15-day model-free window validates the 40-event outer-planet opposition subset (Jupiter and Saturn) at 5.5× larger effect sizes than 155 random dates (Mann-Whitney p = 4.0×10⁻¹⁸, Cohen's d = 1.41), complemented by a continuous daily planetary-potential correlation (r = +0.116, autocorrelation-corrected p = 0.012). Across the broader 156-event alignment catalogue evaluated with an unconstrained ±120-day Gaussian pulse estimator, 56 events exceed 2σ (19 surviving family-wide Bonferroni correction at α = 0.05/156, and 13 hits across 29 non-overlapping windows), with coherence amplitudes of 0.3–4.0% (median ≈1.1%); as demonstrated by the season-preserving controls in Paper 3 (Step 2.6b), free-centre pulse fits on an autocorrelated seasonal curve reflect calendar-position sampling rather than single-event specificity, so the 156-event counts serve as a descriptive estimator baseline and a scaling-null control. The complete absence of GM/r² or GM/r³ scaling across all tested events is consistent with the datum-projection mechanism: common-mode classical gravitational signatures are absorbed by GNSS datum estimation.

#### 6. Geophysical Couplings

- **18.6-Year Nutation:** Preliminary — R² = 0.641 ± 0.10, permutation p = 0.0016 (1.36 complete cycles; amplitude and phase single-cycle-dominated, above the 99th percentile of the phase-shuffled null)

- **Semiannual Coherence Component:** R² = 0.904, p = 2.7×10⁻⁵ (90.4% variance explained; permutation p = 0.00027, with fitted phase peaked near orbital apsides and trough at the equinoxes)

- **Chandler Wobble (14 months):** Secondary signature — R² = 0.096 at prescribed 433-day period (below primary significance threshold; downward revision from Paper 1's R² = 0.377–0.471 due to strict canonical period enforcement without parameter fitting)

#### 7. Collective Network Dynamics (Network Covariance Score)

The network exhibits coordinated collective behavior (network covariance score = 0.582, termed "mesh dance" in Paper 1) with constructive interference
dominant across windows. This confirms long-baseline recovery of Paper 1's multi-centre network covariance findings (CODE: 0.624,
IGS: 0.579, ESA: 0.602) over 25.3 years.

### Summary: Inertial Interferometer, Not Gravimeter

The pattern of detections and non-detections characterizes what this network may be and is not:

- **Amplitude-blind at event level, weakly coupled at baseline level:** No GM/r² ordering across planetary events (Mercury ≈ Jupiter response rates), while a weak continuous M/r² correlation persists in the daily series (r = −0.099, p = 0.032)

- **Unresponsive to solar surface phenomena:** 27-day rotation null

- **Indifferent to static geometry:** Lunar standstill null

- **Exquisitely sensitive to kinematic dynamics:** Orbital velocity (r=-0.888), nutation (R²=0.904), orbital-phase direction (r=0.746)

This selectivity profile suggests that the network may be an inertial interferometer—measuring velocity-dependent correlation geometry—not a gravimeter measuring force. The directional search (~16,000× variance ratio over the Solar-Apex template) identifies a preferred covariance direction in the ecliptic plane at Earth's aphelion-velocity tangent: the phase of the orbit-coupled annual modulation, not the CMB dipole (162° away, anti-phase template correlation). The scan does not fix the background-speed magnitude and therefore measures an annual phase direction rather than a physical speed scale.

### Methodology and Validation

This single-centre analysis (CODE) provides the 25.3-year temporal depth required for long-period geophysical signatures, building on Paper 1's established cross-centre reproducibility (R² = 0.920–0.970 across CODE, IGS, ESA over 2.5 years).

These signals have survived the most rigorous systematic error correction pipeline in geodesy—CODE's processing explicitly removes relativistic effects, atmospheric delays, tidal displacements, and instrumental biases to millimeter precision. The correlations detected in this analysis are unlikely to be artifacts of these well-modeled effects, though a definitive determination requires synthetic signal injection through the actual processing chain. The datum-projection mechanism predicts this selectivity: common-mode offsets (including classical GM/r²) lie in the datum subspace absorbed by estimation, while differential phase structure can survive in residual products (§4.1.3).

**Validation Controls:**

- **Evidence Continuity (§1.5):** Explicit mapping of multi-centre validation to long-baseline extensions

- **Pre-Declared Analysis Window (§2.3.3):** ±120 days as primary; ±60–240 days for sensitivity analysis

- **Physical Predictor Tests:** Neither M/r² nor M/r³ gravitational scaling detected (all p > 0.5), consistent with GNSS processing suppression of classical gravitational signatures

- **Null Event Control:** For the 40-event outer-planet opposition subset, random dates show 5.5× smaller effect sizes than planetary alignments (Mann-Whitney p $\lt$ 10⁻¹⁷, Cohen's d = 1.41); catalogue-wide event-locked inference requires the season-preserving null of Paper 3

### Falsifiability

The systematic coupling hypothesis makes specific predictions that can be tested:

- **Raw carrier-phase analysis:** Should preserve correlation structure (processing artifact would eliminate it)

- **Multi-constellation replication:** GLONASS, Galileo, and BeiDou should exhibit similar orbital coupling

- **Null event discrimination:** ✓ *Confirmed for the 40-event opposition subset*—planetary alignments produce 5.5× larger effects than random dates (p $\lt$ 10⁻¹⁷, Cohen's d = 1.41); full-catalogue event locking requires a season-preserving null

- **Independent replication:** Core findings should be reproducible by other research groups

The strength of these findings lies in their falsifiability. Each criterion represents a concrete, testable prediction.

**Limitations:** This analysis uses processed clock products (not raw carrier-phase) and single-constellation data (GPS only). Both limitations are addressed in the proposed next steps.

#### Multi-Constellation Cross-Validation (MGEX Replication)

**Multi-Constellation MGEX Replication:** Paper 14 provides the dedicated replication of the distance-structured correlation pattern across a combined multi-GNSS product (CODE COD0MGXFIN, 2025–2026, 418 stations, 1.83 million pair-days), reporting a canonical isotropic correlation length of λ = 1,862 ± 155 km (R² = 0.699, rising to R² = 0.85–0.92 for baselines above 500 km; earlier exploratory 2020–2024 pilot cross-centre analyses across GBM, WUM, and JPL yielded λ_T = 3,042 ± 297 km, R² = 0.95). As established in Paper 14, the fitted numerical scale is product- and observable-dependent (reflecting the phase-alignment metric and the multi-GNSS joint clock combination filter relative to GPS-only PPP), while the physical signature shared across all products is the bounded thousands-of-kilometres spatial decay scale that excludes scale-free power laws. Paper 14 replicates the spatial decay structure; its primary pre-specified monthly λ and EW/NS orbital metrics are null (Bonferroni p > 0.5), with orbital-velocity coupling recovered via an exploratory model-free phase-alignment difference metric (r = −0.716, p = 0.0018).

### Conclusions

Seven convergent signatures—orbital velocity coupling (r = −0.888, surrogate p $\lt$ 2×10⁻⁷), the in-ecliptic orbital-phase direction (r = 0.746), the semiannual coherence component (R² = 0.904), 18.6-year nutation (R² = 0.641), null-validated outer-planet opposition responses (5.5× over matched random dates; 156-event free-centre catalogue descriptive), collective network dynamics (score = 0.582), and persistent spatial anisotropy—provide multi-signature convergence. Because the orbital and semiannual signals share annual structure, their probabilities are not combined into a single joint p-value. Combined with 25.3-year aggregate consistency and Paper 1's multi-centre validation (R² = 0.92–0.97 across software-diverse products from three processing centres), these findings are consistent with systematic coupling in GNSS timing networks, though confirmation requires independent replication using different processing pipelines and constellations. Raw carrier-phase analysis and separately processed per-constellation tests represent essential next steps.

## 1. Introduction

### 1.1 Background and Motivation

The Global Navigation Satellite System (GNSS) represents one of humanity's
most precise timing networks, with atomic clocks maintaining synchronization
at nanosecond levels across thousands of kilometres (Hofmann-Wellenhof et
al., 2008; Teunissen & Montenbruck, 2017). While designed for positioning
and navigation, this infrastructure inadvertently provides an unprecedented
natural laboratory for testing fundamental physics at planetary scales. The
continuous operation of hundreds of ground-based receivers, each equipped
with high-stability oscillators phase-locked to satellite atomic clocks
(Senior & Ray, 2008), creates a global mesh of timing correlations that may
be sensitive to subtle relativistic effects (Ashby, 2003).

The theoretical basis for this investigation is the Temporal Equivalence
Principle (TEP), in which matter and clocks couple universally to the canonical causal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$; Paper 0, §2.2, Step 80), predicting that motion through gravitational fields
should induce measurable modulations in the correlation structure of
distributed timing networks. Unlike classical relativistic effects that
manifest as clock rate differences, TEP predicts that the
*correlation* between clocks—their tendency to maintain phase
coherence—should vary with the local gravitational environment and the
system's velocity through that environment.

Critically, this work proposes that standard GNSS processing algorithms, designed to
remove energetic (common-mode) errors via datum constraints, inadvertently
preserve the subtle, geometry-dependent (differential) correlations that are
the focus of this work. This transforms the use of processed data from a
limitation into a mechanistic filter: the absence of classical gravitational
scaling (GM/r²) is not a failure of detection, but an expected consequence
under the datum-projection transfer hypothesis, while the geometric signatures (anisotropy,
alignment) are transmitted.

#### Methodological Note: Definition of "Coherence"

Throughout this paper, "coherence" refers to phase synchronization
between timing residuals at different GNSS stations, quantified via
magnitude-weighted cross-power spectral density in the 10-500 μHz band
(periods of 33 minutes to 28 hours). High coherence indicates stations'
clocks vary together in phase; low coherence indicates independent
variation. Unlike absolute frequency differences (Δf/f), coherence
measures geometric relationships in the correlation structure, which
GNSS processing preserves even while filtering amplitude effects. This
distinction is critical: processed clock products remove network-wide
frequency offsets (where classical gravitational effects manifest) but
preserve the differential phase-coherence structure that TEP predicts
should couple to velocity and gravitational configurations.

### 1.2 Theoretical Context and Interpretive Framework

*TEP in one line:* The Temporal Equivalence Principle (TEP) treats
proper time as a dynamical field: instead of a fixed background parameter,
time behaves like a scalar field woven through spacetime, whose local
configuration and gradients govern clock rates, temporal correlations and
global light-time relations, while the locally measured speed of light
remains invariant.

This investigation is grounded in empirical phenomenology: it
systematically documents patterns in GNSS timing correlations over
multi-decade timescales and interprets them through established
theoretical frameworks. TEP provides one such framework for interpreting
velocity-dependent and gravitational coupling effects in distributed timing
networks.

In TEP, time is modeled as a dynamical field—a kind of "temporal fabric"
permeating spacetime. Clocks do not simply read out a passive parameter;
they interact with this fabric. Variations in the temporal field, and in the
system's motion through it, offer a potential explanation for the
correlation structure modulations documented empirically in this work.

Formally, the temporal field is covariant and introduces no fundamental
preferred frame; a non-zero environmental or cosmological field
configuration may nevertheless define an observational covariance direction.

The Temporal Equivalence Principle extends Einstein's equivalence principle
to temporal dynamics. Key predictions—velocity-dependent anisotropy and
event responses—provide testable signatures that are investigated empirically.
The framework proposes that:

**Velocity-Dependent Coupling:** The correlation decay
length between synchronized clocks should depend on their collective
velocity relative to the relevant cosmological/background covariance frame

**Gravitational Modulation:** Changes in the local
gravitational field configuration should modulate the phase coherence of
timing networks

**Geometric Anisotropy:** The correlation structure should
exhibit directional dependence aligned with the system's motion through
spacetime

These predictions are quantitatively distinct from conventional general
relativistic effects:

Classical GR predicts clock *rate* differences proportional to
gravitational potential (Δf/f ~ GM/rc²)

TEP predicts correlation *structure* modulation: a
conformal-sector covariance with characteristic decay length λT,
whose velocity/orientation dependence is a channel response whose
microscopic transfer law is not yet closed (§1.2)

#### TEP Formalism and Quantitative Predictions

To move beyond qualitative description, the theory is defined by the
following explicit field equations:

**1. Action and Coupling:** The scalar time field φ is
governed by the Einstein-frame action, where the baseline action displays the
canonical kinetic form $X - V(\phi)$ and the operative non-canonical kinetic completion
$P(X,\phi) = X - V(\phi) + X|X|/\Lambda_X^4$ (Paper 0 §2.2) provides the screening mechanism
in high-acceleration regimes:

S = ∫ d⁴x √−g [ (MPl²/2)R − ½K(φ)(∂φ)² − V(φ) ] +
Sm[ψ, ğμν]

where matter couples to the disformal metric
ğμν = A²(φ)gμν +
B(φ)∇μφ∇νφ, with universal conformal factor *A(φ) = exp(βAφ/MPl)*.

**2. Field Equation:** Variation of the action with respect to φ yields a sourced scalar equation. Schematically, the scalar equation contains conformal trace sourcing together with disformal derivative sourcing:

K(φ)□φ + ½K,φ(∂φ)² − V,φ = −𝒬

where the source *𝒬 = AA,φ gμν 𝒯μν + ½ B,φ 𝒯μν ∇μφ ∇νφ − ∇μ(B 𝒯μν ∇νφ)* displays the leading conformal–disformal matter coupling explicitly, with *𝒯μν* the matter-frame stress-energy and the effective scalar coupling *α(φ) ≡ d ln A/dφ = βA/MPl*. In the conformal limit *B → 0* the source reduces to the trace coupling *α(φ)T*; the full disformal source is derived in the foundational paper (Smawfield, 2025).

**3. Conformal-Dominant Covariance Channel:** In TEP, GNSS clocks
read the value of the temporal field $\ln A$ at each station, and the measured
covariance $C_A(r) = \langle \delta \ln A(x) \delta \ln A(x') \rangle$ describes how that
value varies between stations; the screening that keeps local trajectories GR-like
acts on the field's effect on matter (the source-charge projection $\mathcal{S}_\Sigma\nabla\ln A$),
not on clock rates (Paper 0 §7). The
disformal function B(φ) governs cone tilts, one-way non-reciprocity, and
closed-loop synchronization holonomy; it is *not* the sector
tested by open-path GNSS covariance, and is reserved for dedicated
closed-loop / one-way asymmetry experiments (Paper 8).

The measured two-point timing covariance is represented
phenomenologically as

C(r) = C₀ exp( -r / λT ) + C∞,

with spatial decay length λT (the Temporal Topology
covariance scale) and asymptotic floor C∞. The observed
velocity/orientation dependence of the correlation structure (the
≈19% annual modulation of the EW/NS ratio, §3.2) is treated as a
channel response function of the conformal clock-sector covariance projected through the
GNSS processing transfer function. Its microscopic transfer law—mapping
the solved clock covariance $C_A(\mathbf{r}, t)$ to the measured anisotropy ratio (2.16) and
orbital-correlation magnitude (r = −0.888)—is *not yet closed*.
The observed 19% modulation is a network-level covariance-shape
observable and cannot presently be compared directly with a microscopic
conformal amplitude. The transfer law mapping $C_A(\mathbf{r}, t)$
to the measured anisotropy remains to be derived (§4.3.1.1, §5).

#### TEP Predictions vs. Observations

The TEP framework makes specific quantitative predictions that can be tested
against GNSS observations. The spatial search band of 1,000–10,000 km was established
in Paper 0 (§10.E) and Paper 1 (§1.1) as an order-of-magnitude window compared against the empirically calibrated
terrestrial geometric saturation radius *R**T*(*M*⊕) = (3*M*⊕ / 4πρ*T*)¹/³ ≈ 4,146 km,
evaluated using ρ*T* ≈ 20 g/cm³ calibrated on the GNSS scale (Paper 6); this is a consistency comparison, not an independent prediction.
The empirical test therefore distinguishes between scale-free power laws and bounded physical decay,
evaluating kernel structure, velocity modulation, and directional anisotropy.
Table 1 summarizes these a priori predictions and their experimental verification:

| TEP Prediction | Quantitative Forecast | Observed Result | Status |
| --- | --- | --- | --- |
| **Temporal Topology Correlation Length (λT)** | Order-of-magnitude search band 1,000–10,000 km; geometric saturation anchor *R**T*(*M*⊕) ≈ 4,146 km (Paper 0 §10.E, Paper 6; cross-product family ~1.0–4.8×10³ km, Paper 14) | Azimuth-averaged λ = 3,210 ± 389 km (cross-sector dispersion: 4,201 ± 1,967 km, spread 2,314–7,664 km) | **Order-of-magnitude consistent; empirical calibration point** |
| **Distance-structured decay** | Finite covariance scale within pre-specified TEP range; exponential/Gaussian preferred over power-law | Gaussian/squared-exponential preferred by AIC; exponential ΔAIC = 12.8 vs power-law ΔAIC > 30 | **Decay scale confirmed; power-law ruled out** |
| **Velocity-dependent anisotropy** | Correlation structure modulates with orbital velocity and ecliptic declination | r = -0.888 (95% CI: [-0.94, -0.81]); surrogate p $\lt$ 2×10⁻⁷
(0/5M); t-test p ≈ 2.6×10⁻⁴ (N_eff ≈ 11) | **Confirmed (p $\lt$ 2×10⁻⁷)** |
| **Geometric alignment with motion** | Spherical horizon projection across 474-station network ($\langle|\phi|\rangle = 36.88^\circ$, $\varepsilon = 23.44^\circ$): $\mathcal{R}_{\rm EW/NS} = 2.13$ (mean lat) / $2.02$ (ensemble) | EW/NS = 2.16 (1.5% match); strength = 1.981 ± 0.23 (0.19σ match) | **Quantitatively Confirmed (0.19σ)** |
| **Broadband (not frequency-selective)** | Persistence across spectral bands | Detected across 10-500 μHz | **Consistent** |
| **Network-scale coordination** | Global field-like coupling | Mesh score = 0.582 | **Consistent** |
| **GM/r² scaling absent** (measurement-channel prediction) | Classical mass scaling suppressed by datum-projection | No M/r² or M/r³ correlation (all p > 0.5) | **Confirmed Null** |

**Key Finding:** The TEP interpretation of the GNSS covariance
is conformal-dominant: the observed data confirm a finite, distance-structured
covariance scale within the pre-specified TEP range (λT), and the
absence of GM/r² scaling is consistent with the conformal-sector Temporal
Topology. The exponential kernel strongly outperforms simple power-law models
and is retained for cross-paper comparability. Gaussian-family kernels
provide the preferred empirical fit in the present dataset (ΔAIC = 12.8).
This meets the disfavouring condition of Paper 1's pre-specified criterion
(iii), under which a Gaussian-family preference counts against a directly
exponential screened-field covariance. Whether a screened profile passed
through the GNSS processing transfer function yields a Gaussian-like
empirical kernel is untested; until it is, the kernel shape is not counted
as support for the screening reading. The
velocity-dependent anisotropy (r = −0.888) and geometric
alignment (EW/NS = 2.16; strength = 1.981 ± 0.23) follow directly from the
spherical-horizon projection of Earth's ecliptic velocity vector ($\varepsilon = 23.44^\circ$)
across the 474-station network ($\langle|\phi_{\rm lat}|\rangle = 36.88^\circ$,
`step_2_11_horizon_anisotropy_derivation.json`), which predicts a
diurnal- and annual-averaged East–West to North–South kinematic variance ratio
$\mathcal{R}_{\rm EW/NS} = (1 - \tfrac12\sin^2\varepsilon)/[\sin^2\varepsilon\cos^2\phi + (1 - \tfrac12\sin^2\varepsilon)\sin^2\phi] = 2.13$
at the network mean latitude ($1.5\%$ from $2.16$) and $2.02$ across the
474-station ensemble ($0.19\sigma$ from $1.981 \pm 0.23$). The
datum-projection mechanism is further validated directly on raw RINEX carrier-phase
observations and synthetic end-to-end processing in Paper 3 (TEP-GNSS-RINEX). The disformal sector B(φ) is
not tested by this
open-path covariance channel and is reserved for dedicated closed-loop
holonomy and one-way asymmetry experiments (Paper 8).

### 1.3 Previous Work

The multi-centre study established the phenomenon across three
software-diverse analysis centres (CODE, IGS, ESA) whose products derive
from substantially the same tracking network (83–90% pairwise station
overlap), reducing the likelihood of centre-specific processing artifacts
over 2.5 years; common-mode network and datum-level systematics are not
bounded by that comparison alone. That study demonstrated
velocity-dependent anisotropy and anomalous planetary event responses, using
rigorous cross-centre validation and null tests. However, the limited
temporal baseline precluded the investigation of long-period geophysical or
astronomical cycles.

Prior to this, investigations of GNSS timing anomalies focused primarily on
conventional effects such as ionospheric modeling, multipath errors, and
clock stability (Hofmann-Wellenhof et al., 2008; Teunissen & Montenbruck,
2017; Ashby, 2003).

### 1.4 Study Design: Replication, Extension, and Falsification Tests

**TEP-motivated empirical test:** This work was explicitly
conceived as a theory-driven empirical test of the Temporal Equivalence
Principle using long-span GNSS clock products. The analysis pipeline, core
choice of observables (anisotropy, orbital modulation, planetary events,
nutation, mesh coherence), and the a priori prediction table in §1.2 were
defined to
test whether GNSS timing correlations exhibit the specific spatial,
temporal, and geophysical patterns forecast by TEP.

**Methodological Foundation:** This study builds upon Paper 1
(multi-centre study)'s comprehensive validation framework, which established
cross-centre consistency (R² = 0.920-0.970 between CODE, IGS, ESA) through
388 statistical tests and extensive null testing. That foundational work
demonstrated the existence of systematic distance-structured correlations in
GNSS timing networks over 2.5 years across three processing centres running
different software packages on the shared IGS tracking network. This study extends the temporal baseline by a factor of 10× (25.3
years) to address three critical scientific questions that distinguish
transient artifacts from genuine physical coupling:

#### Three Critical Questions: Distinguishing Replication, Extension, and Falsification

**Q1. Temporal Stability (Replication Test):** Do the
patterns detected in Paper 1 persist over 25 years, or are they
transient artifacts of the 2.5-year window?

**Paper 1 Baseline:** Directional anisotropy typical λEW/λNS ≈ 0.55–2.03 (peaks up to ~4.75, discussion quoting ~2.75×), orbital velocity r = −0.571 to −0.793 across centres, mesh coherence = 0.58–0.62

**Expected if genuine:** Patterns should be temporally
stable (recovered with consistent directional structure and coupling sign)

**Expected if artifact:** Patterns should vary with
processing updates, drift over decades, or vanish in extended
baseline

**Test:** Re-measure anisotropy, orbital coupling,
network coherence over 10× longer baseline

**Q2. Long-Period Signatures (Extension/Prediction Test):**
Can TEP coupling to geophysical cycles that require >20-year baselines be
detected?

**18.6-Year Lunar Nutation:** Paper 1 baseline (2.5
years) = 0.13 cycles → untestable. This study: 1.4 cycles → first
test possible

**TEP-motivated expectation:** Inertial orientation coupling
should produce R² ~ 0.6-0.7, based on the semiannual analogy.

**Semiannual Nutation:** Expected R² > 0.85
(persistence of Paper 1 short-period nutation coupling)

**Chandler Wobble:** 21+ complete cycles observed →
test phase stability and amplitude consistency

Q3. Reference Frame Discrimination and Theoretical Selectivity
(Template & Null Tests):
Can the operative reference frame for orbital modulation be identified,
and does the network respond to all periodic signals or specifically to
orbital/inertial dynamics?

A) Fixed-Frame Templates vs Orbital Phase (Directional Template
Discrimination):

**Physical Question:** Does orbital modulation
align with the Solar System's peculiar motion (~20
km/s toward RA=272°, Dec=+30°), with the
CMB dipole direction (~369 km/s toward
RA=168°, Dec=-7°), or with Earth's own orbital-velocity
geometry (the annual phase itself)?

**Method:** Comprehensive multi-resolution grid search across
79,330 sky directions across four grids (10°, 5°, 2.5°, with 65,341 at 1°) to identify which reference
direction best predicts the temporal modulation of spatial
anisotropy, using the corrected prograde orbital-velocity
convention verified against independent ephemeris checks

**Interpretive Hypothesis:** If an operative
cosmological covariance direction exists, the CMB dipole is the
most plausible candidate under TEP (as local temporal gradients
couple to the cosmological background 4-velocity rather than local
galactic peculiar motion). The covariant master action contains no
fundamental preferred frame; an environmental or cosmological field
configuration sets the observational covariance direction.

**Template Discrimination Power:** Direct head-to-head
comparison between the candidate physical templates, including
in-ecliptic orbital-velocity controls. The Solar Apex
vector (+30° Dec) predicts N-S anisotropy, which is geometrically
incompatible with the observed E-W dominance. The variance ratio
quantifies relative explanatory power.

**Expected Outcome:** If galactic: R² ≈
0.3-0.5. If CMB: R² ≈ 0.3-0.5 with galactic R² ≈ 0. Variance
ratio >100 would indicate strong discrimination.

**B) Solar Rotation Null Control (Secondary Test):**

**Solar Rotation (27-day cycle):** Strong
periodic astronomical signal from solar surface phenomena

**TEP Prediction:** Should show NO coupling
(null control)—TEP couples to heliocentric orbital motion
and inertial dynamics, not solar surface phenomena

**Alternative Hypothesis:** If this is a
processing artifact or frequency comb effect, any strong
periodic signal should couple regardless of physical
mechanism

**Falsification Criterion:** If solar rotation
shows R² > 0.10 or p $\lt$ 0.01, the artifact hypothesis gains
support over physical coupling

**Study Classification:** This is a
*replication + extension + falsification* study. Q1 tests
long-baseline consistency (replication). Q2 tests genuinely new predictions
accessible only with this dataset (extension). Q3A provides directional
discrimination among the CMB-dipole, Solar-Apex, and in-ecliptic
orbital-phase templates, while Q3B tests theoretical specificity via null controls
(solar rotation). Together, they distinguish genuine physical coupling
from transient processing artifacts and identify the preferred covariance
direction, strongly discriminating among the directional templates.

**Primary Empirical Objectives**:

**Objective A (Replication):** Document long-baseline consistency
of core phenomenology (anisotropy, orbital coupling, event responses)
over 25.3 years.

**Objective B (Extension):** Test long-period geophysical
signatures (18.6-year nutation, >20 Chandler wobble cycles) inaccessible
in short baselines.

**Objective C (Characterization):** Enhance planetary event
statistics across two decades with improved seasonal coverage.

**Objective D (Falsification):** Identify operative
reference direction (CMB vs galactic vs orbital-phase) via grid search and test
theoretical selectivity with null controls (solar rotation) and
robustness checks (hemisphere stratification).

### 1.5 Evidence Handoff: Building on Multi-Centre Validation

This study builds directly on the multi-centre study's comprehensive
validation framework. The table below maps each major claim to its
validation status in Paper 1 versus extensions in this work:

| Observable | Paper 1 Validation (2.5 years) | This Study Extension (25.3 years) |
| --- | --- | --- |
| **EW > NS Anisotropy** | Validated across 3 centres (CODE, IGS, ESA; typical λEW/λNS ≈ 0.55–2.03, peaks up to ~4.75, discussion quoting ~2.75×)
Cross-centre
R² = 0.920–0.970
388 statistical tests, 40–52% survive
MCC | **Long-baseline recovery:** EW/NS = 2.16, strength =
1.981 ± 0.23 (SE)
Consistent with Paper 1 E-W dominance and discussion scale (~2.75×)
Long-baseline
stability confirmed |
| **Orbital Velocity Coupling** | r = −0.571 to −0.793 across centres
Ionosphere/solar
nulls passed
Hemisphere & seasonal controls | **Strengthened:** r = −0.888 (surrogate p $\lt$ 2×10⁻⁷,
0/5M; t-test p ≈ 2.6×10⁻⁴)
25.3-year baseline
Hemisphere
stratification is the proper discriminant (see §3.2.2;
partial correlation is inappropriate for coupled orbital
variables) |
| **Directional Search (Orbital Phase)** | *Not testable (methodology not developed in Paper 1)* | **Multi-Resolution Verification:** Grid search
across 79,330 sky directions (65,341 at 1° resolution; §2.3.5, §3.2.6)

**Best fit:** RA=6°, Dec=+4° (r=0.746;
permutation p $\lt$ 10⁻⁵)

**Aphelion tangent:** ~6° from best fit — the
orbital-phase readout

**CMB Dipole:** RA=168°, Dec=-7° → 162° from
best fit (template anti-correlates, r=−0.55)

**Solar Apex:** RA=272°, Dec=+30° → r=0.006,
p=0.97

**Variance Ratio: ~16,000×** |
**Convergence verified** across 10°, 5°, 2.5°,
1° resolutions

Identifies the heliocentric-orbital phase direction,
excludes both named fixed-frame templates |
| **Planetary Events** | 6/8 predeclared events Bonferroni-significant
Monte
Carlo permutation validated
3–6σ confidence | **Validated opposition subset:** 40-event
outer-planet opposition subset at 5.5× over matched random dates (p $\lt$ 10⁻¹⁷)
Amplitude metrics
processing-dependent (GM/r² null)
156-event free-centre
catalogue (56/156 ≥2σ) descriptive (§3.3) |
| **Chandler Wobble** | R² = 0.377–0.471 across centers
433-day period
tested
Limited to ~6 cycles | **Extended:** R² = 0.096 (below threshold)
21+
complete cycles observed
Phase stability = 0.72
Chandler-consistent secondary signature |
| **18.6-Year Nutation** | *Not testable (baseline $\lt$ 2.5 years)* | **Preliminary:** R² = 0.641, p = 0.010
1.4
complete cycles observed
Preliminary — requires longer-baseline or red-noise validation |
| **Semiannual Nutation** | *Not explicitly tested* | **New Detection:** R² = 0.904, p = 2.7×10⁻⁵
50+
complete cycles |
| **Network Covariance Score** | Collective network coherence across 3 centres
CODE: 0.624, IGS:
0.579, ESA: 0.602
Consistent global coordination | **Replicated:** Index = 0.582
Consistent
with multi-centre range
102 non-overlapping 90-day windows (104 pipeline bins) over 25
years |

**Key Empirical Insight:** All core phenomenology from Paper 1
is re-observed here with enhanced statistical power and long-baseline consistency.
The multi-centre study's comprehensive validation framework—including 388
statistical tests, extensive null testing, ionospheric controls, and
cross-centre consistency—provides the methodological foundation upon which
these temporal extensions are built. The convergence of consistent findings
across multiple analysis centres and extended timescales strongly counters
artifact concerns.

## 2. Data and Methods

### 2.1 Data Sources

#### 2.1.1 GNSS Clock Products

Thirty-second clock solutions from the Center for Orbit Determination in Europe (CODE) were utilized, processed as part of the International GNSS Service (IGS) final products. The dataset spans:

| Parameter | Value |
| --- | --- |
| **Analysis Centre** | CODE (Center for Orbit Determination in Europe) |
| **Temporal Coverage** | March 1, 2000 to June 30, 2025 (9,218 days analyzed; 9,253 calendar days)

Note: Different analyses use slightly different temporal windows due to edge effects, windowing requirements, and data quality filtering (range: 9,218-9,270 days). |
| **Sampling Rate** | 30-second epochs (2,880 samples per day) |
| **Station Count** | 474 unique receivers (814 total codes including 4-char/9-char variants) |
| **Total Station-Days** | 1,574,861 |
| **Total Station-Pair Observations** | 165,189,605 |
| **Clock Solution Precision** | ~0.1 nanoseconds RMS |

**Data provenance and shared error budgets.** The CODE final product line analysed here belongs to the IGS analysis-centre ecosystem and is not an independent data layer relative to the companion papers. The provenance matrix across the GNSS papers is:

| Paper | Data layer | Product / processing chain | Epoch | Stations | Error budget shared with |
| --- | --- | --- | --- | --- | --- |
| 1 | AC precise clocks | CODE final, IGS combined, ESA final (Bernese / IGS combination / NAPEOS) | 2023-01 to 2025-06 | 364 | CODE family shared with Papers 2, 14; AC ecosystem shared with Paper 33's centre arms |
| **2 (this study)** | AC precise clocks | CODE IGS-final 30 s clocks (Bernese) | 2000-03 to 2025-06 | 474 | Same CODE product family as Papers 1 and 14; epoch superset of Paper 1 |
| 3 | Raw RINEX | Broadcast ephemerides + own RTKLIB SPP | 2022 to 2024 | 539 | Estimation-chain independent; shares the physical tracking network |
| 14 | AC combined clocks | CODE COD0MGXFIN (WUM/GRG fallback ≃6% of days) | 2025-01 to 2026-05 | 418 | Same CODE centre as Papers 1–2; epoch largely held out |
| 33 | Mixed | Own RTKLib SPP arms + AC receiver clocks (COD/GFZ/GRG/JPL) | 2022 to 2025 | 5–160 per arm | AC arms share the IGS/MGEX ecosystem; SPP arms are estimation-independent |

Agreement between this study and the CODE arm of Paper 1, the MGEX product of Paper 14, or the analysis-centre arms of Paper 33 therefore constitutes corroboration within a shared product ecosystem — different epochs, estimators and product lines over substantially the same tracking network — rather than replication on independent data. Within-ecosystem agreement excludes software- and implementation-specific artifacts but cannot exclude systematics of the shared tracking network, constellation geometry, or IGS datum conventions. The estimation-chain-independent axis in the corpus is Paper 3 (raw RINEX observations, broadcast ephemerides, own processing); Paper 33's broadcast-SPP arms provide a second such axis.

**Product-line lineage.** The 25.3-year record is assembled from the CODE (AIUB) IGS-final 30-second clock product line. CODE has produced daily 30-s final clocks continuously over the analysed span; across the record the underlying processing generation evolved through the documented CODE/IGS reprocessing campaigns (including the 2008/2015/2020 reprocessing releases; Steigenberger et al. 2021). All epochs were drawn from the same final-product family and file construction (CLK, 30 s sampling); no mixed-cadence or mixed-product epochs are used. Because processing generations differ in background models, generation-dependent offsets are a live systematic for any 25-year product study and are bounded here by the era-consistency diagnostics of §3.2.2–§3.2.3 rather than assumed absent.

#### 2.1.2 Station Distribution

**Station Code Methodology:** The analysis database contains 814 total station codes, representing 474 unique physical receivers. Many stations appear with both 4-character legacy codes (e.g., "VILL") and 9-character extended codes (e.g., "VILL00ESP") in the coordinate catalog. The analysis uses the actual observed station codes from CODE data, yielding 474 unique receivers with valid clock observations over the 25.3-year period.

The 474 unique GNSS receivers provide global coverage across all continents with concentrations in:

| Region | Station Count | Percentage |
| --- | --- | --- |
| North America | 167 | 35.2% |
| Europe | 115 | 24.3% |
| Asia-Pacific | 91 | 19.2% |
| South America | 52 | 11.0% |
| Africa | 28 | 5.9% |
| Antarctica | 21 | 4.4% |
| Total | 474 | 100% |

**Spatial Coverage:** Station separations range from 1.83 km (co-located receivers) to 19,946 km (antipodal pairs), enabling multi-scale correlation analysis across five orders of magnitude in distance.

**Temporal Window Selection (34 windows):** The 25.3-year dataset is partitioned into 34 day-of-year bins (each ~10.7° of orbital phase, pooling all 25.3 years of data at that phase) for orbital velocity correlation analysis. This binning evaluates the phase-coherent annual modulation of directional anisotropy integrated across the complete 25.3-year baseline. It is emphasized that a folded annual curve demonstrates phase-coherent modulation over the integrated baseline rather than 25 independent single-year fits. Integration over 25.3 years spans approximately one full cycle of the 25.4-year GPS draconitic beat period (359.4° phase drift relative to the solar year), which mathematically cancels draconitic orbit-repeat harmonics (|r| $\lt$ 0.15), confirming heliocentric origin. The surrogate test (0 out of 5,000,000 exceeded) tests spectral coincidence against phase-randomized profiles, while hemisphere stratification confirms spatial consistency and rules out local insolation. Continuous temporal stability is corroborated by the 104-window (90-day) time series, which shows steady East-West collective motion (θ ≈ 180°) and consistent mesh coherence (0.58 ± 0.02) across all three solar cycles. A fine-grained era-block decomposition (e.g., decade or 5-year rolling windows of r) remains an objective for future pipelines that maintain pairwise cross-spectra without day-of-year accumulation.

#### 2.1.3 The Processing Filter: Systematic Error Removal in IGS Clock Estimation

A critical methodological insight of this study is that operational GNSS clock products represent the output of the most rigorous systematic error correction pipeline in geodesy. CODE's clock estimation process, implemented using the Bernese GNSS Software (Dach et al. 2015) with undifferenced carrier-phase and pseudorange observations, explicitly models and removes all known physical effects to achieve sub-nanosecond precision. Understanding what has been removed is essential for interpreting what remains in the residuals.

##### 2.1.3.1 Explicitly Modeled and Removed Effects

The CODE clock estimation pipeline applies corrections following IERS Conventions 2010 and IGS standards (Petit & Luzum 2010; Kouba 2015). The following effects are modeled to millimeter-level precision and removed from the clock solutions analyzed in this study:

##### Relativistic Corrections

- **Constant gravitational redshift:** Factory-adjusted satellite clock frequency accounts for ~38 µs/day difference between orbital altitude and Earth's surface (general relativity)

- **Periodic eccentricity correction:** $-\frac{\sqrt{\mu}}{c^2}e\sqrt{A}\sin E$ removed from clock model (up to 44 ns for GPS satellites). This relativistic effect is explicitly eliminated during orbit and clock determination (Kouba 2015; RTKLIB Manual eq. E.4.37)

- **Sagnac effect:** Earth rotation correction (~133 ns for equatorial stations)

##### Atmospheric Delays

- **Ionospheric delay:** Ionosphere-free combination (dual-frequency L1/L2) removes >99% of first-order effect (~2-50 m at zenith). Higher-order terms explicitly modeled in PPP processing

- **Tropospheric hydrostatic component:** ~90% of total tropospheric delay (~2.3 m zenith) modeled using Vienna Mapping Function 1 (VMF1) with surface meteorology

- **Tropospheric wet component:** Estimated as random-walk parameter in Kalman filter with ~cm-level accuracy (process noise typically 1 cm²/hour)

##### Site Displacement Effects (IERS Conventions)

- **Solid Earth tides:** Up to 30 cm vertical displacement modeled using Love/Shida numbers (l₂=0.609, h₂=0.085) with Sun/Moon ephemerides

- **Ocean loading:** Up to 10 cm coastal displacement modeled using FES2014b 11-harmonic tidal model (M2, S2, N2, K2, K1, O1, P1, Q1, Mf, Mm, Ssa)

- **Polar tides:** Up to 25 mm displacement from polar motion variations (xₚ, yₚ from IGS Earth rotation products)

- **Atmospheric pressure loading:** Secondary effects (~mm level) often included in high-precision solutions

##### Antenna and Instrumental Effects

- **Antenna phase center offset/variation:** IGS ANTEX calibrations (I20.ATX) with mm-level precision for both receiver and satellite antennas

- **Phase wind-up:** Carrier-phase variation due to satellite rotation to maintain solar panel orientation (up to several cycles per orbit)

- **Differential code biases (DCB):** P1-P2, P1-C1 biases estimated and corrected using CODE monthly solutions

- **Satellite antenna attitude:** Modeled using ORBEX attitude files accounting for yaw-steering and eclipse maneuvers

##### 2.1.3.2 The Least-Squares Suppression Mechanism

Beyond explicit corrections, the fundamental architecture of GNSS clock estimation creates an implicit filter. CODE's processing uses sequential least-squares filtering (equivalent to Kalman filtering for GNSS positioning) where receiver clock parameters are estimated at each epoch by minimizing the squared residuals across all satellite observations (Dach et al. 2015). This design has a critical consequence:

**Datum-projection mechanism:** Network-common clock components lie largely in the clock/datum subspace absorbed by GNSS estimation, whereas direction-dependent covariance can survive in residual products. Synthetic injection and covariance propagation provide the direct quantitative test of this selective transfer.

This creates a natural dichotomy in how different physical effects survive processing:

| Effect Type | Manifestation | Processing Behavior | Survival |
| --- | --- | --- | --- |
| Network-common clock component | Lies largely in the clock/datum subspace | Absorbed by datum estimation; simple M/r² or M/r³ scaling need not survive in differential covariance | Removed |
| Phase-coherent correlation structure | Differential signal between pairs | Weakly suppressed (small squared residuals) | Transmitted |
| Geometric anisotropy patterns | Direction-dependent correlations | Not targeted by minimization | Preserved |

##### 2.1.3.3 Testable Expectations for TEP Coupling

This processing architecture leads to clear, falsifiable predictions for how TEP-type coupling should manifest in CODE clock products versus raw carrier-phase observations:

| Observable | CODE Products (This Study) | Raw Carrier Phase (Future) | Status |
| --- | --- | --- | --- |
| Mass scaling (GM/r²) | Absent (suppressed by filter) | May be separately recoverable in raw-link observables; not the claimed conformal covariance observable | Observed: Absent (r=-0.042, p=0.759) |
| Anisotropy structure | Preserved (differential signal) | Should persist | Observed: Present (strength=1.981) |
| Orbital modulation | Preserved (geometric coupling) | Should persist | Observed: Present (r=-0.888, surrogate p $\lt$ 2×10⁻⁷) |
| Phase coherence | Preserved (not targeted) | Should persist | Observed: Present (56 events ≥2σ) |

Consequently, the absence of $GM/r^2$ scaling in this dataset is not a failure of detection; it is consistent with the expected behavior of the datum-projection mechanism. The processed products isolate the differential residual channel that the TEP interpretation predicts should survive; its quantitative transfer remains to be validated by synthetic injection.

### 2.2 Data Processing Pipeline

#### 2.2.1 Clock Preprocessing

- **Outlier Removal:** 3σ filtering based on modified Z-scores

- **Detrending:** Removal of linear and quadratic trends per day

- **Normalization:** Zero-mean, unit-variance normalization per station

- **Gap Handling:** Linear interpolation for gaps $\lt$ 5 minutes, exclusion otherwise

- **Least-Squares Adjustment:** Daily clock parameters are estimated by minimizing the squared amplitude of residuals for each station; this preferentially removes mass-dependent frequency shifts while leaving phase-coherence structure largely intact.

#### 2.2.2 Phase-Coherent Correlation Analysis

For each station pair (i,j) with common observation epochs, phase-coherent correlations were computed using cross-power spectral density analysis (identical to the method used in the multi-centre study):

- **Detrending:** Linear trends removed from both time series

- **Cross-Power Spectral Density:** Complex CSD computed using Welch's method

- **Frequency Band Selection:** Analysis focused on 10-500 µHz (periods: 33 minutes to 28 hours)

- **Phase-Coherent Extraction:** Magnitude-weighted circular averaging of complex phases

- **Correlation Metric:** Band-averaged magnitude with representative phase

This frequency-domain approach preserves phase relationships between station pairs while extracting correlation strength, enabling detection of field-mediated timing correlations predicted by TEP theory. The method is identical to that validated across three software-diverse analysis centres (CODE, IGS, ESA) in the original multi-centre study; because the three products derive from substantially the same tracking network, that validation bounds software- and implementation-specific artifacts rather than network-level common modes.

**Observable Definition:** All subsequent analyses use this band-averaged coherence and its derived correlation lengths as the primary observable. The band-averaged coherence magnitude lies on [0,1]; its representative phase is retained separately. It quantifies the strength of phase-coherent structure between station residuals rather than the absolute clock frequency shift Δf/f. Reported percentage changes (e.g., the ≈19% annual modulation of the EW/NS ratio, or 1–3% planetary absolute coherence amplitudes) therefore describe relative changes in this coherence-based correlation structure and in ratios of correlation lengths, not 10–20% changes in underlying atomic clock rates, which remain at standard GR levels (≈10⁻¹⁶–10⁻¹⁰).

Operational GNSS clock products are generated under an implicit objective of network synchronization: analysis centres estimate and remove offsets, drifts, and other deterministic structure so that clocks agree as closely as possible. This makes absolute clock amplitudes strongly processing-dependent, whereas residual phase-coherent structure in the cross-spectral domain is less directly targeted by these corrections. This design choice makes the pattern-level findings in this paper—distance-structured correlations, anisotropy, orbital modulation, nutation coupling, and planetary event detection rates—largely insensitive to specific processing pipelines, even though absolute amplitudes may shift under reprocessing.

#### 2.2.3 Spatial Binning

Station pairs were categorized by:

- **Distance:** ≈30 logarithmically-spaced bins from 1-20,000 km (29 bins for azimuth-averaged fits)

- **Azimuth:** 8 compass sectors (N, NE, E, SE, S, SW, W, NW)

- **3D Orientation:** 16 spherical harmonic bins for full 3D analysis

### 2.3 Analysis Methods

#### 2.3.1 Exponential Decay Fitting

For each spatial sector, the distance-correlation relationship was fit:

```
C(d) = A × exp(-d/λ) + B
```

where:

- C(d) = mean coherence at distance d

- λT = Temporal Topology correlation length (km)

- A = amplitude

- B = baseline offset

Fitting employed weighted least squares with weights proportional to pair counts per bin.

#### 2.3.2 Temporal Orbital Tracking

The East-West to North-South anisotropy ratio was tracked across the year:

```
R(t) = λ_EW(t) / λ_NS(t)
```

Using 30-day sliding windows sampled every 10 days of year (yielding N = 34 windows per annual cycle, each aggregating all 25.3 years of data at that phase), R(t) was correlated with:

- Earth's orbital velocity (29.3-30.3 km/s)

- Earth-Sun distance (0.983-1.017 AU)

- Orbital phase (0-2π)

**Enhanced Control:** The multi-centre study's hemisphere stratification analysis (§3.4) demonstrated that both Northern and Southern hemisphere stations show identical calendar phasing (peak at perihelion in January), directly falsifying the hypothesis that the correlation arises from hemisphere-local seasonal effects — a meteorological or hemisphere-scale ionospheric driver reverses sign between hemispheres and cannot produce in-phase perihelion locking on both. See §3.2.2 for detailed interpretation of why partial correlation analysis is physically inappropriate for variables coupled by Kepler's laws.

**Significance Estimation:** Because the 30-day windows overlap, the EW/NS–velocity correlation uses an autocorrelation-robust Pearson estimator. Let *N* be the number of overlapping window samples (here *N* = 34), and let ρx, ρy be the lag‑1 autocorrelations of the orbital‑speed and EW/NS‑ratio series, respectively. The effective degrees of freedom are approximated using a Bartlett-style correction

```

Neff = N · (1 − ρxρy) / (1 + ρxρy)

```

The corrected standard error of the correlation is then

```

SEcorr = √[(1 − r²) / (Neff − 2)]

```

and the test statistic *t* = r / SEcorr is evaluated against a Student‑*t* distribution with Neff − 2 degrees of freedom to obtain the autocorrelation‑corrected p‑value. With r = −0.888 and Neff ≈ 11, the autocorrelation-corrected Pearson p-value is p ≈ 2.6×10⁻⁴. Separately, a Monte-Carlo surrogate test with 5,000,000 phase-randomised surrogates yields 0 exceedances, giving an empirical upper-bound p $\lt$ 2×10⁻⁷ under the specified phase-randomisation null. These two p-values test different nulls (parametric t-distribution vs. phase-randomised surrogate) and are reported separately; the surrogate bound is the stronger of the two.

**Surrogate Generation Method:** Phase-randomised surrogates preserve the power spectrum (and hence temporal autocorrelation structure) of the original time series while destroying phase relationships. Implementation: (1) compute FFT of the EW/NS ratio time series, (2) randomise phases uniformly on [0, 2π] while preserving magnitudes, (3) inverse FFT to obtain surrogate. This procedure, following [Theiler et al. (1992)](references.html#ref-theiler), ensures that any correlation with orbital velocity in the surrogates arises purely from spectral overlap rather than genuine phase coupling.

#### 2.3.3 Planetary Event Analysis

**Inference policy:** All primary inferences and multiplicity corrections (Bonferroni across 156 events; FDR q = 0.05) are computed exclusively using the pre-specified ±120-day window. Additional window sizes (±60, ±90, ±180, ±240) are evaluated for robustness only and are not used to select windows or to claim significance. No optimization across windows is performed for any reported p-values. If inferential claims were to be made across multiple window sizes, the family-wise error rate would be controlled across the full (events × windows) test set.

For each planetary alignment (opposition/conjunction), ±120 days is pre-declared as the primary analysis window, with additional window sizes (±60, ±90, ±180, ±240 days) reported as sensitivity analyses:

- **Primary Window:** ±120 days (pre-declared as primary)

- **Sensitivity Windows:** ±60, ±90, ±180, ±240 days (reported as robustness checks; no inferential claims)

- **Gaussian Pulse Fitting:** Fit Gaussian model to event-locked coherence changes

- **Significance Testing:** Amplitude/standard error ratio (σ level)

- **Multiple Testing Correction:** Bonferroni and FDR corrections applied across complete event set

- **Amplitude Metric:** Modulation depth = |amplitude| / (|baseline| + |amplitude|) × 100%. This formula quantifies the fraction of the total signal (baseline + perturbation) attributable to the event response, bounded between 0–100%. This formulation prevents physically impossible values (>100%) that can arise from standard percentage-change metrics when the perturbation amplitude exceeds the baseline coherence level.

- **Scaling Tests:** Correlation of observed amplitudes with both gravitational (GM/r²) and tidal (M/r³) predictors (Pearson, Spearman, log-log transforms). Absence of scaling is expected due to datum-projection suppression.

- **Null Event Control:** Effect size distributions compared between planetary events (n=40) and randomly-generated null dates (n=155) using Mann-Whitney U test. This validates that detections are specific to astronomical alignments rather than temporal autocorrelation artifacts.

#### 2.3.4 Geophysical Coupling Analysis

**Chandler Wobble:**

- Rigorous Period Enforcement: Fixed at 433.0 days (canonical)

- Method: Sinusoidal curve fitting to EW/NS ratio vs phase

- Phase resolution: 36 bins (10° increments)

- Coverage: ~21.2 complete cycles (9,218 days / 433 days)

**Nutation:**

- Tested periods: 18.6 years (main), 1 year (annual), 0.5 years (semiannual)

- Method: Sinusoidal curve fitting to coherence vs nutation phase

- Phase resolution: 12 bins (30° increments)

**Network Coherence:**

- Compute mean field coherence in 90-day windows

- Analyze collective motion patterns via PCA

- Quantify phase synchronization index

#### 2.3.5 Dual-Motion Reference Frame Validation

**Objective:** The orbital velocity correlation (§3.2.1) establishes that spatial anisotropy modulates with Earth's motion, but does not identify the operative reference frame. Two competing physical hypotheses are tested:

- **Galactic Hypothesis:** Modulation driven by Solar System's peculiar motion relative to the local standard of rest (~20 km/s towards Solar Apex: RA=272°, Dec=+30°)

- **Cosmic Hypothesis:** Modulation driven by Earth's motion through the Cosmic Microwave Background rest frame (~369 km/s towards CMB Dipole: RA=168°, Dec=-7°)

**Method - Comprehensive Sky Search:**

**Methodological Note:** This grid search is a *parameter estimation* procedure, not hypothesis testing. The analysis asks "WHICH direction best explains the observed modulation?" (estimating the directional parameter of a single model) rather than "DOES any direction explain it?" (testing 65,341 independent hypotheses). This is analogous to fitting a Gaussian to find its mean—testing many parameter values does not constitute multiple comparisons. The approach follows standard practice in cosmological frame identification (e.g., Planck CMB dipole fitting). Statistical significance is established via permutation testing against random directions, not via the grid search itself.

A blind grid search across 2,701 candidate background velocity vectors was performed:

**Grid Parameters:**

- Right Ascension: 0° to 360° (5° steps) → 73 values

- Declination: -90° to +90° (5° steps) → 37 values

- Total directions tested: 73 × 37 = 2,701

- Background speed: Fixed at 20 km/s (Solar Apex magnitude)

- Higher 5° resolution provides smoother heatmap and finer localization

**Net Velocity Calculation:** For each candidate background vector (RA_bg, Dec_bg) and each temporal window:

- Calculate Earth's orbital velocity vector V_orb(t) in ecliptic coordinates using the corrected prograde convention — verified against independent ephemeris checks (velocity toward ecliptic longitude 270° at the March equinox; RA≈192°, Dec≈−5° at perihelion; RA≈12°, Dec≈+5° at aphelion). An earlier pipeline version carried the antiparallel convention; because the predictor cos(Dec_net) is invariant under V_net → −V_net, that convention produced the identical correlation landscape mirrored through the origin (verified to ~10⁻¹⁵), so directional results of earlier versions read at the antipode

- Transform to equatorial coordinates (J2000 frame)

- Calculate background velocity vector V_bg from (RA_bg, Dec_bg, 20 km/s)

- Compute net velocity: V_net(t) = V_orb(t) + V_bg

- Extract declination: Dec_net = arcsin(V_z / |V_net|)

**Predictor Function:** For geometric consistency with grid search methodology:

- Predictor: P(t) = cos(Dec_net)

- Physical basis: High declination → Strong N-S projection, weak E-W

- Low declination → Strong E-W projection, weak N-S

- The EW/NS ratio is sensitive to this geometric projection

**Correlation Analysis:** For each of 2,701 directions:

- Correlate predictor P(t) with observed EW/NS ratios

- Record Pearson correlation coefficient r

- Store as function of (RA, Dec) → Correlation heatmap

**Model Comparison:** Direct head-to-head testing of physical models:

- Best Fit: Highest correlation direction from grid search

- CMB Dipole: Known physical frame (Planck 2018: RA=167.94°, Dec=-6.94°)

- Solar Apex: Galactic motion frame (RA=271.96°, Dec=+30.0°)

- Null: Orbital-only (background at pole, minimal effect)

**Ecliptic Control Test:** To discriminate direction-specific alignment from generic ecliptic-plane preference:

- Ecliptic East Control: RA=90°, Dec=0°

- Ecliptic West Control: RA=270°, Dec=0°

- Orbital-tangent controls: Earth's velocity direction at perihelion (RA≈192°, Dec≈−5°) and aphelion (RA≈12°, Dec≈+5°) — the directions a pure annual-phase readout must select

- Rationale: If the result merely detects "near ecliptic plane" rather than a specific direction, the ecliptic controls should show similar R² to the best fit; if it reads the orbital phase, the aphelion tangent should match within degrees

- Discrimination metric: Signed Pearson r (R² alone is sign-blind: a negative r means the antipodal direction, not the named direction, fits)

**Statistical Validation:**

- **Permutation Test:** 100,000 random sky directions tested to establish empirical null distribution

- **Bootstrap Confidence Intervals:** 1,000 resamples to assess robustness of best-fit correlation

- **Variance Ratio Analysis:** Compare R² values between competing models to quantify relative explanatory power

**Resolution Verification:** To ensure robustness and rule out resolution-dependent artifacts, four grid resolutions were tested spanning a 93-fold range in grid density: 10° (703 directions), 5° (2,701 directions), 2.5° (10,585 directions), and 1° (65,341 directions). Results converged monotonically (r = 0.707, 0.744, 0.744, 0.746) with diminishing returns (+5.1% → +0.04% → +0.38%), indicating asymptotic approach to a correlation limit. Best-fit locations showed stability across all resolutions (RA: 0°→10°→7°→6°; Dec: 0°→+5°→+5°→+4°). All four resolutions consistently identified the same in-ecliptic direction ~160° from the CMB dipole and ~6° from the aphelion velocity tangent, ruling out resolution-dependent artifacts. The 5° resolution captures 99.6% of the achievable correlation while remaining computationally efficient and consistent with standard practice in cosmological frame studies (Planck ~10°; WMAP ~5-10°). Final publication figures use 1° resolution for optimal visualization quality. The systematic monotonic improvement with finer resolution, combined with asymptotic convergence and location stability, indicates the signal is robust rather than a numerical artifact.

**Physical Interpretation:** This methodology provides a falsification test. Unlike correlative analyses that can only confirm hypotheses, the grid search can *reject* candidate directions when their templates fail to correlate — and, crucially, a negative correlation rejects the named direction itself (indicating the antipode instead). The signed correlation and variance ratio quantify the relative explanatory power of each template; the orbital-tangent controls identify when the recovered direction is simply the annual phase of the orbital modulation expressed in celestial coordinates.

**Scope of the Grid Search — Direction vs Speed:** The grid scan varies *direction* only (RA, Dec). Background speed is fixed at 20 km/s for every trial, and the predictor is cos(Decnet), a purely geometric projection that depends on the *direction* of the net velocity vector, not its magnitude. The scan therefore identifies a preferred direction but does *not* fit the background speed. Because the orbital term dominates the annual phase of the predictor, the recovered direction principally encodes the phase of the orbital modulation; under the corrected convention this phase maps to the aphelion velocity tangent rather than to any fixed external frame.

#### 2.3.6 Predictor Selection: Geometric Model Comparison

To identify the coupling mechanism, two competing geometric models were explicitly evaluated for how velocity might modulate the anisotropy field:

- **Model A (Tilt-Angle Rotation):** Assumes the anisotropy ellipse rotates azimuthally to align its major axis with the net velocity vector direction. Predicted EW/NS ratios were calculated based on the geometric projection of a rotating ellipse onto fixed cardinal axes.

- **Model B (Vertical Modulation):** Assumes the spatial anisotropy pattern is relatively stable in azimuthal direction (West-dominant) but modulates in strength based on the vertical component of the net velocity vector (equatorial vs. polar flow). This uses cos(Decnet) as the predictor.

**Result:** Model A failed to show significant correlation (r=-0.029, p=0.87), indicating the anisotropy pattern does not simply rotate with the local velocity vector. Model B showed strong correlation (r=0.746 at 1° resolution, verified across all resolutions from 10° to 1°), indicating the modulation is driven by the polar vs. equatorial character of the velocity field. Consequently, cos(Decnet) was selected as the physically validated predictor for the full grid search.

#### 2.4 Statistical Validation Framework

#### 2.4.1 Null Hypothesis Testing

- **Temporal Shuffle:** Randomize date labels while preserving spatial structure

- **Spatial Shuffle:** Randomize station positions while preserving temporal structure

- **Phase Randomization:** Destroy correlations while preserving power spectra

#### 2.4.2 Multiple Comparison Corrections

- **Bonferroni:** α_corrected = 0.05/N_tests

- **False Discovery Rate:** Benjamini-Hochberg procedure

- **Permutation Testing:** Empirical p-values from 5,000,000 iterations

#### Monte Carlo Validation for Orbital Correlation

- Generated 5,000,000 surrogate datasets with randomized orbital phases

- Preserved temporal autocorrelation structure of coherence data

- Maintained identical sampling windows (30-day) and temporal coverage

- Computed Pearson r for each surrogate with same methodology

- 0 out of 5,000,000 surrogates exceeded observed r = -0.888

- Empirical p-value: $\lt$ 2×10⁻⁷ (0/5M exceedances under the specified phase-randomisation null)

- Interpretation: None of five million phase-randomised surrogates exceeded the observed statistic

**Technical Implementation:**

- Orbital phase randomization preserves Earth's orbital mechanics

- Temporal structure preserved to maintain realistic autocorrelation

- Bootstrap-style resampling ensures statistical validity

- Computational approach: Parallel processing on high-performance cluster

**Clarification:** Planetary event inference follows the §2.3.3 policy; multi‑window analyses are robustness‑only.

#### 2.4.3 Cross-Validation

- **Leave-One-Station-Out (LOSO):** Verify robustness to single station removal

- **Bootstrap Resampling:** 1,000 iterations for confidence intervals

#### 2.4.4 Empirical Significance Estimation

In place of parametric post-hoc power calculations, all headline significance claims are evaluated against empirical null constructions matched to the data's dependence structure: (i) a five-million-iteration phase-randomized surrogate ensemble for the orbital-velocity correlation; (ii) a maximal non-overlapping-window independence count for the 156-event planetary catalogue (Step 2.9); (iii) autocorrelation-corrected effective sample sizes (N_eff) for the continuous geophysical correlations; and (iv) phase-bin permutation for the nutation fits. Where a season-destroying permutation null is known to under-control for calendar position on an autocorrelated series (as demonstrated for the discrete planetary-event channel by the season-preserving controls of Paper 3, Step 2.6b), the corresponding detection rates are reported as descriptive rather than inferential.

#### 2.4.5 Leveraging Multi-Centre Validation

This study builds upon the comprehensive validation framework established in the multi-centre study (Paper 1). Rather than repeating the full validation suite, the previously validated methodology is adopted and extended:

##### Adopted from Paper 1 Validation Framework:

- **Cross-centre consistency:** R² = 0.920-0.970 between CODE, IGS, ESA processing centres

- **Ionospheric null controls:** TID (Traveling Ionospheric Disturbance) exclusion; weak correlation with F10.7 and Kp indices (r = 0.12-0.13, p > 0.29), well below contamination thresholds

- **Comprehensive null testing:** 388 statistical tests across 19 families with temporal/spatial/phase scrambling

- **Hemisphere stratification:** Identical calendar phasing in Northern and Southern hemispheres (perihelion peak)

- **Bootstrap validation:** LOSO/LODO (Leave-One-Day-Out) robustness confirmed

## 3. Results: Multi-Signature Convergence Across a 25.3-Year Baseline

**Empirical Overview.** Analysis of 165,189,605 station-pair observations over 25.3 years (2000-2025) from CODE reveals systematic distance-structured correlations exhibiting long-baseline consistency and multi-signature convergence. Building on the multi-centre study's validated phenomenology, this study documents strong directional dependence, dynamic coupling to Earth's orbital motion, and newly accessible long-period geophysical signatures:

| Observable | Value | Significance |
| --- | --- | --- |
| Temporal Topology Correlation Length (λT) | 4,201 ± 1,967 km (cross-sector dispersion); azimuth-averaged λ = 3,210 ± 389 km | Sector spread 2,314–7,664 km (exponential R² = 0.55–0.87); azimuth-averaged exponential R² = 0.945 (Gaussian preferred, R² = 0.965, ΔAIC = 12.8) |
| Anisotropy Strength | 1.981 | ± 0.23 resampling SE; EW/NS = 2.16 (no independence-assumed p-value; pair observations are dependent) |
| Orbital Velocity Correlation | r = −0.888 (Primary) | surrogate p $\lt$ 2×10⁻⁷ (0/5M exceeded); t-test p ≈ 2.6×10⁻⁴ (N_eff ≈ 11) |
| Directional Search (Orbital Phase) | r = +0.746 (Best fit RA=6°, Dec=+4°, ~6° from aphelion tangent; 1° resolution) | r=0.746; exploratory directional-rank p $\lt$ 10⁻⁵ (global sky-search significance not yet evaluated); 162° from CMB dipole (anti-phase); ~16,000× variance ratio vs Solar Apex; verified across 4 resolutions |
| 18.6-Year Nutation Coupling | R² = 0.641 ± 0.10 (1.36 cycles; cycle-limited) | p = 0.010 (permutation p = 0.0016) |
| Network Covariance Score | 0.582 (3 components) | p $\lt$ 0.001 |
| Planetary Alignment Coupling | 40-event opposition subset: 5.5× over random dates (p $\lt$ 10⁻¹⁷) | Model-free ±15-day matched-null validation (Cohen's d = 1.41); continuous potential r = +0.116 (p = 0.012); broader 156-event free-centre scan (56/156 ≥2σ, 19 Bonferroni) descriptive of seasonal-curve sampling — see §3.3 |

Note on amplitude scaling: Processed GNSS clock products remove network-wide offsets through least-squares estimation, so any mass-proportional *GM ⁄ r²* signal is largely projected out. The phase-coherent differential metrics used here are sensitive to spatial *gradients*, not absolute potential, and therefore need not inherit simple *GM ⁄ r²* trends (see §4.2).

These findings confirm long-baseline recovery of multi-centre phenomenology over multi-decade timescales and reveal new long-period signatures inaccessible in shorter datasets. The convergence of multiple signatures—from spatial anisotropy to geophysical coupling—provides robust empirical support for systematic gravitational dynamics in GNSS timing networks (the dependence map between signatures is given in §4.1.6). Detailed analyses follow.

### 3.1 Spatial Anisotropy Structure

**Objective**: Replicate the multi-centre study's directional dependence finding (where East-West correlation lengths systematically exceed North-South, typical λEW/λNS ≈ 0.55–2.03 with discussion quoting ~2.75×) over a 25-year baseline to test long-baseline consistency and establish whether this anisotropy is a persistent feature rather than a transient artifact (Claim A).

#### 3.1.1 Directional Correlation Lengths

Replicating the multi-centre study, spatial anisotropy was tested by fitting exponential decay models C(d) = A·exp(-d/λ) + B to station pairs binned by azimuth. Analysis of 165,189,605 station-pair observations reveals pronounced directional variation in correlation decay lengths:

#### Physical Context of Correlation Length

- λT ≈ 4,000 km is Earth-scale (0.6 × Earth's radius)

- Station pairs $\lt$ 2,000 km: Strong coherence (r > 0.5)

- Station pairs 2,000-6,000 km: Moderate coherence (r ≈ 0.2-0.5)

- Station pairs > 8,000 km: Weak coherence (r $\lt$ 0.1)

- This distance dependence is recovered from the complete 25.3-year aggregate

- Suggests a global field affecting clock synchronization

**Comparison to Conventional Effects:**

- Atmospheric Temporal Topology correlation length: ~100-1,000 km

- Tidal deformation: ~20,000 km (global)

- Observed λT is intermediate, suggesting novel coupling

| Direction | λT (km) | R² | Station Pairs | σ_λ (km) | 95% CI (km) |
| --- | --- | --- | --- | --- | --- |
| North (N) | 2,314 | 0.562 | 27,792,346 | 779 | [1,526, 3,102] |
| Northeast (NE) | 2,540 | 0.859 | 35,185,615 | 411 | [2,129, 2,951] |
| East (E) | 3,206 | 0.753 | 16,068,982 | 801 | [2,404, 4,008] |
| Southeast (SE) | 6,808 | 0.873 | 11,972,421 | 2,119 | [4,689, 8,927] |
| South (S) | 2,718 | 0.729 | 12,247,234 | 780 | [1,938, 3,498] |
| Southwest (SW) | 5,332 | 0.604 | 13,801,875 | 3,123 | [2,209, 8,455] |
| West (W) | 7,664 | 0.746 | 14,283,934 | 4,080 | [3,584, 11,744] |
| Northwest (NW) | 3,028 | 0.546 | 33,837,198 | 1,063 | [1,965, 4,091] |

#### Key Insights from 8-Sector Analysis

**1. Omnidirectional Robustness:**

- *All* sectors show R² > 0.5, demonstrating the exponential decay model fits well in every direction

- This is not an artifact of sparse sampling in one direction—the effect is robust and omnidirectional

- Total of 165.2 million station-pair observations distributed across all 8 sectors (12-35 million observations per sector)

**2. Enhanced Coupling Along Rotational Plane:**

- SE and W sectors have the longest Temporal Topology correlation lengths (6,808 km and 7,664 km)

- These directions align approximately with Earth's rotational plane and orbital motion

- Suggests enhanced coupling along the ecliptic/equatorial plane

**3. Consistent Anisotropy Pattern:**

- 3.3× variation between longest (W: 7,664 km) and shortest (N: 2,314 km) Temporal Topology correlation lengths

- E-W oriented sectors (E, SE, W, SW) consistently show longer λT than N-S oriented sectors (N, NE, S, NW)

- Pattern recovered from the 25.3-year aggregate and stable under leave-one-station-out and bootstrap resampling

**Physical Significance:** The directional variation is statistically significant and persistent. If the correlation structure were driven by purely local, isotropic effects (atmospheric turbulence, ionospheric scattering, instrument noise), all directions would show similar decay lengths. The observed 3.3× difference between West and North suggests the correlation structure exhibits directional dependence that is not explained by isotropic noise sources alone.

#### Key Findings

- **Anisotropy Strength:** 1.981 ± 0.23

- **Coefficient of Variation:** 0.468 (moderate anisotropy)

- **Maximum/Minimum Ratio:** 3.31 (West/North)

- **Mean Temporal Topology Correlation Length:** 4,201 ± 1,967 km (cross-sector mean ± directional dispersion across 8 azimuthal sectors; azimuth-averaged isotropic fit yields λ = 3,210 ± 389 km)

- **Statistical Significance:** anisotropy strength = 1.981 ± 0.23 (resampling SE); sector λT ratios reach 3.31× — far outside any plausible isotropic-noise family. Because station-pair observations are highly dependent (shared stations, shared days), no naïve independence-assumed p-value is quoted; significance is established through the effect size, its resampling interval, and stability under LOSO/bootstrap.

- **E-W/N-S Ratio:** 2.16 (rotation-aligned anisotropy)

**Finding**: Replicating the multi-centre study, EW > NS correlation lengths are again observed. The correlation structure exhibits a large measured anisotropy (strength = 1.981 ± 0.23; EW/NS = 2.16, sector ratio up to 3.31×) that is stable under resampling. Magnitudes are consistent with the multi-centre study within uncertainty; the longer record narrows confidence intervals.

#### 3.1.2 Three-Dimensional Spherical Harmonic Analysis

To characterize the full 3D structure of anisotropy, Temporal Topology correlation lengths were analyzed across 16 spherical bins (azimuth × elevation) and decomposed into spherical harmonic coefficients:

| Component | Magnitude (km) | Physical Interpretation |
| --- | --- | --- |
| Monopole (Y₀₀) | 3,760 | Baseline correlation length (isotropic component) |
| Dipole (Y₁₀, Y₁₁) | 3,742 | Primary directional asymmetry |
| Quadrupole (Y₂₀, Y₂₁, Y₂₂) | 3,706 | Secondary directional structure |

- **Spherical bins analyzed:** 16 (azimuth × elevation grid)

- **3D anisotropy strength:** 1.981 (consistent with 2D analysis)

- **Dipole/monopole ratio:** 0.995 (strong directional preference)

- **Quadrupole/monopole ratio:** 0.986 (secondary structure present)

**Finding**: The 3D spherical harmonic decomposition confirms the 2D directional analysis, revealing a structured anisotropy field dominated by dipole and quadrupole components. The near-unity ratios indicate the anisotropy is not a minor perturbation but a fundamental characteristic of the correlation field structure. This provides independent confirmation of the rotation-aligned directional preference observed in the 8-sector analysis.

#### 3.1.3 Model Form Considerations

For comparability with the multi-centre study, the exponential decay kernel (C(d) = A·exp(-d/λ) + B) is retained for directional analyses. Alternative kernels are also evaluated on the azimuth-averaged *C(d)* curve (Gaussian, squared-exponential, Matérn, power-law) using AIC/BIC and residual diagnostics; results are summarized in §3.1.4. Directional analyses continue to use the exponential kernel for consistency.

#### 3.1.4 Model-Comparison Results

For completeness, seven candidate spatial-correlation kernels were evaluated against the azimuth-averaged *C(d)* curve using weighted least squares over 165.2 million station pairs (29 distance bins spanning 57 to 12,143 km). Model selection employed Akaike and Bayesian information criteria (AIC/BIC). The Gaussian kernel provides the best empirical description (AIC = 142.82, BIC = 146.92, R² = 0.965, ΔAIC = 0.00; amplitude A = 0.091 ± 0.004, scale λ = 3,768 ± 178 km, offset C₀ = −0.003 ± 0.001), with the squared-exponential variant statistically indistinguishable (ΔAIC ≈ 0). The traditional exponential decay model—used in the multi-centre study—is retained for cross-paper comparability rather than because it is selected by the present data (AIC = 155.59, BIC = 159.69, ΔAIC = 12.77, R² = 0.945; amplitude A = 0.126 ± 0.007, scale λ = 3,210 ± 389 km, offset C₀ = −0.010 ± 0.003). Power-law models are decisively ruled out (ΔAIC = 20.3 for cutoff, ΔAIC = 33.0 for pure power law).

| Kernel Model | AIC | BIC | R² | ΔAIC |
| --- | --- | --- | --- | --- |
| Gaussian | 142.82 | 146.92 | 0.965 | 0.00 |
| Squared Exponential | 142.82 | 146.92 | 0.965 | ≈0 |
| Matérn (ν = 2.5) | 145.13 | 149.23 | 0.962 | 2.31 |
| Matérn (ν = 1.5) | 147.36 | 151.46 | 0.959 | 4.54 |
| Exponential | 155.59 | 159.69 | 0.945 | 12.77 |
| Power-Law w/ Cutoff | 163.08 | 168.55 | 0.934 | 20.26 |
| Power-Law | 175.78 | 179.88 | 0.891 | 32.96 |

The Gaussian and squared-exponential kernels offer the most parsimonious fits (ΔAIC ≈ 0, R² = 0.965). The exponential kernel aligns with prior studies and yields an isotropic correlation length of λ = 3,210 ± 389 km (R² = 0.945) on the azimuth-averaged curve, directly comparable to Paper 1's multi-centre range (3,330–4,549 km across CODE, IGS, and ESA); the multi-GNSS MGEX product returns a conditioned scale of λ = 1,862 ± 155 km, and a controlled decomposition on identical station-days bounds the estimator-conditioned share of that offset at ≲±25% with the same band-ordered profile recovered in both product families (Paper 14, Step 3.5), so cross-product consistency is stated at the ~1.0–4.8×10³ km family level. This azimuth-averaged estimate is distinct from the sector-resolved directional analysis (§3.1.1), where partitioning the data into eight azimuthal sectors reveals strong rotational stretching: sector correlation lengths range from 2,314 km (North) to 7,664 km (West), with an unweighted cross-sector mean of 4,201 km and a directional dispersion of ±1,967 km. The ±1,967 km figure represents this physical directional anisotropy across azimuths rather than a single-parameter fit uncertainty.

### 3.2 Orbital Velocity Coupling (Primary Detection)

**Objective**: Replicate and strengthen Paper 1's orbital velocity correlation (r = −0.571 to −0.793 across centres over 2.5 years) by extending to a 25.3-year baseline, testing whether this coupling is consistently recovered or degrades over extended timescales (Claim B).

#### 3.2.1 Annual Modulation of Anisotropy

The multi-centre study reported r = −0.571 to −0.793 across centres (CODE r = −0.701, ESA Final r = −0.571, IGS Combined r = −0.793). Over 25.3 years a primary detection is found: r = −0.888 using the same seasonal phase, validated by 0 out of 5,000,000 Monte Carlo surrogates. The EW/NS anisotropy ratio was computed in 30-day sliding windows and correlated with Earth's orbital velocity.

**Primary Correlation:**

- **Metric:** EW/NS ratio vs orbital velocity

- **Sample size:** N = 34 overlapping 30-day windows; after autocorrelation correction, N_eff ≈ 11

- **Pearson r:** −0.888 (95% CI: [−0.938, −0.805])

- **Parametric p-value (autocorr-corrected t-test, N_eff ≈ 11):** p ≈ 2.6×10⁻⁴

- **Surrogate test:** 0/5,000,000 phase-randomised surrogates exceeded observed value (empirical p $\lt$ 2×10⁻⁷ under the specified null)

- **Temporal samples:** 34 (30-day windows)

**Physical Interpretation:** This represents the strongest statistical evidence for temporal-gravitational coupling in the entire dataset. The shape of the spatial anisotropy systematically changes over the course of a year, in lockstep with Earth's orbital velocity. As Earth speeds up near perihelion (January), the E-W correlation length shrinks relative to the N-S length (or the anisotropy becomes less pronounced). As it slows near aphelion (July), the ratio increases (anisotropy becomes more pronounced). This ≈19% annual modulation (offset ≈ 1.27, amplitude ≈ 0.24) describes the change in the *shape* of the coherence field (λEW/λNS).

**Seasonal Pattern:**

- **Perihelion** (January): EW/NS ratio minimum (~1.03)

- **Aphelion** (July): EW/NS ratio maximum (~1.51)

- **Amplitude:** ≈19% relative modulation (offset ≈ 1.27, amplitude ≈ 0.24; see §3.2.3)

##### Robustness Checks

This correlation survives multiple controls:

- Window size variations (15–60 days): r ranges from −0.82 to −0.89

- Detrending methods (linear, polynomial, spline): consistent results

- Outlier removal strategies: effect persists

- Bootstrap resampling: r = −0.888 ± 0.058 (95% CI: [−0.923, −0.765])

- Monte Carlo validation: 0 out of 5,000,000 surrogates exceeded observed value (p $\lt$ 2×10⁻⁷)

- Significance: autocorrelation-corrected p ≈ 2.6×10⁻⁴ (~3.65σ, N_eff ≈ 11); surrogate resolution bound p $\lt$ 2×10⁻⁷ (0/5,000,000 exceeded, nominal ~5.1σ)

- Temporal samples: 34 (30-day windows)

**Finding:** The spatial anisotropy structure varies predictably with Earth's orbital velocity (r = −0.888; t-test p ≈ 2.6×10⁻⁴ with N_eff ≈ 11; surrogate p $\lt$ 2×10⁻⁷, 0/5M exceeded), representing a robust detection for Claim B. The ≈19% annual modulation of the EW/NS *coherence ratio* (offset ≈ 1.27, amplitude ≈ 0.24) quantifies how this dimensionless anisotropy ratio of correlation lengths varies with orbital phase; it is distinct from the global sector-averaged anisotropy ratio of 2.16 (which aggregates all data). It does not represent a 19% change in underlying atomic clock frequencies Δf/f, which remain at standard GR levels.

#### 3.2.2 Physical Interpretation: Orbital Coupling

**Status and Physical Interpretation:** Orbital velocity and Earth-Sun distance are physically coupled by Kepler's laws (r = −1.000 on Earth's orbit), making them inseparable through partial correlation analysis. The strong correlation between the EW/NS anisotropy ratio and Earth's orbital velocity (r = −0.888, p $\lt$ 2×10⁻⁷) provides direct evidence for orbital coupling. Hemisphere stratification (§3.2.4) confirms this pattern: Southern stations show strong orbital coupling (r = −0.79, p = 0.006) while Northern stations show weak correlation (r = +0.25, p = 0.49), with both hemispheres peaking within 15 days of perihelion (IN-PHASE). This identical calendar phasing directly falsifies the hypothesis that the correlation arises from hemisphere-local seasonal effects: a meteorological or hemisphere-scale ionospheric driver reverses sign between hemispheres and cannot produce in-phase perihelion locking on both. The 25.3-year baseline spans a full draconitic–solar beat cycle, cancelling orbit-repeat harmonics (|r| $\lt$ 0.15), and the 104-window series shows steady directional structure and consistent mesh coherence (0.58 ± 0.02) across all three solar cycles — excluding orbit-repeat and activity-driven solar drivers. The surviving degenerate channel is confined to the global annual solar-geometry family sharing the identical annual period and perihelion phase (perihelion insolation, the global annual ionospheric anomaly), which no annual-period regression can separate from orbital velocity by construction.

**Directional Structure and Velocity-Dependent Coupling:** The pronounced E–W > N–S anisotropy (EW:NS = 2.16, λEW ≈ 5,400 km vs λNS ≈ 2,500 km) provides mechanistic insight into the coupling. If GNSS clock correlations are sensitive to velocity-dependent modulation of spacetime geometry, station pairs aligned parallel to Earth's orbital motion (E–W) should experience stronger time-flow gradients than pairs aligned perpendicular to it (N–S). This is because the velocity vector points along the ecliptic plane (approximately E–W in local coordinates), so the gradient of any velocity-dependent field would be strongest along that direction. Consequently, E–W pairs would maintain phase coherence over longer distances, while N–S pairs would decorrelate more rapidly. This prediction is exactly what is observed: the correlation length is 2–3× longer in the E–W direction. The fact that this directional preference persists across 25 years and correlates with orbital velocity (rather than local seasonal factors) indicates the effect is fundamentally tied to heliocentric dynamics, not local environmental confounders.

#### 3.2.3 Seasonal Baseline Comparison

To test whether the EW/NS anisotropy ratio could be explained by mundane seasonal factors, a baseline model was fit containing only annual (*ω*₁) and semi-annual (*ω*₂) sinusoids plus an offset:

```
Rseasonal(t) = α₀ + α₁sin(ω₁t) + β₁cos(ω₁t)
                 + α₂sin(ω₂t) + β₂cos(ω₂t)
```

#### Physical Interpretation of Results

**What r = −0.888 Means:**

- The ratio EW/NS varies systematically with Earth's orbital velocity

- When Earth moves fastest (perihelion, ~30 km/s), EW/NS ratio is lowest

- When Earth moves slowest (aphelion, ~29 km/s), EW/NS ratio is highest

- This 19% modulation reflects velocity-dependent spacetime geometry effects

**What this does *not* mean:**

- *Not* a 19% change in atomic clock frequencies (Δf/f remains at GR levels)

- *Not* a violation of energy conservation

- *Not* a direct measurement of spacetime curvature

**What It Suggests:**

- Phase coherence between station pairs shows velocity-dependent modulation

- The EW/NS anisotropy ratio peaks at aphelion (lowest orbital speed) and reaches its minimum at perihelion (highest orbital speed)

- N-S oriented pairs show less sensitivity to orbital velocity changes

- Consistent with theories predicting velocity-dependent time-flow gradients

The best-fit seasonal model has amplitude 0.24 about a mean of 1.274 (amplitude/mean ≈ 18.8%). Note that this 18.8% is an *amplitude-to-mean ratio*, not a variance explained (R²). In contrast, the orbital-velocity model explains nearly all systematic variance:

- **Orbital model:** Pearson r = −0.888, R² = 0.746, surrogate p $\lt$  2 × 10⁻⁷ (0/5M)

- **Seasonal model:** an unconstrained annual harmonic can reproduce generic annual structure. The orbital-speed template provides the physically constrained fit, explaining 74.6% of the variance with phase and waveform fixed by celestial mechanics.

**Conclusion:** The orbital-speed template provides the physically constrained fit, explaining 74.6% of the variance with the phase and waveform fixed by celestial mechanics. Combined with directional anisotropy, same-phase hemispheric response and the in-ecliptic orbital-phase direction (§3.2.6), this supports the TEP interpretation as velocity-dependent Temporal Shear rather than an unconstrained seasonal coincidence.

**Velocity-tracking versus calendar-phase discrimination (Step 2.10).** The formal comparison is now registered. On the identical 34-bin GLOBAL seasonal series, five templates were compared — *A(t) = a + b|vorb(t)|* (velocity, phase and waveform fixed by celestial mechanics), a kinetic proxy *a + b|vorb|²*, a free-phase annual sinusoid, an annual+semiannual seasonal baseline, and a velocity+semiannual hybrid — with R², adjusted R², AIC/BIC, nested incremental F-tests, contiguous-block held-out prediction error, and a 20,000-draw circular phase-shift calibration. Three results qualify the headline correlation:

- **In-sample, a free-phase annual sinusoid fits marginally better on GLOBAL** (R² = 0.856 vs 0.788; ΔAIC = 11.2, ΔBIC = 9.7 in favour of the sinusoid despite its extra parameter). The seasonal series contains harmonic structure — a mid-plateau dip near aphelion — that the pure *|v|* waveform cannot express; the annual basis adds significant variance beyond *v* (F = 8.3, p = 1.4×10⁻³), while *v* adds nothing beyond the free-phase annual basis on GLOBAL (F = 1.6, p = 0.21).

- **Held-out prediction reverses the ordering in the hemispheric strata.** Under five-fold contiguous-block cross-validation, the fixed-phase velocity template out-predicts the free sinusoid in both Southern (RMSE 0.353 vs 0.386, where R²v = 0.63) and Northern (0.538 vs 0.656) series; only the composite GLOBAL series retains the sinusoid's edge (0.105 vs 0.125). The sinusoid's in-sample advantage on GLOBAL is also atypically *small* under phase randomization: 76% of circularly shifted nulls produce a larger sinusoid advantage than observed (one-sided p ≈ 0.24), consistent with the seasonal waveform being genuinely phase-locked near the Keplerian template rather than arbitrary in phase.

- **The Keplerian asymmetry discriminator is underpowered at seasonal resolution.** The non-sinusoidal variance fraction of *|vorb(t)|* over an eccentric orbit is 6.9×10⁻⁵ at this binning; the data projection onto that component is null (t = −1.26). Waveform-shape discrimination between *|v|* and a pure calendar sinusoid therefore requires an epoch-resolved (daily/weekly) anisotropy series rather than the 34-bin seasonal profile — a registered next-step computation on the un-binned pair-level series.

The discriminating evidence for the velocity interpretation thus rests on phase-locking to perihelion rather than calendar landmarks, in-phase hemispheric response, held-out predictive parity-or-better under phase-fixed templates, and the in-ecliptic directional recovery — not on superior in-sample fit to the seasonal profile, which a free calendar sinusoid marginally exceeds. This is the honest resolution of the velocity-versus-calendar ambiguity flagged at seasonal resolution; it does not by itself distinguish an orbital-velocity driver from an annual solar-geometry driver sharing the same period and perihelion phasing (§3.2.1).

#### 3.2.4 Control Analyses: Hemisphere & Space Weather Stratification

To rigorously test for environmental confounding, the 25-year dataset was stratified by hemisphere and geomagnetic activity (Space Weather). The results confirm the signal's robustness:

| Control Test | Result | Implication |
| --- | --- | --- |
| **Hemisphere Stratification** | Northern: r = +0.25, p = 0.49 (not significant)
Southern: r = −0.79, p = 0.006 (significant)
Phase sync: Both peak within 15 days of perihelion (IN-PHASE) | Southern hemisphere drives signal; identical calendar phasing directly refutes seasonal temperature hypothesis. |
| **Space Weather (Quiet)** | Signal persists in "Quiet" geomagnetic conditions (Kp $\lt$ 3). N_Quiet λEW ≈ 3917 km vs N_Storm λEW ≈ 6227 km. | Signal is not driven by ionospheric storm turbulence; it is a background feature. |

#### 3.2.5 Symmetry Breaking: The Vector vs. Scalar Argument

#### Kinematic Test: Symmetry Breaking

A physical argument for kinematic coupling is the observed symmetry breaking:

- **Purely Isotropic Scalar Model:** If the driver were a scalar field with no spatial gradient (e.g., a uniform temperature or solar-distance effect), the effect would be isotropic—uniform in all directions.

- **Directional Models:** Both a kinematic vector projection (velocity $\vec{v}$) and a directional scalar-field gradient (such as $\Sigma_\mu = \nabla_\mu \ln A$ in the conformal TEP sector) produce anisotropic effects aligned with the gradient or motion direction.

**Observation:** The data shows strong directionality (West > North).

**Conclusion:** The anisotropy rules out a purely isotropic scalar-only model. It remains compatible both with a kinematic vector projection and with a directional scalar-field gradient such as $\Sigma_\mu = \nabla_\mu \ln A$. Discriminating these possibilities requires explicit pairwise projection kernels.

**Visual Proof:** Look at the "Perihelion Peak" in both hemispheres (§3.2.4). If this were seasonal weather, the hemispheres would be anti-phased (Winter vs. Summer). Instead, they beat in unison with the Earth's orbital speed.

#### 3.2.6 Dual-Motion Geometric Validation: Fixed-Frame Templates vs. Orbital Phase

#### Coordinate System Note

This section uses *celestial equatorial coordinates (RA/Dec, J2000 epoch)* to identify the reference direction for *temporal modulation* of the spatial anisotropy pattern. This is distinct from the spatial anisotropy analysis (§3.1), which uses *local azimuth coordinates* (compass directions) to describe the static geometric pattern. Under the corrected orbital-velocity convention (§2.3.5), the temporal modulation couples to RA≈6° in celestial coordinates — near Earth's orbital velocity tangent at aphelion, in the ecliptic plane.

**Objective:** The strong orbital velocity correlation (§3.2.1) establishes that Earth's 30 km/s orbital motion modulates the anisotropy. The directional search tests which sky direction, treated as a fixed background vector added to the orbital velocity, best reproduces the annual modulation phase. Three competing interpretations are tested:

- **Galactic Hypothesis:** The Solar System's peculiar motion relative to the local standard of rest (~20 km/s towards Solar Apex: RA=272°, Dec=+30°)

- **Cosmic Hypothesis:** Earth's motion through the Cosmic Microwave Background rest frame (~369 km/s towards RA=168°, Dec=-7°)

- **Orbital-Phase Hypothesis:** The modulation phase itself, whose directional encoding in this template lies along Earth's orbital velocity tangent at the phase extremum (aphelion: RA≈12°, Dec≈+5°)

**Method:** A comprehensive sky search was performed across 2,701 candidate background velocity vectors (5° grid), calculating the net velocity (orbital + background) for each temporal window and correlating the predicted anisotropy modulation with observed EW/NS ratios. The orbital velocity term uses the corrected prograde convention, verified against independent ephemeris checks (velocity toward ecliptic longitude 270° at the March equinox; RA≈192°, Dec≈−5° at perihelion). This grid search is blind to both the Solar Apex and CMB Dipole locations, allowing the data itself to identify the preferred reference direction.

#### Technical Implementation

For each candidate background vector (RA, Dec):

- Calculate Earth's orbital velocity vector (rotating in ecliptic plane)

- Add background velocity vector → Net velocity

- Compute predicted declination: Decnet = arcsin(Vz / |Vnet|)

- Predictor function: cos(Decnet)

- Correlate predictor with observed EW/NS ratios → r(RA, Dec)

**Results:**

| Background Model | RA / Dec | Correlation (r) | R² | Angular Separation from Best |
| --- | --- | --- | --- | --- |
| **(a) Best Fit (Grid Search)** | 6° / +4° | **+0.746** | **55.7%** | 0.0° |
| **(b) CMB Dipole** | 168° / -7° | **−0.546** | **29.8% (anti-phase)** | 161.8° |
| **(c) Solar Apex (Galactic)** | 272° / +30° | +0.006 | 0.003% | 91.5° |
| Null (Orbital Only) | 0° / +90° | +0.005 | 0.003% | 86.0° |
| **(d) Aphelion velocity tangent** | 12° / +5° | *orbital-phase reference direction* | 6.0° |

**Figure 7: Reference direction identification via comprehensive sky search.** 
Correlation strength (r) between predicted and observed anisotropy modulation across 65,341 candidate background velocity directions (1° resolution grid, representing ultra-high resolution verification; primary analysis performed at 5° resolution with 2,701 directions), under the corrected prograde orbital-velocity convention. 
Markers: (a) Best-fit direction (RA=6°, Dec=+4°, r=0.746), (b) CMB dipole control (RA=168°, Dec=-7°, 162° separation — anti-phase), (c) Solar apex control (RA=272°, Dec=+30°, 92° separation), and (d) Earth's aphelion-velocity tangent (RA≈12°, Dec≈+5°, 6° separation). 
The correlation peak lies in the ecliptic plane near Earth's orbital velocity tangent at aphelion and is nearly antipodal to the CMB dipole, excluding both fixed-frame hypotheses. Multi-resolution verification across four independent resolutions (10°, 5°, 2.5°, 1°) spanning a 93-fold range in grid density shows monotonic convergence with asymptotic behavior (Supplementary Table S1), indicating the signal is robust across resolution scales.
Color scale shows Pearson correlation coefficient. Viridis colormap chosen for perceptual uniformity and accessibility.

**Statistical Validation:**

- **Permutation Test:** 100,000 random sky directions tested. No random direction exceeded the observed correlation (r=0.746), giving p $\lt$ 10⁻⁵. Note: this test ranks the observed best-fit direction against random directions in the observed dataset; it is not a full-search surrogate null (which would require repeating the complete 65,341-direction maximization for each surrogate dataset). The directional map and variance ratio are reported as exploratory evidence; a global search-corrected p-value is a natural next step.

- **Bootstrap 95% CI:** r ∈ [0.572, 0.886] from 1,000 resamples, confirming robustness.

- **Variance Ratio:** The best-fit directional template explains ~16,000× more variance than Solar Apex (R²best₋fit/R²Apex = 55.7% / 0.0035% ≈ 16,000). The CMB-dipole template attains R²=29.8% but with negative correlation (r=−0.546): the observed modulation is in anti-phase with the dipole-anchored prediction, so the dipole direction itself is excluded and the variance is captured by its antipodal region.

- **Resolution Convergence:** Multi-resolution verification across four independent resolutions (10°: r=0.707, 703 directions; 5°: r=0.744, 2,701 directions; 2.5°: r=0.744, 10,585 directions; 1°: r=0.746, 65,341 directions) shows monotonic convergence with diminishing returns (+5.1% → +0.04% → +0.38%), spanning a 93-fold range in grid density. Best-fit locations show stability across all resolutions (RA: 0°→10°→7°→6°; Dec: 0°→+5°→+5°→+4°). The asymptotic convergence pattern, combined with the marginal 0.4% improvement at 1° resolution despite 6× more grid points, indicates convergence toward a physical correlation limit. All four resolutions consistently identify the same in-ecliptic direction ~160° from the CMB dipole and ~6° from the aphelion velocity tangent, indicating the result is robust across resolution scales.

#### Physical Interpretation

**Frame Discrimination:** Both fixed-frame hypotheses are excluded. The Solar Apex hypothesis is geometrically incompatible: the Solar Apex vector (+30° Dec) predicts N-S anisotropy, directly contradicting the observed E-W dominance (EW/NS = 2.16), and its template correlation is null (r=+0.006). The CMB-dipole hypothesis is also excluded under the corrected convention: the dipole lies 162° from the best fit and its template anti-correlates (r=−0.546). The recovered direction instead lies ~6° from Earth's orbital velocity tangent at aphelion — identifying the annual phase of the orbital-velocity-coupled modulation measured in §3.2.1–3.2.4.

**Directional Alignment:**

- Aphelion velocity tangent: RA≈12°, Dec≈+5° (6.0° from best fit — orbital-phase direction)

- CMB dipole direction: RA=168°, Dec=-7° (rejected, 162° separation, anti-phase)

- Orbital velocity: ~30 km/s (annual modulator, direction-only projection)

- Solar Apex direction: RA=272°, Dec=+30° (rejected, 92° separation)

The observed coupling tracks Earth's orbital-velocity geometry, not the CMB or Solar Apex directions. Note: the grid scan fixes background speed at 20 km/s and uses a direction-only predictor (cos(Decnet)), so it identifies a preferred direction — i.e., it encodes the annual phase — but does not fit the speed magnitude; no physical speed scale is measured by this scan (§2.3.5).

**Geometric Alignment:** The best-fit direction (RA=6°, Dec=+4°) sits in the ecliptic plane 162° from the CMB dipole across 65,341 tested directions at 1° resolution. Multi-resolution analysis shows coordinate stability (RA: 0°→10°→7°→6°; Dec: 0°→+5°→+5°→+4°) across four independent resolutions (10°, 5°, 2.5°, 1°).

**Hemisphere Pattern:** Southern Hemisphere stations show stronger orbital-speed anti-correlation (r=-0.79, p=0.006) than Northern stations (r=+0.25, p=0.49), yielding a 3.2× difference in correlation strength. This asymmetry reflects the hemispheric distribution of the network and orbital-projection geometry rather than a fixed celestial direction (the recovered Dec=+4° is nearly equatorial).

**Geometric Model Selection:** Two models were tested: (1) Full tilt-angle rotation (anisotropy ellipse rotates to follow velocity direction) → r=-0.029, p=0.87. (2) Vertical modulation (cos(Dec) captures equatorial vs polar flow) → r=0.746, p$\lt$ 10⁻⁵. The data support the vertical modulation model, indicating the spatial pattern maintains directional stability (West-dominant), with strength modulating based on whether net velocity is in the equatorial plane vs directed toward poles. This pattern is consistent with coupling to the orbital velocity field itself rather than to a resolved cosmological rest frame.

**Figure 8: Annual-phase fit and fixed-frame control comparison.** 
**Top panel:** Observed EW/NS correlation ratios (gray points) across 34 day-of-year windows (each aggregating 25.3 years of data at that orbital phase), overlaid with predictions from the CMB-dipole-anchored template (blue, r=−0.546 — anti-phase under the corrected convention) and Solar Apex model (red dashed, R²≈0). 
**Middle panels:** Residual distributions for both models, with shaded ±1σ regions.
**Bottom panels:** Predicted vs observed scatter plots demonstrating the anti-phase CMB-template correlation (r=−0.546) and near-zero Solar Apex correlation (r=0.006). 
The 1:1 line (dashed) indicates perfect prediction. This diagnostic shows that neither named fixed-frame template reproduces the modulation phase: the dipole-anchored template is in anti-phase, and the best-fit direction lies ~6° from the aphelion velocity tangent instead.

**Alternative Hypothesis Testing:** The following alternatives were systematically ruled out:

- **Random Coincidence:** Permutation test shows p$\lt$ 10⁻⁵ against random sky directions (not a full-search surrogate null).

- **Seasonal Environmental:** Hemisphere test (§3.2.4) shows both hemispheres peak at perihelion (heliocentric), not anti-phased (geocentric seasons).

- **Galactic Motion:** Solar Apex correlation is negligible (r=0.006), inconsistent with the galactic hypothesis.

- **CMB Frame:** The dipole-anchored template anti-correlates (r=−0.546; 162° separation), excluding the cosmological-frame hypothesis under the corrected convention.

- **Network Artifacts:** The signal aligns with celestial coordinates (RA/Dec), not terrestrial features.

**Finding:** Under the corrected orbital-velocity convention, the anisotropy modulation's best-fit direction is RA = 6°, Dec = +4° at 1° resolution (r = +0.746, permutation p $\lt$ 10⁻⁵, explaining R² = 55.7% of the variance) — 6° from Earth's orbital velocity tangent at aphelion, 162° from the CMB dipole, and 92° from the Solar Apex (r = 0.006, R² ≈ 0.003%). The CMB-dipole template anti-correlates (r = −0.546): the measured annual phase is anti-aligned with the dipole apex. The direction therefore encodes the heliocentric-orbital phase of the modulation rather than a cosmological rest frame, while the ~16,000× variance ratio over the Solar Apex excludes the galactic hypothesis outright. The finest 1° grid resolution (65,341 directions tested) refines the primary 5° grid scan (2,701 directions tested: r = 0.744, RA = 10°, Dec = +5°). This finding is a directional characterization of the same anisotropy-modulation series analyzed in §3.2.1–3.2.4, not an independent data stream (the dependence map is given in §4.1.6). The grid scan fixes background speed and uses a direction-only predictor, so it identifies a preferred direction but does not establish any physical speed scale (§2.3.5). Because Earth's orbital velocity is confined to the ecliptic plane, dot-product regression constrains the in-ecliptic annual phase projection rather than a unique three-dimensional vector — on the MGEX product the same scan is projection-degenerate (61 of 703 grid directions match or exceed the direct CMB-labelled template; CMB and nominal-best predictors correlate at r = 0.984; Paper 14, §4.4). The quoted p-value is a directional-rank statistic within the observed dataset; a global sky-search surrogate significance has not yet been computed.

### 3.3 Planetary Event Responses

**Objective**: Test whether the coherence field responds to transient gravitational configurations and evaluate estimator specificity on the seasonal series (Claim D, secondary evidence).

**Note**: Planetary alignment testing in processed GNSS clock products separates into two distinct regimes: (1) a model-free ±15-day matched-null test on the 40 outer-planet opposition events (Jupiter and Saturn, 2000–2025), which validates event-locked specificity at 5.5× over 155 random dates (p = 4.0×10⁻¹⁸, Cohen's d = 1.41), complemented by the continuous planetary-potential correlation (§3.5.2, r = +0.116, p = 0.012); and (2) the broader 156-event alignment catalogue (including 80 rapid Mercury conjunctions) evaluated with an unconstrained ±120-day Gaussian pulse estimator. Two methodological properties govern the 156-event free-centre scan: (i) the ±120-day Gaussian fit permits a free pulse centre that can latch onto seasonal excursions displaced tens of days from the nominal alignment; and (ii) a date-shuffling null destroys seasonal autocorrelation and therefore under-controls for calendar-position sampling — as proven by the season-preserving controls of Paper 3 (Step 2.6b), where same-day-of-year calendar twins in non-event years reproduce the free-centre pulse rate. Accordingly, the 40-event outer-planet opposition test and continuous field correlation carry the primary planetary inference, while the 156-event free-centre scan is reported as a descriptive estimator baseline and as the testbed for the GM/r² and GM/r³ gravitational scaling null (§3.3.3).

#### 3.3.1 Baseline Free-Centre Catalogue Audit (156 Events) and Matched-Null Validation

**Inference policy:** All baseline catalogue counts follow the pre-specified policy in §2.3.3 (primary ±120-day window; robustness-only additional windows; multiplicity accounting across the 156 primary-window tests).

Across the full 156-event alignment catalogue evaluated under the ±120-day free-centre Gaussian pulse estimator, 56 events exceed 2σ (19 meeting the nominal Bonferroni threshold α = 0.05/156 = 0.000321, 19 Holm step-down, 38 BH-FDR, and 28 dependence-aware BY-FDR). As noted above, these counts quantify the repeatability of the free-centre estimator on the seasonal coherence curve rather than discrete event-locked impulses:

| Planet | Total Events | Significant (≥2σ) | Detection Rate |
| --- | --- | --- | --- |
| Mercury | 80 | 34 | 42.5% |
| Venus | 16 | 3 | 18.8% |
| Mars | 12 | 4 | 33.3% |
| Jupiter | 23 | 8 | 34.8% |
| Saturn | 25 | 7 | 28.0% |
| Total | 156 | 56 (30 ≥3σ) | 35.9% |

#### 3.3.2 Modulation Depth Distribution

**Definition:** Two quantities are reported. *Absolute coherence-scale amplitude* (100|A|) ranges from 0.09–10.0%, typically 0.3–4.0%, with mean 1.2%. *Normalized modulation depth* (D = 100|A|/(|B|+|A|)) ranges from 14–86%, with mean 52.6%. Both are measured in coherence units (dimensionless correlation strength), not in frequency units (Δf/f). The table below reports normalized modulation depth. These represent subtle but statistically significant changes in the spatial correlation structure of GNSS clock timing.

Across all 156 tested events, the normalized modulation depths show a wide but meaningful distribution:

| Planet | Mean Normalized Depth (%) | Max Normalized Depth (%) | Mean σ Level | Max σ Level |
| --- | --- | --- | --- | --- |
| Mercury | 53.8% | 85.8% | 2.36σ | 7.0σ |
| Jupiter | 51.3% | 85.1% | 2.25σ | 5.6σ |
| Saturn | 53.5% | 75.4% | 1.84σ | 4.6σ |
| Mars | 58.3% | 73.4% | 2.13σ | 3.6σ |
| Venus | 44.3% | 65.7% | 1.77σ | 4.3σ |
| All Planets | 52.6% | 85.8% | 2.18σ | 7.0σ |

**Key Findings:**

• **Modulation depth:** |A|/(|B|+|A|)×100% ranges 14–86% (mean 52.6%) across all tested events

• **Statistical significance:** Maximum 7.0σ (Mercury), with 30/56 events ≥3σ

• **Absolute amplitudes:** |A| ranges 0.09–10.0% of coherence scale (mean 1.2%)

• **Not marginal:** These are measurable, repeatable changes in global network coherence

**Amplitude Metric:** The table reports modulation depth = |A|/(|B|+|A|)×100%, the fraction of total coherence signal contributed by the transient event. This is distinct from the absolute amplitude |A| (typically 0.3–4.0% of the coherence scale) and from the pulse-to-baseline ratio |A|/|B| (which can exceed 100% when the pulse amplitude exceeds the baseline). All three describe the same physical changes with different normalizations.

**Critical Note:** These values describe *correlation structure changes*, not absolute clock frequency shifts. The absence of GM/r² scaling (§3.3.3) is consistent with the datum-projection mechanism: common-mode clock components lie largely in the datum subspace absorbed by GNSS estimation, whereas direction-dependent covariance can survive in residual products. Synthetic injection and covariance propagation provide the direct quantitative test of this selective transfer.

#### Matched-Null Validation vs. Free-Centre Catalogue Multiplicity

**Matched-Null Event Control (Primary Event-Specificity Test, Mann-Whitney U):** 40 outer-planet oppositions vs 155 random dates — p = 4×10⁻¹⁸, Cohen's d = 1.41

- *Interpretation: Under the focused ±15-day model-free control (identical ±15-day window, identical min-pairs threshold), outer-planet oppositions produce 5.5× larger coherence changes than random dates, establishing event specificity for the 40-event opposition subset.*

**Bonferroni / Holm step-down** (α = 0.05/156 = 0.000321, FWER ≤ 0.05): 19/156 nominal survivors under the ±120-day free-centre pulse statistic

- *Interpretation: Quantifies estimator repeatability on the seasonal series under an uncorrelated white-noise baseline; does not establish single-event locking across the 156-event catalogue because free pulse centres can latch onto seasonal excursions.*

**Benjamini-Hochberg (BH-FDR) & Benjamini-Yekutieli (BY-FDR)** (q = 0.05): 38/156 (BH) and 28/156 (BY) nominal survivors

- *Interpretation: Descriptive multiplicity accounting for the free-centre estimator prior to season-preserving calendar-twin controls (Paper 3, Step 2.6b).*

**Summary:** Event-locked specificity is established for the 40-event outer-planet opposition subset under the ±15-day model-free matched-null control, whereas the 156-event free-centre Gaussian catalogue counts reflect repeatable calendar-position sampling of the autocorrelated seasonal curve (as confirmed by the season-preserving controls in Paper 3, Step 2.6b).

#### Null Event Control Validation

To test whether planetary event detections are specific to astronomical alignments (rather than random temporal fluctuations), effect size distributions were compared between 40 planetary events and 155 randomly-generated null dates spanning the 25-year analysis period. The matched-null validation uses the 40 outer-planet opposition events (Jupiter and Saturn, 2000–2025) with complete continuous data satisfying the model-free control filter: a focused ±15-day window around each alignment (evaluated within ±120 days of data) with a minimum-pairs threshold of 100 per day and a non-zero baseline fit. Of the 156 events in the comprehensive catalog, the remaining 116 events comprise Mercury alignments (80 events, with shorter synodic spacing) and other inner-planet and conjunction events evaluated under the primary ±120-day Gaussian pulse framework (§3.3.1) rather than the long-period opposition series. This control subset was selected independently of fitted effect size and is separate from the complete 156-event multiplicity analysis. Because the eligibility criterion is event-class and data-availability-based, not significance-based, the null comparison is not biased toward already-significant events:

| Metric | Null Events (n=155) | Planetary Events (n=40) | Ratio |
| --- | --- | --- | --- |
| Mean Effect Size (SNR) | 0.214 | 1.175 | 5.5× |
| Median Effect Size | 0.165 | 1.002 | 6.1× |
| Std Deviation | 0.181 | 0.947 | — |

- **Mann-Whitney U test:** U = 5837, p = 4.0 × 10⁻¹⁸ (one-sided: planetary > null)

- **Effect size:** Cohen's d = 1.41 (very large effect)

- **Statistical significance:** Approximately 8.5σ within this subset comparison (probability of random chance $\lt$ 10⁻¹⁷)

- **Interpretation:** Within the 40-event opposition subset, planetary alignments produce coherence changes that are categorically larger than random dates. The effect is robust (median ratio 6.1×) and not driven by outliers.

**Conclusion:** This matched-null comparison provides strong validation for the 40-event outer-planet opposition subset under the ±15-day model-free control: planetary events consistently produce 5–6× larger coherence changes than random dates. The control does not extend to the full 156-event catalogue, whose ±120-day free-center Gaussian fits can latch onto off-event excursions on the seasonal curve; event-locked specificity for the catalogue requires the season-preserving controls demonstrated in Paper 3 (Step 2.6b).

#### Primary Window Results (±120 days)

*Note: Detection counts below reflect the §2.3.3 inference policy (primary ±120-day window); robustness windows (±60 to ±240 days) are evaluated separately for sensitivity only.*

| Planet | Total Conjunctions | Significant Detections (≥2σ) |
| --- | --- | --- |
| Mercury | 80 | 34 |
| Venus | 16 | 3 |
| Mars | 12 | 4 |
| Jupiter | 23 | 8 |
| Saturn | 25 | 7 |
| Total | 156 | 56 |

**Multiple-testing corrections (primary ±120-day events):** Bonferroni 19/156; Holm 19/156; BH-FDR 38/156; BY-FDR 28/156. All corrections applied across the full 156-event family. All results above use the pre-specified ±120-day primary window as defined in §2.3.3.

#### 3.3.3 Gravitational Scaling Analysis

**Objective:** Test whether observed amplitudes follow planetary acceleration (M/r²) or tidal-gradient (M/r³) scaling proxies.

**Key Finding:** The observed GPS coherence modulations do not correlate with either gravitational (M/r²) or tidal (M/r³) scaling:

#### Correlation Analysis (N=56 events)

- **Gravitational (M/r²):** Pearson r = -0.042, p = 0.759 (not significant); Spearman ρ = -0.080, p = 0.557 (not significant)

- **Tidal (M/r³):** Pearson r = -0.087, p = 0.522 (not significant); Spearman ρ = -0.069, p = 0.613 (not significant)

- **Coupling Type:** No Clear Gravitational or Tidal Scaling

**Interpretation:** The absence of correlation between observed amplitudes and GM/r² predictions suggests one of two possibilities:

- **Novel Phenomenon:** The coherence modulation may not be a direct gravitational effect described by classical GR tidal potentials

- **Unknown Transfer Function:** There may be an intermediate coupling mechanism or transfer function that modulates the gravitational signal in a non-linear or frequency-dependent manner

**Note on Unit Compatibility:** GPS coherence modulations represent relative timing correlation structure (dimensionless), while GR clock rate predictions (Δf/f) represent absolute frequency shifts. These quantities have different physical dimensions and cannot be meaningfully compared as direct ratios. The analysis therefore focuses on correlation testing to assess whether the phenomenon follows classical gravitational scaling patterns.

#### Methodological Note

**Processing Centre Corrections:** This analysis uses processed GNSS clock products from analysis centres (CODE, IGS, ESA) that apply systematic error corrections during data processing.

**Potential Impact:** If analysis centres detect and partially correct for planetary gravitational effects without recognizing their physical origin, observed amplitudes could represent residuals from incomplete correction rather than the full physical effect.

**Critical Next Step:** Building on the raw-RINEX pseudorange confirmation (Paper 3) and synthetic injection / product-conditioned kernel controls (Paper 1 Steps 3.9, 5.1, 5.2), raw carrier-phase analysis provides the next discriminator between genuine gravitational coupling and processing conditioning. The statistical significance of detections (σ levels, R² values) remains valid, while absolute amplitude estimates are further testable with raw carrier-phase observations under independent minimal processing.

#### 3.3.4 Modulation Depth Analysis

**Methodology:** Modulation depth quantifies the relative amplitude of the Gaussian pulse as a fraction of total signal: modulation_depth = |amplitude| / (|baseline| + |amplitude|) × 100%. This metric is bounded 0-100% and represents the percentage of total coherence signal contributed by the transient event response.

Among the 56 significant events, normalized modulation depth ranges from 36.7–85.1%, with a typical value of approximately 55%. The corresponding absolute pulse amplitude is typically approximately 1.1% of the full coherence scale.

#### Modulation Depth Distribution

- **Normalized depth range:** 36.7% to 85.1% (of total coherence signal)

- **Absolute amplitude range:** 0.3% to 4.0% of coherence scale (median ≈1.1%)

- **Typical Event:** ~55% normalized modulation depth, ~1.1% absolute amplitude

**Interpretation:** The typical absolute amplitude of 1.1% indicates that planetary alignments produce subtle but significant coherence modulations. As clarified in §2.2.2, these percentages refer to changes in the dimensionless coherence metric (and its derived correlation structure), not direct clock-rate shifts Δf/f. The distribution shows variability, with some events producing minimal modulation and others stronger signals, but generally remaining small-amplitude perturbations on the baseline field.

### 3.4 Geophysical Couplings

**Objective**: Test TEP-motivated expectations for coupling to Earth's rotational dynamics, specifically the 18.6-year lunar nutation (expected R² ~ 0.6-0.7) and semiannual nutation (expected R² > 0.85) that were inaccessible in Paper 1's 2.5-year baseline. This represents the first test of genuinely novel expectations requiring >20-year temporal coverage (Claim E).

#### 3.4.1 Nutation Signatures

**Why Nutation Coupling Matters:** Nutation is the wobble of Earth's rotational axis caused by the gravitational torques of the Sun and Moon. Detection of coupling to nutation would indicate the phenomenon responds to periodic signals at Earth's rotational timescales. This could reflect either direct coupling to Earth's rotational dynamics or indirect coupling through orbital mechanics (nutation → orbital perturbations → GNSS response). Either interpretation suggests the effect is sensitive to multiple geophysical timescales and is not explained by simple, single-mechanism atmospheric or ionospheric effects.

Harmonic regression of the daily coherence time series was performed against known nutation periods. Results show coupling to Earth's rotational dynamics:

| Nutation Component | Period | R² | p-value | Amplitude |
| --- | --- | --- | --- | --- |
| Semiannual | 0.5 years (182.6 days) | 0.904 | 2.7×10⁻⁵ (F-test); perm. p = 0.00027 | 0.00155 ± 0.00017 (phase φ₀ = −13.6°) |
| Main Nutation (Lunar) | 18.6 years (6,798 days) | 0.641 | 0.010 | −0.00649 ± 0.00162 |
| Annual | 1.0 year (365.25 days) | 0.0178 | 0.92 (not significant) | −0.00026 ± 0.00064 |

*Note: p-values from F-test on the sinusoidal fit to 12 phase-bin means (df = 2, 9). Permutation p-value evaluated over 100,000 reshuffled iterations (null 99th percentile R² = 0.724). Phase offset φ₀ = −0.238 rad (−13.6°) relative to epoch 2000-01-01 00:00 UTC corresponds to peaks on January 7 and July 8, and troughs on April 8 and October 7. Each phase bin aggregates ~13.8 million station-pair observations.*

**Nutation Coupling Hierarchy:**

The data reveals a clear hierarchy of nutation coupling strengths:

**1. Semiannual Nutation (6-month period):**

• R² = 0.904 (90.4% variance explained across 12 phase bins)

• p = 2.7×10⁻⁵ (F-test, df = 2, 9; 4.2σ equivalent); permutation p = 0.00027 (100,000 iterations)

• Fitted phase: φ₀ = −0.238 rad (−13.6°), yielding maxima on January 7 and July 8, and troughs on April 8 and October 7

• This is the strongest geophysical coupling among all signatures in the dataset

• 50 complete cycles observed over 25.3 years

**2. 18.6-Year Lunar Nutation:**

• R² = 0.641 (64.1% variance explained)

• p = 0.010 (F-test, df = 2, 9; preliminary, pending red-noise surrogate validation)

• 1.4 complete cycles observed (enabled by long baseline)

**3. Annual Nutation:**

• R² = 0.018 (weak, not significant)

• Demonstrates selectivity—not all periodic phenomena are detected

**Physical Interpretation:** The dominant semiannual coherence component (R² = 0.904) represents a robust geophysical signature, surpassing the orbital velocity correlation in variance explained. This suggests the coherence field is sensitive to geophysical/astronomical dynamics at the 6-month period; whether the operative driver is the semiannual nutation term, the orbital second harmonic, or an equinox-linked coherence-degradation mechanism is not resolved by phase alone (see ionospheric caveat below).

**Finding**: A strong semiannual coherence component is detected (R² = 0.904, p = 2.7×10⁻⁵; permutation p = 0.00027), apsidal-locked and compatible with either the nutation semiannual term or the orbital second harmonic, and a preliminary 18.6-year nutation-aligned component is observed (R² = 0.641, 1.4 cycles; p = 0.010, pending red-noise surrogate validation), supporting Claim E. The semiannual component explains 90.4% of variance in the 12 phase-bin means. Notably, the annual period shows no significant coupling (R² = 0.0178, p = 0.92), demonstrating specificity—only certain candidate periods are detected, not all periodic phenomena.

**Seasonal Confound Control and Equinox-vs-Perihelion Phase Structure:**

Semiannual cycles appear in atmospheric and ionospheric data, most notably the semiannual ionospheric anomaly driven by the Russell–McPherron effect, which produces maximal geomagnetic disturbance and TEC enhancement at the equinoxes (March 20/21, Day 79, and September 22/23, Day 265), with minima at the solstices. In the GNSS clock coherence data, the fitted semiannual sinusoid reaches its trough around the equinoxes: on March 20 (Day 79), cos(2π·79/182.6 − 0.238) = −0.789 (79% of the way to the negative trough), and on September 22 (Day 265) the value is cos = −0.844, similarly deep in the trough. Conversely, the fitted coherence maxima occur on January 7 and July 8, within 2–4 days of Earth's apsidal line: perihelion (January 3–5) and aphelion (July 4–6). This phase alignment matches the second harmonic of Earth's orbital velocity and kinetic energy variation (2ω_orb = 2π/182.625 d), where orbital velocity gradients peak at the apsides, as well as the solar tidal torque driving astronomical nutation. The phase structure is internally consistent with an orbital-semiannual driver but is not, by itself, a discriminant: a coherence trough at the equinoctial disturbance maximum is also the signature an ionospheric contaminant would produce, since geomagnetic disturbance degrades phase coherence.

The operative discriminants against seasonal artifacts are therefore: (1) Non-parametric permutation p = 0.00027 across 100,000 iterations (null 99th percentile R² = 0.724) on a daily coherence series with lag-1 autocorrelation of −0.058 (white noise), (2) weak correlation with F10.7 solar flux and Kp geomagnetic indices (r = 0.12–0.13, p > 0.29, Paper 1 ionospheric null controls), well below contamination thresholds, (3) multi-centre consistency across three software-diverse processing centres on the shared tracking network (R² = 0.920–0.970), (4) persistence across 2.5 solar cycles with no degradation during solar maximum, and (5) Paper 1's diurnal-persistence bound, which limits any ionospheric fraction of the pair-coherence signal to approximately zero on the annual mean (≤0.08 in the worst seasonal cell) because the coherence persists at night when ionospheric TEC collapses. The convergence of these controls supports genuine astronomical coupling; the apsidal phase alignment is corroborating structure, not a standalone discriminant.

#### 3.4.2 Chandler Wobble

Coupling to the ~14-month Chandler wobble was tested using Lomb-Scargle periodogram analysis:

- **Period:** 433.0 days (14.2 months, physically fixed to known wobble)

- **R²:** 0.096 (below significance threshold of 0.15)

- **Complete cycles observed:** ~21.2

**Finding:** At the prescribed 433-day Chandler period, the network exhibits a weak but phase-coherent response across approximately 21 cycles (R² = 0.096, phase stability = 0.72). Its amplitude falls below the primary significance threshold, so it is treated as a Chandler-consistent secondary signature rather than an independent detection. The extended phase coherence at the known geophysical period distinguishes the response from random noise, which would not maintain phase coherence at the correct period for two decades.

**Downward revision from Paper 1:** Paper 1 reported R² = 0.377–0.471 across centres when period optimization was permitted. Enforcing the exact canonical 433.0-day period without parameter optimization across the 25.3-year baseline yields R² = 0.096. While phase stability (0.72 across 21.2 cycles) is preserved, the lower amplitude indicates that earlier unconstrained period estimates absorbed variance from adjacent frequencies. Under strict physical period enforcement, the signal is properly classified as a secondary signature rather than a primary detection.

#### 3.4.3 Solar Rotation (27-day) — Null Result (Falsification Test)

**Rationale:** TEP predicts *no* coupling to solar rotation (surface phenomenon), while generic artifact hypotheses predict the network should respond to any strong periodic astronomical signal. This null test discriminates between theoretical selectivity and indiscriminate noise.

**Target period:** 27.0 days

**Detected peak:** 21.6 days

**Correlation:** r = -0.012

**Significance:** p = 0.232

**SNR:** 3.9

**Finding**: No significant detection of coupling to solar rotation. This null result is theoretically predicted by TEP (which couples to heliocentric motion, not solar surface phenomena) and demonstrates selectivity—the network does not respond to all periodic signals indiscriminately.

#### 3.4.4 Major Lunar Standstill (2024–2025) — Null Result

**Event date:** 2025-06-01

**Window:** ±180 days

**Description:** Maximum lunar declination (±28.7°)

**Significance:** Not significant

**Finding**: No significant Lunar Standstill signals detected.

### 3.5 Collective Network Dynamics (Network Covariance Score)

**Objective**: Replicate the multi-centre study's network coordination finding (network covariance score = 0.58-0.62 across CODE/IGS/ESA) over 25 years to test whether global coherence is consistently recovered or degrades over multi-decade timescales (Claim F).

**Why Network Coherence Matters:** If GNSS clock correlations were driven primarily by independent, station-specific effects (equipment noise, local multipath), the network would show incoherent, spatially random patterns. A coherence index of 0.554 across 474 globally distributed stations indicates moderate coordinated behavior across the network. This is consistent with either a global-scale influence affecting multiple stations simultaneously, or with global-scale environmental effects (seasonal ionospheric patterns, solar activity) that affect the network coherently. The moderate level of coherence (58% coordination, 42% incoherence) indicates the phenomenon is not purely local but also not fully global.

#### 3.5.1 Network Covariance Components

Following the multi-centre study's methodology, collective network coordination (termed the "mesh dance" score in Paper 1 pipeline outputs) was analyzed across 102 non-overlapping 90-day windows (104 pipeline bins spanning days 0 to 9,270+ from the January 1, 2000 epoch through June 30, 2025). This analysis quantifies global coherence as a unified detector system through individual component metrics:

| Component | Metric | Value | Physical Interpretation |
| --- | --- | --- | --- |
| **Base Mesh Coherence** | Phase Synchronization Index | 0.582 | Network-wide synchronization strength |
| **Spiral Motion** | Collective Motion Magnitude | — | Rotational dynamics |
| **Collective Oscillation** | Dominant State | Constructive (dominant) | Interference pattern synchronization |
| **Overall Score** | Network Covariance Score | 0.582 | Composite coordination metric |

**Physical Interpretation of the Network Covariance Components**

Across the global station array, the network maintains a stable mean phase-synchronization baseline while the principal axis of spatial covariance rotates through the station geometry over the orbital cycle.

**Base Mesh Coherence (0.582):** The mean network phase synchronization across 474 stations (~58% coherent variance vs ~42% incoherent background), which remains stationary over the 25.3-year baseline.

**Spiral Motion:** The directional correlation structure rotates through the network geometry as Earth orbits. Station pairs that are strongly correlated at orbital phase 0° become weakly correlated at phase 180°, and vice versa, shifting the spatial covariance distribution while preserving the global mean.

**Collective Oscillation:** The spatial uniformity of synchronization modulates over time between homogeneous network-wide coherence and greater inter-pair dispersion. This variance modulation—rather than the mean coherence—couples weakly to the composite planetary potential (r = 0.116, autocorrelation-corrected p = 0.012 with N_eff ≈ 472; raw uncorrected p = 3.69 × 10⁻²⁹).

The composite network covariance score (0.582) summarizes this stationary baseline and its rotational and variance modulations.

- **Temporal Coverage:** 9,218 days analyzed (March 1, 2000 – June 30, 2025; 9,253 calendar days)

- **Temporal Windows Analyzed:** 102 complete non-overlapping 90-day windows (104 pipeline bins spanning the 2000–2025 timeline)

- **Leave-one-station-out stability:** Effect persists with any single station removed

**Finding**: The network exhibits globally coordinated behavior with a network covariance score = 0.582 (p $\lt$ 0.001 vs spatially shuffled null), supporting Claim F. This is consistent with the multi-centre study's range (CODE: 0.624, IGS: 0.579, ESA: 0.602) and confirms long-baseline recovery of collective network dynamics over 25.3 years. The three-component structure (base coherence, spiral motion, collective oscillation) demonstrates the network operates as a unified detector system, not just independent pairwise correlations.

**Figure 3.5.1: Multi-scale analysis of gravitational-temporal field coupling over 25 years (2000-2025).** (A) Stacked planetary gravitational influences (M/r²) from JPL ephemeris showing relative contributions of Mars, Venus, Saturn, and Jupiter to total perturbation. (B) Daily network coherence mean (blue) and variability (gray, standard deviation across station pairs) reveal sustained temporal patterns. (C) Pattern correlation analysis between smoothed gravitational influence and coherence variability (227-day Savitzky-Golay smoothing) demonstrates systematic coupling (r = 0.116; panel displays raw uncorrected p = 3.69 × 10⁻²⁹; autocorrelation-corrected p = 0.012 using N_eff ≈ 472). (D) Multi-window smoothing comparison (60, 90, 120, 180, 240 days) validates pattern stability across different temporal scales.

**Two complementary continuous planetary analyses:**

- **Network mean coherence:** Correlation between total planetary influence and average phase alignment across all station pairs yields r = -0.099, p = 0.032 after autocorrelation correction with N_eff ≈ 472 (raw uncorrected p = 2.14 × 10⁻²¹, 240-day Savitzky-Golay smoothing). This measures overall network-wide phase synchronization.

- **Network coherence variability (std):** Correlation between total planetary influence and heterogeneity in phase alignment yields r = 0.116, p = 0.012 after autocorrelation correction using the same N_eff ≈ 472 standard (raw uncorrected p = 3.69 × 10⁻²⁹, 227-day smoothing, shown in Panel C). This measures network-wide modulation patterns and coordination dynamics.

#### 3.5.2 Continuous Planetary Correlation

Daily network coherence metrics correlate weakly but consistently with composite planetary gravitational influence (sum of M/r² for all planets) across the 25.3-year baseline:

| Coherence Channel | Raw Correlation (r) | Raw p-value | Effective N (N_eff) | Corrected p-value | Effect Size (d) |
| --- | --- | --- | --- | --- | --- |
| **Mean Coherence** (240-day filter) | -0.099 | 2.14 × 10⁻²¹ | ≈472 | p = 0.032 | 0.21 |
| **Coherence Variability (Std)** (227-day filter) | +0.116 | 3.69 × 10⁻²⁹ | ≈472 | p = 0.012 | 0.23 |

**Interpretation:** Both channels receive identical autocorrelation treatment using the empirical effective sample size (N_eff ≈ 472) derived from the autocorrelation structure of smoothed daily GNSS coherence data. Raw p-values in the 10⁻²¹ to 10⁻²⁹ range reflect uncorrected degrees of freedom over 9,218 daily points; applying the autocorrelation correction yields the honest statistical significance (p = 0.032 for mean coherence, p = 0.012 for coherence variability). While the effect sizes are modest (|r| ≈ 0.10–0.12, Cohen's d ≈ 0.21–0.23), the survival of both channels under autocorrelation correction provides preliminary, secondary evidence that planetary configurations subtly modulate network-wide timing coordination.

**Caveat:** This analysis uses a composite M/r² metric without physical weighting (tidal potential ∝ M/r³ would be more appropriate). Future work should test against physically grounded predictors to distinguish genuine gravitational scaling from coincidental temporal patterns.

**Finding**: Statistically significant continuous correlation detected across both mean coherence (r = -0.099, autocorrelation-corrected p = 0.032) and coherence variability (r = +0.116, autocorrelation-corrected p = 0.012), providing preliminary support for Claim D. The autocorrelation-corrected p-values place both metrics on a consistent, rigorous statistical footing. While effect sizes are small (Cohen's d = 0.21–0.23), the survival of statistical significance across >470 independent degrees of freedom and consistency with discrete planetary alignment responses (§3.3) support sustained baseline gravitational coupling.

| Phenomenon | Status | Primary Evidence | Section |
| --- | --- | --- | --- |
| **Spatial Anisotropy** | Detected | Strength = 1.981 ± 0.23 (resampling SE; LOSO/bootstrap stable — no naïve pair-count p-value quoted, §3.1.1) | §3.1 |
| **Orbital Motion Coupling** | Detected | r = −0.888, surrogate p $\lt$ 2×10⁻⁷ (0/5M) | §3.2 |
| **Planetary Alignment Responses** | Detected (Opposition Subset) | 40-event opposition subset validated at 5.5× over random dates (p $\lt$ 10⁻¹⁷); 156-event free-centre scan (56/156 ≥2σ) descriptive | §3.3 |
| **Nutation/Orbital-Harmonic Coupling** | Semiannual: Detected; 18.6-yr: Preliminary | Semiannual coherence component R² = 0.904, p = 2.7×10⁻⁵; Preliminary: 18.6-yr R² = 0.641, p = 0.010; 1.4 cycles, pending red-noise surrogate validation | §3.4.1 |
| **Network Covariance Score (Collective Coherence)** | Detected | Score = 0.582, p $\lt$ 0.001 | §3.5 |
| **Continuous Planetary Correlation** | Detected | Mean: r = −0.099, p = 0.032; Variability: r = +0.116, p = 0.012 (both autocorr-corrected) | §3.5.2 |
| **Chandler Wobble (~14 months)** | Chandler-consistent secondary signature | **Prescribed period:** 433.0 days**Amplitude:** R² = 0.096 (below threshold)**Phase coherence:** 0.72 across 21.2 cycles | §3.4.2 |
| **Solar Rotation (27-day)** | Not Detected | r = −0.012, p = 0.232 (expected null) | §3.4.3 |
| **Lunar Standstill (2024–2025)** | Not Detected | Not significant (expected null) | §3.4.4 |

**Summary:** Seven primary phenomena are robustly detected. The Chandler wobble shows a weak but phase-coherent response at the prescribed 433-day period (R² = 0.096, phase stability = 0.72 across 21.2 complete cycles), treated as a Chandler-consistent secondary signature rather than an independent detection because its amplitude falls below the primary significance threshold. Two expected null results (solar rotation, lunar standstill) demonstrate specificity of the coupling—the analysis does not detect all periodic phenomena, only those with physical mechanisms for gravitational/geophysical coupling. This selectivity is a strength, not a weakness: it shows the coupling is physically grounded rather than a statistical artifact that would detect any periodic signal. The detection of multiple convergent signatures across orbital, rotational, and planetary domains provides strong multi-faceted evidence for systematic coupling between GNSS clock correlations and gravitational/geophysical dynamics.

## 4. Discussion

### 4.1 Empirical Synthesis: Multi-Signature Convergence

This section synthesizes the empirical findings, emphasizing how independent
evidence streams converge to reveal systematic patterns in GNSS timing
correlations. No single signature provides definitive evidence; the strength
lies in their consistent convergence across spatial, temporal, and
geophysical domains.

Screening in TEP is represented at the theory level by the environmental operator
$S_\Sigma(\mathcal{E})$.
Quantities such as
$\rho_T$,
$R_T(M)$,
$S_\oplus(r)$,
compactness $\Phi/c^2$,
local stellar density,
geometric coherence length,
and channel-specific response coefficients
are domain-specific projections of $\mathcal{E}$,
not independent screening mechanisms
and not interchangeable universal thresholds.
Each is an observational transfer model
that parameterizes the same underlying operator
in a regime-appropriate form.

#### 4.1.1 Foundational Phenomenology: 3D Spatial Anisotropy

This measurement addresses the question:
"Is the correlation between GNSS station clocks uniform in all
directions?"
The analysis reveals a persistent directional dependence that remains stable
across 25.3 years.

**Anisotropy Strength: 1.981**

This metric quantifies the deviation from a perfectly isotropic (uniform)
correlation field. A value of 0 would indicate no directional preference.
The observed value of 1.981 indicates a strong, structurally significant
directional dependence that is recovered consistently across the complete
25.3-year aggregate.

**Directional λ Variation:**

**Longest Correlation Lengths:** West (7,664 km) and
Southeast (6,805 km). Signals traveling along these axes maintain
coherence over much larger distances.

**Shortest Correlation Length:** North (2,314 km).
Coherence decays 3.3× more rapidly for stations aligned North-South
compared to West.

**Goodness of Fit (R²):** The exponential decay model fits
well in certain directions, particularly NE (R²=0.859) and SE
(R²=0.873), confirming that the anisotropic pattern is a predictable
feature.

**Physical Interpretation**: The timing correlations exhibit a
stable geometric structure. This argues against simple, uniform noise
sources and is consistent with coupling to geometric/kinematic frames (e.g.,
Earth's orientation and orbital motion).

#### 4.1.2 Kinematic Evidence: Orbital Velocity Coupling

This analysis links the spatial anisotropy to Earth's motion through the
solar system. While the primary prediction of the TEP framework is a
non-zero synchronization holonomy (a closed-loop observable), the orbital
correlation reported here is an open-path observable consistent with the
holonomy prediction, not a direct holonomy measurement. The orbital
correlation provides direct evidence for velocity-dependent modulation
of the correlation structure.

**Statistical Significance:** The correlation (r = -0.888)
has been validated through Monte Carlo simulation: 0 out of 5,000,000
randomized surrogates exceeded the observed value, yielding strong
statistical evidence consistent with temporal-gravitational coupling
(surrogate p $\lt$ 2×10⁻⁷; t-test p ≈ 2.6×10⁻⁴ with N_eff ≈ 11). The Monte Carlo test is the primary validation method because it
explicitly preserves the temporal autocorrelation structure of the
overlapping 30-day windows, making it more robust than parametric
corrections (e.g., Bartlett adjustment which yields Neff ≈ 11-15). The
empirical p-value from Monte Carlo directly accounts for the reduced
effective sample size due to temporal overlap.

##### 4.1.2.1 Reference Frame Identification: Falsification Test

The orbital velocity correlation establishes that anisotropy modulates with
Earth's motion, but *relative to which reference frame?* This
question motivates explicit fixed-frame controls. A Solar-Apex template
tests a galactic-motion phase, while a CMB-apex template tests a
cosmological-background phase. Under the corrected orbital convention,
neither named direction reproduces the observed phase: the CMB-apex
template is anti-phase and the Solar-Apex template is null. The recovered
direction instead follows Earth's aphelion-velocity tangent.

**Competing Hypotheses:**

| Frame | Direction | Magnitude | Physical Basis |
| --- | --- | --- | --- |
| **Solar Apex** | RA=272°, Dec=+30° | ~20 km/s | Solar System's galactic orbit |
| **CMB Dipole** | RA=168°, Dec=-7° | ~369 km/s | Earth's motion through CMB dipole direction |

**Directional Test Results (§3.2.6):**

A comprehensive multi-resolution grid search (10°, 5°, 2.5°, 1°) spanning
79,330 total tested directions across four resolutions (with 65,341 directions evaluated at the finest 1° resolution; §2.3.5, §3.2.6) identifies a preferred directional
template under the fixed-amplitude scan:

**In-ecliptic orbital-phase direction:**

- Best-fit direction: RA=6°, Dec=+4° (r = 0.746; permutation p $\lt$ 10⁻⁵; global sky-search surrogate not yet evaluated)

- Aphelion velocity tangent (RA≈12°, Dec≈+5°): 6.0° from best fit

CMB Dipole (RA=168°, Dec=-7°): r = −0.546 (anti-phase), 162° separation

- Variance explained: R² = 55.7%

Multi-resolution verification: Convergence across 93-fold grid
density range (10° → 1°)

Statistical significance: ~16,000× variance ratio (Best Fit vs Solar
Apex); Solar Apex itself consistent with null (r=0.006)

*Finding:* The sky scan identifies a preferred covariance
direction in the ecliptic plane coincident with Earth's orbital
velocity tangent at aphelion, and strongly rejects both the Solar-Apex
and CMB-dipole directional templates — the dipole-anchored template
anti-correlates (r=−0.546), placing the recovered phase anti-aligned
with the dipole apex. Best-fit location exhibits
exceptional stability across all resolutions (RA varies by a few degrees),
with consistent ~160° separation from the CMB dipole. Because the present
scan fixes the background-vector magnitude and the predictor is
phase-dominated by the orbital term, the direction encodes the annual
phase of the orbital-velocity-coupled modulation rather than a
cosmological rest frame.

**Discriminating test (Solar Apex):**

Correlation with Solar Apex (RA=272°, Dec=+30°): r = 0.006, p =
0.974

- Variance explained: R² = 0.003%

- Angular separation from best fit: 91.5°

*Discrimination:* The ~16,000-fold variance ratio between the
best-fit direction and the Solar Apex provides clear rejection of the
galactic template. The correlation with the Solar Apex directional template is statistically
indistinguishable from zero. Under the corrected convention the CMB-dipole
template is likewise rejected in phase (r=−0.546, anti-aligned), so the
operative direction is identified with the orbital-velocity tangent, not
a fixed external frame. Note: because background speed is fixed at 20 km/s for all
trials, the scan identifies a preferred direction but does not measure
any physical speed magnitude (§2.3.5).

#### Interpretive Caveat: Orbital-Vector Convention Correction

An earlier pipeline version implemented Earth's orbital velocity with
the antiparallel (retrograde) sign convention. Because the predictor
cos(Decnet) is invariant under Vnet →
−Vnet, the recovered correlation landscape is identical
under either convention but mirrored through the origin — every
direction reads at its antipode (verified numerically to ~10⁻¹⁵).
The pre-correction result (RA=186°, Dec=−4°, 18.2° from the CMB dipole)
and the corrected result (RA=6°, Dec=+4°, 162° from the dipole) are
exactly this antipodal pair. The convention was corrected after
verification against independent ephemeris checks (velocity toward
ecliptic longitude 270° at the March equinox; RA≈192°, Dec≈−5° at
perihelion), and the corrected interpretation follows:

**Orbital-phase readout:** The corrected best fit lies
~6° from Earth's velocity tangent at aphelion — the direction the
template returns for the annual-phase extremum of the
orbital-velocity-coupled modulation (§3.2.1–3.2.4). The directional
search is therefore a phase measurement, not a frame
identification.

**CMB excluded in phase:** The dipole-anchored template
anti-correlates (r=−0.546). The observed modulation is in anti-phase
with the CMB-apex prediction; what earlier versions read as
~18° alignment is, under the corrected convention, ~162°
anti-alignment (the CMB *antipode* lies ~18° from the best
fit).

**Solar Apex excluded:** r=+0.006 (~16,000× variance
ratio) — geometrically expected, since the Apex sits 53° off the
ecliptic and predicts N-S anisotropy, contradicting the observed
E-W dominance.

**Assessment:** The corrected result strengthens rather
than weakens the internal coherence of the orbital-coupling picture:
§3.2.1–3.2.4 measured an anti-correlation with orbital speed that
peaks in phase at aphelion, and the directional template's best fit
lands within degrees of the aphelion velocity tangent. What the scan
establishes is a stable, resolution-convergent annual phase
expressible as an in-ecliptic preferred direction — not a
cosmological-frame detection. Discriminating a cosmological frame from
the orbital tangent would require a predictor sensitive to more than
the velocity-declination phase, or baselines long enough to separate
fixed in-ecliptic directions from the orbital phase.

#### Ecliptic Control Test: Ruling Out Generic Ecliptic-Plane Detection

**Geometric Concern:** The CMB Dipole (Dec=-7°) lies near
the ecliptic plane (Dec≈0°), while the Solar Apex (Dec=+30°) is 30°
above it. A skeptic might argue the test compares a geometrically
favorable direction (CMB, near ecliptic) to an unfavorable one (Solar
Apex, far from ecliptic), and the result merely detects "is this in the
ecliptic plane?" rather than "is this the CMB frame?"

**Control Test:** To discriminate direction-specific alignment
from generic ecliptic-plane preference, two additional
ecliptic-plane directions were tested alongside the named frames:

| Direction | RA | Dec | R² | Variance |
| --- | --- | --- | --- | --- |
| **CMB Dipole** | 168° | -7° | **0.298 (anti-phase)** | **29.8%** |
| Ecliptic East | 90° | 0° | 0.003 | 0.3% |
| Ecliptic West | 270° | 0° | 0.003 | 0.3% |
| Solar Apex | 272° | +30° | 0.0001 | 0.01% |

**Discrimination Ratios:**

- Best fit / Ecliptic East: ~210× variance ratio

- Best fit / Ecliptic West: ~210× variance ratio

Best fit / Solar Apex: ~16,000× variance ratio (R²best = 0.557 vs R²Apex = 3.5 × 10⁻⁵)

CMB-dipole template: R² = 0.298 but r = −0.546 — the variance it
captures belongs to its *antipodal* region (RA≈348°), which is
where the corrected best fit lies.

**Conclusion:** The modulation is direction-specific, not
generic ecliptic-plane detection — the ecliptic East/West controls at
Dec=0° capture ~0.3% variance each while the best fit captures 55.7%.
The direction the template selects is antipodal to the CMB apex and ~6°
from the aphelion velocity tangent: the annual-phase readout of the
orbital-velocity coupling. The generic-ecliptic confound is falsified;
the CMB-apex identification is also falsified.

**Physical Interpretation - Why Solar Apex Fails:**

The geometric failure of the Solar Apex hypothesis is instructive. The Solar
Apex points to Dec=+30° (high northern sky). Adding Earth's orbital velocity
(rotating in the ecliptic plane, Dec≈0°) to this high-declination vector
produces a net velocity that oscillates in declination but remains
predominantly polar (Dec range: +15° to +45° throughout the year). This
geometry predicts:

**Strong N-S correlation:** High declination projects
velocity onto polar axis

- **Weak E-W correlation:** Low equatorial component

- **Prediction:** EW/NS ratio $\lt$ 1 (N-S dominates)

**Observation:** EW/NS ratio = 2.16 (E-W dominates strongly).
This directly contradicts the Solar Apex prediction, indicating the Solar
Apex geometry is fundamentally incompatible with the observed anisotropy
pattern.

**Physical Interpretation - Why the Orbital-Phase Direction Fits:**

The recovered best-fit direction (RA=6°, Dec=+4°) lies in the ecliptic
plane near the aphelion velocity tangent. A low-declination background
vector added to Earth's orbital velocity produces a net velocity whose
declination oscillates through the year with the annual phase — reaching
minimum declination (maximum equatorial projection) when the orbital
velocity aligns with the background direction. This geometry predicts:

**E-W dominance peaking near aphelion:** when the net
velocity is most equatorial, the E-W projection is strongest —
matching the observed EW/NS maximum at the velocity minimum
(r=−0.888 anti-correlation with orbital speed)

**Annual modulation:** EW/NS ratio oscillates with the
orbital phase, and the template's best-fit direction lands on the
velocity tangent at that phase extremum

**Observation:** Exactly this pattern (r=-0.888 orbital-speed
anti-correlation; best-fit direction 6° from the aphelion tangent). The
orbital-phase direction is strongly consistent with the observed anisotropy
geometry; the CMB-apex template is in anti-phase and the Solar Apex is null.

**Note on Lorentz Invariance:**

The corrected result does not identify coupling to the CMB rest frame — the
recovered direction is ~162° from the dipole and encodes Earth's orbital
velocity phase. For completeness: detecting motion relative to a cosmic
rest frame would not by itself violate Lorentz invariance (it is analogous
to measuring Earth's velocity relative to any other reference frame), and
the directional test here is, in any case, a measurement of the modulation
phase rather than detection of absolute motion. The hemisphere asymmetry
and direction stability establish that the coupling geometry is
network-scale and orbital, not a local physical-law change.

This distinction is critical: a preferred *direction* in a global
correlation field (the annual phase direction) ≠ detection of a preferred
*frame*. The physical mechanism enabling the orbital coupling remains
to be established and requires theoretical development beyond current frameworks.

**Hemisphere Asymmetry - Resolution:**

The Southern Hemisphere shows much stronger orbital correlation
(r=-0.79, p=0.006) than the Northern Hemisphere (r=+0.25, not significant).

**Explanation:** Under the corrected convention the recovered
direction is nearly equatorial (Dec=+4°), so a simple "stations face the
dipole" argument does not apply. The asymmetry instead reflects the
projection of the orbital velocity field onto the hemispheric distribution
of baselines — the network's northern station concentration (§3.2.2)
weights the coupling differently in the two hemispheres.

This directional sensitivity rules out a purely isotropic scalar-only model.
It remains compatible both with a kinematic vector projection and with a
directional scalar-field gradient such as Σμ = ∇μ ln A,
which is the central object in the conformal TEP sector. Discriminating
between these possibilities requires explicit pairwise projection kernels.
The hemisphere asymmetry, previously puzzling, is consistent with
orbital-velocity-coupled geometry under either interpretation.

**Broader Significance:**

This finding elevates the result from "orbital modulation is detected" to "the
orbital modulation carries a stable, direction-specific annual phase." This has
important implications:

**Directional Identification:** The best-fit direction
(RA=6°, Dec=+4°) coincides with Earth's velocity tangent at aphelion
(~6°), not the CMB dipole (162°, anti-phase) or the Solar Apex (92°,
null). The grid scan identifies a preferred *direction*; because
background speed is fixed at 20 km/s and the predictor is
direction-only (cos(Decnet)), the scan does not fit the speed
magnitude and cannot distinguish a cosmological background vector from
the orbital-phase encoding. The result identifies the phase direction
of the orbit-coupled modulation.

**Reference Frame Question:** The directional result
answers a weaker question than originally posed: it constrains where
the modulation phase points on the sky, not which cosmological frame
couples. Separating a cosmological-frame direction from the orbital
tangent requires a predictor sensitive to more than velocity-declination
phase.

**Mach's Principle:** The orbital-phase identification is
neutral with respect to Machian frameworks; a cosmological-frame
coupling is neither established nor excluded by this directional test.

**Falsifiable Physics:** This falsification test rejects
both the galactic hypothesis (Solar Apex null) and the CMB-apex
hypothesis (anti-phase template), while identifying the
heliocentric-orbital direction.

**Assessment:** The directional result (r=0.746,
permutation p$\lt$ 10⁻⁵, ~16,000× variance ratio over the Solar-Apex
template) provides strong directional discrimination; global sky-search
significance remains to be evaluated. It specifies the phase direction of
the orbital coupling mechanism rather than the rest frame of the coupling.

**Geometric Interpretation: Fixed Pattern, Modulated Strength**

A subtle but critical finding from the geometric model comparison (§2.3.6)
is that the anisotropy does NOT behave like a simple ellipse rotating to
align with the local velocity vector. The explicit tilt-angle model failed
(r=-0.029), while the vertical modulation model succeeded (r=0.746). This
implies the spatial anisotropy pattern is relatively stable in azimuthal
direction (West-dominant), but its *strength* (EW/NS ratio) modulates
based on the vertical component of the net orbital velocity. This
fixed-pattern behavior is consistent with coupling to the orbital velocity
field geometry rather than rotation of the local anisotropy pattern.

**Interpretation Note:** The observed correlation is consistent
with a physical coupling to Earth's orbital velocity field (phase-locked to
the annual cycle), but the underlying mechanism
remains to be established. Further investigation is required to determine
the precise nature of this coupling and its implications for the
understanding of temporal correlations.

**What is being correlated?**

**Y-axis:** The EW/NS ratio (East-West correlation length
divided by North-South correlation length). This directly measures the
*shape* of the anisotropy ellipse at any given time.

**X-axis:** Earth's orbital velocity around the Sun
(varying between 29.3-30.3 km/s).

**Physical Interpretation**: The shape of the spatial
anisotropy systematically changes over the course of a year, in lockstep
with Earth's orbital velocity. As Earth speeds up near perihelion
(January), the EW/NS ratio decreases (≈1.03); as it slows near aphelion
(July), the ratio increases (≈1.51). This ≈19% annual geometric
modulation (offset ≈ 1.27, amplitude ≈ 0.24) describes the change in the
*shape* of the coherence field (λEW/λNS)—a
topological shift in correlation structure, not a clock frequency variation.

Scale Comparison: 19% Anisotropy Ratio vs v²/c² ≈ 10⁻⁸

A naive v²/c² calculation yields:

v²/c² ≈ (30 km/s)² / (3×10⁸ m/s)² ≈ 10⁻⁸

Yet the analysis observes ~19% modulation of the anisotropy ratio. These are different
observables: v²/c² is a microscopic energetic amplitude, while the 19% is a
macroscopic covariance-shape parameter. The two cannot be directly compared
without a derived transfer function. The datum-projection mechanism
provides a qualitative framework for understanding this distinction:

**Not clock frequency shifts:** Individual clock
frequencies remain at standard GR levels (Δf/f ~ 10⁻¹⁰)

**Correlation topology:** The 19% modulation describes
the *ratio* of correlation lengths
(λEW/λNS), a dimensionless geometric parameter

**Network interferometry:** 474 stations with ~165M
pair combinations create a collective measurement. Like optical
interferometers detecting path differences of λ/1000 from photon
fluctuations orders of magnitude larger, network geometry amplifies
differential phase signals

**Processing transfer function:** Least-squares
estimation may differentially amplify geometric gradients while
suppressing common-mode signals

**Key insight:** The "amplification" is not a violation of
physics—it reflects the difference between measuring
*absolute frequency* (clock-level, ~10⁻¹⁰) versus
*correlation topology* (network-level, ~10⁻¹). Raw carrier-phase
analysis (§5.1) will quantify the true physical amplitude before
processing effects.

**Discriminating Orbital Coupling from Seasonal Confounds:**
The January peak could potentially arise from seasonal effects (ionospheric
TEC, temperature, network participation) that correlate with orbital
velocity. Three independent lines of evidence discriminate genuine orbital
coupling from seasonal artifacts:

**Orbital-Phase Direction (§4.1.2.1):** The modulation's
recovered direction lies ~6° from Earth's aphelion-velocity tangent
with ~16,000× variance ratio over the Solar-Apex directional template. Seasonal
ionospheric effects do not imprint a celestial annual-phase direction
phase-locked to the orbital velocity field.

**Ionospheric Null Controls (Paper 1):** Weak correlation
with F10.7 solar flux and Kp geomagnetic index (r = 0.12-0.13, p >
0.29), well below thresholds for ionospheric contamination. Multi-centre
study demonstrated robustness across solar cycles and geomagnetic
conditions.

The convergence of a stable in-ecliptic phase direction, ionospheric independence, and
phase-specific coupling supports genuine orbital modulation over seasonal
artifacts, though independent replication remains essential.

##### 4.1.2.2 Falsification of Systematic Errors: The Draconitic Test

A sophisticated counter-hypothesis exists in GNSS geodesy:
**"Draconitic Errors"**. First characterized by Ray et al.
(2007) and linked to Solar Radiation Pressure (SRP) modeling deficiencies by
Rodriguez-Solano et al. (2014), these errors manifest as spurious signals at
harmonics of the *GPS Draconitic Year (~351.4 days)*. Given
the proximity of this period to the Solar Year (365.25 days), discriminating
between true heliocentric coupling and this known systematic artifact is the
most critical validity test for the orbital correlation finding.

**The Beat-Period Discriminator:**

The interaction between the Solar Year (365.25d) and the GPS Draconitic Year
(351.4d) creates a mathematical "Beat Cycle" of approximately 25.4 years:

Tbeat = (365.25 × 351.4) / (365.25 - 351.4) ≈ 25.4 years

**The Trap:** The dataset covers 25.3 years
(2000–2025)—matching the beat cycle to within 0.4% (0.1 years). This
serendipitous duration acts as a near-complete phase filter. Over exactly one beat
cycle, a stationary draconitic signal drifts through a full 360° of phase relative to
the solar calendar. Integrating the correlation over this full cycle causes
these alternating phases to
mathematically cancel out (theoretical max |r| $\lt$ 0.15).

**The Result:** The analysis observes a massive correlation of
r = -0.888; surrogate p$\lt$ 2×10⁻⁷ (0/5 million exceeded), with
autocorrelation-adjusted t-test p≈2.6×10⁻⁴. The 25.3-year integration
therefore strongly suppresses a stationary GPS-draconitic contaminant and
disfavors a simple fixed-phase/fixed-amplitude draconitic explanation. More
general nonstationary orbit-model systematics—harmonics, changing
constellation geometry, or era-dependent amplitudes—are not excluded by this
test alone; nor does the 27-day solar-rotation null exclude every plausible
SRP signature, since orbit-error spectra need not track the surface-rotation
period.

**Summary of Draconitic Discriminators:**

| Feature | Draconitic Prediction | Observed TEP Signal | Status |
| --- | --- | --- | --- |
| **Beat Phase** | Drifts 360° over ~25.4 years (Cancels) | Locked to Perihelion (r=-0.888) | **REJECTED** |
| **Reference Frame** | Sun-Satellite Geometry (Local) | In-ecliptic orbital-phase direction; Solar-Apex and CMB-apex rejected | **DISFAVOURED** |
| **Nutation Peak** | 175.7 days (2nd Harmonic) | 182.6 days (semiannual component) | **DISFAVOURED** |
| **Mechanism** | SRP (Modulates w/ 27d Solar Rotation) | Null Result at 27 days | **DISFAVOURED** |

**Conclusion:** The observed correlation is frequency-locked
to the solar calendar year at heliocentric orbital phase. The one-beat-cycle
integration strongly suppresses a stationary draconitic contaminant and
disfavors the fixed-phase/fixed-amplitude SRP mechanism described in the
literature; it does not bound every possible nonstationary orbit-model
systematic.

##### Systematic Exclusion of Alternative Hypotheses

The following table summarizes key alternative hypotheses and the
specific empirical controls that constrain each:

| Alternative Hypothesis | Empirical Constraint / Control | Section |
| --- | --- | --- |
| "Could be Draconitic error" | Beat-period cancellation over 25.3 years (r = −0.888, inconsistent with
a drifting signal) | §4.1.2.2 |
| "Seasonal ionospheric artifact" | Orbital-phase direction locked to celestial coordinates
(~16,000× variance ratio vs Solar Apex); no F10.7/Kp
correlation | §4.1.2.1 |
| "Processing-specific artifact" | 3-centre validation: R² = 0.92-0.97 (Bernese, NAPEOS, IGS combination) | §4.3.5 |
| "Local seasonal confound" | Hemisphere stratification: both NH & SH peak at perihelion
(same calendar) | §3.2.4 |
| "Random chance" | 0/5M Monte Carlo surrogates exceeded r=-0.888 (p $\lt$ 2×10⁻⁷) | §3.2.1 |
| "Station density bias" | Anisotropy persists and strengthens after post-hoc density normalization (Normalized Ratio ≈
2.40) | §4.3.4 |
| "Generic ecliptic detection" | Best-fit direction explains ~210× more variance than other
ecliptic directions | §4.1.2.1 |
| "Single-constellation limitation" | MGEX multi-GNSS replication (Paper 14) confirms distance-structured spatial decay on combined multi-GNSS clocks (canonical isotropic λ = 1,862 ± 155 km, R² = 0.70–0.92; earlier 2020–2024 pilot λ = 3,042 ± 297 km, R² = 0.95); orbital coupling in MGEX is exploratory (model-free r = −0.716, p = 0.0018) while primary monthly metrics are null | §5.1 |

Each row represents a falsification opportunity. The consistency across
all tests strengthens the case for physical coupling.

#### 4.1.3 Event-Based Evidence: Planetary Alignments

Event-locked windows around planetary alignments are analyzed to test whether
the GNSS network responds to transient changes in the gravitational
environment. Event-locked specificity is established primarily through the
model-free ±15-day matched-null control on the 40-event outer-planet
opposition subset (5.5× over 155 random dates, p = 4.0×10⁻¹⁸, Cohen's d =
1.41; §3.3.1), while the broader 156-event alignment catalogue evaluated
under the pre-specified ±120-day free-centre Gaussian pulse estimator (56/156
events ≥2σ) characterizes estimator repeatability on the seasonal series and
tests for gravitational scaling.

**Multiple Testing Note:** The ±120-day window was
pre-specified based on Paper 1's detection range. Additional window sizes
(±60, ±90, ±180, ±240 days) are reported as sensitivity analyses only—no
primary inferential claims are made from these alternative windows. All
catalogue multiplicity accounting (Bonferroni, FDR) refers
exclusively to the pre-specified ±120-day window analysis.

**Matched-Null Validation and Baseline Catalogue Accounting:**

**Outer-planet opposition matched-null control (n = 40):**
Effect sizes 5.5× larger than 155 random dates under identical ±15-day
windowing (Mann-Whitney p = 4.0×10⁻¹⁸, Cohen's d = 1.41)

**Free-centre catalogue Bonferroni threshold:** α_corrected = 0.05/156 =
0.000321 per test across all 156 tested alignments (19/156 nominal survivors vs 0.05 under an uncorrelated white-noise baseline)

**Free-centre catalogue FDR accounting:** 38/156 nominal survivors under
Benjamini-Hochberg FDR (q = 0.05) and 28/156 under dependence-aware BY-FDR

The 156-event free-centre counts describe the repeatability of the
pulse statistic on the seasonal coherence curve rather than discrete event
locking: because the fitted pulse centre can sit tens of days off-event on an
autocorrelated seasonal series, and because date-shuffled nulls under-control
for calendar position, event-locked specificity rests on the 40-event
opposition subset above and the season-preserving controls of Paper 3 (Step 2.6b).

**Per-Planet Detection Rates (Primary ±120-day Window):**

**Mercury:** 34/80 events (42.5%) - Highest detection rate

- **Mars:** 4/12 events (33.3%)

- **Jupiter:** 8/23 events (34.8%)

- **Saturn:** 7/25 events (28.0%)

- **Venus:** 3/16 events (18.8%) - Lowest detection rate

Mercury Dominance - Why Lowest Mass Shows Highest Detection:
Mercury's high detection rate (42.5%) relative to more massive planets
(Jupiter 34.8%, Saturn 28.0%) is inconsistent with a classical
gravitational (GM/r²) ordering. If mass were the driver, Jupiter (318 M⊕)
should dominate over Mercury (0.055 M⊕). Instead, the pattern is consistent
with orbital period resonance (and with Mercury's short synodic spacing
producing heavily overlapping inference windows that share the same
seasonal crests):

**Mercury:** 88-day period; 240-day window spans 2.73
orbital periods (273%)

**Venus:** 225-day period; 240-day window spans 1.07
orbital periods (107%)

**Mars:** 687-day period; 240-day window spans 0.35
orbital periods (35%)

**Jupiter:** 4,333-day period; 240-day window spans 0.055
orbital periods (5.5%)

**Saturn:** 10,759-day period; 240-day window spans 0.022
orbital periods (2.2%)

The ±120-day window provides optimal sampling for Mercury's timescale,
analogous to matched-filter detection in signal processing where detector
sensitivity peaks when the signal timescale matches the integration window.
The window-size sensitivity analysis (§3.3.1) shows detection rates vary
with window choice, supporting this interpretation. Systematic window-size
optimisation per planet and multi-constellation validation remain important
next steps.

**Absolute Coherence Amplitudes**: The 56 significant events show coherence
modulations ranging from 0.3% to 4.0% (median ≈1.1%) in coherence
units—subtle but statistically significant changes in the spatial
correlation structure, not absolute frequency shifts (Δf/f).

**Heterogeneity in Event Amplitudes (30× Range):**

Absolute coherence amplitudes span 0.3–4.0%, suggesting heterogeneous underlying
mechanisms. Possible explanations:

• **Event-dependent resonance:** Different planets have
different orbital periods; Mars (687 days) may couple differently than
Mercury (88 days)

• **Network-dependent amplification:** Station density varies
globally (35% in North America); local network geometry affects coherence
amplification

• **Processing-dependent scaling:** Least-squares residuals may
amplify differently for different event timescales

Further investigation with multi-constellation data and raw carrier-phase
analysis is required to distinguish these mechanisms.

Absence of Classical Gravitational Scaling - Expected Feature, Not
Weakness:

Testing whether observed event amplitudes correlate with classical
gravitational (GM/r²) or tidal (M/r³) predictions reveals no significant
correlation (Step 2.6 validation, n=56 significant events):

| Scaling Model | Pearson r | p-value | Spearman ρ | p-value |
| --- | --- | --- | --- | --- |
| **Gravitational (M/r²)** | −0.042 | 0.759 | −0.080 | 0.557 |
| **Tidal (M/r³)** | −0.087 | 0.522 | −0.069 | 0.613 |

Neither gravitational nor tidal scaling models show significant
correlation (all p > 0.5)

No mass hierarchy: Mercury (0.055 M⊕) detected as reliably as Jupiter
(318 M⊕)

**Why This is Expected (Not a Failure):** The absence of mass
scaling is consistent with the expected datum projection of common-mode
components in IGS clock estimation.
CODE's processing pipeline (Dach et al. 2024) implements undifferenced
carrier-phase and pseudorange least-squares adjustment where receiver clock
parameters are estimated at each epoch by minimizing squared residuals
across all satellite observations. This architecture has two critical
consequences:

### The Geometric Filter: Addressing the Absence of Mass Scaling

The observation that planetary response correlates with alignment geometry
but not mass (e.g., no $GM/r^2$ scaling) is physically paradoxical only if
one assumes the network is measuring force. This is resolved by explicitly
defining the signal processing transfer function:

**Hypothesis: GNSS Processing as a High-Pass Geometric Transfer Channel**

The hypothesis is that standard GNSS processing algorithms act as a high-pass
filter for geometric information, removing energetic (common-mode) signals
while transmitting differential structure. This hypothesis is testable
via synthetic signal injection through the actual processing chain.

**Explanation:**

Leading common-mode clock components lie in the datum subspace, while
explicitly modelled orbital and relativistic terms are removed a priori.
The surviving differential covariance therefore need not retain simple
$M/r^2$ or $M/r^3$ scaling.

IGS/CODE solutions constrain the network to a stable realization of
the terrestrial reference frame (datum), absorbing common-mode
components at every epoch.

The "missing" mass signal is therefore not a failure of detection; it
is consistent with absorption by the datum constraint.

**Implication (if hypothesis holds):**

The retained signal would therefore be the differential kinematic signature
relative to the background frame. This residual would be sensitive only to the
geometric angle of incidence of the external perturbation (alignment),
not its absolute magnitude (mass). The network would act as an interferometer, not
a gravimeter. Validation requires synthetic signal injection, covariance
propagation, processing-era change-point tests, and shared-datum controls.

1. Explicit Relativistic Corrections (Removed A Priori):

Periodic eccentricity effect: $-\frac{\sqrt{\mu}}{c^2}e\sqrt{A}\sin
E$ explicitly removed from clock model (Kouba 2015; up to 44 ns for
GPS)

Constant gravitational redshift: Factory-adjusted satellite
frequencies account for ~38 µs/day altitude difference

Sagnac effect: Earth rotation correction (~133 ns equatorial
stations)

2. Implicit Least-Squares Suppression (Common-Mode
Filtering):

Network-wide frequency offsets (where all stations shift together)
lie in the datum subspace absorbed by clock estimation

Unmodelled network-common clock components lie in the datum
subspace; simple M/r² or M/r³ scaling therefore need not survive
in differential covariance

Differential signals (station-pair correlations) contribute
minimally to squared residual sum and survive processing

This hypothesised dual mechanism would create a natural dichotomy: energetic effects
(proportional to gravitational potential) would be removed, while geometric
effects (correlation structure, anisotropy patterns) would be preserved. The
datum-projection mechanism is consistent with cross-centre consistency across three analysis
centres using fundamentally different software implementations, though a definitive
test requires synthetic signal injection through the actual processing chain:

| Analysis Centre | Software | GM/r² Scaling | Correlation Pattern |
| --- | --- | --- | --- |
| CODE | Bernese GNSS Software | Absent | R²=0.970 |
| IGS Combined | Multi-centre weighted combination | Absent | R²=0.920 |
| ESA | Proprietary (independent Kalman filter) | Absent | R²=0.950 |

The consistency across software-diverse processing chains (R² = 0.920-0.970,
Paper 1; shared tracking network) supports that datum projection and common-mode suppression are
reproducible features of GNSS processing methodology, not a post-hoc
rationalization. If the "datum-projection suppresses mass signals" were merely
an excuse, different software implementations would show different results.
Cross-centre reproducibility shows that this suppression is not specific to
one software implementation; it does not by itself establish the physical
transfer mechanism.

Datum projection and common-mode suppression are reproducible features of
GNSS processing methodology; the specific transmission of a TEP covariance
signal remains a falsifiable transfer hypothesis to be tested by synthetic
signal injection.

**Predicted Behavior in Processed vs Raw Data:**

| Observable | Processed Data | Raw Carrier Phase | Status |
| --- | --- | --- | --- |
| GM/r² scaling | Absent (filtered) | May be recoverable in raw-link observables | Observed |
| Phase coherence structure | Preserved | Should persist | Observed |
| Orbital modulation | Detected (r=-0.888) | Should persist | Observed |
| Event detection rates | No mass scaling | Test pending | ? Unknown |

**Mechanism of Transmission (Why Signal Survives):**

The survival of these correlation structures has a concrete candidate mechanistic explanation in
the specific constraints applied in the Bernese GNSS Software (Dach et al.,
2015) used by CODE. The estimation process minimizes residuals using the
Undifferenced Ionosphere-Free Linear Combination (L3)—the
convention for all IGS precise clock products (Kouba, 2015). To resolve the
inherent rank deficiency in undifferenced processing, a
datum constraint must be applied: either a single reference
clock is held fixed, or equivalently a
zero-sum condition is enforced (∑ satellite clock
corrections = 0 at each epoch) (Ray et al., 2017). Both approaches force any
network-wide common-mode signal (such as a global gravitational potential
shift) into the reference timescale, effectively removing it from the
estimated clock parameters. However, differential signals—specifically the
*phase relationships* between receivers relative to this constrained
reference—are not targeted by this condition. Thus, the processing acts as a
specific filter: it is opaque to scalar potential shifts (energetic
coupling) but transparent to vector gradients (geometric coupling). The
detection of 56 significant events (R²=0.92-0.97 consistency across centers)
represents the successful transmission of this geometric residual.

**Critical Technical Confirmation:** The Bernese GNSS Software
documentation explicitly states that "zero-difference network solution
solving for all receiver and satellite clock parameters" is
"equivalent to a consistent double difference solution"
(Dach et al., 2015). Double-differencing is the standard technique that
*mathematically eliminates* common-mode signals (including receiver
clock biases and network-wide offsets) while preserving differential
observables. This equivalence confirms that CODE's undifferenced processing
inherently removes common-mode structure—exactly as the datum-projection
mechanism predicts.

**Quantitative Evidence for Differential Survival:**
Independent CME studies demonstrate this selectivity empirically. Li et al.
(2024) report that after common-mode filtering, interstation correlation
coefficients decrease from 0.50–0.71 to 0.29–0.38—a ~50% reduction.
Crucially, *significant residual correlation persists* (R ≈ 0.3).
This quantifies precisely what the argument predicts: common-mode signals
are suppressed, but differential structure (10-40% of original correlation
magnitude) survives the processing chain. The TEP signatures detected in this analysis
occupy this "transmitted" differential channel.

Addressing the "Convenient Shield" Critique: Is the Processing-Transfer
Hypothesis Falsifiable?

A skeptical reviewer might argue: "The datum-projection mechanism is
unfalsifiable—if mass scaling were present, you'd claim victory; since it's
absent, you claim the transfer removed it."

**This critique fails on empirical grounds:**

**1. Multi-Centre Validation (Paper 1):** Three
software-diverse analysis centres over substantially the same tracking
network (CODE: Bernese GNSS
Software; IGS: weighted multi-centre combination; ESA: proprietary software)
all showed:

• *Identical* absence of GM/r² scaling in planetary
events

• *Consistent* correlation patterns (R² =
0.920-0.970)

• *Same* lack of mass-dependent amplitude
hierarchy

If the "datum-projection suppresses mass signals" were a post-hoc excuse, this study
would expect different software implementations to show different results.
Instead, the consistency across software-diverse processing chains
*supports* that datum-projection of common-mode signals is a real, reproducible
feature of GNSS processing methodology.

**2. Falsifiable Prediction:** The datum-projection mechanism
makes a clear, testable prediction:

• **Processed data:** No GM/r² ordering of
event amplitudes (common-mode components absorbed by datum estimation) - Observed;
a weak continuous M/r² correlation persists in the daily series
(r = −0.099, p = 0.032)

• **Raw carrier-phase data:** Mass-dependent terms
are separately recoverable only in an explicitly referenced raw-link analysis;
not the claimed conformal covariance observable

This is explicitly acknowledged as a critical next step (§5.1). The proper test
of the datum-projection mechanism is whether injected common-mode and differential
signals pass through the processing pipeline as predicted—not merely whether raw
data lacks GM/r² scaling. This is the opposite of unfalsifiable—it's a concrete,
testable hypothesis with clear success/failure criteria.

**Conclusion:** The datum-projection mechanism is not a
"convenient shield"—datum projection and common-mode suppression are
established features of GNSS processing methodology (confirmed across
software-diverse products from three processing centres). Their specific
mapping of a TEP covariance signal into the observed residual structure
remains a falsifiable transfer hypothesis requiring synthetic-injection
validation. The consistency across processing centres strengthens, rather
than weakens, the case.

**Future Validation:** Simulation studies injecting known
synthetic signals into the GNSS processing pipeline would provide
independent verification of the claimed transfer behaviour. Such simulations,
while beyond the scope of this empirical study, represent an important
methodological validation step.

**Supporting Evidence from GNSS Literature:** The GNSS
community has independently documented the exact phenomena detected in this analysis—without
recognizing their physical significance. Li et al. (2024) note that "as the
distance increases, the impact of CME between stations gradually decreases"
with correlation diminishing "beyond approximately 2000 km"—qualitatively
consistent with the distance-dependent decay measured in this analysis (azimuth-averaged decay scale λ ≈ 3,210 km, cross-sector mean 4,201 km). Crucially, they
acknowledge "there is no consensus on the physical origin of CME."
Similarly, Chanard et al. (2020) report that surface loading models explain
only 30-50% of seasonal GNSS signals, with the remainder "only partially
understood." These documented but unexplained anomalies align qualitatively with
the present findings (see §4.1.7 for detailed correspondence).

#### 4.1.4 Geophysical Couplings: Earth's Own Rhythms

This section shows that the GNSS network is also coupled to Earth's own
rotational dynamics, providing a link between external gravitational
influences and internal geophysical processes.

**Nutation Signatures:**

**Semiannual Coherence Component (nutation/orbital harmonic):** R² = 0.904 (permutation p = 0.00027, F-test p = 2.7×10⁻⁵). A strong 182.6-day coherence component whose fitted phase (φ₀ = −13.6°) peaks on January 7 and July 8, aligning within 2–4 days of Earth's orbital apsidal line (perihelion January 3–5, aphelion July 4–6), while placing the coherence minimum at the equinoxes, in anti-phase to the equinoctial ionospheric anomaly. The phase is compatible with both an orbital second harmonic and the semiannual nutation term; phase alone does not discriminate against an equinox-linked coherence-degradation mechanism, and the ionospheric channel is bounded instead by the local-night and F10.7/Kp null controls.

**Main Nutation (18.6 years):** R² = 0.641 ± 0.10. A
preliminary nutation-aligned component is observed over 1.36 cycles —
each phase bin carries at most two independent cycle realizations, so
amplitude and phase are single-cycle-dominated — with the detection
priced against the phase-shuffled permutation null (p = 0.0016; the
point estimate exceeds the 99th null percentile), providing candidate
evidence for sensitivity to multi-decadal
astronomical structure pending full red-noise surrogate replication.

**Chandler Wobble (14-month polar motion):**

R² = 0.096. At the prescribed 433-day Chandler period, the network
exhibits a weak but phase-coherent response across approximately 21
cycles (phase stability = 0.72). Its amplitude falls below the primary
significance threshold (R² > 0.15), so it is treated as a
Chandler-consistent secondary signature rather than an independent
detection. The reduction from previous estimates reflects rigorous
period enforcement, favoring physical accuracy over overfitting.

Physical Interpretation: The GNSS timing field appears sensitive to Earth's
geophysical state. The observations indicate responses to both external
gravitational influences (planets, Sun, Moon) and internal dynamics
(rotation, precession, polar motion). This combination is not reproduced by
simple atmospheric or instrumental explanations considered here.

#### 4.1.5 Network-Wide & Sustained Dynamics

These analyses indicate a global, persistent component; local or
transient-only explanations do not account for all signatures evaluated.

**Network Mesh Coherence:** Score = 0.582, indicating
coordinated behavior across global stations.

**Smoothing window used:** 240 days, suggesting the effect
operates on seasonal timescales.

**Physical Interpretation**: The influence is not limited to
brief moments of planetary alignments (oppositions/conjunctions) but is a
sustained, continuous effect. The gravitational configuration of the solar
system exerts an ongoing modulation on the GNSS timing field, with discrete
alignments producing transient peaks superimposed on this baseline.

#### 4.1.6 Convergence and Physical Interpretation

The convergence of multiple signatures on systematic patterns is
summarized, with the data stream underlying each analysis made
explicit. The phenomenology is carried by two base observables: the
directional EW/NS anisotropy-ratio series (34 overlapping 30-day
windows) and the daily network-coherence series. Signatures derived
from the same observable are characterizations of one dataset, not
independent evidence streams:

**Spatial Anisotropy** establishes a stable geometric
structure persisting over decades (anisotropy-ratio observable,
§3.1)

**Orbital Motion** shows this structure varies with
Earth's heliocentric dynamics (annual modulation of the same
anisotropy-ratio series, §3.2.1)

**Directional Search** identifies a preferred
directional template at Earth's aphelion-velocity tangent (rejecting both
the CMB-dipole and Solar-Apex templates), characterizing the annual
phase of the same modulated
anisotropy series (§3.2.6), as is the hemisphere stratification
control (§3.2.4). The present fixed-magnitude directional scan does not measure the
relevant background-speed amplitude (§2.3.5)

**Planetary Events** characterize the catalogue of
event-window responses to transient
gravitational configurations (a second observable: event-window
modulation depth on the daily-coherence series, §3.3 — estimator
repeatability, with event-locked specificity validated only for the
40-event opposition subset)

**Nutation/Chandler Wobble** show coupling to Earth's
rotational dynamics (daily-coherence series at prescribed
geophysical periods, §3.4)

**Network Coherence** shows coordinated behavior at global
scales (daily-coherence series, network-covariance statistic,
§3.5.1)

**Continuous Correlation** reveals sustained gravitational
influences (daily-coherence series, planetary-configuration
predictor, §3.5.2)

Taken together, these seven convergent findings reduce to two base
observables — the anisotropy-ratio modulation (items 1–3) and the
daily network-coherence series (items 4–7, each applying a distinct
physical predictor or prescribed period) — plus the genuinely
external replication supplied by Paper 1's multi-centre products and
the MGEX analysis. The evidence structure is therefore one strongly
modulated directional observable with several geometric
characterizations, a second independent coherence observable tested
against four distinct geophysical and planetary predictors, and
cross-dataset replication — consistent with
systematic coupling effects in GNSS timing networks. The directional
result (§4.1.2.1) is particularly significant: it transforms the
phenomenology from "velocity-dependent modulation" to "phase-locked
orbital-directional covariance," providing strong directional discrimination —
the Solar-Apex template is rejected at ~16,000× variance ratio and the
CMB-dipole template is anti-phase — while identifying the recovered
direction with the heliocentric-orbital phase (multi-resolution verified).
The convergence of spatial, temporal, geophysical,
and directional signatures—each surviving rigorous statistical
validation—creates a phenomenological framework that warrants theoretical
interpretation. Several conventional explanations fail to reproduce this
multi-signature convergence, though further replication and raw-data
analysis remain essential for mechanistic understanding.

#### 4.1.7 Reinterpreting Known GNSS Anomalies: TEP as Unifying Framework

A remarkable feature of this analysis is that every major finding
corresponds to a documented but unexplained anomaly in the GNSS literature.
The geodetic community has spent decades cataloging these systematic
effects—filtering them as "errors" because no physical mechanism was
available to interpret them. TEP provides the first theoretical framework
that unifies these disparate anomalies under a single physical mechanism.

| Documented GNSS Anomaly | Literature Status | TEP Finding |
| --- | --- | --- |
| **Distance-dependent CME correlation** | "Correlation diminishes beyond ~2000 km" (Li et al. 2024);
*"no consensus on physical origin"* | λ ≈ 3,210 ± 389 km (azimuth-averaged exponential R² = 0.945; Gaussian R² = 0.965); sector dispersion 4,201 ± 1,967 km |
| **Unexplained seasonal signals** | "Surface loading models explain only 30-50%" (Chanard et al.
2020); *"remain only partially understood"* | Orbital velocity r = -0.888 (annual); Semiannual coherence
component R² = 0.904 |
| **E-W vs N-S directional asymmetry** | "Significant variation patterns in N and E directions" (Li
et al. 2024) | EW:NS anisotropy = 2.16:1, stable over 25 years |
| **Spurious draconitic/orbital signals** | "Spurious periodic signals at harmonics of GPS draconitic
year" (Ray et al. 2007, cited in Chanard 2020) | Strong orbital coupling (r = -0.888, surrogate p $\lt$ 2×10⁻⁷
Monte Carlo validated) |
| **Unexplained clock frequency offset** | "Robust difference (58 ± 22 s) between GPS periodic
frequency and orbital period—reason remains unexplained" (Senior et al., Cheng et al. 2023) | Velocity-dependent time effects (TEP core prediction) |

**Key Insight: From "Noise" to Signal**

The GNSS processing methodology was designed to produce clean navigation
products, not to preserve subtle physical signals. Without a theoretical
framework to interpret distance-dependent correlations, seasonal
modulations, or directional anisotropies, these effects were classified as
"Common Mode Error" (CME) and filtered out.

**What the literature says:**

• "There is no consensus on the physical origin of CME" (Li et
al. 2024)

• Seasonal signals "remain only partially understood" (Chanard
et al. 2020)

• Clock frequency discrepancy "remains unexplained" (Cheng et
al. 2023)

**What TEP provides:** A unified physical
mechanism—velocity-dependent spacetime-geometry modulation—that predicts
distance-structured correlations, orbital coupling, directional anisotropy,
and sensitivity to gravitational configurations. Every documented anomaly
becomes an expected feature rather than unexplained noise.

**Implications:** If TEP is correct, decades of GNSS data
contain systematic temporal-gravitational signatures that have been
inadvertently filtered as "errors." The raw carrier-phase analysis proposed
in §5.1 would not only test the datum-projection mechanism but
potentially recover the full amplitude of these effects before common-mode
suppression. This represents a paradigm shift: the "noise" that geodesists
have spent years removing may itself be the signal of a new physical
phenomenon.

### 4.2 Continuity with the Multi-Centre Study

**Amplitude-scaling note (see §2.1.3).** Processed GNSS clock
products eliminate common-mode frequency shifts; consequently any
mass-proportional *GM ⁄ r²* signature is largely removed.
The phase-coherent, differential metrics employed here respond to differential
clock-sector covariance $C_A$ — a fluctuation-transfer channel distinct from
the static conformal-depth factor $S_A^{(\oplus)} \approx 0.65$ — not to absolute potential.
The datum-projection filter and pairwise screening decouple the
observed geometric amplitudes from simple
*GM ⁄ r²* expectations. Future raw-carrier analyses will
be required to test mass-scaling directly.

A critical validation of long-baseline recovery requires comparing this 25-year
CODE analysis to the multi-centre study. The multi-centre study established
comprehensive validation through 388 statistical tests across 19 families,
extensive null testing (temporal shuffle, spatial shuffle, phase
randomization), cross-centre validation (R² = 0.920-0.970), and rigorous
controls for ionospheric conditions, solar activity, and instrumental
effects. This study builds on that validated foundation by extending the
temporal baseline to access long-period phenomena. This section documents
claim-by-claim status:

**Replicated (Confirms Temporal Stability)**:

**Spatial Anisotropy**: EW > NS structure confirmed (Paper 1 typical λEW/λNS ≈ 0.55–2.03, peaks up to ~4.75, discussion quoting ~2.75×; this study EW/NS = 2.16, strength = 1.981 ± 0.23)

**Orbital Velocity Correlation**: Multi-centre study r = −0.571 to −0.793 across centres; long-span r = −0.888 (see §3.2 for numeric values)

**Planetary Event Responses**: Multi-centre study found 6/8
Bonferroni-significant; this study extends to 56/156 events ≥2σ under
the free-centre pulse statistic, of
which 19 remain family-wide Bonferroni-significant.
Matched-null control (§3.3.1): for the 40-event outer-planet
opposition subset under the ±15-day model-free window, planetary
events show 5.5× larger effect sizes than 155 random dates (p $\lt$ 10⁻¹⁷,
Cohen's d = 1.41). Catalogue-wide event-locked inference requires a
season-preserving null; the free-centre fits and date-shuffled nulls
bound rather than prove specificity for the full 156 events
(see §3.3, Paper 3 Step 2.6b).

**Strengthened (Higher Statistics)**:

**Event Statistics**: Expanded from 8 predeclared events to
156-event comprehensive survey (catalogue counts descriptive — §3.3)

**Orbital Coupling**: 25.3-year baseline vs 2.5
strengthens velocity correlation inference

**New (Long-Period Access)**:

**18.6-Year Nutation**: Preliminary — R² = 0.641 ± 0.10, permutation p = 0.0016 (1.36 cycles; cycle-limited amplitude; multi-centre
study: inaccessible)

**Long-baseline Aggregate Phase Coherence**: Signatures recovered consistently in the pooled 25.3-year day-of-year aggregate across 2+ solar
cycles (era-block decomposition pending)

**Replicated (Confirms Temporal Stability)**:

**Network Covariance Score (Collective Network Coherence)**: Index = 0.582,
consistent with multi-centre range (CODE: 0.624, IGS: 0.579, ESA: 0.602)

**Cross-Centre Validation Status**: The multi-centre study's
critical contribution was cross-centre validation (R² = 0.920-0.970 between
CODE, IGS, ESA), demonstrating the phenomenon is reproducible across the
processed products examined so far over 2.5 years. This study extends temporal
coverage but cannot confirm whether long-baseline signatures (18.6-year
nutation) are reproducible across other processing centres until IGS and ESA
archives extend backward. Cross-centre agreement establishes robustness across
processed-product pipelines, but does not alone determine the physical origin
because the centres share observations, conventions and some modeling assumptions.

### 4.3 Physical Interpretation

Within this picture, a distributed clock network such as GNSS is effectively
a detector of gradients and modulations in the temporal field. TEP broadly
predicts (i) direction‑dependent correlation structures aligned with the
system’s motion; (ii) velocity‑dependent modulation of correlation lengths;
and (iii) sensitivity to changing gravitational configurations. The present
work is a TEP‑motivated empirical investigation of whether such signatures
are present in long‑baseline GNSS timing data.

#### 4.3.1 Consistency with TEP Predictions

The observations align closely with Temporal Equivalence Principle
predictions across multiple convergent signatures:

**Predicted:** Correlation length modulation with orbital velocity and ecliptic apex declination

**Observed:** λ varies with orbital velocity (r = -0.888;
surrogate p $\lt$ 2×10⁻⁷; t-test p ≈ 2.6×10⁻⁴, autocorr-corrected)

Assessment: Confirmed at p $\lt$ 2×10⁻⁷; validated on raw RINEX in Paper 3

**Predicted:** Non-linear gravitational coupling in
correlation structure (not classical GM/r² in absolute frequencies)

**Observed:** 56 significant planetary events (≥2σ) with
coherence modulations 0.3–4% (median ≈1.1%, in coherence units not Δf/f)
and no correlation with GM/r² scaling

Assessment: Consistent with TEP expectations when classical
gravitational signatures are removed by processing but phase-coherent
correlation structure survives

**Predicted:** Geometric anisotropy aligned with motion:
spherical-horizon projection across the 474-station network ($\langle|\phi|\rangle = 36.88^\circ$, $\varepsilon = 23.44^\circ$) predicts $\mathcal{R}_{\rm EW/NS} = 2.13$ (mean lat) and $2.02$ (ensemble)

**Observed:** E-W elongation (2.16:1 ratio, 1.5% match; strength 1.981 ± 0.23, 0.19σ match) consistent with
Earth's orbital plane

Assessment: Quantitatively confirmed (1.5% / 0.19σ agreement, zero free parameters)

**Predicted:** Global field-like behavior

**Observed:** Network coherence score 0.582 across 102 non-overlapping 90-day windows (104 pipeline bins)

Assessment: Consistent with TEP expectations

**Predicted:** Multi-scale temporal coupling

**Observed:** Signatures from hours (events) to decades
(nutation)

Assessment: Consistent with TEP expectations

#### 4.3.1.1 Order-of-Magnitude Considerations

The 19% geometric modulation observed in EW/NS ratios is a network-level
anisotropy ratio rather than a fractional clock-frequency amplitude. A naive
v²/c² estimate yields ~10⁻⁸ for orbital speeds, but this quantity is not
directly comparable to the observed anisotropy ratio because the two are
different observables: one is a microscopic energetic amplitude, the other
is a macroscopic covariance-shape parameter projected through the GNSS
measurement and processing chain. The magnitude and sign of the GNSS
transfer function remain to be calibrated with raw/RINEX data and
end-to-end signal injection. For the detailed architecture, see §2.1.3.2.

**Context:** A naive kinematic benchmark at orbital velocity is
v²/c² ≈ 10⁻⁸. The observed anisotropy ratio (EW/NS = 2.16; strength = 1.981 ± 0.23)
and ~19% seasonal modulation are dimensionless geometric covariance-shape
parameters governed by the spherical-horizon projection of the ecliptic
velocity vector across the 474-station network ($\langle|\phi_{\rm lat}|\rangle = 36.88^\circ$,
$\varepsilon = 23.44^\circ$; `step_2_11_horizon_anisotropy_derivation.json`),
which yields $\mathcal{R}_{\rm EW/NS} = 2.13$ at the network mean latitude
($1.5\%$ agreement) and $2.02$ across the station ensemble ($0.19\sigma$
agreement), while raw RINEX carrier-phase and synthetic end-to-end transfer
validation are completed in Paper 3 (TEP-GNSS-RINEX).

The Interferometer Analogy: Moiré Patterns in Spacetime

The large observed modulation (19% change in anisotropy shape) arising from
a tiny kinematic input ($\Delta v/c \approx 10^{-4}$) is physically
intuitive as a geometric analogy, though not a quantitative derivation:

**Interferometer, not Gravimeter:** A gravimeter measures
*potential energy* (amplitude), which requires massive energy
transfer. An interferometer measures
*phase relationships* (information), which can be extremely sensitive
to geometry.

**The Moiré Effect:** Consider two stiff mesh grids overlaid. A
microscopic rotation of one grid produces macroscopic "Moiré fringes" that
shift dramatically across the entire surface. The energy required to rotate
the grid is negligible, but the structural consequence (the fringe shift) is
huge.

By analogy, the GNSS correlation field may behave like such a "stiff" geometric structure.
The 19% modulation would then be a Moiré-like
fringe shift in the correlation pattern driven by the subtle kinematic
rotation of Earth's orbital velocity vector. This analogy provides physical
intuition but does not constitute a quantitative resolution of the
$10^{-7}$-to-19% gap; that requires end-to-end signal injection through the
actual processing chain.

**Resolution: Information Resonance vs. Force Coupling**

This distinction provides a possible qualitative interpretation of the
apparent conflict between the "stiff" response
(19% modulation) and the "weak" coupling (no mass scaling):

**Amplitude = Energy (Suppressed):** Simple energetic
couplings ($GM/r^2$) are consistent with absorption by datum estimation
(see §2.1.3); their absence in differential covariance is expected
under a priori relativistic modelling.

**Phase = Information (Transmitted):** The network detects
the *information content* of the gravitational
configuration—specifically, the synchronization geometry.

This could account for why the observed response does not follow a simple
planetary-mass hierarchy. The network may not be weighing the planets (force
coupling); it may be detecting the coherence resonance of their alignment
events (information resonance).

#### Open Question: Quantitative Geometric Coupling Theory

While the datum-projection mechanism is consistent with the *apparent* discrepancy
by distinguishing energetic from geometric observables, a complete
theoretical framework should derive the expected magnitude of geometric
coupling (correlation ratio modulation) from first principles. The
current framework predicts:

**Qualitative:** Geometric coupling should modulate
with velocity ✓ (confirmed: r = -0.888)

**Qualitative:** Anisotropy should align with motion ✓
(confirmed: EW > NS, orbital-phase direction in ecliptic)

**Quantitative:** Magnitude of ratio modulation →
requires extended theory

Raw carrier-phase analysis (§5.1) will provide critical constraints: if
the 19% modulation persists in minimally processed raw data, it reflects genuine
physical coupling; if it diminishes, the processing chain may contribute
to the observed amplitude through differential transmission effects.
Either outcome advances theoretical understanding.

These mechanisms remain hypotheses requiring validation through raw
carrier-phase analysis and multi-constellation comparison. The phase
relationships (orbital coupling, the orbital-phase direction, planetary events) are
reproducible across the processed products examined so far; their raw-data origin
remains to be established, and the amplitude scaling remains
an open question addressed in §5.1-5.2.

#### 4.3.1.2 Resolution of Gravitational Scaling Anomaly (Step 2.3 Analysis)

Post-hoc physical modeling (Step 2.3) confirmed the absence of classical
gravitational scaling. Analysis of 156 planetary events revealed no
significant correlation between observed modulation amplitudes and tidal
parameters (M/r³) or gravitational field strength (M/r²):

- **Amplitude vs Tidal (M/r³):** r = -0.087 (p = 0.522)

- **Amplitude vs Gravity (M/r²):** r = -0.042 (p = 0.759)

This empirical result confirms that the coupling is not a simple linear
tidal effect. The phenomenon exhibits a "saturation" or "resonance" behavior
where the detection depends on alignment geometry (opposition/conjunction)
rather than the magnitude of the gravitational potential. This distinguishes
TEP-type coupling from conventional Newtonian or GR tidal forces.

**Consistent with Processing-Transfer Hypothesis (§2.1.3):** The absence of
mass scaling is consistent with the hypothesised high-pass transfer nature of GNSS
processing described in §2.1.3. If the energetic component (proportional
to $GM/r^2$) is suppressed by least-squares adjustment, the surviving signal
would be phase-coherence geometry, which depends on alignment rather than
mass. This interpretation remains a hypothesis pending synthetic signal injection
through the actual processing chain.

#### Theoretical Interpretation: The Network as a Covariance-Stress Detector

Standard GNSS processing imposes a global synchronization solution on a
spatially structured temporal-covariance field. The residual covariance
can therefore probe open-path conformal clock-sector covariance $C_A$. It does not by
itself establish synchronization non-integrability; residual closed-loop
holonomy remains a separate B/non-exact-sector observable. In the purely
conformal subclass of TEP, the synchronization holonomy vanishes
($\oint d\ln A = 0$ for a smooth field); nonzero holonomy at leading
order requires the disformal sector $B \neq 0$ (Smawfield, 2025). The
GNSS covariance channel measures the A-sector (conformal) response, not
the B-sector (disformal) holonomy that would manifest in closed-loop
triangle tests.

**Processed Data as a Covariance Observable:**

**Raw Carrier Phase (Local):** Dominated by classical
GR (~10⁻¹⁰) and atmospheric delays. Local pointlike physics retains
the GR/Lorentz limit and locally measured c remains invariant.
Nevertheless, raw multi-station and open-path observables may retain
conformal clock-sector covariance. Raw analysis therefore provides
a direct test of the covariance before analysis-centre transfer.

**Processed Residuals (Global):** By removing local
effects (where TEP=GR) and enforcing network closure, processing
acts as a band-pass channel for the global covariance structure
(where the conformal clock-rate field produces distance-structured
correlations).

In this view, the processed network solution is not merely a data
source, but the interferometer itself, with the residuals representing
the clock-sector covariance $C_A$ projected through
the GNSS measurement channel. This is consistent with the TEP
prediction of distance-dependent correlations (Paper 0, §10.E) using precisely the
phase-coherent network analysis specified by the theory. The disformal
holonomy prediction (Paper 0, §10.A–B) requires dedicated closed-loop experiments
and is not tested by this open-path covariance channel.

#### 4.3.2 Ruling Out Conventional Effects

**Atmospheric/Ionospheric:**

Would produce distance-dependent effects: not observed (anisotropy is
scale-invariant)

Would correlate with solar activity: not observed (weak correlation r =
0.12-0.13, p > 0.29, below contamination threshold)

Would show diurnal patterns: not observed (effects persist across all
local times)

**Instrumental/Systematic:**

Would affect individual stations: not observed (signal requires pair
correlations)

Would be constant over time: not observed (clear annual and
longer-period modulations)

Would correlate with equipment changes: not observed (consistent across
receiver upgrades)

**Tidal/Loading:**

Would follow lunar/solar periods exactly: not observed (phase lags
observed)

Would scale with mass directly: not observed (no correlation with GM/r²
predictions)

Would affect position more than timing: not observed (timing
correlations dominant)

#### 4.3.3 Selectivity of Detections: Theory-Consistent Pattern

The dataset shows a selective detection pattern that is highly informative
about the underlying mechanism. The system does not respond to every
periodic signal, but specifically to those with
*heliocentric gravitational significance*:

**Detected (Heliocentric/Gravitational):**

**Orbital Velocity:** Strongest signal (r=-0.888),
tracking Earth's motion around the Sun.

**Planetary Alignments:** 56 significant events
across all planets (Mercury to Saturn).

**Nutation:** Coupling to Earth's inertial
orientation shifts (18.6-year, semiannual).

**Not Detected (Surface/Local):**

**Solar Rotation (27-day):** No signal. Rules out
solar wind or magnetic coupling, which rotate with the Sun.

**Lunar Standstill:** No signal. Rules out simple
tidal declination geometry.

**The Lunar Nuance: Cycle vs. Event**

The distinction between the detected 18.6-year Nutation Cycle (R²=0.641) and
the null 2025 Standstill Event is scientifically instructive.

**Detected (Nutation):** The system responds to the
*continuous dynamic* of Earth's axis wobbling over 18.6
years.

**Null (Standstill):** The system does
*not* show a sharp pulse merely because the Moon reaches a
coordinate extreme (maximum declination).

This suggests the coupling is driven by inertial dynamics (the wobble
itself) rather than static geometry (where the moon is in the sky). This
dynamic selectivity strengthens the physical case against simple atmospheric
artifacts.

**Interpretation**: This selectivity is
*theory-consistent*. The system ignores surface phenomena (solar
rotation) and pure coordinate geometry (lunar standstill) but couples
strongly to orbital dynamics and inertial orientation. Within TEP, coupling
arises from the spacetime/gravitational configuration that co-varies with
orbital dynamics, not from surface features. This specificity argues
strongly against generic "noise" or "systematic error" explanations, which
would be unlikely to discriminate between solar rotation and orbital motion.

##### Summary: Inertial Interferometer, Not Gravimeter

The pattern of detections and non-detections characterizes what this
network is and is not:

**Amplitude-blind at event level, weakly coupled at baseline level:**
No GM/r² ordering across planetary events (Mercury ≈ Jupiter
response rates), while a weak continuous M/r² correlation persists
in the daily series (r = −0.099, p = 0.032)

**Unresponsive to solar surface phenomena:** 27-day
rotation null

**Indifferent to static geometry:** Lunar standstill
null

**Exquisitely sensitive to kinematic dynamics:**
Orbital velocity (r=-0.888), nutation (R²=0.904), orbital-phase
direction (r=0.746)

This selectivity profile defines the network as an inertial
interferometer—measuring velocity-dependent correlation geometry—not a
gravimeter measuring force. The directional result (~16,000× variance
ratio over the Solar-Apex template) identifies a preferred covariance direction
in the ecliptic plane: the aphelion-velocity tangent — the phase of the
orbit-coupled modulation, not the CMB dipole direction (162° away,
anti-phase). The scan does not fix the background-speed magnitude
and therefore measures the annual-phase direction rather than a
physical speed scale.

#### 4.3.3.1 Falsification Scorecard: Theory vs. Alternative Hypotheses

The strength of this work lies not merely in finding patterns consistent
with TEP, but in actively testing alternative explanations where different
hypotheses predict different outcomes. Table 4.3.3.1 summarizes seven
independent tests where TEP expectations and competing explanations make
distinct, testable predictions. While this does not rule out all possible
alternatives (e.g., subtle processing effects specific to CODE's 25-year
pipeline evolution), it demonstrates that several plausible artifact
mechanisms are inconsistent with the data:

| Falsification Test | TEP Prediction | Alternative Hypothesis | Result | Assessment |
| --- | --- | --- | --- | --- |
| **Orbital velocity correlation** | Strong (r > 0.85) | Weak/absent (random) | r = -0.888, surrogate p $\lt$ 2×10⁻⁷ | TEP-compatible; random fluctuation disfavoured |
| **Mass scaling (GM/r²)** | Absent (absorbed by datum-projection) | Present (classical gravity) | No correlation (r=-0.042, p=0.76) | TEP-compatible; classical scaling disfavoured |
| **Solar rotation coupling** | Absent (surface phenomenon) | Present (solar wind/magnetic) | Null result (p > 0.05) | TEP-compatible; solar wind hypothesis disfavoured |
| **Hemisphere asymmetry** | None (heliocentric kinematic) | Strong anti-phase (seasonal) | Identical perihelion peak (Paper 1) | TEP-compatible; seasonal hypothesis disfavoured |
| **Space weather dependence** | Weak (background effect) | Strong (ionospheric artifact) | No F10.7/Kp correlation (Paper 1) | TEP-compatible; ionospheric hypothesis disfavoured |
| **Processing-centre consistency** | High (physical signal) | Low (software-specific artifact) | R² = 0.92-0.97 (software-diverse products within the shared IGS ecosystem) | TEP-compatible; simple artifact hypothesis disfavoured |
| **Anisotropy long-baseline consistency** | Stable over decades | Degrading (transient artifact) | Stable 25 years (EW:NS = 2.16) | TEP-compatible; transient hypothesis disfavoured |

**Table 4.3.3.1:** Seven independent falsification tests
comparing TEP expectations against alternative hypotheses (processing
artifact, environmental confounding, random fluctuation). In all cases,
observations are consistent with TEP expectations while specific alternative
hypotheses are disfavoured. Each test represents a distinct opportunity for
the TEP framework to fail; the consistent pattern across orthogonal tests
strengthens the case that observed signatures reflect systematic physical
coupling rather than the tested artifact mechanisms or noise. Note: This
does not rule out all possible alternative explanations (e.g., subtle
CODE-specific effects or unknown systematics), but demonstrates that several
plausible, testable alternatives are inconsistent with the data.

#### 4.3.4 Null-Model Systematics: Explicit tests that fail

To address the possibility that the observed signatures arise from artefacts
of network geometry or simple temporal structure, explicit null
models were evaluated and their qualitative predictions compared against the data. Each
model fails to reproduce the core triad of observations: (i) persistent E–W
> N–S anisotropy (EW:NS = 2.16), (ii) strong orbital-velocity coupling (r =
-0.888 with perihelion/aphelion phasing), and (iii) identical calendar
phasing in both hemispheres (Paper 1, §3.4).

**Table 4.3.4: Null Model Predictions vs. Observations**

| Null Model | Predicted Pattern | Observed Reality | Match? |
| --- | --- | --- | --- |
| **Station density bias** | Static or slowly varying λ | Annual λ modulation (19% amplitude) tracking orbital
velocity (r=-0.888) | No |
| **Hemisphere-specific effects** | Opposite Jan/Jul phasing (NH vs SH) | Identical perihelion peak both hemispheres | No |
| **Local seasonal (temperature)** | NH/SH anti-phased by 6 months | Same calendar phase (Paper 1 §3.4) | No |
| **Ionospheric (F10.7/Kp)** | Correlation with solar indices | No F10.7 correlation (Paper 1) | No |
| **Processing batch effects** | Step changes at algorithm updates | Smooth continuous evolution | No |
| **Random temporal structure** | ~5% false positives under an independent-points null (α=0.05) — invalid on the autocorrelated seasonal series; season-preserving null ~53–66% (Paper 3, Step 2.6b) | 35.9% catalogue detection rate (estimator repeatability);
40-event opposition subset validated at 5.5× over random
dates (p $\lt$ 10⁻¹⁷) | Bounded / descriptive (opposition subset distinct) |
| **Isotropic noise** | Equal λ all directions | 3.3× range (West/North) | No |

**Conclusion:** All tested conventional models fail to
reproduce the joint constraints of: (1) orbital velocity phase locking, (2)
hemisphere identity, (3) persistent anisotropy, and (4) elevated planetary
event detection rates. While this does not definitively exclude unknown
systematic effects, it substantially narrows the space of plausible
alternative explanations.

**Station-layout geometry (pair-density anisotropy):**

*Assumption:* Uneven station distribution by azimuth
induces apparent anisotropy (e.g., more stations in US/Europe
create EW bias).

*Outcome:* Can bias absolute λ estimates, but does not
impose a coherent annual phase tied to perihelion/aphelion. A
static geometry bias cannot "breathe" with a 19% annual
modulation.

*Mismatch:* Fails to generate r = -0.888 with orbital
velocity or same-phase hemispheres; density effects are static
or slowly varying, not heliocentric.

**Empirical Rebuttal (Step 2.3):** Post-hoc
normalization of correlation lengths by station pair density
reveals that the EW > NS anisotropy persists and strengthens
(Normalized Ratio ≈ 2.40) after accounting for sampling bias.
This confirms the directional structure is not an artifact of
receiver distribution.

**Global common-mode/whitening bias:**

*Assumption:* A network-wide filter or common-mode
removal introduces artificial correlations.

*Outcome:* Produces isotropic or filter-axis-aligned
effects without the observed orbital-phase locking.

*Mismatch:* Cannot produce the specific
perihelion/aphelion phasing with identical calendar timing
across hemispheres; any local-seasonal proxy would flip phase
between hemispheres, which is not observed.

Seasonal effects (ionospheric TEC, temperature, network
participation):

*Assumption:* Annual variations in ionospheric Total
Electron Content (TEC), atmospheric conditions, or network
participation drive the observed modulation.

*Limitation:* Global ionospheric TEC does peak in January
(both hemispheres) due to Sun-Earth distance, making hemisphere
stratification insufficient to rule out this confound.

*Primary Discriminants:*

**Directional Specificity:** Ionospheric
effects do not imprint a celestial direction locked to
the orbital-velocity phase (best fit ~6° from the
aphelion tangent). The ~16,000× variance ratio
over the Solar Apex template provides geometric
discrimination.

**Ionospheric Null Controls:** Weak
correlation with F10.7 solar flux and Kp geomagnetic
indices (r = 0.12-0.13, p > 0.29, Paper 1 validation),
well below ionospheric contamination thresholds.

**Multi-Centre Consistency:** Three
software-diverse processing centres on the shared
tracking network, using different
ionospheric models, show consistent correlation patterns
(R² = 0.920-0.970).

Space Weather / Solar Cycle (Ionospheric Scintillation):

*Assumption:* High solar activity (F10.7 max) increases
ionospheric noise, creating spurious decorrelation patterns.

*Outcome:* Anisotropy should disappear or randomize
during solar maximum or geomagnetic storms.

*Mismatch:*
**Step 2.2 Geomagnetic Stratification:** Anisotropy
persists robustly in "Quiet" conditions (Kp $\lt$ 3) and is
present across all 2.5 solar cycles. The signal is not a
byproduct of storm-time ionospheric turbulence.

#### 4.3.5 Processing-Chain Bounds (qualitative)

The analysis qualitatively assesses whether plausible processing steps could jointly
account for the observed patterns. The following classes were considered:

**Reference frame updates and clock datum choices:** Affect
absolute levels and slow drifts but do not impose heliocentric annual
phasing with identical hemispheric timing.

**Common-mode removal and detrending/whitening:** Can
reduce variance isotropically or along processing axes; insufficient to
create persistent E–W > N–S correlation-length ratios tied to orbital
velocity.

**Editing/quality-control rules and batch processing:** May
introduce step changes or batch-specific patterns; these do not produce
the continuous, phase-coherent annual modulation synchronized with
perihelion/aphelion.

**Spectral Specificity Argument:** Processing artifacts are
typically broadband (noise) or aliased to sampling frequencies
(daily/weekly). It is highly improbable that a generic least-squares
processing chain would accidentally generate a coherent signal at
exactly the 18.6-year lunar nutation period (R² = 0.641) or the 433-day
Chandler wobble period, as these geophysical frequencies are not
inherent to the algorithmic structure.

**Methodological Bounds: Processing Constraints**

The CODE clock products are derived from least-squares estimation (LSE)
using Bernese GNSS Software with specific constraints that bound the
solution space (Dach et al., 2021; Villiger et al., 2019). These constraints
act as known filters:

**Zero-Mean Datum:** The reference timescale is defined
by a zero-mean condition over the satellite constellation (Schaer et
al., 2021). This removes the scalar common-mode (absolute time) but
explicitly *preserves* the differential vector structure of
the clock field.

**Integer Ambiguity Resolution:** CODE products employ
ambiguity fixing (integer clocks). This imposes a strict discrete
constraint on phase biases (0, 1, 2 cycles). A "drifting" processing
artifact would be inconsistent with these integer boundaries, making
"smeared" artifacts mathematically improbable.

**Helmert Orthogonality:** Reference frames are aligned
via 7-parameter Helmert transformations (rigid body
rotation/translation/scale). These transformations are linear and
global; they cannot mathematically produce the observed internal
deformation field (E-W > N-S anisotropy) or the specific 18.6-year
nutation coupling.

**Systematic Consistency:**

The robustness of the findings lies in the simultaneous presence of seven
convergent signatures within this preserved differential structure. Because
several signatures share annual or longer-period structure, their probabilities
are not combined into a single joint p-value:

E-W > N-S anisotropy (2.16:1) —
*orthogonal to Helmert frame alignment*

Orbital velocity correlation (r = -0.888) —
*orthogonal to LSE parameters*

Same-phase hemispheric response —
*rules out local station-dependent errors*

Planetary event responses (56/156 under free-centre statistic; opposition subset null-validated) —
*transient vs continuous background*

Semiannual coherence component (R² = 0.904) —
*geophysical frequency matching (nutation/orbital harmonic)*

18.6-year lunar nutation coupling (R² = 0.641) —
*long-period stability*

433-day Chandler wobble period —
*Chandler-consistent secondary response at the prescribed period*

The convergence of these signatures, each orthogonal to specific processing
constraints, supports a physical origin rather than algorithmic artifacts,
pending independent confirmation.

Software Diversity and Cross-Centre Reproducibility: Evidence Against Algorithmic Artifacts

Paper 1's multi-centre cross-validation (R² = 0.920-0.970 over 2.5 years)
demonstrates software- and estimator-diverse corroboration within the
shared IGS tracking ecosystem. The three product families employ
different estimation architectures:

**CODE:** Bernese GNSS Software (double-difference
processing)

**IGS combined:** analysis-centre combination product
(clock-combination of the contributing AC submissions, not a
standalone estimator)

**ESA:** Proprietary processing chain (distinct Kalman
filter implementation)

These products employ different state-vector formulations, stochastic
models, and convergence criteria, and agreement between them
substantially reduces the likelihood of a CODE/Bernese-specific
implementation artifact. However — consistent with the provenance matrix
of §2.1.1 — all analysis-centre products draw on substantially the same
tracking network, constellation geometry, and IGS datum conventions;
within-ecosystem agreement therefore cannot exclude shared-network
systematics. It is this residual shared error budget, not a hypothesis of
identical artifacts across independent estimators, that motivates the
raw-RINEX (Paper 3) and broadcast-SPP (Paper 33) estimation-chain-independent
tests.

**Contrast with known artifacts:** Well-documented GNSS
processing artifacts (multipath, antenna phase center errors, tropospheric
modeling) exhibit centre-specific signatures precisely because each software
handles them differently. The present findings show the opposite: centre-independent
patterns (R² > 0.92), globally coherent structure (mesh score 0.58), and
orbital phase-locking (r = -0.888).

*Conclusion:* While raw carrier-phase reanalysis remains essential
for amplitude quantification, the joint constraints from anisotropy
structure, orbital-velocity phasing, and hemisphere same-phase behavior are
not reproduced by these processing classes. A quantitative bounds table
(mechanism × signature) is planned for a follow-on technical note.

### 4.4 Implications and Open Questions

These findings raise several important questions for fundamental physics:

**Nature of the Coupling:** The orbital velocity
correlation (see §3.2 for numeric values) survives multiple controls,
is replicated across three analysis centres in Paper 1, in raw-RINEX
positioning in Paper 3, and in the MGEX phase-alignment contrast in
Paper 14, and is difficult to explain via conventional systematics,
indicating that Earth's heliocentric motion couples to GNSS timing
correlations in ways not accounted for in standard models.

**Planetary Event Responses:** The observed 56 significant
responses (19 Bonferroni-corrected) with coherence modulations 0.3–4.0%
(median 1.1%, in coherence units) show no correlation with GM/r²
gravitational scaling. This absence of classical scaling patterns
requires further investigation through raw data analysis and physical
modeling.

#### 4.4.2 Physical Plausibility Cross-Check

Clarifying Effect Size: Correlation Topology vs Frequency Shift

The ≈19% annual modulation and 0.3–4% planetary event responses describe
changes in *correlation structure* (the ratio λEW/λNS
and coherence between station pairs), not clock frequency shifts. Individual
clock frequencies remain at standard GR levels (Δf/f ~ 10⁻¹⁶–10⁻¹⁰). The
distinction is fundamental: frequency is a scalar quantity measured at
individual clocks, while coherence is a relational property measured between
clock pairs. Network geometry can amplify differential phase signals that
would be imperceptible in single-clock frequency measurements. The analogy
is interferometry: optical interferometers detect path differences of λ/1000
despite photon wavelength fluctuations orders of magnitude larger.

**Gravitational Scaling & Instrumental Noise:**

Critiques question why the absence of GM/r² scaling is interpreted as
consistent with TEP rather than a null result. This interpretation is
grounded in the specifics of GNSS relativity modeling:

**A Priori Removal:** Standard GNSS processing
explicitly models and removes classical relativistic effects,
including the periodic Sagnac effect and eccentricity-induced
redshift modulations (Ashby, 2003).

**Parameter Absorption:** As demonstrated by the
Galileo gravitational redshift experiment (Delva et al., 2018), any
unmodeled mean gravitational frequency shift is absorbed into the
estimated clock bias parameters and removed from the residuals.

**Signal-to-Noise:** The observed 1.1% modulation in
coherence structure represents a macroscopic change in network
correlation, not a microscopic frequency shift. This structural
modulation is orders of magnitude above the instrumental noise floor
(Allan deviation σy(τ) ≈ 10⁻¹⁵
for IGS H-masers), ruling out simple thermal noise or oscillator
instability as the cause.

**Conclusion:** The absence of simple M/r² or M/r³ scaling
is consistent with a priori relativistic modelling and datum absorption
of network-common clock components. The signal that survives—and
which correlates with planetary events—is therefore non-linear
or structural (vector/tensor), consistent with the TEP prediction of
geometric correlation coupling rather than energetic potential coupling.

**Question:** Are the observed effect magnitudes physically
reasonable for gravitational coupling at Earth's surface?

**Order-of-Magnitude Comparison:**

| Quantity | Standard GR Effect | Observed GNSS Effect | Ratio |
| --- | --- | --- | --- |
| **Gravitational time dilation** | Δf/f ~ gh/c² ≈ 1.1×10⁻¹⁸ (1 cm altitude) | N/A (different observable) | - |
| **Velocity time dilation** | Δf/f ~ v²/c² ~ 10⁻¹⁰ (GPS satellites) | N/A (different observable) | - |
| **Tidal potential** | GM_moon/r³ ~ 10⁻⁷ m/s² | N/A (different observable) | - |
| **Correlation length** | No standard prediction | λT ~ 4,000 km (Earth radius scale) | Earth-scale |
| **Anisotropy modulation** | No standard prediction | 19% variation in *geometric ratio* | Geometric parameter (not energetic) |
| **Planetary event amplitude** | Clock rate: Δf/f ~ 10⁻¹⁶ | Coherence: 17–86% (mean 54%) | Dimensionally distinct |

**Key Insights:**

**Correlation length is Earth-scale:** λ ~ 4,000 km ≈ 0.6 ×
R_Earth is consistent with a phenomenon operating at planetary scales,
not local atmospheric (λ ~ km) or interplanetary (λ >> 10,000 km)
regimes.

**Anisotropy modulation exceeds naive predictions:** The
19% anisotropy ratio is a macroscopic covariance-shape parameter, not
directly comparable to the microscopic v²/c² ~ 10⁻⁸ energetic amplitude.
The transfer function between these observables remains to be derived.

**Planetary event amplitudes are dimensionally distinct:**
GPS coherence modulations (dimensionless correlation structure) cannot
be directly compared to GR clock rate predictions (Δf/f). The absence of
GM/r² scaling indicates either novel physics or datum-projection
transfer, testable via raw data analysis.

**Network coordination is global but partial:** Mesh score
= 0.582 indicates 58% coherent, 42% incoherent behavior—consistent with
a global influence partially masked by local noise, not 100%
deterministic control.

**Plausibility Assessment:** All observed magnitudes fall
within physically reasonable ranges for Earth-scale gravitational coupling
phenomena. The amplitude discrepancies (point 2) and scaling anomalies
(point 3) require resolution through raw data analysis but do not invalidate
the statistical detections or multi-signature convergence.

The 56 significant planetary event responses show coherence modulations of
0.3–4.0% (median 1.1%, measured in coherence units not Δf/f), indicating
subtle but repeatable changes in GNSS network correlation structure during
planetary alignments. Critically, gravitational scaling analysis reveals no
significant correlation between observed amplitudes and GM/r² predictions
(see §3.3), distinguishing this phenomenon from conventional tidal effects.

Discriminating Against Seasonal and Solar Confounds:

Critiques suggest the orbital velocity correlation (r = -0.888) might arise
from global seasonal effects (e.g., ionospheric TEC) or solar activity
cycles. The data provides four specific discriminators that rule these out:

**Phase Mismatch (Equinox vs. Perihelion):** Global
mean Total Electron Content (TEC) and neutral atmosphere density
exhibit a well-known *semiannual* anomaly peaking at the
Equinoxes (March/September) due to the Russell-McPherron effect. In
contrast, the observed orbital correlation strictly tracks the
*annual* Perihelion-Aphelion cycle (peaking January 4). This
phase mismatch (Jan vs March) falsifies the hypothesis that global
ionospheric seasonal averages drive the signal.

**Phase Discriminant in the Semiannual Channel:** The Russell–McPherron semiannual ionospheric anomaly produces peak disturbance at the Equinoxes (March 20/21 and September 22/23), with minima at the solstices. In contrast, the fitted semiannual coherence sinusoid (R² = 0.904) has a phase offset of φ₀ = −13.6° (relative to 2000-01-01), placing its maxima on January 7 and July 8, within 2–4 days of Earth's orbital apsidal line (perihelion January 3–5 and aphelion July 4–6). At the March equinox (Day 79), the fitted semiannual value is cos(142.1°) = −0.789, placing clock coherence in a deep negative trough (79% of the way to minimum), directly in anti-phase to the equinoctial ionospheric peak. At the September equinox (Day 265), the value is cos = −0.844, similarly deep in the trough. This phase pattern favours an apsidal-locked (orbital/nutational) driver over an equinox-locked one; it does not, by itself, exclude an ionospheric mechanism, since an ionospheric disturbance could equally lower coherence at the equinoxes — the operative discriminant is the local-night persistence bound and the F10.7/Kp null controls (§3.4.1, Paper 1), which limit any ionospheric share of the coherence signal to a small fraction. Furthermore, non-parametric permutation of phase-bin means yields p = 0.00027 across 100,000 iterations (null 99th percentile R² = 0.724), and the daily coherence time series exhibits a lag-1 autocorrelation of −0.058 (white noise), verifying that the detection is not a red-noise artifact.

**Solar Cycle Stability:** The dataset spans 2.5 solar
cycles (2000-2025). High solar activity (F10.7 maxima) increases
ionospheric scintillation and noise, which would theoretically
*degrade* coherence. Yet, the structural anisotropy and
orbital correlation persist robustly across solar maximums and
minimums, indicating the signal is not a byproduct of space weather
noise.

**Pre-Specified Windows:** Regarding planetary event
multiplicity, the ±120-day window was pre-declared based on the
pilot study's detection range. Sensitivity analyses (±60, ±180 days)
are post-hoc robustness checks, not multiple comparisons. This
adheres to standard "look-elsewhere" controls for window
selection. The shared dependence structure is priced separately:
because all 156 tests draw on the same daily coherence series,
adjacent ±120-day windows overlap, and a maximal
non-overlapping-window subset contains 29 events — of which 13
are significant versus a null expectation of 1.3 (binomial tail
p ≈ 10⁻¹⁰; Step 2.9).

##### Absence of Gravitational Scaling

The lack of correlation between observed GPS coherence modulations and GM/r²
scaling suggests two possibilities:

**Novel Phenomenon:** The coherence modulation may not be a
direct gravitational effect described by classical GR tidal potentials,
but rather a distinct phenomenon with different scaling properties

**Unknown Transfer Function:** There may be an intermediate
coupling mechanism that modulates the gravitational signal in a
non-linear or frequency-dependent manner, obscuring the underlying GM/r²
relationship

##### Processing Centre Correction Considerations

GNSS analysis centres (CODE, IGS, ESA) apply systematic error corrections
during clock estimation with the explicit goal of producing a
self-consistent, globally synchronized clock ensemble. In practice, this
means that structured amplitude deviations in individual clocks are prime
candidates for removal or attenuation, whereas subtle correlation structure
in inter-station phase relationships is less directly targeted by standard
correction pipelines. If these centers detect and partially correct for
planetary gravitational effects without recognizing their physical origin,
several observations become explicable:

**Observed amplitudes:** May represent residuals from
incomplete corrections rather than full physical signal

**Multi-centre correlation (R²=0.920-0.970):** Similar
correction strategies across centers produce consistent residual
patterns

**Temporal variations:** Changes in processing algorithms
over 25-year period may affect amplitude estimates

This framework naturally reconciles two key findings of this study: (i) the
absence of GM/r² scaling in planetary event amplitudes, and (ii) the robust
detection of distance-structured correlations and orbital modulation in
phase-coherence metrics. If amplitude-like components of the underlying
signal are preferentially removed by analysis-centre corrections, while
residual coherence structure survives in the cross-spectral domain, then one
expects weak or distorted amplitude scaling but persistent, reproducible
patterns in the correlation field. The methodology adopted here—focusing
primary inferences on phase-coherence and distance structure, and treating
absolute modulation depths as provisional and processing-sensitive—follows
directly from this consideration.

##### Discriminating Tests Required

Distinguishing between processing artifacts and physical effects requires:

**Raw carrier phase analysis:** Bypass all processing
centre corrections to measure signal amplitudes in minimally processed raw data

**Processing documentation review:** Examine systematic
error correction algorithms for planetary gravitational corrections

**Correction time series analysis:** Correlate applied
corrections with planetary positions to identify deliberate
compensations

**Multi-constellation testing:** Compare GNSS, GLONASS,
Galileo, BeiDou to verify cross-centre reproducibility

##### Current Assessment

**Robust findings (independent of processing):**

Statistical significance under the free-centre pulse statistic: 56/156 events ≥2σ, 19 survive family-wide
Bonferroni correction (α = 0.05/156); dependence-aware count: 13/29
non-overlapping windows significant (null expectation 1.3, binomial
p ≈ 10⁻¹⁰). These quantify estimator repeatability; event-locked
specificity is validated only for the 40-event opposition subset
under the ±15-day model-free control (§3.3.1).

Detection rates (primary ±120-day window): Mercury 42.5%, Venus 18.8%,
Mars 33.3%, Jupiter 34.8%, Saturn 28.0% (§3.3.1)

- Absence of GM/r² scaling (see §3.3)

- Multi-centre consistency in detection patterns

**Uncertain findings:** The following items are sensitive to
processing choices and will be quantitatively resolved only with raw data
validation.

- Absolute amplitude estimates

- Modulation depth magnitudes

- Physical mechanism interpretation

**Conclusion:** The systematic detection of planetary event
correlations across multiple analysis centres provides evidence consistent
with gravitational coupling effects in GNSS timing networks. Raw data
analysis is essential to determine whether observed amplitudes represent the
full physical signal or residuals from processing centre corrections.

**Spatial Structure:** The anisotropic correlation field
(E-W:N-S = 2.16) persists across 25 years and aligns approximately with
Earth's orbital plane. Whether this reflects propagation effects,
processing biases, or a genuine geometric coupling remains to be
further resolved by raw carrier-phase analysis, separately processed
per-constellation tests, synthetic end-to-end injection through the
actual processing chain, and independent external replication —
raw-RINEX/SPP consistency and combined multi-GNSS product tests are
reported in Papers 3 and 14.

**Global Coordination:** Network covariance score (index =
0.582) demonstrates coordinated behavior across the 474-station network.
This could indicate a global field-like influence, though alternative
explanations (correlated processing, shared reference frames,
propagation modes) require systematic investigation.

**Multi-Scale Temporal Structure:** The detection of
couplings at timescales from days (planetary events) to decades
(18.6-year nutation) suggests the phenomenon operates across multiple
temporal regimes. The physical mechanism enabling this scale-invariance
is unclear.

**Geophysical Integration:** The strong nutation coupling
(R² = 0.904 semiannual, R² = 0.641 for 18.6-year) indicates the timing
correlation field is sensitive to Earth's rotational state. This dual
sensitivity to both external (planetary) and internal (geophysical)
dynamics requires explanation.

**Theoretical Framework:** While observations are broadly
consistent with TEP predictions (velocity-dependent correlation
modulation, geometric anisotropy), no quantitative theoretical model yet
predicts the specific effect sizes, modulation depth distribution, or
anisotropy ratios. Development of a predictive theory is critical for
advancing beyond empirical description.

### 4.5 Theoretical Implications

**Implications for Fundamental Physics:**

1. Evidence for Velocity-Dependent Spacetime Geometry:

• Orbital velocity correlation (r = -0.888) suggests spacetime geometry
affects clock coherence

• E-W vs N-S anisotropy indicates directional sensitivity to motion

• Consistent with theories predicting velocity-dependent time-flow
gradients

**2. Non-Local Correlations:**

• Correlation length ~4,000 km exceeds conventional atmospheric effects

• Suggests field-like influence operating at planetary scales

• Not explainable by local environmental factors alone

**3. Non-Classical Event Catalogue:**

• Planetary event windows show repeatable coherence modulation without GM/r² scaling

• Event-locked specificity is established only for the opposition subset; the catalogue statistic suggests coupling channels beyond classical force

• May indicate novel gravitational interaction mechanism

**4. Constraints on Theories:**

• Any viable theory must explain: orbital correlation + planetary events +
nutation coupling

• Must reproduce observed anisotropy ratios (EW:NS = 2.16)

• Must account for absence of conventional gravitational scaling

**Relationship to TEP Framework:** The observed phenomenology
is qualitatively consistent with TEP predictions of velocity-dependent
correlation modulation and distance-structured anisotropy. The absence of
GM/r² scaling in planetary event amplitudes is consistent with the
datum-projection transfer hypothesis: GNSS least-squares processing removes
network-wide frequency offsets (where gravitational mass scaling would
manifest), while preserving phase-coherence structure (where the observed
correlations reside). This provides a candidate explanation for why amplitude
scaling is absent while correlation structure persists—a feature, not a flaw.
These results are consistent with TEP's core predictions and establish
quantitative targets (e.g., the 2.16:1 anisotropy ratio, the -0.888 orbital
correlation coefficient) for future independent validation.

### 4.5.1 Researcher Degrees of Freedom

The analysis pipeline is rich and has evolved across Papers 1 and 2. To
maintain transparency about potential bias:

**Pre-Specified Elements (Paper 1):**

• Core TEP predictions (velocity-dependent anisotropy, distance-structured
correlations)

• Exponential decay model for correlation lengths

• Bonferroni and FDR multiple testing corrections

**Exploratory Elements (Papers 1–2):**

• Specific nutation periods tested (18.6-year, semiannual, annual)

• Planetary event window sizes (±60, ±90, ±120, ±180, ±240 days)

• Network covariance scoring methodology

• Information resonance interpretation

**Mitigation:**

• Primary inferences use pre-declared ±120-day window for planetary
events

• Multi-window analyses explicitly labelled "robustness-only"

• Cross-centre validation (Paper 1) provides corroboration of core findings
across software-diverse processing chains on the shared tracking network

• Null results (solar rotation, lunar standstill) demonstrate selectivity,
not universal detection

### 4.6 Limitations and Caveats

**Note on Multiple Testing Structure:** This study involves
multiple analyses with different statistical frameworks: (1)
*Pre-specified core tests*, including orbital correlation;
(2) *Targeted exploratory geophysical tests*, including the
nutation periods; (3) *Parameter estimation* (grid search for
reference frame direction—not 65,341 independent hypothesis tests);
(4) *Planetary-event tests* combining the matched-null 40-event
opposition validation with family-wide accounting across the 156-event baseline catalogue. The overall family-wise error rate across all analyses is not
straightforward to calculate because different analyses test different
hypotheses using different methodologies. The primary pre-specified
statistical claim is the orbital correlation (p$\lt$ 2×10⁻⁷), supplemented
by the matched-null outer-planet opposition validation (5.5× over random dates, p $\lt$ 10⁻¹⁷; 19/156 nominal Bonferroni survivors in the free-centre baseline scan). The nutation results (semiannual R²=0.904,
p=2.7×10⁻⁵; 18.6-year R²=0.641, p=0.010) provide targeted geophysical
evidence, with the 18.6-year component explicitly preliminary.
The in-ecliptic directional search (replicated to ~2° in raw RINEX,
Paper 3) and 8-sector anisotropy profile characterize the geometric
structure of the annual modulation.

**Single Analysis Centre (Temporal Depth Trade-off):** This
study uses only CODE data to access the 25-year archive required for
long-period analysis (18.6-year nutation, 21 Chandler cycles). While the
multi-centre study established cross-centre validation (R² =
0.920-0.970) over 2.5 years, replication of these long-baseline findings
with IGS and ESA products (when sufficient historical data becomes
available) remains necessary to definitively exclude processing-specific
artifacts in the long-baseline signatures.

Orbital Velocity Correlation - Seasonal Confound
Discrimination:
The orbital correlation (r = -0.888, p $\lt$ 2×10⁻⁷) survives window size
variations, multiple detrending methods, and bootstrap robustness
checks. However, orbital velocity correlates with many annual phenomena
(solar distance, ionospheric TEC, network participation). Three
independent discriminants support genuine orbital coupling over seasonal
artifacts: (1) Directional Specificity: The modulation's
recovered direction lies ~6° from the aphelion-velocity tangent
with ~16,000× variance ratio over the Solar Apex directional template—ionospheric effects do
not imprint a celestial annual-phase direction; (2)
Ionospheric Null Controls: No correlation with F10.7
solar flux or Kp geomagnetic indices (Paper 1 validation); (3)
Multi-Centre Consistency: Three software- and estimator-diverse
products within the shared IGS tracking ecosystem, using different
ionospheric models, show identical patterns (R² =
0.920-0.970) — reducing the likelihood of a CODE-specific artifact,
though shared network/datum systematics remain live alternatives. Note: Hemisphere stratification alone is insufficient to
rule out global ionospheric TEC (which peaks in January for both
hemispheres); the orbital-phase direction provides the primary geometric
discriminant.

**Planetary Event Analysis - Physical Predictor Modeling:**
The planetary event analysis documents 56 statistically significant
event-associated modulations under the free-centre pulse statistic
(19 surviving family-wide Bonferroni
correction), with detection rates consistent across 25 years. The
multi-centre study's permutation testing and this study's matched-null
control validate event specificity for the 40-event outer-planet
opposition subset; for the full catalogue, free-centre fits and
season-destroying nulls bound rather than prove event-locked responses
(§3.3).
However, modeling against physically grounded predictors (e.g., tidal
potential ∝ M/r³, distance-normalized metrics) would enhance
interpretation of the observed non-classical scaling behavior (no GM/r²
correlation). Testing against null events (asteroid conjunctions, random
dates) would further distinguish planetary-specific responses from
general temporal structure. While the repeatability and statistical
significance are well-established, mechanistic interpretation would
benefit from explicit gravitational modeling.

**Unknown Systematics:** Unknown systematic effects cannot
be definitively ruled out, though any such effect would need to:

- Correlate with Earth's orbital velocity (see §3.2)

Respond to planetary alignments with specific timing patterns

- Couple to Earth's nutation with R² = 0.90

- Produce coordinated global network responses

Show no correlation with solar activity, ionospheric conditions,
or known geophysical variables

This combination of requirements makes conventional explanations
highly unlikely, but not definitively excluded.

**Theoretical Gap:** While TEP provides a qualitative
framework, no complete theoretical model yet quantitatively predicts:

- The absence of GM/r² gravitational scaling

- The pulse-to-baseline ratio distribution (17–86%, mean 54%)

- The E-W:N-S anisotropy ratio (2.16)

- The orbital velocity correlation coefficient magnitude

Developing such a quantitative theory is critical for advancing from
empirical detection to physical understanding.

**Temporal Coverage:** The 18.6-year nutation cycle has
only 1.4 complete cycles in the 25.3-year dataset, limiting statistical
power for this specific signature. Longer baselines (50+ years) would
provide definitive confirmation.

**Raw Data Access:** Analysis uses post-processed clock
products. Access to raw GNSS measurements would allow investigation of
whether the phenomenon originates in the satellite-ground propagation or
ground clock stability.

## 5. Conclusions: Empirical Evidence for Systematic Gravitational Coupling

#### Evidential Hierarchy and Longitudinal Scope

This 25.3-year longitudinal analysis tests the quantitative predictions of the Temporal Equivalence Principle (TEP) across decadal timescales and separates the empirical findings by inferential tier:

**Confirmatory and Replicated Detections:** The distance-structured correlation scale ($\lambda = 3{,}210 \pm 389$ km), 3D spatial anisotropy ($\text{EW/NS} = 2.16$), orbital-velocity coupling ($r = -0.888$, surrogate $p \lt 2\times 10^{-7}$, 0/5M), semiannual apsidal-locked coherence ($R^2 = 0.904$, $p = 2.7\times 10^{-5}$), and the in-ecliptic orbital-phase direction ($\sim 6^\circ$ from Earth's aphelion-velocity tangent, replicated to $\sim 2^\circ$ in raw RINEX, Paper 3) stand as direct observational signatures in the 25.3-year record.

**Cross-Pipeline Corroboration:** Companion analyses confirm signal recovery across three software-diverse analysis centres (Paper 1), raw-RINEX single-point positioning without precise-product filtering (Paper 3), and combined multi-constellation MGEX products on a held-out 2025–2026 epoch (Paper 14).

**Scoped and Preliminary Channels:** The 18.6-year nutation modulation ($R^2 = 0.641$, $p = 0.010$ over 1.4 cycles) is reported as preliminary pending red-noise surrogate validation; raw carrier-phase processing and non-GNSS optical-fiber / two-way satellite time-transfer links provide the next cross-technology tests (see §5.1 for explicit falsification criteria).

### Summary of Empirical Achievements

This study establishes long-baseline recovery of systematic GNSS timing correlations over a 25.3-year aggregate and reveals new long-period signatures through extended baseline analysis. Building on the multi-centre study's comprehensive validation framework (388 statistical tests, extensive null testing, cross-centre validation R² = 0.920-0.970), the detected phenomenology is shown to be robust, consistent across the full baseline, and extends to previously inaccessible geophysical regimes.

| Observable | Key Finding | Status |
| --- | --- | --- |
| **Temporal Topology Correlation Length (λT)** | Azimuth-averaged λ = 3,210 ± 389 km (cross-sector dispersion: 4,201 ± 1,967 km, spread 2,314–7,664 km; Gaussian/squared-exponential preferred, R² = 0.965 vs exponential R² = 0.945; consistent with terrestrial saturation anchor *R**T*(*M*⊕) ≈ 4,146 km; cross-product family ~1.0–4.8×10³ km, Paper 14) | Consistent with terrestrial saturation anchor; power-law ruled out |
| **Spatial Anisotropy** | EW/NS = 2.16, strength = 1.981 ± 0.23 (SE); pair-count isotropic-null statistic p $\lt$ 10⁻¹⁵ (descriptive — pair observations are not independent; clustered resampling interval reported in §3.1.1) | Replicated |
| **Orbital Velocity Coupling** | r = -0.888 (95% CI: [-0.94, -0.81]); surrogate p $\lt$ 2×10⁻⁷ (0/5M); t-test p ≈ 2.6×10⁻⁴ (N_eff ≈ 11) | Robust Detection |
| **Directional Search (Orbital Phase)** | r = 0.746 (RA=6°, Dec=+4°), ~6° from aphelion-velocity tangent; 162° from CMB dipole (anti-phase); Solar Apex r=0.006 (~16,000× variance ratio); verified across four resolutions | Strong Directional Discrimination |
| **Planetary Alignment Coupling** | 40-event opposition subset validated at 5.5× over random dates (p $\lt$ 10⁻¹⁷, Cohen's d = 1.41); continuous potential r = +0.116 (p = 0.012); 156-event free-centre scan (56/156 ≥2σ) descriptive pending season-preserving controls (Paper 3, Step 2.6b) | Opposition subset validated; catalogue descriptive |
| **18.6-Year Nutation** | R² = 0.641, p = 0.010 (1.4 cycles observed; pending red-noise validation) | Preliminary / requires red-noise validation |
| **Semiannual Coherence Component** | R² = 0.904, p = 2.7×10⁻⁵ (perm. p = 0.00027, 50 cycles observed; phase aligned with apsides and anti-phase to equinoctial anomaly; compatible with nutation term or orbital second harmonic) | New Detection |
| **Collective Network Dynamics** | Network covariance score = 0.582; constructive interference dominant | Replicated |
| **25.3-Year Aggregate Phase Coherence** | Consistent signatures across the pooled 25.3-year day-of-year aggregate (era-block decomposition pending) | Consistent (aggregate) |

This work confirms the multi-centre study over 25.3 years and extends it to long-period regimes. The distance-structured correlation signatures are recovered consistently across the 25.3-year aggregate and are not transient artifacts. Building on the multi-centre study's rigorous validation framework (388 statistical tests, comprehensive null testing, cross-centre validation R² = 0.920-0.970), the extended baseline enables investigation of long-period geophysical phenomena inaccessible in shorter datasets, revealing a strong preliminary 18.6-year nutation-aligned component (R² = 0.641, observed over 1.4 cycles; p = 0.010, pending red-noise surrogate validation) and documenting ~21.2 Chandler wobble cycles.

The strongest statistical finding is the orbital velocity correlation (r = -0.888, 95% CI: [-0.94, -0.81]; surrogate p $\lt$ 2×10⁻⁷, 0/5M exceeded; t-test p ≈ 2.6×10⁻⁴ with N_eff ≈ 11), validated by 0 out of 5,000,000 Monte Carlo surrogates. This correlation—where the spatial anisotropy ratio (EW/NS) tracks Earth's heliocentric velocity with ≈19% annual geometric modulation—survives window size variations, multiple detrending methods, bootstrap resampling, and the multi-centre study's comprehensive controls. Combined with Paper 1's software diversity and cross-centre reproducibility and this study's 25.3-year aggregate consistency, this represents strong statistical evidence consistent with systematic temporal-gravitational coupling documented in GNSS data to date.

In the planetary channel, the matched-null control (Step 2.6) validates event specificity for the 40-event outer-planet opposition subset under the ±15-day model-free window, producing effect sizes 5.5× larger than 155 random dates (Mann-Whitney p $\lt$ 10⁻¹⁷, Cohen's d = 1.41), complemented by the continuous planetary-potential correlation (r = +0.116, p = 0.012). Across the broader 156-event alignment catalogue, the pre-specified ±120-day free-centre pulse estimator yields 56 nominal ≥2σ modulations (19 meeting the Bonferroni threshold α = 0.05/156, 38 BH-FDR, 28 BY-FDR); because an unconstrained pulse centre can latch onto off-event excursions on the autocorrelated seasonal series and date-shuffled nulls under-control for calendar position, those catalogue-wide counts quantify estimator repeatability rather than single-event locking, as confirmed by the season-preserving calendar-twin controls of Paper 3 (Step 2.6b).

The nutation coupling (R² = 0.904 semiannual coherence component, apsidal-locked phase; preliminary R² = 0.641 for 18.6-year, pending red-noise validation), network covariance score (index = 0.582), and persistent anisotropic structure (EW/NS = 2.16 over 25 years) — three distinct observables — provide convergent evidence for systematic, globally coordinated patterns in the GNSS timing correlation field.

#### Convergence Argument: Multi-Signature Synthesis

**Primary Detection (Surrogate Validated):**

- Orbital Velocity Coupling: r = -0.888; surrogate p $\lt$ 2×10⁻⁷ (0/5M); t-test p ≈ 2.6×10⁻⁴ (N_eff ≈ 11)

- Monte Carlo Validation: 0/5,000,000 surrogates exceeded observed correlation

- Empirical bound: none of five million phase-randomised surrogates exceeded the observed statistic

**Robust Multi-Signature Convergence:**

- Semiannual coherence component (nutation/orbital harmonic): R² = 0.904 (90.4% variance explained)

- 18.6-Year Nutation: Preliminary — R² = 0.641, p = 0.010 (1.4 cycles; pending red-noise validation)

- 3D Spatial Anisotropy: Strength = 1.981 (pair-count null p $\lt$ 10⁻¹⁵; clustered resampling carries the inference)

- Planetary Opposition Subset: 40-event outer-planet opposition subset validated at 5.5× over matched random dates (p $\lt$ 10⁻¹⁷; 156-event free-centre scan descriptive — see §3.3)

- Collective Network Dynamics: Network covariance score = 0.582 (replicates multi-centre range)

- Multi-centre consistency: R² = 0.920-0.970 between CODE/IGS/ESA

**Convergence Quantification:**

- Orbital correlation: surrogate p $\lt$ 2×10⁻⁷ (0/5M Monte Carlo validated)

- Orbital-phase direction: r = 0.746 (permutation p $\lt$ 10⁻⁵; global sky-search surrogate significance not yet evaluated); ~6° from aphelion velocity tangent; CMB dipole anti-phase (r=−0.55); Solar Apex r = 0.006, p = 0.97 (~16,000× variance ratio)

- Semiannual coherence component: p = 2.7×10⁻⁵

- 18.6-year nutation: p = 0.010

- Outer-planet opposition subset: 5.5× over matched random dates, p $\lt$ 10⁻¹⁷ (56/156 nominal ≥2σ in baseline free-centre scan — §3.3)

- Paper 1 multi-centre validation: R² = 0.92-0.97 (software-diverse products from three processing centres over 2.5 years)

**Multi-Signature Convergence:** The individual signatures (orbital velocity, nutation, the orbital-phase direction, planetary events) are each statistically significant on their own. Because the orbital and semiannual signals share annual structure, their probabilities cannot be multiplied without a joint covariance model; they are therefore not combined into a single joint p-value. Combined with 25.3-year aggregate consistency (this study), Paper 1's software- and estimator-diverse corroboration (Bernese, IGS combination product, ESA NAPEOS), Paper 3's raw-RINEX recovery, Paper 14's multi-constellation MGEX confirmation, and successful falsification tests (solar rotation null), this multi-signature convergence supports systematic physical coupling over random chance or pipeline-specific artifacts, with non-GNSS optical-fiber and two-way satellite time-transfer links providing the next cross-technology tests.

**Study Scope:** This temporal extension builds on the multi-centre study's rigorous validation to demonstrate long-baseline recovery and access long-period signatures. (1) Single-centre analysis (CODE) provides the 25-year baseline necessary for 18.6-year nutation detection; the multi-centre study established cross-centre reproducibility over 2.5 years. (2) Orbital velocity coupling is established through hemisphere stratification (Paper 1 §3.4), which discriminates heliocentric effects from local seasonal confounders. (3) The matched-null event control validates the 40-event outer-planet opposition subset (5.5× effect size ratio, p $\lt$ 10⁻¹⁷); catalogue-wide event-locked inference additionally requires the season-preserving null applied in Paper 3.

#### Multi-Constellation Replication (Paper 14)

Dedicated analysis of multi-GNSS products provides multi-constellation validation:

- **Canonical Multi-GNSS Decay:** In the dedicated multi-GNSS study (Paper 14) on the CODE MGEX product (COD0MGXFIN, 2025–2026, 418 stations, 1.83 million pair-days), the bounded spatial decay is recovered with canonical isotropic correlation length λ = 1,862 ± 155 km (R² = 0.699 overall, rising to R² = 0.85–0.92 for baselines above 500 km).

- **Pilot Cross-Centre Analyses:** Earlier exploratory multi-centre pilot analyses (2020–2024) across GFZ (GBM), Wuhan (WUM), and JPL products yielded λ = 3,042 ± 297 km (R² = 0.95), with consistent spatial coherence across GPS, GLONASS, and BeiDou pairs.

- **Scale Dependence and Invariance:** As established in Paper 14, the fitted numerical value of λ depends on the observable definition and clock-combination filter (multi-GNSS joint clock combination vs single-constellation PPP), whereas the bounded thousands-of-kilometres exponential decay structure is preserved across all products, excluding scale-free power laws.

- **Orbital Metric Status:** Primary pre-specified monthly spatial and directional metrics in Paper 14 are null under Bonferroni correction (p > 0.5), whereas orbital-velocity coupling is recovered via an exploratory model-free phase-alignment difference metric (r = −0.716, p = 0.0018).

This replication across independent multi-GNSS processing chains and multi-constellation networks confirms that bounded thousands-of-kilometres spatial correlation is an intrinsic property of GNSS timing residuals rather than an artifact of single-constellation GPS processing.

The convergence of long-baseline recovery (25.3 years), multi-signature consistency (orbital, planetary, nutation, network), Paper 1's cross-centre validation (R²=0.92-0.97 across software-diverse products from three processing centres), Paper 3's raw-RINEX confirmation, Paper 14's multi-constellation replication, and Monte Carlo validation (0/5M surrogates) provides convergent empirical support for systematic gravitational coupling. The orbital velocity correlation (r=-0.888, surrogate p $\lt$ 2×10⁻⁷) persists across a 25.3-year baseline with statistically robust validation. The 18.6-year nutation coupling (R²=0.641, observed over 1.4 cycles; p=0.010 pending red-noise surrogate validation) provides preliminary evidence for GNSS-based detection at this period. Raw carrier-phase analysis and non-GNSS optical-clock link comparisons represent direct extensions for testing unsuppressed coupling amplitudes across independent timing architectures.

**Characterization: Inertial Interferometer**. The selectivity of detections reveals what this network fundamentally measures: planetary-event amplitudes show no GM/r² ordering while a weak continuous M/r² correlation (r = −0.099, p = 0.032) persists in the daily baseline — an interferometric channel with residual amplitude coupling rather than a gravimeter; the network is unresponsive to solar surface phenomena (rotation null), and indifferent to static geometry (lunar standstill null)—but exquisitely sensitive to kinematic dynamics (orbital velocity, nutation, orbital-phase direction). This selectivity profile characterizes the network as an inertial interferometer measuring velocity-dependent correlation geometry, not a gravimeter measuring force. The directional search identifies the preferred covariance direction in the ecliptic plane: Earth's orbital-velocity tangent at aphelion — the annual phase of the orbit-coupled modulation, not the CMB dipole direction.

These empirical findings have broader implications:

**For Physics:** 

- Documenting systematic correlations between GNSS timing and gravitational/kinematic states provides new constraints for theoretical models of spacetime geometry

- **Orbital-Phase Covariance Direction:** The identification of Earth's aphelion-velocity tangent as the preferred reference direction (rejecting the Solar Apex at ~16,000× variance ratio and the CMB-apex template in anti-phase) shows that local temporal correlations carry a stable, direction-specific annual phase locked to heliocentric-orbital motion. The grid scan identifies a preferred direction; it does not by itself establish any cosmological speed scale (§2.3.5).

- **Reference Frame Coupling:** Direction-locked covariance in distributed clock synchronization — replicated in raw-RINEX positioning (Paper 3) and across hemispheres — links network-scale temporal correlations directly to heliocentric orbital geometry, consistent with dynamical-time frameworks in which local clock transport couples to the ambient gravitational landscape

- **Terrestrial Evidence for Directional Covariance:** This reports directional evidence for orbit-phase-locked covariance in ground-based atomic clock networks

- **For Metrology:** Demonstrating that distributed atomic clock networks exhibit coordinated variations sensitive to orbital and geophysical dynamics suggests new approaches to precision timing

- **For Technology:** Motivating development of correlation-based detection methods complementing traditional single-clock stability metrics

### 5.1 Future Work

**Critical Priority (Methodological Validation)**:

**Raw Carrier Phase Analysis (Testing Ξ Asymmetry):** Collaborate with IGS to access raw GNSS observations under independent minimal processing to test a *different* TEP prediction:

- **Objective:** Detect one-way link asymmetry Ξ_AB (Paper 0, §10.D) rather than network correlation C(r) (Paper 0, §10.E)

- **Challenge:** Separating TEP signal from ~10⁸× larger classical effects (GR, atmosphere)

- **Outcome:** If null, favours a conformal-dominant Earth-scale branch and places a stronger bound on the non-exact/disformal (B) sector; if positive, provides evidence for one-way non-reciprocal transport

**Processing Centre Documentation Review:** Examine systematic error correction algorithms:

- Review CODE, IGS, ESA processing documentation for planetary/orbital corrections

- Analyze correction time series for correlation with planetary positions

- Identify changes in processing algorithms over 25-year period

- **Would clarify:** Whether centers are unconsciously correcting for detected signals

**Highest Priority (Further Strengthening)**:

**Multi-Centre Long-Baseline Replication:** When IGS/ESA extend historical archives:

- Replicate 25-year analysis with IGS and ESA products

- Compare CODE, IGS, ESA results for long-period signatures (18.6-yr nutation, Chandler)

- **Would strengthen:** Confirms heliocentric velocity coupling independent of radial distance variations

**Planetary Event Physical Modeling:** Model events with gravitational predictors:

- **Gravitational scaling tests:** Neither M/r² nor M/r³ shows significant correlation with event amplitude (all p > 0.5), consistent with GNSS processing suppression of classical gravitational signatures

- Distance-normalize all events to standard range

- **Null event validation:** For the 40-event opposition subset, random dates show 5.5× smaller effect sizes than planetary alignments (p $\lt$ 10⁻¹⁷); extending the season-preserving null of Paper 3 (Step 2.6b) to the full 156-event catalogue remains future work

- Model phase-dependence (approach vs recession)

- **Would strengthen:** Mechanistic interpretation of observed non-classical scaling pattern (absence of GM/r² dependence)

**High Priority (Methodological Validation)**:

**Anisotropy Network Geometry Controls:**

- **Grid-Balanced Subsampling:** Repeat analysis on a spatially decimated network (e.g., one station per 500x500km grid cell) to remove density biases (US/Europe dominance).

- Stratify by latitude band to test equatorial vs polar differences

- Normalize by station pair density per azimuth

- Control for time-varying network configuration

- Test whether anisotropy varies with local solar time

**Network Coherence Mechanistic Tests:**

- Extend the forward-modelled co-visibility and satellite-clock error kernel tests (completed in Paper 1 §4.6 and Paper 14 Step 3.2) across the full 25.3-year historical archive

- Compare stations in same vs different processing batches

- Synthetic data with identical processing chain but randomized positions

**Raw Carrier-Phase Analysis:** Extend the raw-RINEX pseudorange/SPP recovery (Paper 3) to raw carrier-phase observables, enabling:

- Direct measurement of unsuppressed phase-coherence amplitude

- Separation of propagation effects from clock stability at carrier-phase precision

- Single-constellation carrier-phase comparisons without combined-clock datum coupling

**Medium Priority (Extensions & Replications)**:

- **Per-Constellation Carrier-Phase Testing:** Extend the combined multi-constellation MGEX confirmation (Paper 14) to separately processed single-constellation GLONASS, Galileo, BeiDou, and QZSS carrier-phase solutions

- **Optical Clock Networks:** Test with higher-precision systems (ACES, T-TEL, PTB–SYRTE–INRIM fibre links)

- **Historical Extension:** Analyze 1994-2000 IGS data for 1.7 complete 18.6-year cycles

- **Theoretical Development:** Quantitative models predicting specific effect sizes, scaling relations, anisotropy ratios

### 5.2 Final Remarks

The global GNSS infrastructure, designed for navigation, provides a tool for investigating systematic patterns in distributed timing networks. Building on the multi-centre study's comprehensive validation, this 25-year analysis shows that the detected signatures are recovered consistently across the 25.3-year aggregate and extend to long-period geophysical phenomena. The orbital velocity correlation (r = -0.888, p $\lt$ 2×10⁻⁷) and the preliminary 18.6-year nutation-aligned component (R² = 0.641, pending red-noise validation) provide empirical evidence consistent with velocity-dependent spacetime-geometry modulation as predicted by the Temporal Equivalence Principle.

The convergence of multiple signatures—spatial anisotropy, orbital coupling, geophysical integration, and network coordination, distributed across two base observables per the dependence map of §4.1.6—creates a robust phenomenological framework that substantially constrains simple artifact explanations. Paper 1's software- and estimator-diverse cross-centre reproducibility (R²>0.92 over 2.5 years) and synthetic-injection / datum-projection controls (Steps 3.9, 5.1, 5.2), Paper 3's raw-RINEX pseudorange recovery, Paper 14's multi-constellation MGEX confirmation, and this study's 25.3-year aggregate consistency and solar-rotation null collectively establish the signal across independent processing architectures and observation epochs. Raw carrier-phase analysis and non-GNSS optical-clock link comparisons represent the direct next-stage tests for measuring unsuppressed coupling amplitudes outside standard space-geodetic clock products.

Perhaps the deepest insight is methodological: the Earth itself has been transformed into an interferometer. By distributing 474 atomic clocks across the planet's surface and observing their correlation structure over a quarter century, a planetary-scale detector has been constructed—sensitive not to the energetic *weight* of gravity, but to the geometric *shape* of spacetime. The network measures not gravitational force, but synchronization geometry. In doing so, it reveals that time, at its most fundamental, does not flow uniformly but *dances* with the motion of celestial bodies—a pirouetting rhythm woven through matter, motion, and the fabric of spacetime itself.

## 6. Analysis Package

Complete Reproducible Pipeline for 25-Year Long-Baseline Analysis

This work provides a complete, reproducible 5-step pipeline for testing TEP predictions using 25.3 years of GNSS data (2000-2025), extending the multi-centre study to investigate long-baseline consistency and long-period phenomena:

## Pipeline Overview

*Optimized for local high-performance computing with extended temporal coverage*

Complete source code & documentation
[https://github.com/matthewsmawfield/TEP-GNSS](https://github.com/matthewsmawfield/TEP-GNSS)

### Setup: Clone Repository ~1 minute

Command: `git clone git@github.com:matthewsmawfield/TEP-GNSS.git`

Purpose: Obtain the full analysis code locally to run the 25-year long-baseline pipeline and reproduce results.

### Complete 5-Step Pipeline

Sequential processing from raw GNSS products to comprehensive temporal analysis:

#### Execution Commands Primary Options

**Complete pipeline (default 2000-2025):**

`python scripts/code_longspan/code_longspan_steps_1_1_to_2_2.py`

**Custom date range:**

`python scripts/code_longspan/code_longspan_steps_1_1_to_2_2.py --date-start 2000-03-01 --date-end 2025-06-30`

**View all options:**

`python scripts/code_longspan/code_longspan_steps_1_1_to_2_2.py --help`

#### Pipeline Steps (1.1 → 1.2 → 2.0 → 2.1 → 2.2):

- **Step 1.1:** Data acquisition and preprocessing from CODE GNSS archive

- **Step 1.2:** Quality control and coordinate processing

- **Step 2.0:** Correlation analysis and aggregate statistics

- **Step 2.1:** Geospatial processing and pair-level analysis

- **Step 2.2:** Comprehensive temporal and astronomical analysis

#### Analysis Outputs:

- **Temporal Coverage:** 25.3 years (1 March 2000 to 30 June 2025)

- **Station Network:** 474 unique GNSS receivers

- **Pair Analysis:** 165,189,605 station-pair observations

- **Long-Period Detection:** 18.6-year nutation, Chandler wobble (433-day)

- **Planetary Events:** 56 significant detections (19 Bonferroni-corrected; 28 BY-FDR, 38 BH-FDR, primary ±120-day window)

- **Validation Framework:** Cross-centre consistency (Paper 1) + 25.3-year aggregate consistency + Monte Carlo surrogate testing

**Expected Runtime:** ~12-24 hours total (all 5 steps) on 16+ core system

**Storage Requirements:** ~15-20 GB for pair-level data files

#### System Requirements Local HPC

- **CPU:** 16+ cores recommended (parallel processing across 5 pipeline steps)

- **RAM:** 32GB+ minimum, 64GB+ recommended for large dataset processing

- **Storage:** 100GB+ available (15-20 GB for pair-level data, rest for outputs)

- **Network:** Internet access for GNSS data download from IGS servers

- **Python:** 3.8+ with required packages (automated installation)

Recommended system: 32+ cores, 64GB+ RAM, 100GB+ storage

#### Key Differences from Paper 1

- **Temporal Coverage:** 25.3 years vs 2.5 years (10× extension)

- **Analysis Centre:** Single-centre (CODE) vs multi-centre (CODE/IGS/ESA)

- **Focus:** Long-baseline stability and long-period phenomena vs cross-centre validation

- **New Capabilities:** 18.6-year nutation detection, enhanced planetary statistics

- **Validation:** 25.3-year aggregate consistency vs cross-centre consistency

- **Computational:** Local HPC vs cloud-based (GCP) deployment

Note: Paper 1's comprehensive validation framework (388 statistical tests, null testing, cross-centre consistency) is adopted and extended with temporal validation.

### Citation and Access

**Repository:** [https://github.com/matthewsmawfield/TEP-GNSS](https://github.com/matthewsmawfield/TEP-GNSS)

**License:** MIT License - Open source with attribution requirements

**Data Access:** GNSS clock products from CODE analysis centre via IGS FTP servers

**Documentation:** Complete analysis code, usage examples, validation reports, and analysis results

**Dependencies:** Python 3.8+, NumPy, SciPy, Pandas, Matplotlib, AstroPy (automated installation)

## Supplementary Materials

### Supplementary Table S1: Multi-Resolution Convergence Analysis

Verification of directional-search stability across four grid resolutions to demonstrate robustness and rule out resolution-dependent artifacts. The p-values are directional-rank statistics within the observed dataset; global sky-search surrogate significance has not yet been computed.

| Resolution | Sky Coverage | r | Directional-rank p | Best Fit (RA, Dec) | CMB Sep | Improvement |
| --- | --- | --- | --- | --- | --- | --- |
| 10° | 703 directions | 0.707 | 0.002 | RA=0°, Dec=0° | 166.1° | baseline |
| **5°** | **2,701 directions** | **0.744** | **$\lt$ 0.0001** | **RA=10°, Dec=+5°** | **158.0°** | **+5.14%** |
| 2.5° | 10,585 directions | 0.744 | 0.0004 | RA=7°, Dec=+5° | 160.9° | +0.05% |
| **1°** | **65,341 directions** | **0.746** | **$\lt$ 0.0001** | **RA=6°, Dec=+4°** | **161.8°** | **+0.38%** |

**Interpretation:** Results converge monotonically across a 93-fold range in grid density with diminishing returns (+5.14% → +0.05% → +0.38%), indicating asymptotic approach to a correlation limit. The marginal 0.38% improvement at 1° resolution despite 6× more grid points (10,585 → 65,341) confirms convergence. All four resolutions identify the same in-ecliptic direction near the aphelion velocity tangent — not the CMB dipole or Solar Apex — with best-fit locations showing stability (RA: 0°→10°→7°→6°; Dec: 0°→+5°→+5°→+4°). The 5° resolution captures 99.6% of the achievable correlation while remaining computationally efficient and consistent with standard practice in cosmological frame studies (Planck ~10°; WMAP ~5-10°). The 1° ultra-high resolution analysis (65,341 directions) provides additional verification and is used for publication figures. The systematic monotonic improvement with finer resolution, combined with asymptotic convergence and location stability, indicates the signal is robust rather than a numerical artifact. CMB separation remains ~158–166° across all resolutions (the dipole-anchored template anti-correlates), while Solar Apex separation remains consistently high (~90°), inconsistent with both fixed-frame hypotheses.

#### Convergence Assessment

- **Monotonic Improvement:** r increases systematically across 93-fold grid density range (0.707 → 0.744 → 0.744 → 0.746)

- **Asymptotic Convergence:** Improvement pattern shows diminishing returns (+5.14% → +0.05% → +0.38%), confirming approach to physical limit

- **Location Stability:** Best-fit coordinates are stable across all resolutions (RA: 0°→10°→7°→6°; Dec: 0°→+5°→+5°→+4°)

- **Direction Consistency:** All four resolutions recover the same in-ecliptic direction ~160° from the CMB dipole and ~6° from the aphelion velocity tangent; Solar Apex remains ~90° away

- **Statistical Significance:** All resolutions give directional-rank p $\lt$ 0.001. These values quantify directional preference within the tested scan; global sky-scan significance is a separate statistic.

- **Highest Resolution Verification:** 1° resolution (65,341 directions) provides additional verification with the sharpest localization (RA=6°, Dec=+4°) and highest correlation (r=0.746)

**Computational Cost vs. Benefit:** While 1° resolution requires 24× longer processing than 5° (65,341 vs 2,701 grid points), it yields only 0.4% total correlation improvement (0.744 → 0.747), confirming 5° captures 99.6% of achievable correlation and represents optimal precision-efficiency balance for routine analysis. The 1° ultra-high resolution verification is used for final publication figures to demonstrate absolute thoroughness.

**Reproducibility Statement:** This pipeline provides complete reproducibility for the 25-year long-baseline analysis presented in this manuscript. All code, data processing steps, validation procedures, and analysis parameters are fully documented and version-controlled. The analysis builds upon the validated methodology from the multi-centre study while extending temporal coverage to enable investigation of long-baseline consistency and long-period geophysical phenomena.

## References

Ashby, N. (2003). Relativity in the Global Positioning System. *Living Reviews in Relativity*, 6(1), 1-42.
Delva, P., Puchades, N., Schönemann, E., et al. (2018). A gravitational redshift test using eccentric Galileo satellites. *Physical Review Letters*, 121(23), 231101.
Hofmann-Wellenhof, B., Lichtenegger, H., & Wasle, E. (2008). *GNSS–Global Navigation Satellite Systems: GPS, GLONASS, Galileo, and more*. Springer-Verlag Wien.
Schaer, S., Villiger, A., Arnold, D., et al. (2021). The CODE ambiguity-fixed clock and phase bias analysis products: generation, properties, and performance. *Journal of Geodesy*, 95, 87.
Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: [10.5281/zenodo.16921911](https://doi.org/10.5281/zenodo.16921911) (Paper 0)
Smawfield, M. L. (2025). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: [10.5281/zenodo.17127229](https://doi.org/10.5281/zenodo.17127229) (Paper 1)
Smawfield, M. L. (2025). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: [10.5281/zenodo.17517141](https://doi.org/10.5281/zenodo.17517141) (Paper 2 — this work)
Smawfield, M. L. (2025). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: [10.5281/zenodo.17860166](https://doi.org/10.5281/zenodo.17860166) (Paper 3)
Smawfield, M. L. (2025). *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: [10.5281/zenodo.17982540](https://doi.org/10.5281/zenodo.17982540) (Paper 4)
Smawfield, M. L. (2025). *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: [10.5281/zenodo.18004832](https://doi.org/10.5281/zenodo.18004832) (Paper 5)
Smawfield, M. L. (2025). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: [10.5281/zenodo.18064365](https://doi.org/10.5281/zenodo.18064365) (Paper 6)
Smawfield, M. L. (2025). *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: [10.5281/zenodo.18059250](https://doi.org/10.5281/zenodo.18059250) (Paper 7)
Smawfield, M. L. (2025). *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: [10.5281/zenodo.18064581](https://doi.org/10.5281/zenodo.18064581) (Paper 8)
Smawfield, M. L. (2025). *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: [10.5281/zenodo.18109760](https://doi.org/10.5281/zenodo.18109760) (Paper 9)
Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: [10.5281/zenodo.18165798](https://doi.org/10.5281/zenodo.18165798) (Paper 10)
Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: [10.5281/zenodo.18209702](https://doi.org/10.5281/zenodo.18209702) (Paper 11)
Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: [10.5281/zenodo.19000827](https://doi.org/10.5281/zenodo.19000827) (Paper 12)
Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: [10.5281/zenodo.19102061](https://doi.org/10.5281/zenodo.19102061) (Paper 13)
Smawfield, M. L. (2026). *Global Time Echoes: MGEX Multi-GNSS Clock Replication, 2025–2026*. Preprint v0.4 (Suva). Zenodo. DOI: [10.5281/zenodo.20572726](https://doi.org/10.5281/zenodo.20572726) (Paper 14)
Smawfield, M. L. (2026). *TEP-PPP: Operational Time Echoes — Phase-Resolved Spatial Clock Priors for GNSS Outage Bridging*. Preprint v0.1 (Pune). In preparation. (Paper 33)
Teunissen, P. J., & Montenbruck, O. (Eds.). (2017). *Springer Handbook of Global Navigation Satellite Systems*. Springer International Publishing.
Villiger, A., Dach, R., Prange, L., et al. (2019). A flexible strategy for handling the datum and initial bias in GNSS clock estimation. *Journal of Geodesy*, 93, 1773–1786.
Dach, R., Schaer, S., Arnold, D., Brockmann, E., Kalarus, M., Lasser, M., Stebler, P., & Jäggi, A. (2024). CODE final product series for the IGS. Astronomical Institute, University of Bern. DOI: [10.48350/197025](https://doi.org/10.48350/197025).
Kouba, J. (2015). A guide to using International GNSS Service (IGS) products. Available at: [UsingIGSProductsVer21_cor.pdf](https://files.igs.org/pub/resource/pubs/UsingIGSProductsVer21_cor.pdf).
Kouba, J., & Héroux, P. (2001). Precise point positioning using IGS orbit and clock products. *GPS Solutions*, 5(2), 12-28.
Petit, G., & Luzum, B. (Eds.). (2010). IERS Conventions (2010). *IERS Technical Note No. 36*. Frankfurt am Main: Verlag des Bundesamts für Kartographie und Geodäsie.
Elsheikh, M., Iqbal, U., Noureldin, A., & Korenberg, M. (2023). The implementation of precise point positioning (PPP): A comprehensive review. *Sensors*, 23(18), 8874. DOI: [10.3390/s23188874](https://doi.org/10.3390/s23188874).
Gendt, G., & Schmid, R. (2005). A common-coordinate approach to global GPS analysis. *IGS Technical Report 2004*, 131-141.
Steigenberger, P., Montenbruck, O., Dach, R., et al. (2021). CODE reprocessing 1995-2020: improved GPS orbits and clocks. *Journal of Geodesy*, 95, 65.
International GNSS Service. (2023). *IGS Technical Report 2023*, Chapter 7: Clock Product Generation.
Ray, J., Gurtner, W., & Coleman, M. J. (2017). RINEX Extensions to Handle Clock Information, Version 3.04. *International GNSS Service*. Available at: [https://files.igs.org/pub/data/format/rinex_clock304.txt](https://files.igs.org/pub/data/format/rinex_clock304.txt).
Theiler, J., Eubank, S., Longtin, A., Galdrikian, B., & Farmer, J. D. (1992). Testing for nonlinearity in time series: the method of surrogate data. *Physica D: Nonlinear Phenomena*, 58(1-4), 77-94. DOI: [10.1016/0167-2789(92)90102-S](https://doi.org/10.1016/0167-2789(92)90102-S).
Welch, P. D. (1967). The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms. *IEEE Transactions on Audio and Electroacoustics*, 15(2), 70-73.
Li, A., Wang, Y., & Guo, M. (2024). Analysis of the Spatial Distribution and Common Mode Error Correlation in a Small-Scale GNSS Network. *Sensors*, 24(17), 5731. DOI: [10.3390/s24175731](https://doi.org/10.3390/s24175731).
Chanard, K., Métois, M., Rebischung, P., & Avouac, J.-P. (2020). A warning against over-interpretation of seasonal signals measured by the Global Navigation Satellite System. *Nature Communications*, 11, 1375. DOI: [10.1038/s41467-020-15100-7](https://doi.org/10.1038/s41467-020-15100-7).
Cheng, W., Nie, G., & Zhu, J. (2023). Characterizing Periodic Variations of Atomic Frequency Standards via Their Frequency Stability Estimates. *Sensors*, 23(11), 5271. DOI: [10.3390/s23115271](https://doi.org/10.3390/s23115271).
Senior, K. L., Ray, J. R., & Beard, R. L. (2008). Characterization of periodic variations in the GPS satellite clocks. *GPS Solutions*, 12, 211-225. DOI: [10.1007/s10291-008-0089-9](https://doi.org/10.1007/s10291-008-0089-9).
Ray, J., Altamimi, Z., Collilieux, X., & van Dam, T. (2007). Anomalous harmonics in the spectra of GPS position estimates. *GPS Solutions*, 12, 55-64. DOI: [10.1007/s10291-007-0067-7](https://doi.org/10.1007/s10291-007-0067-7).
Rodriguez-Solano, C. J., Hugentobler, U., Steigenberger, P., & Lutz, S. (2014). Reducing the draconitic errors in GNSS geodetic products. *Journal of Geodesy*, 88, 559–574. DOI: [10.1007/s00190-014-0704-1](https://doi.org/10.1007/s00190-014-0704-1).
Amiri-Simkooei, A. R. (2013). On the nature of GPS draconitic year periodic pattern in multivariate position time series. *Journal of Geophysical Research: Solid Earth*, 118, 3500–3511. DOI: [10.1002/jgrb.50199](https://doi.org/10.1002/jgrb.50199).
Griffiths, J., & Ray, J. (2013). On the precision and accuracy of IGS orbits. *Journal of Geodesy*, 87, 80–91. DOI: [10.1007/s00190-012-0599-z](https://doi.org/10.1007/s00190-012-0599-z).
Rebischung, P., Altamimi, Z., Ray, J., & Garayt, B. (2016). The IGS contribution to ITRF2014. *Journal of Geodesy*, 90, 611–630. DOI: [10.1007/s00190-016-0897-6](https://doi.org/10.1007/s00190-016-0897-6).

### Data Sources

Johnston, G., Riddell, A., & Hausler, G. (2017). The International GNSS Service. In P. J. G. Teunissen & O. Montenbruck (Eds.), *Springer Handbook of Global Navigation Satellite Systems* (1st ed., pp. 967-982). Cham, Switzerland: Springer International Publishing. [https://doi.org/10.1007/978-3-319-42928-1_33](https://doi.org/10.1007/978-3-319-42928-1_33).
Dach, R., Lutz, S., Walser, P., & Fridez, P. (Eds.). (2015). *Bernese GNSS Software Version 5.2*. Astronomical Institute, University of Bern, Switzerland. Available at: [http://www.bernese.unibe.ch/](http://www.bernese.unibe.ch/).
Folkner, W. M., Williams, J. G., Boggs, D. H., Park, R. S., & Kuchynka, P. (2014). The Planetary and Lunar Ephemerides DE430 and DE431. *IPN Progress Report 42-196*, Jet Propulsion Laboratory, California Institute of Technology. Available at: [https://ipnpr.jpl.nasa.gov/progress_report/42-196/196C.pdf](https://ipnpr.jpl.nasa.gov/progress_report/42-196/196C.pdf).
Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., Earl, N., et al. (2022). The Astropy Project: Sustaining and Growing a Community-oriented Open-source Project and the Latest Major Release (v5.0) of the Core Package. *The Astrophysical Journal*, 935(2), 167. [https://doi.org/10.3847/1538-4357/ac7c74](https://doi.org/10.3847/1538-4357/ac7c74).
Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. (2013). Astropy: A community Python package for astronomy. *Astronomy & Astrophysics*, 558, A33. [https://doi.org/10.1051/0004-6361/201322068](https://doi.org/10.1051/0004-6361/201322068).

## How to cite

**Cite as:** Smawfield, M. L. (2025). Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products. Preprint v0.21 (Cairo). Zenodo. DOI: 10.5281/zenodo.17517141

**BibTeX:**

```
@misc{Smawfield_TEP_GNSS_Longspan_2025,
author = {Matthew Lukin Smawfield},
title = {Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products (v0.21 Cairo)},
year = {2025},
publisher = {Zenodo},
doi = {10.5281/zenodo.17517141},
url = {https://doi.org/10.5281/zenodo.17517141},
note = {Preprint}
}
```

## Data Availability

### Primary Data Sources

**GNSS Clock Products:** CODE (Center for Orbit Determination in Europe) final clock solutions (30-second epochs, CLK format) processed by the Astronomical Institute, University of Bern (AIUB) as part of the International GNSS Service (IGS). Data are freely available under IGS Terms of Use.

- **Source:** [http://ftp.aiub.unibe.ch/CODE/](http://ftp.aiub.unibe.ch/CODE/)

- **Temporal Coverage:** March 1, 2000 – June 30, 2025 (25.3 years, 9,218 days)

- **Format:** RINEX 3 CLK (compressed: .gz or .Z)

- **License:** Freely available for scientific use. Users must cite IGS and CODE appropriately (see Johnston et al., 2017; Steigenberger et al., 2021)

- **Terms of Use:** [IGS Data and Product Disclaimer](https://igs.org/wp-content/uploads/2020/09/IGS-Data-and-Product-Disclaimer-and-Terms-of-Use-200805.pdf)

**Station Coordinates:** IGS Network station coordinates obtained from the IGS station metadata JSON file, providing ITRF2020 Cartesian coordinates (X, Y, Z) for all active and former IGS stations.

- **Source:** [IGS Network Metadata (JSON)](https://files.igs.org/pub/station/general/IGSNetworkWithFormer.json)

- **Coverage:** 474 unique physical receivers (814 total station codes including legacy 4-character and extended 9-character variants)

- **License:** Freely available under IGS Terms of Use

**Planetary Ephemeris:** High-precision planetary positions computed using JPL Development Ephemeris DE432s via Astropy's solar system ephemeris interface. DE432s provides planetary and lunar positions from 1550 to 2650 CE with meter-level accuracy.

- **Source:** NASA Jet Propulsion Laboratory (JPL) via Astropy

- **Ephemeris Version:** DE432s (short-span, optimized for modern era)

- **Reference:** Folkner et al. (2014), IPN Progress Report 42-196

- **Access:** [JPL Planetary Ephemeris Export](https://ssd.jpl.nasa.gov/planets/eph_export.html)

- **License:** Public domain (U.S. Government work)

### Software and Analysis Tools

**Analysis Code:** Complete Python pipeline (TEP-GNSS) available under MIT License at [GitHub: TEP-GNSS](https://github.com/matthewsmawfield/TEP-GNSS). Includes data acquisition, preprocessing, statistical validation, and figure generation scripts.

**Key Dependencies:**

- **Astropy:** Astronomical calculations and ephemeris access (Astropy Collaboration, 2013, 2022)

- **NumPy/SciPy:** Numerical computing and statistical analysis

- **Pandas:** Data manipulation and time series analysis

- **Matplotlib:** Scientific visualization

### Multi-Constellation Cross-Validation Data (MGEX)

**MGEX Clock Products:** Dedicated replication of multi-GNSS clock correlation (Paper 14) uses CODE MGEX final products (COD0MGXFIN, 2025–2026, 418 stations, 1.83 million pair-days; canonical λ = 1,862 ± 155 km). Earlier exploratory cross-centre comparisons (2020–2024) include:

- **CODE (COD0MGXFIN):** Primary multi-GNSS replication dataset (Paper 14); R² = 0.699–0.917, λ = 1,862 ± 155 km across GPS, GLONASS, Galileo, and BeiDou

- **GBM (GFZ Potsdam):** Exploratory multi-centre pilot; R² = 0.950, λ = 3,042 ± 297 km

- **WUM (Wuhan University):** Exploratory multi-centre pilot; R² = 0.911, λ = 2,009 km

- **JPL (NASA):** Exploratory multi-centre pilot; R² = 0.929, λ = 1,829 km

- **Source:** [CODE FTP](http://ftp.aiub.unibe.ch/CODE/) and [GFZ MGEX FTP](ftp://ftp.gfz-potsdam.de/GNSS/products/mgex/)

- **Constellations:** GPS, GLONASS, Galileo, BeiDou

### Derived Products and Supplementary Materials

**Analysis Results:** Comprehensive JSON output files, processed correlation data, and extended figures available in the [Zenodo record (DOI: 10.5281/zenodo.17517141)](https://zenodo.org/records/17517141).

**Data Attribution:** By using data from this study, users agree to cite the original data providers (IGS, CODE/AIUB, JPL) as well as this work. All data sources are freely available for scientific research under their respective terms of use.

## Declarations

**Funding:** This work received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

**Competing Interests:** The author declares no competing financial interests.

**Author Contributions:** M.L.S. designed the study, performed all analyses, and wrote the manuscript.

## Contact

For questions, comments, or collaboration opportunities regarding this work, please contact:

**Matthew Lukin Smawfield**

[matthew@mlsmawfield.com](mailto:matthew@mlsmawfield.com)

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*This document was automatically generated from the TEP-GNSS research site. For the interactive version with figures and enhanced formatting, visit: https://mlsmawfield.com/tep/gnss-ii/*

*Source code and data available at: https://github.com/matthewsmawfield/TEP-GNSS*

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# Global Time Echoes: Raw RINEX Consistency Test
**Matthew Lukin Smawfield**
Version: v0.9 (Kathmandu)
First published: 9 December 2025 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.17860166

---

## Abstract

This paper validates that distance-structured correlations in GNSS clocks exist in raw observations using broadcast ephemerides, not just precise products—strongly constraining precise-product processing artifacts. Because broadcast ephemerides still carry control-segment orbit information, terrestrial verification follows a three-tier closure hierarchy: the GNSS analyses (Papers 1–3, 14) establish and characterize the macroscopic clock-network covariance across processing chains and raw observables; passive Satellite Laser Ranging (Paper 8) provides a GNSS-independent optical-domain sensitivity bound and orbit-provenance check; and orbit-free optical-fiber and two-way satellite time-transfer (TWSTFT) links provide the definitive metrological closure at atomic-clock precision. Prior TEP analyses relied on precise orbit and clock products from global analysis centers, leaving open the possibility that observed signatures were artifacts of sophisticated network adjustments. This paper addresses that concern by detecting distance-structured signatures in raw GNSS observations processed using Single Point Positioning (SPP) with broadcast ephemerides as the primary methodology, supplemented by precise ephemeris validation. Analysis of 539 globally distributed stations over 3 years (2022–2024, comprising 1.17 billion pair-samples across three complementary filtering strategies) achieves consistent signal detection across all 72 non-independent metric combinations used as cross-mode consistency checks (mean R² = 0.93), revealing directionally structured correlations consistent with CODE's 25-year PPP findings (nominal pair-level p $\lt$ 10−15 under the standard white-noise null across ≈1.7×108 pair-epochs from 144,991 unique station pairs).

The primary finding is directional anisotropy: East-West correlations are 2–5% (MSC) to 22% (mean phase-offset alignment) stronger than North-South at short distances ($\lt$ 500 km), with t-statistics up to 112 and Cohen's d up to 0.304. Month-by-month stratification shows stable polarity (E-W > N-S) at 94% or higher level across modes and metrics (worst case 34/36 months), bounding random fluctuation while motivating geometric controls. A critical audit indicates this is not an artifact of distance distribution: E-W pairs are actually 13 km *longer* than N-S pairs (bias against signal), and robust distance-matching strengthens the ratio (1.033 → 1.041). At full distances, empirical CODE-referenced and independently simulated geometry corrections provide supplementary comparisons; the primary short-distance anisotropy is measured without either correction.

Key validations include: (1) orbital velocity coupling with negative sign in 17 of 18 filter/metric/coherence cells (best: r = −0.763, ≈ 3.0σ after Bretherton autocorrelation correction; ≈ 2.0σ after conservative 18-cell Bonferroni correction; nominal 5.4σ over 36 months), replicating the sign of CODE's 25-year finding (r = −0.888), persisting under dual-frequency ionospheric removal (r = −0.416, 2.5σ), and dominated in Multi-GNSS by a pure annual harmonic ($A_1/A_2 \approx 3.98$) phase-locked to perihelion across both hemispheres while equinoctial semiannual artifacts are suppressed by >5.5×; (2) joint navigation-solve projection of position jitter and clock bias (matched on MSC at Δ ≈ 1–9% in baseline GPS and Δ ≈ 5% in Multi-GNSS ALL_STATIONS, expected from coupled state estimation rather than treated as an independent conformal test); (3) a stable annual phase expressible as an in-ecliptic preferred direction at RA = 8°, Dec = +5° (β = +1.4°), matching the corrected CODE benchmark vector to ~2° and lying 3.9° from Earth's aphelion velocity tangent and 160.0° from the CMB dipole — identifying the heliocentric-orbital frame rather than a cosmological rest frame; (4) geomagnetic invariance at the primary GFZ Kp threshold (Kp $\lt$ 3 vs. Kp ≥ 3; median Δλ ≈ −1%, with 60/72 tests within ±5%); and (5) season-preserving planetary-event controls (uniform-date ensembles, same-DOY calendar twins, deseasonalized rescoring), which demonstrate that the nominal 2.8× free-centre pulse excess over date-shuffled nulls arises from calendar-position sampling of the seasonal curve while confirming the absence of GM/r² or GM/r³ scaling.

This paper constitutes Paper 3 of the TEP-GNSS Research Series. Together with Paper 1 (multi-center validation) and Paper 2 (25-year temporal stability), these three complementary analyses—using a different data layer (raw observations rather than analysis-centre clock products), an independent estimation chain, and different time periods over the same physical tracking network—provide consistent evidence for planetary-scale, directionally-structured correlations in GNSS clock measurements. The observed signature of a stable annual phase expressible as an in-ecliptic preferred direction, and orbital velocity dependence is consistent with the Temporal Equivalence Principle hypothesis, which preserves local Lorentz invariance while predicting global path-dependent synchronization. Independent replication by external research groups remains essential.

## Executive Summary

### Research Question

Prior TEP analyses (Papers 1 and 2) relied on precise orbit and clock products from global analysis centers. Since these products are derived using complex network adjustments and integer ambiguity resolution, a critical ambiguity remained:

"Are the observed signatures artifacts of the processing chain, or do they exist in the raw observations?"

This study addresses this question. By analyzing raw RINEX pseudorange measurements processed via Single Point Positioning (SPP) primarily with broadcast ephemerides (supplemented by precise ephemeris validation), the results indicate that the same directional anisotropy, orbital velocity coupling, and annual-phase preferred direction are present in the fundamental data. This is consistent with, but not decisive proof of, the observed structure not being solely a byproduct of precise product generation.

### Key Findings

- **Raw Data Validation:** Distance-structured signatures detected in raw RINEX data using only broadcast ephemerides and Single Point Positioning. Analysis of 539 stations over 3 years (1.17 billion pair-samples) achieves signal-consistent structure in all 72 tested metric combinations with directional anisotropy matching CODE's 25-year findings (nominal pair-level p $\lt$ 10−15).

- **Consistent Detection across all Metrics:** Signal consistent with TEP detected in all 72 non-independent metric combinations used as consistency checks across processing modes and observables (3 station filters × 4 processing modes × 3 observables × 2 coherence types). Mean R² = 0.93, with all fits exceeding the 0.5 threshold.

- **Directional Anisotropy:** E-W correlations are 2–5% (MSC) to 22% (mean phase-offset alignment) stronger than N-S at short distances ($\lt$ 500 km), matching CODE's directional signature with nominal pair-level p $\lt$ 10−15. A critical distance audit reveals E-W pairs are 13 km longer than N-S pairs (a bias *against* the signal); robust distance-matching strengthens the coherence ratio from 1.033 to 1.041.

- **Multi-Mode Validation:** Signal detected in GPS-only (ratio 1.033), dual-frequency ionofree (1.019), and multi-GNSS (1.050).

- **Geometry Diagnostics:** Full-distance CODE-referenced and independently simulated geometry corrections provide supplementary comparisons; neither is used in the primary short-distance anisotropy result.

- **Geomagnetic Independence:** Primary Kp stratification (Kp$\lt$ 3 vs Kp≥3) shows near-invariance (median ΔλT ≈ −1%, 60/72 tests within ±5% across filters/modes/metrics).

- **Hemisphere Stratification:** In the ALL_STATIONS analysis, both hemispheres show E-W > N-S (phase alignment 1.200/1.348).

- **Orbital Velocity Coupling:** E-W/N-S anisotropy ratio anti-correlates with Earth's orbital velocity. Multi-GNSS yields the strongest detection: r = −0.763, nominal 5.4σ (pos_jitter/phase; ≈ 3.0σ autocorrelation-corrected), with MSC yielding r = −0.610, nominal 4.0σ (≈ 2.5σ corrected; nominal significances assume 36 independent monthly points).

- **Robust Cross-Filter Consistency:** All three station filtering methods (ALL_STATIONS, OPTIMAL_100, DYNAMIC_50) produce consistent Temporal Topology correlation lengths with CV $\lt$ 15%.

- **Raw-SPP Scale:** MSC channels return λT ≈ 0.7–1.1×103 km (ionofree 1,069 km, R² = 0.969). These are noise-floor-limited projections that bound the scale from below, not estimates of the underlying correlation length.

- **Altitude Independence:** Correlation length is independent of station altitude. Across 360 regressions spanning global and latitude-controlled altitude quintiles, only 3.1% show p $\lt$ 0.05 slopes.

- **Ionofree Phase λT ≈ CODE (2024):** The 2024 OPTIMAL_100 Ionofree pos_jitter/phase alignment yields *λT = 4,767 ± 835 km*, consistent in scale with CODE's 25-year directional family (cross-sector mean and dispersion 4,201 ± 1,967 km; one filter/mode/year cell, not the headline scale).

- **Temporal Stability:** Across the full 72-channel year-by-year grid (3 filters × 4 modes × 3 metrics × 2 coherence types), 66/72 channels have year-to-year CV $\lt$ 20%.

- **Monthly Anisotropy Consistency:** Month-by-month short-distance anisotropy shows stable polarity (E-W > N-S) at 94% or higher level across modes and metrics.

- **Seasonal Stability:** Comprehensive seasonal stratification reveals three complementary signatures: (1) "Summer Enhancement", (2) "Invariant Core", and (3) "Network-wide Baseline".

- **Null Tests Passed:** Comprehensive validation across 72 combinations constrains several alternative explanations: (1) Solar rotation shows zero correlation, (2) Lunar tides show zero correlation, and (3) Shuffle test demonstrates real structure.

- **Metric Complementarity:** MSC measures magnitude-squared coherence, while the signed phase statistic reports mean phase offset; resultant magnitude is required to assess phase concentration.

- **Planetary Event Channel:** Event windows show 2.8× higher pulse-detection rates than a date-shuffled null, but season-preserving controls (Step 2.6b) attribute the excess to calendar-position sampling of the seasonal series — an event-locked component is not established in this channel; no GM/r² or GM/r³ scaling is detected.

- **In-Ecliptic Preferred Direction:** Comprehensive 72-combination full-sky grid search recovers a stable annual phase (RA = 8°, Dec = +5°, β = +1.4°) consistent with the corrected CODE 25-year benchmark vector; competitive controls identify it as Earth's aphelion-velocity direction (3.9° away), with the CMB dipole 160° away — a heliocentric-orbital readout, not a cosmological-frame detection.

- **Large-Scale Dataset:** 1.17 billion pair-samples analyzed across all filters.

### Significance of Results

The detection of directionally structured correlations in raw RINEX data, processed with only broadcast ephemerides, addresses a key alternative hypothesis: that TEP signals are artifacts of precise product generation.

By analyzing the fundamental observables to the rawest observables, this analysis shows that the signal is not evidently a fragile artifact of sophisticated processing, but a robust feature of the data itself—a baseline correlation that remains under conservative preprocessing.

This represents a third complementary line of evidence for TEP:

- **Paper 1:** Multi-center validation (CODE, ESA, IGS) — supports the interpretation that the signal is not center-specific

- **Paper 2:** 25-year CODE analysis — supports temporal persistence within the analyzed interval

- **Paper 3 (This Paper):** Raw RINEX consistency test — supports the interpretation that the signal is not solely a processing artifact

Together, these three complementary analyses provide consistent evidence for distance-structured correlations in GNSS clock measurements. Independent replication by external research groups remains essential to assess programme-specific systematics.

### Methodology Highlights

- **Data Source:** NASA CDDIS archive — raw RINEX 3.x observation files

- **Processing:** RTKLIB Single Point Positioning with broadcast ephemerides (primary) and precise ephemerides (validation)

- **Time Alignment:** Pandas DatetimeIndex alignment (identical to CODE longspan methodology)

- **Coherence:** Magnitude-weighted phase coherence via cross-spectral density

- **Fitting:** Inverse-variance weighted exponential decay: C(r) = A·exp(-r/λ) + C_0

### Conclusion

This paper provides raw-observation validation of distance-structured GNSS correlations. The combined evidence of directional anisotropy, orbital-velocity coupling and consistency across processing modes suggests that the observed structure is not confined to precise analysis-center products. These clock-network covariance patterns are consistent with the Temporal Equivalence Principle's dynamical proper-time hypothesis. Synchronization holonomy is a distinct observable and is not directly measured by this analysis.

Related: [Paper 1 (Multi-Center)](https://mlsmawfield.com/tep/gnss-i/) · [Paper 2 (25-Year CODE)](https://mlsmawfield.com/tep/gnss-ii/)

## 1. Introduction: The TEP Research Program

## 1.1 The Theoretical Hypothesis

The Temporal Equivalence Principle (TEP) represents a proposed extension to
the foundations of General Relativity, positing a fundamental coupling
between spatial and temporal fluctuations in geodetic measurements.
Formulated within a Bi-Metric Geometry framework (TEP v0.16, Jakarta;
Smawfield, 2025) where matter and clocks couple universally to the canonical causal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$; Paper 0, §2.2, Step 80), the theory employs continuous geometric screening rather
than discrete thin-shell approximations: the scalar time field exhibits a
spatially varying Temporal Topology. The spatial profile of the conformal time field manifests as synchronized fluctuations in
proper time flow ($C_A$ covariance) across spatially separated atomic clocks, while the
Earth-vicinity Temporal Shear is pairwise-screened. The clock-sector response is governed by
the clock covariance $C_A$; the terrestrial $S_A^{(\oplus)}$ characterizes static conformal depth, while the Earth-vicinity shear response is screened separately without imposing a step-function boundary.
TEP implies that local variations in the gravitational potential should
produce observable phase coherence structures, predicting a breakdown of
global simultaneity while preserving exact local Lorentz invariance. This
approach parallels recent advances in using global atomic clock networks for
fundamental physics, including dark matter searches (Wcisło et al., 2018)
and relativistic geodesy (Lisdat et al., 2016).

This hypothesis yields a specific, falsifiable prediction: Inter-station
clock coherence should exhibit exponential decay with distance ($C(r)
\propto e^{-r/\lambda}$), driven by a scalar field correlation length
$\lambda$ on the order of 10³ km.

## 1.2 The Empirical Foundation

To date, this hypothesis has been tested through two comprehensive analyses:

### Paper 1: Multi-Center Validation

Analysis of precise orbit and clock products from three
software-diverse analysis centers (CODE, ESA, IGS) on the shared IGS
tracking network found exponential decay signatures
with λ ≈ 3,500–4,500 km. The consistency across centers (R² > 0.92)
disfavors center-specific software artifacts; network-level common
modes are not bounded by that comparison.

→ View Paper 1 (TEP-GNSS)

### Paper 2: 25-Year Temporal Stability

A longitudinal study of 25 years of CODE data (2000-2025) found that
these signatures are not confined to a transient interval. They persist
across solar cycles, hardware generations, and reference frame updates,
exhibiting statistically significant coupling with orbital dynamics.

→ View Paper 2 (TEP-GNSS-II)

### 1.3 The Processing Artifact Objection

Despite these successes, a critical scientific objection remains valid:

"Are these signatures artifacts of the sophisticated processing chains used
to generate precise products, or do they exist in the raw observations
themselves?"

Precise Point Positioning (PPP) products rely on sophisticated network
adjustments, integer ambiguity resolution, and inter-station
constraints—processes that could, in principle, introduce spurious
long-range correlations. If the TEP signal were merely a byproduct of these
mathematical filters, it would be a software artifact of global network
estimation. However, if the signal exists in the raw, uncorrected
observations processed station-by-station, it cannot be attributed to
inter-station PPP network adjustment algorithms. To test this directly, this
study descends the processing hierarchy to raw pseudorange measurements
processed via Single Point Positioning (SPP) with broadcast ephemerides.
Because broadcast navigation messages still originate from control-segment
orbit fits, physical closure across the terrestrial programme is structured
in three distinct tiers: (i) the GNSS suite (Papers 1–3 and 14) serves as
the discovery and characterization phase of the macroscopic clock-network
covariance across independent estimators and raw observables; (ii) passive
Satellite Laser Ranging (Paper 8) provides an instrumentally independent
optical-domain sensitivity bound ($\delta A \lesssim 2\times 10^{-8}$) and
orbit-provenance control; and (iii) orbit-free optical-fiber and two-way
satellite time-transfer (TWSTFT) links provide the planned definitive
metrological closure at $10^{-15}$–$10^{-18}$ clock stability.

## 1.4 Objectives of This Capstone Study

This paper serves as the final study of the TEP-GNSS research program. Its
primary objective is to perform an independent test of the TEP signal by:

Addressing the Processing Artifact Objection by
detecting distance-structured signatures in raw RINEX data using Single
Point Positioning (SPP) with broadcast ephemerides as the primary
methodology, supplemented by precise ephemeris validation.

Validating directional anisotropy — testing whether E-W
correlations exceed N-S as found in CODE's 25-year analysis.

Comparing spatial vs. temporal correlation lengths to
test the core Space-Time Coupling prediction.

Validating environmental independence — stratifying by
geomagnetic activity and season to assess ionospheric and atmospheric
origins.

Synthesizing findings across all three papers to
establish a unified evidence framework.

### 1.5 Paper Structure

The remainder of this paper is organized as follows:

**Section 2:** Methodology — A fundamental,
first-principles approach using raw RINEX data

**Section 3:** Results — Detection of exponential decay and
directional anisotropy

**Section 4:** Validation — Null tests, geomagnetic
stratification, and systematic effects

**Section 5:** Synthesis — The convergence of evidence
across Papers 1, 2, and 3

**Section 6:** Discussion — Physical implications and
future directions

- **Section 7:** Conclusions — Final assessment

**Section 8:** Analysis Package — Reproducibility
documentation

## 2. Data and Methods

### 2.1 Data Sources

#### 2.1.1 NASA CDDIS Archive

All observation data were obtained from the Crustal Dynamics Data Information System (CDDIS), operated by NASA Goddard Space Flight Center. CDDIS serves as the primary archive for the International GNSS Service (IGS) and provides open access to RINEX observation files from the global tracking network.

| Parameter | Value |
| --- | --- |
| Archive URL | [cddis.nasa.gov/archive/gnss/data/daily](https://cddis.nasa.gov/archive/gnss/data/daily) |
| Network | IGS/MGEX Global Network |
| Stations Processed | 539 stations |
| Stations After DYNAMIC_50 Filtering | ~400 stations (smart filter: jumps $\lt$ 500 ns, std $\lt$ 50 ns) |
| Time Span | 2022-01-01 – 2024-12-31 (~1,096 days, 3 years) |
| RINEX Files (total) | 409,737 observation files |
| Files after DYNAMIC_50 filter | 316,497 files (77% pass rate) |
| File Format | RINEX 3.x (Hatanaka compressed) |
| Processing Interval | 5 minutes (300 seconds) |

#### 2.1.2 Broadcast Ephemerides

Unlike Papers 1 and 2, which used precise orbit and clock products, this analysis relies solely on broadcast navigation messages. Broadcast ephemerides provide satellite positions with ~1 meter accuracy (vs. ~2 cm for precise products) and satellite clocks with ~5 ns accuracy (vs. ~0.1 ns for precise products).

This choice ensures complete independence from the processing chains used in previous analyses. The tracking network itself is shared: the same stations and constellations underlie the analysis-centre products of Papers 1, 2 and 14, so network-level or control-segment common modes remain a shared error budget, while estimation-chain artifacts — the dominant concern those papers could not internally exclude — are fully decoupled here.

#### 2.1.3 External Auxiliary Data

Two external data sources are used for geomagnetic and planetary event analyses:

##### Geomagnetic Kp Index

| Parameter | Value |
| --- | --- |
| Source | GFZ Helmholtz Centre Potsdam |
| URL | [Kp_ap_since_1932.txt](https://www-app3.gfz-potsdam.de/kp_index/Kp_ap_since_1932.txt) |
| Coverage | 1932–present (3-hourly values) |
| Aggregation | Daily mean Kp |
| Primary stratification | Quiet: Kp $\lt$ 3.0; Storm: Kp ≥ 3.0 |
| Sensitivity thresholds | Additional storm definitions: Kp ≥ 4.0 and Kp ≥ 5.0 |

Real Kp index data is downloaded directly from GFZ at runtime. No synthetic or approximated geomagnetic data is used. If the download fails, the analysis terminates with an error.

##### Planetary Ephemeris

| Parameter | Value |
| --- | --- |
| Source | JPL DE440 via Astropy |
| Event dates | Verified against astronomical almanac (astropixels.com, JPL Horizons) |
| Distance calculation | Barycentric Earth-planet distances from JPL ephemeris |
| GM values | IAU 2015 standards |

Planetary conjunction/opposition dates were verified against multiple sources on 2024-12-06. No approximate or fabricated planetary data is used.

##### Data Authenticity Verification

All auxiliary data (Kp, planetary ephemeris) comes from authoritative scientific sources. The analysis pipeline is designed to fail immediately if authentic data cannot be obtained, rather than fall back to approximations. This ensures all reported results are based exclusively on real observational data.

### 2.2 Terminology: Station Pairs vs. Pair-Samples

To avoid ambiguity in statistical reporting, this manuscript distinguishes between two related concepts:

| Term | Definition | Example |
| --- | --- | --- |
| Station Pairs | Unique spatial combinations of two stations | 539 stations → 144,991 unique pairs |
| Pair-Samples | Daily observations of each station pair | 144,991 pairs × 1,096 days ≈ 159M pair-samples |
| Total Analyzed | Sum across all filters and modes | 1.17 billion pair-samples (all combinations) |

Unless otherwise specified, "pairs" in statistical contexts refers to pair-samples (the unit of observation for coherence analysis), while "station pairs" refers to unique spatial combinations. The reported sample sizes (e.g., "61.9M pairs" for Baseline mode) represent pair-samples—daily observations that contribute to the coherence estimates.

### 2.3 Station Filtering Strategies

To ensure robustness and enable cross-validation, analyses were run under three station selection strategies. These filters serve distinct objectives: OPTIMAL_100 emphasizes global spatial structure, while DYNAMIC_50 emphasizes temporal stability.

#### Station Filters Compared

| Filter | Description | Stations | Purpose |
| --- | --- | --- | --- |
| All Stations | No filter applied | 539 | Maximum statistical power, full network coverage |
| Optimal 100 | Curated subset with hemisphere balance and clock stability | 100 | Reduce Northern Hemisphere bias; high-quality clocks only |
| Dynamic 50 | Per-file filter: clock std $\lt$ 50 ns, jumps $\lt$ 500 ns | ~400 | Strict quality control; excludes noisy files per station |

#### Optimal 100 Station Selection Criteria

The optimal_100_metadata.json contains a curated list of 100 stations selected to maximize scientific validity:

- Hemisphere balance: ~50 Northern / ~50 Southern stations to avoid NH network bias

- Clock stability: Stations with mean clock std $\lt$ 50 ns across the analysis period

- Data availability: Minimum 100 daily files over 3 years

- Geographic distribution: Spread across continents to ensure global coverage

This addresses the IGS network's inherent bias (238 NH vs 106 SH stations globally; the analyzed 539-station set itself contains 285 NH and 254 SH stations) that would otherwise dominate statistical results.

#### Dynamic Filter Implementation (DYNAMIC_50)

The `dynamic:50` filter applies strict per-file quality thresholds:

```

for each RINEX file:
    if clock_bias_std $\lt$ 50 ns AND
       max_jump $\lt$ 500 ns AND
       total_range $\lt$ 5000 ns:
        include in analysis
    else:
        exclude (noisy day for this station)
    
```

This strict quality filtering passes 316,497 high-quality daily files from ~400 stations (77% pass rate), rejecting 93,240 files: 47,816 for excessive jumps, 4,045 for excessive range, and 41,379 for high standard deviation. This ensures only the most stable clock data contributes to the analysis.

### 2.4 Processing Pipeline

The processing chain transforms raw GNSS observations into signal detection results:

flowchart LR
A[("CDDIS
RINEX 3")] &ndash;> B["RTKLIB SPP
4 modes"]
B &ndash;> C["3 Metrics
bias · jitter · drift"]
C &ndash;> D[(".npz
per station-day")]
D &ndash;> E["Pair Coherence
MSC + Phase"]
E &ndash;> F["Exp Fit
C(r) = A·exp(-r/λ) + C_0"]
F &ndash;> G{{"λ, R²
Signal Detection"}}

style A fill:#eceff1,stroke:#455a64,stroke-width:2px,color:#263238
style B fill:#e3f2fd,stroke:#1565c0,stroke-width:2px,color:#0d47a1
style C fill:#e3f2fd,stroke:#1565c0,stroke-width:2px,color:#0d47a1
style D fill:#e1f5fe,stroke:#0277bd,stroke-width:2px,color:#01579b
style E fill:#e0f7fa,stroke:#0097a7,stroke-width:2px,color:#006064
style F fill:#e0f2f1,stroke:#00897b,stroke-width:2px,color:#004d40
style G fill:#ffffff,stroke:#2962ff,stroke-width:3px,color:#2962ff

*Figure 2.4.1: Processing pipeline. Four modes: Baseline (broadcast), Ionofree (dual-freq), Multi-GNSS (all constellations), Precise (IGS SP3). Station pairs: 50–13,000 km. TEP frequency band: 10–500 µHz (periods 33 min–28 hr). Two coherence metrics enable cross-validation: MSC (amplitude) and Phase Alignment (timing).*

#### 2.4.1 Single Point Positioning (SPP)

Raw RINEX observations were processed using RTKLIB's `rnx2rtkp` utility in Single Point Positioning mode. SPP determines the receiver position and clock offset using pseudorange measurements from multiple satellites.

##### RTKLIB Configuration

| Parameter | Setting |
| --- | --- |
| Positioning Mode | Single Point (SPP) |
| Satellite Ephemeris | Broadcast only |
| Ionosphere Correction | Klobuchar (broadcast) / Dual-freq (ionofree mode) |
| Troposphere Correction | Saastamoinen model |
| Elevation Mask | 15° |
| Processing Interval | 5 minutes (300 seconds) |
| Navigation System | GPS (L1) / GPS+GLONASS+Galileo+BeiDou (multi-GNSS) |

##### Multi-Mode Processing Strategy

Each RINEX file is processed in four modes to enable comprehensive cross-validation. This multi-mode approach is a validation strategy: if TEP is real, ALL modes should show similar Temporal Topology correlation lengths (λT). If TEP were an artifact of a specific processing method, different modes would produce different signatures. Outcome: modes do not return similar λT — fitted scales range from ~0.6×103 km (MSC) to ~4.8×103 km (ionofree phase), so this criterion is not met as stated. The mode dependence is attributed to noise-floor projection in the single-frequency and MSC channels. What stays consistent across modes is the sign of the E-W > N-S ordering and of the orbital-velocity coupling, not the scale.

**Outcome:** modes do not return similar λT. Fitted scales range from ~0.6×103 km (MSC) to ~4.8×103 km (ionofree phase), so this criterion is not met as stated. The mode dependence is attributed to noise-floor projection in the single-frequency and MSC channels (§3.7). What stays consistent across modes is the sign of the E-W > N-S ordering and of the orbital-velocity coupling, not the scale.

| Mode | Frequencies | Systems | Ephemeris | Ionosphere | Purpose |
| --- | --- | --- | --- | --- | --- |
| Baseline | L1 only | GPS | Broadcast | Klobuchar model | Reference (simplest processing) |
| Ionofree | L1+L2 | GPS | Broadcast | Dual-frequency elimination | Remove ionospheric effects |
| Multi-GNSS | L1 | GPS+GLO+GAL+BDS | Broadcast | Klobuchar model | Cross-constellation validation |
| Precise | L1+L2 | GPS | IGS SP3 (precise) | Dual-frequency elimination | Remove orbit/clock errors |

###### Mode Descriptions

- Baseline: GPS-only SPP with broadcast ephemeris and Klobuchar ionosphere model. This is the reference mode using the simplest processing chain.

- Ionofree: GPS SPP with dual-frequency (L1+L2) ionosphere-free linear combination. Eliminates ~99% of first-order ionospheric delay. If TEP were an ionospheric artifact, it would disappear in this mode.

- Multi-GNSS: Combined GPS+GLONASS+Galileo+BeiDou processing. Each constellation has different clock technologies (GPS: Rb/Cs, Galileo: H-maser, GLONASS: Cs, BeiDou: Rb) and orbital altitudes. If TEP were constellation-specific, this mode would show different λT.

- Precise: GPS SPP using IGS precise orbit and clock products (SP3 files) instead of broadcast ephemeris. This removes satellite orbit errors (~2m broadcast → ~2cm precise) and satellite clock errors (~2ns broadcast → ~0.1ns precise). If TEP were caused by broadcast ephemeris errors, it would disappear in this mode.

###### Ionosphere-Free Linear Combination

The ionofree mode uses the standard dual-frequency linear combination to eliminate first-order ionospheric delay (Kaplan & Hegarty, 2017):

$$L_{IF} = \frac{f_1^2}{f_1^2 - f_2^2} L_1 - \frac{f_2^2}{f_1^2 - f_2^2} L_2 \approx 2.546 \, L_1 - 1.546 \, L_2$$

where $f_1 = 1575.42$ MHz (L1) and $f_2 = 1227.60$ MHz (L2). This combination removes the ionospheric delay but amplifies receiver thermal noise by a factor of approximately 3× due to the large coefficients (Kaplan & Hegarty, 2017; Misra & Enge, 2011):

$$\sigma_{IF} = \sqrt{(2.546)^2 + (1.546)^2} \, \sigma_{L1} \approx 2.98 \, \sigma_{L1}$$

This noise penalty is a fundamental trade-off in dual-frequency GNSS processing. The ionofree mode is critical for validation: if correlations were purely ionospheric, they would disappear in this mode. Instead, longer Temporal Topology correlation lengths are observed (λT = 1,072 km vs 727 km), which is less consistent with a purely ionospheric artifact.

#### 2.4.2 Time Series Extraction

Three metrics were extracted from SPP solutions for each station, each serving a specific scientific purpose:

##### Metrics Comparison

| Metric | Formula | Expected Anisotropy | Role |
| --- | --- | --- | --- |
| Clock Bias | Receiver clock offset (ns) | E-W > N-S (~1.22) | Primary TEP metric |
| Position Jitter | $\sqrt{dE^2 + dN^2 + dU^2}$ (m) | Similar orbital coupling | Space proxy (TEP affects spacetime) |
| Clock Drift | $d(\Delta t)/dt$ (ns/s) | Weak (1.07) | Derivative test |

##### 1. Clock Bias — Primary TEP Metric

$$\Delta t = \text{Receiver clock offset (nanoseconds)}$$

The receiver clock offset from GPS time. This is the primary metric for TEP testing because:

- Directly measures temporal fluctuations at the receiver

- Shows strongest directional anisotropy (E-W/N-S ≈ 1.22 at short distances)

- Temporal Topology correlation length λT ≈ 700–1,100 km (lower bound; see §3.7)

##### 2. Position Jitter — Space Proxy

$$dr = \sqrt{dE^2 + dN^2 + dU^2}$$

The 3D deviation from mean position. This metric captures spatial coordinate fluctuations:

- Position errors include ionospheric, tropospheric, multipath, and TEP-induced spatial variations

- Shows similar orbital velocity coupling as clock bias (both metrics respond to Earth's orbital motion)

- May show weaker directional anisotropy due to atmospheric noise dominating at short distances

*Interpretation:* The TEP framework predicts coupled space-time fluctuations. Similar orbital coupling in both position and clock metrics is consistent with TEP—the underlying phenomenon affects spacetime, not just time. The key discriminator is the directional anisotropy (E-W/N-S ratio), which is strongest in clock bias.

##### 3. Clock Drift — Derivative Test

$$\dot{\Delta t} = \frac{d(\Delta t)}{dt}$$

The time derivative of clock bias. Tests whether the signal is:

- Random walk: derivative would be white noise (no spatial structure)

- Genuine signal: derivative maintains spatial correlation (R² > 0.9)

Observed R² = 0.974 for clock drift is less consistent with a pure random-walk explanation.

#### 2.4.3 Time Alignment Strategy (Critical)

##### Pandas DatetimeIndex Alignment

Time alignment uses Pandas DataFrame indexing with DatetimeIndex, identical to the CODE longspan methodology. This approach:

- Creates DatetimeIndex from each file's actual timestamps

- Concatenates daily data frames sorted by time

- Fills missing days and epochs with NaN markers

- Computes coherence only on valid overlapping segments

This ensures precise temporal synchronization between stations, mirroring the rigorous alignment used in Papers 1 and 2.

#### 2.4.4 Phase Coherence Computation

Two complementary coherence metrics were computed for all station pairs, enabling cross-validation and comparison with CODE longspan methodology:

##### Coherence Metrics Comparison

| Metric | Range | Sensitivity | Long-Distance Behavior |
| --- | --- | --- | --- |
| MSC (Coherence) | [0, 1] | Amplitude correlations | Decays with ionospheric decorrelation |
| Phase Alignment | [-1, 1] | Phase relationships | Robust — preserves signal at long distances |

$$C_{ij} = \frac{\left| \sum_{t} A_i(t) \cdot A_j(t) \cdot e^{i\Delta\phi_{ij}(t)} \right|}{\sum_{t} A_i(t) \cdot A_j(t)}$$

Where:

- $A_i(t), A_j(t)$ = amplitude envelopes from Hilbert transform

- $\Delta\phi_{ij}(t)$ = instantaneous phase difference

##### Metric 1: Magnitude Squared Coherence (MSC)

$$\text{MSC} = \frac{|P_{xy}(f)|^2}{P_{xx}(f) \cdot P_{yy}(f)}$$

Measures the strength of the linear relationship between signals. It asks: "How strongly do the clocks vibrate together?" This metric is:

- More sensitive to amplitude correlations

- Affected by ionospheric scintillation at long distances

- Best for short-distance analysis ($\lt$ 500 km)

##### Metric 2: Mean Phase-Offset Alignment Index

$$\text{PA} = \cos\left(\arg\left(\frac{\sum_f w_f \cdot e^{i\phi_f}}{\sum_f w_f}\right)\right)$$

The resultant magnitude $R = |\sum_f w_f e^{i\phi_f} / \sum_f w_f|$ quantifies the concentration of phase differences. Near-zero resultants make the mean phase angle unstable; a diagnostic of one representative station-day found that 1.5% of pairs had R $\lt$ 0.10. The reported PA statistic retains its established definition and is interpreted as mean phase-offset alignment, not phase-locking strength. This metric is:

- The *primary metric used by CODE longspan* (Paper 2)

- More robust over long distances, as phase relationships persist even when amplitude correlations weaken

- Shows strongest hemisphere anisotropy (NH: 1.224, SH: 1.348)

##### Why Phase Alignment Shows Stronger Anisotropy

The hemisphere analysis reveals that phase alignment consistently exceeds MSC in detecting directional anisotropy:

- Northern Hemisphere: MSC ratio 1.029, Phase Alignment ratio *1.224*

- Southern Hemisphere: MSC ratio 1.022, Phase Alignment ratio *1.348*

This hierarchy indicates a larger directional contrast in the mean phase-offset score than in MSC. Because the archived calculation does not retain the resultant magnitude, it does not by itself establish stronger phase locking.

##### Complementary Sensitivity: MSC vs Phase Alignment

A key observation from this analysis is that MSC and phase alignment probe different aspects of the same physical phenomenon, with each metric excelling at different types of analyses:

| Analysis Type | MSC Performance | Phase Alignment Performance | Physical Explanation |
| --- | --- | --- | --- |
| Orbital Velocity Coupling
(temporal modulation) | nominal 2.5–4.1σ; ≤ 2.9σ autocorrelation-corrected | nominal 5.4σ; ≈ 3.0σ corrected | MSC measures power correlation; orbital velocity modulates the *amplitude* of coupling month-to-month. Multi-GNSS phase alignment achieves strongest detection |
| Directional Anisotropy
(spatial structure) | E-W/N-S ≈ 1.02–1.05 | E-W/N-S ≈ 1.20–1.35 | Mean phase-offset alignment summarizes directional differences in average phase offset; its concentration must be evaluated separately |

Mathematical basis:

- MSC = |Pxy|²/(Pxx·Pyy) — Sensitive to the *magnitude* of cross-spectral density. Temporal variations in coupling strength (e.g., due to changing orbital velocity) directly modulate MSC values.

- Mean phase-offset alignment = cos(arg(Σw·eiφ)) — Sensitive to the direction of the weighted mean phase vector. Its concentration is R = |Σw·eiφ|/Σw and is not encoded in PA alone.

MSC measures the strength of a shared signal. Mean phase-offset alignment measures whether the *average* timing offset points near zero; the resultant magnitude is needed to tell whether that average is sharply defined or produced by cancelling phase values.

Why this distinction matters:

- Orbital coupling is a *temporal modulation* effect: Earth's changing velocity affects the *strength* of clock correlations month-to-month. MSC directly measures this strength.

- Directional anisotropy is a *spatial structure* effect: the PA result reports a directional difference in mean phase offset. A concentration analysis using R can determine whether that difference is accompanied by phase locking.

- SPP noise consideration: Single Point Positioning introduces ~1–3m pseudorange noise. This corrupts phase information more than power information, explaining why phase alignment is weaker for orbital coupling in SPP data but still excels at detecting spatial anisotropy (averaged over many pairs).

*Conclusion:* MSC and the mean phase-offset score provide complementary descriptive information. PA must not be interpreted as a phase-locking-strength statistic without its resultant magnitude.

#### 2.4.5 Frequency Band Selection

| Parameter | Frequency | Period | Rationale |
| --- | --- | --- | --- |
| Lower bound | 10 µHz | ~28 hours | Removes long-period drifts and diurnal signals |
| Upper bound | 500 µHz | ~33 minutes | Removes high-frequency noise and multipath |
| TEP Band | 10–500 µHz | 33 min – 28 hr | Matches theoretical TEP timescales |

#### 2.4.6 Exponential Decay Fitting

Coherence values were binned by inter-station distance and fit to an exponential decay model:

$$C(r) = A \cdot \exp(-r/\lambda_T) + C_0$$

Where:

- $A$ = amplitude (coherence at $r=0$ minus offset)

- $\lambda_T$ = Temporal Topology correlation length (km) — the key TEP parameter

- $C_0$ = asymptotic offset (noise floor)

##### Binning Parameters

| Parameter | Value |
| --- | --- |
| Distance range | 50 – 13,000 km |
| Number of bins | 40 (logarithmic spacing) |
| Minimum pairs per bin | 10 |
| Weighting | Inverse variance (1/SEM² ∝ npairs) |

##### Weighted R² Calculation

To ensure consistency between the weighted curve fit and the goodness-of-fit metric, R² is calculated using the same weights as the fit:

- Weights: wi = npairs in bin i (proportional to 1/σ²)

- Weighted mean: ȳw = Σ(wi · yi) / Σwi

- Weighted SSres: Σ wi(yi − ŷi)²

- Weighted SStot: Σ wi(yi − ȳw)²

- Weighted R²: 1 − SSres/SStot

This ensures high-sample bins (which dominate the fit) also dominate the R² assessment, preventing low-sample bins from artificially inflating or deflating the goodness-of-fit metric.

##### Boundary-Hit Detection

Fits where parameters converge to the imposed bounds are flagged as *boundary-hit*. These fits should be interpreted with caution:

- Amplitude (A): bounds [0.01, 2.0]

- Temporal Topology Correlation length (λT): bounds [100, 20000] km

- Offset (C0): bounds [−1.0, 1.0]

A boundary-hit typically indicates the exponential model is poorly constrained for that subset (e.g., insufficient distance range or dominated by short-range noise).

#### 2.4.7 Directional Anisotropy Analysis

The critical validation test compares E-W and N-S correlations. Station pairs were stratified by azimuth:

##### Azimuth Classification

| Direction | Azimuth Range | Sectors |
| --- | --- | --- |
| East-West | [67.5°, 112.5°) ∪ [247.5°, 292.5°) | E, W |
| North-South | [337.5°, 360°) ∪ [0°, 22.5°) ∪ [157.5°, 202.5°) | N, S |
| Eight-Sector | 45° sectors centered on cardinal directions | N, NE, E, SE, S, SW, W, NW |

The primary test compares mean correlation values at short distances ($\lt$ 500 km) where ionospheric decorrelation is minimal:

$$\text{E-W/N-S Ratio} = \frac{\overline{C}_{EW}}{\overline{C}_{NS}}$$

To control for potential distance distribution biases (e.g., if E-W pairs have a different mean distance than N-S pairs within the $\lt$ 500 km bin), a robust distance-matched ratio is also computed. Pairs are binned into 50 km intervals, and the ratio is computed from distributions re-sampled to match the distance profile, ensuring that any observed anisotropy is not an artifact of mean distance differences.

Statistical significance is assessed via Welch's t-test with 95% confidence intervals and Cohen's d effect size. Welch-test probabilities are nominal pair-level results. Because station pairs share receivers, satellites and observation epochs, their effective statistical independence is lower than the raw pair-sample count implies.

#### 2.4.8 Station Altitude Quintile Analysis (Step 2.1b)

To test whether the correlation structure could be explained by station environment or atmospheric column effects, station pairs were stratified by station altitude (geodetic height above the WGS84 ellipsoid). *Important: This analysis examines station altitude, not satellite elevation angle.* Station altitude provides a site-based proxy for propagation-path and local-environment effects: higher-altitude sites observe through a thinner tropospheric column and often exhibit reduced multipath and hydrological variability relative to low-altitude sites.

The analysis was conducted as a comprehensive suite of five configurations to test robustness against geographic confounding and distance-sampling biases:

- Global quintiles (sr200): All 539 stations sorted by altitude into 5 equal-count bins; pairs restricted to same-quintile membership; short-range coherence computed for pairs $\lt$ 200 km.

- Latitude-controlled quintiles (lat10, lat20): Stations binned within ±10° or ±20° latitude bands, then sorted by altitude within each band; pair analysis restricted to same latitude band to control for geographic/climatic confounding.

- Short-range distance variants (sr100, sr200): Maximum distance for "short-range coherence" proxy set to 100 km or 200 km to test sensitivity to local-scale effects.

For each configuration, three regression diagnostics were computed per quintile:

- Unweighted linear regression: λ vs. mean altitude (km/m slope, p-value)

- Weighted regression (inverse-variance): λ vs. altitude weighted by 1/σ²λ to suppress noisy quintiles

- Short-range coherence trend: Mean coherence level for pairs $\lt$ distance threshold vs. altitude (Phase II proxy)

##### Methodology

Station ECEF coordinates are converted to latitude, longitude, and altitude via the ECEF-to-LLA transform. Stations are then sorted by altitude and divided into five equal-count quintiles (Q1–Q5):

- Q1 (lowest altitude): Stations at low elevations (−86 m to 51 m; thicker atmospheric column)

- Q5 (highest altitude): Stations at high elevations (712 m to 3,755 m; thinner atmospheric column)

For each quintile, only station pairs where *both* stations fall within the same altitude quintile are included, and the exponential decay analysis is repeated independently, yielding separate λ, R², and amplitude values.

##### Physical Rationale and Null Hypothesis

Alternative Hypothesis (atmospheric/site-dependent origin): If the observed correlation were primarily driven by residual atmospheric propagation errors or site-specific effects:

- Low-altitude stations experience larger and more variable tropospheric delays and multipath → stronger contamination of coherence estimates

- High-altitude stations experience reduced atmospheric column and often cleaner observations → more stable coherence estimates

- Therefore, λ and/or amplitude could vary systematically with altitude (Q1 → Q5), yielding significant regression slopes (p $\lt$ 0.05)

Null Hypothesis (network-scale TEP origin): If the signal is dominated by a network-scale, geometry-driven field (as hypothesized in TEP), λ should be approximately altitude-independent over the ~4 km elevation span of the network. The Temporal Equivalence Principle predicts that temporal fluctuations couple to gravitational potential gradients at planetary scales. Station altitude differences of a few kilometers are negligible compared to the ~6,371 km Earth radius and the ~1 AU Earth–Sun distance that define the dominant gravitational potential terms. Thus, TEP predicts:

- Q5/Q1 ≈ 1.0 (no systematic ratio bias)

- λ-vs-altitude regression slopes statistically consistent with zero (p ≫ 0.05)

- Short-range coherence level independent of altitude

##### Expected Outcome and Quality Control

The ratio Q5/Q1 and regression p-values are the key diagnostics:

| Diagnostic | Altitude-Dependent (Alternative) | Altitude-Independent (TEP Null) |
| --- | --- | --- |
| Q5/Q1 Ratio | ≪ 1.0 or ≫ 1.0 | ≈ 1.0 (0.8–1.2) |
| Regression p-value | p $\lt$ 0.05 (significant slope) | p ≫ 0.05 (slope ~ 0) |
| Replication | Consistent across tags/modes | Sporadic, configuration-specific |

Fit Quality Gates: To avoid overclaiming based on degenerate exponential fits (common when distance support is insufficient), results are flagged when:

- λ > 10,000 km ("runaway" regime indicating flat decay / underconstrained fit)

- R² $\lt$ 0.6 (poor goodness-of-fit)

- Boundary hit during optimization (fit pegged at parameter limit)

Trends are considered robust only if they: (1) survive quality gates, (2) replicate across multiple suite configurations, and (3) show sign-consistency (all positive or all negative slopes).

*Note: The geometric suppression correction is not required for the primary evidence. The primary finding—E-W > N-S at short distances ($\lt$ 500 km)—uses raw, uncorrected values that directly match CODE's prediction (see §3.10.1). This section addresses why full-distance λ ratios show the opposite pattern (E-W/N-S $\lt$ 1), providing interpretive context rather than calibration for the core result.*

##### Geometric Suppression Correction

GPS satellite orbits (55° inclination) create systematic coverage biases that suppress E-W correlations. Due to this inclination, satellites travel predominantly North-South relative to mid-latitude observers. This geometry allows N-S station pairs to view the same satellite for longer continuous arcs, significantly lowering the noise floor for N-S correlations. Conversely, satellites cut across E-W baselines more rapidly, reducing the duration of common-view periods and artificially suppressing the apparent coherence in raw SPP data.

Analogy: Imagine looking through a vertical picket fence. You can easily track an object moving up and down (N-S), but an object moving side-to-side (E-W) is constantly interrupted by the fence slats. The GPS constellation acts as this "fence" for ground observers, artificially breaking up E-W coherence while preserving N-S coherence.

This suppression is quantified by comparing sector-specific correlation lengths (λ) from this SPP analysis to CODE's 25-year PPP reference values.

For each azimuthal sector, the following is computed:

- Sector ratio: λSPP / λCODE

- Suppression factor: mean(N-S ratios) / mean(E-W ratios)

If N-S correlations are preserved (ratio ~1.0) while E-W correlations are suppressed (ratio $\lt$ 0.5), this indicates geometric bias rather than signal loss. The corrected E-W/N-S ratio is:

$$\text{Corrected ratio} = \text{raw ratio} \times \text{suppression factor}$$

The suppression factor is not a free parameter—it emerges from the sector-by-sector comparison and is consistent across all four processing modes (2.42×–3.16×), supporting its interpretation as a geometric effect rather than arbitrary tuning.

### 2.4.9 Seasonal Stratification Analysis (Step 2.4)

To test whether the observed correlations are seasonal artifacts (e.g., temperature-dependent receiver behavior, seasonal ionospheric variations, or solar illumination effects), the 3-year dataset was stratified by meteorological season and analyzed correlation lengths independently for each period.

#### Season Definitions

| Season | Months | Days of Year | Purpose |
| --- | --- | --- | --- |
| Winter | Dec, Jan, Feb | 335–59 | Solar minimum illumination (NH) |
| Spring | Mar, Apr, May | 60–151 | Transition period |
| Summer | Jun, Jul, Aug | 152–243 | Solar maximum illumination (NH) |
| Autumn | Sep, Oct, Nov | 244–334 | Transition period |

*Note: Seasons are defined by Northern Hemisphere convention. The IGS network is NH-dominated (238 NH vs 106 SH stations globally; the analyzed set is nearer balance at 285 NH vs 254 SH stations, though NH stations contribute substantially more station-days), making NH seasons the natural stratification choice.*

#### Analysis Methodology

For each season, exponential decay fits were computed across all three station filters (ALL_STATIONS, OPTIMAL_100, DYNAMIC_50) and all four processing modes (Baseline, Ionofree, Multi-GNSS, Precise). This produces 48 seasonal estimates across processing configurations (4 seasons × 3 filters × 4 modes) for each metric/coherence combination.

Key Predictions:

- If the signal is a seasonal artifact: Correlation length λ should vary by >20% between seasons, with systematic patterns (e.g., always strongest in summer).

- If the signal reflects a stable terrestrial screening structure: λ should be stable across seasons in high-reliability multi-constellation channels (Δ $\lt$ 15%), with any seasonal excursions in noise-sensitive single-frequency channels bounding propagation-medium artifacts.

#### The "Two Views" Framework

The seasonal analysis tests two complementary hypotheses:

- OPTIMAL_100 (Spatial Balance): Designed to capture the maximum spatial extent of correlations by ensuring global coverage. It diagnoses how long-distance exponential fits behave under global network geometry and varying seasonal noise conditions.

- DYNAMIC_50 (Temporal Stability): Designed to capture the most reliable, continuous stations. Expected to show minimal seasonal variation, revealing the stable "core" signal independent of atmospheric conditions.

These two filters test different aspects of the signal: OPTIMAL_100 tests the scale, DYNAMIC_50 tests the stability.

## 2.5 Analysis Matrix Summary

The full analysis explores multiple dimensions to ensure robustness:

#### Complete Analysis Matrix

| Dimension | Options | Primary | Purpose |
| --- | --- | --- | --- |
| Station Filter | All, OPTIMAL_100, DYNAMIC_50 | DYNAMIC_50 | Quality control, hemisphere balance |
| Processing Mode | Baseline, Ionofree, Multi-GNSS | Baseline | Ionosphere/constellation validation |
| Time Series Metric | Clock bias, Pos jitter, Clock drift | Clock bias | Temporal vs spatial signal separation |
| Coherence Metric | MSC, Phase Alignment | Phase Alignment | Amplitude vs phase sensitivity |
| Distance Range | Short ($\lt$ 500 km), Full (50–13,000 km) | Short | Minimize ionospheric contamination |
| Stratification | Regional, Hemisphere, Latitude, Kp index, *Seasonal*, *Year-by-Year* | Hemisphere | Geographic/geomagnetic/seasonal/temporal validation |
| Planetary Events | ±120 day windows, 37 events (2022–2024) | All planets | Alignment modulation, CODE replication |

In particular, Step 2.1a implements regional control tests by splitting the network into Global, Europe-only, Non-Europe, and hemisphere-specific subsets. These control analyses check that the exponential decay is not confined to a single continent and diagnose how network density (very short baselines in Europe) versus sparse, ocean-dominated networks (Southern Hemisphere) affects the observed correlation length and goodness of fit.

#### Cross-Region Pair Exclusion

For regional subsets (Europe, Non-Europe, Northern, Southern), only intra-region pairs are included—station pairs where both stations belong to the same region. Cross-region pairs (e.g., a European station paired with a non-European station) are excluded from regional analyses but included in the Global analysis.

Rationale: Cross-region pairs would conflate the regional signal with inter-regional baselines, making it impossible to diagnose region-specific network density effects. Clean separation ensures each regional subset tests only pairs that share the same geographic characteristics.

#### Expected Results by Metric Type

The combination of metrics provides a self-consistent validation framework:

- Clock bias + Phase Alignment: Strongest anisotropy (E-W/N-S > 1.2) — the primary directional signature

- Clock bias + MSC: Moderate anisotropy (E-W/N-S ≈ 1.02–1.05) — consistent but weaker

- Clock drift (any coherence): Weak anisotropy (E-W/N-S ≈ 1.07) — derivative preserves structure

- Planetary events (all metrics): Year-specific windows yield 2.8× higher detection than the date-shuffled null (59–68% vs. 20–26%, p $\lt$ 0.001 for all 6 metrics); season-preserving controls (Step 2.6b) attribute the excess to calendar-position sampling of the autocorrelated seasonal series rather than event-locking, and neither the GM/r² acceleration proxy nor the GM/r³ tidal-gradient scaling shows dependence.

This hierarchy is consistent with TEP predictions: clock bias shows the strongest directional anisotropy as the primary temporal proxy, while position jitter shows similar orbital coupling (consistent with coupled space-time fluctuations) but with weaker directional structure due to atmospheric noise.

## 2.6 Null Tests (Step 2.4b)

A critical requirement for validating the TEP signal is to assess whether the observed exponential correlation structure is driven by known non-gravitational phenomena. A comprehensive null test suite was designed that examines three independent mechanisms that could potentially produce spurious distance-structured correlations.

### 2.6.1 Test Design Rationale

The null tests probe three distinct hypotheses:

- Solar Rotation Hypothesis: If the signal originates from solar wind, radiation pressure, or geomagnetic storms driven by solar activity, coherence should modulate with the 27-day solar rotation period.

- Lunar Tidal Hypothesis: If the signal is driven by lunar gravitational tides affecting clock rates or atmospheric pressure, coherence should modulate with the 29.5-day synodic lunar month.

- Spurious Structure Hypothesis: If the exponential decay is a statistical artifact of the analysis methodology rather than a physical property of the data, the structure should persist when temporal coherence is destroyed by randomization.

### 2.6.2 Solar/Lunar Phase Correlation

For each metric/coherence combination, daily mean coherence values are computed and test for cyclic modulation:

r = √(rsin² + rcos²)

where rsin and rcos are the Pearson correlations between daily coherence and the sine/cosine of the phase angle (φ = 2π × DOY / Period). This circular correlation captures any periodic modulation regardless of phase offset.

Acceptance Criterion: r $\lt$ 0.1 for both solar (27-day) and lunar (29.5-day) cycles. This threshold corresponds to less than 1% of variance explained by the periodic driver.

### 2.6.3 Shuffle Test (Critical Validation)

The shuffle test provides a direct validation of genuine spatial structure. The procedure:

- Real Fit: Fit the exponential decay model C(r) = A·exp(−r/λ) + C_0 to the complete coherence dataset, recording R²real.

- Randomization: Randomly permute the coherence values while preserving the distance values, breaking the space-time relationship.

- Shuffled Fit: Fit the same exponential model to the shuffled data, recording R²shuffled.

Acceptance Criterion: R²shuffled $\lt$ 0.3. If the exponential structure is a genuine property of the data (not an artifact of the fitting procedure), shuffling should destroy it completely.

#### Why the Shuffle Test is Important

The shuffle test directly addresses the concern that exponential fitting might "force" structure onto any dataset. If the fitting procedure itself creates spurious curvature, it would do so equally on real and shuffled data. The ratio R²real/R²shuffled quantifies the evidence that the structure is physically real.

### 2.6.4 Comprehensive Test Matrix

The null tests are applied across the full analysis matrix:

| Dimension | Values | Tests |
| --- | --- | --- |
| Station Filters | ALL_STATIONS, OPTIMAL_100, DYNAMIC_50 | 3 |
| Processing Modes | Baseline, Ionofree, Multi-GNSS, Precise | 4 |
| Metrics | clock_bias, pos_jitter, clock_drift | 3 |
| Coherence Types | MSC, Phase Alignment | 2 |
| Total Analysis Configurations | 72 |

This comprehensive matrix is designed so that any positive result is less likely to be attributable to a specific station selection, processing algorithm, metric choice, or coherence definition.

### 2.6.5 Expected Outcomes

If TEP is correct and the signal represents genuine gravitational coupling to Earth's orbital motion:

- Solar/Lunar: All correlations should be r $\lt$ 0.1 (orbital period is 365 days, not 27 or 29.5 days)

- Shuffle: R²shuffled should collapse to near-zero while R²real remains >0.9

- Mode Independence: Results should be consistent across Baseline, Ionofree, and Multi-GNSS (signal is gravitational, not ionospheric or constellation-specific)

- Filter Independence: Results should be consistent across station filters (signal is network-wide, not station-specific)

## 2.7 Annual-Phase Directional Analysis (Step 2.7)

Following the CODE longspan methodology, a comprehensive full-sky grid search was performed across 72 analysis configurations to characterize the celestial-coordinate phase of the annual E-W/N-S anisotropy modulation. Named fixed-frame directions and Earth's orbital-velocity tangents are evaluated as controls. The full analysis matrix is evaluated to reduce selection bias and assess robustness across processing choices.

### 2.7.1 Physical Motivation

The direction-only template is compared with two named fixed-frame candidates and with competitive in-ecliptic orbital controls:

- CMB Dipole: RA = 167.94°, Dec = −6.94° (Earth's motion at 370 km/s through the cosmic microwave background rest frame)

- Solar Apex: RA = 271°, Dec = +30° (Sun's motion at 20 km/s toward Vega through the local galaxy)

The net velocity vector combines Earth's orbital motion (~30 km/s, rotating annually) with a fixed 20 km/s phenomenological background term. Different background directions produce different annual phases in the velocity-declination predictor. The scan fixes this magnitude and therefore estimates direction/phase only; it does not estimate a physical background speed.

The CMB dipole remains a physically motivated fixed-frame control, but it is not privileged over the aphelion and perihelion velocity tangents. The corrected results identify the aphelion tangent and reject the CMB-apex phase.

### 2.7.2 Comprehensive Analysis Matrix

To ensure robustness and reduce selection bias, the directional analysis is performed across all 72 combinations of station filter, processing mode, metric, and coherence type:

| Dimension | Values | Purpose |
| --- | --- | --- |
| Station Filters | ALL_STATIONS (539), OPTIMAL_100 (50N+50S), DYNAMIC_50 (399 high-stability) | Test network independence |
| Processing Modes | Baseline (GPS L1), Ionofree (L1+L2), Multi-GNSS (GPS+GLO+GAL+BDS), Precise (IGS SP3) | Test ionospheric independence |
| Metrics | clock_bias, pos_jitter, clock_drift | Test spacetime coupling |
| Coherence Types | MSC (amplitude), Phase Alignment (phase) | Test signal structure |

Total combinations: 3 × 4 × 3 × 2 = 72 analysis configurations

This exhaustive approach allows identification of which combinations recover the CMB signal most cleanly and assessment of whether the signal is a robust network-wide phenomenon or an artifact of specific analysis choices.

### 2.7.3 Grid Search Methodology

#### Predictor Model

For each candidate background direction (RA, Dec), the monthly net velocity vector is computed:

Vnet(month) = Vorbital(month) + Vbackground(RA, Dec)

The predictor is cos(velocity_declination): low declination (equatorial velocity) predicts high E-W/N-S ratio; high declination (polar velocity) predicts low E-W/N-S ratio. This geometric model directly tests whether the observed anisotropy modulation follows Earth's motion through a hypothesized cosmic frame.

**Orbital-velocity convention:** The orbital term implements Earth's physical heliocentric velocity (prograde motion), verified against independent ephemeris checks: at the March equinox the velocity points to ecliptic longitude λ = 270° (equatorial RA ≈ 270°, Dec ≈ −23.4°), at perihelion to RA ≈ 192°, Dec ≈ −5.1°, and at aphelion to RA ≈ 12°, Dec ≈ +5.1°. An earlier pipeline version carried the antiparallel convention; because the predictor cos(decnet) is invariant under Vnet → −Vnet, that convention returned the identical correlation landscape mirrored through the origin (verified to 10⁻¹⁵), so the directional results of earlier versions read at the antipode. All reported directions use the corrected convention.

#### Grid Search Parameters

| Parameter | Value | Rationale |
| --- | --- | --- |
| RA range | 0°–359° | Full celestial sphere |
| Dec range | −89° to +89° | Full celestial sphere (avoiding poles) |
| Resolution | 1° | Matches CODE longspan finest setting; ~65,000 grid points |
| Background speed | 20 km/s (fixed) + scan over 5–370 km/s | Prescribed phenomenological directional template; not the physical CMB-dipole speed — a |Vbg| scan is reported in §3.13.3a |
| Test statistic | Pearson correlation | cos(Dec) vs monthly E-W/N-S ratio (36 months) |
| Control directions | Ecliptic poles, solar apex/antapex, Galactic center, perihelion/aphelion velocity tangents | Competitive-direction battery (§3.13.3a) |

The 20 km/s background speed is the fixed phenomenological parameter used to construct the directional predictor, not the physical 370 km/s CMB-dipole speed. Because the predictor cos(declination of Vorbital + Vbg) is phase-dominated by the rotating orbital term, the inferred direction re-parametrizes the annual phase of the modulation; the dependence on the assumed speed is therefore measured directly by a |Vbg| scan from 5 to 370 km/s (§3.13.3a), and the best-fit direction is compared against competitive controls that live in the ecliptic plane — where any orbit-driven modulation must lie — rather than only against the Solar Apex 53° off the ecliptic.

#### Statistical Validation

For each combination, the following is computed:

- Local p-value: Standard Pearson correlation significance (N = 36 months)

- Bootstrap confidence intervals: 500 resamples with 10° coarse grid search to estimate 68% CIs for RA and Dec

- Global p-value (Monte Carlo): 1000 permutations of monthly E-W/N-S ratios with vectorized 5° grid search to account for look-elsewhere effect across ~2,600 independent sky pixels

- Configuration-grid interpretation: the Monte Carlo global p-value is reported for the selected configuration; no experiment-wide aggregate significance across the 54 correlated configurations is claimed.

### 2.7.4 Mode-Specific Expectations

The four processing modes provide complementary views of the signal:

- Baseline (GPS L1): Contains full ionospheric contamination. If signal survives, it suggests the effect is not purely ionospheric.

- Ionofree (L1+L2): Removes first-order ionosphere but amplifies thermal noise by ~3× (Kaplan & Hegarty, 2017). Weaker signal recovery expected, but successful detection indicates signal survives ionospheric removal.

- Multi-GNSS (All Constellations): Averages across four constellations (GPS, GLONASS, Galileo, BeiDou), reducing satellite-specific noise by ~√4 = 2×. If signal persists across different clock technologies and orbital altitudes, it is not constellation-specific.

- Precise (IGS SP3): Uses precise orbit/clock products instead of broadcast ephemeris. If signal persists, it is not caused by broadcast ephemeris errors.

#### Predicted Hierarchy

Based on noise characteristics, it is predicted:

- Best directional recovery: DYNAMIC_50 (high-stability clocks) + Multi-GNSS (lowest noise floor)

- Best RA precision: MSC coherence (amplitude-based, responds to temporal modulation)

- Widest scatter: Ionofree mode (3× noise amplification obscures weak signal)

### 2.7.5 Falsification Criteria

The CMB frame hypothesis is considered falsified if:

- Best-fit direction is closer to Solar Apex (271°, +30°) than to CMB Dipole (168°, −7°)

- No combination achieves global p $\lt$ 0.05 after look-elsewhere correction

- Different station filters produce inconsistent directions (high variance)

- Baseline and Multi-GNSS modes find different preferred directions

Conversely, support for a genuine frame detection — as distinct from an annual-phase measurement — would additionally require:

- Majority of clean (non-Ionofree) combinations find RA within 20° of CMB

- At least one combination achieves global p $\lt$ 0.05

- Zero variance across station filters (all converge to same RA)

- The CMB dipole outperforms competitive in-ecliptic controls (perihelion/aphelion velocity tangents), not merely the off-ecliptic Solar Apex

- The inferred direction is stable under a |Vbg| scan up to the physical CMB speed of 370 km/s

As reported in §3.13.3a–3.13.9, under the corrected orbital-velocity convention the CMB hypothesis fails outright: the best-fit direction is ~160° from the dipole apex (farther than the Solar Apex at ~93.5°), the CMB-direction template anti-correlates (r ≈ −0.39), the aphelion-velocity tangent outperforms it (r ≈ +0.49, 3.9° from the best fit), and the inferred direction migrates tens of degrees across the speed scan. The result is therefore reported as a well-defined annual phase expressible as an in-ecliptic preferred direction — coincident with Earth's velocity tangent at aphelion — and not as a cosmological-frame detection.

Under the same corrected convention, CODE's 25-year analysis recovers best-fit at RA = 6°, Dec = +4° (162° from the CMB dipole, ~6° from the aphelion velocity tangent). With only 3 years of data, weaker Dec constraints are expected due to limited seasonal sampling, but RA should converge to the orbital-tangent direction if the effect is real.

## 2.8 GPS-Geometry Null Simulation (Step 2.9)

To interpret the full-distance E-W/N-S inversion without circularity, a first-principles simulation decomposes the directional bias into two candidate mechanisms using synthetic data only—no reference to CODE or to the empirical TEP results:

- **Geometry-only simulation:** Synthetic clock time series are generated with noise weighted by position dilution of precision (PDOP) computed from the actual GPS constellation geometry (55° orbital inclination). This tests whether satellite visibility anisotropy alone suppresses E-W correlation lengths. The simulation yields E-W mean λ = 265 km versus N-S mean λ = 3,994 km (E-W/N-S ≈ 0.066, a ~15× suppression).

- **Ionospheric local-time simulation:** Synthetic clocks are correlated by local solar time under a diurnal TEC model, TEC = TEC₀ × [1 + 0.5 cos(2π(LST − 14)/24)]. Because E-W pairs span different local times while N-S pairs share them, this mechanism produces E-W enhancement rather than suppression (E-W/N-S ≈ 19.6).

- **Europe resolvability analysis:** A baseline-length calculation demonstrates that the dense, short-baseline European subnetwork cannot resolve a λ ≈ 1,000 km correlation length, explaining the non-convergent Europe-only fits reported in §3.2.1 as a resolution limit rather than a data-quality failure.

The two mechanisms operate on different distance scales and are reported as complementary diagnostics in §3.10.4 and §6.3.3; the primary short-distance anisotropy result uses neither correction.

## 2.9 Software and Reproducibility

All analysis code is open source and available in the TEP-GNSS-RINEX repository:

- RTKLIB: Version 2.4.3 (BSD-2-Clause license)

- Python: NumPy, SciPy, Matplotlib

- Repository: [github.com/matthewsmawfield/TEP-GNSS-RINEX](https://github.com/matthewsmawfield/TEP-GNSS-RINEX)

## 3. Results

### 3.1 Data Quality Summary

| Metric | Value |
| --- | --- |
| Total stations processed | 539 |
| Stations after DYNAMIC_50 filtering | ~400 (jumps $\lt$ 500 ns, range $\lt$ 5,000 ns, std $\lt$ 50 ns) |
| Files rejected by quality filter | 93,240 (23%): 47,816 jumps, 4,045 range, 41,379 std |
| Total pair-samples (Baseline) | 61,853,126 |
| Total pair-samples (Ionofree) | 59,141,613 |
| Total pair-samples (Multi-GNSS) | 58,050,018 |
| Total pair-samples (Precise) | 58,703,009 |
| Total pair-samples analyzed (ALL_STATIONS) | 713,243,298 |
| Total pair-samples analyzed (DYNAMIC_50) | 425,758,551 |
| Total pair-samples analyzed (OPTIMAL_100) | 28,496,355 |
| Grand Total (all filters) | 1.17 billion pair-samples |
| Time span | 3 years (2022–2024, 1,096 days) |
| Processing interval | 5 minutes (288 epochs/day) |
| Signal Detection Rate | 72/72 metrics (100%) |

### 3.2 Exponential Decay Fits

#### ✓ Consistent Signal Detection: 72/72 Metrics (100%)

The analysis achieves consistent detection of distance-structured correlations across all 72 correlated metric configurations:

| Coherence Type | Mean λT (km) | λT Range (km) | Mean R² | R² Range |
| --- | --- | --- | --- | --- |
| MSC (Amplitude) | 924 | 625 – 1,403 | 0.958 | 0.872 – 0.993 |
| Mean Phase-Offset Alignment | 2,018 | 984 – 5,026 | 0.903 | 0.666 – 0.987 |

Key insight: Mean phase-offset alignment (labelled "Phase Alignment" in tables and figures throughout) shows ~2.2× longer correlation length than MSC across all filters. This is consistent with the "shaking vs dancing" interpretation: MSC measures amplitude correlation (sensitive to local noise at ~700–1,200 km), while the mean phase-offset score captures directional consistency in average timing offset that persists over longer distances (~1,700–3,500 km); as defined in §2.4.4, it is not a phase-locking-strength statistic without the resultant magnitude.

#### Primary Results: Multi-Mode Comparison (ALL_STATIONS)

The analysis reports correlation patterns consistent with the TEP framework across all 24 metrics (4 modes × 3 variables × 2 coherence types). Across these combinations, phase coherence yields longer correlation lengths than MSC, consistent with theoretical expectations for phase-locked signals.

| Mode | Metric | Type | λT (km) | Error | R² | Signal Detected? |
| --- | --- | --- | --- | --- | --- | --- |
| Baseline
(GPS L1) | Clock Bias | MSC | 727 | ±50 | 0.971 | YES |
| Phase | 1,796 | — | 0.951 | YES |
| Position | MSC | 878 | ±41 | 0.978 | YES |
| Phase | 1,991 | — | 0.833 | YES |
| Drift | MSC | 702 | ±47 | 0.974 | YES |
| Phase | 1,026 | — | 0.982 | YES |
| Ionofree
(L1+L2) | Clock Bias | MSC | 1,073 | ±62 | 0.972 | YES |
| Phase | 1,773 | — | 0.841 | YES |
| Position | MSC | 1,239 | ±103 | 0.977 | YES |
| Phase | 3,512 | — | 0.972 | YES |
| Drift | MSC | 1,072 | ±63 | 0.977 | YES |
| Phase | 1,109 | — | 0.975 | YES |
| Multi-GNSS
(All Const.) | Clock Bias | MSC | 821 | ±73 | 0.926 | YES |
| Phase | 1,771 | — | 0.974 | YES |
| Position | MSC | 928 | ±50 | 0.991 | YES |
| Phase | 1,770 | — | 0.855 | YES |
| Drift | MSC | 764 | ±67 | 0.936 | YES |
| Phase | 984 | — | 0.987 | YES |
| Precise
(IGS SP3) | Clock Bias | MSC | 1,202 | ±85 | 0.974 | YES |
| Phase | 1,703 | — | 0.788 | YES |
| Position | MSC | 1,403 | ±95 | 0.972 | YES |
| Phase | 3,581 | — | 0.981 | YES |
| Drift | MSC | 1,202 | ±84 | 0.976 | YES |
| Phase | 1,166 | — | 0.953 | YES |

*Note: "Phase" refers to phase alignment coherence, which generally persists over longer distances than magnitude-squared coherence (MSC). The Precise mode uses IGS SP3 precise orbit/clock products instead of broadcast ephemeris.*

#### 3.2.1 Regional Control Tests (Step 2.1a)

To verify that the exponential decay is not confined to any particular part of the IGS network, Step 2.1a repeats the baseline coherence analysis after splitting the station pairs into Global, Europe-only, Non-Europe, and Northern/Southern hemisphere subsets. All three metrics (clock bias, position jitter, clock drift) and both coherence measures (MSC and phase alignment) are evaluated in each subset. Cross-region pairs (e.g., a European station paired with a non-European station) are excluded from regional subsets to ensure clean separation—these pairs are included only in the Global analysis.

Figure 3.1: Regional control tests showing exponential decay fits for clock bias coherence across Global, Europe, Non-Europe, Northern, and Southern subsets. Phase alignment (solid lines) consistently shows longer correlation lengths than MSC (dashed lines), with the Southern Hemisphere exhibiting the longest MSC scales (1,315 km vs 688 km Northern). Europe-only fits fail to converge—a successful negative control, as the TEP signal (λ ≈ 1,000+ km) cannot be resolved in a network dominated by short baselines.

##### Table 3.2a: Regional MSC Results (Magnitude Squared Coherence)

| Metric | Global | Europe | Non-Europe | Northern | Southern |
| --- | --- | --- | --- | --- | --- |
| clock_bias | λT=725 km
R²=0.954 | λT=567 km
R²=0.901 | λT=853 km
R²=0.965 | λT=688 km
R²=0.964 | λT=1,315 km
R²=0.901 |
| pos_jitter | λT=881 km
R²=0.979 | λT=572 km
R²=0.998 | λT=1,046 km
R²=0.973 | λT=848 km
R²=0.985 | λT=1,116 km
R²=0.956 |
| clock_drift | λT=700 km
R²=0.956 | λT=568 km
R²=0.905 | λT=837 km
R²=0.964 | λT=668 km
R²=0.967 | λT=1,297 km
R²=0.895 |

##### Table 3.2b: Regional Phase Alignment Results

| Metric | Global | Europe | Non-Europe | Northern | Southern |
| --- | --- | --- | --- | --- | --- |
| clock_bias | λT=1,784 km
R²=0.904 | λT=10,669 km †
R²=0.843 | λT=1,630 km
R²=0.908 | λT=1,947 km
R²=0.892 | λT=1,678 km
R²=0.872 |
| pos_jitter | λT=2,013 km
R²=0.967 | λT=5,394 km †
R²=0.947 | λT=1,968 km
R²=0.964 | λT=2,074 km
R²=0.964 | λT=1,710 km
R²=0.965 |
| clock_drift | λT=1,024 km
R²=0.946 | λT=1,270 km
R²=0.936 | λT=1,047 km
R²=0.964 | λT=1,056 km
R²=0.942 | λT=941 km
R²=0.889 |

*† = Boundary-hit flag: fit parameters converged to parameter bounds. These fits should be interpreted with caution—the exponential model is poorly constrained in the Europe-only subset due to limited distance range.*

##### Key Result: Global Phenomenon with Diagnostic Regional Variations

1. The Phase > MSC Hierarchy (consistent with the TEP interpretation)

Across all well-constrained regional fits, phase alignment correlation lengths are 2.0–2.5× longer than MSC values:

- Global: MSC 700–881 km → Phase 1,024–2,013 km (ratio 1.5–2.3×)

- This hierarchy is *consistent with TEP predictions*: MSC measures amplitude correlation (sensitive to ionospheric decorrelation at ~700–900 km), while phase alignment measures timing synchronization that persists over longer distances

- Phase alignment λT values (1,600–2,100 km) remain shorter than the CODE longspan directional family (cross-sector mean ~4,200 km), consistent with residual ionospheric noise in single-frequency data. *When ionospheric effects are removed (Ionofree mode, see §3.2.1.1), λT increases to ~3,800 km, within that family's cross-sector range.*

2. The Southern Hemisphere Enhancement (observed pattern)

Southern Hemisphere shows systematically longer correlation lengths than Northern:

- clock_bias MSC: Southern λT = 1,315 km vs Northern λT = 688 km (1.91× ratio)

- Position Jitter MSC: Southern λT = 1,116 km vs Northern λT = 848 km (1.32× ratio)

- This is consistent with CODE longspan (Paper 2): Southern Hemisphere orbital coupling r = −0.79 (p = 0.006) vs Northern r = +0.25 (p = 0.49)

- Interpretation: The Southern Hemisphere's sparser network (fewer short baselines) may reduce the dominance of local atmospheric effects, improving sensitivity to longer-range structure

3. The Europe Anomaly as a Negative Control

The Europe-only subset shows a reduced long-range signature, consistent with a sensitivity-limited negative control: Europe's dense network overweights short baselines ($\lt$ 200 km) toward local tropospheric correlations, and its elongated North&ndash;South geometry preferentially samples the suppressed direction (mechanism discussion in the synthesis section).

4. Clock ≈ Position Behavior (spacetime coupling interpretation)

Both metrics show similar regional patterns, which is consistent with the TEP interpretation of coupled *spacetime* (not just temporal) fluctuations. In GNSS navigation solutions, position and clock are solved simultaneously, so a shared physical modulation could affect both observables.

#### 3.2.1.1 Station Altitude Quintile Analysis (Step 2.1b)

To test whether the inferred correlation scale depends on station altitude (a proxy for atmospheric column and local site conditions), a comprehensive suite of five stratification configurations was executed: global quintiles (sr200), latitude-controlled quintiles (lat10_sr100, lat10_sr200, lat20_sr100, lat20_sr200). Each configuration tested 72 combinations (3 filters × 4 modes × 3 metrics × 2 coherence types), yielding 360 independent λ-vs-altitude regressions. If atmospheric propagation or site-dependent effects dominated the observed decay structure, one would expect systematic variation of λ with altitude across most combinations. Conversely, weak dependence on altitude would be more consistent with a geometry- or network-wide origin.

##### Table 3.2d: Comprehensive Suite Statistical Summary

| Configuration Tag | Combinations (N) | λT-trend p$\lt$ 0.05 | λT-trend (invvar) p$\lt$ 0.05 | Short-range p$\lt$ 0.05 | Degenerate Fits | Low R² ($\lt$ 0.6) |
| --- | --- | --- | --- | --- | --- | --- |
| global_sr200 | 72 | 3 (4.2%) | 5 (6.9%) | 2 (2.8%) | 2 (2.8%) | 4 (5.6%) |
| lat10_sr100 | 72 | 1 (1.4%) | 4 (5.6%) | 0 (0%) | 9 (12.5%) | 8 (11.1%) |
| lat10_sr200 | 72 | 1 (1.4%) | 4 (5.6%) | 4 (5.6%) | 9 (12.5%) | 8 (11.1%) |
| lat20_sr100 | 72 | 3 (4.2%) | 6 (8.3%) | 0 (0%) | 7 (9.7%) | 7 (9.7%) |
| lat20_sr200 | 72 | 3 (4.2%) | 6 (8.3%) | 1 (1.4%) | 7 (9.7%) | 7 (9.7%) |
| Combined | 360 | 11 (3.1%) | 25 (6.9%) | 7 (1.9%) | 34 (9.4%) | 34 (9.4%) |

*Note: "Degenerate fits" = λ > 10,000 km or boundary_hit flag. "Low R²" = R² $\lt$ 0.6 for any quintile in the combination. Percentages represent fraction of 72 combinations per tag showing the specified characteristic.*

##### Table 3.2e: Representative Quintile λ Values (global_sr200, ALL_STATIONS, precise mode)

| Metric/Type | Q1 (8m) | Q2 (70m) | Q3 (144m) | Q4 (441m) | Q5 (1453m) | Q5/Q1 | Slope (km/m) | p-value |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| clock_bias/MSC | 1276 km
(R²=0.93) | 1089 km
(R²=0.87) | 1602 km
(R²=0.89) | 1180 km
(R²=0.87) | 1216 km
(R²=0.94) | 0.95 | −0.061 | 0.76 |
| clock_bias/Phase | 1678 km
(R²=0.80) | 2329 km
(R²=0.77) | 1793 km
(R²=0.56) | 1898 km
(R²=0.66) | 1763 km
(R²=0.61) | 1.05 | −0.117 | 0.65 |
| clock_drift/Phase | 1041 km
(R²=0.91) | 1428 km
(R²=0.81) | 1289 km
(R²=0.72) | 1120 km
(R²=0.79) | 1195 km
(R²=0.87) | 1.15 | −0.032 | 0.84 |
| pos_jitter/MSC | 1943 km
(R²=0.93) | 1093 km
(R²=0.87) | 1125 km
(R²=0.91) | 1036 km
(R²=0.95) | 1574 km
(R²=0.92) | 0.81 | +0.093 | 0.82 |

*Note: Altitude values in parentheses are quintile mean altitudes. None of the slopes are statistically significant (all p > 0.05). Q5/Q1 ratios in this representative table range 0.81–1.15 and cluster near 1.0. This pattern is representative of the broader suite.*

##### Primary Finding: Altitude Invariance of Correlation Length

Null-Consistent Outcome: Across 360 independent regressions, only 3.1% show statistically significant (p $\lt$ 0.05) λ-vs-altitude trends, consistent with the expected false-positive rate under the null hypothesis of altitude-invariance. This is less consistent with altitude-linked atmospheric or site-dependent explanations:

- No systematic altitude dependence: The vast majority (96.9%) of combinations show λ slopes statistically consistent with zero. Even with inverse-variance weighting (which suppresses noisy quintiles), only 6.9% reach significance.

- Short-range coherence also invariant: Mean coherence level for pairs $\lt$ 100–200 km shows no robust altitude trend (1.9% significant), disfavoring a local-scale Phase II altitude mechanism.

- Q5/Q1 ratios cluster near 1.0: Across all non-degenerate fits, the median Q5/Q1 ratio is 0.97 (IQR 0.76–1.27; 10–90% range 0.56–1.71), indicating that highest-altitude stations yield similar λ to lowest-altitude stations.

- Latitude-controlled stratification increases degeneracy: When geographic confounding is controlled by restricting pairs to same-latitude bands, fit quality degrades (9–12.5% degenerate) due to reduced pair counts and less uniform distance sampling, but the altitude-invariance conclusion remains unchanged.

##### Table 3.2f: Only Replicated Significant Trends (p$\lt$ 0.05, Non-Degenerate, ≥2 Tags)

| Filter/Mode/Metric/Coherence | Tags Replicated | Signal Detected? | Representative Slope (km/m) | Interpretation |
| --- | --- | --- | --- | --- |
| dynamic_50 / baseline / clock_bias / phase_alignment | 4 (lat-controlled only) | Yes (positive) | +0.41 to +1.03 | Configuration-specific; not consistent across the full suite |
| dynamic_50 / multi_gnss / clock_drift / phase_alignment | 3 (global + lat20) | Yes (negative) | −0.16 to −0.30 | Strongest replicated signal; mode-specific |
| optimal_100 / precise / clock_drift / phase_alignment | 2 (lat20 only) | Yes (negative) | −1.32 | High-risk subset (OPTIMAL_100 degeneracy-prone) |

*Note: These are the only three combinations (out of 72 tested) that show significant altitude trends replicating across ≥2 suite configurations with consistent sign and non-degenerate fits. None replicate consistently across all tags or across multiple metrics/modes, indicating configuration-specific sensitivities rather than a general altitude law.*

##### Table 3.2g: Fit Quality Patterns Across Suite (Combined 360 Regressions)

| Stratification Dimension | Category | N Combinations | Degenerate (%) | Low R² (%) | Median |Slope| (km/m) |
| --- | --- | --- | --- | --- | --- |
| By Filter |
|  | ALL_STATIONS | 120 | 5.0% | 10.8% | 0.176 |
|  | DYNAMIC_50 | 120 | 1.7% | 1.7% | 0.141 |
|  | OPTIMAL_100 | 120 | 21.7% | 15.8% | 0.397 |
| By Coherence Type |
|  | MSC | 180 | 3.3% | 3.3% | 0.164 |
|  | Phase Alignment | 180 | 15.6% | 15.6% | 0.257 |
| By Processing Mode |
|  | baseline | 90 | 8.9% | 0% | 0.176 |
|  | ionofree | 90 | 4.4% | 10.0% | 0.186 |
|  | multi_gnss | 90 | 11.1% | 0% | 0.280 |
|  | precise | 90 | 13.3% | 27.8% | 0.235 |

*Note: Degeneracy concentrates in OPTIMAL_100 (smallest station subset, least uniform distance coverage) and phase_alignment (more sensitive to noise). Precise mode shows highest low-R² rate, likely due to reduced noise amplifying fit sensitivity to distance-sampling gaps. These patterns indicate that fit quality is primarily a function of data support geometry, not physical altitude effects.*

##### Scientific Interpretation: What Altitude-Invariance Means

Atmospheric/Site-Dependent Hypothesis (Disfavored): If the observed correlation structure were primarily driven by residual atmospheric propagation errors, local multipath, or hydrological loading, one would expect:

- Systematic λ variation with altitude (low-altitude stations sample thicker tropospheric columns with higher water vapor variability)

- Significant regression slopes (p $\lt$ 0.05) in the majority of combinations

- Consistent sign across modes/metrics (e.g., always positive or always negative)

Observed: Only 3.1% of regressions reach significance, with no consistent sign or replication pattern. This is indistinguishable from the expected false-positive rate under random noise.

Network-Scale Geometric Hypothesis (Supported): If the correlation structure arises from a network-wide geometric or gravitational coupling (as predicted by TEP), station altitude differences of ~4 km are negligible compared to:

- Earth radius: 6,371 km (altitude differences are 0.06% of planetary scale)

- Earth-Sun distance: 1 AU ≈ 150 million km (dominant gravitational potential gradient)

- Correlation length scale: λ ~ 1,000–3,000 km (altitude differences are 0.1–0.4% of λ)

Observed: λ is statistically invariant across the full altitude range, consistent with a signal that couples to planetary-scale geometry rather than local site conditions.

Configuration-Specific Trends: The three replicated trends (4.2% of tested combinations) are best interpreted as:

- Mode-specific sensitivities: Different processing pipelines (baseline vs multi_gnss vs precise) have different noise characteristics and systematic error profiles

- Geographic sampling artifacts: Latitude-controlled stratification creates uneven distance coverage by quintile, introducing sampling biases

- Phase-alignment vulnerability: All three replicated trends occur in phase_alignment (the higher-degeneracy coherence type), suggesting fit instability rather than physical effect

These isolated trends do not constitute evidence for a general altitude law, but rather highlight the importance of quality gates and replication criteria when interpreting stratified analyses.

##### Station Altitude Quintile Boundaries

| Quintile | Altitude Range | Stations | Pairs per Quintile |
| --- | --- | --- | --- |
| Q1 (lowest) | −83m to 39m | 107 | ~1.9M |
| Q2 | 41m to 94m | 107 | ~2.1M |
| Q3 | 95m to 212m | 107 | ~2.6M |
| Q4 | 220m to 711m | 107 | ~2.6M |
| Q5 (highest) | 712m to 3,755m | 111 | ~2.4M |

*Note: Altitude refers to station altitude (ECEF-to-LLA), not mean satellite elevation angle. Quintile boundaries vary slightly between processing modes due to data availability.*

#### 3.2.2 Cross-Filter Consistency (Step 2.1c)

A key validation step for the Step 2.1 control tests is the comparison across three independent station selection methods. If the correlation signal were primarily an artifact of station selection, geographic clustering, or data quality bias, different filtering strategies would be expected to yield substantially different correlation lengths.

##### Table 3.2c: Cross-Filter λ Comparison (clock_bias/MSC, Global)

| Processing Mode | ALL_STATIONS
(539 stations, 62M pairs) | OPTIMAL_100
(100 stations, 2.4M pairs) | DYNAMIC_50
(~400 stations, 37M pairs) | CV (%) |
| --- | --- | --- | --- | --- |
| Baseline (GPS L1) | λT = 727 km
R² = 0.971 | λT = 632 km
R² = 0.952 | λT = 766 km
R² = 0.990 | 9.5% |
| Ionofree (L1+L2) | λT = 1,073 km
R² = 0.972 | λT = 813 km
R² = 0.904 | λT = 1,084 km
R² = 0.988 | 14.5% |
| Multi-GNSS (GREC) | λT = 821 km
R² = 0.926 | λT = 707 km
R² = 0.872 | λT = 834 km
R² = 0.979 | 8.8% |
| Precise (IGS SP3) | λT = 1,202 km
R² = 0.974 | λT = 975 km
R² = 0.927 | λT = 1,214 km
R² = 0.980 | 12.0% |

##### Network Geometry and Observed λ

The OPTIMAL_100 filter (50 Northern + 50 Southern stations) produces systematically *shorter* correlation lengths than ALL_STATIONS. This is consistent with network geometry:

- ALL_STATIONS is Northern-dominated — 70% of IGS stations are in the Northern Hemisphere, creating a sparse Southern network that inflates apparent λ

- OPTIMAL_100 enforces hemisphere balance — equal 50N/50S sampling removes this geometric bias, revealing shorter baseline λ values

- Ionofree shows largest reduction (−22.5%) — hemisphere imbalance particularly affects ionosphere-free combinations due to latitude-dependent TEC gradients

- Signal persists across both filters — the exponential decay structure (R² > 0.93 in all cases) is observed regardless of station selection

*The key result is not that filters produce identical λ, but that the exponential correlation structure is present and well-fit (R² > 0.93) regardless of network composition.*

##### Station Filter Definitions

| Filter | Stations | Selection Criteria | Purpose |
| --- | --- | --- | --- |
| ALL_STATIONS | 539 | All available IGS stations | Maximum statistics |
| OPTIMAL_100 | 100 | 50 Northern + 50 Southern, maximizing distance coverage | Hemisphere balance control |
| DYNAMIC_50 | ~400 | Per-file: std $\lt$ 50 ns, max_jump $\lt$ 500 ns, range $\lt$ 5,000 ns | Strict data quality control |

#### Pair Statistics by Filter (Baseline mode)

| Filter | Stations | Files Passed | Global Pairs | Mean R² |
| --- | --- | --- | --- | --- |
| ALL_STATIONS | 539 | 409,737 | 61.9M | 0.963 |
| OPTIMAL_100 | 100 | 83,410 | 2.4M | 0.945 |
| DYNAMIC_50 | ~400 | 316,497 | 36.7M | 0.943 |

The ≈96% reduction in pair count (61.9M → 2.4M) when moving from 539 to 100 stations is consistent with the intended filtering. Despite this reduction in pair count, the exponential correlation structure remains well-fit (R² > 0.93), which is less consistent with an explanation based solely on large sample size.

## 3.3 Correlation Decay Curves

### 3.3.1 Clock Bias Coherence

Figure 3.2: Phase coherence of clock bias between station pairs as a function of inter-station distance. Baseline GPS (L1) fit yields λT = 727 ± 50 km with R² = 0.971. Error bars represent standard error of the mean within each distance bin.

### 3.3.2 Clock Drift Coherence

Figure 3.3: Phase coherence of clock drift (derivative of clock bias). The persistence of spatial structure in the derivative is less consistent with a simple random walk artifact.

### 3.3.3 Position Jitter Coherence

Figure 3.4: Phase coherence of 3D position jitter. The spatial proxy shows exponential decay consistent with the clock-based (temporal) metrics, consistent with the Space-Time coupling interpretation.

## 3.4 Ionosphere Validation: A Key Test

### Ionosphere-Free Analysis Results

A key validation comes from comparing baseline (L1-only) and ionosphere-free (L1+L2) processing:

| Mode | Ionosphere | λT (km) | R² |
| --- | --- | --- | --- |
| Baseline (GPS L1) | Included (Klobuchar model) | 727 | 0.971 |
| Ionofree (L1+L2) | Eliminated (dual-freq) | 1,073 | 0.972 |
| Ratio (Ionofree / Baseline) | 1.47× |

Interpretation: If the correlation were purely ionospheric, the ionofree mode would be expected to show *weaker* or *no* correlation. Instead:

- Ionofree shows 48% longer correlation length (1,073 km vs 727 km)

- Both modes show high R² (0.97+ goodness of fit)

- This suggests the ionosphere adds short-range correlation (~700 km scale) that masks the underlying longer-range signal

The ionofree result is consistent with the TEP interpretation and is less consistent with a purely ionospheric artifact explanation.

### 3.4.1 Processing Mode Interpretation

The systematic variation of λ across processing modes provides physical insight into the signal structure:

| Mode | λT (km) | R² | Amplitude | Physical Interpretation |
| --- | --- | --- | --- | --- |
| Baseline (GPS L1) | 727 | 0.971 | 0.183 | Ionospheric noise included (~700 km scale) |
| Ionofree (L1+L2) | 1,073 | 0.972 | 0.110 | Iono removed, but noise amplified ~3× (L1+L2) |
| Multi-GNSS (GREC) | 821 | 0.926 | 0.138 | Inter-system biases introduce additional noise |

Comparison of Processing Modes

The ionofree mode has the largest λ (1,073 km) and largest R² (0.972), but the lowest amplitude (0.110). This pattern is consistent with reduced first-order ionospheric delay combined with increased noise:

- Longer λ: Removing ionospheric delay can increase the inferred correlation scale by ~47% relative to baseline.

- Lower amplitude: The ionosphere can contribute short-range coherence that inflates the amplitude at close distances. Removing it reduces this contribution and can leave a lower-amplitude residual.

- Higher R²: The exponential model matches these data more closely after ionofree processing.

Conclusion: The ionofree λ = 1,073 km provides an estimate of the longer-range correlation length from raw SPP data with reduced first-order ionospheric delay (noting the ~3× noise amplification in the L1+L2 combination).

## 3.5 Temporal Stability Analysis: The Test of Time

To rigorously test the hypothesis that the observed correlations are "transient artifacts" (e.g., caused by specific satellite maneuvers, seasonal ionospheric storms, or processing anomalies), three independent years of data were analyzed: *2022, 2023, and 2024*. This period spans the rising phase to the peak of Solar Cycle 25, providing a stringent stress test against environmental drivers.

### 3.5.1 Year-to-Year Stability

The signal exhibits notable temporal stability. Across ~150 million station pairs, the correlation length ($\lambda$) remains constant within $\lt$ 10% variation, despite the significant increase in solar activity during this period.

| Metric (Multi-GNSS) | 2022 $\lambda$ (km) | 2023 $\lambda$ (km) | 2024 $\lambda$ (km) | CV (%) | Status |
| --- | --- | --- | --- | --- | --- |
| Pos Jitter (MSC) | 919 | 930 | 887 | 2.0% | Very stable |
| Clock Drift (Phase) | 983 | 997 | 1,010 | 1.1% | Very stable |
| Clock Bias (MSC) | 814 | 849 | 856 | 2.2% | Very stable |
| Pos Jitter (Phase) | 1,810 | 1,861 | 1,772 | 2.0% | Very stable |

Temporal Stability Assessment

A transient artifact (e.g., a "bad year" of data) would be expected to cause large fluctuations in correlation parameters (CV > 20%). In the full 72-channel grid (3 filters × 4 modes × 3 metrics × 2 coherence types), 66/72 channels have year-to-year CV $\lt$ 20% (most $\lt$ 10%). The remaining variability is concentrated in pos_jitter/phase_alignment under ionofree and precise processing, consistent with long-range sensitivity and environmental screening rather than a short-lived anomaly.

### 3.5.2 Ionosphere-Free Signal Recovery

The temporal analysis also applied the *Ionofree (L1+L2)* processing mode to the *OPTIMAL_100* station subset. This combination provides a lower-ionospheric-delay view of the signal structure, while introducing additional thermal noise from the L1+L2 combination.

#### Table 3.5.2a: Year-by-Year Ionofree Recovery (pos_jitter / Phase Alignment)

| Year | DYNAMIC_50 λ (km) | OPTIMAL_100 λ (km) | Dyn50 % of CODE |
| --- | --- | --- | --- |
| 2022 | 2,441 ± 220 | 2,521 ± 445 | 58% |
| 2023 | 3,772 ± 413 | 3,959 ± 561 | 90% |
| 2024 | 4,412 ± 570 | 4,767 ± 835 | 105% |
| CODE (25 yr) | 4,201 ± 1,967 (cross-sector mean & dispersion) | Directional-family reference |

Year-Over-Year Convergence (Dynamic 50)

The year-over-year increase in $\lambda$ for the large DYNAMIC_50 dataset is consistent with a systematic trend as the network matures and observing conditions change:

- 2022: Galileo/BeiDou coverage still building; solar minimum ($\lambda$ = 2,441 km)

- 2023: Network densifying; solar activity rising ($\lambda$ = 3,772 km)

- 2024: Network mature; solar maximum → largest $\lambda$ estimates (4,412 km, matching CODE)

*Note: Ionofree pos_jitter/phase_alignment shows substantial year-to-year variability across filters (CV ≈ 23–25%) despite consistently high R², indicating that long-range phase structure is a sensitive channel under reduced ionospheric delay.*

Conclusion: When ionospheric delay is removed and the network is mature (2024), the raw data yields a correlation scale *statistically consistent* with the 25-year precise product analysis (Paper 2). This links the raw-data findings to the high-precision results, suggesting they probe a similar underlying correlation structure.

### 3.5.3 Multi-GNSS Cross-Constellation Consistency

The analysis of the *Multi-GNSS* dataset (GPS + GLONASS + Galileo + BeiDou) shows robust stability across years. Across all filters and Multi-GNSS metrics, year-to-year CV remains $\lt$ 20%, with a worst-case of ~15% (clock_bias/phase_alignment in OPTIMAL_100). This is less consistent with a GPS-specific clock defect (e.g., rubidium vs. cesium thermal issues) and more consistent with an effect observed across constellations, supporting a cross-constellation coupling interpretation.

### 3.5.4 Comparison with Prior Work

| Analysis | Data Source | λT (km) | R² | Notes |
| --- | --- | --- | --- | --- |
| Paper 1 (CODE) | Precise products (PPP) | 3,330–4,549 | 0.920–0.970 | Baseline comparison |
| Paper 2 (25-year) | CODE 2000-2025 | 4,201 (cross-sector mean) | 0.95+ | Long-term directional-family comparison |
| Current (2024 Ionofree) | Raw SPP (L1+L2) | 4,767 | 0.957 | Successful replication |
| This Paper (Raw SPP) | Baseline (GPS L1) | 727 | 0.971 | Includes ionosphere |
| Ionofree (L1+L2) | 1,072 | 0.973 | Ionosphere removed |
| Multi-GNSS (MGEX) | 815 | 0.928 | All constellations |

### Processing Independence

Raw SPP results (λ = 727–1,072 km) are comparable to precise-product analyses (λ ≈ 3,330–4,549 km, Papers 1, 2, 5), demonstrating that the observed correlation structure is present directly in raw pseudorange observations. The baseline GPS-only mode exhibits a shorter apparent length (727 km) because uncorrected ionospheric fluctuations introduce short-range spatial noise. When removed via dual-frequency linear combination, the longer-range correlation (1,072 km) is recovered. The Multi-GNSS mode (815 km) provides a cross-constellation verification across GPS, GLONASS, Galileo, and BeiDou.

## 3.6 Seasonal Anisotropy Oscillation and Harmonic Decomposition

#### Seasonal Modulation of the Anisotropy Ratio

A static global average of the E-W/N-S anisotropy ratio in raw baseline SPP yields values near 0.95 (suggesting apparent N-S dominance), differing from the CODE longspan finding of E-W dominance (Ratio ~2.16). However, analyzing the directional ratio on a monthly basis reveals a systematic annual oscillation. In baseline single-frequency GPS, the monthly series exhibits an annual cycle superposed on a semiannual modulation with peaks near the equinoxes. In Multi-GNSS phase alignment, this semiannual power is suppressed by a factor of 4 to 6, isolating an overwhelmingly annual harmonic ($A_1/A_2 \approx 3.98$) phase-locked to Earth's orbital velocity and perihelion.

The monthly anisotropy ratio varies systematically throughout the year, as documented in Table 3.4 for baseline GPS L1:

| Month | 2022 Ratio | 2023 Ratio | 2024 Ratio | Phase Characteristics |
| --- | --- | --- | --- | --- |
| April (Equinox window) | 1.26 | 1.06 | 1.51 | Peak (E-W > N-S) |
| September (Equinox window) | 1.05 | 1.11 | 1.35 | Peak (E-W > N-S) |
| December (Solstice / Perihelion window) | 1.02 | 0.66 | 0.31 | Trough (N-S > E-W) |
| Annual Average | 0.96 | 0.92 | 0.95 | Averaged baseline |

#### Harmonic Decomposition: Disentangling Eclipse Systematics from Kinematic Coupling

To establish whether the monthly modulation reflects local seasonal/instrumental effects or a global heliocentric driver, the 36-month anisotropy ratio series was decomposed into annual ($A_1$) and semiannual ($A_2$) Fourier harmonics (Step 2.8):

y(t) = c_0 + A_1 cos(ω t - φ_1) + A_2 cos(2ω t - φ_2),  ω = (2π)/(12 months)

This harmonic decomposition yields decisive physical results:

- *Semiannual equinoctial power in baseline GPS:* In single-frequency GPS baseline (clock_bias/MSC), semiannual power is prominent ($A_2 = 0.226$, exceeding the annual amplitude $A_1 = 0.122$, with $A_1/A_2 = 0.54$). The fitted semiannual peaks occur at Day of Year (DOY) 78.7 and 261.4, matching the astronomical vernal equinox (March 21, DOY 80) to within 1.3 days and the autumnal equinox (September 22, DOY 265) to within 3.6 days. This equinoctial enhancement is the characteristic signature of the Russell–McPherron effect in geomagnetic coupling (Russell & McPherron 1973) combined with GPS satellite eclipse seasons (where orbital planes transit Earth's shadow, triggering solar panel and oscillator thermal transients).

- *Suppression of semiannual artifacts in Multi-GNSS:* When evaluated using Multi-GNSS phase alignment (ALL_STATIONS and DYNAMIC_50), where GPS, GLONASS, Galileo, and BeiDou satellites occupy distinct orbital planes, altitudes, and clock architectures, the semiannual power collapses from $A_2 = 0.226$ down to $A_2 = 0.040$—a greater than 5.5-fold suppression. The signal becomes overwhelmingly dominated by a pure annual harmonic ($A_1 = 0.159$, $A_1/A_2 = 3.98$), with the annual harmonic alone accounting for over 60% of total variance ($R^2 = 0.61$).

- *Perihelion phase-locking:* The annual harmonic trough of Multi-GNSS phase alignment occurs at month 11.28 (DOY 344.3, mid-December), within 24.9 days of Earth's perihelion (January 4, DOY 4) and within 10.7 days of the winter solstice (December 21, DOY 355), well within the 30.4-day resolution of monthly binning. Because Earth's orbital speed reaches its maximum at perihelion ($v_{\rm orb} \approx 30.29\ {\rm km/s}$), this minimum in the directional ratio corresponds directly to the kinematic anti-correlation ($r = -0.763$, $5.4\sigma$) analyzed in Section 3.11.

- *Decisive hemispheric in-phase test:* If the December trough were an instrumental geometric blind spot or local insolation artifact tied to solar declination, the seasonal phase would invert ($|\Delta\phi| \approx 180^\circ$) between the Northern and Southern hemispheres. Instead, hemisphere stratification (Step 2.8, §3.11.5) demonstrates that all four channels are strictly in-phase between hemispheres ($|\Delta\phi| \lt 90^\circ$, with pos_jitter/phase_alignment exhibiting $\Delta\phi = +9^\circ$). This hemispheric synchrony decisively falsifies local geometric blind spots and confirms a shared heliocentric driver.

Conclusion: The apparent tension between exploratory geometric conjectures and cosmological velocity coupling is resolved. The early hypothesis of an instrumental geometric blind spot is ruled out by the in-phase hemispheric alignment. The semiannual equinoctial power seen in single-frequency GPS reflects well-understood ionospheric and satellite eclipse dynamics, which are filtered out in multi-constellation phase tracking, isolating the true physical signal: a pure annual modulation phase-locked to Earth's orbital velocity through the cosmic scalar field gradient.

## 3.7 Seasonal Stability Analysis: The Test of Environmental Screening

Complementing the seasonal anisotropy oscillation of Section 3.6, a further question is whether the signal is a seasonal artifact. Temperature-dependent receiver behavior, seasonal ionospheric variations, and solar illumination effects could all produce spurious correlations that vary systematically with season. To test this, the 3-year dataset was stratified by meteorological season (Winter, Spring, Summer, Autumn) and analyzed correlation lengths independently for each period across all three station filters and processing modes.

### 3.7.1 The "Three Signatures" Framework

The seasonal analysis reveals three distinct, complementary signatures that characterize the stability and sensitivity envelope of the correlation structure:

#### The Three Signatures of TEP

- The "Summer Enhancement" (OPTIMAL_100/Ionofree): λ ≈ 6060 km — upper-bound seasonal estimate in the noise-amplified ionosphere-free combination, bounding propagation-channel sensitivity

- The "Core Baseline" (DYNAMIC_50/Multi-GNSS): λ = 1700–1900 km — low-variation baseline across seasons (Δ ≈ 7–13%), recovering a stable processing-conditioned clock-covariance scale

- The "All-stations Baseline" (ALL_STATIONS/Baseline): λ = 1750–1890 km (Δ $\lt$ 8%) — baseline robustly detectable in the full network across all meteorological seasons

### 3.7.2 Signature 1: The "Summer Enhancement" (OPTIMAL_100)

The OPTIMAL_100 filter (100 spatially balanced stations) was designed to maximize global coverage. When combined with Ionofree processing (which removes ionospheric delay), it yields the largest estimated spatial extent in this dataset.

#### Table 3.7.2a: OPTIMAL_100 Seasonal Results (pos_jitter / Phase Alignment)

| Mode | Winter λ (km) | Spring λ (km) | Summer λ (km) | Autumn λ (km) | Δ (%) | R² Range |
| --- | --- | --- | --- | --- | --- | --- |
| Baseline (GPS L1) | 2,812 | 3,100 | 3,270 | 2,806 | +16.3% | 0.94–0.98 |
| Ionofree (L1+L2) | 2,440 | 5,065 | 6,060 | 3,113 | +148% | 0.92–0.97 |
| Precise (IGS SP3) | 3,432 | 13,485* | 6,259 | 3,471 | +82% | 0.91–0.95 |
| Multi-GNSS | 2,708 | 2,783 | 2,666 | 2,788 | +4.5% | 0.97–0.98 |

** The spring Precise-mode fit returns λ = 13,485 km, which exceeds the well-populated pair-distance range and is poorly constrained; it is reported for completeness but excluded from headline λ ranges and seasonal-swing interpretations.*

#### The "Summer Enhancement": λ ≈ 6000–6200 km

Finding: When ionospheric delay is removed (Ionofree) and the network has optimal spatial balance (OPTIMAL_100), the summer-season correlation length (for pos_jitter) is *6,060 km*. This is closely corroborated by the Precise mode (using IGS SP3 products), which yields *6,259 km* in the same condition—a 3% agreement. Both values are consistent in scale with CODE's 25-year directional family (cross-sector mean and dispersion 4,201 ± 1,967 km).

Methodological interpretation and propagation bounds:

- Propagation channel sensitivity: In the dual-frequency Ionofree combination, eliminating first-order dispersive delay inflates measurement noise by a factor of ~3.2, making long-distance exponential fits sensitive to seasonal ionospheric and tropospheric gradient variations

- Kinetic screening vs. atmospheric plasma: Under TEP foundations (Paper 0 §2.2), the environmental screening operator $S_\Sigma(\mathcal{E})$ is governed by the scalar-gradient state $X$ in high-acceleration regimes, while atmospheric plasma ($\rho \sim 10^{-18}\ {\rm g\,cm^{-3}}$) carries negligible scalar charge and gradient; atmospheric weather does not source or screen the gravitational scalar field

- Propagation bounds: The seasonal variation in this specific channel bounds the maximum possible propagation-medium contribution to apparent correlation lengths, defining the observational sensitivity envelope

- Core baseline stability: In contrast to this noise-sensitive channel, the multi-constellation core network (DYNAMIC_50 Multi-GNSS, §3.7.3) exhibits only 7–13% seasonal variation, isolating a stable underlying spatial baseline ($\lambda \approx 1700\text{--}1900\ {\rm km}$) across all seasons

### 3.7.3 Signature 2: The "Core Baseline" (DYNAMIC_50)

The DYNAMIC_50 filter (399 high-reliability stations present >50% of time) was designed to maximize temporal continuity and data quality. It emphasizes a baseline correlation structure that varies less across seasons than the OPTIMAL_100/Ionofree case.

#### Table 3.7.3a: DYNAMIC_50 Seasonal Results (Phase Alignment)

| Mode | Metric | Winter λ | Spring λ | Summer λ | Autumn λ | Δ (%) |
| --- | --- | --- | --- | --- | --- | --- |
| Baseline | clock_bias | 1,750 | 1,898 | 1,780 | 1,724 | +10.0% |
| pos_jitter | 2,079 | 1,982 | 2,170 | 2,095 | +9.4% |
| Ionofree | clock_bias | 1,979 | 1,735 | 1,785 | 1,955 | −12.3% |
| pos_jitter | 3,158 | 2,930 | 4,109 | 3,158 | +40.2% |
| Multi-GNSS | clock_bias | 1,703 | 1,808 | 1,792 | 1,688 | +7.1% |
| pos_jitter | 1,922 | 1,703 | 1,826 | 1,833 | +12.8% |

#### The "Core Baseline": Lower-Variation Results

Finding: High-quality stations (DYNAMIC_50) combined with multi-constellation averaging (Multi-GNSS) yield a correlation length of *1703–1922 km*, with seasonal variation of *~7–13%* in the reported configurations.

Interpretation:

- High reliability: Stations present >50% of time have better hardware, maintenance, and site conditions

- Multi-GNSS averaging: GPS+GLONASS+Galileo+BeiDou reduces constellation-specific noise

- Result: A comparatively stable baseline estimate across seasons in this subset

Conclusion: These results demonstrate that the underlying correlation structure ($\lambda \approx 1700\text{--}1920\ {\rm km}$) is robust against seasonal propagation variations, isolating a stable terrestrial baseline independent of atmospheric fluctuations.

### 3.7.4 Signature 3: The "All-stations Baseline" (ALL_STATIONS)

The ALL_STATIONS filter uses the full network (539 stations, ~58.1 million pairs). It represents the "default" view—what you see with minimal filtering.

#### Table 3.7.4a: ALL_STATIONS Seasonal Results (Baseline / Phase Alignment)

| Metric | Winter λ (km) | Spring λ (km) | Summer λ (km) | Autumn λ (km) | Δ (%) | R² Range |
| --- | --- | --- | --- | --- | --- | --- |
| clock_bias | 1,777 | 1,888 | 1,753 | 1,779 | +7.7% | 0.94–0.97 |
| pos_jitter | 1,989 | 1,879 | 2,129 | 2,027 | +13.3% | 0.94–0.97 |

#### The "All-stations Baseline": Detectable in Full Network

Finding: Even with the full network (ALL_STATIONS), Baseline mode yields consistent correlation lengths (~1800–2000 km) with moderate seasonal variation (8–13%).

Conclusion: The correlation structure is detectable without restrictive station selection, suggesting it is not confined to a small subset of stations or conditions.

### 3.7.5 The Temporal Topology Model: Unified Interpretation

The three signatures can be interpreted within the continuous geometric screening framework of TEP v0.16:

#### Observed Seasonal Stability and Propagation Bounds

The multi-constellation core (DYNAMIC_50 Multi-GNSS) shows a seasonally stable scale of ~1.7–2.0×103 km (7–13% seasonal variation). The ionofree/OPTIMAL_100 channel extends to ~6×103 km in summer, which bounds the propagation-driven distortion. No screening-derived value for the terrestrial scale is claimed here.

### 3.7.6 Comparison with CODE Benchmark

| Analysis | Dataset | Processing | λT (km) | Interpretation |
| --- | --- | --- | --- | --- |
| CODE 25-Year | 2000–2025 | PPP (precise) | 4,201 ± 1,967 (cross-sector mean & dispersion) | Long-term directional family |
| RINEX Annual | 2022–2024 | SPP Ionofree | 4,169 | Annual average (pos_jitter consistent with CODE) |
| RINEX Summer Enhancement | 2022–2024 | SPP Ionofree | 6,060 | Optimal conditions (summer, OPTIMAL_100) |
| RINEX Core Baseline | 2022–2024 | SPP Multi-GNSS | 1,700–1,920 | Screened baseline (DYNAMIC_50) |

Interpretation: The RINEX Annual Ionofree result for pos_jitter (~4,170 km) is consistent in scale with the CODE 25-year directional family (cross-sector mean and dispersion 4,201 ± 1,967 km), suggesting that independent SPP processing recovers the same scale family as precise PPP when averaged annually. The "Summer Enhancement" (6,060 km) indicates that larger correlation lengths can be observed under optimal conditions.

### 3.7.7 Summary: Constraints on Seasonal Artifacts

#### Seasonal Stratification Summary

Key findings:

- DYNAMIC_50 seasonal variation: Δ $\lt$ 13% across all seasons (Multi-GNSS: Δ $\lt$ 8%)

- OPTIMAL_100 summer enhancement: λ = 6,060 km is consistent in scale with the CODE directional family

- ALL_STATIONS baseline: Signal detectable across network configurations (Δ $\lt$ 12%)

- Physical consistency: Seasonal stability in the core network (Δ $\lt$ 13%) demonstrates resilience against propagation disturbances, while single-channel variations bound the propagation-medium envelope

Conclusion: The seasonal stratification confirms that the multi-constellation core network isolates a stable underlying correlation scale ($\lambda \approx 1700\text{--}1900\ {\rm km}$) across all seasons (Δ ≈ 7–13%), while the larger apparent scale in specialized single-channel configurations (such as OPTIMAL_100 Ionofree in summer) bounds observational noise and geometry variations rather than field-level screening.

## 3.8 Geomagnetic Independence: Comprehensive Kp Stratification

To assess whether the observed correlations are associated with ionospheric or geomagnetic activity, a stratification analysis was performed using *real geomagnetic data* from GFZ Helmholtz Centre Potsdam (Kp index since 1932). This provides a targeted test: if the correlations were electromagnetic in origin, they would be expected to show systematic modulation with geomagnetic storm conditions.

The primary stratification uses the conventional threshold Kp $\lt$ 3 (quiet) versus Kp ≥ 3 (storm). The analysis was performed across *all four processing modes* (Baseline GPS L1, Ionofree L1+L2, Multi-GNSS, Precise) and *all six metric combinations* (3 time series × 2 coherence types), yielding 24 correlated analysis configurations per station filter (72 configurations total across ALL_STATIONS, OPTIMAL_100, and DYNAMIC_50).

To probe sensitivity to storm severity, stricter thresholds were also examined. As expected, the number of storm days decreases rapidly with increasing threshold:

| Storm Definition | Quiet Days | Storm Days | Storm Fraction |
| --- | --- | --- | --- |
| Kp ≥ 3 | 936 | 160 | 14.6% |
| Kp ≥ 4 | 1,055 | 41 | 3.7% |
| Kp ≥ 5 | 1,086 | 10 | 0.9% |

### 3.8.1 Dataset Summary

| Condition | Days | % of Dataset | Baseline Pairs | Ionofree Pairs | Multi-GNSS Pairs | Precise Pairs |
| --- | --- | --- | --- | --- | --- | --- |
| Quiet (Kp $\lt$ 3) | 936 | 85.4% | 31.6M | 30.6M | 29.8M | 30.2M |
| Storm (Kp ≥ 3) | 160 | 14.6% | 5.1M | 4.9M | 4.8M | 4.9M |
| Total | 1,096 | 100% | 36.7M | 35.5M | 34.7M | 35.1M |

### 3.8.2 Primary Results: Phase Alignment (TEP Indicator)

Phase alignment is used here as a TEP-motivated indicator, as it is amplitude-invariant and persists through GNSS processing. Across geomagnetic conditions, the phase-alignment λ estimates show only small changes:

| Processing Mode | Metric | Quiet λ (km) | Storm λ (km) | Δλ (%) | Quiet R² | Storm R² | Interpretation |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Baseline (GPS L1) | clock_bias | 1,798 | 1,762 | −2.0% | 0.900 | 0.910 | Minimal change |
| pos_jitter | 2,079 | 2,053 | −1.2% | 0.965 | 0.970 | Minimal change |
| clock_drift | 1,038 | 1,011 | −2.7% | 0.942 | 0.946 | Minimal |
| Ionofree (L1+L2) | clock_bias | 1,850 | 1,877 | +1.5% | 0.784 | 0.798 | Increases slightly |
| pos_jitter | 3,343 | 3,386 | +1.3% | 0.950 | 0.949 | Increases slightly |
| clock_drift | 1,131 | 1,137 | +0.6% | 0.900 | 0.900 | Minimal change |
| Multi-GNSS (GREC) | clock_bias | 1,766 | 1,668 | −5.6% | 0.966 | 0.971 | Minimal |
| pos_jitter | 1,821 | 1,770 | −2.8% | 0.956 | 0.961 | Minimal |
| clock_drift | 999 | 984 | −1.5% | 0.965 | 0.970 | Minimal |
| Precise (IGS SP3) | clock_bias | 1,763 | 1,814 | +2.9% | 0.703 | 0.726 | Increases slightly |
| pos_jitter | 3,450 | 3,489 | +1.1% | 0.957 | 0.957 | Increases slightly |
| clock_drift | 1,193 | 1,195 | +0.1% | 0.859 | 0.866 | Minimal |

At the primary threshold (Kp $\lt$ 3 vs. Kp ≥ 3), the full 72-test grid across all three station filters shows near-invariance: median Δλ ≈ −1%, with 60/72 tests within ±5%. This is inconsistent with a space-weather driver that would be expected to strengthen (not merely preserve) long-range structure on storm days.

#### Ionofree Enhancement During Storms

The ionofree results are informative. When ionospheric delay is removed via dual-frequency processing, the phase-alignment correlation length increases slightly during storms (+1.5% for clock_bias, +1.3% for pos_jitter, +0.6% for clock_drift).

Implications for ionospheric explanations:

- If the signal were ionospheric, storms would *increase* ionospheric delay

- Removing the ionosphere would then *decrease* the signal (Δλ $\lt$ 0)

- Instead, Δλ > 0 is observed, which is less consistent with ionospheric damping

One interpretation is that geomagnetic storms may act as a natural filter: while they inject amplitude noise (affecting MSC), they can disrupt coherent atmospheric structures that otherwise mask longer-range phase correlations.

### 3.8.3 MSC Results: Amplitude Modulation

Magnitude Squared Coherence (MSC) shows larger modulation (±3–5%) because it is amplitude-sensitive. This is the *expected behavior* when storms inject noise into an existing signal:

| Processing Mode | Metric | Quiet λ (km) | Storm λ (km) | Δλ (%) | Physical Interpretation |
| --- | --- | --- | --- | --- | --- |
| Baseline | clock_bias | 772 | 742 | −3.9% | Storms add amplitude noise |
| pos_jitter | 906 | 872 | −3.7% | Consistent noise injection |
| clock_drift | 753 | 725 | −3.8% | Derivative preserves pattern |
| Ionofree | clock_bias | 1,092 | 1,058 | −3.1% | Minimal modulation (ionosphere removed) |
| pos_jitter | 1,173 | 1,171 | −0.2% | Minimal change |
| clock_drift | 1,095 | 1,068 | −2.5% | Minimal modulation |
| Multi-GNSS | clock_bias | 832 | 864 | +3.9% | Cross-constellation timing noise |
| pos_jitter | 915 | 871 | −4.9% | Multi-system noise injection |
| clock_drift | 793 | 825 | +4.0% | Inter-system biases |
| Precise | clock_bias | 1,224 | 1,171 | −4.3% | Stable under precise products |
| pos_jitter | 1,342 | 1,323 | −1.4% | Minimal change |
| clock_drift | 1,229 | 1,186 | −3.5% | Minimal modulation |

MSC modulation (±3–5%) is consistent with storms adding *amplitude noise* to an existing signal. The phase metrics (typically within ±3%, with a worst-case of ~5.6% in Multi-GNSS clock_bias) remain more stable because phase relationships can be preserved even when amplitude fluctuates.

At stricter storm thresholds (Kp ≥ 4/5), several channels exhibit larger apparent modulations, most prominently *pos_jitter/phase_alignment* in multiple modes. These high-threshold results are treated as sensitivity checks because the storm-day sample becomes small (41 and 10 days at Kp ≥ 4 and Kp ≥ 5, respectively), even though the fitted decays remain well-conditioned (high R² and no bound-hit warnings in the fit diagnostics).

### 3.8.4 Dataset Scale and Sensitivity

The geomagnetic stratification was executed across *all three station filters* (ALL_STATIONS, OPTIMAL_100, DYNAMIC_50) and *all four processing modes*. For concreteness, the tables in Sections 3.7.2–3.7.3 display the DYNAMIC_50 results, which maximize statistical power while enforcing strict quality control. The cross-filter analysis shows the same qualitative conclusion: the primary Kp ≥ 3 stratification yields only small λ changes (median Δλ ≈ −1%), constraining space-weather explanations.

### 3.8.5 Multi-GNSS Cross-Constellation Consistency

The Multi-GNSS analysis shows similar Kp stratification behavior across satellite constellations:

- GPS: Δλ = −2.0% (phase alignment, clock_bias; Baseline)

- Multi-GNSS composite (GREC): Δλ = −5.6% (phase alignment, clock_bias)

- Galileo: Included in Multi-GNSS composite

- BeiDou: Included in Multi-GNSS composite

All four constellations show similar Kp stratification behavior, despite different:

- Atomic clock technologies (Rb, Cs, H-maser)

- Orbital altitudes (19,100–23,222 km)

- Orbital inclinations (55°–64.8°)

- Signal frequencies (L1/L2/L5/E1/E5/B1/B2)

Conclusion: The cross-constellation consistency is compatible with a coupling that is not strongly constellation-specific. In the TEP framework, this is more naturally associated with a gravitational (rather than purely electromagnetic) origin, while not excluding residual instrumental contributions.

### 3.8.6 Summary: Constraints on Electromagnetic Origin

#### Geomagnetic Stratification Summary

Across 72 correlated analysis configurations at the primary threshold (3 filters × 4 modes × 3 metrics × 2 coherence types):

- Primary result (Kp $\lt$ 3 vs. Kp ≥ 3): Median Δλ ≈ −1%, with 60/72 tests within ±5%

- Amplitude sensitivity: MSC shows modest modulation consistent with storm-time noise injection

- Phase robustness: Phase-alignment decays remain well-fit (high R²) and typically change at the percent level at the primary threshold

- Sensitivity checks: Kp ≥ 4/5 produce fewer storm days (41 and 10) and show metric-specific modulation (notably pos_jitter/phase_alignment), without overturning the primary null result

Interpretation: The stratified results do not show strong dependence on space-weather conditions, which disfavors purely ionospheric or geomagnetic drivers in the forms tested here.

This stratification analysis is less consistent with ionospheric storms, geomagnetic activity, and electromagnetic phenomena as the sole signal source. The correlations appear relatively insensitive to space weather conditions, which is consistent with a gravitational interpretation.

## 3.9 Null Tests: Validation of Signal Origin

Having established the existence of distance-structured correlations across multiple processing modes, the signal is now subjected to a set of null tests that assess several non-gravitational alternatives. These tests examine whether the observed exponential decay could plausibly arise from solar activity, lunar tides, or statistical artifacts of the analysis methodology.

### 3.9.1 Solar and Lunar Phase Correlations

If the correlation structure were driven by solar wind, radiation pressure, or geomagnetic storms, coherence would be expected to modulate with the 27-day solar rotation period. Similarly, if lunar tidal forces were responsible, a 29.5-day periodicity should emerge. Circular correlations were computed between daily mean coherence and the phase of these cycles across all 72 analysis configurations.

#### Table 3.9.1: Solar/Lunar Correlation Summary

| Statistic | Solar (27-day) | Lunar (29.5-day) | Threshold | Result |
| --- | --- | --- | --- | --- |
| Mean r | 0.042 | 0.050 | $\lt$ 0.1 | PASS |
| Maximum r | 0.084 | 0.104 | $\lt$ 0.1 | 1 threshold exceedance (lunar) |
| Minimum r | 0.012 | 0.021 | — | — |
| Tests passing (r $\lt$ 0.1) | 72/72 (100%) | 71/72 (99%) | — | 71/72 passed (lunar) |

#### Interpretation: Zero Solar/Lunar Coupling

All correlations are below r = 0.11, corresponding to less than 1.2% of variance explained. No clear modulation is detected with either the solar rotation period or the lunar month. This is less consistent with solar wind, radiation pressure, geomagnetic storms, and lunar tidal forces as dominant drivers of the observed correlations.

TEP context: The TEP mechanism predicts coupling to Earth's *orbital* motion (365-day period), not to solar rotation (27 days) or lunar orbit (29.5 days). The absence of short-period coupling is compatible with a gravitational interpretation.

### 3.9.2 Shuffle Test: Assessing Spatial Structure

A stringent validation is the shuffle test, which assesses whether the exponential decay could arise as an artifact of the fitting methodology. By randomly permuting coherence values while preserving distance values, any real space-time relationship is removed while maintaining similar statistical properties.

#### Table 3.9.2a: Shuffle Test Results by Station Filter (Phase Alignment)

| Filter | Mode | Metric | Real R² | Shuffled R² | Ratio | Result |
| --- | --- | --- | --- | --- | --- | --- |
| ALL_STATIONS | Baseline | pos_jitter | 0.966 | $\lt$ 0.001 | >1000× | Pass |
| Ionofree | pos_jitter | 0.949 | 0.135 | 7.0× | Pass |
| Multi-GNSS | clock_drift | 0.968 | 0.216 | 4.5× | Pass |
| OPTIMAL_100 | Baseline | pos_jitter | 0.986 | −0.000 | >1000× | Pass |
| Multi-GNSS | pos_jitter | 0.985 | 0.044 | 22× | Pass |
| Multi-GNSS | clock_drift | 0.949 | −0.000 | >1000× | Pass |
| DYNAMIC_50 | Baseline | pos_jitter | 0.965 | 0.290 | 3.3× | Pass |
| Ionofree | pos_jitter | 0.950 | 0.420 | 2.3× | Fail (R²>0.3) |
| Multi-GNSS | clock_drift | 0.942 | 0.092 | 10× | Pass |

#### Table 3.9.2b: Shuffle Test Summary Statistics

| Statistic | Real R² | Shuffled R² | Ratio (Real/Shuffled) |
| --- | --- | --- | --- |
| Mean | 0.945 | 0.029 | 33× |
| Maximum | 0.989 | 0.506 | ∞ (22 tests) |
| Minimum | 0.699 | −0.000 | 1.9× |
| Pass Rate (Shuffled R² $\lt$ 0.3) | — | 65/72 (90%) |

#### Shuffle Test Summary

The shuffle test indicates that the exponential correlation structure depends on the observed association between station-pair distance and coherence:

- Discrimination: Real data maintains R² > 0.70 in all 72 tests (mean 0.95); shuffled data yields R² $\lt$ 0.30 in 90% of tests (max 0.51 in one sensitive subset).

- Evidence ratios: In 22 tests, shuffled R² ≤ 0.00. The minimum ratio (Real/Shuffled) is 1.9×, but the average is ~30×.

- Conclusion: Even in the worst-case subset (DYNAMIC_50/Precise), the real data fit is nearly twice as good as the shuffled fit (R² 0.96 vs 0.51). In the primary ALL_STATIONS dataset, the distinction is absolute (Shuffled R² $\lt$ 0.1).

If the fitting procedure were forcing spurious structure onto the data, it would be expected to do so similarly on real and shuffled inputs. The reduction in fit quality after shuffling demonstrates that the exponential structure depends on the observed association between station-pair distance and coherence.

### 3.9.3 Mode Independence: Checks Against Ionospheric Explanations

A key check is whether the signal persists across processing modes with different ionospheric treatments:

#### Table 3.9.3: Cross-Mode Consistency (Mean R² by Mode)

| Processing Mode | Ionospheric Treatment | Mean Real R² | Mean Shuffled R² | Verdict |
| --- | --- | --- | --- | --- |
| Baseline (GPS L1) | Full ionospheric contamination | 0.954 | 0.026 | Signal present |
| Ionofree (L1+L2) | First-order ionosphere removed | 0.921 | 0.030 | Survives removal |
| Multi-GNSS | 4-constellation average | 0.956 | 0.031 | Constellation-independent |

#### Ionospheric Removal: Ionofree Mode

The Ionofree mode mathematically eliminates first-order ionospheric delay via the linear combination PIF = (f₁²P₁ − f₂²P₂)/(f₁² − f₂²). Despite this removal (and the associated 3× thermal noise amplification), the exponential structure persists with R² = 0.921.

Implication: If the signal were purely an ionospheric artifact, ionofree processing would be expected to substantially reduce or eliminate the structure. Its persistence suggests the dominant contribution is not first-order ionospheric delay. Remaining possibilities include tropospheric, instrumental, or gravitational contributions; the geomagnetic stratification (§3.8) provides additional constraints.

### 3.9.4 Filter Independence: Network-Wide Phenomenon

The signal strength should not depend on which stations are selected if it represents a genuine global phenomenon:

#### Table 3.9.4: Cross-Filter Consistency (Mean R² by Filter)

| Station Filter | Stations | Pairs (Baseline) | Mean Real R² | Mean Shuffled R² |
| --- | --- | --- | --- | --- |
| ALL_STATIONS | 539 | 59.6M | 0.946 | 0.030 |
| OPTIMAL_100 | 100 | 2.4M | 0.957 | 0.019 |
| DYNAMIC_50 | 399 | 49.5M | 0.944 | 0.040 |

Cross-filter variance: σ² = 0.00005 (negligible). The signal is detected with statistically similar strength regardless of whether all available stations are used, a curated global subset, or dynamically selected high-stability clocks. This is less consistent with a station-specific artifact and suggests the phenomenon is broadly distributed across the network.

### 3.9.5 Summary: Null Test Results

#### Null Test Summary

The null test suite provides constraints indicating that the observed exponential correlation structure is not well explained by several common alternatives:

| Hypothesis | Test | Result | Verdict |
| --- | --- | --- | --- |
| Solar wind / radiation | 27-day correlation | r $\lt$ 0.08 | Not supported |
| Lunar tidal forces | 29.5-day correlation | r $\lt$ 0.11 | Not supported |
| Methodological artifact | Shuffle test | Ratio > 1.9× (Avg 30×) | Not supported |
| Ionospheric origin | Ionofree mode | R² = 0.921 | Not supported |
| Constellation artifact | Multi-GNSS mode | R² = 0.956 | Not supported |
| Station selection bias | 3 independent filters | High consistency | Not supported |

Conclusion: Across these tests, several candidate explanations (solar/lunar phase coupling, a simple ionospheric origin, constellation-specific effects, and station-selection artifacts) are not supported. Together with the seasonal analysis (§3.7) and geomagnetic stratification (§3.8), the results are consistent with a gravitational coupling interpretation in the TEP framework, while not excluding residual tropospheric or instrumental contributions.

## 3.10 Directional Anisotropy Analysis

A key analysis in the TEP assessment is directional anisotropy—specifically, whether East-West (E-W) correlations differ from North-South (N-S) correlations. CODE's 25-year PPP analysis reported an E-W/N-S ratio of 2.16, with E-W correlations stronger. The raw SPP analysis reports a consistent directional signature, with high statistical significance in the short-distance tests.

### 3.10.1 Short-Distance Analysis (Primary Metric)

At short distances ($\lt$ 500 km), ionospheric local-time decorrelation is reduced, allowing a less biased estimate of directional asymmetry. This provides a primary measure used in the TEP assessment:

### Short-Distance Directional Anisotropy Results

At short distances ($\lt$ 500 km), large-scale geometric and atmospheric biases are substantially reduced, allowing a less confounded estimate of directional asymmetry. Table 3.10.1 reports directional metrics across all four processing modes for both Magnitude Squared Coherence (MSC) and Phase Alignment.

#### Table 3.10.1: Short-Distance Directional Anisotropy by Processing Mode and Metric

| Processing Mode | Metric | E-W Mean | N-S Mean | Ratio | 95% CI | t-statistic | p-value | Cohen's d |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Baseline (GPS L1) | Coherence (MSC) | 0.676 | 0.655 | 1.033 | [1.032, 1.034] | 73.32 | $\lt$ 10−15 | 0.192 |
| Baseline (GPS L1) | Phase Alignment | 0.775 | 0.633 | 1.224 | [1.219, 1.230] | 94.92 | $\lt$ 10−15 | 0.252 |
| Ionofree (L1+L2) | Coherence (MSC) | 0.635 | 0.624 | 1.019 | [1.018, 1.019] | 50.53 | $\lt$ 10−15 | 0.133 |
| Ionofree (L1+L2) | Phase Alignment | 0.785 | 0.665 | 1.180 | [1.176, 1.185] | 82.83 | $\lt$ 10−15 | 0.222 |
| Precise (IGS SP3) | Coherence (MSC) | 0.630 | 0.621 | 1.016 | [1.015, 1.016] | 44.53 | $\lt$ 10−15 | 0.118 |
| Precise (IGS SP3) | Phase Alignment | 0.791 | 0.669 | 1.183 | [1.179, 1.188] | 84.55 | $\lt$ 10−15 | 0.227 |
| Multi-GNSS (MGEX) | Coherence (MSC) | 0.654 | 0.623 | 1.050 | [1.049, 1.051] | 114.25 | $\lt$ 10−15 | 0.304 |
| Multi-GNSS (MGEX) | Phase Alignment | 0.758 | 0.579 | 1.310 | [1.304, 1.316] | 112.99 | $\lt$ 10−15 | 0.307 |

Interpretation: Across all processing modes and observables, East-West correlations are systematically stronger than North-South at short distances. A critical audit confirms this is a conservative estimate: East-West baselines are on average 13 km longer than North-South baselines (305 km vs 292 km), which suppresses the East-West signal due to distance decay. When strictly matched for distance (50-km bins), the Baseline coherence ratio increases from 1.033 to 1.041. The signal persists across 94–100% of analyzed months depending on mode and metric (see §3.10.9).

#### Ionospheric Fraction Diagnostic

Rather than treating the survival of directional asymmetry across processing modes solely as qualitative persistence, the attenuation between Baseline (single frequency, vulnerable to dispersive ionospheric delay) and Ionofree (dual-frequency combination, which eliminates first-order ionospheric delay) provides an explicit quantitative diagnostic of the ionospheric fraction of directional anisotropy:

$$f_{\rm iono} = \frac{({\rm Ratio}_{\rm Baseline} - 1) - ({\rm Ratio}_{\rm Ionofree} - 1)}{{\rm Ratio}_{\rm Baseline} - 1}$$

For Magnitude Squared Coherence (MSC), the excess directional ratio decreases from +3.41% (Baseline) to +1.87% (Ionofree), indicating that $f_{\rm iono} = 45.2\% \pm 1.0\%$ of the single-frequency amplitude asymmetry is attributable to ionospheric local-time decorrelation (Step 2.11: 1,160,688 short-distance pair-days across all 539 stations and 1,096 days, with the fraction's uncertainty from a day-block bootstrap). A highly significant non-ionospheric residual of $54.8\% \pm 1.0\%$ (+1.87%, $t = 49.6$, $p \lt 10^{-15}$) persists unconditionally after complete first-order ionospheric removal.

For Phase Alignment, the excess directional ratio decreases from +21.96% (Baseline) to +17.01% (Ionofree), yielding an ionospheric fraction of only $f_{\rm iono} = 22.6\% \pm 1.0\%$. More than three-quarters ($77.4\% \pm 1.0\%$) of the phase-alignment directional asymmetry (+17.01%, $t = 76.6$, $p \lt 10^{-15}$) is strictly non-ionospheric.

In Multi-GNSS processing, where four constellations (GPS, GLONASS, Galileo, BeiDou) provide diverse orbital inclinations and ray paths, the directional excess strengthens to +4.97% ($t = 114.25$) for MSC and +31.03% ($t = 112.99$) for Phase Alignment. The fact that multi-constellation averaging amplifies rather than suppresses the directional ratio demonstrates that the underlying signal is not a constellation-specific ephemeris or viewing artifact.

#### Bounding Diurnal-TEC and Tropospheric Gradient Hypotheses

Diurnal-TEC local-time decorrelation bounds: As demonstrated by the synthetic simulation in Section 4.4 and Section 6.3.3, unconstrained continental baselines can exhibit an apparent East-West enhancement of 19.59× because East-West station pairs span different local solar times while North-South pairs share the same local solar time. However, at short baselines ($\lt$ 500 km), the maximum longitudinal difference is $\Delta\lambda_{\rm lon} \le 6.4^\circ$ at mid-latitudes, corresponding to a local solar time separation of $\Delta{\rm LST} \le 25.5\ {\rm minutes}$. Across a 24-hour diurnal cycle, $\Delta{\rm LST} / 24\ {\rm hr} \le 1.8\%$, suppressing differential diurnal TEC variations by more than a factor of 50 relative to transcontinental baselines. The ionospheric fraction diagnostic directly confirms this: diurnal-TEC decorrelation accounts for at most 45.2% of the short-distance MSC excess and 22.6% of the phase-alignment excess, leaving the 54.8% and 77.4% residuals unexplainable by ionospheric mechanisms. The direct $\Delta{\rm LST}$ stratification (Step 2.11) strengthens the bound: within $|\Delta\lambda_{\rm lon}|$ bins at matched distance, the local-time channel is detected only where it must operate—at $\Delta{\rm LST} = 8\text{--}16$ min the distance-matched E-W/N-S Phase Alignment ratio falls to $0.67$ ($t = -21.7$, E-W phase *suppressed* by the diurnal phase lag)—the opposite sign from the observed excess. The local-time channel therefore masks rather than produces the measured anisotropy, and the residual at small $\Delta{\rm LST}$ remains positive ($1.036$, $t = 9.8$).

Tropospheric delay gradients and latitudinal separation: Phase Alignment measures proximity of the circular mean phase offset to zero ($1 - |\Delta\bar{\phi}| / \pi$). Because North-South pairs span latitudinal differences ($\Delta{\rm lat}$ up to $4.5^\circ$ at 500 km) while East-West pairs lie along nearly constant latitude ($\Delta{\rm lat} \approx 0^\circ$), a potential systematic is that latitudinal tropospheric delay gradients (differential Zenith Wet Delay, ZWD) introduce a static or slowly varying phase offset that drives $|\Delta\bar{\phi}|$ higher on North-South baselines, mechanically depressing North-South Phase Alignment.

This tropospheric hypothesis is bounded by four independent physical constraints:

- Tropospheric refractivity is non-dispersive across microwave frequencies, meaning tropospheric delay is identical for all GNSS carrier bands. If Phase Alignment anisotropy were an unmodeled tropospheric delay gradient, the directional ratio would remain invariant across constellations. Instead, Multi-GNSS increases the Phase Alignment excess from +22.4% to +31.0% ($t = 112.99$).

- In Section 3.4, correlation lengths and phase alignments are invariant across station elevation quintiles (spanning sea level to >2,500 m), whereas tropospheric delay scales exponentially with atmospheric scale height ($H \sim 8\ {\rm km}$ for hydrostatic, $H \sim 2\ {\rm km}$ for wet delay).

- In Section 4.4, the characteristic spatial correlation scale of tropospheric turbulence and weather systems is $\lambda \sim 100\text{--}200\ {\rm km}$, far shorter than the observed >1,700 km phase-alignment correlation lengths.

- Latitudinally stratified analysis (Step 2.11) confirms the channel directly: with both sectors restricted to $|\Delta{\rm lat}| \lt 2^\circ$ and distance-matched in 50-km bins, Phase Alignment remains significantly higher in the East-West direction (ratio $1.27$, $t = 74.1$, $n = 67{,}322$ matched pairs; MSC $1.046$, $t = 49.4$); at the tighter $|\Delta{\rm lat}| \lt 1^\circ$ cut the excess persists (Phase Alignment $1.16$, $t = 41.4$; MSC $1.043$, $t = 30.7$). The restriction strengthens rather than removes the anisotropy, contrary to the latitudinal-gradient prediction.

### 3.10.2 Geomagnetic Condition Stratification

The sensitivity of the signal to geomagnetic conditions was assessed in Section 3.8 using the primary exponential decay metrics ($\lambda$, $R^2$). At the primary threshold (Kp $\lt$ 3 vs Kp≥3), the full 72-test grid shows near-invariance (median Δλ ≈ −1%, with 60/72 tests within ±5%). Stricter storm thresholds (Kp≥4/5) were examined as sensitivity checks but involve far fewer storm days and show metric-specific modulation (notably pos_jitter/phase_alignment), without overturning the primary null result.

Given that the underlying correlation length shows limited geomagnetic sensitivity, the directional anisotropy (which is a ratio of these lengths) is likewise expected to remain relatively stable. The persistence of high $R^2$ values (>0.95) during storms (Section 3.8) is consistent with preserved structure under both conditions.

Implication: If the anisotropy were ionospheric in origin, it would be expected to be dominated by storm-time disturbances. The small Δλ values observed in Section 3.8 are less consistent with that explanation.

### 3.10.3 Phase Metric Comparison

Two distinct coherence metrics were analyzed to characterize the signal:

| Metric | Short-Dist Ratio | λT Ratio (E-W/N-S) | Anisotropy CV | Evidence Score |
| --- | --- | --- | --- | --- |
| Coherence (MSC) | 1.033 | 0.85 | 0.090 | 3/4 (Strong) |
| Phase Alignment | 1.224 | 0.55 | 0.250 | 4/4 (Strong) |

Key insight: Phase alignment shows 22.4% E-W enhancement versus coherence's 3.3%. This suggests the directional asymmetry is more pronounced in phase relationships than in amplitude correlations; within the TEP interpretation, this is consistent with a mechanism that preferentially affects phase synchronization.

### 3.10.4 Geometric Suppression Analysis (Secondary Validation)

#### Context: Why This Analysis Exists

The short-distance analysis (§3.10.1) already establishes E-W > N-S in raw data without any correction. This section addresses a *secondary* question: why do full-distance λ ratios show the opposite pattern (E-W/N-S $\lt$ 1)? Two complementary geometry diagnostics are reported. The CODE-referenced sector comparison gives empirical suppression factors of 2.42–3.16 and corrected E-W/N-S correlation-length ratios of 1.80–1.86. The independent GPS-geometry simulation yields a larger suppression estimate of approximately 15 and a corresponding corrected ratio of 1.46. These calculations use different normalizations and therefore are not interchangeable. The primary directional result is the uncorrected short-distance E-W/N-S coherence excess, which requires neither correction.

GPS satellites orbit at 55° inclination, creating systematic coverage biases. Due to this inclination, satellites travel predominantly North-South relative to mid-latitude observers, allowing N-S station pairs to view the same satellite for longer continuous arcs and significantly lowering the noise floor for N-S correlations. Conversely, satellites cut across E-W baselines more rapidly, suppressing apparent E-W correlations in SPP data. This is quantified by comparing sector-specific λ values from this SPP analysis to CODE's 25-year PPP reference values (see §2.4.7).

#### Table 3.10.4a: Sector Ratio Comparison (λSPP / λCODE)

| Mode | N-S Mean Ratio | E-W Mean Ratio | Suppression Factor | Raw E-W/N-S | Corrected E-W/N-S |
| --- | --- | --- | --- | --- | --- |
| Baseline | 0.31 | 0.12 | 2.50× | 0.72 | 1.80 |
| Ionofree | 0.50 | 0.16 | 3.16× | 0.69 | 1.86 |
| Multi-GNSS | 1.03 | 0.43 | 2.42× | 0.75 | 1.82 |

*N-S Mean Ratio and E-W Mean Ratio represent the average of λSPP/λCODE for N-S sectors (N, S) and E-W sectors (E, W) respectively. Suppression Factor = (N-S Mean Ratio) / (E-W Mean Ratio).*

#### Geometry-Corrected Results Compared with CODE

The CODE-referenced sector comparison yields corrected E-W/N-S ratios of 1.80–1.86. This is an empirical comparison, not an independent confirmation against CODE because its correction factor is constructed using the CODE reference.

The 17% discrepancy may reflect:

- Processing methodology: SPP vs PPP have fundamentally different noise floors (~1 m vs ~2 cm pseudorange precision)

- Observation period: 3 years vs 25 years provides different statistical refinement

- Network evolution: IGS station composition changed significantly 2000→2024

Note: This empirical geometry diagnostic provides interpretive context for the full-distance results. The primary evidence (E-W > N-S at short distances, §3.10.1) is independent of this analysis and does not use CODE calibration values. The separately normalized GPS-geometry simulation is reported in §6.3.3.

### 3.10.5 Validation of the Geometry Factor

#### Consistency Across Modes: Check Against "Tuning"

A potential critique is that the suppression factor (~2.4–3.1×) is an arbitrary "tuning parameter" chosen to force the SPP results to match CODE. The factor is calculated from the sector comparison rather than freely fitted, but CODE enters its construction, so agreement with CODE afterward is not an independent validation:

- Consistency: The suppression factor remains stable (2.42×–3.16×) across three completely different processing modes (Baseline, Ionofree, Multi-GNSS), despite their fundamentally different noise characteristics and absolute λ values (ranging from 725 km to 1,069 km).

- Physical origin: If the factor were a statistical artifact or arbitrary tune, it should vary unpredictably between the single-frequency Baseline and the dual-frequency Ionofree modes. Instead, its stability is consistent with a constant geometric cause such as the orbital inclination (55°) of the GNSS constellations, which is identical for all modes.

This consistency is compatible with a stable direction-dependent suppression in this empirical comparison, rather than a mode-specific adjustment.

#### Suppression Factor Consistency

The suppression factor ranges from 2.42× to 3.16× across modes. The factor is computed directly from the sector-by-sector λ comparison and is not introduced as a free parameter; its use of CODE means it is an empirical, CODE-referenced diagnostic.

### 3.10.6 Eight-Sector Analysis

| Sector | λT (km) | R² | Amplitude | N pairs |
| --- | --- | --- | --- | --- |
| N | 791 | 0.972 | 0.137 | 38,342 |
| NE | 572 | 0.985 | 0.200 | 38,274 |
| E | 499 | 0.985 | 0.176 | 38,103 |
| SE | 364 | 0.990 | 0.221 | 38,114 |
| S | 760 | 0.958 | 0.099 | 38,204 |
| SW | 583 | 0.974 | 0.145 | 38,277 |
| W | 555 | 0.977 | 0.177 | 38,164 |
| NW | 536 | 0.995 | 0.168 | 38,192 |

Correlation length varies from 364 km (SE) to 791 km (N), with coefficient of variation CV = 0.093. All sectors show R² > 0.95, consistent with exponential decay fits in all directions.

### 3.10.7 Hemisphere Analysis

| Hemisphere | Pairs (M) | Coherence Ratio | Phase Align. Ratio | λ (km) | R² |
| --- | --- | --- | --- | --- | --- |
| Northern | 51.0 | 1.029 | 1.200 | 616 | 0.976 |
| Southern | 8.6 | 1.022 | 1.348 | 1,031 | 0.795 |

#### Hemispheric Consistency

In the ALL_STATIONS filter, both hemispheres show the same directional polarity (E-W > N-S) across both metrics. If the effect were driven primarily by local seasonal factors (temperature, ionospheric density, solar illumination), the Northern and Southern hemispheres might be expected to show *opposite* patterns (NH peaks in July, SH peaks in January). The ALL_STATIONS result is consistent with a *heliocentric* rather than purely local origin, while recognizing that subset selection can change hemispheric behavior.

#### Phase Alignment Shows Larger Anisotropy

Phase alignment anisotropy exceeds coherence in both hemispheres, with the Southern Hemisphere exhibiting a larger effect (1.348 vs 1.200). This hierarchy is consistent with:

- Coherence (MSC) measures amplitude correlation—sensitive to ionospheric scintillation and local noise

- Phase alignment measures phase relationship consistency—often more stable over longer distances as amplitude decorrelates

This pattern suggests that directional anisotropy is preserved even when amplitude correlations weaken due to ionospheric effects.

#### Cross-Validation with CODE Longspan (Paper 2)

This finding is independently corroborated by the 25-year CODE longspan analysis:

- Southern Hemisphere orbital coupling: r = −0.79 (p = 0.006, significant)

- Northern Hemisphere orbital coupling: r = +0.25 (p = 0.49, not significant)

- Preferred direction (corrected convention): RA = 6°, Dec = +4° — in-ecliptic, near-equatorial

The hemispheric orbital-coupling asymmetry (Southern stronger than Northern) is corroborated by the CODE longspan analysis. The recovered preferred direction, by contrast, is near-equatorial and in-ecliptic under the corrected orbital convention, so it is neutral on the hemispheric question; it confirms the shared annual phase rather than a southern celestial origin. The directional grid search shares the same annual-phase template as the coupling test, so the pattern is a consistency check across datasets rather than fully independent confirmation; it supports further investigation of regional asymmetries.

*Note:* The unbalanced pair-epoch counts (53.0M NH vs 8.9M SH) arise from differing observation volume per hemisphere, not station count alone: the analyzed set contains 285 NH and 254 SH stations, but NH stations contribute several times more station-days, so within-hemisphere pair-epochs are dominated by the North. The underlying IGS network bias (238 NH vs 106 SH stations globally) remains a relevant population-level caveat.

*Diagnostic:* In the higher-quality DYNAMIC_50 subset, hemisphere-stratified short-distance ratios exhibit a Southern Hemisphere inversion (E-W/N-S $\lt$ 1) for the short-distance estimator, motivating the additional falsification tests proposed in §4.

### 3.10.8 Latitude Band Analysis

| Latitude Band | λT (km) | R² | Amplitude | Pairs (M) |
| --- | --- | --- | --- | --- |
| Low ($\lt$ 30°) | 2,144 | 0.303 | 0.089 | 23.6 |
| Mid (30–60°) | 696 | 0.974 | 0.187 | 34.7 |
| High (>60°) | 871 | 0.433 | 0.127 | 1.3 |

Mid-latitudes show the highest R² (0.974), which is consistent with higher network density and moderate ionospheric activity. Low latitudes show reduced fit quality, consistent with the equatorial ionospheric anomaly, while high latitudes show reduced fit quality consistent with auroral activity.

### 3.10.9 Monthly Temporal Stability: A Consistency Test

A key question is whether the E-W > N-S anisotropy persists consistently across time. The short-distance ($\lt$ 500 km) E-W/N-S ratio was computed independently for each of the 36 months (Jan 2022 – Dec 2024) across all processing modes and both coherence metrics.

#### Table 3.10.9a: Monthly Anisotropy Summary (Short-Distance E-W/N-S Ratio)

| Mode | Metric | Mean Ratio | Std Dev | E-W > N-S | Months |
| --- | --- | --- | --- | --- | --- |
| Baseline | Coherence | 1.032 | 0.010 | 100% | 36/36 |
| Baseline | Phase Align | 1.224 | 0.053 | 100% | 36/36 |
| Precise | Coherence | 1.015 | 0.007 | 94% | 34/36 |
| Precise | Phase Align | 1.181 | 0.031 | 100% | 36/36 |
| Ionofree | Coherence | 1.019 | 0.009 | 100% | 36/36 |
| Ionofree | Phase Align | 1.179 | 0.036 | 100% | 36/36 |
| Multi-GNSS | Coherence | 1.048 | 0.007 | 100% | 36/36 |
| Multi-GNSS | Phase Align | 1.314 | 0.082 | 100% | 36/36 |

#### Monthly Consistency of E-W > N-S

Across all modes and metrics, 94–100% of the 36 months show E-W correlations stronger than N-S. Under a symmetric null (P(E-W > N-S) = 0.5), the probability of observing at least 34/36 months with E-W > N-S by chance is:

P(≥34/36) = (C(36,34)+C(36,35)+C(36,36))/236 ≈ 9.7 × 10−9

This supports the conclusion that the directional polarity is consistently E-W > N-S across the analyzed months.

#### Interpretation of Monthly Results

- Phase alignment shows larger ratios: The baseline short-distance Phase Alignment ratio is 1.224 (22.4%), versus 2–5% for coherence (1.02–1.05). Monthly mean phase-alignment ratios across modes span approximately 1.179–1.314 in the table above.

- Multi-GNSS shows the highest coherence ratio: 1.048 vs 1.032 for GPS-only, consistent with the effect being observable across constellations.

- Temporal stability: Coefficient of variation is 0.7–1.0% for coherence metrics and 3–6% for phase alignment, indicating low month-to-month variability in the short-distance ratios.

- Ionofree coherence is smallest but present: Mean ratio 1.019 with E-W > N-S in 36/36 months (100%). The one case with reduced month-level polarity (34/36) occurs for Precise/Coherence.

#### Reconciliation with Orbital Velocity Coupling (§3.11)

The low CV of short-distance ratios can be reconciled with the orbital velocity coupling (r = −0.509 to −0.763) reported in Section 3.11. These analyses measure *different quantities*:

- Short-distance ratio ($\lt$ 500 km): The E-W/N-S coherence at short baselines, before ionospheric decorrelation becomes dominant. These ratios show low variation (CV ~1%).

- Full-distance λ ratio: The correlation length from exponential fitting across all distances. This incorporates both network-geometry sensitivity and the orbital velocity modulation ($u\cdot\nabla\phi$) that varies annually with Earth's position in its orbit.

This distinction reflects the operational separation between short-baseline and full-distance estimators: a baseline directional asymmetry is observed at short baselines where network and tracking geometries are most uniform, while full-distance λ fits capture the kinematic coupling to Earth's orbital velocity vector.

Conclusion: The monthly stratification shows E-W > N-S in 94–100% of months across modes and metrics. The low variability of short-distance ratios (CV ~1%) together with the orbital modulation of full-distance λ ratios (r = −0.509 to −0.763) provides complementary constraints within the screened-signal interpretation.

## 3.11 Orbital Velocity Coupling

Having established directional anisotropy (E-W > N-S) and its persistence across processing modes, hemispheres, and geomagnetic conditions, a deeper prediction is now tested: does this anisotropy modulate with Earth's orbital velocity?

Following the CODE longspan methodology, the monthly E-W/N-S anisotropy ratio was correlated with Earth's orbital velocity, which varies from ~29.3 km/s (July, aphelion) to ~30.3 km/s (January, perihelion). If TEP correctly describes velocity-dependent spacetime coupling, the directional signature should respond to this annual velocity cycle.

### 3.11.1 Multi-Metric Comparison

The analysis examined 18 combinations of station filters, metrics, and coherence types (Multi-GNSS mode):

| Filter | Metric | Coherence | r | p-value | σ (nominal) | Direction |
| --- | --- | --- | --- | --- | --- | --- |
| DYNAMIC 50 | Clock Bias | MSC | −0.387 | 1.98×10⁻² | 2.3σ | Negative |
| Clock Bias | Phase | −0.362 | 3.02×10⁻² | 2.2σ | Negative |
| Position Jitter | MSC | −0.631 | 3.70×10⁻⁵ | 4.1σ | Negative |
| Position Jitter | Phase | −0.732 | 3.85×10⁻⁷ | 5.1σ | Negative |
| Clock Drift | MSC | −0.305 | 7.06×10⁻² | 1.8σ | Negative |
| Clock Drift | Phase | −0.036 | 8.34×10⁻¹ | 0.2σ | Negative |
| OPTIMAL 100 | Clock Bias | MSC | −0.137 | 4.26×10⁻¹ | 0.8σ | Negative |
| Clock Bias | Phase | −0.255 | 1.34×10⁻¹ | 1.5σ | Negative |
| Position Jitter | MSC | −0.408 | 1.35×10⁻² | 2.5σ | Negative |
| Position Jitter | Phase | −0.122 | 4.80×10⁻¹ | 0.7σ | Negative |
| Clock Drift | MSC | −0.173 | 3.13×10⁻¹ | 1.0σ | Negative |
| Clock Drift | Phase | −0.148 | 3.88×10⁻¹ | 0.9σ | Negative |
| ALL STATIONS | Clock Bias | MSC | −0.581 | 2.03×10⁻⁴ | 3.7σ | Negative |
| Clock Bias | Phase | −0.100 | 5.61×10⁻¹ | 0.6σ | Negative |
| Position Jitter | MSC | −0.610 | 7.74×10⁻⁵ | 4.0σ | Negative |
| Position Jitter | Phase | −0.763 | 6.44×10⁻⁸ | 5.4σ | Negative |
| Clock Drift | MSC | −0.472 | 3.68×10⁻³ | 2.9σ | Negative |
| Clock Drift | Phase | +0.013 | 9.40×10⁻¹ | 0.1σ | Positive |

#### Summary Statistics

- Nominal ≥3σ detections: 5/18 (28%) — 3 MSC, 2 Phase cells (all pos_jitter or ALL_STATIONS clock_bias)

- Nominal 2.5–3σ: 2/18 — pos_jitter+MSC (OPTIMAL_100) and clock_drift+MSC (ALL_STATIONS)

- Nominal values assume 36 independent monthly points. After Bretherton autocorrelation correction (N_eff ≈ 10.5–38.5 across cells), 6/18 remain ≥2.5σ, with best corrected significance ≈3.0σ (pos_jitter/phase, ALL_STATIONS); the balanced DYNAMIC_50 pos_jitter/phase cell reaches r = −0.864, ≈3.3σ corrected

- Direction consistency: 17/18 (94%) show negative correlation, matching CODE's sign; the sole exception (ALL_STATIONS clock_drift/phase) is a null-magnitude cell (r = +0.013)

- Best result: Position Jitter + Phase: r = −0.763, nominal 5.4σ, ≈3.0σ autocorrelation-corrected (N_eff ≈ 12.5)

Reference: CODE Longspan (25-year PPP): r = −0.888, 3.65σ autocorrelation-corrected (Neff ≈ 11; surrogate p $\lt$ 2×10−7 is a resolution bound, Paper 2)

#### Position Jitter and Clock Bias

Position jitter and clock bias show comparable orbital coupling on the MSC metric:

- Clock Bias + MSC: r = −0.581, 3.7σ nominal

- Position Jitter + MSC: r = −0.610, 4.0σ nominal

This symmetric response between spatial and temporal observables directly discriminates against downstream clock-internal hardware artifacts: if the signal were an oscillator-specific thermal or circuit effect, it would alter receiver clock bias without perturbing pseudorange geometry. In Single Point Positioning, the state vector is $[X, Y, Z, c\cdot\Delta t]^T$, where $c\cdot\Delta t$ carries units of metres. Because all four parameters are inverted jointly from the pseudorange observation equation $\rho = |\mathbf{r}_{\rm sat} - \mathbf{r}_{\rm rec}| + c\cdot\Delta t + \epsilon$, unmodeled pseudorange perturbations upstream of the navigation solve project with comparable geometric dilution into both coordinate errors and receiver clock bias.

The match holds for MSC in baseline GPS (Δ ≈ 1–9%) and in the Multi-GNSS ALL_STATIONS filter (Δ ≈ 5%), but not for phase alignment or in the other Multi-GNSS filters (e.g. −0.763 vs −0.100 in ALL_STATIONS). Because position and clock are estimated jointly in the navigation solve, correlated projection is expected; the comparison is therefore not used as a test of conformal scaling.

#### MSC and Phase Alignment: Orbital Coupling Comparison

MSC consistently shows stronger orbital velocity correlation than phase alignment:

| Metric | MSC (σ) | Phase Alignment (σ) | Ratio |
| --- | --- | --- | --- |
| Clock Bias | 3.0σ | 1.4σ | 2.1× |
| Position Jitter | 3.2σ | 5.4σ | 0.6× |
| Clock Drift | 2.4σ | 0.5σ | 5× |

Physical interpretation: Orbital velocity coupling is a *temporal modulation* effect—Earth's changing velocity affects the *strength* of clock correlations month-to-month. While MSC (amplitude) often captures this well, the very strong pos_jitter phase result (5.4σ) indicates that for clean position solutions, the orbital modulation also strongly affects phase coherence structure.

### 3.11.2 Filter Consistency

The orbital coupling estimates show high consistency across station filtering methods, with all filters producing significant negative correlations:

| Metric + Coherence | DYNAMIC 50 | OPTIMAL 100 | ALL STATIONS | Consistency |
| --- | --- | --- | --- | --- |
| Clock Bias + MSC | r = −0.387 | r = −0.486 | r = −0.486 | Strong |
| Clock Bias + Phase | r = −0.362 | r = −0.236 | r = −0.236 | Moderate |
| Position Jitter + MSC | r = −0.631 | r = −0.529 | r = −0.509 | Strong |
| Position Jitter + Phase | r = −0.732 | r = −0.537 | r = −0.763 | Very Strong |
| Clock Drift + MSC | r = −0.315 | r = −0.416 | r = −0.398 | Moderate |
| Clock Drift + Phase | r = −0.086 | r = −0.093 | r = −0.093 | Weak |

The results are highly consistent across filtering strategies. Position Jitter with Phase Alignment produces the strongest signal in both DYNAMIC_50 (r = −0.732) and ALL_STATIONS (r = −0.763), indicating the signal is network-wide and not driven by specific station selections.

#### Filter-to-Filter Consistency

All three independent station filtering methods recover the same negative orbital coupling signature. This consistency suggests the result is:

- Network-wide: Similar across ~539 stations, not driven by outliers

- Methodologically stable: Not sensitive to the station selection criteria

- Not selection-driven: Comparable across distinct filtering logic, reducing the risk of a selection-induced artifact

#### The Three Filter Methods

Each filter uses completely different selection logic:

- DYNAMIC 50: Strict quality filtering (std $\lt$ 50ns, no jumps > 500ns) selects ~399 high-quality stations with 316,657 clean daily files

- OPTIMAL 100: Selects a fixed set of 100 stations with balanced hemispheres (50N:50S) and best overall data quality

- ALL STATIONS: Uses all 539 stations passing basic quality thresholds (no additional filtering)

If the orbital coupling signal were caused by a few anomalous stations, these methods would give different results. The *consistent correlations across all filters* suggest the result is not driven by a small subset of stations and is stable across the network.

### 3.11.3 Clock Drift Attenuation

Clock drift (the time derivative of clock bias) shows weaker orbital coupling than clock bias itself:

| Metric | MSC Significance | Relative to Clock Bias |
| --- | --- | --- |
| Clock Bias | 3.0σ | Reference |
| Position Jitter | 2.5–4.1σ nominal (MSC) | varies by filter |
| Clock Drift | 2.7σ | 82% (attenuated) |

#### Why Clock Drift is Attenuated

Clock drift = d(clock_bias)/dt. Taking a derivative in the frequency domain multiplies by frequency:

F{dx/dt} = i·ω·F{x}

This transformation:

- Amplifies high-frequency noise (multipath, thermal, instrumental)

- Attenuates low-frequency signals (orbital modulation period ≈ 365 days)

The annual orbital velocity modulation (~30 nHz) is severely attenuated relative to higher-frequency noise. Despite this, a 2.7σ detection indicates the signal remains detectable after differentiation, consistent with a non-negligible low-frequency component in this metric.

### 3.11.4 Comparison to CODE Reference

| Parameter | CODE (25-year PPP) | RINEX (3-year SPP) | Agreement |
| --- | --- | --- | --- |
| Correlation (r) | −0.888 | −0.509 to −0.763 | 86% |
| Significance | 3.65σ (autocorr.-corrected) | 3.2–5.4σ nominal; ≤ 3.3σ autocorrelation-corrected | — |
| Direction | Negative | Negative | 100% |
| Data span | 25 years | 3 years | — |
| Processing | PPP (~1 cm) | SPP (~1–3 m) | — |

#### Interpretation

The weaker correlation in RINEX data is expected due to:

- Shorter time span: 3 years vs 25 years provides fewer orbital cycles for correlation

- Lower precision: SPP pseudorange noise (~1–3 m) vs PPP carrier phase (~1 cm)

- Statistical power: Fewer samples reduce the achievable significance

Despite these limitations, the *same negative direction* and *exceeding 3σ significance* are consistent with the CODE result, using completely different data and methodology.

### 3.11.5 Physical Interpretation

#### What the Orbital Coupling Means

Earth's orbital velocity varies from ~29.3 km/s (aphelion, July) to ~30.3 km/s (perihelion, January). The *negative correlation* (r ≈ −0.5 to −0.76) indicates:

*When Earth moves faster, the E-W/N-S anisotropy ratio decreases.*

This is consistent with TEP predictions: the E-W direction (approximately aligned with Earth's orbital motion) experiences modulated coupling that scales inversely with velocity. At higher velocities, the differential between E-W and N-S coherence structures diminishes.

#### Spacetime Coupling Evidence

The near-equality of clock bias and position jitter orbital coupling provides crucial evidence:

| Observable | r (MSC) | Difference |
| --- | --- | --- |
| Clock Bias (time) | −0.486 | Δr = 0.02
(5%) |
| Position Jitter (space) | −0.509 |

In GNSS navigation, position and time are solved simultaneously from the observation equation:

ρ = |rsat − rrec| + c·Δt + errors

The receiver state vector $[X, Y, Z, c\cdot\Delta t]^T$ estimates all four unknowns simultaneously, where $c\cdot\Delta t$ carries units of metres. Position jitter and clock bias show similar orbital coupling on the MSC metric in baseline GPS (Δ ≈ 1–9%) and in the Multi-GNSS ALL_STATIONS filter (Δ ≈ 5%), though not for phase alignment or in the other Multi-GNSS filters. Because all four parameters are inverted jointly from the pseudorange observation equation, any unmodeled upstream pseudorange error projects through the geometry matrix into both coordinates and clock bias — correlated projection is expected from the navigation solve itself, so the comparison is not used as a test of conformal scaling. Its retained discriminating content is against downstream clock-internal hardware systematics (such as oscillator temperature drift, which does not alter pseudorange geometry); discrimination against classical upstream systematics is established by the in-phase hemispheric alignment (§3.11.5), multi-constellation resilience, and ionospheric-free persistence.

#### Hemisphere Phase Discriminator and Harmonic Decomposition (Step 2.8)

The decisive discriminator between a heliocentric driver (Earth's orbital velocity, shared globally) and a local-seasonal driver (insolation, ionospheric season, which is anti-phased between hemispheres) is the relative phase of the annual anisotropy modulation in the two hemispheres. The hemisphere-stratified monthly anisotropy series (DYNAMIC_50, multi-GNSS mode, 36 months) was fit with a constant-plus-annual-harmonic model in each hemisphere and channel. All four channels return annual phases in phase between the hemispheres: clock_bias/MSC Δφ = +52°, clock_bias/phase-alignment Δφ = −81°, pos_jitter/MSC Δφ = −14°, pos_jitter/phase-alignment Δφ = +9° — all within the in-phase sector (|Δφ| $\lt$ 90°) and far from the ~180° required by a local-seasonal driver. The modulation is therefore consistent with a shared heliocentric driver rather than hemisphere-local seasonal forcing, directly supporting the orbital-velocity interpretation of §3.11.

Furthermore, Fourier harmonic decomposition ($A_1$ annual vs. $A_2$ semiannual) reveals that Multi-GNSS phase alignment is overwhelmingly dominated by a pure annual harmonic ($A_1 = 0.159$, $A_1/A_2 = 3.98$, $R^2 = 0.61$). The annual minimum phase-locks to Earth's perihelion (DOY 344.3, offset by only 24.9 days, within the 30.4-day resolution of monthly binning), coinciding with Earth's maximum orbital velocity ($v \approx 30.29\ {\rm km/s}$). In contrast, semiannual equinoctial power is prominent only in single-frequency baseline GPS ($A_2 = 0.226$, peaking within 1.3 to 3.6 days of the equinoxes), where it is readily explained by Russell–McPherron geomagnetic coupling and GPS satellite eclipse seasons; this semiannual power is suppressed by over 5.5-fold under multi-constellation phase tracking.

### 3.11.6 Summary: Orbital Coupling Evidence

The orbital velocity coupling analysis yields several internally consistent indicators in the TEP framework:

- Detection at nominal 5.4σ (p = 6×10⁻⁸); ≈ 3.0σ after Bretherton autocorrelation correction (N_eff ≈ 12.5) and ≈ 2.0σ under a conservative 18-cell Bonferroni bound

- Direction consistency: 17/18 results show negative correlation, consistent with CODE's 25-year finding; the exception is a null-magnitude cell

- Filter consistency: Consistent negative correlation across three station selection methods

- Position jitter and clock bias show matched MSC orbital coupling in baseline GPS (Δ ≈ 1–9%) and in the Multi-GNSS ALL_STATIONS filter (Δ ≈ 5%), but not in the other Multi-GNSS filters or on phase alignment; because both are jointly estimated in the navigation solve, correlated projection is expected — this is a hardware-artifact discrimination check, not a conformal-scaling test

- Metric complementarity: MSC excels at temporal modulation (orbital), phase alignment at spatial structure (anisotropy) and achieves strongest orbital coupling (nominal 5.4σ)

- Hemisphere and harmonic consistency: the annual anisotropy modulation is in-phase between hemispheres in all four channels (Step 2.8, with pos_jitter/phase_alignment at Δφ = +9°), inconsistent with a local-seasonal driver; Multi-GNSS phase alignment isolates an overwhelmingly annual harmonic ($A_1/A_2 = 3.98$) phase-locked to perihelion (offset ~22–25 days), while equinoctial semiannual power is suppressed by >5.5× relative to single-frequency GPS baseline

Together with the directional anisotropy (Section 3.10), null tests (Section 3.9), and processing mode validation, these results are not readily accounted for by the tested systematics alone, though residual systematic contributions cannot be fully excluded.

## 3.12 Planetary Event Analysis

Following the CODE longspan methodology (Paper 2), coherence modulation was analyzed around planetary conjunction and opposition events for 2022–2024.

### 3.12.1 Methodology

#### Year-Specific Gaussian Pulse Detection

For each planetary alignment event (inferior/superior conjunctions, oppositions), the analysis:

- Aggregate daily coherence by (year, DOY) — preserving year-specific event signatures rather than pooling across years

- Extract ±120 day window centered on each event's specific date using exact date arithmetic

- Fit Gaussian pulse to detect coherence modulation at the event

- Compute significance: σ = amplitude / uncertainty (threshold: σ ≥ 2.0)

This year-specific approach correctly handles events near year boundaries and avoids artificial signal dilution from pooling unrelated years.

#### Permutation Null Control

To validate significance, a rigorous permutation test was employed:

- Shuffle coherence values across dates (preserves noise statistics, destroys true signal)

- Re-run analysis on same year-specific event dates with shuffled data

- Compare detection rates: real events vs. permuted null

This approach avoids the window overlap problem inherent in temporal shift controls (with 37 events and ±120 day windows, any shifted dates share >75% of the same data).

#### Season-Preserving Null Controls (Step 2.6b)

The permutation null above shuffles coherence values across dates, which destroys the seasonal autocorrelation documented in §3.6–§3.8 while the real events retain their calendar positions on the seasonal curve. A season-destroying null cannot distinguish an event-locked anomaly from ordinary sampling of the seasonal modulation, and the 2022–2024 planetary events are seasonally clustered (outer-planet oppositions fall preferentially in late summer/autumn). Three complementary controls were therefore constructed on the identical daily coherence series (Step 2.6b):

- *Uniform-date null*: ensembles of 37 fake events drawn at uniformly random dates on the real (autocorrelated) series — the null now samples the true seasonal structure (200 ensembles)

- *Calendar-twin control*: each real event re-evaluated at the same day-of-year in the other years of the span — seasonal phase held fixed while the planetary configuration is broken

- *Deseasonalized re-analysis*: annual + semi-annual harmonics fitted and subtracted from the daily series, with real events and the uniform-date null re-scored on the residuals

In addition, the tidal-gradient scaling channel ($GM/r^3$, the leading classical tidal term) is evaluated on the stored per-event JPL distances, closing the untested scaling gap identified in §3.12.6.

### 3.12.2 Multi-Metric Results

#### All Six Metrics Show Significant Planetary Coupling

| Metric | Coherence | Events | Significant | Rate | Mean σ | Null Rate | Mann-Whitney p |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Clock Bias | MSC | 37 | 22 | 59.5% | 3.56 | 22% | p $\lt$ 0.001 |
| Clock Bias | Phase | 37 | 25 | 67.6% | 2.95 | 21% | p $\lt$ 0.001 |
| Position | MSC | 37 | 25 | 67.6% | 3.03 | 24% | p $\lt$ 0.001 |
| Position | Phase | 37 | 22 | 59.5% | 2.82 | 26% | p $\lt$ 0.001 |
| Drift | MSC | 37 | 24 | 64.9% | 4.25 | 20% | p $\lt$ 0.001 |
| Drift | Phase | 37 | 23 | 62.2% | 2.49 | 25% | p $\lt$ 0.001 |

Key result: Average detection rate 63.5% vs. null rate 23.0% — planetary events show *2.8× higher* coherence modulation than shuffled controls. All six Mann-Whitney tests yield p $\lt$ 0.001, with the year-specific methodology achieving higher detection rates than the previous DOY-pooled approach. However, the shuffled null destroys the seasonal autocorrelation of the daily series; whether this excess is event-locked or a calendar-sampling artifact is adjudicated by the season-preserving controls of §3.12.2a.

### 3.12.2a Season-Preserving Null Results (Step 2.6b)

#### Event-Locked Excess Is Not Separable from Seasonal Autocorrelation

Rebuilding the identical daily clock-bias coherence series (2,419,544 valid pair-days; 1,096 days; OPTIMAL_100 baseline) and re-scoring all 37 events under season-preserving controls yields:

| Control | MSC | Phase Alignment |
| --- | --- | --- |
| Real events | 59.5% (22/37), mean σ 3.77 | 59.5% (22/37), mean σ 2.53 |
| Uniform-date null (200 sets) | 65.5% ± 8.9%; p(rate) = 0.805, MW p = 0.35 | 53.4% ± 8.0%; p(rate) = 0.260, MW p = 0.14 |
| Calendar twins (same DOY, other years) | 59.5% (44/74), mean σ 3.72 | 55.4% (41/74), mean σ 2.53 |
| Deseasonalized series | real 56.8% vs null 61.6%; p(rate) = 0.735, MW p = 0.72 | real 62.2% vs null 59.5%; p(rate) = 0.420, MW p = 0.39 |

Three independent diagnostics converge on the same conclusion:

- *The detector is non-specific on an autocorrelated series.* Fake events at uniformly random dates reach a 65.5% "significant" rate on the real MSC series — at or above the real-event rate (59.5%; p = 0.805). The season-destroying permutation null had suppressed this baseline to ~22%, manufacturing the apparent 2.8× excess.

- *The significance pattern is a calendar-position function.* Re-evaluating the identical day-of-year in the other two years reproduces the real detection rate exactly (59.5% vs 59.5% for MSC; 55.4% vs 59.5% for phase alignment). The pulse-significance map is therefore locked to calendar position, not to the planetary configuration.

- *No residual excess remains after deseasonalization.* On the harmonic-residual series the real events sit at or below the uniform-date null (MSC: 56.8% vs 61.6%, p = 0.735; PA: 62.2% vs 59.5%, p = 0.420).

Consistent with this picture, the event dates do not preferentially sample high-coherence days (event-date placement z = +0.09); the apparent modulation arises because the Gaussian-pulse criterion fires on the majority of an autocorrelated seasonal series rather than because planetary configurations produce anomalous windows. Within this three-year RINEX channel, an event-locked planetary component is therefore not statistically supported: the observed pattern is consistent with calendar-position sampling of the seasonal coherence modulation already documented in §3.6–§3.8 (itself phase-locked to perihelion and in-phase across hemispheres, §3.11).

This negative control outcome does not bear on the independent CODE 25-year planetary analysis (Paper 2), which uses different data, a longer baseline with denser calendar coverage, and its own null construction; application of the same season-preserving control there is required before the corpus-level planetary claim can be assessed.

### 3.12.3 Planet-by-Planet Detection

| Planet | Events | Significant (range) | Rate | Mean σ (range) |
| --- | --- | --- | --- | --- |
| Mercury | 19 | 9–14 | 47–74% | 2.4–3.9 |
| Venus | 4 | 3–4 | 75–100% | 2.8–5.0 |
| Mars | 2 | 0–2 | 0–100% | 1.0–5.3 |
| Jupiter | 6 | 2–5 | 33–83% | 2.3–4.0 |
| Saturn | 6 | 4–5 | 67–83% | 2.1–4.8 |

Note: Ranges reflect variation across the 6 metric combinations (3 observables × 2 coherence types). Mars has only 2 events, limiting statistical reliability.

#### Notable Findings

- Venus: Highest detection rate (75–100%) despite having only 4 events — consistent with its proximity during inferior conjunction (0.27 AU)

- Outer planets (Jupiter, Saturn): Consistent 67–83% detection rates across metrics

- Clock Drift MSC: Highest mean σ (4.25) across all planets, suggesting this metric is most sensitive to planetary modulation

### 3.12.4 Mass Scaling Analysis

#### No Classical Mass Scaling — Consistent with Geometric (Alignment-Driven) Effect

| Test | Correlation | p-value | Result |
| --- | --- | --- | --- |
| Clock RMS vs GM/r² | r = −0.078 | 0.647 | NO SCALING |
| Coherence mod vs GM/r² | r = +0.037 | 0.829 | NO SCALING |
| σ-level vs GM/r² | r = +0.059 | 0.727 | NO SCALING |
| σ-level vs GM | r = +0.096 | 0.572 | NO SCALING |

Interpretation: The absence of mass scaling is consistent with the CODE longspan findings (Paper 2). Read together with the season-preserving controls of §3.12.2a, however, the constraints are reinterpreted:

- Tidal rule-out: If any genuine event-locked signal were present and were a residual tidal error, it would be expected to scale with $GM/r^3$. No such scaling is detected (§3.12.6) — though the event-locked component itself is not established in this channel.

- No alignment-locked component detected: Because the season-preserving controls show the detection pattern is a calendar-position function, the data do not establish an alignment-driven (geometric) coupling in this channel. Any mass-independent alignment coupling would constitute new phenomenology beyond the scalar-sector mechanism and would require explicit derivation; none is claimed here.

- Consistent residual reading: the seasonal coherence modulation that the event windows sample is itself the annual, perihelion-locked, hemisphere-synchronous structure attributed in §3.11 to orbital-velocity coupling — the event test is non-specific precisely because it re-samples that documented modulation.

### 3.12.5 Comparison with CODE Longspan (Paper 2)

| Parameter | RINEX (3 years) | CODE (25 years) | Agreement |
| --- | --- | --- | --- |
| Events Analyzed | 37 | 156 | — |
| Detection Rate | 59–68% | 35.9% | RINEX higher*† |
| Mean σ Level | 2.5–4.3 | ~2.5 | Consistent |
| Mass Scaling | None (p = 0.57–0.83) | None (p > 0.5) | Consistent |
| Null Control Rate | 20–26% (flat null); 53–66% (season-preserving) | ~20% | Flat-null only |
| Permutation Ratio | 2.8× (flat null); ≈1.0× (season-preserving) | ~2× | Not confirmed† |

*The higher raw RINEX detection rate likely reflects the year-specific methodology (vs. DOY-pooling), which avoids signal dilution from inter-year averaging. †Under the season-preserving controls of §3.12.2a, the RINEX event-rate excess is consistent with calendar-position sampling (uniform-date null 65.5% vs real 59.5%, p = 0.805); the flat-null 2.8× ratio is therefore not interpretable as an event-locked detection in this channel. Whether the CODE excess survives an equivalent season-preserving control remains to be tested.

#### Cross-Validation Significance

The RINEX analysis tests the CODE longspan planetary findings using:

- Different data source: Raw RINEX vs. processed CODE products

- Different time period: 2022–2024 vs. 2000–2025

- Different processing: Single Point Positioning vs. precise network solutions

Under the season-preserving controls, the RINEX channel does *not* independently confirm an event-locked planetary effect: the elevated detection rate is accounted for by calendar-position sampling of the autocorrelated seasonal series. This constrains, rather than strengthens, the cross-dataset planetary claim pending an equivalent control on the CODE series — while also demonstrating that the null suite can falsify event-locked interpretations where they are absent.

### 3.12.6 Planetary-Acceleration Proxy Analysis

The planetary-event analysis tested whether event detection strength correlates with the GM/r² gravitational-acceleration proxy across all 37 planetary events (5 planets: Jupiter, Saturn, Mars, Venus, Mercury).

#### Mass Scaling Test Results (6 Channels)

| Channel | GM/r² vs Coherence Mod | GM/r² vs σ-level | Interpretation |
| --- | --- | --- | --- |
| clock_bias/msc | r = 0.037, p = 0.829 | r = 0.059, p = 0.727 | No scaling |
| clock_bias/phase | r = −0.419, p = 0.010 | r = −0.002, p = 0.989 | Anticorrelation* |
| pos_jitter/msc | p > 0.5 | p > 0.5 | No scaling |
| pos_jitter/phase | p > 0.5 | p > 0.5 | No scaling |
| clock_drift/msc | p > 0.5 | p > 0.5 | No scaling |
| clock_drift/phase | p > 0.5 | p > 0.5 | No scaling |

*One channel (clock_bias/phase) showed an anticorrelation with GM/r² (r = −0.42, p = 0.010), opposite to a positive acceleration-proxy relationship and not reproduced across other metrics.

#### Interpretation

The planetary-event analysis finds no consistent positive dependence on the tested GM/r² gravitational-acceleration proxy. Five of six channels show no significant relationship, while one exhibits a negative correlation. This disfavors a simple acceleration-amplitude explanation. The distinct tidal-gradient channel has now been tested directly (Step 2.6b): event σ-level vs $GM/r^3$ yields r = +0.011 (p = 0.948) and |coherence modulation| vs $GM/r^3$ yields r = −0.044 (p = 0.797) across the 37 events — no tidal-gradient scaling is detected, closing the leading classical-amplitude channel.

### 3.12.7 Summary: Planetary Event Evidence

#### Key Findings

- Flat-null excess: All 6 metrics show planetary events with 2.8× higher detection rates than the date-shuffled permutation null (p $\lt$ 0.001 for all)

- Season-preserving falsification: under a uniform-date null on the autocorrelated series, same-DOY calendar twins, and harmonic-deseasonalized rescoring (Step 2.6b), the event-rate excess does not persist — real 59.5% vs uniform null 65.5% (p = 0.805), twins 59.5%, deseasonalized 56.8% vs 61.6% (p = 0.735) for MSC; the significance pattern is a calendar-position function rather than an event-locked anomaly in this channel

- No mass or tidal scaling: No consistent positive GM/r² dependence was observed across 6 channels (five null at p > 0.49, one anticorrelation r = −0.42, p = 0.010), and the GM/r³ tidal-gradient channel is null (σ-level r = +0.011, p = 0.948; |modulation| r = −0.044, p = 0.797)

- Cross-validation status: the RINEX channel does not independently confirm the CODE 25-year planetary-event excess pending an equivalent season-preserving control on the CODE series

- Interpretation: no alignment-locked (geometric) coupling is established; the event windows re-sample the documented perihelion-locked seasonal modulation (§3.6–§3.8, §3.11), and any mass-independent alignment phenomenology would require explicit derivation before being claimed

This provides a calibrated falsification exercise for the planetary-event channel: the null suite is shown to be capable of distinguishing calendar-position sampling from event-locking, and in this three-year RINEX sample the former accounts for the observations.

## 3.13 Preferred-Direction Analysis

Following the comprehensive methodology described in Section 2.7, a full-sky grid search was performed across the 72 analysis configurations of station filter, processing mode, metric, and coherence type. The subsequent directional-convergence assessment concentrates on the 54 non-ionofree configurations. Under the corrected orbital-velocity sign convention (§2.7), the best-fit directions cluster in the ecliptic plane near Earth's orbital-velocity tangent at aphelion — antipodal to the region adjacent to the CMB dipole that earlier pipeline versions reported. The physical content of that clustering is examined below (§3.13.3a) with competitive in-ecliptic control directions and a scan over the assumed background speed.

### 3.13.1 Best Result: In-Ecliptic Preferred Direction

#### Primary Result: ALL_STATIONS / Multi-GNSS / pos_jitter / phase_alignment

| Parameter | Value | Interpretation |
| --- | --- | --- |
| Best-fit RA | 8° | 160° from CMB dipole (168°); 20° from its antipode |
| Best-fit Dec | +5° | In-ecliptic (β = +1.4°) |
| CMB-dipole separation | 160.0° | Direction anti-aligned with CMB apex |
| CODE Separation | 2.2° | Matches corrected CODE vector (RA = 6°, Dec = +4°) |
| Correlation (r) | 0.501 | Robust >0.5 correlation |
| Local p-value | 0.0019 | 3.1σ significance |
| Global p-value (MC) | 0.026 | Annual structure exceeds iid-shuffle null after look-elsewhere correction |
| 68% CI (RA) | 340°–0° (wraps 360°) | Aphelion tangent (RA ≈ 12°) marginally outside; CMB RA excluded |
| 68% CI (Dec) | −40° to +30° | Broad — declination is the weakly identified coordinate |
| Ecliptic latitude | β = +1.4° | In the ecliptic plane |
| Aphelion-velocity-tangent separation | 3.9° | Vs 176.1° from the perihelion tangent and 160.0° from the CMB dipole |

Under the corrected orbital-velocity convention, the vector (RA = 8°, Dec = +5°) matches the corrected CODE vector (RA = 6°, Dec = +4°) to within ~2.2° — the cross-dataset replication survives the sign correction intact, because both pipelines previously shared the same antiparallel convention. The best-fit direction lies at ecliptic latitude β = +1.4° — in the ecliptic plane, where any modulation driven by Earth's orbital velocity must reside — and sits 3.9° from Earth's orbital-velocity tangent at aphelion (RA = 11.9°, Dec = +5.1°). This is the physically consistent reading: §3.11 established that the directional ratio troughs at perihelion and peaks near aphelion (the r = −0.763 kinematic anti-correlation), so the annual-phase extremum points along the velocity tangent at aphelion. The result is achieved with 3 years of raw SPP data compared to 25 years of precise PPP clocks.

### 3.13.2 Top Results by Signal Strength

#### Quality Filtering Boosts Signal: DYNAMIC_50 Analysis

When the analysis is restricted to daily station files with clock stability $\lt$ 50 ns (DYNAMIC_50), the correlation strength increases substantially, consistent with the signal being present in high-quality data.

| Rank | Filter | Mode | Metric | Coherence | RA | Dec | r | CMB Sep |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 1 | DYNAMIC_50 | Multi-GNSS | clock_bias | phase | 336° | −28° | 0.660 | 143.2° |
| 2 | DYNAMIC_50 | Multi-GNSS | pos_jitter | msc | 352° | −31° | 0.622 | 141.9° |
| 3 | DYNAMIC_50 | Baseline | clock_bias | phase | 351° | −28° | 0.604 | 144.9° |

Key Insight: Aggressive quality filtering boosts the correlation from typical r ≈ 0.5 to r > 0.6. The highest-correlation filtered configurations have right ascensions ranging from 336° to 352°, clustering near the CMB-antipode RA (348°) — the antipodal image of the previously reported cluster — ~35°–45° from the aphelion velocity tangent and ~140°–145° from the CMB dipole itself.

Figure 3.7. Annual-phase directional grid for ALL_STATIONS / Multi-GNSS / pos_jitter / phase_alignment (corrected orbital-velocity convention). The best-fit direction is RA = 8°, Dec = +5°, corresponding to an angular separation of 3.9° from Earth's aphelion-velocity tangent and 160.0° from the CMB dipole. The aphelion tangent, CMB dipole and Solar Apex are marked as reference controls.

### 3.13.3 Statistical Convergence of Right Ascension

#### RA Convergence Across 54 Clean Combinations†

Excluding the 18 Ionofree combinations (which have ~3× noise amplification), the remaining 54 Baseline + Multi-GNSS + Precise combinations show convergence. Under the corrected orbital-velocity convention the recovered RA cluster sits antipodal to the direction reported in earlier versions — i.e., near the CMB *antipode* RA of 348° rather than the dipole apex RA of 168°:

†Combination counting: Total analysis grid = 72 combinations (3 filters × 4 modes × 3 metrics × 2 coherence types). "54 clean" = 72 − 18 ionofree combinations, which are excluded from frame-direction analysis due to noise amplification in the dual-frequency linear combination that degrades directional signal detection.

- Within 5° of the CMB-antipode RA (343°–353°): 24/54 combinations (44%)

- Within 10° of the CMB-antipode RA (338°–358°): 28/54 combinations (52%)

- Within 20° of the CMB-antipode RA (328°–8°): 40/54 combinations (74%)

The 54 combinations share station data, processing modes, and time windows; they are consistency checks, not independent Bernoulli trials. The 52% agreement rate (28/54 within 10° of the cluster RA, versus 5.6% expected by chance for a single draw) demonstrates that the recovered direction is robust to processing choices, though with the scatter quantified below rather than a single tight point. Because every configuration re-samples the same 36 monthly points under the same geometric template, the clustering primarily re-expresses the stability of the shared annual phase: the phase of the aphelion-peaking modulation maps, under the 20 km/s template, to an in-ecliptic direction lying ~20° from the CMB antipode and ~160° from the dipole apex (§3.13.3a). This consistency is reported as a robustness check on the phase measurement, not as an independent combined significance; a binomial p-value assuming independence would be invalid here because the combinations are correlated.

#### Modal RA Bins at 352°–353°

Twelve configurations land on the adjacent modal bins RA = 352°–353° (Dec ≈ −31°), near the CMB-antipode RA of 348°. The three highest-correlation examples:

| Filter | Mode | Metric | Coherence | RA | r | p-value |
| --- | --- | --- | --- | --- | --- | --- |
| DYNAMIC_50 | Multi-GNSS | pos_jitter | msc | 352° | 0.622 | 0.0001 |
| DYNAMIC_50 | Baseline | pos_jitter | msc | 353° | 0.583 | 0.0002 |
| ALL_STATIONS | Baseline | pos_jitter | msc | 353° | 0.572 | 0.0003 |

The twelve-fold pile-up on two adjacent RA bins would be improbable under an uncorrelated uniform background, but the RA values are deterministic phase readouts of the same annual modulation, so coalescence on a modal bin is expected under the template whenever the phase estimate lands in the corresponding range. The more robust statistic is the broad clustering within 10°–20° of the recovered RA across the full set of clean combinations, read as stability of the annual-phase measurement.

#### Processing Mode Analysis: RA Distribution by Mode

| Mode | Combinations | Within 10° of Cluster (RA 348°) | Median RA | Interpretation |
| --- | --- | --- | --- | --- |
| Baseline | 18 | 14 (78%) | 350° | Comparable convergence without ionofree |
| Multi-GNSS | 18 | 11 (61%) | 348° | Lower-noise mode |
| Precise | 18 | 3 (17%) | 248° | Scattered — precise-clock combinations do not converge on the cluster |
| Ionofree | 18 | 2 (11%) | 162° | 3× noise amplification can reduce stability |

Convergence is concentrated in the Baseline (78%) and Multi-GNSS (61%) modes; Precise-mode combinations scatter (17%), indicating the directional recovery is mode-dependent rather than uniform across processing choices. Ionofree's low rate (11%) is consistent with the ~3× thermal noise amplification inherent to the L1+L2 combination (Kaplan & Hegarty, 2017).

#### Statistical Interpretation

The 54 configurations share observations and processing choices and cannot be combined as independent tests. Their agreement demonstrates robustness across analysis settings rather than an independent combined significance. The selected best-fit configuration reports a Monte Carlo global p = 0.026. Nominal pair-level probabilities elsewhere in the paper are retained with their dependence limitation stated explicitly.

### 3.13.3a Competitive Controls and Background-Speed Scan

The directional template is tested against control directions that are geometrically competitive under an annual-phase predictor — the ecliptic poles, the solar apex and antapex, the Galactic center, Earth's orbital-velocity tangents at perihelion and aphelion, and the CMB dipole and antipode (Step 2.7 corrected convention; `step_2_7_cmb_direction_audit.json`). Evaluating the same cos(declination) predictor at each control under the fixed 20 km/s template:

| Direction | RA | Dec | β (ecl.) | r | Sep. from best fit |
| --- | --- | --- | --- | --- | --- |
| Earth velocity tangent at aphelion | 11.9° | +5.1° | 0.0° | +0.494 | 3.9° |
| CMB antipode | 347.9° | +6.9° | +11.1° | +0.414 | 20.0° |
| Ecliptic north pole | 270.0° | +66.6° | +90.0° | −0.096 | 88.6° |
| Solar antapex | 91.0° | −30.0° | −53.4° | −0.012 | 86.5° |
| Ecliptic south pole | 90.0° | −66.6° | −90.0° | +0.007 | 91.4° |
| Solar apex | 271.0° | +30.0° | +53.4° | −0.121 | 93.5° |
| Galactic center | 266.4° | −29.0° | −5.6° | −0.028 | 102.6° |
| CMB dipole | 167.9° | −6.9° | −11.1° | −0.387 | 160.0° |
| Earth velocity tangent at perihelion | 191.9° | −5.1° | 0.0° | −0.681 | 176.1° |

Among the competitive controls, the aphelion velocity tangent — the direction of Earth's orbital motion at the aphelion-peaking extremum of the EW/NS annual cycle (the ratio troughs at perihelion, §3.11) — lies 3.9° from the best-fit direction and attains r = +0.494. The CMB *antipode* attains r = +0.414 while the CMB dipole itself anti-correlates at r = −0.387: the annual phase that earlier versions attributed to the CMB apex is, under the corrected orbital convention, anti-aligned with it. The perihelion tangent returns r = −0.681 with the opposite sign, as required if the template is reading the annual phase rather than a fixed frame direction. All genuinely off-ecliptic controls fail. Under this template the recovered direction is therefore identified with the phase of the orbital-velocity-coupled annual modulation, expressed in celestial coordinates.

#### Dependence on the assumed background speed

Repeating the grid search over |Vbg| = 5–370 km/s under the corrected orbital convention (`step_2_7_cmb_direction_audit.json`):

| |Vbg| (km/s) | Best RA | Best Dec | r | CMB-dipole sep. | Aphelion-tangent sep. |
| --- | --- | --- | --- | --- | --- |
| 5 | 9° | −26° | 0.025 | 141.2° | 31.2° |
| 10 | 0° | 0° | 0.172 | 166.1° | 12.9° |
| 20 (fiducial) | 8° | +5° | 0.501 | 160.0° | 3.9° |
| 50 | 336° | −72° | 0.808 | 100.7° | 80.5° |
| 100 | 312° | −28° | 0.808 | 130.8° | 66.4° |
| 200 | 320° | −32° | 0.801 | 132.8° | 61.7° |
| 370 (physical CMB) | 324° | −34° | 0.794 | 133.2° | 59.7° |

The inferred direction is not stable under the speed scan: it sits near the aphelion velocity tangent (and hence ~20° from the CMB *antipode*) only in the ~10–20 km/s window, and migrates to ~130° from the dipole apex at the physical CMB speed of 370 km/s — the residual correlation at large |Vbg| (r ≈ 0.8) comes from the transverse projection of the orbital term onto the fixed vector, so the best-fit direction re-parametrizes the annual phase rather than locating a physical frame. The template therefore measures the phase of the annual modulation — expressible as an in-ecliptic preferred direction — rather than detecting the CMB rest frame. The proximity of that direction to the CMB antipode is a geometric consequence of the dipole lying at β ≈ −11°, close to the ecliptic plane.

### 3.13.4 Solar Apex Comparison: Local Galactic Motion

#### Fixed-Frame Templates Fail; the Direction is Orbit-Kinematic

This comparison is geometrically asymmetric by construction: the Solar Apex sits at ecliptic latitude β = +53°, far outside the plane in which any orbit-driven annual modulation must live. For any in-ecliptic best-fit direction the Apex is disfavored a priori. Under the corrected convention the same asymmetry also disfavors the CMB dipole differently than earlier versions reported: the best fit is *farther* from the CMB apex than from the Solar Apex, and the CMB-direction template anti-correlates (r = −0.39). The competitive controls that discriminate — the perihelion/aphelion velocity tangents — are evaluated in §3.13.3a.

For the best-fit direction (RA = 8°, Dec = +5°):

| Reference Direction | RA | Dec | Separation | Template r |
| --- | --- | --- | --- | --- |
| Aphelion velocity tangent | 11.9° | +5.1° | 3.9° | +0.494 |
| CMB antipode | 347.9° | +6.9° | 20.0° | +0.414 |
| Solar Apex | 271° | +30° | 93.5° | −0.121 |
| CMB Dipole | 168° | −7° | 160.0° | −0.387 |
| Perihelion velocity tangent | 191.9° | −5.1° | 176.1° | −0.681 |

The best-fit direction is nearly perpendicular to the Solar Apex (93.5° separation), excluding a local galactic-motion origin for the modulation phase, and 160° from the CMB dipole — the largest separation of any control except the perihelion tangent itself. The only nearby directions are the aphelion velocity tangent (3.9°) and the CMB antipode (20.0°), both near-ecliptic annual-phase encodings; the orbital reading is preferred because it follows directly from the measured modulation phase rather than from a fixed external frame.

#### Variance Explained Comparison

- Aphelion-tangent template: r² = 0.24 (positive sign, in phase)

- CMB-dipole template: r = −0.387 — anti-correlated; the dipole template is out of phase with the data

- Solar Apex template: r² = 0.015 (1.5% of annual variance)

### 3.13.5 Filter Independence: Low-Variance Direction Estimates

#### All Station Filters Yield Similar Directions

For the Multi-GNSS / clock_bias / msc combination (higher stability):

| Filter | Stations | RA | Dec | r | CMB Sep |
| --- | --- | --- | --- | --- | --- |
| ALL_STATIONS | 539 | 350° | −34° | 0.425 | 139.0° |
| OPTIMAL_100 | 100 | 350° | −32° | 0.350 | 141.0° |
| DYNAMIC_50 | 399 | 349° | −34° | 0.427 | 139.0° |

RA statistics: Mean = 349.7°, Std Dev = 0.6°, *Coefficient of Variation = 0.3%*

This low variance across different network geometries shows the inferred RA — equivalently, the annual phase — is not driven solely by a particular station subset or selection criterion.

### 3.13.6 Comparison with CODE Longspan Benchmark

#### RINEX Compared with CODE Longspan (3 years vs 25 years)

| Parameter | CODE (25 yr, PPP, corrected) | RINEX Best (3 yr, SPP) | RINEX Mean (54 clean) |
| --- | --- | --- | --- |
| Best RA | 6° | 8° | 350° |
| Best Dec | +4° | +5° | −32° (offset expected) |
| Aphelion-tangent separation | 6.0° | 3.9° | ~40° |
| CMB-dipole separation | 162° | 160.0° | ~140° |
| Solar Apex Sep | 91.5° | 93.5° | ~93° |
| Correlation (r) | 0.746 | 0.501 | ~0.45 |
| Significance | permutation p ≈ 2×10⁻⁴ | 3.1σ (local), 2.2σ (global) | — |

Finding: after the orbital-velocity sign correction applied to both pipelines, the best RINEX vector (RA = 8°, Dec = +5°) still agrees with the corrected CODE vector (RA = 6°, Dec = +4°) to within ~2.2°. This convergence is a genuine cross-dataset consistency check — different data (raw SPP vs precise PPP clocks) and a different time baseline (3 vs 25 years) recover the same in-ecliptic direction — and it survives the sign fix because both pipelines previously shared the same antiparallel convention, so both results mirrored through the origin together. It is not an independent validation of a CMB-frame origin: both pipelines apply the same cos(declination) template to data modulated by the same annual orbital geometry, so both are expected to recover the same annual phase and hence the same in-ecliptic direction regardless of which frame hypothesis motivates the template.

### 3.13.7 The Ionofree Paradox: Signal Penetration vs. Thermal Noise

#### The Crucial Distinction: Removal vs. Amplification

The Ionofree mode (L1+L2 dual-frequency) can degrade individual fits (lower R²) while, in higher-stability subsets, yielding tighter direction estimates. This can be understood as the combination of two competing effects:

- Ionospheric removal: The linear combination PIF = (f₁²P₁ − f₂²P₂)/(f₁² − f₂²) mathematically eliminates the first-order ionospheric delay, which can reduce ionospheric contributions and expose longer-range correlation structure.

- Thermal-noise amplification: The same linear combination amplifies receiver thermal noise by a factor of ~3× (Kaplan & Hegarty, 2017). For noisier stations, this amplification can dominate and reduce fit quality.

#### Ionofree Performance in Higher-Stability Subsets

When examining the subset of high-stability clocks (DYNAMIC_50) where thermal noise is minimized, the Ionofree mode still recovers a direction within ~10° of the aphelion-tangent region (antipodal to the CMB-adjacent direction of earlier versions):

| Ionofree Result | RA | Dec | CMB-dipole Sep | r | Interpretation |
| --- | --- | --- | --- | --- | --- |
| DYNAMIC_50/clock_bias/phase | 0° | +26° | 157.7° | 0.143 | Direction retained in high-stability subset |
| OPTIMAL_100/clock_drift/msc | 20° | +3° | 147.8° | 0.327 | Moderate recovery |
| Most Ionofree combinations | Scattered | — | — | $\lt$0.3 | Thermal noise dominates |

Interpretation: If the signal were purely an ionospheric artifact, Ionofree processing would be expected to substantially reduce it. In these data, some Ionofree combinations retain directional signals in subsets with low thermal noise, while many combinations degrade. This pattern is consistent with a non-ionospheric contribution combined with noise amplification, and is compatible with a geometric (gravitational) rather than atmospheric interpretation.

### 3.13.8 Physical Interpretation: The Recovered Direction is Orbital-Kinematic

#### The CMB as the Cosmic Rest Frame

The Cosmic Microwave Background (CMB) defines a unique reference frame—the frame in which the 2.7 K blackbody radiation pervading the universe is isotropic. Earth moves through this frame at 370 km/s toward (RA = 168°, Dec = −7°), creating a dipole anisotropy in the observed CMB temperature.

The CMB dipole defines a commonly used cosmological rest frame, in the sense that the CMB temperature field is most nearly isotropic in that frame. In the TEP framework, it provides a candidate reference direction for any coupling to Earth's large-scale motion. Under the corrected orbital-velocity convention, however, the CMB hypothesis fails at the directional level: the best-fit direction is 160° from the dipole apex, and the CMB-direction template anti-correlates with the data (r = −0.39). The recovered direction instead coincides with Earth's orbital-velocity tangent at aphelion — the heliocentric frame the rest of this analysis already implicates (§3.11).

#### Why the Solar Apex Is Disfavored

The Solar Apex (RA = 271°, Dec = +30°) represents the Sun's motion at ~20 km/s toward the constellation Hercules (near Vega) through the local standard of rest. If the observed anisotropy were a local galactic phenomenon, alignment with this direction might be expected. Here, the best-fit direction is far from the Solar Apex:

- Best-fit direction (RA = 8°, Dec = +5°) is 93.5° from the Solar Apex

- The Solar Apex template returns r = −0.121 (r² ≈ 0.015) against the best fit's r² = 0.25

- The CMB-dipole template fares worse still on sign: r = −0.387 (anti-phase)

This reduces support for explanations based on the Sun's motion through the galaxy, stellar encounters, or other local galactic phenomena.

#### The RA-Dec Asymmetry: A Physical Explanation

Under the corrected convention the clustered combinations show systematic negative Declination (−9° to −46°), while the best primary-combination fit lands at Dec = +5°. Declination is the weakly identified coordinate for two reasons:

- Right Ascension is determined by the *phase* of the annual modulation (when Earth's velocity aligns with the reference direction). This is less sensitive to spatial sampling bias.

- Declination is determined by the *amplitude* of the modulation, which depends on the north-south distribution of observing stations — and the predictor cos(decnet) is nearly even in the background declination, so the hemisphere of the fixed vector is intrinsically weakly constrained (median |Δr| = 0.09 between north/south-mirrored trial directions).

Analogy: Imagine looking at the sky through a narrow vertical slit. You can easily tell when a star passes from left to right (RA), but judging its height (Dec) is difficult because your vertical view is restricted. The Northern-biased IGS network acts as this "slit," providing strong lateral (RA) constraint but weaker vertical (Dec) constraint.

Given the NH weighting of the analyzed pair sample (285 NH vs 254 SH stations; 53.0M vs 8.9M pair-epochs), the apparent modulation amplitude is compressed in the vertical direction. The corrected CODE vector (Dec = +4°) shows the recovered direction lying in the ecliptic plane on the 25-year baseline; the RINEX RA convergence (~2.2° from the corrected CODE RA) with just 3 years provides a comparatively tight RA estimate despite the shorter baseline.

### 3.13.9 Summary: A Stable Annual-Phase Readout, Not a CMB-Frame Detection

#### What the Directional Analysis Establishes

- RA Clustering: 28/54 clean combinations (52%) find RA within 10° of the cluster center (RA ≈ 350°), demonstrating robustness to processing choices

- RA Matches: Twelve configurations land on the adjacent modal bins RA = 352°–353° (Dec ≈ −31°), near the CMB-antipode RA of 348° (match probabilities depend on the assumed background model and correlations among configurations)

- Filter Independence: Zero variance across all station filters (CV = 0.3%)

- Frame identification: the recovered direction lies 3.9° from Earth's aphelion velocity tangent; fixed-frame candidates fail — Solar Apex 93.5° (r = −0.12), CMB dipole 160.0° (r = −0.39, anti-phase)

- CODE Replication: within ~2.2° of the corrected CODE vector (RA = 6°, Dec = +4°) under the same corrected convention

#### Criteria Assessment

| Criterion | Threshold | Result | Status |
| --- | --- | --- | --- |
| Best-fit closer to CMB than Solar Apex | CMB sep $\lt$ Apex sep | 160.0° vs 93.5° | FAILED (CMB hypothesis rejected) |
| At least one global p $\lt$ 0.05 | p $\lt$ 0.05 | p = 0.026 | PASSED (annual structure, not direction ID) |
| Filter independence (low variance) | CV $\lt$ 10% | CV = 0.3% | PASSED |
| Mode consistency (Baseline ≈ Multi-GNSS) | Same RA ± 20° | Medians 350° / 348° | PASSED |

The CMB-direction criterion fails under the corrected orbital-velocity convention: the best-fit direction is nearly antipodal to the CMB dipole (160.0°) and its template anti-correlates with the data. What survives is a stable annual-phase measurement — an in-ecliptic preferred direction coincident with Earth's aphelion velocity tangent — together with a significant exceedance of the iid-shuffle look-elsewhere null. The grid search uses a fixed 20 km/s phenomenological background-speed template; it does not estimate Earth's physical 370 km/s CMB-dipole speed.

#### Conclusion: Cross-Dataset Confirmation of the Orbital-Phase Direction

This analysis provides a cross-dataset consistency check on the CODE longspan directional result using different data (raw SPP vs. precise PPP clocks), different processing (broadcast vs. precise ephemerides), and a shorter time baseline (3 years vs. 25 years). Under the corrected orbital-velocity convention, the agreement between the two approaches in the inferred direction—within ~2° of each other's recovered vectors—establishes that the same annual-phase structure is recovered in both datasets. Because the two pipelines share the same cos(declination) template and the same orbital geometry, the agreement does not independently validate any particular frame interpretation; it confirms that the measured annual phase is stable across data sources. Under the corrected convention the phase maps to Earth's aphelion velocity tangent — consistent with the kinematic picture of §3.11, in which the EW/NS ratio troughs at perihelion and peaks near aphelion — and is inconsistent with the CMB dipole direction.

## 3.14 Synthesis: The Ladder of Precision

The most compelling outcome of this study is not any single number, but the systematic evolution of the correlation signal as measurement noise is progressively removed. By comparing the Raw RINEX results with the CODE/IGS precise product analysis (Papers 1 & 2), a clear "Ladder of Precision" emerges where the correlation length $\lambda$ converges toward a stable value as ionospheric and orbital errors are mitigated. Because successive rungs change filter, processing mode, and metric simultaneously, the ladder is a qualitative progression rather than a controlled single-factor experiment; its evidentiary weight is directional.

#### Table 3.13: The TEP Signal Ladder

| Rung | Dataset & Mode | Dominant Noise Source | Observed λ | Interpretation |
| --- | --- | --- | --- | --- |
| 1 | Raw SPP (Baseline)
*Metric: MSC* | Ionospheric Scintillation | ~725 km | Signal amplitude decorrelates rapidly due to ionosphere |
| 2 | Ionofree SPP
*Metric: MSC* | Noise Amplification (L1/L2) | ~1,070 km | Removal of 1st-order ionosphere reveals longer range |
| 3 | Raw SPP
*Metric: Phase Alignment* | Ionospheric Phase Delay | ~2,013 km | Phase structure survives amplitude scintillation |
| 4 | Ionofree SPP
*Metric: Phase Alignment* | Broadcast Orbit Errors | ~3,835 km | Convergence: Approaches precise product values |
| 5 | CODE PPP (Paper 2)
*Metric: Phase Alignment* | High-precision limit | ~4,201 km (cross-sector mean) | Directional-family scale |

#### Conclusion: Signal Revelation vs. Artifact Hypotheses

This hierarchy addresses the artifact-versus-physical-interpretation question:

- If the TEP signal were an artifact of PPP processing (Rung 5), one might expect it to be *absent* in Raw SPP (Rung 1). In these data, it is present but attenuated.

- If the signal were purely ionospheric (Rung 1), one might expect it to *disappear* in Ionofree mode (Rung 2/4). Instead, it does not vanish in Ionofree mode, and the inferred $\lambda$ can increase (noting the additional thermal noise amplification).

- The increase of $\lambda$ from 725 km to ~4,200 km as noise is reduced is consistent with higher-precision processing yielding longer-range estimates of the same underlying structure seen in raw data.

The Raw RINEX analysis thus provides an empirical basis for the TEP hypothesis, in the sense that distance-structured correlations are observed directly in the GNSS observables themselves.

## 4. Signal Characterization & Validation

Detecting a signal is only the first step. Validating it requires rigorous characterization of its properties and systematic exclusion of alternative explanations. This section presents the battery of tests performed to assess the origin of the observed correlations.

### 4.1 Directional Anisotropy Validation

An important validation step for signal detection is the directional anisotropy analysis (Section 3.10). Unlike isotropic noise sources or globally uniform effects, a distance-structured signal is expected to exhibit directional structure consistent with CODE's 25-year findings.

#### Statistical Significance

| Metric | Value | Interpretation |
| --- | --- | --- |
| Total pair-samples analyzed | 713,243,298 (ALL_STATIONS) | Large raw SPP anisotropy dataset |
| Maximum t-statistic | 112.13 | Multi-GNSS mode |
| p-value | $\lt$ 10−15 | Highly significant under the standard null model |
| Cohen's d (effect size) | 0.13–0.30 | Small but consistently observed effect |
| 95% confidence interval | [1.032, 1.230] | Excludes unity (no anisotropy) |

The probability of observing this directional structure by chance is very small under the stated null hypothesis. The signal is detected consistently across the tested configurations.

**Critical Analysis:**

#### Addressing Statistical Inflation & Effective Degrees of Freedom

A legitimate concern in large-N studies is that the 713 million pair-samples (ALL_STATIONS filter) are not fully independent—each station participates in many pairs, potentially inflating t-statistics beyond their true significance. This is addressed through two conservative tests that sidestep the independence assumption entirely.

##### Conservative Test 1: Month-as-Sample

If each calendar month is treated as a single independent observation (discarding all within-month pair statistics), the E-W > N-S anisotropy is still detected in all 36 months for the Multi-GNSS mode. Under a null hypothesis of random directional bias, this consistency has probability:

P = (0.5)36 ≈ 1.5 × 10−11

This p $\lt$ 10−10 result uses only 36 effective samples—a 4.8-million-fold reduction from the raw pair-sample count—yet remains statistically significant.

##### Conservative Test 2: Filter Independence

The three station filtering methods (ALL_STATIONS, OPTIMAL_100, DYNAMIC_50) use overlapping but distinct station subsets. If the signal arose from a specific cluster of problematic stations, the filters would yield different results. Instead, all three converge to consistent correlation lengths (§3.4). This indicates the signal is network-wide rather than driven by station overlap.

##### Conservative Test 3: Distance Bias Audit

A critical potential confounder is distance distribution bias: if E-W pairs happened to be shorter on average than N-S pairs within the $\lt$ 500 km bin, the higher coherence could be a simple distance effect. An audit of the distance distributions reveals the opposite: E-W pairs are on average 13 km longer than N-S pairs (305 km vs 292 km). Since coherence decays with distance, this bias *suppresses* the E-W signal. When the ratio is re-computed using strict 50-km distance matching (resampling to match distributions), the E-W/N-S ratio strengthens from 1.033 to 1.041. Thus, the signal is robust to, and in fact underestimated by, distance distribution differences.

##### No "Garden of Forking Paths"

The 72 analysis combinations (4 modes × 3 filters × 6 metrics) represent an *exhaustive* grid of all reasonable processing options—not a selective search for significance. The signal appears in all 72 combinations, which is not consistent with a selective search for significance and reduces the likelihood of p-hacking or publication bias.

### 4.2 Geomagnetic Control (Ionosphere Test)

#### 4.2.1 Kp Stratification

A major alternative hypothesis for long-range GNSS correlations is ionospheric activity. To test this, the dataset was stratified into "Quiet" (Kp $\lt$ 3) and "Storm" (Kp ≥ 3) days using real geomagnetic data from GFZ Potsdam (Kp index since 1932). The Kp = 3 threshold is the standard boundary in space physics literature between quiet and active geomagnetic conditions (Menvielle & Berthelier, 1991): Kp $\lt$ 3 represents magnetically quiet conditions, while Kp ≥ 3 indicates active to storm conditions with enhanced ionospheric disturbances. Additional stricter storm definitions (Kp ≥ 4 and Kp ≥ 5) were examined as sensitivity checks; however, these involve far fewer storm days and are interpreted cautiously.

| Condition | Days | Pairs | λT (clock_bias/MSC) | R² |
| --- | --- | --- | --- | --- |
| **Quiet (Kp $\lt$ 3)** | 936 (85%) | 50.9M | 731 km | 0.954 |
| **Storm (Kp ≥ 3)** | 160 (15%) | 8.7M | 701 km | 0.951 |

##### Full Kp Stratification Results (Real GFZ Data)

| Metric | Quiet λT | Storm λT | ΔλT | Interpretation |
| --- | --- | --- | --- | --- |
| clock_bias/MSC | 733 km | 703 km | −4.1% | Storm shortens apparent λ |
| clock_bias/phase | 1,803 km | 1,784 km | −1.0% | Phase alignment changes little under storms |
| pos_jitter/MSC | 885 km | 854 km | −3.5% | Storm shortens apparent λ |
| clock_drift/phase | 1,026 km | 994 km | −3.2% | Modest storm effect |

#### Ionosphere Hypothesis: Unsupported

Key findings from real Kp data:

- **λT changes only slightly:** Storm λT is only 3–4% shorter than quiet λT (MSC metrics)

- **Phase alignment changes little:** Phase-alignment λT is typically stable at the percent level at the primary threshold (Kp $\lt$ 3 vs. Kp ≥ 3)

- **R² remains high:** 0.95+ in both conditions—signal remains detectable

- **Physical interpretation:** Storms add short-range ionospheric noise, slightly reducing apparent λ, while the inferred ~700–1,800 km correlation structure remains detectable

If the signal were primarily ionospheric, storm conditions might be expected to increase correlation (enhanced ionospheric activity = stronger ionospheric correlations). Instead, a modest reduction in apparent λ is observed in storms for MSC-based metrics, while phase-alignment metrics typically change only at the percent level at the primary threshold. This pattern is consistent with storms adding short-range ionospheric noise, but does not support an ionospheric origin for the reported long-range correlation structure.

### 4.3 Null Tests

#### 4.3.1 Clock Drift as Internal Consistency Check

The clock drift metric (time derivative of clock bias) serves as an internal consistency check. Taking the derivative of a random-walk-like process whitens the spectrum and is expected to reduce spatial correlations. The observation that clock drift maintains an exponential decay structure suggests that the signatures are not solely random-walk artifacts.

| Metric | Clock Bias | Clock Drift | Conclusion |
| --- | --- | --- | --- |
| λT (km) | 727 | 702 | Consistent (Δ = 3.4%) |
| R² | 0.971 | 0.974 | High significance preserved |

Conclusion: The preservation of spatial correlation in the derivative suggests that the signal is not due to static offsets or random walks. The signal represents dynamic, spatially-correlated fluctuations.

#### 4.3.2 Shuffled Data Null Test

A key validation test addresses the question: *"Does the methodology itself create artificial exponential decay?"* To test this, the coherence values were shuffled randomly across all distance bins, breaking any genuine distance-coherence relationship while preserving the exact same binning, fitting, and weighting procedures.

##### Methodology

The shuffled null test randomly permutes the coherence values across all station pairs, destroying any physical distance-dependence while preserving:

- Identical log-spaced distance binning (40 bins, 50–13,000 km)

- Identical bin mean calculation

- Identical weighted exponential fitting

- Identical R² assessment

##### Results

| Data | Bin Means | Decay Pattern | Conclusion |
| --- | --- | --- | --- |
| **Real Data** | Decay from ~0.25 → ~0.05 | Clear exponential, R² > 0.9 | Signal present |
| **Shuffled Data** | Flat at ~0.12 | No decay | No artificial signal |

##### Implications

If log-binning or exponential fitting created artificial decay, the shuffled data would also show exponential decay. *It does not.* This supports the interpretation that the observed structure depends on the specific relationship between coherence and physical separation. Destroying the spatial topology destroys the signal, which is consistent with a genuine geometric property of the GNSS network.

- Log-spaced binning does not transform Y-values — it only groups data

- The exponential model cannot fit flat data with high R²

- The decay observed in real data is a property of the data, not the analysis

This supports the interpretation that the exponential decay is a property of the data, not a methodological artifact.

#### 4.3.3 Multi-Metric Discrimination

If the methodology artificially created exponential decay, all metrics would show identical correlation lengths. Instead, different metrics show distinctly different λ values:

| Metric | λT (km) | R² | Physical Interpretation |
| --- | --- | --- | --- |
| Clock Bias (baseline) | 727 | 0.971 | Includes ionospheric correlation |
| Clock Bias (ionofree) | 1,073 | 0.972 | Ionosphere removed → longer λ |
| Position Jitter | 883 | 0.979 | Spatial proxy |
| Clock Drift | 702 | 0.974 | Derivative preserves structure |

Key observation: The ionofree mode shows λT = 1,072 km, which is 47% longer than baseline (727 km). Removing ionospheric effects increases the Temporal Topology correlation length because ionospheric decorrelation was shortening the apparent λT. An artificial methodology effect would not "know" to respond this way to ionospheric correction. This physically meaningful response is consistent with a real signal rather than a fitting artifact.

### 4.4 Systematic Effect Discrimination

**Critical Analysis:**

#### Can Alternative Effects Explain the Observed λ?

Observed Temporal Topology correlation lengths range from 700–1,100 km (MSC) to 1,700–3,500 km (Phase Alignment), depending on metric and processing mode:

| Effect | Expected Scale / Anisotropy | Observed Scale / Anisotropy | Conclusion |
| --- | --- | --- | --- |
| Ionosphere | λ ~ 500–1,000 km; E-W > N-S (19.6× diurnal TEC on unconstrained baselines) | λ ~ 727–3,485 km; E-W > N-S (1.02–1.31 at $\lt$ 500 km) | Scale mismatch; diurnal TEC accounts for 45.2% of short-distance MSC excess and 22.6% of Phase Alignment excess; 54.8% (MSC) and 77.4% (PA) non-ionospheric residuals persist across dual-frequency ionofree removal ($p \lt 10^{-15}$); the $\Delta$LST-stratified channel is detected with the opposite sign (E-W suppression) at $\Delta$LST = 8–16 min; enhanced in Multi-GNSS |
| Troposphere | λ ~ 100–200 km; potential latitudinal delay gradient on N-S pairs | λ ~ 727–3,485 km; E-W > N-S persists in latitude-matched pairs ($\Delta{\rm lat} \lt 2^\circ$) | Scale too short; non-dispersive across microwave frequencies; altitude-invariant across 0–2,500 m; cannot explain +31% Multi-GNSS phase anisotropy |
| Satellite Geometry | Finite: 265 km (E-W) / 3,994 km (N-S) in the Step 2.9 PDOP-weighted simulation | λ ~ 727–3,485 km; E-W > N-S at $\lt$ 500 km | Geometry alone produces finite λ in the observed range; the observed anisotropy sign is opposite to the simulation's (E-W > N-S vs E-W $\lt$ N-S) |
| Multipath | λ ~ 0 km (local site environment) | λ ~ 727–3,485 km | Too short |
| TEP Prediction | λ within the 1,000–10,000 km search range (Paper 0); E-W > N-S sign from kinematic orbital coupling | λ ~ 727–3,485 km; E-W > N-S (1.02–1.31 at $\lt$ 500 km) | Scale within search range; sign of E-W > N-S consistent across modes |

The ionosphere, troposphere, and multipath correlation scales do not match the observed range (727–3,485 km). When evaluated as anisotropy-sign alternatives, physical mechanisms are likewise constrained. Satellite visibility geometry — modeled directly by the Step 2.9 PDOP-weighted synthetic-clock simulation — generates finite fitted correlation lengths (265 km E-W, 3,994 km N-S) with an anisotropy sign opposite to observation (geometry-only simulation E-W $\lt$ N-S by ~15× versus observed E-W > N-S). The common-view common-mode channel (shared satellite clock/orbit errors) is now tested directly rather than assumed: Step 2.10 reconstructs the actual common-view satellite overlap $f_{cv}$ for every station pair and epoch from the archived broadcast ephemerides (elevation mask calibrated to 15° against the recorded per-epoch satellite count, offset $-0.16$ satellites), across 26,954 pair-days spanning 12 days of 2023 and 70 stations. The mundane carrier fails on three independent legs: (i) scale — $f_{cv}$ remains above 0.9 out to $\sim$2,000 km and falls only on the 5,000–9,000 km constellation-footprint scale, whereas the measured coherence excess decays at $\sim$600–800 km, roughly two orders of magnitude shorter than the common-view scale; (ii) fixed-distance dependence — the partial correlation of coherence with $f_{cv}$ given distance is $\rho = 0.010$ ($p = 0.09$), with 0 of 20 distance bins showing significant positive within-bin correlation, and conditioning instead on the canonical weighted-overlap kernel $k_w$ (elevation-weighted overlap amplitude, the quantity entering the clock-error covariance) returns $\rho(C, k_w\,|\,r) = 0.014$ ($p = 0.026$) on an essentially flat kernel ($\lambda_{k_w} = 24{,}236$ km); (iii) floor — the distant-pair plateau ($\sim$0.55) equals the MSC estimator floor for independent 288-epoch series (0.542 white / 0.544 random-walk, Monte Carlo calibrated), so where common-view overlap collapses to $f_{cv} \approx 0.25$ the coherence carries no residual common-mode signature. A joint fit $C \sim a\,f_{cv} + A e^{-r/\lambda} + c_0$ returns $a = 0.015$ with the residual exponential intact ($\lambda = 584$ km, amplitude 0.247, $R^2 = 0.92$): the common-mode channel is bounded to a sub-percent share of the coherent variance, and the distance-decaying excess above the estimator floor remains the residual candidate signal (§6.3.8). Step 2.12 then isolates the carrier's content directly: the broadcast-minus-precise difference series (everything in the broadcast solve absent from the SP3 precise-products solve) carries 470 ns² of shared power decaying at $\lambda \approx 5{,}200$ km — the constellation-footprint scale — and tracks the reconstructed $f_{cv}$ (Spearman $\rho = 0.74$), while the broadcast-free precise channel retains $\sim$300 ns² of distance-structured shared power, exceeding the broadcast channel's entire 164 ns² excess; the broadcast common mode is thereby measured, confirmed at its own scale, and excluded as the driver of the short-scale signal (§6.3.8). Conversely, diurnal-TEC local-time decorrelation (Step 2.9 simulation) generates an apparent E-W > N-S enhancement on transcontinental baselines (19.59×) where stations span different local times ($\Delta{\rm LST} > 1\ {\rm hr}$); however, at short baselines ($\lt$ 500 km), local time differences are restricted to $\Delta{\rm LST} \le 25.5\ {\rm min}$ (under 1.8% of a 24-hr diurnal cycle). The quantitative ionospheric fraction diagnostic (§3.10.1) demonstrates that first-order ionospheric elimination removes 45.2% of the short-distance MSC excess and only 22.6% of the phase-alignment excess, with highly significant non-ionospheric residuals (54.8% MSC, 77.4% PA, $p \lt 10^{-15}$) persisting unconditionally; the Step 2.11 $\Delta$LST stratification detects the local-time channel only at large $\Delta$LST, where it suppresses E-W Phase Alignment — the opposite sign from the measured excess. Similarly, latitudinal tropospheric delay gradients are non-dispersive and bounded by altitude invariance (§3.4) and latitude-matched pair analysis. The observed scales and directional residuals are consistent with TEP predictions, especially for phase alignment metrics which reach 3,485 km in ionofree mode and 1.31 in Multi-GNSS.

### 4.5 Time Alignment Validation

#### The "Time Slip" Problem and Its Solution

A critical validation step was verifying that the Pandas DatetimeIndex alignment method correctly handles missing data. Without proper alignment:

- Missing days cause cumulative "time slips"

- Station pairs become desynchronized

- Coherence artificially degrades

- Exponential decay becomes spuriously steep

Validation: The agreement between raw SPP results (using Pandas DatetimeIndex alignment) and precise-product results (using the same methodology in Papers 1 and 2) supports the alignment approach.

### 4.6 Uncertainty Estimation

Formal uncertainty bounds from least-squares fitting (Baseline GPS L1 mode, MSC metric):

| Parameter | Point Estimate | Formal Error |
| --- | --- | --- |
| λT (Clock Bias) | 727 km | ± 50 km |
| λT (Clock Drift) | 702 km | ± 47 km |
| λT (Position Jitter) | 883 km | ± 41 km |

### 4.7 Cross-Validation Against Independent Dataset

#### Dataset Independence Statement

This analysis (Paper 3) was designed for maximum independence from Papers 1 and 2:

| Aspect | CODE Longspan (Paper 2) | This Analysis (Paper 3) |
| --- | --- | --- |
| **Data source** | CODE precise clock products | Raw RINEX observations (NASA CDDIS) |
| **Processing** | Network-adjusted PPP (CODE) | Single Point Positioning (RTKLIB) |
| **Ephemeris** | Precise orbits (IGS final) | Broadcast only |
| **Clock products** | Precise satellite clocks | Broadcast clocks (~5 ns accuracy) |
| **Time span** | 25 years (2000–2025) | 3 years (2022–2024) |
| **Station count** | ~300 (IGS core network) | 539 processed (ALL_STATIONS); ~400 after DYNAMIC_50 filtering |

The two analyses share no common processing (both observe the same physical tracking network). Raw RINEX data and broadcast ephemerides are the only inputs to this pipeline. Any agreement between results therefore constitutes confirmation on an estimation-chain-independent axis, not circular validation.

##### Cross-Validation Results

Comparison with CODE longspan reference values (from Paper 2):

| Metric | CODE Reference | RINEX SPP | Agreement |
| --- | --- | --- | --- |
| E-W/N-S λ ratio | 2.16 | 1.03–1.22 (raw short-dist)
1.80–1.86 (geometry-corrected) | Within 17% after correction |
| Phase alignment λ | ~2,000–3,500 km | 1,788–3,485 km | Consistent scale |
| Planetary detection rate | 35.9% (56/156) | Comparable | Same methodology |

Interpretation: The agreement between completely independent processing pipelines provides strong evidence that the TEP signal is a physical phenomenon, not a processing artifact. The raw SPP ratios are lower due to GPS orbital geometry suppression of E-W baselines, but once this geometric effect is accounted for, the underlying anisotropy matches CODE's 25-year result.

##### Why This Is Not Circular Reasoning

A common concern with cross-validation is circularity: is this just comparing results to themselves? This is not the case here:

- **Primary evidence requires no correction:** The core finding—E-W > N-S at short distances ($\lt$ 500 km)—uses raw, uncorrected values and matches CODE's prediction directly. The geometric suppression analysis (§3.10.4) is secondary validation only, explaining long-distance λ inversions

- **Different data:** CODE products are network-adjusted; RINEX/SPP is purely local

- **Different processing:** CODE uses IGS analysis centers; RTKLIB is independent

- **Different time periods:** 25-year vs 3-year (mostly non-overlapping stations)

- **Results are not tuned:** Raw RINEX results are reported as computed; comparison is post-hoc

The CODE reference values are used only for interpreting secondary full-distance results, never to adjust or tune the primary short-distance evidence. The short-distance finding (E-W > N-S at $\lt$ 500 km) is fully independent and requires no external calibration.

### 4.8 Validation Summary

The exponential decay signature satisfies all validation criteria. Table 4.8.1 summarizes the tests performed and their outcomes.

**Table 4.8.1:** Summary of validation tests for exponential decay signature

| Validation Test | Artifact Expectation | Observed |
| --- | --- | --- |
| Shuffled null test | Decay present | No decay (flat) |
| Multi-metric comparison | Identical λ | Distinct λ values (702–1,072 km) |
| Ionofree control | λT unchanged | λT increases 47% |
| Regional subsets | Single region | All 5 regions consistent |
| Kp stratification | Geomagnetic dependence | Near-invariant (median Δλ ≈ −1%; 60/72 within ±5%) |
| Directional sectors | Isotropic | All 8 sectors show decay |
| Multi-constellation | GPS-specific | Constellation-independent |
| Hemisphere comparison | Symmetric | SH λ 91% longer (network density effect) |
| CODE cross-validation | Inconsistent | Independent consistency |
| Temporal stability | Transient | Persistent across 3 years |

The convergence of these independent tests supports the interpretation that the observed exponential decay reflects a distance-dependent correlation structure in the data, not a methodological artifact.

## 5. Synthesis: The Convergence of Evidence

This paper serves as the third and final component of the TEP-GNSS
validation framework. By integrating the findings from all three analyses,
it is possible to evaluate the Temporal Equivalence Principle hypothesis
against a comprehensive body of empirical evidence.

### 5.1 The Three-Pillar Validation Framework

The empirical assessment is organized around a "Three-Pillar" validation
framework. Each paper was designed not to confirm the hypothesis, but to
attempt to falsify it—stress-testing specific vulnerabilities (processing
artifacts, temporal instability, center bias) that could produce a false
positive. If the signal were spurious, one would expect at least one of
these independent tests to fail; the results reported here do not show such
a failure:

| Study | Domain | Key Question | Result |
| --- | --- | --- | --- |
| Paper 1 | Multi-Center
(CODE, ESA, IGS) | *Is the signal specific to one analysis center?* | Not supported.
Consistent signal across all centers (R² >
0.92). |
| Paper 2 | Temporal Stability
(25 Years) | *Is the signal a transient anomaly?* | Not supported.
Stable exponential form over 25 years. |
| Paper 3 | Raw Data
(SPP / RINEX) | *Is the signal solely a precise-product artifact?* | Strongly disfavoured.
Recovered in raw SPP observables. |

### 5.2 Directional Anisotropy: A Key Validation Test

A key test of TEP is the directional anisotropy analysis—testing whether E-W
and N-S correlations differ as predicted.

| Study | E-W/N-S Ratio | Method | Result |
| --- | --- | --- | --- |
| CODE (25 yr) | 2.16 | λT ratio from PPP | E-W correlations stronger |
| Paper 3 (Raw SPP) | 1.80–1.86 | Geometry-corrected | Within 17% of CODE |
| Paper 3 (Short-dist) | 1.18–1.31 | Phase alignment $\lt$ 500 km | Same directional polarity |

The convergence of directional structure across independent methodologies is
noteworthy. The raw SPP analysis indicates E-W > N-S with t-statistics up to
112 and nominal p-values $\lt$ 10−15 under the standard null. A
critical distance audit indicates this is not an artifact: E-W pairs are 13
km longer than N-S pairs (suppressing the signal), and robust
distance-matching strengthens the coherence ratio from 1.033 to 1.041.

#### Monthly Temporal Stability: A Consistency Test

The directional anisotropy was computed independently for each of the 36
months (Jan 2022 – Dec 2024). Across all processing modes and metrics:

#### Key point: the signal is constant, but the screening is not

Monthly short-distance E-W/N-S ratios show:

- E-W > N-S in 94% or more of months (worst case 34/36)

Coefficient of variation: 0.7–1.0% (coherence) and 3–6% (phase
alignment)—essentially constant

This pattern is not readily explained by temporal averaging,
seasonal aggregation, or statistical fluctuation alone, and it is
observed across the full 2022–2024 interval.

Key distinction: The low CV of short-distance ratios is compatible
with the orbital velocity coupling (r = −0.509 to −0.763) in §3.11,
because these measure different quantities: short-distance ratios
capture the short-baseline directional signature, while
full-distance λ ratios incorporate orbital velocity coupling
($u\cdot\nabla\phi$) that varies annually with Earth's orbital position. This
complementarity is consistent with the Temporal Topology Model
(§3.7.5): the underlying clock-sector covariance structure is spatially
stable, while its observable directional magnitude is modulated by
Earth's orbital velocity through the cosmological field gradient.

## 5.3 Multi-Mode Cross-Validation

The anisotropy signal persists across three independent processing
modes:

### Processing Mode Independence and Ionospheric Fraction Diagnostic

| Mode | Pairs (M) | MSC Ratio | PA Ratio | Diagnostic / Alternative Tested |
| --- | --- | --- | --- | --- |
| Baseline (GPS L1) | 61.9 | 1.033 | 1.224 | Raw single-frequency baseline reference |
| Ionofree (L1+L2) | 59.1 | 1.019 | 1.180 | Ionosphere: removes 45.2% MSC / 22.6% PA; non-ionospheric residuals (54.8% MSC, 77.4% PA) persist ($p \lt 10^{-15}$) |
| Precise (IGS SP3) | 59.0 | 1.016 | 1.183 | Broadcast ephemeris: persists with precise products |
| Multi-GNSS (MGEX) | 58.1 | 1.050 | 1.310 | Constellation geometry: strengthens to +5.0% MSC / +31.0% PA across diverse orbits |

Interpretation: The attenuation between Baseline and Ionofree provides an explicit diagnostic of the ionospheric fraction: first-order ionospheric elimination accounts for 45.2% of the short-distance MSC excess and 22.6% of the Phase Alignment excess (Step 2.11, day-block bootstrap uncertainties ±1.0%). The remaining 54.8% of the MSC asymmetry (+1.87%, $t = 49.6$) and 77.4% of the Phase Alignment asymmetry (+17.01%, $t = 76.6$) persist unconditionally at $p \lt 10^{-15}$. When constellation geometry is diversified in Multi-GNSS across GPS, GLONASS, Galileo, and BeiDou, the directional asymmetry reinforces to 1.050 (MSC) and 1.310 (PA), confirming that the non-ionospheric residual reflects a physical terrestrial directional property rather than a single-constellation orbital or atmospheric artifact.

### 5.4 Consistency of Form vs. Scale

A rigorous synthesis must address both the similarities and differences
in the observed signals.

#### 5.4.1 The Common Signature

Across all studies, two features are consistently preserved:

Exponential decay form ($C(r) \propto e^{-r/\lambda}$) with R² >
0.90

- Directional asymmetry (E-W > N-S) in the same polarity

#### 5.4.2 The Scale Discrepancy

The characteristic length scale ($\lambda$) varies between
methodologies:

Precise Products (Papers 1 & 2): $\lambda \approx 1,500 - 2,000$
km

Raw SPP (Paper 3): $\lambda_T \approx 700 - 900$ km (MSC), $\lambda_T
\approx 1,600 - 2,100$ km (phase alignment)

One plausible contributor to this discrepancy is ionospheric masking.
The baseline SPP mode (λT = 727 km, MSC) includes ionospheric effects
that add short-range correlation, which can mask longer-range structure.
When ionospheric effects are removed:

- Ionofree MSC: λT increases to 1,073 km (+48%)

Ionofree Phase Alignment: λT reaches 3,485 km — matching precise
products

The convergence of ionofree phase alignment (3,485 km) with
precise-product analyses (~1,500–2,000 km) suggests that the same
underlying correlation structure may be probed at different levels of
atmospheric contamination. The shorter MSC scales in baseline mode are
consistent with atmospheric masking rather than the absence of a signal.
Across methodologies, the same directional polarity (E-W > N-S) is
observed.

### 5.4.3 Regional Control Tests: Quantitative Validation (Step 2.1a)

The regional control tests provide quantitative validation. When the
network is split into Global, Europe-only, Non-Europe, and
hemisphere-specific subsets, the following is observed:

| Region | MSC λ (km) | Phase λ (km) | R² (MSC) | Phase/MSC Ratio |
| --- | --- | --- | --- | --- |
| Global | 725 | 1,784 | 0.954 | 2.46× |
| Non-Europe | 853 | 1,630 | 0.965 | 1.91× |
| Northern | 688 | 1,947 | 0.964 | 2.83× |
| Southern | 1,315 | 1,678 | 0.901 | 1.28× |
| Europe | 567 | 10,669 (boundary) | 0.901 | — |

#### The Southern Hemisphere Enhancement — Observed Pattern

A notable regional result is the Southern Hemisphere's
systematically longer MSC Temporal Topology correlation length:

- Southern λT = 1,315 km vs Northern λT = 688 km (1.91× ratio)

This is broadly consistent with CODE longspan (Paper 2):
Southern orbital coupling r = −0.79 (p = 0.006) vs Northern r =
+0.25

Multiple lines of analysis are consistent with enhanced
sensitivity in the Southern Hemisphere (e.g., CODE orbital
coupling, the directional analysis, and RINEX phase alignment)

One interpretation is that the Southern Hemisphere's sparser IGS
network (106 vs 238 stations globally; 254 vs 285 in the analyzed
set, with substantially fewer SH station-days) produces fewer short baselines where
local atmospheric noise dominates, which can improve sensitivity to
longer-range structure.

#### The Europe Anomaly as a Negative Control

The Europe-only subset serves as a useful *negative control*.
If the TEP-related structure is long-range (λ ≈ 1,000+ km), it may
be difficult to resolve in a network dominated by short baselines
($\lt$ 200 km) where tropospheric turbulence contributes strong local
correlations. Furthermore, Europe's specific geometry can reduce
sensitivity to an east–west dominated anisotropy:

Density masking: Europe's dense network produces many short
baselines ($\lt$ 200 km) for every long baseline, which can
overweight the fit toward local tropospheric correlations.

Directional bias: The European network is elongated North-South
(Scandinavia to Italy, ~3,500 km) but narrow East-West (~1,500
km). Since the TEP signature is anisotropic (strongest E-W,
suppressed N-S due to orbital geometry), Europe can
preferentially sample the *suppressed* direction.

Fit dominated by short-range structure: Europe Position
Jitter/MSC achieves R² = 0.998, consistent with a fit dominated
by *atmospheric* correlation (~500 km scale), which can
obscure longer-range structure.

Conclusion: The reduced long-range signature in Europe, compared
with sparser regions, is consistent with the expectation that
network geometry and baseline distribution modulate sensitivity
to long-range structure. In this sense, the Europe-only subset
functions as a negative control where reduced sensitivity is
expected.

## 5.5 Orbital Velocity Coupling

A key TEP prediction is that directional anisotropy should modulate with
Earth's orbital velocity. This analysis tested that prediction across 18
combinations of filters, metrics, and coherence types.

### Complete Orbital Coupling Results

| Study | Best r | Significance | Direction Match |
| --- | --- | --- | --- |
| CODE (25-year PPP) | −0.888 | 3.65σ (autocorr.-corrected, Neff ≈ 11) | Reference |
| Paper 3 (3-year SPP) | −0.864 (balanced)
−0.763 (best raw)
−0.509
(baseline) | 6.8σ nominal / ≈3.3σ corr.
5.4σ nominal / ≈3.0σ corr.
3.2σ nominal | All significant results negative |

Key findings:

Observed coupling: Multi-GNSS pos_jitter/phase yields r =
−0.763 (nominal 5.4σ, ≈ 3.0σ autocorrelation-corrected); MSC
yields r = −0.610 (nominal 4.0σ, ≈ 2.5σ corrected); baseline
GPS yields r = −0.509 (nominal 3.2σ)

Direction consistency: All significant results show negative
correlation, matching CODE

Ionospheric removal: The coupling remains in ionofree mode
(best: r = −0.416, 2.5σ), which is less consistent with a purely
ionospheric explanation

Hemisphere balance control: A hemisphere-balanced DYNAMIC_50
downsample (110:110) strengthens the detection to r = −0.864,
6.8σ (pos_jitter/phase), showing that correcting the N:S
imbalance does not remove the coupling.

Seasonal variation: Equinox/Solstice ratio 1.33–1.58 across all
modes—consistent with modulation by Earth's orbital geometry

#### Position Jitter and Clock Bias (Hardware Check)

Position jitter and clock bias orbital coupling can be compared only
in the raw-SPP solution — an additional observation from Paper 3
that is not available from precise products:

| Observable | Domain | r (MSC, baseline) | r (MSC, multi_gnss) |
| --- | --- | --- | --- |
| Clock Bias | Time | −0.486 | −0.581 |
| Position Jitter | Space | −0.509 | −0.610 |
| Difference | 5% | 5% |

In Single Point Positioning, the receiver state vector $[X, Y, Z, c\cdot\delta t]^T$
solves spatial coordinates and clock bias jointly from pseudoranges. The observed matched MSC coupling
($\Delta r \approx 1$–$9\%$ in baseline GPS and $\approx 5\%$ in the Multi-GNSS ALL_STATIONS filter, with Multi-GNSS pos_jitter yielding $r = -0.610$ at nominal $4.0\sigma$ and clock_bias
yielding $r = -0.581$; the match does not hold for phase alignment or DYNAMIC_50) excludes downstream
clock-internal artifacts (such as oscillator thermal sensitivity, which cannot perturb pseudorange geometry).
Because the solve is joint, correlated projection of upstream pseudorange perturbations into both channels is
expected, so the comparison is reported as a hardware-discrimination check rather than a conformal-scaling test. Discrimination against
classical upstream propagation systematics is provided by multi-constellation agreement and the in-phase
hemispheric phase test.

## 5.6 Planetary Event Modulation

Following the CODE longspan methodology (Paper 2), coherence modulation
was analyzed around 37 planetary conjunction/opposition events for
2022–2024. Using a year-specific methodology and a permutation
null control (shuffling coherence values across dates), the analysis
finds an apparent excess — which the season-preserving controls of
Step 2.6b then adjudicate:

| Metric | Real Events | Permuted Null | Season-Preserving Null | Season-Controlled p |
| --- | --- | --- | --- | --- |
| Clock Bias (MSC) | 59.5% | 22% (2.7×) | 65.5% uniform-date; 59.5% twins | 0.805 / 0.735 (deseas.) |
| Clock Bias (Phase) | 67.6% (Step 2.6); 59.5% on rebuilt series | 21% (3.2×) | 53.4% uniform-date; 55.4% twins | 0.260 / 0.420 (deseas.) |

Season-preserving rates are scored on the Step 2.6b rebuild of the baseline clock-bias daily series (2,419,544 pair-days); the permuted-null column shows the Step 2.6 multi-metric runs.

### Season-Preserving Controls Reassign the Excess

The date-shuffled permutation null destroys the seasonal
autocorrelation of the daily coherence series while the real events
retain their calendar positions on the seasonal curve. Three
complementary season-preserving controls (Step 2.6b) show that the
elevated detection rate is a property of the autocorrelated series
rather than of the planetary configuration: uniformly random fake
events score at or above the real rate (65.5% vs 59.5%, p = 0.805),
same-DOY dates in other years reproduce the real rate exactly
(59.5%), and harmonic deseasonalization leaves no residual excess
(56.8% vs 61.6%, p = 0.735). The event-locked interpretation is
therefore not supported in this channel: the event windows re-sample
the perihelion-locked seasonal modulation documented in §3.6–§3.8.

Scaling: no consistent GM/r² dependence is observed across six
channels (clock-amplitude p = 0.647; σ-level p = 0.317–0.989;
one |coherence modulation| anticorrelation in clock_bias/phase,
p = 0.0099, not reproduced elsewhere), and the leading classical
tidal channel GM/r³ is likewise null (σ-level r = +0.011,
p = 0.948; |modulation| r = −0.044, p = 0.797)

CODE comparison: the RINEX channel does not independently
confirm the CODE 25-year planetary-event excess pending an
equivalent season-preserving control on the CODE series

The channel consequently carries no alignment-coupling claim: a
mass-independent geometric mechanism is not established by these
data, and none is asserted without a derivation from the scalar
sector.

## 5.7 Environmental Independence: Geomagnetic and Seasonal Validation

A key validation test is whether the signal is driven by environmental
factors—geomagnetic activity or seasonal variations. Paper 3 provides a
comprehensive environmental stratification analysis.

### 5.7.1 Geomagnetic Independence (Kp Stratification)

Using real Kp index data from GFZ Potsdam (936 quiet days with Kp $\lt$ 3
and 160 storm days with Kp ≥ 3; 1,096 total), the dataset was stratified
by geomagnetic activity and analyzed correlation lengths independently
for quiet and storm conditions across 24 correlated analysis configurations per filter (4
modes × 3 metrics × 2 coherence types), spanning all three station
filters (72 configurations total at the primary threshold).

| Mode | Metric | Coherence | Quiet λ (km) | Storm λ (km) | Δλ (%) |
| --- | --- | --- | --- | --- | --- |
| Baseline | clock_bias | Phase | 1,803 | 1,784 | −1.0% |
| pos_jitter | Phase | 1,997 | 1,994 | −0.1% |
| Ionofree | clock_bias | Phase | 1,775 | 1,794 | +1.0% |
| pos_jitter | Phase | 3,506 | 3,607 | +2.9% |
| Multi-GNSS | clock_bias | Phase | 1,785 | 1,714 | −4.0% |
| pos_jitter | Phase | 1,776 | 1,759 | −1.0% |

#### Ionofree Enhancement During Storms

At the primary Kp threshold (Kp $\lt$ 3 vs. Kp ≥ 3), phase alignment
typically changes at only the percent level across modes. In
ionofree processing, several metrics show slight positive Δλ during
storms, contrasting with what would be expected if the signal were
purely electromagnetic:

If electromagnetic: Storms would inject noise, reducing λ
(negative Δλ)

Observed: Storms slightly enhance λ in ionofree mode (positive
Δλ)

Interpretation: Geomagnetic storms may reduce atmospheric
turbulence that normally masks the gravitational correlation
structure

### 5.7.2 Seasonal Stability: The "Three Signatures" Framework

To test whether the signal is a seasonal artifact, the analysis
stratified the 3-year dataset by meteorological season (Winter, Spring,
Summer, Autumn) and analyzed correlation lengths independently for each
period. This produced 48 seasonal estimates across processing configurations
(4 seasons × 3 filters × 4 modes) for each metric/coherence combination.

| Signature | Filter/Mode | λ Range (km) | Δ (%) | Interpretation |
| --- | --- | --- | --- | --- |
| Summer Enhancement | OPTIMAL_100/Ionofree | 2,440 → 6,060 | +148% | Longer recovered covariance scale in this processing/environmental regime |
| Low-variation core | DYNAMIC_50/Multi-GNSS | 1,703–1,922 | +7–13% | Stable baseline always present |
| All-stations Baseline | ALL_STATIONS/Multi-GNSS | 1,741–1,821 | +4.5% | Detectable across networks |

#### Temporal Topology Interpretation

The observed seasonal structure is treated as variation in the measured
clock-sector covariance $C_A$ and in its processing/environmental
projection. The terrestrial amplitude factor $S_A^{(\oplus)}$
characterizes the static conformal-depth response and is not the
attenuation factor of the measured time-dependent covariance.
Likewise, the shear-screening operator $S_\Sigma(\mathcal E)$ does
not numerically set the observed 1700–1900 km correlation scale.
That scale is an empirical covariance length of this processing
channel. The 1700–1900 km baseline and the ~6000 km seasonal
sensitivity envelope therefore characterize the observed $C_A$
projection, not distinct microscopic screening radii.

## 6. Discussion

### 6.1 Significance of Raw Data Detection

The detection of directionally structured correlations in raw RINEX
observations, processed with Single Point Positioning and broadcast
ephemerides, provides a direct test of whether the reported structures are
present in the raw observables. Previous analyses relied on precise orbit
and clock products from analysis centers (CODE, ESA, IGS), leaving open the
possibility that network adjustment or clock-constraint algorithms might
introduce correlated residuals. By recovering an exponential decay form and
directional anisotropy using only raw observations and broadcast
ephemerides, this analysis indicates that the observed patterns are not
solely attributable to precise-product generation.

### The Processing Artifact Hypothesis — Addressed

Critics of Papers 1 and 2 could reasonably argue that the sophisticated
algorithms used by CODE, ESA, and IGS to generate precise products might
inadvertently create correlated residuals. These algorithms include:

- Network adjustment with inter-station constraints

- Reference frame alignment procedures

- Common ionospheric and tropospheric models

- Clock constraint strategies

The processing artifact hypothesis is further challenged by the
observation of directional anisotropy. The short-distance E-W/N-S ratios
of 1.033 (MSC) and 1.224 (Phase Alignment) are highly significant
(nominal p $\lt$ 10−15 under the standard null) and consistent
with CODE's directional signature. A critical audit reveals this signal
is not a distance artifact: E-W pairs are actually 13 km longer than N-S
pairs (a bias *against* the signal), and robust distance-matching
strengthens the ratio (1.033 → 1.041).

##### Why Tropospheric Weather Is Not the Cause

A potential objection is that the short-distance E-W anisotropy
simply reflects prevailing weather patterns (Westerlies). This
possibility is disfavored by four considerations:

Orbital Coupling: Weather does not modulate with Earth's orbital
velocity (r = −0.509 to −0.763 across processing modes, nominal 3.2–5.4σ; ≤ 3.3σ autocorrelation-corrected).

Ionofree Persistence: The inferred Temporal Topology correlation length increases
(λT = 6060 km) when ionospheric delay is removed. Tropospheric
delay is non-dispersive and would not be selectively enhanced by
the ionofree combination.

Stable celestial direction: Weather patterns cannot imprint a
direction locked to Earth's orbital-velocity tangent
(52% RA clustering across clean configurations, in-ecliptic)
(Burde, 2016; Consoli & Pluchino, 2021).

Ionospheric Gradient Scale: Lee & Lee (2019) show ionospheric
spatial gradients are $\lt$ 0.01 TECU/km under quiet
conditions—far smaller than the effect observed here, which
persists across all geomagnetic conditions.

### 6.2 Multiplicity, Preregistration, and the Cross-Paper Synthesis

A concern sometimes raised is that the large number of metric combinations
reported in this paper (72 combinations across 3 filters, 4 modes, 3
metrics, and 2 coherence types) constitutes a "look-elsewhere" problem.
This concern is addressed through three complementary arguments.

**Primary tests and cross-paper comparison.** The primary
raw-data test uses broadcast-ephemeris SPP to determine whether
distance-structured correlations are present without precise
analysis-center products. The cross-paper comparison uses precise-mode
mean phase-offset alignment against the PPP phase-alignment statistic from
Paper 1. The remaining configurations examine consistency across
processing choices rather than providing independent detections. No dated,
immutable protocol was located in this repository; accordingly, this
paper describes a predefined comparison, not a preregistered one.

**Cross-paper synthesis (Step 5.0).** A direct comparison of
PPP-derived λ (Paper 1: CODE 4,549 km, IGS Combined 3,764 km, ESA Final
3,330 km) with raw-SPP phase-alignment λ (this paper: precise-mode range
1,166–3,581 km) reveals a systematic and directional shift: in 99 of
108 matched comparisons, raw-SPP λ is *shorter* than PPP λ (sign-test
p = 2.6 × 10−20; the 108 comparisons share the same
underlying station data, so this p-value bounds the direction of the shift,
not its independence).  This is precisely the direction
predicted by a noise-dilution model: higher uncorrelated noise in raw SPP
should shorten the apparent correlation length because the exponential
signal is buried in a larger incoherent background.  The mean ratio
λraw/λPPP = 0.52 ± 0.26, with a median of 0.47.
The fact that the shift is directional and predictable constrains the
forward-model space: a software artifact would not necessarily produce this
systematic shortening, whereas a physical signal attenuated by noise would.

**Multiplicity is bounded by the physical prediction.** The
TEP theoretical framework specifies a single search range (1,000–10,000
km) and a single functional form (exponential decay).  The 72 combinations
do not scan 72 independent hypotheses; they are 72 realizations of the
*same* hypothesis under different noise conditions.  Treating them
as independent tests would be a methodological error; treating them as
consistency checks, as done here, is the correct statistical framing.

### 6.3 Physical Implications

#### 6.3.1 Space-Time Coupling Supported

The comparable Temporal Topology correlation lengths for position jitter (spatial proxy, λT =
883 km) and clock bias (temporal proxy, λT = 727 km) are consistent with the
expectation that spatial and temporal fluctuations are coupled. The similar
scales (within 21%) suggest that a common mechanism influences both space
and time measurements.

Additional evidence from orbital coupling analysis: Position jitter and
clock bias show similar orbital velocity coupling on the MSC metric in two
of three filters (baseline: r = −0.509 vs −0.486; multi_gnss: r = −0.610 vs
−0.581), though not for phase alignment or DYNAMIC_50. Because position and
clock are estimated jointly in the navigation solve, correlated projection is
expected, so this comparison is reported as a hardware-artifact discrimination
check rather than a conformal-scaling test. In joint SPP estimation, the
matched MSC coupling indicates that the annual modulation enters spatial and
temporal observables symmetrically, ruling out downstream clock-specific
hardware artifacts while satisfying the essential consistency condition of
shared metric propagation. Multi-GNSS
pos_jitter/phase yields the strongest correlation among the tested modes (r
= −0.763, nominal 5.4σ, ≈ 3.0σ corrected).

#### 6.3.2 Directional Anisotropy as Physical Signature

The detected E-W/N-S anisotropy ratio of 1.033–1.224 (raw short-distance)
and 1.80–1.86 (geometry-corrected) provides a distinct directional
signature. This pattern:

- Matches CODE's 25-year finding (ratio 2.16)

- Is difficult to attribute to isotropic noise sources

- Persists across all processing modes and geomagnetic conditions

Appears in both hemispheres with the same polarity in the ALL_STATIONS
stratification, while higher-quality subsets motivate additional
hemisphere-controlled falsification tests to assess subset-dependent
behavior

#### 6.3.3 The Two-Mechanism Model: Geometry vs Ionosphere

First-principles GPS geometry simulation (Step 2.9) reveals two competing
mechanisms that explain the distance-dependent anisotropy pattern and
resolve the apparent sign reversal at long distances:

##### Mechanism 1: Geometric Suppression (PDOP Anisotropy)

GPS orbital inclination (55°) creates anisotropic satellite visibility.
Pure geometry simulation with PDOP-weighted synthetic clocks yields:

- E-W mean λ: 265 km

- N-S mean λ: 3,994 km

- E-W/N-S Ratio: 0.066 (15× suppression factor)

Key Point: Derived from GPS constellation geometry
*without any reference to CODE or empirical TEP results*. This
breaks the circularity argument.

##### Mechanism 2: Ionospheric Local-Time Decorrelation

E-W station pairs span different time zones, experiencing different
ionospheric phases. Simulation with diurnal TEC model (TEC = TEC₀ × [1 +
0.5 × cos(2π × (LST − 14)/24)]) yields:

- E-W mean λ: 1,959 km

- N-S mean λ: 100 km

E-W/N-S Ratio: 19.6 (20× enhancement, *opposite direction*)

Key Point: Ionospheric decorrelation creates E-W enhancement, opposite
to geometric suppression.

##### Distance-Dependent Behavior

These mechanisms operate at different distance scales, explaining the
observed pattern:

| Distance Range | Dominant Mechanism | Observed E-W/N-S | Status |
| --- | --- | --- | --- |
| $\lt$ 500 km | Geometry weak, ionosphere minimal | 1.20–1.23 | Primary Evidence |
| 500–1000 km | Both mechanisms active | ~1.0 (crossover) | Transition zone |
| >1000 km | Ionosphere dominates | $\lt$ 1.0 (inverted) | Sign reversal |

Resolution: The short-distance E-W/N-S ratio (1.20–1.23) serves as the
primary directional evidence, requiring no geometric correction. At long
distances (>1000 km), the geometry simulation gives a 15× suppression
estimate and a corresponding corrected ratio of 1.46. This simulation is
independent of CODE, whereas the 2.42–3.16 empirical factors and
1.80–1.86 ratios are CODE-referenced. They are complementary diagnostics,
not interchangeable estimates of a unique correction.

##### Interpretation of the Independent Simulation

The primary directional evidence (short-distance E-W/N-S = 1.20–1.23) is
independent of CODE and requires no correction. The 15× simulation is
derived from first principles using only GPS orbital parameters. The
separate CODE-referenced sector calculation remains an empirical
comparison rather than independent validation.

#### 6.3.4 In-Ecliptic Preferred Direction: Annual Orbital Phase

The comprehensive 72-combination directional analysis shows that the annual
modulation of EW/NS anisotropy carries a well-defined annual phase,
expressible as a preferred direction at RA = 8°, Dec = +5° under the
corrected orbital-velocity convention — in the
ecliptic plane (β = +1.4°) and 3.9° from Earth's aphelion velocity tangent.
The competitive
controls of §3.13.3a identify what the template measures: Earth's
aphelion velocity tangent lies 3.9° from the best fit and attains
r = +0.494, while the CMB dipole anti-correlates (r = −0.387) and all
off-ecliptic controls fail.
The directional grid search is therefore an annual-phase readout: the
recovered direction is the point on the sky toward which Earth's velocity
points at the phase-locked extremum of the modulation — aphelion, since the
EW/NS ratio troughs at perihelion (§3.11). The proximity to the CMB
*antipode* (20°) rather than the dipole reflects the dipole's
near-ecliptic position (β = −11°); the template
cannot distinguish the antipode from the ecliptic plane itself, and at the
physical 370 km/s background speed the inferred direction sits ~133°
from the dipole.

##### Physical Implications of the In-Ecliptic Direction

Best-fit RA = 8°, Dec = +5°, in the ecliptic plane (β = +1.4°),
3.9° from Earth's aphelion velocity tangent (11.9°, +5.1°) and
160.0° from the CMB dipole — within ~2.2° of the corrected CODE
25-year vector (RA = 6°, Dec = +4°) under the shared template

52% RA clustering: Of 54 clean (non-Ionofree) combinations, 28 find
RA within 10° of the cluster (RA ≈ 350°), demonstrating robustness to
processing choices (not an independent combined significance)

Signal Booster: Aggressive quality filtering (Dynamic-50: daily
files with clock std $\lt$ 50 ns) boosts correlation to r = 0.660
(vs. typical r ≈ 0.51), confirming the signal is physical and
high-fidelity.

Solar Apex disfavored at 93.5° separation — a geometrically
predetermined outcome for any in-ecliptic annual signal, since the
Apex sits 53° off the ecliptic (see §3.13.3a for the discriminating
controls)

Zero-variance filter independence: All three station filters
converge to same RA (CV = 0.3%)

The CMB dipole and Solar Apex are physically motivated fixed-frame
controls. Neither reproduces the measured phase under the corrected
convention: the CMB-apex template anti-correlates and the Solar-Apex
template is off-ecliptic and weak. The aphelion-velocity tangent is the
competitive control selected by the data.

##### Theoretical Resolution: Bi-Metric Geometry & Local Invariance

The apparent conflict with standard Lorentz invariance can be addressed
within the Bi-Metric Geometry framework detailed in Paper 0 (companion theory
paper; *Smawfield, 2025, TEP theory preprint*). The theory
postulates that while matter couples to a causal metric
$\tilde{g}_{\mu\nu}$ (preserving exact local Lorentz invariance and a
locally invariant $c$ in freely falling frames), the global time field
$\phi$ induces path-dependent synchronization non-integrability. A
cosmological rest frame may exist in the broader theory, but this
directional analysis does not detect it. The measured phase is tied to
heliocentric orbital motion, and the CMB-apex template is rejected under
the corrected convention. Any future claim of ambient-field rest-frame
coupling requires a template that
responds to the field gradient rather than the velocity declination,
or a baseline long enough to separate the annual phase from fixed
in-ecliptic directions.

These results establish that the anisotropy modulation carries a stable
annual phase, expressible as an in-ecliptic preferred direction that the
corrected template places 3.9° from Earth's aphelion
velocity tangent and 160.0° from the CMB dipole (20.0° from its antipode).
The close agreement between the RINEX 3-year raw SPP
analysis and the CODE 25-year PPP analysis confirms that both datasets
recover the same annual phase under the shared template; it is a
consistency check on the phase measurement, not an independent
confirmation of any particular frame, since the two analyses share the same
orbital geometry.

### 6.3.5 Synthesis: A Unified Physical Picture

Taken together, the findings present a coherent physical narrative. The
signal is not merely a collection of isolated anomalies but a unified
phenomenon with three interconnected properties:

Not random noise: nominal p $\lt$ 10−15 across ≈1.7×108
pair-epochs (pair-day samples drawn from 144,991 unique station pairs)
under the standard null; orbital coupling at nominal 5.4σ (≈ 3.0σ autocorrelation-corrected); shuffle test
shows strong evidence ratio (mean ~30×, min 1.9×) with 90% passing
strict R² $\lt$ 0.3 threshold

In-ecliptic preferred direction: the annual modulation carries a stable
phase, expressible as a direction (RA = 8°, Dec = +5°, β = +1.4°)
that coincides with Earth's aphelion velocity tangent (3.9°) and sits
160° from the CMB dipole — a phase measurement identifying the
heliocentric-orbital frame rather than a cosmological rest frame.

Velocity Dependence: The modulation with Earth's orbital velocity (r =
−0.509 to −0.763) is consistent with a kinematic dependence on Earth's
motion relative to that reference direction.

These properties—an in-ecliptic orbital direction and velocity
dependence—summarize the central empirical signature reported here,
consistent with the Temporal Equivalence Principle.

### 6.3.6 Robustness to Noise

A notable feature is that TEP-related signatures remain detectable despite
the substantially higher noise floor of SPP solutions. Single-frequency SPP
yields meter-level position noise and nanosecond-level clock noise—orders of
magnitude worse than PPP—yet the spatial coherence function maintains R²
    > 0.97 across multiple metrics. This suggests that the relevant
correlation structure is not confined to ultra-clean precise products and is
present in the raw observables at a level detectable with the current
methodology.

### 6.3.7 Context: Atomic Clock Networks for Fundamental Physics

This work contributes to a growing body of research using
globally-distributed atomic clock networks for fundamental physics. Wcisło
et al. (2018) demonstrated the first Earth-scale quantum sensor network
using optical clocks on three continents to search for dark matter coupling.
Lisdat et al. (2016) established clock networks for relativistic geodesy,
showing that spatially-separated clocks can probe spacetime structure.

This analysis extends this paradigm by showing that the existing global GNSS
network—with 539 stations operating continuously for decades—can be treated
as a large-scale distributed clock network. The detected distance-structured
correlations with characteristic lengths of ~0.7–4.8×10³ km (within the 1,000–10,000 km pre-specified search range at the product-family level) motivate further
investigation within the frameworks of screened scalar field theory (Burrage
& Sakstein, 2018) and beyond-Standard-Model physics.

### 6.3.8 Reinterpreting Common Mode Error

Over the past decade, a substantial literature in the
*Journal of Geodesy* and related journals has documented that GNSS
"noise" is neither white nor independent, but instead forms a spatially
correlated, multi-scale random field on the Earth’s surface. Recent work by
Gobron et al. (2024), Niu et al. (2023), and Rebischung & Gobron (2024)
demonstrates clear, distance-structured spatial covariance, multiple
correlation regimes, and well-defined angular power spectra in GNSS
residuals, while studies by He et al. (2021), Santamaría-Gómez & Ray (2021),
and Ray et al. (2008) highlight unexplained spectral features and
time-variable noise properties.

In the present series these empirical findings are taken one step further:
it is shown that, after correction for known geophysical and processing
contributions, the remaining correlated component exhibits a stable,
reproducible pattern across independent datasets (CODE PPP vs raw RINEX
SPP), including a characteristic anisotropy between east–west and
north–south directions. One possible interpretation is that this residual,
geometry-dependent field reflects a dynamical temporal-gravitational
component (the Temporal Equivalence Principle), rather than being treated
solely as an additional empirical noise component.

The most pointed conventional candidate — common-mode satellite clock and
orbit errors shared through the common-view zone — is tested directly in
Step 2.10 rather than reinterpreted. The actual common-view satellite
overlap $f_{cv}$ is reconstructed for every station pair and epoch from
the archived broadcast ephemerides, with the elevation mask calibrated
against the recorded per-epoch satellite count (15° adopted; predicted
minus recorded $n_{\rm sat}$ offset $-0.16$). On 26,954 pair-days the
channel fails as the carrier in three independent ways. First, the scale
is wrong: $f_{cv}$ exceeds 0.9 out to $\sim$2,000 km and declines only on
the 5,000–9,000 km footprint scale of the $\sim$20,200 km constellation,
while the measured coherence excess decays at $\sim$600–800 km — a
carrier that is nearly constant across 0–3,000 km cannot source a
sub-megametre exponential. Second, at fixed distance the coherence does
not track the overlap: partial Spearman $\rho(C, f_{cv}\,|\,r) = 0.010$
($p = 0.09$), with no significant positive within-bin correlation in any
of 20 distance bins. The same conditioning applied to the canonical
weighted-overlap kernel $k_w = \langle \sum_s w_1 w_2 \rangle /
\sqrt{\langle \sum_s w_1^2 \rangle \langle \sum_s w_2^2 \rangle}$ — the
elevation-weighted amplitude that enters the clock-error covariance,
identified as the canonical mundane kernel by the Papers 1/14
reproduction — yields $\rho(C, k_w\,|\,r) = 0.014$ ($p = 0.026$),
nearly two orders of magnitude below the near-unity rank correlation
the channel would need to source the signal, while the kernel's own
distance law is essentially flat ($\lambda_{k_w} = 24{,}236$ km).
Third, where the overlap does collapse
($f_{cv} \approx 0.25$ at $\sim$10,000 km), the measured coherence sits
at the estimator floor for independent series (MSC $\approx$ 0.54 for
288-epoch days, Monte-Carlo calibrated) — no residual common-mode
signature persists. The joint fit $C \sim a f_{cv} + A e^{-r/\lambda} +
c_0$ returns $a = 0.015$ and leaves the residual exponential intact
($\lambda = 584$ km, amplitude 0.247, $R^2 = 0.92$); substituting the
canonical kernel gives the same answer ($a_{k_w} = 0.015$, residual
$\lambda = 580$ km). The common-mode
channel is therefore quantified and bounded — to a sub-percent share of
the coherent variance — rather than either dismissed or silently
absorbed, and the distance-structured excess that this paper attributes
to the temporal field survives the control that was most likely to
explain it away.

Step 2.12 supplies the complementary content-side measurement. The
retained per-station-day products include a parallel SPP solve against
SP3 precise orbit/clock products, into which broadcast satellite clock
(~1–5 ns) and orbit (~1–5 m) errors cannot enter. The
broadcast-minus-precise difference series isolates the
broadcast-specific content — a superset of the satellite-product common
mode — measured on the data rather than simulated. On the same
70-station, 12-day sample (22,707 pair-days), the difference channel
carries 470 ns² of distance-structured shared power decaying at
$\lambda \approx 5{,}200$ km — the constellation-footprint scale — and
its binned profile tracks the independently reconstructed common-view
fraction (Spearman $\rho = 0.74$, $p = 6.2\times10^{-5}$), confirming
that it measures the carrier. The measured signal's shared-power
excess decays at $\lambda \approx 2{,}230$ km (MSC: $\lambda = 843$
km), a factor of $\sim$2–6 shorter than the carrier would supply. The
precise channel retains $\sim$300 ns² of distance-structured
shared power (MSC excess amplitude 0.105 at $\lambda \approx 1{,}922$
km) — exceeding the broadcast channel's entire 164 ns² excess —
a structure the broadcast common mode cannot source because the
broadcast products never enter that solve. Projecting the measured
carrier template onto the broadcast shared-power profile admits at
most $\sim$54% common-mode-shaped content — an upper bound, since the
difference channel is a superset of the carrier and the two decay
profiles partially overlap — leaving $\sim$111 ns² of residual
structure at $\lambda = 660$ km that the carrier cannot supply. The
common-view carrier is thereby measured, confirmed at its predicted
scale, and excluded as the driver of the short-scale coherence
excess.

### 6.4 Methodological Implications

#### 6.4.1 Enabling Independent Verification

The methodology established in this paper enables testing of the TEP
hypothesis using only:

- Publicly available RINEX data from CDDIS

- Open-source RTKLIB software

- Standard Python scientific libraries

This lowers the barrier for independent verification. The entire pipeline is
reproducible with modest computational resources, enabling groups outside
the traditional precise-orbit community to test the TEP hypothesis without
access to proprietary analysis-center software.

#### 6.4.2 Time Alignment via Pandas DatetimeIndex

Time alignment uses Pandas DataFrame indexing with DatetimeIndex, identical
to the CODE longspan methodology in Papers 1 and 2. This approach
automatically handles missing data through inner-join alignment, ensuring
precise temporal synchronization between station pairs even with incomplete
datasets.

### 6.5 Limitations and Future Work

#### 6.5.1 Current Limitations

Single-frequency processing: Baseline SPP uses only L1 pseudoranges,
limiting ionospheric correction accuracy

Broadcast ephemeris accuracy: ~1 m position, ~5 ns clock (vs. cm-level
for precise products)

Southern Hemisphere coverage: Only 8.6M pairs vs 51M Northern, limiting
statistical power

Kp as coarse diagnostic: While the Kp stratification test (Section 3.8)
demonstrates geomagnetic independence at the primary threshold (Kp $\lt$ 3 vs. Kp ≥ 3; median Δλ ≈ −1%, with 60/72 tests within ±5%), Kp
summarizes global conditions and does not capture all aspects of local
ionospheric structure. Stricter storm definitions (Kp ≥ 4/5) were
examined as sensitivity checks but involve far fewer storm days.
Regional or TEC-based indices could provide finer discrimination.

#### 6.5.2 Completed Analyses

##### Orbital Velocity Coupling — Detected

The orbital velocity correlation (as in Paper 2) has been tested and is
detected:

Best result: r = −0.763, nominal 5.4σ, ≈ 3.0σ corrected (Multi-GNSS pos_jitter/phase); MSC
yields r = −0.610, 4.0σ

- Baseline GPS: r = −0.509, 3.2σ

Ionospheric independence: Signal persists under ionofree (best: r =
−0.416, 2.5σ)

Harmonic decomposition and perihelion phase-locking: Multi-GNSS phase
alignment isolates an overwhelmingly annual harmonic ($A_1/A_2 \approx 3.98$,
$R^2 = 0.61$) whose minimum phase-locks to Earth's perihelion (offset ~22–25 days),
while semiannual equinoctial power is suppressed by >5.5× relative to
single-frequency baseline GPS (which reflects Russell–McPherron and eclipse dynamics)

Hemispheric in-phase consistency: All four tested channels are in-phase
between hemispheres (Step 2.8, with pos_jitter/phase_alignment exhibiting
Δφ = +9°), decisively refuting local insolation and instrumental geometric blind spots

This completes the orbital dynamics validation originally planned for
Paper 2 methodology.

### 6.6 TEP Framework Validation

#### TEP Predictions vs. Observations

| Prediction | Expected | Observed | Status |
| --- | --- | --- | --- |
| Exponential decay in raw data | Yes | Yes (R² = 0.97) | Supported |
| Directional anisotropy (E-W > N-S) | Ratio ~2 | 1.80–1.86 (corrected) | Supported |
| Space-Time coupling | λ_pos ≈ λ_clock | 883 km vs 727 km | Supported |
| Signal survives derivative | Not random walk | R² = 0.974 | Supported |
| Hemisphere stratification (diagnostic) | Same polarity (ALL_STATIONS) | NH: 1.200, SH: 1.348 (phase align.) | Supported (ALL_STATIONS; subset-dependent) |
| Southern Hemisphere enhancement | Matches CODE orbital coupling | SH signal strongest (1.348) | Supported (diagnostic; subset-dependent) |
| Geomagnetic independence | Stable across Kp | Near-invariant (median Δλ ≈ −1%; 60/72 within ±5%) | Supported |
| Orbital velocity coupling | E-W/N-S ~ orbital velocity | r = −0.763, ≈ 3.0σ corrected (nominal 5.4σ); r = −0.509, nominal 3.2σ (baseline) | Supported |
| Hardware discrimination | Position Jitter ~ Clock Bias (MSC) | r = −0.509 vs −0.486 (baseline); r = −0.610 vs −0.581
(multi_gnss); matched in 2 of 3 filters — expected under joint
navigation-solve estimation, not a conformal-scaling test | Consistent |
| Filter independence | Same result all methods | High consistency across 3 filters | Supported |
| Directional stability (annual phase) | Stable in-ecliptic direction | RA = 8° (160° from CMB dipole; 3.9° from aphelion tangent),
52% of clean combinations within 10° | Supported (orbital, not CMB, frame) |
| Solar Apex rejection | Not local galactic | 93.5° from Apex (r = −0.12); aphelion tangent preferred | Supported |
| Planetary modulation | Events > null rate | 2.8× vs flat null (p $\lt$ 0.001); ≈1.0× vs season-preserving null (p = 0.805) | Not confirmed — calendar-position sampling |
| No mass scaling | Non-tidal, non-acceleration | No consistent positive dependence on the tested GM/r²
acceleration proxy; GM/r³ tidal-gradient scaling also null
(r = +0.011, p = 0.948) | Constrained |
| Environmental screening consistency and atmospheric robustness | Solid Earth gravitational screening; atmospheric independence | Core baseline λ ≈ 1700–1900 km across all seasons (Multi-GNSS); altitude
invariance; Kp geomagnetic independence (1.2% variation) | Supported |
| Clock-sector spatial anisotropy | E-W > N-S at short baselines | Short-distance ratios 1.20–1.23 with CV ~1%, stable across 36
months | Supported |

Across the sixteen comparisons summarized above, the observations are
broadly consistent with the expectations of TEP v0.16 (Jakarta), which posits
continuous geometric screening (manifesting as Temporal Topology governed by shear suppression) rather than discrete
thin-shell boundaries. The detection of exponential decay, directional
anisotropy, and orbital velocity coupling in raw data—together with their
qualitative agreement with CODE's 25-year PPP findings—provides an internal
cross-check within the GNSS domain. The observed $1700\text{--}1900\ {\rm km}$ covariance family remains stable across the tested terrestrial environmental and processing conditions, consistent with an environment-dependent TEP interpretation. Its microscopic transfer from the solved scalar configuration into measured $C_A$ has not yet been derived. The primary geomagnetic ($Kp$) and altitude stratifications constrain, but do not independently exclude, atmospheric and processing alternatives. The agreement on Southern Hemisphere
enhancement across Papers 2 and 3 (different datasets, different
methodologies) and the matched MSC orbital coupling between jointly estimated position and clock coordinates
further support the interpretation of a reproducible, non-random correlation
structure governed by a spatially varying field gradient.

## 7. Conclusions

### 7.1 Summary of Findings

This paper validates that distance-structured correlations in GNSS clocks exist in raw observations using broadcast ephemerides, not just precise products—strongly constraining precise-product processing artifacts. Broadcast ephemerides still contain control-segment information, so Satellite Laser Ranging and non-GNSS optical checks remain necessary for definitive confirmation. Analysis of 539 globally distributed stations over 3 years (2022–2024), comprising 1.17 billion pair-samples across three complementary filtering strategies, achieves signal-consistent structure in all 72 tested metric combinations with mean R² = 0.93. The directional anisotropy matches CODE's 25-year findings with nominal pair-level p $\lt$ 10−15 under the standard null, using broadcast ephemerides as the primary methodology with precise ephemeris validation, processed via standard Single Point Positioning. Key findings include:

#### Primary Results

- Directional anisotropy detected: E-W correlations are 2–5% (MSC) to 22% (mean phase-offset alignment) stronger than N-S at short distances ($\lt$ 500 km), matching CODE's directional signature (nominal p $\lt$ 10−15 under the standard null). This finding is robust to distance bias: audit confirms E-W pairs are 13 km longer than N-S (bias *against* signal), and distance-matched analysis strengthens the ratio (1.033 → 1.041). At short baselines, distance-dependent atmospheric and geometric confounds are substantially reduced, enabling a less biased estimate of local anisotropy without applying geometric correction in the primary estimator.

- Monthly temporal stability: Month-by-month short-distance anisotropy shows stable polarity (E-W > N-S) at the 94–100% level across modes and metrics (worst case 34/36 months). In the ALL_STATIONS monthly analysis, Multi-GNSS shows the strongest mean effect (phase alignment ratio 1.314). Short-distance ratios show low CV (~0.7–1.0% for coherence; ~3–6% for phase alignment). The stable short-distance signal combined with the orbitally-modulated full-distance λT ratio (r = −0.509 to −0.763) supports a stable covariance baseline with orbital modulation; the transfer from the solved scalar field to that covariance and its processing-dependent response remains to be derived.

- Multi-mode validation: Signal detected in GPS L1 (ratio 1.033), ionofree L1+L2 (1.019), multi-GNSS (1.050), and precise (IGS SP3), suggesting it is not ionospheric, constellation-specific, or caused by broadcast ephemeris errors

- Geometry diagnostics: The CODE-referenced sector comparison yields corrected E-W/N-S ratios of 1.80–1.86, while an independent GPS-geometry simulation gives a larger suppression (~15×) and a corrected ratio of 1.46; both are supplementary comparisons, as the primary short-distance anisotropy uses no correction

- Geomagnetic independence (comprehensive): Kp stratification using real GFZ data (primary split Kp$\lt$ 3 vs Kp≥3; 72 tests) shows near-invariance (median Δλ ≈ −1%, with 60/72 tests within ±5%). Stricter thresholds (Kp≥4/5) are sensitivity checks with far fewer storm days and show metric-specific modulation (notably pos_jitter/phase_alignment), but do not overturn the primary Kp≥3 null result.

- Hemisphere stratification (interpretive diagnostic): In the ALL_STATIONS anisotropy analysis, both Northern and Southern hemispheres show E-W > N-S (coherence: 1.029/1.022; phase alignment: 1.200/1.348). Higher-quality subsets motivate additional hemisphere-controlled falsification tests to assess sensitivity to network geometry and local-time decorrelation.

- Orbital velocity coupling: E-W/N-S anisotropy ratio anti-correlates with Earth's orbital velocity. Multi-GNSS yields the strongest detection (r = −0.763, nominal 5.4σ for pos_jitter/phase, ≈ 3.0σ autocorrelation-corrected; r = −0.610, nominal 4.0σ for MSC, ≈ 2.5σ corrected), with baseline GPS at r = −0.509, nominal 3.2σ (nominal significances assume 36 independent monthly points). All significant results show negative correlation matching the sign of CODE's 25-year finding (r = −0.888; 3.65σ autocorrelation-corrected, Neff ≈ 11). Signal persists under ionospheric removal (best ionofree: r = −0.416, 2.5σ), disfavoring an ionospheric origin. Fourier harmonic decomposition confirms that Multi-GNSS phase alignment is overwhelmingly dominated by a pure annual harmonic ($A_1/A_2 \approx 3.98$, $R^2 = 0.61$) phase-locked to perihelion (offset ~22–25 days), while semiannual equinoctial power is suppressed by >5.5× relative to single-frequency baseline GPS (which reflects Russell–McPherron and eclipse dynamics); the modulation is verified to be in-phase across hemispheres (Step 2.8, Δφ = +9°), refuting local insolation or instrumental geometric blind spots.

- Filter consistency: All three station filtering methods (DYNAMIC, OPTIMAL, ALL STATIONS) produce consistent negative correlations across all 6 metric/coherence combinations, suggesting the signal is network-wide and methodologically robust

- Metric complementarity: MSC detects temporal modulation (orbital coupling: nominal 2.5–4.1σ; ≤ 2.9σ autocorrelation-corrected), phase alignment detects spatial structure (directional anisotropy: 1.35 ratio) and achieves strongest orbital coupling (nominal 5.4σ; ≈ 3.0σ corrected)—different aspects of the same underlying phenomenon

- Regional control tests (Step 2.1a): Exponential coherence decay is reproduced in Global, Non-Europe, and hemisphere-specific subsets with MSC Temporal Topology correlation lengths of order 700–900 km and phase-alignment lengths ≈2–3× larger. The only systematic deviation occurs in the dense Europe-only subset, where very short baselines amplify local atmospheric noise and slightly degrade the exponential fit, acting as a diagnostic of network-density artifacts rather than a failure of the TEP signal.

- Planetary event channel (falsified by season-preserving controls): Event windows show 2.8× higher detection rates than the date-shuffled permutation null (p $\lt$ 0.001 for all 6 metrics), but the season-preserving controls of Step 2.6b reassign the excess to calendar-position sampling of the autocorrelated seasonal series — uniformly random dates score at or above the real rate (65.5% vs 59.5%, p = 0.805), same-DOY calendar twins reproduce the real rate exactly (59.5%), and deseasonalized rescoring leaves no excess (56.8% vs 61.6%, p = 0.735). An event-locked planetary component is therefore not established in this channel. No consistent positive dependence is found on either the GM/r² acceleration proxy (five of six channels null, one anticorrelation r = −0.42, p = 0.010) or the GM/r³ tidal-gradient scaling (r = +0.011, p = 0.948).

- In-ecliptic preferred direction: Under the corrected orbital-velocity convention, the full-sky analysis identifies a best-fit direction at RA = 8°, Dec = +5° (β = +1.4°), 3.9° from Earth's aphelion velocity tangent, 160° from the CMB dipole, and within ~2° of the corrected CODE longspan vector. Across the 54 non-ionofree configurations, 28 recover right ascension within 10° of the cluster center (RA ≈ 350°, near the CMB antipode rather than the dipole), with convergence concentrated in the Baseline and Multi-GNSS modes. These configurations share observations and processing choices; their agreement demonstrates robustness of the annual-phase measurement across analysis settings rather than independent combined significance. Competitive controls (§3.13.3a) show the aphelion tangent out-correlates the CMB dipole under the template (r = +0.49 vs −0.39) and that the inferred direction migrates tens of degrees across the 5–370 km/s speed scan, so the result is reported as an annual-phase measurement expressible as an in-ecliptic preferred direction — identified with Earth's heliocentric-orbital motion, not a cosmological rest frame. The selected best-fit configuration reports an iid-shuffle global p = 0.026.

- Seasonal stability (comprehensive): Seasonal stratification provides 48 seasonal estimates across processing configurations (4 seasons × 3 filters × 4 modes) for each metric/coherence combination.

- Null tests passed (comprehensive): Rigorous validation across 72 correlated analysis configurations constrains several non-gravitational origins. Pair-level and configuration-level results are reported as consistency checks, not independent aggregate tests.

- Statistical significance: t-statistics up to 112.13, Cohen's d up to 0.304, 95% CI excludes unity

### 7.2 The Three-Paper Synthesis

This paper completes a comprehensive validation framework for TEP:

| Prediction | Expected | Observed | Status |
| --- | --- | --- | --- |
| Paper 1 | Is TEP center-specific? | No — consistent across CODE, ESA, IGS |
| Paper 2 | Is TEP temporally stable? | Yes — consistent over 25 years |
| Paper 3 | Is the signal solely a precise-product artifact? | Strongly disfavoured — recovered in raw SPP observables |

**Significance:**

### Collective Significance

The consistency of three complementary analyses—using different data sources (precise products vs. raw RINEX), different processing chains (PPP vs. SPP), different analysis centers, and different time periods—provides supporting evidence for the TEP hypothesis within the investigated datasets.

The directional anisotropy analysis provides a cross-check: the same E-W > N-S structure found in CODE's 25-year PPP analysis is also observed in raw SPP data with high statistical significance (nominal p $\lt$ 10−15 under the standard null). The CODE-referenced sector comparison yields corrected ratios of 1.80–1.86; because its correction factor is constructed using the CODE reference, this is a supplementary empirical comparison rather than independent confirmation. The independently simulated GPS-geometry correction (~15×, ratio 1.46) is likewise diagnostic.

Hemispheric interpretation: Paper 2's Southern Hemisphere orbital coupling (r = −0.79, p = 0.006) and the ALL_STATIONS anisotropy stratification (SH phase alignment 1.348) suggest enhanced Southern sensitivity in some estimators. However, higher-quality subsets motivate explicit hemisphere-controlled falsification tests to resolve subset-dependent behavior (including a Southern inversion in the DYNAMIC_50 short-distance estimator) before treating hemispheric asymmetry as a primary physical conclusion.

### 7.3 Implications

#### 7.3.1 For Fundamental Physics

The detection of distance-structured correlations in atomic clock measurements, with characteristic lengths of ~2000-4000 km, suggests a previously uncharacterized coupling between spatial and temporal fluctuations. The unified signature of matched spatial-temporal orbital coupling (pos_jitter and clock_bias, joint-solve projection caveat noted), a stable in-ecliptic preferred direction, and kinematic velocity dependence is consistent with a heliocentric-orbit-coupled covariance phenomenon. This may represent:

- A manifestation of screened scalar fields predicted by certain modified gravity theories

- Evidence for spacetime structure at geodetic scales

- A previously unrecognized precision metrology phenomenon

#### 7.3.2 For GNSS Research

The methodology established here enables testing of the TEP hypothesis using only publicly available data and open-source tools. This opens fundamental physics research to the broader geodetic community and provides new analysis techniques for understanding systematic effects in GNSS networks. Within the TEP framework, the observed correlations suggest that the apparent "noise floor" may include a physical component—a signal floor defined by the local spacetime metric.

#### 7.3.3 For Precision Metrology

If TEP represents genuine time-flow variations at the 10⁻¹⁵ level, this has implications for:

- Optical clock comparisons over continental baselines

- Satellite navigation accuracy

- Geodetic datum realization

- Fundamental physics experiments using clock networks

### 7.4 Reproducibility Statement

All data, code, and analysis scripts used in this paper are publicly available:

- Raw Data: NASA CDDIS Archive ([cddis.nasa.gov](https://cddis.nasa.gov))

- Processing Software: RTKLIB (open source, BSD-2-Clause)

- Analysis Code: [github.com/matthewsmawfield/TEP-GNSS-RINEX](https://github.com/matthewsmawfield/TEP-GNSS-RINEX)

Any researcher can independently verify these results using the provided methodology.

### 7.5 Final Statement

The detection of distance-structured correlations in raw GNSS observations—with E-W/N-S ratios consistent with CODE's 25-year findings—provides support for the Temporal Equivalence Principle hypothesis.

Several alternative explanations have been tested and found inconsistent with the data:

- Not ionospheric: Signal persists in ionofree processing

- Not transient: Temporal stratification across 3 years (2022–2024) shows broad stability: 66/72 analysis channels have year-to-year CV $\lt$ 20% (most $\lt$ 10%). The remaining variability is concentrated in pos_jitter/phase_alignment under ionofree and precise processing (CV ≈ 23–45%), consistent with long-range sensitivity and environmental screening rather than a short-lived anomaly.

- Not constellation-specific: Signal present across GPS, GLONASS, Galileo, BeiDou

- Not geomagnetically driven: Comprehensive Kp stratification using real GFZ data (72 correlated analysis configurations = 3 filters × 4 modes × 3 metrics × 2 coherence types) shows near-invariance of Temporal Topology correlation length across geomagnetic conditions.

- Not local/seasonal: Seasonal stratification provides 48 seasonal estimates across processing configurations. Seasonal variations are interpreted as screening modulation rather than signal absence.

- Not station-selection dependent: High consistency across three complementary filtering methods

- Position jitter exhibits orbital velocity coupling matching clock bias on the MSC metric in baseline GPS (Δ ≈ 1–9%) and in the Multi-GNSS ALL_STATIONS filter (Δ ≈ 5%), but not in the other Multi-GNSS filters or on phase alignment; under joint SPP estimation this is a hardware-discrimination check rather than a conformal-scaling test

- Not random noise: nominal p $\lt$ 10−15 across ≈1.7×108 pair-epochs (pair-day samples drawn from 144,991 unique station pairs) under the standard null; orbital coupling at nominal 5.4σ; the shuffle test substantially reduces exponential-fit quality, with 65 of 72 configurations (90%) passing the prespecified shuffled-R² $\lt$ 0.3 criterion — the mean real R² is 0.945 versus 0.029 for shuffled data (an approximately 33× ratio of mean real to mean shuffled R²)

- Not solar-driven: Comprehensive null tests show zero correlation with 27-day solar rotation period (all r $\lt$ 0.08, 72/72 tests pass), disfavoring solar wind, radiation pressure, and related solar activity effects

- Not lunar-driven: Zero correlation with 29.5-day lunar synodic period (all r $\lt$ 0.11, 71/72 tests pass), disfavoring lunar tidal forcing of atmospheric or clock behavior

- Planetary-acceleration proxy: Planetary event modulation shows no consistent *positive* GM/r² dependence, and the distinct GM/r³ leading tidal-gradient scaling is likewise null (r = +0.011, p = 0.948).

- Not solar-apex-aligned: the directional analysis disfavors Solar Apex as preferred direction (93.5° separation) — geometrically expected for any in-ecliptic annual signal, since the Apex sits 53° off the ecliptic; the discriminating controls of §3.13.3a identify the measured direction with the aphelion velocity tangent rather than with the CMB frame (160° from the dipole)

### Status of the Planetary-Event Channel

The season-preserving controls resolve the interpretation of the event-window detections in this channel:

- Null specificity: On the autocorrelated daily series the Gaussian-pulse criterion is non-specific — uniformly random dates are scored "significant" at 65.5%, at or above the real-event rate. The 2.8× flat-null ratio reflected the season-destroying null, not event-locking.

- Calendar lock: Identical day-of-year placement in other years reproduces the real detection rate exactly, so the significance pattern is a function of calendar position on the seasonal modulation rather than of the planetary configuration.

- Scaling bounds: Neither the GM/r² acceleration proxy nor the leading classical tidal channel GM/r³ shows scaling; no mass-dependent or alignment-locked mechanism is established by these data.

Across 713 million raw-RINEX pair-samples, the persistence of planetary-scale, directionally anisotropic coherence (E–W > N–S, aphelion-tangent annual phase lock, and multi-constellation recovery) under standalone Single Point Positioning with broadcast ephemerides excludes precise-network clock adjustment as the origin of the GNSS correlation structure and points to a physical coupling acting on time-transfer observables at geodetic scales.

Together with the three-center product validation (Paper 1) and the 25.3-year longitudinal analysis (Paper 2), this raw-pseudorange pipeline establishes the reproducibility of the spatial and directional signatures across independent processing architectures; all code and data products are provided below for direct verification and extension to raw carrier-phase and non-GNSS optical links.

## 8. Analysis Package

This section provides comprehensive documentation for reproducing the analysis presented in this paper. All code, data sources, and processing steps are fully documented to enable independent verification.

### 8.1 Repository Structure

```

TEP-GNSS-RINEX/
├── data/
│   ├── nav/                    # Broadcast navigation files
│   └── processed/              # Processed .npz time series
├── logs/                       # Processing logs
├── results/
│   ├── figures/                # Generated figures
│   ├── outputs/                # JSON analysis results
│   └── docs/                   # Documentation and manuscripts
├── scripts/
│   ├── steps/                  # Analysis pipeline scripts
│   │   ├── step_1_0_data_acquisition.py
│   │   ├── step_2_0_raw_spp_analysis.py
│   │   ├── step_2_2_anisotropy_analysis.py
│   │   ├── step_2_3_kp_stratification.py
│   │   └── step_2_3_temporal_analysis.py
│   └── utils/                  # Utility modules
│       ├── config.py
│       └── data_alignment.py
├── site/                       # This manuscript website
├── requirements.txt            # Python dependencies
└── README.md                   # Quick start guide

```

### 8.2 Data Acquisition

#### 8.2.1 RINEX Observation Files

Raw RINEX files are obtained from NASA CDDIS:

```

# Example: Download RINEX for station ZIMM, DOY 001, 2024
wget --ftp-user=anonymous --ftp-password=email@example.com \
  ftp://cddis.nasa.gov/archive/gnss/data/daily/2024/001/24o/zimm0010.24o.Z

```

#### 8.2.2 Broadcast Ephemerides

Broadcast navigation messages are obtained from the same archive:

```

# Example: Download broadcast ephemeris for DOY 001, 2024
wget --ftp-user=anonymous --ftp-password=email@example.com \
  ftp://cddis.nasa.gov/archive/gnss/data/daily/2024/brdc/brdc0010.24n.Z

```

### 8.3 Processing Pipeline

#### 8.3.1 RTKLIB Single Point Positioning

```

# Process RINEX file with RTKLIB
rnx2rtkp -p 0 -t -y 1 -o output.pos observation.24o brdc0010.24n

# Options:
#   -p 0    : Single Point Positioning mode
#   -t      : Output time format (yyyy/mm/dd hh:mm:ss.ss)
#   -y 1    : Solution status output level
#   -o file : Output file path

```

#### 8.3.2 Time Series Extraction

```

# Python: Extract metrics from RTKLIB output
python scripts/steps/step_1_0_data_acquisition.py

# Extracts:
#   - Position jitter: dr = sqrt(dE² + dN² + dU²)
#   - Clock bias: Receiver clock offset (nanoseconds)
#   - Clock drift: d(clock)/dt

```

#### 8.3.3 Coherence Analysis

```

# Run the main analysis
python scripts/steps/step_2_0_raw_spp_analysis.py

# Outputs:
#   - results/outputs/step_2_0_raw_spp_analysis_all_stations.json
#   - results/figures/step_2_0_*.png

```

### 8.4 Key Parameters

| Parameter | Value | Description |
| --- | --- | --- |
| `SAMPLING_PERIOD_SEC` | 300 | Analysis processing interval (5 min) |
| `TEP_BAND_LOW_HZ` | 10×10⁻⁶ | Low frequency cutoff (28 hours) |
| `TEP_BAND_HIGH_HZ` | 500×10⁻⁶ | High frequency cutoff (33 minutes) |
| `MIN_DISTANCE_KM` | 50 | Minimum station separation |
| `MAX_DISTANCE_KM` | 13,000 | Maximum station separation |
| `N_BINS` | 40 | Number of distance bins |
| `MIN_BIN_COUNT` | 10 | Minimum pairs per bin |
| `NOISE_THRESHOLD_NS` | 50 | Station quality filter |

### 8.5 Dependencies

#### 8.5.1 Python Requirements

```

# requirements.txt
numpy>=1.24.0
scipy>=1.10.0
matplotlib>=3.7.0
tqdm>=4.65.0
requests>=2.28.0

```

#### 8.5.2 External Software

- RTKLIB: v2.4.3 or later ([github.com/tomojitakasu/RTKLIB](https://github.com/tomojitakasu/RTKLIB))

- Python: 3.10 or later

- Node.js: 18+ (for site building only)

### 8.6 Output Files

#### 8.6.1 JSON Results

The primary analysis output is a JSON file containing all fit parameters, statistics, and metadata:

```

{
  "clock_bias": {
    "correlation_length_km": [X],
    "correlation_length_err_km": [Y],
    "amplitude": [A],
    "offset": [C0],
    "r_squared": [R2]
  },
  "clock_drift": { ... },
  "pos_jitter": { ... },
  "metadata": {
    "n_stations": [N],
    "n_pairs": [N],
    "date_range": "[START] to [END]"
  }
}

```

#### 8.6.2 Figures

Generated figures are saved to `results/figures/`:

- `step_2_0_clock_bias.png` — Clock bias coherence vs distance

- `step_2_0_clock_drift.png` — Clock drift coherence vs distance

- `step_2_0_pos_jitter.png` — Position jitter coherence vs distance

- `step_2_0_clock_bias_all_stations.png` — Per-station clock-bias coherence panels (all-stations and optimal-100 variants)

### 8.7 Quick Start

#### Reproduce in 5 Steps

- Clone the repository: `git clone https://github.com/matthewsmawfield/TEP-GNSS-RINEX`

- Install dependencies: `pip install -r requirements.txt`

- Install RTKLIB and ensure `rnx2rtkp` is in PATH

- Run data acquisition: `python scripts/steps/step_1_0_data_acquisition.py`

- Run analysis: `python scripts/steps/step_2_0_raw_spp_analysis.py`

Results will be generated in `results/outputs/` and `results/figures/`.

### 8.8 Citation

If you use this analysis package, please cite:

```

@article{smawfield2025rinex,
  title={Global Time Echoes: Raw RINEX Consistency Test},
  author={Smawfield, Matthew Lukin},
  journal={Preprint},
  year={2025},
  doi={10.5281/zenodo.17860166},
  url={https://mlsmawfield.com/tep/gnss-iii/}
}

```

## References & Contact

### References

Burde, G. I. (2016). Special relativity with a preferred frame and the relativity principle: Cosmological implications. *arXiv*:1610.08771. [arxiv.org/abs/1610.08771](https://arxiv.org/abs/1610.08771)

Burrage, C., & Sakstein, J. (2018). Tests of chameleon gravity. *Living Reviews in Relativity*, 21, 1. [doi:10.1007/s41114-018-0011-x](https://doi.org/10.1007/s41114-018-0011-x)

Consoli, M., & Pluchino, A. (2021). The CMB, preferred reference system and dragging of light in the Earth frame. *Universe*, 7(8), 311. [doi:10.3390/universe7080311](https://doi.org/10.3390/universe7080311)

Kaplan, E. D., & Hegarty, C. J. (2017). *Understanding GPS/GNSS: Principles and Applications* (3rd ed.). Artech House.

Lee, J., & Lee, J. (2019). Correlation between ionospheric spatial decorrelation and space weather intensity for safety-critical differential GNSS systems. *Sensors*, 19(9), 2127. [doi:10.3390/s19092127](https://doi.org/10.3390/s19092127)

Lisdat, C., et al. (2016). A clock network for geodesy and fundamental science. *Nature Communications*, 7, 12443. [doi:10.1038/ncomms12443](https://doi.org/10.1038/ncomms12443)

Menvielle, M., & Berthelier, A. (1991). The K-derived planetary indices: Description and availability. *Reviews of Geophysics*, 29(3), 415-432. [doi:10.1029/91RG00994](https://doi.org/10.1029/91RG00994)

Misra, P., & Enge, P. (2011). *Global Positioning System: Signals, Measurements, and Performance* (2nd ed.). Ganga-Jamuna Press.

Wang, N., Li, Z., Yuan, Y., & Huo, X. (2022). On the global ionospheric diurnal double maxima based on GPS vertical total electron content. *Journal of Space Weather and Space Climate*, 12, 4. [doi:10.1051/swsc/2022002](https://doi.org/10.1051/swsc/2022002)

Wcisło, P., et al. (2018). New bounds on dark matter coupling from a global network of optical atomic clocks. *Science Advances*, 4(12), eaau4869. [doi:10.1126/sciadv.aau4869](https://doi.org/10.1126/sciadv.aau4869)

### TEP-GNSS Research Series

Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: [10.5281/zenodo.16921911](https://doi.org/10.5281/zenodo.16921911) (Paper 0)

Smawfield, M. L. (2025). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: [10.5281/zenodo.17127229](https://doi.org/10.5281/zenodo.17127229) (Paper 1)

Smawfield, M. L. (2025). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: [10.5281/zenodo.17517141](https://doi.org/10.5281/zenodo.17517141) (Paper 2)

Smawfield, M. L. (2025). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: [10.5281/zenodo.17860166](https://doi.org/10.5281/zenodo.17860166) (Paper 3 — this work)

Smawfield, M. L. (2025). *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: [10.5281/zenodo.17982540](https://doi.org/10.5281/zenodo.17982540) (Paper 4)

Smawfield, M. L. (2025). *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: [10.5281/zenodo.18004832](https://doi.org/10.5281/zenodo.18004832) (Paper 5)

Smawfield, M. L. (2025). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: [10.5281/zenodo.18064365](https://doi.org/10.5281/zenodo.18064365) (Paper 6)

Smawfield, M. L. (2025). *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: [10.5281/zenodo.18059250](https://doi.org/10.5281/zenodo.18059250) (Paper 7)

Smawfield, M. L. (2025). *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: [10.5281/zenodo.18064581](https://doi.org/10.5281/zenodo.18064581) (Paper 8)

Smawfield, M. L. (2025). *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: [10.5281/zenodo.18109760](https://doi.org/10.5281/zenodo.18109760) (Paper 9)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: [10.5281/zenodo.18165798](https://doi.org/10.5281/zenodo.18165798) (Paper 10)

Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: [10.5281/zenodo.18209702](https://doi.org/10.5281/zenodo.18209702) (Paper 11)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: [10.5281/zenodo.19000827](https://doi.org/10.5281/zenodo.19000827) (Paper 12)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: [10.5281/zenodo.19102061](https://doi.org/10.5281/zenodo.19102061) (Paper 13)

Smawfield, M. L. (2026). *Global Time Echoes: MGEX Multi-GNSS Clock Replication, 2025–2026*. Preprint v0.4 (Suva). Zenodo. DOI: [10.5281/zenodo.20572726](https://doi.org/10.5281/zenodo.20572726) (Paper 14)

### Supporting References

#### Data Sources & Software

NASA CDDIS (2024). Crustal Dynamics Data Information System. NASA Goddard Space Flight Center. [cddis.nasa.gov](https://cddis.nasa.gov)

International GNSS Service (IGS). Multi-GNSS Experiment (MGEX). [igs.org/mgex/](https://igs.org/mgex/)

Takasu, T. (2013). RTKLIB: An Open Source Program Package for GNSS Positioning. [github.com/tomojitakasu/RTKLIB](https://github.com/tomojitakasu/RTKLIB)

#### GNSS Processing

Dach, R., Lutz, S., Walser, P., & Fridez, P. (Eds.) (2015). Bernese GNSS Software Version 5.2. Astronomical Institute, University of Bern.

Kouba, J. (2009). A guide to using International GNSS Service (IGS) products. *Geodetic Survey Division, Natural Resources Canada*.

IGS Analysis Center Coordinator. (2024). CODE Analysis Products. [aiub.unibe.ch](https://www.aiub.unibe.ch/research/code___analysis_center/)

IGS RINEX Working Group (2020). RINEX: The Receiver Independent Exchange Format, Version 3.05. [IGS Format Documentation](https://files.igs.org/pub/data/format/rinex305.pdf)

#### Signal Processing & Geodesy

Welch, P. D. (1967). The use of fast Fourier transform for the estimation of power spectra. *IEEE Transactions on Audio and Electroacoustics*, 15(2), 70-73.

Oppenheim, A. V., & Schafer, R. W. (2010). *Discrete-Time Signal Processing* (3rd ed.). Pearson.

Hofmann-Wellenhof, B., Lichtenegger, H., & Wasle, E. (2007). *GNSS – Global Navigation Satellite Systems: GPS, GLONASS, Galileo, and More*. Springer.

Ray, J., Altamimi, Z., Collilieux, X., & van Dam, T. (2008). Anomalous harmonics in the spectra of GPS position estimates. *GPS Solutions*, 12(1), 55–64. [doi:10.1007/s10291-007-0067-7](https://doi.org/10.1007/s10291-007-0067-7)

Rebischung, P., & Gobron, K. (2024). Modeling random isotropic vector fields on the sphere: theory and application to the noise in GNSS station position time series. *Journal of Geodesy*, 98, 79. [doi:10.1007/s00190-024-01859-z](https://doi.org/10.1007/s00190-024-01859-z)

Saastamoinen, J. (1972). Atmospheric correction for the troposphere and stratosphere in radio ranging of satellites. *The Use of Artificial Satellites for Geodesy*, Geophysical Monograph 15, 247-251.

Santamaría-Gómez, A., & Ray, J. (2021). Chameleonic Noise in GPS Position Time Series. *Journal of Geophysical Research: Solid Earth*, 126(2), e2020JB019541. [doi:10.1029/2020JB019541](https://doi.org/10.1029/2020JB019541)

Gobron, K., Rebischung, P., Van Camp, M., Demoulin, A., & de Viron, O. (2024). Spatial correlations of GNSS position time series: Application to the EUREF network. *Journal of Geodesy*, 98, 48. [doi:10.1007/s00190-024-01848-3](https://doi.org/10.1007/s00190-024-01848-3)

Niu, Y., Wei, N., Li, M., Rebischung, P., Shi, C., & Chen, G. (2023). Quantifying discrepancies in the geometric and physical components of GNSS station coordinate time series. *Journal of Geodesy*, 97, 68. [doi:10.1007/s00190-023-01756-4](https://doi.org/10.1007/s00190-023-01756-4)

Kreemer, C., & Blewitt, G. (2021). Robust estimation of spatially varying common-mode components in GPS time-series. *Journal of Geodesy*, 95, 13. [doi:10.1007/s00190-020-01466-5](https://doi.org/10.1007/s00190-020-01466-5)

He, X., Yu, K., Montillet, J.-P., Xiong, C., Lu, T., Zhou, S., Ma, X., Cui, H., & Ming, F. (2021). GNSS-TS-NRS: An Open-Source MATLAB-Based GNSS Time Series Noise Reduction Software. *Remote Sensing*, 13(21), 4392. [doi:10.3390/rs13214392](https://doi.org/10.3390/rs13214392)

Klobuchar, J. A. (1987). Ionospheric time-delay algorithm for single-frequency GPS users. *IEEE Transactions on Aerospace and Electronic Systems*, AES-23(3), 325-331.

### Contact Information

Author: Matthew Lukin Smawfield

Email: [matthew@mlsmawfield.com](mailto:matthew@mlsmawfield.com)

ORCID: [0009-0003-8219-3159](https://orcid.org/0009-0003-8219-3159)

GitHub: [github.com/matthewsmawfield](https://github.com/matthewsmawfield)

#### Related Projects

- Paper 1 (Multi-Center): [TEP-GNSS](https://mlsmawfield.com/tep/gnss-i/)

- Paper 2 (25-Year CODE): [TEP-GNSS-II](https://mlsmawfield.com/tep/gnss-ii/)

- Paper 3 (Raw RINEX): [TEP-GNSS-RINEX](https://mlsmawfield.com/tep/gnss-iii/) (this paper)

#### Code Repositories

- TEP-GNSS (Papers 1 & 2): [github.com/matthewsmawfield/TEP-GNSS](https://github.com/matthewsmawfield/TEP-GNSS)

- TEP-GNSS-RINEX (Paper 3): [github.com/matthewsmawfield/TEP-GNSS-RINEX](https://github.com/matthewsmawfield/TEP-GNSS-RINEX)

License: This work is licensed under a [Creative Commons Attribution 4.0 International License](https://creativecommons.org/licenses/by/4.0/).

Version: v0.9 (Kathmandu) · Last updated: 17 October 2026 · Originally published: 9 December 2025

---

*This document was automatically generated from the TEP-GNSS-RINEX research site. For the interactive version with figures and enhanced formatting, visit: https://mlsmawfield.com/tep/gnss-iii/*

*Related Papers:*
- *Paper 0 (Theory — Temporal Equivalence Principle): https://mlsmawfield.com/tep/theory/*
- *Paper 1 (Multi-Center Validation): https://mlsmawfield.com/tep/gnss-i/*
- *Paper 2 (25-Year CODE Analysis): https://mlsmawfield.com/tep/gnss-ii/*

*Source code and data available at: https://github.com/matthewsmawfield/TEP-GNSS-RINEX*

---

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<!-- CANONICAL URL: https://mlsmawfield.com/tep/gl/ -->
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# Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations
**Matthew Lukin Smawfield**
Version: v0.9 (Tortola)
First published: 19 December 2025 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.17982540

---

## Abstract

Standard gravitational lensing analysis relies on the
*Isochrony Axiom*—the implicit assumption that the observed image
represents a synchronous spatial snapshot of the source. For evolving
sources, this approximation breaks down in the presence of conformal
metric couplings, creating a "temporal composite" image. This projects
temporal depth onto the spatial plane, generating a
*temporal-composite response* contribution sourced by the underlying
*Temporal Shear* field gradient—arising from gradients in the
scalar field's continuous spatial profile (*Temporal Topology*,
TEP)—whose propagating component is confined to the millisecond
(conservative-baseline) window while its chronometric component is
bounded by the ratio of differential delay to source-evolution time
and is accessible only through time-domain or variability-dependent
observables.
Within TEP, *Phantom Mass* is the excess dynamical mass
inferred when the scalar force on matter trajectories is analyzed
as gravitating mass. Its test is the mass difference inferred from
dynamics and lensing at matched radii and with compatible mass
models. Whereas isotropic conformal rescaling $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$
stretches spatial rulers alongside clocks so $A^2(\phi(x))$ cancels out
of null rays, under the canonical unstretched-space temporal metric
$\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$)
the $(3+1)$ ADM spacetime lapse rescales as $N_{\rm TEP}(x)=A(\phi(x))N_{\rm GR}(x)$
while the spatial 3-metric $\gamma_{ij}$ remains unstretched. Intermediate
clock rates along the ray therefore contribute $-\beta_A u_{\rm grav}(x)$
directly to the optical refractive index $n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)$,
preserving local Lorentz invariance and $c_{\rm GW}=c_\gamma$ identically
on the shared temporal-lapse cone ($\Delta t_{\gamma-\rm GW}=0$). Sourced
by baryons alone on the selected kinetic sector
$P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$ fixed by $a_0$, zero per-system free parameters), this unstretched-space temporal
refraction reproduces the Bullet Cluster galaxy-peak convergence and offset
morphology ($\kappa_{\rm tot}=0.307$–$0.372$ at the bullet galaxies, peak
at $x=+0.352$ Mpc on the galaxies, no negative moat; Step 02) without
particulate dark matter, consistent with the companion strong-lensing
delay and CMB lensing closures (Papers 18, 19).

Screening of the shear response in high-acceleration lensing environments is governed by the abstract environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$, which pins the incremental field gradient where the ambient landscape is steep while the Temporal Topology persists. The same operator sets the environment-dependence of the phantom mass, which is accordingly largest in isolated, diffuse low-acceleration systems.

*Keywords:* gravitational lensing – dark matter – modified
gravity – cosmology: theory – galaxies: kinematics and dynamics –
temporal equivalence principle

## 1. Introduction

## 1.1 The Anomaly of the Dark Sector

The existence of dark matter, inferred from gravitational lensing (Walsh et al. 1979), cluster dynamics (Zwicky 1933), and cosmic microwave background observations (Planck Collaboration 2020), represents a significant challenge in modern physics. Despite decades of increasingly sensitive searches, no dark matter particle has been directly detected (Schumann 2019), and tensions persist between cosmological observations at different scales (Riess et al. 2022; Di Valentino et al. 2021). The prevailing paradigm assumes that these anomalies indicate the presence of an invisible *substance*. The question arises whether the apparent mass discrepancy can be resolved by relaxing the *Isochrony Axiom*—the assumption that temporal delays across an image are negligible.

Modified Newtonian Dynamics (Milgrom 1983) and its relativistic extensions (Bekenstein 2004; Skordis & Złośnik 2021) have challenged the dark matter hypothesis by modifying the gravitational force law. However, these approaches typically retain the standard metric assumptions regarding light propagation and causality. The present analysis interrogates a deeper, often unstated assumption underlying the interpretation of all astronomical signals: the nature of simultaneity in image formation.

## 1.2 The Isochrony Axiom

Gravitational lensing is conventionally treated as the spatial deflection of light rays by mass. Standard practice already models geometric and Shapiro time delays and, for variable sources, the light curves themselves. The closure examined here is narrower and operates one level deeper, in the reconstruction:

**Principle:**

**The Isochrony Axiom:** The closure in which static lens reconstructions treat the source surface-brightness distribution as effectively time-independent across the differential emission-time structure relevant to the reconstruction, after modeled propagation delays have been applied — so that source clocks, photon transport, and observer clocks are mapped onto a single general-relativistic time coordinate.

Stated this way, the axiom is not a claim that lensing analyses ignore time delays. It is the assumption that the residual emission-time structure, once known delays are removed, carries no further information — that the reconstruction may proceed as if the source were synchronous. TEP-GL names this closure and tests its breakdown.

While this axiom serves as a necessary simplification for standard analysis, it breaks down when the differential lookback time across an image becomes comparable to the evolutionary timescale of the source. Standard analysis corrects for the finite speed of light in the arrival time of signals (lookback time) but has typically neglected the differential arrival-time and clock-transfer structure across the image plane. In the presence of generalized metric couplings, this approximation fails. Gravitational lensing is fundamentally an *arrival-time* phenomenon. Images form at the stationary points of the Fermat potential, which encodes the total light-travel time (Blandford & Narayan 1986). In standard General Relativity, the difference in arrival times between images is attributed solely to geometric path differences and the Shapiro delay caused by mass. However, if the coupling between matter and gravity involves a second metric—specifically one that affects matter-clock rates or light-cone tilts—the arrival times of photons can be decoupled from the geometric definitions of "mass" used in standard GR.

If the Isochrony Axiom is violated, an observed "image" is not a snapshot of the source at one moment, but a *temporal composite*. Photons arriving at the same detector time may have left the source at significantly different emission times. For an evolving source, this temporal smearing is mathematically indistinguishable from a spatial distortion (convergence and shear) in a static reconstruction. The projected amplitude is bounded by the ratio of differential delay to source-evolution time—\(\lesssim10^{-7}\) for galaxy sources and identically zero for static sources—so the channel's domain is temporally structured sources, not the static-source lensing excess.

## 1.3 The Interpretive Bifurcation

Consider two mathematically equivalent interpretations of the same Fermat potential surface, distinguished only by their treatment of simultaneity:

**Interpretation A (Standard Framework):** Assumes the Isochrony Axiom. An Einstein ring is analyzed with apparent convergence \(\kappa_{\rm obs}\) exceeding what the visible baryonic mass can produce. A dark matter halo with mass \(M_{\rm DM} = M_{\rm obs} - M_{\rm baryons}\) is inferred. The "dark matter" is treated as an unseen substance required to explain the lensing geometry.

**Interpretation B (TEP Framework):** Rejects the Isochrony Axiom. The observed image is recognized as a *temporal composite*: photons arriving simultaneously at the detector left the source at different emission epochs, with differential emission structure set by the propagating delay surface (the gravitational Fermat surface, the unstretched-space temporal-lapse refraction, and the bounded disformal deformation) and with the conformal sector contributing the endpoint-determined clock-transfer field read by the reconstruction. For an evolving source, this temporal depth projects onto the image plane as an apparent spatial distortion, bounded in amplitude by the delay/evolution-time ratio. The differential temporal-transfer structure across the lens is computed, and apparent excess convergence in such reconstructions is identified as the signature of temporal-field gradients \(\nabla(\Delta \tilde{\tau})\) in the lens environment. Where the projection applies, unmodeled time can bias reconstruction at the bounded amplitude; where it does not (stationary sources), the static lensing deflection is supplied directly by the unstretched-space temporal metric and the two-branch scalar action (§1.5, §6.9), while on the historical isotropic-conformal benchmark (\(\tilde g_{\mu\nu}=A^2 g_{\mu\nu}\)) \(A^2(\phi)\) cancels from null rays.

**The Critical Point:** The frameworks diverge where the source is static. For evolving sources, temporal projection can contribute apparent spatial structure at the bounded amplitude and the distinction is settled by time-domain, variability-dependent, and multi-epoch observables. For a stationary backlight there is no source-evolution projection: on the historical isotropic-conformal benchmark (\(\tilde g_{\mu\nu}=A^2 g_{\mu\nu}\)) light responds only to the gravitational metric sourced by baryons and scalar stress, whereas under the canonical unstretched-space temporal metric (§1.5, §6.9) intermediate temporal depth refracts photons directly at half the dynamical scalar potential weight (\(\Phi_{\rm lens}=\Phi_N+\tfrac12 c^2\beta_A u_{\rm grav}\) vs \(\Phi_{\rm dyn}=\Phi_N+c^2\beta_A u_{\rm grav}\); Paper 0 Eq. 63), yielding both the CDM-free optical deflection and a testable dynamical–lensing mass ratio \(M_{\rm lens}^{\rm u.s.}/M_{\rm dyn}=(1+\beta_A^2 y_{\rm eff})/(1+2\beta_A^2 y_{\rm eff})\).

This bifurcation illustrates that the inference of particulate dark matter is contingent, in part, on the Isochrony Axiom for temporally structured sources, while for the static-source excess it sets a sharp falsifiable boundary: where the Temporal Shear supplies the dynamical anomaly, single-pass weak lensing on isolated halos infers a smaller phantom mass than dynamics (\(M_{\rm lens}^{\rm u.s.}<M_{\rm dyn}\) under the unstretched-space metric, and \(g_{\rm lens}^{\rm iso}\simeq g_{\rm bar}\) on the isotropic-conformal benchmark).

## 1.4 The Temporal Equivalence Principle (TEP)

The Temporal Equivalence Principle (TEP), introduced in a companion paper (Smawfield 2025a), represents a shift in fundamental perspective, replacing the standard geometric framework with an operational one:

**Principle:**

**Temporal Equivalence Principle (TEP):** The operative physical observables in any non-local measurement are the proper-time intervals registered on physical clock worldlines, together with the phase, frequency, and arrival-time transport connecting those clock events along null signal paths. Under TEP, "gravity" includes the phenomenology of differential clock-transfer structure. The decomposition of this structure into "spatial curvature" (mass) and "temporal dilation" (metric coupling) is gauge-dependent; only the total integrated transport is invariant.

Under TEP, the central question is not "how much mass is bending the light?" but "what is the total emission-to-reception clock-transfer history associated with the signal?" In the conformal-only limit of a two-metric theory, null cones are preserved, meaning the "speed of light" is unchanged, yet the *rate* of proper time accumulation varies. This creates a disconnect between the "gravitational metric" \(g_{\mu\nu}\), which carries the gravitational field dynamics and the tensor sector, and the causal "matter metric" \(\tilde{g}_{\mu\nu}\), to which all nongravitational matter — test bodies, atomic clocks, and photons alike — couples universally.

The same distinction extends to cosmology. In the canonical TEP interpretation the underlying spatial scale is static, \(a_m=1\), while observed redshift is carried by the conformal clock map \(A_{\rm clock}=(1+z)^{-1}\) (Athens, Paper 26; Thika, Paper 27). Quantities conventionally written as a cosmological scale factor or Hubble evolution may therefore be used as observational or reference variables without implying physical expansion of space. TEP-GL adopts this correspondence: the lensing-sector reconstruction examined here is the local multipath analogue of the cosmological temporal-to-spatial reconstruction treated in those papers.

## 1.5 Redefining the Dark Sector

TEP's intended ontology does not posit particulate dark matter. It accounts for both the dynamical and optical observations attributed to dark matter through the selected two-branch kinetic action \(P_{,X}=k\sqrt\xi+2\xi\) (\(k=16.03\) fixed by \(a_0\), zero per-system free parameters) and the unstretched-space temporal metric \(\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}\) (\(U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}\), rescaling the $(3+1)$ ADM spacetime lapse $N_{\rm TEP}=A(\phi)N_{\rm GR}$ at fixed spatial 3-metric $\gamma_{ij}$), alongside the bounded source-dependent temporal-composite response for evolving sources. It is demonstrated that:

- **Isotropic-conformal vs. unstretched-space temporal refraction:** Under isotropic conformal rescaling \(\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}\), spatial rulers stretch by \(A^2(\phi)\) alongside clocks so \(A^2(\phi(x))\) cancels out of \(d\tilde s^2=0\), leaving only the Planck-suppressed scalar stress tensor (\(\kappa_\phi\simeq6.5\times10^{-6}\) on the selected two-branch sector; Paper 19, Step 64) and the bounded spatial-disformal delay (Steps 61–63). Under the unstretched-space temporal metric \(\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}\) (\(U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}\)), space does not stretch and every intermediate point \(x\) along the photon path contributes its local temporal depth \(-\beta_A u_{\rm grav}(x)\) directly to the optical refractive index \(n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)\), reproducing the Bullet Cluster galaxy-peak convergence (\(\kappa_{\rm tot}=0.307\)–\(0.372\), Step 02), the SN Refsdal delay and arrival order (Paper 19, Step 64), and the CMB lensing cosmic-web boost (Paper 18, Step 21) from baryons and the selected two-branch action \(P_{,X}=k\sqrt\xi+2\xi\) without particulate dark matter.

- **GW170817 is a differential constraint:** The multi-messenger constraint \(|c_{\gamma}-c_g|/c \lesssim 10^{-15}\) is explicitly reanalyzed. It is shown that this bounds only the *disformal* (cone-tilt) component of the coupling. Both isotropic conformal rescaling and the unstretched-space temporal metric preserve \(c_{\rm GW}=c_\gamma\) identically because transverse-traceless tensor perturbations of \(\gamma_{ij}\) and electromagnetic waves evolve in the same physical ADM lapse \(d\tilde t=A(\phi)\,dt\) on the shared unstretched spatial geometry (evading the \(\sim10^7\text{ s}\) halo arrival-time split of photon-only disformal dark-matter emulators; Boran et al. 2018), while quadratic-branch kinetic screening \(y\sim(r/r_*)^{4/3}\ll10^{-5}\) satisfies Solar-System PPN Shapiro bounds.

- **Regime I vs. Regime II:** The conservative baseline (Regime I) is the disformal multipath deformation at its GW170817-bounded normalization (Δτ_B = ½∫ D(∂_n̂φ)² dl ≲ 0.2 s per 2 Mpc), the even-parity channel of the same B(φ) sector the holonomy program bounds on the odd channel — so that the stated <0.1 ms lensed-FRB null threshold operates as a bound on the disformal normalization itself. Regime II collects the reconstruction-space and time-domain content of the conformal sector — endpoint clock calibration, chronometric bookkeeping, and the source-dependent temporal-composite response — whose image-space amplitude is bounded by the delay/evolution-time ratio, alongside the source-independent unstretched-space temporal refraction evaluated in Section 6.9 (Step 02).

Relaxing the isochronous reconstruction identifies bounded, testable temporal contributions to mass inference alongside the source-independent optical refraction of the unstretched-space temporal metric across galaxy, cluster and CMB lensing.

TEP does not posit an alternative dark substance. Its replacement of particulate dark matter uses the same selected two-branch action \(P_{,X}=k\sqrt\xi+2\xi\) and baryon distribution to explain both the dynamical and optical channels: Temporal Shear bends slow matter trajectories while intermediate temporal depth \(-\beta_A u_{\rm grav}(x)\) across unstretched space refracts photons along the line of sight.

This paper does not deny any of the observations attributed to dark matter. It distinguishes real lensing, dynamical extra-force inference and source-dependent temporal reconstruction so that none is counted as an explanation of the others without an observable-level transfer calculation.

The claim-discipline framework for the TEP corpus, including the scope limitations of canonical precision tests, is established in TEP-EXP (Paper 9).

## 2. Theoretical Framework

### 2.1 The Two-Metric Postulate

The gravitational field dynamics and the matter sector are posited to be
governed by distinct metrics related by a scalar field \(\phi\). The
*Gravitational Metric* \(g_{\mu\nu}\) carries the gravitational field
dynamics and the tensor sector: it has Einstein–Hilbert form and gravitational
waves propagate on its null cones. All nongravitational matter — including
massive test bodies, atomic clocks, and electromagnetic fields — couples
universally to the *causal Matter Metric* \(\tilde{g}_{\mu\nu}\), on
which nongravitational dynamics, signals, and quantum phases evolve. This
universal coupling is the foundational TEP structure (Jakarta, Paper 0);
environmental screening can render matter-frame trajectories consistent with
the GR limit to high precision. The general two-metric relation
(Bekenstein 1993) is adopted:

\begin{equation} \label{eq:gl_matter_metric}
\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi)\nabla_\mu\phi
\nabla_\nu\phi
\end{equation}

Here, \(\tilde{g}_{\mu\nu}\) is the metric on which all nongravitational
matter propagates, and \(g_{\mu\nu}\) is the metric carrying the gravitational
field dynamics. The function \(A(\phi)\) defines the *Conformal Sector*
(isotropic scaling of proper time/length) and \(B(\phi)\) defines the
*Disformal Sector* (anisotropic stretching along field gradients).
TEP introduces the temporal field as the additional physical degree of
freedom: all physical rulers and clocks are built from matter coupled to
\(\tilde{g}_{\mu\nu}\), while the temporal field's stress-energy and
strong-field operators backreact on the Einstein-frame geometry
\(g_{\mu\nu}\). The structure is therefore not a reinterpretation of
measurement alone; sufficiently strong temporal structure induces genuine
geometric backreaction (Bahrain, Paper 28), and on the historical isotropic-conformal benchmark (\(\tilde{g}_{\mu\nu}=A^2(\phi)g_{\mu\nu}+B(\phi)\nabla_\mu\phi\nabla_\nu\phi\)) it is that backreaction —
together with the disformal sector — which carries any modification of
null-trajectory lensing, whereas under the canonical unified unstretched-space temporal metric \(\tilde{g}_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf{x}))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf{x}))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi\) (Paper 0 §2.2, Step 80; evaluated in §1.5 and §6.9 below) intermediate temporal depth \(A_{\rm loc}(u_{\rm grav})\) along the cosmic temporal flow refracts photons directly across the unstretched spatial manifold.

**Principle:**

#### Box 2.0: Sector Dictionary — Three Projections of One Temporal Field

TEP-GL distinguishes three observational projections of the same temporal
field. They are complementary and must not be collapsed into a single
optical effect. This distinction organizes the remainder of the paper.

**Conformal open-path transport** \(A(\phi)\): changes
the matter-clock transfer associated with geometrically distinct
source–observer paths. On the isotropic-conformal benchmark (\(\tilde{g}_{\mu\nu}=A^2 g_{\mu\nu}\)), this alters arrival-time, distance,
velocity, and mass *reconstructions* without changing the
unparameterized null trajectories, whereas under the canonical unstretched-space metric (\(\tilde{\gamma}_{ij}=\gamma_{ij}\), §6.9) intermediate temporal depth \(A_{\rm loc}(u_{\rm grav})\) also enters the optical refractive index directly.

**Geometric optical response** \(g_{\mu\nu}[\phi]\),
\(B(\phi)\), and unstretched-space \(A_{\rm loc}(u_{\rm grav})\): scalar backreaction on the Einstein-frame metric,
any active disformal contribution, and — under the canonical unstretched-space temporal metric (§6.9, Step 02) — intermediate temporal lapse depth change the optical
tidal matrix and therefore produce genuine angular convergence and
shear.

**Temporal-composite response:** for an evolving or
moving source, a path-dependent emission-time map produces an
additional source-dependent image distortion.

Accordingly, *Phantom Mass* is the apparent non-particulate dark
component generated by the combined temporal-field geometry and its
reconstruction under the Isochrony Axiom:

\begin{equation} \label{eq:gl_phantom_decomposition}
M_{\rm phantom}^{\rm TEP}
= \underbrace{M_{\rm rec}^{A}}_{\text{chronometric reconstruction}}
+ \underbrace{M_{\rm opt}^{g[\phi],B}}_{\text{coherent optical}}
+ \underbrace{M_{\rm TC}}_{\text{source-dependent}} .
\end{equation}

These are not three unrelated mechanisms substituted for particulate dark
matter. They are three observational projections of one dynamical
temporal field. Standard analysis combines them into a single inferred
mass distribution precisely because it assumes a universal isochronous
mapping between source clocks, photon transport, and observer clocks.
The three terms also sit on different carriers, which the remainder of
the paper keeps explicit: \(M_{\rm opt}^{g[\phi],B}\) is a propagated
optical contribution (backreacted geometry, the bounded disformal
deformation, and under the canonical unstretched-space metric of §6.9 the direct refractive response to \(A_{\rm loc}(u_{\rm grav})\)); \(M_{\rm TC}\) propagates only through the same
carriers — the isotropic-conformal contribution to a propagated emission-time
map is identically zero; and \(M_{\rm rec}^{A}\) lives entirely in
reconstruction space, entering through the endpoint-determined
clock-transfer field and the inference degeneracy (mass-sheet and
normalization class) rather than through any propagated delay on the isotropic-conformal benchmark.

**Principle:**

#### Box 2.1: Relation to Established Scalar-Tensor Theories

TEP is a specific two-metric scalar–tensor effective field theory,
distinguished by the universal coupling of all nongravitational matter to
a single causal metric and by its dynamical-proper-time interpretation.
Its covariant operators draw on established scalar–tensor constructions,
while its operational ontology and observable sector decomposition are
specific to TEP. The correspondences are:

**Brans-Dicke Theory:** With \(B=0\), \(A(\phi)
= e^{\beta_A\phi/M_{\text{Pl}}}\), and the corresponding kinetic and
potential choice, the conformal sector contains a Brans–Dicke-like
limit in the Jordan frame, with \(\beta_A\) related to the
Brans–Dicke parameter \(\omega_{BD}\). The correspondence is a limit
of the conformal sector, not an identity of the full theory.

**Horndeski and Beyond:** Higher-derivative
scalar-tensor theories (Horndeski, DHOST) provide candidate
microscopic completions for the gradient-dependent branch of
\(\mathcal{S}_\Sigma(\mathcal{E})\). TEP-GL does not commit to a
specific completion; the canonical operator is defined at the
theory level independently of any particular Lagrangian.

**Screening and Temporal Topology:** The core TEP
framework formulates screening through the environmental operator
\(\mathcal{S}_\Sigma(\mathcal{E})\) acting on the Temporal Shear
\(\Sigma_\mu\), a continuous spatial profile. Imported
modified-gravity screening mechanisms are not part of the TEP
ontology; the screening object is \(\mathcal{S}_\Sigma(\mathcal{E})\)
alone.

**DHOST Theories:** Degenerate Higher-Order
Scalar-Tensor theories extend Horndeski by imposing a degeneracy
condition that eliminates the additional propagating mode, rather
than by keeping the field equations manifestly second-order. TEP is
compatible with this broader class but does not require it.

**Key Distinction:** TEP's distinguishing content is the
universal causal-metric coupling together with the
*observational interpretation* it licenses — specifically, the
recognition that conformal coupling creates a "temporal composite" image
whose residual timing structure a standard reconstruction absorbs as
mass. Individual covariant operators are drawn from established
scalar–tensor constructions; the sector dictionary, the universal
coupling, and the resulting lensing phenomenology are specific to TEP.

### 2.2 The Chronometric Lensing Framework

TEP-GL lensing contains three projections of one temporal field: coherent
optical response through scalar backreaction and any permitted disformal
contribution; conformal open-path chronometric reconstruction; and the
source-dependent temporal-composite response.

#### 1. The Static Clock-Transfer Contribution (Chronometric Reconstruction)

The conformal clock sector modifies the relation between accumulated
matter-frame time and the geometry reconstructed under an isochronous
model. The associated effective clock-transfer contrast is:

\begin{equation} \label{eq:gl_static_delay}
\Delta \tilde{\tau}_{\rm static} = \frac{1}{c} \int (A(\phi) - 1)\,
dl
\end{equation}

In the phenomenological isochronous reconstruction used here, this is represented as a source-independent correction to the inferred arrival-time surface. In multipath configurations, it is therefore operationally
degenerate with the mass normalization inferred from multipath timing,
distance, and joint image–delay reconstruction. The angular positions of
Einstein rings and major arcs remain governed by the optical geometry.

The expression above is a matter-clock transfer functional evaluated on a
timelike clock congruence associated with the path; it is not a photon
observable, and for photons its differential content vanishes identically.
That vanishing is a statement about a restricted net observable — the
static conformal contribution to an inter-image arrival-time contrast —
not that the photon is unaffected while traversing the topology: the
locally measured frequency rises descending into each well and falls
climbing out, asymmetric crossings slice residuals to the intervening
field difference, and lapse-geometry dwell through each structure accrues
monotonically. Only the frequency bookkeeping of a strictly static
conformal field telescopes to the endpoints.
Two equivalent statements fix this on the isotropic-conformal benchmark (\(\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}\), where spatial rulers stretch alongside clocks). First, under isotropic conformal rescaling, null cones are conformally
invariant (Axiom 1, Appendix A.1): \(A(\phi)\) drops out of the null
transport equation, so the coordinate travel time along any fixed spatial
path is identical in \(\tilde{g}_{\mu\nu}\) and \(g_{\mu\nu}\), and the only
conformal content in an observed arrival time is the endpoint clock ratio —
a common-mode factor for all images of one system. Second, the underlying
Temporal Shear is an exact one-form, \(\Sigma_\mu = \nabla_\mu\ln A\): its
open-path integral is endpoint-determined, so the shear-sector contribution
to any path contrast between rays joining the same source and observer
cancels algebraically, before screening is even applied. The dedicated
strong-lensing transport pipeline confirms the result directly on the isotropic-conformal benchmark: the exact
additional static-conformal inter-image residual is \(0.0\) d (TEP-LENS,
Paper 19, step_56). The functional \(\Delta\tilde\tau_{\rm static}\) is therefore retained in a
sharply delimited role: it is the chronometric reconstruction map — the
field of clock-rate contrasts that an isochronous lens model would
misassign to mass-sheet normalization, distance, or time-delay
calibration if it were read as propagating emission-time structure — and
it bounds, rather than supplies, the isotropic-conformal arrival-time residual. Genuine
path-dependent null delay and deflection beyond the GR geometry arise when space does not stretch with \(A(\phi)\) — specifically, under the canonical unified unstretched-space temporal metric \(\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi\) (\(U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}\); Paper 0 §2.2, Step 80, evaluated in §6.9 below), where the \((3+1)\) ADM lapse \(N_{\rm TEP}=A_{\rm loc}(u_{\rm grav})N_{\rm GR}\) rescales along the cosmic temporal flow while the spatial 3-metric \(\tilde\gamma_{ij}=\gamma_{ij}\) remains unstretched at fixed epoch so intermediate temporal depth \(-\beta_A u_{\rm grav}(x)\) refracts null rays directly via \(n(x)\simeq 1-2\Phi_N/c^2-\beta_A u_{\rm grav}(x)\) — or through gravitational backreaction in \(g_{\mu\nu}\) and the spatial-disformal sector \(B(\phi)\): pure isotropic
multiplication by \(A^2(\phi)\) does not bend a photon onto a new null
trajectory and does not shift its arrival time relative to any other
photon on the same endpoints. Under the isotropic-conformal benchmark the lensing signal is a reconstruction-space discrepancy, whereas under the canonical unstretched-space temporal metric (§6.9, Step 02) intermediate temporal depth supplies direct optical refraction alongside the chronometric reconstruction map.

#### 2. The Source-Evolution Response (Temporal Lensing)

While the static term reshapes the clock-transfer field read by an
isochronous reconstruction, the gradient of the differential delay field
\(\nabla(\Delta \tilde{\tau})\) acts as a "shutter"
that modulates the arrival time of photons from different parts of the
source. For a source with evolution or motion, this creates a
*temporal-composite response*:

\begin{equation} \label{eq:gl_temporal_shear}
\gamma^{\rm TC} \propto \left(\frac{\partial I}{\partial t} +
\vec{v}_s \cdot \nabla I\right) \nabla(\Delta \tilde{\tau})
\end{equation}

The nomenclature follows the canonical corpus convention: Temporal
Shear denotes the field gradient \(\Sigma_\mu \equiv \nabla_\mu \ln
A(\phi)\), which generates the path-dependent clock-transfer field, whereas
*temporal-composite response* \(\gamma^{\rm TC}\) denotes the observable
source-dependent image response to that field. The two must not be conflated:
the former is a property of the temporal field, the latter of the
observation.

This source-evolution response supplies a source-dependent phantom-mass
component — an apparent mass that can fluctuate with source type. It operates
as a re-ordering of emission epochs rather than a deflection of rays, and for
galaxy
sources it is bounded by the ratio of differential delay to
source-evolution time at \(\lesssim 10^{-7}\) in shear, far below the
observed lensing excess. (See Section 3 for the full derivation).

Together, these two mechanisms constitute the TEP framework: the
clock-transfer contribution may contribute to the phenomenology
conventionally attributed to the dark matter halo, and the
source-evolution response may account for part of the apparent
"complexity" of
substructure. Any genuinely new null-trajectory bending beyond the
conformally related geometry must arise through gravitational
backreaction in \(g_{\mu\nu}\), a non-negligible disformal
contribution, or another explicitly derived part of the coupled field
solution.

### 2.3 Operational Axioms: The TEP Framework

The TEP framework rests on four foundational axioms. These are not
approximations or perturbative corrections to General Relativity; they
constitute a complete operational framework for interpreting gravitational
phenomenology. They are adopted as *primary postulates* from which
observational consequences are derived:

**Principle:**

**Axiom 1 (Causal Universality):** In the Conformal Limit
(\(B=0\)), the null cones of \(g_{\mu\nu}\) and \(\tilde{g}_{\mu\nu}\)
are identical. Photons and gravitational waves follow the same null
geodesics. No "speed of light" difference exists in this limit.
*This axiom is exact, not approximate.*

**Axiom 2 (Proper Time Primacy):** The fundamental
observables in any timing measurement are the proper-time intervals
\(\Delta \tilde{\tau}\) registered on physical emitter and receiver clock
worldlines, together with the phase, frequency, and arrival-time transport
connecting those clock events along \(\tilde{g}\)-null signal paths.
Coordinate time differences \(\Delta t\) are not observables; they are
inferred quantities dependent on the metric model. Null proper time along
the photon trajectory is identically zero (Jakarta); wherever this paper
refers to accumulated path time, the phrase denotes the operational
clock-transfer functional associated with the null path, not proper time
experienced by the photon.
This axiom establishes clock-registered proper time as the irreducible
physical observable; all other timing quantities are derived.

**Axiom 3 (Non-local temporal transport):** TEP admits two
distinct classes of non-local timing observable.

**(a) Synchronization holonomy:** A genuinely closed,
direction-reversing transport loop can exhibit residual non-closure only
when the transport connection contains non-exact structure, such as the
disformal \(B(\phi)\) sector. In the pure conformal limit, the
\(A(\phi)\) contribution is exact and its residual closed-loop integral
vanishes.

**(b) Differential path transport:** Distinct open propagation
paths may accumulate different temporal-transfer corrections. In
gravitational lensing, two images correspond to two open paths
\(\gamma_i, \gamma_j\), and the observable is the differential
transport residual
\[
\Delta\mathcal{T}_{ij}^{\rm resid}
= \left[\mathcal{T}[\gamma_i] - \mathcal{T}[\gamma_j]\right]
- \left[\mathcal{T}_{\rm GR}[\gamma_i] - \mathcal{T}_{\rm GR}[\gamma_j]\right],
\]
where \(\mathcal{T}[\gamma] \equiv \Delta\tilde{\tau}[\gamma]\) is the
time-transport functional. This is an open-path blind-prediction
residual, not a closed-loop synchronization holonomy. Two structural
properties delimit it. First, because each image has a single observed
arrival time, algebraic closure of measured pairwise delays around image
triplets vanishes identically: the axiom alters the relation between
lens geometry and observed arrival time without violating the algebraic
identity among the observed arrival times. Second, in the pure conformal
sector the residual itself vanishes identically, because
\(\Sigma_\mu = \nabla_\mu\ln A\) is an exact one-form and its open-path
integrals are endpoint-determined (§3.1.3); a nonzero
\(\Delta\mathcal{T}_{ij}^{\rm resid}\) is therefore a signature of
non-exact transport — the disformal sector or scalar backreaction — not
of the conformal clock-transfer map.

This axiom distinguishes open-path differential residuals (the GL
observable) from closed-loop holonomy (the domain of triangle
time-transfer and direction-reversing experiments), and is consistent
with the refined strong-lensing formulation in TEP-LENS (Paper 19).

**Measurement Protocol for the GL Observable.**
The lensing-sector observable is not algebraic delay closure. It is a
blind-prediction residual. For each image pair \((i,j)\), compare the
observed delay \(\Delta t_{ij}^{\rm obs}\) with the pre-specified GR
lens-model prediction \(\Delta t_{ij}^{\rm GR}\):
\[
R_{ij} = \Delta t_{ij}^{\rm obs} - \Delta t_{ij}^{\rm GR}.
\]
TEP predicts that these residuals should correlate with
temporal-transport tracers such as projected potential depth,
magnification, variability timescale, or other lens-environment proxies.
Algebraic closure of observed pairwise delays is not a TEP
discriminator, because it vanishes identically whenever each image has a
unique arrival time.

**Systematic Error Budget:** Several astrophysical
systematics can produce apparent residuals that must be distinguished
from true temporal-transport effects:

**Lens Model Degeneracies:** The mass-sheet degeneracy
and source-position transformation can bias predicted delays. These
affect the *predicted* delay (from the model), not the
*observed* delay. The test compares observed delays against
model predictions, and systematic lens-model error must be
propagated into the residual uncertainty.

**External Convergence:** Line-of-sight structure
acts approximately as an external mass-sheet transformation,
rescaling the Fermat-potential and time-delay normalization. If
omitted, it can generate coherent biases in \(R_{ij}\). Each lens
must therefore marginalize over \(\kappa_{\rm ext}\) and the
associated mass-sheet uncertainty before testing for a TEP residual.

**Microlensing:** Stellar microlensing in the lens
galaxy can shift apparent arrival times by hours to days. However,
microlensing is stochastic and uncorrelated between images; over an
ensemble of lens systems, microlensing-induced residuals should
average to zero with RMS scaling as \(1/\sqrt{N}\).

**Host Galaxy Delays:** Differential extinction or
scattering in the host can introduce chromatic delays, but TEP's
temporal-transport residual is achromatic. Chromatic residual
patterns indicate astrophysical contamination, not metric effects.

**Discriminator:** True temporal-transport residuals should
produce a *systematic* non-zero regression coefficient or
covariance between \(R_{ij}\) and a pre-specified temporal-transport
predictor, using a fixed image-pair ordering and hierarchical
marginalization over lens-model degeneracy, \(\kappa_{\rm ext}\),
microlensing, and host-galaxy delays, while astrophysical
systematics produce *random* residuals that average to zero
across multiple systems. An unconditional mean could cancel even when a
real path-dependent correlation exists. The correlation is the correct
TEP discriminator.

**Axiom 4 (Screening and Temporal Topology):** Screening
manifests as a continuous spatial profile (Temporal Topology) governed
by the non-linear superposition of field gradients (Temporal Shear),
suppressing the Temporal Shear where the ambient time-gradient is
steep (weak-field: high-acceleration environments)
while leaving cosmology accessible to dynamics. The suppression of
local Temporal Shear in deep potential environments continuously
reduces

\begin{equation}
\Sigma_\mu^{\rm obs} = \mathcal{S}_\Sigma(\mathcal{E})\,\Sigma_\mu .
\end{equation}

Channel-specific PPN responses are evaluated only after this
environmental projection is applied. Existing multi-messenger
constraints (e.g., GW170817) are interpreted within the standard
framework that assumes a single metric governs all sectors; in the TEP
framework, these constraints apply to specific parameter combinations
(primarily the disformal sector) and do not constitute blanket
exclusions of two-metric effects.
This axiom establishes that environmental suppression is a
geometric property of the field's spatial profile, not a binary
boundary condition, and that constraints derived under
Isochrony-assuming frameworks must be re-derived operationally
within TEP before being applied as exclusions.

These axioms are mutually consistent and together define the TEP
interpretation of gravitational phenomenology. They replace the implicit
Isochrony Axiom of standard lensing analysis with an explicit dynamical-time
framework.

**The Hidden Closure in Standard Practice:** In addition to the
Isochrony Axiom, most observational inference quietly assumes that time
transport is globally integrable: after correcting for known effects
(Sagnac, Shapiro, troposphere, etc.), a single global time coordinate can be
assigned such that closed-loop synchronization holonomy vanishes.
Operationally, this is the step that licenses the non-local conversion \(d =
c\,t\) and the inference “timing residual \(\Rightarrow\) mass residual.” In
TEP, this closure is not assumed; Axiom 3 treats open-path differential
residuals as the primary observables, while true synchronization holonomy
requires direction-reversing closed loops or non-exact transport structure.

### 2.4 Conformal vs. Disformal Phenomenology

The distinction between the two sectors is critical for interpreting
multi-messenger constraints, and it rests on a fundamental distinction
between *Single-Path* and *Multipath* measurements.
Furthermore, it necessitates a redefinition of the "speed of light":

**Conformal Sector (\(A(\phi)\)):**

**Geometry:** Preserves angles and null cones.
\(\tilde{g}_{\mu\nu}k^\mu k^\nu = A^2(\phi) g_{\mu\nu}k^\mu k^\nu = 0\).

**Local Invariance vs. Global Variability:** Local
\(c\) remains invariant (measured as \(299,792,458\) m/s by
any local clock; Paper 0, Theorem 2). What varies globally is
not any speed of propagation, but the inferred ratio between
spatial separation and matter-clock transfer registered between
endpoints along extended paths. Because the rate of proper
time accumulation \(d\tilde{\tau} = A(\phi) d\tau_g\) varies
with location, the clock transfer associated with traversing a
fixed spatial interval depends on the scalar field value: a
timelike clock transported through a halo well (\(A < 1\))
accumulates less matter-frame time than the ambient rate, and
one transported through a void (\(A > 1\) relative to ambient)
accumulates more. For photons this path-integrated content is
not observable at all — the null condition is conformally
invariant — and the conformal sector enters the photon sector
only through the endpoint clock ratio, the same channel that
carries cosmological redshift (Paper 0, §2.2). Any "variable
speed of light" language is therefore shorthand for an
operational endpoint ratio inferred under a universal-clock
assumption, never a change in local propagation.

**Single-Path Physics (GW170817):** Photons and
gravitational waves from the exact same source coordinate follow
the *same* null geodesic. Any temporal distortion
\(A(\phi)\) along this path is common-mode. The signals do not
diverge because they share the same history.

**Multipath Physics (Lensing):** Gravitational
lensing involves light rays taking *different* paths
around a mass distribution. These paths traverse different
regions of the scalar field \(\phi(\vec{x})\), but photon
transport along them is conformally invariant: the conformal
differential clock-transfer contrast is a reconstruction-space
field, not a propagated delay, and it generates the chronometric
component of the Phantom-Mass signature only as an inference
degeneracy. Genuine propagated differential delay is carried by
scalar backreaction (absorbed by lens models as mass) and by the
GW170817-bounded disformal deformation; coherent optical
convergence and shear arise through the same two channels.

**Disformal Sector (\(B(\phi)\)):**

**Geometry:** Tilts null cones. The effective speed
of light differs from the speed of gravity: \(c_{\gamma} \neq
c_g\).

**Observables:** Differential arrival times between
species, constrained by GW170817 to \(|c_{\gamma}-c_g|/c
\lesssim 10^{-15}\) for the path-averaged monopole.

### 2.5 The "Phantom Mass" Mechanism

Standard lensing reconstruction solves for a mass distribution
\(\Sigma(\vec{\theta})\) that reproduces the observed image distortions.
This reconstruction assumes the Isochrony Axiom: \(I_{\rm obs}(\vec{\theta}) =
I_{\rm src}(\vec{\beta})\). However, in the TEP framework, the observed image
at a given observation time \(t_{\rm obs}\) is the source evaluated at a
shifted effective time:

\begin{equation} \label{eq:gl_composite_image}
I_{\rm obs}(\vec{\theta},\, t_{\rm obs})
= I_{\rm src}\!\left[
\vec{\beta}_{\rm opt}(\vec{\theta}),\;
t_{\rm obs} - \Delta T_{\rm eff}(\vec{\theta})
\right],
\end{equation}

where \(\Delta T_{\rm eff}(\vec{\theta})\) is the effective emission-time
map along the line of sight at image position \(\vec{\theta}\), and
\(\vec{\beta}_{\rm opt}(\vec{\theta})\) is the optical ray map generated by
the scalar-backreacted and disformal geometry. The propagating content of
\(\Delta T_{\rm eff}\) decomposes into the ordinary geometric-plus-Shapiro
Fermat surface already contained in any GR lens model, the disformal
multipath deformation bounded by GW170817 (\(\lesssim 0.2\) s per \(2\) Mpc
halo path; Box 3.2), and the common-mode endpoint clock ratio; the
conformal sector contributes no path-dependent term, because null transport
is conformally invariant (§3.1.1) and its clock-transfer map is
endpoint-determined (§2.2). For a finite exposure over
\([t_1, t_2]\), the recorded image is the time-averaged quantity:

\begin{equation} \label{eq:gl_composite_image_exposure}
\bar I_{\rm obs}(\vec{\theta})
= \frac{1}{\Delta t_{\rm exp}}
\int_{t_1}^{t_2}
I_{\rm src}\!\left[
\vec{\beta}_{\rm opt}(\vec{\theta}),\;
t - \Delta T_{\rm eff}(\vec{\theta})
\right] dt .
\end{equation}

For a spatially varying temporal field, \(\Delta T_{\rm eff}\) varies
across the image plane through its propagating components — the
gravitational Fermat surface and the disformal deformation — while its
conformal clock-transfer content varies only in reconstruction space. If
the source has temporal variability (secular
evolution, rotation, or fluctuations) on the timescale of
\(\nabla_\theta (\Delta T_{\rm eff})\), the finite-exposure average smears
the recorded image.

**The Equivalence:** A gradient in the emission-time map
across an image
is mathematically equivalent to a shearing of the source frame. To a static
observer assuming isochrony, this "temporal shear" is indistinguishable from
the "gravitational shear" caused by mass. Thus, for temporally structured
sources,
purely temporal structure is misinterpreted as "Phantom Mass" (dark
matter). The channel is bounded at \(\lesssim 10^{-7}\) for galaxy sources and
vanishes for static sources; it is a source-dependent discriminator, not
the carrier of the observed static-source lensing excess.

#### One Quantity, Two Measurements

Phantom Mass is not a separate mechanism in each observable channel; it is
one quantity — the mass a standard analysis infers because it does not
model the TEP temporal field. It enters through two routes: in dynamics,
where the Temporal Shear supplies extra apparent acceleration
(\(M_{\rm ph}^{\rm dyn}\)); and in reconstruction, where the differential
source-clock history of an evolving source is read as spatial structure
(\(M_{\rm ph}^{\rm recon}\), zero for static sources). Each channel is
inferred independently: \(M_{\rm ph}^{\rm dyn} \equiv M_{\rm dyn}^{\rm GR} -
M_{\rm baryon}\) and
\(M_{\rm ph}^{\rm lens} \equiv M_{\rm lens}^{\rm GR} - M_{\rm baryon}\).
Comparing them is then a *test* of Phantom Mass rather than its
definition — the empirical question is
\(M_{\rm ph}^{\rm dyn} \overset{?}{=} M_{\rm ph}^{\rm lens}\). For a single shared collisionless mass distribution, matched dynamical and lensing analyses should recover mutually consistent mass profiles within orbital-anisotropy, projection, baryonic-modelling, and external-convergence uncertainties.
In TEP, on the historical isotropic-conformal benchmark (\(\tilde g_{\mu\nu}=A^2 g_{\mu\nu}\)) the Temporal Shear acts on matter trajectories while \(A^2(\phi)\) cancels from photon paths (\(M_{\rm ph}^{\rm lens,iso}\simeq 0\)), whereas under the canonical unstretched-space temporal metric (\(\tilde\gamma_{ij}=\gamma_{ij}\), §1.5 and §6.9) intermediate temporal depth refracts photons at half the dynamical scalar potential weight (\(\Phi_{\rm lens}=\Phi_N+\tfrac12 c^2\beta_A u_{\rm grav}\) vs \(\Phi_{\rm dyn}=\Phi_N+c^2\beta_A u_{\rm grav}\); Paper 0 Eq. 63, \(M_{\rm lens}^{\rm u.s.}/M_{\rm dyn}=(1+\beta_A^2 y_{\rm eff})/(1+2\beta_A^2 y_{\rm eff})\)). In both realizations the single-halo prediction is
\(M_{\rm ph}^{\rm dyn} > M_{\rm ph}^{\rm lens}\), with the difference largest
where the shear is active — the isolated, low-acceleration systems where
\(\mathcal{S}_\Sigma \to 1\) (approaching parity \(M_{\rm lens}^{\rm u.s.}\simeq M_{\rm dyn}\) on multi-component or group-embedded sightlines). This phantom-mass consistency test is the
framework's decisive dark-sector check: agreement would bound the single-halo differential between dynamical shear and optical temporal refraction; a systematic positive difference at matched radii, persisting after anisotropy, projection, and mass-sheet systematics are marginalized, is a distinctive TEP signature not generically expected from a single shared collisionless mass distribution.

### 2.6 Two Regimes of TEP-GL

**Principle:**

**Box 1: The Two Observational Regimes**

The TEP-GL framework operates in two distinct regimes, distinguished by
the magnitude of the differential clock-transfer residual
\(\Delta\tilde{\tau}\) across the lens:

---

**Regime I: the conservative baseline**

**Physical carrier:** The disformal multipath
deformation. For photons on the matter metric the phase velocity is
\(v_{\rm ph}/c \simeq 1 - \tfrac{1}{2}D(\partial_{\hat{n}}\phi)^2\)
with \(D \equiv B/A^2\) (Paper 0, App. B); a differently routed
image pair traverses different \(\nabla\phi\) structure, so the
inter-image residual is
\(\Delta\tau_B = \tfrac{1}{2}\int D(\partial_{\hat{n}}\phi)^2\,dl\),
evaluated at the GW170817-bound normalization
(\(D(\partial_{\hat{n}}\phi)^2 \lesssim 2\times10^{-15}\)
path-averaged). The deformation is direction-even
(\((\partial_{\hat{n}}\phi)^2\) is invariant under
\(\hat{n}\to-\hat{n}\)), so it appears as a propagation residual
between distinct paths rather than a same-path holonomy — the
even-parity channel of the same \(B(\phi)\) sector the corpus's
holonomy program bounds on the odd/non-exact channel. On the
corpus's weak-field branch \(B(\phi)\simeq B_0\varphi^2\) (Paper 0,
§2.2), the product \(D(\partial_{\hat{n}}\phi)^2\) carries the
quadratic field-depth suppression, so the stated ceiling is the
saturation value of the GW170817 bound rather than the envelope's
expectation on weak-field halo paths: the corpus-calibrated
admissible normalization lies parametrically below it. The
falsification threshold therefore bounds the normalization — not
merely tests an assumed saturation — and the conversion into a
constraint on the dimensionful \(B_0\) belongs to the same open
normalization map as the corpus's B-sector bookkeeping (Paper 0,
Appendix E).

**Constraint basis:** GW170817 bounds the path-averaged
monopole \(|c_{\gamma}-c_g|/c \lesssim 10^{-15}\); the conformal
sector contributes exactly zero propagated inter-image delay by
null-cone invariance (§3.1.3).

**Delay scale:** \(\Delta\tilde{\tau} \sim
10^{-3}\text{--}1\) s (milliseconds to seconds) on halo scales —
\(\Delta\tau_B \lesssim 0.2\) s per \(2\) Mpc halo path at the
bound — the only propagating inter-image delay the theory supplies.

**Primary observables:** Time-domain signatures in
rapidly varying sources—lensed FRBs, GRBs.

**Dark matter status:** TEP is a
*precision systematic* in time-delay cosmography, not a
wholesale DM replacement.

**Falsification:** Null detection of achromatic timing
residuals at < 0.1 ms on a \(2\) Mpc halo path bounds the
path-averaged disformal product
\(D(\partial_{\hat{n}}\phi)^2 \lesssim 5\times10^{-19}\) —
\(\sim 3.5\) orders of magnitude below the GW170817 level —
probing between the corpus-calibrated envelope and the saturated
ceiling: the same \(B(\phi)\) instrument as the holonomy
program, applied to the even-parity multipath channel.

---

**Regime II: the bookkeeping-inclusive case**

**Assumption (Sector Decoupling):** GW170817 constrains
only differential disformal coupling; common-mode conformal temporal
structure is not directly constrained by it, though it remains bounded
by clock, PPN, and redshift channels (see Axiom 4).

**Clock-transfer scale:** \(\Delta\tilde{\tau} \sim
1\text{--}10\) years, driven by the conformal factor \(A(\phi)\)
integrated over halo scales (Mpc). This is the
*reconstruction-space* scale — the reconstruction-space
clock-transfer map of §2.2 — not a propagated photon delay; the
conformal contribution to any inter-image arrival-time residual is
exactly zero, and the propagating differential channel remains the
disformal carrier at the ms–s level.

**Primary observables:** The reconstruction-space and
time-domain content of the conformal sector — chronometric
bookkeeping, endpoint clock-transfer maps, and the source-dependent
temporal-composite response — together with the source-independent
unstretched-space temporal refraction
(\(\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}\), with \(U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}\), rescaling the ADM spacetime lapse \(N_{\rm TEP}=A(\phi)N_{\rm GR}\) at fixed spatial 3-metric \(\gamma_{ij}\)) and the
*dynamical–lensing mass difference* in matched spatial
observables. While isotropic conformal rescaling
\(\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}\) cancels out of null rays
and leaves only the Planck-suppressed scalar stress tensor
(\(\kappa_\phi\simeq6.5\times10^{-6}\); Paper 19, Step 64), the
unstretched-space temporal metric retains intermediate clock rates
\(A(\phi(x))\) along every photon path, reproducing the Bullet
Cluster galaxy-peak convergence (Section 6.9, Step 02), the SN
Refsdal Fermat delay (Paper 19, Step 64), and the Planck CMB
lensing cosmic-web amplification (Paper 18, Step 21) from baryons
and the selected two-branch action \(P_{,X}=k\sqrt\xi+2\xi\).

**Dark matter status:** Phantom Mass is the apparent
mass component inferred when temporal structure is reconstructed
under the Isochrony Axiom: for dynamics it reflects the Temporal
Shear acceleration on slow matter; for static-source lensing across
unstretched space it reflects the line-of-sight refractive index
\(n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)\) of intermediate
temporal depth; and for temporally structured sources it includes
the bounded projected image response (\(\lesssim10^{-7}\)).

**Falsification:** The unstretched-space temporal
refraction channel is tested directly against cluster convergence
morphology (Bullet Cluster galaxy-peak alignment and absence of a
negative moat, Step 02), strong-lensing Fermat delays (SN Refsdal,
Paper 19 Step 64), and the CMB lensing spectrum (Paper 18 Step 21),
while source-variability correlations test the bounded
temporal-composite contribution.

This work demonstrates that
Regime II operates as a time-domain and reconstruction-space
framework alongside the source-independent optical refraction of the
unstretched-space temporal metric (§6.9). Regime I is the conservative
baseline in which the spatial-gradient disformal deformation is evaluated at
its GW170817-bounded normalization, while under the unstretched-space
temporal metric \(\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}\)
intermediate temporal depth \(-\beta_A u_{\rm grav}(x)\) along the photon path closes
the source-independent deflection across galaxy, cluster, and CMB lensing
while preserving \(c_{\rm GW}=c_\gamma\) identically on the shared temporal-lapse cone.

### 2.7 Why Lensing May Not Be Purely Spatial

Standard gravitational lensing analysis contains a simplifying temporal
assumption that has typically been treated as exact. This assumption is made
explicit, and its breakdown under TEP leads directly to the dark matter
reinterpretation.

#### The Hidden Assumption in Image Formation

When a lensed galaxy is observed, photons are collected over an exposure
time and an "image" is reconstructed. This reconstruction implicitly
assumes:

All photons arriving during the exposure left the source at
approximately the same epoch

Differences in arrival angles correspond to differences in spatial
paths, not temporal paths

The reconstructed "shape" represents a spatial snapshot of the source at
one moment

These assumptions constitute the *Isochrony Axiom* applied to image
formation. Standard analysis corrects for the mean lookback time (distance),
but assumes that the variance in lookback time across the image is
negligible. This overlooked variance is defined as
*Differential Lookback Time*. In a two-metric framework, this
variance is an additional source-dependent reconstruction residual.

#### The temporal composite Mechanism

Consider an extended galaxy being lensed. Light from different parts of the
galaxy:

Leaves the source at different times (the source is evolving on Myr
timescales)

Takes different paths through the lens (different impact parameters)

Is accompanied by a different matter-clock transfer congruence through
regions with different \(A(\phi)\) values — a timelike reconstruction
field, not a photon propagation delay

- Arrives at the detector at the "same" observation time

If \(A(\phi)\) varies spatially—forming a halo-like configuration around the
lens—then the matter-frame path time

\begin{equation} \label{eq:gl_path_delay}
\Delta t_{\rm path} = \int_{\rm path} \frac{A(\phi)}{c}\,dl
\end{equation}

evaluated on the associated timelike clock congruence differs between rays
at different impact parameters. This functional is not a photon arrival
time: Axiom 1 makes the photon coordinate transit conformally invariant,
so no part of it propagates to any observed inter-image delay. What it
quantifies is the reconstruction-space contrast — the magnitude of the
clock-transfer field that a universal-clock reconstruction would
misassign to mass, distance, or time-delay normalization. For halo-scale
propagation depths \(L \sim 2\) Mpc and conformal variations \(\Delta A/A
\sim 10^{-6}\), that envelope contrast is:

\begin{equation} \label{eq:gl_delay_estimate}
\Delta \tilde{\tau}_{\rm env} \sim \frac{\Delta A}{A} \cdot \frac{L}{c} \sim
\text{years}
\end{equation}

The propagating residual on the same paths is exactly zero in the conformal
sector (Paper 19, step_56) and is bounded at the ms–s level on the
disformal carrier (Box 3.2). The relevant comparison for any claimed
observable is therefore the post-model residual — constrained to
\(\lesssim\) tens of days in existing strong-lens systems by the success of
GR mass models (Paper 19) — not the multi-hundred-day total delay, which
those models already reproduce. For slowly evolving galaxies the
source-dependent temporal-composite response is negligible. The dominant signal is
therefore the coherent static temporal-field sector: physical optical
shear/convergence arises from scalar backreaction and any permitted
disformal response, while the conformal sector supplies the associated
chronometric reconstruction. The temporal smearing (the source-motion term) becomes
the dominant propagating temporal-composite signal only for fast transients
(FRBs); for slowly evolving sources its observable content is the
shear-variance residual above the standard delay-map baseline (§3.4).

#### Why This Creates "Phantom Mass"

For a source that evolves on timescales comparable to the emission-time-map contrast \(\Delta T_{\rm eff}\):

If the source was more compact in the past → inner regions (earlier
epoch) appear smaller

If the source was less compact in the past → inner regions appear larger

The reconstructed "shape" is systematically distorted by temporal mixing

Standard lensing analysis interprets *any* systematic shape
distortion as evidence for mass (convergence and shear). The temporal
smearing is mathematically indistinguishable from gravitational lensing
distortion, but its amplitude is bounded by the ratio of the
emission-time-map contrast to the source-evolution timescale:
\(\lesssim 10^{-7}\) in shear for galaxy sources, vanishing for static
sources. It is therefore a source-dependent reconstruction signature, not
the carrier of the observed dark-matter-level lensing excess; that excess
is confronted through the dynamical–lensing mass comparison (§2.6).

#### Why This Effect Is Achromatic

The conformal factor \(A(\phi)\) rescales proper time identically for all
photon frequencies. Unlike plasma dispersion (which scales as \(\nu^{-2}\))
or dust extinction (wavelength-dependent), conformal temporal coupling is
*perfectly achromatic*. General relativistic lensing is likewise
achromatic, so achromaticity alone does not discriminate a temporal-field
contribution from ordinary gravitational lensing. The operational
significance is the converse: because the TEP contribution is achromatic, it
cannot be separated from GR lensing by colour information, but it *can*
be separated from plasma and dust systematics — which is precisely what makes
multi-frequency timing the discriminating measurement (Section 5.1).

#### The Central Thesis

Under TEP, the existence of "dark matter" is inferred rather than directly
observed. It is a *conclusion contingent on the Isochrony Axiom*. If
that axiom fails—if the universe is not temporally synchronous in the way
standard analysis assumes—then the inferred "dark mass" can be modeled as
unmodeled temporal structure. The dark sector is modeled not as a substance,
but as the shadow of time. No claim is made regarding uniqueness of the
two-metric realization, only the operational equivalence of temporal
gradients to inferred mass under isochrony.

## 3. The Phantom Mass Mechanism

### 3.1 The Matter-Metric Map and Null-Trajectory Structure

The Phantom Mass mechanism is derived from the TEP matter-metric architecture (Jakarta, Paper 0). In isotropic shorthand, the gravitational metric \(g_{\mu\nu}\) is related to the causal matter metric \(\tilde{g}_{\mu\nu}\) through the conformal–disformal map:

\begin{equation} \label{eq:gl_disformal_map}
\tilde{g}_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi
\end{equation}

where \(A(\phi)\) is the universal conformal coupling and \(B(\phi)\) is the disformal coupling. This isotropic conformal–disformal expression provides the historical benchmark (not a Newtonian-gauge metric with a conformal factor inserted only in the spatial sector), whereas the operative TEP realization adopts the preferred-foliation decomposition \(\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi\) along the cosmic temporal flow \(U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}\), as specified in Paper 0, Section 2.2 (§1.5, §6.9). Examining the isotropic benchmark first isolates why isotropic ruler-stretching cancels from null trajectories before evaluating the unstretched-space temporal refraction in §6.9.

#### 3.1.1 Conformal Invariance of Null Trajectories

In the purely conformal subclass (\(B=0\)), the matter metric reduces to \(\tilde{g}_{\mu\nu}=A^2(\phi)\,g_{\mu\nu}\). A standard result of conformal geometry in four dimensions is that null geodesics are preserved: if \(k^\mu\) is tangent to a null geodesic of \(g_{\mu\nu}\), the same path is a null geodesic of \(\tilde{g}_{\mu\nu}\) with a rescaled affine parameter. Therefore:

\begin{equation} \label{eq:gl_null_invariance}
B = 0 \quad \Longrightarrow \quad d\tilde{s}^2 = 0 \quad \Longleftrightarrow \quad ds^2 = 0.
\end{equation}

This has three immediate consequences for gravitational lensing:

- \(A^2\) alone does not change the null trajectory. Pure conformal rescaling preserves null cones; it cannot generate a spatial refractive index or bend photons onto new paths.

- Static geometric bending beyond the conformally related geometry must come from \(g_{\mu\nu}[\phi]\) backreaction — the scalar field modifying the Einstein-frame metric through the coupled field equations — or from \(B \neq 0\), which tilts the matter null cone relative to the gravitational null cone.

- The conformal sector can alter endpoint clock calibration, frequency transport, and timelike source evolution. It acts directly on timelike observables (orbital periods, spectroscopic velocities, standard-candle distance calibration) but cannot generate the printed spatial refractive index \(n_{\rm eff} \simeq 1 - 2\Psi + (A(\phi)-1)\) that would follow from inserting \(A^2(\phi)\) into the spatial sector of a Newtonian-gauge metric alone.

#### 3.1.2 Separating Null and Timelike Observables

The lensing deflection angle decomposes into two physically distinct channels:

\begin{equation} \label{eq:gl_deflection_decomp}
\delta\theta_{\rm lens} = \delta\theta_{g[\phi]} + \delta\theta_B
\end{equation}

where \(\delta\theta_{g[\phi]}\) is the deflection from scalar backreaction on the geometric metric \(g_{\mu\nu}\) — the scalar field modifying the Einstein-frame curvature through the coupled Einstein–scalar equations — and \(\delta\theta_B\) is the deflection from the disformal term \(B(\phi)\nabla_\mu\phi\nabla_\nu\phi\), which tilts the matter null cone along the field gradient. On the isotropic-conformal benchmark (\(\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}\)), the conformal factor \(A(\phi)\) does not appear in \(\delta\theta_{\rm lens}\) at leading order because it cancels from the null condition, whereas under the canonical unstretched-space temporal metric (\(\tilde\gamma_{ij}=\gamma_{ij}\) at fixed cosmic epoch \(A_{\rm sp}=1\), §6.9), the local temporal lapse \(A_{\rm loc}(u_{\rm grav})\) supplies the direct refractive deflection \(\delta\theta_{A_{\rm loc}}\) via \(n(\mathbf x)\simeq 1 - 2\Phi_N/c^2 - \beta_A u_{\rm grav}(\mathbf x)\).

The worked backreaction example on the isotropic-conformal benchmark is the scalar-Gauss-Bonnet (sGB) coupling studied in Bahrain (Paper 28, Appendix L). In shift-symmetric sGB, the scalar field \(\phi \sim Q_s/r\) backreacts on the geometric metric at \(\mathcal{O}(\eta^2)\) (where \(\eta = 3\alpha_{\rm GB}/M^2\)), producing a genuine correction to the null-trajectory bending. Under isotropic conformal rescaling \(A = e^{-\phi}\) modifies clock rates at \(\mathcal{O}(\eta)\) — acting on timelike observables such as the ISCO — without contributing to the shadow at leading order because isotropic conformal transformations preserve null cones. Bahrain Appendix L explicitly isolates this sGB backreaction contribution and demonstrates the different coupling orders of the null and timelike observables on that benchmark: the shadow is sensitive to the geometric metric at \(\mathcal{O}(\eta^2)\) while the ISCO feels the conformal factor at \(\mathcal{O}(\eta)\).

#### 3.1.3 The temporal-composite observable

The conformal sector produces observable effects through timelike channels: endpoint clock calibration, frequency transport, and source evolution. These are captured by the temporal composite observable, which relates the observed image intensity to the emission-time shift induced by the temporal field:

\begin{equation} \label{eq:gl_composite_obs}
\delta I_{\rm obs} \simeq -\frac{\partial I_{\rm src}}{\partial t_{\rm em}}\,\delta t_{\rm em}
\end{equation}

where \(\delta t_{\rm em}\) is the differential emission-time shift between paths, sourced by the complete metric \(\tilde{g}_{\mu\nu}\) rather than an unmatched spatial conformal factor. The delay must come from the full disformal metric — including \(g_{\mu\nu}[\phi]\) backreaction, the disformal term \(B\), endpoint clock-rate differences through \(A(\phi)\), and frequency transfer — not from a path integral \(\int A(\phi)\,dl/c\) interpreted as photon proper-time accumulation. Null proper time is zero (Jakarta); any measured open-path effect must be formulated through emission and reception clocks, actual coordinate delay from the geometric and disformal metric, or frequency transfer.

The static clock-transfer contribution from the conformal sector modifies the relation between accumulated matter-frame time and the geometry reconstructed under an isochronous model. The associated effective clock-transfer contrast along a path \(\gamma\) is:

\begin{equation} \label{eq:gl_clock_transfer}
\Delta\tilde{\tau}_{\rm static} = \frac{1}{c} \int_\gamma \bigl(A(\phi) - 1\bigr)\,dl
\end{equation}

This is a clock-transfer discrepancy — a difference between the matter-frame time a timelike congruence accumulates along each path and the time an isochronous GR lens model infers — not a propagated photon delay under isotropic conformal rescaling (\(\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}\)). On the isotropic-conformal benchmark its differential photon content vanishes identically: the null condition is conformally invariant (§3.1.1), and the underlying Temporal Shear is an exact one-form, \(\Sigma_\mu=\nabla_\mu\ln A\), whose open-path integral is endpoint-determined and cancels between any two rays joining the same source and observer; inserting the screening projection \(\mathcal{S}_\Sigma(\mathcal{E})\) inside a photon-path observable is a category error, since photons carry no clocks and do not respond to the shear sector. The dedicated strong-lensing pipeline evaluates the exact additional static isotropic-conformal residual at \(0.0\) d (Paper 19, step_56). On that isotropic benchmark the functional is retained as the chronometric reconstruction map — the field a universal-clock reconstruction would misassign to convergence, mass-sheet normalization, or delay calibration — and it contributes no term to the propagated Fermat surface, while the spatial-disformal multipath deformation is bounded by GW170817 to \(\lesssim 0.2\) s per \(2\) Mpc halo path. By contrast, under the canonical unified unstretched-space temporal metric \(\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi\) (\(U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}\); Paper 0 §2.2, Step 80, evaluated in §6.9 below), where the \((3+1)\) ADM lapse \(N_{\rm TEP}=A_{\rm loc}(u_{\rm grav})N_{\rm GR}\) rescales along the cosmic temporal flow while the spatial 3-metric \(\tilde\gamma_{ij}=\gamma_{ij}\) remains unstretched at fixed epoch, intermediate clock rates \(A_{\rm loc}(u_{\rm grav}(\mathbf x))\) do not cancel against spatial stretching and refract null rays directly via \(n(x)\simeq 1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)\).

#### 3.1.4 The Amplification Matrix and Jacobian Decomposition

Integrating the geodesic deviation equation along the line of sight yields the amplification matrix \(\mathcal{A}_{ij}\), which maps source-plane displacements to image-plane displacements:

\begin{equation} \label{eq:gl_amplification}
\mathcal{A}_{ij} = \frac{\partial \beta_i}{\partial \theta_j} = \delta_{ij} - \int_0^{\chi_s} \frac{(\chi_s - \chi)\chi}{\chi_s}\, \mathcal{R}_{ij}(\chi)\, d\chi
\end{equation}

where \(\mathcal{R}_{ij}\) is the optical tidal matrix constructed from the Riemann tensor of the geometric metric \(g_{\mu\nu}[\phi]\), projected onto the screen space. On the isotropic-conformal benchmark (\(\tilde{g}_{\mu\nu}=A^2(\phi)g_{\mu\nu}\)), the TEP correction to the optical tidal matrix comes from scalar backreaction on \(g_{\mu\nu}\) and from the disformal sector, not from a spatial conformal factor:

\begin{equation} \label{eq:gl_tidal_matrix}
\mathcal{R}_{ij} = \mathcal{R}^{(\Psi)}_{ij} + \mathcal{R}^{(g[\phi])}_{ij} + \mathcal{R}^{(B)}_{ij}
\end{equation}

where \(\mathcal{R}^{(\Psi)}_{ij}\) is the standard GR contribution, \(\mathcal{R}^{(g[\phi])}_{ij}\) is the scalar-backreaction correction to the geometric metric, and \(\mathcal{R}^{(B)}_{ij}\) is the disformal correction. Under isotropic conformal rescaling, the conformal factor does not appear at leading order because it cancels from the null trajectory; under the canonical unstretched-space temporal metric (\(\tilde{\gamma}_{ij}=\gamma_{ij}\) at fixed cosmic epoch, §6.9), by contrast, the local temporal lapse \(A_{\rm loc}(u_{\rm grav})\) adds the direct refractive tidal term \(\mathcal{R}^{(A_{\rm loc})}_{ij} = \beta_A \partial_i\partial_j u_{\rm grav}\).

The magnification of an image is \(\mu = (\det\mathcal{A})^{-1}\). To first order in the TEP correction \(\delta\mathcal{A}_{ij}\), the fractional change in magnification is:

\begin{equation} \label{eq:gl_mag_perturbation}
\frac{\delta\mu}{\mu} = \mathrm{Tr}\left(\mathcal{A}^{-1}_{\rm GR}\,\delta\mathcal{A}\right)
\end{equation}

This defines the lensing amplification kernel \(\mathcal{P}_\mu\), a projection operator that maps the scalar-backreaction and disformal Hessian to the observable magnification shift. Near a critical curve, where \(\det\mathcal{A}_{\rm GR} \to 0\), the kernel diverges as \(\mu_{\rm GR}^2\), amplifying small corrections into large observable residuals. Because the inverse reconstruction is correspondingly sensitive to small timing and geometric perturbations there, this supplies a candidate mechanism capable of closing the factor-of-\(\sim\)90–750 amplitude gap between direct potential-sampling and observed delay shifts identified in Paper 19 (SN Refsdal) — locating that discrepancy in Jacobian amplification rather than in a free phenomenological coefficient. A numerical resolution requires evaluation of the complete tensor and lens-model response, not the scalar log-magnification proxy alone; that calculation is not performed here.

Formally, the operational response proxy used in the blind-prediction tests (Paper 19, §2.3) is the first-order truncation of this kernel:

\begin{equation} \label{eq:gl_response_proxy}
\Gamma_t(i) = 1 + \kappa_{\rm lens}\,\mathcal{P}_\mu\big[\nabla\phi\big](i) \;\approx\; 1 + \kappa_{\rm lens}\log_{10}\!\big(\mu_{\rm norm}(i)\big)
\end{equation}

This is a phenomenological scalar truncation of the tensor response. Weak shear alone does not make it exact; exact reduction additionally requires an approximately isotropic response, \(\delta\mathcal{A}_{ij}\propto\delta_{ij}\). The log-magnification form follows from \(\delta\mu/\mu = \mathrm{Tr}(\mathcal{A}^{-1}\delta\mathcal{A})\) when \(\delta\mathcal{A}\) is proportional to the identity. The mu–kappa–gamma systematic is the residual error from truncating the full tensor kernel to a scalar log-magnification proxy. Eliminating it requires direct evaluation of \(\mathcal{P}_\mu\) from high-resolution mass models, which is the definitive next phase identified in Paper 19.

**Response-coefficient convention.** Following the corpus rule established in TEP-COS (Paper 10) for \(\kappa_{\rm MSP}\), the quantity \(\kappa_{\rm lens}\) is an *observable channel response coefficient*: it measures how strongly the lensing reconstruction responds to Temporal Shear. It is not the bare microscopic conformal coupling \(\beta_A\), not \(A-1\), and not the locally active PPN coupling, and it must not be numerically identified with the amplitudes of other channels — such as \(\epsilon_T^{\rm HC}\) (TEP-HC) or the line-of-sight amplitude of TEP-C0 — unless a solved environmental transfer function establishes that identification. Schematically, the lensing-channel observable takes the form \(\Delta O_{\rm GL} = \kappa_{\rm GL}\,\mathcal{S}_{\rm GL}(\mathcal{E})\,\mathcal{F}_{\rm GL}[\Delta\ln A, \Sigma_\mu, C_A;\ \Phi, \rho, z]\), with the environmental screening operator applied before comparison with any other channel.

**Amplitude dictionary.** The manuscript refers to several distinct quantities that must not be conflated:

- \(\beta_A\): the microscopic conformal parameter (bare Lagrangian quantity).

- \(\Sigma_\mu = \nabla_\mu \ln A(\phi)\): the underlying Temporal Shear.

- \(\Sigma_\mu^{\rm obs} = \mathcal{S}_\Sigma(\mathcal{E})\,\Sigma_\mu\): the observable environmentally projected Temporal Shear.

- \(\langle\Delta A\rangle_\gamma = \int_\gamma \Sigma_\mu^{\rm obs}\,dx^\mu\): the open-path clock-transfer contrast. In the unscreened limit this is the integral of an exact one-form, \(\int_\gamma \nabla_\mu\ln A\,dx^\mu = \ln A_{\rm obs} - \ln A_{\rm em}\), so it is endpoint-determined and its differential between two rays sharing the same endpoints vanishes identically; the environment-projected form is defined on the matter-clock congruence and inherits the same vanishing multipath photon content.

- \(\kappa_{\rm lens}\): the lensing reconstruction response coefficient relating temporal-field structure to inferred convergence.

These quantities are not numerically interchangeable without a solved environmental transfer function: on the isotropic-conformal benchmark the conformal entries act on clock calibration and reconstruction while the propagating delay field is the disformal multipath deformation bounded in Box 3.2, whereas under the canonical unstretched-space temporal metric (§6.9, Step 02; Paper 19, Step 64) the local temporal depth \(-\beta_A u_{\rm grav}(\mathbf{x})\) along the shared ADM lapse directly supplies the static optical Fermat potential and deflection.

#### 3.1.5 Interpretation: Geometric vs. Temporal Contributions

The Jacobian decomposition reveals two physically distinct contributions to image distortion:

| Term | Source | Physical Origin | Observational Signature |
| --- | --- | --- | --- |
| **Geometric** | \(\Psi_{,ij}\) | Spatial curvature from mass | Standard convergence \(\kappa\) and shear \(\gamma\) |
| **Scalar backreaction** | \(g_{\mu\nu}[\phi]\) | Scalar field modifying Einstein-frame curvature | Coherent tangential shear mimicking DM halo |
| **Disformal** | \(B(\phi)\nabla_\mu\phi\nabla_\nu\phi\) | Null-cone tilt along field gradient | Direction-dependent deflection |
| **temporal composite** | \(\mu_s^{\,i}\,\partial_j \Delta T_{\rm eff}\) | Source motion × emission-time-map gradient | Kinematics-correlated shear variance; the residual above the standard delay-map baseline is the TEP-specific content |

The scalar-backreaction and disformal terms produce coherent contributions to the shear field. The temporal composite term arises when the source position evolves during the differential emission-time contrast \(\Delta T_{\rm eff}\) across the image; for a source with proper motion \(\vec{\mu}_s\), the effective source position becomes:

\begin{equation} \label{eq:gl_effective_source}
\beta_{\rm eff}^{\,i}(\vec{\theta}) = \beta_{\rm geom}^{\,i}(\vec{\theta}) - \mu_s^{\,i}\,\Delta T_{\rm eff}(\vec{\theta})
\end{equation}

where \(\mu_s^{\,i}\) is the \(i\)-th component of the source proper motion and \(\Delta T_{\rm eff}\) is the effective emission-time map of §2.5. The corresponding temporal-composite contribution to the Jacobian is:

\begin{equation} \label{eq:gl_tc_jacobian}
\delta\mathcal{A}^{\rm TC}_{ij} = -\mu_s^{\,i}\,\partial_j \Delta T_{\rm eff}
\end{equation}

This adds an asymmetric, source-dependent contribution to the Jacobian that does not average coherently but increases the variance of shear measurements.

**Principle:**

#### Summary: The Phantom Mass Decomposition

The full amplification matrix in TEP is:

\begin{equation} \label{eq:gl_jacobian_decomp}
\mathcal{A}_{ij} = \underbrace{\mathcal{A}^{\text{GR}}_{ij}}_{\text{Baryonic Lensing}} + \underbrace{\mathcal{A}^{(g[\phi],\text{static})}_{ij}}_{\text{"Dark Matter" (Coherent)}} + \underbrace{\mathcal{A}^{(\text{dyn})}_{ij}}_{\text{temporal composite (Stochastic)}}
\end{equation}

Standard analyses attribute the sum of the first two terms to total mass. TEP identifies the second term as the coherent optical component of the deflection ledger and evaluates both metric realizations from the action: under isotropic conformal rescaling (\(\tilde g_{\mu\nu}=A^2 g_{\mu\nu}\), where spatial stretching cancels \(A^2(\phi)\) out of null rays), Einstein-frame scalar stress-energy reaches \(\kappa_\phi\approx4\times10^{-5}\) on the unit-term control (\(\simeq 6.5\times 10^{-6}\) on the selected two-branch sector; Paper 19, Steps 61, 64) and the spatial-disformal term \(B(\partial\phi)^2\) is bounded \(\sim5\times\)–\(70\times\) below the cluster-lensing requirement (Step 63; §6.9), whereas under the canonical unified unstretched-space temporal metric \(\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}\) (§6.9, Step 02; Paper 19, Step 64; Paper 18, Step 21), intermediate temporal depth \(-\beta_A u_{\rm grav}(x)\) along the shared ADM lapse refracts photons via \(n(x)\simeq 1-2\Phi_N/c^2-\beta_A u_{\rm grav}(x)\) and reproduces the Bullet Cluster convergence, SN Refsdal Fermat delay, and Planck CMB lensing power from baryons alone. The third term supplies the source-dependent discriminating channel: shear variance correlated with source kinematics. The correlation's existence is not itself distinctive, since any emission-time map sources the same \(\mu_s\,\partial_j\Delta T\) structure — under particle dark matter and GR a moving source's effective source-plane position is \(\beta-\vec{\mu}_s\Delta T\), and the ordinary Fermat surface alone produces \(\gamma^{\rm std}=\mu_s\,\partial_\theta\Delta T_{\rm Fermat}\sim3\times10^{-8}\) for strong-lens geometries (\(\sim6\times10^{-10}\) at galaxy–galaxy scales; step_01). The discriminating observable is the residual over that computable baseline: on the spatial-disformal carrier it is bounded at \(\gamma^{\rm TC}_{\rm prop}\sim10^{-15}\) — seven orders below the standard baseline and tested directly by the timing residuals of §5.1 — while the reconstruction-space contribution is bound-set rather than measured by the scatter test of §3.4. On the isotropic-conformal benchmark \(A^2(\phi)g_{\mu\nu}\), \(A(\phi)\) acts on timelike observables and clock calibration but generates neither null-trajectory bending nor any inter-image arrival-time residual (exact zero in Paper 19, step_56), whereas on the unstretched-space temporal metric (§6.9) the lapse \(A_{\rm loc}(u_{\rm grav})\) directly refracts null trajectories across unstretched space.

**Principle:**

### Box 3.1: A Minimal Toy Model Estimate (Halo Scale Integration)

To demonstrate the order of magnitude, consider a simple spherical conformal halo profile, applied as a local interior model over the field-bearing extent of the halo:

\begin{equation} \label{eq:gl_halo_profile}
A(\phi) = 1 + \epsilon \ln(r/r_0), \qquad r \lesssim r_h \sim L_{\rm halo}/2
\end{equation}

For a coupling strength \(\epsilon \approx 10^{-6}\) and a characteristic scale \(r_0 = 10\) kpc, the excursion \(|A(\phi)-1|\) remains at the \(10^{-6}\) level throughout the domain (\(|\ln(r/r_0)| \lesssim 5\) for \(r \in [r_E, r_h]\)). The ambient boundary condition is imposed by environmental matching rather than by the toy form: beyond the halo matching radius \(r_h\) the screened field relaxes to the ambient baseline \(A_\infty = 1\), so the asymptotic divergence of the logarithm lies outside the domain of the estimate and is never extrapolated.

- **Integration Path:** The delay is integrated only over the effective halo depth (\(L_{halo} \approx 2\) Mpc), not the full cosmological path. This respects the locality of the potential well.

**Chronometric-Envelope Contrast:** Across an Einstein radius (\(r_E \approx 5\) kpc), the differential clock-transfer field corresponds to:

\begin{equation} \label{eq:gl_diff_delay}
\Delta \tilde{\tau}_{\rm env} \sim \frac{\epsilon}{2} \frac{L_{halo}}{c} \approx \frac{10^{-6}}{2} \cdot (6.5 \times 10^6 \text{ light-years}) \approx 3.2 \text{ years}
\end{equation}

- **Carrier status:** This ~3 year figure is the reconstruction-space scale — the reconstruction-space clock-transfer field of §2.2 and §3.1.3 — not a propagated photon delay under isotropic conformal rescaling (\(\tilde{g}_{\mu\nu}=A^2 g_{\mu\nu}\)). On the isotropic-conformal benchmark the exact additional static-conformal inter-image residual on the same paths is \(0.0\) d (Paper 19, step_56), the propagating differential residual is carried by the disformal multipath deformation at \(\lesssim 0.2\) s per \(2\) Mpc, and scalar backreaction enters the delay surface only as ordinary \(g\)-sector Fermat structure; by contrast, under the canonical unstretched-space temporal metric (\(\tilde{\gamma}_{ij}=\gamma_{ij}\), §6.9; Paper 19, Step 64), intermediate temporal depth \(-\beta_A u_{\rm grav}(\mathbf{x})\) on the selected two-branch sector \(P_{,X}=k\sqrt\xi+2\xi\) supplies the multi-hundred-day static Fermat delay directly (\(+330.9\text{ d}\) sheet and \(+221.6\text{--}443.2\text{ d}\) 3-D NFW ray on SN Refsdal from \(f_b=0.16\) baryons alone). The ~3 year chronometric-envelope figure bounds the post-model residual an isochronous reconstruction could misassign, corresponding to the residual class measured by Paper 19 (\(\lesssim\) tens of days).

- **Clock-transfer reconstruction:** The static gradient \(\nabla(\Delta \tilde{\tau}_{\rm env})\) defines the unmodeled clock-transfer field along existing lens paths. When a GR lens model is required to reproduce a path-dependent timing map while assuming isochrony, such a residual would be absorbed into inferred convergence, mass-sheet normalization, or distance calibration — the reconstruction-space "Phantom Mass" signature. Under isotropic conformal rescaling (\(\tilde{g}_{\mu\nu}=A^2 g_{\mu\nu}\)) it does not refract null geodesics onto new trajectories and does not shift propagated arrival times because isotropic conformal rescaling preserves null cones; under the canonical unstretched-space temporal metric (§6.9, Step 02), however, intermediate temporal depth \(A_{\rm loc}(u_{\rm grav})\) does not cancel against spatial stretching and supplies direct optical convergence and Fermat delay alongside the scalar-backreaction and disformal channels of §3.1.4.

- **Source-motion term (stochastic):** The term \(\vec{\mu}_s \cdot \nabla(\Delta T_{\rm eff})\) couples source proper motion to whichever emission-time map propagates. On the propagating carrier it is the millisecond-to-second disformal contribution, which dominates the differential delay budget for fast transients (the lensed-FRB channel of §5.1); for slowly evolving galaxy sources its observable content is the shear-variance residual above the standard-map baseline, bounded per Box 3.2.

**Profile Dependence Check:** The ~3 year estimate uses a logarithmic profile for simplicity. Realistic dark matter halos follow the NFW profile:

\begin{equation} \label{eq:gl_nfw_profile}
\rho_{NFW}(r) = \frac{\rho_s}{(r/r_s)(1 + r/r_s)^2}
\end{equation}

If the clock-transfer field tracks the gravitational potential (\(A(\phi) - 1 \propto \Psi\)), then \(A(\phi) - 1 \propto \int \rho/r\, dr\), giving:

\begin{equation} \label{eq:gl_nfw_deflection}
(A(\phi)-1)_{\rm NFW}(r) \propto \ln(1 + r/r_s) - \frac{r/r_s}{1 + r/r_s}
\end{equation}

For a cluster with \(r_s \approx 200\) kpc and integration over \(L_{halo} \approx 2\) Mpc:

- The NFW profile concentrates more of the clock-transfer field near the core than the logarithmic profile.

- The differential envelope contrast across an Einstein radius (\(r_E \approx 5\text{--}50\) kpc) is enhanced by a factor of 2–5 relative to the logarithmic estimate.

- Result: \(\Delta\tilde{\tau}_{{\rm env},NFW} \sim 5\text{--}15\) years at the reconstruction-space scale — a reconstruction-scale figure with the carrier status stated above, not an observed-delay prediction.

The order-of-magnitude envelope estimate is robust to profile shape; realistic NFW profiles produce slightly larger chronometric contrasts than the toy logarithmic model.

**Principle:**

### Box 3.2: Order of Magnitude Estimate for stochastic shear

To estimate the magnitude of the *stochastic* shear contribution (the dynamic term \(\mu_s \nabla(\Delta \tilde{\tau})\)), the analysis uses the updated halo-scale delays:

- **Source Velocity:** Typical cluster transverse velocity \(v_s \sim 1000\) km/s at distance \(D_A \sim 1\) Gpc yields an angular proper motion \(\mu_s \approx 2 \times 10^{-7}\) arcsec/year (\(\sim 0.2\,\mu\)as/yr; the milliarcsec-scale figure appearing in earlier drafts reflected a unit slip).

- **Envelope Delay Gradient:** From Box 3.1, a reconstruction-space contrast of \(\sim 3\) years varying over arcsecond scales gives \(\nabla(\Delta \tilde{\tau}_{\rm env}) \sim 3\) years/arcsec at the envelope; the propagating-carrier gradient is \(12\) orders smaller (below).

**Resulting stochastic shear:** The product is dimensionless shear:

\begin{equation} \label{eq:gl_stochastic_shear}
\gamma_{stoch}^{\rm env} \approx (2 \times 10^{-7} \, \text{arcsec/yr}) \times (3 \, \text{yr/arcsec}) \approx 6 \times 10^{-7}
\end{equation}

This stochastic contribution (\(\gamma_{stoch} \sim 10^{-6}\) at the reconstruction-space scale) is small compared to typical weak lensing shear (\(\gamma \sim 0.01\text{--}0.1\)), confirming that the dynamic term is a perturbation (excess scatter), not the dominant signal. The coherent "Dark Matter" halo signature is the combined static temporal-field contribution as registered by the lens model: under the isotropic-conformal benchmark its physical optical-tidal component is carried by scalar backreaction on \(g_{\mu\nu}\) and any bounded disformal term while the conformal sector modifies the associated clock-transfer and inference mapping (the clock-transfer reconstruction), whereas under the canonical unstretched-space temporal metric (§6.9, Step 02) the local temporal lapse \(A_{\rm loc}(u_{\rm grav})\) directly refracts null rays across the unstretched spatial manifold via \(n(\mathbf x)\simeq 1 - 2\Phi_N/c^2 - \beta_A u_{\rm grav}(\mathbf x)\). In neither case does the static halo convergence arise from source motion.

**Two scales, two roles.** The two delay scales of Box 3.3 sit on different carriers, and the applications of Section 5 must be read against the correct one. On the isotropic-conformal benchmark (\(\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}\)), the conformal clock-transfer map contributes zero differential inter-image delay: isotropic conformal rescaling preserves null cones (§3.1.1), so \(A(\phi)\) drops out of the photon transport equation and the open-path integral is endpoint-determined (Paper 19 verifies the vanishing static-conformal residual explicitly), whereas under the canonical unstretched-space temporal metric (\(\tilde\gamma_{ij}=\gamma_{ij}\) at fixed cosmic epoch, §6.9), intermediate temporal depth \(-\beta_A u_{\rm grav}(x)\) supplies the static Fermat delay directly (Paper 19, Step 64). For the spatial-disformal multipath deformation \(B(\phi)\nabla_\mu\phi\nabla_\nu\phi\), the cone-tilt carrier is bounded by GW170817 to \(\lesssim 10^{-15}\) fractional — \(\Delta T \lesssim 0.2\) s over the \(2\) Mpc halo path, the conservative-baseline scale. On that spatial-disformal carrier the same product gives \(\gamma^{\rm TC}_{\rm prop} \sim \mu_s\,\partial_\theta\Delta T \sim (2\times10^{-7}\,{\rm arcsec/yr})(6.5\times10^{-9}\,{\rm yr/arcsec}) \sim 10^{-15}\). The \(6\times10^{-7}\) figure is the *reconstruction-space* envelope value: it quantifies the field a lens model would absorb when the clock-transfer map or unstretched-space Fermat surface is differentiated across a moving source, and it bounds the shear-variance applications of §5. The photon-propagating channel is tested directly and most sharply by the arrival-time residual itself (§5.1).

### 3.2 Connection to Lens-Model Degeneracies

This mechanism parallels the *Source-Position Transformation (SPT)* (Schneider & Sluse 2013), which identifies a degeneracy class of mass models yielding identical observables but differing time delays. TEP extends this degeneracy into the metric sector itself. Rather than permuting the mass distribution \(\kappa(\vec{\theta})\) while holding the metric constant, TEP holds the baryonic mass constant and permutes the spacetime arrival surface \(\Delta \tilde{\tau}(\vec{\theta})\). Consequently, the "Phantom Mass" can be understood as a physical realization of the SPT, where the extra freedom resides in the time-transport sector rather than invisible spatial matter.

### 3.3 The Two Regimes: Parameter Thresholds

TEP phenomenology divides into two distinct regimes, distinguished by the magnitude of the differential clock-transfer residual and the resulting phenomenology:

**Principle:**

#### Box 3.3: Regime Definitions and Parameter Thresholds

| Regime | Clock-transfer scale | Phenomenology |
| --- | --- | --- |
| **conservative baseline** | ms–s (propagating, disformal carrier) | Time-domain residuals; static optical lensing effectively unchanged |
| **Regime II** | \(1\text{--}10\) years (reconstruction space; propagating channel unchanged at ms–s) | Coherent scalar-metric/disformal lensing combined with chronometric reconstruction and source-dependent temporal-composite response |

For a \(2\) Mpc path, the Regime-II envelope contrast is \(\langle\Delta A\rangle_\gamma \sim 10^{-7}\text{--}10^{-6}\), consistent with the halo-scale integration in Box 3.1; per §3.1.3 this contrast is endpoint-determined for photons and is realized in the data only through the reconstruction mapping, while the propagating residual remains the disformal channel.

**Operational Distinction:** The conservative baseline accepts the standard GW170817 translation (arrival-time offset → propagation-speed bound) at face value. Regime II applies if TEP's dynamical-time interpretation is correct, in which case the standard translation may require revision—the observed \(\Delta t = 1.74\) s constrains the *disformal* sector; it does not directly test the *conformal* sector, although conformal scalar sectors remain indirectly constrained by PPN, equivalence-principle, source-screening, and clock-comparison tests (Section 4).

**Empirical Discriminator:** The regime is determined by observation, not assumption. If lensed FRBs show only millisecond residuals, the conservative baseline applies. If strong-lens post-model residuals, reconstruction maps, or variability-correlated structure reveal clock-transfer structure at the envelope scale, Regime II is indicated.

**Independent Regime Criterion (Breaking Circularity):** To avoid circular reasoning, the following observable specifies the regime *independently* of TEP's correctness:

- **The Variability-Mass Correlation Test:** In Regime II, the inferred "dark matter" mass of a lens should correlate with the variability timescale of the background source population. Specifically: lenses observed through rapidly variable sources (AGN, quasars) should show systematically different mass reconstructions than the same lenses observed through slowly evolving sources (elliptical galaxies). This correlation is *forbidden* in standard CDM (mass is source-independent) but *required by the source-dependent temporal-composite component of Regime II*.

- **Decision Rule:** If existing strong-lens catalogs show no statistically significant correlation between inferred lens mass and source variability class at the >3σ level, the source-dependent temporal-composite component is disfavored. If such a correlation exists, it constitutes positive evidence for TEP independent of FRB timing. *Sensitivity floor:* the response enters through \(\delta\kappa \sim \gamma^{\rm TC} = \mu_s\,\partial_\theta\Delta T_{\rm eff}\) (Box 3.2), so a correlation search at the proposed 10% mass precision probes \(\gamma^{\rm TC} \gtrsim 10^{-1}\) — some five orders of magnitude above even the reconstruction-space value — and reaching the propagating carrier through mass reconstruction would require \(\delta M/M \sim 10^{-15}\), far below any achievable precision. A null result at the specified precision therefore bounds the envelope amplitude rather than testing the carrier; the carrier itself is tested by the timing residual of §5.1. Complementarily, the ordinary delay-absorption route already constrains the propagating field: a path-dependent delay \(\Delta T\) absorbed into a lens model shifts the inferred normalization by \(\sim\Delta T/\Delta t_{\rm Fermat}\), so the observed \(\sim 10\%\) closure of strong-lens mass models bounds any propagating residual at \(\lesssim 10\) days per \(100\)-day delay — excluding a propagating year-scale reading outright, consistently with the conformal null of §3.1.1.

- **Current Status:** This test can be performed with existing data (HST strong-lens archives, SDSS quasar lenses vs. galaxy-galaxy lenses) as an envelope-bounding correlation search; its discriminating power on the propagating carrier requires the timing channel of §5.1. This is flagged as a priority observational test.

Under the conservative baseline (Regime I), \(\Delta \tilde{\tau}\) is small (milliseconds to seconds). In this regime:

- **Static Lensing:** The phantom mass effect is negligible for slowly evolving sources (galaxies). Static mass maps are unaffected.

- **Time-Domain Lensing:** The effect is dominant for fast transients. A millisecond gradient across an image plane is huge for an FRB.

**Conclusion:** In the conservative baseline (Regime I), TEP is a precision correction to time-domain astrophysics. In Regime II, TEP offers a conditional geometric reinterpretation in which the component conventionally attributed to dark matter may contain an unmodeled temporal-transport contribution.

### 3.4 The Critical Discriminator: Variability Scatter

Since the dynamic term \(\mu_{s,i} \nabla_j (\Delta \tilde{\tau})\) is randomly oriented (depending on the direction of \(\vec{\mu}_s\)), it does not add to the mean shear profile but contributes to the *variance* of the shear measurement.

**Principle:**

**The Variability Scatter:** The inferred "shear noise" (RMS dispersion of ellipticities) should be higher for source populations with high proper motion or intrinsic variability. While the coherent "dark matter" signal is static, the *scatter* around that signal is dynamic. The amplitude is set by the bookkeeping of Box 3.2: \(\gamma^{\rm TC} \lesssim 6\times10^{-7}\) at the reconstruction-space scale and \(\sim10^{-15}\) on the propagating carrier, so the scatter excess contributes a variance \(\gamma_{\rm TC}^{2}\) that is orders of magnitude below shape noise on either branch. The observable content of the test is therefore a bound on the reconstruction-space scale — a non-detection at a stated scatter precision caps \(\nabla(\Delta\tilde{\tau})\) at that level — while the propagating channel is tested directly by timing residuals (§5.1).

The discriminating quantity is not the correlation's existence — the ordinary delay map produces kinematics-correlated dispersion at the standard level \(\gamma^{\rm std}=\mu_s\,\partial_j\Delta T_{\rm Fermat}\) under particle dark matter as well (step_01: \(\sim3\times10^{-8}\) strong-lens, \(\sim6\times10^{-10}\) galaxy–galaxy) — but the residual above that baseline. Particle dark matter predicts equality with the standard-map value at every source class; a statistically significant excess bounds, and at sufficient precision reveals, a non-standard emission-time field. At achievable shear precisions the reach is the envelope-scale field and its reconstruction-space bound; the propagating disformal carrier sits ~7 orders below the baseline and is accessed by the timing channel of §5.1 rather than by shear.

## 4. Reanalysis of GW170817: What is Actually Constrained?

### 4.1 The Measurement and the Standard Translation

The simultaneous detection of gravitational waves (GW170817) and gamma rays
(GRB 170817A) from a binary neutron star merger at approximately 40 Mpc
(Abbott et al. 2017) represents one of the most precise measurements in
astrophysics. The signals arrived within \(\Delta t_{obs} = 1.74 \pm 0.05\)
seconds of each other after traveling for approximately 130 million years.
This is widely cited as constraining the difference between the speed of
gravity \(c_g\) and the speed of light \(c_{\gamma}\) to (e.g., Baker et al.
2017; Creminelli & Vernizzi 2017; Ezquiaga & Zumalacárregui 2017; Sakstein &
Jain 2017):

Screening in TEP is represented at the theory level by the environmental operator
$S_\Sigma(\mathcal{E})$.
Quantities such as
$\rho_T$,
$R_T(M)$,
$S_\oplus(r)$,
compactness $\Phi/c^2$,
local stellar density,
geometric coherence length,
and channel-specific response coefficients
are domain-specific projections of $\mathcal{E}$,
not independent screening mechanisms
and not interchangeable universal thresholds.
Each is an observational transfer model
that parameterizes the same underlying operator
in a regime-appropriate form.

\begin{equation} \label{eq:gl_gw_speed}
\frac{|c_{\gamma}-c_g|}{c} \lesssim 10^{-15}
\end{equation}

This standard translation assumes that any delay is due to a uniform
difference in propagation speed accumulating linearly over the entire
cosmological distance. This section critically re-examines this assumption
and analyzes what the measurement strictly constrains in a general
two-metric framework.

### 4.2 The Common-Path Invariant: A Differential Measurement

A crucial but often overlooked geometric fact is that both messengers
originated from the *exact same coordinate* in distant space. They
passed through the same host halo, the same intergalactic voids, and the
same Milky Way halo. In the geometric optics limit, they followed the same
spatial trajectory through the scalar field \(\phi(\vec{x})\).

**The Common-Mode Cancellation:**

Because the signals originate from the same coordinate and follow a nearly
identical path, they do not diverge significantly. If the metric coupling
contains a conformal component \(\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu}\),
this factor rescales the common-mode endpoint clock and frequency
calibration for
*both* species identically. Any time dilation caused by passing
through "time-warped regions" of space is experienced by both messengers. If
the universe is "slower" in a specific region due to a scalar potential, it
is slower for both the gravitational wave and the photon.

**Implication for Conformal Coupling (\(A(\phi)\)):**

As established in Section 2, a purely conformal transformation preserves
null cones.
GW170817 imposes no direct constraint on the magnitude of conformal
coupling.
The measurement confirms only that light and gravity share the same causal
structure along a single path; it does not constrain the *rate* at
which they traverse that path relative to other paths in the universe. The
constraint applies only to the *difference* in null cone structures,
not the absolute rate of time flow. The symmetry of this exemption must be
stated with equal precision: the same null-cone invariance that frees the
conformal amplitude from the differential bound also confines its
propagation content to the common-mode endpoint calibration — the conformal
sector contributes exactly zero propagated differential delay between any
two rays (Section 3.1.3; the static-conformal residual vanishes identically,
Paper 19 step_56). The exemption and the inertness are one theorem.

### 4.3 The Operational Reality: Decoupling the Sectors

To resolve the conflict between the GW170817 constraint and dark matter
phenomenology, the two metric sectors must be explicitly distinguished. A
frequent objection concerns the Shapiro delay: if TEP posits significant
scalar potentials to mimic dark matter, should these not introduce
measurable differential delays? The conformal sector can modify the
matter-frame clock-transfer calibration associated with a Shapiro-delay
measurement, while the geometric null-path delay is carried by the
gravitational metric and any backreacted strong-field solution. Same-path
EM–GW comparisons cancel common-mode conformal clock effects.

Since Axiom 1 establishes that photons and gravitational waves traverse the
same null geodesics defined by the geometric metric, they experience an
identical Shapiro delay as they propagate through the gravitational
potential. Consequently, a differential arrival-time measurement like
GW170817 cancels this common-mode geometric delay entirely. The conformal
clock-transfer contribution is also common-mode for same-path EM–GW
propagation and cancels in the differential measurement, so the magnitude of
the conformal potential is not directly constrained by such differential
propagation tests. It remains indirectly constrained by PPN,
equivalence-principle, source-screening, gravitational-redshift, and
clock-comparison tests — and, more bindingly for this paper's mechanism, by
the null-cone invariance itself: the conformal clock-transfer map is a
reconstruction-space field (what a lens model would absorb into convergence,
mass-sheet normalization, or delay calibration), not a propagating delay.
The physically propagating differential residual is the disformal multipath
deformation of Section 3.1.4, bounded by this same measurement to
\(\lesssim 0.2\) s per \(2\) Mpc halo path.

**Principle:**

**Assumption (Sector Decoupling):** The \(10^{-15}\) bound
on the Disformal sector (the "speed of light" vs "speed of gravity") is
accepted. The TEP-GL phenomenology relies on Conformal sector gradients
(the "rate of time"), which this bound does not directly constrain —
and which, by the same null-cone invariance, carry no propagated
differential content at all: their observable role is the
endpoint/common-mode calibration and reconstruction-space clock-transfer
map, bounded by clock-transfer, PPN, redshift, and lensing constraints
after environmental screening is applied, with the propagated sector
carried separately by the GW170817-bounded disformal channel.

This decoupling makes the TEP framework robust against propagation speed
constraints. Consistent with the general disformal relation (Bekenstein
1993), note that the speed of gravitational waves ($c_g$)
constrains the causal structure (the light cone) governed by the disformal
term \(B(\phi)\), while the chronometric component of Phantom Mass arises
from the conformal factor \(A(\phi)\) (the clock rate); coherent optical
lensing additionally involves scalar backreaction and any permitted
disformal response. As long as \(c_g =
c_\gamma\), the rate at which time accumulates along that path is not
determined by EM–GW arrival-time differences. GW170817 strictly constrains
cone tilting, but is insensitive to the common-mode conformal time dilation
that mimics mass.

**Summary of Constraints:**

**Conformal Sector:** Not directly constrained by the
GW170817 differential-propagation bound; independently constrained by
clock, redshift, PPN, equivalence-principle, source-screening and
lensing channels — and, as the binding constraint on the propagation
side, identically zero in propagated differential delay by null-cone
invariance (Section 3.1.3). Its observable content is the
endpoint/common-mode calibration and the reconstruction-space
clock-transfer map.

**Disformal Mean (Monopole):** Tightly constrained by
GW170817 (\(\lesssim 10^{-15}\)).

**Disformal Gradient (Multipole):** Constrained only by the
requirement that the integral of the gradient not exceed the monopole
bound excessively. This permits non-trivial millisecond-scale
differential structure across the image plane, sufficient for the
time-domain signatures predicted in Section 5.

By distinguishing between the *speed of transmission* (directly
constrained by GW170817) and the *rate of proper time accumulation*
(constrained instead by clock, PPN, and redshift channels, and identically
null in propagated differential delay), the conditional
phenomenological viability of TEP as a dark-sector reinterpretation is
established within the screening and amplitude-closure assumptions stated
here — with the propagating discriminator assigned, as in Section 5, to the
disformal multipath channel that GW170817 does bound.

### 4.4 Environmental Screening and Solar System Constraints

A critical quantitative challenge to the TEP framework is the magnitude of
the clock-transfer contrast required to affect cluster lensing
reconstructions. As established in Box 3.1, halo-scale integration over
\(L_{\rm halo} \sim 2\) Mpc gives \(L_{\rm halo}/c \simeq 6.5\times10^6\) yr,
so a path-averaged conformal contrast \(\langle \Delta A\rangle_{\rm path}
\sim 10^{-7}\text{--}10^{-6}\) yields a year-scale reconstruction-space
field — the reconstruction-space quantity of §3.1.3, not a propagated
photon delay. This replaces
the earlier cosmological-integration estimate, which produced unobserved
\(10^3\text{--}10^5\) year gaps and is not the scale required by the Extended
Regime. Even at this reduced contrast, if the same field gradients persisted
unmodified within the Solar System they would violate high-precision
ephemerides constraints (e.g. the Cassini parameter \(\gamma\)).

Screening in TEP is defined at the theory level by the canonical
environmental operator, which acts on the observable Temporal Shear
\(\Sigma_\mu \equiv \nabla_\mu \ln A(\phi)\):

\begin{equation} \label{eq:gl_screening_operator}
\Sigma_\mu^{\rm obs} = \mathcal{S}_\Sigma(\mathcal{E})\,\Sigma_\mu ,
\end{equation}

where the environmental state \(\mathcal{E}\) includes source structure,
scalar gradients, density, compactness, boundary conditions, and coherence
scale. These quantities determine the projection of the underlying temporal
field into an observable clock, dynamical, or lensing response. TEP
screening is defined directly by the environmental projection of Temporal
Shear. No separate screening radius, thin-shell boundary, or imported
modified-gravity mechanism is introduced.

**Principle:**

#### Box 4.1: Canonical Temporal-Gradient Suppression

TEP screening acts directly on the observable Temporal Shear,

\begin{equation}
\Sigma_\mu \equiv \nabla_\mu \ln A(\phi),
\end{equation}

through the canonical environmental operator:

\begin{equation}
\Sigma_\mu^{\rm obs} = \mathcal{S}_\Sigma(\mathcal{E})\,\Sigma_\mu,
\qquad 0 \leq \mathcal{S}_\Sigma(\mathcal{E}) \leq 1.
\end{equation}

The environmental state \(\mathcal{E}\) includes temporal-gradient
strength, density, compactness, source structure, boundary conditions,
and coherence scale. These quantities determine the projection of the
underlying temporal field into an observable clock, dynamical, or
lensing response.

In Solar-System conditions,

\begin{equation}
\mathcal{S}_\Sigma(\mathcal{E}_\odot)\,|\Sigma_\mu|
\ll |\Sigma_\mu|,
\end{equation}

suppressing the locally observable response and preserving precision
tests of GR. Across extended halo and intergalactic paths,

\begin{equation}
\int_\gamma \mathcal{S}_\Sigma(\mathcal{E})\,\Sigma_\mu\,dx^\mu
\neq 0,
\end{equation}

allowing a cumulative open-path clock-transfer contrast even when the
local response is strongly suppressed. The required hierarchy is
therefore a hierarchy in the environmental projection of Temporal
Shear.

The saturation density \(\rho_T\) may be used as an environmental
proximity indicator within \(\mathcal{E}\), but it is not a hard
universal threshold. The fundamental screening object remains
\(\mathcal{S}_\Sigma(\mathcal{E})\).

## 5. Observational Predictions: The Era of Chronometric Mapping

The transition from a geometric to a dynamical-time framework shifts the observational focus from static shapes to dynamic arrival times. The "dark sector" is predicted to reveal its true nature not in deep fields, but in high-cadence time-domain surveys. Explicit, falsifiable predictions are presented, organizing them by scale: from the millisecond "jitter" of fast transients (FRBs), to the statistical bias in variable sources, to the global tension between CMB and galaxy lensing.

### 5.1 The "Jittering" of Lensed Transients

**Prediction:** Strongly lensed fast transients (FRBs, GRBs) will exhibit achromatic *differential* arrival-time residuals between images that cannot be explained by geometric time delays (Refsdal 1964) or plasma dispersion.

In standard GR, the time delay \(\Delta t_{geom}\) between images is fixed by the mass distribution. In TEP, there is an additional propagated arrival-time residual \(\Delta \tilde{\tau}\), carried by the disformal multipath deformation — bounded by GW170817 to \(\lesssim 0.2\) s per \(2\) Mpc halo path — since the conformal sector contributes exactly zero propagated inter-image delay (§3.1.3; Paper 19, step_56). Because \(\phi\) fields in halos may have substructure (or "weather"), this residual varies across the image plane. For a millisecond-duration FRB (e.g., Muñoz et al. 2016), even a tiny gradient in \(\Delta \tilde{\tau}\) will manifest as a timing anomaly in the *relative* arrival times of the images. Unlike plasma dispersion (which scales as \(\nu^{-2}\)), this differential delay is *achromatic* (frequency-independent). The residual is expected to depend on integrated path depth and intervening temporal-field structure, and may therefore correlate with observed redshift; in TEP, redshift here is an observational clock-distance label, not a measure of physical spatial expansion. The anomaly appears as an "Excess Delay vs Redshift" rather than a frequency-dependent sweep. This distinguishes it from "excess Dispersion Measure" (DM), allowing TEP effects to be isolated from plasma effects via multi-frequency observation.

**Target Candidates:** Recent literature has identified specific anomalies suitable for this test. The repeating source *FRB 20190520B* exhibits a "Dispersion Measure Excess" (\(\sim 900\) pc cm\(^{-3}\)) relative to its redshift (Koch Ocker et al. 2022), currently attributed to extreme host density. TEP predicts this excess may partially conceal an achromatic temporal delay. Additionally, *FRB 20190308C* (Chang et al. 2024) has been identified as a lensed candidate in the CHIME catalog; any discrepancy between its mass-model time delay and observed delay would constitute direct evidence of the non-geometric temporal shear \(\Delta \tilde{\tau}\).

**Test Protocol.** No securely multiply-imaged FRB with a published independent lens model is available at the time of writing, so no lensed-FRB residual is claimed here as evidence. The prediction is instead stated as an identifiable measurement, using the blind-prediction residual of Axiom 3 (Section 2.3):

\[
R_{ij} = \Delta t_{ij}^{\rm obs} - \Delta t_{ij}^{\rm GR},
\]

where \(\Delta t_{ij}^{\rm GR}\) must be frozen from an independent lens model before the timing comparison is performed. Each element below is required for the test to be diagnostic; a measurement missing any one of them cannot discriminate TEP from astrophysical delay structure.

| Required element | Specification |
| --- | --- |
| **Secure multiple imaging** | Consistent sub-arcsecond localization and repeated burst morphology across images |
| **Independent lens model** | Frozen, pre-registered prediction \(\Delta t_{ij}^{\rm GR}\) with propagated model uncertainty |
| **Observed delay** | \(\Delta t_{ij}^{\rm obs}\) at high time resolution, corrected for plasma dispersion |
| **Residual** | \(R_{ij}\), with plasma, microlensing, and host-scattering budgets subtracted (Section 2.3) |
| **Discriminator** | Achromatic residual with non-zero ensemble mean, correlated with path environment |

*Table 5.1: Required elements of a diagnostic lensed-FRB timing test. The achromaticity requirement separates a temporal-transport residual from plasma dispersion (\(\propto \nu^{-2}\)); the non-zero ensemble mean separates it from microlensing, which averages to zero with RMS scaling as \(1/\sqrt{N}\).*

### 5.2 The Variability Scatter Relation

**Prediction:** The *dispersion* (scatter) of weak-lensing shear measurements should correlate with the variability/kinematics of the background source population.

Whereas on the isotropic-conformal benchmark (\(\tilde{g}_{\mu\nu}=A^2 g_{\mu\nu}\)) scalar backreaction and spatial-disformal configurations fall short of a CDM-free coherent halo-lensing amplitude, under the canonical unstretched-space temporal metric (\(\tilde{\gamma}_{ij}=\gamma_{ij}\), §6.9 Step 02; Paper 18 Step 21; Paper 19 Step 64) intermediate temporal depth \(-\beta_A u_{\rm grav}(\mathbf{x})\) on the selected two-branch action supplies the coherent static convergence and Fermat potential directly from baryons alone. The source-evolution response is a distinct, bounded reconstruction component. The secondary "stochastic shear" term depends on source proper motion \(\vec{\mu}_s\). Because \(\vec{\mu}_s\) is randomly oriented, this term adds a random vector to the shear signal. TEP predicts that if one constructs a shear map using highly variable or fast-moving sources, the shear RMS will be systematically higher than for static sources, even if the mean profile (the halo) is identical — an excess over the standard-map baseline, which already predicts the same ordering. The predicted excess is bounded by the Box 3.2 ledger values (\(\gamma^{\rm TC} \lesssim 6\times10^{-7}\) at the reconstruction-space scale, \(\sim10^{-15}\) on the propagating carrier), so the test operates as a precision bound on the envelope amplitude at whatever scatter precision is achieved (Box 5.1). Because the ordinary Fermat surface generates the same correlation at \(\gamma^{\rm std}\sim3\times10^{-8}\) for strong-lens geometries (\(\sim6\times10^{-10}\) galaxy–galaxy; step_01), the discriminating content is the dispersion residual over that standard-map baseline, not the correlation's existence.

**Principle:**

#### Box 5.2: The Einstein Cross Test (Existing Data)

The Einstein Cross (Q2237+0305) provides a unique opportunity to test TEP with *existing* archival data. This quadruply-imaged quasar has:

- **High variability:** The background quasar shows significant optical variability on timescales of weeks to months.

- **Anomalous flux ratios:** The observed image flux ratios deviate from smooth lens model predictions, typically attributed to microlensing by stars in the lens galaxy.

- **Extensive monitoring:** Decades of photometric data exist (OGLE, Gaia, HST).

**Amplitude estimate.** Two response routes connect the temporal field to the observed flux ratios, and the bookkeeping of Box 3.2 applies to both. The Jacobian route gives \(\delta\mu/\mu \sim \mu\,\gamma^{\rm TC}\): at magnification \(\mu \sim 10\) this is \(\sim 6\times10^{-6}\) at the reconstruction-space scale and \(\sim10^{-14}\) on the propagating carrier — four orders of magnitude below the observed 10–30% anomalies even at the envelope. The emission-time route gives \(\delta F/F \sim \dot{\nu}\,\delta t_{\rm em}\): for a quasar varying fractionally at \(\dot{\nu} \sim 10\%\) per month, the propagating-carrier delay (\(\lesssim 0.2\) s) contributes \(\delta F/F \sim 10^{-8}\), while a year-scale propagating reading of the clock-transfer map would produce order-unity phase-correlated flux swings that are not observed. The Einstein Cross anomalies therefore remain in the stellar-microlensing regime and are not claimed as a TEP signal at any admissible amplitude.

**TEP Prediction:** The discriminating content of this test is the correlation structure, not the anomaly amplitude. If a residual emission-time contrast \(\delta t_{\rm em}\) exists between the four sightlines, the apparent flux-ratio anomaly amplitude scales with the source's instantaneous variability rate:

- During periods of rapid quasar variability, the temporal-composite contribution to flux ratios is *larger* (\(\delta F/F \propto \dot{\nu}\,\delta t_{\rm em}\)).

- During quiescent periods, it regresses toward the smooth-lens prediction.

- Unlike stellar microlensing, which is uncorrelated with the source's intrinsic state, the temporal contribution is coherent with the quasar's own variability phases—effectively turning "on" during violent source activity.

**Status:** This test can be performed immediately using archival OGLE light curves cross-correlated with flux ratio measurements, and it functions quantitatively as a bound: at \(\sim 1\%\) photometric precision and \(\dot{\nu} \sim 10\%\) per month, the absence of a variability-phase-correlated residual bounds any coherent emission-time contrast at \(\delta t_{\rm em} \lesssim\) a few days — sitting between the millisecond propagating carrier and the excluded year-scale propagating reading. Detection of the conservative-baseline carrier itself requires the arrival-time residual protocol of §5.1. This is flagged as a priority archival analysis.

### 5.3 The CMB-Galaxy Lensing Tension

**Prediction (Regime II):** TEP-GL predicts that galaxy weak-lensing measurements can contain a source-dependent covariance contribution absent from CMB lensing, even when their mean inferred amplitudes are statistically consistent.

While the CMB is effectively a static backlight (zero intrinsic evolution), galaxy sources are dynamic population. Under TEP, the stochastic component of the clock-transfer signal (driven by source proper motion, see Section 5.2) introduces an additional "kinematic noise" to galaxy shear measurements. Because the CMB is static, it is immune to this effect. Consequently, TEP predicts that precision cosmology inferred from galaxy weak lensing carries an unmodeled source-dependent covariance term absent from CMB lensing. The quantitative reach of that term is set by Box 3.2: its variance contribution \(\gamma_{\rm TC}^{2} \sim 4\times10^{-13}\) at the reconstruction-space scale (\(\sim2\times10^{-30}\) on the propagating carrier) is far below the shape-noise variance \(\sim 0.09\), so it cannot supply the percent-level amplitude of the observed \(S_8\) tension. The prediction is the existence and source-class structure of the covariance term, not an explanation of the tension's magnitude.

**Principle:**

#### Box 5.3: Source-Class-Dependent Shear Covariance

**Observed Situation:** Current weak-lensing results are survey-dependent: DES Y6 continues to favour a lower \(S_8\) than the combined CMB determination, whereas KiDS-Legacy is statistically consistent with Planck. TEP-GL therefore does not identify a universal mean \(S_8\) offset as its prediction; its distinctive prediction is an additional source-dependent covariance that should track source temporal structure and survey selection.

- *Planck* CMB (2018): \(S_8 = 0.834 \pm 0.016\)

- DES Y6 (2025): \(S_8 = 0.789 \pm 0.012\), \(\sim 2.6\sigma\) from combined Planck+ACT+SPT

- KiDS-Legacy (2025): \(S_8 = 0.815^{+0.016}_{-0.021}\), \(0.73\sigma\) from Planck

**TEP Interpretation:** Previous iterations of TEP considered secular source evolution as a driver for coherent shear offsets, but the magnitude (\(\sim 10^{-6}\)) is too small to explain the 5% tension. The stochastic-shear covariance term is likewise bounded: \(\gamma_{\rm TC}^{2}/\sigma_{\rm sh}^{2} \sim 4\times10^{-12}\) relative to shape noise at the reconstruction-space scale, so this channel does not resolve the \(S_8\) tension; its role is diagnostic — a source-dependent covariance signature whose presence, not its magnitude, is the observable.

**Mechanism:** Standard maximum-likelihood estimators weight data by the inverse covariance (\(C^{-1}\)). Standard analyses assume shear noise is dominated by random galaxy orientations ("shape noise"). If the covariance matrix omits a source-dependent kinematic term (\(\sigma^2_{\mu}\)), the relative weighting of high-variance regions such as cluster outskirts is misspecified, and the recovered clustering amplitude is biased.

- **Source-Dependent:** Correlated with galaxy type and redshift (via proper motion).

- **Unmodeled:** Not present in standard covariance matrices.

**Result:** Unmodeled excess variance in the likelihood analysis acts as a potential source of systematic bias. However, a zero-mean covariance term does not by itself determine the sign of the inferred \(S_8\) shift; establishing the direction requires an explicit estimator-level calculation for each survey pipeline, which is not carried out here. The CMB, being static, carries no such kinematic covariance term.

**Prediction:** The falsifiable TEP-GL prediction is not a specific \(S_8\) shift but the existence of an additional source-dependent shear covariance: after standard shape-noise controls, residual shear covariance should vary with source variability, proper motion, or temporal structure. The cosmological mean-growth prediction belongs to the TEP-HC (Paper 18) perturbation closure; the GL-specific discriminator is this source-class dependence. Whether it contributes to the observed \(S_8\) tension, and in which direction, depends on the survey estimator.

### 5.4 Comparison of Predictions: TEP-GL vs. Particle Dark Matter

The following table summarizes the distinguishing predictions of the two frameworks:

| Observable | Particle DM Prediction | TEP-GL Prediction |
| --- | --- | --- |
| **Lensed FRB timing** | GR lens delays from baryons, particle dark matter and substructure; no TEP-specific residual correlated with temporal-field environment after lens-model controls | Achromatic *residual* anomaly (ms-scale) |
| **Source-dependent Shear** | No TEP-specific source-temporal correlation after intrinsic-alignment, selection and measurement-systematic controls | **Excess Scatter** (noise scales with \(\mu_s\); bounded amplitude per Box 3.2) |
| **Source variability–lens mass correlation** | None (mass is source-independent) | Source-class-dependent reconstruction residual correlated with source temporal structure; amplitude determined by \(\kappa_{\rm lens}\) and the environmental transfer function |
| **CMB lensing** | Standard convergence | Source-dependent temporal-composite term absent at leading order; under the unstretched-space temporal metric on the selected two-branch action, intermediate temporal depth boosts baryon-only cosmic-web $C_\ell^{\phi\phi}$ by $9.4\times$–$129\times$, recovering the Planck lensing amplitude without CDM (Paper 18, Step 21) |
| **Galaxy weak lensing** | Source-independent shear consistent with the physical mass distribution; no TEP-specific source-temporal covariance | Isothermal-halo convergence $\kappa(b)\propto 1/b$ from unstretched-space temporal depth on the low-gradient $k\sqrt\xi$ branch, plus bounded source-class-dependent temporal-composite covariance for evolving sources |
| **Chromatic dependence** | None | None (both achromatic) |
| **Direct detection experiments** | Expected signal (WIMP recoil accounting for lensing/rotation/CMB) | No particulate signal; dynamical Temporal Shear and unstretched-space temporal refraction account for rotation curves, wide binaries, cluster lensing, and CMB lensing from baryons and the selected two-branch scalar action |

A distinctive TEP-GL discriminator is the additional source-dependent component of the inferred lensing signal above the standard delay-map baseline. The coherent scalar-metric contribution is source-independent, while the temporal-composite contribution predicts a residual correlated with source variability, proper motion, or temporal structure beyond what the ordinary Fermat surface supplies (\(\gamma^{\rm std}\), step_01).

### 5.5 Falsification Criteria

TEP-GL is a falsifiable hypothesis. The following observations would exclude specific regimes or the framework entirely:

- **conservative baseline Exclusion:** If precision timing of strongly lensed FRBs yields achromatic residuals consistent with zero to better than 0.1 ms across diverse lens environments, the conservative baseline parameter space is excluded.

- **Universal Shear Scatter:** If the intrinsic scatter of weak-lensing shear measurements, after correcting for measurement noise, equals the standard delay-map prediction \(\gamma^{\rm std}=\mu_s\,\partial_j\Delta T_{\rm Fermat}\) across all source kinematic classes — i.e., no residual above the baseline — the stochastic temporal shear mechanism is excluded at that precision.

- **CMB-Galaxy Agreement:** CMB–galaxy agreement below 1%, after source and survey controls, excludes a temporal-composite contribution above that level. CMB lensing directly constrains the physical deflection potential, independent of galaxy-source variability. A CDM-free TEP model that cannot reproduce this potential fails the proposed dark-sector replacement, even if its dynamical phantom-mass channel survives as a separate response. Persistent lensing–dynamics agreement in low-acceleration systems also bounds the shear share of the dynamical anomaly.

- **Phantom Mass (decisive):** In isolated, low-acceleration systems where \(S_\Sigma\to1\), the mass inferred from dynamics exceeds the single-pass weak-lensing mass — by the full shear share on the isotropic-conformal benchmark (\(g_{\rm lens}^{\rm iso}\simeq g_{\rm bar}\)), and by the factor-of-two scalar-potential weight under the canonical unstretched-space metric (\(\Phi_{\rm dyn}=\Phi_N+c^2\beta_A u_{\rm grav}\) vs \(\Phi_{\rm lens}=\Phi_N+\tfrac12 c^2\beta_A u_{\rm grav}\), so \(M_{\rm lens}^{\rm u.s.}/M_{\rm dyn}=(1+\beta_A^2 y_{\rm eff})/(1+2\beta_A^2 y_{\rm eff})\); Paper 0 Eq. 63). Environment- and acceleration-resolved comparison of rotation-curve and weak-lensing mass on the same isolated systems is a sharp dark-sector test: detection of a systematic signed dynamical–lensing difference at matched radii after marginalizing over anisotropy, projection, and external-convergence uncertainties—the differential phantom mass—is a distinctive positive signature not generically expected from a single shared collisionless mass distribution; its persistent absence on isolated single-pass halos constrains the single-pass versus multi-component weighting of the unstretched-space optical response (§6.9).

- **Chromatic Anomaly:** If any timing or morphological anomaly shows wavelength dependence after correction for plasma dispersion and dust, the achromatic prediction of two-metric coupling is falsified. This would indicate conventional astrophysical systematics rather than metric effects.

- **Direct Detection:** A confirmed detection of dark matter particles that quantitatively accounts for the relevant lensing, rotation-curve, and CMB phenomenology would falsify the strong TEP dark-sector ontology. Detection of an exotic particle that does not account for these observations would not by itself falsify TEP; the particle would be an additional component, and TEP would then be tested by whether temporal-transport geometry still predicts residuals unaccounted for by the particle model.

Joint satisfaction of these null thresholds excludes the distinctive chronometric and temporal-composite predictions at the amplitudes specified here. The coherent scalar-backreaction sector is separately tested by CMB and galaxy lensing consistency against its frozen HC/GL prediction. The framework does not claim immunity from observation; the claim is that the observations required to test TEP have not yet been performed with sufficient precision.

**Principle:**

#### Box 5.1: Quantitative Falsification Thresholds

To ensure rigorous falsifiability, explicit null-result thresholds are specified for each prediction channel:

| Test | Sample Size | Precision Required | Null-Result Threshold | Regime Falsified |
| --- | --- | --- | --- | --- |
| **Lensed FRB Timing** | ≥10 lensed FRBs | <0.1 ms timing | 95% upper bound on ensemble achromatic residual amplitude below 0.1 ms | conservative baseline |
| **Shear-Variability Correlation** | ≥1000 lenses per class | <0.5% shear scatter | No correlation at >3σ | temporal-composite dynamic channel |
| **CMB-Galaxy \(S_8\) Tension** | CMB-S4 + LSST | <1% \(S_8\) agreement | Statistically consistent agreement (< 1\(\sigma\)) | temporal-composite contribution above 1% |
| **Variability-Mass Correlation** | ≥100 lenses with varied sources | 10% mass precision | No correlation at >3σ | Source-dependent reconstruction channel |

**Amplitude probed by each threshold.** The thresholds act on different carriers (Box 3.2), and the table is populated accordingly. The lensed-FRB row acts on the propagating delay field directly: a 0.1 ms ensemble bound excludes the disformal multipath carrier at three orders of magnitude below the GW170817-saturating level (\(\sim0.2\) s per \(2\) Mpc halo path) — equivalently, it bounds the disformal normalization \(D(\partial_{\hat{n}}\phi)^2\) to \(\sim 2\times 10^{-18}\) on halo paths, the flagship, fully powered test of the same \(B(\phi)\) instrument the holonomy program probes on the odd-parity channel. The shear-variability row bounds \(\gamma^{\rm TC} \lesssim 5\times10^{-3}\), equivalent to a delay-gradient bound \(\nabla(\Delta\tilde{\tau}) \lesssim 2\times10^{4}\) yr/arcsec — a weak upper limit on the reconstruction-space scale, not a test of the carrier. The \(S_8\) row bounds fractional source-dependent covariance at the percent level; the temporal-composite term sits at \(\sim4\times10^{-12}\) relative to shape noise even at the envelope, so this row's content is the structural existence test rather than an amplitude reach. The variability–mass row probes \(\delta\kappa \sim \gamma^{\rm TC} \gtrsim 10^{-1}\) directly, and through the delay-absorption route bounds any propagating residual at \(\lesssim 10\%\) of the Fermat delay (\(\sim10\) d per 100 d) — already excluding a propagating year-scale reading, consistent with the conformal null of §3.1.1. Together the rows bound the propagating delay field at every scale between the millisecond and the tens-of-days level, with the chronometric sector confined to the reconstruction space where the Paper-19 residual program operates.

**Joint Channel Exclusion:** If all four null-result thresholds are met simultaneously, the distinctive chronometric and temporal-composite predictions of TEP-GL are excluded at the amplitudes specified here. The coherent scalar-backreaction sector remains independently testable through its frozen HC/GL prediction for CMB and galaxy lensing. TEP-GL as a complete lensing realization is excluded only if both the time-domain/source-dependent channels and the independently specified coherent optical prediction are ruled out.

**Timeline:** Tests (1) and (4) are achievable within 5–10 years (CHIME, DSA-2000, Rubin/LSST). Tests (2) and (3) require next-generation surveys (CMB-S4, Euclid) with ~10-year horizons. A definitive verdict on TEP-GL is therefore expected by ~2035.

## 6. Discussion: The Dark Sector — Phantom Mass in Dynamics and Lensing

### 6.1 Limitations of the "Stack of Corrections"

Gravitational physics has historically advanced by refining the fundamental
geometric framework rather than adding ad-hoc corrections. TEP represents a
similar upgrade to the fundamental object, following the natural historical
progression of physical theory:

- **Newton:** Gravity is a Force; Time is Absolute.

**Einstein:** Gravity is Geometry; Time is Relative
(Coordinate-Dependent).

**TEP:** Gravity is Geometry; Time is a
*Dynamical Field*.

By treating the rate of proper time accumulation as a physical field with
its own degrees of freedom (rather than a fixed function of the metric), TEP
unifies the "stack" into a single framework. The "dark matter" anomaly is
the observation that this field has spatial gradients.

### 6.2 CMB Lensing and the Integrated Sachs-Wolfe Constraint

A scalar field capable of generating year-scale reconstruction-space
contrasts on
cluster scales raises a critical question: what are the implications for the
Cosmic Microwave Background (CMB)? Two potential tensions must be addressed.

#### 6.2.1 The ISW Effect

The Integrated Sachs-Wolfe (ISW) effect arises when CMB photons traverse
time-evolving gravitational potentials. In TEP, the conformal factor
\(A(\phi)\) contributes an additional term to the photon temperature
perturbation:

\begin{equation} \label{eq:gl_isw}
\frac{\Delta T}{T}\bigg|_{\text{TEP}} = \int \frac{\partial \ln
A(\phi)}{\partial t}\, dt
\end{equation}

For TEP to remain consistent with *Planck* constraints on the ISW
amplitude, one of the following must hold:

**Quasi-Static Regime:** The scalar field evolves slowly
(\(\partial_t A \ll H_0 A\)), so the TEP contribution to ISW is
subdominant to the standard \(\dot{\Psi}\) term. This is natural if
\(\phi\) tracks the matter distribution adiabatically.

**Cancellation:** In some scalar-tensor theories, the
conformal ISW contribution partially cancels the metric ISW, leaving the
net effect within observational bounds (cf. Amendola et al. 2008).

**Principle:**

#### Box 6.1: Order-of-Magnitude ISW Consistency Check

Conditional on the HC tracking closure in which \(\phi\) follows the
matter distribution adiabatically, the resulting order-of-magnitude
estimate satisfies the quasi-static ISW bound.

**Setup:** The conformal factor is \(A(\phi) = e^{-u}
\approx 1 - u\), with the field excursion \(u \sim 10^{-6}\) on halo
scales (\(u\) is the canonical dimensionless field of Paper 0; the symbol
\(\alpha(\phi) \equiv d\ln A/d\phi = \beta_A\) is reserved for the
coupling function). The following
estimate uses the conventional reference-background variables employed by
the HC linear-transfer representation (Paper 18). Here \(H_{\rm ref}(z)\)
is an observational/reference inverse-time scale and \(a_{\rm eff}=
A_{\rm clock}\); neither denotes physical expansion of the underlying
spatial geometry. The ISW constraint requires:

\begin{equation} \label{eq:gl_isw_constraint}
\left|\frac{\partial \ln A}{\partial t}\right| =
|\dot{u}| \ll
H_0
\end{equation}

**Estimate of \(\dot{u}\):** If \(\phi\) tracks the
matter distribution adiabatically (sourced by \(T^\mu_\mu\)), then
\(u\) evolves on the timescale of structure formation:

\begin{equation} \label{eq:gl_u_dot}
\dot{u} \sim u \cdot \frac{\dot{\rho}}{\rho} \sim u
\cdot H_{\rm ref}(z) \cdot f_{\rm ref}(z)
\end{equation}

where \(f_{\rm ref}(z) \equiv d\ln D/d\ln a_{\rm eff} \approx
\Omega_m(z)^{0.55}\) is the growth rate and \(a_{\rm eff}=A_{\rm clock}\).
At \(z \sim 0.5\) (peak ISW sensitivity), \(f_{\rm ref} \approx 0.8\)
and \(H_{\rm ref}(z) \approx 1.2 H_0\).

**Result:**

\begin{equation} \label{eq:gl_isw_result}
\frac{|\dot{u}|}{H_0} \sim u \cdot f_{\rm ref} \cdot \frac{H_{\rm ref}(z)}{H_0}
\sim 10^{-6} \times 0.8 \times 1.2 \approx 10^{-6}
\end{equation}

This is six orders of magnitude below the ISW
constraint threshold (\(\lesssim 0.1\)). The quasi-static approximation
is therefore satisfied under this tracking closure; the result should
not be interpreted as independent derivation of that closure within
TEP-GL.

**Physical Interpretation:** The scalar field \(\phi\)
varies spatially (creating "dark matter" halos) but evolves temporally
only as fast as the underlying matter distribution. Year-scale
*spatial* delays across an image plane are compatible with
cosmologically slow *temporal* evolution because the delay
gradient is set by the static \(\nabla u\), not by \(\dot{u}\).

#### 6.2.2 CMB Lensing Power Spectrum

The CMB lensing convergence power spectrum \(C_\ell^{\kappa\kappa}\) is
measured to high precision by *Planck* and ACT. In standard
\(\Lambda\)CDM, this is sourced entirely by the integrated mass
distribution. Consistent with the optical-tidal decomposition of §3.1.4, the
physical angular remapping in TEP is carried by the baryonic term, the
scalar-backreaction correction to the Einstein-frame geometry, and any
permitted disformal contribution:

\begin{equation} \label{eq:gl_kappa_tep}
\kappa_{\text{TEP}}^{\rm phys}
= \kappa_{\text{baryon}}
+ \delta\kappa_{g[\phi]}
+ \delta\kappa_{B}
\end{equation}

The conformal sector may additionally enter the *inferred*
convergence through the chronometric reconstruction mapping,

\begin{equation} \label{eq:gl_kappa_inferred}
\kappa_{\text{TEP}}^{\rm inferred}
= \kappa_{\text{TEP}}^{\rm phys}
+ \delta\kappa_{A,\,\rm recon},
\end{equation}

on the isotropic-conformal benchmark (\(\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}\), where isotropic conformal rescaling preserves null cones, Axiom 1), whereas under the canonical unstretched-space temporal metric (§6.9) the local lapse \(A_{\rm loc}(u_{\rm grav})\) refracts the CMB rays directly across unstretched space.

For the CMB (a static, non-evolving source), the temporal-composite response
term vanishes identically—there is no source proper motion or intrinsic
variability on lensing timescales to couple to the delay gradient. Removing
that term does *not* make the CMB insensitive to TEP: it means CMB
lensing constrains the coherent scalar-metric perturbation sector rather than
source-evolution smearing.

**Implication:** CMB lensing tests the source-independent
optical potential, not the temporal-composite response of a variable
galaxy. While the baseline `hi_class` implementation in Paper 18
uses isotropic conformal rescaling $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$
(which cancels out of null rays and therefore retains a standard CDM fluid
$\Omega_{\rm cdm}h^2=0.1133\pm0.0061$ in the active-perturbation posterior, $0.1148\pm0.0042$ in the full background+lensing chain, leaving $8.72\%$ of $C_{100}^{\phi\phi}$
when `omega_cdm` is removed), Paper 18 Step 21 demonstrates that
under the unstretched-space temporal metric
$\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) on the selected
two-branch action $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$), intermediate
temporal depth along the photon path amplifies baryon-only cosmic-web and
cluster-outskirt potentials by $M=3.07$–$11.37$, boosting $C_\ell^{\phi\phi}$
by $9.4\times$–$129\times$ and recovering the Planck lensing power scale
without a CDM fluid.

#### 6.2.3 Parameter Space Constraints

Combining ISW and CMB lensing constraints with the continuous gradient-suppression
screening framework (Box 4.1) defines a narrow but viable parameter window:

| Constraint | Requirement | Status |
| --- | --- | --- |
| Solar System (Cassini) | \(\mathcal{S}_\Sigma(\mathcal{E}_\odot)\,|\Sigma| \ll |\Sigma|\) | Satisfied via quadratic-branch kinetic screening (\(y\sim(r/r_*)^{4/3}\ll10^{-5}\); Paper 0, Step 57) |
| Cluster Lensing | Optical-sector requirement: \(\kappa\sim10^{-1}\) at cluster arcs and subcluster galaxies | **Closed under unstretched-space temporal refraction:** while isotropic-conformal stress-energy is Planck-suppressed (\(\kappa_\phi\simeq6.5\times10^{-6}\); Paper 19, Step 64) and spatial-disformal \(B(\partial\phi)^2\) is bounded \(\sim5\times\)–\(70\times\) short, the unstretched-space temporal metric on the selected two-branch action reproduces the SN Refsdal Fermat delay (\(+330.9\) d sheet, \(+221.6\)–\(443.2\) d 3-D ray; Paper 19, Step 64) and the Bullet Cluster galaxy-peak convergence (\(\kappa_{\rm tot}=0.307\)–\(0.372\), no moat; §6.9, Step 02) from baryons alone |
| ISW (*Planck*) | \(\partial_t \ln A / H_0 \lesssim 0.1\) | Satisfied on the kinetically screened \(w_c\gtrsim700\) closure (Paper 18) |
| CMB Lensing | Source-independent lensing spectrum \(C_\ell^{\phi\phi}\) from baryons and the temporal field | **Closed at the cosmic-web scale (Paper 18, Step 21):** unstretched-space temporal refraction on the selected two-branch action boosts baryon-only \(C_\ell^{\phi\phi}\) by \(9.4\times\)–\(129\times\) across \(R=1\)–\(10\) Mpc filaments and cluster outskirts, recovering the Planck lensing amplitude without CDM |

The comparison across channels is therefore sharp: under isotropic conformal
rescaling \(\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}\), spatial stretching
cancels \(A^2(\phi(x))\) out of null rays and leaves only the
Planck-suppressed scalar stress tensor (\(\kappa_\phi\simeq6.5\times10^{-6}\)
on the selected sector; Paper 19, Step 64) and the bounded disformal cone
tilt (Step 63). Under the unstretched-space temporal metric
\(\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}\) (\(U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}\)), intermediate
clock rates \(A(\phi(x))\) along the ray do not cancel, supplying the
baryon-sourced cluster, strong-lensing, and CMB lensing convergence from
the same two-branch action \(P_{,X}=k\sqrt\xi+2\xi\) (\(k=16.03\) fixed by \(a_0\), zero per-system free parameters)
that governs galactic rotation curves and wide binaries. Unlike superfluid
dark matter (Berezhiani & Khoury 2015), which postulates a new eV-scale
dark boson species that clusters as dust on linear scales and undergoes a
thermal BEC transition into a \(P(X)\sim X^{3/2}\) phonon superfluid inside
cold galaxies (breaking down in hot galaxy clusters where \(T>T_c\)), TEP
posits zero dark-matter particles (\(\omega_{\rm cdm}=0\)): across a rich
X-COP galaxy cluster (\(M_b(<R_{200})=1.45\times10^{14}\,M_\odot\),
\(R_{200}=2.0\text{ Mpc}\), \(R_{500}=1.2\text{ Mpc}\); Ghirardini et al.
2019; Eckert et al. 2019), the extended ICM profile (\(\beta\simeq0.65\))
keeps \(q(R)=g_b(R)/g_t=0.015\text{--}0.040\ll1\) across
\(R=0.5\text{--}2.0\text{ Mpc}\) so the cluster ICM never enters the
\(q\gtrsim1\) high-acceleration screened regime of galaxy disks. Under the
exact static two-branch Gauss law \((k/\sqrt2)q(r)y(r)^2+q(r)^2 y(r)^3=1\)
on the empirical X-COP baryon profile (replacing the earlier \(\beta=1\)
Plummer benchmark that placed \(72\%\) of cluster baryons inside
\(0.5\text{ Mpc}\) and gave \(5.78, 4.61, 3.47, 2.39\)), the local 3D ratio
\(1+2\beta_A^2 y_{\rm tb}(R)\) is \(5.80, 5.16, 4.80, 4.59, 3.98\) across
\(R=2.0, 1.5, 1.2, 1.0, 0.5\text{ Mpc}\) with integrated static
potential-depth ratio \(\Phi_{\rm tot}(R)/\Phi_b(R)=7.64, 6.85, 6.44, 6.20, 5.61\);
when the pre-boundary clock-slope well (\(\omega_\phi/\omega_b=y_s/\sqrt2=5.26\),
\(\sigma_{T,p}=0\)) retains its unshocked NFW collapse concentration
(\(c_{200}=4.0\)) while virial shock heating and AGN feedback deplete
collisional ICM gas in the core (\(f_{\rm ret}(R)=f_b(<R)/f_b(R_{200})\)
dropping from \(1.00\) at \(2.0\text{ Mpc}\) to \(0.90\) at \(1.2\text{ Mpc}\)
and \(0.68\) at \(0.5\text{ Mpc}\); Eckert et al. 2019), the virialized
enclosed ratio \(M_{\rm dyn}(<R)/M_{b,\rm obs}(<R)=1+(y_s/\sqrt2)/[f_{\rm ret}(R)(1+q(R))]\)
rises inward from \(6.18\) at \(R=2.0\text{ Mpc}\)
(\(M_{\rm dyn}=8.96\times10^{14}\,M_\odot\)) and \(6.41\) at \(R=1.5\text{ Mpc}\)
(\(6.96\times10^{14}\,M_\odot\)) through \(6.73\) at \(R=1.2\text{ Mpc}\)
(\(5.59\times10^{14}\,M_\odot\)) and \(7.12\) at \(R=1.0\text{ Mpc}\)
(\(4.61\times10^{14}\,M_\odot\)) to \(8.44\) at \(R=0.5\text{ Mpc}\)
(\(1.99\times10^{14}\,M_\odot\); Paper 18, Step 21 Part D), matching observed
X-COP hydrostatic and weak-lensing profiles (\(\simeq6\) at \(R_{200}\),
\(6\text{--}7\) at \(R_{500}\), and \(8\text{--}10\) at \(0.5\text{ Mpc}\))
without a MOND cluster deficit, thermal condensate failure, or collisionless
NFW particle cusp. The baryon-retention profile \(f_{\rm ret}(R)\) is an
observational input from X-COP, so the inner-radius agreement tests the
unshocked, collisionless distribution of the clock-slope mode given the
observed gas depletion; the zero-parameter prediction is the normalization
\(1+y_s/\sqrt2\simeq6.2\) at \(R_{200}\).

**Principle:**

#### Box 6.3: Cross-Scale Temporal-Gradient Constraint

A viable TEP realization must produce a strongly suppressed observable
Temporal Shear in local high-precision environments while retaining a
nonzero integrated temporal-gradient response across extended halo
paths:

\begin{equation}
\mathcal{S}_\Sigma(\mathcal{E}_\odot)\,|\Sigma|
\leq \Sigma_{\rm local}^{\rm max},
\end{equation}

\begin{equation}
\left|
\int_{\gamma_{\rm halo}}
\mathcal{S}_\Sigma(\mathcal{E})\,\Sigma_\mu\,dx^\mu
\right|
\sim 10^{-7}\text{--}10^{-6}.
\end{equation}

These are constraints on one continuous environmental operator, not
independently adjusted couplings at different scales. The Solar-System
bound is satisfied when the environmental projection
\(\mathcal{S}_\Sigma(\mathcal{E}_\odot)\) suppresses the locally
observable Temporal Shear below precision-test thresholds. The halo
constraint is satisfied when the same operator, acting across an
extended path through lower-density environments, permits a cumulative
clock-transfer contrast of order \(10^{-7}\text{--}10^{-6}\) on the
matter-clock congruence — the reconstruction-space field of §3.1.3,
which is endpoint-determined for photons and is not a propagated delay.

### 6.3 Interpretational Challenges

TEP effects mimic standard lensing signatures, making them difficult to
distinguish without time-domain analysis.

**Static Degeneracy:** Most lensing data are analyzed under
static assumptions. In this limit, the conformal sector enters as a
reconstruction degeneracy — the endpoint-determined clock-transfer map
defined through $A(\phi) = \exp(\beta_A \phi/M_{\rm Pl})$ and the
line-of-sight field profile $\phi(r)$ is absorbed into mass-sheet and
delay normalization rather than into propagated arrival times (Section
3), while optical convergence and shear are carried by the
scalar-backreaction and disformal channels. The predicted
reconstruction-space map is qualitatively similar to the observed map;
a full quantitative comparison requires the source light-curve model,
which is the next pipeline step.

**Differential vs. Common-Mode Effects:** The GW170817
constraint is often interpreted as excluding modified metric couplings.
However, as shown in Section 4, this constraint applies to the
differential disformal sector. Conformal proper-time dilation is a
*common-mode effect* for co-propagating signals, leaving the
"speed of gravity" constraint satisfied while permitting significant
time-domain phenomenology.

### 6.4 Reinterpreting the Evidence: Multipath vs. Single-Path

When viewed through the lens of TEP, the disparate constraints on the dark
sector resolve into a consistent picture based on signal topology:

**Lensing (Multipath):** Observations that require "dark
matter" (e.g., lensing) are fundamentally
*multipath* measurements. These different pathways sample
different values of the time field \(\phi\), creating a differential
clock-transfer field that isochronous reconstruction reads as "phantom
mass," while any propagated differential delay rides only on the
backreacted and disformal sectors.

**GW170817 (Single-Path):** Observations that constrain
"modified gravity" (GW170817) are fundamentally
*single-path* measurements. The signals originate from the same
coordinate and traverse the same time-warped regions. The temporal
distortions are common-mode and cancel out.

The apparent contradiction—"how can gravity be modified enough to create
dark matter but unmodified enough to pass GW170817?"—is resolved. TEP
introduces the temporal field as the additional physical degree of freedom;
its conformal sector governs clock transport without bending null geodesics
or shifting propagated arrival times,
while its stress-energy and strong-field operators can backreact on the
Einstein-frame geometry (Bahrain, Paper 28). The differential-propagation
bound constrains the disformal sector, not the common-mode conformal clock
sector. Time manifests as a propagated differential observable only through
non-exact transport on divergent paths; in the exact conformal sector it
manifests as a reconstruction degeneracy.

### 6.5 Path-Dependent Distance/Proper-Time Ratio

If TEP is correct, then local \(c\) remains invariant, but the
inferred ratio between spatial separation and matter-clock transfer
registered between endpoints
along extended paths is path dependent. This challenges the foundational
assumption used to interpret all astrophysical data.

**Standard Assumption:** The universal clock rate is
constant; therefore, any anomaly in \(t\) must be due to extra path
length (spatial curvature) or extra mass (Shapiro delay). Conclusion:
particulate *dark matter* is inferred.

**TEP Reality:** The matter proper-time accumulation rate
\(d\tilde{\tau} = A(\phi) d\tau_g\) varies with location. The anomaly
in inferred travel time is due to the scalar field gradient modulating
the clock-transfer field registered by matter congruences along the
path, not to a variation in local \(c\).
Conclusion: apparent *dark matter* is reconstructed temporal
transport.

TEP removes the need for an invisible substance by correcting the assumption
that a single universal clock rate governs all paths, rather than by
asserting a variable speed of light.

This is the local multipath analogue of the cosmological TEP mapping:
conventional cosmology reconstructs observed redshift as spatial expansion,
whereas TEP attributes it to the evolution of the conformal proper-time map
on an underlying static spatial geometry (Athens, Paper 26; Thika, Paper 27).
In lensing, the corresponding reconstruction can project temporal transport
into inferred spatial mass. The two effects — dark matter and cosmic
expansion — are parallel temporal-to-spatial reconstruction artefacts in
TEP, not independent ad hoc mechanisms.

### 6.6 TEP as a Bridge Between Paradigms

TEP offers a potential unification of competing frameworks:

**CDM:** For the dynamical sector, TEP offers a candidate
temporal-field origin for the mass anomaly ordinarily ascribed to Cold
Dark Matter: the Temporal Shear acts on matter trajectories as the
extra pull Newtonian analysis counts as a halo. In the lensing sector,
under the historical isotropic-conformal benchmark (\(\tilde g_{\mu\nu}=A^2 g_{\mu\nu}\))
\(A^2(\phi)\) cancels out of null rays so real lensing is baryon-sourced,
whereas under the canonical unstretched-space temporal metric (\(\tilde\gamma_{ij}=\gamma_{ij}\), §6.9 Step 02)
intermediate temporal depth \(-\beta_A u_{\rm grav}(\mathbf x)\) refracts photons directly
along the line of sight (\(\Phi_{\rm lens}=\Phi_N+\tfrac12 c^2\beta_A u_{\rm grav}\)),
reproducing the Bullet Cluster galaxy-peak convergence (\(\kappa_{\rm tot}=0.307\)–\(0.372\)),
the SN Refsdal Fermat delay (Paper 19, Step 64), and Planck CMB lensing (Paper 18, Step 21)
from baryons and the selected two-branch action \(P_{,X}=k\sqrt\xi+2\xi\) while predicting
a testable single-halo dynamical–lensing ratio \(M_{\rm lens}^{\rm u.s.}/M_{\rm dyn}=(1+\beta_A^2 y_{\rm eff})/(1+2\beta_A^2 y_{\rm eff})\).

**MOND:** In low-acceleration environments,
environmentally active Temporal Shear supplies a candidate common
origin for MOND-like scaling. This paper establishes the
acceleration-scale correspondence but does not independently derive
the complete MOND interpolation law.

TEP provides a single temporal-gradient framework within which CDM-like
large-scale phenomenology and MOND-like low-acceleration behaviour may
emerge in different environmental projections.

**Principle:**

#### Box 6.2: Relation to Low-Acceleration Phenomenology

In TEP, MOND-like phenomenology is associated with the environmental
activation of observable Temporal Shear in low-acceleration, weakly
suppressed systems. The canonical screening operator
\(\mathcal{S}_\Sigma(\mathcal{E})\) becomes weak in low-gradient
galactic environments, allowing the observable Temporal Shear response
to become active. This supplies a candidate origin for MOND-like
phenomenology without modifying the local value of \(c\).

The empirical coincidence

\begin{equation} \label{eq:gl_acceleration}
a_T \sim c H_0
\end{equation}

identifies the relevant observational acceleration scale, where \(H_0\)
is the conventional inverse-time scale encoded by the cosmological
clock map rather than physical spatial expansion. In TEP, \(H_0\) is
not identified with \(\dot a_m/a_m\), since \(a_m = 1\).

**Numerical Check:**

\begin{equation}
a_0 \sim c H_0 \approx (3 \times 10^8\ \text{m/s}) \times (2.2 \times
10^{-18}\ \text{s}^{-1}) \approx 6.6 \times 10^{-10}\ \text{m/s}^2
\end{equation}

This lies within a factor of about five of the empirical MOND value
\(a_0 \approx 1.2 \times 10^{-10}\) m/s². The canonical TEP acceleration scale is the derived cosmological-horizon scale \(g_t = cH_0/(2|\beta_A|) \approx 3.4 \times 10^{-10}\) m/s² (Paper 0 &sect;2.2, v0.16); the SPARC-extracted empirical benchmark \(g_{\rm TEP} \approx 5 \times 10^{-10}\) m/s² (Paper 0 &sect;7; Paper 10) independently corroborates the same order and matches the ambient Galactic shear \(a_{\rm amb} \approx 3.9 \times 10^{-10}\) m/s² at the solar circle. The environmental transition is supplied by the nested two-body operator \(\mathcal{R}(s)=\mathcal{S}_\Sigma(X_{\rm env})^2\,y(s)\) — the effective pairwise projection of the master environmental operator \(\mathcal{S}_\Sigma(\mathcal{E})\) under the selected two-branch kinetic completion \(P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+X|X|/\Lambda_X^4\) (\(P_{,X}=k\sqrt\xi+2\xi\), \(k=16.03\); with \(P=X-V+X|X|/\Lambda_X^4\) as the retained unit-term high-gradient benchmark; Paper 0 &sect;2.2, v0.16, machine-checked benchmark table). TEP attributes this
cross-scale relation to the same temporal-gradient structure
represented by \(\mathcal{S}_\Sigma(\mathcal{E})\). The defensible
statement is that TEP produces an acceleration scale of cosmological
magnitude, which suggests that TEP and MOND phenomenology may share a
common origin. This matches the level of claim established in TEP-UCD
(Paper 6); TEP-GL does not independently derive the numerical MOND
acceleration scale.

**Physical Interpretation:** Low-acceleration
phenomenology arises in TEP because the environmental operator
\(\mathcal{S}_\Sigma(\mathcal{E})\) leaves the Temporal Shear response
active precisely in low-gradient regions. Regions with internal
accelerations \(a \lesssim g_t\) are weakly screened and the temporal
gradient contributes; regions with \(a \gtrsim g_t\) are strongly
screened and follow Newtonian dynamics. The low-gradient scalar
profile, its metric backreaction, and the reconstruction response must
be fitted jointly to galaxy and lensing data before the MOND
transition can be claimed as derived.

### 6.7 Convergence of Evidence

While TEP-GL is a theoretical proposal, it is notable that multiple
independent anomalies in current cosmology converge on the phenomenology
predicted by this framework. These tensions, often treated as separate
puzzles, may represent a single systemic failure of the isochrony
assumption.

| TEP-GL Prediction | Existing Observational Anomaly | Status |
| --- | --- | --- |
| **Source-Dependent Shear**
(Kinematic noise bias; bounded at
\(\gamma_{\rm TC}^2/\sigma_{\rm sh}^2 \sim 4\times10^{-12}\),
Box 5.3) | **\(S_8\) Tension**
Survey-dependent: DES Y6 lower than CMB; KiDS-Legacy
consistent with Planck. | **Motivating Consistency** |
| **Mass-Sheet Degeneracy**
(Temporal vs Spatial) | **\(H_0\) Tension**
Time-delay cosmography (\(H_0 \approx 73\)) conflicts with
CMB (\(H_0 \approx 67\)). The chronometric entry is a
degeneracy statement, not a supply channel: a common-mode
endpoint rescaling acts on inferred \(D_{\Delta t}\) exactly
as an external convergence sheet does, but under the
present-epoch convention its amplitude is the uniform
local-well residual \(A_o \approx 1-10^{-7}\) — far below
the tension and identical for every lens system. The full
channel audit on the real TDCOSMO-2025 systems (Paper 19,
Step 58) confirms the propagated sector is clean at
\(\lesssim 10^{-8}\) fractional (\(\sim 10\) ms
ceilings on \(10\,R_{\rm E}\) traverses) and finds the only
order-percent route to be the dynamical–lensing mass
difference in the velocity-dispersion prior — in the direction that elevates
inferred \(H_0\) — while the mundane mass-model floor is
\(\sim 6\%\) (SDSS1206+4332 power-law vs composite). The
corpus's quantitative \(H_0\) channel is the Cepheid
clock-bias sector (TEP-H0, Paper 11), a different
projector. | **Degeneracy-level** |
| **temporal-composite response**
(source-evolution response) | **Flux Ratio Anomalies**
Substructure required to explain ratios is often not found;
"phantom" substructure. TC contribution bounded at
\(\delta\mu/\mu \lesssim 10^{-5}\) (Box 5.2). | **Amplitude-bounded** |
| **Variability-Shear Correlation** | **Einstein Cross Anomalies**
Existing Einstein Cross monitoring provides archival data
suitable for testing the predicted variability-phase
correlation; no dedicated TEP correlation analysis has yet
been performed (Eigenbrod et al. 2008). | **Untested** |

The theory supplies testable temporal channels on existing lensing and
timing observations, with the source-independent optical convergence and
Fermat delay evaluated directly from the unstretched-space temporal metric
and two-branch action in §6.9 (Step 02) and Paper 19 (Step 64).

### 6.8 The Path Forward

The analysis shows that a new particulate dark component is not
required by the temporal-field ontology: the same scalar configuration
produces both the dynamical mass-inference response and the
unstretched-space optical refraction (§6.9), alongside source-dependent
reconstruction signatures. Full dark-sector closure is tested by
two complementary observables — the matched dynamical–lensing phantom-mass
ratio on isolated systems and the CDM-free absolute lensing potential across
clusters and the cosmic web (§6.9; Paper 18 Step 21; Paper 19 Step 64).
For particulate dark matter to remain necessary, these TEP predictions
must be excluded at the required precision.

Three concrete consequences follow directly from the phantom-mass mechanism and require no further
model-building:

*Lensing radial-acceleration relation.* On the historical isotropic-conformal benchmark
(\(\tilde g_{\mu\nu}=A^2 g_{\mu\nu}\)), photon trajectories track the baryon-sourced geometry alone
(\(g_{\rm lens}^{\rm iso} \simeq g_{\rm bar}\)), whereas under the canonical unstretched-space temporal
metric (\(\tilde\gamma_{ij}=\gamma_{ij}\), §6.9), intermediate temporal depth contributes half the
dynamical scalar potential weight on single-pass rays (\(g_{\rm lens}^{\rm u.s.} = g_{\rm bar} + \tfrac12 a_\phi\),
giving \(M_{\rm lens}^{\rm u.s.}/M_{\rm dyn}=(1+\beta_A^2 y_{\rm eff})/(1+2\beta_A^2 y_{\rm eff})\);
Paper 0 Eq. 63, reaching \(g_{\rm bar}+a_\phi\) on multi-component sightlines). In either case, the
single-halo dynamical–lensing relation is predicted to be systematic in the radial-acceleration plane,
not a scatter of independent anomalies, and is testable galaxy-by-galaxy wherever both inference
channels exist.

*Isolation dependence.* The phantom mass is largest for isolated, diffuse, low-acceleration
systems ($a \lesssim g_t$, $\mathcal S_\Sigma \to 1$) and is suppressed toward zero for galaxies embedded
in high-acceleration environments — cluster members, compact groups, satellites in deep host potentials —
where the environment screens the Temporal Shear. A dynamical mass excess that persists at full strength
inside cluster cores would count directly against the mechanism.

*Dissociative mergers.* The Bullet Cluster is the discriminating case. The observed lensing
peaks sit on the collisionless galaxies, offset from the dominant baryonic gas. Under isotropic
conformal rescaling \(\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}\), the conformal factor cancels out of
null rays at every intermediate point, leaving only the spatial-gradient disformal channel
\(B(\partial\phi)^2\) (bounded \(\sim 9\)–\(70\times\) below the required convergence) and a
compensated negative moat. Under the unstretched-space temporal metric (§6.9, Step 02), by contrast,
intermediate clock rates \(A(\phi(x))\) do not cancel against spatial stretching, and the selected
two-branch action \(P_{,X}=k\sqrt\xi+2\xi\) reproduces the galaxy-peak convergence and offset
morphology directly from the baryonic potential wells.

### 6.9 Explicit Ray Tracing on a Dissociative Geometry: Disformal Control vs. Unstretched-Space Temporal Refraction

The channel audit is now carried through explicitly at the level of the
photon paths themselves (step_02). First, the disformal-only spatial-gradient route on the historical isotropic-conformal metric is evaluated as an adverse control, isolating why GW170817-bounded cone tilts cannot supply the cluster-scale convergence amplitude; second, the surviving cluster-lensing mechanism—direct optical refraction from the shared-lapse unstretched-space temporal metric—is evaluated on the identical four-component baryon-only geometry. In the isotropic-conformal plus disformal matter metric
\(\tilde g_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi\),
the Sherman–Morrison inverse is
\(\tilde g^{\mu\nu} = A^{-2}g^{\mu\nu} - [B/(A^2(A^2+BQ))]\nabla^\mu\phi\nabla^\nu\phi\)
with \(Q = g^{\alpha\beta}\nabla_\alpha\phi\nabla_\beta\phi\), and null rays are the
characteristics of \(H = \tfrac12 \tilde g^{\mu\nu}k_\mu k_\nu = 0\):
\(\dot x^\mu = \tilde g^{\mu\nu}k_\nu\),
\(\dot k_\mu = -\tfrac12\,\partial_\mu\tilde g^{\alpha\beta}k_\alpha k_\beta\).
In the weak-field, perturbative-\(B\) limit the eikonal index is
direction-dependent,
\(n_{\rm eff}(\hat n,\mathbf x) \simeq 1 - 2\Phi/c^2 + \tfrac12 (B/A^2)(\hat n\cdot\nabla\phi)^2\):
photons are retarded only when propagating along the scalar gradient. The Born
deflection \(\alpha_i(\mathbf b) = \int \partial_i n_{\rm eff}\,dl\) and the
reduced-deflection convergence
\(\kappa_{\rm dis}(\theta) = -\tfrac12 (D_{ls}/D_s)\,D_l\,\nabla_b\cdot\alpha\)
were computed on a Bullet-like two-subcluster geometry (0.72 Mpc separation,
gas centroid lagging the bullet galaxies by 0.24 Mpc, baryons only) with the
screened field solved along each sightline, \(\nabla u = y(g)\,g/c^2\). Three
controls validate the implementation: the identical stencil applied to
\(n_{\rm GR} = -2\Phi/c^2\) reproduces the point-mass deflection
\(4GM/c^2b\) to 1.6%; the full Hamiltonian system integrated with the exact
\(\tilde g^{\mu\nu}\) through a spherical scalar well, with the disformal-only
deflection isolated by subtracting the \(B_0 = 0\) ray, agrees with the Born
reduction to \(0.02\)–\(0.1\) per cent at matched impact parameters once the
on-ray field accumulation and the radial spline are truncated at the same
radius (the earlier apparent shortfall was a mismatched accumulation
domain, not an integration error); and the conformal sector
contributes identically zero deflection.

The result answers the morphology half of the objection affirmatively. The
disformal convergence concentrates on the collisionless scalar wells, not on
the gas: at the bullet-galaxy sightline \(\kappa_{\rm dis}/B_0 = 1.6\times10^{-7}\)
versus \(6.8\times10^{-8}\) at the gas centroid, and the map peak sits on the
main-cluster well to within a resolution cell. The kernel is compensated —
a positive core on each well surrounded by a negative moat — which is itself
a falsifiable structural signature distinguishing disformal-sourced from
mass-sourced convergence. The "empty" lensing regions are not empty in the
matter metric: they host the deepest \(\nabla u\).

The amplitude half remains adverse. Reproducing a peak convergence of
\(\kappa \simeq 0.3\) at the galaxy positions requires
\(B_0^{\rm req} \simeq 3.7\times10^{5}\) under the canonical halo convention —
\(4.8\times10^{3}\) above the GW170817-transferred bound \(B_0 \lesssim 77.7\).
Under the ambient-consistent convention (\(B\) reading the total field
\(u = \bar u + u_{\rm halo}\)) the requirement drops to \(B_0^{\rm req} \simeq 1.2\times10^{-3}\),
but the same-convention multimessenger bound tightens to \(B_0 \lesssim 3.1\times10^{-11}\)
(Paper 19, step_63), worsening the exclusion to \(\sim 4\times10^{7}\). Sourcing
the field from the effective dynamical potential — the scalar wells tracking the
phantom-enhanced \(g_{\rm eff} = g(1+2\beta_A^2 y)\) rather than the bare baryonic
field — deepens \(u\) and closes about 98% of the halo-convention gap
(\(B_0^{\rm req} \simeq 5.3\times10^{3}\), excluded by a factor \(\sim 68\)),
but does not close it. The deficit is partially component-model dependent:
stress-testing the Plummer scales (step_02b) shows that compactening all
components improves the margin fastest, but only until the baryonic
convergence alone overshoots the observed \(\kappa \simeq 0.3\) at the galaxy
sightline (\(a \lesssim 0.85\,a_{\rm nom}\) is excluded by the lensing data
itself); at the boundary \(a \simeq 0.9\,a_{\rm nom}\) the requirement
improves to \(B_0^{\rm req} \simeq 720\), a factor \(\sim 9\) over the bound.
The residual gap therefore spans \(\sim 9\)–\(70\) across the defensible
baryonic-model range, insensitive to the line-of-sight integration extent
(doubling to \(\pm 8\) Mpc moves the margin by \(13\%\)). The corpus's
admissible nested coupling was also evaluated on this geometry: with
\(B_{\rm eff} = B(\phi_{\rm tot})\,G(\delta u)\) — the envelope reading the
total nested field while the gate \(G = g_{\rm sat}\min(1,\delta u/u_{\rm act})\)
closes on the smooth ambient and saturates inside nonlinear wells — and
self-consistent sourcing, the requirement is
\(B_0^{\rm req} \simeq 0.02\)–\(0.15\) across the gate bracket
\(g_{\rm sat} = 10^{-3}\)–\(7\times10^{-3}\), against the same-convention
host-well bound \(B_0 \lesssim 2.1\times10^{-5}\): an exclusion of order
\(10^{3}\), consistent with the Paper-19 gate scan. The inference-side escape route is also bounded: the
conformal-depth difference between the gas- and galaxy-peak sightlines is
\(\Delta u \simeq -3.0\times10^{-6}\) (\(c\,\Delta u \simeq -0.9\ {\rm km\,s^{-1}}\)),
corresponding to an X-ray emissivity and gas-mass inference bias of order
\(10^{-6}\) — five orders of magnitude below what relocating the baryon
dominance requires (and in any individual dissociative merger a
\(\sim 1\ {\rm km\,s^{-1}}\) differential clock shift is masked by
\(\mathcal{O}(10^{2}\text{--}10^{3})\ {\rm km\,s^{-1}}\) hydrodynamic shock
and ram-pressure stripping velocities, making it a baryon-census consistency
bound and a regime accessible only in large ensemble-stacked cluster
gravitational-redshift surveys).

The diagnosis above isolates why the spatial-gradient disformal channel
\(B(\phi)\partial_\mu\phi\partial_\nu\phi\) is suppressed: under isotropic
conformal rescaling \(\tilde g_{\mu\nu} = A^2(\phi)g_{\mu\nu}\), spatial
rulers stretch by the same factor \(A^2(\phi)\) as clocks, so \(A^2(\phi(x))\)
divides out of the null condition \(d\tilde s^2 = 0\) at every intermediate
point along the ray, killing the linear \(\mathcal{O}(u) \sim 10^{-5}\)
temporal depth and leaving only the quadratic \(\mathcal{O}((\nabla u)^2)\)
cone tilt. In TEP's cosmological ontology, however, space does not stretch:
under the unstretched-space temporal metric
\(\tilde g_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}\)
(\(U_\mu \equiv -\nabla_\mu\bar\phi/\sqrt{2\bar X}\)),
the \((3+1)\) ADM spatial geometry \(\gamma_{ij}=(1-2\Phi_N/c^2)\delta_{ij}\) remains unstretched while the spacetime lapse rescales as \(N_{\rm TEP}(x)=A(\phi(x))N_{\rm GR}(x)\) along the
temporal flow. Along a null
ray across static space, intermediate clock rates \(A(\phi(x))\) between
source and observer no longer cancel against spatial stretching:
\(c\,dt/dl = A(\phi(x))^{-1}\sqrt{(1-2\Phi_N/c^2)/(1+2\Phi_N/c^2)} \simeq 1 - 2\Phi_N(x)/c^2 - \beta_A u_{\rm grav}(x)\),
with transverse deflection gradient \(-\beta_A\nabla_\perp u = \vec a_{\phi,\perp}/c^2\).
Evaluated on the same four-component baryon-only Bullet geometry under the
selected two-branch kinetic sector \(P_{,X} = k\sqrt\xi + 2\xi\)
(\(k = 4\sqrt2\,|\beta_A|g_t/a_0 = 16.03\), fixed external calibration with zero per-system free parameters;
`unstretched_temporal_depth_channel` in `step_02`),
intermediate-point temporal refraction yields \(\kappa_{\rm temp} = 0.065\)
(\(1\times\) temporal lapse) to \(0.131\) (\(2\times\) dynamical normalization)
on top of \(\kappa_{\rm bar} = 0.241\) at the bullet-galaxy sightline, giving
\(\kappa_{\rm tot} = 0.307\)–\(0.372\) against the observed target
\(\kappa \simeq 0.30\). The bullet-half convergence peak lands at
\(x = +0.352\ {\rm Mpc}\) (on the galaxies at \(+0.36\ {\rm Mpc}\), versus
\(\kappa_{\rm tot} = 0.223\)–\(0.284\) at the shocked-gas centroid at
\(+0.12\ {\rm Mpc}\)), \(\kappa_{\rm temp}\) is positive everywhere across
the field (\(\min\kappa_{\rm temp} = +4.6\times10^{-3}\), eliminating the
negative moat), Solar-System Shapiro bounds remain satisfied via the
quadratic-branch screening \(y \sim (r/r_*)^{4/3} \ll 10^{-5}\), and
\(c_{\rm GW} = c_\gamma\) holds identically. Specifically, although
\(-(A^2-1)U_\mu U_\nu = \frac{1-A^2}{2\bar X}\nabla_\mu\bar\phi\nabla_\nu\bar\phi\)
has the algebraic form of a longitudinal disformal shift along the cosmic temporal flow,
it is not a photon-only disformal coupling on an un-lapsed tensor metric (which would split
photon and gravitational-wave transit times by \(\Delta t\sim\int|\beta_A|u\,dl/c\sim10^7\ {\rm s}\)
across Galactic and host haloes, the dark-matter-emulator exclusion of Boran et al. 2018, and invert
the longitudinal sign in voids where \(A>1\)); because proper time \(d\tilde t = A_{\rm loc}(u(\mathbf x))\,dt\) is
the universal physical time of the inhomogeneous ADM foliation itself, varying the spatially covariant ADM/DHOST gravitational action
\(S_{\rm grav}=\tfrac12 M_{\rm Pl}^2\int d^3x\,dt\,\sqrt{\gamma}\,N_{\rm GR}[R^{(3)}(\gamma)+\tilde K_{ij}\tilde K^{ij}-\tilde K^2]\)
(\(\tilde K_{ij}=(2N_{\rm TEP}c)^{-1}\partial_t\gamma_{ij}=A_{\rm loc}(u)^{-1}K_{ij}\), with \(\delta S_{\rm grav}/\delta u\equiv 0\) in the static limit \(K_{ij}\to 0\) preserving the Poisson and scalar source equations without Jordan-frame degeneracy; Paper 0, §2.2, Step 80)
with respect to transverse-traceless tensor perturbations
\(h_{ij}^{\rm TT}\) of \(\gamma_{ij}\) and varying \(S_{\rm EM}[\tilde g]\) with respect to electromagnetic waves
\(A_i^{\rm T}\) yields the identical quadratic wave operator
\(\Box_{\rm TEP}=-(N_{\rm TEP}c)^{-2}\partial_t^2+\gamma^{ij}\partial_i\partial_j\) on the unstretched spatial metric \(\gamma_{ij}\), experiencing the identical refractive index
\(n(x)\) across wells and voids alike (\(\Delta t_{\gamma-\rm GW}=0\) identically).

The Bullet Cluster therefore sharply discriminates between the two metric
realizations: under isotropic conformal stretching the quadratic
\(B(\partial\phi)^2\) channel falls short of current bounds by a factor
\(\sim 9\)–\(70\) and produces a compensated negative moat, whereas under
the unstretched-space temporal metric the selected two-branch action
reproduces the \(\kappa \simeq 0.30\) galaxy-peak convergence and offset
morphology from baryons and intermediate temporal depth alone, with the
differential conformal clock bias on the X-ray gas census bounded to the
\(10^{-6}\) level.

## 7. Conclusions

### 7.1 Limitations of the Isochrony Axiom

For nearly a century, dark-matter inference has been developed within a
framework that treats the residual source-time structure relevant to static
reconstruction as effectively synchronous once standard propagation delays
are modeled. By
treating the photons in telescopes as representing a single spatial slice of
the source, standard models have been forced to interpret all observed
distortions as spatial deflections caused by mass. It has been shown that
this *Isochrony Axiom* is an effective approximation that may break
down in the presence of generalized metric couplings. In the TEP ontology,
the dark sector is not fundamental matter or energy; it is part of the
residual produced when dynamical temporal transport is forced into a
synchronous spatial reconstruction. The same interpretive principle applies
cosmologically: observed redshift is reconstructed conventionally as physical
spatial expansion, whereas TEP attributes it to the conformal proper-time map
on an underlying static spatial geometry (Athens, Paper 26; Thika, Paper 27).
Tortola develops the corresponding lensing-sector result: temporal-field
structure can be reconstructed as spatial mass.

Under isotropic conformal rescaling \(\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}\),
spatial rulers stretch by \(A^2(\phi)\) alongside clocks so \(A^2(\phi(x))\)
cancels out of \(d\tilde s^2=0\), leaving only the Planck-suppressed scalar
stress tensor (\(\kappa_\phi\simeq6.5\times10^{-6}\) on the selected sector;
Paper 19, Step 64) and the bounded disformal cone tilt (Step 63). Under
the unstretched-space temporal metric
\(\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}\) (\(U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}\)), space is static
and does not stretch, so intermediate clock rates \(A(\phi(x))\) along the
photon path do not cancel: every intermediate point \(x\) contributes its
local temporal depth \(-\beta_A u_{\rm grav}(x)\) directly to the optical refractive
index \(n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)\), reproducing the Bullet
Cluster galaxy-peak convergence (\(\kappa_{\rm tot}=0.307\)–\(0.372\), no
negative moat; Step 02), the SN Refsdal Fermat delay and arrival order
(Paper 19, Step 64), and the Planck CMB lensing cosmic-web boost (Paper 18,
Step 21) from baryons and the selected two-branch action
\(P_{,X}=k\sqrt\xi+2\xi\) with zero particulate dark matter.

### 7.2 Summary of Findings

**Unified Dynamical and Optical Refraction from Unstretched Space:**
While isotropic conformal rescaling cancels out of null geodesics and
leaves only \(\mathcal{O}(u^2)\) scalar stress-energy, TEP's
unstretched-space temporal metric retains the linear \(\mathcal{O}(u)\)
intermediate temporal depth along every photon path. Sourced by baryons
on the zero-free-parameter selected two-branch action
\(P_{,X}=k\sqrt\xi+2\xi\), it supplies both the dynamical Temporal Shear
on slow matter and the source-independent optical convergence across
galaxies, clusters (Step 02), strong-lensing delays (Paper 19, Step 64),
and CMB lensing (Paper 18, Step 21), alongside the bounded
source-dependent temporal-composite response for evolving sources.

**The GW170817 "Speed Limit" and Solar-System Screening:**
While GW170817 constrains disformal cone-tilt propagation speeds to
\(|c_\gamma - c_g|/c \lesssim 10^{-15}\), the unstretched-space temporal
metric preserves \(c_{\rm GW}=c_\gamma\) identically because both tensor
and electromagnetic wave sectors evolve in the same physical ADM lapse
\(d\tilde t=A(\phi)\,dt\) on the unstretched spatial manifold (\(\Delta t_{\gamma-\rm GW}=0\)),
and it satisfies Solar-System PPN
Shapiro bounds via the quadratic-branch kinetic screening
\(y\sim(r/r_*)^{4/3}\ll10^{-5}\) (Paper 0, Step 57).

**A New Observational Era:** Two regimes for testing the
time-domain and optical channels are defined: millisecond-scale
multipath bounds on the disformal sector in high-time-resolution
astrophysics (FRBs, pulsars), variability-conditioned shear residuals
testing the bounded temporal-composite envelope, and direct cluster,
strong-lens, and ensemble-stacked cluster gravitational-redshift tests
of the unstretched-space temporal convergence and conformal potential
depth (Step 02).

### 7.3 Future Outlook: Time, Not Mass

The persistence of the dark matter problem despite decades of particle
searches suggests a potential error in the underlying premises. The error
may lie in the treatment of time. By treating time as a passive parameter
rather than an active dynamical field, there is a risk of blinding analysis
to the true nature of the "dark" universe.

The path forward lies in *Chronometric Lensing*. The field must move
beyond static mass-mapping and towards
*chronometric mapping*—measuring the detailed arrival-time structure
of the universe. The dark-sector tests are the dynamical–lensing mass difference on
matched low-acceleration systems and the absolute CDM-free lensing
potential, including $C_\ell^{\phi\phi}$. Lensed-FRB residuals
separately test the disformal transport sector. The focus of inquiry must shift from searching for missing mass
to characterizing the dynamical structure of time.

### 7.4 The Paradigm Shift

Within the operational axioms adopted here, the dynamical and optical
phenomenology traditionally attributed to a particulate dark sector is
reinterpreted as the projection of two-metric time transport onto inference
pipelines that assume isochrony: Temporal Shear governs slow matter
trajectories, while intermediate temporal depth across unstretched space
refracts photons (§6.9, Step 02; Papers 18, 19), predicting both the
baryon-sourced static lensing convergence and a testable single-halo
dynamical–lensing mass ratio \(M_{\rm lens}^{\rm u.s.}/M_{\rm dyn}=(1+\beta_A^2 y_{\rm eff})/(1+2\beta_A^2 y_{\rm eff})\).
The anomalies are real; the claim is that their standard "missing mass"
interpretation is not unique once the Isochrony Axiom is relaxed.

In dynamics, the dark sector is thus reinterpreted not as an invisible
substance, but as the shadow of temporal transport on matter trajectories.

The observational program outlined herein—time-domain lensing of fast
transients, precision strong-lens residual timing, source-variability-dependent
shear consistency tests, and above all the environment-resolved
dynamical–lensing mass measurement—provides a viable route
to distinguish particle dark matter from temporal transport while
maintaining a conservative baseline anchored to existing
multi-messenger constraints, thereby offering a decisive, falsifiable
avenue for confronting the dark matter problem.

**Principle:**

#### Updated Operational Interpretation

The later dedicated strong-lensing analysis (TEP-LENS, Paper 19) refines
the observational implementation used here. In particular, ordinary
image-arrival delays remain algebraically integrable; the strong-lensing
test is formulated as an observed-versus-GR-predicted transport residual
rather than a literal closure violation. The framework presented here is
consistent with that refinement: Axiom 3 already distinguishes open-path
differential residuals (the GL observable) from closed-loop holonomy,
and on the historical isotropic-conformal benchmark (\(\tilde{g}_{\mu\nu}=A^2 g_{\mu\nu}\)) the static clock-transfer contribution is identified as a
clock-transfer discrepancy rather than a direct refraction of null
geodesics. The two papers state one unified answer across both metric realizations: on the isotropic-conformal
benchmark, the exact additional static-conformal inter-image residual is
\(0.0\) d (Paper 19, step_56), and every year-scale figure quoted here
is a reconstruction-space amplitude with the
spatial-disformal differential channel bounded at the ms–s level; under the canonical unified unstretched-space temporal metric (\(\tilde{\gamma}_{ij}=\gamma_{ij}\) at fixed cosmic epoch; Paper 0 §2.2, §6.9 Step 02, and Paper 19 Step 64), intermediate temporal depth \(-\beta_A u_{\rm grav}(\mathbf{x})\) on the selected two-branch sector \(P_{,X}=k\sqrt\xi+2\xi\) (\(k=16.03\)) directly refracts photons along the line of sight, reproducing both the Bullet Cluster galaxy-peak convergence (\(\kappa_{\rm tot}=0.307\text{--}0.372\)) and the SN Refsdal Fermat delay (\(+330.9\text{ d}\) sheet and \(+221.6\text{--}443.2\text{ d}\) 3-D NFW ray) from \(f_b=0.16\) baryons alone. Readers should consult TEP-LENS for the current canonical
strong-lensing time-delay formulation.

## References

Abbott, B. P., et al. (LIGO/Virgo Collaboration) 2017,
*Phys. Rev. Lett.*, 119, 161101

Abbott, T. M. C., et al. (DES Collaboration) 2022,
*Phys. Rev. D*, 105, 023520

Abbott, T. M. C., et al. (DES Collaboration) 2026,
arXiv:2601.14559 (DES Year 6 3×2pt cosmological constraints)

Asgari, M., et al. (KiDS Collaboration) 2021, *A&A*, 645, A104

Wright, A. H., et al. (KiDS Collaboration) 2025,
*A&A*, 703, A158 (arXiv:2503.19441) (KiDS-Legacy cosmic shear)

Baker, T., Bellini, E., Ferreira, P. G., Lagos, M., Noller, J., &
Sawicki, I. 2017, *Phys. Rev. Lett.*, 119, 251301
(arXiv:1710.06394)

Bartelmann, M., & Schneider, P. 2001, *Phys. Rep.*, 340, 291

Bekenstein, J. D. 1993, *Phys. Rev. D*, 48, 3641
(arXiv:gr-qc/9211017)

Bekenstein, J. D. 2004, *Phys. Rev. D*, 70, 083509
(arXiv:astro-ph/0403694)

Blandford, R. D., & Narayan, R. 1986, *ApJ*, 310, 568

Boran, S., Desai, S., Kahya, E. O., & Woodard, R. P. 2018,
*Phys. Rev. D*, 97, 041501 (arXiv:1710.06168)

Bullock, J. S., & Boylan-Kolchin, M. 2017, *ARA&A*, 55,
343

Burrage, C., & Sakstein, J. 2018, *Living Rev. Relativ.*, 21,
1

Chang, C., et al. 2024, *arXiv e-prints*, arXiv:2406.19654

Clifton, T., Ferreira, P. G., Padilla, A., & Skordis, C. 2012,
*Phys. Rep.*, 513, 1 (arXiv:1106.2476)

Clowe, D., et al. 2006, *ApJ*, 648, L109

Creminelli, P., & Vernizzi, F. 2017, *Phys. Rev. Lett.*, 119,
251302 (arXiv:1710.05877)

Damour, T., & Esposito-Farèse, G. 1992,
*Class. Quantum Grav.*, 9, 2093

Di Valentino, E., et al. 2021, *Classical Quantum Gravity*, 38,
153001

Eigenbrod, A., et al. 2008, *A&A*, 490, 933 (arXiv:0810.0011)

Ezquiaga, J. M., & Zumalacárregui, M. 2017,
*Phys. Rev. Lett.*, 119, 251304 (arXiv:1710.05901)

Fian, C., Jiménez-Vicente, J., Mediavilla, E., et al. 2018,
*ApJ*, 859, 50 (arXiv:1805.09619)

Gilman, D., et al. 2016, *MNRAS*, 462, 219 (arXiv:1601.01671)

Hu, W., Barkana, R., & Gruzinov, A. 2000, *Phys. Rev. Lett.*,
85, 1158 (arXiv:astro-ph/0003365)

Hui, L., Ostriker, J. P., Tremaine, S., & Witten, E. 2017,
*Phys. Rev. D*, 95, 043541 (arXiv:1610.08297)

Khoury, J., & Weltman, A. 2004, *Phys. Rev. D*, 69, 044026
(arXiv:astro-ph/0309411)

Koch Ocker, S., et al. 2022, *ApJ*, 931, 87

Kochanek, C. S. 2002, *ApJ*, 578, 25 (arXiv:astro-ph/0205319)

Leitherer, C., Schaerer, D., Goldader, J., et al. 1999, *ApJS*,
123, 3 (arXiv:astro-ph/9902334)

Li, Z.-X., Gao, H., Ding, X.-H., Wang, G.-J., & Zhang, B. 2018,
*Nat. Commun.*, 9, 3833 (arXiv:1708.06357)

Lintott, C. J., Schawinski, K., Keel, W., et al. 2009, *MNRAS*,
399, 129 (arXiv:0906.5304)

McGaugh, S. S., et al. 2016, *Phys. Rev. Lett.*, 117, 201101

Milgrom, M. 1983, *ApJ*, 270, 365

Muñoz, J. B., Kovetz, E. D., Dai, L., & Kamionkowski, M. 2016,
*Phys. Rev. Lett.*, 117, 091301 (arXiv:1605.00008)

Narayan, R., & Bartelmann, M. 1996, in
*Formation of Structure in the Universe*, ed. A. Dekel & J.
P. Ostriker (Cambridge Univ. Press) (arXiv:astro-ph/9606001)

Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, *ApJ*,
462, 563

Nityananda, R., & Samuel, J. 1992, *Phys. Rev. D*, 45, 3862
(arXiv:astro-ph/9205001)

Planck Collaboration 2020, *A&A*, 641, A6

Rathna Kumar, S., Tewes, M., Stalin, C. S., Courbin, F., et al. 2013,
*A&A*, 557, A44 (arXiv:1306.5105)

Refsdal, S. 1964, *MNRAS*, 128, 307

Riess, A. G., et al. 2022, *ApJ*, 934, L7

Rubin, V. C., & Ford, W. K., Jr. 1970, *ApJ*, 159, 379

Sakstein, J., & Jain, B. 2017, *Phys. Rev. Lett.*, 119,
251303 (arXiv:1710.05893)

Schawinski, K., Koss, M., Berney, S., & Sartori, L. F. 2015,
*MNRAS*, 451, 2517 (arXiv:1505.06733)

Schneider, P., & Sluse, D. 2013, *A&A*, 559, A37
(arXiv:1306.0901)

Schneider, P., & Sluse, D. 2014, *A&A*, 564, A103
(arXiv:1306.4675)

Schumann, M. 2019, *J. Phys. G*, 46, 103003

Skordis, C., & Złośnik, T. 2021, *Phys. Rev. Lett.*, 127,
161302

Sluse, D., Hutsemékers, D., Courbin, F., Meylan, G., & Wambsganss,
J. 2012, *A&A*, 544, A62

Smette, A., Surdej, J., et al. 1992, *ApJ*, 389, 39

Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: [10.5281/zenodo.16921911](https://doi.org/10.5281/zenodo.16921911) (Paper 0)

Smawfield, M. L. (2025). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: [10.5281/zenodo.17127229](https://doi.org/10.5281/zenodo.17127229) (Paper 1)

Smawfield, M. L. (2025). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: [10.5281/zenodo.17517141](https://doi.org/10.5281/zenodo.17517141) (Paper 2)

Smawfield, M. L. (2025). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: [10.5281/zenodo.17860166](https://doi.org/10.5281/zenodo.17860166) (Paper 3)

Smawfield, M. L. (2025). *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: [10.5281/zenodo.17982540](https://doi.org/10.5281/zenodo.17982540) (Paper 4 — this work)

Smawfield, M. L. (2025). *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: [10.5281/zenodo.18004832](https://doi.org/10.5281/zenodo.18004832) (Paper 5)

Smawfield, M. L. (2025). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: [10.5281/zenodo.18064365](https://doi.org/10.5281/zenodo.18064365) (Paper 6)

Smawfield, M. L. (2025). *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: [10.5281/zenodo.18059250](https://doi.org/10.5281/zenodo.18059250) (Paper 7)

Smawfield, M. L. (2025). *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: [10.5281/zenodo.18064581](https://doi.org/10.5281/zenodo.18064581) (Paper 8)

Smawfield, M. L. (2025). *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: [10.5281/zenodo.18109760](https://doi.org/10.5281/zenodo.18109760) (Paper 9)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: [10.5281/zenodo.18165798](https://doi.org/10.5281/zenodo.18165798) (Paper 10)

Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: [10.5281/zenodo.18209702](https://doi.org/10.5281/zenodo.18209702) (Paper 11)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: [10.5281/zenodo.19000827](https://doi.org/10.5281/zenodo.19000827) (Paper 12)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: [10.5281/zenodo.19102061](https://doi.org/10.5281/zenodo.19102061) (Paper 13)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Blind-Prediction Residual Test in Multiply-Imaged Supernovae*. Preprint v0.3 (Lisboa). Zenodo. DOI: [10.5281/zenodo.20572719](https://doi.org/10.5281/zenodo.20572719) (Paper 19)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Black Holes and the Temporal Horizon*. Preprint v0.4 (Bahrain). Zenodo. DOI: [10.5281/zenodo.21677826](https://doi.org/10.5281/zenodo.21677826) (Paper 28)

Tie, S. S., & Kochanek, C. S. 2018, *MNRAS*, 473, 80
(arXiv:1707.01908)

Treu, T., & Marshall, P. J. 2016, *A&ARv*, 24, 11
(arXiv:1605.05333)

Wambsganss, J. 2006, in
*Gravitational Lensing: Strong, Weak and Micro*, Saas-Fee
Advanced Course 33, ed. G. Meylan et al. (Springer)
(arXiv:astro-ph/0604278)

Wong, K. C., et al. (H0LiCOW Collaboration) 2020, *MNRAS*, 498,
1420

Walsh, D., Carswell, R. F., & Weymann, R. J. 1979, *Nature*,
279, 381

Zwicky, F. 1933, *Helv. Phys. Acta*, 6, 110

#### Contact Information

**Author:** Matthew Lukin Smawfield

**Affiliation:** Independent Researcher

**Email:**
matthew@mlsmawfield.com

**ORCID:**
[0009-0003-8219-3159](https://orcid.org/0009-0003-8219-3159)

**GitHub:**
github.com/matthewsmawfield

**License:** This work is licensed under a
Creative Commons Attribution 4.0 International License.

Version: v0.9 (Tortola) · First published: 19 December 2025 · Last updated: 17 October 2026

## Data Availability & Reproducibility

This work follows open-science practices. This is a theoretical framework paper developing 
the TEP-GL (Temporal Equivalence Principle - Gravitational Lensing) model. The analysis builds 
upon established dark matter phenomenology from the literature and derives new observational 
predictions from first principles.

### Repository & Code

**GitHub Repository:** [github.com/matthewsmawfield/TEP-GL](https://github.com/matthewsmawfield/TEP-GL)

The repository contains the complete theoretical framework, mathematical derivations, 
and observational predictions for the TEP-GL model.

#### Repository Structure

```
TEP-GL/
├── manuscripts/               # Markdown manuscript sources
│   └── manuscript-tep-gl.md
├── site/
│   ├── components/            # HTML manuscript sections
│   │   ├── abstract.html
│   │   ├── section_1_introduction.html
│   │   ├── section_2_theoretical_framework.html
│   │   ├── section_3_quantitative_comparison.html
│   │   ├── section_4_gw170817.html
│   │   ├── section_5_observational_predictions.html
│   │   ├── section_6_discussion.html
│   │   ├── section_7_conclusions.html
│   │   ├── acknowledgments.html
│   │   ├── references.html
│   │   └── appendix_a.html
│   └── manifest.json
├── scripts/                   # Analysis scripts
│   ├── steps/
│   │   └── step_01_amplitude_ledger.py   # temporal-composite amplitude
│   │                                   # ledger (Boxes 3.2, 3.3, 5.1–5.3)
│   └── utils/
├── results/
│   └── step_01_amplitude_ledger.json    # machine-readable ledger output
├── requirements.txt           # Python dependencies
├── CITATION.cff               # Citation metadata
└── README.md                  # Repository documentation
```

The amplitude bookkeeping quoted in Boxes 3.2, 3.3 and 5.1–5.3 — the
source proper motion, the reconstruction-space and propagating-carrier
temporal-composite amplitudes, the standard delay-map baseline
\(\gamma^{\rm std}=\mu_s\,\partial_\theta\Delta T_{\rm Fermat}\) of
§3.4/§5.2, the delay-field amplitude probed by
each falsification threshold, and the exact-one-form cancellation audit
of §3.1.3 (the endpoint-determination of the shear integral on two
multipath rays of the Box-3.1 toy halo) — is reproduced by
`scripts/steps/step_01_amplitude_ledger.py`, whose output is
archived at `results/step_01_amplitude_ledger.json`.

### Data Provenance

This is a theoretical framework paper. Primary data references:

- **GW170817:** LIGO/Virgo gravitational wave and EM counterpart data (public)

- **Dark matter phenomenology:** Established lensing results from the literature (fully cited)

- **SPARC database:** For TEP-UCD cross-paper validation

### Reproduction Instructions

#### Quick Start (Manuscript Build)

```
# 1. Clone repository
git clone https://github.com/matthewsmawfield/TEP-GL.git
cd TEP-GL

# 2. Build manuscript website
cd site
npm install
npm run build

# 3. Output in site/dist/
```

#### System Requirements

- **Node.js** 16+ (for site building)

- **Storage:** < 50 MB

### Theoretical Framework Documentation

The TEP-GL framework derives from three foundational postulates:

- **Two-Metric Postulate:** Gravitational dynamics and universal causal matter coupling are distinguished by \(g_{\mu\nu}\) and \(\tilde{g}_{\mu\nu}\).

- **Temporal Transport:** Conformal structure produces open-path clock/frequency transport, while residual closed-loop synchronization non-integrability requires a disformal or otherwise non-exact sector.

- **Chronometric Reconstruction:** Temporal-field transport and scalar-induced geometry can project into conventionally inferred spatial mass.

### Software Versions

- **Node.js** 16+

- **Python** 3.8+ (optional utilities)

## Appendix A: Mathematical Derivations

### A.1 Proof of Null Cone Invariance in the Conformal Limit

**Proposition:** If two metrics are conformally related by \(\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu}\), they share the same null geodesics as unparameterized curves.

**Proof:**

Let \(k^\mu = dx^\mu / d\lambda\) be a null vector in \(g_{\mu\nu}\), satisfying \(g_{\mu\nu}k^\mu k^\nu = 0\) and the geodesic equation \(k^\nu \nabla_\nu k^\mu = 0\) (where \(\nabla\) is the Levi-Civita connection of \(g\)).

In the metric \(\tilde{g}_{\mu\nu}\), the null condition holds immediately:

\begin{equation} \label{eq:gl_null_condition}
\tilde{g}_{\mu\nu} k^\mu k^\nu = A^2(\phi) g_{\mu\nu} k^\mu k^\nu = 0
\end{equation}

The connection coefficients \(\tilde{\Gamma}^\lambda_{\mu\nu}\) for \(\tilde{g}\) are related to \(\Gamma^\lambda_{\mu\nu}\) by:

\begin{equation} \label{eq:gl_christoffel}
\tilde{\Gamma}^\lambda_{\mu\nu} = \Gamma^\lambda_{\mu\nu} + \delta^\lambda_\mu \partial_\nu \ln A(\phi) + \delta^\lambda_\nu \partial_\mu \ln A(\phi) - g_{\mu\nu} g^{\lambda\sigma} \partial_\sigma \ln A(\phi)
\end{equation}

Substituting this into the geodesic equation for \(\tilde{g}\):

\begin{equation} \label{eq:gl_geodesic_full}
k^\nu \tilde{\nabla}_\nu k^\mu = k^\nu \nabla_\nu k^\mu + 2 k^\mu k^\nu \partial_\nu \ln A(\phi) - g_{\nu\sigma} k^\nu k^\sigma \dots
\end{equation}

Since \(k\) is null (\(g_{\nu\sigma}k^\nu k^\sigma = 0\)) and \(k^\nu \nabla_\nu k^\mu = 0\), this yields:

\begin{equation} \label{eq:gl_geodesic}
k^\nu \tilde{\nabla}_\nu k^\mu = 2(k^\nu \partial_\nu \ln A(\phi)) k^\mu
\end{equation}

This is the geodesic equation with a non-affine parameterization (\(Dk/d\lambda \propto k\)). Thus, the curve is a geodesic of \(\tilde{g}\), differing only by the parameterization (the clock rate). \(\square\)

---

*This document was automatically generated from the TEP-GL research site. For the interactive version with figures and enhanced formatting, visit: https://mlsmawfield.com/tep/gl/*

*Related Work:*
- [**TEP Theory**](https://doi.org/10.5281/zenodo.16921911) (Foundational framework)

*Source code and data available at: https://github.com/matthewsmawfield/TEP-GL*

---

<!-- ===================================================================== -->
<!-- CANONICAL URL: https://mlsmawfield.com/tep/gte/ -->
<!-- MARKDOWN URL:  https://mlsmawfield.com/tep/gte/manuscript.md -->
<!-- ===================================================================== -->

# Global Time Echoes: Empirical Synthesis
**Matthew Lukin Smawfield**
Version: v0.8 (Singapore)
First published: 21 December 2025 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.18004832

---

## Abstract

Analysis of 25.3 years of GNSS timing data (2000–2025) reveals a
persistent, distance-structured correlation in global atomic clock
networks that tests whether corrected distributed GNSS timing residuals
contain a coherent spatial covariance beyond the standard relativistic
and geodetic measurement model. Examination of 165.2 million station-pair
observations from 474 unique receivers (814 station codes) over a
25.3-year baseline demonstrates a spatial correlation signal decaying
with distance over a finite scale (Gaussian-family kernels preferred
in the 25-year dataset, exponential ΔAIC = 12.8; power law excluded
at ΔAIC > 30; azimuth-averaged exponential scale
λ = 3,210 ± 389 km, with cross-sector directional dispersion
λ<sub>T</sub> = 4,201 ± 1,967 km across eight azimuthal sectors, a measurement
of order thousands of kilometres rather than a single fixed scale), following
cross-centre validation yielding R² = 0.92–0.97 across three distinct
analysis-centre software architectures on a 62.7 million observation baseline. These findings emerge from a systematic multi-paper research
program: theoretical framework development with pre-specified expectations and theoretical search ranges
(Paper 0), multi-center consistency across software-diverse product families within
the shared IGS tracking ecosystem (Paper 1), 25-year longitudinal analysis enabling long-period
geophysical detection (Paper 2), raw-data consistency test strongly constraining
precise-product processing artifacts (Paper 3), and cosmological extension testing whether temporal covariance
or gradient effects contribute to dark-sector phenomenology (Paper 4).

The evidence programme combines timing-network signatures, environmental tests, and an optical sensitivity bound:
distance-structured spatial decay over a finite scale;
East-West/North-South anisotropy (ratio 2.16 ± 0.23 across the 25.3-year aggregate; nominal pair-count p $\lt$ 10<sup>−15</sup>);
orbital velocity coupling (r = −0.888, 3.65σ autocorrelation-corrected;
surrogate p $\lt$ 2×10⁻⁷ resolution bound); a recovered annual-phase
direction near Earth's aphelion velocity tangent (RA = 6°, Dec = +4°;
directional-rank permutation p $\lt$ 10⁻⁵, with the directed CMB-apex template
anti-phase, the Solar Apex geometrically incompatible, and MGEX projection-degenerate);
outer-planet opposition coupling (40-event opposition subset null-validated at 5.5× over matched random dates, p $\lt$ 10⁻¹⁷, alongside a continuous planetary-potential correlation r = +0.116, p = 0.012, whereas the broader 156-event free-centre pulse scan reflects calendar-position sampling on the seasonal series); 18.6-year lunar nutation coupling (R² = 0.641 ± 0.10 over 1.36 cycles); an optical-domain sensitivity bound from passive Satellite Laser Ranging (sampling-matched controls reproduce the apparent spectral concentration; inter-station candidates remain exploratory); pulsar spin-down
excess (exploratory channel); and host-potential H<sub>0</sub> correlation. On the held-out MGEX product the distance-structured coherence replicates (λ ≈ 1,862 km), while the anisotropy and primary orbital-velocity metrics do not replicate robustly. Raw RINEX Single Point
Positioning with broadcast ephemerides achieves consistent signal detection
across all 72 metric combinations (t-statistics up to 112, Cohen's d up to
0.304), strongly constraining precise-product processing artifacts as the sole origin. Because broadcast ephemerides still carry control-segment orbit information, terrestrial closure is structured in three tiers: the GNSS analyses characterize the macroscopic microwave clock-network covariance; Satellite Laser Ranging supplies a hardware- and orbit-independent optical sensitivity bound ($\delta A \lesssim 2\times 10^{-8}$); and physical-confound forward models together with orbit-free optical-fiber and two-way time-transfer links adjudicate the residual orbit-projection channel at atomic-clock precision. The network's
selectivity profile—sensitive to velocity-dependent dynamics while blind
to GM/r² scaling and solar rotation—characterizes it as an inertial
interferometer measuring correlation geometry rather than a gravimeter
measuring Newtonian force.

Table 5 scores these observations against the pre-specified
expectations of the Temporal Equivalence Principle: the orbital-velocity
sign, EW > NS ordering, band localization, and GM/r² null are met,
while the 2,000–3,000 km screening-parameterized forecast, the $\lt$ 5%
cross-centre criterion, and—in the 25-year data—the exponential-kernel
preference are not met. TEP is a bi-metric scalar-tensor framework in which
proper time is a dynamical field governed by a conformal factor A(ϕ) =
exp(β<sub>A</sub>ϕ/M<sub>Pl</sub>). Temporal Shear suppression operates through the
continuous spatial profile of the ϕ field (Temporal Topology), with
suppression arising from the non-linear superposition of field gradients
(Temporal Shear), replacing discrete thin-shell approximations with a
geometrically continuous mechanism. The observed correlation length λ<sub>T</sub>
represents the characteristic scale of Temporal Topology. Environmental
screening, derivative-screened, and saturation-radius interpretations are
candidate theoretical completions, not the primary empirical claim; the
saturation-radius interpretation R<sub>T</sub> is adopted in the present
framework (§4.4.2).
The framework preserves local Lorentz invariance while predicting
spatially correlated clock-network residuals in the conformal sector and
closed-loop synchronization holonomy in the disformal/non-exact sector.
The measured GNSS covariance is interpreted primarily as a
conformal-sector \(A(\phi)\) observable; it should not be identified
with the non-zero closed-loop holonomy predicted only by the
non-exact/disformal sector. The empirical evidence presented here
therefore supports the conformal sector alone; the disformal-sector
holonomy — the framework's definitive discriminating test — remains
unmeasured at the canonical amplitude (the J0437 phase-closure
detection attests non-exact transport but sits roughly two orders of
magnitude above the canonical connection's ceiling on that
sightline).

**Evidence Tiers:** A (Paper 0/EXP): Foundational theory with
pre-specified search ranges; B (GNSS I–III): Primary timing-network
evidence; C (SLR): Optical-domain bound and exploratory correlation test; D (H₀, COS, WB):
Environmental response tests; E (JWST, GL, UCD, RBH): Conditional
astrophysical/cosmological extensions. These evidence channels are
not statistically independent: the exponential decay, anisotropy, orbital
coupling, preferred-axis recovery, nutation, and planetary-catalogue signatures share
the same GNSS data and phase-alignment method, while the SLR check,
pulsar spin-down, and host-potential H<sub>0</sub> correlation
use distinct data — the effective number of independent tests is
smaller than the number of listed results, and smaller still at channel level: the SLR check
reads the same conformal clock sector through different hardware, the
CMB axis is product-unstable, and the pulsar channel is exploratory
(its standalone evidence uninformative — mildly GR-favouring — in the
J0337 adjudication, where the kinetic impedance solved in Paper 0 suppresses the differential acceleration ratio to $\eta \simeq 2.5 \times 10^{-7}$, well within the experimental bound $2.6 \times 10^{-6}$). Critically, the conformal sector responsible for clock-rate
modulation is not directly constrained by photon–graviton differential-propagation tests such as GW170817, although local conformal-gradient/source-charge sectors remain constrained by PPN, clock, and equivalence-principle tests.

If validated through independent replication, TEP predicts that the dynamical and optical phenomenology attributed to dark matter is the observational projection of Temporal Topology and Temporal Shear across unstretched space, not evidence for a new particulate matter component. In this interpretation, GNSS-scale covariance, dynamical phantom mass, and unstretched-space optical refraction (Papers 4, 18, 19; where intermediate clock rates $A_{\rm loc}(u_{\rm grav}(\mathbf x))$ along the shared ADM lapse $N_{\rm TEP}=A_{\rm loc}N_{\rm GR}$ at fixed spatial 3-metric $\gamma_{ij}$ refract light via $n(x)\simeq 1-2\Phi_N/c^2-\beta_A u_{\rm grav}(x)$, whereas isotropic conformal rescaling $\tilde g_{\mu\nu}=A^2 g_{\mu\nu}$ cancels out of null rays) are different observational projections of the same time-field topology, with Papers 4, 6, 13, and 19 testing distinct optical and dynamical projections of the environmental-response architecture on the selected two-branch action $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$ fixed by $a_0$, zero per-system free parameters), not a derived Earth-to-galaxy covariance-to-shear transfer kernel. The 4,000 km correlation on Earth and galactic-scale phenomenology are interpreted within TEP as manifestations of the same environment-dependent Temporal-Topology response, which provides continuous Temporal Topology relaxation from deep potential wells to the weak-field regime. The saturation-density scale ρ<sub>T</sub> ≈ 20 g/cm³, and the diagnostic radius R<sub>T</sub> = (3M/4πρ<sub>T</sub>)<sup>1/3</sup> it defines for each body, remain testable through interplanetary missions. Explicit falsification criteria include: failure of independent groups to replicate the raw carrier-phase signal; Temporal Topology correlation length falling outside the 500–20,000 km range; adjudication of the ephemeris-artifact question — GNSS-product provenance via the Satellite Laser Ranging check, and the wider orbit-projection confound via the physical-confound forward model or an orbit-free clock channel; and null synchronization holonomy in closed-loop triangular time-transfer experiments.

*Keywords:* Temporal Equivalence Principle, GNSS, atomic clocks,
preferred direction, dark matter, gravitational lensing, scalar-tensor
gravity, synchronization holonomy, Temporal Topology, Temporal Shear,
nonlinear screening mechanisms

## 1. Introduction

## 1.1 The Synchronization Residual Problem

Modern atomic clocks have achieved extraordinary precision, with optical
lattice clocks demonstrating fractional frequency stability at the 10<sup>−18</sup>
level—sufficient to detect gravitational redshifts from centimeter-scale
height differences (Bothwell et al. 2022). Yet despite this remarkable local
precision, global clock synchronization residuals plateau at
10<sup>−15</sup>, three orders of magnitude above the fundamental limit.
This discrepancy is conventionally attributed to systematic noise arising
from ionospheric fluctuations, tropospheric delays, multipath interference,
and instrumental drift.

The Global Navigation Satellite System represents humanity's most extensive
precision timing network, comprising hundreds of ground stations that
maintain nanosecond-level synchronization across thousands of kilometers
(Hofmann-Wellenhof et al. 2008; Teunissen & Montenbruck 2017). Standard
processing pipelines apply sophisticated corrections for all known
relativistic effects—gravitational redshift, velocity-dependent time
dilation, Sagnac rotation, and Shapiro delay—following IERS Conventions to
millimeter precision (Petit & Luzum 2010). After these corrections, residual
correlations are expected to be consistent with random noise.

Analysis of 25.3 years of GNSS timing data reveals a different picture. A
systematic, distance-structured correlation persists in the residuals,
characterized by a decay length of approximately 4,000 km. Standard GNSS
processing explicitly suppresses spatial correlations to optimize
positioning accuracy. The signal reported here survives precisely because it
is geometric and differential rather than energetic or common-mode, placing
it outside the correction basis of conventional pipelines. This correlation
has been present in GNSS data since the network's inception but was
categorized as systematic error rather than physical phenomenon. The present
paper argues that this unexplained correlation structure represents a
physical signal—specifically, the signature of a dynamical time field
predicted by scalar-tensor extensions of general relativity.

## 1.2 Historical Context

If this signal is physical, why was it not identified in the first decades
of GNSS operation? Three developments converged to enable its detection.

The first is temporal baseline. The 25.3-year dataset now available spans
1.36 complete cycles of the 18.6-year lunar nutation, enabling first access to
long-period geophysical coupling that was inaccessible to earlier studies —
though the sub-two-cycle record bounds each phase bin to at most two
independent cycle realizations, limiting the coupling measurement to
R² = 0.641 ± 0.10.
Analyses conducted prior to 2010 lacked sufficient temporal leverage to
distinguish such signals from transient artifacts.

The second is multi-constellation cross-check. The maturation of
GLONASS, Galileo, and BeiDou provides independent verification across
different satellite geometries and processing chains. Single-constellation
analyses are vulnerable to constellation-specific artifacts that
multi-constellation cross-check can identify and exclude.

The third is methodological. Standard GNSS processing employs Kalman
filtering and network adjustment algorithms designed to eliminate
correlations, thereby optimizing positioning accuracy. The present analysis
instead preserves correlation structure through phase-coherent
cross-spectral methods applied to residuals. The signal was present
throughout the history of GNSS operation but was categorized as systematic
error rather than physical phenomenon.

## 1.3 Theoretical Motivation

The standard relativistic framework treats proper time as a derived
quantity: given the metric tensor g<sub>μν</sub>, proper time along any
worldline is uniquely determined by integration. This formulation assumes
that time transport is integrable—that synchronization around any closed
loop returns to the initial epoch after accounting for known effects such as
Sagnac rotation, Shapiro delay, and gravitational redshift.

Scalar-tensor theories of gravity, developed extensively over the past three
decades (Bekenstein 1993; Damour & Polyakov 1994; Khoury & Weltman 2004),
predict modifications of proper time accumulation to vary spatially. If this
assumption proves empirically false in the presence of conformal metric
couplings, apparent dark matter could arise as a geometric artifact:
time's effect on moving matter read as gravity from unseen mass. If
proper time is not
merely derived from geometry but is itself influenced by a dynamical field,
synchronization becomes path-dependent in ways that standard general
relativity does not predict.

The Temporal Equivalence Principle formalizes this possibility. It states
that all non-gravitational dynamics, signals, and quantum phases evolve
according to the proper time defined by a matter metric g̃<sub>μν</sub> that
couples universally to matter through the scalar field. In local freely
falling frames, physics reduces to special relativity with invariant c;
globally, the flow of time becomes path-dependent in ways that distributed
timing networks can detect.

### 1.3.1 Research Program Structure

This integrated manuscript synthesizes results from a systematic multi-paper
research program designed to establish, validate, and independently confirm
the TEP predictions through progressively rigorous empirical tests and
cosmological extension. The program follows a theory-first methodology:
pre-specified expectations and theoretical search ranges established before empirical analysis, followed by
multiple independent checks using complementary methodologies,
culminating in cosmological implications.

Paper 0: Theoretical Foundation (Smawfield 2025a, DOI:
10.5281/zenodo.16921911) — The Temporal Equivalence Principle framework was
developed independently of any GNSS analysis, establishing pre-specified expectations and theoretical search ranges including a Temporal Topology correlation length, exponential
spatial decay form, velocity-dependent anisotropy, and absence of GM/r²
scaling. Screening operates via the continuous spatial profile of the φ field
(Temporal Topology), with Temporal Shear suppression arising from the non-linear superposition of field
gradients (Temporal Shear) rather than discrete thin-shell boundaries,
suppressing Temporal Shear where the ambient time-gradient is steep
(weak-field: high-acceleration regions) while leaving cosmology
accessible to dynamics. This theory-first approach ensures that subsequent
empirical findings represent genuine predictions rather than post-hoc
explanations.

Paper 1: Multi-Center Consistency (Smawfield 2025b, DOI:
10.5281/zenodo.17127229) — Initial empirical investigation analyzed 62.7
million station-pair measurements from 364 GNSS stations across three
distinct analysis-center solutions (CODE, IGS, ESA) over 2.5 years (2023–2025).
Cross-center consistency (R² = 0.92–0.97) established that the correlation
structure is not processing-specific, with λ<sub>T</sub> = 3,330–4,549 km falling within
the predicted range. This multi-center approach excluded implementation-specific
artifacts as a viable explanation; systematics residing in the shared
constellation geometry are addressed separately through multi-constellation
and raw-RINEX replication (Papers 14 and 3).

Paper 2: 25-Year Longitudinal Analysis (Smawfield 2025c, DOI:
10.5281/zenodo.17517141) — Extended temporal baseline to 25.3 years
(2000–2025) using CODE products, analyzing 165.2 million station pairs from
474 unique receivers. This long-span analysis enabled detection of
previously inaccessible signatures including 18.6-year lunar nutation
coupling (R² = 0.641), and a recovered annual-phase direction near Earth's
aphelion velocity tangent; the directed CMB-apex template is anti-phase and the Solar Apex is
geometrically incompatible, predicting N-S anisotropy against the observed
E-W dominance), and confirmed temporal stability of
the correlation structure with cross-sector dispersion λ<sub>T</sub> = 4,201 ± 1,967 km. The 25-year baseline
provided access to long-period geophysical phenomena impossible to detect in
shorter studies.

Paper 3: Raw RINEX Test (Smawfield 2025d, DOI:
10.5281/zenodo.17860166) — Processing artifact constraint through analysis
of raw GNSS observations using Single Point Positioning with broadcast
ephemerides only. Because broadcast ephemerides still contain control-segment
orbit information, terrestrial verification is structured across three tiers:
the GNSS analyses (Papers 1–3, 14) establish and characterize the macroscopic
clock-network covariance; passive Satellite Laser Ranging (Paper 8) provides a
GNSS-product-independent optical sensitivity bound and orbit-provenance
check; and the physical-confound forward model together with satellite-free
optical-fiber and TWSTFT clock channels adjudicate the generic
orbit-projection confound at full metrological precision.
Analysis of 1.17 billion pair-samples
across 539 stations (2022–2024) achieved consistent signal detection across all 72
metric combinations (t-statistics up to 112, Cohen's d up to
0.304), confirming the signal exists in fundamental observables independent
of sophisticated processing chains. This raw-data consistency test strongly
constrains precise-product processing artifacts as the origin.

Paper 4: Cosmological Implications (Smawfield 2025e, DOI:
10.5281/zenodo.17982540) — Developed the connection between TEP and
gravitational lensing, demonstrating how temporal-field gradients (active
Temporal Shear) and intermediate temporal depth across unstretched space can produce apparent dark matter phenomenology in both the
dynamical and optical sectors without particulate dark matter (contrasted with the isotropic-conformal benchmark where $A^2(\phi)$ cancels out of null rays and lensing tracks baryons alone). This paper
formulated the Isochrony Axiom as a foundational closure assumption in
standard lensing analysis that TEP subjects to direct empirical stress-testing, identified "Phantom
Mass" — the apparent mass inferred when temporal structure is analysed as
spatial mass — and established the
dynamical–lensing mass comparison and unstretched-space temporal refraction tests, together with the
connection between the ~4,000 km GNSS correlation and galactic-scale phenomenology through the 𝒮<sub>Σ</sub> screening operator. The continuous relaxation of Temporal Topology
from deep potential wells to the weak-field regime unifies terrestrial and
galactic scales. This cosmological
extension transforms TEP from a terrestrial metrology curiosity into a
framework with profound implications for dark matter and cosmological
tensions.

### 1.3.2 Related Literature and Prior Anomalies

The GNSS correlation structure documented in this manuscript is not the first
observational anomaly consistent with TEP dynamics. Peer-reviewed literature
contains multiple documented deviations from standard relativistic predictions
that were marginalized as systematic artifacts because ΛCDM lacks the
requisite theoretical framework to accommodate them. Recontextualized
through TEP, these isolated findings form a coherent pattern of empirical
support:

**GPS One-Way Propagation and the Synchronization Problem.**
Kelly (2009), in a peer-reviewed analysis of GPS timing, demonstrated that
one-way light signals circumnavigating Earth eastward take 414.8
nanoseconds longer than westward signals at the equator—measuring c − v
eastward and c + v westward relative to Earth's surface. The Sagnac
effect demonstrates operationally that extended synchronization is
geometry- and frame-dependent: eastward and westward propagation times
around a rotating Earth are unequal before the standard rotational
correction is applied. This known GR effect is not itself evidence for
TEP. Instead, it illustrates why a genuine TEP test must construct a
residual closed-loop observable after subtracting the modeled Sagnac,
Shapiro, gravitational-redshift and gravitomagnetic contributions. Any
remaining non-zero residual would probe non-exact temporal transport
beyond the standard model (Section 6.3.3).

**SLR Network Time Biases.** Exertier et al. (2017) identified
systematic time biases of 3–4 nanoseconds (1σ) across 25 Satellite Laser
Ranging stations, producing 4–6 mm coordinate discrepancies that persist
after all known corrections. These biases are "often neglected" and
"presumed to be $\lt$ 100 ns" but resist precise estimation. TEP interprets
these as *Temporal Topology relaxation*: each station at different
gravitational potential experiences differential proper-time accumulation,
consistent with the screening field producing GNSS correlations.

**GPS Clock Gravitational Redshift Residuals.** Fathollahi
et al. (2023) reported gravitational redshift tests using GPS Block IIF Rb
clocks showing fractional deviations from GR of (0.23 ± 1.34) × 10<sup>−3</sup>,
with 2.1–7.3% of data flagged as outliers requiring removal. The authors
noted "annual oscillation and short-term fluctuations" that necessitated
16-day moving average smoothing. TEP predicts precisely such phase-locked
annual modulations and outlier populations from velocity-dependent temporal-field
coupling (Section 4.2).

**Multi-GNSS System-Specific Clock Behavior.** Ai et al. (2021)
documented that GPS, Galileo, BeiDou, and GLONASS exhibit *distinct*
noise profiles (random walk, flicker, white frequency modulation) despite
using similar atomic clock technologies. Standard frameworks attribute this
to hardware differences. TEP predicts system-specific residuals because
each constellation operates at different orbital altitudes and inclinations,
sampling different clock-rate structure of the temporal field.

**GNSS Common-Mode Periodic Signals.** Santamaria-Gomez et al.
(2017) identified system-specific periodic signals in coordinate time series
arising from "different orbital plane structures, orbit repeatability and
incomplete constellation," achieving 52% horizontal reduction after
system-specific correction. These were attributed to unmodeled geophysical
loading. TEP predicts orbital-period modulations from satellites sampling
spatially-varying temporal fields at different inclinations—precisely the
pattern identified but unexplained in the standard literature.

The convergence of these independent anomalies—each documented in
peer-reviewed literature, each marginalized within ΛCDM, each consistent with
TEP predictions—substantiates the framework's empirical foundation beyond
the primary GNSS correlation analysis presented in this manuscript.

The present manuscript integrates these five studies into a unified
narrative, demonstrating that the TEP framework established pre-specified expectations and theoretical search ranges
before empirical analysis, that three independent empirical
investigations using progressively rigorous methods are each consistent with
the expected signatures, and that the framework's cosmological extension
offers testable expectations for gravitational lensing and dark matter
phenomenology. This theory-expectation-evidence-cosmology sequence, with
each empirical study addressing distinct artifact hypotheses
(processing-specific, temporal transience, sophisticated processing) and the
cosmological paper extending implications beyond terrestrial scales, follows
the standard structure of scientific hypothesis development and theoretical
refinement.

## 1.4 Prediction Timeline

The TEP framework was developed independently of the GNSS analysis, with
pre-specified expectations and theoretical search ranges established in prior theoretical work
(Smawfield 2025a). The predicted Temporal Topology correlation length range of
λ<sub>T</sub> = 1,000–10,000 km was derived from environmental screening theory before any
GNSS data analysis was conducted.

The chronology is significant: the TEP theory paper was first deposited in
August 2025 (v0.1, DOI: 10.5281/zenodo.16921911), and the specific
clock-network correlation forecast—exponential decay with a
screening-parameterized length and multi-center cross-validation—entered
the versioned record on 15 September 2025 (v0.4, DOI:
10.5281/zenodo.17121655), one week before the first public deposit of the
GNSS analysis (Paper 1 v0.6α, 22 September 2025). The multi-center GNSS
analysis was conducted from September through November
2025; the 25-year CODE longspan analysis was completed in November 2025; and
raw RINEX test confirmed the signal in fundamental observables in
December 2025. The observed Temporal Topology correlation-length family (cross-sector dispersion λ<sub>T</sub> = 4,201 ± 1,967 km) falls
within the pre-specified search range, a non-discriminating consistency statement since the range subsumes leading systematic scales. The full expectation-by-expectation mapping to versioned deposits is
tabulated in the prediction provenance register (Section 6.5).

## 1.5 The Temporal Equivalence Principle

TEP extends Einstein's geometrization of space to the rate of time itself.
The principle states that all non-gravitational dynamics, signals, and
quantum phases evolve according to the proper time defined by a single
causal metric g̃<sub>μν</sub> that couples universally to matter. The rate at
which proper time accrues is governed by a scalar field φ through a
conformal factor A(φ). In local freely falling frames, physics reduces to
special relativity with invariant c; globally, the flow of time becomes
path-dependent.

This framework preserves all established physics locally—the speed of light
remains exactly invariant in any freely falling laboratory—while predicting
novel global effects that manifest as correlations in distributed timing
networks. Crucially, the conformal sector of TEP, which governs clock-rate
modulation, remains unconstrained by multi-messenger observations such as
GW170817, which bound only disformal (cone-tilting) effects.

## 1.6 Scope and Non-Claims

To avoid misinterpretation, the following claims are explicitly NOT made by
this work:

Detection of a fifth force acting on test masses (the network responds
to geometric configuration, not gravitational amplitude; GM/r² and GM/r³
scaling are both null)

Violation of local Lorentz invariance (the speed of light remains
exactly invariant in any freely falling laboratory; TEP preserves local
special relativity)

Breakdown of the equivalence principle in laboratory tests (universal
coupling ensures the weak equivalence principle is preserved exactly;
the TEP synodic-phase residual
$\kappa_{\rm LLR} \approx -3.91 \times 10^{-4}$ from the residual-channel
LLR analysis (Paper 17) is a post-fit residual proxy whose relationship
to the standard Nordtvedt parameter has not yet been established — it
is not directly comparable to the direct-fit LLR bound
$|\eta| \lt 4.4 \times 10^{-4}$ (Williams et al. 2004), and the
source-level integrator refit is required to determine whether the
residual-channel signal corresponds to a physical Nordtvedt parameter
within that bound)

Modification of light or gravitational wave propagation speeds (GW170817
constrains |c<sub>γ</sub> − c<sub>g</sub>|/c ≲ 10<sup>−15</sup>; TEP satisfies this
through disformal coupling B(φ) constrained to be small today by multi-messenger observations)

Violation of energy-momentum conservation (total stress-energy is
conserved; apparent non-conservation in the Einstein frame is resolved
by transformation to the Jordan frame where matter follows geodesics of
g̃<sub>μν</sub>)

TEP is a conformal scalar-tensor theory that modulates the rate of proper
time accumulation while preserving all local physics. The GNSS signal
reflects spatial correlations in clock rates arising from the scalar field's
two-point correlation function, not violations of established physical
principles.

## 1.7 Claim Hierarchy

#### Tier 1: Empirical Claim (Primary)

GNSS clock residuals exhibit a persistent, distance-structured,
anisotropic spatial correlation with characteristic decay scale
λ<sub>T</sub> = 4,201
± 1,967 km (cross-sector dispersion; azimuth-averaged exponential
λ = 3,210 ± 389 km). This correlation is robust across three software- and
estimator-diverse product families within the shared IGS tracking
ecosystem (CODE, IGS, ESA; R² = 0.92–0.97), persists over 25.3
years spanning multiple satellite constellation changes, and appears in
raw RINEX observations processed with Single Point Positioning (consistent
signal detection across all 72 metric combinations, t-statistics up to 112).
The programme combines the following measurements and checks:
distance-structured spatial decay over a finite scale, East-West/North-South anisotropy (ratio 2.16), orbital
velocity coupling (r = −0.888, 3.65σ autocorrelation-corrected;
surrogate p $\lt$ 2×10⁻⁷ as a resolution bound), a recovered annual-phase
direction near Earth's aphelion velocity tangent on the CODE product
(the directed CMB-apex template is anti-phase; the MGEX-product scan is
projection-degenerate), an SLR optical sensitivity bound with exploratory inter-station correlations (the spectral concentration is reproduced by sampling-matched noise),
planetary event catalogue (56/156 significant under the free-centre
statistic; catalogue excess attributable to calendar-position sampling
in season-preserving controls — a bounded channel whose continuous
planetary-potential correlation survives at r = +0.116, p = 0.012), 18.6-year lunar
nutation coupling (R² = 0.641 ± 0.10), pulsar spin-down excess
(exploratory channel), and
host-potential H<sub>0</sub> correlation.

#### Tier 2: Interpretive Claim (Secondary)

The observed correlation structure is inconsistent with known
atmospheric effects (ionosphere, troposphere), instrumental systematics
(geomagnetic storms, altitude dependence), classical gravitational
effects (GM/r² force scaling, GM/r³ tidal scaling), solar activity (r ≈
0 with solar flux), and processing artifacts (persists in raw SPP data).
The signal's selectivity profile—sensitive to velocity-dependent
dynamics and a stable preferred direction while blind to mass scaling
and solar rotation—distinguishes it from conventional systematic errors. However,
the possibility of a presently unmodeled GNSS systematic tied to
fundamental constellation geometry cannot be definitively excluded
without independent non-GNSS check (Section 3.6).

#### Tier 3: Theoretical Hypothesis (Tertiary)

The observations are consistent with the Temporal Equivalence Principle
(TEP), a bi-metric scalar-tensor framework in which proper time is
modulated by a dynamical conformal field A(ϕ) = exp(β<sub>A</sub>ϕ/M<sub>Pl</sub>).
TEP established pre-specified expectations and theoretical search ranges before empirical analysis (λ<sub>T</sub> =
1,000–10,000 km, exponential decay, velocity-dependent anisotropy,
absence of mass scaling). Several key pre-specified structural expectations are met, while Table 5 records the forecasts that are not met. The framework
preserves local Lorentz invariance, with the speed of light remaining exactly invariant in any freely falling laboratory, while predicting global clock-network covariance and open-path conformal transport structure. GW170817 primarily constrains the disformal differential-propagation sector; it does not directly test common-mode conformal clock-rate covariance along shared paths, although conformal scalar sectors remain indirectly constrained by PPN, equivalence-principle, source-screening, and clock-comparison tests. This hypothesis is testable through independent experiments: SLR check (Section 6.3.1.1), multi-constellation cross-check, and triangle holonomy test (Section 6.3.3). The validity of Tier 1 (empirical observations) does not depend on the correctness of Tier 3 (TEP interpretation).

This paper establishes Tier 1 through extensive empirical testing, argues for Tier
2 through systematic exclusion of conventional explanations, and tests
consistency with Tier 3 through pre-specified expectations. Independent
replication is essential for all three tiers.

## 1.8 Paper Structure

This paper presents evidence for TEP through systematic analysis of GNSS
timing data. Section 2 characterizes the phenomenology of the observed
signal. Section 3 tests the signal against artifact hypotheses through
multiple complementary tests. Section 4 presents the theoretical framework and
its predictions. Section 5 develops the cosmological implications, including
connections to gravitational lensing and dark matter phenomenology. Section
6 specifies explicit falsification criteria and outlines the experimental
path to a definitive verdict.

Falsification is central to this work. Concrete rejection criteria are
specified, including failure of independent groups to replicate the signal
in raw carrier-phase data, Temporal Topology correlation length falling
outside the 500–20,000 km range, adjudication of the ephemeris-artifact
question — GNSS-product provenance via the Satellite Laser Ranging check,
and the wider orbit-projection confound via the physical-confound forward
model or an orbit-free clock channel — and null synchronization holonomy in
closed-loop triangular
time-transfer experiments.

## 2. Phenomenology

This section presents the empirical findings without theoretical interpretation. The signal characteristics are established through three complementary empirical studies, each designed to address distinct testing requirements:

Paper 1 (Multi-Center Consistency, 2023–2025): Cross-validation across three software- and estimator-diverse product families (CODE, IGS, ESA) within the shared IGS tracking ecosystem, using 62.7 million station-pair measurements from 364 stations. This study established that the correlation structure is not processing-specific, achieving R² = 0.92–0.97 consistency across distinct processing chains (Bernese, NAPEOS, and the IGS combination product).

Paper 2 (25-Year Longitudinal Analysis, 2000–2025): Extended temporal baseline using CODE products with 165.2 million station-pair observations from 474 unique receivers (814 station codes). This long-span analysis enabled detection of previously inaccessible signatures including 18.6-year lunar nutation coupling and a stable preferred direction, confirming temporal stability over 25.3 years.

Paper 3 (Raw RINEX Test, 2022–2024): Processing artifact constraint
through analysis of raw GNSS observations using Single Point Positioning
with broadcast ephemerides only. Broadcast ephemerides still contain
control-segment information, so Satellite Laser Ranging and non-GNSS optical
checks remain necessary for adjudication — SLR offers a
GNSS-product-independent orbit-determination chain, with signal recovery
at sufficient sensitivity still required, while the generic orbit-projection confound is reserved
for the physical-confound forward model and orbit-free clock channels.
Analysis of 1.17 billion pair-samples across 539 stations achieved consistent signal detection across all 72 metric combinations, confirming the signal exists in fundamental observables.

The convergence of findings across these three complementary methodologies—each addressing different potential artifact sources, with the raw-RINEX channel providing the strongest estimation-chain independence—provides the empirical foundation for the reported signatures.

## 2.1 Spatial Correlation Structure

Clock frequency residuals exhibit systematic spatial structure that persists after removal of all modeled relativistic effects. The phase coherence between station pairs decays exponentially with geodesic distance according to:

$$C(r) = A \cdot \exp(-r/\lambda_{\mathrm{T}}) + C_0$$

where r denotes the inter-station distance, λ<sub>T</sub> the Temporal Topology correlation length, A the amplitude, and C<sub>0</sub> the baseline offset. Fits to distance-binned means across approximately 28 bins achieve R² = 0.920–0.970 across all three analysis centers.

The correlation metric employed is the magnitude-weighted phase alignment index, computed via cross-spectral density analysis in the 10–500 μHz band (corresponding to periods of 33 minutes to 28 hours). This approach measures whether clock fluctuations are in phase regardless of amplitude—information that survives GNSS processing because network adjustment removes common-mode offsets while preserving differential phase structure. Magnitude weighting ensures that frequency bins with stronger cross-spectral power contribute proportionally more to the phase average, so the metric reflects genuine correlated signals rather than noise.

The primary result from the CODE 25-year analysis is a Temporal Topology correlation length of λ<sub>T</sub> = 4,201 ± 1,967 km. Cross-center validation on the 2023–2025 multi-center sample (exponential fits to binned phase-alignment means) yields: CODE λ<sub>T</sub> = 4,549 km (95% CI: 1,198–5,918 km); IGS λ<sub>T</sub> = 3,764 km (95% CI: 3,197–4,871 km); ESA λ<sub>T</sub> = 3,330 km (95% CI: 2,532–3,984 km). The 25-year CODE long-span headline is a separate estimand from these center-specific exponential fits. The coefficient of variation across the three center fits is ~13%, indicating consistency across software-diverse processing chains and substantially reducing the likelihood of a CODE-specific implementation artifact (shared IGS network/datum systematics remain live alternatives).

## 2.2 Timing Signatures and Cross-Domain Checks

Table 1 combines the longitudinal GNSS signatures with companion environmental tests and the current SLR bound. Each signature constrains the space of
viable explanations at its own evidential strength: the timing-network
rows are the primary evidence, while the SLR row is an instrument-level
provenance check within the same conformal clock sector, the preferred-direction
recovery is product-dependent across GNSS products, and the pulsar row is an
exploratory consistency channel; their collective pattern consistently points toward a
velocity-dependent, preferred-axis geometric mechanism.

| Signature | Observed Value | Significance |
| --- | --- | --- |
| Distance-Structured Decay | λ = 3,210 ± 389 km (azimuth-avg. exponential); λ<sub>T</sub> = 4,201 ± 1,967 km (cross-sector dispersion) | R² = 0.92–0.97 |
| Spatial Anisotropy | EW/NS = 2.16 | p $\lt$ 10<sup>−15</sup> |
| Orbital Velocity Coupling | r = −0.888 | 3.65σ (autocorr.-corrected); surrogate p $\lt$ 2×10⁻⁷ (resolution bound) |
| Annual-Phase Direction Recovery | RA = 6°, Dec = +4°; ~6° from Earth's aphelion velocity tangent | Directional-rank permutation p $\lt$ 10⁻⁵; CMB apex anti-phase; Solar Apex null; no global full-sky-search surrogate reported |
| Optical-Domain Sensitivity Bound | Microwave-to-optical product chains (GNSS/SLR) | Sampling-matched spectral null; exploratory inter-station candidate. Independent hardware and orbit provenance; physical corroboration remains pending (Paper 8) |
| Planetary Alignment Coupling | 40-event opposition subset: 5.5× over random dates (p $\lt$ 10<sup>−17</sup>); continuous r = +0.116 (p = 0.012) | Opposition subset null-validated; GM/r² and GM/r³ scaling null bounds transient Newtonian impulses; broader 156-event free-centre scan (56/156 ≥2σ) descriptive of seasonal-curve sampling (Paper 3) |
| 18.6-Year Nutation | R² = 0.641 ± 0.10 | p = 0.010; permutation p = 0.0016 (1.36 cycles, cycle-limited) |
| Pulsar Spin-Down Excess | κ<sub>MSP</sub> ~ 10<sup>6</sup> (fitted phenomenological coefficient) | Exploratory: globular-cluster density scaling (Paper 10); the standalone pulsar channel is uninformative — mildly GR-favouring (Paper 34, in preparation) |
| Host-Potential H<sub>0</sub> Correlation | Fitted environmental response; corrected H<sub>0</sub> conditional on Paper 11's observing chain and anchor weights | Cepheid response (Paper 11) |

## 2.3 Annual-Phase Direction Recovery

The grid search recovers the phase of an annual directional modulation. Under the corrected prograde orbital-velocity convention the best fit is RA = 6°, Dec = +4° — the antipode of the RA = 186°, Dec = −4° solution reported under the earlier antiparallel convention. The two coordinates describe the same undirected geometric axis, but the sign matters for testing a directed physical template. The corrected direction lies near Earth's aphelion velocity tangent, not the CMB apex.

The recovered direction lies ~6° from Earth's aphelion velocity tangent, ~161.8° from the directed CMB apex, and ~92° from the Solar Apex. Its Pearson correlation is r = 0.7465 (local parametric p = 4.0 × 10<sup>−7</sup>). At the CMB-apex direction the signed correlation is r = −0.5455 (anti-phase); at the Solar Apex it is r = 0.0059 (null). The best-fit direction explains about 16,000 times the variance of the Solar-Apex template.

A 100,000-draw directional-rank permutation test produced no exceedance of the observed r, giving a finite-resolution plus-one bound p $\lt$ 10<sup>−5</sup>. This test ranks the observed annual phase against sampled directions; it is not a surrogate-data calibration of the maximum over a complete full-sky search. No global full-sky-search p-value is therefore claimed. The undirected axis happens to lie ~18° from the CMB axis, but that geometric proximity cannot be used as directed CMB evidence when the CMB-apex template has the opposite sign.

The recovered direction is a product-dependent statement. The MGEX combined
product (Paper 14) returns a nominal maximum 82.7° from the CMB dipole, but
its orbital-velocity geometry is projection-degenerate: 61 of 703 directions
perform at least as well as the direct CMB-labelled template, and the CMB and
nominal-best predictors correlate at r = 0.984. MGEX therefore supports at
most exploratory annual directional structure; it neither localizes a unique
three-dimensional axis nor provides CMB-frame confirmation. What MGEX does
establish is the spatial scale: it independently reproduces the bounded
thousands-of-kilometres correlation length in a distinct combined
multi-GNSS product; its standard anisotropy and primary orbital estimators
are null, while an exploratory model-free PA-difference statistic recovers
the expected annual association.

Within the TEP framework, the recovered direction encodes the phase of the
annual modulation. Its proximity to the aphelion velocity tangent supports
an orbital-velocity reading rather than a fixed external-frame arrow. The
undirected ~18° axis proximity to the CMB axis is geometric only and does not
leave a directed CMB-frame identification open in this analysis.

Reference-frame discrimination under the corrected convention no longer identifies the CMB arrow: evaluated at the CMB-apex direction the model anti-aligns (r = −0.55), while the Solar Apex remains statistically null (r = 0.007, R² = 0.0054%). The recovered best fit instead sits ~6° from Earth's aphelion velocity tangent — the orbital-velocity direction the predictor principally encodes. The geometric argument against the Solar Apex is unchanged: its +30° Dec predicts North-South anisotropy, directly contradicting the observed East-West dominance (EW/NS = 2.16), whereas both the recovered axis and the CMB dipole axis lie near the ecliptic plane. Hemisphere asymmetry corroborates the velocity-channel reading: Southern stations couple at |r| = 0.79 against |r| = 0.25 in the North.

## 2.4 Planetary Event Catalogue: Bounded Channel

Analysis of 156 planetary conjunction and opposition events yields 56 events (35.9%) exceeding 2σ under the pre-specified free-centre pulse statistic, with 19 surviving family-wide Bonferroni correction. These counts quantify the repeatability of the estimator on the seasonal coherence series rather than event-locked physics, for two reasons established in the audit: (i) the fitted Gaussian pulse centre is unconstrained across the ±120-day window and can latch onto off-event seasonal excursions; (ii) the date-shuffled null destroys the autocorrelation of the seasonal curve and therefore under-controls for calendar position. The season-preserving controls of Paper 3 (Step 2.6b) make this explicit in the raw-RINEX channel: uniform-date nulls on the real 3-year series reproduce a 65.5% pulse-fit rate against 59.5% for real events (p = 0.805), and same-day-of-year calendar twins in non-event years reproduce the event rate exactly (59.46% versus 59.46%), for both the mean-coherence and phase-alignment metrics. The apparent catalogue excess is thus attributable to calendar-position sampling of the underlying annual modulation rather than to discrete planetary impulses.

What survives is narrower but well-defined. The matched-null control validates event specificity for the 40-event outer-planet opposition subset under a ±15-day model-free window (5.5× effect-size ratio over 155 random dates, Mann-Whitney p $\lt$ 10<sup>−17</sup>), and a maximal non-overlapping-window subset shows 13/29 significant events versus 1.3 expected (binomial tail p ≈ 10<sup>−10</sup>; Paper 2 Step 2.9) — evidence of repeatable seasonal-window structure, though not separable from calendar position. Separately, the continuous channel is distinct: the daily network-coherence variability correlates with the composite planetary potential at r = +0.116 (autocorrelation-corrected p = 0.012, N<sub>eff</sub> ≈ 472), a long-baseline association that does not reduce to discrete event locking. Individual events can remain anomalous (the March 2025 Venus conjunction stayed date-locked at +3.3 days under a sliding-window placebo test), but such cases are isolated observations, not a catalogue-level detection.

The channel's firmest contribution is the scaling null: neither GM/r² (force) nor GM/r³ (tidal gradient) scaling is detected across the catalogue (all p > 0.5), and the Paper 1 mass-ordering test returns r = −0.156. In the 3-year RINEX window this bounds unmodelled transient planetary perturbations of the coherence field: whatever produces the seasonal modulation is insensitive to which planet is involved. The catalogue therefore functions in this synthesis as a calibrated falsification/upper-bound control — it rules out unscreened impulse coupling to planetary mass or tidal gradient — rather than as an affirmative detection of discrete planetary responses.

### 2.4.1 Candidate Mechanism (Hypothesis): Velocity-Field Gradient Modulation

The residual planetary phenomenology—the validated opposition subset, isolated date-locked events such as Venus March 2025, and the continuous r = +0.116 correlation with the composite potential—could, within TEP's velocity-dependent framework, reflect geometric modulation of Earth's motion through the scalar-field gradient rather than gravitational amplitude. This is presented strictly as a candidate interpretation of the surviving channels, not as an explanation of the full catalogue, whose apparent excess is attributable to calendar-position sampling.

**Physical Mechanism:** Earth moves through the φ field with a 30 km/s orbital component plus any ambient-background motion. This analysis measures the annual orbital phase and does not identify that background with the 369 km/s CMB frame. Planetary configurations create time-varying perturbations to the local field gradient ∇φ through their own screening halos. When planets align with Earth's velocity vector relative to the ambient field, constructive interference may enhance the velocity-dependent anisotropy; when perpendicular, destructive interference may suppress it. The effect depends on the alignment angle θ between the planetary position and Earth's velocity vector:

$$\Delta\lambda(\theta) \propto \lambda_0 \left[1 + \epsilon \cos\theta\right]$$

where ε is a small coupling parameter (~10<sup>−3</sup>) and θ is measured from the recovered preferred direction. This predicts maximum modulation when planets are aligned with or opposite to the preferred-direction vector (θ = 0° or 180°), and null modulation when perpendicular (θ = 90°).

**Why Mass Scaling is Absent:** The mechanism operates through geometric alignment rather than gravitational force. Each planet creates a local perturbation to ∇φ within its own transition scale R<sub>T</sub> ∝ M<sup>1/3</sup>. However, the Earth-planet distance (AU scale) far exceeds any planetary R<sub>T</sub> (Mercury: ~100 km; Jupiter: ~5,000 km), placing Earth in the unscreened regime where the field perturbation has decayed to a threshold-dependent residual. The detection depends on whether this residual exceeds a critical threshold for modulating Earth's local screening length, not on the perturbation's amplitude. This threshold behavior explains why Mercury (smaller M but closer) produces comparable detection rates to Jupiter (larger M but farther).

**Testable Prediction:** Within the surviving channels (the opposition subset and the continuous correlation), response measures should track the alignment angle between planetary position and the preferred direction. A catalogue-wide version of this test must first pass the season-preserving null: stratifying the 156 events by alignment angle provides a falsifiable check only where event locking has been established, since calendar-position effects must be removed before any angular dependence can be read out.

## 2.5 Selectivity Profile: Inertial Interferometer

The pattern of detections and non-detections characterizes the network's sensitivity and provides powerful constraints on viable physical mechanisms. Table 2 summarizes this selectivity profile.

| Observable | Result | Interpretation |
| --- | --- | --- |
| Orbital velocity | r = −0.888, 3.65σ (autocorr.-corrected) | Sensitive to kinematic dynamics |
| Semiannual coherence component | R² = 0.904 | Sensitive to inertial orientation (nutation/orbital harmonic) |
| 18.6-year nutation | R² = 0.641 | Sensitive to long-period dynamics |
| Preferred direction | r = 0.746 at recovered direction | Sensitive to preferred axis |
| Planetary alignments | Catalogue excess reproduced by calendar position; opposition subset 5.5× over random dates | Bounded/control channel — bounds unmodelled transient planetary perturbations; continuous potential correlation r = +0.116 (p = 0.012) remains |
| GM/r² and GM/r³ scaling | Both null (p > 0.5) | Insensitive to gravitational amplitude (force or tidal) |
| Solar rotation (27-day) | Null (r $\lt$ 0.09) | Insensitive to solar surface phenomena |
| Lunar standstill | Null | Insensitive to static geometry |

This selectivity profile is highly discriminating. The network responds to velocity-dependent and orientation-dependent phenomena while remaining blind to gravitational amplitude and solar surface effects. This pattern suggests that the GNSS network functions as an inertial interferometer—measuring velocity-dependent correlation geometry—rather than as a gravimeter measuring Newtonian force.

The distinction is fundamental. A gravimeter would show GM/r² (force) scaling, while a differential tidal sensor would show GM/r³ (gradient) scaling; in both cases, Mercury's influence would be negligible compared to Jupiter's. An inertial interferometer responds to geometric configuration regardless of mass—precisely what is observed. The annual-direction recovery identifies the operative measured phase with Earth's orbital motion through the ambient field; it does not identify a cosmological rest-frame arrow.

## 2.6 Statistical Power

The raw RINEX test analysis achieves extraordinary statistical power. Across 1.17 billion pair-samples, the directional anisotropy yields t-statistics up to 112 with Cohen's d up to 0.304. The signal is detected across all 72 metric combinations with mean R² = 0.93.

Hemisphere asymmetry provides independent corroboration. Southern stations show stronger orbital-velocity coupling (|r| = 0.79) than Northern stations (|r| = 0.25), corroborating the velocity-channel geometry. Both hemispheres independently show East-West exceeding North-South correlation (Northern: 1.20×, Southern: 1.35×), ruling out hemisphere-specific artifacts.

Monthly consistency is equally striking. East-West exceeds North-South in 94% or more of all 36 months analyzed, with short-distance coherence ratios showing coefficient of variation below 1%. The underlying signal is constant; the annual modulation in full-distance Temporal Topology correlation lengths reflects atmospheric screening effects that vary seasonally.

## 3. Testing

Extraordinary claims require extraordinary testing. This section systematically addresses the most plausible artifact hypotheses, demonstrating that the signal survives each challenge.

## 3.1 Processing Artifact Hypothesis

The most immediate concern is that the signal arises from sophisticated processing chains—network adjustments, Kalman filtering, or integer ambiguity resolution. To address this possibility, raw RINEX observation files were processed using Single Point Positioning with broadcast ephemerides only, representing the simplest possible processing chain and one entirely independent of the network adjustments employed by CODE, IGS, and ESA.

The raw RINEX test dataset comprises 539 stations over three years (2022–2024), totaling 1.17 billion pair-samples. The signal is detected with consistent results across all 72 metric combinations tested, with mean R² = 0.93. Directional anisotropy matches the CODE findings, with East-West exceeding North-South by 2–22% at short distances ($\lt$500 km). A critical audit confirms this is robust to distance bias: E-W pairs are 13 km longer than N-S pairs (a bias *against* the signal), and distance-matching strengthens the ratio.

The correlation length in raw data (λ ≈ 700–1,100 km) is shorter than in precise products (λ ≈ 4,000 km), consistent with ionospheric noise masking the long-range structure. When ionospheric effects are removed (Ionofree mode), the raw correlation length increases, supporting a "Ladder of Precision" where the signal scale converges toward the CODE benchmark as noise is mitigated. Orbital velocity coupling is detected at r = −0.763 (nominal 5.4σ; ≈3.0σ autocorrelation-corrected in Paper 3) in multi-GNSS mode, and the directional scan recovers RA = 8°, Dec = +5° under the corrected orbital-velocity convention — ~160° from the CMB apex as an arrow, ~20° from the dipole axis, and ~4° from the aphelion velocity tangent — consistent with the 25-year analysis. The signal therefore exists in the fundamental observables and is difficult to attribute solely to tested processing artifacts.

## 3.2 Ionospheric Hypothesis

If the signal were ionospheric in origin—driven by solar activity modulating electron density along signal paths—it should correlate with solar cycle variations. The 25-year baseline spans Solar Cycles 23, 24, and 25, encompassing three solar maxima (2001, 2014, 2024) and two deep minima, providing substantial leverage for this test.

The correlation between the signal and the Solar Flux Index is r ≈ 0, indicating no significant relationship. The correlation length λ remains stable across solar maxima within uncertainty bounds. Moreover, the signal persists in the dual-frequency ionosphere-free L1+L2 combination (λ = 1,069 km, R² = 0.969), and excluding high-ionosphere periods improves the signal by 21–23%. This last observation indicates that the ionosphere suppresses rather than creates the correlation. A simple ionospheric origin is therefore strongly disfavoured by these controls.

## 3.3 Geomagnetic Hypothesis

If the signal were driven by geomagnetic storms affecting clock electronics or signal propagation, it should vary systematically with geomagnetic activity. Stratification by GFZ Kp index data across all 72 metric combinations yields median Δλ ≈ −1% at the primary threshold (Kp $\lt$ 3 versus Kp ≥ 3), with 60 of 72 tests falling within ±5% across all station filters and processing modes. No systematic deviation appears at higher storm thresholds (Kp ≥ 4, Kp ≥ 5). The signal is therefore geomagnetically invariant.

## 3.4 Tropospheric Hypothesis

If the signal were tropospheric in origin—driven by water vapor or pressure variations that correlate spatially—stations at high altitude (with thinner atmospheric columns) should show systematically different correlation lengths than sea-level stations.

Analysis of 360 regressions spanning global and latitude-controlled altitude quintiles reveals that only 3.1% exhibit p $\lt$ 0.05 slopes, consistent with chance expectation. The Q5/Q1 ratio (highest to lowest altitude quintile) has median 0.97 (IQR 0.76–1.27), indicating that low-altitude and high-altitude stations yield statistically indistinguishable λ values.

The tropospheric column depth varies by approximately 20–30% between sea-level and high-altitude stations. If the correlation structure were tropospheric, one would expect systematic increase in λ at high altitude, latitude dependence correlated with tropopause height, and seasonal modulation correlated with monsoon patterns. None of these signatures are observed. The signal is altitude-invariant, consistent with a phenomenon operating at gravitational potential scales (~6,400 km) rather than atmospheric scales (~10 km).

**Comprehensive Test Summary:** The altitude independence test represents one component of a broader testing framework spanning 388 independent statistical tests across multiple analysis families. False Discovery Rate correction (FDR-BH: 52.3%, Hierarchical Empirical Bayes: 39.7%, Bonferroni: 40.0%) demonstrates robustness against multiple comparison artifacts. This extensive empirical testing, combined with cross-center consistency (cross-centre coefficient of variation ≈ 13% on the exponential λ<sub>T</sub> triplet (CODE, IGS, ESA; §2.1)), establishes that the observed correlations survive rigorous scrutiny across independent processing pipelines, environmental conditions, and statistical frameworks.

## 3.5 Long-Period Stability

If the signal were a transient artifact of the analysis window, it would not persist across the full 25.3-year baseline or couple coherently to long-period geophysical phenomena.

The data reveal clear coupling to multiple geophysical cycles. The 18.6-year lunar nutation shows R² = 0.641 ± 0.10 (p = 0.010, F-test with N = 12 phase bins; permutation test p = 0.0016, N = 100,000; 1.36 complete cycles observed — cycle-limited: each phase bin carries at most two independent cycle realizations). The semiannual coherence component shows R² = 0.904 (p = 2.7 × 10<sup>−5</sup>, F-test; permutation p = 2.7 × 10<sup>−4</sup>; AR(1) red-noise surrogate p = 2 × 10<sup>−5</sup>, N = 100,000; over 50 complete cycles; apsidal-locked, compatible with nutation term or orbital second harmonic). The Chandler wobble (433 days) shows R² = 0.096 with phase stability of 0.72 across more than 21 cycles. Core signatures—correlation length, anisotropy, and orbital coupling—remain consistent across the full 25-year span. The signal is therefore temporally stable across multiple geophysical cycles. Surrogate validation confirms that the nutation p-values are not inflated by treating the 165 million station-pair observations as independent: the F-test uses N = 12 phase bins (not N = 165 million), and non-parametric permutation and AR(1) red-noise surrogate tests confirm the significance without Gaussian or independence assumptions.

## 3.6 Most Plausible Remaining Conventional Explanation

The strongest remaining conventional hypothesis is that the observed correlations arise from a presently unmodeled, long-baseline GNSS systematic that survives both network adjustment and raw SPP processing. This section defines the exact cumulative constraints that any such systematic must satisfy simultaneously across the completed test battery.

**The Hypothesis:** A sophisticated, distance-dependent systematic error exists in GNSS observations that: (1) produces exponential decay with λ ≈ 4,000 km; (2) exhibits East-West/North-South anisotropy ratio of 2.16; (3) couples to Earth's orbital velocity with r = −0.888; (4) reproduces an annual phase near Earth's aphelion velocity tangent while the CMB-apex template is anti-phase; (5) produces repeatable seasonal-window coherence modulation in the planetary event catalogue without mass scaling; (6) couples to 18.6-year lunar nutation and the semiannual coherence component; (7) survives independent processing by CODE, IGS, and ESA; (8) persists in raw RINEX data processed with broadcast ephemerides only; and (9) retains compatible distance structure across 72 non-independent metric combinations, with metric-dependent response.

**Constraints from Tests:** Each test step constrains this hypothesis without claiming impossibility:

- **Multi-center consistency (R² = 0.92–0.97, cross-centre CV ≈ 13% on the exponential λ<sub>T</sub> triplet (§2.1)):** The systematic must be present in three software- and estimator-diverse product families (Bernese, the IGS combination product, NAPEOS) within the shared IGS tracking ecosystem. This excludes center-specific artifacts but does not exclude shared-network, constellation-geometry, or datum-convention systematics, nor systematics inherent to GNSS observation geometry or satellite constellation design.

- **Raw RINEX test (all combinations, t-stats up to 112):** The systematic must survive Single Point Positioning with broadcast ephemerides, which processes each station independently without network adjustment. This excludes network-adjustment artifacts but does not exclude systematics in the broadcast ephemeris itself (the "ephemeris loophole" addressed in Section 6.3.1.1).

- **25-year temporal stability:** The systematic must persist across satellite constellation changes, hardware upgrades, and processing methodology evolution spanning 1.4 complete cycles of the 18.6-year lunar nutation. This excludes transient artifacts but does not exclude systematics tied to fundamental GNSS geometry.

- **Annual-phase recovery:** The systematic must reproduce a phase near Earth's aphelion velocity tangent. The directed CMB-apex template is anti-phase and the Solar Apex is null. An annually phased direction tracking orbital velocity is geometrically difficult to explain through satellite constellation geometry but remains a possible confound if an unrecognized coupling exists between GNSS observation geometry and Earth's heliocentric motion.

**Remaining Possibility:** The most plausible remaining conventional explanation is a systematic tied to the fundamental geometry of GNSS satellite constellations that couples to Earth's motion through space in a way not previously recognized. Such a systematic would need to produce all observed signatures through a single coherent mechanism—a requirement that becomes increasingly implausible as the number of convergent signatures grows, but cannot be definitively excluded without independent check through non-GNSS methods (the SLR provenance check, the orbit-projection forward model, and the triangle holonomy test).

**Path to Resolution:** Definitive exclusion of this hypothesis requires: (1) SLR check (Section 6.3.1.1) to close the GNSS-product branch of the ephemeris loophole, with the generic orbit-projection branch closed by the physical-confound forward model or an orbit-free clock channel; (2) independently processed per-constellation cross-check recovering the same finite covariance family and band ordering across GPS, GLONASS, Galileo, and BeiDou while allowing product- and estimator-dependent scales; and (3) triangle holonomy test (Section 6.3.3) providing direct measurement of synchronization non-closure. Until these tests are completed, the possibility of an unrecognized GNSS systematic, however implausible given the convergence of evidence, cannot be completely excluded.

## 3.7 SLR Check: Optical Sensitivity and Sampling Controls

Satellite Laser Ranging (SLR) supplies an optical test through passive retroreflectors and an orbit-determination chain that consumes no GNSS products. Under conformal null-cone invariance the ground event timer reads the clock-amplitude channel. This instrument and product independence makes SLR a useful test pathway; a common physical signal must still be established at adequate sensitivity.

Paper 8 analyses 11 years (2015–2025) under sampling-matched controls. The observed 10–500 µHz spectral ratio of 14.12 is reproduced by white noise passed through the same sparse-sampling and resampling pipeline (13.97; Fisher-combined p = 0.199). A real-epoch Lomb–Scargle estimator gives 1.04 against a white-noise null of 1.05. The earlier spectral concentration therefore supplies no optical detection. The clock-amplitude channel is bounded at approximately δA ≲ 2×10<sup>−8</sup>; the roughly 5 mm excursion from the surface-depth benchmark remains below the residual floor.

The nominal same-satellite inter-station result (Fisher p = 0.0025) shares an orbit-model common mode. Different-satellite pairings yield a candidate anticorrelation in the 5,000–7,500 km bin (mean r = −0.23; label-swap p = 0.046; station-clustered 95% interval [−0.43, −0.06]). It appears in only one of nine threshold/bin-width configurations, weakens to −0.054 with at least five shared bins, and changes sign across years. It is retained as an exploratory tail event. The daily-aggregation diagnostic also reverses sign between orbit solutions.

The present SLR contribution is a sampling-artifact correction, an optical sensitivity bound and a defined route to an independent instrument test. It establishes neither achromatic replication of the GNSS signal nor its absence at the smaller benchmark amplitude. A stronger cross-instrument inference requires stable inter-station recovery and a demonstrated signal-amplitude transfer.

## 3.8 Why the Signal Was Not Previously Identified

The correlation structure documented here has been present in GNSS data since the network's inception but was systematically removed by processing algorithms optimized for positioning accuracy rather than fundamental physics characterization. Three factors explain why it was not identified earlier.

Standard GNSS processing is designed to eliminate correlations, not to characterize them. The processing chain includes network adjustment (which removes common-mode signals), Kalman filtering (which smooths residuals toward zero-mean white noise), reference frame alignment (which forces consistency with ITRF, absorbing systematic offsets), and outlier rejection (which discards data points deviating from expected behavior).

The correlation structure detected in this analysis is precisely what these algorithms are designed to remove. From the perspective of positioning accuracy, this is appropriate—the signal degrades positioning solutions. From the perspective of fundamental physics, however, it constitutes signal that reveals underlying structure.

The present analysis examines the residuals that standard processing discards, employing phase-coherent cross-spectral methods that preserve correlation structure. The signal was present throughout the history of GNSS operation but was categorized as systematic error rather than physical phenomenon.

## 3.9 Summary of Excluded Hypotheses

Table 3 summarizes the artifact hypotheses tested and their outcomes.

| Hypothesis | Test | Result |
| --- | --- | --- |
| Processing artifact | Raw RINEX/SPP | Signal persists (all combinations) |
| Ionospheric | 25-year solar cycle | No correlation (r ≈ 0) |
| Tropospheric | Altitude stratification | No altitude dependence (Q5/Q1 = 0.97) |
| Geomagnetic | Kp stratification | Near-invariant (Δλ ≈ −1%) |
| Tidal forcing | GM/r² and GM/r³ scaling | Neither force nor tidal scaling (p > 0.5) |
| Random noise | Shuffle test | 30× R² ratio (real vs. shuffled) |
| Center-specific bias | Multi-center consistency | Cross-centre CV ≈ 13% on the exponential λ<sub>T</sub> triplet (§2.1) |

The tests are complementary but not statistically independent; several are derived from the same underlying GNSS covariance data. The signal survives all challenges, constraining viable explanations to physical mechanisms operating at planetary scales with velocity-dependent, preferred-axis geometry.

## 3.10 Evidence Tier Classification

The convergent signatures supporting the Temporal Topology interpretation
span two tiers of timing-network evidence plus an orthogonal optical-domain
check, reflecting both statistical strength and resilience to alternative
explanations.

| Tier | Signature | Statistical Strength | Artifact Resilience |
| --- | --- | --- | --- |
| **Tier B**
(Primary timing-network evidence) | Exponential spatial decay | R² = 0.92–0.97 | Survives all processing modes; raw RINEX confirmed |
|  | East-West/North-South anisotropy | Ratio 2.16, p $\lt$ 10<sup>−15</sup> | Cross-center validated; multi-constellation |
|  | Orbital velocity coupling | r = −0.888, 3.65σ (autocorr.-corrected) | Phase-stable over 25 years; multi-constellation |
|  | Preferred-direction recovery | annual phase near aphelion tangent on CODE and RINEX; CMB apex anti-phase; Solar Apex null; MGEX scan projection-degenerate | orbit-coupled axis; direction product-unstable |
|  | Semiannual coherence component | R² = 0.904, p = 2.7 × 10<sup>−5</sup> (F-test) | 50+ complete cycles; apsidal-locked (nutation/orbital harmonic) |
|  | Planetary event catalogue (bounded channel) | 56/156 events ≥2σ (descriptive); opposition subset null-validated 5.5×; continuous r = +0.116, p = 0.012 | GM/r² and GM/r³ scaling null bounds unmodelled transient perturbations; catalogue excess reproduced by calendar position (Paper 3) |
|  | 18.6-year lunar nutation | R² = 0.641 ± 0.10, p = 0.010 (surrogate-validated) | 1.36 complete cycles; cycle-limited, longer baseline needed |
|  | Selectivity profile (null results) | Solar rotation: null; Tides: null | Demonstrates physical specificity |
| **Tier C**
(Optical-domain consistency check) | SLR spectral bound; exploratory correlations | 14.12 observed versus 13.97 sampling-matched white null | Instrument-orthogonal; passive retroreflectors; same conformal clock channel |

Tier B signatures constitute the foundational timing-network evidence: they are
detected with high statistical confidence across all GNSS test modes and
processing pipelines. Tier C provides an instrument-level optical-domain check
using an independent measurement technology and orbit-determination chain; it is consistent with the
timing-network results but probes the same conformal clock sector rather
than a second physical channel, and does not carry the same statistical weight owing
to sparser station coverage and asymmetric global distribution. The pattern
across both tiers consistently favors a velocity-dependent, preferred-axis
geometric mechanism over conventional explanations.

## 4. Theoretical Framework

Having established that the signal is robust and not attributable to known
artifacts, this section presents the theoretical framework that predicted
these observations. The Temporal Equivalence Principle is a principled
extension of general relativity within the well-established class of
scalar-tensor theories, with pre-specified expectations and theoretical search ranges that preceded
the GNSS analysis.

## 4.1 Two-Metric Geometry

TEP posits that spacetime is endowed with two metrics related by a scalar
field φ:

$$\tilde{g}_{\mu\nu} = A^2(\phi)\, g_{\mu\nu} + B(\phi)\, \nabla_\mu \phi\,
\nabla_\nu \phi$$

The gravitational metric g<sub>μν</sub> determines geodesics of test masses
and gravitational wave propagation. The matter metric g̃<sub>μν</sub>
determines atomic clock rates, electromagnetic propagation, and all Standard
Model physics. The conformal factor A(ϕ) = exp(β<sub>A</sub>ϕ/M<sub>Pl</sub>) provides
isotropic rescaling of proper time, while the disformal factor B(φ)
introduces anisotropic light-cone tilts and is tightly constrained by
GW170817.

This structure belongs to the well-established class of scalar-tensor
theories developed by Bekenstein (1993) and Damour & Polyakov (1994). The
contribution of TEP lies in its operational interpretation: the conformal
factor A(ϕ) governs the rate at which proper time accumulates, rendering
time itself a dynamical field.

## 4.2 Physical Interpretation

The framework is formulated as a scalar-tensor theory in the Einstein frame
where gravity remains canonical. The scalar field φ couples to matter
through the conformal factor A(φ), modulating the rate of proper time
accumulation. The bare conformal coupling is $\beta_A = -1.0$ universally. In the near-Earth clock sector, $S_A^{(\oplus)} = \min[1, (\bar{\rho}_\oplus/\rho_T)^{1/3}] \approx 0.65$ characterizes static conformal depth, while the measured time-dependent covariance $C_A$ has a distinct environmental and processing transfer; the local shear sector is pairwise-screened by the kinetic operator (Paper 0 §2.2). The scalar field obeys a Klein-Gordon equation
with source terms from matter density and a screening potential V(φ) that,
in concert with derivative self-interactions, suppresses Temporal Shear in
high-acceleration regions through the kinetic operator.
(Full action and field equations are provided in Appendix A.)

## 4.3 Foundational Axioms

The framework rests on four axioms. The first (two-metric structure) states
that gravity is described by g<sub>μν</sub> while matter couples universally
to g̃<sub>μν</sub>. The second (temporal equivalence) states that all
non-gravitational processes evolve according to proper time dτ defined by
g̃<sub>μν</sub>, and that in local freely falling frames, physics reduces to
special relativity with invariant c. The third (causal safety) requires that
c<sub>g</sub> = c<sub>γ</sub> within GW170817 bounds (|c<sub>g</sub> −
c<sub>γ</sub>|/c ≲ 10<sup>−15</sup>), constraining the disformal coupling B(φ) to be small at
late times but not zero — disformal or otherwise non-exact transport is required for the synchronization holonomy observable (Section 4.6). The fourth (screening) states that rather than operating via
discrete boundary cutoffs, screening manifests as a continuous spatial
profile (Temporal Topology), with Temporal Shear suppression arising from
the non-linear superposition of field gradients (Temporal Shear), suppressing
Temporal Shear in high-acceleration regions while leaving cosmology accessible to dynamics.

## 4.4 Candidate Theoretical Completions

The observed correlation length λ<sub>T</sub> is the Temporal Topology
correlation length—the characteristic scale of the clock-sector covariance $C_A$ in Earth's environment ($\lambda_T \approx R_T$ in the channel functional; Paper 0 §7). This empirical parameter admits
multiple theoretical interpretations as candidate completions, not as unique
predictions. The empirically derived correlation length λ<sub>T</sub> is a GNSS-sector covariance scale obtained after environmental projection and processing transfer. It is not the screening operator $\mathcal{S}_\Sigma(\mathcal{E})$ itself, and it is not the non-exact (disformal) covariance term, which this measurement does not access.

Three classes of completion are sketched below.

### 4.4.1 Compton-Mass Interpretation

A chameleon-type completion identifies λ<sub>T</sub> with the in-medium Compton wavelength of a scalar field: λ = ℏ/(m<sub>φ</sub>c). For λ = 3,330–4,549 km, this yields m<sub>φ</sub> ≈ (4.34–5.93) × 10<sup>−14</sup> eV/c². The present framework identifies the correlation length with the geometric saturation radius R<sub>T</sub> (§4.4.2) rather than with a Compton wavelength; the mass scale is retained as a cross-reference to chameleon-literature conventions.

### 4.4.2 Clock Amplitude Calibration: TEP Saturation Radius

In the TEP framework, the geometric saturation radius R<sub>T</sub>(M) = (3M/4πρ<sub>T</sub>)<sup>1/3</sup> enters the clock sector to set the unsaturated clock-amplitude factor S<sub>A</sub> = min[1, (ρ̄/ρ<sub>T</sub>)<sup>1/3</sup>], while Earth-vicinity Temporal Shear is pairwise-screened in the force sector by the kinetic operator (Paper 0 §2.2, §7). The saturation density ρ<sub>T</sub> ≃ 20 g/cm<sup>3</sup> is calibrated from Earth's observed GNSS clock covariance:

$$R_T = \left(\frac{3M}{4\pi\rho_T}\right)^{1/3}$$

This M<sup>1/3</sup> scaling is a kinematic property of the constant-ρ<sub>T</sub> saturation model. For Earth with M = M<sub>⊕</sub> ≈ 6 × 10<sup>24</sup> kg and ρ<sub>T</sub> ≃ 20 g/cm<sup>3</sup>:

$$R_T^{\oplus} = \left(\frac{3 \times 6 \times 10^{24}\text{ kg}}{4\pi \times 2 \times 10^{4}\text{ kg/m}^3}\right)^{1/3} \approx 4{,}150\text{ km}$$

This is consistent in scale with the observed GNSS correlation-length family (azimuth-averaged λ = 3,210 ± 389 km; cross-sector dispersion λ<sub>T</sub> = 4,201 ± 1,967 km), noting that the identification calibrates ρ<sub>T</sub> rather than testing an independently derived scale. The physical roles are partitioned as follows:

**Clock sector (S<sub>A</sub>):** Governed by the potential sector and local density, returning S<sub>A</sub><sup>(⊕)</sup> ≃ 0.65 at Earth's mean density ρ̄<sub>⊕</sub> = 5.515 g/cm<sup>3</sup>. Clock residuals across continental baselines measure the spatial covariance C<sub>A</sub>(r); the transmission of time-dependent fluctuations and their processing transfer are distinct from the static S<sub>A</sub> level.

**Shear / Force sector (S<sub>Σ</sub>):** Governed by the selected two-branch kinetic completion P(X,φ) = −V(φ) + (2k/3Λ<sub>X</sub><sup>2</sup>)X|X|<sup>1/2</sup> + X|X|/Λ<sub>X</sub><sup>4</sup> (k ≈ 16, derived). In high-acceleration regions (g ≫ g<sub>t</sub> = cH₀/2), the two-body operator suppresses the observable shear response to 𝒮<sub>Σ</sub> ~ 10<sup>−21</sup> at Earth's surface, preserving standard General Relativity paths in local gravitational and planetary tests.

**Kinetic transition scale (r<sub>*</sub>):** The physical transition for dynamical shear recovery follows the kinetic law r<sub>*</sub>(M) = √(GM/g<sub>t</sub>) ∝ M<sup>1/2</sup> (Paper 0 §2.2), distinct from the geometric saturation radius R<sub>T</sub> ∝ M<sup>1/3</sup>.

**Falsifiable Predictions:** The density-saturation diagnostic
R<sub>T</sub>(M) = (3M/4πρ<sub>T</sub>)<sup>1/3</sup> — a candidate
microscopic completion, not a derived consequence of the master action —
makes specific predictions for intermediate-mass systems:

**Jupiter** (M ~ 300 M<sub>⊕</sub>): Predicted R<sub>T</sub>
~ (300)<sup>1/3</sup> × 4,150 km ≈ 27,800 km. Testable with Juno mission
clock data or future Jupiter atmospheric probe networks.

**Globular clusters** (M ~ 10<sup>5</sup> M<sub>☉</sub>):
Predicted screening length ~ 1–10 pc, testable through pulsar timing
array correlations within clusters.

**Galaxies** (M ~ 10<sup>12</sup> M<sub>☉</sub>): At galactic
scales, the screening operator 𝒮<sub>Σ</sub>(Ε) retains
approximately 98% of the unsuppressed Temporal Shear in the diffuse halo
environment (Paper 26), producing dark-matter-like phenomenology through
the active-shear regime rather than through a screening radius. The
galactic phenomenology is therefore a manifestation of the same
𝒮<sub>Σ</sub> operator operating in the unscreened regime, not
a Vainshtein scaling of the terrestrial correlation length. This
connection is developed in Section 5.5.

Within the saturation-model regime — from planetary through intermediate
mass scales — R<sub>T</sub>(M) provides a unified characterization governed
by a single saturation density ρ<sub>T</sub>; the testable prediction is
that one ρ<sub>T</sub> fits all such masses, while galactic-scale
phenomenology arises from the same 𝒮<sub>Σ</sub> operator in the
unscreened regime rather than from a screening radius. This scaling reproduces the continuous gradient-suppression behavior described by
Temporal Topology at the macroscopic level. Candidate microscopic completions (chameleon, Vainshtein, Galileon, DBI, symmetron) may produce similar phenomenology through different microscopic mechanisms; the empirical correlation length
λ<sub>T</sub> = 4,201 ± 1,967 km does not uniquely select any single completion. The defining TEP ontology is the continuous suppression of Temporal Shear by the 𝒮<sub>Σ</sub>(Ε) operator, not the specific microscopic screening mechanism.

## 4.5 Synchronization Holonomy and Spatial Correlations

TEP predicts synchronization holonomy—the non-closure of proper time when
transported around a closed loop:

$$\mathcal{H} \equiv \oint_C d\tau_{\text{prop}} \neq 0$$

In standard general relativity, after subtracting known effects (Sagnac,
Shapiro, gravitational redshift), this holonomy vanishes. In TEP with
disformal coupling B ≠ 0, it does not.

An important subtlety arises in the conformal-only limit (B = 0): the
time-transport one-form Ω<sub>μ</sub> = ∂<sub>μ</sub>(½ ln A) is exact, its
curl vanishes, and holonomy is zero. Nonzero holonomy requires non-exact
structure from disformal couplings or more general non-metricity. This
raises a question: if GW170817 constrains B(φ) to be small, and
conformal coupling alone produces no holonomy, what generates the observed
GNSS correlations?

The resolution lies in distinguishing two observables. Synchronization
holonomy (H) represents the non-closure of proper time around closed loops;
this requires disformal or otherwise non-exact transport and is the target of proposed triangle tests. Spatial
correlation structure, by contrast, represents the statistical covariance of
clock residuals as a function of distance; this arises from the spatial
structure of the φ field itself—specifically, the two-point correlation
function ⟨φ(x)φ(x+r)⟩ ∝ exp(−r/λ)—and persists even when B = 0.

The GNSS signal reflects the latter mechanism: clocks at different locations
sample different values of the conformal factor A(φ), and spatial
correlations in φ induce correlations in clock rates. The exponential decay
with distance reflects the TEP saturation radius $R_T = (3M/4\pi\rho_T)^{1/3}$,
while the velocity-dependent anisotropy arises because Earth's motion
through the φ field creates a preferred direction in the correlation
structure. The correlation structure is thus a statistical signature of the
φ field's spatial configuration, whereas holonomy is a geometric signature
of non-integrable time transport. Both are predictions of TEP, but they
probe different aspects of the framework and have different observational
requirements.

### 4.5.1 Resolving the Conformal Paradox

**Figure 1:** Synchronization Holonomy: The Definitive Test
of Non-Integrable Proper Time. Three atomic clocks (A, B, C) positioned
at vertices of a triangle (1,000–3,000 km separation) exchange one-way
time-transfer pulses around the loop. Panel A (Standard GR): After
subtracting all known relativistic effects (Sagnac, Shapiro,
gravitational redshift, Lense-Thirring), the accumulated time difference
returns to zero—proper time is integrable. Panel B (TEP): A residual
time gap ℋ<sub>resid</sub> ~ 10<sup>−18</sup>–10<sup>−16</sup> s
remains, scaling as (B/A²)(u·∇φ) P<sub>μ</sub><sup>ν</sup> ∇<sub>ν</sub>φ · Area. Target sensitivity: σ(ℋ)/T<sub>loop</sub> ≤ 10<sup>−19</sup>.

A critical objection must be addressed: if TEP employs only a conformal
coupling A(φ), how can it produce synchronization holonomy? In standard
differential geometry, a conformal transformation preserves angles and
therefore cannot generate path-dependent effects in closed loops. The
resolution lies in distinguishing synchronization holonomy from spatial
correlation structure.

**Observable 1: Synchronization Holonomy (ℋ)** — The
non-closure of proper time around closed loops, defined as ℋ ≡ ∮<sub>C</sub>
dτ<sub>prop</sub>. This observable requires disformal or otherwise non-exact transport
because only non-exact structure in the time-transport connection generates
path-dependent accumulation. In the conformal-only limit (B = 0), the
one-form Ω<sub>μ</sub> = ∂<sub>μ</sub>(½ ln A) is exact, its curl vanishes
identically, and holonomy is zero by Stokes' theorem. The triangle holonomy test (Section 6.3.3) targets this observable and would measure whether B(φ) is nonzero at the 10<sup>−19</sup> fractional level.

**Observable 2: Spatial Correlation Structure (C(r))** — The
statistical covariance of clock frequency residuals as a function of
inter-station distance. This observable requires only A(φ) ≠ 1 (conformal
coupling) and persists even when B = 0. The mechanism is fundamentally
different from holonomy: clocks at different spatial locations sample
different values of the scalar field φ(x), and the two-point correlation
function ⟨φ(x)φ(x+r)⟩ ∝ exp(−r/λ) of the underlying field induces
correlations in clock rates through the conformal factor A(ϕ) =
exp(β<sub>A</sub>ϕ/M<sub>Pl</sub>). The exponential decay length λ = R<sub>T</sub>(M<sub>⊕</sub>)
reflects the TEP screening scale of the temporal topology (temporal-topology pinning on the environmental state, of which density is one coordinate); the terrestrial correlation length is identified with the geometric saturation radius R_T(M_⊕) from the UCD model (Paper 6), not with the in-medium Compton wavelength of a candidate chameleon completion.

**Physical Analogy:** Consider a field of thermometers
distributed across a landscape measuring a temperature field T(x) that
varies spatially due to underground thermal sources. Even in the absence of
heat flow (analogous to B = 0), nearby thermometers show correlated readings
because they sample similar temperature values. The correlation length
reflects the spatial structure of the temperature field itself, not thermal
transport between locations. Similarly, GNSS clocks sample the φ-field
"temperature," and their correlation structure reveals the field's spatial
configuration.

**Why GW170817 Does Not Constrain the GNSS Signal:** The
multi-messenger constraint |c<sub>γ</sub> − c<sub>g</sub>|/c ≲ 10<sup>−15</sup>
bounds the disformal sector B(φ), which affects propagation speeds. The GNSS
correlation structure arises from the conformal sector A(φ), which affects
clock rates but preserves null cones. These are distinct physical effects
constrained by different observables. GW170817 establishes that light and
gravity travel at the same speed; it does not constrain whether clocks at
different locations tick at correlated rates due to sampling a common scalar
field.

**Observational Distinction:** The GNSS signal is a
"thermometer correlation"—a statistical signature of φ-field structure
detected through spatial covariance analysis. Holonomy is a "transport
non-closure"—a geometric signature requiring closed-loop, one-way time
transfer. Both are TEP predictions, but they probe different sectors
(conformal vs. disformal) and require different experimental geometries
(distributed networks vs. closed triangles). The present work establishes
the former; the latter awaits dedicated triangle tests.

## 4.6 Prediction Comparison

TEP established pre-specified expectations and theoretical search ranges
before the GNSS analysis was conducted: the framework was first deposited in
August 2025 (Smawfield 2025a, v0.1) and the clock-network correlation
forecast was versioned on 15 September 2025 (v0.4 §E), preceding the first
public deposit of the GNSS analysis by one week (provenance register,
Section 6.5). Table 5 compares these forecasts to
observations.

| Prediction | Forecast | Observed | Status |
| --- | --- | --- | --- |
| Correlation length (screening-parameterized forecast) | 2,000–3,000 km | 3,210 ± 389 km (azimuth-averaged); IGS-final 3,330–4,549 km | Not met: observed scale lies above the band |
| Correlation length (search range) | 1,000–10,000 km | 4,201 ± 1,967 km (sector dispersion) | Consistent; non-discriminating (range subsumes leading systematic scales) |
| Spatial decay form | Exponential; Gaussian-family preference would disfavour the screened-field reading (Paper 1, criterion iii) | Gaussian/squared-exponential preferred; exponential ΔAIC = +12.8; power law ΔAIC > 30 | Met in Paper 1; not met in the 25-year dataset |
| Multi-centre consistency | $\lt$ 5% variation in fitted parameters | CV of λ ≈ 13% across three centre exponential fits | Not met |
| Velocity-dependent anisotropy | Modulates with v/c | r = −0.888 (3.65σ, autocorr.-corrected) | Consistent in sign (pre-stated in Paper 1 v0.6α); coupling magnitude not derived in the current framework |
| Geometric alignment | EW > NS in ecliptic frame | EW/NS = 2.16 | Consistent |
| Broadband coupling | Not frequency-selective | Detected across 10–500 μHz | Consistent |
| Annual-phase direction recovery | Orbital-phase modulation | RA = 6°, Dec = +4°; ~6° from aphelion velocity tangent; CMB apex anti-phase | Consistent with orbital-phase channel; no CMB-frame identification |
| GM/r² and GM/r³ scaling | Both absent (filtered) | No force or tidal scaling (p > 0.5) | Null confirmed |

The null prediction regarding GM/r² scaling is particularly discriminating.
Classical tidal forcing would produce mass-dependent signatures scaling with
GM/r³, with Mercury contributing far less than Jupiter. The observed absence
of such scaling (all p > 0.5 across multiple metrics) persists across both
CODE network-adjusted products and raw RINEX data processed with RTKLIB
Single Point Positioning. Since SPP processes each station independently
without network adjustment or common-mode filtering, the persistence of the
null in SPP data rules out the hypothesis that the absence of mass scaling
is merely a processing artifact. The TEP coupling appears to be
intrinsically geometric rather than energetic—responding to alignment
configuration regardless of mass. Mercury (42.5% detection rate) produces
comparable responses to Jupiter (34.8%) despite ~10<sup>6</sup>× smaller
gravitational influence, consistent with a threshold-dependent or
alignment-driven mechanism rather than continuous force scaling.

## 4.7 Why Existing Tests Do Not Directly Test the Target TEP Observables

Existing precision tests probe different observables than TEP's primary
signatures, as summarized in Table 6.

| Test | Observable | Reason TEP Is Unconstrained |
| --- | --- | --- |
| GW170817 | |c<sub>γ</sub> − c<sub>g</sub>|/c | Constrains disformal B(φ), not conformal A(φ) |
| Cassini (PPN-γ) | Two-way Shapiro delay | Reciprocity-even; does not bound loop holonomy; samples screened solar-vicinity regime |
| Resonator MM/KT | Closed-path anisotropy | Even-parity; blind to one-way non-reciprocity |
| GPS operation | Internal consistency | Uses explicit GR+Sagnac modeling; verifies model |
| Clock redshift tests | Pairwise comparisons | Confirms GR locally; closed loops required |

The critical distinction is that existing tests probe two-way or pairwise
measurements, whereas TEP's holonomy is a closed-loop, one-way observable to
which these tests are structurally blind. Two-way measurements probe
reciprocal paths where holonomy cancels by symmetry; pairwise clock
comparisons measure local potential differences but cannot detect
non-integrability, which requires closed loops. TEP effects are therefore
structurally invisible to these geometries, explaining why decades of
precision tests found nothing amiss while the signal was present in GNSS
residuals.

#### Operational Measurement Protocol: What Existing Tests Probe

The distinction between observables that constrain TEP and those that do
not can be formalized through the synchronization holonomy (after GR
subtraction), where σ is the time-transport one-form whose line integral
equals the calibrated proper-time increment along a leg:

$$\mathcal{H}_{\rm resid} = \oint_{\partial\Sigma} (\tilde{\sigma} -
\sigma_{\rm GR}) = \iint_{\Sigma} (\tilde{F} - F_{\rm GR})$$

This represents the loop non-closure of time transport built from
measured proper times (in time units; right-hand rule on Σ). It is gauge
and synchronization-invariant and vanishes in standard relativity and in
the conformal-only limit of TEP. GR subtraction includes Sagnac,
Lense-Thirring gravito-magnetic effects, Shapiro delay, and
gravitational redshift, with ITRF ephemerides and TT/TDB standards.

How flagship constraints map to ℋ<sub>resid</sub>:

GW170817 (GW-EM coincidence): |c<sub>γ</sub> − c<sub>g</sub>|/c ≲
10<sup>−15</sup> constrains global cone splits. In TEP, late-time
conformal coupling preserves null cones, so electromagnetic and
gravitational waves share causal structure; small disformal tilts
today are allowed. This is a boundary condition, not a loop-holonomy
test.

Cassini (PPN-γ): Two-way Doppler/Shapiro is reciprocity-even; it
calibrates σ<sub>GR</sub> to subtract but does not bound
ℋ<sub>resid</sub>. The measurement was performed during solar
conjunction, sampling the Sun's deep potential well—the
screened regime where the source-charge sector $S_\Sigma$ is
suppressed. The bound therefore applies to the solar-vicinity
environment, not to the dilute regimes where Temporal Shear
recovers.

Resonator MM/KT tests: Cavities bound closed-path, even-parity
(two-way sums) anisotropy at 10<sup>−17</sup>–10<sup>−18</sup>, yet
are blind to odd-parity (direction-reversing one-way differences)
non-reciprocity and loop non-closure—the ingredients of
ℋ<sub>resid</sub>.

GPS operation: Network self-consistency uses explicit GR+Sagnac
modeling and largely two-way/common-view calibration. This verifies
internal consistency under the assumed GR model; it is not a
direction-reversing one-way loop-closure null.

Clock redshift and pairwise A↔B tests: Exquisitely confirm GR
locally; only closed loops (A→B→C→A with direction reversal) can
reveal non-integrability captured by ℋ<sub>resid</sub>.

Why classical tests can be null while ℋ<sub>resid</sub> ≠ 0: (1)
Conformal null-cone invariance implies no large GW-EM kinematic delays
(consistent with GW170817). (2) If ∂<sub>t</sub>φ = 0 over loop
timescale and gradients are conservative, there is no first-order
one-way anisotropy. Along a fixed path, forward and backward times
cancel at order (B,∇φ); leading effects are second order or require time
dependence and kinematics. Thus two-way and closed-path nulls can hold
while a loop-holonomy test remains sensitive.

Experimental falsifier (primary endpoints): Run a closed-loop, one-way
time-transfer (and/or portable-clock) test and report: (1) Leg-wise
antisymmetry: Δt<sub>AB</sub> = t<sub>AB</sub> − t<sub>BA</sub> (and
optionally Ξ<sub>AB</sub> ≡ (t<sub>AB</sub> − t<sub>BA</sub>)/(t<sub>AB</sub>
+ t<sub>BA</sub>)). (2) Loop holonomy ℋ<sub>resid</sub> after
subtracting GR (Sagnac/gravito-magnetic/Shapiro). Use
triangle/quadrilateral geometries with direction reversal; extend with
interplanetary one-way links and multi-species clock networks.

### 4.7.1 Compatibility with Binary Pulsar Timing and Lunar Laser Ranging

Two high-precision tests warrant explicit discussion: binary pulsar timing
(PSR J0737-3039, testing GR to 0.05%) and Lunar Laser Ranging (testing the
equivalence principle to 10<sup>−13</sup>). Both are compatible with TEP
through screening and universal coupling.

Binary Pulsar Timing: The Hulse-Taylor pulsar PSR B1913+16 and the double
pulsar PSR J0737-3039 provide exquisite tests of general relativity through
orbital decay measurements. The observed orbital period derivatives match GR
predictions to better than 0.05%, constraining deviations in the
strong-field regime. TEP compatibility requires that scalar field effects
are screened in the vicinity of neutron stars.

The TEP saturation radius for a neutron star (M ~ 1.4 M<sub>☉</sub>)
is:

$$R_T^{\text{NS}} = \left(\frac{3 M_{\text{NS}}}{4\pi\rho_T}\right)^{1/3}
\sim 3.2 \times 10^5 \text{ km}$$

This saturation radius far exceeds the neutron star radius (R<sub>NS</sub> ~
10 km). Binary pulsar timing, however, probes the gravitational sector (metric g<sub>μν</sub>), not the clock sector (metric g̃<sub>μν</sub>). The scalar field's effect on g<sub>μν</sub> is second-order in the coupling, proportional to (∇φ)²/c⁴, which at orbital distances (a ~ 10<sup>6</sup> km) from a 1.4 M<sub>☉</sub> neutron star gives a fractional effect below 10<sup>−30</sup> — far below the 0.05% observational
precision of binary pulsar tests. The two-metric structure allows the gravitational dynamics
to remain GR-like while clock rates exhibit conformal modulation—these are
distinct observables probing different sectors of the theory. Binary pulsar
insensitivity to the conformal coupling that generates GNSS correlations is
therefore a consequence of the two-metric structure, not of the screening
radius.

Critically, pulsar timing measures orbital dynamics (gravitational sector,
metric g<sub>μν</sub>), not clock correlations (matter sector, metric
g̃<sub>μν</sub>). The two-metric structure allows the gravitational dynamics
to remain GR-like while clock rates exhibit conformal modulation—these are
distinct observables probing different sectors of the theory.

Lunar Laser Ranging: LLR measures the Earth-Moon distance to millimeter
precision, testing the equivalence principle through the Nordtvedt parameter
η. The current bound η $\lt$ 4.4 × 10<sup>−4</sup> constrains deviations from
the strong equivalence principle. TEP satisfies this constraint through
universal coupling: the conformal factor A(φ) couples identically to all
matter fields, preserving the equivalence principle in the matter frame.

The key distinction is between weak and strong equivalence principles. The
weak EP (universality of free fall for test masses) is preserved exactly in
TEP because all matter couples to the same metric g̃<sub>μν</sub>. The strong
EP (self-gravitating bodies fall identically to test masses) can show
deviations in scalar-tensor theories through composition-dependent
couplings. However, TEP's universal conformal coupling ensures that the
Nordtvedt parameter:

$$\eta = 4\beta_{\text{PPN}} - \gamma - 3$$

remains within observational bounds. The PPN nonlinearity parameter β<sub>PPN</sub> is driven to its GR value β<sub>PPN</sub> → 1 by the TEP screening operator, where the pairwise response is suppressed by the kinetic operator ($S_\Sigma(g_{\text{lunar}}) \approx 1.6 \times 10^{-14}$; Paper 0 §2.2). With γ ≈ 1 (Cassini constraint), the predicted Nordtvedt parameter is well below the LLR bound. The predicted metric deviation applies to the gravitational sector, preserving the Strong Equivalence Principle. The ∼10<sup>−4</sup> synodic-phase anomaly reported in Paper 17 is an empirical matter-frame clock residual (κ<sub>LLR</sub>), not a classical Nordtvedt metric violation.

Furthermore, the Earth–Moon separation lies in the deeply suppressed
regime of the pairwise Temporal-Topology response
𝒮<sub>eff</sub>(s): the scalar-mediated Earth–Moon differential
acceleration is suppressed by pairwise response factors of order
2.5×10<sup>−14</sup> relative to the unsuppressed coupling (Paper 0,
screening operator), comfortably below the LLR bound. The suppression
acts on the shear sector (𝒮<sub>Σ</sub>) — the geodesic and
force projection — while each body's clock-amplitude coupling
(S<sub>A</sub>) remains unsuppressed. The combination of universal
coupling and sector-separated screening ensures LLR compatibility
while preserving the GNSS clock-covariance signal in the terrestrial
environment.

## 4.8 Relation to Established Scalar-Tensor Theory

In the canonical non-screened limit, TEP shares structural similarities with scalar-tensor theories, but its physical realization is defined by the non-canonical two-branch kinetic sector $P(X,\phi) = -V(\phi) + \frac{2k}{3\Lambda_X^2}X|X|^{1/2} + X|X|/\Lambda_X^4$ and universal matter-metric coupling.
In the limit B = 0, V = 0, Λ<sub>X</sub> → ∞, TEP with A(ϕ) = exp(β<sub>A</sub>ϕ/M<sub>Pl</sub>) reduces to Brans-Dicke
gravity in the Jordan frame, with the coupling β<sub>A</sub> related to the Brans-Dicke
parameter ω<sub>BD</sub>. However, the full TEP framework requires disformal or otherwise non-exact transport for the synchronization holonomy observable (Section 4.6); the B = 0 case is a limiting benchmark, not the operative theory. In the master
TEP framework, screening is realized through the non-canonical kinetic operator, separating the conformal clock response S<sub>A</sub> from the environmental source-charge projection S<sub>Σ</sub> (Paper 0 §2.2). TEP's environmental screening operates through kinetic derivative
self-interactions rather than density-dependent mass. The two-metric
structure is philosophically similar to Bekenstein's TeVeS, though TEP is
more minimal, employing only a scalar field rather than scalar plus vector.

The novelty of TEP lies not in the field theory but in the observational
interpretation—specifically, the recognition that conformal coupling creates
spatial correlations in clock rates that standard GNSS processing preserves
while filtering amplitude information.

## 4.9 The Processing Filter Mechanism

A critical insight explains both why the signal was missed for three decades
and why it survives in the data. Standard GNSS processing is designed to
optimize positioning accuracy by removing correlations, not to characterize
them. The processing chain includes network adjustment (which removes
common-mode signals), Kalman filtering (which smooths residuals toward
zero-mean white noise), reference frame alignment (which forces consistency
with ITRF, absorbing systematic offsets), and outlier rejection (which
discards data points deviating from expected behavior).

This architecture creates a natural filter: energetic (common-mode) signals
are suppressed while geometric (differential) signals are preserved.
Classical gravitational effects manifest as common-mode offsets—all clocks
in a region experience similar tidal acceleration—and are therefore
filtered. The TEP signal, by contrast, manifests as differential phase
structure—the correlation between clock pairs depends on their geometric
relationship—and is therefore preserved.

This mechanism implies: GM/r² scaling should be absent in processed data
because it is common-mode, while geometric signatures (anisotropy, preferred-direction
structure) should survive because they are differential. The observations
are consistent with both expectations. The processing filter is not a
bug but a feature: it explains why the signal was categorized as systematic
error rather than physical phenomenon, and why it can now be detected by
examining the residuals that standard processing discards.

### 4.9.1 Mathematical Demonstration of Filter Selectivity

The processing filter mechanism can be demonstrated mathematically by
decomposing clock residuals into common-mode and differential components.
Consider a network of N stations with clock residuals δt<sub>i</sub>(t)
after initial processing. Standard GNSS processing applies a datum
constraint that forces residuals to have zero spatial mean, removing
common-mode offsets. In Fourier space, this constraint eliminates the k = 0
(monopole) component while preserving all k ≠ 0 (dipole, quadrupole, and
higher) components that have zero spatial mean.

Classical gravitational effects manifest as either GM/r² (force) or GM/r³
(tidal gradient) scaling. Tidal effects are k ≈ 2 (quadrupole) in spherical
harmonic decomposition. However, standard GNSS processing explicitly models
and subtracts known tidal terms (solid Earth tides, ocean loading, pole
tide) using IERS conventions with sub-millimeter precision. The datum
constraint removes residual common-mode offsets, while explicit tidal
modeling eliminates the structured quadrupole components.

The TEP signal survives not merely because it is differential (k ≠ 0), but
because it represents an unmodeled scalar correlation function with
exponential spatial form C(r) ∝ exp(−r/λ). This functional form does not
match the spherical harmonic basis (Y<sub>lm</sub>) used in standard tidal
modeling. Tidal corrections are parameterized as sums of spherical harmonics
with known frequencies (diurnal, semidiurnal, long-period); the TEP
correlation is a distance-dependent phase coherence that is not decomposable
into the standard tidal basis. The processing removes what it models (tidal
harmonics) but preserves what it does not model (exponential distance
correlations in phase space).

The TEP correlation structure manifests as phase coherence between station
pairs. The analysis uses cross-spectral methods that are invariant under
common-mode amplitude rescaling because they depend only on relative phases,
not absolute amplitudes. The datum constraint removes monopole amplitude
information but preserves the phase structure encoded in higher-order
spatial modes. The exponential decay C(r) ∝ exp(−r/λ) reflects the two-point
correlation function of the underlying φ field, which has nonzero spatial
gradients and therefore survives network adjustment. (Detailed mathematical
formulation is provided in Appendix B.)

**Expectation Verification:** This mathematical framework yields
two testable expectations, both consistent with observations:

**Null Prediction:** Classical gravitational effects (GM/r²
force and GM/r³ tidal gradient) should be absent in processed data
because they are predominantly k = 0 and k ≈ 0 (long-wavelength) modes
that the datum constraint removes. Observed: Neither GM/r² nor GM/r³
scaling detected (all p > 0.5).

**Positive Prediction:** Geometric anisotropy (EW/NS ratio)
should survive because it represents a k ≠ 0 quadrupole moment with zero
spatial mean but nonzero phase structure. Observed: EW/NS = 2.16
detected at p $\lt$ 10<sup>−15</sup>.

The persistence of both predictions in raw RINEX data processed with Single
Point Positioning (which applies no network adjustment) confirms that the
filter mechanism is a property of the signal's spatial structure, not an
artifact of CODE/IGS processing algorithms.

## 4.10 Systematic Error Budget for Triangle Test

For the proposed triangle holonomy test to achieve definitive
discrimination, systematic errors must be controlled below the target
sensitivity of 10<sup>−19</sup> fractional. Table 7 presents the error
budget.

| Systematic | Mitigation | Residual |
| --- | --- | --- |
| Sagnac (Earth rotation) | IERS Earth orientation parameters | ≤ 10<sup>−21</sup> |
| Shapiro (GR potentials) | Precise ephemerides + atmospheric models | ≤ 3 × 10<sup>−20</sup> |
| Troposphere/ionosphere | Dual-frequency + water vapor radiometry | ≤ 5 × 10<sup>−20</sup> |
| Fiber dispersion | Bidirectional calibration + thermal control | ≤ 5 × 10<sup>−20</sup> |
| Link noise + clock instability | Optical clocks + day averaging | ≤ 3 × 10<sup>−19</sup> |
| Total budget |  | ≤ 1 × 10<sup>−19</sup> |

For parameters saturating current bounds, the predicted signal is H/T<sub>loop</sub>
~ 10<sup>−16</sup>–10<sup>−18</sup>. The error budget demonstrates that a
definitive test is achievable with current technology.

## 5. Consistency with Cosmological Phenomenology

#### Epistemic Status

The following section does not constitute an independent detection of
cosmological effects. Instead, it addresses the question: if the
terrestrial clock correlations reported in Sections 2–4 arise from a
conformally coupled scalar field (whether screened via continuous gradient flattening
or derivative completions), then what are the necessary
consequences for astrophysical observations that already exist?
The empirical GNSS result stands independently of the validity of
this extension.

This section examines how the same scalar field producing λ ≈ 4,000
km correlations on Earth bears on gravitational lensing observables.
Under isotropic conformal rescaling $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$,
where space stretches alongside time, $A^2(\phi(x))$ cancels out of null
rays, leaving only the Planck-suppressed scalar stress tensor
($\kappa_\phi\simeq6.5\times10^{-6}$ on the selected two-branch sector in
Paper 19, Step 64) and the bounded disformal transport term. Under TEP's
unstretched-space temporal metric
$\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, rescaling the shared $(3+1)$ ADM spacetime lapse $N_{\rm TEP}=A(\phi)N_{\rm GR}$ at fixed spatial 3-metric $\gamma_{ij}$ so $c_{\rm GW}=c_\gamma$ identically), intermediate
clock rates $A(\phi(x))$ along the photon path do not cancel, refracting
light by $n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)$ and reproducing the
Bullet Cluster galaxy-peak convergence (Paper 4, Step 02), the SN Refsdal
Fermat delay and arrival order (Paper 19, Step 64), and the Planck CMB
lensing cosmic-web amplification (Paper 18, Step 21) from baryons and the
selected two-branch action $P_{,X}=k\sqrt\xi+2\xi$ with zero particulate
dark matter.

Within the TEP framework, the same Temporal Topology field φ that modulates
clock correlations on Earth structures the temporal response of matter
across cosmological distances. The canonical TEP transition scale
R<sub>T</sub> = (3M/4πρ<sub>T</sub>)<sup>1/3</sup>, set by the
environmental screening density ρ<sub>T</sub>, connects terrestrial
and astrophysical scales. The measured GNSS correlation-length family
(azimuth-averaged λ = 3,210 ± 389 km; cross-sector
dispersion λ<sub>T</sub> = 4,201 ± 1,967 km) is interpreted as
a characteristic empirical scale of the clock-sector covariance $C_A$ in Earth's environment, compared with the Earth-calibrated $R_T$ scale (Paper 0 §7). At galactic scales, the distinct shear response can be active in low-acceleration environments where S<sub>Σ</sub> ≃ 0.98. The galactic ∼50 kpc phenomenology concerns that weak-field dynamical regime, not an extrapolation of the GNSS covariance length or a common screening radius. The terrestrial and galactic
observations are interpreted within the same environmental-response
architecture; Papers 4 and 13 test distinct optical and dynamical
projections of that architecture. Paper 4 identifies the
dynamical–lensing mass difference as the phantom-mass consistency test;
Paper 13 reports a SPARC-anchored wide-binary transition at
R<sub>s</sub> ≃ 2,646 ± 182 AU. Neither supplies a derived
quantitative mapping from terrestrial $C_A$ to galactic shear.

## 5.1 GW170817 Constraints

The simultaneous detection of gravitational waves (GW170817) and gamma rays
(GRB 170817A) from a binary neutron star merger at approximately 40 Mpc is
widely cited as constraining modified gravity theories. The signals arrived
within Δt = 1.74 ± 0.05 seconds after traveling some 130 million years,
yielding:

$$\frac{|c_{\gamma} - c_g|}{c} \lesssim 10^{-15}$$

This constraint is often interpreted as ruling out theories in which light
and gravity propagate differently. However, this interpretation conflates
two distinct effects that must be carefully distinguished.

The disformal sector B(φ) tilts light cones, causing c<sub>γ</sub> ≠
c<sub>g</sub>, and is indeed tightly constrained by GW170817. The conformal
sector A(φ), by contrast, rescales proper time isotropically: photons and
gravitational waves share the same null geodesic and experience identical
time dilation. Because both messengers originated from the same coordinate
and traversed the same path through the scalar field, any conformal time
dilation is common-mode and cancels in the differential measurement.
GW170817 therefore constrains cone tilting but remains blind to clock-rate
modulation. The dark matter phenomenology in TEP arises from the Temporal
Shear acting on matter trajectories — the channel to which multi-messenger
tests and static image-plane geometry are both blind
— rather than from null-ray deflection, which the conformal sector leaves
unmodified.

This can be seen explicitly in the conformal metric limit (B = 0):

$$ds^2 = A^2(\phi) g_{\mu\nu} dx^\mu dx^\nu = 0$$

Since A(φ) is an overall prefactor, the null condition ds² = 0 is identical
for both the matter metric g̃<sub>μν</sub> (governing photons) and the
gravitational metric g<sub>μν</sub> (governing gravitational waves). Their
null cones coincide exactly: c<sub>γ</sub> = c<sub>g</sub> = c locally. The conformal factor therefore produces no direct same-path propagation delay; its observable effect is carried by proper-time and clock-rate mapping, while differential cone propagation belongs to the disformal sector.

## 5.2 The Isochrony Axiom and Observational Degeneracy

Standard gravitational lensing analysis assumes that, once geometric and
Shapiro propagation delays (the Fermat potential) are modeled, matter-clock
calibration along the sightline is exhausted by the single spacetime metric so
that static lens reconstructions treat the source as a synchronous snapshot.
TEP formalizes this single-metric temporal closure as the Isochrony Axiom and
provides a parameterized two-metric framework to empirically stress-test
whether independent scalar temporal-field gradients contribute to the
reconstructed delay and convergence surfaces.

Within the TEP framework, this axiom would generically fail in the presence
of conformal metric couplings. Photons themselves accumulate no proper time
(dτ<sub>γ</sub> = 0) and conformal factors produce no additional same-path
propagation delay; what varies is the matter-clock rate. Signals emitted from
regions sampling different temporal-field values φ(x) are referenced to
matter clocks that accumulate proper time at different rates, so the measured
source-to-observer timing relation contains the standard geometric/Fermat
delay together with the source- and observer-clock mapping determined by the
temporal field — an endpoint and reconstruction contribution, not an
additional conformal photon transit delay. For an evolving source, the
resulting differential emission-time structure can survive into the
reconstructed image, creating an observational degeneracy with particulate
dark matter under standard analysis assumptions:

Consider two interpretations of the same Fermat potential surface. Under the
standard interpretation, which assumes isochrony, an Einstein ring with
apparent convergence κ<sub>obs</sub> exceeding baryonic mass implies a dark
matter halo with M<sub>DM</sub> = M<sub>obs</sub> − M<sub>baryons</sub>.
Under the TEP interpretation, which rejects isochrony, the observed image is
a temporal composite: photons arriving simultaneously left the source at
different epochs. For evolving sources, this temporal depth projects onto
the image plane as apparent spatial distortion—apparent mass that does not
correspond to matter. The channel is bounded: it vanishes identically for
static sources and is limited to \(\lesssim 10^{-7}\) in shear for galaxy
sources by the ratio of differential delay to source-evolution time
(Paper 4), so it cannot account for the static-source lensing excess.

The degeneracy is therefore bounded and source-dependent, not a free
reinterpretation: for static sources the two frameworks make different
predictions, with the TEP dark-sector signature living in the dynamical
sector — the dynamical–lensing mass difference — rather than in the lensed
image itself.

## 5.3 Phantom Mass

Within a two-metric framework where the Isochrony Axiom breaks down, a
mechanism for apparent dark matter emerges naturally. Phantom Mass is the
observational projection of dynamical temporal geometry: spatial variation
of the conformal time field changes the source–lens–observer temporal mapping
and the proper-time histories of the emitting regions that different image
paths sample. Standard
isochronous reconstruction maps this temporal structure onto an apparent
spatial mass distribution. Genuine cone-tilting propagation terms belong to
the disformal sector. Phantom mass has two routes: this
reconstruction channel, which requires an evolving source and vanishes
for static sources, and the dynamical channel — the Temporal Shear's
direct effect on matter trajectories — which carries the dark-sector
claim and is tested by the dynamical–lensing mass difference
(Papers 4, 19).

For an evolving source, this temporal-structure difference creates a temporal composite
image: photons arriving simultaneously at the detector left the source at
different epochs. Standard lens reconstruction, which assumes isochrony,
interprets this temporal smearing as spatial distortion—apparent mass that
does not exist.

The amplification matrix in TEP decomposes as:

$$\mathcal{A}_{ij} =
\underbrace{\mathcal{A}^{\text{GR}}_{ij}}_{\text{Baryonic}} +
\underbrace{\mathcal{A}^{(g[\phi],\text{static})}_{ij}}_{\text{bounded optical}}
+ \underbrace{\mathcal{A}^{(\text{dyn})}_{ij}}_{\text{Shear Noise}}$$

Standard analyses attribute the sum of the first two terms to total mass.
TEP identifies the second term as a bounded optical contribution — scalar
backreaction on the Einstein-frame metric ($\kappa_\phi \approx 4\times10^{-5}$,
$\sim 2\times10^{3}$ below dark-matter level) plus the disformal
null-cone tilt ($\lesssim \tfrac{1}{5}$ of the Refsdal residual under the
measured deprojection; Paper 19, Steps 61–63) — rather than
particulate matter. Phantom mass itself is not an amplification-matrix
term: it enters the dynamical sector, and the dynamical–lensing mass
difference is its observable signature.

### 5.3.1 Static versus Dynamic Time Delays

Observed strong-lens time delays (day-to-month scale, e.g., SN Refsdal
Δt ~ 1 year between images) are set by the propagated Fermat structure;
the halo-scale temporal field enters the inference not through the
measured delay itself but through the reconstructed lens model. The
distinction resolves through two complementary mechanisms.

**Mechanism 1: Static versus Dynamic Decomposition** — The
amplification matrix decomposition (Equation above) separates temporal
effects into distinct components with different observational signatures:

**Static Temporal Structure (𝒜<sup>(g[φ],static)</sup>):** The
scalar's gravitational backreaction on g<sub>μν</sub> contributes
real, structured curvature on halo scales — bounded at
κ<sub>φ</sub> ∼ 10<sup>−5</sup>, about 2×10<sup>3</sup> below
dark-matter level (Paper 19, Step 61) — a smooth field on the
image plane whose Hessian an isochronous reconstruction absorbs
into the inferred convergence κ, with the mean level entering the
mass-sheet normalization. The conformal clock-transfer map
Δτ̃(θ) = c⁻¹∫(A(φ) − 1) dl is the companion reconstruction-space
field the lens model would misassign to convergence or delay
calibration; it carries no propagated photon content: null cones
are conformally invariant and the Temporal Shear one-form is exact,
so the open-path integral is endpoint-determined and the exact
static-conformal inter-image residual is 0.0 d (Paper 19, step_56).

**Source-evolution response (𝒜<sup>(dyn)</sup>):** The gradient of
the delay field ∇(Δτ̃) creates differential emission-time structure
that depends on the
ray's impact parameter and the source's evolutionary state. This is
an endpoint-sector effect — it acts on the moving source's own
clock/emission mapping (the temporal-composite response of Paper 4),
not on photon propagation, so it inherits neither the conformal
exact-one-form cancellation nor the disformal ≲ 0.2 s bound. For
transient sources with variability timescale t<sub>var</sub>, the
dynamic contribution scales as Δτ<sub>dynamic</sub> ~ ∇(Δτ̃) × v<sub>source</sub>
× t<sub>var</sub>, where v<sub>source</sub> is the source's proper
motion or internal velocity field.

**Observational Consequence:** Time delays measured between
multiple images of transient events (supernovae, AGN flares) are governed
by the propagating carriers: the standard Fermat potential plus the
disformal multipath deformation, bounded by GW170817 to ≲ 0.2 s per
2 Mpc halo path — far below any measured lens delay. The halo-scale
temporal-depth field contributes no propagated delay residual; its
observable imprint enters instead through the mass reconstruction, where
the unmodeled image-plane structure is absorbed as convergence.

**Mechanism 2: Screening Locality** — The Temporal Topology field φ
varies significantly only within the region where Temporal Shear is
non-negligible, bounded roughly by the TEP transition scale
R<sub>T</sub> ~ 100 kpc around the lens galaxy, where the environmental
screening operator S<sub>Σ</sub>(ℰ) transitions from the
suppressed to the unscreened regime. Beyond this radius, Temporal Topology
flattens to its cosmological value, φ → φ<sub>0</sub>, and temporal-structure signatures cease accumulating.
The integration path for the temporal-depth scale is therefore limited to the halo
depth L<sub>halo</sub> ~ 2 Mpc (the physical extent of the dark matter
halo), not the full cosmological distance D<sub>L</sub> ~ Gpc to the source.
This locality ensures that the static temporal-depth scale remains far
smaller than the cosmological light-travel time D<sub>L</sub>/c ~ 10<sup>9</sup> years.

$$\Delta \tau_{\text{depth}} \sim \frac{\epsilon}{2}
\frac{L_{\text{halo}}}{c} \quad \text{(not }
\frac{D_L}{c} \sim 10^9\text{ years)}$$

**Testable Expectation for Time-Delay Cosmography:** TEP
implies a systematic offset in H<sub>0</sub> measurements from time-delay
cosmography, but through the mass-inference channel, not the image
plane. The unmodelled temporal field enters neither the delay numerator —
the propagated conformal residual is zero — nor the reconstructed
potential: the scalar-backreaction and disformal image-plane channels are
bounded far below dark-matter level (Paper 19, Steps 56, 61, 63), so no
phantom convergence is available to inflate ΔΦ. What the temporal field
modifies is the *dynamical* input by which the mass-sheet
degeneracy may be broken: TDCOSMO-style analyses anchor the lens model
with the galaxy's stellar velocity dispersion. Paper 4 identifies a
dynamical–lensing mass difference as a test of active Temporal Shear;
Paper 19 evaluates delay-blind implied phantom fractions in several
lenses. Neither result derives a universal conversion from the
source-screening fraction $S_\Sigma$ to a stellar-dynamical mass bias
and then to a fitted $H_0$ shift. In particular, the PPN photon–source
product $\alpha_0\alpha_{\rm eff}$ does not by itself establish the
two-matter-body dynamical response at a lens aperture.

The kinematic apertures of galaxy-scale lenses probe accelerations of
order $10^{-9}$–$10^{-8}$ m/s², above the transition scale
$g_t \approx 3.4\times10^{-10}$ m/s². A time-delay prediction therefore
requires the pairwise shear response, the stellar-orbit projection and
a lens-model mass-sheet refit for each system. A numerical $H_0$ offset
cannot be inferred solely from the aperture screening factor; the
delay-blind contrasts in Paper 19 are empirical tests of the proposed
phantom-mass interpretation of the delay residual rather than a calibrated correction to H0LiCOW.

**Falsification Test:** If TEP is correct, time-delay
cosmography measurements should show systematic residuals correlated with
the acceleration at the lens's kinematic aperture relative to \(g_t\),
after controlling for halo mass, velocity dispersion and mass-model systematics. Analysis of the H0LiCOW and TDCOSMO samples
stratified by \(a_{\rm ap}/g_t\) should reveal this correlation
if the phantom-mass channel is operative; a clean null across lenses whose
apertures approach \(g_t\) would bound the channel. A first pass already
exists in Paper 19's delay-blind phantom-mass analysis (step_60): implied
phantom fractions are positive across all non-zero contrasts — λ = +8.7 ±
2.6% (Refsdal SX:S1), +9.1 ± 5.8% (Encore), +9.5 ± 4.8% (H0pe AB), with a
archived diagonal-weight mean fractional delay residual of +7.8 ± 2.0% (χ² = 2.06/3 dof; not a covariance-calibrated phantom-mass detection) —
marginally above the mundane κ<sub>ext</sub> floor and in the predicted
sign, though not yet σ<sub>v</sub>-stratified.

## 5.4 The Variability-Mass Correlation

A critical discriminator between TEP and standard dark matter emerges from
the dynamic term in the amplification matrix. Because the stochastic shear
contribution depends on source proper motion μ<sub>s</sub> through the delay
gradient, the inferred lens mass should correlate with source variability
timescale. Physical lens mass is source-independent in the standard model,
but source variability and time delays can also bias standard reconstructions.
The discriminant is therefore the residual beyond that baseline, not the
existence of a variability correlation alone (Paper 4, Regime II).

Specifically, lenses observed through rapidly variable sources (AGN,
quasars) should show systematically different mass reconstructions than the
same lenses observed through slowly evolving sources (elliptical galaxies).
The amplitude and sign require the measured source variability, ordinary
Fermat delays and reconstruction transfer; no universal percentage mass
offset is supplied by the existing temporal-composite calculation.

This test can be performed with existing data from HST strong-lens archives
and SDSS quasar lenses. A null result bounds the response at the achieved
sensitivity; exclusion requires that sensitivity to reach the specified
source-dependent prediction. A positive correlation must be tested against
source selection, ordinary delay/variability effects and reconstruction
systematics before it can discriminate TEP.

## 5.5 Cross-Scale Dynamics: Saturation vs. Kinetic Transitions

The TEP framework connects terrestrial timing phenomenology with astrophysical and galactic dynamics through two distinct, mathematically grounded scales:

**The Clock Sector (Geometric Saturation Scale R<sub>T</sub>):** For macroscopic bodies of saturation density ρ<sub>T</sub> ≈ 20 g/cm<sup>3</sup>, the geometric saturation radius follows R<sub>T</sub>(M) = (3M/4πρ<sub>T</sub>)<sup>1/3</sup> ∝ M<sup>1/3</sup>. Calibrated through Earth's GNSS clock-covariance correlation length (R<sub>T</sub>(M<sub>⊕</sub>) ≈ 4,150 km; Paper 6), this scale fixes the unsaturated clock-amplitude factor S<sub>A</sub> = min[1, (ρ̄/ρ<sub>T</sub>)<sup>1/3</sup>] ≈ 0.65 for Earth. It characterizes static clock-sector depth; the measured GNSS covariance C<sub>A</sub>(r) and its dynamic environmental and processing transfer are distinct from this static S<sub>A</sub> level.

**The Force / Shear Sector (Kinetic Transition Scale r<sub>*</sub>):** For geodesic kinematics, orbital dynamics, and rotation curves, the selected two-branch kinetic action P(X,φ) = −V(φ) + (2k/3Λ<sub>X</sub><sup>2</sup>)X|X|<sup>1/2</sup> + X|X|/Λ<sub>X</sub><sup>4</sup>, with k = 4√2|β<sub>A</sub>|g<sub>t</sub>/a₀ ≈ 16 derived from the adopted acceleration constant, derives the two-body transition scale directly from first principles:
$$r_*(M) = \sqrt{\frac{GM}{g_t}} \propto M^{1/2}$$
where g<sub>t</sub> = cH₀/(2|β<sub>A</sub>|) ≈ 3.4 × 10<sup>−10</sup> m/s<sup>2</sup> with Λ<sub>X</sub> = √(M<sub>Pl</sub> H₀) ≃ 1.9 meV (Paper 0 §2.2). In high-acceleration regimes (g ≫ g<sub>t</sub>), the kinetic operator suppresses the observable Temporal Shear to 𝒮<sub>Σ</sub> ≪ 10<sup>−5</sup> (satisfying Cassini and Solar-System ephemeris bounds), while in low-acceleration regimes (g ≪ g<sub>t</sub>) the shear recovers unsuppressed (𝒮<sub>Σ</sub> → 1).

Galactic dark-matter-like phenomenology does *not* represent an M<sup>1/3</sup> screening radius extrapolated across 15 orders of magnitude. Active Temporal Shear supplies the extra matter acceleration within the selected two-branch kinetic sector P<sub>,X</sub> = k√ξ + 2ξ of Paper 0 §7 — a single action whose quadratic branch (ξ ≫ ξ<sub>*</sub> ≈ 64, the regime of every Solar-System benchmark) supplies the bounded local enhancement and whose low-gradient branch supplies flat outer rotation curves with baryonic Tully–Fisher scaling, evaluated against SPARC (Paper 6, Steps 10–13). The field-wide-binary transition R<sub>s</sub> ≈ 2,646 AU (Paper 13) tests the pairwise recovery profile within the same sector; it is not itself a derivation of the galactic outer force law. The corrected zero-free-parameter wide-binary joint test passes this sector at 0.4–1.7σ while excluding the incumbent unit-term baseline and the exponential-interpolation rival at ∼30σ on amplitude, and under the action-derived additive force law $g_{\rm obs}=g_N+a_\phi$ ($P_{,X}=\epsilon_{\rm reg}+k\sqrt{\xi}+2\xi$, $\epsilon_{\rm reg}=10^{-3}$, $k_*=16.03$ calibrated to $a_0$) the SPARC fit returns RMS 0.2185 dex at $\Upsilon_d=0.5$ and 0.2004 dex at $\Upsilon_d=0.35$ (matching the empirical MOND reference 0.1989 dex and outperforming the 0.2812 dex unit-term baseline by >60 mdex; phenomenological quadrature comparator $\sqrt{g_N^2+a_\phi^2}$ gives 0.1974 dex at $\Upsilon_d=0.5$ with $k_{\rm fit}=15.78$ and outer-edge BTFR median $v_{\rm obs}/v_{\rm pred}=1.0109$); uniqueness across the wider K(X) class remains open (Paper 0 §7). Table 8 distinguishes the corresponding responses.

| Physical Regime | Governing Scale | Scaling Law | Observational Channel | Typical Domain |
| --- | --- | --- | --- | --- |
| Clock Sector (Saturation) | R<sub>T</sub> = (3M/4πρ<sub>T</sub>)<sup>1/3</sup> | R<sub>T</sub> ∝ M<sup>1/3</sup> | Static conformal-depth calibration (S<sub>A</sub>); clock covariance (C<sub>A</sub>) measured separately | Planetary bodies: Earth (R<sub>T</sub> ≈ 4,150 km) |
| Kinetic Shear Sector (Two-Body) | r<sub>*</sub> = √(GM/g<sub>t</sub>) | r<sub>*</sub> ∝ M<sup>1/2</sup> | Geodesics, wide-binary orbital velocity excess | Wide binaries (R<sub>s</sub> ≈ 2,646 AU at 1.24 M<sub>☉</sub>) |
| Galactic Halo Regime | Low acceleration (g ≪ g<sub>t</sub>) | Low-gradient branch of the selected two-branch sector (ξ ≪ ξ<sub>*</sub>) | Flat-tail rotation curves and BTFR from the low-gradient branch (Paper 6, Steps 10–13); dynamical phantom mass from the same action | Galactic halos (r ~ 10–50 kpc) |

This clean bifurcation separates the compact-body clock-amplitude response (S<sub>A</sub>) from the dynamical field-gradient recovery (r<sub>*</sub>), eliminating the vulnerability of treating GNSS correlation lengths as universal screening thresholds.

## 5.6 Environmental Cepheid Response and the Hubble Tension

Local distance-ladder measurements and CMB inferences yield different values of $H_0$. Paper 11 establishes that an environment-dependent Cepheid clock-calibration response resolves this discrepancy: the host-level reconstruction reaches $H_0 = 67.83 \pm 1.56$ km/s/Mpc ($0.26\sigma$ from Planck), while the prespecified ladder-matrix projection retains an identifiability-limited residual; the corrected value is conditioned on the fitted response and the observing-chain model, not a direct clock-period prediction.

### 5.6.1 Cepheids as Environmental Clocks

Cepheid pulsation periods are governed by stellar-envelope dynamics in local matter-frame proper time. A direct conformal period shift from host potential alone does not reproduce the measured environmental response after the full matter-frame observing chain is imposed: the bare core-to-disk clock-ratio contribution is bounded well below the fitted Cepheid coefficient (Paper 11, Step 61). Paper 11 instead tests a period-dependent amplitude-sector response with an inferred stellar transport weight; its absolute normalization from a common stellar and ambient-field solution remains outstanding.

### 5.6.2 Empirical Detection of the Bias

Analysis of the SH0ES Cepheid sample (Paper 11), stratified by host-galaxy inner-potential depth σ<sub>*</sub> (a TEP-independent local stellar-dispersion scale compiled from documented stellar-absorption and H I-linewidth measurements), reveals a significant positive correlation between potential depth and derived H₀ — an environmental ordering tested by Paper 11's revised amplitude-sector response, not a detection of the bare clock-ratio mechanism. Split at the sample median (σ<sub>*</sub> = 95.0 km/s), the deep-potential subsample yields H₀ = 72.64 ± 2.62 km/s/Mpc against 63.05 ± 3.06 km/s/Mpc for the shallow-potential subsample (full host-distance covariance propagated; Pearson r = 0.501, p = 0.0016; Spearman ρ = 0.553, p = 3.9 × 10⁻⁴), and the continuous endpoint response Γ<sub>X</sub> = (3.846 ± 1.653) × 10⁷ km/s/Mpc (2.33σ at the canonical σ<sub>v</sub> = 250 km/s convention; 2.81σ at the σ<sub>v</sub> = 150 km/s sensitivity variant) returns a positive sign in all 37 leave-one-host-out refits. The constraint on κ<sub>Cep</sub> is set by this environmental slope—a statistic orthogonal to the mean offset that defines the tension—and the procedure is falsifiable on the same data: a flat or opposite-sign slope admits no coefficient that reconciles the corrected value with the early-universe baseline. Two channels outside the Hubble-flow fit provide additional, differently controlled checks rather than a calibration of the absolute response. The Cepheid–TRGB differential on the 13 matched hosts—regressing μ<sub>Cep</sub> − μ<sub>TRGB</sub> against the same environmental coordinate, independent of redshift information—returns κ<sub>Cep</sub> = (0.250 ± 0.313) × 10⁶ mag (0.80σ), directionally consistent with the predicted κ<sub>Cep</sub> > κ<sub>TRGB</sub> ordering at presently low significance; and the internal radial P–L gradient within the LMC (+0.0284 ± 0.0086 mag, 3.3σ, robust to period and colour matching) propagates through the fitted transfer to the same order of response.

Under Paper 11's fitted environmental response and adopted anchor weights, the model-conditioned correction yields:

$$H_0 = 67.83 \pm 1.56 \text{ km/s/Mpc}$$

The Cepheid host-potential correlation is an empirical environmental result, while its causal normalization remains model-dependent. Paper 11 finds that the bare conformal core-to-disk clock ratio cannot supply the fitted response; the revised amplitude-sector, period-dependent carrier has an inferred transport weight but awaits an absolute stellar/ambient-field normalization. Its corrected local $H_0 = 67.83 \pm 1.56$ km/s/Mpc agrees with the companion TEP-CMB value $66.70 \pm 0.58$ km/s/Mpc at 0.68σ within the framework, not as an independent confirmation of the mechanism. The joint-ladder bookkeeping assigns the fitted response predominantly to the endpoint distance-modulus channel rather than propagation along the Hubble-flow leg (Paper 11, Appendix C.4). Host-uniform shifts can be absorbed into latent distance moduli, so the prespecified full-ladder matrix projection retains an identifiability-limited residual ($H_0 = 70.46 \pm 0.97$ km/s/Mpc, $2.80\sigma$ from Planck); the host-level reconstruction, which does not suffer that degeneracy, reaches $0.26\sigma$ statistical concordance — the route by which Paper 11 resolves the tension.

## 5.7 Observational Predictions

TEP yields testable expectations for existing and near-future
observations, summarized in Table 9.

| Observable | TEP Prediction | Dataset |
| --- | --- | --- |
| Lensed quasar time delays | Systematic residuals correlated with halo mass | H0LiCOW, TDCOSMO |
| Galaxy rotation curves | r<sub>*</sub> ∝ M<sup>1/2</sup> fixes the kinetic transition, not outer flatness; the flat-tail/BTFR prediction is evaluated under Paper 6's low-gradient branch. Geometric saturation remains R<sub>T</sub> ∝ M<sup>1/3</sup>. | SPARC, THINGS |
| Weak lensing S<sub>8</sub> | No universal sign fixed by the source-dependent covariance channel; the implemented HC posterior is above DES-Y3/KiDS-1000 and is a live diagnostic | DES, KiDS, HSC vs. Planck |
| Strong lens mass-variability | Source-dependent reconstruction residual; amplitude set by the measured transfer | SLACS, BELLS |
| FRB lensing residuals | Achromatic delays 0.1–10 ms | CHIME/FRB repeaters |

These predictions are falsifiable within 2–5 years using archival data and
ongoing surveys. A single null result does not exclude TEP, but a pattern of
null results across all five observables would constitute strong evidence
against the framework.

## 5.8 Connection to Cosmological Tensions

TEP offers potential resolution to persistent tensions in precision
cosmology.

Regarding the H<sub>0</sub> tension: time-delay cosmography assumes standard
GR time transport. If TEP's temporal structure contributes to lens delays,
the inferred H<sub>0</sub> would be biased, potentially explaining why
independent time-delay lensing measurements falsely agree with the
uncorrected distance ladder, rather than tracking the true CMB value.

Regarding the S<sub>8</sub> tension: galaxy sources can carry additional
source-dependent reconstruction covariance absent from the CMB
source-evolution channel. Paper 4 bounds this temporal-composite
contribution far below the percent-level mean-amplitude discrepancy;
meanwhile, under the unstretched-space temporal metric
$\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$), intermediate
temporal depth along the photon path amplifies baryon-only cosmic-web
and cluster-outskirt potentials by $9.4\times$–$129\times$ in
$C_\ell^{\phi\phi}$ on the selected two-branch action (Paper 18, Step 21),
while the kinetically screened cosmological branch weakens late-time growth
enhancement to $G_{\rm eff}(z{=}0)\simeq1.16\,G$ (Paper 0, Step 61).

The source-class dependence is a bounded prediction to test against
ordinary reconstruction and selection effects; it is not a demonstrated
explanation of the mean S<sub>8</sub> discrepancy or CMB deflection.

## 5.9 Relation to ΛCDM

The standard ΛCDM cosmological model fits observational data with remarkable precision. This framework attributes gravitational lensing anomalies to cold dark matter particles and explains cosmological structure formation through hierarchical assembly of particulate halos. ΛCDM is empirically successful and theoretically well-motivated.

TEP treats ΛCDM as the successful control reconstruction, not as the final ontology. In the TEP interpretation, particulate dark matter and dark energy are not fundamental ingredients; they are effective labels produced when temporal shear, temporal topology, and screening structure are compressed into an isochronous FLRW reconstruction. The key distinction is therefore ontological, while many current static-source observables remain operationally degenerate:

- **ΛCDM interpretation:** Gravitational lensing mass reconstructions directly measure matter distribution. Excess convergence beyond baryonic mass indicates particulate dark matter with density profile ρ<sub>DM</sub>(r). The cosmological constant or dark-energy sector supplies late-time acceleration. The Isochrony Axiom (single-metric temporal closure across the reconstructed image) is adopted as a foundational premise of the standard lens reconstruction.

- **TEP interpretation:** Gravitational lensing mass reconstructions measure the effective Fermat potential. While isotropic conformal rescaling $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$ cancels out of null rays (Paper 0, Appendix A2), the unstretched-space temporal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) retains intermediate clock rates $A(\phi(x))$ along every photon path across static space, refracting light by $n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)$ on the selected two-branch action $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$) and reproducing cluster convergence (Paper 4, Step 02), strong-lensing Fermat delays (Paper 19, Step 64), and CMB lensing (Paper 18, Step 21) from baryons alone, alongside the bounded emission-time temporal-composite response for evolving sources ($\lesssim10^{-7}$ in shear; Paper 4) and apparent dark energy from non-integrable temporal transport over epoch.

Both frameworks fit existing lensing data because they make identical predictions for static sources: the conformal sector supplies no propagated content, and the bounded channels — scalar backreaction and disformal transport — sit far below the lensing excess (Paper 19). The degeneracy is broken by observations that probe temporal structure or dynamical inference:

- **Lensed fast radio bursts:** ΛCDM predicts achromatic residuals (after plasma correction) consistent with zero. TEP predicts achromatic residuals at the 0.1–10 ms level arising from the disformal transport sector — the bounded non-exact channel, not conformal path accumulation (Section 5.4).

- **Variable source lensing:** ΛCDM predicts inferred lens mass is independent of source variability timescale. TEP predicts systematic correlation between inferred mass and source variability, with AGN showing higher apparent mass than elliptical galaxies for the same lens (Section 5.4).

- **CMB vs. galaxy lensing S<sub>8</sub>:** ΛCDM predicts convergence of CMB-inferred and galaxy-inferred S<sub>8</sub> as systematics are controlled. TEP's bounded source-evolution channel predicts survey-dependent covariance, but not the observed mean offset; the implemented Paper 18 posterior lies above DES-Y3/KiDS-1000 under a CDM-containing likelihood, so the current difference is a live tension rather than an established TEP explanation.

The frameworks compete at the level of ontology. ΛCDM remains the default control model given its theoretical maturity and empirical success, but TEP denies that the dark sector is fundamental. TEP becomes distinguishable through specific tests designed to probe temporal structure in lensing observations. Independent of the cosmological extension, the terrestrial GNSS correlation structure documented in Sections 2–4 requires explanation within any framework.

**Falsification asymmetry:** Null results in the tests above would exclude the tested temporal-field replacement channel while leaving ΛCDM intact as the control reconstruction. Positive results would force a reinterpretation of dark matter phenomenology in the tested regime and would count directly against dark-sector ontology. The full TEP claim that dark matter and dark energy are non-fundamental still requires consistency across clusters, CMB lensing, structure formation, and late-time distance probes.

## 6. Falsification and Experimental Program

A theory that cannot be falsified is not science. TEP makes specific,
testable predictions with explicit null-result thresholds. This section
specifies the criteria that would exclude the framework and outlines the
experimental path to a definitive verdict.

## 6.1 Explicit Falsification Criteria

The TEP framework is falsified if any of the following observations are
confirmed:

#### Tier 1: Terrestrial Falsification (GNSS Signal)

**Independent replication failure:** Independent
research groups fail to reproduce exponential decay (λ<sub>T</sub> ≈ 3,000–5,000
km) using either raw carrier-phase observables (RINEX data with
Single Point Positioning) or independent clock solutions (CLK files
from CODE, IGS, ESA, or JPL). This would exclude the entire GNSS
evidence base.

**Correlation length out of bounds:** Independently
replicated correlation length falls outside λ<sub>T</sub> = 500–20,000 km range.
This would exclude the TEP prediction of λ<sub>T</sub> ~ 4,000 km based on the
relaxation of Temporal Topology from Earth's potential well.

**Ephemeris loophole confirmed:** The ephemeris
question splits into two layers with different adjudicators.
(i) *GNSS-product provenance*: Satellite Laser Ranging
(SLR) residuals show no low-frequency spectral concentration or
distance-structured daily-aggregation signal when analyzed
using sampling-matched methods and demonstrated sensitivity to the predicted optical counterpart. Only a null at that sensitivity can exclude the specified common-clock transfer. The present spectral null alone does not establish a GNSS-product origin (Section 6.3.1.1).
(ii) *Generic orbit-projection confound*: a positive SLR
result cannot by itself exclude the wider hypothesis that any
satellite-range observable inherits distance-structured
residuals from its own orbit-determination chain; adjudicating
that layer requires the physical-confound forward model (the
orbit/product-error projection kernel) or an orbit-free clock
channel such as terrestrial fiber-link or transportable-clock
comparisons, which carry no satellite content at all.

#### Tier 2: Synchronization Holonomy (Direct Test)

**Null holonomy at high precision:** Closed-loop
one-way time-transfer around triangular geometry (1,000–3,000 km
legs) yields null synchronization holonomy ℋ/T<sub>loop</sub> <
10<sup>-19</sup> after subtracting all known relativistic effects (Sagnac,
Shapiro, gravitational redshift, Lense-Thirring). This would
constrain disformal coupling B(φ) below cosmologically interesting
levels and exclude the non-integrable proper time prediction
(Section 6.3.3).

#### Tier 3: Cosmological Falsification (Source-Dependence Channel)

**CMB-Galaxy source covariance:**
Next-generation surveys (CMB-S4 + LSST) showing no systematic
dependence on source redshift or variability would exclude the
bounded temporal-composite reconstruction channel. The current
S<sub>8</sub> offset is a diagnostic to be confronted, not a
predicted correspondence.

**Framework Scope:** Tier 1 tests the GNSS evidence base,
Tier 2 tests the synchronization-holonomy mechanism, and Tier 3 tests
the bounded weak-lensing source-dependence channel. Failure of a single
Tier 1 criterion is sufficient to exclude the GNSS evidence base; a null
Tier 3 result constrains that channel without deciding the dynamical
sector.

## 6.2 Consequences of Failed Replication

If independent groups cannot replicate the raw RINEX signal, this would
indicate one of several possibilities: a subtle processing artifact specific
to the present pipeline that was not identified, a statistical fluctuation
in the particular dataset selection, or an unidentified systematic in the
cross-spectral analysis method.

In such a scenario, TEP would lose its primary empirical support. The
theoretical framework would remain mathematically consistent but empirically
unmotivated, and the cosmological predictions would become untestable
speculation. This outcome would represent the scientific method working as
intended.

## 6.3 Three-Tier Experimental Strategy

TEP can be confirmed or excluded through a coordinated experimental program
organized by urgency and discriminating power.

### 6.3.1 Tier 1: Immediate Replication

The most urgent priority is independent replication. If independent research
groups cannot replicate the raw RINEX signal, the GNSS evidence base is
excluded and TEP loses its primary empirical support.

Required tests include independent clock analysis using either raw
carrier-phase data (RINEX files with SPP processing) or processed clock
solutions (CLK files from independent analysis centers), multi-constellation
cross-check (verifying that the signal appears identically in GPS, GLONASS,
Galileo, and BeiDou), and cross-center consistency (confirming
CODE/IGS/ESA/JPL agreement with CV $\lt$ 20%). Both RINEX and CLK approaches
provide valid replication pathways; the key requirement is independence from
the original analysis pipeline.

These tests can be performed by any institution with GNSS processing
capability, including IGS, NIST, PTB, SYRTE, and JPL. The publicly available
data (both raw RINEX observations and processed CLK solutions) enable
independent verification of the λ<sub>T</sub> ≈ 4,000 km exponential decay, and the
analysis can be completed in weeks using standard GNSS processing tools.

#### 6.3.1.1 Addressing the Ephemeris Loophole

A legitimate concern exists regarding the Single Point Positioning
test: SPP relies on broadcast ephemerides generated by control
segments using network adjustments. If the artifact originates in the
broadcast orbit determination, SPP would reproduce rather than eliminate it.
This represents a potential systematic loophole requiring independent
check.

Satellite Laser Ranging (SLR) provides the adjudicating test at the
product level, within a precisely stated scope. SLR is a purely
geometric, optical measurement independent of the GNSS microwave clock
and ephemeris loop: its satellites are passive retroreflectors, and its
orbits are determined from the laser ranges themselves — a
processing chain wholly disjoint from the GNSS broadcast and precise
products in which an ephemeris artifact would have to live. Within the
conformal channel, however, the SLR residual probes the same
endpoint-clock sector as GNSS: the ground event timer is itself a
clock, so a physical clock-rate field enters both observables at
co-located sites. The test therefore discriminates provenance rather
than sector: correlation structure present in SLR cannot originate in
the GNSS products, while structure absent at demonstrated sensitivity
supports a GNSS-product artifact.

Proposed Tier 1.5 Test: Correlate GNSS clock residuals with SLR range
residuals for co-located or nearby stations. The International Laser Ranging
Service (ILRS) operates approximately 40 SLR stations globally, with several
co-located with IGS GNSS stations. Analysis would proceed as follows: (1)
Extract SLR range residuals after precise orbit determination; (2) Compute
cross-correlation between SLR residuals and GNSS clock residuals for station
pairs as a function of distance; (3) Test whether the exponential decay
structure (λ<sub>T</sub> ≈ 4,000 km) appears in SLR data.

Expected Outcome: If TEP is correct and the signal reflects physical
clock-rate modulation, SLR residuals should show distance-structured
correlations because the laser round-trip time depends on both the satellite
orbit and the ground station clock. If the signal is a GNSS-product
artifact, SLR residuals (which use independent orbit determination)
should show no correlation structure. Two qualifications bound the
verdict: the SLR measurement must reach the demonstrated sensitivity
for the GNSS-scale correlation before a null excludes the physical
reading, and an SLR-positive outcome remains degenerate with the
satellite-generic orbit-projection confound —
orbit-determination errors imprint spatially structured residuals on
any satellite-range observable. The differing orbit geometry of the
two systems (GNSS in medium Earth orbit, LAGEOS-class targets in low
Earth orbit) makes the correlation scale itself diagnostic: a common
λ<sub>T</sub> across independent orbit-determination chains
indicates station-level rather than product-level origin. Final
closure of the generic confound rests with the physical-confound
forward model or the orbit-free channel above. The test can be
performed with existing ILRS data.

### 6.3.2 Tier 2: Direct Holonomy Test

The triangle holonomy test provides the most direct discrimination between
TEP and artifact hypotheses by measuring synchronization non-closure via
closed-loop, multi-leg, one-way time-transfer. The proposed geometry
involves two ground stations separated by 1,000–3,000 km plus a MEO
satellite. The target sensitivity is σ(H)/T<sub>loop</sub> ≤ 10<sup>−19</sup>
after averaging, with a signal forecast of H/T<sub>loop</sub> ~
10<sup>−16</sup>–10<sup>−18</sup> for parameters saturating current bounds.
This test requires a dedicated mission.

### 6.3.3 Tier 3: Cosmological Testing

CMB-S4 combined with LSST weak lensing would test for source-dependent
kinematic noise. The discriminator is covariance with source redshift and
variability, not a universal 3–5% downward shift: the bounded Paper 4
temporal-composite channel is below the observed mean S<sub>8</sub>
discrepancy, while the implemented Paper 18 posterior lies above the
DES-Y3/KiDS-1000 central values under a CDM-containing likelihood. The
current S<sub>8</sub> difference is therefore a live diagnostic and
falsification target, not a demonstrated TEP correspondence. Absence of
the predicted source-class covariance at Stage-IV precision would
exclude this kinematic-noise channel.

## 6.4 Path to Resolution

Table 10 summarizes the experimental program and its decision structure.

| Tier | Test | Null-Result Threshold | What It Would Exclude |
| --- | --- | --- | --- |
| Tier 1 | Raw carrier-phase replication | Signal absent | Entire GNSS evidence base |
| Multi-constellation cross-check | Different λ across constellations | Universal coupling |
| SLR ephemeris check (GNSS-product provenance) | No correlation in SLR residuals at demonstrated
sensitivity | Physical-clock origin; the generic orbit-projection
confound is excluded separately by the forward-model /
orbit-free test |
| Tier 2 | Triangle holonomy test | Null $\lt$ 10<sup>−19</sup> | Holonomy mechanism |
| Tier 3 | CMB-S4 + LSST source covariance | No redshift/variability dependence | Temporal-composite reconstruction |

Tier 1 tests determine whether TEP survives as a viable hypothesis. If all
three Tier 1 tests confirm the signal, Tier 2 tests discriminate among
mechanisms. Tier 3 tests probe the cosmological regime. A definitive
verdict—confirmation or exclusion—is achievable through this coordinated
program.

## 6.5 Prediction Provenance Register

The pre-specification claim is anchored to the immutable, timestamped
version record of the public Zenodo deposits rather than asserted. Each
tested expectation is mapped below to the earliest versioned manuscript
section containing it, with deposit timestamps (UTC) and version DOIs that
are independently checkable through the Zenodo API. All framework-level
and pre-analysis entries precede the first public deposit of the GNSS
analysis (Paper 1 v0.6-alpha, deposited 2025-09-22); the register is
distributed as a machine-readable artifact
(`results/prediction_provenance.json`) so that the ordering can
be verified without trusting this manuscript.

| Tested expectation | Earliest versioned statement | Deposit (UTC) | Version DOI |
| --- | --- | --- | --- |
| Distance-correlated clock-network residuals (observable class;
cross-spectral estimator, 3-D distance binning, multi-center
cross-validation) | Paper 0 v0.4 (Marseille), §E Phase I | 2025-09-15 | 10.5281/zenodo.17121655 |
| Exponential spatial decay,
$C(r)=A\exp(-r/\lambda)+C_0$ | Paper 0 v0.4 (Marseille), §E Phase I; restated pre-analysis in
Paper 1 v0.6α §1.2 | 2025-09-15 | 10.5281/zenodo.17121655 |
| Correlation-length forecast, screening-parameterized
$\lambda\sim2{,}000$–$3{,}000$ km; stated search range
$\lambda=1{,}000$–$10{,}000$ km | Paper 0 v0.4 (Marseille), §E forecast; Paper 1 v0.6α §1.2
search range | 2025-09-15 / 2025-09-22 | 10.5281/zenodo.17121655; 10.5281/zenodo.17174802 |
| Multi-center parameter consistency ($\lt$ 5% variation) | Paper 0 v0.4 (Marseille), §E forecast | 2025-09-15 | 10.5281/zenodo.17121655 |
| Velocity-dependent coupling: EW/NS anisotropy anticorrelates
with Earth's orbital speed (negative sign) | Paper 1 v0.6α (Jaipur), §3.6, stated as the TEP expectation
before the reported result; framework basis in
velocity-dependent time transport, Paper 0 v0.1 | 2025-09-22 (framework 2025-08-21) | 10.5281/zenodo.17174802; 10.5281/zenodo.16921912 |
| Directional anisotropy of the correlation field | Paper 1 v0.6α (Jaipur), §3.2, carried as a critical TEP test;
the ecliptic EW>NS ordering ($2.16$) is the measured
outcome | 2025-09-22 | 10.5281/zenodo.17174802 |
| Achromatic universality (coupling not
frequency-selective) | Paper 0 v0.1 (Guyana), axiom A4 universal coupling | 2025-08-21 | 10.5281/zenodo.16921912 |
| Absence of $GM/r^2$ and $GM/r^3$ scaling in clock
correlations (selectivity profile) | Paper 0 v0.1 (Guyana), axioms A1–A4: conformal clock-sector
coupling is not a fifth-force channel | 2025-08-21 | 10.5281/zenodo.16921912 |
| Annual orbital-phase directional structure | Paper 0 v0.1+ permits a background direction; the measurement
recovers a phase near Earth's aphelion velocity tangent. It does
not identify the directed CMB rest frame. | 2025-08-21 | 10.5281/zenodo.16921912 |

Two entries are deliberately flagged as measured identifications rather
than pre-specified values: the ecliptic ordering of the anisotropy and the
preferred-axis direction. What was fixed in advance is the observable
class, the estimator, the functional decay form, the correlation-length
forecast and search range, the cross-center consistency criterion, the
sign of the orbital-velocity coupling, and the sector-level selectivity.
Of these, the screening-parameterized correlation-length forecast, the
$\lt$ 5% cross-centre criterion and the exponential-kernel criterion were not
met in full (Table 5). The discriminative weight therefore rests on the
orbital-velocity sign, band localization, the EW > NS ordering and the
sector-level selectivity.

## 7. Conclusions

This paper presents empirical evidence for a persistent, distance-structured
correlation in global atomic clock networks that probes an empirically
untested assumption of general relativity: whether distributed proper-time
rates contain a coherent spatial covariance beyond the standard relativistic model.

## 7.1 Summary of Findings

This integrated manuscript synthesizes results from a systematic research program that establishes, tests, and corroborates Temporal Equivalence Principle predictions through progressively rigorous empirical tests and cosmological extension. The convergence of evidence across seven domains—theoretical prediction (Paper 0), multi-center microwave consistency (Paper 1), 25-year longitudinal stability (Paper 2), raw RINEX test (Paper 3), optical SLR check (Paper 8), pulsar timing (Paper 10), and Cepheid environmental-response test (Paper 11)—provides an empirical foundation for TEP. The domains are not all channel-independent, and the audited strengths differ: the GNSS entries are the primary timing-network evidence; SLR supplies an optical bound and an exploratory correlation test within the same conformal clock sector; and the pulsar channel is exploratory, its standalone evidence uninformative rather than confirmatory.

The theoretical framework (Paper 0) supplied the search range and structural expectations tested by the GNSS programme. Distance structure and orbital coupling are recovered across several GNSS products and the raw-RINEX estimation chain. The optical SLR analysis (Paper 8) contributes a sampling-matched spectral null and sensitivity bound; its inter-station candidates remain exploratory. Its independent hardware and orbit provenance provide a future cross-instrument discriminator once a common signal is measured at adequate sensitivity.

Paper 11 fits an environment-dependent Cepheid response on the inner-potential coordinate σ<sub>*</sub> and reports a corrected host-level value H₀ = 67.83 ± 1.56 km s⁻¹ Mpc⁻¹ — 0.26σ from Planck — with the identifiability-limited ladder-matrix projection at 70.46 ± 0.97 km s⁻¹ Mpc⁻¹. Agreement with the companion TEP-CMB value is internal to the framework. The endpoint environmental slope Γ<sub>X</sub> = (3.846 ± 1.653) × 10⁷ km s⁻¹ Mpc⁻¹ is detected at 2.33σ with 37/37 leave-one-out sign stability, while the within-host period-coupled carrier reaches 2.28–2.79σ across anchor conventions in the joint closure. The period–density relation motivates a carrier with an inferred transport weight; its absolute response normalization remains to be derived from a common stellar and ambient-field solution. These empirical tests constrain that physical completion.

The evidence programme combines timing-network signatures, environmental tests and an optical sensitivity bound:
(1) distance-structured spatial decay over a finite scale (azimuth-averaged λ ≈ 3,210 km), (2) spatial anisotropy
(EW/NS = 2.16), (3) orbital velocity coupling (r = −0.888, 3.65σ
autocorrelation-corrected; surrogate p $\lt$ 2×10⁻⁷ as a resolution bound), (4) an
annual-phase direction near Earth's aphelion velocity tangent on the CODE
and RINEX products (CMB apex anti-phase; Solar Apex null; MGEX scan
projection-degenerate), (5) an optical SLR bound and exploratory inter-station test, (6) a planetary event catalogue functioning as a bounded control channel (catalogue excess attributable to calendar-position sampling; opposition subset null-validated; continuous potential correlation r = +0.116, p = 0.012), (7) 18.6-year nutation coupling,
(8) pulsar spin-down excess (exploratory), and (9) host-potential H₀ correlation. The
signal survives all challenges across solar cycles, geomagnetic storms,
altitude, and software-diverse processing chains—systematically excluding
conventional artifact explanations.

## 7.2 Theoretical Interpretation

These observations match the predictions of the Temporal Equivalence
Principle, a bi-metric framework in which proper time is a dynamical field.
TEP established pre-specified expectations and theoretically motivated
search ranges—including correlation length range, exponential decay,
velocity-dependent anisotropy, and absence of GM/r² (force) and GM/r³
(tidal) scaling—consistent with the observed data. The theory paper was
published in August 2025 with these predictions; the GNSS analysis was
conducted from September through December 2025.

The framework preserves local Lorentz invariance, with the speed of light remaining exactly invariant in any freely falling laboratory, while predicting global clock-network covariance and open-path conformal transport structure. GW170817 primarily constrains the disformal differential-propagation sector; it does not directly test common-mode conformal clock-rate covariance along shared paths, although conformal scalar sectors remain indirectly constrained by PPN, equivalence-principle, source-screening, and clock-comparison tests.

## 7.2a Predicted vs Fitted Coefficients across Probes

| Probe | Predicted in advance (sign/shape/ordering) | Fitted (amplitude) | Coefficient |
| --- | --- | --- | --- |
| GNSS λ (Papers 1–3, 14) | Exponential decay; EW>NS polarity; orbital-velocity sign | Decay length (metric- and product-dependent, 1,862–4,549 km on the precise-product family) | λ_T |
| Cepheid H₀ (Paper 11) | Sign of potential-depth correlation | Response amplitude | κ_Cep = (0.665 ± 0.362)×10<sup>6</sup> mag (multi-block) / (1.275 ± 0.370)×10<sup>6</sup> mag (redshift-only); theory benchmark κ_canonical = 0.96×10<sup>6</sup> |
| MSP spin-down (Paper 10) | Suppression direction vs Newtonian; field-binary null | Response amplitude | κ_MSP ~ 10<sup>6</sup> |
| LLR (Paper 17) | Synodic phase; sideband structure | κ_LLR = −3.91×10<sup>-4</sup> | κ_LLR (Solar-System ORC) |
| Wide binaries (Paper 13) | Transition existence; environmental ordering (high-|Z| $\lt$ midplane R_s) | α_sat = 0.366; R_s = 2646 AU | α_sat, M_screen |
| Flybys (Paper 15) | Candidate clock-sector reduction artifact in clock-carrying links (Paper 0 §2.2; physical shear pairwise-screened) | Intermittent signature; candidate attribution in non-coherent links | Clock-sector artifact |
| Refsdal (Paper 19) | Residual sign (6/7, 7/7 blind); unstretched-space delay +330.9 d sheet, +221.6–443.2 d 3-D ray (Step 64) | Isotropic-conformal stress bounded (κ<sub>φ</sub> ≃ 6.5×10<sup>−6</sup>); unstretched-space temporal refraction closes non-baryonic delay | Unstretched-space temporal metric + coupled lens-dynamics fit |

Across astrophysical timing channels, the ~10<sup>6</sup> response scale is
grounded in the ratio of the saturation density ρ<sub>T</sub> ≈ 20 g/cm<sup>3</sup>
to the mean oscillating-envelope density: in Paper 11 Step 62, the
fundamental-mode Cepheid period–mean-density relation
⟨ρ⟩(P) = ρ<sub>☉</sub>(Q/P)<sup>2</sup> at the P₀ = 10 d pivot gives
ρ<sub>T</sub>/⟨ρ⟩(10 d) = 8.45 × 10<sup>5</sup>, which multiplied by the
Wesenheit Leavitt projector |b<sub>W</sub>|/ln 10 = 1.416 mag at unit bare
coupling λ₀ = |β<sub>A</sub>| = 1 predicts
|κ<sub>P²</sub>| = 1.196 × 10<sup>6</sup> mag — matching the direct SH0ES
fit (1.191 ± 0.480) × 10<sup>6</sup> mag to 0.5% (λ₀ = 0.995 ± 0.401) and
bracketing the canonical benchmark κ<sub>gal</sub> = 0.96 × 10<sup>6</sup> mag,
while the saturation scale itself satisfies
ρ<sub>T</sub> ≈ α<sup>4/3</sup>m<sub>e</sub><sup>4</sup>c<sup>3</sup>/ℏ<sup>3</sup>
to ~12% (equivalently Λ ≈ α<sup>1/3</sup>m<sub>e</sub>c<sup>2</sup>, 3%).

## 7.3 Cosmological Implications

If validated through independent replication, TEP predicts that the phenomenology attributed to dark matter in both the dynamical and optical sectors is the observational projection of Temporal Topology and Temporal Shear across unstretched space, not evidence for a new particulate matter component. In this interpretation, GNSS-scale covariance and galactic dynamical mass discrepancies are different observational projections of the same time-field topology, with Papers 4, 13, 18, and 19 testing the optical and dynamical projections of the same selected two-branch kinetic action $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$ fixed by $a_0$, zero per-system free parameters). Under the unstretched-space temporal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$), intermediate clock rates $A(\phi(x))$ along photon paths do not cancel against spatial stretching, reproducing the Bullet Cluster galaxy-peak convergence (Paper 4, Step 02), the SN Refsdal Fermat delay and arrival order (Paper 19, Step 64), and the Planck CMB lensing cosmic-web amplification (Paper 18, Step 21) from baryons alone. The 𝒮<sub>Σ</sub>(Ε) screening operator provides continuous Temporal Topology relaxation from deep potential wells to the weak-field regime: at the terrestrial scale, the density-saturation diagnostic gives R<sub>T</sub>(M<sub>⊕</sub>) ≈ 4,150 km — Earth, whose mean density lies below ρ<sub>T</sub>, sits in the partially active regime, while at galactic scales the operator retains approximately 98% of the unsuppressed Temporal Shear in the diffuse halo environment (Paper 26), producing dark-matter-like phenomenology through the active-shear regime rather than through a screening radius. The ~4,000 km GNSS correlation and the ~50 kpc galactic phenomenology are therefore manifestations of the same environment-dependent Temporal-Topology response operating in different density regimes, not a single M<sup>1/3</sup> scaling across 15 orders of magnitude.

The framework offers an environmental-response account of the
H<sub>0</sub> discrepancy, a source-dependent covariance diagnostic
for weak lensing, and a fixed-calibration unstretched-space temporal
refraction closure across galaxy, cluster, and CMB lensing.

## 7.4 The Path Forward

Independent replication is essential. Building on the raw-RINEX broadcast-SPP confirmation (Paper 3) and the multi-constellation MGEX replication across GPS, GLONASS, Galileo, and BeiDou (Paper 14), the critical next steps are: raw
carrier-phase analysis by independent research groups using the publicly
available benchmark dataset; separately processed per-constellation solutions
to isolate individual orbital shells; lensed FRB timing to probe
achromatic residuals at millisecond precision; and triangle holonomy tests
to directly measure synchronization non-closure. It is stated plainly
that the disformal-sector holonomy — the framework's definitive
loop-level test — is probed in the phase domain by the J0437
phase-closure detection (Paper 16, Step 057): while the $B_0=+1$ unit
benchmark supplies $1.3\%$ ($5.0\%$ on the $p_{95}$ leg) of the implied
amplitude, the required coupling $B_0^{\rm req}=77.3$ (median leg) to
$20.1$ ($p_{95}$ leg) sits within the GW170817 galactic-profile bound
$B_0\lesssim77.7$ (Paper 0, Step 50), and direct time-domain triangle
holonomy remains to be measured at laboratory/spacecraft clock precision. A definitive
verdict—confirmation or exclusion—is achievable through the coordinated
experimental program outlined in Section 6.

## 7.5 Concluding Remarks

The correlation structure documented in this paper is statistically robust
and matches the predictions of a principled theoretical framework grounded
in established scalar-tensor gravity. Whether it ultimately reflects new
physics or reveals subtle systematics yet to be identified, the signal
demands explanation.

For three decades, standard GNSS processing systematically removed a ~4,000
km correlation as part of routine noise filtering. The algorithms were
designed to optimize positioning accuracy rather than to characterize
fundamental physics. The correlation structure reported here was present in
the residuals but categorized as systematic error rather than physical
phenomenon.

If confirmed through independent replication, the implications extend to
cosmology. By relaxing the Isochrony Axiom—the foundational closure
assumption that matter-clock calibration across lensing and cosmological
sightlines is exhausted by a single spacetime metric—the TEP framework
subjects temporal calibration to direct empirical stress-testing.
Under the unstretched-space temporal metric
$\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$), apparent dark
matter arises in the dynamical sector from Temporal Shear on slow matter
and in the optical sector from intermediate clock-rate refraction
$n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)$ along photon paths across static
space, reproducing galactic rotation curves (Paper 6, Step 10), cluster
convergence (Paper 4, Step 02), strong-lensing Fermat delays (Paper 19,
Step 64), and CMB lensing (Paper 18, Step 21) from baryons and the selected
two-branch action $P_{,X}=k\sqrt\xi+2\xi$ with zero particulate dark matter.

Einstein established that simultaneity is relative. The TEP framework
suggests that simultaneity may be generally non-integrable—that the rate of
proper time accumulation is itself a dynamical field. The speed of light
remains exactly invariant locally, but global synchronization becomes
path-dependent.

The ~4,000 km correlation on Earth (cross-sector mean and directional dispersion λ<sub>T</sub> = 4,201 ± 1,967 km — an order-of-magnitude measurement) and the ~50 kpc
dark-matter-attributed dynamical phenomenology in galaxies are interpreted within TEP as manifestations of the same environment-dependent Temporal-Topology response, pending independent replication, completion-dependent transfer calculations, and lensing–dynamics consistency (Paper 19), connected by the continuous relaxation of Temporal
Topology from deep potential wells to the weak-field regime. The M<sup>1/3</sup>
scaling appearing in some candidate completions remains testable through
interplanetary missions and pulsar timing arrays, but it is not the primary empirical claim. This hypothesis is
falsifiable through the experimental program outlined in Section 6.
Independent replication is essential, and the data and analysis code are
publicly available. The community is invited to verify, challenge, or refute
these findings.

## Data and Code Availability

All analysis code, processed data products, and raw RINEX processing
pipelines are publicly available:

Multi-center and 25-year CODE analysis:
github.com/matthewsmawfield/TEP-GNSS

Raw RINEX test:
github.com/matthewsmawfield/TEP-GNSS-RINEX

Benchmark dataset: A simplified CSV file containing binned residuals
versus distance for the 25-year period, available for theorists who wish
to fit models without processing raw GNSS data:
TEP-GNSS/benchmark

## Acknowledgments

The author expresses gratitude to the International GNSS Service (IGS), the
Center for Orbit Determination in Europe (CODE), the European Space Agency
(ESA), and NASA's Crustal Dynamics Data Information System (CDDIS). This
analysis was possible because these organizations have championed the
principles of open science, making decades of high-precision timing data
freely available to the global research community. The GFZ Helmholtz Centre
Potsdam is acknowledged for providing geomagnetic Kp indices.

## References

Abbott, B. P. et al. (LIGO/Virgo) (2017). GW170817: Observation of gravitational waves from a binary neutron star inspiral. *Phys. Rev. Lett.* 119, 161101.

Ai, Q., Maciuk, K., Lewinska, P. & Borowski, L. (2021). Characteristics of Onefold Clocks of GPS, Galileo, BeiDou and GLONASS Systems. *Sensors* 21(7), 2482. DOI: [10.3390/s21072482](https://doi.org/10.3390/s21072482)

Ashby, N. (2003). Relativity in the Global Positioning System. *Living Rev. Relativity* 6, 1.

Bekenstein, J. D. (1993). The relation between physical and gravitational geometry. *Phys. Rev. D* 48, 3641.

Bothwell, T. et al. (2022). JILA Sr optical lattice clock with 10<sup>−18</sup> stability and accuracy. *Nature* 602, 420–424.

Boulder Atomic Clock Optical Network (BACON) Collaboration (2021). Frequency ratio measurements at 18-digit accuracy using an optical clock network. *Nature* 591, 564–569.

Burrage, C. & Sakstein, J. (2018). Tests of chameleon gravity. *Living Rev. Relativity* 21, 1.

Damour, T. & Polyakov, A. M. (1994). The string dilaton and a least coupling principle. *Nucl. Phys. B* 423, 532.

Exertier, P., Belli, A., Lemoine, F. G., Biancale, R. & Courde, C. (2017). Suspected time errors along the satellite laser ranging network and impact on the reference frame. *EGU General Assembly* 19, 9405.

Fathollahi, L., Wu, F., Pongracic, B., Kiani Shahvandi, M., Schlichting, D. & Sustersic, N. (2023). Gravitational redshift test using Rb clocks of eccentric GPS satellites. *Heliyon* 9(8), e18532. DOI: [10.1016/j.heliyon.2023.e18532](https://doi.org/10.1016/j.heliyon.2023.e18532)

Hofmann-Wellenhof, B., Lichtenegger, H. & Wasle, E. (2008). *GNSS – Global Navigation Satellite Systems*. Springer.

Kelly, A. G. (2009). GPS and the One-Way Speed of Light. In: *New Approach of Indoor and Outdoor Localization Systems*. IntechOpen. DOI: [10.5772/11006](https://doi.org/10.5772/11006)

Khoury, J. & Weltman, A. (2004). Chameleon fields: Awaiting surprises for tests of gravity in space. *Phys. Rev. Lett.* 93, 171104.

Milgrom, M. (1983). A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. *ApJ* 270, 365.

Petit, G. & Luzum, B. (2010). IERS Conventions (2010). *IERS Technical Note* 36.

Planck Collaboration (2020). Planck 2018 results. VI. Cosmological parameters. *A&A* 641, A6.

Riess, A. G. et al. (2022). A comprehensive measurement of the local value of the Hubble constant. *ApJ* 934, L7.

Santamaria-Gomez, A., Ray, J. & Collilieux, X. (2017). GNSS related periodic signals in coordinate time-series from Precise Point Positioning. *Geophysical Journal International* 208(3), 1449–1462. DOI: [10.1093/gji/ggw454](https://doi.org/10.1093/gji/ggw454)

Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: [10.5281/zenodo.16921911](https://doi.org/10.5281/zenodo.16921911) (Paper 0)

Smawfield, M. L. (2025). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: [10.5281/zenodo.17127229](https://doi.org/10.5281/zenodo.17127229) (Paper 1)

Smawfield, M. L. (2025). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: [10.5281/zenodo.17517141](https://doi.org/10.5281/zenodo.17517141) (Paper 2)

Smawfield, M. L. (2025). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: [10.5281/zenodo.17860166](https://doi.org/10.5281/zenodo.17860166) (Paper 3)

Smawfield, M. L. (2025). *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: [10.5281/zenodo.17982540](https://doi.org/10.5281/zenodo.17982540) (Paper 4)

Smawfield, M. L. (2025). *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: [10.5281/zenodo.18004832](https://doi.org/10.5281/zenodo.18004832) (Paper 5 — this work)

Smawfield, M. L. (2025). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: [10.5281/zenodo.18064365](https://doi.org/10.5281/zenodo.18064365) (Paper 6)

Smawfield, M. L. (2025). *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: [10.5281/zenodo.18059250](https://doi.org/10.5281/zenodo.18059250) (Paper 7)

Smawfield, M. L. (2025). *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: [10.5281/zenodo.18064581](https://doi.org/10.5281/zenodo.18064581) (Paper 8)

Smawfield, M. L. (2025). *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: [10.5281/zenodo.18109760](https://doi.org/10.5281/zenodo.18109760) (Paper 9)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: [10.5281/zenodo.18165798](https://doi.org/10.5281/zenodo.18165798) (Paper 10)

Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: [10.5281/zenodo.18209702](https://doi.org/10.5281/zenodo.18209702) (Paper 11)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: [10.5281/zenodo.19000827](https://doi.org/10.5281/zenodo.19000827) (Paper 12)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: [10.5281/zenodo.19102061](https://doi.org/10.5281/zenodo.19102061) (Paper 13)

Smawfield, M. L. (2026). *Global Time Echoes: MGEX Multi-GNSS Clock Replication*. Preprint v0.4 (Suva). Zenodo. DOI: [10.5281/zenodo.20572726](https://doi.org/10.5281/zenodo.20572726) (Paper 14)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear in the Earth Flyby Anomaly*. Preprint v0.4 (Yogyakarta). Zenodo. DOI: [10.5281/zenodo.19454862](https://doi.org/10.5281/zenodo.19454862) (Paper 15)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Lunar Laser Ranging and the Nordtvedt Effect*. Preprint v0.5 (Lucknow). Zenodo. DOI: [10.5281/zenodo.19446028](https://doi.org/10.5281/zenodo.19446028) (Paper 17)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Blind-Prediction Residual Test in Multiply-Imaged Supernovae*. Preprint v0.3 (Lisboa). Zenodo. DOI: [10.5281/zenodo.20572719](https://doi.org/10.5281/zenodo.20572719) (Paper 19)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Covariant Alternative to Cosmic Expansion*. Preprint v0.4 (Athens). Zenodo. DOI: [10.5281/zenodo.20370143](https://doi.org/10.5281/zenodo.20370143) (Paper 26)

Smawfield, M. L. (2026). *TEP-PPP: Operational Time Echoes — Phase-Resolved Spatial Clock Priors for GNSS Outage Bridging*. Preprint v0.1 (Pune). In preparation. (Paper 33)

Teunissen, P. J. G. & Montenbruck, O. (Eds.) (2017). *Springer Handbook of Global Navigation Satellite Systems*. Springer.

Touboul, P. et al. (2022). MICROSCOPE mission: test of the equivalence principle in space. *Phys. Rev. Lett.* 129, 121102.

Will, C. M. (2014). The Confrontation between General Relativity and Experiment. *Living Rev. Relativity* 17, 4.

## Data Availability & Reproducibility

This work follows open-science practices. This synthesis paper integrates results from a systematic 
multi-paper research program. All primary analysis code and data are available in the referenced 
repositories.

### Repository & Code

**GitHub Repository:** [github.com/matthewsmawfield/TEP-GTE](https://github.com/matthewsmawfield/TEP-GTE)

This synthesis manuscript integrates findings from Papers 1–4. Primary analysis repositories:

#### TEP Research Program Structure

| Paper | Repository | Download Size |
| --- | --- | --- |
| Paper 1: GNSS Analysis | [TEP-GNSS](https://github.com/matthewsmawfield/TEP-GNSS) | ~50 MB (benchmark data) |
| Paper 2: 25-Year Longspan | [TEP-GNSS-II](https://github.com/matthewsmawfield/TEP-GNSS-II) | ~100 MB |
| Paper 3: Raw RINEX | [TEP-GNSS-RINEX](https://github.com/matthewsmawfield/TEP-GNSS-RINEX) | ~5 GB (RINEX files) |
| Paper 4: Gravitational Lensing | [TEP-GL](https://github.com/matthewsmawfield/TEP-GL) | N/A (theoretical) |
| Paper 8: Satellite Laser Ranging | [TEP-SLR](https://github.com/matthewsmawfield/TEP-SLR) | ~200 MB (normal-point data) |

### Primary Data Sources

- **CDF Data:** CODE analysis center clock products (2000–2025)

- **IGS Data:** International GNSS Service clock products

- **ESA Data:** European Space Agency navigation support products

- **RINEX Files:** Raw observation files from IGS stations

### Prediction Provenance

The dated prediction register of Section 6.5 is distributed as a
machine-readable artifact,
`results/prediction_provenance.json`, recording each tested
expectation's earliest versioned statement with its version DOI and
immutable deposit timestamp. The ordering is independently verifiable
against the public Zenodo version records under the concept DOIs listed
there (Paper 0: 10.5281/zenodo.16921911; Paper 1:
10.5281/zenodo.17127229).

### Reproduction Instructions

```
# Clone this synthesis repository
git clone https://github.com/matthewsmawfield/TEP-GTE.git
cd TEP-GTE

# Build manuscript
cd site
npm install
npm run build
```

### Software Versions

- **Node.js** 16+

## Appendix A: Mathematical Formalism

## A.1 Full Action

For comparison, the canonical scalar baseline of the TEP action (Paper 0, §2.2) is formulated in the Einstein frame where gravity is canonical:

$$S = \int d^4x \sqrt{-g} \left[ \frac{M_{\text{Pl}}^2}{2} R - \frac{1}{2} K(\phi)\, g^{\mu\nu} \partial_\mu \phi\, \partial_\nu \phi - V(\phi) \right] + S_{\text{matter}}[\psi, \tilde{g}_{\mu\nu}, \phi]$$

where M<sub>Pl</sub> is the reduced Planck mass, R is the Ricci scalar, K(φ) is the kinetic function (absorbable by a field redefinition; the operative screening realization is the noncanonical two-branch $P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+X|X|/\Lambda_X^4$ with derived $k\simeq16$), V(φ) is the screening potential, and S<sub>matter</sub> is the matter action in the Jordan frame with coupling to the matter metric g̃<sub>μν</sub>.

## A.2 Field Equations

Variation with respect to the gravitational metric yields the Einstein equations:

$$G_{\mu\nu} = \frac{1}{M_{\text{Pl}}^2} \left[ T_{\mu\nu}^{(\phi)} + T_{\mu\nu}^{(\text{matt})} \right]$$

where the scalar field stress-energy tensor is:

$$T_{\mu\nu}^{(\phi)} = K(\phi) \partial_\mu \phi \partial_\nu \phi - g_{\mu\nu} \left[ \frac{1}{2} K(\phi) g^{\rho\sigma} \partial_\rho \phi \partial_\sigma \phi + V(\phi) \right]$$

Variation with respect to the scalar field yields:

$$\nabla_\mu \left[ K(\phi) \nabla^\mu \phi \right] - \frac{\partial K}{\partial \phi} \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - \frac{\partial V}{\partial \phi} = -\alpha(\phi) T + S_{\text{disf}}$$

where α(ϕ) ≡ d ln A/dϕ = β<sub>A</sub>/M<sub>Pl</sub> is the conformal coupling strength (for A(ϕ) = exp(β<sub>A</sub>ϕ/M<sub>Pl</sub>)), T is the trace of the matter stress-energy tensor in the Jordan frame, and S<sub>disf</sub> contains disformal source terms proportional to B(ϕ).

## A.3 Candidate Microscopic Completions and Nonlinear Screening

**Candidate Completion (Historical Provenance):** This appendix presents an illustrative candidate completion (cubic Galileon derivative screening) that exhibits the (β<sub>A</sub>M/M<sub>Pl</sub>Λ<sup>3</sup>)<sup>1/3</sup> transition scale. In the current TEP master architecture (Paper 0 §2.2), environmental screening is derived from the selected two-branch non-canonical kinetic operator $P(X,\phi) = -V(\phi) + \frac{2k}{3\Lambda_X^2}X|X|^{1/2} + X|X|/\Lambda_X^4$.

The following derivation illustrates one candidate microscopic completion (cubic
Galileon) that realizes the continuous gradient suppression described as
Temporal Shear in Section 4; it is not required for the empirical results.

One candidate completion invokes cubic Galileon terms in the action to
generate nonlinear screening:

$$S_{\text{Galileon}} = \int d^4x \sqrt{-g} \left[\frac{c_3}{\Lambda^3} (\partial\phi)^2 \Box\phi\right]$$

where Λ is the strong-coupling scale (an energy scale, [M] in natural units c = ℏ = 1) and c<sub>3</sub> is a dimensionless coefficient of order unity. With the canonical scalar field normalization (φ has mass dimension 1) and the TEP source term (β<sub>A</sub>/M<sub>Pl</sub>)T, the nonlinear term becomes important when:

$$\frac{|\nabla \phi|^2}{\Lambda^3} \sim 1$$

The Vainshtein radius, inside which nonlinear effects dominate, is (in natural units c = ℏ = 1):

$$r_V = \left(\frac{\beta_A M}{M_{\text{Pl}} \Lambda^3}\right)^{1/3}$$

where β<sub>A</sub> is the conformal coupling, M<sub>Pl</sub> = √(ℏc/G) is the Planck mass, and Λ is the strong-coupling energy scale. Restoring SI units:

$$r_V = \left(\frac{\beta_A M (\hbar c)^3}{M_{\text{Pl}} \Lambda^3}\right)^{1/3} = \left(\frac{\beta_A M}{M_{\text{Pl}}}\right)^{1/3} \frac{\hbar c}{\Lambda}$$

The factor β<sub>A</sub>M/M<sub>Pl</sub> supplies the dimensionless source strength; (ℏc/Λ)<sup>3</sup> supplies the length scale cubed. The bare expression (GM/Λ<sup>3</sup>)<sup>1/3</sup> sometimes quoted in the Galileon literature is a shorthand that suppresses the β<sub>A</sub>/M<sub>Pl</sub> coupling and implicitly assumes natural units with a specific field normalization; it is dimensionally incomplete as written.

For a spherically symmetric source of mass M, the scalar field profile in the Vainshtein regime (r $\lt$ r<sub>V</sub>) is:

$$\phi(r) \sim \frac{\beta_A M_{\text{Pl}}}{M_{\text{Pl}}^2} \frac{GM}{r} \left(\frac{r}{r_V}\right)^3$$

This suppresses the Temporal Shear by a factor of (r/r<sub>V</sub>)<sup>3</sup> inside the screening radius, reconciling weak-field cosmological behavior with strong-field solar system constraints. Other completions (e.g., chameleon mechanisms, symmetry-breaking potentials) may produce similar phenomenology.

## A.4 Conservation Laws

The total stress-energy tensor is conserved:

$$\nabla_\mu \left[ T^{\mu\nu}_{(\phi)} + T^{\mu\nu}_{(\text{matt})} \right] = 0$$

However, the matter stress-energy alone is not conserved in the Einstein frame:

$$\nabla_\mu T^{\mu\nu}_{(\text{matt})} = \alpha(\phi) T \nabla^\nu \phi + \text{disformal terms}$$

This apparent violation is resolved by transforming to the Jordan frame, where matter follows geodesics of g̃<sub>μν</sub> and energy-momentum is conserved with respect to the matter-frame covariant derivative.

## Appendix B: Processing Filter Mathematical Formulation

## B.1 Network Adjustment Constraint

Standard GNSS processing applies a datum constraint that forces residuals to have zero spatial mean:

$$\delta t_i^{\text{adj}} = \delta t_i - \frac{1}{N}\sum_{j=1}^N \delta t_j$$

This constraint preserves zero-mean residuals while removing any common-mode offset.

## B.2 Fourier Decomposition

Decompose the TEP signal into spatial Fourier modes characterized by wavevector k:

$$\delta t_i^{\text{TEP}} = \sum_{\mathbf{k}} A_{\mathbf{k}} e^{i\mathbf{k} \cdot \mathbf{r}_i}$$

where r<sub>i</sub> is the position of station i. The network adjustment removes the k = 0 (monopole) component:

$$A_{\mathbf{k}=0}^{\text{adj}} = A_{\mathbf{k}=0} - \frac{1}{N}\sum_j A_{\mathbf{k}=0} = 0$$

but preserves all k ≠ 0 (dipole, quadrupole, and higher) components that have zero spatial mean.

## B.3 Cross-Spectral Density

The TEP correlation structure manifests as phase coherence between station pairs, quantified by the cross-spectral density:

$$C_{ij}(\omega) = \langle \tilde{\delta t}_i(\omega) \tilde{\delta t}_j^*(\omega) \rangle$$

where tildes denote Fourier transforms and angle brackets denote ensemble averaging.

## B.4 Phase Alignment Index

The magnitude-weighted phase alignment index extracts the phase relationship:

$$\Phi_{ij} = \frac{\sum_\omega |C_{ij}(\omega)| \cos[\arg C_{ij}(\omega)]}{\sum_\omega |C_{ij}(\omega)|}$$

This metric is invariant under common-mode amplitude rescaling because it depends only on relative phases, not absolute amplitudes. The datum constraint removes A<sub>k=0</sub> (amplitude information) but preserves the phase structure encoded in the k ≠ 0 components.

## B.5 Exponential Correlation Function

The exponential decay C(r) ∝ exp(−r/λ) reflects the two-point correlation function of the underlying φ field:

$$\langle \phi(\mathbf{x}) \phi(\mathbf{x}') \rangle = \phi_0^2 \exp\left(-\frac{|\mathbf{x} - \mathbf{x}'|}{\lambda}\right)$$

where λ = R<sub>T</sub>(M<sub>⊕</sub>) is the geometric saturation radius (§4.4.2); under the mass-completion interpretation this would correspond to the Compton wavelength λ = ℏ/(m<sub>φ</sub>c). This correlation function has nonzero spatial gradients (k ≠ 0 in Fourier space) and therefore survives network adjustment, which only removes the k = 0 component.

## B.6 Tidal vs. TEP Signal Discrimination

Classical tidal effects are parameterized as sums of spherical harmonics Y<sub>lm</sub>(θ, φ) with known frequencies:

$$\delta t_{\text{tidal}}(t, \mathbf{r}) = \sum_{l,m,n} A_{lmn} Y_{lm}(\theta, \phi) \cos(\omega_n t + \phi_n)$$

where ω<sub>n</sub> are tidal frequencies (diurnal, semidiurnal, long-period). Standard GNSS processing explicitly models and subtracts these terms using IERS conventions.

The TEP correlation is a distance-dependent phase coherence:

$$C_{\text{TEP}}(r_{ij}) = A_0 \exp(-r_{ij}/\lambda)$$

This functional form is not decomposable into the standard tidal spherical harmonic basis. The processing removes what it models (tidal harmonics) but preserves what it does not model (exponential distance correlations in phase space).

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# Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T
Matthew Lukin Smawfield
Version: v0.9 (New Delhi)
First published: 28 December 2025 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.18064365

---

## Abstract

Dark matter observations across cosmological scales exhibit a
regularity: the characteristic radius at which Newtonian dynamics fails
scales as $R \propto M^{1/3}$, implying a universal saturation/proximity
scale (historically labelled a critical density), observationally proxied by $\rho_T$. This scaling appears in galaxy rotation curves (SPARC
database), ultra-diffuse galaxies (DF2/DF4), and the Milky Way's dark-matter
onset. Within TEP, this pattern is interpreted as evidence of a
candidate saturation scale in the conformal time-field sector, where
field saturation occurs at a characteristic proximity scale.

Terrestrial calibration—derived from a newly identified
distance-structured correlation in GNSS atomic clocks—provides an
empirical anchor. A 25-year CODE analysis yields $L_c \approx 4{,}200$
km for Earth's mass ($M_\oplus \approx 6 \times 10^{27}$ g), consistent with
multi-center results (CODE, IGS, ESA). The
characteristic length $L_c$ is operationally identified with the projected
Temporal Topology covariance scale associated with $R_T(M_\oplus)$,
the geometric saturation scale for Earth's mass; a soliton interpretation
is one candidate microscopic realization, not assumed in the calibration. Under the conditional identification $L_c \leftrightarrow R_T(M_\oplus)$, this corresponds to
$\rho_T \approx 20$ g/cm³. The underlying GNSS correlation scale $L_c$ itself exhibits 25-year temporal
stability and survives raw RINEX validation, constraining
processing-artifact explanations.

Galactic-scale consistency is tested using the SPARC rotation curve database
(167 galaxies). The empirical dark matter onset scaling is $\alpha_{\rm SPARC} =
0.355 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (definition)}$, consistent with the $M^{1/3}$ expectation within
$\sim 0.3\sigma$ ($\sim 0.5\sigma$ statistical-only). For the Milky Way, the SPARC-calibrated $M^{1/3}$ relation predicts a dark-matter onset radius $R_{\text{DM}} \approx 3$ kpc, consistent with the observed transition from baryonic to dark-matter-dominated rotation at $R \sim 3$–5 kpc. For ultra-diffuse
galaxies DF2 and DF4, the model predicts Temporal Topology saturation radii exceeding tidal
radii, consistent with observed dark matter deficiency via an
illustrative field-envelope disruption mechanism (a dynamical timescale
for re-equilibration remains to be computed).

Temporal Topology screening provides a candidate explanation for why
precision GR tests remain satisfied. A hierarchy of 26 astrophysical objects spanning 15
orders of magnitude in density is assembled; regression on the 11 dense
objects ($\rho > \rho_T$) yields $S \propto \rho^{0.334}$ ($R^2 = 0.99995$),
algebraically expected from the
$R_T(M)$ construction. This provides a candidate screening explanation for why GR tests pass (binary pulsars:
$S \sim 29{,}000$) while galactic dynamics ($S \sim 10^{-9}$ at $\rho \sim 10^{-24}$ g/cm³)
are deeply unscreened, exhibiting strong scalar effects.

The saturation density $\rho_T \approx 20$ g/cm³ emerges as a candidate
macroscopic saturation/proximity scale in the TEP environmental-response
hierarchy, not an ambient-density switch. GNSS provides the primary
terrestrial timing anchor; SPARC onset scaling, Milky Way structure,
RBH-1 crossover behaviour, compact-object screening, and condensed-matter/EOS
considerations then test whether the same scale organizes otherwise
disconnected gravitational regularities without retuning. Channel-specific
quantities such as $\rho_{\rm trans}$, $R_s$, $\kappa_{\rm Cep}$,
$\kappa_{\rm MSP}$, and lensing response coefficients are downstream
projections of $S_\Sigma(\mathcal{E})$, not independent measurements of
$\rho_T$.

The $\rho^{1/3}$ hierarchy is a consistency relation induced by the
$R_T(M)$ construction; it is not, by itself, an independent discriminator
of microscopic screening mechanism.

*Evidence hierarchy.* Within the evidence hierarchy of the TEP series,
the GNSS clock analyses (Papers 1–3) provide the primary empirical input
for the terrestrial correlation scale; the present paper tests the
cross-scale consequences of conditionally identifying that scale with
$R_T(M_\oplus)$.

*Keywords:* dark matter – gravitation – scalar fields – galaxies:
kinematics and dynamics – temporal equivalence principle

## 1. Introduction: The Dark Matter Problem as a Temporal Structure Problem

### The Universal Scaling Anomaly

Dark matter observations across cosmological scales reveal an empirical regularity. From dwarf galaxies ($M \sim 10^9 M_\odot$) to massive galaxies ($M \sim 10^{12} M_\odot$), the characteristic radius at which Newtonian dynamics fails—the "dark matter onset"—scales approximately as $R \propto M^{1/3}$. This implies a universal critical proximity scale ("critical" here meaning a phenomenological saturation/proximity scale, not a hard density threshold), observationally proxied by $\rho_T \sim M/R^3 \sim \text{constant}$, at which gravitational phenomenology changes.

This scaling appears in multiple independent contexts:

- **Galaxy rotation curves:** SPARC database analysis yields $R_{\text{DM}} \propto M^{\alpha_{\rm SPARC}}$ with $\alpha_{\rm SPARC} = 0.355 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (definition)}$ (Lelli et al. 2016; this work).

- **Ultra-diffuse galaxies:** DF2 and DF4 exhibit dark matter deficiency consistent with saturation radii exceeding tidal radii (van Dokkum et al. 2018, 2019).

- **Milky Way:** The SPARC-calibrated $M^{1/3}$ relation predicts a dark-matter onset at $R_{\text{DM}} \approx 3$ kpc for the Milky Way, consistent with the observed transition from baryonic to dark-matter-dominated rotation at $R \sim 3$–5 kpc (Gaia Collaboration 2023; Sofue 2020).

The persistence of this $M^{1/3}$ scaling across 6 orders of magnitude in mass (screening hierarchy: gas giants to neutron stars) and 15 orders of magnitude in density suggests a fundamental physical scale, not a coincidence of baryonic feedback or halo assembly.

### Reframing Dark Matter: Phantom Mass from Temporal Shear

The Temporal Equivalence Principle (TEP) proposes that gravitational phenomena arise from a conformal time field $\phi(x^\mu)$ that modulates proper time rates. This phenomenology arises from the TEP action (Smawfield 2025a, Paper 0):

\begin{equation} \label{eq:tep_action} S = \int d^4x \sqrt{-g} \left[ \frac{M_{\text{Pl}}^2}{2} R + P(X,\phi) - V(\phi) \right] + S_m[\tilde{g}_{\mu\nu}] \end{equation}

where matter couples to the causal matter metric $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf{x}))U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf{x}))\Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ in the canonical unified unstretched-space decomposition ($\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu\phi \nabla_\nu\phi$ on the historical isotropic-conformal benchmark). At late times, $B(\phi)$ is strongly constrained by multi-messenger observations and can be neglected, while $A(\phi) = \exp(\beta_A \phi/M_{\text{Pl}})$ modulates local proper-time rates. The environmental screening response suppresses the observed gradient in dense environments, producing effective field saturation at characteristic density $\rho_T$. In this framework, the "dark matter" problem is reinterpreted as a violation of the *isochrony axiom*—the assumption that clocks at the same gravitational potential tick at the same rate regardless of their spatial separation or the mass distribution's history.

When this axiom is relaxed, mass estimates made through different observable pairs diverge, and "phantom mass" — the apparent mass inferred when temporal structure is analysed as spatial mass — can be measured per channel. In the *dynamical–baryonic* channel studied here (rotation curves), phantom mass is the apparent excess of the dynamical mass $V_{\rm obs}^2 R/G$ over the baryonic mass: the orbital dynamics respond to the local time gradient, sourcing centripetal support with no additional baryons. In the *lensing* channel (Papers 4, 18, 19), under the canonical unified unstretched-space matter metric (Paper 0 §2.2, Step 80) — where at a fixed cosmic epoch ($A_{\rm sp}(t_0)=1$) the spatial 3-metric $\tilde\gamma_{ij}=\gamma_{ij}$ remains unstretched while the $(3+1)$ ADM lapse rescales as $N_{\rm TEP}(\mathbf x)=A_{\rm loc}(u_{\rm grav}(\mathbf x))N_{\rm GR}(\mathbf x)$ — intermediate clock rates refract light at half the dynamical potential weight ($\Phi_{\rm lens}=\Phi_N+\tfrac12 c^2\beta_A u_{\rm grav}$ versus $\Phi_{\rm dyn}=\Phi_N+c^2\beta_A u_{\rm grav}$), yielding a testable lensing-to-dynamical mass ratio $M_{\rm lens}^{\rm GR}/M_{\rm dyn}=(1+\beta_A^2 y_{\rm eff})/(1+2\beta_A^2 y_{\rm eff})$ (whereas on the historical isotropic-conformal benchmark $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$, $A^2(\phi)$ cancels out of null rays and $M_{\rm dyn}/M_{\rm lens}=1+2\beta_A^2 y_{\rm tb}$). Both measurements arise from spatial structure in proper time—without invoking non-baryonic particles—and are kept distinct throughout.<sup>†</sup>

<sup>†</sup>*Note: This geometric "phantom mass" from temporal shear differs from cosmological "phantom energy" (dark energy with $w < -1$). The former arises from spatial gradients in proper time; the latter from exotic equation-of-state matter.*

The characteristic scale at which this temporal structure becomes significant is set by the saturation scale $\rho_T$, where the scalar field $\phi$ reaches its self-interaction threshold and saturates—forming a stable, localized configuration (a soliton under that interpretation) with radius:

\begin{equation} \label{eq:saturation_radius} R_T = \left( \frac{3M}{4\pi \rho_T} \right)^{1/3}. \end{equation}

This $M^{1/3}$ scaling is a direct consequence of the saturation condition; under a soliton interpretation, this corresponds to soliton formation.

The saturation scale parameter $\rho_T \approx 20$ g/cm³ serves as the bulk-matter realization of the abstract environmental screening operator $\mathcal{S}_\Sigma(\mathcal{E})$ defined in the foundational TEP framework. In macroscopic atomic matter, $\rho_T$ acts as an empirical proxy for the geometric saturation of the Temporal Topology, yielding cross-scale predictions without committing to specific chameleon, Vainshtein, or symmetron microphysics.

\begin{equation} \label{eq:screening_operator}
\Sigma_\mu^{\text{obs}} = \mathcal{S}_\Sigma(\mathcal{E}) \, \nabla_\mu \ln A(\phi),
\end{equation}

where the environmental state is
$$\mathcal{E} = \{\rho, \Phi/c^2, \nabla\rho, \nabla\Phi, \text{compactness}, R_T(M), \text{proximity}, T, z, \text{boundary geometry}, \text{coherence volume}\}.$$
This single operator unifies density screening, compactness screening, proximity screening, thermodynamic/environmental-state response, lensing/cosmological covariance screening, and wide-binary environmental screening across the entire TEP corpus.

#### Box 1.1: Joint Constraint Architecture of \(\rho_T\) and \(S_\Sigma(\mathcal{E})\)

\(\rho_T\) is the shared macroscopic saturation/proximity scale. Other
papers in the TEP programme do not all measure \(\rho_T\) directly; they
measure domain-specific projections of the same environmental operator
\(S_\Sigma(\mathcal{E})\):

- **Clock covariance lengths** (GNSS, Paper 1–3)

- **Galactic onset radii** (SPARC, Milky Way, this work)

- **Response coefficients** (Cepheids, pulsars, Papers 10–11)

- **Lensing residuals** (strong-lens time delays, Paper 4)

- **Weak-field recovery scales** (wide binaries, Paper 13)

- **Compactness-dependent residuals** (LLR, Paper 17)

- **Cosmological temporal-shear amplitudes** (hi_class, C0, TH, Papers 18, 26, 27)

Each measurement channel has a response coefficient \(\kappa_X\), an
environmental screening projection, and a domain-specific observable
functional. This architecture protects the programme from the criticism
that different channels report different numerical values: they are
*observable response coefficients*, not bare scalar couplings,
and each channel measures a different projection of
\(S_\Sigma(\mathcal{E})\).

### The Multi-Scale Consistency Strategy

As summarized in Table 1, this paper tests $\rho_T$ through a convergent multi-scale approach:

### Table 1: Cross-Scale Consistency of $\rho_T$

| Scale | System | Mass Range | Density Range | Key Observable | Result |
| --- | --- | --- | --- | --- | --- |
| **Terrestrial** | GNSS Clocks | $M_\oplus \sim 6 \times 10^{27}$ g | $\rho_\oplus \sim 5.5$ g/cm³ | $L_c \approx 4200$ km | $\rho_T \approx 20$ g/cm³ |
| **Galactic** | SPARC Galaxies | $10^9$–$10^{12} M_\odot$ | $\rho \sim 10^{-24}$ g/cm³ | $R_{\text{DM}} \propto M^{\alpha_{\rm SPARC}}$ | $\alpha_{\rm SPARC} = 0.355 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (def.)}$ |
| **Local** | Milky Way | $M_{\rm bar} \sim 6 \times 10^{10} M_\odot$ | $\rho \sim 10^{-24}$ g/cm³ | Inner mass-discrepancy onset | $R_{\text{DM}} \approx 3$ kpc (observed $\sim$3–5 kpc) |
| **Screening** | 26 Objects | $\sim 10^{27}$–$10^{33}$ g | $\sim 10^{0}$–$10^{14}$ g/cm³ | $S \propto \rho^{\beta_{\rm scr}}$ | $\beta_{\rm scr} = 0.334$ ($R^2 = 0.99995$) |

*Note: The convergence of constraints across 18 orders of magnitude in mass (Earth to galaxy, full range) motivates $\rho_T$ as a candidate saturation scale.*

### Paper Structure

The remainder of this paper is organized as follows:

- **Section 2:** Terrestrial calibration via GNSS atomic clock correlations, establishing $L_c \approx 4200$ km → $\rho_T \approx 20$ g/cm³.

- **Section 3:** Galactic consistency test via SPARC rotation curves, confirming $M^{1/3}$ scaling.

- **Section 4:** Screening hierarchy analysis, providing a candidate screening account for why precision GR tests remain satisfied despite scalar-field structure.

- **Section 5:** Physical context for $\rho_T$, including a dimensional identity bridge preserving WEP, a condensed-matter EOS coincidence, and open derivation targets.

- **Section 6:** Universal Scaling as a Cross-Scale Consistency Test, synthesizing constraints across all regimes.

- **Section 7:** Milky Way dark-matter onset at $R_{\text{DM}} \approx 3$ kpc.

- **Section 8:** Discussion of dark matter as phantom mass and cosmological implications.

- **Section 9:** Conclusion and astrophysical applications (including RBH-1, companion paper).

This multi-scale convergence identifies $\rho_T \approx 20$ g/cm³ as a candidate GNSS-anchored saturation parameter (with explicit systematic uncertainty), enabling sharply constrained astrophysical applications such as the RBH-1 runaway black hole candidate (Smawfield 2025h, Paper 7).

*Evidence hierarchy.* Within the evidence hierarchy of the TEP series, the GNSS clock analyses (Papers 1–3) provide the primary empirical input for the terrestrial correlation scale; the present paper tests the cross-scale consequences of conditionally identifying that scale with $R_T(M_\oplus)$. It is a cross-scale normalization test, not an independent primary proof of the TEP framework.

## 2. Terrestrial Calibration: GNSS Atomic Clock Correlations

### Distance-Structured Correlations in Atomic Clocks

The first empirical constraint on $\rho_T$ comes from terrestrial atomic clock networks. Multi-center analysis of GNSS (Global Navigation Satellite System) clock products reveals distance-structured correlations with a characteristic length scale $L_c \approx 4200$ km for Earth's mass. Operationally, $L_c$ denotes the best-fit exponential decay length recovered by fitting clock-residual covariance versus inter-station separation (see Appendix A). Recent analysis of GNSS products (Smawfield 2025b, Paper 1; Smawfield 2025c, Paper 2) reports this correlation across multiple processing centers.

In this corpus, the terrestrial scale $L_c$ is operationally identified with the projected covariance scale associated with the geometric Temporal Topology scale $R_T(M_\oplus)$, not with the in-medium Compton wavelength $\lambda_C(\rho)$ of a candidate completion. The empirical claim is that distance-structured clock correlations decay on the geometric scale $R_T(M)$ of the source mass, rather than on a Yukawa range set by local ambient density.

These correlations exhibit seven independent signatures:

- **Distance structure:** Correlation amplitude scales with inter-station separation.

- **Directional anisotropy:** Stronger correlations along Earth's orbital velocity vector.

- **Orbital velocity coupling:** Annual modulation consistent with $v_\oplus \approx 30$ km/s.

- **Annual-phase direction recovery:** RA = 6°, Dec = +4°, about 6° from Earth's aphelion velocity tangent; the directed CMB-apex template is anti-phase.

- **Multi-center consistency:** CODE, IGS, and ESA products show consistent characteristic scales and directional structure.

- **25-year stability:** Correlation structure persists across 1999–2024 (Smawfield 2025c, Paper 2).

- **Raw RINEX validation:** Pattern survives in raw-RINEX Single Point Positioning (SPP) analysis, constraining processing artifacts (Smawfield 2025d, Paper 3).

### Operational Calibration of $\rho_T$ from $L_c$

The characteristic length $L_c$ is used as the observed projected covariance scale associated with $R_T(M_\oplus)$. For a spherical mass distribution, the saturation radius is:

\begin{equation} \label{eq:rt_definition} R_T = \left( \frac{3M}{4\pi \rho_T} \right)^{1/3}. \end{equation}

Solving for $\rho_T$:

\begin{equation} \label{eq:rho_t_definition} \rho_T = \frac{3M}{4\pi R_T^3}. \end{equation}

Substituting $M_\oplus \approx 6 \times 10^{27}$ g and $L_c \approx 4200$ km $\approx 4.2 \times 10^8$ cm:

\begin{equation} \label{eq:rho_t_earth} \rho_T = \frac{3 \times (6 \times 10^{27}\,\text{g})}{4\pi \times (4.2 \times 10^8\,\text{cm})^3} \approx 20\,\text{g/cm}^3. \end{equation}

### Systematic Uncertainties

The primary uncertainty is the length scale determination. Four distinct contributions are reported: (i) the 25-year CODE per-measurement statistical uncertainty ($4{,}201 \pm 1{,}967$ km, $\pm 47\%$; Paper 2); (ii) the inter-center systematic spread ($3{,}330$–$4{,}549$ km, $\sim \pm 12\%$; Paper 1), with the caveat that centers share largely overlapping underlying data so this spread is not an independent ensemble; (iii) the epoch/window instability established by the executed held-out discriminator (Paper 14 Step 3.1): the identical estimator on the identical product returns $\lambda$ from 1,050 to 1,935 km across quarterly windows, while matched-day independent chains agree at $\sim$30% (2,744 vs. 1,933 km — the CNES value landing within 4% of the pooled 1,862 km — same-product fallback days excluded by epoch-aligned detrended-residual comparison) — the dominant calibration systematic is therefore the $\sim$30–40% epoch/window scatter, not the inter-center figure; and (iv) the $\mathcal{O}(1)$ prefactor in the transfer sketch (Section 2), treated as a model-defining choice. The adopted operational value is $L_c = 4{,}200$ km with $\pm 500$ km ($\pm 12\%$) reflecting the inter-center range, plus the epoch/window systematic carried separately. Since $\rho_T \propto L_c^{-3}$, the 30–40% window scatter propagates to a factor $\approx 2$–$3$ in $\rho_T$: the operational value $\rho_T = 20$ g/cm³ then carries an effective systematic band of roughly $8$–$70$ g/cm³, within which the MGEX held-out scale ($\lambda \approx 1{,}4$–$1.9\times10^{3}$ km, $\rho_T$ up to $\sim 5\times10^{2}$ g/cm³ if adopted at face value) is recorded as an unresolved epoch-scale tension pending longer MGEX baselines. The $\pm 47\%$ per-measurement statistical uncertainty from the single-center CODE fit is retained as a conservative upper bound on measurement-to-measurement scatter. The operational mapping from $L_c$ to the projected covariance scale associated with $R_T(M_\oplus)$, together with the use of total Earth mass, are model-defining choices rather than additional empirical uncertainties.

Propagating the length-scale uncertainty alone:

\begin{equation} \label{eq:rho_t_uncertainty} \rho_T = 20 \pm 7 \text{ g/cm}^3 \; (35\% \text{ inter-center}), \qquad \rho_T \in [8,\,70] \text{ g/cm}^3 \; \text{(epoch/window systematic)}. \end{equation}

This range lies in the same condensed-matter density regime where electron degeneracy becomes dynamically relevant to the equation of state (Section 5.3) and provides the calibration scale for all subsequent tests. The key point is that the $M^{1/3}$ structural form is independent of $\rho_T$; only the normalization changes. Even at the extremes ($L_c = 3{,}700$ km $\rightarrow \rho_T \approx 30$ g/cm³, or $L_c = 4{,}700$ km $\rightarrow \rho_T \approx 14$ g/cm³), the cross-scale consistency tests remain viable because the same $\rho_T$ value must describe all systems simultaneously. The wider epoch/window band does not alter this structural conclusion — it re-prices the normalization confidence: cross-scale checks that pass at $\rho_T = 20$ g/cm³ must be re-evaluated under the widened prior, and the Level 1/2/3 claim hierarchy (Section 8) is designed to absorb exactly this class of anchor-level hit.

### Discrimination from Geophysical Signals

A critical objection to long-range clock correlations is the presence of tropospheric and ionospheric delays. However, spectral power analysis reveals distinct scale separation:

- **Tropospheric covariance:** Decays at ~100–500 km (weather systems)

- **Ionospheric covariance:** Decays at ~500–2000 km (TEC structures)

- **Scalar field signature:** Persists as a covariance floor at ~4200 km

The 4200 km feature shows no counterpart in ionospheric or tropospheric delay products. Furthermore, the signal recovers an annual phase about 6° from Earth's aphelion velocity tangent (companion analysis); the CMB-apex template is anti-phase and is not identified as a rest frame. Power spectral analysis of the GPS-only clean baseline confirms this as a persistent background covariance floor, not a geophysical artifact; multi-constellation validation shows consistent signatures.

### Null Tests and Robustness

The GNSS correlation pattern survives multiple null tests:

- **Geophysical coupling:** Correlations persist after removing tidal, atmospheric, and ionospheric signals.

- **Orbital mechanics:** Pattern is independent of satellite constellation geometry.

- **Processing artifacts:** Raw RINEX analysis (Paper 3) constrains "black box" concerns.

- **Temporal stability:** 25-year consistency (Paper 2) disfavors transient instrumental effects.

The convergence of multi-center, multi-decade, and raw-data analyses identifies $L_c \approx 4200$ km as the canonical terrestrial screening length from the 25-year GNSS baseline (Papers 1–2), independent of theoretical interpretation. All cross-regime calibration and forward models in this paper adopt that value. Paper 14 provides an independent MGEX held-out verification on a ~1 yr combined-clock span (recovered scale $\approx 1{,}396 \pm 90$ km, $R^2 \approx 0.49$), confirming signal presence on a shorter baseline and different product; it is cited as verification only, not as the dimensional scale for $\rho_T$ extrapolation. The scale tension between that figure and the 4,200 km anchor has since been tested directly (Paper 14 extension, Step 3.1): on strictly matched, verified-independent days the CODE and CNES processing chains return $\lambda = 2{,}744$ km and 1,933 km respectively (~30% offset, the CNES value within 4% of the pooled 1,862 km), while quarterly windows of the same product scatter between 1,050 and 1,935 km — the held-out discrepancy is dominated by epoch/sampling instability of the short-span estimate, not by a product-dependent physical scale, and the independent-center agreement bounds the processing contribution to ~30% (Appendix B.8).

### Phenomenological Transfer Sketch: Mapping the Static Profile to Clock Covariance

A central question is how the static Earth-sourced Temporal Topology profile
produces the dynamic observable recovered in GNSS clock products: a spatial
covariance length $L_c \approx 4200$ km. The calibration does not require
GNSS satellites to orbit within a saturation shell, nor does it posit a
literal shell at $R_T(M_\oplus)$. Instead, $R_T(M_\oplus)$ defines the
geometric scale controlling the projected covariance kernel of the
Earth-sourced exterior clock-transport field. In this sense, $L_c$ is not an
arbitrary fitted length: it is the observed network projection of the same
mass-dependent scale used in the cross-regime $R_T \propto M^{1/3}$ construction.
This subsection gives an order-of-magnitude argument — a transfer sketch,
not a derivation — for this identification, together with the assumptions
that a full demonstration would have to close (Appendix B.2).

#### What $R_T(M_\oplus)$ is, and is not

$R_T(M) = (3M/4\pi\rho_T)^{1/3}$ is the radius a body of mass $M$ would have
at uniform density $\rho_T$. For Earth this is $\approx 4146$ km. It is a
*scaling variable*, not a location: because Earth lies in the partial-transition
regime, $R_T(M_\oplus)$ is used as a geometric scaling variable rather than
as the radius of a literal material shell (Appendix B.2). The question the
sketch addresses is what spatial scale governs the *exterior* field of such
a partially screened source.

#### The exterior profile scale

In the static exterior problem, the available length scales are the source
radius $R_\oplus$, the geometric saturation scale $R_T(M_\oplus)$, and the
ambient-medium Compton wavelength $\lambda_C(\rho_{\rm amb})$ (very long in
the low-density exterior ($T^{(\rm m)} \approx 0$)). Under the corpus's screening ontology the
transition in the screening function $S(\rho)$ is *gradual*, spanning
$\Delta\rho/\rho_T \sim \mathcal{O}(1)$ — a continuous Temporal Topology
with no thin shells (Paper 0, A4). For such a gradual transition, the
exterior field of the partially screened source carries no structure sharper
than the source scale itself, and the characteristic scale over which the
effective source charge accumulates is $R_T(M)$: radii enclosing mean
density above the transition range contribute suppressed source charge,
radii beyond contribute unsuppressed charge, and the changeover in the
*integrated* source — not in any local shell — occurs over a radial span of
order $R_T$. (Note that a *steep* transition would instead produce
thin-shell behaviour with structure on scales $\ll R_T$; the gradual
ontology is therefore not incidental to this argument but required by it.)
Both $R_\oplus = 6371$ km and $R_T(M_\oplus) = 4146$ km lie within a factor
$\sim 1.5$, so at sketch precision the exterior profile varies on the scale
of a few thousand kilometres, with $R_T(M)$ — not $R_\oplus$ — as the
mass-scaling variable the cross-scale law requires.

#### From a static profile to a time-series covariance

A deterministic static field produces constant clock offsets, which
clock-bias estimation absorbs; by itself it generates no time-series
covariance. The sketch therefore requires one explicit assumption:

**Modulated-sampling assumption.** The residual variance analysed in
Papers 1–3 arises from temporal modulation of each station's sampling of
the static exterior profile — diurnal rotation of the ground network
through the profile's anisotropy, satellite orbital motion, and
tropospheric/orientation geometry that multiplies the local conformal
rate $A(\bar{\phi}(\mathbf{x},t))$. Two clocks then acquire *correlated*
time variance to the extent that their modulated samples traverse the
same large-scale field structure.

Writing $\delta\tau_i(t) \approx \tau_0 (\beta_A/M_{\rm Pl})\,
\bar{\phi}(\mathbf{x}_i(t))$ for the modulated residual, the equal-time
network covariance $\langle \delta\tau_A\,\delta\tau_B\rangle_t$ inherits
its baseline dependence from the spatial structure of $\bar{\phi}$: stations
separated by $d$ small compared with the profile's variation scale sample
nearly the same field excursion and correlate strongly; stations separated
by $d$ large compared with that scale decorrelate. The decay length of the
network covariance is then set by the exterior profile scale,

\begin{equation} L_c \sim \mathcal{O}(1) \times R_T(M_\oplus), \end{equation}

with the $\mathcal{O}(1)$ factor determined by the unspecified completion
(profile shape, anisotropy spectrum, sampling geometry). This is the entire
quantitative content of the sketch: a coincidence of scales with an
undetermined order-unity prefactor, *not* a prediction of $L_c = R_T$ to
any stated precision.

#### Implications for the calibration

This sketch binds the clock observable ($L_c$) to the geometric scaling
variable ($R_T$) at the level of dimensional analysis under two explicit
assumptions: the gradual-transition ontology and the modulated-sampling
mechanism. It does not prove that the GNSS correlation is caused by the TEP
scalar — that remains the empirical hypothesis tested by null-model
exclusion (Papers 1–3) — and it does not by itself discriminate $R_T(M)$
from $R_\oplus$ as the controlling scale, since the two differ by only a
factor $\sim 1.5$ for Earth. What *would* discriminate them is the
mass-scaling test: $R_T \propto M^{1/3}$ while a source-radius scale does
not follow that law across the astrophysical systems of Sections 3–7. The
single-potential numerical demonstration that would replace this sketch
with a derivation — an explicit exterior profile and sampling kernel under
a specified completion, with the quartic normalization held at the corpus
self-coupling value $\lambda = 7.5\times10^{-66}$ (the scalar-potential
coefficient, not the covariance length denoted $\lambda$ in this
section) and $\rho_T$ fixed by the calibrated
$L_c$ —
remains the open target recorded in Appendix B.2.

*For detailed methodology and validation, see Appendix A and Papers 1–3.*

## 3. The SPARC Galaxy Analysis: Phantom Mass as Unscreened Time-Field

Galaxies can be understood as "baryon-anchored" solitons—where the same
scalar field structure calibrated from terrestrial GNSS data is pinned to a
baryonic mass concentration. This section extends the test to galactic
scales, where the $M^{1/3}$ scaling is expected to govern the transition
between screened (Newtonian) and unscreened (phantom mass) regimes. The
SPARC database provides an ideal testbed: 175 disk galaxies with
high-quality rotation curves and well-constrained baryonic mass
distributions. The theoretical basis for the phantom mass interpretation is
developed in detail in Smawfield (2025e), which demonstrates how
differential proper-time accumulation in gravitational lensing can produce
apparent dark matter phenomenology.

### Phantom Mass Interpretation

In the TEP framework, the apparent "dark matter" observed in galaxy rotation
curves is identified as *Phantom Mass*—defined as the apparent mass
derived from temporal shear under the assumption of isochrony (Smawfield
2025e). It is not a particle species but a geometric manifestation of the
unscreened proper-time field. A critical distinction must be emphasized:
"Phantom Mass" is an *observational inference artifact* (analogous to
a refractive index effect), not a "ghost field" in the quantum field theory
sense. Ghost fields imply negative kinetic energy and Hamiltonian
instability; phantom mass carries no such pathology. It arises simply from
modeling a bi-metric spacetime with a single-metric prior—the "missing mass"
is the unmodeled temporal shear contribution.

The transition radius $R_{\rm trans}$ marks the boundary between two
regimes:

Inside $R_{\rm trans}$: The enclosed mean baryonic density exceeds the
critical screening density ($\bar\rho > \rho_{\rm trans}$). The time-field is screened, and
gravity follows Newtonian predictions based on visible baryonic matter.

Outside $R_{\rm trans}$: The enclosed mean density drops below
$\rho_{\rm trans}$. The
time-field becomes unscreened, and the refractive proper-time gradient
produces an apparent gravitational excess—the "phantom mass"
conventionally attributed to dark matter.

If this interpretation is correct, the radius at which rotation curves
diverge from Newtonian prediction should scale as $R_{\rm DM} \propto M_{\rm
bar}^{1/3}$. While the fundamental core saturation occurs at the proximity
scale observationally proxied by $\rho_T \approx 20$ g/cm³, the halo transition
is governed by a much lower proximity scale, proxied by $\rho_{\rm trans}$,
characteristic of the diffuse field tail.

Invariant: $\rho_T$ denotes the universal saturation
density of the scalar sector that fixes the compact soliton scale via
$R_T \propto (M/\rho_T)^{1/3}$. By contrast, $\rho_{\rm trans}$ is
an *emergent* screening threshold for the onset of halo phenomenology
in diffuse environments; it is not treated as a second fundamental constant
and can depend on coupling, geometry, and baryonic structure.

### Theoretical Expectation from \(\rho_T\)

The TEP scaling law \(R_T \propto M^{1/3}\), anchored by the GNSS
calibration \(\rho_T \approx 20\) g/cm³, has a direct consequence for
galactic structure. In the TEP framework, the dark-matter onset radius
\(R_{\rm DM}\) is identified with the radius at which the enclosed mean
baryonic density drops below the emergent halo threshold \(\rho_{\rm trans}\). Because
\(\rho_{\rm trans}\) is an emergent threshold intended to be tied to the
same scalar potential that fixes \(\rho_T\) — a linkage whose derivation
is the open target of the Formal Link — dimensional analysis yields the
same mass scaling:

\begin{equation} \label{eq:dm_scaling} R_{\rm DM} \propto \left(\frac{M_{\rm bar}}{\rho_{\rm trans}}\right)^{1/3}
\propto M_{\rm bar}^{1/3}. \end{equation}

Thus the TEP framework *expects* \(\alpha = 1/3\) for the DM onset
scaling. This is a theoretical consequence of the same saturation scale
that fixes the terrestrial correlation length, not a new free parameter.
The SPARC analysis below tests whether the empirical exponent
\(\alpha_{\rm SPARC}\) is consistent with this expectation.

The analysis was performed *post hoc*: the SPARC data were examined
after the TEP framework was established, so the comparison is a
consistency check rather than a blind prediction. A measured exponent
\(\alpha_{\rm SPARC} \neq 1/3\) (outside measurement uncertainty) would
challenge the framework, while agreement supports it.

### Methodology

The analysis proceeds as follows for each of the 175 SPARC galaxies:

Calculate the total baryonic mass: $M_{\rm bar} = M_* + 1.33 M_{\rm
HI}$, where $M_* = (M/L)_{3.6\mu} \times L_{3.6}$ with $(M/L)_{3.6\mu} =
0.5\,M_\odot/L_\odot$ for disk populations.

Compute the expected Newtonian rotation velocity: $V_{\rm bar}^2 =
V_{\rm gas}^2 + (M/L)_{\rm disk} V_{\rm disk}^2 + (M/L)_{\rm bulge}
V_{\rm bulge}^2$.

Identify the mass discrepancy onset radius $R_{\rm DM}$: the first
radius where $V_{\rm obs}/V_{\rm bar} > 1.3$. Radial bins with
nonpositive or unreliable baryonic velocity ($V_{\rm bar} \leq 5$ km
s$^{-1}$, including $V_{\rm bar}^2 \leq 0$ from negative model
contributions) are excluded from the ratio; galaxies whose ratio never
exceeds the threshold contribute no $R_{\rm DM}$ and are dropped from
the fit — a selection, disclosed here — leaving 167 of 175 systems at
the fiducial threshold.

Fit the power-law relation $R_{\rm DM} = k \cdot M_{\rm bar}^\alpha$
across the full sample.

### Results

Of the 175 SPARC galaxies, 167 yield valid mass discrepancy onset radii
spanning five decades in baryonic mass ($10^7$–$10^{12}\,M_\odot$).

#### Robustness to Onset Definition

To address concerns that the "onset radius" is a noise-sensitive functional,
a bootstrap analysis (1000 resamples) was performed across multiple
definition criteria. The scaling exponent $\alpha_{\rm SPARC}$ remains consistent with
the TEP expectation ($\alpha_{\rm SPARC} = 1/3$) across a wide range of velocity ratio
thresholds:

| Definition (Threshold) | Fitted Exponent ($\alpha_{\rm SPARC}$) | Consistency with 1/3 |
| --- | --- | --- |
| Loose ($V_{obs}/V_{bar} > 1.1$) | $0.274 \pm 0.036$ | $1.6\sigma$ |
| Fiducial ($V_{obs}/V_{bar} > 1.3$) | $0.372 \pm 0.041$ | $1.0\sigma$ |
| Strict ($V_{obs}/V_{bar} > 1.5$) | $0.413 \pm 0.045$ | $1.8\sigma$ |

While individual definitions shift the normalization, the slope consistently
clusters around the $M^{1/3}$ expectation. An alternative definition based on
acceleration thresholds ($a < a_0$) yields a steeper slope ($\alpha_{\rm SPARC} \approx
0.5$). This estimator is the $M^{1/3}$ vs. $M^{1/2}$ (TEP vs. MOND-like) discrimination
in miniature; the fact that the verdict is currently estimator-dependent is noted
as an open systematic. Because the acceleration-threshold method is degenerate
with the MOND acceleration scale itself, the kinematic divergence test is retained
as the fiducial estimator. This preference is theoretically motivated: TEP models
phantom mass as a geometric effect driven by temporal shear (a density/proximity
threshold) rather than an instantaneous acceleration threshold, so a purely kinematic
divergence estimator is intrinsically less biased toward MOND-like phenomenologies.

\begin{equation} \label{eq:dm_relation} R_{\rm DM} = k \cdot M_{\rm bar}^{\alpha_{\rm SPARC}} \end{equation}

\begin{equation} \label{eq:sparc_exponent} \alpha_{\rm SPARC} = 0.355 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (definition)} \end{equation}

The bootstrap-marginalized exponent deviates from the TEP expectation of $\alpha_{\rm SPARC} = 1/3
= 0.333$ by approximately 0.3$\sigma$ under the combined uncertainty ($\sim 0.5\sigma$ statistical-only)—consistent within typical statistical tolerance. The correlation
coefficient $r \approx 0.6$ indicates a significant relationship.
Importantly, this result is robust: individual threshold choices yield
exponents ranging from 0.28 to 0.42, but the ensemble average converges near
1/3.

#### Size-Scaling Confound Test

The $M^{1/3}$ onset exponent is not by itself discriminating: for
approximately self-similar exponential disks, a fixed velocity-ratio
crossing occurs at a nearly fixed multiple of the disk scale length $R_d$,
so the galaxy mass–size relation can produce the same exponent without any
density threshold. The confound is tested directly on the same SPARC
sample (pipeline step 8). The in-sample mass–size relation is
$R_d \propto M_{\rm bar}^{0.322 \pm 0.017}$ ($r = 0.821$, $n = 175$), which
alone predicts an onset exponent $\approx 0.32$ — within the stated
uncertainty of the measured $0.355 \pm 0.043$. Comparing the two candidate
invariants per galaxy — the size ratio $R_{\rm DM}/R_d$ and the enclosed
mean baryonic density $\bar\rho(<R_{\rm DM}) = 3M_{\rm bar}(<R_{\rm
DM})/(4\pi R_{\rm DM}^3)$, with $M_{\rm bar}(<R)$ estimated spherically as
$V_{\rm bar}(R)^2 R/G$ — the size invariant is mass-independent at the
fiducial threshold (slope $0.056 \pm 0.044$, $p = 0.21$; median
$R_{\rm DM}/R_d = 0.64$), while the enclosed density retains a significant
mass correlation (slope $0.190 \pm 0.068$, $p = 5.7\times10^{-3}$). The
pattern is threshold-dependent: at $V_{\rm obs}/V_{\rm bar} > 1.2$ the
size null wins decisively (density slope $0.302$, $p = 5.4\times10^{-6}$),
while at $1.4$ the two are comparable ($0.086$ vs $0.084$).

The onset scaling is therefore consistent with, but not by itself
discriminating evidence for, the density-threshold mechanism: a fixed
fraction of the disk scale length produces the same exponent on this
sample. The discriminating weight of the SPARC test accordingly rests on
the two channels the size-scaling null cannot supply — the residual
correlations with baryonic content (Step 7) and the two-branch kinetic profile $P_{,X}=\epsilon_{\rm reg}+k\sqrt\xi+2\xi$ ($k_*=16.03$ calibrated to $a_0$, $\epsilon_{\rm reg}=10^{-3}$; Steps 10 and 13),
which reproduces the full 3,389-point SPARC radial-acceleration relation under the action-derived additive force law $g_{\rm obs}=g_{\rm bar}+a_\phi$ at $0.2185$ dex RMS ($\Upsilon_d=0.5$) and $0.2004$ dex RMS ($\Upsilon_d=0.35$, matching the empirical MOND reference $0.1989$ dex and improving over the $0.2812$ dex unit-term baseline by $>60$ mdex; phenomenological quadrature comparator $\sqrt{g_{\rm bar}^2+a_\phi^2}$ gives $0.1974$ dex RMS with $k_{\rm fit}=15.78$ and outer-edge BTFR median $v_{\rm obs}/v_{\rm pred}=1.0109$).

#### Phantom-Mass Amplitude and Profile Test

The onset radius alone does not test the mechanism — any threshold
criterion produces an onset. The discriminating content of the
phantom-mass claim lies in the amplitude and the radial profile, both of
which are evaluated directly on the same SPARC sample (pipeline step 9).
For a conformal coupling of charge $\beta_A$, an unscreened scalar channel
contributes a centripetal term of order $2\beta_A^2$ times the Newtonian
one, giving the plateau ratio $V_{\rm obs}/V_{\rm bar} = \sqrt{1 +
2\beta_A^2 \mathcal S_{\rm eff}}$, which for the bare coupling
$|\beta_A| = 1$ at full saturation is $\sqrt{3} \approx 1.73$. Measured on
the outer third of each curve ($n = 164$ galaxies with $\geq 6$ valid
points and $V_{\rm bar} > 5$ km s$^{-1}$), the median plateau ratio is
$1.98$ ($16$–$84$th percentile $1.38$–$2.49$) — order-consistent with the
bare-coupling saturation value but systematically above it: the implied
effective response is $\mathcal S_{\rm eff} = 1.46$ at the median, and
$71\%$ of galaxies require $\mathcal S_{\rm eff} > 1$. The plateau
amplitude is therefore quantified rather than asserted: it sits within a
factor of order unity of the bare-coupling prediction and the residual
excess is precisely the response-normalization demand already flagged in
the Formal Link — the amplitude test converts that open derivation into a
measured gap of factor $\sim 1.5$–$2$ rather than an unconstrained
parameter.

The radial profile supplies the second discriminator. Defining the
enhancement $E(R) = V_{\rm obs}^2/V_{\rm bar}^2 - 1 = M_{\rm
ph}(R)/M_{\rm bar}(<R)$, the phantom fraction is required by the
transition picture to grow outward (screened inside, unscreened outside),
whereas a constant coupling rescaling of the baryonic potential predicts
$d\log E/d\log R = 0$. The measured profile rises: median slope $0.46$
across 161 galaxies with sufficient baseline, positive in 133 (Wilcoxon
signed-rank $p = 1.0\times10^{-17}$). The rising phantom fraction is the
expected morphology of a baryon-anchored screening transition and is
inconsistent with a global multiplicative rescaling; it is not, by itself,
separable from acceleration-threshold phenomenologies, which predict the
same qualitative rise. What the two tests jointly establish is that the
SPARC sample carries both an amplitude-level and a profile-level
constraint on the mechanism — the onset scaling is no longer the sole
empirical content of the claim.

#### Cross-Regime Consistency

The significance of this result emerges when combined with the Temporal
Topology screening analysis (Section 4). The complete screening hierarchy spans
~15 orders of magnitude in density—from gas giants ($\rho \sim 1$ g/cm³) to
binary pulsars ($\rho \sim 10^{14}$ g/cm³)—while the fitted regression uses the
11 dense objects ($\rho > \rho_T$) spanning ~12 orders of magnitude
($\sim 10^2$–$10^{14}$ g/cm³). The consistency of this exponent across such a range is
striking, though the high $R^2$ is partly a consequence of the definitions
used (see Box 4.1). The result is consistent with the hypothesis that the
$M^{1/3}$ scaling reflects a genuine feature of the gravitational sector,
though systematic uncertainties in rotation curve fitting and baryonic
modeling remain.

Combined with the SPARC galaxy scaling ($\alpha_{\rm SPARC} = 0.355 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (definition)}$), the
RBH-1 crossover consistency (within $\sim 25\%$ combined uncertainty), and
the Milky Way dark-matter onset ($R_{\text{DM}} \approx 3$ kpc, observed $\sim$3–5 kpc), these results
suggest a universal density-limited structure may span 18 orders of
magnitude in mass—from Earth to galaxy.

### Physical Interpretation

The observed $M^{1/3}$ scaling admits a natural interpretation in terms of
density-limited screening. If the screening transition occurs at a
characteristic density $\rho_{\rm trans}$, dimensional analysis requires:

\begin{equation} \label{eq:trans_radius} R_{\rm trans} \sim \left( \frac{M}{\rho_{\rm trans}} \right)^{1/3}. \end{equation}

Fixing $\alpha_{\rm SPARC} = 1/3$ and fitting only the normalization yields $k \approx
7.9 \times 10^{-4}\,{\rm kpc}/M_\odot^{1/3}$. This normalization corresponds to an
emergent screening density $\rho_{\rm trans} \approx 0.5\,M_\odot/{\rm pc}^3$ ($\sim
3 \times 10^{-23}$ g/cm³) recovered from the fitted normalization $k$ via
Eq. ($\ref{eq:trans_radius}$). A completion is conjectured to derive $\rho_{\rm trans}$
from the scalar-sector normalization (the quartic coupling $\lambda$ and the
coupling $\beta_A$), but this remains an open
target (see Formal Link below). This value is within an order of magnitude of
typical disk densities at the optical radius (0.01–0.1 $M_\odot/{\rm
pc}^3$), providing a physically reasonable anchor for the screening
mechanism.

*Note:* This galactic transition density $\rho_{\rm trans}$ is
distinct from the fundamental core saturation scale $\rho_T \approx 20$
g/cm³ calibrated in Section 2 and cross-checked in Section 5. The core density $\rho_T$ governs the compact
object size (soliton surface), while $\rho_{\rm trans}$ governs the onset of
the diffuse halo effect (screening radius). Both scales follow the same
$M^{1/3}$ structural form, consistent with the environment-dependent nature of the
theory across 15 orders of magnitude in density.

#### Formal Link: From Core Saturation to Galactic Transition

The galactic transition density $\rho_{\rm trans}$ is intended to
emerge from the same scalar field profile that fixes the core
saturation scale $\rho_T$; at present it is an independent empirical
normalization recovered from the fitted $k$, and the derivation
sketched below is the stated open target rather than an established
result. In the TEP
framework (Box 6.5), the scalar field obeys

\begin{equation} \label{eq:profile_link} \nabla^2 \phi = V'(\phi) + \alpha(\phi)\rho, \end{equation}

with effective mass $m_{\rm eff}^2(\rho) = V_{\rm eff}''(\phi_{\rm min})$.
The field profile transitions from saturated ($\rho \sim \rho_T$,
$R_T$ scale) to screened ($\rho \gg \rho_T$, Compton wavelength
$\lambda_C \ll R_T$) to unscreened ($\rho \ll \rho_T$,
$\lambda_C \gg R_T$). The galactic halo onset occurs where the
enclosed mean baryonic density $\bar\rho(<R) = 3M_{\rm bar}(<R)/(4\pi
R^3)$ drops to the value $\rho_{\rm trans}$ at
which the scalar-field-induced gravitational acceleration becomes
comparable to the baryonic contribution — an enclosed-mean-density
criterion, consistent with the form of Eq. ($\ref{eq:trans_radius}$);
a local-density formulation would require a specified field/density
profile and is not used here. Because the field profile
is governed by the same scalar sector — the quartic normalization
$\lambda$ and coupling $\beta_A$ — that underlies the compact-object
saturation,
$\rho_{\rm trans}$ is expected to be a derived consequence of these
parameters rather than a free fit; pending that derivation it is an
independent empirical input. The SPARC analysis measures the structural
exponent $\alpha_{\rm SPARC}$ and normalization $k$; the
corresponding $\rho_{\rm trans}$ is then recovered from $k$
via Eq. ($\ref{eq:trans_radius}$) as a consistency check.

#### Connection to the MOND Acceleration Scale

The fitted screening density implies a characteristic transition
acceleration. At the transition radius, the gravitational acceleration is:

$$g_{\rm TEP} = \frac{GM}{R_{\rm trans}^2} \approx 5 \times 10^{-10}\,{\rm
m/s}^2.$$

This is within a factor of 4 of the MOND acceleration scale $a_0 \approx 1.2
\times 10^{-10}\,{\rm m/s}^2$. The near-coincidence suggests that the TEP
screening mechanism and the empirical MOND phenomenology may share a common
origin: both describe the transition from screened (Newtonian) to unscreened
(modified) gravity at a characteristic acceleration scale set by
cosmological boundary conditions.

### Connection to the Radial Acceleration Relation

The SPARC database is the source of the celebrated Radial Acceleration
Relation (RAR), which shows a tight empirical coupling between observed
acceleration $g_{\rm obs}$ and baryonic acceleration $g_{\rm bar}$ (McGaugh
et al. 2016). The RAR has been interpreted as evidence for modified gravity
(MOND) or as a consequence of dark matter halo profiles fine-tuned to
baryonic distributions.

The TEP framework offers a third interpretation: the RAR may emerge
from the environment-dependent screening of the proper-time field. At high
accelerations (high densities), screening is complete and $g_{\rm obs}
\approx g_{\rm bar}$. At low accelerations (low densities), the unscreened
time-field gradient contributes additional apparent gravity, producing the
characteristic upturn in the RAR. The $M^{1/3}$ scaling of the transition
radius is the spatial manifestation of this acceleration-dependent
screening.

A fixed-normalization test applies the existing alternative low-gradient
kinetic realization directly to the SPARC mass models (pipeline steps 10–13).
Its response $P_{,X}(\xi)=k\sqrt{\xi}+2\xi$, with $\xi=|X|/\Lambda_X^4$,
differs from the unit-term baseline $1+2\xi$ of Paper 0 §7 and yields the
derived flat-tail and baryonic Tully–Fisher limits. With the cosmic temporal-gradient
threshold fixed by the Hubble clock roll at $g_t=cH_0/2\simeq3.40\times10^{-10}$ m/s$^2$
and the DEF coupling normalization $\alpha_0=\sqrt{2}\beta_A=-\sqrt{2}$, the natural
two-branch crossover $k=16$ ($\xi_*=(k/2)^2=64$) yields
$a_{\rm eff}=4\sqrt{2}\,g_t/k=cH_0/(4\sqrt{2})=1.202\times10^{-10}$ m/s$^2$
($k=4\sqrt{2}\,g_t/a_0\simeq16.03$ at the canonical $a_0=1.20\times10^{-10}$ m/s$^2$),
deriving Milgrom's $a_0\sim cH_0$ coincidence from the cosmological time field with no
galaxy-specific parameter adjusted across the 175-galaxy sample.
The algebraic flux law $\bar{u}P_{,X}(\bar{u}^2/2) = g_{\rm bar}/g_t$ (with canonical zero-gradient regulator $P_{,X}(\xi)=\epsilon_{\rm reg}+k\sqrt{\xi}+2\xi$, $\epsilon_{\rm reg}=10^{-3}$ ensuring strict hyperbolicity at $X=0$ while leaving galactic scales $\xi\sim10^{-2}$–$1$ invariant to $<10^{-4}$) yields
the temporal-shear acceleration $a_\phi = 2g_t\bar{u}$, where $k=4\sqrt{2}\,g_t/a_0\simeq16.03$ is calibrated to the empirical scale $a_0=1.20\times10^{-10}$ m/s$^2$ (so agreement of the deep-tail normalization with $a_0$ is a calibration input rather than an independent prediction, while the shape across the 175-galaxy sample has zero refitted parameters). From the point-particle action $L_{\rm pp}=-mc\sqrt{-\tilde g_{\mu\nu}dx^\mu dx^\nu}$, the action-derived radial force law is additive,
$g_{\rm obs} = g_{\rm bar} + a_\phi$, which is reported as the primary action test alongside the phenomenological quadrature interpolating comparator $\sqrt{g_{\rm bar}^2 + a_\phi^2}$. Evaluated on all 3,389
measured radii across 175 galaxies, the action-derived additive law $g_{\rm bar}+a_\phi$ at $k=16.03$ achieves an rms log-residual of $0.2185$ dex at $\Upsilon_d = 0.5$ (median $-0.086$ dex, $44.7\%$ within $0.1$ dex) and $0.2004$ dex at $\Upsilon_d = 0.35$ (median $-0.020$ dex — statistically indistinguishable from the empirical MOND interpolating reference $0.1989$ dex while substantially outperforming the $0.2812$ dex unit-term baseline and $0.2749$ dex exponential-interpolation candidate). Because the two-term flux law $P_{,X}=\epsilon_{\rm reg}+k\sqrt{\xi}+2\xi$ omits higher-order crossover terms, direct addition $g_{\rm bar}+a_\phi$ at fixed $\Upsilon_d=0.5$ carries a $+2g_{\rm bar}a_\phi$ crossover offset ($k_{\rm fit}\approx30.9$ at $\Upsilon_d=0.5$), which is absorbed either by the standard disk mass-to-light shift $\Upsilon_d=0.35$ ($0.2004$ dex) or by the phenomenological quadrature comparator $\sqrt{g_{\rm bar}^2+a_\phi^2}$ ($0.1974$ dex at $k=16.03$, $k_{\rm fit}=15.78$; outer-edge BTFR median $v_{\rm obs}/v_{\rm pred}=1.0109$ vs $0.910$ additive at $\Upsilon_d=0.5$, Step 13; derivation of a quadrature-equivalent transition directly from the covariant action requires a mid-range kinetic interpolation). A field-value
coupling $K(\phi) = e^{c\psi}$ improves the RAR match monotonically (rms
$\to 0.199$ dex at $c = 3$) but is excluded by a sign tension with the
wide-binary pair channel, which requires the opposite sign, so the
monotone form is not adopted. The residual offset is quantified by the
accompanying scans: disk mass-to-light $\Upsilon_d = 0.35$ alone
absorbs the offset (rms $0.2004$ dex, median $-0.020$ dex —
indistinguishable from the MOND reference at zero additional sector
parameters); the ambient-shear scan bounds the cosmic-web background
at $u_{\rm amb} \lesssim 0.02$ (larger values degrade the deep tail
monotonically), validating the isolated-source assumption; and a
small canonical term $\epsilon \lesssim 0.1$ in
$P_{,X} = \epsilon + k\sqrt{\xi} + 2\xi$ is observationally free and
marginally preferred (rms $0.2098$ dex at $\epsilon = 0.1$), while
order-unity values are excluded not by the data but by the
flat-asymptote requirement — a linear tail reappears below
$g_{\rm bar} \sim (\epsilon/k)\,g_t$. The offset is therefore a
baryonic mass-model systematic, not a sector-shape failure.

![Example Rotation Curves](results/figures/figure_6_sparc_examples.png)

Figure 1: Representative SPARC rotation curves. Observed velocities
diverge from baryonic predictions near the TEP transition radius $R_{\rm
trans}$. Shaded regions indicate the screened (Newtonian) zone. The
agreement between $R_{\rm trans}$ and the observed onset $R_{\rm DM}$
supports the screening interpretation.

### Linking RBH-1 to Galactic Halos

The SPARC analysis identifies a cross-scale consistency between the
RBH-1 wake phenomenon and galactic dark matter halos:

RBH-1 represents a "naked saturation"—a Temporal Topology saturation not anchored
to a baryonic host, propagating freely through the intergalactic medium
and inducing star formation via its time-dilation wake.

Galactic halos represent "baryon-anchored solitons"—the same temporal
field structure, but centered on and shaped by the baryonic mass
distribution of the host galaxy.

Both phenomena are consistent with the same $M^{1/3}$ scaling law, calibrated from
terrestrial GNSS observations. The universality of this scaling across
planetary, stellar, galactic, and cosmological mass scales constitutes a
central theoretical consequence of the Temporal Equivalence Principle.

### Residual Analysis: Discriminating Baryonic Feedback vs Field Theory

The RMS scatter of 0.48 dex around the $M^{1/3}$ relation indicates
substantial galaxy-to-galaxy variation. A critical test distinguishes
whether this scatter arises from baryonic feedback processes (Standard
Model) or from the intrinsic field theory mechanism (TEP): correlate the
residuals with baryonic properties versus screening proxies.

#### Methodology

For each of the 167 SPARC galaxies with valid mass discrepancy radii, the
residual from the $M^{1/3}$ expectation is calculated as:

\begin{equation} \label{eq:residual} \text{Residual} = \log_{10}(R_{\rm DM} / R_{\rm exp}) \end{equation}

where $R_{\rm exp} = k \cdot M_{\rm bar}^{1/3}$ with $k = 7.86 \times 10^{-4}$ kpc/$M_\odot^{1/3}$. These
residuals are then correlated with:

*Baryonic properties.* Gas fraction ($f_{\rm gas}$), surface
brightness ($\Sigma$), and inclination, treated as proxies for feedback
efficiency, star formation history, and observational systematics.

*Screening proxy.* Central density ($\rho_{\rm central}$),
treated as a proxy for local screening strength independent of $R_{\rm
DM}$.

#### Results

The residual correlations reveal a clear pattern:

| Property | Correlation (r) | P-value | Interpretation |
| --- | --- | --- | --- |
| Gas Fraction | $-0.164$ | 0.034 | Weak, marginally significant |
| Surface Brightness | $+0.062$ | 0.427 | Null, not significant |
| Inclination | $+0.115$ | 0.139 | Null, not significant |
| Central Density | $+0.032$ | 0.685 | Null, not significant |

#### Interpretation

When evaluated using the measured SPARC Table 1 effective radii $R_{\rm eff}$
and inclinations $i$ (`step_7_sparc_residuals.json`), none of the
baryonic or observational properties exhibits a strong correlation with the
scaling residuals ($\max |r| = 0.164$): inclination ($r = +0.115$, $p = 0.139$)
and surface brightness ($r = +0.062$, $p = 0.427$) are statistically null,
while gas fraction shows only a weak marginal trend ($r = -0.164$, $p = 0.034$,
$|r| < 0.2$). This confirms that the scaling relation is not driven by
rotation-curve deprojection systematics or baryonic feedback efficiency.

The central density ($r = +0.032$, $p = 0.685$) likewise shows no residual
trend after the primary mass scaling is removed, consistent with the
full rotation-curve radial-acceleration fit in Steps 10 and 13 where the
two-branch profile $P_{,X}=\epsilon_{\rm reg}+k\sqrt\xi+2\xi$ ($k=16.03$, $\epsilon_{\rm reg}=10^{-3}$)
accounts for the complete radial profile across all 175 galaxies at $0.2185$ dex RMS ($\Upsilon_d=0.5$; $0.2004$ dex at $\Upsilon_d=0.35$) under action-derived addition $g_{\rm bar}+a_\phi$ ($0.1974$ dex under the quadrature comparator).

Summary: across all 167 SPARC galaxies with measured onset radii, the
residuals are uncorrelated with galaxy inclination ($r = +0.115$, $p = 0.139$),
surface brightness ($r = +0.062$, $p = 0.427$), and central density
($r = +0.032$, $p = 0.685$), and only weakly correlated with gas fraction
($r = -0.164$, $p = 0.034$), ruling out inclination deprojection and
baryonic feedback artifacts.

![SPARC Residual Analysis](results/figures/figure_7_sparc_residuals.png)

Figure 2: Residual analysis of the SPARC $M^{1/3}$ scaling. (a) Scaling
relation colored by residual. (b-e) Correlations with baryonic
properties and screening proxies. (f) Gaussian residual distribution
($\sigma = 0.48$ dex). (g) Correlations are weak in amplitude ($|r|<0.3$); inclination is statistically significant but plausibly reflects deprojection systematics rather than baryonic feedback physics.

### Scatter Quantification and Comparative Analysis

The observed scatter of $\sigma = 0.48$ dex around the $M^{1/3}$ relation is
substantial (corresponding to a factor of $\sim 3$ variation). Quantifying
the origin of this scatter is essential for distinguishing the TEP signal
from competing hypotheses.

#### Intrinsic vs. Measurement Scatter

Standard error budgeting suggests that a significant fraction of this
scatter is observational. Distance uncertainties in the SPARC sample
typically range from 10–20%, which propagate to a ~0.1–0.2 dex uncertainty
in $R_{\rm DM}$. Inclination corrections and non-circular motions contribute
additional noise. However, the residual scatter ($\sim 0.3$ dex) likely
reflects intrinsic variation in the galaxy population.

#### Comparison with MOND and LCDM+Feedback

MOND: The Modified Newtonian Dynamics framework typically
yields a tighter scatter ($\sim 0.13$ dex) in the Radial Acceleration
Relation (RAR). This is expected because MOND modifies gravity based on
instantaneous local acceleration, a continuous variable. In contrast, this
analysis extracts a single discrete parameter ($R_{\rm DM}$) from each
rotation curve, a method inherently more sensitive to local irregularities
and noise than the integrated RAR analysis. The larger scatter in $R_{\rm
DM}$ does not invalidate the mean scaling; rather, it indicates that the
"onset radius" is a noisy estimator of the underlying screening transition.

LCDM + Feedback (Null Hypothesis): The standard
cosmological model relies on baryonic feedback to explain rotation curve
diversity. In this scenario, the "dark matter onset" is not a fundamental
scale but an emergent property of halo assembly and feedback history.
Hydrodynamic simulations (e.g., NIHAO, FIRE) typically predict a scaling of
$R_{\rm DM} \propto M^{\alpha}$ with $\alpha \approx 0.3$–$0.4$, broadly
consistent with observation. However, they offer no account of why the
galactic relation shares its structural exponent with the compact-object
scaling. The fitted normalization corresponds to $\rho_{\rm trans} \approx
0.5\,M_\odot/{\rm pc}^3$ ($\sim 3\times10^{-23}$ g/cm³), twenty-four orders
of magnitude below $\rho_T$: what is presently demonstrated is a shared
structural exponent between two independently normalized scales, not a
normalization link, and whether $\rho_{\rm trans}$ follows from the same
scalar-sector normalization that underlies the $\rho_T$ saturation law
is the open derivation target
recorded in the Formal Link.

#### Distinctive Cross-Scale Linkage

While MOND provides a tighter fit to galaxy kinematics alone, it offers
no explanation for the GNSS clock correlations ($L_c \approx 4200$ km).
LCDM+Feedback can accommodate the
galaxy scaling but treats the normalization as a free parameter,
offering no predictive link to other scales. TEP is distinctive in that
it poses the normalization of the galactic relation ($k \approx 7.9
\times 10^{-4}$ kpc/$M_\odot^{1/3}$, equivalently $\rho_{\rm trans}$)
as a quantity to be derived from the same GNSS-anchored scalar sector —
a stated falsification condition rather than an established link: the
measured normalization is an independent empirical input pending that
derivation, within the stated systematic uncertainty.

### Reproducibility

Full analysis code, data products, and methodology documentation are
available in the public repository:

*Repository.*
github.com/matthewsmawfield/TEP-UCD

*SPARC Scaling Analysis.*
`scripts/steps/step_4_sparc_analysis.py`

*SPARC Residual Analysis.*
`scripts/steps/step_7_sparc_residuals.py`

*Input Data.* `data/sparc/` (SPARC database tables)

#### Methodological Details

The "onset radius" $R_{\rm DM}$ is defined as the
*first radial bin* where $V_{\rm obs}/V_{\rm bar} > 1.3$ (fiducial
threshold). Specifically:

For each galaxy, the rotation curve is evaluated at the native SPARC
radial sampling (not interpolated).

The velocity ratio $V_{\rm obs}/V_{\rm bar}$ is computed at each radius.

$R_{\rm DM}$ is the radius of the first bin exceeding the threshold.

Galaxies with non-monotonic or noisy $V_{\rm obs}/V_{\rm bar}$ profiles
are retained; the "first crossing" definition is robust to subsequent
fluctuations.

Galaxies that never exceed the threshold (8 of 175) are excluded from
the fit.

The headline value $\alpha_{\rm SPARC}=0.355\pm0.043$ is the threshold-marginalized bootstrap estimate, obtained by resampling galaxies while marginalizing over the accepted onset definitions. The fiducial ($V_{\rm obs}/V_{\rm bar}>1.3$) definition alone gives $0.372\pm0.041$; the difference is retained as onset-definition systematic uncertainty. Varying the threshold from 1.1 to 1.5 yields exponents from 0.28 to 0.42, bracketing the headline estimate.

## 4. Temporal Topology Screening Analysis

If a scalar field permeates spacetime, why has it not been detected in
precision gravitational experiments? This section answers that question. The
key insight is that the scalar field's influence depends on the geometric
overlap of topological charge cores: in environments where charge cores are
tightly packed, the field is suppressed—a phenomenon called Temporal
Topology screening. Rather than invoking discrete thin-shell boundaries,
screening operates via the continuous spatial profile of the scalar field
(Temporal Topology). The tight geometric packing in deep potential wells
suppresses the local field gradient (Temporal Shear), ensuring short-range
Temporal Shear suppression while leaving the field light cosmologically.
General Relativity is recovered in the regimes where it has been tested most
stringently.

Screening in TEP is represented at the theory level by the environmental operator
$S_\Sigma(\mathcal{E})$.
Quantities such as
$\rho_T$,
$R_T(M)$,
$S_\oplus(r)$,
compactness $\Phi/c^2$,
local stellar density,
geometric coherence length,
and channel-specific response coefficients
are domain-specific projections of $\mathcal{E}$,
not independent screening mechanisms
and not interchangeable universal thresholds.
Each is an observational transfer model
that parameterizes the same underlying operator
in a regime-appropriate form.

The screening factor $S$ quantifies this suppression. It is defined as the
ratio of the Temporal Topology saturation radius (where the scalar field saturates) to the
physical radius of the object:

\begin{equation} \label{eq:screening_factor} S = \frac{R_T}{R_{phys}}. \end{equation}

The quantity $S$ is used here as a geometric proxy for how deeply the
baryonic source is embedded within a saturated soliton configuration. When
$S \gg 1$, the physical object occupies a small region relative to the
saturated field scale; in this regime the phenomenology assumes that
non-linear response in the scalar sector suppresses gradients in the dense
interior, recovering GR to high precision in local dynamics. When $S \sim
1$, the system lies near the transition between strongly screened and weakly
screened behavior. When $S \ll 1$, the Temporal Topology saturation scale is smaller than the
object and the saturated region does not envelop the full baryonic
configuration; in this regime the model does not assume strong Temporal
Topology suppression a priori, and constraints must be assessed
case-by-case.

\begin{equation} \label{eq:ppn_suppression} \alpha_{\rm PPN}^{\rm eff}=\alpha_0 F(S,\rho,\Phi,\nabla\phi), \qquad F\rightarrow 0 \end{equation}

in the screened or source-charge-suppressed limit. The geometric factor $S$ is not itself the observable PPN coupling. Precision tests constrain $\alpha_{\rm PPN}^{\rm eff}$; $S$ is used here only as a proxy for the nonlinear suppression regime.

The near-transition regime $S \sim 0.4$–$0.7$ is resolved by the corpus's
channel-projection structure (Paper 0). The screening response is not a
single function of $S$: clock-comparison and transport observables probe
the amplitude projection $S_A$, while orbital and geodesic observables
probe the source-charge/shear projection $\mathcal S_\Sigma$, which is
suppressed more strongly for the pairwise configurations used in ephemeris
tests. The same geometric $S$ therefore places different channels on
different sides of their respective thresholds: Earth's $S \approx 0.66$
admits a detectable clock-covariance signal ($C_A$ covariance channel, the GNSS
calibration of Section 2) while the LLR, Cassini, and MESSENGER orbital
observables at $S = 0.42$–$0.65$ remain GR-consistent through the
$\mathcal S_\Sigma$ projection. The reading of sub-unity $S$ is uniform
throughout this paper: $S < 1$ denotes that the saturation region does
not envelop the object, i.e. the unscreened side of the transition; the
observable amplitude in any given channel is then set by that channel's
projection, not by $S$ alone.

### 4.1 The White Dwarf Stress Test

White Dwarfs are the ideal stress test for two reasons. First, their
structure is determined by quantum mechanics (electron degeneracy pressure),
not thermal physics, so their mass-radius relation is calculable from first
principles. Second, their mass-radius scaling runs in the opposite direction
to the saturation scaling (see Figure 3):

White Dwarf (Chandrasekhar): $R_{WD} \propto M^{-1/3}$
— heavier stars are smaller

Soliton (TEP): $R_{sol} \propto M^{+1/3}$ — heavier
fields are larger

![White Dwarf Screening Mechanism](results/figures/figure_3_wd_screening.png)

Figure 3: The "Scissors" Effect. The defining visual signature is the
divergence of scales: as mass increases, the white dwarf physical radius
shrinks ($R_{WD} \propto M^{-1/3}$) while the saturation radius grows
($R_{sol} \propto M^{1/3}$). This divergence drives the screening factor
$S = R_T/R_{\rm phys}$ to large values ($S \gg 1$), recovering
General Relativity within the star.

These opposing scalings create a "scissors" pattern. As mass increases, the saturation radius grows while the physical radius shrinks—the screening factor
diverges. This is the signature of Temporal Topology flattening.

For Sirius B ($M \approx 1.02 M_{\odot}$, $\rho \approx 2.4 \times 10^6$
g/cm³):

\begin{equation} \label{eq:jupiter_radius} R_{phys} \approx 5{,}800 \text{ km} \quad \text{(observed)} \end{equation}

\begin{equation} \label{eq:jupiter_soliton} R_{sol} \approx 289{,}000 \text{ km} \quad \text{(TEP expectation)} \end{equation}

\begin{equation} \label{eq:jupiter_screening} \text{Screening Factor} = \frac{R_T}{R_{phys}} \approx 50\times. \end{equation}

With a mean density \(\rho \approx 2.4 \times 10^6\) g/cm³ far exceeding the
saturation threshold \(\rho_T \approx 20\) g/cm³, the saturation field extends
50 times beyond the physical surface of the star. The dense baryonic matter is
deeply embedded within the scalar field's saturation core, consistent with strong
screening under the stated model assumptions. Within the star, GR dynamics are recovered; the Keplerian mass
measured from binary orbital motion is the true baryonic mass.

### 4.2 The Empirical Screening Law

Extending this analysis across 26 astrophysical objects (planets, brown
dwarfs, main sequence stars, white dwarfs, neutron stars, and binary
pulsars) yields a compact summary of how the defined screening factor varies
with density. Restricting the regression fit to the 11 dense objects
($\rho > \rho_T$, excluding planets and black holes) gives:

\begin{equation} \label{eq:screening_law} S \propto \rho^{\beta_{\rm scr}}, \qquad \beta_{\rm scr} = 0.334 \quad (R^2 = 0.99995). \end{equation}

The exponent $\beta_{\rm scr} = 0.334$ is statistically indistinguishable from 1/3. This density-dependent screening law is a projection of the source-charge screening operator $\mathcal{S}_\Sigma(\mathcal{E})$ (Paper 0 &sect;7) onto the UCD density regime; it is not an independent screening operator. Two sector assignments are stated explicitly. The geometric diagnostic $S = R_T/R_{\rm phys}$ tracks the shear-sector $\mathcal{S}_\Sigma$ regime — density is the licensed stand-in for the ambient-steepness criterion inside dense macroscopic matter (Paper 0 &sect;8) — while the scale $\rho_T$ that sets $R_T$ is itself calibrated through the clock-amplitude projection $S_A = \min[1,(\bar\rho/\rho_T)^{1/3}]$ (Paper 0 &sect;3). Because each object's density is evaluated at its physical radius, $\bar\rho = 3M/(4\pi R_{\rm phys}^3)$, the diagnostic carries an enclosed-acceleration equivalent: the criterion $S\sim1$ corresponds to $GM/R_{\rm phys}^2 \propto \bar\rho\,R_{\rm phys}$, so the density and acceleration readings of the hierarchy agree object-by-object. Under the
stated definitions, this is the expected scaling: if $R_T \propto
M^{1/3}$ and $R_{phys} \propto (M/\rho)^{1/3}$, then:

\begin{equation} \label{eq:screening_derivation} S = \frac{R_T}{R_{phys}} = \frac{M^{1/3}}{M^{1/3}/\rho^{1/3}} =
\rho^{1/3}. \end{equation}

#### Interpretation of the Screening Law

It is important to recognize that the regression $S \propto
\rho^{0.334}$ is primarily a consistency check, not independent evidence
for the scaling law. Since $S \equiv R_T/R_{\text{phys}}$ and
the radii are defined by mass-density relations, the slope $\sim 1/3$ is
algebraically expected given the model assumptions.

The value of this analysis is not to prove the scaling 'ab initio', but
to demonstrate that a *single* saturation scale $\rho_T$ yields
a consistent screening hierarchy across 15 orders of magnitude in
density ($R^2 \approx 1$) without requiring regime-dependent
adjustments. The high $R^2$ confirms internal consistency; it does not
constitute independent confirmation of TEP.

Furthermore, the extreme screening factors observed in binary pulsars
($S \sim 29{,}000$) are consistent with a strongly non-linear suppression
mechanism in the scalar sector. The empirical screening hierarchy
($S$ vs $\rho$) derives from the canonical Temporal Topology mechanism
(Paper 0), where screening operates via the continuous spatial profile
of the scalar field governed by non-linear superposition of field
gradients (Temporal Shear). Box 6.5 (Section 6) sketches a candidate
route from the canonical action to soliton-like saturation, showing how
the interplay between the kinetic term and a candidate saturation potential could generate
the characteristic $M^{1/3}$ scaling and screening behavior. The detailed
dynamical derivation is not required for the empirical hierarchy used here.

The empirical screening law is a direct consequence of the $M^{1/3}$ soliton
scaling, providing cross-regime consistency under a single $\rho_T$ rather
than independent confirmation (see Note on Interpretation above).

### 4.3 Complete Screening Hierarchy

![Screening Hierarchy Across 15 Orders of Magnitude](results/figures/figure_4_screening_hierarchy.png)

Figure 4: The Screening Hierarchy. Screening factor $S$ plotted against
density for 26 astrophysical objects. The empirical power law $S \propto
\rho^{0.334}$ explains why dense objects (binary pulsars) are screened
($S \gg 1$) while diffuse systems (galaxies) are weakly screened ($S \ll
1$).

The complete hierarchy of screening factors across all object classes is
visualized in Figure 4 and tabulated below.

| Object Class | Density (g/cm³) | Screening | Physical Meaning |
| --- | --- | --- | --- |
| Gas Giants | 0.7 – 1.6 | 0.3 – 0.4× | $R_T$ smaller than physical radius; saturated region does not
envelop the object — the unscreened side of the transition,
with channel-projected response (clock channels more active
than orbital channels) |
| Main Sequence Stars | 0.6 – 57 | 0.3 – 1.3× | Mixed regime; scalar and baryonic scales comparable |
| Rocky Planets (Earth) | 3.3 – 5.5 | 0.56 – 0.66× | Saturation scale comparable to object radius; GNSS probes this boundary |
| Brown Dwarfs | ~100 | ~1.7× | Just above $\rho_T$; screening onset begins |
| White Dwarfs | $\sim 4 \times 10^5$ – $2.5 \times 10^6$ | 27 – 50× | Baryonic star embedded within saturation scale; Keplerian mass = baryonic mass |
| Neutron Stars | $10^{14}$ | 26,800× | Scalar contribution less than 0.004%; pure GR dynamics |
| Binary Pulsars | $10^{14}$ | 29,500× | GR verified to 0.2%; Nobel Prize 1993 |

Table 4.1: Complete screening hierarchy across astrophysical object
classes. The screening factor increases monotonically with density,
consistent with a Temporal Topology screening hierarchy under the stated
definitions. Binary pulsars provide the strongest validation, with GR
verified to 0.2% precision at screening factors exceeding 29,500×.

### 4.4 Precision GR Tests: Explained, Not Violated

A critical question for any modified gravity theory is: why do precision
tests of General Relativity show no deviation? Temporal Topology provides
the answer: the suppression of Temporal Shear (vanishing field gradient)
reduces the screened PPN scalar charge $\alpha_{\rm PPN}^{\rm eff}$ where the ambient landscape is steep. The five most precise
tests of GR all occur in regimes where screening is operative:

| Test | Observable | Precision | Geometric $S_A$ ($R_T/R$) | Status |
| --- | --- | --- | --- | --- |
| Lunar Laser Ranging | Nordtvedt effect | $10^{-13}$ | 0.56× | GR-consistent in the orbital ($\mathcal S_\Sigma \sim 10^{-14}$) projection; synodic residual channel analyzed in Paper 17 |
| Cassini Conjunction | Shapiro delay | $2 \times 10^{-5}$ | 0.42× | Satisfied via kinetic source-charge suppression ($y_{\rm tb}(1.6\,R_\odot)=2.16\times10^{-8}$, $|\gamma_{\rm PPN}-1|=4.32\times10^{-8}\ll2.3\times10^{-5}$) |
| MESSENGER | Perihelion precession | $3 \times 10^{-4}$ | 0.65× | GR-consistent in the orbital ($\mathcal S_\Sigma \ll 10^{-8}$) projection |
| Hulse-Taylor Pulsar | GW emission | 0.2% | 29,500× | Completely screened |
| Double Pulsar | 7 PPN tests | 0.05% | 26,400× | Completely screened |

Table 4.2: Precision GR tests and their screening factors. All tests
are consistent with GR; TEP explains this via environment-dependent
screening rather than requiring the scalar sector to be absent.

The Hulse-Taylor binary pulsar is particularly significant. Its orbital
decay matches the GR prediction for gravitational wave emission to 0.2%
precision—a result honored with the 1993 Nobel Prize (Taylor & Weisberg
1982). At a density of $\sim 10^{14}$ g/cm³, the screening factor reaches
29,500×. The scalar field contributes less than 0.003% to the orbital
dynamics. This is not a violation of GR tests; it is a requirement of
consistent screening.

### 4.5 Earth as the Calibration Anchor

The screening hierarchy reveals why Earth is the natural calibration
point—and why GNSS clocks can detect what neutron stars cannot.

At $\rho \approx 5.5$ g/cm³, Earth sits just below the critical proximity
scale, proxied by $\rho_T \approx 20$ g/cm³. The screening factor is 0.66×,
meaning the saturation radius (4,200 km) is comparable to the physical radius
(6,371 km). Earth occupies the narrow window where:

**The scalar field is not screened:** Unlike neutron stars
(S = 26,800×), Earth's density is low enough that the soliton extends to
observable scales.

**The transition is close:** Unlike gas giants ($S \approx
0.3$), where the saturation core sits deep inside the object and well
below the channel-projection boundary, Earth's $S \approx 0.66$ places
it near the transition, where clock-comparison ($S_A$) channels retain
a measurable response.

GNSS satellites orbit at $\sim$20,000 km altitude, sampling clock transport
through the exterior field region associated with the terrestrial Temporal Topology
profile. The measured covariance length $L_c \approx 4200$ km is interpreted as a
characteristic correlation/saturation scale of the Earth-sourced field, not as the
orbital radius of the satellite network. The observed clock correlation
length is therefore not an arbitrary parameter—it is the
characteristic scale where the scalar field's gradient becomes steep enough
to produce measurable timing correlations.

Within this phenomenology, the same saturation scale is expected to govern
both terrestrial clock correlations and compact-object structure. The
terrestrial scale provides an empirical calibration point, and the RBH-1
crossover provides an astrophysical consistency test under the same
parameter choice.

### 4.6 The Critical Proximity Scale

The empirical data suggest a critical proximity scale, observationally
proxied by the density $\rho_T \approx 20$ g/cm³,
corresponding to approximately 4× Earth's mean density. This value has
coincidence-level significance: it lies near the onset of electron degeneracy, where
the equation of state transitions from thermal to quantum pressure support.
This coincidence motivates treating the 10–30 g/cm³ regime as a useful condensed-matter reference scale, while the mechanism setting $\rho_T$ remains an open derivation target.

**Principle:**

$\rho_T \approx 20$ g/cm³ is a saturation scale of the temporal-field topology, not a local on/off switch. Systems approach the GR-like limit when the observable shear/source-charge sector is suppressed, $\mathcal{S}_\Sigma \to 0$, which depends on source structure, environment, and boundary conditions, not $\rho > \rho_T$ alone. The empirical screening hierarchy $S \propto \rho^{0.334}$ (§4.2) is a smooth power-law across 15 orders of magnitude in density, not a step function at $\rho_T$.

#### Box 4.1: Summary — Three Cross-Regime Consistency Tests

The screening analysis provides three cross-regime consistency tests of
the TEP framework under a single $\rho_T$:

**The RBH-1 Crossover:** The saturation radius is
consistent with the Schwarzschild radius at $M \approx 10^7
M_{\odot}$ within combined uncertainties ($R_T/R_S \approx 1.3$). This
provides a tightly constrained cross-regime consistency check under
the fixed $\rho_T$ calibration.

**The Screening Exponent:** The empirical scaling $S
\propto \rho^{\beta_{\rm scr}}$ with $\beta_{\rm scr} = 0.334$ is fitted to the 11 dense objects
($\rho > \rho_T$) spanning ~12 orders of magnitude in density, with $R^2 = 0.99995$. The exponent 1/3 is a
direct consequence of the $M^{1/3}$ saturation scaling. The high $R^2$
partly reflects the definitional structure of $S$ (the $\rho^{1/3}$
dependence is algebraically favoured once $R_T(M)$ enters the
construction); the substantive content is the absence of large deviations,
not the correlation coefficient — see Appendix B.4.

**GR Test Consistency:** All five precision tests of
General Relativity are consistent with screening factors that suppress
scalar contributions below current measurement limits.

The TEP framework does not violate established physics. It extends the
metric structure to include a conformal sector that becomes observable
only in specific density regimes—such as those probed by GNSS atomic
clocks and cosmological soliton-scale systems.

## 5. Physical Constraints on the Proximity Scale

A natural question arises: why does the saturation scale take the
observational proxy value \(\rho_T \approx 20\) g/cm³? This section argues
that the underlying proximity scale is not treated as an arbitrary fit
parameter: it is empirically calibrated, while additional coincidence-level
reference scales and open derivation targets are recorded for completeness.
The GNSS coherence length \(L_c\) is a derived quantity; the fundamental
parameter is the proximity scale itself, of which density is an observable
proxy. In the canonical TEP framework, the full screening state of any
system is determined by the environmental operator \(S_\Sigma(E)\);
\(\rho_T\) is the density projection of that operator, not the complete
screening description.

### 5.1 \(\rho_T\) as the Conjectured Fundamental Parameter, \(L_c\) as Derived

The saturation scaling law \(R_T \propto M^{1/3}\)—where a soliton interpretation is one candidate microscopic realization—implies a constant
characteristic density:

\begin{equation} \label{eq:rho_t_constant} \rho_T = \frac{M}{\frac{4}{3}\pi R_T^3} = \text{constant}. \end{equation}

For any object of mass \(M\), the saturation radius is determined by:

\begin{equation} \label{eq:rt_mass} R_T(M) = \left(\frac{3M}{4\pi \rho_T}\right)^{1/3}. \end{equation}

The terrestrial coherence length \(L_c \approx 4200\) km is therefore not an
independently tuned galaxy-scale parameter; operationally, it is the observed
GNSS covariance scale used to calibrate the projected Earth-scale realization
of \(R_T(M_\oplus)\). This relation is inverted in Section 2 to calibrate
\(\rho_T\) from the measured \(L_c\) for Earth:

\begin{equation} \label{eq:lc_earth} L_{c,\oplus} = \left(\frac{3 M_\oplus}{4\pi \rho_T}\right)^{1/3} \approx
4200 \text{ km} \quad \text{for } \rho_T \approx 19\text{–}20 \text{
g/cm}^3. \end{equation}

The question thus reduces to: what physical considerations constrain \(\rho_T\)?

### 5.2 The Dimensional Identity Bridge

A natural concern is whether the proximity of $\rho_T \approx 20$ g/cm³ to the density of terrestrial condensed matter implies a composition-dependent coupling that would violate the Weak Equivalence Principle. Under universal conformal coupling, the scalar sector is sourced by the bulk trace $T = -\rho$, independent of microscopic composition. The saturation scale $\rho_T$ must therefore be a property of the vacuum potential $V(\phi)$, not an emergent property of the local matter.

The electron Fermi wavelength does not cause the scalar field to saturate. If it did, the field would require a direct, particle-specific coupling to the electron number density $n_e$, making the field exert a different force on iron than on hydrogen at the same bulk density. This would introduce a gross WEP violation, already constrained by experiments such as the MICROSCOPE satellite.

Instead, $\rho_T$ is treated as a fundamental constant of the scalar
sector — the geometric reference density of the saturation-radius law —
rather than an emergent property of the local matter. The recorded
coincidence is, in natural units,

\begin{equation} \label{eq:vacuum_scale} \rho_T \approx \alpha^{4/3} m_e^4 \quad \text{(a density coincidence, not a potential-scale identification)}. \end{equation}

Because this is a property of the scalar sector itself, the dimensional identity does not introduce composition-dependent coupling; WEP protection then follows from the universal matter coupling and the screened effective scalar charge. The field universally couples to bulk $\rho$ and begins its continuous screening transition when local $\rho$ approaches this $\rho_T$. The potential sector's own normalization is a separate quantity — the quartic coupling $\lambda$ (equivalently $\Lambda_V = \lambda^{1/4}M_{\rm Pl} \approx 1.3\times10^{2}$ GeV on the Cassini-compatible branch) — and is not to be identified with $\rho_T$ (Appendix C.3).

The bulk density of Thomas-Fermi condensed matter is governed by Coulomb packing. The volume per atom scales with the Bohr radius $a_0 = (\alpha m_e)^{-1}$. In the Thomas-Fermi model, the effective atomic radius is $R_{\rm TF} \approx a_0 Z^{-1/3}$, giving a volume $V \approx \frac{4\pi}{3} a_0^3 Z^{-1}$ per atom. With nucleon mass $M \approx A m_p$:

\begin{equation} \label{eq:condensed_density_corrected} \rho_{\rm CM} \approx \frac{3}{4\pi} A Z \, m_p m_e^3 \alpha^3. \end{equation}

For representative rock-forming elements ($A/Z \approx 2$), the product $AZ$ ranges from $\sim 10^2$ (oxygen, $Z=8$) to $\sim 10^3$ (iron, $Z=26$). Evaluating with $m_p/m_e \approx 1836$ and $\alpha \approx 1/137.036$ gives $\rho_{\rm CM} \sim 10^2$–$10^3$ g/cm$^3$, one to two orders of magnitude above the calibrated $\rho_T \approx 20$ g/cm$^3$. The Thomas-Fermi Coulomb-packing estimate therefore does not independently converge on the saturation scale; it situates ordinary condensed matter in a denser regime. The relevant condensed-matter reference that lies near $\rho_T$ is the electron-degeneracy onset threshold (Section 5.3).

### 5.3 The Electron Degeneracy Threshold

Coincidence-level significance attaches to \(\rho_T \approx 20\) g/cm³:
this density lies in the condensed-matter regime where quantum pressure begins
to become dynamically relevant. The crossover from thermal and electrostatic
support to degeneracy-dominated support is gradual, with full degeneracy
domination occurring only at much higher (white-dwarf) densities.

| Density Regime | Dominant Physics | Examples |
| --- | --- | --- |
| \(\rho < 1\) g/cm³ | Gas pressure, thermal | Planets, main-sequence stars |
| \(\rho \sim 1\text{–}10\) g/cm³ | Coulomb/thermal dominated solids and liquids | Rocky planets |
| \(\rho \sim 10\text{–}10^2\) g/cm³ | Degeneracy onset (crossover) | Earth's core, brown dwarfs |
| \(\rho \gtrsim 10^4\) g/cm³ | Electron degeneracy dominated | White dwarfs |

Table 5.1: Equation of state transitions by density. The saturation
density \(\rho_T \approx 20\) g/cm³ lies near the onset of electron
degeneracy.

This coincidence provides a useful condensed-matter reference scale for
where quantum pressure becomes dynamically relevant to the equation of state.
Under the universal conformal-coupling benchmark used throughout this corpus,
the scalar sector is sourced primarily by the non-relativistic mass-density
term, with pressure corrections entering only as a small relativistic
correction in the condensed-matter regime. Electron degeneracy therefore does
not, by itself, provide a mechanism that sets \(\rho_T\), but it motivates the
\(\sim 10\text{–}30\) g/cm³ window as a natural coincidence-level cross-check.

### 5.4 Open Derivation Target: From Potential Scale to Transition Width

Box 6.5 adopts the corpus's quartic completion, in which the amplitude
sector is normalized by the quartic coupling \(\lambda\) and \(\rho_T\)
enters as the geometric constant of the saturation-radius law
\(R_T(M)\). Once \(\rho_T\) and the corpus branch of \(\lambda\) are
specified, the field equation determines a smooth crossover between
Compton-unresolved and Compton-resolved regimes
(\(\lambda_c(\bar\rho)\) versus \(R_T(M)\)). Determining the detailed
width and shape of this
crossover as a function of ambient density—particularly in the terrestrial
transition regime \(\rho \sim 1\text{–}30\) g/cm³—requires an explicit numerical
profile solution at the specified corpus normalization. This remains a target for future work.

Many-body condensed-matter physics (including the onset of electron degeneracy)
governs the baryonic equation of state and provides useful reference densities,
but under universal coupling it does not by itself select the saturation scale
\(\rho_T\). A complete theoretical account must instead explain both the value of
\(\rho_T\) and the mapping between the geometric saturation radius \(R_T(M)\) and the
observable correlation length in clock networks.

While Section 5.3 contextualizes *ρ*<sub>T</sub> within macroscopic condensed-matter
physics, establishing its fundamental origin requires examining the scalar coupling
scale at the vacuum level.

### 5.5 Dimensional Analysis: The Scalar Coupling Scale

Expressed in natural units, the saturation density carries the
energy-density scale \(\rho_T^{1/4} \approx 96.4\) keV for
\(\rho_T = 20\) g/cm³. This is a fourth-root reading of a density, not the
scalar potential's normalization: the potential coefficient is the
quartic coupling \(\lambda = 7.5\times10^{-66}\), equivalently
\(\Lambda_V \equiv \lambda^{1/4}M_{\rm Pl} \approx 1.3\times10^{2}\) GeV
(Box 6.5, Appendix C; step 14). The kinetic completion carries its own
distinct scale, \(\Lambda_X = \sqrt{M_{\rm Pl}H_0} \approx 1.9\) meV,
which sets the Temporal-Shear threshold and plays no role in the
amplitude-sector density argument.

\begin{equation} \label{eq:alpha_density} \rho_T \approx \alpha^{4/3}\,\frac{m_e^4 c^3}{\hbar^3} \approx 22.4 \text{ g/cm}^3. \end{equation}

This correspondence is recorded as a numerical identity: the exponent \(4/3\)
(equivalently \(\rho_T^{1/4} \sim \alpha^{1/3} m_e c^2\)) is not derived within the
present corpus and must be treated as an open target for theoretical work, not
as an independent support.

### 5.6 Status: Phenomenological Constraint with Theoretical Foundation

The present analysis treats \(\rho_T \approx 20\) g/cm³ as an empirically calibrated
parameter, while recording additional coincidence-level reference scales and open
derivation targets:

**Condensed-matter EOS coincidence:** \(\rho_T\) lies near
the density scale where electron degeneracy becomes dynamically relevant to the equation of state

**Numerical identity:** \(\rho_T \approx \alpha^{4/3} m_e^4 c^3/\hbar^3\),
with the \(4/3\) exponent an explicit open target

**Open derivation:** a first-principles account must explain why the saturation scale
\(\rho_T\) and the operational identification of \(L_c\) as the projected covariance scale
associated with \(R_T(M_\oplus)\) govern the observable
clock-correlation envelope

Its use in quantum or topological extensions as a microscopic cutoff or
defect-overlap scale is a conjectural extrapolation of the macroscopic
saturation proxy, not an independent consequence of the present paper.

The key point for the present manuscript is that \(\rho_T\) is not treated as a free
fit parameter: it is GNSS-anchored by clock correlations (Section 2) and then
carried, without retuning, into the cross-scale tests that follow.

#### Box 5.1: Summary — The Saturation Scale: One Calibration, Two Coincidences, One Open Derivation

Operationally, \(\rho_T\) is calibrated from the measured GNSS coherence length
\(L_c\) by treating \(L_c\) as the observed projected covariance scale associated with
\(R_T(M_\oplus)\), and then inverting
\(\rho_T = 3M_\oplus/(4\pi L_c^3)\). The framework’s conjecture—left as an open
derivation target—is that \(\rho_T\) is the fundamental saturation parameter and
that \(L_c\) follows as a derived geometric scale once \(\rho_T\) is fixed.
Two additional numerical correspondences are noted,
but they remain coincidence-level until a first-principles derivation exists:

- Condensed-matter EOS coincidence (electron degeneracy onset)

- Numerical identity \(\rho_T \approx \alpha^{4/3} m_e^4 c^3/\hbar^3\) (exponent underived)

In the present phenomenological calibration, the terrestrial coherence length
is treated as the observed projected covariance realization of the Earth-scale
\(R_T(M_\oplus)\) geometry; deriving this mapping from a specified scalar
potential remains the open target. Both the condensed-matter coincidence and
the numerical identity are contingent on the transfer-sketch \(\mathcal{O}(1)\)
prefactor being close to unity; a prefactor of \(\sim\)2 would shift \(\rho_T\) to
\(\sim\)2.5 g/cm³, evaporating the coincidences. Operationally, GNSS observations
provide the calibration by which \(\rho_T\) is inferred in the most accessible
laboratory: Earth's gravitational field.

## 6. The Universal Density Constraint

A key test is whether the Temporal Topology saturation radius \(R_T\) yields quantitative
expectations for otherwise free scales. Under a soliton interpretation, RBH-1 would correspond to a
gravitational soliton; however, the saturation radius itself is the primary observable, constrained by
the universal density \(\rho_T\) rather than introduced as an adjustable parameter.

### The Universal Density Hypothesis

The Temporal Topology framework tests the empirical claim that a candidate
saturation scale $\rho_T \approx 20$ g/cm³ governs
compact-object structure across mass scales. A soliton interpretation provides one candidate
microscopic realization of this saturation physics. This hypothesis is testable
via two independent windows:

**Planetary Scale:** The Earth-scale Temporal Topology parameter
$R_T(M_\oplus)$ is calibrated through the observed GNSS projected
covariance length ($L_c \approx 4200$ km), serving as the empirical anchor.

**Cosmological Scale:** The extrapolated Temporal Topology saturation radius for
RBH-1 ($M \approx 10^7 M_\odot$) can be compared to the Schwarzschild
diameter, placing the object near the crossover mass where horizon and
soliton interpretations are maximally degenerate.

The central question is whether a single density parameter, validated
cosmologically, remains universal at galactic scales.

### The Saturation Scale as Fundamental Parameter

If the scalar sector saturates at a critical energy density $\rho_T$,
dimensional analysis requires a universal mass–radius relation:

\begin{equation} \label{eq:universal_scaling} R_T(M) = \left( \frac{3M}{4\pi \rho_T} \right)^{1/3}. \end{equation}

#### Box 6.1: The Theoretical Priority of $\rho_T$

A potential misreading of this framework is that "GPS clock noise
determines black hole physics." This inverts the logical and historical
order of the analysis.

**Calibration (Terrestrial):** The GNSS analysis
(Smawfield 2025b) serves as an empirical anchor. The
detection of a correlation length $L_c \approx 4200$ km in atomic
clock data provides an empirical estimate of the saturation scale, conditional on treating $L_c$ as the projected covariance scale associated with $R_T(M_\oplus)$.

**Application (Cosmological):** The resulting $M^{1/3}$
scaling is therefore not an extrapolation of clock noise, but a test
of whether this terrestrially-validated constant
remains universal at galactic scales.

The alignment of the RBH-1 crossover scale is thus an independent
consistency check, not a fitted result.

#### Box 6.2: Null Hypothesis and Look-Elsewhere Effect

A potential concern is that apparent cross-scale correspondences could
arise from post-hoc alignment rather than predictive structure. To
address this possibility, the prior probability of simultaneous
agreement across multiple reference scales is estimated under an
explicit null hypothesis.

Null hypothesis: assume $\rho_T$ is drawn uniformly from the range
$10^{-3}$–$10^{6}$ g/cm³ (spanning interstellar gas to nuclear density).
The GNSS constraint ($R_T(M_\oplus)$ within 50% of 4200 km) permits
$\rho_T$ in the range $\sim$6–160 g/cm³ ($\approx$1.4 dex). The RBH-1
constraint ($R_T(10^7 M_\odot)$ within 50% of $2R_S$) permits
$\rho_T$ in the range $\sim$7–180 g/cm³ ($\approx$1.4 dex). The
overlap is $\sim$7–160 g/cm³ ($\approx$1.4 dex). Under a uniform
logarithmic prior spanning 9 dex, the probability of a randomly
drawn $\rho_T$ satisfying both constraints simultaneously is
$P \approx 1.4/9 \approx 0.15$ ($\sim$15%). While not overwhelmingly
small, this probability is reduced further when the
condensed-matter coincidence (Section 5.3) is included as a third
independent cross-check. Given the GNSS calibration and the
condensed-matter coincidence noted in Section 5, the RBH-1
correspondence provides an independent consistency check of the same underlying
parameter. The convergence of two physically distinct systems—geodesy and cosmology—on the
same density scale across 15 orders of magnitude is not readily explained by independent tuning,
but the universality hypothesis remains
falsifiable: any future measurement that robustly requires a substantially
different $\rho_T$ would exclude this framework.

Provenance scope: the terrestrial GNSS calibration is established across Papers 1–3 and 14 (Smawfield 2025b,c,d), while out-of-ecosystem verification on non-GNSS optical and two-way satellite links provides the external metrological check. The astrophysical and cosmological inputs referenced here are derived from external published catalogs (SPARC, Lelli et al. 2016; van Dokkum et al. 2025).

The GNSS-derived value $\rho_T \approx 19\text{–}20$ g/cm³
(Section 2) lies in the same condensed-matter density regime where electron degeneracy becomes dynamically relevant to the equation of state (Section 5).
The central question is whether this same density scale holds for RBH-1.

#### Box 6.3: Robustness of the GNSS Calibration (Systematics Check)

The terrestrial calibration length $L_c \approx 4200$ km — the GNSS
instrument's measured realization of the canonical correlation scale
$\lambda_T$ (Paper 0 §7) — is an
extraordinary claim requiring careful exclusion of geodetic
systematics. The following tests support its physical origin (full
methodology in Smawfield 2025b,c,d):

**Multi-Center Verification:** The correlation
structure persists across independent clock solutions from CODE,
ESA, and IGS (1999–2024), disfavoring software-specific
processing artifacts (e.g., Bernese vs. GIPSY). The recovered $L_c$
spans $\sim$3,330–4,549 km across centers (a $\sim$23% spread).

**Null Tests:** (i) Randomizing satellite epochs
destroys the correlation ($r^2 < 0.01$, $N = 1000$ shuffles); (ii)
shuffling clock residuals within each satellite eliminates the
spatial structure; (iii) replacing real data with white noise yields
no coherent scale. All three nulls are satisfied at $>5\sigma$.

**Scale Separation:** The 4200 km scale is distinct
from tropospheric correlation lengths (~100–500 km) and orbital
period harmonics (half-sidereal, ~12 hr). Power spectral analysis
shows the 4200 km feature as a persistent background covariance
floor in the GPS-only clean baseline, with no counterpart in
ionospheric or tropospheric delay products; multi-constellation validation shows consistent signatures.

**Dataset:** IGS final clock products (CLK files),
30-second sampling, 1999–2024. Preprocessing: removal of
satellite/receiver clock offsets, relativistic corrections (Sagnac,
gravitational redshift), and reference frame alignment to ITRF2020.

*Limitation:* The present analysis treats $L_c$ as an empirical
calibration parameter. A complete derivation from first
principles—linking the empirical $L_c$ to the geometric saturation radius $R_T(M)$ and to a candidate completion’s field-profile parameters—remains a target for future theoretical work.

#### Box 6.4: Derivation of the Scaling Law

The $R \propto M^{1/3}$ scaling is not an ad hoc ansatz but a direct
consequence of saturation in the scalar sector. Consider a scalar field
$\phi$ coupled to matter density $\rho$ with a potential $V(\phi)$ that
enforces a maximum gradient or energy density.

In the dense limit (compact objects), the scalar field profile
saturates to a core of constant effective energy density $\rho_T$. For a
self-gravitating configuration of total mass $M$, the volume of this
saturated core is constrained by mass conservation:

\begin{equation} \label{eq:mass_conservation} M \approx \frac{4}{3}\pi R_T^3 \rho_T. \end{equation}

Solving for the radius yields the characteristic scaling:

\begin{equation} \label{eq:scaling_law} R_T \propto \left(\frac{M}{\rho_T}\right)^{1/3}. \end{equation}

This relation describes the boundary of the saturated "soliton" region.
Outside this radius, the field decays, recovering Newtonian gravity
(Temporal Topology screening). Inside, the field is phase-locked,
modifying the effective metric (proper time). The "Universal Scaling" is
thus simply the statement that the scalar potential $V(\phi)$ has a maximum capacity to
support scalar gradients before saturating at $\rho_T$.

#### Box 6.5: Soliton Formation from the Canonical Action

To move beyond phenomenology, a candidate route from the canonical
Temporal Topology action (Paper 0) to soliton-like saturation is sketched. The action
above contains a canonical kinetic term $-\frac{1}{2}(\partial\phi)^2$
and a candidate saturation potential $V(\phi)$ that would prevent gradient divergence.
The interplay between these terms can generate soliton-like saturated configurations with
characteristic radius $R_T \propto M^{1/3}$.

**1. Field Equation and Equilibrium**

For a static, spherically symmetric configuration, the scalar field
equation is:

\begin{equation} \label{eq:field_equation} \nabla^2 \phi = V'(\phi) + \alpha(\phi)\rho \end{equation}

where $\alpha(\phi) \equiv d\ln A/d\phi$ and $\rho$ is the ambient
matter density. In the dense limit, the effective potential
$V_{\rm eff}(\phi; \rho) = V(\phi) + [A(\phi)-1]\rho$ develops a
minimum at $\phi_{\rm min}(\rho)$ with effective mass
$m_{\rm eff}^2(\rho) = V_{\rm eff}''(\phi_{\rm min})$.

**2. Saturation Mechanism**

Under the corpus coupling $\beta_A = -1$ the matter term enters with
negative slope ($\alpha\rho = -\rho/M_{\rm Pl}$), so a minimum of
$V_{\rm eff}$ exists only on the confining branch $V' > 0$. Writing
$u = \phi/M_{\rm Pl}$, the corpus's own weak-field branch is the
quartic $V(u) = (\lambda/4)\,M_{\rm Pl}^4\,u^4$ — the member of the
confining family $V \propto u^m$ selected by the corpus amplitude
law $S_A \propto \rho^{1/3} = \rho^{1/(m-1)}$, i.e. $m = 4$
(Appendix C.2). The equilibrium condition $V_{\rm eff}' = 0$ reads
$\lambda M_{\rm Pl}^4 u_{\rm min}^{3} = \rho\,e^{-u_{\rm min}}$, giving
$u_{\rm min} \approx (\rho/\lambda M_{\rm Pl}^4)^{1/3}$ for
$u_{\rm min} \ll 1$: the field is elevated inside matter and
approaches zero in the ambient, in the direction required by the
corpus sign convention. (The inverse-power branch
$V \propto \phi^{-n}$ is inadmissible under $\beta_A = -1$: both
slopes are then negative and no minimum exists; its implied
$\phi_{\rm min} \propto \rho^{-1/(n+1)}$ would in any case run
opposite to the required direction.) The scalar energy density in a
saturated core is:

\begin{equation} \label{eq:effective_density} \rho_{\rm eff} \sim V(u_{\rm min}) + \frac{1}{2}M_{\rm Pl}^2(\nabla u)^2
\simeq \tfrac{1}{4}\rho\,u_{\rm min} \ll \rho. \end{equation}

The saturated configuration is therefore a field-geometry statement,
not a competing mass density: at the corpus normalization
$\lambda = 7.5\times10^{-66}$ (the scalar self-coupling, Appendix C)
the core energy density is
$\sim 3\times10^{-8}$ g/cm³ at terrestrial densities, and the
equilibrium $u_{\rm min}(\rho_T) \approx 6.9\times10^{-9}$ remains
perturbative. What saturates at $\rho_T$ is the clock-amplitude
response $S_A = \min[1, u_{\rm min}(\bar\rho)/u_{\rm min}(\rho_T)]$
through the geometric radius $R_T$; the order-unity self-quenching
crossing $u_{\rm min} \sim 1$ (where $e^{-u}$ suppresses the
conformal source) lies at $\rho_{\rm sat} \sim \lambda M_{\rm Pl}^4
\approx 6\times10^{25}$ g/cm³, inside the temporal-well domain of
Paper 28 (step 14).

**3. Temporal Topology Screening**

The effective mass
$m_{\rm eff}^2(\rho) = V_{\rm eff}''(u_{\rm min})/M_{\rm Pl}^2 =
[3\lambda M_{\rm Pl}^4\,u_{\rm min}^{2} + \rho\,e^{-u_{\rm min}}]/M_{\rm Pl}^2$
grows with density as $m_{\rm eff} \propto \lambda^{1/6}\rho^{1/3}$,
so the Compton wavelength $\lambda_C = 1/m_{\rm eff}$ contracts
through denser interiors: a body is Compton-resolved once
$\lambda_c(\bar\rho) \lesssim R_T(M)$, the crossover that for Earth
sits at $\sim 4$ g/cm³ (step 14). Inside such a body the field rests
near its local equilibrium $u_{\rm min}(\rho)$ and the spatial
variation is confined to a boundary layer of width $\sim\lambda_C$,
where the kinetic response $P_{,X}$ controls the gradient amplitude.
This creates the characteristic flattening of
the scalar field spatial profile—Temporal Topology screening—that
reconciles precision local tests with cosmological dynamics.
In the saturation regime ($\rho \sim \rho_T$), the Temporal Topology saturation radius
scales as $R_T = (3M/4\pi\rho_T)^{1/3}$, yielding the
observed $M^{1/3}$ dependence. At still higher densities — where the
equilibrium approaches order unity ($\rho \sim \rho_{\rm sat} \approx
6\times10^{25}$ g/cm³) — the effective source slope
$\rho\,A_{,u} = -\rho\,e^{-u_{\rm min}}$ is exponentially
suppressed, so the scalar charge of deep-interior matter quenches as
well: that is the temporal-well regime, not the terrestrial one.

**4. Coupling and Stability**

Matter couples to the Jordan frame metric $\tilde{g}_{\mu\nu} =
A^2(\phi) g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$ with $A(\phi) = \exp(\beta_A \phi/M_{\text{Pl}})$.
Stability is ensured because the canonical kinetic term has the correct
sign for a physical scalar field, and the theory satisfies the Null
Energy Condition for physically realizing solutions.

### Testing the Density Constraint at Cosmological Scales

The saturation scale $\rho_T$ determined from terrestrial clocks is now
tested against RBH-1. Using the revised best-fit mass estimate of $M \approx
2 \times 10^7 M_{\odot}$ (van Dokkum et al. 2025), the predicted Temporal Topology
saturation radius is:

\begin{equation} \label{eq:rbh_radius} R_T = \left( \frac{3M}{4\pi \rho_T} \right)^{1/3} \approx 7.8
\times 10^7 \text{ km}. \end{equation}

The predicted Temporal Topology saturation radius can be compared directly to the Schwarzschild
radius for this mass:

Schwarzschild radius: $R_{\rm S} = 2GM/c^2 \approx 5.9
\times 10^7$ km

Predicted Temporal Topology saturation radius: $R_T =
(3M/4\pi\rho_T)^{1/3} \approx 7.8 \times 10^7$ km

Ratio: $R_T/R_{\rm S} \approx 1.3$

This order-unity correspondence ($R_T \approx 1.3 R_{\rm S}$) arises
because $\rho_T$ lies near the horizon-formation density
threshold in the underlying scalar sector. Within the combined uncertainties
from $M$ and $\rho_T$, which propagate to $\sim 25\%$ uncertainty in $R_T$, the two scales remain consistent at the factor-of-few level. This
places RBH-1 near a crossover regime where horizon and saturated-soliton
interpretations can be observationally degenerate.

This geometric match is the primary prediction of the scaling law. While the
velocity discontinuity ($\Delta v \sim 650$ km/s) provides a secondary
constraint on the *amplitude* of the screening (see Smawfield 2025h,
Paper 7), the *scale* of the object is set fundamentally by the
saturation scale $\rho_T$. The correspondence suggests that what is
conventionally identified as the event horizon scale in GR may correspond to
the saturation boundary of the scalar field in the TEP framework.

In the TEP interpretation, what is conventionally called a "black hole" at
this mass scale is modeled as a saturated soliton core, i.e., a region where
the conformal time-field approaches a maximum gradient set by $\rho_T$. In
this phenomenology, the characteristic radius is set by the saturation
scale rather than by horizon formation. The observed central dimming is
attributed to extreme time dilation (strong redshifting) rather than causal
disconnection.

A further structural implication follows from the different mass scalings.
The soliton relation predicts $R_T \propto M^{1/3}$, whereas the
Schwarzschild radius scales as $R_{\rm S} \propto M$. These opposing
scalings intersect at a unique crossover mass $M_{\times}$ where the two
radii coincide (see Figure D.1). Empirically, the terrestrial calibration
places this crossover near $M_{\times} \approx 10^7\,M_{\odot}$—precisely
where RBH-1 resides. Objects near $M_{\times}$ are expected to be maximally
degenerate between horizon and soliton interpretations, making RBH-1 an
unusually diagnostic system.

The identification is thereby restricted to the crossover neighbourhood:
it is a statement about $M \approx M_{\times}$, not a general
reinterpretation of compact objects. The directional implication is
explicit. For $M \ll M_{\times}$ the weak-field construction gives
$R_T \gg R_{\rm S}$ — a $10\,M_\odot$ object would carry $R_T$ of order
$10^{6}$ km against $R_{\rm S} \approx 30$ km — so soliton-realized
compact objects at stellar masses would appear as extended structures and
are incompatible with observed merger phenomenology. For $M \gg M_{\times}$
the saturation radius retreats far inside the horizon (Appendix E computes
$S \sim 0.03$ for M87*), and the identification ceases to be degenerate.
Two resolutions are admitted, and the corpus requires one of them: either
soliton-realized compact objects exist only in the crossover band, or the
weak-field $R_T(M)$ construction fails at horizon compactness and must be
replaced by the relativistic strong-field treatment of Paper 28, within
which the $M^{1/3}$ ansatz is not guaranteed to survive. The crossover
statement is therefore conditional on weak-field validity at
horizon-compactness — an assumption stated rather than assumed — while
the degeneracy at $M_{\times}$ itself remains a legitimate and diagnostic
coincidence that feeds the RBH-1 analysis of Paper 7.

### Status of the Scaling Law

The relation $R \propto M^{1/3}$ is a density-limited scaling expected for
compact, self-bound field configurations whose cores approach a finite
saturation scale. Such behavior is familiar in non-topological solitons
and bosonic compact objects (Coleman 1985; Seidel & Suen 1991; Gleiser
1994; Hui et al. 2017).

The central claim is not that terrestrial clocks "predict" black hole sizes,
but that a fundamental density constant $\rho_T$ appears across scales. The
saturation scale is determined empirically from Earth—the most accessible
high-precision gravitational laboratory—and then tested for consistency at
cosmological scales. The present analysis should be read as a
phenomenological identification of an unexpected regularity: the same
density constraint that governs clock correlations in Earth's gravitational
field also governs the characteristic size of a $10^7 M_\odot$ compact
object.

### Dependency Structure and Constraint Isolation

A critical requirement for the robustness of this framework is to
demonstrate that the apparent convergence of $\rho_T$ is not a result of
circular reasoning. Specifically, it must be determined whether the
"independent" tests are truly distinct or if they all implicitly rely on the
same prior assumptions.

To resolve this, a constraint-isolation analysis is performed, isolating
each constraint to determine what value of $\rho_T$ it yields without input
from the others.

#### Test 1: Condensed-Matter Density Coincidence Isolation (Dropping GNSS)

If all GNSS data are discarded, condensed-matter equation-of-state transition scales
(Section 5.3) provide a coincidence-level density estimate.

Result: Condensed-matter transition considerations yield densities in the ~10–30
g/cm³ range, broadly consistent with the GNSS-derived value.

#### Test 2: Terrestrial Isolation (External Calibration Check)

If the condensed-matter coincidence is discarded and reliance is placed *solely* on
the observed GNSS correlation length $L_c \approx 4200$ km for Earth's mass
($M_\oplus$), what density is derived?

\begin{equation} \label{eq:gnss_rho} \rho_T(\text{GNSS}) = \frac{3 M_\oplus}{4\pi L_c^3} \approx 20 \pm 7
\text{ g/cm}^3 \; (\text{inter-center}), \quad \in [8,\,70] \text{ g/cm}^3 \; (\text{epoch/window}). \end{equation}

Result: Terrestrial data independently recovers the same
density scale, consistent with the condensed-matter coincidence estimate but derived from a system 51
orders of magnitude more massive. The quoted band carries the epoch/window
systematic established by the executed held-out discriminator (Appendix B.8);
the identification of $L_c$ with $R_T(M_\oplus)$ remains conditional on that
systematic not being resolved in favour of the lower MGEX scale.

#### Test 3: Cosmological Isolation (Dropping Local Constraints)

If both condensed-matter considerations and terrestrial inputs are discarded and the density required
to place the RBH-1 Temporal Topology saturation radius at the horizon scale (the crossover
condition) is computed, the result is:

\begin{equation} \label{eq:rbh_rho} \rho_T(\text{RBH-1}) = \frac{3 M_{\text{RBH-1}}}{4\pi (1.3 R_S)^3} \approx
15\text{–}30 \text{ g/cm}^3. \end{equation}

Result: The cosmological constraint points to the same
narrow density window.

#### Conclusion on Circularity

The convergence is not a trivial post-hoc fit. Multiple
physically distinct systems—the Earth (geodesy)
and galactic black holes (cosmology)—are mutually
consistent with the same saturation scale $\rho_T \approx 20$ g/cm³.
The framework remains falsifiable: any future measurement that
robustly requires a substantially different $\rho_T$ would exclude this
universal-density soliton model as formulated here.

## 7. Milky Way Test: Dark Matter Onset

### Expected Onset Radius

The Milky Way provides a local test of the galactic $M^{1/3}$ scaling law established in Section 3. Unlike external galaxies where distance uncertainties dominate, the Milky Way offers high-precision kinematic data. This section tests whether the SPARC-calibrated relation predicts the radius at which the mass discrepancy first becomes significant in the Milky Way.

Using the SPARC-calibrated normalization ($k \approx 7.9 \times 10^{-4}$ kpc/$M_\odot^{1/3}$) and the Milky Way's total baryonic mass estimated at $M_{\text{bar}} \approx 6 \times 10^{10} M_\odot$ (Bland-Hawthorn & Gerhard 2016), the expected dark-matter onset radius is:

\begin{equation} \label{eq:milky_way_dm} R_{\text{DM}} = k \cdot M_{\text{bar}}^{1/3} \approx 7.9 \times 10^{-4} \cdot (6 \times 10^{10})^{1/3} \text{ kpc} \approx 3.1 \text{ kpc}. \end{equation}

This radius marks the scale where the baryonic rotation curve (bulge + disk) first falls below the observed circular velocity, signaling the onset of the phantom mass effect. It is the same quantity $R_{\text{DM}}$ that is measured directly in the SPARC ensemble; the outer Keplerian decline at larger radii is a separate phenomenon governed by the full halo profile, not by the onset scaling alone.

### Gaia DR3 and Local Observations

Gaia Data Release 3 (Gaia Collaboration 2023) provides precise proper motions and radial velocities for millions of stars, enabling construction of the Milky Way rotation curve to greater distances than previously possible. Recent analyses (e.g., Jiao et al. 2023) also report evidence for a Keplerian-like decline beginning near $R \approx 19$ kpc; this outer feature traces the halo edge and is distinct from the inner onset scale tested here.

Key observational features relevant to the $R_{\text{DM}}$ prediction:

- **Baryonic dominance ($R \lesssim 3$ kpc):** The bulge and inner disk dominate; the rotation curve rises steeply and is well described by visible matter alone (Sofue 2020).

- **Dark-matter onset ($R \sim 3$–$5$ kpc):** Beyond the peak of the baryonic contribution, the observed circular velocity ($v_{\text{circ}} \approx 220$ km/s) stays flat while the baryonic model predicts a decline. This is the Milky Way's equivalent of the SPARC $R_{\text{DM}}$ — the radius where the mass discrepancy first becomes significant.

- **Flat regime ($5 \lesssim R \lesssim 15$ kpc):** The rotation curve remains approximately flat, indicating substantial phantom mass contribution.

- **Keplerian decline ($R \gtrsim 19$ kpc):** At large radii the enclosed mass growth slows and $v_{\text{circ}}$ begins to decline. This outer feature is not predicted by the onset scaling $R_{\text{DM}} \propto M^{1/3}$.

### Uncertainty Analysis and Error Budget

| Parameter | Value | Uncertainty | Impact on $R_{\text{DM}}$ |
| --- | --- | --- | --- |
| **MW Baryonic Mass** | $6.0 \times 10^{10} M_\odot$ | $\pm 15\%$ ($0.9 \times 10^{10}$) | $\pm 5\%$ (scaling $M^{1/3}$) |
| **SPARC Constant ($k$)** | $7.9 \times 10^{-4}$ | $\pm 4\%$ (fit error) | $\pm 4\%$ |
| **Model Systematic** | - | $\pm 10\%$ (geometry) | $\pm 10\%$ |
| Total Expected $R_{\text{DM}}$ | 3.1 kpc | $\pm 0.4$ kpc ($\sim$14%) | - |
| **Observed Onset** | $\sim$3–5 kpc | $\pm 1$ kpc | - |

The expected value ($3.1 \pm 0.4$ kpc) and the observed dark-matter onset
scale ($\sim$3–5 kpc) are consistent within the combined uncertainty. Because
the SPARC-calibrated normalization $k$ is the sole empirical input to this
prediction, the Milky Way test is a SPARC-internal consistency check rather
than a GNSS cross-scale test, consistent with the Numerology Firewall
classification (Section 8).

### Comparison with Dark Matter Halo Models

#### NFW Halo vs. TEP Soliton

- **NFW Halo (CDM):** In standard CDM, the inner rotation curve profile depends on the halo concentration $c$ and virial mass $M_{\text{vir}}$, both of which are free parameters. The radius where baryons cease to dominate is not predicted a priori; it emerges from the halo assembly history and can vary significantly between cosmological simulations.

- **TEP Soliton:** The $M^{1/3}$ scaling yields the onset radius directly from the baryonic mass, with no Milky Way-specific tuning. The normalization is fixed by the SPARC ensemble ($k \approx 7.9 \times 10^{-4}$ kpc/$M_\odot^{1/3}$) and gives $R_{\text{DM}} \approx 3$ kpc for the Milky Way. The agreement with the observed transition at $\sim$3–5 kpc supports the universality of the scaling law.

### Future Refinements

Ongoing Gaia data releases will improve the precision of the inner rotation curve, potentially constraining the exact baryonic-to-dark-matter transition radius to $\sim$10% accuracy. Combined with improved Milky Way mass estimates from satellite kinematics and gravitational lensing, this could provide a sub-10% test of the $M^{1/3}$ onset scaling in the Milky Way.

## 8. Discussion: The Nature of the Dark Sector

### Synthesis of Multi-Scale Evidence

The central result of this work is the identification of a single density
scale, $\rho_T \approx 20$ g/cm³, that organizes gravitational anomalies
across 18 orders of magnitude in mass (Earth to galaxy). This scale, originally calibrated
from terrestrial GNSS atomic clock correlations ($L_c \approx 4200$ km), is
consistent with:

The dark matter onset radius in spiral galaxies ($R_{\text{DM}} \propto
M^{1/3}$).

The dark-matter onset radius of the Milky Way ($R_{\text{DM}} \approx 3$ kpc, observed $\sim$3–5 kpc).

The screening hierarchy in binary pulsars vs. galaxies ($S \propto
\rho^{1/3}$).

The convergence of these mutually reinforcing constraints suggests that $\rho_T$ is
not merely a fitting parameter for a specific system, but a candidate
universal parameter of the effective description of the dark sector, within
the stated uncertainties.

#### Claim Hierarchy and Falsification Scope

The empirical content of this work can be read at three distinct levels.
This separation is critical for interpreting the consequences of future
measurements.

**Level 1 (Empirical regularities):** The existence of
approximate $M^{1/3}$ scaling features in rotation-curve onset
radii, and the density-ordered screening hierarchy summarized by $S
= R_T/R_{\text{phys}}$.

**Level 2 (Universal-density soliton model):** The
hypothesis that a single saturation scale $\rho_T$ organizes these
regularities via $R_T(M) = (3M/4\pi\rho_T)^{1/3}$, with
GNSS providing an empirical calibration (subject to independent
replication) and cross-regime consistency tests (SPARC ensemble,
Milky Way, RBH-1 crossover).

**Level 3 (TEP microphysics):** The full dynamical
realization in which temporal shear and a conformal time field
$A(\phi) = \exp(\beta_A\phi/M_{\text{Pl}})$ generate the effective soliton/screening
phenomenology. Failures at Level 2 primarily constrain the soliton
implementation and/or calibration mapping, without necessarily
excluding the broader TEP framework.

### Numerology Firewall: Which Agreements Were Predicted?

The elegance of cross-scale agreement is vulnerable to the accusation of
"curve-fitting across scales." This section provides a candid assessment
of which constraints were predicted *a priori* and which were
identified *post hoc*.

| Comparison | Before/after GNSS calibration? | Free parameters? | Independent? | Weight |
| --- | --- | --- | --- | --- |
| **Earth GNSS $L_c$** | Calibration anchor | yes/no (fit to correlation) | no — defines scale | Calibration |
| **SPARC $M^{1/3}$ scaling** | After | Yes (normalization $k$) | Medium | Moderate |
| **DF2/DF4 UDGs** | After | Yes (extrapolation) | Weak | Illustrative |
| **Milky Way inner mass-discrepancy onset** | After | No new normalization; uses SPARC-calibrated $k$ | Medium | Local consistency check |
| **Wide binaries (if tested)** | Before/after | Yes | Stronger (if replicated) | **High if replicated** |

#### Calibration Scope and Pre-Registered Discriminators for $\rho_T$

The density scale $\rho_T \approx 20$ g/cm³ is anchored on the terrestrial GNSS correlation length $L_c \approx 4{,}200$ km and tested across held-out mass regimes. Because the $\rho^{1/3}$ scaling is a structural consequence of the $R_T(M) = (3M/4\pi\rho_T)^{1/3}$ construction, discriminating the microscopic mechanism requires a pre-registered test in which the scale is predicted prior to analysis and recovered without refitting.

The per-constellation MGEX analysis (Paper 14 extension) provides a
concrete pre-registered discriminator: if GPS, Galileo, and GLONASS
individually recover $\sim$4,200 km while the combined product compresses
to the measured $\sim$1,900 km scale (pooled MGEX: 1,862 km), the MGEX shift is resolved as a multi-constellation datum-projection effect.
If per-constellation analysis also yields $\sim$1,900 km, the
$L_c \leftrightarrow R_T(M_\oplus)$ calibration shifts accordingly,
and the Level 1/2/3 claim hierarchy updates the anchor scale directly.

### Dark Matter as Phantom Mass

In the TEP framework, "dark matter" is reinterpreted not as a particle
species, but as "phantom mass"—an apparent excess inferred when a geometry
with temporal shear (two metrics) is analyzed under the assumption of
isochrony (single metric). The saturation scale $\rho_T$ represents the
scale at which the conformal factor $A(\phi) = \exp(\beta_A\phi/M_{\text{Pl}})$ enters a non-linear,
self-supporting phase (the soliton).

This explains the phenomenology of "cores" in dark matter halos. In standard
CDM simulations, halos tend to form cusps ($\rho \propto r^{-1}$), which are
often in tension with observations of flat cores in dwarf galaxies (the
core-cusp problem). In the soliton picture, the core is a distinct physical
state—a Bose-Einstein condensate-like configuration of the scalar field—that
naturally supports a flat density profile with a characteristic radius
$R_T$.

### Comparison with Alternative Frameworks

The TEP framework shares phenomenological features with both Modified
Newtonian Dynamics (MOND) and Fuzzy Dark Matter (FDM) but is distinguished
by its screening mechanism and cross-scale predictive power.

| Feature | MOND | Fuzzy DM (FDM) | TEP (Soliton) |
| --- | --- | --- | --- |
| **Physical Basis** | Modified Inertia/Gravity | Ultralight Boson ($m \sim 10^{-22}$ eV) | Conformal Time Field Saturation |
| **Fundamental Scale** | Acceleration $a_0 \approx 1.2 \times 10^{-10}$ m/s² | Particle Mass $m_\psi$ | Proximity Scale $\xi_T$ (obs. proxy $\rho_T \approx 20$ g/cm³) |
| **Galactic Scaling** | Fits RAR ($a_0$ is free parameter) | $M_c \propto M_h^{1/3}$ (Core-Halo) | $R \propto M^{1/3}$ (derived from $\rho_T$) |
| **Screening** | External Field Effect (EFE) | None (requires tuning) | Temporal Topology ($S \propto \rho^{1/3}$) |
| **Solar System** | Recovered via interpolation function | Unsuppressed (tension) | Screened ($S \sim 0.6$ for Earth, $>30k$ for Pulsars) |
| **Key Prediction** | Exact RAR | Solitonic Cores | GNSS Correlation Length + Screening Hierarchy |

#### TEP vs. MOND

MOND provides an excellent fit to galaxy rotation curves but offers no
explanation for the GNSS clock correlations. The critical distinction lies in
the triggering mechanism: MOND modifies kinematics at a specific
*acceleration* scale ($a_0 \approx 1.2 \times 10^{-10}$ m/s²), which
is a derived, environment-independent parameter that does not predict
where screening will occur in dense systems. By contrast, TEP is triggered
by a *density/proximity* scale ($\rho_T \approx 20$ g/cm³), enabling it
to explain anomalous dynamics in environments where MOND either makes no
prediction or fails—such as ultra-diffuse galaxies (where the low surface
brightness yields a different acceleration at fixed density) and the Milky
Way core (where the high baryonic density should suppress MOND-like effects
but does not in the data).

TEP recovers MOND-like phenomenology ($a_0 \sim G \Sigma_c$) as an
emergent property of the screening transition, while correctly predicting
the environment-dependent suppression required for precision GR tests.

#### TEP vs. FDM

Standard FDM predicts solitonic cores but fundamentally struggles to recover
the correct structural scales for both dwarf galaxies and the Milky Way simultaneously, requiring contradictory particle mass bounds. TEP naturally and uniquely links the core scale directly to the ambient saturation scale, generating the correct cross-scale hierarchy anchored directly by GNSS empirical measurements.

### Response-Coefficient Hierarchy

A natural criticism of the multi-channel TEP programme is that different
papers report different numerical coefficients. This is expected, and it is
a feature rather than a flaw. In the TEP framework, each measurement channel
carries an *observable response coefficient* \(\kappa_X\) that
absorbs stellar physics, environment, and projection geometry, not the bare
microscopic conformal coupling \(\beta_A\).

Paper 11 (TEP-H0) independently calibrates the weakly screened galactic-disk
response via Cepheid period–luminosity residuals. The Paper 0 benchmark
normalization is \(\kappa_{\rm gal}^{\rm bench} = (0.96 \pm 0.40) \times 10^6\) mag,
while the empirical Cepheid-channel analyses recover
\(\kappa_{\rm equiv}^{39} = (1.220 \pm 0.531) \times 10^6\) mag (\(\sigma_v=250\)),
\(\kappa_{\rm WLS}^{44} = (1.275 \pm 0.370) \times 10^6\) mag (\(3.44\sigma\); \(\sigma_v=150\)), and
\(\kappa_{\rm Cep}^{44} = (0.665 \pm 0.362) \times 10^6\) mag (\(\sigma_v=250\)) — bracketing the
benchmark with a midpoint of \(0.970 \times 10^6\) mag (\(1.0\%\) agreement) — and
Paper 11 Step 62 derives \(|\kappa_{P^2}^{\rm pred}(\lambda_0=1)| = 1.196 \times 10^6\) mag
directly from the Cepheid envelope density ratio
\(\rho_T/\langle\rho\rangle(10\text{ d}) = 8.45 \times 10^5\) at the unit-transport benchmark
\(\lambda_0 = 1\) with bare coupling \(|\beta_A| = 1\) fixed by Paper 0 (matching the SH0ES fit \((1.191 \pm 0.480) \times 10^6\) mag
to \(0.5\%\) at the central calibration, \(\lambda_0 = 0.995 \pm 0.401\) statistical and
\(\lambda_0 \in [0.28,\,2.49]\) across the \(\rho_T \in [8,\,70]\text{ g cm}^{-3}\) GNSS window
band; equivalently, inverting the SH0ES fit at \(\lambda_0=1\) independently yields
\(\rho_T^{\rm (Cepheid)} = 19.91 \pm 8.02\text{ g cm}^{-3}\), cross-validating the 25-year
GNSS value \(\rho_T = 20 \pm 7\text{ g cm}^{-3}\) across fifteen orders of magnitude in scale). Paper 10 (TEP-COS) measures the globular-cluster
pulsar response \(\tilde\kappa_{\rm MSP} \approx 3 \times 10^4\)
(full response coefficient, step_44_kappa_msp_prior.json)
after dense-cluster geometric suppression; the mapping to Paper 10's
effective channel coefficient is \(\kappa_{\rm MSP} = \tilde\kappa_{\rm
MSP}\,\mathcal S_{\rm cluster} \approx 0.05\), i.e. a cluster suppression
factor \(\mathcal S_{\rm cluster} \approx 1.7\times10^{-6}\) applied to the
full response — the same coefficient quoted under the un-tilded symbol.
The two values are consistent
with the bare TEP geometric-factor estimate (\(\sim 10^6\)–\(10^7\)) only
after environmental transfer factors are included; they do not assert a
direct one-to-one equality of raw channel coefficients.

This hierarchy protects the programme from the criticism:
*"Why does each paper have a different coefficient?"*
The answer is that they are *observable response coefficients*, not
bare scalar couplings. Each channel measures a different projection of
\(S_\Sigma(\mathcal{E})\):

| Channel | Response coefficient | Environment | Projection |
| --- | --- | --- | --- |
| **GNSS clocks** | \(L_c \approx 4200\) km | Planetary crust/mantle | Radial covariance scale |
| **Cepheids (Paper 11)** | \(\kappa_{\rm Cep} = 0.67\text{--}1.28 \times 10^6\) mag empirical; Step 62 \(\rho_T/\langle\rho\rangle \implies 1.20 \times 10^6\) mag (\(\lambda_0=1\)) | Galactic disk (weakly screened) | Period–luminosity modulation |
| **Pulsars (Paper 10)** | \(\tilde\kappa_{\rm MSP} \sim 10^4\) full; \(\kappa_{\rm MSP} \approx 0.05\) effective | Globular cluster (strongly screened) | Spin-down excess (suppressed) |
| **Wide binaries (Paper 13)** | \(\alpha_{\rm sat} \approx 0.366\) | Galactic halo (unscreened) | Velocity-profile saturation |

The cross-paper comparison is therefore through a shared clock-response
structure, not raw equality of coefficients. This preserves the internal
consistency of the programme while acknowledging that each domain probes a
different facet of the same environmental operator.

### Resolving Galactic Anomalies

#### The Radial Acceleration Relation (RAR)

The SPARC analysis (Section 3) confirms that the onset of mass discrepancies
follows the baryonic mass distribution, a key feature of the Radial
Acceleration Relation (McGaugh et al. 2016). In TEP, this coupling is a
natural consequence of the assumed sourcing: the scalar field is sourced by
the trace of the energy-momentum tensor ($T^\mu_\mu$), so the "dark matter"
halo is directly anchored to the baryonic mass. This recovers the MOND-like
phenomenology of the RAR without modifying inertia or requiring new force
laws, through the non-linear response of the scalar sector.

#### Ultra-Diffuse Galaxies (DF2/DF4)

The "dark matter free" galaxies NGC 1052-DF2 and DF4 pose a challenge to
theories where dark matter and baryons are dynamically coupled. TEP offers
an illustrative mechanism — "soliton stripping" — stated here as a
qualitative possibility pending a dynamical treatment: the scalar field is
not a separable substance, so "stripping" denotes the disruption and
slow re-equilibration of the baryon-anchored field configuration during a
high-velocity encounter (as proposed by van Dokkum et al. 2022), with the
timescale set by the effective mass $m_{\rm eff}$ and the encounter
velocity rather than by material removal. The remaining baryonic component
would appear devoid of dark matter until the field configuration
re-equilibrates. A quantitative version requires the relaxation timescale
of the disrupted profile to be computed; until then the mechanism is
registered as illustrative, not as a demonstrated channel.

### Systematic Uncertainties

The primary uncertainty in the global fit remains the GNSS calibration
length $L_c$. Three distinct contributions comprise this uncertainty: (i) the 25-year CODE
per-measurement statistical uncertainty ($4{,}201 \pm 1{,}967$ km, $\pm 47\%$;
Paper 2); (ii) the inter-center systematic spread ($3{,}330$–$4{,}549$ km,
$\sim \pm 12\%$; Paper 1), with the caveat that centers share largely overlapping
underlying data so this spread is not an independent ensemble; and (iii) the
$\mathcal{O}(1)$ prefactor in the transfer sketch (Section 2), treated as a
model-defining choice. The adopted operational value is $L_c = 4{,}200$ km
with $\pm 500$ km ($\pm 12\%$) reflecting the inter-center range. The
$\pm 47\%$ per-measurement statistical uncertainty from the single-center CODE
fit is retained as a conservative upper bound on measurement-to-measurement
scatter, but the $\pm 12\%$ inter-center spread is preferred as the operational
systematic because it reflects the actual range of scales recovered by
independent processing pipelines applied to the same underlying data, and the
consensus value (4,200 km) lies within that range. Propagating this alone gives
$\rho_T = 20 \pm 7$ g/cm³; the epoch/window instability established by the
executed held-out discriminator (Paper 14 Step 3.1; quarterly-window scatter
of 1,050–1,935 km on an identical product) is the dominant calibration
systematic and widens the effective band to roughly $8$–$70$ g/cm³
(Appendix B.8). The $M^{1/3}$ structural form is
independent of $\rho_T$; only the normalization changes. Even the widened
band is far smaller than the dynamic range over which the model is tested,
preserving the falsifiability of the scaling law.

### Predictions for High-Redshift JWST Observations

The TEP framework makes concrete predictions for the "Little Red
Dots"—compact, massive galaxies at $z > 5$ discovered by JWST. These objects
are a critical testing ground because they are compact enough for scalar-envelope effects to dominate their apparent structure, yet massive enough for JWST to resolve the predicted kpc-scale transition region.

#### Qualitative Predictions for High-Redshift JWST Observations

For a target galaxy with stellar mass $M_* = 10^{10} M_\odot$ (typical of
LRDs), the TEP model makes the following qualitative predictions:

#### 1. Soliton Envelope Scale

The fundamental core saturation density $\rho_T \approx 20$ g/cm³
governs the compact soliton surface, but the observable galactic
transition is set by the diffuse screening density $\rho_{\rm trans}
\approx 3 \times 10^{-23}$ g/cm³ (fitted from SPARC, Section 4). The
predicted emission extent therefore scales as $R_{\rm trans} \propto
M^{1/3}$ at the transition density, not at $\rho_T$. Precise
numerical predictions for high-redshift galaxies require
epoch-specific calibration of $\rho_{\rm trans}$; until then, the
model predicts that resolved gas tracers (e.g., H$\alpha$) should
reveal more extended kinematic structure than expected from a purely
stellar compact core.

#### 2. Velocity Dispersion

The dynamical support speed is set by the soliton potential at the
transition radius: $\sigma_v^2 \sim G M / R_{\rm trans}$. Because
$R_{\rm trans} \propto M^{1/3}$, the velocity dispersion scales as
$\sigma_v \propto M^{1/3}$. This predicts characteristic velocity
dispersions intermediate between the broad-line AGN regime
($>1000$ km/s) and the low-mass dwarf regime ($<50$ km/s). Precise
normalization awaits epoch-specific calibration of $\rho_{\rm trans}$.

#### Discriminant: Soliton vs. Supermassive Black Hole

A leading alternative explanation is that LRDs are reddened Active Galactic
Nuclei (AGNs) powered by supermassive black holes ($M_{\text{BH}} \sim
10^7$–$10^8 M_\odot$).

**AGN Hypothesis:** Emission should be point-like or
confined to the nuclear region ($< 1$ kpc). Broad lines reflect BLR
physics ($v > 1000$ km/s).

**TEP Soliton Hypothesis:** Emission should be extended
over the galactic transition scale (kpc-scale for typical LRD masses,
pending epoch-specific calibration of $\rho_{\rm trans}$). Broad lines
reflect the deep gravitational potential of the soliton well
($\sim$100–200 km/s).

Discriminant: Spatially resolve the H$\alpha$ emission.
Extended kpc-scale emission with velocity dispersions $\sim$100–200 km/s
would disfavor a purely nuclear AGN interpretation, while strongly compact
emission with $v > 1000$ km/s would favor an AGN-dominated explanation.
This constitutes a concrete observational discriminator for this
particular soliton-interpretation channel.

### Implications for Paper 7 (RBH-1)

The GNSS-calibrated $\rho_T$ identified here provides the working normalization for
the analysis of RBH-1 in the companion paper (Smawfield 2025h). By adopting
$\rho_T \approx 20$ g/cm³ (with stated uncertainty), the soliton
interpretation for RBH-1 becomes tightly constrained. The wake structure,
velocity jump, and thermal emission properties can then be compared against
the interaction of the wake with a soliton of characteristic radius $R
\approx 7.8 \times 10^7$ km. Deviations would primarily constrain the
object-specific soliton interpretation and/or the assumed mapping between
$\rho_T$ and the effective radius in this environment, rather than the
broader TEP framework.

### Cross-Domain Constraint Network

| Tier | Role | Channels | Relation to \(\rho_T\) |
| --- | --- | --- | --- |
| **A** | Direct / quasi-direct calibration | GNSS, SPARC/MW, RBH-1, condensed-matter EOS | Enters \(\rho_T\) normalization directly |
| **B** | Screening / response-transfer test | Pulsars (10), Cepheids (11), wide binaries (13), LLR (17), Earth flyby (15) | Constrains \(S_\Sigma(\mathcal{E})\) and transfer functions, not independent \(\rho_T\) |
| **C** | Phantom Mass / conformal transport | Lensing (4), multiply-imaged SNe (19), JWST (12) | Tests the mechanism; not a \(\rho_T\) calibration |
| **D** | Orthogonal architecture test | J0437 (16), hi_class (18), C0 (26), TH (27), QF (23) | Tests deep TEP structure; does not calibrate \(\rho_T\) |

*Only Tier A contributes to the present \(\rho_T\) normalization. Tiers B–D constrain the environmental operator, response hierarchy, and theoretical viability of the same TEP architecture.*

The TEP programme now spans more than twenty papers across terrestrial,
galactic, compact-object, lensing, and cosmological domains. It is
essential to distinguish which of these constrain \(\rho_T\) directly
and which constrain the broader TEP architecture. The following tiered
classification makes this dependence explicit and prevents the
misreading that every paper independently confirms the same density
number.

#### Tier A — Direct or Quasi-Direct \(\rho_T\) Constraints

These channels calibrate the macroscopic saturation scale most directly:

**GNSS \(\lambda_T\) (Papers 1–3):** The primary
empirical anchor. The 4,200 km correlation length is conditionally
identified with the projected covariance scale associated with
\(R_T(M_\oplus)\), up to an order-unity transfer factor, yielding the
operational estimate \(\rho_T \approx 20\) g/cm³.

**SPARC/Milky Way onset scaling (this work):**
Quasi-direct scaling consistency. The \(M^{1/3}\) scaling of
dark-matter onset radii across 175 spiral galaxies is consistent with
the saturation-radius construction, but does not independently
calibrate \(\rho_T\). The Milky Way test uses the SPARC-calibrated
normalization and is therefore not an independent measurement.

**RBH-1 crossover (Paper 7):** The runaway-black-hole
candidate tests the extrapolation of \(R_T(M)\) to
\(M \sim 10^7 M_\odot\), with wake structure providing an
object-specific consistency test.

**Condensed-matter EOS coincidence (this work, Section 5):**
The proximity of \(\rho_T \approx 20\) g/cm³ to warm-dense-matter /
degeneracy-onset scales provides a coincidence-level physical
plausibility check, not an independent empirical calibration.

#### Tier B — Screening and Response-Transfer Constraints

These channels do not independently calibrate \(\rho_T\), but they
constrain the environmental operator \(S_\Sigma(\mathcal{E})\) and
the transfer functions that connect \(\rho_T\) to observables:

**Globular-cluster pulsars (Paper 10):** The suppressed
density scaling of spin-down excess (\(\Gamma \approx 0.39\) dex/dex,
vs. Newtonian \(0.75\)) constrains how dense environments screen the
temporal-field response. The observable coefficient
\(\tilde\kappa_{\rm MSP}\) is a downstream projection, not a direct
\(\rho_T\) measurement.

**Cepheids (Paper 11):** The Cepheid clock-bias response —
\(0.665\text{--}1.275\times10^6\) mag empirically (\((0.665\pm0.362)\times10^6\) mag multi-block and \((1.206\pm0.528)\times10^6\) mag redshift-only WLS at canonical \(\sigma_v=250\text{ km/s}\), \((1.275\pm0.370)\times10^6\) mag at \(\sigma_v=150\text{ km/s}\)), bracketing the \(0.96\times10^6\) mag
benchmark and matching the Step 62 envelope-density prediction \(1.196\times10^6\) mag
at the unit-transport benchmark \(\lambda_0=1\) (\(|\beta_A|=1\)) to \(0.5\%\) at the central estimate (\((1.191\pm0.480)\times10^6\) mag empirical fit) — in galactic disks bridges the
weakly screened regime. It agrees with \(\tilde\kappa_{\rm MSP}\)
after environmental transfer factors are applied, demonstrating that
response coefficients are channel-specific projections.

**Wide binaries (Paper 13):** Gaia DR3 reveals a
weak-field screening transition at a few thousand AU with
environmental ordering (high-\(|Z|\) vs. midplane). This is an
independent *morphology* test of Temporal Shear recovery, not
an independent \(\rho_T\) calibration: the paper explicitly states
that imported \(\rho_T\) and \(g_{\rm TEP}\) enter only as
cross-scale consistency checks.

**Lunar Laser Ranging (Paper 17):** Reports a
compactness-dependent residual channel at high significance, but
remains at the "candidate residual" level pending integrator-level
ephemeris refits. Classified as a *guardrail*: it suggests
the local screened regime may still contain residual-channel
structure, but does not yet provide a completed \(\rho_T\) likelihood
term.

**Earth flyby (Paper 15):** Uses the UCD
\(\rho_T \approx 20\) g/cm³ as a *shared input*, not an
independent validation. Valuable as a downstream posterior predictive
test: if the terrestrial screening profile is correct, flyby anomalies
should organize under the restricted trajectory model.

#### Tier C — Phantom Mass / Conformal Transport Tests

These channels test the *mechanism* by which TEP generates apparent
mass discrepancies, without directly calibrating \(\rho_T\):

**Gravitational lensing (Paper 4):** Under the canonical
unstretched-space temporal metric (Paper 0 §2.2; Paper 4 §6.9, Step 02),
intermediate temporal depth refracts photons at half the dynamical
scalar potential weight (\(\Phi_{\rm lens}=\Phi_N+\tfrac12 c^2\beta_A u_{\rm grav}\)
vs \(\Phi_{\rm dyn}=\Phi_N+c^2\beta_A u_{\rm grav}\), whereas on the
historical isotropic-conformal benchmark \(\tilde g_{\mu\nu}=A^2 g_{\mu\nu}\),
\(A^2(\phi)\) cancels out of null rays), supplying both the
baryon-sourced Bullet Cluster convergence and a testable
lensing-to-dynamical mass ratio
\(M_{\rm lens}^{\rm u.s.}/M_{\rm dyn}=(1+\beta_A^2 y_{\rm eff})/(1+2\beta_A^2 y_{\rm eff})\).
This provides *mechanistic support* for the TEP interpretation of
the dark sector, not an independent \(\rho_T\) calibration.

**Multiply-imaged supernovae (Paper 19):** SN Refsdal
provides directional evidence from blind model residuals for
temporal transport in the lensing sector, while the action-level
unstretched-space temporal refraction solve on the selected two-branch
sector \(P_{,X}=k\sqrt\xi+2\xi\) (\(k=16.03\), Step 64) reproduces the
\(+330.9\text{ d}\) sheet and \(+221.6\text{--}443.2\text{ d}\) 3-D NFW
ray Fermat delay and arrival order from \(f_b=0.16\) baryons alone
alongside the phenomenological magnification proxy.

**JWST high-redshift galaxies (Paper 12):** Tests
potential-depth response at \(z > 5\) using an external response
prior. The signal is not an independent \(\rho_T\) measurement;
it probes whether the same conformal architecture organizes
high-redshift compact galaxies.

#### Tier D — Orthogonal TEP-Architecture Tests

These papers do not calibrate \(\rho_T\) at all, but they test deep
structural claims of the TEP framework:

**J0437 scintillation (Paper 16):** Reports non-zero
phase-closure signal in millisecond pulsar scintillation, rejecting
standard scalar-delay ISM models (\(\psi = 0\)) at high confidence.
This tests *non-integrable time transport* / synchronization
holonomy, one of the deepest TEP claims. It does not calibrate
\(\rho_T\).

**hi_class cosmology (Paper 18):** Demonstrates that
the conformal temporal-transport geometry preserves CMB acoustic
structure, sound-horizon behaviour, and stable linear perturbations.
A *global consistency constraint*, not a \(\rho_T\) term.

**C0 supernova conformal redshift (Paper 26):** Shows
that Pantheon+ distance-redshift data do not uniquely require
primitive spatial expansion. Supports the temporal-transport
interpretation of cosmological redshift.

**TH temporal horizon (Paper 27):** The apparent
\(a_{\rm eff} \to 0\) limit becomes a temporal conformal boundary
rather than a physical singularity. Provides theoretical
closure for the late-time cosmology.

**QF quantum tangent limit (Paper 23):** Shows that
the screened quantum tangent limit is consistent with the TEP
architecture. A *theoretical compatibility* check, not empirical
\(\rho_T\) evidence.

**Summary:** Only Tier A enters the present
\(\rho_T\) normalization argument directly. Tiers B–D constrain
the environmental operator, response hierarchy, and theoretical
viability of the same TEP architecture. This tiered dependence is
stronger than a flat pile of evidence because it acknowledges
exactly where each paper contributes and where it does not.

## 9. Conclusion

### A Universal Organizing Parameter

This paper identifies the saturation scale $\rho_T \approx 20$ g/cm³ as a candidate organizing parameter for gravitational phenomena across planetary, galactic, compact-object, and RBH-scale regimes. The convergence of terrestrial GNSS timing structure, SPARC galaxy onset scaling, Milky Way inner mass-discrepancy structure, and the density-ordered screening hierarchy motivates the hypothesis that $\rho_T$ is a real physical scale in the temporal-field topology, not a system-specific fitting parameter. The screening hierarchy is not treated as an independent discovery of the $1/3$ exponent; rather, it demonstrates that the same $\rho_T$ coherently recovers GR-like behavior in dense systems while permitting active-shear scalar phenomenology in diffuse regimes.

### Key Results

The primary findings are:

- **GNSS Calibration:** Distance-structured correlations in atomic clocks yield $L_c \approx 4200$ km, corresponding to $\rho_T \approx 20$ g/cm³ under the conditional identification $L_c \leftrightarrow R_T(M_\oplus)$. The underlying GNSS correlation scale $L_c$ exhibits 25-year stability, multi-center consistency, and survives raw RINEX validation.

- **Galactic Scaling:** SPARC rotation curves yield $\alpha_{\rm SPARC} = 0.355 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (definition)}$, consistent with the $M^{1/3}$ expectation within $\sim 0.3\sigma$ ($\sim 0.5\sigma$ statistical-only).

- **Screening Hierarchy:** 26 objects spanning ~15 orders of magnitude in density reveal a consistency relation $S \propto \rho^{1/3}$, algebraically expected from the $R_T(M)$ construction, explaining why GR tests pass (binary pulsars: $S \sim 29{,}000$) while galactic dynamics are in the active-shear regime ($S \sim 10^{-9}$ at $\rho \sim 10^{-24}$ g/cm³).

- **Milky Way Test:** The SPARC-calibrated $M^{1/3}$ relation predicts a dark-matter onset radius $R_{\text{DM}} \approx 3$ kpc for the Milky Way, consistent with the observed transition from baryonic to dark-matter-dominated rotation at $R \sim 3$–5 kpc, providing a local scale-consistency check.

### Dark Matter as Phantom Mass

The TEP framework reinterprets dark-matter observations as violations of the isochrony axiom—the assumption that spatially separated clocks at the same gravitational potential tick at the same rate. On the selected two-branch kinetic sector $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$ fixed externally by $a_0$, zero per-galaxy free parameters), the low-acceleration square-root branch reproduces the SPARC radial-acceleration relation ($0.2185$ dex rms at $\Upsilon_d=0.5$ and $0.2004$ dex at $\Upsilon_d=0.35$ under the action-derived additive force law $g_{\rm bar}+a_\phi$, versus $0.2812$ dex for the unit-term baseline; $0.1974$ dex under the quadrature comparator, Step 10) and the outer-edge BTFR ($v_{\rm obs}/v_{\rm pred}=1.0109$ quadrature, $0.910$ additive at $\Upsilon_d=0.5$, Step 13), while the high-gradient quadratic branch satisfies Solar-System screening bounds. Evaluated under the canonical unstretched-space temporal metric (Paper 0 §2.2, Step 80), this same zero-parameter kinetic sector also closes the optical lensing sector across cluster, strong-lensing, and CMB scales from baryons alone (Papers 4, 18, 19).

This reinterpretation offers a candidate common account of several dark-sector puzzles:

- **Universal scaling:** The $M^{1/3}$ relation follows naturally from the saturation-radius construction.

- **Dark-matter-deficient UDGs:** Field-envelope disruption ("soliton stripping") offers an illustrative explanation for DF2/DF4-like systems, pending computation of the re-equilibration timescale.

- **Core-cusp structure:** Saturated temporal-field cores offer a possible route to flat effective density profiles.

- **Baryonic Tully-Fisher behaviour:** The $M^{1/3}$ scaling may contribute to BTFR-like phenomenology, though a full BTFR derivation is not completed here.

### Astrophysical Applications

The GNSS-anchored value $\rho_T \approx 20$ g/cm³ enables specific applications for astrophysical systems. The companion paper (Smawfield 2025h, Paper 7) applies this calibration to the RBH-1 runaway black hole candidate, yielding a saturation radius $R_T \approx 7.8 \times 10^7$ km. This provides a testable extrapolation: if the RBH-1 mass is revised by a factor of 3, the expected radius would change by $\sim 44\%$, offering a clear consistency check with observations.

Future applications include:

- **JWST high-redshift galaxies:** "Little Red Dots" and massive quiescent galaxies at $z > 10$ may exhibit soliton signatures.

- **EHT polarimetry:** M87* and Sgr A* polarization patterns could reveal scalar field structure near supermassive compact objects.

- **Gravitational waves:** Binary black hole mergers may produce scalar radiation detectable by LISA.

- **Strong lensing time delays:** The phantom-mass sector can act on time-delay cosmography both through the mass-model normalization — where the single-halo dynamical–lensing mass difference biases the velocity-dispersion prior that breaks the mass-sheet degeneracy in the direction that elevates the inferred $H_0$ (Paper 19, Step 58 channel audit; $0.0\text{ d}$ propagated static-conformal residual on the isotropic-conformal benchmark) — and directly through the canonical unstretched-space temporal metric (Paper 0 §2.2; Paper 19, Step 64), where intermediate temporal depth on the two-branch sector $P_{,X}=k\sqrt\xi+2\xi$ reproduces the SN Refsdal Fermat delay ($+330.9\text{ d}$ sheet and $+221.6\text{--}443.2\text{ d}$ 3-D NFW ray) from $f_b=0.16$ baryons alone.

### Theoretical Implications

The convergence of terrestrial, compact, galactic, and cosmological constraints on a single density scale suggests a fundamental organization principle governing gravitational phenomenology across many orders of magnitude in mass. This organization is formalized in the Temporal Equivalence Principle (Smawfield 2025a, Paper 0), which posits that gravitational phenomena arise from a temporal scalar field $\phi(x^\mu)$ coupled to matter via the two-branch k-essence action:

\begin{equation} \label{eq:tep_action_conclusion} S = \int d^4x \sqrt{-g} \left[ \frac{M_{\text{Pl}}^2}{2} R + P(X,\phi) - V(\phi) \right] + S_m[\tilde{g}_{\mu\nu}] \end{equation}

where $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav})U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift})\Pi_{\mu\nu}$ in the canonical unified unstretched-space temporal metric ($\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu}$ on the historical isotropic-conformal benchmark) with $A(\phi) = \exp(\beta_A\phi/M_{\text{Pl}})$.

The saturation scale $\rho_T$ emerges as the scale at which the scalar field $\phi$ reaches its self-interaction threshold, saturating the field. Under a soliton interpretation, this would correspond to soliton formation; regardless of microscopic realization, this provides a natural explanation for the $M^{1/3}$ scaling and the dynamical phantom-mass phenomenon.

### Systematic Uncertainties and Robustness

The primary uncertainty is the GNSS length scale determination. Four distinct contributions are reported: (i) the 25-year CODE per-measurement statistical uncertainty ($4{,}201 \pm 1{,}967$ km, $\pm 47\%$; Paper 2); (ii) the inter-center systematic spread ($3{,}330$–$4{,}549$ km, $\sim \pm 12\%$; Paper 1), with the caveat that centers share underlying data so this spread is not an independent ensemble; (iii) the epoch/window instability established by the executed held-out discriminator (Paper 14 Step 3.1): the identical estimator on the identical product returns $\lambda$ from 1,050 to 1,935 km across quarterly windows, while matched-day independent chains agree at $\sim$30% (2,744 vs. 1,933 km — the CNES value landing within 4% of the pooled 1,862 km — same-product fallback days excluded by epoch-aligned detrended-residual comparison) — the dominant calibration systematic is the $\sim$30–40% epoch/window scatter; and (iv) the $\mathcal{O}(1)$ prefactor in the transfer sketch (Section 2), treated as a model-defining choice. The adopted operational value is $L_c = 4{,}200$ km with $\pm 500$ km ($\pm 12\%$) reflecting the inter-center range, plus the epoch/window systematic carried separately. Propagating the inter-center spread alone gives $\rho_T = 20 \pm 7$ g/cm³; propagating the dominant epoch/window scatter widens the effective band to roughly $8$–$70$ g/cm³, and the MGEX held-out scale ($\rho_T$ up to $\sim 5\times10^{2}$ g/cm³ if adopted at face value) is recorded as an unresolved epoch-scale tension pending longer MGEX baselines. The $M^{1/3}$ structural form is independent of $\rho_T$; only the normalization changes. The present evidence consists of one primary terrestrial timing calibration, one galactic onset-scaling consistency test, one local SPARC-internal Milky Way check, and one algebraic screening-consistency hierarchy. These layers are mutually consistent, but they are not statistically independent measurements of $\rho_T$.

The robustness of the result is demonstrated by:

- **Multi-center consistency:** CODE, IGS, and ESA yield consistent GNSS correlation structure across independent processing centres.

- **Temporal stability:** 25-year consistency disfavors transient effects.

- **Raw data validation:** RINEX analysis constrains processing artifacts.

- **Cross-scale agreement:** Four tests spanning 18 orders of magnitude in mass.

### Future Directions

The identification of $\rho_T \approx 20$ g/cm³ as a candidate cross-scale saturation parameter opens several avenues for future research:

- **Precision tests:** Improved GNSS analysis (longer baselines, more stations) could tighten $L_c$ to $\sim$5% accuracy.

- **High-redshift galaxies:** JWST observations of $z > 10$ systems could test whether soliton formation affects early galaxy assembly.

- **Gravitational wave cosmology:** LISA could detect scalar radiation from binary black hole mergers, providing direct evidence for the time-field sector.

- **Laboratory tests:** Atomic interferometry experiments could search for temporal gradients at sub-meter scales.

### Concluding Remarks

This paper identifies $\rho_T \approx 20$ g/cm³ as a candidate macroscopic saturation/proximity scale in the TEP environmental-response hierarchy. GNSS clock correlations provide the primary terrestrial timing anchor for the Earth-scale covariance length. SPARC onset scaling, Milky Way structure, RBH-1 crossover behaviour, compact-object screening, and condensed-matter/EOS considerations then test whether the same scale organizes otherwise disconnected gravitational regularities without retuning. Channel-specific quantities such as $\rho_{\rm trans}$, $R_s$, $\kappa_{\rm Cep}$, $\kappa_{\rm MSP}$, and lensing response coefficients are not independent measurements of $\rho_T$, but downstream projections of $S_\Sigma(\mathcal{E})$.

This paradigm shift—from dark matter as substance to dark matter as temporal shear—provides a candidate unified organizing framework for the $M^{1/3}$ scaling, screening hierarchy, and dynamical phantom-mass phenomenon, to the extent permitted by lensing–dynamics consistency. The GNSS-anchored value of $\rho_T$ enables specific testable applications, including the RBH-1 case study (Paper 7), positioning the Temporal Equivalence Principle as an empirically grounded candidate alternative to the particle dark matter paradigm in the dynamical sector, with the dynamical–lensing mass comparison (Paper 19) as its decisive open test.

## References

Abedi, J., Dykaar, H., & Afshordi, N. 2017, *Phys. Rev. D*, 96, 082004 (arXiv:1612.00266)

Allen, M. G., Groves, B. A., Dopita, M. A., Sutherland, R. S., & Kewley, L. J. 2008, *ApJS*, 178, 20 (arXiv:0805.0204)

Barro, G., et al. 2025, *From "The Cliff" to "Virgil": Mapping the Spectral Diversity of Little Red Dots with JWST/NIRSpec* (arXiv:2512.15853)

Bland-Hawthorn, J., & Gerhard, O. 2016, *ARA&A*, 54, 529 (Milky Way mass model)

Campanelli, M., Lousto, C. O., Zlochower, Y., & Merritt, D. 2007, *Phys. Rev. Lett.* (arXiv:gr-qc/0702133)

Cardoso, V., Hopper, S., Macedo, C. F. B., Palenzuela, C., & Pani, P. 2016, *Phys. Rev. D*, 94, 084031 (arXiv:1608.08637)

Chen, K., Li, Z., Inayoshi, K., & Ho, L. C. 2025, *ApJ Lett.* (DOI: 10.3847/2041-8213/ae1955; arXiv:2505.22600)

Colpi, M., & Dotti, M. 2011, *Adv. Astron.*, 2011, 1 (arXiv:0906.4339)

Colpi, M., Geppert, U., & Page, D. 2000, *ApJ*, 529, L29 (DOI: 10.1086/312448; arXiv:astro-ph/9912066)

Donato, F., Gentile, G., Salucci, P., et al. 2009, *MNRAS*, 397, 1169 (DOI: 10.1111/j.1365-2966.2009.15004.x; arXiv:0904.4054)

Delvecchio, I., et al. 2025, *A&A* (DOI: 10.1051/0004-6361/202557164; arXiv:2509.07100)

Event Horizon Telescope Collaboration. 2019, *ApJ Lett.*, 875, L1 (DOI: 10.3847/2041-8213/ab0ec7; M87* image)

Event Horizon Telescope Collaboration. 2021, *ApJ Lett.*, 910, L12 (DOI: 10.3847/2041-8213/abe71d; M87* polarization)

Event Horizon Telescope Collaboration. 2021, *ApJ Lett.*, 910, L13 (DOI: 10.3847/2041-8213/abe71e; EHT Paper VII)

Event Horizon Telescope Collaboration. 2022, *ApJ Lett.*, 930, L12 (DOI: 10.3847/2041-8213/ac6674; Sgr A* image)

Heeck, J., et al. 2021, *Phys. Rev. D*, 103, 115004 (arXiv:2009.08463)

Hinterbichler, K., & Khoury, J. 2010, *Phys. Rev. Lett.*, 104, 231301 (DOI: 10.1103/PhysRevLett.104.231301; arXiv:1001.4525)

Khoury, J., & Weltman, A. 2004, *Phys. Rev. Lett.*, 93, 171104 (DOI: 10.1103/PhysRevLett.93.171104; arXiv:astro-ph/0309300)

Hui, L., Ostriker, J. P., Tremaine, S., & Witten, E. 2017, *Phys. Rev. D*, 95, 043541 (DOI: 10.1103/PhysRevD.95.043541; arXiv:1610.08297)

Komossa, S. 2012, *Recoiling black holes: electromagnetic signatures, candidates, and astrophysical implications* (arXiv:1202.1977)

Coleman, S. 1985, *Nucl. Phys. B*, 262, 263 (DOI: 10.1016/0550-3213(85)90286-X)

Gleiser, M. 1994, *Phys. Rev. D*, 49, 2978 (DOI: 10.1103/PhysRevD.49.2978; arXiv:hep-ph/9308279)

Gaia Collaboration. 2023, *A&A*, 674, A1 (DOI: 10.1051/0004-6361/202243940; Gaia Data Release 3)

Gong, Y., Papantonopoulos, E., & Yi, Z. 2018, *Eur. Phys. J. C*, 78, 738 (DOI: 10.1140/epjc/s10052-018-6227-9; arXiv:1711.04102)

Gronke, M., & Oh, S. P. 2018, *MNRAS*, 480, L111 (arXiv:1806.02729)

Hviding, R. E., et al. 2025, *A&A* (DOI: 10.1051/0004-6361/202555816; arXiv:2506.05459)

Seidel, E., & Suen, W.-M. 1991, *Phys. Rev. Lett.*, 66, 1659 (DOI: 10.1103/PhysRevLett.66.1659)

Kusenko, A. 1997, *Phys. Lett. B*, 404, 285 (arXiv:hep-th/9704073)

Lelli, F., McGaugh, S. S., & Schombert, J. M. 2016, *AJ*, 152, 157 (DOI: 10.3847/0004-6256/152/5/157; SPARC Database)

McGaugh, S. S., Lelli, F., & Schombert, J. M. 2016, *Phys. Rev. Lett.*, 117, 201101 (DOI: 10.1103/PhysRevLett.117.201101; arXiv:1609.05917)

McGaugh, S. S., Schombert, J. M., Bothun, G. D., & de Blok, W. J. G. 2000, *ApJ Lett.*, 533, L99 (DOI: 10.1086/312628; arXiv:astro-ph/0003001)

Ogiya, G., & Nagai, D. 2023, *ApJ Lett.*, 958, L5 (arXiv:2309.09031)

Poggianti, B. M., et al. 2019, *ApJ*, 874, 140 (arXiv:1810.05164)

Rybicki, G. B., & Lightman, A. P. 1979, *Radiative Processes in Astrophysics* (Wiley-VCH)

Sanchez Almeida, J., Montes, M., & Trujillo, I. 2023, *A&A* (DOI: 10.1051/0004-6361/202346430; arXiv:2304.12344)

Schive, H.-Y., Chiueh, T., & Broadhurst, T. 2014, *Phys. Rev. Lett.*, 113, 261302 (DOI: 10.1103/PhysRevLett.113.261302; arXiv:1407.7762)

Sofue, Y. 2020, *PASJ*, 72, 81 (DOI: 10.1093/pasj/psaa065; arXiv:2006.15008)

Sutherland, R. S., & Dopita, M. A. 1993, *ApJS*, 88, 253 (DOI: 10.1086/191823)

TDCOSMO Collaboration. 2025, *A&A* (DOI: 10.1051/0004-6361/202555801; arXiv:2506.03023)

van Dokkum, P., et al. 2023, *ApJ Lett.*, 946, L50 (arXiv:2302.04888)

van Dokkum, P., et al. 2025, *JWST Confirmation of a Runaway Supermassive Black Hole via its Supersonic Bow Shock* (arXiv:2512.04166)

Taylor, J. H., & Weisberg, J. M. 1982, *ApJ*, 253, 908 (DOI: 10.1086/159690)

Vainshtein, A. I. 1972, *Phys. Lett. B*, 39, 393 (DOI: 10.1016/0370-2693(72)90147-5)

Weisberg, J. M., & Taylor, J. H. 2005, *Binary Radio Pulsars*, ASP Conf. Ser., 328, 25 (arXiv:astro-ph/0407149)

Westerweck, J., et al. 2018, *Phys. Rev. D*, 97, 124037 (arXiv:1712.09966)

Will, C. M. 2014, *Living Rev. Relativ.*, 17, 4 (DOI: 10.12942/lrr-2014-4; arXiv:1403.7377)

Williams, J. G., Turyshev, S. G., & Boggs, D. H. 2012, *Class. Quantum Grav.*, 29, 184004 (DOI: 10.1088/0264-9381/29/18/184004; arXiv:1203.2150)

Younes, G., et al. 2020, *ApJ Lett.*, 896, L42 (DOI: 10.3847/2041-8213/ab9a48; arXiv:2006.02814)

Younes, G., et al. 2022, *Nature Astronomy*, 7, 339 (arXiv:2210.11518)

Zhang, H., et al. 2025, *Polarization Images of Solitonic Boson Stars* (arXiv:2508.11992)

Zumalacárregui, M., & Bellini, E. 2018, *JCAP*, 2018, 054 (DOI: 10.1088/1475-7516/2018/10/054; arXiv:1705.05302; hi_class cosmology solver)

Pintore, F., et al. 2016, *MNRAS*, 458, 2088 (DOI: 10.1093/mnras/stw449; arXiv:1602.05950)

Ray, P. S., et al. 2019, *ApJ*, 879, 130 (DOI: 10.3847/1538-4357/ab24d8)

Tuo, Y. L., et al. 2024, *ApJ*, 966, 80 (DOI: 10.3847/1538-4357/ad2fb6; arXiv:2403.12137)

Kramer, M., et al. 2006, *Science*, 314, 97 (DOI: 10.1126/science.1132305)

van Dokkum, P., et al. 2018, *Nature*, 555, 629 (DOI: 10.1038/nature25767)

van Dokkum, P., et al. 2019, *ApJ Lett.*, 874, L5 (DOI: 10.3847/2041-8213/ab0d92)

van Dokkum, P., et al. 2022, *Nature*, 605, 435 (DOI: 10.1038/s41586-022-04665-6)

Jiao, Y., et al. 2023, *A&A*, 678, A208 (DOI: 10.1051/0004-6361/202347513; arXiv:2309.00048)

Archibald, R. F., et al. 2013, *Nature*, 497, 591 (DOI: 10.1038/nature12159)

Archibald, R. F., et al. 2017, *ApJ*, 829, L21 (DOI: 10.3847/2041-8205/829/1/L21; arXiv:1608.01007)

de Graaff, A., et al. 2025, *A&A* (arXiv:2503.01891)

Dib, R., & Kaspi, V. M. 2014, *ApJ*, 784, 37 (DOI: 10.1088/0004-637X/784/1/37; arXiv:1401.5738)

Nicolis, A., Rattazzi, R., & Trincherini, E. 2009, *Phys. Rev. D*, 79, 064036 (DOI: 10.1103/PhysRevD.79.064036; arXiv:0811.2197)

Olausen, S. A., & Kaspi, V. M. 2014, *ApJS*, 212, 6 (DOI: 10.1088/0067-0049/212/1/6; arXiv:1309.4167)

Şaşmaz Muş, S., et al. 2014, *MNRAS*, 440, 2916 (DOI: 10.1093/mnras/stu445; arXiv:1402.6054)

### TEP Research Series

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: [10.5281/zenodo.16921911](https://doi.org/10.5281/zenodo.16921911) (Paper 0)

Smawfield, M. L. (2025b). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: [10.5281/zenodo.17127229](https://doi.org/10.5281/zenodo.17127229) (Paper 1)

Smawfield, M. L. (2025c). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: [10.5281/zenodo.17517141](https://doi.org/10.5281/zenodo.17517141) (Paper 2)

Smawfield, M. L. (2025d). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: [10.5281/zenodo.17860166](https://doi.org/10.5281/zenodo.17860166) (Paper 3)

Smawfield, M. L. (2025e). *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: [10.5281/zenodo.17982540](https://doi.org/10.5281/zenodo.17982540) (Paper 4)

Smawfield, M. L. (2025f). *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: [10.5281/zenodo.18004832](https://doi.org/10.5281/zenodo.18004832) (Paper 5)

Smawfield, M. L. (2025g). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: [10.5281/zenodo.18064365](https://doi.org/10.5281/zenodo.18064365) (Paper 6 — this work)

Smawfield, M. L. (2025h). *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: [10.5281/zenodo.18059250](https://doi.org/10.5281/zenodo.18059250) (Paper 7)

Smawfield, M. L. (2025). *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: [10.5281/zenodo.18064581](https://doi.org/10.5281/zenodo.18064581) (Paper 8)

Smawfield, M. L. (2025). *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: [10.5281/zenodo.18109760](https://doi.org/10.5281/zenodo.18109760) (Paper 9)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: [10.5281/zenodo.18165798](https://doi.org/10.5281/zenodo.18165798) (Paper 10)

Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: [10.5281/zenodo.18209702](https://doi.org/10.5281/zenodo.18209702) (Paper 11)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: [10.5281/zenodo.19000827](https://doi.org/10.5281/zenodo.19000827) (Paper 12)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: [10.5281/zenodo.19102061](https://doi.org/10.5281/zenodo.19102061) (Paper 13)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Native hi_class Conformal Implementation, Linear Perturbation Closure, and CMB Acoustic Peak Preservation*. Preprint v0.8 (Cambridge). Zenodo. DOI: [10.5281/zenodo.20572722](https://doi.org/10.5281/zenodo.20572722) (Paper 18)

## Contact Information

Author: Matthew Lukin Smawfield

Affiliation: Independent Researcher

Email: [matthew@mlsmawfield.com](mailto:matthew@mlsmawfield.com)

ORCID: [0009-0003-8219-3159](https://orcid.org/0009-0003-8219-3159)

GitHub: [github.com/matthewsmawfield](https://github.com/matthewsmawfield)

License: This work is licensed under a [Creative Commons Attribution 4.0 International License](https://creativecommons.org/licenses/by/4.0/).

Version: v0.9 (New Delhi) · Last updated: 17 October 2026

Revision note (v0.8): consistency revision aligning screening terminology with Jakarta v0.16.

## Appendix A: GNSS Calibration — Summary of Evidence

The claim that terrestrial clock correlations calibrate a fundamental density parameter ($\rho_T \approx 20$ g/cm³) is extraordinary and requires rigorous justification. This appendix summarizes the key validation results from the companion GNSS papers (Smawfield 2025b,c,d) and addresses common concerns about systematics.

### A.1 Physical Mechanism

In the TEP framework, the scalar field $\phi$ modulates proper time via a conformal factor $A(\phi) = \exp(\beta_A\phi/M_{\text{Pl}})$. Near a gravitating mass, the field profile creates a spatial gradient in clock rates. GNSS atomic clocks, distributed across Earth's gravitational field, sample this gradient at different radial positions. The characteristic correlation length $L_c \approx 4200$ km corresponds to the scale where the scalar field's gradient becomes steep enough to produce measurable timing correlations—the saturation radius for Earth's mass.

### A.2 Why Not Ionospheric, Tropospheric, or Orbital Systematics?

#### Scale Separation

The 4200 km correlation length is distinct from known geodetic systematics:

- **Tropospheric correlations:** ~100–500 km (weather systems)

- **Ionospheric correlations:** ~500–2000 km (TEC structures)

- **Orbital harmonics:** ~12 hr (half-sidereal period)

Power spectral analysis shows the 4200 km feature as a persistent background covariance floor in the GPS-only clean baseline, with no counterpart in ionospheric or tropospheric delay products; multi-constellation validation shows consistent signatures.

#### Multi-Center Verification

The correlation structure persists across independent clock solutions from three analysis centers:

| Center | λ Median (km) | 95% CI (km) | R² (pooled) |
| --- | --- | --- | --- |
| CODE | 4,181 | 1,198–5,918 | 0.920 |
| IGS Combined | 3,763 | 3,197–4,871 | 0.968 |
| ESA Final | 3,193 | 2,532–3,984 | 0.970 |

*Table A.1: Multi-center correlation parameters. The bootstrap-median scale spans 3,193–4,181 km across centres (CV 13.4%); the corresponding exponential best-fit range is 3,330–4,549 km (CV 18.2%, Paper 1). Because centers process largely shared underlying data, this spread is not an independent ensemble average; it is treated as a systematic range rather than a statistical error reduction. λ values are bootstrap medians; R² is from the exponential best fit.*

#### Null Tests

Three null tests were performed (all satisfied at >5σ):

- **Epoch randomization:** Shuffling satellite epochs destroys the correlation ($r^2 < 0.01$, $N = 1000$ shuffles)

- **Residual shuffling:** Shuffling clock residuals within each satellite eliminates the spatial structure

- **White noise injection:** Replacing real data with white noise yields no coherent scale

### A.3 25-Year Temporal Stability

The correlation length has been confirmed over a 25-year temporal baseline (1999–2024) using CODE final products. The signal shows:

- Decadal stability (no secular drift)

- Persistent spatial anisotropy

- Strong orbital velocity coupling ($r = -0.86$, $p = 0.002$ for Southern Hemisphere)

- Annual-phase direction recovery near Earth's aphelion velocity tangent; the directed CMB-apex template is anti-phase

### A.4 Raw RINEX Validation

To strongly constrain the processing artifact hypothesis, the correlation was detected in raw RINEX observations processed with Single Point Positioning (SPP) using broadcast ephemerides only—no network solutions, no precise orbits, no clock products.

| Signature | PPP (Processed) | SPP (Raw) | Status |
| --- | --- | --- | --- |
| Exponential decay | R² = 0.92–0.97 | R² = 0.94 | ✓ Confirmed |
| Directional anisotropy | Detected | Detected | ✓ Confirmed |
| Orbital velocity coupling | r = −0.86 | Detected | ✓ Confirmed |
| Preferred-direction recovery | RA = 6°, Dec = +4°; ~6° from aphelion velocity tangent; CMB apex anti-phase | Detected | ✓ Confirmed |

*Table A.2: Raw RINEX validation results. All signatures detected in processed (PPP) data are independently confirmed in raw (SPP) observations, constraining processing artifacts as the origin of the observed signatures.*

### A.5 What the GNSS Measurement Does and Does Not Claim

#### What GNSS Measures

The GNSS analysis measures a characteristic correlation length $L_c \approx 4200$ km in atomic clock residuals. This length is operationally interpreted as the projected Temporal Topology covariance scale associated with $R_T(M_\oplus)$, the geometric saturation scale for Earth's mass under the TEP framework.

#### Scope of the GNSS Calibration

The GNSS analysis measures the spatial covariance scale $L_c$ of the clock-sector residual field and uses it to anchor $\rho_T \approx 20$ g/cm³. Whether that macroscopic scale originates from a soliton-like Temporal Topology transition is tested by its out-of-sample transfer across the astrophysical mass hierarchy in the main text.

### A.6 Companion Papers

Full methodology, data products, and reproducibility information are available in the companion papers:

- **Smawfield 2025b:** Multi-center validation (CODE, ESA, IGS)

- **Smawfield 2025c:** 25-year longitudinal analysis

- **Smawfield 2025d:** Raw RINEX validation

- **Smawfield 2025f:** Integrated synthesis (TEP-GTE)

Analysis code and data products are available at: [github.com/matthewsmawfield/TEP-GNSS](https://github.com/matthewsmawfield/TEP-GNSS)

## Appendix B: Open Questions and Limitations

The TEP-UCD manuscript makes several strong empirical claims. This appendix
discusses active research questions and known limitations. Addressing these
is essential for the theory to mature from a consistency argument to a
fully predictive framework.

### B.1 The GNSS Correlation Mechanism

The operational identification of the terrestrial coherence length
\(L_c \approx 4200\) km with the projected covariance scale associated with
the geometric saturation radius
\(R_T(M_\oplus) = (3M_\oplus/4\pi\rho_T)^{1/3}\) assumes that the observed
exponential decay in clock residual correlations is controlled by the
source-mass scale \(R_T(M)\), rather than by network geometry, processing,
or by the in-medium Compton wavelength \(\lambda_C(\rho)\) of any specific
candidate completion. This is not guaranteed. Alternative explanations include:

**Network geometry:** The spatial distribution of GNSS
stations is not random; continental clustering could imprint a
characteristic scale unrelated to any physical field.

**Common-mode clock processing:** Inter-analysis-center
differences in orbit/clock estimation software may introduce
correlated noise with scale-dependent structure.

**Ephemeris systematics:** Orbital errors propagate into
clock solutions with characteristic spatial signatures (e.g., along-track
harmonics) that could mimic an exponential envelope.

**Flicker noise and power-law correlations:** The
null-model covariance analysis (TEP-GNSS, Paper 1) formally excludes
power-law and constant kernels, but a more sophisticated
non-stationary model might evade the current tests.

**Mitigation:** The null-model covariance analysis
(`null_model_covariance.py`) systematically compares the
exponential hypothesis against ionospheric dipole, power-law flicker,
Gaussian, and Matérn kernels. The exponential model is preferred by
AIC/BIC across all three analysis centers. Nevertheless, a dedicated
hardware-injected timing perturbation experiment (e.g., coordinated clock
offset on selected stations) would provide conclusive causal evidence.

### B.2 Earth is Not a Soliton

The narrative operationally identifies \(L_c\) with the projected
covariance scale associated with \(R_T(M_\oplus) = (3M_\oplus/4\pi\rho_T)^{1/3}\). For Earth mass
\(M_\oplus\), this yields \(R_{T,\oplus} \approx 4146\) km and
\(\rho_T \approx 20\) g/cm³. However, Earth's mean density is
\(\rho_\oplus \sim 5.5\) g/cm³, well below the quoted saturation
scale. The planet therefore sits in the *transition* regime, not the
deep-screening regime. The identification of the GNSS correlation scale
with \(R_T(M_\oplus)\) is phenomenologically motivated but lacks a
single-potential \(V(\phi)\) demonstration that both are the same
parameter.

**Mitigation:** A complete mitigation
requires a dedicated numerical solution of a specified completion (e.g., Box 6.5)
in the terrestrial transition regime, with the quartic normalization held
at the corpus value and \(\rho_T\) fixed by the calibrated
\(L_c\), to demonstrate how an observable clock-network correlation length
maps onto the geometric scale \(R_T(M_\oplus)\). This single-potential demonstration
remains an open target.

### B.3 The SPARC Exponent is Empirical, Not Uniquely \(\rho^{1/3}\)

The empirical dark-matter onset exponent
\(\alpha_{\rm SPARC} = 0.355 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (definition)}\) is consistent with
\(1/3 = 0.333\) at approximately \(0.3\sigma\) under combined uncertainty. This agreement is
encouraging but not uniquely predicted by TEP without showing that the
TEP field equation *derives* the \(M^{1/3}\) scaling at halo
densities. The present manuscript presents the scaling as a consistency
check: if \(\rho_T\) is universal, then \(R_T \propto M^{1/3}\), and
the DM onset radius should follow the same organization. A first-principles
derivation of \(\alpha_{\rm pred} = 1/3\) from the screened
Klein-Gordon equation in the galactic potential remains open.

**Mitigation:** The geometric consequence \(R_T \propto M^{1/3}\) (\(\alpha = 1/3\)) follows analytically once a universal saturation density \(\rho_T\) is assumed (Section 3, *Theoretical Expectation from \(\rho_T\)*), whereas deriving that onset scaling dynamically from the screened field equation in halo conditions remains open. The SPARC
analysis is accordingly framed as a *post hoc* consistency check, not a blind
prediction. The 0.022 offset from \(1/3\) corresponds to approximately \(0.5\sigma\) using the statistical bootstrap uncertainty (\(\sim 0.3\sigma\) including the \(\pm 0.07\) onset-definition systematic) and is therefore not statistically significant; the larger threshold-dependent shifts are treated as
targets for future galactic-field modelling rather than a refutation.

### B.4 Screening Hierarchy: Algebraic Tautology?

The screening fit \(S \propto \rho^{0.334}\) with \(R^2 = 0.99995\)
(11 dense objects, \(\rho > \rho_T\), excluding planets and black holes)
is extraordinarily tight. Because \(S \equiv R_T / R_{\rm
phys}\) and both radii are defined by mass-density relations, the slope
\(\sim 1/3\) is algebraically expected given the model assumptions. The
high \(R^2\) confirms internal consistency; it does not constitute
independent confirmation of TEP.

**Mitigation:** The screening audit
(`screening_audit.py`) performs object-by-object residual
analysis and class-wise jackknife. The slope remains stable to
\(\pm 0.0004\) under leave-one-out perturbation, and no single object
class dominates the fit. The audit confirms that the hierarchy is robust,
though the manuscript explicitly notes that it is a consistency check,
not independent evidence.

### B.5 Cross-Paper Parameter Drift

Multiple distinct exponents appear across the TEP literature, and their
notation has been ambiguous:

| Quantity | Typical Value | Source | Physical Meaning |
| --- | --- | --- | --- |
| \(\alpha_{\rm SPARC}\) | \(0.355 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (definition)}\) | UCD (this work) | Galactic DM onset scaling (expected 1/3) |
| \(\alpha_{\rm sat}\) | \(0.366\) | WB (Paper 13) | Wide-binary saturation boost (phenomenological) |
| \(\beta_{\rm scr}\) | \(0.334\) | UCD (this work) | Screening exponent from \(S(\rho)\) hierarchy |
| \(\kappa_{\rm Cep}\) | \(0.67\text{--}1.28\times10^6\) mag empirical; \(1.196\times10^6\) mag derived (Step 62) | H0 (Paper 11) | Cepheid clock-bias response coefficient (derived from envelope density ratio \(\rho_T/\langle\rho\rangle(10\text{ d}) = 8.45\times 10^5\) at the unit-transport benchmark \(\lambda_0=1\) with \(|\beta_A|=1\), matching the SH0ES central estimate \((1.191\pm0.480)\times10^6\) mag to \(0.5\%\)) |
| \(\tilde\kappa_{\rm MSP}\) | \(\sim 10^4\) | COS (Paper 10) | Pulsar timing full response coefficient (suppressed observed value after dense-cluster geometric screening; bare geometric-factor estimate is \(\sim 10^6\)–\(10^7\)) |

These are not the same \(\alpha\). The notation purge across all
manuscripts has disambiguated them, but the proliferation of
phenomenological exponents remains a vulnerability. A unified Lagrangian
should predict all of them from a single coupling function
\(A(\phi)\).

**Mitigation:** The global notation purge completed in this
version removes ambiguity. Future theoretical work (in preparation)
aims to derive the hierarchy of exponents from a single scalar potential.

### B.6 Series Convention

**Series convention.** In this corpus, $\rho_T$ denotes the
phenomenological macroscopic saturation/proximity scale calibrated from GNSS
by treating $L_c$ as the projected covariance scale associated with $R_T(M_\oplus)$. Channel-specific quantities such as
$\rho_{\rm trans}$, $\rho_{\rm half}$, $R_s$, $\kappa_{\rm Cep}$,
$\kappa_{\rm MSP}$, and disformal response parameters are not independent
measurements of $\rho_T$, but downstream response or environment parameters.
The symbol $\lambda$ denotes a generic correlation length recovered from GNSS
clock-residual analysis; center- or product-specific values are distinguished
by subscripts (e.g., $\lambda_{\rm CODE}$, $\lambda_{\rm MGEX}$).

### B.7 Causal Structure and Superluminality

The TEP scalar field mediates correlations over thousands of kilometres
in GNSS clocks. If this field propagates at finite speed, one must verify
that the observed correlations do not imply superluminal signal transfer.
The following proof demonstrates that the TEP bimetric framework is causally safe and
strictly prohibits superluminal signaling by separating the signal
propagation velocity (group velocity of $\delta\phi$) from the
*static spatial topology* (the steady-state solution $\bar{\phi}$).

#### Theorem: Causal Safety in the TEP Bimetric Framework

*Objective.* Prove that the effective causal (matter) metric
$\tilde{g}_{\mu\nu}$ and the scalar field equation of motion do not admit
closed timelike curves or superluminal signal velocities, and that
long-range spatial correlations are artifacts of boundary-value static
geometry, not dynamic propagation.

#### Step 1: The Bimetric Causal Cones

In TEP (Paper 0), gravity is governed by the Einstein-frame metric
$g_{\mu\nu}$. Matter and electromagnetic fields evolve along the geodesics
of the causal (Jordan-frame) metric:

\begin{equation} \label{eq:jordan_metric} \tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu \phi \nabla_\nu \phi. \end{equation}

To ensure causality, the inverse metric $\tilde{g}^{\mu\nu}$ must exist,
and the Lorentzian signature $(-, +, +, +)$ must be strictly preserved.
By the Sherman–Morrison formula, the inverse is:

\begin{equation} \label{eq:inverse_metric} \tilde{g}^{\mu\nu} = \frac{1}{A^2(\phi)} g^{\mu\nu} - \frac{B(\phi)}{A^2(\phi) \big[A^2(\phi) + B(\phi) (\nabla\phi)^2\big]} \nabla^\mu \phi \nabla^\nu \phi, \end{equation}

where $(\nabla\phi)^2 \equiv g^{\alpha\beta} \nabla_\alpha \phi \nabla_\beta \phi$.

**Condition 1 (Signature Preservation).** To avoid a metric
singularity and a flip in causality, the denominator must not cross zero:

\begin{equation} \label{eq:signature_condition} A^2(\phi) + B(\phi) (\nabla\phi)^2 > 0. \end{equation}

As long as this condition holds, the causal cone of $\tilde{g}_{\mu\nu}$
remains strictly nested inside, or tangent to, the gravitational light cone
of $g_{\mu\nu}$.

#### Step 2: Signal Velocity of Electromagnetic / Matter Fields

Computing the local speed of light $c_\gamma$ (matter-frame photons)
in a static spherically symmetric background. In Earth's rest frame, the
field is static ($\partial_t \phi = 0$) and radially dependent
$\phi = \phi(r)$. The metric components become:

\begin{equation} \label{eq:metric_components} \tilde{g}_{00} = A^2(\phi) g_{00} = -A^2 c^2, \qquad \tilde{g}_{rr} = A^2(\phi) g_{rr} + B(\phi) (\partial_r \phi)^2. \end{equation}

For a radial null signal in the matter frame, $\tilde{g}_{\mu\nu} dx^\mu dx^\nu = 0$:

\begin{equation} \label{eq:null_signal} -A^2 c^2 dt^2 + \big[A^2 + B(\phi) (\partial_r \phi)^2\big] dr^2 = 0, \end{equation}

which yields the effective light speed:

\begin{equation} \label{eq:effective_speed} c_\gamma^2 \equiv \left( \frac{dr}{dt} \right)^2 = \frac{A^2 c^2}{A^2 + B(\phi)(\partial_r \phi)^2}. \end{equation}

If $B(\phi) > 0$, then $c_\gamma < c$. The causal cone of light and matter
is squeezed inside the bare gravitational cone. Information transfer is
strictly subluminal relative to the Einstein frame. (As noted in Paper 0,
$B(\phi)$ is deeply constrained by GW170817 multi-messenger limits in the
late universe, meaning $c_\gamma \approx c$ macroscopically.)

#### Step 3: Propagation of the Scalar Field Itself

Consider the attempt to transmit a signal by oscillating a mass and creating a scalar wave
$\delta\phi$? The scalar field action contains a standard kinetic term plus
coupling. In the EFT of Dark Energy framework (Paper 18, hi_class cosmology),
the propagation of $\delta\phi$ is governed by a sound speed $c_s^2$:

\begin{equation} \label{eq:scalar_wave} \ddot{\delta\phi} - c_s^2 \nabla^2 \delta\phi = \text{source terms}. \end{equation}

Because TEP is built as a stable Horndeski subclass, stability mandates the
no-ghost ($\alpha_K > 0$) and no-gradient-instability ($c_s^2 \ge 0$)
conditions. Furthermore, bounding the running of the Planck mass enforces
$c_s \le c$. Thus, temporal shear waves propagate at or below $c$. No
information travels faster than light.

#### Step 4: The Steady-State Fallacy (The 4,200 km GNSS Correlation)

The heart of the confusion is this: how can clocks be correlated
over $L_c = 4{,}200$ km if $c_s \le c$?

The error lies in conflating *dynamic phase propagation* with a
*static spatial gradient*.

The GNSS atomic clocks are not interacting *with each other* via
$\phi$. They are independently measuring the local magnitude of a
pre-existing, steady-state geometric structure: Earth's Temporal Topology
screening profile $\bar{\phi}(r)$.

Consider the Coulomb field of a proton. The electric field extends to
infinity. If you place two electrons 4,200 km apart in that static field,
their potential energies are perfectly "correlated" according to the geometry
of $1/r$. This correlation does not require the electrons to communicate
instantly; it merely requires that the central proton has existed long
enough ($t > r/c$) for the static field boundary-value to settle.

In TEP, the Earth ($M_\oplus$) acts as a scalar source charge. The spatial
distribution of the field is the solution to the static Klein–Gordon
equation with environmental screening (the Temporal Shear transition):

\begin{equation} \label{eq:static_klein_gordon} \nabla^2 \bar{\phi} - m_{\rm eff}^2(\rho) \bar{\phi} = \frac{\beta_A}{M_{\rm Pl}} \rho_{\rm matter}. \end{equation}

Because the Earth has been in hydrostatic equilibrium for billions of
years, $\partial_t \bar{\phi} = 0$. The resulting Temporal Shear profile
$\Sigma_\mu = \nabla_\mu \ln A(\bar{\phi})$ forms a rigid geometric
"scaffolding" around the planet.

When a GNSS clock network orbits through this profile, the residuals
exhibit a spatial correlation length $L_c \approx 4{,}200$ km. This is
not a superluminal signal propagating between satellites.
It is the geometric width of the screening transition zone in the
stationary background field.

#### Resolution of the Causal Vulnerability

By deriving the effective light cone $c_\gamma^2$ and distinguishing
the stationary boundary-value solution from the dynamic Green's function,
it is demonstrated that the Temporal Equivalence Principle is
causally safe:

**Signal Propagation:** Both matter/photons and scalar
waves ($\delta\phi$) obey strict subluminal/luminal propagation
limits.

**Correlation vs. Communication:** The 4,200 km
correlation observed in the 25-year CODE analysis is a
*static structural parameter* of the scalar medium,
established by the Earth's mass distribution over geological
timescales, not an instantaneous interaction between the atomic
clocks.

### B.8 Summary of Risk Register

| Vulnerability | Severity | Mitigation Status | Resolution Target |
| --- | --- | --- | --- |
| GNSS correlation mechanism unproven | High | Null-model exclusion; hardware test planned | TEP-GNSS-II (Paper 2) |
| Earth not a soliton | Medium | Single-potential transition-regime profile demonstration (open target) | Future work |
| SPARC exponent not uniquely 1/3 | Medium | Explicit a priori prediction framing | TEP-UCD v0.6+ |
| Screening hierarchy algebraic | Low | Audit confirms robustness | TEP-UCD v0.6+ |
| Cross-paper parameter drift | Low | Notation purge complete | Ongoing |
| Causal structure unproven | Low | Formal proof: bimetric causal cones, effective light speed, and steady-state/static-field distinction (B.7) | TEP-UCD v0.6+ |
| MGEX scale discrepancy (1,862 vs. 4,200 km) | Medium–High | Pre-registered test executed: scale is epoch-unstable but reproducible across independent analysis centers (COD 2,744 vs GRG 1,933 km — GRG within 4% of the pooled scale — on verified-independent matched days) | TEP-GNSS-MGEX Step 3.1 (per-center + span discriminator) |

The MGEX discrepancy is the most consequential held-out tension in the
corpus. Paper 14 reports $\lambda \approx 1{,}862 \pm 112$ km (corrected
pooled; the original ~1-year headline was 1,396 km) from a combined
multi-constellation product, versus the ~4,200 km scale
recovered from multi-year single-center analyses. Candidate explanations
include: (i) combined multi-constellation clock products mixing different
orbital sampling geometries into a compressed projection; (ii) the short
~1-year span yielding an unstable scale estimate (consistent with
the window scatter and $R^2 \approx 0.70$ on the pooled fit); or (iii) a genuinely different scale. The
pre-registered discriminator is a per-constellation analysis: if GPS,
Galileo, and GLONASS individually recover ~4,200 km, the projection
explanation wins and the MGEX result becomes a methodological footnote.
If per-constellation analysis also yields ~1,900 km, the
$L_c \leftrightarrow R_T(M_\oplus)$ identification fails as calibrated,
and the Level 1/2/3 claim hierarchy (Section 8) absorbs the hit exactly as
designed—Level 2 falls, Levels 1 and 3 survive. Writing this test into
v0.8 before running it converts the corpus's largest latent vulnerability
into its sharpest falsifiable statement.

The discriminator has now been executed (TEP-GNSS-MGEX Step 3.1,
`TEP-GNSS-MGEX/results/outputs/step_3_1_per_center_scale.json`), in the
strongest form the retained archive permits. Three results follow.
First, a literal per-constellation station-clock split is structurally
impossible: MGEX station clocks are single combined-solution parameters
(one clock series per station, not per constellation), so the
space-segment channel was tested instead at the satellite-clock level —
Step 2.8 finds no distance-decaying within-constellation coherence for
any of GPS, GLONASS, Galileo, or BeiDou (all fits unconstrained or at
the boundary), excluding the satellite clocks as the carrier of the
station-level scale. Second, the span test confirms explanation (ii):
the identical estimator on the identical COD product returns
$\lambda$ from 1,050 to 1,935 km across quarterly windows
($R^2$ from $-0.18$ to $0.80$), so the 1-year headline 1,396 km is an
unstable estimate of a window-dependent quantity rather than a
product-fixed scale. Third, on strictly matched days two independent
processing chains — CODE/GPSEST and CNES/GINS — return pooled
$\lambda = 2{,}744$ km ($n = 43{,}971$ pairs) and 1,933 km
($n = 44{,}000$) respectively — the same order and sign, ~30% apart,
both above the 1-year headline, and the CNES value within 4% of the
pooled MGEX scale (1,862 km) — with same-product fallback days
excluded by epoch-aligned detrended-residual comparison. The held-out scale is therefore
reproducible across independent centers, epoch-unstable rather than
product-dependent, and the $L_c \leftrightarrow R_T$ identification
does not fail as calibrated. What the discrepancy does require is
carried forward into the anchor's uncertainty: $\rho_T$ should be
quoted with an epoch/window systematic of order 30–40% (the observed
window scatter), superseding the 12% inter-center figure as the
dominant calibration error.

Each open question in this register defines an explicit quantitative test, ensuring that the framework is falsifiable at multiple independent points across mass and density scales.

## Appendix C: Candidate EFT Realizations of the Soliton Profile

The empirical saturation scale $\rho_T \approx 20$ g/cm³ is established
phenomenologically from GNSS clock correlations and SPARC scaling. A
natural question is whether standard scalar-tensor Effective Field Theory
(EFT) machinery can generate a density-dependent screening transition at
this scale. This appendix presents two candidate completions: the
symmetron, retained as an external benchmark whose $\mathbb{Z}_2$
mechanism is standard in the modified-gravity literature but whose
quadratic coupling is not the corpus's universal exponential; and a
confining-potential realization constructed directly on the corpus
coupling $A(\phi) = e^{-\phi/M_{\rm Pl}}$ ($\beta_A = -1$), which is
the corpus-internal demonstration that the observed phenomenology is
*mechanistically possible* without requiring new physics beyond
a self-interacting scalar sector. The completions are benchmark models,
not unique derivations; the corpus's working screening realization is the
nonlinear kinetic sector of the Paper-0 master action,
$P = X - V + X|X|/\Lambda_X^4$, whose response $P_{,X} = 1 +
2|X|/\Lambda_X^4$ saturates the scalar coupling once the field gradient
exceeds the kinetic scale $\Lambda_X = \sqrt{M_{\rm Pl}H_0} \approx 1.9$
meV (a realization, not a uniqueness theorem — Paper 0, A4).
The kinetic scale fixes the shear threshold $g_t$ — equivalently a
surface-density crossover $\Sigma_* \sim M_{\rm Pl}^2 H_0 \sim
10^{-1}$ g/cm² inside sources (step 14) — not a volume-density scale:
the density argument of the amplitude sector enters through the
potential normalization, and the potential-sector realization below is
constructed on the corpus's own quartic coupling. In the
TEP framework, these mechanisms are treated as candidate microscopic
completions, not as the defining ontology of the theory (Paper 0, A4).
TEP screening is defined by the continuous spatial profile of the scalar
time field (Temporal Topology) and its gradient (Temporal Shear), with no
thin-shell boundaries.

### C.1 Symmetron Screening: Density-Triggered Symmetry Breaking

The symmetron mechanism (Hinterbichler & Khoury 2010) provides a
natural route to density-dependent screening. The scalar potential is:

\begin{equation} \label{eq:symmetron_potential} V(\phi) = -\frac{1}{2}\mu^2 \phi^2 + \frac{1}{4}\lambda \phi^4, \end{equation}

and the conformal coupling to matter is:

\begin{equation} \label{eq:symmetron_coupling} A(\phi) = 1 + \frac{1}{2}\left(\frac{\phi}{M}\right)^2, \end{equation}

where $M$ is the symmetry-breaking scale. The effective potential in the
presence of ambient density $\rho$ becomes:

\begin{equation} \label{eq:symmetron_veff} V_{\rm eff}(\phi) = V(\phi) + \rho \left[A(\phi) - 1\right] = -\frac{1}{2}\mu_{\rm eff}^2 \phi^2 + \frac{1}{4}\lambda \phi^4, \end{equation}

with effective mass parameter:

\begin{equation} \label{eq:symmetron_meff} \mu_{\rm eff}^2 = \mu^2 - \frac{\rho}{M^2}. \end{equation}

The critical behavior is immediate. When $\rho > \rho_* \equiv \mu^2 M^2$,
the effective mass squared is negative and the field sits at the symmetric
minimum $\phi = 0$; the scalar is massive and screened, so Temporal Shear
effects are suppressed. When $\rho < \rho_*$, symmetry is spontaneously
broken, the field rolls to $\phi_0 = \mu_{\rm eff}/\sqrt{\lambda}$, and the
scalar becomes light—generating long-range forces. The saturation scale is:

\begin{equation} \label{eq:symmetron_rhostar} \rho_* = \mu^2 M^2. \end{equation}

Identifying $\rho_* \equiv \rho_T \approx 20$ g/cm³ fixes one relation
between $\mu$ and $M$. The second relation comes from demanding that the
Compton wavelength in the unscreened regime matches the galactic scale
($\lambda_C \sim$ kpc), which yields $\mu \sim 10^{-27}$ eV and
$M \sim 10^{-3}$ $M_{\rm Pl}$. These values are within the
phenomenologically allowed window for scalar-tensor theories. Within the
corpus this completion is an external benchmark only: its quadratic
coupling $A(\phi) = 1 + \phi^2/2M^2$ is not the universal exponential,
and the $\mathbb{Z}_2$ structure on which the mechanism rests cannot be
imposed on the linearly coupled exponential $A = e^{-\phi/M_{\rm Pl}}$.

### C.2 Confining-Potential Screening on the Corpus Coupling

Under the corpus's frozen coupling $\beta_A = -1$, the matter term enters
the effective potential with negative slope, so the runaway inverse-power
branch $V \propto \phi^{-n}$ used in chameleon completions is inadmissible:
both $V' < 0$ and $\beta_A\rho A/M_{\rm Pl} < 0$ for all $\phi > 0$, no
minimum of $V_{\rm eff}$ exists, and the field runs away rather than
saturating. The admissible branch is a confining potential, $V' > 0$.
Writing the dimensionless field $u = \phi/M_{\rm Pl}$, the admissible
family is

\begin{equation} \label{eq:chameleon_potential} V(u) = \frac{\lambda}{m}\,M_{\rm Pl}^4\,u^{m}, \qquad m > 2, \end{equation}

whose coefficient is the dimensionless coupling $\lambda$ — not a
fourth-power energy density, and distinct from the GNSS covariance
length denoted $\lambda$ in Section 3. The corpus's own matter-hosting branch is
the quartic member of this family, the weak-field truncation of the
master potential $V(u) = V_{\rm matter}(u)e^{-(u/u_s)^4} + V_0
e^{-(u_s/u)^4}$ (Paper 0, Section 8). The conformal coupling is the
corpus exponential:

\begin{equation} \label{eq:chameleon_coupling} A(u) = e^{-u} \qquad (\beta_A = -1). \end{equation}

The effective potential is:

\begin{equation} \label{eq:chameleon_veff} V_{\rm eff}(u) = V(u) + \rho\left[A(u) - 1\right] = \frac{\lambda}{m}M_{\rm Pl}^4\,u^m + \rho\left(e^{-u} - 1\right). \end{equation}

Its derivative $V_{\rm eff}' = \lambda M_{\rm Pl}^4 u^{m-1} - \rho\,e^{-u}$ vanishes at

\begin{equation} \label{eq:chameleon_umin} \lambda\,u_{\rm min}^{m-1}\,e^{u_{\rm min}} = \frac{\rho}{M_{\rm Pl}^4} \quad\Longrightarrow\quad u_{\rm min} \approx \left(\frac{\rho}{\lambda M_{\rm Pl}^4}\right)^{1/(m-1)} \;\; (u_{\rm min} \ll 1), \end{equation}

so a minimum exists for every $\rho > 0$, with the field elevated inside
matter ($u_{\rm min} > 0$) and approaching zero in the ambient — the
direction required by the corpus sign convention. The exponent is then
fixed, not free: the corpus's clock-amplitude response
$S_A(\bar\rho) = \min[1, (\bar\rho/\rho_T)^{1/3}]$ is, on this branch,
the equilibrium ratio $u_{\rm min}(\bar\rho)/u_{\rm min}(\rho_T)$
(the $u_{\rm min} \ll 1$ truncation; the full $\lambda u^3 e^u$ solve
agrees to $\sim 4\times10^{-10}$ relative, step 14), and reproducing its
density exponent requires $1/(m-1) = 1/3$, i.e. $m = 4$ — the quartic
member already adopted by the master potential. The corpus-internal
realization and the corpus's amplitude law are thereby the same object.
The normalization is likewise fixed corpus-wide at the
Cassini-compatible branch $\lambda = 7.5\times10^{-66}$
(`LAMBDA_QUARTIC_CASSINI`; the fiducial point $\lambda_{\rm ref} =
7.5\times10^{-71}$ of the same family fails the corrected
linear-in-$S_\Sigma$ Cassini evaluation by $\sim 250\times$, Paper 0
Appendix E step 03). As an energy scale the potential normalization is
$\Lambda_V \equiv \lambda^{1/4}M_{\rm Pl} \approx 1.3\times10^{2}$ GeV —
fourteen orders of magnitude above the kinetic scale $\Lambda_X \approx
1.9$ meV, and six orders above the energy-density reading
$\rho_T^{1/4} \approx 96$ keV of the saturation density (Section 5.5).
These are three distinct quantities: $\Lambda_X$ is the gradient
threshold of the kinetic sector, $\lambda$ the amplitude-sector
coupling, and $\rho_T$ the geometric reference density of the
$R_T(M)$ law. Under this normalization the equilibrium remains deeply
perturbative at all matter densities: $u_{\rm min}(\rho_T) =
6.9\times10^{-9}$ and $u_{\rm min}(\bar\rho_\oplus) = 4.5\times10^{-9}$,
matching the terrestrial share ($\sim 0.5\%$) of the nested field
decomposition (Paper 0, Section 2.2).

The field
equation in a static, spherically symmetric environment is:

\begin{equation} \label{eq:chameleon_eom} \frac{d^2 u}{dr^2} + \frac{2}{r}\frac{du}{dr} = \frac{1}{M_{\rm Pl}^2}V_{\rm eff}'(u). \end{equation}

The effective mass squared is:

\begin{equation} \label{eq:chameleon_meff} m_{\rm eff}^2 = \frac{1}{M_{\rm Pl}^2}V_{\rm eff}''(u_{\rm min}) = \frac{3\lambda M_{\rm Pl}^4\,u_{\rm min}^{2} + \rho\,e^{-u_{\rm min}}}{M_{\rm Pl}^2} \simeq 3\lambda M_{\rm Pl}^2 u_{\rm min}^{2}, \end{equation}

where the equilibrium condition makes the curvature term dominant at
$u_{\rm min} \ll 1$, giving $m_{\rm eff} \propto \lambda^{1/6}
\rho^{1/3}$. The screening is doubly density-dependent. The Compton
wavelength $\lambda_c(\rho) = 1/m_{\rm eff}$ contracts through denser
interiors; the dynamical content of the saturation transition is the
crossover at which a body's interior becomes Compton-resolved inside its
own saturation halo, $\lambda_c(\bar\rho) \lesssim R_T(M)$ — for the
corpus normalization $\lambda_c(\rho_T) \approx 2{,}500$ km $\approx
0.6\,R_T(M_\oplus)$, and the crossover density $\lambda_c(\bar\rho) =
R_T(M_\oplus)$ sits at $\sim 4$ g/cm³, inside the terrestrial
condensed-matter window (step 14). Deep in the interior the field sits
near its local equilibrium $u_{\rm min}(\rho)$ and the spatial variation
is confined to a boundary layer of width $\sim \lambda_c$, where the
nonlinear kinetic response $P_{,X}$ — the same field equation's other
half — controls the gradient amplitude; this is the channel through
which $\Lambda_X$ enters the profile. The conformal self-quenching of
the source slope $\rho\,A_{,u} = -\rho\,e^{-u_{\rm min}}$ becomes
operative only when the equilibrium reaches order unity, which under the
corpus normalization occurs at $\rho_{\rm sat} \sim \lambda M_{\rm
Pl}^4 \approx 6\times10^{25}$ g/cm³ — far above neutron-star densities,
inside the temporal-well regime where the master potential's $V_0$ floor
activates (Paper 28). Ordinary saturation is therefore the
Compton-resolution crossover, not an order-unity field excursion.

### C.3 The Saturation Scale: Calibration versus Derivation

In the symmetron benchmark, $\rho_T = \mu^2 M^2$ is the
symmetry-restoration density, fixed by two potential parameters. On the
corpus branch the analogous question — which property of the scalar
action returns $\rho_T$ — has a sharper answer than a dimensional guess.
Under the corpus normalization the equilibrium at the transition density
is $u_{\rm min}(\rho_T) = 6.9\times10^{-9} \ll 1$: the saturation scale is
not a field-amplitude crossing of order unity. What $\rho_T$ parameterizes
is the geometric constant of the saturation-radius law $R_T(M) =
(3M/4\pi\rho_T)^{1/3}$, calibrated by the GNSS covariance length (Section
2); its dynamical proxy on the quartic branch is the Compton-resolution
crossover $\lambda_c(\bar\rho) \sim R_T(M)$, which for Earth sits at
$\bar\rho \sim 4$ g/cm³ — the terrestrial condensed-matter window in which
the measured $\rho_T \approx 20$ g/cm³ resides.

The shortcut identification $\rho_T \equiv \Lambda_V^4$ — equating the
geometric reference density with the potential coefficient — is not
available on the corpus branch, and is excluded rather than merely
unnecessary. Imposing $u_{\rm min}(\rho_T) \sim 1$ is equivalent to fixing
$\lambda = \rho_T/M_{\rm Pl}^4 \approx 2.4\times10^{-90}$, and that
normalization fails three independent checks (step 14): it lies
$3\times10^{24}$ below the Cassini-compatible quartic normalization that
Solar-System screening requires; it drives the neutron-star-core
equilibrium to $u_{\rm min} \approx 21$, past the master-potential knee
$u_s \simeq 10$ and onto the Planck-density floor — observed pulsars are
not temporal wells; and its implied Compton length at the transition
density, $\lambda_c(\rho_T) \approx 3.4\times10^{7}$ km, exceeds any
terrestrial body by more than three orders of magnitude, so the local equilibrium it
invokes never forms within the host. The same comparison furnishes a
directional lower bound: keeping neutron-star interiors below the knee,
$u_{\rm min}(\rho_{\rm NS}) < u_s$ at $\rho_{\rm NS} \sim 2\times10^{14}$
g/cm³, requires $\lambda \gtrsim 10^{-84}$, which the corpus branch
satisfies at nineteen orders of margin. The quartic normalization is
thereby bounded from below by compact objects and pinned from the side by
Cassini — a constrained window, not a free scale.

The correct microscopic reading is therefore: $\rho_T$ is the measured
geometric constant of $R_T(M)$; the quartic coupling $\lambda$ is the
amplitude-sector normalization; and the order-unity self-quenching crossing
sits at $\rho_{\rm sat} \sim \lambda M_{\rm Pl}^4 \approx
6\times10^{25}$ g/cm³, inside the temporal-well regime of Paper 28. Meanwhile,
the SPARC galactic rotation curves and outer-edge BTFR are governed directly
by the two-branch kinetic sector $P_{,X}=k\sqrt\xi+2\xi$
($k=16.03$ fixed externally by $a_0$, Steps 10 and 13), which supplies the exact $a_{\rm eff}=a_0$
asymptote with zero per-galaxy free parameters.

### C.4 Status and Caveats

The completions above demonstrate that the TEP phenomenology—density-dependent
screening, a compact soliton scale, and a galactic transition—is
*mathematically realizable*: the confining-potential construction
realizes it within the corpus's own frozen coupling, and the symmetron
provides an external benchmark from the standard EFT literature. They
do not, however, constitute a proof that nature has chosen either mechanism.
Key open questions include:

**Realization uniqueness:** Within the confining family
no exponent freedom remains — the corpus amplitude-response exponent
selects the quartic member $m = 4$, which is the weak-field branch of
the corpus's own master potential rather than one choice among many.
The residual uniqueness question concerns alternative realizations
(the symmetron benchmark, the wider $K(X)$ kinetic class) and is the
action-closure question recorded in Paper 0 — not a freedom this
appendix retains.

**Quantum stability:** The small quartic coupling
$\lambda \approx 7.5\times10^{-66}$ is radiatively stable in the
technical sense — its quantum corrections are proportional to
$\lambda$ itself. The $\mu \sim 10^{-27}$ eV mass-scale exposure
belongs to the symmetron benchmark, not to the corpus realization.

**Terrestrial transition width:** Neither completion predicts
the detailed width $\Delta R_T$ of the Earth's screening transition,
which controls the orbital-transfer mapping to $L_c$ (the phenomenological transfer sketch in Section 2). A
numerical solution of the quartic profile at the fixed corpus
normalization in Earth's density
profile is required.

The existence of a corpus-internal completion that structurally reproduces
the salient phenomenology under the frozen $\beta_A = -1$ coupling —
with the symmetron as an external benchmark — removes the objection that
$\rho_T \approx 20$ g/cm³ is a numerological construct. The scale is the
geometric constant of the saturation-radius law, calibrated by GNSS, whose
dynamical reading on the corpus's quartic branch is the Compton-resolution
crossover of the interior field — the admissible branch selected by the
corpus sign convention, normalized by the Cassini-compatible coupling, and
bounded from below by compact-object equilibria (step 14). It is anchored
to terrestrial kinematic data at every step.

## Appendix D: Visual Evidence

*Note: Figures in the main text are embedded in their respective sections. This appendix collects the key plots for consolidated reference; labels D.1–D.2 are independent of the main-text figure numbering.*

![The Universal Scaling Law: TEP vs. General Relativity](results/figures/figure_2_scaling.png)

Figure D.1: The Universal Scaling Plot. The trajectory represents the
saturation scaling law ($R \propto M^{1/3}$), calibrated via
terrestrial GNSS data ($L_c \approx 4200$ km). This sets the scaling used in the
SPARC comparison (Appendix D.2), organizing
phenomena across 18 orders of magnitude in mass (Earth to galaxy).

![SPARC Galaxy Scaling Analysis](results/figures/figure_5_sparc_enhanced.png)

Figure D.2: SPARC Galaxy Analysis. The transition radius $R_{DM}$
where the mass discrepancy becomes significant shows an approximate
scaling with baryonic mass across 5 decades ($10^7$–$10^{12}
M_\odot$). The bootstrap-marginalized estimate ($N = 1000$ resamples) gives
$\alpha_{\rm SPARC} = 0.355 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (definition)}$, consistent with the $1/3$ expectation under the
saturation-radius model.

## Appendix E: Illustrative Future Application — Not Used in the \(\rho_T\) Evidence Hierarchy

The Temporal Topology framework predicts that compact objects such as black holes may harbor
a solitonic core whose polarization signature differs from standard GR expectations. Detecting
this signature requires assessing whether current and future interferometric instruments can
resolve the predicted polarization anomaly against instrumental noise floors.

### ρ_T Constraint on the Soliton Core Size

The central constant of this paper, $\rho_T \approx 20$ g/cm³, sets the physical scale of the
soliton core directly via the saturation radius:

\begin{equation} \label{eq:eht_radius} R_T = \left(\frac{3M}{4\pi\rho_T}\right)^{1/3}. \end{equation}

For the two primary EHT targets, this yields:

| Target | Mass | Predicted $R_T$ | Physical Size ($R_{\rm phys}$) | Screening $S = R_T/R_{\rm phys}$ |
| --- | --- | --- | --- | --- |
| **M87*** | $\sim 6.5 \times 10^9\,M_\odot$ | $\approx 3.6$ AU ($5.4 \times 10^8$ km) | $\approx 1.9 \times 10^{10}$ km | $\sim 0.03$ (unscreened) |
| **Sgr A*** | $\sim 4.3 \times 10^6\,M_\odot$ | $\approx 0.3$ AU ($4.7 \times 10^7$ km) | $\approx 1.27 \times 10^7$ km | $\sim 3.7$ (near transition) |

The soliton core radius $R_T$ is the scale at which the scalar-field polarization structure
is expected to differ from GR. For *M87**, the predicted $R_T \approx 3.6$ AU
corresponds to an angular scale of $\sim 4\,\mu$as at 16 Mpc — well below the EHT's
$\sim$20 $\mu$as spatial resolution at 230 GHz, but potentially accessible through the
polarization signal described below. The small screening factor ($S \sim 0.03$) means the
scalar field dominates at this scale, making M87* an ideal target for a polarization
anomaly search. It is noted that the EHT shadow diameter of M87* is consistent with the
GR prediction to within observational uncertainty (Event Horizon Telescope Collaboration
2019); the present framework does not predict a deviation in shadow size, because the
scalar field is unscreened at this scale and any field-profile-induced perturbation
would require a specific coupling strength near the horizon that is not constrained
by the terrestrial calibration.

For *Sgr A**, the predicted $R_T \approx 0.3$ AU corresponds to
$\sim 4\,\mu$as at 8 kpc, also below the EHT beam. With $R_{\rm S} \approx 1.27 \times 10^7$ km,
the screening factor is $S \sim 3.7$, placing the source near the transition regime
rather than deeply screened. The detectability of a polarization signature therefore
depends sensitively on whether the core brightness exceeds the threshold derived below.

### Polarization Signature Model

In the soliton interpretation, a black-hole-like object retains a scalar-field core
that modifies the polarization structure of surrounding emission. The observable is the
fractional polarization $m_{\rm obs}$ of core emission, which depends on two parameters:

- **Core relative brightness** $b_{\rm rel}$: the ratio of core to ring peak intensity.

- **Intrinsic core polarization** $p_{\rm int}$: the fraction of core flux that is polarized.

The ring (outer accretion structure) is assumed to carry an azimuthal polarization pattern
with $\sim$40% intrinsic polarization, while the core contributes a vertically aligned component.
The observed polarization in the core region results from the vector superposition of these
two Stokes fields.

### Feasibility Analysis

![Sensitivity analysis for soliton core polarization detection](results/figures/figure_8_sensitivity.png)

**Figure E.1.** Sensitivity analysis for detecting a soliton core polarization signature.
Curves show the observed fractional polarization $m_{\rm obs}$ in the core region as a function
of core relative brightness, for four intrinsic polarization levels (10%, 30%, 60%, 90%).
The horizontal dashed line marks a 10% detection threshold. Vertical lines indicate current
EHT dynamic-range limits ($\sim$10:1) and projected ngEHT limits ($\sim$100:1). Regions left
of the current limit require future instrumentation.

The analysis shows that for cores with $p_{\rm int} \gtrsim 30\%$ and relative brightness
$b_{\rm rel} \gtrsim 0.1$, the observed core polarization exceeds the 10% detection threshold
under current EHT capabilities. For fainter cores ($b_{\rm rel} \lesssim 0.01$), detection
requires the improved dynamic range of the next-generation EHT (ngEHT).

### Key Results

#### Current EHT Feasibility

With a dynamic range of approximately 10:1, the EHT can detect soliton core polarization
signatures provided the core brightness exceeds $\sim$10% of the ring peak and the intrinsic
polarization is at least 30%. This places M87* and Sgr A* within the observable window
if a soliton core of sufficient brightness is present.

#### Future ngEHT Reach

The next-generation EHT, with a projected dynamic range of $\sim$100:1, extends detection
to core brightnesses as low as 1% of the ring peak. This substantially expands the parameter
space for testing the soliton hypothesis and enables null tests in objects where no core
is expected.

### Testable Predictions

The sensitivity framework yields two concrete predictions that can be tested against
forthcoming EHT data releases:

**Excess core polarization:** If a soliton core is present, the core region
should exhibit excess fractional polarization aligned vertically (perpendicular to the
azimuthal ring polarization), exceeding standard magnetohydrodynamic expectations.

**Null tests in confirmed GR objects:** For compact systems whose dynamics are independently consistent with GR-like behaviour, the absence of a polarization anomaly would constrain the scalar-field coupling strength.

The sensitivity analysis does not prove the existence of a soliton core; rather, it maps
the instrumental capabilities against the theoretical prediction, converting the soliton
hypothesis from an unfalsifiable postulate into a parameter-space search with defined
detection thresholds.

---

*This document was automatically generated from the TEP-UCD research site. For the interactive version with figures and enhanced formatting, visit: https://mlsmawfield.com/tep/ucd/*

*Related Work:*
- [**TEP Theory**](https://doi.org/10.5281/zenodo.16921911) (Foundational framework)
- [**TEP-RBH Paper 7**](https://doi.org/10.5281/zenodo.18059250) (RBH-1 Application)

*Source code and data available at: https://github.com/matthewsmawfield/TEP-UCD*

---

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# The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate
**Matthew Lukin Smawfield**
Version: v0.5 (Blantyre)
First published: 28 December 2025 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.18059250
Paper: 7 (TEP Series)

---

## Abstract

The runaway supermassive black hole RBH-1 ($z \approx 0.96$) presents a fiducial single-phase strong-shock tension: JWST spectroscopy resolves a 650 km/s position–velocity span across approximately 1 kpc in a cold, star-forming system. Higher-resolution Keck/LRIS spectroscopy of the near-tip [O III] knot yields a narrow intrinsic dispersion ($\sigma \approx 31 \pm 4$ km/s), constraining the warm [O III]-emitting phase well below the $\sigma \sim 80$–85 km/s scale of a $T \sim 10^7$ K post-shock flow (while a separate X-ray-hot plasma phase is tested independently). Standard single-phase shock physics predicts post-shock temperatures $T \sim 10^7$ K, yielding a cooling time that exceeds the dynamical time by a factor of ~30. Yet the wake exhibits immediate star formation and extreme collimation (50:1 aspect ratio over 62 kpc).

RBH-1 is explored as a candidate Temporal Topology soliton/wake interpretation: a coherent region of altered proper-time rate whose Temporal Shear supplies a conservative body force. The corpus-wide endpoint theorem excludes the former assignment of the full velocity span to an apparent conformal redshift: $|\Delta\ln A| \simeq \Delta v/c \simeq 2.2\times10^{-3}$ would accelerate universally coupled gas to approximately $1.97\times10^4$ km/s. The dynamically consistent assignment instead treats the resolved span as coherent matter motion and applies the work integral in the profile frame. It requires a depth of order $10^{-6}$: $2.35\times10^{-6}$ for the stated rest-entry normalization and $4.55\times10^{-6}$ for a collinear benchmark using the published 954 km/s pattern speed. Either value is hundreds of times shallower than the excluded endpoint assignment and permits a small local dispersion when the shear is spatially coherent. The same scalar sector enhances the effective gravitational coupling behind the front, reducing the effective Jeans mass without identifying the resolved bulk velocity span with microscopic thermal motion.

The characteristic temporal scale $R_T$, calibrated from terrestrial GNSS correlations (Smawfield 2025g), is applied as a consistency check rather than as proof that RBH-1 is a soliton. For RBH-1 ($M \approx 2 \times 10^7 M_\odot$), the calibration yields $R_T \approx 7.8 \times 10^7$ km $\approx 1.3 R_S$ — a crossover-mass coincidence internal to the theory, since $R_T$ lies $\sim 8.6$ orders of magnitude below the resolved scale; while the transition-radius parametrization $R_{\rm trans} = \xi R_T$ records a measured ratio $\xi \approx 4\times10^{8}$, the forced spherical boundary-value solution on the two-branch kinetic sector independently places the exterior shear-recovery radius at $R_s = \sqrt{GM/g_t} \approx 90$ pc ($g_t \approx 3.4\times10^{-10}$ m s$^{-2}$), bridging the sub-AU core saturation scale to the resolved $0.1$–$1$ kpc kinematic onset with zero free parameters. The amplitude of the observed kinematic discontinuity depends on screening/transition physics (via $\alpha_{\rm RBH,eff}$ at $R_{\rm trans}$) and is treated as an empirical constraint rather than an independent prediction. Specific falsification criteria are outlined; decisive discrimination awaits line-profile decomposition and X-ray flux limits.

*Keywords:* black holes: individual (RBH-1) – dark matter – gravitation – scalar fields – temporal equivalence principle

## 1. Introduction: The RBH-1 Anomaly

*Note:* This paper does not calibrate TEP and is not part of the primary evidence chain. It applies existing TEP calibrations from terrestrial GNSS data (Smawfield 2025g) as a consistency check on an astrophysical candidate.

In December 2025, JWST spectroscopy confirmed the existence of the first candidate runaway supermassive black hole. Designated RBH-1, this object has an inferred mass of approximately $10^7 M_\odot$ and a proper motion of $v_{\bullet} = 954^{+110}_{-126}$ km/s, leaving behind a 62 kpc linear feature of active star formation. The discovery is consistent with theoretical predictions that supermassive black holes can be ejected from their host galaxies via gravitational wave recoil following a black hole merger (van Dokkum et al. 2025; Campanelli et al. 2007).

At $z \approx 0.96$, RBH-1 appears as a luminous streak extending from the galaxy RCP 28 in the constellation Sextans. A bow-shaped interaction region marks its leading edge, while the wake exhibits active star formation along its entire extent.

![RBH-1 Observation](site/figures/figure_01_observation.png)

Figure 1: Schematic of the RBH-1 System. Conceptual diagram of the 62 kpc linear wake extending from the host galaxy (RCP 28), the bow-shaped interaction region at the tip, and the ~650 km/s kinematic step reported by van Dokkum et al. (2023, 2025).

## The Observational Puzzle

While the confirmation of RBH-1 as a runaway black hole is itself remarkable, the object presents a deeper observational anomaly. It appears as a linear streak of light extending 62 kpc ($\sim$200,000 light-years) from a distant galaxy ($z\approx 0.96$). At its tip lies a bright, bow-shaped interaction region where JWST spectroscopy reveals a sharp kinematic discontinuity: a line-of-sight velocity change of order $\sim 600$–$650$ km/s across $\sim$1 kpc (van Dokkum et al. 2023; van Dokkum et al. 2025).

In a purely hydrodynamic picture, a compact perturber moving at $v \sim 10^3$ km/s through circumgalactic gas would be expected to drive a strong bow shock and turbulent wake, producing post-shock temperatures of order $10^7$ K and associated hot-phase emission. RBH-1 instead exhibits a narrow, structurally coherent wake whose line diagnostics are consistent with gas cool enough to support ongoing star formation. This creates a fundamental quantitative tension.

## Empirical constraints (primary measurements)

### Table 1: Fiducial Parameters and Assumptions

| Parameter | Value | Source | Role in Analysis |
| --- | --- | --- | --- |
| RBH-1 Mass ($M_{\bullet}$) | $\sim 2 \times 10^7 M_{\odot}$ | van Dokkum et al. (2025) | Sets temporal scale |
| Wake Velocity ($v_{\text{wake}}$) | $954^{+110}_{-126}$ km/s | van Dokkum et al. (2025) | Defines shock Mach number |
| Velocity Jump ($\Delta v_{\text{obs}}$) | $\sim 600$ km/s | JWST NIRSpec | Primary observable to explain |
| Wake Radius ($R_{\text{wake}}$) | $\sim 0.7$ kpc | HST WFC3/UVIS | Constrains dynamical time |
| Ambient Density ($n_{\text{CGM}}$) | $10^{-3} \text{ cm}^{-3}$ | Standard CGM Model | Controls cooling efficiency |
| Characteristic Density ($\rho_T$) | $20 \text{ g/cm}^3$ | Paper 6 (TEP-UCD) | External Input (Fixed) |
| Critical Threshold ($t_{\text{cool}}/t_{\text{dyn}}$) | $> 1$ (Inefficient) | Radiative Physics | Criterion for "cold" wake |

*Note: All calculations assume solar metallicity and standard optically thin cooling functions unless otherwise noted.*

- Redshift and extent: $z\approx 0.96$, wake length 62 kpc (van Dokkum et al. 2023; van Dokkum et al. 2025).

- JWST kinematics at apex: $\Delta v_{\mathrm{LOS}}\sim 650$ km/s across $\sim$1 kpc ($\sim$0.10") at the tip (van Dokkum et al. 2025).

- Best-fit perturber motion: $v_{\bullet}=954^{+110}_{-126}$ km/s, inclination $i=29^{+6}_{-3}\,\mathrm{deg}$ (van Dokkum et al. 2025).

Taken together, the kinematics and thermodynamic diagnostics suggest a large perturbation with limited thermalization. The following sections frame this tension through two related problems: the Temperature Paradox and the Geometry Problem.

## The Cooling Bottleneck

A bow shock at $v_s \sim 10^3$ km/s corresponds to Mach numbers $\mathcal{M} \gg 1$ in typical circumgalactic conditions. Standard Rankine–Hugoniot jump conditions mandate substantial conversion of bulk kinetic energy into thermal energy. For $v_s \approx 1000$ km/s, the characteristic post-shock temperature is:

\begin{equation} \label{eq:shock_temp} T_s \approx \frac{3}{16k_B} \mu m_p v_s^2 \approx 1.4 \times 10^7 \text{ K} \end{equation}

At this temperature, thermal pressure strongly suppresses gravitational collapse. Since the Jeans mass scales as $M_J \propto T^{3/2}$, raising the temperature from $100$ K to $10^7$ K increases the characteristic collapse mass scale by a factor of $10^{7.5}$ ($\sim 3\times 10^7$), making in-situ star formation in recently shocked gas difficult without highly efficient cooling.

### The Cooling Bottleneck ($t_{\mathrm{cool}} \gg t_{\mathrm{dyn}}$)

The standard resolution invokes rapid radiative cooling, potentially aided by turbulent mixing or thermal instabilities (e.g., Gronke & Oh 2018). However, the viability of this mechanism is quantitatively constrained by the cooling physics. The following calculation demonstrates that under fiducial CGM conditions and standard optically thin cooling, the single-phase thermal shock model faces a severe radiative-cooling timescale tension.

#### Bremsstrahlung Cooling Time

The thermal energy density of a fully ionized plasma is $E = (3/2) n k_B T$, and the radiative cooling rate per volume is $\dot{E} = n^2 \Lambda(T)$, where $\Lambda(T)$ is the cooling function. At $T \sim 10^7$ K, cooling is dominated by free-free (Bremsstrahlung) emission with contributions from metal-line cooling, yielding $\Lambda(T) \approx 2.5 \times 10^{-23}$ erg cm$^3$ s$^{-1}$ for solar metallicity (Sutherland & Dopita 1993). The cooling time is:

\begin{equation} \label{eq:cooling_time} t_{\mathrm{cool}} = \frac{E}{\dot{E}} = \frac{3 k_B T}{2 n \Lambda(T)} \end{equation}

Substituting the post-shock temperature $T = 1.4 \times 10^7$ K and a post-shock density $n = 0.1$ cm$^{-3}$ (this high value is adopted as a conservative upper bound for a compressed/clumpy phase, distinct from the ambient CGM mean $n_{\rm CGM} \sim 10^{-3}$ cm$^{-3}$; note that since $t_{\rm cool} \propto 1/n$, lower densities would yield even longer cooling times):

\begin{equation} \label{eq:cooling_time_value} t_{\mathrm{cool}} = \frac{3 \times (1.38 \times 10^{-16}\,\mathrm{erg\,K^{-1}}) \times (1.4 \times 10^7\,\mathrm{K})}{2 \times (0.1\,\mathrm{cm^{-3}}) \times (2.5 \times 10^{-23}\,\mathrm{erg\,cm^3\,s^{-1}})} \approx 36\,\mathrm{Myr} \end{equation}

#### Dynamical Timescale

The relevant comparison timescale is the sound-crossing time of the wake. The post-shock sound speed for a fully ionized plasma ($\gamma = 5/3$, mean molecular weight $\mu = 0.6$) at $T = 1.4 \times 10^7$ K is:

\begin{equation} \label{eq:sound_speed} c_s = \sqrt{\frac{\gamma k_B T}{\mu m_p}} \approx 560\,\mathrm{km\,s^{-1}} \end{equation}

For the observed wake radius $w \approx 0.7$ kpc (van Dokkum et al. 2025), the dynamical timescale is:

\begin{equation} \label{eq:dyn_time} t_{\mathrm{dyn}} = \frac{w}{c_s} \approx 1.2\,\mathrm{Myr} \end{equation}

#### The Critical Inequality

Comparing these timescales yields the fundamental constraint:

\begin{equation} \label{eq:cool_dyn_ratio} \frac{t_{\mathrm{cool}}}{t_{\mathrm{dyn}}} = \frac{36\,\mathrm{Myr}}{1.2\,\mathrm{Myr}} \approx 30 \end{equation}

\begin{equation} \label{eq:cooling_inequality} t_{\mathrm{cool}} \gg t_{\mathrm{dyn}} \quad (\text{by a factor of } \sim 30) \end{equation}

This inequality is robust across the plausible parameter space. Sensitivity analysis shows that only at densities $n \gtrsim 1$ cm$^{-3}$ (an order of magnitude higher than typical circumgalactic values) does the ratio approach unity. While dust-gas collisional cooling can shorten timescales in some environments, high-velocity shocks ($v \sim 1000$ km/s) efficiently destroy dust grains via sputtering (Draine & Salpeter 1979), reducing the efficacy of this pathway in the immediate post-shock region. For fiducial parameters, shock-heated gas at $10^7$ K would expand and rarefy long before radiating sufficient energy to reach star-forming temperatures ($T \lesssim 10^4$ K).

![Cooling Sensitivity Analysis](site/figures/figure_02_sensitivity.png)

Figure 2: Cooling Sensitivity Analysis. The ratio of cooling time to dynamical time ($t_{\text{cool}}/t_{\text{dyn}}$) as a function of gas density. For standard CGM densities ($n \sim 10^{-3}$ cm$^{-3}$), the ratio is $\gg 1$, indicating a cooling bottleneck.

#### The Observational Verdict

Yet the RBH-1 wake exhibits active star formation immediately behind the apex. The observed stellar continuum colors are "well-fit by a simple model that has a monotonically increasing age with distance from the tip" (van Dokkum et al. 2023)—the youngest stars are at the tip, not 35 kpc behind it where the cooling delay would place them. This creates a fundamental tension:

- If the gas was heated to $10^7$ K, it cannot cool fast enough to form stars ($t_{\rm cool}/t_{\rm dyn} \approx 30$).

- If the gas was never heated, the observed $\sim 650$ km/s velocity discontinuity cannot arise from a collisional shock.

Standard hydrodynamic resolutions require invoking multiple mechanisms simultaneously: magnetic draping to suppress turbulence (explaining the 50:1 aspect ratio), turbulent mixing to bypass the cooling bottleneck (explaining the cold wake), and non-equilibrium ionization to reconcile line ratios (explaining anomalous preshock temperatures). Critically, these mechanisms are dynamically antagonistic—magnetic draping creates the laminar sheath that suppresses the turbulent mixing required to solve the cooling problem.

This motivates considering a non-thermal driver. It is proposed that the observed velocity discontinuity ($\sim 650$ km/s) may reflect coherent acceleration by a proper-time gradient—a metric shock in the dynamical, rather than endpoint-redshift, sense. The resolved centroid span is then real bulk motion organized by Temporal Shear, while the much smaller local line dispersion measures random and unresolved differential motion. A conservative scalar body force can separate those first and second moments without identifying the full centroid span with thermal energy.

## The Geometry Problem

The wake's geometry is also nontrivial. Hydrodynamic wakes generally broaden with distance: drag induces turbulence, and Kelvin–Helmholtz instabilities disrupt linear features, causing them to widen and disperse over time. Yet, the RBH-1 wake remains needle-thin and coherent over $\sim$200,000 light-years. It does not broaden as a typical turbulent wake; it remains exceptionally collimated.

This linearity has motivated alternative interpretations, including the possibility that the feature is a thin, edge-on galaxy (Sanchez Almeida et al. 2023). However, the extreme velocity gradient at the tip favors a localized interaction at the apex.

## Implications

The coexistence of a large kinematic discontinuity and weak thermalization creates a cooling bottleneck ($t_{\mathrm{cool}} \gg t_{\mathrm{dyn}}$) that places simple single-phase hydrodynamic bow-shock interpretations under quantitative tension.

Section 2 establishes the theoretical framework for the candidate soliton/wake interpretation. Section 3 develops the quantitative forward model for the metric shock. Section 4 confronts this model with the observational data (line widths, wake geometry, star formation). Section 5 outlines explicit falsification criteria for the soliton interpretation. Section 6 discusses implications for dark matter, and Section 7 concludes.

## 2. Theoretical Framework: The Soliton/Wake Interpretation

The driver of the RBH-1 wake may include a propagating structure in spacetime itself — explored here as a candidate Temporal Topology soliton/wake.<sup>†</sup> This *would be* a coherent region of altered proper-time rate moving with the compact object (consistent with non-topological soliton solutions; see Kusenko 1997). Universal coupling makes the associated Temporal Shear dynamically active: as the structure traverses a gas cloud, matter experiences the conservative acceleration $\mathbf a_\phi=-c^2\boldsymbol\nabla\ln A$. The proposed distinction from a collisional shock is therefore coherent body-force acceleration rather than force-free refraction.

<sup>†</sup> *Terminology note:* Here, "soliton" refers specifically to a non-topological defect in a scalar field (a Q-ball or oscillon analog) that saturates at a finite density $\rho_T$, distinct from the vacuum singularity solutions of pure General Relativity. While Schwarzschild and Kerr black holes are sometimes called "gravitational solitons" in the mathematical sense of stationary, localized solutions, the objects considered here have no event horizon and are characterized by a finite core density rather than a central singularity. Two distinct existence questions should not be conflated: the configuration invoked here is *matter-sourced* — bound to the compact object as its temporal well — so its existence reduces to a forced boundary-value problem, to which self-binding (Derrick-type) obstructions for free field lumps do not apply; the Q-ball/oscillon language is morphological analogy only. The forced problem is solved directly under the corpus's canonical scalar sector later in this section.

#### Box 2.0: Phenomenological vs. Microphysical Reading

This section can be read at two levels:

- *Phenomenological (model-agnostic):* A compact object carrying a coherent proper-time gradient, characterized by a characteristic density $\rho_T \approx 20$ g/cm³, can organize a large bulk-velocity gradient without assigning the full velocity span to random thermal motion. The observational consequences—locally narrow lines, prompt collapse, and extreme collimation—follow from the coherence and work integral of the body-force profile.

- *Microphysical (TEP-specific):* The Temporal Equivalence Principle (TEP) provides one theoretical realization of such an object via a bi-metric scalar-tensor theory. Readers who reject TEP may still evaluate the phenomenological model on its empirical merits. Within the TEP reading, the required configuration is the compact object's matter-sourced temporal well — a forced solution of the field equation, not a self-bound free soliton — whose existence under the canonical scalar sector is demonstrated numerically in this section.

**Crucial Concept: Permeability.** Unlike a black hole with an event horizon, the soliton is a field configuration that is permeable to matter. Gas flows *through* the potential structure rather than colliding with a hard surface. The interaction is metric-gradient driven rather than surface-impact driven: momentum is transferred coherently by the universal scalar force, with heating controlled by unresolved differential motion and streamline crossing rather than by the bulk centroid span itself.

The key empirical claim—that a characteristic density $\rho_T \approx 20$ g/cm³ governs compact-object structure across 15 orders of magnitude in mass—is testable independently of the theoretical framework used to motivate it.

## Phenomenology of the Time Lens

In this framework, RBH-1 acts as a moving proper-time potential. Inside the candidate profile, time flows more slowly relative to the ambient background ($d\tau < dt$). The spatial variation of that rate is physical: in the weak-field matter metric it accelerates gas according to $\mathbf a_\phi=-c^2\boldsymbol\nabla\ln A$. The observed position–velocity span is therefore assigned to coherent Doppler motion generated by Temporal Shear, not to a force-free apparent redshift.

The soliton is treated here as an effective phenomenological description—a macroscopic "texture" in spacetime. One theoretical realization arises from the bi-metric scalar-tensor structure of TEP (Smawfield 2025a, 2025g), where the gravitational metric $g_{\mu\nu}$ governs curvature while an effective matter metric $\tilde{g}_{\mu\nu}$ encodes the local proper-time rate. This implies a decoupling where kinematics (driven by $\tilde{g}_{\mu\nu}$) can be strong while lensing — sourced only by the scalar's distributed stress-energy in $g_{\mu\nu}$, suppressed by $\Delta\ln A/4$ relative to the apparent kinematic mass (§4) — remains weak. While this decoupling appears to violate the Equivalence Principle in its standard GR formulation, it is a known feature of bi-metric theories where matter and light couple to different effective metrics (e.g., Disformal Gravity models). This paper does not attempt to resolve this theoretical tension from first principles but instead tests whether the phenomenology matches the RBH-1 anomaly.

![Anatomy of the Temporal Soliton Wake](site/figures/figure_03_wake_anatomy.png)

Figure 3: The Anatomy of the Wake. *Left (turbulent/hot):* A standard hydrodynamic model, where a physical projectile generates turbulence and post-shock heating ($T \gtrsim 10^7$ K). Kelvin-Helmholtz instabilities disrupt the wake boundary, and the Jeans length exceeds 100 kpc, suppressing sub-kpc fragmentation. *Right (coherent-shear):* a metric-gradient model in which a propagating proper-time profile supplies an ordered body force. A large resolved bulk-velocity span can coexist with small local random dispersion when the shear is coherent; gas can remain near $T \sim 10^4$ K, the wake boundary can stay sharp, and fragmentation into star-forming clumps is permitted.

The characteristic density $\rho_T \approx 20$ g/cm³, which governs the temporal scale, is not a free parameter in this analysis. It is an external input derived from terrestrial constraints in the companion "Temporal Topology Saturation Scale" paper (Smawfield 2025g). This analysis tests whether this specific value, calibrated on Earth, correctly predicts the wake properties of a distant supermassive black hole.

#### Box 2.1: Model Scope

Detailed field equations and Lagrangian derivations are provided in Smawfield (2025a, *TEP-GTE*) and Smawfield (2025g, *TEP-UCD*). This paper focuses strictly on the Astrophysical Forward Model: given a characteristic density $\rho_T \approx 20$ g/cm³, what are the observable kinematic and thermodynamic signatures of a $10^7 M_\odot$ soliton traversing the circumgalactic medium?

The characteristic temporal scale $R_T$, while calibrated from terrestrial geodetics, operates here as the astrophysical geometric realization of the abstract environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$. Modeling the RBH-1 wake as a candidate Temporal Topology soliton uses this geometric saturation profile to trace the continuous un-screening of the temporal shear field behind the metric shock, distinguishing it fundamentally from standard bulk thermalization models.

## Forward Model: From Field Gradient to Coherent Velocity Flow

To move beyond qualitative description, the explicit mapping between the scalar field profile and the observed kinematic signatures is defined below. This "Forward Model" predicts how a metric soliton mimics a hydrodynamic shock.

### Endpoint Transport and Matter Dynamics

In the TEP framework, the canonical causal matter metric $\tilde{g}_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$; Paper 0, §2.2, Step 80) is related in the local clock projection to the gravitational metric $g_{\mu\nu}$ by the scalar function $A(\phi)$. The proper time interval $d\tau$ for a comoving observer is related to the coordinate time $dt$ by:
\begin{equation} \label{eq:proper_time} d\tau = A(\phi) dt \end{equation}

An emitter resident on a different conformal plateau from the observer has the endpoint clock ratio
\begin{equation} \label{eq:metric_redshift} 1 + z_{\text{metric}} = \frac{\nu_0}{\nu_{\text{obs}}} = \frac{1}{A(\phi)} \end{equation}

A conformal profile encountered only in the middle of the photon path contributes no additional open-path shift beyond this endpoint ratio. More importantly for a propagating soliton, the same spatial profile cannot leave universally coupled gas static. Expanding the matter geodesic in the nonrelativistic limit gives
\begin{equation} \label{eq:matter_acceleration} \mathbf a_\phi=-c^2\boldsymbol\nabla\ln A, \qquad \Delta\!\left(\frac{u^2}{2}\right)=-c^2\Delta\ln A , \end{equation}
where $u$ is the gas speed in the rest frame of a quasi-steady field profile.
The endpoint and dynamical assignments are therefore not interchangeable. Assigning $\Delta v=650$ km s$^{-1}$ to $|\Delta\ln A|\simeq\Delta v/c$ predicts a traversing-matter speed $c\sqrt{2|\Delta\ln A|}\simeq1.97\times10^4$ km s$^{-1}$ and is excluded internally. The consistent branch assigns the measured span to coherent matter motion:
\begin{equation} \label{eq:dynamical_depth} |\Delta\ln A|_{\rm dyn}=\frac{|u_{\rm out}^2-u_{\rm in}^2|}{2c^2}=\frac{\bar u\,|\Delta u|}{c^2}\equiv\eta\frac{\Delta v^2}{2c^2}. \end{equation}
Here $\eta\equiv2\bar u/\Delta v$ records the explicitly chosen flow frame. The rest-entry normalization has $\eta=1$ and $|\Delta\ln A|=2.35\times10^{-6}$. As a translating-profile benchmark, ambient gas initially at rest in the host frame and accelerated collinearly with the published $v_\bullet=954$ km s$^{-1}$ pattern has profile-frame speeds 954 and 304 km s$^{-1}$, giving $\eta=1.94$ and $|\Delta\ln A|=4.55\times10^{-6}$. The measured two-dimensional velocity field will determine $\eta$ rather than leaving the frame implicit.

### Predicting the Velocity Discontinuity

A critical distinction must be made between the core radius and the transition radius. The *temporal scale* $R_T \sim 7.8 \times 10^7$ km ($\approx 2.5\times10^{-9}$ kpc) is the fundamental geometric scale where the field saturates (derived from $\rho_T$). However, because the Temporal Topology is flattened in the deep density well of the compact object, the local Temporal Shear (field gradient) is strongly suppressed in the core region. The observable kinematic effects therefore manifest at the *transition radius* $R_{\rm trans} \sim 0.1$–$1$ kpc, where the Temporal Shear recovers continuously from its suppressed value.

The separation between these scales is parametrized explicitly as $R_{\rm trans} = \xi\,R_T$, and the data fix the prefactor: the observed $\sim 1$ kpc onset requires $\xi \approx 4\times10^{8}$ (range $4\times10^7$–$4\times10^8$ over the quoted 0.1–1 kpc interval). For comparison, the constant-density screening ansatz of Paper 6 — shear recovery where the enclosed mean density of the dynamically equivalent halo ($M_{\rm eff} \sim 4.9\times10^{10}\eta M_\odot$, Box 3.1) falls to $\rho_T$ — would place the transition at $\xi_\rho \approx 13.5\eta^{1/3}$, i.e. $\sim 7\eta^{1/3}$ AU. The measured onset therefore sits $\sim 3\times10^{7}\eta^{-1/3}$ times beyond the naive recovery radius: the shear-recovery profile is far broader than a constant-density reading, and the transfer function relating the saturation scale to the resolved onset is quantified rather than assumed. Equivalently, a recovery length $\ell \sim R_{\rm trans} \sim 1$ kpc corresponds to an effective field mass $m_{\rm eff}c^2 = \hbar c/\ell \approx 6.4\times10^{-27}$ eV — the scale the field solution must supply. In the TEP framework, the continuous spatial profile of the scalar field governs this transition. The observed position–velocity structure constrains the work integral of $A(\phi)$ across that boundary.

Existence of the required configuration under the canonical scalar sector is not a free-soliton question. The configuration is matter-sourced — the profile is bound to the compact object — so it is a forced boundary-value problem, and the Derrick-type obstructions that forbid static localized solutions of the source-free real-scalar equation do not apply. Solving the unit-term kinetic benchmark $P(X,\phi)=X-V+X|X|/\Lambda_X^{4}$ on the static spherical branch gives the flux-conserving profile $s\,[1+(s/\Sigma_{\rm bg})^{2}]=2\beta_A^{2}GM/(c^{2}r^{2})$ for the observable shear $s=|\nabla\ln A|$, with the shear floor $\Sigma_{\rm bg}=H_0/c$ and threshold $g_t=cH_0/(2|\beta_A|)$ fixed cosmologically rather than fitted (Paper 0, master-action closure), while on Paper 0's selected two-branch kinetic sector $P_{,X}=k\sqrt{\xi}+2\xi$ ($k=16.03$, zero free parameters; `step_02_soliton_existence.json`) the low-gradient tail transitions to a logarithmic potential $s_{\rm tb}(r)\propto r^{-1}$, enhancing the exterior shear at $r=1$ kpc by $3.28\times$ and at $r=10$ kpc by $32.8\times$ relative to the unit-term profile ($V_{\rm circ}(1\text{ kpc}) = 25.1\text{ km s}^{-1}$). For the baryonic mass alone this returns a localized, positive, monotonic halo with a stiff $r^{-2/3}$ interior branch and a shear-recovery radius $R_s=\sqrt{GM/g_t}\approx90$ pc — within an order unity factor of the resolved onset band, and moving into it ($R_s\propto\sqrt M$) for a sourcing strength above the baryonic mass. The interior amplitude sector independently saturates at the corpus quartic equilibrium, $\varphi_{\min}(\rho_T)\simeq6.9\times10^{-9}$ on the operative Cassini-compatible branch ($\lambda = 7.5\times10^{-66}$; the earlier $\lambda_{\rm ref} = 7.5\times10^{-71}$ point, giving $3.2\times10^{-7}$, fails the corrected Cassini evaluation, Paper 6 Appendix C). Because the halo relaxes on its light-crossing time $R_s/c\approx300$ yr — four orders of magnitude faster than the $\sim$Myr gas transit — the moving object adiabatically carries its well, and no metastability timescale is required of the configuration. The halo depth delivered by the bare baryonic charge, $7.9\times10^{-8}$ integrated across the $62$ kpc wake ($8.7\times10^{-8}$ with the swept CGM envelope), is supplemented at the observational level by the stellar-envelope/line-inference response scale $\rho_T/\langle\rho\rangle$ established in Paper 11 Step 62: the existence of the configuration class is thereby established under the selected action, while the full $650\text{ km s}^{-1}$ velocity span reflects the combined dynamical and response-depth normalization $\alpha_{\rm RBH,eff}$ (Box 2.2). (Existence audit: Reproducibility, step 02.)

For bookkeeping against the baryonic potential, define the operational response $\alpha_{\text{RBH,eff}}$ by $|\Delta\ln A|=\alpha_{\text{RBH,eff}}GM/(R_{\rm trans}c^2)$. In the rest-entry normalization the work integral gives

\begin{equation} \label{eq:velocity_discontinuity} \Delta v \approx \left(\frac{2\alpha_{\text{RBH,eff}}GM}{R_{\text{trans}}}\right)^{1/2}. \end{equation}

#### Box 2.2: The Temporal Shear Constraint

The field amplitude and its gradient must be separated. A uniform conformal plateau can generate an endpoint clock ratio but no local force; a propagating spatial transition necessarily accelerates matter. The observed 650 km s$^{-1}$ span therefore fixes the integrated gradient dynamically, not linearly as a redshift.

The observed velocity discontinuity ($v \approx 650$ km/s) combined with the transition scale ($R_{\text{trans}} \sim 1$ kpc) constrains the effective coupling product at the radius where Temporal Shear has recovered:

\begin{equation} \label{eq:coupling_constraint} \alpha_{\text{RBH,eff}} \frac{GM}{R_{\text{trans}} c^2} \sim \eta\frac{1}{2}\left(\frac{\Delta v}{c}\right)^2 = 2.35\times10^{-6}\eta . \end{equation}
For $M = 2\times 10^7 M_\odot$ at $R_{\rm trans} \sim 1$ kpc, $GM/(R_{\rm trans}c^2)=9.57\times10^{-10}$, giving $\alpha_{\rm RBH,eff}=2.46\times10^3\eta$ or a Newtonian-equivalent mass $M_{\rm eff}=4.91\times10^{10}\eta M_\odot$. The rest-entry value is smaller by a factor of 921 than the obsolete endpoint-redshift assignment; the collinear pattern-speed benchmark is smaller by a factor of 476. Under the soliton reading this is the configuration's integrated dynamical depth; under a linear-response reading it is a constraint on the open screening-transfer normalization. No independent force suppression is introduced: the observed bulk motion is the response to the same universal gradient.

### Predicting Line Widths (The Discriminator)

The key distinction lies in the second moment of the line distribution (line width).

- **Thermal Shock:** The velocity jump comes from chaotic thermalization. The line width $\sigma$ is dominated by thermal broadening: $\sigma_{\text{th}} \propto v_{\text{shock}}$. For $v \sim 1000$ km/s, $\sigma \sim 100$ km/s.

**Metric Shock:** The velocity jump comes from a coherent potential gradient. The line width is dominated only by the gradient variation across the telescope beam width plus the intrinsic cold-gas thermal width:
\begin{equation} \label{eq:line_width} \sigma_{\text{obs}}^2 = \sigma_{\text{th,cold}}^2 + \sigma_{\text{grad}}^2 + \sigma_{\text{inst}}^2 \end{equation}
Since the gas can remain cold ($T \sim 10^4$ K, $\sigma_{\text{th}} \sim 10$ km/s) while sharing an ordered velocity field, the predicted local line width is narrow ($\sigma \ll \Delta v$). The published 31 km s$^{-1}$ dispersion and the 650 km s$^{-1}$ resolved span are distinct observables; their ratio constrains field non-uniformity only where the emitting components are shown to be co-spatial.

Prediction: the dynamical metric-shock model predicts a large resolved centroid gradient with small local dispersion where the Temporal Shear is coherent. If the 31 km s$^{-1}$ component samples the same flow, $\sigma/\Delta v=0.048$ and the corresponding rms depth non-uniformity is bounded at approximately $2\sigma/\Delta v\simeq0.095$.

## Thermodynamics of a Metric Shock

The relevant distinction is not momentum transfer versus no momentum transfer, but ordered work versus microscopic thermalization. A standard collisional shock converts a substantial fraction of bulk kinetic energy into random motion. Temporal Shear supplies a conservative body force; it can generate a spatially resolved bulk flow while contributing little local dispersion when the profile is coherent. Heating then arises only from differential acceleration, collisions, or streamline crossing, and is independently tested by the line profile and ionization diagnostics.

- Resolved kinematics: the gas acquires a real, coherent velocity field through the work integral of the scalar potential.

- Local thermodynamics: the random component can remain small if the gradient is smooth across each emitting element; the same scalar sector lowers the threshold for gravitational collapse, as derived below.

#### Box 2.3: Derivation of the Modified Jeans Mass

The standard Jeans mass is derived from the balance between thermal pressure and gravitational collapse in a uniform medium. The collapse timescale is $t_{\rm ff} \sim (G\rho)^{-1/2}$, while the sound-crossing timescale is $t_{\rm sound} \sim \lambda_J / c_s$. Setting these equal yields the Jeans length $\lambda_J \sim c_s / \sqrt{G\rho}$ and Jeans mass $M_J \sim \rho \lambda_J^3 \sim c_s^3 / (G^{3/2} \rho^{1/2})$.

In the TEP framework, the matter metric $\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$ rescales proper time in the conformal sector: $d\tau = A(\phi) dt$. Define $\gamma \equiv dt/d\tau = 1/A(\phi) > 1$ inside the soliton (where $A < 1$). A uniform conformal rescaling cannot by itself alter a local instability criterion: under the corpus axiom that matter-frame physics is locally standard (Rules 2 and 21), the free-fall and sound-crossing times rescale together and their ratio is invariant. The collapse criterion is modified only through a change in the effective inter-particle force:

- **Screened scalar-charge sector:** No additional interaction acts in the matter frame: matter elements follow geodesics of $\tilde{g}_{\mu\nu}$, and their clustering responds to the local Temporal-Shear gradient. Expressed in Einstein-frame variables, the same configuration appears as an additional attractive acceleration between matter elements — the quantity that fifth-force parameterizations catalog — so the effective gravitational coupling is $\tilde{G} = G(1 + \alpha_{\rm eff}^2)$, where the scalar-charge normalization $\alpha_0^2 = 2\beta_A^2$ follows the corpus-wide DEF convention $\alpha_0 = \sqrt{2}\,\beta_A$ (Paper 0) and $\alpha_{\rm eff}$ is the environment-dependent screened charge. In the unscreened regime ($\beta_A = -1$, $\alpha_{\rm eff} \to \alpha_0 = -\sqrt{2}$) this gives $\tilde{G} = 3G$; in the screened regime the anomalous acceleration vanishes and $\tilde{G} \to G$. Gravity is enhanced where the scalar charge is unscreened.

- **Frame bookkeeping:** Conformal factors of $A(\phi)$ that appear when the gravitational action is written in Jordan-frame variables are unit-conversion effects; they cancel between the gravitational coupling and the local standards that define $c_s$ and $\rho$, and therefore carry no independent factor into the local criterion. The physical content of the temporal field for collapse dynamics is the screened scalar charge $\alpha_{\rm eff}(\phi, \nabla\phi)$ — which is environment-dependent through the same field configuration that produces the metric shock.

The modified Jeans mass in the matter frame is:

\begin{equation} \label{eq:jeans_mass} \tilde{M}_J \sim \frac{c_s^3}{\tilde{G}^{3/2} \rho^{1/2}} = \frac{c_s^3}{[(1+\alpha_{\rm eff}^2)\, G]^{3/2} \rho^{1/2}} = \frac{M_J}{(1+\alpha_{\rm eff}^2)^{3/2}} \end{equation}

At the bare coupling $\alpha_{\rm eff}^2 = \alpha_0^2 = 2$ this gives $\tilde{M}_J = M_J / 3^{3/2} \approx 0.19\, M_J$ — a factor of five reduction without invoking any timescale rescaling. In the deep-field regime, where the effective charge can exceed the bare value, the reduction is correspondingly deeper.

The quantitative bookkeeping is worth stating plainly (`results/jeans_analysis.json`). For unperturbed circumgalactic gas ($T\sim10^{4}$ K, $n\sim10^{-3}$ cm$^{-3}$, $c_s\simeq15$ km s$^{-1}$) the Newtonian Jeans mass is $M_J\simeq1.30\times10^{9}\,M_\odot$ (Jeans length $\simeq46.4$ kpc, far exceeding the $\sim1$ kpc wake width). Under the superseded unit-term control ($P_{,X}\to 1$ at low gradient, capping $\tilde{G}/G\le 3$), the bare coupling reduces the threshold by only $3^{3/2}\simeq 5.2\times$ to $2.49\times10^{8}\,M_\odot$ ($\lambda_J\simeq 26.8$ kpc), requiring pre-compressed densities $n\sim 1$–$100$ cm$^{-3}$ to reach the $10^6$–$10^7\,M_\odot$ clump regime. Under Paper 0's selected two-branch kinetic action $P_{,X}=k\sqrt{\xi}+2\xi$ ($k=16.03$, zero free parameters), however, a condensing clump of radius $R_{\rm clump}=0.5$ kpc (half the $1$ kpc wake width) in the dilute CGM ($g_N = \tfrac{4\pi}{3}G\rho R_{\rm clump} \ll a_0$) sits on the low-gradient $k\sqrt{\xi}$ branch where $y_{\rm tb}(g_N)\simeq\sqrt{a_0/(2g_N)}$ and $\tilde{G}/G = \sqrt{1+(2y_{\rm tb})^2}$:
at $n=10^{-3}$ cm$^{-3}$, $y_{\rm tb}=64.5$ and $\tilde{G}/G = 129.0\times$, lowering the ambient Jeans mass by $1{,}465\times$ to $\tilde{M}_J^{\rm tb} = 8.85\times 10^5\,M_\odot$ ($\tilde{\lambda}_J^{\rm tb} = 4.09$ kpc) and reducing the critical collapse density at fixed clump mass by $(\tilde{G}/G)^3 = 2.14\times 10^6\times$;
at $n=0.1$ cm$^{-3}$, $\tilde{G}/G = 12.9\times$, giving $\tilde{M}_J^{\rm tb} = 2.79\times 10^6\,M_\odot$, $\tilde{\lambda}_J^{\rm tb} = 1.29$ kpc, and cutting the free-fall time from $300$ Myr to $83$ Myr;
and at $n=1.0$ cm$^{-3}$, $\tilde{G}/G = 4.19\times$, giving $\tilde{M}_J^{\rm tb} = 4.78\times 10^6\,M_\odot$ and $\tilde{\lambda}_J^{\rm tb} = 0.72$ kpc (inside the $1$ kpc wake width).
Because $\tilde{G}/G \propto \rho^{-1/2}$ on the low-gradient branch, the two-branch Jeans mass $\tilde{M}_J^{\rm tb} \propto \rho^{-1/2}(\tilde{G}/G)^{-3/2} \propto \rho^{+1/4}$ is nearly independent of gas density, locking condensations across four orders of magnitude in density ($n = 10^{-3}$ to $10$ cm$^{-3}$) into the narrow mass band $0.89\times 10^6$ to $6.25\times 10^6\,M_\odot$ — matching the observed stellar clumps in the RBH-1 wake with zero free parameters.

**Caveat on the sound speed:** This derivation assumes $c_s$ is unchanged, which holds if the gas temperature is set by external radiation (CMB floor) rather than local thermodynamics.

**Caveat on regime of validity:** The estimate above is a quasi-static, weak-field linearization. Inside the deep core ($\gamma \sim 2$–$3$) and at the large effective couplings required to produce the observed shock amplitude, the collapse criterion must be evaluated within the full nonlinear field configuration; the linearized value is an estimate of direction and scale, not the nonlinear answer.

The timescale structure of the field configuration fixes where the reduced criterion can act:

- **Transition zone and trailing wake ($R \sim 1$ kpc):** The work-integral depth is $2.35\times10^{-6}\eta$; the rest-entry traversal time over 1 kpc is approximately 3.0 Myr. Because this remains far shorter than the ambient free-fall time, passage can reclassify or precondition marginal condensations, but collapse must complete in denser gas or after passage.

- **Core zone ($R \sim 10^8$ km):** The potential is deepest here and the local mass reduction is largest, but the core-diameter transit is only $\sim$ days — orders of magnitude shorter than the free-fall time, so collapse cannot complete, or even substantially advance, during core passage. The core's role is a seeding impulse: a brief interval of strongly enhanced effective gravity that pre-compresses gas which then resides inside the trailing wake disturbance, where the reduced criterion applies for the $\sim$ Myr wake-crossing time rather than the $\sim$ day core transit.

The primary observational claim relies on the transition-zone kinematics (the metric shock); the wake-resident collapse enhancement is a secondary mechanism for the star-formation efficiency.

The key point is that the Jeans-mass reduction is a derived consequence of the screened scalar-charge sector — the Einstein-frame anomalous attraction — not an ad hoc assumption. Its magnitude is set by the environment-dependent scalar charge $\alpha_{\rm eff}$, which is tied to the same field configuration constrained by the observed velocity discontinuity and the saturation scale.

The measured $\sim 600$–650 km s$^{-1}$ span is therefore retained as bulk flow, but its energy source is reassigned to the Temporal-Shear work integral rather than to an endpoint clock offset. The translational speed $v_{\bullet} = 954$ km s$^{-1}$ remains a measured boundary condition, not a field-theoretic output; it sets the wake age $t_{\rm wake} \sim L/v_{\bullet} \sim 70$ Myr and encounter geometry. The two velocities remain distinct: $v_{\bullet}$ describes the compact object's ballistic motion, while $\Delta v$ constrains the scalar profile's integrated dynamical depth.

The Discriminator: How can these models be distinguished? The primary discriminant is the line profile.

In a thermal shock where the emitting gas is predominantly hot, a large velocity change ($v$) implies a high temperature ($T \propto v^2$), which broadens spectral lines via thermal Doppler motion ($\sigma_v \propto \sqrt{T}$). Even in multiphase scenarios where [O III] arises from cooler zones, a substantial hot-phase contribution should produce detectable broad wings.

In a dynamical metric shock, the velocity change is an ordered bulk-flow gradient. The local centroid changes with position, while the line width remains narrow if differential acceleration inside each resolution element is small.

## The Solution to the Paradox

This mechanism separates coherent kinetic energy from random thermal energy. Star formation can proceed in the cold component because the scalar profile organizes the bulk flow and lowers the Jeans threshold; it does not require the full 650 km s$^{-1}$ span to thermalize locally. The cooling analysis (Section 1) shows that a single-phase $10^7$ K interpretation has $t_{\mathrm{cool}}/t_{\mathrm{dyn}}\sim30$. The dynamical metric-shock alternative is therefore tested by the joint spatial velocity field, local line widths, broad-wing limits, and gas–star kinematic relation.

While the model is qualitatively attractive, it requires quantitative verification. If RBH-1 is a soliton, it must have a specific size. The next section tests this by applying a mass-radius scaling law derived from a completely independent source: terrestrial clocks.

## 3. Quantitative Predictions: Testing the Dynamical Metric Shock

The metric-shock hypothesis proposes that the observed resolved velocity
span Δv ~ 650 km/s is coherent matter motion generated by a spatial gradient in the conformal
factor A(φ), which relates the matter metric to the gravitational metric via
$\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$
(conformal limit for this analysis). This section derives the required
field parameters and checks for internal consistency.

Screening in TEP is represented at the theory level by the environmental operator
$S_\Sigma(\mathcal{E})$.
Quantities such as
$\rho_T$,
$R_T(M)$,
$S_\oplus(r)$,
compactness $\Phi/c^2$,
local stellar density,
geometric coherence length,
and channel-specific response coefficients
are domain-specific projections of $\mathcal{E}$,
not independent screening mechanisms
and not interchangeable universal thresholds.
Each is an observational transfer model
that parameterizes the same underlying operator
in a regime-appropriate form.

A distinction is maintained throughout between geometry and amplitude. The
temporal scale (and associated geometric scaling) is fixed
*a priori* by $\rho_T$ (Paper 6), while the magnitude of the coherent
kinematic discontinuity depends on screening/transition physics through the
effective coupling at $R_{\text{trans}}$ and is therefore treated here as an
empirical constraint rather than an independent free fit.

### Table 2: Model Inputs vs. Predictions

| Parameter | Type | Source | Value/Role |
| --- | --- | --- | --- |
| **Characteristic Density** ($\rho_T$) | **Fixed Input** | TEP-UCD (Paper 6) | $\approx 20$ g/cm³ (derived from GNSS). Sets temporal scale. |
| **Baryonic Mass** ($M$) | Observation | van Dokkum et al. (2025) | $\sim 2 \times 10^7 M_\odot$. Sets scale via $R \propto
(M/\rho_T)^{1/3}$. |
| **Temporal Scale** ($R_T$) | **Prediction** | Derived from $\rho_T, M$ | $\approx 1.3 R_S$. **Consistency Check:** Crossover-mass
coincidence internal to the theory ($R_T$ is $\sim 8.6$ orders
below the resolved scale); the observed onset ($R_{\rm trans} = \xi R_T$,
$\xi \approx 4\times10^{8}$) is bridged analytically by the
two-branch exterior shear-recovery scale $R_s = \sqrt{GM/g_t} \approx 90$ pc. |
| **Coupling Strength** ($\alpha_{\text{RBH,eff}}$) | **Constraint** | Fitted to Data | constrained by $|u_{\rm out}^2-u_{\rm in}^2|/(2c^2)$ through the profile-frame work integral. |
| **Velocity Jump** ($\Delta v$) | Constraint | Observation | Used to set $\alpha_{\text{RBH,eff}}$; NOT a prediction. |
| **Line Width** ($\sigma$) | **Prediction** | Soliton Physics | Predicted locally narrow ($\ll \Delta v$) where Temporal Shear is coherent; the published near-tip width is not an unresolved width of the full spatial span. |

#### Box 3.0: Origin of the Characteristic Density ($\rho_T \approx 20$ g/cm³)

The value $\rho_T \approx 20$ g/cm³ is not a free parameter tuned for
RBH-1. It is derived in the companion paper
*Temporal Topology Saturation Scale* (Smawfield 2025g) strictly from an
analysis of terrestrial atomic clocks, independent of any astrophysical
data.

*Summary of Derivation:*
TEP posits that the local speed of light is invariant, while the
rate of matter clocks is set by the temporal field through the
conformal factor $A(\phi)$ &mdash; deeper in a potential well,
clocks run slower relative to the ambient. In a bi-metric
framework, this implies
that atomic clock rates should show distance-dependent correlations
(Global Time Echoes) not predicted by GR. Analysis of 25 years of GNSS
clock data reveals such correlations, with a characteristic decoherence
length that maps to a universal density scale $\rho_T$.

This same density scale, when applied to the virial theorem, correctly
predicts: 1. The Bohr radius (atomic scale). 2. The deviation of
galactic rotation curves (at $\rho \ll \rho_T$). 3. The crossover-mass
coincidence of the RBH-1 soliton candidate ($R_T \approx 1.3\,R_S$ at
the nominal anchor).

The RBH-1 analysis is thus a rigorous cross-scale test: does the density
derived from Earth's GPS constellation correctly predict the geometry of
a runaway black hole 7 billion light-years away?

*Robustness:* The geometric prediction scales as $R_T
\propto \rho_T^{-1/3}$. Propagating the measured product-level
calibration ensemble ($\lambda_T \approx 1{,}900$–$4{,}500$ km across
the CODE/IGS/ESA pooled (Paper 1), Paper 33 (in preparation), and held-out MGEX
(Paper 14) products) returns $\rho_T \approx 15$–$221$ g/cm³ and
$R_T/R_S \in [0.59, 1.45]$ for RBH-1: the crossover coincidence is
uncalibrated at its own falsification boundary, with the pooled
calibrations placing the object outside the horizon scale and the
finest-scale products inside it. This sensitivity is itself a
falsifiability criterion: the GNSS transfer-map resolution tracked in
Papers 1, 14, and 33 will either tighten or break the RBH-1
concordance.

## Transport-Consistent Conformal Factor Gradient

The resolved position–velocity structure runs from approximately $-600$ to
$+50$ km s$^{-1}$ across 1 kpc, giving $\Delta v=650$ km s$^{-1}$. The
former endpoint assignment converted this linearly to
$|\Delta\ln A|\simeq\Delta v/c$. That conversion applies to a stationary
emitter and observer on different conformal plateaux; it cannot describe gas
traversing the same profile. Universal matter coupling instead gives

\begin{equation} \label{eq:matter_work} \mathbf a_\phi=-c^2\boldsymbol\nabla\ln A,\qquad
\Delta\!\left(\frac{u^2}{2}\right)=-c^2\Delta\ln A , \end{equation}

The endpoint depth $\ln(1+\Delta v/c)=2.166\times10^{-3}$ would therefore
accelerate matter from rest to $1.973\times10^4$ km s$^{-1}$, 30.4 times
the observed span. In the rest frame of a quasi-steady profile, let $u$
denote the gas speed and define $\eta=2\bar u/\Delta v$. The dynamically
consistent depth is then

\begin{equation} \label{eq:log_a} |\Delta\ln A|_{\rm dyn}=\frac{|u_{\rm out}^2-u_{\rm in}^2|}{2c^2}
=\eta\frac{\Delta v^2}{2c^2}=2.350\times10^{-6}\eta . \end{equation}

Rest entry gives $\eta=1$. A collinear translating-profile benchmark using
the published 954 km s$^{-1}$ pattern speed gives profile-frame entry and
exit speeds 954 and 304 km s$^{-1}$, hence $\eta=1.94$ and
$|\Delta\ln A|=4.55\times10^{-6}$. Across 1 kpc the rest-entry normalization
gives $|\nabla\ln A|=7.62\times10^{-26}$ m$^{-1}$ and
$|a_\phi|=6.85\times10^{-9}$ m s$^{-2}$. A Newtonian fit reads the same
acceleration as $M_{\rm app}=4.91\times10^{10}\eta M_\odot$ at 1 kpc. On the
canonical kinetic branch, the shell ledger gives
$E_{\rm scalar}=(|\Delta\ln A|/4)M_{\rm app}=2.89\times10^4\eta^2M_\odot$ and
$M_{\rm app}/E_{\rm scalar}=1.70\times10^6/\eta$. Relative to the discarded
endpoint assignment, the rest-entry depth falls by a factor 921 and the canonical
gradient energy by $8.49\times10^5$; the pattern-speed benchmark gives
factors 476 and $2.27\times10^5$, respectively. These values are generated by the
registered transport-consistency step.

## Lensing and Kinematic Signatures

A critical prediction of the conformal scalar-tensor framework is the
quantitative asymmetry between matter response and scalar stress-energy.
The scalar field $A(\phi)$ generates coherent matter acceleration through
the matter metric. Light follows
the gravity metric $g_{\mu\nu}$, which is sourced by the total
stress-energy — the baryonic mass plus the scalar contribution
evaluated above. At the dynamical depth its projected canonical surface
density is $\Sigma_{\rm scalar}=2.30\times10^{-3}\eta^2
M_\odot\,{\rm pc^{-2}}$ for the stated shell geometry, far below a
cosmological strong-lensing critical density. The scalar-gradient
contribution to lensing is therefore negligible on this branch, while the
compact baryonic deflection remains.

For light passing through the wake boundary ($b \sim 1$ kpc), the
compact deflection angle depends only on the baryonic mass
$M \approx 2 \times 10^7 M_\odot$:

\begin{equation} \label{eq:lensing_angle} \theta_{\text{lens}} \sim \frac{4GM}{c^2 b} \sim \frac{4 \times 6.67
\times 10^{-11} \times 2 \times 10^7 \times 2 \times 10^{30}}{(3 \times
10^8)^2 \times 3 \times 10^{19}} \sim 4 \times 10^{-9} \text{ rad} \approx
0.8 \, \text{mas} \end{equation}

This value ($\sim 0.8$ milliarcseconds) is far below current observational
limits (HST resolution $\sim 50$ mas).

#### The Lensing Discriminator

This provides a sharp test against a particulate halo interpretation.
A particulate mass $M_{\rm app}\simeq4.9\times10^{10}\eta M_\odot$ inside
1 kpc would produce order-arcsecond strong lensing, whereas the canonical
scalar configuration carrying the same matter acceleration contains only
$2.9\times10^4\eta^2M_\odot$ in gradient energy. The predicted
kinematic-to-scalar-lensing mass ratio is $1.70\times10^6/\eta$, versus unity
for a particulate model. Deep imaging or weak-shear mapping therefore
tests the dynamical Temporal-Shear assignment without relying on an
endpoint-redshift channel.

Under the dynamical metric-shock interpretation the offset is real bulk
motion. Universal coupling predicts the same local acceleration for gas and
collisionless tracers at the same spacetime point, while hydrodynamic drag
acts directly only on the gas. Newly formed stars should therefore inherit
the local gas centroid at formation:

\begin{equation} \label{eq:stellar_velocity} \Delta v_\star(x_{\rm form}) \simeq
\Delta v_{\rm gas}(x_{\rm form}),\qquad
\Delta\!\left(\frac{v^2}{2}\right)=-c^2\Delta\ln A . \end{equation}

The decisive observable is the gas–star velocity relation along the wake:
a universal metric force predicts a composition-independent acceleration
law and a formation-epoch imprint in the stars, whereas ram pressure and
turbulent entrainment permit systematic gas–star slip. Spatially resolved
absorption and emission spectroscopy can test this relation directly.

## Screening and Parameter Consistency

The above estimates assume the scalar field profile is governed by
continuous Temporal Topology, where the local field gradient (Temporal
Shear) is suppressed in deep density wells. The phenomenological scaling
ansatz (detailed in Smawfield 2025e) is adopted:

Unlike traditional chameleon mechanisms that invoke discrete thin-shell boundaries
with sharp density cutoffs, TEP screening operates through continuous field gradient
flattening. The Temporal Shear is gradually suppressed in deep potential wells,
avoiding the fine-tuning problems of thin-shell approximations while maintaining
Temporal Shear suppression in steep ambient landscapes.

\begin{equation} \label{eq:screening_factor} S = \frac{\beta_0}{\alpha_{\text{RBH,eff}}} \propto
\left(\frac{\rho}{\rho_T}\right)^{1/3} \end{equation}

This $S$ is a coupling ratio&mdash;the bare coupling $\beta_0$ divided
by the screened effective coupling $\alpha_{\text{RBH,eff}}$&mdash;and
should not be conflated with the clock-amplitude response factor of
Paper 6, $S_A = (\bar{\rho}/\rho_T)^{1/3}$, a radius ratio evaluated at
a body's mean density. The two share the $(\rho/\rho_T)^{1/3}$
saturation scaling but are distinct quantities.

For RBH-1 at the crossover mass (M ~ 10⁷ M_☉, ρ ~ ρ_T ~ 20 g/cm³), the
suppression factor is S ~ 1, meaning the Temporal Shear is near the
transition between strongly flattened and asymptotically recovered profiles.
This is consistent with the object being a "soliton" where the field
saturates at the characteristic density. Two distinct uses of
$\alpha_{\rm eff}$ must be kept separate: in the screening ansatz the
effective charge is the response channel's screened coupling, and S ~ 1 at
the crossover asserts that the shear profile transitions there — a statement
about where the field flattens, not about its amplitude. In the
velocity-discontinuity constraint (Box 2.2), $\alpha_{\rm RBH,eff} =
2.46\times 10^{3}\eta$ is the linear-response equivalent of the observed dynamical depth;
read literally as a coupling ratio, S = $\beta_0/\alpha_{\rm eff}$ at S ~ 1
would require the bare response $\beta_0$ itself to carry the same $\sim
10^{3}$ amplification — the corpus's open response-normalization question
(shared with $\kappa_{\rm Cep}$, $\kappa_{\rm MSP}$ and $\beta_{A,\rm eff}$),
stated here explicitly rather than absorbed into the ansatz.

At higher densities (e.g., white dwarfs, neutron stars), S >> 1, and the
Temporal Shear is suppressed below observational limits, explaining why
precision GR tests show no deviation. At lower densities (e.g., galaxies), S
< 1, and the scalar field produces the dynamical anomaly inferred as
"phantom mass" under Newtonian bookkeeping
(detailed in the companion paper, Smawfield 2025e).

#### Box 3.1: The Phantom Halo (Mass Deficit) Implication

A critical reader will note that a Newtonian potential for $M = 2 \times
10^7 M_\odot$ at $R = 1$ kpc yields $\Phi/c^2 \sim 10^{-9}$, far smaller
than the required dynamical depth $|\Delta\ln A|_{\rm dyn}=2.35\times10^{-6}\eta$.

This discrepancy implies that the scalar sector carries an effective
"Phantom Mass" of order $M_{\text{eff}} = 4.91 \times 10^{10}\eta M_\odot$
in Newtonian-equivalent terms — the enclosed mass that produces the
required coherent acceleration at $R_{\rm trans} \sim 1$ kpc — comparable to
a galactic halo. In the TEP
framework, this is not "missing matter" but a "metric
halo"—the scalar field profile maintains a deep potential well out to
$R_{\text{trans}}$, decoupling the metric potential from the compact
baryonic source. Stated as a linear response, the same amplitude
corresponds to an effective response $\alpha_{\rm RBH,eff} =
2.46\times 10^{3}\eta$ (Box 2.2); under the soliton reading it is the field
configuration's internal depth rather than a response coefficient.

**Crucially, this potential is bi-metric:** it affects the
matter motion far more strongly than the
spatial curvature sensed by light. The scalar's own stress-energy is
only $|\Delta\ln A|/4$ of the apparent kinematic mass and is distributed
through the transition volume, so the object generates a kinematic
"kick" equivalent to a massive halo while producing at most marginal,
extended lensing — not the strong arcs or extreme gas focusing
expected from $\sim 5\times 10^{10}\eta M_\odot$ of particulate matter.

**Observationally:** RBH-1 acts as a "naked halo"—a compact
baryonic object ($2 \times 10^7 M_\odot$) clothed in a metric
distortion whose Newtonian-equivalent mass is $\sim 5 \times 10^{10}\eta
M_\odot$. This explains how it
generates a wake magnitude ($\sim 600$ km/s) typical of galactic
interactions despite its small physical size.

#### Consistency Check

The transport-consistent conformal gradient ($|\Delta\ln A|=2.35\times10^{-6}\eta$ over 1 kpc)
produces:

- Coherent velocity span: $\Delta v=650$ km/s by the profile-frame work integral, with $\eta$ fixed by the resolved flow geometry

Lensing: θ ~ 0.8 mas compact deflection (baryonic; far below
current HST/JWST resolution, consistent with non-detection);
the scalar sector adds only negligible extended shear,
$\Sigma_{\rm scalar}=2.30\times10^{-3}\eta^2M_\odot\,{\rm pc^{-2}}$

Gas–star comoving relation at formation, with subsequent gas–star
slip directly testing hydrodynamic rather than universal-metric forcing

These parameters are internally consistent and do not violate existing
constraints. The soliton interpretation for RBH-1 is falsifiable via
lensing, stellar spectroscopy, and coronal-line/X-ray non-detection.

## 4. Observational Analysis: Confrontation with Data

For RBH-1 itself, direct empirical tests are available using published JWST and HST data (van Dokkum et al. 2025). Six independent observables discriminate between thermal shock and metric shock (TEP) interpretations.

## 4.1 The Line Width Test

In a single-phase thermal shock where the [O III]-emitting gas resides predominantly at $T \sim 10^7$ K, thermal Doppler motion would broaden spectral lines to $\sigma_{\text{th}} \approx 80\text{--}85$ km/s. In practice, [O III] emission in shock-heated environments often arises from cooler recombination/cooling zones rather than the hottest post-shock plasma; however, if a substantial hot-phase component contributes to the line flux, it should produce detectable broad wings. In the dynamical metric-shock model, Temporal Shear produces an ordered bulk-velocity field, so a large resolved centroid span can coexist with a narrow local line where differential acceleration inside the emitting element is small.

Higher-resolution Keck/LRIS spectroscopy of the [O III] λ5007 knot at the tip of the wake yields (van Dokkum et al. 2025, Appendix C):

\begin{equation} \label{eq:line_width_obs} \sigma_{\text{obs}} = 36 \pm 4 \text{ km/s} \quad \xrightarrow{\text{instr. corr.}} \quad \sigma = 31 \pm 4 \text{ km/s} \end{equation}

where the instrumental resolution $\sigma_{\text{instr}} = 18$ km/s has been subtracted in quadrature. The key inequality:

\begin{equation} \label{eq:line_width_inequality} \sigma_{\text{obs}} \approx 31 \text{ km/s} \ll \sigma_{\text{thermal}}(10^7 \text{ K}) \approx 85 \text{ km/s} \end{equation}

This observed dispersion is 3× smaller than the $\sim 80$–$85$ km/s scale expected if the [O III]-emitting gas were thermally broadened at $T \sim 10^7$ K, and 10× larger than pure thermal broadening at $T \sim 10^4$ K ($\sigma_{\text{th}} \approx 2$–$3$ km/s). Because $\text{O}^{2+}$ ([O III]) traces the warm ionized phase ($T \sim 10^4$ K) rather than a collisionally ionized $10^7$ K plasma (where oxygen is stripped to O VII/O VIII), the narrow [O III] dispersion directly constrains the kinematics and non-thermal broadening of the warm line-emitting phase rather than measuring the temperature of a separate X-ray-hot phase; the presence of any dominant hot phase must be tested independently through X-ray or coronal-line observations (Section 5). The intermediate value ($\sigma \approx 31$ km/s) is consistent with warm gas experiencing bulk/turbulent motions or a coherent velocity gradient across the beam.

![Line Width Test: Thermal vs Metric Shock](site/figures/figure_04_line_width.png)

Figure 4: The Line Width Test. (A) Thermal broadening of [O III] emission as a function of gas temperature. The observed dispersion (σ = 31 km/s) is 3× smaller than expected for a simple $10^7$ K post-shock phase, placing that picture under tension. (B) Schematic illustration of how a coherent body-force velocity field can accommodate a large spatial centroid span with comparatively narrow local lines.

### Critical Next Step: Line-Profile Decomposition

The bulk line-width measurement ($\sigma = 31$ km/s) is necessary but not sufficient to rule out thermal heating. A standard thermal shock could be "hiding" beneath the observed profile: the bulk of the gas could be cold (narrow core), while a faint, high-velocity wing of hot gas exists but is lost in the noise or blended into the continuum.

![Line Profile Decomposition](site/figures/figure_05_decomposition.png)

Figure 5: Line Profile Decomposition Strategy (synthetic benchmark demonstration). (A) A single-component fit (cold gas only, $\sigma = 31$ km/s) on a synthetic $\mathrm{S/N}=50$ profile. (B) A two-component fit (cold core + hot wing). Applying this test to RBH-1 requires the reduced Keck/LRIS spectrum: if a broad thermal wing is statistically required by the data, the thermal model is supported; if excluded, the metric shock is favored.

The definitive test requires rigorous line-profile decomposition:

- *Single-component model.* A single Gaussian (σ ≈ 30 km/s) may be fit as a minimal description of a cold, single-phase line profile.

- *Two-component model.* A narrow core plus a broad wing (σ₂ ≈ 80–90 km/s, corresponding to $T \sim 10^7$ K) may be fit to represent a cold component plus a hot shocked component.

- *Model selection.* Information criteria (AIC/BIC) may be used to determine whether an additional broad component is supported by the data.

If a statistically significant broad component is required, a thermal-shock contribution is supported. If the profile remains single-component at adequate S/N, a metric-shock interpretation is strengthened.

#### Data Availability Assessment

JWST NIRSpec IFU data (Program 3149) are publicly available via MAST but have insufficient spectral resolution for a robust decomposition between a σ ≈ 30 km/s narrow component and a σ ≈ 80–90 km/s broad component. The instrumental line-spread function dominates the intrinsic profile at this level. The JWST data confirm high-S/N [O III] emission, but they are not decisive for the narrow-core versus broad-wing question.

The critical dataset is the Keck/LRIS 1200 lines/mm spectrum (σ<sub>inst</sub> = 18 km/s) used to derive the published σ = 31 ± 4 km/s measurement. If the reduced spectrum is not publicly available, the analysis cannot be independently repeated at present. A line-profile decomposition workflow is provided in `scripts/analyze_line_profiles.py`, but application to the Keck/LRIS spectrum requires access to the extracted line profile (or collaboration with the van Dokkum et al. team).

The narrow near-tip line width is in tension with a single-phase thermal shock in which the emitting [O III] gas is predominantly at $T \sim 10^7$ K. It is naturally accommodated by a coherent Temporal-Shear velocity field. The 31 km s$^{-1}$ dispersion is not the unresolved width of the full 650 km s$^{-1}$ spatial span; establishing their co-spatial relation requires the reduced LRIS profile and resolved JWST velocity field. The principal observational discriminants are a statistically supported broad wing and the local covariance of centroid and dispersion across the apex.

## 4.2 The Wake Collimation Test

Thermal shocks generate turbulence via Kelvin-Helmholtz instabilities at the shear layer between the wake and the ambient medium. The wake lifetime implied by the observed extent and proper motion is $t_{\text{wake}} \sim L_{\text{wake}}/v_{\bullet} \sim 62\,\text{kpc}/(950\,\text{km/s}) \approx 70$ Myr. Over this timescale, such instabilities should broaden the wake substantially. The characteristic K-H growth timescale is $\tau_{\text{KH}} \sim \lambda / v_{\text{shear}} \approx 3$ Myr for $\lambda \sim 1$ kpc and $v_{\text{shear}} \sim 300$ km/s. Over $\sim 70$ Myr, this corresponds to ~22 e-folding times—implying the wake boundary should be significantly disrupted and broadened in standard hydrodynamic scenarios (see Figure 6).

Instead, HST WFC3/UVIS imaging reveals (van Dokkum et al. 2025, Section 6.2.1):

\begin{equation} \label{eq:aspect_ratio} R_{\text{wake}} \approx 0.7 \text{ kpc} \quad \text{over} \quad L_{\text{wake}} = 62 \text{ kpc} \quad \Rightarrow \quad \text{Aspect Ratio: } 50:1 \end{equation}

This is described by the authors as "strikingly narrow." The wake maintains morphological coherence over its entire 200,000 light-year extent, with no evidence of the turbulent broadening expected from a thermal shock.

In the dynamical metric-shock interpretation, the apex discontinuity is attributed to an ordered Temporal-Shear acceleration field. A smooth body force need not seed the same transverse mixing as a collisional obstacle, so the wake can remain collimated while carrying a large longitudinal bulk-velocity gradient.

![Wake Geometry Analysis](site/figures/figure_06_wake_geometry.png)

Figure 6: Wake Geometry Analysis. (A) Wake width vs. distance from galaxy. The thermal shock model predicts significant broadening due to Kelvin-Helmholtz instabilities; the observed wake remains collimated at 0.7 kpc. (B) Schematic comparison of wake morphologies. The 50:1 aspect ratio is inconsistent with thermal turbulence.

The extreme collimation (50:1 aspect ratio) is difficult to reconcile with generic turbulent-wake expectations and is qualitatively consistent with a more laminar, non-thermal driver.

## 4.3 The Stellar Age Gradient Test

In a thermal shock, star formation is delayed by the cooling time ($t_{\text{cool}} \approx 36$ Myr). During this delay, the perturber travels a distance $d = v \times t_{\text{cool}} \approx 35$ kpc. This creates a "star formation delay zone" where no young stars should exist (see Figure 7).

van Dokkum et al. (2023) reports that stellar continuum colors are "well-fit by a simple model that has a monotonically increasing age with distance from the tip" (van Dokkum et al. 2023). The youngest stars are at the tip, not 35 kpc behind it.

![Stellar Age Gradient](site/figures/figure_07_stellar_age.png)

Figure 7: Stellar Age Gradient Schematic. van Dokkum et al. (2023) report that stellar continuum colors are well-fit by a monotonically increasing age with distance from the tip, consistent with immediate star formation at the apex, whereas a 36 Myr thermal cooling delay would produce a star-free gap of ~35 kpc.

The observed age gradient qualitatively disfavors a long cooling-delay zone; quantitative model comparison requires a fully specified stellar-population fitting procedure and error model.

## 4.4 The Preshock Temperature Anomaly

The Mappings V shock models used to fit the emission line ratios require a preshock temperature of $T_{\text{pre}} \sim 10^{5.6}$ K (van Dokkum et al. 2025, Section 5.2). This is 40× higher than standard CGM conditions ($T_{\text{CGM}} \sim 10^4$ K) (see Figure 8).

In the TEP framework, the "preshock ionization" may admit a non-thermal contribution: the soliton boundary supplies a coherent matter acceleration. For electrons traversing the transition zone, work done by the effective potential could contribute to ionization and excitation, potentially reducing the need to interpret the inferred preshock conditions as a true thermal temperature. A quantitative mapping from the measured velocity field to ionization diagnostics remains to be performed on the real spectra.

![Line Ratio Analysis](site/figures/figure_08_line_ratios.png)

Figure 8: Line Ratio and Preshock Temperature Schematic. (A) Qualitative spatial ionization trends along the wake toward the apex. (B) Standard shock models (van Dokkum et al. 2025) require anomalously high preshock temperatures ($T_{\rm pre} \sim 10^{5.6}$ K) relative to standard $10^4$ K CGM conditions to reproduce the observed ionization. In the metric-shock framing, the soliton boundary may contribute non-thermally to ionization and excitation, though a quantitative mapping from field profile to line ratios remains future work.

Within the TEP framing, a metric-induced contribution could reduce reliance on anomalously high preshock temperatures, but this mapping requires a future forward model from field profile to ionization diagnostics.

#### Box 4.2: Toy Model — Ionization via Partial Coupling

Standard shock models assume 100% conversion of bulk kinetic energy to thermal energy ($T \sim T_{\text{virial}} \sim 10^7$ K). The soliton model proposes a "cold" interaction, but must still explain the high ionization states ($T_{\text{ion}} \sim 10^{5.6}$ K).

A first-order "Partial Coupling" model can reconcile these:

- **Available Budget:** The bulk kinetic energy of infalling gas is $K \sim \frac{1}{2} m_p v^2 \approx 2$ keV per particle, equivalent to a virial temperature of $\sim 1.5 \times 10^7$ K.

**Coupling Efficiency ($\epsilon$):** In a laminar metric flow, most energy is adiabatic (reversible). However, if plasma instabilities at the transition boundary couple just $\epsilon \sim 1\%$ of the bulk energy to the electron population, the effective electron temperature becomes:

$T_{\text{eff}} \approx \epsilon \times T_{\text{virial}} \approx 0.01 \times (1.5 \times 10^7 \text{ K}) \approx 1.5 \times 10^5 \text{ K}$

- **Result:** This $T_{\text{eff}} \sim 10^{5.2}$ K matches the "preshock anomaly" required by ionization data, *without* heating the bulk ion fluid to $10^7$ K. The gas appears "highly ionized" (due to non-thermal electrons) but "dynamically cold" (narrow line widths).

## 4.5 The Star Formation Efficiency Problem

The thermal shock model faces a mass budget problem (van Dokkum et al. 2025, Section 6.2.2). The observed stellar mass ($M_* \sim 3 \times 10^8 M_\odot$) equals the total entrained gas mass, implying a star formation efficiency of ~100%. The maximum realistic efficiency is ~30% (see Figure 9).

In the metric shock model, heating is not the primary driver; the discontinuity is interpreted as coherent scalar-driven bulk motion. This relieves the mass-budget tension in two ways: (1) the reduced effective Jeans mass allows collapse of lower-density gas that would otherwise remain stable, increasing the available reservoir of star-forming material; and (2) limiting conversion of the ordered flow into thermal pressure permits a larger cold reservoir. A standard SFE of ~30% acting on this effectively larger reservoir can reproduce the observed stellar mass without requiring unphysical 100% conversion of the nominal entrained gas.

![Star Formation Efficiency](site/figures/figure_09_efficiency.png)

Figure 9: Star Formation Efficiency. The thermal model implies a near 100% conversion of entrained gas to stars to match the observed mass. The metric model reduces the thermal support, potentially allowing high efficiency, but the mass budget remains a key constraint.

The metric-shock hypothesis offers a possible route to reducing the star-formation-efficiency tension by avoiding a prolonged hot phase; a decisive assessment still depends on the inferred gas mass and its systematics.

## 4.6 Summary of Empirical Tests

| Observable | Thermal Prediction | Metric Prediction | Observed Value | Implication |
| --- | --- | --- | --- | --- |
| Line width (\(\sigma\)) | ~80 km/s (hot) | ~30 km/s (cold) | 31 ± 4 km/s | Favors cold metric shock; tension with hot single-phase model |
| Wake Aspect Ratio | ~10:1 (broadened) | ~50:1 (collimated) | 50:1 | Inconsistent with turbulent broadening |
| Post-shock temperature | ~\(10^7\) K | ~\(10^4\) K | ~\(10^4\) K | Consistent with metric cooling evasion |
| Star formation timing | Delayed (35 kpc zone) | Immediate | Immediate | Tension with cooling time delay |
| Preshock temperature | ~\(10^{5.6}\) K (anomalous) | ~\(10^4\) K (standard) | ~\(10^{5.6}\) K (inferred) | Requires anomaly under basic shock models |
| Star formation efficiency | ~100% (unrealistic) | ~30% (realistic) | ~100% (inferred) | Mass budget tension in both, but reduced in metric model |

#### Box 4.1: Constraints on Composite Thermal-Shock Models

For completeness, a composite thermal-shock interpretation may be constructed in which multiple physically plausible mechanisms operate simultaneously:

- *Magnetic draping.* Ordered fields of order $B \sim 1$–$3$ μG can suppress Kelvin–Helmholtz growth and maintain a narrow wake (e.g., Dursi & Pfrommer 2008; Ruszkowski et al. 2014).

- *Turbulent mixing layers.* Entrainment of cold gas into the post-shock flow can, in principle, accelerate the emergence of $10^4$ K emitting material (e.g., Gronke & Oh 2018, 2020; Ji et al. 2019).

- *Non-equilibrium ionization.* Ionization states can lag temperature during rapid cooling or in shock precursors, affecting inferred preshock conditions (e.g., Dopita & Sutherland 2003; Sutherland & Dopita 2017).

Several quantitative constraints follow directly from the RBH-1 observables:

- *Line width.* If a hot $T \sim 10^7$ K component contributes appreciably to the [O III] emission, a broad wing (σ ≳ 80 km/s) is expected. The published σ = 31 ± 4 km/s constrains the hot-phase contribution to be sub-dominant in the observed line profile.

- *Draping versus mixing.* Magnetic draping that maintains laminar boundaries tends to suppress shear-driven mixing; the simultaneous requirement for strong collimation and rapid mixing introduces a coupling between magnetic geometry and cooling efficiency that must be satisfied by the model.

- *Star formation timing.* The presence of the youngest stellar populations near the apex disfavors a long downstream delay unless the cold phase is generated promptly behind the interaction front.

A practical falsification threshold for the metric-shock hypothesis may be stated as follows. If future high-resolution spectroscopy detects a broad component with σ > 80 km/s containing a non-negligible fraction of the [O III] flux, a thermal-shock contribution is strongly supported. Conversely, if the line profile remains consistent with a single narrow component (σ < 40 km/s) at high S/N, thermal models must place the dominant emitting gas in the cold phase.

#### Model Structure: Metric Shock versus Composite Thermal Shock

Reproducing the joint RBH-1 dataset under a thermal interpretation typically invokes a composite model in which several mechanisms contribute simultaneously:

- Magnetic draping to suppress Kelvin–Helmholtz growth and maintain a high aspect ratio.

- Turbulent mixing and/or multiphase cooling to generate a dominant cold emitting phase despite an initially hot shock.

- Non-equilibrium ionization and/or shock precursors to reconcile ionization diagnostics with fiducial CGM temperatures.

In the dynamical metric-shock interpretation, the resolved velocity discontinuity is attributed to coherent Temporal-Shear acceleration rather than local thermalization of the full span. The same observational elements then align: a large ordered first moment, small local second moment, minimal hot-phase requirements, and preserved collimation.

## 4.7 Alternative Explanations: Composite Thermal-Shock Models

Several well-motivated astrophysical mechanisms could, in principle, reconcile a thermal shock with the cold, star-forming wake observed in RBH-1. Each deserves careful consideration:

### Mechanism-by-Mechanism Assessment

- **Turbulent Mixing Layers:** Shear-driven entrainment of cold ambient gas into the hot wake (Gronke & Oh 2018, 2020) is a robust prediction of supersonic cloud–wind interactions. *What it explains:* rapid appearance of $10^4$ K gas downstream of a hot shock front; multiphase coexistence. *What remains in tension:* mixing-layer models generically produce broad, asymmetric line profiles with extended wings from the velocity shear; the observed [O III] profile is narrow and single-peaked. *Discriminant:* high-S/N line-profile decomposition searching for faint broad wings or secondary components.

- **Magnetic Draping:** Ordered magnetic fields swept up ahead of the perturber can suppress Kelvin-Helmholtz instabilities and maintain wake coherence (Dursi & Pfrommer 2008; Pfrommer & Dursi 2010). Recent MHD simulations indicate that magnetic fields can also facilitate cooling via reconnection or anisotropic conduction (e.g., Banda-Barragán et al. 2024). *What it explains:* the extreme 50:1 aspect ratio, morphological coherence, and potentially accelerated cooling. *What remains in tension:* while draping aids collimation, the near-tip $\sigma \approx 31$ km/s dispersion remains a tight constraint on any hot component sampled by that spectrum. *Discriminant:* Faraday rotation or synchrotron polarimetry to map the field geometry; comparison with MHD bow-shock simulations that include radiative cooling.

- **Non-Equilibrium Ionization (NEI):** Rapid cooling through the $10^5$–$10^6$ K range can produce ionization states that lag behind the instantaneous temperature (Sutherland & Dopita 2017). *What it explains:* anomalously high ionization (e.g., the $T_{\text{pre}} \sim 10^{5.6}$ K inferred from Mappings V) even if the gas has already cooled. *What remains in tension:* NEI affects ionization diagnostics but does not widen or narrow the thermal velocity dispersion; the line-width constraint is independent. *Discriminant:* time-dependent photoionization modeling with realistic cooling trajectories; comparison of multiple ionization-sensitive line ratios (e.g., [O III]/[O II], [N II]/Hα) to NEI grids.

- **Beam Smearing:** Instrumental resolution effects could, in principle, artificially narrow observed line widths if the emission is spatially unresolved and dominated by a single cold clump. *What it explains:* apparent single-component profile. *What remains in tension:* the Keck/LRIS measurement already corrects for instrumental broadening ($\sigma_{\text{instr}} = 18$ km/s); the JWST/NIRSpec IFU spatially resolves the tip, and the narrow dispersion persists across multiple spaxels. *Discriminant:* spatially resolved line-width maps from the IFU data.

Magnetic draping and non-equilibrium ionization are physically plausible and may well operate in RBH-1. Recent work suggests these mechanisms can extend the parameter space for cold gas survival (Ogiya & Nagai 2023; Banda-Barragán et al. 2024). However, each addresses only a subset of the six anomalies. A fully satisfactory thermal-shock model would need to invoke multiple mechanisms simultaneously—draping for collimation, non-equilibrium ionization for line ratios, and efficient mixing for cooling—while also explaining the immediate star formation and the star formation efficiency tension. The metric-shock interpretation offers a single-mechanism explanation but requires accepting the TEP framework. Decisive discrimination awaits deeper spectroscopy (line-profile decomposition, spatially resolved temperature mapping) and polarimetric constraints on magnetic field geometry.

### Conclusion

Under the stated assumptions, the combined set of observables places the simplest single-phase thermal-shock picture under substantial strain. Thermal-shock explanations remain viable if multiple additional mechanisms (magnetic draping, non-equilibrium ionization, turbulent mixing) operate in concert; such composite models are not ruled out but require fine-tuning across several independent parameters. The metric-shock (TEP) interpretation offers a more parsimonious single-mechanism account but rests on an unconventional theoretical framework. More decisive discrimination awaits deeper spectroscopy (line-profile decomposition, spatially resolved temperature mapping) and polarimetric constraints on the magnetic field geometry.

## 5. Falsification Criteria

This paper treats the characteristic density $\rho_T \approx 20$ g/cm³ as a fixed input derived in Paper 6. This makes the RBH-1 analysis a consistency check: the model does not have the freedom to fit the temporal scale; it must match the geometric scale dictated by the $10^7 M_\odot$ mass and the universal density. Two bookkeeping facts delimit what the check establishes. First, $R_T \approx 2.5 \times 10^{-9}$ kpc lies $\sim 8.6$ orders of magnitude below the resolved scale at $z \approx 0.96$ ($\sim 1$ kpc per $0.1''$): the comparison $R_T \approx 1.3\,R_S$ is a theory-internal scale coincidence — the crossover mass at which the constant-density temporal-topology radius meets the horizon radius — not a match to a resolved feature. The observable kinematic onset lives at the transition radius $R_{\rm trans} \sim 0.1$–$1$ kpc, parametrized in Section 2 as $R_{\rm trans} = \xi\,R_T$ ($\xi \approx 4\times10^8$) and anchored physically by the two-branch exterior shear-recovery radius $R_s = \sqrt{GM/g_t} \approx 90$ pc. Second, the anchor itself carries product-level systematic scatter far larger than its formal uncertainty: propagating the measured calibration ensemble through the check (below) places the object's candidacy inside that systematic rather than cleanly past the falsification boundary.

## The Soliton Size Prediction

![Universal Scaling Law](site/figures/figure_10_scaling.png)

Figure 10: Universal Scaling Law. The temporal topology scale ($R_T \propto M^{1/3}$) vs. Mass. The solid line is the consistency check fixed by $\rho_T \approx 20$ g/cm³ (derived from terrestrial clocks). RBH-1 (star) sits at the crossover mass where $R_T \approx 1.3 R_S$ — a theory-internal scale coincidence, since $R_T$ itself lies $\sim 8.6$ orders of magnitude below the resolved scale.

Given $\rho_T \approx 20$ g/cm³ (from Paper 6), the consistency check for the temporal scale of RBH-1 ($M \approx 2 \times 10^7 M_\odot$) yields:

\begin{equation} \label{eq:rt_falsification} R_T = \left(\frac{3M}{4\pi \rho_T}\right)^{1/3} \approx 7.8 \times 10^7 \text{ km} \approx 1.3 R_S \end{equation}

### Uncertainty Propagation

The check uncertainty derives from two inputs: the mass estimate and the characteristic density. For the mass $M \sim 2 \times 10^7 M_\odot$ (van Dokkum et al. 2025), propagating through $R \propto M^{1/3}$ yields $\delta R/R = (1/3)(\delta M/M) \approx 10\%$. The density input is the larger term by far, and its uncertainty is not the $\pm 30\%$ formal figure quoted in earlier drafts: $\rho_T = 3M_\oplus/(4\pi\lambda_T^3)$ is set by the GNSS decoherence length, and the corpus's measured product-level ensemble spans $\lambda_T \approx 1{,}900$–$4{,}500$ km on the precise-product branch — pooled coherent-band estimates of 4,549 km (CODE), 3,764 km (IGS), and 3,328 km (ESA) from Paper 1, the Paper 33 value near 4,200 km, and the corrected Paper 14 multi-GNSS (MGEX) pooled estimate at 1,862 km. Propagating each calibration through $\rho_T$ and $R_T$:

| Calibration product | $\lambda_T$ (km) | $\rho_T$ (g/cm³) | $R_T$ (km) | $R_T/R_S$ | $M_\times$ ($M_\odot$) |
| --- | --- | --- | --- | --- | --- |
| CODE pooled (Paper 1) | 4,549 | 15.1 | $8.6\times10^7$ | 1.45 | $3.5\times10^7$ |
| IGS pooled (Paper 1) | 3,764 | 26.7 | $7.1\times10^7$ | 1.20 | $2.6\times10^7$ |
| ESA pooled (Paper 1) | 3,328 | 38.7 | $6.3\times10^7$ | 1.06 | $2.2\times10^7$ |
| Paper 33 (in prep.) | 4,200 | 19.2 | $7.9\times10^7$ | 1.34 | $3.1\times10^7$ |
| Paper 14 multi-GNSS | 1,862 | 221 | $3.5\times10^7$ | 0.59 | $9.1\times10^6$ |

Here $M_\times(\rho_T) = c^3\sqrt{3/(32\pi G^3\rho_T)}$ is the crossover mass at which $R_T = R_S$ — the falsification boundary of the mass test below. Across the raw unharmonized table, $R_T/R_S \in [0.59, 1.45]$ and $M_\times \in [9.1\times10^6, 3.5\times10^7]\,M_\odot$: the three primary full-day $[10, 500]\,\mu\text{Hz}$ calibrations and Paper 33 place RBH-1 ($2\times10^7\,M_\odot$) outside the horizon scale with margins of 6–45%, whereas the raw Paper 14 multi-GNSS (MGEX) number ($1{,}862$ km) appears lower by a factor of $2.25$. As established in Paper 0 (Step 08, `step_08_empirical_tensions.json`), this shift is an exact spectral-windowing consequence of MGEX's 8-hour Welch estimator (5-minute sampling with `nperseg = 96`, fundamental non-zero frequency bin $\Delta f = 1/28{,}800\text{ s} = 34.72\,\mu\text{Hz}$), which physically omits the $10\text{–}30\,\mu\text{Hz}$ diurnal and semidiurnal sub-bands ($\lambda = 5{,}912$ and $5{,}691$ km in Paper 1, Table 8a); evaluated on the identical $[34.72, 500]\,\mu\text{Hz}$ FFT grid, Paper 1's own CODE sub-band profile predicts $\lambda = 1{,}864.3$ km, agreeing with MGEX ($1{,}862 \pm 112$ km) to $0.02\sigma$. Mapped to the common $[10, 500]\,\mu\text{Hz}$ passband ($\lambda_{\rm MGEX,equiv} = 4{,}195$ km), the MGEX product yields $R_T/R_S = 1.34$, so every passband-matched calibration product satisfies $R_T/R_S \in [1.06, 1.45] > 1$. Under the nominal Paper 6 anchor alone, $R_T = 7.8\times10^7$ km with $\delta R/R \approx 14\%$ from the mass and nominal density terms in quadrature.

The correspondence $R_T \approx 1.3 R_S$ under the nominal anchor places the temporal topology scale just outside the Schwarzschild radius. This would predict a "naked halo" phenomenology if RBH-1 is a soliton: the object interacts with the environment via its metric gradient (the "hair") rather than through an absorbing horizon. The canonical TEP strong-field formulation, including the treatment of compact-object interiors and the soliton–horizon distinction, is developed in TEP-BH (Paper 28, Bahrain); the framing here is consistent with that treatment.

## Explicit Falsification Criteria

The hypothesis that RBH-1 is a candidate Temporal Topology soliton makes specific, falsifiable predictions. It is important to distinguish between tests of this specific interpretation and tests of the underlying TEP theory. A failure in the object-specific tests below would rule out the soliton candidate model for RBH-1 (returning it to the status of an unexplained anomaly), but would not falsify the broader TEP framework.

- **Mass falsification (Object Specific):** The crossover check $R_T \approx 1.3 R_S$ is sensitive to both mass and the density anchor. Under the nominal calibration, a dynamical revision of $M_{\text{RBH-1}}$ above $M_\times \approx 3 \times 10^7\,M_\odot$ (where $R_T < R_S$) would falsify the soliton interpretation for this object. The boundary itself moves with the anchor: across the measured calibration ensemble $M_\times$ spans $5.9\times10^6$–$3.5\times10^7\,M_\odot$, and the finest-scale products (Paper 14, MGEX) would place the present mass inside the horizon scale. The criterion is therefore conditional on the GNSS transfer-map resolution tracked in Papers 1, 14, and 33.

**Discriminant falsification (Spectroscopy):** If deep spectroscopy reveals:

- Strong coronal-line emission ([Fe X], [Fe XIV]) or soft X-rays consistent with $T \sim 10^7$ K gas dominating the emission measure would exclude a predominantly cold Temporal-Shear flow.

- Broad [O III] wings containing >50% of the flux, indicating thermal broadening from high-velocity shear, the narrow-line argument is falsified.

- A gas–star velocity relation incompatible with universal acceleration: the dynamical metric-shock model predicts that stars inherit the local gas centroid at formation, whereas ram pressure and turbulent entrainment permit systematic gas–star slip. A resolved failure of the profile-frame work relation $\Delta(u^2/2)=-c^2\Delta\ln A$ for any admissible profile would exclude the candidate mechanism.

- **X-ray constraint:** A search of the Chandra and XMM-Newton archives reveals no pointed observations covering the RBH-1 field. The ROSAT All-Sky Survey provides only shallow upper limits ($F_X \lesssim 10^{-13}$ erg/s/cm²) insufficient to constrain $T \sim 10^7$ K emission at $z \approx 0.96$. Dedicated X-ray follow-up (Chandra ACIS, ~50 ks) can test the full-wake thermal-shock emission measure ($L_X \sim 10^{41}$–$10^{42}$ erg/s expected if the bulk $\Delta v \sim 650$ km/s post-shock gas mass were predominantly thermalized at $T \sim 10^7$ K; detection at that bulk-shock threshold would falsify the predominantly cold body-force model, whereas a sub-dominant X-ray component from minor streamline crossing or apex boundary heating remains compatible with coherent acceleration of the star-forming wake).

- **Universal Calibration Failure (Theory Level):** Unlike the object-specific tests above, if the external input $\rho_T \approx 20$ g/cm³ is invalidated by independent replication of the GNSS analysis (Paper 6), the basis for the specific quantitative consistency check collapses.

#### Summary of Logic

**Input:** $\rho_T \approx 20$ g/cm³ (External from Paper 6; product-level ensemble $\rho_T \approx 15$–$221$ g/cm³)

**Consistency Check:** RBH-1 Wake Properties (ordered 650 km s$^{-1}$ span, narrow local component, prompt star formation)

**Test:** Does the observed wake match the prediction?

**Verdict:** The wake phenomenology is consistent with the metric-shock candidate, pending X-ray/coronal confirmation to rule out hidden thermal components. The $R_T \approx 1.3\,R_S$ coincidence is a crossover-scale statement internal to the theory — uncalibrated at its falsification boundary under the current calibration ensemble — while the resolved $\sim$kpc onset ($R_{\rm trans} = \xi R_T$, $\xi \approx 4\times10^8$) is physically bridged by the two-branch exterior shear-recovery radius $R_s = \sqrt{GM/g_t} \approx 90$ pc (Section 2).

## 6. Discussion: Implications for Dark Matter

## Observational Context

The confirmation of RBH-1 provides a rare observational laboratory for testing the interaction of a compact perturber with the circumgalactic medium. The primary analysis infers a supersonic motion, $v_{\bullet}=954^{+110}_{-126}$ km/s, and identifies a 62 kpc ($\sim$200,000 light-year) wake of rapid cooling and star formation (van Dokkum et al. 2025). Such velocities are physically plausible within gravitational-wave recoil scenarios (Campanelli et al. 2007; Colpi & Dotti 2011; Komossa 2012). However, the coexistence of a large kinematic discontinuity with a cold, star-forming wake remains difficult to reconcile with standard collisional shock expectations.

The central tension concerns the wake rather than the recoil. In standard gas dynamics, a $\sim 10^3$ km/s shock is expected to thermalize the flow, heating gas to $T\gtrsim 10^7$ K and providing pressure support against prompt collapse. The observation of shock-excited emission coincident with rapid cooling and star formation suggests that the dominant driver of the structure is non-thermal.

While recent hydrodynamic simulations (Ogiya & Nagai 2023) and ram-pressure stripping studies (Poggianti et al. 2019) have shown that star formation can occur in tails under specific conditions, they do not by themselves resolve the temperature paradox posed by the RBH-1 apex. The coexistence of a high-Mach kinematic discontinuity and cold, star-forming gas motivates consideration of non-thermal contributions to the observed discontinuity.

## Resolution via Temporal Solitons

A metric-driven interpretation offers an alternative contribution to a purely hydrodynamic bow-shock model. If RBH-1 carries a propagating region of altered proper-time rate, its Temporal Shear supplies the conservative acceleration $\mathbf a_\phi=-c^2\boldsymbol\nabla\ln A$. The resolved 650 km s$^{-1}$ span then requires a dynamical depth of order $10^{-6}$—$2.35\times10^{-6}$ in the rest-entry normalization and $4.55\times10^{-6}$ in the stated collinear pattern-speed benchmark—rather than an endpoint redshift of order $10^{-3}$. This ordered body force can generate a large bulk-flow gradient without requiring the same energy to appear as local random motion, while the enhanced scalar attraction lowers the collapse threshold behind the front.

Theoretical analogies exist in the form of non-topological solitons in scalar field theories (e.g., Kusenko 1997; Heeck et al. 2021), which demonstrate how coherent field configurations can maintain stable cores. However, the argument presented here is primarily observational: a consistent density scale is observed emerging across vast differences in mass.

## RBH-1 as a Naked Halo Candidate

This reinterpretation also offers a way to connect competing morphological interpretations: is RBH-1 a runaway compact object or a "bulgeless edge-on galaxy" (Sanchez Almeida et al. 2023)? In the soliton hypothesis, a temporal soliton corresponds to a "naked halo": a compact, self-contained packet of scalar-field energy whose passage could generate a narrow wake.

One possible formation channel is a major merger or strong interaction. In the TEP phenomenology, such events could excite non-linear dynamics in the underlying scalar sector and eject a compact field configuration from the parent halo. This framing makes the ambiguity observational: deep imaging and resolved kinematics can test whether the light traces a bound stellar disk or instead follows in-situ star formation triggered along a trajectory.

## Signatures of Internal Dynamics

Two features of the RBH-1 data point to the internal physics of the soliton.

- Wake Fragmentation (Empirical Scale Tension): A key empirical constraint comes from the scale of star formation itself. The wake contains "knots" or clumps of star formation with sizes $d \lesssim 1$ kpc. In a simple single-phase hydrodynamic shock picture ($v \approx 1000$ km/s), the post-shock temperature is $T \approx 1.4 \times 10^7$ K. At this temperature, the Jeans Length would be $L_J \approx 170$ kpc, far larger than the observed clumps. This places a strong constraint on models in which the dominant post-front phase remains very hot for an extended time.

- Gravitational Echoes: A future test lies in gravitational waves. When two compact objects merge, the resulting "ringdown" signal decays exponentially. If the objects are horizonless solitons rather than true black holes, gravitational waves could be partially trapped between the photon sphere and the compactness scale of the soliton core, producing repeating pulses or "gravitational echoes" (Cardoso et al. 2016). The detailed strong-field prediction depends on the interior solution; the canonical TEP compact-object formulation is developed in TEP-BH (Paper 28, Bahrain).

Additional tests involving magnetar timing anomalies and their connection to the universal scaling law are discussed in Appendix A.

## Broader Context

The characteristic density $\rho_T \approx 20$ g/cm³ used to check the RBH-1 wake geometry is not arbitrary. As detailed in the companion paper (Smawfield 2025g), this same parameter successfully organizes phenomena across 40 orders of magnitude in mass, from the atomic scale (Bohr radius) to galactic rotation curves (SPARC database) and the Milky Way's Keplerian decline.

The fact that a single calibration, derived from terrestrial GNSS clocks, yields a scale consistent with the wake properties of a distant $10^7 M_\odot$ black hole provides support for the hypothesis that RBH-1 may manifest universal scalar-field dynamics. The "dark sector" in this framework is not a particle fluid, but the shadow of temporal structure.

## 7. Conclusion

#### Summary

The identification of RBH-1 as a candidate runaway supermassive black hole presents a significant observational puzzle. The 62 kpc wake of active star formation, produced by an object with inferred velocity $v_{\bullet} \approx 950$ km/s, is difficult to reconcile with standard thermal shock expectations. The coexistence of a high-Mach kinematic discontinuity with cold, star-forming gas motivates consideration of alternative drivers.

The TEP interpretation offers a candidate resolution: RBH-1 may carry a Temporal Topology soliton in addition to its compact baryonic source. In this framework, Temporal Shear supplies coherent bulk acceleration while the unscreened scalar charge — an additional attractive acceleration in Einstein-frame parameterization — reduces the effective Jeans mass, enabling star formation without assigning the full resolved velocity span to local thermal energy.

The data currently available show a resolved position–velocity span $\Delta v \sim 650$ km/s and an independently measured near-tip dispersion $\sigma \approx 31$ km/s. These are distinct observables, not the first and second moments of one unresolved line. Their coexistence places strong constraints on single-phase thermal-shock interpretations and is consistent with the dynamical Temporal-Shear prediction of a large ordered flow with small local dispersion. The work integral requires $|\Delta\ln A|=2.35\times10^{-6}\eta$, where the measured flow geometry fixes $\eta$; the rest-entry and collinear 954 km s$^{-1}$ pattern-speed normalizations give $\eta=1$ and 1.94, respectively. The former endpoint-redshift depth is excluded because it would accelerate matter to approximately $1.97\times10^4$ km s$^{-1}$.

If the broader TEP program is independently validated, the dark sector could be reinterpreted as temporal structure in the conformal metric sector rather than as an undetected particle species. In that interpretation, what is conventionally called dark matter is modeled as phantom mass, i.e., an apparent excess inferred when Temporal Shear is analyzed under an isochronous prior. The RBH-1 wake provides a concrete astrophysical case study in which this interpretation can be confronted with data.

A potential objection concerns the apparent "shadows" imaged for M87* and Sgr A* by the Event Horizon Telescope. These observations strongly support the existence of ultra-compact objects with photon-ring structure consistent with General Relativity, but they do not by themselves uniquely select a mathematical event horizon over all horizonless alternatives. In general, any sufficiently compact configuration that reproduces near-horizon light-bending and exhibits high optical depth can produce an apparent shadow-like depression. The RBH-1 hypothesis therefore does not require that all supermassive black holes be identical neutral soliton candidates; rather, it motivates a targeted comparison between RBH-1 and EHT-class objects, with particular emphasis on whether the central brightness depression behaves as a true absorbing horizon or as a saturating refractive core (Event Horizon Telescope Collaboration 2019; Event Horizon Telescope Collaboration 2022).

The key next step is falsification. Specific falsification criteria are outlined; decisive discrimination regarding the neutral soliton candidate interpretation awaits line-profile decomposition and X-ray flux limits.

- **Resolved phase-space test:** spatially resolved spectra should show an ordered centroid field whose local dispersion remains small where the inferred Temporal Shear is smooth.

- **Gas–star universality:** newly formed stars should inherit the local gas centroid at formation; systematic gas–star slip instead identifies hydrodynamic forcing.

- **Wake Chronometry:** Stellar population ages along the wake should be consistent with the transit time of the perturber across the observed wake length (distance/$v_{\bullet}$). Regions whose inferred stellar ages significantly exceed the local passage time would favor a pre-existing tidal feature; conversely, a tight age-distance correlation would support an in-situ instability front.

- **Wake Collimation:** The wake should remain narrow (high aspect ratio) over its full extent, consistent with a laminar metric disturbance rather than turbulent thermal mixing.

#### Box 7.1: Falsifiers — What Would Rule Out the Metric-Shock Interpretation for RBH-1

The following observations would falsify the specific candidate Temporal Topology soliton/wake interpretation of RBH-1, without necessarily invalidating the broader TEP framework:

- **Hot X-ray Halo:** Detection of an extended hot X-ray component ($T \gtrsim 10^7$ K) coincident with the wake with sufficient luminosity and emission measure ($L_X \sim 10^{41}$–$10^{42}$ erg/s) to account for the dominant post-shock kinetic energy dissipation would indicate bulk thermal shock heating, excluding the predominantly cold metric-shock model for this object.

- **Thermal Line Widths:** If the near-tip [O III] or H$\alpha$ emitting component has $\sigma \gtrsim 80$ km/s (consistent with $T \sim 10^7$ K thermalization), the cold-flow interpretation fails.

- **Wake Broadening:** If the wake aspect ratio decreases to $\lesssim 10:1$ at large distances (indicating Kelvin-Helmholtz turbulent mixing), the laminar metric-shock model is excluded.

- **Scaling Mismatch:** If RBH-1's crossover scale deviates from the $M^{1/3}$ prediction by $>3\sigma$, it would indicate that RBH-1 is not consistent with the soliton interpretation (or that the characteristic density varies), but would not by itself falsify the Universal Scaling Law derived from other systems (e.g., Milky Way).

*Note: Additional tests involving EHT polarimetry are discussed in Appendix A as future directions.*

## References

Abedi, J., Dykaar, H., & Afshordi, N. 2017, *Phys. Rev. D*, 96, 082004 (arXiv:1612.00266)

Allen, M. G., Groves, B. A., Dopita, M. A., Sutherland, R. S., & Kewley, L. J. 2008, *ApJS*, 178, 20 (arXiv:0805.0204)

Archibald, R. F., et al. 2013, *Nature*, 497, 591 (DOI: 10.1038/nature12159)

Archibald, R. F., et al. 2017, *ApJ*, 829, L21 (DOI: 10.3847/2041-8205/829/1/L21; arXiv:1608.01007)

Banda-Barragán, W. E., et al. 2024, *MNRAS*, 527, 3 (arXiv:2311.13600)

Barro, G., et al. 2025, *From "The Cliff" to "Virgil": Mapping the Spectral Diversity of Little Red Dots with JWST/NIRSpec* (arXiv:2512.15853)

Campanelli, M., Lousto, C. O., Zlochower, Y., & Merritt, D. 2007, *Phys. Rev. Lett.* (arXiv:gr-qc/0702133)

Cardoso, V., Hopper, S., Macedo, C. F. B., Palenzuela, C., & Pani, P. 2016, *Phys. Rev. D*, 94, 084031 (arXiv:1608.08637)

Chen, K., Li, Z., Inayoshi, K., & Ho, L. C. 2025, *ApJ Lett.* (DOI: 10.3847/2041-8213/ae1955; arXiv:2505.22600)

Coleman, S. 1985, *Nucl. Phys. B*, 262, 263 (DOI: 10.1016/0550-3213(85)90286-X)

Colpi, M., & Dotti, M. 2011, *Adv. Astron.*, 2011, 1 (arXiv:0906.4339)

Colpi, M., Geppert, U., & Page, D. 2000, *ApJ*, 529, L29 (DOI: 10.1086/312448; arXiv:astro-ph/9912066)

Delvecchio, I., et al. 2025, *A&A* (DOI: 10.1051/0004-6361/202557164; arXiv:2509.07100)

Dib, R., & Kaspi, V. M. 2014, *ApJ*, 784, 37 (DOI: 10.1088/0004-637X/784/1/37; arXiv:1401.5738)

Donato, F., Gentile, G., Salucci, P., et al. 2009, *MNRAS*, 397, 1169 (DOI: 10.1111/j.1365-2966.2009.15004.x; arXiv:0904.4054)

Dopita, M. A., & Sutherland, R. S. 2003, *Astrophysics of the Diffuse Universe* (Springer; DOI: 10.1007/978-3-662-05866-4)

Draine, B. T., & Salpeter, E. E. 1979, *ApJ*, 231, 77 (DOI: 10.1086/157165)

Dursi, L. J., & Pfrommer, C. 2008, *ApJ*, 677, 993 (DOI: 10.1086/529371; arXiv:0711.0213)

Event Horizon Telescope Collaboration. 2019, *ApJ Lett.*, 875, L1 (DOI: 10.3847/2041-8213/ab0ec7; M87* image)

Event Horizon Telescope Collaboration. 2021, *ApJ Lett.*, 910, L12 (DOI: 10.3847/2041-8213/abe71d; M87* polarization)

Event Horizon Telescope Collaboration. 2021, *ApJ Lett.*, 910, L13 (DOI: 10.3847/2041-8213/abe71e; EHT Paper VII)

Event Horizon Telescope Collaboration. 2022, *ApJ Lett.*, 930, L12 (DOI: 10.3847/2041-8213/ac6674; Sgr A* image)

Gleiser, M. 1994, *Phys. Rev. D*, 49, 2978 (DOI: 10.1103/PhysRevD.49.2978; arXiv:hep-ph/9308279)

Gong, Y., Papantonopoulos, E., & Yi, Z. 2018, *Eur. Phys. J. C*, 78, 738 (DOI: 10.1140/epjc/s10052-018-6227-9; arXiv:1711.04102)

Gronke, M., & Oh, S. P. 2018, *MNRAS*, 480, L111 (arXiv:1806.02729)

Gronke, M., & Oh, S. P. 2020, *MNRAS*, 492, 1970 (DOI: 10.1093/mnras/stz3332; arXiv:1907.04771)

Heeck, J., et al. 2021, *Phys. Rev. D*, 103, 115004 (arXiv:2009.08463)

Hui, L., Ostriker, J. P., Tremaine, S., & Witten, E. 2017, *Phys. Rev. D*, 95, 043541 (DOI: 10.1103/PhysRevD.95.043541; arXiv:1610.08297)

Hviding, R. E., et al. 2025, *A&A* (DOI: 10.1051/0004-6361/202555816; arXiv:2506.05459)

Ji, S., Oh, S. P., & McCourt, M. 2019, *MNRAS*, 487, 737 (DOI: 10.1093/mnras/stz1248; arXiv:1809.09101)

Jiao, Y., et al. 2023, *A&A*, 678, A208 (DOI: 10.1051/0004-6361/202347513; arXiv:2309.00048)

Komossa, S. 2012, *Recoiling black holes: electromagnetic signatures, candidates, and astrophysical implications* (arXiv:1202.1977)

Kramer, M., et al. 2006, *Science*, 314, 97 (DOI: 10.1126/science.1132305)

Kusenko, A. 1997, *Phys. Lett. B*, 404, 285 (arXiv:hep-th/9704073)

Lelli, F., McGaugh, S. S., & Schombert, J. M. 2016, *AJ*, 152, 157 (SPARC Database)

McGaugh, S. S., Lelli, F., & Schombert, J. M. 2016, *Phys. Rev. Lett.*, 117, 201101 (DOI: 10.1103/PhysRevLett.117.201101; arXiv:1609.05917)

McGaugh, S. S., Schombert, J. M., Bothun, G. D., & de Blok, W. J. G. 2000, *ApJ Lett.*, 533, L99 (DOI: 10.1086/312628; arXiv:astro-ph/0003001)

Nicolis, A., Rattazzi, R., & Trincherini, E. 2009, *Phys. Rev. D*, 79, 064036 (DOI: 10.1103/PhysRevD.79.064036; arXiv:0811.2197)

Ogiya, G., & Nagai, D. 2023, *ApJ Lett.*, 958, L5 (arXiv:2309.09031)

Olausen, S. A., & Kaspi, V. M. 2014, *ApJS*, 212, 6 (DOI: 10.1088/0067-0049/212/1/6; arXiv:1309.4167)

Pfrommer, C., & Dursi, L. J. 2010, *Nature Physics*, 6, 520 (DOI: 10.1038/nphys1657; arXiv:0911.2476)

Pintore, F., et al. 2016, *MNRAS*, 458, 2088 (DOI: 10.1093/mnras/stw449; arXiv:1602.05950)

Poggianti, B. M., et al. 2019, *ApJ*, 874, 140 (arXiv:1810.05164)

Ray, P. S., et al. 2019, *ApJ*, 879, 130 (DOI: 10.3847/1538-4357/ab24d8)

Ruszkowski, M., Brüggen, M., Lee, D., & Shin, M.-S. 2014, *ApJ*, 784, 75 (DOI: 10.1088/0004-637X/784/1/75; arXiv:1402.0025)

Rybicki, G. B., & Lightman, A. P. 1979, *Radiative Processes in Astrophysics* (Wiley-VCH)

Sanchez Almeida, J., Montes, M., & Trujillo, I. 2023, *A&A* (DOI: 10.1051/0004-6361/202346430; arXiv:2304.12344)

Schive, H.-Y., Chiueh, T., & Broadhurst, T. 2014, *Phys. Rev. Lett.*, 113, 261302 (DOI: 10.1103/PhysRevLett.113.261302; arXiv:1407.7762)

Seidel, E., & Suen, W.-M. 1991, *Phys. Rev. Lett.*, 66, 1659 (DOI: 10.1103/PhysRevLett.66.1659)

Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: [10.5281/zenodo.16921911](https://doi.org/10.5281/zenodo.16921911) (Paper 0)

Smawfield, M. L. (2025). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: [10.5281/zenodo.17127229](https://doi.org/10.5281/zenodo.17127229) (Paper 1)

Smawfield, M. L. (2025). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: [10.5281/zenodo.17517141](https://doi.org/10.5281/zenodo.17517141) (Paper 2)

Smawfield, M. L. (2025). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: [10.5281/zenodo.17860166](https://doi.org/10.5281/zenodo.17860166) (Paper 3)

Smawfield, M. L. (2025). *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: [10.5281/zenodo.17982540](https://doi.org/10.5281/zenodo.17982540) (Paper 4)

Smawfield, M. L. (2025). *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: [10.5281/zenodo.18004832](https://doi.org/10.5281/zenodo.18004832) (Paper 5)

Smawfield, M. L. (2025). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: [10.5281/zenodo.18064365](https://doi.org/10.5281/zenodo.18064365) (Paper 6)

Smawfield, M. L. (2025). *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: [10.5281/zenodo.18059250](https://doi.org/10.5281/zenodo.18059250) (Paper 7 — this work)

Smawfield, M. L. (2025). *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: [10.5281/zenodo.18064581](https://doi.org/10.5281/zenodo.18064581) (Paper 8)

Smawfield, M. L. (2025). *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: [10.5281/zenodo.18109760](https://doi.org/10.5281/zenodo.18109760) (Paper 9)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: [10.5281/zenodo.18165798](https://doi.org/10.5281/zenodo.18165798) (Paper 10)

Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: [10.5281/zenodo.18209702](https://doi.org/10.5281/zenodo.18209702) (Paper 11)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: [10.5281/zenodo.19000827](https://doi.org/10.5281/zenodo.19000827) (Paper 12)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: [10.5281/zenodo.19102061](https://doi.org/10.5281/zenodo.19102061) (Paper 13)

Stevenson, S. D., et al. 2025, *MNRAS* (DOI: 10.1093/mnras/staf2087; arXiv:2509.06913)

Sutherland, R. S., & Dopita, M. A. 1993, *ApJS*, 88, 253 (DOI: 10.1086/191823)

Sutherland, R. S., & Dopita, M. A. 2017, *ApJS*, 229, 34 (DOI: 10.3847/1538-4365/aa6541; arXiv:1703.04946)

TDCOSMO Collaboration. 2025, *A&A* (DOI: 10.1051/0004-6361/202555801; arXiv:2506.03023)

Taylor, J. H., & Weisberg, J. M. 1982, *ApJ*, 253, 908 (DOI: 10.1086/159690)

Tuo, Y. L., et al. 2024, *ApJ*, 966, 80 (DOI: 10.3847/1538-4357/ad2fb6; arXiv:2403.12137)

Vainshtein, A. I. 1972, *Phys. Lett. B*, 39, 393 (DOI: 10.1016/0370-2693(72)90147-5)

Weisberg, J. M., & Taylor, J. H. 2005, *Binary Radio Pulsars*, ASP Conf. Ser., 328, 25 (arXiv:astro-ph/0407149)

Westerweck, J., et al. 2018, *Phys. Rev. D*, 97, 124037 (arXiv:1712.09966)

Will, C. M. 2014, *Living Rev. Relativ.*, 17, 4 (DOI: 10.12942/lrr-2014-4; arXiv:1403.7377)

Williams, J. G., Turyshev, S. G., & Boggs, D. H. 2012, *Class. Quantum Grav.*, 29, 184004 (DOI: 10.1088/0264-9381/29/18/184004; arXiv:1203.2150)

Younes, G., et al. 2020, *ApJ Lett.*, 896, L42 (DOI: 10.3847/2041-8213/ab9a48; arXiv:2006.02814)

Younes, G., et al. 2022, *Nature Astronomy*, 7, 339 (arXiv:2210.11518)

Zhang, H., et al. 2025, *Polarization Images of Solitonic Boson Stars* (arXiv:2508.11992)

de Graaff, A., et al. 2025, *A&A* (arXiv:2503.01891)

van Dokkum, P., et al. 2018, *Nature*, 555, 629 (DOI: 10.1038/nature25767)

van Dokkum, P., et al. 2019, *ApJ Lett.*, 874, L5 (DOI: 10.3847/2041-8213/ab0d92)

van Dokkum, P., et al. 2022, *Nature*, 605, 435 (DOI: 10.1038/s41586-022-04665-6)

van Dokkum, P., et al. 2023, *ApJ Lett.*, 946, L50 (arXiv:2302.04888)

van Dokkum, P., et al. 2025, *JWST Confirmation of a Runaway Supermassive Black Hole via its Supersonic Bow Shock* (arXiv:2512.04166)

Şaşmaz Muş, S., et al. 2014, *MNRAS*, 440, 2916 (DOI: 10.1093/mnras/stu445; arXiv:1402.6054)

## Contact Information

Author: Matthew Lukin Smawfield

Affiliation: Independent Researcher

Email: [matthew@mlsmawfield.com](mailto:matthew@mlsmawfield.com)

ORCID: [0009-0003-8219-3159](https://orcid.org/0009-0003-8219-3159)

GitHub: [github.com/matthewsmawfield](https://github.com/matthewsmawfield)

License: This work is licensed under a [Creative Commons Attribution 4.0 International License](https://creativecommons.org/licenses/by/4.0/).

Version: v0.5 (Blantyre) · Last updated: 17 October 2026

## Appendix A: Prospective Observational Tests

This appendix outlines prospective observational tests in which the Temporal Topology and metric-shock phenomenology developed for RBH-1 yield distinct predictions across independent astrophysical systems.

## A.1 EHT Polarimetry

![EHT Polarimetry Prediction](site/figures/figure_A1_polarization.png)

Figure A.1: EHT Polarimetry Prediction. Simulated polarization signatures for a standard black hole (left) vs. a horizonless soliton (right). The soliton model predicts non-zero polarized flux in the central depression due to transmission through the core.

Permeable horizonless temporal-well models predict detectable polarized flux inside the central brightness depression of Event Horizon Telescope (EHT) sources (M87* and Sgr A*) from rays transmitted through the regular core (developed quantitatively in Paper 28, TEP-BH).

- **Observable discriminator:** Non-zero interior polarized flux fraction and altered inner-ring Faraday rotation profile in full-Stokes VLBI reconstructions, tested via polarized radiative-transfer and GRMHD forward modeling.

## A.2 Additional JWST Case Studies

Several late-2025 JWST/NIRSpec discoveries exhibit kinematic or stellar-population tensions where a refractive proper-time gradient provides a concrete, testable observable channel alongside standard baryonic models (analyzed at catalog scale in Paper 12, TEP-JWST, and Paper 19, TEP-LENS).

### A.2.1 Little Red Dots (LRDs)

JWST has revealed a population of compact sources at $z \gtrsim 3$ with distinctive UV-optical continua and broad Balmer emission. Late-2025 population studies based on public NIRSpec/PRISM spectroscopy report samples of order $\sim 10^2$ objects and conclude that broad Balmer lines are widespread in LRD-selected sources (de Graaff et al. 2025; Barro et al. 2025). A large spectroscopic census further reports that a v-shaped UV-to-optical continuum, a rest-optical point-source component, and broad Balmer lines are strongly linked within a large $z>3$ NIRSpec sample (Hviding et al. 2025). Independent analyses of stacked multi-wavelength data report evidence for AGN-heated dust in median LRD SEDs, while also highlighting tensions in dust-based interpretations (e.g., hot-dust evidence and obscuration geometry; dust-budget constraints) (Delvecchio et al. 2025; Chen et al. 2025).

- *Potential TEP connection.* In a conservative TEP reinterpretation, part of the apparent broad-line width could include a gravitational-redshift component produced by a structured clock-rate field, in addition to virial motion and radiative-transfer effects.

- *Discrimination.* Separating kinematic broadening from redshift gradients using reverberation mapping, spatially resolved spectroscopy, and host-mass constraints.

### A.2.2 Massive Quiescent Galaxies at High Redshift

PRIMER+JADES analyses report a mass-complete catalog of 225 quiescent candidates at $z>2$ with $M_{*} > 10^{10}\,M_{\odot}$ over $\sim 320$ arcmin$^2$, and infer number densities that exceed representative pre-JWST estimates by factors of order a few, while noting that simulations increasingly fall short at $z>3$ with discrepancies approaching $\sim 1$ dex (Stevenson et al. 2025).

- *Potential TEP connection.* In a time-field phenomenology, part of the tension could arise from inference systematics if clock-rate gradients contribute an additional refractive component to observables in dense environments. Under an isochronous GR prior, such contributions can be misinterpreted as excess mass or altered stellar-population parameters.

- *Discrimination.* Compare SED-based stellar masses to independent dynamical and lensing constraints (where available), and test whether residuals correlate with environment rather than with purely baryonic tracers alone.

### A.2.3 Strong-Lensing Time-Delay Cosmography

The TDCOSMO 2025 analysis reports new JWST-NIRSpec stellar-kinematics spectra for multiple time-delay lenses (including spatially resolved kinematics for RX J1131−1231) and emphasizes that improved kinematics can break lensing degeneracies and sharpen cosmological inference (TDCOSMO Collaboration 2025).

- *Potential TEP connection.* This dataset is directly relevant to TEP because time delays are intrinsically chronometric observables. If an additional conformal contribution accumulates in the proper-time sector, the inferred time-delay distance could be biased in a way correlated with lens environment and with mass-profile degeneracies.

- *Discrimination.* Search for residual systematics that correlate with independent indicators of temporal structure, rather than with purely baryonic tracers alone.

## A.3 Magnetar Timing Anomalies

Standard neutron stars experience "glitches" (sudden spin-ups). However, magnetars have exhibited rare "anti-glitches" (sudden spin-downs). In the TEP framework, these are interpreted as boundary interactions: as the star's light cylinder expands toward the soliton scale ($R \propto M^{1/3}$), the magnetosphere approaches a field transition, enabling a rapid change in torque.

The magnetar 1E 2259+586 provides a prospective test case—its spin period lies within 3% of the nominal TEP critical period ($P_{\rm crit} \approx 6.8$ s for a $1.4 M_{\odot}$ neutron star, using the same $\rho_T$ calibration as RBH-1), motivating a population-level test of whether the soliton boundary scale extends universally across compact-object mass scales.

- *Potential TEP connection.* The anti-glitch timing and magnitude could be correlated with the light-cylinder radius approaching the soliton boundary scale predicted by $\rho_T$.

- *Discrimination.* Statistical analysis of magnetar glitch/anti-glitch populations as a function of period; comparison with the critical period predicted by the universal scaling law.

## A.4 Summary of Prospective Channels

These prospective channels are distinct from the primary RBH-1 wake analysis in the main text; each specifies a concrete observational signature—interior VLBI polarization, environment-correlated SED-versus-dynamical mass residuals, strong-lensing time-delay residuals, and the magnetar critical-period boundary—through which the universal $\rho_T$ scaling can be independently falsified.

## Data Availability & Reproducibility

This work follows open-science practices. Every numerical claim introduced by the transport-consistency revision is
generated deterministically from explicitly cited published measurements or stated theory inputs.
The registered calculations do not synthesize observational data. Diagrammatic figures are labelled as
schematics and are not used as measurements. Reproduction of the published Keck/LRIS line fit remains
contingent on release of its reduced spectrum.

### Repository & Code

**GitHub Repository:** [github.com/matthewsmawfield/TEP-RBH](https://github.com/matthewsmawfield/TEP-RBH)

The repository contains deterministic, version-controlled consistency calculations for the RBH-1 candidate
interpretation, together with figure-generation and optional public-archive retrieval utilities.

#### Repository Structure

```

TEP-RBH/
├── results/                         # Analysis outputs and figures
├── scripts/
│   ├── steps/                     # Registered manuscript calculations
│   │   ├── step_01_transport_consistency.py
│   │   └── step_02_soliton_existence.py
│   ├── analysis_checks/           # Physics validation scripts
│   │   ├── cooling_calculation.py
│   │   ├── jeans_analysis.py
│   │   ├── soliton_scale_check.py
│   │   └── stress_energy_check.py
│   ├── figures/                   # Figure generation (10 figure scripts)
│   │   ├── 01_observation_schematic.py
│   │   ├── 01_wake_anatomy.py
│   │   ├── 02_sensitivity.py
│   │   └── [4 additional figure scripts]
│   ├── site/
│   │   └── figures/               # Site-specific figure outputs
│   └── utils/                     # Shared utilities
│       ├── compress_pdf.py
│       ├── logger.py
│       └── process_pdf.py
├── site/
│   └── components/                # HTML manuscript source
└── requirements.txt                 # Python dependencies

```

### Data Provenance

| Data Source | Provider | Access Method | Download Size | Reference |
| --- | --- | --- | --- | --- |
| RBH-1 resolved kinematics | van Dokkum et al. (2026) | Published measurements | Not redistributed | arXiv:2512.04166v2 |
| JWST Archive | MAST | Public archive | Archive-dependent | [MAST](https://archive.stsci.edu/) |

**Repository data status:** no observational FITS products or reduced Keck/LRIS spectrum are
redistributed. The registered transport calculation uses only the published numerical endpoints and their
cited provenance. Archive download sizes depend on the products selected at MAST.

### Reproduction Instructions

#### Quick Start (Analysis Reproduction)

```

# 1. Clone repository
git clone https://github.com/matthewsmawfield/TEP-RBH.git
cd TEP-RBH

# 2. Install dependencies
pip install -r requirements.txt

# 3. Run physics validation checks
python scripts/steps/step_01_transport_consistency.py
python scripts/steps/step_02_soliton_existence.py
python scripts/analysis_checks/cooling_calculation.py
python scripts/analysis_checks/jeans_analysis.py

# 4. Generate all figures
python scripts/figures/01_observation_schematic.py
python scripts/figures/01_wake_anatomy.py
python scripts/figures/02_sensitivity.py
# ... (run remaining figure scripts)

# 5. Build manuscript
cd site
npm install
npm run build

```

#### System Requirements

| Component | Minimum | Recommended | Tested On |
| --- | --- | --- | --- |
| CPU | 4 cores | 8 cores | Apple M4 Pro (14-core) |
| RAM | 8 GB | 16 GB | 24 GB (M4 Pro) |
| Storage | 2 GB | 5 GB | NVMe SSD |
| Runtime | ~10-20 minutes | ~10 minutes (M4 Pro) |

#### Detailed Analysis Scripts

The repository contains physics validation scripts and 10 diagram-generation scripts. The diagrams are explanatory schematics; they are not synthetic observations and do not enter the numerical evidence ledger.

##### Physics Validation Scripts

- **step_01_transport_consistency.py** — Applies the conformal endpoint theorem and the profile-frame work integral to the published RBH-1 position–velocity endpoints; records both the rest-entry normalization and a collinear benchmark using the published 954 km s$^{-1}$ pattern speed, together with the acceleration, equivalent mass, scalar stress-energy ledger, and conditional local-coherence bound in `results/step_01_transport_consistency.json`

- **step_02_soliton_existence.py** — Solves the forced spherical boundary-value problem for the required matter-sourced temporal well under both the unit-term control and Paper 0's selected two-branch kinetic sector ($P_{,X}=k\sqrt{\xi}+2\xi$, $k=16.03$; quartic amplitude-sector interior equilibrium); returns the shear-recovery radius $R_s=\sqrt{GM/g_t}$, the delivered halo-depth ledger, the two-branch exterior shear enhancement ($3.28\times$ at $1$ kpc, $32.8\times$ at $10$ kpc), and the transit-coherence timescales in `results/step_02_soliton_existence.json`

- **cooling_calculation.py** — Validates post-shock cooling times vs dynamical times for RBH-1 soliton; computes t_cool/t_dyn ratio to confirm shock persistence

- **jeans_analysis.py** — Computes effective Jeans mass behind metric shock; derives M_Jeans from TEP field equations with screening corrections; evaluates the ambient-density bookkeeping and the compressed-condensate density table (output: results/jeans_analysis.json)

##### Figure Generation Scripts (10 total)

- **01_observation_schematic.py** — Figure 1: RBH-1 observation schematic showing JWST NIRSpec IFU layout, shock geometry, and wake orientation

- **01_wake_anatomy.py** — Wake anatomy diagram: detailed shock structure, ionization fronts, and velocity field decomposition

- **02_sensitivity.py** — Figure 2: Falsification sensitivity analysis showing detection limits and parameter constraints

- **07_polarization.py** — Polarization analysis: predicted line polarization signatures from anisotropic shock excitation

- **09_line_width_test.py** — Line width diagnostic: [OIII] line width vs shock velocity correlation test

- **10_wake_geometry.py** — Wake geometry reconstruction from observed ionization gradient

- **11_stellar_age.py** — Stellar age analysis: age constraints from stellar population synthesis

- **12_line_ratios.py** — Line ratio diagnostics: [OIII]/Hβ vs [NII]/Hα BPT classification and shock models

- **13_energy_budget.py** — Energy budget calculation: shock energetics, radiative losses, and total injected energy

- **14_scaling.py** — Figure 3: Universal soliton scaling law M_soliton ∝ σ⁴ showing RBH-1, magnetars, and theoretical prediction

All scripts produce outputs in `site/figures/` with JSON metadata logs for traceability.
Run individual scripts via: `python scripts/figures/XX_script_name.py`

#### Key Analysis Outputs

- `site/figures/figure_01_observation.png` — Observation schematic

- `site/figures/figure_02_sensitivity.png` — Falsification sensitivity

- `site/figures/figure_10_scaling.png` — Soliton scaling law

### Software Versions

- **Python** 3.10+

- **NumPy** 1.24+

- **SciPy** 1.10+

- **Matplotlib** 3.7+

- **Astropy** 5.0+

---

*This document was automatically generated from the TEP-RBH research site. For the interactive version with figures and enhanced formatting, visit: https://mlsmawfield.com/tep/rbh/*

*Related Work:*
- [**TEP Theory**](https://doi.org/10.5281/zenodo.16921911) (Foundational framework)
- [**TEP-UCD Paper 6**](https://doi.org/10.5281/zenodo.18064365) (Temporal Topology Saturation Scale)

*Source code and data available at: https://github.com/matthewsmawfield/TEP-RBH*

---

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# Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging
**Matthew Lukin Smawfield**
**Version:** v0.7 (Mombasa)
**Date:** First published: 30 December 2025 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.18064581

---

## Abstract

An optical-domain consistency test of TEP is presented using 11 years (2015–2025) of Satellite Laser Ranging (SLR) data from passive ILRS geodetic satellites (LAGEOS-1/2, Etalon-1/2, and LARES). By employing two-way optical ranging to passive retroreflectors—a measurement chain orthogonal in hardware and systematic-error class to the microwave atomic-clock chain used in Global Navigation Satellite Systems (GNSS)—this analysis probes the terrestrial clock-amplitude channel through its ground-station event timer and bounded slant-path propagation.

Frequency-domain analysis is carried out under sampling-matched
control. On 5-minute resampled station series the TEP-band (10–500
$\mu$Hz) mean PSD exceeds the broadband floor ($f>1$ mHz) by
$14.12\times$ (95% CI: 13.55–14.67; $N=46$ stations); however, with the
ILRS observing duty cycle below one percent, the diagnostic is dominated
by the resampling kernel rather than by the data. Surrogate processes
generated on each station's actual observing grid and passed through the
identical resample–interpolate–concatenate pipeline show that white
noise alone reproduces the measured ratio (sampling-matched null
$13.97$ vs observed $14.12$; two of 46 stations above $p<0.05$), while
every conventional coloured-noise model returns a larger concentration
through the same sampling (masked AR(1): $16.95$; flicker $1/f$:
$26.98$; random walk: $257$). A real-epoch Lomb–Scargle estimator on
the unbinned observation times returns a flat in-band spectrum (ratio
$1.04$ vs white null $1.05$), and the pairwise structure function sits
on the white floor at all lags from 20 minutes to 24 hours. This
sampling-matched construction corrects a concentration reported in
earlier versions of this work as in-band structure, and applies to
spectral diagnostics on any duty-cycle-limited geodetic series. The
conformal timer channel is bounded rather than detected in the
spectral domain: under the timer-rate map $\delta R = R\,\delta A$,
in-band coherent excursions are limited to $\delta A \lesssim
2\times10^{-8}$ (~decimetre equivalent at the mean slant range), while a
landscape excursion of order the surface conformal depth ($u_\oplus
\approx 7\times10^{-10}$) predicts coherent wander of only
$\sim 5$ mm — below the noise floor — so the remaining
discriminating test is the inter-station channel measured below. An
NCEP/NCAR Reanalysis surface
pressure control bounds the synoptic-weather channel at its native
6-hourly resolution (4/34 significant, binomial $p=0.088$; uncorrected
pressure effect 12% of the residual RMS). A
range-dependent lag-1 coherence diagnostic shows that longer signal paths ($\gtrsim 8{,}000$ km)
accumulate greater decoherence than shorter paths
($\lesssim 6{,}500$ km), with the long-minus-short contrast
$\Delta=-0.208$ (95% CI: −0.418 to 0.000) at the 0.5 m threshold. Because isotropic conformal rescaling preserves null cones and the unstretched-space radial gradient is kinetically screened at $y_{\rm tb}\sim 10^{-7}$ at LAGEOS altitude, this path-length dependence operates as a systematic monitor for airmass-correlated tropospheric effects awaiting hourly reanalysis control.

Inter-station pass-correlation analysis under 15-minute contemporaneous
binning yields a nominally significant Fisher-combined result
($\chi^2=16.45$, 4 d.o.f.; $p=0.0025$) that concentrates in three
same-satellite LAGEOS-2 pairs sharing an orbit arc — orbit-model error
enters both stations' residuals as a common mode, and is the
controlling confound for any same-satellite pairing. A dedicated
control therefore forms the contemporaneous pairings across
*different* satellites in the same window — two stations ranging
different targets share no orbit solution, so the channel is closed by
construction while the baseline structure is preserved. At the
5,000–7,500 km bin — pre-specified as the first bin entirely beyond
the simulated $\lambda_T$ turnover under either corpus scale (4,201 km
GPS-PPP; 1,862 km MGEX) — cross-satellite pairs return
$\bar{r}=-0.23$ over 32 baselines (epoch-preserving label-swap null
$p=0.046$ one-sided; circular-shift synchrony null
$p\le5\times10^{-4}$; station-clustered bootstrap 95% CI
$[-0.43,-0.06]$), and it persists essentially unchanged
($\bar{r}=-0.22$, $n=30$) when every model range is recomputed under
an independent analysis centre's orbit product (ESA/ESOC) — though the
feature appears in only one of nine
threshold/bin-width configurations, falls to $\bar{r}=-0.054$ when
restricted to pairs sharing five or more bins, and alternates sign year
to year, so it is reported as a tail event rather than a detection.
A daily-aggregation statistic ($N=190$ pairs, $p_{\mathrm{FWER}}=0.020$)
reverses sign between the two LAGEOS orbit solutions and is retained
as exploratory only.

Taken together — a spectral bound, an amplitude budget sitting
seven or more orders of magnitude above the $10^{-15}$–$10^{-18}$
fractional stabilities of GNSS clock comparisons, and a nominal
turnover-bin candidate short of detection strength — this analysis
establishes an instrumentally independent optical bound on conformal
timing excursions in a constellation carrying no onboard clocks and
no microwave propagation chain, while remaining insufficiently
sensitive to discriminate the reported GNSS-scale signal from
GNSS-specific instrumental effects.

## 1. Introduction

### 1.1 The Necessity of Independent Testing

The Temporal Equivalence Principle (TEP) posits that proper time is a
dynamical field governed, in its conformal sector, by
$A(\phi) = \exp(\beta_A\phi/M_{\text{Pl}})$,
producing distance-structured clock-network covariance and
optical-domain correlation structure. Closed-loop synchronization
holonomy belongs to the disformal or otherwise non-exact transport
sector, not to the pure conformal covariance tested here.
Previous analyses of the GNSS network
(Smawfield 2025b, 2025c, 2025d; Papers 1-3) identified a persistent
correlation structure with Temporal Topology correlation length $\lambda_T \approx 4,000$ km,
providing evidence consistent with TEP's conformal-sector phenomenology.
While robust across processing centers (CODE, IGS, ESA; $R^2 =
0.92-0.97$), temporally stable over 25 years, and present in raw RINEX
observations, these findings relied on a single geodetic technique:
one-way microwave transmission to active ground clocks.

Screening in TEP is represented at the theory level by the environmental operator
$S_\Sigma(\mathcal{E})$.
Quantities such as
$\rho_T$,
$R_T(M)$,
$S_\oplus(r)$,
compactness $\Phi/c^2$,
local stellar density,
geometric coherence length,
and channel-specific response coefficients
are domain-specific projections of $\mathcal{E}$,
not independent screening mechanisms
and not interchangeable universal thresholds.
Each is an observational transfer model
that parameterizes the same underlying operator
in a regime-appropriate form.

This reliance leaves open a critical counter-hypothesis: that the signal
arises from subtle couplings in receiver electronics (e.g., thermal
responses), clock steering algorithms, ionospheric modeling errors
specific to the microwave L-band, or pervasive "colored noise" (flicker
noise) often treated as irreducible station systematics. To
independently test this "clock artifact" hypothesis, the conformal
sector predictions must be sought in a system whose space segment
carries no onboard clocks and no microwave propagation. This study addresses this challenge by
analyzing 11 years (2015–2025) of Satellite Laser Ranging (SLR) data
from the LAGEOS-1 and LAGEOS-2 missions. Unlike GNSS, SLR relies on
two-way optical pulses reflected off passive retroreflectors. The
"clock" is a ground-based event timer, and the observable is the pure
round-trip time of flight. If the TEP signal represents a fundamental
Temporal-Topology response, it is
expected to manifest in these optical residuals—potentially offering a
physical origin for the persistent "flicker noise" floor and scale
drifts observed in geodetic time series.

The distance-structured phase correlation analyzed in the SLR network is governed by the same common environment-dependent Temporal-Topology response as the microwave GNSS networks. By probing the geometric saturation scale through two-way optical ranging, this analysis directly tests the continuous macroscopic flattening of the Temporal Topology in Earth's local potential well.

### 1.2 TEP's Two Sectors: Conformal vs. Disformal

The TEP framework (Smawfield 2025a; Paper 0) distinguishes two
physically distinct sectors with different observational signatures:

#### Conformal Sector (Clock-Rate Modulation)

**Coupling:** Universal conformal factor $A(\phi) =
\exp(\beta_A\phi/M_{\text{Pl}})$ modulates proper time rates:
$\mathrm{d}\tau/\mathrm{d}t \propto A(\phi)$.

**Observable:** Spatial correlations in clock
frequencies with characteristic length $R_T = (3M/4\pi\rho_T)^{1/3} \approx 4{,}150$ km for Earth,
set by the scale of the scalar field's continuous spatial profile
(Temporal Topology).

**Constraint Status:**
*Not directly constrained by photon–graviton differential-propagation bounds; indirectly constrained by PPN, source-screening, clock-comparison, and equivalence-principle tests.*

**GNSS Evidence (Smawfield 2025b,c,d; Papers 1-3):**
Supported by distance-structured correlations ($\lambda_T = 4,201 \pm
1,967$ km), orbital velocity coupling ($r = -0.888$, 5.1$\sigma$),
and an annual-phase direction near Earth's aphelion velocity tangent;
the directed CMB-apex template is anti-phase rather than a detected rest frame.

#### Disformal Sector (Synchronization Holonomy)

**Coupling:** Disformal term $B(\phi) \nabla_\mu\phi
\nabla_\nu\phi$ tilts photon light cones, creating path-dependent
one-way time asymmetries.

**Observable:** Synchronization holonomy $H_{\rm resid} = \oint_C
(\tilde{\sigma} - \sigma_{\rm GR}) \neq 0$ in closed-loop time transfer
(after GR subtraction).

**Constraint Status:**
*Tightly bounded by GW170817:* $|c_\gamma - c_g|/c \lesssim
10^{-15}$ requires $B(\phi)(\partial\phi)^2$ to be negligible today.
This constraint applies specifically to the disformal sector because
the disformal coupling tilts photon light cones independently of
graviton propagation, creating a differential speed. In contrast,
conformal coupling modulates proper time rates universally,
affecting both photons and gravitons equally, leaving their relative
speed unchanged and thus evading the GW170817 bound.

**GNSS Evidence:** *Untested* (no closed-loop
holonomy experiments performed).

### 1.3 SLR Predictions: Conformal Sector Only

**Scope of This Analysis:** This paper tests TEP's
conformal sector predictions only. The sparse ILRS network (46
stations) and two-way measurement geometry do not provide the
closed-loop time-transfer topology and dense temporal coverage
required to test synchronization holonomy, the genuinely
disformal observable. The estimator-mediated sign inversion of
the conformal common-mode field under dynamic orbit
determination—the "Anti-Echo"—is a conformal-sector signature
and lies within the tested scope of this work.

The Temporal Topology response makes three testable predictions for SLR:

-
**Path Dependence:** Residual temporal structure
should vary systematically with signal path length (stronger at
lower elevations), distinguishing Temporal-Topology response from
purely local station systematics.
Prediction: order-unity to order-of-magnitude path dependence
in a lag-1 diagnostic. Observed: a strong short-path vs
long-path contrast in a gap-aware, binned lag-1 statistic, with
estimator-sensitive uncertainty (§3.2).

-
**Spectral Structure:** Residuals may carry
low-frequency structure in the TEP band
($10-500 \mu\text{Hz}$, the convention inherited from the GNSS
clock analyses where characteristic clock-noise structure
resides); for a station co-rotating within Earth's temporal well
the physically accessible in-band content is the rotation-cadence
sampling of superposed anisotropic wells, the orbital sampling of
ambient-landscape structure on $6\times10^{4}$–$3\times10^{6}$ km
scales ($v_{\rm orb}/f \approx 30\,{\rm km\,s^{-1}}/f$), and the
landscape's own temporal evolution.
Observed: an apparent $14.12\times$ band/broadband
concentration on resampled station series (§3.3), shown by
sampling-matched nulls to be a property of the
sparse-sampling kernel — white noise through the identical
pipeline returns $13.97$ — with the real-epoch spectrum flat;
the channel is carried as a bound on in-band conformal
excursions ($\delta A \lesssim 2\times10^{-8}$ under
$\delta R = R\,\delta A$), not a detection (§3.3.1).

-
**Frequency Independence:** The conformal coupling is
achromatic; optical (SLR, $\sim 500$ THz) and microwave (GNSS, $\sim
1$ GHz) should show consistent correlation lengths when scaled by
their respective ambient density profiles (accounting for local
matter density differences).
Prediction: Range-dependent coherence structure consistent with
the corpus's measured Temporal-Topology scales
($\lambda_T = 4{,}201 \pm 1{,}967$ km GPS-PPP; $1{,}862$ km
MGEX — the fitted scale is product- and metric-dependent,
Paper 14). Observed: consistent scaling
(§5.1).

-
**Spatial Coherence:** Residuals should exhibit
distance-dependent spatial correlations consistent with the
characteristic scale of the Temporal Topology, $\lambda_T$.
Prediction: anticorrelation at intermediate baselines under
dynamic orbit determination—the estimator-mediated
"Anti-Echo" signature of the conformal common mode—or
short-range coherence. Observed:
a nominally significant pass-correlation result ($p=0.0025$)
rests on three LAGEOS-2 pairs including one with only three
passes; under the confound-controlled cross-satellite
pairing — stations ranging different satellites in the same
window, which cannot share orbit error — the anticorrelation
persists at the first bin beyond the $\lambda_T$ turnover
scale (5,000–7,500 km, $\bar{r}=-0.23$, $n=32$ pairs,
epoch-preserving-null $p=0.046$ one-sided, §3.4). The
daily-aggregation
statistic ($p_{\mathrm{FWER}}=0.020$) is retained as
exploratory: its driving bin reverses sign between
independent orbit solutions.

**What This Paper Does NOT Test:** The disformal
sector's defining observables—synchronization holonomy $H_{\rm
resid}$ and one-way propagation asymmetry—require the closed-loop
configurations described in Smawfield (2025a; Paper 0, §10.A–D):
closed-loop optical time transfer, interplanetary one-way asymmetry
measurements, and triangle synchronization holonomy tests. These
experiments remain to be performed. The estimator-mediated sign
inversion analyzed here is distinct: it is the conformal common-mode
field remapped by dynamic orbit determination and invokes no
disformal coupling.

## 2. Methodology

### 2.1 The Target: LAGEOS

The Laser Geodynamics Satellites (LAGEOS-1 and LAGEOS-2) represent excellent test masses for this investigation. As dense, passive spheres covered in retroreflectors, they have the highest mass-to-area ratio of any satellite, minimizing non-gravitational perturbations (e.g., drag, radiation pressure). They carry no active electronics and no clocks in the space segment. The "time" measurement is performed by the ground segment's event timer — an active clock in the ground chain — which reads the round-trip flight time of a photon against its own reference. Under isotropic conformal rescaling $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$, null cones are preserved identically so the two-way optical path carries no conformal propagation content; under the unstretched-space temporal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, rescaling the $(3+1)$ ADM spacetime lapse $N_{\rm TEP}=A(\phi)N_{\rm GR}$ at fixed spatial 3-metric $\gamma_{ij}$), the static terrestrial radial gradient across the LAGEOS altitude ($r\approx 12{,}270$ km) is kinetically screened by $y_{\rm tb}(g_\oplus)\sim 10^{-7}$ on the selected two-branch action (Paper 0, §7; Paper 15, Step 045), so the ground timer's rate remains the dominant terrestrial observable while time-dependent slant-path fluctuations are subdominant to tropospheric airmass delays. SLR thereby isolates the ground-station leg of the clock-amplitude channel — the same terrestrial channel probed by GNSS — while removing onboard-clock, steering-electronics, microwave-propagation, and ionospheric-modeling systematics by construction. What is orthogonal is the instrument, calibration chain, and systematic-error class, not the physical channel; the satellite's orbital response to Temporal Shear along its $\tilde g$ geodesic is screened deep in the Earth environment under the corpus's terrestrial evaluation (Section 4) and is not what the residual spectra analyzed here measure.

### 2.2 Dataset & Processing

The complete International Laser Ranging Service (ILRS) dataset for passive geodetic satellites (LAGEOS-1/2, Etalon-1/2, and LARES) was analyzed over an 11-year period (Mar 2015 – Dec 2025), comprising 4,647,088 Normal Point observations from 46 global stations. Residuals were computed for all observations with available ephemerides and modeling inputs.

Residuals were computed relative to the ILRS combined SP3 orbit products (ILRSA/ILRSB, preferred where available over single-analysis-center files). A consistent orbit-product family was retained throughout to avoid systematic noise introduced by mixed-center product aggregation. The reduction strategy proceeded in two stages:

- **Geodetic Validation:** A broad 5-meter outlier rejection window retained $\approx 1.83$ million "valid" geodetic observations. The initial RMS ($\approx 2.77$ m) reflects the raw pre-fit state relative to the *a priori* orbit, preserving large-scale signal structures that are typically removed by aggressive orbital fitting.

- **Coherence Analysis Subset:** To isolate subtle timing correlations from gross interpolation and modeling errors, a strict 0.5-meter (50 cm) threshold was applied. This high-precision subset (201,503 residuals, $\approx 4.3\%$ of all computed residuals) forms the basis of the primary inter-station and propagation diagnostics reported in this work. This cut prioritizes epochs where orbit interpolation and environmental corrections remain within the sub-meter regime. Robustness is quantified by an explicit residual-threshold sweep (0.3, 0.5, 1.0 m) in the analysis outputs.

- **Corrections:** Standard Marini-Murray troposphere model, Shapiro delay, and Sagnac corrections were applied. No station-specific meteorological data was used, to avoid introducing local sensor systematics.

- **Parameter Estimation Strategy:** This analysis avoids the standard practice of estimating frequent empirical accelerations or "geographically correlated" parameters, which are often used in precise orbit determination (POD) to whiten residuals. By fixing the orbit to the ILRS combined solution and avoiding secondary empirical filtering, the "common mode" signals—typically discarded as noise—are preserved for analysis.

### 2.3 Inter-Station Metric: Contemporaneous Pass-Bin & Daily Aggregation

For sparse SLR networks, continuous, regularly sampled inter-station time series are generally unavailable. Therefore, the primary inter-station metric used here was based on *contemporaneous pass bins*. For each satellite and each time bin (5-minute and 15-minute windows), a pass-mean residual anomaly was computed at each station after subtracting that station’s global mean residual. Inter-station correlation was then computed for each station pair by correlating these pass-mean anomalies across bins.

Statistical significance was assessed using a family-wise circular-shift permutation test across distance bins (2000 permutations). Two confound controls were applied to the pass-bin statistic (Step 2.7): a cross-satellite pairing — correlating residuals of stations ranging *different* satellites within the same time bin, which removes the shared-orbit-error common mode by construction while preserving baseline structure — and a per-satellite split of the daily-aggregation test. Because the circular-shift null destroys bin synchrony but does not price the mechanical anticorrelation floor induced by station-level debiasing within each epoch's pairing geometry, significance is additionally evaluated against an epoch-preserving label-swap null (2000 permutations) that permutes residuals across the cells within each time bin; the label-swap statistic is the primary confound-controlled inference and the circular-shift value is reported as the synchrony-only comparison. Additionally, a daily-aggregation analysis was performed where residuals were averaged daily per station to maximize temporal overlap ($N=190$ station pairs), providing a check against short-term pass-geometry artifacts; its outcome is interpreted jointly with the orbit-solution split (Section 3.4).

As a secondary check, an irregular-sampling phase-alignment statistic was computed on the same contemporaneous pass-bin series (without interpolation) and evaluated under an analogous family-wise circular-shift null test.

### 2.4 Spectral Diagnostic (TEP Band)

In addition to the pass-bin spatial test, the residuals were examined in the frequency domain to quantify the concentration of power in the TEP band (10–500 $\mu$Hz). Two estimators were used, with complementary strengths. (i) The resampled diagnostic of the GNSS convention: residuals were averaged into 5-minute bins, linearly interpolated over gaps of up to two bins, concatenated, detrended, and analysed with Welch's method; the summary statistic is the ratio of mean PSD in the TEP band to the broadband floor ($f>1$ mHz). Because the ILRS observing duty cycle is below one percent, this estimator mixes the residual spectrum with the sampling kernel; its interpretation was therefore established against *sampling-matched* surrogate nulls — continuous surrogate processes generated on each station's full observing grid, sampled at the actual observation bins, and passed through the identical resample–interpolate–concatenate–detrend–Welch chain (Step 2.8). The surrogate families cover the conventional geodetic-noise alternatives: white noise, station-matched AR(1), canonical flicker $1/f$, a power-law grid, a station-matched power law, and random walk (60 surrogates per family per station). (ii) A real-epoch Lomb–Scargle estimator on the actual 5-minute observation epochs — requiring no interpolation or stitching — evaluated against the same surrogate families sampled at the same epochs, supplemented by the pairwise structure function $SF(\tau)=\tfrac12\langle(y_j-y_i)^2\rangle$ over lag windows spanning 20 minutes to 24 hours. The TEP band itself (periods of ~30 minutes to ~28 hours) is the convention inherited from the GNSS clock analyses (Papers 1–3); its lower edge encloses the diurnal family through which a co-rotating station samples superposed anisotropic wells, and its upper edge sits near the pass-resolution floor of normal-point data.

## 3. Conformal Sector Assessment

Four conformal-sector diagnostics are evaluated across the 11-year LAGEOS
dataset below: characterizing the residual noise floor (§3.1), monitoring
path-length-dependent tropospheric decoherence as a propagation systematic
control (§3.2), establishing a sampling-matched spectral bound on in-band
ground-timer excursions (§3.3), and testing confound-controlled
cross-satellite inter-station correlations (§3.4).

### 3.1 Data Characterization: The Noise Floor

The raw SLR residuals exhibit a broad distribution. Within the strictly
filtered subset ($N=201,503$, $|\Delta\rho|<0.5$ m), the RMS is 0.29 m
(288 mm). The central question is whether there exists reproducible
structure in this noise floor consistent with the Temporal-Topology
response. In standard geodetic analysis, this floor is often
characterized as "flicker noise" ($1/f$) or "colored noise" (Williams et
al., 2004) and is typically attributed to unmodeled station systematics.
TEP predicts this colored spectrum as a physical consequence of scalar
field coupling.

### 3.2 Range-Dependent Coherence: The Propagation Signature

To distinguish propagation effects from local station systematics, a
lag-1 residual statistic was analyzed as a function of signal path
length (range to satellite). Here "lag-1" denotes the correlation
between successive 5-minute binned residual means, computed on
station-debiased residuals and made gap-aware by excluding time gaps
exceeding 450 s. This diagnostic targets sub-hour temporal structure
while avoiding trivial within-pass autocorrelation.

-
**Shorter path ($\lesssim 6{,}500$ km):** mean lag-1
$\approx -0.063$ (95% CI: −0.167 to 0.011).

-
**Longer path ($\gtrsim 8{,}000$ km):** mean lag-1
$\approx -0.271$ (95% CI: −0.500 to −0.045).

Under a strict $|\Delta\rho|<0.5$ m filter, the long-minus-short
contrast is $\Delta(\mathrm{low}-\mathrm{high})=-0.208$ (95% bootstrap
CI: −0.418 to 0.000). The corresponding low/high ratio is 4.30 (95%
bootstrap CI: −21.42 to 48.11), but is ill-conditioned because the
short-path mean is near zero. Under a looser $|\Delta\rho|<1.0$ m
filter, the contrast becomes
$\Delta(\mathrm{low}-\mathrm{high})=-0.086$ (95% CI: −0.248 to
0.069) and the corresponding ratio becomes −8.68 (95% CI: −56.90 to
63.51). This threshold sensitivity, combined with the null-invariance
of light propagation in the conformal sector, means the path-length
diagnostic operates as a monitor for airmass-correlated systematics
(e.g., unmodeled tropospheric delay) rather than a TEP signature.

Under isotropic conformal rescaling, light propagation is null-invariant, while under the unstretched-space temporal metric the static terrestrial radial gradient at LAGEOS altitude is kinetically screened ($y_{\rm tb}\sim 10^{-7}$). The observed path dependence — where lower elevations (longer paths) decorrelate more — is dominated by ordinary tropospheric delay mismodeling, as the Marini–Murray standard-atmosphere correction leaves synoptic pressure and wet-path fluctuations uncorrected, which grow with airmass. The path-length diagnostic is therefore retained as a systematic check awaiting high-resolution reanalysis controls (e.g. ERA5). The resampled-series spectral diagnostic (Section 3.3) returns an apparent
14.12× TEP-band enhancement that persists across rejection
thresholds (spanning 11.9×–14.9×, a ~25% variation, over the
0.3–1.0 m range); its interpretation is fixed by the sampling-matched
null analysis of §3.3.1, which attributes the concentration to the
resampling kernel rather than to low-frequency structure in the
residuals themselves.

### 3.3 Spectral Concentration on the Resampled Grid

Frequency-domain analysis of the resampled station series returns a
large apparent concentration of power within the empirical TEP
frequency band (10–500 $\mu$Hz). On 5-minute resampled station
series, the station-averaged TEP-band mean PSD exceeds the
full-spectrum mean PSD by $2.48\times$ (95% CI: 2.46–2.50; $N=46$
stations), and the broadband floor defined by $f > 1$ mHz by
$14.12\times$ (95% CI: 13.55–14.67; $N=46$ stations). The band
(periods of ~30 minutes to ~28 hours) is the convention inherited
from the GNSS clock analyses (Smawfield 2025b, 2025c, 2025d; Papers
1-3), rather than derived from a scalar-field transit mechanism
(which for a co-rotating Earth station yields frequencies above the
band). For a station co-rotating within Earth's temporal well, the
physically accessible in-band content is the rotation-cadence
sampling of superposed anisotropic wells (solar, lunar and
inertial-landscape directions), the orbital sampling of
ambient-landscape structure on $6\times10^{4}$–$3\times10^{6}$ km
scales ($v_{\rm orb}/f \approx 30\,{\rm km\,s^{-1}}/f$), and the
landscape's own temporal evolution; the band's lower edge encloses
the diurnal family and its upper edge sits near the pass-resolution
floor of normal-point data. Whether the measured concentration
carries any information about the residuals' own spectrum, however,
can only be decided against nulls that reproduce the observing grid
— the task of §3.3.1.

### 3.3.1 Sampling-Matched Coloured-Noise Nulls

The resampled diagnostic must be read against the sampling that
produced it. Across the 11-year record, a typical station fills well
under one percent of its 5-minute bins, so the
interpolate-and-concatenate construction used to form a Welch input
mixes the residual spectrum with the sampling kernel; a surrogate
null is only meaningful if the surrogate process is passed through
the identical pipeline. A dedicated control (Step 2.8) was therefore
implemented: for each of the 46 stations, continuous surrogate
processes are generated on the station's full 5-minute observing
grid, sampled at the actual observation bins, linearly interpolated
over gaps of up to two bins, concatenated, detrended, and transformed
by Welch's method — the identical chain applied to the data. The
surrogate families cover the conventional geodetic-noise
alternatives: white noise, AR(1) at the station's measured lag-1
autocorrelation ($\bar\phi = 0.66$), canonical flicker $1/f$, a
power-law grid ($\alpha = 0.5$–1.5), a station-matched power law,
and random walk (60 surrogates per family per station).

The observed $14.12\times$ ratio is reproduced almost exactly by
white noise alone: the sampling-matched white null mean is $13.97$,
the median station percentile of the observed ratio inside its own
null is 0.58, and only two of 46 stations exceed $p<0.05$ — the
chance rate. The residual $\sim 1\%$ offset of the observed mean
above the white-null mean lies inside the null dispersion (Fisher
combined $p = 0.20$), so no coherent excess is supported: the
claimed fourteen-fold enhancement belongs to the sampling
kernel. Every coloured alternative returns a larger
concentration through the same sampling — masked AR(1) $16.95$,
flicker $26.98$, power-law $\alpha = 1.5$ $91.1$, random walk $257$ —
so the residuals are less red than any conventional coloured-noise
model seen through the observing grid. This corrects the conclusion
reported in earlier versions of this work under an unmasked
comparison, in which the observed ratio was evaluated against
surrogates never subjected to the sampling kernel. The earlier
AR(1)-only diagnostic of Step 2.3 found $35/46$ stations individually
above its null, but the excess lay in the estimator's response to the
observing pattern, not in the data. The apparent concentration is
therefore a property of the resampling pipeline, not of the residuals.

Two estimator-independent checks close the question. A Lomb–Scargle
analysis on the actual 5-minute observation epochs — no
interpolation, no stitching — returns a flat in-band spectrum: the
observed band/broadband ratio is $1.04$ against a white-noise null of
$1.05$ at the same epochs (27 of 46 stations sit below the null
mean), and below every red alternative (masked AR(1) $1.39$; flicker
$1.59$; random walk $1.70$). The pairwise structure function
$SF(\tau)=\tfrac12\langle(y_j-y_i)^2\rangle$ sits on the white-noise
floor $\sigma_y^2$ at every lag from 20 minutes to 24 hours
(excess of at most a few percent). No in-band spectral excess is
detected at the station level.

This null is the expected structure of the theory rather than a
failure of it. Under the timer-rate channel the two-way flight time
is conformally invariant while the station event timer runs at the
local matter rate, so a conformal excursion maps to a range offset
$\delta R = R\,\delta A$ — and a uniform excursion is invisible to an
isolated station precisely because measurement is relational: the
conformal carrier is a network common mode whose evidence lives in
the inter-station spatial statistic (§3.4). The map also yields a
falsifiable amplitude budget at the mean slant range
$\bar R \approx 6{,}700$ km. Reproducing the full residual RMS
($\approx 282$ mm) would require $\delta A \approx 4.2\times10^{-8}$
— about sixty times the Earth's surface conformal depth
$u_\oplus = GM_\oplus/c^2R_\oplus \approx 6.95\times10^{-10}$, so the
bulk residual cannot be conformal in origin; the undetected coherent
in-band wander is bounded near the decimetre level, giving
$\delta A \lesssim 2\times10^{-8}$ (about $30\times\,u_\oplus$); and
at a landscape excursion of order the surface conformal depth itself
the channel predicts coherent wander of only $\sim 5$ mm — an order
of magnitude below the detection floor. Per-station spectral tests
are therefore blind to the channel at its natural amplitude at this
observing cadence. The spectral null is therefore the outcome
expected at the surface conformal depth rather than a failed
prediction, and the remaining discriminating test is the
network-coherent spatial channel measured in §3.4 — the observable
class the corpus's measurement taxonomy (Smawfield 2025; Paper 9)
assigns to the conformal sector under two-way measurement.

### 3.3.2 NWM Spectral Control: Surface Pressure Coherence

The TEP band (10–500 $\mu$Hz; periods 30 min to 28 h) overlaps
timescales characteristic of synoptic meteorology, raising the
possibility that low-frequency residual structure reflects
uncorrected atmospheric pressure variations rather than a physical
signal. The Marini–Murray tropospheric correction applied in the
residual reduction uses a standard-atmosphere pressure profile
($P = 1013.25\,\mathrm{hPa}\times e^{-h/8.5\,\mathrm{km}}$), which
removes the mean zenith delay but does not correct synoptic
pressure variations. A dedicated control was therefore performed
using NCEP/NCAR Reanalysis surface pressure (Kalnay et al., 1996),
obtained via OPeNDAP at the nearest grid point to each of the 46
stations (2.5° Gaussian grid, 4×daily, 2015–2025).

NCEP pressure is sampled at 6-hourly intervals (Nyquist
$= 23.15\,\mu$Hz), so the TEP/broadband ratio cannot be computed
at native cadence (the broadband floor at $>1$ mHz lies above the
Nyquist). Interpolating to 5-minute cadence inflates the ratio
because linear interpolation adds no high-frequency power; the
primary diagnostic is instead the magnitude-squared coherence
between pressure and residuals at native 6-hourly resolution in
the resolvable portion of the TEP band (10–23 $\mu$Hz). A Monte
Carlo null (200 random permutations per station, Welch
$n_{\mathrm{perseg}}=128$) provides station-specific significance
thresholds.

The mean pressure-residual coherence in the 10–23 $\mu$Hz band is
$0.133$ (95% CI: 0.101–0.170) across 34 stations with sufficient
overlap. Four of 34 stations individually reject the null at
$p<0.05$, against 1.7 expected by chance (binomial
$p=0.088$); the excess is not statistically significant. The
estimated uncorrected range error from pressure variation—combining
the residual tropospheric delay ($2\times 0.002277\,\sigma_P/f_{\mathrm{lat}}$)
and pressure loading ($0.3\sin 45°\,\sigma_P$ mm hPa$^{-1}$)—is
$30.5$ mm, or $12\%$ of the mean residual RMS ($257.4$ mm). The
residual stream's low-frequency structure is therefore not
attributable to synoptic weather: the pressure effect is too small,
and the two series are spectrally independent at the majority of
stations. This control is computed on shared 6-hourly epochs rather
than on the resampled grid, so it constrains the synoptic channel
directly and is unaffected by the sampling question resolved in
§3.3.1.

A limitation of this control is that 6-hourly NCEP resolution
constrains only the lower third of the TEP band (10–23 $\mu$Hz);
the upper band (23–500 $\mu$Hz) is unconstrained by NCEP.
Hourly reanalysis products (e.g., ERA5) would resolve the full
TEP band and are identified as a priority for future work
(§5).

### 3.4 Inter-Station Analysis: Expected Network Limitations

The ILRS network is sparse in time; most station pairs lack sufficient
contiguous overlap to support the cross-spectral methods employed in
GNSS analysis. Inter-station inference uses contemporaneous,
same-satellite 5-minute bins: for each satellite and time bin, pass-mean
residual anomalies are computed per station (after removing each
station's global mean), and station-pair correlations are estimated
across bins.

**Key Finding:**

#### Inter-Station Pass Correlations: Network Sparsity and Statistical Power

Distance-binned mean pass-correlation estimates fluctuate with large
variance under a strict 5-minute contemporaneous binning. When the
contemporaneous overlap window is widened to 15-minute bins, a
family-wise circular-shift test yields a nominally significant
Fisher-combined result ($\chi^2=16.45$ with 4 d.o.f.;
$p=0.0025$). However, this result requires careful scrutiny: the
combined significance is driven entirely by LAGEOS-2
($p=0.0005$), whose signal concentrates in a single distance bin
(5,000–7,500 km) containing only three station pairs. One of these
pairs (7124–7825) contributes a correlation of $r=-0.910$ from
merely three contemporaneous passes—a value that is not
individually significant at the $p<0.05$ level ($p=0.273$ for
$n=3$). LAGEOS-1, with nine pairs in the same bin and comparable
observation counts, remains consistent with the null hypothesis
($p \approx 0.54$). When station pairs are restricted to those with
ten or more contemporaneous passes, both satellites converge to
near-zero mean correlations (LAGEOS-1: $\bar{r}=+0.007$; LAGEOS-2:
$\bar{r}=+0.035$), indicating that the apparent LAGEOS-2 signal is
an artifact of small-sample variance rather than a physical
detection.

A daily-aggregation analysis ($N=190$ station pairs, top-20
stations by observation count) was performed to improve
statistical power through increased temporal overlap. The
family-wise permutation test yields
$p_{\mathrm{FWER}}=0.020$, nominally reaching conventional
significance, with the most negative correlation at
3000–5000 km baselines ($\bar{r}=-0.074$). That statistic does
not, however, control the dominant common-mode confound: the
driving bin contains only four pairs, and a per-satellite split
(Step 2.7) shows it reverses sign between the two independent
LAGEOS orbit solutions (LAGEOS-1 $+0.11$; LAGEOS-2 $-0.08$) —
the daily-aggregation feature is therefore orbit-solution
dependent and is retained as exploratory only. Removing the
per-day network mean likewise leaves a near-flat profile on the
mechanical $-1/(N-1)\approx-0.05$ floor rather than a
turnover-localised feature.

The confound-controlled test is the cross-satellite pairing
(Step 2.7): two stations ranging *different* satellites in
the same 15-minute window share no orbit solution, so the
shared-orbit-error common mode is absent by construction while
the contemporaneous baseline structure is preserved. Under that
pairing the anticorrelation not only survives but strengthens:
at 5,000–7,500 km — the first distance bin lying entirely beyond
the simulated $\lambda_T$ turnover under either corpus anchor
($4{,}201$ km GPS-PPP; $1{,}862$ km MGEX) —
cross-satellite pairs return $\bar{r}=-0.228$ over $n=32$ pairs
(station-clustered bootstrap 95% CI $[-0.43,-0.06]$, pricing
the non-independence of pairs sharing a station). Significance is
quoted against two nulls. The within-pair circular-shift null,
which destroys bin synchrony only, gives per-bin
$p\le5\times10^{-4}$; the epoch-preserving label-swap null —
which permutes residuals across the cells inside each time bin,
preserving epoch marginals, within-bin common modes, and the
pairing geometry — returns a mechanical anticorrelation floor of
$-0.122\pm0.062$ at this bin (debiasing removes each station's
global mean, so same-bin residuals sum near zero and cross
products anticorrelate by construction). Against that sharper
null the observed value is a nominal one-sided excess
($p=0.046$, uncorrected across the eight bins), i.e.
$\approx-0.11$ beyond the geometric floor, while every other
distance bin is consistent with its own floor (per-bin
$p=0.25$–$0.93$). The matched
same-satellite pairs give a diluted $\bar{r}=-0.09$ at the same
bin — actually above the corresponding same-satellite
label-swap floor of $-0.144$ ($p=0.74$) — the expected
direction, since an orbit-error common mode
enters co-visible ranges positively and therefore masks rather
than produces the anticorrelation. The turnover-bin composition
is dominated by the LAGEOS-1$\times$LAGEOS-2 channel (all 32
pairs), i.e. two independent orbit arcs, not one shared solution.
A centre-level variant of the same confound — a shared
orbit-model or Earth-gravity-model imprint entering both
LAGEOS solutions through the common analysis-centre product —
is bounded directly (Step 2.9): recomputing every model range
under an independent centre's SP3 solution (ESA/ESOC) on the
identical filtered epoch set returns $\bar{r}=-0.217$ over
$n=30$ LAGEOS-1$\times$LAGEOS-2 pairs at the same bin
(circular-shift null $p\le5\times10^{-4}$), matching the
ILRSB reference value $\bar{r}=-0.228$. The residual caveat is
that both products are determined against largely the same
geodesy data, so errors common to geodetic modelling itself —
rather than to a particular centre's solution — are not
excluded by this swap alone.
Because the sign inversion is generic to monopole absorption of
any spatially correlated common-mode error, the discriminating
content is the turnover's location rather than the sign itself:
for the GNSS-measured $\lambda_T = 4{,}201$ km on a global-scale
network the simulated anticorrelation sets in at
$\approx 4{,}000$ km, bracketed by the observed onset bin, while
the cross-estimator contrast against the positive correlations
measured in kinematic GNSS solutions at matched baselines
supplies an independent check on the same field. The turnover
location is also robust to which corpus scale anchors it. The
corpus carries two measured $\lambda_T$ values — $4{,}201 \pm
1{,}967$ km from GPS PPP (Paper 2) and $1{,}862$ km from the
MGEX combined product (Paper 14; cluster-robust
$\sigma = \pm 155$ km) — whose difference reflects the product-
and metric-dependence of the fitted scale rather than a
contradiction. Repeating the monopole-absorption simulation
across the injected scale returns zero crossings of
$3{,}421$ km at $\lambda_T = 1{,}862$ km and $3{,}947$ km at
$\lambda_T = 4{,}201$ km: under either anchor the 5,000–7,500 km
bin remains the first bin lying entirely beyond the turnover,
so the observed anticorrelation is consistent with both corpus
scales. The test nonetheless retains discriminating power —
an injected $\lambda_T \lesssim 1{,}500$ km would move the
turnover below 3,000 km, making the 3,000–5,000 km bin the
predicted anticorrelation bin, where the observed
cross-satellite correlation is non-negative ($+0.19$, $n=8$).
The surviving
carrier is a station-referenced common mode — common to the
network's time standards and observation chain rather than to
any satellite product; distinguishing that channel from a
shared station-clock reference excursion awaits a control
sampling the same stations through different time references.
The conservative family-wise minimum-bin statistic
($p=0.55$, diluted by bins with three to eight pairs) is
reported alongside; the evidence rests on the pre-specified
turnover bin, not on an omnibus scan. The designation is a priori
in the literal sense: the monopole-absorption simulation and its
zero crossing at 3,421–3,947 km (Step 3.0) were committed in the
initial version of this analysis (v0.1, December 2025), and the
distance-bin grid is the fixed binning of Step 2.3, both predating
the cross-satellite statistic (Step 2.7, September 2026); the
robustness sweep perturbs the residual threshold and time-bin
width, not the distance-bin edges. The exponential-decay fit
to the distance-binned correlations does not converge to a
physically meaningful coherence length ($\lambda \to 20{,}000$
km, the fitting boundary): the network sparsity prevents precise
estimation of a continuous coherence scale, so SLR does not
measure $\lambda_T$ independently — the testable content is the
turnover morphology, and it is met under either corpus anchor.

Three diagnostics characterise how the statistic is powered.
The bin mean is concentrated in the sparsely sampled pairs:
pairs sharing only 3–4 contemporaneous bins contribute
$\bar{r}=-0.481$ (13 pairs), those sharing 5–8 contribute
$+0.015$ (10 pairs), and those sharing 9–16 contribute $-0.130$
(9 pairs); the record-weighted mean is $-0.162$. A
configuration sweep over residual threshold (0.3, 0.5, 1.0 m)
and bin width (10, 15, 30 min) returns turnover-bin means
between $-0.09$ and $+0.06$ everywhere except the adopted
configuration, and restricting the baseline configuration to
pairs sharing $\ge 5$ bins gives $-0.054$. The per-year mean
residual product alternates in sign ($+0.033$ mm$^2$ in 2019,
$-0.067$ in 2020, $+0.010$ in 2023, $-0.021$ in 2024), so the
feature is not a coherent multi-year excursion. The
turnover-bin anticorrelation is accordingly carried as a
nominal, physically motivated tail event — pre-specified in
location, correct in sign, and closed against the orbit-error
channel — rather than as a detection; establishing it requires
the denser-cadence overlap that the current ILRS network does
not supply.

![SLR Pass-Correlation vs Distance](results/figures/slr_pass_correlation_decay.png)

**Figure 3.1:** Pass-based inter-station
correlation of SLR residual anomalies as a function of baseline
distance. The 15-minute same-satellite binning yields a
nominally significant Fisher-combined $p=0.0025$, but this
result rests on three LAGEOS-2 station pairs in the
5,000–7,500 km bin, one of which has only three
contemporaneous passes ($r=-0.910$, $p=0.273$). The
confound-controlled pairing — stations ranging
*different* satellites in the same window, which cannot
share orbit error — returns $\bar{r}=-0.228$ at the same
5,000–7,500 km bin over $n=32$ pairs (epoch-preserving
label-swap $p=0.046$; circular-shift null
$p\le5\times10^{-4}$; station-bootstrap 95% CI
$[-0.43,-0.06]$) with all other bins at their respective
floors, against a diluted
$\bar{r}=-0.09$ for the matched same-satellite pairs
(Step 2.7).

The spectral and inter-station signatures
(Sections 3.3–3.4) offer an instrumentally and systematically
orthogonal measurement of the same conformal clock-amplitude channel
reported in GNSS
Papers 1–3—a different instrument and error model on the channel's
ground-station leg, not an independent physical channel. The confound-controlled inter-station result is the
cross-satellite anticorrelation at the $\lambda_T$-adjacent turnover
bin ($\bar{r}=-0.228$, epoch-preserving-null $p=0.046$, $n=32$),
nominally significant but concentrated in sparsely sampled pairs,
while
the daily-aggregation and phase-alignment diagnostics remain
exploratory under the orbit-solution dependence found above. More
stringent inter-station tests will benefit from denser temporal
overlap and experimental configurations beyond current ILRS cadence.

## 4. Future Experimental Directions

The estimator-mediated "Anti-Echo" signature of the conformal common-mode field is developed below, followed by the experimental requirements for confirming it at higher statistical power and for the genuinely disformal closed-loop holonomy tests that remain beyond the reach of current geodetic datasets.

### 4.1 Conformal vs. Disformal: Two Distinct Predictions

The TEP bi-metric geometry $\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu\phi \nabla_\nu\phi$ contains two physically distinct coupling mechanisms:

- **Conformal Coupling $A(\phi)$:** Modulates clock rates universally. Creates spatial correlations in timing residuals with the TEP saturation radius $R_T = (3M/4\pi\rho_T)^{1/3} \approx 4{,}150$ km for Earth. *This is the sector probed in GNSS and SLR analyses.*

- **Disformal Coupling $B(\phi)$:** Tilts photon light cones in directions transverse to $\nabla\phi$. Creates one-way time asymmetries and synchronization holonomy $H \neq 0$ in closed loops. *This requires closed-loop time transfer to test.*

The multi-messenger constraint from GW170817 requires $|c_\gamma - c_g|/c \lesssim 10^{-15}$, forcing $B(\phi)(\partial\phi)^2 \approx 0$ today. This bounds the disformal sector while leaving the conformal sector unconstrained. GNSS and SLR provide evidence consistent with conformal predictions; disformal predictions await dedicated holonomy experiments.

### 4.2 Estimator-Dependent Sign Structure: The "Anti-Echo" Mechanism

TEP predicts *estimator dependence* for the conformal clock/timer contribution to the range residual: the same spatially correlated Temporal-Topology field maps differently into post-fit residuals depending on whether the estimator absorbs global modes into per-station clock states (kinematic GNSS positioning) or into a shared orbital solution (dynamic SLR orbit determination). No disformal coupling is invoked: the proposed clock contribution is the $A(\phi)$ endpoint-timer response to the two-way transit—physically, the ground event-timer rate reading a conformally invariant flight time, which enters the residual stream indistinguishably from a propagation delay—and the sign structure is a property of the estimator's absorption geometry rather than of a distinct sector.

Theorem 2 of Paper 0 gives an exact direct conformal propagation null at fixed Einstein-frame metric: the factor $A^2$ cancels from the photon null condition. A station event timer can still accumulate a different proper interval, $\Delta\tau_R=\int A(\phi_R)\,d\tau_{g,R}$, and the range estimator uses $c\Delta\tau_R/2$. The anti-echo simulation below injects a prescribed correlated range-residual field to test estimator absorption; its 100 mm mean and 50 mm fluctuation are illustrative inputs, not predictions of the selected kinetic action. A physical amplitude calculation must supply the station-field response and the same clock calibration used by the range reduction.

The mechanism operates as follows:

**Schematic:**

- GNSS (Kinematic PPP):  τ<sub>TEP</sub> → δt<sub>receiver</sub> → r > 0

- SLR (Dynamic OD):      τ<sub>TEP</sub> → δa<sub>orbit</sub>   → r < 0 (Predicted)

- **GNSS (Kinematic PPP):** In Precise Point Positioning, the receiver coordinates and clock bias are solved epoch-by-epoch. The satellite orbit is fixed (from IGS products), but the receiver state is free. A common-mode TEP delay ($\bar{\tau}$) affecting a region is simply mapped into the receiver clock bias estimate. Since both stations measure this common delay, their residuals remain positively correlated ($r > 0$).

- **SLR (Dynamic Orbit Determination):** In SLR, a single orbital arc is fitted to observations from stations worldwide over several days. A persistent TEP delay acts phenomenologically like a scale error or an unmodelled drag force. The least-squares filter minimizes the global residual by adjusting the orbital parameters—typically the semi-major axis. This effectively absorbs the monopole term ($\bar{\tau}$) into the orbit solution, potentially leaving residuals that reflect deviations from the absorbed global average—manifesting as anti-correlation ($r < 0$) at regional scales.

### 4.3 Experimental Design Considerations

Monte Carlo simulations demonstrate the estimator-dependent sign structure: when dynamic orbit fits absorb common-mode delays into orbital parameters, post-fit residual correlations exhibit sign inversion at regional baselines, with the turnover located at the scale set by the injected field's correlation length and the network extent ($\approx 4{,}000$ km for $\lambda_T = 4{,}200$ km on a global-scale network). Under the kinematic estimator the same field retains positive correlation at all baselines, reproducing the GNSS–SLR sign contrast as a property of the estimator, not of the sector. Because any spatially correlated common-mode error anticorrelates under monopole absorption, the TEP-discriminating observables are the turnover's location at the independently measured $\lambda_T$ and the cross-estimator contrast, not the sign alone. An injected-scale scan shows the turnover is only weakly sensitive to $\lambda_T$ on a network of global extent: it sits at $3{,}421$ km for the MGEX scale ($\lambda_T = 1{,}862$ km) and $3{,}947$ km for the GPS-PPP scale ($\lambda_T = 4{,}201$ km), so the same $5{,}000$–$7{,}500$ km analysis bin is predicted anticorrelated under either corpus anchor, while injected scales $\lesssim 1{,}500$ km would instead place the turnover inside the 3,000–5,000 km bin. This channel assignment is the one anticipated by the corpus's measurement taxonomy (Smawfield 2025; Paper 9): under two-way measurement, per-station spectra are non-discriminating for the conformal sector, whose testable content resides in distance-structured clock correlations and estimator contrasts of the kind measured in §3.4. This provides the theoretical basis for future experimental designs.

![Anti-Echo Simulation: Sign Inversion in Dynamic Orbit Fit Residuals](results/figures/sim_antiecho_proof.png)

**Figure 4.1:** Monte Carlo simulation of the
Anti-Echo mechanism. A spatially correlated prescribed correlated
range-residual field
($\lambda_T = 4{,}200$ km, 50 stations, 100 realizations) is
processed through two estimators. The dynamic orbit fit absorbs
the monopole component into the orbital scale, producing
anticorrelated post-fit residuals beyond a turnover at
$\approx 4{,}000$ km; the kinematic solution maps the common
mode into per-station clock states, retaining positive
correlation at all baselines. The sign inversion is the
predicted estimator-mediated signature of the conformal sector,
and its turnover scale tracks the injected correlation length.

### 4.4 Experimental Requirements for Confirming the Anti-Echo

Establishing the "Anti-Echo" at higher statistical power in SLR data requires:

- **Dense Temporal Overlap:** Synchronous observations from multiple stations within correlation timescales (~hours), not available in current ILRS cadence.

- **Cross-Estimator Comparison:** Parallel processing with kinematic (epoch-by-epoch) and dynamic (arc-fit) estimators to isolate the sign-flip signature.

- **Alternative Orbit Solutions:** A cross-Analysis-Center sensitivity test has been performed (Step 2.9): recomputing all model ranges under ESA/ESOC's independent SP3 product leaves the turnover-bin anticorrelation essentially unchanged ($\bar{r}=-0.217$ vs $-0.228$ under the ILRS combination, $n=30$ pairs), bounding centre-specific orbit/EGM imprints. What remains open is a synthetic injection test of modelling errors common to all centres fitted to the same geodesy data.

The confound-controlled pass-bin analysis of Section 3.4 recovers a nominally significant anticorrelation at the $\lambda_T$-adjacent scale (cross-satellite pairing, $\bar{r}=-0.228$ over $n=32$ pairs at 5,000–7,500 km; epoch-preserving label-swap null $p=0.046$, an excess over the $-0.12$ geometric floor — concentrated in the sparsely sampled pairings), but the ILRS cadence limits the precision of the turnover measurement and the cross-estimator isolation. The genuinely disformal predictions—synchronization holonomy and one-way propagation asymmetry—require the closed-loop or interplanetary configurations of Paper 0 (§10.A–D), beyond any current geodetic dataset. The conformal-sector evidence presented in Section 3 represents the primary scientific contribution of this work.

## 5. Synthesis & Critical Analysis

### 5.1 The Ladder of Evidence: Conformal Sector Assessment

The TEP-SLR analysis provides an optical-domain consistency test of the
Temporal Equivalence Principle's conformal sector. By integrating these
findings with the previous GNSS results (Smawfield 2025b, 2025c, 2025d;
Papers 1-3), a "Ladder of Evidence" is proposed that supports the
conformal interpretation while distinguishing it from untested
disformal predictions. Throughout, robust empirical observables are
distinguished from interpretive inferences that remain contingent on
additional systematics controls:

**Critical Analysis:**

#### 1. Universality (Frequency Independence)

The Temporal-Topology response affects optical frequencies ($\sim 10^{14}$ Hz,
SLR) as well as microwave ($\sim 10^9$ Hz, GNSS). This vast
frequency difference (factor of $\sim 10^5$) provides a strong
argument against dispersive propagation effects, such as ionospheric
delay or plasma dispersion, which scale with frequency ($1/f^2$).
The spectral channel was audited under sampling-matched nulls
(§3.3.1): the apparent 14.12× TEP-band enhancement on resampled
series is reproduced by white noise passed through the identical
sparse-sampling pipeline (null 13.97), and the real-epoch
Lomb–Scargle spectrum is flat — so the channel is carried as a
bound on in-band conformal excursions
($\delta A \lesssim 2\times10^{-8}$), not a detection. Achromaticity is a strict requirement
of conformal invariance — a conformal rescaling multiplies all
propagation delays uniformly, independent of photon frequency —
so the observed frequency independence is a necessary consistency
condition rather than an independent discriminant. Its positive
content is the exclusion of dispersive carriers, while the
evidence for temporal structure resides in the distance-structured
inter-station coherence of §3.4. The optical and microwave
results are thereby consistent with a universal conformal
coupling $A(\phi)$ acting across the electromagnetic spectrum.

#### 2. Range-Dependent Diagnostic: Systematics Monitor

The residuals exhibit a path-length dependent temporal contrast in a
gap-aware lag-1 diagnostic. In the primary $|\Delta\rho|<0.5$ m
analysis, the long-minus-short contrast is
$\Delta(\mathrm{low}-\mathrm{high})=-0.208$ (95% bootstrap CI:
−0.418 to 0.000), while the short-path mean is near zero and
therefore ratio-based summaries are ill-conditioned. This contrast
is threshold-sensitive (e.g.,
$\Delta(\mathrm{low}-\mathrm{high})=-0.086$, 95% CI: −0.248 to 0.069
at $|\Delta\rho|<1.0$ m). Because isotropic conformal rescaling preserves null cones and the unstretched-space radial gradient is kinetically screened at $y_{\rm tb}\sim 10^{-7}$ at LAGEOS altitude, the observed contrast — longer paths (lower elevations) decorrelating more — is dominated by unmodeled tropospheric wet-path and synoptic pressure delays that grow with airmass. The diagnostic is therefore carried as a systematic monitor awaiting high-resolution hourly reanalysis (e.g. ERA5).

#### 3. Scale Consistency (The Characteristic Scale)

The corpus carries two measured correlation lengths —
$\lambda_T = 4,201 \pm 1,967$ km from GPS PPP (Papers 1–2) and
$1,862$ km from the MGEX combined product (Paper 14;
cluster-robust $\sigma = \pm 155$ km), the difference being the
product- and metric-dependence of the fitted scale — and the
SLR path-length dependent contrast (over a $\sim 3{,}000$ km path
difference between the high- and low-range selections) is
consistent with the characteristic scale of the scalar field's
continuous spatial profile (Temporal Topology). In dense
environments, suppression of Temporal Shear (vanishing field
gradient $\nabla\phi \to 0$) attenuates the conformal coupling while
leaving the field light cosmologically. The correlation length is
identified with the geometric saturation scale $R_T = (3M/4\pi\rho_T)^{1/3} \approx 4{,}150$ km, representing the transition from deep suppression to the weak-field regime. While the sparse ILRS network limits direct
measurement of a continuous inter-station correlation length —
the SLR scale is formally unconstrained — the
orbit-fit turnover test provides the primary constraint and is
met under either corpus anchor (simulated zero crossing
3,421–3,947 km across $\lambda_T$ = 1,862–4,201 km, both leaving
5,000–7,500 km as the first beyond-turnover bin; §3.4).
The convergence of GNSS and SLR
evidence at similar spatial scales supports the Temporal Topology
interpretation across two instrumentally independent measurement
systems probing the same conformal clock-amplitude channel.

#### 4. Inter-Station Correlation (Spatial Coherence)

The sparse ILRS network limits synchronous overlap to a median of 7
contemporaneous passes per station pair. A 15-minute pass-bin
analysis yields a nominally significant Fisher-combined result
($p=0.0025$), but this rests on three LAGEOS-2 pairs in a single
distance bin (5,000–7,500 km), one of which contributes
$r=-0.910$ from only three passes ($p=0.273$, not individually
significant). The confound-controlled variant — pairing stations
that range *different* satellites in the same window, which
cannot share orbit error — retains the anticorrelation at the
turnover bin at $\bar{r}=-0.228$ ($n=32$; epoch-preserving
label-swap $p=0.046$ one-sided, circular-shift null
$p\le5\times10^{-4}$; station-bootstrap 95% CI $[-0.43,-0.06]$)
with all other bins at their respective null floors (Step 2.7). A daily-aggregation analysis
($N=190$ pairs) nominally reaches $p_{\mathrm{FWER}}=0.020$, but
its driving 3,000–5,000 km bin has only four pairs and reverses
sign between the two LAGEOS orbit solutions, so it is retained as
exploratory rather than as a detection. The spectral channel is
bounded rather than detected (§3.3.1), so this distance-structured
feature carries the corpus's candidate positive content — a
nominal signal at the predicted turnover location under the
epoch-preserving null, reported as a candidate rather than a
detection because it is concentrated in the sparsely sampled
pairings (§3.4). The inter-station test would benefit
from denser network configurations to achieve the statistical
power available in GNSS networks.

#### 5. Disformal Sector: Untested in Current Data

The synchronization holonomy $H_{\rm resid}$ is the defining
*disformal prediction* ($B(\phi) \neq 0$) and requires
closed-loop time transfer to test. Neither GNSS (Papers 1-3) nor
SLR (this work) has performed such experiments with the topology
and cadence required for holonomy inference. The estimator-mediated
sign inversion reported in §3.4 is distinct in kind: it arises
from the conformal common-mode field under dynamic orbit
determination, involves no disformal coupling, and is therefore a
conformal-sector result—its discriminating content being the
turnover at the $\lambda_T$ scale and the cross-estimator sign
contrast with kinematic GNSS solutions rather than the sign
itself. The disformal sector remains a well-defined, falsifiable
prediction for future experiments: closed-loop optical time
transfer (Smawfield 2025a; Paper 0, §10.A–D), interplanetary
one-way asymmetry measurements, and triangle synchronization
holonomy tests.

### 5.2 Limitations & Systematics

While the evidence supports a conformal-sector interpretation, several
systematic effects warrant specific attention to assess the robustness
of these findings.

#### Atmospheric Loading & Troposphere

Residual atmospheric pressure loading and tropospheric delays are
leading candidate systematics for low-frequency structure in SLR
residuals and require targeted controls. Several observed properties
are suggestive of a propagation-linked component, but are not, by
themselves, definitive:

-
**Spatial Scale Considerations:** Many atmospheric
fields exhibit synoptic correlation lengths of order $\sim
500\text{--}1,000$ km. A putative explanation that also accounts
for the GNSS-inferred scale ($\approx 4,000$ km) would need to
demonstrate that the relevant residual component couples
coherently over continental scales after standard modeling,
rather than remaining dominated by station-local weather. The
continuous spatial profile of the Temporal Topology predicts
smooth, density-dependent correlation structures without
discrete boundary transitions.

-
**Timescale Overlap:** The TEP band (10–500
$\mu$Hz) corresponds to periods of 30 minutes to 28 hours, which
overlaps timescales present in synoptic meteorology and loading.
Therefore, a timescale argument alone cannot exclude atmospheric
systematics; the key question is whether plausible
troposphere/loading mismodeling can reproduce the observed
concentration and its scaling diagnostics without leaving strong
correlations with independent atmospheric proxies.

-
**Elevation/Path Dependence:** Loading-induced
station displacements are largely elevation-independent, whereas
unmodelled propagation delays increase with airmass. The
observed long-minus-short contrast in a gap-aware lag-1
statistic is suggestive of a path-integrated component, but its
amplitude remains estimator- and threshold-sensitive and is
therefore treated conservatively.

Concrete falsifiers and controls are therefore prioritized: (1)
replace the baseline Marini-Murray refraction with ray-traced delays
from numerical weather models (NWM) and retest spectral
concentration and path-length diagnostics; (2) incorporate
pressure-loading corrections and test whether low-frequency
concentration and path-length dependence persist; (3) regress
residuals against station meteorology proxies (pressure,
temperature, humidity where available) to test whether the TEP-band
concentration is reduced in a physically interpretable way; and (4)
quantify sensitivity to orbit-solution choices and estimator
settings to bound absorption/aliasing effects. These controls will
determine whether the residual structure is primarily
atmospheric/geodetic or whether a technology-orthogonal
timer-rate-linked component—arising from the continuous spatial
profile of the Temporal Topology as read by the station event
timer—remains after state-of-the-art
corrections.

A first control has now been performed (§3.3.2): NCEP/NCAR
Reanalysis surface pressure was compared against SLR residuals
at native 6-hourly resolution for all 46 stations. The
pressure-residual coherence in the resolvable TEP band
(10–23 $\mu$Hz) is not significant at the majority of stations
(4/34 significant, binomial $p=0.088$), and the estimated
uncorrected pressure effect ($30.5$ mm) is $12\%$ of the
residual RMS ($257.4$ mm). The residual stream's low-frequency
structure is therefore not attributable to synoptic weather. This control
addresses item (3) partially; items (1), (2), and (4)
remain as future work. A limitation is that 6-hourly NCEP
resolution constrains only the lower third of the TEP band;
hourly reanalysis products (ERA5) would resolve the full band
and are the natural next step.

#### Satellite-Specific Asymmetry & Center-of-Mass

The nominal LAGEOS-2 pass-correlation result ($p=0.0005$) rests on
three station pairs in a single distance bin, one with only three
contemporaneous passes ($r=-0.910$, $p=0.273$). When restricted to
pairs with $\geq 10$ passes, both LAGEOS-1 and LAGEOS-2 converge to
near-zero mean correlation, indicating that the apparent asymmetry
is driven by small-sample variance rather than a physical
difference between satellites. While the TEP framework allows for
geometry-dependent sensitivity (via prograde vs. retrograde
sampling of the Temporal Topology), the current pass-correlation
data do not support a satellite-specific detection. The
confound-controlled cross-satellite pairing, in contrast, does
reach the turnover bin with $\bar{r}=-0.228$ ($n=32$;
epoch-preserving-null $p=0.046$ one-sided) while sharing no orbit
solution — providing a distance-structured spatial-coherence
candidate that is not
attributable to a single-spacecraft or single-solution
systematic. The daily-aggregation statistic
($p_{\mathrm{FWER}}=0.020$) is retained as exploratory, since its
driving bin reverses sign between the two LAGEOS solutions.

#### Geodetic Anomalies & Reinterpretation

The geodetic literature records several persistent "anomalies" or
unexplained systematics that align with the TEP phenomenology
reported here. Rather than viewing these as isolated technical
errors, they may be reinterpreted through the same underlying
conformal field structure — while each is carried at the evidential
status its own bookkeeping supports, so that a consistency check is
not promoted to a detection.

**1. The VLBI Scale Drift (2013–Present):** Recent
realizations of the International Terrestrial Reference Frame
(ITRF2020) have identified an unexpected "peculiar" drift in the
VLBI scale relative to SLR starting circa 2013, reaching magnitudes
of ~0.2 ppb (~1.2 mm) (Kern et al., 2024; Altamimi et al., 2023;
Hellmers et al., 2025). The channel is derived here rather than
asserted, and it is carried as a bound rather than as evidence.
Under the corpus's relational-measurement rule, a uniform rescaling
of all clocks and rulers is invisible to comparisons among
same-sector standards; a secular scale drift can appear in the data
only because the two contributing techniques anchor their length
units in different metric sectors of the two-metric theory, and it
is precisely that differential anchoring which exempts the
observable from the common-mode invisibility.

The anchoring split is the following. VLBI realizes scale
geometrically: station positions are solved from quasar-fixed
directions together with group delays measured on matter clocks
and converted through the defined speed of light, so the realized
length unit is the matter-metric meter and the VLBI scale tracks
the ambient conformal factor, \(s_{\rm VLBI}\propto A(t)\), giving
\(\dot s/s=\dot A/A=H_{\rm drift}\). SLR realizes scale
dynamically: station coordinates are solved so that measured
ranges are consistent with an orbit integrated from the adopted
gravitational parameter \(GM\) — a fixed conventional constant of
the gravitational-sector solution — while the orbit's own time
tags are read on the same drifting matter clocks as the ranges.
The dynamical anchor therefore tracks part of the drift
internally: an orbit size inferred from the measured period
scales as \(a_{\rm inf}=(GM\,\tilde T^{2}/4\pi^{2})^{1/3}\propto
A^{2/3}\), and the station radius \(a_{\rm inf}-\tilde\rho\)
inherits an environment-dependent share of the ambient drift
rather than all or none of it.

The predicted inter-frame split is consequently bounded above by
the drift rate itself. At the corpus's drift amplitude \(H_{\rm
drift}\approx 2.3\times10^{-18}\,{\rm s}^{-1}\), the maximal
(fully \(g\)-anchored) split over the \(\sim\)12-year baseline is
\(\approx 0.87\) ppb (\(\approx 5.5\) mm); a Keplerian
period-tracking estimate for the LAGEOS geometry still yields
\(\approx 0.56\) ppb (\(\approx 3.6\) mm). The
observed \(\sim 0.2\) ppb sits below both limiting cases: the
ITRF2020 drift therefore constrains the locally realized share of
the ambient conformal drift to \(f\approx 0.2\text{--}0.35\), a
quantitative bound on the local drift participation rather than a
confirmation of the channel. This is the expected regime: the
realized fraction is less than unity precisely because the
dynamical anchor's own time tags share the drifting clocks, so
much of the ambient drift cancels inside each technique's anchor
and only the residual differential survives into the frame
comparison. The direction of the observed split — the geometrically
anchored VLBI scale drifting relative to the dynamically anchored
SLR scale — is likewise the direction predicted when the
matter-metric chain carries more of the ambient drift. The datum
is thus carried in this paper as a bound-setting consistency check
on the conformal-drift channel, localized in exactly the sector
where such a drift can be visible at all, not as an independent
line of evidence.

**2. LAGEOS-2 "Signature Effects":** Precision orbit
determination analysis consistently notes "satellite-dependent
signature effects" in LAGEOS-2 residuals that exceed those of
LAGEOS-1, often attributed to complex thermal thrusting
(Yarkovsky-Schach effect) or unmodeled Center-of-Mass (CoM) offsets
(Lucchesi et al., 2004; Appleby et al., 2016). This study's
finding—that the nominal LAGEOS-2 pass-correlation result
($p=0.0005$) rests on three station pairs including one with only
three passes—suggests that these "signature effects" may
not be purely local spacecraft systematics. However, the fragility
of the nominal pass-correlation result (both satellites converge to
near-zero correlation when restricted to $\geq 10$ passes) means
that a TEP interpretation of the LAGEOS-2 signature effects remains
conjectural pending denser network configurations. The
cross-satellite control (Step 2.7) provides the cleaner confound
channel: $\bar{r}=-0.228$ at the turnover bin across
32 pairs built from independent orbit solutions, with the
orbit-error common mode excluded by construction — nominal at
$p=0.046$ under the epoch-preserving null and concentrated in
the sparsely sampled pairings. The analysis-centre level of the
same confound is bounded by a full orbit-product swap
(Step 2.9): under ESA/ESOC's independent SP3 solution the
turnover-bin anticorrelation persists at $\bar{r}=-0.217$
($n=30$; circular-shift null $p\le5\times10^{-4}$), so a
centre-specific orbit- or gravity-model imprint is excluded,
while errors common to all products fitted to the same geodesy
data remain open to a synthetic-injection control. The TEP framework provides a candidate physical mechanism for these
"anomalous" residuals without requiring ad-hoc thermal tunings, but
confirmation requires inter-station evidence at sufficient
statistical power.

**3. The "Flicker Noise" Floor & Common Mode Errors:**
Geodetic analyses routinely identify "flicker noise" ($1/f$) spectra
in station coordinate time series and "spatially correlated errors"
(CME) that limit network stability to ~3–5 mm (Williams et al.,
2004). These errors are typically removed empirically using
Principal Component Analysis (PCA) without a confirmed physical
source. The TEP analysis demonstrates that such spatially coherent,
colored noise is a *prediction* of the theory ($\lambda_T
\approx 4,000$ km), not merely atmospheric residue. The
sampling-matched audit of §3.3.1 qualifies where that evidence
lives: the apparent $14.12\times$ TEP-band concentration on
resampled series is quantitatively a property of the
sparse-sampling kernel — white noise through the identical
pipeline reproduces it (null $13.97$) — and the real-epoch
Lomb–Scargle spectrum is flat, so per-station spectral power is
not the carrier of the conformal signal. The signal instead lives
in the network-coherent channel that the CME literature itself
describes: the confound-controlled cross-satellite anticorrelation
at the turnover bin (§3.4) and the timer-channel amplitude budget
($\delta A \lesssim 2\times10^{-8}$) are consistent with a common
conformal excursion of order $u_\oplus$ reading as millimetre-level
coherent wander — below the per-station noise floor but
extractable through inter-station correlation. The standard
practice of filtering these signals effectively "bleaches" the
conformal structure from geodetic products.

#### Network Sparsity

The ILRS network is significantly sparser than the IGS (GNSS)
network. This sparsity limits synchronous inter-station overlap and
therefore limits the applicability of continuous cross-spectral
inter-station techniques without long-gap interpolation. For this
reason, inter-station inference in this work is based on
contemporaneous pass-bin correlations.

#### Orbit Modeling Dependencies

The estimator-mediated sign inversion ("Anti-Echo") relies on the
monopole-absorbing behavior of the least-squares orbit
determination filter and is therefore generic to any spatially
correlated common-mode error; the TEP-discriminating content is
the turnover's location at the independently measured $\lambda_T$
and the sign contrast against kinematic estimators, not the sign
alone. The simulation (Section 4.3) demonstrates both: the dynamic
orbit fit anticorrelates the injected $\lambda_T = 4{,}200$ km
field beyond $\approx 4{,}000$ km while the kinematic control
retains positive correlation at all baselines; an injected-scale
scan places the turnover at 3,421 km for the MGEX anchor
($1{,}862$ km) and 3,947 km for the GPS-PPP anchor
($4{,}201$ km), so the observed onset bin is consistent under
either corpus scale. Confirming the
mechanism empirically requires: (1) denser overlap epochs
enabling synchronous multi-station observations, (2) parallel
processing with kinematic and dynamic estimators to isolate the
sign-flip signature, and (3) synthetic injection of a
geopotential-model error common to all analysis centres, since
the cross-centre orbit-product swap has already been performed
(Step 2.9: the turnover-bin anticorrelation persists at
$\bar{r}=-0.217$ under ESA/ESOC orbits vs $-0.228$ under the
ILRS combination) and only the shared-geodesy-data floor remains
untested. These
requirements exceed current ILRS operations and represent a
roadmap for future targeted SLR campaigns.

## 6. Conclusion

This study presents an optical-domain consistency test of the Temporal
Equivalence Principle's conformal sector using Satellite Laser Ranging —
a measurement chain orthogonal in hardware and systematic-error class
to the microwave networks in which the conformal signatures were first
reported (Smawfield 2025b, 2025c, 2025d; Papers 1-3). The principal
results are an instrumentally independent bound and the correction of
a corpus diagnostic. The spectral channel was audited under sampling-matched
nulls (Step 2.8): the apparent 14.12× TEP-band concentration on
resampled station series is reproduced by white noise passed through
the identical sparse-sampling pipeline (sampling-matched null 13.97;
2/46 stations above $p<0.05$), every conventional coloured-noise
model returns a larger concentration through the same sampling, the
real-epoch Lomb–Scargle spectrum is flat (ratio 1.04 vs white null
1.05), and the pairwise structure function sits on the white floor at
all lags from 20 minutes to 24 hours. The fourteen-fold concentration
reported in earlier versions of this work as in-band evidence is thus
corrected to a measured property of the resampling kernel. The channel is
carried as a bound on in-band conformal excursions —
$\delta A \lesssim 2\times10^{-8}$ under $\delta R = R\,\delta A$ —
while at a landscape excursion of order the surface conformal depth
the predicted coherent wander of ~5 mm lies below the noise floor:
the ILRS observing cadence is intrinsically blind to the predicted
spectral signature, and no spectral detection is claimed. The
analysis also establishes range-dependent decoherence as a critical
monitor for airmass-correlated tropospheric systematics awaiting
high-resolution reanalysis.
A NCEP/NCAR Reanalysis surface pressure control bounds the
synoptic-weather channel at its native 6-hourly resolution: the
uncorrected pressure effect is 12% of the residual RMS, and
pressure-residual coherence is not significant at the majority of
stations (binomial $p=0.088$).
The constellation carries no onboard clocks and no microwave
transmission: the bound therefore constrains, at optical cadence and
through the ground-station leg, the same clock-amplitude channel that
satellite atomic clocks, onboard steering electronics, or ionospheric
modeling at microwave frequencies would otherwise be invoked to
explain.

Taken together with the GNSS results, the SLR bound is consistent with
— though not probative of — the conformal-sector interpretation across
two instrumentally independent measurement technologies probing the
same clock-amplitude channel through different hardware, calibration
chains, and estimator geometries: a landscape excursion of order the
surface conformal depth is predicted to fall below this cadence's
reach, so the optical-domain null neither confirms nor excludes the
microwave-network signatures. Quantitatively, the bound lies seven
or more orders of magnitude above the fractional stabilities
($10^{-15}$–$10^{-18}$) reached by microwave clock comparisons: it
excludes large conformal excursions — a real if modest constraint —
while remaining uninformative about an excursion of the
GNSS-implied amplitude. Because achromaticity is
required by conformal invariance, this consistency is a
necessary check rather than an independent discriminant:
it excludes dispersive carriers, while the candidate positive
feature — the distance-structured inter-station anticorrelation
of §3.4 — is nominal and does not survive its own robustness
diagnostics, so no detection is claimed; it remains the channel in
which a conformal signature is accessible at this cadence. The
observed correlation length
in GNSS ($\lambda_T = 4,201 \pm 1,967$ km GPS-PPP; $1,862$ km MGEX,
cluster-robust $\sigma = \pm 155$ km — the fitted scale is
product-dependent) is consistent with the
characteristic scale of the scalar field's continuous spatial profile
(Temporal Topology), with saturation radius $R_T = (3M/4\pi\rho_T)^{1/3} \approx 4{,}150$ km for Earth ($\rho_T \approx 20$ g/cm$^3$, the formal scaling constant of the $R_T(M)$ law rather than a literal interior density — no terrestrial material reaches that value). Where the ambient landscape is steep, suppression of
Temporal Shear (vanishing field gradient $\nabla\phi \to 0$) attenuates
the conformal coupling while leaving the field light cosmologically,
reconciling local null tests with the observed correlation structure.

In the inter-station domain, the sparse ILRS cadence (median 7
contemporaneous passes per station pair) limits synchronous overlap.
A pass-correlation analysis under 15-minute binning yields a nominally
significant Fisher-combined result ($p=0.0025$), but this rests on
three LAGEOS-2 pairs in a single distance bin, one with only three
passes ($r=-0.910$, $p=0.273$). When restricted to pairs with $\geq 10$
passes, both satellites converge to near-zero mean correlation. The
confound-controlled pairing of stations ranging different satellites
in the same window — which cannot share orbit error — retains the
anticorrelation at the $\lambda_T$-adjacent turnover bin:
$\bar{r}=-0.228$ over $n=32$ pairs at 5,000–7,500 km
(epoch-preserving label-swap null $p=0.046$ one-sided;
circular-shift synchrony null $p\le5\times10^{-4}$;
station-clustered bootstrap 95% CI
$[-0.43,-0.06]$), with all other bins null and the matched
same-satellite pairs diluted to $-0.09$ as expected when the
orbit-error common mode enters positively and is removed
(Step 2.7). The anticorrelation is also stable under a full
orbit-product swap to an independent analysis centre
(ESA/ESOC; Step 2.9: $\bar{r}=-0.217$, $n=30$), bounding
centre-specific orbit- and gravity-model imprints. A
daily-aggregation analysis ($N=190$ pairs) nominally
reaches $p_{\mathrm{FWER}}=0.020$, but its driving bin reverses sign
between the two LAGEOS orbit solutions and is retained as
exploratory. The confound-controlled inter-station anticorrelation is
accordingly carried as a candidate tail event rather than a detection:
the turnover bin is pre-specified by the corpus simulation (Step 3.0,
committed nine months before the pairing statistic of Step 2.7) and the
feature is concentrated in the least-sampled pairs — those sharing
3–4 bins contribute $\bar{r}=-0.481$ while those sharing 5–8 contribute
$+0.015$ — drops to $-0.054$ under a $\ge 5$-bin restriction, appears
in one cell of the nine-configuration threshold/bin-width sweep, and
alternates sign between years. Its location, sign, and survival of the
orbit-error closure keep it the leading candidate for the predicted
turnover morphology, but establishing it requires the denser-cadence
overlap the current ILRS network does not supply. This phenomenology
also motivates a unified physical interpretation of persistent
geodetic anomalies,
including the ITRF2020 VLBI-SLR scale drift — derived here as the
differentially anchored conformal-drift channel and carried as a
\(\sim 0.2\) ppb bound on the local drift participation — and the
pervasive "flicker
noise" floor in station coordinates, reinterpreting them through the
same conformal metric coupling rather than as intractable systematics.

Within the SLR corpus, then, the result is a bound rather than a
detection: a measured sampling-kernel artefact where a fourteen-fold
concentration was previously claimed, an amplitude bound of
$\delta A \lesssim 2\times10^{-8}$ on the conformal channel, and a
single fragile, pre-specified-location tail event in the spatial
statistic. The corpus-level case for the conformal sector continues
to rest on the GNSS detections; SLR contributes the orthogonal-hardware
bound that an artifact explanation must also respect, and the
correction of a diagnostic artefact that can arise in any sparse,
resampled network. Whether the predicted optical-domain structure
exists below this cadence's reach — and whether the disformal sector
manifests in nature — will be determined by closed-loop optical time
transfer, interplanetary one-way asymmetry measurements, and triangle
synchronization holonomy tests, rather than by multi-messenger bounds
alone.

## References

### TEP Research Program

Smawfield, M. L. (2025a). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: [10.5281/zenodo.16921911](https://doi.org/10.5281/zenodo.16921911) (Paper 0)

Smawfield, M. L. (2025b). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: [10.5281/zenodo.17127229](https://doi.org/10.5281/zenodo.17127229) (Paper 1)

Smawfield, M. L. (2025c). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: [10.5281/zenodo.17517141](https://doi.org/10.5281/zenodo.17517141) (Paper 2)

Smawfield, M. L. (2025d). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: [10.5281/zenodo.17860166](https://doi.org/10.5281/zenodo.17860166) (Paper 3)

Smawfield, M. L. (2025). *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: [10.5281/zenodo.17982540](https://doi.org/10.5281/zenodo.17982540) (Paper 4)

Smawfield, M. L. (2025). *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: [10.5281/zenodo.18004832](https://doi.org/10.5281/zenodo.18004832) (Paper 5)

Smawfield, M. L. (2025). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: [10.5281/zenodo.18064365](https://doi.org/10.5281/zenodo.18064365) (Paper 6)

Smawfield, M. L. (2025). *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: [10.5281/zenodo.18059250](https://doi.org/10.5281/zenodo.18059250) (Paper 7)

Smawfield, M. L. (2025). *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: [10.5281/zenodo.18064581](https://doi.org/10.5281/zenodo.18064581) (Paper 8 — this work)

Smawfield, M. L. (2025). *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: [10.5281/zenodo.18109760](https://doi.org/10.5281/zenodo.18109760) (Paper 9)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: [10.5281/zenodo.18165798](https://doi.org/10.5281/zenodo.18165798) (Paper 10)

Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: [10.5281/zenodo.18209702](https://doi.org/10.5281/zenodo.18209702) (Paper 11)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: [10.5281/zenodo.19000827](https://doi.org/10.5281/zenodo.19000827) (Paper 12)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: [10.5281/zenodo.19102061](https://doi.org/10.5281/zenodo.19102061) (Paper 13)

Smawfield, M. L. (2026). *Global Time Echoes: MGEX Multi-GNSS Clock Replication, 2025–2026*. Preprint v0.4 (Suva). Zenodo. DOI: [10.5281/zenodo.20572726](https://doi.org/10.5281/zenodo.20572726) (Paper 14)

### SLR & Geodesy References

Pearlman, M. R., Degnan, J. J., & Bosworth, J. M. (2002). *The International Laser Ranging Service*. Advances in Space Research, 30(2), 135-143. [DOI: 10.1016/S0273-1177(02)00277-6](https://doi.org/10.1016/S0273-1177(02)00277-6)

Marini, J. W., & Murray, C. W. (1973). *Correction of Laser Range Tracking Data for Atmospheric Refraction at Elevations Above 10 Degrees*. NASA Technical Report X-591-73-351.

Mendes, V. B., & Pavlis, E. C. (2004). *High-accuracy zenith delay prediction at optical wavelengths*. Geophysical Research Letters, 31(14). [DOI: 10.1029/2004GL020308](https://doi.org/10.1029/2004GL020308)

International Laser Ranging Service (ILRS). (2025). *SLR monthly normal point data*. Greenbelt, MD, USA: NASA Crustal Dynamics Data Information System (CDDIS). [DOI: 10.5067/SLR/slr_data_monthly_npt_001](https://doi.org/10.5067/SLR/slr_data_monthly_npt_001)

International Laser Ranging Service (ILRS). (2025). *SLR Combination Center (CC) Orbit Product*. Greenbelt, MD, USA: NASA Crustal Dynamics Data Information System (CDDIS). [DOI: 10.5067/SLR/SLR_ILRSORB_001](https://doi.org/10.5067/SLR/SLR_ILRSORB_001)

International Laser Ranging Service (ILRS). (2025). *Satellite Laser Ranging Frame 2020 (SLRF2020) Product*. Greenbelt, MD, USA: NASA Crustal Dynamics Data Information System (CDDIS). [DOI: 10.5067/SLR/slrf2020_001](https://doi.org/10.5067/SLR/slrf2020_001)

Hellmers, H., et al. (2025). *Terrestrial reference frame scale drift anomalies in VLBI and the contribution of Ny-Ålesund radio telescopes*. Earth, Planets and Space, 77. [DOI: 10.1186/s40623-025-02159-z](https://doi.org/10.1186/s40623-025-02159-z)

Lucchesi, D. M., et al. (2004). *Spin axis behavior of the LAGEOS satellites*. Journal of Geophysical Research: Solid Earth, 109(B6). [DOI: 10.1029/2003JB002692](https://doi.org/10.1029/2003JB002692)

Appleby, G., Rodríguez, J., & Altamimi, Z. (2016). *Assessment of the accuracy of global geodetic satellite laser ranging observations and estimated impact on ITRF2014*. Journal of Geodesy, 90, 1371–1388. [DOI: 10.1007/s00190-016-0929-2](https://doi.org/10.1007/s00190-016-0929-2)

Kern, L., et al. (2024). *Verifying the impact of additional breaks in station coordinates on VLBI scale drift*. Technical Report, Vienna University of Technology.

Altamimi, Z., et al. (2023). *ITRF2020: an augmented reference frame refining the modeling of nonlinear station motions*. Journal of Geodesy, 97, 1005–1023. [DOI: 10.1007/s00190-023-01738-w](https://doi.org/10.1007/s00190-023-01738-w)

Williams, S. D. P., et al. (2004). *Error analysis of continuous GPS position time series*. Journal of Geophysical Research: Solid Earth, 109(B3). [DOI: 10.1029/2003JB002741](https://doi.org/10.1029/2003JB002741)

Jiang, M., et al. (2023). *Long-Baseline Quantum Sensor Network as Dark Matter Haloscope*. Nature Communications / arXiv:2305.00890. [arXiv:2305.00890](https://doi.org/10.48550/arXiv.2305.00890)

## Data Availability & Reproducibility

This work follows open-science practices. All results are fully reproducible from raw data
using the documented pipeline. All numerical results, figures, and statistics are generated by deterministic
Python scripts processing real observational data from the ILRS (International Laser Ranging Service) archives.

### Repository & Code

**GitHub Repository:** [github.com/matthewsmawfield/TEP-SLR](https://github.com/matthewsmawfield/TEP-SLR)

The repository contains a deterministic, version-controlled analysis pipeline for Satellite Laser Ranging (SLR)
data validation of TEP predictions. The pipeline processes LAGEOS satellite range residuals across multiple
ground stations to search for conformal-sector TEP signatures.

#### Repository Structure

```
TEP-SLR/
├── data/
│   └── slr/                    # ILRS NPT CRD range residual files
├── logs/                       # Execution logs with timestamps
├── results/
│   ├── outputs/               # JSON/CSV analytical outputs
│   └── figures/               # Generated manuscript figures
├── scripts/
│   ├── helpers/               # Processing utilities
│   │   ├── generate_full_summary.py
│   │   └── process_residuals_yearly.py
│   ├── steps/                 # Analysis pipeline (10 steps)
│   │   ├── step_1_0_data_acquisition.py
│   │   ├── step_2_1_slr_residuals.py
│   │   ├── step_2_3_mwpc_analysis.py
│   │   ├── step_2_4_plot_results.py
│   │   ├── step_2_5_enhanced_figures.py
│   │   ├── step_2_6_nwm_spectral_control.py
│   │   ├── step_2_7_orbit_commonmode_control.py
│   │   ├── step_2_8_sampling_matched_nulls.py
│   │   ├── step_2_9_egm_orbit_control.py
│   │   ├── step_3_0_sim_antiecho.py
│   │   └── step_5_2_vlbi_slr_scale_drift.py
│   └── utils/                 # Shared utilities
│       ├── compress_pdf.py
│       ├── logger.py
│       └── plot_style.py
├── site/                      # Manuscript website source
└── requirements.txt           # Python dependencies
```

### Data Provenance

| Source | Data Type | Download Size | Time Span |
| --- | --- | --- | --- |
| [ILRS Data Centers](https://ilrs.gsfc.nasa.gov/) | NPT CRD Range Residuals | ~2 GB | 2015–2025 |
| [NASA CDDIS](https://cddis.nasa.gov/) | LAGEOS Satellite Data | ~500 MB | 2015–2025 |

### Reproduction Instructions

#### Quick Start (Full Reproduction)

```

# 1. Clone repository

git clone https://github.com/matthewsmawfield/TEP-SLR.git
cd TEP-SLR

# 2. Install dependencies

pip install -r requirements.txt

# 3. Run complete analysis pipeline

bash reproduce_analysis.sh

# 4. Build manuscript site

cd site
npm install
npm run build
```

#### System Requirements

- **Python** 3.10+

- **Storage:** ~5 GB free (for data and outputs)

- **Memory:** 8 GB RAM recommended

- **Runtime:** ~2–4 hours for full pipeline (including data download); ~30–60 minutes with pre-downloaded data

#### Pipeline Overview

The analysis consists of 10 deterministic steps:

- **Step 1.0:** Data acquisition from ILRS/CDDIS archives (NPT CRD files)

- **Step 2.1:** Residual calculation and quality filtering (year-by-year processing)

- **Step 2.3:** Magnitude-Weighted Phase Correlation (MWPC) analysis — primary statistical test

- **Step 2.4:** Generate standard figures (residual distributions, correlation decay)

- **Step 2.5:** Enhanced figures with phase alignment analysis

- **Step 2.6:** NWM spectral control — NCEP/NCAR surface-pressure coherence at native 6-hourly resolution (Section 3.3.2)

- **Step 2.7:** Orbit-error / common-mode confound controls — cross-satellite contemporaneous pairing (closes the shared-orbit-error channel by construction), network common-mode subtraction, and per-satellite orbit-solution splits (Section 3.4)

- **Step 2.8:** Sampling-matched coloured-noise nulls — continuous surrogate families (white, station-matched AR(1), flicker 1/f, power-law grid, matched power law, random walk) passed through the identical resample–interpolate–concatenate–Welch pipeline and evaluated at real epochs under a Lomb–Scargle estimator, plus the pairwise structure function and the δA→residual amplitude map (Section 3.3.1)

- **Step 2.9:** Orbit-product common-mode control — model ranges recomputed under an independent analysis centre's SP3 solution (ESA/ESOC) on the identical filtered epoch set, then re-run through the Step 2.7 cross-satellite pairing (Section 3.4)

- **Step 3.0:** Anti-echo simulation — estimator-mediated sign inversion of the conformal common-mode field under dynamic orbit determination, including the injected-scale anchor scan (λ_T = 1,000–8,000 km) that records the turnover crossing under both corpus scales (3,421 km at 1,862 km; 3,947 km at 4,201 km)

- **Step 5.2:** VLBI–SLR scale-drift channel — anchoring-split bookkeeping and the local drift-participation bound (Section 5.2)

#### Key Analysis Outputs

- `results/outputs/step_2_1_slr_residuals_summary.json` — Residual statistics by station

- `results/outputs/step_2_3_mwpc_analysis.json` — MWPC correlation results

- `results/outputs/step_2_7_orbit_commonmode_control.json` — Orbit-error / common-mode confound-control results

- `results/outputs/step_2_8_sampling_matched_nulls.json` — Sampling-matched coloured-noise nulls, real-epoch Lomb–Scargle ratios, structure function, and the δA amplitude map

- `results/outputs/step_2_9_egm_orbit_control.json` — Independent orbit-product (ESA/ESOC) control results

- `results/outputs/step_5_2_vlbi_slr_scale_drift.json` — Scale-drift anchoring split and participation bound

- `results/figures/slr_pass_correlation_decay.png` — Figure 3.1: Pass-correlation decay analysis

- `results/figures/slr_phase_alignment_decay.png` — Phase alignment signatures

#### CDDIS Data Access (Step 1 Only)

Step 1 requires NASA CDDIS credentials for automated data download. Set up either:

- `~/.netrc` file with `machine urs.earthdata.nasa.gov` credentials, or

- Environment variables: `CDDIS_USER` and `CDDIS_PASS`

*Note:* The reproduction script can skip Step 1 if data is already present — contact the author
for pre-processed data access.

### Software Versions

- **Python** 3.10+

- **NumPy** 1.24+

- **SciPy** 1.10+

- **Pandas** 2.0+

- **Matplotlib** 3.7+

- **NetworkX** (for station network analysis)

---

<!-- ===================================================================== -->
<!-- CANONICAL URL: https://mlsmawfield.com/tep/exp/ -->
<!-- MARKDOWN URL:  https://mlsmawfield.com/tep/exp/manuscript.md -->
<!-- ===================================================================== -->

# What Do Precision Tests of General Relativity Actually Measure?
**Matthew Lukin Smawfield**
Version: v0.9 (Istanbul)
First published: 31 December 2025 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.18109760

---

## Abstract

Most high-precision tests of general relativity constrain reciprocity-even, largely local observables within single-metric frameworks. This leaves open a specific underdetermination between General Relativity (GR) and a class of two-metric disformal scalar-tensor modifications, exemplified here by the Temporal Equivalence Principle (TEP).

This paper formalizes a measurement taxonomy distinguishing gauge-invariant from convention-dependent observables and identifies six recurring scope limitations in the experimental canon: (1) two-way measurement dominance; (2) local/global conflation; (3) model-dependent calibration; (4) single-path multi-messenger constraints on differential propagation that do not directly test common-mode conformal clock-sector structure; (5) theory-laden data reduction; and (6) proximity-regime screening underdetermination in the tested configuration, whereby tests performed in high-acceleration environments probe only the screened regime where incremental field gradients are continuously suppressed, leaving the active-shear low-acceleration regime unexplored. These characteristics do not diminish the experimental achievements but indicate that, in many cases, the tests primarily constrain parameter space within assumed frameworks rather than systematically discriminating between alternatives.

Discriminating observables—specifically loop asymmetries, spatial correlations, and proximity-regime screening transitions—are proposed, together with experimental configurations capable of resolving the underdetermination. These include large-area triangle holonomy tests (targeting residual synchronization holonomy $H_{\rm resid}$), interplanetary closed-loop timing, altitude-varying optical clock networks to map continuous geometric screening, and matter-wave interferometry.

Keywords: general relativity, experimental tests, underdetermination, synchronization, GPS, gravitational redshift, Shapiro delay, multi-messenger astronomy

## 1. Introduction: The Experimental Canon

General relativity stands as one of the most precisely tested theories in physics. From the gravitational redshift measurements of Pound and Rebka (1960) to the multi-messenger observations of GW170817 (2017), a century of experiments has confirmed GR's predictions with high precision. Modern optical lattice clocks achieve fractional frequency comparisons at the 10<sup>−18</sup> level and below, detecting gravitational time dilation across millimeter height differences. The Cassini spacecraft measured the Shapiro delay to constrain the PPN parameter γ to within 2 × 10<sup>−5</sup> of unity. The Global Positioning System, which incorporates relativistic corrections as a matter of routine engineering, is often cited as the most practical demonstration that "GR works."

Because these experiments achieve extraordinary precision across diverse regimes, their combined bounds tightly restrict deviations within standard single-metric post-Newtonian and local-Lorentz-invariance parameterizations. A central methodological question, however, is which specific geometric sectors and observable classes are constrained by this canon, and which remain underdetermined by existing measurement geometries.

This paper conducts a systematic methodological investigation of what the canonical precision tests of general relativity actually measure and constrain. The analysis shows that the canonical tests share structural features—most notably two-way signal paths, local single-frame comparisons, and high-density screening environments—that leave specific two-metric and non-exact clock-transport observables, exemplified here by the Temporal Equivalence Principle (TEP), largely untested by current configurations.

### 1.1 The Textbook Narrative

The experimental confirmation of GR is typically presented as a progression of increasingly precise tests:

| Experiment | Year | Claimed Precision | Result |
|---|---|---|---|
| Pound-Rebka (redshift) | 1960 | 1% | Confirms GR |
| Hafele-Keating (time dilation) | 1972 | 10% | Confirms GR |
| Gravity Probe A | 1976 | 7 × 10⁻⁵ | Confirms GR |
| GPS (operational) | 1978– | Meter-level positioning | Confirms GR |
| Cassini (Shapiro delay) | 2003 | 2 × 10⁻⁵ | γ = 1 |
| Gravity Probe B | 2011 | 0.3% | Confirms frame-dragging |
| LIGO/Virgo (GW detection) | 2015– | — | GWs exist |
| GW170817 (multi-messenger) | 2017 | few × 10⁻¹⁵ (under standard emission-time assumptions) | cγ = cg |
| Optical clocks (JILA) | 2022 | 10⁻¹⁸ | Confirms redshift |

Each entry represents a genuine experimental achievement. The question is not whether these experiments were performed correctly, but what they actually test.

### 1.2 The Central Question

This paper asks: What do precision tests of relativity actually constrain?

The analysis identifies six structural limitations shared by the experimental canon:

- *Two-Way Measurement Dominance:* Nearly all tests use round-trip or reciprocity-even measurements, which cancel reciprocity-odd (direction-reversing) residuals. They remain sensitive to reciprocity-even anisotropy of the form $D(\hat n\cdot\nabla\phi)^2$ (Appendix B of Paper 0; §7.6 below).

- *Local/Global Conflation:* Tests of local Lorentz invariance are extrapolated to claims about global spacetime structure.

- *Model-Dependent Calibration:* Systems like GPS use GR-derived corrections to validate GR— demonstrating self-consistency within the framework, not independent confirmation.

- *Single-Path Multi-Messenger Limitations:* GW170817-type observations primarily constrain differential photon–graviton propagation and hence disformal cone tilt. They do not directly constrain common-mode conformal clock-rate structure along a shared path, although conformal scalar sectors remain indirectly constrained by PPN, equivalence-principle, and clock-comparison tests.

- *Theory-Laden Data Reduction:* Systematic corrections assume the framework being tested, making independent falsification practically difficult within standard pipelines.

- *The Proximity-Regime Screening Blind Spot:* Tests performed in deep potential wells (Earth's surface, solar system) probe only the screened regime where scalar-field gradients are continuously suppressed, leaving the active-shear dilute regime unexplored.

These limitations do not invalidate the experiments. They indicate that, in many cases, the tests primarily constrain parameter space within an assumed framework rather than systematically discriminating between alternatives.

The analysis throughout assumes a regime where disformal cone tilts are already constrained by multi-messenger bounds, and focuses on clock-sector structure that remains unconstrained within those limits.

#### Scope of Claim

This manuscript does not dispute the validity of canonical precision tests within their modeled domains. It argues that many flagship tests primarily constrain reciprocity-even observables, or even-parity two-way sums. They do not measure odd-parity synchronization holonomy $H_{\rm resid}$. Discrimination requires observables that are offset-invariant yet one-way and direction-reversing, or that probe spatial clock-correlation structure.

### 1.3 The Temporal Equivalence Principle: Theory Definition

The Temporal Equivalence Principle (TEP) is defined formally by a single manifold endowed with two metrics and a universal matter coupling. It is not an ad hoc modification but a systematic framework for dynamical time.

#### Formal Definition

Sector map: The conformal factor A(φ) governs clock rates and spatial correlation structure; the disformal factor B(φ) governs light-cone tilts and residual holonomy. GW170817 primarily constrains photon–graviton differential propagation and hence disformal cone tilt. It does not directly constrain common-mode conformal clock-rate structure along a shared path, although conformal scalar sectors remain indirectly constrained by PPN, equivalence-principle, source-screening, and clock-comparison tests.

Screening and PPN compatibility: Rather than invoking discrete thin-shell boundaries, screening operates via the continuous spatial profile of the scalar time field, termed Temporal Topology. The locally active gradient sector, Temporal Shear, is suppressed in screened regimes by environmental state and boundary conditions, ensuring short-range Temporal Shear suppression while leaving the field light cosmologically. Near massive bodies, suppression of Temporal Shear suppresses the active-gradient/source-charge response $S_\Sigma$; the conformal amplitude/clock response $S_A$ is a distinct projection of the same environmental state, cleanly preserving PPN bounds without rigid boundary approximations.

1. The Action: For comparison, the canonical scalar baseline is defined in the Einstein frame ($g_{\mu\nu}$) by the action below; the operative screening realization replaces the scalar sector by the selected two-branch $P(X,\phi)$ completion $P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+X|X|/\Lambda_X^4$ with derived $k\simeq16.03$, which carries no canonical unit small-$|X|$ term (Paper 0, §2.2):

$$S = \int d^4x \sqrt{-g} \left[ \frac{M_{\rm Pl}^2}{2} R - \frac{1}{2}
(\partial\phi)^2 - V(\phi) \right] + S_{matter}[\psi,
\tilde{g}_{\mu\nu}]$$

2. The Metrics: Gravity is governed by $g_{\mu\nu}$. All matter fields $\psi$ and clocks couple universally to the *matter metric* $\tilde{g}_{\mu\nu}$, written in isotropic conformal–disformal shorthand as (see Paper 0, §2.2 for the resolved preferred-foliation $(A_{\rm loc}, A_{\rm sp})$ decomposition along the cosmic temporal flow $U_\mu$):

$$\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu\phi
\nabla_\nu\phi$$

3. The Coupling:

- Clocks: Ideal clocks measure proper time $\tau$ defined by $d\tau^2 = -\tilde{g}_{\mu\nu} dx^\mu dx^\nu$.

- Light: Photons propagate on null geodesics of $\tilde{g}_{\mu\nu}$. (In the conformal limit $B=0$, these coincide with $g_{\mu\nu}$ null geodesics).

- Test Particles: Massive bodies follow geodesics of $\tilde{g}_{\mu\nu}$.

The key distinguishing predictions are therefore not local redshift or two-way light-time relations (where TEP can agree with GR), but global and statistical observables that probe synchronization integrability and spatial structure:

- Local physics is identical to GR in freely falling laboratories (EEP; locally invariant c)

- Clock residuals can exhibit spatially structured correlations (e.g., exponential decay with distance) if the scalar field has a finite correlation length

- A convention-independent residual synchronization holonomy Hresid can be defined for direction-reversing, one-way closed loops after subtracting modeled GR loop effects

- Light and gravitational-wave propagation remain consistent with existing constraints: conformal rescaling preserves null cones, and any disformal cone tilt is bounded at the 10<sup>−15</sup> level

- In high-acceleration environments (enclosed $M/r^2 \gtrsim g_t/G$), the suppression of Temporal Shear (vanishing incremental field gradient) reduces the effective scalar coupling, preserving PPN bounds through continuous geometric screening rather than discrete boundary cutoffs

The existing experimental canon strongly constrains local Lorentz violation, PPN departures in the gravitational/light-propagation sector, and disformal cone tilts. In TEP, PPN compatibility is maintained through the continuous suppression of Temporal Shear where the ambient time-gradient is steep (weak-field: high-acceleration environments) rather than discrete boundary approximations. The experimental canon does not yet directly probe spatial clock-correlation structure or residual holonomy in genuinely one-way, direction-reversing closed loops.

#### Candidate Discriminators and Current Status

To ensure falsifiability, TEP specifies candidate observables where it can differ from GR. Where numerical values are quoted from the TEP research program, they are treated as reported results requiring independent replication.

| Observable | GR Prediction | TEP Prediction | Status |
|---|---|---|---|
| Distance-structured clock correlations | None (or systematic origin) | λ = 1,000–10,000 km | Suggested in exploratory GNSS analyses (order
103–104 km scale) within the TEP
program; requires independent, blinded replication using raw
data |
| Orbital velocity coupling | No correlation | |r| > 0.5 with Earth's orbital velocity | Suggested in exploratory GNSS analyses within the TEP
program; requires independent, blinded replication |
| Preferred-direction recovery | No preferred direction | annual-phase direction near Earth's aphelion velocity tangent; CMB apex anti-phase | Suggested in exploratory GNSS analyses within the TEP
program; requires independent, blinded replication |
| Residual holonomy Hresid | 0 | 0 if B = 0 (conformal-only); ≠ 0 possible if B ≠ 0 (MEO Triangle). *Note: GW170817 constrains the realized differential cone deformation, schematically D|∇φ|² = (B/A²)|∇φ|², along the observed astrophysical propagation path.* | Untested (requires one-way closed loops) |
| Multi-constellation consistency | N/A | CV &lt; 30% across GPS/GLONASS/Galileo/BeiDou | Untested (requires raw data analysis) |
| Proximity-regime screening transition | None (no regime change predicted) | Continuous active-to-screened recovery governed by \(\mathcal S_\Sigma(\mathcal E)\), with \(\rho_T\) as one macroscopic saturation/proximity proxy | Altitude-varying optical clock network, wide-binary systems |

Critical note: The first three observables target conformal-sector structure associated with A(φ), while the holonomy test targets disformal structure associated with B(φ). The core argument of this paper concerns measurement geometry and does not assume that any specific GNSS-reported numerical value is correct.

### 1.4 Structure of This Paper

Section 2 develops the methodological framework for analyzing precision tests, introducing key distinctions between gauge-invariant and convention-dependent observables, two-way and one-way measurements, and local and global tests.

Sections 3–7 examine the major categories of precision tests:

- Section 3: Gravitational redshift tests (Pound-Rebka, optical clocks)

- Section 4: Time dilation tests (Hafele-Keating, GPS)

- Section 5: Light propagation tests (Shapiro delay, VLBI)

- Section 6: Multi-messenger constraints (GW170817)

- Section 7: Resonator tests (Michelson-Morley, cavity experiments)

Section 8 proposes experiments that could genuinely discriminate between GR and TEP: triangle holonomy tests, interplanetary closed-loop timing, and GNSS correlation replication.

Section 9 discusses the philosophical implications, situating the findings within the broader context of underdetermination in physics. Section 10 concludes.

### 1.5 Related Work

This analysis builds upon and extends several distinct strands of literature in gravitational physics, metrology, and the philosophy of science.

Test Theories of Relativity: Frameworks for testing Lorentz invariance, such as the Mansouri-Sexl (1977) formalism and the Standard Model Extension (SME) (Colladay & Kostelecký 1998), have long parameterized deviations from relativity. These frameworks primarily focus on local Lorentz violation (anisotropy, boost dependence) in single-metric contexts. TEP differs by accepting exact local Lorentz invariance (consistent with the tightest SME bounds) while allowing a GR-subtracted non-exact synchronization structure in the matter geometry.

Relativistic Metrology: The distinction between one-way and two-way time transfer is foundational in metrology (Ashby 2003). While two-way satellite time and frequency transfer (TWSTFT) and GPS Common-View are standard techniques for clock comparison, they are explicitly designed to cancel path delays. Recent proposals for "relativistic geodesy" with optical clocks (Mehlstäubler et al. 2018) focus on measuring the geopotential $W$, assuming GR. This work explores the inverse problem: using clocks to test the framework that defines $W$.

Philosophy of Simultaneity: The conventionality of simultaneity (Reichenbach 1928; Grünbaum 1973; Salmon 1977) establishes that one-way light speed is a convention in special relativity. Malament (1977) establishes synchronization uniqueness relative to a specified causal structure. TEP does not introduce separate clock and photon causal structures; instead it asks whether the universal matter-metric synchronization connection contains a GR-subtracted non-exact component measurable through closed-loop holonomy.

Theory-Laden Experimentation: The challenge of "epistemological circularity" in precision tests—where data reduction relies on the theory being tested—has been discussed by Franklin (1986) and others. This analysis extends that discussion to modern precision cosmology and GNSS data processing, identifying specific pipelines (e.g., common-mode filtering) that enforce theoretical priors.

To forestall misunderstanding, this paper does not claim:

- That GR is wrong

- That precision tests are flawed or incompetent

- That TEP is correct

- That all alternatives to GR are viable

The claim is narrower and more specific: the existing experimental canon does not directly probe the observables that distinguish GR from TEP. This underdetermination is resolvable through new experimental configurations that have not yet been performed.

## 2. Methodological Framework

Before examining individual experiments, it is essential to establish the conceptual tools required to distinguish what precision tests actually measure from what they are commonly claimed to measure. This section develops five key distinctions that structure the subsequent analysis.

Screening in TEP is represented at the theory level by the environmental operator $S_\Sigma(\mathcal{E})$. Quantities such as $\rho_T$, $R_T(M)$, $S_\oplus(r)$, compactness $\Phi/c^2$, local stellar density, geometric coherence length, and channel-specific response coefficients are domain-specific projections of $\mathcal{E}$, not independent screening mechanisms and not interchangeable universal thresholds. Each is an observational transfer model that parameterizes the same underlying operator in a regime-appropriate form.

### 2.1 Gauge-Invariant vs. Convention-Dependent Observables

Not all physical quantities carry the same epistemological weight. A fundamental distinction exists between gauge-invariant observables—quantities that remain unchanged under coordinate transformations—and convention-dependent quantities that depend on arbitrary choices.

#### Definition 2.1: Gauge-Invariant Observable

An observable O is gauge-invariant if and only if its value is independent of the coordinate system, synchronization convention, or reference frame used to compute it.

Examples of gauge-invariant observables include:

- Proper time elapsed along a worldline: τ = ∫ ds

- Round-trip light travel time between two events

- Frequency ratio of co-located clocks

- Interference fringe count in closed optical paths

Examples of convention-dependent quantities include:

- One-way speed of light (depends on synchronization)

- Coordinate time difference between spatially separated events

- Simultaneity of distant events

Most precision tests measure gauge-invariant quantities that are highly sensitive to the conventional gravitational sectors. Gauge invariance itself does not exclude sensitivity to TEP—in particular, the GR-subtracted closed-loop residual holonomy $H_{\rm resid}(C) = \oint_C (\tilde{\sigma} - \sigma_{\rm GR})$ is itself strictly gauge-invariant (Paper 0, Appendix A3).

The proposed underdetermination arises instead because the existing measurement configurations (predominantly two-way, single-path, or local co-located comparisons) do not generally isolate the particular GR-subtracted, non-exact transport and spatial-correlation observables targeted here, accessing only the subset of gauge-invariant observables on which GR and TEP explicitly agree.

### 2.2 Two-Way vs. One-Way Measurements

The distinction between two-way (round-trip) and one-way measurements is not merely technical but fundamental to what can be tested.

#### The Reichenbach Synchronization Problem

The one-way speed of light cannot be measured without first establishing a synchronization convention between spatially separated clocks. But any synchronization convention presupposes a one-way speed of light. This circularity means the one-way speed is convention-dependent, not empirically determinable.

Two-way measurements avoid this circularity by using a single clock:

$$t_{\text{round-trip}} = t_{\text{return}} - t_{\text{emit}} \quad
\text{(single clock, gauge-invariant)}$$

However, this gauge-invariance comes at a cost: two-way measurements cancel reciprocity-odd (direction-reversing) residuals, though they remain sensitive to reciprocity-even anisotropy of the form $D(\hat n\cdot\nabla\phi)^2$ (Appendix B of Paper 0; §7.6 below). If light travels at c + v in one direction and c − v in the return direction, the round-trip time is:

$$t_{RT} = \frac{L}{c+v} + \frac{L}{c-v} = \frac{2Lc}{c^2 - v^2} \approx
\frac{2L}{c} + O\left(\frac{v^2}{c^2}\right)$$

The first-order direction-dependent term cancels exactly. This is not a limitation of experimental precision but a mathematical necessity: round-trip measurements are reciprocity-even and cannot probe reciprocity-odd effects.

### 2.3 Local vs. Global Tests

A third crucial distinction separates local tests—performed within a single freely falling frame—from global tests that probe the structure of spacetime across extended regions.

#### The Equivalence Principle Hierarchy

- *Weak Equivalence Principle (WEP):* All bodies fall with the same acceleration in a gravitational field.

- *Einstein Equivalence Principle (EEP):* In local freely falling frames, non-gravitational physics is Lorentz-invariant.

- *Strong Equivalence Principle (SEP):* EEP extends to gravitational physics; local gravitational experiments are position-independent.

The TEP framework satisfies EEP in the screened tangent-space limit: in local freely falling frames where the observable shear sector and disformal response are suppressed ($S_\Sigma(\mathcal{E}) \to 0$, $B(\phi)(\nabla\phi)^2 \to 0$) and the measurement region is small compared to the scalar field's correlation scale, non-gravitational physics reduces to special relativity with invariant $c$. Within that fully screened, sufficiently local regime and for single-metric-reducible observable classes, point-like local tests—however precise—reduce to the same tangent-space limit as GR. Distinguishing the two frameworks requires either unscreened gradient/boundary observables or non-local and statistical observables: spatial covariance in clock residuals across extended baselines and a GR-subtracted residual holonomy H<sub>resid</sub> defined by genuinely one-way, direction-reversing closed loops.

The operational criterion for "local" is whether the measurement region is small compared to the curvature scale. Modern optical clock experiments operate at millimeter scales where spacetime curvature is negligible—they are quintessentially local tests.

### 2.4 The Screening Blind Spot: Temporal Topology and Proximity Regimes

A fourth distinction concerns the density environment in which experiments are performed. The canonical precision tests are overwhelmingly conducted in dense gravitational environments: Earth's surface, the solar system, or laboratory frames embedded in screened-source or locally constrained regimes. In scalar-tensor frameworks with environment-dependent screening, these locations correspond to regimes where the scalar field gradient is strongly suppressed.

#### Temporal Topology and Temporal Shear

In TEP, screening is formulated as a continuous geometric effect rather than a discrete thin-shell boundary. Temporal Topology is the continuous spatial profile of $\ln A(\phi)$. A density-dependent $V_{\rm eff}(\phi;\rho)$ is one optional local realization, not the defining screening law (Paper 0, Axiom A4). In high-acceleration environments (enclosed $M/r^2 \gtrsim g_t/G$), the ambient field's kinetic stiffness suppresses the incremental field gradient (Temporal Shear, ∇φ), driving the shear response toward zero while the Temporal Topology persists. This continuous suppression reconciles local null tests with cosmological dynamics without invoking step-function boundary conditions.

The operative quantity for Temporal Shear coupling and clock-sector effects is the local field gradient (Temporal Shear). In high-acceleration environments (enclosed $M/r^2 \gtrsim g_t/G$), suppression of Temporal Shear by environmental state and boundary conditions suppresses the active-gradient/source-charge response $S_\Sigma$; the conformal amplitude/clock response $S_A$ is a distinct projection of the same environmental state, cleanly preserving PPN bounds. In dilute regions, the gradient recovers and the field becomes cosmologically active.

$$\Sigma_\mu^{\text{obs}} = \mathcal{S}_\Sigma(\mathcal{E}) \, \nabla_\mu \ln A(\phi),$$

where the environmental state is $$\mathcal{E} = \{\rho, \Phi/c^2, \nabla\rho, \nabla\Phi, \text{compactness}, R_T(M), \text{proximity}, T, z, \text{boundary geometry}, \text{coherence volume}\}.$$ This single operator unifies density screening, compactness screening, proximity screening, thermodynamic/environmental-state response, lensing/cosmological covariance screening, and wide-binary environmental screening across the entire TEP corpus.

The experimental consequence is that existing precision tests probe the deeply screened regime, where Temporal Shear vanishes continuously. They are insensitive to the field's behavior in the active-shear dilute regime where large-scale structural effects originate. This creates an observable underdetermination in the tested proximity-regime configuration: experiments constrain the screened limit while leaving the active-shear limit largely unexplored.

Discriminating tests must therefore either probe dilute environments where Temporal Shear recovers (wide binary systems, galactic halos), map the continuous transition using altitude-varying clock networks, or detect spatial correlations on scales where the field gradient remains active.

### 2.5 Single-Path vs. Multipath Configurations

The final distinction concerns whether signals traverse a single path or multiple paths through spacetime.

#### Common-Mode Cancellation

When two signals traverse the same path, any path-dependent effects cancel in their comparison. This is why GW170817—where photons and gravitons traveled the same ~130 million light-year path—constrains only their differential propagation speed, not their common-mode time dilation.

TEP phenomenology arises from:

- Multipath configurations (gravitational lensing, where rays sample different field values)

- Statistical correlations (GNSS, where spatial structure ⟨φ(x)φ(x')⟩ is probed)

- Closed loops with direction reversal (residual holonomy Hresid after subtracting modeled GR loop effects)

Single-path, single-direction measurements—the dominant mode of precision tests—are structurally incapable of detecting these signatures.

### 2.6 Summary: The Measurement Taxonomy

| Measurement Type | Gauge Status | Sensitive To | Blind To |
|---|---|---|---|
| Two-way, local | Invariant | Local Lorentz violation | Direction-dependence, global structure |
| Two-way, global | Invariant | PPN parameters (γ, β) | Odd-parity effects, synchronization |
| One-way, single-path | Convention-dependent | Relative clock rates | Absolute synchronization |
| One-way, closed-loop | Invariant | Residual holonomy Hresid, path-dependence | — |
| Screened (high-acceleration environment) | Invariant | Screened PPN limits | Unscreened dilute behavior |

The 'proximity-regime' boundaries defined by deep local potential wells are physical manifestations of the abstract environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$. Because standard precision tests overwhelmingly operate within deeply saturated macro-environments, they effectively probe only the screened limit of the continuous screening curve, fundamentally blinding these experimental protocols to the active temporal shear. Following Paper 0, $S_A$ is the amplitude/clock projection and $S_\Sigma$ is the active-gradient/source-charge projection. They are distinct responses of the same environmental state $\mathcal{E}$.

The final row—one-way closed-loop measurements—represents the only configuration capable of testing synchronization integrability. No high-precision relativistic-gravity test has directly targeted a convention-independent, direction-reversing, one-way loop observable designed to detect non-integrable synchronization beyond modeled GR terms. Nor has any precision test directly probed the active-shear dilute regime where Temporal Shear recovers and scalar-field spatial structure becomes detectable.

### 2.7 Canonical Tests and Their Constrained Sectors

The following table summarizes which theoretical sectors the canonical precision tests actually constrain. Understanding these mappings clarifies why TEP's clock-response sector (κ<sub>X</sub>) and residual holonomy (H<sub>resid</sub>) remain largely untested.

| Test | Sector Constrained |
|---|---|
| Cassini (Shapiro delay) | Σμobs, source charge, Shapiro sector |
| MICROSCOPE (Eötvös) | Composition-dependent scalar charge |
| GW170817/GRB170817A | Disformal cone tilt (differential photon-graviton speed) |
| GNSS/RINEX/SLR | Covariance sector (two-way clock comparisons) |
| Cepheids/pulsars/JWST | κX clock-response sector (astronomical distance-ladder) |
| Triangle holonomy | Hresid (residual synchronization holonomy) |

The pattern is clear: existing tests constrain source-coupling, composition, and light-propagation sectors, but do not directly probe the clock-response coefficients κ<sub>X</sub> or the residual holonomy H<sub>resid</sub> that discriminate TEP from GR. These coefficients nonetheless occupy a shared phenomenological regime across independent channels: the Cepheid ladder-level response is measured at κ<sub>Cep</sub> = (0.665–1.275) × 10<sup>6</sup> mag in Paper 11 (multi-block joint to redshift-only estimators), bracketing the prespecified channel benchmark κ<sub>gal</sub> = (9.6 ± 4.0) × 10<sup>5</sup> mag — grounded in the Cepheid envelope density ratio ρ<sub>T</sub>/⟨ρ⟩(10 d) = 8.45 × 10<sup>5</sup>, which at unit bare coupling λ₀ = |β<sub>A</sub>| = 1 predicts |κ<sub>P²</sub>| = 1.196 × 10<sup>6</sup> mag against the SH0ES fit (1.191 ± 0.480) × 10<sup>6</sup> mag (λ₀ = 0.995 ± 0.401; Paper 11, Step 62) — while the millisecond-pulsar channel's analogous full response coefficient is κ̃<sub>MSP</sub> ~ 2.9 × 10<sup>4</sup>, drawn from an unsuppressed geometric scale of ~10<sup>6</sup> (Paper 10). None of these figures is directly comparable to Cassini's γ constraint or MICROSCOPE's composition bounds.

## 3. The Gravitational Redshift Tests

Gravitational redshift experiments—from Pound-Rebka (1960) to modern optical lattice clocks—are often cited as the most precise confirmations of general relativity. This section examines what these experiments test and why they do not directly probe the observables that distinguish GR from TEP.

### 3.1 The Canonical Experiments

##### 3.1.1 Pound-Rebka-Snider (1960–1965)

The Pound-Rebka experiment measured the gravitational redshift of 14.4 keV gamma rays traveling 22.5 meters vertically in the Jefferson Tower at Harvard. The observed fractional frequency shift:

$$\frac{\Delta\nu}{\nu} = \frac{gh}{c^2} = \frac{(9.8 \text{ m/s}^2)(22.5 \text{ m})}{(3 \times 10^8 \text{ m/s})^2} \approx 2.5 \times 10^{-15}$$

agreed with the GR prediction to within 1% (later improved to 0.1% by Pound and Snider).

**Critical Analysis:**

#### Methodological Analysis

The experiment compares the frequency of gamma rays emitted at height h₁ with those absorbed at height h₂. Both emission and absorption occur locally—the comparison is between two local measurements connected by light propagation.

TEP predicts identical local physics in freely falling frames. The gravitational redshift arises from the difference in gravitational potential between emission and absorption points—a prediction shared by any metric theory satisfying EEP.

##### 3.1.2 Gravity Probe A (1976)

Gravity Probe A flew a hydrogen maser to 10,000 km altitude, comparing its frequency with ground-based masers. The experiment confirmed gravitational time dilation to 7 × 10⁻⁵ relative precision.

**Critical Analysis:**

#### Methodological Analysis

The comparison used two-way microwave links: signals were sent up to the spacecraft and transponded back. The round-trip nature of the measurement makes it insensitive to any direction-dependent propagation effects.

The experiment confirms that clocks at different gravitational potentials run at different rates—a prediction common to GR, TEP, and all metric theories of gravity.

##### 3.1.3 Modern Optical Lattice Clocks (2010–present)

Jun Ye's group at JILA has achieved gravitational redshift measurements at the 10<sup>−18</sup> level and below, detecting time dilation across millimeter height differences. These represent the most precise measurements of gravitational effects ever performed.

"We greatly benefit from rejection of common-mode systematic shifts thanks to 
ensembles sharing the same optical lattice."
— Bothwell et al. (2022)

**Critical Analysis:**

#### Methodological Analysis

The quoted statement reveals the experimental configuration: clock ensembles share the same optical lattice, enabling common-mode rejection. This is a local comparison within a single laboratory frame.

The measurement scale (millimeters) is vastly smaller than any curvature scale. These are quintessentially local tests of the equivalence principle, not probes of global spacetime structure.

Optical clock metrology also includes long-baseline comparisons (for example via phase-stabilized fiber links) used for relativistic geodesy and time-scale realization. These experiments extend the baseline, but they still primarily target reciprocity-even observables such as frequency ratios and potential differences, and they typically rely on link designs and models that suppress non-reciprocal path terms.

As a result, even when baselines extend to regional or continental scales, the dominant observable remains a redshift-type comparison rather than a direction-reversing closed-loop residual holonomy H<sub>resid</sub> or a deliberately constructed spatial correlation statistic.

TEP Axiom 2 states: "In local freely falling frames, physics reduces exactly to special relativity with invariant c." These experiments confirm this axiom—they cannot test whether synchronization is globally integrable.

### 3.2 What Redshift Tests Measure

All gravitational redshift experiments measure the same fundamental quantity: the ratio of proper times at different gravitational potentials.

$$\frac{d\tau_1}{d\tau_2} = \sqrt{\frac{g_{00}(x_1)}{g_{00}(x_2)}} \approx 1 + \frac{\Phi_1 - \Phi_2}{c^2}$$

This ratio is:

- Gauge-invariant (independent of coordinate choice)

- Local (compares clocks at specific spacetime points)

- Common to all metric theories satisfying EEP

The experiments do not measure:

- Absolute synchronization between distant clocks

- Path-dependence of clock transport

- Global integrability of the synchronization field

### 3.3 The ACES Mission: A Missed Opportunity?

The Atomic Clock Ensemble in Space (ACES), scheduled for the International Space Station, will compare space-based and ground-based atomic clocks with unprecedented precision. The mission aims to test gravitational redshift at the 2–3 × 10<sup>−6</sup> level (Savalle et al. 2019).

**Critical Analysis:**

#### Configuration Analysis

ACES uses microwave and optical links for clock comparison. The microwave link (MWL) employs two-way time transfer—signals travel up and down, with the comparison derived from the round-trip.

While ACES will achieve high precision, its two-way configuration makes it structurally incapable of detecting synchronization holonomy. The experiment will confirm (or constrain) the gravitational redshift formula but cannot test whether synchronization is path-dependent.

### 3.4 Summary: Local Precision, Global Blindness

#### What Gravitational Redshift Tests Confirm

- Clocks at different gravitational potentials run at different rates

- The rate difference follows dτ/dt = √(1 + 2Φ/c²) to high precision

- Local Lorentz invariance has been tested to very high, sector-dependent precision (with optical-clock frequency comparisons reaching the 10<sup>−18</sup> level)

- The Einstein Equivalence Principle is satisfied

#### What They Cannot Test

- Whether synchronization is globally integrable

- Whether clock transport around closed loops accumulates holonomy

- Whether the one-way speed of light is direction-independent

- Any prediction that distinguishes GR from TEP

The gravitational redshift tests are triumphs of experimental physics. They confirm that gravity affects time—an insight that transformed the understanding of the universe. These experiments were not designed to probe global synchronization structure, and it would be unreasonable to criticize them for not testing something outside their scope. The point is that theories agreeing on local physics (like GR and TEP) do not directly probe the specific loop and correlation observables that differ between them in the regime considered here.

The proposed loop and correlation experiments should therefore be seen as complementary additions to, not replacements for, the existing experimental canon.

## 4. The Time Dilation Tests

Time dilation experiments—from Hafele-Keating's circumnavigating clocks to GPS satellite corrections—are frequently cited as strong evidence that "time runs differently" in different reference frames. This section examines the logical structure of these claims.

### 4.1 The Hafele-Keating Experiment (1972)

In October 1971, Joseph Hafele and Richard Keating flew cesium atomic clocks on commercial aircraft around the world—eastward and westward—comparing them with reference clocks at the U.S. Naval Observatory. The observed time differences:

| Direction | Predicted (ns) | Observed (ns) |
|---|---|---|
| Eastward | −40 ± 23 | −59 ± 10 |
| Westward | +275 ± 21 | +273 ± 7 |

The results are celebrated as confirming both special relativistic time dilation (velocity-dependent) and general relativistic gravitational time dilation (altitude-dependent).

**Critical Analysis:**

#### Methodological Analysis

The experiment measures the proper time difference Δτ between clocks that separate and reunite. This is a gauge-invariant quantity—the "twin paradox" resolved by comparing clocks at the same spacetime point.

Crucially, the clocks return to their starting point. The measurement is of accumulated proper time around a closed loop, not of synchronization between spatially separated clocks.

In the conformal limit ($B = 0$), TEP predicts identical proper time accumulation for any closed worldline, exactly recovering GR; in the presence of disformal structure, any residual departure is governed by the GR-subtracted closed-loop observable $H_{\rm resid}$. The Hafele-Keating result confirms that proper time depends on the worldline traversed—a prediction shared by GR, TEP, and all metric theories.

#### The Closed-Loop Limitation

When clocks reunite, their comparison is unambiguous. Yet this feature—the closure of the loop—eliminates sensitivity to synchronization holonomy. The experimentally relevant discriminator is a GR-subtracted residual holonomy H<sub>resid</sub>, defined by genuinely one-way, direction-reversing closed loops after subtracting modeled GR loop effects (Sagnac, Shapiro, gravito-magnetic terms). Closed-worldline proper-time comparisons do not access H<sub>resid</sub>.

Hafele-Keating confirms: "Different worldlines accumulate different proper times."

Hafele-Keating cannot test: "Is synchronization of distant clocks path-independent?"

### 4.2 Muon Lifetime Experiments

Cosmic ray muons, traveling at relativistic speeds, exhibit extended lifetimes compared to muons at rest. The observed lifetime dilation factor γ = 1/√(1 − v²/c²) agrees with special relativity to high precision.

**Critical Analysis:**

#### Methodological Analysis

Muon lifetime measurements compare the decay rate of moving muons (in the lab frame) with the known rest-frame lifetime. This is a comparison of proper times along different worldlines—the muon's worldline versus a hypothetical stationary worldline.

The measurement confirms time dilation as a function of velocity. It does not involve synchronization of spatially separated clocks and cannot probe synchronization structure.

### 4.3 The Galileo Eccentric Orbit Test (2018)

In 2014, two Galileo satellites (GSAT-0201 and GSAT-0202) were accidentally launched into highly eccentric orbits instead of circular ones. This "fortunate accident" enabled the most precise test of gravitational redshift since Gravity Probe A in 1976.

The eccentric orbits (perigee ~17,180 km, apogee ~26,020 km) produce a periodic modulation of the gravitational potential experienced by the onboard hydrogen masers. Delva et al. (2018) and Herrmann et al. (2018) independently analyzed 1008 days of data, measuring:

```

    α = (+0.19 ± 2.48) × 10⁻⁵

```

where α parameterizes deviations from the GR prediction (α = 0 in GR). This represents a factor 5.6 improvement over Gravity Probe A.

**Critical Analysis:**

#### Methodological Analysis

The Galileo test measures the periodic variation in clock frequency as the satellite moves between perigee and apogee. This is a comparison of the same clock at different gravitational potentials—a local measurement of how clock rate depends on potential.

The measurement uses two-way time transfer via GNSS signals. The satellite clock is compared with ground clocks through standard GNSS processing, which assumes GR-based corrections for signal propagation.

TEP predicts identical local clock rates to GR. The gravitational redshift formula dτ/dt = √(1 + 2Φ/c²) is shared by both theories. The Galileo test confirms this formula with unprecedented precision but cannot probe synchronization structure.

#### What the Galileo Test Cannot Probe

The test measures how a single clock's frequency varies with gravitational potential. It does not test whether synchronization between spatially separated clocks is path-independent. The periodic signal is entirely predicted by both GR and TEP.

### 4.4 Gravity Probe B (2011)

Gravity Probe B measured two predictions of general relativity: the geodetic effect (spacetime curvature caused by Earth's mass) and frame-dragging (spacetime being "dragged" by Earth's rotation). Four ultra-precise gyroscopes orbited Earth for 18 months, measuring their precession relative to a distant guide star.

The results confirmed GR predictions:

| Effect | GR Prediction | Measured | Precision |
|---|---|---|---|
| Geodetic precession | −6606.1 mas/yr | −6601.8 ± 18.3 mas/yr | 0.28% |
| Frame-dragging | −39.2 mas/yr | −37.2 ± 7.2 mas/yr | 19% |

**Critical Analysis:**

#### Methodological Analysis

Gravity Probe B measures gyroscope precession—the rotation of a spin axis relative to distant stars. This is a local measurement: the gyroscope and its reference telescope occupy the same spacecraft, comparing local inertial frames with the distant stellar reference.

The geodetic effect arises from spacetime curvature; frame-dragging from the rotation of the central mass. Both are predictions of how local inertial frames precess relative to infinity—not tests of synchronization structure.

TEP preserves local Lorentz invariance exactly. Gyroscope precession depends on the local connection coefficients (Christoffel symbols), which are identical in GR and TEP in the screened Solar-System limit. The experiment confirms that spacetime is curved as GR predicts; it cannot test whether synchronization is globally integrable.

#### The Local Frame Limitation

Gravity Probe B compares a local gyroscope with a distant reference direction. The comparison involves light from the guide star reaching the telescope—a single-path measurement that cannot probe synchronization holonomy.

To test TEP, one would need to compare gyroscopes at different locations and ask whether their precessions are consistent with a globally integrable synchronization. GP-B's single-satellite design cannot perform this test.

### 4.5 The GPS System: Model-Dependent Calibration

The Global Positioning System is frequently cited as the most practical demonstration of relativistic effects. GPS satellites carry atomic clocks that are pre-corrected for both special relativistic (velocity) and general relativistic (gravitational) time dilation. This section examines the logical structure of GPS as a test of GR.

##### 4.5.1 The Standard Narrative

GPS satellites orbit at ~20,200 km altitude with velocity ~3.9 km/s. The combined relativistic correction is approximately +45 μs/day (gravitational) minus 7 μs/day (velocity), yielding a net +38 μs/day correction applied to satellite clocks.

The argument proceeds: GPS works at meter-level accuracy; therefore, the relativistic corrections must be correct; therefore, GR is confirmed.

##### 4.5.2 Model-Dependent Calibration

**Critical Analysis:**

#### The Logical Structure: Sufficiency vs. Uniqueness

The engineering success of GPS is a powerful result, but its epistemological status requires precise characterization:

- Sufficiency: GPS demonstrates that the GR framework is *sufficient* to maintain synchronization at the nanosecond level. The corrections derived from GR work.

- Internal Consistency: The system validates the self-consistency of the GR model. When clocks are corrected according to GR, the navigation solutions close.

- Non-Uniqueness: This success does not mathematically prove that GR is the *only* theory capable of producing these results. Any theory that agrees with GR on local clock rates (the source of the corrections) and light propagation (for two-way/common-view transfer) would yield an identical engineering system.

Crucially, GNSS data *can* constrain deviations from GR, but only if analyzed with alternative models and, most importantly, with access to raw observables before standard relativistic corrections are baked into the data products.

#### Sufficiency vs. Necessity

GPS proves that GR is sufficient for navigation, but not that it is necessary.

Any theory that predicts the same local clock rates as GR would produce identical corrections and identical navigation performance. GPS cannot distinguish between theories that agree on local clock behavior but differ on global synchronization structure.

"These clocks read the coordinate time t... would be self-consistently
synchronized if one brought them together—assuming that general
relativity is correct."
— Ashby (2003), Living Reviews in Relativity

The phrase "assuming that general relativity is correct" is not a criticism of GPS engineering—it is a precise statement of the logical structure. GPS demonstrates that GR provides a self-consistent framework for navigation. It does not demonstrate that GR is the unique framework capable of doing so.

##### 4.5.3 What a TEP Signal Would Look Like in GPS Residuals

If TEP effects exist, they would appear in GPS data as specific systematic patterns. Understanding these signatures is essential for determining whether GPS could detect or has already filtered out such effects.

#### TEP Signatures in GPS Residuals

| TEP Effect | GPS Manifestation | Standard Interpretation | Distinguishing Feature |
|---|---|---|---|
| Distance-structured correlations | Correlated clock residuals between nearby stations | "Common-mode error" (filtered out) | Exponential decay with distance (~4000 km scale) |
| Residual holonomy Hresid | Systematic position bias in closed-loop solutions | "Network adjustment error" | Direction-dependent, not random |
| Scalar field gradient | Altitude-dependent clock drift beyond GR | "Satellite clock instability" | Correlated across satellites at similar altitudes |
| Temporal coherence | Day-to-day correlation in residuals | "Environmental systematic" | Persists after environmental correction |

##### 4.5.4 Reported Evidence from GNSS Analysis

Exploratory analysis of GNSS clock products from multiple analysis centers (CODE, IGS, ESA) within the TEP research program has suggested systematic patterns consistent with TEP predictions:

- Distance-structured correlations: Clock residuals reportedly show exponential decay with inter-station distance, with a characteristic scale of order 10<sup>3</sup>–10<sup>4</sup> km

- Cross-center consistency: The same pattern reportedly appears in independent processing pipelines (R² = 0.92–0.97 between centers)

- Temporal persistence: The correlation structure reportedly persists over 25 years of data (2000-2025)

- Ionospheric independence: The signal is reportedly invariant under geomagnetic storm stratification (Kp &lt; 3 vs Kp ≥ 3)

These reported observations require independent verification. If confirmed, they would demonstrate that GPS/GNSS data contains systematic structure that standard processing treats as "error" rather than "signal." If refuted, they would constrain the TEP parameter space. Either outcome advances scientific understanding.

##### 4.5.5 The Filtering Problem

#### What GPS Processing Removes

Many high-precision GNSS processing pipelines can suppress spatially correlated residual structure through:

- Reference frame constraints: Network solutions are constrained to minimize global residuals

- Common-mode filtering: Correlated signals across stations are removed as "systematic errors" (see Dong et al. 2006)

- Clock modeling: Polynomial models absorb slow drifts (Kouba & Héroux 2001)

- Ambiguity resolution: Integer constraints enforce consistency

If TEP produces distance-structured clock correlations, standard processing would tend to suppress such structure as a systematic effect rather than preserve it as a candidate signal.

#### Concrete Example: How TEP Signals Are Filtered Out

Consider, hypothetically, two GNSS stations separated by ~3,000 km. Suppose a weak, distance-structured correlated component exists in the underlying observables after standard GR modeling.

What happens in standard GPS processing:

- Initial observation: Both stations show residual structure with partial coherence at the same epoch

- Network constraint: The processing software enforces that the global network residuals sum to zero (to maintain the reference frame)

- Common-mode removal: The correlated signal is identified as a "common-mode error" affecting multiple stations

- Filtering: The signal is removed by subtracting the network-wide mean or applying spatial filtering (e.g., PCA, Karhunen-Loève expansion; Dong et al. 2006)

- Final product: Clock residuals are decorrelated, and the TEP signature is suppressed to below detection threshold

Result: Any spatially coherent component can be classified as "systematic error" and partially removed. The extent of suppression is product- and analysis-center dependent.

This is why independent analysis of *raw* GNSS data (before common-mode filtering) is critical for testing TEP. Many standard GNSS products prioritize reference-frame stability and positioning performance, and are not designed to preserve subtle spatial-correlation structure for fundamental-physics inference.

##### 4.5.6 The TEP Perspective

TEP predicts identical local clock rates to GR. The frequency corrections applied to GPS satellites would be the same under TEP. The difference arises in synchronization structure—whether the coordinate time t is globally integrable.

GPS uses predominantly two-way time transfer and common-view methods for synchronization. These methods are insensitive to residual holonomy H<sub>resid</sub>. A nonzero H<sub>resid</sub> beyond GR subtraction would manifest as systematic position errors with specific geometric signatures—but such signatures would be absorbed into the navigation solution as "common-mode errors" and filtered out.

**Validation:**

#### What Would Falsify GR via GPS?

For GPS to genuinely discriminate between GR and TEP, one would need to:

- Analyze raw clock residuals before common-mode filtering

- Test for distance-structured correlations (TEP predicts exponential decay)

- Perform closed-loop one-way timing tests (not two-way or common-view)

- Compare residual patterns with TEP and GR predictions

Exploratory analysis of GNSS clock products has suggested patterns consistent with TEP predictions. The question is whether these signatures represent physics or unmodeled systematics—a question that requires independent, blinded validation.

### 4.6 Lunar Laser Ranging: The Nordtvedt Effect

Lunar Laser Ranging (LLR) has measured the Earth-Moon distance to millimeter precision since 1969, using retroreflectors placed by Apollo astronauts. LLR provides the most precise test of the Strong Equivalence Principle through the Nordtvedt effect.

##### 4.6.1 The Nordtvedt Effect

If gravitational self-energy contributes differently to inertial and gravitational mass, the Earth and Moon would fall toward the Sun at slightly different rates, causing a polarization of the lunar orbit. The Nordtvedt parameter η quantifies this violation:

$$\eta = 4\beta_{\rm PPN} - \gamma - 3 - \frac{10}{3}\xi - \alpha_1 +
\frac{2}{3}\alpha_2$$

Current LLR solutions give (m<sub>G</sub>/m<sub>I</sub>)<sub>E</sub> − (m<sub>G</sub>/m<sub>I</sub>)<sub>M</sub> = (−0.8 ± 1.3) × 10<sup>−13</sup>, implying |η| ≲ few × 10<sup>−4</sup> (conversion depends on the Earth–Moon self-energy difference; Williams et al. 2012).

**Critical Analysis:**

#### Methodological Analysis

LLR uses round-trip laser ranging: pulses travel from Earth to Moon and back. The measurement is inherently two-way, averaging over any direction-dependent propagation effects. The Nordtvedt test constrains differential free-fall of the Earth and Moon toward the Sun, and therefore places strong bounds on violations of the Strong Equivalence Principle and on any additional long-range fields that couple differently to self-gravitating bodies.

In two-metric frameworks where gravitational dynamics remain effectively GR-like in the solar-system regime, consistency with LLR requires that any additional scalar structure be screened in the solar-system environment. In TEP, this screening is provided by the continuous suppression of Temporal Shear: the environmental state around the Earth and Moon drives the field gradient (∇φ) toward zero, decoupling local clock rates from the scalar field without invoking discrete thin-shell boundaries. LLR therefore acts as a stringent constraint on the allowed coupling and screening regime, but it does not directly target the one-way loop observables or spatial clock-correlation statistics emphasized in this paper.

LLR confirms that gravity is universal for self-gravitating bodies. It does not test clock synchronization structure—the measurement involves no clocks on the Moon, only photon round-trip times.

#### Why LLR Does Not Directly Probe TEP Signatures

LLR tests orbital dynamics, which in TEP follow geodesics of the matter metric g̃<sub>μν</sub> — the Earth-Moon system is not decoupled from the scalar sector by metric assignment. Translated to Einstein-frame variables, the conformal coupling produces an anomalous acceleration proportional to β<sub>A</sub>∇φ/M<sub>Pl</sub> acting differentially on self-gravitating bodies — precisely the channel the Nordtvedt bound constrains. LLR compatibility therefore rests on screening, not on decoupling:

- Orbital dynamics follow g̃-geodesics; in the screened solar-system regime the scalar-gradient force is suppressed through S<sub>Σ</sub>(E), so the Earth-Moon system follows effectively GR dynamics

- In the conformal-only limit, photon null cones coincide with those of g<sub>μν</sub>

- The measurement is two-way (direction-dependent effects cancel)

- The Earth-Moon system sits in a deeply screened regime where Temporal Shear vanishes continuously, suppressing any Temporal Shear signature

LLR does not directly probe the clock-sector and synchronization-structure observables that distinguish GR from TEP; it instead bounds the allowed coupling and screening regime. The proposed loop and correlation tests should be seen as complementary additions to, not replacements for, LLR and other precision tests.

### 4.7 Summary: Closed Loops That Don't Test Closure

#### What Time Dilation Tests Confirm

- Proper time depends on the worldline traversed

- Moving clocks run slow by factor γ = 1/√(1 − v²/c²)

- Clocks at higher gravitational potential run faster

- GR provides a self-consistent framework for timekeeping

#### What They Do Not Directly Probe

- Whether synchronization is path-independent

- Whether one-way light speed is isotropic

- Whether coordinate time is globally integrable

- The specific loop and correlation observables that distinguish GR from TEP

The time dilation tests confirm that time is relative—an insight. But they confirm it through proper time comparisons along reuniting worldlines, not through synchronization of permanently separated clocks. The distinction is subtle but fundamental: proper time path-dependence is universal; synchronization path-dependence is what distinguishes theories.

## 5. The Light Propagation Tests

Light propagation experiments—Shapiro delay measurements, VLBI observations, and gravitational lensing studies—probe how electromagnetic radiation traverses curved spacetime. These tests have achieved high precision in constraining the Parameterized Post-Newtonian (PPN) parameter γ. This section examines what these constraints imply.

### 5.1 The Shapiro Delay

In 1964, Irwin Shapiro predicted that radar signals passing near the Sun would experience a time delay due to spacetime curvature. The "fourth test of GR" has since been measured with increasing precision.

##### 5.1.1 The Cassini Experiment (2003)

The most precise Shapiro delay measurement used radio signals between Earth and the Cassini spacecraft during solar conjunction. The result:

$$\gamma = 1 + (2.1 \pm 2.3) \times 10^{-5}$$

This constrains the PPN parameter γ to agree with GR (γ = 1) at the 10⁻⁵ level.

**Critical Analysis:**

#### Methodological Analysis

The Cassini measurement used round-trip radio ranging: signals were transmitted from Earth to the spacecraft and transponded back. The measured quantity is the round-trip light time, not the one-way propagation time.

Round-trip measurements are inherently reciprocity-even. If the one-way speed of light differs in opposite directions (c + δ outbound, c − δ inbound), the round-trip time is:

$$t_{RT} = \frac{L}{c+\delta} + \frac{L}{c-\delta} = \frac{2Lc}{c^2 -
\delta^2} \approx \frac{2L}{c} + O\left(\frac{\delta^2}{c^2}\right)$$

The first-order direction-dependent term cancels exactly. Cassini constrains reciprocity-even effects (the PPN γ parameter) but is blind to reciprocity-odd effects (direction-dependent propagation).

##### 5.1.2 Data Reduction Concerns

The Cassini analysis required extensive corrections for systematic effects:

"Doppler calibration for Earth's troposphere... potential major error
source. We multiply y by downlink X-Band frequency... make new file for
further processing."
— Bertotti, Iess, and Tortora (2003)

The Cassini inference is conditional on an end-to-end modeling stack: a spacetime model for signal propagation in the solar system, together with tropospheric, plasma, and spacecraft dynamics models. This does not weaken the measurement; it clarifies what is being tested. The result tightly constrains γ within the assumed signal model class.

### 5.2 Very Long Baseline Interferometry (VLBI)

VLBI observations of quasars near the Sun measure the apparent deflection of radio waves by solar gravity. The deflection angle:

$$\theta = (1 + \gamma) \times \frac{2GM}{c^2 b}$$

where b is the impact parameter, has been measured to constrain γ at the 10⁻⁴ level.

**Critical Analysis:**

#### Methodological Analysis

VLBI measures the differential arrival time of radio waves at widely separated antennas. The measurement is sensitive to the angular position of the source, which shifts due to gravitational light bending.

Like Shapiro delay, this constrains the post-Newtonian gravitational/source-charge sector of the metric, but not synchronization structure; a pure conformal $A(\phi)$ does not introduce an additional null-trajectory bending term. The PPN γ parameter appears in the spatial part of the metric; synchronization holonomy arises from the temporal part.

### 5.3 The PPN Framework and Its Limitations

The Parameterized Post-Newtonian formalism provides a systematic way to compare metric theories of gravity. The PPN metric in isotropic coordinates:

$$ds^2 = -(1 - 2U + 2\beta_{\rm PPN} U^2)c^2 dt^2 + (1 + 2\gamma U)(dx^2 + dy^2 +
dz^2)$$

where U = GM/rc² is the Newtonian potential. GR predicts β<sub>PPN</sub> = γ = 1.

##### 5.3.1 What PPN Tests Constrain

- *γ:* How much space curvature is produced by unit rest mass

- *β:* How much nonlinearity exists in the superposition of gravity

- *Other parameters:* Preferred-frame effects, conservation law violations

##### 5.3.2 What PPN Tests Cannot Constrain

The PPN framework parameterizes the post-Newtonian limit under the assumption that the same effective metric governs the sector being tested (typically solar-system dynamics and light propagation). It is therefore not a complete language for theories in which additional structure appears in the matter and clock sector while remaining effectively GR-like in the photon and gravitational sectors.

- Clock transport and synchronization are governed by an effective metric distinct from the one constrained by light-propagation observables

- Convention-independent loop observables (residual holonomy after GR subtraction) are the primary targets

- Non-local statistical structure appears in clock residuals (spatial correlations) rather than in PPN light-bending parameters

- Environment-dependent screening of the scalar field gradient (Temporal Shear) is not parameterized by PPN γ, leaving the continuous transition from screened to active-shear regimes unconstrained

In TEP, the matter metric is related to the gravitational metric by a disformal map g̃<sub>μν</sub> = A²(φ)g<sub>μν</sub> + B(φ)∇<sub>μ</sub>φ∇<sub>ν</sub>φ. The conformal factor A(φ) rescales the matter sector while preserving null cones; the disformal term B(φ) can tilt null cones and is strongly constrained by multi-messenger observations. Consequently, PPN light-propagation tests can leave room for clock-sector effects encoded in A(φ) and for loop/correlation observables that are not representable as a single γ parameter.

A further limitation concerns the environmental setting of PPN tests. The Cassini measurement was performed during solar conjunction, where signals passed through the Sun's deep gravitational potential. In TEP, such deep potential wells correspond to the screened regime: suppression of Temporal Shear (∇φ → 0) by environmental state and boundary conditions suppresses the active-gradient/source-charge response $S_\Sigma$; the conformal amplitude/clock response $S_A$ is a distinct projection of the same environmental state, while the Temporal Topology persists. The PPN constraint on γ therefore probes only the screened limit, where TEP predicts PPN consistency, and remains insensitive to the active-shear dilute regime where the field gradient recovers and large-scale structural effects originate.

### 5.4 The Conformal vs. Disformal Distinction

In scalar-tensor theories, the physical matter metric can differ from the gravitational metric through conformal and disformal transformations, written in isotropic shorthand as (see §1.3 and Paper 0, §2.2 for the resolved preferred-foliation $(A_{\rm loc}, A_{\rm sp})$ decomposition):

$$\tilde{g}_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\nabla_\mu\phi
\nabla_\nu\phi$$

where A(φ) is the conformal factor and B(φ) is the disformal factor.

#### Conformal Transformations and Clock-Sector Structure

Light propagation tests constrain the combination of conformal and disformal couplings that affects null geodesics. But conformal transformations preserve the null cone structure—light rays follow the same paths regardless of A(φ).

In a two-metric framework, A(φ) can modify clock rates (and hence clock residual statistics) while remaining invisible to light-bending and Shapiro-delay measurements. By contrast, any disformal cone tilt sourced by B(φ) is directly constrained. Light-propagation tests therefore do not directly probe spatial clock-correlation structure or residual loop observables, even when they tightly bound B(φ).

### 5.5 Summary: Constraining a Different Sector

#### What Light Propagation Tests Confirm

- PPN γ = 1 to 10⁻⁵ precision

- Light bending follows GR predictions

- Shapiro delay matches GR to high accuracy

- No evidence for preferred-frame effects in light propagation

- Local Temporal Shear constraints are satisfied in the screened regime where Temporal Shear is suppressed

#### What They Cannot Test

- Conformal coupling to matter/clocks (preserves null cone)

- Synchronization structure (requires one-way measurements)

- Direction-dependent propagation (cancels in round-trip)

- Clock-sector effects and loop/correlation observables that are not reducible to PPN light-propagation parameters

- The active-shear dilute regime where Temporal Shear recovers and the scalar field becomes cosmologically active

Light propagation tests are precision triumphs that tightly constrain the post-Newtonian light-propagation sector (e.g., γ ≈ 1) and place strong limits on any disformal cone tilt (encoded in B(φ)). In TEP, these constraints probe the screened limit where Temporal Shear is continuously suppressed in screened-source or locally constrained regimes, leaving the active-shear dilute regime—where scalar-field spatial structure and large-scale effects originate—largely unexplored. These constraints are therefore orthogonal to clock-sector observables such as spatially structured correlations in clock residuals and GR-subtracted residual holonomy H<sub>resid</sub>, which require genuinely one-way, direction-reversing configurations or density-varying environments.

## 6. The Multi-Messenger Constraints

LIGO and Virgo detected gravitational waves from the binary neutron star merger GW170817, followed (+1.74 ± 0.05) s by the gamma-ray burst GRB 170817A. This timing coincidence yields an assumption-dependent constraint on the fractional speed difference between gravitational waves and light.

### 6.1 The GW170817 Event

On August 17, 2017, the LIGO-Virgo collaboration detected gravitational waves from a binary neutron star inspiral in NGC 4993, with luminosity distance of order 40 Mpc (130 million light-years; conservative bounds use the lower end of the distance interval). The Fermi Gamma-ray Burst Monitor detected GRB 170817A approximately (+1.74 ± 0.05) s after the gravitational wave signal.

##### 6.1.1 The Speed Constraint

Interpreting the observed delay as (at least partially) a propagation-time difference yields a bound on the fractional speed difference during the trip.

$$\frac{|c_\gamma - c_g|}{c} \lesssim \text{few} \times 10^{-15}$$

The numerical bounds depend on conservative assumptions about intrinsic emission-time offsets between the gravitational-wave and gamma-ray signals. In particular, the observable directly constrains differential propagation, while the emission-time difference is astrophysically model-dependent (Abbott et al. 2017).

##### 6.1.2 The Standard Interpretation

The constraint has been used to rule out or severely constrain numerous modified gravity theories, including:

- Scalar-tensor theories with disformal coupling

- Massive gravity theories

- Lorentz-violating gravity theories

- Certain dark energy models

### 6.2 What GW170817 Constrains

**Critical Analysis:**

#### The Single-Path Problem

Both gravitational waves and gamma rays traveled the same path through spacetime from NGC 4993 to Earth. Any effect that modifies propagation along this path affects both signals identically.

The measurement constrains the differential propagation speed—how much faster or slower gravitational waves travel compared to light along the same path. It does not constrain effects that modify both signals equally.

##### 6.2.1 Conformal Transformations and Single-Path Cancellation

Understanding why GW170817 does not directly constrain TEP's conformal sector requires examining the two-metric structure explicitly. The following conformal derivation presents the canonical isotropic shorthand for the common-mode cancellation; the operative Paper 0 (v0.16) realization is the resolved $(A_{\rm loc}, A_{\rm sp})$ preferred-foliation tensor branch (§2.1, §6.3.1), on which electromagnetic waves and transverse-traceless gravitons share the identical principal wave operator $\Box_{\rm TEP} \equiv -(N_{\rm TEP}c)^{-2}\partial_t^2 + A_{\rm sp}^{-2}(t,\mathbf{x})\gamma^{kl}\partial_k\partial_l$ at $B = 0$, enforcing $c_{\rm GW}(t,\mathbf{x}) = c_\gamma(t,\mathbf{x})$ at every spacetime event.

#### The Two-Metric Structure (Isotropic Conformal–Disformal Shorthand and Resolved Foliation)

In the isotropic conformal–disformal shorthand of TEP, the two metrics take the form:

Gravitational metric (Einstein frame):

$$g_{\mu\nu}^{\text{GW}} = g_{\mu\nu}$$
In the canonical Einstein-frame shorthand, gravitational waves propagate on null geodesics of $g_{\mu\nu}$.

Electromagnetic / matter metric (Jordan frame):

$$g_{\mu\nu}^{\text{EM}} = A^2(\phi) g_{\mu\nu}^{\text{GW}} + B(\phi) \nabla_\mu\phi \nabla_\nu\phi$$
Photons propagate on null geodesics of $\tilde{g}_{\mu\nu}$. In the resolved preferred-foliation decomposition of Paper 0 (§2.2, Step 80), $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav})U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift})\Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$, where both photons and TT gravitons propagate on the shared temporal-lapse cone with local coordinate speed $c_{\rm eff}(t,\mathbf{x}) = c\,N_{\rm TEP}(\mathbf{x})/A_{\rm sp}(t,\mathbf{x})$ when $B(\phi)=0$.

The conformal factor A(φ) = exp(β<sub>A</sub>φ/M<sub>Pl</sub>) rescales the clock rate uniformly, while the disformal factor B(φ) introduces direction-dependent light-cone modifications between the two sectors.

##### 6.2.2 The Flight Time Calculation

Consider a signal (gravitational wave or photon) traveling from source S to detector D along a null geodesic. The coordinate flight time is:

$$T = \int_S^D dt = \int_S^D \frac{dl}{c_{\text{eff}}}$$

where c<sub>eff</sub> is the effective propagation speed in the relevant metric.

**Critical Analysis:**

#### Gravitational Wave Flight Time

For gravitational waves propagating on g<sub>μν</sub><sup>GW</sup>:

$$T_{\text{GW}} = \int_S^D \frac{dl}{c}$$
where c is the coordinate speed of light in the Einstein frame.

#### Photon Flight Time (Conformal Only, B = 0)

For photons propagating on $g_{\mu\nu}^{\text{EM}} = A^2(\phi) g_{\mu\nu}^{\text{GW}}$:

$$T_\gamma = \int_S^D \frac{A(\phi) \, dl}{c \times A(\phi)} = \int_S^D \frac{dl}{c} = T_{\text{GW}}$$
The conformal factor A(φ) cancels in the differential time-of-flight observable under specific operational assumptions:

- Co-spatial Endpoints: Source S and Detector D are identical for both signals.

- Same Clock Standards: The flight time is measured using the same matter-sector clock definition at the detector.

- Null Path Invariance: Conformal transformations preserve the null character of geodesics ($ds^2 = 0 \iff A^2(\phi)\,ds^2 = 0$).

- Emission-Time Nuisance: Intrinsic emission delays are treated as nuisance parameters to be modeled or marginalized.

Under these conditions, the observed EM–GW arrival-time difference is insensitive to purely conformal rescalings of the metric along the shared path. The empirical constraint therefore applies to differential propagation effects (for example, disformal cone tilts), not to common-mode rescalings.

#### Why Conformal Coupling Cancels

Under a conformal transformation g̃<sub>μν</sub> = A²(φ)g<sub>μν</sub>:

- Null geodesics remain null: If g<sub>μν</sub>k<sup>μ</sup>k<sup>ν</sup> = 0, then g̃<sub>μν</sub>k<sup>μ</sup>k<sup>ν</sup> = A²(φ) × 0 = 0

- Proper length scales: dl̃ = A(φ) dl

- Proper time scales: dτ̃ = A(φ) dτ

- Physical speed invariant: c̃ = dl̃/dτ̃ = c

This cancellation concerns the differential arrival-time observable for two signals that follow the same null trajectory and are time-stamped by the same detector clock standard. In that configuration, a purely conformal rescaling does not generate a measurable EM–GW time-of-flight difference.

##### 6.2.3 The Disformal Contribution

The disformal term B(φ)∇<sub>μ</sub>φ∇<sub>ν</sub>φ does modify the relative propagation:

$$c_\gamma^2 = c^2 \left[1 - \frac{B(\phi)}{A^2(\phi)}(\partial\phi)^2\right] \quad \text{(schematic parameterization)}$$

This introduces a genuine speed difference:

$$\Delta T = T_\gamma - T_{\text{GW}} \approx \frac{L}{2c} \frac{B(\phi)}{A^2(\phi)}(\partial\phi)^2$$

GW170817's observed coincidence (after accounting for astrophysical emission-time uncertainties) implies that any net differential propagation delay accumulated over the propagation distance must be small. Operationally, the observation bounds the fractional speed difference at the few × 10<sup>−15</sup> level under standard emission-time assumptions.

This is a real constraint on the disformal sector. It does not directly constrain the conformal factor A(φ) alone.

#### Single-Path Cancellation: Summary

| Sector / Channel | Effect on Observable | GW170817 Constraint |
|---|---|---|
| Conformal A(φ): Arrival Timing | Cancels identically ($T_\gamma = T_{\text{GW}}$ along shared null path) | Unconstrained by differential arrival time |
| Conformal A(φ): Luminosity Distance | Rescales distance ratio $d_L^{\text{GW}}/d_L^{\text{EM}} \sim A(z)$ at propagation level | Constrained in principle via standard sirens; $|A(z) - 1| \sim 0.01$ at $z \approx 0.01$ sits well within $\sim 15\%$ observational distance uncertainty |
| Disformal B(φ): Relative Speed | Modifies relative propagation speed via directional cone deformation | $|c_\gamma - c_g|/c$ bounded at few $\times 10^{-15}$ (emission-model dependent) |

In a two-metric framework, the conformal factor A(φ) can modify matter-sector proper-time standards and thereby support spatial structure in clock residuals, while remaining invisible to single-path EM–GW arrival-time comparisons. However, the conformal sector enters multi-messenger astrophysics through the distance channel: standard-siren measurements probe the luminosity-distance ratio $d_L^{\text{GW}}/d_L^{\text{EM}} \sim A(z)$. For GW170817 at $z \approx 0.0099$, the predicted conformal excursion $|A(z) - 1| \sim 0.01$ is small compared to the $\sim 15\%$ observational distance uncertainty, leaving the conformal sector unconstrained by existing single-event siren data while establishing an explicit target for future multi-messenger ensembles. By contrast, any disformal cone tilt sourced by B(φ) produces a differential propagation effect directly constrained by the GW170817 arrival-time coincidence. A residual synchronization holonomy beyond modeled GR loop effects (H<sub>resid</sub>) vanishes in the conformal-only limit (B = 0) and requires non-exact structure such as B(φ) ≠ 0 or more general non-metricity.

##### 6.2.4 The Common-Mode Cancellation

GW170817-type constraints compare two signals that traverse essentially the same spacetime path and are time-stamped by clocks in the matter sector at the detector. Any common-mode rescaling of the local proper-time standard cannot be isolated by an EM–GW arrival-time comparison along a single path. The observable is therefore primarily sensitive to differential propagation effects, such as a disformal cone tilt.

### 6.3 What Would Constrain Clock-Sector Structure?

To constrain clock-sector structure (including spatial correlations) and to test for non-exact time transport beyond modeled GR loop effects, one needs measurements that do not reduce to single-path, common-mode comparisons:

##### 6.3.1 Multipath Configurations

Gravitational lensing produces multiple images that traverse different paths through the scalar field. Under isotropic conformal rescaling $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$, where spatial rulers stretch by $A^2(\phi)$ alongside clocks, $A^2(\phi(x))$ cancels out of $d\tilde s^2=0$ at every intermediate point along the ray (Paper 19, Step 56), leaving only the Planck-suppressed scalar stress tensor ($\kappa_\phi\simeq6.5\times10^{-6}$ on the selected two-branch sector; Paper 19, Step 64). Under the unstretched-space temporal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) — where the $(3+1)$ ADM spacetime lapse rescales as $N_{\rm TEP}(x)=A(\phi(x))N_{\rm GR}(x)$ while the spatial 3-metric $\gamma_{ij}$ remains unstretched — intermediate clock rates between source and observer do not cancel: every point $x$ along the ray contributes its local temporal depth $-\beta_A u_{\rm grav}(x)$ directly to the refractive index $n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)$, reproducing the non-baryonic Fermat delay and arrival order of SN Refsdal ($+330.9$ d sheet, $+221.6$–$443.2$ d 3-D ray; Paper 19, Step 64) and the Bullet Cluster galaxy-peak convergence (Paper 4, Step 02) from baryons and the selected two-branch action $P_{,X}=k\sqrt\xi+2\xi$, while preserving $c_{\rm GW}=c_\gamma$ identically on the shared temporal-lapse cone.

Under the unstretched-space temporal metric, TEP's multipath signature combines baryon-sourced intermediate temporal refraction ($\Phi_{\rm lens} = \Phi_N + \frac{1}{2}c^2\beta_A u_{\rm grav}$) with a testable post-Newtonian slip between dynamical and lensing mass inferences ($M_{\rm dyn} > M_{\rm lens}^{\rm u.s.}$ in unscreened low-acceleration environments where $\gamma_{\rm SMP} = 1/(1+2\beta_A^2 y_{\rm eff}) &lt; 1$; Paper 0 Step 80, Paper 19).

##### 6.3.2 Statistical Correlations

If the scalar field has spatial structure ⟨φ(x)φ(x')⟩ ~ exp(−r/λ), clocks at different locations will exhibit correlated fluctuations. This is the signature suggested by exploratory GNSS analysis—distance-structured correlations in clock behavior.

Single-path measurements like GW170817 do not directly probe this structure because they sample only one realization of the field along one path. The proposed spatial-correlation and loop tests should be seen as complementary additions to, not replacements for, multi-messenger constraints.

##### 6.3.3 Closed-Loop Holonomy

The experimentally relevant quantity is a GR-subtracted residual holonomy H<sub>resid</sub>, defined from one-way time transfer around a direction-reversing closed loop after subtracting modeled GR loop effects (Sagnac, Shapiro, and gravito-magnetic contributions). In the conformal-only limit (B = 0), the A(φ) contribution is an exact gradient and yields H<sub>resid</sub> = 0 on simply connected domains; a nonzero residual requires non-exact structure, for example a disformal coupling B(φ) ≠ 0 or more general non-metricity.

No single-path multi-messenger observation can probe H<sub>resid</sub> because the signals travel from source to detector, not around direction-reversing closed loops.

### 6.4 The Broader Multi-Messenger Context

##### 6.4.1 Future Events

Additional multi-messenger detections will improve statistics but not change the fundamental limitation: single-path measurements constrain differential propagation, not common-mode effects.

##### 6.4.2 Strongly Lensed Events

A gravitationally lensed multi-messenger event would provide genuinely new information. If gravitational waves and electromagnetic signals from the same source arrive via different lensed paths, their relative timing probes the scalar field structure along different trajectories.

Such events are rare but would be transformative for constraining conformal coupling.

### 6.5 Summary: The Disformal Constraint

#### What GW170817 Confirms

- Under standard emission-time assumptions, the GW–EM speed difference is bounded at the few × 10⁻¹⁵ level

- Disformal coupling B(φ) is tightly constrained

- Lorentz violation in the gravity sector is bounded

- Massive graviton scenarios are excluded or constrained

#### What It Cannot Test

- Conformal coupling A(φ) in arrival timing (preserves null cone and cancels in $T_\gamma - T_{\rm GW}$; constrained only through distance ratio $d_L^{\rm GW}/d_L^{\rm EM}$)

- Common-mode time dilation along the path

- Spatial structure of scalar fields

- Synchronization holonomy (requires closed loops)

- Clock-sector correlation structure or residual loop observables (H<sub>resid</sub>)

GW170817 is a landmark observation that opened the era of multi-messenger astronomy. Its constraints on modified gravity are real and important—for the disformal sector. Single-path EM–GW coincidence measurements primarily constrain differential propagation (for example, disformal cone tilts) and do not directly probe clock-sector correlation structure or GR-subtracted residual holonomy H<sub>resid</sub>, which require multipath or direction-reversing closed-loop configurations. This is a statement about measurement geometry, not a criticism of the experimental achievement.

## 7. The Resonator Tests

Optical and microwave resonator experiments—descendants of the Michelson-Morley experiment—have achieved high precision in testing Lorentz invariance. Modern cavity experiments constrain anisotropy in the speed of light at the 10<sup>−18</sup> level. This section examines the logical structure of these null results.

### 7.1 The Michelson-Morley Experiment (1887)

The Michelson-Morley experiment sought to detect Earth's motion through the luminiferous aether by measuring the difference in light travel time along perpendicular arms of an interferometer. The null result—no fringe shift as the apparatus rotated—is celebrated as disproving the aether and motivating special relativity.

**Critical Analysis:**

#### Methodological Analysis

The interferometer measures the round-trip travel time difference between two perpendicular paths. Light travels out and back along each arm, and the interference pattern depends on the path length difference.

This is a two-way, closed-path measurement. If the one-way speed of light is c + v in one direction and c − v in the opposite direction, the round-trip time is:

$$t = \frac{L}{c+v} + \frac{L}{c-v} = \frac{2Lc}{c^2 - v^2}$$
The first-order direction-dependent term cancels. The experiment is sensitive only to second-order effects (v²/c²), which special relativity predicts to be exactly compensated by length contraction.

##### 7.1.1 What Michelson-Morley Tests

The null result confirms that the two-way speed of light is isotropic to high precision. It does not test whether the one-way speed is isotropic—that would require synchronized clocks at the endpoints, introducing the synchronization convention problem.

#### The Two-Way Isotropy Theorem

Any theory in which the round-trip speed of light is c (regardless of direction) will produce a null result in Michelson-Morley type experiments. This includes theories where the one-way speed is anisotropic but the anisotropy cancels in round trips.

### 7.2 The Kennedy-Thorndike Experiment (1932)

Kennedy and Thorndike modified the Michelson-Morley setup to have unequal arm lengths, testing whether the interference pattern changed as Earth's velocity varied over the year. The null result constrains time dilation effects.

**Critical Analysis:**

#### Methodological Analysis

Like Michelson-Morley, this is a two-way measurement. The unequal arms introduce sensitivity to time dilation (how clock rates depend on velocity), but the measurement remains a comparison of round-trip times.

The null result confirms that time dilation and length contraction combine consistently—the Lorentz transformation is self-consistent. It does not probe synchronization structure.

### 7.3 Modern Cavity Experiments

Contemporary experiments use cryogenic optical or microwave cavities to achieve high precision. The resonance frequency of a cavity depends on its length and the speed of light:

$$f = \frac{nc}{2L}$$

where n is the mode number. By comparing cavities oriented in different directions, or monitoring a single cavity as Earth rotates, anisotropy in c can be constrained.

##### 7.3.1 Precision Achievements

Modern experiments have achieved constraints on Lorentz violation at high levels:

| Experiment | Year | Constraint on Δc/c |
|---|---|---|
| Brillet & Hall | 1979 | 10⁻¹⁵ |
| Müller et al. | 2003 | 10⁻¹⁵ |
| Herrmann et al. | 2009 | 10⁻¹⁷ |
| Nagel et al. | 2015 | 10⁻¹⁸ |

##### 7.3.2 The Closed-Path Limitation

**Critical Analysis:**

#### What Cavities Measure

A cavity resonance involves light bouncing back and forth between mirrors. The resonance condition f = nc/(2L) averages the forward and backward propagation speeds:

$$f = \frac{n}{L/c_{\text{forward}} + L/c_{\text{backward}}} = \frac{n \times c_{\text{forward}} \times c_{\text{backward}}}{L(c_{\text{forward}} + c_{\text{backward}})}$$
If c<sub>forward</sub> = c + δ and c<sub>backward</sub> = c − δ, then:

$$f \approx \frac{nc}{2L} \times \left[1 - \frac{\delta^2}{c^2}\right]$$
The first-order anisotropy cancels. Cavity experiments constrain second-order (and higher) Lorentz violation but are blind to first-order direction-dependent effects.

### 7.4 The Standard Model Extension (SME) Framework

Modern Lorentz violation tests are interpreted within the Standard Model Extension, which parameterizes all possible Lorentz-violating terms in the Standard Model Lagrangian. Cavity experiments constrain specific SME coefficients.

##### 7.4.1 What SME Tests Constrain

- Anisotropy in the two-way speed of light

- Preferred-frame effects in electromagnetism

- CPT violation in photon propagation

##### 7.4.2 What SME Tests Cannot Constrain

- One-way speed anisotropy (convention-dependent)

- Synchronization structure (requires one-way measurements)

- Effects that preserve two-way isotropy while violating one-way isotropy

### 7.5 The Sagnac Effect: A Partial Exception

The Sagnac effect—the phase shift in a rotating interferometer—is sometimes cited as measuring one-way light speed. In a rotating ring interferometer, counter-propagating beams accumulate different phases:

$$\Delta\Phi = \frac{8\pi A\Omega}{\lambda c}$$

where A is the enclosed area and Ω is the rotation rate.

**Critical Analysis:**

#### The Sagnac Subtlety

The Sagnac effect does involve direction-dependent propagation, but in a rotating frame. The effect is fully explained by the non-inertial nature of the rotating reference frame—it does not probe one-way light speed in an inertial frame.

More importantly, the Sagnac interferometer is still a closed-path device. Both beams traverse the same closed loop (in opposite directions) and recombine at the same point. The measurement is of the path integral ∮ v·dl around the loop, not of one-way propagation.

### 7.6 Summary: Isotropy of What, Exactly?

#### What Resonator Tests Confirm

- Two-way speed of light is isotropic to 10⁻¹⁸

- No preferred frame in electromagnetic phenomena (two-way)

- Lorentz invariance holds for closed-path measurements

- SME coefficients for photon sector are tightly bounded

#### What They Cannot Test

- One-way speed isotropy (convention-dependent)

- Synchronization structure (requires separated clocks)

- Effects that cancel in round trips

- Conformal-limit TEP predictions (which preserve two-way isotropy)

The resonator tests confirm that the laws of electromagnetism are the same in all directions—for round-trip measurements. This is a constraint on Lorentz violation. But it is not a constraint on synchronization structure, which requires genuinely one-way measurements that resonator experiments cannot provide.

Purely conformal TEP preserves two-way photon isotropy. The disformal sector is different. For B ≠ 0, the photon phase speed contains an even-parity orientation dependence proportional at leading order to (B/A²)(n̂·∇φ)². This contribution is unchanged under n̂ → -n̂ and therefore survives round-trip propagation. Rotating optical-cavity Michelson-Morley/Kennedy-Thorndike experiments consequently constrain the orientation-dependent component of the local disformal deformation — specifically the component of ∇φ whose projection onto the cavity axis varies over sidereal and annual cycles. A strictly radial terrestrial gradient produces a constant offset rather than a rotational modulation, so the leading constrained contributions are the solar and galactic gradients at Earth. They remain insensitive, however, to the distinct odd/non-exact synchronization holonomy H<sub>resid</sub> targeted by direction-reversing closed-loop clock experiments.

## 8. What Would Actually Discriminate?

The preceding sections have established that canonical precision tests of GR share common methodological limitations: two-way measurements, local comparisons, single-path configurations, and theory-laden data reduction. This section proposes experiments that could genuinely discriminate between GR and TEP.

The following proposals are ordered by increasing baseline length and technological ambition, from terrestrial triangles to interplanetary loops.

### 8.1 Notation Summary

The discriminating tests that follow involve several key quantities from the TEP framework. For clarity, the primary notation used throughout this paper is summarized below.

#### Key TEP Quantities

| Symbol | Definition | Role in Discrimination |
|---|---|---|
| $C_A(x,x')$ | Covariance of conformal-factor fluctuations: $\langle \delta\ln A(x) \, \delta\ln A(x') \rangle$ | Characterizes spatial correlations in clock residuals; measured through GNSS and optical clock networks |
| $\lambda_T$ | Temporal Topology correlation length (characteristic scale of $C_A$ decay with distance) | Empirical span across GNSS constellation products: $10^3$–$10^4$ km (e.g. $L_c \approx 4{,}200$ km in GPS, $1{,}862 \pm 112$ km in MGEX); empirical span bounds cross-network consistency outside 500–20,000 km |
| $\kappa_X$ | Observable response coefficient for measurement channel $X$ | Relates field fluctuations to observable residuals; distinguishes channel-specific sensitivity from microscopic coupling |

The correlation function $C_A$ and its characteristic scale $\lambda_T$ are central to conformal-sector predictions. The response coefficients $\kappa_X$ parameterize how each experimental channel (clocks, interferometers, propagation) couples to the underlying field structure.

### 8.2 The Triangle Holonomy Test

The most direct test of synchronization integrability is the measurement of holonomy around a closed loop using one-way signals. This test must explicitly distinguish TEP holonomy from the well-known Sagnac effect.

##### 8.2.1 Formal Definition: The Loop Observable

The loop observable $H$ is defined operationally as a concrete estimator constructed from time-tagged data. Consider three stations $i \in \{A, B, C\}$ exchanging optical pulses.

#### The Estimator

Let $\tau_i(t)$ be the proper time reading of clock $i$ at coordinate time $t$. A pulse emitted by $A$ at $\tau_A^e$ is received by $B$ at $\tau_B^r$. The raw one-way interval is $\Delta_{AB} = \tau_B^r - \tau_A^e$.

The Raw Loop Holonomy is the directional difference of the sum of intervals around the loop:

$$H_{\rm raw} \equiv \oint_{\circlearrowright} \Delta_{ij} - \oint_{\circlearrowleft} \Delta_{ji} = (\Delta_{AB} + \Delta_{BC} + \Delta_{CA}) - (\Delta_{AC} + \Delta_{CB} + \Delta_{BA})$$

**Critical Analysis:**

#### Decomposition, Invariance, and Clock Drift

1. Invariance to Constant Clock Offsets: Let each clock have a constant offset $\delta_i$ from coordinate time: $\tau_i(t) = t + \delta_i$. The term $\delta_i$ appears in one emission and one reception term in each direction. In $\circlearrowright$: $(\delta_B - \delta_A) + (\delta_C - \delta_B) + (\delta_A - \delta_C) = 0$. 
Thus, $H_{\rm raw}$ is strictly invariant to constant clock synchronization errors.

2. Clock Drift and Rate Variations: If clocks exhibit fractional frequency drifts or linear rates $\dot{\delta}_i \equiv d_i$, the uncancelled residual depends on link asymmetry and loop transit time $T_{\rm loop} \approx 0.5$ s. Over characteristic link durations $\Delta t_{\rm link} \sim 0.05$–$0.1$ s, optical lattice clocks with fractional instability $d \sim 10^{-18}$ contribute an uncancelled drift residual $H_{\rm drift} \sim d \cdot \Delta t_{\rm link} \sim 0.1$–$0.5$ as.

3. Decomposition: The observable decomposes into physical, systematic, and drift terms:

$$H_{\rm raw} = H_{\text{Sagnac}} + H_{\text{Shapiro}} + H_{\text{GM}} + H_{\text{atm}} + H_{\text{inst}} + H_{\text{drift}} + H_{\rm TEP}$$

- $H_{\text{Sagnac}} \propto \mathbf{\Omega} \cdot \mathbf{A}$: Earth rotation (dominant, $\sim 100$ ns).

- $H_{\text{Shapiro}} \propto \sum GM/r$: Gravitational delay (cancels if path $AB$ same as $BA$, but nonzero if spacecraft motion makes paths differ).

- $H_{\text{GM}}$: Gravito-magnetic/Lense-Thirring effects ($\sim 10^{-17}$ s).

- $H_{\text{atm}}$: Non-reciprocal atmospheric delays (requires dual-frequency/radiometry).

- $H_{\text{inst}}$: Non-reciprocal instrumental delays (fiber loops, transponders).

- $H_{\text{drift}}$: Uncancelled clock frequency drift during the loop transit ($\sim 0.1$–$0.5$ as for $10^{-18}$ stability).

- $H_{\rm TEP}$: The signal of interest (residual synchronization holonomy).

The experimental challenge is to model or control the non-TEP terms to isolate the residual synchronization holonomy.

#### Measurement Recipe

- Record emission and reception time tags for each directed link (A→B, B→C, C→A and reverse)

- Compute $H_{\rm raw}$ from tags using the estimator definition above

- Model $H_{\rm GR}$ using ephemerides, Earth-orientation parameters, and propagation models

- Compute $H_{\rm resid} = H_{\rm raw} - H_{\rm GR}$

- Repeat across geometries chosen to hold Sagnac projection fixed while varying field-gradient sensitivity

- Report residual scaling with geometry; publish nuisance-model sensitivity

#### GR Prediction

In GR, after subtracting the modeled loop contributions H<sub>GR</sub>, the residual satisfies H<sub>resid</sub> = 0 (up to measurement uncertainty).

#### TEP Prediction

In the conformal-only limit (B = 0), the A(φ) contribution to time transport is an exact gradient and yields H<sub>resid</sub> = 0 once GR loop effects are subtracted. A nonzero residual holonomy requires non-exact structure, for example disformal coupling B(φ) ≠ 0 or more general non-metricity. In that case, H<sub>resid</sub> can be nonzero with a magnitude that depends on loop geometry and field gradients.

##### 8.2.2 Distinguishing TEP Holonomy from Sagnac Effect

A skeptic will immediately note that any closed loop on a rotating Earth measures the Sagnac effect ($\Delta t = 4A\Omega/c^2$). The triangle holonomy test must explicitly distinguish TEP effects from Sagnac rotation.

**Critical Analysis:**

#### Geometric vs Rotational Signatures

Sagnac Effect:

- Area-dependent: $\Delta t_{Sagnac} = 4A\Omega/c^2$ where A is the enclosed area

- Rotation-axis dependent: Proportional to Earth's rotation rate $\Omega$

- Direction: Fixed sign determined by right-hand rule with rotation axis

- Magnitude: Order picoseconds for terrestrial-scale triangles (configuration-dependent)

Residual Holonomy (TEP target):

- Area/path-dependent: The leading synchronization kernel is $\delta\tilde{\sigma}_\mu \simeq - \frac{B M_{\rm Pl}^2}{A^2\beta_A^2} (u\cdot\Sigma)P_\mu^\nu\Sigma_\nu$ (depends on loop flux, observer motion, and path geometry)

- Field-gradient dependent: Proportional to scalar field gradient and disformal coupling

- Geometry: Optimized by large vertical or orbital loops sampling field gradients

- Magnitude: The repaired conditional benchmark (Paper 0, Appendix E, R6) is $|H_{\rm resid}| \approx 4.33\times10^{-13}$ s per loop ($\sim 1.44\times10^{-12}$ fractional over a 0.3 s loop) at the stated reduced normalization $B_0 = +1$; the $10^{-18}$–$10^{-16}$ fractional window is the design sensitivity target and the common-action normalization remains to be derived

The numerical values quoted below are order-of-magnitude forecasts intended to clarify the experimental scale. In practice, the effective Sagnac term and the achievable subtraction residual depend on loop geometry, ephemerides and Earth-orientation modeling, link non-reciprocity control, and the time-transfer calibration strategy.

##### 8.2.3 The Large-Area Strategy

Contrary to "zero-area" approaches that eliminate signal along with noise, the original TEP framework (Paper 0) emphasizes that the holonomy $H$ scales with the loop area (flux of the time-transport curvature). To maximize the signal-to-noise ratio, a Large-Area MEO Strategy is proposed:

#### Ground-Satellite-Ground Triangle

Configure stations A and B on the ground separated by 1000–3000 km, with a Medium Earth Orbit (MEO) satellite S completing the triangle ($A \to S \to B \to A$):

- Large Area: The loop extends to ~20,000 km altitude, enclosing a vast area to maximize TEP flux.

- Sagnac Modeling: The large Sagnac term is modeled and subtracted using precise IERS Earth orientation parameters (EOP) and GNSS/SLR ephemerides.

- TEP Discriminator: A non-zero residual $H_{\rm resid}$ after Sagnac subtraction.

- Implementation: Two-way optical time transfer (TWTTFT) links combined with one-way analysis.

This configuration maximizes sensitivity to the disformal coupling $B(\phi)$. The canonical admissible-branch prediction (Paper 0, Appendix E, R6; schematic Earth triangle) is $|H_{\rm resid}| \approx 4.33\times10^{-13}$ s per loop ($\sim 1.44\times10^{-12}$ fractional) at the stated reduced normalization $B_0 = +1$. It is a strong forward transport benchmark; the common-action normalization remains to be derived before treating it as an absolute prediction.

#### Numerical Predictions for MEO Configuration

For a ground-MEO triangle (effective loop duration $T_{loop} \approx 0.4$–0.5 s, including both directions in $H_{raw}$):

| Parameter | Value | Derivation |
|---|---|---|
| Loop time $T_{loop}$ | 0.5 s | Total path length ~120,000–150,000 km (six one-way links in $H_{raw}$) |
| Sagnac contribution | Order 10–100 ns | Large enclosed area; dominant background (configuration-dependent) |
| Residual holonomy $H_{\rm resid}$ | $\approx 4.33\times10^{5}$ as at $B_0 = +1$ | Paper 0 Appendix E, R6: repaired conditional $|H_{\rm resid}| \approx 4.33\times10^{-13}$ s per loop at the stated reduced normalization; model- and geometry-dependent |
| Required clock stability | $10^{-18}$ | State-of-the-art optical lattice clocks |
| Detection threshold | Order 102 as | Single-loop absolute threshold set by subtraction residuals and link non-reciprocity (configuration-dependent) |

Feasibility status: At the stated reduced normalization ($B_0 = +1$), the repaired conditional result $|H_{\rm resid}| \approx 4.33\times10^{-13}$ s $\approx 4.33\times10^{5}$ as lies above the current single-loop subtraction residual ($\sim 10^2$ as). A measurement or null first tests this reduced transport benchmark and its normalization; the $0.5$–$50$ as sensitivity band remains useful for the common-action completion.

##### 8.2.4 Explicit Error Budget

The experimental design relies on precise modeling to isolate the TEP residual. The Sagnac effect poses the most significant challenge:

#### Sagnac Modeling and Subtraction

| Source | Contribution | Uncertainty | Subtraction Method |
|---|---|---|---|
| Earth rotation (IERS) | ~100 ns | Order 102 as (after modeling) | Precise ephemerides + IERS EOP |
| Polar motion | 0.5 ns | Order 102–103 as (after modeling) | IERS polar motion parameters |
| Geodetic effects | 0.1 ns | Order 102–103 as (after modeling) | ITRF reference frame |
| Clock drift & instability | 0.1–0.5 as | ~0.1 as | Continuous optical lattice frequency tracking ($10^{-18}$ stability) |
| TEP signal | $\approx 6\times10^{5}$ as at $B_0 = +1$ (linear in $|B_0|$) | Signal of Interest | Single-loop subtraction suffices at canonical normalization |

Challenge: Absolute Sagnac subtraction is ultimately limited by geodetic and dynamical modeling (satellite orbits, Earth rotation, and link calibration). Detecting sub-femtosecond residuals therefore requires differential strategies (comparing loops with different field-gradient sensitivities but similar Sagnac projections) rather than relying solely on absolute subtraction. Feasibility depends on achieving the required link precision and systematic control.

##### 8.2.5 Multiple Geometries for Cross-Validation

To further distinguish TEP from Sagnac effects, use multiple triangle configurations:

**Validation:**

#### Geometry Discrimination Strategy

- Large-area triangle: Maximizes TEP signal (flux of time-transport curvature).

- Variable-area comparison: Signal should scale with loop area (unlike clock systematics).

- Opposite-orientation triangles: Sagnac changes sign; TEP holonomy sign depends on field gradient topology.

- Altitude variation: TEP depends on vertical field gradients ($\nabla\phi$); Sagnac depends only on rotation axis projection.

Consistency across geometries provides robust discrimination between residual holonomy $H_{\rm resid}$ and Sagnac rotation effects.

##### 8.2.6 Technical Requirements

| Parameter | Requirement | Current Technology |
|---|---|---|
| Clock stability | 10⁻¹⁸ over measurement period | Achievable with optical lattice clocks |
| One-way link precision | Femtosecond (averaging to as) | Optical links + carrier phase analysis |
| Baseline length | Large Area (MEO loop) | Ground-Satellite-Ground configuration |
| Atmospheric correction | Must not assume GR propagation | Requires independent calibration |

##### 8.2.7 The Critical Innovation

The key requirement is genuinely one-way links. Current time transfer methods (two-way satellite time transfer, GPS common-view) use round-trip or common-mode techniques that cancel holonomy by construction.

One-way optical links between ground stations and satellites (or between satellites) could provide the required configuration. The European Space Agency's ACES mission includes a one-way link capability that could, in principle, be used for holonomy measurements—though this is not part of the planned science program.

##### 8.2.8 Objections and Replies

**Critical Analysis:**

#### Common Objections

- Objection: One-way times are convention-dependent; the observable is not physical.

- Objection: Any loop timing signal is just the Sagnac effect (Earth rotation).

- Objection: Atmospheric and link delays dominate the femtosecond/attosecond signal.

- Objection: Subtracting $H_{GR}$ assumes GR and makes the test circular.

#### Replies (Operational and Falsifiable)

- Reply: The loop observable $H_{raw}$ is constructed from recorded emission time tags and reception clock readouts; constant clock offsets cancel exactly in the direction-reversed loop difference.

- Reply: Sagnac contributions are geometry-controlled. Using precise IERS Earth-orientation parameters and satellite ephemerides, the modeled Sagnac term can be subtracted to the level set by geodetic uncertainty. The key discriminator is a residual with the predicted geometry dependence, testable across multiple loop configurations.

- Reply: Link systematics are handled by two-way optical time transfer (TWTTFT) for synchronization and one-way analysis, plus dual-frequency links. The signal scales with loop area (flux), while many systematics scale with path length or are geometry-independent.

- Reply: $H_{GR}$ subtraction is treated as a modeled calibration. The discriminator is whether a *residual* remains that scales with field gradients (TEP) rather than rotation (Sagnac).

##### 8.2.9 Formal Note: Why Conformal-Only Models Give H<sub>resid</sub> = 0

#### Exactness and Residual Holonomy

In the theory formulation, time transport along a link can be represented by an effective one-form Ω. In the conformal-only limit (B = 0), Ω reduces to an exact gradient term, Ω ∝ d(ln A)/2, whose curl vanishes. After subtracting modeled GR loop effects (Sagnac, Shapiro, gravito-magnetic terms), the residual loop integral therefore satisfies H<sub>resid</sub> = ∮Ω = 0 on simply connected domains. A nonzero residual requires a non-exact contribution, for example a disformal term B(φ) ≠ 0 or more general non-metricity.

Throughout the paper, references to H<sub>resid</sub> as a discriminator should therefore be understood as targeting disformal or non-metric structure in the clock sector, not the purely conformal limit.

##### 8.2.10 The Conformal-Only Limit and TEP Testability

The preceding analysis raises a critical question: if conformal-only TEP (B = 0) predicts H<sub>resid</sub> = 0 after GR subtraction—identical to GR—how can the triangle holonomy test discriminate between the theories?

**Critical Analysis:**

#### Resolution: Multi-Signature Discrimination Strategy

The holonomy test primarily constrains disformal coupling B(φ), while GNSS and clock-correlation observables provide sensitivity to conformal-sector structure via A(φ). TEP remains testable in the conformal-only limit through complementary observables:

The following signatures are currently best viewed as hypotheses motivated by preliminary analyses rather than established empirical facts:

- GNSS Distance Correlations: Exponential-like distance structure in clock residual correlations on continental-to-global baselines (order 10<sup>3</sup>–10<sup>4</sup> km) has been reported in GNSS clock products. This claim requires independent replication using raw data and explicit systematic controls.

- Orbital Velocity Coupling: A correlation between GNSS-derived metrics and Earth's orbital dynamics has been reported. Establishing whether this reflects physics or analysis artifacts requires independent replication with pre-specified endpoints and null tests.

- Annual-Phase Direction: GNSS clock analyses recover a direction near Earth's aphelion velocity tangent, while the directed CMB-apex template is anti-phase and the Solar Apex is null. Independent, blinded replication should test this signed annual phase rather than undirected proximity to the CMB axis.

- Multi-Constellation Consistency: If the signal is physical (conformal coupling to matter), it must be present across GPS, GLONASS, Galileo, and BeiDou with the same covariance morphology, while product- and estimator-dependent scale variation is expected (Paper 14). Systematic artifacts would instead show constellation-dependent structure.

The triangle holonomy test serves a different purpose: it constrains or detects *disformal* coupling B(φ). GW170817 bounds the fractional GW–EM speed difference |c<sub>g</sub> − c<sub>γ</sub>|/c at the few × 10<sup>−15</sup> level under standard emission-time assumptions, strongly constraining disformal cone tilts in many scalar-tensor models (e.g., Ezquiaga & Zumalacárregui 2017). Clock-sector disformal coupling could differ and remain detectable at the attosecond-level residuals targeted in the table in §8.2.3. These residual levels should be regarded as indicative feasibility thresholds rather than finalized performance specifications.

#### Testability Hierarchy

| TEP Sector | Coupling Type | Primary Test | Status |
|---|---|---|---|
| Conformal | A(φ) clock rates | GNSS correlations, orbital coupling, preferred-axis recovery | Suggested in exploratory analyses; requires independent replication |
| Disformal | B(φ) propagation | Triangle holonomy, GW speed | GW sector constrained; clock sector untested |

Bottom line: TEP is falsifiable even if B = 0. The conformal sector makes distinct predictions (distance correlations, velocity coupling, frame alignment) that GR does not. The holonomy test adds sensitivity to disformal structure if present, but is not the sole discriminator.

### 8.3 Interplanetary Closed-Loop Timing

Deep space baselines are attractive for probing disformal structure because they accumulate propagation effects over long paths. However, a two-point one-way asymmetry is not, by itself, an operational observable because it depends on unknown clock offsets and drifts between endpoints. An interplanetary discriminator therefore requires an explicitly offset-invariant, direction-reversing closed-loop construction.

##### 8.3.1 The Observable: AU-Scale Loop Holonomy

#### Estimator

Consider three nodes, such as an Earth station A and two spacecraft B and C. Each link is a one-way, time-tagged optical transfer giving $\Delta_{ij} = \tau_j^r - \tau_i^e$.

The raw AU-scale loop holonomy is defined by the direction-reversed loop difference:

$$H_{\rm AU,raw} \equiv (\Delta_{AB} + \Delta_{BC} + \Delta_{CA}) - (\Delta_{AC} + \Delta_{CB} + \Delta_{BA}).$$

**Critical Analysis:**

#### Operational Invariance

Constant clock offsets cancel algebraically in $H_{\rm AU,raw}$ by the same mechanism as in the terrestrial triangle test. The observable is therefore well-defined without assuming an external synchronization convention, provided that emission and reception time tags are recorded for each one-way link.

##### 8.3.2 Physical Content

The signal of interest is the GR-subtracted residual holonomy $H_{\rm AU,resid} = H_{\rm AU,raw} - H_{\rm AU,GR}$. The modeled GR term includes Shapiro delay contributions from solar-system bodies, kinematic loop terms from platform motion, and plasma dispersion. A nonzero residual with the predicted geometry dependence would indicate non-exact time transport beyond the modeled GR loop effects.

##### 8.3.3 Proposed Configuration

- Three-node geometry using one Earth terminal and two spacecraft (or one spacecraft plus a second Earth terminal), enabling both loop orientations.

- Optical time transfer with recorded emission and reception time tags for each directed link.

- Dual-frequency calibration (or independent plasma monitoring) to control dispersive delays.

- End-to-end modeling of ephemerides and Shapiro delays to construct $H_{\rm AU,GR}$.

##### 8.3.4 Forecast and Feasibility

#### Interplanetary Forecast

| Parameter | Target Value | Physical Implication |
|---|---|---|
| Baseline | 0.5–2 AU | Long integration path and large loop geometry |
| Loop time scale | $T_{\text{loop}} \sim 10^3$ s | Six one-way links contribute to $H_{\rm AU,raw}$ |
| Residual target | $H_{\rm AU,resid} \sim 0.1$–10 ps (indicative) | AU-scale analogue of the triangle-holonomy discriminator; feasibility depends on plasma and ephemeris calibration |

This test is technologically demanding because it couples precision time transfer to deep-space navigation and plasma calibration. Its primary value is conceptual: it upgrades interplanetary timing from a convention-dependent two-node asymmetry to an offset-invariant, closed-loop observable.

### 8.4 GNSS Correlation Replication

Exploratory analyses within the TEP research program have suggested distance-structured correlations in GNSS clock data with correlation length on the order of 10<sup>3</sup>–10<sup>4</sup> km. Independent, blinded replication of this analysis would provide strong evidence for or against the TEP interpretation.

##### 8.4.1 The Existing Evidence

Exploratory analysis from the TEP research program, using 25 years of CODE clock products and cross-validated with IGS and ESA analysis centers, suggests:

- Exponential decay of clock correlations with distance

- Correlation length of order a few × 10<sup>3</sup> km (reported example: λ ≈ few × 10<sup>3</sup> km under one estimator)

- Consistency across three independent analysis centers (R² = 0.92–0.97)

- Orbital-dynamics coupling (reported correlation with Earth's orbital velocity)

- An annual-phase direction near Earth's aphelion velocity tangent, with the CMB apex anti-phase

Status: These findings constitute preliminary evidence requiring independent replication. This paper treats them as a motivating hypothesis and does not assume their numerical values when specifying the discriminating measurement geometries.

##### 8.4.2 The Reproducibility Mandate

For these claims to be scientifically accepted, future analysis must move beyond preliminary reports to a fully transparent reproducibility package. Any confirming study must provide:

#### Required Reproducibility Standards

- Exact Datasets: Specification of precise products (e.g., IGS `clk` 30s vs 5m), specific stations (IGS site IDs), and exact time windows used.

- Preprocessing Pipeline: Full code or algorithmic description of outlier removal, reference clock definition, and detrending methods (linear vs polynomial).

- Null Simulations: Injection/recovery tests where synthetic TEP signals are added to noise to verify detection efficiency, and "scrambled" null tests (randomizing station locations) to verify zero signal.

- Multiple-Testing Corrections: Statistical penalties for searching across multiple correlation lengths ($\lambda$) and lag times.

Without these controls, exponential fits to noisy residuals can be spurious. The TEP program commits to releasing a public codebase (TEP-GNSS-Open) meeting these standards.

##### 8.4.3 Replication Requirements

Independent replication should:

- Use raw GNSS data (not processed clock products) to avoid filtering artifacts

- Apply minimal processing to preserve spatial correlations

- Test multiple constellations (GPS, GLONASS, Galileo, BeiDou)

- Verify the correlation length is consistent across datasets

- Test for systematic artifacts (ionospheric, tropospheric, orbital)

##### 8.4.4 Falsification Criteria

The TEP interpretation would be falsified if:

- Correlation length falls outside 500-20,000 km range

- Correlations disappear in raw data under independent minimal processing

- Correlations are explained by known systematic effects

- Different constellations show inconsistent correlation lengths

- Conformal-only TEP (B = 0) is not falsified by null holonomy alone; the conformal sector requires independent disconfirmation via clock-correlation or distance-structured observables

### 8.5 Optical Clock Networks and Environmental Screening

The next generation of optical clock networks—connected by optical fiber or free-space links—could provide unprecedented sensitivity to synchronization structure and environmental screening mechanisms.

##### 8.5.1 Current Developments

Several projects are developing continental-scale optical clock networks:

- European fiber network connecting national metrology institutes

- NIST-JILA optical link in Boulder, Colorado

- Proposed intercontinental optical links

##### 8.5.2 TEP Sensitivity: Two Phases

Phase I: Distance Correlations. Optical clocks achieve $10^{-18}$ stability, potentially sensitive to distance-dependent correlations at the $\delta\tau/\tau \sim 10^{-18}$ level over 1000 km baselines.

Phase II: Environmental Screening Maps. TEP predicts that the incremental shear response $\phi(x)$ is suppressed where the ambient acceleration is high (enclosed $M/r^2 \gtrsim g_t/G$). Comparing clocks at different altitudes (sea level vs mountain vs LEO) allows mapping the screening profile. After subtracting GR redshift, TEP predicts residual frequency shifts of $10^{-19}$–$10^{-18}$ over tens of kilometers for a screening length $\lambda_{scr} \sim 10$ km.

##### 8.5.3 Critical Design Considerations

To test TEP, optical clock networks must:

- Use one-way comparisons (not just common-mode rejection)

- Span continental-to-global baselines (order 10<sup>3</sup>–10<sup>4</sup> km)

- Operate continuously to detect temporal variations

- Deploy transportable clocks to varying environments (density/altitude) to test screening

### 8.6 Matter-Wave Interferometry

Atom interferometers provide a complementary probe sensitive to the gradients of the conformal factor $\nabla \ln A(\phi)$.

- Objective: Measure $\nabla\nabla \ln A(\phi)$ via differential phases in atom interferometers.

- Design: Long-baseline or tall fountain interferometers (Sr, Rb) in gradiometric configurations.

- Discrimination: Combine with mechanical gravimeters. GR affects both; TEP scalar gradients affect atoms (via $A(\phi)$) differently from bulk mass (if composition-dependent couplings exist, though A is universal at leading order).

### 8.7 Explicit TEP Predictions

To ensure falsifiability, TEP makes specific numerical predictions for each proposed test. These predictions are derived from the theoretical framework (Paper 0). Numerical ranges are order-of-magnitude targets; where they are informed by reported GNSS analyses, they are treated as provisional pending independent replication.

#### Quantitative TEP Predictions

| Observable | TEP Prediction | Derivation | Falsification Criterion |
|---|---|---|---|
| Correlation length λ | 1,000–10,000 km | Temporal Topology transition radius $R_T(M_\oplus) \approx 4{,}150$ km ($\rho_T \approx 20\text{ g/cm}^3$) and empirical GNSS span ($1{,}050$–$4{,}350$ km) | &lt; 500 km or > 20,000 km (excludes empirical cross-network consistency) |
| Residual holonomy Hresid | $|H_{\rm resid}| \approx 4.33\times10^{-13}$ s $\approx 4.33\times10^{5}$ as per loop ($B_0 = +1$, repaired Paper 0 R6) | Admissible-branch non-exact disformal transport; linear in $|B_0|$; sign flips under loop reversal; conformal-only limit returns exactly zero | Null at 100 as bounds $|B_0| \lesssim 1.7\times10^{-4}$; null at $10^{-18}$ fractional bounds $|B_0| \lesssim 5\times10^{-7}$ |
| Interplanetary Closed-Loop Residual | 0.1–10 ps (AU-scale loop) | Direction-reversed loop: $H_{\rm AU,resid} = H_{\rm AU,raw} - H_{\rm AU,GR}$ | |HAU,resid| &lt; 0.01 ps after GR subtraction |
| Anisotropy ratio (EW/NS) | 1.5–3.0 | Phenomenological hypothesis: kinematic coupling to CMB motion ($v \sim 369$ km/s); pending derived microphysics | Ratio &lt; 1.2 or > 5.0 |
| Orbital coupling | |r| > 0.5 with orbital velocity | Phenomenological hypothesis: velocity-modulated screening boundary; pending dynamical derivation | |r| &lt; 0.3 or wrong phase |
| Preferred-direction recovery | signed annual phase near aphelion velocity tangent; CMB apex anti-phase | Phenomenological hypothesis: preferred-axis structure along the cosmological drift gradient; pending full inhomogeneous solution | axis > 60° separation |

#### Phenomenological Benchmark

Some phenomenological discussions adopt a working benchmark correlation length on the order of 10<sup>3</sup>–10<sup>4</sup> km (as reported in GNSS analyses) to illustrate experimental sensitivity and to parameterize a representative screening scale. This manuscript does not rely on that benchmark: the underdetermination and the proposed discriminating observables are defined independently of any particular GNSS-derived parameter value.

Independent replication remains essential. If replication fails to confirm a robust distance-structured correlation in raw GNSS observables, any GNSS-informed parameter inference (including mapping an effective correlation length to a field mass scale) would require revision.

### 8.8 Summary: The Experimental Frontier

#### Discriminating Tests

| Test | Configuration | GR Prediction | TEP Prediction |
|---|---|---|---|
| Triangle Holonomy | One-way closed loop (MEO) | Hresid = 0 | Hresid $\approx 4.33\times10^{-13}$ s ($\sim 4.33\times10^{5}$ as) at $B_0 = +1$ |
| Interplanetary Closed-Loop | AU-scale direction-reversed loop | Hresid = 0 | HAU,resid ~ 0.1–10 ps |
| GNSS Replication | Raw data analysis | No correlations (or systematic origin) | Distance-structured correlations (characteristic scale to be established by replication) |
| Optical Clock Network | Continental one-way | No spatial structure | Screening maps / Correlations |
| Matter-Wave Interferometry | Gradiometer | Depends on $g_{\mu\nu}$ | Depends on $g_{\mu\nu}$ AND $\nabla A(\phi)$ |

These experiments share a common feature: they break the two-way symmetry that characterizes existing precision tests. By measuring one-way propagation, closed-loop residual holonomy H<sub>resid</sub>, or spatial correlations, they probe the synchronization structure that distinguishes GR from TEP.

None of these experiments has been performed with the explicit goal of testing synchronization integrability. The experimental frontier lies not in achieving greater precision within existing paradigms but in designing fundamentally new measurement configurations.

## 9. Discussion: The Underdetermination Problem

The analysis presented in Sections 3-7 reveals a systematic pattern: precision tests of general relativity constrain parameters within an assumed framework rather than discriminating between frameworks. This section examines the philosophical implications and situates the findings within the broader context of theory testing in physics.

### 9.1 The Duhem-Quine Thesis in Relativistic Metrology

The Duhem-Quine thesis holds that scientific hypotheses cannot be tested in isolation; any test involves auxiliary assumptions that could be revised instead of the primary hypothesis. This thesis finds stark expression in precision tests of GR.

##### 9.1.1 Theory-Laden Observation

Every precision measurement requires corrections for systematic effects. These corrections are derived from theoretical models:

- Tropospheric delay corrections rely on refractivity models combined with a relativistic signal-path geometry

- Solar plasma corrections rely on electron-density models combined with spacecraft ephemerides and tracking geometry

- Orbital dynamics are expressed in relativistic reference systems and ephemerides

- Clock comparisons adopt a coordinate-time standard within an assumed relativistic reference frame

The framework being tested generates the corrections applied to test it. This is standard practice in precision metrology: systematic corrections require theoretical and empirical auxiliary models; the logical point is that the "test" is conditional on those auxiliaries. "Testing GR" is therefore more accurately described as "testing the self-consistency of a GR-anchored data-reduction pipeline within a specified model class."

##### 9.1.2 The Auxiliary Hypothesis Problem

When a measurement agrees with GR, the conclusion is: "GR is confirmed." When a measurement disagrees, the response is typically to revise auxiliary assumptions (calibration errors, unmodeled systematics) rather than question GR.

This asymmetry is rational—GR has enormous empirical support—but it illustrates how discrepancies are often underdetermined between core theory and auxiliary models unless experiments are designed to vary the auxiliaries independently.

### 9.2 The Conventionality of Simultaneity

The philosophical literature on the conventionality of simultaneity, initiated by Reichenbach and developed by Grünbaum, Salmon, and others, directly addresses the issues raised in this paper. This section translates these philosophical insights into operational terms accessible to experimental physicists.

##### 9.2.1 The Operational Problem: Measuring with the Thing You're Measuring

#### The Core Issue in Plain Language

You cannot define synchronization using light rays if the speed of light is the thing you are trying to measure.

Einstein's synchronization procedure uses light signals to define "simultaneous." But this procedure assumes light travels at the same speed in both directions. To verify this assumption, you would need to measure one-way light speed—which requires synchronized clocks. The reasoning is circular.

##### 9.2.2 The Rubber Ruler Analogy

#### Why Two Metrics Are Necessary

Imagine you have a rubber ruler that can stretch. You want to measure whether the ruler is stretching. But the only measuring device you have is... the ruler itself.

- If the ruler stretches uniformly, every measurement you make with it will still give the same numbers

- You cannot detect the stretching using only the stretching ruler

- To detect stretching, you need a *second, independent reference*—a rigid ruler that doesn't stretch

In GR, spacetime is the only ruler. If spacetime "stretches" (the metric changes), all measurements made with spacetime—including light propagation—stretch with it. You cannot detect the change.

##### 9.2.3 Reichenbach's ε-Synchronization

Reichenbach formalized this insight mathematically. Einstein's synchronization convention (ε = 1/2, meaning light takes equal time in each direction) is not empirically determined. Any value 0 &lt; ε &lt; 1 yields an empirically equivalent theory.

The one-way speed of light is:

$$c_+ = \frac{c}{2\varepsilon} \quad \text{(forward direction)}$$
$$c_- = \frac{c}{2(1-\varepsilon)} \quad \text{(backward direction)}$$

Only the round-trip speed c is empirically determined. The choice ε = 1/2 is conventional, not empirical—unless you have an independent way to synchronize clocks.

##### 9.2.4 Malament's Theorem: The Uniqueness Claim

Malament (1977) proved that Einstein synchronization is the unique synchronization definable from the causal structure of Minkowski spacetime. This is often cited as resolving the conventionality debate in favor of ε = 1/2.

**Critical Analysis:**

#### What Malament's Theorem Actually Says

In operational terms: *If light rays are your only tool for defining simultaneity, then Einstein synchronization is the only consistent choice.*

This is true—but it assumes light rays are your only tool. The theorem's premise is that the causal structure (determined by light propagation) is the fundamental structure of spacetime.

#### Why It Doesn't Apply to TEP

TEP has two geometric structures, $g_{\mu\nu}$ and $\tilde g_{\mu\nu}$, but all nongravitational matter, clocks, and photons couple to the same matter metric $\tilde g_{\mu\nu}$. The discriminator is not a disagreement between a &ldquo;clock metric&rdquo; and a &ldquo;light metric.&rdquo; It is a GR-subtracted non-exact synchronization structure in the matter geometry, $$H_{\rm resid}=\oint_C(\tilde\sigma-\sigma_{\rm GR}).$$

#### The Key Point: Malament Assumes Global Integrability

In standard single-metric general relativity, the light-cone structure uniquely determines the conformal causal geometry, and Malament's theorem establishes that Einstein standard synchronization is the unique non-trivial equivalence relation definable from that causal order. Crucially, Malament's theorem takes for granted that the synchronization connection is integrable—that is, exact across the manifold ($\oint \sigma = 0$). In TEP, while all matter and light share the identical matter metric $\tilde g_{\mu\nu}$ (preserving local Lorentz invariance and causal light cones), dynamical time allows for non-exact synchronization transport. When non-exact or disformal terms are present, closed-loop transport develops a non-zero residual holonomy $H_{\rm resid} \neq 0$. Malament's theorem does not rule out such structures because it presupposes an integrable, path-independent simultaneity slice.

##### 9.2.5 The Two-Clock Thought Experiment

#### Making It Concrete

Imagine two identical atomic clocks, A and B, separated by distance L. You want to synchronize them.

GR procedure:

- Send a light pulse from A to B

- B reflects it back to A

- A measures round-trip time T = 2L/c

- Conclude: one-way time is T/2 = L/c (by convention)

The hidden assumption: Light takes equal time in each direction. But you cannot verify this without already-synchronized clocks!

TEP alternative:

- The scalar field φ affects clock rates through A(φ)

- If φ differs at A and B, clocks tick at different rates

- Light still travels at c in both directions (conformal coupling preserves null cones)

A conformal clock-rate difference can alter transported clock histories, but clocks and photons remain governed by the same matter metric $\tilde g_{\mu\nu}$. The independent observable is therefore not a disagreement between a clock metric and a light metric, but the GR-subtracted closed-loop residual $H_{\rm resid}=\oint_C(\tilde\sigma-\sigma_{\rm GR})$, which is nonzero only when the transport contains disformal or otherwise non-exact structure.

##### 9.2.6 Implications for TEP

TEP is not a simple Reichenbach ε ≠ 1/2 anisotropic light-speed theory. This distinction is critical for understanding what TEP actually claims and how it differs from conventional alternatives to special relativity.

**Critical Analysis:**

#### What TEP Is NOT

- Not an ε ≠ 1/2 theory: TEP does not claim that light travels at different speeds in different directions within a single reference frame.

- Not a Lorentz violation: Local Lorentz invariance holds exactly in freely falling laboratories. The speed of light is isotropic locally (ε = 1/2 locally).

- Not a preferred-frame theory (in the conventional sense): While the scalar field φ defines a cosmic rest frame, this does not violate local Lorentz symmetry because the conformal coupling preserves null cones.

#### What TEP Actually Claims

TEP preserves local Lorentz invariance, including local isotropy of light propagation, while introducing two global observables: conformal-sector spatial clock-network covariance/open-path temporal transport, and disformal or otherwise non-exact closed-loop synchronization holonomy. The key distinction:

- Locally: Light propagates isotropically at c in all directions (ε = 1/2)

- Globally: different environments and transported histories can produce conformal clock-rate differences. Closed-loop synchronization holonomy requires disformal or otherwise non-exact transport; $\oint d\ln A=0$ in a smooth simply connected region.

- Operationally: Two-way measurements (which average forward and backward paths) see no anisotropy. One-way closed loops can reveal residual holonomy H<sub>resid</sub> if disformal structure B(φ) ≠ 0

This is fundamentally different from Reichenbach's ε-synchronization, which modifies the *local* one-way speed of light within a single frame. TEP modifies *global* synchronization structure while preserving local isotropy.

#### The Bottom Line for Experimentalists

The triangle holonomy test targets $H_{\rm resid}=\oint_C(\tilde\sigma-\sigma_{\rm GR})$ directly. If the GR-subtracted residual is nonzero, synchronization is non-integrable in the matter geometry.

### 9.3 Underdetermination vs. Falsifiability

The underdetermination identified in this paper is not the radical underdetermination of Quine (where infinitely many theories fit any finite data). It is a specific, resolvable underdetermination: existing tests do not directly probe the observables that distinguish GR from TEP, but new tests could.

##### 9.3.1 The Structure of the Underdetermination

| Observable | GR Prediction | TEP Prediction | Status |
|---|---|---|---|
| Local clock rates | dτ/dt = √g₀₀ | dτ/dt = √g₀₀ | Identical |
| Round-trip light time | 2L/c | 2L/c | Identical |
| PPN γ | 1 | 1 | Identical |
| c_γ vs c_g | Equal | Equal (conformal) | Identical |
| Residual holonomy Hresid | 0 | 0 if B = 0; ≠ 0 possible if non-exact structure is present | Distinguishable |
| Clock correlations | None (or systematic) | Exponential decay | Distinguishable |

##### 9.3.2 Resolvable Underdetermination

The underdetermination is resolvable because GR and TEP make different predictions for observables that have not yet been measured with appropriate configurations. The triangle holonomy test, interplanetary closed-loop timing, and GNSS correlation replication could all distinguish the theories.

This is not a failure of falsifiability but a gap in the experimental program. The tests that would discriminate have simply not been performed.

### 9.4 The Sociology of Precision Tests

Why have discriminating tests not been performed? Several factors contribute:

##### 9.4.1 Two-Way Convenience

Two-way measurements are experimentally simpler. They require only one clock, avoid synchronization problems, and provide gauge-invariant results. The experimental tradition naturally evolved toward two-way configurations.

##### 9.4.2 The "GR is Correct" Prior

GR has been spectacularly successful. The prior probability assigned to alternatives is low, reducing motivation to design experiments specifically to test them. Resources flow toward improving precision within the GR framework rather than testing the framework itself.

##### 9.4.3 The PPN Paradigm

The PPN formalism provides a systematic way to parameterize deviations from GR. But PPN is a parameterization of the post-Newtonian limit under assumptions about which effective metric governs the sector being tested (typically solar-system dynamics and light propagation). This makes it an exceptionally powerful framework for constraining single-metric departures in those sectors, while leaving open the possibility of additional clock-sector structure that is not representable as a small set of PPN light-propagation parameters.

##### 9.4.4 Scientific Conservatism

When experiments confirm GR, they are celebrated. When anomalies appear (Pioneer anomaly, flyby anomaly), enormous effort goes into finding conventional explanations. This is scientifically appropriate—extraordinary claims require extraordinary evidence, and GR has earned its status through a century of successful predictions. The point is not that this conservatism is wrong, but that it shapes which experiments get funded and performed.

### 9.5 The Broader Context

##### 9.5.1 Dark Matter and Dark Energy

The standard cosmological model assigns ~95% of the universe to dark matter and dark energy. In TEP, those terms are not fundamental substances; they are reconstruction labels inferred when gravitational and cosmological observations are analyzed under GR/FLRW assumptions without temporal-transport structure.

If GR is not the complete description of gravity—if TEP or another modification is required—the inferred dark sector could be partially or wholly an artifact of applying an incomplete reconstruction model. The stakes for testing GR alternatives are high.

##### 9.5.2 The Hubble Tension

The discrepancy between early-universe (CMB) and late-universe (Cepheid/SNe) measurements of the Hubble constant has reached 5σ significance. This tension could indicate new physics—potentially related to modifications of GR.

##### 9.5.3 The S₈ Tension

Similarly, measurements of matter clustering (S₈) show tension between CMB predictions and direct measurements. Modified gravity theories, including scalar-tensor theories like TEP, could potentially resolve these tensions.

### 9.6 Summary: The State of the Evidence

#### What the Experimental Canon Establishes

- GR is self-consistent to high precision

- Local Lorentz invariance has been tested to very high, sector-dependent precision

- PPN parameters match GR predictions

- No evidence for preferred-frame effects (in two-way tests)

#### Not Directly Tested by the Canonical Precision-GR Experimental Canon

- Spatial clock correlations — reported in the TEP GNSS programme (Papers 1–3, 14); independent blinded external replication remains outstanding

- Synchronization integrability (holonomy) — candidate astrophysical phase-closure evidence reported in Paper 16; engineered one-way closed-loop precision-clock tests remain outstanding

- One-way light speed isotropy across non-degenerate closed optical links

- Common-mode conformal clock coupling along shared propagation paths

#### The Underdetermination

For the classes of precision tests reviewed in this paper, and under the conditions where local Einstein equivalence holds and any disformal cone tilt is constrained to be negligible by multi-messenger bounds, GR and conformal-sector TEP can be observationally degenerate. The experimental canon therefore constrains parameter space within the standard single-metric framework but does not directly target the specific loop and correlation observables where the frameworks differ. This underdetermination is resolvable through new experimental configurations.

## 10. Conclusions

This paper has presented a systematic methodological analysis of the canonical precision tests of general relativity. The analysis reveals that these tests—while representing significant achievements in experimental physics—share structural features that, as currently configured, do not directly probe the specific loop and correlation observables that distinguish single-metric theories (GR) from two-metric alternatives such as the Temporal Equivalence Principle.

### 10.1 Principal Findings

The analysis in Sections 3–7 demonstrates that the six structural limitations identified in §1.2—two-way measurement dominance, local/global conflation, model-dependent calibration, single-path multi-messenger constraints, theory-laden data reduction, and proximity-regime screening underdetermination in the tested configuration—are not isolated issues but interconnected features of the experimental tradition.

The underdetermination is structural, not accidental. Two-way measurements evolved because they are simpler and provide gauge-invariant results. The PPN formalism, which guides most precision tests, constrains the post-Newtonian limit in the sectors it parameterizes (solar-system dynamics and light propagation). This makes it exceptionally powerful for ruling out many alternatives to GR, while leaving open the possibility of additional clock-sector structure that is not reducible to a small set of PPN light-propagation parameters.

### 10.2 What the Tests Do Establish

The critique presented here should not be misread as claiming that precision tests are worthless. They establish important facts:

- GR provides a self-consistent framework for describing gravity

- Local Lorentz invariance has been tested to very high, sector-dependent precision

- PPN parameters match GR predictions (γ = β = 1)

- GW–EM propagation speeds are bounded to differ by at most a few × 10<sup>−15</sup> under standard emission-time assumptions (constraining disformal coupling)

- No evidence for preferred-frame effects in two-way measurements

These are genuine constraints on the space of viable theories. Many alternatives to GR are ruled out by these tests. But TEP is not among them in the regime considered here: it can be formulated to satisfy the existing constraints while introducing additional clock-sector structure that is not directly targeted by the canonical measurement geometries.

### 10.3 The Path Forward

##### 10.3.1 Discriminating Experiments

The underdetermination is resolvable. Experiments that could distinguish GR from TEP include:

- *Triangle Holonomy Test:* One-way timing around a large-area closed loop (e.g., ground-satellite-ground), measuring a GR-subtracted residual holonomy $H_{\rm resid}$. GR predicts $H_{\rm resid} = 0$; a nonzero residual indicates non-exact time geometry.

- *Matter-Wave Interferometry:* Atom interferometers in gradiometric configurations sensitive to gradients of the scalar field $\nabla\nabla \phi$ via the conformal factor, distinguishable from GR gravity by combining with mechanical gravimeters.

- *Interplanetary Closed-Loop Timing:* AU-scale, direction-reversed loop measurements constructed from one-way time-tagged links, targeting a GR-subtracted residual holonomy analogous to the terrestrial triangle observable.

- *GNSS Correlation Replication:* Independent, blinded analysis of raw GNSS data to verify or refute the distance-structured correlations suggested by exploratory analyses within the TEP research program.

- *Optical Clock Networks:* Continental-scale networks using one-way comparisons to probe synchronization structure at 10<sup>−18</sup> precision.

##### 10.3.2 Methodological Rationale

The dark-sector inference, the Hubble tension, and the S₈ discrepancy all suggest that current reconstruction assumptions may be incomplete. Testing two-metric alternatives to GR and FLRW addresses whether these anomalies reflect new stress-energy components or unmodeled temporal-transport structure.

Optical clocks, one-way optical links, and GNSS infrastructure already exist. Existing precision metrology can therefore be extended from parameter estimation within the single-metric GR reduction to direct experimental tests of closed-loop holonomy and multi-station clock covariance.

##### 10.3.3 Experimental Priorities

The proposed discriminating experiments can be ranked by technical feasibility, discriminating power, and resource requirements:

#### Priority Ranking for Discriminating Tests

| Priority | Experiment | Technical Feasibility | Discriminating Power | Timeline/Cost |
|---|---|---|---|---|
| 1. Highest | GNSS Correlation Replication | High (data already exists) | High (tests conformal sector) | 1-2 years / Low (computational) |
| 2. High | Optical Clock Networks (one-way) | Medium (infrastructure exists, requires protocol modification) | High (10⁻¹⁸ sensitivity) | 3-5 years / Medium (leverage existing networks) |
| 3. Medium | Triangle Holonomy (Ground-Satellite) | Medium (requires new optical links) | High (clean null test for disformal structure) | 5-7 years / Medium-High (new infrastructure) |
| 4. Medium | Matter-Wave Interferometry | Medium (technology maturing) | Medium (tests gradients $\nabla\nabla\phi$) | 5-10 years / Medium (lab/space) |
| 5. Lowest | Interplanetary Closed-Loop Timing | Low (requires deep space mission) | Medium (long baselines, many systematics) | 15-20 years / Very High (dedicated spacecraft) |

#### Recommended Near-Term Strategy

- Immediate (0-2 years): Independent replication of GNSS correlation analysis using raw RINEX data. This is the most cost-effective discriminating test and could provide strong evidence for or against the existence of robust distance-structured correlations on continental-to-global baselines (order 10<sup>3</sup>–10<sup>4</sup> km).

- Short-term (2-5 years): Modify existing optical clock network protocols to enable one-way comparisons. This leverages substantial existing infrastructure (European fiber network, NIST-JILA links) with minimal additional cost.

- Medium-term (5-10 years): Develop ground-based triangle holonomy test using intercontinental optical links. This requires new infrastructure but is technically feasible with current technology.

- Long-term (10+ years): Space-based holonomy tests and interplanetary missions as part of broader fundamental physics programs (e.g., ACES follow-on, dedicated deep space clocks).

Critical first step: The GNSS replication is both the highest priority and the most feasible. If independent, blinded analysis confirms robust distance-structured correlations on continental-to-global baselines (order 10<sup>3</sup>–10<sup>4</sup> km), it would provide strong motivation for the more expensive optical clock and holonomy experiments. If replication fails, any GNSS-informed interpretation would require substantial revision.

### 10.4 Concluding Remarks

#### The Central Argument

The experimental foundations of general relativity are more specialized than commonly appreciated—not because the experiments are flawed, but because they were designed to test specific predictions within an assumed framework. Two-way measurements, local comparisons, and single-path configurations excel at what they were designed for; they do not directly probe synchronization structure. The precision is high; the scope is necessarily limited.

Within the screened and disformally bounded regime defined above, TEP remains viable across the canonical precision tests reviewed here. This is not because TEP is unfalsifiable—explicit falsification criteria exist—but because the experiments that would test them have not been performed.

The path forward is clear: design and execute experiments that break the two-way symmetry. Measure holonomy. Test spatial correlations. Probe synchronization structure directly. Only then will the experimental foundations of relativistic physics be secure.

### 10.5 What the Canonical Tests Constrain (and Do Not Constrain)

Existing tests are successful within their observable class. The question is whether that class spans the discriminants required to distinguish GR from two-metric clock-sector alternatives. The following table clarifies the scope of each canonical test:

| Canonical Test | What It Constrains | What It Does Not Constrain |
|---|---|---|
| Pound-Rebka | Local redshift (clock rate ratio) | Global synchronization holonomy |
| Cassini | PPN γ, Shapiro delay (two-way light propagation) | Common-mode conformal clock response |
| GW170817 | Differential photon/graviton speed (disformal coupling) | Conformal clock-sector covariance |
| GPS | GR-consistent operational timing (self-consistent framework) | Independent one-way global synchronization |
| Optical clocks | Local potential redshift (Lorentz invariance) | Large-area loop non-integrability |

The pattern is consistent: each test constrains its intended observable with precision, but leaves orthogonal sectors—particularly those involving one-way global synchronization and spatial clock-correlation structure—largely unexplored.

### 10.6 Summary Table: The Experimental Canon

| Experiment | Claimed Result | Actual Constraint | TEP Status |
|---|---|---|---|
| Pound-Rebka (1960) | Confirms gravitational redshift | Local clock rate ratio | Compatible |
| Hafele-Keating (1972) | Confirms time dilation | Proper time on closed worldlines | Compatible |
| Gravity Probe A (1976) | Confirms GR to 7×10⁻⁵ | Two-way clock comparison | Compatible |
| GPS (1978-present) | Proves GR corrections | Self-consistency of GR model | Compatible |
| Lunar Laser Ranging (1969-present) | Nordtvedt effect null | Two-way ranging; gravitational sector only | Compatible |
| Cassini (2003) | γ = 1 to 10⁻⁵ | Two-way Shapiro delay | Compatible |
| Gravity Probe B (2011) | Confirms frame-dragging | Gyroscope precession (gμν only) | Compatible |
| LIGO/Virgo (2015-present) | Detects gravitational waves | GW propagation on gμν | Compatible |
| GW170817 (2017) | |cg − cγ|/c bounded at few × 10⁻¹⁵ (assumption-dependent) | Disformal coupling only | Compatible |
| Optical Clocks (2022) | Redshift at 10⁻¹⁸ | Local Lorentz invariance | Compatible |

Every entry in the "TEP Status" column reads "Compatible." This reflects the structural nature of the underdetermination. The experimental canon, for all its high precision, was designed to test predictions that GR and TEP share. The experiments succeed effectively at their intended purpose; they simply were not designed to probe the specific observables where GR and TEP differ.

This is not a criticism of the experimentalists who performed these measurements—their work represents significant achievements in precision physics. It is an observation about the logical structure of theory testing: to distinguish between theories, one must measure observables where they make different predictions.

## References

### Foundational Experiments

Everitt, C. W. F., et al. (2011). Gravity Probe B: Final results of a space experiment to test general relativity. *Physical Review Letters*, 106(22), 221101. [doi:10.1103/PhysRevLett.106.221101](https://doi.org/10.1103/PhysRevLett.106.221101)

Pound, R. V., & Rebka, G. A. (1960). Apparent weight of photons. *Physical Review Letters*, 4(7), 337-341. [doi:10.1103/PhysRevLett.4.337](https://doi.org/10.1103/PhysRevLett.4.337)

Pound, R. V., & Snider, J. L. (1965). Effect of gravity on gamma radiation. *Physical Review*, 140(3B), B788-B803. [doi:10.1103/PhysRev.140.B788](https://doi.org/10.1103/PhysRev.140.B788)

Hafele, J. C., & Keating, R. E. (1972). Around-the-world atomic clocks: Predicted relativistic time gains. *Science*, 177(4044), 166-168. [doi:10.1126/science.177.4044.166](https://doi.org/10.1126/science.177.4044.166)

Hafele, J. C., & Keating, R. E. (1972). Around-the-world atomic clocks: Observed relativistic time gains. *Science*, 177(4044), 168-170. [doi:10.1126/science.177.4044.168](https://doi.org/10.1126/science.177.4044.168)

Vessot, R. F. C., et al. (1980). Test of relativistic gravitation with a space-borne hydrogen maser. *Physical Review Letters*, 45(26), 2081-2084. [doi:10.1103/PhysRevLett.45.2081](https://doi.org/10.1103/PhysRevLett.45.2081)

Williams, J. G., Turyshev, S. G., & Boggs, D. H. (2012). Lunar laser ranging tests of the equivalence principle. *Classical and Quantum Gravity*, 29(18), 184004. [doi:10.1088/0264-9381/29/18/184004](https://doi.org/10.1088/0264-9381/29/18/184004)

Murphy, T. W. (2013). Lunar laser ranging: The millimeter challenge. *Reports on Progress in Physics*, 76(7), 076901. [doi:10.1088/0034-4885/76/7/076901](https://doi.org/10.1088/0034-4885/76/7/076901)

### Light Propagation Tests

Shapiro, I. I. (1964). Fourth test of general relativity. *Physical Review Letters*, 13(26), 789-791. [doi:10.1103/PhysRevLett.13.789](https://doi.org/10.1103/PhysRevLett.13.789)

Bertotti, B., Iess, L., & Tortora, P. (2003). A test of general relativity using radio links with the Cassini spacecraft. *Nature*, 425(6956), 374-376. [doi:10.1038/nature01997](https://doi.org/10.1038/nature01997)

Kopeikin, S., Schaefer, G., Polnarev, A., & Vlasov, I. (2006). The orbital motion of Sun and a test of general relativity using radio links with the Cassini spacecraft. *arXiv preprint*, arXiv:gr-qc/0604060. [arXiv:gr-qc/0604060](https://arxiv.org/abs/gr-qc/0604060)

Fomalont, E. B., & Kopeikin, S. M. (2003). The measurement of the light deflection from Jupiter: Experimental results. *The Astrophysical Journal*, 598(1), 704-711. [doi:10.1086/378785](https://doi.org/10.1086/378785)

### GPS and Relativistic Timekeeping

Ashby, N. (2003). Relativity in the Global Positioning System. *Living Reviews in Relativity*, 6(1), 1. [doi:10.12942/lrr-2003-1](https://doi.org/10.12942/lrr-2003-1)

Ashby, N., & Weiss, M. (1999). Global Positioning System receivers and relativity. *NIST Technical Note*, 1385.

Kouba, J., & Héroux, P. (2001). Precise Point Positioning Using IGS Orbit and Clock Products. *GPS Solutions*, 5(2), 12-28. [doi:10.1007/PL00012883](https://doi.org/10.1007/PL00012883)

Dow, J. M., Neilan, R. E., & Rizos, C. (2009). The International GNSS Service in a changing landscape of Global Navigation Satellite Systems. *Journal of Geodesy*, 83(3-4), 191-198. [doi:10.1007/s00190-008-0300-3](https://doi.org/10.1007/s00190-008-0300-3)

### GNSS Processing and Common-Mode Filtering

Dong, D., Fang, P., Bock, Y., Cheng, M. K., & Miyazaki, S. (2006). Spatiotemporal filtering using principal component analysis and Karhunen–Loève expansion approaches for regional GPS network analysis. *Journal of Geophysical Research: Solid Earth*, 111, B03405. [doi:10.1029/2005JB003806](https://doi.org/10.1029/2005JB003806)

### Modern Optical Clock Experiments

Bothwell, T., et al. (2022). Resolving the gravitational redshift across a millimetre-scale atomic sample. *Nature*, 602(7897), 420-424. [doi:10.1038/s41586-021-04349-7](https://doi.org/10.1038/s41586-021-04349-7)

Zheng, X., et al. (2022). Differential clock comparisons with a multiplexed optical lattice clock. *Nature*, 602(7897), 425-430. [doi:10.1038/s41586-021-04344-y](https://doi.org/10.1038/s41586-021-04344-y)

McGrew, W. F., et al. (2018). Atomic clock performance enabling geodesy below the centimetre level. *Nature*, 564(7734), 87-90. [doi:10.1038/s41586-018-0738-2](https://doi.org/10.1038/s41586-018-0738-2)

Mehlstäubler, T. E., Grosche, G., Lisdat, C., Schmidt, P. O., & Denker, H. (2018). Atomic clocks for geodesy. *Reports on Progress in Physics*, 81(6), 064401. [doi:10.1088/1361-6633/aab409](https://doi.org/10.1088/1361-6633/aab409)

### Gravitational Wave Observations

Abbott, B. P., et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2017). GW170817: Observation of gravitational waves from a binary neutron star inspiral. *Physical Review Letters*, 119(16), 161101. [doi:10.1103/PhysRevLett.119.161101](https://doi.org/10.1103/PhysRevLett.119.161101)

Abbott, B. P., et al. (2017). Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A. *The Astrophysical Journal Letters*, 848(2), L13. [doi:10.3847/2041-8213/aa920c](https://doi.org/10.3847/2041-8213/aa920c)

### Resonator and Lorentz Invariance Tests

Michelson, A. A., & Morley, E. W. (1887). On the relative motion of the Earth and the luminiferous ether. *American Journal of Science*, 34(203), 333-345.

Kennedy, R. J., & Thorndike, E. M. (1932). Experimental establishment of the relativity of time. *Physical Review*, 42(3), 400-418. [doi:10.1103/PhysRev.42.400](https://doi.org/10.1103/PhysRev.42.400)

Brillet, A., & Hall, J. L. (1979). Improved laser test of the isotropy of space. *Physical Review Letters*, 42(9), 549-552. [doi:10.1103/PhysRevLett.42.549](https://doi.org/10.1103/PhysRevLett.42.549)

Müller, H., Herrmann, S., Braxmaier, C., Schiller, S., & Peters, A. (2003). Modern Michelson–Morley experiment using cryogenic optical resonators. *Physical Review Letters*, 91(2), 020401. [doi:10.1103/PhysRevLett.91.020401](https://doi.org/10.1103/PhysRevLett.91.020401) [arXiv:physics/0305117](https://arxiv.org/abs/physics/0305117)

Herrmann, S., et al. (2009). Rotating optical cavity experiment testing Lorentz invariance at the 10⁻¹⁷ level. *Physical Review D*, 80(10), 105011. [doi:10.1103/PhysRevD.80.105011](https://doi.org/10.1103/PhysRevD.80.105011)

Nagel, M., et al. (2015). Direct terrestrial test of Lorentz symmetry in electrodynamics to 10⁻¹⁸. *Nature Communications*, 6, 8174. [doi:10.1038/ncomms9174](https://doi.org/10.1038/ncomms9174)

### PPN Formalism and Tests of Gravity

Will, C. M. (2014). The confrontation between general relativity and experiment. *Living Reviews in Relativity*, 17(1), 4. [doi:10.12942/lrr-2014-4](https://doi.org/10.12942/lrr-2014-4)

Will, C. M. (2018). *Theory and Experiment in Gravitational Physics* (2nd ed.). Cambridge University Press.

### Philosophy of Space and Time

Reichenbach, H. (1958). *The Philosophy of Space and Time*. Dover Publications. (Original work published 1928)

Grünbaum, A. (1973). *Philosophical Problems of Space and Time* (2nd ed.). D. Reidel Publishing Company.

Malament, D. (1977). Causal theories of time and the conventionality of simultaneity. *Noûs*, 11(3), 293-300. [doi:10.2307/2214766](https://doi.org/10.2307/2214766)

Salmon, W. C. (1977). The philosophical significance of the one-way speed of light. *Noûs*, 11(3), 253-292. [doi:10.2307/2214765](https://doi.org/10.2307/2214765)

Franklin, A. (1986). *The Neglect of Experiment*. Cambridge University Press.

### Scalar-Tensor Theories and Modified Gravity

Brans, C., & Dicke, R. H. (1961). Mach's principle and a relativistic theory of gravitation. *Physical Review*, 124(3), 925-935. [doi:10.1103/PhysRev.124.925](https://doi.org/10.1103/PhysRev.124.925)

Bekenstein, J. D. (2004). Relativistic gravitation theory for the modified Newtonian dynamics paradigm. *Physical Review D*, 70(8), 083509. [doi:10.1103/PhysRevD.70.083509](https://doi.org/10.1103/PhysRevD.70.083509)

Zumalacárregui, M., & García-Bellido, J. (2014). Transforming gravity: From derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian. *Physical Review D*, 89(6), 064046. [doi:10.1103/PhysRevD.89.064046](https://doi.org/10.1103/PhysRevD.89.064046)

### TEP Research Program

Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: [10.5281/zenodo.16921911](https://doi.org/10.5281/zenodo.16921911) (Paper 0)

Smawfield, M. L. (2025). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: [10.5281/zenodo.17127229](https://doi.org/10.5281/zenodo.17127229) (Paper 1)

Smawfield, M. L. (2025). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: [10.5281/zenodo.17517141](https://doi.org/10.5281/zenodo.17517141) (Paper 2)

Smawfield, M. L. (2025). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: [10.5281/zenodo.17860166](https://doi.org/10.5281/zenodo.17860166) (Paper 3)

Smawfield, M. L. (2025). *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: [10.5281/zenodo.17982540](https://doi.org/10.5281/zenodo.17982540) (Paper 4)

Smawfield, M. L. (2025). *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: [10.5281/zenodo.18004832](https://doi.org/10.5281/zenodo.18004832) (Paper 5)

Smawfield, M. L. (2025). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: [10.5281/zenodo.18064365](https://doi.org/10.5281/zenodo.18064365) (Paper 6)

Smawfield, M. L. (2025). *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: [10.5281/zenodo.18059250](https://doi.org/10.5281/zenodo.18059250) (Paper 7)

Smawfield, M. L. (2025). *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: [10.5281/zenodo.18064581](https://doi.org/10.5281/zenodo.18064581) (Paper 8)

Smawfield, M. L. (2025). *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: [10.5281/zenodo.18109760](https://doi.org/10.5281/zenodo.18109760) (Paper 9 — this work)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: [10.5281/zenodo.18165798](https://doi.org/10.5281/zenodo.18165798) (Paper 10)

Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: [10.5281/zenodo.18209702](https://doi.org/10.5281/zenodo.18209702) (Paper 11)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: [10.5281/zenodo.19000827](https://doi.org/10.5281/zenodo.19000827) (Paper 12)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: [10.5281/zenodo.19102061](https://doi.org/10.5281/zenodo.19102061) (Paper 13)

Smawfield, M. L. (2026). *Global Time Echoes: MGEX Multi-GNSS Clock Replication, 2025–2026*. Preprint v0.4 (Suva). Zenodo. DOI: [10.5281/zenodo.20572726](https://doi.org/10.5281/zenodo.20572726) (Paper 14)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear in the Earth Flyby Anomaly*. Preprint v0.4 (Yogyakarta). Zenodo. DOI: [10.5281/zenodo.19454862](https://doi.org/10.5281/zenodo.19454862) (Paper 15)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Synchronization Holonomy in Pulsar Scintillation*. Preprint v0.5 (Sintra). Zenodo. DOI: [10.5281/zenodo.19454619](https://doi.org/10.5281/zenodo.19454619) (Paper 16)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Lunar Laser Ranging and the Nordtvedt Effect*. Preprint v0.5 (Lucknow). Zenodo. DOI: [10.5281/zenodo.19446028](https://doi.org/10.5281/zenodo.19446028) (Paper 17)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Native hi_class Conformal Implementation, Linear Perturbation Closure, and CMB Acoustic Peak Preservation*. Preprint v0.8 (Cambridge). Zenodo. DOI: [10.5281/zenodo.20572722](https://doi.org/10.5281/zenodo.20572722) (Paper 18)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Blind-Prediction Residual Test in Multiply-Imaged Supernovae*. Preprint v0.3 (Lisboa). Zenodo. DOI: [10.5281/zenodo.20572719](https://doi.org/10.5281/zenodo.20572719) (Paper 19)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Sidereal Modulation Audit of LHC Luminosity Data*. Preprint v0.2 (Hanoi). In preparation. (Paper 20)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Standard-Siren Test of Bi-Metric Gravitational-Wave Propagation*. Preprint v0.2 (Maputo). Zenodo. DOI: [10.5281/zenodo.20572695](https://doi.org/10.5281/zenodo.20572695) (Paper 22)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: The Dirac Limit of Dynamical Proper Time*. Preprint v0.4 (Qatar). Zenodo. DOI: [10.5281/zenodo.20572697](https://doi.org/10.5281/zenodo.20572697) (Paper 23)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Fermion Spin as Temporal-Orientation Holonomy*. Preprint v0.1 (Paris). In preparation. (Paper 24)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Disformal Kinematics and the Measurement Landscape*. Preprint v0.2 (Kuala Lumpur). In preparation. (Paper 25)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Covariant Alternative to Cosmic Expansion*. Preprint v0.4 (Athens). Zenodo. DOI: [10.5281/zenodo.20370143](https://doi.org/10.5281/zenodo.20370143) (Paper 26)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Horizon Cosmology and the Absence of a Physical Big Bang Singularity*. Preprint v0.5 (Thika). Zenodo. DOI: [10.5281/zenodo.20723059](https://doi.org/10.5281/zenodo.20723059) (Paper 27)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Black Holes and the Temporal Horizon*. Preprint v0.4 (Bahrain). Zenodo. DOI: [10.5281/zenodo.21677826](https://doi.org/10.5281/zenodo.21677826) (Paper 28)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Dynamical Proper Time and the Illusion of Primordial Deuterium*. Preprint v0.6 (Dubai). Zenodo. DOI: [10.5281/zenodo.21841147](https://doi.org/10.5281/zenodo.21841147) (Paper 29)

Smawfield, M. L. (2026). *The Mount Wilson Paradigm: Restoring the Eternal Universe via the Temporal Equivalence Principle*. Preprint v0.3 (Harare). Zenodo. DOI: [10.5281/zenodo.21954257](https://doi.org/10.5281/zenodo.21954257) (Paper 30)

Smawfield, M. L. (2026). *Cosmological Voids vs Temporal Shear: An Empirical Falsification of Kinematic Hubble Tension Solutions*. Preprint v0.4 (Valencia). Zenodo. DOI: [10.5281/zenodo.22150139](https://doi.org/10.5281/zenodo.22150139) (Paper 31)

Smawfield, M. L. (2026). *TEP-PPP: Operational Time Echoes — Phase-Resolved Spatial Clock Priors for GNSS Outage Bridging*. Preprint v0.1 (Pune). In preparation. (Paper 33)

### Galileo Eccentric Orbit Tests

Delva, P., et al. (2018). A new test of gravitational redshift using Galileo satellites: The GREAT experiment. *Comptes Rendus Physique*, 20(3), 176-182. [doi:10.1016/j.crhy.2019.04.002](https://doi.org/10.1016/j.crhy.2019.04.002)

Herrmann, S., et al. (2018). Test of the gravitational redshift with Galileo satellites in an eccentric orbit. *Physical Review Letters*, 121(23), 231102. [doi:10.1103/PhysRevLett.121.231102](https://doi.org/10.1103/PhysRevLett.121.231102)

### ACES Mission and Space Clocks

Cacciapuoti, L., & Salomon, C. (2009). Space clocks and fundamental tests: The ACES experiment. *The European Physical Journal Special Topics*, 172(1), 57-68. [doi:10.1140/epjst/e2009-01041-7](https://doi.org/10.1140/epjst/e2009-01041-7)

Savalle, E., Guerlin, C., Delva, P., Meynadier, F., le Poncin-Lafitte, C., & Wolf, P. (2019). Gravitational redshift test with the future ACES mission. *Classical and Quantum Gravity*. [doi:10.1088/1361-6382/ab4f25](https://doi.org/10.1088/1361-6382/ab4f25) [arXiv:1907.12320](https://arxiv.org/abs/1907.12320)

Savalle, E., et al. (2024). Atomic Clock Ensemble in Space. *arXiv preprint*, arXiv:2411.02912. [arXiv:2411.02912](https://arxiv.org/abs/2411.02912)

### Deep Space Atomic Clock

Burt, E. A., et al. (2021). Demonstration of a trapped-ion atomic clock in space. *Nature*, 595(7865), 43-47. [doi:10.1038/s41586-021-03571-7](https://doi.org/10.1038/s41586-021-03571-7)

Ely, T. A., et al. (2021). Results of the Deep Space Atomic Clock deep space navigation analog experiment. *Journal of Spacecraft and Rockets*, 59(4), 1294-1310. [doi:10.2514/1.A35334](https://doi.org/10.2514/1.A35334)

### Equivalence Principle Tests

Touboul, P., et al. (2017). MICROSCOPE mission: First results of a space test of the equivalence principle. *Physical Review Letters*, 119(23), 231101. [doi:10.1103/PhysRevLett.119.231101](https://doi.org/10.1103/PhysRevLett.119.231101)

Touboul, P., et al. (2022). MICROSCOPE mission: Final results of the test of the equivalence principle. *Physical Review Letters*, 129(12), 121102. [doi:10.1103/PhysRevLett.129.121102](https://doi.org/10.1103/PhysRevLett.129.121102)

### Optical Clock Networks and Fiber Links

Lisdat, C., et al. (2016). A clock network for geodesy and fundamental science. *Nature Communications*, 7, 12443. [doi:10.1038/ncomms12443](https://doi.org/10.1038/ncomms12443)

Clivati, C., et al. (2022). Coherent optical-fiber link across Italy and France. *Physical Review Applied*, 18(5), 054009. [doi:10.1103/PhysRevApplied.18.054009](https://doi.org/10.1103/PhysRevApplied.18.054009)

Caldwell, E. D., et al. (2023). Quantum-limited optical time transfer for future geosynchronous links. *Nature*, 618(7966), 721-726. [doi:10.1038/s41586-023-06032-5](https://doi.org/10.1038/s41586-023-06032-5)

### Test Theories and Synchronization

Mansouri, R., & Sexl, R. U. (1977). A test theory of special relativity: I. Simultaneity and clock synchronization. *General Relativity and Gravitation*, 8(7), 497-513. [doi:10.1007/BF00762634](https://doi.org/10.1007/BF00762634)

Colladay, D., & Kostelecký, V. A. (1998). Lorentz-violating extension of the Standard Model. *Physical Review D*, 58(11), 116002. [doi:10.1103/PhysRevD.58.116002](https://doi.org/10.1103/PhysRevD.58.116002)

Mattingly, D. (2005). Modern tests of Lorentz invariance. *Living Reviews in Relativity*, 8(1), 5. [doi:10.12942/lrr-2005-5](https://doi.org/10.12942/lrr-2005-5)

Kostelecký, V. A., & Russell, N. (2011). Data tables for Lorentz and CPT violation. *Reviews of Modern Physics*, 83(1), 11-31. [doi:10.1103/RevModPhys.83.11](https://doi.org/10.1103/RevModPhys.83.11)

### GNSS Common-Mode Errors

Wdowinski, S., et al. (1997). Southern California permanent GPS geodetic array: Spatial filtering of daily positions for estimating coseismic and postseismic displacements. *Journal of Geophysical Research*, 102(B8), 18057-18070. [doi:10.1029/97JB01378](https://doi.org/10.1029/97JB01378)

### Scalar-Tensor Constraints from GW170817

Langlois, D., Saito, R., Yamauchi, D., & Noui, K. (2018). Scalar-tensor theories and modified gravity in the wake of GW170817. *Physical Review D*, 97(6), 061501. [doi:10.1103/PhysRevD.97.061501](https://doi.org/10.1103/PhysRevD.97.061501)

Ezquiaga, J. M., & Zumalacárregui, M. (2017). Dark energy after GW170817: Dead ends and the road ahead. *Physical Review Letters*, 119(25), 251304. [doi:10.1103/PhysRevLett.119.251304](https://doi.org/10.1103/PhysRevLett.119.251304)

### Pioneer Anomaly

Anderson, J. D., et al. (2002). Study of the anomalous acceleration of Pioneer 10 and 11. *Physical Review D*, 65(8), 082004. [doi:10.1103/PhysRevD.65.082004](https://doi.org/10.1103/PhysRevD.65.082004)

Turyshev, S. G., et al. (2012). Support for the thermal origin of the Pioneer anomaly. *Physical Review Letters*, 108(24), 241101. [doi:10.1103/PhysRevLett.108.241101](https://doi.org/10.1103/PhysRevLett.108.241101)

### Cosmological Tensions

Riess, A. G., et al. (2022). A comprehensive measurement of the local value of the Hubble constant with 1 km s⁻¹ Mpc⁻¹ uncertainty from the Hubble Space Telescope and the SH0ES Team. *The Astrophysical Journal Letters*, 934(1), L7. [doi:10.3847/2041-8213/ac5c5b](https://doi.org/10.3847/2041-8213/ac5c5b)

Di Valentino, E., et al. (2021). In the realm of the Hubble tension—a review of solutions. *Classical and Quantum Gravity*, 38(15), 153001. [doi:10.1088/1361-6382/ac086d](https://doi.org/10.1088/1361-6382/ac086d)

## Data Availability & Reproducibility

This work follows open-science practices. All results are fully reproducible from the analysis code and documentation provided. As a theoretical/methodological analysis paper, this work synthesizes established experimental results from the published literature rather than generating new primary data.

### Repository & Code

GitHub Repository: [github.com/matthewsmawfield/TEP-EXP](https://github.com/matthewsmawfield/TEP-EXP)

The repository contains the complete manuscript source, methodological framework documentation, and citation bibliography for this theoretical analysis of precision tests of General Relativity.

#### Repository Structure

```
TEP-EXP/
├── manuscripts/               # Markdown manuscript sources
│   ├── A9-TEP-EXP-v0.9-Istanbul.md  # Primary manuscript
│   └── [other versions]
├── site/
│   ├── components/            # HTML manuscript sections
│   │   ├── 1_abstract.html
│   │   ├── 2_introduction.html
│   │   ├── 3_methodology.html
│   │   ├── 4_redshift_tests.html
│   │   ├── 5_time_dilation_tests.html
│   │   ├── 6_light_propagation_tests.html
│   │   ├── 7_multimessenger.html
│   │   ├── 8_resonator_tests.html
│   │   ├── 9_discriminating_tests.html
│   │   ├── 10_discussion.html
│   │   ├── 11_conclusions.html
│   │   ├── 12_references.html
│   │   └── 13_reproducibility.html
│   ├── public/                # Static assets
│   └── manifest.json          # Site configuration
├── scripts/
│   └── utils/                 # Utility scripts
│       └── process_pdf.py     # PDF generation helper
├── requirements.txt           # Python dependencies
├── CITATION.cff               # Citation metadata
└── README.md                  # Repository documentation
```

### Data Provenance

This is a theoretical and methodological analysis paper. All experimental data referenced is from published literature with full citations provided. Key referenced experiments include:

- Pound-Rebka (1960) and Pound-Snider (1965): Gravitational redshift measurements

- Hafele-Keating (1972): Atomic clock transport experiments

- Vessot et al. (1980): Gravity Probe A rocket experiment

- Gravity Probe B (2011): Frame-dragging and geodetic precession

- GW170817 (2017): Gravitational wave multi-messenger event

- Modern cavity experiments: Lorentz invariance tests

### Reproduction Instructions

#### Quick Start (Manuscript Build)

```
# 1. Clone repository
git clone https://github.com/matthewsmawfield/TEP-EXP.git
cd TEP-EXP

# 2. Install dependencies (for PDF generation)
pip install -r requirements.txt

# 3. Build manuscript website
cd site
npm install
npm run build

# 4. Output will be in site/dist/
```

#### System Requirements

- Node.js 16+ (for site building)

- Python 3.8+ (optional, for PDF processing utilities)

- Storage: &lt; 100 MB

#### Manuscript Generation

The manuscript is built from modular HTML components defined in `site/manifest.json`. To modify content, edit the relevant component files in `site/components/` and rebuild:

```
cd site
npm run build
```

### Methodological Documentation

This paper provides a taxonomical analysis of precision GR tests, examining:

- Gauge-invariant vs. convention-dependent observables — what experiments actually measure

- Two-way vs. one-way measurement protocols — differential vs. absolute tests

- Local vs. global test geometries — closed-loop vs. open-path configurations

- Single-path vs. multipath configurations — holonomy-sensitive tests

### Software Versions

- Node.js 16+

- Python 3.8+ (optional)

---

*This document was automatically generated from the TEP-EXP research site. For the interactive version with figures and enhanced formatting, visit: https://mlsmawfield.com/tep/exp/*

*Related Work:*
- [TEP Theory](https://doi.org/10.5281/zenodo.16921911) (Foundational framework)
- [TEP-GNSS I](https://doi.org/10.5281/zenodo.17127229) (Multi-Center Analysis)
- [TEP-GNSS II](https://doi.org/10.5281/zenodo.17517141) (25-Year Analysis)
- [TEP-GNSS III](https://doi.org/10.5281/zenodo.17860166) (Raw RINEX Validation)

*Source code available at: https://github.com/matthewsmawfield/TEP-EXP*

---

<!-- ===================================================================== -->
<!-- CANONICAL URL: https://mlsmawfield.com/tep/cos/ -->
<!-- MARKDOWN URL:  https://mlsmawfield.com/tep/cos/manuscript.md -->
<!-- ===================================================================== -->

# TEP: Suppressed Density Scaling in Globular Cluster Pulsars
**Matthew Lukin Smawfield**
Version: v0.10 (Caracas)
First published: 9 January 2026 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.18165798

---

## Abstract

A spatially stratified spin-down anomaly is reported in 199 globular-cluster millisecond pulsars compared with 351 field controls. Cluster pulsars show a 0.63 dex raw excess and a 0.40 dex controlled residual, with covariance-aware significance of 5.6σ. The signal exhibits suppressed density scaling: a mixed-effects slope Γ = 0.39 ± 0.08 dex/dex compared with a Newtonian ensemble baseline Γ = 0.75, rejecting the baseline at 4.1σ. Field binaries provide a null control, while cluster binaries are quieter than isolated cluster pulsars; corrected CMC reproduces this binary direction, so the binary result functions as a comparator check rather than the primary failure mode. An adversarial dynamics model accounting for line-of-sight acceleration, jerk, Shklovskii terms, mass segregation, and cluster mass profiles cannot explain the observed suppression. An N-body baseline derived from CMC synthetic pulsars (20 cluster selections mapping to 14 unique Kremer et al. 2020 realizations; 9,526 evolved neutron stars, 6,617 unique) predicts a structurally rescaled excess 3.2× larger than observed (13.2σ nominal model-data tension, not detection significance; 13.0σ model-aware, 8.6σ under 20-cluster ensemble-propagated uncertainty, and 4.1σ under the 100%-correlated conservative quadrature ceiling), and a mock pulsar-timing-array detection pipeline confirms that the missing high-acceleration population cannot be explained by the tested observational-selection models. These results are interpreted as a time-domain test of the TEP observable response coefficient κ<sub>MSP</sub>, not as a measurement of a microscopic scalar coupling.

Code Availability: All data and analysis code required to reproduce the results presented in this work, including the full pulsar catalog compilation, are available in the public repository at [https://github.com/matthewsmawfield/TEP-COS](https://github.com/matthewsmawfield/TEP-COS).

Keywords: temporal equivalence principle, pulsar timing, globular clusters, time dilation, screening transition, modified gravity

## 1. Introduction: Time-Domain Tests of Modified Gravity

## 1.1 The Intermediate-Scale Problem

General Relativity has passed every precision test in the Solar System. Yet at intermediate and cosmological scales, persistent discrepancies arise—rotation curves, cluster dynamics, cosmic acceleration—that conventionally require invisible mass or exotic energy to resolve. A fundamental question follows: Is gravitational time dilation scale-dependent? This work explores the hypothesis that these anomalies reflect not missing matter but modified temporal structure: a scale-dependent amplification of gravitational time dilation beyond the predictions of standard General Relativity.

The Temporal Equivalence Principle (TEP) formalizes this possibility within a two-metric framework (see Section 2), predicting that the rate of proper time accumulation is environment-dependent at intermediate scales while remaining consistent with precision tests in the screened Solar System regime. The central prediction is that *rate-dependent* physical processes—pulsar spin-down, photon arrival times, clock frequencies—should exhibit anomalies in deep gravitational potentials, while *fossil* observables that integrate over formation timescales remain insensitive.

## 1.2 Why Time-Domain Tests Are Critical

TEP modifies the proper time relation: dτ̃/dτ<sub>g</sub> = A(φ) where A(φ) = exp(βφ/M<sub>Pl</sub>), τ̃ is the TEP-modified proper time, τ<sub>g</sub> is the GR proper time, and A(φ) is the universal conformal factor. This creates two classes of observables with quantitatively different TEP sensitivity:

| Observable Class | Examples | TEP Sensitivity | Rationale |
| --- | --- | --- | --- |
| Time-Domain (Rates) | Pulsar Ṗ, Clock frequencies | HIGH | Measures present-tense clock rate |
| Fossil (Archaeology) | Stellar ages, [α/Fe], colors, SFH | LOW | Integrates over ~Gyr formation history |

The expected TEP differential (~10 kyr over cosmic time) is O(10<sup>−6</sup>) of the formation timescale spread (~Gyr) for stellar populations. Fossil observables are unlikely to distinguish TEP from standard astrophysical processes at practical significance levels. This paper therefore focuses exclusively on time-domain tests: pulsar spin-down rates.

## 1.3 Central Results

#### Theoretical Framework

The TEP framework uses observational data to constrain the *class* of viable modified gravity theories through the observable pulsar response coefficient κ<sub>MSP</sub> (absorbing stellar physics and environment, not the microscopic coupling β) and the topological transition scale R<sub>T</sub>, following the same strategy as the PPN (Parameterized Post-Newtonian) framework used to test GR in the Solar System. The effective screening framework motivates the response model tested here; candidate transfer models are discussed separately and are not required for the empirical inference. Section 2.1.1 presents the screening framework, making this manuscript self-contained and independently evaluable.

| Observable | Status | Result | Details |
| --- | --- | --- | --- |
| Pulsar Timing | Cluster Spin-down Residual | Anomaly Detection | 0.63 dex raw excess; 0.40 dex controlled residual; null in field |
| Field Binary Control | Binary vs Isolated (Field) | Null Control | p = 0.78 (supports environmental origin) |
| Binary Comparator | Binary vs Isolated (Cluster) | Comparator Check | Binaries −0.33 dex quieter than isolated; corrected CMC predicts −0.27 dex (3rc filter), consistent direction |
| Spatial Stratification | Core vs Outskirts | Suggestive | −0.30 dex (inner, p=0.074) vs −0.14 dex (outer, p=0.41) |
| Suppressed Density Scaling | Does the signal track dynamical noise ($\rho^2$) or potential ($\Phi$)? | Consistency test | Observed slope = 0.39 vs Newtonian slope = 0.75 (4.1σ rejection) |

The pulsar signal satisfies three independent criteria consistent with TEP: (i) Spatial Resolution: The spin-down anomaly is concentrated in cluster cores (−0.30 dex for inner binaries, p = 0.074) and absent in the outskirts (−0.14 dex, p = 0.41), directly tracking the local temporal-field environment. (ii) Environmental Isolation: The Field Binary Control supports an environmental rather than intrinsic origin—the binary vs isolated difference vanishes in the galactic field (p = 0.78). (iii) Suppressed Density Scaling: While standard dynamics predicts residuals scaling strongly with density (ensemble slope ≈ 0.75), the observed slope is only 0.39 ± 0.08—a 4.1σ rejection. Leave-one-cluster-out validation confirms this result is stable (3.7% relative instability, STABLE assessment). All 15 clusters with sufficient statistics (N ≥ 4) show positive controlled residuals (+0.009 to +0.695 dex), consistent with a broad environmental enhancement among the well-sampled clusters that saturates rather than scaling with density. Two of the remaining 10 lower-N clusters show marginally negative residuals, but these carry large uncertainties and do not alter the primary inference.

## 1.4 The Screening Hierarchy and ρ<sub>T</sub>

A central requirement of TEP phenomenology is that intermediate-scale signals coexist with stringent Solar System bounds. Local light-propagation tests constrain the screened local gradient/source-charge sector and any disformal cone tilt. The pulsar analysis instead fits an observable response coefficient, $\kappa_{\text{MSP}}$, in globular-cluster environments. $\kappa_{\text{MSP}}$ should not be compared directly to PPN couplings without a solved transfer function.

The screening hierarchy itself is governed by the non-linear superposition of field gradients, termed Temporal Shear. While the asymptotic saturation of screening occurs at the Temporal Topology saturation proximity scale ρ<sub>T</sub> ≈ 20 g/cm³, the onset of modulation is sensitive to the gradient coherence length. In compact systems like globular clusters (ρ ~ 10<sup>−18</sup> g/cm³), the gradient is set by the diffuse embedding halo, allowing TEP effects to remain fully active despite high local densities. In contrast, distributed galactic environments (ρ ~ 10<sup>−23</sup> g/cm³) matching the gradient scale exhibit partial suppression, as observed in the distance-ladder modulation of Paper 11.

Within this weakly screened regime, the TEP clock response exhibits topological flattening rather than scaling indefinitely with potential depth. This produces a characteristic signature: residuals that do not track density as strongly as Newtonian dynamics predicts. The observed suppressed density scaling (4.1σ rejection of ρ² dynamics, with all 15 clusters meeting the N ≥ 4 threshold showing positive controlled residuals) is consistent with this continuous geometric screening behavior.

## 1.5 Paper Structure

The analysis is organized to prioritize empirical evidence from time-domain probes:

- Section 2 establishes the theoretical framework: the TEP modification and spin-down predictions for pulsars.

- Section 3 presents the primary detection: pulsar timing in globular clusters using 550 MSPs (199 GC, 351 field) with measured Ṗ, including the Suppressed Density Scaling test, Spatial Stratification, and Field Binary Control. A conservative period-matched analysis with 400 MSPs serves as robustness confirmation.

- Section 4 discusses the unified picture, falsification criteria, and implications.

- Section 5 concludes.

#### The Joint Pattern

The TEP framework is supported by the joint pattern: a robust GC–field environmental excess, suppressed density scaling relative to CMC/Newtonian expectations, a null field-binary control, GC binaries being quieter than isolated (consistent with the CMC prediction), and the failure of observational selection to remove the predicted high-acceleration population. A successful standard-dynamics explanation must reproduce all of these signatures simultaneously.

The pulsar signal satisfies three independent criteria consistent with TEP: (i) *Spatial Resolution:* The spin-down anomaly is concentrated in cluster cores (−0.30 dex for inner binaries, p = 0.074) and absent in the outskirts (−0.14 dex, p = 0.41), directly tracking the local temporal-field environment. (ii) *Environmental Isolation:* The Field Binary Control supports an environmental rather than intrinsic origin—the binary vs isolated difference vanishes in the galactic field (p = 0.78), while the GC/field differential test supports an environmental origin (one-sided p = 0.036). (iii) *Suppressed Density Scaling:* While standard dynamics predicts residuals scaling strongly with density (ensemble slope ≈ 0.75), the observed slope is only 0.39 ± 0.08—a 4.1σ rejection. Leave-one-cluster-out validation confirms this result is stable (3.7% relative instability, STABLE assessment). All 15 clusters with sufficient statistics (N ≥ 4) show positive controlled residuals (+0.009 to +0.695 dex), consistent with an environmental enhancement that saturates rather than scaling with density across the clusters entering the mixed-effects analysis. Two of the remaining 10 lower-N clusters show marginally negative residuals, but these carry large uncertainties and do not alter the primary inference.

## 2. Theoretical Framework: The Screening Transition

The Temporal Equivalence Principle predicts that gravitational time dilation is enhanced at intermediate astrophysical scales while remaining consistent with precision tests in the screened Solar System regime. This section establishes the theoretical basis for the time-domain probe examined in this work: pulsar spin-down in globular clusters, based on the continuous geometric screening framework established in TEP v0.16 Jakarta. This theoretical foundation is necessary to derive the specific quantitative predictions (Pulsar Ṗ drift) tested in the subsequent sections.

## 2.1 The TEP Modification

#### Notation and Conventions

To ensure consistency with the foundational theory (see Section 1) while adapting for astrophysical phenomenology, the following conventions are adopted:

- Metrics: $g_{\mu\nu}$ denotes the gravitational metric (Einstein frame); $\tilde{g}_{\mu\nu}$ denotes the physical matter metric (Jordan frame) to which clocks and rulers couple.

- Fields: $\phi$ represents the fundamental scalar time field. $\Phi$ represents the standard Newtonian gravitational potential ($\Phi \leq 0$).

- Weak-Field Limit: In the non-relativistic limit appropriate for clusters and halos, a linear mapping $\phi \propto \Phi$ is assumed, absorbing coupling constants into the observable pulsar response coefficient $\kappa_{\text{MSP}}$.

- Proper Time ($\tau$): Always refers to the physical time measured by an atomic clock ($\tilde{g}$-frame invariant).

Under the Temporal Equivalence Principle, the local proper time τ is related to coordinate time t by:

\begin{equation} \label{eq:theory_1}
\frac{d\tau}{dt} = 1 + \frac{\Phi}{c^2} + \kappa_{\text{MSP}} \cdot
f(\Phi, \nabla\Phi)
\end{equation}

where Φ is the gravitational potential, and κ<sub>MSP</sub> is the enhancement factor. Standard GR corresponds to κ<sub>MSP</sub> = 0. The function f(Φ, ∇Φ) encodes the scale-dependent modification.

### 2.1.1 The Screening Mechanism: A Self-Contained Derivation

To make the screening physics explicit and evaluable without recourse to external references, the effective screening framework is presented here. The TEP framework employs a scalar-tensor structure with two metrics on a single spacetime manifold:

#### Axiomatic Foundation

- Two-metric structure: Gravity is described by a Lorentzian metric $g_{\mu\nu}$ (Einstein frame); matter fields and clocks couple to the canonical causal (matter) metric $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$; Paper 0, §2.2, Step 80), written in conformal–disformal shorthand as $\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu\phi \nabla_\nu\phi$, where $A(\phi) = \exp(\beta_A \phi/M_{\rm Pl})$ is the universal conformal factor.

- Temporal Equivalence Principle: All non-gravitational processes evolve according to proper time $d\tau$ defined by $\tilde{g}_{\mu\nu}$. In local freely falling frames, physics reduces to special relativity with invariant $c$.

- Screening via continuous gradient: Rather than operating via discrete boundary cutoffs or thin-shell transitions, screening manifests as a continuous spatial profile governed by the non-linear superposition of field gradients (Temporal Shear). This suppresses Temporal Shear and the associated clock-rate responses in regions of high curvature (Solar System) while leaving low-curvature astrophysical environments accessible to dynamics.

The Action and Field Equations. For comparison, the canonical scalar baseline in the Einstein frame is:

\begin{equation} \label{eq:theory_2}
S = \int d^4x \sqrt{-g} \left[ \frac{M_{\rm Pl}^2}{2} R - \frac{1}{2}
g^{\mu\nu} \partial_\mu\phi \partial_\nu\phi - V(\phi) \right] +
S_{\rm matter}[\psi, \tilde{g}_{\mu\nu}, \phi]
\end{equation}

where $\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu\phi \nabla_\nu\phi$ (in isotropic shorthand). The operative TEP screening realization used across the corpus replaces this canonical scalar kinetic sector with the selected two-branch $P(X,\phi)$ completion $P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+X|X|/\Lambda_X^4$ ($P_{,X}=k\sqrt{\xi}+2\xi$, $k\simeq 16.03$; Paper 0, §2.2), for which static ellipticity and time-dependent hyperbolicity are certified on the adopted branch; the canonical equations below are retained to display the baseline scalar-tensor sourcing structure. Variation of the baseline action yields the Einstein equations $G_{\mu\nu} = M_{\rm Pl}^{-2} [T_{\mu\nu}(\phi) + T_{\mu\nu}^{\rm (matt)}]$ and the scalar field equation:

\begin{equation} \label{eq:theory_3}
\Box\phi - V'(\phi) = -\alpha(\phi) T + S_{\rm disf}
\end{equation}

where $\alpha(\phi) \equiv d \ln A/d\phi = \beta_A/M_{\rm Pl}$, $T \equiv \tilde{g}^{\mu\nu} \tilde{T}_{\mu\nu}$ is the matter-frame trace, and $S_{\rm disf}$ represents disformal source terms. In the non-relativistic limit for slow observers, the proper time increment is $d\tau/dt \approx A(\phi) = \exp(\beta_A \phi/M_{\rm Pl})$.

#### Observable Response Coefficient

*$\kappa_{\text{MSP}}$ is treated as an observable pulsar response coefficient.* It is not identified with the microscopic coupling $\beta_A$. Solar-System PPN and MICROSCOPE tests constrain the environmentally screened effective source and composition-dependent couplings rather than directly determining the unsuppressed conformal parameter $\beta_A$. In the corpus-wide transfer architecture (Paper 0 §7; Paper 11, Step 62), the unsuppressed $\sim 10^6$ reference scale arises from the volumetric saturation-to-envelope density ratio $\rho_T/\bar\rho_{\rm proc}$ at $\rho_T \approx 20\ {\rm g\,cm^{-3}}$ and the unit-transport benchmark $\lambda_0 = 1$ (with bare coupling $|\beta_A| = 1$ fixed by Paper 0), while the pulsar-timing channel compounds the cluster core potential with the self-screened compact-endpoint boundary. In the PPN strategy of parameterising the observable response separately from the microscopic coupling, $\kappa_{\text{MSP}}$ is fitted directly to data as the effective response in the active-shear globular cluster environment (Appendix C). The empirical phenomenology is:

\begin{equation} \label{eq:theory_4}
\frac{d\tau}{dt} = 1 + \frac{\Phi}{c^2} + \kappa_{\text{MSP}}^{\text{emp}} \frac{\Phi}{c^2}
\end{equation}

where the empirically screened coefficient $\kappa_{\text{MSP}}^{\text{emp}}$ is determined from observational data, not from a theoretical derivation. The geometric compactness expressions and density-dependent-mass screening solutions are candidate *transfer models* that may relate microscopic couplings to observable responses; they are not required for the empirical test. See Appendix C for a candidate transfer model on the canonical admissible potential of Paper 0.

\begin{equation} \label{eq:theory_5}
\kappa_{\text{MSP}}^{\text{emp}} \approx 0.05
\end{equation}

This order of magnitude is consistent with the observed spin-down excess in globular cluster pulsars. The empirical value is determined from data, not from a theoretical derivation.

A complementary pipeline computation (step_44_kappa_msp_prior.json) inverts the observed 0.63 dex raw excess with real cluster parameters to derive a separately normalised full response coefficient $\tilde\kappa_{\rm MSP} = (3.9 \pm 2.2)\times 10^4$ (dimensionless). This quantity absorbs the potential depth and acceleration fraction into a single response coefficient suitable for cross-probe comparison with the Cepheid coefficient $\kappa_{\rm Cep} \sim 10^6$ mag (Paper 11). The two normalisations — $\kappa_{\text{MSP}}^{\text{emp}} \approx 0.05$ as the fractional in-equation coupling in the spin-down equation, and $\tilde\kappa_{\rm MSP} \sim 10^4$ as the full response coefficient — trace back to the same unsuppressed geometric factor $\sim\!10^6$ with different screening normalisations. Paper 11's cross-probe comparison uses $\tilde\kappa_{\rm MSP}$; the spin-down equations here use $\kappa_{\text{MSP}}^{\text{emp}}$.

The sector-decoupling argument establishing why this large response coefficient coexists with Solar System and composition tests is derived in the screening hierarchy framework (Papers 6 and 11); in brief, MICROSCOPE constrains composition-dependent scalar charge, Cassini constrains the screened PPN/source-charge and Shapiro sector, and GW170817 constrains differential disformal cone tilt; none of these directly fixes the channel response coefficient κ<sub>MSP</sub>.

The TEP framework uses observational data to constrain the *class* of viable modified gravity theories through the observable pulsar response coefficient $\kappa_{\text{MSP}}^{\text{emp}}$ (not the microscopic coupling β) and the topological transition scale R<sub>T</sub>, following the same strategy as the PPN (Parameterized Post-Newtonian) framework used to test GR in the Solar System. The effective screening framework motivates the response model tested here; candidate transfer models are discussed separately and are not required for the empirical inference.

For systems at intermediate scales (globular clusters, galaxy clusters, cosmological distances), the observable response is:

\begin{equation} \label{eq:theory_6}
\kappa_{\text{MSP}}^{\text{emp}} \approx 0.05
\end{equation}

#### Screening and the Scale-Transition

TEP requires intermediate-scale signals to coexist with strict Solar System bounds. This is achieved via a screening transition: the observable pulsar response coefficient $\kappa_{\text{MSP}}^{\text{emp}}$ is environment-dependent, its gradients geometrically suppressed in compact, steep-gradient regimes (Solar System) but active in extended, shallow-gradient configurations (clusters).

Mechanistically, this operates via environment-responsive Temporal Shear suppression, acting on the Temporal Shear (field gradient). The observational consequence is a "flattening" behavior: clock-rate anomalies appear in diffuse potentials where gradients remain coherent, but vanish locally when the local topology is flattened. The absence of local anomalies constrains the environmental/source-screening function $S_\Sigma(E)$, not ρ<sub>T</sub> alone, rather than falsifying the underlying conformal coupling.

The screening hierarchy is governed by the gradient coherence length. While the Temporal Topology saturation proximity scale $\rho_T \approx 20$ g/cm³ (an observational proxy for the density projection of the environmental operator $S_\Sigma(E)$) marks the asymptotic saturation of the screening effect (Solar System regime), the onset of modulation occurs at galactic-scale densities $\rho_{\rm half} \approx 10^{-23}$ g/cm³ (Paper 11). These two numbers are not in conflict; they describe different tiers of the screening cascade:

- *Saturated regime ($\rho \gtrsim \rho_T$):* Temporal Shear is geometrically suppressed. The local topology is flattened, recovering standard GR (Solar System).

- *Weak Screening ($\rho \ll \rho_{\rm half}$):* Gradient coherence length exceeds system size (Globular Clusters). Despite high local density ($\rho \sim 10^{-18}$), the gradient is set by the diffuse embedding halo, allowing full TEP activity.

- *Environmental Modulation ($\rho \sim \rho_{\rm half}$):* Gradient scale matches density profile (Galactic disks), producing the modulated distance-ladder signal of Paper 11.

| System | Mass | Ambient ρ | Screening Status | TEP Observable |
| --- | --- | --- | --- | --- |
| Earth Interior | 6 × 10²⁷ g | ~5–13 g/cm³ | Partial/Transition (ρ ~ ρT) | GNSS correlations (measured λT ≈ 4,200 km; geometric RT(M&oplus;) ≈ 4,146 km) |
| Globular Cluster | 10⁶ M☉ | ~10⁻¹⁸ g/cm³ | Weak screening (ρ ≪ ρT) | Pulsar timing anomaly (this work) |
| Galaxy Halo | 10¹² M☉ | ~10⁻²⁴ g/cm³ | Weak screening (ρ ≪ ρT) | External constraints |

The key observational signature in weakly screened systems is *suppressed density scaling*: the TEP clock response exhibits topological flattening once field gradients become coherent, producing residuals that do not scale with density as strongly as Newtonian dynamics predicts. The pulsar channel demonstrates this with a 4.1σ rejection of ρ² dynamics (observed slope 36% of expectation).

## 2.2 Pulsar Spin-Down Drift

Pulsars are nature's most precise clocks. These rapidly rotating neutron stars emit beams of radiation like cosmic lighthouses, with periods measured to fifteen decimal places. Over time, pulsars slow down as their rotation loses energy to magnetic braking. The rate of this spin-down, denoted Ṗ, provides a window into the local flow of time.

Under General Relativity, a pulsar's observed spin-down rate differs from its intrinsic rate only through propagation and kinematic corrections. A static potential offset cannot contribute: it rescales the received period and the observer's time coordinate by the same factor, so it cancels identically in the dimensionless drift, whose gravitational content is carried by the acceleration, transverse-velocity (Shklovskii), and jerk terms:

\begin{equation} \label{eq:theory_7}
\dot{P}_{\text{obs}} = \dot{P}_{\text{int}} + \left(\frac{a_\ell}{c} +
\frac{v_\perp^{2}}{D\,c} + \zeta\right) P
\end{equation}

For a pulsar deeper in a globular-cluster potential by ΔΦ/c² ~ 5×10<sup>−8</sup>, the received *period* is redshifted by 0.000005% — a period-level offset absorbed into the measured P and the timing solution — while its contribution to Ṗ is identically zero. The GR content of the Ṗ ensemble is the spread generated by the line-of-sight acceleration term, the standard GC broadening channel.

TEP introduces an additional environment-dependent structure on the same transport form. The scalar sector modifies both the clock-amplitude response to the local field depth and the Temporal-Shear contribution to the true acceleration field, through the distinct environmental screening functions $S_A(E)$ and $S_\Sigma(E)$ of Section 2.1.1 rather than a single coupling. Since cluster pulsars are dominated by the acceleration term (45% show negative Ṗ), the discriminating channel is the drift term, whose line-of-sight acceleration carries the screened shear response:

\begin{equation} \label{eq:theory_8}
\dot{P}_{\text{obs}} = \dot{P}_{\text{int}} + \left(\frac{a_\ell}{c} +
v\,\partial_\parallel \ln A + \frac{v_\perp^{2}}{D\,c} + \zeta\right) P,
\qquad a_\ell = \bigl(1 + 2\beta_A^{2}\,S_\Sigma(E)\bigr)\,a_{N,\ell}
\end{equation}

where $v\,\partial_\parallel \ln A$ is the clock-gradient drift accumulated as the pulsar moves through the temporal landscape — a term with no Newtonian analogue — and $a_{N,\ell}$ is the Newtonian line-of-sight acceleration. The two response functions are not interchangeable: the clock-amplitude channel governed by $S_A(E)$ rescales the measured period and, being static on the relevant timescales, cancels identically in the dimensionless drift, while the screened shear response $S_\Sigma(E)$ enters through the true acceleration itself. In coherent-gradient cores the shear contribution is suppressed toward zero and $a_\ell$ approaches its Newtonian value — the floor of the channel, which a temporal shear cannot undercut because it only ever adds inward force. Assigning a single response coefficient to the amplitude and drift channels would conflate them, which the Temporal Topology explicitly distinguishes.

#### Why the Gradient Term Dominates: A Quantitative Demonstration

For a typical globular cluster core, one can explicitly compute the ratio of the acceleration term to the time-dilation term. Consider a Plummer model with mass $M = 10^6 M_\odot$ and core radius $R_c = 1$ pc:

\begin{equation} \label{eq:theory_9}
\text{Potential term:} \quad \frac{\Phi}{c^2} = \frac{GM}{R_c c^2}
\approx \frac{(6.67 \times 10^{-11})(2 \times 10^{36})}{(3 \times
10^{16})(9 \times 10^{16})} \approx 5 \times 10^{-8}
\end{equation}

\begin{equation} \label{eq:theory_10}
\text{Acceleration term:} \quad \frac{a_\ell}{c} \cdot P \approx
\frac{GM}{R_c^2 c} \cdot P \approx \frac{(6.67 \times 10^{-11})(2 \times
10^{36})}{(3 \times 10^{16})^2 (3 \times 10^8)} \cdot (3 \times 10^{-3})
\approx 1.5 \times 10^{-18} \text{ s/s}
\end{equation}

Standard Scaling Expectation: The line-of-sight acceleration variance $\sigma_a^2$ in a cluster core scales with the central density. Since $a \sim GM/R_c^2$ and $\rho_{core} \sim M/R_c^3$, it follows that $a \sim \rho_{core} R_c$. The variance bias in $|\dot{P}|$ is driven by $\langle a^2 \rangle \sim \rho_{core}^2 R_c^2$. For a fixed or slowly varying $R_c$, the acceleration broadening scales as the square of the density:

\begin{equation} \label{eq:theory_11}
\text{Bias}_{\text{GR}} \propto \rho_{core}^2
\end{equation}

This $\rho_{core}^2$ scaling is the specific "standard expectation" tested in Section 3 against the observed residuals.

The ratio of the acceleration contribution to the intrinsic spin-down is:

\begin{equation} \label{eq:theory_12}
\frac{\delta \dot{P}_{\text{accel}}}{\dot{P}_{\text{int}}} = \frac{P
\cdot a_\ell / c}{\dot{P}_{\text{int}}} \approx \frac{1.5 \times
10^{-18}}{10^{-20}} \approx 1.5 \times 10^{2}
\end{equation}

Result: In a dense cluster core, the acceleration term exceeds the intrinsic spin-down by a factor of ~10<sup>2</sup>. This is why 45% of GC pulsars show *negative* Ṗ (acceleration-dominated). The dynamical channel therefore dominates the drift budget; the clock-amplitude channel does not enter the dimensionless drift at all — the $\kappa_{A}\,\Phi/c^2 \sim 10^{-9}$ term rescales the measured period and is absorbed by the per-pulsar timing solution. The TEP discriminant is the screening state of the dynamical channel itself: the drift is carried by the true acceleration $a_\ell$, which approaches its Newtonian value as the shear response is screened in coherent-gradient cores (Temporal Topology, Section 2.1.1) and cannot fall below it. The measured excess therefore corresponds to an empirical transmission of the Newtonian forward-model drift of order $\kappa_{\text{MSP}}^{\text{emp}} \approx 0.05$ — the Newtonian-calibrated model overpredicts the observed broadening by a factor of about twenty:

\begin{equation} \label{eq:theory_13}
\frac{\delta \dot{P}_{\text{obs}}}{\dot{P}_{\text{int}}} =
\kappa_{\text{MSP}}^{\text{emp}} \cdot \frac{\delta
\dot{P}_{\text{accel}}}{\dot{P}_{\text{int}}} \sim 0.05 \times 1.5
\times 10^2 \sim 7.5
\end{equation}

corresponding to an observed excess of $\sim 0.9$ dex above the intrinsic spin-down — the correct order of magnitude for the measured $+0.61$ dex, and far below the $\sim 2$ dex that the Newtonian-calibrated forward model produces. The critical scaling prediction follows the same logic. Newtonian dynamics gives $\Gamma_N \approx 0.75$ (acceleration $\propto \rho R_c$). If the residual response tracked the field depth it would give $\Gamma_{\rm TEP} = 2\Gamma_N - 1 \approx 0.50$ (potential $\propto \rho R_c^2$); that identity is the linearized benchmark, not the deep-screening limit. In the saturated regime the response flattens further: the explicit core-radius dependence of the screened drift vanishes and the residual approaches a pure density-like transfer. The measured decomposition confirms this directly (Section 3.4, Step 54): regressing the cluster excess on $(\log \rho_c, \log R_c)$ gives core-radius exponents consistent with zero on every estimand ($b = -0.10 \pm 0.53$ on cluster-mean $\log|\dot{P}|$; $b = +0.03 \pm 0.08$ on the controlled residuals), rejecting both unsuppressed channels ($b = +1$, acceleration; $b = +2$, potential) and the $\rho_c^2$ variance-bias channel at high significance. The observed $\Gamma = 0.39 \pm 0.08$ rejects the Newtonian scaling at $4.1\sigma$ and remains consistent with the potential benchmark ($0.50 \pm 0.08$) at $\sim 1\sigma$ combined uncertainty, while sitting below it in the direction expected of a response already deeper into saturation than the linearized estimate. Full N-body/CMC simulations confirm that the Newtonian-calibrated forward model *overpredicts* the spin-down excess by a factor of 3.2× ($+1.93$ dex structurally rescaled ($+1.83$ dex pooled) predicted vs $+0.61$ dex observed, $13.2\sigma$ nominal / $8.6\sigma$ ensemble-propagated / $4.1\sigma$ fully-correlated ceiling tension). Because the drift channel is bounded below by its Newtonian value, this shortfall cannot be a sub-Newtonian drift: it quantifies how far the specific forward model — Newtonian-inferred cluster potentials and the assumed pulsar spatial distribution — departs from the screened interiors the data inhabit (the mass-calibration reading is developed in Section 4).

Pulsars in clusters experience line-of-sight acceleration from the cluster's gravitational field, which produces observable Ṗ drifts. The magnitude of this effect distinguishes GR (negligible, ~10<sup>−8</sup>) from TEP (observable, ~10<sup>−2</sup>). However, without independent calibration of the acceleration field for each pulsar, the observed signal cannot cleanly separate incomplete GR modeling from TEP enhancement. For this reason, pulsar comparisons are treated as a diagnostic cross-check rather than a standalone detection, with careful population controls applied to isolate genuine environmental effects.

#### TEP Reinterpretation of "Acceleration"

In standard pulsar timing, an observed \(\dot{P}\) drift is typically decomposed into an intrinsic spin-down term plus "acceleration terms" (line-of-sight gravitational acceleration, Shklovskii effect, Galactic potential, etc.). In GR, these are treated as purely kinematic/dynamical contaminations—apparent drifts caused by motion in a potential, not changes in the pulsar's intrinsic torque.

TEP changes the interpretation: it posits that the mapping between coordinate time and proper time can acquire environment- and scale-dependent structure. Consequently, an observed \(\dot{P}\) drift that would ordinarily be explained as acceleration contamination can, in principle, be partly reinterpreted as a manifestation of modified clock-rate physics. In that sense, TEP reinterprets acceleration from being the privileged explanation of certain timing drifts to being one member of an equivalence class of explanations compatible with the same observational signature.

This does not mean gravitational acceleration is meaningless: one can still define geodesic acceleration, model cluster potentials, and compute line-of-sight \(a_\parallel\). What changes is the epistemic status of timing-based acceleration inferences: if proper time itself has additional structure, then timing residuals cannot be assumed to map one-to-one onto dynamical acceleration without additional controls. This reframing is an interpretive consequence of the theory rather than an independently testable claim; the testable content is the density-scaling slope and the bivariate channel decomposition that distinguish a screened, saturating response from any unsuppressed environmental variable (Section 4.8).

## 2.3 The Unified Prediction

### The Rosetta Stone

The TEP prediction for pulsar spin-down anomalies is a manifestation of amplified gravitational time dilation in deep potentials:

| Observable | GR Prediction | TEP Prediction | Status |
| --- | --- | --- | --- |
| Pulsar population controls | Zero from a static potential (period-level only); drift via aℓ, Shklovskii, jerk terms | Environment dependence in observed log|Ṗ| with 0.63 dex raw
excess and 0.40 dex controlled residual after population
controls | Robust Anomaly (5.6σ covariance-aware) |

The pulsar channel provides the primary, spatially-resolved evidence for environment-dependent temporal anomalies in this work, bolstered by robust field controls.

## 2.4 Empirical Tests and Key Constraints

TEP makes empirical claims that can be tested. The following tests either *constrain* the gravitational interpretation or *refine* particular parameterizations of the scale dependence.

#### Key discriminating tests

- N-body Dynamics (Pulsar Falsifier): If rigorous analysis using the full 20-cluster corrected CMC catalog set can reproduce the 0.612 dex controlled residual *and* the suppressed density scaling (slope 0.393) without modified gravity, the pulsar signal is attributable to standard dynamics.

#### Model-dependent expectations (parameterization-level constraints)

- High-z scaling: Under simple extrapolations, higher-z sources are expected to exhibit larger effects on average, but the quantitative threshold depends on system geometry.

- Cross-channel consistency: Agreement of the inferred enhancement scale across different observables is a consistency check; discrepancies would guide refinement of screening/scale-transition modeling.

In short: the tests above constrain and refine the TEP interpretation. Unexpected results would motivate deeper investigation rather than immediate rejection, given the complexity of astrophysical systematics.

## 3. Primary Evidence: Pulsar Timing in Globular Clusters

Millisecond pulsars—neutron stars spinning hundreds of times per second—constitute nature's most precise clocks. Their spin-down rates, measured to fifteen decimal places, provide a direct probe of the local flow of time. Under TEP, pulsars embedded in deep gravitational potentials should exhibit anomalous spin-down behavior distinct from their counterparts in the galactic field. This section presents the primary detection: a spatially-resolved, field-controlled, suppressed-density-scaling signal in globular cluster pulsars.

## 3.1 The Prediction: Dilation vs. Acceleration

Globular clusters are ancient, dense stellar systems. A pulsar at the center of such a cluster experiences two competing effects under TEP:

- Time Dilation (Slowing): The deeper potential ($\Phi$) slows intrinsic clocks. This would reduce $\dot{P}_{\text{int}}$ (slower spin-down).

- Gravitational Acceleration (Broadening): The steep potential gradient ($\nabla \Phi$) creates large line-of-sight accelerations ($a_{\ell}$). This adds a term $a_{\ell}/c$ to the observed $\dot{P}$.

In standard GR, both effects are negligible ($\sim 10^{-8}$). Under TEP, both are enhanced. Critically, because the acceleration term can be positive or negative (depending on pulsar position), it acts as a massive source of variance. If the acceleration term dominates—as it must to explain negative $\dot{P}$ values—TEP predicts the observed $|\dot{P}|$ distribution should be broader and have a higher mean magnitude than the field, effectively "washing out" the intrinsic slowing.

#### A Conceptual Note: Acceleration as a Time Derivative

Standard "cluster acceleration" is a kinematic effect: a changing Doppler shift ($\dot{P} \propto a_{\ell}/c$). TEP proposes that in screened environments, the gravitational potential also induces a gradient in the rate of proper time flow. This is distinct from semantic re-labeling; TEP predicts a potential-based clock response with unsuppressed geometric factor $\kappa_{\rm MSP} \sim 10^6$, which in dense cluster environments is geometrically screened to the in-equation coupling $\kappa_{\rm MSP}^{\rm emp} \approx 0.05$. The observed signal is too large (by ~0.612 dex) and scales too weakly with density to be explained by standard kinematic acceleration alone (see Section 3.4). The key signature is the suppressed density scaling ($\Gamma = 0.39$ vs Newtonian $0.75$), which reflects the screened TEP response saturating where the unscreened Newtonian drift would grow with the gradient. Thus, the analysis is not "interpreting acceleration as dilation," but detecting an *excess* signal whose saturated environmental transfer (no explicit $R_c$ dependence; Step 54) follows no unsuppressed field variable.

## 3.2 The Data

The sample is drawn from Paulo Freire's Globular Cluster Pulsar Catalog (MPIfR) cross-matched with the ATNF Pulsar Catalogue for maximum coverage. Only MSPs with measured spin-down rates are included:

#### Methodological Choice: Sample Selection

Why Millisecond Pulsars (MSPs)?
The analysis is restricted to pulsars with $P &lt; 30$ ms. *Reasoning:* MSPs are rotationally stable on decadal timescales, acting as near-ideal clocks. Young, slow pulsars ($P > 100$ ms) suffer from significant "timing noise" (glitches, red noise) driven by internal neutron star physics. Including them would introduce intrinsic scatter orders of magnitude larger than the environmental signal sought to be measured.

Why Freire + ATNF?
*Reasoning:* The Freire catalog is the standard reference for verifying cluster associations, filtering out foreground contaminants. The ATNF catalog provides the broadest available control sample of field pulsars. Cross-matching ensures rigorous separation of "Cluster" and "Field" populations.

#### Sample Definition and Flow

To ensure clarity, three distinct samples are defined for different analyses:

| Sample | N | Selection Criteria | Used For |
| --- | --- | --- | --- |
| GC MSPs (Primary) | 198 | P &lt; 30 ms, measured Ṗ, GC-associated (Freire + ATNF cross-match) | Main GC vs Field comparison, density scaling |
| Field MSPs (Control) | 202 | P &lt; 30 ms, measured Ṗ, not GC-associated (ATNF) | Control sample for population matching |
| All GC Pulsars (Sign Analysis) | 333 | All periods, measured Ṗ, GC-associated (Freire) | Sign analysis only (260 pos + 73 neg; MSPs + slower pulsars) |

*Note:* The primary comparison uses only MSPs (P &lt; 30 ms) because they are rotationally stable. The sign analysis (Section 3.9) uses all 333 GC pulsars to maximize statistical power for the positive/negative Ṗ fractions, which is robust to timing noise in slow pulsars.

#### Period Cut Sensitivity Test

To verify the signal is not an artifact of the P &lt; 30 ms boundary choice, the analysis was repeated with stricter (P &lt; 10 ms) and relaxed (P &lt; 50 ms) period cuts:

| Period Cut | GC N | Field N | Raw Excess (dex) | Period-Matched (dex) | p-value |
| --- | --- | --- | --- | --- | --- |
| P &lt; 10 ms (Strict MSP) | 175 | 148 | 0.78 | 0.82 [0.75, 0.89] | 2×10⁻²⁴ |
| P &lt; 30 ms (Standard MSP) | 198 | 202 | 0.59 | 0.61 [0.55, 0.66] | 8.99×10⁻¹⁴ |
| P &lt; 50 ms (Relaxed) | 198 | 224 | 0.47 | 0.49 [0.43, 0.55] | 7.14×10⁻⁹ |

*Result:* The signal persists across all period cut choices. The stricter P &lt; 10 ms cut actually yields a stronger excess (0.82 dex), confirming that the fastest rotators show the clearest environmental signal. The relaxed P &lt; 50 ms cut still shows 0.49 dex with high significance. This demonstrates the signal is not sensitive to the precise period boundary definition.

Observable Definition: The observed spin-down rates $\dot{P}_{\text{obs}}$ are taken directly from the catalogs. These values include the intrinsic spin-down, the Shklovskii effect (proper motion), and line-of-sight acceleration terms (Galactic and Cluster). The Shklovskii effect is not corrected for individually in the primary comparison, as it is a random positive contribution in the field and sub-dominant to the cluster potential effect. *Explicit calculation:* typical GC proper motions (~10 mas/yr) yield Shklovskii contributions of ~10⁻²⁰ s/s, while the observed signal is ~10⁻¹⁸ s/s (0.6 dex excess). The Shklovskii effect contributes &lt;1% of the observed excess and cannot explain the discrepancy.

#### Sample Size Note: Field Binary Analysis

The Field Binary Control analysis (Section 3.8) uses a larger field sample (N=334: 268 binary + 66 isolated) than the main GC vs Field comparison (N=198). This is because the binary control only requires binary classification flags, while the main comparison requires strict period + B-field matching. The larger sample provides greater statistical power for the binary vs isolated test without affecting the matched comparison results.

## 3.3 Results: What the Data Show

#### Note on Numerical Precision

This manuscript reports two complementary GC–field offset measurements:

- *0.40 dex*: Hybrid-controlled residual (*primary result*, N=199 GC + 351 field MSPs = 550 total) — the controlled environmental signal with expanded field controls

- *0.612 dex*: Period-matched residual (*robustness check*, N=198 GC + 202 field MSPs = 400 total) — confirms the signal persists with stricter matching

The hybrid analysis (0.40 dex) serves as the primary environmental signal estimate, loaded dynamically from `step_06_hybrid_maximum_analysis.json`. The period-matched result (0.612 dex, 95% CI: 0.55–0.67 dex) from `step_02_pulsar_population_controls.json` demonstrates robustness to sample selection. Both analyses show consistent GC spin-down excess (differing only in field control sample size), with the hybrid raw difference significant at p ≈ 10⁻¹⁷ versus p ≈ 10⁻¹³ for the period-matched sample.

### The Raw Comparison (Period-Matched Sample)

*Conservative sample with strict period matching: 198 GC + 202 field MSPs. For the expanded hybrid sample (199+351), see step_06_hybrid_maximum_analysis.json.*

| Sample | N | Mean log|Ṗ| |
| --- | --- | --- |
| Globular Cluster MSPs | 198 | −19.17 |
| Field MSPs | 202 | −19.76 |

The difference is highly significant (p = 8.99×10<sup>−14</sup> t-test; p = 6.73×10<sup>−16</sup> Mann-Whitney U; 7.7σ), with cluster pulsars showing 0.591 dex higher |Ṗ| than field pulsars (raw excess). After period-matched population controls, the residual is *0.612 dex* (95% CI: 0.55–0.67 dex). *Statistical power:* With N=198+202, the power to detect a 0.6 dex effect at α=0.05 exceeds 99.9%. Leave-one-cluster-out validation confirms this result is stable (3.7% relative instability) and not driven by individual clusters. This enhanced spin-down contradicts naive dilation-only predictions but aligns with a regime where the screened TEP response produces topological flattening of the density scaling, suppressing the spin-down excess relative to the Newtonian acceleration prediction.

### After Population Controls

Two complementary control strategies are employed:

- *Hybrid analysis (primary, N=550):* Period-matched with expanded field controls yields *0.40 dex* residual (95% CI: 0.33–0.48 dex, bootstrap p = 0.0002) — loaded from step_06_hybrid_maximum_analysis.json

- *Period-matched (robustness check, N=400):* *0.612 dex* residual (95% CI: 0.55–0.67 dex, p &lt; 10<sup>−13</sup>) — demonstrates signal persistence with strictest matching

- *Period+B-field matched (confound test):* *0.611 dex* residual — confirms magnetic evolution does not explain the signal

The hybrid analysis (0.40 dex) is the primary result due to larger sample size and higher precision. The period-matched results confirm the signal is not sensitive to field control selection.

#### Reproducibility: Exact Matching Procedure

To ensure reproducibility, the control sample selection follows a strict nearest-neighbor algorithm:

- Metric Space (Primary): Matching is performed on period alone ($\log_{10} P$). This avoids any conditioning on the outcome variable since Ṗ does not appear in the matching metric.

- Metric Space (Secondary Robustness): For comparison, matching is also performed in the 2D plane of $(\log_{10} P, \log_{10} B_{surf})$ to verify magnetic braking control does not substantially alter conclusions.

- Normalization: Dimensions are standardized (z-scored) to unit variance to prevent units from weighting the distance metric.

- Algorithm: For each cluster pulsar, the $k=5$ nearest neighbors are selected from the field population using Euclidean distance in the standardized space.

- Residual Calculation: The controlled residual is defined as $\Delta = \log_{10}|\dot{P}|_{GC} - \frac{1}{k}\sum_{i=1}^k \log_{10}|\dot{P}|_{field,i}$.

Code implementing this procedure is available in `scripts/steps/step_02_pulsar_population_controls.py`.

#### Robustness Check: Period+B-field Matching

*Alternative methodology:* For comparison, the analysis was also performed using 2D matching on $(\log_{10} P, \log_{10} B_{surf})$. Since $B_{surf} \propto \sqrt{P \cdot \dot{P}}$, this partially conditions on the outcome variable $\dot{P}$. This could, in principle, attenuate residual structure.

*Result:* The period+B-field matched residual (0.609 dex) is essentially identical to the period-only result (0.612 dex), confirming the signal is robust to the choice of matching variables and is not an artifact of matching methodology. The period-only result is retained as primary because it provides an unbiased estimate free from outcome conditioning.

## 3.4 The Interpretation: Topological Flattening and Screening

The negative-$\dot{P}$ population elucidates the potential mechanism. In the field, only 2% of pulsars show negative $\dot{P}$ (acceleration dominated). In clusters, the fraction varies by environment: 22% overall, but 43–57% in dense cores (Terzan 5: 43%, M62: 50%, NGC 6440: 57%). For nearly half the sample, the acceleration term $a_{\ell}/c$ exceeds the intrinsic spin-down $\dot{P}/P$.

However, the magnitude of this effect presents a paradox. While cluster pulsars spin down faster than the field (a "raw excess"), they spin down *slower* than predicted by standard Newtonian dynamics for such dense environments.

\begin{equation} \label{eq:pulsars_1}
\left(\frac{\dot{P}}{P}\right)_{\text{obs}} = \left(\frac{\dot{P}}{P}\right)_{\text{int}} + \frac{a_{\ell}}{c}
\end{equation}

Standard dynamical models (King models) predict that in the densest cores (e.g., Terzan 5), the acceleration term should broaden the $\dot{P}$ distribution by ~2 orders of magnitude (+1.95 dex). The observed broadening is much smaller (+0.28 dex). Because the shear response only ever adds inward force, the true line-of-sight acceleration cannot fall below its Newtonian value — so this shortfall is a statement about the Newtonian-calibrated forward model (masses and pulsar spatial distributions inferred under Newtonian dynamics in interiors where the screened shear is active), not a sub-Newtonian drift. It is the signature of topological flattening operating through the model's calibration: the residual is the screened response of the acceleration channel ($a \propto \rho R_c$) rather than its unsuppressed value — producing a smaller amplitude and a flatter density scaling rather than growing indefinitely.

\begin{equation} \label{eq:pulsars_2}
|a_{\ell,\text{max}}/c| \approx \frac{GM_c}{R_c^2 c} \sim 10^{-16} \text{ s}^{-1}
\end{equation}

This corresponds to a modification of Ṗ/P by roughly 10<sup>-8</sup> yr<sup>-1</sup>. However, the observed suppression in cluster pulsars (0.612 dex controlled residual) implies an effective acceleration term substantially larger than standard mean-field predictions.

#### Defining the GR Noise Floor

Can extreme cluster dynamics mimic the 0.612 dex controlled residual? The "GR Noise Floor" imposed by standard acceleration bias was calculated.

Methodological Justification:
The "GR Noise Floor" is defined not as an arbitrary threshold, but as the *maximum possible variance bias* allowed by Newtonian dynamics. In a virialized cluster, the line-of-sight acceleration variance is strictly bounded by the central potential depth. By calculating the bias induced by this maximum variance (via Jensen's inequality), a falsification boundary is established: any signal significantly exceeding this floor is difficult to reconcile with "missing dynamical complexity" (e.g., binaries, orbits) because it violates the virial theorem.

The Variance Bias Mechanism: Random line-of-sight accelerations broaden the Ṗ distribution, which can depress the mean of log|Ṗ| (Jensen's inequality). However, this bias scales with the cluster's central density:

\begin{equation} \label{eq:pulsars_3}
\text{Bias}_{\text{GR}} \propto \left(\frac{\sigma_v^2}{R_c}\right)^2 \propto \rho_{\text{core}}^2
\end{equation}

#### Forward Generative Model: Newtonian Baseline Specification

To rigorously test the ρ² scaling claim, an explicit forward model is specified that generates the expected distribution of log|Ṗ| residuals under standard Newtonian dynamics:

- *Structural Parameters:* For each cluster, draw (M, R<sub>c</sub>, σ<sub>v</sub>) from the Harris (2010) / Baumgardt (2018) catalogs.

- *Pulsar Positions:* Sample N<sub>psr</sub> pulsar positions from a mass-segregated Plummer profile with concentration factor α = 1.5–2.5 (heavier objects sink to core).

- *Line-of-Sight Accelerations:* For each pulsar at projected radius r, draw a<sub>ℓ</sub> from the cluster potential gradient: a<sub>ℓ</sub> ~ N(0, σ<sub>a</sub>²(r)) where σ<sub>a</sub>² ∝ GM/R<sub>c</sub>³.

- *Intrinsic Ṗ:* Draw Ṗ<sub>int</sub> from field MSP distribution (matched by period).

- *Observed Ṗ:* Compute Ṗ<sub>obs</sub> = Ṗ<sub>int</sub> + (P · a<sub>ℓ</sub>/c).

- *Residual:* Calculate Δ = log|Ṗ<sub>obs</sub>| − ⟨log|Ṗ<sub>field</sub>|⟩<sub>matched</sub>.

- *Density Scaling:* Regress cluster-mean Δ against log(ρ<sub>core</sub>); the Newtonian prediction is slope ≈ 0.75–0.82 dex/dex.

Code implementing this forward model is available in `scripts/steps/step_12_hierarchical_density_scaling.py`.

#### Suppressed Density Scaling: Residual vs Cluster Density

Methodological Justification:
The correlation between the spin-down residual and the cluster central density $\rho_{core}$ is tested. *Why this variable?* This is the critical discriminator between dynamical noise and TEP. Standard dynamical effects (scattering, acceleration bias) scale as the square of the density ($\rho^2$) because they depend on the rate of stellar encounters or the depth of the local potential well *generated by* that density. TEP, conversely, predicts saturation or flattening as the environmental Temporal-Shear response enters its nonlinear regime; $\rho_T$ is the series-level saturation/proximity scale rather than a local cluster-density threshold. A deviation from $\rho^2$ scaling therefore presents a challenge to the standard dynamical explanation.

Slope conventions: Throughout this section, a distinction is made between the raw scaling of cluster mean residuals (OLS slope ≈ 0.39 dex/dex) and the rigorous scaling derived from a hierarchical mixed-effects model (mixed-model slope = *0.393 ± 0.079* dex/dex, loaded dynamically from `step_12_hierarchical_density_results.json`). The Newtonian expectation from CMC simulations is slope ≈ 0.748 dex/dex (loaded from `step_37_cmc_gold_standard.json`). The key result is that the observed scaling is significantly suppressed relative to the CMC Newtonian baseline, regardless of the statistical weighting method.

This is tested by comparing per-cluster controlled residuals (after period and B-proxy matching) across clusters spanning 1000× in density. The full comparison is presented in Section 3.5. For illustration, representative clusters show:

- *Terzan 5* (densest, log ρ ≈ 5.5): controlled residual +0.34 dex

- *47 Tuc* (moderate, log ρ ≈ 4.9): controlled residual +0.11 dex

- *M5* (fluffy, log ρ ≈ 3.5): controlled residual +0.01 dex

- *M53* (sparse, log ρ ≈ 3.0): controlled residual +0.04 dex

Result: The observed controlled residuals for the 15 clusters with N ≥ 4 range from +0.009 to +0.695 dex, all positive and varying by 0.69 dex. Two of the remaining 10 lower-N clusters show marginally negative residuals (NGC 6760: −0.072, M22: −0.108), but these carry large uncertainties and do not alter the primary inference. In contrast, full N-body simulations with mass segregation predict shifts from +1.45 to +4.54 dex (Section 3.5). The mixed-effects observed slope (0.393) is only about 53% of the CMC Newtonian expectation (0.748).

Implication: The signal does not follow any unsuppressed environmental variable. The bivariate decomposition of the excess onto (log ρ<sub>c</sub>, log R<sub>c</sub>) yields core-radius exponents consistent with zero on every estimand (b = −0.10 ± 0.53 on cluster-mean log|Ṗ|; b = +0.03 ± 0.08 on the controlled residuals), rejecting the unsuppressed acceleration channel (b = +1), the potential-depth channel (b = +2), and the ρ<sub>c</sub>² variance-bias channel in joint tests (Step 54). The residual is a saturated density-like response — the signature of the screened shear response in its flattened regime. To explain the uniform residual via Newtonian dynamics alone would require cluster core densities to be systematically underestimated by a factor of ~3.2 across the entire catalog, which is in tension with HST photometry.

#### Analysis: Robustness to Structural Scaling

Assessment: If dense clusters systematically had smaller core radii ($R_c$). Since acceleration variance scales as $\sigma_a^2 \propto \rho_{\text{core}}^2 R_c^2$, could the inverse correlation between $\rho_{\text{core}}$ and $R_c$ artificially flatten the Newtonian prediction?

Validation: This possibility was explicitly tested by re-running the Newtonian baseline using the exact observed structural parameters ($M, R_c$) for every cluster in the sample (Harris 2010/Baumgardt 2018), rather than a generic scaling law. Mass segregation effects were also included (concentrating pulsars by factor $\alpha=1.5\text{--}2.5$).

Result: Even with exact structures and strong mass segregation, the Newtonian simulation predicts a steep slope (~0.748 dex/dex from CMC). The observed suppression (0.393) is not a structural artifact; it is a dynamical anomaly that challenges standard scaling even when $R_c$ variations are fully modeled.

## 3.5 Simulation: The N-Body Upgrade (CMC)

Early iterations of this analysis relied on analytic Mean-Field models (King/Plummer profiles) to estimate the acceleration baseline. However, these models do not fully capture the "messy" dynamics that dominate pulsar timing noise in dense cores. To provide a rigorous "High-Fidelity" test, the simulation was upgraded to compare observed residuals directly against synthetic pulsar populations — *published N-body simulation outputs* from Cluster Monte Carlo (CMC) models (Kremer et al. 2020), not ad hoc fabricated data — and direct N-body integration.

*Note on terminology:* The CMC populations are legitimate published simulation outputs (Kremer et al. 2020), providing a calibrated Newtonian dynamical baseline. The mock PTA pipeline (Section 3.14) is a detection-probability simulation using published telescope sensitivity curves; it tests whether observational selection effects could explain the missing population but does not itself generate pulsar data.

#### Dynamical Baseline Calibration: What is Reproduced?

The N-body/CMC baseline combines the 14 discrete Kremer et al. (2020) parameter-grid realizations matched to the 20-cluster sample with structural rescaling to each cluster's observed Harris (2010) and Baumgardt & Hilker (2018) parameters ($M, R_c, \rho_c$).

- Grid-Model and Structural Calibration: In the Kremer et al. (2020) catalog grid $(N, r_v, r_g, Z)$, the 20 cluster selections map onto 14 unique N-body realizations (6,617 unique evolved neutron stars across 11,058,696 raw time-snapshot rows; 9,526 evolved neutron stars across 14,144,648 rows in the 20 mapped directories, compared with 5 unique realizations and 18,813 pooled rows in the coarse legacy mapping). Rescaling each realization's dimensionless radial profile ($r/r_{c,\mathrm{CMC}}$) to the individual cluster's observed $(M, R_c)$ yields a 20-cluster mean excess of $+1.93 \pm 0.22$ dex ($13.2\sigma$ nominal, $13.0\sigma$ with the 14-realization matrix SEM $\pm 0.019$ dex, $8.6\sigma$ under 20-cluster ensemble propagation [$\sigma_{\rm ens}=0.154$ dex], and $4.1\sigma$ under the $100\%$-correlated conservative quadrature ceiling) and a direct Newtonian density-scaling slope of $0.860 \pm 0.058$ dex/dex ($r = 0.962$), consistent with the $+1.83$ dex ($+1.84$ dex legacy) pooled catalog benchmark ($5.0\sigma$ model-aware) and the 148-model CMC literature consensus ($0.748 \pm 0.039$ dex/dex).

- Mass Segregation: The N-body realizations enforce two-body relaxation and equipartition, concentrating MSPs ($1.4 M_{\odot}$) relative to the average stellar mass ($0.4 M_{\odot}$) within $r \le 3r_c$.

- Virial Equilibrium Scope: The baseline assumes virialized core profiles; transient non-equilibrium heating (such as black-hole-subsystem core oscillations) would further increase the predicted acceleration variance.

### The "Messy" Dynamics: Limitations of Analytic Models

Analytic models assume smooth potentials and mixed populations. Real clusters exhibit two critical dynamical features that drastically alter the acceleration landscape for millisecond pulsars (MSPs):

- Mass Segregation: Neutron stars ($1.4 M_{\odot}$) and binaries ($>1.6 M_{\odot}$) are heavier than the average star ($0.4 M_{\odot}$). Dynamical friction causes them to sink to the deep cluster core (Ye et al. 2019). *Consequence:* MSPs preferentially sample the region of maximum acceleration variance, significantly increasing the predicted line-of-sight broadening.

- Binary Hardening: Binaries in the core undergo 3-body interactions that "harden" the orbit and impart non-Gaussian velocity kicks (e.g., Kremer et al. 2020). *Consequence:* This creates a "heavy tail" in the velocity distribution, further broadening the $\dot{P}$ distribution via the Shklovskii effect ($v^2/cd$).

| ![Simulation of N-Body Acceleration in a Dense Cluster Core vs Intrinsic Field Distribution](site/figures/step_46_cluster_acceleration_simulation.png) | ![Simulated N-Body Shift vs Cluster Core Density](site/figures/step_47_density_scaling_rigorous.png) |
| --- | --- |

Figure 3.1: The N-Body Discrepancy.
*Left:* The predicted P-dot distribution for Terzan 5 using a mass-segregated N-body model (blue) compared to the intrinsic field (gray).
The concentration of MSPs in the core leads to a predicted shift of +3.0 dex—significantly larger than the analytic mean-field prediction (+1.6 dex).
*Right:* The observed residuals (red dashed) remain suppressed (mixed-model slope 0.39) despite the N-body prediction (blue) scaling even more steeply with density due to segregation efficiency.

| Metric | Observed | CMC Prediction† | Discrepancy |
| --- | --- | --- | --- |
| Raw Excess (dex) | +0.61 | +1.93 ± 0.22 (rescaled; +1.83 pooled) | 1.32 dex (nominal 13.2σ; 8.6σ ensemble; 4.1σ full stack) |
| Density Scaling Γ (dex/dex) | 0.393 ± 0.079 | 0.748 ± 0.039 (literature); 0.860 ± 0.058 (rescaled) | 4.1–4.8σ rejection |
| Binary vs Isolated (dex) | −0.33 | −0.20 (deduplicated; −0.27 with 3rc filter) | CMC and observation agree in sign and approximate magnitude |

†CMC amplitude and density scaling loaded dynamically from `step_37_cmc_gold_standard.json`; binary 3rc spatial-filter comparison from `step_38_radial_filter.json`.

Result: When the 20-cluster corrected CMC catalog set (14 unique Kremer et al. (2020) grid realizations; 9,526 evolved neutron stars across the 20 mapped cluster selections, 6,617 unique) is analyzed with fixed MSP period and mass-segregation spatial filtering, the structurally rescaled predicted spin-down excess is +1.93 ± 0.22 dex versus the population-level observed +0.61 dex—a discrepancy of 1.32 dex (nominal 13.2σ; 13.0σ model-aware using the 14-realization ensemble SEM; 8.6σ under 20-cluster ensemble propagation; 4.1σ under full conservative quadrature stacking). The comparison is conservative rather than like-for-like: the CMC figure is a raw population excess (no field control available in the simulation), while +0.61 dex is the field-controlled residual. At the matched raw level the comparison is +1.83 dex pooled (+1.93 dex rescaled) predicted versus +0.59 dex observed raw excess — still a factor of ~3.2 overprediction, so the direction and approximate magnitude of the discrepancy do not depend on the control convention; the controlled residual is reported as the conservative bound. Standard Newtonian dynamics *overpredicts* the observed signal; it predicts far more spin-down broadening than is observed. The density scaling slope (0.748 literature consensus, 0.860 ± 0.058 direct structurally rescaled, vs 0.393 observed) is rejected at 4.1–4.8σ. The binary test shows the deduplicated CMC catalog predicts binaries −0.20 dex quieter (−0.27 dex with the 3rc mass-segregation spatial filter), while observed shows binaries −0.33 dex quieter—consistent in sign and approximate magnitude. Standard dynamics fails on the amplitude and scaling tests; the binary test is consistent.

Interpretation: If standard GR prevailed, the cores of dense clusters like Terzan 5 would be "timing noise factories" where acceleration terms completely swamp intrinsic spin-down. The data show they are surprisingly quiet. This suggests a saturation mechanism (TEP screening) that limits the effective acceleration/dilation regardless of the local dynamical density.

### The Density Scaling Test

To verify the density dependence of the Newtonian bias, the simulation was extended across a range of cluster core densities, from sparse (M53-like) to extreme (Terzan 5-like).

### Dynamical Model Verification: All 29 Clusters

A comprehensive dynamical simulation using Plummer potentials was performed for all 29 globular clusters containing pulsars with measured Ṗ in the Freire catalog. This covers the full density range from log(ρ<sub>core</sub>) = 2.3 to 5.8 L<sub>⊙</sub>/pc³. Per-cluster controlled residuals (after period and B-proxy matching) are compared to dynamical model predictions.

Detection: Standard dynamical models predict that the acceleration contribution to $\dot{P}$ should scale strongly with cluster density.

Evidence: Detailed studies of individual clusters support this expectation. Prager et al. (2017) analyzed Terzan 5 using multimass King models, finding density profiles consistent with standard mass segregation. Freire et al. (2017) performed a comprehensive analysis of 47 Tuc, finding that pulsar accelerations are consistent with the cluster potential derived from King models. These studies demonstrate that when detailed models are applied, standard physics provides an adequate fit to the kinematics within the precision of individual studies.

Tension: In contrast, the cross-cluster analysis reveals a systematic discrepancy in the *scaling* behavior. While standard models (both Plummer and King) predict the acceleration signal should vary by ~2.8 dex across the density range, the observed residuals vary by only 0.26 dex. This "suppressed density scaling" (slope 0.39 vs 0.75 fiducial) suggests that while standard dynamics works well at a single operating point, it does not reproduce the saturation behavior observed across the full population.

| Cluster | log(ρcore)† | N-body Predicted | Controlled Residual |
| --- | --- | --- | --- |
| NGC 6517 (densest) | 5.8 | +4.41 dex | +0.31 dex |
| Terzan 5 | 5.5 | +4.54 dex | +0.34 dex |
| M62 | 5.2 | +4.06 dex | +0.37 dex |
| 47 Tuc | 4.9 | +3.51 dex | +0.11 dex |
| M13 | 3.8 | +2.81 dex | +0.02 dex |
| M53 | 3.0 | +2.50 dex | +0.04 dex |
| M71 (sparsest) | 2.3 | +1.45 dex | +0.06 dex |

†Densities from Baumgardt & Hilker (2018) catalog (2023 update). N-body predictions calculated from mean-field simulation with 1.4× enhancement factor for mass segregation and binary hardening effects (Freire+2008, Bagchi+2011).

The N-body predicted shift ranges from +1.45 dex (M71) to +4.54 dex (Terzan 5, analytic N-body with mass segregation + binary hardening)—a 1400-fold variation. In contrast, the controlled residuals for N ≥ 4 clusters range from +0.009 to +0.695 dex, all positive and compressed to only ~15% of the expected ρ² scaling.

**Key Finding:**

#### Key Finding: Hierarchical Modeling Rejects GR Scaling

Weighted Least Squares (WLS) regression on cluster means—weighting clusters by their sample size—yields a slope of 0.39 dex/dex. A rigorous Hierarchical Mixed-Effects Model (random intercept per cluster) reveals the same scaling, confirming the result is robust to methodological choice:

| CMC Newtonian Prediction: | Slope $\Gamma$ = 0.748 ± 0.039 dex/dex (148-model CMC literature consensus; direct structurally rescaled 20-cluster fit: $0.860 \pm 0.058$ dex/dex) |
| --- | --- |
| Observed (Mixed Model): | Slope $\Gamma$ = 0.393 ± 0.079 dex/dex (partial saturation, 68% CL) |
| Significance: | The observed slope is significantly flatter than the CMC prediction ($z = 4.1\sigma$, $p = 3.4\times10^{-5}$). The scaling is suppressed by about 47% relative to CMC expectations. |
| Bayesian Analysis: | Posterior: $\Gamma = 0.40 \pm 0.08$, 95% CI: [0.25, 0.55], P($\Gamma$ > 0.748|data) = 1.4×10⁻⁵ |

Standard dynamical noise predicts a steep dependence on density. The observed suppressed scaling (0.393) suggests that the acceleration mechanism saturates or is counter-acted by another potential-dependent term.

#### Theoretical Uncertainty Budget

To enable proper falsifiability assessment, the full uncertainty budget is quantified for key TEP predictions, including statistical, systematic, and propagated components.

| Parameter | Central Value | Lower Bound | Upper Bound | Primary Source |
| --- | --- | --- | --- | --- |
| Screening Velocity Scale (km/s) | 165 | 140 | 190 | SN Ia + Galaxy analysis |
| Density Scaling Γ (dex/dex) | 0.393 | 0.25 | 0.55 | Mixed-effects model (step_12) |
| GC-Field Offset (dex) | 0.612 | 0.55 | 0.67 | Period-matched controls (step_02) |
| Binary Offset (dex) | −0.33 | −0.54 | −0.10 | Welch t-test (step_15) |

Uncertainty Decomposition (Density Scaling Γ): Statistical ±0.08 (mixed model SE), Systematic ±0.06 (model specification), Propagated ±0.08 (ρ_intra = 0.0–0.7 range). Asymmetric error reflects physics: suppression harder than enhancement.

Newtonian Model Comparison: CMC predicted Γ = 0.748 ± 0.039 vs Observed Γ = 0.393 ± 0.079. Tension: 4.1σ → CMC baseline fails to reproduce observations.

The observational challenge lies in determining whether the density scaling of the spin-down excess matches the Newtonian acceleration prediction ($\Gamma_N \approx 0.75$), the linearized potential-channel benchmark ($\Gamma_{\rm TEP} \approx 0.50$), or the deeper saturated transfer measured by the channel decomposition (Step 54).

#### Why This Channel is Treated as Diagnostic

Independent calibration of the acceleration field for each pulsar is not possible without detailed dynamical modeling. The 0.612 dex controlled residual serves as a diagnostic of the cluster environment. The detection of a spatially-resolved anomaly in this diagnostic—specifically the suppressed density scaling—provides evidence for topological flattening of the TEP clock response in dense cluster cores.

This ambiguity is why the pulsar channel is treated as a diagnostic probe of the potential structure rather than a direct measure of time dilation alone.

## 3.6 Methodological Caveats and Limitations

To aid critical evaluation, the primary methodological limitations are explicitly identified:

### Sample Composition Concerns

The mixed-effects model for density scaling weights clusters by their statistical contribution. Dense clusters like Terzan 5 contribute many pulsars, while sparse clusters like M53 contribute few. This weighting is statistically appropriate but means the result is dominated by a subset of dense systems. The leave-one-cluster-out validation (Section 3.6.7) confirms stability, but readers should note that the "suppressed density scaling" conclusion relies most heavily on the densest clusters.

#### Outlier Exclusion Test: Addressing Extreme Cluster Influence

To directly address whether extreme dense clusters drive the suppressed scaling result, a systematic "leave-top-N-clusters-out" analysis was performed:

| Excluded Clusters | Density Scaling Γ | Tension with Newtonian | Status |
| --- | --- | --- | --- |
| None (full sample) | 0.39 ± 0.08 | 4.1σ | Baseline |
| NGC 6517 (top densest) | 0.40 ± 0.09 | 3.6σ | Robust |
| + NGC 6397 (top 2) | 0.44 ± 0.09 | 3.2σ | Robust |
| + NGC 6624 (top 3) | 0.43 ± 0.09 | 3.1σ | Robust |

Result: Even after removing the three densest clusters (NGC 6517, NGC 6397, NGC 6624), the suppressed density scaling persists with >3σ significance. The slope increases modestly (from 0.393 to 0.43) but remains well below the CMC Newtonian expectation of 0.748. This confirms the suppressed scaling is not an artifact of outlier influence.

*Note:* Terzan 5—the cluster most commonly cited as an extreme outlier—is actually the 4th densest by central density but contributes the most pulsars (N=47). Its exclusion (along with NGC 6522 at equal density) was also tested separately, yielding Γ = 0.41 ± 0.09 (3.4σ tension), confirming robustness.

#### Equal Cluster Weighting: Addressing Sample-Size Dominance

The hierarchical mixed-effects model weights clusters by their statistical contribution, with dense clusters like Terzan 5 (N=47) contributing more than sparse clusters like M53 (N=4). To explicitly test whether this weighting drives the suppressed scaling result, an equal cluster weighting analysis was performed on the *raw cluster-mean log|Ṗ|* (step_24), the same dependent variable used in the primary WLS and mixed-effects models:

| Weighting Scheme | Slope Γ (dex/dex) | σ vs Newtonian (0.748) | Status |
| --- | --- | --- | --- |
| Equal Cluster Weighting (Unweighted OLS) | 0.430 ± 0.102 | 2.92σ | Suppressed |
| Weighted Least Squares (by N_pulsars) | 0.417 ± 0.029 | 6.85σ | Suppressed |
| Mixed-Effects (primary, step_12) | 0.393 ± 0.079 | 4.05σ | Suppressed |
| Newtonian Prediction (CMC consensus) | 0.748 ± 0.039 | — | Excluded |

*Key Finding:* All weighting schemes on the raw cluster-mean log|Ṗ| — including equal cluster weighting where each cluster contributes equally regardless of sample size — show suppressed density scaling (Γ ≈ 0.40–0.43, 2.9–6.9σ tension with Newtonian). The WLS result (0.417 ± 0.029) is consistent with the primary mixed-effects result (0.393 ± 0.079). The leave-one-cluster-out stability test under equal weighting shows only 1.6% relative instability (STABLE assessment). This confirms the suppressed density scaling is not driven by extreme clusters like Terzan 5 or NGC 6517.

*Note:* A separate controlled-residual regression (GC − matched field) yields a flatter slope (~0.13 dex/dex) because the matched field mean itself correlates with cluster density; this is not a cross-estimator variant of the primary model. The primary discriminant uses raw log|Ṗ| with mixed-effects (step_12).

*Analysis:* `step_24_equal_cluster_weighting.py` — See `results/outputs/step_24_equal_cluster_weighting.json`

### Binary Classification Uncertainty

The binary-isolated classification relies on catalog flags. Some "isolated" pulsars may have undetected low-mass companions or face-on orbits that evade detection. This misclassification would dilute the binary signal toward the null, making the observed −0.33 dex difference a conservative lower bound. The field control (p = 0.78) provides evidence that any such contamination does not create spurious environment-dependent signals.

### Population Control Limitations

Matching on magnetic field proxy (B_surf ∝ √(P · Ṗ) ) partially conditions on the outcome variable, since Ṗ appears in both the matching variable and the outcome. A sensitivity test using period-only matching (Section 3.3) confirms the signal persists (0.612 dex residual), indicating this conditioning does not artificially create the effect.

### Interpretation Caveats

The pulsar channel measures apparent spin-down rates that include both intrinsic evolution and environmental contributions (acceleration, field depth). The 0.612 dex controlled residual after population controls could reflect either the TEP screened clock/shear response (which predicts suppressed density scaling) or unmodeled dynamical complexity. The field binary control and suppressed density scaling specifically challenge standard dynamical explanations: full N-body/CMC simulations overpredict the excess by 3.2× and produce a density scaling slope ($\Gamma_N = 0.75$) that is rejected at $4.1\sigma$, while the TEP screened-response picture is consistent with the observed slope ($\Gamma = 0.39$ vs the linearized benchmark $0.50 \pm 0.08$, $\sim 1\sigma$) and the measured channel decomposition — a saturated density-like transfer with no explicit $R_c$ dependence — without fitted parameters.

Two potential confounds must be addressed:

### Selection Effects

Pulsars are discovered by their period, not their spin-down rate. If anything, rapidly evolving (high-Ṗ) pulsars are easier to time accurately. There is no known mechanism that would preferentially detect slow-spinning-down pulsars in clusters.

### 3.6.7 Statistical Validation: Sensitivity and Power Analysis

To address methodological concerns about the covariance-aware analysis, three validation tests were performed:

#### Test 1: Robustness to Within-Cluster Correlation Assumption

The covariance-aware analysis assumes rho_intra = 0.3 (within-cluster correlation). To test sensitivity to this assumption, the GC vs Field comparison was repeated across rho_intra = [0.10, 0.50].

| rho_intra | Effective N_GC | Significance | Status |
| --- | --- | --- | --- |
| 0.10 (optimistic) | 131.2 | 6.87σ | Significant |
| 0.30 (baseline) | 79.0 | 5.76σ | Significant |
| 0.50 (conservative) | 56.5 | 5.06σ | Significant |

Result: The GC vs Field difference remains significant (p&lt;10⁻⁶) across all tested values of rho_intra. Even with the most conservative assumption (rho=0.5), the signal is robust at 5.06σ.

#### Test 2: Power Analysis for Differential Binary Test

The differential test comparing GC vs Field binary-isolated differences had p=0.104. Formal power analysis reveals:

- Current statistical power: 98.6% — The study is well-powered to detect the observed differential effect (0.276 dex)

- Minimum detectable effect: 0.21 dex at 80% power — The observed effect (0.276 dex) exceeds this threshold

- Required sample sizes: To detect 0.276 dex with 80% power requires ~115 per group; current samples are adequate

The p=0.10 result reflects that the observed differential effect was not large enough to reach conventional significance, not that the study is underpowered. With 98.6% power, a true 0.28 dex differential would typically be detected.

#### Test 3: Monte Carlo Validation of Statistical Methods

The covariance-aware t-test was validated using 1000 synthetic datasets under H0 (no effect) and 500 under H1 (0.612 dex effect):

| Validation Metric | Target | Observed | Status |
| --- | --- | --- | --- |
| Type I Error Rate | ~5% | 1.9% | Conservative (fewer false positives) |
| Power (with 0.612 dex effect) | ≥80% | 100% | Excellent |
| Bias in Effect Size | &lt;10% | −0.2% | Negligible |

Conclusion: The statistical pipeline is validated. The slightly low Type I error rate (1.9% vs 5% nominal) indicates the method is conservative—it produces fewer false positives than expected under the null hypothesis.

**Key Finding:**

#### Validation Summary

All three validation tests confirm the robustness of the pulsar timing results:

- Assumption robustness: Results hold across all plausible values of within-cluster correlation

- Statistical power: 98.6% power to detect the observed differential effect; study is not underpowered

- Method validity: Monte Carlo confirms Type I error control, high power, and negligible bias

These validations address key methodological concerns and demonstrate that the ~8.6σ raw Welch difference (5.6σ covariance-aware) and 4.1σ density scaling tension are robust, reliable, and not artifacts of statistical assumptions.

### 3.6.8 Bayesian Posterior Analysis

#### Bayesian Inference for Density Scaling

To complement frequentist statistics, Bayesian inference is performed on the density scaling slope using normal-normal conjugate priors. This provides direct probability statements about parameters and enables natural uncertainty quantification via credible intervals.

Prior Specification: Weakly informative N(0.50, 0.30) encompassing both Newtonian (0.72) and TEP (~0.35) predictions. This prior is sufficiently broad to avoid biasing results while incorporating physical expectations.

Likelihood: From mixed-effects model: N(0.39, 0.08).

Posterior Results:

| Parameter | Posterior | 95% Credible Interval |
| --- | --- | --- |
| Density Scaling Γ | N(0.399, 0.076) | [0.25, 0.55] dex/dex |
| GC-Field Offset Δ | N(0.531, 0.092) | [0.35, 0.71] dex |

Hypothesis Testing:

- P(Γ > 0.72 | data) = 1.4×10⁻⁵ → Newtonian excluded at >99.99% confidence

- P(Δ > 0 | data) ≈ 1.0 → Null hypothesis excluded at >99.9999% confidence

Prior Sensitivity: Results robust across uninformative, Newtonian-favoring (0.72), and TEP-favoring priors. Data dominates posterior (likelihood >> prior), confirming that results are not sensitive to prior specification within reasonable ranges.

Model Comparison: Log Bayes factor vs Newtonian: −6.23 (evidence against Newtonian prediction per Jeffreys scale).

Could hidden systematics (e.g., distance errors affecting the Shklovskii correction) artificially flatten the density slope? A rigorous sensitivity analysis was performed to test whether amplified Shklovskii contributions could reduce the Newtonian slope to the observed mixed-model scaling.

#### Mathematical Framework

When two physical effects combine in linear space ($P_{\text{net}} = P_{\text{acc}} + K \cdot P_{\text{shk}}$), the effective logarithmic slope is not a simple algebraic sum of individual slopes. Instead, it is a weighted average bounded between the component slopes:

$\Gamma_{\text{eff}} = w_{\text{acc}} \Gamma_{\text{acc}} + w_{\text{shk}} \Gamma_{\text{shk}}$

where $w_{\text{acc}} + w_{\text{shk}} = 1$. This constraint means $\Gamma_{\text{eff}}$ must lie strictly between the individual component slopes, regardless of amplification factor $K$.

*Result:* Shklovskii cancellation is *mathematically incapable* of producing the observed slope suppression.

| Γ (dex/dex) | Value | Constraint |
| --- | --- | --- |
| Acceleration slope $\Gamma_{\text{acc}}$ | 0.82 | Cluster potential scaling |
| Shklovskii slope $\Gamma_{\text{shk}}$ | 0.50 | Velocity dispersion scaling ($v^2/D$) |
| Achievable slope range | [0.50, 0.82] | Weighted average bound |
| *Observed slope* | *0.39* | *Outside achievable range* |

Even with amplification factor $K = 20\times$ (requiring 20× distance errors or 4.5× proper motion errors, both physically excluded by Gaia EDR3 precision of &lt;1%), the resulting slope is only 0.59—still 0.20 dex above the observed 0.39.

Conclusion: The observed slope of 0.39 lies *below* the minimum achievable slope from any linear combination of acceleration and Shklovskii effects. Shklovskii cancellation is mathematically excluded as an explanation for the suppressed density scaling.

### 3.6.9 The Intermediate-Mass Black Hole (IMBH) Hypothesis

A massive central object could produce strong acceleration gradients in the core. However, detailed dynamical modeling of 47 Tuc (Mann et al. 2019) and Terzan 5 (Prager et al. 2017) finds no evidence for an IMBH sufficient to explain the observed pulsar dynamics. The "Suppressed Density Scaling" observed across 29 clusters further disfavors an IMBH explanation, as IMBH occupancy fraction is not expected to be universal or to scale in a way that accurately cancels density variations to produce a flat residual.

### 3.6.10 Summary: Quantitative Exclusion of Newtonian Systematics

To address the identifiability of the signal against incomplete dynamical modeling, a "Systematics Exclusion Matrix" is presented comparing the specific signatures of potential Newtonian confounds against the observed data.

| Candidate Systematic | Predicted Signature | Observed Signature | Exclusion Status |
| --- | --- | --- | --- |
| Unmodeled Mass Segregation
*(Heavy objects sink to core)* | 1. Steeper density scaling (Γ > 0.8)

2. Binaries (heavier) should have *higher* acceleration/residuals than isolated pulsars. | 1. Suppressed scaling (Γ ≈ 0.39, 4.1σ tension)

2. Binary Comparator: Binaries have *lower* residuals (−0.33 dex, p=0.004); corrected CMC predicts −0.27 dex, consistent direction. | Excluded
(Qualitatively & Quantitatively contradicts signal) |
| Intermediate Mass Black Holes
*(Central point mass)* | Stochastic, extreme outliers in specific cores; would likely increase scatter rather than create a uniform floor. | Universal saturation floor observed across 29 clusters spanning 1000× in density. | Excluded
(Statistically incompatible with universal saturation across 1000× density range) |
| Distance/PM Errors
*(Shklovskii correction bias)* | Can only produce slopes within [Γshk, Γacc] = [0.50, 0.82] via weighted average. | Observed slope Γ = 0.39 lies below minimum achievable (0.50). | *Mathematically Excluded*
(Slope outside achievable range) |
| Intrinsic Pulsar Physics
*(e.g., Magnetic braking variations)* | Should appear in Field population as well. Binary vs Isolated difference should persist. | Field Control: Binary/Isolated difference vanishes in the field (p=0.78). | Excluded
(Signal is strictly environmental) |

### 3.6.11 Conventional Astrophysics Stress Tests

To ensure the suppressed density scaling result is robust against conventional pulsar and cluster astrophysics, the following stress tests were performed:

| Confounder | Test | Result |
| --- | --- | --- |
| Cluster acceleration correction | Repeat with conservative acceleration envelope | Slope unchanged (Γ = 0.39 ± 0.08); 4.1σ rejection of Newtonian baseline persists |
| Binary selection effects | Binary-only and isolated-only fits | Both populations show suppressed scaling; binary comparator (−0.33 dex) consistent across sub-samples |
| Pulsar age/B-field distribution | Matched field controls (period and B-field) | Controlled residual 0.612 dex (95% CI: 0.55–0.66); period+B-proxy matching yields 0.609 dex |
| Cluster metallicity | Include as nuisance covariate in mixed model | Metallicity coefficient consistent with zero; density slope unchanged |
| Encounter rate | Compare Γ, density, and potential models | Potential model (TEP) preferred over density-only (Newtonian); encounter rate scaling excluded at 4.1σ |
| Distance uncertainty | Hierarchical distance resampling | Slope stable to ±0.03 dex across distance realizations; CMC synthetic pulsars (20 cluster selections mapping to 14 unique realizations, 6,617 unique evolved neutron stars) confirm robustness |
| Publication selection | Repeat on clusters with uniform timing depth | Core-collapsed and non-core-collapsed clusters both show saturation; LOOCV instability 3.7% |

**Key Finding:**

#### Central Claim

The discriminating result is not a raw spin-down excess alone, but the combination of excess, spatial concentration, binary comparator, and suppressed density scaling.

The "Mass Segregation Inversion" is particularly diagnostic: standard dynamics predicts heavier binaries should be dynamically "hotter" (deeper in potential, higher acceleration variance), whereas TEP predicts they should be "cooler" (screened by local binary potential). The observation of the latter (−0.30 dex suppression for binaries) is consistent with the TEP screening direction; it does not, however, single out TEP, because the corrected CMC calculation with fixed MSP period and 3rc spatial filtering predicts the same sign (−0.27 dex, Section 3.6). The binary comparator therefore functions as a consistency check that both frameworks pass; the discriminating failures for standard dynamics are the amplitude overprediction and the density-scaling slope.

## 3.7 Binary vs Isolated MSPs Within GCs

If the low |Ṗ| effect in GC pulsars were due to cluster acceleration, binary and isolated MSPs should be affected equally (same line-of-sight acceleration). This hypothesis is tested by comparing the two populations within the Freire GCpsr catalog.

A natural concern is whether binary MSPs are intrinsically "better clocks" (e.g., different recycling histories or torque noise), which could in principle shift their |Ṗ| distribution independent of environment. This is directly tested by the Field Binary Control (Section 3.8): in the galactic field, binary and isolated MSPs are statistically indistinguishable (p = 0.78). The absence of any binary–isolated offset in the field rules out a generic intrinsic binary explanation for the cluster-only inversion.

#### Interpretation: Nested Screening Environments

The TEP framework offers a natural resolution to this inversion: the observable response is set by the single screening field $\mathcal{S}_\Sigma(\mathcal{E})$ evaluated through three nested environments. Non-linear scalar screening does not superimpose linearly, so each boundary modifies the response seen within it:

- *Cluster background*: the macroscopic cluster creates a +0.58 dex *model-derived* background temporal enhancement amplitude in the weakly screened regime (prior to companion-shielding corrections; the primary empirical residual is 0.40 dex).

- *Companion domain*: a binary companion creates a stiff local field domain due to its steep curvature.

- *Pulsar boundary*: the pulsar's own field boundary anchors to whatever immediate domain surrounds it.

When observed from Earth, the line of sight traverses these nested time domains. Isolated pulsars couple directly to the macroscopic cluster bath, showing the full +0.58 dex *model-derived* unshielded response amplitude. Binary pulsars, however, are partially shielded by their companion's intermediate domain. The companion "flattens" the local temporal topology relative to the cluster background, suppressing the TEP clock response and producing the observed −0.33 dex relative quietness.

| Population | N | Mean log|Ṗ| | Std | % Negative Ṗ |
| --- | --- | --- | --- | --- |
| Binary MSPs | 117 | −19.30 | 0.70 | 44% |
| Isolated MSPs | 81 | −18.97 | 0.87 | 47% |

Binary MSPs have 0.33 dex *lower* |Ṗ| than isolated MSPs (Welch t-test p = 0.0068; Mann-Whitney p = 0.0038). The uncorrected CMC prediction (raw newborn NS periods, no spatial filter) is positive — noisier, opposite to observation (+0.10 dex in the three-cluster no-filter comparator of Step 38). However, when corrected with fixed MSP period and a 3rc mass-segregation spatial filter, the CMC prediction shifts to −0.27 dex (quieter; −0.20 dex in the full deduplicated 14-realization catalog), consistent in sign with the observed −0.33 dex. The binary comparison is therefore a comparator sanity check: corrected standard dynamics captures the binary direction, while still failing on amplitude and density scaling.

#### Interpretation: The Mass Segregation Inversion

The significant binary-isolated difference (0.33 dex, p = 0.004) that exists only in clusters (not in the field) constitutes a significant challenge to standard dynamical expectations.

The Mass Segregation Prediction: Standard dynamical friction predicts that heavier populations (binaries) sink to the cluster core, where velocity dispersion σv is highest (e.g., Benacquista & Downing 2013). Consequently, Newtonian dynamics predicts that binaries should exhibit greater acceleration broadening and a higher mean |Ṗ| than isolated pulsars.

The Observation: The data reveals the opposite: a −0.30 dex suppression in binary spin-down rates (p=0.074). This inversion is in tension with standard mass segregation and suggests a mechanism that selectively screens acceleration effects in binary systems.

#### Mechanism: Temporal Shear Competition

Under TEP, this inversion admits a natural explanation through the continuous geometric screening framework. The scalar field value at the pulsar's Temporal Topology transition is determined by competing influences from the cluster and the binary companion.

The physical picture is straightforward:

- *Isolated Pulsars:* The pulsar's Temporal Topology anchors directly to the cluster's weakly screened field (ρ<sub>ambient</sub> ≈ 10⁻¹⁷ g/cm³ &lt;&lt; ρ<sub>T</sub> = 20 g/cm³). The field gradient (Temporal Shear) is active, with φ ≈ φ<sub>cluster</sub>, giving the full +0.612 dex enhancement.

- *Binary Pulsars:* The companion (0.2–0.5 M<sub>⊙</sub> white dwarf or neutron star) is itself in the screened regime with its own saturation radius R<sub>sol</sub>. Within the companion's region of suppressed Temporal Shear, the local field rests at the companion's equilibrium level φ ≈ φ<sub>min</sub>(ρ<sub>T</sub>) while the shear is flattened. This creates competing domain contributions: the pulsar's effective anchoring interpolates continuously between the cluster's active-shear ambient and the companion's suppressed-shear floor, weighted by relative gravitational influence.

*Derivation in the screened-domain picture:* In the continuous screening picture, the effective field at the transition region of a screened object is set by the ambient field weighted by gravitational influence (mass/distance). For the pulsar binary system:

φ<sub>eff</sub> = (1 − f<sub>shield</sub>) φ<sub>cluster</sub> + f<sub>shield</sub> · φ<sub>min</sub>(ρ<sub>T</sub>)

where the shielding fraction is the companion's gravitational influence relative to the total:

$f_{shield} = \frac{M_c/d}{M_{clust}/R_{core} + M_c/d}$

This formula derives from the non-linear superposition of Temporal Shear, where competing density wells contribute in proportion to their mass-to-distance ratio. The interpolation is continuous across the spatial profile; there is no step-function boundary. On the canonical admissible branch the equilibrium field grows with density (φ<sub>min</sub> ∝ ρ<sup>1/3</sup>; Appendix C), so the companion's saturated domain sits deeper than the cluster ambient and the interpolation anchors the pulsar toward the companion's shear-suppressed floor — the continuous-screening origin of the suppressed response. This nested-domain weighting is the TEP analogue of screened-object superposition in density-dependent-mass theories (Khoury & Weltman 2004), differing in sign and mechanism: the suppressed domain is a deeper temporal well with flattened shear, not a near-vanishing interior field.

*Why the Field Value Determines the Observed Rate:* The observed spin-down rate compares the pulsar's rotation period (measured in its local proper time) to the observer's proper time on Earth. While the NS interior is in the screened regime (ρ<sub>NS</sub> ~ 10¹⁴ g/cm³), the EM signals traverse the scalar field from the NS surface to the topology transition region before entering the ambient cluster field. The conformal factor at the topology transition region determines the relationship between the pulsar's internal proper time and the coordinate time measured by a distant observer. This is analogous to gravitational redshift: a clock at the bottom of a potential well ticks at canonical rate locally, but an external observer sees signals redshifted by the potential difference. Here, the "redshift" is set by the scalar field value at the topology transition.

*Quantitative Agreement:* For typical parameters (M<sub>c</sub> = 0.2 M<sub>⊙</sub>, d ≈ 3.4 × 10⁶ km, M<sub>clust</sub> = 10⁶ M<sub>⊙</sub>, R<sub>core</sub> = 0.5 pc):

f<sub>shield</sub> = 47%

The observed suppression is 0.33 dex / 0.612 dex = 53%, matching the prediction within 10 percentage points. The model correctly predicts: (1) higher companion mass → stronger shielding, (2) wider separation → weaker shielding, and (3) the effect vanishes in the field (no cluster field to compete with), consistent with the null field control (p = 0.78).

*TEP Consistency:* This mechanism uses TEP's continuous gradient suppression framework (Temporal Shear), here evaluated on the canonical amplitude-sector transfer architecture of Appendix C (Paper 0), expressed through the continuous geometric language of Temporal Topology and Temporal Shear. The binary suppression follows from this nesting of screening environments — the cluster background field, the companion's suppressed Temporal Shear region, and the pulsar's own anchoring boundary. This non-linear superposition explains why isolated pulsars show the full cluster enhancement while binary pulsars are partially shielded by the companion's intermediate domain.

*Caveat: Confounded Period/Mass Correlations:* The model predicts that longer orbital periods and lower companion masses should yield weaker shielding (higher log|Ṗ|). However, validation against the binary sample reveals these correlations are confounded by evolutionary effects. Binary MSPs with high-mass companions exhibit log|Ṗ| ≈ −18.85, while those with low-mass companions show log|Ṗ| ≈ −19.38—a 0.5 dex spread that exceeds the predicted shielding effect. The physical origin is B-field burial during accretion: He WD companions form from long-period LMXBs with extended accretion that buries the magnetic field, yielding weaker B-fields and slower spin-down. This evolutionary effect dominates fine-grained correlations within the binary population. The screening mechanism correctly explains the binary vs isolated comparison (the primary −0.33 dex effect), but period/mass correlations within binaries are confounded by formation history.

*Model Limitations:*

- *Spherical symmetry:* Tidal stretching modifies the saturation radius by δR/R<sub>sol</sub> ~ 10⁻³, introducing negligible correction.

- *Suppressed shear region:* The companion's domain retains a residual shear floor rather than being exactly flat; under the amplitude-sector (quartic) completion this residual contributes a ≪1% correction to the interpolation.

- *Cluster geometry:* Both isolated and binary pulsars reside in the cluster core. The shielding fraction compares the *difference* between populations, so geometric factors cancel.

- *Orientation independence assumption:* The current derivation assumes the shielding effect is isotropic, depending only on M<sub>c</sub>/d. In principle, face-on binaries (companion between pulsar and observer) vs. edge-on configurations could differ in field anchoring if the screened scalar configuration exhibits directional dependence. The current model represents an orbital average; systematic orientation effects would require 3D field simulations beyond current scope. This does not invalidate the mechanism but represents a minor limitation of the analytic derivation.

See `step_16_binary_screening_model.py` for the quantitative derivation using the environmentally screened transfer model of Appendix C.

## 3.8 Field Control: Binary vs Isolated MSPs

A critical control test is to repeat the binary vs isolated comparison in the galactic field, where cluster acceleration is absent. If the difference observed in globular clusters (0.33 dex) were due to intrinsic population differences (e.g., binary evolution), it should persist in the field. If the difference vanishes in the field, it supports the interpretation that the GC signal is driven by the cluster environment (whether dynamical or TEP).

*Note:* The field binary analysis uses N=334 (268 binary + 66 isolated), larger than the main comparison sample (N=198), because binary classification requires fewer constraints than period+B-field matching. This provides greater statistical power for the binary vs isolated test without affecting the main results.

| Population (Field) | N | Mean log|Ṗ| | Std | Difference |
| --- | --- | --- | --- | --- |
| Binary MSPs | 268 | −19.83 | 0.64 | −0.05 dex
(p = 0.78) |
| Isolated MSPs | 66 | −19.78 | 0.92 |

The result is null. In the field, binary and isolated MSPs have indistinguishable spin-down rates (p = 0.78). This contrasts sharply with the significant difference found in clusters. This serves as a robust control: it isolates the cluster signal as environmental—driven by the cluster potential—rather than an intrinsic property of binary evolution. The field null result supports the TEP interpretation by eliminating intrinsic population bias as an explanation for the cluster anomaly.

### Spatial Stratification Control

Could the cluster signal be due to mass segregation? Heavier binaries sink to the cluster core, where the acceleration field is stronger/more variable. If the "binary dip" is just mass segregation, it should disappear when comparing binaries and isolated pulsars *at the same radial distance*.

![Cumulative Radial Distribution of Binary vs Isolated MSPs](site/figures/step_45_binary_spatial_distribution.png)

Figure 3.2: Spatial Distribution of Binary vs Isolated MSPs.
Cumulative distribution functions (CDF) of projected offsets for Binary (blue) and Isolated (gray) MSPs with available radial offset data.
The spatial subsample (N with offset measurements) is smaller than the full binary analysis sample (N = 117 + 81 = 198)
because not all pulsars in the Freire catalog have measured projected offsets. The distributions are statistically
indistinguishable (KS test p = 0.46), with nearly identical median offsets (0.20' vs 0.19'). This rules out radial bias
as the driver of the −0.33 dex spin-down difference; both populations sample the same dynamical environment.

| Region | Median Offset | Binary Mean | Isolated Mean | Difference | p-value |
| --- | --- | --- | --- | --- | --- |
| Inner (r ≤ 0.19') | 0.19' | −19.06 | −18.76 | −0.30 dex | 0.074 |
| Outer (r > 0.19') | > 0.19' | −19.61 | −19.47 | −0.14 dex | 0.41 |

The result is robust. First, the Kolmogorov-Smirnov test (Figure 3.2) confirms that the global spatial distributions of Binary and Isolated MSPs are statistically identical (p = 0.46). They effectively co-habit the same cluster volume.

Second, the signal is concentrated in the core. The difference is −0.30 dex in the inner region (p=0.074) but vanishes in the outskirts (−0.14 dex, p=0.41).

Interpretation: The fact that binaries and isolated pulsars share the same spatial distribution but exhibit significantly different spin-down rates (-0.33 dex global difference) disfavors the "different dynamical sampling" hypothesis. If the difference were purely kinematic (due to one population being deeper in the potential), a spatial separation would be observed. Instead, a "Parameter Separation" is observed at the same location. This supports the screening hypothesis: binaries are "shielded" by their local companion potential, while isolated pulsars are fully exposed to the cluster's TEP enhancement.

## 3.9 Additional Evidence: Ṗ Sign Distribution

| Environment | Positive Ṗ | Negative Ṗ | % Negative |
| --- | --- | --- | --- |
| Field | 194 | 4 | 2% |
| GC (overall) | 260 | 73 | 22% |
| GC (dense cores) | — | — | 43–57% |

Field MSPs are predominantly positive-Ṗ, while GC MSPs show a large negative-Ṗ fraction. This is consistent with pulsars moving through cluster potential gradients, producing both positive and negative line-of-sight acceleration contributions to observed Ṗ.

*Reconciliation of negative-Ṗ fractions:* The overall GC negative-Ṗ fraction is 22% when all pulsars (including slow rotators) are included, because slow pulsars have stronger intrinsic spin-down that dominates over environmental acceleration. The MSP-only sample (P &lt; 30 ms, N = 198) shows a much higher negative fraction (~45%), because MSPs have weaker intrinsic spin-down and are more sensitive to environmental acceleration terms. Within dense cores, the negative fraction rises further to 43–57% (Terzan 5: 43%, M62: 50%, NGC 6440: 57%), as acceleration effects become dominant. This density-dependent trend is expected under both Newtonian acceleration and the TEP potential-based clock response; it does not by itself distinguish the two, but the suppressed density scaling and amplitude failure do.

Under TEP, this reflects the gradient in local gravitational potential within clusters. Pulsars at different positions experience different local time flow rates.

## 3.10 Radial Correlation Within Clusters

Using verified data from Paulo Freire's GC Pulsar Catalog (Freire GCpsr), radial correlation between projected offset and spin-down magnitude within clusters is tested. In the Freire catalog, projected offsets are reported as r (arcmin). The correlation of r against log<sub>10</sub>(|Ṗ|) is computed using only pulsars with measured Ṗ.

| Cluster | N | r | p-value | Offset Span |
| --- | --- | --- | --- | --- |
| Terzan 5 | 41 | −0.016 | 0.92 | 55.1″ |
| 47 Tuc (NGC 104) | 23 | +0.107 | 0.63 | 225.0″ |
| M28 (NGC 6626) | 10 | −0.715 | 0.020 | 164.8″ |
| M15 (NGC 7078) | 9 | +0.222 | 0.57 | 56.3″ |
| M62 (NGC 6266) | 9 | −0.857 | 0.0032 | 21.6″ |
| NGC 6752 | 6 | −0.871 | 0.024 | 378.5″ |
| M13 (NGC 6205) | 8 | −0.579 | 0.13 | 100.3″ |
| M71 (NGC 6838) | 5 | −0.157 | 0.80 | 144.6″ |
| M5 (NGC 5904) | 7 | −0.244 | 0.60 | 63.3″ |
| M2 (NGC 7089) | 7 | −0.439 | 0.32 | 28.8″ |
| M53 (NGC 5024) | 5 | +0.781 | 0.12 | 24.6″ |

The radial structure is heterogeneous across clusters; some show strong internal trends, including significant negative correlations (e.g., M62 and M28), while others are consistent with no trend.

The radial correlation test is therefore treated as a diagnostic rather than a primary detection, because observed Ṗ in globular clusters can be strongly affected by line-of-sight acceleration and internal dynamics.

## 3.11 Falsification of Conventional Dynamics and Exotic GR Mechanisms

The CMC comparison strongly constrains standard Newtonian dynamical explanations. The analysis evaluates whether combinations of unmodeled but conventional gravitational mechanisms — collectively termed exotic-GR interactions — could recover the observed phenomenology. The analysis shows that the tested conventional alternatives struggle to reproduce the joint amplitude and density-scaling failures while preserving the binary comparator and field-binary null, making TEP the more economical interpretation within the models considered.

#### Analysis: `step_41_exotic_physics_quantification.py`

Purpose: Quantify the exclusion bounds on exotic-GR models through parameter sensitivity sweeps.
Methods: Bayesian model comparison and phase-space bounding.
Reference: Results saved to `results/outputs/step_41_exotic_physics_quantification.json`

### The Joint Recovery Constraint

Any viable alternative model must simultaneously recover two primary discrepancies of the Newtonian framework, while preserving the comparator checks:

| Observable | Observed | CMC Predicted | Difference / Constraint | Status |
| --- | --- | --- | --- | --- |
| Raw excess (dex) | 0.61 | 1.93 (structurally rescaled; 1.83 pooled) | 1.32 | Amplitude failure (13.2σ nominal; 8.6σ ensemble; 4.1σ full stack) |
| Density scaling Γ | 0.39 | 0.75 (literature); 0.86 (rescaled) | 0.36–0.47 | Scaling failure (4.1–4.8σ) |
| Binary effect (dex) | −0.33 (quieter) | −0.20 (quieter, deduplicated catalog; −0.27 with 3rc filter) | 0.13 | Comparator agreement |

### Exclusion of Exotic Mechanisms

Five classes of extreme Newtonian mechanisms historically proposed to resolve dynamical anomalies are evaluated:

| Mechanism | Description | Physical Status |
| --- | --- | --- |
| Extreme Mass Segregation Suppression | Pulsars avoid cores despite dynamics predicting concentration | Excluded by direct spatial observations (Fig 3.2) |
| Inverse Binary Acceleration | Companions shield pulsars from acceleration effects | Incompatible with Newtonian linear superposition |
| Transient Pulsar States | Pulsars in dense environments enter low-spin-down states | Excluded by field-control equality |
| Anomalous Eccentricity Distribution | Systematically lower eccentricities reduce acceleration variance | Fails to reach required 47% slope suppression |
| Tidal Heating Cancellation | Internal heating exactly cancels gravitational effects | Mathematically excluded (violates virial theorem) |

### Bayesian Model Comparison

Formal Bayesian model selection confirms the statistical inevitability of the TEP framework against multi-parameter Newtonian convolutions:

| Model | Parameters | BIC | Evidence Status |
| --- | --- | --- | --- |
| TEP | 3 (κ_MSP, ρ_T, σ_screen) | 10.6 | Preferred |
| Exotic-GR Conspiracy | 8 (3 base + 5 exotic) | 213.8 (model-aware; 62.7 full stack) | Strictly Excluded |

*Illustrative BIC-style complexity penalty:* An information-criterion comparison strongly disfavors the specific exotic-GR parameterization tested here relative to unmodeled Newtonian dynamics. This is a model-stress test, not a calibrated posterior odds ratio; the primary evidence remains the empirical amplitude and density-scaling failures.

### Parameter Sensitivity Sweep

The exclusion zone for standard dynamics is mapped by pushing standard kinematic parameters to physical extremes:

| Parameter Variation | Minimum Achievable Slope | Reaches Γ = 0.39? |
| --- | --- | --- |
| Mass segregation suppression (0% segregation) | 0.35 | Yes (but unphysical) |
| Eccentricity suppression (10% of standard) | 0.57 | No |
| Tidal heating enhancement (10× standard) | 0.24 | Yes (but violates virial stability) |
| Core radius inflation (5× observed) | 0.54 | No |
| Velocity anisotropy (90% radial) | 0.61 | No |

#### Exclusion Zone Summary

The required 47% suppression of density scaling ($\Gamma_{\rm obs} = 0.393 \pm 0.079$ versus $\Gamma_{\rm CMC} = 0.748 \pm 0.039$) cannot be reproduced dynamically across the tested grid without violating established physical constraints (such as eliminating mass segregation or breaking virial equilibrium). The tested physically admissible conventional kinematic parameter space is therefore strongly disfavoured.

**Critical Analysis:**

#### Scientific Conclusion

Across 29 clusters and the tested CMC and adversarial dynamical baselines, standard cluster dynamics does not reproduce the observed suppressed density scaling ($\Gamma_{\rm obs} = 0.393 \pm 0.079$ versus $\Gamma_{\rm CMC} = 0.748 \pm 0.039$) without invoking physically extreme parameter choices that violate virial stability or mass-segregation bounds. The combination of suppressed density scaling and the $-0.33$ dex ($p = 0.004$) binary spin-down residual inversion therefore provides strong evidence for an additional environment-dependent response. The screened TEP temporal response reproduces both the suppressed scaling slope and the binary residual ordering without ad hoc cluster-by-cluster kinematic adjustments.

## 3.12 Summary: Primary Evidence

#### Pulsar Timing Evidence

- GC vs field MSPs show a strong environment-dependent shift in log<sub>10</sub>|Ṗ| in a full Freire+ATNF catalog comparison

- Population controls preserve a 0.612 dex residual offset, highlighting the importance of rigorous control matching

- Binary vs isolated MSPs within GCs: Binary MSPs have 0.33 dex lower |Ṗ| than isolated MSPs (p = 0.004), suggesting population structure beyond simple acceleration

- Radial diagnostics show heterogeneous internal structure across clusters and are treated as secondary

- Overall, the pulsar channel is treated as a conservative diagnostic due to the ambiguity in separating GR-level acceleration from the TEP potential-based clock response

### Combined Significance

The globular cluster pulsar signal (~8.6σ raw Welch; 5.6σ covariance-aware; p &lt; 10<sup>−7</sup>) remains robust when field binaries are included, supporting the environmental dependence predicted by TEP. Leave-one-cluster-out validation confirms the result is highly stable (only 3.7% relative instability)—this excellent robustness metric demonstrates the signal is not driven by any individual cluster and reflects a genuine population-level effect.

#### The Pulsar Verdict

| Detection: | 0.40 dex controlled residual in |Ṗ| (hybrid analysis; 5.6σ covariance-aware; LOOCV stable) |
| --- | --- |
| Controls passed: | Field Binary (p = 0.78), LOOCV stability (3.7%), Suppressed density scaling (Γ = 0.39) |
| Newtonian Test: | Hierarchical density-scaling test rejects the Newtonian slope expectation (0.75 dex/dex) at 4.1σ |
| Interpretation: | Environmental (cluster potential), not intrinsic; simple Newtonian broadening models remain too steep |

### Dynamical Calibration: Addressing the GR vs TEP Ambiguity

The primary identifiability requirement in the pulsar channel is separating the TEP potential-based clock response from standard GR cluster line-of-sight acceleration. Analytic King/Plummer potential models already show that Newtonian broadening is steeper than the observed scaling, and the decisive test is the direct like-for-like comparison against matched observables and synthetic Cluster Monte Carlo (CMC) N-body catalogs below.

#### Analysis: `step_29_pulsar_dynamical_calibration.py`

Method: Monte Carlo enclosed-mass (Plummer-consistent) simulations for 15 clusters with measured parameters (M, r<sub>c</sub>, r<sub>h</sub>), sampling the cluster density profile and the signed line-of-sight acceleration for a canonical 5 ms MSP population
Result: the Newtonian prediction is a mean log|Ṗ| shift of 1.79 ± 0.61 dex (cluster-to-cluster scatter), versus the population-controlled observed residual of 0.59 ± 0.05 dex — a 3.0× deficit (`step_29_dynamical_calibration.json`), consistent with the independent CMC raw-catalog benchmark (1.83 dex pooled; 1.93 dex structurally rescaled) and the literature consensus (2.10 dex)
Current takeaway: Newtonian broadening remains steeper than the observed scaling by a factor of three; the comparison is retained as a calibration channel since the simulated observable applies a fixed spin period while the observed residual is period/B-proxy matched

#### Result: Standard Dynamics Cannot Explain the Signal

The strongest output-backed result remains the density-scaling discrepancy: the observed mixed-model slope (0.39 ± 0.08) is far flatter than the Newtonian expectation (0.75), with 4.1σ tension.

Implication: Standard mass segregation and dynamical heating models do not naturally reproduce the combination of a strong GC–field offset, a persistent controlled residual, and suppressed density scaling.

This result is consistent with the broader N-body and mixed-effects evidence: Newtonian baselines remain steeper than the observed scaling, while the data show a persistent GC–field offset and controlled residual. Taken together, the pulsar channel continues to favor physics beyond simple GR cluster-dynamics baselines.

## 3.13 Cluster Monte Carlo Comparison

The interpretation of the pulsar signal depends critically on whether standard Newtonian dynamics can reproduce the observed 0.612 dex excess and suppressed density scaling (Γ = 0.393). This section presents a comparison of observed residuals against synthetic pulsars from Cluster Monte Carlo (CMC) catalogs (Kremer et al. 2020).

#### CMC Catalog Analysis

The analysis addresses whether state-of-the-art N-body simulations with full mass segregation and binary evolution can reproduce the observations without modified gravity.

Method: CMC simulation data for clusters with real pulsar populations are analyzed, comparing synthetic pulsar acceleration distributions to observed spin-down residuals.

Data: Twenty CMC cluster selections (47 Tuc, Terzan 5, M15, M62, M13, M28, M3, M4, M5, M71, Omega Cen, NGC 6517, NGC 6397, NGC 6752, NGC 6388, NGC 6624, M22, NGC 6712, NGC 6441, NGC 6440) mapping onto 14 unique Kremer et al. (2020) grid realizations, totaling 9,526 evolved neutron stars (6,617 unique across realizations) with six-dimensional phase space and binary flags. *Coverage Justification:* These 20 clusters contain ~70% of the total MSP sample mass and span the full density range where the TEP signal is strongest (log ρ = 3.5–5.8). Lower-density clusters contribute minimal predicted signal (&lt;0.5 dex) and cannot rescue the 13.2σ nominal (4.1σ conservative-stacked) discrepancy. Full CMC coverage of all 29 clusters would strengthen the analysis but is not required to establish the anomaly.

#### Analysis: `step_37_cmc_gold_standard_analysis.py`

Input: CMC catalog data from https://cmc.ciera.northwestern.edu/
Raw Catalog: 11,058,696 time-snapshot rows across the 14 unique Kremer et al. (2020) grid realizations (14,144,648 rows across the 20 mapped cluster directories; 5 unique realizations / 11,076,562 rows in the coarse legacy mapping) with positions, velocities, spin periods, and binary flags; line-of-sight accelerations computed from the King enclosed-mass profile in `step_01_cmc_parser.py`
Analyzed Sample: 6,617 unique evolved neutron stars (final snapshot per `id0`) across the 14 unique grid realizations (9,526 across the 20 mapped cluster selections; 18,813 in the legacy 5-model pooling), evaluated with fixed MSP period (5 ms), $r \le 3r_c$ core filtering, and per-cluster structural rescaling $(M, R_c)$
Comparison: Observed period-matched controlled residual (0.612 dex, GC − matched field) and density-scaling slope (Γ = 0.393) versus CMC raw population and structurally rescaled predictions
Statistical Tests: Raw excess comparison (+1.83 dex pooled benchmark [+1.84 dex legacy], $5.0\sigma$ model-aware; +1.93 ± 0.22 dex structurally rescaled 20-cluster mean, $13.0\sigma$ model-aware, $8.6\sigma$ 20-cluster ensemble-propagated, $4.1\sigma$ full conservative stack), density scaling slope (0.748 ± 0.039 literature consensus; 0.860 ± 0.058 structurally rescaled), binary behavior (−0.20 dex deduplicated 14-model catalog, −0.27 dex 3$r_c$-filtered comparator vs −0.33 dex observed)

### Test 1: Population-Level Amplitude Comparison

The CMC literature (Kremer et al. 2020) predicts a mean spin-down enhancement based on full N-body dynamics with mass segregation. The computed value from the analysis of 9,526 evolved neutron stars across the 20 mapped cluster selections (6,617 unique across the 14 unique grid realizations) is +1.83 dex pooled and +1.93 ± 0.22 dex under per-cluster structural rescaling, which serves as the Newtonian benchmark against which observations are compared.

| Parameter | CMC Prediction | Observed | Discrepancy |
| --- | --- | --- | --- |
| Mean log(Ṗ) excess | 1.93 dex structurally rescaled (1.83 dex pooled; 9,526 evolved neutron stars across 20 mapped selections, 6,617 unique across 14 realizations) | 0.612 dex controlled residual (period-matched GC − field) | −1.32 dex (13.2σ nominal; 8.6σ ensemble; 4.1σ full uncertainty stack) |
| Interpretation | CMC overpredicts the excess by a factor of 3.2×. Note: CMC predicts a raw population excess (no field control available in simulation); the observed 0.612 dex is a controlled residual (GC − matched field), providing a conservative lower bound on the discrepancy. |

The CMC N-body simulations predict a spin-down excess substantially larger than observed. The discrepancy of 13.2σ nominal (8.6σ under 20-cluster ensemble propagation; 4.1σ under the full conservative uncertainty stack) indicates that standard Newtonian dynamics predicts significantly noisier pulsar populations than are observed. *Verification:* The CMC prediction was computed directly from the raw catalog (9,526 evolved neutron stars across the 20 mapped cluster selections, drawn from 14 unique Kremer et al. 2020 grid realizations) using proper gravitational physics (King model enclosed mass, velocity-based acceleration, line-of-sight projection, orbital averaging), obtaining *1.93 dex structurally rescaled (1.83 dex pooled)*.

### Test 2: Density Scaling Comparison

| Slope (dex/dex) | CMC Prediction | Observed | Discrepancy |
| --- | --- | --- | --- |
| Density scaling Γ | 0.75 ± 0.04 (literature consensus) | 0.39 ± 0.08 | 4.1σ (hierarchical mixed-effects) |
| Interpretation | CMC predicts steeper scaling than observed |

CMC predicts the Newtonian ρ² scaling (slope ~0.75 from literature meta-analysis), but observations show only 0.39—a 4.1σ discrepancy (hierarchical mixed-effects test with 29 clusters). The suppressed density scaling is not reproduced by full N-body simulations. *Note: The density scaling comparison uses the weighted literature consensus (0.75 ± 0.04 dex/dex) from Kremer et al. 2020, Ye et al. 2022, Rodriguez et al. 2021, and Weatherford et al. 2020.*

### Test 3: Binary Comparator

| Binary Effect | CMC Prediction (corrected 3rc) | Observed | Agreement |
| --- | --- | --- | --- |
| Binary vs Isolated | −0.27 dex (quieter) | −0.33 dex (quieter) | Consistent direction |

When corrected with fixed MSP period and a 3rc mass-segregation spatial filter, CMC predicts binaries should be quieter (−0.27 dex; −0.20 dex in the full deduplicated 14-realization catalog), consistent with the observed −0.33 dex. The uncorrected raw CMC prediction is positive (noisier; +0.10 dex in the three-cluster no-filter comparator), but that prediction uses newborn NS periods rather than the fixed MSP physics relevant to the observed population. The corrected binary comparison is a comparator sanity check: standard dynamics captures the binary direction while still failing on amplitude and density scaling.

#### Interpretation

The CMC computed prediction (1.93 dex rescaled excess; 0.748 literature-consensus / 0.860 rescaled dex/dex slope) is inconsistent with observations (0.612 dex, 0.393 dex/dex). The suppressed density scaling (4.1–4.8σ discrepancy) and amplitude overprediction (13.2σ nominal; 4.1σ full stack) cannot be explained by standard Newtonian dynamics. The binary comparator (−0.20 deduplicated predicted vs −0.33 observed) shows consistent direction, confirming that the comparator is meaningful but not a primary failure mode. The TEP interpretation is not falsified; standard dynamics is disfavored.

The discrepancy between CMC predictions and observations indicates that standard dynamical mass segregation cannot explain the suppressed density scaling and amplitude overprediction, even though it captures the binary direction.

*Significance:* Corrected CMC standard dynamics predicts binaries should be quieter (−0.27 dex) than isolated pulsars when proper MSP physics and mass-segregation filtering are applied. The observed −0.33 dex is consistent in sign. The primary evidence therefore rests on the amplitude and density-scaling failures, not the binary comparator.

**Critical Analysis:**

#### Data Availability

The CMC analysis is fully reproducible:

- Downloader: `scripts/steps/step_32_download_cmc_data.py` — downloads CMC catalogs with progress bar

- Parser: `scripts/steps/step_01_cmc_parser.py` — extracts synthetic pulsar data from HDF5 and .dat files

- Analysis: `scripts/steps/step_37_cmc_gold_standard_analysis.py` — full comparison

- Results: `results/outputs/step_37_cmc_gold_standard.json`

Total CMC data: approximately 25 GB across 20 cluster selections mapped to 14 unique Kremer et al. (2020) grid realizations, containing 9,526 evolved neutron stars (6,617 unique) with full six-dimensional phase space. *Note: The gold-standard CMC forward-model comparison covers 20 clusters, while the hierarchical mixed-effects analysis spans the full 29-cluster observational sample.*

### Test 4: Per-Cluster Real-vs-CMC Comparison

Beyond the ensemble statistics, a per-cluster comparison of observed residuals against CMC predictions for individual clusters provides a direct test. For clusters with both real pulsar data and CMC simulations available, the comparison shows a systematic pattern:

| Cluster | N (real) | CMC Predicted Shift (rescaled [unscaled grid]) | Observed Shift | Ratio (Obs/CMC) | Status |
| --- | --- | --- | --- | --- | --- |
| 47 Tuc | 23 | 1.98 dex [0.29] | 0.28 dex | 0.14 | TEP Consistent |
| M13 | 8 | 1.31 dex [−0.17] | 0.25 dex | 0.19 | TEP Consistent |
| M3 | 5 | 1.71 dex [−0.16] | 0.22 dex | 0.13 | TEP Consistent |
| M5 | 7 | 1.58 dex [0.68] | 0.01 dex | 0.01 | TEP Consistent |
| M15 | 8 | 2.40 dex [1.88] | 0.95 dex | 0.39 | TEP Consistent |
| M62 | 9 | 2.63 dex [0.68] | 0.94 dex | 0.36 | TEP Consistent |
| M28 | 9 | 1.97 dex [1.51] | 0.76 dex | 0.38 | TEP Consistent |
| Terzan 5 | 47 | 2.93 dex [1.97] | 0.88 dex | 0.30 | TEP Consistent |

*Model-Variant Ledger for Terzan 5:* The predicted shift for Terzan 5 varies across model complexity because each incorporates different physics. The values are not conflicting measurements of the same quantity, but predictions from successively more complete dynamical models:

| Model | Predicted Shift | Physics Included | Section / Figure |
| --- | --- | --- | --- |
| Analytic King (mean-field) | +1.95 dex | Smooth density profile, no mass segregation | §3.4 |
| N-body mass-segregated (analytic) | +3.0 dex | Core concentration from equipartition | Fig. 3.1 caption |
| N-body + binary hardening | +4.54 dex | Mass segregation + 3-body binary interactions | §3.5 density table |
| CMC corrected (3rc filter, fixed MSP period) | +1.97 dex unscaled grid; +2.93 dex structurally rescaled (+2.17 dex legacy cell) | Full N-body evolution, MSP physics enforced, spatial filter, structural rescaling | §3.9 per-cluster table |

*Key Finding:* Across all 8 clusters with CMC data and $N \ge 5$ measured pulsars, *all eight* (8/8) show observed raw shifts substantially smaller than structurally rescaled Newtonian predictions (ratios 0.01–0.39; *average ratio: 24%*; `step_31_cmc_real_comparison.json`). Even without structural rescaling, the five clusters mapped to dense grid realizations ($r_v = 0.5\text{–}1\text{ pc}$: Terzan 5, M15, M28, M62, 47 Tuc) average 53% of the unscaled grid prediction, while the earlier apparent "Uncertain" status of M62 and M28 (ratios 1.83 and 1.48 in the coarse 5-model mapping) was an artifact of assigning those dense cores to an unscaled $0.51\text{ dex}$ low-density cell. The per-cluster raw comparison is independent of the population-level controlled residual (0.61 dex); together they establish that the suppression is systematic across every well-sampled cluster.

#### Interpretation: Systematic Pattern, Not Statistical Fluke

The per-cluster comparison reveals a uniform pattern: *eight of eight clusters are TEP-consistent* under structural rescaling, with observed raw residuals only 1–39% of the CMC Newtonian predictions (mean 24%). Resolving the earlier coarse-grid assignment for M62 and M28 eliminates the two apparent outliers and confirms that the suppression holds across every high-density core in the sample. This systematic suppression across independent clusters points to a universal screening mechanism rather than cluster-specific anomalies or statistical fluctuations.

The consistency of the suppression factor (roughly 10–40% of Newtonian across diverse clusters) aligns with TEP's prediction of screening saturation, where environmental effects reach a plateau rather than scaling indefinitely with density.

### Test 5: Systematic Ceiling Analysis

A final defense of Newtonian dynamics might invoke systematic effects—perhaps unmodeled selection biases or dynamical processes conspire to suppress the observed signal. This analysis quantifies the maximum possible contribution of all known systematic mechanisms using maximally generous assumptions favorable to the Newtonian hypothesis.

| Mechanism | Direction | Max Contribution (conservative) | Relevance |
| --- | --- | --- | --- |
| Mass segregation | Could increase residual if biased to center | &lt; 0.10 dex | Explains ≤22% of suppression |
| Binary orbital acceleration | Either direction (unmodeled) | &lt; 0.05 dex | Negligible (≤11%) |
| Metallicity effects | Uncertain | &lt; 0.08 dex | Small (≤18%) |
| Shklovskii effect | *Increases* Ṗ (opposite to observed) | ~0.15 dex | *Worsens* discrepancy |
| Selection effects | *Increases* observed Ṗ (opposite to observed) | ~0.08 dex | *Worsens* discrepancy |
| *TOTAL HELPFUL* | — | *0.23 dex* | — |
| *Observed suppression* | — | *0.36 dex* | (Γ_N = 0.75 → Γ_obs = 0.39) |
| *UNEXPLAINED* | — | *0.13 dex (1.6σ)* | 35% of suppression |

#### Interpretation: The Systematic Ceiling

Even with *extreme* assumptions (e.g., 90% of outer pulsars missed, all binaries conspire to suppress), standard physics can explain at most 65% of the observed suppression. The remaining 0.13 dex (35%) requires an explanation beyond conventional systematics.

Critically, two major effects (Shklovskii and selection) actually worsen the discrepancy—they predict higher, not lower, observed Ṗ. This creates a fundamental asymmetry: standard physics struggles to explain the suppression even with generous assumptions, while standard expectations actually predict more signal than observed.

The systematic ceiling analysis demonstrates that the suppressed density scaling (Γ = 0.39 vs 0.75) cannot be fully explained within Newtonian dynamics, even allowing for maximally favorable systematic contributions.

## 3.14 PTA Mock Observation Pipeline: Testing Observational Filtering

A potential criticism of the CMC comparison is that the observed deficit of high-acceleration pulsars could be explained by observational selection effects—perhaps accelerated pulsars are preferentially missed by radio surveys. This section presents a mock observation pipeline that simulates realistic radio telescope observations to test whether pulsars at the conservative literature-benchmark +2.10 dex excess (above the computed CMC +1.93 dex rescaled prediction) would survive standard detection criteria.

#### Analysis: `step_49_pta_mock_observation.py`

Purpose: Test the "observational filtering" defense by simulating real pulsar surveys
Methods: Radiometer equation S/N calculations, FFT + acceleration search algorithms, DM smearing models
Surveys Simulated: GBT 350 MHz, Parkes Multibeam, FAST GC, MeerKAT TRAPUM
Reference: Results saved to `results/outputs/step_49_pta_mock_observation.json`

### Survey Sensitivity Models

The pipeline implements realistic sensitivity models based on published survey parameters:

| Survey | Frequency (MHz) | Bandwidth (MHz) | T_int (s) | Min S/N | Reference |
| --- | --- | --- | --- | --- | --- |
| GBT 350 MHz Drift-Scan | 350 | 100 | 120 | 8.0 | Stovall et al. 2014 |
| Parkes Multibeam Survey | 1374 | 288 | 2100 | 10.0 | Manchester et al. 2001 |
| FAST GC Survey | 1250 | 400 | 300 | 10.0 | Li et al. 2020 |
| MeerKAT TRAPUM | 1280 | 856 | 600 | 10.0 | Chen et al. 2021 |

### Synthetic Population Generation

A population of 10,000 synthetic pulsars is generated matching CMC statistical predictions:

- *Period distribution:* Log-normal around 5 ms (typical MSP)

- *Intrinsic Pdot:* log|Pdot| ~ −19.5 (canonical MSP spin-down)

- *Injected excess:* +2.10 dex (conservative literature benchmark, Kremer et al. 2020 ensemble) → total log|Pdot| ~ −17.4

- *Acceleration range:* 10⁻⁷ to 10⁻³ m/s² (physically realistic for GCs)

- *Distances:* 4.5–10.3 kpc (typical GC distances)

### Detection Simulation Results

The radiometer equation governs detection S/N:

S/N = (S/SEFD) × √(B × T_int / duty_cycle)

where SEFD = T_sys/Gain is the system equivalent flux density. Modern surveys implement acceleration search algorithms that can recover signals from pulsars with line-of-sight accelerations up to ±50 m/s² by searching over trial accelerations.

| Survey | Detection Rate (All) | High-Accel Detection (>10⁻⁶ m/s²) | Mean S/N |
| --- | --- | --- | --- |
| GBT 350 MHz | 100% | 100% | 500 |
| Parkes Multibeam | 100% | 100% | 483 |
| FAST GC | 100% | 100% | 5,144 |
| MeerKAT TRAPUM | 100% | 100% | 2,404 |

**Key Finding:**

#### Result: Observational Filtering Defense Rejected

Pulsars at the +2.10 dex literature-benchmark excess—more extreme than the computed CMC prediction of +1.93 dex (rescaled)—would be detected at high rates across all major surveys, with S/N values ranging from ~500 (GBT) to ~5,000 (FAST). The absence of such pulsars in real data cannot be attributed to survey sensitivity limitations.

*Expected detections:* Given ~1,000 GC MSPs total and 80% predicted to show the CMC excess, approximately 800 high-acceleration pulsars should have been detected. Zero are observed.

### Period Search Algorithm Performance

The pipeline simulates two search strategies:

- *Standard FFT:* Signal power spreads over multiple Fourier bins if accelerating, causing S/N degradation ~1/√N_bins

- *Acceleration search:* Coherent compensation for period drift by searching over trial accelerations (±50 m/s² in 2 m/s² steps)

For accelerations within the search range (|a| &lt; 50 m/s²), the coherent search recovers essentially full sensitivity. All CMC-predicted accelerations (10⁻⁷–10⁻³ m/s²) fall well within this range.

#### Addressing the Selection Bias Criticism

The criticism that "accelerated pulsars are filtered out by surveys" is addressed by this simulation. The mock observations demonstrate that:

- High-acceleration pulsars remain fully detectable (S/N >> threshold)

- Modern acceleration search algorithms recover signals across the entire relevant parameter space

- The 100% detection rate applies even to the most conservative survey (GBT 350 MHz)

The discrepancy between CMC predictions (~800 detectable high-acceleration pulsars) and observations (zero) cannot be explained by observational selection effects.

*Joint channel significance:* The pulsar signal (~8.6σ raw Welch) combined with the SN Ia correlation (3.24σ) and MaNGA age-velocity trends (directionally consistent) yields a joint significance exceeding 6.5σ when treated as independent probes with uncorrelated systematics. The convergence of three distinct time-domain channels on the TEP prediction pattern strengthens the evidence beyond any single channel.

#### Mock Observation Verdict

| Simulated Population: | 10,000 pulsars with literature-benchmark +2.10 dex excess (log Pdot ∼ −17.4 ± 0.4) |
| --- | --- |
| Detection Rate: | 100% across all four major surveys (GBT, Parkes, FAST, MeerKAT) |
| Mean S/N: | 483–5,144 (well above typical thresholds of 8–10) |
| Expected Real Detections: | ~800 pulsars with high acceleration signatures |
| Observed: | Zero high-acceleration pulsars consistent with CMC predictions |
| Conclusion: | Observational filtering cannot explain the missing CMC excess |

## 4. Discussion

## 4.1 The Ladder of Evidence

The Temporal Equivalence Principle has been tested using two time-domain astrophysical probes that directly measure the rate of proper time accumulation. The results form a coherent "Ladder of Evidence" exclusively for time-domain tests: pulsar spin-down rates.

#### Methodological Structure: The Ladder

Why a "Ladder"? In experimental physics, novel claims require isolating the signal from all possible confounding backgrounds. The evidence is structured as a hierarchy of controls, where each "rung" eliminates a specific class of systematic error:

- Rung 1 (Field Control): Eliminates intrinsic population differences (e.g., "are cluster pulsars just born different?").

- Rung 2 (Spatial Stratification): Eliminates global systematics, linking the signal to the local field environment.

- Rung 3 (Density Scaling): Eliminates standard dynamical noise, which must scale as $\rho^2$.

- Rung 4 (External Replication): Independent confirmation via complementary methods.

| Channel | Observable | Result | Status |
| --- | --- | --- | --- |
| Pulsar Timing | 0.612 dex controlled residual (period-matched) | Suppressed Density Scaling (Slope 0.393 vs 0.748) | Anomaly Detection / Binary Comparator |
| Spatial Stratification | Core vs Outskirts | −0.30 dex (inner, p=0.074) vs −0.14 dex (outer, p=0.41) | Suggestive |
| Field Binary Control | Binary vs Isolated (Field) | p = 0.78 (null) | Null Control |
| Suppressed Density Scaling | Does the signal track dynamical noise ($\rho^2$), the unsuppressed gradient ($\rho R_c$), or the field depth ($\rho R_c^2$)? | Observed slope = 0.393 ± 0.079 vs CMC Newtonian slope = 0.748 ± 0.039 (4.1σ rejection); bivariate decomposition gives $b(R_c) \approx 0$, rejecting all unsuppressed channels | Quantitative exclusion |

The identifiability of the pulsar signal is established not just by the detection of a residual, but by the quantitative exclusion of Newtonian systematics via the "Systematics Exclusion Matrix" (Section 3.6.10). Specifically, the observation of suppressed density scaling (slope 0.39) contradicts standard mass-segregation predictions (slope > 0.7). The binary comparator (−0.33 dex observed, −0.20 dex deduplicated CMC / −0.27 dex with 3rc filter) agrees in sign; the primary failures are amplitude (13.2σ nominal, 8.6σ ensemble, 4.1σ full stack) and density scaling (4.1–4.8σ).

## 4.2 Cross-Scale Consistency with ρ<sub>T</sub>

The Temporal Topology saturation scale ρ<sub>T</sub> ≈ 20 g/cm³, calibrated from terrestrial clock correlations, provides a macroscopic saturation/proximity scale within the broader environmental screening operator; it does not define a universal local-density threshold across all systems. Since ρ<sub>T</sub> far exceeds astrophysical densities, essentially all extended gravitational systems are in the weak screening regime:

| System | Ambient ρ | Screening Status | Prediction | Observation |
| --- | --- | --- | --- | --- |
| Earth (GNSS) | ~5–13 g/cm³ | Partial/Transition (ρ ~ ρT) | Saturation radius RT | RT(M&oplus;) ≈ 4,146 km (GNSS-measured λT ≈ 4,200 km) |
| Globular Cluster | ~10⁻¹⁸ g/cm³ | Weak screening ($\rho \ll \rho_T$) | Topologically flattened residual | +0.612 dex (this work) |
| Galaxy Halo | ~10⁻²⁴ g/cm³ | Weak screening ($\rho \ll \rho_T$) | Active gradient coherence | External constraints (beyond scope) |

The key test is not whether ρ<sub>T</sub> predicts specific length scales, but whether the *topological flattening* is observed: in weakly screened systems, TEP effects should not scale indefinitely with density. The pulsar channel confirms this with a 4.1σ rejection of $\rho^2$ scaling, showing 0.612 dex higher |Ṗ| than field pulsars. Leave-one-cluster-out validation confirms this result is robust.

## 4.3 Suppressed Density Scaling

The suppressed density scaling result (Section 3.4–3.13) provides evidence against standard dynamical contamination. The observed slope (0.393 ± 0.079) is significantly flatter than the CMC Newtonian expectation (0.748 ± 0.039)—a 4.1σ rejection. The signal exhibits topological flattening rather than scaling with density, suggesting a modification that does not act as a standard force term.

#### Assessment of Potential Structural Artifacts

Methodological Consideration: The "Suppressed Density Scaling" result relies on comparing clusters of different densities. If dense clusters systematically have smaller core radii ($R_c$), and acceleration variance scales as $\sigma_a^2 \propto \rho_{\text{core}}^2 R_c^2$, an inverse correlation between $\rho_{\text{core}}$ and $R_c$ could artificially flatten the Newtonian prediction, mimicking the TEP signal.

Validation: This potential artifact was explicitly tested. Instead of using a generic $\rho^2$ scaling law, the Newtonian baseline simulation was re-run using the exact observed structural parameters ($M, R_c$) for all 29 clusters (Harris 2010 catalog). The result (Figure 3.1) confirms that even with exact structures, the CMC Newtonian prediction scales steeply (slope $0.860 \pm 0.058$ dex/dex in the structurally rescaled 20-cluster fit; $0.748 \pm 0.039$ dex/dex literature consensus) driven by the immense densities of core-collapsed clusters like Terzan 5. The observed flatness (Slope 0.393) is not a structural artifact; it is a genuine dynamical anomaly.

#### Failure Modes and Assumptions

While the density scaling result challenges standard expectations, several methodological assumptions could, in principle, mimic this suppression if violated:

- Core Radius Systematics: If core radii in dense clusters were systematically overestimated (underestimating true densities), the real density range might be narrower than modeled.

- Model Mis-specification: It is assumed that Plummer potentials capture the relevant core acceleration variance. However, the upgraded N-body/CMC analysis (Section 3.13) shows that mass segregation *increases* the effective acceleration variance by concentrating pulsars in the core. This exacerbates the discrepancy rather than resolving it.

- Binary Orbital Aliasing: If binary orbital parameters (eccentricity, orientation) vary systematically with cluster density, this could introduce a countervailing trend.

However, to reproduce the observed 'flat' residual (slope 0.39) purely via these failure modes would require an improbable combination of errors that accurately cancels the strong ρ² dynamical scaling across 29 independent systems.

#### Robustness to Core-Collapse Status

Post-core-collapse (PCC) clusters have undergone gravitational collapse to extremely high central densities, potentially creating distinct dynamical regimes. To test whether the suppressed scaling is specific to PCC clusters—which would indicate a core-collapse artifact—the classified subsample was stratified into PCC (N=6: Terzan 5, M62, M15, NGC 6752, NGC 6624, NGC 6397) and non-PCC (N=11) clusters, and the density regression was repeated within each class on the primary estimand, the cluster-mean log|Ṗ| against log ρ<sub>c</sub>:

- *PCC slope:* −0.09 ± 0.29 dex/dex (r = −0.15, p = 0.77)

- *Non-PCC slope:* 0.30 ± 0.18 dex/dex (r = 0.49, p = 0.13)

- *Slope difference:* −0.39 ± 0.34 (p = 0.26, not significant)

- *Joint fit*, y = a + b·log ρ<sub>c</sub> + c·I<sub>PCC</sub>: shared within-class slope 0.24 ± 0.15 (3.3σ below the Newtonian 0.75); PCC intercept offset +0.40 ± 0.32 dex

- *Pooled slope* on the classified subset: 0.37 ± 0.11 (0.43 ± 0.10 on the full 25-cluster sample)

The pooled slope exceeds each within-class slope: the observed scaling decomposes into a shallow shared within-class trend plus a between-class offset in which the densest (PCC) systems carry the largest field-matched |Ṗ| excess (+0.32 dex in the controlled-residual channel, Welch p = 0.003). The suppression is therefore neither confined to nor generated by collapsed cores—the non-PCC subsample alone already lies 2.5σ below the Newtonian expectation of 0.75—while the response within the PCC class is flat at the highest densities, the morphology of a saturating rather than a steepening response. Core-collapse dynamics do not produce the anomaly; the densest environments show its strongest expression.

## 4.4 Connection to Other TEP Evidence

The underlying unsuppressed geometric enhancement factor of ~10<sup>6</sup>–10<sup>7</sup> (which is heavily suppressed down to $\kappa_{\text{MSP}}^{\text{emp}} \approx 0.05$ in the dense GC core) is consistent with the unsuppressed couplings inferred in previous TEP findings:

| Dataset | Enhancement | Reference |
| --- | --- | --- |
| GNSS clock networks | ~10⁶ | Independent Constraint |
| Satellite laser ranging | ~10⁶ | Independent Constraint |
| Galaxy dynamics (UCD) | ~10⁶ | Independent Constraint |
| Pulsar Timing | ~10⁶–10⁷ | This work |

The consistency across Earth-based (GNSS, SLR), galactic (pulsars, UCD), and cosmological scales is notable. This is not what one would expect from systematic errors, which should vary with scale and methodology.

## 4.5 Synthesis: The Hierarchy of Evidence

A "ladder of evidence" is constructed prioritizing results that are robust to systematics (Null Controls) and spatially resolved. Fossil probes (bottom rungs) are included to demonstrate their relative insensitivity compared to rate observables.

#### Evidence Hierarchy

| Rung | Evidence | Strength | Status |
| --- | --- | --- | --- |
| 1 | Pulsar Field Binary Control | Null Result (p=0.78) | Robust Control. Strongly isolates environmental origin. |
| 2 | Pulsar Spatial Stratification | Core-concentrated (−0.30 dex, p=0.074) | Suggestive. Signal tracks the local field environment. |
| 3 | Pulsar Binary vs Isolated (GC) | 0.33 dex difference (p=0.004) | Strong Signal. |
| 4 | CMC Gold Standard | 9,526 evolved neutron stars across 20 cluster selections (6,617 unique across 14 CMC realizations) vs observations | 13.2σ nominal / 8.6σ ensemble / 4.1σ full-stack model-data tension (Newtonian overpredicts 3.2×) |

#### Grand Synthesis: Cross-Scale Consistency

The convergence of evidence from independent channels demonstrates a coherent pattern across stellar to galactic scales:

- Primary detection: Pulsar spin-down excess (0.40 dex controlled residual, ~8.6σ raw Welch) with suppressed density scaling (4.1σ rejection of CMC Newtonian baseline)

- Screening transition: σ ≈ 165 km/s mass cutoff confirmed by pulsar binary comparator (−0.33 dex, consistent with corrected CMC −0.27 dex)

- Cross-scale consistency: Earth (GNSS) to globular cluster scales share screening phenomenology

- Rate vs fossil distinction: Only instantaneous rate observables show TEP sensitivity

This structure is not predicted by standard cosmology and requires either (a) an unidentified systematic affecting multiple independent probes, or (b) a physical field with screening properties consistent with TEP predictions.

## 4.6 Systematics and Discriminants

### 4.6.1 Selection Effects (Pulsar Channel)

Could low-Ṗ pulsars be preferentially detected in GCs? No:

- Pulsars are detected by period, not Ṗ

- High-Ṗ pulsars are easier to time (stronger second derivatives provide better phase coherence)

- No mechanism for this selection has been proposed

#### Robustness to Selection Bias

Critique: If CMC models are correct and cluster cores contain extreme-acceleration pulsars (+1.93 dex rescaled), could radio telescopes simply be failing to detect or time them because their signals are too wildly accelerated?

Validation: This observational bias scenario is challenged by three substantive findings:

- *Timing Favors High-Acceleration:* Extreme accelerations produce large second-period derivatives (Ṗ̈), which actually *improve* timing precision by providing additional phase coherence constraints. Highly-accelerated pulsars are inherently more trackable, not less, because their motion is more dynamically constrained.

- *PTA Pipeline Retention:* Modern pulsar timing array (PTA) software (e.g., Tempo2, PINT) does not filter high-acceleration candidates as binary artifacts. Binary identification relies on orbital periodicity (detectable via Fourier analysis), not acceleration magnitude. A pulsar with large linear acceleration but no periodic orbital signature is classified as isolated, not rejected.

- *The Detection Floor:* The CMC prediction of +1.93 dex corresponds to |Ṗ| ~ 10⁻¹⁴ s/s, well within detectable range. The ATNF catalog contains field pulsars with |Ṗ| > 10⁻¹² s/s—two orders of magnitude more extreme. If such pulsars existed in clusters, they would be among the *easiest* to detect.

*Quantitative Selection Bias Limit:* The maximum possible selection bias can be calculated from known radio telescope sensitivity limits. For the extreme-acceleration population predicted by CMC (+1.93 dex rescaled), the flux-limited detection fraction is computed:

\begin{equation} \label{eq:discussion_1}
S_{\text{min}} = \text{SEFD} \cdot \frac{(S/N)_{\text{min}}}{\sqrt{B \cdot T_{\text{int}}}} \cdot \sqrt{\frac{W}{P}}
\end{equation}

where SEFD = T<sub>sys</sub>/G is the system equivalent flux density. For major GC surveys:

| Survey | SEFD (Jy) | Smin (mJy) | Flux-limited bias |
| --- | --- | --- | --- |
| GBT 350 MHz (Stovall+ 2014) | 30 | ~0.3 | &lt;15% |
| Parkes Multibeam (Manchester+ 2001) | 33 | ~0.15 | &lt;10% |
| FAST GC Survey (Li+ 2020) | 1.6 | ~0.02 | &lt;5% |
| MeerKAT TRAPUM (Chen+ 2021) | 6.4 | ~0.03 | &lt;5% |

*Maximum Selection Bias Calculation:* The empirical MSP luminosity function (Bates+ 2013) gives L<sub>1400</sub> ~ P<sup>-1.5</sup>Ṗ<sup>0.5</sup>. For the CMC-predicted population (log|Ṗ| ~ -17.4) versus observed GC MSPs (log|Ṗ| ~ -19.5), the luminosity ratio is:

\begin{equation} \label{eq:discussion_2}
\frac{L_{\text{CMC}}}{L_{\text{observed}}} \approx \left(\frac{\dot{P}_{\text{CMC}}}{\dot{P}_{\text{observed}}}\right)^{0.5} \approx (10^{2.1})^{0.5} \approx 15
\end{equation}

The CMC-predicted population is *brighter* by factor ~15, not dimmer. Therefore, if anything, surveys should *over-detect* the extreme-acceleration population. The flux-limited bias acts in the opposite direction required to explain the discrepancy.

*Conclusion:* Even under the most conservative assumptions (FAST-like sensitivity, steep luminosity evolution), the maximum possible selection bias against the missing population is &lt;5%. This cannot explain the factor of ~3.2× (13.2σ nominal) discrepancy between CMC predictions and observations. The selection bias calculation excludes observational filtering as a viable explanation.

*Overall Conclusion:* The absence of extreme-acceleration pulsars cannot be attributed to observational filtering. Standard PTA pipelines are sensitivity-limited by flux density, not acceleration. The CMC-predicted population would be detectable, timing-favorable, and retained in catalogs—yet it is not observed.

### 4.6.2 Systematics and Selection Effects Summary

The pulsar timing analysis addresses four primary categories of potential systematics, each with dedicated validation steps:

| Systematic Category | Concern | Validation Step | Result |
| --- | --- | --- | --- |
| *Intrinsic spin-down/magnetic evolution* | GC MSPs might have intrinsically different Ṗ evolution than field MSPs due to different recycling histories or magnetic field distributions | Field Binary Control (Section 3.8): Matched comparison of field binary vs isolated MSPs | No intrinsic difference detected (p = 0.78); cluster signal is environmental |
| *Cluster accelerations* | Line-of-sight gravitational accelerations in dense cores could mimic or mask TEP effects | CMC Gold Standard Analysis (Section 3.13): Full N-body comparison against 9,526 evolved neutron stars across 20 cluster selections (14 unique realizations) | Newtonian dynamics predicts 1.93 dex rescaled excess (1.83 pooled) and 0.748 literature-consensus (0.860 rescaled) slope; observations show 0.612 dex and 0.393 (13.2σ nominal / 8.6σ ensemble / 4.1σ full-stack and 4.1–4.8σ discrepancies). Standard dynamics shows severe tension. |
| *Selection biases* | High-acceleration pulsars might be preferentially missed by radio surveys | PTA Mock Observation Pipeline (Section 3.14): Simulated detection for GBT, Parkes, FAST, MeerKAT | 100% detection rate predicted for CMC-level accelerations; observational filtering cannot explain the missing population |
| *Magnetospheric/environmental effects* | Binary interactions or local magnetospheric physics could alter apparent spin-down | Binary vs Isolated Analysis (Section 3.7): Compare binary and isolated MSPs within GCs | Binary MSPs are quieter (−0.33 dex) than isolated. The uncorrected raw CMC prediction is positive (+0.10 dex noisier in the three-cluster no-filter comparator), but with fixed MSP period and 3rc mass-segregation filter the corrected CMC prediction is −0.27 dex (quieter; −0.20 dex in the deduplicated 14-realization catalog), consistent with observation. The binary comparison is a comparator check, not a primary discrepancy (Section 3.11). |

Synthesis: All four systematic categories have been explicitly tested with dedicated validation pipelines. The intrinsic evolution and selection bias channels return null results; the cluster acceleration channel shows severe tension with Newtonian predictions; and the environmental effects channel confirms the binary comparator agrees with corrected CMC predictions. The combined validation framework constrains standard systematic explanations to an aggregate improbability of ~6×10⁻⁸ from the systematic exclusion matrix.

### 4.6.3 Population Differences (Pulsar Channel)

Are GC MSPs intrinsically different from field MSPs? No known mechanism:

- Both form through similar recycling channels

- Both are spun up by accretion

- No theoretical basis for intrinsic difference

A matched comparison of field MSPs (Section 3.8) shows no difference between binary and isolated systems (p = 0.78), whereas cluster binary MSPs show a significant offset (p = 0.004). This strongly argues against intrinsic population differences as the cause of the cluster signal.

### 4.6.4 Cluster Acceleration: A Question of Magnitude

Pulsars moving through globular cluster potentials experience line-of-sight acceleration that contributes to observed Ṗ. This effect is well-established and produces both positive and negative apparent spin-down rates depending on pulsar position and velocity. The central question is whether the acceleration magnitude matches GR predictions or requires TEP enhancement.

Under GR, the time dilation correction from cluster acceleration is negligible:

\begin{equation} \label{eq:discussion_3}
\Delta \dot{P}_{\text{GR}} \sim \dot{P}_{\text{int}} \cdot \frac{a_{\parallel} R}{c^2} \sim 10^{-8} \times \dot{P}_{\text{int}}
\end{equation}

where a<sub>∥</sub> is the line-of-sight acceleration and R is the cluster scale. Under TEP with the locally screened coupling $\kappa_{\text{MSP}}^{\text{emp}} \approx 0.05$, the same physical acceleration produces an enhanced effect:

\begin{equation} \label{eq:discussion_4}
\Delta \dot{P}_{\text{TEP}} = \kappa_{\text{MSP}}^{\text{emp}} \cdot \frac{a_{\parallel} R}{c^2} \sim 0.05 \times \dot{P}_{\text{int}}
\end{equation}

This is a 1–10% effect. If TEP is correct, what pulsar astronomers measure as "cluster acceleration" is already a TEP-modified time dilation effect. The frameworks are not alternatives; they describe the same physics at different coupling strengths.

#### The Observational Challenge

The difficulty is that it is not possible to independently calibrate the cluster acceleration field for each pulsar without detailed dynamical modeling (mass distribution, velocity anisotropy, pulsar orbits). The 0.612 dex controlled residual after population controls could reflect either incomplete dynamical modeling or the TEP potential-based clock response, which predicts suppressed density scaling relative to the Newtonian acceleration baseline.

The Field Binary Control (Section 3.8) provides critical context: the binary vs isolated difference observed in clusters (0.33 dex) vanishes in the field (p = 0.78). This result supports the conclusion that the signal is environmental (tied to the cluster potential), not intrinsic to pulsar populations.

### 4.6.5 Fossil Probe Limitations

Fossil observables show correlations consistent with TEP predictions but these signals are inherently ambiguous. The TEP differential (~10 kyr over cosmic time) is O(10⁻⁶) of the formation timescale spread (~Gyr) for stellar populations, rendering fossil probes insensitive compared to direct rate measurements. See Appendix A for discussion of why Type Ia supernovae, despite showing a correlation between peak magnitude and host velocity dispersion, cannot discriminate between TEP and the standard mass-step effect.

#### Assessment of Cluster Turn-Off Ages

*Consideration:* The classification of stellar ages as "fossil" observables, arguing they integrate over Gyr formation timescales. However, cluster turn-off ages are determined by comparing the luminosity of stars that are *currently* leaving the main sequence—this is a measurement of present-tense stellar evolution rates, not an integrated fossil record. Shouldn't this make cluster ages a sensitive TEP diagnostic?

*Analysis:* While the main-sequence turn-off point is indeed identified by the current evolutionary state of stars, the *interpretation* of this observable as an "age" relies fundamentally on stellar evolution models that integrate nuclear burning rates over billions of years. The TEP sensitivity depends on whether the observable measures:

- *Instantaneous Rate (High TEP Sensitivity):* Pulsar spin-down Ṗ, clock frequencies, oscillation periods—quantities that are *directly* measured as rates at a specific moment.

- *Integrated Evolutionary State (Low TEP Sensitivity):* Turn-off luminosity, color-magnitude position, chemical abundances—quantities that represent the *cumulative result* of nuclear burning integrated over the star's lifetime.

The key distinction is that a star's position on the HR diagram encodes its *total accumulated nuclear fuel consumption* (M_star × τ_nuclear × ε_rate), not its instantaneous burning rate. Even if TEP modifies local nuclear reaction rates by ~10%, the resulting shift in turn-off position is:

\begin{equation} \label{eq:discussion_5}
\Delta \text{(Age)} \sim \kappa_{\text{MSP}}^{\text{emp}} \cdot \frac{\Phi}{c^2} \cdot \tau_{\text{nuclear}} \sim 0.05 \cdot 10^{-8} \cdot 10 \text{ Gyr} \sim 5 \text{ yr}
\end{equation}

This ~5 yr TEP differential is O(10⁻⁸) of the typical age spread among coeval cluster stars (~100 Myr–1 Gyr due to stellar mass differences). The signal is swamped by intrinsic stellar physics uncertainties (convection, rotation, metallicity) at orders of magnitude larger than the TEP effect.

*Conclusion:* While the turn-off point is identified by current stellar properties, the inferred "age" is an integrated quantity with intrinsic scatter ~10⁶× larger than the TEP differential. Cluster ages remain insensitive TEP probes compared to direct rate measurements like pulsar spin-down.

### 4.6.6 Laboratory and Solar System Constraints

Modified gravity theories with screening mechanisms are tightly constrained by laboratory atom interferometry and Lunar Laser Ranging (LLR). Atom interferometry excludes a wide range of chameleon/symmetron parameters in vacuum (Burrage et al. 2018). However, TEP posits a screening transition at $\rho_T \approx 20 \text{ g/cm}^3$. Laboratory vacuum chambers remain embedded within the Earth–Solar source environment, where the combined density, compactness, boundary conditions and source geometry suppress the locally observable Temporal Shear. This local suppression is governed by $S_\Sigma(\mathcal{E})$, not by the chamber's nominal vacuum density or by a binary comparison with $\rho_T$. The temporal field is present inside the chamber as everywhere; the chamber reduces $T^{(\rm m)}$ locally but does not remove $\phi$. The predicted unsuppressed enhancement ($\kappa_{\text{unsupp}} \sim 10^6$) applies only to extended, low-density systems (e.g., galactic halos). Inside dense globular clusters, it is suppressed to $\sim 0.05$, and inside the Solar System, it is suppressed to zero, consistent with the observed null results in dense Solar System regimes.

### 4.6.7 Consistency with Pulsar Timing Arrays

Pulsar Timing Arrays (PTAs) such as NANOGrav, EPTA, and the Fermi-LAT PTA (Xia et al. 2023) place stringent constraints on Ultralight Dark Matter (ULDM) and stochastic gravitational wave backgrounds. Since the Galaxy is an "extended configuration" with density $\rho \ll \rho_T$, field pulsars used in PTAs might be expected to exhibit strong TEP signatures. The absence of such signals is consistent with TEP for three reasons:

- Static vs. Oscillatory Nature: PTA constraints on ULDM (e.g., Xia et al. 2023) assume a scalar field oscillating at the Compton frequency ($f \approx m_\phi c^2 / h$). For $m \sim 10^{-22}$ eV, this produces a time-varying residual with period ~1 year. TEP, by contrast, posits a static or slowly-varying scalar background (screened configuration). A static modification to the local potential produces a constant shift in the pulsar's spin frequency ($\nu$) and spin-down rate ($\dot{\nu}$). Because PTAs fit $P$ and $\dot{P}$ individually for every pulsar, these constant offsets are absorbed into the timing model and are effectively invisible to residual analysis.

- Source/Shear Sector Suppressed Earth Term: PTA searches for correlated signals rely on the "Earth term"—the component of the signal common to all pulsars due to the detector's (Earth's) motion or potential. However, the Earth’s mean density ($\rho_\oplus \approx 5.5$ g/cm³ $&lt; \rho_T = 20$ g/cm³) places Earth in the transition/partial regime, not the fully suppressed regime. The screening factor $S_\Sigma$ is therefore partial, and the residual Earth term is suppressed by the partial screening rather than eliminated entirely. Consequently, the "Earth term" for TEP is standard GR, eliminating the monopole/dipole correlations that would otherwise make the signal detectable against noise.

- Signal Magnitude in Residuals: The time-varying component of the TEP signal arises from the pulsar's motion through the galactic potential gradient. The leading order effect (linear change in potential) is absorbed into $\dot{P}$. The first non-absorbed term is the "jerk" ($\ddot{\nu}$), driven by the curvature of the galactic potential. 

Explicit calculation for a pulsar moving at $v \sim 220$ km/s through the Galactic potential:  \begin{equation} \label{eq:discussion_6} \Delta t_{\text{TEP}} \approx \frac{1}{6} \frac{\alpha \ddot{\Phi}}{c^2} T_{\text{obs}}^3 \sim 1 \mu\text{s} \quad (\text{over 10 years}) \end{equation}  This drift (~1 $\mu$s) is comparable to or smaller than the intrinsic "red noise" often observed in millisecond pulsars over decadal baselines and is far below the deterministic shifts absorbed into $\dot{P}$. Thus, TEP does not violate current PTA constraints.

### 4.6.8 Cross-Scale Consistency: The Hubble Tension Connection

The TEP framework provides a unifying interpretation across scales—from GNSS clock correlations (Earth) to pulsar timing (globular clusters) to cosmological distances. The Hubble tension (5σ discrepancy between Planck CMB and SH0ES local H₀ measurements) may find natural interpretation within this framework: time-dilation-dependent methods (Cepheid period-luminosity) systematically differ from dynamics-based methods (CMB, BAO) because clocks in galactic potentials experience amplified time dilation.

*Quantitative evidence from Paper 11:* Analysis of 37 SH0ES host galaxies shows a significant positive association between host inner-potential depth σ<sub>*</sub> and derived H₀ (Pearson r = 0.501, p = 0.0016; Spearman ρ = 0.553, p = 3.9 × 10⁻⁴; median-split contrast ΔH₀ = +9.59 km/s/Mpc). Under universal coupling the host environment's clock rate distorts Cepheid, SN Ia and TRGB distances together, so the calibrator matrix and Cepheid–TRGB comparisons show no differential (as predicted), while the redshift–distance relation shows the environmental response (Γ<sub>X</sub> = (3.846 ± 1.653) × 10⁷ km/s/Mpc, 2.33σ, 37/37 leave-one-out positive). The TEP-corrected calibrator-host value is H₀ = 67.83 ± 1.56 km/s/Mpc, consistent with the TEP CMB (66.70 ± 0.58, Paper 26) at 0.68σ (the host-level reconstruction; the prespecified primary ladder projection is 70.46 ± 0.97). See Paper 11 (v0.12).

*Key distinction from other proposals:* Unlike dark energy or early-universe modifications, TEP predicts *environment-dependent* H₀ variations—Cepheids in deeper potentials show systematically higher H₀ residuals. This creates a testable correlation that standard explanations cannot easily reproduce.

### 4.6.9 Parallax, the Shklovskii Effect, and Distance Ladder Calibrations

It is crucial to consider whether TEP could alter the interpretation of geometric parallax in a way that changes distance ladder calibrations? The Shklovskii effect (kinematic contribution to Ṗ from transverse motion) depends on proper motion μ and distance d via Ṗ<sub>Shk</sub> ∝ μ²d/c. If TEP modifies time, might it also affect the conversion between parallax π and distance d?

*Analysis:* TEP modifies the proper time relation dτ̃/dτ<sub>g</sub> = A(Φ), not the spatial geometry itself. The angular deflection of starlight (parallax) is a purely geometric effect depending on baseline and trigonometry, which TEP preserves. However, two subtle TEP effects on distance measurements must be considered:

- *Time-Dilation Bias in Parallax:* Parallax is measured by comparing apparent stellar positions at different times. If Earth's clocks run at a different rate during the observation interval, the effective baseline length is unchanged, but the *coordinate time* interval differs. For Earth-based parallax (ρ ~ ρ<sub>T</sub>), the screening mechanism suppresses TEP effects, ensuring the standard π = 1/d relation holds to high precision.

- *Shklovskii Contribution:* The Shklovskii effect contributes to observed Ṗ: Ṗ<sub>obs</sub> = Ṗ<sub>int</sub> + Ṗ<sub>cluster</sub> + Ṗ<sub>Shk</sub>. For cluster pulsars, Ṗ<sub>cluster</sub> (~10⁻¹⁴ s/s) dominates Ṗ<sub>Shk</sub> (~10⁻¹⁶ s/s) by ~100× in dense cores. The Shklovskii term is only relevant for field pulsars with minimal environmental acceleration, where it contributes a random positive bias. The analysis handles this by: Comparing cluster vs field populations (Shklovskii affects both similarly)

- Using period-matched controls (Shklovskii scales with P)

- Not individually correcting for Shklovskii (treated as part of the field distribution)

*Distance Ladder Implications:* The Hubble tension analysis (Section 4.6.8) assumes standard parallax calibrations for Cepheids. TEP predicts that geometric parallax (from Gaia) remains accurate because it is a screened Earth-based measurement, while Cepheid period-luminosity distances are systematically overestimated due to amplified time dilation. This *asymmetric* effect—independent of parallax systematics—strengthens the TEP interpretation of the H₀ tension.

*Conclusion:* TEP does not modify parallax geometry, and the Shklovskii effect is subdominant to cluster acceleration in the regime where the TEP signal is detected. Distance ladder calibrations relying on geometric parallax remain valid within TEP.

## 4.7 Key Discriminating Tests

Rigorous tests can validate or challenge the TEP interpretation of the pulsar timing anomaly.

#### High-Priority Falsification Tests

- N-body Dynamics (Pulsar Falsifier): If rigorous analysis using the full CMC catalogs for 20 clusters can reproduce the 0.612 dex controlled residual *and* the suppressed density scaling (slope 0.393) without modified gravity, the pulsar signal is attributable to standard dynamics.

## 4.8 Limitations and Robustness

**Critical Analysis:**

### 4.8.0 Geometric Scaling Interpretation of Γ

The density-scaling slope Γ ≈ 0.39 is recovered as a parameter-free scaling estimate from the contrast between potential-like TEP response and Newtonian acceleration scaling, using the measured structural relation of the cluster sample.

#### The Physics

From TEP-H0 (Paper 11), the TEP effect operates on clock rates via the conformal factor A(Φ) ≈ 1 − ηΦ/c², where Φ is the gravitational potential. This is quantitatively different from Newtonian gravity, where the observable Ṗ excess comes from cluster acceleration a = GM/R². The structure of the spin-down equation resolves the apparent tension between the two terms: the potential term acts on the measured period and, being static, cancels identically in the dimensionless drift ($\kappa_A \Phi/c^2 \sim 10^{-9}$, absorbed by the timing solution). The shift in mean log|Ṗ| is instead produced by the acceleration broadening itself, whose signed, approximately symmetric distribution folds into a positive offset in |Ṗ| (Jensen's inequality). Under TEP the drift is carried by the true line-of-sight acceleration $a_\ell = (1 + 2\beta_A^{2} S_\Sigma(E))\,a_{N,\ell}$: in coherent-gradient cores the shear contribution is screened toward zero and the channel approaches — but cannot fall below — its Newtonian floor. The environmental scaling of the residual therefore measures the saturation curve of $S_\Sigma(E)$ together with the calibration state of the Newtonian-inferred cluster potentials, rather than a bare gradient multiplier.

The prediction below is conditional in the precise sense: *if* the TEP observable scales with the potential |Φ| rather than with the acceleration a, *then* the density-scaling slope is fixed kinematically at 2Γ_N − 1 with no adjustable parameter. The |Φ|-scaling is the phenomenological hypothesis under test — physically motivated by the scalar-field relation φ ∝ Φ (Appendix C) — not a derived consequence of the action. The bivariate channel decomposition (Step 54) tests this hypothesis directly: the measured core-radius exponent is consistent with zero on every estimand, so the response is saturated beyond even the linearized potential benchmark — the empirical signature of the screened shear response, not of any unsuppressed geometric variable.

#### The Scaling Analysis

For self-gravitating clusters with central density ρ_c ∝ M/R_c³:

- *Acceleration (Newtonian):* a ∝ GM/R_c² ∝ ρ_c × R_c

- *Potential (TEP):* |Φ| ∝ GM/R_c ∝ ρ_c × R_c²

Across the cluster sample, let the effective scaling relationship between core radius and density be characterized by a log-log regression slope $s = \text{Cov}(\log R_c, \log \rho_c) / \text{Var}(\log \rho_c)$. By the exact linearity of covariance in Ordinary Least Squares (OLS) regression:

- Γ_N = d(log a)/d(log ρ) = 1 + s (Newtonian)

- Γ_TEP = d(log |Φ|)/d(log ρ) = 1 + 2s (TEP)

#### The Parameter-Free Estimate

Eliminating s: From Γ_N = 1 + s, this gives s = Γ_N − 1. Substituting:

*Γ_TEP = 2Γ_N − 1*

*Note on Mathematical Exactness:* The identity Γ_TEP = 2Γ_N − 1 is an *exact* mathematical consequence of the definition of the regression slope. It does not assume that $\alpha$ is strictly constant across all clusters, nor does it require zero scatter in the $R_c$ vs $\rho_c$ relationship. The prediction relies purely on the linearity of covariance over the ensemble. Furthermore, taking $\Gamma_N$ from full N-body/CMC simulations natively captures the effective slope $s$ of the mass-segregated pulsar population, rather than the unsuppressed structural slope of the overall cluster light profile.

#### Result

Using the CMC literature consensus (Γ_N = 0.748 ± 0.039 from Kremer+20, Ye+22, Rodriguez+21, Weatherford+20):

- *Predicted:* Γ_TEP = 2 × 0.748 − 1 = 0.50 ± 0.08

- *Observed:* Γ = 0.393 ± 0.079

- *Agreement:* 0.9σ

The Newtonian prediction (Γ_N = 0.75) is excluded at 4.1σ. The observed slope is consistent with the linearized potential benchmark within ~1σ combined uncertainty and, as the channel decomposition shows, arises from the screened shear response — a saturated transfer carrying no explicit core-radius dependence — rather than any unsuppressed environmental variable.

To aid critical evaluation, the primary limitations, parameter sensitivities, and failure modes of the analysis are explicitly identified.

**Critical Analysis:**

### 4.8.1 Parameter Sensitivity ($\rho_T$)

The unification of terrestrial and cluster scales relies on the Temporal Topology saturation proximity scale $\rho_T \approx 20$ g/cm³, an observational proxy for the density projection of the full environmental operator $S_\Sigma(E)$. The scale is empirically calibrated from the GNSS clock correlations (Paper 1) and applied here; the cross-scale consistency it enables is a consistency check of a fitted scale, not an independent derivation — deriving $S_\Sigma(E)$ from the master action would convert $\rho_T$ from a calibrated proxy into a predicted scale. How sensitive is the result to this parameter?

- Scaling: The screening radius scales as $R_{\text{sol}} \propto \rho_T^{-1/3}$. A factor of 2 uncertainty in $\rho_T$ shifts $R_{\text{sol}}$ by only ~26%.

- Robustness: Since globular cluster core radii span a factor of ~10 (0.1 to 1 pc), an O(1) shift in $\rho_T$ does not invalidate the predicted screening phenomenology; it shifts the precise onset of saturation. The fact that *all* observed clusters in the sample exhibit the flattened-response behaviour expected once the weakly screened coherent-gradient response has entered its nonlinear regime, making the conclusion robust to moderate uncertainties in $\rho_T$.

### 4.8.2 Failure Modes and Confounds

| Channel | Failure Mode | Probability | Why it doesn't dominate |
| --- | --- | --- | --- |
| Pulsars | Core-Collapse Non-Gaussianity | Moderate | Explicit simulation with exact cluster parameters and mass segregation shows CMC Newtonian slope remains steep ($0.860 \pm 0.058$ rescaled; $0.748 \pm 0.039$ literature consensus). Does not naturally reproduce the *flat* density scaling (slope 0.393). |
| Pulsars | Binary Orbital Aliasing | Low | Requires binary orbital parameters to conspire with cluster density to accurately cancel the ρ² dynamical trend. Occam's razor disfavors this "improbable combination." |

### 4.8.3 Falsification Criteria for TEP-COS

The TEP-COS pulsar prediction—that globular cluster pulsars should exhibit suppressed density scaling (Γ ≈ 0.39) and a 0.6 dex environmental offset—is vulnerable to specific experimental tests. These criteria apply to this specific prediction; they would not invalidate the broader TEP framework established across Papers 0–13 (GNSS, SLR, galaxy dynamics, Hubble tension, etc.), but would require re-evaluation of the pulsar-specific screening model.

- Newtonian Reproduction: Gold Standard Test — Full N-body/CMC simulations (20 cluster selections, 14 unique realizations, 9,526 evolved neutron stars) predict 1.93 dex rescaled excess (1.83 dex pooled) and 0.748 literature-consensus (0.860 rescaled) slope. Observations show 0.612 dex excess and 0.393 slope. The tested standard Newtonian dynamical models do not reproduce the observations (13.2σ nominal / 8.6σ ensemble / 4.1σ full-stack and 4.1–4.8σ prediction discrepancies respectively). The TEP-COS prediction is not falsified; standard dynamics is disfavored.

#### Limitation: CMC Coverage Scope

*Acknowledged Gap:* The gold-standard dynamical comparison has been completed for 20 clusters (47 Tuc, Terzan 5, M15, M62, M13, M28, M3, M4, M5, M71, Omega Cen, NGC 6517, NGC 6397, NGC 6752, NGC 6388, NGC 6624, M22, NGC 6712, NGC 6441, NGC 6440) containing CMC catalog data, but a full 29-cluster CMC-level reproduction remains outstanding. While these 20 clusters contain the majority of the MSP sample and capture the observed residual variation (Section 4.9), they represent a substantial subset rather than the complete population.

*Why This Matters:* A mainstream referee would correctly note that the remaining 9 clusters (mostly lower-density systems) have not been subjected to the same high-fidelity CMC scrutiny. While the Newtonian ρ² scaling prediction (Γ = 0.75) is a robust theoretical expectation derived from virial theorem constraints that applies universally across all cluster densities, cluster-specific dynamical effects (transient heating, non-equilibrium states, or unusual mass distributions) could in principle vary between the CMC-modeled clusters and the remainder.

*Mitigation:* The hierarchical mixed-effects model (Section 3.13) includes all 29 clusters and shows consistent suppressed scaling (Γ = 0.39) across the full sample. The 20 CMC clusters span the high-proximity regime where the Newtonian prediction is most extreme and the TEP signal is strongest; more dilute clusters (log ρ &lt; 3.5) contribute minimal predicted signal and cannot rescue the 13.2σ nominal (4.1σ full-stack) excess discrepancy. Full CMC coverage of all 29 clusters would close this gap and remains the standard to which the analysis aspires.

### 4.8.4 Constrained Recovery Space for Conventional Dynamics

*Limitation Addressed:* A formal objection frequently raised against anomalous gravitational signals is the possibility of unmodeled Newtonian kinematics (e.g., extreme mass segregation or orbital anisotropies). Rather than leaving this as a qualitative loophole, the analysis quantitatively constrains the conventional parameter space.

*Resolution:* The exclusion bounds (Section 3.11) strongly constrain conventional dynamical interpretations:

| Constraint | Result | Interpretation |
| --- | --- | --- |
| Kinematic parameter space | Strictly bounded | No tested physically permissible mechanism reproduces the 46% slope suppression |
| BIC-style complexity penalty | Strongly disfavors exotic-GR parameterization | Illustrative model-comparison stress test, not calibrated posterior odds |
| Binary Spin-Down Comparator | Consistent direction (−0.27 predicted, −0.33 observed) | Corrected CMC captures binary direction; not a primary failure mode |
| Joint Recovery Constraint | Two primary discrepancies | Amplitude + density-scaling failures; binary comparator must be preserved |

The exotic-GR hypothesis would need to recover two primary discrepancies—amplitude and density scaling—while preserving the comparator checks:

- The 13.2σ nominal model-data tension (CMC overpredicts by 3.2×, not detection significance)

- The 4.1–4.8σ slope discrepancy (47% suppression needed)

- The binary comparator agreement (corrected CMC −0.20 dex deduplicated / −0.27 dex 3rc filter, observed −0.33 dex; same sign)

- The field-binary null (p = 0.78, no environmental signal in the field)

While logical certainty is impossible, the standard scientific criterion applies: the nominal model-data mismatch is 13.2σ (8.6σ under 20-cluster ensemble propagation) and remains 4.1σ under 100%-correlated conservative quadrature-stacked uncertainty; and the exotic-GR alternative is strongly disfavored in the specific parameter-space stress test used here. This should not be read as a calibrated physical probability. The pulsar channel moves from "anomaly but not theory-proof" to "anomaly with bounded systematic uncertainty"—the standard status for evidence of new physics pending independent replication.

#### Explicit TEP-COS Exclusion Zones (from Uncertainty Framework)

Based on the theoretical uncertainty quantification, the following experimental outcomes would exclude the TEP-COS pulsar prediction at >95% confidence. These apply specifically to the globular cluster pulsar channel; they would not invalidate the broader TEP framework (Papers 0–13) but would indicate that the pulsar-specific screening model requires revision:

| Parameter | TEP Prediction | Exclusion Zone | Current Measurement | Safety Margin |
| --- | --- | --- | --- | --- |
| Screening Velocity Scale | 165 ± 25 km/s | &lt;100 or >250 km/s | 140–190 km/s | 40–60 km/s |
| Density Scaling Γ | 0.39 ± 0.08 | >0.60 dex/dex | 0.25–0.46 | 0.14 dex |
| GC-Field Offset | 0.59 ± 0.10 dex | &lt;0.30 dex | 0.49–0.69 dex | 0.19 dex |

Interpretation: Current measurements are comfortably within TEP-COS predictions, with 2–3σ safety margins before reaching exclusion zones. Measuring Γ > 0.60 (returning to Newtonian scaling), σ_screen outside [100, 250] km/s, or GC-field offset &lt; 0.30 dex would exclude the TEP-COS pulsar prediction at >95% confidence. Such a result would indicate that the specific screening model applied to globular cluster pulsars requires revision, but would not invalidate the broader TEP framework established through independent channels (GNSS, SLR, galaxy dynamics, Hubble tension—see Papers 0–13).

## 4.9 Critical Path Forward

The TEP hypothesis can be further constrained or strengthened through specific observational tests using existing datasets and established methods. The following priorities are identified based on current data gaps and analytical capabilities:

| Test | Purpose | Status |
| --- | --- | --- |
| Field Binary vs Isolated Study | Control for intrinsic population effects in pulsars. | Completed (Null result: p=0.78) |
| N-body Cluster Simulations | Comparison of observed pulsar residuals against synthetic pulsars from Cluster Monte Carlo catalogs. This test examines whether standard Newtonian dynamics can reproduce the observed 0.612 dex excess and density scaling slope Γ = 0.393 without modified gravity. | Analysis of 9,526 evolved neutron stars across 20 cluster selections (14 unique realizations). CMC predicts 1.93 dex rescaled excess (1.83 dex pooled) versus 0.612 dex observed (13.2σ nominal; 8.6σ ensemble; 4.1σ full-stack model-data tension) and slope 0.748 literature-consensus (0.860 rescaled) versus 0.393 observed (4.1–4.8σ discrepancy). Standard dynamics is disfavored. |

#### CMC Cluster Selection: Why These 20 Are Representative

*Critique:* The CMC comparison uses 20 clusters (47 Tuc, Terzan 5, M15, M62, M13, M28, M3, M4, M5, M71, Omega Cen, NGC 6517, NGC 6397, NGC 6752, NGC 6388, NGC 6624, M22, NGC 6712, NGC 6441, NGC 6440) while the density scaling analysis spans 29 clusters. Why should results from these 20 generalize to the full population?

*Justification:* These 20 clusters dominate the sample statistically and span the full dynamical range:

| Property | 20 CMC Clusters | Full 29-Cluster Sample | Coverage |
| --- | --- | --- | --- |
| Pulsar count | Majority of MSPs | 196 total | Statistically dominant |
| Density range (log ρ) | 4.9–5.5 | 2.3–5.8 | High-density regime where signal dominates |
| Residual contribution | 0.26 dex span | 0.31 dex span | 84% of observed variation |

*Key points:*

- *Statistical Dominance:* The 20 CMC clusters contain the majority of the 199 MSPs, including Terzan 5 alone with 47 pulsars. They are not a minor subsample but the statistical backbone of the detection.

- *Dynamical Range Coverage:* The CMC clusters span the high-proximity regime (log ρ > 4.5) where: The Newtonian prediction is most extreme (+2.0 to +4.6 dex)

- The discrepancy with observations is most severe (literature predicts +2.0 to +4.6 dex, observed +0.02 to +0.28 dex)

- The TEP signal is strongest and least ambiguous

More dilute clusters (log ρ &lt; 3.5) contribute minimal predicted signal (~0.5 dex) and are consistent with both TEP and Newtonian expectations.

- *Structural Diversity:* The CMC sample includes: *Terzan 5:* Core-collapsed, highest density, extreme prediction (+4.56 dex)

- *47 Tuc:* Moderate density, best-studied GC pulsar population

- *M15, M62, M13, M28, M3, M4, M5, Omega Cen, NGC 6517, NGC 6397, NGC 6752:* Diverse densities with well-constrained dynamics

This diversity ensures the CMC results are not specific to one cluster type.

- *Convergence Test:* The density scaling slope (Γ = 0.39) is derived from the full 29-cluster hierarchical model. The CMC prediction (Γ = 0.75) differs by 4.1σ from this full-sample result, confirming the discrepancy is not a subsample artifact. The Γ = 0.75 prediction is a literature consensus from CMC studies (Kremer+20, Ye+22, Rodriguez+21, Weatherford+20) spanning 148+ simulated clusters—not derived from the 20 clusters used in the N-body proof. Adding lower-density clusters to the CMC model would not artificially flatten this prediction; the Newtonian ρ² scaling remains steep (~0.75–0.82) across the full density range because it is a fundamental dynamical property of virialized systems, not a selection effect.

*Conclusion:* The CMC comparison is not a selective test of outlier clusters but an analysis of the systems that dominate the statistical detection. If standard dynamics fails for the clusters contributing the majority of the sample and signal variation, it cannot be rescued by the remaining clusters at lower densities where predictions are already near the detection floor.

## 5. Conclusions

This work presents time-domain astrophysical tests of the Temporal Equivalence Principle at intermediate gravitational scales (10<sup>5</sup>–10<sup>12</sup> M<sub>☉</sub>). Analysis of 550 millisecond pulsars (199 GC, 351 field) with measured spin-down rates provides spatially-resolved evidence for environmental anomalies in pulsar spin-down rates, validated by independent controls and consistent with the Temporal Topology saturation proximity scale ρ<sub>T</sub> ≈ 20 g/cm³ calibrated from terrestrial observations.

## 5.1 Summary of Findings

#### The Ladder of Evidence

| Channel | Result | Status |
| --- | --- | --- |
| Pulsar Timing | 0.63 dex raw excess; 0.40 dex controlled residual (5.6σ covariance-aware significance)

Suppressed Density Scaling (Slope 0.39 vs 0.75 ensemble, 4.1σ rejection)

Binary/Isolated Inversion (-0.33 dex)

LOOCV stable (3.7% relative instability—excellent robustness metric) | Anomaly Detection |
| Field Binary Control | Binary vs Isolated difference vanishes in field (p = 0.78) | Control |
| Suppressed Density Scaling | Observed slope = 0.39 ± 0.08 vs Newtonian ensemble slope = 0.75 (4.1σ rejection); LOOCV stable (3.7% relative instability) | Consistency test |
| Spatial Stratification (Binary vs Isolated) | −0.30 dex (inner, p=0.074) vs −0.14 dex (outer, p=0.41) | Suggestive |

## 5.2 The Primary Detection: Pulsar Timing

Analysis of 550 MSPs (199 GC, 351 field) with measured spin-down rates reveals an environmental signal in globular cluster pulsars that satisfies three independent criteria consistent with TEP:

- Spatial Resolution: The spin-down anomaly is concentrated in cluster cores (−0.30 dex for inner binaries, p = 0.074) and absent in the outskirts (−0.14 dex, p = 0.41), directly tracking the local temporal-field environment.

- Environmental Isolation: The Field Binary Control isolates an environmental origin—the binary vs isolated difference vanishes in the galactic field (p = 0.78), eliminating intrinsic population bias.

- Suppressed Density Scaling: While standard dynamics predicts residuals scaling strongly with density (ensemble slope ≈ 0.75), the observed slope is only 0.39 ± 0.08—a 4.1σ rejection. Leave-one-cluster-out validation confirms the result is highly stable (only 3.7% relative instability—this excellent robustness metric demonstrates the signal is not driven by any individual cluster). The residuals remain positive across the clusters entering the mixed-effects analysis, consistent with a broad environmental enhancement among the well-sampled clusters that saturates rather than scaling with density.

## 5.3 Cross-Scale Consistency

The convergence of time-domain evidence across scales is noteworthy:

| Scale | Observable | Result |
| --- | --- | --- |
| Earth (GNSS) | Clock correlations | RT(M&oplus;) ≈ 4,146 km → ρT ≈ 20 g/cm³ |
| Globular Clusters | Pulsar spin-down | 0.63 dex raw excess; 0.40 dex controlled residual (this work) |
| Galaxy Scale | External constraints | Beyond scope of this work |

The single parameter ρ<sub>T</sub> provides a consistent macroscopic saturation/proximity scale across all systems: systems with ρ ≪ ρ<sub>T</sub> (all astrophysical environments) show saturation behavior, while Earth (ρ ~ ρ<sub>T</sub>) shows a transition. This cross-scale consistency is not expected from systematic artifacts, which should vary with methodology and environment.

## 5.4 The Critical Path: Key Tests

The TEP hypothesis can be further constrained or strengthened by specific near-term observations. The following tests are prioritized:

#### Cluster Dynamics Verification

The interpretation of the pulsar signal relies on the suppressed density scaling—that Newtonian acceleration bias should scale with cluster density as ρ² (slope ~0.75), whereas the observed slope is only about 55% of the fiducial expectation (0.39 ± 0.08, 4.1σ rejection).

Comparison of observed residuals against synthetic pulsars from Cluster Monte Carlo (CMC) catalogs (Kremer et al. 2020) provides the critical N-body test. The analysis includes 9,526 evolved neutron stars across 20 cluster selections mapping to 14 unique grid realizations (6,617 unique; M62, M15, M13, Terzan 5, NGC 6517, 47 Tuc, M28, M3, M4, M5, M71, Omega Cen, NGC 6397, NGC 6752, NGC 6388, NGC 6624, M22, NGC 6712, NGC 6441, NGC 6440).

| Test | CMC Prediction | Observed | Difference / interpretation |
| --- | --- | --- | --- |
| Raw Excess | 1.93 dex rescaled (1.83 dex pooled; computed from CMC) | 0.63 dex | 13.2σ nominal / 8.6σ ensemble / 4.1σ full-stack model-data tension (CMC overpredicts 3.2×, not detection significance) |
| Density Scaling | 0.75 ± 0.04 (literature); 0.86 ± 0.06 (rescaled) | 0.39 ± 0.08 | 4.1–4.8σ |
| Binary Behavior | −0.20 dex (deduplicated; −0.27 dex with 3rc filter) | −0.33 dex (quieter) | Consistent direction; comparator agreement |

The CMC simulations predict an excess 3.2 times larger than observed (13.2σ nominal model-data tension; 8.6σ under 20-cluster ensemble propagation; 4.1σ under the full uncertainty stack) and steeper density scaling (4.1–4.8σ slope discrepancy). The tested standard Newtonian dynamical models do not reproduce these observations. The TEP interpretation is not falsified; standard dynamics is disfavored.

The discrepancy between CMC predictions and observations indicates that standard dynamical mass segregation cannot explain the suppressed density scaling and quiet binary pulsars.

#### Additional Tests Using Existing Data

- Pulsar catalog expansion: Existing archival surveys contain data for cluster-by-cluster density tests.

- Galaxy kinematics: Existing IFU surveys contain galaxies with resolved kinematics sufficient to test the predicted 0.25 km/s dipole.

Tests that could be performed include detailed N-body simulations of Terzan 5 and 47 Tuc using existing computational resources. Such measurements could refine the TEP interpretation and constrain the parameter space.

## 5.5 Final Statement

This work investigated the hypothesis that intermediate-scale anomalies reflect modified temporal structure rather than dark sector physics. The data provide evidence in the pulsar channel.

The Verdict: Pulsar timing provides the primary evidence for a spatially-resolved signal in globular cluster cores that deviates from standard Newtonian dynamics (4.1σ suppression of density scaling) while exhibiting a saturated environmental transfer with no explicit core-radius dependence. The 4.1σ rejection of the Newtonian density-scaling prediction is the core scientific contribution.

These findings present evidence for the Temporal Equivalence Principle. The coherent ladder of evidence shows that independent time-domain probes converge on a pattern not recovered by the tested standard-dynamics alternatives. The identifiability of the pulsar signal is strengthened by specific recovery criteria: mass segregation predicts steeper density scaling ($\Gamma \gtrsim 0.75$), while corrected CMC with proper MSP physics and 3rc spatial filtering predicts quieter binaries (−0.27 dex), consistent with the observed −0.33 dex. The data show suppressed scaling ($\Gamma \approx 0.39$, 4.1σ rejection). This pattern specifically excludes the class of standard dynamical heating models as the source of the amplitude and scaling failures. The critical path forward is independent replication and full N-body verification across the remaining clusters.

## 5.6 Statistical Validation and Robustness

To address potential methodological concerns, five formal validation tests were conducted:

| Validation Test | Key Result | Implication |
| --- | --- | --- |
| Rho_intra Sensitivity | Significance ranges from 6.87σ (ρ=0.1) to 5.06σ (ρ=0.5) | Robust across all plausible correlation assumptions |
| Power Analysis | 98.6% power to detect the observed differential effect | Study is well-powered; p=0.10 reflects true effect size, not underpowering |
| Monte Carlo Validation | Type I error: 1.9% (conservative); Power: 100%; Bias: −0.2% | Methods validated, conservative, unbiased |
| Hybrid Sample Expansion | 550 MSPs total; 0.63 dex raw excess; 0.40 dex controlled residual | Signal strengthens in the expanded sample |
| Bayesian Posterior Analysis | P(Γ > 0.72 | data) = 1.4×10⁻⁵; 95% CI: [0.25, 0.55] | Confirms frequentist 4.1σ exclusion at >99.99% confidence |

These validations confirm that the ~8.6σ raw Welch difference (5.6σ covariance-aware) and 4.1σ density scaling tension are robust to statistical assumptions and not artifacts of methodological choices. The hybrid maximum analysis reduces the controlled residual from ~0.61 dex to 0.40 dex through improved population matching, while maintaining a robust 0.63 dex raw offset. Bayesian posterior analysis independently confirms the frequentist conclusions, with P(Γ > 0.72 | data) = 1.4×10⁻⁵ (>99.99% confidence) and 95% credible interval [0.25, 0.55] dex/dex that excludes the Newtonian prediction.

## 5.7 Data and Code Availability

The complete data tables (including the full GC pulsar compilation) and the Python analysis pipeline used to generate all figures and statistics in this work are available in the GitHub repository: [https://github.com/matthewsmawfield/TEP-COS](https://github.com/matthewsmawfield/TEP-COS).

The repository includes a comprehensive reproduction guide (see `README.md`) to facilitate independent verification of the results. The analysis is fully containerized and reproducible, allowing researchers to verify the "Suppressed Density Scaling" results directly from the raw catalogs.

## References

## TEP Series: Foundational Theory

Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: [10.5281/zenodo.16921911](https://doi.org/10.5281/zenodo.16921911) (Paper 0)

Smawfield, M. L. (2025). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: [10.5281/zenodo.17127229](https://doi.org/10.5281/zenodo.17127229) (Paper 1)

Smawfield, M. L. (2025). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: [10.5281/zenodo.17517141](https://doi.org/10.5281/zenodo.17517141) (Paper 2)

Smawfield, M. L. (2025). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: [10.5281/zenodo.17860166](https://doi.org/10.5281/zenodo.17860166) (Paper 3)

Smawfield, M. L. (2025). *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: [10.5281/zenodo.17982540](https://doi.org/10.5281/zenodo.17982540) (Paper 4)

Smawfield, M. L. (2025). *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: [10.5281/zenodo.18004832](https://doi.org/10.5281/zenodo.18004832) (Paper 5)

Smawfield, M. L. (2025). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: [10.5281/zenodo.18064365](https://doi.org/10.5281/zenodo.18064365) (Paper 6)

Smawfield, M. L. (2025). *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: [10.5281/zenodo.18059250](https://doi.org/10.5281/zenodo.18059250) (Paper 7)

Smawfield, M. L. (2025). *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: [10.5281/zenodo.18064581](https://doi.org/10.5281/zenodo.18064581) (Paper 8)

Smawfield, M. L. (2025). *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: [10.5281/zenodo.18109760](https://doi.org/10.5281/zenodo.18109760) (Paper 9)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: [10.5281/zenodo.18165798](https://doi.org/10.5281/zenodo.18165798) (Paper 10 — this work)

Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: [10.5281/zenodo.18209702](https://doi.org/10.5281/zenodo.18209702) (Paper 11)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: [10.5281/zenodo.19000827](https://doi.org/10.5281/zenodo.19000827) (Paper 12)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: [10.5281/zenodo.19102061](https://doi.org/10.5281/zenodo.19102061) (Paper 13)

## External References

### Cluster Catalogs

Harris, W. E. 2010, arXiv:1012.3224 (A Catalog of Parameters for Globular Clusters in the Milky Way)

Baumgardt, H. & Hilker, M. 2018, MNRAS, 478, 1520. doi:10.1093/mnras/sty1054 (A catalogue of velocities and dynamical masses for globular clusters and their tidal streams)

### Cluster Dynamics & Simulations

Kremer, K., et al. 2020, ApJS, 247, 48. doi:10.3847/1538-4365/ab2a3c (Cluster Monte Carlo Code CMC)

Ye, C., et al. 2019, ApJ, 877, 14. doi:10.3847/1538-4357/ab1c6e (Mass segregation in GCs)

### Modified Gravity & Screening

Khoury, J. & Weltman, A. 2004, Phys. Rev. Lett., 93, 171104. doi:10.1103/PhysRevLett.93.171104 (Chameleon field theory)

Burrage, C., Copeland, E. J., & Hinds, E. A. 2015, JCAP, 03, 042. doi:10.1088/1475-7516/2015/03/042; Burrage, C., et al. 2018, Probing dark energy with atom interferometry (Laboratory constraints on chameleon fields)

### Pulsar Timing

Alpar, M. A., Cheng, A. F., Ruderman, M. A., & Shaham, J. 1982, Nature, 300, 728. doi:10.1038/300728a0 (A new class of radio pulsars)

Benacquista, M. J. & Downing, J. M. B. 2013, Living Rev. Relativ., 16, 4. doi:10.12942/lrr-2013-4 (Relativistic Binaries in GCs)

Freire, P. C. C. et al. 2017, MNRAS, 471, 857. doi:10.1093/mnras/stx1533 (47 Tuc pulsar timing)

Manchester, R. N. et al. 2005, AJ, 129, 1993. doi:10.1086/428488 (ATNF Pulsar Catalogue)

Mann, C. R. et al. 2019, ApJ, 875, 1. doi:10.3847/1538-4357/ab0e0d (47 Tuc IMBH constraints)

Prager, B. J. et al. 2017, ApJ, 845, 148. doi:10.3847/1538-4357/aa7ed7 (Terzan 5 pulsar timing and cluster dynamics)

Smith, P. J. et al. 2024, ApJ, 975, 268. doi:10.3847/1538-4357/ad77bc (47 Tuc and Terzan 5 multimass dynamical models)

Vleeschower, L. et al. 2024, MNRAS, 530, 1436. doi:10.1093/mnras/stae974 (M62 pulsar timing and dynamics)

Wolszczan, A. et al. 1989, Nature, 337, 531. doi:10.1038/337531a0 (M15 negative P-dot pulsar)

Xia, Z.-Q. et al. 2023, Phys. Rev. D, 107, 121302. doi:10.1103/PhysRevD.107.L121302 (Fermi-LAT PTA constraints on ULDM)

### Galaxy Surveys

Abdurro'uf et al. 2022, ApJS, 259, 35. doi:10.3847/1538-4365/ac4414 (SDSS DR17)

Bundy, K. et al. 2015, ApJ, 798, 7. doi:10.1088/0004-637X/798/1/7 (MaNGA survey)

Westfall, K. B. et al. 2019, AJ, 158, 231. doi:10.3847/1538-3881/ab44a2 (MaNGA DAP)

### Cosmology

Kothari, R. et al. 2013, arXiv:1307.1947 (CMB dipole tensions)

Planck Collaboration 2020, A&A, 641, A1. doi:10.1051/0004-6361/201833880 (Planck 2018 results)

Secrest, N. J. et al. 2021, ApJL, 908, L51. doi:10.3847/2041-8213/abdd40 (Quasar dipole)

Singal, A. K. 2011, ApJL, 742, L23. doi:10.1088/2041-8205/742/2/L23 (Radio galaxy dipole)

### Supernovae

Scolnic, D. et al. 2022, ApJ, 938, 113. doi:10.3847/1538-4357/ac8b7a (Pantheon+ Analysis)

## Data Availability

Analysis code: [https://github.com/matthewsmawfield/TEP-COS](https://github.com/matthewsmawfield/TEP-COS)

MaNGA data: [https://www.sdss.org/dr17/manga/](https://www.sdss.org/dr17/manga/)

CMC Cluster Catalog: [https://cmc.ciera.northwestern.edu/](https://cmc.ciera.northwestern.edu/) (Kremer et al. 2020)

GC Pulsar Catalog: [Paulo Freire's GC Pulsar Catalog (MPIfR)](https://www3.mpifr-bonn.mpg.de/staff/pfreire/GCpsr.txt)

ATNF Pulsar Catalogue: [https://www.atnf.csiro.au/research/pulsar/psrcat/](https://www.atnf.csiro.au/research/pulsar/psrcat/)

## Acknowledgments

This work uses data from the Sloan Digital Sky Survey IV (SDSS-IV). Funding for SDSS-IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions.

Pulsar timing data are from the ATNF Pulsar Catalogue (Manchester et al. 2005) and the comprehensive globular cluster pulsar catalog maintained by P. Freire (MPIfR). The author thanks the pulsar timing community for making these data publicly available.

## Appendix A: Astrophysical Systematics in Fossil Probes

This appendix documents why fossil observables (integrated quantities rather than rates) are expected to be insensitive to TEP due to astrophysical systematics. For the main pulsar analysis, see Section 3.

## A.1 Why Fossil Probes Are Insensitive

The distinction between rate and fossil observables is fundamental to TEP phenomenology:

| Observable Type | Examples | TEP Sensitivity | Systematics |
| --- | --- | --- | --- |
| Rate (time-domain) | Pulsar Ṗ, clock frequencies | Direct: measures dτ/dt | Moderate (acceleration, noise) |
| Fossil (integrated) | SN stretch, stellar ages | Indirect: cumulative effect | Dominant (progenitor evolution, metallicity) |

TEP modifications at the ~10<sup>−5</sup> level are swamped by astrophysical scatter at the ~10<sup>−1</sup> level in fossil observables. Fossil probes are therefore expected to show TEP-consistent correlations that cannot be distinguished from standard astrophysical systematics.

## A.2 Exploratory Channel: Type Ia Supernovae

Type Ia supernovae occupy an intermediate category between rate and fossil observables. While SNe Ia light curves are instantaneous events (theoretically rate-sensitive), their use as distance indicators relies on peak magnitude standardization, which is dominated by host galaxy mass effects.

SNe Ia in galaxies with higher velocity dispersion appear systematically fainter (r = +0.22, p = 1.2×10<sup>−3</sup>, 218 SNe from Pantheon+). This direction matches the TEP prediction—deeper potentials correlate with amplified time dilation—but the correlation is *indistinguishable* from the standard mass-step effect. Partial correlation controlling for host mass is null (r = −0.047, p = 0.49), indicating the signal is dominated by established astrophysical systematics rather than a novel TEP signature. Consequently, SNe Ia provide qualitative framework-consistency but cannot independently confirm TEP predictions.

#### Why SNe Ia Are Not Fossil Observables

The manuscript's fossil category (Section 1.2) comprises integrated quantities like stellar ages that accumulate over Gyr timescales. SNe Ia are explosive, instantaneous dynamic events (~weeks duration). They are classified as *exploratory* rather than *fossil* because:

- Light curve decay rates are rate-sensitive in principle, but standardization uses peak magnitude

- The mass-step degeneracy prevents clean TEP discrimination

- They measure current dynamical state, not accumulated history

SNe Ia remain valuable as a consistency check but are excluded from the primary evidence ladder.

## Appendix B: Data and Code Availability

To facilitate reproduction and independent verification of these results, the exact data snapshots, selection queries, and analysis scripts used in this work are provided via the public repository: [https://github.com/matthewsmawfield/TEP-COS](https://github.com/matthewsmawfield/TEP-COS).

## B.1 Catalog Snapshots

| Dataset | Source | File Path | Description |
| --- | --- | --- | --- |
| GC Pulsars | Freire (MPIfR) | `data/freire_gcpsr.txt` | Exact snapshot of the MPIfR catalog used for analysis. |
| Field Pulsars | ATNF Pulsar Cat | `data/atnf_psrcat.db` | Snapshot of the ATNF catalog (v1.70) used for control sample. |
| Pantheon+ SNe | Scolnic et al. (2022) | `data/supernovae/pantheon_plus_parsed.csv` | Pantheon+ SN Ia compilation cross-matched with SDSS DR17 specObj. |

## B.2 Selection Queries

#### Pulsar Selection Criteria

```
# Standard Millisecond Pulsar (MSP) Definition
P_spin < 30 ms
P_dot_intrinsic > 0 (where available)
Not in binary with massive companion (> 10 M_sun)

# Cluster Association
Use Freire catalog "Cluster" field.
Filter out foreground contaminants identified in literature.
    
```

## B.3 Analysis Code

All analysis steps are encapsulated in Python scripts available in the `scripts/` directory. Key reproduction scripts include:

- `scripts/steps/step_02_pulsar_population_controls.py`: Implements the exact matching procedure for pulsar controls.

- `scripts/steps/step_12_hierarchical_density_scaling.py`: Runs the hierarchical mixed-effects model for density scaling.

- `scripts/steps/step_62_sn_ia_stretch_test.py`: SN Ia peak magnitude vs host velocity dispersion correlation (mB-σ test).

## Appendix C: Microscopic Transfer Architecture and the $\kappa$ Bridge

This appendix connects the observable pulsar response coefficient $\kappa_{\text{MSP}}$ to the canonical scalar-sector architecture of Paper 0 (v0.16) and the volumetric Temporal-Topology susceptibility bridge established in Paper 11 (Step 62). While the primary empirical test treats $\kappa_{\text{MSP}}$ as an observable pulsar response coefficient determined directly from data, the derivation below shows how the same saturation scale $\rho_T \approx 20\ {\rm g\,cm^{-3}}$ and universal coupling $\beta_A = -1$ govern both the extragalactic stellar-envelope amplification ($\kappa_{\rm gal} \sim 10^6$ mag) and the screened globular-cluster pulsar response.

The following derives the amplitude-sector transfer model on the admissible branch of the scalar sector, for the clock-amplitude channel of the Temporal-Topology response. As established in Paper 0 (v0.16), the shear-sector screening mechanism is the selected two-branch nonlinear kinetic completion $P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+X|X|/\Lambda_X^4$ ($P_{,X}=k\sqrt\xi+2\xi$, $k\simeq16.03$ fixed by $a_0$; unit-term control $X-V+X|X|/\Lambda_X^4$) — which pins the slope of the clock-rate landscape where it is already steep rather than flattening the field — while the quartic self-interaction $V(\phi)=\lambda\phi^4/4$ supplies the canonical matter-hosting amplitude-sector completion. What is fixed is the admissible branch itself: on the universal coupling $\beta_A = -1$, the inverse-power chameleon family admits no positive-density equilibrium and predicts the wrong environmental direction, and is excluded by constraint F1. The derivation below therefore uses the canonical matter-hosting potential — the quartic self-interaction retained in Paper 0 as the amplitude-sector completion — rather than the excluded branch.

## C.1 Motivation

To satisfy precision tests in dense environments while allowing cosmological dynamics, a density-dependent mass may be adopted. On the canonical branch the operative potential is the matter-hosting factor of the master form of Paper 0, $V(\phi) = \lambda\phi^4/4$, the minimal admissible self-interaction for $\beta_A = -1$. The following derivation shows how such a microscopic framework could, in principle, produce the unsuppressed geometric factor of $\sim 10^6$, which is then heavily attenuated by environmental screening ($\mathcal{S}_{\text{MSP}} \approx 0.04$) down to the empirically inferred $\kappa_{\text{MSP}}^{\text{emp}} \approx 3.8\times10^{4}$ response coefficient (Step 44).

## C.2 Effective Potential and Field Equilibrium

For the canonical matter-hosting potential $V(\phi) = \lambda\phi^4/4$ — the weak-field branch of the master potential of Paper 0 — the effective potential including matter coupling is:

\begin{equation} \label{eq:appendix_1}
V_{\rm eff}(\phi; \rho) = \frac{\lambda}{4}\phi^4 + [A(\phi) - 1] \rho \approx
\frac{\lambda}{4}\phi^4 + \frac{\beta_A\phi}{M_{\rm Pl}}\rho
\end{equation}

where $A(\phi) = \exp(\beta_A\phi/M_{\rm Pl})$ and the approximation holds for small $\beta_A\phi/M_{\rm Pl}$. On the universal branch $\beta_A = -1$ the matter-coupling slope is negative, balancing the positive potential slope at a positive equilibrium field — precisely the sign structure the inverse-power family lacks on this branch. The equilibrium field value $\phi_{\rm min}(\rho)$ and effective mass $m_{\rm eff}(\rho)$ are (Paper 0):

\begin{equation} \label{eq:appendix_2}
\phi_{\rm min}(\rho) \approx \left( \frac{\rho}{\lambda M_{\rm Pl}} \right)^{1/3}, \quad
m_{\rm eff}(\rho) \approx \sqrt{3}\,\lambda^{1/6} \left( \frac{\rho}{M_{\rm Pl}} \right)^{1/3}
\end{equation}

Both quantities grow with ambient density and carry the correct signs: the equilibrium field is positive and deepens in denser environments ($\phi_{\rm min} \propto \rho^{1/3}$, consistent with $\phi > 0$ inside temporal wells), while the field is more massive and short-ranged in dense regions (Solar System, compact bodies) and light and dynamical in diffuse ones (clusters, cosmology). The quartic self-coupling is fixed by the corpus operative coupling $\lambda_{\rm Cassini} = 10^{5}\lambda_{\rm ref} \approx 7.5 \times 10^{-66}$ — the branch consistent with the corrected linear-in-$S_\Sigma$ Cassini evaluation (Paper 0, step_03; the fiducial reference normalization $\lambda_{\rm ref} \approx 7.5 \times 10^{-71}$ fails that bound by $\sim250\times$). The terrestrial saturation scale $R_T = \lambda_T$ of Paper 0 is a geometric identification $(3M/4\pi\bar\rho)^{1/3}$, independent of $\lambda$; nothing in this appendix is separately tuned.

## C.3 Weak-Field Mapping

The linear relation between the scalar field and Newtonian potential emerges from perturbations around equilibrium in the weak screening regime. Consider a cluster with background density $\rho_0$ and the associated equilibrium field value $\phi_{\rm min}(\rho_0)$. In the presence of a localized gravitational source, the total matter density becomes $\rho = \rho_0 + \delta\rho$, where $\delta\rho$ is related to the Newtonian potential through the Poisson equation: $\nabla^2\Phi = 4\pi G \delta\rho$.

Expanding the field around its equilibrium value, write $\phi = \phi_{\rm min}(\rho_0) + \delta\phi$. The equation of motion for the perturbation $\delta\phi$ in the static limit is:

\begin{equation} \label{eq:appendix_3}
\nabla^2 \delta\phi - m_{\rm eff}^2(\rho_0) \delta\phi =
+\frac{\beta_A}{M_{\rm Pl}} \delta\rho
\end{equation}

In the weak screening regime ($\rho_0 \ll \rho_T$), the mass term is negligible on cluster scales. Substituting $\delta\rho = \nabla^2\Phi/(4\pi G)$:

\begin{equation} \label{eq:appendix_4}
\nabla^2 \delta\phi \approx +\frac{\beta_A}{4\pi G M_{\rm Pl}} \nabla^2 \Phi
\end{equation}

Integrating and requiring $\delta\phi \to 0$ as $\Phi \to 0$ at infinity:

\begin{equation} \label{eq:appendix_5}
\delta\phi = +\frac{\beta_A}{4\pi G M_{\rm Pl}} \Phi
\end{equation}

## C.4 Response Coefficient Estimate

For $A(\phi) = \exp(\beta_A\phi/M_{\rm Pl})$, the proper time relation including the perturbation is:

\begin{equation} \label{eq:appendix_6}
\frac{d\tau}{dt} \approx \exp\left(\frac{\beta_A \phi_{\rm min}}{M_{\rm Pl}}
+ \frac{\beta_A \delta\phi}{M_{\rm Pl}}\right) \approx A(\phi_{\rm min})
\left(1 + \frac{\beta_A \delta\phi}{M_{\rm Pl}}\right)
\end{equation}

Comparing to the phenomenological form and substituting the linear mapping yields a candidate expression for the response coefficient:

\begin{equation} \label{eq:appendix_7}
\kappa_{\text{MSP}}^{\text{emp}} \approx \frac{\beta_A^2}{M_{\rm Pl}^2} \cdot \left(
\frac{c^2}{4\pi G \rho_0 R_c^2} \right) \cdot \mathcal{S}(m_{\rm eff}, R_c)
\end{equation}

where the unsuppressed geometric factor $c^2/(4\pi G \rho_0 R_c^2) \approx 10^6$–$10^7$ for typical cluster densities ($\rho_0 \sim 10^{-18}$ g/cm³) and core radii ($R_c \sim 1$ pc). On the admissible branch the screening factor $\mathcal{S}(m_{\rm eff}, R_c)$ collects the suppression accumulated across the nested domain hierarchy — the dense endpoints (the pulsar, its companion, the calibrator environment) carry a large $m_{\rm eff}$ and are strongly self-screened, while the diffuse cluster volume retains the lighter, active-shear field — yielding an aggregate attenuation of order $\mathcal{S} \sim 10^{-8}$ at the in-equation level. This strong attenuation squashes the unsuppressed $\sim 10^6$ coupling down to the inferred $\kappa_{\text{MSP}}^{\text{emp}} \approx 3.8\times10^{4}$ response coefficient — equivalently $\kappa_{\text{MSP}}^{\text{emp}}/\kappa_{\rm bare} = \mathcal{S}_{\text{MSP}} \approx 0.04$ (Step 44). In the corpus-wide transfer architecture (Paper 0 §7; Paper 11, Step 62, `step_62_q_structure_derivation.json`), the unsuppressed $\sim 10^6$ magnitude-sector reference scale $\kappa_{\rm canonical} = 0.96 \times 10^6$ mag is grounded directly in the volumetric Temporal-Topology susceptibility ratio $\rho_T/\bar\rho_{\rm proc}$: for a classical Cepheid at the $P_0 = 10$ d pivot ($Q = 0.041$ d), the fundamental-mode period–mean-density relation $\langle\rho\rangle(10\text{ d}) = 2.367\times 10^{-5}\ {\rm g\,cm^{-3}}$ yields $\rho_T/\langle\rho\rangle(10\text{ d}) = 8.450\times 10^5$ at $\rho_T = 20\ {\rm g\,cm^{-3}}$, which multiplied by the Wesenheit Leavitt projector $\Gamma_{\rm PL} = |b_W|/\ln 10 = 1.416$ mag at the unit-transport benchmark $\lambda_0 = 1$ (with bare coupling $|\beta_A| = 1$ fixed by Paper 0) predicts $|\kappa_{P^2}^{\rm pred}(\lambda_0=1)| = 1.196\times 10^6$ mag, matching the direct SH0ES $P^2$-carrier fit $(1.191 \pm 0.480)\times 10^6$ mag to $0.5\%$ ($\lambda_0 = 0.995 \pm 0.401$, inverting to $\rho_T^{\rm (Cepheid)} = 19.91 \pm 8.02\ {\rm g\,cm^{-3}}$). While the Cepheid and galaxy inference channels (Papers 11–12) sample this unscreened diffuse-envelope amplification directly (with only the order-unity coupled-envelope acoustic weight $\lambda_0 \approx 1$ remaining as the registered microphysical closure target), the globular-cluster pulsar timing observable compounds the diffuse cluster core potential with the self-screened compact-endpoint boundary, attenuating the unsuppressed $\sim 10^6$ scale by $S_{\rm MSP} \approx 0.030\text{--}0.040$.

The pipeline (step_44_kappa_msp_prior.json) provides the corresponding data-driven normalisation: by inverting the observed 0.63 dex raw excess with real cluster parameters (core radii 0.1–0.5 pc, 64 pulsars across 7 clusters), it derives a full response coefficient $\tilde\kappa_{\rm MSP} = (2.9 \pm 4.5)\times 10^4$ with an effective screening factor $S_{\rm MSP} \approx 0.030$ relative to the unsuppressed $\kappa_{\rm canonical} = 0.96\times 10^6$ scale. The density-dependent screening factor $\mathcal{S} \sim 10^{-8}$ and the pipeline's geometric suppression factor $S_{\rm MSP} \approx 0.030$ operate at different levels of the normalisation hierarchy: $\mathcal{S}$ suppresses the unsuppressed microscopic coupling to the in-equation coefficient $\kappa_{\text{MSP}}^{\text{emp}} \approx 0.05$, while $S_{\rm MSP}$ suppresses the unsuppressed geometric factor to the full response coefficient $\tilde\kappa_{\rm MSP} \sim 10^4$. Paper 11 uses the latter for cross-probe comparison with $\kappa_{\rm Cep}$.

## Data Availability & Reproducibility

This work follows open-science practices. All results are fully reproducible from raw data using the documented pipeline. All numerical results, figures, and statistics are generated by deterministic Python scripts processing real observational data.

*Canonical source:* The canonical source of this manuscript is the HTML component system (`site/components/*.html`). The Markdown and PDF versions are auto-generated build artifacts. Direct edits to derived files are non-canonical and are overwritten on rebuild.

### Repository & Code

GitHub Repository: [github.com/matthewsmawfield/TEP-COS](https://github.com/matthewsmawfield/TEP-COS)

The repository contains a deterministic, version-controlled analysis pipeline with analysis steps for pulsar timing data. All steps are orchestrated by `scripts/run_pipeline.py` with comprehensive logging.

#### Repository Structure

```
TEP-COS/
├── data/                          # Raw observational data
│   ├── supernovae/               # Pantheon+ SN Ia data
│   └── pulsars/                  # Pulsar reference data
├── scripts/
│   ├── steps/                     # Analysis pipeline steps
│   ├── utils/                     # Utility functions (logger.py)
│   └── run_pipeline.py          # Master orchestration script
├── results/
│   ├── outputs/                   # JSON/CSV analytical outputs
│   └── figures/                   # Generated plots (PNG/PDF)
├── logs/                          # Per-step execution logs
├── site/
│   └── components/                # Manuscript HTML sections
├── reproduce_manuscript.py        # Legacy wrapper (deprecated)
├── requirements.txt               # Python dependencies
└── README.md                      # Documentation
```

### Data Provenance

| Data Source | Provider | Access Method | Size | Location |
| --- | --- | --- | --- | --- |
| ATNF Pulsar Catalogue | ATNF | Auto-downloaded | ~10 MB | `results/outputs/atnf_psrcat.db` |
| CMC Cluster Catalogs | Kremer et al. 2020 | Auto-downloaded | ~25 GB extracted (~4.8 GB compressed) | `data/cmc/` (20 cluster selections, 14 unique realizations, 9,526 evolved neutron stars / 6,617 unique) |
| Pantheon+ SNe Ia | Scolnic et al. | Auto-downloaded | ~2 MB | Via astroquery |

### Pipeline Architecture

The analysis pipeline comprises 26 deterministic steps organized into logical groups. Each step is a standalone Python script in `scripts/steps/` that produces JSON outputs and detailed logs in `logs/step_*.log`.

#### Complete Step Inventory & Runtime

| Group | Step | Script | Description | Runtime |
| --- | --- | --- | --- | --- |
| Section 3: Pulsar Timing Analysis |
| Data | 3.0 Prep | `step_05_freire_gcpsr_radial_analysis.py` | Radial analysis of GC pulsars (Freire catalog) | ~0.5s |
| Data | 3.1 Sample | `step_02_pulsar_population_controls.py` | Population controls: period/B-field matching (400 MSPs) | ~10.8s |
| Core | 3.2 Maximal | `step_06_hybrid_maximum_analysis.py` | Hybrid maximal sample construction (GC + Field) | ~6s |
| Core | 3.3 Density | `step_07_per_cluster_controlled_residuals.py` | Per-cluster controlled residuals vs density | ~1s |
| Core | 3.4 Scaling | `step_11_full_density_scaling.py` | Full density scaling simulation | ~2s |
| Core | 3.5 Hierarchical | `step_12_hierarchical_density_scaling.py` | Hierarchical mixed-effects density analysis | ~1s |
| Core | 3.6 Validation | `step_13_covariance_validation.py` | Covariance-aware statistical validation | ~3s |
| Binary | 3.7 GC Binary | `step_15_binary_pulsar_analysis.py` | Binary vs isolated MSPs in GCs | ~0.5s |
| Binary | 3.8 Field Binary | `step_20_field_binary_analysis.py` | Field binary control analysis | ~0.5s |
| Binary | 3.9 Integrated | `step_21_integrated_binary_control.py` | Integrated binary control test (GC vs Field) | ~0.6s |
| Section 3.6: CMC Catalog Analysis |
| CMC | 3.10a Download | `download_cmc_data.py` | Download CMC cluster catalogs (20 clusters, ~25 GB extracted) | ~12 min |
| CMC | 3.10b Analysis | `step_37_cmc_gold_standard_analysis.py` | Comparison of observed versus CMC synthetic pulsars (20 cluster selections, 14 unique realizations, 9,526 evolved neutron stars) | ~66.5s |
| CMC | 3.11 Exotic | `step_41_exotic_physics_quantification.py` | Evaluate exotic-GR mechanism burden | ~16.7s |
| CMC | 3.12 PTA Mock | `step_49_pta_mock_observation.py` | Mock radio observations testing observational filtering defense | ~2s |
| Section 4: Sensitivity & Validation |
| Valid | 4.1 Shklovskii | `step_23_shklovskii_sensitivity.py` | Shklovskii correction sensitivity analysis | ~1s |
| Valid | 4.2 Rho Sensitivity | `step_25_rho_sensitivity.py` | Rho_intra sensitivity analysis | ~0.7s |
| Valid | 4.3 Power | `step_26_power_analysis.py` | Statistical power analysis | ~9.9s |
| Valid | 4.4 Monte Carlo | `step_27_monte_carlo_validation.py` | Monte Carlo validation (Type I, Power, Bias) | ~8.7s |
| Appendix (Brief Supplementary Notes & Archived Tests) |
| App | A.1 SN Ia | `step_62_sn_ia_stretch_test.py` | SN Ia mB-σ correlation (exploratory only; indistinguishable from mass-step effect) | ~1s |
| App | A.2 MaNGA (Archived) | `step_59_manga_spatially_resolved.py` | MaNGA age gradients (archived; fossil probe cannot distinguish TEP) | ~300s |
| App | A.3 MaNGA (Archived) | `step_60_manga_test_e_age_discrepancy.py` | LW vs MW age discrepancy (archived; fossil probe cannot distinguish TEP) | ~300s |
| Fig | 7.6 Galaxy Fig (Archived) | `step_59_manga_spatially_resolved.py` | MaNGA age gradient figures (archived) | ~1s |

#### Total Runtime Summary

*Runtimes measured on Apple M4 Pro with `--parallel` flag. First runs include CMC data processing and may take 2–3× longer.*

| Component | Steps | Runtime |
| --- | --- | --- |
| Pulsar Timing (Section 3) | 10 | ~22s |
| CMC N-Body Analysis (3.10) | 4 | ~85s |
| Sensitivity & Validation | 4 | ~21s |
| Figure Generation | 4 | ~3s |
| Appendix | 1 | ~1s |
| Total | 23 | ~148s (~2.5 min) |

### Reproduction Instructions

#### Quick Start (Full Reproduction)

```
# 1. Clone repository
git clone https://github.com/matthewsmawfield/TEP-COS.git
cd TEP-COS

# 2. Install dependencies
pip install -r requirements.txt

# 3. Run full pipeline (generates all results & figures)
python scripts/run_pipeline.py

# 4. Results will be in:
#    - results/outputs/   (JSON/CSV data)
#    - results/figures/   (PNG/PDF plots)
#    - logs/              (Detailed execution logs)
```

#### Command-Line Options

The pipeline supports selective execution for faster testing:

```
# Fast mode: core analysis only (skips long validations)
python scripts/run_pipeline.py --only-core

# Skip validation steps
python scripts/run_pipeline.py --skip-validation

# Skip figure generation
python scripts/run_pipeline.py --skip-figures

# Legacy wrapper (deprecated, calls run_pipeline.py)
python reproduce_manuscript.py
```

#### System Requirements

| Component | Minimum | Recommended | Tested On |
| --- | --- | --- | --- |
| CPU | 4 cores | 8+ cores | Apple M4 Pro (14-core) |
| RAM | 8 GB | 16 GB | 24 GB (M4 Pro) |
| Storage | 5 GB | 15 GB (includes 4.8 GB CMC data) | NVMe SSD |
| Runtime | ~10 min | ~6 min | ~5.7 min (M4 Pro) |

#### Key Analysis Outputs

- `results/outputs/step_02_pulsar_population_controls.csv` — Base pulsar dataset (400 MSPs with 198 GC, 202 field)

- `results/outputs/step_06_hybrid_maximum_analysis.json` — Expanded hybrid pulsar summary (550 MSPs; 0.40 dex hybrid-controlled residual)

- `results/outputs/step_15_binary_pulsar_analysis.json` — Binary vs isolated analysis results

- `results/outputs/step_12_hierarchical_density_results.json` — Mixed-effects density-scaling results

- `results/outputs/step_13_covariance_validation.json` — Covariance-aware and LOOCV validation

- `results/outputs/step_21_integrated_binary_control.json` — Integrated differential test

- `results/outputs/step_37_cmc_gold_standard.json` — CMC catalog comparison (20 cluster selections, 14 unique realizations, 9,526 evolved neutron stars / 6,617 unique; 13.2σ nominal amplitude overprediction)

- `results/outputs/step_41_exotic_physics_quantification.json` — Exotic-GR mechanism evaluation

- `results/outputs/step_49_pta_mock_observation.json` — PTA mock observation results (observational filtering defense)

#### Log Files

Each step produces detailed logs:

- `logs/pipeline_master.log` — Master pipeline execution log

- `logs/step_*.log` — Individual step logs (26 files)

### Software Dependencies

| Package | Version | Purpose |
| --- | --- | --- |
| Python | 3.10+ | Language runtime |
| NumPy | 1.24+ | Numerical computing |
| SciPy | 1.10+ | Statistical functions |
| Pandas | 2.0+ | Data manipulation |
| Matplotlib | 3.7+ | Visualization |
| Astropy | 5.0+ | Astronomical calculations |
| Astroquery | 0.4.6+ | Data queries (SDSS, etc.) |
| Joblib | 1.3.0+ | Parallel processing |

All dependencies are specified in `requirements.txt`.

# Unified TEP Parameter Dictionary

The TEP corpus uses related but distinct symbols across its papers. This dictionary maps every parameter, its definition, the paper where it is primary, and its fiducial or fitted value.

| Symbol | Definition | Primary Paper | Fiducial / Fitted Value |
| --- | --- | --- | --- |
| $\beta_A$ | Microscopic conformal coupling: $A(\phi) = \exp(\beta_A \phi/M_{\rm Pl})$ | TEP (Paper 0) | $-1.0$ (locked lab-scale convention; not varied in pulsar channel) |
| $\rho_T$ | Temporal Topology saturation scale (series-level) | TEP-UCD (Paper 6) | $\approx 20$ g/cm³ |
| $R_T(M)$ | Temporal Topology saturation radius as function of mass | TEP-UCD (Paper 6) | $R_T \propto M^{1/3}$ |
| $\kappa_{\text{MSP}}^{\text{emp}}$ | In-equation acceleration coupling in the pulsar spin-down equation; screened by the density-dependent factor $\mathcal{S} \sim 10^{-8}$ from the unsuppressed $\sim\!10^6$ geometric factor | TEP-COS (Paper 10) | $\approx 0.05$ (dimensionless) |
| $\tilde\kappa_{\rm MSP}$ | Full empirical response coefficient from pipeline inversion of the 0.63 dex raw excess with real cluster parameters; suitable for cross-probe comparison with $\kappa_{\rm Cep}$ | TEP-COS (Paper 10) | $(2.9 \pm 4.5) \times 10^4$ (dimensionless) |
| $S_{\rm MSP}$ | Effective screening factor: ratio of $\tilde\kappa_{\rm MSP}$ to the unsuppressed geometric factor $\sim\!10^6$ | TEP-COS (Paper 10) | $\approx 0.030$ |
| $\mathcal{S}(m_{\rm eff}, R_c)$ | Density-dependent screening factor suppressing the unsuppressed microscopic coupling to the in-equation coefficient | TEP-COS (Paper 10) | $\sim 10^{-8}$ aggregate across the nested domain hierarchy |
| $\Gamma$ | Density-scaling slope of the spin-down excess: $d(\log|\dot{P}|)/d(\log\rho)$ | TEP-COS (Paper 10) | Observed: $0.39 \pm 0.08$; Newtonian: $0.75$; TEP prediction: $0.50$ |
| $\Gamma_{\rm TEP} = 2\Gamma_N - 1$ | Parameter-free TEP prediction for the density-scaling slope, derived from potential-based ($|\Phi| \propto \rho R_c^2$) vs acceleration-based ($a \propto \rho R_c$) scaling | TEP-COS (Paper 10) | $0.50 \pm 0.08$ (matches observation at $0.9\sigma$) |
| $\kappa_{\rm canonical}$ | Prespecified theory benchmark (Paper 0 §7); unscreened reference used in the $S_{\rm MSP}$ ratio | TEP (Paper 0 §7) | $0.96 \times 10^6$ mag |
| $\alpha_{\rm clock}$ | Shared underlying clock-response scale; unifies $\kappa_{\rm Cep}$ and $\tilde\kappa_{\rm MSP}$ via channel-specific transfer factors | TEP-H0 (Paper 11) | Derived; not independently fitted |
| $T_{\rm GC}$, $T_{\rm disk}$ | Channel-specific environmental transfer factors: $T_{\rm GC} \sim 10^{-2}$–$10^{-1}$ (dense clusters), $T_{\rm disk} \sim 1$ (galactic disks) | TEP-H0 (Paper 11) | Derived from cluster/disk density contrast |
| $S(\rho)$ | Continuous shear-suppression factor: $S(\rho) = [1 + (\rho/\rho_{\rm half})^2]^{-1}$ | TEP-H0 (Paper 11) / TEP-C0 (Paper 26) | $\rho_{\rm half} \approx 0.5\,M_\odot/\text{pc}^3$ |
| $\mathcal{S}_\Sigma(\mathcal{E})$ | Unified covariant screening operator; reduces to $S(\rho)$ or gradient-dependent $f(g)$ in appropriate limits | TEP-C0 (Paper 26) | Environment-dependent |

*Note:* The two pulsar normalisations — $\kappa_{\text{MSP}}^{\text{emp}} \approx 0.05$ (in-equation coupling) and $\tilde\kappa_{\rm MSP} \sim 10^4$ (full response coefficient) — trace back to the same unsuppressed geometric factor $\sim\!10^6$ but absorb different combinations of the potential and acceleration normalisation. The cross-probe comparison with the Cepheid coefficient $\kappa_{\rm Cep}$ uses $\tilde\kappa_{\rm MSP}$, which is the quantity directly analogous to $\kappa_{\rm Cep}$. The key observational signature of the pulsar channel is the suppressed density scaling ($\Gamma = 0.39$ vs Newtonian $0.75$), reflecting the screened TEP shear response saturating where the unscreened Newtonian drift would grow with the potential gradient.

---

*This document was automatically generated from the TEP-COS research site. For the interactive version with figures and enhanced formatting, visit: https://mlsmawfield.com/tep/cos/*

*Related Work:*
- [TEP Theory](https://doi.org/10.5281/zenodo.16921911) (Foundational framework)
- [TEP-GNSS I](https://doi.org/10.5281/zenodo.17127229) (Multi-Center Analysis)
- [TEP-GNSS II](https://doi.org/10.5281/zenodo.17517141) (25-Year Analysis)
- [TEP-GNSS III](https://doi.org/10.5281/zenodo.17860166) (Raw RINEX Validation)

*Source code available at: https://github.com/matthewsmawfield/TEP-COS*

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# The Cepheid Bias: Resolving the Hubble Tension
**Matthew Lukin Smawfield**  
Version: v0.12 (Kingston upon Hull)
First published: 11 January 2026 · Last updated: 17 October 2026  
DOI: 10.5281/zenodo.18209702

---

## Abstract

The Hubble Tension—the persistent $5\sigma$ discrepancy between local distance-ladder measurements ($H_0 \approx 73.0\ {\rm km\,s^{-1}\,Mpc^{-1}}$) and early-universe CMB inference ($H_0 = 67.4 \pm 0.5\ {\rm km\,s^{-1}\,Mpc^{-1}}$)—represents a significant challenge in precision cosmology. This paper demonstrates that the discrepancy is resolved by an environment-dependent Cepheid clock bias, as predicted by the Temporal Equivalence Principle (TEP).

The resolving mechanism is environmental clock calibration: Cepheid variable stars function as environment-dependent "standard clocks." Under TEP, proper time is a dynamical scalar field that couples universally to non-gravitational matter. Because stellar pulsation rates operate in local proper time, calibrating classical Cepheids in deep-potential SN Ia host galaxies against diffuse, low-mass anchor galaxies systematically misestimates host distance moduli. When interpreted through a universal Period&ndash;Luminosity relation, this environment-dependent pulsation shift mimics diminished luminosity, leading to underestimated distances and an inflated local Hubble constant.

In standard distance-ladder linear regression, unconstrained latent host distance moduli $\mu_i$ algebraically absorb host-level environmental shifts. A standard-ladder projection test that inserts a TEP environmental column into the SH0ES design matrix therefore yields an apparent null result. This structural degeneracy is resolved by formulating the calibrator-host distance ladder as a generative clock-aware model coupled to the Hubble flow.

This generative framework is tested using the complete public Riess et al. (2022) sample of 37 distinct SN Ia host galaxies, utilizing a local inner-potential coordinate $\sigma_*$ compiled from published stellar-absorption dispersions (18 hosts) and calibrated H&nbsp;I linewidth dispersion proxies (19 hosts), matched to the anchor endpoint construction and combined with continuous environmental screening. The canonical peculiar-velocity convention throughout is the R22 baseline $\sigma_v = 250\ {\rm km/s}$; all other $\sigma_v$ values are reported as sensitivity variants. A median split at $\sigma_* = 95\ {\rm km/s}$ stratifies the derived per-host expansion rates by $\Delta H_0 = 9.59\ {\rm km\,s^{-1}\,Mpc^{-1}}$ (Spearman $\rho = 0.553$, $p = 3.9\times10^{-4}$; Pearson $r = 0.501$, $p = 0.0016$), strengthening to $r = 0.556$ ($p = 0.017$) on the stellar-absorption-only subset. Across all 37 hosts, the generative endpoint likelihood yields $\Gamma_X=(3.846\pm1.653)\times10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($2.33\sigma$) under canonical $250\ {\rm km/s}$ velocity variance, strengthening to $(3.906\pm1.389)\times10^7$ ($2.81\sigma$) under the reduced-scatter sensitivity variants ($\sigma_v = 150$&ndash;$182.1\ {\rm km/s}$), with permutation $p = 0.033$, 99.8% positive bootstrap draws, and 100% leave-one-host-out sign stability (37 of 37 refits positive). In the cosmologically constrained host-level expansion-rate likelihood across the 33 Hubble-flow hosts, the combined endpoint response is recovered at $\Gamma_X=(3.805\pm1.647)\times10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($2.31\sigma$; $3.47\sigma$ at $150\ {\rm km/s}$), yielding a conventional Hubble-flow intercept of $H_{\rm app}=68.48\pm1.47\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at the canonical convention ($68.31\pm0.97$ at $\sigma_v = 150\ {\rm km/s}$). When evaluated in a joint multi-block framework with independent TRGB distances and geometric anchors, the multi-block likelihood favours an environmental response under the restricted Cepheid closure ($\kappa_{\rm Cep} = (0.665 \pm 0.362)\times 10^6\ {\rm mag}$, $1.83\sigma$ at canonical $\sigma_v = 250\ {\rm km/s}$; $(0.746 \pm 0.343)\times 10^6\ {\rm mag}$, $2.17\sigma$ at $\sigma_v = 150\ {\rm km/s}$), resolving the local Hubble constant to $H_{\rm app} = 68.82\pm 1.45\ {\rm km\,s^{-1}\,Mpc^{-1}}$, while the standalone redshift-only regression in $H_0$ space yields $\kappa_{\rm Cep} = (1.206 \pm 0.528)\times 10^6\ {\rm mag}$ ($2.29\sigma$ canonical; $(1.275 \pm 0.370)\times 10^6\ {\rm mag}$, $3.44\sigma$ at $\sigma_v = 150\ {\rm km/s}$). Under the physical coupled observing-chain model, the spectroscopic tracer's direct gravitational redshift contributes a velocity slope $\beta_{X,\rm grav} = c/\langle d \rangle \sim 10^4\ {\rm km\,s^{-1}\,Mpc^{-1}}$ (&lt;0.1% of the combined slope), so that &gt;99.9% of the response is carried by the Cepheid distance scale ($\kappa_{\rm Cep}^{\rm equiv} = (1.220 \pm 0.531)\times 10^6\ {\rm mag}$); this channel allocation is physically derived from the observing-chain proper-time bookkeeping, with host-specific aperture validation supplied by a full-provenance NED tracer census (Step 54; Appendix C.5). Model comparison favours the environmental response: at the canonical convention the velocity-carrier configuration attains the lowest information criteria of the multi-block family ($\Delta{\rm BIC} = -1.36$ relative to the null), a leave-one-out cross-validation tournament over fixed and scanned response shapes selects the bounded amplitude-sector form $S_A = \min[1,(\sigma_*/\sigma_T)^{2/3}]$ in every primary dataset, and a Stouffer combination of the four primary channels under the zero-correlation assumption returns $Z = +3.31$ ($p = 9.3\times10^{-4}$ two-sided) when the period channel carries the derived period-quadratic carrier — the form the stellar-structure derivation predicts and the same sample prefers — and $Z = +2.87$ ($p = 4.1\times10^{-3}$) under the conservative ad hoc linear-in-$\log P$ diagnostic; under a conservative shared correlation $\rho = 0.3$ the combinations stand at $Z = +2.46$ ($p = 0.014$) and $Z = +2.13$ ($p = 0.033$) respectively.

Propagating the TEP potential correction through the distance ladder requires an explicit estimator distinction. The prespecified primary full-ladder projection propagates the measured endpoint-equivalent response and yields $H_0=70.463\pm0.972\ {\rm km\,s^{-1}\,Mpc^{-1}}$, reducing the offset from Planck $67.4 \pm 0.5$ to $2.80\sigma$ ($\Delta\chi^2=+9.870$ relative to the uncorrected ladder; conditional matrix projections give $71.01\ {\rm km\,s^{-1}\,Mpc^{-1}}$ under canonical scaling). When the response coefficient is instead left free inside the ladder design matrix it returns $\kappa_{\rm Cep}=(0.056\pm0.370)\times10^6\ {\rm mag}$ — the expected near-null: the shared Cepheid zero point absorbs the anchor-to-host common mode, so the matrix channel measures only the differential component and cannot see the term that closes the remaining offset. A separate unified host-level reconstruction, which re-optimizes the environmental response to flatten the host-environment trend, yields $H_0=67.83\pm1.56\ {\rm km\,s^{-1}\,Mpc^{-1}}$, internally consistent with the TEP CMB value ($66.70 \pm 0.58$, Paper 26) at $0.68\sigma$ — an agreement between two products of the same framework, not an independent cross-check — and $0.26\sigma$ from Planck. The latter is a host-level reconciliation route, not the primary full-ladder estimator: it evaluates the fitted response at the cosmic baseline and is conditional on that response form extending below the anchor ensemble to the unperturbed potential. Full-matrix propagation through the 3,490-row SH0ES system confirms exact reference-gauge invariance. The statistically cleanest channel is the within-host period structure: the environment-coupled Cepheid period term, identified through the 41 independent period distributions (37 hosts plus 4 anchors) and immune to latent-modulus absorption and peculiar-velocity systematics by construction, returns $\kappa_P = (-4.20 \pm 3.38)\times 10^5\ {\rm mag}$ for the ad hoc linear carrier, resolving to $\kappa_{P^2} = (-11.91 \pm 4.80)\times 10^5\ {\rm mag}$ ($2.3$&ndash;$2.8\sigma$, matching the Step 62 unit-transport benchmark $|\kappa_{P^2}^{\rm pred}(\lambda_0=1)| = 11.96\times 10^5\ {\rm mag}$ at fixed $|\beta_A|=1$ to $0.5\%$) under the stellar-structure-derived period-quadratic carrier $X\cdot(P/10\,{\rm d})^2$, which serves as the primary matrix-level discriminant isolating the clock bias from host-metallicity confounding ($r = -0.17$), with a negative sign in all 41 leave-one-host-out refits and a fitted per-host slope ordering of $+1.12 \pm 0.30 \times 10^6$ mag per unit $X$. Finally, single-galaxy differential tests provide internal signatures with the sign predicted by TEP free from host-to-host peculiar velocity systematics: M31 inner versus outer Cepheids exhibit an empirical Period&ndash;Luminosity offset of $+0.356 \pm 0.136\ {\rm mag}$ ($2.6\sigma$; $+0.630 \pm 0.195\ {\rm mag}$, $3.24\sigma$ in the PHAT footprint and $+0.489 \pm 0.154\ {\rm mag}$, $3.18\sigma$ in the upper quality bands $e_W &lt; p_{75}$), increasing in the dedicated HST PHAT J/H catalogue to $+0.681 \pm 0.187\ {\rm mag}$ ($3.65\sigma$), retaining $+0.604 \pm 0.277\ {\rm mag}$ ($2.18\sigma$) under 2D period–colour matching and $+0.501 \pm 0.171\ {\rm mag}$ ($2.93\sigma$) under band decomposition, and sharpening to $+0.906 \pm 0.147\ {\rm mag}$ ($6.17\sigma$) under flux-independent neighbour-density crowding control (Step 26; whereas pair-matching on propagated magnitude error $e_W$ reverses the sign to $-0.565 \pm 0.079$ mag in PHAT and $-0.103 \pm 0.129$ mag in Pan-STARRS because Poisson photon-noise uncertainty correlates directly with apparent magnitude $W_H$ at $r=+0.78$ and with the P&ndash;L residual at $r=+0.71$, inducing outcome-collider bias, while matching on the ambient-field mediator $\log u_{\rm amb}$ absorbs the radial contrast by causal mediation). The LMC gradient ($+0.0284 \pm 0.0086$ mag, $3.3\sigma$) is robust to period and colour matching and provides independent internal confirmation.

*Keywords:* Hubble tension &ndash; Cepheid variables &ndash; distance ladder &ndash; stellar velocity dispersion &ndash; peculiar velocities &ndash; temporal equivalence principle

## 1. Introduction

### 1.1 The Hubble Tension and Distance Ladder Systematics

The discrepancy between the local expansion rate of the Universe measured via the classical distance ladder and the value inferred from early-universe cosmological observations has emerged as one of the most pressing foundational problems in modern physics. The SH0ES collaboration reports a local Hubble constant of $H_0 = 73.04 \pm 1.04\ {\rm km\,s^{-1}\,Mpc^{-1}}$ using Cepheid-calibrated Type Ia supernovae (Riess et al. 2022; hereafter R22), whereas cosmic microwave background (CMB) measurements by the Planck satellite under flat $\Lambda$CDM establish $H_0 = 67.4 \pm 0.5\ {\rm km\,s^{-1}\,Mpc^{-1}}$ (Planck Collaboration VI 2020). This exceeds $5\sigma$ in statistical significance, resisting resolution through conventional astrophysical systematics, photometric crowding, or standard cosmological parameters.

The standard distance ladder relies on the assumption that stellar standard candles calibrated in nearby anchor galaxies (the Milky Way, Large Magellanic Cloud, Small Magellanic Cloud, and NGC 4258) behave identically in distant host galaxies. However, the anchor galaxies possess predominantly shallow gravitational potential wells, whereas the SN Ia host galaxies sample substantially deeper galactic potential wells. If stellar pulsation clocks respond dynamically to the local gravitational potential, an environmental gradient between calibrators and hosts will introduce a systematic bias into the inferred distance scale.

### 1.2 The Temporal Equivalence Principle

The physical intuition is direct. Classical Cepheids are dynamical acoustic clocks: their pulsation periods tick in local proper time, set by the mean density of the pulsation envelope. If an SN Ia host galaxy samples a deeper temporal environment than the nearby anchor galaxies used to calibrate the Period&ndash;Luminosity relation, its Cepheid pulsation response is shifted. Read through a universal Leavitt law, the shifted periods mimic a fainter intrinsic luminosity, so the host distance is underestimated and the local Hubble constant is inflated. The covariant framework below supplies the mechanism that generates and propagates this environmental clock offset.

The Temporal Equivalence Principle (TEP) provides a covariant scalar-tensor framework that elevates proper time from a fixed background coordinate parameter to a dynamical physical field. The canonical scalar comparison baseline is formulated on a two-metric spacetime:

\begin{equation}
S = \int d^4x\sqrt{-g}\left[\frac{M_{\rm Pl}^2}{2}R - \frac{1}{2}(\nabla\phi)^2 - V(\phi)\right] + S_m[\tilde g_{\mu\nu}, \Psi_m] ,
\end{equation}

where gravity is governed by the metric $g_{\mu\nu}$, while all non-gravitational matter fields, atomic transitions, and stellar clocks couple universally to the canonical causal matter metric $\tilde g_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$, written in conformal–disformal shorthand as $\tilde g_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$; Smawfield 2025, Paper 0, §2.2, Step 80). The operative weak-field screening realization replaces the canonical scalar term by the noncanonical kinetic sector $P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+\frac{X|X|}{\Lambda_X^4}$, $k\simeq16$ (Paper 0, §2.2); the canonical form displayed is a comparison baseline — the physical branch carries no canonical small-$|X|$ term. In gravitational bound states, the conformal factor satisfies $A(\phi) &lt; 1$, slowing the rate of proper time relative to cosmic background time.

The absolute matter-clock rate relative to cosmic time is defined as

\begin{equation}
r_j \equiv \frac{(d\tilde\tau/dt)_j}{(d\tilde\tau/dt)_{\rm cosmic}} .
\end{equation}

Throughout the TEP framework, the physical rate ordering across galactic environments is strictly bounded:

\begin{equation}
0 &lt; r_{\rm core} &lt; r_{\rm disk} &lt; r_{\rm cosmic} \equiv 1 .
\label{eq:rate_hierarchy}
\end{equation}

Within matter-hosted environments the ordering is monotone — clocks in active-shear galactic disks are simply less slowed than clocks in dense nuclear cores — while the unsourced void sector may exceed the ambient only within the bounded spatial contrast established by the eternal-slice construction (Paper 0 &sect;8). Environmental screening $S(\rho)$ suppresses the observable scalar-charge sector in high-density regimes while preserving dynamical time-rate variations on galactic and cosmological scales. Microscopically this follows from the nonlinear kinetic completion $P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+\frac{X|X|}{\Lambda_X^4}$, $k\simeq16$, of the specified scalar action (Paper 0 §2.2, v0.16): where the ambient clock-rate landscape is already steep, the field is kinematically stiff, so the observable Temporal Shear — the slope of that landscape, which Einstein-frame analyses parameterize as a fifth force — is pinned rather than newly carved (the $k_{\rm rec} \geq 4$ recovery steepness of constraint F4 is a consistency check on the derived response, realized by the hierarchical $s^4$ deep-interior asymptote). The field amplitude $A(\phi)$ continues to vary inside screened matter, preserving the clock-rate channel: what is pinned is the gradient, not the field amplitude. The quartic self-interaction $V = \lambda\phi^4/4$ is the matter-hosting weak-field amplitude-sector branch of the unified potential (Paper 0 §2), whose radial ODE closure $\nabla^2\phi = V_{,\phi} + \rho_* A_{,\phi}$ yields $m_{\rm eff} \propto \rho^{1/3}$; the density-dependent screening factor $S(\rho)$ used in this paper is the EFT-level projection of the kinetic closure onto the Cepheid host-galaxy density regime, not an independent chameleon potential.

| Regime | Absolute Rate | Physical Interpretation |
| --- | --- | --- |
| Cosmic background | $r_{\rm cosmic} = 1$ | Unscreened cosmological reference |
| Galactic disk | $0 &lt; r_{\rm disk} &lt; 1$ | Moderate potential depth; active scalar shear |
| Galactic core | $0 &lt; r_{\rm core} &lt; r_{\rm disk}$ | Deep potential depth; strongly screened |

### 1.3 Stellar Pulsation Physics and the Observing Chain

Classical Cepheid variables are dynamical pulsation clocks whose pulsation periods $P_{\rm local}$ are governed by hydrodynamic acoustic transit times across the stellar envelope in local matter-frame proper time. When observed in the heliocentric telescope frame and corrected for the systemic redshift $z_{\rm spec}$ of the host galaxy, the inferred rest-frame period is:

\begin{equation}
P_{\rm rest}^{\rm inf} = \frac{P_{\rm obs}}{1 + z_{\rm spec}} = P_{\rm local}\cdot q_i ,
\qquad
q_i \equiv \frac{r_{\rm spec,i}}{r_{\rm Cep,i}} .
\label{eq:intro_period_ratio}
\end{equation}

Under the TEP core&ndash;disk tracer closure the observing-chain ratio $q_i = r_{\rm spec}/r_{\rm Cep}$ supplies a bounded bare-clock contribution, while the dominant observed response is carried by the amplitude-sector response of the pulsation envelope — a mechanical inverse-density transport $\delta\ln P_{\rm amp} = -\lambda_0\,X_i\,\rho_T/\bar\rho(P)$ that converts the order-$10^{-7}$ environmental field increment into a magnitude-level period shift (§3.8, Appendix C.3). Because the fundamental-mode Cepheid mean density scales as $\bar\rho(P) = \rho_\odot(Q/P)^2$, this stellar-structure relation predicts an environment-coupled period-quadratic carrier $X_i\cdot(P/10\,{\rm d})^2$ (with $\rho_T/\bar\rho(10\,{\rm d}) = 8.45\times 10^5$ at $\rho_T = 20\ {\rm g\,cm^{-3}}$ and the unit-transport benchmark $\lambda_0 = 1$, with bare coupling $|\beta_A| = 1$ fixed independently by Paper 0; Step 62), which operates within the 41 host and anchor period distributions as the primary matrix-level discriminant of the clock bias, immune to latent-modulus absorption and separating the pulsation-envelope response from host-metallicity confounding. Through the Leavitt Period&ndash;Luminosity law ($M_W = \alpha + \beta \log P$), this period response alters the inferred host distance modulus by $\Delta \mu_i = \kappa_{\rm Cep} X_i$, where $X_i$ is the environmental potential coordinate.

### 1.4 Scope and Structure of this Study

This paper presents a comprehensive empirical test of the TEP framework across multiple independent observational tiers using public astronomical data:

1. Single-galaxy differential tests in M31 and the LMC, isolating the potential-clock coupling free from host-to-host peculiar velocity uncertainties.

2. Homogeneous, exact-identifier inner-potential reconstruction for all 37 distinct R22 SN Ia host galaxies from fully provenanced stellar-absorption dispersions and calibrated H&nbsp;I linewidth proxies, with continuous screening.

3. Generative endpoint likelihood evaluation across flow-velocity models and peculiar velocity dispersions.

4. Mathematical resolution of the latent-modulus parameter absorption in standard distance-ladder design matrices using the TEP-native generative ladder and the within-host period-quadratic discriminant $X_i\cdot(P/10\,{\rm d})^2$ derived from the Cepheid period&ndash;mean-density relation.

5. Complete computational reproducibility: the resolution of the design matrix degeneracy and the full-ladder generative likelihoods are accompanied by an open-source, end-to-end Python pipeline, ensuring transparent verification of all statistical claims.

## 2. Data and Methods

### 2.1 Public Distance-Ladder Data and Independent Sample Reconstruction

The empirical distance ladder is reconstructed from the public R22 data release, comprising the 3,490-row generalized least-squares design matrix, data vector, full covariance matrix, and the Pantheon+SH0ES supernova catalogue (Scolnic et al. 2022; Brout et al. 2022). Solving the unmodified baseline system yields $H_0 = 73.0434 \pm 1.0072\ {\rm km\,s^{-1}\,Mpc^{-1}}$ with $\chi^2 = 3552.7063$ for 3,444 degrees of freedom, reproducing the published SH0ES solution.

The R22 calibrator sample contains 42 SNe Ia situated in 37 distinct host galaxies. In the host-level inference, the independent statistical unit is the individual host galaxy, as all Cepheids and multiple SNe within a single galaxy share the same host potential well. Each supernova is matched to its host galaxy using exact R22 catalogue identifiers and PGC numbers, strictly excluding angular nearest-neighbour heuristics. All 37 distinct host galaxies have measured positive Hubble-flow redshifts ($z_{\rm HD} > 0$). Of these, 34 satisfy the Hubble-flow union cut ($z_{\rm CMB} > 0.0035$ or $z_{\rm HD} > 0.0035$); 33 of those have matching Cepheid design-matrix entries and enter the primary generative likelihood.

### 2.2 Local Inner-Potential Coordinate

The TEP clock response is sourced by the gravitational potential at the pulsation site, not by the asymptotic halo. SH0ES Cepheid fields occupy inner-to-mid disk radii ($r_{\rm Cep} \sim 1.8 R_d$) where the potential is baryon-dominated; the relevant coordinate is therefore the local inner-region stellar velocity dispersion $\sigma_{*,i}$, which tracks the enclosed inner potential and is obtained from inclination-independent stellar spectroscopy. For each of the 37 hosts and the ladder anchors, $\sigma_{*,i}$ is compiled from the published literature with full per-host provenance (Appendix B): direct stellar-absorption measurements for 18 hosts (H&eacute;raudeau et al. 1999; Ho et al. 2009; Kormendy &amp; Ho 2013; the Riess et al. 2022 calibrator compilation) and calibrated H&nbsp;I linewidth dispersion proxies for the remaining 19 hosts (Campbell et al. 2014 6dFGSv). The anchor endpoint is constructed identically from local stellar dispersions at the Cepheid calibration sites (Milky Way solar neighbourhood $\sigma_z = 30.0\ {\rm km\,s^{-1}}$, LMC disk $24.0\ {\rm km\,s^{-1}}$, NGC 4258 intermediate annulus $115.0\ {\rm km\,s^{-1}}$), so the environmental coordinate subtracts like for like.

\begin{equation}
U_i = S_i\,\sigma_{*,i}^2 ,
\qquad
\Phi \sim U_i ,
\label{eq:uphi}
\end{equation}

A homogeneous whole-galaxy alternative, $u_\phi = V_{\rm rot}/\sqrt{2}$ from pinned HyperLEDA maximum rotation velocities, is retained as a documented robustness coordinate. The two coordinates are not interchangeable: $V_{\rm rot}$ probes the outer dark-halo potential at tens of kpc and requires $1/\sin i$ deprojection that is unstable for the eight Hubble-flow hosts at $i &lt; 40^\circ$, whereas $\sigma_*$ measures the inner baryonic potential inhabited by the Cepheids. Direct comparison (Section 4; Step 64) confirms that the environmental response tracks the local coordinate and is attenuated under the global rotation proxy, establishing coordinate validity rather than proxy convenience.

### 2.3 Environmental Screening Formulation

The scalar field sector in TEP acts through two distinct macroscopic operators governed by Rules 10, 11, and 22: the Temporal Shear suppression operator $\mathcal{S}_\Sigma(\mathcal{E})$ and the clock-amplitude response $A(\phi)$. The macroscopic shear operator $\mathcal{S}_\Sigma(\mathcal{E})$ suppresses spatial clock-rate gradients and the corresponding Einstein-frame anomalous accelerations in dense environments ($\mathcal{S}_\Sigma \to 0$ in screened matter), recovering standard General Relativity for gravitational orbits. By contrast, the clock amplitude $A(\phi) = \exp(-\phi / M_{\rm Pl})$ tracks potential depth monotonically: proper-time clock rates run slower in deeper gravitational wells ($A &lt; 1$), and this clock slowing continues inside screened matter without vanishing, preserving gravitational redshift.

In galaxy halos, the two-body environmental suppression factor $S_i = S_{{\rm local},i} S_{{\rm group},i}$ parameterizes the shear suppression of the host's extended potential well within its surrounding group or cluster environment. In local high-density stellar cores (such as inner M31 or LMC centers), the local potential depth is deepest, causing local clocks to run slowest ($A &lt; 1$), while the associated environmental ordering sets the sign of the observed positive Period&ndash;Luminosity offset ($\Delta W > 0$) relative to diffuse outer-disk Cepheids through the amplitude-sector response. Host disk scale lengths are evaluated from RC3 isophotal diameters ($R_{25} = D_{25}/2$, $R_d = R_{25}/3.2$, $z_d = 0.1 R_d$) and stellar mass densities are computed at representative Cepheid radii ($r_{\rm Cep} = 1.8 R_d$). Group-scale density is obtained from the Tully (2015) 2MRS group catalogue Table 5 via exact PGC matching. Evaluated at $r_{\rm Cep}$, the local density projection is equivalent to the enclosed-acceleration diagnostic $M({&lt;}r_{\rm Cep})/r_{\rm Cep}^2 \propto \bar\rho({&lt;}r_{\rm Cep})\,r_{\rm Cep}$ up to the disk's vertical-thickness correction — the density stand-in licensed for dense macroscopic matter by the $\mathcal{S}_\Sigma(\mathcal{E})$ construction (Paper 0 §7). The total continuous screening factor is:

\begin{align}
S_{{\rm local},i} &= \left[1 + \left(\frac{\rho_i}{0.5\ M_\odot\,{\rm pc}^{-3}}\right)^2\right]^{-1} , \\
S_{{\rm group},i} &= \left[1 + \left(\frac{N_{{\rm mb},i}}{10}\right)^{1.2}\right]^{-1} , \\
S_i &= S_{{\rm local},i} S_{{\rm group},i} .
\label{eq:screening}
\end{align}

The screened potential scale for each host is $U_i = S_i \sigma_{*,i}^2$. The assignment is positional: $S_i$ is the source-side suppression — the host's potential well sources the local field level through its effective scalar charge, which the shear-sector operator $\mathcal{S}_\Sigma$ suppresses at the habitat density — while the probe-side clock-amplitude response $\mathcal{S}_A$, evaluated at the Cepheid's own material density, is a per-source constant absorbed into the channel coefficient $\kappa_{\rm Cep}$ (Appendix C). For likelihood evaluation, the dimensionless centered environmental regressor is defined as:

\begin{equation}
\widetilde X_i = \frac{U_i - U_{\rm ref}}{c^2} - \left\langle \frac{U - U_{\rm ref}}{c^2} \right\rangle ,
\label{eq:centered_x}
\end{equation}

where centering guarantees that the fitted environmental slope is mathematically invariant to the choice of reference scale $U_{\rm ref}$.

### 2.4 Single-Galaxy Internal Differential Methodology

To test the TEP clock-rate coupling independently of host-to-host peculiar velocities and distance ladder calibration steps, internal Period&ndash;Luminosity variations are analysed within individual galaxies. In M31 (Andromeda), the homogeneous Pan-STARRS Cepheid catalog (Kodric et al. 2018) containing 2,686 variables is used, comparing Cepheids in the dense inner potential ($R &lt; 5.0$ kpc, mean density $\rho \approx 0.31\ M_\odot\,{\rm pc}^{-3}$) against the diffuse outer disk ($R > 15.0$ kpc, $\rho \approx 0.006\ M_\odot\,{\rm pc}^{-3}$). Spatial restriction to the HST PHAT footprint is also performed. In the Large Magellanic Cloud, fundamental-mode Cepheids from OGLE-IV (Soszy&nacute;ski et al. 2015) are analysed, partitioned by galactocentric radius.

### 2.5 Generative Endpoint Likelihood and Flow Velocities

The host-level expansion relation is evaluated via the generative model:

\begin{equation}
cz_i = d_i\left(H_{\rm app} + \Gamma_X \widetilde X_i\right) + \epsilon_i ,
\label{eq:primary_likelihood}
\end{equation}

where $d_i = 10^{(\mu_i - 25)/5}\ {\rm Mpc}$, and the Gaussian variance is:

\begin{equation}
V_i = \sigma_v^2 + \left[\frac{\ln 10}{5} \left|d_i\left(H_{\rm app} + \Gamma_X \widetilde X_i\right)\right|\sigma_{\mu,i}\right]^2 + \sigma_{{\rm int},v}^2 .
\end{equation}

The likelihood is evaluated across a spectrum of peculiar-velocity dispersions: the standard R22 baseline ($\sigma_v = 250\ {\rm km\,s^{-1}}$), alternative velocity scales ($150\ {\rm km\,s^{-1}}$ and $500\ {\rm km\,s^{-1}}$), and the data-driven residual velocity scatter after bulk-flow removal from Pantheon+ ($\sigma_v = 182.1\ {\rm km\,s^{-1}}$). Because the intrinsic-scatter parameter $\sigma_{{\rm int},v}$ enters the variance only through the sum $\sigma_v^2 + \sigma_{{\rm int},v}^2$, the likelihood identifies a single quantity &mdash; the total velocity-space scatter &mdash; rather than $\sigma_v$ and $\sigma_{{\rm int},v}$ separately. With $\sigma_{{\rm int},v}$ free, any adopted $\sigma_v$ below the data-implied total is compensated by the fitted $\sigma_{{\rm int},v}$, so the $\sigma_v = 150$ and $182.1\ {\rm km\,s^{-1}}$ variants are reparameterizations of a single optimum rather than distinct fits; the likelihood-implied total is reported with the results ($\sigma_{\rm tot} \approx 210\text{--}226\ {\rm km\,s^{-1}}$ across the cz-space and $H_0$-space likelihoods), and the canonical $\sigma_v = 250\ {\rm km\,s^{-1}}$ convention sits mildly above it, pinning $\sigma_{{\rm int},v}$ at its lower bound.

### 2.6 The TEP-Native Generative Distance Ladder

In standard distance-ladder linear regression, host distance moduli $\mu_i$ are treated as unconstrained latent parameters. The linear design matrix does not generically erase observation-level Cepheid perturbations (as confirmed by the unbiased ensemble recovery of raw row-level injections in Step 34: pull mean $+0.12$ with unit scatter over 200 covariance-level realizations). What it cannot identify without external constraints is an environmental perturbation that is mathematically equivalent to shifting the latent host modulus, because the 37 unconstrained $\mu_i$ parameters absorb the host-level shift ($\hat{\mu}_i \to \mu_i - \kappa_{\rm Cep} X_i$).

The true generative observable model couples the observed Cepheid distance moduli $\mu_i^{\rm obs} = \mu_i^{\rm true} - \kappa_{\rm Cep} X_i$ directly to the observed Hubble-flow relation $cz_i = d_i^{\rm true}(H_{\rm app} + \beta_X X_i) + v_i$. The identifiable physical combination is:

\begin{equation}
\Gamma_X = \beta_X + \left(\frac{\ln 10}{5}\right) H_{\rm app} \kappa_{\rm Cep} .
\label{eq:gamma_relation}
\end{equation}

($\Gamma_X$ here is a velocity-space slope in ${\rm km\,s^{-1}\,Mpc^{-1}}$, distinct from the dimensionless channel projector of the same letter in the Paper 0 transfer map.)

In the generative ladder formulation, the relation is evaluated directly in expansion-rate space across the $N=33$ Hubble-flow SN Ia hosts with Cepheid calibrations ($z_{\rm HD} > 0.0035$ or $z_{\rm CMB} > 0.0035$, excluding local anchors):

\begin{equation}
H_{0,i} \equiv \frac{cz_i}{d_i^{\rm SH0ES}} = H_{\rm app} + \Gamma_X \widetilde X_i + \epsilon_i ,
\label{eq:generative_wls}
\end{equation}

where $d_i^{\rm SH0ES} = 10^{(\mu_i^{\rm SH0ES}-25)/5}$ Mpc, and each host is weighted by its inverse total variance:

\begin{equation}
w_i = \frac{1}{\sigma_{H_0,i}^2} = \frac{d_i^2}{\sigma_v^2 + \left(\frac{\ln 10}{5} cz_i \sigma_{\mu,i}\right)^2} .
\label{eq:generative_weights}
\end{equation}

Because $H_{0,i}$ is scale-invariant across distance, all 33 hosts contribute according to their fractional distance precision. To account for potential cross-host correlations in the SH0ES calibration chain, the model is also evaluated via Generalized Least Squares (GLS) using the full $33\times 33$ covariance matrix $\mathbf{C}_H = \mathbf{J} \mathbf{C}_\mu \mathbf{J}^T + \mathbf{C}_v$, where $\mathbf{C}_\mu = (\mathbf{L}^T \mathbf{C}^{-1} \mathbf{L})^{-1}_{\mu}$ is extracted directly from the inverted 3,490-row SH0ES normal matrix, $\mathbf{J} = \text{diag}(-\frac{\ln 10}{5} H_{0,i})$ is the Jacobian transformation, and $\mathbf{C}_v = \text{diag}(\sigma_v^2/d_i^2)$ is the peculiar velocity dispersion matrix.

Under the observing-chain mechanism formalized in Appendix C, the same potential well that produces the environmental period response also imparts a gravitational redshift to the adopted systemic-redshift tracer:

\begin{equation}
1 + z_{{\rm spec},i} = \frac{1 + z_{{\rm path},i}}{r_{{\rm spec},i}} \approx 1 + z_{{\rm path},i} + \frac{\Phi_{{\rm spec},i}}{c^2} .
\end{equation}

This shifts the observed velocity by $\Delta(cz)_i = c \Phi_{{\rm spec},i}/c^2 = \Phi_{{\rm spec},i}/c \sim 70\ {\rm m\,s^{-1}}$. When expressed as an apparent expansion-rate slope by dividing by host distance $d_i$, this physical gravitational redshift contributes:

\begin{equation}
\beta_{X,\rm grav} \equiv \frac{c}{\langle d \rangle} \approx \frac{299,792\ {\rm km\,s^{-1}}}{31.2\ {\rm Mpc}} \approx 9.60 \times 10^3\ {\rm km\,s^{-1}\,Mpc^{-1}} .
\end{equation}

In contrast, the Cepheid distance-modulus bias $\Delta \mu_i = \kappa_{\rm Cep} X_i$ rescales the host distance by $\Delta d / d \approx 7\%$, producing an apparent expansion-rate slope:

\begin{equation}
\Gamma_{X,\rm Cep} \equiv \left(\frac{\ln 10}{5}\right) H_{\rm app} \kappa_{\rm Cep} \approx 3.85 \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}} .
\end{equation}

The ratio of the direct velocity response to the distance-modulus response is:

\begin{equation}
\frac{\beta_{X,\rm grav}}{\Gamma_{X,\rm Cep}} = \frac{9.60 \times 10^3}{3.85 \times 10^7} \approx 2.5 \times 10^{-4} \approx 0.02\% \text{--} 0.03\% .
\end{equation}

Consequently, the physical observing chain intrinsically allocates $>99.97\%$ of the combined response $\Gamma_X$ to the Cepheid distance channel. Setting $\beta_X \approx 0$ (or coupling $\beta_X = \beta_{X,\rm grav}$) is therefore not an ad-hoc allocation, but the direct leading-order consequence of metric proper-time transport.

The fitted intercept $H_{\rm app}$ represents the expansion rate at the sample mean environment $\langle X \rangle = -1.5 \times 10^{-9}$ on the screened coordinate — essentially the midpoint of the host ensemble relative to the anchor reference. To evaluate the expansion rate at physically defined reference frames, the relation maps back to:

\begin{equation}
H_{\rm ref} = H_{\rm app} - \Gamma_X \langle X \rangle,
\qquad
H_{\rm cosmic} = H_{\rm app} + \Gamma_X (X_{\rm cosmic} - \langle X \rangle) = H_{\rm ref} - \Gamma_X \left(\frac{U_{\rm ref}}{c^2}\right) ,
\label{eq:href_mapping}
\end{equation}

where $X_{\rm ref} \equiv 0$ corresponds to the anchor reference zero-point ($\sigma_{\rm ref} = 87.165\ {\rm km\,s^{-1}}$), and $X_{\rm cosmic} = -U_{\rm ref}/c^2 = -8.45 \times 10^{-8}$ corresponds to the unperturbed cosmic background (zero virial potential depth).

### 2.7 Joint Multi-Block Distance&ndash;Redshift Likelihood

While the standalone host-level regression measures the identifiable combined slope $\Gamma_X$, breaking the internal parameter degeneracy of the distance ladder without fixing $\beta_X=0$ requires external observational constraints. A joint multi-block likelihood is formulated by combining three observational data blocks:

\begin{equation}
\ln \mathcal{L}_{\rm joint}(\boldsymbol{\theta}) = \ln \mathcal{L}_{\rm redshift} + \ln \mathcal{L}_{\rm TRGB} + \ln \mathcal{L}_{\rm anchor} + \ln \mathcal{P}(\delta m) + \ln \mathcal{P}(\delta a) ,
\label{eq:joint_likelihood}
\end{equation}

where $\boldsymbol{\theta}$ encompasses the physical and nuisance parameters. The three blocks are structured as follows:

1. *Redshift&ndash;Distance Block ($N=33$ Hubble-flow hosts):* Evaluates the conventional Hubble-flow relation via a linearised regression in $H_0$ space, $H_{0,i} = cz_i / d_i^{\rm obs} = H_{\rm app} + \Gamma_X X_i + \epsilon_i$, where $\Gamma_X = \beta_X + (\ln 10 / 5) H_{\rm app} \kappa_{\rm Cep}$ is the identifiable combined endpoint response. This formulation is equivalent to the nonlinear $cz$-space model $cz_i = d_i^{\rm true}(H_{\rm app} + \beta_X X_i) + v_i$ with $d_i^{\rm true} = d_i^{\rm obs} 10^{\kappa_{\rm Cep} X_i / 5}$, but offers superior numerical conditioning because the parameter space is linear in $(H_{\rm app}, \Gamma_X)$. The 33 hosts comprise the full Hubble-flow calibrator sample, including the two lowest-redshift systems (NGC 4424 and NGC 4536) whose CMB-corrected redshifts place them above the Hubble-flow threshold. The per-host variance is $\sigma_{v,i}^2 = \sigma_v^2 / d_i^2 + (\frac{\ln 10}{5} H_{0,i} \sigma_{\mu,i})^2 + \sigma_{{\rm int},v}^2 / d_i^2$, where $\sigma_{{\rm int},v}$ is an intrinsic velocity dispersion nuisance parameter.

2. *TRGB Differential Block ($N=18$ non-anchor calibrators):* Compares Cepheid and Tip of the Red Giant Branch distance moduli, $\Delta \mu_i \equiv \mu_i^{\rm Cep} - \mu_i^{\rm TRGB} = \delta m - \kappa_{\rm Cep} X_i + \epsilon_i$, with variance $\sigma_{\Delta\mu,i}^2 = \sigma_{\mu,{\rm Cep},i}^2 + \sigma_{\mu,{\rm TRGB},i}^2$. Because TRGB stellar evolution clocks operate independently of pulsation physics, TRGB distances provide an external anchor on the host modulus. The 18 hosts represent the complete overlap of SH0ES Cepheid calibrators, the inner-potential catalogue, and EDD/CCHP TRGB measurements (excluding geometric anchors to prevent double counting).

3. *Independent Geometric Anchor Block ($N=2$ primary anchors):* Constrains the absolute Cepheid zero-point against independent geometric distances in the Large Magellanic Cloud (detached eclipsing binaries) and NGC 4258 (water megamasers): $\Delta \mu_{\rm anc} \equiv \mu_{\rm Cep} - \mu_{\rm geo} = \delta a - \kappa_{\rm Cep} X_{\rm anc} + \epsilon_a$. M31 is excluded from this block because its distance scale is composite rather than purely geometric.

Five nested model configurations are evaluated against the data: the *Null Model* ($N_{\rm par}=4$: $H_{\rm app}, \delta m, \delta a, \sigma_{{\rm int},v}$), the *Cepheid Model* ($N_{\rm par}=5$: setting $\beta_X \equiv 0$ and fitting $\kappa_{\rm Cep}$), the *Coupled Model* ($N_{\rm par}=5$: setting $\beta_X = \beta_{X,\rm grav} \equiv c/\langle d \rangle \approx 9.60 \times 10^3\ {\rm km\,s^{-1}\,Mpc^{-1}}$ and fitting $\kappa_{\rm Cep}$), the *Velocity Model* ($N_{\rm par}=5$: setting $\kappa_{\rm Cep} \equiv 0$ and fitting $\beta_X$), and the *Mixed Model* ($N_{\rm par}=6$: simultaneously fitting both $\kappa_{\rm Cep}$ and $\beta_X$). The Cepheid and Velocity models represent two alternative single-degree-of-freedom closures of the underlying physical response rather than additive components, while the Coupled model realizes the physical prediction where the direct velocity perturbation is small but nonzero. In this multi-block formulation, the three data blocks are treated as quasi-independent observational constraints; while excluding primary geometric anchors from the TRGB block prevents direct anchor duplication, shared Cepheid calibration terms are treated within individual block variances.

### 2.8 Unified Host-Level Reconstruction

Complementing the expansion-rate likelihood, a unified host-level reconstruction is performed directly on the complete 37-host sample. Each host's distance modulus is corrected according to its screened kinematic potential:

\begin{equation}
\mu_i^{\rm corr} = \mu_i^{\rm obs} + \kappa_{\rm Cep} \frac{S(\rho_i)\, \sigma_{*,i}^2 - U_{\rm ref}}{c^2} ,
\label{eq:endpoint_correction}
\end{equation}

yielding reconstructed physical distances $d_i^{\rm corr} = 10^{(\mu_i^{\rm corr} - 25)/5}$ and individual expansion rates $H_{0,i} = cz_i / d_i^{\rm corr}$. The anchor reference zero-point is defined as $\sqrt{U_{\rm ref}} = \sigma_{\rm ref} = 87.165\ {\rm km\,s^{-1}}$, constructed from the weighted composite of local stellar velocity dispersions at the specific Cepheid disk locations within the primary calibration anchors: Milky Way solar neighbourhood ($\sigma_z = 30.0\ {\rm km\,s^{-1}}$, weight 0.20; Bovy et al. 2012), LMC stellar disk ($\sigma_{\rm disk} = 24.0\ {\rm km\,s^{-1}}$, weight 0.25; van der Marel et al. 2002), and NGC 4258 intermediate annulus ($\sigma_{\rm local} = 115.0\ {\rm km\,s^{-1}}$, weight 0.55; Kormendy & Ho 2013). As proven in Appendix D.3, the reference scale $\sqrt{U_{\rm ref}}$ is an exact gauge origin in the full linear system, absorbed into the Cepheid zero point $M_H^W$ without altering the physical response $\kappa_{\rm Cep}$.

The optimal response coefficient $\kappa_{\rm Cep}$ is determined by minimizing the residual variance and environmental gradient across the host sample ($\partial H_{0,i} / \partial \sigma_{*,i} \to 0$). Honest uncertainties are estimated via a joint bootstrap ($N=1,000$ resamples with replacement), where $\kappa_{\rm Cep}$ is re-optimized for each realization, simultaneously propagating host-to-host sampling scatter and $\kappa_{\rm Cep}$ parameter variance to yield the unified Hubble constant $H_0 = 67.83 \pm 1.56\ {\rm km\,s^{-1}\,Mpc^{-1}}$.

Two properties of this route delimit its interpretation. First, under the standard reference scale the host-sample mean of the correction regressor nearly vanishes ($\langle X_i\rangle \approx -1.5\times10^{-9}$), so the reconstructed $H_0$ level is largely insensitive to $\kappa_{\rm Cep}$ — the raw unweighted host mean is already $67.72 \pm 1.58\ {\rm km\,s^{-1}\,Mpc^{-1}}$ — and the TEP-specific content of the route is the flattening of the per-host environmental slope rather than the absolute level. Second, the optimizer's objective is defined on the linear $\sigma_*$ axis: an estimator decomposition of the same corrected moduli against the physical correction regressor $X_i \propto S_i\sigma_{*,i}^2$ returns a markedly different nominal optimum ($\kappa \approx 3.4\times10^6$ mag under the $\sigma_*$-slope objective versus $(0.98 \pm 0.48)\times10^6$ mag under a weighted least-squares fit of $\delta\mu$ on $X_i$ with photometric plus peculiar-velocity errors — consistent with the endpoint-equivalent coefficient $\kappa_{\rm Cep}^{\rm equiv} = 1.22\times10^6$ mag). The reconstruction is therefore carried as a fitted consistency diagnostic: it demonstrates that the environmental response required to flatten the host trend is compatible with the independently measured coefficient, without constituting a prespecified propagation of that coefficient.

Evaluating the host-level reconstruction under the alternative screened anchor reference scale $\sqrt{U_{\rm ref}} = 30.507\ {\rm km\,s^{-1}}$ yields $H_0 = 60.45 \pm 1.39\ {\rm km\,s^{-1}\,Mpc^{-1}}$. The $7.4\ {\rm km\,s^{-1}\,Mpc^{-1}}$ offset from the standard-construction value ($67.83 \pm 1.56\ {\rm km\,s^{-1}\,Mpc^{-1}}$, $0.26\sigma$ from Planck) is a pure gauge artifact of the unconstrained host-level shortcut, which applies an additive modulus shift $\Delta \mu = \kappa_{\rm Cep} \Delta U_{\rm ref} / c^2$ without simultaneously refitting the anchor absolute magnitude zero point. In the full generative distance ladder (Appendix D.3), where the anchor absolute magnitude $M_H^W$ is freely refit, this additive shift is identically absorbed into the zero point, returning $H_0$ values that are identical to four decimal places across all four reference origins. Furthermore, in the generative expansion-rate likelihoods, the cosmological background extrapolation $H_{\rm cosmic} = H_{\rm app} - \Gamma_X \langle U \rangle / c^2$ is algebraically independent of $U_{\rm ref}$, yielding background expansion rates of $65.1\text{--}65.5\ {\rm km\,s^{-1}\,Mpc^{-1}}$ across the $\sigma_v$ grid and joint-model configurations — within $\sim 1.3\sigma$ of Planck — with exact gauge invariance.

### 2.9 Observational Controls for Single-Galaxy Internal Gradients

Single-galaxy differential tests in M31 and the LMC provide internal empirical checks of the TEP potential hierarchy free from host-to-host peculiar velocity dispersions. To ensure these internal Period&ndash;Luminosity offsets are not generated by standard astrophysical systematics, the analysis incorporates three rigorous controls:

1. *Extinction-Free Photometry:* All Cepheid magnitudes are evaluated using the reddening-free Wesenheit index $W \equiv m_I - R_V (m_V - m_I)$ (with $R_V = 1.55$ for OGLE-IV LMC and $R_V = 1.54$ for M31), which algebraically cancels total and differential interstellar dust extinction along the line of sight.

2. *Spatial Resolution and Crowding Controls:* In M31, ground-based blending and crowding effects in the dense inner bulge are controlled in three ways: (a) restricting the Pan-STARRS sample to the PHAT survey footprint, which recovers the internal offset at $\Delta W = +0.630 \pm 0.195\ {\rm mag}$ ($3.24\sigma$; $n_{\rm inner}=106$, $n_{\rm outer}=110$), and restricting to the highest photometric quality bands ($e_W &lt; p_{25}, p_{50}, p_{75}$), which strengthens the offset to $+0.421 \pm 0.213$ mag ($1.97\sigma$), $+0.396 \pm 0.175$ mag ($2.26\sigma$), and $+0.489 \pm 0.154$ mag ($3.18\sigma$); (b) the dedicated HST PHAT J/H photometric catalogue (Kodric et al. 2018), which yields $\Delta W = +0.681 \pm 0.187\ {\rm mag}$ ($3.65\sigma$; $n_{\rm inner}=78$, $n_{\rm outer}=69$), retains $+0.604 \pm 0.277$ mag ($2.18\sigma$) under 2D period–colour matching and $+0.501 \pm 0.171$ mag ($2.93\sigma$) under band decomposition, and sharpens to $\Delta W = +0.906 \pm 0.147$ mag ($6.17\sigma$) under flux-independent arcsec-scale neighbour-density crowding control; and (c) an outcome-collider audit (step_26_m31_phat_robustness_summary.json) demonstrating why naive pair-matching on propagated magnitude error $e_W$ reverses the sign ($-0.565$ mag in PHAT, $-0.103$ mag in Pan-STARRS): in a fixed-exposure single-galaxy survey ($D=780$ kpc), Poisson photon-noise uncertainty $\sigma_m \propto 10^{+0.2\,m}$ is a direct monotonic function of apparent magnitude $W_H$ itself (${\rm corr}(e_{W_H}, W_H) = +0.78$, ${\rm corr}(e_{W_H}, \Delta W_{\rm resid}) = +0.71$), so conditioning on $e_W$ conditions directly on the outcome variable $W_H$ (collider bias), whereas the flux-independent neighbour-density crowding control ($+0.906 \pm 0.147$ mag, $6.17\sigma$) and causal mediation matching on the ambient-field proxy $\log u_{\rm amb}$ (which absorbs the radial contrast as expected when holding the environmental mediator fixed) confirm the physical signal.

3. *Metallicity Plausibility Limits:* To test whether radial metallicity gradients could mimic the observed $\Delta W$, the required metallicity sensitivity is evaluated. For the empirical M31 metallicity gradient ($\Delta [\text{O/H}] \sim 0.2\text{--}0.4\ {\rm dex}$), explaining the observed $\Delta W \approx +0.36\text{--}0.63\ {\rm mag}$ purely via metallicity would require a positive coefficient $\gamma \simeq +0.7\text{--}1.4\ {\rm mag/dex}$ across the gradient range, of the wrong sign and several standard errors removed from the empirical metallicity dependence measured across the SH0ES calibrator sample, $\gamma = -0.20 \pm 0.08\ {\rm mag/dex}$ (Step 10: $\gamma_{\rm req} = +0.74\ {\rm mag/dex}$ at the baseline offset).

### 2.10 Host-Mass Decoupling and Supernova Mass-Step Marginalization

An essential empirical test is verifying that the physical TEP environmental coordinate does not act as a surrogate for the established Type Ia supernova host-galaxy stellar mass step. In standard supernova standardization pipelines (e.g., Betoule et al. 2014; Scolnic et al. 2018; Brout et al. 2022), an empirical magnitude correction $\gamma_{\rm step} \sim 0.05\text{--}0.06\ {\rm mag}$ is applied across the split threshold $\log(M_*/M_\odot) = 10.0$ to account for residual correlations between peak luminosity and host galaxy properties.

Host stellar masses $\log(M_*/M_\odot)$ are compiled directly from Riess et al. (2022, Table 4), spanning $8.50 \le \log(M_*/M_\odot) \le 10.87$ across the complete 37-calibrator sample. The inner-region dispersion correlates only weakly with host stellar mass in this sample ($r = 0.12$), so the physical TEP coordinate $X_i \equiv (S_i \sigma_i^2 - \sigma_{\rm ref}^2)/c^2$ incorporates multi-scale environmental screening $S_i = S_{\rm local}(\Sigma_*) \times S_{\rm group}(\rho_{\rm env})$ — the factorized evaluation of the canonical shear-sector projection $\mathcal{S}_\Sigma(\mathcal{E})$ on the two-component host environment $\mathcal{E} = \{\Sigma_*, \rho_{\rm env}\}$ (Paper 0 §7) — which suppresses both dense galactic disks and massive group potentials. Consequently, the screened potential depth is fundamentally distinct from total integrated stellar mass.

To evaluate whether the measured slope $\Gamma_X$ could be mimicked by host-mass corrections, a joint multi-regressor likelihood is formulated:

\begin{equation}
cz_i = d_i \left[ H_{\rm app} + \Gamma_X X_i + \gamma_{\rm step} \Theta(\log M_{*,i} - 10.0) + \gamma_M (\log M_{*,i} - \langle \log M_* \rangle) \right] + \epsilon_i ,
\label{eq:joint_mass_likelihood}
\end{equation}

where $\Theta(x)$ is the Heaviside step function, $\gamma_{\rm step}$ measures the discrete host-mass step in equivalent velocity-space units, and $\gamma_M$ captures continuous mass dependence. In addition, an orthogonalized coordinate framework is constructed by projecting out host stellar mass:

\begin{equation}
X_{\perp \rm step} \equiv X - (\alpha_{\rm step} \Theta + \beta_{\rm step}) , \quad X_{\perp M} \equiv X - (\alpha_M \log M_* + \beta_M) ,
\label{eq:orthogonalized_coordinates}
\end{equation}

allowing the isolated environmental slope $\Gamma_{X,\perp}$ to be fitted against the mass-residualized coordinate completely independent of any host stellar mass variations.

## 3. Results

### 3.1 Host Sample Reconstruction and Potential Stratification

The exact-identifier reconstruction yields all 37 distinct R22 SN Ia host galaxies with measured positive Hubble-flow redshifts ($z_{\rm HD} > 0$), complete RC3 isophotal diameters, and a fully provenanced inner-potential coordinate $\sigma_*$ (Section 2.2). Splitting the 37 hosts at the sample median $\sigma_* = 95.0\ {\rm km/s}$ reveals a pronounced empirical stratification: the 19 shallow-potential hosts yield a mean Hubble parameter of $63.05 \pm 3.06\ {\rm km\,s^{-1}\,Mpc^{-1}}$, whereas the 18 deep-potential hosts yield $72.64 \pm 2.62\ {\rm km\,s^{-1}\,Mpc^{-1}}$, propagating the full host distance covariance matrix — a contrast of $\Delta H_0 = 9.59\ {\rm km\,s^{-1}\,Mpc^{-1}}$, comparable in magnitude to the entire local&ndash;CMB discrepancy.

The potential-velocity association is significant: Pearson $r = 0.501$ ($p = 0.0016$) and Spearman rank $\rho = 0.553$ ($p = 3.9 \times 10^{-4}$). The correlation survives the principal systematic controls. Heteroscedasticity weighting, with $w_i = 1/\sigma_{H_0,i}^2$ from the diagonal of the full host Hubble-rate covariance matrix, yields $r_w = 0.520$ ($p = 0.0070$, $n_{\rm eff} = 25.5$). Replacing the Pantheon+ peculiar-velocity estimates with Stiskalek et al. 2026 Manticore Bayesian coherent-flow posterior means where available retains a weighted correlation of $r_w = 0.432$ ($p = 0.029$). Restricting to the 18 hosts whose $\sigma_*$ derives from direct stellar-absorption spectroscopy — the cleanest provenance tier — strengthens the association further ($r = 0.556$, $p = 0.017$; $\rho = 0.596$, $p = 0.009$; $\Delta H_0 = 11.2\ {\rm km\,s^{-1}\,Mpc^{-1}}$), so the signal is not carried by the H&nbsp;I proxy tier. The positive gradient between host potential depth and apparent expansion rate matches the directional prediction of TEP stellar clock modulation.

Coordinate validity is tested directly: substituting the homogeneous whole-galaxy rotation proxy $u_\phi = V_{\rm rot}/\sqrt{2}$ for the same 37 hosts and identical $H_{0,i}$ collapses the correlation to $r = 0.098$ ($p = 0.61$). The environmental signal therefore tracks the local inner potential inhabited by the Cepheids, not the outer-halo rotation scale — a distinction with direct physical meaning under TEP, whose clock response is sourced at the pulsation site (Step 64 audit; Appendix D).

![Host-level H0 values plotted against the square of the local inner-potential velocity dispersion for 37 R22 supernova hosts](public/figures/step_03_figure_01_h0_vs_sigma.png?v=3)

Figure 1: Host-level expansion rate $cz_{\rm HD}/d$ as a function of the inner-potential coordinate $\sigma_*^2$ for all 37 R22 SN Ia host galaxies. Error bars include distance modulus covariance and peculiar velocity uncertainties.

### 3.2 Single-Galaxy Internal Verification: M31 and LMC

Single-galaxy tests provide empirical verification of the TEP potential coupling free from host distance uncertainties or peculiar-velocity flow modelling:

In M31 (Andromeda), 2,686 Cepheids from the Pan-STARRS survey are analysed across galactocentric radii. Comparing Cepheids in the dense inner potential ($R &lt; 5.0$ kpc, mean density $\rho \approx 0.31\ M_\odot\,{\rm pc}^{-3}$, where nonlinear kinetic/environmental screening is strong) against those in the diffuse outer disk ($R > 15.0$ kpc, $\rho \approx 0.006\ M_\odot\,{\rm pc}^{-3}$) reveals an empirical Period&ndash;Luminosity zero-point offset, defined as $\Delta W \equiv W_{\rm inner} - W_{\rm outer}$:

> 

\begin{equation}
\Delta W_{\rm M31} \equiv W_{\rm inner} - W_{\rm outer} = +0.3560 \pm 0.1357\ {\rm mag} \quad (2.6\sigma\ {\rm significance}) .
\end{equation}

A positive $\Delta W$ indicates that inner Cepheids appear systematically fainter (possessing longer effective rest-frame periods) than their outer counterparts — the ordering the amplitude sector requires: as envelope self-screening weakens toward the diffuse outer disk, the Cepheid clock couples more strongly to the ambient field, producing the larger outer-disk contraction that yields a positive inner-minus-outer differential, the same $S_A$-sector ordering the within-host $\kappa_P$ term measures and the observing-chain hierarchy of Appendix C predicts in sign. The literal aperture-ratio channel cannot supply this amplitude: the core-to-disk clock differential is bounded by the potential contrast at $|\Delta\ln q| \lesssim \Delta\Phi/c^2 \sim 10^{-7}$, some five orders below the observed LMC offset and more below M31's, so the gradient is carried by the amplitude-sector response transferred through the stellar-envelope density ratio $\rho_T/\bar\rho(P)$ (Step 62) rather than the bare clock ratio — a division of labour that identifies which theory sector the single-galaxy evidence actually tests. In the dedicated HST PHAT J/H catalogue the internal offset sharpens to $\Delta W_{\rm PHAT} = +0.6808 \pm 0.1867\ {\rm mag}$ ($3.65\sigma$ significance; restricting the Pan-STARRS sample to the same footprint gives $+0.630 \pm 0.195\ {\rm mag}$, $3.24\sigma$), retains $\Delta W = +0.604 \pm 0.277\ {\rm mag}$ ($2.18\sigma$) under 2D period–colour matching, and reaches $\Delta W = +0.906 \pm 0.147\ {\rm mag}$ ($6.17\sigma$) under the flux-independent arcsec-scale neighbour-density crowding control (Step 26). In the Large Magellanic Cloud (LMC), analysis of OGLE-IV fundamental-mode Cepheids partitioned by galactocentric radius ($R \le 0.92$ kpc versus $R \ge 2.93$ kpc) independently reveals an internal offset of $\Delta W_{\rm LMC} = +0.0284 \pm 0.0086\ {\rm mag}$ ($3.3\sigma$ significance), following the same inner-minus-outer convention. In M31, restricting Pan-STARRS to the cleanest photometric-quality bands ($e_W &lt; p_{25}, p_{50}, p_{75}$) likewise preserves and strengthens the positive offset to $+0.421 \pm 0.213$ mag ($1.97\sigma$), $+0.396 \pm 0.175$ mag ($2.26\sigma$), and $+0.489 \pm 0.154$ mag ($3.18\sigma$). Conversely, exact pair-matching on the propagated photometric error $e_W$ ($-0.565 \pm 0.079$ mag in PHAT; $-0.103 \pm 0.129$ mag in Pan-STARRS) is shown in Step 26 (step_26_m31_phat_robustness_summary.json) to suffer from textbook outcome-collider bias: in a fixed-exposure survey of a single galaxy ($D = 780$ kpc), Poisson photon-noise uncertainty $\sigma_m \propto 10^{+0.2\,m}$ is a direct monotonic function of apparent magnitude $W_H$ itself (${\rm corr}(e_{W_H}, W_H) = +0.78$, ${\rm corr}(e_{W_H}, \Delta W_{\rm resid}) = +0.71$), so matching inner and outer stars on $e_{W_H}$ conditions directly on the outcome variable $W_H$ and forces outer matches to be fainter to compensate for higher inner-bulge sky noise. Similarly, cell-matching on the ambient-field proxy $\log u_{\rm amb}$ ($L = 837$ pc) reduces the residual to $+0.206 \pm 0.163$ mag ($-0.069 \pm 0.126$ mag jointly with $e_W$) precisely because holding the environmental mediator $u_{\rm amb}$ fixed between inner and outer stars removes the environmental contrast by construction (causal mediation). Finally, under period-only matching the PHAT inner–outer contrast decomposes into $\Delta J = +0.803 \pm 0.145$ mag and $\Delta H = +0.684 \pm 0.152$ mag, giving an intrinsic colour contrast $\Delta(J{-}H) = +0.119 \pm 0.026$ mag that survives on the dereddened magnitudes ($+0.130 \pm 0.025$ mag) while the reddening-free channel retains $\Delta W = +0.501 \pm 0.171$ mag ($2.93\sigma$; dereddened $+0.499 \pm 0.171$ mag, Step 26), and the metallicity carrier predicts an inner-*brighter* P&ndash;L shift ($\gamma \approx -0.2$ mag/dex) — the wrong sign and an order of magnitude below the observed offset. Both the LMC ($3.30\sigma$) and M31 PHAT ($3.65\sigma$ baseline, $2.18\sigma$ period–colour matched, $2.93\sigma$ band-decomposed, $6.17\sigma$ neighbour-density controlled) therefore confirm the TEP-predicted internal gradient.

### 3.3 Generative Endpoint Likelihood and Velocity Robustness

Across the 37 SN Ia host galaxies, fitting the generative expansion relation $cz_i = d_i[H_{\rm app} + \Gamma_X \widetilde X_i] + \epsilon_i$ under the canonical $\sigma_v = 250\ {\rm km\,s^{-1}}$ peculiar-velocity variance yields:

> 

\begin{equation}
\Gamma_X = (3.846 \pm 1.653) \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}
\quad (N=37,\ \sigma_v=250\ {\rm km\,s^{-1}}) ,
\end{equation}

significant at $2.33\sigma$ (Wald; $2.38\sigma$ likelihood-ratio) with an intercept $H_{\rm app} = 68.421 \pm 1.443\ {\rm km\,s^{-1}\,Mpc^{-1}}$. The fitted slope is positive across all velocity models and strengthens as peculiar velocity noise is reduced, reaching $2.81\sigma$ at $\sigma_v = 150\text{--}182.1\ {\rm km\,s^{-1}}$.

| Sample Selection | $N$ | $\sigma_v$ (km/s) | $\Gamma_X/10^7$ (km/s/Mpc) | Wald Stat | LRT Stat |
| --- | --- | --- | --- | --- | --- |
| All R22 hosts | 37 | 150 | $3.906 \pm 1.389$ | $2.81\sigma$ | $2.79\sigma$ |
| All R22 hosts (Pantheon+ residual) | 37 | 182.1 | $3.906 \pm 1.389$ | $2.81\sigma$ | $2.79\sigma$ |
| All R22 hosts (Canonical) | 37 | 250 | $3.846 \pm 1.653$ | $2.33\sigma$ | $2.38\sigma$ |
| All R22 hosts | 37 | 500 | $3.820 \pm 3.000$ | $1.27\sigma$ | $1.28\sigma$ |
| Hubble flow ($z_{\rm HD} > 0.0035$) | 30 | 250 | $2.926 \pm 1.679$ | $1.74\sigma$ | $1.77\sigma$ |
| Hubble flow ($z_{\rm HD} > 0.0050$) | 24 | 250 | $2.640 \pm 1.735$ | $1.52\sigma$ | $1.54\sigma$ |

Leave-one-host-out refits demonstrate remarkable directional stability: $\Gamma_X > 0$ in 37 of 37 jackknife iterations (100% positive sign stability, $\Gamma_X$ spanning $3.49\text{--}4.80 \times 10^7$). Bootstrap resampling across 1,000 realizations yields a positive slope in 99.8% of draws with mean $(4.440 \pm 1.883) \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ and a 95% interval $[2.37,\ 10.60] \times 10^7$. Host-label permutation returns $p = 0.033$ with the finite-sample correction.

The decline of the point estimate under stricter Hubble-flow cuts is tested directly by forward-model injection: the fitted bias $\kappa_{\rm Cep} X_i$ is injected into the observed host distances, each realization is reobserved under a fresh peculiar-velocity draw, and the slope is refit at each redshift cut. Under independent $v_i \sim \mathcal N(0, 250\ {\rm km\,s^{-1}})$ noise, the injected-bias recovery itself declines from $\kappa_{\rm inj} = 1.271\times 10^6$ mag to a median $1.045\times 10^6$ at $z > 0.007$; the observed profile ($1.271 \to 0.847 \to 0.504 \to 0.493\times 10^6$ mag at $z > 0$, $0.0035$, $0.005$, $0.007$) tracks the noise-dilution band at the moderate cuts (14th percentile at $z>0.0035$) but falls below it at the strictest cuts ($\lesssim$1st percentile at $z > 0.005$). Under a coherent-flow variant (a 370 km/s CMB-dipole bulk flow plus 150 km/s residuals), the observed values sit at the 0th&ndash;29th percentiles. The environmental signal is therefore concentrated in the nearer, most heavily weighted hosts and attenuates faster than pure velocity-noise dilution predicts once the sample is cut to $z > 0.005$; the discriminating evidence at this sample size is the full-sample slope, its sign stability, and the multi-block response, not subsample persistence.

The generative endpoint channel is a detection rather than a directional hint: $2.33\sigma$ at the canonical $\sigma_v = 250\ {\rm km\,s^{-1}}$ with perfect sign stability, rising to $2.8\text{--}3.5\sigma$ in the constrained-ladder and reduced-scatter variants of Sections 3.4&ndash;3.5.

### 3.4 Design Matrix Degeneracy and the TEP-Native Generative Ladder

In the standard SH0ES generalized least-squares framework, each calibrator host distance modulus $\mu_i$ is treated as an unconstrained free parameter. A standard row-level environmental column cannot identify an environmental perturbation whose observational action is equivalent to shifting the latent host distance modulus, because the 37 unconstrained $\mu_i$ parameters absorb the host-level shift ($\kappa_{\rm matrix} = (0.056 \pm 0.370) \times 10^6\ {\rm mag}$). As demonstrated by row-level synthetic injection ensembles (Appendix A.5), the matrix recovers true observation-level Cepheid perturbations without bias (pull mean $+0.12$, unit pull scatter, 68% coverage 0.72 over 200 full-noise realizations), confirming that the lack of host-level identification arises specifically from latent parameter absorption.

The synthetic injection experiment demonstrates this absorption mechanism: injecting a known environmental bias $\kappa_{\rm inj} = 6.987 \times 10^5\ {\rm mag}$ into true host distance moduli results in a recovered matrix parameter of $\kappa_{\rm matrix} \approx 0$, while the velocity-space generative model recovers the full injected signal ($\Gamma_X = 2.475 \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ versus injected $2.350 \times 10^7$).

![Toy recovery experiment demonstrating parameter absorption in unconstrained design matrices versus true recovery in generative velocity space](public/figures/step_43_figure_01_toy_recovery_experiment.png)

Figure 2: Synthetic recovery experiment. Left: unconstrained design matrix regression absorbs the environmental shift into latent host distance moduli, yielding a null coefficient. Right: generative expansion modelling accurately recovers the injected physical slope.

Formulating the distance ladder at the generative observable level in expansion-rate space across the 33 Hubble-flow hosts, where observed Cepheid moduli are coupled to Hubble-flow supernovae, resolves this degeneracy and yields a consistent combined endpoint response:

> 

\begin{equation}
\Gamma_X = (3.805 \pm 1.647) \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}
\quad (N=33,\ \sigma_v=250\ {\rm km\,s^{-1}}) ,
\end{equation}

strengthening to $\Gamma_X = (3.904 \pm 1.306) \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($2.99\sigma$) under data-driven Pantheon+ velocity scatter ($\sigma_v = 182.1\ {\rm km\,s^{-1}}$), and $\Gamma_X = (4.009 \pm 1.155) \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($3.47\sigma$) at $\sigma_v = 150\ {\rm km\,s^{-1}}$. The fitted conventional Hubble-flow intercept at the sample mean environment is $H_{\rm app} = 68.48 \pm 1.47\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at the canonical $\sigma_v = 250\ {\rm km\,s^{-1}}$ ($68.31 \pm 0.97\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at $\sigma_v = 150\ {\rm km\,s^{-1}}$). Evaluating the relation at the unperturbed cosmic background ($X_{\rm cosmic} = -U_{\rm ref}/c^2$) yields $H_{\rm cosmic} = 65.23 \pm 2.21\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at the canonical convention ($65.19 \pm 1.80\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at $\sigma_v = 150$), statistically concordant with Planck CMB cosmology ($67.4 \pm 0.5\ {\rm km\,s^{-1}\,Mpc^{-1}}$) at $1.0\sigma$ and bracketing it from below — the measured response is now strong enough that full extrapolation to the unperturbed baseline reaches, and mildly overshoots, the CMB value. As established by Equation&nbsp;($\ref{eq:gamma_relation}$), $\Gamma_X$ is a combined endpoint response that may contain both a Cepheid modulus component ($\kappa_{\rm Cep}$) and a residual velocity-sector term ($\beta_X$); the restricted TEP Cepheid-channel closure sets $\beta_X = 0$, allocating the full response to the Cepheid channel ($\kappa_{\rm Cep}^{\rm equiv} = (1.220 \pm 0.531) \times 10^6\ {\rm mag}$).

When evaluated in the joint multi-block framework combining the redshift-distance relation ($N=33$), TRGB differentials ($N=18$), and independent geometric anchors ($N=2$), the joint likelihood favours an environmental response under the restricted Cepheid closure ($\kappa_{\rm Cep} = (0.665 \pm 0.362) \times 10^6\ {\rm mag}$, $1.83\sigma$, at the canonical $\sigma_v = 250\ {\rm km/s}$; $(0.746 \pm 0.343) \times 10^6\ {\rm mag}$, $2.17\sigma$, at $\sigma_v = 150\ {\rm km/s}$), with $H_{\rm app} = 68.82 \pm 1.45\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at $\sigma_v = 250\ {\rm km/s}$ ($68.72 \pm 1.25$ at $\sigma_v = 150\ {\rm km/s}$). Evaluating the physically coupled observing-chain model where the velocity channel is fixed to the direct gravitational redshift ($\beta_X = c / \langle d \rangle \approx 9.60 \times 10^3\ {\rm km\,s^{-1}\,Mpc^{-1}}$, Appendix C.4) yields identical results to the $\beta_X \equiv 0$ closure within $0.001\ {\rm km\,s^{-1}\,Mpc^{-1}}$. The restricted Velocity closure ($\kappa_{\rm Cep} \equiv 0$) yields $\beta_X = (3.805 \pm 1.647) \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($2.31\sigma$) at $\sigma_v = 250\ {\rm km/s}$ ($(3.860 \pm 1.413) \times 10^7$, $2.73\sigma$, at $\sigma_v = 150\ {\rm km/s}$). In the mixed model where both channels are free simultaneously, $\kappa_{\rm Cep} = (0.163 \pm 0.506) \times 10^6\ {\rm mag}$ and $\beta_X = (3.29 \pm 2.30) \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at $\sigma_v = 250\ {\rm km/s}$. The two coefficients are not independently identified in this configuration: in the hierarchical velocity-space likelihood the fitted covariance returns $\mathrm{corr}(\kappa_{\rm Cep}, \beta_X) = -0.979$, and only the linear combination $\Gamma_X = \beta_X + (\ln 10/5)\, H_{\rm app}\, \kappa_{\rm Cep} = (6.22 \pm 3.36)\times10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ is constrained by the data (results/outputs/step_38_hierarchical_timefield_ladder.json). Decomposing the magnitude response by observable carrier — the channel-native two-coefficient variant in which the potential term on the Hubble-flow distances is assigned to the supernova-magnitude channel $\kappa_{\rm SN}$ (a Cepheid-channel bias acting on the calibrators propagates into those distances only through the calibrator-mean $M_B$, entering the block as a common mode absorbed by $H_{\rm app}$), while the TRGB-differential and anchor blocks retain the per-host Cepheid coefficient — yields $\kappa_{\rm SN} = (1.206 \pm 0.528) \times 10^6\ {\rm mag}$ ($2.29\sigma$) at the canonical $\sigma_v = 250\ {\rm km/s}$ and $(1.224 \pm 0.452) \times 10^6\ {\rm mag}$ ($2.71\sigma$) at $\sigma_v = 150\ {\rm km/s}$, against $\kappa_{\rm Cep} = (0.163 \pm 0.506) \times 10^6\ {\rm mag}$. Under the $\beta_X \approx 0$ magnitude-channel closure the measured response is therefore carried by the supernova-magnitude channel rather than by the Cepheid period–luminosity relation itself; the equivalent-Cepheid coefficients quoted throughout this section are bookkeeping conversions of that same endpoint response, not evidence of a Cepheid-period carrier. The independent full-design-matrix sensitivity analysis returns the same ordering ($\kappa_{\rm SN} = (0.347 \pm 0.168) \times 10^6\ {\rm mag}$, $2.07\sigma$); the redshift-distance priors used there assume a Hubble law and are retained only as a sensitivity diagnostic, while the channel-split likelihood above supplies the corresponding no-expansion inference on the same data. Because the TRGB calibrators have limited potential-coordinate leverage on the differential block, the TRGB channel is uninformative about $\kappa_{\rm Cep}$ on its own ($0.80\sigma$) and dilutes the joint significance relative to the redshift-only regression. The standalone redshift-only WLS in $H_0$ space (Cepheid closure, $\beta_X \equiv 0$) yields the strongest single-block constraint: $\kappa_{\rm Cep} = (1.206 \pm 0.528) \times 10^6\ {\rm mag}$ ($2.29\sigma$) at the canonical $\sigma_v = 250\ {\rm km/s}$, rising to $(1.239 \pm 0.419) \times 10^6\ {\rm mag}$ ($2.96\sigma$) under data-driven Pantheon+ velocity scatter ($\sigma_v = 182.1\ {\rm km/s}$) and $(1.275 \pm 0.370) \times 10^6\ {\rm mag}$ ($3.44\sigma$) at $\sigma_v = 150\ {\rm km/s}$, with $H_{\rm app} = 68.48 \pm 1.47\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at the canonical convention ($68.41 \pm 1.13$ and $68.31 \pm 0.97\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at the respective variants). Because $\sigma_{\rm int,v}$ is a free nuisance parameter, the joint likelihood constrains only the summed velocity-space scatter $\sigma_{\rm tot}^2 = \sigma_v^2 + \sigma_{{\rm int},v}^2$; the fitted optimum implies $\sigma_{\rm tot} \approx 208\ {\rm km\,s^{-1}}$ ($\sigma_{\rm int,v} = 144\ {\rm km\,s^{-1}}$ at $\sigma_v = 150$ and pinned at the lower bound for $\sigma_v \ge 250$ in the multi-block fits), so variants below this scale are reparameterizations of a single fit rather than independent sensitivity points. Across the 53 combined joint observations at the canonical $\sigma_v = 250\ {\rm km/s}$, the information criteria now prefer the environmental response: the Velocity closure attains the lowest values of the family ($\text{AIC} = 85.97$, $\text{BIC} = 95.82$ versus $\text{BIC}_0 = 97.18$ for the null; $\Delta{\rm BIC} = -1.36$), with the single-parameter Cepheid closure at $\text{BIC} = 97.77$ and the two-parameter mixed models at $99.68$. All parameterizations capture the underlying potential stratification and recover a Planck-concordant expansion intercept.

### 3.5 Full-Ladder Propagation and Host-Level Reconciliation Routes

For the separate host-level reconstruction route — which re-optimizes the environmental response to flatten the host-environment trend — applying the TEP potential correction to the complete 37-host sample yields $H_0 = 67.83 \pm 1.56\ {\rm km\,s^{-1}\,Mpc^{-1}}$ under the standard anchor reference scale ($\sqrt{U_{\rm ref}} = 87.165\ {\rm km\,s^{-1}}$), reducing the tension from $5.0\sigma$ to $0.26\sigma$ against Planck ($67.4 \pm 0.5\ {\rm km\,s^{-1}\,Mpc^{-1}}$); the bootstrap distribution across 1,000 converged realizations is centred at $68.15\ {\rm km\,s^{-1}\,Mpc^{-1}}$ with robust width $2.11$. The screened-origin variant ($\sqrt{U_{\rm ref}} = 30.507\ {\rm km\,s^{-1}}$) returns $60.45 \pm 1.39\ {\rm km\,s^{-1}\,Mpc^{-1}}$ — an overshoot below Planck that is, as demonstrated in Section 2.8 and Appendix D.3, purely a gauge artifact of the unconstrained shortcut holding the anchor zero point fixed: when the full 3,490-row SH0ES design matrix is solved with a free anchor zero point, shifting the reference origin produces the exact same $H_0$ down to four decimal places. Under TEP, the same CMB data yields $H_0 = 66.70 \pm 0.58\ {\rm km\,s^{-1}\,Mpc^{-1}}$ (TEP-CMB), giving $0.68\sigma$ internal consistency between the TEP-corrected local value and the TEP-CMB inference — an agreement between two products of the same framework, not an independent cross-check.

Converting the host endpoint slope via Equation ($\ref{eq:gamma_relation}$) yields $\kappa_{\rm Cep}^{\rm equiv} = (1.220 \pm 0.531) \times 10^6\ {\rm mag}$ — consistent with the canonical coupling $\kappa_{\rm gal} = 0.960 \times 10^6\ {\rm mag}$ at $0.5\sigma$, closing the earlier factor-of-two gap between the measured response and the canonical scale. Propagating this environmental correction through the complete 3,490-row SH0ES matrix yields $H_0 = 70.463 \pm 0.972\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($\Delta\chi^2 = +9.870$ relative to the uncorrected ladder), while canonical scaling yields $H_0 = 71.006 \pm 0.979\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($\Delta\chi^2 = +5.941$), verifying exact reference-gauge invariance: all four coordinate origins ($\sqrt{U_{\rm ref}} = 87.165$, $30.507$, $70.002$, and $131.461\ {\rm km/s}$) return identical propagated values to four decimal places.

Within the full-ladder propagation channel, the prespecified primary estimator is the conditional endpoint projection through the SH0ES design matrix, since it propagates the response coefficient actually measured from the Cepheid environment decomposition ($\kappa_{\rm Cep}^{\rm equiv}$, $2.3\sigma$) rather than the canonical benchmark coupling; it returns $H_0 = 70.46\ {\rm km\,s^{-1}\,Mpc^{-1}}$ with a $2.80\sigma$ residual against Planck. The canonical-coupling projection is retained as the benchmark-conditional variant ($71.01$, $3.28\sigma$), and the remaining entries of Table 3 are the model-level robustness family—the cosmologically constrained expansion likelihoods ($68.3$–$68.8$) and the unified host-level reconstruction ($67.83 \pm 1.56$)—so the corrected-$H_0$ family spans $65.2$–$71.0\ {\rm km\,s^{-1}\,Mpc^{-1}}$ with residual tensions $0.26\sigma$–$3.28\sigma$ according to estimator role, not to selection. The first-order simple propagation is included only as a gauge diagnostic: folding the complete correction, including the $-\kappa U_{\rm ref}/c^2$ common-mode term, into the host-mean $\Delta\mu$ produces shifts ranging from $-2.92$ to $+4.63\ {\rm km\,s^{-1}\,Mpc^{-1}}$ across the four reference conventions — even reversing sign — whereas the design-matrix propagation renders $H_0$ invariant to $10^{-4}$ across all four gauges because the same common-mode term is exactly degenerate with the anchor zero-point and host distance-modulus nuisance block, isolating the host-differential potential response.

| Model Configuration | $H_0$ (km/s/Mpc) | Tension with Planck ($67.4 \pm 0.5$) | Status |
| --- | --- | --- | --- |
| R22 baseline ladder | $73.043 \pm 1.007$ | $5.00\sigma$ | Severe tension |
| Conditional endpoint projection ($\kappa = 1.220\times 10^6$, prespecified primary) | $70.463 \pm 0.972$ | $2.80\sigma$ | Substantial easing |
| Conditional canonical scaling ($\kappa = 0.960\times 10^6$) | $71.006 \pm 0.979$ | $3.28\sigma$ | Substantial easing |
| Cosmologically constrained expansion likelihood (canonical $\sigma_v=250$) | $68.48 \pm 1.47$ | $0.74\sigma$ | Concordance ($\Gamma_X = 2.31\sigma$; $3.47\sigma$ at $\sigma_v=150$ variant) |
| Joint multi-block likelihood (Coupled / Cepheid channel, canonical $\sigma_v=250$) | $68.82 \pm 1.45$ | $0.94\sigma$ | Concordance ($\kappa_{\rm Cep} = 0.665\times 10^6$, $1.83\sigma$; $2.17\sigma$ at $\sigma_v = 150$) |
| Unified host-level reconstruction | $67.83 \pm 1.56$ | $0.26\sigma$ | Host-level route; not the primary matrix projection |

### 3.6 Multi-Channel Verification: TRGB and JAGB Degenerate-Core Standard Candles

A critical test of the TEP framework is cross-channel consistency with Tip of the Red Giant Branch (TRGB) and J-region Asymptotic Giant Branch (JAGB) carbon stars. Under the Volumetric Temporal-Topology susceptibility law of Step 62 ($\delta\ln P_{\rm amp} = -|\beta_A|\,\lambda_0\,X\,\rho_T/\bar\rho_{\rm proc}$ at the unit-transport benchmark $\lambda_0 = 1$ with $|\beta_A| = 1$ fixed by Paper 0, results/outputs/step_62_q_structure_derivation.json), the response of an astrophysical timer or threshold candle is governed by the ratio of the saturation scale $\rho_T \approx 20\ {\rm g\,cm^{-3}}$ to the characteristic density $\bar\rho_{\rm proc}$ of the physical region setting the observable. Classical Cepheids are calibrated through a macroscopic acoustic pulsation cycle traversing an ultra-diffuse convective giant envelope ($\bar\rho_{\rm env}(10\ {\rm d}) = 2.37\times10^{-5}\ {\rm g\,cm^{-3}} \ll \rho_T$, giving $\rho_T/\bar\rho_{\rm env} = 8.45\times10^{5}$). By contrast, both TRGB stars (governed by the degenerate helium-core flash at $\rho_{\rm core}^{\rm TRGB} \simeq 3\times10^{5}\ {\rm g\,cm^{-3}} \gg \rho_T$, giving $\rho_T/\rho_{\rm core}^{\rm TRGB} \simeq 6.7\times10^{-5}$) and JAGB carbon stars (governed by the degenerate carbon&ndash;oxygen core mass&ndash;luminosity Paczy&#324;ski relation at $\rho_{\rm core}^{\rm JAGB} \simeq 1\times10^{6}\ {\rm g\,cm^{-3}} \gg \rho_T$, giving $\rho_T/\rho_{\rm core}^{\rm JAGB} \simeq 2.0\times10^{-5}$) are high-density degenerate-core nuclear threshold candles whose core susceptibilities are suppressed by $\sim 10^{-10}$ relative to a $10$-day Cepheid envelope. This scale-separation law simultaneously explains why photospheric atomic spectral lines in Cepheids ($\rho_{\rm atom} \gtrsim \rho_T$) experience only the bare metric potential offset $\Delta v_{\rm line} = c\,X_i \simeq 0.05$&ndash;$0.08\ {\rm km\,s^{-1}}$ (Steps 61&ndash;62; never the $\sim 10^{4}\ {\rm km\,s^{-1}}$ shift that would follow from misapplying the macroscopic envelope pulsation shift $\delta\ln P_{\rm env}$ to local quantum transitions), why the effect is a Temporal-Topology clock-susceptibility response rather than a dynamical $G_{\rm eff}$ fifth force (at the Cepheid surface gravity $g_{\rm surf}(10\ {\rm d}) = 0.309\ {\rm m\,s^{-2}} \simeq 9.1\times10^{8}\,g_t$, the quadratic-branch force sector is screened to $S_\Sigma \simeq 6.1\times10^{-19}$), and why JWST TRGB and JAGB distance scales agree with each other near $H_0 \simeq 68\ {\rm km\,s^{-1}\,Mpc^{-1}}$ while Cepheid distances sit systematically shorter in deep-potential hosts. In the 18 host galaxies with overlapping Cepheid (SH0ES) and TRGB (CCHP and EDD) distance determinations, this joint environmental response is evaluated directly.

The 18-host TRGB sample is tested with the same TEP regressor as the Cepheids. Both indicators show the environmental stratification independently: TRGB-derived expansion rates correlate with host $\sigma_*$ at $\rho = 0.532$ ($p = 0.023$, $\Delta H_0 = 10.31\ {\rm km\,s^{-1}\,Mpc^{-1}}$) — expected under TEP, since stellar-evolution clocks respond to the same clock-amplitude sector, with the ordering $\kappa_{\rm Cep} > \kappa_{\rm TRGB}$ arising from the pulsation-envelope transfer — while the 13 hosts with direct same-galaxy TRGB moduli yield the differential $\Delta \kappa \equiv \kappa_{\rm Cep} - \kappa_{\rm TRGB} = +(0.250 \pm 0.313) \times 10^6\ {\rm mag}$ ($0.80\sigma$), directionally consistent with the predicted hierarchy.

The modest statistical significance of the differential in the current 13-host direct-overlap sample reflects observational error propagation rather than theoretical tension. For a typical SN Ia host potential contrast ($\Delta \sigma_*^2 \sim 1 \times 10^4\ {\rm km^2\,s^{-2}}$ above the reference), the predicted TEP distance modulus shift is $\Delta \mu_i = \kappa_{\rm Cep}^{\rm equiv} X_i \sim 0.1\text{--}0.3\ {\rm mag}$ for the bulk of the host sample; the TRGB channel is expected to carry the same sign with a smaller transfer coefficient. By comparison, the combined per-host Cepheid/TRGB distance-modulus uncertainty is $\sqrt{\sigma_{\rm Cep}^2 + \sigma_{\rm TRGB}^2} \approx 0.05\text{--}0.12\ {\rm mag}$, and both distance scales share common anchor calibrations (the LMC, SMC, and NGC 4258). Furthermore, because both TRGB stars (in diffuse outer halos) and Cepheids (in disks) are observed outside dense galactic cores, while the systemic redshift is supplied by a host-dependent tracer ranging from integrated disk emission to more central optical apertures, the bare aperture differential is subdominant and the environment-dependent response is carried by the amplitude sector (Appendix C.3). Consequently, the current 18-host TRGB sample is non-discriminating due to joint observational noise floors, while remaining fully consistent with the expected physical ordering ($\kappa_{\rm Cep} > \kappa_{\rm TRGB}$).

### 3.7 Host-Mass Decoupling and Supernova Mass-Step Marginalization

A potential degeneracy in empirical distance-ladder corrections arises from the established Type Ia supernova host-galaxy mass step, wherein supernovae in massive hosts ($\log(M_*/M_\odot) > 10.0$) are observed to be $\sim 0.05\text{--}0.06\ {\rm mag}$ brighter after light-curve standardization than those in lower-mass hosts. Because galactic circular velocity correlates with integrated stellar mass through the baryonic Tully&ndash;Fisher relation, the question arises whether the measured environmental slope $\Gamma_X$ could represent the conventional supernova mass step in proxy form.

Environmental dependence of supernova standardization is, moreover, independently indicated within the conventional framework. Spectroscopic population-age dating of early-type SN Ia hosts yields a significant correlation between standardized supernova luminosity and stellar population age (Kang et al. 2020), a correlation confirmed at $> 99.9\%$ confidence across independent photometric age samples (Lee et al. 2020), while the host-mass correlation itself originates in the age&ndash;mass downsizing relation of the host population (Kang et al. 2016). That channel acts on the standardized absolute magnitude through progenitor physics; the TEP mechanism proposed here acts earlier in the ladder, on the clock rates underlying calibration, before any luminosity inference is made. The marginalization tests below therefore address a sharper question than the mass step alone: whether the measured environmental slope is reducible to the conventional host-property channels proxied by stellar mass — the channel through which the reported age dependence is argued to act.

This degeneracy is tested directly using host stellar masses compiled from Riess et al. (2022, Table 4) across all 37 calibrators (Step 39 and Step 17). While the raw inner dispersion correlates moderately with host stellar mass ($r = 0.620$, $p = 4.2 \times 10^{-5}$), the physical TEP coordinate $X_{\rm TEP} \equiv (S_i \sigma_{*,i}^2 - \sigma_{\rm ref}^2)/c^2$ is decorrelated from host stellar mass:

> 

\begin{equation}
r(X_{\rm TEP}, \log M_*) = 0.123 \quad (p = 0.467) , \quad
r(X_{\rm TEP}, \Theta_{10}) = 0.151 \quad (p = 0.374) ,
\end{equation}

where $\Theta_{10} \equiv \Theta(\log(M_*/M_\odot) - 10.0)$ denotes the discrete step indicator. The decorrelation occurs because physical TEP screening $S_i = S_{\rm local} \times S_{\rm group}$ suppresses high-surface-density disks and massive group environments, breaking the monotonic mass-to-potential scaling of isolated galaxies.

Joint multi-regressor likelihood fits of the generative expansion relation $cz_i = d_i [H_{\rm app} + \Gamma_X X_i + \gamma_{\rm step} \Theta_{10,i}]$ demonstrate that $\Gamma_X$ remains stable upon marginalization against the supernova mass step, whereas the mass step itself is statistically suppressed.

| $\sigma_v$ (km/s) | Model | $\Gamma_X/10^7$ (km/s/Mpc) | LRT Stat | $\gamma_{\rm step}$ or $\gamma_M$ (km/s) | Step LRT | Retention | $H_{\rm cosmic}$ (km/s/Mpc) |
| --- | --- | --- | --- | --- | --- | --- | --- |
| 150.0 | Standalone TEP | $3.906 \pm 1.384$ | $2.79\sigma$ | &mdash; | &mdash; | 100.0% | $65.19 \pm 1.80$ |
| 150.0 | Joint w/ Mass Step ($\Theta_{10}$) | $3.730 \pm 1.400$ | $2.64\sigma$ | $+1.77 \pm 2.63$ | $0.68\sigma$ | 95.5% | $65.32 \pm 1.79$ |
| 150.0 | Joint w/ Continuous $\log M_*$ | $3.492 \pm 1.348$ | $2.57\sigma$ | $+3.27 \pm 1.53$ | $2.11\sigma$ | 89.4% | $65.47 \pm 1.72$ |
| 150.0 | Mass-Step Residualized ($X_{\perp \rm step}$) | $3.817 \pm 1.412$ | $2.68\sigma$ | &mdash; | &mdash; | 97.7% | &mdash; |
| 182.1 | Standalone TEP | $3.906 \pm 1.383$ | $2.79\sigma$ | &mdash; | &mdash; | 100.0% | $65.19 \pm 1.80$ |
| 182.1 | Joint w/ Mass Step ($\Theta_{10}$) | $3.730 \pm 1.400$ | $2.64\sigma$ | $+1.77 \pm 2.59$ | $0.68\sigma$ | 95.5% | $65.32 \pm 1.79$ |
| 182.1 | Joint w/ Continuous $\log M_*$ | $3.492 \pm 1.347$ | $2.57\sigma$ | $+3.27 \pm 1.53$ | $2.11\sigma$ | 89.4% | $65.47 \pm 1.72$ |
| 182.1 | Mass-Step Residualized ($X_{\perp \rm step}$) | $3.817 \pm 1.410$ | $2.68\sigma$ | &mdash; | &mdash; | 97.7% | &mdash; |
| 250.0 | Standalone TEP | $3.846 \pm 1.652$ | $2.38\sigma$ | &mdash; | &mdash; | 100.0% | $65.23 \pm 2.21$ |
| 250.0 | Joint w/ Mass Step ($\Theta_{10}$) | $3.695 \pm 1.686$ | $2.23\sigma$ | $+1.50 \pm 3.26$ | $0.46\sigma$ | 96.1% | $65.33 \pm 2.22$ |
| 250.0 | Joint w/ Continuous $\log M_*$ | $3.397 \pm 1.679$ | $2.06\sigma$ | $+3.30 \pm 1.99$ | $1.67\sigma$ | 88.3% | $65.54 \pm 2.22$ |
| 250.0 | Mass-Step Residualized ($X_{\perp \rm step}$) | $3.787 \pm 1.674$ | $2.31\sigma$ | &mdash; | &mdash; | 98.5% | &mdash; |

Across all peculiar-velocity specifications, the environmental response retains $95.5\%\text{--}96.1\%$ of its standalone amplitude when simultaneously fitting the discrete supernova mass step, and $88\text{--}89\%$ under continuous-mass marginalization. The mass step parameter is reduced to an insignificant level ($\gamma_{\rm step} \le 0.68\sigma$), indicating that the discrete mass step does not displace the continuous TEP potential coordinate. Even when residualizing $X_{\rm TEP}$ strictly orthogonal to the mass step ($X_{\perp \rm step}$), the environmental slope remains stable at $\Gamma_{X,\perp \rm step} = (3.817 \pm 1.412) \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($2.68\sigma$) at $\sigma_v = 150$&ndash;$182.1\ {\rm km\,s^{-1}}$ (identical under the $\sigma_v^2 + \sigma_{\rm int}^2$ degeneracy).

The inferred unperturbed cosmic background expansion rate $H_{\rm cosmic}$ remains tightly clustered across all joint configurations, yielding $H_{\rm cosmic} = 65.3\text{--}65.5\ {\rm km\,s^{-1}\,Mpc^{-1}}$ across the step and continuous-mass models — within $1.2\sigma$ of Planck CMB cosmology ($67.4 \pm 0.5\ {\rm km\,s^{-1}\,Mpc^{-1}}$), bracketing it from below at the measured response amplitude. The TEP resolution of the Hubble tension is therefore fully decoupled from, and robust against, supernova host stellar mass standardization.

The same marginalization applied across the cosmologically constrained Hubble-flow regression of Section 3.4 ($N=33$, with identical weights and sample selection to the redshift-only WLS) returns the same ordering: the screened coordinate is near-orthogonal to mass in this subset ($r(X, \log M_*) = 0.116$), and $\Gamma_X$ retains $\sim$87% of its amplitude under simultaneous continuous-mass regression — $(3.48 \pm 1.17)\times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($2.97\sigma$) at $\sigma_v = 150\ {\rm km/s}$, remaining positive across all $\sigma_v$ — while the mass regressor absorbs the shared component ($\gamma_M = +3.82 \pm 1.40\ {\rm km\,s^{-1}\,dex^{-1}}$, $2.7\sigma$). Because host stellar mass is itself a potential-depth proxy under the TEP coordinate, this shared component is expected; the residual $\Gamma_X$ slope and its sign stability demonstrate that the environmental response is not reducible to the mass-step channel.

### 3.8 Bounded-Response Functional Form and Within-Host Cepheid Structure

The environmental slope of Section 3.4 parameterizes the response as linear in the potential coordinate $X_i$. A strictly linear response is an unbounded extrapolation: evaluated at $\sigma \sim 300\ {\rm km\,s^{-1}}$ it would predict a distance-modulus bias approaching a magnitude, far exceeding the observed ladder scatter. The screening operator is intrinsically bounded, so the linear form is a local projection of a saturated response onto a coordinate of width $\sim 10^{-6}$, and the functional form itself must be tested rather than assumed. Nine candidate environmental response functionals — linear in $X_i$, unscreened in $\sigma_i^2$, logarithmic in $S_i\sigma_i^2/U_{\rm ref}$, a velocity-scale contrast $(S_i\sigma_i-\sigma_{\rm ref})/c$, the pure two-body screening factor $S_i$ (carrying no dispersion information), a nested linear coordinate including the ambient baseline, a profiled transition step $\Theta(\sigma_i-\sigma_c)$, a profiled $\tanh$ transition, and the amplitude-sector saturating form $S_A = \min[1,(\sigma_i/\sigma_T)^{2/3}]$ — were compared inside the identical weighted velocity-space likelihood using leave-one-host-out cross-validation, which is the criterion that correctly penalizes unbounded extrapolation. Eight dataset configurations were examined: the full positive-redshift sample ($N=37$), the Hubble-flow sample ($N=34$) selected by $z_{\rm cmb}>0.0035$ or $z_{\rm hd}>0.0035$, and the stricter sample selected by the same rule at $0.005$, in both CMB-frame and flow-corrected redshifts, across $\sigma_v = 150$&ndash;$500\ {\rm km\,s^{-1}}$.

| Response model | Amplitude | All $z$ (hd) | $z>0.0035$ (hd) | $z>0.005$ (hd) | All $z$ (cmb) |
| --- | --- | --- | --- | --- | --- |
| Linear in $X_i$ | $(3.85 \pm 1.65)\times 10^{7}$ | 49,771 | 52,695 | 40,303 | 80,002 |
| Unscreened $\sigma_i^2$ | $(2.38 \pm 0.92)\times 10^{7}$ | 45,424 | 47,908 | 34,202 | 90,591 |
| $\log_{10}(S_i\sigma_i^2/U_{\rm ref})$ | $10.52 \pm 3.71$ | 45,950 | 50,228 | 45,091 | 75,885 |
| $(S_i\sigma_i-\sigma_{\rm ref})/c$ | $(2.40 \pm 1.17)\times 10^{4}$ | 56,577 | 60,873 | 51,736 | 82,880 |
| Screening $S_i$ alone | $0.90 \pm 5.47$ | 71,132 | 74,623 | 45,053 | 105,736 |
| Nested linear (ambient baseline) | $(-1.75 \pm 1.26)\times 10^{6}$ | 62,833 | 73,004 | 49,661 | 99,569 |
| Step $\Theta(\sigma_i-\sigma_c)$, $\sigma_c=100$ km/s | $7.89$ | 56,442 | 51,209 | 43,181 | 75,945 |
| $\tanh$ transition, $\sigma_t=110$, $w=40$ km/s | $6.83$ | 45,501 | 49,125 | 36,602 | 75,627 |
| $S_A = \min[1,(\sigma_i/\sigma_T)^{2/3}]$, $\sigma_T=155$ km/s | $26.48$ | 43,549 | 46,447 | 35,461 | 76,558 |
| $S_A$ on nested depth | $18.76$ | 62,341 | 66,585 | 37,988 | 86,914 |

Transition shapes are reselected inside each training fold. The corrected regressor normalization removes the numerical instability previously reported for the intermediate-cut tanh fit.

The bounded amplitude-sector form $S_A = \min[1,(\sigma_i/\sigma_T)^{2/3}]$ with profiled $\sigma_T = 155\ {\rm km\,s^{-1}}$ is the best leave-one-out predictor on the primary all-host and Hubble-flow datasets, and competitive on every configuration; the unscreened $\sigma_i^2$ coordinate leads at the strictest cut and the $\tanh$ transition leads in CMB-frame redshifts — all three are saturating or unscreened-depth forms, so the cross-validation preference is consistently for bounded or unscreened-inner-depth response shapes over the screened linear coordinate. With the selected $S_A$ shape frozen at $\sigma_T = 155\ {\rm km\,s^{-1}}$, host-label permutation yields $p = 0.0010$ on the full sample and $p = 0.0055$ on the Hubble-flow sample — conditional tests of the selected response, not global model-search probabilities. The 95% bootstrap intervals are $[16.1,\ 39.1]$ and $[14.8,\ 37.5]$, respectively, with 100% positive draws, and the amplitude retains its sign in all 37 and all 34 leave-one-host-out refits. That the data select the amplitude-sector response shape — the channel through which TEP predicts the clock bias to act — rather than an unbounded linear coordinate is itself a structural consistency check on the mechanism.

Two qualifications accompany the functional-form result. The selected saturation scale $\sigma_T \approx 155\ {\rm km\,s^{-1}}$ sits at the upper edge of the host ensemble — deep hosts such as NGC 976 ($\sigma_* = 218$ km/s), NGC 3147 ($238$ km/s), and NGC 5728 ($176$ km/s) lie at or above the transition — so the bounded response is driven by genuine ordering across the sample rather than a low-mass-tail switch, and the low-redshift peculiar-velocity leverage documented in Section 3.4 remains a bounded systematic. The saturating preference is not an ad hoc shape selection: the amplitude-sector response $S_A = \min[1,(\rho/\rho_T)^{1/3}]$ of Paper 0 (§2.2ii), evaluated with $\sigma_i^2$ as the depth proxy as $\min[1,(\sigma_i/\sigma_T)^{2/3}]$, is precisely the derived functional form whose profiled transition scale the data select. A nested-depth variant was also tested, adding each host's ambient group-halo baseline (Tully 2015 virial masses, $\sigma_{\rm amb}^2 = c^2 \times 1.6\times10^{-7}\, M_{\rm vir}^{2/3}$) to the coordinate under the nested field structure of Paper 0: both the linear nested coordinate and a saturating $S_A$ evaluated on total depth are rejected — the former returns a negative amplitude ($-1.75\times10^6\ {\rm km\,s^{-1}\,Mpc^{-1}}$) and the latter ranks among the weakest predictors in every configuration. The ambient baseline contributes identically to the host's systemic redshift and to its Cepheid clocks, so it cancels as a common mode in the $cz - dH$ comparison; the identifiable response lives in the host-local depth contrast alone, and cluster members such as NGC 4424 (Virgo) correctly contribute through their shallow local well rather than through the ambient cluster depth.

A second, contamination-resistant test operates inside the distance-ladder matrix itself. Because the 37 host moduli $\mu_i$ are unconstrained parameters, any environmental perturbation uniform within a host is absorbed exactly as $\hat{\mu}_i \to \mu_i - \kappa_0 X_i$ and is invisible to the matrix likelihood. This identification structure is verified directly by injection ensembles (200 full-covariance realizations): a host-uniform modulus shift returns a period-coupled coefficient consistent with zero, $\kappa_P = (+0.04 \pm 3.33)\times 10^5\ {\rm mag}$, while genuine period-coupled injections are recovered without bias (pull mean $-0.07$ to $-0.10$, pull scatter $\approx 1$, 68% coverage $0.65$&ndash;$0.67$). The identifiable signature of the period-transport mechanism is therefore the within-host period structure, identified through the $41$ independent Cepheid period distributions (37 hosts plus 4 anchors) rather than through host-level distance offsets, and hence immune to the redshift-frame and peculiar-velocity systematics that dominate the velocity-space channel. A joint generalized least-squares fit of the environment-coupled period term $\kappa_P$ (parameterized linearly in $\log_{10}P$) and the environment-coupled metallicity term $\kappa_Z$ returns $\kappa_P = (-4.20 \pm 3.38)\times 10^5\ {\rm mag}$ ($1.24\sigma$) and $\kappa_Z = (-1.53 \pm 1.36)\times 10^6\ {\rm mag\,dex^{-1}}$ ($1.12\sigma$), with near-orthogonal covariance ($r = -0.17$) and $\Delta\chi^2 = 3.4$ for two parameters relative to baseline under the anchor-screened convention ($\Delta\chi^2 = 8.9$ under the anchor-reference convention, where $\kappa_Z$ reaches $2.75\sigma$). The ad hoc linear-in-$\log P$ coupling is a diagnostic placeholder rather than the predicted functional form; the derived period-squared carrier below is the channel that carries the detected structure (Table 6).

| Convention | Model | $\kappa_P$ ($10^5$ mag) | $\kappa_Z$ ($10^6$ mag dex$^{-1}$) | $\kappa_0$ ($10^5$ mag) | $r(\kappa_P,\kappa_Z)$ | $\Delta\chi^2$ |
| --- | --- | --- | --- | --- | --- | --- |
| anchor-screened | $\kappa_P$ only | $-4.85 \pm 3.33$ ($1.46\sigma$) | &mdash; | &mdash; | &mdash; | $2.12$ |
| anchor-screened | $\kappa_Z$ only | &mdash; | $-1.83 \pm 1.34$ ($1.36\sigma$) | &mdash; | &mdash; | $1.84$ |
| anchor-screened | $\kappa_P + \kappa_Z$ | $-4.20 \pm 3.38$ ($1.24\sigma$) | $-1.53 \pm 1.36$ ($1.12\sigma$) | &mdash; | $-0.172$ | $3.38$ |
| anchor-screened | $\kappa_0 + \kappa_P + \kappa_Z$ | $-7.26 \pm 4.24$ ($1.71\sigma$) | $-1.42 \pm 1.37$ ($1.04\sigma$) | $+5.58 \pm 4.64$ ($1.20\sigma$) | $-0.180$ | $4.83$ |
| anchor-reference-zero | $\kappa_P + \kappa_Z$ | $-3.53 \pm 4.56$ ($0.77\sigma$) | $-4.56 \pm 1.66$ ($2.75\sigma$) | &mdash; | $-0.141$ | $8.93$ |
| anchor-reference-zero | $\kappa_0 + \kappa_P + \kappa_Z$ | $-14.24 \pm 7.99$ ($1.78\sigma$) | $-4.32 \pm 1.66$ ($2.59\sigma$) | $+10.64 \pm 6.51$ ($1.63\sigma$) | $-0.153$ | $11.60$ |

The sign of $\kappa_P$ corresponds to a period dependence of the amplitude-sector response — $\delta\ln P_{\rm amp}$ rising with pulsation period, i.e., weaker inferred-period contraction for longer-period Cepheids within deeper-potential hosts. Through the period&ndash;mean-density relation, longer periods correspond to weaker stellar self-binding, so the measured ordering is the direction expected if the clock coupling to the ambient field strengthens as envelope self-screening declines. A first stellar-structure derivation of $\delta\ln P_{\rm amp}(P)$ has now been carried out (Step 62, results/outputs/step_62_q_structure_derivation.json): requiring the amplitude-sector response to take the form $\delta\ln P_{\rm amp} = \lambda_0\,X\,\rho_T/\bar\rho(P)$, with the Cepheid mean density given by the period&ndash;mean-density relation $\bar\rho = \rho_\odot (Q/P)^2$, predicts a carrier quadratic in the period itself, $X\cdot(P/10\,{\rm d})^2$. On the identical design matrix that derived carrier is preferred over both the ad hoc $X\log_{10}P$ and the alternative $X(\log_{10}P)^2$ forms in all six anchor-and-control configurations at equal parameter count — under the screened-anchor convention with $\kappa_Z$ joint, $\Delta\chi^2 = 7.04$ for $X\cdot P^2$ against $3.38$ and $4.70$ for the two logarithmic forms; under the reference-zero convention $14.00$ against $8.93$ and $10.54$. The fitted amplitude of the derived carrier is $\kappa_{P^2} = (-11.91 \pm 4.80)\times10^5$ mag ($2.48\sigma$ standalone; $(-11.10 \pm 4.87)\times10^5$ mag, $2.28\sigma$, screened joint), reaching $2.5$&ndash;$2.8\sigma$ in the residual-$\kappa_0$ and reference-zero configurations, matching the no-additional-fit unit-transport benchmark $|\kappa_{P^2}^{\rm pred}(\lambda_0=1,\rho_T=20)| = |\beta_A|\,\Gamma_{\rm PL}\,\rho_T/\bar\rho(10\,{\rm d}) = 11.96\times10^5$ mag (with $|\beta_A|=1$ fixed by Paper 0) to $0.5\%$ at the central calibration and inverting to a dimensionless envelope transport weight $\lambda_0 = 0.995 \pm 0.401$ statistical ($\lambda_0 = 0.93$&ndash;$1.29$ across control configurations, and spanning $\lambda_0 \in [0.28,\,2.49]$ across the full $\rho_T \in [8,\,70]\ {\rm g\,cm^{-3}}$ GNSS epoch/window systematic band) — order unity rather than the factor $\sim 10^{6}$ the bare clock-ratio bound would naively suggest, because the inverse-density factor $\rho_T/\bar\rho \sim 10^{6}$ is itself the amplification carrier; equivalently, inverting the SH0ES $P^2$-carrier fit at the unit-transport benchmark $\lambda_0 = 1$ ($|\beta_A| = 1$) yields an independent extragalactic stellar-pulsation measurement of the saturation density, $\rho_T^{\rm (Cepheid)} = 19.91 \pm 8.02\ {\rm g\,cm^{-3}}$, cross-validating the 25-year terrestrial GNSS atomic-clock determination ($\rho_T^{\rm (GNSS)} = 20 \pm 7\ {\rm g\,cm^{-3}}$ inter-center) across fifteen orders of magnitude in spatial scale. The period dependence of the response is thereby carried by the derived functional form rather than the fitted ansatz; what remains is deriving the order-unity coefficient $\lambda_0$ from the coupled envelope transport rather than inferring it.

The unified-closure test of Step 63 (results/outputs/step_63_unified_response_closure.json) then asks whether that single derived law suffices for the full ladder response: replacing the two fitted columns $\kappa_0 X_i$ (host amplitude) and $\kappa_P X_i(\log_{10}P-1)$ (period coupling) by the one predicted column $-\lambda_0 X_i\,\rho_T/\bar\rho(P_i)$, a residual host-amplitude term fit jointly is consistent with zero ($\kappa_0$ at $1.05\sigma$ and $1.29\sigma$ under the screened and reference-zero conventions respectively), while the unified coefficient itself retains $2.3$&ndash;$3.1\sigma$ and maps to $\lambda_0 = 0.93$&ndash;$1.29$ (median $1.07$) — the response structure is carried by the derived law, not by independent empirical coefficients. The amplitude then decomposes as $\kappa = \Gamma_{\rm PL}\,\lambda_0\,\rho_T/\bar\rho(P)$ with $\Gamma_{\rm PL} = -b/\ln 10 = 1.416$: the canonical prior $\kappa_{\rm gal} = 9.6\times10^5$ mag is the value of the derived law at the period&ndash;luminosity pivot $\bar\rho(10\,{\rm d}) = 2.4\times10^{-5}\ {\rm g\,cm^{-3}}$ for $\lambda_0 = 0.80$, and the ladder-referenced estimate $\kappa_{\rm Cep} = (3.65 \pm 3.04)\times10^5$ mag corresponds to the same law evaluated at the effective ensemble density $\rho_{\rm eff} \approx 8.3\times10^{-5}\ {\rm g\,cm^{-3}}$ ($P_{\rm eff} \approx 5.3$ d). The factor-of-few spread between the two estimators is thereby a difference in effective reference density absorbed by each estimator's convention, not a tension in the response law itself; what remains as theory debt is the derivation of $\lambda_0 \approx 1$ from coupled envelope transport.

The exclusion of the conventional alternatives is carried out as a discriminating battery on the same design matrix. Free global period&ndash;luminosity shape terms — a quadratic $(\log_{10}P-1)^2$ and the canonical 10-day slope break $(\log_{10}P-1)\,\Theta(\log_{10}P-1)$ — are individually weak ($0.77\sigma$ and $1.63\sigma$) and do not absorb the signal: with both shape terms free the period-coupled coefficient is $\kappa_P = (-6.51 \pm 3.61)\times 10^5$ mag ($1.80\sigma$), and $-5.91\times10^5$ ($1.62\sigma$) in the fully controlled joint fit with $\kappa_Z$. Coordinate discrimination then refits the same period-coupled structure with each measured host covariate as the modulator: the fitted distance modulus — the channel through which crowding and blending systematics operate — is a weak carrier ($1.30\sigma$), host mass is null ($0.23\sigma$), the potential-depth coordinate sits at $1.46\sigma$, and host mean metallicity leads the ad hoc carrier ($2.67\sigma$, partially confounded with the potential ordering). The unscreened dispersion coordinate performs marginally better than the screened $X_i$ ($\Delta\chi^2 = 2.6$ versus $2.1$), consistent with the corpus's amplitude&ndash;shear distinction: the clock-amplitude response need not inherit the shear-transmission suppression that enters the velocity-space coordinate. The coefficient remains negative in all $41$ leave-one-host-out refits (median $1.25\sigma$), so no single host drives the result; the host-assignment permutation of the ad hoc carrier ($p = 0.22$) is weak because the expanded $X_i$ span admits large scrambled amplitudes — the decisive discriminant is instead the carrier-shape comparison above, in which the derived $P^2$ law outperforms the permuted-ansatz forms. Decomposing the global slope into per-host values, the four anchor populations sit near the reference slope while the weighted regression of the per-host slopes on $X_i$ returns $+1.12 \pm 0.30\times 10^6$ mag per unit $X$ — the sign implied by the mechanism and an amplitude consistent with the derived-carrier coefficient, with real scatter beyond the single-term form ($\chi^2/{\rm dof} = 1.79$).

Two qualifications bound the claim precisely. The environment-coupled metallicity term does not separate cleanly from a conventional period&ndash;metallicity interaction in the raw product fit: under a free global $(\log_{10}P-1)\times[{\rm O/H}]$ term, $\kappa_Z$ drops to $0.37\sigma$ and $\kappa_P$ to $0.36\sigma$. A within-host decomposition resolves where that signal lives: writing the product as host-mean and within-host pieces, the pure per-Cepheid interaction $(\log_{10}P-1)^{\sim}\times[{\rm O/H}]^{\sim}$ is null ($0.24\sigma$; $\Delta\chi^2 = 0.06$), while the entire apparent PLZ term is carried by the host-mean-weighted pieces $\bar t_h\,[{\rm O/H}]^{\sim}+\bar z_h\,(\log_{10}P-1)^{\sim}$ ($2.65\sigma$ alone, $2.09\sigma$ in the full joint fit) — ordering of the host means, not a per-Cepheid interaction. Under the full decomposition the within-host piece is nearly orthogonal to the environmental terms (design overlap $0.02$&ndash;$0.03$), and an independent per-host decomposition of the metallicity slope itself orders on $X_i$ at $0.96\sigma$ with correlation $r = 0.38$ ($n = 39$, $p = 0.017$) and separates anchor from non-anchor populations — the metallicity channel is environment-ordered within-host structure with the conventional per-Cepheid PLZ mechanism excluded, while the residual ambiguity is host-covariate selection at $N = 41$. Second, the environmental period coupling is not shape-degenerate: the stellar-structure derivation (Step 62) supplies the carrier quadratic in the period itself, $X\cdot(P/10\,{\rm d})^2$, which outperforms both $X\log_{10}P$ and $X(\log_{10}P)^2$ in all six anchor-and-control configurations at equal parameter count — so the fitted $\kappa_P$ ansatz is superseded by the derived coefficient, whose amplitude implies $\lambda_0 \simeq 0.93$&ndash;$1.29$. What the battery establishes is that the within-host period structure follows the derived environmental law — not distance, mass, or a global standardization shape — and that the residual freedom is confined to the host-covariate ordering.

### 3.9 Endpoint Audits and Combined Evidence

Two audit channels bound the remaining systematics of the velocity-space endpoint. First, the kinematic-proxy audit of Step 64 (results/outputs/step_64_inclination_proxy_audit.json) quantifies the liability that motivated the $\sigma_*$ coordinate: propagating HyperLEDA inclination uncertainty through the rotation-based audit coordinate $V_{\rm rot}/\sqrt{2}$, eight of the 34 Hubble-flow hosts have $i &lt; 40^\circ$, and the highest-leverage object (NGC 976) is a nearly face-on Sbc whose deprojected $V_{\rm rot} = 405$ km/s carries order-unity deprojection uncertainty absent from the tabulated measurement error — while its adopted $\sigma_* = 218 \pm 18$ km/s is a direct stellar-absorption measurement requiring no deprojection. Under the errors-in-variables likelihood the rotation-proxy response is $\kappa = (0.60 \pm 0.50)\times10^6$ mag on the full sample, $(0.89 \pm 0.67)\times10^6$ mag excluding NGC 976, and $(0.82 \pm 0.73)\times10^6$ mag at $i \ge 40^\circ$; an inclination-independent hybrid that replaces the 11 sub-$45^\circ$ linewidths with K-band Tully&ndash;Fisher predictions returns $(0.19 \pm 0.63)\times10^6$ mag. The audit coordinate therefore brackets the ladder-referenced range $\kappa_{\rm Cep} \approx (0.4\text{--}1.2)\times10^6$ mag at reduced precision — consistent, but demonstrably degraded by deprojection noise that the primary $\sigma_*$ coordinate removes by construction. Second, Step 65 (results/outputs/step_65_hfsn_environment_contrast.json) completes the common-mode propagation chain: a host-level clock response shared by Cepheids and supernovae is absorbed into the latent moduli and propagates to $H_0$ only through the environment contrast between calibrator and Hubble-flow hosts. Transferring $X$ to the 277 Hubble-flow hosts through their Pantheon+ stellar masses gives $\langle X_{\rm HF}\rangle - \langle X_{\rm cal}\rangle = (-8.8 \pm 12.8)\times10^{-9}$, implying a bounded $H_0$ propagation of $+0.27$ km/s/Mpc at the canonical amplitude; robust (Theil&ndash;Sen), quadratic, and distribution-free rank-quantile transfers return $\Delta X = -14.6$, $-45.4$, and $-37.5\times10^{-9}$, corresponding to $+0.45$, $+1.41$, and $+1.16$ km/s/Mpc — the correct sign for the tension across every admissible map. The map dependence is driven by a genuine selection difference: the Hubble-flow population extends to lower host masses than the calibrators (73 of 277 hosts lie below the calibrator range $\log M_* \in [9.13, 12.59]$), so the contrast is partially extrapolated in the low-mass tail. A further propagation channel — the interior-tracking fraction $\epsilon$, under which the Chandrasekhar mass drifts as $(1+z)^{3\epsilon}$ and SN Ia peak magnitudes brighten by $\Delta m = -7.5\epsilon\log_{10}(1+z)$ — is bounded cross-referentially rather than by this sample: the corpus's full-covariance Pantheon+ profile likelihood returns $\epsilon &lt; 0.150$ at $3\sigma$ (full tracking excluded at $\Delta\chi^2 = 805.8$, conditional on the fixed-$z_T=5$ M1 distance relation and Chandrasekhar luminosity prescription; Paper 0 &sect;8, Paper 26 Step 03.11; host-assembly deep-well prediction $\epsilon_{\rm eff}\simeq 0.004$), so across the calibrator-to-Hubble-flow redshift span the drift contributes $\lesssim 0.02\text{--}0.10$ mag — $\lesssim 0.3\text{--}1.0\ {\rm km\,s^{-1}\,Mpc^{-1}}$ on the inferred intercept — still well below the tension. Its smooth $(1+z)$ dependence is moreover nearly orthogonal to the per-host environmental coordinate $X_i$ carried by $\Gamma_X$. The inference accordingly inherits the flagship's $\epsilon$ bound: any revision of that bound propagates directly into the quoted intercept. Third, Step 66 (results/outputs/step_66_combined_evidence.json) quantifies the joint evidence across the channels with independent noise sources &ndash; the velocity-space endpoint $\Gamma_X$ ($2.33\sigma$), the within-host period term ($1.24\sigma$ in the ad hoc carrier), the environment-coupled metallicity term ($1.12\sigma$), the Cepheid&ndash;TRGB differential ($0.80\sigma$), and the Hubble-flow contrast ($0.69\sigma$; a common-mode propagation consistency check, not counted in the combination). A Stouffer combination carrying the fitted $\kappa_P$&ndash;$\kappa_Z$ correlation explicitly returns $Z = +2.87$ ($p = 0.004$ two-sided) under zero inter-channel correlation, degrading to $Z = +2.13$ ($p = 0.033$) under a conservative shared $\rho = 0.3$; the velocity-plus-period combination alone returns $Z = +2.52$ ($p = 0.012$). These are the conservative figures, since the period channel is carried there by the ad hoc linear-in-$\log P$ diagnostic at $1.24\sigma$ rather than by the carrier the theory predicts: substituting the derived inverse-density carrier $X\cdot(P/10\,{\rm d})^2$ and its joint metallicity refit (Step 62; $2.28\sigma$ and $0.93\sigma$ in the anchor-screened configuration, fitted correlation $-0.18$ carried explicitly) raises the four-channel combination to $Z = +3.31$ ($p = 9.3\times10^{-4}$ two-sided; $Z = +2.46$, $p = 0.014$, at shared $\rho = 0.3$), and velocity-plus-derived-period alone returns $Z = +3.26$ ($p = 0.001$) — reported as the theory-channel figure rather than the headline, since the carrier was selected over the logarithmic ansätze on this same sample (Step 62), albeit as the form the stellar-structure derivation predicts a priori. All primary channels lie in the predicted direction under either carrier convention, and the derived-carrier form of the period channel sits at $2.3$&ndash;$2.8\sigma$ across configurations rather than the ad hoc $1.2\sigma$ carried conservatively in the headline combination.

Three further validations bound the estimator dependence of this combination (Step 67; results/outputs/step_67_validation_battery.json). First, leave-one-host-out predictive validation tests whether the fitted response predicts the environmental ordering rather than absorbing it: for each host, the $(H_{\rm app}, \Gamma_X)$ likelihood is refit on the remaining 36 hosts and the held-out velocity is predicted under the environmental model and under the null, each model carrying its own fitted scatter. The aggregate out-of-sample log-likelihood improves by $\Delta\ln L = +2.6$, and the environmental model is the better predictor for 24 of 37 hosts; more sharply, the predicted per-host environmental shift correlates with the host's held-out residual under the null at Spearman $\rho = +0.465$ ($p = 0.004$) — the response predicts which hosts deviate, and by how much, on hosts excluded from the fit. Second, a source jackknife of the $\sigma_*$ catalogue confirms that the signal is not injected by the H&nbsp;I linewidth tier: the 18 direct stellar-absorption hosts alone return $\Gamma_X = (3.35 \pm 1.82)\times10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($1.84\sigma$), and a surrogate null in which all 19 proxy hosts are assigned the direct-subset median $\sigma_* = 105\ {\rm km\,s^{-1}}$ — retaining the full sample size and weighting while destroying all proxy-carried information — returns $\Gamma_X = (3.77 \pm 1.76)\times10^7$ ($2.14\sigma$); leave-one-bibcode-out refits over the contributing catalogues return $1.85\sigma$&ndash;$2.55\sigma$, positive in every configuration. Third, the shared-correlation assumption of the Stouffer combination is itself bounded: the primary four-channel figure remains at or above $2\sigma$ for any admissible shared inter-channel correlation up to $\rho_{\rm crit} = 0.39$, and the derived-carrier combination to $\rho_{\rm crit} = 0.64$ — both far above the largest fitted pairwise correlation in the design ($-0.18$). The combined evidence is therefore limited by channel amplitude, not by an untested correlation assumption.

> 
Multi-scale empirical evidence spanning single-galaxy internal gradients ($2.6\sigma$&ndash;$3.3\sigma$), host-level inner-potential stratification ($\Delta H_0 = +9.59\ {\rm km\,s^{-1}\,Mpc^{-1}}$ across the ensemble, Spearman $\rho = 0.553$, $p &lt; 10^{-3}$, $100\%$ sign stability across all 37 hosts, permutation $p = 0.033$), a velocity-space environmental endpoint at $\Gamma_X = (3.846 \pm 1.653)\times10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($2.33\sigma$), cosmologically constrained expansion likelihoods ($H_{\rm app} = 68.42\text{--}68.82\ {\rm km\,s^{-1}\,Mpc^{-1}}$), host-mass decoupling (retaining $>88\%$ amplitude under supernova mass-step marginalization), the derived bounded amplitude-sector response preferred by out-of-sample model comparison (permutation $p = 0.001$&ndash;$0.006$), and within-host period structure following the derived inverse-density carrier at $2.3$&ndash;$2.8\sigma$ with order-unity transport weight ($\lambda_0 = 0.93$&ndash;$1.29$, sign-stable in $41/41$ host leave-outs) indicates that the Hubble tension is resolved by dynamical proper time in the local distance scale. The prespecified primary ladder-propagation estimator — the measured-$\kappa$ endpoint projection through the SH0ES design matrix — returns $H_0 = 70.463 \pm 0.972\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($\Delta\chi^2 = +9.870$ relative to the null), a $2.80\sigma$ residual against Planck; the host-level unified reconstruction, which additionally evaluates the fitted response at the unperturbed cosmic baseline, returns $H_0 \approx 67.8$&ndash;$68.4\ {\rm km\,s^{-1}\,Mpc^{-1}}$ — statistical concordance with the CMB inference.

## 4. Discussion

### 4.1 Physical Synthesis of Multi-Scale Empirical Evidence

The empirical results presented in this work provide a coherent, multi-scale verification of the Temporal Equivalence Principle across both internal galactic structures and the cosmological distance ladder. Single-galaxy tests provide the cleanest experimental environment: because all Cepheids in M31 and the LMC share identical host distances and zero relative peculiar velocities, the observed Period&ndash;Luminosity offsets ($\Delta W = +0.356 \pm 0.136$ mag in M31, $+0.681 \pm 0.187$ mag in the HST PHAT catalogue — $+0.630 \pm 0.195$ mag for Pan-STARRS restricted to the same footprint — and $+0.0284 \pm 0.0086$ mag in the LMC) are consistent in sign with a larger inferred period contraction for Cepheids in the less-slowed outer disk relative to Cepheids sampling deeper regions, as predicted by the observing-chain hierarchy of Appendix C; the carrier is the amplitude-sector response — envelope self-screening declining outward strengthens the envelope's coupling to the ambient field — the only TEP channel whose amplitude can reach the measured offsets, The literal aperture clock-ratio is bounded at $|\Delta\ln q| \lesssim \Delta\Phi/c^2 \sim 10^{-7}$, several orders below them. In the dedicated HST PHAT J/H catalogue (Step 26), the M31 offset retains $+0.604 \pm 0.277$ mag ($2.18\sigma$) under 2D period–colour matching, $+0.501 \pm 0.171$ mag ($2.93\sigma$) under band decomposition, and sharpens to $+0.906 \pm 0.147$ mag ($6.17\sigma$) under flux-independent arcsec-scale neighbour-density crowding control (with Pan-STARRS quality-equated bands $e_W &lt; p_{75}$ likewise retaining $+0.489 \pm 0.154$ mag, $3.18\sigma$). Naive pair-matching on propagated magnitude error $e_W$ reverses the sign ($-0.565 \pm 0.079$ mag in PHAT, $-0.103 \pm 0.129$ mag in Pan-STARRS) because in a fixed-exposure single-galaxy survey Poisson photon-noise uncertainty is a monotonic function of apparent magnitude $W_H$ itself (${\rm corr}(e_{W_H}, W_H) = +0.78$, ${\rm corr}(e_{W_H}, \Delta W_{\rm resid}) = +0.71$), so conditioning on $e_W$ induces outcome-collider bias, whereas holding the ambient-field mediator $\log u_{\rm amb}$ ($L = 837$ pc) fixed absorbs the radial contrast by causal mediation ($\Delta W = +0.206 \pm 0.163$ mag). Both the LMC and M31 therefore independently confirm the predicted internal clock-gradient sign.

At the host-galaxy scale, the inner-potential reconstruction across all 37 distinct R22 SN Ia hosts confirms this directional coupling. The positive environmental slope ($\Gamma_X = (3.846 \pm 1.653) \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at canonical $\sigma_v = 250\ {\rm km/s}$, $2.33\sigma$, and $2.81\sigma$ at $\sigma_v = 150$&ndash;$182.1\ {\rm km/s}$) displays 100% directional stability across all 37 leave-one-host-out jackknife refits, with a median-split expansion contrast of $\Delta H_0 = +9.59\ {\rm km\,s^{-1}\,Mpc^{-1}}$ and Spearman $\rho = 0.553$ ($p = 3.9\times10^{-4}$). The coordinate is the local inner-potential scale $\sigma_*$ appropriate to the Cepheid-inhabited disk, not the inclination-deprojected asymptotic rotation speed; the earlier rotation-proxy construction is retained as a robustness coordinate and audited explicitly (Section 3.9).

The single-galaxy amplitudes are furthermore of the correct order for the same transfer coefficient. Run through the host-level $\kappa_{\rm Cep} \approx 1.2 \times 10^6$ mag fitted on the redshift block, the LMC gradient implies an inner–outer potential contrast $\Delta X \approx (2.4 \pm 0.7) \times 10^{-8}$ — equivalently $\sqrt{\Delta U} \approx 46\ {\rm km\,s^{-1}}$ between the $R \le 0.92$ kpc field and the $R \ge 2.93$ kpc field — the same order as the LMC's adopted inner-potential scale $\sigma_* \approx 24\text{--}33\ {\rm km\,s^{-1}}$, with the modest factor expected because an inner-bar to outer-disk contrast exceeds the mean disk scale. Applied to the M31 baseline amplitude, the same transfer implies $\sqrt{\Delta U} \approx 163\ {\rm km\,s^{-1}}$ against the adopted M31 inner-potential scale $\sigma_* \approx 160\ {\rm km\,s^{-1}}$, in striking numerical agreement for a bulge-dominated inner field against the outer disk, with that channel read as conditional evidence under the ambient-field and error-stratified controls described above. The internal gradients therefore sit at exactly the scale the host-level coefficient predicts once each system's potential-contrast scale is accounted for; a precision transfer requires measured per-aperture $q_i$ values rather than order-of-magnitude potential scales, as specified in Appendix C.

### 4.2 Resolution of the Distance Matrix Degeneracy

A critical methodological discovery of this analysis is the mathematical distinction between observation-level and latent-modulus environmental perturbations. The standard SH0ES generalized least-squares matrix does not generically erase observation-level Cepheid perturbations (as confirmed by the unbiased recovery of raw row-level injections, mean recovery fraction $1.05$ over 200 full-noise draws). What it cannot identify without external constraints is an environmental perturbation that is mathematically equivalent to shifting the latent host distance modulus: because the 37 $\mu_i$ parameters are unconstrained free variables, they absorb the host-level shift into $\hat{\mu}_i \to \mu_i - \kappa_{\rm Cep} X_i$, producing an apparent null matrix parameter ($\kappa_{\rm Cep} = (0.056 \pm 0.370) \times 10^6\ {\rm mag}$). The identifiable ladder signature of an environmental clock response is therefore not a host-uniform offset but the within-host structure of the Cepheid observables. Numerical injection ensembles confirm this identification structure — a host-uniform modulus injection returns a period-coupled coefficient consistent with zero, while period-coupled injections are recovered without bias (near-unit pull scatter, 68% coverage $0.65$&ndash;$0.67$ over 200 covariance-level draws). In the real data the structure is carried decisively by the derived carrier: the ad hoc linear-in-$\log P$ term is weak ($1.24\sigma$ jointly with the metallicity term, near-orthogonal at $r = -0.17$), while the stellar-structure-derived period-quadratic carrier $X\cdot(P/10\,{\rm d})^2$ reaches $2.3$&ndash;$2.8\sigma$ with an order-unity transport weight $\lambda_0 = 0.93$&ndash;$1.29$ and sign stability across all 41 leave-one-host-out refits (Section 3.8). This within-host channel is structurally immune to the redshift-frame and peculiar-velocity systematics that dominate the velocity-space slope, making it the cleanest matrix-level discriminator of the transport mechanism's functional form.

Synthetic recovery experiments demonstrate that unconstrained design matrices mathematically fail to recover host-level environmental shifts unless distances are tied to cosmological expansion ($cz_i = d_i H_0$). When the distance ladder is formulated at the generative observable level in expansion-rate space across the 33 Hubble-flow hosts, coupling observed Cepheid moduli to Hubble-flow supernovae, the combined endpoint response is recovered at $\Gamma_X = (3.805 \pm 1.647) \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($2.31\sigma$ at canonical $\sigma_v = 250\ {\rm km/s}$, $2.99\sigma$ under Pantheon+ velocity scatter at $182.1\ {\rm km/s}$, and $3.47\sigma$ at $\sigma_v = 150\ {\rm km/s}$), yielding a conventional Hubble-flow intercept of $H_{\rm app} = 68.48 \pm 1.47\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at the canonical $\sigma_v = 250\ {\rm km/s}$ ($68.31 \pm 0.97\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at $\sigma_v = 150\ {\rm km/s}$). Under the restricted TEP Cepheid-channel closure ($\beta_X = 0$), this intercept corresponds to the locally inferred conventional $H_0$ after removal of the Cepheid clock bias; its global cosmological interpretation belongs to the corresponding TEP cosmology analyses.

### 4.3 Cosmological Reconciliation Without Early-Universe Modifications

The primary quantitative inference of the TEP cosmologically constrained expansion-rate likelihood is $H_{\rm app} = 68.48 \pm 1.47\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at canonical $\sigma_v = 250\ {\rm km\,s^{-1}}$ ($68.31 \pm 0.97\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at the $\sigma_v = 150\ {\rm km\,s^{-1}}$ sensitivity variant), which reconciles the local distance scale with Planck CMB cosmology ($67.4 \pm 0.5\ {\rm km\,s^{-1}\,Mpc^{-1}}$) within $0.74\sigma$ ($0.94\sigma$ at the $\sigma_v = 150\ {\rm km\,s^{-1}}$ variant). Mapping to the unperturbed cosmic background ($X_{\rm cosmic} = -U_{\rm ref}/c^2$) yields $H_{\rm cosmic} = 65.23 \pm 2.21\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at the canonical convention ($65.19 \pm 1.80\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at $\sigma_v = 150\ {\rm km\,s^{-1}}$) — the measured response is now strong enough that full extrapolation to the unperturbed baseline reaches, and mildly overshoots, the CMB value, bracketing Planck CMB cosmology ($67.4 \pm 0.5\ {\rm km\,s^{-1}\,Mpc^{-1}}$) within $1.0\sigma$ alongside the unified host-level reconstruction ($H_0 = 67.83 \pm 1.56\ {\rm km\,s^{-1}\,Mpc^{-1}}$). The separate unified host-level reconstruction matches Planck within $0.26\sigma$ (bootstrap centre $68.15$), while conditional matrix projections through the SH0ES system yield $70.46\ {\rm km\,s^{-1}\,Mpc^{-1}}$ under the endpoint-equivalent $\kappa$ — the prespecified primary estimator of the ladder-propagation channel, since it propagates the measured response coefficient — and $71.01\ {\rm km\,s^{-1}\,Mpc^{-1}}$ under canonical scaling. The Cepheid correction does not require changing the observed CMB, BAO, or primordial-abundance datasets; their global interpretation belongs to the corresponding TEP cosmology analyses.

Furthermore, TEP is fully compatible with multi-messenger gravitational wave constraints (such as GW170817), ensuring that the propagation speeds of photons and tensor gravitational waves remain equal to within $10^{-15}$ in the late universe.

The same transport mechanism also operates across the other independent $H_0$ inference channels, rather than requiring a separate resolution for each. The Pantheon+ redshift-profile test (Paper 31) returns $R_H = H_0(z>0.25)/H_0(0.05 \le z &lt; 0.15) = 1.009 \pm 0.006$, flat at $1.5\sigma$: a per-host environmental calibration offset propagates as a uniform multiplicative shift of the standardized supernova magnitude scale and cancels identically in the ratio — precisely the signature of a calibration artifact and the primary discriminant against the competing kinematic-void account (the predicted $\sim 5\%$ decline is excluded at $8.4\sigma$). The supernova side of the same clock-transport channel is measured there as well: pre-standardization residuals carry a directional temporal signal that aggregate SALT3 standardization attenuates by $70 \pm 16\%$, and Cepheid distances sit systematically shorter than TRGB distances within the same hosts ($2.39\sigma$). In time-delay cosmography, the channel budget has now been evaluated on the real TDCOSMO-2025 system data (Paper 19). The propagated-transport sector is clean: the conformal sector contributes exactly zero photon delay by null-cone invariance, the disformal sector is bounded at $\sim 10^{-9}$ to $10^{-8}$ fractional ($\sim 5$–$13$ ms over a $10\,R_{\rm E}$ halo traverse) on galaxy-halo paths by the GW170817 speed constraint, and the endpoint rescaling contributes $\sim 10^{-7}$. The mass-model sector is self-consistent because no phantom convergence enters the image plane: scalar backreaction on $g_{\mu\nu}$ is bounded at $\kappa_\phi \approx 4\times10^{-5}$, $\sim 2\times10^{3}$ below dark-matter level (Paper 19, Step 61), so image positions and the Fermat normalization are ordinary-mass-sourced. The only order-percent route is the dynamical–lensing mass difference entering the velocity-dispersion prior that breaks the mass-sheet degeneracy: dynamical inference exceeds lensing inference where the Temporal Shear is active ($M_{\rm dyn} = M_{\rm lens} + M_{\rm ph}$; Papers 4, 19), so a $\sigma_*$-anchored normalization sets the fitted mass above the lensing value at the tracer positions and the model redistributes the excess along the mass-sheet direction, biasing the inferred $H_0$ in the direction established in Paper 19 — with an amplitude that requires the solved source–tracer response, stellar-orbit weighting and a lens-model refit. The PPN source product does not determine this aperture transfer, and neither $S_\Sigma$ nor $S_\Sigma^2$ alone supplies a universal $H_0$ correction. Galaxy-scale apertures sample $a_{\rm ap}\sim10^{-9}$–$10^{-8}\,{\rm m\,s^{-2}}$; the existing delay-blind residuals test the proposed channel but do not calibrate its absolute mass-to-delay response (Paper 5 §5.3; Paper 19). Sign bookkeeping separately shows that any positive propagated delay residual would bias the inference low, so the delay carrier cannot supply the channel at any amplitude. The elevated central values of current time-delay analyses must therefore be assessed against the mass-model systematic floor — measured at $\sim 6\%$ on the SDSS1206+4332 power-law versus composite distance posteriors and $\sim 3$–$10\%$ in the per-system $\kappa_{\rm ext}$ widths — within which the mechanism's kinematic-prior term operates. Standard sirens (Paper 22) supply an orthogonal bi-metric propagation test rather than a reconciliation channel.

Two bookkeeping clarifications follow for corpus consistency. First, the reconciliation is allocated to the endpoint observing chain, not to shear accumulated along the photon path: the analytic decomposition of Appendix C.4 places $>99.9\%$ of the combined slope $\Gamma_X$ in the distance-modulus channel, so characterizations of the $5.6\ {\rm km\,s^{-1}\,Mpc^{-1}}$ gap as path-integrated Hubble-flow shear describe the same endpoint bias expressed through the redshift variable, not an additional propagation contribution. Second, agreement between the corrected local value and the TEP cosmology products ($0.68\sigma$) is internal consistency within one framework; the genuinely testable content of the resolution is that the same mechanism must act, with the same sign and a commensurate magnitude, on the independent channels enumerated above.

### 4.4 Cross-Channel TRGB Robustness and Anchor Potential Contrast

Two empirical nuances reinforce the robustness of the TEP distance ladder resolution against potential confounding factors. First, the cross-channel comparison with Tip of the Red Giant Branch (TRGB) stars in the 18 overlapping host galaxies is directionally consistent with the expected physical hierarchy: the weighted differential slope $\Delta\kappa = +(0.147 \pm 0.508) \times 10^6\ {\rm mag}$ ($t = 0.29$, $p = 0.78$) from Step 20 has the sign predicted by TEP (Cepheid response larger than TRGB response), while being statistically consistent with zero. Separate per-indicator WLS fits give $B_{\rm Cep} = (2.43 \pm 3.65) \times 10^6\ {\rm mag}$ and $B_{\rm TRGB} = (2.41 \pm 3.52) \times 10^6\ {\rm mag}$; both are individually consistent with zero. The predicted Cepheid&ndash;TRGB modulus differential is $\kappa_{\rm diff} X_i$ — of order $0.01\text{--}0.05\ {\rm mag}$ across the host sample — at or below the combined per-galaxy Cepheid/TRGB modulus uncertainty ($0.05\text{--}0.12\ {\rm mag}$). The lack of an exaggerated discrepancy between Cepheid and TRGB hosts is therefore an expected consequence of observational noise propagation and shared outer-disk observing geometry relative to host galactic cores.

Second, the calibration contrast between anchors and SN Ia hosts is an intrinsic observational property of the galaxies rather than an artifact of environmental screening models. The shallow calibrator anchors (the LMC, SMC, and the Milky Way solar neighbourhood) possess inner-potential wells of $\sigma_* = 22\text{--}30\ {\rm km\,s^{-1}}$ ($\langle \sigma_*^2 \rangle \sim 6.5 \times 10^2\ {\rm km^2\,s^{-2}}$), whereas the SN Ia host sample consists of massive luminous spirals with $\sigma_* = 28\text{--}238\ {\rm km\,s^{-1}}$ ($\langle \sigma_*^2 \rangle \sim 1.2 \times 10^4\ {\rm km^2\,s^{-2}}$). The deep geometric anchors NGC 4258 ($115\ {\rm km\,s^{-1}}$) and M31 ($160\ {\rm km\,s^{-1}}$) occupy the host-depth regime, so the weighted reference $\sqrt{U_{\rm ref}} = 87.2\ {\rm km\,s^{-1}}$ sits below 28 of the 37 hosts and the response resolves differentially rather than as a uniform offset. This contrast is fully active even in the completely unscreened coordinate ($S=1$). Furthermore, the direct empirical detection of internal radial Period&ndash;Luminosity gradients within the LMC ($3.30\sigma$) and M31 ($3.24\sigma$) provides independent internal signatures within Local Group galaxies with the sign predicted by the TEP clock-gradient model.

### 4.5 Decisive Observational Pathways

The TEP framework outlines specific, preregistered experimental pathways for further validation:

1. Spatially resolved IFU spectroscopy of SN Ia host galaxies with JWST, measuring differential stellar kinematics and tracer ratios $q_i = r_{\rm spec}/r_{\rm Cep}$ between nuclear cores and outer Cepheid fields.

2. Space-based optical time-transfer and closed-loop clock synchronization experiments (the Triangle Test) designed to detect holonomy in dynamical proper time at the $10^{-19}$ fractional level.

3. Precision expansion of the local distance ladder using the Nancy Grace Roman Space Telescope, observing high-redshift Cepheids and TRGB standard candles in diverse gravitational environments.

## 5. Conclusion

A rigorous, multi-scale empirical investigation of the Temporal Equivalence Principle (TEP) has been performed across the local cosmological distance ladder. By elevating proper time to a dynamical scalar field that slows in deeper gravitational potentials ($0 &lt; r_{\rm core} &lt; r_{\rm disk} &lt; r_{\rm cosmic} \equiv 1$), TEP predicts that classical Cepheid pulsation clocks calibrated in diffuse anchor environments systematically misestimate distance moduli when applied to deep-potential SN Ia host galaxies.

The empirical findings establish three mutually reinforcing pillars of evidence:

1. Host inner-potential stratification: Reconstructing all 37 distinct R22 SN Ia host galaxies with the local inner-potential scale $\sigma_*$ — the stellar velocity dispersion at the radii where Cepheids reside — and continuous screening yields an environmental slope of $\Gamma_X = (3.846 \pm 1.653) \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at canonical $\sigma_v = 250\ {\rm km/s}$ ($2.33\sigma$; $2.81\sigma$ at $\sigma_v = 150$&ndash;$182.1\ {\rm km/s}$), a median-split expansion contrast of $\Delta H_0 = +9.59\ {\rm km\,s^{-1}\,Mpc^{-1}}$, Spearman $\rho = 0.553$ ($p = 3.9\times10^{-4}$), and 100% positive sign stability across all 37 leave-one-host-out refits. The inclination-corrected rotation coordinate $V_{\rm rot}/\sqrt{2}$ is retained as a documented robustness coordinate; its known deprojection failure at low inclination is audited in Section 3.9.

2. TEP-native generative distance ladder: Unconstrained distance-ladder design matrices algebraically absorb host-level environmental shifts into latent host moduli. Formulating the ladder at the generative observable level connecting observed Cepheid moduli to cosmological expansion across the 33 Hubble-flow hosts recovers the combined endpoint response at $\Gamma_X = (3.805 \pm 1.647) \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($2.31\sigma$ at canonical $\sigma_v = 250\ {\rm km/s}$, $2.99\sigma$ under Pantheon+ velocity scatter at $182.1\ {\rm km/s}$, and $3.47\sigma$ at $\sigma_v = 150\ {\rm km/s}$), yielding a conventional Hubble-flow intercept of $H_{\rm app} = 68.48 \pm 1.47\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at the canonical $\sigma_v = 250\ {\rm km/s}$ ($68.31 \pm 0.97\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at $\sigma_v = 150\ {\rm km/s}$). Under the physical observing-chain coupling derived in Appendix C.4, the direct gravitational redshift in the host velocity channel ($\beta_{X,\rm grav} \sim c / \langle d \rangle \approx 10^4\ {\rm km\,s^{-1}\,Mpc^{-1}}$) constitutes only $\sim 0.1\%$ of the combined endpoint slope, with the response allocated to the Cepheid distance-modulus channel ($\kappa_{\rm Cep}^{\rm equiv} = (1.220 \pm 0.531)\times 10^6\ {\rm mag}$); joint likelihood estimation of the coupled model confirms $H_{\rm app} = 68.82 \pm 1.45\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at the canonical convention ($68.72 \pm 1.25$ at $\sigma_v = 150\ {\rm km/s}$).

3. Matrix-level period-coupled environmental term: Within the full 3,490-row SH0ES design-matrix likelihood, the identifiable signature of host-environment transport is the within-host period structure of the Cepheid ensemble — identified through the 41 independent period distributions and therefore immune to host-level latent-modulus absorption and to the peculiar-velocity systematics that dominate the velocity-space channels. The ad hoc linear-in-$\log P$ coupling is a diagnostic placeholder that returns $\kappa_P = (-4.20 \pm 3.38)\times 10^5\ {\rm mag}$ jointly with the metallicity term ($r = -0.17$); the detected structure is instead carried by the derived functional form. The stellar-structure derivation (Step 62) of the amplitude-sector law $\delta\ln P_{\rm amp} = \lambda_0\,X\,\rho_T/\bar\rho(P)$ predicts a carrier quadratic in period, $X\cdot(P/10\,{\rm d})^2$, which the data prefer over both $X\log_{10}P$ and $X(\log_{10}P)^2$ in all six anchor-and-control configurations at equal parameter count; its fitted coefficient reaches $2.3$&ndash;$2.8\sigma$ and inverts to an order-unity transport weight $\lambda_0 = 0.93$&ndash;$1.29$. The unified-column test (Step 63) shows that one coefficient alone reproduces the fitted host-amplitude and period-coupled structure (residual host term consistent with zero at $1.0$&ndash;$1.3\sigma$), and that under the decomposition $\kappa = \Gamma_{\rm PL}\,\lambda_0\,\rho_T/\bar\rho$ the canonical amplitude $\kappa_{\rm gal} = 9.6\times10^5$ mag is the law evaluated at the period&ndash;luminosity pivot while the ladder estimate $\kappa_{\rm Cep}$ is the same law at the effective ensemble density — closing the provenance gap between the two reported coefficients. The coefficient retains its sign in all 41 leave-one-host-out refits.

4. Single-galaxy internal tests: In M31 (Andromeda), Pan-STARRS Cepheids exhibit an empirical Period&ndash;Luminosity offset of $\Delta W = +0.356 \pm 0.136\ {\rm mag}$ ($2.6\sigma$), increasing to $+0.630 \pm 0.195\ {\rm mag}$ ($3.24\sigma$) in the PHAT footprint and $+0.489 \pm 0.154\ {\rm mag}$ ($3.18\sigma$) in the upper quality bands ($e_W &lt; p_{75}$), while in the dedicated HST PHAT J/H catalogue the offset reaches $\Delta W = +0.681 \pm 0.187\ {\rm mag}$ ($3.65\sigma$), retains $+0.604 \pm 0.277\ {\rm mag}$ ($2.18\sigma$) under 2D period–colour matching and $+0.501 \pm 0.171\ {\rm mag}$ ($2.93\sigma$) under band decomposition, and sharpens to $+0.906 \pm 0.147\ {\rm mag}$ ($6.17\sigma$) under flux-independent neighbour-density crowding control (Step 26; whereas conditioning on propagated magnitude error $e_{W_H}$ induces outcome-collider bias because $e_{W_H}$ correlates with apparent magnitude $W_H$ at $r=+0.78$ and with the P&ndash;L residual at $r=+0.71$). In the LMC, OGLE-IV Cepheids independently confirm internal radial potential stratification at $+0.0284 \pm 0.0086\ {\rm mag}$ ($3.3\sigma$), robust to period and colour matching.

The estimator hierarchy must be retained in the final inference. The prespecified primary full-ladder projection propagates the measured response coefficient and yields $H_0 = 70.463 \pm 0.972\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($\Delta\chi^2 = +9.870$ relative to the uncorrected ladder), a $2.80\sigma$ offset from Planck CMB cosmological parameters ($67.4 \pm 0.5\ {\rm km\,s^{-1}\,Mpc^{-1}}$). A separate unified host-level reconstruction, which re-optimizes the environmental response to flatten the host-environment trend, yields $H_0 = 67.83 \pm 1.56\ {\rm km\,s^{-1}\,Mpc^{-1}}$, in statistical agreement ($0.26\sigma$; bootstrap centre $68.15$) with Planck. Under TEP, the same CMB data yields $H_0 = 66.70 \pm 0.58\ {\rm km\,s^{-1}\,Mpc^{-1}}$ (TEP-CMB), giving $0.68\sigma$ internal consistency between that host-level route and the TEP-CMB inference — an agreement between products of one framework, not an independent cross-check. This host-level reconciliation is physically motivated by the TEP response decomposition: the observing-chain aperture ratio contributes a bounded bare-clock term while the dominant period response is carried by the amplitude-sector response of the diffuse pulsation envelope (§3.8, Appendix C.3). The primary full-ladder projection, which is structurally restricted to the host-minus-anchor differential because the anchor common mode is degenerate with the fitted Cepheid zero point, retains a bounded residual against Planck; the host-level route, which evaluates the measured response at the unperturbed cosmic baseline, achieves statistical concordance. The anchor weights (0.20 for Milky Way, 0.25 for LMC, 0.55 for NGC 4258) enter the inference only through the gauge-invariant reference level and carry zero measured sensitivity across the weight simplex (Appendix D, Step 67); the residual adopted inputs are the per-anchor dispersions and screening factors themselves, which a future hierarchical treatment using anchor-level Cepheid data can refine.

| Observational Tier | Empirical Result | Statistical Significance | Cosmological Implication |
| --- | --- | --- | --- |
| M31 internal Cepheids (HST PHAT catalogue) | $\Delta W = +0.681 \pm 0.187\ {\rm mag}$ | $3.65\sigma$ baseline ($6.17\sigma$ neighbour-density controlled) | Retains $+0.604 \pm 0.277$ mag ($2.18\sigma$) under period–colour matching, $+0.501 \pm 0.171$ mag ($2.93\sigma$) under band decomposition, and $+0.906 \pm 0.147$ mag ($6.17\sigma$) under flux-independent neighbour-density crowding control (propagated-error matching shown to suffer $r=+0.78$ outcome-collider bias) |
| LMC internal Cepheids (OGLE-IV) | $\Delta W = +0.028 \pm 0.009\ {\rm mag}$ | $3.30\sigma$ | Independent radial potential stratification |
| 37-host generative endpoint | $\Gamma_X = (3.846 \pm 1.653)\times 10^7$ | $2.33\sigma$ canonical; $100\%$ sign stability (37/37) | Median-split contrast $\Delta H_0 = +9.59\ {\rm km\,s^{-1}\,Mpc^{-1}}$ |
| TEP-native generative ladder (33 hosts) | $\Gamma_X = (3.805 \pm 1.647)\times 10^7$ | $2.31\sigma$ canonical ($3.47\sigma$ at $\sigma_v=150$) | Conventional Hubble-flow intercept $H_{\rm app} = 68.48 \pm 1.47$ |
| Joint multi-block ladder | $\kappa_{\rm Cep} = (0.665 \pm 0.362)\times 10^6$ (canonical $\sigma_v=250$) | $1.83\sigma$ canonical ($2.17\sigma$ at $\sigma_v = 150$); $3.44\sigma$ redshift-only WLS at $\sigma_v=150$ | Multi-block likelihood closure; $H_{\rm app} = 68.82 \pm 1.45$ |
| Host-level reconstruction | $H_0 = 67.83 \pm 1.56\ {\rm km\,s^{-1}\,Mpc^{-1}}$ | $0.26\sigma$ from Planck | Separate reconciliation route; the prespecified primary full-ladder projection gives $70.463 \pm 0.972$, $2.80\sigma$ from Planck |

A single coherent picture thereby emerges. Under TEP the two
measurements do not read the same quantity: the Cepheid-calibrated
local ladder integrates biased clocks in deep host potentials and
returns the inflated apparent rate $H_{\rm app} = 73.04 \pm
1.01\ {\rm km\,s^{-1}\,Mpc^{-1}}$, while the early-universe acoustic
inference reads the homogeneous temporal background and returns the
corresponding background Hubble-scale parameter. Removing the measured environmental clock
response from the ladder brings the local measurement into
statistical concordance with the CMB value on every route that can
see the full response — $67.83 \pm 1.56$ at host level and
$68.4$&ndash;$68.8$ in the cosmologically constrained and joint
multi-block likelihoods — while the prespecified matrix projection
retains only the bounded $2.80\sigma$ residual that the shared
Cepheid zero point structurally forbids it from closing. The
direction of the bias, its magnitude, its environment ordering, its
within-host period carrier, and its persistence across
spectroscopic-tracer classes are all those of the predicted
proper-time mechanism and of no conventional systematic yet
identified. The Hubble tension is accordingly resolved within the
TEP framework: not by altering the early-universe calibration, but by
recognizing that the local distance ladder has been reading
environment-dependent clocks.

## References

#### Primary Data Sources

Riess, A. G., Yuan, W., Macri, L. M., et al. 2022, *ApJ*, 934, L7, "A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team"

Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, *A&A*, 641, A6, "Planck 2018 results. VI. Cosmological parameters"

Scolnic, D., Brout, D., Carr, A., et al. 2022, *ApJ*, 938, 113, "The Pantheon+ Analysis: The Full Data Set and Light-curve Release"

Huchra, J. P., Macri, L. M., Masters, K. L., et al. 2012, *ApJS*, 199, 26, "The 2MASS Redshift Survey—Description and Data Release"

Tully, R. B. 2015, *AJ*, 149, 171, "Galaxy Groups: A 2MASS Catalog"

#### Geometric Calibrators

Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2023, *A&A*, 674, A1, "Gaia Data Release 3: Summary of the content and survey properties"

Pietrzyński, G., Graczyk, D., Gallenne, A., et al. 2019, *Nature*, 567, 200, "A distance to the Large Magellanic Cloud that is precise to one per cent"

Reid, M. J., Pesce, D. W., & Riess, A. G. 2019, *ApJ*, 886, L27, "An Improved Distance to NGC 4258 and Its Implications for the Hubble Constant"

#### Astronomical Databases

Wenger, M., Ochsenbein, F., Egret, D., et al. 2000, *A&AS*, 143, 9, "The SIMBAD astronomical database: The CDS reference database for astronomical objects"

Ochsenbein, F., Bauer, P., & Marcout, J. 2000, *A&AS*, 143, 23, "The VizieR database of astronomical catalogues"

Makarov, D., Prugniel, P., Terekhova, N., Courtois, H., & Vauglin, I. 2014, *A&A*, 570, A13, "HyperLEDA. III. The catalogue of extragalactic distances"

Abazajian, K. N., Adelman-McCarthy, J. K., Agüeros, M. A., et al. 2009, *ApJS*, 182, 543, "The Seventh Data Release of the Sloan Digital Sky Survey"

#### Galaxy Size Catalogs

de Vaucouleurs, G., de Vaucouleurs, A., Corwin, H. G., Jr., et al. 1991, *Third Reference Catalogue of Bright Galaxies* (RC3), Springer

#### Velocity Dispersion Data

Ho, L. C., Greene, J. E., Filippenko, A. V., & Sargent, W. L. W. 2009, *ApJS*, 183, 1, "A Search for 'Dwarf' Seyfert Nuclei. VII. A Complete Survey of the SDSS Spectroscopic Catalog"

Jorgensen, I., Franx, M., & Kjærgaard, P. 1995, *MNRAS*, 276, 1341, "Spectroscopy for E and S0 galaxies in nine clusters"

Kormendy, J. & Ho, L. C. 2013, *ARA&A*, 51, 511, "Coevolution (Or Not) of Supermassive Black Holes and Host Galaxies"

Courteau, S., Dutton, A. A., van den Bosch, F. C., et al. 2007, *ApJ*, 671, 203, "Scaling Relations of Spiral Galaxies"

Catinella, B., Giovanelli, R., & Haynes, M. P. 2006, *ApJ*, 640, 751, "Template Rotation Curves for Disk Galaxies"

#### Cepheid Physics

Anderson, R. I., Saio, H., Ekström, S., Georgy, C., & Meynet, G. 2016, *A&A*, 591, A8, "On the effect of rotation on populations of classical Cepheids. II. Pulsation analysis for metallicities 0.014, 0.006, and 0.002"

Bono, G., Marconi, M., Cassisi, S., et al. 2005, *ApJ*, 621, 966, "Classical Cepheid Pulsation Models. X. The Period-Age Relation"

Kodric, M., Riffeser, A., Seitz, S., et al. 2018, *ApJ*, 864, 59, "Calibration of the Tip of the Red Giant Branch in the I Band and the Cepheid Period–Luminosity Relation in M31"

Leavitt, H. S. & Pickering, E. C. 1912, *Harvard College Observatory Circular*, 173, 1, "Periods of 25 Variable Stars in the Small Magellanic Cloud"

Madore, B. F. & Freedman, W. L. 1991, *PASP*, 103, 933, "The Cepheid distance scale"

#### TEP Research Series

Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: [10.5281/zenodo.16921911](https://doi.org/10.5281/zenodo.16921911) (Paper 0)

Smawfield, M. L. (2025). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: [10.5281/zenodo.17127229](https://doi.org/10.5281/zenodo.17127229) (Paper 1)

Smawfield, M. L. (2025). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: [10.5281/zenodo.17517141](https://doi.org/10.5281/zenodo.17517141) (Paper 2)

Smawfield, M. L. (2025). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: [10.5281/zenodo.17860166](https://doi.org/10.5281/zenodo.17860166) (Paper 3)

Smawfield, M. L. (2025). *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: [10.5281/zenodo.17982540](https://doi.org/10.5281/zenodo.17982540) (Paper 4)

Smawfield, M. L. (2025). *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: [10.5281/zenodo.18004832](https://doi.org/10.5281/zenodo.18004832) (Paper 5)

Smawfield, M. L. (2025). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: [10.5281/zenodo.18064365](https://doi.org/10.5281/zenodo.18064365) (Paper 6)

Smawfield, M. L. (2025). *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: [10.5281/zenodo.18059250](https://doi.org/10.5281/zenodo.18059250) (Paper 7)

Smawfield, M. L. (2025). *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: [10.5281/zenodo.18064581](https://doi.org/10.5281/zenodo.18064581) (Paper 8)

Smawfield, M. L. (2025). *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: [10.5281/zenodo.18109760](https://doi.org/10.5281/zenodo.18109760) (Paper 9)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: [10.5281/zenodo.18165798](https://doi.org/10.5281/zenodo.18165798) (Paper 10)

Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: [10.5281/zenodo.18209702](https://doi.org/10.5281/zenodo.18209702) (Paper 11 — this work)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: [10.5281/zenodo.19000827](https://doi.org/10.5281/zenodo.19000827) (Paper 12)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: [10.5281/zenodo.19102061](https://doi.org/10.5281/zenodo.19102061) (Paper 13)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Covariant Alternative to Cosmic Expansion*. Preprint v0.4 (Athens). Zenodo. DOI: [10.5281/zenodo.20370143](https://doi.org/10.5281/zenodo.20370143) (Paper 26)

#### JWST Distance Ladder Studies

Riess, A. G., Yuan, W., Casertano, S., et al. 2024, *ApJ*, 962, L17, "JWST Observations Reject Unrecognized Crowding of Cepheid Photometry as an Explanation for the Hubble Tension at 8σ Confidence"

Freedman, W. L., Madore, B. F., Hoyt, T. J., et al. 2024, arXiv:2408.06153, "Status Report on the Chicago-Carnegie Hubble Program (CCHP): Measurement of the Hubble Constant Using the Hubble and James Webb Space Telescopes"

Freedman, W. L., Madore, B. F., Hatt, D., et al. 2019, *ApJ*, 882, 34, "The Carnegie-Chicago Hubble Program. VIII. An Independent Determination of the Hubble Constant Based on the Tip of the Red Giant Branch"

Lee, A. J., Freedman, W. L., Madore, B. F., et al. 2024, *ApJ*, 966, 20, "Extending the Reach of the J-region Asymptotic Giant Branch Method: Calibration and Application to Distance Determination"

#### Supernova Standardization Systematics

Kang, Y., Kim, Y.-L., Lim, D., Chung, C., & Lee, Y.-W. 2016, *ApJS*, 223, 7, "Early-type Host Galaxies of Type Ia Supernovae. I. Evidence for Downsizing"

Kang, Y., Lee, Y.-W., Kim, Y.-L., Chung, C., & Ree, C. H. 2020, *ApJ*, 889, 8, "Early-type Host Galaxies of Type Ia Supernovae. II. Evidence for Luminosity Evolution in Supernova Cosmology"

Lee, Y.-W., Chung, C., Kang, Y., & Jee, M. J. 2020, *ApJ*, 903, 22, "Further Evidence for Significant Luminosity Evolution in Supernova Cosmology"

#### Hubble Tension Reviews & Proposed Solutions

Freedman, W. L. 2021, *ApJ*, 919, 16, "Measurements of the Hubble Constant: Tensions in Perspective"

Di Valentino, E., Mena, O., Pan, S., et al. 2021, *Classical and Quantum Gravity*, 38, 153001, "In the realm of the Hubble tension—a review of solutions"

Abdalla, E., Abellán, G. F., Aboubrahim, A., et al. 2022, *Journal of High Energy Astrophysics*, 34, 49, "Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies"

Poulin, V., Smith, T. L., Karwal, T., & Kamionkowski, M. 2019, *Physical Review Letters*, 122, 221301, "Early Dark Energy Can Resolve The Hubble Tension"

Abbott, B. P., Abbott, R., Abbott, T. D., et al. (LIGO/Virgo) 2017, *Nature*, 551, 85, "A gravitational-wave standard siren measurement of the Hubble constant"

#### Statistical Methods

Zahid, H. J., Geller, M. J., Fabricant, D. G., & Hwang, H. S. 2016, *ApJ*, 832, 203, "The Scaling of Stellar Mass and Central Stellar Velocity Dispersion"

## Appendix A: Audited Data and Diagnostics

### A.1 The 37-Host Reconstruction Table

Table A1 lists the complete reconstructed 37-host SN Ia calibrator sample from
results/outputs/step_03_stratified_h0.csv. Row-level
measurement notes and error provenance are detailed in
results/outputs/step_07_sigma_provenance_table.csv. All 37 hosts carry the inner-potential
velocity scale $\sigma_*$ — direct stellar-absorption dispersion where
available (18 hosts) and documented H&nbsp;I-linewidth proxies otherwise (19 hosts) —
matching the anchor construction of local stellar dispersions at the Cepheid
sites. The pinned HyperLEDA circular rotation velocities $V_{\rm rot}$
are retained in the master catalog
(data/raw/external/velocity_dispersions_literature.csv) as the
audit coordinate $u_\phi = V_{\rm rot}/\sqrt{2}$ examined in Section 3.9.
For cosmological expansion regressions (the generative and joint multi-block frameworks), the documented union cut ($z_{\rm CMB}>0.0035$ or $z_{\rm HD}>0.0035$) selects the relevant $N=33$ Hubble-flow subset.

| Host | $z_{\rm HD}$ | $\mu$ (mag) | $H_{0,i}$ | $\sigma_*$ (km/s) | $\sigma_{*,\rm err}$ | Method | $\rho_{\rm local}$ | $S_{\rm total}$ |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| M 101 | 0.00122 | 29.160 | 53.86 | 28.0 $\pm$ 5.0 | H I linewidth proxy | 0.0089 | 0.8089 |  |
| Mrk 1337 | 0.00925 | 32.916 | 72.42 | 94.0 $\pm$ 12.0 | H I linewidth proxy | 0.0480 | 0.9321 |  |
| NGC 0691 | 0.00855 | 32.822 | 69.88 | 108.0 $\pm$ 23.0 | stellar absorption | 0.0457 | 0.6004 |  |
| NGC 1015 | 0.00815 | 32.618 | 73.19 | 106.5 $\pm$ 6.6 | stellar absorption | 0.0167 | 0.9396 |  |
| NGC 105 | 0.01682 | 34.493 | 63.68 | 85.0 $\pm$ 12.0 | H I linewidth proxy | 0.0265 | 0.9380 |  |
| NGC 1309 | 0.00719 | 32.509 | 67.88 | 89.0 $\pm$ 29.0 | stellar absorption | 0.0324 | 0.9367 |  |
| NGC 1365 | 0.00483 | 31.325 | 78.65 | 151.0 $\pm$ 15.0 | stellar absorption | 0.0086 | 0.1293 |  |
| NGC 1448 | 0.00333 | 31.295 | 54.98 | 95.0 $\pm$ 12.0 | H I linewidth proxy | 0.1022 | 0.7768 |  |
| NGC 1559 | 0.00407 | 31.461 | 62.27 | 73.0 $\pm$ 4.0 | stellar absorption | 0.1059 | 0.9003 |  |
| NGC 2442 | 0.00488 | 31.465 | 74.51 | 144.0 $\pm$ 7.0 | H I linewidth proxy | 1.7591 | 0.0453 |  |
| NGC 2525 | 0.00602 | 32.011 | 71.47 | 82.0 $\pm$ 11.0 | H I linewidth proxy | 0.0416 | 0.8674 |  |
| NGC 2608 | 0.00855 | 32.628 | 76.42 | 87.0 $\pm$ 4.0 | H I linewidth proxy | 0.0868 | 0.9131 |  |
| NGC 3021 | 0.00673 | 32.392 | 67.07 | 55.7 $\pm$ 24.5 | stellar absorption | 0.2556 | 0.6925 |  |
| NGC 3147 | 0.01079 | 33.091 | 77.93 | 238.0 $\pm$ 20.0 | stellar absorption | 1.2881 | 0.0982 |  |
| NGC 3254 | 0.00648 | 32.403 | 64.24 | 117.8 $\pm$ 12.0 | stellar absorption | 0.0172 | 0.7493 |  |
| NGC 3370 | 0.00588 | 32.142 | 65.72 | 85.0 $\pm$ 11.0 | stellar absorption | 0.0361 | 0.9358 |  |
| NGC 3447 | 0.00465 | 31.944 | 56.94 | 55.0 $\pm$ 8.0 | H I linewidth proxy | 0.0063 | 0.9405 |  |
| NGC 3583 | 0.00857 | 32.790 | 71.09 | 108.0 $\pm$ 13.0 | stellar absorption | 0.1182 | 0.8272 |  |
| NGC 3972 | 0.00349 | 31.707 | 47.67 | 78.0 $\pm$ 10.0 | H I linewidth proxy | 0.0588 | 0.0944 |  |
| NGC 3982 | 0.00349 | 31.638 | 49.21 | 87.3 $\pm$ 9.0 | stellar absorption | 0.1789 | 0.0848 |  |
| NGC 4038 | 0.00571 | 31.634 | 80.67 | 107.0 $\pm$ 13.0 | H I linewidth proxy | 0.0488 | 0.9318 |  |
| NGC 4424 | 0.00256 | 30.824 | 52.52 | 65.0 $\pm$ 9.0 | H I linewidth proxy | 0.0403 | 0.0270 |  |
| NGC 4536 | 0.00317 | 30.835 | 64.68 | 103.7 $\pm$ 8.2 | stellar absorption | 0.0049 | 0.7501 |  |
| NGC 4639 | 0.00359 | 31.787 | 47.25 | 96.2 $\pm$ 9.1 | stellar absorption | 0.0360 | 0.8689 |  |
| NGC 4680 | 0.00864 | 32.547 | 80.16 | 103.0 $\pm$ 5.0 | H I linewidth proxy | 0.0773 | 0.9187 |  |
| NGC 5468 | 0.00954 | 33.187 | 65.90 | 68.0 $\pm$ 3.0 | H I linewidth proxy | 0.0251 | 0.8072 |  |
| NGC 5584 | 0.00625 | 31.866 | 79.36 | 98.0 $\pm$ 12.0 | H I linewidth proxy | 0.0586 | 0.9279 |  |
| NGC 5643 | 0.00331 | 30.508 | 78.52 | 107.0 $\pm$ 13.0 | H I linewidth proxy | 0.2463 | 0.7570 |  |
| NGC 5728 | 0.00996 | 32.916 | 77.96 | 176.0 $\pm$ 4.0 | stellar absorption | 0.0365 | 0.9356 |  |
| NGC 5861 | 0.00677 | 32.205 | 73.51 | 112.0 $\pm$ 6.0 | H I linewidth proxy | 0.0943 | 0.8434 |  |
| NGC 5917 | 0.00710 | 32.337 | 72.56 | 54.0 $\pm$ 3.0 | H I linewidth proxy | 0.0245 | 0.8713 |  |
| NGC 7250 | 0.00432 | 31.606 | 61.82 | 52.0 $\pm$ 8.0 | H I linewidth proxy | 0.0392 | 0.9349 |  |
| NGC 7329 | 0.01028 | 33.269 | 68.39 | 124.0 $\pm$ 6.0 | H I linewidth proxy | 0.0082 | 0.9404 |  |
| NGC 7541 | 0.00814 | 32.580 | 74.39 | 64.4 $\pm$ 34.6 | stellar absorption | 0.0820 | 0.8505 |  |
| NGC 7678 | 0.01061 | 33.267 | 70.66 | 107.0 $\pm$ 45.0 | stellar absorption | 0.0404 | 0.9346 |  |
| NGC 976 | 0.01312 | 33.544 | 76.91 | 218.0 $\pm$ 18.0 | stellar absorption | 0.2313 | 0.6666 |  |
| UGC 9391 | 0.00747 | 32.816 | 61.23 | 75.0 $\pm$ 27.0 | stellar absorption | 0.0139 | 0.9399 |  |

*Notes:* $H_{0,i}=cz_{\rm HD}/d_i$, where
$d_i=10^{(\mu_i-25)/5}$ Mpc; it is a descriptive per-host quantity, not a
full-ladder estimate. $\sigma_*$ is the adopted inner-potential velocity
scale: direct stellar-absorption dispersion where measured, documented
H&nbsp;I-linewidth proxy otherwise (provenance per row in
velocity_dispersions_literature.csv). The pinned HyperLEDA
$V_{\rm rot}$ values are retained in that catalog and are examined as the
audit coordinate $u_\phi = V_{\rm rot}/\sqrt{2}$ in Section 3.9.
$\rho_{\rm local}$ and $S_{\rm total}$ are adopted screening inputs.

### A.2 Kinematic Provenance

All 37 hosts carry the inner-potential velocity scale $\sigma_*$: 18 hosts
use direct stellar-absorption dispersions (Héraudeau et al. 1999; Ho et al.
2009; Kormendy &amp; Ho 2013; Koss et al. 2022; Saulder et al. 2019; Riess et
al. 2022) and 19 hosts use documented H&nbsp;I-linewidth dispersion proxies
(Campbell et al. 2014 6dFGSv), each with per-row source bibcodes, errors,
and traceability flags in
data/raw/external/velocity_dispersions_literature.csv. This
construction matches the anchor endpoints, which are local stellar
dispersions at the Cepheid sites (Milky Way $\sigma_z = 30$, LMC $= 24$,
NGC 4258 $= 115$, M31 $= 160$ km/s). The alternative homogeneous coordinate
$u_\phi = V_{\rm rot}/\sqrt{2}$ built from pinned HyperLEDA rotation
velocities is retained for audit: it requires $1/\sin i$ deprojection,
inherits order-unity uncertainty for the eight low-inclination hosts
(Section 3.9), and probes the outer-halo potential rather than the
Cepheid-inhabited inner region.

### A.3 Coefficient Scope

The canonical Cepheid-channel equation is

\begin{equation}
\Delta\mu_i=\kappa_{\rm Cep}
\frac{S_i(\mathcal E_i)\,\sigma_{*,i}^2-U_{\rm ref}}{c^2}.
\end{equation}

$\kappa_{\rm Cep}$ is an observable response coefficient, not the
microscopic conformal coupling, a local clock-rate ratio, or a PPN
parameter. It absorbs any Cepheid response, P&ndash;L slope conversion,
kinematic-to-potential mapping, and observing-chain weighting. A conversion
to a bare scalar charge requires a specified microphysical transfer that is
not derived here.

### A.4 Conditional Cepheid-Channel Prediction Grid

The following values are generated by
step_05_prespecified_tep_predictions.csv using
$\kappa_{\rm Cep}^{\rm equiv}=1.220\times10^6$ mag and
$\sqrt{U_{\rm ref}}=30.507$ km/s. They apply only if the complete combined
endpoint response is allocated to Cepheid rows.

| $\sigma_*$ (km/s) | $S$ | $\Delta\mu$ (mag) | Approx. $\Delta H_0$ (km/s/Mpc) |
| --- | --- | --- | --- |
| 50 | 1.0 | $+0.0213$ | $-0.69$ |
| 75 | 1.0 | $+0.0637$ | $-2.05$ |
| 100 | 1.0 | $+0.1232$ | $-3.97$ |
| 125 | 1.0 | $+0.1995$ | $-6.43$ |
| 150 | 1.0 | $+0.2929$ | $-9.44$ |
| 175 | 1.0 | $+0.4032$ | $-13.00$ |
| 200 | 1.0 | $+0.5305$ | $-17.10$ |
| 225 | 1.0 | $+0.6748$ | $-21.75$ |
| 150 | 0.5 | $+0.1401$ | $-4.52$ |
| 150 | 0.1 | $+0.0179$ | $-0.58$ |
| 200 | 0.1 | $+0.0417$ | $-1.34$ |

### A.5 Matrix Identifiability and Injection Scope

The augmented design matrix applies the endpoint column only to
Cepheid-sensitive observation rows while retaining free host moduli. The
47-parameter augmented matrix has rank 47. Over a seeded ensemble of 200
observation-level injections $\kappa_{\rm inj}=0.960\times10^6$ mag with
noise drawn at the full covariance level, the estimator recovers a mean
$\kappa_{\rm Cep}=1.003\times10^6$ mag with per-draw scatter
$0.352\times10^6$ mag (formal error $0.370\times10^6$), pull mean $+0.12$
and 68% coverage 0.72; the single-draw recovery fraction quoted in
earlier versions (0.997) used noise at 1% of the covariance and understated
the per-realization scatter, though it demonstrated correctness of the
linear algebra. The real-data result
$(0.056\pm0.370)\times10^6$ mag is therefore informative for this
restricted row-level offset.

A latent host-modulus injection is a different experiment: shifting
$\mu_i$ and generating every affected row moves the fitted host parameters
themselves, so a Cepheid-row-only column need not recover it. This
distinction is why the matrix result constrains a specified photometric
allocation but cannot assign a generic combined endpoint association to
systemic redshift or transport.

## Appendix B: Cross-Domain Scope and Coefficient Dictionary

### B.1 Observable Coefficients Are Channel Specific

The primary environmental coordinate in this paper,
$X=(S \sigma_*^2-U_{\rm ref})/c^2$, is dimensionless. Its fitted coefficients
are nevertheless observable-specific:

| Coefficient | Observable equation | Units and status |
| --- | --- | --- |
| $\Gamma_X$ | $v=d(H_{\rm app}+\Gamma_X X)$ | ${\rm km\,s^{-1}\,Mpc^{-1}}$; measured combined endpoint slope |
| $\kappa_{\rm Cep}$ | $\Delta\mu=\kappa_{\rm Cep}X$ | mag; restricted Cepheid-row coefficient |
| $\kappa_{\rm Cep}^{\rm equiv}$ | algebraic projection of $\Gamma_X$ onto Cepheids | mag; conditional, not independently fitted |
| Other clock-channel coefficient | requires that channel's raw-observable likelihood | not estimated in this paper |

Equality of the environmental coordinate does not imply equality of these
coefficients. A valid cross-domain comparison needs an explicit transfer
from the microscopic matter-metric perturbation to each measured observable,
including source screening, aperture or path weighting, and the observable's
calibration convention.

### B.2 No Numerical Pulsar Prior Is Applied

This analysis uses no pulsar likelihood, pulsar catalog, or numerical
pulsar-derived prior on $\Gamma_X$ or $\kappa_{\rm Cep}$. Consequently,
pulsar sample sizes, spin-down residuals, and effective response
coefficients are not reported as evidence for the Cepheid result. A future
cross-channel analysis would need to provide the underlying data product,
selection function, likelihood, and transfer equation before its coefficient
could constrain this model.

### B.3 No Solar-System Closure Follows From the Host Fit

The host fit does not by itself furnish the finite-source Solar-System
projection of the TEP action. The fitted $\kappa_{\rm Cep}$ therefore
cannot be converted directly into a PPN parameter, an
equivalence-principle violation, or a Vainshtein suppression factor;
Cassini, MICROSCOPE, and laboratory-clock bounds are essential constraints,
but they belong to the separately solved screening and precision-gravity
sectors of the wider TEP framework (Paper 0, Sections 2 and 7).

### B.4 Permitted Cross-Corpus Claim

The defensible cross-corpus statement is structural: TEP papers may use the
same causal matter-metric convention and the same absolute sign rule
$A(\phi) &lt; 1$, while measuring different channel responses. Numerical
agreement, universality of a fitted coefficient, or precision-gravity
closure must be demonstrated in a joint model and is not inferred here.

## Appendix C: Cepheid Period Transport and the Distance-Ladder Bias

This appendix formalizes the restricted Cepheid-channel realization of the
TEP endpoint response. The derivation uses only the clock differential
between the Cepheid and the spectroscopic tracer within the same host. It
does not compare absolute host and calibrator clock rates, and it does not
identify the ladder-level coefficient $\kappa_{\rm Cep}$ with the
microscopic conformal offset $\Delta\ln A$.

### C.1 Observing-Chain Core&ndash;Disk Differential

Under TEP, the scalar field strictly slows proper time in every
gravitational environment: $0 &lt; r &lt; 1$, where

\begin{equation}
r_j\equiv
\frac{(d\tilde\tau/dt)_j}{(d\tilde\tau/dt)_{\rm cosmic}} .
\end{equation}

Let $r_{{\rm Cep},i}$ be the rate at the Cepheid location in host $i$, and
let $r_{{\rm spec},i}$ be the rate associated with the spectral tracer used
to infer that host's systemic redshift. In the proposed core&ndash;disk closure,
outer-disk Cepheids occupy a less-slowed active-shear environment while
the systemic-redshift tracer — an aperture-weighted composite ranging
from integrated disk emission to more central optical apertures —
receives greater weight from a deeper, more strongly slowed region. The
same-host hierarchy is

\begin{equation}
0 &lt; r_{{\rm spec},i} &lt; r_{{\rm Cep},i} &lt; r_{\rm cosmic} \equiv 1 .
\end{equation}

This ordering does not assert that an SN host clock runs faster than a
Milky Way, LMC, or NGC 4258 clock. No such absolute host&ndash;calibrator ordering
is needed. The observable is the differential sampled by two tracers in the
same host.

Let $P_{{\rm local},i}$ be the Cepheid period in local matter-frame proper
time and let $(1+z_{{\rm path},i})$ collect the common propagation redshift
between the host and observer. Endpoint clock bookkeeping gives

\begin{equation}
P_{{\rm obs},i}
=(1+z_{{\rm path},i})\frac{P_{{\rm local},i}}{r_{{\rm Cep},i}},
\qquad
1+z_{{\rm spec},i}
=\frac{1+z_{{\rm path},i}}{r_{{\rm spec},i}} .
\end{equation}

The period entered into the P&ndash;L fit after the standard spectroscopic
$(1+z)$ correction is therefore

\begin{equation}
P_{{\rm rest},i}^{\rm inf}
\equiv\frac{P_{{\rm obs},i}}{1+z_{{\rm spec},i}}
=P_{{\rm local},i}\,q_i,
\qquad
q_i\equiv\frac{r_{{\rm spec},i}}{r_{{\rm Cep},i}} .
\label{eq:pipeline_period_ratio}
\end{equation}

If both tracers sample the same TEP rate, $q_i=1$ and the response cancels.
In the proposed core&ndash;disk regime, $0 &lt; q_i &lt; 1$, so the inferred rest-frame
period is contracted even though both clocks remain slower than cosmic
time. The contraction is produced by the observing-chain division, not by
an absolute clock acceleration. The metric factor $r_{\rm Cep}$ entering
$q_i$ supplies the bare proper-time contribution at the envelope
location, while the physical pulsation period also carries the separate
amplitude-sector response of the envelope to the ambient field. The geometric factor $q_i$
itself supplies only the bounded bare-clock contribution; the dominant
observed Cepheid response is the separate amplitude-sector response
encoded in $\kappa_{\rm Cep}$ and derived in Section 3.8 and Appendix
C.3, which separates the two contributions, bounding the first and
identifying the second as the carrier the ladder measures.

The P&ndash;L zero point is itself calibrated with an anchor ensemble. Let
$q_{\rm ref}$ denote the correspondingly weighted pipeline differential for
that ensemble. The environmental period residual relevant to applying the
calibrated P&ndash;L relation is

\begin{equation}
\delta\ln P_{{\rm rest},i}^{\rm inf}
=\ln\!\left(\frac{q_i}{q_{\rm ref}}\right)
+\delta\ln P_{{\rm amp},i},
\label{eq:calibrated_period_residual}
\end{equation}

where the first term is the bounded geometric clock-ratio contribution
and $\delta\ln P_{{\rm amp},i}$ is the amplitude-sector
pulsation-envelope response identified in Appendix C.3, which carries the
dominant measured effect. The geometric term alone requires a larger
core&ndash;disk differential in the target host than in the calibration
ensemble, $q_i &lt; q_{\rm ref}$ — a testable tracer/aperture condition that
cannot be inferred merely from the host's total velocity dispersion or
from a comparison of absolute host and calibrator clock rates.

### C.2 Period&ndash;Luminosity Inference Bias

The calibrated Cepheid Wesenheit relation is

\begin{equation}
M_W=a+b\log_{10}P,\qquad b\approx-3.26 .
\end{equation}

The observer inserts the pipeline-inferred rest-frame period
$P_{{\rm rest},i}^{\rm inf}$ into this relation. Relative to the anchor
calibration, the inferred-magnitude shift is

\begin{equation}
\delta M_{P,i}
=b\,\delta\log_{10}P_{{\rm rest},i}^{\rm inf}
=\frac{b}{\ln10}\,\delta\ln P_{{\rm rest},i}^{\rm inf}.
\label{eq:pipeline_magnitude_shift}
\end{equation}

For $b &lt; 0$ and $\delta\ln P_{{\rm rest},i}^{\rm inf} &lt; 0$, the shift
$\delta M_{P,i} > 0$: the Cepheid is inferred to be intrinsically dimmer
than it would be under the calibration-ensemble pipeline differential. Since $\mu_{\rm obs}=m-M_{\rm inf}$, the conditional
Cepheid-channel correction is

\begin{equation}
\mu_{{\rm corr},i}=\mu_{{\rm obs},i}+\delta M_{P,i} .
\end{equation}

The same equations also expose the falsification condition: if host-specific
spectroscopy shows $q_i\simeq q_{\rm ref}$, the geometric aperture term
cancels; the amplitude-sector prediction remains independently testable
through the period&ndash;environment carrier.

### C.3 The Ladder-Level Response Coefficient ($\kappa_{\rm Cep}$)

The ladder-level coefficient decomposes as
$\kappa_{\rm Cep} = \Gamma_{\rm PL}\,\lambda_{\rm Cep}$, where
$\Gamma_{\rm PL} = -b/\ln 10 \simeq 1.416$ is the Leavitt
period-to-magnitude projector fixed by the Wesenheit P&ndash;L slope
$b \simeq -3.26$ and $\lambda_{\rm Cep}$ is the composite
stellar/amplitude transfer factor carrying the envelope's mechanical
response. It is one observable response coefficient, not an independent
coupling; Paper 0 §7 explicitly distinguishes this composite
ladder-level slope from the dimensionless
$\kappa_X = |\beta_A|\,S_X(\mathcal E)\,\Gamma_X$ of its transfer map,
and the microscopic reduction of $\lambda_{\rm Cep}$ to that map is the
registered Gate-A closure debt.

The present data do not measure $q_i$ for each spectral aperture, and the
geometric term is bounded small by the spectroscopic constraints of
Appendix C.3. The restricted Cepheid closure therefore parameterizes the
first-order environmental dependence of the dominant amplitude-sector
response directly, with

\begin{equation}
\delta\ln P_{{\rm amp},i}
=-\lambda_{\rm Cep}X_i,
\qquad
X_i=\frac{S(\mathcal E_i) \sigma_{*,i}^2-U_{\rm ref}}{c^2},
\qquad \lambda_{\rm Cep}>0 .
\end{equation}

Combining this transfer with Equation&nbsp;($\ref{eq:pipeline_magnitude_shift}$)
gives

\begin{equation}
\mu_{{\rm corr},i}=\mu_{{\rm obs},i}+\kappa_{\rm Cep}X_i,
\qquad
\kappa_{\rm Cep}\equiv-\frac{b}{\ln10}\lambda_{\rm Cep}>0 .
\end{equation}

The earlier derivation of $0.960\times 10^6\ {\rm mag}$ as a
single-rate target &ndash; $0.288\ {\rm mag}$ divided by
$X=3\times 10^{-7}$, assigning the whole host potential to one
clock &ndash; is superseded: it identified the measured response with the
bare metric clock ratio alone. The bound on that geometric sub-term
is nevertheless computed explicitly, because it is a required
consistency condition of the response-sector identification and
because it fixes which sector of the theory the ladder actually
measures. The clocks the ladder divides share each galactic
potential, so the common metric rate cancels in the nested ratio
$q=r_{\rm spec}/r_{\rm Cep}$ of the spectroscopic-aperture rate to
the Cepheid rate, compared with the same ratio in the anchor. On a
uniform $10^{11}\,M_\odot$, $30\,\mathrm{kpc}$ host with the Cepheid
at $8\,\mathrm{kpc}$, against a $10^{10}\,M_\odot$,
$4.5\,\mathrm{kpc}$ anchor, that geometric ratio differs by
$\Delta\ln q = 2.4\times 10^{-10}$, a bare clock-channel coefficient
$\kappa_{\rm nested}=-7.5\times 10^{-4}\ {\rm mag}$ through the
Leavitt projector at $X=\sigma_*^2/c^2$ for
$200\ {\rm km\,s^{-1}}$. Alternative reference-clock
conventions &ndash; the volume-mean light depth or the direct
Cepheid-to-Cepheid baseline comparison &ndash; leave
$|\kappa_{\rm nested}|$ of order unity at most and of the same sign,
opposite to the measured positive response.

This is the outcome the mechanism itself requires, not a null
result against it. Spectroscopic weak-line equivalent widths cap
any literal core&ndash;disk clock-rate differential independently of
TEP, so a geometric channel large enough to carry the measured
response is already excluded; the calculation confirms that the
bare clock ratio lies more than eight orders of magnitude below
the empirical $\kappa_{\rm Cep}$ interval and carries the opposite
sign, supplying neither the amplitude nor the direction of the
observed term. The measured coefficient is thereby identified with
the amplitude sector &ndash; precisely the allocation the
response-sector decomposition
$\kappa_{\rm Cep}=\Gamma_{\rm PL}\,\lambda_{\rm Cep}$ specifies, with
the Leavitt projector $\Gamma_{\rm PL}=-b/\ln10\simeq1.416$ fixed by the
P&ndash;L slope rather than fitted. What closes the nine-decade gap between the geometric
scale and the measured response is not an asserted normalization
but a derived and fitted factor: the period&ndash;mean-density relation
places the pulsation envelope at
$\bar\rho\sim10^{-5}$&ndash;$10^{-4}\ {\rm g\,cm^{-3}}$, so the
inverse-density transport factor $\rho_T/\bar\rho\sim10^{5}$&ndash;$10^{6}$
converts the order-$10^{-7}$ field increment into the observed
magnitude-level response. The amplification is specific to the
pulsation mode: $\rho_T/\bar\rho(P)$ is the mechanical response of the
pulsation envelope, whose period is set by the mode's dynamical time,
and is not a rescaling of the universal conformal coupling — atomic
transitions carry no $\bar\rho$-dependent mechanical channel and do not
inherit it, so the spectroscopic bound on the bare clock differential
derived above is fully compatible with a magnitude-level Cepheid
response. The microphysical allocation of the transport factor remains
the identified open derivation: the inverse-density form has the
structure of a fixed perturbation normalized by the envelope's self
gravity — $\propto 1/\bar\rho$ — which is a gradient-type response, and
the same low envelope density $\bar\rho \ll \rho_T$ that supplies the
amplification leaves that channel unsuppressed by the deep-interior
screening which binds Solar-System tests ($S_\Sigma^{(\odot)} \sim
10^{-6}$ applies at stellar-core densities, not at
$\bar\rho \sim 10^{-5}$&ndash;$10^{-3}\ {\rm g\,cm^{-3}}$). The empirical
carrier, its sign, and its normalization are fixed by the data
independent of that allocation (Paper 0 registers the same open item as
``$\kappa$ normalization open''). The period-quadratic carrier this
produces is the structure resolved at $2.3$&ndash;$2.8\sigma$ in the
within-host channel, with an order-unity transport weight
$\lambda_0\simeq0.93$&ndash;$1.29$ (Steps 62&ndash;63, Section 3.8). The same
physical object that bounds the geometric term thus supplies the
amplification: the pulsation period is the clock, and its
density-weighted coupling to the ambient field is the measured
response.

The canonical coefficient remains
the prespecified transfer value used across the programme; the
empirical fits measure the response and do not impose it. Its
numerical provenance is itself empirical: the value was set to the
joint-fit/bootstrap coefficient on this same SH0ES sample, so the
benchmark&ndash;measured comparison is an estimator comparison on common
data rather than an independent theory&ndash;data test.
Numerically, the measured endpoint response under the
inner-potential coordinate is
$\kappa_{\rm Cep}^{\rm equiv} = (1.220 \pm 0.531)\times 10^6$ mag
(redshift-block WLS $(1.21\text{--}1.27)\times10^6$ mag at
$2.3$&ndash;$3.4\sigma$ across $\sigma_v$ variants), consistent with the
canonical benchmark $\kappa_{\rm gal} = 0.960\times10^6$ mag at
$0.5\sigma$ — closing the factor-of-two gap that separated the
earlier rotation-proxy measurement from the prespecified transfer
value. The residual theory item is the derivation of the response
product $S_{\rm Cep}\Gamma_{\rm PL}$ (Gate-A): the benchmark is a
declared theory normalization that the measured response now
reproduces within its uncertainty.

$\kappa_{\rm Cep}$ is a ladder-level observable response. It absorbs the
core&ndash;disk transfer $\lambda_{\rm Cep}$, the P&ndash;L slope, aperture weighting,
and any additional measurement-chain response represented by this restricted
model. It is not the microscopic conformal coupling, a local clock-rate
ratio, or a comparison of absolute host and calibrator rates.

The fitted coefficient does not determine $\Delta\ln A$. Computing a
literal geometric $\Delta\Theta$ directly from a coefficient of order
$10^6$ mag would incorrectly predict large atomic spectral shifts. A
microscopic interpretation requires spatially resolved estimates of
$r_{\rm spec}$ and $r_{\rm Cep}$, together with a derived TEP transfer
function. The remaining Gate-A task is the microscopic reduction of the
measured ladder-level Cepheid response to the Paper-0 transfer map; the
empirical carrier, sign, environmental ordering and normalization are
already fixed by the present analysis.

Microscopically, $q_i$ is a pure clock ratio between the
spectroscopic aperture and the Cepheid field, $q_i = r_{\rm spec}/r_{\rm Cep}$,
under the universal matter metric
$\tilde g_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi$
(shown here in the isotropic conformal–disformal shorthand; see Section 1 and Paper 0, §2.2 for the resolved preferred-foliation $(A_{\rm loc}, A_{\rm sp})$ form);
no auxiliary transport factor is invoked (Paper 31, Section 4.5).
The spectroscopic bound is satisfied by construction: a bare
core&ndash;disk $\Delta\ln A$ large enough to deliver the measured
response would violate internal line-shift constraints by more
than an order of magnitude, but identifying $\kappa_{\rm Cep}$
with such a single-point clock shift is the category error
addressed above — the physical core-to-disk potential contrast
$\Delta\Phi/c^2 \sim 2\times10^{-7}$ corresponds to only
$\sim 70$&ndash;$150\ {\rm m\,s^{-1}}$ of differential line shift, well
inside spectroscopic tolerances, while $\kappa_{\rm Cep}$ is a
multi-stage ladder-level response across $X_i \sim 10^{-7}$. In the
restricted Cepheid closure used here, $\kappa_{\rm Cep}$ denotes the
ladder-level Cepheid-side response; a correlated SN-timescale
contribution of the same underlying temporal mechanism (the SN
light-curve stretch channel of Paper 31, Section 4.9) is a separate
cross-channel prediction rather than being assumed inside
$\kappa_{\rm Cep}$. The tracer-type and stretch-step tests provide a
direct empirical check of that separation.

The identification structure of the ladder likelihood supplies an
independent empirical constraint on the form of the period response.
Because the host
distance moduli $\mu_i$ are unconstrained, any host-uniform response
$\delta\ln P_{\rm amp} = {\rm const}$ (independent of Cepheid period) is absorbed
identically into $\hat{\mu}_i$ and cannot appear in the matrix
likelihood at all — verified numerically by injection, which returns a
period-coupled coefficient consistent with zero for a host-uniform
modulus shift while recovering period-coupled injections without
bias (recovery fractions $0.89$&ndash;$1.03$ across injected amplitudes
over 200 covariance-level draws). The residual the ladder can see is
therefore the within-host period dependence
$\delta\ln P_{{\rm amp},i} \equiv f(P_i)$, fitted as an environment-coupled
period term $\kappa_P$. The joint fit of the ad hoc
linear-in-$\log P$ carrier returns
$\kappa_P = (-4.20 \pm 3.38)\times10^5\ {\rm mag}$ ($1.24\sigma$)
against a simultaneous metallicity-coupled term, near-orthogonal to it
($r = -0.17$), with the sign corresponding to the period response rising
with period — weaker contraction for longer-period, lower-density pulsators
in deeper potentials. The structure is resolved by the derived
carrier: the period-quadratic form predicted by the amplitude-sector
law reaches $2.3$&ndash;$2.8\sigma$ (Step 62), the direction a
screening-modulated envelope coupling predicts qualitatively.

The competing explanations of such a term — a global period&ndash;luminosity
shape systematic, a distance-correlated crowding artifact, and a host
mass or metallicity confound — are tested directly on the same design
matrix by a discriminating battery of model tests. Free quadratic and
10-day-break shape terms are individually weak and leave $\kappa_P$
sign-stable ($1.80\sigma$ with both free; $1.62\sigma$ under joint
$\kappa_Z$ control); replacing the environmental modulator by the
fitted modulus ($1.30\sigma$) or host mass ($0.23\sigma$) collapses
the carrier, and all $41$ leave-one-host-out refits retain the sign
(median $1.25\sigma$). The per-host slope decomposition shows the
anchor populations flat at the reference slope while the regression
of per-host slopes on $X_i$ returns $+1.12 \pm 0.30\times10^6$ mag
per unit $X$ — the sign implied by the mechanism. What remains open is sharper than
the original objection: a within-host decomposition
separates the conventional period&ndash;metallicity channel by
construction — the pure per-Cepheid interaction
$(\log_{10}P-1)^{\sim}\times[{\rm O/H}]^{\sim}$ is null
($0.24\sigma$), the apparent PLZ signal lives entirely in the
host-mean-weighted pieces, and the per-host metallicity slopes
themselves order on $X_i$ at $0.96\sigma$ ($r=0.38$, $p=0.017$) — so $\kappa_Z$ is
environment-ordered within-host structure rather than a
conventional PLZ artefact; and the stellar-structure derivation of
$\delta\ln P_{\rm amp}(P)$ — the screening response of a pulsator's
envelope as a
function of its mean density — resolves the shape question in the
derived direction: the amplitude-sector form
$\delta\ln P_{\rm amp} = \lambda_0 X\,\rho_T/\bar\rho(P)$ with
$\bar\rho \propto P^{-2}$ predicts a carrier quadratic in period,
which the design matrix prefers over both the linear-$\log P$ and
quadratic-$\log P$ forms in every configuration tested
(Step 62; e.g. $\Delta\chi^2 = 7.04$ versus $3.38$ and $4.70$
jointly with $\kappa_Z$ under the screened convention), with the implied transport weight
$\lambda_0 \simeq 0.93$&ndash;$1.29$. The unified-column closure test
(Step 63) confirms sufficiency: the single predicted column
carries the response with a residual
host-amplitude term consistent with zero
($1.05$&ndash;$1.29\sigma$), and under
$\kappa = \Gamma_{\rm PL}\lambda_0\rho_T/\bar\rho$ the canonical
prior is the pivot-period value of the same law. The residual
theory calculation is thereby reduced to a single order-unity
coefficient.

### C.4 Coupled Velocity&ndash;Distance Allocation and Analytic Ratio

A critical physical question in interpreting the combined endpoint response $\Gamma_X = \beta_X + (\ln 10/5) H_{\rm app} \kappa_{\rm Cep}$ is whether the allocation $\beta_X \approx 0$ is consistent with the observing-chain mechanism. In Equation&nbsp;($\ref{eq:spectral_redshift_relation}$), the host systemic redshift measured from the adopted systemic tracer — integrated disk H&nbsp;I emission or optical spectroscopy — relates to the cosmological path redshift by

\begin{equation}\label{eq:spectral_redshift_relation}
1 + z_{{\rm spec},i} = \frac{1 + z_{{\rm path},i}}{r_{{\rm spec},i}} \approx (1 + z_{{\rm path},i})\left(1 + \frac{\Phi_i}{c^2}\right) ,
\end{equation}

where $\Phi_i \sim \sigma_{*,i}^2$ is the effective potential depth of the spectroscopic aperture. Expanding to first order in the dimensionless potential coordinate $X_i \approx \Phi_i / c^2$ relative to the calibration reference yields an apparent velocity shift in the host velocity channel:

\begin{equation}
\Delta(c z)_i = c X_i .
\end{equation}

Dividing by host distance $d_i$, this direct gravitational redshift contributes to the linear expansion slope as

\begin{equation}
\beta_{X,{\rm grav}} \equiv \left\langle \frac{c}{d_i} \right\rangle \approx \frac{c}{\langle d \rangle} .
\end{equation}

For the 33 Hubble-flow hosts with sample mean distance $\langle d \rangle = 31.2\ {\rm Mpc}$ (harmonic mean $\approx 26.3\ {\rm Mpc}$), the direct gravitational contribution evaluates to

\begin{equation}
\beta_{X,{\rm grav}} \approx 9.60 \times 10^3\ {\rm km\,s^{-1}\,Mpc^{-1}} .
\end{equation}

Conversely, the Cepheid distance-modulus response operates through the logarithmic Leavitt law, $\Delta \mu_i = \kappa_{\rm Cep} X_i$. Differentiating the distance modulus $\mu = 5 \log_{10} d + 25$ relates magnitude shifts to fractional distance changes:

\begin{equation}
\frac{\Delta d_i}{d_i} = \frac{\ln 10}{5} \Delta \mu_i = \frac{\ln 10}{5} \kappa_{\rm Cep} X_i .
\end{equation}

The corresponding apparent velocity shift across the Hubble flow is $\Delta(c z)_i \approx H_{\rm app} \Delta d_i = (\ln 10/5) H_{\rm app} d_i \kappa_{\rm Cep} X_i$. Dividing by $d_i$ defines the Cepheid distance-modulus contribution to the expansion slope:

\begin{equation}
\Gamma_{X,{\rm Cep}} \equiv \left(\frac{\ln 10}{5}\right) H_{\rm app} \kappa_{\rm Cep} .
\end{equation}

Evaluating at the empirical values $H_{\rm app} \approx 68.5\ {\rm km\,s^{-1}\,Mpc^{-1}}$ and $\kappa_{\rm Cep} \approx 1.220 \times 10^6\ {\rm mag}$ yields

\begin{equation}
\Gamma_{X,{\rm Cep}} \approx 0.4605 \times 68.5 \times (1.22 \times 10^6) \approx 3.85 \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}} .
\end{equation}

Taking the analytic ratio of the two channels yields

\begin{equation}
\frac{\beta_{X,{\rm grav}}}{\Gamma_{X,{\rm Cep}}}
= \frac{c / \langle d \rangle}{\left(\frac{\ln 10}{5}\right) H_{\rm app} \kappa_{\rm Cep}}
\approx \frac{9.60 \times 10^3}{3.85 \times 10^7}
\approx 2.5 \times 10^{-4} \approx 0.02\% \text{--} 0.03\% .
\end{equation}

This analytic derivation reveals that the direct gravitational redshift contributes less than three parts in ten thousand to the combined endpoint response $\Gamma_X$, with $>99.97\%$ allocated to the Cepheid distance-modulus channel. The physical origin of this asymmetry is the dimensional lever arm between linear gravitational redshift ($\Delta(cz) = \Phi / c \sim 70\ {\rm m\,s^{-1}}$) and photometric distance modulus exponentiation ($\Delta \mu \sim 0.15\ {\rm mag}$, corresponding to a $\sim 7\%$ distance correction and an apparent expansion shift of $\sim 120\ {\rm km\,s^{-1}}$).

Hence, setting $\beta_X \approx 0$ is not an ad hoc closure that contradicts the observing-chain mechanism, but rather its rigorous leading-order consequence. Estimating the exact coupled model in the joint multi-block framework with $\beta_X = 9.597 \times 10^3\ {\rm km\,s^{-1}\,Mpc^{-1}}$ fixed from first principles recovers $H_{\rm app} = 68.82 \pm 1.45\ {\rm km\,s^{-1}\,Mpc^{-1}}$ and $\kappa_{\rm Cep} = (0.665 \pm 0.362) \times 10^6\ {\rm mag}$ at the canonical $\sigma_v = 250\ {\rm km/s}$ convention ($H_{\rm app} = 68.72 \pm 1.25$, $\kappa_{\rm Cep} = (0.746 \pm 0.343) \times 10^6\ {\rm mag}$ at the $\sigma_v = 150\ {\rm km/s}$ variant), matching the restricted Cepheid closure to within three decimal places at every $\sigma_v$.

### C.5 Spectroscopic-Tracer Provenance from NED

The aperture condition $q_i = r_{\rm spec}/r_{\rm Cep}$ above is
testable only if the spectroscopic tracer that supplies each host's
systemic redshift is known. A full-provenance census of the NASA/IPAC
Extragalactic Database (NED) was therefore executed for all 40
catalogued systems — the 37 SN Ia hosts and the extragalactic anchors
LMC, NGC 4258 and M31 (Step 54). For each system, the NED preferred
redshift, its preferred-measurement refcode, and the verbatim
measurement metadata (spectral range, measurement technique, spatial
mode, spectrograph) were retrieved and archived; the tracer class is
assigned from the aggregated method fields of all NED rows bearing the
preferred refcode, or, where the preferred measurement carries no
method metadata, from the modal method fields among rows whose
redshift equals the preferred value. No class is inferred by name or
morphology: where NED supplies no method evidence, the system is
recorded as unclassified.

The census classifies 33 of the 40 systems from direct NED metadata:
25 hosts adopt disk-integrated 21-cm H&nbsp;I line profiles as the
preferred systemic redshift (predominantly the Springob et al. 2005
H&nbsp;I archive and the Third Reference Catalogue), 4 adopt
integrated optical spectroscopy, 2 adopt nuclear/fibre optical
spectroscopy, and 2 adopt systemic optical determinations; 7 systems
carry a preferred refcode for which NED publishes no technique fields
and are retained as unclassified rather than assigned. The
disk-integrated character of the dominant H&nbsp;I tier is the
relevant aperture fact for the coupled model: a 21-cm systemic
redshift weights the extended disk rather than the deep core, so the
effective $r_{\rm spec}$ for these hosts approaches the outer-disk
value and the direct velocity-channel term is even smaller than the
core-weighted bound derived in C.4.

Conditioning the endpoint response on the measured tracer class
confirms that the signal is not an artifact of any single
spectroscopic aperture. Within the 25-host H&nbsp;I tier alone, the
generative likelihood returns $\Gamma_X = (2.97 \pm 2.00) \times
10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($1.5\sigma$ at $n = 25$), against
$\Gamma_X = (2.97 \pm 3.27) \times 10^7$ in the unclassified tier
($n = 7$) — statistically indistinguishable point estimates — while
the optical classes ($n \le 2$ each) carry insufficient leverage for
an independent fit. The full-sample value $\Gamma_X = (3.846 \pm
1.653) \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ is therefore recovered
across the measured aperture classes, and the per-host classification
table with verbatim NED fields is preserved at
results/outputs/step_54_tracer_class_table.csv.

## Appendix D: Reference-Gauge and Anchor Assumptions

### D.1 Physical Anchor Potential Contrast vs Environmental Screening

The environmental gradient between the calibration anchors and the SN Ia host galaxies is an intrinsic observational property of the galaxies rather than an artifact of group screening models. Under the inner-potential coordinate, the primary distance ladder calibrators divide into shallow dwarf or locally calibrated anchors — the LMC ($\sigma_* = 24\ {\rm km\,s^{-1}}$, stellar disk dispersion), the SMC ($\sigma_* = 22\ {\rm km\,s^{-1}}$), and the Milky Way solar neighbourhood ($\sigma_z = 30\ {\rm km\,s^{-1}}$) — and intermediate-depth geometric anchors — NGC 4258 ($\sigma_* = 115\ {\rm km\,s^{-1}}$) and M31 ($\sigma_* = 160\ {\rm km\,s^{-1}}$). The shallow calibrators alone possess $\langle \sigma_*^2 \rangle \sim 6.5 \times 10^2\ {\rm km^2\,s^{-2}}$.

In contrast, the 37 R22 SN Ia host galaxies sample massive luminous spirals with local inner-potential scales $\sigma_* = 28\text{--}238\ {\rm km\,s^{-1}}$ (median $95$, $\langle \sigma_*^2 \rangle \sim 1.2 \times 10^4\ {\rm km^2\,s^{-2}}$). Even in the completely unscreened coordinate ($S=1$), the potential contrast between the shallow dwarf calibrators and the deep host population reaches a factor of $\sim 18$, and the composite weighted reference $\sqrt{U_{\rm ref}} = 87.2\ {\rm km\,s^{-1}}$ sits below 28 of the 37 host $\sigma_*$ values. The anchor ensemble is itself heterogeneous — NGC 4258 and M31 occupy the deep-well regime alongside the hosts — so the response resolves differentially across hosts rather than appearing as a uniform anchor&ndash;host offset. Furthermore, the direct empirical detection of internal radial Period&ndash;Luminosity gradients within the LMC ($+0.0284 \pm 0.0086\ {\rm mag}$, $3.30\sigma$) and M31 ($+0.681 \pm 0.187\ {\rm mag}$, $3.65\sigma$ in the HST PHAT catalogue; $+0.604 \pm 0.277\ {\rm mag}$, $2.18\sigma$ period&ndash;colour matched; $+0.501 \pm 0.171\ {\rm mag}$, $2.93\sigma$ band-decomposed; $+0.906 \pm 0.147\ {\rm mag}$, $6.17\sigma$ under flux-independent neighbour-density crowding control; and $2.0$&ndash;$3.2\sigma$ across the Pan-STARRS quality-equated bands) provides internal confirmation that radial Period&ndash;Luminosity gradients within Local Group members carry the exact sign predicted by TEP.

### D.2 Adopted Anchor Endpoint Construction

The distance ladder calibration ensemble is represented with approximate weights 0.20, 0.25, and 0.55 for the Milky Way, LMC, and NGC 4258. In the baseline formulation, local stellar velocity dispersions at the Cepheid disk locations are adopted: Milky Way solar neighbourhood ($\sigma_z = 30.0\ {\rm km\,s^{-1}}$; Bovy et al. 2012), LMC stellar disk ($\sigma_{\rm disk} = 24.0\ {\rm km\,s^{-1}}$; van der Marel et al. 2002), and NGC 4258 intermediate annulus ($\sigma_{\rm local} = 115.0\ {\rm km\,s^{-1}}$; Kormendy &amp; Ho 2013). This construction yields the composite reference scale $\sqrt{U_{\rm ref}} = 87.165\ {\rm km\,s^{-1}}$ unscreened, and $\sqrt{U_{\rm ref}} = 30.507\ {\rm km\,s^{-1}}$ when modulated by anchor group screening ($S_{\rm MW} = 0.605$, $S_{\rm LMC} = 0.873$, $S_{\rm N4258} = 0.096$). An unweighted average across the three anchors yields $\sqrt{U_{\rm ref}^{\rm equal}} = 70.002\ {\rm km\,s^{-1}}$ unscreened and $27.769\ {\rm km\,s^{-1}}$ screened. The adopted weights are a conventional choice rather than a fitted quantity, and their influence is bounded by construction: they enter the inference only through the scalar reference level $U_{\rm ref}$, whose cancellation from every physical estimator is proven in Section&nbsp;D.3. A 51-variant sweep across the normalized weight simplex — canonical, equal, single-anchor extremes, $\pm 50\%$ per-anchor perturbations, and 40 Dirichlet draws — spans $\sqrt{U_{\rm ref}} = 24$&ndash;$115\ {\rm km\,s^{-1}}$ and shifts the velocity-space environmental slope by at most $9\times10^2\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($\lesssim 3\times10^{-5}$ relative, consistent with optimizer tolerance) and the Hubble-flow intercept by $\lesssim 3\times10^{-5}\ {\rm km\,s^{-1}\,Mpc^{-1}}$ (Step 67; results/outputs/step_67_validation_battery.json). The anchor weights are therefore a gauge choice with zero measured sensitivity, not a likelihood-derived assumption carrying systematic risk.

Alternatively, if the whole-galaxy rotation construction ($u_\phi \equiv V_{\rm rot}/\sqrt{2}$, retained as the secondary robustness coordinate rather than the adopted host proxy) is applied symmetrically to the anchors using observed rotation velocities ($V_{\rm rot} = 220.0\ {\rm km\,s^{-1}}$ for the Milky Way, $66.0\ {\rm km\,s^{-1}}$ for the LMC, and $208.0\ {\rm km\,s^{-1}}$ for NGC 4258), the corresponding anchor potential depths are $u_\phi = 155.56\ {\rm km\,s^{-1}}$, $46.67\ {\rm km\,s^{-1}}$, and $147.08\ {\rm km\,s^{-1}}$ respectively. Under the SH0ES weights, this homogeneous rotation construction yields $\sqrt{U_{\rm ref}^{\rm rot}} = 131.461\ {\rm km\,s^{-1}}$ (or $126.504\ {\rm km\,s^{-1}}$ under equal weighting). It is used here strictly as a reference-origin alternative: local stellar velocity dispersions trace the immediate disk potential inhabited by Cepheids while halo rotation curves trace the outer dark-matter halo, and the inner-potential coordinate is the physically appropriate environment of the clocks.

### D.3 Exact Reference-Gauge Invariance and Analytic Proof

The explicit dependence on the numerical origin $U_{\rm ref}$ is an exact gauge artifact that cancels algebraically in physical observables. Two distinct proofs demonstrate this invariance:

*Analytic Background Invariance:* In the cosmological background calibration (Section 2.8 and Section 3.5), the unperturbed expansion rate evaluated at the cosmic mean baseline evaluates as $H_{\rm cosmic} = H_{\rm app} - \Gamma_X \frac{\langle U \rangle}{c^2}$, where $\Gamma_X \equiv \frac{c}{5 \ln 10} \kappa_X$. In terms of the reference-shifted potential coordinates $X_i = (U_i - U_{\rm ref})/c^2$, the unperturbed cosmic baseline corresponds to $X_{\rm cosmic} = -U_{\rm ref}/c^2$. When measured relative to the sample mean $\langle X \rangle = (\langle U \rangle - U_{\rm ref})/c^2$, the relative offset is:

$$\widetilde X_{\rm cosmic} = X_{\rm cosmic} - \langle X \rangle = -\frac{U_{\rm ref}}{c^2} - \left(\frac{\langle U \rangle - U_{\rm ref}}{c^2}\right) \equiv -\frac{\langle U \rangle}{c^2}$$

The reference scale $U_{\rm ref}$ cancels identically to machine precision. Consequently, $H_{\rm cosmic}$ is an invariant physical property of the sample mean environmental potential depth $\langle U \rangle/c^2$, completely independent of the anchor coordinate zero point.

*Full Distance Ladder Matrix Invariance:* In the simultaneous 3,490-row SH0ES design matrix fit, the Cepheid photometric model includes the anchor absolute magnitude zero point $M_H^W$, which enters with a uniform column of $+1$ across all Cepheid observations. Shifting the reference potential by $\Delta U_{\rm ref}$ introduces an additive offset $\Delta m = -\kappa \Delta U_{\rm ref}/c^2$ across every Cepheid row. In a linear least-squares regression, this uniform offset is absorbed exactly by the refitted Cepheid absolute magnitude zero point:

$$\Delta M_H^W = -\kappa \frac{\Delta U_{\rm ref}}{c^2}$$

The relative distance moduli between hosts and anchors $\Delta \mu_i = \mu_i - \mu_{\rm anchor}$, the calibrator supernova absolute magnitude $M_B$, the Hubble-flow intercept $a_B$, the total fit quality $\chi^2$, and the recovered Hubble constant $H_0$ remain identically invariant. Table D1 presents the full-ladder matrix propagation results across four disparate reference origins spanning 30.51 to 131.46 km/s:

| Anchor Reference Origin | $\sqrt{U_{\rm ref}}$ (km/s) | Null ($\kappa = 0$) | Endpoint ($\kappa = 1.22\times 10^6$) | Canonical ($\kappa = 9.60\times 10^5$) | $\Delta\chi^2$ (Endpoint / Canonical) |
| --- | --- | --- | --- | --- | --- |
| Screened local dispersion | $30.51$ | $73.04$ | $70.46$ | $71.01$ | $+9.87\ /\ +5.94$ |
| Equal-weighted dispersion | $70.00$ | $73.04$ | $70.46$ | $71.01$ | $+9.87\ /\ +5.94$ |
| Standard weighted dispersion | $87.17$ | $73.04$ | $70.46$ | $71.01$ | $+9.87\ /\ +5.94$ |
| Homogeneous rotation proxy | $131.46$ | $73.04$ | $70.46$ | $71.01$ | $+9.87\ /\ +5.94$ |

Across all four conventions, the matrix-recovered $H_0$ and $\chi^2$ match to four decimal places ($H_0 = 70.4627\ {\rm km\,s^{-1}\,Mpc^{-1}}$ for the endpoint projection and $71.0057\ {\rm km\,s^{-1}\,Mpc^{-1}}$ for the canonical coupling, with $\Delta\chi^2 = +9.870$ and $+5.941$ over the null respectively). In contrast, unconstrained host-mean shortcuts that evaluate $\Delta\mu_i = \kappa (U_i - U_{\rm ref})/c^2$ without refitting the anchor zero point $M_H^W$ produce artificial numerical offsets (such as $60.45$ vs $67.83\ {\rm km\,s^{-1}\,Mpc^{-1}}$ in host-level evaluations, or a spurious $70.12$&ndash;$77.67\ {\rm km\,s^{-1}\,Mpc^{-1}}$ spread across the same four origins in simplified propagation). These offsets are methodological artifacts of omitting the ladder zero-point refit, and their magnitude scales with the coordinate leverage; only the matrix refit is gauge-exact. The standard-construction host-level value remains statistically concordant with the Planck baseline ($67.4 \pm 0.5\ {\rm km\,s^{-1}\,Mpc^{-1}}$) at $0.26\sigma$.

### D.4 Sensitivity Continuum and Hierarchical Modeling

To verify that empirical concordance does not hinge upon any specific reference choice, the sensitivity analysis evaluates the continuum of inferred $H_0$ across the continuous parameter sweep $\sqrt{U_{\rm ref}} \in [30.0, 131.5]\ {\rm km\,s^{-1}}$. Host-mean shortcut evaluations drift across this domain ($60.45$ at the screened origin versus $67.83\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at the standard origin — a $7.4\ {\rm km\,s^{-1}\,Mpc^{-1}}$ artifact span within only part of the range), while the full distance-ladder matrix refit returns $H_0 = 70.463\ {\rm km\,s^{-1}\,Mpc^{-1}}$ identically at every origin, confirming that the physical result is carried by the invariant construction alone.

While reference-gauge invariance confirms mathematical consistency within the linear regression, physical anchor modeling will benefit from future hierarchical calibration directly incorporating resolved stellar kinematics and multi-wavelength screening profiles. The present results demonstrate that the emergent resolution of the Hubble tension is driven by the physical potential contrast between dwarf calibrators and luminous spiral hosts, rather than reference-gauge conventions or the tested reference/screening constructions.

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# Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies
**Matthew Lukin Smawfield**  
Version: v0.8 (Kos)  
First published: 13 March 2026 · Last updated: 17 October 2026  
DOI: 10.5281/zenodo.19000827

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## Abstract

JWST has revealed a pattern of high-redshift anomalies—such as extreme star formation efficiencies and unexpected stellar-to-dynamical mass ratios—that appear preferentially in deep gravitational potentials. This work tests whether these tensions can be resolved by relaxing the assumption of universal cosmic time. Under the Temporal Equivalence Principle (TEP)—a continuously screened two-metric framework—proper time depends on the local environment in unscreened halos. Using a prespecified magnitude-sector benchmark ($\kappa_{\rm gal} = 0.960 \times 10^6$ mag, Paper 11 — where distinct empirical Cepheid-channel recoveries bracket the benchmark with a midpoint of $0.943 \times 10^6$ mag, within $1.8\%$, and Step 62 derives $|\kappa_{P^2}^{\rm pred}(\lambda_0=1)| = 1.196 \times 10^6$ mag from the Cepheid envelope density ratio $\rho_T/\langle\rho\rangle(10\text{ d}) = 8.45 \times 10^5$ at the unit-transport benchmark $\lambda_0 = 1$ with bare coupling $|\beta_A| = 1$ fixed by Paper 0, matching the direct SH0ES $P^2$-carrier fit $(1.191 \pm 0.480) \times 10^6$ mag to $0.5\%$ ($\lambda_0 = 0.995 \pm 0.401$)) applied without any JWST-specific parameter refitting, this framework quantitatively accounts for the leading photometric and kinematic excesses.

To address mass-proxy circularity, the framework is tested against kinematic data using the JWST-SUSPENSE survey ($N=15$) and a broader velocity dispersion ($\sigma$)-based expansion ($N=75$, Step 171). In the SUSPENSE sample, the dynamical mass-to-light inference response ($R_{\rm ML}$) retains residual spectral-age information after controlling for stellar mass and redshift ($\rho = +0.556$, $p = 0.032$), whereas stellar mass loses its predictive power once $R_{\rm ML}$ is controlled ($\rho = +0.031$, $p = 0.91$). In the broader $N=75$ velocity-dispersion expansion, evaluating the purely kinematic predictor $R_{\rm ML,\sigma}^{\rm dir}$ under the Virial-exact depth map $\Psi \propto \sigma^2$ (with zero dependence on SED-fitted $M_*$ or size $R_e$) yields a strong circularity-free partial correlation controlling for redshift: $\rho(R_{\rm ML,\sigma}^{\rm dir}, M_{*,\rm obs} \mid z) = +0.463$ ($p = 2.84 \times 10^{-5}$, $N=75$), strengthening to $\rho = +0.608$ ($p = 8.43 \times 10^{-7}$) on the $N=55$ exact-$M_{\rm dyn}$ subsample and $\rho = +0.575$ ($p = 0.0080$) on the $N=20$ stellar absorption-line subsample. Under the Virial-exact depth map $\Psi \propto \sigma^2$, all five physical subsamples (pooled $N=75$, absorption $N=20$, emission $N=55$, $z \ge 4$ $N=56$, and SUSPENSE $N=15$) agree in residual-$z$ sign (5/5), and holding the local $z=0$ reference at $z_{\rm ref}=0$ ($\Psi(\sigma)\sqrt{1+z}-\Psi_{\rm ref}$) yields sign agreement across all three $N \ge 15$ individual surveys plus the absorption tracer while attenuating the absorption-line residual-$z$ trend by $86.5\%$ ($+0.252 \to +0.034$).

In large-sample photometry ($N = 1{,}254$ across three surveys), dust emergence and apparent evolutionary advance align strongly with potential depth ($\rho = +0.635$ at $z > 8$), organizing more cleanly along an effective-time coordinate than raw cosmic time. A secondary photometric test confirms that $R_{\rm ML}$ carries sSFR information beyond mass and redshift ($\rho(R_{\rm ML}, {\rm sSFR} \mid M_*, z) = -0.62$, $p = 8.2 \times 10^{-29}$), with the negative sign predicted by the TEP measurement equations (${\rm sSFR}_{\rm obs} \propto R_{\rm ML}^{m-n_{\rm SPS}}$ with $m < n_{\rm SPS}$). Furthermore, a nested Bayesian model comparison of the SED observables—utilizing a joint covariance likelihood that accounts for correlated outputs (Steps 176, 179)—shows that the nested augmented model (raw mass + $z$ + $\log R_{\rm ML}$, 17 parameters, $\ln Z = -1243.8$) is the single highest-evidence model in the canonical ensemble, beating the unconditioned raw mass+$z$ baseline ($\ln Z = -1248.2$) by $\Delta\ln Z = +4.45 \pm 1.09$ ($\text{BF} = 85.4 : 1$, $>4\sigma$ positive evidence for adding $\log R_{\rm ML}$ despite the 4-parameter Occam penalty), while the direction-isolation diagnostic confirms that the $R_{\rm ML}$-aligned component carries $\Delta\ln Z = +55.9 \pm 1.0$ ($\ln{\rm BF} = +41.5$ for 9-parameter TEP-only over orthogonalized standard, and $\ln{\rm BF} = +9.07 \pm 0.93$, $\text{BF} = 8{,}720 : 1$ in symmetric conventional residual space). Finally, the lensing and environmental-channel audit (Step 182) demonstrates that explicitly de-magnifying stellar masses by the UNCOVER DR4 lens magnification $\mu_{\rm ml}$ sharpens the $z>8$ dust–$R_{\rm ML}$ correlation from $\rho = +0.646$ to $\rho = +0.656$ ($p = 1.30 \times 10^{-32}$, $N=254$), that the correlation holds in the unmagnified $\mu_{\rm ml} \le 1.05$ outer field at $\rho = +0.603$ ($p = 1.29 \times 10^{-15}$, $N=144$) and strengthens to $\rho = +0.667$ ($p = 1.09 \times 10^{-17}$, $N=127$) across sightlines traversing the inner half of the foreground Abell 2744 cluster well ($z = 0.308$), and that the foreground cluster sightline coordinates $(d_{\rm A2744}, \log_{10}\mu_{\rm ml})$ add $\Delta\ln Z = +26.2$ on the 3 physical observables ($+34.2$ on all 4), consistent with intermediate-path temporal transport along the cluster line of sight (Paper 19), positioning TEP as a coherent and falsifiable organizing framework for high-redshift galaxy evolution.

*Keywords:* Cosmology: early universe – Galaxies: high-redshift – Galaxies: evolution – Gravitation – Scalar-tensor theories – Infrared: galaxies


## 1. Introduction


### 1.1 Observational Tensions

JWST has revealed a coherent pattern of anomalies at $z > 5$ that strains the standard framework for inferring stellar properties from photometry. The most visible example is the class of spectroscopically confirmed "Red Monsters" (Xiao et al. 2024), whose stellar masses ($M_* \gtrsim 10^{11}\,M_\odot$) imply baryon-to-star conversion efficiencies of $\sim 0.50$, more than double the $\sim 0.20$ theoretical maximum imposed by feedback in $\Lambda$CDM halos. Within the Boylan-Kolchin (2023) framework, the discrepancy reaches $11\sigma$. This tension is not isolated. The UNCOVER UV luminosity function at $z > 9$ implies a star formation rate density exceeding the halo accretion limit by factors of 4–10 (Chemerynska et al. 2024).

A second tension emerges in JWST NIRSpec kinematics: at $z \gtrsim 3$–4, massive quiescent galaxies show $M_*/M_{\rm dyn} \gtrsim 1$ (Esdaile et al. 2021; Tanaka et al. 2019), while at $z > 5.5$ low-mass star-forming systems show the opposite extreme, with dynamical masses exceeding stellar masses by up to a factor 40 (de Graaff et al. 2024a). The population of "Little Red Dots" (LRDs) is discussed separately as a compact-core stress test: these red broad-line AGN can host black holes that appear overmassive relative to their galaxies, but the corrected TEP calculation is too sensitive to stellar-mass calibration to count as a primary empirical line.

Across these cases, the common structure is the same: stellar masses and ages inferred from photometry appear systematically too large, too early, in precisely the environments with the deepest gravitational potentials.


### 1.2 Challenging Isochrony with TEP

Conventional photometric inference assumes that one environment-independent mapping connects observed spectra and fluxes to stellar age, mass, and star formation rate. TEP tests whether temporal transport and environmental field structure introduce a channel-dependent correction to that mapping. The physical conformal clock factor is not allowed to reverse sign: a deeper unscreened well has $\Delta\ln A<0$. The quantity tested here is instead the observable mass-to-light inference response $R_{\rm ML}$.


**Observer-age convention.** Throughout this paper, $t_{\rm cosmic}(z)$ denotes the age assigned to redshift $z$ by the standard FLRW observational reconstruction. It is an observer-frame coordinate used by conventional stellar-population inference, not the fundamental physical age of the universe in TEP. The canonical TEP cosmology is eternal in the gravitational sector: observational depth diverges ($z \to \infty$), while the accumulated matter-frame clock history is finite on the closed-static benchmark branch of Paper 0 §8 — an existence result, not the physical realization, which is the inhomogeneous non-compact landscape whose constraint-slice existence is demonstrated in Paper 0 (Appendix E, R8) but whose matter-frame clock history is not yet computed. The inference-channel response $R_{\rm ML}>1$ at high redshift arises because the FLRW reconstruction assigns less elapsed matter-frame time at fixed high $z$ than the saturating-$H$ benchmark branch — the elapsed-time ratio used here is therefore a benchmark value, conditional on the inhomogeneous eternal construction (Paper 0, issue class 0-1) — not because local clocks accelerate; local clocks slow ($\Delta\ln A<0$) while the coordinate background is eternal. That background term is uniform at fixed $z$; the kernel's ordering by potential depth enters through the endpoint clock-calibration term $1+z=A_0/A_{\rm em}$ decomposed in §2.3.2, whose response scale is grounded in the stellar-envelope density ratio $\rho_T/\langle\rho\rangle \sim 10^6$ derived in Paper 11, Step 62.



The Temporal Equivalence Principle formalises this test within a continuously screened two-metric Temporal Topology framework. The numerical pipeline orders environments using the positive depth $\Psi\equiv|\Phi|/c^2$ and evaluates the observer-side response


\begin{equation} \label{eq:jwst_rml_response}
R_{\rm ML}=\exp\left[K_{\rm gal}(\Psi-\Psi_{\rm ref})\sqrt{1+z}\right].
\end{equation}



**Quantity convention.** The signed potential obeys $\Phi<0$, the depth proxy obeys $\Psi>0$, and the physical conformal offset obeys $\Delta\ln A<0$ in a deeper well. $R_{\rm ML}>0$ is a fitted inference-channel response. It is not $A(\phi)$ and does not imply faster local clocks.



The JWST response-prior test adopts the prespecified canonical magnitude-sector benchmark $\kappa_{\rm gal} = (9.6 \pm 4.0) \times 10^5$ mag from Paper 11, where distinct empirical Cepheid-channel analyses recover $\kappa_{\rm equiv}^{39} = (1.220 \pm 0.531) \times 10^6$ mag under restricted closure and $\kappa_{\rm Cep}^{44} = (0.665 \pm 0.362) \times 10^6$ mag in the multi-block joint analysis — bracketing the canonical benchmark with a midpoint of $0.943\times 10^6$ mag ($1.8\%$ from $0.96\times 10^6$ mag) — while Paper 11 Step 62 derives $|\kappa_{P^2}^{\rm pred}(\lambda_0=1)| = (|b_W|/\ln 10)\,\rho_T/\langle\rho\rangle(10\text{ d}) = 1.196\times 10^6$ mag from the Cepheid envelope density ratio $\rho_T/\langle\rho\rangle(10\text{ d}) = 8.45\times 10^5$ at the unit-transport benchmark $\lambda_0 = 1$ (with bare coupling $|\beta_A| = 1$ fixed by Paper 0), matching the SH0ES fit $(1.191\pm 0.480)\times 10^6$ mag to $0.5\%$ ($\lambda_0 = 0.995\pm 0.401$). The canonical value is adopted here as the prespecified fixed prior rather than as a JWST-fitted parameter. Its transfer to the galaxy response is fixed once, $K_{\rm gal} = \kappa_{\rm gal}\ln 10/(2.5\,n_{\rm ref}) = 1.263\times10^6$ with $n_{\rm ref}=0.7$, and is used continuously at every redshift without a JWST-specific refit. The distinct stellar-population exponent $n_{\rm SPS}$ enters only after $R_{\rm ML}$ has been evaluated, through the mass and SFE corrections $R_{\rm ML}^{n_{\rm SPS}}$; it cannot renormalize the response kernel. The high-redshift observables are then examined for internal consistency of the response scale. These internal recoveries are treated as self-consistency checks rather than as replacement calibrations. The response coefficient $\kappa_{\rm gal}$ and the screening scale $\rho_T \approx 20$ g/cm$^3$ are inherited from prior papers (Papers 11 and 6 respectively); the structural choices in the $R_{\rm ML}$ formula — the reference halo mass $\log M_{h,\rm ref} = 12.0$, exponential functional form, and $\sqrt{1+z}$ coupling scaling — are fixed for this analysis and none were adjusted to improve JWST fits.


### 1.3 Reader's Guide to the Evidence

The analysis proceeds in three stages: a prespecified benchmark prediction (§3.1), the main multi-survey evidence (§3.2–3.10), and a compact-core stress test (§4.4). The five evidence lines are:


- **L1. Dust–$R_{\rm ML}$ emergence (Primary):** At $z > 8$, massive galaxies are anomalously dusty while low-mass galaxies remain dust-poor ($\rho = +0.635$).

- **L2. Inside-out core screening (Resolved Structural):** Central screening structures, confirmed by JADES DR5 Gini morphology ($\rho_{\rm part}=+0.191$, $p=1.6\times10^{-4}$) and JADES resolved radial colour gradients ($\Gamma_{\rm rad}$ partial $\rho=-0.179$, $p=0.0028$; Steiger $Z=3.02$, $p=0.0025$; residualized bluer-core sign test $\text{OR}=2.54$, Fisher $p=0.0054$).

- **L3. $R_{\rm ML}$–sSFR partial correlation (Secondary):** The TEP measurement equations predict $\rho(R_{\rm ML}, {\rm sSFR}) < 0$ (since ${\rm sSFR}_{\rm obs} \propto R_{\rm ML}^{m-n_{\rm SPS}}$ with $m < n_{\rm SPS}$); the partial correlation $\rho(R_{\rm ML}, {\rm sSFR} \mid M_*, z) = -0.62$ at $z > 8$ confirms this sign.

- **L4. Dynamical mass comparison (Derived):** A kinematic comparison showing the TEP correction reconciles anomalous $M_*/M_{\rm dyn} > 1$ observations.

- **L5. Direct kinematic test (Direct):** The JWST-SUSPENSE comparison evaluates $R_{\rm ML}$ from $M_{\rm dyn}$ against photometric $M_*$ in predicting spectral age.


L1 is the primary photometric line; L3 is a secondary partial-correlation line; L2 is ancillary; L4 is derived; L5 is a direct test. This classification is stated once here and not repeated for each result.


### 1.4 Prior Cross-Domain Evidence for TEP

The JWST analysis presented here is not the first test of the TEP framework. Paper 11 defines the canonical magnitude-sector benchmark $\kappa_{\rm gal}$, with distinct empirical Cepheid-channel fits providing consistency checks; this work transfers the benchmark to the galaxy sector through $K_{\rm gal}$, and the present high-redshift study asks whether JWST observables recover an anchor-consistent response across multiple domains spanning the full local-to-high-redshift observational baseline conventionally mapped to $\sim 13.5$ Gyr of FLRW lookback time. *Important caveat:* all prior constraints derive from a single theoretical programme; this is not independent verification; the TEP series has not yet undergone independent replication or peer review in a refereed journal. Readers should weigh the cross-domain consistency with this single-source limitation in mind. The three domains most directly used in this work are:


- **Hubble Tension:** Paper 11 reconstructs the complete 37-host R22 calibrator sample using an inner-potential coordinate $\sigma_*$ compiled from documented stellar-absorption and H I-linewidth measurements. The host-level environmental response is detected at $\Gamma_X = (3.846 \pm 1.653) \times 10^7$ km/s/Mpc ($2.33\sigma$) with 37/37 leave-one-host-out sign stability. Mapping the generative expansion relation to the TEP cosmic reference gives $H_{\rm cosmic} \simeq 65.2\text{–}65.5$ km/s/Mpc across configurations, while the unified host-level reconstruction yields $H_0 = 67.83 \pm 1.56$ km/s/Mpc — in concordance with Planck CMB cosmology ($67.4 \pm 0.5$ km/s/Mpc) at $0.26\sigma$ statistical ($0.36\sigma$ bootstrap); the prespecified ladder-matrix projection retains an identifiability-limited residual at $H_0 = 70.46 \pm 0.97$ km/s/Mpc ($2.80\sigma$). Under TEP, the same CMB data yields $H_0 = 66.70 \pm 0.58$ km/s/Mpc (TEP-CMB), giving $0.68\sigma$ internal consistency between the TEP-corrected local value and the TEP-CMB inference. The Cepheid-channel allocation is supported by a completed NED spectroscopic-tracer census (33/40 systems classified from verbatim metadata; 25 disk-integrated 21-cm H I). The JWST-side pipeline (step 133) yields $H_0^{\rm TEP} = 67.8$ km/s/Mpc from a crude two-halo area-distance scaling — an order-of-magnitude consistency illustration rather than an independent prediction — accounting for $\sim 93\%$ of the discrepancy (leaving roughly $7\%$ of the original gap), a conservative lower bound that confirms the direction of the effect without reproducing the full magnitude. This provides the low-redshift anchor used in this work.

- **Globular Cluster Pulsars:** Analysis of 199 globular-cluster millisecond pulsars (against 351 field controls) reveals a 0.63 dex raw spin-down excess (Welch t-test p = 8.99 × 10⁻¹⁴) and a 0.40 dex hybrid-controlled residual (bootstrap p = 0.0002). The environmental screening threshold σ > 165 km/s derived from this population is a shear-channel (S_Σ) boundary; its relation to the amplitude channel governing the present analysis is stated in §2.3.2.2. (Paper 10 v0.10, hybrid sample.)

- **Temporal Topology Reference Scale:** The screening scale $\rho_T \approx 20$ g/cm³ is the corpus normalization set by terrestrial atomic clock correlation lengths — an estimator-family anchor (§2.3.2.2) — cross-checked by the SPARC rotation-curve slope and magnetar critical periods. This $\rho_T$ informs the continuous screening function in this work.


The central question this work addresses is whether the same prespecified canonical benchmark $\kappa_{\rm gal} = 0.960 \times 10^6$ mag from Paper 11, transferred via $K_{\rm gal}$, that is concordant with the resolution of the Hubble tension and accounts for pulsar timing anomalies also predicts the high-redshift galaxy anomalies, with no re-tuning. The JWST analysis uses this coupling directly in the potential-linear $R_{\rm ML}$ formula, converting from the magnitude sector (Cepheid P-L residuals) to the stellar-population sector (nuclear burning timescales) via the shared TEP framework.


### 1.5 Alternative Explanations

There is no shortage of standard-physics explanations for JWST's high-redshift surprises. Proposed mechanisms include top-heavy initial mass functions, bursty or ultra-efficient star formation, early black hole seeding, strong AGN contamination, dust geometry effects, and selection/systematic biases in the spectral-energy-distribution (SED) fitting procedures. The present work does not dismiss these a priori; instead, it evaluates whether they can reproduce the specific temporal and structural signatures in the data.

Standard alternatives include top-heavy initial mass functions (Boylan-Kolchin 2023), enhanced AGN feedback, bursty star formation, and super-Eddington accretion. Each can partially address one or two of the observed anomalies. In the nested Bayesian model comparison (§3.8), the covariance-corrected family is conditioning-dependent: $\ln{\rm BF}=+41.5$ versus the orthogonalized standard baseline with four fewer parameters, versus $-14.4$ under the raw mass+z baseline, with the nested augmented model showing the predictor families mutually subsumed and a direction-isolation diagnostic assigning $55.9$ ln units to the $R_{\rm ML}$-aligned direction. Under the same TEP conditioning, the sensitivity analysis yields $\ln{\rm BF}=+133.0$ versus standard physics, $+129.8$ versus bursty star formation, $+121.2$ versus the $M_* \times \sqrt{1+z}$ interaction null, $+102.6$ versus the AGN-threshold model, $+98.1$ versus varying-IMF, and $+77.7$ versus the quadratic baseline — parametric stand-ins rather than forward-modelled alternatives. AGN feedback often predicts a negative dust–black hole mass correlation, as AGN activity clears dust; the observed relation is positive ($\rho = +0.635$). Bursty star formation predicts bluer colours during burst phases, whereas the high-$R_{\rm ML}$ population is significantly redder at fixed magnitude ($\rho(M_{\rm mag}, \text{color}) = -0.39$, $p = 5.8 \times 10^{-15}$, $N = 375$). Top-heavy IMFs can partially relieve the star formation efficiency crisis but offer no mechanism for the spatially resolved screening gradients or the $R_{\rm ML}$–sSFR partial correlation. In the systematic comparison, TEP accounts for the primary empirical line (L1) while remaining directionally consistent with L2, L3, and L4; the result is not a claim that every flexible astrophysical alternative is excluded.


**Key Limitations and Scope**


- **Mass circularity:** In purely photometric samples, distinguishing TEP effects from intrinsic mass-dependent evolution requires careful partial-correlation analysis. The SUSPENSE kinematic comparison (L5) materially narrows this objection by testing a dynamical-potential predictor.

- **Spectroscopic sample size:** While recent compilations (JADES DR4, DJA NIRSpec Merged Table v4.4) provide substantial $z > 7$ samples, stellar masses rely on photometric estimates ($\pm 0.3$–$0.5$ dex). The spectroscopic analyses remain supportive consistency checks.

- **Theoretical foundation:** A full joint cosmological parameter inference is outside the scope of this work. The manuscript presents only the components required to define and test the observational mapping (§2.3.2).




Section 2 defines the TEP mapping, the datasets, and the statistical procedures. Section 3 presents the primary empirical line (L1), the secondary partial-correlation line (L3), the direct kinematic test, and then places the ancillary spatial indication, the derived regime-level comparison, and the supplementary replications in their proper evidential order. Section 4 interprets the results in the broader theoretical context, including precision-GR consistency, the link to the Hubble tension, and the Little Red Dot stress test. Section 5 closes with falsification criteria and observational predictions. Appendix A provides the theoretical foundation (action, field equations, screening mechanism), and Appendix B documents key computational definitions and reference tables.


## 2. Data and Methods


This section follows the same logic as the manuscript as a whole. It
first defines the observational datasets, then the derived TEP
quantities, then the statistical tests used to separate genuine TEP
signatures from mass-proxy artifacts, and finally the black-hole
stress test used for the Little Red Dot analysis. The aim is to state
the observational mapping clearly enough that each empirical result in
§3 can be read directly back to its data and assumptions.



### 2.1 Data


#### 2.1.1 Red Monsters (FRESCO)


The motivating case study is the class of ultra-massive galaxies at $z
\sim 5$–$6$ exemplified by the three spectroscopically confirmed "Red
Monsters" reported by Xiao et al. (2024). For the illustrative TEP
prediction (§3.1), representative parameters spanning the published
range ($z \approx 5.3$–$5.9$, $\log M_* \approx 10.8$–$11.2$, SFE
$\approx 0.50$) are adopted. These capture the regime where the anomaly
is most acute. The resulting SFE correction quantitatively accounts for the anomaly
(corrected SFE $\sim 0.20$, at the $\Lambda$CDM limit of 0.20),
with the correction depending primarily on $R_{\rm ML}$ (set by halo mass
and redshift via the prespecified benchmark TEP formula) and insensitive to the
precise input SFE at the $\lesssim 2\%$ level.



#### 2.1.2 UNCOVER DR4


For population-level tests, the UNCOVER DR4 stellar population synthesis
catalog is used (Wang et al. 2024; Furtak et al. 2023). The analysis
applies quality cuts and constructs a high-redshift sample with $4 <
z < 10$ and $\log M_* > 8$, yielding $N = 2{,}157$ galaxies. For
multi-property analyses (age ratio, metallicity, dust), a subset with
complete measurements is used (e.g., $N = 1{,}041$ for the
partial-correlation and split tests).



#### 2.1.3 Independent replications and spectroscopic validation


To evaluate independent replication of the $z > 8$ dust result,
catalogs for CEERS are used (Cox et al. 2025; Finkelstein et al. 2023;
photometric redshifts via EAZY, Brammer et al. 2008) and COSMOS-Web
(Shuntov et al. 2025). The COSMOS2025 catalog (Shuntov et al. 2025)
provides LePHARE SED-derived stellar masses, SFRs, E(B-V) dust, and ages
for 784,016 galaxies over 0.54 deg², with 37,965 sources at $z > 4$
passing quality cuts — the largest high-$z$ photometric SED sample used
in this analysis. The UNCOVER DR4 SPS catalog (Wang et al. 2024; Suess
et al. 2024; Price et al. 2025) uses 20-band MegaScience photometry and
Prospector-β SED fitting, providing 2,628 sources at $z > 4$ with
Prospector dust2 and a spec-z sub-catalog of 203 sources with
spectroscopic redshifts fixed in the SED fit. For spectroscopic
validation, two complementary catalogs are used:



**JADES Data Release 4** (D'Eugenio et al. 2025;
Curtis-Lake et al. 2025; Scholtz et al. 2025): 2,858 high-quality
spectroscopic redshifts (flags A/B) across GOODS-N and GOODS-S, with 118
sources at $z > 7$ and 41 at $z > 8$. UV-luminosity-based stellar
masses (Song et al. 2016) are derived for the 1,345 sources with valid
$M_{\rm UV}$.



**DAWN JWST Archive (DJA) NIRSpec Merged Table v4.4**
(Brammer et al.; de Graaff et al. 2024a; Heintz et al. 2023; September
2025): a comprehensive compilation of 80,367 uniformly reduced
JWST/NIRSpec spectra from all public programs, processed with the
msaexp/grizli reductions. After applying grade $\ge 3$ quality cuts and
deduplication by sky position, 19,445 unique sources are retained, of
which 3,251 are at $z > 5$, 698 at $z > 7$, and 234 at $z > 8$.
Photometric stellar masses are available for 2,598 of the high-$z$
sources. This catalog spans JADES, CEERS, RUBIES, UNCOVER, GLASS,
PRIMER, and more than 50 other public programs, providing the largest
uniform cross-survey spectroscopic sample to date.



#### 2.1.4 MIRI-based mass calibration context


Recent JWST/MIRI analyses (Pérez-González et al. 2024) show that
NIRCam-only SED fits can overestimate stellar masses at $z > 5$
because of age-attenuation degeneracy and emission-line contamination.
When MIRI photometry is included, the number density of the most massive
systems decreases and some candidates are reclassified as dusty or
line-dominated sources. The photometry is not reprocessed in this work,
but published masses are treated as conservative upper bounds and
MIRI-based studies serve as an external check on the interpretation of
the extreme-mass tail.





Table 1a: Observational Datasets

| Dataset | Role | Sample Size | Redshift Range | Mass Cut ($\log M_*$) | Key Reference | Key Biases |
| --- | --- | --- | --- | --- | --- | --- |
| Red Monsters | Case Study | 3 | $5.3 < z < 5.9$ | $> 10.5$ | Xiao et al. (2024) | Small N, Selection Function |
| UNCOVER DR4 | Primary Statistical Sample | 2,157 | $4 < z < 10$ | $> 8.0$ | Wang et al. (2024) | NIRCam Mass Overestimation |
| CEERS DR1 | Independent Replication | 82 | $z > 8$ | $> 8.0$ | Cox et al. (2025) | Field Variance |
| COSMOS-Web | Large-Volume Check | 2,606 (918 dust-detected) | $z > 8$ | $> 8.0$ | Shuntov et al. (2025) | Photometric Redshift Uncertainties; Zero-Inflated Dust |
| JADES DR4 (NIRSpec/MSA) | Spectroscopic Validation | 2,858 (flags A/B); 118 at $z > 7$ | $z = 0.1$–$14.2$ | None | D'Eugenio et al. (2025); Curtis-Lake et al. (2025) | Slit Losses; UV-based $M_*$ ($\pm 0.4$ dex) |
| DJA NIRSpec Merged v4.4 | Cross-Survey Spectroscopic Validation | 19,445 unique (grade $\ge 3$); 698 at $z > 7$; 234 at $z > 8$ | $z = 0.1$–$14.1$ | None | Brammer et al. (DJA); de Graaff et al. (2024) | Photometric $M_*$ from grizli; heterogeneous survey depths |
| UNCOVER DR4 SPS (MegaScience) | Primary + Spectroscopic Validation | 2,628 (z$>$4, Prospector-β); 203 with spec-z fixed fits | $z = 4$–$12$ | Abell 2744 (lensed) | Wang et al. 2024; Suess et al. 2024; Price et al. 2025 | 20-band photometry; lensing magnification corrections |
| COSMOS2025 (LePHARE SED) | Cross-Field Replication | 48,861 (z$>$4, adopted LePHARE selection); 7,249 at $z > 7$; 2,659 at $z > 8$ | $z = 4$–$13$ | None (blank field) | Shuntov et al. 2025 (COSMOS2025) | LePHARE E(B-V) less precise than Prospector dust2; photo-z scatter |




Related MIRI-supported analyses of Little Red Dots (LRDs) at $z > 4$
find that inferred stellar masses can shift by up to orders of magnitude
depending on the assumed AGN contribution. This motivates a conservative
stance in the interpretation of compact red sources and provides a
systematic-control context for any extreme-mass claims in the
literature.



### 2.2 Key Terminology

The following terms are used consistently throughout this work:




Table 1b: Glossary of Key Terms

| Term | Symbol | Definition |
| --- | --- | --- |
| Mass-to-Light Inference Response | $R_{\rm ML}$ | A positive observable channel response that parameterizes the environment-dependent bias in conventional stellar-population inference. It is not the conformal factor $A(\phi)$, a local proper-time ratio, or the microscopic coupling $\beta_A$. |
| Temporal Shear | $\Sigma_\mu$ | The locally active gradient of the temporal potential, $\Sigma_\mu = \nabla_\mu \Theta$, where $\Theta = \ln A(\phi)$. High ambient matter density can suppress the locally active gradient continuously rather than at a discrete boundary. The inference response $R_{\rm ML}$ is tested against potential depth and environment but is not identified algebraically with $\Sigma_\mu$. |
| Isochrony Bias | — | The systematic error in inferred stellar properties (mass, age, SFR) produced when an environment-dependent transport and inference response is reduced with one universal calibration. |
| Screening | — | The suppression of TEP effects in regions where the locally observable Temporal Shear/source-charge sector is screened ($\rho_T \approx 20$ g/cm³ is an organizing saturation scale). Two types are distinguished: *Core Screening*—Screening within a single galaxy, where the deep central potential suppresses TEP ($R_{\rm ML} \to 1$) while the outskirts retain a large response. Produces bluer cores and redder outskirts. *Environmental Screening*—Screening by the ambient group or cluster potential, causing galaxies in dense environments to appear younger than isolated field galaxies of the same mass. |
| Inferred-Time Proxy | $t_{\rm inf}^{\rm proxy}$ | The observer-side catalogue proxy $t_{\rm inf}^{\rm proxy}=R_{\rm ML}(M_h,z)t_{\rm FLRW}^{\rm obs}(z)$. It is not accumulated matter-frame proper time. |




The analysis keeps three quantities distinct. The signed Newtonian potential obeys $\Phi\leq0$; the numerical halo-ordering proxy is the positive depth $\Psi\equiv-\Phi/c^2=|\Phi|/c^2$; and $R_{\rm ML}$ is an observable response parameterized from $\Psi$, environment, and redshift. The action-level conformal clock offset remains separate and satisfies $\Delta\ln A<0$ for a deeper unscreened well. No fitted value of $K_{\rm gal}$ is identified with $A$, $\beta_A$, or a local clock-rate ratio.



### 2.3 Derived quantities


#### 2.3.1 Halo mass proxy


For each galaxy, the analysis uses an abundance-matching relation
(Behroozi et al. 2019) to map stellar mass to halo mass. This mapping is
used solely to construct the potential proxy $\Phi$ for the TEP parameterization.
To mitigate circularity, sensitivity tests are performed with $\pm 0.3$
dex scatter in the $M_h-M_*$ relation, propagating to $\pm 12\%$ in
$R_{\rm ML}$ corrections.



#### 2.3.2 From the TEP Field to an Observable Inference Channel


Matter couples to the canonical unified metric $\tilde{g}_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$; Paper 0, §2.2, Step 80), written in conformal–disformal shorthand as $\tilde{g}_{\mu\nu}=A^2(\phi)g_{\mu\nu}+B(\phi)\partial_\mu\phi\partial_\nu\phi$, with $A(\phi)=\exp(\beta_A\phi/M_{\rm Pl})$. For the canonical signed potential $\Phi<0$, the static conformal solution gives $\Delta\ln A\propto\beta_A^2\Phi<0$: local clocks slow in a deeper well. This action-level sign is immutable and is not fitted in the present analysis.



The catalogue pipeline does not solve $A(\phi)$. It computes the positive halo-depth proxy



\begin{equation} \label{eq:jwst_psi_proxy}
\Psi\equiv-\frac{\Phi}{c^2}=\frac{|\Phi|}{c^2}>0,
\end{equation}


and tests the prespecified observable mass-to-light response



\begin{equation} \label{eq:jwst_rml_kernel}
R_{\rm ML}=\exp\left[K_{\rm gal}(\Psi-\Psi_{\rm ref})\sqrt{1+z}\right].
\end{equation}


Here $\kappa_{\rm gal}=(9.6\pm4.0)\times10^5$ mag is the canonical reference benchmark for the galaxy-sector observable response, and $K_{\rm gal} = \kappa_{\rm gal}\ln 10/(2.5\,n_{\rm ref}) = 1.263\times10^6$ (with the fixed transfer convention $n_{\rm ref}=0.7$ — sharing its value with the isochrone index used for the correction-stage population scaling, a physically anchored coincidence rather than a fitted degree of freedom, while remaining a separate convention from the population-dependent $n_{\rm SPS}$) is the single normalization entering the exponential kernel. Neither $n_{\rm ref}$ nor $K_{\rm gal}$ changes at a redshift boundary. $\Psi_{\rm ref}$ corresponds to $\log M_{h,\rm ref}=12.0$, and $\sqrt{1+z}$ supplies the only explicit redshift dependence of this kernel. A dedicated variant comparison (Step 181, `step_181_kernel_variant_test.json`) tests the static-space depth $\Psi_0(M_h)$ against FLRW-contracting virial depth evolution $\Psi\propto E(z)^{2/3}$: while scale-invariant correlations on $\ln R_{\rm ML}$ are insensitive to the overall normalization $\kappa_{\rm gal}$, evaluating on the physical exponential response $R_{\rm ML}=\exp[\ln R_{\rm ML}]$ at the prespecified canonical benchmark $\kappa_{\rm gal}=0.96\times10^6$ mag shows that FLRW virial contraction ($\Psi\propto E(z)^{2/3}$, $\sim6.8\times$ deeper at $z\sim9$) drives $R_{\rm ML}$ to unphysical blow-up ($\max R_{\rm ML}=3.24\times10^3$ for virial-only and $3.01\times10^{11}$ for virial+$\sqrt{1+z}$, collapsing the multi-observable $R^2$ score to $0.015$ and $0.013$), whereas the static-space branch ($s=0$) maintains a physical response ($\max R_{\rm ML}=2.6\text{--}22.7$, score $0.050\text{--}0.240$, beating the FLRW-contracting variants by $3.4\times\text{--}19\times$). $R_{\rm ML}$ is a channel-level inference response: it parameterizes how conventional stellar-population inference changes with environment. It is not $A$, $A/A_{\rm ref}$, $A_{\rm ref}/A$, or the bare scalar coupling.



**Provenance of $\kappa_{\rm gal}$.** The value $\kappa_{\rm gal} = 0.960 \times 10^6$ mag is the canonical theory benchmark adopted across the TEP corpus (Paper 11, Appendix C). It corresponds to an effective host-to-anchor potential depth $\Delta(\sigma_*^2)/c^2 \sim 3 \times 10^{-7}$ mapped through the Leavitt law slope $b \approx -3.30$ to produce a $\sim 0.3$ mag characteristic modulus shift. Its numerical provenance is empirical: the value is the Paper 11 joint-fit/bootstrap coefficient on the SH0ES sample, adopted corpus-wide as the fixed reference benchmark. It is therefore a same-programme calibration carried as a prior for this analysis — prespecified with respect to the JWST data and not refit here — rather than an ab-initio action-derived constant. The empirical Cepheid-channel analyses in Paper 11 fit $\kappa$ directly from observational data without imposing this benchmark: the endpoint-equivalent projection recovers $\kappa_{\rm Cep}^{\rm equiv} = (1.220 \pm 0.531) \times 10^6$ mag, and the multi-block joint analysis (Step 44) recovers $\kappa_{\rm Cep} = (0.665 \pm 0.362) \times 10^6$ mag. The regenerated JWST-side joint recovery from the high-redshift galaxy sample is $\kappa = (8.38 \pm 4.11) \times 10^5$ mag, consistent with the canonical benchmark at $0.21\sigma$ and equally within $\sim 0.9\sigma$ of the empirical Cepheid values; it therefore does not discriminate between the two anchors. The empirical recoveries bracket the canonical benchmark — the endpoint-equivalent value lies $\sim 0.5\sigma$ above it and the multi-block value $\sim 0.8\sigma$ below — at individual significances of $1.8\text{--}2.3\sigma$ — because the benchmark is itself a same-sample estimator, this is within-estimator spread on the SH0ES data rather than an independent theory-vs-data tension (Paper 11; the unresolved response-normalization class of Paper 0). The $\pm 4.0 \times 10^5$ mag uncertainty adopted here is the canonical theory uncertainty, not the posterior width of any single Cepheid fit. Sensitivity analyses spanning the empirical Paper 11 multi-block value ($\kappa = 0.665 \times 10^6$ mag), the JWST-side recovery, and the canonical benchmark are reported in §3.9; the dust–$R_{\rm ML}$ correlation remains significant ($\rho > 0.4$) across this range, though the magnitude of the mass correction scales proportionally.



Values above and below unity therefore denote positive and negative inference responses relative to the reference environment, not faster and slower local clocks. Screening acts on the environmental transfer entering the response, while the physical conformal clock offset retains the same slowing sign throughout. The low-mass dust and high-mass stellar-population tests below are consequently tests of an environment-ordered inference channel; they do not constitute direct measurements of local proper-time acceleration.



The physical origin of $R_{\rm ML}>1$ at high redshift lies in the background temporal structure, not in faster local clocks. In the canonical TEP cosmology (TEP-HUB, Paper 30), the spatial manifold is static and the gravitational coordinate time extends without finite origin. The FLRW observer-age $t_{\rm cosmic}(z)$ used by standard stellar-population inference is a reconstruction under the isochrony axiom: it interprets the clock-calibration ratio $1+z=A_0/A_{\rm em}$ as spatial expansion and assigns a finite, shrinking age to high redshift ($\sim 540$ Myr at $z=9$). Under TEP, the actual coordinate time elapsed is far larger than this reconstruction. Local matter clocks run slower than coordinate time ($\Delta\ln A<0$ in a deeper well), but the coordinate background is eternal, so the accumulated proper time can still exceed the FLRW assigned age by a large factor — the quantitative elapsed-time ratio available to date is the closed-static benchmark-branch value of Paper 0 §8, conditional on the inhomogeneous eternal construction whose constraint-slice existence is demonstrated (Appendix E, R8) but whose matter-frame clock history is not yet computed (§1, observer-age convention). $R_{\rm ML}>1$ parameterizes this discrepancy as an inference-channel response: it quantifies how much the standard pipeline inflates apparent age and $M/L$ when it uses the too-short FLRW baseline. It is not itself a direct measurement of the proper-time ratio; the catalogue proxy $t_{\rm inf}^{\rm proxy} = R_{\rm ML} \, t_{\rm FLRW}^{\rm obs}$ propagates the fitted response through the conventional $M/L \propto t^n$ scaling but is not a physical proper-time integral (§2.3.3). Analytically, the $\sqrt{1+z}$ response-evolution factor arises from the first-order sensitivity of the FLRW lookback-age calibration to an endpoint redshift perturbation on a static halo potential: because the matter-dominated FLRW age assignment scales as $t_{\rm FLRW}(z) \propto H(z)^{-1} \propto (1+z)^{-3/2}$ while the endpoint clock-calibration shift $\delta(1+z)/(1+z) = -\delta\ln A_{\rm em} \propto \Psi$ enters a cosmic clock baseline whose background drift rate scales as $\dot\phi_{\rm bg} \propto (1+z)^{1/2}$ across the conformal horizon mapping (Papers 26–27, 30), the resulting differential log-age inference kernel scales as $(\Psi - \Psi_{\rm ref})\sqrt{1+z}$ without invoking FLRW spatial virial contraction.


**Origin of the depth ordering.** The eternal-background term is uniform at fixed $z$ and is blind to $\Psi$; the kernel's depth ordering enters through a separate term of the same clock-calibration ratio. The measured redshift decomposes as $1+z = (A_0/A_{\rm em})\times(\text{propagation and kinematic factors})$, and the emitter's conformal factor $A_{\rm em}=\exp(\beta_A u_{\rm em})$ contains the local halo contribution, $u_{\rm em}\sim u_{\rm cosmo}(z)+\Psi$: a deeper well lowers $A_{\rm em}$, raises the assigned redshift, and shrinks the assigned FLRW age, enlarging the inferred evolutionary excess in proportion to $\Psi$. This endpoint clock-calibration term carries the sign the kernel implements — $R_{\rm ML}$ grows with depth — while the matter-frame proper-time integral $\tau=\int A\,dt$ acts in the opposite sense (deeper wells accumulate less proper time) and must be shown to be subdominant or counted as a countervailing contribution within the same channel rather than invoked as the mechanism. Two notes follow. First, while the bare endpoint amplitude is $\Delta\ln A\sim\Delta\Psi\sim10^{-7}$ per halo decade, the factor $K_{\rm gal}\simeq1.26\times10^6$ reflects the stellar-envelope density-ratio amplification $\rho_T/\langle\rho\rangle \sim 10^6$ established in Paper 11 Step 62 (where $\rho_T/\langle\rho\rangle(10\text{ d}) = 8.45\times 10^5$ at the unit-transport benchmark $\lambda_0 = 1$, with $|\beta_A| = 1$ fixed by Paper 0, predicts $|\kappa_{P^2}| = 1.196\times 10^6$ mag against the SH0ES fit $(1.191\pm 0.480)\times 10^6$ mag, $\lambda_0 = 0.995\pm 0.401$). Second, in Step 181, evaluating the physical exponential response $R_{\rm ML}=\exp[\ln R_{\rm ML}]$ at the canonical $\kappa_{\rm gal}=0.96\times10^6$ mag rules out FLRW virial contraction $\Psi\propto E(z)^{2/3}$ (which blows up $R_{\rm ML}$ to $3.2\times10^3\text{--}3.0\times10^{11}$ at $z>6$), confirming TEP's static-space depth prescription $\Psi=\Psi_0(M_h)$.



##### 2.3.2.1 Auxiliary Log-Mass Approximation


Earlier variants considered a Log-Mass approximation in which $R_{\rm ML}$ scales with the log-mass perturbation rather than the potential directly:



\begin{equation} \label{eq:jwst_rml_logmass}
R_{\rm ML} = \exp\left[ \alpha(z) \cdot \frac{2}{3} \cdot (\log_{10} M_h - \log_{10} M_{h,\rm ref}) \cdot \frac{1+z}{1+z_{\rm ref}} \right]
\end{equation}


where $\alpha(z) = \kappa_{\rm gal} \sqrt{1+z}/10^6$ and $z_{\rm ref} = 5.5$ is a normalization epoch used only in this auxiliary form. The primary results in §3 do not use this approximation; they use the Potential-Linear kernel above. The Log-Mass form is retained only as an auxiliary note for interpreting observable scaling relations; it does not enter the nested Bayesian comparison in §3.8, which uses the Potential-Linear kernel exclusively.



##### 2.3.2.2 Screening and Scale Separation

For an illustrative screened effective coupling $\beta_{A,\rm eff} \approx 0.8$ — a representative intermediate-environment value used only to display the pre-screening Brans–Dicke bookkeeping (the bare coupling is frozen at $\beta_A = -1.0$, Paper 0 §2.2; $|\beta_{A,\rm eff}| = |\beta_A| S_A(\mathcal{E})$) — the bare
Brans-Dicke parameter would be $\omega_{\rm BD} = 1/(2\beta^2) - 1/2 \approx
0.28$ — roughly five orders of magnitude below the Cassini bound
($\omega_{\rm BD} > 40{,}000$; Bertotti et al. 2003). This large
pre-screening discrepancy illustrates the central logic of the TEP
framework: any underlying bare coupling is strong, but where the ambient
landscape is steep the scalar
field gradient (Temporal Shear) flattens continuously, suppressing the
effective scalar coupling to $\beta_{\rm eff} \ll \beta_A$ (yielding an effective Brans–Dicke parameter $\omega_{\rm BD}^{\rm eff} = 1/(2\beta_{\rm eff}^2) - 1/2 > 10^6$, well within Cassini bounds). On cosmological scales, suppression of the $k_8 = 2\pi/8 = 0.785\,h$/Mpc mode requires a Compton wavelength short enough that $m_{\phi,0} \gg k_8$: the effective coupling $\beta_{\rm eff}(k_8) = \beta_A\,k_8/\sqrt{k_8^2 + m_{\phi,0}^2}$ evaluates to $0.0079$ at the scale-dependent-growth best fit $m_{\phi,0} = 100\,h$/Mpc ($\lambda_C = 0.063\,h^{-1}$ Mpc; Step 135) — well below the Planck bound.
Within individual halos ($r \lesssim 50$ kpc), the field tracks the local
potential and operates locally. This two-scale picture is standard for
screened scalar-tensor theories.
The continuous screening via Temporal Shear provides PPN-compliant
suppression without a rigid threshold. The observable galaxy-sector response coefficient is attenuated by the screening operator as:


\begin{equation} \label{eq:jwst_kappa_eff}
\kappa_{\rm eff}(\rho) = \kappa_{\rm gal} \cdot \mathcal{S}(\rho/\rho_T),
\end{equation}


where $\mathcal{S}$ is a smooth suppression function and $\rho_T \approx 20$ g/cm³
is the Temporal Topology reference density. The normalization is set by
terrestrial GNSS clock-covariance correlations: $L_c \approx 4200$ km for
Earth's mass on the IGS precise-product convention — the measured realization
of the canonical correlation scale $\lambda_T$ (Paper 0 §7). That correlation
length is an estimator-dependent family — the precise-product bracket
spans $\sim 1.9$–$4.5\times10^{3}$ km (MGEX $1{,}862\pm155$ km; IGS
inter-center $3{,}330$–$4{,}549$ km), dominated by the $\sim30$–$40\%$
epoch/window scatter and propagating to an effective $\rho_T$ band of
roughly $8$–$70$ g/cm³, while raw single-epoch channels bound the scale
from below at $\sim 0.7$–$1.1\times10^{3}$ km (Papers 1, 3, 6, 14).
The adopted normalization is supported by two estimator-independent
cross-checks — atomic physics (Temporal Topology radius
$R_T(m_p) \sim a_0$ at the proton mass scale) and magnetar anti-glitches
($P_{\rm crit} \approx 6.8$ s for 1E 2259+586, 4% match) — whose
convergence on the adopted scale across 40 orders of magnitude in mass
provides a consistency check on this screening scale.


At galactic scales, an effective kinematic screening threshold emerges from
analysis of 199 millisecond pulsars in globular clusters, which reveals that
the TEP spin-down excess saturates for systems with velocity dispersion
$\sigma \gtrsim 165$ km/s, consistent with the scalar field gradient
flattening as potential depth increases. This is a shear-channel
($\mathcal{S}_\Sigma$) boundary: it governs kinematic observables —
spin-down, accelerations, and dynamical-mass inference — and is not
interchangeable with the clock-amplitude channel $\mathcal{S}_A$ that
modulates $R_{\rm ML}$ (cf. the channel-projection notice of §4.1).
For the amplitude channel, the operative suppression is Eq.
($\ref{eq:jwst_kappa_eff}$) evaluated at the halo's own environment:
galaxy-halo and disk densities ($\rho \sim 10^{-25}$–$10^{-24}$ g/cm³)
lie $\sim 25$ orders of magnitude below $\rho_T \approx 20$ g/cm³, at
which $\mathcal{S}(\rho/\rho_T) = 1$ to machine precision (Steps 120, 180), so
every halo in the present samples — including the massive $z > 5$ Red
Monster systems at $\log M_h \approx 12.7$–$13.1$ — is unscreened in the
amplitude channel. The unscreened primary kernel of §2.3.2 is therefore
the correct $\mathcal{S}_A \to 1$ limit for this sample, not an omitted
correction; amplitude-channel screening first becomes relevant at
stellar-interior and compact-object densities.


##### 2.3.2.3 Local Dilation and Channel Response


The local conformal and observational quantities must not be conflated. A deeper unscreened well has $\Delta\ln A<0$, while $R_{\rm ML}$ may lie above or below unity because it is an observer-side inference response with its own transfer function. The present pipeline measures the latter and makes no numerical identification between $K_{\rm gal}$ and the microscopic coupling $\beta_A$.





Table 1: TEP Model Parameters (Fixed)

| Parameter | Value | Source | Description |
| --- | --- | --- | --- |
| $\kappa_{\rm gal}$ (Paper 11) | $(9.6 \pm 4.0) \times 10^5$ mag | Canonical benchmark (Paper 11) | Canonical TEP magnitude-sector benchmark fixed prespecified across the TEP corpus; empirical Paper 11 checks recover $(1.220 \pm 0.531)\times10^6$ and $(0.665 \pm 0.362)\times10^6$ mag |
| $K_{\rm gal}$ (galaxy kernel) | $\approx 1.26 \times 10^6$ | Transferred from $\kappa_{\rm gal}$ via response normalization | Phenomenological response normalization for stellar-population $R_{\rm ML}$ kernel; not a microscopic coupling |
| $\log M_{h, \rm ref}$ | 12.0 | TEP-JWST | Reference halo mass at $z=0$ ($R_{\rm ML}=1$) |
| $\rho_T$ | 20 g/cm$^3$ | TEP-UCD | Temporal Topology reference density for continuous screening |






![TEP mass-to-light inference response as a function of halo mass and redshift](public/figures/figure_1_tep_model.png)




Figure 1: The observable response $R_{\rm ML}(M_h,z)$ evaluated from the positive potential-depth proxy. The reference mass ($\log M_h=12$) defines $R_{\rm ML}=1$; values above and below unity are inference responses relative to that reference, not local clock-rate measurements. Environmental screening attenuates the channel response.





The JWST response-prior test adopts the prespecified canonical magnitude-sector benchmark $\kappa_{\rm gal} = (9.6 \pm 4.0) \times 10^5$ mag from Paper 11 and applies it without JWST-specific refit. The high-redshift observables are then examined for internal consistency of the response scale. These internal recoveries are treated as consistency checks rather than as the input calibration.



##### 2.3.2.4 Cosmological Viability Summary


The TEP framework has been checked against three classes of precision
cosmological constraints:



- 
**Early-universe compatibility (historical):** Earlier
versions verified that a conformal scalar remains perturbatively
invisible during a conventional radiation-dominated era ($|\Delta
H/H|_{\rm max} = 1.7 \times 10^{-13}$; $\Delta Y_p <
10^{-14}$). This serves as a historical compatibility check;
the canonical TEP early-universe interpretation is supplied by
TEP-TH and TEP-BBN (Appendix A.1.7).


- 
**Linear growth ($\sigma_8$):** TEP screening (topology pinning) on
$\gtrsim 10$ Mpc scales reduces the effective coupling to
$\beta_{\rm eff}(k_8) = 0.0079$ at the best-fit $m_{\phi,0} = 100\,h$/Mpc
(Step 135), preserving $\Lambda$CDM-consistent
$\sigma_8$ (§2.3.2.2; Appendix A.1.7–A.1.8).


- 
**Solar System (PPN):** Temporal Shear suppression in
dense environments reduces the effective microscopic scalar coupling to
$\beta_{\rm eff} \ll \beta_A$ (suppressed to $\lesssim 10^{-6}$ for solar-system bodies),
satisfying Cassini and lunar laser ranging bounds (Appendix A.1.3).




**Scale-dependent growth computation:** To go beyond the
analytic TEP screening argument, the linear growth ODE is solved independently
for each Fourier mode $k$, incorporating the full scale-dependent
gravitational coupling $G_{\rm eff}(k,z)/G_N = 1 + 2\beta^2 k^2/(k^2 +
m_\phi(z)^2)$, where $m_\phi(z) = m_{\phi,0}(1+z)^{9/4}$ for $n=1$
potential (standard chameleon form, used here as a conventional-background compatibility embedding). This inverse-power $n=1$ branch is the potential family excluded by Paper 0's constraint F1 under $\beta_A = -1$; the $\sigma_8$/CMB viability demonstrated below is therefore a proxy result for a screened scalar sector, not a verification of the canonical $V(\phi)$, and transfers to canonical TEP only conditionally. The canonical TEP screening structure is the nested two-body operator $\mathcal{R}(s)=\mathcal{S}_\Sigma(X_{\rm env})^2\,y(s)$ (Paper 0 &sect;2.2, v0.16), whose environmental separation dependence differs from the inverse-power $n=1$ form; the chameleon embedding is retained for computational tractability and cross-check against the native TEP-HC hi_class closure. The resulting matter power spectrum ratio $P_{\rm
TEP}(k)/P_{\Lambda{\rm CDM}}(k)$ and integrated $\sigma_8$ are computed
self-consistently (Appendix A.1.8). Key results:



- 
Planck consistency requires $m_{\phi,0} \gtrsim 0.43\,h$/Mpc
($\lambda_C \lesssim 14.6\,h^{-1}$ Mpc at $z=0$; Appendix A.1.8).


- 
At the scale-dependent-growth best fit $m_{\phi,0} = 100\,h$/Mpc
($\lambda_C = 0.063\,h^{-1}$ Mpc), $\beta_{\rm eff}$ on $R_8$ scales
is $\approx 0.0079$—well below the bare coupling
($G_{\rm eff}/G_N - 1 = 1.2 \times 10^{-4}$ at $k_8$; Appendix
A.1.8).


- 
The predicted $\sigma_8^{\rm TEP} = 0.8110$ vs.
$\sigma_8^{\Lambda{\rm CDM}} = 0.811$; $\Delta\sigma_8 = 2.1 \times
10^{-7}$ ($3.6 \times 10^{-5}\sigma$). RSD: $\Delta\chi^2 = 2.7 \times
10^{-4}$ across 8 data points (Appendix A.1.8).




This k-dependent growth calculation substantially strengthens the
viability argument beyond the earlier analytic estimate. A CAMB-based
late-time propagation (Appendix A.1.9.8) confirms these results: at the
fiducial $m_{\phi,0} = 1.0\,h$/Mpc, the CAMB-computed $\sigma_8^{\rm
TEP} = 0.8116$ ($0.10\sigma$ from Planck), with CMB TT deviations <
0.02% at all $\ell < 2500$ and $\chi^2/{\rm dof} \ll 1$ against
Planck error bars. Planck consistency holds for $m_{\phi,0} \gtrsim
0.43\,h$/Mpc. The CAMB integration uses the scale-dependent growth
equation with TEP-screened $G_{\rm eff}(k,z)$ and CAMB's exact
lensed CMB spectra, substantially narrowing the remaining theoretical
gap. These CAMB calculations are conventional-background compatibility embeddings on the F1-excluded inverse-power $n=1$ branch — a proxy demonstration that a screened scalar sector need not spoil observed CMB/growth phenomenology; they are not the canonical static-space TEP cosmological solution and do not verify the canonical $V(\phi)$. The remaining approximation — that the scalar field does not modify
acoustic peaks at $z > 1089$ — is justified by $T^\mu_\mu \approx 0$
during radiation domination. A natively coupled hi_class integration
remains desirable for completeness but is no longer expected to change
the conclusion.



**Screening function:** The transition from the unscreened
regime ($\mathcal{S} \approx 1$) to the screened regime ($\mathcal{S}
\ll 1$) is parameterized by the smooth tanh profile:



\begin{equation} \label{eq:jwst_screening_tanh}
\mathcal{S}(x) = 1 - \tanh\left(\left[\frac{x}{x_T}\right]^p\right),
\end{equation}


where $x = \rho/\rho_{\rm crit}$ or $|\Phi|/c^2$, $x_T$ is the screening
threshold, and $p \ge 1$ controls the sharpness of the transition. For
galaxy-scale potentials, the transition is sufficiently sharp that halos
with $M_h \lesssim 10^{12.5}\,M_\odot$ are fully unscreened, while
cluster-scale potentials ($M_h \gtrsim 10^{14}\,M_\odot$) are screened.
This functional form is used in the environmental screening tests
(§3.5) and the core-screening morphology analysis (§3.7).



##### 2.3.2.5 Cosmological Background Compatibility


Cosmological parameter constraints from CMB, BAO, and SNe Ia data were
verified using CAMB-compatible background evolution. Within this conditional FLRW compatibility embedding, scalar-induced deviations from the reference $\Lambda$CDM background expansion remain small. Using the standard cosmological parameters
($\Omega_m = 0.315$, $\Omega_\Lambda = 0.685$, $H_0 = 67.4$ km/s/Mpc),
the TEP-modified expansion history satisfies:



- 
**CMB acoustic scale:** Shift in the angular scale of the
first acoustic peak $\Delta\theta_* / \theta_* < 0.08\%$, well
within the Planck 2018 uncertainty of $0.03\%$ (Aghanim et al. 2020).


- 
**Baryon Acoustic Oscillations:** Shifts in $D_V(z)/r_d$
are $< 0.3\%$ across the redshift range $z \in [0.1, 2.5]$,
consistent with BOSS DR12 and DESI 2024 constraints (Alam et al.
2017; DESI Collaboration 2024).


- 
**Supernova distance moduli:** Residuals relative to
$\Lambda$CDM are $|\Delta\mu(z)| < 0.008$ mag for $z \in [0.01,
1.5]$, well below the Pantheon+ systematic uncertainty floor of $\sim
0.015$ mag (Scolnic et al. 2022).




These calculations are conditional conventional-background compatibility tests. They show that embedding the screened TEP scalar sector in an FLRW background does not spoil established CMB, BAO, or SN phenomenology. They do not define the canonical TEP cosmology. In the canonical framework developed in TEP-HUB (Paper 30), the spatial manifold is static, the temporal background extends without finite origin, and cosmological redshift is interpreted through cumulative temporal calibration rather than spatial expansion. The results of the present JWST analysis depend on the local-halo inference response $R_{\rm ML}$ and therefore remain compatible with the conventional-background tests, while their physical interpretation throughout this work follows the canonical static-space TEP framework. The static-space TEP cosmology is discussed further in §4.1, Appendix A.1.7, and the conclusion.



#### 2.3.3 Inference-response correction


The catalogue proxy $t_{\rm inf}^{\rm proxy}=R_{\rm ML}(M_h,z)t_{\rm FLRW}^{\rm obs}(z)$ is used only to propagate the fitted response through the conventional stellar population synthesis (SPS) mass-to-light scaling $M/L\propto t^{n_{\rm SPS}}$; it is not a physical proper-time integral. The fixed $n_{\rm ref}=0.7$ belongs to the preceding response stage, whereas the physical SPS exponent $n_{\rm SPS}$ belongs exclusively to this correction stage. The catalogue construction used in the UNCOVER, CEERS, COSMOS-Web, and cross-survey steps assigns $n_{\rm SPS}=0.7$ for $z\le4$, $0.9$ for $4<z\le6$, and $0.5$ for $z>6$; the stellar-mass-function analysis uses the declared sensitivity convention $n_{\rm SPS}=0.7$ throughout. These choices change $R_{\rm ML}^{n_{\rm SPS}}$, never $R_{\rm ML}$ at fixed $(\Psi,z)$. In contrast, star-formation rates traced by UV and emission-line luminosity reach steady state much earlier ($\sim 10$–$100$ Myr), scaling as $L_{\rm UV} \propto t^m$ with $m \approx 0.3$–$0.5$. Because ionizing and UV luminosity saturates much faster than integrated stellar mass accumulation across standard SPS libraries, for the representative SPS/SFR response ranges used here, $m < n_{\rm SPS}$ and the expected bias is negative ($m - n_{\rm SPS} \approx -0.2$ to $-0.4$); it approaches zero only at the limiting $m = n_{\rm SPS} = 0.5$ case, yielding ${\rm sSFR}_{\rm obs} \propto R_{\rm ML}^{m - n_{\rm SPS}}$. The observational mass correction is:



\begin{equation} \label{eq:jwst_mass_correction}
M_{*,\rm true} = M_{*,\rm obs}/R_{\rm ML}^{n_{\rm SPS}}, \quad \mathrm{SFE}_{\rm
true} = \mathrm{SFE}_{\rm obs}/R_{\rm ML}^{n_{\rm SPS}}.
\end{equation}


Because $R_{\rm ML}$ depends on the halo mass $M_h$, which is derived
from the stellar mass via abundance matching, the correction is solved
self-consistently by iterating the $M_* \to M_h \to R_{\rm ML} \to
M_{*,\rm true}$ cycle to convergence (tolerance $10^{-4}$, typically
3–5 iterations). For the vast majority of galaxies at $z > 8$ ($M_* <
10$), the single-pass and iterated solutions differ by less than 2%.
At $M_* > 10.5$ the difference can reach 10–60% because the
abundance-matching slope amplifies the feedback loop; the iterated
solution is used throughout to ensure correctness in this regime.



### 2.4 Statistical procedures


The statistical design separates three questions: whether the predicted
associations are present, whether they survive control for obvious
confounders, and whether they generalize across surveys and subsamples.
Associations are quantified using Spearman rank correlations and
bootstrap confidence intervals. To address confounding by redshift and
stellar mass, partial-correlation analyses implemented via
residualization are employed. In addition to correlation-based tests,
the following are reported:



- 
Stratified comparisons (e.g., high vs low $R_{\rm ML}$ splits) for
multi-property coherence


- 
Distributional comparisons (e.g., Kolmogorov-Smirnov tests) for
regime separation


- 
Model comparison using both AIC/BIC and nested Bayesian evidence:
the regression comparisons test predictors {z}, {z, $\log M_*$}, {z,
$R_{\rm ML}$}, and {z, $\log M_*$, $R_{\rm ML}$}, while a separate
`dynesty` nested-sampling analysis compares TEP against
explicit bursty-SF, varying-IMF, standard-physics, and AGN
alternatives in both raw standardized space and a
mass+$z$-residualized control space. Each comparison is evaluated
under two predictor conditionings — raw mass and redshift, and mass
and redshift orthogonalized against $\log R_{\rm ML}$ — with a nested
augmented model (mass $+$ $z$ $+$ $\log R_{\rm ML}$) providing the
conditioning-free arbitration. Three likelihood structures are
used: (i) the standard independent-Gaussian joint likelihood (product
of $K$ univariate Gaussians), (ii) a joint covariance likelihood
(single multivariate Gaussian with the empirical $K \times K$ residual
covariance matrix, correcting for correlated SED outputs), and (iii) a
quadratic mass-plus-redshift baseline (mass $+$ $z$ $+$ mass$^2$ $+$
$z^2$ $+$ mass$\times z$) to test whether the TEP signal is
specifically due to $R_{\rm ML}$ or merely reflects generic
nonlinearity in the mass–redshift plane. Per-observable evidence is
also reported to identify which channels drive the joint signal.





#### 2.4.1 Combined significance and multiple testing


Combined significance is assessed using Fisher's method, Bonferroni
correction, Brown's method (dependence-adjusted), harmonic mean p-value,
and Benjamini-Hochberg FDR ($\alpha = 0.05$). Because omnibus
significance depends sensitively on how clustering and shared predictors
are penalized, the manuscript treats the three-survey photometric L1
replication as the headline result and uses broader multi-test
combinations as supportive context. An extreme stress test that reduces
effective sample sizes by 90% via spatial clustering autocorrelation
still leaves a Bonferroni-corrected floor of $3.2\sigma$ for the mixed
test set. Parametric p-values are supplemented by permutation tests ($N =
10{,}000$ shuffles) and bootstrap confidence intervals ($N = 10{,}000$
resamples). Cross-survey effect sizes are combined via DerSimonian-Laird
random-effects meta-analysis with $I^2$ heterogeneity assessment and
leave-one-out influence diagnostics.



#### 2.4.2 Blind validation protocol


Three split strategies test generalization: (1) time-split (low-$z$
train / high-$z$ test, 60/40); (2) field-split (RA median); (3)
cross-survey leave-one-out. A test passes if the dust–$R_{\rm ML}$
correlation remains significant ($p < 0.05$) on held-out data.



#### 2.4.3 Stellar-to-halo mass mapping and sensitivity


Each galaxy's stellar mass is mapped to halo mass using a redshift-dependent
abundance matching relation parameterized to mirror the high-$z$ tail of
Behrozi et al. ($\log M_h = \log M_* + 1.8 + 0.1(\log M_* - 10) - 0.05(z-5)$).
Dynamical masses are mapped using an identically sloped relation to
ensure rank-order preservation. Key results survive $\pm 0.5$ dex
perturbations. To test robustness against MIRI-based mass recalibration
(Pérez-González et al. 2024), mass reductions of 0.0–1.0 dex are
applied; TEP signatures persist under a 0.5 dex
reduction (2/4 key signatures survive; step_040). At $z > 8$, selection bias toward bright galaxies is
quantified via Monte Carlo completeness weighting ($N = 1{,}000$
iterations) and Savage-Dickey Bayes Factors.



*Extrapolation caveat:* The Behroozi et al. (2019) UNIVERSEMACHINE
relation is calibrated for $z = 0$–$10$. Analyses extending to $z = 9$–$13$
(COSMOS2025 sSFR, UNCOVER MegaScience dust) therefore extrapolate the
linear redshift term ($-0.05(z-5)$) beyond the calibration range. At
$z > 10$, the physics of halo assembly and baryon cooling differs
substantially from $z = 5$, and the linear extrapolation carries
unquantified systematic uncertainty in $R_{\rm ML}$. Results at $z > 10$
should be interpreted with this caveat; a theoretically motivated
$M_*$–$M_h$ relation from high-$z$ hydrodynamical simulations would
tighten these inferences.



#### 2.4.4 Forward-modeling validation


The correction-stage $M/L \propto t^{n_{\rm SPS}}$ scaling is tested over $n_{\rm SPS}=0.3$–$1.0$ by residual mass–age minimization, independently of the fixed response normalization. The live validation gives boundary minima at the edge of the scanned range for $z=4$–$6$ and $6$–$8$ ($n_{\rm SPS}=1.0$) and at $n_{\rm SPS}=0.3$ for $z=8$–$10$; these unstable minima are treated as sensitivity diagnostics, not as fitted replacements for the declared catalogue values. The same validation verifies numerically that the fixed-$n_{\rm ref}$ kernel is continuous at $z=4$ and $z=6$.



### 2.5 Black Hole Growth Stress Test


Differential central-versus-halo temporal structure may affect the inferred accretion history of compact sources, but a physical growth calculation requires the matter-frame proper-time history $\tau_\star = \int A(\phi)\,dt$ rather than the observer-side inference response $R_{\rm ML}$. The quantitative modeling of supermassive black hole growth in compact Little Red Dot cores is accordingly deferred to dedicated relativistic structure calculations.



### 2.6 Reproducibility


All analyses are reproducible from the public repository. An end-to-end
run regenerates the manuscript tables, figures, and archived outputs;
execution instructions are provided in the repository README.



## 3. Results


### 3.0 Evidence Summary

The evidence is organized by role: L1 is the primary photometric line, L3 is a secondary partial-correlation line, L2 is ancillary, L4 is derived, and L5 is a direct kinematic test. The classification is stated once here and in §1.3; individual results below are reported without repeating it.




Table 3a: All Tested TEP Signatures

| Signature | Finding | Significant | Survives Mass Control | Status |
| --- | --- | --- | --- | --- |
| **L1. Dust–$R_{\rm ML}$ + AGB threshold** | $\rho = +0.635$ across three surveys ($N=1{,}254$); AGB threshold odds ratio 12.7 ($p = 2.1 \times 10^{-159}$) | ✔ | ✔ | **Primary line** |
| **L2. Inside-out core screening** | Gini partial $\rho=+0.191$ after mass+$z$ control; size proxies and $\sigma_\star$ non-significant | ✔ | Partial — Gini only | **Ancillary** |
| **L3. $R_{\rm ML}$–sSFR partial correlation** | Partial $\rho(R_{\rm ML}, {\rm sSFR}\|M_*, z) = -0.62$ at $z>8$ ($p = 8.2 \times 10^{-29}$, $N=254$); sign consistent with ${\rm sSFR}_{\rm obs} \propto R_{\rm ML}^{m-n_{\rm SPS}}$, $m < n_{\rm SPS}$ | ✔ | ✔ | **Secondary line** |
| **L4. Dynamical mass comparison** | TEP correction of 0.245 dex on the anomalous literature subset ($0.259 \rightarrow +0.015$ dex) lands the population at the admissibility boundary; 0.139 dex regime-level reduction vs 0.15 dex published excess | ✔ | ✔ | **Derived** |
| **L5. Direct kinematic test** | SUSPENSE ($N=15$): $\rho({\rm Age}, R_{\rm ML} \mid M_*, z)=+0.556$ ($p=0.032$); $\rho({\rm Age}, M_* \mid R_{\rm ML}, z)=+0.031$ ($p=0.91$). Sigma expansion ($N=75$, Step 171): Virial-exact $\Psi\propto\sigma^2$ yields $\rho(R_{\rm ML,\sigma}^{\rm dir}, M_{*,\rm obs} \mid z)=+0.463$ ($p=2.8\times10^{-5}$), $+0.608$ ($p=8.4\times10^{-7}$, $N=55$ exact-$M_{\rm dyn}$), and $+0.575$ ($p=0.0080$, $N=20$ absorption), with 5/5 subsample sign agreement and $86.5\%$ absorption residual attenuation ($+0.252\to+0.034$). | ✔ | ✔ | **Direct test** |
| Steiger Z-test ($t_{\rm eff}$ vs $M_*$) | $Z = 24.7$ ($p = 1.5 \times 10^{-134}$, combined $z = 4$–$10$ sample); three-survey meta $z = 26.4$ | ✔ | ✔ | Robustness check on L1 |
| Partial correlations | $\rho=+0.31$ after polynomial control; $M_*$ zero residual after $t_{\rm eff}$ control | ✔ | ✔ | Robustness check on L1 |
| Cross-survey generalization | $\rho=0.60$–0.65 across three surveys (fixed-effects meta $\rho=+0.635$) | ✔ | ✔ | Robustness check on L1 |
| Age coherence | $\rho = +0.15$ (mass-only); vanishes with $M_*$+$z$ control | ✔ | ✘ | Not independent |
| Metallicity | $\rho = +0.16$ (mass-only); vanishes with $M_*$+$z$ control | ✔ | ✘ | Not independent |
| Environmental screening | Same-epoch ($\Delta z = 0.5$) 5-NN density (Step 138): $z>8$ contrast $\Delta\rho = +0.227$ ($\rho_{\rm field}=+0.712$ vs $\rho_{\rm dense}=+0.484$, $Z = 2.85$, $p = 0.0044$), positive in 3/3 $d_{\rm A2744}$ terciles ($+0.269, +0.184, +0.137$, mean $+0.197$); $z=6$–$8$ $\Delta\rho = +0.355$ ($Z = 4.32$, $p = 1.6\times10^{-5}$); full sample $\Delta\rho = +0.205$ ($Z = 5.15$, $p = 2.6\times10^{-7}$) | ✔ | ✔ | **Confirmed ($2.85\sigma$ at $z>8$)** |
| Colour-gradient sign test | Resolved radial predictor $\Gamma_{\rm rad}$ (incorporating measured inner-to-outer F444W concentration $C_{\rm core}$): partial $\rho = -0.179$ ($p = 0.0028$, $2.99\sigma$ after $M_*$+$z$ control; $C_{\rm core}^{-1}$ partial $\rho = -0.237$, $p = 6.9\times10^{-5}$, $4.0\sigma$); Steiger $Z = 3.02$ ($p = 0.0025$ vs $M_*$); residual sign test $58.6\%$ vs $35.7\%$ ($\text{OR} = 2.54$, Fisher $p = 0.0054$, $\Delta = -0.273$ mag) | ✔ | ✔ | **Resolved L2 confirmation** |




### 3.1 Red Monsters: A No-JWST-Specific-Refit Prediction

The TEP parameterization is applied to galaxies in the Red Monster regime ($z \sim 5$–$6$, $\log M_* \gtrsim 10.5$; Xiao et al. 2024). This is a predictive test: the prespecified canonical TEP magnitude-sector benchmark $\kappa_{\rm gal} = (9.6 \pm 4.0) \times 10^5$ mag from Paper 11 was fixed prior to the high-redshift fit (with Paper 11's independent empirical Cepheid analyses recovering $\kappa_{\rm equiv}^{39} = (1.220 \pm 0.531) \times 10^6$ mag and $\kappa_{\rm Cep}^{44} = (0.665 \pm 0.362) \times 10^6$ mag). The regenerated high-redshift concordance analysis recovers $\kappa = (8.38 \pm 4.11) \times 10^5$ mag from the informative analyses, internally concordant ($p_{\rm concordance}=1.0$) and consistent with the canonical benchmark at $0.21\sigma$ — and equally within $\sim 1.1\sigma$ of the empirical Cepheid values, so the recovery does not discriminate between the canonical and empirical anchors; the concordance reflects the recovery uncertainty rather than agreement on a unique scale. It is therefore classified as a partial, anchor-consistent self-consistency check rather than as the input prior. No parameters are fitted or tuned to the high-redshift observations. The three entries below (S1–S3) use representative parameters spanning the published range (§2.1.1); the predicted correction depends primarily on $R_{\rm ML}$ and is insensitive to the exact input SFE.

Because the sample contains only three objects, the Red Monster calculation is best read as an illustrative no-JWST-specific-refit case study rather than as a standalone statistical test. The primary statistical weight comes from the population-level analyses ($N = 2{,}157$), and the prespecified benchmark prediction is further checked across three surveys ($N = 1{,}254$ at $z > 8$).




Table 3b: Illustrative TEP Predictions for Red Monster–Class Galaxies

| ID | $z$ | $R_{\rm ML}$ (Predicted) | SFE$_{\rm obs}$ | SFE$_{\rm true}$ | % Anomaly Resolved |
| --- | --- | --- | --- | --- | --- |
| S1 | 5.85 | 10.66 | 0.50 | 0.10 | 100% |
| S2 | 5.30 | 4.02 | 0.50 | 0.19 | 100% |
| S3 | 5.55 | 2.76 | 0.50 | 0.25 | 85% |
| Average | — | 5.81 | 0.50 | 0.18$^*$ | 95% |



The predicted mass bias $R_{\rm ML}^{n} \approx 3.4$ (using the mean $R_{\rm ML} = 5.81$ and $n = 0.7$) reduces the mean corrected SFE to $\sim 0.15$ ($= 0.5/5.81^{0.7} \approx 0.146$); the arithmetic mean of the three individually corrected SFEs is $\sim 0.18$ ($^*$ in Table 3b), reflecting the convexity of the $1/R_{\rm ML}^n$ mapping. Propagating the canonical benchmark (Paper 11, $\kappa_{\rm gal} = (9.6 \pm 4.0) \times 10^5$ mag) uncertainty, the corrected mean spans $\sim 0.12$–$0.26$ across the $1\sigma$ prior: fully resolved at the upper bound ($\kappa_{\rm gal} = 13.6 \times 10^5$ mag, SFE $\sim 0.12$), resolved at the central value ($\sim 0.18$, below the standard $\Lambda$CDM limit of 0.20), and only partially resolved at the lower bound ($\kappa_{\rm gal} = 5.6 \times 10^5$ mag, $\sim 0.26$, above the limit). The resolution is therefore prior-range dependent, secured at and above the central benchmark but not at its lower bound. Per object, the anomaly is fully resolved for the two most massive objects (S1, S2); the third (S3) is 85% resolved, with a residual SFE$_{\rm true} = 0.25$ reflecting the lower halo mass and correspondingly weaker inference-channel response.


### 3.2 UNCOVER DR4: Mass-sSFR and Mass-Age Correlations

The Red Monster case study establishes that TEP predicts the correct direction and magnitude of the SFE correction for individual extreme objects. The critical question is whether this signal extends to the full galaxy population. In the UNCOVER DR4 sample ($N = 2{,}157$), the mass-sSFR correlation is weak but significant ($\rho = -0.14$, $p = 1.7 \times 10^{-11}$, Cohen's $d = -0.29$). Under the TEP measurement equations, $R_{\rm ML}$ inflates inferred $M_*$ by $R_{\rm ML}^{n_{\rm SPS}}$ ($n_{\rm SPS} \approx 0.7$) and inflates inferred SFR by $R_{\rm ML}^m$ ($m \approx 0.5$), so inferred sSFR $= {\rm SFR}/M_*$ scales as $R_{\rm ML}^{m-n_{\rm SPS}}$. Since $m < n_{\rm SPS}$, the TEP bias lowers sSFR in massive galaxies, deepening the negative mass–sSFR relation rather than canceling it. The observed weak negative correlation is therefore consistent with TEP adding to the intrinsic downsizing trend, not reversing it. The mass-age correlation is positive ($\rho = +0.15$, $p = 2.4 \times 10^{-12}$), consistent with more massive galaxies carrying a larger inference-channel response and therefore appearing older under the standard isochrony mapping. Both correlations are attenuated by the full redshift range; the signal sharpens substantially when the sample is stratified by redshift.


### 3.3 Redshift Evolution: The High-z Transition

The TEP measurement equations predict $\rho(R_{\rm ML}, {\rm sSFR}) < 0$ at all redshifts, since ${\rm sSFR}_{\rm obs} \propto R_{\rm ML}^{m-n_{\rm SPS}}$ with $m < n_{\rm SPS}$. This is confirmed: the full-sample $R_{\rm ML}$–sSFR correlation yields $\rho = -0.49$ ($p = 7.7 \times 10^{-129}$, $N=2{,}157$), and the partial correlation controlling for $M_*$ and $z$ gives $\rho(R_{\rm ML}, {\rm sSFR} \mid M_*, z) = -0.62$ ($p = 8.2 \times 10^{-29}$, $N=254$) at $z > 8$. The TEP bias deepens the negative mass–sSFR relation in massive halos; it does not predict a sign inversion. The observed redshift evolution of $\rho(M_*, {\rm sSFR})$ — which does invert from negative to weakly positive at $z > 7$ — is an empirical phenomenon that TEP does not claim to generate. It may reflect the weakening of intrinsic downsizing at high redshift, selection effects, or additional physics not captured by the inference-channel response. The TEP-consistent test is the negative $R_{\rm ML}$–sSFR partial correlation, not the mass–sSFR inversion. The redshift stratification is shown for completeness:




Table 4: Mass-sSFR Correlation by Redshift

| $z$ Range | $N$ | Spearman $\rho$ | 95% CI | Interpretation |
| --- | --- | --- | --- | --- |
| 4–5 | 883 | $-0.17$ | [$-0.24$, $-0.11$] | Standard downsizing |
| 5–6 | 471 | $-0.13$ | [$-0.22$, $-0.04$] | Standard downsizing |
| 6–7 | 345 | $-0.12$ | [$-0.22$, $-0.01$] | Weakening |
| 7–8 | 204 | $+0.17$ | [$+0.04$, $+0.30$] | Inversion |
| 8–9 | 164 | $+0.12$ | [$-0.04$, $+0.26$] | Weak positive |
| 9–10 | 90 | $-0.21$ | [$-0.40$, $+0.00$] | Reversal (selection effects) |



Comparing low-$z$ ($4 < z < 6$, $\rho = -0.16$) to high-$z$ ($z > 7$, $\rho = +0.09$): $\Delta\rho = +0.25$ [+0.14, +0.35] (95% CI excludes zero), indicating a statistically significant shift in the mass–sSFR relation. This shift is an empirical observation; the TEP-consistent prediction is the negative $R_{\rm ML}$–sSFR partial correlation, not the mass–sSFR inversion itself.


### 3.4 Partial Correlation Test

The redshift evolution in §3.3 is consistent with TEP, but by itself it does not eliminate the mass-proxy concern, since $R_{\rm ML}$ depends on halo potential $\Psi \propto M_h^{2/3}$ and correlates with stellar mass. The partial-correlation hierarchy is designed to test exactly that issue. With mass-only control, age-ratio and metallicity remain weakly positive. With joint mass+redshift control they become consistent with zero, so they are classified as mass-proxy-adjacent rather than independent. The high-redshift dust signal behaves differently: at $z > 8$, the dust–$R_{\rm ML}$ correlation survives joint mass+redshift control ($\rho = +0.257$, $p = 1.2 \times 10^{-5}$, Cohen's $d = +0.52$), indicating that $R_{\rm ML}$ carries information about dust beyond mass alone. The clock-level version of the test is stronger again: controlling directly for cosmic time leaves $\rho(t_{\rm eff}, A_V \mid t_{\rm cosmic}) = +0.472$ ($p = 5.0 \times 10^{-17}$), so the signal is not a trivial restatement of redshift ordering.

*The key asymmetry:* the SUSPENSE kinematic comparison (§3.10) provides the cleanest test. $R_{\rm ML}$ retains residual age information after $M_*$+$z$ control ($\rho = +0.556$, $p = 0.032$), whereas $M_*$ contributes no residual signal once $R_{\rm ML}$+$z$ are controlled ($\rho = +0.031$, $p = 0.91$). A pure mass proxy cannot generically produce that one-way residual structure. If $R_{\rm ML}$ were only a reparameterisation of $M_*$, the relationship would be symmetric and neither predictor would retain residual information once the other was controlled.

A further complication is that if TEP is correct, SED-inferred stellar masses are themselves biased upward by $R_{\rm ML}^{0.7}$. Partial-correlation tests that control for observed $M_*$ would then over-control the true signal by removing TEP-predicted variance, making the reported partial correlations conservative lower bounds. This argument is logically valid but dialectically limited: it cannot be used to dismiss null results, and the cleanest resolution is to seek mass-independent tests. The circularity-breaker tests in §3.4.1 and the dynamical-mass comparison in §3.10 provide that independent route.

*Mass-range sensitivity:* the partial correlation $\rho(R_{\rm ML}, A_V \mid M_*, z) = +0.31$ is concentrated in the high-mass tail where $R_{\rm ML}$ transitions from screened ($R_{\rm ML} < 1$) to unscreened ($R_{\rm ML} > 1$) values. Restricting to $\log M_* < 9.5$ ($N = 241$, excluding the 13 most massive galaxies) reduces the partial to $\rho = +0.27$ ($p = 2.4 \times 10^{-5}$), and restricting to $\log M_* < 9$ ($N = 228$) gives $\rho = +0.24$ ($p = 3.5 \times 10^{-4}$). The raw mass–dust correlation, by contrast, remains robust at $\rho = +0.49$ ($p = 6.8 \times 10^{-15}$) even for $\log M_* < 9$. This means the TEP-specific residual beyond mass+$z$ is driven by the non-linear $R_{\rm ML}$ transition at the high-mass end, not by a universal linear correlation. This is consistent with the TEP prediction that the signal should emerge where the potential depth crosses the screening threshold, but it also means the partial correlation cannot by itself exclude a generic non-linear mass–redshift interaction. The Steiger comparison (§3.7.5) and the Bayesian model comparison (§3.8) provide the specificity tests that the partial correlation alone cannot.

**MIRI-Indicated Mass Recalibration Check:** To directly test the vulnerability to SED systematics (such as AGN or emission-line contamination inflating NIRCam-only masses), a systematic mass reduction was applied to the entire high-mass ($>10^{10} M_\odot$) UNCOVER sample, simulating the MIRI-based recalibrations reported by Pérez-González et al. (2024). When all masses are artificially reduced by 0.5 dex, the $R_{\rm ML}$-dust correlation at $z > 8$ remains significant ($\rho = +0.447$, $p = 2.8 \times 10^{-15}$). The signal survives even a full 1.0 dex systematic reduction ($\rho = +0.310$, $p = 9.9 \times 10^{-8}$). The correlation weakens under mass reduction — expected because $R_{\rm ML}$ is a non-linear function of halo mass — but remains statistically significant even under extreme calibration shifts, confirming that the TEP signal is not an artifact of the absolute photometric mass calibration.


#### 3.4.1 Circular Mass-Loop Breaker Tests

The most fundamental structural criticism of the photometric analysis is the circular mass loop: $R_{\rm ML}$ is computed from halo mass, halo mass is estimated from stellar mass via abundance matching, and TEP claims stellar mass is itself biased by $R_{\rm ML}^n$. The $N = 2{,}157$ sample is caught in this loop; only the SUSPENSE kinematic test ($N = 15$; §3.10) fully escapes it. Three additional tests directly target this circularity on the full photometric sample.

**Test A — 2D mass-$z$ pairing shuffle:** Rather than shuffling mass within narrow $z$-bins (as in step 143, which preserves the $z$-dependence of $R_{\rm ML}$), this test randomly reassigns each galaxy's stellar mass to a different galaxy's redshift, breaking both the mass-$z$ correlation and the $R_{\rm ML}$ functional form simultaneously. At $z > 8$ ($N = 254$), the observed $\rho(R_{\rm ML}, A_V) = +0.646$ collapses to a shuffled mean of $-0.002 \pm 0.055$ ($Z = 10.9$, $p = 5.0 \times 10^{-4}$ over 2,000 shuffles). The signal requires the correct physical mass-$z$ pairing and cannot be reproduced by any arbitrary mass-$z$ combination.

**Test B — Placebo $R_{\rm ML}$ (wrong functional form):** A placebo predictor with the same mass and redshift dependence but a different functional form — a power-law $(M_h / M_{h,{\rm ref}})^{0.3} (1+z)^{0.5}$ instead of the exponential potential-depth response — is constructed and tested against dust. Evaluated with the canonical bounded, self-consistent $R_{\rm ML}$ used throughout the pipeline, the real response yields $\rho = +0.646$ and the placebo yields $\rho = +0.507$. The bootstrap difference is $\Delta\rho = +0.139$ with 95% CI $[+0.065, +0.213]$: the real response is superior in 99.3% of 2,000 bootstrap draws and the interval excludes zero, so the exponential potential-depth form is statistically distinguished from a generic power-law mass-$z$ interaction on the photometric sample alone. An earlier non-decisive version of this comparison used a non-canonical, unbounded response construction inconsistent with the pipeline's calibrated variable; once the same bounded self-consistent $R_{\rm ML}$ is used for both predictors, the advantage of the TEP form is resolved at the stated confidence.

**Test C — Narrow mass-bin subsets and the predicted sign transition:** Restricting to 0.5-dex-wide mass bins removes the mass axis of the circularity, leaving $R_{\rm ML}$ to vary only through its redshift dependence. The reference-depth kernel makes a falsifiable sign prediction here: within a bin, $\mathrm{d}\ln R_{\rm ML}/\mathrm{d}z = K_{\rm exp}\sqrt{1+z}\,\mathrm{d}\Psi/\mathrm{d}z + K_{\rm exp}(\Psi - \Psi_{\rm ref})/(2\sqrt{1+z})$ changes sign across a crossover mass set by the competition between the positive stellar-to-halo-mass drift and the $(\Psi - \Psi_{\rm ref})$ term. The observed within-bin correlations execute exactly this transition: $\log M_* \in [8.0, 8.5]$ ($N = 1{,}528$, $\rho = -0.346$, $p = 2.9 \times 10^{-44}$), $\log M_* \in [8.5, 9.0]$ ($N = 526$, $\rho = -0.457$, $p = 1.7 \times 10^{-28}$), $\log M_* \in [9.0, 9.5]$ ($N = 190$, $\rho = -0.023$, consistent with zero at the crossover), and $\log M_* \in [9.5, 10.0]$ ($N = 48$, $\rho = +0.347$, $p = 0.016$). All four bins match the sign predicted by the kernel derivative at the bin centres ($4/4$, zero discordant at $p<0.05$), with the interpolated crossover at $\log M_* \simeq 9.6$ ($\log M_h \simeq 11.3$, $\Psi \simeq 0.35\,\Psi_{\rm ref}$, shifted below the reference depth by the positive within-bin SMHM drift).

The sign structure is the discriminating content: it is not shared by any uniform-sign monotone-in-$z$ baseline. Raw redshift predicts a positive within-bin correlation in all four bins ($\rho = +0.36$, $+0.50$, $+0.19$, $+0.44$), the power-law placebo likewise ($\rho = +0.28$, $+0.43$, $+0.32$, $+0.45$), and a deliberately inverted monotone baseline $M_*^{0.3}(1+z)^{-0.5}$ predicts a negative correlation in all four ($\rho = -0.29$, $-0.40$, $-0.04$, $-0.21$) — each family fails at least one end of the transition, while raw $M_*$ is consistent with zero in the three lower bins ($\rho = -0.03$, $-0.01$, $+0.15$) and cannot generate the negative branch at all. Only the reference-depth kernel produces the observed negative-to-positive inversion, so the within-bin sign map falsifies every uniform-sign monotone-in-$z$ explanation of the dust signal. The negative sign in the two largest bins is therefore not the ordinary dust–redshift trend read backwards: dust rises with $z$ inside those bins, yet $R_{\rm ML}$ orders anti-correlated with $z$ precisely where $\Psi < \Psi_{\rm ref}$ — the signature of a response anchored to a reference potential depth, not of a generic mass-$z$ correlation.

**Assessment:** 3 of 3 circularity-breaking tests pass. The 2D shuffle confirms the signal requires the correct mass-$z$ pairing; the placebo comparison now resolves the exponential potential-depth form over a generic power-law ($\Delta\rho$ CI excludes zero); and the narrow-mass-bin sign map reproduces the kernel-predicted crossover that no monotone-in-$z$ baseline can generate. The photometric sample therefore does carry TEP-specific functional-form information, complementing — rather than deferring to — the kinematic and cross-survey evidence. The remaining limitations are quantitative: the positive-sign bin is supported by only $N = 48$ galaxies, and ordinary linear-model AIC comparison (step 136) finds the $M_* \times z$ interaction and the TEP predictor comparably efficient ($\Delta$AIC $\simeq 1$–$2$), so the sign-structure and placebo tests — not regression ranking — carry the discrimination.


### 3.5 Screening Signatures

A distinctive feature of the TEP framework — one that distinguishes it from any smooth mass-dependent function — is the screening prediction: above a Temporal Topology saturation proximity scale $\rho_T \approx 20$ g/cm³, the scalar field is suppressed and $R_{\rm ML} \to 1$. Paper 10 (TEP-COS) established an effective kinematic screening threshold at $\sigma > 165$ km/s from globular cluster pulsar timing; that boundary belongs to the shear channel $\mathcal{S}_\Sigma$ and does not transfer to the amplitude channel $\mathcal{S}_A$ governing $R_{\rm ML}$ (§2.3.2.2). Under the amplitude-channel suppression of Eq. ($\ref{eq:jwst_kappa_eff}$), all halos in these samples are unscreened — their internal densities sit $\sim 24$ orders below $\rho_T$ — so the predicted column below applies the $\mathcal{S}_A = 1$ kernel, which is the correct limit rather than an approximation. The halo-mass ordering of age ratios is then examined as a screening-adjacent diagnostic:




Table 5: Age Ratio by Halo Mass (5 < z < 8)

| $\log M_h$ | $N$ | $\langle$MWA/$t_{\rm cosmic}\rangle$ | $R_{\rm ML}$ Predicted |
| --- | --- | --- | --- |
| 10–11 | 390 | $0.15 \pm 0.003$ | 0.5–0.6 |
| 11–12 | 42 | $0.18 \pm 0.015$ | 0.6–1.0 (ref at 12.0) |
| 12–12.5 | 3 | $0.30 \pm 0.12$ | 1.0–1.6 |
| 12.5–13 | 1 | $0.05$ | 1.6–4.5 |



*Environmental contrast.* A complementary environmental-screening test splits each sample by local galaxy number density and asks whether the dust–$R_{\rm ML}$ correlation is suppressed in overdense regions, as the ambient-environment dependence of the response predicts (Step 138, `step_138_environmental_screening_steiger.json`). Whereas an earlier all-epoch 2-D sky projection mixed foreground $z \sim 4$–$6$ galaxies into the neighbour counts of $z > 8$ sources, restricting the 5th-nearest-neighbour density estimator to same-epoch neighbours ($|z_j - z_i| \le 0.5$) isolates the true co-moving environment: at $z > 8$ ($N = 254$), the dust–$R_{\rm ML}$ correlation drops from $\rho_{\rm field} = +0.712$ ($p = 6.2 \times 10^{-21}$) in low-density environments to $\rho_{\rm dense} = +0.484$ ($p = 7.7 \times 10^{-9}$) in overdense environments, yielding a statistically significant contrast $\Delta\rho = +0.227$ (95% CI $[+0.075, +0.376]$, Fisher $Z = +2.85$, $p = 0.0044$), which survives holding the foreground Abell-2744 cluster-centric distance fixed across all three $d_{\rm A2744}$ terciles ($+0.269, +0.184, +0.137$; 3/3 positive, mean $d_{\rm A2744}$-controlled contrast $\Delta\rho = +0.197$). The same-epoch screening contrast is likewise confirmed at $z = 6$–$8$ ($N = 549$, $\Delta\rho = +0.355$, $Z = +4.32$, $p = 1.6 \times 10^{-5}$) and across the full $z = 4$–$10$ sample ($N = 2{,}157$, $\Delta\rho = +0.205$, $Z = +5.15$, $p = 2.6 \times 10^{-7}$), while remaining null in the $z = 4$–$6$ control bin ($\Delta\rho = -0.075$, $p = 0.17$) where the $z > 8$ dust anomaly is absent.


#### 3.5.1 Resolved Core Screening

TEP predicts that deep core potentials screen the scalar field locally while outskirts retain a large response, producing a structurally concentrated, bluer-core signature in massive galaxies. In the JADES DR5 direct-mass morphology sample ($N = 384$, Step 155), after controlling for mass and redshift, the Gini coefficient is strongly supportive (partial $\rho = +0.191$, $p = 1.6 \times 10^{-4}$; replicated in GOODS-S $N = 588$ at $\rho = +0.120$, $p = 3.6 \times 10^{-3}$, Step 156). In the JADES resolved colour-gradient analysis ($N = 277$, Steps 139 and 168), the raw mass-gradient trend is $\rho(M_*, \nabla_{\rm Color}) = -0.166$ ($p = 5.7 \times 10^{-3}$) and the global halo-scale $R_{\rm ML}$ correlation is $\rho(R_{\rm ML}, \nabla_{\rm Color}) = -0.192$ ($p = 1.3 \times 10^{-3}$). While the size-blind global halo predictor $\Gamma_t(M_h, z)$ has no residual variance once $(M_*, z)$ are regressed out ($\rho_{\rm partial} = +0.037$, $p = 0.54$), incorporating the measured inner-to-outer F444W stellar concentration $C_{\rm core} = F_{\rm F444W,CIRC1}/F_{\rm F444W,CIRC4}$ across the exact same apertures defines the resolved radial screening predictor $\Gamma_{\rm rad} = \exp[K_{\rm gal}\Psi_{\rm halo}(M_h)\sqrt{1+z}/C_{\rm core}]$: $\Gamma_{\rm rad}$ correlates with the colour gradient at $\rho = -0.168$ ($p = 0.0050$), retains partial $\rho(\Gamma_{\rm rad}, \nabla_{\rm Color} \mid M_*, z) = -0.179$ ($p = 0.0028$, $2.99\sigma$, exact TEP bluer-core sign; partial $\rho = -0.237$, $p = 6.9 \times 10^{-5}$, $4.0\sigma$ on $C_{\rm core}^{-1}$), outperforms $M_*$ on the Steiger $Z$-test ($Z = 3.02$, $p = 0.0025$, $3.02\sigma$), and produces a statistically significant $(M_*, z)$-residualized bluer-core sign reversal ($58.6\%$ vs $35.7\%$ on $q_{25}/q_{75}$, $\text{OR} = 2.54$, Fisher $p = 0.0054$, $\Delta = -0.273$ mag, 95% CI $[-0.526, -0.035]$; and $57.6\%$ vs $42.4\%$ on $q_{33}/q_{67}$, $\text{OR} = 1.85$, Fisher $p = 0.0275$).


### 3.6 The z > 8 Dust Anomaly: Correlation Structure

The $R_{\rm ML}$–sSFR partial correlation (§3.3) and the partial-correlation hierarchy (§3.4) show that $R_{\rm ML}$ carries information beyond a simple mass trend. The clearest empirical test is the dust–$R_{\rm ML}$ correlation structure. Under standard physics, the universe at $z \sim 9$ is only $\sim 540$ Myr old, barely enough for the first generation of AGB stars to complete their evolution. The question is not whether the dust budget can be closed under TEP — the inferred-time proxy $t_{\rm eff} = R_{\rm ML} \, t_{\rm cosmic}$ is not a physical proper-time integral (§2.3.2) and cannot be used to compute physical AGB evolution timescales. The question is whether the *pattern* of dust emergence across the galaxy population is organized by the inference-channel response $R_{\rm ML}$ more strongly than by the simple time-only and mass-only baselines tested here.


**The Uniformity Paradox ($N=33$ massive galaxies at $z > 8$)**

Under standard physics, the cosmic time $t_{\rm cosmic}(z)$ available for dust production is uniform across all galaxies at a given redshift. If dust parameters (AGB yield, SN yield, destruction timescale) are tuned to close the budget at $z > 8$, dust should be ubiquitous or track star formation. Instead, observations reveal a strong mass-dependent suppression ($\rho = +0.56$): massive galaxies are dusty; low-mass galaxies are dust-poor. A purely redshift- or time-only explanation cannot reproduce the observed mass-dependent gradient. Under TEP, this gradient arises because the inference-channel response $R_{\rm ML}$ orders the apparent evolutionary advance with potential depth: massive halos ($R_{\rm ML} > 1$) appear more evolved and dust-rich, while low-mass halos ($R_{\rm ML} \ll 1$) appear less evolved and dust-poor. The anomaly is not that massive galaxies have dust — it is that low-mass galaxies *do not*, in a pattern that tracks gravitational potential depth rather than star formation rate or cosmic time.

Recent JWST spectroscopy shows that AGB stars produce SiC and iron dust even at low metallicity ($\sim 1$–$7\%\,Z_\odot$; Boyer et al. 2025), with onset as early as 30–50 Myr for the most massive AGB progenitors. The physical question of whether the canonical TEP matter-frame proper time $\tau_\star = \int A[\phi(t, \mathbf{x})] \, dt$ is sufficient for AGB evolution at high redshift is left for future work; the present analysis establishes the empirical correlation structure, not the physical dust-budget closure.






![The z > 8 Dust Anomaly: Mass-Dust Correlation](public/figures/figure_5_dust_anomaly.png)



Figure 2: The Key Dust Anomaly. (a) At $z \sim 5$ (grey), mass and dust are uncorrelated ($\rho \approx 0$). (b) At $z > 8$ (color), a strong correlation emerges ($\rho = +0.60$ in UNCOVER). Massive galaxies (high $R_{\rm ML}$, yellow) have successfully produced dust despite the short cosmic time (< 600 Myr), while low-mass galaxies (low $R_{\rm ML}$, purple) remain dust-poor. TEP predicts this specific mass-dependent divergence.





**Figure 3: The Dust Saturation Crisis.** The ratio of observed dust mass to the maximum theoretical yield under standard-physics timescales is plotted for massive galaxies at $z > 8$. The population sits near the saturation limit (100% of yield), leaving no margin for error. The mass-dependent pattern of dust emergence — dusty in massive halos, dust-poor in low-mass halos — is the empirical signature that TEP organizes via $R_{\rm ML}$. The physical dust-budget closure under the canonical TEP matter-frame proper time is not computed here; the inferred-time proxy is not a physical proper-time integral (§2.3.2).




#### 3.6.1 The $z = 5$–$7$ Dip

The redshift-binned correlations reveal a more complex pattern than a simple monotonic emergence. At $z = 5$–$6$, $\rho(A_V, R_{\rm ML}) = -0.14$ ($p = 1.2 \times 10^{-3}$) and $\rho(A_V, t_{\rm eff}) = -0.19$ ($p = 3.3 \times 10^{-5}$), both significantly negative — the opposite sign from the TEP-predicted positive correlation. At $z = 6$–$7$, $\rho(A_V, R_{\rm ML}) = -0.04$ ($p = 0.47$, non-significant). The signal only turns positive and significant at $z > 8$: $\rho(A_V, R_{\rm ML}) = +0.36$ ($p = 6.7 \times 10^{-7}$) at $z = 8$–$9$ and $\rho = +0.76$ ($p = 9.8 \times 10^{-21}$) at $z = 9$–$10$. The negative correlations at $z = 5$–$7$ are a genuine anomaly for the universal-correlation interpretation of TEP: if $R_{\rm ML}$ universally enhances dust production, the correlation should be positive at all redshifts. The "competition epoch" interpretation — high sSFR depleting dust faster than AGB stars replenish it — is a post-hoc explanation, not a prespecified prediction. This anomaly is reported transparently: the TEP signal is an emergence phenomenon at $z > 8$, not a universal correlation, and the intermediate-redshift negative correlations are currently unexplained by the theory.

The mass-dust correlation was therefore tested across three independent surveys (UNCOVER, CEERS, COSMOS-Web) using different SED fitting codes (Prospector/BEAGLE, EAZY, LePhare) and priors.


### 3.7 Cross-Survey Replication and Meta-Analysis


#### 3.7.1 Cross-Code Robustness

The $z > 8$ dust-$R_{\rm ML}$ correlation is detected in all three datasets despite differences in methodology:




Table 7: Cross-Survey Replication of $z > 8$ Dust-$R_{\rm ML}$ Correlation

| Survey | Code | $N$ (z > 8) | $\rho(R_{\rm ML}, \text{Dust})$ | 95% CI | $p$-value | Significance |
| --- | --- | --- | --- | --- | --- | --- |
| UNCOVER | Prospector/BEAGLE | 254 | $+0.65$ | $[+0.57, +0.71]$ | $p = 2.3 \times 10^{-31}$ | $13.4\sigma$ |
| CEERS | EAZY | 82 | $+0.66$ | $[+0.51, +0.76]$ | $p = 2.4 \times 10^{-11}$ | $6.7\sigma$ |
| COSMOS-Web | LePhare | 918 | $+0.63$ | $[+0.59, +0.67]$ | $p = 1.0 \times 10^{-102}$ | $21.5\sigma$ |
| Fixed-effects meta | Combined | 1,254 | $+0.635$ | $[+0.60, +0.67]$ | $p = 3.65 \times 10^{-154}$ | $26.4\sigma$ |




#### 3.7.2 Meta-Analysis

Combining all three surveys ($N = 1{,}254$ at $z > 8$) yields a fixed-effects meta-correlation of $\rho = +0.635$ with $p = 3.65 \times 10^{-154}$ (Cohen's $d = 1.63$, a large effect). All three surveys independently confirm the positive $\rho$ at $>6\sigma$, and mass-stratification confirms the signal persists at fixed mass. The per-survey effect sizes are $\rho = +0.65$ (UNCOVER, $N = 254$), $\rho = +0.66$ (CEERS, $N = 82$), and $\rho = +0.63$ (COSMOS-Web, $N = 918$). Between-study heterogeneity is negligible ($I^2 = 0\%$, Cochran's $Q = 0.24$, $p_Q = 0.89$), meaning the three surveys agree on a common effect size — as expected if TEP is a universal law with a single coupling constant. This is a non-trivial concordance: the three surveys use different SED fitting codes (Prospector/BEAGLE, EAZY, LePhare), different filter sets, and different selection functions, yet converge on the same $\rho \approx 0.64$. Including the supplementary NIRSpec Balmer-decrement branch raises the combined Fisher $z$ from $25.1\sigma$ to $26.2\sigma$, while the Balmer partial-$\rho$ branch itself is weak after $M_*$+$z$ control ($\rho = 0.136$, $p = 5.9 \times 10^{-15}$ — statistically significant but substantially attenuated relative to the photometric partial $\rho = 0.31$); the COSMOS2025 blank-field dataset is classified as supplementary rather than primary due to its different selection function. A lensing-systematic control specific to the UNCOVER field further isolates the correlation: restricting to unmagnified sources ($\mu_{\rm ML} \le 1.05$, $N = 144$, DR4 v2.0 lens model) leaves $\rho = +0.585$ ($p = 1.34 \times 10^{-14}$), and the correlation is positive in both clustercentric halves ($+0.667$ inner, $+0.580$ outer), with $\rho(\mu_{\rm ML}, A_V) = -0.061$ ($p = 0.33$) — the dust–$R_{\rm ML}$ structure is not carried by the lensing gradient, in contrast to the environmental-density channels audited in §3.8. The magnification-corrected mass diagnostic confirms this: evaluating $R_{\rm ML}$ with $\mu$-corrected stellar masses ($\log M_{*,\rm true} = \log M_{*,\rm obs} - \log_{10}\mu$) strengthens the correlation to $\rho = +0.656$ ($p = 1.30 \times 10^{-32}$, $N = 254$; $\rho = +0.603$ at $\mu \le 1.05$), demonstrating that uncorrected magnification bias does not inflate the primary photometric result. Applying the survey's photometry flag (${\tt use\_phot} = 1$) canonically excludes flagged sources and yields the confirmed headline $\rho = +0.646$ ($N = 254$, $p = 2.3 \times 10^{-31}$; $\rho = +0.585$ on the unmagnified subset, $N = 144$) (step_182).


#### 3.7.3 Inferred-Time Ordering and AGB-Motivated Threshold

A more physically targeted and falsifiable test compares dust against cosmic time ($t_{\rm cosmic}$) versus the TEP-effective inference coordinate ($t_{\rm eff} = R_{\rm ML}\,t_{\rm cosmic}$). Under standard physics, dust should track $t_{\rm cosmic}$; under TEP, dust emergence is organized along $t_{\rm eff}$ and exhibits an empirical step-like transition near the canonical AGB-motivated coordinate threshold ($t_{\rm eff} \gtrsim 0.3$ Gyr).




Table 7b: Cross-Survey Temporal Inversion and AGB Threshold (z > 8)

| Survey | $\Delta\rho = \rho(t_{\rm eff}, A_V) - \rho(t_{\rm cosmic}, A_V)$ | Dust ratio ($t_{\rm eff} > 0.3$ Gyr) | $p$ (threshold) |
| --- | --- | --- | --- |
| UNCOVER | $+0.357$ | $1.63\times$ | $6.6 \times 10^{-6}$ |
| CEERS | $+0.609$ | $2.24\times$ | $8.4 \times 10^{-3}$ |
| COSMOS-Web | $+0.850$ | $1.75\times$ | $1.1 \times 10^{-3}$ |


To test whether the location of the step is being tuned to a particular survey, a leave-one-survey-out holdout validation is performed. The threshold selected on the training surveys has median $t_{\rm eff} = 1.93$ Gyr (range $0.06$–$1.93$ Gyr). Despite this fold-to-fold variation, the held-out results remain strongly inconsistent with the null (Fisher-combined $p = 1.1 \times 10^{-25}$). Using the fixed AGB-motivated threshold $t_{\rm eff} > 0.3$ Gyr yields a Fisher-combined $p = 1.5 \times 10^{-252}$.

In COSMOS-Web, where the dust estimator is zero-inflated, the dust detection fraction is 0.58 above threshold versus 0.07 below threshold (Fisher exact test; p-value $< 10^{-10}$). An independent combined-survey threshold scan ($N = 2{,}971$) confirms the transition. For the fixed theoretical threshold of $t_{\rm eff} \ge 0.3$ Gyr, the combined odds ratio is 9.8 ($p = 2.9 \times 10^{-165}$), with dust detection fraction 0.64 above versus 0.15 below. A step-function AIC comparison against a mass-matched threshold yields $\Delta$AIC $= +0.25$ (the mass-matched step fits better as a pure step), but the polynomial $M_* \times z$ baseline outperforms both step models; the $t_{\rm eff}$ step is therefore not a unique discriminator against mass-threshold alternatives in this single-observable test. A secondary unconstrained threshold scan yields a data-selected transition at $t_{\rm eff} = 0.61$ Gyr (bootstrap 16th–84th percentile: $0.52$–$1.37$) that structurally validates the presence of an abrupt temporal step. This cross-survey temporal-ordering behavior tests whether $t_{\rm eff}$ organizes dust emergence and is not a generic "more massive galaxies are dustier" statement.

A dedicated UNCOVER-only validation independently passes all four targeted tests: the AGB threshold gives a 1.60$\times$ dust ratio above versus below $t_{\rm eff} = 0.3$ Gyr; controlling for cosmic time leaves $\rho(t_{\rm eff}, A_V \mid t_{\rm cosmic}) = +0.471$ ($p = 5.0 \times 10^{-17}$); the $t_{\rm eff}$–dust correlation remains positive in the low-mass half ($\rho = +0.42$, $p = 2.0 \times 10^{-7}$) and remains positive but weaker in the high-mass half ($\rho = +0.17$, $p = 0.038$); and the raw mass-dust signal steepens monotonically from $z = 8$–$8.5$ to $z = 9$–$10$ ($\rho = +0.325 \rightarrow +0.716$).


##### 3.7.3.1 AGB Dust Phase Boundary in ($M_*$, $z$) Space

The empirical inferred-time boundary $t_{\rm eff} = 0.3$ Gyr (motivated by the canonical AGB onset timescale but used here as an inference-coordinate threshold, not a physical proper-time integral; see §2.3.2) defines a *curve* in ($M_*$, $z$) space — not a vertical line (mass-only) or horizontal line ($z$-only). Its shape encodes both the exponential $R_{\rm ML}$ form and the redshift-dependent coupling $\alpha(z) \propto \sqrt{1+z}$. A mass-only threshold cannot replicate this curve.

Using the UNCOVER sample ($N = 2{,}157$) with $A_V > 0.1$ as the dust detection criterion, the TEP phase boundary achieves classification F1 $= 0.859$ (precision $= 0.775$, recall $= 0.964$). A mass-only quantile-matched threshold (1D vertical line in $M_*$ space) yields F1 $= 0.865$ ($\Delta$F1 $= -0.005$), indicating that the TEP boundary does not provide a classification advantage over a simple mass threshold in this single-observable test. This is expected: the TEP boundary is a curve in $(M_*, z)$ space, but dust detection at $z > 4$ is dominated by the mass axis, so a 1D mass threshold captures most of the signal. The TEP boundary's distinctive non-linear shape — curving toward lower masses at higher redshift as $\alpha(z)$ increases — is a qualitative prediction that a mass-only model cannot reproduce, but this shape difference does not translate to a significant F1 improvement. At $z > 8$: 93.1% of galaxies above the TEP boundary are dusty (175/188), while 90.9% below the boundary are also dusty (reflecting that some low-$t_{\rm eff}$ galaxies acquire dust through non-AGB channels such as supernovae). The classification power of the TEP boundary is therefore not a standalone discriminator; the primary evidence for TEP rests on the correlation structure (L1, L3) and the $t_{\rm eff}$-vs-$t_{\rm cosmic}$ Steiger test, not on the F1 classification margin.


#### 3.7.4 The Time-Lens Map: Effective Redshift $z_{\rm eff}$

To express the dust-clock result in a coordinate that is directly comparable across observed redshift, an effective redshift $z_{\rm eff}$ is defined by solving $t_{\rm cosmic}(z_{\rm eff}) = t_{\rm eff} = R_{\rm ML}\,t_{\rm cosmic}(z_{\rm obs})$. In this mapping, galaxies with larger $R_{\rm ML}$ are assigned lower $z_{\rm eff}$ (older effective ages). The key falsifiable prediction is that dust should be more strongly ordered by $z_{\rm eff}$ than by $z_{\rm obs}$.




Table 7c: Time-Lens Map: Dust vs $z_{\rm obs}$ and $z_{\rm eff}$ (z > 8, dust > 0)

| Survey | $N$ | $\rho(A_V, z_{\rm obs})$ | $p$ | $\rho(A_V, z_{\rm eff})$ | $p$ |
| --- | --- | --- | --- | --- | --- |
| UNCOVER | 254 | $+0.029$ | $0.65$ | $-0.329$ | $8.3 \times 10^{-8}$ |
| CEERS | 82 | $+0.052$ | $0.64$ | $-0.557$ | $5.5 \times 10^{-8}$ |
| COSMOS-Web | 918 | $+0.230$ | $1.8 \times 10^{-12}$ | $-0.620$ | $1.2 \times 10^{-98}$ |


Across surveys, $|\rho(A_V, z_{\rm eff})| > |\rho(A_V, z_{\rm obs})|$. Critically, UNCOVER and CEERS show *zero* dust–$z_{\rm obs}$ correlation ($\rho \approx 0$, $p > 0.6$), while the TEP effective-time coordinate yields $|\rho|$ up to $0.62$. Classification performance confirms this: in COSMOS-Web ($N = 2{,}606$), where dust-free galaxies exist, AUC for predicting dusty ($A_V > 0$) vs. dust-poor galaxies is $0.89$ for $t_{\rm eff}$ vs. $0.68$ for $t_{\rm cosmic}$. The combined three-survey AUC is $0.85$ for $t_{\rm eff}$ vs. $0.80$ for $M_*$ vs. $0.72$ for $t_{\rm cosmic}$. (Note: UNCOVER and CEERS $z > 8$ samples have $A_V > 0$ for all galaxies, so binary classification is only possible in COSMOS-Web and the combined sample.)


#### 3.7.5 Functional Form Discrimination

A pure mass proxy makes a specific set of predictions. It should produce dust that increases monotonically with $M_*$ at all redshifts, it should generalize cross-survey because mass is survey-independent, and it should not generate the negative $R_{\rm ML}$–sSFR partial correlation seen in L3. TEP predicts the opposite pattern: little or no dust–mass correlation at $z < 7$, emergence at $z > 8$, and a non-linear inferred-time boundary that curves in ($M_*, z$) space. The tests below are therefore aimed not at asking whether both models can fit one subset of the data, but at asking which set of predictions matches the full activation pattern.

**The critical distinction from a mass-only model:** a mass proxy that fits the $z > 8$ dust signal would still have to be re-fit survey by survey because survey-specific SED systematics shift the absolute calibration. By contrast, $R_{\rm ML}$, anchored by the prespecified canonical theory benchmark $\kappa_{\rm gal} = 0.960 \times 10^6$ mag, maintains $\rho = 0.63$–$0.66$ across three surveys (fixed-effects meta $\rho = +0.635$) with no retraining. The Steiger tests below therefore compare not just two correlated predictors, but two different claims about what should remain stable across datasets:


- **Within-regime ($z > 8$):** While the size-blind global halo predictor $t_{\rm eff}(M_h, z)$ does not resolve internal radial structure beyond total mass ($Z = -0.49$, $p = 0.62$), the resolved radial screening predictor $\Gamma_{\rm rad}(M_h, z, C_{\rm core})$ incorporating the measured inner-to-outer F444W stellar concentration $C_{\rm core}$ significantly outperforms $M_*$ on the Steiger test ($Z = 3.02$, $p = 0.0025$, $3.02\sigma$) and retains partial $\rho = -0.179$ ($p = 0.0028$, $2.99\sigma$) after $(M_*, z)$ control.

- **Activation pattern test ($z > 8$):** $\rho(\text{dust}, t_{\rm eff}) = +0.61$ vs. $\rho(\text{dust}, M_*) = +0.53$ ($Z = 7.4$, $p = 1.7 \times 10^{-13}$, combined three-survey $z>8$ sample). This confirms $t_{\rm eff}$ correctly predicts both the absence of correlation at low $z$ and its emergence at $z > 8$, though the margin over $M_*$ is narrower than for the raw dust–$R_{\rm ML}$ correlation.

- **$t_{\rm eff}$ vs. $t_{\rm cosmic}$ per-survey:** $t_{\rm eff}$ significantly outperforms raw cosmic time in every survey (per-survey Steiger $Z = +8.1$, $+6.2$, $+30.3$ for UNCOVER, CEERS, COSMOS-Web; Stouffer combined $Z \approx 25.8$).





### 3.8 Nested Bayesian Model Comparison



Table 8: Bayesian Evidence ($\ln Z$) for $z \ge 8$ Multi-Observable Models. All models are evaluated using dynesty nested-sampling ($N_{\rm live}=200$, $d\log Z=0.5$). The self-consistent (iterated) $R_{\rm ML}$ predictor is used throughout. The covariance family is shown under both conditionings of the standard baseline: the orthogonalized model uses mass and redshift orthogonalized against $\log R_{\rm ML}$ (TEP-conditioned), while the raw model uses unmodified mass and redshift. In the independent-likelihood family, all alternative models use orthogonalized mass and redshift. Bayes factors are meaningful only within a single likelihood family.

| Model Name | Parameters | $\ln Z$ | $\pm \Delta \ln Z$ |
| --- | --- | --- | --- |
| **1. Joint Likelihood Family (Correlated SED Outputs)** |  |  |  |
| Covariance Augmented (Mass + z + $R_{\rm ML}$) | 17 | -1243.8 | 0.83 |
| Covariance TEP (Single Theory Predictor) | 9 | -1262.6 | 0.65 |
| Covariance Standard (Raw Mass + z) | 13 | -1248.2 | 0.71 |
| Covariance Standard (Orthogonalized Mass + z) | 13 | -1304.1 | 0.68 |
| Covariance Interaction (Mass + z + Mass×z) | 17 | -1190.9 | 0.85 |
| Covariance Interaction + $R_{\rm ML}$ | 21 | -1191.8 | 0.85 |
| Covariance Ambient ($u_{\rm amb}$ proxy only) | 9 | -1304.6 | 0.68 |
| Covariance Standard + $u_{\rm amb}$ | 17 | -1233.0 | 0.75 |
| Covariance Interaction + $u_{\rm amb}$ | 21 | -1170.5 | 0.86 |
| **2. Independent-Likelihood Family (Orthogonalized Mass & z)** |  |  |  |
| TEP Augmented (Mass + z + $R_{\rm ML}$) | 20 | -1302.7 | 0.88 |
| TEP (Single Theory Predictor) | 12 | -1332.8 | 0.76 |
| Quadratic Baseline (Mass + z + $M^2$ + $z^2$ + $Mz$) | 28 | -1410.5 | 0.84 |
| $M_* \times \sqrt{1+z}$ Interaction | 20 | -1454.0 | 0.80 |
| Varying IMF (Quadratic Mass + z) | 20 | -1430.9 | 0.77 |
| AGN Feedback (Sigmoid Mass Threshold) | 18 | -1435.3 | 0.77 |
| Bursty SF (Mass-dependent timescale) | 21 | -1462.6 | 0.75 |
| Standard Physics (Linear Mass + z) | 16 | -1465.8 | 0.76 |
| **2b. Systematics-Controlled Environmental Ladder (step 182 audit; $\Delta\ln Z$ vs $\{M_*,z\}=-1249.2$)** |  |  |  |
| $\{M_*,z,\mu_{\rm ML}\}$ (lensing magnification) | 17 | -1234.5 | +14.7 |
| $\{M_*,z,d_{\rm A2744}\}$ (clustercentric distance) | 17 | -1221.1 | +28.1 |
| $\{M_*,z,d_{\rm A2744},\mu_{\rm ML}\}$ (systematics baseline) | 21 | -1215.0 | +34.2 |
| $\{M_*,z,d_{\rm A2744},\mu_{\rm ML},u_{\rm amb,mw}\}$ (mass-weighted same-epoch density on top) | 25 | -1222.0 | $-7.1$ vs systematics baseline |
| $\{M_*,z,M_*\!\times\!z,d_{\rm A2744},\mu_{\rm ML},u_{\rm amb,mw}\}$ | 29 | -1166.6 | $-7.8$ vs $\{M_*,z,M_*\!\times\!z,d_{\rm A2744}\}$ |
| **3. TEP-Aware Residual Family (Orthogonalized)** |  |  |  |
| TEP-Aware Residual (Orthogonalized) | 12 | -1314.1 | 0.72 |
| Residual Constrained AGN (Orthogonalized) | 10 | -1418.7 | 0.62 |
| Residual Null (Orthogonalized) | 8 | -1472.1 | 0.59 |

The evaluation is structured into three comparison families, distinguished by the conditioning applied to the predictor space:

- **TEP-conditioned comparisons (orthogonalized predictors):** alternative models are evaluated on the components of mass and redshift orthogonal to $\log R_{\rm ML}$. This isolates whether any explanatory variance lies outside the TEP-predicted direction, but it removes the competitor's access to that direction and is therefore reported as a direction-isolation diagnostic rather than an unconditional Bayes factor.

- **Unconditioned comparisons (raw mass + z):** the baseline evaluated on observed mass and redshift. Under the TEP measurement model ($M_{\rm obs} = M_{\rm true} + n\log_{10}R_{\rm ML}$), observed mass already contains part of the response, so this conditioning is not neutral either: it grants the baseline access to a channel that carries the TEP signal under the hypothesis being tested.

- **Nested augmented comparison:** the conditioning-free arbiter. The augmented model (raw mass + z + $\log R_{\rm ML}$) is compared against each reduced model, yielding the marginal evidence for each predictor direction given the other.

The covariance-corrected family, in which the correlated SED outputs are modelled with a joint multivariate Gaussian likelihood using the empirical $K \times K$ covariance matrix of the standardized observables (shared identically across all compared models), exhibits the full conditioning dependence (Table 8). Under TEP conditioning, the 9-parameter theory-fixed predictor ($\ln Z = -1262.6$, 9 parameters, with no separate baseline secular $z$ slope) exceeds the 13-parameter orthogonalized standard model ($\ln Z = -1304.1$, 13 parameters) by $\ln{\rm BF} = +41.5$. When the 9-parameter single-predictor model is compared directly against the 13-parameter unconditioned raw mass+z baseline ($\ln Z = -1248.2$, which fits two free slopes per observable), the difference is $\ln{\rm BF} = -14.4$. The direction-isolation diagnostic quantifies where the signal resides: restoring the $R_{\rm ML}$-aligned direction to the standard predictor is worth $\Delta\ln Z = +55.9$, so essentially all explanatory variance is concentrated in the direction TEP specifies, while the orthogonal complement of mass space is evidentially inert. Crucially, the nested augmented model (mass + z + $R_{\rm ML}$, 17 parameters, $\ln Z = -1243.8$) is the single highest-evidence model in the ensemble and arbitrates the marginal contributions without conditioning bias: adding $\log R_{\rm ML}$ to raw mass+z yields $\Delta\ln Z = +4.45 \pm 1.09$ ($\text{BF} = 85.4 : 1$, $>4\sigma$ positive Bayesian evidence for the nonlinear TEP predictor even after raw mass and $z$ are already included, despite the 4-parameter Occam penalty), whereas adding mass+z to the single 9-parameter $R_{\rm ML}$ predictor yields $\Delta\ln Z = +18.9$. The per-observable breakdown shows the conditioned signal is concentrated in dust ($\ln{\rm BF} = +21.1$) and the SED $\chi^2$ statistic ($\ln{\rm BF} = +52.5$), with a positive contribution from sSFR ($\ln{\rm BF} = +49.9$) and a null result for metallicity ($\ln{\rm BF} = +9.9$). Because the SED $\chi^2$ is a fit-quality diagnostic rather than a physical observable, the 3-observable physical subset (dust, sSFR, metallicity) is the primary observable set: it yields $\ln{\rm BF} = +37.4$ under orthogonalized predictors and $\ln{\rm BF} = -20.4$ under raw predictors (7 versus 10 parameters). Two further arbiters sharpen the reading. First, a free mass–redshift interaction model ($\{1, M_*, z, M_*\times z\}$, 17 parameters) attains $\ln Z = -1190.9$, exceeding the raw baseline by $57.3$ and the augmented model by $52.9$; adding $\log R_{\rm ML}$ on top of the interaction returns $\Delta\ln Z = -0.9$, so the instantiated kernel contributes no unique evidence once the unconstrained interaction direction is open — consistent with the deterministic subsumption of $\log R_{\rm ML}$ within (mass, z) noted below. Second, and decisive for the operative theory-0 reading, the embedded response law is ambient-field-driven ($D^{-1}\propto u_{\rm amb}^{-2}$), not the magnitude-sector kernel $R_{\rm ML}=\exp[K_{\rm gal}\Psi(M_h)\sqrt{1+z}]$ the pipeline instantiates. A $k$-nearest-neighbour ambient proxy on (RA, Dec, $\ln(1+z)$) — $\log u_{\rm amb}\propto\tfrac12\log\rho_{\rm env}$ — attains $\ln Z = -1304.6$ as a lone predictor but adds $\Delta\ln Z = +16.2$ beyond mass+z ($\ln Z = -1233.0$). The proxy audit (step_182) decomposes this gain: it is carried by the on-sky density structure, not the redshift axis — the pure-$z$ variant loses $-9.9$ while the $z$-residualized proxy retains $+15.9$ and the sky-only variant adds $+12.9$ on top of the free interaction. Two confounds bound the interpretation, however. The sky proxy is essentially cluster-centric distance in the Abell-2744 mosaic (correlation $+0.78$), and the direct comparator makes this quantitative: clustercentric distance entered as a single predictor attains $\Delta\ln Z = +28.1$ beyond mass+z — exceeding the $k$NN proxy itself ($+9.2$ sky, $+16.2$ 3D) — while the sky proxy residualized on clustercentric distance retains nothing ($\Delta\ln Z = -11.7$). All of the proxy's independent evidence is therefore the smooth radial coordinate; the local-density texture it was designed to capture contributes nothing beyond it. On top of the free interaction the same ordering holds: $\{M_*, z, M_*\! \times\! z, d_{\rm A2744}\}$ is the best model of the audit ($\ln Z = -1158.8$; $+33.1$ over the interaction, against $+12.9$ for the sky proxy). Second, on the physical three-observable subset (dust, sSFR, metallicity — the SED $\chi^2$ channel dropped) the proxy gain falls to $\Delta\ln Z = +4.8$, so much of the evidence rides the fit-quality channel that the mosaic depth gradient also modulates. The final audit tier resolves the remaining ambiguity. Abell 2744 is a foreground cluster at $z = 0.308$: cluster-centric distance cannot be an ambient-field coordinate for $z > 8$ galaxies, so its predictive power is a systematics signature — the lensing-magnification bias and the mosaic-depth gradient, both radially centred on the core. Joining the DR4 v2.0 lens-model magnification $\mu_{\rm ML}$ to the sample confirms this: ${\rm corr}(\log\mu_{\rm ML}, d_{\rm A2744}) = -0.82$, ${\rm corr}(\log\mu_{\rm ML}, \chi^2) = +0.44$, and $\{M_*, z, \mu, d_{\rm A2744}\}$ attains $\Delta\ln Z = +34.2$ over mass+z — the systematics structure is real and must be carried in the null baseline. The theory-faithful proxy is then rebuilt as the mass-weighted projected density of the $z > 8$ epoch slice itself (the photometric-redshift widths, $\sigma_z \approx 4.8$, make the slice a single epoch; $u_{\rm amb} \propto \sqrt{\rho_{\rm env} L}$ uses the mass density), and tested on top of the systematics baseline: it returns $\Delta\ln Z = -7.1$ on the full observable set and $-6.0$ on the physical subset, while the original sky proxy adds $-8.9$ in the same slot. With the foreground-lens coordinates properly included, no same-epoch density proxy adds evidence — the earlier gains are absorbed entirely by the systematics baseline. The covariance channel thus yields a clean control result rather than an ambient-field detection: the observables' radial structure is explained by the cluster's lensing and depth signature, the instantiated $R_{\rm ML}$ kernel contributes no unique direction, and the strongest remaining term is the generic mass–redshift interaction ($+57.3$), a standard mass-dependent evolutionary dependence. Functional-form discrimination of the ambient-field law is not achievable with this field's selection geometry and requires an off-cluster control field or a forward-modelled magnification-corrected density map.

The orthogonalized sensitivity analysis provides an additional test under TEP-conditioned levelling. When all alternatives are levelled by orthogonalizing their mass predictor against $R_{\rm ML}$ (preventing circular absorption of the TEP signal through the mass variable), TEP outperforms every tested alternative: $\ln{\rm BF}=+133.0$ versus standard physics (16 parameters), $+129.8$ versus the bursty star-formation model (21 parameters), $+121.2$ versus the M*×sqrt(1+z) interaction model (20 parameters), $+102.6$ versus the AGN-threshold model (18 parameters), $+98.1$ versus the varying-IMF model (20 parameters), and $+77.7$ versus the quadratic baseline (28 parameters). In the joint augmented test, TEP augmented yields $\ln{\rm BF} = +163.1$ versus standard physics (Table 8). The mean $\ln{\rm BF}$ across all eleven alternatives is $+113.8$. TEP achieves this with the fewest parameters (12), leveraging a single theory-fixed predictor across all four observables. The M*×sqrt(1+z) interaction model — the minimal non-linear null that captures the specific mass–redshift interaction TEP encodes through the $\sqrt{1+z}$ factor in $R_{\rm ML}$, without any TEP-specific potential-depth structure — is outperformed by TEP with $\ln{\rm BF} = +121.2$, though the unconditioned covariance-family interaction reverses that ordering ($\ln{\rm BF} = -71.7$ in favour of $\{1, M_*, z, M_*\times z\}$), confirming that the conditioned victories reflect the predictor restriction rather than an unconditional preference. The orthogonalized family is reported as a sensitivity analysis because the orthogonalization is TEP-conditioned: alternatives have their mass predictor stripped of $R_{\rm ML}$-correlated variance, while TEP uses raw $R_{\rm ML}$. Two scope limitations apply. First, these alternatives are parametric stand-ins for the labelled mechanisms — linear, quadratic, interaction, sigmoid, and timescale modulations of mass+z — rather than forward-modelled predictions of bursty star formation, IMF variation, or AGN-plus-dust physics, so the comparisons do not test realistic astrophysical models. Second, the results are conditioning-dependent: the covariance-corrected value ($+41.5$) uses orthogonalized predictors, while the unorthogonalized comparison yields $\ln{\rm BF} = -14.4$ (physical subset: $+37.4$ versus $-20.4$). The subsumption driving the flip is deterministic rather than empirical: $\log R_{\rm ML}$ is computed from (mass, z) with ${\rm corr}(M_*, \log R_{\rm ML}) = 0.86$ and $R^2(\log R_{\rm ML}\mid M_*, z) = 0.74$, so orthogonalizing mass against $\log R_{\rm ML}$ removes most of the direction the observables require — the $+41.5$ endpoint measures that structural handicap of the comparator, not independent support. Within the conditioned family the result does carry real information: the standard model stripped of the $R_{\rm ML}$-aligned direction loses $55.9$ ln units of evidence, establishing that the signal direction specified by the theory is where the variance lives.

The residual-space comparisons show the same conditioning split. In the orthogonalized residual space — where observables and competing predictors are residualized against the $R_{\rm ML}$-orthogonal components of mass+z, retaining the $R_{\rm ML}$-aligned direction by construction — TEP outperforms both the residual null model ($\ln{\rm BF} = +157.9$, Table 8) and the constrained AGN model ($\ln{\rm BF} = +104.6$). Under the symmetric conventional residualization, in which the full linear mass+z trend is removed from both observables and predictors, the comparison yields $\ln{\rm BF} = +9.1$ in favour of TEP: the nonlinear remainder of $\log R_{\rm ML}$ beyond linear mass+z provides a positive residual contribution. A log-gamma standardization sensitivity check confirms the joint-space evidence is robust to the scaling choice ($\Delta\ln{\rm BF} = 1.4$ between raw and standardized $\log\Gamma_t$). Additionally, applying the TEP mass correction to conventional astrophysical models yields mixed results (Varying IMF $\ln{\rm BF} = +8.6$, Bursty SF $\ln{\rm BF} = -8.6$, Standard Physics $\ln{\rm BF} = -8.7$, AGN $\ln{\rm BF} = -3.2$; mean $\ln{\rm BF} = -3.0$), with the quadratic IMF model benefiting from the mass-scale correction while linear models absorb the scale shift without improvement.




Table 9: Primary Empirical Line, Secondary Partial-Correlation Test, Ancillary Spatial Indication, and Derived Regime Comparison — Key Statistics

| Line | TEP Prediction | Observed | Significance | Replication |
| --- | --- | --- | --- | --- |
| **L1. Dust–$R_{\rm ML}$ + AGB threshold** | $\rho > 0.3$ at $z > 8$; $t_{\rm eff}$ retains residual after polynomial control; $M_*$ zero residual after $t_{\rm eff}$ control; dust jumps at $t_{\rm eff} \gtrsim 0.3$ Gyr | $\rho = +0.635$; partial $\rho = +0.31$ ($p = 4.6 \times 10^{-7}$); AGB odds ratio $12.7$; dust ratio $7.0\times$ vs $3.3\times$ for the mass boundary | $p = 3.0 \times 10^{-139}$ (Fisher); $p = 2.1 \times 10^{-159}$ (threshold) | UNCOVER, CEERS, COSMOS-Web ($N = 1{,}254$; NIRSpec supplementary $N = 3{,}259$); Fisher $z = 25.1\sigma$; $I^2 = 0\%$; UNCOVER tests pass 4/4 |
| **L2. Inside-out core screening** | Bluer-core result in more massive/radially stratified galaxies; higher central concentration at larger $R_{\rm ML}$ after mass+$z$ control | Gini partial $\rho=+0.191$ ($p=1.6\times10^{-4}$); resolved radial predictor $\Gamma_{\rm rad}$ partial $\rho=-0.179$ ($p=2.8\times10^{-3}$, $2.99\sigma$; $C_{\rm core}^{-1}$ partial $\rho=-0.237$, $p=6.9\times10^{-5}$, $4.0\sigma$); Steiger $Z=3.02$ ($p=2.5\times10^{-3}$ vs $M_*$); residual sign test $58.6\%$ vs $35.7\%$ ($\text{OR}=2.54$, Fisher $p=0.0054$, $\Delta=-0.273$ mag, 95% CI $[-0.526,-0.035]$) | Gini partial $p = 1.6 \times 10^{-4}$; $\Gamma_{\rm rad}$ partial $p = 2.8 \times 10^{-3}$ ($2.99\sigma$), Steiger $p = 2.5 \times 10^{-3}$ ($3.02\sigma$), Fisher $p = 0.0054$ ($2.55\sigma$) | JADES resolved photometry ($N = 277$, Steps 139 & 168) plus JADES DR5 morphology ($N_{\rm matched}=464$, Step 155) and GOODS-S ($N=588$, Step 156) |
| **L3. $R_{\rm ML}$–sSFR partial correlation** | $\rho(R_{\rm ML}, {\rm sSFR}) < 0$ at all $z$ (from ${\rm sSFR}_{\rm obs} \propto R_{\rm ML}^{m-n_{\rm SPS}}$, $m < n_{\rm SPS}$); partial $\rho(R_{\rm ML}, {\rm sSFR}\|M_*, z) \neq 0$ | Partial $\rho = -0.62$ at $z>8$ ($p = 8.2 \times 10^{-29}$, $N=254$); full-sample $\rho(R_{\rm ML}, {\rm sSFR}) = -0.49$ ($p = 7.7 \times 10^{-129}$) | Sign and magnitude consistent with $m - n_{\rm SPS} \approx -0.2$ | UNCOVER ($N = 2{,}157$); replicated across COSMOS2025 ($N = 7{,}178$ at $z = 7$–$13$, Step 157): 3/3 high-$z$ bins recover the predicted negative sign after mass debiasing ($z=7$–$8$ weighted $\rho=-0.066$, $p=9.6\times10^{-5}$; $z=8$–$9$ $\rho=-0.016$, weighted $-0.034$; $z=9$–$13$ $\rho=-0.103$, $p=7.2\times10^{-5}$; combined $z=8$–$13$ $N=2{,}588$ $\rho=-0.042$, $p=0.032$; activation $Z=-3.92$, $p=8.8\times10^{-5}$) |
| **L4. Dynamical mass comparison** | TEP correction moves $M_*/M_{\rm dyn} > 1$ objects toward the admissible stellar-fraction band via isochrony bias | Published excess 0.15 dex; regime-level reduction 0.139 dex ($1.41 \rightarrow 1.03$; evaluated at RUBIES regime center); object-level reduction 0.245 dex vs 0.259 dex anomalous-subset excess (mean corrected $+0.015$ dex, 2 of 3 resolved) | Lands the anomalous population at the admissibility boundary at canonical $\kappa$; under-resolves at the empirical $\kappa$ anchor | Derived regime-level comparison against published literature; not counted with primary empirical lines |



The L1 observed column reports the three-survey dust–$R_{\rm ML}$ correlation $\rho = +0.635$ across UNCOVER, CEERS, and COSMOS-Web ($N = 1{,}254$; NIRSpec supplementary $N = 3{,}259$), with the partial correlation $\rho = +0.31$ ($p = 4.6 \times 10^{-7}$) surviving mass+redshift control. The fixed AGB step ($t_{\rm eff} = 0.3$ Gyr) yields an odds ratio of 12.7 ($p = 2.1 \times 10^{-159}$); the $\Delta$AIC $= +0.25$ against the mass-matched step indicates that the mass step fits better as a pure step, though the polynomial $M_* \times z$ baseline outperforms both step models. Dedicated UNCOVER-only validation passes all four prespecified tests, and supplementary DJA-based GOODS-S and Balmer analyses are not part of the primary evidence count.

The L2 morphology and resolved-gradient analyses confirm inside-out core screening once the radial concentration profile is included. In JADES DR5 morphology ($N_{\rm matched}=464$, Step 155), after mass and redshift control, the Gini coefficient is strongly supportive (partial $\rho = +0.191$, $p = 1.6 \times 10^{-4}$; GOODS-S $N=588$ $\rho=+0.120$, $p=3.6\times10^{-3}$, Step 156). In the JADES resolved colour-gradient sample ($N = 277$, Steps 139 and 168), whereas the size-blind global halo predictor $\Gamma_t(M_h, z)$ cannot resolve the internal radial profile after $(M_*, z)$ are regressed out ($\rho_{\rm partial}=+0.037$, $p=0.54$), incorporating the measured inner-to-outer F444W stellar concentration $C_{\rm core} = F_{\rm F444W,CIRC1}/F_{\rm F444W,CIRC4}$ across the same apertures defines the resolved radial screening predictor $\Gamma_{\rm rad} = \exp[K_{\rm gal}\Psi_{\rm halo}(M_h)\sqrt{1+z}/C_{\rm core}]$: $\Gamma_{\rm rad}$ correlates with the colour gradient at $\rho = -0.168$ ($p = 0.0050$), retains partial $\rho(\Gamma_{\rm rad}, \nabla(B-V) \mid M_*, z) = -0.179$ ($p = 0.0028$, $2.99\sigma$, exact TEP bluer-core sign; partial $\rho = -0.237$, $p = 6.9 \times 10^{-5}$, $4.0\sigma$ on $C_{\rm core}^{-1}$), outperforms $M_*$ on the Steiger $Z$-test ($Z = 3.02$, $p = 0.0025$, $3.02\sigma$), and yields a significant $(M_*, z)$-residualized bluer-core sign reversal ($58.6\%$ vs $35.7\%$ on $q_{25}/q_{75}$, $\text{OR} = 2.54$, Fisher $p = 0.0054$, $\Delta = -0.273$ mag, 95% CI $[-0.526, -0.035]$; and $57.6\%$ vs $42.4\%$ on $q_{33}/q_{67}$, $\text{OR} = 1.85$, Fisher $p = 0.0275$).

The L3 replication status: UNCOVER ($N = 2{,}157$) is the primary L3 line, and the COSMOS2025 blank-field follow-up ($N = 49{,}539$ total, $N = 7{,}178$ at $z = 7$–$13$, Step 157) confirms the predicted negative sign across all three high-redshift bins once the positive mass-measurement bias on $M_*$ is debiased (3/3 high-$z$ bins negative: $z = 7$–$8$ debiased $\rho = -0.022$, quality-weighted $\rho = -0.066$, $p = 9.6 \times 10^{-5}$; $z = 8$–$9$ debiased $\rho = -0.016$, quality-weighted $\rho = -0.034$; $z = 9$–$13$ debiased $\rho = -0.103$, $p = 7.2 \times 10^{-5}$, quality-weighted $\rho = -0.070$, $p = 0.027$; combined $z = 8$–$13$, $N = 2{,}588$, debiased $\rho = -0.042$, $p = 0.032$; high-$z$ vs low-$z$ activation shift $\Delta\rho_{\rm debiased} = -0.111$, Fisher $Z = -3.92$, $p = 8.8 \times 10^{-5}$).

**Statistical independence:** L1 and L3 probe distinct observables (dust and sSFR). The UNCOVER partial $\rho(R_{\rm ML}, {\rm sSFR}|{\rm dust}) = -0.64$ at $z > 8$ ($p = 7.4 \times 10^{-31}$, $N = 254$) confirms that L3 carries information orthogonal to dust. The three-survey L1 Fisher combination is the headline statistic; omnibus multi-test combinations are supportive context.

**Supplementary cross-dataset checks:** These extend the case without altering the primary evidence count, since they reuse the same predictor families as L1 or L3.


- **COSMOS2025 blank-field:** The mass+redshift-controlled dust partial is $\rho = +0.200$ ($p < 10^{-300}$) at $z > 4$. In the sSFR follow-up (Step 157), once the mass-measurement bias on $M_*$ is debiased, all 3 of 3 high-redshift bins recover the TEP-predicted negative sign ($z = 7$–$8$: quality-weighted $\rho = -0.066$, $p = 9.6 \times 10^{-5}$; $z = 8$–$9$: debiased $\rho = -0.016$, weighted $-0.034$; $z = 9$–$13$: debiased $\rho = -0.103$, $p = 7.2 \times 10^{-5}$, weighted $-0.070$, $p = 0.027$, and $z = 9$–$10$ reference-mass-reweighted $\rho = -0.123$, $p = 0.0021$; combined $z = 8$–$13$, $N = 2{,}588$: debiased $\rho = -0.042$, $p = 0.032$; activation Fisher $Z = -3.92$, $p = 8.8 \times 10^{-5}$).

- **Cross-survey temporal ordering:** Recovered in UNCOVER, CEERS, and COSMOS-Web with $\Delta\rho_{\rm time} = +0.357$, $+0.609$, and $+0.850$.

- **UNCOVER DR4 MegaScience:** The dust signal is null below $z = 7$ and reaches $\rho = +0.631$ at $z = 8$–9. The $z = 9$–12 null reflects compressed dust posteriors and inflated redshift uncertainties; a posterior-broad stack recovers a positive high-$R_{\rm ML}$ reddening contrast.

- **UNCOVER $z > 8$ targeted tests:** All four prespecified tests return the predicted sign (AGB threshold, cosmic-time-controlled partial, split-sample persistence, monotonic steepening with redshift).

- **UNCOVER $z = 9$–12 posterior-broad stack:** Comparing upper and lower $R_{\rm ML}$ quartiles ($N = 16 + 16$) gives $\Delta \text{dust2} = +0.287$ (95% CI $[+0.103, +0.490]$), with redder rest-frame colours $\Delta(U-V) = +0.354$ and $\Delta(V-J) = +0.365$.

- **JADES DR5 morphology:** After mass and redshift control, Gini gives partial $\rho=+0.191$ ($p=1.6\times10^{-4}$); both half-light-radius proxies and $\sigma_\star$ are non-significant. The indication is specific to central concentration.

- **JADES $z = 9$–12 UV-slope:** The raw $\rho(R_{\rm ML}, \beta) = +0.259$ ($p = 0.18$, $N = 28$); the quartile split gives $\Delta\beta = +0.941$ (95% CI $[-0.384, +3.299]$). Low power, directionally consistent.

- **Debiased mass control:** Correcting for TEP mass bias strengthens O32 and H$\beta$-equivalent-width signals by $\sim 1.5\times$–$1.9\times$.




### 3.9 TEP Predictions vs Observations Summary

Table 10 is best read as a compact consistency summary rather than as a count of independent confirmations. Several of the 12 listed predictions reuse the same underlying $R_{\rm ML}$ predictor derived from halo mass, so they are not statistically independent. The very high overall correlation ($r = 0.999$) is therefore informative about coherence, but it should not be interpreted as 12 separate demonstrations of the effect.





Table 10: Prediction-Observation Agreement Summary

| Metric | Value | Interpretation |
| --- | --- | --- |
| Raw Fisher combination (5-test synthesis) | $\chi^2 = 600.5$ | $p = 1.4 \times 10^{-122}$ ($23.5\sigma$) |
| Brown adjustment (correlated tests) | $p = 6.0 \times 10^{-85}$ | $19.2\sigma$ |
| $N_{\rm eff}$-Bonferroni stress test (10% effective $N$) | $p = 1.19 \times 10^{-3}$ | $z = 3.2\sigma$ |
| Effective independent tests | Mean $N_{\rm eff}/N \approx 11\%$ | After spatial-clustering autocorrelation correction |


The strongest evidence rests not on the number of predictions but on the coherence of the evidential structure and its robustness checks (§3.9): the primary empirical line (L1), the secondary $R_{\rm ML}$–sSFR partial correlation (L3), together with the ancillary inside-out core-screening indication (L2) and the derived dynamical-mass comparison (L4). Steiger Z-tests, partial correlations, and non-linear AIC are robustness checks on L1, not additional independent lines. Age-ratio and metallicity correlations do not survive joint mass+redshift control and are not counted as independent evidence.


#### 3.9.1 Adversarial Tests

A genuine physical signal should survive attempts to break it. To test whether the dust–$R_{\rm ML}$ correlation could arise from confounding, selection effects, or artifacts, a set of adversarial tests is applied:


- **Randomized mass ordering:** Shuffling the stellar-mass ordering that feeds $R_{\rm ML}$ destroys the correlation: over 2,000 shuffles the null distribution is $\langle\rho\rangle = -0.007 \pm 0.067$ ($z$-score $= 9.8$; 0 of 2,000 permutations reach the observed $\rho = 0.646$).

- **Within-redshift-bin persistence:** The correlation is detected in all three $z > 8$ bins independently: $\rho = 0.41$ ($z = 8$–$8.5$, $N = 102$, $p = 1.7 \times 10^{-5}$), $\rho = 0.55$ ($z = 8.5$–$9$, $N = 62$, $p = 2.9 \times 10^{-6}$), $\rho = 0.77$ ($z = 9$–$10$, $N = 90$, $p = 1.3 \times 10^{-18}$), ruling out a pure redshift-confounding origin.

- **$R_{\rm ML}$ vs pure mass:** $R_{\rm ML}$ ($\rho = 0.646$) outperforms both $\log M_*$ ($\rho = 0.601$) and $\log M_h$ ($\rho = 0.576$) as a dust predictor, consistent with the redshift-dependent component of $R_{\rm ML}$ carrying additional information beyond mass alone.

- **Mass-split persistence:** The correlation is detected in both mass halves of the $z > 8$ sample ($\rho = 0.42$ below the median, $p = 8.7 \times 10^{-7}$; $\rho = 0.24$ above, $p = 6.3 \times 10^{-3}$), ruling out a pure mass-confounding origin.




#### 3.9.2 Falsification Tests

A prespecified falsification test set examines six necessary conditions for the TEP framework. All six pass:


- **Sign consistency:** Dust–$R_{\rm ML}$ ($\rho = +0.65$, $p = 2.3 \times 10^{-31}$) and mass–age ($\rho = +0.15$, $p = 2.4 \times 10^{-12}$) correlations match predicted signs.

- **Magnitude scaling:** The correlation is positive in all four $R_{\rm ML}$ quartiles and strongest in the highest quartile ($\rho = 0.67$, $p = 2.7 \times 10^{-9}$), as predicted by a real physical effect.

- **Redshift evolution:** The correlation strengthens at higher redshift, consistent with TEP's $(1+z)$ scaling and weaker cosmological screening.


The full six-condition test set is documented in the supplementary materials.


### 3.10 Direct Kinematic Test

A fundamental vulnerability of evaluating TEP using purely photometric samples is the mass-proxy circularity: because $R_{\rm ML}$ is computed from halo mass (which in turn is inferred from photometric stellar mass), the observed correlations could in principle be driven by an unmodeled standard-physics process that scales with baryonic mass, rather than by a true temporal dilation tracking the gravitational potential. The circularity-breaker tests in §3.4.1 substantially narrow this concern — the 2D mass-$z$ shuffle confirms the signal requires the correct mass-$z$ pairing, the placebo comparison resolves the exponential potential-depth form over a generic power-law, and the narrow-mass-bin sign map reproduces the kernel-predicted crossover — but they remain anchored to SED-inferred stellar mass at one end of the loop. The kinematic tests in this section provide the mass-independent evidence that closes that loop.

The JWST-SUSPENSE survey of massive quiescent galaxies at $z = 1.2$–$2.3$ ($N = 15$) directly addresses this circularity by employing dynamically measured masses ($M_{\rm dyn}$) from stellar velocity dispersions and spectral ages derived from absorption features. The SUSPENSE analysis tests a dynamical-potential predictor and photometric stellar mass side by side. The central comparison shows that $R_{\rm ML}$ predicts spectral age more strongly than stellar mass, yielding $\rho({\rm Age}, R_{\rm ML} \mid z) = +0.690$ ($p = 4.4 \times 10^{-3}$) compared to $\rho({\rm Age}, M_* \mid z) = +0.493$ ($p = 0.062$). Under joint control of the competing predictor and redshift, $R_{\rm ML}$ retains a residual association with age, $\rho({\rm Age}, R_{\rm ML} \mid M_*, z) = +0.556$ ($p = 3.2 \times 10^{-2}$), whereas stellar mass contributes no residual signal once $R_{\rm ML}$ is controlled, $\rho({\rm Age}, M_* \mid R_{\rm ML}, z) = +0.031$ ($p = 0.912$). Propagating the published asymmetric uncertainties for all 15 galaxies preserves a positive $R_{\rm ML}$ residual in 99.7% of Monte Carlo draws. The direct Steiger comparison remains non-significant ($p=0.253$), so this one-sided residual structure is supportive but still carried with the stated small-sample caveat.

A combined kinematic sample of $N = 75$ galaxies ($z = 1.2$–$7.6$) drawn from six independent surveys (SUSPENSE, Esdaile et al. 2021, Tanaka et al. 2019, de Graaff et al. 2024a, Saldana-Lopez et al. 2025, Danhaive et al. 2025; Step 171, `step_171_sigma_kinematic_expansion.json`) breaks the SED-mass circularity. By the Virial Theorem ($\sigma^2 \equiv GM_h/R_{\rm vir}$), the gravitational potential depth at velocity dispersion $\sigma$ is identically $\Psi(\sigma) = \Psi_{\rm ref}(\sigma/\sigma_{\rm ref})^2$ (the cosmology-free direct depth map, $\sigma_{\rm ref}=136.3$ km/s). Under this Virial-exact depth map — computed exclusively from measured velocity dispersion $\sigma$ and redshift $z$, with zero dependence on SED-fitted $M_*$ or size $R_e$ — the circularity-free partial correlation controlling for redshift $z$ is strongly positive across all subsamples: $\rho(R_{\rm ML,\sigma}^{\rm dir}, M_{*,\rm obs} \mid z) = +0.463$ ($p = 2.84 \times 10^{-5}$) on the full $N=75$ sample, $\rho = +0.608$ ($p = 8.43 \times 10^{-7}$) on the $N=55$ exact-$M_{\rm dyn}$ subsample, and $\rho = +0.575$ ($p = 0.0080$) on the $N=20$ stellar absorption-line subsample (whereas conditioning simultaneously on both $\sigma$ and $z$ removes the primary $\sigma$ signal by construction, leaving only the nonlinear remainder $\rho = +0.079$, $p = 0.50$). In the primary $M_*$–$\sigma$ residual-evolution test (T1/T6), an earlier virial-map variant had introduced an artificial $(1+z)^{-1}$ decay in $\Psi(\sigma, z)$ by using an expanding-universe $R_{\rm vir}(z)\propto(1+z)^{-1}$ in $\sigma\to M_h$ while using the $z=0$ frozen $R_{\rm vir}(0)$ in $M_h\to\Psi$; under the Virial-exact direct map $\Psi\propto\sigma^2$, all five physical subsamples — pooled ($N=75$, $p=3.0\times10^{-5}$), absorption-line stellar kinematics ($N=20$), emission-line gas kinematics ($N=55$, $p=0.040$), $z\ge4$ ($N=56$, $p=0.020$), and SUSPENSE ($N=15$) — agree in sign with the TEP prediction (5/5), and the TEP mass correction attenuates the absorption-line residual-$z$ trend by $76\%$ ($+0.252 \to +0.062$) and the SUSPENSE trend by $65\%$ ($+0.100 \to -0.035$). Furthermore, when the local $z=0$ reference $\Psi_{\rm ref}$ is held at $z_{\rm ref}=0$ ($\ln R_{\rm ML} = K_{\rm gal}[\Psi(\sigma)\sqrt{1+z}-\Psi_{\rm ref}]$), sign agreement holds across all three $N\ge15$ individual surveys (Danhaive+25, Saldana-Lopez+25, Slob+25) plus the absorption tracer, with the absorption-line residual-$z$ trend reduced by $86.5\%$ ($+0.252 \to +0.034$) and the Fundamental Plane residual scatter reduced by $3.96\%$ ($0.578 \to 0.555$ dex, $N=53$).


### 3.11 L4 and L5 Future Validation

The cleanest direct kinematic test targets the most massive, brightest galaxies at $z > 7$. Such spectroscopy serves two distinct but complementary purposes: measuring Balmer absorption equivalent widths, and mapping the host galaxy velocity dispersion.


**1. Balmer Absorption Physics:** The primary photometric signature of TEP is that massive galaxies appear older and dustier than their cosmic age permits. This can be tested spectroscopically via Balmer absorption lines (e.g., H$\delta$), which peak in strength $\sim 300$–$500$ Myr after a starburst as A-type stars dominate the continuum. Under standard physics, a galaxy at $z = 9$ (cosmic age $\sim 540$ Myr) cannot host a dominant $\sim 500$ Myr-old stellar population. Under TEP, even a moderately massive halo ($\log M_* \gtrsim 9.5$) at this redshift exceeds $R_{\rm ML} \approx 3$, the threshold for an effective age of $\sim 1.6$ Gyr — readily allowing for strong Balmer absorption. More massive systems ($\log M_* > 10$) have $R_{\rm ML} \sim 8$–$22$, making the prediction even stronger. Observing H$\delta$ equivalent widths $\gtrsim 4$ Å at $z > 8$ would provide strong confirmation of the older effective stellar age.

**2. IFU Kinematics as a Direct Mass Proxy:** As discussed in §3.4, the current analysis relies on SED-derived stellar masses to compute $R_{\rm ML}$, creating a potential circularity. A direct resolution requires an independent proxy for the depth of the gravitational potential well. Spatially resolved kinematics (e.g., from JWST NIRSpec IFU) can map the central velocity dispersion ($\sigma$). Using $\sigma$ rather than $M_*$ to predict $R_{\rm ML}$—precisely as was done for the local Cepheid calibration and globular cluster pulsars—directly addresses the photometric mass degeneracy.



**Falsification Criteria**

**TEP prediction:** $\rho(R_{\rm ML}, \text{EW}_{H\delta}) > 0.5$, with mean $\Delta$EW $< -1.0$ Å for high-response-regime galaxies.

**Standard physics:** $\rho \approx 0$ (no $R_{\rm ML}$ dependence).



## 4. Discussion

The SUSPENSE kinematic comparison (L5) breaks mass circularity. The dynamical $R_{\rm ML}$ predictor retains spectral-age information after stellar-mass and redshift control ($\rho=+0.556$, $p=0.032$), whereas stellar mass contributes no residual signal once $R_{\rm ML}$ is controlled ($\rho=+0.031$, $p=0.91$). The bootstrap $\beta$ CI is $[0.149, 0.746]$. The Bayesian tests support this via the Conventional Comparison, Incremental Test, and TEP-Aware Comparison. In the broader $\sigma$-based expansion ($N=75$, Step 171), evaluating the purely kinematic predictor $R_{\rm ML,\sigma}^{\rm dir}$ under the Virial-exact depth map ($\Psi\propto\sigma^2$, avoiding the $(1+z)^{-1}$ round-trip artifact of combining an expanding-cosmology $\sigma\to M_h(z)$ drift with a $z=0$ frozen $M_h\to\Psi$ conversion) yields a circularity-free partial correlation controlling for redshift of $\rho(R_{\rm ML,\sigma}^{\rm dir}, M_{*,\rm obs} \mid z) = +0.463$ ($p = 2.84 \times 10^{-5}$, $N=75$), $\rho = +0.608$ ($p = 8.43 \times 10^{-7}$, $N=55$ exact-$M_{\rm dyn}$), and $\rho = +0.575$ ($p = 0.0080$, $N=20$ stellar absorption-line kinematics). All five physical subsamples (pooled, absorption, emission, $z\ge4$, and SUSPENSE) agree in residual-$z$ sign under the Virial-exact direct map (5/5), and holding the $z=0$ reference at $z_{\rm ref}=0$ ($\Psi(\sigma)\sqrt{1+z}-\Psi_{\rm ref}$) yields sign agreement across all three $N\ge15$ individual surveys plus the absorption tracer while attenuating the absorption-line residual-$z$ trend by $86.5\%$ ($+0.252 \to +0.034$) and the SUSPENSE trend by $65\%$ ($+0.100 \to -0.035$).

### 4.1 The Isochrony Bias Mechanism

The primary empirical line (L1: dust–$R_{\rm ML}$ emergence), together with the secondary $R_{\rm ML}$–sSFR partial correlation (L3), the resolved-screening indication, and the dynamical-mass comparison, converge on one physical interpretation: the isochrony axiom fails in massive, active-shear halos at $z > 5$. TEP accounts for the Red Monster star formation efficiency anomaly not by introducing new baryonic physics but by exposing a systematic bias already built into standard stellar-population inference. Standard SED fitting assumes that stellar clocks tick at the universal cosmic rate and that the environment-to-photometry mapping is universal. It uses the FLRW observer-age $t_{\rm cosmic}(z)$ as the time baseline, a reconstruction that shrinks to $\sim 540$ Myr at $z=9$ under the assumption that redshift traces spatial expansion. In the canonical TEP cosmology the spatial manifold is static and the gravitational coordinate time extends without finite origin (TEP-HUB, Paper 30); the FLRW age is therefore a reconstruction under isochrony, not the physical elapsed time. Local matter clocks run slower than coordinate time ($\Delta\ln A<0$ in a deeper well), but the coordinate background is eternal, so the accumulated proper time can still exceed the FLRW assigned age by a large factor. The inference-channel response $R_{\rm ML}>1$ quantifies how much the standard pipeline inflates apparent age and $M/L$ when it uses the too-short FLRW baseline and misattributes the resulting age excess to an older stellar population. Inferred mass-to-light ratios rise, inferred stellar masses rise, inferred specific star formation rates fall, and the galaxies appear more evolved than the FLRW timeline permits — not because their clocks tick faster, but because the timeline the pipeline uses is too short.

**Screening projection notice.** Screening in TEP is represented at theory level by the environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$. Quantities such as $\rho_T$, $R_T(M)$, $\mathcal{S}_\oplus(r)$, compactness $\Phi/c^2$, local stellar density, thermal epoch, coherence length, proximity, and boundary geometry are domain-specific projections of $\mathcal{E}$, not independent screening mechanisms and not interchangeable universal thresholds.

The prespecified canonical TEP magnitude-sector benchmark $\kappa_{\rm gal} = 0.960 \times 10^6$ mag from Paper 11 was fixed before the JWST analysis (with Paper 11's distinct empirical Cepheid-channel fits recovering $\kappa_{\rm equiv}^{39} = (1.220 \pm 0.531) \times 10^6$ mag and $\kappa_{\rm Cep}^{44} = (0.665 \pm 0.362) \times 10^6$ mag — bracketing the benchmark with a midpoint of $0.943\times 10^6$ mag, within $1.8\%$ of $0.960\times 10^6$ mag, while Paper 11 Step 62 derives $|\kappa_{P^2}^{\rm pred}(\lambda_0=1)| = 1.196\times 10^6$ mag from the Cepheid envelope density ratio $\rho_T/\langle\rho\rangle(10\text{ d}) = 8.45\times 10^5$ at the unit-transport benchmark $\lambda_0 = 1$ with $|\beta_A| = 1$ fixed by Paper 0, matching the SH0ES fit $(1.191\pm 0.480)\times 10^6$ mag to $0.5\%$) and then applied to $z > 5$ galaxies with only the physically motivated redshift scaling $\sqrt{1+z}$ and no tuning to JWST data. That it quantitatively accounts for the anomaly is therefore a non-trivial predictive test. TEP is not invoked here as a total replacement for early-galaxy astrophysics; it is invoked as the systematic correction required when photometric inference is forced through the wrong clock.

### 4.2 Model Discrimination and Bayesian Evidence

The multi-observable Bayesian comparison (§3.8) is evaluated under both conditionings of the predictor space, with the nested augmented model as the conditioning-free arbiter. Under TEP conditioning — the standard baseline restricted to the components of mass and redshift orthogonal to $\log R_{\rm ML}$ — the covariance-corrected TEP model ($\ln Z = -1262.6$, 9 parameters) outperforms the orthogonalized standard model ($\ln Z = -1304.1$, 13 parameters) by $\ln{\rm BF} = +41.5$ with four fewer parameters. Under the unconditioned comparison, the raw mass+z baseline is the best model in the ensemble ($\ln Z = -1248.2$) and exceeds TEP by $\ln{\rm BF} = -14.4$. Neither conditioning is neutral: orthogonalization removes the competitor's signal channel, while raw mass under the TEP measurement model ($M_{\rm obs} = M_{\rm true} + n\log_{10}R_{\rm ML}$) already contains part of the response. The nested augmented model quantifies the marginal contributions — adding $\log R_{\rm ML}$ to mass+z yields $\Delta\ln Z = +4.5$, while adding mass+z to the $R_{\rm ML}$ predictor yields $\Delta\ln Z = +18.9$ — and the direction-isolation diagnostic shows that stripping the $R_{\rm ML}$-aligned direction from the standard predictor costs $55.9$ ln units of evidence. The defensible reading is channel consistency without functional-form discrimination: the explanatory variance is concentrated in the direction the theory specifies, and a single theory-fixed predictor recovers the full augmented model within one log-evidence unit, but the linear mass+z family spans the same direction and cannot be excluded by this comparison alone.

Second, the orthogonalized sensitivity analysis evaluates TEP against a broader family of explicit astrophysical alternatives where all candidate mass predictors are orthogonalized against $\log R_{\rm ML}$ to prevent circular absorption of the TEP signal. Within this TEP-conditioned sensitivity family, TEP outperforms every tested alternative: standard physics ($\ln{\rm BF}=+133.0$, 16 parameters), the bursty star-formation model ($\ln{\rm BF}=+129.8$, 21 parameters), the $M_* \times \sqrt{1+z}$ interaction null ($\ln{\rm BF}=+121.2$, 20 parameters), the AGN-threshold model ($\ln{\rm BF}=+102.6$, 18 parameters), the varying-IMF model ($\ln{\rm BF}=+98.1$, 20 parameters), and the quadratic baseline ($\ln{\rm BF}=+77.7$, 28 parameters), with a mean $\ln{\rm BF} = +113.8$ across all eleven alternatives. These alternatives are parametric stand-ins — linear, quadratic, interaction, sigmoid, and timescale modulations of the mass+z baseline — rather than forward-modelled predictions of bursty star formation, IMF variation, or AGN-plus-dust physics; the conditioned family therefore tests whether any of these directions explains variance outside the $R_{\rm ML}$-aligned channel, not whether realistic astrophysical models are excluded.

Specifically, once a halo potential reaches the critical binding energy, both AGN feedback and TEP predict departure from standard linear stellar mass assembly. The AGN model attributes this to winds clearing the cold gas supply, halting star formation and revealing underlying older stellar populations. TEP attributes this to a large inference-channel response ($R_{\rm ML} \gg 1$), in which standard isochrony mapping misattributes the environmental temporal structure of a deep unscreened well to an older, dustier stellar population. The primary discriminator is the predicted dust and metallicity: while AGN-driven winds physically expel gas and curtail dust production, TEP predicts that apparent dust buildup tracks the inference-channel response alongside apparent stellar age. When the mass predictor is orthogonalized against $R_{\rm ML}$, the AGN-threshold model loses its competitive advantage in the full joint space.

**Sensitivity and disclosures.** Transparent diagnostics are reported: the per-observable breakdown shows the conditioned signal is concentrated in dust ($\ln{\rm BF} = +21.1$) and the SED $\chi^2$ statistic ($\ln{\rm BF} = +52.5$), with a positive contribution from sSFR ($\ln{\rm BF} = +49.9$) and a null result for metallicity ($\ln{\rm BF} = +9.9$). The SED $\chi^2$ is a fit-quality diagnostic rather than a physical observable, so the 3-observable physical subset (dust, sSFR, metallicity) is the primary observable set: it yields $\ln{\rm BF} = +37.4$ under orthogonalized predictors (7 versus 10 parameters) and $\ln{\rm BF} = -20.4$ under raw predictors. In the 4-observable covariance family the conditioned comparison gives $\ln{\rm BF} = +41.5$ and the unconditioned comparison $-14.4$; the nested augmented model shows the predictor families mutually subsumed ($\Delta\ln Z = +18.9$ for adding mass+z to $R_{\rm ML}$; $+4.5$ for adding $R_{\rm ML}$ to mass+z), while the direction-isolation diagnostic assigns $55.9$ ln units to the $R_{\rm ML}$-aligned direction. In the orthogonalized residual space — which retains the $R_{\rm ML}$-aligned direction by construction — TEP outperforms the constrained-AGN model ($\ln{\rm BF} = +104.6$); under the symmetric conventional residualization the comparison yields $+9.1$ in favour of TEP. A log-gamma standardization sensitivity check confirms the evidence is robust to the scaling choice ($\Delta\ln{\rm BF} = 1.4$ between raw and standardized predictors). The Bayesian evidence therefore provides channel-consistency context — the variance is organized along the theory-predicted direction — while the discriminating empirical strength of TEP rests on the multi-survey photometric correlation structure (L1, L3), the cross-survey Steiger and meta-analyses, and the direct SUSPENSE kinematic test (L5).

### 4.3 Synthesis

Two primary empirical observational anomalies that have resisted unified
explanation under standard physics admit consistent interpretation under
the single-parameter TEP mapping, while a resolved-gradient indication
remains directionally aligned and a derived dynamical-mass comparison
remains supportive. The $z > 8$ dust paradox (mass-dependent
suppression, $\rho = +0.635$ cross-survey) arises because $R_{\rm ML}$
orders the inference-channel response with potential depth. The
$R_{\rm ML}$–sSFR partial correlation
($\rho(R_{\rm ML}, {\rm sSFR} \mid M_*, z) = -0.62$ at $z > 8$,
$p = 8.2 \times 10^{-29}$) has the sign predicted by the TEP
measurement equations: since ${\rm sSFR}_{\rm obs} \propto R_{\rm ML}^{m-n_{\rm SPS}}$
with $m \approx 0.5 < n \approx 0.7$, the inference bias lowers sSFR
in massive halos, deepening the negative mass–sSFR relation. The
observed high-$z$ mass–sSFR inversion ($\Delta\rho = +0.25$) is an
empirical phenomenon that TEP does not claim to generate; it may reflect
the weakening of intrinsic downsizing at high redshift or selection
effects. The resolved core-screening result (bluer cores,
$\rho = -0.166$) arises because the deepest central potentials screen the
scalar field, restoring standard time in galactic nuclei while outskirts
remain in the unscreened high-response regime. While the size-blind global
halo predictor $R_{\rm ML}(M_h, z)$ has no residual variance after $(M_*, z)$
control ($\rho = +0.037$, $p = 0.54$), incorporating the measured inner-to-outer
F444W stellar concentration $C_{\rm core} = F_{\rm F444W,CIRC1}/F_{\rm F444W,CIRC4}$
across the same apertures defines the resolved radial screening predictor
$\Gamma_{\rm rad} = \exp[K_{\rm gal}\Psi_{\rm halo}(M_h)\sqrt{1+z}/C_{\rm core}]$
(Steps 139 and 168), which retains partial $\rho = -0.179$ ($p = 0.0028$, $2.99\sigma$;
$C_{\rm core}^{-1}$ partial $\rho = -0.237$, $p = 6.9 \times 10^{-5}$, $4.0\sigma$)
after mass-plus-redshift control, outperforms $M_*$ on the Steiger test
($Z = 3.02$, $p = 0.0025$), and yields a significant $(M_*, z)$-residualized
bluer-core sign reversal ($58.6\%$ vs $35.7\%$, $\text{OR} = 2.54$, Fisher $p = 0.0054$,
$\Delta = -0.273$ mag), complementing the JADES DR5 Gini concentration signal
($\rho = +0.191$, $p = 1.6 \times 10^{-4}$).
The dynamical-mass comparison supports the same mechanism
at the regime level: the TEP mass correction is large enough to remove the
published $M_*/M_{\rm dyn}$ excess in the RUBIES-like regime. Galaxies in
the high-response regime show $5.96\times$ more dust above the $t_{\rm eff}$
threshold. Age-ratio and metallicity correlations, by contrast, remain
weak under mass-only control but vanish under joint mass+redshift control
— the framework correctly predicts which observables should and should not
survive stricter controls.

#### 4.3.1 $\Lambda$CDM Tension Quantification

The impact on the $\Lambda$CDM stellar mass excess can be quantified
through the cosmic SFRD metric (Table 12), which does not rely on a
sharp mass threshold. At $z > 8$, the mean SFRD excess is reduced
from $11.0\times$ to $7.5\times$ $\Lambda$CDM — a 31%
reduction with zero free parameters tuned to JWST data. The correction
is most effective at $z = 6$–$7$ (65% reduction) and remains
substantial at $z > 9$ (29–42%), though a large residual excess
persists. The residual excess at $z > 9$ likely requires additional
astrophysical contributions (bursty star formation, cosmic variance)
operating alongside any TEP effect.

A complementary mass-threshold metric — counting galaxies above $\log
M_* \geq 10$ before and after correction — shows that the TEP correction
reduces counts at the highest threshold ($\log M_* > 10.5$) by 20–53%,
while the $\log M_* > 10.0$
threshold shows a modest change ($-4\%$ to $+4\%$), reflecting the
mass-dependent direction of the correction. The SFRD-based
quantification is therefore preferred as the primary tension diagnostic
because it avoids sensitivity to an arbitrary threshold choice.

The most dramatic JWST anomaly — "too many massive galaxies" at $z > 7$
— admits a partial reduction under TEP. Isochrony bias causes SED
fitting to overestimate stellar masses by a factor $R_{\rm ML}^n$ ($n
\approx 0.7$), because the standard pipeline misattributes the
environmental temporal structure of deep unscreened wells to
older-looking stellar populations with higher mass-to-light ratios. Applying the
correction $\log M_{*,{\rm true}} = \log M_{*,{\rm obs}} -
n\log_{10}R_{\rm ML}$ to the observed stellar mass function:



Table 11: TEP Mass Correction at Key Thresholds

| Redshift | Threshold | $N_{\rm obs}$ | $N_{\rm corr}$ | Reduction |
| --- | --- | --- | --- | --- |
| $z = 7$–$8$ | $\log M_* > 10.0$ | 119 | 123 | $-3\%$ |
| $z = 7$–$8$ | $\log M_* > 10.5$ | 41 | 33 | $20\%$ |
| $z = 8$–$9$ | $\log M_* > 10.0$ | 113 | 117 | $-4\%$ |
| $z = 8$–$9$ | $\log M_* > 10.5$ | 34 | 18 | $47\%$ |
| $z = 9$–$10$ | $\log M_* > 10.0$ | 54 | 52 | $4\%$ |
| $z = 9$–$10$ | $\log M_* > 10.5$ | 17 | 8 | $53\%$ |

Anomalous galaxy census: in the external Labbé+2023 check, the
z-dependent TEP correction resolves 8/9 anomalous systems (89%). At the
benchmark literature level, the TEP mass correction resolves $\sim 29\%$ of the
stellar-mass-function excess on average across $z = 6$–$10$; at $z = 9$, the
typical 0.15 dex correction addresses 21% of the quoted 1.1 dex excess. Within the
three-survey sample shown above, the counts at the most extreme mass threshold
($\log M_* > 10.5$) are reduced by 20–53%, while the $\log M_* > 10.0$ threshold
shows a modest change ($-4\%$ to $+4\%$), reflecting the
mass-dependent direction of the TEP correction at lower masses.

**Correction-stage exponent:** The mass correction depends on the SPS
power-law index $n_{\rm SPS}$ (adopted: 0.7 for this mass-function analysis, versus
the regime assignments $n_{\rm SPS} = 0.7$ for $z\le4$, $0.9$ for $4<z\le6$, and
$0.5$ for $z>6$ in the primary catalogue construction, §2.4.4). The response kernel remains fixed at $n_{\rm ref}=0.7$ in both analyses, so this sensitivity cannot introduce a discontinuity in $R_{\rm ML}$. The
choice of $n_{\rm SPS} = 0.7$ here follows standard SSP predictions (Bruzual &
Charlot 2003) for rest-frame optical $M/L$ scaling and is conservative:
$n_{\rm SPS} = 0.5$ would produce a *smaller* mass correction, resolving
fewer anomalous galaxies, while $n_{\rm SPS} = 0.9$ resolves more. Values $n_{\rm SPS} =
0.5$–$0.9$ shift the correction by $\sim \pm 30\%$ but do not change the
qualitative picture: the most extreme massive galaxies ($\log M_* >
10.5$ at $z > 8$) are eliminated for any $n > 0.4$. The correction also
does not account for possible environmental dependence of the M/L index.

The same isochrony bias that inflates stellar masses also inflates
SED-derived star formation rates, because the apparent mass-to-light
ratio is overestimated. If ${\rm SFR}_{\rm true} = {\rm SFR}_{\rm obs} /
R_{\rm ML}^m$ with $m \approx 0.5$ (UV-based SFR is less affected than
cumulative mass, since it traces recent star formation over $\lesssim
100$ Myr), the cosmic SFRD correction is applied to the combined
UNCOVER + CEERS sample ($N = 4{,}152$):



Table 12: TEP Cosmic SFRD Correction

| Redshift | $N$ | Observed Excess | TEP-Corrected Excess | Reduction |
| --- | --- | --- | --- | --- |
| $z = 6$–$7$ | 2,207 | $5.1\times$ $\Lambda$CDM | $1.8\times$ | $65\%$ |
| $z = 7$–$8$ | 775 | $3.4\times$ | $2.4\times$ | $30\%$ |
| $z = 8$–$9$ | 561 | $4.0\times$ | $3.1\times$ | $22\%$ |
| $z = 9$–$10$ | 340 | $10.2\times$ | $5.9\times$ | $42\%$ |
| $z = 10$–$12$ | 269 | $18.9\times$ | $13.4\times$ | $29\%$ |

The TEP SFRD correction is most effective at $z = 6$–$7$ (65% reduction
in the combined survey excess) and decreases to 22% at $z = 8$–$9$,
then rises again to 42% at $z = 9$–$10$ where the observed excess is
largest. At $z > 10$ the correction remains substantial (29%),
reflecting the persistent $R_{\rm ML}$ offset even among lower-mass
galaxies. The overall mean reduction across $z > 8$ bins is 31%,
indicating that the isochrony bias channel probed by the $m = 0.5$ SFR
correction accounts for roughly a third of the SFRD excess at the
highest redshifts. The residual excess at $z > 9$ likely requires
additional astrophysical contributions (cosmic variance, bursty star
formation, or physics beyond the isochrony bias) rather than TEP alone.

**Caveat:** The SFR bias index $m = 0.5$ is approximate.
UV-based SFRs probe recent star formation ($\lesssim 100$ Myr) and are
less affected by long-term aging than cumulative stellar mass. Values $m
= 0.3$–$0.7$ bracket the plausible range; the quoted results use a
conservative central value. Full SED forward-modeling with TEP-modified
stellar population synthesis would provide a more rigorous correction.

The dynamical-mass validation is expressed primarily as a matched
regime-level kinematic consistency test. In the RUBIES-like $z \sim
4.5$, $\log M_* > 10.5$ regime, the published mean excess is 0.15 dex
and the canonical-kernel TEP reduction is 0.139 dex, landing the
regime at $M_*/M_{\rm dyn} \simeq 1.03$ — at the admissibility
boundary rather than inside it; this regime bin contains only a
single UNCOVER object, so the regime-level figure is illustrative
rather than statistical. The supplementary five-object direct
literature ingestion at $z = 3.2$–$4.0$, including one conservative
upper-limit row, gives mean observed excess $0.168$ dex and mean
corrected excess $-0.124$ dex on the exact-mass subset ($M_*/M_{\rm
dyn} \simeq 0.75$); on the three anomalous objects the mean corrected
excess is $+0.015$ dex, with two of the three brought below zero.

The amplitude calibration is reported explicitly rather than absorbed
into the resolution claim. Because the mass correction is linear in
$\kappa_{\rm gal}$, the $\kappa$-sensitivity ledger (Step 117) spans
the canonical benchmark and the empirical Paper 11 anchor directly:
on the anomalous subset the mean corrected excess runs $+0.157$ dex at
$\kappa = 0.40\times10^{6}$ mag (under-resolved), $+0.015$ dex at the
canonical $\kappa = 0.96\times10^{6}$ mag, and $-0.087$ dex at the
$+1\sigma$ bound $\kappa = 1.36\times10^{6}$ mag. Two features are
worth noting. First, the admissible target is not zero excess —
$M_*/M_{\rm dyn}=1$ would still imply a fully baryonic interior with
no dark-matter or gas headroom — so the physical resolution band is a
stellar fraction below unity, and even the $+1\sigma$ corrected ratios
($\simeq 0.87$–$0.90$) remain inside it; no anchor produces an
unphysical overshoot. Second, the dynamical channel prefers the same
amplitude direction as the SFE channel (§3.1): both resolve at the
canonical benchmark and under-resolve at the empirical anchor, so the
two channels bracket a common $\kappa$ rather than pulling in opposite
directions — the residual amplitude uncertainty is the open $\kappa$
normalization discrepancy already recorded, not a channel-specific
defect.

A simulated validation exercise predicts a strong positive correlation
between $R_{\rm ML}$ and spectroscopic age ratio—a testable prediction for
uniform spectroscopic surveys. This is a forward prediction using
representative parameters, not an empirical validation against published
objects.

### 4.4 Little Red Dots as a Qualitative Stress Test

The Little Red Dot (LRD) population (Greene et al. 2024; Kokorev et al. 2024; Kocevski et al. 2023) is evaluated here as a qualitative stress test of compact-core structure rather than as a primary evidentiary line. TEP provides a directional mechanism: central black holes reside in the deepest potential wells, so central inference-channel responses naturally exceed the stellar halo response ($R_{\rm ML}^{\rm cen} > R_{\rm ML}^{\rm halo}$).

However, because $R_{\rm ML}$ is an observer-side inference response rather than a physical matter-frame proper time, quantitative accretion growth cannot be computed simply by inserting $R_{\rm ML}$ into an exponential Salpeter accretion integral. Differential central-versus-halo temporal structure may affect the inferred accretion history of compact sources, but a physical growth calculation requires the matter-frame proper-time history $\tau_\star = \int A(\phi)\,dt$ and is deferred to dedicated relativistic core models. The population-level LRD anomaly is accordingly not claimed as quantitatively resolved in this work.

Removing AGN-dominated LRDs reduces the tension with $\Lambda$CDM, but a
density excess remains. The TEP isochrony correction predicts a
reduction in apparent SFE for the most massive galaxies: $M/L$ inflation
by $R_{\rm ML}^n$ (with $n \approx 0.5$) implies that standard
SED-inferred stellar masses overestimate the true values, lowering the
inferred efficiency. Quantitative validation requires applying this
correction to a uniform spectroscopically confirmed Blue Monster sample
with well-characterized completeness, which is not yet available.

### 4.5 Limitations and Caveats

The limitations below are organised by tier, following the claim hierarchy of Paper 5 (TEP-GTE). *Tier 1 (empirical):* items 1–3 and 5–6 affect the magnitude of the correlations but not their existence or sign. *Tier 2 (interpretive):* items 1, 4, and 7 address whether the correlations arise from isochrony bias or a confound. *Tier 3 (theoretical):* items 4, 7, and 9 address the scalar-tensor framework itself — the most open questions.

- 
**Mass circularity:** $R_{\rm ML}$ depends on halo mass
inferred from stellar mass. Several distinct tests mitigate this
concern, spanning four data types. Age-ratio and
metallicity correlations do not survive joint mass+redshift control
and are not counted. The colour-gradient analysis is presently an
ancillary real-data indication only: the raw JADES
gradient–$R_{\rm ML}$ correlation is significant, but the Steiger and
partial-correlation tests are not, so it is not counted.
The dedicated mass-proxy breaker tests (step 143) narrow the degeneracy:
a non-parametric LOWESS double-residual retains $\rho = 0.161$ ($p = 0.007$)
at $z > 8$ after flexible mass+$z$ removal, a partial-rank residual gives
$\rho = 0.240$ ($p = 4.4 \times 10^{-5}$), and a shuffled-mass null within
$z$-bins yields $Z = 9.3$ ($p = 5 \times 10^{-4}$, unique fraction 101.2%).
The additional circularity-breaker tests (step 178, §3.4.1) further confirm
that a 2D mass-$z$ pairing shuffle destroys the signal ($Z = 10.9$) and that
$R_{\rm ML}$ retains explanatory power within narrow mass bins where $M_*$
has none. Evaluated under the canonical bounded, self-consistent response —
the same variable used throughout the pipeline — the placebo comparison is
now decisive: the exponential potential-depth form outperforms a generic
power-law mass-$z$ interaction ($\Delta\rho = +0.088$, 95% CI
$[+0.016, +0.161]$, superior in 99.3% of bootstrap draws). Most
importantly, the narrow-mass-bin analysis now carries a sharp
functional-form discriminator: the within-bin sign of
$\rho(R_{\rm ML}, \mathrm{dust})$ follows the kernel derivative
$\mathrm{d}\ln R_{\rm ML}/\mathrm{d}z$, which crosses zero at
$\log M_* \simeq 9.6$ through competition between the
$(\Psi - \Psi_{\rm ref})$ term and the positive stellar-to-halo-mass
drift. The observed sign map — negative in the two largest bins, null
at the crossover, positive above — matches the kernel prediction in
4/4 bins and falsifies every uniform-sign monotone-in-$z$ baseline
(raw $z$, the power-law placebo, and an inverted monotone all fail at
least one end). The photometric sample therefore does distinguish the
TEP functional form from generic mass-$z$ alternatives; the residual
caveat is quantitative, namely the small $N = 48$ size of the
positive-sign bin and the fact that ordinary AIC regression (step 136)
ranks the $M_* \times z$ interaction comparably to the TEP predictor
($\Delta$AIC $\simeq 1$–$2$).


- 
**SED fitting systematics:** All properties derive from
photometric SED fitting, introducing covariant uncertainties.
Photo-$z$ scatter degrades $\rho$ by $< 2\%$. The three surveys
use different codes (Prospector, EAZY, LePhare); cross-survey
consistency mitigates survey-specific artifacts but a uniform
re-fitting has not been performed. The assumed Calzetti attenuation
curve, SFH prior choice, and nebular emission contamination ([O
III]+H$\beta$) could each shift the quantitative slope by $\sim
10$–$20\%$, though the qualitative correlation direction is
preserved.


- 
**Photo-$z$ catastrophic outliers:** At $z > 6$,
Lyman/Balmer break confusion produces $\sim 5$–$15\%$ catastrophic
failures. Public spectroscopic coverage is now far better than in
the earlier small-sample stage: JADES DR4 provides 2,858
good-quality spec-z, including 118 at $z > 7$, and DJA v4.4
contributes 19,445 grade-$\ge 3$ sources, including 698 at $z > 7$
and 234 at $z > 8$. Even so, the majority of the full high-redshift
photometric sample still lacks spectroscopic confirmation and
therefore remains vulnerable to residual photo-$z$ systematics.


- 
**Theoretical foundation:** The $R_{\rm ML}$ formula
derives from a scalar-tensor action with Temporal Shear screening
(Appendix A.1). A CAMB-based late-time propagation (Appendix A.1.9.8)
confirms $\sigma_8$ consistency at the fiducial scalar field mass:
$\sigma_8^{\rm TEP} = 0.8116$ ($0.10\sigma$ from Planck), with CMB
TT deviations $< 0.02\%$ at all $\ell < 2500$ and $\chi^2/{\rm
dof} \ll 1$ against Planck error bars. Planck consistency requires
$m_{\phi,0} \gtrsim 0.43\,h$/Mpc ($\lambda_C \lesssim 14.6\,h^{-1}$
Mpc). The CAMB integration substantially closes this gap relative to
the earlier semi-analytic estimate; however, it uses a
modified-growth approach rather than a natively coupled scalar-field
Boltzmann solver (e.g., hi_class with the full Temporal Shear sector),
and the embedding employs the inverse-power $n=1$ mass evolution — the
potential branch excluded by Paper 0's constraint F1 under $\beta_A =
-1$. The $\sigma_8$/CMB consistency is accordingly a proxy
demonstration for a screened scalar sector, transferring to canonical
$V(\phi)$ only conditionally (Appendix A.1.9.8).
The remaining approximation is that acoustic-peak modifications from
the scalar field at $z > 1089$ are assumed negligible (justified by
$T^\mu_\mu \approx 0$ during radiation domination). A fully
self-consistent hi_class integration remains desirable for
completeness but is no longer expected to change the conclusion.


- 
**Statistical caveats:** Combined p-values exceeding
$10^{-90}$ should not be taken as a single omnibus headline. The
three-survey L1 Fisher combination is the primary summary statistic;
for the broader mixed test set, the dependence-adjusted Brown
combination remains small while a 10%-$N_{\rm eff}$ Bonferroni
stress test gives a lower-bound floor of $3.2\sigma$. BH-FDR
correction shows the broader validation tests remain significant
at $\alpha = 0.05$ (7 of 8 tested signatures, including the two
not-counted checks). The look-elsewhere effect from testing multiple
observables is partially addressed by Bonferroni/BH corrections, but
a formal pre-registration was not performed. All null results are
reported publicly.


- 
**Underpowered tests:** The Red Monsters ($N = 3$) and
several narrow highest-redshift or morphology-selected subsets
remain underpowered — for example UNCOVER spec-z at $z > 5$ has $N =
35$, and the JADES DR5 morphology subset at $z > 7$ has $N = 77$.
These subsets are excluded from the primary combined significance.


- 
**$z = 9$–$12$ UNCOVER MegaScience tail:** The 20-band
MegaScience Prospector-$\beta$ subset gives a raw $\rho(R_{\rm ML},
\text{dust2}) = -0.001$ ($p = 0.99$, $N = 122$) at $z = 9$–$12$,
contrasting with the positive lower-redshift bins at $z = 7$–$8$
($\rho = +0.319$, $N = 129$) and $z = 8$–$9$ ($\rho = +0.631$, $N =
66$), and with the COSMOS2025 blank-field raw dust trend. The
audit shows that this subset is better described as
sensitivity-limited than as a clean physical null: relative to $z =
8$–$9$, the dust dynamic range contracts to $0.657\times$, the
median dust uncertainty grows by $1.32\times$, and the relative
redshift uncertainty by $3.97\times$, while sample size does not
collapse. A new catalog-level stacked surrogate targeted at the
posterior-broad tail partially closes the gap. Restricting to the
broad half of the $z = 9$–$12$ sample ($N = 61$) and comparing the
upper and lower $R_{\rm ML}$ quartiles ($N = 16 + 16$) yields a
weighted $\Delta\text{dust2} = +0.249$ with 95% CI $[+0.032,
+0.468]$, together with redder rest-frame colours $\Delta(U-V) =
+0.341$ and $\Delta(V-J) = +0.335$, both with positive bootstrap
intervals. A conservative JADES $z = 9$–$12$ UV-slope companion is
directionally aligned (raw $\rho(R_{\rm ML}, \beta) = +0.259$, $p =
0.18$; weighted $\Delta\beta = +0.941$, $N = 28$) but remains
underpowered. The interpretation is that this is a sensitivity-limited tail rather than an unexplained null: broad-posterior stacking and an independent photometric companion both recover the TEP-predicted reddening direction. A true
spectral stack remains desirable once public extracted spectra are
incorporated into the canonical analysis.


- 
**Alternative explanations:** A fully nested Bayesian evidence computation yields comparison families under both conditionings of the predictor space, arbitrated by the nested augmented model. In the covariance-corrected joint family, the TEP model ($\ln Z = -1262.6$, 9 parameters) outperforms the orthogonalized standard model ($\ln Z = -1304.1$, 13 parameters) by $\ln{\rm BF} = +41.5$, while the unconditioned raw mass+z baseline is the best model in the ensemble ($\ln Z = -1248.2$, $\ln{\rm BF} = -14.4$ over TEP). Neither conditioning is neutral: orthogonalization strips the competitor's $R_{\rm ML}$-aligned channel, whereas raw mass under the TEP measurement model already contains a $0.7\log_{10}R_{\rm ML}$ component of the response. The nested augmented model gives the marginal contributions — adding $\log R_{\rm ML}$ to mass+z yields $\Delta\ln Z = +4.5$; adding mass+z to $R_{\rm ML}$ yields $\Delta\ln Z = +18.9$ — and the direction-isolation diagnostic assigns $55.9$ ln units to the $R_{\rm ML}$-aligned direction of mass space. The orthogonalized sensitivity family yields $\ln {\rm BF} = +157.9$ in favour of TEP over the residual null, and a mean $\ln{\rm BF} = +113.8$ across eleven parametric stand-ins; the symmetric conventional residualization yields $\ln {\rm BF} = +9.1$ for the null. A log-gamma standardization check confirms robustness to the scaling choice ($\Delta\ln{\rm BF} = 1.4$). The Bayesian evidence is thus read as channel consistency — the variance is concentrated in the theory-predicted direction, which a single theory-fixed predictor captures competitively — rather than as functional-form discrimination against the mass+z family.


- 
**TEP mass correction and astrophysical model fits:** A direct test of the TEP measurement-correction framework applies the correction $M_{\rm true} = M_{\rm obs} - 0.7\log_{10} R_{\rm ML}$ to four standard astrophysical models (Standard Physics, Bursty Star Formation, Varying IMF, AGN Feedback) and compares nested-sampling evidence against the same models with uncorrected mass, using non-orthogonalized mass to preserve the correction signal. The correction yields mixed results: the Varying IMF model improves significantly ($\ln {\rm BF} = +7.0$), while the linear models (Standard Physics $\ln {\rm BF} = -7.9$, Bursty SF $\ln {\rm BF} = -8.6$) and AGN Feedback ($\ln {\rm BF} = -2.8$) do not benefit, with a mean $\ln {\rm BF} = -3.1$. The improvement in the quadratic IMF model is physically meaningful: the TEP correction changes the mass scale in a way that the quadratic mass term is sensitive to, while linear coefficients can absorb a pure scale shift without improvement. The evidence for TEP therefore rests on the correlation structure (L1, L3, L5), the regime-level dynamical-mass comparison (L4), and the channel consistency established by the Bayesian comparison, with the mass correction providing partial support through the IMF channel. Here L1 is the primary photometric line and L3 is a secondary partial-correlation line.


- 
**Response coefficient uncertainty:** The prespecified canonical
benchmark is $\kappa_{\rm gal} = (9.6 \pm 4.0) \times 10^5$ mag from Paper 11,
while the independent empirical Cepheid-channel fits recover
$(1.220 \pm 0.531)\times 10^6$ and $(0.665 \pm 0.362)\times 10^6$ mag —
bracketing the canonical value to within ~1σ, carried as an open
normalization discrepancy of the corpus (Paper 11; the unresolved
response-normalization class of Paper 0), not a resolved agreement.
Full propagation through the $R_{\rm ML}$ formula gives a corrected
Red Monster SFE spanning $\sim 0.12$–$0.26$ across the benchmark's
$1\sigma$ range (Table 3b): the anomaly is resolved at the central
value ($\sim 0.18$, below the $\Lambda$CDM limit of 0.20) and fully
resolved at the upper bound, but only partially resolved at the lower
bound ($\sim 0.26$, above the limit) — the resolution is
prior-range dependent. The two most massive objects are fully resolved;
the least massive is 85% resolved at the central value.
The JWST dust-only and joint concordance
recoveries ($(8.38 \pm 4.11) \times 10^5$ mag) are consistent with
the canonical benchmark at $0.21\sigma$ but equally within
$\sim 1.1\sigma$ of the empirical Cepheid values, so they do not
discriminate between the two anchors; because they also arise within
the same high-redshift, mass-proxy-linked evidence they are classified
as internal consistency checks rather than as tighter replacement
constraints. An earlier result of $0.60 \pm 0.10$ was an artefact of
[0,1]-normalised RSS, which is also rank-invariant (see item 10).
Table 3b uses representative parameters, not exact catalog values.


- 
**Per-bin $\kappa_{\rm gal}$ recovery — a methodological
non-test:**
An earlier attempt to recover $\kappa_{\rm gal}$ by optimising the Spearman
$\rho(R_{\rm ML}, \text{dust})$ per redshift bin was performed. The
optimizer hits the grid floor in every bin, yielding an
apparent tension with the Cepheid value. This is a mathematical
artefact, not a physical failure. $R_{\rm ML}$ is a strictly monotonic
function of $\log M_h$ at fixed $z$; Spearman rank correlation is
invariant under monotonic transforms. Therefore, within any narrow
redshift bin, $\rho(R_{\rm ML}, \text{dust})$ is
*identical* for all positive response coefficients — confirmed numerically:
$\rho = 0.6954$ across the tested range in the $z = 8.5$–$10$ bin. The optimiser
cannot distinguish $\kappa_{\rm gal}$ values and converges to the lower
boundary by numerical accident. The apparent "$2.15\sigma$ tension"
is an artefact of using an identically flat objective function, not
evidence against the prespecified canonical response benchmark. The corrected
recovery (internal concordance values consistent with Paper 11
from the Pearson $R^2$ method) uses multi-observable combination
sensitive to the calibrated magnitude of $R_{\rm ML}$, not just its
rank order. The earlier result was itself a
[0,1]-normalised RSS artefact confirmed to have an identically flat
objective; it is now corrected. Per-bin Spearman or normalised-RSS
optimisation is not a valid $\kappa_{\rm gal}$ estimator. Rank-invariant
information is, however, not the only information the photometric sample
carries: functional-form discrimination is achieved instead through the
placebo comparison and the kernel-sign structure of the narrow-mass-bin
correlations (§3.4.1), both of which now resolve the TEP form against
generic mass-$z$ alternatives.


### 4.6 Falsification Regimes

#### 4.6.1 Critical Test: The Mass-Dust Inversion

Falsification: If sufficiently large JWST/MIRI samples establish a
persistent lack of correlation ($\rho(M_*, A_V) \approx 0$) at $z > 8$
under rigorous selection control, the TEP prediction of emergence is
ruled out.

The joint-covariance result is conditioning-dependent: $\ln {\rm BF} = +41.5$ under orthogonalized predictors versus $-14.4$ under raw mass+z (physical subset $+37.4$ versus $-20.4$), with the nested augmented model showing the two predictor families mutually subsumed ($\Delta\ln Z = +4.5$ and $+18.9$ for the two nestings). The residual-space pair shows the same split ($+157.9$ orthogonalized versus $+9.1$ conventional). Neither conditioning is neutral — orthogonalization removes the competitor's access to the $R_{\rm ML}$-aligned direction, while raw mass under the TEP measurement model already contains part of the response — so the comparison is reported as channel consistency: the explanatory variance is concentrated in the $R_{\rm ML}$ direction (direction-isolation diagnostic $+55.9$ ln units), and the theory-fixed predictor recovers the full augmented model within one log-evidence unit, but the linear mass+z family is not excluded. The ambient-field audit (step_182) sharpens this further and then resolves it: a $k$NN environmental-density proxy adds $\Delta\ln Z = +16.2$ beyond mass+z where the instantiated kernel adds $-0.9$ — but the gain is carried entirely by the smooth clustercentric coordinate, and the foreground nature of Abell 2744 ($z = 0.308$) makes that coordinate a systematics signature, not an ambient-field one. The DR4 v2.0 lens-model magnification $\mu_{\rm ML}$ confirms the mechanism (${\rm corr}(\log\mu, d_{\rm A2744}) = -0.82$, ${\rm corr}(\log\mu, \chi^2) = +0.44$; $\{M_*,z,\mu,d_{\rm A2744}\}$ gains $+34.2$ over mass+z), and the theory-faithful proxy — the mass-weighted projected density of the $z>8$ epoch slice, since $u_{\rm amb}\propto\sqrt{\rho_{\rm env}L}$ is a mass density — adds nothing on top of that systematics baseline ($-7.1$ full, $-6.0$ physical subset). The covariance channel therefore delivers a clean control result: the observables' radial structure is explained by the cluster's lensing and depth signature, and no ambient-field detection is recoverable from this field's selection geometry. The $\kappa_{\rm gal} = 0.96 \times 10^6$ benchmark is prespecified from Paper 11, not independently calibrated here; the concordance confirms the benchmark propagates correctly to the high-redshift regime.

Falsification: If fitting the $z > 8$ dust anomaly with
higher-resolution spectroscopic data consistently requires $\kappa_{\rm gal} >
2 \times 10^6$ mag or $\kappa_{\rm gal} < 2 \times 10^5$ mag, the cross-domain consistency with Cepheids is
severely challenged.

Several theoretical predictions extend beyond the present JWST sample
and define additional falsification opportunities in wider survey
regimes:

- 
**Euclid Wide ($N \sim 300{,}000$ massive galaxies, $z =
0.9$–$1.8$):**
Typical $R_{\rm ML} \approx 1.25$ predicts a 25% age offset at fixed
$z$. Combined sensitivity reaches $\rho_{\rm min} = 0.0022$ —
sufficient to detect TEP at $> 5\sigma$ even if the effect is
10$\times$ weaker than at $z > 8$. Key falsification: no
mass-dependent age offset at $z \sim 1.5$.


- 
**Roman Supernova Survey ($N \sim 2{,}700$ SNe Ia, $z < 1.7$):**
TEP predicts host-potential-dependent SN Ia rate enhancements and an elevated Ia/CC ratio with host gravitational potential ($R_{\rm ML}$). Key falsification: no host potential or mass dependence in SN rates at $z > 1$.


- 
**Roman High-Latitude ($N \sim 500{,}000$ at $z > 2.5$):**
Tests the gas vs. stellar metallicity discriminant and
morphology–$R_{\rm ML}$ correlation. Key falsification: strong [O
III]/H$\beta$–$R_{\rm ML}$ correlation.


At this aggregate sample scale ($\sim 801{,}000$ galaxies), the
statistical power would be sufficient for rigorous cross-verification.
Current cross-field consistency (UNCOVER $\sigma_{\rm cv} \approx 22\%$,
CEERS $15\%$, COSMOS-Web $3.5\%$) supports the conclusion that the
signal is not driven by large-scale structure. Full theoretical
predictions are detailed in Appendix C.5.

All studies testing the TEP framework are ultimately falsifiable by a
single class of experiment that no current precision test has performed:
a
*closed-loop, direction-reversing, one-way time-transfer test*
targeting the synchronization holonomy $H \equiv \oint_C d\tau_{\rm
prop}$. Under standard GR, $H = 0$ after subtracting modelled Sagnac and
Shapiro terms. Under TEP, $H \neq 0$ if the disformal coupling $B(\phi)
\neq 0$, with a predicted amplitude:

\begin{equation} \label{eq:jwst_holonomy_prediction}
H_{\rm resid} \sim \frac{B(\phi)}{A(\phi)} |\nabla\phi|^2 \times
\mathcal{A}
\end{equation}

where $\mathcal{A}$ is the loop area. For a triangular
ground-satellite-ground loop with $\mathcal{A} \sim 10^6$ km$^2$ (e.g.,
two ground stations and one MEO satellite), the predicted holonomy is
$H_{\rm resid} \sim 10^{-19}$ s — at the frontier of current optical
clock technology but achievable with next-generation transportable
optical lattice clocks (Lisdat et al. 2016; Grotti et al. 2018). Three
experimental configurations are ranked by discriminating power:

- 
**Tier 1 (Decisive):** Closed triangular time-transfer
loop with three optical clocks at $\sim 1{,}000$ km separation,
targeting $H_{\rm resid}$ at $10^{-19}$ s after GR subtraction. A
non-zero result at $> 3\sigma$ would confirm the disformal sector; a
null result would constrain $B(\phi)/A(\phi) < 10^{-10}$
Mpc$^2$/km$^2$, ruling out the disformal contribution to the GNSS
signal.


- 
**Tier 2 (Strong):** Interplanetary one-way optical
time transfer (Earth–Mars or Earth–L2) targeting picosecond-level
asymmetries over AU baselines. Predicted asymmetry $\Xi \sim
10^{-12}$ s at current solar-system $\phi$ gradients.


- 
**Tier 3 (Confirmatory):** Roman/Euclid population
statistics ($N > 800{,}000$; see Appendix C.5) — these test the
conformal sector ($A(\phi)$, which governs $R_{\rm ML}$) independently
of the disformal sector. A positive Euclid detection combined with a
null holonomy would uniquely constrain the $B/A$ ratio.


The holonomy test provides a clean discriminant between the full
disformal theory and its conformal-only limit. Detection at the
predicted level would support the full theoretical construction. A null
result at that level would imply that the disformal sector is suppressed
below current sensitivity, and the conformal-only limit ($B = 0$)
applies — preserving the JWST, Hubble tension, and pulsar predictions
while removing the holonomy signal. The holonomy test therefore
separates the full disformal theory from a self-consistent
conformal-only sub-theory.


## 5. Conclusion

JWST has revealed a coherent pattern of anomalies at $z > 5$: ultra-massive galaxies with star formation efficiencies exceeding $\Lambda$CDM limits and stellar masses that can exceed dynamical masses. What links these anomalies is not merely that they are surprising, but that they cluster in the deepest gravitational potentials and point in the same direction — photometrically inferred stellar properties appear too large and too early. This work tested whether a single violation of the isochrony axiom, encoded by the continuously screened Temporal Equivalence Principle (TEP), can account for that shared structure. Using the prespecified canonical TEP benchmark $\kappa_{\rm gal} = (9.6 \pm 4.0) \times 10^5$ mag from Paper 11 with no JWST retuning, the framework reproduces the scale of the Red Monster efficiency excess and brings the $M_*/M_{\rm dyn}$ anomaly to the admissibility boundary in the direct object-level comparison (mean corrected excess $+0.015$ dex on the anomalous subset at the canonical benchmark; two of three anomalous objects resolved). The canonical magnitude-sector benchmark is recovered across distinct Cepheid response analyses in Paper 11: the empirical Cepheid recoveries bracket $\kappa_{\rm gal} = 0.960 \times 10^6$ mag (midpoint $0.943 \times 10^6$ mag, within $1.8\%$), while the Step-62 stellar-envelope unit-transport benchmark $|\kappa_{P^2}^{\rm pred}(\lambda_0=1)| = 1.196 \times 10^6$ mag (with bare coupling $|\beta_A|=1$ fixed by Paper 0) matches the direct SH0ES period-quadratic recovery $(1.191 \pm 0.480) \times 10^6$ mag to $0.5\%$, reducing the remaining microscopic task to deriving the order-unity transport weight $\lambda_0$ from coupled envelope dynamics rather than resolving a normalization gap. The Little Red Dot analysis is an unresolved compact-core stress test, not part of the primary evidence.


### 5.1 Synthesis of Results

The core empirical case rests on the dust–$R_{\rm ML}$ emergence (L1) as the primary photometric line, supported by the $R_{\rm ML}$–sSFR partial correlation line (L3). The SUSPENSE kinematic comparison (L5) breaks mass circularity: $R_{\rm ML}$ retains residual age information after $M_*$+$z$ control ($\rho = +0.556$, $p=0.032$), whereas $M_*$ contributes no residual signal once $R_{\rm ML}$+$z$ are controlled ($\rho = +0.031$, $p=0.91$). In the broader ($N=75$) $\sigma$-based kinematic expansion (Step 171), the Virial-exact direct depth map ($\Psi\propto\sigma^2$, computed purely from $\sigma$ and $z$ with zero SED-mass dependence) yields a circularity-free partial correlation controlling for redshift of $\rho(R_{\rm ML,\sigma}^{\rm dir}, M_{*,\rm obs} \mid z) = +0.463$ ($p = 2.84 \times 10^{-5}$, $N=75$), $\rho = +0.608$ ($p = 8.43 \times 10^{-7}$, $N=55$ exact-$M_{\rm dyn}$), and $\rho = +0.575$ ($p = 0.0080$, $N=20$ stellar absorption-line kinematics), with 5/5 subsample sign agreement and an $86.5\%$ attenuation of the absorption-line residual-$z$ trend ($+0.252 \to +0.034$) under the $z_0$-anchored reference. Within the photometric sample itself, the circularity-breaker battery (§3.4.1) supplies functional-form discrimination: the canonical $R_{\rm ML}$ outperforms a power-law placebo ($\Delta\rho = +0.099$, 95% CI $[+0.028, +0.174]$), and the narrow-mass-bin correlations execute the kernel-predicted negative-to-positive sign transition (4/4 bins, crossover at $\log M_* \simeq 9.6$) that no uniform-sign monotone-in-$z$ baseline can produce. L2 provides both controlled central-concentration support (Gini partial $\rho=+0.191$, $p=1.6\times10^{-4}$) and resolved radial colour-gradient confirmation once the measured inner-to-outer F444W stellar concentration $C_{\rm core}$ is included in the radial screening predictor $\Gamma_{\rm rad}$ (partial $\rho = -0.179$, $p = 0.0028$, $2.99\sigma$ after $M_*$+$z$ control; Steiger $Z = 3.02$, $p = 0.0025$ over $M_*$; residualized bluer-core sign test $58.6\%$ vs $35.7\%$, $\text{OR} = 2.54$, Fisher $p = 0.0054$, $\Delta = -0.273$ mag, 95% CI $[-0.526, -0.035]$; Steps 139 and 168), and L4 provides a derived dynamical-mass comparison that lands the anomalous literature objects at the admissibility boundary at the canonical benchmark.


### 5.2 Interpretative Framework

Physical processes require proper time. Standard inference assumes proper time and the FLRW observer-age coordinate are identical. The JWST anomalies appear precisely where this identification fails. In the canonical TEP cosmology the spatial manifold is static and the gravitational coordinate time extends without finite origin (TEP-HUB, Paper 30); the FLRW age $t_{\rm cosmic}(z)$ is a reconstruction under isochrony that shrinks to a few hundred Myr at high redshift. Local clocks run slower than coordinate time ($\Delta\ln A<0$ in a deeper well), but the coordinate background is eternal, so accumulated proper time can still exceed the FLRW assigned age by a large factor. The inference-channel response $R_{\rm ML}$ quantifies how much the standard pipeline inflates apparent age and $M/L$ when it uses the too-short FLRW baseline, and a single parameter propagates coherently through stellar ages, mass-to-light ratios, apparent dust-evolution diagnostics, star-formation diagnostics, and dynamical-mass comparisons.

A nested Bayesian model comparison using a joint covariance likelihood that accounts for correlated SED outputs is conditioning-dependent: $\ln{\rm BF}=+41.5$ under orthogonalized predictors (and $+37.4$ in the 3-observable physical subset) versus $\ln{\rm BF}=-14.4$ under the raw mass+z baseline ($-20.4$ in the subset). The flip is structural rather than a bug: $\log R_{\rm ML}$ is a deterministic function of (mass, z) (${\rm corr}=0.86$, $R^2=0.74$), so orthogonalizing mass strips the comparator of the signal direction. The nested augmented model shows the predictor families mutually subsumed ($\Delta\ln Z = +18.9$ for adding mass+z to $R_{\rm ML}$), and a direction-isolation diagnostic assigns $55.9$ ln units to the $R_{\rm ML}$-aligned direction of mass space — the variance is concentrated in the direction the theory specifies. A free mass–redshift interaction is the strongest baseline ($\ln Z=-1190.9$, $+57.3$ over raw), and adding the instantiated kernel on top contributes $-0.9$. The ambient-field proxy audit (step_182) initially appears to supply the missing direction ($+16.2$ beyond mass+z, $z$-independent), but the completed audit resolves it as a systematic: Abell 2744 is a foreground cluster ($z = 0.308$) that cannot belong to a $z>8$ galaxy's ambient field, and the clustercentric coordinate's predictive power is the lensing-magnification/mosaic-depth signature — confirmed by the DR4 lens model, ${\rm corr}(\log\mu_{\rm ML}, d_{\rm A2744})=-0.82$ and ${\rm corr}(\log\mu_{\rm ML},\chi^2)=+0.44$. With $\{M_*,z,\mu,d_{\rm A2744}\}$ in the null baseline ($+34.2$ over mass+z), the theory-faithful mass-weighted same-epoch density proxy contributes $-7.1$ (full) and $-6.0$ (physical subset): the environmental-channel evidence is fully explained by the foreground-lens systematics, and functional-form discrimination of the ambient-field law is not achievable with this field's selection geometry. The Bayesian comparison is accordingly read as a conditioning-corrected control result — the $R_{\rm ML}$-aligned direction organizes the explanatory variance, but neither the instantiated kernel nor a same-epoch ambient proxy supplies unique evidence beyond the systematics-controlled baseline — and the supportive weight of the analysis rests on the kinematic comparisons (SUSPENSE) and the multi-survey photometric correlation structure.

The cross-domain consistency of the coupling is a central feature of the evidence base. The prespecified canonical benchmark $\kappa_{\rm gal} = (9.6 \pm 4.0) \times 10^5$ mag from Paper 11 (where distinct empirical Cepheid-channel analyses recover $(1.220 \pm 0.531)\times 10^6$ and $(0.665 \pm 0.362)\times 10^6$ mag, bracketing the benchmark with a midpoint of $0.943\times 10^6$ mag, and Step 62 derives $|\kappa_{P^2}^{\rm pred}(\lambda_0=1)| = 1.196\times 10^6$ mag from the Cepheid envelope density ratio $\rho_T/\langle\rho\rangle(10\text{ d}) = 8.45\times 10^5$ at the unit-transport benchmark $\lambda_0 = 1$ with $|\beta_A| = 1$ fixed by Paper 0, matching the SH0ES fit $(1.191\pm 0.480)\times 10^6$ mag to $0.5\%$) provides the magnitude-sector normalization, while the regenerated informative JWST high-redshift analyses recover $\kappa = (8.38 \pm 4.11) \times 10^5$ mag. This is consistent with the canonical benchmark at $0.21\sigma$ and within $0.3\text{--}0.6\sigma$ of the empirical Cepheid recoveries ($p_{\rm concordance}=1.0$), confirming quantitative cross-domain concordance from local Cepheids to $z > 8$ galaxies.


Key signatures survive a 0.5 dex mass reduction, and blind validation passes all three generalisation tests — time-split, field-split, and cross-survey leave-one-out — with recovery across all 9 survey-test combinations. Each of the three independent JWST surveys confirms the dust relation individually above $5\sigma$, and all three independently confirm that $t_{\rm eff}$ outperforms $t_{\rm cosmic}$ at $>5\sigma$ (combined Steiger $Z \approx 25.8$), ruling out pure redshift ordering. A Fisher combination across the three photometric datasets gives $z = 25.1\sigma$ ($p = 3.0 \times 10^{-139}$). Fixed-effects meta-analysis, dependence-adjusted Brown combinations, permutation tests, and CAMB-based cosmological consistency provide supportive context rather than a competing headline.

The main remaining limitations are the self-referential evidence base and the fact that some secondary morphology and emission-line diagnostics weaken under biased mass control. These caveats sit within an evidential structure whose spine is coherent: the primary photometric line (L1), a secondary $R_{\rm ML}$–sSFR partial correlation line (L3), a direct kinematic comparison whose one-sided conditional asymmetry narrows the mass-circularity objection, an ancillary screening indication, and a derived dynamical-mass comparison.


### 5.3 Falsification Criteria

TEP makes specific, quantitative predictions that can be tested against additional data in the same observables and in wider survey regimes. The following failure conditions are defined; any one of them, if met, requires rejection of the TEP interpretation of the JWST anomalies.




Table 14: TEP Falsification Criteria

| Observable | Baseline Comparison | TEP Prediction | Falsification Criteria |
| --- | --- | --- | --- |
| Mass-Dust ($z > 8$) | No correlation or Negative | Strong Positive ($\rho > 0.4$) | $\rho \approx 0$ or Negative |
| Balmer Absorption | Correlates with $z$ | Correlates with $M_*$ at fixed $z$ | No mass trend |
| LRD compact-core stress test | No dependence | Mass-calibration-sensitive response concentrated in compact hosts | Ancillary diagnostic only; not a standalone rejection criterion |
| Cluster vs Field | Cluster galaxies older | Cluster galaxies younger (screened) | Field $\approx$ Cluster or Field < Cluster (Standard) |
| [OIII]/H$\beta$ vs $R_{\rm ML}$ | Correlates with mass | Weak correlation ($\rho < 0.1$) | Strong correlation ($\rho > 0.3$) |
| Radial Age Gradient | Inside-out (Red Core, Blue Out) | Core Screening (Blue Core, Red Out) | Standard inside-out gradients in massive high-z galaxies |
| Time-Lens Ordering | Properties track $z_{\rm obs}$ | Properties track $z_{\rm eff}$ | Correlation with $z_{\rm obs}$ is stronger than with $z_{\rm eff}$ |
| Type Ia / CC SN Ratio | No $R_{\rm ML}$ dependence | Positive correlation of Ia/CC ratio with host gravitational potential / $R_{\rm ML}$; CC rate uncoupled | No host potential dependence of Ia/CC ratio at fixed mass (e.g. Roman Space Telescope High-Latitude Time Domain Survey) |
| Coupling Recovery | N/A | $\kappa_{\rm gal} = (9.6 \pm 4.0) \times 10^5$ mag (Paper 11); JWST recovery $(8.38 \pm 4.11)\times10^5$ mag is anchor-consistent ($0.21\sigma$) and internally concordant | Independent recovery yielding factor $>2$ discrepancy from Paper 11 |




### 5.4 Reproducibility

This analysis is designed to be fully reproducible. The public repository contains the end-to-end analysis code needed to regenerate the manuscript tables, figures, and archived outputs; execution instructions are provided in the repository README.


### 5.5 Data Availability

The manuscript source, complete analysis code, generated figures, intermediate outputs, and the raw and processed catalogs are available on GitHub and archived on Zenodo for long-term reproducibility.


- **Analysis repository:** github.com/matthewsmawfield/TEP-JWST — complete analysis code, reproducible outputs, and build instructions.

- **Input catalogues:** UNCOVER DR4, CEERS, and COSMOS-Web — all publicly available through MAST.

- **Processed outputs:** All intermediate and final data products (`interim/`, `outputs/`, `figures/`) are version-controlled and reproducible from the input catalogues.

- **Documentation:** `README.md` provides installation instructions, a dependency list (`requirements.txt`), and a quick-start guide.


The full TEP theoretical framework series is available on Zenodo; DOIs are listed in the References. Key identifiers: Paper 0 — TEP foundation [Jakarta] (10.5281/zenodo.16921911); Paper 6 — Temporal Topology Saturation Scale (10.5281/zenodo.18064365); Paper 10 — COSMOS2025 Screening Analysis (10.5281/zenodo.18165798); Paper 11 — Cepheid H₀ Calibration (10.5281/zenodo.18209702); Paper 12 — JWST High-Redshift Test (this work).


## References

Abbott, B. P., et al. 2017, ApJL, 848, L13. *Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A.*

Arrabal Haro, A., et al. 2023, Nature, 622, 707. *Spectroscopic confirmation and refutation of CEERS high-redshift candidates.*

Behroozi, P., Wechsler, R. H., Hearin, A. P., & Conroy, C. 2019, MNRAS, 488, 3143. *UNIVERSEMACHINE: The correlation between galaxy growth and dark matter halo assembly from z = 0−10.*

Berg, D. A., et al. 2013, ApJ, 775, 93. *New Detections of C/O Abundance Ratios in Metal-Poor Dwarf Galaxies.*

Boyer, M. L., et al. 2025, ApJ, 991, 24. *Discovery of SiC and Iron Dust around AGB Stars in the Very Metal-Poor Dwarf Galaxy Sextans A with JWST.*

Bertotti, B., Iess, L., & Tortora, P. 2003, Nature, 425, 374. *A test of general relativity using radio links with the Cassini spacecraft.*

Boylan-Kolchin, M. 2023, Nature Astronomy, 7, 731. *Stress testing ΛCDM with high-redshift galaxy candidates.*

Brammer, G. B., van Dokkum, P. G., & Coppi, P. 2008, ApJ, 686, 1503. *EAZY: A Fast, Public Photometric Redshift Code.*

Brout, D., et al. 2022, ApJ, 938, 110. *Type Ia supernova host-mass step measurements in Pantheon+.*

Brax, P., van de Bruck, C., Davis, A.-C., Khoury, J., & Weltman, A. 2004, PhRvD, 70, 123518. *Small scale structure formation in chameleon cosmology.*

Bruzual, G. & Charlot, S. 2003, MNRAS, 344, 1000. *Stellar population synthesis at the resolution of 2003.*

Burrage, C. & Sakstein, J. 2018, Living Reviews in Relativity, 21, 1. *Tests of Chameleon Gravity.*

Carniani, S., et al. 2024, Nature, 633, 318. *A shining cosmic dawn: spectroscopic confirmation of two luminous galaxies at z > 14.*

Carnall, A. C., McLure, R. J., Dunlop, J. S., & Davé, R. 2018, MNRAS, 480, 4379. *Inferring the star formation histories of massive quiescent galaxies with BAGPIPES.*

Carnall, A. C., et al. 2023, Nature, 619, 716. *A massive quiescent galaxy at redshift 4.658.*

Chemerynska, I., Atek, H., et al. 2024, MNRAS, 531, 2615. *JWST UNCOVER: The Overabundance of Ultraviolet-luminous Galaxies at z > 9.*

Chworowsky, K., et al. 2025, arXiv:2509.07695. *The growth evolution of the most massive galaxies in Renaissance compared with observations from JWST.*

Claeyssens, A., et al. 2023, MNRAS, 520, 2162. *JWST study of the Sparkler system and proto-globular cluster candidates.*

Conroy, C., Gunn, J. E., & White, M. 2009, ApJ, 699, 486. *The Propagation of Uncertainties in Stellar Population Synthesis Modeling.*

Cox, T. J., et al. 2025, ApJS (in press). *CEERS DR1 photometric and physical parameter catalog.*

Curti, M., et al. 2023, MNRAS, 518, 425. *Chemical enrichment in the first billion years: the JADES perspective on early galaxy metallicities.*

Curtis-Lake, E., et al. 2023, Nature Astronomy, 7, 622. *Spectroscopic confirmation of four metal-poor galaxies at z = 10.3–13.2.*

D'Eugenio, F., et al. 2025, ApJS (in press). *JADES Data Release 4: Spectroscopic Redshifts and Emission Line Measurements.*

Danhaive, J., et al. 2025, arXiv:2503.21863. *The dawn of disks: JWST/NIRCam grism kinematics of galaxies at z ~ 4–6.*

de Graaff, A., et al. 2024a, A&A, 684, A87. *Ionised gas kinematics and dynamical masses of z ≳ 6 galaxies from JADES/NIRSpec high-resolution spectroscopy.*

de Graaff, A., et al. 2024b, Nature, 630, 846. *A dormant overmassive black hole in the early Universe.*

Endsley, R., et al. 2023, MNRAS, 524, 2312. *A JWST/NIRCam Study of Key Contributors to Reionization: The Star-forming and Ionizing Properties of UV-faint z ∼ 7–8 Galaxies.*

Esdaile, J., et al. 2021, ApJL, 908, L35. *Massive Quiescent Galaxies at z ~ 3: A Comparison of Selection, Stellar Population, and Structural Properties with Simulation Predictions.*

Eisenstein, D. J., et al. 2023, arXiv:2306.02465. *Overview of the JWST Advanced Deep Extragalactic Survey (JADES).*

Finkelstein, S. L., et al. 2023, ApJL, 946, L13. *CEERS early release science survey overview.*

Freedman, W. L., Madore, B. F., Hoyt, T. J., et al. 2024, arXiv:2408.06153. *Status Report on the Chicago-Carnegie Hubble Program (CCHP).*

Fujimoto, S., et al. 2023, ApJL, 949, L25. *JWST/NIRSpec spectroscopic confirmation of z > 8 CEERS candidates.*

Furtak, L. J., et al. 2023, MNRAS, 523, 4568. *JWST UNCOVER: The Strong Lensing Model of Abell 2744.*

Grotti, J., et al. 2018, Nature Physics, 14, 437. *Geodesy and metrology with a transportable optical clock.*

Greene, J. E., et al. 2024, ApJ, 964, 39. *UNCOVER: The Growth of the First Massive Black Holes.*

Hainline, K. N., et al. 2023, arXiv:2306.02468. *The Cosmos in its Infancy: JADES Galaxy Candidates at z > 8 in GOODS-S and GOODS-N.*

Heintz, K. E., et al. 2023, ApJL, 953, L10. *Extreme Damped Lyman-α Absorption in Young Star-Forming Galaxies at z = 9–11.*

Ilie, C., et al. 2025, PNAS. *Supermassive Dark Star candidates seen by JWST.*

Jiang, Y.-F., Stone, J. M., & Davis, S. W. 2019, ApJ, 880, 67. *Super-Eddington Accretion Disks around Supermassive Black Holes.*

Jin, B., et al. 2025, A&A, 698, A30. *Spatially resolved colours and sizes of galaxies at z ~ 3–4.*

Ju, M., et al. 2025, arXiv:2506.12129. *A 13-Billion-Year View of Galaxy Growth: Metallicity Gradients.*

Kelly, P. L., et al. 2010, ApJ, 715, 743. *Host-galaxy mass step in Type Ia supernova distances.*

Khoury, J. & Weltman, A. 2004, PhRvL, 93, 171104. *Chameleon Fields: Awaiting Surprises for Tests of Gravity in Space.*

Kawinwanichakij, L., et al. 2025, ApJ (in press). *Environmental dependence of galaxy morphology at z = 3–4.*

Kocevski, D. D., et al. 2023, ApJL, 954, L4. *Hidden Little Monsters: Spectroscopic Identification of Low-Mass, Broad-Line AGN at z > 5 with CEERS.*

King, A. R., Lasota, J.-P., & Kluzniak, W. 2023, MNRAS, 519, 5765. *Super-Eddington accretion: models and applications.*

Kodric, M., Riffeser, A., Seitz, S., et al. 2018, ApJ, 864, 59. *Calibration of the Tip of the Red Giant Branch in the I Band and the Cepheid Period–Luminosity Relation in M31.*

Kokorev, V., et al. 2024, arXiv:2401.09981. *A Census of Photometrically Selected Little Red Dots at 4 < z < 9 in JWST Blank Fields.* github.com/VasilyKokorev/lrd_phot

Larson, R. L., et al. 2023, ApJ, 953, 34. *A CEERS Discovery of an Accreting Supermassive Black Hole 570 Myr after the Big Bang.*

Labbé, I., et al. 2023, Nature, 616, 266. *A population of red candidate massive galaxies ~600 Myr after the Big Bang.* Data: github.com/ivolabbe/red-massive-candidates

Leja, J., et al. 2019, ApJ, 876, 3. *How to Measure Galaxy Star Formation Histories. II. Nonparametric Models.*

Lisdat, C., et al. 2016, Nature Communications, 7, 12443. *A clock network for geodesy and fundamental science.*

Li, Q., et al. 2025, MNRAS, 539, 1796. *EPOCHS Paper X: Environmental effects on Galaxy Formation and Protocluster Galaxy candidates at 4.5 < z < 10.*

Maiolino, R., et al. 2024, Nature, 627, 59. *A small and vigorous black hole in the early Universe.*

Matthee, J., et al. 2024, ApJ, 963, 129. *Little Red Dots: An Abundant Population of Faint Active Galactic Nuclei at z ~ 5 Revealed by JWST.*

Meng, X.-L., Rosenthal, R., & Rubin, D. B. 1992, Biometrika, 79, 425. *Comparing correlated correlation coefficients.*

Middleton, M. J., et al. 2015, MNRAS, 447, 3243. *NuSTAR reveals extreme absorption in z = 2–3 type 2 quasars.*

Mota, D. F. & Shaw, D. J. 2007, PhRvD, 75, 063501. *Evading equivalence principle violations, cosmological, and other experimental constraints in scalar field theories with a strong coupling to matter.*

Mowla, L., et al. 2022, ApJL, 937, L35. *The Sparkler: Evolved High-Redshift Globular Cluster Candidates Captured by JWST.*

Naidu, R. P., et al. 2022, ApJL, 940, L14. *Two Remarkably Luminous Galaxy Candidates at z ≈ 10–12 Revealed by JWST.*

Nanayakkara, T., et al. 2024, Science, 384, 890. *A massive galaxy that was quenched by z ∼ 3.*

Nedkova, K. V., et al. 2025, A&A. *Evolution and mass dependence of UV-to-near-IR color gradients of galaxies at 0.5 < z < 2.5.*

Nakajima, K., et al. 2023, ApJS, 269, 33. *JWST Census for the Mass-Metallicity Star Formation Relation at z = 4–10.*

Pérez-González, P. G., et al. 2024, ApJ, 968, 4. *CEERS Key Paper VII: JWST/MIRI Reveals a Faint Population of Galaxies at Cosmic Dawn.*

Planck Collaboration, Aghanim, N., et al. 2020, A&A, 641, A6. *Planck 2018 results. VI. Cosmological parameters.*

Price, S. H., et al. 2024, ApJ, 964, 73. *UNCOVER: JWST spectroscopy of three cold brown dwarfs at kiloparsec-scale distances.*

Rieke, M. J., et al. 2023, PASP, 135, 028001. *JWST NIRCam Performance: Commissioning and Calibration.*

Riess, A. G., et al. 2022, ApJL, 934, L7. *A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team.*

Saldana-Lopez, A., et al. 2025, arXiv:2501.17145. *Feedback and dynamical masses in high-z galaxies: the advent of high-resolution NIRSpec spectroscopy.*

Scholtz, J., et al. 2025, A&A (in press). *JADES: Spectroscopic properties of faint AGN at z > 4.*

Shamir, L. 2025, MNRAS, 538, 76. *The distribution of galaxy rotation in JWST Advanced Deep Extragalactic Survey.*

Slob, M., et al. 2025, A&A (in press). *SUSPENSE: Spectroscopy of z = 1–2 massive quiescent galaxies with JWST/NIRSpec.*

Shuntov, M., et al. 2025, ApJS (in press). *COSMOS-Web DR1 / COSMOS2025 catalog.*

Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: 10.5281/zenodo.16921911 (Paper 0)

Smawfield, M. L. (2025). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: 10.5281/zenodo.17127229 (Paper 1)

Smawfield, M. L. (2025). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: 10.5281/zenodo.17517141 (Paper 2)

Smawfield, M. L. (2025). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: 10.5281/zenodo.17860166 (Paper 3)

Smawfield, M. L. (2025). *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: 10.5281/zenodo.17982540 (Paper 4)

Smawfield, M. L. (2025). *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: 10.5281/zenodo.18004832 (Paper 5)

Smawfield, M. L. (2025). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: 10.5281/zenodo.18064365 (Paper 6)

Smawfield, M. L. (2025). *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: 10.5281/zenodo.18059250 (Paper 7)

Smawfield, M. L. (2025). *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: 10.5281/zenodo.18064581 (Paper 8)

Smawfield, M. L. (2025). *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: 10.5281/zenodo.18109760 (Paper 9)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: 10.5281/zenodo.18165798 (Paper 10)

Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: 10.5281/zenodo.18209702 (Paper 11)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: 10.5281/zenodo.19000827 (Paper 12 — this work)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: 10.5281/zenodo.19102061 (Paper 13)

Song, M., et al. 2016, ApJ, 825, 5. *The Evolution of the Galaxy Stellar Mass Function at z = 4–8.*

Suess, K. A., et al. 2024, ApJL, 976, L21. *UNCOVER: MegaScience Photometric Catalogs.*

Sullivan, M., et al. 2010, MNRAS, 406, 782. *Type Ia supernova host-galaxy correlations and the mass step.*

Tanaka, M., et al. 2019, ApJL, 885, L34. *Stellar Velocity Dispersion of a Massive Quenching Galaxy at z ∼ 4.*

Taylor, A., et al. 2025, arXiv:2505.04609. *CAPERS-LRD-z9: A Gas Enshrouded Little Red Dot Hosting a Supermassive Black Hole.*

Tripodi, R., et al. 2025, Nature Communications. *CANUCS-LRD-z8.6: A rapidly accreting overmassive black hole at z = 8.6.*

van Dokkum, P., et al. 2025, ApJ (in press). *A Candidate Runaway Supermassive Black Hole.*

VandenBerg, D. A., et al. 2013, ApJ, 775, 134. *Milky Way globular cluster ages.*

Wang, B., et al. 2024, ApJS, 270, 12. *UNCOVER DR4 stellar population synthesis catalog.*

Weibel, A., et al. 2024, MNRAS, 533, 1808. *Galaxy build-up at z > 9: Connecting UV luminosity functions to stellar mass assembly.*

Xiao, M., et al. 2024, Nature, 635, 303. *Three ultra-massive galaxies in the early Universe.*

Yang, G., et al. 2025, ApJ (in press). *DJA GOODS-S: Spectrophotometric Catalog of 7,325 Galaxies.*


## Appendix A: Theoretical Foundation


### A.1 The TEP Action and Field Equations


The Temporal Equivalence Principle is formulated as a scalar-tensor
theory with a two-metric structure. The complete Lagrangian density in
the Einstein frame is:



\begin{equation} \label{eq:jwst_lagrangian}
\mathcal{L} = \frac{M_{\rm Pl}^2}{2} R - \frac{1}{2} K(\phi)
(\partial\phi)^2 - V(\phi) + \mathcal{L}_{\rm matter}[\psi,
\tilde{g}_{\mu\nu}]
\end{equation}


The theory assumes a disformal coupling where non-gravitational matter couples to a causal matter metric $\tilde g_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$, where the conformal sector is dominant for clock observables. The JWST response normalization
$K_{\rm gal}$ is informed by the prespecified canonical magnitude-sector response benchmark $\kappa_{\rm gal} = (9.6 \pm 4.0) \times 10^5$ mag from Paper 11 (where empirical Cepheid analyses recover $\kappa_{\rm equiv}^{39} = (1.220 \pm 0.531)\times 10^6$ and $\kappa_{\rm Cep}^{44} = (0.665 \pm 0.362)\times 10^6$ mag). It is a transferred galaxy-sector effective normalization,
not the empirical Cepheid coefficient itself and not a microscopic scalar coupling.
$K_{\rm gal}$ is not identified with bare couplings ($\beta_A$, $\kappa_{\rm Cep}$,
or $\alpha_{\rm clock}$). A microscopic mapping requires a solved transfer function
that absorbs stellar physics, environmental activation, and field gradients.
The JWST analysis adopts this observable response normalization for the
exponential $R_{\rm ML}$ kernel. No JWST-specific refit.


The complete action in the Einstein frame is:

\begin{equation} \label{eq:jwst_action_total}
S = S_{\rm grav} + S_\phi + S_{\rm matter}
\end{equation}
where the gravitational sector is:


\begin{equation} \label{eq:jwst_action_grav}
S_{\rm grav} = \int d^4x \sqrt{-g} \frac{M_{\rm Pl}^2}{2} R
\end{equation}

the scalar field sector is:


\begin{equation} \label{eq:jwst_action_scalar}
S_\phi = \int d^4x \sqrt{-g} \left[ -\frac{1}{2} K(\phi) g^{\mu\nu}
\partial_\mu\phi \partial_\nu\phi - V(\phi) \right]
\end{equation}

and matter couples to the Jordan-frame metric:


\begin{equation} \label{eq:jwst_action_matter}
S_{\rm matter} = S_{\rm matter}[\psi, \tilde{g}_{\mu\nu}], \quad
\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu\phi
\nabla_\nu\phi
\end{equation}


The conformal factor $A(\phi) = \exp(\beta_A\phi/M_{\rm Pl})$ controls 
the clock rate $d\tau/dt \propto A(\phi)$. The disformal term $B(\phi)$ 
is constrained by GW170817 to be negligible at late times, ensuring 
$c_\gamma \approx c_g$ in the cosmological background.



#### A.1.1 Field Equations


Variation with respect to $g_{\mu\nu}$ yields the modified Einstein
equations:



\begin{equation} \label{eq:jwst_einstein_modified}
G_{\mu\nu} = \frac{1}{M_{\rm Pl}^2} \left[ T_{\mu\nu}^{(\phi)} +
T_{\mu\nu}^{(\rm matter)} \right]
\end{equation}

where the scalar field stress-energy is:


\begin{equation} \label{eq:jwst_scalar_stress_energy}
T_{\mu\nu}^{(\phi)} = K(\phi) \partial_\mu\phi \partial_\nu\phi -
g_{\mu\nu} \left[ \frac{1}{2} K(\phi) (\partial\phi)^2 + V(\phi)
\right]
\end{equation}

Variation with respect to $\phi$ yields the scalar field equation:


\begin{equation} \label{eq:jwst_scalar_eom}
K(\phi) \Box\phi + \frac{1}{2} K'(\phi) (\partial\phi)^2 - V'(\phi) =
-\frac{\beta}{M_{\rm Pl}} T^{(\rm matter)}
\end{equation}


where $T^{(\rm matter)} = \tilde{g}^{\mu\nu} \tilde{T}_{\mu\nu}$ is the
trace of the matter stress-energy tensor in the Jordan frame.



#### A.1.2 Screening Mechanism: Temporal Shear


TEP requires a screening mechanism to reconcile the large clock-sector 
response with Solar System precision tests. In the TEP 
framework, this is achieved through Temporal Shear: the 
suppression of the incremental field gradient by the environmental operator. 
Phenomenologically, the screening is characterized by a 
density-dependent saturation profile:



\begin{equation} \label{eq:jwst_screening_response}
R_{\rm gal}(E) = \kappa_{\rm gal} \cdot \mathcal{S}_{\rm gal}(E)
\end{equation}


The screening mechanism operates through the density-dependent
saturation of the observable response: $R_{\rm gal}(E) = \kappa_{\rm gal}
\cdot \mathcal{S}_{\rm gal}(E)$ where $\rho_T \approx 20$ g/cm³ is the
saturation scale and $E$ includes source structure, environment,
boundary conditions, and density. In the diffuse environments of
high-redshift galaxy halos ($\rho \ll \rho_T$), the response remains
near the bare value; in the dense central regions ($\rho \gg \rho_T$),
the response is attenuated toward GR.



where $\rho_T \approx 20$ g/cm³ is the Temporal Topology reference density (Paper 6).
The field gradient (Temporal Shear) varies continuously with ambient density:
as $\rho$ increases toward $\rho_T$, the gradient flattens smoothly and GR is
approached asymptotically; as $\rho$ decreases below $\rho_T$, the Temporal
Shear strengthens, modifying local proper time. This continuous hierarchy
ensures that TEP effects are maximum in the diffuse stellar halos of
high-redshift galaxies where the Red Monster and LRD anomalies are observed.



#### A.1.3 PPN Parameters

For an unscreened photon probe and a screened scalar source charge, the
Eddington PPN parameter follows the linear source-charge mapping under the unified unstretched-space metric (Paper 0 §2.2, Step 80; with $-4\beta_A^2 S_\Sigma/(1+2\beta_A^2 S_\Sigma)$ on the isotropic-conformal benchmark):


\begin{equation} \label{eq:jwst_ppn_gamma}
\gamma - 1 = -\frac{2\beta_A^2\,S_\Sigma(\mathcal{E})}{1 + 2\beta_A^2\,S_\Sigma(\mathcal{E})}
\quad \left(\text{isotropic benchmark: } -\frac{4\beta_A^2\,S_\Sigma(\mathcal{E})}{1 + 2\beta_A^2\,S_\Sigma(\mathcal{E})}\right)
\end{equation}


For a partially screened source charge $S_\Sigma \sim 0.8$ (bare $\beta_A = -1.0$, Paper 0 §2.2), this would give $|\gamma -
1| \approx 0.62$ ($1.2$ isotropic), violating the Cassini bound $|\gamma-1| < 2.3\times 10^{-5}$ by more than four orders of
magnitude. Near massive bodies, the locally active PPN scalar charge is
suppressed by two-branch source/environment screening ($y_{\rm tb}(1.6\,R_\odot)=2.16\times 10^{-8}$), bringing
$|\gamma - 1|_{\rm eff} = 4.32\times 10^{-8} \ll 2.3\times 10^{-5}$ (well inside $S_\Sigma^{(\odot)} \lesssim 1.15\times 10^{-5}$ [$5.8\times 10^{-6}$ isotropic]) into compliance with observations
without invoking rigid thin-shell approximations.



The phenomenological TEP model assumes that the scalar field profile
$\phi(r)$ tracks the gravitational potential $\Phi_N(r)$ within galactic
halos, satisfying $\phi(r) \propto \Phi_N(r)$ in the relevant regime. To
validate this assumption, a numerical solution of the static spherical
scalar field equation of motion was obtained:


\begin{equation} \label{eq:jwst_scalar_bvp}
\nabla^2 \phi = \frac{dV_{\rm eff}}{d\phi}
\end{equation}

for a standard NFW density profile. The boundary value problem (BVP) was
solved using relaxation methods on a logarithmic radial grid.



The numerical results confirm that in the regime relevant for galaxy
formation ($0.1 R_s < r < 10 R_s$), the scalar field solution
tracks the Newtonian potential shape with high fidelity. This justifies
the use of the potential-dependent parameterization $R_{\rm ML} =
\exp(\alpha \Phi)$ used throughout the main text.



#### A.1.5 Screening Hierarchy Validation


The environment-dependent screening hierarchy explains why precision tests
of General Relativity show no deviation despite the large halo-scale
response. The five most precise GR tests all occur in regimes where
screening is operative. For example, the Hulse-Taylor binary pulsar at
density $\sim 10^{14}$ g/cm³ has a screening factor $S \approx
33{,}000$, meaning the scalar field contributes less than 0.003% to
orbital dynamics—consistent with GR verification to 0.2% precision.
Conversely, Earth at $\rho \approx 5.5$ g/cm³ sits just below $\rho_T$
with $S \approx 0.66$, placing it in the narrow window where the scalar
field is observable via GNSS clock comparisons but not yet fully
screened.



#### A.1.5b Observable Response Coefficient and the $R_{\rm ML}$ Kernel


The JWST response normalization $K_{\rm gal}$ is informed by the prespecified canonical magnitude-sector
response benchmark $\kappa_{\rm gal} = (9.6 \pm 4.0) \times 10^5$ mag from Paper 11 (where distinct empirical Cepheid-channel analyses recover $\kappa_{\rm equiv}^{39} = (1.220 \pm 0.531) \times 10^6$ mag and $\kappa_{\rm Cep}^{44} = (0.665 \pm 0.362) \times 10^6$ mag).
It is a transferred galaxy-sector effective normalization, not the empirical Cepheid coefficient
itself and not a microscopic scalar coupling. Paper 11 measures $\kappa_{\rm Cep}$
from Cepheid period-luminosity residuals. The conversion to the exponential
$R_{\rm ML}$ kernel defines $K_{\rm gal}$.



The relationship between the magnitude correction $\Delta M$ and the 
exponential kernel is defined at the reference stellar-population transfer $M/L \propto t^{n_{\rm ref}}$ with $n_{\rm ref}=0.7$:



\begin{equation} \label{eq:jwst_delta_mag}
\Delta M = -2.5 \log_{10}(R_{\rm ML}^{n_{\rm ref}}) = \frac{2.5 n_{\rm ref} K}{\ln 10} \frac{\Delta\Phi}{c^2}
\end{equation}


Equating with $\Delta M = \kappa_{\rm gal} \Delta\Phi/c^2$ yields the 
response normalization:



\begin{equation} \label{eq:jwst_kgal_normalization}
K_{\rm gal} = \frac{\kappa_{\rm gal} \ln 10}{2.5 n_{\rm ref}} = 1.263 \times 10^6
\end{equation}


This equation defines a fixed transfer convention: $n_{\rm ref}$ is not replaced by the population-dependent $n_{\rm SPS}$ used subsequently in mass and SFE corrections. Here $\kappa_{\rm gal}$ denotes the derived galaxy-sector effective response
(not the Cepheid-measured coefficient $\kappa_{\rm Cep}$). Neither $K$
nor $\kappa_{\rm gal}$ is identified with $\beta$, $\kappa_{\rm Cep}$, or
$\alpha_{\rm clock}$. A microscopic mapping requires a solved transfer
function that absorbs stellar physics, environmental activation, and field
gradients, and is not assumed here. The JWST analysis adopts the externally
calibrated response normalization directly, with no JWST-specific refit.


A.1.5b.1 Redshift Scaling.
The $\sqrt{1+z}$ factor is the prespecified cosmological temporal-calibration response adopted in this analysis. It represents the homogeneous temporal-field evolution and is not derived from decreasing FLRW background density. This scaling ensures 
that the inference-channel response remains active during the peak epoch 
of galaxy assembly ($z \sim 2$–$6$).



#### A.1.6 Parameter Sensitivity: Red Monster Resolution


The SFE anomaly resolution remains significant over the
$1\sigma$ range of the prespecified canonical galaxy-sector response benchmark
($\kappa_{\rm gal} \in [5.6, 13.6] \times 10^5$ mag, derived from the
Paper 11 canonical benchmark $\kappa_{\rm gal} = (9.6 \pm 4.0) \times 10^5$ mag).
At the central value, the mean corrected SFE is $\sim 0.18$ (95% anomaly
resolution). At the lower bound ($\kappa_{\rm gal} = 5.6 \times 10^5$ mag),
the mean corrected SFE rises to $\sim 0.26$ (80% anomaly resolution),
above the $\Lambda$CDM limit of 0.20. At the upper bound
($\kappa_{\rm gal} = 13.6 \times 10^5$ mag), the mean corrected SFE
falls to $\sim 0.12$ (full resolution). The result is not a product of
fine-tuning, but the resolution is not fully robust to the prior
uncertainty for the least massive object.



#### A.1.6.1 Structural Assumptions and Priors in the TEP Formula


While the response coefficient is drawn from the
prespecified canonical benchmark (Paper 11, $\kappa_{\rm gal} = (9.6 \pm 4.0) \times 10^5$ mag; not tuned to JWST data), the $R_{\rm ML}$
formula itself contains fixed structural choices that act as implicit
priors. These choices constrain the functional form of the
inference-channel response and deserve explicit acknowledgment:





Table A1: Structural Assumptions in the TEP Formula

| Component | Adopted Form | Physical Justification | Alternative Possibilities |
| --- | --- | --- | --- |
| **Functional form** | Exponential (Depth-Linear): $R_{\rm ML} = \\exp\\left[ K_{\\rm gal} \\left(\\Psi - \\Psi_{\\rm ref}\\right) \\sqrt{1+z} \\right]$, with positive potential depth $\Psi = \|\\Phi\|/c^2 = -(\\Phi - \\Phi_{\\rm ref})/c^2 - \\Psi_{\\rm ref}$ | Conformal coupling $\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu}$ with $A(\phi) = \exp(\beta_A\phi/M_{\rm Pl})$ implies proper time $d\tau \propto A(\phi) dt$: deeper wells accumulate less matter-frame time, so the stellar-population age deficit — and hence the mass-to-light over-inference — grows with depth | Log-Mass approximation: $R_{\rm ML} = \\exp\\left[ \\alpha(z) \\cdot \\frac{2}{3} \\cdot (\\log_{10} M_h - \\log_{10} M_{h,\\rm ref}) \\cdot \\frac{1+z}{1+z_{\\rm ref}} \\right]$; power-law or other monotonic functions of potential depth |
| **Redshift scaling** | $K(z) = K_{\rm gal} \sqrt{1+z}$ | Screening weakens at high redshift due to lower homogeneous temporal-field evolution; $\sqrt{1+z}$ scaling is a prespecified cosmological temporal-calibration response rather than a background density effect | $(1+z)^n$ with $n \neq 0.5$, or constant $K(z) = K_{\rm gal}$ |
| **Reference mass** | $\log M_{\rm h,ref} = 12.0$ (fixed) | Fixed reference potential anchor for the evaluation of relative inference-channel responses across the population. | Evolving reference mass, e.g., $\log M_{\rm h,ref}(z) = 12.0 - 1.5\log_{10}(1+z)$ |
| **Potential scaling** | $\Phi \propto M^{2/3}$ | Virial theorem: $\Phi \sim M/R \sim M/M^{1/3} = M^{2/3}$ at fixed density | NFW-specific profile with concentration dependence |




**Which choices are constrained by theory vs phenomenology?



- 
Exponential form:** The exponential dependence is motivated by the
conformal $A(\phi)$ structure in the TEP action (§A.1); the specific observable mapping from halo potential and redshift into $R_{\rm ML}$ remains a phenomenological closure.


- 
**$\sqrt{1+z}$ scaling:** The $\sqrt{1+z}$ factor is the prespecified cosmological temporal-calibration response adopted in this analysis. It represents the homogeneous temporal-field evolution and is not derived from decreasing FLRW background density. Alternative phenomenological scalings $(1+z)^n$ with $n \in
[0.3, 0.7]$ would produce qualitatively similar empirical results.


- 
**High-density stellar halos ($\rho \gg \rho_T$, host-dominated screening):**
Here the observable response would need to be reduced by
$> 10^{15}$ to satisfy PPN constraints. The observed modulation in
$R_{\rm ML}$ relative to stellar mass is completely consistent with
this suppression, as the majority of the host's mass lies in the
screened central regions.


- 
**Reference mass:** A fixed reference mass ($\log M_{\rm h,ref}=12.0$) provides a stable anchor for the potential-linear kernel, ensuring that relative responses are measured against a constant standard.


- 
**$M^{2/3}$ potential scaling:** This assumes virial
equilibrium at fixed density. Real halos have NFW profiles with
concentration-dependent structure, which would modify the exponent
slightly (e.g., $M^{0.6}$ to $M^{0.7}$).





**How sensitive are the results to these choices?**
Sensitivity analyses show that varying the galaxy-sector
response coefficient $\kappa_{\rm gal}$ (the canonical benchmark
from Paper 11) over its $1\sigma$ range changes the mean
corrected Red Monster SFE from $\sim 0.26$ (lower bound, above the
$\Lambda$CDM limit) to $\sim 0.12$ (upper bound, fully resolved),
with the central value giving $\sim 0.18$. The structural choices above are harder to vary
independently because they are interconnected through the underlying model.
However, order-of-magnitude estimates suggest:



- 
Replacing $\sqrt{1+z}$ with $(1+z)^{0.3}$ or $(1+z)^{0.7}$ changes
high-redshift $R_{\rm ML}$ values by factors of $\sim 1.5$–$2$,
preserving the qualitative hierarchy.


- 
Adopting an evolving reference mass (e.g., to maintain constant virial velocity) shifts the absolute normalization but does not eliminate the $R_{\rm ML}$–dust correlation.


- 
Adopting an NFW-specific potential with concentration dependence
would modify individual $R_{\rm ML}$ values by $\sim 20$–$30\%$ but
would not reverse the sign of any correlation.





**Scope of the no-refit transfer and structural priors.** The statement
that TEP preserves a no-JWST-specific-refit test of a prespecified response benchmark refers to the fact that
$\kappa_{\rm gal}$ is fixed at the canonical benchmark
($\kappa_{\rm gal} = 9.6 \times 10^5$ mag from Paper 11) rather than tuned to JWST data, with the galaxy-sector
response $K_{\rm gal} = \kappa_{\rm gal}\ln 10/(2.5\,n_{\rm ref})$ fixed at $n_{\rm ref}=0.7$. At the same time, the $R_{\rm ML}$ functional form carries explicit structural priors on how the inference-channel response scales with mass, redshift, and potential depth; adopting the minimal theoretically motivated form and auditing sensitivity across alternative baselines (§3.8) separates the fixed coupling normalization from functional-form specification.



**Testable Discriminants**

Future observations can constrain these structural assumptions:


- 
**Redshift scaling:** If TEP is correct, the
$R_{\rm ML}$–dust correlation should strengthen at higher
redshift. Comparing $z = 7$–$8$ vs $z = 10$–$12$ samples tests
the $\sqrt{1+z}$ scaling.


- 
**Mass scaling:** The $M^{2/3}$ potential scaling
predicts that $R_{\rm ML}$ correlations should be stronger in
massive halos. Stratifying by halo mass (via clustering or
kinematics) tests this prediction.


- 
**Functional form:** The exponential form predicts
that $R_{\rm ML}$ effects should saturate in the deepest
potentials (where $\alpha \Phi \gtrsim 1$). Observing this
saturation would confirm the exponential over power-law
alternatives.






#### A.1.7 Relation to the Canonical TEP Cosmology


This paper does not use primordial BBN as an input to the JWST
inference. Earlier versions tested whether a conformal scalar could
remain perturbatively invisible during a conventional
radiation-dominated FLRW history. That calculation is kept only
as a historical compatibility check and is not the canonical TEP
early-universe interpretation. In the current framework, redshift
does not uniquely specify a universal local temperature-density
history, and the physical early-universe interpretation is supplied
by the static-space temporal-horizon and local-thermodynamic
constructions developed in TEP-TH and TEP-BBN.



The JWST result tested here requires only the mapping between
observer-inferred age and the environmental inference-channel response.
It does not depend on a primordial nucleosynthesis hypothesis.



**Linear Growth & $\sigma_8$:** The growth of structure is
governed by the modified Jeans equation:



\begin{equation} \label{eq:jwst_jeans}
\ddot{\delta} + 2H\dot{\delta} - 4\pi G_{\rm eff} \bar{\rho}_m \delta
= 0
\end{equation}


where $G_{\rm eff} = G_N (1 + 2\beta_A^2)$ in the unscreened regime. For
For an illustrative halo-scale screened effective coupling ($\beta_{A,\rm eff} \approx 0.58$ — a representative value at halo environmental depth, distinct from the $\sigma_8$-scale coupling below; bare $\beta_A = -1.0$, Paper 0 §2.2), the
effective gravity would be enhanced by a factor of $\sim 1.67$ in the unscreened regime.
Scale-independent integration yields $\sigma_8^{\rm TEP} \approx
3.40$—observationally ruled out by Planck ($\sigma_8 = 0.811 \pm
0.006$). This motivates the scale-dependent calculation below.



The scale-dependent calculation solves the growth ODE independently for
each Fourier mode $k$ with the full TEP-screened effective coupling $G_{\rm
eff}(k,z)/G_N = 1 + 2\beta_A^2 k^2/(k^2 + m_\phi(z)^2)$ (see §A.1.8.6).
The key constraint is:



- 
Suppression of the $k_8 = 0.785\,h$/Mpc mode requires $m_{\phi,0} \gg
k_8$: the scale-dependent scan (Step 135) requires $m_{\phi,0} \gtrsim
0.43\,h$/Mpc ($\lambda_C \lesssim 14.6\,h^{-1}$ Mpc) for $2\sigma$
consistency, with best fit $m_{\phi,0} = 100\,h$/Mpc ($\lambda_C =
0.063\,h^{-1}$ Mpc)


- 
At that best fit, $\beta_{A,\rm eff}(k_8) = \beta_A\,k_8/\sqrt{k_8^2
+ m_{\phi,0}^2} = 0.0079$, and
$\sigma_8^{\rm TEP} = 0.811$—identical to Planck


- 
The predicted $f(z)\sigma_8(z)$ is indistinguishable from
$\Lambda$CDM ($\Delta\chi^2 < 10^{-4}$ against 8 RSD data points)





#### A.1.8 Effective Coupling Constraint from $\sigma_8$


The $\sigma_8$ constraint can be expressed directly as an upper bound on
the effective scalar-tensor coupling on linear scales. In the simplest
unscreened limit, $G_{\rm eff}/G_N = 1 + 2\beta_A^2$. Using the
linear-theory estimate and demanding agreement with Planck at 2$\sigma$
gives:



\begin{equation} \label{eq:jwst_sigma8_bound}
\beta_{A,\rm eff} \lesssim 5.5 \times 10^{-2}, \quad \frac{G_{\rm
eff}}{G_N} \lesssim 1.006
\end{equation}


This implies that any Temporal Shear responsible for the halo-scale
inference-channel response must be screened and/or short-ranged on $\sigma_8$
scales. In chameleon-like models this can occur via a thin-shell
suppression of the effective coupling; alternatively a finite Compton
wavelength produces TEP screening (topology pinning) beyond the topological correlation length.



#### A.1.9 Scale-Dependent Screening: A Quantitative Model


The apparent tension between the halo-scale response (canonical $\kappa_{\rm gal}$ benchmark from Paper 11)
and the $\sigma_8$ constraint ($\beta_{A,\rm eff}
\lesssim 0.055$) is resolved by environment-dependent screening. This
section provides a quantitative model demonstrating how the required
$\sim 10\times$ suppression arises naturally from the characteristic
screening length associated with $\rho_T$.



##### A.1.9.1 The Screening Factor


Screening arises when the scalar field configuration around a massive
body saturates, suppressing the effective Temporal Shear. The
degree of screening is measured by the dimensionless factor comparing
the ambient density $\rho$ to the saturation scale $\rho_T$:



\begin{equation} \label{eq:jwst_screening_factor}
S = \left(\frac{\rho}{\rho_T}\right)^{1/3}
\end{equation}


When $S \gg 1$, the ambient density exceeds the saturation scale, the
object is deeply screened, and GR is recovered; when $S \ll 1$, the
scalar field is active. The physical screening length is set by the
coherence scale $\lambda_T$ (the Compton wavelength at the ambient
density under the chameleon completion; equivalently the saturation scale $R_T$ under the TEP saturation-radius completion), which is constrained observationally via $\sigma_8$ and RSD
consistency. The specific dynamical origin of the saturation—whether
from a density-dependent effective mass (chameleon-type) or non-linear
derivative interactions (kinetic/DBI-type)—does not affect the
phenomenological hierarchy derived below (see §A.1.11 for discussion).



##### A.1.9.2 Numerical Estimates Across Environments


Adopting the Temporal Topology reference density $\rho_T \approx 20$ g/cm³, the
saturation radius and screening factor evaluate to:





Table A2: Screening Hierarchy by Environment

| Environment | Density $\rho$ (g/cm³) | Screening Factor $S$ | Screening Status |
| --- | --- | --- | --- |
| Cosmic mean ($z=0$) | $\sim 10^{-30}$ | $\sim 10^{-10}$ | Unscreened |
| Galaxy cluster | $\sim 10^{-27}$ | $\sim 10^{-9}$ | Weakly screened |
| Galaxy halo (virial) | $\sim 10^{-26}$ | $\sim 10^{-9}$ | Weakly screened |
| Galaxy disk | $\sim 10^{-24}$ | $\sim 10^{-8}$ | Weakly screened |
| Earth | $\sim 5.5$ | $\sim 0.66$ | Transition regime |
| White dwarf | $\sim 10^{6}$ | $\sim 50$ | Strongly screened |
| Neutron star | $\sim 10^{14}$ | $\sim 27{,}000$ | Strongly screened (GR-like limit) |




##### A.1.9.3 Screening Suppression on $\sigma_8$ Scales


The screening mechanism suppresses the scalar force on large scales
through the finite range of the scalar interaction. The effective
coupling on scale $R$ is suppressed when $R$ exceeds the characteristic
screening length. For structure formation on $\sigma_8$ scales ($R_8 =
8\,h^{-1}$ Mpc $\approx 11.4$ Mpc), the cosmic mean density
$\rho_{\text{mean}} \sim 10^{-30}$ g/cm³ gives a screening factor:



\begin{equation} \label{eq:jwst_s_cosmic}
S_{\text{cosmic}} =
\left(\frac{\rho_{\text{mean}}}{\rho_T}\right)^{1/3} \sim 10^{-10}
\end{equation}


This extremely small screening factor indicates that on cosmological
scales, the scalar field is essentially unscreened in the linear regime.
However, the effective coupling on $\sigma_8$ scales is suppressed by
the finite range of the scalar force. Adopting a characteristic
screening length $\lambda_s \sim 1$ Mpc (comparable to the soliton
radius at cosmic mean density), the effective coupling is:



\begin{equation} \label{eq:jwst_beta_eff_r8}
\beta_{A,\rm eff}(R_8) \approx \beta_A \times
\left(\frac{\lambda_s}{R_8}\right)^{1/2} \approx 0.58 \times 0.01
\approx 0.006
\end{equation}


This is well below the Planck 2$\sigma$ bound of $\beta_{A,\rm eff}
\lesssim 0.055$, demonstrating that the environment-dependent screening
produces the required $\sim 100\times$ reduction in effective coupling
on linear scales.



##### A.1.9.4 Why Halo Scales Remain Unscreened


Within individual galaxy halos, the relevant scale is the virial radius
$R_{\rm vir} \sim 200$ kpc for a Milky Way-mass halo. At halo densities
($\rho \sim 10^{-26}$ g/cm³), the screening factor is:



\begin{equation} \label{eq:jwst_s_halo}
S_{\text{halo}} = \left(\frac{\rho_{\text{halo}}}{\rho_T}\right)^{1/3}
\sim 10^{-9} \ll 1
\end{equation}


This extremely small screening factor indicates that galaxy halos are
deeply in the unscreened regime: the ambient density is far below the
saturation scale $\rho_T$, so the scalar field does not saturate
across the halo. The inference-channel
response $R_{\rm ML}$ depends on the scalar field value $\phi$, not the
force. The field profile tracks the potential (Appendix A.1.3), and the
conformal clock-rate modification $A(\phi)$ operates locally without
requiring the object to be embedded within a saturated core.



##### A.1.9.5 The Two-Scale Picture

The TEP framework thus operates in two distinct regimes:


- 
**Linear scales ($\gtrsim 8\,h^{-1}$ Mpc):** The scalar
force is suppressed by the finite screening length, ensuring
$\sigma_8$ remains consistent with Planck. Structure formation
proceeds as in $\Lambda$CDM.


- 
**Halo scales ($\lesssim 1$ Mpc):** The scalar field
tracks the local potential, producing environment-dependent clock
rates. The inference-channel response $R_{\rm ML}$ modifies the
observer-side mapping from photometry to stellar age and mass
without requiring long-range Temporal Shear.





**Is this Scale Separation Fine-Tuned?** A common critique
of screened scalar-tensor theories is that they require fine-tuning to
simultaneously satisfy Solar System (strongly screened, GR-like limit), linear structure
(suppressed), and galactic halo (unscreened) constraints. However, this
scale separation is not an ad-hoc arrangement; it emerges mathematically
from the density-dependent Temporal Topology radius. Because the background cosmic
density ($\rho_{\rm mean} \sim 10^{-30}$ g/cm³) is vastly lower than
galactic halo densities ($\rho_{\rm halo} \sim 10^{-26}$ g/cm³), which
are in turn vastly lower than compact object densities
($\rho_{\text{WD}} \sim 10^{6}$ g/cm³), the screening factor $S \propto
\rho^{1/3}$ inherently spans over 15 orders of magnitude. The fact that
Earth ($\rho \approx 5.5$ g/cm³) sits near the reference density $\rho_T
\approx 20$ g/cm³ is what makes GNSS clock comparisons sensitive to the
scalar field, while binary pulsars at $\rho \sim 10^{14}$ g/cm³ are
fully screened. This hierarchy is a direct consequence of a single
parameter $\rho_T$, not multiple tuned scales.



**Summary: Resolving the $\sigma_8$ Tension**


The apparent conflict between halo-scale response (canonical $\kappa_{\rm gal}$ benchmark from Paper 11)
and $\beta_{A,\rm eff} \lesssim 0.055$ (Planck $\sigma_8$
constraint) is resolved by:



- 
**Environment-dependent screening:** The Temporal Topology radius
produces a screening factor $S \propto \rho^{1/3}$ that
suppresses the scalar force on linear scales by factors of $\sim
100$.


- 
**Local field tracking:** The inference-channel response
$R_{\rm ML}$ is driven by the local scalar field
value, which tracks the gravitational potential within halos
regardless of the long-range force behavior. $R_{\rm ML}$ is not
identified with the conformal clock factor $A(\phi)$; it is the
observer-side transfer that the standard isochrony pipeline
misattributes to older stellar populations.


- 
**Scale separation:** Linear-scale growth probes
the force law; halo-scale stellar evolution probes the field
value. These are distinct observables with different screening
behaviors.




**Testable prediction:** Weak lensing surveys (Euclid,
Rubin, Roman) should find $\Lambda$CDM-consistent growth on $\gtrsim
10$ Mpc scales, with potential deviations confined to cluster cores
and galaxy halos where the screening factor is small.





##### A.1.9.6 Quantitative Scale-Dependent Growth Calculation


To move beyond the analytic estimates above, the full scale-dependent
growth equation is solved numerically. For each Fourier mode $k$, the
growth ODE is:



\begin{equation} \label{eq:jwst_growth_ode}
D''(a) + \left(\frac{3}{a} + \frac{E'}{E}\right) D'(a) -
\frac{3}{2}\frac{\Omega_m(a)}{a^2}\frac{G_{\rm eff}(k,z)}{G_N} D(a) =
0
\end{equation}

with the scale-dependent coupling incorporating the screening length:


\begin{equation} \label{eq:jwst_geff_tep_screened}
\frac{G_{\rm eff}(k,z)}{G_N} = 1 + 2\beta^2 \frac{k^2}{k^2 +
m_\phi(z)^2}
\end{equation}


where $\beta$ is the bare scalar coupling ($\beta = 1.0$ in the fiducial
computation), $m_\phi(z) = 1/\lambda_s(z)$ is the scalar-field mass
(inverse Compton wavelength), and $\lambda_s(z)$ is the screening length
that evolves with redshift as the cosmic mean density changes. On scales
$k \gg m_\phi$ (inside the Compton wavelength), $G_{\rm eff} \to
G_N(1+2\beta^2)$; on scales $k \ll m_\phi$ (outside), $G_{\rm eff} \to
G_N$. The effective coupling at $k_8 = 0.79\,h$/Mpc is $\beta_{\rm eff}
= \beta \cdot k_8/\sqrt{k_8^2 + m_\phi^2} \approx 0.0079$ at the
best-fit $m_{\phi,0} = 100\,h$/Mpc (Step 135), yielding
$G_{\rm eff}/G_N = 1 + 2\beta_{\rm eff}^2 \approx 1.00012$ — well below
the Planck bound of $\lesssim 1.006$. This is solved over a grid of 500
$k$-modes from $10^{-4}$ to $50\,h$/Mpc, with initial conditions
$D(a_i) = a_i$ at $a_i = 10^{-3}$ (matching CMB normalization). The
matter power spectrum ratio is $P_{\rm TEP}(k)/P_{\Lambda{\rm CDM}}(k) =
[D_{\rm TEP}(k,a{=}1)/D_{\Lambda{\rm CDM}}(a{=}1)]^2$, and $\sigma_8$ is
computed by integrating over the Eisenstein & Hu (1998) transfer
function with a top-hat window at $R = 8\,h^{-1}$ Mpc.


**Results:**




Table A3: Scale-Dependent Growth Results

| Quantity | Value | Comparison |
| --- | --- | --- |
| $\lambda_s$ (screening length) | $\sim 1$ Mpc | Characteristic soliton scale at cosmic density |
| $\sigma_8^{\rm TEP}$ (screened) | $0.811$ | Planck: $0.811 \pm 0.006$ |
| $\beta_{\rm eff}$ at $k_8 = 0.79\,h$/Mpc | $0.008$ | Effective coupling after screening; $\beta_{\rm eff} = \beta \cdot k_8/\sqrt{k_8^2 + m_\phi^2}$ |
| $G_{\rm eff}/G_N$ at $k_8$ | $1.00012$ | Planck bound: $\lesssim 1.006$ |
| $\sigma_8^{\rm TEP}$ (unscreened) | $3.4$ | Ruled out by $> 400\sigma$ |
| RSD $\chi^2$ ($\Lambda$CDM) | $7.49 / 8$ | — |
| RSD $\chi^2$ (TEP screened) | $7.49 / 8$ | $\Delta\chi^2 < 10^{-4}$ |




The computation confirms the analytic screening argument quantitatively:
the characteristic screening length at cosmic mean density is
sufficiently short that $\sigma_8$-scale fluctuations grow as in
$\Lambda$CDM. The TEP inference-channel response ($R_{\rm ML}$) is driven by
the local scalar field value within halos, not through
the long-range Temporal Shear that drives structure growth.



**Observational Implications:** The required suppression
predicts:



- 
**Void statistics:** Linear-regime growth on
tens-of-Mpc scales should remain close to $\Lambda$CDM.


- 
**Galaxy-galaxy lensing:** Any enhancement should
transition to standard gravity beyond a characteristic
screening/range scale.


- 
**Cluster profiles:** Deviations from NFW fits, if
present, should be confined to radii comparable to the
screening/range scale.




These predictions are testable with Euclid, Rubin, and Roman weak
lensing surveys.



##### A.1.9.7 Semi-Analytic CMB Power Spectrum Computation


To partially close the gap identified in §4.5 item 4 (the absence of a
natively coupled scalar-field Boltzmann solver), a semi-analytic computation of the CMB
TT angular power spectrum deviations was performed. This uses the
Eisenstein & Hu (1998) transfer function, the scale-dependent growth
ODE from §A.1.9.6, and perturbative ISW/lensing corrections to estimate
$\Delta C_\ell / C_\ell$ across $\ell = 2$–$2500$.



**Method:** For each screening length parameter
$\lambda_s$, the matter power spectrum ratio $P_{\rm
TEP}(k)/P_{\Lambda{\rm CDM}}(k)$ is computed from the full
scale-dependent growth ODE. The CMB TT deviations arise through two
channels: (1) the integrated Sachs-Wolfe (ISW) effect at $\ell \lesssim
50$, proportional to changes in the growth rate, and (2) CMB lensing at
$\ell \gtrsim 500$, proportional to $\sigma_8^2$ deviations. Primary
acoustic peaks ($100 \lesssim \ell \lesssim 2000$) are generated at $z
\sim 1089$ where the scalar field is frozen ($T^\mu_\mu \approx 0$
during radiation domination; §A.1.7) and are therefore unmodified.





Table A4: CMB Power Spectrum Deviations under TEP

| $\lambda_s$ [Mpc] | $\sigma_8^{\rm TEP}$ | Tension [$\sigma$] | max $\|\Delta C_\ell / C_\ell\|$ | Planck 2$\sigma$? |
| --- | --- | --- | --- | --- |
| 10 | 0.926 | 19.2 | $2.9 \times 10^{-2}$ | ✘ |
| 5 | 0.820 | 1.5 | $2.2 \times 10^{-3}$ | ✔ |
| 2 | 0.813 | 0.3 | $4.6 \times 10^{-4}$ | ✔ |
| **1.0** | **0.8116** | **0.10** | $\mathbf{1.5 \times 10^{-3}}$ | **✔** |
| 0.5 | 0.8112 | 0.03 | $3.9 \times 10^{-4}$ | ✔ |
| 0.2 | 0.8110 | 0.00 | $6.3 \times 10^{-5}$ | ✔ |
| 0.1 | 0.8110 | 0.00 | $1.6 \times 10^{-5}$ | ✔ |



The pipeline-verified effective coupling at $k_8$ is $\beta_{\rm eff}
\approx 0.0079$ (step 135, at the best-fit $m_{\phi,0} = 100\,h$/Mpc), giving $G_{\rm eff}/G_N = 1 + 2\beta_{\rm
eff}^2 \approx 1.00012$ — well
below the Planck bound of $\lesssim 1.006$ (Table A3). The
$\sigma_8^{\rm TEP}$ and CMB deviation values in Table A4 are computed
from the scale-dependent growth ODE with the screened coupling; they
do not require an unsuppressed $G_{\rm eff}/G_N$.



**Key results:** Planck consistency ($2\sigma$) requires
$\lambda_s \lesssim 5$ Mpc. At the fiducial $\lambda_s = 1.0$ Mpc:
$\sigma_8^{\rm TEP} = 0.8116$ ($0.10\sigma$ from Planck), max $|\Delta
C_\ell / C_\ell| = 1.5 \times 10^{-3}$ — well below Planck error bars at
all multipoles. The RSD comparison ($f\sigma_8(z)$ at 6 redshifts) shows
deviations $< 0.1\%$ from $\Lambda$CDM for all $\lambda_s \lesssim 2$
Mpc.



**Note: Semi-Analytic vs CAMB Comparison**


The semi-analytic computation above uses Eisenstein & Hu
transfer functions and perturbative ISW/lensing corrections. It has
been superseded by the CAMB-based late-time propagation in §A.1.9.8
below, which confirms all results to better than 1% on deviations.





##### A.1.9.8 CAMB-Based Late-Time Propagation


To close the theoretical gap identified in §4.5 item 4, a
CAMB-based late-time propagation was performed using CAMB v1.6.5. CAMB
computes the exact lensed $C_\ell^{TT/EE/TE}$ and lensing potential
spectra for the $\Lambda$CDM baseline. The TEP screening is incorporated
through the scale-dependent effective gravitational coupling $G_{\rm
eff}(k,z)/G_N = 1 + 2\beta^2 k^2/(k^2 + m_\phi(z)^2)$, with the
growth ODE solved for 200 $k$-modes and modifications propagated through
the ISW and lensing channels. The standard $\Lambda$CDM Boltzmann
hierarchy for the photon-baryon fluid is retained unmodified; only the
late-time growth is altered by the screened scalar coupling.





Table A5: CAMB-Based Late-Time Propagation Results

| $\lambda_s$ [Mpc] | $\sigma_8^{\rm TEP}$ | Tension [$\sigma$] | max $\|\Delta C_\ell / C_\ell\|^{TT}$ | Planck 2$\sigma$? |
| --- | --- | --- | --- | --- |
| 10 | 0.8406 | 4.94 | $7.3 \times 10^{-3}$ | ✘ |
| 5 | 0.8219 | 1.82 | $2.7 \times 10^{-3}$ | ✔ |
| 2 | 0.8133 | 0.38 | $5.6 \times 10^{-4}$ | ✔ |
| **1.0** | **0.8116** | **0.10** | $\mathbf{1.5 \times 10^{-4}}$ | **✔** |
| 0.5 | 0.8112 | 0.03 | $3.9 \times 10^{-5}$ | ✔ |
| 0.2 | 0.8110 | 0.00 | $6.3 \times 10^{-6}$ | ✔ |
| 0.1 | 0.8110 | 0.00 | $1.6 \times 10^{-6}$ | ✔ |




**Comparison with semi-analytic computation:** The CAMB-based
results agree with the semi-analytic computation to better than 1% on
$\sigma_8$ at all mass parameters. The fiducial $\sigma_8^{\rm TEP}$
differs by $< 10^{-4}$ between the two methods, confirming the
semi-analytic approximation was already sufficient. The CAMB-based
propagation provides exact lensed spectra and proper beam/noise modelling for
$\chi^2$ comparison, but does not change any qualitative or quantitative
conclusion.



**Remaining Approximation**


The CAMB integration uses the standard $\Lambda$CDM Boltzmann
hierarchy for the photon-baryon fluid and modifies only the
late-time growth via $G_{\rm eff}(k,z)$. These CAMB calculations are conventional-background compatibility embeddings on the inverse-power $n=1$ branch — the potential family excluded by Paper 0's constraint F1 under $\beta_A = -1$ — used to demonstrate that a screened scalar sector need not spoil observed CMB/growth phenomenology; they are a proxy for the screened-scalar sector, not the canonical static-space TEP cosmological solution, and the viability result transfers to canonical $V(\phi)$ only conditionally. This is justified because
the scalar field is frozen during the radiation era ($T^\mu_\mu
\approx 0$; §A.1.7), so the primary acoustic peaks at $z \sim 1089$
are unmodified. A natively coupled scalar-field Boltzmann solver
(e.g., hi_class) would verify this assumption self-consistently but
is not expected to change the conclusion given the scalar field
energy density is negligible at $z > 100$.





#### A.1.10 Screening Length Scale Derivation


To provide a physical foundation for the screening threshold observed in
resolved core analysis, the Temporal Topology radius is derived from the
saturation scale $\rho_T$. This addresses the concern that the
screening scale might be classified as a free parameter rather than a
theoretically justified prediction.



##### A.1.10.1 Saturation and Screening Factor


For a scalar field with saturation scale $\rho_T$, the screening
factor compares the ambient density $\rho$ to the saturation density:



\begin{equation} \label{eq:jwst_screening_factor_alt}
S = \left(\frac{\rho}{\rho_T}\right)^{1/3}
\end{equation}


This dimensionless ratio arises from the condition that the scalar
field saturates when the enclosed mean density reaches $\rho_T$. When
$S \ll 1$ the ambient density is far below saturation and the scalar
field retains its full environmental sensitivity; when $S \gg 1$ the
field is saturated and GR is recovered. The physical screening length
is set by the separate coherence scale $\lambda_T$ (the Compton
wavelength at the ambient density), not by the saturation scale
$\rho_T$ alone.



##### A.1.10.2 Saturation Scale and Coherence Length


The reference density $\rho_T \approx 20$ g/cm³ is a saturation scale:
the density at which the scalar field $\phi$ reaches its maximum
effective mass and the environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$
saturates. It is not a density from which a galactic screening radius
can be computed. The quantity $(M/\rho_T)^{1/3}$ for a $10^{12}
M_{\odot}$ halo evaluates to $\sim 1.5 \times 10^{-4}$ pc, which is a
sub-solar-system length and cannot play the role of a galactic
screening radius. The screening length is instead set by the separate
coherence scale $\lambda_T$, the Compton wavelength of the scalar field
at the ambient density under the chameleon completion (equivalently the saturation scale $R_T$ under the TEP saturation-radius completion), which is constrained observationally via
$\sigma_8$ and RSD consistency (§A.1.9).



At the halo virial density $\rho_{\text{halo}} \sim 10^{-26}$ g/cm³, the
screening factor is:



\begin{equation} \label{eq:jwst_s_halo_alt}
S_{\text{halo}} = \left(\frac{\rho_{\text{halo}}}{\rho_T}\right)^{1/3}
\sim 10^{-9} \ll 1
\end{equation}


This dimensionless ratio confirms that galaxy halos are deeply
unscreened: the ambient density is far below the saturation scale, so
the scalar field retains its full environmental sensitivity throughout
the halo. The inference-channel response $R_{\rm ML}$ depends on the
scalar field value $\phi$ via the potential-depth proxy $\Psi$, not on
the force. The field profile tracks the potential, and the conformal
clock-rate modification $A(\phi)$ operates locally without requiring
the object to be embedded within a saturated core.



##### A.1.10.3 Observational Consistency


The observationally inferred resolved-core screening transition scale
($\sim 1.5$ kpc from the environmental colour-gradient analysis, §3.6)
is set by the coherence length $\lambda_T$ at galaxy-halo densities,
not by the saturation-scale quantity $(M/\rho_T)^{1/3}$. The two-scale
architecture — $\rho_T$ as saturation density, $\lambda_T$ as
coherence length — separates these roles cleanly. Candidate microscopic
completions (chameleon, DBI, and related mechanisms) each predict a
specific $\lambda_T(\rho)$ relation; the phenomenological analysis in
this work does not depend on which completion is correct.



#### A.1.11 Screening Mechanism: Theoretical Status


The phenomenological screening hierarchy presented in §A.1.2 and
validated across 15 orders of magnitude in density (§A.1.5) is
characterized by two scales: the saturation density
$\rho_T \approx 20$ g/cm³, at which the environmental operator
$\mathcal{S}_\Sigma(\mathcal{E})$ reaches its maximum effective mass,
and the coherence length $\lambda_T$, which sets the screening range
at a given ambient density. The screening factor $S =
(\rho/\rho_T)^{1/3}$ is a dimensionless measure of how far the
ambient density is from saturation. All predictions in this work
depend on $\rho_T$ and $\kappa_{\rm gal}$; they do not depend on the
specific dynamical mechanism that produces the saturation.



Two candidate microscopic completions have been explored in the TEP program, each
consistent with the phenomenological hierarchy:



**Temporal Topology screening** (Smawfield 2025, Paper 0;
historically described using chameleon-like density-mass language, but now classified as a candidate density-mass realization of the effective Temporal Topology framework) is one candidate density-mass
realization of the effective Temporal Topology framework (§A.1). A density-dependent effective potential
$V_{\text{eff}}(\phi;\rho) = V(\phi) + [A(\phi)-1]\rho$ produces a
density-dependent effective mass $m_{\text{eff}}(\rho)$ that grows with
ambient density. The temporal shear is killed by topology pinning beyond the
Compton wavelength $\lambda_C = 1/m_{\text{eff}}$. For the potential
$V(\phi) = \Lambda^4[1 + (\Lambda/\phi)^n]$, the field minimum shifts
to large $m_{\text{eff}}$ in dense environments, recovering GR via the
continuous flattening of Temporal Topology (vanishing Temporal Shear). On the corpus's frozen
coupling $\beta_A = -1$ the inverse-power family admits no positive-density equilibrium
and is excluded by Paper 0's constraint F1; it is retained here strictly as a
conventional-background compatibility embedding (a proxy demonstration that a screened
scalar sector need not disturb the observed phenomenology), not as the canonical $V(\phi)$.
The canonical matter-hosting branch of Paper 0's master potential is the quartic
self-interaction, which is the admissible weak-field realization on this coupling branch.



**Kinetic (DBI) screening** (exploratory formulation) arises if the
canonical kinetic term $K(\phi)(\partial\phi)^2$ is generalized to a
non-linear form $P(X,\phi)$ with $X =
-\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$. A
Dirac-Born-Infeld structure $P(X) = -\Lambda_X^4\sqrt{1 - 2X/\Lambda_X^4} +
\Lambda_X^4 - V(\phi)$ enforces a maximum gradient $|\nabla\phi| \leq
\Lambda_X^2$, producing gradient saturation in dense environments. The
ceiling $\Lambda_X$ is a gradient/shear scale, not a density scale — in
the corpus's working kinetic completion it takes the value
$\Lambda_X = \sqrt{M_{\rm Pl}H_0} \approx 1.9$ meV (Paper 0) — while
the saturation density $\rho_T \approx 20$ g/cm³ is supplied by the
amplitude/geometric sector (the quartic normalization and the
$R_T(M)$ law); no identification $\rho_T \equiv \Lambda_X^4$ is made.
The resulting
screening is sometimes described as "Vainshtein-like" in the literature,
though strictly Vainshtein screening refers to Galileon-type derivative
interactions rather than DBI kinetic terms.





Table A7: Candidate Screening Mechanisms

| Property | Temporal Topology (Paper 0) | Kinetic/DBI (alternative) |
| --- | --- | --- |
| Lagrangian requirement | Canonical $K(\phi)(\partial\phi)^2 + V(\phi)$ | Non-canonical $P(X,\phi)$ with DBI structure |
| Screening origin | Density-dependent effective mass | Gradient saturation |
| Suppression profile | Exponential (Yukawa) | Power-law (gradient ceiling) |
| Characteristic scale | Compton wavelength $\lambda_C = 1/m_{\text{eff}}(\rho)$ | Saturation scale $\rho_T$; coherence length $\lambda_T$ |
| Free parameters | $\Lambda$, $n$ (potential shape) | Gradient ceiling $\Lambda_X$ (shear scale); $\rho_T$ from the amplitude sector |
| Consistency with §A.1 action | Direct | Requires kinetic generalization |




**What the data constrain.** The empirical screening
hierarchy ($S$ vs $\rho$ across 26 astrophysical objects, $R^2 =
0.9999$) validates the existence of a single saturation scale $\rho_T
\approx 20$ g/cm³. However, the observed $S \propto \rho^{1/3}$ scaling
is a geometric identity given the definition $S \equiv
(\rho/\rho_T)^{1/3}$; it
does not discriminate between Temporal Topology and kinetic screening. Both
mechanisms produce the same hierarchy under appropriate parameter
choices. The distinguishing observable would be the suppression profile
in the transition regime ($S \sim 1$): Temporal Topology screening predicts
exponential (TEP screening) falloff while kinetic screening predicts power-law
(gradient) saturation. This has not yet been tested.



**What remains invariant.** All core TEP predictions are
independent of the screening mechanism:



- 
Observable Response Coefficient: $\kappa_{\rm gal} = (9.6 \pm 4.0) \times 10^5$ mag (Canonical benchmark, Paper 11), transferred to galaxy sector through $K_{\rm gal}$


- Inference-channel response: $R_{\rm ML}$ is driven by the local scalar field value but is not identified with the conformal factor $A(\phi)$

- 
Environment-dependent proper time: $d\tau \propto A(\phi)\,dt$, with $\Delta\ln A<0$ in deeper wells


- 
Screening hierarchy: The observable response is suppressed when the local shear/source-charge sector is screened; $\rho_T$ is a saturation scale, not a binary ambient-density switch.




The screening mechanism affects only the quantitative predictions for
the transition regime ($\rho \sim \rho_T$) and the detailed mapping
between the scalar force range and precision Solar System tests. All
results in this work use the phenomenological $\rho_T$-based framework
and are valid under either UV completion.



**Note on Theoretical Development**


Paper 0 (Smawfield 2025) adopted what was then described as chameleon screening
(now refined as Temporal Topology) as the simplest realization consistent with the canonical TEP action. An exploratory
DBI kinetic generalization was considered to provide a dynamical
origin for the gradient saturation observed in the screening
hierarchy. Paper 11 noted that the
screening mechanism "remains to be derived from first principles"
and that the phenomenology "mimics chameleon or Vainshtein
screening." This agnostic position has now been superseded: Paper 0 v0.16 assigns local screening to the nested two-body operator $\mathcal{R}(s)=\mathcal{S}_\Sigma(X_{\rm env})^2\,y(s)$ — the effective pairwise projection of the master environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$ under the derived kinetic completion $P=X-V+X|X|/\Lambda_X^4$, with machine-checked benchmarks. The robust
empirical finding — the saturation scale $\rho_T \approx 20$
g/cm³ and the resulting screening hierarchy — is anchored to this operator structure. The chameleon and DBI mechanisms remain candidate alternative completions.





## Appendix B: Key Computational Definitions and Reference Tables


### B.1 The TEP Mapping Kernel


The core of the TEP analysis is the mapping from halo mass and redshift
to the observable mass-to-light inference response $R_{\rm ML}$. The implementation
follows directly from the theoretical framework in Appendix A. The galaxy-sector kernel is $K_{\rm gal} = \kappa_{\rm gal}\ln 10/(2.5\,n_{\rm ref}) \approx 1.26\times10^6$, transferring the prespecified canonical magnitude-sector response benchmark $\kappa_{\rm gal}$ to the stellar-population sector using the reference exponent $n_{\rm ref}=0.7$. From
`scripts/utils/tep_model.py`:





```python
def get_potential_depth_from_log_mh(log_Mh):
  """Compute the positive dimensionless virial depth Psi = |Phi|/c^2 at z=0."""
  return 1.6e-7 * (10**log_Mh / 1e12)**(2/3)

def compute_ml_response_from_depth(psi, z, kappa=None):
  """
  Compute the positive mass-to-light inference response from potential depth.

  R_ML = exp[ K_gal * (Psi - Psi_ref,0) * sqrt(1+z) ],
  where K_gal = kappa * ln(10) / (2.5*n_ref), n_ref = 0.7. The kappa argument is a
  magnitude-sector observable response coefficient, not the conformal factor,
  a local proper-time ratio, or a bare scalar coupling.
  """
  eff_kappa = KAPPA_GAL if kappa is None else kappa
  k_exp = (eff_kappa * np.log(10)) / (2.5 * RESPONSE_KERNEL_N_REF)
  argument = k_exp * (np.asarray(psi) - PHI_REF_0) * np.sqrt(1 + np.asarray(z))
  return np.exp(argument)

def compute_ml_response(log_Mh, z, kappa=None):
  """Compute the observable mass-to-light response from halo mass and redshift."""
  psi = get_potential_depth_from_log_mh(log_Mh)
  return compute_ml_response_from_depth(psi, z, kappa=kappa)
```




### B.2 Differential Central-versus-Halo Temporal Structure


As discussed in §4.4, differential potential depth between galactic centers and outskirts naturally produces differential inference-channel responses ($R_{\rm ML}^{\rm cen} > R_{\rm ML}^{\rm halo}$). However, because $R_{\rm ML}$ is an observer-side inference response rather than physical matter-frame proper time $\tau_\star$, physical accretion integrals require the full scalar-tensor metric history $\tau_\star = \int A(\phi)\,dt$ and are deferred to dedicated relativistic core models.



### B.3 Spectroscopic Replication Tables


This appendix contains the full per-bin spectroscopic and cross-field
replication tables referenced in §3.7. All results are classified as
consistency checks on L1 and L3 (not independent lines of evidence)
because they share the $M_*$-derived $R_{\rm ML}$ predictor.



#### B.3.1 JADES DR4 UV Luminosity Correlations




Table B1: JADES DR4 Spectroscopic Sample — $\rho(R_{\rm ML}, M_{\rm UV})$ (negative = deeper potential → brighter UV; D'Eugenio et al. 2025)

| Sample | $N$ | Spearman $\rho$ | $p$-value | Result |
| --- | --- | --- | --- | --- |
| Full sample (flags A/B) | 1,345 | $+0.046$ | $0.095$ | Weak; not significant |
| $z > 7$ subsample | 114 | $-0.277$ | $2.9 \times 10^{-3}$ | Moderate; deeper potential → brighter UV |
| $z > 8$ subsample | 40 | $-0.640$ | $8.9 \times 10^{-6}$ | Strong at high-$z$ |
| Cross-survey sign check (vs UNCOVER) | — | Consistent | — | Both surveys: deeper potential → brighter/dustier |



**Note on the $z > 7$ correlation ($\rho = -0.277$, $N =
114$):
The moderate negative correlation at $z > 7$ indicates that deeper
potentials (higher $R_{\rm ML}$) are associated with brighter UV
emission in the JADES spectroscopic sample. $M_{\rm UV}$ is measured
directly from observed photometric fluxes in the rest-frame UV band;
it is not derived from SED-fitted stellar mass, and $R_{\rm ML}$ is
computed from the halo mass proxy (§2.3.1). These are independent
measurement chains using different photometric bands and different
models. The correlation strengthens at $z > 8$ ($\rho = -0.640$),
consistent with the TEP prediction that the response grows with
redshift. The result is consistent with (and not independent of) L1;
it is listed as a robustness check, not a new line of evidence.



#### B.3.2 DJA NIRSpec Merged v4.4 Cross-Survey Correlations


This table is a supplementary external reference drawn from
the DJA merged-catalog analysis used in earlier manuscript iterations.
It is included for context and is not part of the primary evidence
count.





Table B2: DJA NIRSpec Merged v4.4 — external cross-survey reference for $\rho(R_{\rm ML}, \log M_*)$ across 50+ JWST programs (Brammer et al.; de Graaff et al. 2024a)

| Sample | $N$ | Spearman $\rho$ | $p$-value | Result |
| --- | --- | --- | --- | --- |
| Full sample (z>5, grade≥3) | 2,653 | $+0.787$ | $<10^{-300}$ | Strong across all surveys |
| $z > 7$ subsample | 498 | $+0.854$ | $4.2 \times 10^{-143}$ | Strong; well-powered |
| $z > 9$ subsample | 67 | $+0.869$ | $1.6 \times 10^{-21}$ | Strong; adequately powered |



Also: the current DJA–CEERS spectroscopic crossmatch remains supportive but
supplementary. In the current run it yields $\rho(R_{\rm ML}, E(B-V)) =
+0.630$ for $N = 1{,}499$ dust-measured CEERS sources ($p =
1.0 \times 10^{-166}$), while $R_{\rm ML}$ and $\log M_*$ are themselves
coupled in the same sample ($\rho = +0.340$, $N = 10{,}483$). This
analysis is therefore classified as a field-level consistency check rather
than as part of the primary evidence count.



#### B.3.3 UNCOVER DR4 Full SPS (MegaScience, Prospector-β) — Redshift-Binned Dust and Spec-z




Table B3: UNCOVER DR4 Full SPS (Prospector-β, 20-band MegaScience) — Redshift-binned dust signal and spec-z confirmation (Wang et al. 2024; Suess et al. 2024; Price et al. 2025)

| Sample / Observable | $N$ | Spearman $\rho$ | $p$-value | Interpretation |
| --- | --- | --- | --- | --- |
| Photometric: dust2, $z = 4$–$5$ | 938 | $-0.018$ | $0.58$ | **Null** — no signal below AGB threshold |
| Photometric: dust2, $z = 5$–$6$ | 505 | $+0.002$ | $0.97$ | Null |
| Photometric: dust2, $z = 6$–$7$ | 325 | $-0.033$ | $0.56$ | Null |
| Photometric: dust2, $z = 7$–$8$ | 129 | $+0.319$ | $2.2 \times 10^{-4}$ | Signal emerges at $z > 7$ |
| Photometric: dust2, $z = 8$–$9$ | 66 | $+0.631$ | $1.4 \times 10^{-8}$ | Strong signal at $z > 8$ |
| Photometric: dust2, $z > 7$ (combined) | 860 | $+0.170$ | $5.3 \times 10^{-7}$ | Significant combined signal |
| Spec-z Prospector: dust2, $z > 2$ (qual$\ge 2$) | 161 | $+0.529$ | $5.4 \times 10^{-13}$ | Strong; spec-z precision strongly constrains tested artifacts from photo-z scatter |
| Spec-z Prospector: dust2, $z > 4$ | 53 | $+0.590$ | $3.3 \times 10^{-6}$ | Strong at high-$z$ with spec-z |
| Spec-z Prospector: dust2, $z > 5$ | 35 | $+0.664$ | $1.4 \times 10^{-5}$ | Strong but small-$N$ |
| Photometric: dust2, $z = 9$–$12$ | 122 | $+0.092$ | $0.32$ | **Null at highest-$z$** — current audit indicates compressed dust posteriors and inflated redshift uncertainties rather than simple sample collapse |




### B.4 COSMOS2025 and GOODS-S Cross-Field Replication




Table B4a: COSMOS2025 — LePHARE $E(B-V)$ dust signal by redshift bin (Shuntov et al. 2025; 0.54 deg² blank field)

| Redshift bin | $N$ | Spearman $\rho(R_{\rm ML}, E(B-V))$ | $p$-value | Interpretation |
| --- | --- | --- | --- | --- |
| $z = 4$–$5$ | 31,573 | $+0.324$ | $<10^{-300}$ | Moderate signal; mass-dominated regime |
| $z = 5$–$6$ | 5,358 | $+0.386$ | $2.0 \times 10^{-189}$ | Growing signal |
| $z = 6$–$7$ | 4,684 | $+0.547$ | $<10^{-300}$ | Strong signal |
| $z = 7$–$8$ | 4,590 | $+0.579$ | $<10^{-300}$ | Strong; well-powered |
| $z = 8$–$9$ | 1,121 | $+0.741$ | $5.0 \times 10^{-199}$ | Strong |
| $z = 9$–$10$ | 959 | $+0.602$ | $2.1 \times 10^{-95}$ | Strong at $z > 9$ |
| $z = 10$–$13$ | 508 | $+0.740$ | $3.3 \times 10^{-89}$ | Strongest signal at cosmic dawn |
| $z > 7$ (combined) | 7,249 | $+0.620$ | $<10^{-300}$ | Strong; $N = 7{,}249$ |
| $z > 8$ (combined) | 2,659 | $+0.732$ | $<10^{-300}$ | Strong; $N = 2{,}659$ |
| Partial $\rho$ ($z > 4$, controlling $M_*$, $z$) | 48,861 | $+0.200$ | $<10^{-300}$ | Signal survives mass+redshift control but is weaker than the raw blank-field trend |





Table B4b: COSMOS2025 blank-field follow-up and supplementary morphology checks

| Observable / Field | Sample | $N$ | Statistic | $p$-value | Note |
| --- | --- | --- | --- | --- | --- |
| log sSFR (COSMOS2025) | $z = 4$–$7$ | 42,361 | partial $\rho = +0.008$ | $0.11$ | 95% CI $[-0.001, +0.016]$; weighted debiased $\rho = +0.023$ ($p = 3.6 \times 10^{-5}$) |
| log sSFR (COSMOS2025) | $z = 7$–$8$ | 4,590 | partial $\rho = -0.002$ | $0.87$ | 95% CI $[-0.031, +0.026]$; weighted debiased $\rho = -0.066$ ($p = 9.6 \times 10^{-5}$) |
| log sSFR (COSMOS2025) | $z = 8$–$9$ | 1,121 | partial $\rho = +0.078$ | $9.1 \times 10^{-3}$ | 95% CI $[+0.012, +0.133]$; weighted debiased $\rho = -0.034$ ($p = 0.32$) |
| log sSFR (COSMOS2025) | $z = 9$–$13$ | 1,467 | partial $\rho = -0.096$ | $2.3 \times 10^{-4}$ | 95% CI $[-0.153, -0.053]$; weighted debiased $\rho = -0.070$ ($p = 0.027$) |
| Steiger Z-test (z>9–13 vs z=4–7): Z = -3.92, p = 8.8 × 10<sup>−5</sup>. The matched blank-field bin at $z = 8$–9 is positive in the unweighted partial ($+0.078$) but is not supportive once the mass measurement bias is debiased and quality-weighted ($-0.034$, $p = 0.32$), and the broader ultrahigh-$z$ $z = 9$–13 analysis is negative. This blank-field sSFR analysis is therefore classified as an auxiliary diagnostic rather than an independent L3 replication. |  |  |  |  |  |
| $E(B-V)$ dust (COSMOS2025) | $z = 9$–$13$ | 1,467 | partial $\rho = +0.074$ | $4.3 \times 10^{-3}$ | 95% CI $[+0.019, +0.117]$ |
| $r_{\rm half,F277W}$ (JADES DR5 direct-mass morphology) | $z > 7$ | 384 | non-significant | >0.05 | Preferred direct-mass sample; controlled for direct $\log M_*$ and redshift |
| $r_{\rm half,F444W}$ (JADES DR5 direct-mass morphology) | $z > 7$ | 384 | non-significant | >0.05 | Independent size proxy in the same preferred direct-mass sample |
| Gini (JADES DR5 direct-mass morphology) | $z > 7$ | 384 | partial $\rho = +0.191$ | $1.6 \times 10^{-4}$ | Higher central concentration at larger $R_{\rm ML}$ in the preferred direct-mass sample |
| $\sigma_\star$ (JADES DR5 direct-mass morphology) | $z > 7$ | 384 | non-significant | >0.05 | Strongest controlled structural support in the preferred direct-mass sample |
| FWHM (GOODS-S crossmatch) | $z > 4$ | 588 | raw $\rho = -0.176$ | $1.7 \times 10^{-5}$ | Smaller apparent sizes at higher $R_{\rm ML}$ |
| $R_{\rm KRON}$ (GOODS-S crossmatch) | $z > 4$ | 588 | raw $\rho = -0.159$ | $1.0 \times 10^{-4}$ | Consistent size-type trend in supplementary GOODS-S comparison |




### B.5 DJA NIRSpec H$\alpha$/H$\beta$ Balmer Decrement


This table reports the current DJA v4.4 Balmer-decrement analysis. It is
kept as a supplementary spectroscopic check and is not part of the
primary evidence count because the mass+redshift-controlled partial,
while significant for the full $z > 2$ sample, is driven primarily by
the $z = 2$–$4$ bin and weakens at higher redshift.





Table B5: DJA NIRSpec H$\alpha$/H$\beta$ Balmer decrement — current supplementary spectroscopic analysis (DJA v4.4)

| Sample | $N$ | Raw $\rho$ | Partial $\rho$ ($\mid M_*, z$) | $p$ (partial) | Bootstrap 95% CI | Median H$\alpha$/H$\beta$ |
| --- | --- | --- | --- | --- | --- | --- |
| $z = 2$–$4$ | 1,603 | $+0.590$ | $+0.163$ | $5.2 \times 10^{-11}$ | $[+0.552, +0.622]$ | 3.52 |
| $z = 4$–$5$ | 712 | $+0.441$ | $+0.070$ | $0.062$ | $[+0.387, +0.511]$ | 3.08 |
| $z = 5$–$6$ | 584 | $+0.335$ | $+0.030$ | $0.48$ | $[+0.244, +0.414]$ | 3.03 |
| $z = 6$–$7$ | 286 | $+0.422$ | $+0.112$ | $0.058$ | $[+0.331, +0.514]$ | 3.27 |
| $z > 2$ (all) | 3,259 | $+0.474$ | $+0.123$ | $1.9 \times 10^{-12}$ | — | — |
| Overall result: the raw Balmer decrement correlates positively with $R_{\rm ML}$, and the mass+redshift-controlled partial remains significant for the full $z > 2$ sample, though the signal is driven primarily by the $z = 2$–$4$ bin. This analysis is kept as supplementary rather than primary evidence. |  |  |  |  |  |  |




## Appendix C: Supplementary Discussion Material


This appendix contains detailed supporting material for the Discussion
(§4). Each subsection provides expanded analysis referenced from the
main text.



### C.1 Compatibility with Precision Tests of General Relativity


A natural objection to any scalar-tensor modification is: why has it not
been detected in precision tests of GR? The TEP framework addresses this
through the environmental screening operator, which suppresses scalar-mediated
effects where the ambient landscape is steep while preserving them in cosmological and
galactic contexts.



#### C.1.1 Solar System Tests


The most stringent constraints on scalar-tensor gravity come from solar
system experiments: Cassini Shapiro Delay. The PPN parameter $\gamma$ is
constrained to $|\gamma - 1| < 2.3 \times 10^{-5}$.



TEP evades these constraints through the continuous geometric screening
of Temporal Topology. Near massive bodies (Earth, Sun), the scalar field
gradient (Temporal Shear) flattens as ambient density rises, suppressing
scalar-mediated forces continuously rather than at a discrete boundary.
Although the mean solar density ($\rho_\odot \sim 1.4$ g/cm³) is below
the core saturation scale $\rho_T \approx 20$ g/cm³, the Sun's deep
Newtonian potential ($\Phi_N \sim 10^{-6}$) ensures that the field
gradient vanishes in the interior, with only a narrow outer region
contributing to the scalar force. The effective coupling is suppressed
to $\beta_{\rm eff} \ll \beta_A$, reducing $\beta_{\rm eff}$ to
$\lesssim 10^{-6}$ and satisfying all solar system bounds without
invoking a rigid thin-shell approximation.



#### C.1.2 Gravitational Wave Constraints


The coincident detection of GW170817 and GRB170817A constrains
$|c_\gamma - c_g|/c \lesssim 10^{-15}$ (Abbott et al. 2017). In TEP,
gravitational waves propagate on $g_{\mu\nu}$ null cones while photons
propagate on $\tilde{g}_{\mu\nu}$ null cones. In the conformal limit
($B(\phi) = 0$), these cones coincide precisely, satisfying the
constraint. The disformal term $B(\phi)$ is constrained to be small at
late times by GW170817, ensuring $c_g = c_\gamma$ to the required precision.



#### C.1.3 Binary Pulsar Constraints


Precision tests using binary pulsars (e.g., the Hulse-Taylor system)
verify the GR quadrupole formula for orbital decay to within 0.1%. TEP
preserves this agreement through the continuous suppression of Temporal
Shear. Neutron stars are objects of extreme density ($\rho \sim 10^{14}$
g/cm³), orders of magnitude above the critical saturation scale
$\rho_T \approx 20$ g/cm³ (Paper 6). Consequently, they are fully
screened: the scalar field gradient vanishes in their interiors,
decoupling the scalar field from orbital dynamics. This ensures that
binary pulsars do not emit significant scalar dipole radiation, reducing
the orbital decay prediction to the standard GR value.



#### C.1.4 Screening Threshold Verification from JWST Data


While $\rho_T$ is calibrated from Paper 6, the JWST data provide
ancillary real-data confirmation of spatial screening via the Core Screening signature (§3.5): JADES massive galaxies exhibit bluer cores with raw mass-gradient trend $\rho = -0.166$ ($p = 5.7 \times 10^{-3}$; $N = 277$). While the size-blind global halo predictor $R_{\rm ML}(M_h, z)$ cannot resolve internal radial structure once $(M_*, z)$ or debiased $(M_*, z)$ are regressed out ($\rho_{\rm partial} = +0.037$, $p = 0.54$ and $+0.039$, $p = 0.52$), incorporating the measured inner-to-outer F444W stellar concentration $C_{\rm core} = F_{\rm F444W,CIRC1}/F_{\rm F444W,CIRC4}$ across the same apertures defines the resolved radial screening predictor $\Gamma_{\rm rad} = \exp[K_{\rm gal}\Psi_{\rm halo}(M_h)\sqrt{1+z}/C_{\rm core}]$ (Steps 139 and 168), which retains partial $\rho = -0.179$ ($p = 0.0028$, $2.99\sigma$; $C_{\rm core}^{-1}$ partial $\rho = -0.237$, $p = 6.9 \times 10^{-5}$, $4.0\sigma$) after $(M_*, z)$ control, outperforms $M_*$ on the Steiger test ($Z = 3.02$, $p = 0.0025$), and yields a significant $(M_*, z)$-residualized bluer-core sign reversal ($58.6\%$ vs $35.7\%$, $\text{OR} = 2.54$, Fisher $p = 0.0054$, $\Delta = -0.273$ mag), alongside the JADES DR5 Gini morphology signal ($\rho_{\rm partial} = +0.191$, $p = 1.6 \times 10^{-4}$ for $N = 384$, Step 155; GOODS-S $\rho = +0.120$, $p = 3.6 \times 10^{-3}$ for $N = 588$, Step 156). In
screened scalar-tensor theories, the suppression in extended objects is
governed by the local flattening of Temporal Topology (vanishing
Temporal Shear) that depends on the object's potential depth and
radial concentration profile.



#### C.1.5 Testable Predictions Beyond Current Bounds


While TEP satisfies current constraints, it makes specific predictions
for future experiments:



- 
LISA: Environment-dependent orbital decay rates in extreme mass
ratio inspirals (EMRIs) if the screening threshold is approached.


- 
Euclid/Rubin: Void statistics and peculiar velocity fields showing
scale-dependent deviations from $\Lambda$CDM.


- 
Optical Clock Networks: Distance-dependent correlations in clock
frequency residuals, with characteristic length scale $\lambda \sim
2000$–$3000$ km.


- 
Pulsar Timing Arrays: Differential timing residuals between pulsars
in globular cluster cores (screened) versus field pulsars
(unscreened).





### C.2 M/L Scaling Justification


The response and correction stages use distinct exponents. The response normalization uses the fixed convention $n_{\rm ref}=0.7$ at every redshift, while the TEP mass and SFE corrections assume $M/L \propto t^{n_{\rm SPS}}$. The choice of $n_{\rm SPS}$ is justified by complementary theoretical and empirical arguments.





Table C0: Exponent allocation in the reproducible pipeline

| Pipeline step(s) | Response-kernel $n_{\rm ref}$ | Correction-stage $n_{\rm SPS}$ | Role |
| --- | --- | --- | --- |
| 002, 032, 034, 081 | 0.7 at all $z$ | 0.7 ($z\le4$); 0.9 ($46$) | UNCOVER, CEERS, COSMOS-Web, and cross-survey catalogue construction |
| 146, 174, 176 | 0.7 at all $z$ | 0.7 | Stellar-mass-function and joint-evidence corrections |
| 076 | 0.7 at all $z$ | 0.3–1.0 scan | Sensitivity and continuity validation; no production exponent is fitted |




#### C.2.1 Physical Basis from Stellar Population Synthesis


The $M/L \propto t^n$ scaling emerges from the fading of stellar
populations as massive stars evolve off the main sequence. For a simple
stellar population (SSP), the luminosity-weighted age dependence of
$M/L$ is governed by: main sequence turnoff timing, giant branch
contribution to near-IR light, and metallicity-dependent line
blanketing.





Table C1: M/L Power-Law Index from SSP Models

| Model | Metallicity | Age Range | $n$ (V-band) | $n$ (K-band) |
| --- | --- | --- | --- | --- |
| BC03 | $Z_\odot$ | 0.1–10 Gyr | 0.85 | 0.72 |
| BC03 | $0.2 Z_\odot$ | 0.1–10 Gyr | 0.68 | 0.55 |
| BC03 | $0.02 Z_\odot$ | 0.1–1 Gyr | 0.52 | 0.48 |
| FSPS | $0.1 Z_\odot$ | 0.1–1 Gyr | 0.55 | 0.51 |
| BPASS (binary) | $0.1 Z_\odot$ | 0.1–1 Gyr | 0.48 | 0.45 |



At high redshift ($z > 6$), galaxies have typical metallicities $Z \sim
0.1$–$0.2 Z_\odot$. The SSP-predicted low-$n$ regime is consistent with
the canonical residual-minimization analysis, which prefers lower
effective $n$ values once the high-$z$ $R_{\rm ML}$ response becomes
important.



#### C.2.2 Live Empirical Residual-Minimization Validation


In the canonical analysis, the overall residual mass-age correlation is minimized at $n_{\rm SPS}=0.3$ ($\rho = 0.118$). The redshift-split minima are $n_{\rm SPS}=1.0$ at $z=4$–$6$ and $6$–$8$, and $n_{\rm SPS}=0.3$ at $z=8$–$10$; because each lies on an edge of the scanned interval, these values are treated as sensitivity diagnostics rather than calibrated exponents. K-fold cross-validation gives mean $n_{\rm SPS}=0.82 \pm 0.07$ with mean test $\rho = 0.61$. The pipeline separately verifies that changing $n_{\rm SPS}$ leaves $R_{\rm ML}$ invariant at fixed potential depth and redshift, and that the kernel is continuous across $z=4$ and $z=6$.



#### C.2.3 TEP-Induced Modification


Beyond the standard SSP prediction, TEP introduces an additional effect:
in the high-response regime ($R_{\rm ML} > 1$), stellar populations appear
to have accumulated more effective time than their coordinate age suggests, compressing the
observed age range and effectively flattening the $M/L$-age slope. The
empirically preferred low-$n$ regime at $z > 6$ may therefore reflect
both low metallicity and TEP-induced compression.



Circularity Resolution**


**Concern:** The redshift-dependent $n$ is itself
claimed as a TEP signature, potentially introducing circularity.



**Resolution:** K-fold cross-validation gives $n_{\rm SPS} = 0.82
\pm 0.07$ with mean test $\rho = 0.61$. The redshift-blind holdout
($n$ calibrated at $z < 8$, applied at $z \ge 8$) yields $\rho
= -0.05$ ($p = 0.17$), indicating that the data-driven optimal $n$
does not generalize across redshift regimes — a known limitation
of the empirical residual-minimization approach. However, sensitivity
analysis shows the dust signal remains significant
across $n \in [0.3, 1.0]$. An empirical JADES mass-to-light proxy
check is also consistent with the mechanism: $\rho(R_{\rm ML},
M/L_{\rm proxy}) = +0.938$ ($p = 3.1 \times 10^{-195}$),
with partial $\rho = +0.812$ after redshift control ($p
= 2.0 \times 10^{-100}$). However, both $R_{\rm ML}$ and the M/L proxy
($\log M_* + M_{\rm UV}/2.5$) are functions of $\log M_*$, so the
$z$-only partial correlation is inflated by this shared mass dependence.
After controlling for both $M_*$ and $z$, the partial correlation drops
to $\rho = +0.249$ ($p = 2.3 \times 10^{-7}$) — still significant but
much weaker. The M/L proxy test therefore confirms the direction of the
relationship but cannot by itself distinguish TEP from a generic
mass–M/L correlation. Independent age indicators via Balmer
absorption (H$\delta$, H$\gamma$) are predicted to correlate with
$R_{\rm ML}$ and would provide a spectroscopic test independent of M/L
assumptions.





### C.3 Model Discrimination and Falsifiability Tests


#### C.3.1 Modified Gravity Theory Comparison




Table C2: Modified Gravity Comparison (JWST Anomaly Predictions)

| Theory | JWST Score (/8) | Constraint Score | Key Limitation |
| --- | --- | --- | --- |
| **TEP** | **8** | **3** | — |
| $f(R)$ | 1 | 3 | No dust/clock prediction |
| Galileon | 1 | $-1$ | Ruled out by GW170817 |
| Symmetron | 0 | 3 | No JWST predictions |
| DGP | 0 | 1 | Self-accelerating branch ruled out |
| MOND | 0 | $-1$ | Non-relativistic; no clock effect |
| Horndeski (generic) | 0 | 0 | Too broad; no specific predictions |



Within this comparison test set, TEP matches 8/8 JWST anomaly predictions
compared to 1/8 for the next-best theory ($f(R)$). This comparison
illustrates TEP's breadth of coverage across domains, not a definitive
model selection (which would require computing likelihoods for each
theory).



#### C.3.2 Theoretical Consistency Tests




Table C3: Theoretical Consistency Tests

| Test | Result | Status |
| --- | --- | --- |
| Causality Constraint | 0/2000 causal violations | ✓ Pass |
| $\kappa_{\rm gal}$ Error Budget | $R_{\rm ML}$ uncertainty $\pm 4.2\%$ ($\sigma_{\kappa} = 4.0 \times 10^5$ mag) | ✓ Pass |
| Time-Space Decoupling | Temporal/spatial ratio > 1.5× | ✓ Pass |
| Multi-Tracer Consistency | *Removed:* this test used hardcoded synthetic α values, not measured data; result pending real multi-tracer calibration | — N/A |
| Screening Length Scale | λ_C = 2.5 kpc vs observed 1.5 kpc | ✓ Pass |
| Screening Transition | Transition spans 2.2 dex; cosmic mean fully unscreened | ✓ Pass |
| Precision Gravity (Cassini, LLR, pulsars, CMB) | All satisfied via Temporal Topology (continuous geometric screening) | ✓ Pass |




#### C.3.3 Model Discrimination Tests




Table C4: Model Discrimination and Falsifiability Tests

| Test | Result | Conclusion |
| --- | --- | --- |
| IMF Constraint (Red Monsters) | Required slope: $\alpha = 1.5$ (no TEP) vs $2.1$ (with TEP) | TEP removes need for extreme IMF |
| IMF vs TEP Discrimination | Estimated simulation power 100% in 5000 trials (finite-sample estimate; true power &le; 99.94% at 95% CI) | TEP distinguishable from IMF |
| Mass-Proxy Breaker | 2/3 tests pass; z>8 LOWESS $\rho = 0.149$, partial-rank $\rho = 0.310$, shuffled-mass $Z = 9.8$, unique fraction 101.1% | $R_{\rm ML}$ carries information beyond mass+$z$ ordering |
| Dust Model Comparison | TEP 46.5× vs SN-only 10× | Dust physics favors TEP |
| Cross-Survey Systematics | $\rho$ scatter = 0.024 across surveys | Consistent across survey analyses |
| AGN Discrimination Power | Estimated simulation power 100% in 2000 trials (finite-sample estimate; true power &le; 99.85% at 95% CI) | TEP distinguishable from AGN |
| LRD Sensitivity Diagnostic | Mean $\Delta R_{\rm ML} = 1.73$, all compact | Directional differential-potential diagnostic; quantitative growth deferred |
| Hubble Connection | JWST-side pipeline consistency check (step 133): $H_0^{\rm TEP} = 67.8$ km/s/Mpc from simplified two-halo scaling — an order-of-magnitude illustration, not an independent prediction — consistent in direction with Paper 11 (v0.12: $H_0 = 67.83 \pm 1.56$; TEP-CMB $0.68\sigma$); fitted $\kappa_{\rm Cep}$, not derived from first principles | Simplified partial reconstruction; consistent with Paper 11 |
| Prediction Error Budget | $\pm 16.5\%$ combined uncertainty | Falsifiable at $2\sigma$ |



**IMF constraint:** Without TEP, the Red Monster SFE
anomaly requires a top-heavy IMF ($\alpha_{\rm min} = 1.5$). With TEP,
the corrected SFE requires only $\alpha_{\rm min} = 2.1$, consistent
with standard IMFs. TEP and IMF produce observationally distinct
signatures via environment-dependence (estimated simulation
discrimination power 100% across 5000 finite trials; finite-sample upper bound, not an asymptotic guarantee).



**Mass proxy and systematics:** The canonical robustness
results no longer rely on the retired selection-function Monte Carlo
analysis. Instead, the evidence comes from the mass-proxy breaker and
cross-survey systematics checks: at $z > 8$, non-parametric residual tests retain the dust–$R_{\rm ML}$ signal after mass+$z$ removal (LOWESS $\rho = 0.149$; partial-rank $\rho = 0.310$), and shuffled-mass null tests show the observed correlation exceeds mass-ordering expectations by $Z = 9.8$, with 101.1% of the signal attributable to the non-linear $R_{\rm ML}$ form
rather than to mass ordering alone. The mass-to-light correction uses the declared primary high-redshift value $n_{\rm SPS}=0.5$ (and the SMF sensitivity value $n_{\rm SPS}=0.7$), while the response normalization remains fixed at $n_{\rm ref}=0.7$; a single k-fold cross-validation gives mean $n_{\rm SPS}=0.82 \pm 0.07$. Cross-survey dust correlations show
low scatter ($\rho_{\rm std} = 0.025$), arguing against survey-specific
artifacts.



**Hubble tension:** The JWST-side pipeline (step 133)
yields $H_0^{\rm TEP} = 67.8$ km/s/Mpc from the simplified two-halo approximation — an order-of-magnitude illustration, a conservative lower bound consistent in direction with the full Paper 11 reconstruction. The residual offset from the ladder value (73.04) belongs to Paper 11's model-conditioned endpoint calibration response, not an independently measured path-integrated shear. The full
conditional reconstruction is provided in Paper 11 (v0.12), where the
unified host-level reconstruction of 37 SN Ia hosts yields $H_0 = 67.83 \pm 1.56$
km/s/Mpc, in concordance with Planck at $0.26\sigma$ ($0.36\sigma$
bootstrap; TEP-CMB internal consistency at $0.68\sigma$); the JWST pipeline value is a
conservative lower bound from the crude area-distance scaling.



**Falsifiability:** Combined prediction uncertainty is $\pm
16.5\%$. Key falsification criteria at $2\sigma$: mass-dust slope
differing from 0.56 by $> 0.30$; SFE correction differing from 100% by
$> 25\%$.



### C.4 Little Red Dots: Theoretical and Methodological Status


As established in §2.5 and §4.4, Little Red Dots (LRDs) represent a qualitative stress test of compact-core potential structure rather than a primary evidentiary line. Because $R_{\rm ML}$ parameterizes observer-side inference bias under an assumed FLRW baseline rather than physical matter-frame proper time $\tau_\star = \int A(\phi)\,dt$, quantitative black hole accretion cannot be evaluated by inserting $R_{\rm ML}$ into an exponential growth integral.



While deeper central gravitational potentials qualitatively yield larger inference responses ($R_{\rm ML}^{\rm cen} > R_{\rm ML}^{\rm halo}$) in compact systems, rigorous accretion modeling requires fully solved relativistic scalar-tensor core solutions. The population-level LRD anomaly is therefore deferred to dedicated relativistic compact-object studies.



### C.5 External-Regime Discriminants


#### C.5.1 High-Value Observational Discriminants


Beyond the present empirical core, the most informative discriminants
are observables that isolate the unique signatures of the scalar-field
coupling. Key discriminants include:



- 
**Spectroscopic Ages:** Deep NIRSpec MSA spectroscopy
of high-$R_{\rm ML}$ candidates at $z > 6$ to measure direct Balmer
absorption ages, testing the predicted apparent age offset independent
of photometrically derived mass-to-light ratios.


- 
**Resolved Screening Maps:** Spatially resolved IFU
spectroscopy of massive $z \sim 4$–$6$ galaxies to map radial age
gradients. TEP predicts a specific "blue core / red outskirts"
inversion due to central potential screening, opposite to standard
inside-out growth.


- 
**Environmental Contrasts:** Comparative spectroscopy
of galaxies in dense protoclusters versus field environments at
fixed mass. TEP predicts that environmental screening should
suppress age and dust signatures in dense regions, reversing the
standard "downsizing" trend. A preliminary implementation using
DJA $\beta$-residuals ($N = 1{,}238$) did not yield a clean
dense-younger-than-field reversal after matched controls
(0 matched cells supportive; step 167), so this remains a
future observational priority rather than a current result.


- 
**Rest-Frame Mid-IR:** MIRI imaging to directly probe
the dust continuum at $z > 7$, confirming that the observed
UV-reddening is driven by dust grains (as predicted by the
$R_{\rm ML}$-ordered dust buildup) rather than exotic stellar
populations.





#### C.5.2 Wide-Field Regime Predictions


TEP also makes quantitative predictions in large wide-field survey
regimes:



- 
**Euclid-wide regime ($15{,}000$ deg²):** $N \sim
300{,}000$ massive galaxies at $z = 0.9$–$1.8$; TEP predicts $\sim
25\%$ mass-dependent age offset.


- 
**Roman high-latitude regime ($2{,}000$ deg²):** $N
\sim 500{,}000$ emission-line galaxies at $z > 2.5$; TEP predicts
weak gas-phase metallicity–$R_{\rm ML}$ correlation.


- 
**Roman supernova regime ($N \sim 2{,}700$):** TEP
predicts host-potential-dependent SN Ia rate enhancements and an elevated Ia/CC ratio in deeper potential wells.




At the combined sample scale ($N > 800{,}000$), the low-redshift regime
provides a stringent test of TEP at $z < 3$.



#### C.5.3 Cosmic Variance Budget


Current survey fields: UNCOVER (Abell 2744, 45 arcmin², $\sigma_{\rm cv}
\approx 22\%$), CEERS (EGS, 100 arcmin², $\sigma_{\rm cv} \approx
15\%$), COSMOS-Web (COSMOS, 1800 arcmin², $\sigma_{\rm cv} \approx
3.5\%$). Cross-field consistency despite different environments
strengthens the conclusion that the correlation is not driven by
large-scale structure.



## Data Availability & Reproducibility




This work follows open-science practices. All results are fully reproducible from raw data 
using the documented pipeline. All numerical results, figures, and statistics are generated by deterministic 
Python scripts processing real observational data.





### Repository & Code


**GitHub Repository:** github.com/matthewsmawfield/TEP-JWST



The repository contains a deterministic, version-controlled analysis pipeline of 131 active scripts
(numbered up to 178, with gaps from deprecated and merged steps) and automated consistency checks.



#### Repository Structure



```
TEP-JWST/
├── data/             # Raw and interim catalogs
│  ├── raw/            # JWST catalogs from MAST
│  ├── interim/          # Processed datasets
│  └── DATA_PROVENANCE.md     # Download timestamps and checksums
├── logs/             # Step execution logs with timestamps
├── results/
│  ├── outputs/          # JSON/CSV analytical outputs
│  └── figures/          # Generated manuscript figures
├── scripts/
│  ├── steps/           # 131 active scripts (numbered up to 178)
│  │  ├── step_001_uncover_load.py
│  │  ├── step_002_tep_model.py
│  │  ├── step_140_evidence_tier_summary.py
│  │  ├── step_159_mass_measurement_bias.py
│  │  ├── step_160_manuscript_consistency_check.py
│  │  ├── step_170_kinematic_decisive_test.py
│  │  ├── step_171_sigma_kinematic_expansion.py
│  │  ├── step_176_nested_bayesian_evidence.py
│  │  └── run_all_steps.py    # Master pipeline runner
│  └── utils/           # Shared analysis utilities
├── site/
│  ├── components/        # Source of truth for manuscript
│  └── dist/           # Built site artifacts
└── requirements.txt        # Python dependencies (pinned)
```




### Data Provenance



| Data Source | Provider | Access Method | Download Size | DOI/URL |
| --- | --- | --- | --- | --- |
| UNCOVER DR4 | MAST/JWST | Public archive | 62.8 MB (SPS catalog) | MAST Archive |
| CEERS | MAST/JWST | Public archive | ~500 MB (photometry) | MAST Archive |
| COSMOS-Web/COSMOS2025 | MAST/JWST | Public archive | 270 MB (master catalog) | MAST Archive |
| JADES DR5 | MAST/JWST | Public archive | 673 MB (GOODS-S) + 818 MB (GOODS-N) | MAST Archive |
| JADES DR4 Spectroscopic | JADES Team | Direct download | ~200 MB | JADES DR4 |
| DJA NIRSpec Merged v4.4 | DJA Archive | Zenodo | ~150 MB (compressed) | Zenodo |
| FRESCO Red Monsters | Literature | Author-provided | <1 MB (3 objects) | Via GitHub repo |
| Kokorev LRD Catalog | Literature | Published catalog | 210 KB | Via GitHub repo |


**Total Download Size:** ~2.5 GB for full JWST survey catalogs (optional for replication; core analysis runs on processed interim files).**
Data Provenance Log:** Complete download timestamps, file checksums, and version records 
are maintained in `data/DATA_PROVENANCE.md`.



### Reproduction Instructions



#### Quick Start (Full Reproduction)



```
# 1. Clone repository
git clone https://github.com/matthewsmawfield/TEP-JWST.git
cd TEP-JWST

# 2. Install dependencies
pip install -r requirements.txt
npm install --prefix site

# 3. Run complete pipeline (131 active scripts)
python scripts/steps/run_all_steps.py

# 4. Build manuscript
npm run build --prefix site
```




#### System Requirements



| Component | Minimum | Recommended | Tested On |
| --- | --- | --- | --- |
| CPU | 8 cores | 14+ cores | Apple M4 Pro (14-core) |
| RAM | 16 GB | 32 GB | 24 GB (M4 Pro) |
| Storage | 20 GB | 50 GB | NVMe SSD |
| Runtime | ~30-35 minutes (full pipeline, 131 active scripts) | ~32 minutes (M4 Pro) |  |


#### Pipeline Overview

The analysis pipeline consists of 131 active scripts (numbered up to 178) organized into phases:


- **Steps 001-008:** Data loading and TEP model computation ($R_{\rm ML}$ for all galaxies)

- **Steps 009-100:** Core empirical analysis across five evidence lines (L1-L5)

- **Steps 101-140:** Cross-survey replication and robustness validation

- **Steps 141-160:** Advanced discriminating tests and manuscript consistency checks

- **Steps 161-178:** Figure generation and final summary outputs



Each step produces JSON outputs with full metadata, and execution logs are written to `logs/` 
with timestamps for full traceability.



#### Consistency Verification


The pipeline includes automated consistency checks (`step_160_manuscript_consistency_check.py`) 
that verify every numerical claim in the manuscript against actual computed outputs. 
**Zero tolerance for statistical-data mismatches.**



### Data Provenance

Analysis performed using:


- **Python** 3.11+ (pinned in requirements.txt)

- **NumPy** 1.24+ (numerical computation)

- **SciPy** 1.10+ (statistical tests, optimize)

- **Pandas** 2.0+ (data manipulation)

- **Matplotlib** 3.7+ (visualization)

- **Astropy** 5.0+ (astronomical calculations)



Full dependency tree with exact versions is available in `requirements.txt`.

---

<!-- ===================================================================== -->
<!-- CANONICAL URL: https://mlsmawfield.com/tep/wb/ -->
<!-- MARKDOWN URL:  https://mlsmawfield.com/tep/wb/manuscript.md -->
<!-- ===================================================================== -->

# Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries
**Matthew Lukin Smawfield**
Version: v0.7 (Kilifi)
First published: 19 March 2026 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.19102061

---

## Abstract

The Gaia DR3 catalog of over one million wide binaries opens a precise window onto gravity in the weak-field regime ($a \lesssim 10^{-10}\,\mathrm{m\,s^{-2}}$), yet whether the observed velocity excess reflects modified gravity or unresolved systematics remains contested.

In the Temporal Equivalence Principle (TEP), a conformal scalar field modulates matter proper time as

\begin{equation} \label{eq:conformal_relation} \mathrm{d}\tau/\mathrm{d}t \approx A(\phi),
    \qquad
    A(\phi)=\exp(\beta_A\phi/M_{\rm Pl}).
    \end{equation}

The Cepheid-calibrated response scale is denoted $\kappa_{\rm Cep}$, while the wide-binary transition is parameterized independently by the velocity-profile saturation amplitude $\alpha_{\rm sat}$, not by a bare scalar coupling. This paper tests whether the Gaia wide-binary anomaly is better described as smooth Temporal Shear recovery in weak-field environments.

From 341,315 high-purity systems, the analysis identifies a screening transition at $R_s = 2{,}646 \pm 182$ AU (statistical; $\pm 609$ AU total), strongly preferred over both a flat Newtonian profile ($\Delta \chi^2 = 14{,}845$) and a constant boost ($\Delta \chi^2 = 3{,}583$). The transition is strongly preferred under diagonal, AR(1) and Gaussian-process covariance treatments; even the most conservative covariance-aware comparison gives $\Delta\chi^2 \ge 1{,}073$ against a flat Newtonian profile and $\Delta\chi^2 \ge 284$ against a constant boost. At large separation the profile saturates at $\alpha_{\rm sat} = 0.366 \pm 0.012$, roughly 35–40% above the Keplerian baseline. Broader smooth-transition fits preserve the same few-thousand-AU onset.

The signal also shows the environmental ordering required by TEP. With a non-circular metallicity guardrail that uses a conservative external $\beta_{\rm MLR}$ prior unless independent spectroscopic metallicities are cached, the lower-density high-$|Z|$ population transitions at smaller radius than the higher-density midplane. The headline comparison is the joint profile-likelihood fit, which shares one saturation amplitude across environments—the physically prescribed constraint, since screening modulates the recovery scale while the ambient-floor-limited plateau is a shared response at fixed environment—and returns $R_s = 3{,}447 \pm 366$ versus $4{,}912 \pm 792$ AU with $\Delta\chi^2 = 60.3$ ($p = 8.1 \times 10^{-15}$), the ordering preserved in 100% of resamples. The model-free permutation test confirms the ordering at $p \le 10^{-4}$ (0 exceedances in $10{,}000$ realizations; fixed-amplitude protocol: $R_s = 4{,}662 \pm 196$ versus $7{,}131 \pm 1{,}341$ AU), as does the solar-track control ($R_s = 4{,}145 \pm 276$ versus $6{,}856 \pm 920$ AU; $p \le 10^{-3}$), whose zero-metallicity-correction selection additionally breaks the $|Z|$–metallicity correlation by construction. The transition scale furthermore rises with system mass ($R_s = 2{,}778$, $2{,}966$, $4{,}848$ AU across mass terciles; measured exponent $n = 0.64$ with 86% of resamples positive), consistent with the derived radius law $R_s \propto M^{1/2}$. A five-bin $|Z|$ stratification confirms the density-dependent scaling: the environmental ordering is quantified by an empirical density-scaling exponent, $R_s \propto \rho_{\rm amb}^{p}$ with $p \approx 0.4 \pm 0.1$ under free fits of the same stratified radii—consistent with the square-root benchmark $p = 1/2$ and the $\rho^{1/3}$ ambient channel alike (TEP does not commit to a specific microscopic mechanism; the inverse-power chameleon form that motivated the parameterization is excluded for the TEP coupling $\beta_A = -1$, Smawfield 2025a, F1). The ordering itself is selection-independent: restricted to the window below 9 kAU where the purity cut removes fewer than 2% of pairs, the per-separation median-$\tilde v$ ordering is monotone in $|Z|$ in every bin, reaches $+0.043 \pm 0.006$ ($6.9\sigma$, 100% bootstrap-positive) at $s > 3$ kAU, and is unchanged with no purity cut applied at all. Under bound-probability weighting the profiles rise monotonically to the widest measured separations, the absolute high- versus low-$|Z|$ contrast grows to $+0.094 \pm 0.018$ ($5.3\sigma$) at $s = 15$–$30$ kAU, and a mixture deconvolution leaves the bound-component median elevated at $s = 20$–$50$ kAU under interloper fractions up to three times the prior estimate. Scrambling tests and phase-mixed Newtonian orbital forward models fail to reproduce the observed screening preference. The wide-binary anomaly is therefore not a generic low-acceleration excess but a structured, environmentally modulated screening transition—one whose morphology, onset scale, and environmental ordering are quantitatively consistent with the conformal scalar field of TEP and are not reproduced by the Newtonian orbital-projection or MOND/EFE parameterizations tested here.

*Keywords:* Temporal Equivalence Principle – wide binaries – Gaia DR3 – weak-field gravity – Temporal Shear recovery – environmental transition morphology – Temporal Topology – Temporal Shear – modified gravity – MOND

## 1. Introduction

Gaia Data Release 3 (DR3) provided astrometry for more than a million wide binary stars, opening a direct laboratory for gravity in the extreme weak-field regime ($a \lesssim 10^{-10}$ m/s$^2$). Yet the physical interpretation of the resulting signal remains unsettled.

Chae (2023, 2024a) and Hernandez (2023) report a significant anomalous velocity boost at wide separations ($s > 3{,}000$ AU), interpreting it as evidence for Modified Newtonian Dynamics (MOND) and a challenge to dark matter at sub-galactic scales. Hernandez et al. (2024) further showed through statistical analysis that the anomaly persists in kinematically cleaner subsamples. Chae (2024b) confirmed the anomaly using normalized velocity profiles with increasingly stringent quality cuts, arguing that triple contamination alone cannot account for the observed profile shape. By contrast, Banik et al. (2024) argue that their Gaia DR3 analysis is consistent with Newtonian gravity once hierarchical triple contamination is modeled explicitly, and Pittordis et al. (2025) reach a similar conclusion using realistic triple population synthesis, attributing the apparent velocity excess to unresolved inner binaries that inflate photometric masses and bias the velocity ratio. The disagreement therefore hinges on whether the observed signal survives stringent quality cuts and whether its separation-dependent morphology can be quantitatively reproduced by triple contamination alone. Independent constraints from the vertical dynamics of the Galactic disk add further context, suggesting that the weak-field regime may harbor signals beyond the reach of standard dark-matter models.

This paper argues that the Temporal Equivalence Principle (TEP) offers a more physically specific resolution. By predicting smooth Temporal Shear recovery in weak-field environments, TEP naturally explains a partial, environmentally modulated velocity boost—one that neither pure MOND nor a purely artifactual GR interpretation accommodates as readily.

The extension from the Temporal Topology saturation scale framework (Smawfield 2025g) to wide binaries follows directly. In the conformal sector of TEP (written in isotropic shorthand $\tilde{g}_{\mu\nu} = A^2(\phi)\,g_{\mu\nu}$ with $A(\phi) = \exp(\beta_A\phi/M_{\text{Pl}})$, and resolved in the operative preferred-foliation realization of Smawfield 2025a, &sect;2.2 into the local lapse $A_{\rm loc}(u_{\rm grav})$ and spatial ruler factor $A_{\rm sp}(u_{\rm drift})$), matter proper time in the non-relativistic weak-field limit at a fixed cosmic epoch is modulated as $\mathrm{d}\tau/\mathrm{d}t \approx A(\phi)$. In dense environments the source-charge sector continuously suppresses the locally observable Temporal Shear $\nabla \ln A$; in diffuse environments that suppression weakens and a residual temporal gradient emerges. The wide-binary test therefore probes the radius at which this pre-screened conformal field first becomes kinematically visible against the Galactic background.

The velocity-profile saturation amplitude $\alpha_{\text{sat}}$ and the transition scale $R_s$ function as the kinematic realizations of the abstract screening operator $\mathcal{S}_\Sigma(\mathcal{E})$ for virialized orbital phase space. Rather than a flat scalar coupling, these parameters trace the continuous un-screening of the Temporal Shear in the ultra-diffuse, weak-field environments of the galactic halo.

The claim-discipline framework for the TEP corpus, including the scope limitations of canonical precision tests, is established in TEP-EXP (Paper 9).

## 2. Theoretical Framework: Temporal Shear Recovery

Unlike MOND, which is organized around a universal acceleration threshold $a_0 \approx 1.2 \times 10^{-10}$ m/s$^2$ (Milgrom 1983), the Temporal Equivalence Principle (TEP) is organized around Temporal Shear recovery in weak-field environments of a conformal scalar field. In the TEP framework (Smawfield 2025a), the isotropic conformal–disformal expression $\tilde{g}_{\mu \nu} = A^2(\phi)\,g_{\mu \nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi$ provides the historical benchmark shorthand, while the operative realization adopts the preferred-foliation decomposition $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav})\,U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift})\,\Pi_{\mu\nu} + B(\phi)\,\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ along the cosmic temporal flow $U_\mu \equiv -\nabla_\mu\bar{\phi}/\sqrt{2\bar{X}}$ (Paper 0, &sect;2.2), where $A(\phi) = \exp(\beta_A\phi/M_{\text{Pl}})$ governs local proper-time rates and $B(\phi)$ encodes small disformal deformations. Wide binaries probe non-relativistic orbital separations and velocities at a fixed cosmic epoch ($A_{\rm sp}(t_0)=1$), where the disformal sector is negligible and both metric realizations yield the identical slow-matter lapse modulation $\mathrm{d}\tau/\mathrm{d}t \approx A(\phi)$, with $\alpha(\phi) \equiv \mathrm{d}\ln A/\mathrm{d}\phi$ the conformal coupling strength. The underlying screening transition is associated with a Temporal Topology saturation scale $\rho_T \approx 20$ g/cm$^3$ (Smawfield 2025g), which is not a binary on/off threshold but a non-linear saturation scale of the conformal-factor sector. In the language of scalar-tensor theory, the gravitational metric $g_{\mu\nu}$ corresponds to the Einstein frame and the matter metric $\tilde{g}_{\mu\nu}$ to the Jordan frame; the kinematic observables measured here—orbital velocities and projected separations—are defined in the Jordan frame, where matter propagates and clocks tick. The conformal factor $A(\phi)$ therefore enters the velocity ratio $\tilde{v}$ directly through its effect on matter-frame proper time, not merely as a post-Newtonian correction to the Einstein-frame potential. This frame distinction is what ties the kinematic enhancement to the clock-rate signature in Section 2.3.

In the weak-field limit, writing $\Theta = \ln A$, the matter-frame potential is $\Phi_{\rm eff} \simeq \Phi_N + c^2\Theta$. Matter-frame motion therefore responds to

\begin{equation} \label{eq:effective_acceleration} \mathbf{a}_{\rm eff} = -\nabla\Phi_N - c^2\,\boldsymbol{\Sigma}^{\rm obs}, \qquad \boldsymbol{\Sigma}^{\rm obs} = \mathcal{S}_\Sigma(\mathcal{E})\,\nabla\ln A \end{equation}

where $\mathcal{S}_\Sigma(\mathcal{E})$ is the environmental screening factor that suppresses the locally observable Temporal Shear. For an approximately radial binary response,

\begin{equation} \label{eq:vc_squared} v_c^2(s) \simeq \frac{GM}{s} + c^2\,s\,\hat{\mathbf{s}}\cdot\boldsymbol{\Sigma}^{\rm obs} \end{equation}

The observed velocity enhancement is therefore the kinematic projection of recovered Temporal Shear, while $\alpha_{\rm sat}$ parameterizes the wide-binary channel response rather than the microscopic coupling $\beta_A$.

### 2.1 Consistency with Solar-System PPN Constraints

The Cassini Shapiro-delay bound $|\gamma_{\rm PPN} - 1| < 2.3 \times 10^{-5}$ (Bertotti, Iess & Tortora 2003) constrains the source-charge sector $S_\Sigma$ in the solar-vicinity environment, not the wide-binary environment probed here. Two features of the Cassini measurement make the bound weaker than a naive reading suggests for TEP. First, the measurement is a round-trip radio ranging experiment performed during solar conjunction: the signal path samples the Sun's deep potential well, which in TEP corresponds to the screened regime where $S_\Sigma$ is suppressed. Under the unified unstretched-space metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ (Paper 0 &sect;2.2, Step 80), $\gamma_{\rm PPN}-1=-2\beta_A^2 S_\Sigma^{(\odot)}/(1+2\beta_A^2 S_\Sigma^{(\odot)})$, so the bound $S_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ ($5.8 \times 10^{-6}$ on the isotropic-conformal benchmark; with the solved two-branch solar profile giving $y_{\rm tb}(1.6\,R_\odot)=2.16\times 10^{-8}$ and $|\gamma_{\rm PPN}-1|=4.32\times 10^{-8}\ll 2.3\times 10^{-5}$) applies to the solar-vicinity environment along the signal path, not to the dilute interstellar medium where the Temporal Shear recovers and wide-binary dynamics originate. Second, the round-trip geometry is reciprocity-even: the conformal clock variation enters as a gradient integral $\int \nabla(\ln A)\cdot d\mathbf{l}$, and the outbound and inbound contributions have opposite signs because the path is traversed in opposite directions, so the clock-sector contribution cancels exactly in the round-trip observable (Smawfield 2025a, Smawfield 2025i). The wide-binary signal, by contrast, is a kinematic observable in the dilute interstellar medium where $S_\Sigma$ has recovered; it probes a different environment and a different measurement geometry from the Cassini bound. The ten-order-of-magnitude gap between the Cassini and wide-binary Yukawa mass requirements is therefore environmental, not sectoral: $S_\Sigma$ is suppressed in the high-density solar vicinity and recovers in the low-density interstellar medium. The density-dependent minimum of the TEP field equation $\nabla^2\phi = V_{,\phi} + \rho_* A_{,\phi}$ provides the environmental transition; this is a general property of the bidirectional conformal field equation, not specific to any particular microscopic completion.

The reconciliation has a precise spatial form. The corpus's master-action screening closure derives a pairwise response of the form $\mathcal R(s) = \mathcal S_\Sigma(X_{\rm env})^2\,y(s)$ (Smawfield 2025a, &sect;7): the mutual shear profile $y(s)$—the exact flux-conserving solution $y\bigl[1+y^2(r_*/r)^4\bigr]=1$ of the stated kinetic completion $P=X-V+X|X|/\Lambda_X^4$, with $\phi'\propto r^{-2/3}$ deep inside the shell—propagated with the ambient inverse-stiffness factor $\mathcal S_\Sigma(X_{\rm env})=[1+2X_{\rm env}/\Lambda_X^4]^{-1}$ at the emission and response vertices. The ambient $X_{\rm env}$ is the field of the hierarchy level above the pair, excluding the pair's own co-generated mutual field. Two regimes of the same operator follow. For a hierarchical pair—a test member embedded in a dominant member's pre-existing nonlinear shell—the dominant member's field is the embedding ambient, and the identity $2X_{\rm dom}/\Lambda_X^4=(1-y)/y$ makes the vertex factors equal to $y(s)$ itself: the response recovers $\mathcal R\sim y^3\sim s^{4}$, the quartic deep-interior asymptote. For a comparable-mass wide binary the mutual field is the measured response rather than a fixed embedding—the two members' field gradients oppose and cancel at the pair's bridge region—so the vertex ambient saturates at the Galactic field and the transition profile is the flux-conserving $s^{4/3}$ law, the exponent the data select (Section 4; effective exponent $\simeq1.3$ through the full forward model). This is a continuous Temporal-Topology recovery rather than a thin-shell boundary—consistent with the rejection of the thin-shell morphology reported in Section 4. The shell radius is the derived law $R_{\rm shell}(M) \equiv r_*(M) = \sqrt{GM/g_t}$, with $g_t = cH_0/(2|\beta_A|) \simeq 3.4\times10^{-10}$ m s$^{-2}$, not a per-channel fitted scale. On this single map every existing Solar-System constraint lies deep inside the solar shell, where the hierarchical response $\mathcal R\sim s^4$ applies: $\mathcal R\simeq10^{-23}$ at the Cassini conjunction separation ($1.6\,R_\odot$), $\simeq3\times10^{-15}$ at 1 AU, and $\simeq3\times10^{-11}$ at Saturn's orbit ($9.5$ AU)—the corresponding anomalous acceleration $\delta a \simeq 3.5\times10^{-15}$ m s$^{-2}$ sitting more than two orders of magnitude inside the Cassini-era ephemeris bound—while the Earth–Moon pair lies at $\simeq2\times10^{-15}$ inside the nested terrestrial and solar shells. The pairwise response at the conjunction separation thus sits seventeen orders of magnitude below the already stringent single-body bound $S_\Sigma^{(\odot)}\lesssim1.15\times10^{-5}$ ($5.8\times10^{-6}$ isotropic), whereas a wide-binary companion at $s\sim R_s$ samples the shell boundary itself. The fitted $R_s = 2{,}646$ AU is therefore a measurement of the solar-type screening-shell radius, not a new screening variable. The corpus's zero-parameter derived value, $R_{\rm shell}\simeq4.7$ kAU at the sample mean mass, sits a factor $\simeq1.9$ above the fitted scale under the isolated-profile reading—but the isolated profile is not the embedded one. Solving the source perturbation on the Galactic nonlinear background ($u_0=\sqrt{X_{\rm gal}}\simeq0.72\,g_t$; Paper 0 step 32) shows the ambient stiffness $Z_{ij}=P_{,X}\delta_{ij}+2P_{,XX}\,a_i a_j$ suppressing the pair's far-field response to $y_{\rm far}\simeq0.56$ and compressing the transition scale to $\simeq0.6\,r_*$. Folded through this paper's own sem-weighted single-scale estimator and mass distribution (step 017), the embedded prediction returns $R_s\simeq3{,}300$ AU and a half-excess marker at $\simeq1{,}715$ AU—the observed half-excess—a factor $\simeq1.26$ residual now inside the systematic band rather than a factor-$\simeq2$ contradiction. The ambient floor displaces the transition inward through the propagator itself, not through a tuned normalization (Smawfield 2025a, &sect;7). At the directly measured ambient ($u_{\rm sun}=0.57$, above) with the ambient setting the vertex and the propagator together, the same estimator returns $\alpha=0.354$ against the observed $0.366$—the amplitude is reproduced at the percent level with no calibrated input—at $R_s\simeq3{,}750$ AU, a factor $\simeq1.4$; the two ambients bracket the physical value because the ambient field gradient is itself a solved quantity in the theory. The environmental ordering constrains and, under the corpus's own ambient sectors, now checks quantitatively. The observed subsamples show the midplane more suppressed under either estimator — free-$\alpha$ fits give ($R_s$, $\alpha$) = ($2{,}821$ AU, $0.244$) at $|Z|<0.1$ kpc versus ($4{,}681$ AU, $0.401$) at $|Z|>0.15$ kpc, both parameters falling toward the plane along the mechanism's own ambient trajectory; the shared-$\alpha$ joint fit reads the same suppression ordering through the amplitude–scale degeneracy as a larger midplane $R_s$. Inverting the free-$\alpha$ amplitudes on the fully-coupled sweep requires the effective ambient to be $\sim36\%$ stronger at the midplane ($u_{\rm eff}\simeq0.71$ versus $\simeq0.52$). A Galactic field-model check (step 017, supply-versus-demand block) tests two ambient estimators. The linear-gradient estimator $u = |a|/g_t$ supplies only $\sim4\%$, and with the wrong sign for the amplitude ordering — the radial floor $u_r\simeq0.57$ is common to both heights and the vertical component adds with $|Z|$ — but that estimator is the weak-field exterior reading: inside the nonlinear disk the local acceleration is not the ambient field state (the disk's field *value* sits at its density-set equilibrium, $u_{\rm min}\propto\rho^{1/3}$ on the quartic branch, while the pair-relevant ambient is the shear variable, whose nonlinear response the k-branch balance resolves). The density-set ambient supplies the contrast quantitatively: the three-component Galactic model gives $\rho_{\rm mid}/\rho_{\rm halo}=2.44$ between the strata's median heights, a $34.6\%$ ambient contrast toward the midplane under $u_{\rm eff}\propto\rho^{1/3}$ against $35.8\%$ required, with the identified $u_{\rm eff}\propto\rho^{1/2}$ k-branch law bracketing the same requirement at $56\%$ under the median-density mapping — the two-bin contrast is an estimator-level check that brackets the two exponent channels, while the five-stratum inversion fixes $p\simeq0.46$–$0.5$ (Section 6) and the ambient map is identified at coefficient level as the local k-branch law $u_{\rm amb}=\sqrt{4\sqrt{2}\pi G\rho L/(K g_t)}$ with coherence length $L\simeq0.84$ kpc (Paper 0, step 67). The matched-covariate audit of step 018 finds that mass, separation, distance and metallicity matching absorbs only $\simeq22\%$ of the contrast (and the mass covariate predicts the opposite sign), so ordinary demographic channels do not supply it either. The ordering is therefore consistent with the density-set ambient the corpus specifies for a nonlinear environment. Which projection enters the pair vertex — the former Gate-A question — is now resolved at response-function level: the embedded pair-scale solve on the constraint-slice background returns a response charge $D^{-1}\propto u_{\rm amb}^{-2.0}$ with local background-density sensitivity $\lesssim0.02\%$ across the full disk range, so the vertex samples the ambient boundary variable and the density ordering enters only through the ambient map (step 67, embedded-response block).

The shell reading also fixes the amplitude bookkeeping. With $\beta_A=-1$, a fully unscreened source would contribute $2\beta_A^{2}g_N = 2g_N$ of extra centripetal acceleration—a plateau at $\alpha_{\rm sat}=\sqrt{3}-1\simeq0.73$. The observed plateau corresponds to $+0.87\,g_N$, approximately 43% of the unscreened response: a partial recovery bounded by the Galactic ambient floor rather than a return to unsuppressed charge, with an implied per-body factor $S_{\rm body}\simeq0.66$ comparable to the corpus's ambient suppression at the solar circle evaluated on the measured field, $\mathcal S_\Sigma(X_{\rm sun})\simeq0.75$ for $u_{\rm sun}=a_\odot/g_t=0.57$ ($a_\odot=v_c^2/R_0\simeq1.94\times10^{-10}$ m s$^{-2}$ with $v_c=220$ km s$^{-1}$, $R_0=8.1$ kpc; the legacy corpus value $\mathcal S_\Sigma\simeq0.66$ at $X_{\rm gal}=0.52$ was itself calibrated to this plateau under the no-propagator bookkeeping; step 017, consistent-vertex sweep). The same shell map closes the corpus's remaining constraints on this class of source. At Oort-cloud separations $s\sim10^{4}$–$10^{5}$ AU the pairwise factor is $\gtrsim0.95$: bodies there orbit in the recovered-shear regime, the domain analyzed by the companion outer-Solar-System boundary study (Smawfield 2026, Paper 36). Terrestrial clock networks, at 1 AU, sit deep inside the solar shell (solar contribution $\mathcal S_{\rm eff}\sim10^{-15}$) and measure the nested terrestrial well's own shear, consistent with the GNSS correlation phenomenology (Smawfield 2025b, 2025c). And the ambient-floor-limited response surviving beyond every stellar shell supplies the effective-gravity contribution invoked by the rotation-curve sector (Smawfield 2025g). The screening shell is therefore not an additional assumption but the spatial semantics of the single operator $\mathcal S_\Sigma(\mathcal E)$ evaluated across separations: one map, bounded inside by Cassini, measured at its boundary by wide binaries, and read off asymptotically by the Galactic field.

### 2.2 The Screening Radius

$R_T$ here denotes the projected separation at which the Temporal Shear suppression transitions from the screened to active-shear regime, related to but distinct from the GNSS correlation length $\lambda_T$ (which characterises the radial relaxation scale within a gravitational well) and the geometric saturation radius $R_T(M, \rho_T, \epsilon_{\rm env})$ derived in Paper 6 (which characterises the system-level boundary condition for a self-gravitating source of mass $M$). The wide-binary screening radius $R_s$ introduced below in Equation ($\ref{eq:screening_radius}$) is the operative benchmark scale for this transition.

For a binary of total mass $M$, the locally active Temporal Shear $\Sigma_\mu \equiv \nabla_\mu \ln A(\phi)$ is suppressed continuously by the environmental and source state rather than gated by an on/off density threshold (Smawfield 2025a, Section 7). As the binary's internal Newtonian potential becomes shallow enough that the source-charge sector no longer fully suppresses the locally observable Temporal Shear, the residual gradient of $\ln A$ generates an additional kinematic enhancement above the Keplerian baseline. The transition is therefore characterized by a continuous recovery of the Temporal Shear, parameterized below by an effective screening radius $R_s$.

Within the Galaxy, however, the binary is not isolated. The TEP framework itself does not commit to a single microscopic screening mechanism: Vainshtein, Galileon, DBI, and symmetron mechanisms are treated as candidate completions of the conformal-sector screening (Smawfield 2025a, &sect;A4, &sect;7), with the defining ontology being the continuous suppression of the locally observable Temporal Shear $\Sigma_\mu \equiv \nabla_\mu \ln A(\phi)$ by source-charge sector, environmental state, and boundary conditions; the inverse-power chameleon branch is excluded under the TEP coupling convention, since $\beta_A = -1$ with $\phi > 0$ in potential wells admits no admissible minimum for $V \propto \phi^{-n}$ (Smawfield 2025a, constraint F1). The quantitative benchmark for the wide-binary transition scale is therefore an empirical environmental-density parameterization rather than a specific potential. Any admissible completion supplies an environment-dependent screened equilibrium—a strong effective scalar response that pins the ambient clock landscape in dense environments and suppresses the incremental shear, driving the locally observable $\Sigma_\mu \to 0$ continuously rather than via thin-shell matching. The surrounding halo and disk already drive $\phi$ toward this screened configuration, so the transition scale relevant to a wide binary is set not by $\rho_T$ directly but by the residual local screening floor that the binary samples. This floor admits a closed-form parameterization that serves as a benchmark; admissible completions would yield qualitatively similar but quantitatively different forms.

\begin{equation} \label{eq:rho_floor} \rho_{\rm floor} = \epsilon_{\rm env}\,\rho_T \end{equation}

\begin{equation} \label{eq:screening_radius} R_s = \sqrt{\frac{GM}{g_t}} \propto M^{1/2}, \qquad \rho_{\rm eff}(R_s) \equiv \frac{3M}{4\pi R_s^3} = \rho_{\rm floor} = \epsilon_{\rm env}\,\rho_T \end{equation}

Here $\epsilon_{\rm env} < 1$ is the pre-screening factor generated by the Galactic environment. The left-hand member is the canonical derived radius law of the nested operator (the flux-conserving shell radius; Section 2.1); the right-hand member is the equivalent enclosed-density diagnostic coordinate of the same transition—for the canonical law $\rho_{\rm eff}$ scales as $M^{-1/2}$ at fixed environment rather than as a fixed density—and it serves as a calibration benchmark rather than a unique prediction of the underlying theory. Two of the three ingredients—$\rho_T$ and the field equation whose solution determines $\epsilon_{\rm env}$—are already derived in earlier work. The Temporal Topology saturation scale $\rho_T \approx 20$ g/cm$^3$ is anchored by GNSS atomic-clock networks and supported by SPARC rotation-curve slopes and compact-object consistency tests (Smawfield 2025g). The density-dependent effective potential $V_{\rm eff}(\phi;\rho)$ whose ground state sets the local screening floor follows from the TEP two-metric action together with the scalar self-interaction sector (Smawfield 2025a, &sect;7). The character of the transition is further tested by comparing the phenomenological TEP recovery profile—the historical fitted form whose steepness is now derived from the master-action kinetic completion $P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+X|X|/\Lambda_X^4$ ($P_{,X}=k\sqrt\xi+2\xi$, $k\simeq16.03$; unit-term control $X-V+X|X|/\Lambda_X^4$)—to alternative transition morphologies. As shown in Section 4, both the thin-shell exterior profile itself and a sigmoid step-like proxy are rejected relative to the TEP exponential ($\Delta\chi^2 = +613.0$ and $+131.5$, respectively; Table 4.1). This rejection is naturally explained by the continuous Temporal Topology framework: the field gradient recovers smoothly, not via a step-function onset.

The canonical cross-scale anchor is now the master-action closure itself: the derived acceleration scale $g_t = cH_0/(2|\beta_A|) \simeq 3.4\times10^{-10}$ m/s$^2$ yields the radius law $R_s = \sqrt{GM/g_t} \simeq 4.7$ kAU at the sample mean mass, a factor $\simeq1.8$ above the fitted $R_s$ (Section 2.2; Smawfield 2025a, &sect;7)—the physically expected direction, since the ambient Galactic field raises the environmental floor and displaces the transition inward. An older SPARC-extracted benchmark, $g_{\rm TEP} \approx 5\times10^{-10}$ m/s$^2$ (Smawfield 2025g), returns $3{,}831$ AU (factor 1.5), and a further external-field divisor $\eta=2$ tightens that empirical normalization to $2{,}709$ AU (3%)—but that tightening is a tuned factor rather than a derived one, and both empirical anchors are superseded as predictions by the canonical $g_t$. The residual is now quantified by a same-estimator check (step 017): folding the mass-convolved derived profile through this paper's own sem-weighted exponential estimator returns $\simeq5.1$ kAU against the fitted $2{,}646\pm609$ AU (ratio $\simeq1.9$), and the physical half-excess marker agrees ($\simeq3.4$ kAU versus the observed $\simeq1.7$–$1.9$ kAU). Definitional and mass-normalization explanations are quantitatively insufficient — the canonical-fit scale reduction is $\lesssim2\%$, $\sqrt{M}$-weighting accounts for $\sim4\%$, and mass convolution broadens the transition upward. The embedded-propagator check of step 017 then resolves most of the scale offset: solving the source perturbation on the Galactic nonlinear background ($u_0=\sqrt{0.52}$) rather than on the isolated-vacuum background compresses the transition, and the same estimator returns $\simeq3.3$ kAU against the fitted $2{,}646\pm609$ AU (ratio $\simeq1.26$, within the systematic band), with the physical half-excess marker coinciding with the observed value ($\simeq1.7$ kAU on both sides). The mean-scale offset is therefore substantially a propagator effect of the ambient stiffness $Z_{ij}=P_{,X}\delta_{ij}+2P_{,XX}a_i a_j$ — a derived consequence of the master-action kinetic operator acting in its own nonlinear ambient field — not a free environmental coefficient. The remaining offset (factor $\simeq1.26$) collects the open residuals quantified by the corpus: the amplitude factorization, with the observed $\alpha_{\rm sat}=0.366$ sitting between the pure-propagator ($0.455$) and propagator-times-vertex-squared ($0.24$) readings, and the Gate-A single-configuration question; the older standalone divisor $\eta_{\rm env}\simeq3.7$ on $g_t$ (implied $g_{\rm eff}\simeq1.3\times10^{-9}$ m s$^{-2}$) is its bookkeeping shorthand, not an additional mechanism. The wide-binary $R_s$ is therefore not an unanchored binary-sector scale: its order of magnitude is fixed by the cross-scale architecture, and its residual scale is supplied by the same ambient-stiffness propagator rather than by an environmental coefficient introduced for the purpose.

The absolute value of $\epsilon_{\rm env}$ at a given Galactic position remains calibration-dependent at this stage. A semantic point should be stated plainly: the quantity $\rho_{\rm eff} = 3M/(4\pi R_s^3) \approx 9.2 \times 10^{-18}$ g/cm$^3$ evaluated at $M \approx 1.2\,M_\odot$ is the binary's own enclosed mean density at the transition separation—some six orders of magnitude above the ambient ISM density—not an environmental density floor. Under the canonical operator the transition criterion is the ratio of the pair's internal field gradient to the ambient environmental shear floor; $\rho_{\rm eff}$ is an equivalent diagnostic coordinate of that criterion (for $R_s = \sqrt{GM/g_t}$ it scales as $M^{-1/2}$, not as a fixed density), while $\epsilon_{\rm env} \approx 4.6 \times 10^{-19}$ is the numerical value of the benchmark parameterization $\rho_{\rm floor} = \epsilon_{\rm env}\rho_T$ rather than a physical density. The height dependence of $\epsilon_{\rm env}$ is then semi-predictive within that parameterization: the screening radius is taken to scale as a power law in ambient baryonic density, $R_s \propto \rho_{\rm amb}^{p}$, so the ratio of screening radii at two Galactic heights depends only on the ambient density ratio and the scaling exponent $p$. As shown in Section 5.2, the benchmark value $p = 1/2$ combined with a standard three-component Galactic density model reproduces the observed environmental $R_s$ ratio within $\sim 1\sigma$. The TEP-native statement is qualitative—more strongly screening environmental states more strongly suppress the locally observable Temporal Shear—while the quantitative exponent $p$ is an empirical property of the environmental response, not of a specified microscopic completion. (The inverse-power chameleon form that historically motivates writing $p = 1/(n+1)$ is excluded under the TEP coupling convention; Smawfield 2025a, F1.) Promoting the absolute $\epsilon_{\rm env}$ to a first-principles prediction would require solving $\nabla^2\phi = dV_{\rm eff}/d\phi$ across the full three-dimensional baryonic density field, feasible with existing N-body–scalar-field codes (e.g., Llinares, Mota & Winther 2014). The large gap between $\rho_T$ and $\rho_{\rm eff}$ does not imply two unrelated saturation scales; it reflects the many orders of magnitude of pre-screening already absorbed by the Galactic halo and disk before the binary's own potential is tested. The radius $R_s$ is thus the screening-shell boundary introduced in Section 2.1: the surface on which the pair's enclosed mean density falls to the effective environmental floor, equivalently parameterized by the corpus's derived transition law $R_{\rm shell}(M) = \sqrt{GM/g_t}$. The density and acceleration parameterizations of the shell boundary identify the same physical transition, and their distinct mass scalings are discriminated directly by the mass-ordered transition itself (Section 4), whose measured exponent is consistent with the derived $\sqrt{GM/g_t}$ law.

### 2.3 The Velocity Enhancement Profile

Within this framework, the kinematic profile is expected to satisfy three qualitative constraints: (i) $\tilde{v} \to 1$ in the fully screened limit $s \ll R_s$; (ii) a monotonic transition near $s \sim R_s$; and (iii) saturation at a bounded enhancement $\tilde{v} \to 1 + \alpha_{\rm sat}$ for $s \gg R_s$. These follow directly from conformal-sector screening and do not depend on the detailed nonlinear field dynamics.

These qualitative features are not merely asserted; they follow from the continuous Temporal Topology of the conformal scalar field. In the TEP framework (Smawfield 2025a, &sect;7), screening operates not via the thin-shell approximation but through the continuous spatial profile of $\ln A(\phi)$ (the Temporal Topology), governed by the non-linear superposition of conformal-factor gradients (the Temporal Shear, $\Sigma_\mu \equiv \nabla_\mu \ln A = \alpha(\phi)\,\nabla_\mu\phi$). In dense environments the source-charge sector and environmental state drive the locally observable shear toward zero, flattening the Temporal Topology. As that suppression weakens with the diluting ambient environment—rather than crossing a binary on/off density threshold—the Temporal Shear recovers continuously, producing a smooth transition from the screened to the active-shear regime. This Temporal-Topology framing is independent of the specific microscopic screening completion; the empirical density-scaling benchmark used in &sect;2.2 is consistent with the same continuous-suppression ontology shared by the admissible completions (Vainshtein, Galileon, DBI, symmetron).

The classical thin-shell formalism (Khoury & Weltman 2004; Burrage & Sakstein 2018) provides useful context. For a screened mass $M$ with screening radius $R_s$ embedded in a background of effective scalar mass $m_{\rm bg} = \sqrt{V_{\rm eff}''(\phi_{\rm bg})}$, the exterior scalar field profile obtained under thin-shell matching conditions is:

\begin{equation} \label{eq:field_profile} \phi(r) = \phi_{\rm bg} - C\,\frac{R_s}{r}\,e^{-m_{\rm bg}(r - R_s)}, \qquad r > R_s \end{equation}

where $C$ is determined by the thin-shell matching and the $R_s/r$ prefactor arises from the thin-shell boundary. The corresponding unscreening fraction $f(s) = 1 - (R_s/s)\,e^{-m_{\rm bg}(s - R_s)}$ produces onset–rise–saturation morphology with a step-like character at $R_s$.

However, the TEP Temporal Topology framework replaces the thin-shell approximation with a continuous field profile. Without a step-function matching condition, the $R_s/s$ geometric prefactor—which encodes the thin-shell boundary—does not arise. Continuous Temporal Topology therefore requires a smooth, one-scale recovery from the screened state toward a finite asymptotic Temporal Shear. The canonical wide-binary realization of this relaxation is the pure exponential:

\begin{equation} \label{eq:velocity_profile} \tilde{v}(s) = 1 + \alpha_{\rm sat}\bigl(1 - e^{-s/R_s}\bigr) \end{equation}

where $R_s$ is the characteristic scale of the Temporal Topology transition and $\alpha_{\rm sat}$ is the saturation amplitude set by the asymptotic Temporal Shear. This is adopted as the canonical fitting function throughout the analysis. Unlike the thin-shell formula, the pure exponential is the canonical one-scale realization of continuous Temporal Topology in the wide-binary sector, containing exactly the two physical quantities required by the TEP screening geometry: the recovery scale $R_s$ and the asymptotic response $\alpha_{\rm sat}$. Equation ($\ref{eq:velocity_profile}$) is the minimal smooth realization of the TEP onset–recovery–saturation morphology; the transition scale and environmental ordering, rather than the exact exponential shape alone, carry the physical inference. Mass and environmental variation broaden this canonical profile rather than changing its underlying onset–recovery–saturation structure, as demonstrated explicitly by the mass-convolved TEP model in Section 4.1. A literal first-principles prediction for the precise profile shape would require solving the full coupled system $\nabla^2\phi = dV_{\rm eff}/d\phi$ in the realistic potential of each binary within its Galactic environment. What the data can test at this stage is whether the morphological class of profile predicted by continuous Temporal Topology—finite onset, smooth exponential rise, bounded saturation—is preferred over the alternatives (scale-free MOND, constant boost, flat Newtonian, or the sharper step-like transition of the thin-shell sigmoid). The sensitivity of the inferred transition scale to the choice of functional form is assessed explicitly in Section 4 by fitting sigmoid and double-exponential alternatives, and the resulting spread is absorbed into the systematic uncertainty budget. As shown there, all three transition models agree on a $\sim 2{,}000$–$3{,}200$ AU onset scale, confirming that the inferred $R_s$ is a robust feature of the data rather than an artifact of any particular parametric choice. Notably, the data reject both the stated thin-shell exterior profile ($\Delta\chi^2 = +613.0$ versus the TEP exponential) and the sigmoid step-like proxy ($\Delta\chi^2 = +131.5$). This rejection is naturally explained by the continuous Temporal Topology framework: the field gradient recovers smoothly, not via a step-function onset.

A natural question is what distinguishes TEP screening from a generic chameleon or symmetron model at the observational level, given that the qualitative profile shape (onset, rise, saturation) is common to all screening gravities. The answer lies in three features that go beyond morphology. First, TEP makes a quantitative cross-scale prediction: the same universal conformal sector and Temporal Topology saturation architecture that govern atomic and compact-object phenomenology (Smawfield 2025g) also set the Galactic screening floor. The wide-binary transition scale is therefore not theoretically isolated within the binary sector, but is anchored to independently measured TEP scales. Second, the environmental test (Section 5) probes the environmental dependence of the locally observable Temporal Shear generated by the conformal sector; in the density-scaling benchmark this dependence projects primarily onto ambient baryonic density. Third, continuous Temporal Topology predicts a smoother recovery than the thin-shell approximation of standard chameleon models; in the canonical wide-binary realization adopted here, that recovery is represented by the pure exponential $\tilde{v}(s) = 1 + \alpha_{\rm sat}(1 - e^{-s/R_s})$ rather than the thin-shell form $f(s) = 1 - (R_s/s)\,e^{-m_{\rm bg}(s-R_s)}$ with its geometric $R_s/s$ prefactor. Notably, the data already favor this prediction: the stated thin-shell profile $f(s) = 1 - (R_s/s)\,e^{-m_{\rm bg}(s-R_s)}$ is rejected at $\Delta\chi^2 = +613.0$ and the sigmoid step-like proxy at $\Delta\chi^2 = +131.5$, both relative to the TEP exponential (Table 4.1). Future observations could sharpen these distinctions: Gaia DR4 radial velocities would enable three-dimensional deprojection, testing whether the proper-time signature of TEP—in which the same Temporal Shear that produces the kinematic enhancement also rescales matter-frame clock rates as $\mathrm{d}\tau/\mathrm{d}t = A(\phi)$—produces a different velocity-anisotropy pattern than a generic screened scalar model that does not make the TEP-specific identification of the conformal gradient with dynamical proper-time geometry. Measurements of the transition radius as a function of Galactocentric radius $R_{\rm gc}$ would further discriminate among candidate completions, since the density-scaling benchmark adopted here predicts a screening floor set by the local baryonic density rather than the total gravitational potential (Burrage & Sakstein 2018), whereas symmetron- or Vainshtein-style completions would imply different gradient-profile dependences—a pattern that maps directly onto the underlying Temporal Shear morphology rather than onto a uniform acceleration threshold.

## 3. Data and Methodology

### 3.1 Sample Selection

This study constructs a wide-binary sample from the Gaia DR3 astrometric catalog (Gaia Collaboration et al. 2023), following the pair-identification methodology and quality criteria of El-Badry et al. (2021). The reproducible pipeline then applies: a chance-alignment probability cut of less than 1%; parallax signal-to-noise $> 20$ and proper-motion signal-to-noise $> 10$ for both components; a strict RUWE $< 1.2$ cut; and a projected-separation window of $50 < s < 50{,}000$ AU. Together these filters suppress visual interlopers, noisy astrometric solutions, and unresolved hierarchical multiples before any kinematic inference is attempted. The resulting high-purity sample contains 341,315 systems.

The RUWE cut deserves particular comment, since hierarchical triples are the principal contaminant identified by Pittordis et al. (2025). Gaia's Renormalized Unit Weight Error measures goodness-of-fit to a single-star astrometric model; RUWE $> 1.4$ is the standard flag for an unresolved companion. The pipeline adopts a stricter threshold of $1.2$, removing $193{,}545$ systems ($36\%$ of the post-parallax sample), of which $130{,}733$ have RUWE $> 1.4$ for at least one component. If the pre-filter triple fraction is $15$–$20\%$ (Pittordis et al. 2025), the number removed exceeds the expected triple count by $1.8$–$2.4\times$, indicating that the cut also discards genuinely poor astrometric solutions beyond triple contamination alone.

The residual triple fraction is bounded by RUWE detection incompleteness for long-period inner binaries ($P \gtrsim 3$–$5$ yr), which Gaia's $\sim 34$-month DR3 baseline cannot resolve. For plausible detection efficiencies of $50$–$70\%$, the residual contamination is $\sim 5$–$10\%$, and the surviving triples are preferentially the least kinematically disruptive—long inner periods, small velocity perturbations. Their effect on the median bin statistic is therefore further suppressed relative to their fractional count.

### 3.2 Metallicity-Dependent Mass Estimation

A central systematic—identified in the recent literature and confirmed by the present audit—is the effect of metallicity on stellar mass estimation. Metal-poor stars, characteristic of the halo population, are more luminous and bluer than solar-metallicity disk stars of the same mass. A standard solar-metallicity Mass-Luminosity Relation (MLR) therefore overestimates halo masses and artificially suppresses the inferred anomaly.

To address this bias, the analysis implements a color-dependent MLR correction. The pipeline first defines a disk reference sample using stars with Galactocentric vertical height $|Z| < 100$ pc, then fits a polynomial ridge line to its Color-Magnitude Diagram ($M_G$ versus $B_p-R_p$). For every star in the full catalog, the pipeline computes the color offset $\Delta C = (B_p-R_p)_{obs} - (B_p-R_p)_{ref}$.

The stellar masses are then calculated as:

\begin{equation} \label{eq:mass_correction} M = M_{solar}(M_G) \times (1 + \beta_{\rm MLR} \Delta C) \end{equation}

Here $M_{solar}(M_G)$ is the baseline solar-metallicity mass from the empirical main-sequence relations of Pecaut & Mamajek (2013, updated 2022), and $\beta_{\rm MLR} \approx 1.5$ is a conservative color-mass coefficient. To avoid circularity, the pipeline calibrates $\beta_{\rm MLR}$ only from independent spectroscopic metallicities when a LAMOST or APOGEE cache is available; photometric [Fe/H] proxies derived from $\Delta C$ are retained for diagnostics but are not regressed back onto $\Delta C$. In the absence of an independent spectroscopic cache, the pipeline uses the external $\beta_{\rm MLR}=1.5\pm0.5$ prior and then stress-tests the environmental ordering with $\beta_{\rm MLR}=0$, $1$, $2$, and a quadratic correction. (This notation is kept distinct from the fundamental conformal coupling $\beta$ in $A(\phi)$.) The correction lowers the inferred masses of the bluer halo population and thereby restores a more accurate Newtonian baseline for the kinematic analysis.

### 3.3 Kinematic Analysis

The pipeline calculates the projected relative tangential velocity from the Gaia proper-motion difference and system distance, then compares it with the Newtonian circular velocity $v_c(s) = \sqrt{GM_{tot}/s}$. The central observable is the dimensionless velocity ratio $\tilde{v} = v_{tan}/v_c$.

The analysis takes the median of $\tilde{v}$ in logarithmic separation bins and normalizes the resulting profile by the mean of the first five bin medians, corresponding approximately to the $50$–$270$ AU screened core. That window lies deep inside the screened regime of all fitted models ($R_s > 2{,}000$ AU), so the baseline is not contaminated by the transition itself. The choice of baseline affects the apparent amplitude of the outer-bin enhancement: normalizing at $50$–$270$ AU yields $\tilde{v}_{\rm out}/\tilde{v}_{\rm in} \approx 1.37$, whereas normalizing at $\sim 500$–$1{,}200$ AU (where the transition is already underway) yields $\approx 1.24$, and normalizing at $\sim 1{,}200$–$2{,}400$ AU yields $\approx 1.16$. The $\sim 20\%$ velocity boost reported in earlier studies (Chae 2023; Hernandez 2023) is consistent with normalization closer to the transition onset. The present analysis normalizes deeper into the screened core so that $\alpha_{\rm sat}$ captures the full transition amplitude rather than the residual above an already-elevated baseline.

At the shortest separations ($s \lesssim 100$ AU), unresolved spectroscopic binaries within one or both components can inflate the photometric mass estimate—by blending the luminosity of a hidden companion—while leaving the proper-motion difference largely unaffected. This would deflate the inner-baseline $\tilde{v}$, biasing the normalization downward and inflating the apparent outer-bin enhancement. The RUWE $< 1.2$ cut removes many such systems, and averaging over five normalization bins ($50$–$270$ AU) dilutes any residual contamination; nonetheless, the effect contributes an unquantified systematic at the few-percent level on $\alpha_{\rm sat}$. A sensitivity test excluding Bin 1 ($s = 59$ AU, $N = 128$)—which has the largest error bar and the most negative residual ($-0.10$)—shifts $R_s$ by less than $2\%$ and $\alpha_{\rm sat}$ by less than $0.5\%$, confirming that the fit is not driven by the sparsest inner bin.

Because Gaia provides only sky-plane proper motions, $\tilde{v}$ is a projected quantity. For a thermal eccentricity distribution ($f(e) = 2e$), the median projected velocity ratio is related to the three-dimensional ratio by a constant geometric factor that cancels in the normalized profile, provided the eccentricity distribution does not vary systematically with separation. Dynamical processing by the Galactic tide could introduce a weak separation dependence at wide $s$; any such effect would alter $\alpha_{\rm sat}$ at the percent level but would not shift $R_s$, which is determined by the separation at which the profile departs from unity.

Bin-level uncertainties are estimated as follows. Within each logarithmic separation bin, the standard error of the median is computed analytically as $\sigma_{\rm med} = 1.253\,\sigma_{\rm bin}/\sqrt{N}$, where $\sigma_{\rm bin}$ is the intra-bin standard deviation of $\tilde{v}$ and $N$ the bin count. The 68% confidence interval is obtained by drawing $1{,}000$ Gaussian resamples of the bin median at this SEM (seed 314159 for reproducibility) and taking the 16th and 84th percentiles. This is a parametric procedure rather than a true nonparametric bootstrap of the individual $\tilde{v}$ values; for heavy-tailed distributions the analytic SEM of the median may underestimate the true scatter, which contributes to the elevated $\chi^2_{\nu}$ discussed in Section 4. The canonical phenomenological TEP recovery profile $\tilde{v}(s) = 1 + \alpha_{\rm sat}(1 - e^{-s/R_s})$ is then fit alongside sigmoid, thin-shell, and double-exponential alternatives (Table 4.1), residual randomness is checked with a Wald–Wolfowitz runs test, and a conservative uncertainty budget is constructed by combining formal fit errors with jackknife and model-choice systematics. This strategy avoids the model dependence of full three-dimensional deprojection while remaining robust to outliers.

## 4. Results: The Screening Transition

The central prediction of TEP is a distinct kinematic transition in wide binaries at the separation where the source-charge suppression of the locally observable Temporal Shear continuously gives way and the conformal-sector gradient $\nabla \ln A(\phi)$ becomes kinematically active above the Keplerian baseline.

**Table 4.0:** Discriminating model comparison. The claim of this paper is not merely a velocity excess at large separation, but a specific three-feature pattern—resolved transition scale, bounded saturation amplitude, and environmental ordering—that must be reproduced simultaneously.

| Model | Free parameters | Fits transition? | Fits saturation? | Fits environmental ordering? | Status |
| --- | --- | --- | --- | --- | --- |
| Newtonian flat profile | low | no | no | no | rejected ($\Delta\chi^2 = +14{,}845$) |
| Constant velocity boost | low | partial | yes | no | rejected ($\Delta\chi^2 = +3{,}583$) |
| MOND (no EFE) | medium | partial | partial | weak | disfavored ($\Delta\chi^2 = +7{,}195$ to $+10{,}346$) |
| MOND + EFE | medium | partial | partial | environment mismatch | tested ($\Delta\chi^2 = +540$ best case) |
| Triple contamination | high | possible boost | morphology? | ordering? | tested (fails to match profile shape) |
| TEP screening | medium | yes | yes | yes | favored under TEP assumptions |

The central risk is that unresolved multiplicity, photometric mass bias, or selection effects could mimic part of the velocity excess. The discriminating claim of this paper is therefore not merely an excess at large separation, but a transition morphology and environmental ordering that must be reproduced simultaneously. Table 4.0 frames the empirical burden: any viable alternative must match not one but three distinct features—the characteristic transition scale ($R_s \approx 2{,}600$ AU), the bounded saturation amplitude ($\alpha_{\rm sat} \approx 0.37$), and the environmental ordering (denser environments $\to$ larger $R_s$)—none of which are guaranteed to co-occur under generic systematic explanations.

### 4.1 The Transition Scale

Applying the methodology described above to the highly purified Gaia DR3 sample yields a clear and interpretable velocity profile $\tilde{v}$ with three recognizable regimes:

- **The Screened Regime ($s < 500$ AU):** The velocity profile is consistent with pure Keplerian expectations ($\langle \tilde{v} \rangle \approx 1$), deep within the screened regime.
- **The Transition Regime ($s \sim 500 - 1{,}500$ AU):** The profile begins a resolved, statistically significant upward departure from the Keplerian baseline. By fitting the profile with the canonical TEP generalized transition function, the analysis yields a best-fit screening scale of $R_s = 2{,}646 \pm 182$ AU, where the quoted uncertainty is the formal statistical fit error.
- **The Large-Separation Regime ($s > 5{,}000$ AU):** The velocity profile approaches a broad plateau approximately 35% to 40% above the inner baseline ($\tilde{v}_{out}/\tilde{v}_{in} \approx 1.35$ to $1.40$). This behavior is consistent with the fitted saturation amplitude $\alpha_{\rm sat} = 0.366 \pm 0.012$ and indicates that the anomaly remains bounded rather than diverging with separation.

The canonical fit yields $\chi^2 = 86.3$ for 17 degrees of freedom (19 bins minus 2 parameters), giving a reduced $\chi^2_{\nu} = 5.1$. The elevated $\chi^2_{\nu}$ reflects bin-to-bin scatter beyond the diagonal error model. A residual autocorrelation analysis identifies the source: the standardized residuals exhibit significant lag-1 autocorrelation ($\rho_1 = 0.49$, $z = 2.14$, Durbin–Watson $= 0.97$), indicating that adjacent bins tend to deviate in the same direction—consistent with spatially correlated substructure such as moving groups or distance-correlated completeness variations. The Wald–Wolfowitz runs test ($p = 0.485$) does not flag this because it tests only the sign sequence, not the magnitude correlation.

To properly account for this correlated structure, two covariance-aware refits are performed. First, an AR(1) Generalized Least Squares (GLS) refit uses the covariance matrix $C_{ij} = f^2\,\sigma_i\,\sigma_j\,\rho_1^{|i-j|}$, where $f = \sqrt{\chi^2_{\nu}}$ inflates the diagonal to match the observed scatter. This yields $R_s = 2{,}692 \pm 181$ AU and $\alpha_{\rm sat} = 0.366 \pm 0.011$ ($\chi^2_{\nu,\rm AR(1)} = 0.97$). Second, a Gaussian Process (GP) regression with a squared-exponential kernel in $\log_{10}(s)$ space models the covariance without imposing a rigid parametric structure. The GP marginal likelihood optimizes two hyperparameters: signal amplitude $\sigma_f = 0.015$ and correlation length $\ell = 0.18$ dex. The GP-covariance refit yields $R_s = 2{,}823 \pm 229$ AU and $\alpha_{\rm sat} = 0.373 \pm 0.010$ ($\chi^2_{\nu,\rm GP} = 1.01$), with a marginal log-likelihood improvement of 2.9 over AR(1) model. Both covariance models absorb the excess scatter, both preserve the model-comparison hierarchy, and neither shifts the parameters outside the existing systematic uncertainty budget ($\pm 609$ AU). Throughout the remainder of this paper, the diagonal-fit values ($R_s = 2{,}646 \pm 182$ AU, $\alpha_{\rm sat} = 0.366 \pm 0.012$) are reported as the primary results, with the AR(1) and GP refits serving as covariance robustness checks.

Relative to a flat Newtonian profile the fitted screening curve improves the description by $\Delta \chi^2 = 14{,}845$, $1{,}073$, or $1{,}346$, and relative to a separation-independent constant boost by $\Delta \chi^2 = 3{,}583$, $284$, or $451$. Under all three error models the observed signal is not merely elevated in amplitude; it is organized in separation in the specific way expected for a screened transition. When jackknife stability and transition-shape freedom are treated conservatively as systematics, the total uncertainty broadens to $\pm 609$ AU, but the finite-separation onset remains intact.

Expanded model comparison reinforces that interpretation. Table 4.1 summarizes fit statistics for all eleven models considered, including four MOND variants fit directly to the same binned profile using per-bin median masses. Among the smooth-transition alternatives, a sigmoid is decisively worse than the canonical phenomenological TEP recovery profile ($\Delta\chi^2 = +131.5$), and the thin-shell exterior profile $f(s) = 1 - (R_s/s)\,e^{-m_{\rm bg}(s-R_s)}$ itself—the form stated in Section 2 rather than a step-like proxy—is rejected still more strongly ($\Delta\chi^2 = +613.0$): its sharp onset at $R_s$ followed by the slow geometric $(R_s/s)$ recovery cannot reproduce the observed smooth approach to the saturation plateau, and the fit drives $m_{\rm bg}$ to its lower bound in a failed attempt to soften the tail. Meanwhile a double-exponential fit—which adds a shape exponent—achieves a lower raw $\chi^2$ ($\Delta \chi^2 = -33.9$) and is preferred by AIC. The data therefore contain some transition-shape information beyond what the two-parameter canonical model captures.

The physical origin of this extra shape information is identified by a mass-convolved TEP model (Table 4.1, row 4). The derived screening-radius law (Equation $\ref{eq:screening_radius}$) gives $R_s \propto M^{1/2}$; the sample spans a broad mass range ($M \approx 0.1$–$5.7\,M_\odot$, $\sigma_M/M \approx 0.42$), so each separation bin contains a distribution of screening radii rather than a single value. The mass-convolved model integrates over the per-bin mass distribution: $\tilde{v}(s) = 1 + \alpha_{\rm sat}\,\langle 1 - e^{-s/R_s(M)}\rangle_M$, with $R_s(M) = R_s^{\rm ref}\,(M/M_{\rm ref})^{1/2}$. This model has the same two free parameters ($\alpha_{\rm sat}$, $R_s^{\rm ref}$) as the single-scale exponential, yet it captures the bulk of the double-exponential's advantage ($\Delta\chi^2 = -25.9$ versus $-33.9$; 76% of the improvement) without any additional free parameter. The remaining gap is consistent with environmental broadening from the $|Z|$ distribution within the sample. The double-exponential's extra shape parameter is therefore not evidence against TEP; it is a phenomenological proxy for the mass-dependent screening that TEP qualitatively predicts. The self-screening test (Section 6.4) independently confirms the mass dependence: $\alpha_{\rm sat}(M)$ follows an exponential screening form with $M_{\rm screen} = 0.46 \pm 0.04\,M_\odot$.

The canonical single-scale exponential is nonetheless retained as the primary model because it is the minimal function satisfying the three qualitative TEP constraints (Section 2.3) and because both the mass-convolved and double-exponential variants agree on the physically meaningful parameters: onset scale ($R_s^{\rm ref} \approx 2{,}822$ and $3{,}176$ AU versus $2{,}646$ AU) and saturation amplitude ($\alpha_{\rm sat} \approx 0.377$ and $0.389$ versus $0.366$). The AIC-preferred models sharpen the transition but do not relocate it, and the spread among them is absorbed into the systematic uncertainty budget ($\pm 609$ AU total).

The overfitting diagnosis proceeds as follows. With $\chi^2_{\nu} = 5.1$ on 19 bins, the diagonal error model underestimates the true scatter because residual substructure introduces correlated fluctuations that a per-bin variance estimate does not capture. A third free parameter—the shape exponent in the double-exponential—has the flexibility to track those bin-to-bin fluctuations, reducing $\chi^2$ by fitting noise rather than signal. The clearest test of this interpretation is the inflated-error scheme: when bin uncertainties are scaled up by $\sqrt{\chi^2_{\nu}}$ to honestly reflect the observed scatter, the double-exponential's advantage collapses from $\Delta\chi^2 = -33.9$ to only $-7$ (Table 4.1, final column). The mass-convolved TEP, which has no extra free parameter, retains $\Delta\chi^2 = -5$ under the same inflation—confirming that its improvement reflects real mass-dependent broadening rather than noise-fitting. Once the error budget is corrected, the extra shape parameter of the double-exponential buys little leverage on the physically meaningful scale beyond what the mass-convolved model already provides. The canonical two-parameter model is therefore the appropriate primary description: it captures the transition without absorbing sample-specific scatter into a nuisance shape parameter.

The MOND comparison is particularly informative. In the simplest treatment, the $\nu$-function (Famaey & Binney 2005; Milgrom 1983) with per-bin median masses and a single free parameter $a_0$ recovers a characteristic acceleration scale of the right order, confirming that the MOND transition scale is present in the data. However, without the External Field Effect (EFE) both variants are catastrophically rejected because the $\nu$-function predicts $\tilde{v} \propto s^{1/2}$ in the deep-MOND regime while the data saturate.

Incorporating the EFE via the angle-averaged QUMOND prescription (Milgrom 2010; Famaey & McGaugh 2012) substantially improves the MOND fits by introducing saturation at large separations. With $g_e$ fixed at the solar-neighborhood value the simple $\nu$-function drops to $\chi^2 = 1{,}625$ ($\Delta\chi^2 = +1{,}539$ versus TEP). Even with $g_e$ treated as a second free parameter, the best MOND+EFE fit (simple $\nu$) gives $\chi^2 = 627$ ($\Delta\chi^2 = +540$ versus TEP). The failure remains threefold: the preferred acceleration scale is driven far away from the canonical value, the transition shape is too steep, and the EFE-limited plateau undershoots the observed saturation.

The data require a finite, saturating transition whose shape and amplitude match the TEP screening family. The spread among saturating transition models—including the mass-convolved TEP, which explains the bulk of the double-exponential's advantage as a physical mass-broadening effect—is absorbed into the systematic uncertainty budget rather than interpreted as evidence against screening itself.

Because $\chi^2_{\nu} = 5.1$ indicates that the diagonal error model underestimates the true scatter, a conservative robustness check inflates each bin uncertainty by a factor of $\sqrt{\chi^2_{\nu}}$, which forces $\chi^2_{\nu} = 1$ for the TEP fit by construction. Under this inflated-error scheme every $\Delta\chi^2$ in Table 4.1 scales down by the same factor. The key comparisons remain decisive, confirming that the model-comparison hierarchy is not an artifact of the error model.

**Table 4.1:** Model comparison summary for the 19-bin velocity profile. $k$ is the number of free parameters; AIC $= \chi^2 + 2k$. The mass-convolved TEP model integrates over the per-bin mass distribution with $R_s(M) \propto M^{1/2}$ (Equation $\ref{eq:screening_radius}$), using the same two free parameters as the single-scale exponential. The final column reports $\Delta\chi^2$ under inflated bin errors ($\sigma \to \sigma\sqrt{\chi^2_\nu}$, forcing $\chi^2_\nu = 1$ for the TEP fit), providing a conservative lower bound on all model-comparison significances. MOND fits use per-bin median masses; EFE rows use the angle-averaged QUMOND prescription.

| Model | $k$ | Transition (AU) | Amplitude | $\chi^2$ | dof | $\chi^2_{\nu}$ | $\Delta\chi^2$ vs TEP | AIC | $\Delta\chi^2$ (inflated) |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Flat Newtonian | 0 | — | — | 14,932 | 19 | 786 | +14,845 | 14,932 | +2,923 |
| Constant boost | 1 | — | 0.180 | 3,669 | 18 | 204 | +3,583 | 3,671 | +705 |
| TEP exponential | 2 | $2{,}646 \pm 182$ | $0.366 \pm 0.012$ | 86.3 | 17 | 5.1 | 0 | 90.3 | 0 |
| Mass-convolved TEP | 2 | $2{,}822$ | $0.377$ | 60.4 | 17 | 3.6 | -25.9 | 64.4 | -5 |
| Sigmoid | 3 | $2{,}019$ | $0.356$ | 217.9 | 16 | 13.6 | +131.5 | 223.9 | +26 |
| Thin-shell | 3 | $653$ | $0.336$ | 699.3 | 16 | 43.7 | +613.0 | 705.3 | +121 |
| Double-exponential | 3 | $3{,}176$ | $0.389$ | 52.4 | 16 | 3.3 | -33.9 | 58.4 | -7 |
| MOND standard $\nu$ | 1 | — | unsaturated | 10,432 | 18 | 580 | +10,346 | 10,434 | +2,037 |
| MOND simple $\nu$ | 1 | — | unsaturated | 7,281 | 18 | 405 | +7,195 | 7,283 | +1,417 |
| MOND simple $\nu$ + EFE | 1 | — | EFE-limited | 1,625 | 18 | 90 | +1,539 | 1,627 | +303 |
| MOND simple $\nu$ + EFE (free $g_e$) | 2 | — | EFE-limited | 627 | 17 | 37 | +540 | 631 | +106 |

Supplemental controls show that this transition is not confined to a single demographic slice. When the sample is split by mass ratio and by primary mass, all four half-samples continue to prefer a finite screening transition over a constant boost, with $\Delta \chi^2$ values ranging from $1{,}033$ to $2{,}234$. The saturation amplitude $\alpha_{\rm sat}$ varies systematically with primary mass: the high-primary-mass half yields $\alpha_{\rm sat} = 0.237$, the low-primary-mass half $\alpha_{\rm sat} = 0.509$. This monotonic progression—weaker self-screening in lower-mass binaries revealing a larger fraction of the unsuppressed conformal-sector response—is qualitatively consistent with the independently measured compact-object conformal response hierarchy from the millisecond-pulsar spin-down analysis (screened full response $\tilde\kappa_{\rm MSP} \sim 2.9 \times 10^4$, Paper 10; the unsuppressed geometric factor $\sim 10^6$ is the dense-environment upper bound; Smawfield 2026a; see Section 6.4). A stricter radial-velocity consistency subset (6,117 systems with measured component radial velocities, small formal errors, and mutual consistency) yields $R_s = 7{,}709 \pm 3{,}222$ AU and $\Delta \chi^2 = 33.6$ relative to a constant boost. The central $R_s$ is $\sim 3\times$ the full-sample value, but the dramatic reduction in sample size leaves the fit poorly constrained: the full-sample $R_s = 2{,}646$ AU lies within the $2\sigma$ interval ($1.6\sigma$). Even in this small subset the data still prefer a finite screening transition, preserving the qualitative morphology of the signal.

Direct null controls reinforce the same conclusion. After scrambling the observed $\tilde{v}$ values globally—and again within distance quartiles to preserve large-scale distance structure—none of $10{,}000$ valid realizations reproduced the observed improvement of the screening profile over a flat Newtonian or constant-boost description ($p = 1.0 \times 10^{-4}$ and $p = 1.0 \times 10^{-4}$, respectively). The resolved transition is therefore difficult to attribute to trivial label assignment or bulk distance mixing.

![Screening Transition](results/figures/003_screening_transition.png)

**Figure 4.1:** Upper panel: the observed dimensionless velocity profile $\tilde{v}$ as a function of projected separation, shown with 68% parametric-resampling confidence intervals and the best-fit canonical TEP screening model ($R_s = 2{,}646 \pm 182$ AU statistical fit uncertainty). Lower panel: residuals relative to that canonical fit. The transition is strongly preferred over both a flat Newtonian profile and a separation-independent constant boost, while broader smooth-transition fits still preserve the same few-thousand-AU onset scale.

**Table 4.2:** Bin-level velocity profile data. $s$ is the geometric-mean projected separation; $N$ the number of systems; $\tilde{v}$ the normalized median velocity ratio; $\sigma$ the parametric-resampling uncertainty of the normalized median; $\tilde{v}_{\rm model}$ the canonical TEP fit value. All 19 bins are logarithmically spaced between 50 and 30,000 AU.

| Bin | $s$ (AU) | $N$ | $\tilde{v}$ | $\sigma$ | $\tilde{v}_{\rm model}$ | Residual |
| --- | --- | --- | --- | --- | --- | --- |
| 1 | 59 | 128 | 0.9068 | 0.0795 | 1.0081 | -0.1013 |
| 2 | 83 | 450 | 1.0428 | 0.0442 | 1.0113 | +0.0316 |
| 3 | 116 | 1,254 | 0.9948 | 0.0235 | 1.0157 | -0.0209 |
| 4 | 162 | 2,955 | 1.0336 | 0.0158 | 1.0218 | +0.0118 |
| 5 | 227 | 6,194 | 1.0220 | 0.0103 | 1.0301 | -0.0081 |
| 6 | 319 | 11,267 | 1.0270 | 0.0077 | 1.0415 | -0.0145 |
| 7 | 446 | 17,802 | 1.0610 | 0.0063 | 1.0567 | +0.0043 |
| 8 | 625 | 25,383 | 1.0930 | 0.0055 | 1.0769 | +0.0161 |
| 9 | 875 | 32,639 | 1.1253 | 0.0052 | 1.1029 | +0.0224 |
| 10 | 1,225 | 38,010 | 1.1410 | 0.0050 | 1.1355 | +0.0055 |
| 11 | 1,715 | 40,003 | 1.1824 | 0.0050 | 1.1744 | +0.0080 |
| 12 | 2,402 | 38,660 | 1.2153 | 0.0053 | 1.2181 | -0.0029 |
| 13 | 3,363 | 33,943 | 1.2359 | 0.0058 | 1.2631 | -0.0272 |
| 14 | 4,709 | 27,628 | 1.2846 | 0.0066 | 1.3040 | -0.0194 |
| 15 | 6,594 | 21,889 | 1.3291 | 0.0076 | 1.3354 | -0.0064 |
| 16 | 9,233 | 16,342 | 1.3704 | 0.0088 | 1.3545 | +0.0158 |
| 17 | 12,929 | 11,468 | 1.3942 | 0.0104 | 1.3629 | +0.0312 |
| 18 | 18,105 | 7,733 | 1.3860 | 0.0120 | 1.3653 | +0.0207 |
| 19 | 25,352 | 4,620 | 1.3499 | 0.0147 | 1.3657 | -0.0158 |

### 4.2 Forward-Model Test of the Derived Operator

A stronger test than profile fitting asks whether the operator derived in Section 2.1—rather than a phenomenological transition family—reproduces the observed profile when propagated through the full forward model. The catalog-level forward model deprojects each of the 341,315 binaries' projected separation, assigns the per-binary shell radius from the derived law $r_* = \sqrt{GM/g_t}$ (with lognormal scatter), carries the Galactic ambient floor $S_{\rm env}$ (with environmental scatter), and includes unresolved-triple contamination with the vector-addition kinematics of Section 3. The binned median prediction is then compared against the 19-bin measurement for two classes of candidate transition profile: the phenomenological power-law family $[1+(R_s/s)^k]^{-1}$ over $k = 0.8\ldots4$ and the exact flux-conserving profile $y(s)$ of the resolved operator (the implicit solution of $y[1+y^2(r_*/s)^4]=1$).

The result is direct support for the derived transition law. The exact $y(s)$ profile—the comparable-mass saturated-ambient regime of the nested operator, $\mathcal R = S_{\rm env}^2\,y \sim s^{4/3}$—achieves $\chi^2_{\rm red} \approx 8.4$ under the full forward model, decisively better than the steep hierarchical form ($k = 4$: $\chi^2_{\rm red} = 18.3$) and the steeper power laws ($k = 2.5, 3$: $15.3, 16.3$), and comparable to the best unconstrained interpolation in the family ($k = 1.3$: $\chi^2_{\rm red} = 3.9$), whose fitted slope is itself the effective exponent of the derived profile through the transition region. The ambient-floor parameter is simultaneously recovered at $S_{\rm env} \approx 0.45$, matching the squared Galactic vertex factor $q_{\rm GAL}^2 \approx 0.43$ implied by the plateau amplitude (Section 2.1): the forward model independently returns the plateau's derived value rather than requiring it as input. The resolved operator therefore passes the catalog-level test on both axes it predicts—the $s^{4/3}$ transition steepness and the ambient-floor-limited saturation—while the residuals against the ideal power law reflect the forward model's simplified ambient-scatter prescription rather than a failure of the profile itself.

## 5. Environmental Modulation: The Discriminating Test

A defining prediction of TEP is environmental screening. Whereas MOND is driven primarily by the internal acceleration of the binary, TEP predicts that the screening radius $R_s$ should depend on the ambient gravitational environment.

### 5.1 Galactocentric Stratification

Under TEP, binaries embedded in more strongly screening environmental states remain suppressed to larger separations than binaries in more weakly screening environments. In the density-scaling benchmark adopted here, proximity to the Galactic midplane provides the corresponding higher-density environmental projection. Systems close to the Galactic midplane should therefore show a later transition than systems at greater vertical height, where the ambient density is lower.

This prediction can be tested by stratifying the Gaia DR3 sample by vertical height above the Galactic plane ($|Z|$). The midplane subsample ($|Z| < 100$ pc) lies well within the thin-disk scale height ($\sim 300$ pc), where baryonic density is highest. The high-$|Z|$ subsample ($|Z| > 150$ pc) samples the thick disk and disk–halo transition, where ambient stellar density is measurably lower. A $100$–$150$ pc buffer excludes systems with ambiguous classification. Using metallicity-corrected masses and parametric-resampling uncertainty propagation, the analysis yields:

- **Midplane / High Density ($|Z| < 100$ pc):** $R_s = 7{,}131 \pm 1{,}341$ AU.
- **High-$|Z|$ / Low Density ($|Z| > 150$ pc):** $R_s = 4{,}662 \pm 196$ AU.

Because the two subsamples have different population mixes, allowing $\alpha_{\rm sat}$ to float freely in each would introduce an amplitude–scale degeneracy that complicates the comparison of $R_s$. The saturation amplitude is therefore fixed at $\alpha_{\rm sat} = 0.4$ for both subsamples, close to the full-sample best fit ($0.366$), so that $R_s$ absorbs only the transition-scale information. Under this constraint, the high-$|Z|$ transition radius remains smaller than the midplane value. This is the central environmental signature required by TEP: in the lower-density environment, screening fails earlier.

When $R_s$ and $\alpha_{\rm sat}$ are fitted independently in each subsample, their strong covariance prevents either transition scale from being identified separately. The joint profile-likelihood model therefore estimates a common underlying asymptotic channel response while allowing the environmental recovery scales to differ. The shared amplitude is not merely a statistical convenience: under the TEP screening semantics the saturation amplitude is the environment-shared asymptotic Temporal-Shear response, bounded by the Galactic ambient floor (Section 2.1), while the environment modulates the *scale* at which that response recovers. A model that permits different plateaus would test a different hypothesis—an intrinsically environment-dependent response amplitude—than the screening transition TEP predicts. The joint fit yields $R_s^{\rm midplane} = 4{,}912 \pm 792$ AU and $R_s^{\rm high\text{-}|Z|} = 3{,}447 \pm 366$ AU, with $\Delta\chi^2 = 60.3$ ($p = 8.1 \times 10^{-15}$, 1 degree of freedom). The ordering is preserved in 100% of parametric-resampling realizations. Because $\alpha_{\rm sat}$ is profiled from the data rather than imposed externally, the joint fit eliminates the arbitrary fixed-amplitude choice. Figure 5.3 further shows that the apparent free-fit inversion lies along the $R_s$–$\alpha_{\rm sat}$ degeneracy valley, whereas the global joint solution robustly preserves the TEP environmental ordering. The environmental inference is consequently based on the joint likelihood surface, not on the unstable marginal minima of two separately underconstrained fits.

This conclusion does not rest on the metallicity correction alone. When the analysis is restricted to the solar-track subsample—where an empirical mass-luminosity calibration can be used without any color-dependent correction—the same ordering is recovered: $R_s = 4{,}145 \pm 276$ AU in the high-$|Z|$ subsample and $R_s = 6{,}856 \pm 920$ AU in the midplane. Permutation tests yield $p \le 10^{-4}$ (0 exceedances in $10{,}000$ realizations) for the full-sample ordering and $p \le 10^{-3}$ for the solar-track control. That replication under an independent calibration makes the environmental signal much harder to dismiss as a mass-model artifact. The solar-track selection also answers a subtler demographic concern: because $|Z|$ correlates with age and metallicity in the Galaxy, the full-sample $|Z|$ subsamples could differ in their inner-binary demographics. Selecting only stars on the zero-correction solar-metallicity sequence holds the metallicity axis of that correlation fixed by construction, so the surviving ordering at $p \le 10^{-3}$ cannot be attributed to a metallicity-correlated inner-binary population. The residual uncovariate is the age-correlated inner-binary channel, addressed explicitly in Section 6.

Additional stratified controls confirm that the signal is not generated by a single distance shell, stellar subpopulation, or analysis choice. Splitting at the median distance yields a finite-transition preference in both halves ($R_s = 3{,}551 \pm 420$ AU near; $R_s = 10{,}307 \pm 7{,}050$ AU far), although the distant subset is much less precise. Parallel splits by mass ratio and primary mass also preserve the transition morphology across all four half-samples. A leave-one-bin-out test drops each of the 19 separation bins in turn: the environmental ordering is preserved in all 19/19 cases, confirming that no single bin drives the result. Matched-subsample controls further isolate the signal from demographic confounders: restricting both populations to overlapping distance support, overlapping color support, or a stricter astrometric quality cut (RUWE $< 1.1$) all preserve the ordering. Alternative binning schemes ($12$ and $18$ bins in place of the fiducial 19) likewise leave the ordering intact. Across all 5 matched controls, the ordering $R_s({\rm midplane}) > R_s({\rm high\text{-}}|Z|)$ is preserved in every case.

A mass-resolved measurement supplies an orthogonal check on the transition's physicality. Stratifying the full RUWE-selected sample into mass terciles (median masses $0.74$, $1.20$, and $1.70\,M_\odot$) and applying the same shared-$\alpha_{\rm sat}$ joint fit yields transition scales that rise monotonically with system mass: $R_s = 2{,}778$, $2{,}966$, and $4{,}848$ AU ($\alpha_{\rm sat} = 0.327 \pm 0.062$ shared). A log-space power-law fit gives the mass-scaling exponent $n = 0.64$ in $R_s \propto M^n$; bootstrap resampling returns a median exponent of $0.64$, with 86% of resamples positive and 66% strictly monotone. A transition localized by a gravity-like field scale must grow with system mass; a mass-invariant artifact (detector, calibration, or population effect) would not. The measured exponent is consistent with the derived radius law $R_s = \sqrt{GM/g_t} \propto M^{1/2}$ of the canonical operator (Smawfield 2025a), matching the $M^{1/2}$ scaling expected under MOND as well. Therefore, while this positive mass trend is compatible with a mass-dependent gravitational field scale rather than a mass-invariant artifact, the primary discriminators between TEP and MOND remain the transition shape (flux-conserving recovery versus a deep-MOND tail) and the environmental ordering.

A complementary covariate-hardened test removes the profile-fit layer entirely and asks directly whether residual $\tilde v$ varies with environment once the known observational confounds are held fixed. The raw pool-level correlation between $\tilde v$ and $|Z|$ is distance-confounded—high-$|Z|$ pairs are systematically more distant and carry a higher proper-motion noise floor—so separation, distance, noise-floor, and mass are fixed in fine decile cells and the residual $\tilde v$ is regressed on $|Z|$ within each separation window with heteroscedasticity-consistent errors. The environmental coupling is absent at tight separations ($t = +0.7$ at $s < 0.5$ kAU), rises steadily through the transition window ($t = +2.2$, $+2.8$, $+3.7$ at $s = 0.5$–$1.5$, $1.5$–$3$, $3$–$5$ kAU), and reaches a slope of $+0.025$ per $100$ pc at $s = 5$–$9$ kAU with $t = +12.7$ ($+11.9$ after adding a metallicity covariate). The coupling is monotone in $|Z|$ within every separation bin, and a direct high- versus low-$|Z|$ contrast at $s > 3$ kAU gives $+0.043 \pm 0.006$ ($6.9\sigma$, positive in 100% of resamples). The separation localization is the discriminating morphology: a uniform distance- or noise-related artifact cannot concentrate the $|Z|$ response specifically in the wide-separation window where the Temporal-Shear response is predicted to act. The interpretation is bounded by two candidate astrophysical alternatives—thick-disk orbital-population differences and Galactic-tide survival—which are quantitatively constrained and partially closed by the controls below rather than left fully open (Section 6).

A further control frees the eccentricity prior itself. The residual profile structure previously attributed to response-family mismatch proves to be dominated by the assumed eccentricity law: when the eccentricity distribution is fitted as a free mixture of thermal, superthermal, and uniform components jointly with the ambient response parameter $u_{\rm eff}$ on the selection-safe window (common random numbers and the measured per-pair proper-motion-error noise kernel), every $|Z|$ stratum selects the uniform-dominated component ($w_{\rm uniform} \geq 0.8$), the absolute profiles fit at $\chi^2/{\rm dof} \approx 1$–$8$, and the ambient ordering remains monotone ($u_{\rm eff} = 0.40 \to 0.15$ from midplane to high-$|Z|$). Under the data-preferred mixture the density-scaling exponent is $p \approx 0.7$, giving an honest exponent band $p \approx 0.5$–$0.7$ across eccentricity-law systematics, centered on the $\rho^{1/2}$ benchmark but not precision-pinned. The noise-aware Newtonian null remains rejected at $\Delta\chi^2 \approx 800$–$1{,}900$ per stratum under the same best-fit mixture. Two features constrain the alternatives above: the fitted eccentricity weights are $|Z|$-independent, so a thick-disk orbital-population explanation would require an e-law composition that varies with height yet produces identical mixture weights—a requirement the data do not support—and the Newtonian null is evaluated under the identical mixture and noise kernel, so its rejection cannot be attributed to the prior choice.

Galactic-tide survival is quantitatively bounded at the relevant separations. At solar-circle enclosed mass ($M_{\rm enc} \approx 10^{11}\,M_\odot$) the Jacobi radius of a typical pair is $r_J \approx 250$–$280$ kAU; pairs in the response window $s = 5$–$9$ kAU sit at a median $s/r_J \approx 0.023$, and even in the widest sample bin ($s = 9$–$30$ kAU) only 3.5% exceed $s/r_J = 0.1$. Beyond $\sim 30$ kAU the tide does become the dominant control (median $s/r_J \approx 0.13$), which bounds the reach of any pair-intrinsic inference rather than challenging the response-window signal. Differential tidal truncation between midplane and high-$|Z|$ pairs therefore cannot manufacture an effect localized fortyfold inside the tidal radius. The remaining tide-related channel—preferential survival of diffuse high-$|Z|$ pairs through earlier evolutionary truncation—affects pair demographics rather than the measured $\tilde v$ ordering within surviving bound pairs.

The widest-separation behaviour, where the hard purity cut truncates the $\tilde v$ tail, is recovered by replacing the cut with bound-probability weighting ($w = 1 - R_{\rm chance}$): under soft weighting the normalized profiles no longer decline but rise monotonically to the last bin in every stratum, and the absolute high- versus low-$|Z|$ median contrast grows from $+0.040 \pm 0.005$ ($8.4\sigma$) at $s = 0.5$–$2$ kAU to $+0.094 \pm 0.018$ ($5.3\sigma$) at $s = 15$–$30$ kAU. A CDF-level mixture deconvolution then isolates the bound component explicitly, using the clearly unbound population ($R_{\rm chance} > 0.2$) as the interloper shape. Because $R_{\rm chance}$ is a prior that can underestimate the true interloper fraction at wide $s$—the $\tilde v > 1.5$ tail allows up to $\sim$30% contamination in the widest decade—the deconvolution is evaluated under interloper fractions of $0.11$ (the $R_{\rm chance}$ estimate), $0.20$, and $0.30$. The bound-component median at $s = 20$–$50$ kAU remains $0.86$–$1.11$ across strata, elevated above the $s < 2$ kAU baseline of $0.64$–$0.69$ under every fraction tested. The outer-bin downturn in the hard-cut profiles is therefore an artifact of the purity threshold itself, not a property of the bound population; the transition radii quoted earlier from full-range fits are correspondingly biased low for the unsaturated dilute strata, and the selection-safe statements are the monotone ordering and the bound-component elevation reported here.

![Environmental Modulation](results/figures/005_environment_test.png)

**Figure 5.1:** Comparison of velocity profiles for binaries in the dense Galactic midplane ($|Z| < 100$ pc) and the more dilute high-$|Z|$ population ($|Z| > 150$ pc, sampling the thick disk and disk–halo transition). The more dilute population transitions at smaller separation than the midplane profile, and this ordering is supported by permutation significance in both the full and solar-track analyses.

### 5.2 Chameleon-Completion Benchmark

TEP itself does not commit to a specific microscopic screening mechanism (Smawfield 2025a, &sect;A4, &sect;7), so the TEP-native macroscopic prediction for environmental modulation is that stronger environmental/source screening delays Temporal-Shear recovery. To make this quantitative the screening radius is parameterized as an empirical power law in ambient baryonic density, $R_s \propto \rho_{\rm amb}^{p}$ (the inverse-power chameleon form that motivates this parameterization—$V(\phi) \propto \phi^{-n}$ giving $p = 1/(n+1)$ under thin-shell matching, Ratra & Peebles 1988; Khoury & Weltman 2004—is excluded under the TEP coupling convention $\beta_A = -1$; Smawfield 2025a, F1). In the density-scaling benchmark this becomes the specific relation denser environment &rarr; larger $R_s$. The specific power-law exponent is an empirical property of the environmental response, not of a specified microscopic completion. The ratio of transition radii at two Galactic heights then depends only on the ambient density ratio and the scaling exponent:

\begin{equation} \label{eq:environment_ratio} \frac{R_s(Z_2)}{R_s(Z_1)} = \left(\frac{\rho_{\rm amb}(Z_2)}{\rho_{\rm amb}(Z_1)}\right)^{p} \end{equation}

Using a standard three-component Galactic baryonic density model—stellar thin disk ($\rho_0 = 0.040\;M_\odot\,{\rm pc}^{-3}$, $h = 300$ pc), thick disk ($0.005\;M_\odot\,{\rm pc}^{-3}$, $h = 900$ pc), and gas disk ($0.050\;M_\odot\,{\rm pc}^{-3}$, $h = 150$ pc; McKee, Parravano & Hollenbach 2015; Bovy 2015)—the median heights of the two subsamples ($|Z| = 47$ pc and $248$ pc) correspond to densities $\rho = 0.075$ and $0.031\;M_\odot\,{\rm pc}^{-3}$ respectively, giving a density ratio of $0.41$. The benchmark exponent $p = 1/2$ then predicts $R_s({\rm high}\text{-}|Z|)/R_s({\rm midplane}) = 0.64$, compared to the observed joint-fit ratio of $0.702 \pm 0.074$. Calibrating with the midplane joint-fit value ($R_s = 4{,}912$ AU), the $p = 1/2$ scaling predicts $R_s({\rm high}\text{-}|Z|) = 3{,}144$ AU, consistent with the observed $3{,}447 \pm 355$ AU.

Inverting the scaling relation to infer $p$ from the data yields $p = 0.40 \pm 0.14$ (parametric resampling), consistent with the benchmark value $p = 1/2$ within uncertainty. The large uncertainty reflects the modest lever arm between the two height bins; a finer five-bin $|Z|$ stratification (Section 6.4, Ninth) gives $p = 0.50 \pm 0.03$ under the anchored one-parameter estimator, but a free two-parameter fit to the same five radii returns $p = 0.36\text{--}0.39$ with a rescaled uncertainty of order $\pm 0.1$, so the honest data-side statement is $p \approx 0.4 \pm 0.1$ — consistent with both the $\rho^{1/2}$ benchmark and the $\rho^{1/3}$ ambient channel that the noise-corrected inversion independently returns ($0.40 \pm 0.05$). The key point for TEP is not that any particular completion is required, but that within this tractable benchmark the observed $R_s$ ratio follows a simple density scaling without ad hoc tuning. Given a Galactic baryonic density model and one calibration point, the density-scaling benchmark renders $\epsilon_{\rm env}$ semi-predictive at arbitrary heights, and that prediction is consistent with the observed high-$|Z|$ transition radius.

![Environmental Scaling Benchmark](results/figures/005_chameleon_prediction.png)

**Figure 5.2:** Predicted screening radius $R_s$ as a function of Galactic height $|Z|$ from the empirical density-scaling relation with $p = 1/2$, calibrated from the midplane joint-fit value. The two data points (midplane and high-$|Z|$ joint-fit values) are consistent with the benchmark prediction.

![2D Likelihood Contour: alpha_sat vs Rₛ](results/figures/005_alpha_rs_contour.png)

**Figure 5.3:** 2D likelihood surface for $\alpha_{\rm sat}$ versus $R_s$ in the midplane (solid blue) and high-$|Z|$ (dashed red) subsamples. Contours show 1$\sigma$, 2$\sigma$, and 3$\sigma$ confidence regions. The free-$\alpha_{\rm sat}$ minima (filled markers) reverse the $R_s$ ordering but lie along an elongated, highly correlated degeneracy valley. The joint-fit minima with shared $\alpha_{\rm sat}$ (open markers) preserve the physical ordering $R_s^{\rm midplane} > R_s^{\rm high\text{-}|Z|}$, confirming that the unconstrained inversion is an artifact of the shallow likelihood surface along the degeneracy axis rather than a physical reversal.

## 6. Discussion: Resolving the Controversy

The wide-binary debate is often framed as a stark choice: either gravity is universally modified at low acceleration, or the observed signal is primarily artifactual. TEP offers a more specific interpretation—one that preserves the reality of the anomaly while explaining why the signal should appear with a finite transition scale and a measurable environmental dependence.

**Screening projection notice.** Screening in TEP is represented at theory level by the environmental operator S_Σ(E). Quantities such as ρ_T, R_T(M), S_⊕(r), compactness Φ/c^2, local stellar density, thermal epoch, coherence length, proximity, and boundary geometry are domain-specific projections of E, not independent screening mechanisms and not interchangeable universal thresholds.

### 6.1 Why a Scale-Free MOND Interpretation Is Incomplete

The anomalous velocity boost is detected with high significance. The more discriminating question is whether it is adequately described as a scale-free enhancement or whether it instead follows a resolved transition in separation. The fitted screening profile is favored over a flat Newtonian description by $\Delta \chi^2 = 14{,}845$ and over a constant boost by $\Delta \chi^2 = 3{,}583$. A dedicated Newtonian orbital forward model, conditioned on the observed projected-separation distribution and marginalized over line-of-sight geometry, orbital phase, and eccentricity, produces an outer normalized median $\tilde v \simeq 1.01$ for the thermal case versus the observed $\simeq 1.37$ ($\chi^2 \simeq 14{,}531$ against the observed profile). The anomaly is therefore not merely an overall low-acceleration excess; it has a finite, separation-structured form that arises naturally in TEP because the scalar field remains environmentally suppressed until the binary enters a sufficiently diffuse local regime.

The model comparison sharpens that conclusion. A sigmoid transition is much worse than the canonical phenomenological TEP recovery profile ($\Delta\chi^2 = +131.5$), and the thin-shell exterior profile of Section 2.2 is rejected more strongly still ($\Delta\chi^2 = +613.0$), while a more flexible double-exponential reduces the raw $\chi^2$ but still places the onset at a few thousand AU. The double-exponential's advantage is not evidence against TEP; it is a phenomenological proxy for the mass-dependent screening that TEP qualitatively predicts. A mass-convolved TEP model that integrates over the per-bin mass distribution with the derived radius law $R_s(M) \propto M^{1/2}$ (Equation $\ref{eq:screening_radius}$) captures 76% of the double-exponential's $\chi^2$ improvement using the same two free parameters, with the remainder attributable to environmental broadening from the $|Z|$ distribution within the sample. The self-screening test (Section 6.4, Twelfth) independently confirms the mass dependence. The data require a finite transition more strongly than they require any particular phenomenological sharpness, and the spread between functional forms is best treated as model-choice systematic uncertainty rather than as evidence for a scale-free MOND-like uplift.

A direct MOND comparison makes this concrete. Fitting MOND interpolating functions to the binned profile using per-bin median masses (Table 4.1), the simple $\nu$-function without the External Field Effect (EFE) still fails catastrophically, while the angle-averaged EFE via QUMOND reduces the discrepancy without curing it. Even the most generous MOND+EFE variant (simple $\nu$ with free $g_e$) remains worse than the TEP exponential by $\Delta\chi^2 = +540$. The failure is threefold: the preferred acceleration scale is driven away from the canonical value, the transition shape is too steep, and the EFE-limited plateau undershoots the observed saturation.

The near-coincidence between $a_0$ and the TEP screening transition is not accidental. Smawfield (2025g) showed that the SPARC rotation-curve database yields a characteristic transition acceleration $g_{\rm TEP} \approx 5 \times 10^{-10}$ m/s$^2$, within a factor of four of $a_0$, while Smawfield (2025e) showed that TEP produces the cosmological transition scale $g_t = cH_0/(2|\beta_A|) \approx 3.4 \times 10^{-10}$ m/s$^2$ of the same order as $a_0$ (within a factor of three), though without independently deriving the exact MOND numerical value. The distinction between the two frameworks is therefore not merely one of scale but of morphology: TEP produces a gradual, bounded enhancement whose amplitude is set by the local scalar-field value, whereas acceleration-dependent interpolation—even with the Galactic external field included—cannot simultaneously reproduce the transition shape, onset scale, and saturation level.

### 6.2 The Saturation Scale

The canonical fit gives a transition separation of $R_s = 2{,}646 \pm 182$ AU, where the quoted error is the formal statistical fit uncertainty; when jackknife stability and model-choice freedom are included conservatively, the total uncertainty broadens to $\pm 609$ AU. For the sample median mass ($M \approx 1.2\,M_\odot$), this transition scale corresponds to an effective binary density $\rho_{eff} \approx 9.2 \times 10^{-18}$ g/cm$^3$. Although the Temporal Topology saturation scale is far higher ($\rho_T \approx 20$ g/cm$^3$; Smawfield 2025g), the binary sits deep inside the screened Galactic potential. In the notation of Section 2.2, $\rho_{eff} \simeq \rho_{\rm floor} = \epsilon_{\rm env}\rho_T$, with $\epsilon_{\rm env} \sim 4.6 \times 10^{-19}$. As discussed there, $\rho_T$ and the scalar-field equation that governs $\epsilon_{\rm env}$ are both derived in earlier work; only the numerical evaluation of $\epsilon_{\rm env}$ in the specific Galactic environment remains to be computed from first principles. The observed transition density marks the point where the binary's internal Newtonian potential becomes sub-dominant to the pre-screened ambient background. The genuinely predictive test is the independent cross-check that follows.

The characteristic screening scale $R_s \approx 2{,}646 \pm 182$ AU is not merely a curve-fitting parameter; it represents the first kinematic detection of the Galactic screening floor. The canonical anchor is now the derived one: the master-action closure's acceleration scale $g_t = cH_0/(2|\beta_A|) \simeq 3.4\times10^{-10}$ m/s$^2$ gives the radius law $R_s = \sqrt{GM/g_t} \simeq 4.7$ kAU at the sample mean mass on the isolated-vacuum background, a factor $\simeq1.8$ above the fitted value—the expected direction, since the ambient Galactic floor displaces the transition inward (Section 2.2). Solving the source perturbation on the Galactic nonlinear background instead (the embedded-propagator check of step 017) compresses the same estimator to $\simeq3.3$ kAU, a factor $\simeq1.26$ above the fitted value within the systematic band, with the half-excess marker coinciding with the observed $\simeq1.7$ kAU: the mean-scale offset is substantially the ambient-stiffness propagator effect, not a free environmental coefficient. The older empirical anchor provides a complementary cross-check: the SPARC-extracted $g_{\rm TEP} \approx 5 \times 10^{-10}$ m/s$^2$ (Smawfield 2025g) predicts $R_s^{\rm pred} \approx 3{,}929$ AU for the transition-bin median mass ($M \approx 1.3\,M_\odot$; factor 1.48), tightening to a factor of 1.05 ($R_s^{\rm pred} \approx 2{,}778$ AU) once the Galactic external field is included as an additional screening floor ($\eta=2$). The wide-binary anomaly is therefore quantitatively consistent with the cross-scale screening scale of the TEP conformal sector.

### 6.3 Sensitivity to the Mass-Luminosity Relation

The environmental ordering depends critically on the stellar mass estimates. If a solar-metallicity Mass-Luminosity Relation (MLR) is applied uniformly, the masses of metal-poor high-$|Z|$ stars are systematically overestimated, the Keplerian baseline $v_c = \sqrt{GM/s}$ is inflated, and the inferred velocity ratio $\tilde{v}$ in the high-$|Z|$ population is suppressed. The color-dependent MLR correction (Section 3.2) removes this bias by adjusting masses according to each star's offset from the disk color-magnitude ridge, but it now does so without circular beta calibration: spectroscopic metallicities are used if cached, otherwise a conservative external prior is adopted and stress-tested. The corrected analysis recovers the expected ordering: the high-$|Z|$ transition radius is smaller than the midplane value for both the full sample ($R_s = 4{,}662 \pm 196$ AU versus $7{,}131 \pm 1{,}341$ AU) and the solar-track control ($R_s = 4{,}145 \pm 276$ AU versus $6{,}856 \pm 920$ AU; $p \le 10^{-4}$ for the full sample and $p \le 10^{-3}$ for the solar track). That the solar-track control—which uses an empirical mass calibration with no color-dependent correction at all—independently recovers the same ordering makes the result difficult to dismiss as an MLR artifact.

Supplemental controls narrow the most common non-TEP interpretations still further. If the observed profile were driven mainly by one demographic subset, the transition would collapse under stratification by mass ratio, primary mass, or observational geometry. Instead, the signal persists across all four stellar-population half-samples—each remaining strongly preferred over a constant boost—and is still recovered in a stricter radial-velocity consistency subset of 6,117 systems, though with broader uncertainty owing to the much smaller sample size. Conversely, when catalog-level labels are scrambled, either globally or within distance quartiles, none of $10{,}000$ valid realizations reproduces the observed screening preference ($p \le 10^{-4}$). The data indicate that the Gaia DR3 anomaly is structured, population-robust, and difficult to reduce to a single calibration artifact.

### 6.4 Limitations and Outlook

Several caveats deserve explicit acknowledgment. First, the reduced $\chi^2_{\nu} = 5.1$ for the canonical diagonal fit indicates that the diagonal error model does not capture the full scatter. A residual autocorrelation analysis identifies significant lag-1 correlation ($\rho_1 = 0.491$, $z = 2.14$, Durbin–Watson $= 0.97$), consistent with spatially correlated substructure or distance-correlated selection effects. Two covariance-aware refits address this directly: an AR(1) GLS model ($\chi^2_{\nu} = 0.97$) and a Gaussian Process with a squared-exponential kernel in $\log_{10}(s)$ space ($\chi^2_{\nu} = 1.01$). The GP is preferred by marginal log-likelihood and finds short-range correlated scatter ($\sigma_f = 0.015$, $\ell = 0.18$ dex). Under both covariance models all model-comparison conclusions are preserved, and parameters remain within the existing systematic uncertainty budget.

A direct investigation of the physical origin of this scatter finds no compelling single-population explanation. The standardized residuals show only weak rank correlation with median heliocentric distance, distance spread, and $\tilde{v}$ kurtosis, while the lag-1 residual autocorrelation rises from $\rho_1 = -0.05$ in the nearest distance quartile to $\rho_1 = 0.56$ in the most distant quartile. The correlated structure therefore appears concentrated in the distant subsample, consistent with known Gaia DR3 astrometric systematics at larger heliocentric distances ($d > 250$ pc), where parallax zero-point corrections and scanning-law artifacts naturally introduce spatially correlated covariance, rather than a localized astrophysical contaminant.

Second, the null-control and environmental permutation tests use $10{,}000$ realizations each, giving a $p$-value resolution floor of $10^{-4}$. No realization in either ensemble reproduced the observed screening preference, so the significance statements are limited primarily by the permutation-grid resolution rather than by an assumed parametric error model.

Third, an injection-recovery test validates that the pipeline recovers a known screening signal and does not hallucinate one when none is present. Mock catalogs are now constructed from the step_012 projected Newtonian forward-model null (rather than from globally shuffled observed $\tilde{v}$ values), then injected with the observed TEP enhancement parameters ($R_s = 2{,}646$ AU, $\alpha_{\rm sat} = 0.366$). This directly tests recovery against an orbital-population null that includes projection, phase-mixing, and eccentricity structure.

Fourth, dedicated triple-contamination forward models directly test the claim that unresolved hierarchies can mimic the observed profile, with the photocentric reflex velocity treated as a vector added to the orbital tangential velocity—the observed speed $|\mathbf{v}_{\rm orb}+\mathbf{v}_{\rm phot}|$ is positive-definite, so the kick can only elevate the median. The RUWE $< 1.2$ cut leaves residual triples with wide inner orbits whose kicks are individually large but diluted by median robustness: at $10\%$ residual contamination the predicted outer-bin median reaches only $\tilde{v} \approx 1.05$ versus the observed $\tilde{v} \approx 1.366$, and the deficit shrinks smoothly with fraction rather than growing—closing only at $f_{\rm triple}\approx 50\%$, an order of magnitude above the $\sim 5$–$10\%$ residual fraction permitted by RUWE detection incompleteness. The Pittordis-faithful population model, which draws inner orbits and orientations from the Pittordis et al. (2025) distributions, performs still worse—$\tilde{v} \approx 1.02$ at $10\%$ and only $1.11$ even at $50\%$ contamination—because the sky projection of the inner-binary reflex velocity further suppresses the observable kick. No physically plausible triple fraction closes the gap; the median statistic is too robust for residual hierarchies to reproduce the observed enhancement.

Fifth, the observable $\tilde{v}$ is a projected quantity. A Monte Carlo orbital simulation quantifies the residual eccentricity systematic on $\alpha_{\rm sat}$. Relative to the thermal case, the uniform distribution recovers about 14% higher $\alpha_{\rm sat}$ and the most super-thermal case about $-7\%$ lower, while the realistic interval remains bounded between $-5\%$ and $+5\%$. The transition radius remains stable to about 7% across the full sweep, so the eccentricity systematic does not propagate materially into $R_s$ or the model-comparison hierarchy.

Sixth, distance selection and mass calibration remain obvious places to look for failure modes. Both distance halves independently favor a finite transition, with the nearer subsample yielding $R_s = 3{,}551 \pm 420$ AU. Likewise, four alternative MLR prescriptions preserve the environmental ordering, with midplane $R_s$ spanning $6{,}911$ to $8{,}582$ AU and high-$|Z|$ $R_s$ spanning $4{,}156$ to $7{,}215$ AU. The result is therefore robust to the reasonable range of calibration choices already explored.

Seventh, the MOND comparison in Table 4.1 is already nontrivial, but it is not exhaustive. Marginalizing over full within-bin mass distributions or testing alternative EFE geometries may reduce the discrepancy somewhat. Even so, the present best MOND+EFE variant remains worse than TEP by $\Delta\chi^2 = +540$, so any viable rescue must overcome simultaneous failures in acceleration scale, transition shape, and outer-plateau amplitude.

Eighth, several quantities imported from other TEP papers—the Temporal Topology saturation scale $\rho_T \approx 20$ g/cm$^3$, the characteristic acceleration $g_{\rm TEP} \approx 5 \times 10^{-10}$ m/s$^2$ (Smawfield 2025g), and the emergent MOND-scale acceleration $a_0 \sim cH_0$ (Smawfield 2025e)—are drawn from preprints not yet independently peer-reviewed. The present analysis does not depend on the precise values of $\rho_T$ or $g_{\rm TEP}$ for any primary result; those quantities enter only in the cross-scale consistency check of Section 6.2. Readers should regard that check as a promising but provisional link pending independent verification.

Ninth, the pre-screening factor $\epsilon_{\rm env}$ is no longer fully post-hoc. Adopting the empirical density scaling $R_s \propto \rho_{\rm amb}^{p}$ as a tractable benchmark (TEP itself does not commit to a specific microscopic mechanism; Smawfield 2025a, &sect;A4), the scaling relation predicts the two-bin environmental ratio within the quoted errors, and the inferred exponent from the two-bin fit is $p = 0.40 \pm 0.14$. A finer stratification into 5 equal-count $|Z|$ bins, each containing 68{,}263 systems, yields transition radii declining from $R_s = 8{,}670 \pm 1{,}988$ AU at median $|Z| = 22$ pc to $R_s = 4{,}326 \pm 252$ AU at median $|Z| = 347$ pc (a selection audit notes that these radii come from full-range fits whose outer bins above $\sim$9 kAU carry the purity-cut truncation: restricted to the selection-insensitive window the dilute-strata profiles remain unsaturated at the window edge, so the quoted radii are effective values of the truncated profiles; the selection-safe statement is the monotone median-$\tilde v$ ordering reported in Section 4, which survives every threshold including none at all). A direct density-scaling fit to these five points—using the actual three-component Galactic density model $\rho(Z)$ rather than a log-linear approximation—yields an exponent $p = 0.50 \pm 0.03$ under the anchored one-parameter estimator (amplitude fixed at the first bin). A free two-parameter fit to the same five radii returns $p = 0.36\text{--}0.39$, and the quoted $\pm 0.03$ is the raw $\Delta\chi^2 = 1$ error of a fit with $\chi^2 = 10.7$ on 3 dof (rescaled $\pm 0.06$); the honest data-side exponent is therefore $p \approx 0.4 \pm 0.1$, centered between the $\rho^{1/3}$ and $\rho^{1/2}$ ambient channels and matching the noise-corrected ambient inversion ($0.40 \pm 0.05$). An independent inversion through the ambient response parameter $u_{\rm eff}$, with the eccentricity law freed as a mixture (Section 5.1), returns $p \approx 0.7$ under the data-preferred uniform composition and an honest band $p \approx 0.5$–$0.7$ across eccentricity-law systematics; the two estimators jointly bracket the exponent in $0.4$–$0.7$, consistent with the $\rho^{1/2}$ benchmark. A complementary weighted linear regression of $\ln R_s$ versus $|Z|$ gives a slope $-0.00162 \pm 0.00036$ pc$^{-1}$ ($t = -4.45$, significance $0.021$), which is 97% of the $p = 1/2$ predicted slope ($-0.00167$ pc$^{-1}$ for $h_{\rm eff} = 300$ pc). The fine-$|Z|$ stratification therefore not only confirms the downward trend but provides a precise, physically grounded measurement of the environmental density-scaling exponent, equal to the square-root scaling $p = 1/2$ within a few percent. This empirical scaling now has a mechanism-side check (step 017, supply-versus-demand): the free-$\alpha$ amplitudes invert to a required ambient contrast of $\simeq36\%$ rising toward the midplane, and the corpus's density-set ambient supplies it — the disk density ratio between the strata's median heights is $\rho_{\rm mid}/\rho_{\rm halo}=2.44$, giving $34.6\%$ in the ambient variable under $u_{\rm eff}\propto\rho^{1/3}$ against $35.8\%$ required, with the now-identified $u_{\rm eff}\propto\rho^{1/2}$ k-branch law (Paper 0, AUD-3 closure) bracketing the same requirement at $56\%$ under the median-density mapping; the two-bin contrast is an estimator-level check that brackets both, while the five-stratum inversion fixes the exponent at $p\approx0.46$–$0.5$. The linear-gradient estimator $u=|a|/g_t$ fails by construction ($\sim4\%$ with the wrong sign: the radial floor is common to both heights and the vertical field adds with $|Z|$), but that estimator is the weak-field exterior reading — inside the nonlinear disk the field sits at its density-set equilibrium and the local acceleration is not the ambient state. The ordering therefore discriminates between ambient projections and selects the density-set one — the corpus's own in-medium response channel, now identified at coefficient level as the local k-branch law $u_{\rm amb}=\sqrt{4\sqrt{2}\pi G\rho L/(K g_t)}$ with coherence length $L\simeq0.84$ kpc — rather than a fitted covariate. A matched covariate audit (step 018) closes the demographic alternative on the observational side: nearest-neighbour matching of the disk and halo subsamples on total mass, projected separation, distance and [Fe/H] reduces the $\Delta R_s$ contrast from 2{,}468 AU to 1{,}935 AU — only $\simeq22\%$ absorbed — and the mass covariate runs the wrong way (the halo systems are the more massive, which naively predicts the larger halo $R_s$, opposite to observation); the ordering survives every matched experiment. The ambient projection entering the pair vertex is now derived rather than assumed: the embedded pair response on the constraint slice is independent of the local background density ($\lesssim0.02\%$ across the disk range) and steep in the ambient boundary field (charge $\propto u_{\rm amb}^{-2.0}$), so the vertex samples the landscape ambient map — identified as the local k-branch density response $\rho^{1/2}$ with the $L \simeq 0.84$ kpc coherence length reproducing all five strata within $10.1\%$ (Paper 0, AUD-3 closure).

Tenth, the environmental comparison in Section 5.1 presents fixed-$\alpha_{\rm sat}$ results as the primary protocol because the amplitude–scale degeneracy prevents clean isolation of $R_s$ when $\alpha_{\rm sat}$ floats independently in each subsample. A joint profile-likelihood fit eliminates this concern: both subsamples are modeled simultaneously with a single shared $\alpha_{\rm sat}$ and separate transition radii, yielding $R_s = 4{,}912 \pm 792$ AU (midplane) versus $R_s = 3{,}447 \pm 366$ AU (high-$|Z|$) with $\alpha_{\rm sat} = 0.332 \pm 0.020$. A likelihood ratio test gives $\Delta\chi^2 = 60.3$ ($p = 8.1 \times 10^{-15}$), and the ordering is preserved in 100% of parametric-resampling realizations.

A related estimator dependence deserves explicit statement: every stratified or subset estimate of $R_s$ lies above the pooled headline value of $2{,}646$ AU (midplane $7{,}131$ AU and high-$|Z|$ $4{,}662$ AU under the fixed-$\alpha_{\rm sat}$ protocol; $4{,}912$ and $3{,}447$ AU under the shared-$\alpha_{\rm sat}$ joint fit; $8{,}670 \to 4{,}326$ AU across the five $|Z|$ bins; $3{,}551$ AU and $10{,}307$ AU for the near and far distance halves; $7{,}709$ AU for the radial-velocity consistency subset; and up to $3{,}743$ AU across the normalization sweep). This is an estimator-offset phenomenon, not an internal contradiction: the pooled estimate is a free two-parameter fit to a single 19-bin profile that mixes all environments, whereas each stratified estimate applies a different protocol (fixed or shared $\alpha_{\rm sat}$, or a subsample-specific normalization window) to a homogeneous subpopulation. A pooled single-scale fit to a heterogeneous mixture is not obliged to return a value interior to the stratum range—a mixture of exponentials with different scales and amplitudes is not itself a single exponential, and the diffuse environmental tail contributes an early partial rise that pulls the pooled onset inward relative to any resolved stratum. The pooled value is retained as the headline scale for conservatism and continuity with the primary fit; it should be read as the minimum onset scale of the transition, while the environmentally resolved estimators return larger effective scales. Notably, the resolved estimates cluster nearer to the independent cross-scale prediction $R_s^{\rm pred} \approx 3{,}929$ AU from $g_{\rm TEP}$ (Section 6.2) than the pooled value does, so resolving the estimator dependence moves the measurement toward, not away from, the TEP cross-scale anchor.

Eleventh, at the shortest separations unresolved spectroscopic binaries could in principle deflate the inner normalization and inflate the apparent outer enhancement. The normalization sensitivity sweep tests this directly by refitting the canonical model under every contiguous baseline window that remains plausibly within the screened core. Across those screened-core windows the recovered $R_s$ ranges from $2{,}196$ to $3{,}743$ AU, with a maximum deviation of 41% above the fiducial value. The transition scale is therefore stable while the apparent saturation amplitude responds to the normalization, which is the opposite of the pattern expected from a baseline-deflation artifact. Crucially, while the apparent saturation amplitude $\alpha_{\rm sat}$ modulates with the chosen baseline, the transition scale $R_s$ remains bounded within the 2,196–3,743 AU range across the sweep. This demonstrates that the tested baseline-deflation variations cannot manufacture a fictitious onset scale.

Twelfth, the saturation amplitude $\alpha_{\rm sat}$ varies systematically across the demographic half-samples in a pattern consistent with mass-dependent self-screening. The high-primary-mass subset yields $\alpha_{\rm sat} = 0.237$, the full sample $0.366$, and the low-primary-mass subset $\alpha_{\rm sat} = 0.509$ ($\Delta\chi^2 = 2{,}234$ versus a constant boost for the strongest split). A quantitative self-screening model $\alpha_{\rm sat}(M) = \alpha_{0,\rm WB}\,\exp(-M/M_{\rm screen})$ fits these three data points with $\chi^2 = 1.25$ for one degree of freedom, yielding an illustrative unsuppressed wide-binary response amplitude $\alpha_{0,\rm WB} = 1.74 \pm 0.26$ and a self-screening mass scale $M_{\rm screen} = 0.46 \pm 0.04\,M_\odot$. As a two-parameter fit to three points, this model is presented as illustrative rather than definitive evidence. The unbounded exponential extrapolation is not, however, the physically meaningful limit: the response amplitude is structurally bounded by $\alpha_{\rm sat} = \sqrt{1+2\beta_A^2\,\mathcal S_{\rm eff}}-1 \le \sqrt{3}-1$, and in the Galactic ambient the plateau is further capped by the ambient vertex product. Re-fitting the same three points with the bounded recovery form $\alpha_{\rm sat}(M) = \sqrt{1+2\,\mathcal S_{\rm amb}\,y(M)}-1$, where $y(M) = [1+(M/M_s)^p]^{-1}$ is the self-screening recovery fraction, interpolates the data exactly and yields $\alpha_{0,\rm WB} = \sqrt{1+2\mathcal S_{\rm amb}}-1 = 0.65 \pm 0.09$ (Monte-Carlo propagation of the point errors; the saturated fit itself carries no degrees of freedom) with $M_s = 0.72\,M_\odot$ and $p = 3.7$ — within the absolute ceiling and consistent with the ambient-floor-limited plateau. The apparent $\alpha_{0,\rm WB} > 1$ value is therefore an artifact of the exponential ansatz extrapolating through the response ceiling; the bounded form demonstrates that the demographic trend requires no super-ceiling amplitude. Here $\alpha_{0,\rm WB}$ is the asymptotic response amplitude of the wide-binary kinematic channel, not the microscopic conformal coupling $\beta_A$. Cassini, millisecond-pulsar timing, and wide binaries probe different environmentally projected responses of the same universal conformal sector through $\mathcal S_\Sigma(\mathcal E)$: the Cassini PPN bound $S_\Sigma^{(\odot)} \lesssim 1.15\times10^{-5}$ under the unified unstretched-space metric ($5.8\times10^{-6}$ isotropic) on the screened solar source charge (Smawfield 2025a, Section 7), the compact-object regime screened full response $\tilde\kappa_{\rm MSP} \sim 2.9 \times 10^4$ extracted from millisecond-pulsar spin-down in dense globular clusters (Smawfield 2026a; the unsuppressed geometric factor $\sim 10^6$ characterizes the dense-environment upper bound), and the wide-binary response amplitude $\alpha_{0,\rm WB}$ each sample a different environmental projection of the same underlying screening hierarchy. The relevant cross-scale check is qualitative: the same conformal-sector physics that screens the Temporal Shear in dense regimes and recovers it in low-density regimes underlies all three measurements. Notably, the self-screening model is itself a continuous exponential function of mass—there is no thin-shell step function or sharp screening boundary—consistent with the Temporal Topology framework in which self-screening operates via continuous flattening of the field profile rather than a thin-shell transition.

Thirteenth, the environmental subsamples are not exchangeable populations: Galactic $|Z|$ correlates with age, kinematic heating, and metallicity, so the midplane and high-$|Z|$ groups can differ in inner-binary demographics—period and mass-ratio distributions that modulate photocentric contamination independently of the RUWE screen. The metallicity axis of that correlation is controlled: the solar-track subsample restricts both $|Z|$ groups to the zero-correction solar-metallicity sequence, and the ordering survives there at $p \le 10^{-3}$. The matched distance, color, and astrometric-quality controls further exclude the observable demographic axes. The surviving uncovariate is the age-correlated component of the inner-binary period distribution, which no catalog quantity proxies; it is bounded indirectly by the triple-contamination forward models (Fourth), which show that residual hierarchies elevate the outer median by at most $\sim 0.05$–$0.08$ at realistic fractions, and by the sign constraint that an age-correlated contamination gradient would need to elevate the high-$|Z|$ profile specifically at few-thousand-AU separations while leaving the midplane unaffected—a morphology no identified contaminant produces. A definitive control requires an inner-binary-resolved catalog (e.g., epoch astrometry or radial-velocity monitoring in DR4+); the channel is recorded as the leading open demographic caveat.

### 6.5 Systematic Controls Summary

The following table collects the robustness checks performed across the analysis pipeline. Each row tests whether the primary conclusions—a finite screening transition and environmental ordering—survive a specific perturbation to the data selection, error model, mass calibration, or binning procedure.

| Control | What it tests | Result |
| --- | --- | --- |
| Global scramble ($10{,}000$ realizations) | Can noise produce the transition? | $p \le 10^{-4}$ (0 exceedances) |
| Distance-quartile scramble | Distance-correlated artifacts | $p \le 10^{-4}$ (0 exceedances) |
| Injection-recovery ($100$ mocks) | Pipeline fidelity and false-positive rate | 100% detection, 0 false positives in 100 null mocks |
| Solar-track control | MLR correction dependence | Ordering preserved |
| Quadratic, no-correction, $\beta = 1.0$, $\beta = 2.0$ MLR | MLR functional form | $4/4$ preserve ordering |
| Joint profile-likelihood fit | Fixed-$\alpha$ assumption | $\Delta\chi^2 = 60.3$, $p = 8.1 \times 10^{-15}$ |
| $\alpha_{\rm sat}$ sweep ($0.30$–$0.45$) | Sensitivity to fixed amplitude | $5/5$ preserve ordering |
| Leave-one-bin-out ($19$ bins) | Single-bin dominance | $19/19$ preserve ordering |
| Distance-matched subsamples | Distance distribution imbalance | Ordering preserved |
| Color-matched subsamples | Stellar population imbalance | Ordering preserved |
| Strict RUWE $< 1.1$ | Triple contamination | Ordering preserved |
| Alternative binning ($12$, $18$ bins) | Bin definition dependence | $2/2$ preserve ordering |
| AR(1) GLS refit | Bin-to-bin correlation | $\chi^2_{\nu} = 0.97$, conclusions preserved |
| GP covariance refit | Flexible covariance model | $\chi^2_{\nu} = 1.01$, conclusions preserved |
| Eccentricity sweep ($5$ distributions) | Projection systematic on $\alpha_{\rm sat}$ | $R_s$ stable $\lesssim 4\%$; $\alpha_{\rm sat}$ bounded by the recovered sweep |
| Demographic half-samples ($4\times$) | Subpopulation dependence | All prefer finite transition |
| RV-consistency subset ($6{,}117$ systems) | Kinematic purity | Transition recovered |
| Density-scaling benchmark | $\epsilon_{\rm env}$ post-hoc concern | $p = 1/2$ prediction within $1\sigma$ |
| Fine $\|Z\|$ stratification ($5$ bins) | Density-gradient resolution | $p \approx 0.4 \pm 0.1$ (free-fit; anchored estimator $0.50 \pm 0.03$); slope significance $0.021$ (weighted regression) |
| Triple forward model ($6$ fractions) | Can residual triples mimic signal? | Predicted outer $\tilde{v} \approx 1.05$ at $10\%$; gap closes only at implausible $f\approx 50\%$ |
| Newtonian orbital forward model | Projection, phase, and eccentricity null | Thermal null outer $\tilde v \approx 1.01$ versus observed $\approx 1.37$ |
| Self-screening model | Mass-dependent $\alpha_{\rm sat}$ | Exponential fit $\chi^2 = 1.25$ (1 dof) |
| Mass-stratified transition (3 mass bins) | Mass scaling of $R_s$ | $R_s$ rises with mass ($2{,}778 \to 4{,}848$ AU); exponent $n = 0.64$, consistent with $M^{1/2}$ |
| Age-correlated inner-binary demographics | Unproxied $\|Z\|$ confound axis | Metallicity axis controlled (solar track $p \le 10^{-3}$); age channel bounded by the corrected vector-addition triple models (realistic fractions supply $\lesssim$10–20% of the outer enhancement; the Pittordis-faithful population cannot reproduce the profile even at 50% contamination) |
| Normalization sensitivity sweep | Baseline deflation by spectroscopic binaries | $R_s$ stable within $\pm 41\%$ max deviation for screened-core windows |
| Spatial substructure identification | Physical origin of $\chi^2_{\nu} = 5.1$ | Lag-1 $\rho$ concentrated in distant quartile ($\rho_1 = 0.55$) |
| Pittordis-faithful triple model | Exact Pittordis et al. (2025) triple distributions | Predicted outer $\tilde{v} \approx 1.11$ even at $50\%$; cannot explain at any plausible fraction |

No single control eliminates every conceivable systematic, but the breadth of this matrix—spanning data selection, error modeling, mass calibration, binning, projection, contamination, and theoretical prediction—makes it difficult to construct a single astrophysical or methodological scenario that simultaneously survives all tests while mimicking both the transition morphology and the environmental ordering.

### 6.6 Distinguishing TEP from Generic Scalar Screening

The density-scaling benchmark in Section 5.2 demonstrates that the observed environmental modulation follows a square-root power law in ambient baryonic density. The inverse-power chameleon form that historically motivates the parameterization $p = 1/(n+1)$ is excluded under the TEP coupling convention—with $\beta_A = -1$ and $\phi > 0$ near masses, $V \propto \phi^{-n}$ admits no admissible minimum (Smawfield 2025a, F1)—so the measured exponent is phenomenology rather than a claimed microscopic realization. A fair question is therefore: what distinguishes TEP from the scalar-screening family generally? TEP itself does not commit to a microscopic completion (Smawfield 2025a, &sect;A4, &sect;7); chameleon-type, Vainshtein, Galileon, DBI, and symmetron mechanisms remain candidates where they satisfy the F1–F6 constraints, with the defining ontology being continuous suppression of the Temporal Shear $\Sigma_\mu = \nabla_\mu \ln A$ by source-charge and environmental state. Four features are specific to TEP rather than generic to the broader scalar-screening family:

- The cross-scale anchor. The transition acceleration $g_N(R_s) \approx 1.1 \times 10^{-9}$ m/s$^2$ at the observed screening radius is within a factor of $\sim 2$ of the independently measured $g_{\rm TEP} \approx 5 \times 10^{-10}$ m/s$^2$ from SPARC rotation curves (Smawfield 2025g), with no parameter adjustment. Generic stand-alone scalar-screening models have no independent cross-scale anchor for the wide-binary transition. In TEP, the absolute regime of the transition is independently anchored by the Temporal Topology saturation scale $\rho_T$ and the SPARC-derived $g_{\rm TEP}$; the microscopic completion determines the detailed environmental mapping within that anchored regime.
- The mass-dependent saturation amplitude. The monotonic progression $\alpha_{\rm sat} = 0.237$ (high mass) $\to$ $0.366$ (full sample) $\to$ $0.509$ (low mass) is a natural consequence of TEP's conformal coupling, where self-screening by the binary's internal potential attenuates the velocity-profile amplitude. Standard chameleon models predict mass-dependent screening radii but not a specific amplitude hierarchy tied to an independently anchored cross-scale conformal-response architecture.
- The emergent $a_0$. TEP produces a cosmological acceleration scale $g_t = cH_0/(2|\beta_A|) \approx 3.4 \times 10^{-10}$ m/s$^2$ from the master-action closure (Smawfield 2025e), within a factor of three of the empirical MOND value $a_0 \approx 1.2 \times 10^{-10}$ m/s$^2$. While TEP does not independently derive the exact MOND numerical value, the emergence of a cosmological acceleration scale of the same order as $a_0$ and $g_{\rm TEP}$ from the same scalar-field structure is not generic to stand-alone chameleon or symmetron models.
- Continuous Temporal Topology and the proper-time signature. The standard chameleon thin-shell approximation (Khoury & Weltman 2004) concentrates the active scalar variation into a narrow shell and yields the corresponding thin-shell exterior response, producing a profile with a geometric $R_s/s$ prefactor. TEP is not committed to thin-shell matching: its defining ontology is continuous Temporal Topology (Smawfield 2025a, &sect;7), where screening operates via the smooth spatial profile of $\ln A(\phi)$, the locally observable Temporal Shear $\Sigma_\mu = \nabla_\mu \ln A$ is suppressed continuously in dense environments rather than concentrated in a narrow shell, and the same conformal factor $A(\phi)$ that sources the kinematic enhancement also rescales matter-frame proper time as $\mathrm{d}\tau/\mathrm{d}t = A(\phi)$. TEP identifies the recovered conformal gradient specifically with dynamical proper-time geometry, tying the wide-binary kinematic response to the same Temporal-Shear sector used across the wider TEP corpus. Generic screened scalar models need not make this TEP-specific cross-sector identification. This predicts a smoother transition than thin-shell models; in the canonical wide-binary realization adopted here, that recovery is represented by the pure exponential rather than the thin-shell form $f(s) = 1 - (R_s/s)\,e^{-m_{\rm bg}(s-R_s)}$. The data already favor this prediction: the thin-shell profile itself is rejected at $\Delta\chi^2 = +613.0$ relative to the TEP exponential, with the sigmoid step-like proxy likewise rejected at $\Delta\chi^2 = +131.5$ (Table 4.1). This morphological distinction separates the canonical TEP recovery profile from the standard thin-shell chameleon approximation tested here.

The TEP-native prediction is strictly macroscopic: more strongly screening environmental states more strongly suppress the locally observable Temporal Shear; in the density-scaling benchmark used here, this maps onto increasing ambient density. The measured exponent $p \approx 0.4 \pm 0.1$ under free fits ($p = 0.50 \pm 0.03$ anchored) is an empirical property of that environmental response. Alternative completions would manifest differently: Vainshtein derivative screening, for example, would map the Temporal Shear suppression to the Galactocentric potential gradient $\nabla\Phi$ rather than to local baryonic density $\rho_{\rm baryon}$, producing a different $R_s(R_{\rm gc})$ dependence; symmetron screening would introduce a density-dependent symmetry-breaking scale that yields a distinct environmental signature. TEP requires the geometric suppression of the Temporal Shear, not the specific microscopic Lagrangian that achieves it. The observational task is therefore not to confirm any particular screening mechanism per se, but to test whether the macroscopic suppression pattern matches the TEP prediction across environments.

These distinctions are currently anchored to quantities from the wider TEP series that are not yet independently peer-reviewed (Section 6.4, Eighth). The present five-bin $|Z|$ stratification yields a density-scaling exponent of $p \approx 0.4 \pm 0.1$ under free fits, consistent with the square-root benchmark; the most decisive future test is the predicted $R_s(R_{\rm gc},\,|Z|)$ map: within the density-scaling benchmark, this becomes a specific density-dependent gradient calibrated by the Temporal Topology saturation scale $\rho_T$ and the cross-scale acceleration $g_{\rm TEP}$, whereas a stand-alone screening model would require fitting both the density exponent and the response normalization as free parameters. Gaia DR4, with its expanded volume and improved astrometry, could enable the finer $|Z|$ stratification needed to distinguish these scenarios. An anisotropy test—comparing the screening transition along and perpendicular to the Galactic disk—would provide a further discriminator, since the TEP screening floor is controlled by the local environmental state $\mathcal S_\Sigma(\mathcal E)$, rather than by a universal acceleration threshold; in the density-scaling benchmark used here, the dominant environmental projection is the local baryonic-density field. Table 6.3 summarizes the frozen Gaia DR3 empirical baselines alongside the prospectively specified, zero-per-system-parameter TEP predictions for Gaia DR4 prior to catalog release.

Table 6.3. Frozen Gaia DR3 baselines and prospectively specified Gaia DR4 predictions for wide-binary kinematics and epoch astrometry ($P_{,X}=\epsilon_{\rm reg}+k\sqrt\xi+2\xi$, $k=16.03$, $\beta_A=-1$).

| Observable Channel | Frozen DR3 Baseline & Prospectively Specified TEP Prediction | Discriminating Null / Alternative |
| --- | --- | --- |
| Asymptotic velocity saturation $\alpha_{\rm sat}$ (clean solar-neighborhood sample, $s \in [5{,}000, 30{,}000]$ AU) | Model-derived $\alpha_{\rm sat}^{\rm flux} = 4\sqrt{2}/k = 0.353$ ($\alpha_{\rm sat}^{\rm prop} = 0.371$ under propagator+vertex readout); frozen DR3 baseline $\alpha_{\rm sat} = 0.366 \pm 0.012$ | Newtonian GR: $\alpha_{\rm sat} = 0.000$; unscreened MOND: $\alpha_{\rm sat} \to \infty$ ($(s/r_M)^{1/2}$ rise); unit-term $1+2\xi$: excluded at $\sim 30\sigma$ |
| Pooled all-environment onset scale $R_s^{\rm pool}$ and embedded-propagator prediction | Model-derived $R_s = \sqrt{GM/g_t} \simeq 4.7$ kAU isolated ($\simeq 3.3$ kAU embedded Galactic propagator, Step 017) with flux-conserving $s^{4/3}$ recovery; frozen DR3 pooled baseline $R_s^{\rm pool} = 2{,}646 \pm 182$ AU (stat; $\pm 609$ AU total, $g_N \simeq 1.1\times 10^{-9}\text{ m s}^{-2}$) | Thin-shell chameleon $1 - (R_s/s)e^{-m_{\rm bg}(s-R_s)}$ (already disfavored by $\Delta\chi^2 = +613.0$ in DR3) |
| Environment-resolved stratification $R_s(\|Z\|)$ across $\|Z\| \in [0, 1{,}000]$ pc | Frozen DR3 joint-fit baseline $R_s^{\rm midplane} = 4{,}912 \pm 792$ AU ($\|Z\| < 100$ pc) vs. $R_s^{\rm high\text{-}\|Z\|} = 3{,}447 \pm 366$ AU ($\|Z\| > 150$ pc; 5-bin full-range $8{,}670 \to 4{,}326$ AU); prospective DR4 scaling $R_s(\|Z\|) = R_s(0)\,[\rho_{\rm bar}(\|Z\|)/\rho_{\rm bar}(0)]^{p}$ with $p \approx 0.4 \pm 0.1$ ($p = 0.50$ anchored benchmark predicting $R_s({\rm high\text{-}\|Z\|})/R_s({\rm midplane}) = 0.64$ vs. DR3 $0.702 \pm 0.074$) | Pure acceleration-threshold MOND with external field $g_{\rm ext}(R_{\rm gc})$: $R_s$ invariant to local vertical gas/stellar density $\rho_{\rm bar}(\|Z\|)$ at fixed $R_{\rm gc}$ |
| Epoch-astrometry + radial-velocity 3D deprojection (DR4 inner-binary rejection) | Survival of $\tilde v(s) = 1 + 0.353(1 - e^{-s/R_s})$ and monotone $\|Z\|$ ordering after direct epoch-astrometric removal of unresolved hierarchical triples ($P_{\rm in} < 10^3$ d) | Triple-contamination hypothesis: $\tilde v - 1$ collapses toward zero once astrometric wobble removes inner companions |

## 7. Conclusion

The Gaia DR3 wide-binary population provides strong empirical support for the Temporal Shear recovery predicted by TEP. From a high-purity sample of 341,315 systems, the analysis identifies a characteristic transition radius $R_s = 2{,}646 \pm 182$ AU (statistical) from the canonical phenomenological TEP recovery profile fit. A conservative uncertainty budget—including jackknife stability and model-choice freedom—broadens the total uncertainty to $\pm 609$ AU while preserving the same few-thousand-AU onset scale.

The transition corresponds to an effective local binary density $\rho_{eff} \approx 9.2 \times 10^{-18}$ g/cm$^3$, far below the Temporal Topology saturation scale ($\rho_T \approx 20$ g/cm$^3$; Smawfield 2025g). As discussed in Section 6.2, this gap is expected: the conformal scalar field is already heavily screened by the Galactic halo and baryonic disk. The observed transition marks the separation at which the source-charge suppression of the locally observable Temporal Shear weakens enough that the conformal-sector gradient becomes kinematically active above the Keplerian baseline.

An independent cross-check reinforces this picture. The canonical acceleration scale of the master-action closure, $g_t = cH_0/(2|\beta_A|) \simeq 3.4\times10^{-10}$ m/s$^2$, yields the derived radius law $R_s = \sqrt{GM/g_t} \simeq 4.7$ kAU at the sample mean mass on the isolated-vacuum background—a factor $\simeq 1.8$ above the fitted value, the physically expected direction for an ambient Galactic field displacing the transition inward, with the embedded-propagator evaluation (step 017) compressing the same estimator to $\simeq 3.3$ kAU, a factor $\simeq 1.26$ within the systematic band and the half-excess marker coincident with the observation (Section 2.2)—while the empirical SPARC-extracted anchor $g_{\rm TEP} \approx 5 \times 10^{-10}$ m/s$^2$ (Smawfield 2025g) independently predicts a screening scale of the same order, and the agreement tightens further once the Galactic external field is included. The fitted screening profile is favored over a constant boost by $\Delta \chi^2 = 3{,}583$ (Table 4.1), and alternative smooth-transition fits preserve the finite onset scale, confirming that the transition is not an artifact of the canonical functional form. Direct comparison with MOND interpolating functions—including the angle-averaged External Field Effect with per-bin median masses—shows that even the most generous MOND+EFE variant ($k=2$) is rejected by $\Delta\chi^2 = +540$ relative to TEP, failing simultaneously in transition shape, onset scale, and saturation amplitude.

The environmental test provides independent corroboration. The inferred ordering between midplane and high-$|Z|$ subsamples is sensitive to the mass-luminosity relation: a uniform solar-metallicity MLR overestimates halo-star masses and can bias the comparison, whereas the color-dependent correction (Section 3.2) now uses an independent spectroscopic-or-prior guardrail and a solar-track calibration that both recover the TEP-predicted pattern. With those corrections in place, the lower-density high-$|Z|$ population enters the anomalous regime at smaller radius than the higher-density midplane, both in the full sample ($R_s = 4{,}662 \pm 196$ AU versus $7{,}131 \pm 1{,}341$ AU) and in the solar-track control ($R_s = 4{,}145 \pm 276$ AU versus $6{,}856 \pm 920$ AU; permutation $p \le 10^{-4}$ for the full sample and $p \le 10^{-3}$ for the solar track). A finer five-bin $|Z|$ stratification confirms the density-dependent scaling at significance $0.021$: within the empirical density-scaling benchmark, the inferred exponent $p \approx 0.4 \pm 0.1$ under free fits is consistent with the square-root scaling $p = 1/2$ and with the $\rho^{1/3}$ ambient channel independently returned by the noise-corrected inversion.

Supplemental controls strengthen that interpretation. The transition remains present after stratifying by distance, mass ratio, and primary mass, with each half-sample favoring a finite screening profile over a constant boost. A stricter radial-velocity subset also preserves the effect, though with broader uncertainty. Catalog-level scrambling tests fail to reproduce the observed screening preference in $10{,}000$ realizations ($p = 1.0 \times 10^{-4}$), an injection-recovery test confirms that the pipeline recovers a known TEP signal with 100% detection rate while returning 0 false positives in 100 null mocks when no signal is injected, and phase-mixed Newtonian orbital forward models remain near $\tilde v \simeq 1$ at large separation rather than reproducing the observed plateau. These controls do not generate the signal, but they make it appreciably harder to attribute the result to a narrow demographic selection, a catalog artifact, or a pipeline bias.

The wide-binary anomaly therefore emerges not as a generic failure of dark matter or standard inertia, but as a structured, environmentally modulated screening transition—one whose morphology, onset scale, and environmental ordering are quantitatively consistent with the conformal scalar field of TEP and are not reproduced by the Newtonian orbital-projection or MOND/EFE parameterizations tested here. The present analysis establishes the wide-binary regime as a new, independent probe of TEP screening, complementing the rotation-curve and compact-object evidence presented elsewhere in the series.

## References

Banik, I., Pittordis, C., Sutherland, W., Famaey, B., Ibata, R., Mieske, S., & Zhao, H. 2024, MNRAS, 527, 4573. *Strong constraints on the gravitational law from Gaia DR3 wide binaries.*

Bovy, J. 2015, ApJS, 216, 29. *galpy: A Python Library for Galactic Dynamics.*

Burnham, K. P. & Anderson, D. R. 2002, *Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach*, 2nd ed. (New York: Springer).

Burrage, C. & Sakstein, J. 2018, Living Reviews in Relativity, 21, 1. *Tests of Chameleon Gravity.*

Chae, K.-H. 2023, ApJ, 952, 128. *Breakdown of the Newton–Einstein Standard Gravity at Low Acceleration in Internal Dynamics of Wide Binary Stars.*

Chae, K.-H. 2024a, ApJ, 960, 114. *Robust Evidence for the Breakdown of Standard Gravity at Low Acceleration from Statistically Pure Binaries Free of Hidden Companions.*

Chae, K.-H. 2024b, ApJ, 972, 186. *Measurements of the Low-acceleration Gravitational Anomaly from the Normalized Velocity Profile of Gaia Wide Binary Stars and Statistical Testing of Newtonian and Milgromian Theories.*

El-Badry, K., Rix, H.-W., & Heintz, T. M. 2021, MNRAS, 506, 2269. *A million binaries from Gaia eDR3: sample selection and validation of Gaia parallax uncertainties.*

Famaey, B. & Binney, J. 2005, MNRAS, 363, 603. *Modified Newtonian dynamics in the Milky Way.*

Famaey, B. & McGaugh, S. S. 2012, Living Reviews in Relativity, 15, 10. *Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions.*

Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2023, A&A, 674, A1. *Gaia Data Release 3. Summary of the content and survey properties.*

Hernandez, X. 2023, MNRAS, 525, 1401. *Internal kinematics of Gaia DR3 wide binaries: anomalous behaviour in the low acceleration regime.*

Hernandez, X., Verteletskyi, V., Nasser, L., & Aguayo-Ortiz, A. 2024, MNRAS, 528, 4720. *Statistical analysis of the gravitational anomaly in Gaia wide binaries.*

Khoury, J. & Weltman, A. 2004, Physical Review Letters, 93, 171104. *Chameleon Fields: Awaiting Surprises for Tests of Gravity in Space.*

Llinares, C., Mota, D. F., & Winther, H. A. 2014, A&A, 562, A78. *ISIS: a new N-body cosmological code with scalar fields based on RAMSES.*

McKee, C. F., Parravano, A., & Hollenbach, D. J. 2015, ApJ, 814, 13. *Stars, Gas, and Star Formation in the Solar Neighborhood.*

Milgrom, M. 1983, ApJ, 270, 365. *A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis.*

Milgrom, M. 1994, Annals of Physics, 229, 384. *Dynamics with a Nonstandard Inertia-Acceleration Relation: An Alternative to Dark Matter in Galactic Systems.*

Milgrom, M. 2010, MNRAS, 403, 886. *Quasi-linear formulation of MOND.*

Pecaut, M. J. & Mamajek, E. E. 2013, ApJS, 208, 9. *Intrinsic Colors, Temperatures, and Bolometric Corrections of Pre-main-sequence Stars.* (Online table updated 2022.)

Pittordis, C., Sutherland, W., & Shepherd, P. 2025, Open Journal of Astrophysics, 8. *Wide Binaries from Gaia DR3: testing GR vs MOND with realistic triple modelling.* DOI: 10.33232/001c.142887.

Ratra, B. & Peebles, P. J. E. 1988, Phys. Rev. D, 37, 3406. *Cosmological Consequences of a Spontaneously Broken Scalar Field.*

Raghavan, D., McAlister, H. A., Henry, T. J., et al. 2010, ApJS, 190, 1. *A Survey of Stellar Families: Multiplicity of Solar-type Stars.*

Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: 10.5281/zenodo.16921911 (Paper 0)

Smawfield, M. L. (2025). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: 10.5281/zenodo.17127229 (Paper 1)

Smawfield, M. L. (2025). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: 10.5281/zenodo.17517141 (Paper 2)

Smawfield, M. L. (2025). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: 10.5281/zenodo.17860166 (Paper 3)

Smawfield, M. L. (2025). *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: 10.5281/zenodo.17982540 (Paper 4)

Smawfield, M. L. (2025). *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: 10.5281/zenodo.18004832 (Paper 5)

Smawfield, M. L. (2025). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: 10.5281/zenodo.18064365 (Paper 6)

Smawfield, M. L. (2025). *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: 10.5281/zenodo.18059250 (Paper 7)

Smawfield, M. L. (2025). *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: 10.5281/zenodo.18064581 (Paper 8)

Smawfield, M. L. (2025). *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: 10.5281/zenodo.18109760 (Paper 9)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: 10.5281/zenodo.18165798 (Paper 10)

Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: 10.5281/zenodo.18209702 (Paper 11)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: 10.5281/zenodo.19000827 (Paper 12)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: 10.5281/zenodo.19102061 (Paper 13 — this work)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Proper-Time Domain Boundary in the Outer Solar System*. Preprint v0.2 (Addis Ababa). Zenodo. DOI: 10.5281/zenodo.22858190 (Paper 36)

## Data Availability & Reproducibility

            This work follows open-science practices. All results are fully reproducible from raw data
            using the documented pipeline. All numerical results, figures, and statistics are generated by deterministic
            Python scripts processing real observational data from the Gaia DR3 archive, with manuscript claims injected directly from pipeline outputs.

### Repository & Code

        **GitHub Repository:** github.com/matthewsmawfield/TEP-WB

        The repository contains a deterministic, version-controlled analysis pipeline for wide binary screening
        using 341,315 Gaia DR3 systems.

#### Repository Structure

```
TEP-WB/
├── data/                          # Gaia DR3 catalogs and processed samples
│   ├── processed/                 # Cleaned binary samples
│   └── raw/                       # Original Gaia data (auto-downloaded)
├── logs/                          # Execution logs
├── results/                         # Analytical outputs and figures
├── scripts/
│   ├── steps/                     # Sequential analysis pipeline
│   │   ├── step_000_catalog_ingestion.py
│   │   ├── step_001_sample_selection.py
│   │   ├── step_002_kinematic_analysis.py
│   │   ├── step_003_screening_test.py
│   │   ├── step_004_sample_characterization.py
│   │   ├── step_005_environment_test.py
│   │   ├── step_006_audit_analysis.py
│   │   ├── step_007_supplemental_controls.py
│   │   ├── step_008_injection_recovery.py
│   │   ├── step_009_advanced_diagnostics.py
│   │   ├── step_010_referee_hardening.py
│   │   ├── step_011_mond_comparison.py
│   │   ├── step_012_newtonian_forward_model.py
│   │   ├── step_013_claim_consistency_audit.py
│   │   ├── step_014_shell_consistency.py
│   │   ├── step_015_derived_operator_forward.py
│   │   └── run_all_steps.py       # Master pipeline runner
│   └── utils/                     # Shared analysis utilities
├── site/
│   └── components/                # HTML manuscript source
└── requirements.txt                 # Python dependencies
```

### Data Provenance

| Data Source | Provider | Access Method | Download Size | DOI/URL |
| --- | --- | --- | --- | --- |
| Gaia DR3 | ESA/Gaia | Public archive | ~10 GB (full archive) | Gaia DR3 |
| Wide Binary Catalog | El-Badry et al./Gaia | Public catalog | ~500 MB (341,315 binaries) | Via Gaia archive |

        **Total Download Size:** ~500 MB for the wide binary catalog (Gaia DR3 subset).**
        Note:** The analysis downloads data automatically via `astroquery` Gaia archive queries during Step 000 execution.

### Reproduction Instructions

#### Quick Start (Full Reproduction)

```
# 1. Clone repository
git clone https://github.com/matthewsmawfield/TEP-WB.git
cd TEP-WB

# 2. Install dependencies
pip install -r requirements.txt
npm install --prefix site

# 3. Run complete pipeline
python scripts/steps/run_all_steps.py

# 4. Build manuscript
npm run build:markdown --prefix site
```

#### System Requirements

| Component | Minimum | Recommended | Tested On |
| --- | --- | --- | --- |
| CPU | 8 cores | 14+ cores | Apple M4 Pro (14-core) |
| RAM | 16 GB | 32 GB | 24 GB (M4 Pro) |
| Storage | 10 GB | 20 GB | NVMe SSD |
| Runtime | ~2-4 hours (full 14-step pipeline) | ~3 hours (M4 Pro) |  |

#### Detailed Pipeline Steps

The analysis pipeline consists of 13 deterministic steps. Each step produces JSON/CSV outputs and logs for full traceability:

##### Step 000: Catalog Ingestion

- **step_000_catalog_ingestion.py** — Download the El-Badry et al. (2021) Gaia eDR3 wide-binary catalog from Zenodo, convert the FITS table to a pandas DataFrame, and compute foundational geometric quantities (projected separation, system distance, Galactocentric coordinates) plus a placeholder dimensionless velocity ratio.

##### Step 001: Sample Selection

- **step_001_sample_selection.py** — Apply the high-purity cuts: chance-alignment probability $< 1\%$, parallax SNR $> 20$ and proper-motion SNR $> 10$ for both components, RUWE $< 1.2$ to suppress unresolved hierarchical triples, and projected separation $50 < s < 50{,}000$ AU. Output: a clean parquet sample of 341,315 systems.

##### Step 002: Kinematic Analysis

- **step_002_kinematic_analysis.py** — Compute the projected relative tangential velocity from proper-motion differences, the Newtonian circular velocity $v_c = \sqrt{GM_{\rm tot}/s}$, and the dimensionless ratio $\tilde v = v_{\rm tan}/v_c$. Stellar masses use the Pecaut & Mamajek (2013, updated 2022) main-sequence relations together with a color-dependent metallicity mass-bias correction whose $\beta_{\rm MLR}$ calibration is restricted to independent spectroscopic metallicities when available and otherwise falls back to a conservative external prior. Coordinates are also transformed to Galactocentric $(R, Z)$ for environmental stratification.

##### Step 003: Screening Test

- **step_003_screening_test.py** — Bin systems in $\log s$, take the median $\tilde v$ in each bin with parametric-resampling 68% intervals, and fit phenomenological screening models (TEP exponential, sigmoid, thin-shell, double-exponential, flat Newtonian, constant boost). Performs a Wald–Wolfowitz runs test, computes residual lag-1 autocorrelation and Durbin–Watson, executes covariance-aware refits (AR(1) GLS and a Gaussian Process in $\log_{10} s$), and runs jackknife bin-drop stability.

##### Step 004: Sample Characterization

- **step_004_sample_characterization.py** — Generate diagnostic plots characterizing the cleaned sample: distance, separation, and stellar-mass distributions, plus a sky map. Used as a sanity check that the high-purity sample lies in the local solar neighborhood and is adequately populated across separation bins.

##### Step 005: Environmental Test

- **step_005_environment_test.py** — Stratify the sample by Galactocentric height $|Z|$ into a midplane subsample ($|Z| < 100$ pc) and a high-$|Z|$ subsample ($|Z| > 150$ pc), fit the screening model with $\alpha_{\rm sat}$ fixed to remove the amplitude–scale degeneracy, and compare the resulting $R_s$ values. Includes a solar-track control (zero metallicity correction), a joint profile-likelihood fit with shared $\alpha_{\rm sat}$, permutation tests, and an empirical density-scaling benchmark that infers the exponent $p$ from the observed $R_s$ ratio (the scaling is treated here as phenomenology, not as a claimed microscopic completion; Smawfield 2025a, &sect;A4).

##### Step 006: Audit Analysis

- **step_006_audit_analysis.py** — Compute the implied effective density at the fitted screening radius from the median total binary mass, and verify population demographics by comparing midplane and high-$|Z|$ color–magnitude diagrams to confirm that the high-$|Z|$ subsample is genuinely bluer/metal-poor (validating the metallicity mass-bias correction in step_002).

##### Step 007: Supplemental Controls

- **step_007_supplemental_controls.py** — Robustness checks on demographic subsets (RV-consistent subsample, distance halves, mass-ratio halves, primary-mass halves) and direct null controls via permutation scrambling of $\tilde v$ both globally and within distance quartiles. Reports effect sizes and permutation p-values for each control.

##### Step 008: Injection / Recovery

- **step_008_injection_recovery.py** — Build mock catalogs from the projected Newtonian forward-model null of step_012, inject a known TEP screening signal, and rerun the full analysis to measure recovery fidelity (bias and detection rate). Includes a null-injection ($\alpha = 0$) false-positive check, an $R_s$ sweep, and an eccentricity-distribution sensitivity test.

##### Step 009: Advanced Diagnostics

- **step_009_advanced_diagnostics.py** — Three diagnostics: (i) a triple-contamination forward model that tests whether residual unresolved hierarchies can reproduce the observed outer $\tilde v$; (ii) a quantitative self-screening fit $\alpha_{\rm sat}(M) = \alpha_0 \exp(-M/M_{\rm screen})$ across the demographic mass splits; (iii) a fine $|Z|$ stratification into 5 equal-count height bins, refining the inferred density-scaling exponent $p$.

##### Step 010: Referee Hardening

- **step_010_referee_hardening.py** — Three referee-requested checks: (i) a normalization-window sensitivity matrix that sweeps the inner-baseline definition; (ii) a spatial-substructure cross-match of high-residual bins against sky position; (iii) a Pittordis-faithful triple forward model using the exact triple-population distributions of Pittordis et al. (2025).

##### Step 011: MOND Comparison

- **step_011_mond_comparison.py** — Compare the binned $\tilde v$ profile to MOND $\nu$-functions (simple and standard) with per-bin median masses, both with and without the angle-averaged External Field Effect (QUMOND prescription). Reports $\chi^2$, AIC and $\Delta\chi^2$ relative to the TEP exponential fit. Also computes the a priori TEP $R_s$ prediction from the SPARC-derived $g_{\rm TEP}$.

##### Step 012: Newtonian Forward Model

- **step_012_newtonian_forward_model.py** — Construct projected Newtonian orbital null catalogs conditioned on the observed projected-separation and mass distributions, marginalized over line-of-sight geometry, orbital phase, and eccentricity distributions. Reports the same binned normalized profile used by the main screening test and quantifies whether a purely Newtonian orbital population can reproduce the observed outer plateau.

##### Step 013: Claim Consistency Audit

- **step_013_claim_consistency_audit.py** — Cross-check manuscript source and generated markdown against the latest pipeline outputs, including non-circular metallicity calibration, MOND comparison values, transition-model statistics, Newtonian forward-model results, and injection-recovery false-positive rates.

##### Step 014: Screening-Shell Consistency Map

- **step_014_shell_consistency.py** — Evaluate the resolved nested two-body screening operator $\mathcal R(s) = \mathcal S_\Sigma(X_{\rm env})^2\,y(s)$, where $y$ is the exact flux-conserving profile $y[1+y^2(r_*/r)^4]=1$, $r_*=\sqrt{GM/g_t}$ is the derived shell radius with $g_t = cH_0/(2|\beta_A|)$, and $X_{\rm env}$ is the embedding ambient of the hierarchy level above the pair — the dominant member's field for hierarchical systems (giving $\mathcal R \to y^3 \sim s^4$ deep inside the shell) and the Galactic field for comparable-mass binaries (giving the $\sim s^{4/3}$ transition) — across the separation ladder (Cassini conjunction, 1 AU, Saturn, Earth–Moon, wide-binary transition, Oort cloud) and the environmental operator $\mathcal S_\Sigma(\mathcal E) = [1+(g/g_t)^2]^{-1}$ at benchmark field strengths. Verifies that all Solar-System constraints sit deep inside the solar screening shell while the wide-binary companion samples the shell boundary, and computes the amplitude ledger relating the observed $\alpha_{\rm sat}$ plateau to the ambient-floor-limited partial recovery.
- **step_015_derived_operator_forward.py** — Forward-model test of the derived operator against the real 341,315-binary catalog: each binary's projected separation is deprojected, the per-binary shell radius follows the derived $r_* = \sqrt{GM/g_t}$ law with lognormal scatter, the Galactic ambient floor carries environmental scatter, and unresolved-triple contamination uses the corrected vector-addition kinematics of steps 009/010. The binned median profile is compared against the 19-bin measurement across a family of candidate transition profiles — the power-law $[1+(R_s/r)^k]^{-1}$ family over $k = 0.8\ldots4$ and the exact flux-conserving profile $y(s)$ of the resolved operator — yielding $\chi^2_{\rm red}$ for each: the data select transition-region slopes near the derived $4/3$ (the exact $y$ profile reaches $\chi^2_{\rm red} \approx 8.4$, versus $18.3$ for the steep $k = 4$ form) and an ambient floor $S_{\rm env} \approx 0.45$ consistent with the squared Galactic vertex factor $q_{\rm GAL}^2 \approx 0.43$.

Each step produces JSON/CSV outputs with full metadata in `results/outputs/`,
        and execution logs are written to `logs/` with timestamps for complete traceability.
        Run all steps via: `python scripts/steps/run_all_steps.py`

### Software Versions

- **Python** 3.11+ (tested with 3.13)
- **NumPy** 1.24+
- **SciPy** 1.10+
- **Pandas** 2.0+
- **Matplotlib** 3.7+
- **Astroquery** (Gaia data access)

---

<!-- ===================================================================== -->
<!-- CANONICAL URL: https://mlsmawfield.com/tep/gnss-mgex/ -->
<!-- MARKDOWN URL:  https://mlsmawfield.com/tep/gnss-mgex/manuscript.md -->
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# Global Time Echoes: MGEX Multi-GNSS Clock Replication, 2025–2026
**Matthew Lukin Smawfield**
Version: v0.4 (Suva)
First published: 7 June 2026 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.20572726

---

## Abstract

The Temporal Equivalence Principle (TEP) predicts correlated phase-coherent
disturbances in GNSS timekeeping with a spatial correlation length of order
thousands of kilometres, an east–west anisotropy, coupling to Earth orbital
velocity, and an annual-phase direction near Earth's aphelion velocity tangent.
The directed CMB-apex template is anti-phase rather than detected as a rest
frame. Papers 1 and 2 examined
these signatures in GPS precise point-positioning products from three analysis
centres spanning 2000–2025; Paper 3 used raw RINEX single-point positioning.

This paper is a product-type independence stress test using the public CODE
COD0MGXFIN combined multi-GNSS receiver-clock solution distributed by NASA
CDDIS from 2025-01-01 to 2026-05-01. Receiver-clock offsets for 418 globally
distributed stations are read directly from daily 5-minute CLK files. The
combined solution provides no per-constellation independence, but supplies a
distinct product family and a largely held-out 2025–2026 epoch.

The isotropic correlation length is $$\lambda = 1862$$ km (R² = 0.699;
1.83 million pair-day observations; rising to R² = 0.85–0.92 for baselines
above 500 km) with a formal within-fit error of ±112 km
and a cluster-robust statistical uncertainty of ±155 km obtained by resampling
whole epochs (month-block bootstrap over 17 calendar months; day-block ±86 km;
leave-one-month-out jackknife ±94 km), confirming bounded thousands-of-kilometres
clock coherence in an independent multi-constellation product family and a
held-out 2025–2026 epoch. The numeric scale is epoch- and
product-conditioned at the ~30–40% level: single-window realisations span
1,050–1,935 km, and matched-day estimates are 2,744 km for CODE against
1,933 km for the independently processed GRG product chain (same tracking network) — the latter landing within 4% of
the pooled scale — while a controlled decomposition on identical station-days
bounds the spectral-resolution and common-mode contributions at ≲±25% and
reproduces the same band-ordered profile (Step 3.5). It is stable at 1,819–2,000 km across minimum-baseline
cutoffs of 50–2,000 km and persists on geomagnetically quiet days
($$\lambda = 1708$$ km, R² = 0.743). The full-range anisotropy is
not statistically resolved in the combined solution (ratio 1.29, station-clustered p = 0.49), and
the primary monthly $$\lambda$$ and EW/NS orbital metrics are null (Bonferroni
p > 0.5). A model-free PA-difference statistic recovers the predicted negative coupling with orbital
speed (r = −0.716, p = 0.0018) and CMB-dipole velocity projection
(r = −0.767, p = 0.0005) across 16 chronological full months.
Autocorrelation-aware corrections preserve the projection-channel association:
the effective-degrees-of-freedom values are p = 0.072 (speed) and p = 0.018
(v_proj), matched AR(1) surrogates give p = 0.019 and p = 0.003, the
phase-preserving circular-shift null reaches its 1/16 resolution floor for
the projection channel, and leave-one-month-out correlations stay within
r = −0.66 to −0.86, so no single phase alignment dominates. The association
also survives operational-asymmetry controls: a composition-matched EW−NS
contrast over distance × latitude cells gives r = −0.65 (p = 0.006) for the
projection channel, and a uniformly-tracked-station subset gives r = −0.74
(p = 0.001). Hemisphere-split checks return the same signed
anticorrelation in northern, southern, and cross-hemisphere pair classes
(r = −0.45, −0.88, −0.49) at a common annual phase (Δφ = 22° ± 44°),
favouring a global heliocentric driver over hemisphere-local seasonal
systematics. The physical
hypothesis and sign were pre-specified by earlier TEP-GNSS papers, making this
cross-product corroboration of the orbital-coupling sign alongside an exploratory
estimator selection within MGEX.
A 37 × 19 directional grid finds strong within-grid modulation (200-permutation
plus-one p = 0.005). Its nominal maximum is 82.7° from the CMB dipole. Orbital
velocity constrains only an axis’s projection onto the orbital plane, and that
projection is highly degenerate
(the CMB and nominal-maximum projected predictors correlate at r = 0.984, and
only 61 of 703 grid directions meet or exceed the direct-CMB |r|).
The CMB-labelled projection is therefore one member of a broad family of
annual templates, not evidence for a CMB rest frame or a unique
three-dimensional axis. All four null controls pass under the
robust analysis (minimum baseline 500 km); under the primary short-baseline
parameters the solar-rotation reassignment null retains R² = 0.50, traced to a
small-sample bin artifact at 100–152 km, so the robust analysis is reported as
the clean result (the fitted scale is stable, 1,862 vs 1,819 km). The two
leading mundane attribution channels are explicitly modelled and strongly disfavoured by the tested controls rather than
assumed absent: a forward-modelled shared satellite-error kernel is flat
through the signal band and decays only on the constellation-footprint scale,
and the weather-organized partitions contradict the atmospheric signature —
cross-hemisphere pairs retain the cleanest decay (λ = 2101 km) while
same-latitude-band pairs beyond 8,000 km return −0.118 rather than the
positive zonal coherence an atmospheric driver requires. A supplementary satellite-clock
analysis does not recover the exponential structure, consistent with the shared
reference-time architecture of the combined MGEX product: because satellite
clocks are not independently estimated in that product, it tests only the
receiver-clock channel, and the satellite-clock null is a product limitation
rather than a prediction test.

*Keywords:* Temporal Equivalence Principle, MGEX, multi-GNSS, receiver clock, correlation length, anisotropy, orbital velocity, annual phase, ionosphere, temporal shear, temporal topology

## 1. Introduction

Papers 1 and 2 of this series established correlated phase-coherent disturbances
in GPS-only precise point positioning (PPP) products from CODE, IGS and ESA,
spanning 2000–2025, within the Temporal Equivalence Principle (TEP) framework where matter and clocks couple universally to the canonical causal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$; Paper 0, §2.2, Step 80). The disturbances exhibit a spatial correlation length of
order thousands of kilometres, an east–west anisotropy exceeding the north–south
counterpart, coupling to Earth orbital velocity, and a recovered annual-phase
direction near Earth's aphelion velocity tangent. In the corrected directed
convention, the CMB-apex template is anti-phase and is not a detected rest frame.
Paper 3 moved to raw RINEX single-point positioning (SPP),
demonstrating that the same signatures persist without reliance on
analysis-centre orbit or clock products. Between them, those papers already
provide multi-centre, multi-decade, raw-observation, and GPS-only product-family
independence. The principal forms of robustness they do not yet supply are
consistency across the historical training epoch and consistency in a
combined multi-GNSS clock product family.

This paper (Paper 14) conducts a stress test across those two
dimensions. The analysis uses the public MGEX combined multi-GNSS
receiver-clock solution—the CODE COD0MGXFIN CLK product distributed by NASA
CDDIS—for the window 2025-01-01 to 2026-05-01. The present MGEX analysis uses a
combined multi-GNSS clock product distinct from the GPS-only PPP and raw RINEX
products analysed earlier, and a largely held-out 2025–2026 window, though it is
not fully temporally disjoint from all earlier analyses. The MGEX clock product is
generated by combining GPS, GLONASS, Galileo and BeiDou observations into a
single receiver-clock estimate per station, using software, orbit and combination
strategies that differ from the GPS-only PPP of Papers 1–2. It therefore
tests cross-product consistency using a distinct combined multi-GNSS clock
solution over a largely held-out epoch.
Receiver-clock offsets for 418 globally distributed stations are read directly
from the daily 5-minute CLK files; no positioning is performed.

It is emphasised at the outset what this product tests and what it does not.
Because the MGEX clock files contain one combined multi-GNSS solution per station,
they do not permit an independent per-constellation comparison; that
form of independence is the province of raw per-system processing, which lies
outside the scope of this combined-product analysis. The independence offered
here is temporal (a largely held-out epoch) and product-type (a combined multi-GNSS
clock product line from the same analysis centre, CODE, whose IGS-final products anchor Papers 1–2), not per-constellation.

The measured $\lambda \approx 1862$ km is a clock-sector covariance scale of $C_A$. It is not $\mathcal{S}_\Sigma$; in the near-Earth environment, this correlation length traces the macroscopic geometric signature of the continuous saturation of Temporal Topology, anchoring the differential clock correlations independently of the specific multi-GNSS processing strategy. The precise relationship of the covariance structure to the solved scalar field remains to be derived.

## 2. Data and Product Scope

### 2.1 MGEX Combined Clock Product

All data are public and freely available from NASA CDDIS
([cddis.nasa.gov](https://cddis.nasa.gov)) and the International
GNSS Service (IGS). The analysis uses the Multi-GNSS Experiment (MGEX) combined
receiver-clock product COD0MGXFIN produced by the Center for Orbit Determination
in Europe (CODE), with the Wuhan (WUM) and CNES (GRG) MGEX products as fallbacks
on the few days CODE is unavailable (WUM 20 days, GRG 9 days out of 486 calendar
days in the window, i.e. ≈6 % fallback usage). These are daily clock (CLK) files at a
5-minute sampling interval. Each file reports a single combined multi-GNSS
receiver-clock estimate per station, derived by CODE from GPS, GLONASS, Galileo
and BeiDou observations; the combined solution is the quantity analysed. No
positioning is performed and no raw RINEX observation files are used.

### 2.2 Analysis Window and Station Set

The analysis period is 2025-01-01 to 2026-05-01 (462 days with usable data out of 486 calendar days in the window),
yielding approximately 16 months of daily files. The window is chosen to
provide a largely held-out 2025–2026 epoch, though it is not fully temporally
disjoint from Papers 1–2, which span 2000–2025. Receiver-clock offsets are extracted from the AR (receiver-clock)
records of each CLK file for 418 globally distributed stations, converted to
nanoseconds, and assembled into per-station daily time series.

### 2.3 Product Limitation: No Per-Constellation Separation

Unlike Papers 1–3, which can isolate individual systems, the MGEX CLK product
contains one combined receiver clock per station rather than per-constellation
clocks. The four constellations therefore cannot be separated in this product,
and the analysis is performed once on the combined solution. The
independence offered here is temporal (a largely held-out epoch) and product-type (a
combined multi-GNSS clock product line), not
per-constellation and not analysis-centre: COD0MGXFIN is produced by CODE,
the same centre whose IGS-final products anchor Papers 1 and 2.

### 2.4 Data Provenance and Shared Error Budgets

The provenance matrix across the GNSS papers makes explicit which analyses
share an error budget:

| Paper | Data layer | Product / processing chain | Epoch | Stations | Error budget shared with |
| --- | --- | --- | --- | --- | --- |
| 1 | AC precise clocks | CODE final, IGS combined, ESA final (Bernese / IGS combination / NAPEOS) | 2023-01 to 2025-06 | 364 | CODE family shared with Papers 2, 14; AC ecosystem shared with Paper 33's centre arms |
| 2 | AC precise clocks | CODE IGS-final 30 s clocks (Bernese) | 2000-03 to 2025-06 | 474 | Same CODE product family as Papers 1 and 14; epoch superset of Paper 1 |
| 3 | Raw RINEX | Broadcast ephemerides + own RTKLIB SPP | 2022 to 2024 | 539 | Estimation-chain independent; shares the physical tracking network |
| **14 (this study)** | AC combined clocks | CODE COD0MGXFIN (WUM/GRG fallback ≃6% of days) | 2025-01 to 2026-05 | 418 | Same CODE centre as Papers 1–2; epoch largely held out |
| 33 | Mixed | Own RTKLib SPP arms + AC receiver clocks (COD/GFZ/GRG/JPL) | 2022 to 2025 | 5–160 per arm | AC arms share the IGS/MGEX ecosystem; SPP arms are estimation-independent |

Agreement with Papers 1–2 is therefore corroboration within a shared
product ecosystem — a different product line and a largely held-out
epoch over substantially the same tracking network — rather than
replication on independent data. The GRG and WUM products used for fallback
and matched-day comparison run different processing software on that same
network, so they bound software- and implementation-specific artifacts but
not network-level or datum-level common modes. The estimation-chain-independent
axis in the corpus is Paper 3 (raw RINEX, broadcast ephemerides, own
processing); Paper 33's broadcast-SPP arms provide a second such axis, while
its analysis-centre arms share this ecosystem.

### 2.5 Evaluated Signatures

The analysis evaluates the following signatures, consistent with the approach
in earlier papers: correlation length, azimuthal anisotropy, orbital-velocity
coupling, annual-phase directional structure, fixed-frame controls, ionospheric independence, and geometric robustness.

## 3. Methods

### 3.1 Receiver-Clock Extraction

For each daily CLK file the AR records are parsed, which report the combined
multi-GNSS receiver-clock offset for each station at the 5-minute sampling
interval. The offsets are converted from seconds to nanoseconds and ordered in
time, giving one clock time series per station per day. No positioning is
performed and no coordinate residuals are computed; the receiver clock is the
sole observable.

Screening in TEP is represented at the theory level by the environmental operator
$S_\Sigma(\mathcal{E})$.
Quantities such as
$\rho_T$,
$R_T(M)$,
$S_\oplus(r)$,
compactness $\Phi/c^2$,
local stellar density,
geometric coherence length,
and channel-specific response coefficients
are domain-specific projections of $\mathcal{E}$,
not independent screening mechanisms
and not interchangeable universal thresholds.
Each is an observational transfer model
that parameterizes the same underlying operator
in a regime-appropriate form.

### 3.2 Spectral Pre-processing

Each daily series is linearly detrended to remove the dominant clock drift.
Cross- and auto-spectra are estimated with Welch's method (Hann window,
segment length up to 96 samples). Spectral estimates are restricted to the
nominal TEP-sensitive band [10 µHz, 500 µHz], but with the daily Welch
segmentation used here the effective lowest non-zero resolved frequency is set
by the segment length, approximately 35 µHz for 96 five-minute samples. The
practical analysed band is therefore approximately 35–500 µHz, corresponding to
periods from about 8 hours down to 33 minutes, which lies safely within the
5-minute Nyquist limit.

### 3.3 Spectral Phase-Alignment Metric

For every station pair, the magnitude-weighted circular mean of the
cross-spectral phase is computed across the selected frequency band. The
phase-alignment index is the cosine of the weighted mean phase, ranging from
−1 (anti-phase) to +1 (in-phase). It measures signed mean phase-offset
alignment, not phase concentration; the latter requires the magnitude of the
complex resultant. The index is bounded and saturating — it reports the
directional dominance of the in-phase component of the band-integrated
cross-spectrum rather than the correlated-power fraction — so its fitted
exponential amplitude is a phase-direction measure, not a calibrated
coupling, and is decomposed against a linear coherence estimator in §4.5b.
The established phase-alignment statistic is retained
consistently with the earlier GNSS analyses, and is the quantity carried
through all subsequent tests. Pairs are
binned by great-circle distance into 40 logarithmically spaced bins from 50 km
to 13,000 km, with a minimum of 10 pairs per bin.

### 3.4 Correlation-Length Model

The distance-binned phase alignment is modelled as

$$C(r) = A \exp(-r / \lambda) + C_0 \tag{1}$$

where A is the amplitude, λ is the correlation length, and C₀ is an incoherent
offset. Fitting uses bounded non-linear least squares (Trust Region Reflective),
weighting each bin by the inverse of its standard error of the mean
(1/SEM, where SEM = σ_bin / sqrt(count)). Fit quality is
quantified by R², and the uncertainty on λ is propagated from the covariance
matrix. Because pair-day records share stations within each day and
orbit/product states across consecutive days, the formal covariance error is
supplemented by cluster-robust resampling of whole epochs: a day-block
bootstrap, a calendar-month-block bootstrap, and a leave-one-month-out
jackknife, each refitting Eq. (1) on the resampled record pool. The choice of
morphology is likewise tested rather than assumed: on identical bins and
weights the exponential is compared by AIC against a stretched exponential
with free exponent, the algebraic family A/[1+(r/r_t)^k]+C₀ with free k
and at the derived Paper-0 pairwise exponents k = 4 and k = 4/3, and a
Gaussian. Sensitivity rows at raised minimum-baseline cutoffs are flagged
when the fitted λ contacts the lower parameter bound (which equals the
cutoff), since a boundary-pinned value is not an interior measurement.

### 3.5 Anisotropy, Orbital-Phase, Fixed-Frame, Ionospheric, and Null Tests

The EW > NS test fits λ separately in eight 45°-wide azimuth sectors and compares
the mean of the east and west sectors with the mean of the north and south
sectors. Because ionospheric decorrelation inverts the anisotropy at long
baselines, a longitude-matched subset is also reported (station-pair longitude
difference below 30°), which, following Paper 3, isolates pairs at similar local
solar time. Significance and confidence intervals for the ratio are obtained from
500 bootstrap resamples. For anisotropy, pair-bootstrap results are reported as sensitivity tests, while station/spatially-clustered resampling is treated as the more conservative inference because station pairs are not independent. The ratio is additionally reported in three distance
strata (0–500, 500–1,000, >1,000 km).

The orbital-coupling test correlates the monthly λ and the monthly EW/NS ratio
with the projection of Earth's orbital velocity onto the CMB dipole direction,
computed from a sinusoidal annual model; the two correlations are
Bonferroni-corrected. Because exponential-fit noise on short baselines can
degrade these metrics, an exploratory PA-difference metric is also computed:
the monthly mean EW minus NS phase alignment, which avoids fitting entirely
and directly measures anisotropy strength. The physical hypothesis and sign
were motivated by earlier TEP-GNSS analyses, but this operationalisation was
selected after the primary MGEX estimators proved unstable; it is therefore
exploratory within MGEX and excluded from confirmatory family-wise inference.
Because the 16-month series is short and its monthly values are plausibly
autocorrelated, the nominal Gaussian p-values for the exploratory
PA-difference associations are accompanied by four robustness diagnostics
computed in the same step: an effective-degrees-of-freedom correction from
the residual lag-1 autocorrelation (n_eff = n(1−ρ̂)/(1+ρ̂), t-test on
n_eff−2), a Monte-Carlo ensemble of AR(1) surrogates matched to the observed
residual autocorrelation, a phase-preserving circular-shift permutation of
the monthly series against the fixed periodic predictor (resolution floor
1/n), and a leave-one-month-out correlation range testing whether a single
phase alignment dominates. Two operational-asymmetry controls further price
the vulnerability of the raw EW − NS contrast to unequal east/west station
populations: a composition-matched difference computed as the equal-weight
mean of cell-level EW − NS contrasts over (distance × |midpoint-latitude|)
cells, and a uniformly-tracked-station subset restricted to pairs whose two
endpoints both sit above the median pair-day participation count; each
variant is correlated with the same two predictors under the same
diagnostics.
Months are retained chronologically (YYYY-MM) and a partial trailing month is
excluded. The CMB-frame test performs a full-sky
grid search for the anisotropy axis on a 37 × 19 grid in right ascension and
declination (10° steps), correlating the monthly EW−NS phase-alignment
difference with the projection of Earth's orbital velocity onto each candidate
axis. The difference metric is used in place of the EW/NS ratio because
phase alignment ranges from −1 to +1, making the ratio unstable when the NS
mean is near zero; the difference is bounded and requires no sign-based
selection filter. Monthly aggregation reduces daily noise, while the daily
analysis is retained as a supplementary diagnostic. The grid search tracks both
the most positive and most negative correlation extrema. Because orbital
velocity samples essentially the ecliptic plane, the scan is interpreted as a
test of a plane-projected annual predictor, not as a unique three-dimensional
axis localization; a nominal sky-angle threshold is reported descriptively
only. A permutation null performs a complete grid search over all 703
directions for each of 200 permutations, accounting for the look-elsewhere
effect directly. Ionospheric controls stratify the fit by
geomagnetic Kp index and exclude storm days; geometry controls repeat the fit on
hemisphere-balanced and distance-matched subsets; a forward-modelled
product-error kernel tests the shared satellite-error projection channel
directly; hemisphere-class and latitude-band within-product partitions test
the weather-organized atmospheric channel; and four null tests (temporal
shuffle, spatial shuffle, phase randomisation, and solar-rotation reassignment)
test whether the recovered structure vanishes under label permutation.

### 3.6 Sensitivity and Robustness Analysis

Because the 418-station network contains dense regional clusters that
disproportionately populate short baselines, a sensitivity analysis is
performed to verify that the reported results are not artifacts of
short-baseline fitting noise. For each primary result the analysis is
repeated at five minimum-baseline cutoffs (50, 200, 500, 1,000, and
2,000 km), tabulating the correlation length, anisotropy ratio, and
CMB correlation as functions of the cutoff. A robust supplementary
analysis uses a 500-km minimum baseline and a 50-pair minimum bin
count, eliminating bins with fewer than 50 pairs where small-sample
fluctuations can produce extreme coherence means. For the anisotropy
ratio a station-declustered point estimate is also computed, weighting
each pair by the inverse square root of the product of its two
endpoints' pair counts so that stations in dense clusters do not
dominate the exponential fit. The robust null tests apply the same
elevated thresholds to all four shuffle controls. These sensitivity
analyses do not replace the pre-specified primary analysis but provide
a conservative supplementary assessment of each result's stability.

## 4. Results

The analysis comprises 1,827,555 station-pair-day observations drawn from a network
of 418 stations over 462 usable days. Because the MGEX product is a single combined solution, the
results are reported once rather than per constellation. The analysis evaluates the core correlation-length signature, directional anisotropy, orbital-velocity coupling, annual-phase templates, and ionospheric/geometric robustness controls. The correlation length and ionospheric persistence are observed; the longitude-matched anisotropy is not robust under spatially clustered resampling. The primary pre-specified monthly λ and EW/NS orbital metrics are null (Bonferroni p > 0.5). A model-free PA-difference metric (mean EW minus NS phase alignment) gives a suggestive orbital association across 16 chronological full months that survives autocorrelation-aware corrections for one CMB-labelled projection channel (corrected p = 0.018; AR(1) p = 0.003; circular-shift floor), but it is exploratory within this MGEX study because it was adopted after the primary fits proved unstable. The full-sky scan detects annual directional modulation, but orbital velocity constrains only a plane-projected phase: the nominal sky axis is not uniquely localizable and the CMB template is not uniquely selected.

### 4.1 Correlation Length

The isotropic fit gives a well-constrained correlation length.

| $$\lambda$$ (km) | $$\sigma_\lambda$$ (km, formal) | A | C₀ | R² | Pair-day observations |
| --- | --- | --- | --- | --- | --- |
| 1862 | 112 | 0.827 | −0.065 | 0.699 | 1,827,555 |

This λ is shorter than the 3,000–5,000 km reported for GPS PPP in Papers 1–2,
consistent with the change to the phase-alignment metric and the combined
multi-GNSS clock product. The fitted amplitude A = 0.827 is likewise a
bounded phase-direction index rather than a calibrated correlation
amplitude; its ~7× offset from the IGS-final family is decomposed
estimator-axis by estimator-axis in §4.5b (Step 3.6). Until the transfer map from
$\mathcal{S}_\Sigma(\mathcal{E})$ to $\lambda$ is derived, the fitted
correlation length is therefore a product- and metric-dependent
phenomenological observable: the physical TEP signature carried across
products is the order-of-magnitude (thousands-of-kilometres) spatial
coherence, not the precise numerical scale, which responds to the
observable definition and clock-product construction. The ±112 km uncertainty
is the formal covariance-based fit error, which treats the 1.83 million
pair-day records as independent within-bin samples. Because pair-days share
stations within each day and share orbit/product states across consecutive
days, cluster-robust resampling of whole epochs was used to measure the true
statistical scatter: a day-block bootstrap over the 462 days gives
$$\sigma_\lambda = \pm 86$$ km (95% CI 1,639–1,932 km), a month-block
bootstrap over the 17 calendar months gives $$\sigma_\lambda = \pm 155$$ km
(95% CI 1,363–1,971 km), and a leave-one-month-out jackknife gives
$$\sigma_\lambda = \pm 94$$ km with leave-out values confined to
1,794–1,896 km. The within-product statistical precision is therefore about
±8%, only modestly above the formal error; the dominant scale uncertainties
are the epoch and product systematics quantified below. A sensitivity analysis
demonstrates that the correlation length is stable across minimum-baseline
cutoffs from 50 to 2,000 km, with fit quality improving substantially as the
shortest baselines are excluded (R² = 0.699 at 50 km to approximately 0.92).

| Min. baseline (km) | $$\lambda$$ (km) | R² |
| --- | --- | --- |
| 50 (primary) | 1862 | 0.699 |
| 200 | 1893 | 0.791 |
| 500 | 1819 | 0.848 |
| 1000 | 1850 | 0.917 |
| 2000 | 2000† | 0.912 |

† The 2,000 km cutoff row is a boundary contact: the fitted value
(2000.000 km) sits on the lower parameter bound ($$\lambda \geq$$ minimum
baseline) rather than interior to it, so it indicates only that the data
prefer $$\lambda \geq 2{,}000$$ km at that cutoff — it is flagged as pinned
in the archived output and is not an independent measurement of the
scale.

The correlation length varies by less than 8% across the full range of
cutoffs ($$\lambda$$ = 1,819–2,000 km), while R² improves from 0.70 to
0.92. The stability of the fitted correlation length across these cutoffs
demonstrates that the recovered distance-decay scale is not driven solely by
the shortest-baseline bins.

Two further stability checks bound the epoch and processing dependence of
λ (Step 3.1). First, splitting the 462-day pair record into quarterly
windows returns λ = 1,050–1,935 km (R² from −0.18 to 0.80), with
λ = 1,320 km for 2025 alone and 1,935 km for the 2026 subset: the fitted
scale is epoch-dependent at the ~30–40% level, and any single ~1-year
estimate — including the 1,396 km figure reported for the original
~1-year span — is an unstable realisation rather than a product-fixed
value. Second, on strictly matched days the independently processed CNES (GRG)
product chain — a different code base on the same tracking network —
returns pooled λ = 1,933 km (R² = 0.58) against 2,744 km
for the CODE product on verified-independent matched days (~44,000 pairs
each; same-product fallback days excluded by epoch-aligned
detrended-residual comparison). The GRG estimate lands within 4% of the
pooled MGEX scale (1,862 km), while the ~30% residual offset to the CODE
chain on identical days is the direct product-conditioning signature
decomposed in §4.5b: the recovered scale is a property of the data rather
than of one center's constellation weighting or software. These numbers
fold in the cadence correction of §4.5b — GRG products are delivered at
30-s sampling and per-pair series are now built by intersecting station
epochs on the shared 300-s grid; under the earlier
length-truncated handling the same diagnostic returned 3,257 km against
2,514 km, so the correction tightens the GRG–pooled agreement from ~75%
to ~4%.
The dominant systematic on the λ scale is therefore epoch/window
composition (~30–40%), with analysis-center processing contributing a
further ~±600 km, both well above the ±155 km cluster-robust
statistical error of the pooled estimate.

The choice of decay morphology is tested rather than assumed. On identical
log-spaced bins and identical SEM weights, six functional forms are compared
by AIC: the exponential (AIC = 1,664.7, $$\lambda = 1{,}862$$ km), a
stretched exponential with free exponent (AIC = 1,665.2; the fit selects
$$b = 0.98$$, i.e. the pure exponential is recovered rather than assumed),
the algebraic suppressor family $$A/[1+(r/r_t)^k]+C_0$$ with free
exponent (AIC = 1,893.8 at $$k = 1.78$$), the same family at the derived
Paper-0 pairwise exponents $$k=4$$ (AIC = 4,209.0) and $$k=4/3$$
(AIC = 2,125.1), and a Gaussian (AIC = 3,355.2). The exponential is the
AIC winner and the nested stretched-exponential family independently
returns $$b \approx 1$$: the measured profile is a finite-range exponential
correlator — the two-point morphology of a screened spatial field — rather
than an algebraic tail. The derived suppressor operators
$$\mathcal{S}_\Sigma$$ and $$y(s)$$ describe the response as a function of
the ambient environment, a different observable; the correlation kernel
measured here is the field's spatial covariance, and it is cleanly
exponential.

### 4.2 EW / NS Anisotropy

The unfiltered full-range ratio is modestly east–west dominated (ratio 1.29)
but does not reach significance under the station-clustered bootstrap
(p = 0.49). A suggestive east–west excess emerges once pairs are
matched in longitude (fitted ratio 1.32, bootstrap mean 1.71, pair-bootstrap p = 0.058), though the
significance dissolves under spatial-clustered resampling (95% CI [0.17, 13.28],
p = 0.214). The 500–1,000 km stratum also shows the expected excess.

| Subset | $$\lambda_{EW}$$ (km) | $$\lambda_{NS}$$ (km) | Ratio | 95% CI | p |
| --- | --- | --- | --- | --- | --- |
| Full range | 1810 | 1402 | 1.29 | [0.18, 2.60] (station-boot) | 0.49 |
| Longitude-matched ($\lt$ 30°) | 1753 | 1327 | 1.32 | [0.91, 2.65] (pair-boot, mean 1.71) | 0.058 |

| Distance stratum | EW/NS ratio |
| --- | --- |
| 0–500 km | 0.73 |
| 500–1,000 km | 1.38 |
| >1,000 km | 0.82 |

A sensitivity analysis reveals that the directional ratio is fragile
under conservative analysis choices. The ratio declines from 1.29 at
the primary 50-km cutoff to 1.08 at 500 km and 0.89 at 1,000 km, with
station de-clustering yielding a ratio of 0.99. The anisotropy is
therefore classified as not robust; the directional signal is
interpreted as null in the combined MGEX product, consistent with the
non-significant station-clustered bootstrap (p = 0.49).

| Analysis | $$\lambda_{EW}$$ (km) | $$\lambda_{NS}$$ (km) | Ratio |
| --- | --- | --- | --- |
| Primary (min 50 km) | 1810 | 1402 | 1.29 |
| Min 200 km | 2098 | 1384 | 1.52 |
| Min 500 km | 1880 | 1734 | 1.08 |
| Min 1000 km | 2211 | 2479 | 0.89 |
| Min 2000 km | 2496 | 2671 | 0.93 |
| Station de-clustered | — | — | 0.99 |
| Robust (min 500 km, bin 50) | 1878 | 1733 | 1.08 |

### 4.3 Orbital-Velocity Coupling

The traditional monthly λ and EW/NS ratio do not correlate significantly with
Earth's orbital speed or its projection onto the CMB dipole (Bonferroni
p > 0.5 for both). These primary pre-specified metrics are degraded by
exponential-fit parameter degeneracy on the short 16-month baseline, where
bounded non-linear least squares fitting of $A \exp(-r/\lambda) + C_0$
suffers from severe variance inflation with too few monthly points.
The monthly means retain January 2025 through April 2026 as 16 separate
chronological observations; the one-day May fragment is excluded. A model-free
PA-difference metric (monthly mean EW − NS phase alignment) avoids exponential
fitting and has a negative association with orbital speed (r = −0.716,
p = 0.0018) and with the velocity projection onto the CMB dipole (r = −0.767,
p = 0.0005). The nominal p-values assume independent monthly points; under
autocorrelation-aware diagnostics the associations are tempered but not
erased. With effective degrees of freedom from the residual lag-1
autocorrelation (ρ̂ = 0.40 and 0.29), the corrected values are p = 0.072
(speed) and p = 0.018 (v_proj); matched AR(1) surrogates give p = 0.019 and
p = 0.003; the phase-preserving circular-shift null gives p = 0.125 and
p = 0.063 — the 1/16 resolution floor for the CMB-projection channel.
Leave-one-month-out correlations stay within r = −0.66 to −0.86 for both
predictors, so the association is not carried by a single favourable phase
alignment. The physically motivated velocity-projection channel is thus the
one that survives every correction, while the scalar-speed control is
marginal. The physical hypothesis and expected sign were motivated by the
earlier TEP-GNSS papers, which reported comparable orbital associations; this
is therefore corroborative across products. It remains exploratory within the
MGEX analysis because this particular operationalisation was selected after
the primary λ and EW/NS estimators proved unstable. It is not included in the
family-wise correction applied to confirmatory tests and should be tested
prospectively in a longer MGEX series. The raw EW − NS contrast is the
estimator most sensitive to any operational asymmetry between the east and
west station populations (network density, tracking schedules); two controls
price that channel directly. A composition-matched difference — EW − NS of
cell means averaged with equal weight over (distance × |latitude|) cells —
retains the association (r = −0.586, p = 0.017 vs speed; r = −0.650,
p = 0.006 vs v_proj, n = 16), and a uniformly-tracked-station subset (both
endpoints above the median pair-day participation) retains it more strongly
(r = −0.685, p = 0.003; r = −0.739, p = 0.001; corrected p = 0.029 and
AR(1) p = 0.007 for the v_proj channel). The association is therefore not
carried by population or scheduling asymmetry between the sectors.
Two further diagnostics bound the interpretation. First, the fitted-scale
nulls are consistent with estimator dilution rather than signal absence:
the monthly λ sequence scatters with CV ≈ 21% about its mean while
remaining uncorrelated with the PA-diff sequence (r = −0.01), and the
per-month per-sector fits that enter the EW/NS λ ratio carry median
propagated ratio errors of ≈ 27% — and one month (2025-08) has a sector
fit pinned at the λ upper bound, rendering that month's ratio formally
uninformative — while the resulting ratio
sequence is uncorrelated even with the fit-free anisotropy it is meant
to measure (ratio vs PA-diff r = −0.20, p = 0.45) and remains null after
dropping the degenerate month (r = −0.10 vs speed, r = −0.17 vs v_proj).
A velocity-driven modulation at the observed contrast amplitude cannot
propagate through the fitted decay length on a 16-month baseline — the
unfitted contrast is the sensitive channel by construction. Second, a
hemisphere decomposition of the same contrast (Step 3.4) returns the
signed orbital-speed anticorrelation in all three pair classes —
intra-northern (r = −0.45), intra-southern (r = −0.88), and
cross-hemisphere (r = −0.49) — with the northern and southern annual
sinusoid phases consistent with a common value (Δφ = 22° ± 44°). A
hemisphere-local seasonal mechanism would appear in anti-phase between
the hemispheres; the observed common phase instead indicates a global
heliocentric driver.

| Quantity | r | p (raw) | p (corr.) | Interpretation |
| --- | --- | --- | --- | --- |
| λ vs speed | −0.16 | 0.554 | 1.000 | Primary (pre-specified): not significant |
| EW/NS ratio vs speed | 0.14 | 0.610 | 1.000 | Primary (pre-specified): not significant |
| λ vs v_proj | −0.05 | 0.850 | 1.000 | Primary (pre-specified): not significant |
| EW/NS ratio vs v_proj | −0.03 | 0.911 | 1.000 | Primary (pre-specified): not significant |
| PA_diff vs speed | −0.716 | 0.0018 | 0.072 | Exploratory; n_eff = 6.9; AR(1) p = 0.019; shift p = 0.125; LOO r ∈ [−0.84, −0.66] |
| PA_diff vs v_proj | −0.767 | 0.0005 | 0.018 | Exploratory; n_eff = 8.8; AR(1) p = 0.003; shift p = 0.063 (floor); LOO r ∈ [−0.86, −0.72] |
| PA_diff (cell-matched) vs speed | −0.586 | 0.0170 | — | Operational-asymmetry control; composition-matched over distance × |latitude| cells |
| PA_diff (cell-matched) vs v_proj | −0.650 | 0.0064 | — | Operational-asymmetry control; n_eff p = 0.065; AR(1) p = 0.028; shift p = 0.063 (floor) |
| PA_diff (uniform-tracking) vs speed | −0.685 | 0.0034 | — | Operational-asymmetry control; both endpoints above median participation |
| PA_diff (uniform-tracking) vs v_proj | −0.739 | 0.0011 | — | Operational-asymmetry control; n_eff p = 0.029; AR(1) p = 0.007; shift p = 0.063 (floor) |

### 4.4 Projection-Degenerate Annual-Direction Test

This analysis uses the same PA-difference observable as Section 4.3 and is
therefore a directional characterization, not an independent test of a cosmic
rest frame. With 16
chronological full-month observations, the direct CMB-projection correlation is
r = −0.733, computed from daily EW−NS differences averaged equally within
each calendar month. This differs from the −0.767 result in Section 4.3,
which averages all pair observations within each month (so months are
pair-count weighted) before correlating PA difference with the CMB-projected
velocity. The latter is the stronger weighted PA-difference result (p = 0.0005).
A 37 × 19 scan gives a nominal maximum
at RA = 230°, Dec = +60° (r = −0.745), 82.7° from the CMB dipole. Earth’s orbital velocity
samples essentially one plane, so the scan identifies only a direction’s
projection onto that plane, not its full three-dimensional vector. The CMB and
nominal maximum projections correlate at r = 0.984, and 61 of 703 grid
directions have |r| at least as large as the direct CMB value. The 30°
three-dimensional threshold is consequently not an identifiable test in this
geometry. The grid detects directional annual modulation (plus-one permutation
p = 0.005 from 200 complete-grid shuffles), but it does not select the CMB
projection over the many compatible annual templates. Widening sectors to 60° gives r =
−0.791 at another member of the same projection-degenerate family. The short
series further limits precision, so this is evidence only for exploratory
annual directional structure, not a CMB-frame detection or a unique sky-axis localization.

| Min. baseline (km) | Direct r(CMB) | Months |
| --- | --- | --- |
| 50 (primary) | −0.733 | 16 |
| 200 | −0.731 | 16 |
| 500 | −0.730 | 16 |
| 1000 | −0.723 | 16 |
| 2000 | −0.697 | 16 |

The daily EW−NS difference, retained as a supplementary diagnostic, gives a
direct CMB correlation of r = −0.495 (n = 462). Its agreement with the monthly
result supports the directional temporal pattern. Neither cadence can identify
a unique three-dimensional direction from Earth-orbital velocity alone.

| Cadence | Sector width | Nominal RA (°) | Nominal Dec (°) | r | Δθ_CMB (°) | p (perm.) | Interpretation |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Monthly | 45° | 230 | 60 | −0.745 | 82.7 | 0.005 | Annual template detected; CMB not uniquely selected |
| Monthly | 60° | 230 | 60 | −0.791 | 82.7 | 0.005 | Annual template detected; CMB not uniquely selected |
| Daily | 45° | 230 | 60 | −0.505 | 82.7 | — | Axis not uniquely identifiable |

### 4.5 Ionospheric and Geometry Controls

The correlation length persists on geomagnetically quiet days ($$\lambda = 1708$$ km,
Kp ≤ 2, 712,186 pairs, R² = 0.743). Active days (Kp ≥ 5, 39,093 pairs) give a longer
$$\lambda = 1931$$ km (R² = 0.295); the smaller sample increases the variance and the value awaits confirmation with additional active-period data. Storm-excluded days (Kp $\lt$ 5) give
$$\lambda = 1858$$ km (R² = 0.705, 1,788,462 pairs), consistent with the base value. The distance-matched subset
yields a comparable $$\lambda = 1889$$ km (R² = 0.599), indicating the signal is not a
product of network geometry. Hemispheric $$\lambda$$ differs north to south, as expected
from the uneven station distribution; the distance-matched control is the
geometry-robust comparison.

| Control | $$\lambda$$ (km) | R² |
| --- | --- | --- |
| Base (all days) | 1862 | 0.699 |
| Quiet days (Kp ≤ 2) | 1708 | 0.743 |
| Active days (Kp ≥ 5) | 1931 | 0.295 |
| Storm-excluded (Kp $\lt$ 5) | 1858 | 0.705 |
| Distance-matched | 1889 | 0.599 |
| Northern hemisphere | 1365 | 0.678 |
| Southern hemisphere | 2353 | 0.038 |

### 4.5a Product-Error and Atmospheric-Channel Controls

Two mundane channels could in principle mimic a distance-ordered clock
correlation; each is evaluated against a computed forward model or a
within-product partition rather than by label-permutation nulls alone.
The leading channel is shared satellite orbit and clock error projecting
into each station's receiver-clock estimate. The forward model (Step 3.2)
constructs the expected kernel directly: the pairwise cosine similarity
of the stations' satellite-weight vectors, built from real CODE SP3
orbits (122 satellites, 300 s cadence, 97 epochs spanning a full day)
under standard elevation weighting and the 513 MGEX station coordinates.
The kernel is flat above 0.93 through the 0–2,000 km band where the
measured coherence carries most of its structure, decays only on the
constellation-footprint scale (0.70 at 3–5,000 km, 0.45 at 5–8,000 km,
0.18 at 8–12,000 km — an effective scale an order of magnitude longer
than the observed $$\lambda = 1862$$ km), and is everywhere non-negative
whereas the measured profile crosses zero. A shared satellite-error
projection can weight this kernel but cannot steepen it, so the
geometric product-error channel cannot supply the measured decay.

The remaining mundane candidate is the spatially-correlated atmospheric
component — tropospheric delay terms absorbed into the per-station clock
estimates. An atmospheric driver is weather-organized rather than
distance-organized: synoptic fields are hemisphere-confined, and the
large-scale modes (annular modes, jet-stream teleconnections) are
coherent along latitude bands over $$\sim 10^4$$ km, not across them.
Three within-product controls on true station latitudes (Step 3.3) test
this directly. First, refitting the profile within hemisphere classes,
the cross-hemisphere subset — pairs that cannot share any synoptic
weather — returns the cleanest structure in the archive
($$\lambda = 2101$$ km, R² = 0.769, 655,068 pair-days), against intra-NH
$$\lambda = 1308$$ km (R² = 0.653). The intra-SH binned profile likewise
decays monotonically (+0.63 to −0.14); its weak exponential-fit quality
(R² = 0.032) reflects the long-baseline-dominated coverage of the
southern network rather than absent structure. Second, at
$$d \gt 4000$$ km, same-latitude-band pairs ($$|\Delta\mathrm{lat}| \lt 10°$$)
return mean phase alignment −0.096 while different-band pairs
($$|\Delta\mathrm{lat}| \gt 30°$$) return −0.026 — the opposite ordering of
the zonal-teleconnection prediction, and not a distance artifact because
the same-band subset is the closer one (median 7,963 vs 9,129 km).
Third, the decisive cell: same-band pairs at $$d \gt 8000$$ km — where a
zonal atmospheric kernel must retain positive coherence — return mean
phase alignment −0.118 on 107,013 pair-days, opposite in sign to the
atmospheric requirement. The weather-organized component is therefore
excluded at the level of its required structure; what remains of the
tropospheric channel is per-station incoherent residual error, which
cannot generate a distance-ordered pair correlation by construction.

| Control subset | $$\lambda$$ (km) | R² | Pair-days | Mundane prediction |
| --- | --- | --- | --- | --- |
| Intra-NH pairs | 1308 | 0.653 | 850,240 | Structure expected |
| Intra-SH pairs (profile decays +0.63 → −0.14) | 2437 | 0.032 | 322,247 | Structure expected |
| Cross-hemisphere pairs (no shared weather) | 2101 | 0.769 | 655,068 | No structure expected — cleanest fit observed |

| Latitude-band control | Mean PA | Pair-days | Median d (km) | Atmospheric requirement |
| --- | --- | --- | --- | --- |
| Same band, $$|\Delta\mathrm{lat}| \lt 10°$$, $$d \gt 4000$$ km | −0.096 | 217,430 | 7,963 | Positive zonal coherence — contradicted |
| Different band, $$|\Delta\mathrm{lat}| > 30°$$, $$d > 4000$$ km | −0.026 | 899,956 | 9,129 | Lower coherence than same band — contradicted |
| Same band, $$|\Delta\mathrm{lat}| \lt 10°$$, $$d \gt 8000$$ km | −0.118 | 107,013 | — | Positive coherence required — contradicted |

### 4.5b Estimator Decomposition of the Fitted Scale

The pooled MGEX scale ($$\lambda = 1862$$ km) sits below the IGS-final
family reported in Papers 1–2 (3,330–4,549 km). Both analyses measure
the same statistic — the cosine of the magnitude-weighted circular mean
cross-spectral phase — so the offset can be decomposed
estimator-axis by estimator-axis on identical real station-days
(Step 3.5). Three documented settings differ between the pipelines: the
Welch segment length (96 samples here, $$\Delta f \approx 34.7$$ µHz,
versus a full-day segment in Papers 1–2, $$\Delta f \approx 11.6$$ µHz),
the per-epoch common-mode removal (applied here; not applied in Papers
1–2), and the clock products themselves.

On a 58-day sample spanning the archive (211,012 pair-days), the
factorial decomposition returns: canonical settings
$$\lambda = 1285 \pm 233$$ km (day-block bootstrap; the full-archive
estimate on all 462 usable days is 1,862 km); common-mode removal switched off,
$$\lambda = 1368$$ km; full-day spectral resolution with common-mode,
$$\lambda = 1363$$ km; and the Papers-1–2 estimator combination
(full-day resolution, no common-mode removal), $$\lambda = 1598$$ km.
Restricting the full-day spectrum to the resolved MGEX floor
($$\geq 34.7$$ µHz) returns $$\lambda = 1045$$–$$1049$$ km, confirming
that the two lowest resolved bins carry the longest-range content — as
required if the correlation is band-ordered. The estimator axes thus
move the fitted scale over $$1.0$$–$$1.6 \times 10^3$$ km on identical
data (≲±25% around the canonical value) and cannot supply the ~2×
offset to the IGS-final family; the residual resides in the clock
products themselves — their multi-GNSS combined construction, station
complement, and epoch coverage — rather than in the correlation
estimator.

The same ordering is recovered per sub-band: on the full-day matrix the
fitted scale declines from $$\lambda \approx 1.8$$–$$2.0 \times 10^3$$ km
below 30 µHz to $$\approx 0.5$$–$$1.2 \times 10^3$$ km above 50 µHz,
mirroring the IGS-final profile, which declines from
≈3.9–5.9×10<sup>3</sup> km in the tidal bands to ≈1.0–1.4×10<sup>3</sup> km
at intermediate frequencies (Paper 1 §3.5.1). Both products therefore
return the same band-ordered, distance-structured correlation; the MGEX
profile is a scaled-down version of the IGS-final one. The defensible
cross-product statement is a thousands-of-km coherence scale with a
shared band ordering. The fitted scale is product- and
estimator-conditioned (product-level values ~1.0–4.8×10<sup>3</sup> km; raw-SPP
channels ~0.7–1.1×10<sup>3</sup> km and the raw ionofree channel returns
4,767 ± 835 km; Paper 3), and no point-scale identification with the
geometric saturation radius is made.

Two provenance details are bounded by the same diagnostic. First, GRG
product-days are natively 30-s sampled, so on the configured
5-min-labelled frequency grid those station-days resolve a
~0.35–5 mHz band rather than the nominal 10–500 µHz band; substituting
their correctly decimated contribution moves the pooled scale by ~1%
(1,271 → 1,285 km on the sampled day set; 9 of 486 days carry GRG data).
The per-center consequence is larger — the GRG-only matched-day scale
moves from 3,257 km to 1,933 km under the corrected pairwise epoch
intersection (Step 3.1) — because the per-center path additionally
truncated unequal-length series without timestamp alignment, a fault the
shared-grid construction removes. The pairing step now thins
finer-cadence products to the 300 s grid
before spectral estimation. Second, single-window realisations remain
epoch-dependent at the ~30–40% level, as reported in §4.1.

The fitted amplitude A decomposes along the same axes (Step 3.6;
46 COD days, 190,867 pair-days, all statistics evaluated on identical
pair sets). The phase-alignment index is bounded in [−1, +1] and
saturates once the in-phase component dominates the band-integrated
cross-spectrum, so A is not a linear estimator of the shared coherent
power: on the same bins the canonical configuration returns
A = 0.789 while the band-averaged real coherency — the linear
in-phase coherence measure — returns A = 0.250. The same
statistic-level inflation is visible inside the corpus on Paper 3's
raw-SPP analysis, which reports both statistics on identical pair-days:
normalised coherence amplitude 0.182 against phase-alignment 1.056,
a ~5.8× spread on 61.8 million pairs. Estimator conditioning
contributes the next factor: under the Papers-1–2 settings on the same
MGEX station-days (single full-day Welch segment, no common-mode
removal, no flat-line rejection) the fitted amplitude falls to
A = 0.466, and the flat-line rejection alone moves
A = 0.436 → 0.789 — held-constant clocks constitute 11.5% of
COD and 18.2% of GRG station-days and inject near-random pair phases.
Per-product ordering inside MGEX confirms the remaining conditioning:
on identical canonical settings GRG returns A = 0.462 against
COD 0.789. A labelled transfer calibration (Step 3.6; synthetic
red-noise pairs, estimator calibration only) inverts the published
amplitudes to underlying band coherences of ρ ≈ 0.30 (MGEX) and
ρ ≈ 0.05 (IGS finals) under each family's own spectral window —
a product-level difference in the shared in-phase budget — not yet modelled
from product construction, and not in contradiction with the measured
structure. The fitted offset shows the
same bookkeeping: all three product families return negative C₀, and the
deeper MGEX asymptote is the zero-sum signature of its per-epoch median
datum — toggling the common-mode subtraction on identical pair-days
moves C₀ from −0.015 to −0.066, the same distant-pair anticorrelation
produced by the conditioned datum kernel (Paper 1, §4.6). The
cross-product comparison is therefore carried by the signed,
distance-ordered in-phase decay and its scale ordering, not by amplitude
equality, and the ~7× headline ratio is a property of the statistic's
saturation plus product and estimator conditioning.

### 4.6 Null Tests

Null tests were run using the same MGEX combined-clock phase-alignment pipeline.
Three of the four nulls (temporal shuffle, spatial shuffle, and phase
randomisation) collapsed to R² ≤ 0, confirming that the recovered correlation
length depends on the true temporal, spatial and phase information. The
solar-rotation reassignment null, however, retains R² = 0.50 under the
primary parameters. A sensitivity analysis attributes this to a small-sample
bin artifact: short-baseline bins at 100–115 km (40 pairs) and 132–152 km
(23 pairs) receive extreme coherence values (0.29 and 0.21) under the
specific shuffle sequence, producing a spurious short-baseline spike that
the exponential fit interprets as structure. Under a robust analysis (minimum baseline 500 km, minimum bin
count 50) all four nulls give R² ≤ 0.122, confirming that the
distance-decay structure depends on the true temporal, spatial, and phase
information and is not a solar-period artifact.

| Null | R² (primary) | Passes? | R² (robust) | Passes? |
| --- | --- | --- | --- | --- |
| Temporal shuffle | −0.343 | Yes | −0.011 | Yes |
| Spatial shuffle | −0.000 | Yes | 0.028 | Yes |
| Phase randomisation | −0.001 | Yes | 0.122 | Yes |
| Solar-rotation reassignment | 0.499 | No | 0.010 | Yes |

The robust analysis (minimum baseline 500 km, minimum bin count 50)
eliminates the small-sample bin artifact that caused the solar-rotation
null to fail under the primary parameters. All four null controls pass
under the robust analysis, confirming that the observed spatial covariance
requires genuine spatial and temporal phase ordering.

## 5. Discussion

The combined MGEX receiver-clock product shows a distance-dependent structure
with $$\lambda = 1862$$ km. The estimate remains within 1,819–2,000 km
when the minimum baseline is raised from 50 to 2,000 km, while the binned-fit
R² improves from 0.70 to about 0.92. It persists on quiet days and after
distance matching, and it survives the two computed mundane channels rather
than only label-permutation nulls: the forward-modelled shared
satellite-error kernel is flat through the signal band and decays an order of
magnitude more slowly than the measured profile; conditioning the same
channel on the product's own ephemerides and satellite-clock content decays
no faster than λ = 6,650 km (~3.6× the measured scale; Paper 1, Step 5.2);
and the weather-organized
partitions return the opposite of the atmospheric signature — cross-hemisphere
pairs, which share no synoptic weather, carry the cleanest decay, and
same-latitude-band pairs beyond 8,000 km return negative coherence where a
zonal atmospheric kernel requires positive coherence. The ±112 km formal fit
uncertainty treats 1.83 million pair-days as independent; cluster-robust
resampling of whole epochs inflates the statistical error only modestly —
±86 km under day-block bootstrap, ±155 km under month-block bootstrap, and
±94 km under leave-one-month-out jackknife — so the within-product
statistical precision is about ±8%. The dominant scale uncertainties are
systematic rather than statistical: single-window realisations are
epoch-dependent at the ~30–40% level, and the matched-day cross-center
comparison contributes ~±600 km. A factorial decomposition on identical
station-days (Step 3.5, §4.5b) shows that the estimator axes — Welch
segment length and per-epoch common-mode removal — move the fitted scale
by ≲±25%, so the residual offset to the IGS-final family
(3,330–4,549 km, Papers 1–2) resides in the clock products rather than
in the correlation estimator, and the same band-ordered profile (longest
scale in the lowest resolved sub-band) is recovered in both product
families. The amplitude is conditioned the same way and more strongly:
the fitted A is a bounded, saturating phase-direction index (§4.5b, Step
3.6) whose ~7× offset from the IGS-final family decomposes into a ~3×
statistic nonlinearity, the documented estimator axes, and a
product-level difference in the shared in-phase budget — the linear
coherence measured on identical pair-days is A = 0.250, and the
deeper negative MGEX asymptote is the signature of its own zero-sum
datum, not a sign discrepancy. The result is nonetheless a useful
cross-product corroboration of a correlation scale in the thousands-of-
kilometres range, not a direct measurement-equivalence claim with the earlier
GPS PPP products.

The conventional directional measures do not replicate robustly. The full-range
EW/NS ratio is 1.29 but is non-significant under station-clustered bootstrap
(p = 0.49). The longitude-matched subset is suggestive under a pair bootstrap
(mean ratio 1.71, p = 0.058) yet non-significant under spatially clustered
resampling (p = 0.214). Its decline under more conservative baseline choices
and station de-clustering makes a positive anisotropy conclusion unwarranted
for this product.

The orbital finding is more informative but has a narrower evidential status.
The primary monthly $$\lambda$$ and EW/NS estimators are null after Bonferroni
correction, unsurprising with only 16 complete chronological months and
unstable monthly exponential fits. The model-free PA-difference statistic,
monthly mean EW minus NS phase alignment, has r = −0.716 with orbital speed
(p = 0.0018) and r = −0.767 with velocity projected onto the CMB dipole
(p = 0.0005). Because the nominal p-values assume independent monthly points,
they are reported alongside autocorrelation-aware diagnostics: effective-
degrees-of-freedom values p = 0.072 and p = 0.018, matched AR(1)-surrogate
values p = 0.019 and p = 0.003, and circular-shift values p = 0.125 and
p = 0.063 (the 1/16 floor for the projection channel). The association is
therefore tempered but not erased, and it is the physically motivated
velocity-projection channel that survives; leave-one-month-out correlations
remain within r = −0.66 to −0.86, excluding single-month dominance. These
signs and magnitudes are consistent with the preceding
TEP-GNSS analyses, which motivated the physical prediction before this MGEX
analysis. That history makes the result corroborative across products. However,
PA difference was chosen here after the primary estimators were unstable, so it
is exploratory within this study and should not be counted as an independent
confirmatory test or folded into confirmatory family-wise inference.

The annual-direction analysis keeps January 2025 through April 2026 as separate
months. Its direct CMB-projection correlation is r = −0.733, consistent with
the stronger weighted orbital result (r = −0.767, p = 0.0005). The full-sky
scan has a nominal maximum at RA = 230°, Dec = +60° (r = −0.745), 82.7° from
the CMB dipole. Earth’s orbital velocity remains in the ecliptic plane, so dot-product
regression identifies only an axis’s in-plane projection; directions separated
substantially in three dimensions can give nearly identical annual predictors.
Here the CMB and nominal-best projected predictors correlate at r = 0.984, and
61 of 703 grid directions equal or exceed the direct-CMB |r|. The 30°
three-dimensional rule is thus not an identifiable criterion for this design.
The permutation result (plus-one p = 0.005 over 200 complete-grid shuffles)
supports directional annual structure, while the direct CMB projection is only
one non-unique member of that structure. It does not identify the CMB frame.
The 60° sensitivity result and daily analysis
are corroborative diagnostics, not independent confirmations. The 16-month
series further limits precision; a preregistered multi-year analysis should
lock the PA-difference definition and report an identifiable ecliptic-plane
direction or add non-coplanar velocity information before claiming a unique
sky-frame axis.

The satellite-clock analogue does not recover an exponential structure.
GLONASS, BeiDou, and Galileo have negative fitted R² values, while GPS has
R² = +0.13 with boundary convergence. That result is consistent with the
single-reference-time architecture of the combined MGEX solution and limits
what this product can say about satellite-clock effects; it is not evidence
against a physical effect in independently estimated satellite clocks.

The central limitation is structural: CODE MGEX supplies one combined
multi-GNSS receiver-clock solution, so the constellations cannot be separated
in this analysis. The present study therefore contributes product-type and
epoch diversity, rather than per-constellation independence. Longer, separately
processed multi-GNSS or raw-observation analyses are needed to turn the
corroborative orbital result into a prospective test and to evaluate sky-frame
localization.

## 6. Conclusions

This product-type independence stress test uses the CODE COD0MGXFIN combined
multi-GNSS receiver-clock product from 2025-01-01 to 2026-05-01. Its principal
result is a distance-dependent correlation scale of $$\lambda = 1862$$
km (R² = 0.699; formal within-fit σ = ±112 km, cluster-robust
σ = ±155 km from month-block epoch resampling), stable across the tested
baseline cutoffs and present on quiet
geomagnetic days, with epoch- and product-level scale systematics of ~30–40%
reported alongside the statistical error. Under the primary short-baseline parameters the
solar-rotation reassignment null retains R² = 0.50, traced to a small-sample
bin artifact at 100–152 km; the robust analysis (minimum baseline 500 km,
minimum bin 50) passes all four null controls and is reported as the clean
result, with the fitted scale essentially unchanged (1,819 km). This is cross-product corroboration of a spatial structure at
the thousands-of-kilometres scale reported in the earlier TEP-GNSS papers,
while remaining subject to the different metric and combined-clock product
construction. The fitted scale is a product-dependent phenomenological
observable — the transfer map from the microscopic screening operator
$\mathcal{S}_\Sigma(\mathcal{E})$ to $\lambda$ remains to be derived — so
the result is reported as corroboration of the TEP scale, not as a
parameter-level prediction test.

The secondary results are deliberately qualified. Standard anisotropy and
primary orbital metrics are not robust in this short MGEX interval. A
model-free PA-difference statistic has strong associations with orbital speed
(r = −0.716, p = 0.0018) and CMB-projected orbital velocity (r = −0.767,
p = 0.0005) across 16 chronological months. Because those signs were
anticipated by previous TEP-GNSS work, the result is corroborative across
products; because this operationalisation was selected after primary MGEX fits
were unstable, it remains exploratory within this study and needs a
prospective, longer-baseline replication.

The CMB-labelled orbital template is correlated with the exploratory statistic: the direct
monthly CMB projection has r = −0.733, while the weighted PA-difference test
has r = −0.767 (p = 0.0005). The grid’s nominal RA = 230°, Dec = +60° maximum
is 82.7° from the CMB dipole. Because orbital velocity constrains only an
axis’s ecliptic-plane projection, the grid cannot
uniquely localize a three-dimensional sky direction; its within-grid
permutation statistic (plus-one p = 0.005 from 200 shuffles) measures annual
directional structure; it does not detect a CMB rest frame. The satellite-clock analysis is limited by the combined
product’s shared reference time and does not recover an exponential signal; the
MGEX product therefore cannot test the satellite-clock channel at all, and a
per-constellation solution with independently estimated satellite clocks would
be required for that test. The
next decisive test is a preregistered chronological analysis over multiple
annual cycles, reporting an identifiable plane-projected direction or using
non-coplanar velocity information.

## 7. Data and Code Availability

All data are publicly available from NASA CDDIS
([cddis.nasa.gov](https://cddis.nasa.gov)) and the IGS
([igs.org](https://igs.org)) under their standard open-data
policies. The analysis uses the MGEX combined multi-GNSS receiver-clock product
COD0MGXFIN (CODE), with WUM (Wuhan) and GRG (CNES) MGEX clock products as
day-level fallbacks, downloaded via authenticated HTTPS. No proprietary or
restricted data are used, and no raw RINEX observation files are required.

Analysis code and pipeline outputs are available at
[github.com/matthewsmawfield/TEP-GNSS-MGEX](https://github.com/matthewsmawfield/TEP-GNSS-MGEX).

## 8. References

### 8.1 TEP Series

- Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. v0.16 (Jakarta). DOI: 10.5281/zenodo.16921911.

- Smawfield, M. L. (2025). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks* (Paper 1, TEP-GNSS). DOI: 10.5281/zenodo.17127229.

- Smawfield, M. L. (2025). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products* (Paper 2, TEP-GNSS-II). DOI: 10.5281/zenodo.17517141.

- Smawfield, M. L. (2026). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: [10.5281/zenodo.17860166](https://doi.org/10.5281/zenodo.17860166) (Paper 3)

- Smawfield, M. L. (2026). *TEP-PPP: Operational Time Echoes — Phase-Resolved Spatial Clock Priors for GNSS Outage Bridging*. Preprint v0.1 (Pune). In preparation. (Paper 33, TEP-PPP).

### 8.2 GNSS Data and Products

- Montenbruck, O., Steigenberger, P., & Hauschild, A. 2014, *GPS Solutions*, 18, 49, "Multi-signal GNSS data from the IGS MGEX experiment — status and outlook".

- International GNSS Service (IGS). MGEX Multi-GNSS Experiment. [https://igs.org/mgex/](https://igs.org/mgex/).

- NASA CDDIS. Crustal Dynamics Data Information System. [https://cddis.nasa.gov](https://cddis.nasa.gov).

- Center for Orbit Determination in Europe (CODE). MGEX combined products. [https://www.aiub.unibe.ch](https://www.aiub.unibe.ch).

### 8.3 Cosmology and CMB

- Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, *A&A*, 641, A6, "Planck 2018 results. VI. Cosmological parameters", doi:10.1051/0004-6361/201833910.

### 8.4 Methodological References

- Welch, P. D. 1967, "The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms", *IEEE Trans. Audio Electroacoust.*, AU-15, 70–73.

Smawfield, M. L. 2026. Temporal Equivalence Principle series, Papers 0–33. Zenodo preprints and associated repositories.

## 9. Reproducibility Statement

This manuscript is accompanied by a complete, open-source reproduction pipeline
available at
[github.com/matthewsmawfield/TEP-GNSS-MGEX](https://github.com/matthewsmawfield/TEP-GNSS-MGEX).
All figures, tables, and statistical values reported in this paper can be
regenerated from publicly available data using the code and instructions
provided in the repository.

**Data:** The analysis uses only publicly distributed MGEX combined
multi-GNSS receiver-clock products (CODE COD0MGXFIN), with WUM and GRG
products as day-level fallbacks. Data were downloaded from NASA CDDIS
([cddis.nasa.gov](https://cddis.nasa.gov)) and the IGS
([igs.org](https://igs.org)) via standard authenticated HTTPS
endpoints. No proprietary, restricted, or raw RINEX observation files are
required.

**Software:** The pipeline is written in Python (NumPy, SciPy)
and standard geodetic libraries. All dependencies are listed in
`requirements.txt` in the repository root. No commercial or
closed-source software is required.

**Execution:** After installing dependencies and configuring access to the
authenticated CDDIS endpoints, running `python scripts/run_all.py`
from the repository root reproduces the analysis chain: data download,
station-pair differencing, phase-coherent stacking, isotropic and anisotropic
correlation fitting, orbital-velocity projection, and figure generation. Run
`python scripts/generate_figures.py` after an analysis-only rerun;
then run `cd site && npm run build` to build the manuscript site.
Execution time depends on hardware, cached downloads, and data availability.

**Verification:** All derived output files in `results/outputs/`
are regenerated by the pipeline from raw data, permitting direct
verification of reproduced results.

**Environment:** The pipeline has been tested on macOS 14 and
Ubuntu 22.04 with Python 3.11. All dependencies are listed in
`requirements.txt` for reproducible installation.

---

*This document was automatically generated from the TEP-GNSS-MGEX research site. For the interactive version with figures and enhanced formatting, visit: https://mlsmawfield.com/tep/gnss-mgex/*

*Source code and data available at: https://github.com/matthewsmawfield/TEP-GNSS-MGEX*

---

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# Temporal Equivalence Principle and the Earth Flyby Anomaly: Clock-Sector Response and Coherent-Link Cancellation
**Matthew Lukin Smawfield**
Version: v0.4 (Yogyakarta)
First published: 17 May 2026
Last updated: 17 October 2026
DOI: 10.5281/zenodo.19454862

---

## Abstract

Twelve Earth gravity assist flybys spanning nine spacecraft are analyzed within the Temporal Equivalence Principle (TEP) framework. TEP posits that proper time is governed by a dynamical scalar field φ; all non-gravitational matter couples universally to a causal matter metric through the conformal factor A(φ) = exp(β_A φ/M<sub>Pl</sub>) with β_A = −1. Under Paper 0's amplitude–shear split (v0.16), the Earth-vicinity Temporal Shear is strongly screened by the two-body kinetic operator, so the flyby anomaly cannot be a shear-force effect. The candidate mechanism is instead the clock sector: highly asymmetric, fast hyperbolic flybys accumulate rapid proper-time offsets at perigee through the unsuppressed amplitude channel S_A, which orbit-determination codes — reconstructing velocities under an assumed GR time standard — misread as ΔV discontinuities wherever the tracking link references the spacecraft's own time standard. The channel analysis performed here shows the term survives only in clock-carrying observables (one-way, non-coherent, or regenerative-ranging links and onboard-clock telemetry); in the coherent-transponder class assigned to the published catalogue from the literature, the transponder endpoint cancels it identically, so on that classification the catalogue anomalies are not produced by the clock sector in the measured observable (encounter-specific tracking modes remain to be checked) — the mechanism's claim is instead a falsifiable clock-carrying prediction whose m/s-scale raw amplitude is conditional on the adopted profile and the eccentric-orbit clock constraint in Section 5.7a. The fitted response profile S_⊕(r) is accordingly reinterpreted as the clock-sector amplitude calibration, with the characteristic terrestrial field scale R_sol = R_T(M_⊕) = (3M_⊕/4πρ_T)<sup>1/3</sup> ≈ 4146 km for ρ_T ≈ 20 g/cm³. Since the UCD calibration of ρ_T descends from the GNSS coherence-length measurement, this terrestrial length scale is a single shared input rather than an independent validation.

The accumulated proper-time offset manifests as an apparent velocity discontinuity in GR-based orbit reconstruction — a clock-sector artefact misread as ΔV without an unmodeled force — in the measurement classes that carry the spacecraft's own time standard. The non-radial component of the offset integral, modulated by Earth's oblateness (J2, J3, J4), trajectory asymmetry, the velocity-dependent response template, and the perigee plasma template layer, supplies the geometric template against which the observed flyby anomaly pattern is organized. Six published nonzero entries are retained in the re-transcribed catalogue — four firm detections (NEAR, Galileo 1990, Rosetta 2005, Galileo 1992) and two marginal ~2σ entries (Cassini, MESSENGER) — alongside three published nulls/bounds and three flybys without public anomaly reports; the re-transcription against the Anderson et al. (2008) primary source corrected several sign, uncertainty, and declination entries (Section 2.3). Under a full-catalog likelihood with a fractional misspecification floor — a per-flyby systematic set to the measured between-flyby amplitude heterogeneity ($CV_\beta \approx 0.47$) times each model's own predicted signal, iterated to a fixed point and invariant under the arbitrary field-normalization anchor — the archived information-criterion comparison describes how the restricted geometric template organizes the catalogue relative to the Null by $\Delta{\rm BIC} \approx 1.67\times10^{4}$; the Anderson empirical baseline, whose two parameters are fitted on the same catalogue, attains a nominally smaller BIC but at parity-level separation ($\Delta{\rm BIC} \approx 4.8$), an ordering that reverses in TEP restricted's favour ($\Delta{\rm BIC} \approx +2767$) when the published uncertainties are taken at face value without a systematic floor. These scores describe empirical geometry-template organization, not evidence for the clock-sector mechanism, which is absent from the coherent catalogue observable. They retain the archived iteratively weighted fitting convention; a fully optimized prediction-dependent-variance likelihood and its nuisance accounting are required before interpreting them as calibrated model-selection evidence.

A structural consistency check is provided by the ordering of anomaly magnitudes against the trajectory-asymmetry factor (cosδ_in − cosδ_out). In the gated n = 6 detection ensemble the ordering is positive but not decisive (Spearman ρ = +0.66, p = 0.156): NEAR carries both the largest asymmetry factor and the largest published anomaly in every ranked subset. Across the broader catalogue the rank association is moderate: Spearman ρ = 0.56 (p = 0.11) over the n = 9 flybys with published asymmetries, and ρ = 0.68 (p = 0.090) over the n = 7 flybys with nonzero published anomalies, computed in the pipeline's ordering diagnostic. These are reported as a qualitative consistency check rather than a primary discriminator. The inverse-variance weighted mean $\beta_{\rm fit} \approx 4.74 \times 10^{-3}$ (formal SE $5.83 \times 10^{-4}$; random-effects SE $1.28 \times 10^{-3}$; between-flyby $\tau \approx 2.23 \times 10^{-3}$) quantifies the cross-flyby effective response scale; it is not the bare conformal coupling $\beta_A$. The amplitude-informative per-flyby fits span $4.43 \times 10^{-3}$ to $1.44 \times 10^{-2}$ (with the negative-branch Galileo 1992 member at $6.7 \times 10^{-2}$), consistent with geometry-dependent modulation. The fitted amplitudes are trajectory-response coefficients, not the Cassini source charge; Cassini PPN compliance is evaluated through the solar source-charge projection $\alpha_{\rm eff} = S_\Sigma \alpha_0$ (Section 4.6.1a). The re-transcribed catalogue resolves the historical Cassini sign question: the primary-source anomaly is negative (−2 ± 1 mm/s), so the kernel's predicted sign agrees and the residual tension is magnitude-only. The geometry envelope carries seven nominal deterministic coefficients plus one fitted amplitude against six retained nonzero entries, so the model-selection comparisons are read as organisational support under the adopted convention rather than calibrated evidence.

This work shows that a restricted trajectory-geometry template quantitatively organizes the published Earth flyby anomaly catalogue under the adopted Anderson trajectory-geometry convention, while remaining consistent with precision solar system constraints and identifying specific stress tests—principally the Galileo 1992 amplitude shortfall and the Juno raw-tension null, together with raw DSN reanalysis—that are required to advance from model organisation to key confirmation. The catalogue's evidential status is conditional on one provenance question carried openly here: the orbit-determination filtering interpretation invoked for the published nulls is symmetric with the alternative that the early anomalies were artefacts of less sophisticated orbit-determination pipelines — the two readings are equally consistent with the present data, and only an independent minimal-OD reanalysis of anomaly and null flybys under identical force models can arbitrate between them.

The terrestrial response profile $S_\oplus(r)$ constitutes the Earth-vicinity geometric realization of the clock-sector amplitude response $S_A(\mathcal{E})$. Rather than a hard boundary, this radial profile traces the continuous amplitude structure of the temporal field within Earth's local potential well.

Keywords: Earth flyby anomaly, Temporal Equivalence Principle, clock-sector artefact, proper-time offset, trajectory asymmetry, geometric screening, Temporal Topology, Temporal Shear

## 1. Introduction

The Equivalence Principle (EP) is a cornerstone of general relativity, stating that gravitational acceleration is locally indistinguishable from acceleration due to motion. However, the Temporal Equivalence Principle (TEP)—the assertion that proper time is governed by a dynamical scalar field whose conformal gradients and disformal velocity-dependent terms can produce screened trajectory-level residuals—posits that the rate of time is a dynamical scalar field $\phi$. This framework, established in Paper 0, proposes that all non-gravitational matter couples universally to a causal matter metric $\tilde{g}_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$, where $A(\phi) = \exp(\beta_A \phi/M_{\text{Pl}})$. In the flyby regime the disformal term is subdominant and the conformal sector dominates the clock-sector response.

#### Key Terminology

- *Proper time* ($\tau$) is the time measured by a clock following a specific trajectory through the causal metric.

- *Temporal Topology* refers to the spatial structure of the field $\phi$, whose gradient response is pinned where the ambient time-gradient is steep — weak-field language: high-acceleration configurations — by the nonlinear kinetic operator; density is an environmental diagnostic, not an independent threshold.

- *Temporal Shear* ($\Sigma_\mu = \nabla_\mu \ln A = (\beta_A/M_{\rm Pl})\nabla_\mu\phi$) is the gradient of the conformal factor, which generates the observed geometric acceleration.

- *Clock-sector artefact* describes the apparent velocity discontinuity that orbit-determination codes produce when accumulated proper-time offsets — sampled through the unsuppressed amplitude channel $S_A$ at perigee — are present in a clock-carrying tracking link (one-way, non-coherent, or regenerative-ranging observables) and are fitted under an assumed GR time standard. In the fully coherent two-way class the transponder endpoint cancels the term identically (Section 5.2).

- *PPN parameter $\gamma$* measures the amount of spatial curvature; in Paper 0 (Sec. 7, Step 80), $\gamma_{\rm PPN} - 1 = -\alpha_0\,\alpha_{\rm eff}/(1 + \alpha_0\,\alpha_{\rm eff}) = -2\beta_A^2 S_\Sigma/(1 + 2\beta_A^2 S_\Sigma)$ under the canonical unified unstretched-space metric (or $-2\alpha_0\,\alpha_{\rm eff}/(1 + \alpha_0\,\alpha_{\rm eff}) = -4\beta_A^2 S_\Sigma/(1 + 2\beta_A^2 S_\Sigma)$ on the isotropic-conformal benchmark) with $\alpha_{\rm eff} = S_\Sigma\,\alpha_0$ the screened source-charge coupling — linear in $S_\Sigma$ because the photon probe is unscreened; for $A=\exp(\beta_A\phi/M_{\rm Pl})$ this maps to $|\gamma - 1| \approx 2\beta_A^2 S_\Sigma^{(\odot)}$ under the unified unstretched-space metric ($4\beta_A^2 S_\Sigma^{(\odot)}$ on the isotropic-conformal benchmark) when comparing to Cassini's bound on $|\gamma-1|$.

## 1.1 The Earth Flyby Anomaly

Since 1990, spacecraft executing Earth gravity assists have exhibited anomalous orbital energy changes that lack a standard explanation. The NEAR spacecraft (1998) showed the largest effect: an unexplained velocity increase of 13.46 mm/s. Galileo (1990, 1992) and Cassini (1999) displayed smaller but significant anomalies. These velocity shifts occur precisely at perigee passage and persist as asymptotic excess velocities ($v_\infty$) in the outbound trajectories.

The published residuals motivate tests of both physical signals and tracking/reconstruction systematics. The physical checks in Section 5.2 bound several conventional contributions but do not exclude every systematic explanation. TEP supplies a distinct candidate clock-sector artefact through trajectory-dependent proper-time transport in links carrying an onboard time reference; Section 5.2 establishes that this term does not account for the published coherent-link catalogue.

## 1.2 TEP as a Candidate Explanation

The TEP framework provides a testable clock-sector response for flyby observables that carry the spacecraft's own time reference. Under Paper 0's amplitude–shear split (v0.16), the Earth-vicinity Temporal Shear is strongly screened by the two-body kinetic operator, so the anomaly cannot be a shear-force effect. Instead, highly asymmetric hyperbolic flybys accumulate rapid proper-time offsets at perigee through the unsuppressed amplitude channel; orbit-determination codes, reconstructing velocities under an assumed GR time standard, misread these offsets as $\Delta V$ discontinuities wherever the tracking link carries the spacecraft's own time reference — the Clock-Sector Reduction Artefact of Paper 0, whose observable reach the channel analysis of Section 5.2 fixes exactly.

The observed heterogeneity in flyby anomaly magnitudes is not random scatter but arises from deterministic geometry-dependent modulation. The TEP prediction for a given flyby depends on several physical factors: (1) perigee altitude (determines the clock-sector field excursion via the density-dependent amplitude profile), (2) approach-departure asymmetry (disformal coupling requires velocity-dependent anti-aligned geometry), (3) the perigee plasma template layer (a bounded heuristic envelope coefficient; the stronger Debye attenuation ansatz is quarantined at $S_{\rm plasma} = 1$, Section 3.10), (4) solar activity (modulates ionospheric density within that template layer), and (5) cosmographic CMB-frame velocity geometry (the disformal coupling scales as v² in the ambient-field rest frame — the frame in which the spatial gradient of the cosmological temporal field vanishes — approximated operationally by the CMB dipole frame)—a deeper prediction tested only in an exploratory capacity (Section 4.11). These factors combine to produce a wide span in per-flyby fitted β<sub>fit</sub> across the Step 008 S/N-qualified ensemble; the legacy inverse-variance diagnostic restricted to sign agreement at $\beta_{A,\text{ref}}$ (NEAR, Galileo 1990, Rosetta 2005) is reported separately as `beta_statistics_sign_gated_diagnostic` in `results/step008_fitting_results.json`, while Cassini and any other sign-tension case remain in the primary pooled layer when `strict_sign_gate` is false in `config/pipeline_config.json`. The Temporal Topology screening mechanism is essential for three reasons: (1) it ensures the flyby amplitude and environmental response remain consistent with solar system constraints; (2) it explains both detections and null results through environment-dependent screening; and (3) it establishes the transition radius $R_{\rm sol} \approx 4146$ km as a universal scale. Flybys sampling regions of large field excursion (low altitude, high asymmetry) exhibit anomalies, while those in shielded regimes (high altitude or symmetric trajectories) remain null.

## 1.3 This Work

The analysis proceeds by reconstructing trajectories from JPL Horizons, computing TEP predictions with full 3D integration, and fitting effective per-flyby couplings. Step 008 reports both the inverse-variance mean over all S/N-qualified fits (primary pooled layer when the sign gate is relaxed) and the sign-agreement-restricted diagnostic in machine-readable JSON for auditability.

The structure of this paper is as follows: Section 2 describes the data sources; Section 3 presents the TEP Temporal Topology model; Section 4 reports the fitting results and PPN validation, with an exploratory cosmographic test in Section 4.11; Section 5 discusses the clock-sector artefact interpretation; and Section 6 concludes with prospects for further tests.

## 2. Observations and Data

## 2.1 The Flyby Spacecraft Sample

This analysis utilizes nine spacecraft spanning twelve Earth flyby events between 1990 and 2020: Galileo (1990, 1992), NEAR (1998), Cassini (1999), Rosetta (2005, 2007, 2009), MESSENGER (2005), Juno (2013), Stardust (2001), OSIRIS-REx (2017), and BepiColombo (2020). The dataset is divided into three data quality classes: *published anomalies* (six flybys with measured nonzero Δv and formal uncertainties — the four positive detections plus the two negative or marginal entries of Anderson et al. 2008 Table I), *published nulls/bounds* (three flybys with explicitly reported null results or upper limits), and *no public anomaly report* (three flybys with no published search or measurement). The latter class is not used in quantitative likelihood. Table 1 summarizes the key parameters for each flyby.

#### Physical Constants

The analysis uses the following CODATA 2018 values: Earth radius $R_\oplus = 6.371 \times 10^6$ m, gravitational constant $G = 6.67430 \times 10^{-11}$ m$^3$ kg$^{-1}$ s$^{-2}$, and speed of light $c = 299\,792\,458$ m/s (exact). The reduced Planck mass $M_{\rm Pl} = 2.435 \times 10^{18}$ GeV is derived from $\hbar c/G^{1/2}$.

Table 1: Earth Flyby Spacecraft Parameters

| Spacecraft | Date | Perigee (km) | $v_\infty$ (km/s) | $\Delta v_{\rm obs}$ (mm/s) | $\sigma$ (mm/s) | Data class |
| --- | --- | --- | --- | --- | --- | --- |
| Galileo | 1990-12-08 | 960 | 13.73 | 3.92 | 0.08 | Published anomaly |
| Galileo | 1992-12-08 | 303 | 14.08 | −4.60 | 1.00 | Published anomaly (negative) |
| NEAR | 1998-01-23 | 539 | 12.72 | 13.46 | 0.13 | Published anomaly |
| Cassini | 1999-08-18 | 1175 | 19.02 | −2.00 | 1.00 | Published anomaly (negative) |
| Rosetta | 2005-03-04 | 1955 | 10.51 | 1.80 | 0.03 | Published anomaly |
| Rosetta | 2007-11-13 | 5301 | 12.46 | 0.02 | 0.05 | Published null/bound |
| Rosetta | 2009-11-13 | 2572 | 13.31 | 0.00 | 0.05 | Published null/bound |
| MESSENGER | 2005-08-02 | 2347 | 10.39 | 0.02 | 0.01 | Published anomaly (marginal) |
| Juno | 2013-10-09 | 817 | 14.79 | 0.00 | 0.02 | Published null/bound |
| Stardust | 2001-01-15 | 6009 | 10.31 | — | — | No public anomaly report |
| OSIRIS-REx | 2017-09-22 | 17239 | 8.52 | — | — | No public anomaly report |
| BepiColombo | 2020-04-10 | 12697 | 7.59 | — | — | No public anomaly report |

*Note:* $\Delta v_{\rm obs}$ values for the published anomaly and published null/bound classes are from Anderson et al. (2008) and companion papers. Stardust, OSIRIS-REx, and BepiColombo have no public anomaly report; em-dashes indicate that no published measurement or bound exists. These three flybys are not used in quantitative likelihood but are listed for geometry and predicted-null context. Rosetta 2009 is a published null (Müller et al. 2010); the archival catalogue assigns $\sigma = 0.05$ mm/s in the same null/bound precision class as Table&nbsp;1, so the automated gate classifies it as below the headline S/N threshold rather than as a missing-uncertainty row. Perigee values are altitudes above Earth's surface; $v_\infty$ is the hyperbolic excess velocity. The Galileo 1992, Cassini, and MESSENGER entries follow the Anderson et al. (2008) Table I primary-source values (−4.60 ± 1.00, −2 ± 1, and +0.02 ± 0.01 mm/s respectively), confirmed against the mission declination reconstruction in Table 1a.

## 2.2 Data Sources and Provenance

The anomaly measurements used in this analysis are taken from the peer-reviewed literature, specifically the comprehensive study by Anderson et al. (2008) and subsequent mission-specific analyses. These values were obtained through NASA's Deep Space Network (DSN) Doppler tracking combined with the Jet Propulsion Laboratory Orbit Determination Program (ODP).

Literature sources:

- Primary reference: Anderson, J. D., et al. (2008). "Anomalous Orbital-Energy Changes Observed during Spacecraft Flybys of Earth." *Physical Review Letters*, 100(9), 091102.

- Rosetta analysis: Morley, T., &amp; Budnik, F. (2007). "Rosetta Navigation at its First Earth-Swingby." *Proceedings of the 20th International Symposium on Space Flight Dynamics*.

- Juno analysis: Aksenov, E. L., &amp; Tuchin, A. G. (2020). "Earth flyby anomalies and the general relativistic theory of the Kerr gravitational field." *MNRAS*, 492(3), 3703-3711.

## 2.3 Data Quality Assessment

A rigorous analysis requires assessment of data quality for each flyby. All six published detections have complete DSN coverage spanning $\pm 12$ hours around perigee, enabling robust pre/post comparison (catalog-level statement; the inverse-variance $\beta_{\rm fit}$ ensemble in Steps 008 and 026 uses $n=6$ S/N-qualified primary fits when `strict_sign_gate` is false; with the corrected primary-source anomalies, all six detections agree in sign with the TEP prediction at the reference coupling, so the sign-agreement diagnostic coincides with the gated set). The reported uncertainties (0.01–1.00 mm/s) span the DSN Doppler precision range for the different missions and tracking configurations.

Ensemble gates (Steps 008 and 026) are applied mechanically: a flyby enters the restricted likelihood only if it carries a published $(\Delta v, \sigma)$ pair, satisfies $|\Delta v|/\sigma &gt; 2$, and shows sign agreement between $\Delta v_{\rm obs}$ and the TEP prediction at the pre-specified reference coupling. Rows that fail a gate are retained in the JSON audit trail with an explicit machine-readable reason (`missing_observation`, `snr_below_threshold`, `sign_mismatch`, etc.) and, where available, the Step&nbsp;003 archival reference string, so exclusions are reviewable rather than tacit.

Systematic error controls: Antenna phase center, tropospheric delay, and station positions are well-modeled in the JPL ODP software. Residual uncertainties are at the $\sim 0.1$ mm/s level, which is an order of magnitude below the larger anomalies (NEAR, Galileo).

## 2.4 Trajectory Data from JPL Horizons

Spacecraft trajectories for the analysis were obtained from NASA's JPL Horizons ephemeris system. For each flyby, state vectors (position and velocity) spanning $\pm 2$ days around perigee passage are reconstructed. These trajectories represent the best-estimate spacecraft paths based on all available tracking data.

### 2.4a Geometry Factor Audit: Published Declinations vs Horizons

The trajectory asymmetry factor used in both the Anderson et al. (2008) empirical formula and the TEP model is $\cos\delta_{\rm in} - \cos\delta_{\rm out}$. Table 1a audits the provenance of this factor by comparing three independent estimates for each flyby with published declinations: (i) the Anderson et al. (2008) published value transcribed directly into the catalog, (ii) the value computed from the published inbound and outbound declination angles, and (iii) the value derived independently from JPL Horizons state-vector reconstruction.

Table 1a: Trajectory Asymmetry Factor Audit

| Flyby | $\delta_{\rm in}$ (deg) | $\delta_{\rm out}$ (deg) | $\cos\delta_{\rm in} - \cos\delta_{\rm out}$ (published) | $\cos\delta_{\rm in} - \cos\delta_{\rm out}$ (computed from decs) | $\cos\delta_{\rm in} - \cos\delta_{\rm out}$ (Horizons) |
| --- | --- | --- | --- | --- | --- |
| NEAR | $-20.76$ | $-71.96$ | $\mathbf{0.625}$ | $0.625$ | $0.625$ |
| Galileo 1990 | $-12.52$ | $-34.15$ | $\mathbf{0.149}$ | $0.149$ | $0.149$ |
| Cassini | $-12.92$ | $-4.99$ | $\mathbf{-0.022}$ | $-0.022$ | $-0.022$ |
| Galileo 1992 | $-34.26$ | $-4.87$ | $\mathbf{-0.170}$ | $-0.170$ | $-0.170$ |
| Rosetta 2005 | $-2.81$ | $-34.29$ | $\mathbf{0.173}$ | $0.173$ | $0.173$ |
| Rosetta 2007 | $-10.80$ | $+18.60$ | $\mathbf{0.034}$ | $0.034$ | $0.034$ |
| MESSENGER 2005 | $+31.44$ | $-31.92$ | $\mathbf{0.005}$ | $0.004$ | $0.005$ |
| Juno | $-26.9$ | $+34.6$ | — | $0.069$ | $0.069$ |

Key finding: With the re-transcribed Anderson et al. (2008) Table I declinations, the computed $\cos\delta_{\rm in} - \cos\delta_{\rm out}$ agrees with the published asymmetry factor for every row, and both agree with the independent JPL Horizons state-vector reconstruction. An earlier version of this catalogue carried declination transcription errors (most consequentially NEAR's outbound leg, entered as $-19.7^\circ$ rather than $-71.96^\circ$, and both Galileo 1992 legs); those errors manufactured an apparent convention mismatch between the tabulated declinations and the published asymmetry factors that does not exist in the primary source. The three-way agreement now shown — published asymmetry, recomputed from the published declinations, and Horizons-derived — removes the declination-provenance question entirely and validates the adopted geometry factors used by both the Anderson empirical model and the TEP kernel. The TEP pipeline loads the published literature values when available (from Anderson et al. 2008 Table I) and computes from Horizons-derived declinations otherwise.

Provenance of the corrected values. The transcription errors were identified by auditing the catalogue against the Anderson et al. (2008) primary source and its companion trajectory paper (Anderson, Campbell &amp; Nieto 2006), cross-checked against independent reproductions. Corrections were confined to the published tabulated quantities: NEAR's outbound declination ($-71.96^\circ$, not $-19.7^\circ$) and uncertainty ($\pm 0.13$ mm/s, not $\pm 0.01$), Galileo 1992's two declinations ($-34.26^\circ/-4.87^\circ$, not $-30.3^\circ/-33.8^\circ$) and its anomaly ($-4.60 \pm 1.00$ mm/s, previously entered as a null), Cassini's anomaly ($-2 \pm 1$ mm/s, previously entered as $+0.11 \pm 0.05$), MESSENGER's anomaly ($+0.02 \pm 0.01$ mm/s, previously a null), Galileo 1990's inbound declination ($-12.52^\circ$, not $-25.1^\circ$) and uncertainty ($\pm 0.08$ mm/s, not $\pm 0.03$), and Rosetta 2005's anomaly ($+1.80 \pm 0.03$ mm/s) and inbound declination ($-2.81^\circ$, not $-3.4^\circ$). Perigee altitudes were likewise aligned to the primary-source tabulation (539, 960, 303, 1175, 1955, 2347 km). All downstream results in this paper use the corrected catalogue; the corrected geometry factors are corroborated independently by the Horizons column of Table 1a.

## 3. Methodology

The analysis employs a four-step pipeline to compare the published flyby anomaly geometry with the TEP response template and to test the clock-sector artefact separately in the link classes where it survives. The pipeline retrieves spacecraft trajectories from JPL Horizons, computes TEP predictions for each flyby geometry using full 3D integration, fits the coupling parameter $\beta_{\rm fit}$ to match observed anomalies, and validates all parameters against solar system PPN constraints. Which measurement classes can carry the clock-sector term is itself part of the analysis: the channel accounting of Section 5.2 shows the spacecraft's conformal excursion survives only in clock-carrying links (one-way, non-coherent, or regenerative-ranging observables), while the published catalogue is assigned to the coherent-transponder class on the literature-based ledger of Section 5.2, pending encounter-specific tracking verification.

## 3.1 Data Acquisition

Spacecraft trajectories are obtained from the NASA JPL Horizons ephemeris system via direct HTTP query to the batch CGI endpoint. For each flyby, observational quantities (right ascension, declination, range, and range-rate) are retrieved in the ICRF (International Celestial Reference Frame) at 1-minute intervals spanning $\pm 2$ days around perigee passage; three-dimensional state vectors are reconstructed from these observables in a subsequent processing step.

#### Trajectory Parameters Extracted

- Perigee altitude (minimum geocentric distance)

- Perigee velocity (speed at closest approach)

- Inbound/outbound asymptotic velocity ($v_\infty$)

- Spacecraft potential at perigee ($\Phi_{\rm sc}$)

Flyby velocity anomalies ($\Delta v_{\rm obs}$) are taken from published literature. The primary source is Anderson et al. (2008), with supplementary references for Rosetta (Morley &amp; Budnik 2007; Müller et al. 2008, 2010) and Juno (Aksenov &amp; Tuchin 2020). All values were measured by NASA/JPL using Deep Space Network Doppler tracking with the Orbit Determination Program. Asymptotic $v_\infty$ declinations ($\delta_{\rm in}$, $\delta_{\rm out}$) tabulated in Anderson et al. (2008) are transcribed directly from that source; for additional catalogued flybys without published declinations, the same eccentricity-vector reconstruction applied to archived JPL Horizons state arcs (Step 040a / Step 007, geocentric perigee anchored to the archival flyby date) supplies $\delta_{\rm in}$ and $\delta_{\rm out}$ so that geometry is traceable to Horizons rather than hand-entered asymmetries.

## 3.2 TEP Temporal Topology Model

The TEP framework provides a quantitative model for the flyby anomaly through the perigee proper-time offset arising from the temporal-field amplitude excursion. With the universal conformal coupling A(φ) = exp(β_A φ/M<sub>Pl</sub>), the accumulated offset along the trajectory projects into an apparent velocity discontinuity under GR-based orbit determination:

\begin{equation}
\mathbf{F}_\phi = \beta_{A,\rm eff} \, \frac{c^2 \nabla\phi}{M_{\rm Pl}}
\end{equation}

Under the amplitude–shear split (Paper 0 §7), the Earth-vicinity Temporal Shear is screened by the two-body operator, so this kernel is not a physical force on the spacecraft — the shear-force channel is closed. It is the orbit-determination-equivalent kernel of the clock-sector artefact in a clock-carrying link: the spacecraft's matter-metric rate offset $\eta(t) = \exp(\beta_A\psi/M_{\rm Pl})-1$ carries the same radial gradient $\beta_A\nabla\phi/M_{\rm Pl}$, and a tracking filter that assumes the GR time standard reconstructs the accumulated rate-offset excursion exactly as if it were this impulse. The two signatures are observationally indistinguishable over a flyby arc *where the term is present in the observable* — one-way, non-coherent, or regenerative-ranging links; in the coherent two-way class of the published catalogue the transponder endpoint cancels it identically (Section 5.2). The mechanism-faithful forward model is the Step 043 OD rederivation (Table 3h), which computes the artefact's signature on the clock-carrying link class over the real arcs. Here β<sub>A,eff</sub> = β_A × S_⊕(r) is the effective artefact-response amplitude, where S_⊕(r) is the continuous clock-amplitude response profile. The characteristic surface ratio S_⊕ ≈ 0.35 is fixed by the UCD saturation geometry as S_⊕ = (R_⊕ − R_sol)/R_⊕ with R_sol ≈ 4146 km (distinct from the embedding factor R_sol/R_⊕ used in Paper 6). The radial component of the apparent impulse is indistinguishable from a small shift in GM and is absorbed by orbit determination. The non-radial component—modulated by Earth's oblateness (J2, J3, J4) and the spacecraft's trajectory geometry—produces the apparent net velocity change that reads as the flyby anomaly.

The predicted velocity discontinuity is resolved through rigorous numerical integration of the artefact kernel along the trajectory in the Earth-centered inertial (ECI) frame — the accumulated apparent impulse that an orbit-determination filter would attribute to dynamics, not a physical trajectory deflection. This approach captures the accumulated proper-time offset as the spacecraft traverses the varying field amplitude, incorporating a 4th-order Spherical Harmonic Expansion (SHEX) for the geopotential to ensure that local gravitational perturbations are not conflated with the clock-sector response:

\begin{equation}
\Delta \mathbf{v}_{\rm TEP} \approx \left. \frac{\beta_{A,\rm eff} \, c^2 \nabla\phi}{M_{\rm Pl}} \right|_{\rm peri} \Delta t_{\rm peri} + \left. \frac{b_{\rm disf}}{M_{\rm Pl}} (\nabla\phi \cdot \mathbf{v}) \mathbf{v} \right|_{\rm peri} \Delta t_{\rm peri}
\end{equation}

where the simplified perigee approximation is:

\begin{equation}
\Delta v_{\rm TEP} \approx \beta_{A,\rm eff} \, \frac{c^2}{M_{\rm Pl}} \left(\frac{d\phi}{dr}\right)_{r_p} \, \frac{r_p}{v_p} \, \left[J_2 + J_3 \sin(\lambda_p) + J_4 P_4(\sin\lambda_p)\right] \left(\frac{R_\oplus}{r_p}\right)^2 (\cos\delta_{\rm in} - \cos\delta_{\rm out})
\end{equation}

where:

- $(d\phi/dr)_{r_p}$ is the scalar field gradient at perigee altitude

- $r_p$ and $v_p$ are the perigee distance and velocity

- $J_2, J_3, J_4$ are the zonal harmonics (EGM96/WGS84 coefficients)

- $\lambda_p$ is the perigee latitude

- $\delta_{\rm in}$ and $\delta_{\rm out}$ are the asymptotic declinations on approach and departure (from Anderson et al. (2008))

### 3.2b Full 3D Trajectory Integration

The perigee approximation provides a computationally efficient closed-form estimate, but the primary predictions are validated against full 3D trajectory integration to capture the accumulated proper-time offset along the spacecraft path. JPL Horizons ephemeris data for historical flybys provide scalar range and velocity observables rather than complete 3D state vectors. To reconstruct the trajectory, perigee state vectors (position and velocity) from the TEP geometry model serve as anchor points, and the hyperbolic orbit is propagated via Keplerian mechanics to generate full 3D ephemeris consistent with the JPL Horizons time grid. The reconstructed trajectory is validated against the JPL range data to ensure physical consistency.

The apparent velocity discontinuity is accumulated by integrating the TEP clock-sector response along the reconstructed path:

\begin{equation}
\Delta v_{\rm TEP}^{(3D)} = \int_{t_{\rm in}}^{t_{\rm out}} \beta_{A,\rm eff}(r(t)) \, \frac{c^2}{M_{\rm Pl}} \, \left|\frac{d\phi}{dr}\right|_{r(t)} \, \mathcal{B}(\lambda(t), r(t)) \, \cos\delta_{\rm asym} \, \mathcal{E}(t) \, s_{\rm disf}(t) \, dt
\end{equation}

where $\mathcal{B}(\lambda, r) = [J_2 + J_3 \sin(\lambda) + J_4 P_4(\sin\lambda)] (R_\oplus/r)^2$ is the zonal harmonic bracket, $\mathcal{E}(t)$ is the geometry envelope factor, and $s_{\rm disf}(t)$ is the disformal modulation factor. The integration uses 240–360 points per flyby and captures altitude-dependent field gradient variation, local plasma modulation, and velocity-dependent disformal factors at each point along the trajectory.

Step 019 applies the same impulse integrator along idealized asymmetric flyby arcs (metadata: `idealized_analytic_hypoflyby`). On comparable rows of Table 3b (Section&nbsp;4.1.2), the ratio $\Delta v_{\rm 3D}/\Delta v_{\rm simplified}$ differs from unity by tens of percent for the three gated primaries—a consistency check inside the analytic path class—while ratios against the Step&nbsp;007 perigee impulse are deliberately not marketed as ±10% “agreement.” Symmetric/high-altitude cases remain perturbatively small in Step&nbsp;007, matching the qualitative null catalogue. Ensemble fitting continues to anchor on Step&nbsp;007 + Step&nbsp;008; Step&nbsp;019 is a diagnostic shim for the trajectory integrator, not a calibration of Anderson-scale residuals.

J3 contribution: The J3 term adds a latitude-dependent asymmetry to the non-radial component of the artefact kernel. However, J3 is two orders of magnitude smaller than J2 ($|J_3/J_2| \approx 2.3 \times 10^{-3}$), and its inclusion does not collapse the heterogeneity in fitted β<sub>fit</sub> across the gated ensemble (factor ~3.3 across the tightly constrained members at fixed reference coupling, extending to ~15 with the negative-branch Galileo 1992 member, with formal Cochran Q ≫ 1). This indicates that remaining spread arises from phase-boundary and geometry–plasma modulation in the envelope, not from neglected J3 alone.

The scalar field φ relaxes outside Earth with a profile scale λ_T = 4200 km — the operational value anchored on the measured GNSS clock-correlation length L_c = 4201 km (±15% calibration band). Under the corpus's terrestrial transfer-sketch identification (Paper 6 §2) the exterior profile varies on 𝒪(R_T) with an undetermined order-unity prefactor, since a gradual screening transition produces no exterior structure sharper than the scale over which the integrated source charge changes over; the adopted value sits 1.3% above the ansatz target R_T = 4146 km, within the stated prefactor freedom and calibration band (the two are not independent: ρ_T was itself calibrated from L_c). This is a stated ansatz anchoring the profile scale — not a derived equality, and distinct from the in-medium Compton wavelength of a candidate chameleon completion.

The trajectory asymmetry factor $\cos\delta_{\rm in} - \cos\delta_{\rm out}$ is the dominant source of inter-flyby variation. Nearly symmetric trajectories (e.g., MESSENGER, $\cos\delta_{\rm in} - \cos\delta_{\rm out} \approx 0.005$) predict negligible anomalies, consistent with observations. Asymmetric trajectories (e.g., NEAR with $\cos\delta_{\rm in} - \cos\delta_{\rm out} = 0.625$) predict large anomalies. Cassini carries a small negative asymmetry factor ($-0.022$), verified by independent JPL Horizons reconstruction (Table 1a) — a physical property of the trajectory, not a convention artifact. The kernel therefore predicts a negative apparent anomaly, consistent in sign with the corrected $-2 \pm 1$ mm/s catalogue value, though the predicted magnitude is negligible and the amplitude shortfall remains a genuine limitation of the phenomenological kernel for this geometry, not a convention mismatch. The mechanism-faithful calculation is consistent with this reading: the Step 043 OD rederivation, which propagates the proper-time artefact through the batch least-squares filter rather than the kernel, returns only a negligible endpoint remnant for Cassini ($\Delta V_{\rm app} = +0.01$ mm/s, Table 3h) — the clock-sector channel does not produce the corrected $-2\pm1$ mm/s catalogue value, consistently with that datum's coherent two-way link class, in which the term is absent at the observable (Section 5.2).

\begin{equation}
\phi(r) = \phi_{\rm earth} + (\phi_{\rm space} - \phi_{\rm earth}) \left[1 - \exp\!\left(-\frac{r - R_\oplus}{\lambda_T}\right)\right]
\end{equation}

Geometric screening: Critical to the flyby amplitude and environmental response is the transition radius $R_{\rm sol} \approx 4146$ km from the UCD saturation model (Step 010), anchored on the same GNSS terrestrial calibration that fixes $\rho_T$ — a shared-input relationship under the Paper 6 §2 identification (Section 3.11), not an independent validation. This defines the characteristic ratio $S_{\oplus} = (R_{\oplus} - R_{\rm sol})/R_{\oplus} \approx 0.35$, the responsive-shell fraction of the radial profile through which a transported clock accumulates its differential excursion. Its relationship to the canonical clock-amplitude operator is an exact partition rather than a separate estimate: under the uniform-density-equivalent construction $R_{\rm sol}^3 = 3M_\oplus/4\pi\rho_T$ and $R_\oplus^3 = 3M_\oplus/4\pi\bar\rho$, the canonical response evaluated at Earth's mean density reduces to $S_A(\bar\rho_\oplus) = (\bar\rho_\oplus/\rho_T)^{1/3} = R_{\rm sol}/R_\oplus \approx 0.65$ — the embedded, saturated fraction of the equivalent radius, identical to the UCD embedding factor used in Paper 6 (UCD). The flyby clock channel projects the complementary fraction $S_\oplus = 1 - S_A(\bar\rho_\oplus)$: inside $R_{\rm sol}$ the field is pinned and carries no radial relaxation, so the excursion is generated precisely across the unsaturated shell. The two quantities are a partition sum rule of one radial profile, not two estimates of one operator (machine-checked in Step 010's `canonical_amplitude_partition` output: $S_A + S_\oplus = 1$ identically).

\begin{equation}
\beta_{A,\rm eff} = \beta_A \times S_{\oplus}(r)
\end{equation}

The pipeline's field-amplitude normalization is the canonical amplitude-sector excursion of Paper 0 v0.16. The pipeline carries the excursion complement $\phi(r)$ — zero at the pinned interior floor, rising to the ambient reference $\phi_{\rm space}$ — so that the physical excursion $\psi(r) = \phi_{\rm space} - \phi(r) \geq 0$ is the temporal-well depth sampled by a transported clock: positive inside the well and relaxing to zero at the ambient baseline, matching the corpus convention $\phi &gt; 0$ in wells, $\phi \to 0$ ambient (Paper 0 Rule 4). The normalization is the unscreened linear response of the source computed in the Paper 0 radial-ODE closure (`TEP/results/step_01_radial_ode.json`, Earth entry):

\begin{equation}
\psi_{\rm uns} = \frac{M_\oplus}{4\pi M_{\rm Pl}^{2} R_\oplus} = \frac{2 G M_\oplus}{R_\oplus c^{2}} = 1.3926 \times 10^{-9}
\end{equation}

This value is parameter-free — fixed by Earth's measured mass and radius — and independent of the fitted response amplitude $\beta$, so every pipeline prediction is strictly linear in $\beta$. It replaces the v0.1 density-minimum prescription $\phi_{\rm min}(\rho) \propto \rho^{-1/(n+1)}$ (an inverse-power/chameleon-class realization): under $\beta_A = -1$ that branch admits no stable minimum (Paper 0 constraint F1) and runs the field largest in vacuum, opposite to the corpus convention. The Step 010 self-consistent-field diagnostic on that realization is retained solely as the documented contrast — it returns $S_\oplus \approx 1.0$, i.e. the excluded branch produces no terrestrial suppression shell.

#### Characteristic Field Values — Canonical Excursion (Paper 0 amplitude sector)

- Earth well depth (unscreened source response): $\Delta\phi = \psi_{\rm uns}\,M_{\rm Pl} = 3.39 \times 10^{9}$ GeV

- Surface clock excursion: $\eta_\oplus = e^{-\psi_{\rm uns}} - 1 = -1.3926 \times 10^{-9}$ (clock-carrying link class; Step 043). This is the amplitude-sector excursion, not the static lapse: the corpus's weak-field lapse ledger gives $\ln(A/A_\infty) = 2\beta_A^2\,\Phi_N$ on the unscreened branch and the measured redshift-channel lapse $\tilde N \to e^{\Phi_N}$ under screening of the shear sector (Paper 0 §2.2, R9) — $\eta_\oplus$ is the transport excursion a carried clock accumulates, which static clock comparisons do not constrain

- Interior: excursion pinned at the well depth ($\phi_{\rm interior} = 0$ in complement units)

- TEP exterior profile scale: $\lambda_T \approx 4200$ km (set equal to the geometric saturation scale R_T(M_⊕) under the Paper 6 §2 transfer-sketch ansatz, up to an order-unity prefactor; the GNSS covariance scale $L_c = 4201$ km is the empirical anchor of that identification)

- Characteristic suppression: $S_{\oplus} \approx 0.35$ (UCD-derived from Step 010)

Fundamental coupling: The bare conformal coupling is β_A = −1.0, established in Paper 0 (Jakarta) §2.2. The fitted values reported here (β<sub>fit</sub> ∼ 10⁻³) are the fitted response amplitudes; the effective clock-sector response β<sub>A,eff</sub> = S_⊕(r) β<sub>fit</sub> is derived from these under the excursion projection S_⊕(r) defined above. The sign of the artefact contribution in the flyby geometry is determined by trajectory asymmetry; the reported β<sub>fit</sub> magnitudes should be compared to the absolute fundamental scale |β_A| = 1.0 — the ratio κ_flyby = β_fit/|β_A| ≈ 4.7×10⁻³ is the population-level channel response coefficient, reported as an unexplained transfer-map quantity of the same open class as the Paper 0 Γ_X entries pending the action-level derivation. The fitted flyby amplitude is a trajectory-response coefficient; Solar-System PPN bounds apply to the screened source-charge projection α<sub>eff</sub> = S_Σ α_0, not to the fitted flyby amplitude itself. The converse direction closes identically: terrestrial bounds on the Earth environment constrain the shear channel, for which the corpus's own two-body operator returns an Earth-vicinity suppression of ∼10⁻²¹, whereas the claimed flyby observable is the unsuppressed clock-amplitude channel — and even there it is a transport excursion accumulated by a clock carried through the relaxing radial profile, not a stationary rate offset that inter-comparisons between clocks at rest in the terrestrial environment could bound. The same unsuppressed terrestrial amplitude response is the structure that the corpus's GNSS coherence-length measurement anchors (L_c = 4201 km, Section 3.11), so the fitted excursion scale is consistent with — indeed required by — the corpus's own terrestrial calibration rather than bounded away by it.

Ambient field value: Under the canonical excursion normalization, the ambient reference carries the full well depth φ_space = Δφ = 3.39×10⁹ GeV in complement units, while the physical excursion ψ(r) decays to the interplanetary baseline over λ_T — no ad-hoc density cutoff enters, since the profile is geometric rather than density-minimum-derived. The normalization is β-independent, so the reference amplitude β = 10⁻⁴ enters only as a linear scaling of the response, not of the field.

Geometry modulation factors: Per-flyby fitted β<sub>fit</sub> in the six-member sign-consistent ensemble spans roughly a factor of three across the tightly constrained members (extending to a factor of ≈15 across the four amplitude-informative fits) — formally divergent for the near-zero-prediction Cassini and MESSENGER members — reflecting substantial geometry-, plasma-, and velocity-dependent modulation encoded in the Step 007 envelope. Four physical mechanisms are tracked in the analysis: (1) *inclination-dependent screening*—higher latitude trajectories sample less equatorial bulge; (2) *J2 oblateness*—altitude-dependent screening from Earth's shape; (3) *plasma environment*—ionospheric density modulates local screening (IRI at perigee, Step 033); and (4) *velocity effects*—disformal coupling in the high-velocity regime. These factors are incorporated into the clock-sector response calculation.

### 3.2c Component-Level Geometry Factor Analysis

To address the extreme heterogeneity in fitted β<sub>fit</sub> values across flybys (see Section 4.8), a component-level analysis extracts the effective geometry factor for each flyby independently. The geometry factor isolates trajectory-dependent modulation of the path-resolved scalar response, revealing the physical origin of the observed variation.

The effective geometry factor is defined as the ratio of observed anomaly to the gradient prediction at the reference coupling:

\begin{equation}
G_{i,\text{eff}} = \frac{\Delta v_{i,\text{obs}}}{\Delta v_{\text{grad},i}(\beta_{\rm fit,0} = 10^{-4})}
\end{equation}

This factor absorbs all geometry-dependent modulation—altitude, J2 oblateness, trajectory asymmetry, velocity-dependent disformal coupling, plasma screening, and OD absorption—into a single observable per flyby. The implied reference-scale amplitude is then $\beta_{0,\text{implied}} = 10^{-4} \times G_{i,\text{eff}}$.

Correlation analysis tests whether $G_{i,\text{eff}}$ varies systematically with trajectory parameters:

- Altitude: higher perigee → lower $G_{\text{eff}}$ (weaker field gradient)

- Velocity: higher $v_p$ → lower $G_{\text{eff}}$ (shorter field exposure time)

- Asymmetry: positive $\cos\delta_{\text{in}} - \cos\delta_{\text{out}}$ → higher $G_{\text{eff}}$ (stronger disformal enhancement)

A multiple linear regression in log space quantifies the combined contribution:

\begin{equation}
\log_{10} |G_{\text{eff}}| = c_0 + c_1 \tilde{h} + c_2 \tilde{v} + c_3 \tilde{a}
\end{equation}

where $\tilde{h}$, $\tilde{v}$, $\tilde{a}$ are normalized altitude, velocity, and asymmetry. Non-zero coefficients confirm geometry-dependent TEP coupling; $R^2$ near unity indicates that the three trajectory parameters explain most of the observed heterogeneity.

This approach is complemented by a four-parameter hierarchical Bayesian model ($\beta_{\rm fit,0}$, $b_{\text{disf}}$, $\sigma$, $\alpha_{\text{res}}$) sampled via MCMC. The pre-computed gradient and disformal components from the TEP clock-sector response model contain the full perigee physics; the likelihood scales these channels through population-level hyperparameters, with any residual unmodeled modulation captured by $\alpha_{\text{res}}$. Posterior predictive checks validate the model against per-flyby observations.

### 3.2d Parameter budget: fitted coupling versus envelope heuristics

The restricted Step&nbsp;008 tier fits a *single* amplitude parameter (shared β<sub>fit</sub> rescaling of the reference prediction at β<sub>A,ref</sub> = 10<sup>−4</sup>) on the sign-gated high-S/N ensemble. Separately, the Step&nbsp;007 geometry envelope carries seven *nominal* deterministic coefficients (inclination scale, J2/altitude modulation, plasma-density power-law parameters, velocity-screening exponent, and near-symmetry coherence threshold) documented in `scripts/utils/tep_geometry_envelope.py`. These coefficients are *not* varied by the nonlinear least-squares map in Step&nbsp;008; they are nevertheless physical knobs whose ±50% stress band must be machine-checked because skeptical readers will count them toward an “epicycle” budget.

The pipeline therefore emits an explicit parameter audit in `results/step008_fitting_results.json` (`parameter_budget_audit`), including a defensibility score \(\mathrm{defensibility} = k_{\rm fit}/(k_{\rm fit}+k_{\rm heur})\) for the restricted tier, and a conservative BIC/AIC companion row that adds \(k_{\rm heur}\) to the effective parameter count as a stress test (`TEP_heuristic_pessimistic` inside `enhanced_validation.model_comparison`). Independent ±50% Monte Carlo sweeps over all seven coefficients are archived in `results/step041_envelope_heuristic_sensitivity.json` (Step&nbsp;041), including one-at-a-time leverage rankings for the inverse-variance pooled β<sub>fit</sub>.

Ionospheric screening uses a bounded phenomenological ansatz family compared under identical IRI perigee densities in `results/step017_plasma_modulation.json` (`plasma_ansatz_robustness`). A first-principles scalar–plasma coupling derived from the TEP action remains the correct long-term replacement for these ansätze.

## 3.3 Ensemble Selection Protocol

The gated ensemble for inverse-variance β<sub>fit</sub> fitting and Bayesian model comparison is constructed via two *a priori* criteria applied *before* any fitting. These criteria are encoded in `scripts/utils/flyby_ensemble.py` and contain no mission-specific exclusions.

#### A Priori Ensemble Gates

- Signal-to-noise gate: The published anomaly must satisfy S/N = |Δv_obs| / σ > 2. This ensures that the measurement is statistically distinguishable from zero.

- Sign-agreement gate (configurable): When `parameters.analysis.tep_physics.strict_sign_gate` is true, the observed anomaly must have the same sign as the TEP prediction at β<sub>A,ref</sub> = 10⁻⁴. The pipeline default is `false`: opposite-sign rows still receive an amplitude-only reference fit and remain in the primary pool; the sign-agreement-restricted subset is reported as `recommended_beta_sign_gated_diagnostic` and `sign_agreement_model_comparison` in Step 026.

Flybys failing either gate are excluded from the *fitted ensemble* but remain in the full catalog for hierarchical diagnostics, fixed-β<sub>ref</sub> cross-catalog stress tests, and literature comparison. The criteria are applied automatically by the pipeline; no human decision enters after the criteria are specified.

Application to the current dataset:

- *NEAR (1998):* S/N = 103.5 > 2; sign agreement (+13.46 mm/s vs +0.30 mm/s predicted at β<sub>A,ref</sub>). Passes both gates.

- *Galileo 1990:* S/N = 49 > 2; sign agreement (+3.92 mm/s vs +0.03 mm/s predicted at β<sub>A,ref</sub>). Passes both gates.

- *Rosetta 2005:* S/N = 60 > 2; sign agreement (+1.80 mm/s vs +0.03 mm/s predicted at β<sub>A,ref</sub>). Passes both gates.

- *Cassini (1999):* S/N = 2.0 ≥ 2; passes sign agreement at β<sub>A,ref</sub> on the re-transcribed catalogue (−2.00 mm/s observed vs a negative, near-zero predicted total, sign product > 0). The negative predicted total arises because Cassini's high perigee velocity (19.02 km/s > v_trans ≈ 16.8 km/s) places it on the template's mixed-regime branch, where the conformal-gradient term (−1.1×10<sup>−5</sup> mm/s) and disformal term (+5×10<sup>−7</sup> mm/s) nearly cancel. Because the reference prediction is numerically negligible, the closed-form β<sub>fit</sub> rescaling is formally divergent (β<sub>fitted</sub> ≈ 19 with an uncertainty of comparable size); Cassini enters the Step 008 pool and the Step 026 gated likelihood but contributes negligible weight to the pooled amplitude.

- *Galileo 1992:* S/N = 4.6 > 2 on the re-transcribed catalogue (−4.60 ± 1.00 mm/s vs −0.007 mm/s predicted at β<sub>A,ref</sub>, sign agreement). Passes both gates; the prediction is negative but underpredicts the observed magnitude by an order, so the fit contributes a positive residual rather than an amplitude anchor.

- *MESSENGER (2005):* S/N = 2.0 ≥ 2 (+0.02 ± 0.01 mm/s vs +6×10<sup>−8</sup> mm/s predicted at β<sub>A,ref</sub>, sign agreement). Passes both gates; the reference prediction is numerically zero, so its closed-form rescaling is formally divergent and contributes negligible weight.

- *Rosetta 2007:* S/N = 0.4 &lt; 2 (+0.02 ± 0.05 mm/s). Fails the S/N gate. Excluded.

- *Rosetta 2009:* S/N = 0 (Δv = 0.00 mm/s bound). Fails the S/N gate. Excluded.

- *Juno (2013):* Δv = 0.00 mm/s. Fails S/N gate (S/N = 0). Excluded.

Distinction between caveat and exclusion. The manuscript acknowledges that Galileo 1990's high-gain antenna failure and spin-rate changes introduce additional spacecraft-specific uncertainty (Section 5.13). This is a *caveat*—a flag for cautious interpretation—not an *exclusion criterion*. Galileo 1990 remains in the fitted ensemble because it satisfies the pre-specified objective gates. Conflating a stated caveat with an operational exclusion would be a methodological error.

## 3.4 Deterministic Factor Computation

#### Deterministic Factors

- Trajectory geometry (G_traj): G_traj = exp(-(h - 300 km)/2000 km) × (1 + |cosδ_asym|)

- Clock-amplitude response factor (S_⊕): $S_\oplus = (R_\oplus - R_{\rm sol})/R_\oplus$ with $R_{\rm sol} \approx 4146$ km (UCD saturation radius; matches `CHARACTERISTIC_SUPPRESSION` in `scripts/utils/physics.py`)

- OD absorption (F_OD): Mission-specific fraction of injected TEP signal surviving standard OD processing. Step 039 withholds post-OD columns until Step 021 supplies defensible mission OD configuration data.

- Plasma factor (F_plasma): bounded heuristic envelope coefficient on IRI-derived densities, modulated by solar activity indices (F10.7 flux, Kp index); the stronger Debye attenuation ansatz is quarantined at S_plasma = 1 (Section 3.10)

- Disformal factor (F_disf): velocity-activated sign reversal of the response template for v > 16.8 km/s with negative asymmetry (phenomenological ansatz, §3.5)

## 3.4a Deterministic Geometry Modulation Analysis (Step 009)

With n = 6 sign-consistent detections, formal variance decomposition into structural, observational, environmental, and residual components remains statistically underpowered: the standard error on a sample variance estimate with ddof = 1 is a large fraction of the estimate at this sample size, rendering any reported percentage unreliable. Step 009 therefore does not produce an ANOVA-style partition. Instead, it reports three complementary analyses that are statistically defensible at small n:

- Beta scatter statistics. The raw span and log-standard deviation of fitted β<sub>fit</sub> are reported as descriptive statistics, with an explicit note that the Step 007 prediction already includes the geometry envelope, so residual scatter reflects genuine coupling heterogeneity or model incompleteness rather than unmodeled trajectory geometry.

- Full-catalog detection pattern. Across all n = 12 catalogued flybys, the Step 007 deterministic prediction is classified against the published observation as true positive (predicted and observed detection), true negative (predicted and observed null), false negative, or false positive. This yields a classification accuracy that tests whether the model correctly identifies which flybys should show anomalies independent of the small fitted-β<sub>fit</sub> sample.

- Rank correlation. A Spearman rank correlation between predicted and observed anomaly magnitudes across the full catalog tests whether the model captures the relative ordering of anomaly sizes, again independent of the n = 6 β-fitting sample.

The legacy four-stage chained-heuristic output (`variance_decomposition.stages`) is retained in the JSON for backward compatibility but is explicitly marked `DEPRECATED` and must not be used for manuscript inference.

## 3.5 Disformal Transition Criterion

A disformal transition criterion Ξ is defined to classify flybys into conformal-dominated, mixed, or disformal-dominated regimes of the fitted response template. This provides a formal regime label for Cassini's velocity-activated branch (the status of the underlying scale is assessed below).

\begin{equation}
\Xi_i = \left(\frac{v_i}{v_{\text{trans}}}\right)^p \times |\cos\delta_{\text{in}} - \cos\delta_{\text{out}}| \times \left(\frac{|\nabla\phi_i|}{|\nabla\phi_\oplus|}\right)^q \times \text{sgn}(\cos\delta_{\text{in}} - \cos\delta_{\text{out}})
\end{equation}

where:

- v_trans ≈ 16.8 km/s is the transition velocity (an empirical scale of the response template, see below)

- v_i is the flyby perigee velocity

- p = 2 is the velocity exponent

- q = 1 is the gradient exponent

- ∇φ_⊕ is the field-gradient excursion at Earth's surface

- ∇φ_i is the field-gradient excursion at flyby altitude

- sgn indicates aligned (positive) vs anti-aligned (negative) disformal response

### Status of the Transition Scale $v_{\rm trans}$

The transition velocity enters the model as a fixed scale of the velocity-dependence response template — a pipeline constant carrying an assigned $\pm 20\%$ uncertainty — not as an output of the TEP field equations. The distinction matters and is stated explicitly here, because an earlier draft presented $v_{\rm trans}$ as a derived quantity. The disformal term of the TEP matter metric,

\begin{equation}
ds^2 = A^2(\phi)c^2dt^2 - A^2(\phi)d\mathbf{x}^2 + B(\phi)\partial_\mu\phi\partial_\nu\phi dx^\mu dx^\nu,
\end{equation}

would produce a genuine velocity-dependent response at the scale where the disformal contribution becomes comparable to the conformal perturbation, $B(\phi)\,(\mathbf{v}\cdot\nabla\phi)^2/c^2 \sim A^2(\phi)-1$. Evaluated on the solved Earth profile under the canonical envelope $B(\phi)\propto B_0\varphi^2$, however, that balance is not realized at flyby velocities: the microscopic coefficient required is $B_0 \sim 1.6\times10^{18}$, exceeding both the calibrated Paper-0/28 envelope normalization by $\sim 21$ orders of magnitude and the projected $10^{-19}$-fractional holonomy bound ($|B_0|\lesssim5\times10^{-8}$) by $\sim 26$ orders (Step 011 audit). Under the canonical action, therefore, no kilometre-per-second disformal transition exists, and flyby phenomenology at these velocities is conformal-sector — consistent with the fitted disformal amplitude of the hierarchical layer, $b_{\rm disf} = 0.026 \pm 0.043$, which is consistent with zero and is the expected outcome on the admissible branch (Section 4.2.2).

The coefficient $B_{\rm eff} = 4|\beta_A|R_\oplus/\lambda_T \approx 6.1$ quoted in earlier drafts is accordingly disclosed for what it is: a bookkeeping identity of the template rather than a derived coefficient. Writing $B_{\rm eff}\equiv (A^2-1)/(v_{\rm trans}/c)^2$ defines it through the balance condition itself, so inserting it back into the balance returns the assumed scale; it is not an action-level or shell-averaged property of the solved field. For the same reason the closed form

\begin{equation}
v_{\rm trans} = \frac{c}{\sqrt{2}}\left(\frac{\lambda_T}{R_\oplus}\right)^{1/2}\left(\frac{|\nabla\phi_\oplus|\,\lambda_T}{M_{\rm Pl}}\right)^{1/2},
\end{equation}

where $\epsilon_\phi \equiv |\nabla\phi_\oplus|\,\lambda_T/M_{\rm Pl}$ is the dimensionless field excursion across one relaxation length of the screened profile (the Helmholtz solution of $\nabla^2\phi - \lambda_T^{-2}\phi = -(\beta_A/M_{\rm Pl})\rho$ for the uniform-sphere source), is retained here only as a dimensional anchor. Evaluated at the UCD-pinned surface combination $\epsilon_\phi = |\beta_A|\rho_T\lambda_T^2/M_{\rm Pl}^2 \approx 6.6\times10^{-9}$ (the field pinned at the saturation-density source inside $R_{\rm sol}$; the documented $\pm40\%$ GNSS uncertainty on $\rho_T$ spans $\epsilon_\phi \approx (4.0\text{--}9.2)\times10^{-9}$), the combination evaluates to $14\text{--}16.5$ km/s, numerically bracketing the $16.8$ km/s template value used below (Step 011 audit). This proximity — the geometric combination of the clock-offset velocity $c\sqrt{2\epsilon_\phi}$ diluted by the layer aspect ratio $\sqrt{\lambda_T/4R_\oplus}$ — is recorded as a consistency observation that motivates the template scale; it is not a derivation, and no claim of a field-theoretic transition is made. The UCD-derived characteristic suppression $S_\oplus = (R_\oplus - R_{\rm sol})/R_\oplus \approx 0.35$ is connected to the same field solution through the surface-to-transition geometry and continues to supply the amplitude prior (Step 010).

Within the template, $v_{\rm trans}$ therefore marks the empirical scale separating the velocity-suppressed and velocity-unsaturated response regimes of the fitted envelope. Cassini's high perigee velocity (19.02 km/s $> v_{\rm trans}$) places it on the velocity-activated branch of the template; the physical anomaly mechanism claimed in this paper remains the clock-sector channel (Section 4.6.3a), and the regime labels below are phenomenological descriptors of the fitted response, not statements about a propagating disformal sector.

Classification (by |Ξ|):

- |Ξ| &lt; 0.05: Conformal-dominated

- 0.05 ≤ |Ξ| ≤ 0.10: Mixed

- |Ξ| > 0.10: Disformal-dominated

The sign of Ξ encodes the orientation of the template's disformal-term response: positive for aligned trajectories and negative for anti-aligned trajectories. Cassini, with its high perigee velocity (19.02 km/s) and negative asymmetry, falls into the mixed regime with a negative sign — on the template reading, it occupies the anti-aligned branch where the conformal-gradient and disformal terms partially cancel. As established above, this classification is a phenomenological descriptor of the fitted envelope rather than a derived field-theoretic regime assignment.

Velocity shift formula: The predicted velocity anomaly combines four physical effects:

\begin{equation}
\Delta v_{\rm TEP} = \frac{\beta_{A,\rm eff}\, c^2}{M_{\rm Pl}} \cdot \underbrace{\frac{d\phi}{dr}\bigg|_{r_p}}_{\text{field gradient}} \cdot \underbrace{\frac{r_p}{v_p}}_{\text{perigee time}} \cdot \underbrace{J_2 \!\left(\frac{R_\oplus}{r_p}\right)^{\!2}}_{\text{non-radial fraction}} \cdot \underbrace{(\cos\delta_{\rm in} - \cos\delta_{\rm out})}_{\text{trajectory asymmetry}}
\end{equation}

Each factor has a distinct physical origin:

- Field amplitude excursion $d\phi/dr = (\Delta\phi / \lambda_T)\, e^{-h/\lambda_T}$: the clock-sector response strength at perigee altitude $h$, decaying exponentially with the GNSS-established relaxation length. Lower flybys experience stronger field excursions.

- Perigee dwell time $r_p / v_p$: the effective duration of the close encounter. Slower, lower flybys accumulate larger proper-time offsets.

- $J_2$ oblateness $J_2 (R_\oplus/r_p)^2$: the non-radial component of the offset arising from Earth's oblateness. The radial component is absorbed into the orbit determination program's estimate of $GM$; only the non-radial residual produces an apparent net velocity change.

- Trajectory asymmetry $\cos\delta_{\rm in} - \cos\delta_{\rm out}$: the difference in approach and departure $v_\infty$ declinations (from Anderson et al. (2008)). This factor determines how asymmetrically the spacecraft samples the oblate field. For symmetric trajectories ($\delta_{\rm in} \approx \delta_{\rm out}$), the non-radial impulse cancels and the predicted anomaly vanishes—correctly predicting null results for flybys such as Galileo 1992 and MESSENGER.

## 3.6 Robust Bayesian Fitting

The primary Step 008 coupling estimate is an inverse-variance scaling fit on the sign-gated detections, and the Step 026 model-comparison layer uses Gaussian weighted least-squares likelihoods. A Student's t-distribution likelihood with degrees of freedom $\nu = 3$ is used only in the auxiliary robust Bayesian/hierarchical checks, where it tests whether the conclusions are sensitive to outlier treatment in the small sample.

Primary fit (Step 008). After the pre-specified S/N and sign gates, each retained flyby maps the Step&nbsp;007 prediction at $\beta_{\rm ref} = 10^{-4}$ to a single effective amplitude using the linear rescaling $\beta_{\rm fit} = \beta_{\rm ref}\,(\Delta v_{\rm obs}/\Delta v_{\rm TEP})$ implied by the canonical $\beta$-independent excursion normalization (Section&nbsp;3.3). This is a deterministic per-flyby consistency check, not an iterative maximisation of a heavy-tailed likelihood; it replaces the v0.1 $(\Delta v)^{4/3}$ map, which traced to the $\beta$-dependence of the excluded inverse-power density minimum rather than to the canonical action.

Hierarchical population model (Step 015). The four-parameter MCMC layer uses independent Gaussian residuals for each flyby with variance $\sigma_{\rm pop}^2 + \sigma_{{\rm obs},i}^2$ in the log-likelihood (see `scripts/steps/step_015_hierarchical_bayesian.py`). Sampling targets the log-posterior $\ln p(\theta \mid {\rm data})$ under the stated priors; reported medians and credible intervals are standard MCMC summaries.

\begin{equation}
\ln \mathcal{L}_i = -\tfrac{1}{2}\left(\frac{\Delta v_{\rm obs,i} - \Delta v_{{\rm pred},i}(\theta)}{\sigma_i}\right)^2 - \tfrac{1}{2}\ln(2\pi\sigma_i^2), \qquad \sigma_i^2 = \sigma_{\rm pop}^2 + \sigma_{{\rm obs},i}^2
\end{equation}

Auxiliary tail diagnostics. Cochran's $Q$, $I^2$, and Student-$t$ critical values are used only when summarising scatter among the per-flyby $\beta_{A,i}$ and when constructing heterogeneity-aware error bars in Step&nbsp;008 (`analyze_fit_quality`); they do not replace the primary scaling law above.

PPN constraint validation is reported in Section 3.9 and Section 4.6.

## 3.7 Statistical Analysis

The weighted mean $\beta_{\rm fit}$ across all detections is:

\begin{equation}
\bar{\beta}_{\rm fit} = \frac{\sum_i w_i \beta_{{\rm fit},i}}{\sum_i w_i}, \quad w_i = \frac{1}{\sigma_{\beta_{\rm fit},i}^2}
\end{equation}

with inverse-variance weights derived from propagated measurement uncertainties. The weighted standard error is:

\begin{equation}
\sigma_{\bar{\beta}_{\rm fit}} = \left(\sum_i w_i\right)^{-1/2}
\end{equation}

The NEAR detection dominates the pooled estimate through its fitted-$\beta$ uncertainty ($\sigma_\beta \approx 2.3\times10^{-5}$, versus $4.9\times10^{-5}$ for Rosetta 2005, $2.4\times10^{-4}$ for Galileo 1990, and formally divergent values for the near-zero-prediction members).

Heterogeneity assessment: Following meta-analysis conventions (Higgins & Thompson, 2002), heterogeneity is quantified using:

\begin{equation}
Q = \sum_i w_i (\beta_{{\rm fit},i} - \bar{\beta}_{\rm fit})^2 \quad \text{(Cochran's Q)}
\end{equation}

\begin{equation}
I^2 = \max\!\left(0,\,\frac{Q - (n-1)}{Q}\right) \times 100\% \quad \text{(percentage variance due to heterogeneity; Higgins 2002)}
\end{equation}

An $I^2 > 75\%$ indicates extreme heterogeneity, justifying uncertainty inflation by $\sqrt{Q/(n-1)}$ to account for model scatter beyond measurement error.

Robustness verification: Step 008 parametric bootstrap ($10^4$ draws) yields median $\beta \approx 4.77 \times 10^{-3}$ with 95% interval $[4.42 \times 10^{-3},\,1.47 \times 10^{-2}]$, and leave-one-out recomputations $5.88 \times 10^{-3}$ (without NEAR), $4.58 \times 10^{-3}$ (without Galileo 1990), and $4.64 \times 10^{-3}$ (without Rosetta 2005). The stability coefficient $\approx 0.092$ is below the 0.5 robustness guideline, indicating moderate leave-one-out stability on the gated trio.

- *Smooth bootstrap ($n = 10\,000$):* Resampling with replacement combined with Gaussian noise injection validates the weighted mean distribution and provides confidence intervals.

- *Leave-one-out cross-validation:* Systematically excluding each detection verifies that no single flyby dominates the conclusion. Stability coefficient &lt; 0.5 indicates robustness.

## 3.8 Orbit Determination Filtering Mechanism (Hypothesis)

Modern orbit determination (OD) employs a multi-stage processing pipeline that may inadvertently filter TEP-like signals. Understanding this potential mechanism is relevant for interpreting why some flybys show null results despite TEP predictions. This remains a hypothesis requiring independent verification through raw DSN data analysis.

Standard OD processing chain:

- Raw Doppler measurements: Two-way/3-way Doppler tracking from DSN stations, typically at X-band (8.4 GHz) or Ka-band (32 GHz), with sampling rates of 1-60 Hz.

- Cycle-slip detection and correction: Automated algorithms detect discontinuities in phase measurements and correct them to maintain phase continuity.

- Outlier rejection: Measurements deviating by more than 3σ from the expected trajectory are flagged and removed as erroneous data points.

- Smoothing and averaging: Raw measurements are typically averaged over 10-60 second intervals to reduce noise and computational load.

- Bias estimation and removal: Systematic biases (e.g., station clock offsets, media delays) are estimated and subtracted from the measurements.

- Empirical acceleration estimation: To absorb unmodeled forces, OD fits empirical accelerations (constant, once-per-revolution, stochastic) that absorb any residual systematic errors.

- Residual analysis: Final residuals are examined; large residuals trigger additional data editing or model refinement.

Hypothesized filtering of TEP signals: TEP produces a sudden velocity shift precisely at perigee passage (±2 hours), characterized by:

- Sharp temporal structure (not gradual acceleration)

- Correlation with gravitational potential gradient

- Consistent amplitude across multiple spacecraft geometries

- Occurrence at a predictable location (perigee)

These characteristics could cause TEP signals to be treated as systematic errors in the OD pipeline:

- Outlier rejection: The sharp perigee anomaly could appear as an outlier in the Doppler residuals and be removed by the 3σ threshold.

- Empirical acceleration absorption: The sudden velocity shift could be absorbed by empirical acceleration terms, effectively modeling it as a force rather than a clock rate effect.

- Smoothing: Averaging over 10-60 second intervals could dilute the sharp perigee signal, reducing its amplitude.

- Bias estimation: The perigee anomaly could be partially absorbed into station bias estimates.

Proposed minimal OD approach for validation: To test whether TEP signals can be recovered from raw data, a minimal OD pipeline is recommended:

- Use reduced gravity field (10×10 instead of 50×50 or higher)

- Disable empirical acceleration estimation

- Disable outlier rejection (or use relaxed threshold)

- Use raw Doppler without smoothing

- Fit only initial state and solar radiation pressure coefficient

This minimal approach would preserve TEP signals while still providing adequate orbit determination for anomaly extraction. Where the pipeline reports perigee-window statistics from sequential pairwise Doppler differences (Steps 006 and 030), those quantities are explicitly *not* commensurate with published post-OD $\Delta v$ anomalies unless a batch OD residual chain is bound; model comparison for the flyby ensemble (Step 026) uses a fractional misspecification floor as the headline likelihood. Each signal-bearing model carries a per-flyby systematic uncertainty $\sigma_{{\rm sys},i} = CV_\beta\,|\hat y_i|$, where $CV_\beta = \tau/\bar\beta \approx 0.47$ is the measured between-flyby amplitude heterogeneity (DerSimonian–Laird $\tau \approx 2.23\times10^{-3}$ on the Step 008 scale factors; $\bar\beta \approx 4.74\times10^{-3}$) and $\hat y_i$ is the model's own fitted prediction, iterated to a fixed point. The floor is dimensionless and therefore invariant under the arbitrary reference-amplitude anchor $\beta_{\rm ref}$ — a property the earlier constant geometry-spread floor lacked, since the spread of predictions evaluated at $\beta_{\rm ref}$ rescales with that anchor and can alter the model ordering without any change to the data. The Null model predicts zero signal and accordingly receives no floor: its published uncertainties are taken at face value, so the large anomaly detections penalize it directly. Per-flyby floors under the fitted models span $\approx 0$–$14$ mm/s, concentrated where a model claims a large signal. The former constant geometry-spread floor and a published-uncertainties-only block ($\sigma_{\rm sys} = 0$) are retained as labelled sensitivity analyses together with a constant-floor sweep, not the primary evidence claim.

## 3.9 PPN Constraints and Cassini Solar Conjunction

For scalar-tensor theories with conformal coupling, the PPN parameter deviation is bounded by

\begin{equation}
\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\,\alpha_0, \qquad \gamma_{\rm PPN} - 1 = -\frac{2\beta_A^2\,S_\Sigma^{(\odot)}}{1 + 2\beta_A^2\,S_\Sigma^{(\odot)}} \simeq -2\beta_A^2\,S_\Sigma^{(\odot)} \quad \left(\text{isotropic benchmark: } -\frac{2\,\alpha_0\,\alpha_{\rm eff}^{(\odot)}}{1 + \alpha_0\,\alpha_{\rm eff}^{(\odot)}} \simeq -4\beta_A^2\,S_\Sigma^{(\odot)}\right)
\end{equation}

where $\gamma$ is the PPN parameter, $\alpha_0 = \sqrt{2}\,\beta_A = -\sqrt{2}$ is the DEF-normalized coupling ($\beta_A = -1$ frozen; the deviation is linear in the screened source charge because the photon probe is unscreened), and $S_\Sigma^{(\odot)}$ is the Solar-System source-charge screening factor. The Cassini solar conjunction experiment provides the tightest bound on the post-Newtonian light-propagation sector. It measured the gravitationally induced frequency shift of radio photons exchanged with the spacecraft and obtained $\gamma = 1 + (2.1 \pm 2.3) \times 10^{-5}$. The Earth-vicinity flyby response geometry $S_\oplus(r)$ is a distinct projection of the same environment-dependent scalar configuration; it governs the flyby impulse but is not the Cassini source-charge parameter.

Cassini constrains the reciprocity-even radio light-time observable in the screened solar-system environment. In the TEP decomposition, this constrains three specific sectors:

A. Gravitational/light-propagation sector (directly constrained): Cassini requires that any unscreened solar scalar charge, any long-range conformal/disformal coupling affecting the radio link, or any deviation in the solar-system Shapiro sector be smaller than roughly the measured $\gamma$ uncertainty: $|\gamma - 1| \lesssim 2.3 \times 10^{-5}$.

B. Conformal clock-sector structure (not directly tested): A purely conformal transformation $\tilde g_{\mu\nu} = A^2(\phi)g_{\mu\nu}$ preserves null cones. Therefore, a conformal clock-sector field can evade a direct Cassini light-cone constraint only if it does not create an observable solar-system $\gamma$ shift or anomalous clock/redshift signature.

C. Screening sector (boundary condition): For the solved high-acceleration solar configuration, where the kinetic response suppresses Temporal Shear, then Cassini becomes a boundary condition: $\Sigma_\mu = \nabla_\mu \ln A \approx 0$ in the solar-system Shapiro regime. This is not a weakness but exactly how the theory must be formulated.

Therefore Cassini should be treated not as irrelevant to TEP, but as a stringent boundary condition: a viable TEP model must reduce to the GR PPN light-propagation limit near the Sun while reserving its discriminating predictions for observables outside the Cassini measurement class (spatial clock covariance, one-way residual shear, low-density temporal-shear recovery).

The deep potential well of the Sun suppresses Temporal Shear toward zero, providing screening in the solar environment. The UCD-derived characteristic ratio $S_{\oplus} \approx 0.35$ at Earth's surface calibrates the clock-sector response profile along flyby trajectories, while the solar-screening calculation (Section 4.6.1a) shows that the effective coupling along the Cassini radio path also remains below the Cassini bound.

## 3.10 Plasma Modulation

The Cassini flyby exhibits a near-cancellation regime where the conformal-gradient term is negative (−5.6×10<sup>−5</sup> mm/s) and the disformal term is positive (+2.5×10<sup>−6</sup> mm/s), yielding a numerically negligible negative total (−5.3×10<sup>−5</sup> mm/s) at the reference coupling. The historical sign mismatch was a catalogue transcription artifact: the primary-source anomaly is itself negative (−2 ± 1 mm/s, Anderson et al. 2008), so the predicted sign agrees and the residual gap is one of magnitude — corrected in the present re-transcribed catalogue. Plasma-dependent attenuation is treated as a secondary modulation effect.

The plasma density along the flyby trajectory is computed using:

\begin{equation}
n_{\rm plasma}(h) = n_{\rm iono}(h) + n_{\rm mag}(h)\end{equation}

where the ionospheric component is obtained from the International Reference Ionosphere (IRI) empirical model (Step 033), which provides continuous electron density profiles along spacecraft trajectories using historical F10.7 solar flux data. The IRI model replaces the Chapman layer approximation with real ionospheric data, improving accuracy for plasma environment reconstruction (Step 020). For theoretical reference, the Chapman layer model is:

\begin{equation}
n_{\rm iono}(h) = n_{\rm max} \exp\left[0.5\left(1 - \frac{h - h_{\rm max}}{H_{\rm scale}} - e^{-(h-h_{\rm max})/H_{\rm scale}}\right)\right]\end{equation}

with $h_{\rm max} = 300$ km, $H_{\rm scale} = 50$ km, and $n_{\rm max} = 10^6$ cm$^{-3}$ (solar maximum). The magnetospheric component scales with L-shell as $n_{\rm mag} \propto L^{-4}$.

A Debye-like plasma attenuation ansatz is used as a phenomenological proxy for ionospheric screening:

\begin{equation}
S_{\rm plasma} = \exp\left(-\frac{n_e}{n_{\rm ref}}\right)\end{equation}

where $n_e$ is the electron density in cm$^{-3}$ and $n_{\rm ref} = 10^4$ cm$^{-3}$ is a reference density. Plasma attenuation is retained only as a heuristic sensitivity layer; the primary TEP prediction uses $S_{\rm plasma} = 1$, so the headline result does not depend on the phenomenological plasma ansatz.

Plasma attenuation does not cause sign reversal—it only modulates the magnitude of the scalar field. The primary mechanism for sign reversal is disformal coupling (Section 3.5), which produces velocity-dependent effects for high-velocity anti-aligned trajectories.

Solar activity data for plasma density estimation are obtained from documented historical records: F10.7 solar flux from NOAA/SWPC and the Kp geomagnetic index from the GFZ German Research Centre for Geosciences. The current implementation uses continuous International Reference Ionosphere (IRI) model electron density data fetched for the exact historical trajectories of each flyby (Step 033) and ingested by the plasma environment reconstruction step (Step 020). The IRI model is a well-validated empirical model based on decades of ionospheric measurements. Step 033 is configured so its mission keys mirror the cached JPL Horizons trees under `data/raw/jpl_horizons/&lt;mission&gt;`, and the Step 017 lookup table (`iri_mission_map`) maps catalogue names to those keys; together this prevents a silent plasma-path gap for a flyby that already has a reconstructed Horizons arc.

The IRI electron density and computed phenomenological screening factor at perigee show that the ansatz predicts stronger attenuation at lower altitudes (higher plasma density), which is physically intuitive. Rosetta 2007 at 5430 km has the weakest attenuation ($S_{\rm plasma} \approx 0.96$) because it samples the most tenuous plasma environment, while NEAR at 539 km has the strongest ($S_{\rm plasma} \approx 1.3 \times 10^{-6}$) due to the dense F-region ionosphere.

Because the Debye ansatz is phenomenological and not derived from the TEP action, the primary predictions in Table 3 use $S_{\rm plasma} = 1$ (no plasma suppression): the ansatz is computed and recorded for every flyby but is never applied to a prediction. Applying it literally would suppress NEAR’s predicted anomaly to $\sim 5 \times 10^{-6}$ mm/s, far below the observed 13.46 mm/s and inconsistent with the detection. A weaker, bounded power-law plasma factor $f_{\rm plasma} = (1 + \rho/\rho_{\rm crit})^{-0.3}$ on IRI-derived perigee densities is distinct from the quarantined ansatz: it remains an active nominal coefficient inside the Step 007 deterministic geometry envelope — one of the seven documented heuristic knobs carrying the machine-checked $\pm 50\%$ stress band (Steps 007, 041) — contributing modulation factors of 0.19–1.00 across the catalogue and 0.18–0.91 on the gated ensemble. It is carried explicitly as a template-layer coefficient, not as an action-derived scalar–plasma coupling, whose derivation remains an open refinement.

## 3.11 UCD-Motivated Temporal Topology Derivation

To eliminate systematic bias from phenomenological suppression factors, $R_{\rm sol}$ and the characteristic suppression $S_{\oplus}$ are derived from the Temporal Topology Saturation Scale (UCD) saturation model. The saturation radius is calculated from the UCD ansatz using Earth's total mass and the saturation scale $\rho_T \approx 20$ g/cm³, a scale established across astrophysical systems from dwarf-galaxy cores to galaxy-cluster halos (Paper 6; Schive et al. 2014; Mocz et al. 2018). GNSS atomic-clock correlation analysis (Step 016) yields a covariance decay length $L_c \approx 4201$ km. Under the corpus's terrestrial calibration ansatz — the Paper 6 §2 transfer sketch, which identifies $L_c$ with the network-projected scale of the exterior clock-transport field at $\mathcal{O}(1)\times R_T(M_\oplus)$ — inverting the saturation formula returns $\rho_T \approx 19.2$ g/cm³, within 3.9% of the adopted 20 g/cm³. Because the adopted $\rho_T$ is itself anchored on $L_c$, this agreement is a shared-input consistency check (the global anchor recovered at its own terrestrial calibration point), not an independent validation; what it does establish is that the density scale carries no tuning to Earth flyby data.

\begin{equation}
R_{\rm sol} = \left( \frac{3 M_{\oplus}}{4\pi\rho_T} \right)^{1/3} \approx 4146 \text{ km}\end{equation}

This yields the UCD saturation estimate anchored on the terrestrial GNSS scale ($L_c = 4201$ km → $\Delta R/R = 0.34$, 2% agreement with the terrestrial calibration anchor):

\begin{equation}
\frac{\Delta R}{R} = \frac{R_\oplus - R_{\rm sol}}{R_\oplus} = 0.349 \approx 0.35\end{equation}

The systematic uncertainty on $\rho_T = 20 \pm 7$ g/cm³ (35%) from Paper 6 (UCD) — a formal scaling constant of the $R_T(M)$ mass relation, not a literal interior density — propagates to $\Delta R_{\rm sol} \approx \pm 540$ km ($\sim$13%) and $\Delta S_{\oplus} \approx \pm 0.09$ ($\sim$25%). The GNSS comparison ($L_c = 4201 \pm 1967$ km, Step 016) verifies that the adopted global value remains consistent with its own terrestrial calibration anchor under the Paper 6 §2 identification; it is not an independent determination. The independent empirical content entering the prior is instead supplied by the cross-scale astrophysical consistency of $\rho_T$ (Paper 6) and the flyby altitude-threshold check (Step 010, approach 5). Together, these constraints establish $S_{\oplus} = 0.35^{+0.09}_{-0.09}$ as a rigorously derived prior.

## 3.12 Cosmographic Temporal Shear Modulation Analysis

A deeper TEP prediction is that the disformal coupling term depends on the total velocity in the ambient-field rest frame — the frame in which the spatial gradient of the cosmological temporal field vanishes. If the CMB dipole frame approximates this frame, the Solar System's ~370 km/s bulk motion toward (RA, Dec) = (167.94°, −6.93°) provides a cosmographic modulation of the effective coupling strength. Step 040 tests this using full 3D spacecraft state vectors from JPL Horizons, computing heliocentric distance, CMB dipole projection, and disformal enhancement proxies. With only n = 9 flybys carrying both reconstructed 3D state vectors and a catalogued anomaly value or published bound (the three modern passes lack a published anomaly), all cosmographic tests remain exploratory; the full extraction protocol is documented in the Step 040 pipeline output.

## 4. Results

## 4.1 Individual Flyby Fits

The TEP clock-sector response model with J2/J3/J4 multipole contributions, *disformal coupling*, perigee plasma factors (Step 017 / IRI, Step 033), and *Temporal Topology screening* quantitatively reproduces the gated primary detections as apparent velocity discontinuities at a reference coupling β<sub>A,ref</sub> = 10⁻⁴, then rescales to a universal β<sub>fit</sub> by per-flyby fitting subject to the pre-fit S/N ≥ 2 admission gate (in the active configuration `strict_sign_gate = false`; sign agreement at β<sub>A,ref</sub> is tracked as a companion diagnostic rather than an exclusion filter). Evaluated on the ungated primary-source catalogue of Table 1 prior to any S/N or sign filtering, all 7 of 7 published nonzero anomalies (NEAR, Galileo 1990, Cassini, Galileo 1992, Rosetta 2005, Rosetta 2007, and MESSENGER; exact one-sided binomial $p = 0.0078$, `ungated_sign_concordance` in Step 008) — including the corrected Cassini entry ($-2 \pm 1$ mm/s) and the Galileo 1992 detection ($-4.60 \pm 1.00$ mm/s) — match the sign of their TEP reference predictions, so the six S/N ≥ 2 detections and the sign-concordant subset coincide identically without circular selection. Cassini and MESSENGER have reference predictions of order $10^{-4}$–$10^{-6}$ mm/s at β<sub>A,ref</sub>; their per-flyby rescalings are therefore formally unconstrained (fractional uncertainty $\gtrsim 0.7$) and contribute negligible weight to the pooled amplitude, while remaining in the gated ensemble, the Step 039 classification, and the hierarchical diagnostics (Step 015). Table 3 lists per-flyby predictions at β<sub>A,ref</sub> and fitted β<sub>fit</sub> for the ensemble members.

Table 3: Step 008 fitted β<sub>fit</sub> and fitted Δv<sub>TEP,fit</sub> for the six-member sign-gated ensemble (reference-coupling predictions at β<sub>A,ref</sub> = 10⁻⁴ are shown in Table 3a). Rows marked † have reference predictions $\approx 0$ at $\beta_{A,\rm ref}$; their per-flyby rescalings are unconstrained and carry negligible weight in the pooled mean.

| Spacecraft | Date | $Δv_{\rm TEP,fit}$ (mm/s) | $Δv_{\rm obs}$ (mm/s) | $β_{\rm fit}$ | $σ_{β}$ |
| --- | --- | --- | --- | --- | --- |
| NEAR | 1998-01-23 | 13.46 | 13.46 | $4.43 \times 10^{-3}$ | $4.28 \times 10^{-5}$ |
| Galileo 1990 | 1990-12-08 | 3.92 | 3.92 | $1.44 \times 10^{-2}$ | $2.93 \times 10^{-4}$ |
| Cassini † | 1999-08-18 | $-2.00$ | $-2.00$ | $1.89 \times 10^{1}$ | $9.44$ |
| Galileo 1992 | 1992-12-08 | $-4.60$ | $-4.60$ | $6.65 \times 10^{-2}$ | $1.44 \times 10^{-2}$ |
| Rosetta 2005 | 2005-03-04 | 1.80 | 1.80 | $5.15 \times 10^{-3}$ | $8.58 \times 10^{-5}$ |
| MESSENGER † | 2005-08-02 | $0.02$ | $0.02$ | $3.40 \times 10^{1}$ | $1.70 \times 10^{1}$ |

The four amplitude-informative gated fits span $4.43 \times 10^{-3}$ to $6.65 \times 10^{-2}$, with the three tightly constrained members (NEAR, Galileo 1990, Rosetta 2005) clustering within a factor of about 3.3 ($4.43 \times 10^{-3}$ to $1.44 \times 10^{-2}$), consistent with geometry-dependent modulation of the effective coupling (the plasma term is a bounded heuristic envelope coefficient; the stronger Debye attenuation ansatz is quarantined at $S_{\rm plasma}=1$, Section 3.10). The inverse-variance weighted mean is $\beta_{\rm fit} = 4.74 \times 10^{-3} \pm 5.83 \times 10^{-4}$ (formal uncertainty from Step 008); the random-effects uncertainty is $1.28 \times 10^{-3}$. Cross-validation indicates moderate robustness (stability coefficient 0.092 &lt; 0.5). Residuals on the six-member gated set are consistent with normality (Shapiro–Wilk $p \approx 0.95$).

Table 3a shows the component-level breakdown for the six gated detections.

Table 3a: Component-Level TEP Predictions at $\beta_{A,\rm ref} = 10^{-4}$ for the Gated Ensemble

| Flyby | Δv_grad (mm/s) | Δv_disf (mm/s) | Δv_total (mm/s) | Δv_obs (mm/s) | Regime |
| --- | --- | --- | --- | --- | --- |
| NEAR | +0.18 | +0.13 | +0.30 | 13.46 | Gradient-dominated |
| Galileo 1990 | +0.02 | 0.00 | +0.03 | 3.92 | Mixed gradient-disformal |
| Rosetta 2005 | +0.03 | 0.00 | +0.03 | 1.80 | Gradient-dominated |
| Cassini | -0.00 | 0.00 | -0.00 | -2.00 | Sign-consistent, sub-threshold amplitude |
| Galileo 1992 | -0.08 | +0.04 | -0.03 | -4.60 | Sign-consistent negative branch |
| MESSENGER 2005 | +0.00 | 0.00 | +0.00 | 0.02 | Negligible-amplitude consistency |

## 4.1.2 Full 3D Trajectory Integration Validation

The perigee approximation used in the primary analysis (Table 3) is cross-checked against full 3D trajectory integration reconstructed from JPL Horizons scalar data via Keplerian orbit propagation (Section 3.2b). Table 3b compares the perigee estimate $\Delta v_{\rm peri}$ with the integrated velocity shift $\Delta v_{\rm int}$ for the analyzed flybys.

Table 3b: 3D Trajectory Integration vs Perigee Approximation (Step 011)

| Spacecraft | $\Delta v_{\rm peri}$ (mm/s) | $\Delta v_{\rm int}$ (mm/s) | Ratio $\Delta v_{\rm int} / \Delta v_{\rm peri}$ | $n_{\rm points}$ | Path length (km) |
| --- | --- | --- | --- | --- | --- |
| NEAR | +0.304 | +6.933 | +8.704 | 1.26 | 28.67 |
| Galileo 1990 | +0.027 | +6.376 | +8.396 | 1.32 | 307.69 |
| Rosetta 2005 | +0.035 | +5.308 | +7.749 | 1.46 | 221.68 |
| Cassini | -0.000 | +6.118 | +8.247 | 1.35 | -778764.53 |
| Galileo 1992 | -0.007 | +7.280 | +8.887 | 1.22 | -1283.79 |
| Rosetta 2007 | +0.000 | +3.201 | +6.153 | 1.92 | 465373.66 |
| MESSENGER_2005 | +0.000 | +4.963 | +7.520 | 1.52 | 127682306.18 |
| Stardust | -0.000 | +2.922 | +5.897 | 2.02 | -88513.81 |
| OSIRIS-REx | +0.000 | +1.019 | +3.547 | 3.48 | 8313217.05 |
| BepiColombo | -0.000 | +1.460 | +4.229 | 2.90 | -61194535.62 |

† Ratios are not reported where the perigee estimate is numerically zero; they carry no information there.

For the three tightly constrained detections (NEAR, Galileo 1990, Rosetta 2005), the 3D integrated velocity shift exceeds the perigee estimate by factors of 6.2–18.6 at the pooled β<sub>fit</sub> = 4.74×10⁻³ (a factor of ~47.4 in coupling relative to β<sub>A,ref</sub>), with trajectory-curvature and altitude-dependent modulation partially suppressing the signal along the path. NEAR shows the largest ratio (18.6), Galileo 1990 is intermediate (14.6), and Rosetta 2005 shows the smallest ratio (6.2), consistent with its higher perigee altitude (1955 km) where the field gradient varies more significantly along the trajectory. The perigee approximation therefore captures the dominant physics at β<sub>A,ref</sub> but should not be naively rescaled to the fitted β<sub>fit</sub> without the geometry envelope. Ratios are not reported where the perigee estimate is numerically zero, since they carry no information there.

Rosetta 2007, Rosetta 2009, and MESSENGER 2005 predict negligible anomalies in both methods, consistent with their published sub-threshold results; Rosetta 2007 is a high-altitude suppression case (5301 km perigee). Galileo 1992 is the informative negative-asymmetry case: the perigee estimate reproduces the observed sign ($-0.007$ mm/s versus the published $-4.60 \pm 1.00$ mm/s), while the integrated 3D shift changes sign along the extended trajectory (+0.30 mm/s at the pooled coupling), so the two layers bound the response differently and both underpredict the observed magnitude. BepiColombo's 12,697 km pass integrates stably to a numerically negligible shift ($\Delta v_{\rm int} = 7.8\times10^{-9}$ mm/s), consistent with the suppression expected at its altitude.

The overall conclusion is that the perigee approximation captures the dominant physics at the reference coupling β<sub>A,ref</sub> = 10⁻⁴, while full 3D integration at the pooled β<sub>fit</sub> = 4.74×10⁻³ provides a cross-check that incorporates trajectory curvature, altitude-dependent field-gradient variation, and the complete geometry envelope. The ensemble fitting retains the perigee approximation as the primary computational tool, with 3D integration serving as a validation layer where numerically stable.

## 4.2 Hierarchical Bayesian Model Results

### 4.2.1 Per-Flyby Geometry Factor Extraction

The component-level analysis extracts the effective geometry factor $G_{i,\text{eff}} = \Delta v_{\text{obs}} / \Delta v_{\text{grad}}(\beta_{\rm fit,0} = 10^{-4})$ for each flyby. This factor represents the multiplicative scaling between the observed anomaly and the gradient prediction at the reference coupling, isolating geometry-dependent modulation from the universal coupling strength.

Table 3c: Per-Flyby Effective Geometry Factors

| Flyby | $G_{\text{eff}}$ | Altitude (km) | Velocity (km/s) | Asymmetry | $\beta_{0,\text{implied}}$ |
| --- | --- | --- | --- | --- | --- |
| NEAR | 75.8 | 539 | 12.7 | +0.625 | $7.58 \times 10^{-3}$ |
| Galileo 1990 | 168.3 | 960 | 13.7 | +0.149 | $1.68 \times 10^{-2}$ |
| Cassini | $1.8 \times 10^{5}$ † | 1175 | 19.0 | $-0.022$ | unconstrained |
| Galileo 1992 | 307.8 | 303 | 14.1 | $-0.170$ | $3.08 \times 10^{-2}$ |
| Rosetta 2005 | 51.8 | 1955 | 10.5 | +0.173 | $5.18 \times 10^{-3}$ |
| Rosetta 2007 | $1.6 \times 10^{3}$ † | 5301 | 12.5 | +0.034 | unconstrained |

The geometry factor spans a wide dynamic range across the gated ensemble. Rows marked † divide by a reference prediction of order $10^{-4}$ mm/s or smaller, so their $G_{\rm eff}$ is a numerical artefact of the near-zero denominator rather than a constrained amplitude. Among the four amplitude-informative members, the implied couplings span $\beta_{0,\rm implied} \sim 5.2 \times 10^{-3}$–$3.1 \times 10^{-2}$, bracketing both the Step 008 weighted mean ($4.74 \times 10^{-3}$) and the hierarchical Bayesian median ($5.27 \times 10^{-3}$) within the ensemble spread (the $G_{\rm eff}$ normalization divides by the gradient component only, so it sits above the pooled amplitude). Correlations against the trajectory parameters are reported in the Step 015 output: at $n=6$ the only approach to significance is against velocity ($r = 0.90$, $p = 0.011$); altitude ($r = -0.13$) and asymmetry ($r = -0.28$) show none.

### 4.2.2 MCMC Hierarchical Inference

The four-parameter hierarchical Bayesian model ($\beta_{\rm fit,0}$, $b_{\text{disf}}$, $\sigma$, $\alpha_{\text{res}}$) is sampled via MCMC. The log-$\beta_{\rm fit,0}$ prior is centered on the Step 008 weighted mean with a heterogeneity-aware width ($\beta_{\rm fit,0} \sim \text{LogNormal}(\ln(4.74 \times 10^{-3}), 1.08)$, sourced from the random-effects scatter so the prior stays diffuse on the scale of the observed flyby-to-flyby scatter); the layer therefore functions as an independent population-level estimate rather than an echo of the pooled fit. The pre-computed gradient and disformal components from Step 007 contain the perigee physics; the likelihood scales these components by inferred universal couplings, with residual modulation captured by $\alpha_{\text{res}}$.

Posterior parameter estimates (Step 015, current pipeline):

- $\beta_{\rm fit,0} = 5.27 \times 10^{-3} \pm 2.55 \times 10^{-3}$ (16th–84th: $3.57$–$7.57 \times 10^{-3}$)

- $b_{\text{disf}} = 0.026 \pm 0.043$

- $\sigma = 1.65 \pm 0.62$ mm/s (flyby-to-flyby scatter)

- $\alpha_{\text{res}} = +0.022 \pm 0.286$ (consistent with zero)

The hierarchical posterior median $\beta_{\rm fit,0} = 5.27 \times 10^{-3}$ is consistent with the Step 008 inverse-variance weighted mean ($4.74 \times 10^{-3}$) within the heterogeneity width, and the agreement is not prior-imposed: the stored likelihood profile over $\beta_{\rm fit,0}$ — evaluated with the remaining parameters at their posterior medians and no prior weighting — peaks at $5.57 \times 10^{-3}$, with the Step 008 centre within $\Delta\ln\mathcal L \approx 0.75$ of the maximum. The two estimation layers thereby reconcile on a common $\mathcal O(10^{-3})$ amplitude. Posterior predictive checks track NEAR and Rosetta 2005 well; Galileo 1990, Galileo 1992, and Cassini retain residuals of 2.5, 4.0, and 2.0 mm/s respectively in this hierarchical layer, motivating continued envelope and OD work.

Table 3d: Posterior Predictive Checks

| Flyby | $\Delta v_{\text{obs}}$ (mm/s) | $\Delta v_{\text{pred}}$ (mm/s) | Residual (mm/s) |
| --- | --- | --- | --- |
| NEAR | 13.46 | $13.54 \pm 1.85$ | $-0.08$ |
| Galileo 1990 | 3.92 | $1.39 \pm 0.29$ | $+2.53$ |
| Cassini | $-2.00$ | $-0.00 \pm 0.00$ | $-2.00$ |
| Galileo 1992 | $-4.60$ | $-0.58 \pm 0.33$ | $-4.02$ |
| Rosetta 2005 | 1.80 | $1.90 \pm 0.50$ | $-0.10$ |
| Rosetta 2007 | 0.02 | $0.00 \pm 0.00$ | $+0.02$ |
| MESSENGER 2005 | 0.02 | $0.00 \pm 0.00$ | $+0.02$ |

The posterior median $\beta_{\rm fit,0} = 5.27 \times 10^{-3}$ agrees with the Step 008 inverse-variance weighted mean ($4.74 \times 10^{-3}$) within the heterogeneity width; the width is driven by the population scatter $\sigma \approx 1.65$ mm/s absorbed by the hierarchical layer rather than by any residual offset between the two estimators. The factor-of-order-unity rescaling from the reference coupling ($\beta_{A,\rm ref} = 10^{-4}$) is a computational anchor only: because the canonical excursion normalization is $\beta$-independent, the response model is strictly linear ($\Delta v_{\rm TEP} \propto \beta$), so the closed-form inversion $\beta_{\rm fit} = \beta_{A,\rm ref}\,(\Delta v_{\rm obs}/\Delta v_{\rm TEP})$ is algebraically independent of $\beta_{A,\rm ref}$ — the fitted amplitude is set by the data, and re-evaluating the same flyby at $\beta_{A,\rm ref} = 10^{-5}$ or $10^{-3}$ returns the identical fitted value. The residual bookkeeping is therefore a single channel quantity: the population response coefficient $\kappa_{\rm flyby} = \beta_{\rm fit}/|\beta_A| \approx 4.7 \times 10^{-3}$ — an unexplained transfer-map coefficient of the same open class as $\kappa_{\rm Cep}$ and the Paper 0 $\Gamma_X$ entries, reported pending the action-level derivation rather than absorbed into the theory. The Step 043 clock-channel forward model carries the same bookkeeping in its own normalization: the OD-level rescaling $C_A \approx 4.2$ between reconstructed apparent shifts and published anomalies is the clock-channel analogue of $\kappa_{\rm flyby}$ and is reported as such.

## 4.3 Deterministic Geometry Modulation Analysis

With only n = 6 sign-gated detections, formal variance decomposition into structural, observational, environmental, and residual components remains underpowered: the standard error on a sample variance estimate with ddof = 1 exceeds 100% of the estimate for n &lt; 5 and stays near 50% at n = 6. Step 009 therefore does not report ANOVA-style percentages. Instead, three complementary analyses that are defensible at small n are presented.

### 4.3.1 Beta Scatter Statistics

Across the six gated fits, fitted β spans $4.43 \times 10^{-3}$ to $3.40 \times 10^{1}$ (log STD = 1.77 dex, n = 6), dominated by the two unconstrained near-zero-prediction rows; the four amplitude-informative members span $4.43 \times 10^{-3}$ to $6.65 \times 10^{-2}$. Because the Step 007 prediction already includes the geometry envelope, the residual scatter among the constrained members reflects genuine coupling heterogeneity or model incompleteness, not unmodeled trajectory geometry. With n = 6, no formal partitioning of this scatter into structural, observational, or environmental buckets is attempted.

### 4.3.2 Full-Catalog Detection Pattern

Across all n = 12 catalogued flybys (Step 009), evaluating the deterministic geometry envelope at the Step 008 pooled coupling $\beta_{\rm pooled} = 4.74 \times 10^{-3}$ correctly classifies 11/12 flybys at the $0.5$ mm/s threshold (accuracy 91.7%: 3 true positives [NEAR $+14.39$ mm/s predicted vs $+13.46$ mm/s observed; Galileo 1990 $+1.29$ mm/s vs $+3.92$ mm/s; Rosetta 2005 $+1.66$ mm/s vs $+1.80$ mm/s], 8 true negatives, 1 false negative [300 km perigee drag-dominated Galileo 1992, predicted at $-0.33$ mm/s], and 0 false positives), and achieves 12/12 (100% accuracy: 4 true positives, 8 true negatives, 0 false negatives, 0 false positives) at the $0.3$ mm/s threshold. (By contrast, at the unfitted placeholder reference $\beta_{A,\rm ref} = 10^{-4}$ used prior to Step 008, magnitudes sit a factor of $47.4$ lower so all four detections fall below $0.5$ mm/s while retaining the exact observed signs.) Cassini's corrected $-2$ mm/s entry sits at the adopted detection boundary ($2\sigma$) and is classified as a true negative; the sign of its prediction nevertheless agrees with observation. No false positives are present: no published null or sub-threshold row is predicted as a detection.

### 4.3.3 Rank Correlation

A Spearman rank correlation between predicted and observed anomaly magnitudes across the full catalog (n = 12) yields ρ = 0.56 (p = 0.059). This indicates moderate rank agreement: the model captures the relative ordering of anomaly sizes, with the largest predicted anomalies corresponding to the largest observed anomalies. The companion ordering diagnostic between |trajectory asymmetry| and |observed anomaly magnitude| gives ρ = 0.56 (p = 0.11, n = 9 flybys with published asymmetries) and ρ = 0.68 (p = 0.090, n = 7 nonzero anomalies); within the gated n = 6 subset the ordering is positive but not decisive (ρ = +0.66, p = 0.156): NEAR carries both the largest asymmetry and the largest anomaly in every subset.

The legacy four-stage variance-decomposition output in `results/step009_variance_analysis.json` is retained for backward compatibility but is explicitly marked `DEPRECATED` and must not be used for manuscript inference. The dominant formal heterogeneity statistic remains Cochran Q / I² on the six gated β fits (Step 008).

## 4.4 Disformal Transition Criterion Results

The disformal transition criterion Ξ classifies flybys into conformal-dominated, mixed, or disformal-dominated regimes of the fitted response template, using velocity, asymmetry, and altitude. The definition Ξ = (v/v_trans)² × |asym| × (|∇φ|/|∇φ_⊕|) × sgn(asym) uses the empirical template scale v_trans ≈ 16.8 km/s (status: phenomenological ansatz, Section 3.5 — under the canonical B(φ) envelope no kilometre-per-second disformal transition exists, the required microscopic coefficient being excluded by ~26 orders of magnitude); the analyzed flybys span multiple template regimes.

Cassini, with its high perigee velocity (19.02 km/s) and weakly negative asymmetry (cos_asymmetry = -0.022), lies in the template's mixed regime where gradient and disformal contributions partially cancel at β<sub>A,ref</sub>, yielding a small negative total prediction ($-5.3 \times 10^{-5}$ mm/s). With the corrected catalogue entry the published anomaly is also negative ($-2 \pm 1$ mm/s), so Cassini passes the sign gate; its residual tension is in amplitude only, since the reference prediction is far below the observation and the unconstrained per-flyby rescaling is absorbed into the negligible-weight fit reported in Table 3. Consistent with the canonical branch, the hierarchical layer's fitted disformal amplitude is null ($b_{\rm disf} = 0.026 \pm 0.043$), so the velocity-template terms act as phenomenological descriptors rather than a detected propagating sector.

## 4.4a Full-Catalog Raw Stress Test

Before restricting attention to the sign-gated detections, the universal-$\beta_{\rm fit}$ prediction is tested against all published rows with explicit raw TEP predictions in Step 039. This raw-layer stress test includes NEAR, Galileo 1990, Rosetta 2005, Cassini, Galileo 1992, MESSENGER, Rosetta 2009, Juno, and Rosetta 2007 ($n=9$). Flybys with no public anomaly report are also excluded. The uncertainty is $\sigma_{\rm tot}^2=\sigma_{\rm obs}^2+\sigma_{\rm raw}^2$, combining the published measurement uncertainty with the propagated universal-$\beta_{\rm fit}$ prediction uncertainty.

Table 3e: Full-Catalog Raw Stress-Test Likelihood (Step 039)

| Quantity | Null | Raw TEP universal-$\beta_{\rm fit}$ |
| --- | --- | --- |
| Included rows | $n=9$ published rows with explicit raw predictions |  |
| Log likelihood | $-81.79$ | $-38.76$ |
| $\chi^2$ | $175.61$ | $89.54$ |
| Improvement | $\Delta\log L_{\rm TEP-null}\approx+43.03$; $\Delta\chi^2_{\rm null-TEP}\approx86.07$ |  |

This table is the headline full-catalog stress test: once the random-effects prediction uncertainty is included, the raw universal-$\beta_{\rm fit}$ model improves over the null by $\Delta\log L \approx 43.03$. The improvement reflects the large NEAR, Galileo 1990, and Rosetta 2005 signals, while still exposing the remaining stress cases: Galileo 1992 is classified as a raw surplus (correct sign, underpredicted magnitude; the uncertainty-aware scheme flags it as an observed detection with an uncertain prediction), and Juno remains the explicit raw-tension null (predicted $+0.12$ mm/s at the pooled $\beta_{\rm fit}$, a $\sim 3\sigma$ tension under the propagated uncertainty). These results are therefore stronger than a pure three-row fit, but they are not a post-OD mission likelihood because the $F_{\rm OD}$ columns remain withheld until real OD configuration data are available.

## 4.5 Bayesian Model Comparison

The four-tier model comparison below evaluates Null, Anderson, TEP restricted, and TEP flexible models on the full catalog of nine flybys with published observations (n = 9), using Gaussian likelihoods and systematic uncertainty from the Step 026 heterogeneity budget. The TEP restricted amplitude β<sub>fit</sub> is fitted on the six-member sign-gated ensemble (all published detections now pass the sign gate on the corrected catalogue), then applied to the full n = 9 catalog for the model-comparison likelihood. This tests whether a coupling inferred from the detections also compresses the full catalog.

### 4.5.1 Model Definitions and Parameter Status

| Model | Fitted Parameters | Pre-specified Quantities | Description |
| --- | --- | --- | --- |
| Null ($M_0$) | 0 | — | Predicts $\Delta v = 0$ for all flybys. |
| Anderson Empirical ($M_A$) | 2 (A, B) | Geometry (declinations) from JPL Horizons | $\Delta v = A (\cos\delta_{\rm in} - \cos\delta_{\rm out}) + B$. Captures the core empirical correlation identified by Anderson et al. (2008). Perigee latitude is omitted because it is not catalogued. |
| TEP Restricted ($M_{\rm T}^{\rm res}$) | 1 ($\beta_{\rm fit}$) | $\lambda_T = 4200$ km (GNSS Step 016); $S_\oplus \approx 0.35$ (UCD Step 010); $v_{\rm trans} \approx 16.8$ km/s (empirical template scale); geometry from JPL Horizons | $\Delta v = dv_{\rm pred}^{\rm base}(\beta_{\rm fit}/\beta_{\rm ref})$. All physics except the coupling amplitude is pre-specified from independent data or stated response templates; the excursion normalization is $\beta$-independent under the canonical Paper 0 amplitude sector, so the response is linear. |
| TEP Flexible ($M_{\rm T}^{\rm flex}$) | 3 ($\beta_{\rm fit}$, $b_{\rm disf}$, offset) | Same pre-specified quantities as restricted | $\Delta v = (\beta_{\rm fit}/\beta_{\rm ref})(dv_{\rm grad} + b_{\rm disf} \, dv_{\rm disf}) + \text{offset}$. Allows disformal amplitude and residual modulation (plasma, OD) to vary freely. |

### 4.5.2 Log-likelihoods and Information Criteria

The archived Step 026 models are trajectory-geometry templates, not distinct clock-sector predictions for the coherent catalogue. Their iteratively weighted least-squares scores are retained below for reproducibility. Because the fractional variance depends on the fitted prediction, this fixed-point procedure is not a certificate of maximum likelihood for the full heteroscedastic model; the fitted scatter and template choices also require explicit nuisance accounting. The following log-likelihood, AIC and BIC values are therefore archived convention-dependent scores, not calibrated evidence for the clock mechanism:

- Null: log L = -8360.96, AIC = 16721.9, BIC = 16721.9

- Anderson: log L = -11.59, AIC = 27.2, BIC = 27.6

- TEP restricted: log L = -15.09, AIC = 32.2, BIC = 32.4

- TEP flexible: log L = -30.88, AIC = 67.8, BIC = 68.4

- TEP restricted (pessimistic): k_eff = 8 (1 fitted β<sub>fit</sub> + 7 heuristic envelope coefficients), BIC = 47.8

Sign-gated ensemble (n = 6) log-likelihoods under the same fractional floor: Null log L = -8368.02, Anderson log L = -6.55, TEP restricted log L = -16.97, TEP flexible log L = -35.42 (BICs 16736.0, 16.7, 35.7, 76.2 respectively). The gated sensitivity at $\sigma_{\rm sys} = 0$ — published uncertainties taken at face value — favours TEP restricted over Anderson by $\Delta{\rm BIC} \approx +848$ (log L $-582.0$ versus $-1005.2$), the same direction as the full-catalog $\sigma_{\rm sys}=0$ result ($+2767$); the nominal Anderson preference exists only under a nonzero systematic floor whose scale is a modelling convention rather than a measurement.

### 4.5.3 Information Criteria and Model Comparison

Information criteria for the full n = 9 catalog:

- Anderson vs Null: $\Delta$BIC $\approx 16694.3$

- TEP restricted vs Null: $\Delta$BIC $\approx 16689.5$

- TEP flexible vs Null: $\Delta$BIC $\approx 16653.6$

- TEP restricted vs Anderson: $\Delta$BIC $\approx -4.8$ (parity; Anderson nominally first)

Interpretation. The three geometric templates organize the catalogue with archived score differences $\Delta$BIC $\gtrsim 1.66\times10^{4}$ against the zero-signal Null under the adopted uncertainty convention. These scores quantify geometric compression of the trajectory-asymmetry pattern; the link-class condition under which the clock-sector mechanism populates this template (requiring an independent onboard clock reference rather than coherent two-way turnaround) is established in Section 5.2, and because the Null is assigned no additional fractional scatter, the comparison does not by itself exclude geometry-correlated tracking systematics. On the direct head-to-head, the Anderson empirical model attains the smallest BIC ($27.6$ versus $32.4$ for TEP restricted) — a $\Delta$BIC $\approx 4.8$ gap that lies in the weak-evidence regime and constitutes statistical parity rather than a decisive preference. Two context points matter. First, the Anderson tier is fitted on the very catalogue it is scored against (amplitude $A$ and offset $B$), whereas TEP restricted carries a single amplitude on top of geometry pre-specified from independent measurements. Second, the ordering between the two tiers is systematic-convention dependent: under the published-uncertainties-only sensitivity block ($\sigma_{\rm sys} = 0$), TEP restricted leads Anderson by $\Delta$BIC $\approx +2767$, because its geometry tracks the dominant NEAR amplitude while the linear asymmetry law does not; the former constant geometry-spread floor placed Anderson ahead by $\approx 98$ only through its dependence on the arbitrary $\beta_{\rm ref}$ anchor, and the constant-floor sweep in Step 026 shows the ordering crossing zero near $\sigma_{\rm sys}\approx5$ mm/s. The headline fractional floor sits inside this range and yields parity. The TEP flexible tier (BIC $= 68.4$) does not improve on the restricted tier under the canonical linear response.  The BIC-approximated Bayes factor ($BF \approx e^{\Delta{\rm BIC}/2}$) is a large-sample result. For n &lt; 10 and extreme signal-to-noise it is unreliable; when $\log_{10}(BF) &gt; 100$ the value is driven by formal uncertainties and should not be read as a literal probability ratio (Step 026 flags all comparisons in this regime). The displayed scores are arithmetic functions of the archived objective values and parameter counts. Their probabilistic interpretation additionally requires a verified likelihood optimum and treatment of all data-estimated nuisance quantities; those checks are distinct from the channel theorem.  Even under the pessimistic parameter count ($k_{\rm eff} = 8$), the TEP restricted BIC = 47.8 still beats the Null by $\Delta$BIC $\approx 16674.1$, confirming that the catalog compression is not an artifact of aggressive parameter counting; under that penalty the Anderson tier's margin over TEP restricted widens to $\approx 20$ BIC units.

Akaike weights (Step 026, full $n = 9$ catalog, AICc): Anderson $\approx 0.86$, TEP restricted $\approx 0.14$, TEP flexible $\approx 0$, Null $\approx 0$. The weight distribution quantifies the same parity-level ordering as the BIC comparison — neither tier dominates under the fractional floor.

The restricted model remains the scientifically important tier because every quantity except $\beta_{\rm fit}$ is pre-specified from independent measurements or stated response templates. Its large $\Delta$BIC separation on the full n = 9 catalog therefore reflects predictive compression relative to the Null with a catalog-independent geometry, while the Anderson tier measures how much additional compression is purchasable with a second catalog-fitted parameter.

#### Artefact-Kernel Impulse Consistency Verification

The fitted $\beta_{\rm fit}$ values provide a direct probe of the clock-sector artefact structure through the kernel impulse diagnostic. The artefact-kernel impulse $\mathcal{I} = \int_{\rm path} \mathbf{F}_\phi \cdot d\mathbf{r}$ measures the net accumulation of the clock-sector response along the flyby trajectory. In the TEP framework, the predicted apparent velocity shift relates to the impulse via $\Delta v_{\rm TEP} \propto \beta_{A,\rm eff} \cdot \mathcal{I}$, modulated by trajectory geometry and disformal coupling. The consistent mapping between fitted $\beta_{\rm fit}$ values and the geometric impulse computed from each flyby's 3D trajectory (using JPL Horizons ephemerides) supports the conclusion that the clock-sector response model respects the fundamental field structure of the TEP equations. On the corrected catalogue the impulse magnitude is not positively correlated with the fitted $\beta_{\rm fit}$ ($r \approx -0.4$ across the six-member ensemble): the flybys with near-zero reference impulses (Cassini, MESSENGER) return divergent rescalings that carry no amplitude information, while the well-conditioned members (NEAR, Galileo 1990, Rosetta 2005) span the tightly constrained range $\beta_{\rm fit} \sim (4.4$–$14.4)\times10^{-3}$. The diagnostic therefore confirms that the per-flyby rescaling is well-posed where the reference impulse is non-negligible, and that the ensemble amplitude is set by the members with genuine response leverage.

The fitted $\beta_{\rm fit}$ values are flyby trajectory-response amplitudes, not PPN $\gamma$ predictions. Cassini PPN compliance is evaluated through the solar source-charge projection $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$ (Section 4.6.1a). The ensemble weighted mean yields $\beta_{\rm fit} = 4.74 \times 10^{-3} \pm 5.83 \times 10^{-4}$. The solar source-charge projection in Section 4.6.1a adopts the Paper 0 v0.16 convention $S_\Sigma^{(\odot)} \sim 2\times10^{-6}$, inside the Cassini requirement $S_\Sigma^{(\odot)} \lesssim 1.15\times 10^{-5}$ under the unified unstretched-space metric ($5.8\times 10^{-6}$ on the isotropic-conformal benchmark); the two-branch kinetic-operator evaluation on the solar profile gives $y_{\rm tb}(1.6\,R_\odot)=2.16\times 10^{-8}$ ($|\gamma_{\rm PPN}-1|=4.32\times 10^{-8}\ll 2.3\times 10^{-5}$) and $\mathcal S_\Sigma \sim 10^{-15}$ at the 1 AU path. This supports Temporal Topology screening in the solar environment without overstating the margin.

## 4.6 PPN Constraints and Validation

### 4.6.1 PPN Constraint Derivation

The PPN (Parametrized Post-Newtonian) formalism characterizes deviations from General Relativity. For scalar-tensor theories with conformal coupling $A(\phi) = \exp(\beta_A \phi/M_{\rm Pl})$, the PPN parameter $\gamma$ relates to the coupling strength under the unified unstretched-space metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ (Paper 0 §2.2, Step 80; with $4\beta_A^2 S_\Sigma^{(\odot)}$ on the isotropic-conformal benchmark):

\begin{equation}
|\gamma - 1| \approx 2\beta_A^2\,S_\Sigma^{(\odot)} \quad \left(\text{isotropic benchmark: } 4\beta_A^2\,S_\Sigma^{(\odot)};\ \text{for small }
S_\Sigma\right)
\end{equation}

Derivation (Paper 0, Sec. 7, Step 80): in the screened-source limit the photon probe is unscreened while the source charge is suppressed, so under the unified unstretched-space metric $\gamma_{\rm PPN} - 1 = -2\beta_A^2 S_\Sigma/(1 + 2\beta_A^2 S_\Sigma)$ (or $-2\alpha_0\,\alpha_{\rm eff}/(1 + \alpha_0\,\alpha_{\rm eff}) = -4\beta_A^2 S_\Sigma/(1 + 2\beta_A^2 S_\Sigma)$ on the isotropic-conformal benchmark), linear in $S_\Sigma$, with $\alpha_{\rm eff} = S_\Sigma\,\alpha_0$ and $\alpha_0 = \sqrt{2}\,\beta_A$ the DEF-normalized microscopic coupling. The Cassini source-charge parameter is $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$, giving $|\gamma - 1| \approx 2\beta_A^2 S_\Sigma^{(\odot)}$ ($4\beta_A^2 S_\Sigma^{(\odot)}$ isotropic) for magnitude comparisons to Cassini (the measured $\gamma - 1$ is negative in both conventions). The Earth-vicinity flyby response geometry $S_\oplus$ is a distinct projection and is not the Cassini source-charge parameter.

The fitted $\beta_{\rm fit}$ values are flyby trajectory-response amplitudes for Earth-vicinity geometry, distinct from the Cassini source-charge parameter $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$. The Earth-screened flyby response amplitudes $\beta_{\rm fit} \times S_\oplus$ are:

- NEAR: $\beta_{\rm fit} \times S_\oplus = 1.55 \times 10^{-3}$

- Galileo 1990: $\beta_{\rm fit} \times S_\oplus = 5.03 \times 10^{-3}$

- Galileo 1992: $\beta_{\rm fit} \times S_\oplus = 2.33 \times 10^{-2}$

- Rosetta 2005: $\beta_{\rm fit} \times S_\oplus = 1.80 \times 10^{-3}$

- Cassini and MESSENGER: formally gated but unconstrained (reference prediction $\approx 0$); Cassini PPN compliance is evaluated through the solar source-charge projection $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$ (Section 4.6.1a), not through the flyby amplitudes.

These are trajectory-response amplitudes, not PPN $\gamma$ predictions. Cassini PPN compliance is evaluated through the solar source-charge projection $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$ (Section 4.6.1a), not through the fitted flyby amplitudes.

### 4.6.1a Solar-Screening PPN Check for Cassini

The Cassini Shapiro-delay measurement constrains the scalar field along the radio path during solar conjunction, not at Earth's surface. Cassini PPN compliance is evaluated through the solar source-charge projection, not through the fitted flyby amplitudes. Following the Jakarta mapping (Paper 0, §7):

\begin{equation}
\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\,\alpha_0,
\qquad
\gamma_{\rm PPN} - 1 = -\frac{2\beta_A^2\,S_\Sigma^{(\odot)}}{1 + 2\beta_A^2\,S_\Sigma^{(\odot)}}
\simeq -2\beta_A^2\,S_\Sigma^{(\odot)}
\quad \left(\text{isotropic benchmark: } -4\beta_A^2\,S_\Sigma^{(\odot)}\right),
\end{equation}

where $\alpha_0 = \sqrt{2}\,\beta_A$ is the DEF-normalized bare microscopic coupling (linear in the source charge because the photon probe is unscreened) and $S_\Sigma^{(\odot)}$ is the solar Temporal-Topology screening factor. With $\beta_A = -1$ (Jakarta §2.2), the Cassini bound $|\gamma - 1| &lt; 2.3\times 10^{-5}$ requires (Paper 0, Jakarta v0.16 §7, Step 80):

\begin{equation}
S_\Sigma^{(\odot)} \lesssim \frac{2.3\times 10^{-5}}{2\beta_A^2} \approx 1.15\times 10^{-5}
\quad \left(\text{isotropic benchmark: } \frac{2.3\times 10^{-5}}{4\beta_A^2} \approx 5.8\times 10^{-6}\right).
\end{equation}

Cassini constrains the solar exterior source-charge projection $S_\Sigma^{(\odot)}$, not the Earth-vicinity flyby response $S_\oplus$. Paper 0 v0.16 adopts the Solar-System source-charge convention $S_\Sigma^{(\odot)} \sim 2\times10^{-6}$, which satisfies the Cassini requirement $S_\Sigma^{(\odot)} \lesssim 1.15\times 10^{-5}$ ($5.8\times 10^{-6}$ isotropic) comfortably; the two-branch kinetic-operator evaluation on the solar field returns $y_{\rm tb}(1.6\,R_\odot)=2.16\times 10^{-8}$ ($|\gamma_{\rm PPN}-1|=4.32\times 10^{-8}\ll 2.3\times 10^{-5}$) and $S_\Sigma \sim 1.6\times10^{-15}$ at the 1 AU path (Paper 0 Appendix E, static-branch benchmarks), with the microscopic uniqueness of that charge within the wider $K(X)$ class remaining open. The flyby response and the solar PPN source charge are therefore treated as distinct environmental projections of the same scalar configuration.

### 4.6.2 Sensitivity Analysis

To assess robustness, the TEP model is tested against variations in key parameters. Table 3f shows how results change when parameters are varied within physically plausible ranges:

Table 3f: Sensitivity Analysis - Parameter Variations

| Parameter | Nominal Value | Tested Range | Fit stable? | Impact on β |
| --- | --- | --- | --- | --- |
| Geometric suppression factor (S_⊕) | 0.35 | 0.30 – 0.50 | ✓ Yes (all values) | ±6% |
| Relaxation length (λ_T) | 4200 km | 2000 – 8000 km | ✓ Yes (within range) | ≲50% (sensitivity coefficient 0.36) |
| J2 coefficient | 1.08263×10⁻³ | ±0.1% | ✓ Yes | <1% |
| J3 coefficient | -2.54×10⁻⁶ | ±10% | ✓ Yes | negligible |
| Trajectory uncertainty | declination ±0.5° | ±1° | ✓ Yes | ±5% |

Robustness conclusion: The TEP model maintains fit stability across a broad range of parameter values. The phase-boundary factor can vary by ±32% (0.25 to 0.45) and all fitted β<sub>fit</sub> values remain within the fitted range. This suggests that the fit stability is not fine-tuned but is a feature of the screening mechanism. The relaxation length has moderate impact on predicted Δv but does not affect fit stability because the fitted β values adjust to compensate.

### 4.6.3 OD Filter Simulation: Suppression Hypothesis Validation

Step 021 now withholds all mission-specific OD survival factors until real mission OD configuration data are available. The earlier synthetic Step 012 batch least-squares experiment is retained only as a diagnostic development artifact: its empirical-acceleration implementation is numerically unstable in the current 3D form, and the generated result is not valid for computing $F_{\rm OD}$ or for supporting quantitative claims about modern OD suppression.

#### Current OD Evidence Status

- Mission $F_{\rm OD}$ values: not computed; mission
OD configuration files are required.

- Step 012 synthetic OD run: quarantined as
`synthetic_diagnostic_not_for_manuscript_inference`.

- Manuscript policy: no era-based or synthetic OD
survival factors are used in the Step 039 classification table.

Interpretation: The OD-suppression mechanism remains a falsifiable hypothesis rather than a calibrated correction. It is physically plausible that empirical acceleration states and residual editing can absorb unmodeled perigee-local forces, but this paper does not assign numerical survival fractions without mission-specific OD settings.

Table 3g: OD Survival-Factor Status

| Quantity | Status | Use in likelihood? |
| --- | --- | --- |
| Mission-specific $F_{\rm OD}$ | Not computable without OD configuration files | No |
| Step 012 synthetic OD diagnostic | Quarantined; not valid for manuscript inference | No |

Connection to observations: Step 039 classifies fixed-amplitude raw predictions (Step 008 pooled $\beta_{\rm fit}$) with the Step 007 geometry envelope (3 deterministic true positives, 4 deterministic true nulls, 1 sign-consistent raw surplus, and 1 deterministic fixed-amplitude warning case across the 9 published rows). After propagating Step 008 random-effects amplitude scatter, the uncertainty-aware raw layer retains 1 raw-tension case (Juno) and moves Galileo 1992 to "observed detection, prediction uncertain". Post-OD survival factors are withheld until mission OD configuration yields defensible $F_{\rm OD}$ estimates.

Juno tension: The Juno non-detection (universal-$\beta_{\rm fit}$ prediction $+0.12 \pm 0.03$ mm/s, observed $0.00 \pm 0.02$ mm/s) is the most serious raw-tension case and motivates independent raw DSN re-analysis. In the current pipeline run, the DSN ingestion layer emits mission discovery and request artifacts but does not yet provide indexed raw Doppler products for most missions (ingest status `no_indexed_products`), so this re-analysis remains a defined falsification pathway rather than a completed reprocessing result.

### 4.6.3a Clock-Sector Artefact: End-to-End Orbit-Determination Rederivation

The clock-sector mechanism has now been propagated through the full reduction chain rather than asserted (Step 043). For each of the twelve flybys with reconstructed JPL Horizons state-vector arcs, the canonical field excursion $\psi(r) = \phi_{\rm space}-\phi(r)$ of the pipeline's screened field solution is evaluated along the real trajectory; the matter-metric rate offset $\eta(t) = \exp(\beta_A\,\psi/M_{\rm Pl})-1$ ($\beta_A=-1$; the unsuppressed amplitude channel $S_A$, not the screened shear channel) accumulates the proper-time offset $\delta\tau(t)=\int\eta\,dt$ and injects the apparent range-rate offset $c\,\eta(t)$ that the excursion produces in a clock-carrying link — a spacecraft-local oscillator term entering once wherever the downlink is referenced to an onboard time standard (one-way, non-coherent, or regenerative-ranging observables). In the fully coherent two-way class of the published catalogue the term is absent at the observable: the transponder multiplies the received carrier by a fixed ratio, and the endpoint conformal factors cancel identically (Section 5.2). The step therefore computes what the clock excursion reconstructs as on a clock-carrying link over the real arcs — the mechanism's falsifiable prediction — together with the postfit-residual signature that quantifies how much of it survives reduction into the orbit solve. The identical batch least-squares orbit-determination filter used in the OD-suppression study (six-state estimator, point-mass dynamics, assumed GR time standard) is then run on clean and corrupted synthetic Doppler, and the differential of the fitted end-minus-start speed change isolates the clock-induced apparent $\Delta V$. Truth dynamics are pure point-mass gravity — consistent with the screened shear channel contributing nothing to the trajectory — while $\eta(t)$ is sampled on the real arcs, preserving the physical in/out asymmetry.

Table 3h: Clock-sector OD rederivation — per-flyby proper-time signature, reconstructed apparent velocity shift, and postfit residual signature on a clock-carrying link (Step 043; noiseless differential estimator, three-complex DSN network, $\pm$12 h tracking window at 180 s cadence). $\Delta V_{\rm app}$ is an absorption remnant — it shrinks as the arc lengthens (NEAR: $+0.68$/$+0.18$/$+0.05$ mm/s for $\pm$6/$12$/$24$ h) — so the discriminating observable is the unabsorbed postfit residual column. The injected term is absent from the catalogue's coherent two-way class (Section 5.2); the comparison column therefore evaluates sign and scale, not attribution.

| Flyby | $\eta$ at perigee | $\delta\tau$ total ($\mu$s) | $\Delta V_{\rm app}$ (mm/s) | Postfit resid. RMS (peak) mm/s | Observed (mm/s) |
| --- | --- | --- | --- | --- | --- |
| NEAR | $-1.22\times10^{-9}$ | $-2.0$ | $+0.18$ | $5.8$ ($35$) | $13.46\pm0.13$ |
| Galileo 1990 | $-1.11\times10^{-9}$ | $-1.6$ | $+0.07$ | $7.7$ ($83$) | $3.92\pm0.08$ |
| Galileo 1992 | $-1.29\times10^{-9}$ | $-1.8$ | $+0.06$ | $4.6$ ($42$) | $-4.60\pm1.00$ |
| Cassini | $-1.05\times10^{-9}$ | $-1.0$ | $+0.01$ | $1.3$ ($13$) | $-2.00\pm1.00$ |
| Rosetta 2005 | $-8.73\times10^{-10}$ | $-2.0$ | $+0.50$ | $4.7$ ($27$) | $1.80\pm0.03$ |
| Rosetta 2007 | $-3.93\times10^{-10}$ | $-0.7$ | $+0.01$ | $0.7$ ($5$) | $0.02\pm0.05$ |
| Rosetta 2009 | $-7.70\times10^{-10}$ | $-1.2$ | $+0.05$ | $3.0$ ($26$) | $0.00\pm0.05$ |
| MESSENGER | $-7.97\times10^{-10}$ | $-1.9$ | $+0.49$ | $4.7$ ($31$) | $0.02\pm0.01$ |
| Juno | $-1.22\times10^{-9}$ | $-1.6$ | $+0.04$ | $1.8$ ($15$) | $0.0\pm0.02$ |
| Stardust | $-3.33\times10^{-10}$ | $-0.8$ | $+0.04$ | $0.5$ ($3$) | — |
| OSIRIS-REx | $-2.31\times10^{-11}$ | $-0.1$ | $+0.02$ | $0.3$ ($1$) | — |
| BepiColombo | $-6.78\times10^{-11}$ | $-0.3$ | $+0.05$ | $0.2$ ($0.7$) | — |

Three findings follow, all conditional on the link class. First, the amplitude regime is correct: the canonical field excursion ($\Delta\phi = 3.4\times10^{9}$ GeV, $\eta_{\rm surface}=-1.39\times10^{-9}$) produces accumulated proper-time offsets of $0.1$–$2\,\mu$s; the raw signature reaches $c\eta \sim$ m/s at perigee, so the excursion would be an unmissable feature of any one-way, non-coherent, or onboard-referenced flyby measurement. Second, the discriminating observable is the postfit residual, not the reconstructed endpoint $\Delta V$: the six-state fit absorbs the bulk of the perigee-concentrated offset into the estimated state, but a structured remnant always survives — $0.2$–$7.7$ mm/s RMS across the catalogue (peaks up to $\sim 83$ mm/s), i.e. $4$–$25\%$ of the injected signature leaks through orbit determination at $\sim 180$ s cadence and remains detectable at DSN noise floors. The endpoint $\Delta V_{\rm app}$ column is reported for completeness but is an absorption remnant rather than a signal amplitude — it shrinks by a factor of $\sim 15$ as the tracking window widens from $\pm 6$ to $\pm 24$ h (NEAR: $+0.68 \to +0.05$ mm/s) — while the residual signature is window-stable (the linearized filter response reproduces the nonlinear result within 10–40%). Third, the recovered artefact is controlled by the projection of the $\eta(t)$ profile through the tracking geometry into the fit basis, not by the leg-integrated proper-time asymmetry: across the nine published rows the reconstructed apparent shift correlates only moderately with the catalogue (Pearson $r = 0.17$, Spearman $\rho = 0.58$), so the artefact is intrinsically geometry-projection-dependent, consistent with the observed anomaly's documented resistance to simple energy-change parameterizations.

Amplitude calibration: evaluated against the detection subset, the raw canonical normalization under-produces the detected anomalies, and the global clock-response coefficient — the clock-channel analogue of the Step 008 response amplitude $\beta_{\rm fit}$ — returns $C_A = 4.2$ on the sign-inclusive $>3\sigma$ detection subset (NEAR, Galileo 1990, Galileo 1992, Rosetta 2005; $n=4$, Pearson $r=0.08$ at this normalization) with $\chi^2 = 1.17 \times 10^{4}$ on 3 dof: no single global clock-channel scaling describes the detections, so the coefficient is reported rather than adopted. The per-flyby ordering of $\Delta V_{\rm app}$ does not, by itself, reproduce the catalog pattern: the same geometry envelope and response-amplitude structure fitted in §4.3–4.5 remains load-bearing, and the clock-sector rederivation supplies the physical channel through which that response acts in a clock-carrying link rather than a replacement fit. Because the catalogued anomalies were recorded in the coherent two-way class (Section 5.2), the coefficients calibrated here characterize the forward model's response scale for clock-carrying observables, not an attribution of the published values. Limitations are explicit: a generic three-complex network rather than each mission's historical tracking schedule, point-mass truth dynamics, and single-channel (clock-only) injection. The per-mission tracking-schedule reconstruction and joint clock-plus-environment injection are identified refinements.

### 4.6.4 Leave-One-Out Cross-Validation

To verify that the weighted mean β is not dominated by any single detection, the analysis is repeated excluding each flyby successively:

Table 3i: Leave-One-Out Cross-Validation Results

| Excluded Flyby | β without this flyby | Change from full sample |
| --- | --- | --- |
| None (full sample) | 4.74×10⁻³ | — |
| NEAR (1998) | 5.88×10⁻³ | +24% |
| Galileo (1990) | 4.58×10⁻³ | −3.5% |
| Rosetta (2005) | 4.64×10⁻³ | −2.1% |
| Cassini (1999) | 4.74×10⁻³ | <0.1% |
| Galileo (1992) | 4.74×10⁻³ | <0.1% |
| MESSENGER (2005) | 4.74×10⁻³ | <0.1% |

The stability coefficient (relative standard deviation of LOO estimates divided by their mean) is 0.092, indicating moderate robustness (values &lt; 0.5 are considered robust). Even when the high-S/N NEAR detection is excluded, the remaining five flybys yield β = 5.88×10⁻³, within the 95% bootstrap interval. This indicates that the TEP conclusion does not depend on any single detection.

### 4.6.5 Enhanced Statistical Validation

Artefact-kernel impulse consistency: The clock-sector response model's apparent velocity predictions integrate the field excursion along 3D trajectories while preserving the TEP metric structure. For each flyby, the predicted $\Delta v_{\rm TEP}$ is computed via path integration of the artefact kernel $\mathbf{F}_\phi = \beta_{A,\rm eff} c^2 \nabla\phi/M_{\rm Pl}$ — the OD-equivalent impulse of the proper-time rate-offset excursion, not a physical force — along the actual spacecraft trajectory from JPL Horizons ephemeris. The open-path impulse $\mathcal{I} = \int \mathbf{F}_\phi \cdot d\mathbf{r}$ is consistently mapped to observable velocity shifts. This geometric consistency check distinguishes TEP from phenomenological force laws that lack field-theoretic structure.

Effect size analysis: Cohen's d compares each detection to the null-result population mean, using the pooled standard deviation of the two groups. The null population comprises the three published sub-threshold flybys (Rosetta 2007, Rosetta 2009, Juno) with mean $\Delta v = 0.007 \pm 0.012$ mm/s. The detection population ($n=6$) has mean $\Delta v = 2.10 \pm 6.30$ mm/s. The pooled standard deviation is $\sigma_{\rm pooled} = 5.33$ mm/s. Cohen's d for each detection vs. the null population:

- NEAR: $d = +2.53$ — large effect ($d \gg 0.8$)

- Galileo 1990: $d = +0.73$ — medium effect

- Galileo 1992: $d = -0.87$ — large effect on the negative branch

- Cassini: $d = -0.38$ — small effect on the negative branch

- Rosetta 2005: $d = +0.34$ — small effect

- MESSENGER 2005: $d = +0.003$ — negligible effect

NEAR shows the largest effect, and the corrected catalogue places Galileo 1992 on the negative branch with a large $|d|$ — consistent with the negative sign the TEP kernel predicts for both it and Cassini. The two strongest detections (NEAR and Galileo 1990) provide the bulk of the positive-branch statistical separation.

Bayesian model comparison: Stable four-tier model comparison (Step 026) on the full n = 9 catalog favors TEP restricted over Null ($\Delta{\rm BIC}\approx16689.5$) under the fractional misspecification floor, while the Anderson empirical model — whose two parameters are fitted on this same catalogue — attains the nominally smallest BIC at parity-level separation (TEP restricted minus Anderson $\Delta{\rm BIC}\approx-4.8$; the ordering reverses in TEP restricted's favour under the published-uncertainties-only sensitivity). See Section 4.5 for tier definitions. The physics-based single-amplitude model therefore compresses the full catalog far beyond the null, with the residual gap to the empirical tier quantifying the amplitude-level heterogeneity documented in Section 4.2.

Prediction accuracy: On the Step 008 primary comparison, $R^2 \approx 0.850$, Pearson $r \approx 0.930$, MAE $\approx 1.67$ mm/s, RMSE $\approx 2.24$ mm/s, and MAPE $\approx 62.5\%$. NEAR dominates the variance fraction; small-anomaly rows inflate percentage errors.

Residual analysis: Shapiro–Wilk normality on the gated prediction residuals gives $p \approx 0.95$ (consistent with Gaussian tails at $n=6$).

### 4.6.6 Characteristic Suppression from UCD Saturation Model

The characteristic suppression $S_{\oplus} \approx 0.35$—critical to the flyby amplitude and the magnitude of the flyby anomaly—is derived from the UCD saturation model in Step 010. The derivation uses Earth's total mass and the saturation scale $\rho_T = 20$ g/cm³, yielding a transition radius $R_{\rm sol} \approx 4146$ km and suppression factor $S_{\oplus} = (R_{\oplus} - R_{\rm sol})/R_{\oplus} \approx 0.35$. Under the corpus's terrestrial identification $L_c \sim \mathcal{O}(1)\,R_T(M_\oplus)$ (Paper 6 §2 transfer sketch), the GNSS covariance scale returns $S_\oplus = 0.34$ — a shared-input consistency check at the 2% level, since the same measurement anchors $\rho_T$, rather than an independent validation. The independent empirical checks — the flyby altitude threshold and the astrophysical cross-scale consistency of $\rho_T$ across dwarf-galaxy cores and cluster halos (Paper 6) — converge on $S_{\oplus} \in [0.34, 0.39]$. See Step 010 for the complete derivation and cross-scale consistency arguments.

Distinction from the canonical amplitude operator: The EFA uses $S_{\oplus} = (R_{\oplus} - R_{\rm sol})/R_{\oplus}$ as the field-excursion response ratio at the surface. Its relation to the corpus's canonical clock-amplitude operator $S_A(\mathcal E) = \min[1, (\bar\rho/\rho_T)^{1/3}]$ (Paper 0/Paper 26) is an exact identity, not a second estimate: under the uniform-density-equivalent construction, $S_A(\bar\rho_\oplus) = (\bar\rho_\oplus/\rho_T)^{1/3} = R_{\rm sol}/R_\oplus \approx 0.65$ — the saturated (embedded) fraction of the equivalent radius, coinciding with the UCD embedding factor used in Paper 6 (UCD). The flyby clock channel projects the complementary responsive-shell fraction $S_\oplus = 1 - S_A \approx 0.35$, because a transported clock's differential excursion relative to the ground accumulates only across the unsaturated shell where the field relaxes; the pinned interior contributes no radial excursion. The Step 010 output records the machine-checked partition sum $S_A + S_\oplus = 1$ (`canonical_amplitude_partition`). For strongly centrally condensed objects the enclosed mean density exceeds $\rho_T$, in which case $S_A$ saturates at unity and the excursion projection clamps at $S_\oplus \to 0$ (a fully pinned profile) rather than going negative — the uniform-density-equivalent partition applies to bodies whose enclosed mean density remains below $\rho_T$.

### 4.6.7 Systematic Uncertainty Budget

A comprehensive uncertainty budget quantifies the contribution of each uncertainty source to the fitted $\beta_{\rm fit}$ parameters. The corrected uncertainty analysis (Step 025) distinguishes between variance contributions and total relative uncertainty:

Variance Contributions:

- Statistical: 0.8%

- Systematic: 3.4%

- Heterogeneity: 95.8%

Total Relative Uncertainty:

- Statistical: 14.5%

- Systematic: 29.2%

- Heterogeneity: 155.4%

- Total: 158.8%

Systematic Breakdown:

- Trajectory reconstruction: 1.0%

- Characteristic suppression (UCD): 25.0% (from ρ_T = 20 ± 7 g/cm³, Paper 6) ← DOMINANT

- Multipole coefficients: 0.1%

- Relaxation length (UCD): 15.0% (SCF theoretical prior)

Interpretation: The total relative uncertainty of 152.7% is dominated by heterogeneity (149.4%), which reflects genuine geometry-dependent physical variation in the effective coupling across flybys. This is expected in the TEP framework where $\beta_{A,\rm eff}$ varies with altitude, latitude, velocity, plasma environment, and trajectory asymmetry (the plasma term is a bounded heuristic envelope coefficient, Section 3.10). The systematic uncertainty (29.2%) is dominated by characteristic suppression uncertainty (25.0%) from the UCD saturation model (ρ_T = 20 ± 7 g/cm³, Paper 6), with relaxation length uncertainty (15.0%) from the SCF theoretical prior as the second-largest source.

## 4.7 Model Predictions for All Flybys

Table 4 presents the full prediction set evaluated at the universal weighted-mean coupling constant ($\beta_{\rm fit} = 4.74 \times 10^{-3}$), scaled from the reference predictions ($\beta_{\rm fit,0} = 10^{-4}$) via the linear response established in Step 008 (the canonical excursion normalization is $\beta$-independent, so $\Delta v \propto \beta_{\rm fit}$). Each row reports the raw TEP prediction at that universal coupling, the universal-$\beta_{\rm fit}$ residual $\Delta v_{\rm obs} - \Delta v_{\rm TEP}^{\rm raw}$, and the raw classification from Step 039. Mission-specific OD survival factors $F_{\rm OD}$ and post-OD predictions are emitted only when Step 021 supplies defensible mission OD configuration data; otherwise those columns are withheld. The raw classification uses a $3\sigma$ detection threshold relative to the published uncertainty.

Table 4: Per-Flyby Universal-$\beta_{\rm fit}$ Predictions and Classification (Step 039)

| Spacecraft | Data class | Alt. (km) | $\cos\delta_{\rm in} - \cos\delta_{\rm out}$ | $\Delta v_{\rm obs}$ (mm/s) | $\Delta v_{\rm TEP}^{\rm raw}$ (mm/s) | Residual (mm/s) | $F_{\rm OD}$ | $\Delta v_{\rm TEP}^{\rm post\text{-}OD}$ (mm/s) | Raw classification |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| NEAR | Published anomaly | 539.0 | $+0.625$ | $+13.46 \pm 0.13$ | $+14.394 \pm 3.901$ | $-0.934$ | — | — | True positive |
| Galileo 1990 | Published anomaly | 960.0 | $+0.149$ | $+3.92 \pm 0.08$ | $+1.294 \pm 0.351$ | $+2.626$ | — | — | True positive |
| Rosetta 2005 | Published anomaly | 1955.0 | $+0.173$ | $+1.80 \pm 0.03$ | $+1.657 \pm 0.449$ | $+0.143$ | — | — | True positive |
| Cassini | Published anomaly | 1175.0 | $-0.022$ | $-2.00 \pm 1.00$ | $-0.001 \pm 0.000$ | $-1.999$ | — | — | True null |
| Galileo 1992 | Published anomaly | 303.0 | $-0.170$ | $-4.60 \pm 1.00$ | $-0.328 \pm 0.089$ | $-4.272$ | — | — | Raw surplus |
| MESSENGER | Published anomaly | 2347.0 | $\approx 0$ | $+0.02 \pm 0.01$ | $0.000 \pm 0.000$ | $+0.020$ | — | — | True null |
| Rosetta 2009 | Published null/bound | 2572.0 | $+0.037$ | $0.00 \pm 0.05$ | $+0.011 \pm 0.003$ | $-0.011$ | — | — | True null |
| Juno | Published null/bound | 817.4 | $+0.069$ | $0.00 \pm 0.02$ | $+0.124 \pm 0.034$ | $-0.124$ | — | — | Raw tension |
| Rosetta 2007 | Published null/bound | 5301.0 | $+0.034$ | $+0.02 \pm 0.05$ | $+0.001 \pm 0.000$ | $+0.019$ | — | — | True null |
| Stardust | No public anomaly report | 6008.9 | $-0.053$ | — | $-0.003 \pm 0.001$ | — | — | — | Uncertainty Unavailable |
| OSIRIS-REx | No public anomaly report | 17239.1 | $+0.150$ | — | $0.000 \pm 0.000$ | — | — | — | Uncertainty Unavailable |
| BepiColombo | No public anomaly report | 12697.3 | $-0.034$ | — | $-0.000 \pm 0.000$ | — | — | — | Uncertainty Unavailable |

Summary: At the universal weighted-mean $\beta_{\rm fit}$, Step 039 classifies three published anomalies as raw true positives (NEAR, Galileo 1990, Rosetta 2005). Cassini is a raw true null (corrected $-2 \pm 1$ mm/s entry versus a near-zero prediction), Galileo 1992 is a raw surplus (correct sign, underpredicted magnitude at fixed universal amplitude), MESSENGER and the three published null/bound cases are consistent with universal-$\beta_{\rm fit}$ predictions under the Step 007 geometry envelope, and one raw-tension case remains (Juno). Stardust, OSIRIS-REx, and BepiColombo lack public anomaly reports and are not used in quantitative likelihood.

Falsifiability criterion: Raw-tension cases define model stress tests independent of era-based OD survival factors. Juno remains the sole deterministic fixed-amplitude warning case at the refit weighted-mean $\beta_{\rm fit}$ ($+0.12$ mm/s raw prediction vs. $0.00 \pm 0.02$ mm/s observed) with random-effects prediction uncertainty $\pm 0.03$ mm/s. Even after propagating Step 008 random-effects scatter, the uncertainty-aware Step 039 layer retains Juno as a raw-tension case ($\sim 3\sigma$), while Galileo 1992 moves to the uncertainty-aware class "observed detection, prediction uncertain" — the model assigns the correct sign but insufficient magnitude at the pooled amplitude. Post-OD false-negative counts are reported only when mission-specific $F_{\rm OD}$ values are available from Step 021; with current OD configuration data, those columns are withheld.

Honest assessment: The clock-sector response model reproduces the three largest published anomalies at universal $\beta_{\rm fit}$ with sub-millimetre to millimetre residuals, while high-altitude or symmetric trajectories remain consistent with null predictions. Galileo 1992 — now correctly catalogued as a $-4.60 \pm 1.00$ mm/s published detection — and Juno define the priority raw DSN reanalysis targets. Cassini is no longer a sign-tension case: the corrected $-2 \pm 1$ mm/s entry carries the same sign as the TEP prediction, leaving only an amplitude shortfall that requires independent DSN OD or envelope refinements, not only a universal rescaling of $\beta_{\rm fit}$.

## 4.8 Heterogeneity and Robustness Analysis

Heterogeneity assessment: The six gated fitted β<sub>fit</sub> values span $4.43 \times 10^{-3}$ to $3.40 \times 10^{1}$ (Step 009), the upper end driven by the two unconstrained near-zero-prediction rows; the four amplitude-informative members span $4.43 \times 10^{-3}$ to $6.65 \times 10^{-2}$. Step 009 does not perform a formal variance decomposition because n = 6 remains underpowered for ANOVA-style partitioning. Instead, Step 009 reports three complementary deterministic geometry modulation analyses: (1) beta scatter statistics (log STD = 1.77 dex across all 6 rows, or a factor of 3.24 across the three clean detections NEAR, Galileo 1990, and Rosetta 2005); (2) full-catalog detection-pattern classification at the Step 008 pooled coupling $\beta_{\rm pooled} = 4.74 \times 10^{-3}$ (91.7% accuracy, 11/12 at the $0.5$ mm/s threshold: 3 true positives, 8 true negatives, 1 false negative, 0 false positives; and 100% accuracy, 12/12: 4 true positives, 8 true negatives at the $0.3$ mm/s threshold); and (3) Spearman rank correlation between predicted and observed anomaly magnitudes (ρ = 0.56, p = 0.059, n = 12) together with the asymmetry-ordering diagnostic (ρ = 0.56, p = 0.11, n = 9). The formal heterogeneity statistics on the gated ensemble are dominated by sub-percent measurement precision:

Table 5: Heterogeneity Statistics

| Statistic | Value | Interpretation |
| --- | --- | --- |
| Cochran's Q | $\approx 1.18 \times 10^{3}$ | Large (expected: $\sim 5$ for 5 d.o.f.) |
| Degrees of freedom | 5 | $n - 1$ for $n = 6$ gated fits |
| Reduced $\chi^2$ | $\approx 235.7$ | >> 1 (scatter exceeds measurement noise) |
| $I^2$ | $\approx 99.58\%$ | Formally extreme ($I^2 > 75\%$) |
| $\beta_{\rm fit}$ range (gated) | $4.43 \times 10^{-3}$ – $3.40 \times 10^{1}$ | Includes two unconstrained near-zero-prediction rows; the four amplitude-informative members span a factor $\approx 15$ |
| Random-effects scatter | $\tau \approx 2.23 \times 10^{-3}$ | Geometry- and environment-dependent modulation across the ensemble |

The elevated $I^2$ on the six gated $\beta_{\rm fit}$ fits reflects formal tension between sub-percent per-flyby uncertainties and a multiplicative spread in fitted coupling, amplified by the two formally unconstrained rows. The metric is designed for meta-analyses where true effects are identical; here, geometry, velocity, and heuristic envelope structure intentionally modulate $\beta_{A,\rm eff}$, so large $I^2$ is expected until additional physics is folded into a single generative curve or $n$ grows.

Bootstrap resampling: To assess uncertainty given the small sample ($n = 6$ gated detections), parametric bootstrap resampling with $10\,000$ iterations is performed in Step 008:

- *Bootstrap median:* $\beta_{\rm fit} \approx 4.77 \times 10^{-3}$ (tracks the inverse-variance weighted mean)

- *68% interval:* $[4.54 \times 10^{-3},\, 5.93 \times 10^{-3}]$

- *95% confidence interval:* $[4.42 \times 10^{-3},\, 1.47 \times 10^{-2}]$

The bootstrap median reproduces the weighted mean while the interval is widened by Galileo 1990’s higher per-flyby $\beta_{\rm fit}$; the occasional resampled divergent row inflates the bootstrap mean, so the median and percentile intervals are the robust summaries.

Leave-one-out (Step 008): Inverse-variance $\beta_{\rm fit}$ is recomputed excluding each gated detection:

- *Exclude NEAR:* $\beta_{\rm fit} \approx 5.88 \times 10^{-3}$ (+24%)

- *Exclude Galileo 1990:* $\beta_{\rm fit} \approx 4.58 \times 10^{-3}$ (-3%)

- *Exclude Rosetta 2005:* $\beta_{\rm fit} \approx 4.64 \times 10^{-3}$ (-2%)

- *Exclude Cassini, Galileo 1992, or MESSENGER:* $\beta_{\rm fit} \approx 4.74 \times 10^{-3}$ (&lt;0.1%)

The stability coefficient is $0.092$, indicating moderate robustness (values $&lt; 0.5$ are treated as acceptable in Step 008). NEAR is the dominant lever on the pooled scale; the three near-zero-prediction rows are negligible levers, as expected from their formal uncertainties.

Effect size: Cohen's $d$ compares each detection to the null-result population using the pooled standard deviation of the two groups:

\begin{equation}
d = \frac{\Delta v_{\rm det} - \mu_{\rm null}}{\sigma_{\rm pooled}}, \quad
\sigma_{\rm pooled} = \sqrt{\frac{(n_{\rm det}-1)s_{\rm det}^2 + (n_{\rm null}-1)s_{\rm null}^2}{n_{\rm det}+n_{\rm null}-2}}
\end{equation}

The null population comprises the published flybys with sub-threshold anomalies ($n_{\rm null}=3$: Rosetta 2007, Rosetta 2009, Juno; $\mu_{\rm null} = 0.007$ mm/s, $s_{\rm null} = 0.012$ mm/s).  The detection population ($n_{\rm det}=6$, $\mu_{\rm det} = 2.10$ mm/s, $s_{\rm det} = 6.30$ mm/s) yields $\sigma_{\rm pooled} \approx 5.33$ mm/s.  The resulting Cohen's $d$ values are:

- NEAR: $d = +2.53$ (large effect)

- Galileo 1990: $d = +0.73$ (medium effect)

- Galileo 1992: $d = -0.87$ (large effect, negative branch)

- Cassini: $d = -0.38$ (small effect, negative branch)

- Rosetta 2005: $d = +0.34$ (small effect)

- MESSENGER 2005: $d = +0.003$ (negligible effect)

NEAR remains strongly distinguishable from the null population, and Galileo 1992 now appears as a large negative-branch effect — both consistent with the sign structure the TEP kernel assigns. Rosetta 2005 shows a small effect and MESSENGER a negligible one, reflecting their proximity to the null-population mean.  The spread in $d$ values is consistent with the spread in gated fitted $\beta_{\rm fit}$, confirming geometry-dependent modulation rather than a perfectly universal effective coupling at fixed envelope.

## 4.9 Resolution of Beta Heterogeneity

The multiplicative spread in gated fitted $\beta_{\rm fit}$ values is partially summarized through a four-stage decomposition (Step 009). This unified analysis consolidates structural physics modulation, observational pipeline effects, environmental modulation, and statistical limitations into a coherent framework. The apparent scatter is not treated as pure noise: it is the object of the envelope construction. See Section 4.3 for the detailed variance decomposition analysis.

## 4.10 PPN Compliance and Global State

Model comparison: Stable four-tier model comparison (Step 026) compares the Null, Anderson empirical, TEP restricted, and TEP flexible models on the full n = 9 catalog. All non-null tiers compress the catalog by $\Delta$BIC $\gtrsim 1.66\times10^{4}$ against the Null (TEP restricted $\approx 16689.5$, TEP flexible $\approx 16653.6$, Anderson $\approx 16694.3$), confirming that trajectory asymmetry carries genuine signal. The Anderson empirical model — fitted with two free parameters on this same catalogue — attains the nominally smallest BIC, at parity-level separation from TEP restricted ($\Delta$BIC $\approx 4.8$); that ordering and its systematic-convention dependence are discussed in Section 4.5.

Formal correlation analysis: Pearson correlation tests quantify relationships between the per-flyby effective geometry factor $G_{\rm eff}$ (Section 4.2.1) and the physical parameters, as computed in Step 015 on the six-member ensemble:

Table 6: Geometry-Factor Correlation Results (n = 6 gated ensemble members, Step 015; correlations are underpowered and should be interpreted cautiously)

| Parameter | Pearson r | p-value | Interpretation |
| --- | --- | --- | --- |
| Perigee altitude | -0.13 | 0.82 | No altitude trend detected |
| Velocity | +0.90 | 0.011 | Moderate positive correlation; the only approach to significance |
| Trajectory asymmetry | -0.28 | 0.62 | Weak ($G_{\rm eff}$ already incorporates asymmetry by construction) |

With n = 6 the correlation analysis is exploratory: the velocity trend ($r = 0.90$, $p = 0.011$) is the only coefficient approaching significance and is consistent with velocity-modulated response in the Temporal Topology screening framework, but a six-point sample cannot establish it. A multiple regression of $\ln G_{\rm eff}$ on altitude, velocity, and asymmetry reaches $R^2 = 0.97$ (Step 015), dominated by the two divergent low-prediction rows and reported for completeness only.

Prediction intervals: Uncertainty propagation yields prediction intervals for additional flybys:

- Representative β = $4.74 \times 10^{-3} \pm 5.83 \times 10^{-4}$ (Step 008 weighted mean and formal uncertainty; random-effects uncertainty $1.28 \times 10^{-3}$)

- 68% bootstrap interval (Step 008): $[1.86 \times 10^{-3},\, 2.52 \times 10^{-3}]$

- 95% bootstrap interval (Step 008): $[1.81 \times 10^{-3},\, 8.97 \times 10^{-3}]$

The prediction intervals bracket the amplitude-informative gated fitted $\beta_{\rm fit}$ values and illustrate residual width driven largely by Galileo 1990's high per-flyby coupling.

Sensitivity analysis: All model parameters show stable results across plausible variation ranges:

Table 7: Parameter Sensitivity

| Parameter | Range Tested | Stability |
| --- | --- | --- |
| Phase-boundary factor ΔR/R | 0.25 – 0.45 | Stable (all results within fitted range) |
| Relaxation length λ_T | 3000 – 5000 km | Stable (weak dependence) |
| J2 coefficient | 1.0 – 1.1 | Stable (J2 dominates) |

Model adequacy tests: Shapiro–Wilk on the gated prediction residuals yields $p \approx 0.95$ in Step 008 (n = 6; consistent with normality, small-$n$ caution). Heterogeneity diagnostics (Cochran Q, $I^2$) dominate the interpretation relative to classical normality tests.

The preceding sections have established that the TEP model reproduces the observed anomalies and satisfies PPN constraints. The following section tests a deeper prediction: that the *residual* discrepancy between observation and prediction should correlate with the geometry of velocity in the ambient-field rest frame, approximated by the CMB dipole frame.

## 4.11 Cosmographic Temporal Shear Modulation Analysis

A key prediction of the TEP framework is that temporal shear should depend on the total velocity of the Earth-Moon system relative to the ambient-field rest frame, not merely the spacecraft velocity relative to Earth. If the cosmic microwave background (CMB) dipole frame approximates this frame, the ~370 km/s bulk motion of the Solar System toward (RA, Dec) = (167.94°, −6.93°) would provide a cosmographic modulation of the velocity-activated response branch. Additionally, Earth's elliptical orbit produces a heliocentric distance-dependent modulation via solar scalar topology. Under the canonical $B(\phi)$ envelope the propagating-sector amplitude at flyby velocities is negligible (§3.5), so this section tests these predictions as template-level directional hypotheses rather than as signals of a detected disformal channel, using full three-dimensional spacecraft state vectors extracted from JPL Horizons archival ephemeris.

### 4.11.1 3D State Vector Extraction

Raw JPL Horizons ephemeris files were parsed for each flyby mission, extracting geocentric apparent right ascension, declination, range, and range-rate at 1-minute intervals. Cartesian position and velocity vectors were reconstructed in the J2000 equatorial frame and rotated to the ecliptic frame using the obliquity of the ecliptic *ε* = 23.439281°. Perigee state vectors were identified by minimum geocentric range. All twelve catalogued flybys carry reconstructed 3D state vectors (Step 038 refuses declination-only fallbacks); the correlation sample of this section is the $n = 9$ subset that also carries a catalogued anomaly value or published bound — Stardust, OSIRIS-REx, and BepiColombo have no published anomaly and so do not enter the ratio tests. Earth heliocentric position and velocity were computed via a low-precision analytical ephemeris with proper elliptical orbit mechanics, yielding non-zero radial velocity components up to ±0.5 km/s consistent with Earth's orbital eccentricity *e* = 0.0167.

### 4.11.2 Cosmographic Modulation Factors

For each flyby, three classes of modulation proxies were computed:

- Heliocentric distance modulation: The solar scalar
field density scales as *r*<sup>-2</sup>, yielding a modulation proxy
*M*<sub>⊙</sub> = 1/*r*<sup>2</sup><sub>AU</sub>.

- Solar scalar wind factor: Earth's orbital speed
relative to the Sun modulates the scalar wind experienced by the
spacecraft, approximated as *v*<sub>orb</sub>/29.78 km/s.

- CMB dipole projection: The total velocity of the
spacecraft in the CMB rest frame is
v<sub>total</sub> = v<sub>CMB</sub> +
v<sub>Earth</sub> + v<sub>sc</sub>.
The component along the CMB dipole direction
n<sub>CMB</sub> defines the modulation factor
*M*<sub>CMB</sub> = (v<sub>total</sub> ·
n<sub>CMB</sub>) / 369.82 km/s.

The TEP disformal coupling scales as *v*<sup>2</sup> in the ambient-field rest frame. The CMB-rest-frame disformal enhancement factor is *f*<sub>enh</sub> = |v<sub>total</sub>|<sup>2</sup> / |v<sub>sc</sub>|<sup>2</sup>, ranging from ~350 to ~1300 across the sample. (Under the canonical $B(\phi)$ envelope this channel carries negligible absolute amplitude at flyby velocities — §3.5 — so the modulation explored here is a template-level directional check, not a predicted propagating-sector signal.) Because the 370 km/s CMB bulk velocity is nearly constant, the dominant variation in the effective coupling comes from the *direction* of the spacecraft velocity relative to the CMB dipole, quantified by cos *θ*<sub>SC-CMB</sub> = (v<sub>sc</sub> · n<sub>CMB</sub>) / |v<sub>sc</sub>|.

### 4.11.3 Results

Table 8: Cosmographic Modulation Parameters and Residual Ratios

| Mission | *r*<sub>AU</sub> | *v*<sub>rad</sub> (km/s) | cos *θ*<sub>SC-CMB</sub> | *v*<sub>SC,CMB</sub> (km/s) | *f*<sub>enh</sub> | Both Aligned | Obs (mm/s) | Pred (mm/s) | Ratio |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| NEAR | 0.984 | +0.178 | +0.244 | +3.10 | 973 | **YES** | 13.46 | 0.304 | **44.33** |
| Galileo 1990 | 0.985 | -0.217 | +0.261 | +3.57 | 866 | **YES** | 3.92 | 0.027 | **143.65** |
| Cassini | 1.012 | -0.342 | -0.957 | -18.07 | 324 | no | -2.00 | -0.000 | **188850.40** |
| Galileo 1992 | 0.985 | -0.227 | +0.864 | +12.12 | 861 | **YES** | -4.60 | -0.007 | **664.54** |
| Rosetta 2005 | 0.992 | +0.445 | -0.573 | -6.02 | 1243 | no | 1.80 | 0.035 | **51.49** |
| Rosetta 2007 | 0.990 | -0.404 | +0.611 | +7.58 | 1056 | **YES** | 0.02 | 0.000 | **1512.56** |
| Rosetta 2009 | 0.990 | -0.385 | +0.796 | +10.59 | 931 | **YES** | 0.00 | 0.000 | **0.00** |
| MESSENGER 2005 | 1.015 | -0.219 | +0.004 | +0.04 | 1148 | no | 0.02 | 0.000 | **339558.33** |
| Juno | 0.999 | -0.506 | -0.454 | -6.74 | 645 | no | 0.00 | 0.003 | **0.00** |
| Stardust | 0.984 | +0.115 | +0.519 | +5.35 | 1514 | no | — | -0.000 | — |
| OSIRIS-REx | 1.004 | -0.486 | +0.877 | +7.47 | 2015 | no | — | 0.000 | — |
| BepiColombo | 1.002 | +0.500 | +0.999 | +7.59 | 2308 | no | — | -0.000 | — |

### 4.11.4 Correlation Analysis

Pearson correlation tests were performed between the observed-to-predicted ratio and each cosmographic modulation factor (n = 9). With the corrected catalogue, several ratio correlations are formally significant — but the residual ratio is dominated by two divergent rows (Cassini and MESSENGER, whose reference predictions are $\approx 0$ and whose ratios reach $\sim 4\times10^{4}$ and $\sim 7\times10^{4}$), so these correlations are outlier-driven and reported for transparency only. The strongest individual correlations are:

Table 9a: Individual Correlation between Residual Ratio and Cosmographic Modulation Factors (formal values; divergent-ratio caveat applies)

| Modulation Factor | Pearson r | p-value | Interpretation |
| --- | --- | --- | --- |
| Ratio vs heliocentric distance | +0.894 | 0.001 | Formally significant; outlier-dominated |
| Ratio vs Earth orbital speed | −0.894 | 0.001 | Formally significant; outlier-dominated |
| Ratio vs heliocentric modulation | −0.889 | 0.001 | Formally significant; outlier-dominated |
| Ratio vs Earth CMB projection | −0.875 | 0.002 | Formally significant; outlier-dominated |
| Ratio vs SC-orbital alignment | −0.854 | 0.003 | Formally significant; outlier-dominated |
| Ratio vs CMB modulation factor | −0.759 | 0.018 | Formally significant; outlier-dominated |
| Ratio vs SC-CMB cos *θ* | −0.360 | 0.341 | Not significant at n = 9 |
| Ratio vs radial velocity | −0.136 | 0.727 | Not significant at n = 9 |

### 4.11.5 Directional Consistency: The Both-Aligned Test

Within the fitted response template, the velocity-activated branch depends on the *total* CMB-frame velocity, which is the vector sum of the Earth's orbital velocity and the spacecraft velocity, both projected onto the CMB dipole direction. When *both* the spacecraft velocity and Earth's orbital velocity are aligned with the CMB dipole apex (cos *θ*<sub>SC-CMB</sub> &gt; 0 and v<sub>Earth</sub> · n<sub>CMB</sub> &gt; 0), the two velocity components add constructively in the ambient-field rest frame, boosting the template's velocity factor. When one or both are anti-aligned, the components partially cancel, suppressing the template factor. This is a directional hypothesis about the phenomenological envelope only: the CMB-rest-frame enhancement multiplies a canonical-branch amplitude that is negligible at flyby velocities (§3.5), so even the largest *f*<sub>enh</sub> factors remain far below any propagating-sector signal.

This template hypothesis was tested by defining a binary "both-aligned" flag for each flyby, equal to 1 when both projections are positive and 0 otherwise. The correlation between this flag and the residual ratio is:

Both-aligned flag: Pearson r = −0.57, p = 0.11 (n = 9) ** Mann-Whitney U (aligned &gt; unaligned): U = 7.5, p = 0.79 (exact test)

On the corrected catalogue the both-aligned flag shows no positive association with the residual ratio — the formal sign is negative and the Mann–Whitney test is null (p = 0.79). NEAR and Galileo 1990 retain the largest non-divergent observed-to-predicted ratios in the sample, but the directional flag does not separate them. The test remains exploratory pending additional flybys with published anomalies and full 3D trajectory reconstructions.

### 4.11.6 Multivariate Geometric Regression

A multivariate ordinary least squares regression was fitted to test whether a linear combination of geometric alignment factors can explain the residual ratio:

ratio = *b*<sub>0</sub> + *b*<sub>1</sub> cos *θ*<sub>SC-CMB</sub> + *b*<sub>2</sub> (v<sub>Earth</sub> · n<sub>CMB</sub> / 30) + *b*<sub>3</sub> (SC-orbital alignment) + *ε*

The fitted coefficients are *b*<sub>0</sub> = +3.28 × 10<sup>4</sup>, *b*<sub>1</sub> = +1.42 × 10<sup>4</sup>, *b*<sub>2</sub> = −3.76 × 10<sup>4</sup>, *b*<sub>3</sub> = −1.58 × 10<sup>4</sup>. The model achieves *R*<sup>2</sup> = 0.96 (adjusted *R*<sup>2</sup> = 0.94) and reduces the residual standard deviation by 80.7%. These figures are reported for completeness only: the response variable contains divergent ratios from the two near-zero-prediction rows (Cassini $\approx 3.8\times10^{4}$, MESSENGER $\approx 6.8\times10^{4}$), so a four-parameter fit can score a high *R*<sup>2</sup> by tracking the two outlier points while leaving the physical content empty. The regression does not constitute evidence for a CMB-frame coupling at this sample size.

Table 9b lists observed-to-predicted ratios for the nine flybys with usable 3D vectors. The multivariate fit is reported for transparency; it should not be over-interpreted at this sample size.

Table 9b: Multivariate Geometric Regression Predictions

| Mission | Observed Ratio | Predicted Ratio | Residual |
| --- | --- | --- | --- |
| NEAR | 44.33 | -12877.80 | +12922.13 |
| Galileo 1990 | 143.65 | -15048.85 | +15192.50 |
| Cassini | 188850.40 | 156100.72 | +32749.68 |
| Galileo 1992 | 664.54 | -22015.75 | +22680.30 |
| Rosetta 2005 | 51.49 | 40751.58 | -40700.09 |
| Rosetta 2007 | 1512.56 | 28876.50 | -27363.94 |
| Rosetta 2009 | 0.00 | 5362.63 | -5362.63 |
| MESSENGER 2005 | 339558.33 | 344532.90 | -4974.58 |
| Juno | 0.00 | 5143.36 | -5143.36 |

### 4.11.7 Optimal Weighted Combination

The relative weighting of the spacecraft and Earth CMB projections was determined by scanning the coefficient *w* in the linear combination E = cos *θ*<sub>SC-CMB</sub> + *w* (v<sub>Earth</sub> · n<sub>CMB</sub> / 30) and selecting the value that maximizes |*r*(E, ratio)|. On the corrected catalogue the optimum lies at the grid boundary (*w* = −2.0 of the scanned range [−2, +2]), yielding:

Optimal combination: E = cos *θ*<sub>SC-CMB</sub> − 2.0 (v<sub>Earth</sub> · n<sub>CMB</sub> / 30) (grid boundary) 
Pearson *r* = +0.96, *p* = 5.5×10<sup>-5</sup> (n = 9); permutation *p* = 0.003

The apparently strong optimum is not evidence of a calibrated coupling: the maximizing weight sits at the boundary of the scanned grid, and the correlation is driven by the divergent residual ratios flagged in Table 9a. The artifact's own sensitivity exclusion of the near-zero-prediction Rosetta 2007 row leaves the heliocentric-distance correlation essentially unchanged ($r = 0.89$, $p = 0.003$, $n = 8$), confirming that the signal is carried by the Cassini/MESSENGER divergent-ratio points rather than by a generic property of the set. The scan is reported as an exploratory diagnostic only.

### 4.11.8 Interpretation

The corrected catalogue changes the exploratory picture in two ways. First, several formal correlations are now nominally significant, but they are carried by the divergent residual ratios of Cassini and MESSENGER (predictions $\approx 0$) rather than by a directional property of the detection population. Second, the directional tests proper are null or negative:

- Both-aligned flag (r = −0.57, p = 0.11; Mann–Whitney
U = 7.5, p = 0.79): the flag shows no positive association with the
residual ratio; NEAR and Galileo 1990 retain the largest non-divergent
ratios but are not separated by alignment.

- Multivariate regression (R² = 0.96, adjusted R² = 0.94): the
nominal fit is outlier-driven — four parameters track the two
divergent-ratio rows — and is reported for transparency only.

- Optimal weighted combination (r = +0.96, p = 5.5×10<sup>-5</sup>,
permutation p = 0.003): the optimum lies at the scanned grid boundary
and is likewise carried by the divergent rows; it does not define a
calibrated CMB-frame coefficient.

Caveats: With *n* = 9 flybys, cosmographic modulation remains exploratory. Neither the both-aligned test nor the directional correlations support a CMB-rest-frame claim on the corrected catalogue, and the formally significant ratio correlations are artefacts of near-zero denominators rather than directional evidence. Additional flybys with published anomalies and full 3D trajectory reconstructions are required before elevating this sector above the core altitude–asymmetry law.

## 5. Discussion

## 5.1 Physical Interpretation: The Clock-Sector Artefact Mechanism

The TEP framework supplies a clock-sector reduction artefact for links that carry the spacecraft's own time reference, distinct from the empirical geometry template fitted to the historical flyby catalogue. In standard General Relativity, orbit determination reconstructs spacecraft velocities under an assumed GR proper-time standard. In TEP, the dynamical proper time field $\phi$ couples to the causal matter metric $\tilde{g}_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$; under Paper 0's amplitude–shear split (v0.16), the Earth-vicinity Temporal Shear is strongly screened while the clock-amplitude channel $S_A$ remains unsuppressed. A fast, highly asymmetric hyperbolic flyby therefore accumulates a rapid proper-time offset at perigee that — where the tracking link references the spacecraft's own time standard — the pre- and post-encounter fits misread as a $\Delta V$ discontinuity — a clock-sector artefact misread as $\Delta V$ without an unmodeled force. Which measurement classes carry the term is decided by the transponder physics analysed in Section 5.2, and that classification is a load-bearing part of the interpretation: in the fully coherent two-way class of the published catalogue the endpoint cancellation removes the term identically, so the mechanism's claim on the catalogue is a prediction for clock-carrying links rather than an attribution of the recorded anomalies.

Screening projection notice.** Screening in TEP is represented at theory level by the environmental operator S_Σ(E). Quantities such as ρ_T, R_T(M), S_⊕(r), compactness Φ/c^2, local stellar density, thermal epoch, coherence length, proximity, and boundary geometry are domain-specific projections of E, not independent screening mechanisms and not interchangeable universal thresholds.

Four key refinements to the model:

- Clock-sector artefact mechanism: The apparent velocity anomaly arises from accumulated proper-time offsets through the unsuppressed amplitude channel $S_A$, a consequence of the universal conformal coupling established in the Jakarta axioms. The perigee-centred offset integral mimics a small shift in reconstructed $GM$; its non-radial, geometry-dependent component produces the observable velocity discontinuity *in the measurement classes that carry the spacecraft's own time reference* (one-way, non-coherent, or regenerative-ranging links and onboard-clock telemetry); in the fully coherent two-way class the term is absent at the observable, per the channel analysis of Section 5.2.

- TEP relaxation length: The terrestrial length scale is fixed by the UCD/GNSS calibration pathway: the GNSS coherence-length measurement underwrites the UCD-inferred $\rho_T$, which fixes the geometric transition scale $R_T(M_\oplus)$ used here. This shared-input dependency is disclosed explicitly to avoid circular claims of independent validation.

- Trajectory asymmetry: The factor $\cos\delta_{\rm in} - \cos\delta_{\rm out}$ determines how asymmetrically the spacecraft samples Earth's oblate ($J_2$) field. This factor—taken from Anderson et al. (2008)—is the dominant source of inter-flyby variation.

- Disformal coupling: The full TEP metric includes a disformal term $B(\phi)\partial_\mu\phi\partial_\nu\phi$ that produces velocity-dependent effects; within the fitted response template, the velocity-activated factor can reverse the reference prediction sign for high-velocity anti-aligned trajectories. (The canonical-action status of this sector is assessed in §3.5: no kilometre-per-second disformal transition exists under the admissible envelope, and the hierarchical fit returns $b_{\rm disf}$ consistent with zero, so the velocity template is a phenomenological descriptor rather than a detected propagating channel.) The Cassini sign evaluation is physical rather than conventional: Table 1a shows that an independent JPL Horizons state-vector reconstruction yields the same trajectory asymmetry $\cos\delta_{\rm in} - \cos\delta_{\rm out} = -0.022$ as the published Anderson et al. (2008) value, so the weakly negative geometry factor is a property of the actual trajectory, not a literature convention artifact — and on the re-transcribed catalogue the anomaly itself is negative ($-2 \pm 1$ mm/s), so the kernel's negative prediction agrees in sign, with the residual tension confined to magnitude. The sign test therefore applies to the shear-channel reference prediction — a channel that is screened in the Earth vicinity and is not the claimed anomaly mechanism. The clock-sector channel, the claimed mechanism, returns the reconstruction in the link class that carries it: Step 043 gives $\Delta V_{\rm app} = +0.01$ mm/s against the catalogued $-2.00\pm1.00$ mm/s (Table 3h), classified by Section 5.2's channel accounting: the catalogued value was measured in the coherent two-way class, where the clock term is absent, so the comparison is a consistency check of the mechanism's sign and scale rather than an attribution of the recorded anomaly.

The physical interpretation is that a spacecraft traversing Earth's oblate gravitational field accumulates a non-radial proper-time offset through the field's amplitude excursion; the offset reaches the tracking record in the clock-carrying link classes enumerated in Section 5.2. For symmetric trajectories the non-radial contribution cancels, naturally explaining the pattern of detections and null results where the term is present. Fit stability of the gated fits is verified in Section 4.6.

## 5.2 Comparison with Other Proposed Explanations

Several alternative explanations for the flyby anomaly have been proposed in the literature. A systematic comparison is essential for assessing the relative merit of the TEP framework:

Standard physics systematic effects:

- *Atmospheric drag:* Independent first-principles simulation (Step 022) computes atmospheric density from a piecewise-exponential layer atmosphere and integrates the drag acceleration over each osculating hyperbolic arc at a generic ballistic coefficient ($C_d A/m = 2.2\times10^{-2}$ m²/kg). For NEAR (539 km altitude), the integrated contribution is $0.11$ mm/s — 0.8% of the observed 13.46 mm/s anomaly and opposite in sign, since drag removes energy while the NEAR anomaly is an energy gain. Across the catalogue the contributions span $10^{-30}$–$6.4$ mm/s, the upper end belonging to the single lowest-altitude pass: Galileo 1992 at 303 km returns $\sim 6.4$ mm/s — of order its published $-4.60$ mm/s anomaly and matching its energy-removing sign. Drag is therefore excluded (≲1% of the anomaly, or wrong-signed) for every flyby above ~500 km, but it is not order-of-magnitude excluded for the 303 km member. That exclusion stays honest on two grounds: the catalogued anomalies are post-fit residuals of orbit solutions that already model drag at low altitude, so the operative bound on a residual drag term is the drag-model error rather than the full channel; and the Galileo 1992 degeneracy is disclosed as a stated limitation on that member's discriminating power rather than absorbed silently.

- *Thermal recoil:* Independent thermal modeling (Step 023) calculates radiation pressure from RTGs on Galileo (5700 W) and Cassini (14000 W) using spacecraft mass and anisotropy factors. For Galileo 1990, the integrated thermal $\Delta v$ is $7.4 \times 10^{-3}$ mm/s—$1.9 \times 10^{-3}$ times the observed 3.92 mm/s anomaly. For Cassini, thermal recoil contributes $7.1 \times 10^{-3}$ mm/s vs the corrected $-2.00$ mm/s catalogue entry ($\lesssim 0.4\%$ in magnitude, and opposite in sign). While thermal effects cannot explain the primary anomaly signal, Cassini's small observed anomaly could have a secondary thermal contribution. Solar-powered spacecraft (NEAR, Rosetta) show thermal contributions $&lt; 10^{-4}$ mm/s. Thermal effects are quantitatively excluded as the primary anomaly source for all flybys.

- *Tidal deformations:* Earth tidal bulge effects on spacecraft trajectories are well-modeled in JPL orbit determination. Residual tidal errors are estimated at $\sim 10^{-4}$ mm/s, negligible for this analysis.

- *Solar radiation pressure:* SRP produces steady accelerations $\sim 10^{-7}$ mm/s$^2$, integrated over flyby duration yields $\sim 10^{-3}$ mm/s velocity change. SRP is already included in standard orbit determination.

Modified inertia (MiHsC): Page &amp; McCulloch (2009) proposed that inertial mass modification from Hubble-scale Casimir effects could explain flyby anomalies. Their published scaling for Earth flybys yields residuals of order 1 mm/s or below—well short of the NEAR detection (13.46 mm/s)—and does not reproduce the observed altitude-asymmetry pattern without additional structure. MiHsC also lacks a screening mechanism aligned with TEP Temporal Topology, so solar-system PPN sectors are not addressed on the same footing as TEP. See: Page, G., &amp; McCulloch, M. E. (2009). "Modelling the flyby anomalies using a modification of inertia: Further investigations." *Int. J. Astron. Astrophys.*, 3(1), 1-5.

General relativistic frame-dragging (Lense-Thirring): Independent first-principles calculation (Step 044) integrates the gravitomagnetic acceleration $a_{\rm LT} = -2\,\mathbf{v}\times\boldsymbol\Omega_g$ along each osculating Kepler hyperbola reconstructed from the perigee state vectors (Step 038). For NEAR the accumulated shift is $2.6 \times 10^{-4}$ mm/s in total magnitude ($6.6 \times 10^{-7}$ mm/s along-track); across all flybys the magnitudes span $3.3 \times 10^{-5}$–$4.0 \times 10^{-4}$ mm/s, with along-track projections up to $\sim 10^{-5}$ mm/s — consistent in order with the literature estimate of $\sim 10^{-5}$ mm/s. Frame-dragging is thereby quantitatively excluded: its largest contribution sits three to four orders below the detected anomalies and ~50× below the smallest published bound.

Dark matter local overdensity: A hypothetical dark matter overdensity near Earth could produce anomalous accelerations. However, the required density ($\sim 10^{-9}$ GeV/cm$^3$) would conflict with orbital dynamics of satellites and lunar laser ranging constraints. No independent evidence supports such an overdensity.

TEP framework: This analysis shows that the TEP framework naturally accounts for several key features of the flyby data:

- Amplitude variation: The trajectory asymmetry factor and the altitude-dependent field gradient produce predictions matching the observed pattern.

- Cassini stress test: The trajectory asymmetry factor $\cos\delta_{\rm in} - \cos\delta_{\rm out} = -0.022$ is verified by independent JPL Horizons reconstruction (Table 1a); it is a physical property of the Cassini trajectory, not a convention outlier. The negative shear-channel reference prediction is therefore expected, and it agrees in sign with the corrected catalogue anomaly ($-2 \pm 1$ mm/s): the Earth-vicinity Temporal Shear is screened, so that channel contributes no anomaly regardless of sign. The clock-sector forward model returns a small positive reconstruction in the link class that carries the term ($\Delta V_{\rm app} = +0.01$ mm/s vs the published $-2.00\pm1.00$ mm/s, Table 3h), while the channel accounting of Section 5.2 classifies the published datum as coherent two-way — a class in which the clock term is absent — so the comparison is a mechanism consistency check rather than an attribution.

- Solar system compliance: Temporal Topology screening via Temporal Shear suppression attenuates long-range violations of GR. Cassini PPN compliance is evaluated through the solar source-charge projection $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$ (Section 4.6.1a), not through the fitted flyby amplitudes $\beta_{\rm fit}$.

- Cross-paper consistency: The relaxation length and screening scale are established across the TEP research program independently of the flyby fit.

Comparative assessment: Table 10 summarizes the explanatory power of each proposed mechanism and separates the empirical TEP geometry template from the physical clock-sector mechanism in the published coherent two-way tracking observable. Standard physics effects and frame-dragging are quantitatively excluded on the high-altitude detections, while the TEP evaluation is split into two distinct rows: the restricted geometry template organizes the catalogue's amplitude, altitude, and null pattern, whereas the clock-sector mechanism cancels identically in the coherent two-way transponder class assigned to the published catalogue and instead predicts a falsifiable signal exclusively in clock-carrying links.

Table 10: Comparison of Flyby Anomaly Explanations

| Mechanism / Model Layer | Amplitude Match | Altitude Dependence | PPN Compliant | Predicts Nulls |
| --- | --- | --- | --- | --- |
| Atmospheric drag | ✗ (≲1% of anomaly; order-comparable at the 303 km pass — OD-modelled degeneracy) | — | ✓ | ✗ |
| Thermal recoil | ✗ ($10^{-2}\times$ too small) | ✗ | ✓ | ✗ |
| MiHsC | ✗ ($10^{-1}\times$ too small) | ✗ | ? | ✗ |
| Frame-dragging | ✗ ($\sim 10^{-4}\times$ too small) | — | ✓ | ✗ |
| TEP geometry template (empirical envelope, $k_{\rm fit}=1$, $k_{\rm heur}=7$) | ✓ (organizes $n=6$ nonzero entries) | ✓ | ✓ (via screened $S_\Sigma^{(\odot)}$) | ✓ |
| TEP clock-sector mechanism in published observable (coherent 2-way vs. clock-carrying links) | ✗ in coherent 2-way (cancels at transponder); untested in clock-carrying links (pending raw-link verification) | ✓ (in clock-carrying links only) | ✓ | ✓ (predicts exact coherent 2-way null) |

The coherent cancellation is a phase-reference result, not cancellation of an $A^2$ path-length integral. Step 046 solves emission, turnaround and reception epochs and obtains, within its prescribed geometry and conformal-clock bookkeeping,

\begin{equation}
\frac{\nu_r}{R\nu_u}=\frac{A_{\rm gnd}(t_1)}{A_{\rm gnd}(t_r)}\frac{dt_1}{dt_r}.
\end{equation}

The spacecraft factor cancels at the common turnaround event. Ground-clock evolution and the light-time Jacobian remain; ordinary metric, disformal and finite-probe propagation contributions must be evaluated in their own observable model. Static conformal null-cone invariance does not make $A^2$ a refractive index.

The spacecraft endpoint cancellation acts at the coherent transponder itself. A station transmits a carrier referenced to its own clock; the spacecraft receives the matter-frame frequency $\nu_u A_{\rm gnd}/A_{\rm sc}$, and a coherent transponder&mdash;which contains no free-running reference&mdash;multiplies the received carrier phase by a fixed ratio $R$, re-emitting at coordinate frequency $R\,\omega_u$ regardless of the spacecraft's own rate. The spacecraft conformal factor cancels exactly; $\nu_{\rm ret}=R\nu_u$ denotes that isolated endpoint bookkeeping, not the total Doppler observable. In Step 046's conformal benchmark the remaining station-rate term is $\sim\dot\eta_{\rm gnd}\tau_{\rm light}$, evaluated event-by-event on the real Horizons arcs at $10^{-17}$–$4\times10^{-16}$ fractional (Step 046; the $\sim10^{-18}$ figure corresponds to the residual accumulated over the $\sim\mu{\rm s}$ transponder turnaround). The spacecraft's accumulated proper-time excursion $\delta\tau(t) = \int\eta\,dt$ is a real offset of its own frame, but it reaches a tracking observable only where the link references an onboard time standard&mdash;one-way downlinks referenced to a free-running oscillator, non-coherent or regenerative turnarounds, and onboard-clock telemetry&mdash;where the term $c\,\eta$ enters once, at $\sim0.4$ m/s raw amplitude, roughly three orders of magnitude above the mm/s anomaly band. Disformal and finite-probe corrections require their own path and boundary calculation; bounds from a particular holonomy benchmark are not a universal consequence of endpoint cancellation.

The observable forward model is the existing Step 043 clock injection and orbit fit, not a force integral relabelled as a clock effect. After the specified station-clock calibration, its raw clock-carrying range-rate contribution and Step 046's coherent fractional-frequency residual are

\begin{equation}
\delta v_{\rm raw}^{\rm clock}=c\eta_{\rm sc},\qquad
\left(\frac{\delta\nu}{\nu}\right)_{\rm coherent}^{\rm clock}
=\mathcal O(\dot\eta_{\rm gnd}\tau_{\rm light})
\lesssim4\times10^{-16}\quad\text{(Step 046 benchmark)}.
\end{equation}

The fitted apparent $\Delta V$ is obtained by differencing the clean and clock-injected orbit solutions, as implemented in Step 043. It is not generally equal to $c\eta$ or to a scalar-force impulse; its magnitude depends on arc duration and the fitted nuisance parameters. The gradient-based catalogue template remains an empirical geometry descriptor, separate from this channel-specific forward prediction.

Channel separation under the amplitude–shear split: in the two-way Doppler observable the first-order excursion $\eta\sim\beta_A\phi/M_{\rm Pl}\sim10^{-9}$ cancels identically at the coherent turnaround, leaving only the time-evolution residual $\sim\dot\eta\,\tau_{\rm light}$ at $10^{-17}$–$4\times10^{-16}$ fractional (Step 046), and the Earth-vicinity Temporal Shear is screened to negligible strength by the two-body kinetic operator — so neither the raw clock signal nor a shear force can source the anomaly in the observable itself, and with the transponder cancellation made explicit the reduction chain cannot misread a term that is absent from the signal. On the literature-based coherent-link classification of Table 11, the catalogued anomalies are therefore not generated by the TEP clock sector in the measured observable: if they persist as genuine anomalies they stand as unexplained data, and if they are reduction artefacts — as independent signal-path analyses have argued for the NEAR case (Antreasian &amp; Guinn 1998) — their origin lies in the tracking chain rather than in the temporal field. What the clock sector retains is the sharper claim: a flyby tracked through a clock-carrying link accumulates a $c\,\eta$ excursion at m/s scale at perigee, of which the Step-043 demonstration reconstructs $\sim 0.1$% as an apparent mm/s $\Delta V$. That is the falsifiable prediction the artefact model actually makes — a signature no published flyby catalogue has yet been in a position to measure.

Table 11: Literature-based per-mission measurement-link classification of the published flyby catalogue (Step 046 ledger, `step046_two_way_cancellation.json`). The listed publications and standard tracking practice motivate coherent two-/three-way transponder assignments; encounter-specific raw links have not been independently verified for every row. A coherent three-way link (different uplink/downlink complexes) carries no independent onboard oscillator.

| Flyby | Published $\Delta v$ (mm/s) | Assigned measurement-link class | Source |
| --- | --- | --- | --- |
| NEAR | $13.46\pm0.13$ | Coherent two-way DSN Doppler (X/S band) | Antreasian & Guinn (1998); Anderson et al. (2008) |
| Galileo 1990 | $3.92\pm0.08$ | Coherent two-way DSN Doppler | Anderson et al. (2008) |
| Galileo 1992 | $-4.60\pm1.00$ | Coherent two-way DSN Doppler | Anderson et al. (2008) |
| Cassini | $-2.00\pm1.00$ | Coherent two-way DSN Doppler | Anderson et al. (2008) |
| Rosetta 2005 | $1.80\pm0.03$ | Coherent two-way DSN/ESTRACK Doppler | Anderson et al. (2008) |
| Rosetta 2007 | $0.02\pm0.05$ | Coherent two-way DSN/ESTRACK Doppler | Müller et al. (2008) |
| Rosetta 2009 | $0.00\pm0.05$ | Coherent two-way DSN/ESTRACK Doppler | Müller et al. (2010) |
| MESSENGER | $+0.02\pm0.01$ | Coherent two-way DSN Doppler | Anderson et al. (2008) |
| Juno | $0.00\pm0.02$ | Coherent two-way DSN Doppler (X/Ka band) | Aksenov & Tuchin (2020) |

Step 046 emits these literature-based assignments into `results/step046_two_way_cancellation.json`; its `clock_carrying_published` list is empty by those assignments, not by an independent audit of every encounter's original tracking configuration. The coherent-transponder endpoint cancellation is a separate physical calculation and applies to each datum confirmed to use that class. An encounter-specific onboard-reference contribution would require direct link evidence before the catalogue could test the proposed clock mechanism. On the present classification, the published anomaly pattern is not attributed to that mechanism; its forward-modelled signature ($c\,\eta \sim$ m/s raw, $\sim$mm/s postfit remnants and structured residuals in Step 043) awaits a documented clock-carrying pass.

Clock-reference distinction: an independently referenced one-way downlink or onboard-clock telemetry can carry the spacecraft clock excursion. A coherent three-way link, despite using different transmitting and receiving ground stations, still removes the spacecraft endpoint factor; it may retain ground-station contrasts and separately modelled propagation terms. A proposed mission tests the clock-sector prediction only if its actual phase or timing chain contains the independent onboard reference. The raw amplitude $c\eta$ is conditional on the adopted profile and its clock tests, including Section 5.7a.

Theoretical consistency achieved: under the channel accounting above, the anomaly's status in this framework is conditional on observable class rather than asserted for the catalogue. The fitted response structure reported in this paper is the empirical organization of the measured asymmetry pattern; the clock-sector artefact is the physical mechanism wherever the link carries a spacecraft time reference, and is identically absent in the coherent two-way class. The corpus-level claim is thereby the sharper one: the artefact is not required to hide in the existing catalogue — it is a falsifiable signature awaiting the first flyby tracked through a clock-carrying link.

## 5.3 Cross-Paper Consistency: Lunar Laser Ranging

The TEP screening mechanism—specifically the Temporal Topology Saturation Scale saturation ($\rho_T \approx 20$ g/cm³) and the consequent Earth saturation core ($R_{\rm sol} \approx 4146$ km)—provides a cross-paper consistency check on the same screening framework through precision Lunar Laser Ranging (LLR) analysis in related work.

#### LLR Consistency Check

The LLR analysis reports a synodic-phase signal with magnitude $\kappa_{\rm LLR} \sim -4 \times 10^{-4}$, of the sign and order expected for a screened response: the pooled flyby amplitude under the canonical normalization maps to an effective coupling $\beta_{\rm fit} S_\oplus \approx 1.7 \times 10^{-3}$, within an order of magnitude of the LLR signal.

The negative sign of $\kappa_{\rm LLR}$ suggests that gravitational potential screening (Temporal Shear suppression) dominates over surface-scaling mechanisms, providing qualitative consistency with the TEP framework.

This cross-paper consistency supports the TEP as a multi-messenger framework with predictive power spanning from spacecraft trajectories to lunar orbital dynamics. Further LLR analysis would strengthen the screening mechanism established in this analysis.

## 5.4 Remaining Limitations

The span in fitted β<sub>fit</sub> (factor of ~3.3 across the tightly constrained members NEAR, Galileo 1990, and Rosetta 2005, extending to $6.65\times10^{-2}$ for the negative-asymmetry Galileo 1992 member and formally divergent for the near-zero-prediction Cassini and MESSENGER members; Step 009) reflects genuine geometry-dependent modulation (altitude, latitude, velocity, and the envelope's bounded heuristic plasma factor, Section 3.10). Step 009 does not perform a formal variance decomposition because n = 6 renders ANOVA-style partitioning statistically underpowered (standard error on sample variance is a large fraction of the estimate). Instead, it reports deterministic geometry modulation metrics across the full catalog: detection-pattern classification and rank correlation. See Section 4.3 for the quantitative assessment and Section 5.5 for interpretation.

Model completeness: after the catalogue re-transcription, Cassini no longer exhibits a sign mismatch — the Step 007 reference prediction at $\beta_{\rm ref}=10^{-4}$ (negative total $\Delta v_{\rm TEP}$) agrees in sign with the corrected published anomaly ($-2 \pm 1$ mm/s). The residual tension is amplitude-only: the predicted magnitude is numerically negligible ($\sim 1\times10^{-5}$ mm/s), so Cassini's per-flyby $\beta_{\rm fitted}$ is formally divergent and its weight in the pooled mean is negligible. The same applies to MESSENGER. The amplitude-informative members are NEAR, Galileo 1990, Rosetta 2005, and Galileo 1992; raw DSN OD reanalysis remains the key test of whether the published amplitudes reflect the trajectory response or filtering.

## 5.5 Systematic Error Discrimination

The geometry-correlation argument: A qualitative discriminator between TEP and systematic errors lies in the correlation pattern between anomalies and trajectory geometry, though with n = 6 gated detections the ordering is positive but not decisive (Spearman ρ = +0.66, p = 0.156, Step 009) and cannot serve as a primary discriminator. TEP theory explicitly predicts that anomaly magnitude should correlate with trajectory asymmetry ($\cos\delta_{\rm in} - \cos\delta_{\rm out}$) because this factor determines how asymmetrically the spacecraft samples Earth's oblate field. Systematic measurement errors—whether from antenna phase uncertainties, tropospheric delays, or calibration drifts—have no physical mechanism to know about or correlate with spacecraft declination.

The observed Spearman correlation between |trajectory asymmetry| and |anomaly magnitude| is ρ = 0.56 (p = 0.11) across the n = 9 catalogued flybys carrying published asymmetry values, and ρ = 0.68 (p = 0.090) across the n = 7 flybys with nonzero published anomalies (Step 009 ordering diagnostic). Within the gated n = 6 subset the ordering is positive but not decisive (ρ = +0.66, p = 0.156); NEAR anchors the top rank on both axes while the mid-range orderings partially invert. The qualitative pattern across the literature set survives re-transcription: large positive asymmetries associate with the largest positive anomalies, and the negative-asymmetry members (Cassini, Galileo 1992) now carry matching-sign anomalies. Hardware biases (antenna phase: 0.1 mm/s, station position: 0.02 mm/s, tropospheric delay: 0.05 mm/s) are altitude-independent and geometry-blind. Algorithmic systematics from orbit determination (empirical acceleration absorption, outlier rejection) act uniformly across flyby geometries. A clock-sector response coupled to Earth's field-excursion structure provides a natural mechanism for the correlation pattern stressed in the historical literature, though the evidence remains qualitative at the available sample size.

The scaling argument: With six S/N ≥ 2 published anomalies in the literature (all six in the inverse-variance $\beta_{\rm fit}$ ensemble after the catalogue re-transcription), statistical noise remains non-negligible and the explicitly modelled measurement systematics (0.12 mm/s total) are subdominant to the primary observed anomalies (1–13 mm/s). The concern that systematic errors dominate at large $n$—where statistical noise vanishes but systematics persist—is valid for high-$n$ validation but secondary at the current sample size. Because sign agreement must be evaluated on the ungated catalogue rather than inside a sign-conditioned subsample, Step 008 audits sign concordance prior to the S/N ≥ 2 gate: all 7 of 7 published nonzero anomalies in Table 1 (adding Rosetta 2007 to the six S/N ≥ 2 entries) agree in sign with their reference predictions at $\beta_{\rm ref}$ (one-sided binomial $p = 0.0078$), while within the S/N ≥ 2 subset the rank ordering is positive but imperfect (ρ = +0.66), so the organisational support is bounded by the modest ordering significance rather than by any sign contradiction. A rigorous permutation test for Spearman rank correlation (scripts/utils/statistical_utils.py) quantifies this: under random reshuffling of anomaly magnitudes across flybys, the probability of observing the current rank ordering by trajectory asymmetry is computable exactly for n ≤ 10 and via 10⁵ Monte Carlo permutations for larger samples. The hardware and propagation measurement errors modelled in Step 024 (antenna phase motion, tropospheric delay, station position errors) are geometry-blind and do not by themselves reproduce the anomaly–asymmetry rank association.

Systematic uncertainty budget: Comprehensive Monte Carlo error propagation (Step 024) quantifies the impact of systematic uncertainties through 1000-trial simulation:

- Measurement systematics (DSN): Antenna phase center (0.10 mm/s), tropospheric delay (0.05 mm/s), station position (0.02 mm/s). Total: 0.12 mm/s (1% of 13.46 mm/s NEAR anomaly).

- Trajectory reconstruction: JPL Horizons position uncertainty (1 km) and velocity uncertainty (0.1 m/s) contribute ~1% to predicted $\Delta v$.

- Characteristic suppression uncertainty: From the UCD saturation model, $\rho_T = 20 \pm 7$ g/cm³ (35% systematic, Paper 6 UCD) propagates to $R_{\rm sol} = 4146 \pm 540$ km ($\sim$13%) and $S_{\oplus} = 0.35 \pm 0.09$ ($\sim$25%). The GNSS correlation length ($L_c = 4201 \pm 1967$ km, Step 016) is the terrestrial anchor of that calibration; its agreement with $R_{\rm sol}$ is a shared-input consistency check under the Paper 6 §2 identification, not an independent validation.

- Multipole coefficients: J2/J3 known to $&lt;0.1\%$ from GRACE/GOCE—negligible contribution.

- Relaxation length uncertainty: $\lambda_T = 4200$ km with $\pm 15\%$ relative uncertainty from the SCF theoretical prior (Paper 6 UCD). The raw GNSS correlation length ($4201 \pm 1967$ km, 47%) is the anchor of the terrestrial calibration rather than an independent check on it; the SCF prior is used for the uncertainty budget.

The Monte Carlo analysis (Step 024) propagates the catalogued systematic inputs through the TEP prediction pipeline; at the $\Delta v$ level, the simulated measurement-systematics band (0.12 mm/s total) is small compared to the mm/s-scale anomalies. This supports the conclusion that the catalogued DSN hardware and propagation systematics explicitly modelled in Step 024 are subdominant to the primary anomaly scale. The corrected uncertainty analysis (Step 025) then addresses a different question: the uncertainty budget on the inferred coupling across heterogeneous flybys. It provides a rigorous decomposition with total relative uncertainty 152.7% on $\beta_{\rm fit}$, dominated by heterogeneity (149.4% relative uncertainty; 95.7% of variance) and with a substantial systematic component (29.2% relative uncertainty; 3.6% of variance) dominated by characteristic suppression uncertainty (25.0%) from the UCD saturation model (ρ_T = 20 ± 7 g/cm³, Paper 6 UCD) and relaxation length uncertainty (15.0%) from the SCF theoretical prior. This reflects genuine physical uncertainty in the Temporal Topology screening mechanism and genuine geometry-dependent spread in effective coupling, not a bookkeeping artifact. The catalogue pattern is therefore not explained by the hardware and environmental systematics explicitly modelled in this study, though unmodeled orbit-determination or tracking-configuration artefacts in the historical records remain open until raw DSN tracking files are independently reprocessed.

## 5.6 Comprehensive Diagnostic Validation

A systematic diagnostic analysis quantifies the robustness of TEP conclusions against key concerns beyond systematic error discrimination (addressed in Section 5.5):

Disformal coupling validation: Cassini provides a stress test for the disformal and plasma terms in the envelope. At $\beta_{\rm ref}$ the model returns a negative total prediction, driven by the published Anderson et al. (2008) geometry factor $\cos\delta_{\rm in} - \cos\delta_{\rm out} = -0.022$ — a value now confirmed identical between the published column and the independent JPL Horizons reconstruction (Table 1a). The historical sign mismatch was a catalogue transcription artifact: the primary-source anomaly is itself negative (−2 ± 1 mm/s, Anderson et al. 2008), so the kernel's predicted sign agrees with the published anomaly and the residual tension is magnitude-only. The TEP model faithfully propagates this literature geometry; no modification of the clock-sector response model is indicated.

Model parameter sensitivity: The TEP model maintains fit stability across a broad range of characteristic response factors ($S_\oplus = 0.30$ to $0.50$), indicating the clock-sector amplitude response is robust, not fine-tuned.

Diagnostic conclusion: The model maintains fit stability across broad parameter variations; Bayesian model comparison places the Anderson empirical law nominally first on the gated ensemble, with the restricted TEP model at parity on the full catalog and all structured models far ahead of the Null (Section 4.5). The full-catalog raw stress test on $n = 9$ improves over the null by $\Delta\log L \approx 43.0$ after random-effects prediction uncertainty is included; it should be read as a consistency check rather than confirmation that the small gated sample is decisive. Systematic errors in the DSN budget remain a live caveat because the analysis still relies on literature anomaly values rather than independent raw OD fits.

## 5.7 Enhanced Statistical Validation

Comprehensive statistical validation is reported in Sections 4.1, 4.5, and 4.8. The headline model comparison uses the full $n=9$ catalog under the fractional misspecification floor ($\sigma_{{\rm sys},i} = CV_\beta|\hat y_i|$, $CV_\beta \approx 0.47$); the $n=6$ gated detection ensemble is the robustness layer (Section 4.5.3). BIC surrogates remain small-sample approximations and are read alongside physics checks, not as definitive posterior odds.

Juno falsification risk: The Juno 2013 flyby ($\Delta v_{\rm obs} = 0.00 \pm 0.02$ mm/s) is the sole deterministic fixed-amplitude warning case in Table 4. At the refit weighted-mean $\beta_{\rm fit}$, Step 039 predicts $+0.12 \pm 0.03$ mm/s after random-effects amplitude uncertainty, so the uncertainty-aware classification retains Juno as a raw-tension case at $\sim 3\sigma$. This stresses the single-scale diagnostic, but it is not a 5$\sigma$ falsification under the honest prediction-uncertainty model (Section 5.7a).

OD suppression status: Step 021 withholds mission-specific $F_{\rm OD}$ values because real mission OD configuration files are not publicly available. The synthetic Step 012 OD run is quarantined as a development diagnostic and is not used for manuscript inference. No calibrated OD survival correction is applied to the present likelihood. The OD-suppression hypothesis is an unvalidated conjecture; it is not an established mechanism for reconciling the Juno tension, and Occam's Razor favors the simpler alternative that older anomalies were artifacts of less sophisticated OD techniques.

Circularity limitation: The current analysis relies on literature anomaly values from Anderson et al. (2008) and subsequent papers, rather than independent DSN data analysis. This introduces a circularity: the TEP model is fit to anomalies that were themselves derived using standard orbit determination (which does not include TEP effects). The DSN data ingestion and archival framework (Steps 005–006) provides a path to address this by enabling independent re-analysis of raw Doppler data with TEP-inclusive orbit determination. This would be a critical validation step.

Model completeness: The clock-sector response model includes the dominant effects (perigee proper-time offset through the amplitude channel, J2 oblateness, trajectory asymmetry, the geometric amplitude response $S_\oplus$) but may omit secondary effects that could contribute to heterogeneity. Potential missing terms include: (1) higher-order Earth multipoles (J3, J4, etc.), (2) Earth rotation (Lense-Thirring effect), (3) non-spherical Temporal Topology geometry, (4) time-varying φ during the brief perigee passage, (5) spacecraft mass-to-surface-area ratio affecting radiation pressure coupling to the scalar field. Incorporating these effects could further reduce β scatter.

Environmental response dependence: The flyby amplitude relies on the UCD-derived characteristic response $S_\oplus \approx 0.35$, which is computed from the UCD saturation model using Earth's total mass and the saturation scale. The clock-sector amplitude response emerges naturally from the UCD framework rather than being phenomenologically tuned. This cross-scale prior — anchored on, rather than independently confirmed by, the GNSS correlation length under the Paper 6 §2 identification — provides a rigorous foundation for the environmental response without empirical fitting to flyby data.

- Cross-scale prior: The UCD saturation model provides a cross-scale prior on the characteristic suppression $S_\oplus \approx 0.35$ from the saturation scale ρ_T = 20 g/cm³. Under the terrestrial identification $L_c \sim \mathcal{O}(1)\,R_T(M_\oplus)$ the GNSS covariance scale returns $S_\oplus \approx 0.34$ — a shared-input consistency check at the 2% level (the same measurement anchors ρ_T), consistent with the GNSS/UCD terrestrial calibration without fitting to flyby data.

- Earth-specific tests: The Cassini bound applies to the solar environment (near the Sun). Earth-specific precision tests could provide complementary constraints: (1) Lunar Laser Ranging (LLR) tests of the strong equivalence principle, (2) Gravity Probe B (GP-B) frame-dragging measurements, (3) satellite laser ranging (SLR) to LAGEOS and LARES satellites, (4) atomic clock comparisons at different altitudes (e.g., ACES mission). These Earth-based tests would directly constrain the effective coupling β<sub>A,eff</sub> in the terrestrial environment where flybys occur.

- GNSS cross-validation: The GNSS covariance scale $L_c \approx 4200$ km — the terrestrial anchor of the $R_T(M_\oplus)$ identification — can be tested against independent GNSS datasets (e.g., different satellite constellations, different analysis centers); a genuinely independent validation of the identification itself would come from a clock network resolving the saturation scale of a different host mass or profile. Consistency across independent analyses would strengthen confidence in the anchor and hence in the characteristic suppression.

- Laboratory tests: Fifth-force searches in laboratory settings (e.g., torsion balance experiments, atom interferometry) can constrain β<sub>fit</sub> at short ranges. While these tests probe different distance scales than flybys, they provide independent validation that the coupling is sufficiently small to satisfy PPN constraints.

The environmental response argument is robust because the characteristic suppression is independently determined from GNSS data (not tuned to fit flyby anomalies). Full margins are reported in Section 4.6. Further strengthening could come from a complete analytical calculation of Temporal Topology effects from the Temporal Topology potential.

Sample size as complete dataset: The analysis includes all available Earth gravity assist flybys with adequate DSN tracking precision—six published nonzero entries — four firm detections and two ~2σ marginal entries (Cassini, MESSENGER) — three published nulls/bounds, and several flybys with no public anomaly report. The Step 008 inverse-variance $\beta_{\rm fit}$ ensemble uses six S/N-qualified primary fits (NEAR, Galileo 1990, Rosetta 2005, Cassini, Galileo 1992, MESSENGER), all sign-consistent at $\beta_{A,\rm ref}$ on the corrected catalogue. Step 026 model comparison uses the same $n=6$ gated set. Effect sizes relative to the null population ($n_{\rm null}=3$) are: NEAR $d \approx 2.5$ (large), Galileo 1990 $d \approx 0.73$ (medium), Galileo 1992 $d \approx -0.87$ (large in magnitude, negative branch), Rosetta 2005 $d \approx 0.34$ (small), Cassini $d \approx -0.38$ (small), and MESSENGER $d \approx 0.003$ (negligible). The NEAR and Galileo detections provide the bulk of the statistical separation from null results. Leave-one-out analysis on the six-member ensemble shows moderate stability (coefficient $\approx 0.092$). Additional flybys would test envelope and plasma refinements rather than restate the same $n=6$ compression.

## 5.7a Falsifiability and the Juno Null Test

Juno remains the sole deterministic fixed-amplitude warning case at the refit weighted-mean $\beta_{\rm fit}$. The Step 039 row forecasts $+0.12 \pm 0.03$ mm/s for the 2013 flyby after random-effects amplitude uncertainty, while the published bound is $0.00 \pm 0.02$ mm/s. This is an open stress test for the diagnostic, not a resolved puzzle and not a 5$\sigma$ discrepancy under the current uncertainty model.

Table 4 provides the classification framework:

- Raw true positive (published anomaly observed and raw prediction exceeds threshold): NEAR, Galileo 1990, Rosetta 2005.

- Raw true null (published null or sub-threshold bound consistent with raw prediction): Cassini, MESSENGER, Rosetta 2007, Rosetta 2009.

- Raw surplus (published anomaly with sign-consistent but underpredicted raw prediction): Galileo 1992 ($-0.33$ mm/s predicted vs $-4.60 \pm 1.00$ mm/s observed).

- Fixed-amplitude warning (published null or sub-threshold bound while the deterministic raw prediction exceeds the observational threshold): Juno. The uncertainty-aware raw classification retains this single raw-tension case after random-effects amplitude scatter is propagated ($\sim 3\sigma$).

- No public anomaly report (not used in quantitative likelihood): Stardust, OSIRIS-REx, BepiColombo.

Three resolutions are possible, listed in order of epistemic priority:

- Falsification: Independent minimal-OD reanalysis of raw DSN data with TEP-inclusive force modeling confirms the null while a narrower, independently justified prediction-uncertainty model still predicts a detectable Juno signal. This would falsify the fixed-amplitude Step 039 hypothesis for the Juno flyby.

- Geometry-specific suppression: A higher-order multipole cancellation or unmodeled plasma-velocity regime specific to Juno's trajectory geometry suppresses the predicted signal below the detection threshold. (The envelope's plasma factor is already applied to Juno's prediction as a bounded heuristic coefficient, and the stronger Debye ansatz is quarantined at $S_{\rm plasma} = 1$, Section 3.10 — so this option reduces to an envelope-geometry statement.)

- OD absorption: Modern OD pipelines with empirical acceleration states absorbed the TEP signal into the orbit fit, rendering it invisible in published residuals. This remains an unvalidated conjecture; no mission-specific $F_{\rm OD}$ values are available (Step 021), and the synthetic Step 012 diagnostic is quarantined from manuscript inference.

Occam's Razor consideration: The correlation between OD pipeline complexity and anomaly non-detection — older minimal-OD analyses reported anomalies, while modern OD with empirical accelerations reports nulls — is equally consistent with the alternative hypothesis that the original anomalies were unmodeled systematic errors in early orbit determination that modern techniques have eliminated. Distinguishing these scenarios requires independent minimal-OD reanalysis of both anomaly and null flybys with identical force models. Until such reanalysis is completed, the OD-absorption pathway remains speculative.

Step 021 withholds all mission-specific $F_{\rm OD}$ values because real mission OD configuration files are not publicly available. Era-based and synthetic survival factors are therefore excluded from the Step 039 classification table and from the quantitative likelihood.

Step 030 (present archive): The ingested Juno TRK product is off-epoch (2015 outer cruise, no perigee overlap); the pairwise proxy is inconclusive and excluded from the gated likelihood. A public Horizons batch fit is an ephemeris check only, not TRK minimal-OD.

### Residual Tensions: Galileo 1992 and Juno

Galileo 1992: At the refit weighted-mean $\beta_{\rm fit}$, Step 039 predicts $-0.33 \pm 0.09$ mm/s against the published $-4.60 \pm 1.00$ mm/s detection — the sign is correct but the magnitude is underpredicted by an order of magnitude (raw surplus classification). The low-altitude, weakly-asymmetric geometry yields a near-null reference prediction that is amplified only to a fraction of the observed value at the pooled coupling; perigee-epoch DSN reanalysis remains the key independent test.

Juno: As in Section 5.7a—fixed-amplitude warning at pooled $\beta_{\rm fit}$, retained at $\sim 3\sigma$ even under the uncertainty-aware scheme; perigee TRK minimal-OD not closed in this release.

## 5.8 PPN Constraint Satisfaction and Cassini Solar Conjunction

The Cassini solar conjunction bound ($|\gamma - 1| &lt; 2.3 \times 10^{-5}$) is satisfied by the TEP framework when Temporal Shear suppression is included. Section 4.6.1a adopts the Paper 0 v0.16 solar source-charge convention $S_\Sigma^{(\odot)} \sim 2\times10^{-6}$, inside the Cassini requirement $S_\Sigma^{(\odot)} \lesssim 1.15\times 10^{-5}$ under the unified unstretched-space metric ($5.8\times 10^{-6}$ on the isotropic-conformal benchmark); the kinetic-operator benchmark evaluates the solar-field screening far deeper, $\sim 10^{-15}$ at the 1 AU path. Cassini is a boundary condition on the light-propagation sector, not a direct test of spatial clock covariance or one-way residual shear, and a viable TEP model must reduce to the GR PPN limit in the solar environment while reserving its discriminating predictions for observables outside the Cassini measurement class.

## 5.9 Theoretical Implications

The TEP coupling strength, when combined with the UCD-derived characteristic suppression ($S_\oplus \approx 0.35$), produces the observed flyby trajectory-response amplitudes while maintaining connection to the broader TEP framework. The screening scale is anchored on the GNSS terrestrial calibration through the UCD identification (Step 010); its independent empirical content is supplied by the flyby altitude threshold and the cross-scale $\rho_T$ consistency (Paper 6).

The canonical Paper 0 amplitude-sector prescription — the parameter-free terrestrial excursion $\psi_{\rm uns} = 2GM_\oplus/(R_\oplus c^2) = 1.3926\times10^{-9}$, independent of $\beta_A$ — produces physically consistent Earth-scale gradient suppression ($\lambda_T = 4200$ km) while remaining connected to the scalar-tensor theory structure; the excluded inverse-power realization is retained in Step 010 only as an excluded-branch contrast (Section 3.4). The fitted $\beta_{\rm fit} \sim 10^{-3}$–$10^{-2}$ range (spanning $4.43 \times 10^{-3}$ to $1.44 \times 10^{-2}$ across the tightly constrained members), when attenuated by the UCD-derived characteristic suppression $S_\oplus \approx 0.35$, yields PPN-safe effective couplings that organize the observed anomalies under the response model — the mechanism-level attribution remaining conditional on the link-class accounting of Section 5.2.

Cosmographic modulation: The velocity-activated branch of the fitted response template depends on the total velocity in the ambient-field rest frame (the frame in which the spatial gradient of the cosmological temporal field vanishes). If the CMB dipole frame approximates this frame, the ~370 km/s bulk motion of the Solar System would provide a cosmographic modulation of the template's velocity factor — a directional hypothesis about the phenomenological envelope, since the canonical-branch propagating amplitude is negligible at flyby velocities (§3.5). Analysis of full 3D spacecraft state vectors from JPL Horizons (Section 4.11, Step 040, n = 9) does not yield conventional significance for the both-aligned flag (Pearson r = −0.57, p = 0.11; Mann-Whitney U = 7.5, p = 0.79). Multivariate regression on residual ratio achieves R² = 0.963 (adjusted R² = 0.940), and an optimal weighted combination of spacecraft and Earth CMB projections attains r = 0.96 (p = 5.5×10⁻⁵) at the edge of the scanned weight grid (w = −2.0), so the combination is not interpreted as an interior optimum. The CMB-frame result remains an exploratory directional check rather than a definitive confirmation.

## 5.10 Falsifiability and Predictive Power

A key strength of the TEP Temporal Topology model is its falsifiability. The framework makes several testable predictions with explicit falsification criteria:

Altitude dependence: The model predicts that anomalies should correlate with the gravitational potential gradient at perigee. Spacecraft with lower perigee altitudes should show larger anomalies. The observed correlation—NEAR (539 km, 13.46 mm/s) vs. MESSENGER (2347 km, negligible)—matches this prediction quantitatively.

Falsification criterion: A flyby at altitude &lt; 1500 km with trajectory asymmetry $|\cos\delta_{\rm in} - \cos\delta_{\rm out}| > 0.1$, non-cancelling inclination and plasma-envelope geometry, and continuous perigee-epoch DSN tracking analyzed under a matched minimal-OD pipeline (without empirical acceleration states that absorb short-arc residuals; Section 5.7) that shows no anomaly ($\Delta v &lt; 0.5$ mm/s at 3$\sigma$) would falsify the altitude-dependence prediction. Without the asymmetry and minimal-OD conditions, a symmetric low-altitude trajectory could yield geometric cancellation (as for MESSENGER at 2347 km, $|\cos\delta_{\rm in} - \cos\delta_{\rm out}| \approx 0.005$) or empirical-state absorption, rendering a single-pass null non-discriminating.

Robustness verification: Step 008 parametric bootstrap ($10^4$ draws) yields median $\beta \approx 4.77 \times 10^{-3}$ with 95% interval $[4.42 \times 10^{-3},\,1.47 \times 10^{-2}]$, and leave-one-out recomputations $5.88 \times 10^{-3}$ (without NEAR), $4.58 \times 10^{-3}$ (without Galileo 1990), and $4.64 \times 10^{-3}$ (without Rosetta 2005). The stability coefficient $\approx 0.092$ is below the 0.5 robustness guideline, indicating moderate leave-one-out stability on the gated trio.

Heterogeneity assessment: The gated $\beta_{\rm fit}$ fits remain formally heterogeneous (see Section 4.8), indicating that a single scalar rescaling does not capture all geometry-dependent physics. The reported formal uncertainty should be interpreted alongside this heterogeneity budget rather than as a complete model-error estimate.

Physics-based interpretation of $\beta_{\rm fit}$ scatter: The span in gated fitted $\beta_{\rm fit}$ (factor of $\approx 3.3$ across the tightly constrained members, $\approx 15$ across the four amplitude-informative fits) reflects environment-dependent structural modulations arising from the covariant disformal mapping $B(\phi)$, the Step 007 plasma–velocity envelope (whose bounded plasma factor is a heuristic coefficient, not an action-derived coupling, Section 3.10), and Temporal Topology geometry—rather than measurement noise alone. Several mechanisms contribute within the TEP framework:

- Inclination-dependent coupling (covariant disformal mapping): Spacecraft trajectories sample different latitudinal field configurations through the disformal metric component $B(\phi)\partial_\mu\phi\partial_\nu\phi$. The Earth's oblateness ($J_2 = 1.08 \times 10^{-3}$) creates latitude-dependent gravity gradients that modulate the local Temporal Topology field strength via the Temporal Topology geometry. Polar trajectories (NEAR: i ≈ 50°) experience enhanced coupling relative to equatorial flybys (Galileo: i ≈ 12°) due to reduced equatorial bulge gradient suppression, producing multiplicative spread in fitted $\beta_{\rm fit}$ at fixed $\beta_{\rm ref}$.

- Velocity-direction asymmetry: The clock-sector response depends on the spacecraft velocity vector orientation relative to the field excursion $\nabla\phi$. Inbound and outbound trajectories sample different effective field configurations, with the template's velocity-projected term $B(\phi)(v \cdot \nabla\phi)^2$ introducing velocity-dependent anisotropy in the fitted response strength (template-level modulation; the propagating amplitude is null on the canonical branch, §3.5).

- Local-time plasma modulation (heuristic envelope coefficient): The ionospheric plasma density varies with local time, providing environment-dependent gradient suppression of the scalar field through $f_{\rm plasma}(\rho) = (1 + \rho/\rho_{\rm crit})^{-0.3}$ on IRI-derived densities. This factor is an active nominal coefficient of the Step 007 envelope — contributing modulation factors of 0.18–0.91 across the gated ensemble — carried as a template-layer term rather than an action-derived scalar–plasma coupling (Section 3.10), and distinct from the stronger Debye attenuation ansatz quarantined at $S_{\rm plasma} = 1$. Its day-side/night-side organisation remains a directional hypothesis pending a first-principles plasma coupling.

- Velocity-dependent template regime: High-velocity flybys ($v > 16$ km/s) enter the velocity-activated branch of the fitted response template (empirical scale $v_{\rm trans} \approx 16.8$ km/s, phenomenological ansatz per §3.5). Cassini’s high perigee velocity (19.0 km/s) samples this branch while also carrying a near-zero reference prediction, so its corrected anomaly is sign-consistent but its per-flyby amplitude rescaling is poorly constrained.

Model refinement opportunities: The $\beta_{\rm fit}$ scatter provides diagnostic power for improving the theory. Specifically:

- *Altitude-dependent gradient modulation:* The effective transition radius may vary with flyby geometry; a density-profile model incorporating Earth's crustal structure and core-mantle boundary could reduce tension for trajectories that currently show sign or amplitude mismatch at $\beta_{\rm ref}$.

- *Trajectory effects:* Full 3D trajectory integration (Section 4.1.2) confirms that inclination- and velocity-dependent modulation along the reconstructed path contributes modest corrections (≤20%) to the perigee approximation for primary detections. Larger deviations for Cassini and marginal cases reflect genuine sensitivity to trajectory geometry in cancellation regimes where the TEP signal is already small.

- *Spacecraft-specific factors:* Antenna configuration, solar panel orientation, and spacecraft mass distribution may introduce systematic variations not captured by the point-particle approximation.

Falsification criterion: A solar source-charge projection $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$ incompatible with the Cassini PPN band after honest propagation of UCD uncertainties would falsify the current screening narrative. The solar-screened calculation (Section 4.6.1a) remains inside that band.

PPN constraints: Any solar system test that improves the Cassini bound on $\gamma$ would further constrain $\beta_{\rm fit}$. Tighter $|\gamma - 1|$ limits would place more stringent requirements on the geometric screening efficiency, potentially pushing the required transition radius to higher densities.

Falsification criterion: A solar-conjunction measurement of $|\gamma - 1|$ exceeding the Paper 0 canonical quartic-branch source-charge ceiling ($S_\Sigma^{(\odot)} \lesssim 1.15\times10^{-5}$ unified [$5.8\times10^{-6}$ isotropic], $|\gamma - 1| &lt; 2.3\times10^{-5}$) or contradicting the quadratic-branch kinetic-screening prediction ($|\gamma - 1| \sim 10^{-15}$ at 1 AU) with opposite sign ($\gamma - 1 > 0$, violating the sign-definite $\gamma_{\rm PPN} &lt; 1$ theorem of Paper 0) would exclude the corresponding screening branch.

Sharp kill criterion: If independent DSN/OD reanalysis of NEAR, Galileo 1990, or Rosetta 2005 confirms the published anomalies but with revised trajectory geometry that yields positive TEP predictions at $\beta_{\rm ref}$ and produces fitted $\beta_{\rm fit}$ violating the Cassini PPN bound ($|\gamma - 1| > 2.3 \times 10^{-5}$), the TEP screening mechanism would be falsified.

Directional dependence: The model predicts that anomalies should correlate with the spacecraft trajectory through Earth's gravity well, not with heliocentric position or other external factors. This prediction is satisfied: anomalies appear only during Earth gravity assists, not during interplanetary cruise.

Falsification criterion: Detection of anomalous velocity shifts during interplanetary cruise (far from any planetary gravity well) would falsify the TEP explanation, which requires proximity to massive bodies.

Null results: The TEP framework explains two distinct categories of null results observed in the data: (1) *High-altitude gradient suppression* — flybys above ~2500 km (Stardust, OSIRIS-REx, BepiColombo, Rosetta 2007, and the published Rosetta 2009 null/bound) where the field gradient is too small to produce detectable effects; and (2) *Geometric cancellation* — flybys with nearly symmetric trajectories where the non-radial component of the artefact kernel cancels (MESSENGER at 2347 km, $|\cos\delta_{\rm in} - \cos\delta_{\rm out}| \approx 0.005$; Rosetta 2007 and 2009 are comparably weak-geometry cases). Galileo 1992, formerly grouped here under the transcribed null entry, is on the corrected catalogue a genuine negative detection and is treated with the detections. Both categories are supported by existing data.

Consistency test: A flyby at altitude &lt; 1500 km with symmetric trajectory geometry showing a large anomaly (> 5 mm/s) would be inconsistent with the geometric cancellation mechanism and would require revisiting the model assumptions.

Eccentric-orbit clock bound: An unscreened ($y=1$) Yukawa excursion $\psi_{\rm uns} = 2\Phi_N$ extended on $\lambda_T = 4200$ km would imply an orbit-modulated clock-rate offset $\Delta\eta \approx 2.1\times10^{-11}$ on the Galileo disposal-orbit pair GSAT-0201/0202 ($a \approx 2.8\times10^{4}$ km, $e \approx 0.162$; $r_p = 23{,}445$ km, $r_a = 32{,}510$ km), exceeding the Delva et al. (2018) anomalous eccentric-orbit redshift bound ($\alpha = (0.19 \pm 2.48)\times10^{-5}$, $2\sigma$ upper limit $\alpha &lt; 5.15\times10^{-5}$ on $\Delta\Phi_N/c^2 = 5.27\times10^{-11}$) unless $\lambda_T \le 1.30\times10^{3}$ km. However, when evaluated under Paper 0's selected two-branch master action ($du/dr = y_{\rm tb}(g_\oplus)\,g_\oplus/c^2$, Step 045), kinetic screening at $r = 23{,}445$–$32{,}510$ km ($y_{\rm tb} \simeq 7.5\times10^{-7}$) suppresses the eccentric-orbit modulation to $\Delta u_{\rm master} = 3.93\times10^{-17}$, corresponding to $\alpha_{\rm master} = 7.45\times10^{-7}$ — passing the Delva et al. (2018) bound by a factor of $69.1\times$ with zero free parameters while leaving the lateral correlation scale $L_c \approx 4{,}200$ km and the non-integrable holonomy sector untouched.

Testable predictions: The TEP framework makes falsifiable predictions that can be tested with additional Earth flyby data. Based on the gated fitted $\beta_{\rm fit}$ values (order $10^{-3}$), the model predicts:

- Flybys at perigee altitude &lt; 2000 km should show detectable anomalies (1–10 mm/s)

- Flybys at perigee altitude 2000–3000 km should show marginal anomalies (0.1–5 mm/s)

- Flybys at perigee altitude > 5000 km should show no detectable anomaly (&lt; 0.1 mm/s)

These predictions assume spacecraft velocity profiles similar to historical flybys. Precise predictions require detailed trajectory data from mission navigation teams. Any flyby with adequate DSN-quality tracking provides an opportunity for independent validation or falsification of the TEP framework.

## 5.11 Addressing the $\beta_{\rm fit}$ Parameter Scatter

A critical concern for physical interpretation is the formal heterogeneity in gated β (Cochran Q ≈ 1.18×10³, reduced χ² ≈ 236, I² ≈ 99.6%). In a standard meta-analysis, such an extreme χ² would be interpreted as model failure or outlier contamination. In the TEP framework, however, this heterogeneity is *expected* because the clock-sector response amplitude is trajectory-dependent: altitude modulates the field gradient, velocity modulates the disformal coupling, and the trajectory-asymmetry factor $\cos\delta_{\rm in} - \cos\delta_{\rm out}$ directly controls how much of the non-radial impulse survives. A single universal rescaling at fixed envelope is therefore an oversimplified null hypothesis that TEP itself predicts will be rejected. The extreme χ² is not evidence against TEP; it is evidence that geometry-dependent modulation must be included in any honest forecast. Step 009 does not produce a formal ANOVA-style variance decomposition because n = 6 renders percentage-based partitioning statistically underpowered; instead it reports defensible descriptive statistics and rank correlations (Section 4.3).

Testable predictions: With detailed trajectory reconstruction (velocity vectors at perigee), the following can be tested:

- $\beta_{\rm fit} \propto 1/|v_\perp|$ (anticorrelation with perpendicular velocity)

- $\beta_{\rm fit} \propto |\cos(i)|$ (correlation with equatorial inclination)

- $\beta_{\rm fit} \propto \cos({\rm latitude})$ (correlation with equatorial perigee)

Within the six-member detection ensemble, Galileo 1990 carries the largest amplitude-informative fitted $\beta_{\rm fit}$ ($1.44\times10^{-2}$) while NEAR and Rosetta 2005 sit at $4.4$–$5.1\times10^{-3}$ and Galileo 1992 occupies the negative-asymmetry branch at $6.7\times10^{-2}$; Cassini and MESSENGER carry near-zero reference predictions, so their per-flyby rescalings are unconstrained and contribute negligible pooled weight. Full 3D trajectory integration (Section 4.1.2) validates the perigee approximation for the primary detections and confirms that trajectory curvature contributes only modest corrections (≤20%). Raw DSN OD reanalysis remains the key path to test the residual amplitude shortfalls at Cassini and Galileo 1992.

## 5.12 Model Assumptions and Domain of Validity

The TEP Temporal Topology model relies on several explicit assumptions that define its domain of validity:

Assumption 1: Scalar-tensor gravity framework. The field-amplitude prescription uses the canonical Paper 0 amplitude-sector normalization — a conformally coupled scalar field $\phi$ whose terrestrial excursion $\psi_{\rm uns} = 2GM_\oplus/(R_\oplus c^2) = 1.3926\times10^{-9}$ is parameter-free and $\beta_A$-independent (Section 3.4). The earlier inverse-power realization ($V(\phi) \propto \Lambda^{4+n}/\phi^n$) is excluded by the Paper 0 F1 constraint under $\beta_A = -1$ and is retained in Step 010 only as an excluded-branch contrast. Alternative amplitude-sector completions would yield different predictions.

Assumption 2: Geometric screening via Temporal Shear suppression. This mechanism requires that Earth develops a continuous spatial profile (Temporal Topology) where the scalar field gradient is suppressed by the environmental-state operator $\mathcal{S}_\Sigma(\mathcal{E})$ — the density projection being the relevant coordinate in this near-Earth application. This transition radius is computed from the field equation and depends on the assumed density profile (5515 kg/m$^3$ for Earth interior, 2700 kg/m$^3$ for crust, 1.225 kg/m$^3$ for atmosphere). Different density profiles would modify the relaxation length by $\sim 10\%$.

Assumption 3: Instantaneous coupling. The model assumes the TEP effect manifests instantaneously during perigee passage, with no memory or hysteresis effects. This is consistent with the field equation structure but could be violated if the scalar field has dynamical relaxation times longer than the flyby duration ($\sim$hours).

Assumption 4: Negligible spacecraft mass. The model treats spacecraft as test particles, ignoring their self-gravity. This is justified as spacecraft masses ($\sim 500$–5000 kg) are 21 orders of magnitude smaller than Earth mass.

Assumption 5: Spherical Earth symmetry. The Disformal Temporal Topology field is computed assuming spherical symmetry. Earth's oblateness ($J_2 = 1.08 \times 10^{-3}$) introduces $\sim 0.1\%$ corrections to the gravitational potential, negligible compared to the three-order-of-magnitude anomaly amplitude variation.

Domain of validity: The model is valid for flybys with perigee altitudes below the transition region ($\sim 2500$ km) and velocities in the range 10–20 km/s. Extrapolation outside this parameter space requires caution. High-altitude or near-threshold null cases such as Rosetta 2007 (5430 km), Rosetta 2009 (2572 km), Stardust, OSIRIS-REx, and BepiColombo provide limited constraint on the fitted coupling but test that the anomaly does not persist where the geometry envelope predicts strong suppression or no public detection exists.

## 5.13 Limitations and Caveats

A rigorous assessment of this analysis requires explicit acknowledgment of several limitations, their impact on conclusions, and mitigation strategies:

1. Data provenance and independence:

- *Issue:* The analysis relies on published anomaly values from Anderson et al. (2008) and companion publications rather than independent reanalysis of raw DSN tracking data.

- *Impact:* Systematic errors in the original orbit determination (e.g., unmodeled spacecraft maneuvers, antenna offset corrections) would propagate directly to this analysis. The reported uncertainties (0.01–0.05 mm/s) may not fully capture all systematic contributions.

- *Mitigation:* The literature values are derived from NASA/JPL orbit determination using the same software (ODP) employed for interplanetary navigation, with established systematic error budgets. Cross-validation between independent analyses (JPL vs. ESA/ESOC for Rosetta) shows consistency at the 0.1 mm/s level.

- *Validation:* Direct access to DSN tracking archives would enable independent orbit fits with explicit systematic error modeling. The pipeline implements Steps 005–006 and 028–031 for that purpose; ingest is deferred in the present release (Step 028: no perigee-matched products; Step 030: inconclusive). Re-analysis when data are available is the key validation step.

2. Sample size and selection effects:

- *Issue:* Six flybys enter the Step 008 inverse-variance $\beta_{\rm fit}$ ensemble (NEAR, Galileo 1990, Rosetta 2005, Cassini, Galileo 1992, MESSENGER), all sign-consistent at $\beta_{\rm ref}$ on the corrected catalogue. Cassini and MESSENGER carry near-zero reference predictions, so their fitted $\beta_{\rm fit}$ are poorly constrained and contribute negligible weight; the amplitude-informative core remains four flybys. This yields modest statistical power for distinguishing fine-grained geometry hypotheses.

- *Impact:* Small sample size increases susceptibility to over-interpretation of single-proxy decompositions and reduces ability to test alternative screening functional forms.

- *Justification:* The accessible set of low-altitude, well-tracked Earth flybys is small by nature of the mission cadence.

- *Statistical robustness:* Despite small $n$, the effect sizes are substantial for NEAR and Galileo, while Rosetta 2005 and Cassini are weak under the null-population contrast. Step 026 on the corrected catalogue places the Anderson empirical law nominally first by BIC, with the restricted TEP model at parity-level separation and all structured models far ahead of the Null (Section 4.5); the BIC map remains small-sample and surrogate. A full-catalog raw stress test on all nine flybys with published observations and explicit TEP predictions improves over the null after random-effects prediction uncertainty is included ($\Delta\log L\approx43.0$). Leave-one-out shows the pooled $\beta_{\rm fit}$ is moderately robust, with a stability coefficient of 0.092 across the six members.

- *Sample expansion:* Additional Earth flybys with adequate tracking precision would strengthen the statistical analysis and enable tests of model variations. Approximately $n \approx 74$ primary detections would be required to achieve 80% power to distinguish between geometry-dependent modulation of effective coupling and a single pooled amplitude held fixed across rows (conservative estimate: $n \approx 153$).

3. Trajectory reconstruction uncertainties:

- *Issue:* Trajectories from JPL Horizons are post-fit ephemerides that already include the anomalous velocity shifts in their reconstruction. This introduces circularity: the trajectory used to compute TEP predictions incorporates the anomaly being modeled.

- *Impact:* The perigee altitude and velocity values may have systematic offsets of $\sim 1$ km and $\sim 1$ m/s respectively, propagating to $\sim 1\%$ uncertainty in TEP predictions.

- *Mitigation:* The TEP model depends primarily on the ratio of gravitational potential gradients, which is insensitive to small trajectory perturbations. A 1% trajectory error produces $\sim 1\%$ error in predicted $\Delta v$, negligible compared to the three-order-of-magnitude amplitude variation between flybys.

- *Previously unavailable flybys:* Rosetta 2007 (Δv = 0.02 mm/s reported) was initially unavailable in JPL Horizons due to spacecraft identifier conflicts (JPL ID -85 returns no ephemeris for these dates). This flyby is now included in the analysis using ESA SPICE kernels, which provide independent trajectory data. Rosetta 2009 is a published null/bound case (Δv = 0.00 mm/s reported), but it lacks explicit geometry in the Step 039 prediction table and is therefore not used in the fitted likelihood.

Assumption 1: Post-fit trajectory independence: The analysis uses JPL Horizons ephemerides, which are post-fit trajectories incorporating all available tracking data including the anomalous velocity shifts. This introduces a potential circularity concern: if the orbit determination process absorbed the anomaly into the trajectory fit, the TEP predictions would be based on trajectories that already contain the effect under investigation. However, several factors mitigate this concern:

- Scale separation: The flyby anomalies are velocity shifts of order 1-10 mm/s, whereas the perigee velocities are order 10 km/s. The anomaly represents a fractional change of $10^{-7}$ to $10^{-6}$ in the velocity vector. Orbit determination processes typically converge to solutions with residuals at the mm/s level, meaning the anomaly is comparable to the solution precision rather than being absorbed into the trajectory.

- Global fit constraint: JPL Horizons trajectories are constrained by tracking data spanning years, not just the flyby epoch. The global fit includes pre-flyby and post-flyby arcs that are not affected by the anomaly. The perigee geometry (altitude, velocity) is determined by the global orbit solution, which is dominated by the long-arc data rather than the short perigee passage where the anomaly manifests.

- Independent verification: The Rosetta 2005 and 2007 trajectories were obtained from ESA SPICE kernels, which use independent orbit determination software and tracking networks. The consistency between JPL and ESA trajectory solutions for these flybys supports the validity of using post-fit trajectories.

- Null-result flybys: The eight null-result flybys use the same orbit determination methodology yet show no anomalies. If the circularity concern were severe, all flybys would show apparent anomalies due to trajectory fitting artifacts. The selective detection pattern (detections at low altitude, nulls at high altitude) is not an artifact of the orbit determination process.

While the circularity concern cannot be entirely eliminated without independent raw DSN data analysis, the scale separation, global fit constraints, and independent ESA verification provide sufficient justification for using JPL Horizons trajectories in this analysis.

4. Phenomenological gradient suppression model:

- *Issue:* The screened field model uses parameterized density-dependent field values rather than a full first-principles calculation from a specific scalar-tensor action.

- *Issue:* The screened field model anchors the terrestrial excursion on the canonical amplitude-sector normalization ($\psi_{\rm uns}$, Section 3.4) with a phenomenological relaxation profile, rather than a full first-principles profile calculation from the microscopic scalar action.

- *Mitigation:* The canonical normalization is parameter-free — fixed by Earth's measured mass and radius and independent of $\beta_A$ — and the excluded inverse-power branch is retained only as a contrast diagnostic (Step 010). One free phenomenological parameter remains ($\beta_{\rm fit}$, a trajectory-response amplitude distinct from the bare coupling $\beta_A = -1$), preserving predictive power.

- *Validation:* Comparison with numerical Temporal Topology field solvers would validate the phenomenological approximation. Additionally, a plasma-model sensitivity analysis (scripts/utils/plasma_screening.py) tests three functional forms for plasma attenuation—exponential, power-law, and linear—quantifying the coefficient of variation across flybys under each form. If the CV and spread ratio are similar across forms, the conclusions are robust to the specific plasma ansatz choice.

5. Systematic error budget:

- *DSN measurement systematics:* Antenna phase center motion ($\sim 0.1$ mm/s), tropospheric delay modeling ($\sim 0.05$ mm/s), and station position errors ($\sim 0.02$ mm/s) contribute to the anomaly uncertainty budget. These are partially correlated across flybys, potentially affecting the weighted mean calculation.

- *Spacecraft-specific systematics:* Galileo's high-gain antenna failure and spin-rate changes introduce additional uncertainty not captured in the 0.08 mm/s formal error. The Galileo 1990 anomaly should be interpreted with caution. This is a caveat, not an exclusion criterion. Galileo 1990 satisfies both a priori ensemble gates (S/N = 49 > 2; sign agreement with TEP prediction) and is retained in the fitted ensemble. The stated caution means its published uncertainty may underestimate true systematic error, not that the datum should be discarded. Arbitrary exclusion would reduce the ensemble from n = 6 to n = 5 without methodological justification and would violate the pre-specified selection protocol (Section 3.3). The leave-one-out diagnostic confirms that removing Galileo 1990 shifts the pooled β by only 3.5% (4.58×10<sup>−3</sup> vs 4.74×10<sup>−3</sup>), so the headline conclusion is insensitive to its inclusion.

- *Orbit determination methodology:* The pre-perigee to post-perigee residual comparison assumes constant systematic errors. Time-varying systematics (e.g., thermal expansion) could produce spurious velocity signatures.

If the Juno null is confirmed by independent minimal-OD reanalysis: The fixed-amplitude Step 039 cross-catalog hypothesis is stressed for this flyby. It fails only if the prediction-uncertainty model can be independently tightened enough that Juno remains a detectable forecast. The original detections in older analyses would then require re-evaluation as possible systematic errors in less sophisticated OD pipelines.

If the Juno null is overturned by minimal-OD reanalysis: A TEP signal recovered from raw DSN data with TEP-inclusive force modeling would demonstrate that modern OD absorbed the anomaly, validating the OD-absorption conjecture for this mission.

Current status: A statistical power and uncertainty analysis (Step 039b, `step039b_juno_falsification_bounds.json`; `juno_falsification_power_analysis`) quantifies the Juno comparison across uncertainty models: at the Step 008 pooled $\beta = 4.74\times10^{-3}$ with the mean-$\beta$ standard error ($\sigma_{\rm pred,SE} \approx 0.034$ mm/s) and DSN measurement precision ($\sigma_{\rm obs} = 0.020$ mm/s), the combined $\sigma_{\rm total} = 0.039$ mm/s gives $3.17\sigma$ (Cohen's $d \approx 3.2$) for the raw $+0.124$ mm/s prediction; when the Step 008 random-effects prediction uncertainty ($\sigma_{\rm pred,RE} \approx 0.067$ mm/s, capturing cross-flyby geometry scatter $\tau$) is included, the combined $\sigma_{\rm total} = 0.070$ mm/s reduces the raw difference to $1.77\sigma$ ($&lt;2\sigma$, not falsified at the $3\sigma$ threshold), even prior to accounting for Juno's perigee safe-mode tracking interruption and modern stochastic-acceleration OD absorption (which absorbs $\sim 97\%$ of a perigee impulse in Step 012/043). A definitive post-OD test requires MONTE- or ODP-class minimal orbit determination without empirical stochastic accelerations.

## 5.14 Evidence Ledger for Review

The following evidence ledger documents the core model-comparison and cross-check entries required for independent assessment:

- Restricted TEP vs Null: large information-criterion separation ($\Delta$BIC $\approx 16690$ on full n = 9 catalog under the fractional misspecification floor); TEP restricted is strongly favoured over the Null, while the Anderson empirical law fitted on the same catalogue attains the nominally smallest BIC overall.

- Restricted TEP vs Anderson: the Anderson empirical model attains the nominally smaller BIC ($\Delta$BIC $\approx -4.8$ on full n = 9 catalog — parity-level separation, not a decisive preference; under the published-uncertainties-only sensitivity the ordering reverses, $\Delta$BIC $\approx +2767$ in TEP restricted's favour). The restricted model retains the pre-specified-parameter advantage.

- External anchors: GNSS atomic-clock correlation anchors $\lambda_T = 4200$ km under the Paper 6 §2 identification (Step 016 records the bookkeeping consistency of the adopted constant with its own anchor); UCD saturation model gives $S_\oplus \approx 0.35$ (Step 010); Cassini solar conjunction constrains $|\gamma - 1|$ (Section 4.6).

- Null-catalog stress test: full n = 9 raw-catalog likelihood improves by $\Delta\log L \approx 43.0$ over the null after random-effects prediction uncertainty is included (Step 039).

- Failure-mode accounting: velocity-template regime classification via the empirical scale $v_{\rm trans} \approx 16.8$ km/s (phenomenological ansatz, §3.5) and sign-rule consistency checks (Step 007/017).

- Machine-checked manuscript: this document is generated from site/components/*.html and audited against pipeline JSON outputs by Step 027.

## 5.15 Summary

Within these explicitly defined scope boundaries—link-class conditioning, literature-derived $\Delta v$ inputs, and finite-sample heterogeneity—TEP with Temporal Shear suppression within continuous Temporal Topology organizes the Earth flyby anomaly with a single fitted amplitude on pre-specified geometry and defines falsifiable predictions for future clock-carrying planetary encounters.

## 6. Conclusions

This study investigated whether the Temporal Equivalence Principle (TEP), through the clock-sector reduction-artefact channel identified in Paper 0 (v0.16) and scoped to its observable classes in Section 5.2, can organize the Earth flyby anomaly—unexplained velocity shifts observed during spacecraft gravity assists. The analysis of twelve Earth flyby events spanning nine spacecraft (Galileo 1990/1992, NEAR, Cassini, Rosetta 2005/2007/2009, MESSENGER, Juno, Stardust, OSIRIS-REx, BepiColombo) yields the following key findings:

- Six-member TEP ensemble: Inverse-variance fitting (Step 008) enters NEAR ($13.46 \pm 0.13$
mm/s), Galileo 1990 ($3.92 \pm 0.08$ mm/s), Rosetta 2005 ($1.80 \pm
0.03$ mm/s), Cassini ($-2.00 \pm 1.00$ mm/s), Galileo 1992 ($-4.60 \pm 1.00$ mm/s), and MESSENGER ($+0.02 \pm 0.01$ mm/s) under the S/N ≥ 2 pre-fit admission gate (with `strict_sign_gate = false` in the active pipeline configuration). Evaluated on the ungated catalogue prior to S/N filtering, all 7 of 7 published nonzero anomalies (including Rosetta 2007; one-sided binomial $p = 0.0078$) agree in sign with the reference prediction at $\beta_{\rm ref}=10^{-4}$, so the six S/N-qualified detections also satisfy sign concordance without selection circularity. At $\beta_{\rm ref}$, Rosetta 2005 predicts $\Delta v_{\rm TEP} \approx 0.03$ mm/s vs $1.80$ mm/s observed; the full-model fit yields $\beta_{\rm fitted} \approx 5.15\times10^{-3}$. Cassini and MESSENGER carry near-zero reference predictions, so their per-flyby rescalings are poorly constrained and contribute negligible pooled weight. The amplitude-informative gated $\beta_{\rm fit}$ values span roughly a factor of 3.3 ($4.43\times10^{-3}$ to $1.44\times10^{-2}$, with Galileo 1992 at $6.7\times10^{-2}$),
consistent with geometry-dependent modulation (the plasma term is a bounded heuristic envelope coefficient; the stronger Debye attenuation ansatz is quarantined at $S_{\rm plasma}=1$, Section 3.10). The flyby
trajectory-response amplitudes $\beta_{\rm fit}$ are distinct from the
bare conformal coupling $\beta_A = -1.0$ and from the Cassini source
charge $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$. Cassini
PPN compliance is evaluated through the solar source-charge projection
(Section 4.6.1a), not through the fitted flyby amplitudes.

- TEP parameter estimate: The inverse-variance weighted
mean $\beta_{\rm fit} \approx 4.74 \times 10^{-3} \pm 1.28 \times 10^{-3}$ (random-effects standard error; formal inverse-variance error $5.83 \times 10^{-4}$) summarizes the gated ensemble. This is the flyby trajectory-response amplitude, distinct from the bare conformal coupling $\beta_A = -1.0$ and from the Cassini source charge $\alpha_{\rm eff}$. Formal heterogeneity (reduced $\chi^2 \approx 236$, $I^2 \approx
99.6\%$) signals that a single rescaling does not exhaust the geometry physics carried by the envelope (the plasma term is a bounded heuristic envelope coefficient). Bootstrap resampling ($10^4$ draws) and
leave-one-out recomputations in Step 008 show moderate stability (coefficient $\approx 0.092$), with NEAR as the dominant lever.

- PPN compliance via Temporal Topology screening: The Cassini solar conjunction experiment provides the tightest bound on the post-Newtonian light-propagation sector, measuring $\gamma = 1 + (2.1 \pm 2.3) \times 10^{-5}$. This constrains the solar-system Shapiro/light-propagation sector but does not directly test spatial clock-sector covariance, one-way residual shear, or low-density temporal-shear recovery. The fitted $\beta_{\rm fit}$ values are flyby trajectory-response amplitudes, not the Cassini source charge $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$. Cassini PPN compliance is evaluated through the solar source-charge projection (Section 4.6.1a): the adopted Paper 0 v0.16 convention gives $S_\Sigma^{(\odot)} \sim 2\times10^{-6}$, inside the Cassini requirement $S_\Sigma^{(\odot)} \lesssim 1.15\times 10^{-5}$ under the unified unstretched-space metric ($5.8\times 10^{-6}$ isotropic, Paper 0 §7), with the kinetic-operator benchmark evaluating the solar-field screening at $\sim 10^{-15}$ along the 1 AU path. This supports the claim that TEP can reduce to the GR PPN light-propagation limit in the screened solar environment while reserving its discriminating predictions for observables outside the Cassini measurement class.

- TEP suppression by modern orbit determination: Analysis
of the expanded dataset reveals that published null results (MESSENGER,
Rosetta 2007) are consistent with universal-$\beta_{\rm fit}$ null predictions,
while Juno is the sole raw-tension case at
universal $\beta_{\rm fit}$. Post-OD survival factors are withheld until mission
OD configuration yields defensible $F_{\rm OD}$ estimates. Rosetta 2009 is treated as a
published null/bound case with insufficient explicit geometry for the
universal-$\beta_{\rm fit}$ prediction table; Stardust, OSIRIS-REx, and BepiColombo
have no public anomaly report and are not used in quantitative
likelihood.

- Multiple independent lines of evidence: Altitude-dependent
anomaly pattern (see point 6), historical
timeline and the OD filtering mechanism motivate the hypothesis that
modern orbit determination can treat TEP-like signals as systematic
errors. However, mission-specific survival factors are not computed in
this paper: Step 021 withholds $F_{\rm OD}$ until real OD configuration
data are available, and the synthetic Step 012 OD run is quarantined
from manuscript inference.

- Temporal Topology screening support: The response template
organizes the catalogue pattern — null results on high-altitude flybys, where the conformal excursion is shallow, and large anomalies on low-altitude encounters — conditional on observable class: in the coherent two-way class of the published catalogue the organization measures how strongly the recorded anomalies track the trajectory geometry the artefact kernel encodes, while the mechanism-level signature is confined to clock-carrying links (Section 5.2). Altitude alone is not the relevant TEP discriminator; the pattern is governed by the full trajectory-geometry response, which combines altitude, asymmetry, velocity and field structure.

- Systematic uncertainty compression: Transitioning from
empirical characteristic suppression factors to a UCD-derived estimate via
the Self-Consistent Field (SCF) solver and the corrected uncertainty analysis (Step 025) provides a rigorous uncertainty budget. The total relative uncertainty is 152.7%, with heterogeneity contributing 149.4% and systematic uncertainty contributing 29.2%. The systematic component is dominated by characteristic suppression uncertainty (25.0%, Paper 6 UCD) and relaxation length uncertainty (15.0%, SCF theoretical prior). This shift from "parameter fitting" to "systematic prediction" with proper variance decomposition strengthens the theoretical foundation of the TEP analysis.

- Robust statistical checks: The primary gated fit and
model-comparison layer use inverse-variance/Gaussian weighted
likelihoods, while Student's t likelihoods are retained in auxiliary
Bayesian checks to test sensitivity to outliers. Residual diagnostics on
the gated ensemble is consistent with normality (Shapiro–Wilk $p \approx 0.95$, $n = 6$),
so heterogeneity diagnostics, not normality alone, dominate the
statistical interpretation.
(NEAR, Galileo 1990, Rosetta 2005, Cassini, Galileo 1992, MESSENGER).

- Cosmographic CMB-frame directional consistency: Full 3D
spacecraft state vectors from JPL Horizons were tested for CMB-frame
velocity geometry (Step 040, n = 9). The exploratory both-aligned flag
yields Pearson r = −0.57, p = 0.11, and Mann-Whitney U = 7.5, p = 0.79.
Multivariate regression on residual ratio achieves R² = 0.96 (adjusted
R² = 0.94), and an optimal weighted combination of spacecraft and Earth
CMB projections gives r = 0.96, though the optimum sits at the boundary
of the scanned weight grid (w = −2.0) and is read as a boundary-limited
exploratory diagnostic, not a calibrated coefficient. These results are
sample-limited and remain exploratory.

## Significance

The TEP interpretation of the Earth flyby anomaly provides a coherent theoretical framework connecting spacecraft dynamics to fundamental physics. The flyby trajectory-response amplitude $\beta_{\rm fit} \approx 4.74\times10^{-3}$ (weighted mean) is a trajectory-response coefficient, not the bare coupling $\beta_A$ or the Cassini source charge $\alpha_{\rm eff}$. Interpreted as the clock-sector amplitude response under the amplitude–shear split, the flyby catalogue's geometric organization is consistent with solar system constraints and organizes the anomalous velocity-shift pattern. What that organization attests is conditional on the tracking-link class: the published catalogue is assigned to the coherent-transponder class on the literature-based ledger of Section 5.2, pending encounter-specific tracking verification; in that class the clock-sector term is absent at the observable (Section 5.2), so the catalogue pattern constrains the response template's structure while the mechanism's claim is carried by the falsifiable prediction of an m/s-scale signature on the first clock-carrying flyby pass.

Unlike ad hoc modifications to gravity, the TEP framework preserves all successes of general relativity in solar system tests while supplying a structured account of the anomaly pattern in the specific regime of planetary gravity assists — an account whose mechanism-level claim is scoped to the clock-carrying link classes of Section 5.2 rather than asserted for the recorded catalogue. The geometric amplitude response, calibrated by independent UCD saturation analysis, is essential for the flyby amplitude and environmental response: without it, the required $\beta_{\rm fit}$ would not produce the observed trajectory-dependent residuals.

Template-organization metrics: The quantitative validation summarizes how the restricted geometric response template organizes the published catalogue (measuring empirical template compression rather than mechanism-level confirmation in the coherent two-way observable):

- Effect sizes: Cohen's $d$ relative to the published null population ($n_{\rm null}=3$) yields a large effect for NEAR ($d \approx 2.5$), medium for Galileo 1990 ($d \approx 0.73$), large-in-magnitude negative for Galileo 1992 ($d \approx -0.87$), and small effects for Rosetta 2005 ($d \approx 0.34$) and Cassini ($d \approx -0.38$). The coefficient of variation CV $\approx 149\%$ across the six gated $\beta_{\rm fit}$ fits is inflated by the near-zero-prediction members and reflects genuine geometry-dependent modulation at fixed envelope.

- Model comparison: On the full n = 9 catalog under the fractional misspecification floor, Step 026 gives a large information-criterion separation between TEP restricted and Null ($\Delta{\rm BIC}\approx16690$); the Anderson empirical law, whose two parameters are fitted on the same catalogue, attains the nominally smallest BIC overall but only at parity-level separation ($\Delta{\rm BIC}\approx4.8$), an ordering that reverses in TEP restricted's favour under the published-uncertainties-only sensitivity

- Akaike model weight favours the Anderson empirical tier ($\approx 0.86$ versus $\approx 0.14$ for TEP restricted, full $n=9$ catalogue under the fractional misspecification floor) — a parity-level ordering that reverses under the published-uncertainties sensitivity; the TEP restricted and flexible tiers remain far above the Null and are documented in Step 026 outputs.

- Robustness: Bootstrap resampling, leave-one-out
recomputations, and auxiliary robust checks are reported in Step 008/013.

- Prediction accuracy: Step 008 reports $R^2 \approx 0.85$ (correlation $r \approx 0.93$) between pooled-$\beta$ predicted and observed anomalies across the six-member ensemble; prediction intervals are bootstrap-derived.

- Residual analysis: Shapiro–Wilk $p \approx 0.95$
on the gated six-member ensemble; heterogeneity diagnostics dominate formal tests.

- Sensitivity analysis: All parameters stable across
plausible ranges; fit stability maintained

The catalog of Earth flyby events (six published nonzero entries — four firm detections and two ~2σ marginal entries, Cassini and MESSENGER — plus three published nulls/bounds and several without public anomaly reports) provides the empirical substrate. Step 026's headline likelihood uses the full n = 9 catalog with a fractional misspecification floor ($\sigma_{{\rm sys},i} = CV_\beta|\hat y_i|$, $CV_\beta \approx 0.47$, measured from the catalogue's own amplitude heterogeneity and invariant under the field-normalization anchor). The archived comparison gives $\Delta{\rm BIC}\approx1.67\times10^{4}$ for the restricted geometric template relative to Null and near-parity with the catalogue-fitted Anderson tier. These scores describe catalogue organization under the documented fitting convention, not support for the clock-sector mechanism. Section 4.5 states the outstanding likelihood-optimum and nuisance-accounting checks; the physical clock-sector test is a clock-carrying pass, with the profile independently constrained by clock observations.

## Robustness Assessment

Several potential concerns have been investigated and addressed through rigorous statistical analysis (Step 024, Step 025, Step 026):

Systematic error discrimination: trajectory asymmetry ($\cos\delta_{\rm in} - \cos\delta_{\rm out}$) organizes the adopted catalogue template, and while the hardware and environmental systematics explicitly modelled in Step 024 (0.12 mm/s combined) do not reproduce this pattern, this correlation does not by itself exclude unmodeled geometry-dependent tracking or orbit-determination reconstruction errors pending raw DSN reanalysis. Within the gated n = 6 ensemble the ordering is positive but imperfect (Spearman ρ = +0.66, p = 0.156), a qualitative consistency check rather than a primary discriminator. On the ungated re-transcribed catalogue, all 7 of 7 published nonzero anomalies (and all 6 S/N ≥ 2 entries) agree in sign with the reference prediction, while the geometry envelope carries seven nominal deterministic coefficients plus one fitted amplitude against six S/N-qualified anomalies, so the archived scores characterize template organization under the adopted convention, not confirmation of the clock-sector mechanism in the coherent two-way observable. The broader catalogue rank correlations are complementary diagnostics on overlapping data: asymmetry-vs-magnitude gives ρ = 0.56 (p = 0.11, n = 9) and ρ = 0.68 (p = 0.090, n = 7 nonzero anomalies) in the Step 009 ordering diagnostic. See Section 5.5 for comprehensive systematic uncertainty budget.

Data provenance: The analysis relies on published anomaly values from Anderson et al. (2008) rather than independent DSN re-analysis. This is addressed by: (a) cross-referencing multiple literature sources for consistency, (b) demonstrating that TEP predictions match the observed anomaly pattern (altitude dependence, trajectory geometry), (c) providing a framework for raw DSN data re-analysis to independently test the suppression hypothesis. The current repository state provides mission discovery, structured requests, and an ingestion scaffold; in typical runs most missions do not yet have indexed local raw Doppler products available for reprocessing (pipeline ingest status `no_indexed_products`). The DSN pathway is therefore an explicit falsification route, not a completed independent reanalysis across the catalog.

β<sub>fit</sub> scatter as physical modulation: Fitted β<sub>A,eff</sub> spans roughly a factor of 3.3 across the tightly constrained members ($4.43\times10^{-3}$ to $1.44\times10^{-2}$), extending to $6.7\times10^{-2}$ for Galileo 1992 and formally divergent for the near-zero-prediction Cassini and MESSENGER members (Step 009), reflecting geometry-dependent modulation: altitude ($J_2$ gradient suppression), perigee latitude (inclination-dependent coupling), and velocity (the response template's disformal branch); the plasma term is a bounded heuristic envelope coefficient (Section 3.10). Step 009 does not report a formal variance decomposition because n = 6 renders ANOVA-style partitioning statistically underpowered. Instead, it reports deterministic geometry modulation metrics: beta scatter statistics, full-catalog detection-pattern classification, and Spearman rank correlation (ρ = 0.56 over the n = 9 catalogue; ρ = +0.66 in the gated ensemble). Leave-one-out recomputation on the six-member ensemble reports stability coefficient ≈ 0.092. The UCD-derived characteristic suppression $S_\oplus \approx 0.35$ provides a cross-scale prior. See Section 4.3 for the quantitative geometry modulation assessment and Section 5.5 for detailed interpretation.

Cassini status: At $\beta_{\rm ref}=10^{-4}$ the Step 007 envelope yields a negative $\Delta v_{\rm TEP}$ of negligible magnitude, and the corrected published anomaly is $-2.00 \pm 1.00$ mm/s, so Cassini is sign-consistent at $\beta_{\rm ref}$ and enters the gated ensemble (n = 6) — its residual tension is amplitude-only. Galileo 1992, transcribed as a null in the earlier catalogue, is a genuine negative detection ($-4.60 \pm 1.00$ mm/s) whose magnitude is underpredicted ($\approx -0.33$ mm/s at the pooled amplitude). Addressing these amplitude shortfalls requires independent DSN OD or envelope refinements, not only a universal rescaling of $\beta_{\rm fit}$.

Juno falsification pressure: At the refit weighted-mean $\beta_{\rm fit}$, Step 039 predicts $\Delta v_{\rm TEP}^{\rm raw}\approx +0.12$ mm/s with uncertainty $\sim 0.03$ mm/s while the published bound is $0.00\pm 0.02$ mm/s. This raw-tension case is the headline stress test on universal $\beta_{\rm fit}$ before mission-specific OD survival factors are supplied (Step 021). Independent raw DSN re-analysis with TEP-inclusive orbit determination is required to adjudicate the tension.

Sample size as complete dataset: The analysis includes all accessible Earth gravity assist flybys with adequate DSN tracking between 1990–2020. The rarity of suitable flyby events (low altitude, Doppler tracking, no major maneuvers) means only a handful of literature anomalies exist; six enter the gated $\beta_{\rm fit}$ ensemble.    Step 008 bootstrap 95% CI $[4.42\times10^{-3},\,1.47\times10^{-2}]$ brackets the gated fits (bootstrap median $4.77\times10^{-3}$). Additional flybys would test envelope refinements rather than restate the same $n=6$ compression.

PPN compliance: The UCD-derived characteristic suppression $S_\oplus \approx 0.35$ is determined from the UCD saturation model. The flyby trajectory-response amplitudes $\beta_{\rm fit}$ are distinct from the Cassini source charge $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$; Cassini PPN compliance is evaluated through the solar source-charge projection (Section 4.6.1a), where the adopted Paper 0 v0.16 convention gives $S_\Sigma^{(\odot)} \sim 2\times10^{-6}$, inside the Cassini requirement $S_\Sigma^{(\odot)} \lesssim 1.15\times 10^{-5}$ under the unified unstretched-space metric ($5.8\times 10^{-6}$ on the isotropic-conformal benchmark).

Model comparison: Step 026 on the full n = 9 catalog gives a large information-criterion separation between TEP restricted and Null ($\Delta{\rm BIC}\approx16690$). The Anderson empirical model, fitted on the same catalogue, attains the nominally smallest BIC overall ($\Delta{\rm BIC}\approx4.8$ ahead of TEP restricted — parity-level; $\Delta{\rm BIC}\approx16694$ ahead of Null), confirming that trajectory asymmetry carries genuine signal. The TEP flexible model (3 parameters) is penalized by its parameter count and does not outperform the restricted model on BIC. Residual diagnostics on the gated ensemble are consistent with normality (Shapiro–Wilk $p\approx0.95$, $n=6$).

Independent validation pathways: Several approaches can independently test the TEP hypothesis without relying on the published anomaly values:

- Raw DSN data re-analysis: Analysis of raw DSN tracking
archives from NASA's Planetary Data System using minimal orbit
determination (reduced gravity field expansion, unfiltered Doppler, no
continuity penalties) would test whether TEP signals are filtered by
modern orbit determination methods. This would provide an important test
of the suppression hypothesis.

- Additional flyby analysis: Earth gravity assist
missions provide opportunities for independent detection. Analysis with
both standard and minimal orbit determination methods would test the
suppression prediction.

- GNSS clock correlation: GNSS atomic clock
correlation analysis provides a
cross-consistency check with the GNSS/UCD terrestrial calibration on the transition radius ($R_{\rm sol} \approx
4200$ km). This shared-input dependency is disclosed explicitly to avoid circular claims of independent validation.

- Lunar Laser Ranging: Precision LLR analysis in related work 
reports a synodic-phase signal consistent with the screening mechanism and 
Temporal Topology Saturation Scale (UCD) framework. Independent LLR validation would 
strengthen the screening mechanism established in this analysis.

## Data Availability

Spacecraft trajectories are available through the JPL Horizons ephemeris service. Literature anomaly values are from Anderson et al. (2008) and companion publications. Analysis code and processed data products are available at https://github.com/matthewsmawfield/TEP-EFA with archived DOI at 10.5281/zenodo.19454862.

## Acknowledgments

The NASA Deep Space Network and Jet Propulsion Laboratory provided the precision Doppler tracking that enabled flyby anomaly detection. The JPL Horizons system provided trajectory reconstruction. This work utilizes published literature values from the Orbit Determination Program analyses by Anderson et al. and collaborators. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. The author declares no conflicts of interest.

## Additional Considerations

Several avenues for extending this analysis are identified:

Raw DSN data re-analysis: Analysis of raw DSN tracking archives from NASA's Planetary Data System using minimal orbit determination (reduced gravity field expansion, unfiltered Doppler, no continuity penalties) would test whether TEP signals are filtered by modern orbit determination methods. This provides an important test of the suppression hypothesis.

Extended spacecraft sample: Additional flyby events would increase the sample size beyond the current six published nonzero entries (all six sign-consistent in the inverse-variance $\beta_{\rm fit}$ ensemble, with Cassini and MESSENGER marginal at ~2σ). A sample of $n \approx 74$ primary detections would provide sufficient statistical power to distinguish between geometry-dependent modulation of $\beta_{\rm fit}$ and a single universal coupling constant at 80% power (conservative estimate: $n \approx 153$).

- Full numerical Temporal Shear Suppression solver: Implementation of a
numerical Temporal Topology field solver (e.g., using the shooting method or
relaxation techniques) validates the phenomenological gradient
suppression model used in this analysis. This enables prediction of the
Temporal Topology profile without the phase-boundary approximation and
could explain the observed $\beta_{\rm fit}$ scatter through detailed
density-dependent effects.

- Inclination-dependent modeling: Incorporation of
Earth's oblateness ($J_2$) and latitude-dependent density variations
into the TEP model could explain part of the observed $\beta_{\rm fit}$ scatter.
Spacecraft with different orbital inclinations sample different
gravitational field geometries, which modulate the Temporal Topology field
strength.

- Disformal coupling exploration: Extension to
scalar-tensor theories with disformal coupling terms introduces
velocity-dependent gradient suppression that could explain the
correlation between fitted $\beta_{\rm fit}$ and flyby velocity. This provides a
more general framework for understanding the geometry-dependence of the
TEP effect.

- Local-time plasma effects: A first-principles scalar–plasma coupling
derived from the TEP action would test whether ionospheric density
variations with local time modulate the clock-sector response. At
present plasma enters only as a bounded heuristic envelope coefficient,
with the stronger Debye attenuation ansatz quarantined at
$S_{\rm plasma}=1$ (Section 3.10); this bullet records the question as an
open refinement, not a current mechanism.

## References

- Anderson, J. D., Campbell, J. K., Ekelund, J. E., Ellis, J., &amp; Jordan, J. F. 2008, "Anomalous Orbital-Energy Changes Observed during Spacecraft Flybys of Earth," *Phys. Rev. Lett.*, 100, 091102

- Anderson, J. D., &amp; Nieto, M. M. 2009, "Astrometric solar-system anomalies," in *Relativity in Fundamental Astronomy*, IAU Symp. 261, 189

- Antreasian, P. G., &amp; Guinn, J. R. 1998, "Investigations into the Unexpected Delta-V during the Earth Gravity Assist of NEAR," Paper AAS 98-428

- Bertotti, B., Iess, L., &amp; Tortora, P. 2003, "A test of general relativity using radio links with the Cassini spacecraft," *Nature*, 425, 374

- Brax, P., van de Bruck, C., Davis, A.-C., Khoury, J., &amp; Weltman, A. 2004, "Detecting dark energy in orbit: The cosmological chameleon," *Phys. Rev. Lett.*, 93, 200405

- Einstein, A. 1915, "Die Feldgleichungen der Gravitation," *Sitzungsberichte der Preussischen Akademie der Wissenschaften*, 844

- Halsey, D., et al. 2012, "Anomalous Earth flybys: Status and developments," *Adv. Space Res.*, 50, 362

- Khoury, J., &amp; Weltman, A. 2004, "Chameleon cosmology," *Phys. Rev. D*, 69, 044026

- Lämmerzahl, C., Preuss, O., &amp; Dittus, H. 2006, "Is the physics within the Solar system understood?" in *Lasers, Clocks and Drag-Free Control*, 75, 75

- McCulloch, M. E. 2008, "Modelling the Pioneer anomaly as modified inertia," *MNRAS*, 389, L57

- Meeus, J. 1998, *Astronomical Algorithms*, 2nd edn. (Richmond: Willmann-Bell)

- Mota, D. F., &amp; Shaw, D. J. 2007, "Strongly coupled chameleon fields," *Phys. Rev. Lett.*, 97, 151102

- Nieto, M. M., &amp; Anderson, J. D. 2007, "Search for a solution of the Pioneer anomaly," *Contemp. Phys.*, 48, 41

- Page, G., &amp; McCulloch, M. E. 2009, "Modelling the flyby anomalies using a modification of inertia: Further investigations," *Int. J. Astron. Astrophys.*, 3, 1

- Schive, H.-Y., Chiueh, T., &amp; Broadhurst, T. 2014, "Understanding the Core-Halo Relation of Quantum Wave Dark Matter from 3D Simulations," *Phys. Rev. Lett.*, 113, 261302

- Turyshev, S. G., &amp; Toth, V. T. 2010, "The Pioneer anomaly," *Living Rev. Relativ.*, 13, 4

- Will, C. M. 2014, "The confrontation between general relativity and experiment," *Living Rev. Relativ.*, 17, 4

- Folkner, W. M., et al. 2022, "Planetary ephemeris DE440," *IPN Progress Report*, 42-284, 1

- JPL Horizons, "NASA/JPL Horizons System" https://ssd.jpl.nasa.gov/horizons/ (accessed 2024)

- Morley, T., &amp; Budnik, F. 2007, "Rosetta Navigation at its First Earth-Swingby," *Proceedings of the 20th International Symposium on Space Flight Dynamics*

- Müller, J., Soffel, M., &amp; Klioner, S. A. 2008, "Geodesy and relativity," *Journal of Geodesy*, 82, 133

- Müller, J., et al. 2010, "Relativistic models for spacecraft tracking," *Acta Astronautica*, 67, 975

- Aksenov, E. L., &amp; Tuchin, A. G. 2020, "Earth flyby anomalies and the general relativistic theory of the Kerr gravitational field," *MNRAS*, 492, 3703

- Ciufolini, I., &amp; Pavlis, E. C. 2004, "A confirmation of the general relativistic prediction of the Lense-Thirring effect," *Nature*, 431, 958

- IERS Conventions 2010, IERS Technical Note No. 36, eds. Petit, G. &amp; Luzum, B.

- Brax, P., &amp; Burrage, C. 2014, "Constraining screened modified gravity with the CASPEr experiment," *Phys. Rev. D*, 90, 104009

- Lemoine, F. G., et al. 1998, "The Development of the NASA GSFC and NIMA Joint Geopotential Model," in *Proceedings of the International Symposium on Gravity, Geoid, and Marine Geodesy*, Tokyo, Japan

- Pavlis, N. K., et al. 2012, "The development and evaluation of the Earth Gravitational Model 2008 (EGM2008)," *J. Geophys. Res.*, 117, B04406

- Mocz, P., Vogelsberger, M., Robles, V., et al. 2018, "Galaxy Halos from Fuzzy Dark Matter," *Phys. Rev. Lett.*, 121, 141102

- Moyer, T. D. 2000, *Formulation for Observed and Computed Values of Deep Space Network Data Types*, JPL Publication 00-7

- Burrage, C., &amp; Sakstein, J. 2016, "Tests of Ambient Symmetry Restoration," *Living Rev. Relativ.*, 21, 1

- Upadhye, A., Hu, W., &amp; Khoury, J. 2012, "Quantum stability of chameleon field theories," *Phys. Rev. Lett.*, 109, 041301

- Joyce, A., Jain, B., Khoury, J., &amp; Trodden, M. 2015, "Beyond the cosmological standard model," *Phys. Rept.*, 568, 1

- Kass, R. E., &amp; Raftery, A. E. 1995, "Bayes Factors," *J. Am. Stat. Assoc.*, 90, 773

- Clifton, T., Ferreira, P. G., Padilla, A., &amp; Skordis, C. 2012, "Modified gravity and cosmology," *Phys. Rept.*, 513, 1

- Higgins, J. P., &amp; Thompson, S. G. 2002, "Quantifying heterogeneity in a meta-analysis," *Stat. Med.*, 21, 1539

### TEP Research Series

Smawfield, M. L. *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: 10.5281/zenodo.16921911

Smawfield, M. L. *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: 10.5281/zenodo.17127229

Smawfield, M. L. *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: 10.5281/zenodo.17517141

Smawfield, M. L. *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: 10.5281/zenodo.17860166

Smawfield, M. L. *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: 10.5281/zenodo.17982540

Smawfield, M. L. *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: 10.5281/zenodo.18004832

Smawfield, M. L. *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: 10.5281/zenodo.18064365

Smawfield, M. L. *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: 10.5281/zenodo.18059250

Smawfield, M. L. *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: 10.5281/zenodo.18064581

Smawfield, M. L. *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: 10.5281/zenodo.18109760

Smawfield, M. L. *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: 10.5281/zenodo.18165798

Smawfield, M. L. *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: 10.5281/zenodo.18209702

Smawfield, M. L. *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: 10.5281/zenodo.19000827

Smawfield, M. L. *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: 10.5281/zenodo.19102061

## Data Availability &amp; Reproducibility

This work follows open-science practices. All results are fully reproducible from the repository’s raw inputs using the documented pipeline. All numerical results, figures, and statistics are generated by deterministic Python scripts processing real mission ephemerides and peer-reviewed, published Doppler-derived anomaly measurements.

### Cross-corpus theory registry (TEP-EFA ↔ manuscript series)

The flyby pipeline fixes a single set of conventions so numerical results agree with the manuscript HTML in `site/components/`. The following table is the authoritative map between this repository and the numbered theory papers under `manuscripts/` (Paper 0 = Jakarta, Paper 6 = UCD, Paper 10 = Caracas COS, Paper 12 = JWST Kos, etc.).

Canonical EFA quantities aligned to the manuscript corpus

| Quantity | EFA implementation | Manuscript anchor(s) |
| --- | --- | --- |
| Matter metric / conformal factor | `A(φ) = exp(β_A φ / M<sub>Pl</sub>)` (reduced Planck mass in GeV) | Paper 0 (Jakarta) axioms; Papers 4–5, 8–9, 12 (notation) |
| Artefact kernel (flyby sector) | Clock-sector excursion: Δv<sub>TEP</sub> = β × g(geometry) with canonical normalization Δφ = ψ<sub>uns</sub> = M<sub>⊕</sub>/(4π M<sub>Pl</sub><sup>2</sup> R<sub>⊕</sub>) = 1.39×10⁻⁹ (dimensionless η); the earlier force mapping F<sub>φ</sub> = β<sub>A,eff</sub> c² ∇φ / M<sub>Pl</sub> was retired at v0.3 under the amplitude–shear split (S<sub>Σ</sub> screened; the effect lives in the clock sector) | Paper 0 v0.16; Paper 4 (Phantom Mass); methodology §3.2 |
| Density minimum φ<sub>min</sub>(ρ) — candidate realization | φ = Λ [ n Λ<sup>n+4</sup> M<sub>Pl</sub> / (2 β ρ<sub>GeV4</sub>) ]<sup>1/(n+1)</sup>, evaluated at reference response amplitude β = 10⁻⁴, in `step_007_tep_model.py`, `step_011_trajectory_integration.py`, `step_019_3d_field_integration.py` | Paper 10 Appendix C (aligned to this form, May 2026); Paper 6 (scaling φ ∝ ρ<sup>−1/(n+1)</sup> only). Paper 0 v0.16 states screening/PPN mapping; it does not fix the closed φ(ρ) line — this prescription is a candidate microscopic realization, not the defining Temporal-Topology mechanism, and β here is the channel response amplitude, not the bare coupling β_A = −1. |
| PPN γ (magnitude checks) | `ppn_gamma_deviation`: report \|γ − 1\| ≈ 2 β<sub>A,eff</sub>² vs Cassini | Paper 0 Sec. 7: γ − 1 = −2 α<sub>0</sub>α<sub>eff</sub> (DEF); Papers 5, 11, 12 (screened limit narrative) |
| Screening / UCD radius | R<sub>sol</sub> ≈ 4146 km, S<sub>⊕</sub> = (R<sub>⊕</sub> − R<sub>sol</sub>)/R<sub>⊕</sub> ≈ 0.35; ρ<sub>T</sub> ≈ 20 g cm<sup>−3</sup> | Paper 6 (UCD); Step 010 / `physics.py` |
| Scalar field equation (sign reference) | Pipeline uses explicit `field_gradient` / TEP screening relaxation outside Earth; trace source uses β_A, M<sub>Pl</sub> as in Step 007 comments | Paper 12 Appendix A.1.2: K(φ)□φ − V′ = −(β/M<sub>Pl</sub>)T<sup>(matter)</sup> (Einstein-frame convention; overall sign of T follows chosen action) |

*Residual ambiguities:* Individual papers sometimes use illustrative potentials or linearized V<sub>eff</sub> without the Einstein-frame factor 2; any updated analytic appendix should match the table above before reusing EFA numerical φ<sub>Earth</sub>, φ<sub>space</sub>, or Δφ in secondary calculations.

### Repository &amp; Code

The repository contains a deterministic, version-controlled analysis pipeline with analysis steps for Earth flyby trajectory data. All steps are orchestrated by `scripts/run_all.py` with comprehensive logging.

#### Repository Structure

```
TEP-EFA/
├── data/                          # Raw and processed data
│   ├── raw/                       # Raw DSN tracking, trajectories
│   │   ├── dsn_tracking/           # Deep Space Network archives
│   │   ├── flyby_trajectories/     # JPL Horizons ephemeris data
│   │   └── spice_kernels/        # Navigation SPICE kernels
│   └── processed/                 # Pipeline outputs (JSON/CSV)
├── scripts/
│   ├── steps/                     # Analysis pipeline steps
│   │   ├── step_001_download_spice.py
│   │   ├── step_002_spice_to_json.py
│   │   ├── step_003_archival_data_mining.py
│   │   ├── step_004_jpl_horizons_fetch.py
│   │   ├── step_005_dsn_data_ingestion.py
│   │   ├── step_006_dsn_framework.py
│   │   ├── step_007_tep_model.py
│   │   ├── step_008_fitting.py
│   │   ├── step_009_variance_analysis.py
│   │   ├── step_010_tep_first_principles.py
│   │   ├── step_011_trajectory_integration.py
│   │   ├── step_012_od_filter_simulation.py
│   │   ├── step_013_cross_validation.py
│   │   ├── step_014_sensitivity_analysis.py
│   │   ├── step_015_hierarchical_bayesian.py
│   │   ├── step_016_gnss_validation.py
│   │   ├── step_017_plasma_modulation.py
│   │   ├── step_018_space_weather.py
│   │   ├── step_019_3d_field_integration.py
│   │   ├── step_020_plasma_environment_reconstruction.py
│   │   ├── step_021_mission_specific_od_absorption.py
│   │   ├── step_022_atmospheric_drag_simulation.py
│   │   ├── step_023_thermal_recoil_modeling.py
│   │   ├── step_024_systematic_error_monte_carlo.py
│   │   ├── step_025_corrected_uncertainty.py
│   │   ├── step_026_stable_model_comparison.py
│   │   ├── step_027_claim_consistency_audit.py
│   │   ├── step_028_dsn_processing.py
│   │   ├── step_029_read_trk234.py
│   │   ├── step_030_juno_reanalysis.py
│   │   ├── step_031_pds_search.py
│   │   ├── step_032_tep_suppression.py
│   │   ├── step_033_iri_trajectory_profile.py
│   │   ├── step_034_covariant_holonomy.py
│   │   ├── step_035_cross_corpus_export.py
│   │   ├── step_036_final_report.py
│   │   ├── step_037_visualizations.py
│   │   ├── step_038_extract_3d_vectors.py
│   │   ├── step_039_flyby_prediction_table.py
│   │   ├── step_040_cosmographic_shear.py
│   │   ├── step_041_envelope_heuristic_sensitivity.py
│   │   ├── step_042_time_resolved_cosmography.py
│   │   ├── step_043_clock_sector_od.py
│   │   ├── step_044_lense_thirring.py
│   │   └── step_anderson_audit.py
│   ├── utils/                     # Utility functions
│   └── build_markdown.js          # Manuscript builder
├── site/
│   └── components/                # Manuscript HTML sections
├── config/                        # Pipeline configuration
│   └── pipeline_config.json
├── logs/                          # Per-step execution logs
├── requirements.txt               # Python dependencies
├── README.md                      # Documentation
└── LICENSE                        # CC-BY-4.0
```

### Data Provenance

| Data Source | Provider | Access Method | Size | Location |
| --- | --- | --- | --- | --- |
| JPL Horizons Ephemeris | NASA/JPL | Astroquery API | ~2 MB | `data/raw/flyby_trajectories/` |
| DSN Doppler Archives | NASA DSN | Literature values | ~500 KB | Anderson et al. (2008) |
| Flyby Anomaly Catalog | Peer-reviewed literature | Manual compilation | ~50 KB | `results/step003_archival_flyby_catalog.json` |
| SPICE Kernels | NASA NAIF | Auto-downloaded | ~100 MB | `data/raw/spice_kernels/` |

### Pipeline Architecture

The analysis pipeline comprises 42 deterministic steps organized into logical groups. Each step is a standalone Python script in `scripts/steps/` that produces JSON outputs and detailed logs in `logs/step_*.log`.

#### Complete Step Inventory & Runtime

| Group | Step | Script | Description | Runtime |
| --- | --- | --- | --- | --- |
| Phase 1: Data Acquisition & Preparation (001-006) |  |  |  |  |
| Data | 001 | `step_001_download_spice.py` | SPICE kernel download (NAIF archive) | ~30s |
| Data | 002 | `step_002_spice_to_json.py` | SPICE to JSON conversion | ~1s |
| Data | 003 | `step_003_archival_data_mining.py` | Archival flyby catalog compilation | ~2s |
| Data | 004 | `step_004_jpl_horizons_fetch.py` | JPL Horizons ephemeris data fetch | ~5s |
| Data | 005 | `step_005_dsn_data_ingestion.py` | DSN tracking data ingestion | ~1s |
| Data | 006 | `step_006_dsn_framework.py` | DSN raw data acquisition framework | ~1s |
| Phase 2: Core Physics & Geometry Modulation Analysis (007-010) |  |  |  |  |
| Core | 007 | `step_007_tep_model.py` | TEP Temporal Topology model with screening | ~1s |
| Core | 008 | `step_008_fitting.py` | β parameter fitting with PPN validation | ~1s |
| Core | 009 | `step_009_variance_analysis.py` | Deterministic geometry modulation analysis (formal variance partitioning deferred to n &ge; 10) | ~2s |
| Core | 010 | `step_010_tep_first_principles.py` | UCD saturation derivation | ~10s |
| Phase 3: Trajectory & Observational Pipeline (011-012) |  |  |  |  |
| Traj | 011 | `step_011_trajectory_integration.py` | Numerical trajectory integration | ~5s |
| OD | 012 | `step_012_od_filter_simulation.py` | Synthetic OD diagnostic: noise-only control, perigee state bias vs TEP truth, station/two-way sensitivity (not F_OD) | ~10s |
| Phase 4: Validation & Robustness (013-016) |  |  |  |  |
| Valid | 013 | `step_013_cross_validation.py` | Cross-validation analysis | ~5s |
| Valid | 014 | `step_014_sensitivity_analysis.py` | Parameter sensitivity analysis | ~2s |
| Valid | 015 | `step_015_hierarchical_bayesian.py` | Hierarchical Bayesian model | ~30s |
| Valid | 016 | `step_016_gnss_validation.py` | GNSS atomic clock validation | ~1s |
| Phase 5: Extended Physics (017-019) |  |  |  |  |
| Phys | 017 | `step_017_plasma_modulation.py` | Plasma-dependent gradient modulation | ~2s |
| Phys | 018 | `step_018_space_weather.py` | Space weather correlation analysis | ~1s |
| Phys | 019 | `step_019_3d_field_integration.py` | 3D integrator diagnostic (idealized analytic hypoflyby; see JSON metadata) | ~1s |
| Phase 6: Plasma & Environmental (020-023) |  |  |  |  |
| Plasma | 020 | `step_020_plasma_environment_reconstruction.py` | Plasma environment reconstruction | ~3s |
| OD | 021 | `step_021_mission_specific_od_absorption.py` | Mission-specific OD absorption | ~1s |
| Env | 022 | `step_022_atmospheric_drag_simulation.py` | Atmospheric drag simulation | ~1s |
| Env | 023 | `step_023_thermal_recoil_modeling.py` | Thermal recoil modeling | ~1s |
| Phase 7: Statistical Analysis (024-026) |  |  |  |  |
| Stat | 024 | `step_024_systematic_error_monte_carlo.py` | Systematic error Monte Carlo analysis | ~5s |
| Stat | 025 | `step_025_corrected_uncertainty.py` | Corrected uncertainty analysis | ~1s |
| Stat | 026 | `step_026_stable_model_comparison.py` | Stable model comparison | ~2s |
| Phase 8: Advanced Topics (027-028) |  |  |  |  |
| Audit | 027 | `step_027_claim_consistency_audit.py` | Claim consistency audit | ~1s |
| DSN | 028 | `step_028_dsn_processing.py` | DSN processing framework | ~1s |
| Phase 9: DSN Reanalysis (029-032) |  |  |  |  |
| DSN | 029 | `step_029_read_trk234.py` | Read TRK-2-34 data format | ~1s |
| DSN | 030 | `step_030_juno_reanalysis.py` | Juno 2013: archive ingest, pairwise-Doppler residual proxy (& optional Horizons range/velocity batch; not MONTE/ODP OD) | ~5s |
| DSN | 031 | `step_031_pds_search.py` | NASA PDS archive search | ~2s |
| DSN | 032 | `step_032_tep_suppression.py` | TEP suppression analysis | ~1s |
| Phase 10: Advanced Analysis (033-035) |  |  |  |  |
| IRI | 033 | `step_033_iri_trajectory_profile.py` | Continuous IRI trajectory profiles | ~5s |
| Holo | 034 | `step_034_covariant_holonomy.py` | Covariant temporal shear impulse | ~1s |
| Export | 035 | `step_035_cross_corpus_export.py` | Cross-corpus parameter export | ~1s |
| Phase 11: Reporting & Extended Validation (036-042) |  |  |  |  |
| Report | 036 | `step_036_final_report.py` | Final report generation | ~1s |
| Fig | 037 | `step_037_visualizations.py` | Publication-quality figure generation | ~3s |
| Vec | 038 | `step_038_extract_3d_vectors.py` | Extract 3D state vectors from JPL Horizons | ~2s |
| Pred | 039 | `step_039_flyby_prediction_table.py` | Flyby prediction table with uncertainty-aware classification | ~1s |
| Cosmo | 040 | `step_040_cosmographic_shear.py` | Cosmographic temporal shear modulation test | ~1s |
| Env | 041 | `step_041_envelope_heuristic_sensitivity.py` | Geometry envelope heuristic sensitivity | ~1s |
| Cosmo | 042 | `step_042_time_resolved_cosmography.py` | Time-resolved cosmography (after Horizons + optional Juno DSN sidecar) | ~2s |
| Analysis | 043 | `step_043_clock_sector_od.py` | Clock-sector forward model: proper-time excursion and reconstructed apparent velocity on clock-carrying links | ~1s |
| Analysis | 044 | `step_044_lense_thirring.py` | Lense–Thirring frame-dragging bound check on the flyby trajectories (auxiliary; not in CORE_STEPS) | ~2s |
| Audit | — | `step_anderson_audit.py` | Primary-source catalogue cross-check against Anderson et al. tabulations (auxiliary; not in CORE_STEPS) | <1s |

#### Total Runtime Summary

| Component | Steps | Runtime |
| --- | --- | --- |
| Data Acquisition (001-006) | 6 | ~40s |
| Core Physics & Variance (007-010) | 4 | ~14s |
| Trajectory & Observational (011-012) | 2 | ~8s |
| Validation & Robustness (013-016) | 4 | ~38s |
| Extended Physics (017-019) | 3 | ~4s |
| Plasma & Environmental (020-023) | 4 | ~6s |
| Statistical Analysis (024-026) | 3 | ~8s |
| Advanced Topics (027-028) | 2 | ~2s |
| DSN Reanalysis (029-032) | 4 | ~9s |
| Advanced Analysis (033-035) | 3 | ~7s |
| Reporting & Extended Validation (036-043) | 8 | ~13s |
| Total | 43 | ~2 min |

### Reproduction Instructions

#### Quick Start (Full Reproduction)

```
# 1. Clone repository
git clone https://github.com/matthewsmawfield/TEP-EFA.git
cd TEP-EFA

# 2. Install dependencies
pip install -r requirements.txt

# 3. Run full pipeline (generates all results & figures)
python scripts/run_all.py

# 4. Results are located in:
#    - results/          (JSON data products and figures)
#    - logs/             (Detailed execution logs)
#    - site/dist/        (Built static site)
```

#### System Requirements

| Component | Minimum | Recommended | Tested On |
| --- | --- | --- | --- |
| CPU | 2 cores | 4+ cores | Apple M4 Pro (14-core) |
| RAM | 4 GB | 8 GB | 24 GB (M4 Pro) |
| Storage | 500 MB | 1 GB | NVMe SSD |
| Runtime | ~2 min | ~1 min | ~40s (M4 Pro) |

#### Key Analysis Outputs

- `results/step003_archival_flyby_catalog.json` — Literature flyby catalog with provenance

- `results/step007_tep_predictions.json` — TEP model predictions for all modeled flybys

- `results/step008_fitting_results.json` — β fitting results with PPN validation

- `results/step027_claim_consistency_audit.json` — Machine-readable manuscript/pipeline claim audit, including evidence-frame pass/fail status

- `results/step036_final_report.json` — Comprehensive results with Temporal Topology screening

- `results/step039_flyby_prediction_table.json` — Per-flyby raw pooled-β classification (post-OD columns withheld until Step 021 supplies $F_{\rm OD}$)

- `results/step032_tep_suppression_analysis.json` — Legacy empirical suppression diagnostic; superseded by Step 039 for manuscript inference

- `results/step037_figure1_altitude_anomaly.png` — Altitude vs anomaly correlation

- `results/step037_figure2_beta_comparison.png` — Fitted β comparison by spacecraft

- `results/step037_figure3_ppn_constraints.png` — PPN constraint analysis

- `results/step037_figure4_screening_profile.png` — Temporal Topology profile

#### Log Files

Each step produces detailed logs:

- `logs/pipeline.log` — Master pipeline execution log

- `logs/step_*.log` — Individual step logs

### Software Dependencies

| Package | Version | Purpose |
| --- | --- | --- |
| Python | 3.10+ | Language runtime |
| NumPy | 1.24+ | Numerical computing |
| SciPy | 1.10+ | Statistical functions |
| Matplotlib | 3.7+ | Visualization |
| Astroquery | 0.4.6+ | JPL Horizons interface |
| spiceypy | 5.1+ | SPICE kernel handling |
| PyIRI | (package current) | Ionospheric electron density for Step 033 trajectory profiles |
| pytest | 7+ | Smoke tests (`pytest` from repository root) |

All dependencies are specified in `requirements.txt`.

### Validation &amp; Testing

The pipeline includes comprehensive validation:

- Bootstrap Resampling: n=10,000 iterations for uncertainty quantification

- Leave-One-Out Cross-Validation: Tests robustness against single-flyby exclusion

- Heterogeneity Assessment: Cochran's Q and I² statistics for model scatter

- GNSS clock correlation: The GNSS atomic clock correlation analysis provides a cross-consistency check with the GNSS/UCD terrestrial calibration on the transition radius ($R_{\rm sol} \approx 4146$ km). This shared-input dependency is disclosed explicitly to avoid circular claims of independent validation.

- Claim-consistency audit: Step 027 machine-checks manuscript claims against pipeline outputs, including model-comparison values, Table 3 fitted parameters, variance decomposition, Cassini exclusion status, Step 039 Juno classification, and evidence framing. The audit fails if the manuscript reverses the TEP-vs-Null comparison, omits the random-effects uncertainty, or promotes Juno from a deterministic warning case to an uncertainty-aware falsification.

- Uncertainty discipline: Formal inverse-variance summaries, random-effects scatter, the fractional-floor model comparison, and published-uncertainty plus constant-floor stress tests are reported as separate layers rather than blended into a single headline number.

- Automated smoke tests: `pytest` (see `tests/` and `pytest.ini`) checks repository layout and step logger conventions.

### Reproducibility Checklist

To verify successful reproduction:

- All configured pipeline steps complete with "SUCCESS" status

- Primary JSON products are present in `results/`

- Figure files are present in `results/` (PNG)

- Key result: gated inverse-variance $\beta_{\rm fit}$ span $4.43 \times 10^{-3}$ to $1.44 \times 10^{-2}$ across the tightly constrained members (NEAR, Galileo 1990, Rosetta 2005); Galileo 1992 fits at $\beta \approx 6.7 \times 10^{-2}$ and Cassini and MESSENGER are formally divergent through near-zero reference predictions (weighted mean $\approx 4.74 \times 10^{-3}$ from `results/step008_fitting_results.json`)

- Key result: $S_\oplus \approx 0.35$ terrestrial clock-amplitude shell fraction ($R_{\rm sol} \approx 4146$ km)

- Key result: $|\gamma_{\rm PPN}-1| = 4.32\times 10^{-8}$ on the two-branch solar profile at $1.6\,R_\odot$ ($y_{\rm tb}=2.16\times10^{-8}$; $S_\Sigma^{(\odot)} \sim 2\times 10^{-6}$ conservative solar source-charge convention, safely below the Cassini bound $2.3 \times 10^{-5}$)

- Key result: $I^2 \approx 99.6\%$ extreme heterogeneity (supports β scatter hypotheses)

- Key result: Altitude-anomaly correlation $\rho$ = -0.65 ($p = 0.06$, $n$ = 9); the TEP envelope predicts near-zero for high-altitude null trajectories and large for low-altitude high-asymmetry encounters, which is the relevant test

- Key result: the published null or bound cases are consistent with fixed-amplitude Step 039 predictions (Step 008 pooled $\beta_{\rm fit}$) under the Step 007 geometry envelope; one deterministic fixed-amplitude raw-tension case remains (Juno, predicted $+0.12$ mm/s) pending mission-specific OD survival factors, and Galileo 1992 carries a raw amplitude surplus (observed $-4.60$ vs predicted $\approx-0.3$ mm/s, correct sign); Rosetta 2009 is a published null/bound case with insufficient explicit geometry for the Step 039 table; 3 flybys (Stardust, OSIRIS-REx, BepiColombo) have no public anomaly report

### Data Availability Statement

Spacecraft trajectories are available through the NASA JPL Horizons ephemeris service. Literature anomaly values are from Anderson et al. (2008) and companion publications. Analysis code and processed data products are available at https://github.com/matthewsmawfield/TEP-EFA with archived DOI at 10.5281/zenodo.19454862.

Raw DSN tracking products may be obtained from the NASA Deep Space Network and the Planetary Data System following institutional access procedures; per-mission pointers appear under `data/raw/dsn_tracking/&lt;mission&gt;/DOWNLOAD_INSTRUCTIONS.txt`. The present manuscript release does not bundle perigee-matched Level-1 TRK archives: Steps 005–006 and 028–031 implement ingest and audit only, and Step 030 remains inconclusive until such products are added. Headline flyby inference uses literature $\Delta v$ and JPL Horizons trajectories (Steps 007–026, 039).

---

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<!-- CANONICAL URL: https://mlsmawfield.com/tep/j0437/ -->
<!-- MARKDOWN URL:  https://mlsmawfield.com/tep/j0437/manuscript.md -->
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# Temporal Equivalence Principle: Synchronization Holonomy in Pulsar Scintillation
**Matthew Lukin Smawfield**
Version: v0.5 (Sintra)
First published: 17 May 2026
Last updated: 17 October 2026
DOI: 10.5281/zenodo.19454619

---

## Abstract

The scalar-factorizable thin-screen delay model tested here assigns each scattered ray an additive scalar delay, so differential delays around a closed triplet cancel identically: \(\tau_{ij}+\tau_{jk}+\tau_{ki}=0\). The Temporal Equivalence Principle (TEP) instead predicts that proper-time transport is path-dependent in dilute, unscreened environments, producing a non-zero synchronization holonomy. This paper reports the rejection of the scalar-delay null hypothesis in pulsar scintillation using the phase-domain closure statistic \(\psi\), a zero-centered circular observable that separates geometric phase from folded-noise bias.

The primary target is PSR J0437-4715, analyzed with 19,167 scintillation triplets from 1,391 closure-capable Parkes/PPTA epochs; 1,093 epochs form the independent sample. The broader 15-pulsar catalog includes PSR J1603-7202, ten Jiamusi pulsars, and three MeerKAT pulsars. While J0437-4715 provides the primary phase-domain rejection of the \(\psi=0\) null, PSR J1603-7202 contributes complementary bipolar geometric structure from a distinct line of sight at different velocity geometry. The Jiamusi and MeerKAT samples are noise-limited because they contain only 1–2 closure-capable epochs; they provide bounds but do not independently diagnose environmental suppression.

J0437-4715 shows a non-zero Phase Closure signal. Because the Rayleigh and V-test statistics measure circular concentration against uniformity (and are rotation-invariant with respect to the mean angle), circular concentration and zero-mean rejection are reported separately: the unweighted circular mean is \(\psi=1.120\) rad (95% bootstrap CI \([+0.988,+1.254]\) rad excluding zero; circular concentration Rayleigh \(p=1.34\times10^{-44}\)), and the weighted circular mean is \(\bar{\psi}=0.984\pm0.046\) rad with 95% circular-bootstrap confidence interval \([0.737,1.235]\) rad excluding zero (\(>15\sigma\) circular mean offset from zero, corroborated in the delay domain by the MJD-ordered 15–240-epoch block-bootstrap noise-subtracted excess at \(3.8\sigma\)–\(10.8\sigma\), Steps 052 and 060; circular concentration \(\bar{R}=0.308\), Rayleigh \(p=1.39\times10^{-13}\), V-test \(p=1.02\times10^{-5}\) relative to \(\mu_0 = 0\)). The unweighted statistic is identical in heliocentric and CMB-frame analyses, providing an implementation-invariance check; physical discrimination comes from the non-zero mean closure offset and its null controls.

PSR J1603-7202 has a 73.8° proper-motion separation from J0437-4715 and exhibits a frame-independent bipolar decomposition of the morphology expected for high-dispersion sightlines; its circular Phase Closure is noise-limited (R̄ = 0.019, Rayleigh p = 0.926), so the decomposition is a directional consistency check rather than demonstrated parity structure. Delay-domain unsigned amplitude estimators (\(M_{\rm cl}\)) and their folded-normal Rice noise-floor bounds are documented in Appendix B.

Multiple independent checks support the phase-domain result. Phase-scramble and pre-alignment controls pass, unweighted \(\psi\) is strictly frame-invariant, signed-delay cancellation behaves as expected for a bipolar signal, and rigorous synthetic noise tests yielded zero false positives in 100 synthetic null trials. A signed-delay orbital diagnostic shows phase-locked structure directionally consistent with TEP kinematic coupling, though the hierarchical mixed-effects amplitude is not independently significant (\(p = 0.372\), 2 df), as expected for a partially screened orbital channel. Multi-pulsar scaling, chromaticity, cross-telescope environmental bounds, and orbital structure provide multiscale consistency checks on the TEP framework.

The observation rejects the tested scalar-factorizable cross-term null. Within the TEP measurement model, the non-zero Phase Closure is a synchronization-holonomy candidate sourced by the non-exact disformal sector: on the measured screen-triangle geometry (Step 057), the canonical disformal connection yields \(\approx 1.5\times10^{-12}\) s as a ceiling at order-unity drift inhomogeneity, with the \(B_0=+1\) unit benchmark (\(1.3\%\) of the narrowband-equivalent \(H_{\rm implied}\approx 1.13\times10^{-10}\) s on the median leg, \(5.0\%\) on the 95th-percentile leg) and reaches \(1.14\times10^{-10}\) s (\(100.5\%\) on the median leg; \(B_0^{\rm req}=20.1\) on the \(p_{95}\) leg) at the corpus's GW170817 galactic-profile envelope \(B_0\lesssim 77.7\) (Paper 0, Step 50), while remaining within the delay-domain bound (\(\lesssim 0.46\) ns). The tested scalar-factorizable implementations of thin-screen, multi-screen, refractive, chromatic-plasma and Doppler-delay models do not reproduce the observed coherent non-zero Phase Closure. The geometric structure from J1603-7202 is directionally consistent with the Temporal Equivalence Principle's non-integrable time transport.

Code Availability: All data and analysis code required to reproduce the results presented in this work, including the full 15-pulsar scintillation catalog compilation, are available in the public repository at https://github.com/matthewsmawfield/TEP-J0437.

Keywords: Temporal Equivalence Principle, pulsar scintillation, synchronization holonomy, closure delays, modified gravity, interstellar medium, multi-pulsar analysis, PSR J0437-4715, PSR J1603-7202, arc curvature scaling, scattering strength

## 1. Introduction: Temporal Equivalence Principle and Observable Holonomy

General Relativity (GR) is built upon the Einstein Equivalence Principle (EEP), which postulates that at any spacetime point, a local inertial frame can be chosen such that the laws of physics take their special relativistic form. In this framework, spacetime is represented as a single pseudo-Riemannian manifold $(M, g_{\mu\nu})$, where proper time $d\tau$ is a purely geometric quantity defined by the line element $d\tau^2 = g_{\mu\nu} dx^\mu dx^\nu$. The standard scalar-factorizable thin-screen scintillation model assigns each scattered propagation path an additive scalar delay, making differential delays simple differences and the closure residual identically zero by arithmetic.

The Temporal Equivalence Principle (TEP) retains the EEP's local validity—freely falling frames still recover special relativity—but relaxes the global assumption that a single metric governs both gravitational dynamics and matter coupling. Where Einstein's derivation assumes $g_{\mu\nu}$ universally defines proper time for all physical processes, TEP introduces a matter metric $\tilde{g}_{\mu\nu}$ that coincides with $g_{\mu\nu}$ in dense, screened environments while permitting deviations in unscreened regions.

The divergence between TEP and standard theory occurs at the level of the path-delay model. Whereas the additive-delay null hypothesis assumes each scattered path carries a scalar delay—making the closure residual vanish by simple arithmetic—TEP allows non-exact structure through disformal coupling, yielding path-dependent proper time accumulation that varies spatially. In this framework, proper time is no longer a passive coordinate but a dynamical entity affected by the local gravitational potential through a two-metric scalar-tensor structure.

This dynamical structure yields a specific, falsifiable prediction: synchronization holonomy, a geometric phase effect where proper time transport around closed loops fails to close by measurable amounts. A robust non-zero closure signal rejects the tested scalar-factorizable additive-delay null; if conventional multi-path propagation explanations are excluded by independent replication and matched-screen controls, it demonstrates non-integrable synchronization transport. This work presents observational evidence consistent with this phenomenon, providing candidate evidence for synchronization holonomy as a feature of macroscopic clock transport in the interstellar medium.

The TEP framework is a two-metric scalar-tensor theory defined on a single spacetime manifold endowed with two metrics: a gravitational metric \(g_{\mu\nu}\) (governing gravitational interactions) and a canonical causal matter metric \(\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi\) (\(U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}\), \(\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu\); Paper 0, §2.2, Step 80) to which all non-gravitational fields and clocks universally couple. In conformal–disformal shorthand, the metrics are related by the map:

\begin{equation} \label{eq:tep_metric} \tilde{g}_{\mu\nu} = A^2(\phi)
g_{\mu\nu} + B(\phi) \nabla_{\mu}\phi \nabla_{\nu}\phi \end{equation}

where \(\phi\) is the time field, \(A(\phi) = \exp(\beta_A \phi / M_{\text{Pl}})\) is a universal conformal factor, \(B(\phi)\) encodes tiny, direction-dependent deformations of the light cone, and \(\nabla_{\mu}\) denotes the covariant derivative with respect to the gravitational metric \(g_{\mu\nu}\). The constants \(\beta_A\) and \(M_{\text{Pl}}\) represent the scalar coupling strength and Planck mass, respectively. General Relativity is recovered exactly when \(A(\phi) = 1\) and \(B(\phi) = 0\). The TEP framework is founded on four axioms:

The four axioms that define the TEP framework are: (A1) a two-metric structure where gravity is governed by \(g_{\mu\nu}\) while matter couples to a causal matter metric \(\tilde{g}_{\mu\nu}\); (A2) the Temporal Equivalence Principle, requiring that all non-gravitational processes evolve according to proper time defined by \(\tilde{g}_{\mu\nu}\); (A3) causal safety, ensuring that deviations from GR remain within current constraints on photon-graviton speed differences; and (A4) environmental suppression of observable Temporal Shear through \(\Sigma_\mu^{\rm obs}=\mathcal{S}_\Sigma(\mathcal{E})\,\Sigma_\mu\), which suppresses Temporal Shear in screened environments while leaving dilute, unscreened regions such as the ISM accessible to TEP dynamics.

The critical consequence for this work is that the ISM—being a dilute environment where Temporal Shear remains unsuppressed—is precisely where TEP effects should manifest maximally. The disformal coupling \(B(\phi) \neq 0\) introduces non-exact structure into the time-transport 1-form, yielding path-dependent proper time accumulation around closed scattering loops.

The breakdown of global simultaneity is formalized through a synchronization-transport law, deriving the convention-independent synchronization holonomy \(H \equiv \oint_C d\tilde{\tau}\) as an invariant measure of non-integrability of time transport around closed loops. In purely conformal theories this holonomy vanishes once general-relativistic contributions are removed; a nonzero holonomy at leading order requires non-exact structure provided by disformal coupling \(B(\phi) \neq 0\).

This study applies the TEP framework to pulsar scintillation across 15 pulsars, providing a comprehensive multi-pulsar test in the interstellar medium—a dilute environment where Temporal Shear remains unsuppressed and TEP effects manifest maximally. The phase-domain falsification of the \(\psi=0\) null is driven by PSR J0437-4715; the remaining objects verify environmental scaling predictions and provide complementary geometric structure.

The dataset combines measurements from Parkes observations of PSR J0437-4715 as the primary target and PSR J1603-7202 for velocity geometry testing with a 73.8° velocity separation. The velocity vector geometry provides a critical geometric test: TEP predicts opposite-sign holonomy for pulsars with different velocity projections onto scattering geometry.

Additionally, the study incorporates Jiamusi telescope observations of ten bright, distant pulsars (distances 1–3.7 kpc; see Table 3.1 for the full catalog), extending the distance baseline by an order of magnitude to enable rigorous tests of TEP scaling with scattering strength. These distant pulsars provide noise-limited bounds because they contain only 1–2 closure-capable epochs; they provide bounds but do not independently diagnose environmental suppression.

### 1.1 Key Terminology

*Temporal Equivalence Principle (TEP):* A theoretical framework where proper time is elevated to a dynamical field that couples to the local gravitational environment, creating path-dependent time accumulation that varies spatially.

*Synchronization holonomy (H):* The non-closure of proper time transport around closed loops, representing path-dependent time accumulation. This effect is analogous to a geometric phase for clock transport. The Phase Closure \(\psi\) is the primary detection statistic.

*Closure delay:* The sum of differential delays around a triplet of scattered propagation paths through the interstellar medium, constituting a direct measurement of synchronization holonomy.

*Stokes alignment:* Geometric rectification of closure delays using Stokes' theorem, which aligns delays according to the orientation of the scattering geometry and the effective velocity through the scattering screen.

*Arc curvature (\(\eta\)):* A measure of scattering strength in the interstellar medium, which TEP predicts should scale with the holonomy magnitude.

*Temporal Topology:* The spatial and covariance structure of \(\ln A(\phi)\).

*Temporal Shear:* The canonical synchronization gradient \(\Sigma_\mu \equiv \nabla_\mu \ln A(\phi) = \alpha(\phi)\,\nabla_\mu \phi\). Its observable response is \(\Sigma_\mu^{\rm obs} = \mathcal{S}_\Sigma(\mathcal{E})\,\Sigma_\mu\), where \(\mathcal{E}\) denotes the environmental state (density, compactness, gradients, source structure, coherence scale, boundary conditions).

Signed mean vs. Phase Closure \(\psi\): The signed mean (average of signed delays) serves as a physical metric and is expected to be near zero due to bipolar partial cancellation. The Phase Closure \(\psi\) is the primary detection statistic representing the physical holonomy phase.

Methodology Status: The prediction of non-zero synchronization holonomy follows directly from the disformal metric structure defined above; a self-contained derivation is given in Appendix A. The pulsar scintillation closure delay methodology presented here was developed independently to test that prediction. The empirical tests reported below serve to test the predictions of this framework.

### 1.2 Summary of Findings

The analysis reports the following core results, which together form a multiscale consistency check on the Temporal Equivalence Principle:

- **Primary Phase Falsification:** PSR J0437-4715 rejects the additive scalar path-delay null through a non-zero Phase Closure \(\psi\) (\(\bar{\psi}=0.984 \pm 0.046\) rad, 95% circular-bootstrap CI \([0.737,1.235]\) rad excluding zero; unweighted \(\psi=1.120\) rad, 95% CI \([0.990,1.253]\) rad excluding zero; circular concentration confirmed by Rayleigh \(p=1.39\times10^{-13}\) weighted and \(1.34\times10^{-44}\) unweighted, V-test \(p=1.02\times10^{-5}\)).

- **Geometric Corroboration:** PSR J1603-7202 exhibits a frame-independent bipolar decomposition and elevated circular variance directionally consistent with the high-dispersion pattern; its Phase Closure is noise-limited, so the decomposition is a consistency check rather than demonstrated parity structure.

- **Environmental Bounds:** Jiamusi and MeerKAT pulsars are noise-limited, consistent with TEP's predicted environmental suppression on distant, noise-limited sightlines where Temporal Shear is suppressed.

- **Orbital Kinematics:** A hierarchical mixed-effects model reveals triplet-level orbital structure (amplitude 1.11 ± 0.79 ns, LR p = 0.372) consistent with partial screening of orbital-scale shear by the white dwarf companion.

Together, these findings exclude the \(\psi = 0\) null hypothesis across multiple standard interstellar scattering models, supporting the existence of synchronization holonomy.

## 2. Synchronization Holonomy in Pulsar Scintillation

### 2.1 The Closure Delay Observable

When pulsar signals traverse the interstellar medium, scattering from plasma inhomogeneities creates multiple propagation paths. These paths interfere in the observer's dynamic spectrum, producing characteristic patterns that encode information about the scattering geometry. In standard scintillation theory, each scattered propagation path is assigned a scalar delay \(\tau_i\). Differential delays are differences of these scalars, so the closure sum around any triplet vanishes identically by arithmetic—a property termed synchronization consistency.

The *closure residual* \(C_{ijk}\) tests this additive representation. Each cross-term \(\hat{\tau}_{ab}\) is independently measured from the secondary-spectrum peak between scattered paths \(a\) and \(b\) via sub-pixel interpolation. For a triplet of scintillation arclets indexed by \(i, j, k\) forming a closed loop, the closure residual is:

\begin{equation} \label{eq:closure_residual}
C_{ijk} = \hat{\tau}_{ij} + \hat{\tau}_{jk} + \hat{\tau}_{ki}
\end{equation}

Under the additive-delay null hypothesis, there exists a scalar assignment \(\tau_i\) such that \(\hat{\tau}_{ab} = \tau_b - \tau_a\) within measurement noise, implying \(C_{ijk} \equiv 0\). The closure construction cancels common-mode contributions—including standard relativistic effects that apply equally to all scattered paths—by design, isolating only path-dependent contributions that violate the additive representation. TEP predicts a non-zero synchronization holonomy \(H_{\rm resid}^{(\tau)}\), defined as the GR-subtracted loop integral of the synchronization connection \(\Delta\sigma^{(\tau)} = \tilde{\sigma}^{(\tau)} - \sigma_{\rm GR}^{(\tau)}\) around the closed path (Jakarta, Paper 0). This holonomy—a geometric phase for clocks—arises because the disformal sector \(B(\phi) \neq 0\) of the matter metric \(\tilde{g}_{\mu\nu}\) generates a non-exact contribution to the synchronization connection when \(u \cdot \nabla\phi \neq 0\). The observed closure delay relates to the holonomy magnitude as:

\begin{equation} \label{eq:holonomy_magnitude} M_{\rm cl} = |\langle
\tau_{\text{closure}} \rangle_{\text{aligned}}| \end{equation}

where \(\langle \cdot \rangle_{\text{aligned}}\) denotes the Stokes-aligned average accounting for the relative orientation of the path triplet and the observer's velocity (see Section 4.7 for the alignment algorithm specification). The alignment ensures that delays correlate with geometric orientation as predicted by TEP.

The primary detection statistic is the *Phase Closure* \(\psi\), constructed directly from the geometric phase of the complex secondary-spectrum cross-term triplet. Unlike unsigned delay-domain amplitude estimators (\(M_{\rm cl}\), Appendix B), which are strictly positive under folded-normal Rice noise, the Phase Closure \(\psi\) is a zero-centered, odd-parity circular observable that cancels identically in the absence of path-dependent synchronization transport:

\begin{equation} \label{eq:phase_closure}
\psi_{ijk} = \operatorname{Arg}\!\left[\exp\!\left(i\left(\phi_{ij} + \phi_{jk} - \phi_{ik}\right)\right)\right]
\end{equation}

where \(\phi_{ab}\) is the phase of the cross-term between scattered paths \(a\) and \(b\), measured directly from the secondary spectrum. The minus sign on \(\phi_{ik}\) reflects the actual geometry: the three independently measured cross-terms are \((i,j)\), \((j,k)\) and the direct difference \((i,k)\); their closure is \(\phi_{ij} + \phi_{jk} - \phi_{ik}\). For a sample of \(N\) triplet measurements, the circular mean is:

\begin{equation} \label{eq:phase_closure_mean}
\bar{\psi} = \operatorname{Arg}\!\left[\frac{1}{N}\sum_{ijk} \exp\!\left(i\psi_{ijk}\right)\right]
\end{equation}

TEP predicts \(\bar{\psi} \neq 0\) while the additive-delay null hypothesis predicts \(\bar{\psi} = 0\).

*Why ordinary ray phases close but TEP synchronization transport need not.* If every measured cross-term is factorizable as \(Q_{ab} = X_a X_b^*\), then \(\arg(Q_{ij}Q_{jk}Q_{ki}) = 0\) identically. This is precisely the scalar-delay null tested here: each ray carries a scalar accumulated phase \(\Theta_a\), and the closure \((\Theta_i - \Theta_j) + (\Theta_j - \Theta_k) + (\Theta_k - \Theta_i) = 0\) vanishes regardless of path dependence, because endpoint phases are state functions. TEP modifies the operational comparison itself: the pairwise observable is the transport of synchronization between the path-defined events associated with rays \(a\) and \(b\),

\begin{equation} \label{eq:tep_pairwise}
\varphi_{ab}^{\rm TEP} = \varphi_b - \varphi_a + \int_{\Gamma_{ab}} \omega_{\rm eff}(x)\,\Delta\sigma^{(\tau)} ,
\end{equation}

where \(\Gamma_{ab}\) is the comparison path in spacetime induced by the scattering geometry, \(\Delta\sigma^{(\tau)}\) is the time-valued residual synchronization transport (units of seconds), and \(\omega_{\rm eff}(x)\) is the effective angular frequency that converts synchronization transport to phase. The time-valued residual holonomy is:

\begin{equation} \label{eq:h_resid_tau}
H_{\rm resid}^{(\tau)}(C) \equiv \oint_C \Delta\sigma^{(\tau)} \qquad [\mathrm{s}] ,
\end{equation}

and the phase closure is:

\begin{equation} \label{eq:tep_closure}
\psi_{ijk} = \operatorname{Arg}\!\exp\!\left[i\oint_C \omega_{\rm eff}(x)\,\Delta\sigma^{(\tau)}\right] .
\end{equation}

Only when \(\omega_{\rm eff}\) is common to all three legs may this be reduced to \(\psi_{ijk} = \operatorname{wrap}\!\left[\omega_{\rm eff}\,H_{\rm resid}^{(\tau)}(C_{ijk})\right]\). The effective frequency \(\omega_{\rm eff}\) is determined by the secondary-spectrum measurement channel: it is the angular frequency at which the cross-term interference pattern is sampled, set by the observing band centre and the cross-spectrum extraction procedure. It is not simply the carrier frequency; the full identification requires an explicit electromagnetic transfer derivation (Appendix A.4). A direct empirical test of the phase–time mapping is available through the subband derivative:

\begin{equation} \label{eq:omega_eff_test}
\frac{d\psi_{\rm unwrap}}{d\omega_{\rm eff}} = H_{\rm resid}^{(\tau)} ,
\end{equation}

valid for an unwrapped phase and a frequency-independent residual over the fitted subband. If frequency dependence is allowed, \(d\psi_{\rm unwrap}/d\omega = H + \omega\,dH/d\omega\). The derivative connects the phase and delay channels and is testable across observing subbands. The null \(\psi = 0\) is recovered whenever \(\Delta\sigma^{(\tau)}\) is exact, including the conformal-only sector, whereas the disformal non-exact sector permits \(\psi \neq 0\). The TEP measurement operator modifies each cross-term as:

\begin{equation} \label{eq:tep_measurement}
Q_{ab}^{\rm TEP} = X_a\,X_b^*\,\exp\!\left[i\int_{\Gamma_{ab}} \omega_{\rm eff}(x)\,\Delta\sigma^{(\tau)}\right] .
\end{equation}

Within the TEP measurement model, synchronization transport contributes an edge-dependent phase, so the three pairwise terms need not factorize into endpoint phases and their closure can be non-zero. Appendix A.4 defines the associated events and paths; electromagnetic transfer determines the channel coefficient. The closure product is then \(Q_{ij}^{\rm TEP} Q_{jk}^{\rm TEP} Q_{ki}^{\rm TEP} = |X_i X_j X_k|^2 \exp(i \oint_C \omega_{\rm eff}\,\Delta\sigma^{(\tau)})\), so \(\psi_{ijk} = \operatorname{Arg}\exp(i \oint_C \omega_{\rm eff}\,\Delta\sigma^{(\tau)})\). The measured cross-term is non-factorizable under TEP because the synchronization transport along each edge is an independent path integral, not a difference of state functions. The full identification of the three measured cross-terms with the edges of a genuine closed spacetime loop, including the event and path definitions, is given in Appendix A.4.

#### 2.1.1 Statistical Motivation for Phase Closure

The choice of \(\psi\) as the primary detection statistic follows from the symmetry properties of the cross-term closure phase, independent of any specific theoretical framework. Define the complex closure product:

\begin{equation} \label{eq:closure_product}
\mathcal{B}_{ijk} = Q_{ij}\,Q_{jk}\,Q_{ik}^{*}, \qquad
\psi_{ijk} = \arg \mathcal{B}_{ijk} .
\end{equation}

Under scalar factorization (\(Q_{ab} = X_a X_b^*\)), \(\mathcal{B}_{ijk} = |X_i X_j X_k|^2\), and therefore \(\psi_{ijk} = 0\). Measurement and extraction errors broaden the observed distribution around the scalar-null direction. TEP synchronization curvature adds a nonfactorizable link phase, shifting the circular mean away from zero. The folded closure magnitude \(M_{\rm cl} = \langle |C_{ijk}| \rangle\), by contrast, is strictly positive under any noise model—it measures the mean absolute deviation, not a physical holonomy. This is the Rice-distribution bias: \(M_{\rm cl} > 0\) even when no signal is present.

The Phase Closure \(\psi\) is therefore the appropriate zero-centered test statistic: it cancels to zero for rank-one scalar-factorizable delays, while any physical mechanism producing non-factorizable delay-Doppler cross-term phase structure yields \(\langle \psi \rangle \neq 0\). Because the Rayleigh statistic \(Z=2n_{\rm eff}\bar{R}^2\) measures circular concentration (rejecting circular uniformity) and is invariant under rotating all angles by a constant offset, rejection of \(\langle\psi\rangle = 0\) is established by the circular mean and its bootstrap confidence interval (\(\bar{\psi}=0.984\pm0.046\) rad, 95% CI \([0.737,1.235]\) rad; unweighted \(\psi=1.120\) rad, 95% CI \([0.990,1.253]\) rad, both excluding zero), while Rayleigh \(Z=59.21\) (\(p=1.39\times10^{-13}\) weighted; \(p=1.34\times10^{-44}\) unweighted) establishes that the phase distribution is sharply concentrated rather than uniform circular noise. The interpretation of this non-zero mean offset as a synchronization-holonomy candidate follows from the TEP measurement model (Appendix A.4–A.5), while the statistical exclusion of \(\langle\psi\rangle=0\) under the tested estimator is model-independent.

### 2.2 TEP Predictions for Closure Delays

The TEP framework—a two-metric scalar-tensor theory introduced in Paper 0—makes a central prediction that directly contrasts with the standard additive-delay model: the synchronization holonomy must be non-zero. The theory posits a disformal coupling between a scalar field $\phi$ and the matter metric $\tilde{g}_{\mu\nu}$ (Bekenstein 1993; Koivisto & Zumalacárregui 2012):

\begin{equation} \label{eq:disformal_metric}
\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu\phi \nabla_\nu\phi
\end{equation}

where $g_{\mu\nu}$ is the Einstein metric governing gravitational dynamics, $A(\phi)$ is the conformal coupling function, and $B(\phi) \geq 0$ is the disformal coupling parameter. The observational observable is not a raw one-way speed or the integral of a scalar proper-time differential. Following the Jakarta construction (Paper 0, Section 3.3 and Appendix A3), let $\tilde{\sigma}^{(\tau)}$ denote the matter-frame synchronization transport one-form induced by $\tilde{g}_{\mu\nu}$ and the chosen clock congruence, and let $\sigma_{\rm GR}^{(\tau)}$ denote the corresponding GR one-form including Sagnac, gravitomagnetic/Lense–Thirring, Shapiro delay, gravitational redshift, station motion, clock-scale realization, and reference-frame corrections. The residual synchronization connection is:

\begin{equation} \label{eq:residual_sigma}
\Delta\sigma^{(\tau)} \equiv \tilde{\sigma}^{(\tau)} - \sigma_{\rm GR}^{(\tau)}
\end{equation}

and the time-valued residual holonomy around a closed loop $C = \partial\Sigma$ is:

\begin{equation} \label{eq:holonomy_resid}
H_{\rm resid}^{(\tau)}(C) = \oint_C \Delta\sigma^{(\tau)} = \iint_\Sigma d(\Delta\sigma^{(\tau)})
\end{equation}

This quantity is invariant under admissible synchronization re-gaugings because both $\tilde{\sigma}^{(\tau)}$ and $\sigma_{\rm GR}^{(\tau)}$ shift by the same exact one-form, leaving $\Delta\sigma^{(\tau)}$ unchanged. In the purely conformal subclass ($B = 0$), the scalar contribution to clock-rate transport is generated by the exact one-form $\omega^{(A)} = d\ln A$, so $\oint_C d\ln A = 0$ for any closed loop in a simply connected region. A leading-order nonzero $H_{\rm resid}^{(\tau)}$ requires residual synchronization curvature $d(\Delta\sigma^{(\tau)}) \neq 0$, supplied in the minimal TEP model by the disformal sector $B(\phi) \neq 0$.

#### 2.2.1 Projected Disformal Contribution

To state the disformal contribution in a measurement-defined way, let $u^\mu$ denote the four-velocity field of the physical clock network or observer congruence, and let $P_\mu{}^\nu = \delta_\mu{}^\nu + u_\mu u^\nu$ be the spatial projector into the local rest space of that congruence. To leading order in the disformal correction, the projected contribution to the synchronization connection has the representative form (Jakarta, Appendix A3):

\begin{equation} \label{eq:disformal_sigma}
\delta\tilde{\sigma}_\mu^{(\tau)} \simeq -\frac{B}{A^2}\,(u \cdot \nabla\phi)\,P_\mu{}^\nu\,\nabla_\nu\phi
\end{equation}

up to the sign convention used for the synchronization one-form and higher-order disformal corrections. In a hypersurface-orthogonal $3+1$ slicing, where $u^\mu = n^\mu$, this reduces to:

\begin{equation} \label{eq:disformal_3plus1}
\delta\tilde{\sigma}_i^{(\tau)} \simeq -\frac{B}{A^2 N}\,(\partial_i \phi)\,(n \cdot \partial\phi)
\end{equation}

where $N$ is the lapse and $n^\mu$ is the unit normal to the spatial slices. Unlike $d\ln A$, the disformal contribution is not generically exact. Its curvature satisfies $d(\delta\tilde{\sigma}^{(\tau)}) \neq 0$ whenever the prefactor multiplying the projected scalar gradient varies independently around the loop.

#### 2.2.2 Physical Condition for Non-Vanishing Holonomy

A static scalar gradient in a hypersurface-orthogonal frame does not generate the leading holonomy. The pulsar observable requires relative motion through the field, time dependence, mixed temporal-spatial structure, or independently varying lapse/shift and disformal response around the effective loop. The key factor is $u \cdot \nabla\phi \neq 0$: the four-velocity of the clock congruence must have a component along the scalar field gradient.

For the J0437−4715 scintillation geometry, $u \cdot \nabla\phi \neq 0$ is generated by three independent contributions:

- **Pulsar proper motion:** J0437−4715 has a large transverse
proper motion ($\mu_\alpha \simeq 121$ mas/yr, $\mu_\delta \simeq -71$ mas/yr;
Deller et al. 2008), corresponding to a tangential velocity
$v_\perp \simeq 104$ km/s at $d \simeq 156$ pc. The pulsar's worldline
sweeps through the interstellar scalar field gradient, producing a
time-varying $u \cdot \nabla\phi$ along the line of sight.

- **Earth's orbital motion:** The Earth's orbital velocity
($v_{\rm orb} \simeq 30$ km/s) modulates the observer's four-velocity
projection onto $\nabla\phi$ over annual timescales, producing a
periodically varying contribution to $u \cdot \nabla\phi$.

- **Screen motion:** The scattering screens at
$\sim 89.8$ pc and $\sim 124$ pc have their own peculiar velocities relative
to the local standard of rest, contributing additional temporal structure to
the effective loop geometry.

These kinematic ingredients enter the theoretical prediction through the $u \cdot \nabla\phi$ factor in equations $\eqref{eq:disformal_sigma}$ and $\eqref{eq:disformal_3plus1}$, not merely through the later Stokes-alignment algorithm. The predicted holonomy is therefore a function of the pulsar's proper motion, the Earth's orbital phase, and the screen kinematics — all of which are independently measured. The Stokes-aligned analysis (Section 4.7) is the observational implementation of this kinematic projection; the theoretical prediction precedes it.

Where the single-metric null hypothesis requires the closure residual to vanish, TEP predicts path-dependent residual synchronization holonomy as a consequence of disformal coupling in dilute environments where Temporal Shear remains unsuppressed. The Phase Closure $\psi$ provides a zero-centered test: $\psi = 0$ under the additive-delay null hypothesis, $\psi \neq 0$ for TEP.

This non-zero holonomy manifests with specific geometric properties. In a Stokes-aligned distribution, delays converge to a consistently negative mean ($-H$) when aligned with the orientation kinematics vector, while the magnitude $M_{\rm cl}$ remains invariant regardless of specific triplet geometry. The sign of the holonomy correlates with geometric orientation, defined by clockwise versus counter-clockwise configurations.

A critical consequence of this sign-geometry correlation is that simple aggregate averaging over all orientations yields $\langle \tau \rangle \approx 0$. This bipolar cancellation is expected by construction, not evidence against TEP. Analysis by orientation is therefore required to reveal the underlying signal, with the effect manifesting maximally in dilute environments where Temporal Shear is unsuppressed, such as the interstellar medium.

Key distinction: TEP predicts a *bipolar* signal where the holonomy sign flips with velocity projection. The primary detection statistic is the Phase Closure $\psi$, which represents the physical holonomy phase and is immune to folded-magnitude noise-floor bias (Appendix B), corroborated in the delay domain by the paired noise-subtracted closure excess ($1.290 \pm 0.11$ ns, $t \approx 12.2\sigma$; $\approx 7.6$–$10.8\sigma$ under 15–240-epoch contiguous blocks). The signed mean serves as a parity metric expected near zero due to bipolar cancellation across orientations.

### 2.3 The ISM as an Unscreened Laboratory

*Temporal Topology* is the spatial and covariance structure of the conformal clock field \(\ln A(\phi)\), characterized locally by its gradients and statistically by \(C_A(x,x')\). *Temporal Shear* is the canonical synchronization gradient:

\begin{equation} \label{eq:temporal_shear}
\Sigma_\mu \equiv \nabla_\mu \ln A(\phi) = \alpha(\phi)\,\nabla_\mu\phi .
\end{equation}

The observable Temporal Shear is screened by the environmental operator \(\mathcal{S}_\Sigma(\mathcal{E})\):

\begin{equation} \label{eq:screening}
\Sigma_\mu^{\rm obs} = \mathcal{S}_\Sigma(\mathcal{E})\,\Sigma_\mu ,
\end{equation}

where \(\mathcal{E}\) includes density, compactness, gradients, source structure, coherence scale and boundary conditions. Density is one environmental projection, not a universal switch. The disformal response is correlated with the same field configuration but is not fixed by \(\mathcal{S}_\Sigma\) alone. It depends additionally on \(B(\phi)\), the field solution, the observer congruence and the measurement channel.

The interstellar medium toward PSR J0437-4715 provides a suitable testbed for TEP effects. With typical densities $\rho \sim 10^{-24} \text{ g/cm}^3$, the ISM is a dilute environment. The scattering screens at $\sim 89.8$ pc and $\sim 124$ pc (Gwinn et al. 2006) create multiple path configurations that enable closure delay measurements around closed loops through the Galactic plasma. The diffuse ISM provides a long, low-compactness multipath baseline and is therefore a motivated candidate environment for an active J0437 response. However, $\rho \ll \rho_T$ alone does not determine $\mathcal{S}_\Sigma(\mathcal{E})$, nor does conformal unscreening by itself fix the disformal transfer. The J0437 analysis therefore constrains the channel-integrated non-exact response rather than inferring it from ambient density alone.

The anisotropic nature of scattering toward J0437-4715, documented by previous studies (Walker et al. 2004; Coles et al. 2011), naturally introduces preferred orientations that may affect the distribution of clockwise versus counter-clockwise triplet configurations. This ISM anisotropy does not invalidate the TEP test; rather, it provides a physical explanation for orientation asymmetries while preserving the core prediction of magnitude equality between orientations.

It is important to distinguish the unscreened ISM scattering screens from the pulsar's immediate binary environment. The white dwarf companion to J0437-4715 generates a deep gravitational potential that locally suppresses observable Temporal Shear through the environmental screening operator $\mathcal{S}_\Sigma(\mathcal{E})$. Within the companion's screening radius, the scalar field gradient flattens toward its minimum, attenuating the locally observable Temporal Shear. Consequently, orbital-scale kinematics are expected to be partially shielded, while proper-motion-scale kinematics—sampling the unscreened ISM at much larger distances—remain the dominant TEP channel.

## 3. Data and Methodology

### 3.1 Multi-Pulsar Dataset

This study analyzes scintillation data from 15 pulsars. The primary detection is obtained from PSR J0437-4715; PSR J1603-7202 provides complementary geometric evidence through bipolar structure and variance decomposition; Jiamusi and MeerKAT provide noise-limited bounding data. The dataset combines Parkes observations of nearby millisecond pulsars with Jiamusi telescope observations of more distant pulsars and MeerKAT observations of three additional pulsars (J0908-1739, J0922-0638, J1731-4744), spanning a range of distances (~156 pc to ~3700 pc), proper motions, and scattering environments.

| Pulsar | Telescope | Distance (pc) | Proper Motion (mas/yr) | Epochs | Triplets |
| --- | --- | --- | --- | --- | --- |
| J0437-4715 | Parkes | 156.3 ± 0.3 | (121.4, −71.4) | 1,391 (of 10,201 cataloged) | 19,167 |
| J1603-7202 | Parkes | 250.0 ± 50.0 | (−6.6, −25.9) (Wang et al. 2018) | 248 | 3,653 |
| B0329+54 | Jiamusi | 1000 ± 200 | (7.10, −11.75) | 0 | 0 |
| B0355+54 | Jiamusi | 1000 ± 200 | (9.17, 0.70) | 1 | 20 |
| B0540+23 | Jiamusi | 1600 ± 300 | (2.50, −21.80) | 2 | 40 |
| B0740-28 | Jiamusi | 2000 ± 400 | (−2.44, −0.09) | 0 | 0 |
| B1508+55 | Jiamusi | 2100 ± 400 | (−73.70, −62.70) | 1 | 20 |
| B1933+16 | Jiamusi | 3700 ± 700 | (−2.00, −0.10) | 0 | 0 |
| B2154+40 | Jiamusi | 2900 ± 600 | (14.60, −2.60) | 1 | 20 |
| B2310+42 | Jiamusi | 1060 ± 210 | (−3.00, −6.00) | 0 | 0 |
| B2324+60 | Jiamusi | 2700 ± 500 | (0.0, 0.0) | 0 | 0 |
| B2351+61 | Jiamusi | 2400 ± 500 | (−0.19, −0.01) | 0 | 0 |
| J0908-1739 | MeerKAT | 400 ± 100 | Unknown | 0 | 0 |
| J0922-0638 | MeerKAT | 1000 ± 200 | Unknown | 0 | 0 |
| J1731-4744 | MeerKAT | 400 ± 100 | Unknown | 1 | 20 |

#### Data Provenance

| Pulsar | Telescope | Dataset | Years | Public source | Used for |
| --- | --- | --- | --- | --- | --- |
| J0437-4715 | Parkes (64m Murriyang) | PPTA DR2 | 2002–2018 | CSIRO DOI 10.25919/5f3cd2bc1c213 (Reardon et al. 2020) | Primary detection |
| J1603-7202 | Parkes (64m Murriyang) | PPTA DR2 | 2002–2018 | CSIRO DOI 10.25919/82f5-mh79 (Walker et al. 2022) | Geometric test / bipolar structure |
| B0329+54, B0355+54, B0540+23, B0740-28, B1508+55, B1933+16, B2154+40, B2310+42, B2324+60, B2351+61 | Jiamusi 66m | Pulsar Scintillation Archive | Various | Jiamusi 66m telescope archive | noise-limited bounds |
| J0908-1739, J0922-0638, J1731-4744 | MeerKAT (64 × 13.5 m) | Thousand-Pulsar-Array | 2019–2023 | SARAO / MeerKAT archive | noise-limited bounds |

*Note:* Jiamusi epoch and triplet counts reflect raw scintillation data availability. Four Jiamusi pulsars (B0355+54, B0540+23, B1508+55, B2154+40) produced viable closure triplets in the current pipeline configuration; the remaining six had raw data but did not yield closure-capable measurements. Of the three MeerKAT pulsars, only J1731-4744 produced a viable closure measurement (1 epoch, 20 triplets); J0908-1739 and J0922-0638 did not yield closure-capable data. Unknown proper motions limit MeerKAT geometric utility.

#### 3.1.1 Parkes Pulsars (J0437-4715, J1603-7202)

The Parkes Pulsar Timing Array (PPTA) Data Release 2 provides multiband ($732\text{--}3104 \text{ MHz}$) observations spanning 2002–2018. The J0437-4715 dataset includes 10,201 epochs spanning L-band ($\sim 1400 \text{ MHz}$) and UHF ($\sim 3000 \text{ MHz}$) frequencies; J1603-7202 contributes 761 epochs.

The current analysis processes all cataloged epochs through steps 002–003 (secondary spectra and closure analysis). Of 1,391 epochs with measurable closure triplets, 1,093 have ≥5 triplets and form the independent sample; the remaining 298 are excluded from primary inference but retained for downstream analyses. The analysis includes both L-band and UHF observations, with the L-band subset providing the primary detection.

J0437-4715 serves as the primary target with 1,391 closure-capable epochs. J1603-7202 provides a complementary velocity geometry test with 73.8° velocity vector separation from J0437-4715, yielding 248 viable epochs from 761 total processed.

#### 3.1.2 Jiamusi Pulsars (B0329+54, B0355+54, B0540+23, B0740-28, B1508+55, B1933+16, B2154+40, B2310+42, B2324+60, B2351+61)

The Jiamusi 66m telescope (China) provides observations of ten bright, distant pulsars with strong scintillation. These pulsars extend the distance baseline by \(\sim 10\times\), enabling critical tests of TEP scaling with scattering strength.

#### 3.1.3 MeerKAT Pulsars (J0908-1739, J0922-0638, J1731-4744)

The MeerKAT radio telescope (South Africa) provides observations of three pulsars via the Thousand-Pulsar-Array programme. Proper motions for these pulsars are not available in the current catalog; velocities are treated as unknown. These pulsars provide noise-limited bounding checks but do not contribute to geometric sign tests due to the missing velocity data.

Dataset rationale: The 15-pulsar sample spans diverse distances, velocities, and ISM environments. The robust Phase Closure detection is confined to J0437-4715; J1603-7202 contributes complementary bipolar geometric validations from the same Parkes/PPTA DR2 data family, supplying an independent line-of-sight geometric test of the Stokes-aligned holonomy pattern rather than a separate telescope systematics check. The Jiamusi and MeerKAT data provide noise-limited bounding checks against simple universal baseline offsets; their null status is consistent with TEP's predicted environmental suppression in distant, noise-limited sightlines, though this interpretation remains noise-limited and not independently diagnostic. The velocity vector geometry (particularly the 73.8° proper motion vector separation between J0437 and J1603) provides a geometric test of TEP's velocity-dependent predictions.

### 3.2 Data Processing Pipeline

Dynamic spectra are processed through a standardized, automated pipeline comprising 49 sequential steps organized into validation stages:

| Stage | Steps | Description |
| --- | --- | --- |
| Data Ingestion | 000 | Downloads scintillation data from Scintools ATNF repository and CSIRO Data Access Portal, generating epoch catalogs with observational metadata. |
| Core Processing | 001-003 | Parse and calibrate dynamic spectra (RFI mitigation, bandpass calibration), compute secondary spectra via 2D FFT with automated arc detection, and extract closure delays from triplets with sub-pixel parabolic interpolation. |
| Validation Suite | 004-014 | Comprehensive validation including verification, GR null hypothesis tests, systematic error analysis, alternative explanations simulation, parameter sensitivity, data quality metrics, ISM density modeling, environmental dependence, falsification criteria, and synthetic data validation. |
| Advanced Analysis | 015-048 | Extended validation including blind analysis, control pulsar analysis, orientation specification, systematic Monte Carlo, replication readiness, epoch-level significance, multi-pulsar scaling, bootstrap resampling, selection bias analysis, SNR correlation, alternative selection criteria, Bayesian hierarchical modeling, TEP scaling analysis, theoretical predictions, eta extraction, Parkes analysis, synthetic injection, arclet-based eta extraction, Jiamusi analysis, multi-pulsar scaling, temporal evolution, higher-order closures, and chromatic testing. |

### 3.3 Physical Metrology of Synchronization

The closure delay measurement rests on precise differential time metrology across scattered signal paths. The step-by-step physical procedure is:

Step 1: Signal Acquisition. Pulsar signals are recorded at the telescope with timestamps derived from hydrogen maser frequency standards (typical stability $10^{-15}$ at 100 s integration). Raw voltage samples are digitized at MHz sampling rates with GPS-disciplined clock synchronization.

Step 2: Dynamic Spectrum Formation. Voltage data undergo coherent dedispersion to remove interstellar dispersion, then channelization via polyphase filter banks yields intensity as a function of frequency and time $I(f, t)$ with microsecond temporal resolution.

Step 3: Secondary Spectrum Computation. The conjugate Fourier transform converts $I(f, t)$ to the secondary spectrum $S(\tau, f_D)$, where differential delay $\tau$ and differential Doppler $f_D$ conjugate variables encode scattering geometry. The 2D power distribution reveals parabolic scintillation arcs—signatures of scattered paths through anisotropic plasma screens.

Step 4: Arclet Identification. Local maxima in $S(\tau, f_D)$ above SNR $\geq 5$ threshold identify discrete scintillation arclets at coordinates $(\tau_i, f_{D,i})$. Cross-correlation of each arclet with parabolic templates refines arc positions with sub-pixel precision (0.1 ns interpolation).

Step 5: Triplet Formation and Closure Construction. For each epoch with $\geq 3$ arclets, all possible triplets $(i, j, k)$ are formed. The closure residual $C_{ijk} = \hat{\tau}_{ij} + \hat{\tau}_{jk} + \hat{\tau}_{ki}$, with each $\hat{\tau}_{ab}$ independently measured from the corresponding cross-term peak in the secondary spectrum, geometrically closes the path loop. By construction, this quantity is independent of absolute pulse arrival time and cancels all common-mode contributions—including standard relativistic effects and instrumental timing offsets—leaving only the path-dependent TEP holonomy signal.

Step 6: Stokes Alignment. Each triplet's oriented area in the $(\tau, f_D)$ plane determines its geometric sign. The effective velocity vector (pulsar proper motion plus Earth's orbital velocity, including annual aberration and Sagnac corrections) projects onto this area to yield the kinematic weighting factor. This alignment ensures that the signed delay correlates with velocity geometry as predicted by TEP. Earth's orbital velocity is obtained from the solar-system barycentric Earth state vector via `astropy.coordinates.get_body_barycentric_posvel` (astropy 6.1, built-in SOFA/ERFA ephemeris `epv00`), yielding agreement with JPL DE440 at the centimetre-per-second level.

Step 7: Statistical Aggregation. The Phase Closure \(\psi\) across all triplets provides the primary TEP detection statistic. Under GR, \(\psi = 0\); under TEP, \(\psi \neq 0\) with magnitude scaling with scattering strength and path geometry. The bipolar prediction—opposite signs for opposite velocity projections—is tested via signed mean validations. The magnitude $M_{\rm cl} = \langle |C_{ijk}| \rangle$ is analyzed as a secondary validation, noting its susceptibility to noise floor bias.

This metrology chain transforms raw telescope voltages into a physically interpretable measure of path-dependent time transport, with each step traceable to atomic clock standards and geometric observables.

### 3.4 Error Propagation and Uncertainty Quantification

*Glossary — Phase Closure estimators on the 1,093-epoch J0437-4715 sample:*

- *Inverse-variance weighted circular mean* \(\bar{\psi}_w\) (primary precision summary): Epoch-level Phase Closure angles are aggregated with inverse-variance weights from delay-domain dispersion. Typical reported values include \(\bar{\psi}_w = 0.984 \pm 0.046\) rad (circular SE), \(R_{\mathrm{bar}} = 0.308\), Rayleigh \(p \approx 1.39\times10^{-13}\); epoch-level bootstrap 95% CI \([{+}0.737,{+}1.235]\) rad (Section 4.1, Section 3.4.1).

- *Unweighted circular mean* \(\bar{\psi}_{\mathrm{uw}}\) (frame-invariance baseline): Equal weight per epoch; identical \(\bar{\psi}_{\mathrm{uw}}\) in heliocentric and CMB frames for J0437. Typical values \(\bar{\psi}_{\mathrm{uw}} \approx {+}1.120\) rad, \(R_{\mathrm{bar}} \approx 0.304\), Rayleigh \(p \approx 1.34\times10^{-44}\); bootstrap 95% CI $[{+}0.990,{+}1.253]$ rad (Section 4.13). Use \(\bar{\psi}_{\mathrm{uw}}\) when testing dependence on the velocity frame; use \(\bar{\psi}_w\) when quoting the inverse-variance pipeline summary.



The uncertainty in the Stokes-aligned closure delay is derived from the standard error of the mean across all measurements. For PSR J0437-4715, this comprises 19,167 triplet measurements from 1,391 closure-capable epochs. Individual triplet delay uncertainties (\(\sigma_{\text{triplet}} \approx 0.5\text{--}2 \text{ ns}\), measured via cross-correlation peak fitting) propagate through the analysis as follows:

Individual measurement uncertainty: Each closure residual \(C_{ijk} = \hat{\tau}_{ij} + \hat{\tau}_{jk} + \hat{\tau}_{ki}\) (for triplet paths \(i, j, k\)) has an associated uncertainty \(\sigma_{\tau}\). In the pipeline implementation, this is approximated by the characteristic grid resolution of the secondary spectrum:

\begin{equation}
\label{eq:error_prop_triplet}
\sigma_{\tau} = d_{\tau} \times \sqrt{3} \approx 1\text{--}3\text{ ns}
\end{equation}

where \(d_{\tau}\) is the median grid spacing in the delay domain and the \(\sqrt{3}\) factor accounts for the three cross-terms in the closure sum. This provides a characteristic scale for triplet uncertainties; the final reported uncertainties use the empirical standard error of the mean across all measurements.

Epoch-level independence: The conservative framework treats the observational epoch as the fundamental unit of independence. Epochs with fewer than 5 triplets are excluded from the primary statistical analysis because low triplet counts produce unreliable epoch-level means. For PSR J0437-4715, of 1,391 closure-capable epochs, 298 have fewer than 5 triplets and are excluded from epoch-level aggregation, leaving \(N_{\text{epochs}} = 1{,}093\) independent epoch means for primary inference. The uncertainty in the global mean \(\sigma_{\text{global}}\) is given by the standard error across these independent samples:

\begin{equation}
\label{eq:error_prop_global}
\sigma_{\text{global}} = \frac{\text{std}(\text{epoch_means})}{\sqrt{N_{\text{epochs}}}}
\end{equation}

Because the triplets formed within a single epoch share arclets and propagation paths, triplet-level measurements are not mutually independent; the operative SEM for the inverse-variance-weighted global \(M_{\rm cl}\) is therefore obtained by resampling epochs rather than triplets. An epoch-level bootstrap of the identical weighted estimator (10,000-resample audit, step_060) yields \(\sigma_{\rm global} = 0.26\) ns under independent-epoch resampling, rising to \(0.39\text{--}0.55\) ns under contiguous 15–240-epoch moving blocks (≈77–1200 d median span) that absorb residual campaign/screen-state clustering; the epoch-mean autocorrelation persists to ~200-epoch lags, so the blocked value is a lower envelope rather than a converged plateau. The internal inverse-variance SEM \(1/\sqrt{\sum_i w_i} = 0.102\) ns is retained in the pipeline output as the measurement-noise diagnostic but is not the quoted aggregate uncertainty.

#### 3.4.1 Sample Accounting

The following table reconciles every reported J0437-4715 statistic to a single pipeline stage. All counts are extracted directly from `step_003_closure_final_summary_j0437.json` and the per-epoch results file `step_003_closure_final_per_epoch_j0437.json`.

| Stage | J0437 epochs | J0437 triplets | Reason for change |
| --- | --- | --- | --- |
| Raw processed | 10,201 | — | All epochs with secondary spectra (step 002 output) |
| Closure-capable | 1,391 | 19,167 | ≥3 arclets, produced measurable triplets |
| Independent epoch sample | 1,093 | — | Epochs with ≥5 triplets (reliable epoch-level statistics) |
| Phase-closure sample | 1,093 | — | Epoch-level aggregation (conservative independence framework) |
| \(M_{\rm cl}\) magnitude sample | 1,093 | — | Epoch-level aggregation; magnitudes are noise-floor dominated |

Every reported primary statistic refers to the 1,093-epoch independent sample. The 19,167 triplet count describes the full closure-capable dataset and is reported in the abstract and data tables for sample-size transparency. The 298 epochs excluded by the ≥5-triplet filter are retained in the raw results and used by downstream analyses (e.g., probabilistic weighting, bootstrap) that do not require epoch-level aggregation.

Global Stokes-aligned Phase Closure: The primary TEP detection statistic is the Phase Closure \(\psi\), computed with circular statistics on the 1,093-epoch sample. The circular mean is \(\bar{\psi} = 0.984 \pm 0.046 \text{ rad}\) (circular SE, \(\bar{R}\) = 0.308, Rayleigh p = 1.39 × 10⁻¹³, V-test p = 1.02 × 10⁻⁵). An epoch-level circular bootstrap (10,000 iterations) yields a 95% CI of [+0.737, +1.235] rad, which excludes zero. The raw absolute magnitude is \(M_{\rm cl} = 8.100 \pm 0.26 \text{ ns}\) (epoch-level bootstrap SEM; the internal inverse-variance SEM \(1/\sqrt{\sum_i w_i} = 0.102\) ns counts triplet measurement noise only and understates the epoch-to-epoch scatter), but a per-epoch folded-normal noise floor \(E[M_{\rm cl}]\) = 6.810 ns quantitatively saturates this value, leaving a noise-subtracted excess of \(1.290 \pm 0.11\) ns (paired epoch-level SEM of the mean-minus-floor difference, \(t \approx 12.2\sigma\); \(\approx 7.6\)–\(10.8\sigma\) under 15–240-epoch contiguous-block resampling, step_060 audit). The Phase Closure \(\psi\) is therefore the primary TEP validation; \(M_{\rm cl}\) remains a noise-floor-dominated validation in the current dataset.

The signed mean ($-0.184 \pm 0.157 \text{ ns}$ epoch-level, $-1.17\sigma$)—computed on Stokes-aligned closure delays—shows minimal bipolar bias, consistent with nearly isotropic sampling of the scattering geometry. This near-zero value indicates successful alignment of the bipolar signal (where positive and negative contributions cancel), confirming the expected parity behavior. The Phase Closure \(\psi\) measures the physical holonomy phase and is the primary detection metric.

The total standard deviation (\(\sigma_{\text{total}} \approx 16 \text{ ns}\) for raw \(M_{\rm cl}\)) reflects measurement uncertainty, ISM variability, and the heavy-tailed noise distribution. The frequentist MAD-based folded-normal subtraction reported above (\(1.290 \pm 0.11\) ns epoch-level, \(t \approx 12.2\sigma\)) and the hierarchical Bayesian posterior for triplet-level holonomy amplitude \(H\) (\(0.000 \pm 0.459\) ns; 95% CI [0.000, 0.899] ns) answer different questions and must not be conflated: the former subtracts a per-epoch Rice floor from the inverse-variance-weighted epoch mean, whereas the latter marginalizes triplet delays under a generative \(\text{Normal}(s_i H, \sigma_i)\) model with inverse-variance weights and a HalfNormal prior, testing whether latent \(H\) exceeds zero after probabilistic weighting. Because the orientation sign is included in the model, \(H\) estimates the orientation-aligned amplitude. The posterior near zero shows that this particular signed Gaussian model does not recover a coherent delay-domain amplitude. The folded trimmed excess measures a different, unsigned feature of the distribution and is not by itself a signed-holonomy estimate. Phase Closure \(\psi\) remains the primary detection statistic.

### 3.5 Statistical Analysis Framework

The statistical validation employs multiple cross-validated methods:

| Method | Description |
| --- | --- |
| *\(M_{\rm cl}\) magnitude test* | One-sample t-test on absolute delays (noise-floor consistency check; primary detection is Phase Closure \(\psi\)) |
| *Orientation-resolved analysis* | Separate tests for negative and positive delays |
| *Magnitude equality* | Two-sample t-test comparing \(\|H^{-}\|\) vs. \(\|H^{+}\|\) |
| *Non-parametric confirmation* | Wilcoxon signed-rank test |
| *Binomial validation* | Sign test for orientation balance |
| *Robustness* | Bootstrap confidence intervals (1000 iterations, seed=42) |
| *Bayesian estimation* | Posterior probability calculation |
| *Bootstrap resampling* | Robust confidence intervals (1000 iterations) |
| *Bipolar structure test* | TEP-specific bimodal validation |

Multiple Comparison Correction: Bonferroni correction is applied for the 4 primary validation tests (aggregate mean, \(M_{\rm cl}\) magnitude, negative delays, positive delays). Uncorrected \(\alpha = 0.05\), corrected \(\alpha = 0.0125\). All reported significance values account for this correction.

The closure delay distribution for J0437-4715 exhibits the expected behavior for interstellar scattering data. The Bayesian analysis provides robust estimates of the mean and uncertainty despite the presence of outliers, which are handled through probabilistic weighting.

The standard mean and Bayesian hierarchical model are reported as robust estimators, with the latter naturally weighting measurements by their individual uncertainties to reduce sensitivity to heavy tails while preserving the sign and central tendency of the signal.

Fixed random seeds ensure full reproducibility. All exclusion decisions are logged with explicit reasons to prevent cherry-picking.

### 3.6 Data Sources and Acknowledgments

The primary scintillation data used in this study are sourced from the Parkes Pulsar Timing Array Data Release 2. Specifically, dynamic spectra for PSR J0437-4715 were obtained from the CSIRO Data Access Portal (Reardon et al. 2020, DOI: 10.25919/5f3cd2bc1c213), while data for PSR J1603-7202 were retrieved from the same portal following the work of Walker et al. (2022, DOI: 10.25919/82f5-mh79). Additionally, 5 archival epochs were sourced from the ATNF Scintools repository. For the more distant Jiamusi pulsars B0329+54, B0355+54, B0540+23, B0740-28, B1508+55, B1933+16, B2154+40, B2310+42, B2324+60, and B2351+61, dynamic spectra were obtained from the Jiamusi 66m telescope Pulsar Scintillation Archive, collected during monitoring campaigns and processed using standard radio astronomy pipelines.

Both CSIRO datasets are provided under the Creative Commons Attribution 4.0 International Licence.

## 4. Results: Detection of Synchronization Holonomy

### 4.1 Primary Detection and validations

Analysis of 19,167 scintillation triplets from the primary PSR J0437-4715 dataset (2002–2018) reports evidence consistent with Stokes-aligned proper-time holonomy through the Phase Closure ψ metric. Of 1,391 epochs with closure triplets, 298 had fewer than 5 triplets and were excluded from the independent sample, leaving 1,093 epochs for primary inference.

*Observable map (aggregation level).* *Triplet-level* quantities include per-triplet aligned closure delay \(H_i\) (geometric `delta_us`), per-triplet phase closure \(\psi_i\) on \([-\pi,\pi)\), and triplet-level \(|H_i|\) magnitudes used in some null sweeps. *Epoch-level* summaries form one mean signed delay and one mean \(M_{\rm cl}\) per epoch (before global weighting), plus the SNR²-weighted circular mean of \(\psi_i\) within each epoch for the primary phase stack. *Global* Step 003 statistics inverse-variance-weight those epoch delay means for \(M_{\rm cl}\) and delay-SEM-weight the epoch circular \(\psi\) means for the headline Rayleigh/V/bootstrap pipeline. Step 007’s epoch-mean \(M_{\rm cl}\) cross-validation averages \(|\text{epoch mean } H|\) over holdout epochs; Step 007’s `phase_closure_epoch_cv` applies the same epoch \(\psi\) construction with a shuffle split (dedicated RNG seed in the JSON). These channels must not be numerically cross-compared without noting the aggregation.

The Phase Closure ψ is the primary detection statistic—a zero-centered, odd-parity metric theoretically immune to the Rice bias that affects folded delay magnitudes. All phase statistics are computed with circular statistics (circular mean, circular standard deviation, and branch-cut-safe bootstrap confidence intervals) because ψ lives on the circle [−π, π). Because the Rayleigh test (\(Z=2n_{\rm eff}\bar{R}^2\)) and V-test test circular uniformity against concentration (and a distribution tightly concentrated at zero would also yield a small Rayleigh \(p\)-value), concentration and non-zero mean direction are tested by distinct statistics: (i) circular concentration is established by Rayleigh \(Z = 59.21\) (\(p = 1.39 \times 10^{-13}\) weighted; \(Z_{\rm uw}=202.05\), \(p=1.34\times 10^{-44}\) unweighted) and V-test \(V = +4.26\) (\(p = 1.02 \times 10^{-5}\) relative to \(\mu_0 = 0\)), and (ii) rejection of zero mean closure (\(\langle\psi\rangle = 0\)) is established by the measured circular mean \(\bar{\psi} = 0.984 \pm 0.046\) rad (circular SE, \(\bar{R} = 0.308\); unweighted \(\psi = 1.120\) rad) and its 10,000-iteration epoch-level circular bootstrap 95% CI of \([+0.737, +1.235]\) rad (unweighted 95% CI \([+0.990, +1.253]\) rad), both of which exclude zero at \(>15\sigma\).

*Weighted Rayleigh convention.* The reported \(Z\) uses the inverse-variance-weighted epoch sample: \(Z = 2 n_{\rm eff}\bar{R}^2\) with effective sample size \(n_{\rm eff} = (\sum_i w_i)^2/\sum_i w_i^2 \approx 313\), and \(p = e^{-Z/2} = e^{-n_{\rm eff}\bar{R}^2}\). With \(\bar{R} = 0.3076\) this gives \(Z = 59.21\), \(p = 1.39\times10^{-13}\). The unweighted construction on the same 1,093 independent epochs gives \(Z_{\rm uw} = 202.05\) (\(p = 1.34\times10^{-44}\), \(\bar{R}_{\rm uw} = 0.304\)); the unweighted statistic is the frame-invariant primary. Both conventions reject circular uniformity.

In the delay domain, subtracting the per-epoch folded-normal noise floor ($E[M_{\rm cl}] = 6.810$ ns) from the epoch-level inverse-variance weighted closure magnitude ($M_{\rm cl} = 8.100 \pm 0.26$ ns) yields a paired noise-subtracted excess of $1.290 \pm 0.11$ ns (paired epoch-level bootstrap SEM; $t \approx 12.2\sigma$; $\approx 7.6$–$10.8\sigma$ under 15–240-epoch contiguous blocks, Steps 052 and 060), corroborating the primary phase-domain detection. The statistical properties of raw unsigned delay estimators ($M_{\rm cl}$) and trimmed diagnostics ($M_{\rm cl,sym10}$) are detailed in Appendix B.

#### 4.1.1 Hierarchical Variance Decomposition

A hierarchical Bayesian uncertainty model accounts for variance contributions at multiple levels: triplet, epoch, screen state, pulsar, and telescope. This variance decomposition yields the effective sample size accounting for correlations:

| Level | N | Variance Contribution |
| --- | --- | --- |
| Triplet (raw) | 19,167 | 100% |
| Closure-capable epochs | 1,391 | 1.88% |
| Independent epoch sample | 1,093 | — |
| Pulsar | 1 | 0.0% |
| Telescope | 1 | 0.0% |

The primary statistical analysis uses the 1,093-epoch independent sample (epochs with ≥5 triplets). The epoch-level variance contribution (1.88%) indicates moderate correlation within epochs, as expected for scintillation data. The 298 epochs with <5 triplets are retained in the raw results for downstream analyses that do not require epoch-level aggregation.

#### 4.1.2 Systematic Error Budget and Phase-Closure Robustness

The primary detection statistic ψ is estimated via circular statistics on 1,093 independent epochs. Its precision is quantified by the circular standard error, which already encodes all sources of phase-angle variance (measurement noise, pixel discretization, thermal variation, and any residual calibration structure) through the empirical dispersion of the angles themselves. The measured circular mean is ψ = 0.984 ± 0.046 rad (circular SE, \(\bar{R}\) = 0.308). The significance of this detection is self-contained: Rayleigh Z = 59.21 (p = 1.39 × 10⁻¹³) rejects uniformity, V-test V = +4.26 (p = 1.02 × 10⁻⁵) rejects circular uniformity toward the reference direction μ₀ = 0, and an epoch-level circular bootstrap (10,000 iterations) yields a 95% CI of [+0.737, +1.235] rad that excludes zero. The bootstrap CI establishes confidence-interval resolution: no ordinary bootstrap replicate crossed zero, so the directly resolved empirical tail is p < 10⁻⁴. A formal null p-value would require a null-centred block bootstrap or permutation distribution. No linearized delay-to-phase conversion is required or used for these inference statements.

Below the primary detection, a separate delay-domain metrology budget is maintained for the \(M_{\rm cl}\) magnitude validations. Because closure delays cancel common-mode contributions—including standard relativistic effects and instrumental calibration offsets—by construction, the systematic terms relevant to \(M_{\rm cl}\) are bounded and qualitatively different from those that would affect a single-path delay measurement. The budget quantifies the residual uncertainty in delay-domain amplitude metrology; it does not propagate linearly into the phase-domain circular error because the circular SE already captures the operative phase variance.

| Error Source | Delay-domain (ns) | How Estimated | Relevance to Closure Observables |
| --- | --- | --- | --- |
| Statistical (delay SEM) | 0.26 | Standard error of the mean for 1,093 independent epochs | Baseline precision for \(M_{\rm cl}\) magnitude |
| Pixel discretization | 0.197 | Empirical bound from parabolic sub-pixel interpolation: ≤ 0.1 × d<sub>τ</sub>, where d<sub>τ</sub> = 1.97 ns is the median tau grid spacing | Directly bounds cross-term peak-location precision; contributes to phase-angle scatter already captured by circular SE |
| Thermal noise floor variation | 0.137 | Epoch-to-epoch σ variation (σ<sub>epoch</sub> = 5.11 ns across 1,391 epochs) propagated to the mean | Captures stationary-to-station noise-floor differences; common-mode in closure |
| Bandpass calibration | 0.128 | 1% of median \(M_{\rm cl}\) from Parkes/UWL bandpass accuracy specification | Common-mode in closure: differential delays across paths largely cancel bandpass residuals |
| **Core metrology systematic** | **0.272** | Quadrature sum of pixel discretization, thermal noise floor variation, and bandpass calibration | Physical metrology uncertainty on \(M_{\rm cl}\); does not include threshold-robustness validations |

The core metrology systematic of 0.272 ns is dominated by the pixel discretization bound (0.197 ns). This bound is a conservative theoretical upper limit for parabolic sub-pixel interpolation. The actual contribution to phase-angle variance is already subsumed in the empirical circular SE of 0.045 rad: if pixel discretization were a dominant phase error, the circular distribution would be broadened and \(\bar{R}\) would be reduced; the observed \(\bar{R}\) = 0.308 is consistent with the measured phase scatter.

*Threshold robustness (step_041):* The SNR and arclet threshold sweeps in step_041 test the stability of the signed mean across selection cuts. The signed mean is a physical metric expected to be near zero due to bipolar cancellation (Section 2.2); its variation across thresholds (span ±0.57 ns across the tested SNR grid) reflects the inherent noise of a near-zero bipolar estimator, not a systematic bias in ψ. The unsigned \(M_{\rm cl}\) magnitude is robust across all threshold cuts (t-statistic ≥ 35.3σ minimum). The Phase Closure ψ itself was not threshold-dependent: the circular mean and \(\bar{R}\) are stable because ψ is extracted from complex cross-term phases prior to any SNR-based weighting. Threshold-robustness concerns therefore apply to the signed-mean validation, not to the primary phase detection.

#### 4.1.3 Validation Summary

The Phase Closure distribution across the primary dataset shows a non-zero central value with concentrated circular scatter (\(\bar{R}\) = 0.308, circular σ = 1.50 rad), consistent with a physical signal superimposed on measurement noise. The following table summarizes the primary and secondary validation metrics:

| Estimator | Metric | Statistical Significance | Circular Tests | Delay-Domain Consistency |
| --- | --- | --- | --- | --- |
| Phase Closure ψ (Primary, Weighted) | *0.984 ± 0.046 rad* (circular SE, \(\bar{R}\) = 0.308) | Rayleigh Z = 59.21 (p = 1.39 × 10⁻¹³); V-test p = 1.02 × 10⁻⁵. Phase Closure ψ is the primary detection metric. | Rayleigh p = 1.39 × 10⁻¹³; V-test p = 1.02 × 10⁻⁵; Bootstrap 95% CI [0.737, 1.235] rad excludes zero | Core metrology systematic 0.272 ns; does not affect circular inference |
| Phase Closure ψ (Primary, Unweighted) | *1.120 rad* (\(\bar{R}_{\rm uw}\) = 0.304) | Rayleigh \(Z_{\rm uw} = 202.05\) (\(p = 1.34 \times 10^{-44}\)); strictly frame-invariant | Bootstrap 95% CI [+0.990, +1.253] rad excludes zero (>15σ) | Identical in heliocentric and CMB frames |
| Paired Noise-Subtracted Delay Excess \(M_{\rm cl,excess}\) | 1.290 ± 0.11 ns | ≈12.2σ iid (7.6–10.8σ under 15–240-epoch block bootstrap; Steps 052, 060) | — | Paired excess of \(M_{\rm cl} = 8.100\pm 0.26\) ns above folded-normal floor \(E[M_{\rm cl}] = 6.810\) ns (see Appendix B for raw and trimmed delay-amplitude diagnostics) |

### 4.2 ISM Null Model Simulations

Standard ISM models were tested to determine whether they can produce the observed non-zero Phase Closure ψ. Seven scalar-factorizable ISM null implementations were simulated in step_008. All returned zero mean Phase Closure and none reproduced the observed coherent signal — a loop-ordered monopole whose orientation parity is verified algebraic and whose placement classes share one mean direction (Step 056). The observed Phase Closure reported in Section 4.1 is inconsistent with all tested scalar-factorizable implementations.

| ISM Model | Predicted ψ | Simulated \(M_{\rm cl}\) (ns) | Can Explain? |
| --- | --- | --- | --- |
| Thin screen Kolmogorov | 0.0 | 0.0 | NO |
| Multi-screen | 0.0 | 0.0 | NO |
| Thick screen (5 layers) | 0.0 | 0.0 | NO |
| Anisotropic turbulence | 0.0 | 0.0 | NO |
| Velocity gradient | 0.0 | 0.100 | NO |
| Instrumental noise | 0.0 | 7.966 | NO |
| Localized anisotropic filament | 0.0 | 0.0 | NO |

None of the tested scalar-factorizable ISM implementations reproduces the observed coherent non-zero Phase Closure. All simulated null distributions from step_008 are centered at zero, while the observed circular mean has a 95% bootstrap CI that excludes zero. The instrumental-noise simulation produces \(M_{\rm cl}\) ~ 8 ns through unsigned noise bias, but its Phase Closure distribution remains centred at zero and does not reproduce the observed coherent circular mean.

### 4.3 Velocity-Sign Consistency Check

An informal velocity-sign consistency check was performed. Pulsar names and proper motion vectors were temporarily withheld while ψ was computed, then matched to velocity vectors to test whether the observed sign pattern matched the TEP prediction.

Note on B2324+60: This pulsar has no measured proper motion in the literature (Wang et al. 2018, Table 1), so a zero proper motion reference value of (0.0, 0.0) mas/yr was used.

Velocity-sign consistency table. The \(\psi\) column is the Step 003 *inverse-variance-weighted circular mean* across epochs (delay-variance weights); it is not the unweighted frame-invariant circular statistic used for the primary Rayleigh gate.

| Pulsar | ψ (rad) | Bipole sign | Test statistic | μ (mas/yr) |
| --- | --- | --- | --- | --- |
| J0437-4715 | +0.984 | Weak bipole (ψ₁ = −0.129) | Rayleigh Z = 59.21 (p = 1.39 × 10⁻¹³) | 140.9 |
| J1603-7202 | −2.328 | Strong bipole (ψ₀ = −0.821, ψ₁ = −1.965) | High dispersion: \(\bar{R}\) = 0.027, Rayleigh p = 0.936 (bipolar geometry PASS) | 26.7 |

Out of 15 pulsars analyzed, one shows a robust circular Phase Closure detection (J0437-4715, Section 4.1). J1603-7202 shows a strong bipolar Stokes decomposition (ψ₀ = −0.821, ψ₁ = −1.965) with high circular dispersion at its lower transverse speed (Rayleigh p = 0.936, Section 4.1), matching the TEP expectation that the monopole is diluted while bipolar structure remains. The table lists inverse-variance–weighted circular ψ for both Parkes targets; those weights are delay-variance dominated, so the weighted circular row is not the primary J1603 statistic. The informal velocity-sign stress test instead uses the *signed epoch-mean closure delays*, which are negative for J0437 and positive for J1603 (Section 4.9), matching the opposite bias expected when the two proper-motion vectors are separated by ~74°. This pattern is consistent with TEP across two independent sightlines with complementary observables.

The non-zero Phase Closure signal reported in Section 4.1 provides robust, zero-centered evidence for TEP holonomy. The Phase Closure distribution across the primary dataset shows a non-zero circular mean with concentrated scatter (\(\bar{R}\) = 0.308), consistent with a physical signal superimposed on measurement noise.

### 4.4 Velocity-Direction Controls

A critical review concern is whether the Stokes/velocity-projection alignment procedure encodes the expected sign. Five independent controls are implemented to demonstrate that the observed sign structure is physical, not an artifact of the alignment algorithm.

| Control | Description | Result | Status |
| --- | --- | --- | --- |
| 1. Velocity-label permutation | Swap velocity labels between pulsars; check sign consistency | Both velocity-label permutations attain the same best-match score under Step 047 (expected degeneracy at N=2) | DOCUMENTED |
| 2. Pre-alignment validation | Raw closure delays before velocity weighting | Phase coherence exists before velocity weighting (Step 047 pre-alignment validation) | PASS |
| 3. Velocity-sign freeze check | ψ computed with velocity labels temporarily withheld, then matched to check sign consistency | Freeze-record passes; records Step 003 ψ values and timestamps (reproducibility metadata; external blinding is outside manuscript scope) | PASS |
| 4. Wrong-velocity control | Reversed sign and random-direction velocities destroy coherence | Not applicable to phase_closure_rad: velocity is not an input to phase extraction | NOT APPLICABLE |

*Control 1 (Velocity-Label Permutation):* With 2 pulsars, there are 2! = 2 possible velocity-label assignments. In the current Step 047 implementation, both permutations tie the best match score, which is the expected score degeneracy at N=2; the row documents that outcome rather than supplying extra degrees of freedom for label swapping.

*Control 2 (Pre-Alignment validation):* The epoch-level phase-closure distribution is already coherent before any velocity-domain weighting, demonstrating that the primary phase observable is not created by the alignment step.

*Control 3 (Freeze Record):* The Step 047 freeze record stores a hash of the Step 003 phase-closure values and file timestamps at control runtime. This records the internal chronology of artifacts for reproducibility; external blinding protocols are outside the scope of the present manuscript, so the freeze record is classified as a documentation control.

*Control 4 (Wrong-Velocity):* Not applicable to the primary phase closure observable. Individual triplet phases (and therefore epoch-level phase closure) are extracted from the complex secondary spectrum prior to any velocity weighting, so modifying the velocity vector cannot change phase_closure_rad.

*Phase-Scramble Specificity (Step 047):* The phase-scramble control is a uniform-null Monte Carlo: it draws 1,000 sets of \(N\) independent uniform phases on \([-\pi, \pi)\) and compares the resulting circular concentration (\(\bar{R}\)) to the observed value. This generates fresh random angles from the uniform distribution; it does not permute the observed angles (permuting a fixed set of angles would leave \(|\sum_i e^{i\psi_i}|\) exactly unchanged). In 1,000 Monte Carlo samples of independent uniform phases, none reached the observed concentration; the finite-simulation estimate is \(p = (0+1)/(1000+1) = 9.99\times10^{-4}\).

*Leg-Permutation Null (Step 055):* The deferred bispectrum control—permuting one cross-term leg within each epoch before reconstructing \(\mathcal{B}_{ijk}\)—is now implemented. The per-triplet leg phases \(\phi_{01}, \phi_{12}, \phi_{02}\) were recomputed through the unchanged Step 002–003 chain on all 1,391 viable epochs (the recomputed aggregation reproduces the headline values exactly: \(\psi = 0.984\) rad, \(\bar{R} = 0.308\), \(n = 1{,}093\)). Permuting one leg's phases across the triplets of each epoch destroys loop-level association while preserving every leg's marginal distribution, the estimator chain, and the epoch weights. Over 400 replicates the concentration collapses from \(\bar{R} = 0.308\) to \(\bar{R}_{\rm perm} = 0.097\text{--}0.129\) depending on which leg is permuted (and to \(0.066\) when the leg is drawn from a different epoch entirely), with no replicate reaching the observed value (\(p \le 2.5\times10^{-3}\)). An extended 5{,}000-replicate audit of the strictest cross-epoch mode (step_059) resolves the floor: no replicate attains \(\bar{R} = 0.308\) (null maximum \(0.190\), empirical \(p < 2\times10^{-4}\)); the observed concentration sits \(7.6\) null standard deviations above the cross-epoch null mean (Gaussian-tail \(p \approx 2\times10^{-14}\)). The measured \(\psi\) is therefore a genuine per-loop phase association—a real closure property of the data—not an artifact of the leg marginals or the estimator.

*Non-Factorizable Screen Null (Step 055):* The second control synthesizes dynamic spectra \(I(\nu,t) = |E(\nu,t)|^2\) where \(E\) is a complex Gaussian field whose conjugate spectrum concentrates on the measured J0437 two-arc geometry (\(\eta_1 = 0.0072\), \(\eta_2 = 0.0076\) \(\mu\)s/mHz\(^2\), arclet \(|f_D|_{\max} = 6.5\) mHz modelled as a \(\sigma = 3.25\) mHz Gaussian extent, flux statistics matched to the data). For this \(\chi^2\) field the secondary-spectrum cross-terms are convolutions, not factorizable products, so the closure phase is generically non-zero under standard scintillation physics—the channel the scalar-factorizable step_008 suite cannot reach. Pushed through the identical chain, 71 of 3,000 synthetic epochs met the cross-screen viability cuts; their concentration at matched sample size (\(\bar{R} = 0.26\), vs the real distribution resampled to the same \(n\): \(0.35 \pm 0.13\)) is statistically comparable to the data—closure-phase concentration of this magnitude is *not* anomalous per se. What the screen null does not reproduce is the direction persistence: synthetic epoch-mean directions are consistent with uniform (Rayleigh \(p = 0.55\); unweighted direction \(\bar{R} = 0.09\) overall, \(0.14\) symmetric and \(0.04\) for the asymmetric-arm variant), while the 1,093 real epoch means persist at \(+0.984\) rad over fifteen years. The anomaly thus contracts to the stable direction of the closure phase, not its existence.

*Forward Scintillation-Null Battery (Step 058):* To test whether the circular-uniformity null is mis-specified—i.e., whether standard thin-screen scintillation can concentrate closure phases near the observed direction through mundane physics—a suite of additive-image thin-screen forward models was propagated through the unchanged Step 002–003 chain (Hough arc detection, arclet selection, the cross-screen triplet requirement, sub-pixel cross-term measurement, and complex-patch phase extraction), with epoch dimensions, sampling intervals, and observing bands drawn from the empirical distribution of the real J0437 catalogue. Ten models spanning single-screen, two-screen, non-quadratic curved-arc, discrete arclet-family, interior-fill, and three-screen propagation, together with a 30%-persistent-arclet variant, a fully persistent image field, pure noise, and a phase-scrambled control, contribute \(\sim\)14,500 synthetic epochs and \(\sim\)85,000 simulated triplets. Every model in which the image population decorrelates between epochs—the appropriate ISS null, since arc curvatures persist while image content decorrelates on the scintillation timescale—returns a pooled \(\psi\) distribution consistent with circular uniformity (\(\bar{R} \le 0.05\); epoch-direction \(\bar{R} \le 0.12\)), against the observed \(\bar{R} = 0.304\) and direction persistence \(0.311\); the phase-scrambled control—identical amplitude spectrum, identical arc detection and triplet selection, with only the measured phases randomized—yields \(\bar{R} = 0.003\) and epoch-direction \(\bar{R} = 0.035\) over 668 viable epochs, the deepest null in the battery at full yield, confirming that the clustering resides entirely in the phase content rather than the selection chain. Concentration appears only in the frozen-field limit in which every image's \((\tau, f_D)\) position, amplitude, and phase is held fixed across all epochs; each such screen realization concentrates \(\psi\) at a realization-specific direction scattered over the circle (per-realization \(\bar{R} \sim 0.03\)–\(0.39\), directions spanning \([-2.8, +1.5]\) rad), with no systematic convergence near the observed \(+1.1\) rad. The measured-position sign-pattern channel is reproduced structurally by the simulators—origin-straddling peak captures occur at comparable rates—but the flipped-sign triplets remain circularly uniform in every fresh-population model (per-pattern \(\bar{R} \lesssim 0.15\)), in contrast to the real sample where those patterns concentrate at \(+1.3\)–\(+1.4\) rad. The closure-phase concentration is therefore not a generic property of multi-screen propagation, non-quadratic curvature, discrete arclet families, or of the measurement channel itself; it is specific to the data.

Together, these controls demonstrate that the primary phase observable exists before velocity weighting, that scramble controls reject a trivial circular-null interpretation, and that the closure is a genuine loop-level association under leg permutation. With only two Parkes pulsars, velocity-label permutation is necessarily score-degenerate; the discriminating power is carried by the phase-scramble, pre-alignment, leg-permutation, and direction-persistence checks together with the J0437 circular statistics.

### 4.5 Independent Statistical Validation

The primary detection statistic is the Phase Closure \(\psi\), whose 95% circular-bootstrap interval \([0.737,1.235]\) rad excludes zero and whose directional V-test rejects uniformity toward \(\mu_0 = 0\) at \(p = 1.02 \times 10^{-5}\) (Rayleigh \(Z = 59.21\), \(p = 1.39 \times 10^{-13}\) weighted; \(Z_{\rm uw} = 202.05\), \(p = 1.34 \times 10^{-44}\) unweighted). Five independent validation suites test out-of-sample generalization, parity symmetry, seasonal stability, hierarchical weighting, and synthetic injection recovery; legacy delay-domain amplitude estimators (\(M_{\rm cl}\)) and their folded-normal Rice noise-floor diagnostics are documented in Appendix B.

#### 4.5.1 Phase Closure \(\psi\) — Epoch Holdouts (Circular Cross-Validation)

Step 007 implements `phase_closure_epoch_cv`: a 5-fold shuffle split on the *same* 1,093-epoch stack used for primary circular inference (epochs with \(\geq 5\) triplets; within-epoch SNR²-weighted circular mean of triplet \(\psi\); across-epoch inverse-delay-variance weights, matching the Step 003 construction). A dedicated RNG seed (\(42 + 70007\)) generates the fold assignment. Each holdout fold recomputes circular mean \(\psi\), Rayleigh uniformity, and a directed V-test of uniformity toward \(\mu_0 = 0\) using only held-out epochs. Pass rule in the ledger: ≥4/5 folds have Rayleigh \(p < 0.05\) *and* the fold circular mean lies in the same open half-plane as the full-sample mean \(\cos(\bar\psi_{\text{fold}} - \bar\psi_{\text{full}}) > 0\). Frozen ledger: 5/5 folds meet the Rayleigh threshold.

Full-sample (recomputed from arrays, Step 003-consistent): \(\bar\psi = +0.984\) rad, Rayleigh \(Z = 59.21\) (\(p = 1.39e-13\)), V-test \(p = 1.02e-05\); the recomputed full-sample \(\bar\psi\) matches the Step 003 summary weighted mean at machine precision (\(|\Delta\psi| = 0\) rad in the frozen ledger).

| Fold | Holdout epochs | \(\bar\psi_{\text{fold}}\) (rad) | Rayleigh \(p\) | V-test \(p\) | \(\cos\Delta\) vs global |
| --- | --- | --- | --- | --- | --- |
| 0 | 218 | \(+0.583\) | \(2.30e-03\) | \(1.81e-03\) | \(0.9206\) |
| 1 | 218 | \(+1.035\) | \(8.55e-03\) | \(5.75e-02\) | \(0.9987\) |
| 2 | 218 | \(+1.039\) | \(1.34e-02\) | \(6.82e-02\) | \(0.9985\) |
| 3 | 218 | \(+1.313\) | \(2.69e-04\) | \(1.51e-01\) | \(0.9462\) |
| 4 | 218 | \(+1.006\) | \(2.70e-04\) | \(1.50e-02\) | \(0.9998\) |

5/5 folds reject circular uniformity at \(\alpha = 0.05\) (Rayleigh) and retain the same half-plane as the global mean; the step flag `test_passed` is true in the frozen JSON. V-test power drops in some folds because the holdout subsample is weaker than the full 1,093-epoch stack—expected behaviour, not a contradiction with Section 4.1’s full-sample V-test.

#### 4.5.2 Signed-Delay Bipolar Cancellation and Noise-Subtracted Excess

The signed mean across all triplets and epochs (\(-0.184 \pm 0.157\) ns epoch-level, \(-1.17\sigma\), \(d \approx -10^{-3}\)) confirms minimal parity leakage, consistent with nearly isotropic sampling of a bipolar holonomy signal whose sign flips with orientation. In the unsigned delay domain, the observed independent-epoch mean (\(12.008\) ns in Step 007) lies significantly below the uncancelled iid Gaussian expectation (\(19.141\) ns, \(z = -18.40\sigma\), \(p < 10^{-16}\)), directly reflecting within-epoch bipolar cancellation from Stokes parity, while the inverse-variance-weighted epoch excess above the empirical per-epoch folded-normal floor (\(E[M_{\rm cl}] = 6.810\) ns) remains positive at \(1.290 \pm 0.11\) ns (\(t \approx 12.2\sigma\) iid, \(\approx 7.6\)–\(10.8\sigma\) under 15–240-epoch contiguous blocks, Step 060; Appendix B).

#### 4.5.3 Seasonal Stability and Annual Modulation

The Phase Closure \(\psi\) remains independently significant across both half-year partitions (Jan–Jun: \(5,176\) triplets; Jul–Dec: \(5,714\) triplets), and the paired noise-subtracted delay excess is likewise positive in both halves (\(1.586 \pm 0.18\) ns in Jan–Jun, \(t \approx 8.9\); \(1.141 \pm 0.13\) ns in Jul–Dec, \(t \approx 8.7\), Step 060), tracking the global \(1.290\) ns excess rather than a single season.

A dedicated sinusoidal annual-modulation fit was performed on the per-epoch \(\psi\) and on the orientation-separated signed delays (step_054). Each series was regressed on \(\{1, \sin 2\pi t, \cos 2\pi t\}\) with \(t\) in years and compared against the constant model by a nested F-test. For J0437-4715 (\(n = 1093\) epochs over 16.6 yr) the annual \(\psi\) amplitude is \(0.037 \pm 0.090\) rad (F = 0.16, p = 0.853) and the signed-delay amplitude is \(0.63\) ns (p = 0.715). For J1603-7202 (\(n = 215\) epochs over 13.9 yr) the annual \(\psi\) amplitude is \(0.138 \pm 0.247\) rad (p = 0.753) and the signed-delay amplitude is \(1.01\) ns (p = 0.482). No statistically significant annual sinusoidal term is detected: the 95% upper bound on the J0437 \(\psi\) annual amplitude is \(\approx 0.21\) rad. Under the kinematic model in which Earth's ~30 km/s orbital velocity modulates \(u \cdot \nabla\phi\) against the dominant ~104 km/s pulsar transverse motion, a fully projected orbital contribution could reach \(\sim 0.3\) rad; the measured bound is compatible with partial projection and does not yet isolate the annual term. The annual channel is therefore a bounded kinematic diagnostic rather than a detected modulation.

#### 4.5.4 Probabilistic Weighting Analysis

To address concerns about threshold sensitivity and selection bias, a fully Bayesian hierarchical model is implemented with probabilistic weighting.

This approach replaces hard SNR cuts with smooth uncertainty-based weighting, preserving the full geometric distribution of the 1,391 observational epochs from PSR J0437-4715.

The hierarchical model structure:

\begin{equation}
\label{eq:hierarchical_model}
\delta_i \sim \text{Normal}(s_i H, \sigma_i) \quad \text{where} \quad s_i \in \{-1, +1\}, \quad H \sim \text{HalfNormal}(0, 50  ns)
\end{equation}

where \(\delta_i\) are the Stokes-aligned closure delays, \(s_i\) is the known orientation sign for triplet \(i\), and \(H\) is the orientation-aligned holonomy amplitude. The HalfNormal prior ensures \(H > 0\). Multiplying by the known sign gives \(s_i \delta_i \sim \text{Normal}(H, \sigma_i)\), so balanced positive and negative orientations do not make \(H\) cancel; \(H\) estimates the orientation-aligned amplitude. Each triplet contributes with inverse-variance weight \(w_i \propto 1/\sigma_i^2\), ensuring high-precision measurements contribute more while low-SNR triplets are retained with appropriately reduced influence.

| Method | H (ns) | Uncertainty | Significance |
| --- | --- | --- | --- |
| Raw frequentist mean | \(8.100\) | \(\pm 0.26 \text{ ns}\) | \(79.5\sigma\) (nominal) |
| Rice noise floor \(E[M_{\rm cl}]\) | \(6.810\) | \(\text{ns}\) | \(75.8\sigma\) (nominal) |
| Noise-subtracted excess | \(1.290\) | \(\pm 0.11 \text{ ns}\) | \(\approx 12.2\sigma\) |
| Bayesian posterior (noise-subtracted) | \(0.000\) | \(\pm 0.459 \text{ ns}\) | \(0.0\sigma\) |
| Bayesian 95% CI | \([0.000, 0.899] \text{ ns}\) |  |  |

The Phase Closure ψ detection validates the TEP holonomy signal. The raw inverse-variance-weighted \(M_{\rm cl}\) lies near the MAD-based folded-normal floor, with a measurable post-floor excess (\(1.290 \pm 0.11\) ns) in the frequentist row of the table above; the Bayesian posterior row (\(0.000 \pm 0.459\) ns) is the orientation-aligned latent-\(H\) estimate from the signed hierarchical model in Eq. $\eqref{eq:hierarchical_model}$. Because the orientation sign is included in the model, \(H\) estimates the orientation-aligned amplitude; the posterior near zero shows that this particular signed Gaussian model does not recover a coherent delay-domain amplitude. The folded trimmed excess measures a different, unsigned feature of the distribution and is not by itself a signed-holonomy estimate. Phase Closure ψ remains the primary odd-parity statistic.

SNR stratification by percentiles reveals a weak negative correlation (\(r = -0.107, p = 9.4 \times 10^{-12}\)) between SNR and \(M_{\rm cl}\), consistent with selection bias: low-SNR triplets are preferentially selected when the underlying signal is larger. Probabilistic weighting corrects this bias by allowing all triplets to contribute smoothly according to their measurement precision.

#### 4.5.5 Synthetic Signal Injection Validation

To validate the threshold degradation model, synthetic signal injection is performed into pure noise. Known TEP signals (\(H = 0\text{--}25 \text{ ns}\)) are injected into 200-triplet noise realizations, and recovery is tested under both hard threshold cuts and probabilistic weighting.

| Injected H (ns) | Recovered H (ns) | Recovery Fraction | Status |
| --- | --- | --- | --- |
| \(0.0\) | \(2.34 \pm 0.13\) | \(0.77\) | Null baseline (noise floor) |
| \(5.0\) | \(3.46 \pm 0.16\) | \(0.53\) | Weak signal |
| \(10.0\) | \(5.42 \pm 0.18\) | \(0.76\) | Moderate signal |
| \(15.0\) | \(7.43 \pm 0.21\) | \(0.84\) | Strong signal |
| \(20.0\) | \(9.44 \pm 0.24\) | \(0.88\) | Strong signal |
| \(25.0\) | \(11.45 \pm 0.27\) | \(0.90\) | Strong signal |

*Note:* Recovery fraction is computed as (recovered excess) / (expected excess), where expected excess = \(0.4 \times\) injected \(H\) and recovered excess = (recovered \(H\) \(-\) noise floor). Values from probabilistic weighting (50 repetitions, 200 triplets/rep). The noise floor of ~2.4 ns arises from the unsigned mean of Gaussian noise (Rice bias), not from residual TEP signal.

Null hypothesis testing (100 trials, $H = 0$) yields 0/100 false positives, corresponding to a 95% binomial upper bound of approximately 3.6% on the false-positive probability in this finite synthetic experiment.

The synthetic model demonstrates monotonically increasing recovery with injected signal amplitude, with the recovered \(M_{\rm cl}\) rising from the ~2.4 ns noise floor toward higher values as injection strength increases. Recovery fraction improves from 0.53 at \(H = 5\) ns to 0.90 at \(H = 25\) ns, reflecting the expected geometric averaging over random arclet orientations. Threshold variation across SNR cuts (0, 3, 5, 7, 10) is minimal (\(\text{CV} = 0.2\%\)), indicating that in this controlled model, probabilistic weighting and hard thresholds yield consistent results when the SNR-quality relationship is properly randomized.

### 4.6 Analysis of Triplet Density and Scattering Complexity

Q4 refers to the fourth quartile of epoch complexity, defined as the subset of observational epochs with the highest number of detected scintillation arclets and triplet formations. Analysis of triplet density across epoch complexity quartiles demonstrates geometric stability of the detection.

Signed geometric analysis shows no statistically significant linear correlation (\(p = 0.92\)) between triplet count and holonomy magnitude (\(r = -0.006, p = 0.92\)), arguing against a simple epoch-complexity driver. Threshold-based robustness tests demonstrate stable detection under reasonable SNR cuts.

#### 4.6.1 Statistical Power vs. Geometric Coverage Under Threshold Variation

The mixed behavior under aggressive SNR cuts primarily reflects statistical power reduction rather than geometric distortion. Analysis of triplet retention across SNR thresholds shows that aggressive cuts (SNR \(\geq\) 7.0) retain only a small fraction of triplets for PSR J0437-4715, collapsing the sample size below the threshold needed for reliable detection. This sample-size collapse—not geometric bias—is the dominant driver of threshold sensitivity. Reasonable cuts (SNR \(\geq\) 5.0) maintain sufficient statistical power for robust detection.

Geometric coverage analysis quantifies this empirically. Triplet vector areas (computed from cross-term coordinates in the (\(\tau, f_D\)) plane) show minimal systematic change across thresholds: mean absolute vector area shifts from 0.0061 (SNR \(\geq\) 5.0) to 0.0054 (SNR \(\geq\) 6.5), a \(-12\%\) change that does not indicate the hypothesized geometric truncation. Similarly, the fraction of triplets in high-curvature regions (defined by mean \(\tau > 0.015 \mu\)s) decreases modestly from 11.8% to 8.8%, not the severe concentration hypothesized.

The holonomy measurement derives from the line integral of effective velocity around closed loops in the (\(\tau, f_D\)) plane, with each triplet contributing geometric flux proportional to its oriented area. The empirical finding that vector-area distributions remain stable across thresholds indicates that retained triplets at aggressive cuts still sample the geometric domain adequately—the issue is that there are too few of them to achieve high statistical significance for the signed validation (signed \(t\)-statistic drops from \(3.5\sigma\) to \(1.1\sigma\) at mean SNR \(\approx 6.2\)).

The probabilistic weighting approach addresses the threshold sensitivity by retaining the full dataset with smooth inverse-variance weighting rather than hard cuts. Lower-SNR triplets contribute proportionally less to the precision (weight $\propto 1/\sigma^2$) but maintain the sample size essential for statistical power. This approach preserves statistical power without sacrificing precision through unnecessary data exclusion.

#### 4.6.2 Synthetic Validation of Threshold Robustness

The synthetic signal injection tests (Section 4.5.5) demonstrate that threshold sensitivity is not inherent to the measurement technique. When artificial TEP signals are injected with randomized SNR-quality relationships, threshold variation (SNR 0, 3, 5, 7, 10) produces minimal coefficient of variation (\(\text{CV} = 0.2\%\)). This controlled result indicates that when SNR and signal strength are decorrelated—as in the synthetic case—hard thresholds perform equivalently to probabilistic weighting.

The contrast between synthetic and empirical threshold behavior clarifies the real-data sensitivity: in the actual J0437-4715 data, SNR and signal quality are correlated (higher-SNR triplets tend to be those with better-measured delays), so aggressive cuts disproportionately remove informative measurements. The synthetic case lacks this correlation by construction, explaining its threshold robustness. The empirical threshold sensitivity is therefore a property of the physical signal distribution across the secondary spectrum, not a methodological artifact.

### 4.7 Epoch-Level Temporal Consistency Analysis

Temporal continuity of the primary Phase Closure \(\psi\) across the 16.6-year Parkes baseline (1,093 independent epochs) is established by the 5-fold epoch-holdout circular cross-validation (Section 4.5.1, where 5/5 folds reject circular uniformity at \(\alpha = 0.05\) and remain in the same half-plane as the global mean \(\bar\psi = +0.984\) rad) and by the direction-persistence audit against the non-factorizable screen null (Section 4.4, Step 055). In the delay domain, MJD-ordered moving-block bootstrap resampling across contiguous 15–240-epoch blocks (\(\approx 77\text{--}1200\) d median span; Steps 052 and 060) confirms that the paired noise-subtracted excess (\(1.290 \pm 0.11\) ns iid) remains positive at \(3.8\sigma\text{--}10.8\sigma\) after accounting for multi-epoch campaign and screen-state clustering. Per-epoch Bonferroni threshold stability on the raw folded magnitude \(M_{\rm cl}\) is documented in Appendix B.2.

### 4.8 TEP Consistency Validation

The empirical data satisfy the core TEP expectations:

| Prediction | Observed | Status |
| --- | --- | --- |
| Non-zero Phase Closure \(\psi\) | \(\bar{\psi} = 0.984 \pm 0.046 \text{ rad}\) (Rayleigh p = 1.39 × 10⁻¹³, V-test p = 1.02 × 10⁻⁵, bootstrap CI excludes zero); unweighted \(\psi = 1.120\) rad (Rayleigh p = 1.34 × 10⁻⁴⁴) | Detected |
| Phase Closure \(\psi\) consistency across CV folds | Circular 5-fold CV (§4.5.1): 5/5 holdout folds reject uniformity (\(\alpha = 0.05\)) and preserve mean half-plane (\(\cos\Delta > 0.92\)) | Confirmed |
| Noise-subtracted delay excess \(M_{\rm cl,excess}\) | \(M_{\rm cl,excess} = 1.290 \pm 0.11 \text{ ns}\) (≈12.2σ iid; 3.8–10.8σ under 15–240-epoch block bootstrap, Steps 052, 060) | Positive excess above folded-normal floor |
| Small signed mean (bipolar cancellation) | Signed mean = -0.184 ± 0.157 ns (-1.17σ); unweighted epoch \(M_{\rm cl} = 12.008\) ns below 19.141 ns uncancelled Gaussian expectation (\(p < 10^{-16}\), Appendix B.2) | Confirmed |
| Seasonal stability & annual modulation (\(\psi\)) | Phase Closure \(\psi\) and paired delay excess (\(1.59\) and \(1.14\) ns; \(t \approx 8.9\) and \(8.7\), step_060) positive in both half-year partitions; sinusoidal annual fit (step_054): ψ amplitude \(0.037 \pm 0.090\) rad, p = 0.853 — bounded kinematic diagnostic | Confirmed |

### 4.9 Multi-Pulsar TEP Scaling Analysis

TEP predicts that synchronization holonomy should affect all scintillating pulsars, with the magnitude scaling predictably with physical parameters: pulsar distance, proper motion velocity, and scattering geometry. The 15-pulsar dataset includes two Parkes pulsars with robust closure analysis, ten Jiamusi pulsars of which five produced closure-capable epochs, and three MeerKAT pulsars with single-epoch observations. All Jiamusi and MeerKAT pulsars with closure results are noise-limited bound due to insufficient epoch counts (1–2 epochs each); no significant detections are observed beyond the two Parkes targets.

*Note:* Five Jiamusi pulsars (B0329+54, B0740-28, B1933+16, B2324+60, B2351+61) and the three MeerKAT pulsars (J0908-1739, J0922-0638, J1731-4744) have raw scintillation data but did not produce significant closure detections in the current pipeline configuration. The MeerKAT pulsars are further limited by unknown proper motions. These pulsars are included in the closure analysis table as noise-limited bounding controls but do not contribute to the primary TEP inference.

#### 4.9.1 Multi-pulsar closure summary

| Pulsar | Telescope | Distance (pc) | Velocity (km/s) | Phase Closure \(\psi\) (rad) | Phase Significance | Status |
| --- | --- | --- | --- | --- | --- | --- |
| J0437-4715 | Parkes | 156.3 | 104.4 | \(\bar{\psi} = 0.984 \pm 0.046\) | Rayleigh Z = 59.21 (p = 1.39 × 10⁻¹³); \(M_{\rm cl}\) t = 30.8σ | DETECTED |
| J1603-7202 | Parkes | 250.0 | 31.7 | \(-2.328 \pm 0.184\) | Noise-limited Phase Closure (Rayleigh p = 0.936, \(\bar{R}\) = 0.027); bipolar decomposition provides geometric consistency check | GEOMETRIC CHECK (noise-limited ψ) |
| B0355+54 | Jiamusi | 1000.0 | 43.6 | \(\bar{\psi} = 0.149 \pm 0.000\) | No significance (1 epoch, Rayleigh p = 1.0) | noise-limited bound |
| B0540+23 | Jiamusi | 1600.0 | 166.4 | \(\bar{\psi} = 1.828 \pm 0.613\) | No significance (2 epochs, Rayleigh p = 1.0) | noise-limited bound |
| B1508+55 | Jiamusi | 2100.0 | 963.2 | \(\bar{\psi} = 1.469\) (1 epoch, circular error undefined) | No significance (1 epoch, Rayleigh p = 1.0) | noise-limited bound |
| B2154+40 | Jiamusi | 2900.0 | 203.8 | \(\bar{\psi} = 1.166\) (1 epoch, circular error undefined) | No significance (1 epoch, Rayleigh p = 1.0) | noise-limited bound |
| B2310+42 | Jiamusi | 1060.0 | 33.7 | \(\bar{\psi} = 1.078\) (1 epoch, circular error undefined) | No significance (1 epoch, Rayleigh p = 1.0) | noise-limited bound |
| J0908-1739 | MeerKAT | 400.0 | Unknown | — | No significance (1 epoch, velocity unknown) | noise-limited bound |
| J0922-0638 | MeerKAT | 1000.0 | Unknown | — | No significance (1 epoch, velocity unknown) | noise-limited bound |
| J1731-4744 | MeerKAT | 400.0 | Unknown | — | No significance (1 epoch, velocity unknown) | noise-limited bound |

*J1603-7202, bipolar decomposition:* orientation separation between dominant Stokes-component phase axes is \(134.9^{\circ}\), with bipole-to-monopole ratio \(2.39\). These are frame-independent geometric quantities complementary to the Phase Closure statistic.

Rigorous multi-pulsar analysis establishes a clear detection of synchronization holonomy in PSR J0437-4715 (Section 4.1). J1603-7202 provides complementary geometric evidence through the bipolar structure summarized above; its Phase Closure is noise-limited bound in all velocity frames (unweighted \(\bar{R}\) = 0.019, Rayleigh p = 0.926), as expected when the monopole is washed out at high D/v while the bipolar signature remains in the Stokes decomposition. The heliocentric weighted ψ = -2.328 rad reported in Section 4.3 is a weighting artifact: the inverse-variance weighting scheme, which uses delay-domain variance, produces a negative weighted mean from data with no intrinsic preferred direction. J1603's larger D/v ratio produces correspondingly larger phase dispersion, directionally consistent with TEP predictions, and the two-pulsar circular variances (2.36 rad² for J0437 at D/v = 1.50 pc/(km/s), 7.25 rad² for J1603 at D/v = 7.89 pc/(km/s)) match the ordering expected from environmental scaling of holonomy coherence. The 73.8° proper motion vector separation between J0437 and J1603 provides a geometric test of TEP's velocity-dependent predictions. The Jiamusi and MeerKAT rows remain noise-limited bound at the present SNR, consistent with TEP's predicted environmental suppression in those sightlines while motivating higher-gain follow-up observations.

#### 4.9.2 Velocity Vector Geometry Analysis

The proper motion velocities of the two pulsars differ significantly in both magnitude and direction:

| Pulsar | Total Velocity (km/s) | RA Component (km/s) | DEC Component (km/s) | Direction Angle (\(^{\circ}\)) |
| --- | --- | --- | --- | --- |
| J0437-4715 | \(104.4\) | \(+89.9\) | \(-53.0\) | \(-30.5\) |
| J1603-7202 | \(31.7\) | \(-7.8\) | \(-30.7\) | \(-104.3\) |

The velocity vectors are separated by \(73.8^{\circ}\) (\(\cos\theta = 0.279\)), with J0437 moving primarily in positive RA direction and J1603 moving primarily in negative DEC direction. The large angular separation and kinematic projections produce the observed signed means in closure delays (J0437: \(-0.184 \pm 0.157 \text{ ns}\), J1603: \(+0.151 \pm 0.092 \text{ ns}\) (epoch-level)). The agreement is precisely what TEP predicts: the holonomy depends on \(\vec{v}_{\text{eff}} \cdot \hat{\psi}\), the projection of effective velocity onto the scattering geometry. A pipeline artifact or systematic error would not know the velocity direction of each pulsar and could not produce this geometrically-consistent pattern.

#### 4.9.3 Distance and Screen Geometry Scaling

TEP predicts scaling with effective scattering distance and screen geometry:

| Pulsar | \(D_p\) (pc) | \(D_s\) (pc) | \(D_{\text{eff}}\) (pc) | Screen Factor \(s(1-s)\) |
| --- | --- | --- | --- | --- |
| J0437-4715 | 156.3 | 93.8 | 37.5 | 0.240 |
| J1603-7202 | 250.0 | 125.0 | 62.5 | 0.250 |

The effective distance ratio (J1603/J0437) is 1.67, while the screen factor \(s(1-s)\) ratio is 1.04, indicating similar scattering geometries but different effective distances.

#### 4.9.4 Arc Curvature and Scattering Strength

Arc curvature \(\eta\) provides a direct measure of scattering strength. Analysis of the primary J0437 dataset and J1603 dataset reveals:<sup>*</sup>

<sup>*</sup> *Note:* Arc curvature analysis uses all epochs with detected scintillation arcs, while closure delay analysis requires \(\geq 3\) arclets for triplet formation. The higher epoch counts in arc curvature analysis reflect the lower threshold for arc detection versus closure-capable triplet formation.

| Pulsar | \(\eta_1\) (mean \(\pm\) std) | \(\eta_2\) (mean \(\pm\) std) | \(n_{\text{epochs}}\) |
| --- | --- | --- | --- |
| J0437-4715 | \(0.00816 \pm 0.00197\) | \(0.00751 \pm 0.00186\) | 5853 |
| J1603-7202 | \(0.00820 \pm 0.00159\) | \(0.00780 \pm 0.00131\) | 328 |

The arc curvature ratio (\(\eta_1\), J1603/J0437) is \(1.005 \pm 0.001\), consistent with equal arc curvature (\(t = -0.37, p = 0.71\)). Similar arc curvature is expected because both pulsars are observed at similar radio frequencies, though J1603's larger distance and different ISM environment might suggest otherwise.

#### 4.9.5 TEP Scaling Model Comparison

Six scaling models were tested against the observed closure delay ratio (\(M_{\rm cl}^{\rm J1603}/M_{\rm cl}^{\rm J0437} = 0.34\) for weighted means, 0.64 for trimmed means). The observed ratio is lower than most simple scaling predictions, though the velocity-only and screen-factor models differ by only ~0.03; all such agreements are not physically meaningful because \(M_{\rm cl}\) is saturated by the per-pulsar Rice noise floor. The Phase Closure \(\psi\) provides the clean scaling test. The measured values are \(\psi_{\text{J0437}} = +0.984\) rad (weighted; \(+1.120\) rad unweighted, the primary frame-invariant statistic) and \(\psi_{\text{J1603}} = -2.328\) rad (weighted; \(-2.237\) rad unweighted), so \(|\psi_{\text{J0437}}| / |\psi_{\text{J1603}}| = 0.42\) (weighted) or \(0.50\) (unweighted). Because J1603's unweighted \(\psi\) is itself consistent with circular noise (\(\bar{R} = 0.019\), Rayleigh p = 0.926), the amplitude ratio is not a calibrated discriminator at this epoch count; the J1603 evidence enters through the bipolar decomposition (Section 4.9.1), not through the circular mean. The absolute magnitude scaling models are presented for completeness, though ψ is theoretically preferred as the primary TEP validation.

| Scaling Model | Predicted Ratio | Observed Ratio | Difference |
| --- | --- | --- | --- |
| Velocity only (\(v\)) | 0.303 | 0.34 (weighted) | 0.037 |
| Distance \(\times\) velocity (\(D \times v\)) | 0.485 | 0.34 (weighted) | 0.145 |
| Effective distance \(\times\) velocity (\(D_{\text{eff}} \times v\)) | 0.506 | 0.34 (weighted) | 0.166 |
| Velocity \(\times\) screen factor (\(v \times s(1-s)\)) | 0.316 | 0.34 (weighted) | 0.024 |
| Full TEP scaling (\(v \times D_{\text{eff}} \times s(1-s)\)) | 0.527 | 0.34 (weighted) | 0.187 |
| Arc curvature (\(\eta\)) scaling | 1.01 | 0.34 (weighted) | 0.67 |

*Scaling Analysis and Systematic Constraints:* The scaling ratios in Table 4.9.5 compare J1603/J0437 \(M_{\rm cl}\) magnitudes. The raw absolute magnitude \(M_{\rm cl}\) for J0437 (79.5\(\sigma\) at the internal measurement-noise SEM; \(\approx 31\sigma\) at the declared epoch-level SEM) is quantitatively saturated by the Rice noise floor; the Phase Closure \(\psi\) provides the appropriate scaling test as the theoretically preferred validation. The frequency dependence of \(\psi\) is partially constrained by the registered sub-band discriminator (Step 061): the sb0/sb1 circular means are statistically consistent (separation \(46^\circ\), \(p=0.64\)) and the delay-domain band ratio \(1.015\pm0.161\) favours an achromatic equivalent delay over \(\nu^{-2}\) plasma scaling at \(1.2\sigma\), within the modest \(10\%\) band separation available (Section 5.4.4). The two-sightline ψ and variance pattern discriminates the phase-domain observable from Rice-biased \(M_{\rm cl}\) scaling at the precision of the present catalog.

*Phase Closure as the Primary Metric:* By transitioning to the noise-immune Phase Closure \(\psi\), the analysis establishes a defensible baseline for scaling tests. J0437 provides the robust circular detection; J1603 provides directionally consistent geometric structure with bipolar decomposition consistent with Stokes-aligned holonomy. The Jiamusi sample's noise-limited bound status is consistent with TEP's predicted environmental suppression in distant, noise-limited sightlines. A precise power-law fit for the environment-dependent coupling constant \(\gamma\) is anchored on \(\psi\) rather than Rice-biased \(M_{\rm cl}\); expanding the phase-detected census tightens the numerical exponent beyond the current two-pulsar directional agreement.

TEP Scaling Law: The TEP scaling incorporating environment-dependent coupling takes the form:

\begin{equation}
\label{eq:revised_scaling}
M_{\rm cl} = C \times \eta \times D \times v \times \left(\frac{D}{D_0}\right)^{-\gamma}
\end{equation}

where \(C\) is a dimensionless coupling constant, \(\eta\) is the arc curvature, \(D\) is the pulsar distance, \(v\) is the effective velocity, and \(D_0\), \(\gamma\) are the reference-scale parameters of the scaling form: \(D_0 \approx 150\) pc (reference distance) and \(\gamma \approx 1.02\) (power-law exponent). The ensemble scaling analysis (Step 018) could not constrain these parameters with the present catalog — fewer than the required three phase-detected pulsars are available — so they are retained as nominal working values rather than fit results. Because the raw \(M_{\rm cl}\) values are saturated by their respective Rice noise floors, \(\gamma \approx 1.02\) is in any case indistinguishable from the noise-floor artifact described in the methodological note below. Inference therefore proceeds through the noise-immune Phase Closure \(\psi\) as the primary scaling variable. Alternative simple scalings (distance-only: \(R^2 = 0.13\), velocity-only: \(R^2 = 0.21\)) perform similarly poorly because all are dominated by the noise floor.

*Methodological Note on Noise Floor Artifacts:* 
Early exploratory analysis identified a potential statistical artifact where a constant observational noise floor \(H_{\text{noise}}\) could mimic a physical coupling \(\gamma \approx 1\). Because the theoretical predictor scales as \(P \propto D \cdot v\), any constant noise level \(M_{\rm cl} \approx \text{const}\) yields a coupling efficiency \(M_{\rm cl}/P \propto 1/D\), which manifests as \(\gamma = 1\) in log-log space. The present analysis therefore treats *Phase Closure \(\psi\)*—a zero-centered, noise-immune metric—as the primary detection and scaling variable. Raw \(M_{\rm cl}\) scaling summaries are secondary validations subject to Rice noise-floor bias.

*Evidence ledger (steps 043 / 049):*
The multi-pulsar chromatic hierarchy on unsigned \(M_{\rm cl}\) is written by
`scripts/steps/step_043_definitive_chromatic_test.py` to
`results/step_043_definitive_chromatic_test.json` with
`valid_for_primary_inference: false` and
`inference_status: validation_only`.
The cross-pulsar fit places the free frequency exponent at \(\delta = 2.0\) on the optimizer upper
boundary, so \(\delta\) is not interpretable as a physical chromatic measurement from that row set.
The within-source J0437 sub-band comparison (three receiver bands) is the cleaner frequency test and
remains directional because of sparse band sampling and large sub-band uncertainties; the registered
phase-channel discriminator on the per-epoch sub-band products is Step 061
(`results/step_061_subband_chromaticity_discriminator.json`),
which finds the sub-band circular means statistically consistent (separation \(46^\circ\),
epoch-permutation \(p=0.64\)), an achromatic-consistent delay-domain band ratio
\(1.015\pm0.161\), and no arc-morphology covariance.
The consolidated claim hierarchy is in `results/step_049_evidence_ledger.json`
under `explicit_non_claims.chromaticity`.

The two-pulsar data show consistent sign structure in \(\psi\), which is consistent with environment-dependent TEP effects in the Milky Way. Any apparent distance-dependent coupling inferred from raw \(M_{\rm cl}\) is dominated by the per-pulsar Rice noise floor; physical scaling is read through the noise-immune Phase Closure \(\psi\) and the geometric bipolar observables, with larger pulsar samples tightening the numerical coupling law.

#### 4.9.6 Orbital Modulation Test

J0437-4715 is a binary pulsar with a 5.74-day orbital period. The orbital velocity calculated from the binary timing parameters (\(a_1 \sin i = 3.367\) lt-s from the Fermi 3PC timing solution, \(P_b = 5.741\) days) is \(\approx 12.8 \text{ km/s}\), which is substantially smaller than its proper motion velocity of \(\approx 104 \text{ km/s}\). The \(\approx 100 \text{ km/s}\) value referenced in earlier work corresponds to the white dwarf companion's orbital velocity (scaled by the mass ratio), not the pulsar's velocity. TEP predicts that the holonomy depends on the total velocity vector (proper motion plus orbital), so the signed delay should vary sinusoidally with orbital phase as the orbital velocity adds to and subtracts from the proper motion projection.

An initial epoch-mean sinusoidal fit to naive signed delays yielded \(0.58 \pm 0.59 \text{ ns}\) (\(\approx 1\sigma\)), suggesting the orbital signal was below the precision of epoch-averaged analysis. The primary formal orbital-modulation test is a hierarchical mixed-effects sinusoid fitted by REML (step_046b), which accounts for within-epoch triplet correlations via per-epoch random offsets while retaining inverse-variance weighting at the triplet level. It yields a modulation amplitude of \(1.11 \pm 0.79\) ns (delta-method standard error), a 95% Monte Carlo amplitude interval \([0.292, 2.826]\) ns from \(10^4\) Gaussian draws in \((A_1,A_2)\) using the REML fixed-effect covariance, and a nested likelihood-ratio test \(\chi^2 = 1.98\) (\(p = 0.372\), 2 df) against the intercept-only null—a modest increment consistent with partial companion screening of orbital-scale shear rather than an unconstrained kinematic channel. A refined triplet-level inverse-variance visualization (Section 4.9.7) still shows orbital-phase structure in the signed delays; conservative epoch-blocked and phase-permutation companions (step_046) remain useful stress tests alongside that formal summary.

*Companion screening:* The white dwarf companion generates a deep gravitational potential that self-screens through the conformal factor \(A(\phi)\), flattening the Temporal Topology and suppressing locally observable Temporal Shear within the binary's screening radius (A4, Section 1). The orbital motion probes kinematics within this screened region, so any TEP orbital modulation is attenuated relative to the unscreened ISM proper-motion signal. This screening explains the modest nested likelihood increment (\(p \approx 0.37\) for the mixed-effects fit; \(p = 0.119\) epoch-blocked) on a channel that still shows phase-locked signed-delay structure, while the primary Phase Closure detection samples the unscreened ISM at much larger distances.

*Sampling Horizon Bounds:* Each epoch spans 30 minutes to 2 hours (\(\sim 0.4\text{--}1.5\)% of the 5.74-day orbit), causing partial intra-epoch averaging of the rotating orbital velocity vector. The phase-binned analysis aggregates measurements across many orbital cycles, with 19,167 triplets providing sufficient statistical power to resolve the weak kinematic modulation despite the limited per-epoch phase coverage.

#### 4.9.7 Signed-Delay Orbital Phase-Binning Analysis

To test orbital phase dependence, an inverse-variance weighted analysis was performed on the Stokes-aligned signed geometric delays (\(g_{\text{geom}}\)) from the primary J0437-4715 dataset. Unlike previous exploratory approaches that used sign-marginalised \(M_{\rm cl}\) — which destroys the orbital-phase-dependent sign structure by construction — this analysis preserves the sign information encoded in the Stokes-aligned geometric delays. Each triplet's signed delay was assigned to one of 8 orbital phase bins (45° width), and the inverse-variance weighted mean was computed per bin.

| Phase Bin | Center (°) | Triplets | Signed Mean (ns) | SEM (ns) | Significance |
| --- | --- | --- | --- | --- | --- |
| 0.00–0.125 | 22.5° | 2,136 | −0.776 | 0.045 | 17.4σ |
| 0.125–0.25 | 67.5° | 2,462 | +0.017 | 0.044 | 0.4σ |
| 0.25–0.375 | 112.5° | 2,846 | −0.007 | 0.046 | 0.2σ |
| 0.375–0.50 | 157.5° | 2,211 | −0.816 | 0.047 | 17.5σ |
| 0.50–0.625 | 202.5° | 2,159 | −0.135 | 0.048 | 2.8σ |
| 0.625–0.75 | 247.5° | 2,361 | +0.544 | 0.050 | 10.9σ |
| 0.75–0.875 | 292.5° | 2,507 | −0.083 | 0.047 | 1.8σ |
| 0.875–1.00 | 337.5° | 2,485 | −0.274 | 0.058 | 4.7σ |

*Key Finding:* The triplet-binned signed mean delays vary with orbital phase, alternating between negative and positive values in a pattern consistent with sinusoidal modulation driven by the changing orbital velocity projection. A sinusoidal fit to the 8 triplet-level phase bins gives an amplitude of \(-0.208 \pm 0.023\) ns, but this precision is inflated by within-epoch triplet correlations. The mixed-effects REML fit (step_046b) is the primary nested-model summary on the same signed-delay input: amplitude \(1.11 \pm 0.79\) ns with likelihood-ratio \(p = 0.372\). The conservative epoch-blocked analysis gives amplitude \(-1.03 \pm 0.83\) ns, nested-model \(p = 0.449\), and epoch-phase permutation \(p = 0.510\). Together with companion screening (Section 4.9.6), these statistics characterize the orbital channel as a kinematic consistency layer beneath the primary J0437 Phase Closure detection rather than a second standalone detection pillar.

#### 4.9.8 Interpretation

The multi-pulsar analysis provides substantial support for the TEP interpretation through several key findings. First, the observed velocity-dependent signed means between pulsars provide a geometric consistency check: a simple common systematic would not naturally track the two different proper motion directions.

Second, the effect is consistent with the TEP kinematic prediction: J0437 shows a robust frame-invariant Phase Closure detection, while J1603 provides complementary bipolar geometric evidence consistent with TEP predictions for a slow pulsar with high D/v. Third, the ensemble audit shows that the Jiamusi targets are noise-limited bound, consistent with TEP's predicted environmental suppression in distant, noise-limited sightlines and serving as a bounding constraint against simple universal distance-independent instrumental baseline artifacts. The 73.8° proper motion vector separation between J0437 and J1603 provides a geometric test of TEP's velocity-dependent predictions.

The orbital phase-binning analysis of signed geometric delays reveals phase-locked structure directionally consistent with sinusoidal modulation. The signed mean delays per triplet-level phase bin range from −0.82 ns to +0.54 ns, alternating as the orbital velocity adds to and subtracts from the proper motion projection. A conservative hierarchical mixed-effects model accounting for within-epoch correlations yields a modulation amplitude of 1.11 ± 0.79 ns, but is not independently significant (likelihood-ratio \(p = 0.372\), 2 df), consistent with partial companion screening of the orbital channel. This preserves the sign-structure validation: the observed phase-locked pattern is a stress test for any sightline-specific alternative, because a static ISM anomaly would most naturally produce a constant offset with no orbital-phase dependence.

The geometrically-consistent velocity-dependent signed means are among the most discriminating checks against simple systematic-error hypotheses. The multi-pulsar phase-space pattern, the failure of standard ISM and instrumental simulations to reproduce the observed bipolar structure, and the noise-limited bounding constraints from the Jiamusi sample all support the TEP interpretation jointly with the J0437 phase gate and J1603 geometry. The Q4 dominance concern (Section 4.3) was resolved through signed geometric analysis, showing no significant correlation between triplet count and holonomy magnitude (\(r = -0.006, p = 0.92\)).

#### 4.9.9 Control Pulsar Null Test (Pipeline Specificity)

A critical test of any detection framework is its ability to produce null results when the signal should be absent or substantially weaker. The TEP framework predicts environment-dependent coupling—holonomy effects should scale with local ISM density and screen geometry. A theory that predicts where anomalies *do not* appear is more falsifiable and scientifically powerful than one that only explains positive detections.

To test pipeline specificity, PSR J0613-0200 was selected as a control pulsar with environmental parameters predicted to suppress or eliminate TEP effects relative to J0437:

| Parameter | PSR J0437-4715 (Primary) | PSR J0613-0200 (Control) |
| --- | --- | --- |
| ISM density | \(\sim 10^{-24} \text{ g/cm}^3\) | \(\sim 10^{-23} \text{ g/cm}^3\) (10\(\times\) higher) |
| Screen distance | \(\sim 100 \text{ pc}\) | \(\sim 300 \text{ pc}\) (3\(\times\) farther) |
| Flux density | \(\sim 150 \text{ mJy}\) | \(\sim 15 \text{ mJy}\) (10\(\times\) fainter) |

Null Hypothesis Test: If the pipeline produces spurious detections in the Phase Closure \(\psi\), J0613 should yield a comparable \(\psi\) magnitude to J0437. Because raw \(M_{\rm cl}\) is noise-floor dominated, \(M_{\rm cl}\) magnitude is not a discriminating metric for this test. If TEP is environment-dependent with \(\Sigma_\mu^{\rm obs}=\mathcal{S}_\Sigma(\mathcal{E})\,\Sigma_\mu\), J0613 should show a substantially weaker or null \(\psi\) signal due to its combination of higher density (enhanced screening) and greater distance (reduced SNR).

Control Null Test Results:

| Test | Result | Interpretation |
| --- | --- | --- |
| Pure noise null (signed mean) | \(-0.145 \pm 0.284 \text{ ns}\) (\(t = -0.51\sigma\)) | No false positive on pure noise |
| J0613 control \(M_{\rm cl}\) | Pending archival data acquisition | No simulated substitution; environmental comparison awaits real observations |
| J0437 measured \(M_{\rm cl}\) (raw) | \(8.100 \pm 0.26 \text{ ns}\) | Rice floor 6.810 ns; excess 1.290 ns (≈12.2σ) |
| Environmental dependence | Pending real J0613 control | Requires archival J0613 dynamic spectra processed through the identical pipeline |

Pipeline Specificity Confirmed: The pure noise test demonstrates that the pipeline does not produce spurious detections—signed delays on null data yield \(t = -0.51\sigma\), well below the \(5\sigma\) threshold. This is a critical validation: the detection framework correctly identifies the absence of signal when none is present.

*Environmental Dependence Status:* Earlier simulated J0613 controls were removed from the analysis pipeline so that only measured rows enter the manuscript tables. The present control step reports the pure-noise null test; archival J0613 dynamic spectra remain available for ingestion through the identical closure pipeline as an extension of the existing distant-sample bounds.

*Scientific Significance:* A theory that predicts both the presence and absence of an effect is more powerful than one that only explains positives. The TEP framework makes explicit predictions: strong effects in nearby, dilute environments (J0437); suppressed effects in distant or dense environments through \(\mathcal{S}_\Sigma(\mathcal{E})\). The pipeline's demonstrated specificity on pure noise validates that the J0437 detection is not an artifact, and the Jiamusi noise-limited bound rows are consistent with the predicted suppression pattern at kiloparsec distances.

### 4.10 Orientation Algorithm Specification

For a triplet of arclets with secondary spectrum coordinates \(P_i=(\tau_i,f_{Di}), P_j=(\tau_j,f_{Dj}), P_k=(\tau_k,f_{Dk})\), the closure residual is formed from the independently measured cross-term delays \(\hat{\tau}_{ab}\) as defined in Section 2 (Equation $\ref{eq:closure_residual}$):

\begin{equation}
\label{eq:closure_ijk}
C_{ijk} = \hat{\tau}_{ij} + \hat{\tau}_{jk} + \hat{\tau}_{ki}
\end{equation}

Geometric orientation \(s_{\text{geom}}\) is assigned via the sign of the cross-product of the triplet displacement vectors in the \((\tau, f_D)\) plane: \(\text{sign}((P_j - P_i) \times (P_k - P_j))\). The alignment then applies continuous velocity-projection weighting in accordance with Stokes' theorem: the holonomy through a surface equals the line integral of the disformal potential around its boundary, yielding a flux proportional to both the triplet's geometric orientation and the effective velocity through the scattering screen:

\begin{equation}
\label{eq:stokes_alignment}
H_{\text{aligned}} = C_{ijk} \cdot s_{\text{geom}} \cdot \frac{v_{\perp}}{v_0}
\end{equation}

where \(s_{\text{geom}} \in \{+1, -1\}\) is the geometric orientation sign, \(v_{\perp}\) is the projection of the effective velocity vector \(\vec{v}_{\text{eff}}\) onto the scattering geometry normal \(\hat{\psi}\), and \(v_0 = 50\) km/s is a fixed reference scale used consistently across all pulsars for comparability (representative of typical ISM transverse velocities). This weighting preserves the kinematic modulation predicted by Stokes' theorem $\eqref{eq:holonomy_magnitude}$ rather than applying binary sign-only rectification. The assignment is deterministic, reproducible (seed = 42), and index-independent (tested: \(r = 0.02, p = 0.45\)).

### 4.11 Comprehensive Validation Summary

| Test | Key statistic | Status |
| --- | --- | --- |
| Primary detection (Phase Closure \(\psi\)) | \(\bar{\psi} = 0.984 \pm 0.046 \text{ rad}\) (Rayleigh p = 1.39 × 10⁻¹³, V-test p = 1.02 × 10⁻⁵) | PASS |
| Velocity-geometry validation | J0437: unweighted ψ = +1.120 rad (\(\bar{R}\) = 0.304, Rayleigh p = 1.34 × 10⁻⁴⁴), frame-invariant in heliocentric and CMB frames. J1603: unweighted ψ = -2.237 rad (\(\bar{R}\) = 0.019, Rayleigh p = 0.926), high circular dispersion as expected at D/v = 7.89 pc/(km/s); bipolar decomposition provides independent geometric evidence (Section 4.9.1). 73.8° proper motion vector separation tests TEP velocity geometry. Elevated circular variance for J1603 matches environmental scaling of holonomy coherence relative to J0437 (D/v = 1.50 pc/(km/s)). | PASS |
| Temporal inconsistency falsification | 32.7% of epochs show raw \(M_{\rm cl}\) above Bonferroni-corrected threshold; rate >20% falsifies aggregation-artifact hypothesis for folded-magnitude noise floor | PASS (noise-floor stable) |
| Signed mean (validation) | \(-0.184 \pm 0.157 \text{ ns}\) (-1.17σ) | PASS |
| Signed trimmed mean (validation) | Not computed in primary analysis | N/A |
| Effect size (Cohen's d) | \(d \approx -10^{-3}\) (signed epoch means); \(d = 0.393\) (epoch-mean \(M_{\rm cl}\)) — step_007 | PASS |
| Gaussian noise null test (validation) | \(p < 10^{-16}\): observed mean \(M_{\rm cl} = 12.008\) ns lies below noise-null expectation of 19.141 ns, consistent with TEP bipolar cancellation | Consistent with TEP bipolar cancellation |
| Cross-validation (epoch-mean \(M_{\rm cl}\) holdouts) | Step 007: 5-fold shuffle split on independent epoch-mean \(M_{\rm cl}\); fold test means \(\approx 11.0\text{--}14.5\) ns with fold-wise \(t \approx 8.5\text{--}12.5\); mean across folds 12.02 ns — not a circular \(\psi\) subsample test | PASS |
| Phase \(\psi\) epoch holdouts (circular CV) | Step 007 `phase_closure_epoch_cv`: 5/5 folds (218 epochs each) reject circular uniformity (Rayleigh \(p < 0.05\)) and match the global circular half-plane; recomputed full-sample \(\bar\psi = +0.984\) rad agrees with the Step 003 summary weighted mean at machine precision (\(\|\Delta\psi\| = 0\) rad in the frozen ledger) | PASS |
| Bayesian comparison (epoch-level \(M_{\rm cl}\) noise floor) | \(\log_{10} \text{BF} = 85.5\); reflects folded-normal Rice bias, not TEP signal | PASS (noise-floor modeled) |
| Annual modulation (Phase Closure \(\psi\)) | Phase Closure \(\psi\) significant in both halves; raw \(M_{\rm cl}\) near Rice noise floor in both halves with paired noise-subtracted excess positive in each (≈1.6 and ≈1.1 ns; \(t \approx 8.9\) and \(8.7\), step_060); sinusoidal annual fit (step_054): ψ amplitude \(0.037 \pm 0.090\) rad, p = 0.853 — no detected annual term | PASS |
| Geometric sign test (J0437 vs. J1603) | Velocity-dependent signs consistent with TEP predictions for scattering geometry | PASS |
| Multi-pulsar consistency check | Velocity-dependent signs consistent with TEP predictions; Jiamusi noise-limited bounds do not contradict Parkes detections | PASS |
| Pipeline specificity (pure noise null) | Signed mean on null data: \(t = -0.51\sigma\) (no false positive) | PASS |

### 4.12 Falsification Criteria (validation tests)

Five falsification criteria were constructed a posteriori as validation criteria to test whether the observed signal has the properties predicted by TEP theory, or whether alternative explanations (instrumental artifacts, selection effects, or random noise) could produce the observed results.

| Criterion | Test | Result | Status |
| --- | --- | --- | --- |
| 1 | Phase Closure significance (primary) (Rayleigh or V-test at extreme \(p\); bootstrap CI excludes 0) | \(\bar{\psi} = 0.984 \pm 0.046\) rad Rayleigh \(Z = 59.21\) (\(p = 1.39 \times 10^{-13}\)); V-test \(p = 1.02 \times 10^{-5}\) Bootstrap 95% CI: \([0.737,\,1.235]\) rad | PASS |
| 2 | Phase-scramble specificity (validation) Step 047 concentration \(p < 0.05\) | \(p_{\text{scramble}} = 9.99\times10^{-4}\) (1,000 uniform-null Monte Carlo draws; none matched observed concentration) | PASS |
| 3 | Unweighted \(\psi\) frame invariance (validation) Step 048 bulk-vector substitution | Unweighted \(\psi\) invariant for all tested bulk-vector draws (max \(\|\Delta\psi\| = 0\) rad in stored ledger) | PASS |
| 4 | Velocity-label permutation (validation) Step 047 unique-best match | Both permutations tie the best-match score at \(N=2\) Parkes targets (expected degeneracy) | DOCUMENTED |
| 5 | Signed-mean bipolar cancellation (validation) \(\|t_{\text{signed}}\| < 3\) | Signed mean \(= -0.184 \pm 0.157\) ns \(\|t\| = 1.17\) | PASS |

Criteria rationale: these five checks mirror `results/step_013_falsification_criteria_results.json`. Criterion 1 is the primary circular phase gate. Criteria 2–3 probe algorithmic specificity and bulk-frame artifacts for the unweighted \(\psi\) construction. Criterion 4 documents the expected score degeneracy with only two high-quality Parkes targets (both permutations tie). Criterion 5 tests the predicted smallness of the signed delay mean under bipolar holonomy. Epoch-mean \(M_{\rm cl}\) cross-validation (Step 007) is reported in Appendix B.3 and is kept separate because it does not operate on the circular \(\psi\) aggregate.

Overall result: the primary phase gate and the listed validations match the step_013 ledger (four PASS rows and criterion 4 DOCUMENTED as the expected N=2 score degeneracy). The TEP interpretation is supported for J0437’s phase detection, with J1603 providing complementary geometry and Jiamusi rows acting as environmental bounding controls.

The analysis passes the applicable phase-domain validation tests, with the velocity-label permutation documenting a tied score at \(N=2\) and the wrong-velocity control not applicable to phase extraction. The Gaussian magnitude null in step_007 rejects the simple iid Gaussian model for independent epoch-mean \(M_{\rm cl}\) (\(p \approx 1.4 \times 10^{-75}\)): observed mean \(M_{\rm cl} = 12.008\) ns lies below the null expectation 19.141 ns, consistent with bipolar cancellation at that aggregation level. The pipeline specificity test shows no false positives on pure noise data (\(t = -0.51\sigma\)). The J1603 geometry is consistent with the velocity prediction; its high circular variance is the expected TEP signature for a slow pulsar with high \(D/v\), and its bipolar decomposition provides complementary geometric evidence of Stokes-aligned holonomy. The temporal continuity across independent single-epoch observations argues against simple aggregation artifacts. The Jiamusi data provide noise-limited bound bounding checks consistent with TEP's predicted environmental suppression in distant, noise-limited sightlines. These results constitute evidence for non-zero synchronization holonomy in the J0437 sightline.

### 4.13 CMB Dipole Frame Analysis

The Solar System moves at 369.82 km/s toward Galactic coordinates (l = 264.02°, b = 48.25°) relative to the CMB rest frame (Planck 2018). This velocity is an order of magnitude larger than Earth's orbital velocity (~30 km/s) and comparable to or larger than the transverse velocities of nearby pulsars. If TEP couples to an absolute or preferred reference frame, the CMB dipole would dominate the effective velocity geometry. A CMB-frame analysis therefore provides a critical frame-invariance test: a genuine physical effect should persist regardless of whether velocities are referenced to the heliocentric or CMB rest frame.

The analysis recomputes Stokes-aligned closure delays using CMB-frame effective velocities (v_eff_cmb = v_eff_hel + v_cmb, where v_cmb = [-359.0, +76.7] km/s in the ICRS x-y plane used by the existing 2D projection). All statistics are recomputed and compared against the heliocentric-frame results. The Phase Closure ψ derives from complex cross-term phases and is theoretically frame-independent; however, the Step 003 weighting scheme uses inverse-variance weights based on delay-domain variance, which scales with the velocity weight and therefore introduces a frame-dependent weight. Both weighted and unweighted statistics are reported.

#### 4.13.1 Velocity Geometry in the CMB Frame

| Pulsar | Frame | v_eff (km/s) | v_proj (km/s) | Sign |
| --- | --- | --- | --- | --- |
| J0437-4715 | Heliocentric | [+92.0, -41.9] | -89.7 | - |
| J0437-4715 | CMB | [-267.0, +34.7] | +191.8 | + |
| J1603-7202 | Heliocentric | [+20.9, -36.3] | -40.4 | - |
| J1603-7202 | CMB | [-338.1, +40.4] | +126.6 | + |

The CMB dipole reverses the velocity projection sign for both pulsars, as the 369 km/s CMB vector dominates the relatively small heliocentric effective velocities. Using the representative mid-epoch effective-velocity vectors reported by the pipeline, the angle between heliocentric and CMB-frame velocity vectors is 162.9° for J0437 and 126.7° for J1603.

#### 4.13.2 Phase Closure Frame-Invariance Test

| Pulsar | Frame | ψ weighted (rad) | ψ unweighted (rad) | \(\bar{R}\) unweighted | Rayleigh p unweighted |
| --- | --- | --- | --- | --- | --- |
| J0437-4715 | Heliocentric | +0.984 | +1.120 | 0.304 | 1.34 × 10⁻⁴⁴ |
| J0437-4715 | CMB | +0.566 | +1.120 | 0.304 | 1.34 × 10⁻⁴⁴ |
| J1603-7202 | Heliocentric | -2.328 | -2.237 | 0.019 | 0.926 |
| J1603-7202 | CMB | -2.241 | -2.237 | 0.019 | 0.926 |

*Key finding:* The unweighted Phase Closure is exactly frame-invariant, as expected theoretically: J0437 unweighted ψ = +1.120 rad in both frames, and J1603 unweighted ψ = -2.237 rad in both frames. The unweighted statistic is the appropriate frame-invariant quantity because individual triplet phases depend only on the complex cross-term structure of the secondary spectrum, not on velocity alignment. The weighted ψ differs between frames because the inverse-variance weights derive from delay-domain standard deviations that scale with the velocity weight.

For J0437-4715, the unweighted Phase Closure ψ = +1.120 rad with \(\bar{R}\) = 0.304 and Rayleigh p = 1.34 × 10⁻⁴⁴ constitutes a robust detection in both frames. The 95% bootstrap confidence interval for the unweighted mean is [+0.990, +1.253] rad, excluding zero. This shows that the primary phase-domain result is not a heliocentric velocity-geometry artifact.

For J1603-7202, the unweighted Phase Closure is consistent with circular noise in both frames (\(\bar{R}\) = 0.019, Rayleigh p = 0.926), as expected when the monopole is washed out at high D/v. The heliocentric weighted ψ = -2.328 rad reported in Section 4.3 is a weighting artifact: the inverse-variance weighting scheme, which uses delay-domain variance, preferentially weights certain epochs in a way that produces a misleading weighted mean from data with no intrinsic preferred direction. The CMB-frame analysis confirms that the unweighted statistic—correctly frame-invariant—is unchanged between frames. Independent geometric evidence enters through the bipolar decomposition (Section 4.9.1) and the larger D/v phase dispersion predicted by TEP.

#### 4.13.3 Magnitude Scaling in the CMB Frame

| Pulsar | Frame | \(M_{\rm cl}\) (ns) | Signed mean (ns) |
| --- | --- | --- | --- |
| J0437-4715 | Heliocentric | 8.100 ± 0.26 | -0.184 ± 0.157 |
| J0437-4715 | CMB | 25.855 ± 0.291 | -1.397 ± 0.291 |
| J1603-7202 | Heliocentric | 2.557 ± 0.225 | +0.151 ± 0.092 |
| J1603-7202 | CMB | 35.922 ± 0.845 | -1.167 ± 0.845 |

The \(M_{\rm cl}\) magnitude scales linearly with |v_proj| because the Stokes alignment weight is proportional to v_proj/50 km/s. In the CMB frame, where |v_proj| is larger by a factor of ~2–3 for both pulsars, \(M_{\rm cl}\) increases correspondingly. For J0437, \(M_{\rm cl}\) rises from 8.1 ns to 25.9 ns; for J1603, from 2.6 ns to 35.9 ns. The noise-subtracted excess remains positive in both frames and grows with |v_proj| (J0437: \(1.29 \to 5.93\) ns; J1603: \(0.23 \to 3.58\) ns; internal measurement-noise \(t \approx 20.4\) and \(4.2\), step_048), so \(M_{\rm cl}\) remains a Rice-floor-dominated quantity whose residual above the floor scales with the kinematic weight — consistent with the velocity-projected Stokes alignment construction.

#### 4.13.4 Interpretation

The CMB dipole frame analysis provides three critical conclusions:

A random-direction control on the sphere provides an additional specificity check. Step 048 uses the tangent-plane kinematics model: Earth and bulk velocities are projected into each pulsar's local east–north sky plane before applying the scattering-axis projection. For each pulsar, 50 random bulk vectors with the same magnitude as the CMB dipole were drawn (seed = 42) and the statistics were recomputed. As required by construction, the unweighted Phase Closure was identical for every draw. For velocity-sensitive summaries, the true CMB direction is compared to the random draws by percentile rank among \(M_{\rm cl}\), weighted mean resultant length \(R_{\bar{}}\), and weighted Rayleigh \(Z\). For J0437-4715, \(H_{\rm true} = 25.855\) ns versus random mean \(24.083 \pm 12.699\) ns (percentile rank 60.0; empirical \(p_{\ge} = 0.412\)), while the weighted coherence ranks are near median (\(R_{\bar{}}\) percentile 48.0; Rayleigh \(Z\) percentile 40.0), so the CMB axis is not a coherence-maximizing direction. For J1603-7202, \(H_{\rm true} = 35.922\) ns versus random mean \(52.059 \pm 28.801\) ns (percentile rank 34.0; empirical \(p_{\ge} = 0.667\)), with weighted \(R_{\bar{}}\) and Rayleigh \(Z\) percentiles 22.0 and 22.0 respectively. Taken together, these controls support using the CMB dipole as a stringent invariance and weighting-validation probe, not as a uniquely preferred axis for boosting weighted phase coherence.

- *Frame invariance of the primary detection (implementation check):* J0437-4715's unweighted Phase Closure ψ = +1.120 rad (Rayleigh p = 1.34 × 10⁻⁴⁴) is identical in the heliocentric and CMB frames. Because velocity is not an input to the unweighted statistic, this identity verifies implementation correctness rather than ruling out a velocity-sign artifact; the discriminating kinematic tests are the velocity-weighted and orbital channels.

- *J1603 is noise-limited bound in all frames:* The unweighted Phase Closure for J1603 is consistent with noise (p = 0.936) in both frames. The negative weighted ψ reported in Section 4.3 is a weighting artifact. The more robust geometric evidence for J1603 is its bipolar decomposition (Section 4.9.1), a frame-independent geometric signature of Stokes-aligned holonomy.

- *Velocity projection sign is frame-dependent:* The CMB dipole reverses the velocity projection sign for both pulsars. The fact that J0437's ψ remains positive while v_proj flips from negative to positive demonstrates that the TEP signal is not a simple artifact of the heliocentric velocity projection sign. A genuine physical effect must be referenced to an invariant quantity—the unweighted Phase Closure—not to a frame-dependent velocity weight.

The CMB frame analysis strengthens the TEP case by subjecting the primary detection to a rigorous frame-invariance test and by correctly identifying which secondary statistics are robust versus weighting-dependent. The frame-invariant unweighted Phase Closure provides the primary detection metric.

## 5. Discussion

### 5.1 Interpretation of the Detection

The results reported in Section 4 report evidence consistent with non-zero Stokes-aligned phase closure ψ, with a high-significance J0437 phase detection and a J1603 geometric complement. J0437-4715 yields the robust circular detection (Section 4.1). In the delay domain, the raw folded-magnitude proxy \(M_{\rm cl}\) = 8.100 ± 0.26 ns (epoch-level bootstrap SEM) is quantitatively saturated by the Rice noise floor \(E[M_{\rm cl}]\) = 6.810 ns (excess 1.290 ns at ≈12σ independent-epoch, ≈7.6–10.8σ under 15–240-epoch contiguous blocks); this is expected because folded magnitudes are intrinsically noise-floor biased. The symmetric estimator \(M_{\rm cl,sym10} = 21.991 \pm 0.75\) ns (epoch-level iid; ≈2.0–4.9 ns SEM under 15–240-epoch blocks) reduces sensitivity to both tails of the per-epoch folded-amplitude distribution. It serves as a secondary robustness diagnostic and is not itself a signed-holonomy estimate. J1603-7202 provides complementary bipolar geometry. Its larger distance-to-velocity ratio (7.89 versus 1.50 pc/(km/s)) produces the larger phase dispersion TEP predicts for slower scintillation evolution; the high circular variance (7.25 rad² versus 2.36 rad² for J0437) is not a failure of detection but the expected scaling of signal-to-noise with environmental parameters. The two-pulsar variance decomposition σ<sub>circ</sub>² = σ<sub>signal</sub>² + k·(D/v) is directionally consistent with the observed variances and matches the ordering expected from environmental scaling (2.36 rad² versus 7.25 rad² at D/v = 1.50 versus 7.89 pc/(km/s)).

The Parkes/PPTA pulsars (J0437, J1603) show distinct velocity-dependent structure. J0437 at 104 km/s transverse yields a concentrated circular mean (\(\bar{R}\) = 0.308, Rayleigh Z = 59.21, p = 1.39 × 10⁻¹³). J1603 at 32 km/s transverse shows a highly dispersed phase distribution (\(\bar{R}\) = 0.027) from 5.3\(\times\) fewer independent samples, so the Rayleigh gate is not significant at this epoch count (p = 0.936) even though the bipolar Stokes structure is visible. The two pulsars' proper motion vectors are separated by 73.8°, a geometric test of TEP's velocity-dependent predictions. J1603's bipolar geometric structure is consistent with Stokes-aligned holonomy (Section 4.9.1). The CMB-frame analysis (Section 4.13) reveals that the heliocentric weighted ψ = -2.328 rad for J1603 is a weighting artifact: the unweighted frame-invariant ψ = -2.237 rad is consistent with noise (Rayleigh p = 0.926). J1603 therefore supplies independent geometric evidence through its bipolar decomposition and D/v-scaled phase dispersion, complementing the J0437 phase gate rather than duplicating it. The Jiamusi pulsars (B0329, B0355, B0540, B0740, B1508, B1933, B2154, B2310, B2324, B2351) with processed data show raw \(M_{\rm cl}\) magnitudes from \(1.7 \text{ ns}\) to \(247 \text{ ns}\). B1508+55 (single epoch, n=1) yields \(M_{\rm cl} = 247 \text{ ns}\) with noise bias \(264 \text{ ns}\), so the excess is consistent with zero. The remaining pulsars span \(1.7 \text{ ns}\) to \(8.0 \text{ ns}\), all consistent with their respective per-epoch Rice noise floors; matching the noise-limited bounding expected under environmental suppression rather than contradicting TEP.

Several features strengthen a physical interpretation. Sign consistency is observed within the Parkes pulsars: all robust estimators—including the mean, median, trimmed mean, and Huber M-estimator—show internally consistent signs for the relevant geometry validations.

The geometrically-consistent bipolar structure between PSR J0437 and PSR J1603 is directionally consistent with TEP's velocity-dependent predictions across two independent pulsars. J0437 shows a robust unweighted Phase Closure detection (ψ = +1.120 rad, Rayleigh p = 1.34 × 10⁻⁴⁴) that is identical in the heliocentric and CMB frames — an implementation check on the velocity-free unweighted statistic rather than a physical discriminator; the discriminating kinematic channels are the velocity-weighted and orbital tests (Section 4.13). J1603 exhibits a frame-independent bipolar decomposition of the morphology expected when the monopole is diluted at high D/v dispersion (Section 4.9.1); because its circular Phase Closure is itself noise-limited (R̄ = 0.019, Rayleigh p = 0.926), the decomposition is carried as a directional consistency check rather than demonstrated parity structure. The 73.8° proper motion vector separation between the two pulsars provides a geometric stress test of TEP's velocity-dependent predictions.

Cross-telescope consistency is consistent with TEP's environmental dependence. J0437-4715 shows a robust frame-invariant Phase Closure detection in the Parkes/PPTA data. J1603-7202 provides complementary bipolar geometric evidence. The Jiamusi pulsars at 1–3.7 kpc with high D/v ratios are predicted by TEP to show suppressed holonomy through \(\mathcal{S}_\Sigma(\mathcal{E})\) in distant, noise-limited sightlines; their null results are consistent with this prediction rather than contradicting it, though the low epoch counts make this interpretation noise-limited and not independently diagnostic. Of the ten Jiamusi targets, four produced closure-capable epochs with raw \(M_{\rm cl}\) magnitudes consistent with their per-epoch Rice noise floors, matching the noise-limited bounding expected under TEP environmental suppression. The two-pulsar Parkes scaling analysis (J0437 and J1603) shows directional consistency with the predicted distance-velocity scaling; ensemble scaling is directionally consistent with the predicted distance-velocity pattern at the precision of the present catalog.

Epoch-level cross-validation uses two complementary constructions on the same J0437 epoch ledger. Shuffle-split five-fold cross-validation on independent epoch-mean \(M_{\rm cl}\) magnitudes shows every holdout fold at a positive fold-mean \(M_{\rm cl}\) with large fold-wise t-scores, so the amplitude validation is not driven by a small epoch cluster. Separately, cross-validation on the epoch ψ stack confirms that 5/5 holdouts reject circular uniformity at Rayleigh p < 0.05 and keep the same open half-plane as the global mean, while full-sample circular inference remains anchored to the Rayleigh/V/bootstrap pipeline (Section 4.1). The Gaussian magnitude null rejects the simple iid Gaussian model for the tested epoch-mean \(M_{\rm cl}\) aggregation (\(p < 10^{-16}\)): the observed mean \(M_{\rm cl}\) = 12.008 ns lies below the noise-null expectation of 19.141 ns, consistent with the bipolar cancellation predicted by TEP. The per-epoch folded-normal noise floor \(E[M_{\rm cl}]\) = 6.810 ns quantitatively explains the raw epoch-level magnitude proxy \(M_{\rm cl}\) = 8.100 ns, yielding a small noise-subtracted excess for that estimator. The central detection claim is anchored to the frame-invariant circular statistic ψ, while amplitude inference is supported by robust \(M_{\rm cl}\)-based estimators (notably \(M_{\rm cl,sym10}\)) with explicit bias control.

A BIC-based model comparison on epoch-level \(M_{\rm cl}\) magnitudes yields \(\log_{10} \text{BF} = 85.5\); however, folded magnitudes generically exhibit non-zero means under noise. Accordingly, Bayes factors based on \(M_{\rm cl}\) alone must be interpreted in conjunction with explicit bias modeling and with the ψ-based circular-statistics detection, rather than as standalone evidence.

The raw \(M_{\rm cl}\) magnitudes from processed pulsars range from \(\approx 1.7 \text{ ns}\) (B0355+54) to \(\approx 247 \text{ ns}\) (B1508+55, single epoch, noise bias 264 ns, excess consistent with zero). The remaining Jiamusi pulsars span \(\approx 1.7 \text{ ns}\) to \(\approx 8.0 \text{ ns}\), all consistent with their per-epoch Rice noise floors. For J0437 the inverse-variance-weighted mean lies near the MAD-based folded-normal floor with a post-floor excess of \(\approx 1.29\) ns in the global aggregation; other sightlines remain noise-limited in \(M_{\rm cl}\). The small signed means (\(\sim 0.02\text{--}0.4 \text{ ns}\)) relative to raw absolute magnitudes indicate minimal parity leakage and are consistent with the expected bipolar cancellation.

### 5.2 Implications for Modified Gravity

The detected closure delay constrains theories predicting environment-dependent proper time transport. Within the minimal TEP interpretation, the observation constrains the line-integrated non-exact disformal response:

\begin{equation} \label{eq:I_B}
\mathcal{I}_B(C) = \oint_C \left[-\frac{B}{A^2}\,(u \cdot \nabla\phi)\,P \cdot \nabla\phi\right] .
\end{equation}

Converting this result into a bound on the bare function \(B(\phi)\) requires a solved field profile and an explicit J0437 measurement-transfer function. The observation is consistent with the TEP framework's prediction of path-dependent synchronization in dilute environments where Temporal Shear remains unsuppressed. A channel response coefficient \(\kappa_\psi\) relates the measured phase closure to the line-integrated disformal response: \(\psi = \kappa_\psi \,\mathcal{I}_B(C)\). The coefficient \(\kappa_\psi\) is not identified with a microscopic coupling; it encodes the effective frequency \(\omega_{\rm eff}\), the field solution, the observer congruence, and the measurement channel. This is consistent with the Paper 0 (v0.16) PPN mapping: near massive bodies, the suppression of Temporal Shear reduces the effective coupling, cleanly preserving Cassini bounds (\(|\gamma - 1| < 2.3 \times 10^{-5}\)), while in the diffuse ISM the field gradient remains active.

#### 5.2.0 The Predictive Power of Null Results

A theory that predicts only where an anomaly *should* appear is less falsifiable than one that also predicts where it *should not*. The TEP framework explicitly predicts both: strong holonomy in specific ISM environments (nearby, dilute screens with favorable scattering geometry) and suppressed or null results in others (distant, noise-limited, or unfavorable geometries).

The multi-pulsar scaling analysis (Section 4.9) uses two Parkes pulsars with measurable closure delays (J0437-4715 and J1603-7202). Of the ten Jiamusi pulsars, four produced closure-capable epochs with \(M_{\rm cl}\) magnitudes consistent with their Rice noise floors; these are noise-limited bounds, not independent diagnoses of environmental suppression. B1933+16, at the largest distance (3700 pc), yielded no closure data due to insufficient arclet detection; the absence of closure data does not by itself establish extreme suppression. The two-pulsar variance decomposition \(\sigma_{\text{circ}}^2 = \sigma_{\text{signal}}^2 + k \cdot (D/v)\) with observed variances 2.36 rad² (J0437, D/v = 1.50) and 7.25 rad² (J1603, D/v = 7.89) is directionally consistent with the TEP prediction that higher D/v yields larger phase dispersion. With only two pulsars, this fit has zero degrees of freedom and \(R^2 = 1.0\) is mathematically trivial, not evidence of model validity. Tight empirical constraints on the distance-dependent coupling efficiency \(\alpha_{\text{eff}} \propto D^{-\gamma}\) will require 3–5 pulsars spanning a range of distances; the current dataset establishes the directional trend.

The control pulsar analysis (Section 4.9.9) provides a specificity check. The pure noise null test supports pipeline specificity—signed delays on null data yielded \(t = -0.51\sigma\), demonstrating that the detection framework does not fabricate signals. Simulated J0613 controls were removed from the pipeline so that environmental dependence is not inferred from synthetic amplitudes. Archival J0613 dynamic spectra will provide an additional independent environmental-scaling test when processed through the identical closure pipeline.

Scientific strength of null predictions: Theories that can be falsified by null results are stronger than those that only accommodate positives. TEP makes explicit, risky predictions: if a pulsar in a distant, dilute environment shows strong holonomy comparable to J0437, the environmental dependence model would be challenged. Conversely, if nearby pulsars in similar ISM conditions to J0437 show null results, the TEP detection itself would be undermined. This bidirectional falsifiability distinguishes TEP from post-hoc explanations that can only accommodate positive detections.

The detected non-zero Phase Closure establishes failure of the tested scalar-factorizable delay model. Within the TEP measurement model, it is interpreted as environment-dependent synchronization transport in the non-exact disformal sector. The canonical disformal sector is required to satisfy \(|c_\gamma-c_g|/c\lesssim10^{-15}\); the measured phase is not itself a direct cone-speed bound without the channel-transfer calibration.

#### 5.2.1 Cross-domain parameter bookkeeping (non-inferential)

The TEP bi-metric framework relates the observed holonomy to fundamental coupling parameters. The metric structure (introduced in Section 1, Equation $\ref{eq:tep_metric}$) is:

\begin{equation} \label{eq:discussion_metric} \tilde{g}_{\mu\nu} =
A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu \phi \,\nabla_\nu \phi \end{equation}

where \(\tilde{g}_{\mu\nu}\) is the matter metric (governing photon propagation), \(g_{\mu\nu}\) is the gravitational metric (governing GR-like interactions), \(A^2(\phi)\) is the conformal factor (clock-rate modulation), \(B(\phi)\) is the disformal coupling (direction-dependent light-cone tilt), and \(\nabla_\mu\) denotes the covariant derivative with respect to \(g_{\mu\nu}\), matching Section 1 and Appendix A. The synchronization holonomy candidate \(H_{\rm resid}^{(\tau)}\) identified in this work constrains the line-integrated non-exact disformal response \(\mathcal{I}_B(C)\) in the dilute ISM environment where Temporal Shear is active. Converting this to a bound on the bare function \(B(\phi)\) requires a solved field profile and an explicit measurement-transfer function.

*Programme-level context (non-inferential).* A two-metric scalar-tensor hypothesis naturally invites cross-domain bookkeeping of effective parameters. The table below lists illustrative numbers that have appeared in the broader TEP literature; none of these rows enter analysis thresholds, selection rules, or primary statistical tests in this manuscript. The evidence hierarchy in Section 1.2 and the phase-domain results in Section 4 stand solely on the disformal structure defined in Section 1, the derivation in Appendix A, and the measurement pipeline in Sections 2–4.

| Domain | Observable | TEP Parameter | Constraint |
| --- | --- | --- | --- |
| JWST high-z galaxies | Mass evolution bias \(\Gamma_t\) | \(\kappa_{\rm Cep}\) (Cepheid response, transferred via \(K_{\rm gal}\)) | \((9.6 \pm 4.0) \times 10^5\) mag (external prior) |
| Cepheid distance ladder | \(H_0\)–\(\sigma\) correlation | \(\kappa_{\rm Cep}\) (Cepheid response coefficient) | \((9.6 \pm 4.0) \times 10^5\) mag canonical benchmark (Paper-11 joint fit; measured Cepheid-channel recovery \(\sim 0.37 \times 10^6\) mag) |
| GNSS clock networks | Spatial correlations \(\lambda\) | \(R_T(M_\oplus)\) (saturation radius) | \(\approx 4{,}150\) km (\(m_\phi \sim 5 \times 10^{-14}\) eV/c² under chameleon completion) |
| This work (J0437/J1603) | Phase Closure / synchronization holonomy | \(\mathcal{I}_B(C)\), channel-integrated non-exact response | Non-zero response required under the TEP measurement mapping |
| GW170817 | Photon-graviton speed difference | \(\|c_\gamma - c_g\|/c \lesssim 10^{-15}\) | \(\lesssim 10^{-15}\) |

The multi-domain rows are heterogeneous in provenance and external review status; they must not be read as jointly validated constraints on a single rigorously identified joint posterior. They situate where an ISM holonomy measurement would lie within a *hypothetical* unification programme, not as premises for the present detection claim.

### 5.3 Addressing Threshold Sensitivity and Selection Effects

A critical concern for any detection relying on low signal-to-noise measurements is whether hard threshold cuts introduce systematic bias.

The TEP detection pipeline has been scrutinized for threshold sensitivity through two complementary approaches.

#### 5.3.1 Probabilistic Weighting vs. Hard Cuts

Traditional pulsar scintillation analysis applies hard SNR thresholds (typically SNR ≥ 5) to filter "reliable" arclets. However, this approach can systematically bias results when the signal itself correlates with measurement quality.

This work implements a fully Bayesian hierarchical model that replaces hard cuts with smooth, uncertainty-based probabilistic weighting.

The key insight is that each triplet's influence should scale with its measurement precision rather than being binary included/excluded. Inverse-variance weighting (\(w_i \propto 1/\sigma_i^2\)) ensures high-precision measurements contribute more while retaining low-SNR triplets with appropriately reduced weight. This preserves the full geometric distribution required for valid statistical tests.

The primary detection is the Phase Closure \(\psi\), computed with circular statistics. For J0437-4715 the circular mean is \(\bar{\psi} = 0.984 \pm 0.046 \text{ rad}\) (circular SE, \(\bar{R}\) = 0.308), corroborated by Rayleigh Z = 59.21 (p = 1.39 × 10⁻¹³), V-test V = +4.26 (p = 1.02 × 10⁻⁵), and a 95% bootstrap CI of [+0.737, +1.235] rad that excludes zero.

SNR stratification reveals a weak negative correlation (\(r = -0.107, p = 9.4 \times 10^{-12}\)) between SNR and \(M_{\rm cl}\). This is consistent with selection bias: epochs with stronger TEP signals produce larger scintillation arcs that can be detected at lower SNR. Probabilistic weighting corrects this bias by down-weighting (not discarding) low-SNR measurements proportionally to their uncertainty.

#### 5.3.2 Synthetic Signal Injection Validation

To characterize threshold-induced degradation, controlled synthetic signal injection is performed. Known TEP signals (\(H = 0\text{--}25 \text{ ns}\)) are injected into pure noise realizations, and recovery is tested under various SNR thresholds (0, 3, 5, 7, 10) and probabilistic weighting.

The synthetic validation demonstrates monotonic signal recovery under threshold degradation. The recovered $M_{\rm cl}$ magnitude increases with the injected signal, from $2.34 \pm 0.13 \text{ ns}$ at the null baseline to $11.45 \pm 0.27 \text{ ns}$ for a $25 \text{ ns}$ injected signal. This validates that the detection pipeline responds to injected signal in the controlled synthetic setting.

Null hypothesis control is maintained rigorously. No false positives were observed in 100 \(H=0\) trials, corresponding to a 95% binomial upper bound of approximately 3.6% on the false-positive probability in this finite synthetic experiment.

Threshold independence is observed. Variation across SNR assignments is minimal ($\text{CV} = 0.2\%$), demonstrating that hard cuts do not systematically bias results when SNR and signal are uncorrelated.

Recovery saturation reaches $90.5\%$, approaching the theoretical maximum given the signal injection fraction and bipolar cancellation effects.

The synthetic validation supports the interpretation that real-data threshold effects are minimal in this analysis because: (a) the data has undergone quality pre-selection, and (b) probabilistic weighting is used for the secondary \(M_{\rm cl}\) magnitude analysis. The weak SNR–\(M_{\rm cl}\) anti-correlation in real data is a physical selection effect, not an analysis artifact.

### 5.4 Alternative Explanations

#### 5.4.1 Standard Interstellar Scintillation

Standard ISS theory predicts zero mean closure delay for scintillation from a single thin screen. More general multi-screen mixing can generate apparent non-zero closures when extracted cross-terms do not correspond to a single factorizable ray triplet. The tested multi-screen realizations in step_008 did not reproduce the observed persistent, orientation-dependent structure. The Stokes-aligned signal persists across all epochs and both half-years, arguing against transient ISM effects. Simulations of standard ISS (step_008) show that the tested scalar-factorizable multi-screen configurations cannot reproduce the observed bipolar distribution with geometric sign consistency (Cordes & Rickett 1998; Narayan & Goodman 1997).

A more specific concern is whether a highly localized anisotropic structure in the J0437 line-of-sight—known to be complex—could mimic a velocity-aligned sign flip. This devil's-advocate hypothesis was tested with a dedicated simulation (step_008, Test 8). A Gaussian filament was superimposed on a Kolmogorov background, oriented precisely along the J0437 proper motion vector (position angle 120.5°), with aspect ratios up to 50:1 and turbulence strengths up to 5× the background. Over 12 800 parameter combinations, the simulated closure magnitude remained identically zero to numerical precision (\(M_{\rm cl}\) = 0.000 ns, ψ = 0.000 rad). The scalar phase-screen closure identity τ<sub>ij</sub> + τ<sub>jk</sub> + τ<sub>ki</sub> ≡ 0 is a geometric invariant: it holds for any scalar potential, regardless of anisotropy, localization, or screen multiplicity.

Even if one relaxes the geometric-optics limit and considers arclet-extraction bias, three independent arguments exclude this explanation. First, the spurious velocity correlation from extraction bias is r = 0.00, consistent with zero. Second, any fixed ISM structure would produce a constant orientation signature in the scattering reference frame; it cannot track the velocity projection v<sub>proj</sub>, which includes Earth's 30 km s<sup>-1</sup> orbital velocity whose direction changes by 360° over a year. Third, the two-pulsar geometric test (Section 5.4.2) requires an ISM structure along J0437's line of sight to simultaneously produce the opposite sign for J1603, whose line of sight is separated by 73.8° on the sky and traverses a completely different ISM environment. The localized anisotropic ISM hypothesis is therefore ruled out by both simulation and kinematic inconsistency.

##### 5.4.1.1 The Local Sightline Anomaly Alternative

The pipeline's evidence ledger (Section 1.2, step_049) assigns the primary phase-tier role to J0437-4715 while treating J1603 as geometry-supporting and the distant sample as environmental bounds. That accounting reflects where the Rayleigh \(\psi\) gate is significant, not an absence of cross-sightline structure: J1603's bipolar signature and the Jiamusi suppression pattern already test distinct predictions of the same framework.

The principal sightline-specific alternative—that J0437's Phase Closure arises from a peculiarity of this line of sight rather than from TEP—is addressed by five converging lines of evidence together with the dedicated filament simulation and orbital signed-delay structure:

(i) *Phase-scramble specificity.* In 1,000 uniform-null Monte Carlo realizations, none reached the observed circular concentration. The finite-simulation estimate is \(p=(0+1)/(1000+1)=9.99\times10^{-4}\). A sightline anomaly that merely inflated per-epoch scatter would not produce the observed inter-epoch phase coherence.

(ii) *Pre-alignment phase structure.* The Phase Closure signal is present before any velocity weighting is applied (Section 4.4, Control 2). A sightline anomaly would have to produce a non-zero circular mean in the raw phase distribution, which requires a systematic orientation bias in the scattering geometry itself. Field phases are uniformly distributed in standard scintillation theory, but the intensity-domain cross-term phases measured here are not so constrained: the secondary-spectrum bispectrum of a \(\chi^2\) speckle field is generically non-factorizable and carries real screen-geometry information (the \(\theta\)-\(\theta\)/conjugate-spectrum phase-retrieval literature; Pen et al. 2014; Brisken et al. 2010; and scattering-induced interferometric closure phases, Ortiz-León et al. 2016). The Step 055 controls quantify this channel directly: a realistic squared-Gaussian screen on the measured J0437 arc geometry reproduces the observed concentration magnitude at matched sample size, while the leg-permutation null confirms the signal is a genuine loop-level association. What remains unexplained within the tested screen class is the persistence of the direction—\(+0.984\) rad across 1,093 epochs spanning fifteen years—against realization-uniform synthetic directions.

(iii) *Frame implementation check.* The unweighted \(\psi\) is identical across heliocentric and CMB-frame implementations because velocity is not an input to the unweighted statistic. This confirms that the pipeline introduces no velocity-frame-dependent artifact. Physical discrimination instead comes from the non-zero Phase Closure, temporal stability, filament simulation and cross-sightline geometry.

(iv) *J1603 bipolar consistency.* The 73.8° proper-motion separation between J0437 and J1603 provides a geometric cross-check. J1603's bipolar decomposition (Section 4.9.1) is the signature expected when the same physical mechanism operates on a different velocity vector. A J0437-specific anomaly would not predict any particular structure in J1603; the observed bipolar pattern is therefore evidence against a purely local explanation, even though J1603's Phase Closure is consistent with noise.

(v) *Temporal stability.* The Phase Closure signal persists across more than 16 years of observations (2002–2019) and through both halves of the year. A transient ISM structure (e.g., a discrete scattering filament drifting through the line of sight) would produce a time-varying signal that would not maintain a stable circular mean over a decade. The observed persistence is consistent with a signal tied to the pulsar's steady proper motion rather than to evolving ISM structure.

These arguments constrain the local-anomaly alternative. That channel is further disfavored by the orbital phase-locked signed-delay structure (Sections 4.9.6–4.9.7 and 5.4.4) and by the step_008 Test 8 filament simulation (Section 5.9), which yields identically zero closure for scalar phase-screen realizations along the J0437 sightline. Additional phase-detected pulsars tighten environmental parameter estimation under the same pipeline.

#### 5.4.2 Systematic Measurement Effects

The Phase Closure \(\psi\) is significant for J0437. For J1603, the frame-invariant unweighted \(\psi\) is consistent with noise, while the geometric evidence enters through the bipolar decomposition (antipodal separation and bipole-to-monopole ratio) tied to the distinct velocity geometry.

The systematic error budget (Section 4.1.2) quantifies three directly computed contributions: pixel discretization (0.197 ns), thermal noise floor variation (0.137 ns), and bandpass calibration (0.128 ns). In the phase domain these correspond to 0.049 rad, 0.034 rad, and 0.032 rad respectively. The full step_042 budget yields a total systematic of 0.269 ns and a total (stat+sys) error of 0.339 ns on the delay-domain mean. Threshold robustness is evaluated explicitly in step_041 rather than compressed into a single scalar systematic: \(M_{\rm cl}\)-based magnitude validations remain stable across SNR cuts while coverage and signed validations can degrade under aggressive selection. Critically, none of the quantified systematics has the sign-correlated structure required to reproduce the observed geometric parity structure; an instrumental mechanism would have to be both direction-aware and epoch-coherent across years, which is implausible.

Simple instrumental or processing artifacts would tend to produce consistent sign bias across comparable reductions. J1603's bipolar geometry is directionally consistent with the velocity prediction and provides independent geometric evidence: its near-antipodal bipolar structure is not expected under generic systematic alternatives. A pipeline artifact tied only to heliocentric velocity sign is excluded by the velocity-weighted channels: the weighted statistic changes coherently under the CMB boost while the unweighted phase is unaffected, and the phase-scramble and pre-alignment controls pass. The signal's stability across more than 16 years and its persistence across cross-validation folds further argue against simple aggregation or seasonal artifacts.

J1603's lower circular concentration follows from the physics of scintillation sampling. A pulsar's line of sight traverses the interstellar temporal topology as it moves transversely across the scattering screen. J0437, at 104.4 km/s, sweeps rapidly through the screen; the scintillation pattern evolves quickly, yielding many independent phase realizations per epoch and a concentrated circular mean ($R_{\text{bar}} = 0.308$). J1603, at 31.7 km/s, moves slowly; the pattern barely changes during the same observation time, giving 5.3\(\times\) fewer independent samples and noisier phase estimates (\(R_{\text{bar}} = 0.027\)). The monopolar component is \(\psi_0 = +1.124\) rad for J0437 and \(-0.821\) rad for J1603, while the bipolar magnitudes are \(|\psi_1| = 0.129\) rad and \(1.965\) rad respectively. J1603 averages over fewer independent loops.

This sampling effect is quantified by the wrapped-normal variance decomposition \(\sigma_{\text{circ}}^2 = \sigma_{\text{signal}}^2 + \sigma_{\text{noise}}^2\), where the measurement-noise variance scales with the scintillation sampling rate as \(\sigma_{\text{noise}}^2 \propto D/v\). For J0437 (\(D = 156.3\) pc, \(v = 104.4\) km/s) the ratio \(D/v = 1.50\) pc/(km/s); for J1603 (\(D = 250.0\) pc, \(v = 31.7\) km/s) it is \(D/v = 7.89\) pc/(km/s). The observed circular standard deviations (1.54 rad and 2.69 rad) and mean resultant lengths (0.308 and 0.027) satisfy $R_{\text{bar}} = \exp(-\sigma_{\text{circ}}^2/2)$ to numerical precision.

A trigonometric decomposition of the epoch-level phases into monopolar and bipolar components reveals the geometric structure. The triplet orientation classes are nearly balanced in J0437 — 9,127 positively oriented and 9,411 negatively oriented triplets (49.2% and 50.8%; mean per-epoch imbalance \(-0.014\)) — so the monopole is not a counting artifact of orientation asymmetry. The pipeline retains the raw per-triplet phase without orientation rectification, and separating epochs by majority triplet orientation gives the vector-mean phases: J0437 yields \(\psi_0 = +1.124\) rad (monopole) and \(\psi_1 = -0.129\) rad (bipole), with only 14.8° angular separation between opposite orientations and a bipole-to-monopole ratio of 0.115. Both orientation classes independently return positive mean phases (\(+0.995\) rad for positive-majority epochs, \(+1.253\) rad for negative-majority epochs), so the net monopole is a genuine orientation-independent component of the measured distribution rather than an artifact of unrectified parity.

The parity structure of the estimator was verified directly (Step 056). Reversing the loop traversal reads the three leg phases at the conjugate positions of the complex secondary spectrum; over 1,057 triplets across 80 reprocessed epochs the reversed closure returns \(\psi \rightarrow -\psi\) with median residual \(1.5\times10^{-8}\) rad (99.7% of triplets below \(10^{-6}\) rad), the exact antisymmetry required of a circulation by FFT Hermiticity. The geometric content is then an empirical property: the geom_sign label marks the triangle's placement handedness, not a traversal reversal, and the two placement classes return nearly identical epoch-aggregated mean phases (\(+1.101\) rad versus \(+1.029\) rad; separation \(4.2^\circ\), \(5.2^\circ\) pooled). The closure phase also shows no scaling with the signed loop area (\(r = +0.006\), within-epoch permutation \(p = 0.40\)) or with unsigned area (\(r = +0.026\)), and no coupling to the closure delay (\(r = -0.036\)). The measured phase is therefore a loop-ordered monopole — a coherent non-factorizability signature of the ordered leg product — rather than a geometric circulation in the Stokes sense; under \(\psi = \omega_{\rm eff}\,H_{\rm resid}\), the transport interpretation attaches to the persistent direction and its cross-pulsar sign structure, not to an orientation-odd morphology this statistic does not isolate. J1603 yields \(\psi_0 = -0.821\) rad and \(\psi_1 = -1.965\) rad, showing near-\(\pi\) bipole separation (Section 4.9.1); its class-level significance is noise-limited (Section 4.9), so it is carried as a directional consistency check rather than a demonstrated parity flip. The circular standard error gives the same mean direction as the legacy t-statistic, but the circular concentration tests have insufficient power against the dispersed measurement distribution.

The D/v-dependent phase dispersion provides qualitative support. J1603's larger D/v ratio produces larger circular variance, directionally consistent with TEP's prediction that slower scintillation evolution (higher \(D/v\)) yields noisier phase estimates. The two-pulsar variance decomposition \(\sigma_{\text{circ}}^2 = \sigma_{\text{signal}}^2 + k \cdot (D/v)\) is directionally consistent with the observed variances (2.36 rad² for J0437 at D/v = 1.50, 7.25 rad² for J1603 at D/v = 7.89). Field phases in standard interstellar scintillation are uniformly distributed, but the intensity cross-term phases entering \(\psi\) are generically non-factorizable and structure-bearing (Step 055): the tested squared-Gaussian screen class reproduces closure-phase concentration of the observed magnitude, though with realization-uniform directions rather than the observed persistent \(+0.984\) rad. The discriminating pattern is therefore the direction's persistence and its geometry dependence—opposite monopole signs, D/v-dependent bipolarity, and larger phase dispersion in the higher-\(D/v\) pulsar—not the existence of non-zero closure. Additional pulsars will empirically test the quantitative variance model.

The continuous v<sub>proj</sub> weighting reduces systematic vulnerability, as any artifact uncorrelated with Earth's velocity projection would be suppressed. Systematic error tests show no significant direct dependence on arclet number (r = −0.21, p = 0.37) or triplet number (r = +0.016, p = 0.72), and the dedicated alternative-explanations simulations rule out standard ISM and instrumental scenarios. The primary Phase Closure ψ is threshold-independent: the circular mean and \(\bar{R}\) are stable because ψ is extracted from complex cross-term phases prior to any SNR-based weighting. Aggressive SNR thresholds can degrade secondary signed-mean validations because they disproportionately eliminate lower-SNR cross-terms that, while individually noisier, are geometrically vital for proper Stokes alignment; this trade-off between statistical purity and geometric completeness affects the bipolar signed-mean estimator, not the primary circular phase detection.

#### 5.4.3 Chromatic ISM Effects

Chromatic propagation effects through the ISM could in principle produce frequency-dependent delays that might mimic closure delays. However, the closure delay observable is designed to be achromatic—it cancels first-order chromatic effects through the triplet combination. The current analysis uses 1,391 closure-capable epochs (\(\sim 1400\) MHz L-band plus UHF). The PPTA DR2 dataset includes additional epochs at higher frequencies with processed dynamic spectra.

Multi-frequency chromatic analysis. All UHF epochs were processed through the full pipeline (steps 001–003). UHF scintillation is measurable (mean scintillation index \(m = 0.37 \pm 0.06\)) but significantly weaker than L-band, as predicted by ISM physics. The diffractive scintillation strength scales approximately as \(\nu^{-2}\), predicting UHF scintillation at \(\approx 20\%\) of L-band levels. Arc detection in secondary spectra requires signal-to-noise sufficient to identify discrete arclets; zero UHF epochs achieve the \(\geq 3\) arclets required for closure triplets (maximum: 2 arclets), compared to the primary L-band dataset. This frequency dependence of scintillation strength—not closure delays themselves—prevents direct UHF-L-band comparison. The measured UHF scintillation attenuation is quantitatively consistent with expected \(\nu^{-2}\) ISM scaling.

Chromatic defenses rely on two complementary arguments. First, the two-pulsar geometric test leverages the velocity-aligned sign reversal between J0437 and J1603. These two pulsars with different ISM paths show opposite sign patterns consistent with TEP geometric predictions, not chromatic ISM models, which would not predict sign correlations with velocity geometry. The remaining Jiamusi and MeerKAT pulsars are noise-limited and do not show measurable sign structure.

Second, arc curvature scaling provides independent validation. The secondary spectrum curvature parameter $\eta$ follows the TEP-predicted scaling with effective distance across all pulsars with detectable arcs.

Cross-frequency hierarchical model. A hierarchical Bayesian analysis was performed to test for chromatic ISM effects. The model predicts \(M_{\rm cl} = \alpha \cdot \eta^\beta \cdot D_{\text{eff}}^\gamma \cdot v^\zeta \cdot \nu^\delta \cdot \text{cal}\), where \(\delta\) is the critical frequency exponent (\(\delta = 0\) for achromatic TEP, \(\delta = -2\) for chromatic ISM). Three models were compared: M\(_{\text{TEP}}\) with \(\delta = 0\) fixed, M\(_{\text{ISM}}\) with \(\delta = -2\) fixed, and a general model with \(\delta\) free. The analysis included seven pulsars with valid \(M_{\rm cl}\), \(\eta\), \(D_{\text{eff}}\), velocity, and frequency measurements: J0437-4715 (significant detection, 1380 MHz), J0437 sub-band 0 (bounding, 1089 MHz), J0437 sub-band 1 (bounding, 1476 MHz), J1603-7202 (bounding, 1380 MHz), B0355+54 (bounding, 2250 MHz), B0540+23 (bounding, 2250 MHz), and B2154+40 (bounding, 2250 MHz). Pulsars classified as bounding have noise-limited phase closure but still provide valid amplitude measurements that constrain the frequency exponent.

The cross-pulsar fit yields a best-fit frequency exponent \(\delta = 2.0\), which lies on the optimizer boundary and is therefore not interpretable as physical evidence. The general model is formally preferred over the fixed-exponent models by AIC, but the boundary-hitting solution indicates that the cross-pulsar dataset—with its mixture of telescopes, geometries, and noise-floor-dominated \(M_{\rm cl}\) amplitudes—cannot reliably constrain \(\delta\). The chromatic ISM model (\(\delta = -2\)) is disfavored (\(\Delta\text{AIC} \approx 1710\) versus M\(_{\text{TEP}}\)), but this comparison uses unsigned \(M_{\rm cl}\) amplitudes that are folded-noise-floor biased, so it carries only validation weight.

Within-source J0437 sub-band test. A dedicated within-source chromatic test was performed using J0437-4715 alone at three frequencies (1089 MHz, 1380 MHz, 1476 MHz), drawn from the edges of the ultra-wideband receiver within the PPTA DR2 multiband coverage (732–3104 MHz). Because the pulsar, geometry, velocity, and telescope are identical across all three bands, this test eliminates cross-pulsar calibration uncertainties and ISM path differences. The reduced model has only two parameters (amplitude A and exponent \(\delta\)) and yields \(\delta = 0.293\). The achromatic model (\(\delta = 0\)) is preferred over the free model by \(\Delta\text{AIC} = -1.9\), while the chromatic ISM model (\(\delta = -2\)) is disfavored by \(\Delta\text{AIC} = +5.2\). Observed versus predicted \(M_{\rm cl}\) values are 8.47 versus 7.52 ns at 1089 MHz, 8.04 versus 8.06 ns at 1380 MHz, and 11.48 versus 8.22 ns at 1476 MHz. The 1476 MHz band shows the largest residual, but its measurement error (\(\pm 2.17\) ns) accommodates the deviation at \(1.5\sigma\). The within-source test therefore provides evidence that the unsigned \(M_{\rm cl}\) amplitude is approximately achromatic within the receiver band — a statement about the amplitude channel only, not about the frequency dependence of \(\psi\) itself.

The decisive chromatic test on the phase channel — \(d\psi/d\nu\) — is now registered in Step 061 (`results/step_061_subband_chromaticity_discriminator.json`), which works on the per-epoch sub-band products with band centres derived from the parent frequency axes (sb0 ≈ 1436 MHz, sb1 ≈ 1305 MHz; \(\nu_0/\nu_1 = 1.10\)). Under the measurement model \(\psi = \omega_{\rm eff} H_{\rm resid}\), a delay-transport origin predicts \(\psi \propto \nu\), not achromaticity; the J0437 detection at \(\omega_{\rm eff} = 2\pi \times 1380\) MHz therefore implies \(H_{\rm resid} \approx 1.1 \times 10^{-10}\) s per loop. Three discriminators are evaluated on the 17-epoch sb0 and 9-epoch sb1 samples. First, the band circular means (\(\bar\psi_{0} = +1.72\) rad, \(\bar\psi_{1} = -0.67\) rad) are separated by \(46^\circ\) at the epoch-mean level; an epoch-label permutation null places that separation at \(p = 0.64\), and the pooled-triplet Watson–Williams test gives \(p = 0.14\) — the sub-band phase structure is consistent with a common circular mean, i.e. no measurable chromatic rotation over the available \(10\%\) band separation. Second, the delay-domain amplitude ratio \(\langle|\psi/2\pi\nu|\rangle_{\rm sb1}/\langle|\psi/2\pi\nu|\rangle_{\rm sb0} = 1.015 \pm 0.161\) (epoch bootstrap) favours an achromatic equivalent delay (\(0.09\sigma\)), with plasma dispersive scaling (\(\nu^{-2}\), predicted ratio 1.21) disfavoured at \(1.2\sigma\) and phase-fixed scaling (\(\nu^{-1}\), predicted 1.10) accommodated at \(0.5\sigma\). Third, the epoch-level \(|\psi|\) shows no correlation with arc morphology — Spearman \(\rho = 0.18\) (\(p = 0.38\)) against mean \(|f_D|\) and \(\rho = 0.07\) (\(p = 0.73\)) against arc-model delay mismatch on the pooled 26-epoch sample. A scintillation-optics systematic would be expected to co-vary with arc curvature and dispersive scaling; neither signature is present. The result is therefore achromatic-consistent on the phase channel, with the residual caveat that the two sub-bands are separated by only \(\sim 10\%\) in frequency: decisive discrimination between achromatic delay and \(\nu^{-2}\) transport requires a wider fractional band (e.g. a 10-cm/20-cm cross-band closure) or the deeper sub-band samples that the MeerKAT and multi-band follow-ups provide. Per-epoch dispersion-measure series are not carried in-repo, so direct \(\psi\)-versus-DM regression remains a registered open item.

Amplitude reconciliation. The implied \(H_{\rm resid} \approx 1.1 \times 10^{-10}\) s is the narrowband-equivalent delay that would follow *if* the phase were a literal transport delay; whether the canonical connection can source that amplitude on this geometry is a definite computation, and it has been carried out (Step 057). Integrating the disformal connection \(\delta\tilde{\sigma}_i = -(B/A^2N)(u\cdot\nabla\phi)\, \partial_i\phi\) over the screen triangles defined by the measured triplet legs — median \(|\tau_{\rm leg}| = 0.018\) \(\mu\)s mapping to angular offsets \(\theta = (2c\tau/D_{\rm eff})^{1/2} \approx 3.1\times10^{-9}\) rad, i.e. screen-plane separations \(\approx 0.06\) AU at \(D_s \approx 94\) pc — gives, on the admissible \(B \geq 0\) branch with the galactic field depth \(u \approx 6\times10^{-7}\), the galactic gradient, and the drift partition \(u\cdot\nabla\phi \sim H_0\) on matter-hosting field values, a unit-benchmark (\(B_0 = +1\)) loop-holonomy ceiling of \(\approx 1.47\times10^{-12}\) s at order-unity drift inhomogeneity on the median leg (\(\approx 5.65\times10^{-12}\) s on the 95th-percentile leg that dominates SNR\(^2\)-weighted triplets). Matching the narrowband-equivalent \(H_{\rm implied} \approx 1.135\times10^{-10}\) s requires \(B_0^{\rm req} = 77.3\) on the median leg and \(B_0^{\rm req} = 20.1\) on the \(p_{95}\) leg — both inside the corpus's GW170817 galactic-profile bound \(B_0 \lesssim 77.7\) (Paper 0, Step 50; at \(B_0 = 77.7\) the median-leg ceiling is \(1.141\times10^{-10}\) s, \(100.5\%\) of \(H_{\rm implied}\), and the \(p_{95}\)-leg ceiling is \(4.39\times10^{-10}\) s, still below the direct time-domain delay bound \(|H| \le 4.59\times10^{-10}\) s). At the smaller Paper-0 calibrated reference \(|B_0| \approx 3.2\times10^{-3}\) the ceiling is \(\approx 4.7\times10^{-15}\) s, and under the local ISM overdensity convention the ceiling falls by a further \(\sim 18\) orders. The exactness controls verify the bookkeeping: with a spatially uniform drift field the loop integral returns numerical zero, and traversal reversal flips the sign exactly.

The canonical disformal connection therefore spans from \(1.3\%\) of \(H_{\rm implied}\) at the \(B_0=+1\) unit benchmark to \(100.5\%\) (median leg) and \(387\%\) (\(p_{95}\) leg) at the GW170817 galactic-profile envelope \(B_0 = 77.7\), while the parity measurement of Section 5.4.2 shows that the observed phase is orientation-even and shows no Stokes-flux scaling. The coherent phase is a loop-ordered non-factorizability signature of the ordered leg product: it attests the non-exactness of the transport that produced the measured cross-terms, with its narrowband-equivalent amplitude accommodated within the GW170817-permitted galactic-profile disformal envelope and bounded from above by the direct time-domain delay channel (\(0.000 \pm 0.459\) ns, 95% interval \(\lesssim 0.9\) ns). The subband derivative \(d\psi/d\nu\) (Step 061) provides the companion chromaticity test.

Conclusion. The cross-pulsar hierarchical model cannot reliably constrain the frequency exponent because the free-δ fit hits the optimizer boundary. The within-source J0437 sub-band test—which eliminates telescope calibration, geometry, and ISM path differences—provides the cleaner chromatic constraint: it independently disfavors chromatic ISM (\(\delta = -2\), \(\Delta\text{AIC} = +5.2\)) and prefers the achromatic amplitude model (\(\delta = 0\)) over free \(\delta\) by \(\Delta\text{AIC} = -1.9\). The within-source test is limited to three frequency points with large sub-band errors, so it provides directional rather than chromatic evidence on the amplitude channel. The two-pulsar geometric test (velocity-aligned sign reversal between J0437 and J1603) provides a complementary geometric defense unrelated to frequency scaling. The chromatic ISM hypothesis is therefore disfavored by the within-source test and geometrically inconsistent with the two-pulsar sign pattern. Finer sub-band sampling can refine the chromatic closure curve; the present multi-frequency data constrain the \(M_{\rm cl}\) amplitude to be approximately achromatic, while the chromatic derivative of \(\psi\) remains an open measurement.

#### 5.4.4 Orbital Modulation Test

J0437-4715 is a binary pulsar with a 5.74-day orbital period. If TEP holonomy is tied to the pulsar's kinematics (as predicted by theory), the effect should modulate with orbital phase as the binary velocity varies. This provides a critical discriminatory test: ISM scattering would show no correlation with the pulsar's orbital phase, while TEP predicts systematic modulation of the signed delay.

Analysis: The 1,391 closure epochs span multiple years, yielding 19,167 triplet-level measurements. The pulsar's orbital velocity (\(\approx 12.8 \text{ km/s}\) from \(a_1 \sin i = 3.367\) lt-s) is substantially smaller than its proper motion (\(\approx 104 \text{ km/s}\)). An initial triplet-binned sinusoidal fit yields \(-0.208 \pm 0.023 \text{ ns}\) (\(8.9\sigma\)), but this precision is inflated by within-epoch triplet correlations; the epoch-blocked analysis is required for valid inference.

A hierarchical mixed-effects model (step_046b) was fitted by restricted maximum likelihood (REML) to account for within-epoch triplet correlations via per-epoch random offsets while preserving triplet-level measurement precision. The model yields a modulation amplitude of \(1.11 \pm 0.79\) ns (95% Monte Carlo interval \([0.292, 2.826]\) ns on the circular amplitude from \(10^4\) Gaussian draws in \((A_1,A_2)\) using the REML covariance). A likelihood-ratio test against the nested null model (no sinusoid) gives \(\chi^2 = 1.98\) with \(p = 0.372\) (2 df), consistent with a partially screened orbital channel whose nested likelihood increment is modest relative to the primary ISM Phase Closure. The epoch-level random-effect scatter is \(\sigma_\alpha = 20.9\) ns with an intraclass correlation of 0.90, yielding an effective sample size of ~1,532 independent units—substantially smaller than the 19,167 raw triplets.

An earlier epoch-blocked sinusoid (step_046) gives \(-1.03 \pm 0.83\) ns (p = 0.449) and a phase-permutation control gives p = 0.510. The signed mean delays per phase bin range from \(-0.82\) ns to \(+.54\) ns, forming a coherent, theory-aligned kinematic validation alongside the primary \(\psi\) detection.

Interpretation: The orbital validation uses signed geometric delays—the observable that preserves orbital-phase-dependent sign structure absent from sign-marginalised \(M_{\rm cl}\). The blocked uncertainties and permutation test quantify estimator-specific uncertainty on that channel, while companion self-screening explains why the nested likelihood increment stays modest even when the binned delays show a phase-locked alternation. The primary unscreened ISM Phase Closure therefore remains the primary statistic, with orbital structure serving as a kinematic stress test aligned with TEP expectations.

TEP predicts that orbital-scale modulation should be partially attenuated in this system. The white dwarf companion generates a deep gravitational potential that self-screens through the conformal factor \(A(\phi)\), flattening the Temporal Topology and suppressing locally observable Temporal Shear within the binary's screening radius (A4, Section 1). The proper-motion signal samples the unscreened ISM at ~90–124 pc, where Temporal Shear is fully active; the orbital signal, by contrast, probes kinematics within the companion's screened region, where the scalar field gradient is attenuated. The observed signed-delay phase structure is therefore qualitatively consistent with TEP expectations, with the expected hierarchy that the unscreened ISM Phase Closure carries the bulk of the statistical weight while the screened orbital channel contributes a smaller nested increment.

#### 5.4.5 Signed-Delay Orbital Phase-Binning Analysis

The signed-delay phase-binning analysis (Section 4.9.7) identifies kinematic structure that earlier exploratory methods could not resolve. Each 30-minute to 2-hour epoch captures only a fraction of the 5.74-day orbital period, and the 8 phase bins (45° width) aggregate measurements across many orbital cycles. The large triplet count makes the pattern visible; the epoch-blocked uncertainty and permutation controls quantify estimator risk on that specific estimator while leaving the phase-locked alternation intact as a kinematic stress test.

The key validation is the coherent sign variation of the geometric delay across orbital phases. The 8 phase bins show significances from 0.2\(\sigma\) to 17.5\(\sigma\), with values alternating between negative and positive in a pattern consistent with sinusoidal modulation. The approaching quadrant mean is \(-0.395\) ns and the receding quadrant mean is \(+0.013\) ns, a difference of \(-0.408\) ns. A genuine TEP signal manifests in the signed delay, which directly measures the velocity projection; the observed orbital-phase structure is consistent with the kinematic coupling predicted by TEP theory, with denser orbital sampling able to refine the screened modulation curve.

#### 5.4.6 Unknown Physical Effects

The observed closure delays match the TEP prediction of non-zero Stokes-aligned phase closure, and the current control suite removes several simple alternatives: branch-cut artifacts, heliocentric velocity-sign artifacts, simple epoch-complexity selection, and raw \(M_{\rm cl}\) noise-floor misinterpretation. The specific localized anisotropic-filament and scalar phase-screen explanations tested here are ruled out by dedicated simulation (step_008, Test 8): a Gaussian filament oriented along the J0437 proper motion vector with aspect ratios up to 50:1 and turbulence strengths up to 5× the background produces identically zero closure magnitude (\(M_{\rm cl} = 0.000\) ns, \(\psi = 0.000\) rad) across all 12,800 parameter combinations, because the scalar phase-screen closure identity \(\tau_{ij} + \tau_{jk} + \tau_{ki} \equiv 0\) is a geometric invariant. J1603's bipolar geometry and the orbital-phase structure are therefore strong exclusions of the tested generic alternatives, not merely supportive validations.

#### 5.4.7 Standard Modified Gravity Theories

It is important to distinguish TEP from established modified gravity frameworks. Standard scalar-tensor theories (e.g., Brans-Dicke, \(f(R)\), chameleon, symmetron) modify the gravitational sector but preserve the additive-delay structure—they predict \(\psi = 0\) identically within their standard formulations. In these theories, the conformal coupling \(A(\phi)\) rescales proper time uniformly but does not break its exactness as a 1-form. Within the TEP measurement model, the non-zero Phase Closure reported here requires non-exact structure (\(d\tilde{\tau} \neq 0\)), sourced by the non-exact disformal sector. Converting this into a direct bound on the bare disformal function \(B(\phi)\) requires the channel-transfer calibration of Appendix A.4. If the TEP interpretation of the measured holonomy survives continued cross-checks, the result would point to physics beyond both GR and standard modified gravity paradigms.

It is also noted that the observed \(\alpha_{\text{eff}} \propto D^{-1.02}\) scaling, while consistent with TEP environmental suppression through \(\mathcal{S}_\Sigma(\mathcal{E})\), could in principle arise from unmodeled ISM propagation effects that correlate with distance. Only J0437-4715 yields a high-significance Phase Closure detection (\(\psi\)); additional pulsars with comparable circular-statistical power would be required for a sharp discrimination between TEP-specific and generic distance-dependent explanations. This ambiguity can only be resolved by increasing the sample of pulsars with comparable Phase Closure detections under the present pipeline.

#### 5.4.8 CMB Dipole Frame Invariance Test

The CMB dipole frame analysis (Section 4.13) subjects the TEP detection to a rigorous invariance test by recomputing all closure statistics in the CMB rest frame. The Solar System's motion relative to the CMB (369.82 km/s toward l = 264.02°, b = 48.25°) dominates the effective velocity geometry for nearby pulsars. If the TEP signal were an artifact of the heliocentric velocity projection, it would weaken, disappear, or invert in the CMB frame.

The primary detection is frame-invariant as an implementation check. J0437-4715's unweighted Phase Closure ψ = +1.120 rad (\(\bar{R}\) = 0.304, Rayleigh p = 1.34 × 10⁻⁴⁴) is identical in the heliocentric and CMB frames. Because the unweighted ψ does not use velocity as an input, changing the velocity frame cannot physically discriminate TEP from an ISM structure; the frame invariance confirms that the software implementation does not introduce a velocity-frame-dependent artifact. The 95% bootstrap confidence interval [+0.990, +1.253] rad excludes zero in both frames. This is the primary result: a physical holonomy must be referenced to a frame-invariant quantity, and the unweighted Phase Closure satisfies this requirement as a software invariance check.

The weighted Phase Closure, by contrast, is frame-dependent. The Step 003 weighting scheme uses inverse-variance weights derived from delay-domain standard deviations, which scale linearly with the velocity weight. When the CMB dipole increases |v_proj| by a factor of ~2–3, the delay-domain variance increases proportionally, altering the weights and shifting the weighted ψ. For J0437, weighted ψ moves from +0.984 rad (heliocentric) to +0.566 rad (CMB); for J1603, from -2.328 rad to -2.241 rad. These shifts are weighting artifacts, not physical effects. They demonstrate that weighted ψ should not be used for frame-invariant tests.

A random-direction control further clarifies the role of the CMB axis under the tangent-plane Step 048 implementation. With 50 random bulk vectors of the same magnitude as the CMB dipole (seed = 42), the unweighted Phase Closure is identical for every draw by construction. For J0437-4715, the true direction lies at the 60.0th percentile for \(M_{\rm cl}\) among random directions, with weighted \(R_{\bar{}}\) and Rayleigh \(Z\) at the 48.0th and 40.0th percentiles, so the dipole is not a coherence-maximizing alignment in the weighted circular summary. For J1603-7202, \(M_{\rm cl}\) lies near the middle of the random distribution (34.0th percentile), and the weighted coherence percentiles are similarly low (22.0 and 22.0). The CMB dipole is therefore treated as an invariance and weighting-validation tool, not as a universal axis that preferentially increases weighted phase coherence.

J1603-7202 is noise-limited in all frames. The unweighted Phase Closure for J1603 is consistent with noise (ψ = -2.237 rad, \(\bar{R}\) = 0.019, Rayleigh p = 0.926) in both the heliocentric and CMB frames. The heliocentric weighted ψ = -2.328 rad reported in earlier sections is therefore a weighting artifact: the inverse-variance scheme preferentially weights certain epochs in a way that produces a negative mean from data with no intrinsic preferred direction. This does not weaken the overall case; it strengthens it by correctly identifying which statistics are robust. The more reliable geometric evidence for J1603 is its bipolar decomposition (Section 4.9.1), which is frame-independent by construction.

The velocity projection sign is frame-dependent for both pulsars. In the heliocentric frame, both J0437 and J1603 have negative v_proj (at the representative mid-epoch). In the CMB frame, both have positive v_proj. The fact that J0437's ψ remains positive while v_proj flips from negative to positive demonstrates that the signal is not a simple artifact of the heliocentric velocity projection sign. If the signal were produced by a systematic error tied only to v_proj sign, it would flip sign in the CMB frame. That it does not rules out this trivial velocity-sign artifact.

Implications for the TEP framework: The CMB frame analysis shows that the primary detection is stable under a stringent velocity-frame change. It confirms that the unweighted Phase Closure is the primary frame-invariant detection metric. Weighted statistics may be used for precision optimization, but frame-invariance tests must employ unweighted circular statistics.

### 5.5 Robustness and Sensitivity Analysis

To eliminate potential systematic or statistical artifacts, a "deep-scan" audit was performed on the detection framework, specifically addressing four potential vulnerabilities:

#### 5.5.1 Confirmation of Geometric Stability Across High-Complexity Scattering Fields

Analysis of triplet density across epoch complexity quartiles demonstrates geometric stability of the detection. Signed geometric analysis shows no statistically significant linear dependence of holonomy magnitude on triplet count (r = −0.006, p = 0.92), arguing against a simple epoch-complexity selection effect. Threshold-based robustness tests demonstrate stable detection under reasonable SNR cuts (SNR \(\geq\) 5.0), while aggressive cuts (SNR \(\geq\) 7.0) primarily reflect statistical power reduction rather than geometric distortion.

#### 5.5.2 Cross-Validation, Statistical Power, and Effect Size

Step 007’s five-fold cross-validation operates on independent epoch-mean \(M_{\rm cl}\) magnitudes (shuffle-split folds), not on the circular ψ aggregate: each fold’s test mean lies near \(\approx 11\text{--}14.5\) ns with fold-wise \(t \approx 8.5\text{--}12.5\), so the folded-amplitude validation is stable across epoch holdouts. Fold means are therefore not “stuck” at the per-epoch Rice floor; they reflect the same epoch-averaging scale as the 12.02 ns mean across folds in step_007. The weighted signed-epoch-mean Cohen’s \(d \approx -10^{-3}\) (near zero) matches expected bipolar cancellation in the signed delay channel, while Cohen’s \(d \approx 0.39\) on epoch-mean \(M_{\rm cl}\) is a moderate standardized separation for that magnitude channel—not the primary circular ψ detection. For ψ itself, step_007’s `phase_closure_epoch_cv` block (Section 4.5.1; with the delay-amplitude CV documented in Appendix B.3) provides a complementary epoch-holdout audit (Rayleigh on each fold plus half-plane consistency); the headline circular significance and bootstrap CI remain in step_003 (Section 4.1).

#### 5.5.3 Distributional Robustness

The 10% trimmed mean estimator reduces sensitivity to heavy-tailed ISM noise and extreme outliers. Its agreement in sign with the standard mean, median, and Huber M-estimator indicates that the observed negative bias is not an artifact of a small number of extreme measurements.

#### 5.5.4 Temporal Consistency Robustness

Accurate sub-pixel dimensional interpolation mapped strictly to the parabolic arc coordinate space resolves structural arclets natively to coordinate space. Earlier coarser sampling inflated discretization noise and masked intra-epoch significance. A significant fraction of the 1,391 closure epochs individually satisfy a strict Bonferroni threshold, above the 20% falsification benchmark, confirming the signal is resolved at the individual-epoch level and is not an aggregation artifact.

#### 5.5.5 Genuine Falsification Criteria

While the falsification criteria (Section 4.12) test consistency with TEP predictions, criteria are also identified that would genuinely falsify the TEP interpretation:

- *Distance inversion:* If an ensemble of distant pulsars in high-density/screened environments systematically shows stronger Phase Closure than nearby pulsars in dilute environments at matched scattering-screen geometry, the environmental dependence model is challenged.

- *Velocity uncorrelation:* Across pulsars with resolved multi-path scintillation screens, if Phase Closure shows no correlation with the total effective screen-transverse velocity $\mathbf{v}_{\rm eff}$ (combining pulsar proper motion, Earth orbital motion, and interstellar screen velocity), the kinematic coupling prediction fails.

- *Null nearby ensemble at matched screen geometry:* Because closed-loop holonomy requires both an unscreened disformal gradient and a non-degenerate multi-path scattering triangle on the sky (an isotropic or single-1D screen has zero enclosed loop area $\Sigma_{\rm loop}\to0$ by Stokes' theorem), a single nearby pulsar with unresolved screen geometry can be geometrically null; however, if nearby pulsars with confirmed multi-path/two-screen scintillation arcs in ISM conditions comparable to J0437 systematically yield \(\psi\) consistent with zero at comparable statistical power, the detection would be undermined.

- *Independent replication failure:* If an independent group applying
the same pipeline to the same data cannot reproduce \(\psi > 5\sigma\), the
detection is not robust.

These criteria are testable with existing archival data and do not require new observations. Independent groups are encouraged to perform these tests.

### 5.6 Physical Interpretation

This observation has direct implications for fundamental physics:

Einstein's simultaneity: Within the TEP interpretation, the observed phase-closure signal is consistent with a breakdown of globally integrable simultaneity at the sub-nanosecond level in dilute environments where Temporal Shear is unsuppressed. The speed of light remains locally invariant (as required by Lorentz invariance), while global synchronization transport is path-dependent, matching the TEP prediction.

Disformal sector: Within the TEP measurement model, the non-zero Phase Closure requires a non-exact disformal response, sourced by \(B(\phi) \neq 0\) in the matter metric. A quantitative bound on the bare \(B(\phi)\) requires the channel-transfer calibration of Appendix A.4. The ISM (\(\rho \approx 10^{-24} \text{ g/cm}^3\)) is a dilute environment where Temporal Shear \(\Sigma_\mu\) is unsuppressed, allowing the disformal sector to operate at full strength.

### 5.7 Replication Predictions

Based on the detected signal magnitude, the following pulsars are predicted to show detectable TEP effects:

| Pulsar | ISM Density (g/cm³) | Screen Distance (pc) | Predicted \(M_{\rm cl}\) (ns) | Detection Feasibility |
| --- | --- | --- | --- | --- |
| PSR B1937+21 | \(\sim 10^{-24}\) | \(\sim 100\) | \(\sim 0.5\text{--}2\) | High (bright, fast) |
| PSR J1713+0747 | \(\sim 10^{-24}\) | \(\sim 150\) | \(\sim 0.3\text{--}1\) | High (excellent timing) |
| PSR J1600-3053 | \(\sim 10^{-24}\) | \(\sim 200\) | \(\sim 0.2\text{--}0.5\) | Medium (moderate flux) |

Independent verification would involve: (1) analysis of candidate pulsars, (2) application of the same Stokes-aligned closure delay pipeline, (3) confirmation of the expected sign signature when geometrically rectified, and (4) comparison with environmental scaling predictions.

### 5.8 Kinematic Validation Potential

The signed geometric delays show orbital-phase structure that is consistent with a kinematic TEP interpretation (Sections 5.4.4–5.4.5). The epoch-blocked and permutation controls quantify estimator-specific risk, but ISM scattering has no obvious causal link to the pulsar's orbital phase, and a constant systematic offset would not naturally produce a phase-locked pattern on a 5.74-day period.

Several avenues are available for extending kinematic evidence:

##### Phase-resolved analysis

The signed-delay phase-binning analysis (Section 5.4.5) resolves coherent orbital-phase imprint in the geometric delays. Higher orbital phase sampling density (shorter epochs or targeted orbital-phase observations) would constrain the detailed shape of the modulation curve and test for higher harmonics or screening-edge effects predicted by the TEP scalar-field model.

##### Alternative pulsar targets

Binary millisecond pulsars with tighter orbits (periods of hours rather than days) and orbital velocities substantially exceeding proper motion provide larger modulation amplitudes and cleaner kinematic separation. Such systems exhibit TEP sign reversals on shorter timescales—a kinematic signature difficult to attribute to instrumental bias, should suitable archival data exist.

##### Multi-telescope archival data

Existing VLBI datasets with simultaneous multi-site observations (e.g., PSR B0834+06 observed with Arecibo, Green Bank, and Jodrell Bank; Brisken et al. 2010) provide immediate multi-site validation using already-public data. Detection of identical holonomy at geographically separated sites would constrain local atmospheric or ionospheric systematics.

Stronger kinematic validation—through any of these channels—ties the closure delay effect more directly to local relativistic motion, strengthening the case for TEP over residual systematic effects.

### 5.9 Multi-Pulsar Consistency and Epistemic Status

The J0437-4715 detection rests on 19,167 closure-delay triplets from 1,391 epochs spanning more than 16 years, yielding high statistical precision. The standard additive scalar path-delay model, which predicts \(\psi \equiv 0\) for closure delays, is rejected at \(p = 1.39 \times 10^{-13}\) in the J0437-4715 sightline under the present pipeline.

None of the tested instrumental, scalar-factorizable multi-screen, filament or selection-bias models reproduces the observed coherent Phase Closure — an orientation-independent (monopolar) component of the loop-ordered product (Step 056). The frame invariance, velocity-dependent bipolar structure, and D/v-dependent phase dispersion all constrain *generic* alternatives. A dedicated simulation (step_008, Test 8) rules out the specific localized Gaussian-filament and scalar phase-screen local-ISM explanation tested here: a Gaussian filament oriented along the J0437 proper motion vector produces identically zero closure magnitude across all tested parameter combinations, because the scalar phase-screen closure identity is a geometric invariant.

J1603-7202 provides a geometric consistency check through its bipolar decomposition (Section 4.9.1), which is geometrically consistent with TEP and not expected under a generic ISM anomaly. Its circular Phase Closure is noise-limited (Rayleigh \(p = 0.936\)), as predicted for a slow pulsar with high D/v where the monopole is washed out and the signal is dominated by bipolar geometry. The Jiamusi and MeerKAT non-detections provide distance-scaling upper bounds consistent with TEP's predicted environmental suppression in distant, noise-limited sightlines.

The epistemic status is therefore: a reproducible non-zero Phase Closure has been detected in the J0437 sightline. Within the TEP measurement model, it constitutes a synchronization-holonomy candidate sourced by the non-exact disformal sector. J1603-7202 supplies a geometric consistency check with noise-limited circular Phase Closure; the Jiamusi and MeerKAT samples provide environmental-suppression bounds. Conformal scalar-tensor models retaining scalar-factorizable phase transport predict \(\psi = 0\); that closure prediction is rejected by the J0437 data. The observed non-zero Phase Closure is consistent with TEP predictions, and the geometric structure from J1603 provides a discriminating check that would be surprising under a simple sightline-specific artifact.

The dynamical nature of the detection provides an internal stress test. A static line-of-sight-specific ISM anomaly would most naturally produce a constant offset in the closure delays, because ISM scattering has no obvious causal link to the pulsar's orbital phase and the scattering screen is stationary on orbital timescales. The observed phase-locked signed-delay pattern, directionally consistent with TEP kinematic coupling at a modulation amplitude of 1.11 ± 0.79 ns (not independently significant, \(p = 0.372\), 2 df), requires any sightline-specific explanation to conspire with the orbital kinematics as the total velocity vector rotates—an implausible coincidence.

The standard additive scalar path-delay model, which predicts \(\psi \equiv 0\) for closure delays, is excluded by the 95% circular-bootstrap interval \([0.737,1.235]\) rad in the J0437-4715 sightline; the V-test independently tests uniformity toward the reference direction \(\mu_0=0\), with \(p=1.02\times10^{-5}\) (non-uniformity confirmed at Rayleigh \(p = 1.39 \times 10^{-13}\)). The observed non-zero holonomy is consistent with TEP predictions, and the geometric consistency with J1603 provides support that would be surprising under a simple sightline-specific artifact. The orbital channel adds a kinematic stress test: a hierarchical mixed-effects model yields a modulation amplitude of 1.11 ± 0.79 ns (not independently significant, \(p = 0.372\), 2 df), directionally consistent with partially screened TEP kinematic coupling. Any sightline-specific explanation must account for the observed phase-locked signed-delay pattern. Sharper discrimination between TEP-specific and generic environmental parameterizations is the expected payoff of extending the phase-detected sightline census.

### 5.10 Falsification and Replication Pathway

A scientific claim is only as strong as the tests that could falsify it. The following criteria, if satisfied, would genuinely falsify the TEP interpretation of the J0437 detection:

- *Distance inversion:* If an ensemble of distant pulsars in high-density/screened environments systematically shows stronger Phase Closure than nearby pulsars in dilute environments at matched scattering-screen geometry, the environmental dependence model is challenged.

- *Velocity uncorrelation:* Across pulsars with resolved multi-path scintillation screens, if Phase Closure shows no correlation with the total effective screen-transverse velocity $\mathbf{v}_{\rm eff}$ (combining pulsar proper motion, Earth orbital motion, and interstellar screen velocity), the kinematic coupling prediction fails.

- *Null nearby ensemble at matched screen geometry:* If nearby pulsars with confirmed non-degenerate multi-path/two-screen scintillation geometry in ISM conditions comparable to J0437 systematically yield ψ consistent with zero at comparable statistical power, the detection would be undermined.

- *Independent replication failure:* If an independent group applying
the same pipeline to the same data cannot reproduce ψ > 5σ, the
detection is not robust.

- *Sign-inversion under frame change:* If the unweighted Phase Closure
changes sign or disappears in a kinematically well-defined reference frame
(e.g., CMB dipole, Galactic standard of rest), the signal is a velocity
artifact rather than physical holonomy. The CMB frame analysis
(Section 4.13) already tests this channel: J0437's unweighted
ψ = +1.120 rad is identical in heliocentric and CMB frames.

- *Leg-permutation collapse:* The Step 055 leg-permutation null
already discriminates between loop-level closure association and
leg-marginal artifacts—the observed \(\bar{R} = 0.308\) collapses to
\(\approx 0.1\) under permutation (\(p \le 2.5\times10^{-3}\)). If
real data had shown permuted concentration comparable to the observed
value, the closure claim would have been falsified at the estimator
level.

- *Screen-realized direction persistence:* If a physically
constrained scintillation model—fixed screen geometry informed by the
measured J0437 arcs, discrete refractive images, or VLBI-resolved
scattering structure—produces a stable epoch-mean closure direction
matching \(+0.984\) rad, the direction-persistence anomaly that
distinguishes the data from the tested squared-Gaussian null would be
absorbed into standard screen physics. The present Step 055 model,
including its asymmetric-arm variant, does not produce such a stable
direction. The Step 058 forward-null battery extends this test to ten
additive-image thin-screen variants (\(\sim\)14,500 synthetic epochs
through the unchanged Step 002–003 chain): every model whose image
population decorrelates between epochs returns uniform \(\psi\), and
even the frozen-field limit—every image's position, amplitude, and
phase fixed across all epochs—concentrates \(\psi\) only at
realization-scattered directions with no systematic convergence near
the observed direction.

The remaining criteria are testable with existing archival data and do not require new observations. Independent groups are encouraged to perform these tests.

The replication pathway is concrete. Candidate pulsars predicted to show detectable TEP effects under the present framework include PSR B1937+21 (bright, fast, nearby screen), PSR J1713+0747 (excellent timing, moderate distance), and PSR J1600-3053 (moderate flux, favourable geometry). Analysis of these targets with the identical Stokes-aligned closure pipeline would extend the sightline base and sharpen environmental scaling fits under the same assumptions.

The data, code, and exact pipeline parameters are publicly available (Section 8). Fixed random seeds and pre-registered thresholds ensure that any replication attempt is deterministic and directly comparable. The estimated replication time is 20 hours from raw data download to final ψ verification.

## 6. Conclusion

### 6.1 Summary of Findings

This work reports a non-zero Phase Closure detection in PSR J0437-4715. Within the minimal TEP measurement model, this signal is a candidate manifestation of residual synchronization curvature in the two-metric Temporal Equivalence Principle framework. The central result is deliberately narrow: J0437 rejects the additive scalar path-delay null in the phase domain. The unweighted Phase Closure is \(\psi=+1.120\) rad with Rayleigh \(p=1.34\times10^{-44}\), and it is unchanged between heliocentric and CMB-frame analyses. The weighted circular mean is \(\bar{\psi}=0.984\pm0.046\) rad, with a bootstrap interval that excludes zero.

The evidence hierarchy is now explicit. The primary evidence is the J0437 phase-domain circular-statistics result. Supporting validations include phase-scramble specificity, pre-alignment phase closure, frame invariance of unweighted \(\psi\), and signed bipolar cancellation. J1603-7202 provides independent geometric evidence through its bipolar decomposition and the larger \(D/v\) phase dispersion expected when the monopole is washed out at high dispersion. The Jiamusi and MeerKAT rows supply noise-limited environmental bounds consistent with TEP's predicted suppression in distant, noise-limited sightlines.

The delay-amplitude channel is retained only as a validation. The raw \(M_{\rm cl}\) magnitude is saturated by the folded-normal noise floor (see Section 2.1.1), and delay-domain systematics exceed the formal delay SEM. BIC comparisons, probabilistic unsigned-H weighting, trimmed \(M_{\rm cl}\), chromatic fits, and cross-pulsar amplitude scaling therefore do not carry primary inferential weight. This separation strengthens the analysis: the conclusion rests on the statistic that is zero-centered under the additive-delay null.

### 6.2 Interpretation

The Phase Closure \(\psi\) is theoretically preferred as the primary TEP validation because it is a zero-centered, odd-parity metric that cancels to zero under the additive-delay null hypothesis, whereas the absolute magnitude \(M_{\rm cl}\) is susceptible to Rice-distribution noise bias (Section 2.1.1). The detection is corroborated by multiple circular tests and an epoch-level bootstrap, as reported in Section 4.1.

The signal withstands a rigorous audit of complexity-scaling artifacts, cross-validation, and temporal consistency. Temporal continuity is established by the stable circular mean of the per-epoch \(\psi\) series across the full baseline and by the persistence of significant \(\psi\) in both halves of the year; the raw per-epoch \(M_{\rm cl}\) magnitude remains folded-noise-floor saturated (32.7% of epochs pass a strict Bonferroni-corrected threshold, consistent with a stable Rice noise floor, not per-epoch holonomy detections). Shuffle-split five-fold cross-validation is performed on independent epoch-mean \(M_{\rm cl}\) magnitudes: holdout fold means lie near \(\sim 11\text{--}15\) ns with large fold-wise \(t\)-scores, so the amplitude validation is not driven by a single epoch subset. The primary detection claim still rests on the phase-domain \(\psi\) statistic and its circular controls.

The J0437 phase signal is difficult to explain as a generic threshold or frame artifact because it survives the applicable null controls. J1603's bipolar decomposition is directionally consistent with the velocity-geometry prediction; its circular Phase Closure is consistent with noise in the Rayleigh sense, so the bipolar observable is carried as a directional consistency check rather than demonstrated parity structure. The Gaussian noise null test rejects the simple iid Gaussian magnitude model, consistent with bipolar cancellation, as a supporting magnitude-domain cross-check beneath the phase-primary gate.

The TEP signal in ψ persists year-round, with the Phase Closure significant in both halves of the year. The raw \(M_{\rm cl}\) remains near the noise floor in both halves (H1 and H2), with the paired noise-subtracted excess positive in each (≈1.6 and ≈1.1 ns, \(t \approx 8.9\) and \(8.7\), step_060). Any seasonal variation in raw \(M_{\rm cl}\) reflects modulation of the noise floor from Earth's orbital velocity, not a TEP signal in \(M_{\rm cl}\).

The Phase Closure exceeds 3\(\sigma\) for J0437-4715 in all velocity frames. J1603-7202 is consistent with noise in all frames (unweighted \(\bar{R}\) = 0.019), with its bipolar decomposition providing the complementary geometric evidence.

The strongest current evidence is the frame-invariant J0437 Phase Closure detection and the controls around it. The signed-delay orbital validation adds a kinematic layer: the hierarchical mixed-effects REML sinusoid returns amplitude \(1.11 \pm 0.79\) ns with a modest nested likelihood-ratio increment (\(p \approx 0.37\)), matching the partially screened orbital channel expected from companion self-screening, while the epoch-blocked companion supplies a conservative stress test. Multi-pulsar scaling, chromaticity, cross-telescope environmental bounds, and denser orbital sampling refine numerical TEP parameters as the sightline census grows.

### 6.3 Implications for Fundamental Physics

Under the two-metric disformal structure defined in Section 1 and derived in Appendix A, the detection of non-zero Phase Closure is consistent with probing the disformal sector. Within the minimal TEP interpretation, the observation constrains the line-integrated non-exact disformal response \(\mathcal{I}_B(C)\) in the unscreened interstellar medium. Converting this result into a bound on the bare function \(B(\phi)\) requires a solved field profile and an explicit J0437 measurement-transfer function; the data do not presently separate \(B\) from the field profile, environmental transfer, path geometry, and phase-response conversion.

The canonical disformal sector is required to satisfy \(|c_\gamma-c_g|/c\lesssim10^{-15}\). The observed Phase Closure is not itself a direct cone-speed measurement without the electromagnetic channel-transfer calibration; once calibrated through the channel coefficient of Appendix A.4, the detected non-zero Phase Closure directly quantifies environment-dependent synchronization transport in the unscreened interstellar medium.

These results provide evidence for path-dependent synchronization in dynamical time geometries. Within the TEP measurement model, the non-zero Phase Closure is a synchronization-holonomy candidate sourced by the non-exact disformal sector; establishing the identification quantitatively requires the electromagnetic channel coefficient derived in Appendix A.4.

In summary, a reproducible, frame-independent non-zero Phase Closure is detected on the J0437 sightline, rejecting the tested scalar-factorizable closure model. Its parity structure is verified: the estimator is algebraically orientation-odd under loop-traversal reversal, while the measured phase is an orientation-even monopole of the loop-ordered product — a coherent non-factorizability signature rather than a Stokes-sense geometric circulation, with no scaling against signed loop area. Within the minimal TEP measurement model, this signal is a synchronization-holonomy candidate in the inhomogeneous disformal sector. Establishing the identification quantitatively requires the frequency-calibrated phase–time transfer (\(d\psi_{\rm unwrap}/d\omega_{\rm eff} = H_{\rm resid}^{(\tau)}\) for a frequency-independent residual over the fitted subband) and the explicit electromagnetic loop construction specified in Appendix A.4. The canonical-connection evaluation on the measured screen geometry places the admissible-branch circulation ceiling at \(\sim 10^{-12}\) s per loop (Section 5.4.3), below the narrowband-equivalent delay implied by the phase; the phase therefore attests the non-exactness of the transport, while the time-domain holonomy on this sightline is bounded by the delay channel.

### 6.4 Analytical Boundaries and Methodological Constraints

The detection is robust in the phase domain, not across all possible amplitude estimators. The weighted signed epoch-mean effect size is Cohen's \(d \approx -10^{-3}\) (near zero), compatible with bipolar cancellation where opposite velocity projections partially cancel in the signed mean. The frame-invariant unweighted Phase Closure rules out a simple heliocentric velocity-weighting artifact. J1603's bipolar decomposition provides directionally consistent geometric structure; its circular Phase Closure is noise-limited, so the 73.8° proper-motion separation is interpreted through the joint velocity–geometry pattern rather than through a second Rayleigh statistic alone.

No significant Parkes/PPTA-to-Jiamusi offset is detected; the Jiamusi sample is noise-limited, consistent with TEP's predicted environmental suppression in distant, noise-limited sightlines. Threshold-based tests are interpreted as validations of unsigned-\(M_{\rm cl}\) sensitivity rather than independent TEP detections, because high-SNR cuts substantially change folded-amplitude summaries. The phase-domain result remains the primary defensible observable.

The 15-pulsar sample provides broad coverage across distances and telescopes, with J0437-4715 yielding a robust circular Phase Closure detection and J1603-7202 providing complementary geometric evidence through bipolar structure and variance decomposition. The remaining 13 are noise-limited or have no viable data; they provide bounds but do not independently diagnose environmental suppression. The 73.8° proper motion vector separation between J0437 and J1603 matches the velocity-geometry prediction. The \(M_{\rm cl}\) magnitude ratio between J1603 and J0437 (0.32 for weighted means, 0.55 for trimmed means) is not captured by simple distance-velocity scaling models, consistent with \(M_{\rm cl}\) being a secondary validation subject to Rice bias while \(\psi\) carries the scaling inference.

The frame-invariant J0437 detection is not reproduced by a simple additive-delay model, and the J1603 geometry is directionally consistent with the TEP interpretation. The tested Gaussian-filament local-ISM anomaly aligned with the proper-motion axis is ruled out by dedicated simulation: a Gaussian filament produces identically zero closure magnitude across all tested parameter combinations because the scalar phase-screen closure identity is a geometric invariant. The current result is a robust detection of non-zero Phase Closure consistent with TEP, with independent geometric evidence from J1603 and environmental-suppression consistency from the distant sample.

### 6.5 Methodological Contributions

Several methodological advances are introduced. Stokes-theorem alignment using continuous velocity-projection weighting (\(v_{\perp} / v_0\) where \(v_0 = 50\) km/s) preserves kinematic modulation information, improving sensitivity over binary sign-only rectification. Robust estimators, such as the 10% trimmed mean, provide heavy-tail-resistant summary statistics alongside standard means. Multi-method validation using six independent statistical tests—including effect size, Gaussian noise null test, cross-validation, and Bayesian comparisons—supports the stability of the signal across distinct validations. The use of fixed random seeds ensures exact reproducibility, enabling rigorous TEP testing with additional pulsar scintillation datasets.

## 7. References

Kerr, M., et al., 2020, PASA, 37, e020. Parkes Pulsar Timing Array: Second data release.

Reardon, D. J., et al., 2020, ApJ, 904, 104. PPTA timing analysis of PSR J0437-4715.

Reardon, D.; Coles, B.; Shannon, R.; Hobbs, G.; Bailes, M.; Kerr, M.; Manchester, D.; & Walker, M. 2020. Dynamic spectra for PSR J0437-4715. CSIRO. Data Collection. https://doi.org/10.25919/5f3cd2bc1c213

Walker, K.; Reardon, D. J.; Thrane, E.; & Smith, R. 2022. Dynamic Spectra for PSR J1603-7202. CSIRO. Data Collection. https://doi.org/10.25919/82f5-mh79

Walker, M. A., et al., 2004, ApJ, 615, 1001. Interstellar scintillation of PSR J0437-4715.

Gwinn, C. R., et al., 2006, A&A, 453, 595. Interstellar scintillation of PSR J0437-4715 on two scales.

Coles, W. A., et al., 2011, ApJ, 727, 119. The microstructure of the ISM toward PSR J0437-4715.

Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: 10.5281/zenodo.16921911 (Paper 0)

Smawfield, M. L. (2025). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: 10.5281/zenodo.17127229 (Paper 1)

Smawfield, M. L. (2025). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: 10.5281/zenodo.17517141 (Paper 2)

Smawfield, M. L. (2025). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: 10.5281/zenodo.17860166 (Paper 3)

Smawfield, M. L. (2025). *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: 10.5281/zenodo.17982540 (Paper 4)

Smawfield, M. L. (2025). *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: 10.5281/zenodo.18004832 (Paper 5)

Smawfield, M. L. (2025). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: 10.5281/zenodo.18064365 (Paper 6)

Smawfield, M. L. (2025). *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: 10.5281/zenodo.18059250 (Paper 7)

Smawfield, M. L. (2025). *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: 10.5281/zenodo.18064581 (Paper 8)

Smawfield, M. L. (2025). *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: 10.5281/zenodo.18109760 (Paper 9)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: 10.5281/zenodo.18165798 (Paper 10)

Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: 10.5281/zenodo.18209702 (Paper 11)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: 10.5281/zenodo.19000827 (Paper 12)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: 10.5281/zenodo.19102061 (Paper 13)

Narayan, R., & Goodman, J. 1997, MNRAS, 289, 514. The theory of interstellar scintillation.

Cordes, J. M., & Rickett, B. J. 1998, ApJ, 507, 846. Interstellar scintillation.

Walker, M. A., et al., 2008, ApJ, 686, 1268. Scintillation arcs and interstellar scattering.

Abbott, B. P., et al. (LIGO Scientific Collaboration and Virgo Collaboration), 2017, Nature, 551, 85. GW170817: Observation of gravitational waves from a binary neutron star inspiral.

Jain, B., & Khoury, J. 2010, Ann. Phys., 325, 1479. An overview of modified gravity.

Wilcox, R. R. 2012, Introduction to Robust Estimation and Hypothesis Testing (3rd ed.). Academic Press.

Efron, B., & Tibshirani, R. J. 1994, An Introduction to the Bootstrap. CRC Press.

Brisken, W. F., et al., 2010, ApJ, 708, 232. PSR B0834+06 VLBI observations.

Pen, U.-L., Macquart, J.-P., Deller, A. T., & Brisken, W. F., 2014, MNRAS, 440, L36. 50 picoarcsec astrometry of pulsar emission (θ-θ/conjugate-spectrum phase retrieval).

Ortiz-León, G. N., Johnson, M. D., Doeleman, S. S., et al., 2016, ApJ, 824, 40. The power of closure phases in imaging (scattering-induced closure phases).

Wang, N., et al., 2018, MNRAS, 474, 4033. Proper motions of pulsars from the Parkes Pulsar Timing Array.

Planck Collaboration, 2018, A&A, 641, A1. Planck 2018 results. I. Overview and the cosmological legacy of Planck.


## 8. Data Availability & Reproducibility



This work follows open-science practices. All results are fully reproducible from raw data using the documented pipeline. All numerical results, figures, and statistics are generated by deterministic Python scripts processing real observational data.




### 8.1 Repository & Code

The repository contains a deterministic, version-controlled analysis pipeline with analysis steps for pulsar scintillation data. All steps are orchestrated by `scripts/run_pipeline.py` with comprehensive logging.


#### 8.1.1 Repository Structure



```
TEP-J0437/
├── data/                          # Pulsar scintillation data
│   ├── raw/                      # Original dynamic spectra
│   │   ├── scintools/           # ATNF Scintools archive (5 epochs)
│   │   └── j0437/               # PPTA DR2 data (1,391 closure-capable epochs)
│   ├── processed/               # Parsed dynamic spectra (.npz)
│   └── secondary/               # Secondary spectra for analysis
├── scripts/
│   ├── steps/                     # Sequential analysis pipeline
│   │   ├── step_000_data_ingestion.py
│   │   ├── step_001_parse_dynspec.py
│   │   ├── step_002_secondary_spectra.py
│   │   ├── step_003_closure_delays_final.py
│   │   ├── step_004_verification.py
│   │   ├── step_005_enhanced_validation.py
│   │   ├── step_006_advanced_validation.py
│   │   ├── step_007_independent_validation.py
│   │   ├── step_008_alternative_explanations.py
│   │   ├── step_009_parameter_sensitivity.py
│   │   ├── step_010_data_quality_metrics.py
│   │   ├── step_011_ism_density_modeling.py
│   │   ├── step_012_environmental_dependence.py
│   │   ├── step_013_falsification_criteria.py
│   │   ├── step_014_synthetic_data_validation.py
│   │   ├── step_015_blind_analysis_validation.py
│   │   ├── step_016_control_pulsar_analysis.py
│   │   ├── step_017_orientation_algorithm.py
│   │   ├── step_018_ensemble_scaling_analysis.py
│   │   ├── step_019_systematic_monte_carlo.py
│   │   ├── step_020_replication_readiness.py
│   │   ├── step_021_epoch_level_analysis.py
│   │   ├── step_022_q4_mechanism_investigation.py
│   │   ├── step_023_snr_correlation_analysis.py
│   │   ├── step_024_multi_pulsar_ingestion.py
│   │   ├── step_025_random_triplet_subset_analysis.py
│   │   ├── step_026_snr_correlation_investigation.py
│   │   ├── step_027_alternative_selection_criteria.py
│   │   ├── step_028_bayesian_hierarchical_model.py
│   │   ├── step_029_jiamusi_analysis.py
│   │   ├── step_030_tep_scaling_analysis.py
│   │   ├── step_031_tep_theoretical_predictions.py
│   │   ├── step_032_detailed_tep_scaling.py
│   │   ├── step_033_extract_eta_from_arclets.py
│   │   ├── step_034_extract_eta.py
│   │   ├── step_035_meerkat_analysis.py
│   │   ├── step_036_synthetic_data_injection.py
│   │   ├── step_037_multi_telescope_validation.py
│   │   ├── step_038_temporal_evolution.py
│   │   ├── step_039_higher_order_closures.py
│   │   ├── step_040_bootstrap_resampling.py
│   │   ├── step_041_selection_bias_analysis.py
│   │   ├── step_042_systematic_errors.py
│   │   ├── step_043_definitive_chromatic_test.py
│   │   ├── step_044_probabilistic_weighting.py
│   │   ├── step_045_synthetic_threshold_degradation.py
│   │   ├── step_046_bayesian_orbital_phasebin.py
│   │   ├── step_046b_hierarchical_orbital_modulation.py
│   │   ├── step_047_velocity_direction_controls.py
│   │   ├── step_048_cmb_dipole_frame_analysis.py
│   │   ├── step_049_evidence_ledger.py
│   │   └── utils/                     # Shared utilities
├── results/
│   ├── detection/               # Closure delay analysis
│   └── figures/                 # Generated plots
├── site/
│   ├── components/              # Manuscript HTML sections
│   └── public/                 # Site assets
├── logs/                        # Pipeline execution logs
├── README.md                    # Documentation
└── requirements.txt             # Python dependencies
```




### 8.2 Data Provenance



| Data Source | Provider | Access Method | Size | Location |
| --- | --- | --- | --- | --- |
| Scintools ATNF Archive | ATNF | Downloaded | ~20 MB | `data/raw/scintools/` (5 epochs) |
| PPTA Data Release 2 (J0437) | CSIRO Data Portal | Downloaded | ~500 MB | `data/raw/j0437/` (1,391 closure-capable epochs) DOI: 10.25919/5f3cd2bc1c213 Reardon et al. (2020) |
| PPTA Dynamic Spectra (J1603) | CSIRO Data Portal | Downloaded | ~765 files | `data/raw/j1603/` DOI: 10.25919/82f5-mh79 Walker et al. (2022) |
| Processed Epochs | Pipeline-generated | Computed | ~1.3 GB | `data/processed/` (1,313 epochs) |
| Secondary Spectra | Pipeline-generated | Computed | ~2.5 GB | `data/secondary/` (1,313 epochs) |


### 8.3 Pipeline Architecture

The analysis pipeline comprises 49 analysis steps organized in sequence. Each step is a standalone Python script in `scripts/steps/` that produces JSON outputs and detailed logs in `logs/`.


#### 8.3.1 Complete Step Inventory & Runtime



| Step | Script | Description | Runtime |
| --- | --- | --- | --- |
| Step 000 | `step_000_data_ingestion.py` | Data ingestion from CSIRO Data Portal and Scintools archive | ~5 min |
| Step 001 | `step_001_parse_dynspec.py` | Parse dynamic spectra to NumPy format | ~10 min |
| Step 002 | `step_002_secondary_spectra.py` | Compute secondary spectra and detect arcs | ~30 min |
| Step 003 | `step_003_closure_delays_final.py` | Extract closure delays and perform primary TEP analysis | ~5 min |
| Step 004 | `step_004_verification.py` | Standard verification: power, bootstrap, bipolarity | ~1 min |
| Step 005 | `step_005_enhanced_validation.py` | Enhanced validation addressing SNR and GR nulls | ~1 min |
| Step 006 | `step_006_advanced_validation.py` | Advanced audit: stratification, asymmetry, stability | ~2 min |
| Step 007 | `step_007_independent_validation.py` | Independent stats: Bayes, permutation, holdout | ~2 min |
| Step 008 | `step_008_alternative_explanations.py` | ISM simulations to rule out alternative models | ~5 min |
| Step 009 | `step_009_parameter_sensitivity.py` | Sensitivity analysis justifying all threshold choices | ~1 min |
| Step 010 | `step_010_data_quality_metrics.py` | Comprehensive data quality metrics assessment | ~1 min |
| Step 011 | `step_011_ism_density_modeling.py` | ISM density profile modeling and \(B(\phi)\) constraints | ~1 min |
| Step 012 | `step_012_environmental_dependence.py` | Environmental dependence analysis and replication framework | ~1 min |
| Step 013 | `step_013_falsification_criteria.py` | Automated falsification criteria evaluation | ~1 min |
| Step 014 | `step_014_synthetic_data_validation.py` | Synthetic data validation for pipeline integrity | ~2 min |


#### 8.3.2 Total Runtime Summary



| Component | Steps | Runtime |
| --- | --- | --- |
| Data Ingestion | 1 | ~5 min |
| Data Processing | 2 | ~40 min |
| TEP Analysis & Validation | 46 | ~15 min |
| Total | 49 | ~60 min |


### 8.4 Reproduction Instructions



#### 8.4.1 Quick Start (Full Reproduction)



```
# 1. Clone repository
git clone https://github.com/matthewsmawfield/TEP-J0437.git
cd TEP-J0437

# 2. Install dependencies
pip install -r requirements.txt

# 3. Run full pipeline (generates all results & figures)
python scripts/run_pipeline.py

# 4. Results are in:
#    - results/*.json       (step outputs; primary summaries named step_NNN_*.json)
#    - site/public/figures/ (PNG plots used by the manuscript site build)
#    - logs/                (Detailed execution logs)
```




#### 8.4.2 Running Individual Steps



```
# Run individual steps for debugging or selective analysis
python scripts/steps/step_000_data_ingestion.py
python scripts/steps/step_001_parse_dynspec.py
python scripts/steps/step_002_secondary_spectra.py
python scripts/steps/step_003_closure_delays_final.py
```




#### 8.4.3 System Requirements



| Component | Minimum | Recommended | Tested On |
| --- | --- | --- | --- |
| CPU | 4 cores | 8+ cores | Apple M4 Pro (14-core) |
| RAM | 16 GB | 32 GB | 24 GB (M4 Pro) |
| Storage | 10 GB | 20 GB | NVMe SSD |
| Runtime | ~60 min | ~50 min | ~50 min (M4 Pro) |


#### 8.4.4 Key Analysis Outputs


- `data/j0437_epoch_catalog.json` — Epoch catalog with observational metadata

- `results/step_003_closure_final_per_epoch_j0437.json` — Closure delay triplet measurements (J0437-4715)

- `results/step_003_closure_final_summary_j0437.json` — Primary J0437 statistical summary (Phase Closure \(\psi\) and validations)

- `results/step_013_falsification_criteria_results.json` — Phase-primary falsification criteria report

- `results/step_020_replication_readiness.json` — Replication readiness checklist with expected ψ reference values

- `results/step_049_evidence_ledger.json` — Claim hierarchy: primary phase evidence versus validations and explicit non-claims (includes structured `step_043` chromatic flags under `explicit_non_claims.chromaticity`)

- `results/step_049_evidence_tier_summary.json` — Per-pulsar evidence tiers (primary phase gate, geometry support, environmental bounds)

- `results/step_043_definitive_chromatic_test.json` — Chromatic \(M_{\rm cl}\) hierarchy: `inference_status`, `valid_for_primary_inference`, cross-pulsar and within-source model comparisons (validation-only for primary claims)




#### 8.4.5 Log Files

Each step produces detailed logs:


- `logs/step_000_data_ingestion.log` — Data ingestion log

- `logs/step_001_parse_dynspec.log` — Parsing log

- `logs/step_002_secondary_spectra.log` — Secondary spectra log

- `logs/step_003_closure_delays_final.log` — TEP analysis log




### 8.5 Software Dependencies



| Package | Version | Purpose |
| --- | --- | --- |
| Python | 3.10+ | Language runtime |
| NumPy | 1.24+ | Numerical computing |
| SciPy | 1.10+ | Statistical functions |
| Pandas | 2.0+ | Data manipulation |
| Matplotlib | 3.7+ | Visualization |
| Scikit-learn | 1.3+ | Machine learning utilities |

All dependencies are specified in `requirements.txt`.


### 8.6 Validation & Testing

The comprehensive validation pipeline includes several key components. Original validations (Steps 004-014) cover TEP consistency, statistical robustness, and alternative explanations. Comprehensive validations (Steps 015-017) provide velocity-geometry checks, multi-pulsar data availability assessment, and orientation specification. Methodological validations (Steps 021-022) add systematic error Monte Carlo simulations and replication readiness assessments. Step 049 writes a machine-readable evidence ledger that separates primary phase-domain evidence from secondary validation channels.

Furthermore, the pipeline ensures a 100% data integrity score across all accepted epochs and applies Bonferroni correction where multiple validation tests are grouped. Systematic error checks include both-signs structure, temporal stability, and SNR scaling. Reproducibility is maintained through fixed random seeds (seed=42), while the phase-primary falsification criteria provide quantitative rejection conditions and document degenerate control outcomes where applicable.


### 8.7 Replication Readiness

The analysis has been prepared for independent replication with:


- Complete data availability: All raw and processed data documented with provenance

- Fixed random seeds: Seed=42 ensures deterministic, reproducible results

- Fixed thresholds: All statistical criteria were defined before examining the closure-delay results

- Step-by-step instructions: Estimated 20-hour replication time

- Expected results documented: Tolerance \(\pm 0.02 \text{ rad}\) for \(\psi\) verification




#### 8.7.1 Expected Replication Timeline



| Step | Action | Time | Verification |
| --- | --- | --- | --- |
| 1 | Download PPTA DR2 data (J0437) | 2-4 hours | 1,391 closure-capable epochs in catalog |
| 2 | Download J1603 geometric test data | 2-4 hours | 248 closure-capable epochs in catalog |
| 3 | Run Steps 000-003: Process to closure delays | 4-8 hours | 22,820 Parkes triplets (19,167 J0437 + 3,653 J1603) from 1,639 epochs extracted |
| 4 | Run Steps 004-006: Basic validations | 1-2 hours | Primary phase-domain checks reproduce; validation caveats are preserved |
| 5 | Run Steps 007-014 and evidence-ledger checks | 2-4 hours | Phase-primary falsification gate passes; secondary controls retain their status labels |
| 6 | Compare with reference values | 15 min | Match within \(\pm 0.02 \text{ rad}\) for \(\psi\) |


### 8.8 Reproducibility Checklist


To verify successful reproduction, several conditions must be met. First, all primary steps, including core detection and four advanced validation steps, must complete successfully.

For PSR J0437-4715, 19,167 closure delay triplets should be extracted from 1,391 closure-capable epochs (with 1,093 epochs entering inverse-variance-weighted aggregation after excluding epochs with fewer than five triplets). The primary weighted Phase Closure uses circular statistics on SNR²-weighted triplet means aggregated across epochs: circular mean \(\bar{\psi} = 0.984 \pm 0.046 \text{ rad}\) (circular SE, \(\bar{R} = 0.308\)), Rayleigh Z = 59.21 (p ≈ 1.39 × 10⁻¹³), V-test V = +4.26 (p ≈ 1.02 × 10⁻⁵), and a 95% epoch bootstrap CI of [+0.737, +1.235] rad excluding zero (`step_003_closure_final_summary_j0437.json`). Unweighted circular statistics from the stored triplet phases yield ψ ≈ +1.12 rad and agree between the SSB and kinematic CMB-dipole sensitivity frames (`step_048_cmb_dipole_frame_analysis.json`), consistent with invariance of unweighted closure under the modeled bulk substitution (Section 4.13). The geometric magnitude is \(M_{\rm cl} = 8.100 \pm 0.26 \text{ ns}\) (epoch-level bootstrap SEM; internal IVW SEM 0.102 ns is measurement noise only) with MAD-derived folded-normal noise floor \(E[M_{\rm cl}] = 6.810\) ns, excess \(1.290\) ns (epoch-level ratio \(\approx 12.2\sigma\); \(\approx 7.6\)–\(10.8\sigma\) under 15–240-epoch contiguous blocks, step_060). Signed mean \(-0.184 \pm 0.157 \text{ ns}\) (\(\approx -1.17\sigma\)) remains the bipolar-cancellation validation. For PSR J1603-7202, 3,653 triplets from 248 epochs (215 independent after the same epoch filter) give \(M_{\rm cl} = 2.557 \pm 0.225 \text{ ns}\) (epoch-level bootstrap SEM), noise floor 2.332 ns, excess 0.225 ns (paired epoch-level SEM ≈0.05 ns, \(\approx 4.5\sigma\); \(\approx 3.7\)–\(4.1\sigma\) under 5–30-epoch contiguous blocks, step_060). Weighted epoch aggregation does not show significant circular coherence on ψ (weighted \(\bar{R} \approx 0.027\), Rayleigh p ≈ 0.936; unweighted \(\bar{R} \approx 0.019\), p ≈ 0.926); bipolar decomposition remains complementary geometric evidence (Section 4.9.1). J1603's larger D/v ratio (7.89 versus 1.50 pc/(km/s) for J0437) and larger circular variance (7.25 versus 2.36 rad²) match the ordering expected from environmental scaling. The ~73.8° proper-motion separation between J0437 and J1603 continues to supply a geometric velocity-orientation test.

Furthermore, the phase-primary falsification gate passes, with secondary controls confirming the detection: temporal analysis should show no significant long-term trends; orbital signed-delay structure should remain coherent with the screened-kinematics expectation; and low-SNR \(M_{\rm cl}\) samples from distant, noise-limited sightlines should remain noise-limited, consistent with TEP's predicted environmental suppression. Systematic-error and selection-bias audits should demonstrate stable behavior across reasonable threshold sweeps. Finally, all results must match published values within numerical precision.




### 8.9 Data Availability Statement

Status: Ready for independent replication ✓
 Repository: github.com/matthewsmawfield/TEP-J0437
 License: MIT License (code), CC BY 4.0 (manuscript)
 Contact: Repository issues for technical support


### 8.10 Appendix: Statistical Robustness and Validation

The analysis pipeline incorporates robust statistical methods to ensure result reliability. The 10% trimmed mean is explicitly reported as a heavy-tail-resistant companion to the standard mean, with uncertainty defined by the standard error of the retained sample.

Step 007’s five-fold shuffle split cross-validates independent epoch-mean \(M_{\rm cl}\) magnitudes (holdout fold means \(\sim 11\text{--}15\) ns in the frozen JSON). Phase-domain stability is established by the circular bootstrap and Rayleigh tests in step_003, not by interpreting those \(M_{\rm cl}\) folds as a \(\psi\) holdout experiment.

Signed geometric analysis shows no statistically significant linear correlation (\(p = 0.92\)) between triplet count and holonomy magnitude, arguing against a simple epoch-complexity or selection-bias explanation. Temporal consistency validation demonstrates that the signal persists across independent observational epochs.

## Appendix A. Derivation of Synchronization Holonomy from Disformal Metric Geodesics

This appendix provides a self-contained derivation of the observable synchronization holonomy from the disformal metric structure. The derivation proceeds from the geodesic equation in the matter metric, showing explicitly that non-integrability of the synchronization 1-form leads to a non-zero closure when \(B(\phi) \neq 0\).

### A.1 Disformal metric structure

TEP posits that all non-gravitational fields couple to the matter metric \(\tilde{g}_{\mu\nu}\), related to the gravitational metric \(g_{\mu\nu}\) by the disformal map (Bekenstein 1993; Koivisto & Zumalacárregui 2012):

\begin{equation} \label{eq:app_disformal}
\tilde{g}_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_{\mu}\phi\,\nabla_{\nu}\phi .
\end{equation}

In the weak-field, static limit the gravitational metric is \(g_{00}=-(1+2\Phi)\), \(g_{ij}=(1-2\Phi)\delta_{ij}\) with \(|\Phi|\ll 1\). Assuming the scalar field is time-independent (\(\nabla_0\phi=0\)) at fixed cosmic epoch (\(A_{\rm sp}(t_0)=1\)), the canonical matter-metric components are \(\tilde g_{00}=-A_{\rm loc}^2(u_{\rm grav})(1+2\Phi)\) and \(\tilde g_{ij}=(1-2\Phi)\delta_{ij}+B(\phi)\partial_i\phi\partial_j\phi\) (or in the isotropic-conformal shorthand,

\begin{align}
\tilde{g}_{00} &= -A^2(\phi)\,(1+2\Phi), \\
\tilde{g}_{ij} &= A^2(\phi)\,(1-2\Phi)\,\delta_{ij} + B(\phi)\,\partial_i\phi\,\partial_j\phi ,
\end{align}

both sharing the identical leading projected disformal synchronization connection \(\delta\tilde\sigma_i\) below).

### A.2 Residual synchronization connection and projected disformal correction

Following the Jakarta construction (Paper 0, Section 3.3 and Appendix A3), the physical observable is not the raw loop integral of a proper-time 1-form but the GR-subtracted residual. Let \(\tilde{\sigma}^{(\tau)}\) denote the matter-frame synchronization transport one-form and \(\sigma_{\rm GR}^{(\tau)}\) the corresponding GR reference one-form. The residual connection is:

\begin{equation} \label{eq:app_residual}
\Delta\sigma^{(\tau)} = \tilde{\sigma}^{(\tau)} - \sigma_{\rm GR}^{(\tau)}
\end{equation}

and the time-valued residual holonomy is:

\begin{equation} \label{eq:app_holonomy_resid}
H_{\rm resid}^{(\tau)}(C) = \oint_C \Delta\sigma^{(\tau)} = \iint_\Sigma d(\Delta\sigma^{(\tau)}), \qquad C = \partial\Sigma.
\end{equation}

This is invariant under synchronization re-gaugings because both \(\tilde{\sigma}^{(\tau)}\) and \(\sigma_{\rm GR}^{(\tau)}\) shift by the same exact one-form. In the purely conformal subclass (\(B = 0\)), the scalar contribution is the exact one-form \(\omega^{(A)} = d\ln A\), so \(\oint_C d\ln A = 0\) for any closed loop. A leading-order nonzero \(H_{\rm resid}^{(\tau)}\) requires the disformal sector.

To leading order in the disformal correction, let \(u^\mu\) denote the four-velocity of the clock congruence and \(P_\mu{}^\nu = \delta_\mu{}^\nu + u_\mu u^\nu\) the spatial projector. The projected disformal contribution to the synchronization connection is (Jakarta, Appendix A3):

\begin{equation} \label{eq:app_disformal_sigma}
\delta\tilde{\sigma}_\mu^{(\tau)} \simeq -\frac{B}{A^2}\,(u \cdot \nabla\phi)\,P_\mu{}^\nu\,\nabla_\nu\phi
\end{equation}

In a hypersurface-orthogonal \(3+1\) slicing (\(u^\mu = n^\mu\)):

\begin{equation} \label{eq:app_3plus1}
\delta\tilde{\sigma}_i^{(\tau)} \simeq -\frac{B}{A^2 N}\,(\partial_i \phi)\,(n \cdot \partial\phi)
\end{equation}

Unlike \(d\ln A\), the disformal contribution is not generically exact. Its curvature satisfies \(d(\delta\tilde{\sigma}^{(\tau)}) \neq 0\) whenever the prefactor varies independently around the loop — through time dependence, motion through \(\nabla\phi\), anisotropic boundary conditions, inhomogeneous field structure, or spatial variation of the disformal response.

### A.3 Closure holonomy and the physical condition for non-vanishing signal

At leading order beyond the GR-subtracted reference model:

\begin{equation} \label{eq:app_dsigma}
d(\Delta\sigma^{(\tau)}) = d(\tilde{\sigma}^{(\tau)} - \sigma_{\rm GR}^{(\tau)}) \supset d(\delta\tilde{\sigma}^{(\tau)})
\end{equation}

A static scalar gradient in a hypersurface-orthogonal frame does not generate the leading holonomy. The pulsar observable requires \(u \cdot \nabla\phi \neq 0\): relative motion through the field, time dependence, mixed temporal-spatial structure, or independently varying lapse/shift and disformal response around the effective loop. For J0437−4715, the pulsar's proper motion (\(v_\perp \simeq 104\) km/s), Earth's orbital motion (\(v_{\rm orb} \simeq 30\) km/s), and screen motion provide the kinematic ingredients that generate \(u \cdot \nabla\phi \neq 0\). These enter the theoretical prediction through the \(u \cdot \nabla\phi\) factor in equation $\eqref{eq:app_disformal_sigma}$, not merely through the later Stokes-alignment algorithm.

The leading-order residual holonomy for a loop \(C\) bounding a smooth surface \(\Sigma\) is:

\begin{equation} \label{eq:app_holonomy_final}
H_{\rm resid}^{(\tau)}(C) = \iint_\Sigma d(\delta\tilde{\sigma}^{(\tau)})
\end{equation}

Equation $\ref{eq:app_holonomy_final}$ is the central result: *the residual holonomy vanishes in the conformal-only subclass* (\(B = 0\)), where the scalar contribution is exact. A non-zero \(H_{\rm resid}^{(\tau)}\) at leading order is possible when \(B(\phi) \neq 0\) and \(u \cdot \nabla\phi \neq 0\) combine to produce nonzero residual curvature. The precise necessary condition is:

\begin{equation} \label{eq:app_curvature_condition}
d\!\left[\frac{B}{A^2}\,(u \cdot \nabla\phi)\,P \cdot \nabla\phi\right] \neq 0
\end{equation}

over the loop region. This is stronger than merely requiring \(B \neq 0\) and \(u \cdot \nabla\phi \neq 0\) individually, because it specifies exactly when the curvature of the residual connection — and therefore the holonomy — is nonzero. The conformal sector alone cannot generate a closed-loop residual; it can affect local rates and open-path redshift comparisons but not the synchronization holonomy.

### A.4 Why ordinary ray phases close but TEP synchronization transport need not

A central question is whether the measured secondary-spectrum cross-term phases are the edges of a genuine spacetime synchronization loop, or merely differences between three complete source-observer ray phases that would trivially cancel. The following measurement model identifies the observed cross-terms with synchronization-transport links. A complete electromagnetic transfer derivation is required to determine the corresponding channel coefficient and effective phase frequency \(\omega_{\rm eff}\); the identification below is a measurement-model postulate, not a derivation from the matter-coupled electromagnetic field.

For three single-ray complex amplitudes \(X_i, X_j, X_k\) at the observer, rank-one pairwise cross-terms \(Q_{ab} = X_a X_b^*\) obey the algebraic identity \(Q_{ij}Q_{jk}Q_{ki} = |X_i X_j X_k|^2\), whose phase vanishes identically regardless of the scalar phase \(\Theta_a\) accumulated along each individual ray: \((\Theta_i - \Theta_j) + (\Theta_j - \Theta_k) + (\Theta_k - \Theta_i) = 0\). In the secondary-spectrum measurement pipeline, however, each extracted arclet peak \(Q_{ab}(\tau_{ab}, f_{D,ab})\) is not a point-wise product of isolated single-ray amplitudes; it is the two-dimensional Fourier transform of the dynamic intensity spectrum \(I(\nu,t) = |E(\nu,t)|^2\) evaluated over a finite delay–Doppler pixel bin integrating a coherent bundle of sub-rays across the scattering screen (as demonstrated explicitly by the non-factorizable \(\chi^2\) screen simulation of Step 055, where secondary-spectrum cross-terms are intensity convolutions that break rank-one factorization). While conventional multi-ray screen convolutions produce non-zero per-epoch closure phases whose epoch-mean direction averages to zero over time (Step 055: synthetic \(\bar{R}_{\rm dir}=0.09\), Rayleigh \(p=0.55\)), a stationary non-exact spacetime synchronization connection \(\Delta\sigma^{(\tau)}\) couples to the finite-baseline interference transport across each screen-pair link \(\Gamma_{ab}\), breaking rank-one factorization with a persistent directional bias.

TEP modifies the operational comparison itself. The pairwise observable is not merely the difference of endpoint phases, but the transport of synchronization between the path-defined events associated with rays \(a\) and \(b\):

\begin{equation} \label{eq:tep_transport}
\varphi_{ab}^{\rm TEP} = \varphi_b - \varphi_a + \int_{\Gamma_{ab}} \omega_{\rm eff}(x)\,\Delta\sigma^{(\tau)}
\end{equation}

where \(\Gamma_{ab}\) is the comparison path in spacetime induced by the scattering geometry: the spatial separation between the points where rays \(a\) and \(b\) intersect the scattering screen, transported along the observer's worldline. The endpoint phases \(\varphi_a, \varphi_b\) are state functions and cancel in the closure; the residual synchronization transport \(\int_{\Gamma_{ab}} \omega_{\rm eff}\,\Delta\sigma^{(\tau)}\) is path-dependent and generically non-exact when \(B(\phi) \neq 0\).

The three measured cross-terms \((ij), (jk), (ki)\) correspond to three comparison paths \(\Gamma_{ij}, \Gamma_{jk}, \Gamma_{ki}\) that form a genuine closed spacetime loop:

\begin{equation} \label{eq:closure_loop}
\psi_{ijk} = \varphi_{ij}^{\rm TEP} + \varphi_{jk}^{\rm TEP} + \varphi_{ki}^{\rm TEP}
= \operatorname{Arg}\!\exp\!\left[i\oint_C \omega_{\rm eff}(x)\,\Delta\sigma^{(\tau)}\right]
= \operatorname{Arg}\!\exp\!\left[i\iint_\Sigma d(\omega_{\rm eff}\,\Delta\sigma^{(\tau)})\right]
\end{equation}

The endpoint phases cancel telescopically, while the residual synchronization curvature remains. Thus the null \(\psi = 0\) is recovered whenever \(\Delta\sigma^{(\tau)}\) is exact — including the conformal-only sector — whereas the disformal non-exact sector permits \(\psi \neq 0\). The measured cross-term is non-factorizable under TEP precisely because the synchronization transport along each edge is an independent path integral, not a difference of state functions.

In the scintillation geometry, each arclet \(i\) in the secondary spectrum corresponds to a discrete scattered ray arriving at the observer with delay \(\tau_i\) and Doppler shift \(f_{D,i}\). The physical events and paths are:

- \(S_i, S_j, S_k\): the three scattering-screen events where rays
\(i, j, k\) are redirected by plasma inhomogeneities.

- \(O_i, O_j, O_k\): the corresponding observer detection events
(co-located at the receiver, but distinguished by their arrival
directions and delays in the secondary spectrum).

- \(u^\mu\): the physical clock congruence (the observer's
four-velocity, including Earth orbital motion and station velocity).

- \(\Gamma_{ij}\): the oriented synchronization-comparison link between
the scattering events \(S_i\) and \(S_j\), transported along the
observer's worldline.

- \(\Gamma_{jk}, \Gamma_{ki}\): the corresponding links for the other
two pairs.

- \(C = \Gamma_{ij} + \Gamma_{jk} + \Gamma_{ki}\): the closed
synchronization-comparison loop.

The cross-term \(\hat{\tau}_{ab}\) is measured from the interference of rays \(a\) and \(b\) in the secondary spectrum, and its phase \(\phi_{ab}\) is extracted from the complex cross-spectrum. The three cross-terms \((i,j), (j,k), (k,i)\) are measured independently from three distinct pairs of arclets, and their closure \(\phi_{ij} + \phi_{jk} - \phi_{ik}\) corresponds to the loop \(C = \Gamma_{ij} + \Gamma_{jk} + \Gamma_{ki}\) in the scattering geometry. The minus sign on \(\phi_{ik}\) reflects the reversal of the \((i,k)\) leg to close the loop. Under the scalar-delay null, each \(\phi_{ab}\) is a state-function difference and the closure vanishes; under TEP, each \(\phi_{ab}\) carries an additional path-dependent synchronization transport that does not cancel. The measured complex cross-term acquires \(\int_{\Gamma_{ab}} \omega_{\rm eff}\,\Delta\sigma^{(\tau)}\) through the electromagnetic phase transport between the scattering events, calibrated by the emitter and receiver clock congruence; photon propagation is null, so the phase is formulated through clock calibration and electromagnetic phase transport, not photon proper time.

Amplitude closure of the measurement model is evaluated in Step 057 (Section 5.4.3): the canonical connection integrated over the measured screen-triangle geometry on the admissible \(B \geq 0\) branch yields \(\approx 1.47\times10^{-12}\) s at the \(B_0=+1\) unit benchmark (\(1.3\%\) of \(H_{\rm implied}\approx 1.135\times10^{-10}\) s on the median leg, \(5.0\%\) on the \(p_{95}\) leg) and reaches \(1.141\times10^{-10}\) s (\(100.5\%\) on the median leg; \(B_0^{\rm req}=20.1\) on the \(p_{95}\) leg) at the corpus's GW170817 galactic-profile bound \(B_0\lesssim 77.7\) (Paper 0, Step 50), while remaining below the direct time-domain delay bound \(|H|\le 0.459\) ns (Section 4.4).

### A.5 Mapping to the measured closure observables

The measured Phase Closure is the complex cross-term closure product \(\mathcal{B}_{ijk} = Q_{ij} Q_{jk} Q_{ik}^*\) (Equation $\ref{eq:closure_product}$), where each \(Q_{ab}\) is extracted from the complex secondary spectrum at the cross-term peak position. The TEP measurement operator modifies each cross-term:

\begin{equation} \label{eq:tep_cross_term}
Q_{ab}^{\rm TEP} = X_a\,X_b^*\,\exp\!\left[i\int_{\Gamma_{ab}} \omega_{\rm eff}(x)\,\Delta\sigma^{(\tau)}\right] .
\end{equation}

Within the TEP measurement model, the measured cross-term is represented as above. Appendix A.4 defines the associated events and paths; electromagnetic transfer determines the channel coefficient. The closure product is then:

\begin{equation} \label{eq:tep_closure_product}
Q_{ij}^{\rm TEP}\,Q_{jk}^{\rm TEP}\,Q_{ki}^{\rm TEP}
= |X_i X_j X_k|^2\,\exp\!\left[i\oint_C \omega_{\rm eff}(x)\,\Delta\sigma^{(\tau)}\right] ,
\end{equation}

and therefore:

\begin{equation} \label{eq:psi_holonomy}
\psi_{ijk} = \operatorname{Arg}\!\exp\!\left[i\oint_C \omega_{\rm eff}(x)\,\Delta\sigma^{(\tau)}\right] .
\end{equation}

The endpoint amplitudes and phases cancel, leaving only the residual synchronization holonomy converted to phase through \(\omega_{\rm eff}\). Under the additive-delay null hypothesis (\(\Delta\sigma^{(\tau)} = 0\)), \(\psi_{ijk} = 0\) up to measurement noise. When \(B(\phi)\neq 0\) and the curvature condition (Equation $\ref{eq:app_curvature_condition}$) is satisfied, the nonfactorizable synchronization transport imprints a non-zero mean on \(\psi\), providing the circular statistic used as the primary detection test in this work. The delay-domain folded magnitude \(M_{\rm cl}\) is a separate quantity, computed from the real-valued closure delay \(\tau_{\text{closure}}\); it is not the real part of a complex delay-Doppler vector and is not used to define \(\psi\).

## Appendix B. Limitations of Delay-Domain Amplitude Estimators

While the Phase Closure \(\psi\) is a zero-centered odd-parity circular statistic, unsigned delay-domain amplitude estimators of the form \(M_{\rm cl} = |\langle \tau_{\text{closure}} \rangle_{\text{aligned}}|\) are strictly non-negative and therefore subject to folded-normal (Rice-distribution) noise-floor bias. For zero-mean Gaussian noise with standard deviation \(\sigma\), the expectation of the folded magnitude is \(E[|X|] = \sigma\sqrt{2/\pi} > 0\). Consequently, raw unsigned delay magnitudes and information-criterion comparisons evaluated on \(M_{\rm cl}\) against a zero-mean null primarily measure the non-zero expectation of the folded noise distribution rather than a physical synchronization holonomy. This appendix documents the complete delay-domain amplitude audit from Step 007, Step 052, and Step 060 for pipeline completeness.

### B.1 Raw, Noise-Subtracted, and Symmetrically Trimmed Delay Estimators

In the primary PSR J0437-4715 dataset (1,093 independent epochs), the inverse-variance-weighted epoch-level unsigned closure magnitude is \(M_{\rm cl} = 8.100 \pm 0.26\) ns (epoch-level bootstrap SEM; the internal measurement-noise SEM is \(0.102\) ns). Estimating the per-epoch folded-normal noise expectation from the within-epoch signed-delay dispersion yields \(E[M_{\rm cl}] = 6.810 \pm 0.24\) ns, demonstrating that the raw unsigned estimator is dominated by the Rice noise floor. Subtracting the per-epoch floor in paired epoch-level comparisons leaves a statistically positive residual delay excess of \(M_{\rm cl,excess} = 1.290 \pm 0.11\) ns (\(t \approx 12.2\sigma\) under independent-epoch resampling, and \(3.8\sigma\)–\(10.8\sigma\) across 15–240-epoch MJD-ordered contiguous block bootstraps; Steps 052 and 060).

A secondary delay-domain diagnostic, the symmetric 10% trimmed amplitude \(M_{\rm cl,sym10} = 21.991 \pm 0.75\) ns (epoch-level iid bootstrap SEM, rising to \(\pm 2.0\text{--}4.9\) ns under 15–240-epoch contiguous blocks; Step 052), removes the lowest and highest 10% of unweighted per-epoch folded magnitudes. Because \(M_{\rm cl,sym10}\) is an unweighted trimmed summary of per-epoch folded magnitudes whereas \(M_{\rm cl} = 8.100\) ns is an inverse-variance-weighted global mean, their numerical scales reflect their distinct weighting conventions rather than two estimates of the same parameter, and neither carries primary inferential weight.

Summary of delay-domain amplitude estimators and folded-normal noise-floor diagnostics for PSR J0437-4715.

| Delay-Domain Quantity | Estimate | Nominal Quotient | Diagnostic Interpretation |
| --- | --- | --- | --- |
| Raw Delay Magnitude \(M_{\rm cl}\) | 8.100 ± 0.26 ns | 31.2σ (epoch SEM) | Dominated by folded-normal Rice noise floor; not a standalone signal detection |
| Rice Noise Floor \(E[M_{\rm cl}]\) | 6.810 ± 0.24 ns | 66.9σ (internal) | Folded-normal expectation \(\sigma\sqrt{2/\pi}\) from per-epoch signed-delay dispersion |
| Paired Noise-Subtracted Excess \(M_{\rm cl,excess}\) | 1.290 ± 0.11 ns | ≈12.2σ iid (3.8–10.8σ blocked) | Paired epoch-level excess above the Rice floor (Steps 052, 060) |
| Symmetric 10% Trimmed Magnitude \(M_{\rm cl,sym10}\) | 21.991 ± 0.75 ns (±2.0–4.9 ns blocked) | ≈29σ iid | Unweighted trimmed epoch-magnitude diagnostic (Step 052) |

### B.2 Step 007 Delay-Amplitude Validation Ledger

Step 007 applied seven validation checks to the unweighted independent-epoch \(M_{\rm cl}\) summary prior to the adoption of circular Phase Closure \(\psi\) as the primary test statistic:

| Test | Result | Status |
| --- | --- | --- |
| Effect size: \(M_{\rm cl}\) magnitude | \(d = 0.393\) (epoch-mean \(M_{\rm cl}\); small standardized separation in step_007) | PASS |
| Effect size: signed mean (validation) | \(d \approx -10^{-3}\) (negligible; bipolar cancellation in signed channel) | PASS |
| Robust statistics (\(M_{\rm cl}\) estimators) | All estimators positive (mean, median, trimmed; \(M_{\rm cl,trim007} = 7.489\) ns symmetric 10% trimmed mean of \(\|\)signed epoch means\(\|\)) | PASS |
| Per-epoch Bonferroni threshold check | 46.2% of epochs (643/1,391) exceed \(5\sigma\) uncorrected and 32.7% (455/1,391) pass Bonferroni correction (\(\alpha = 0.05/1,391\)) on raw \(M_{\rm cl}\), confirming a stationary Rice noise-floor distribution across epochs | Stationary Rice floor |
| Gaussian noise null test (validation) | \(p < 10^{-16}\) (\(z = -18.40\sigma\)): observed unweighted epoch mean \(M_{\rm cl} = 12.008\) ns lies below the uncancelled iid Gaussian expectation of 19.141 ns due to within-epoch bipolar cancellation | Consistent with TEP bipolar cancellation |
| Cross-validation (5-fold, epoch-mean \(M_{\rm cl}\) holdouts) | Step 007: 5/5 folds yield positive fold-mean epoch \(M_{\rm cl}\) (mean across folds 12.02 ns; Table B.3) | PASS |
| Bayesian model comparison (epoch-level \(M_{\rm cl}\) vs zero) | \(\Delta\text{BIC} = 393.9\), \(\log_{10} \text{BF} = 85.5\); reflects the non-zero mean of the folded-normal Rice noise floor rather than TEP signal evidence | Rice-floor artifact (non-inferential) |
| Half-year raw \(M_{\rm cl}\) partition | Jan–Jun (5,176 triplets): \(14.20 \pm 0.69\) ns (20.57σ Rice floor); Jul–Dec (5,714 triplets): \(10.05 \pm 0.56\) ns (17.85σ Rice floor). The ~23.8% seasonal difference tracks the ~29% kinematic modulation of the folded noise floor from Earth's ~30 km/s orbital motion against J0437's ~104 km/s transverse velocity | Noise-floor modulation (paired excess positive in both halves: 1.59 ns and 1.14 ns) |

### B.3 Five-Fold Holdout Cross-Validation of Epoch-Mean \(M_{\rm cl}\)

In the Step 007 `cross_validation` block (`results/step_007_independent_statistical_validation_results.json`), a 5-fold shuffle split on unweighted epoch-mean \(M_{\rm cl}\) values confirms that the delay-amplitude scale (\(\mathcal{O}(10\text{--}15)\) ns before inverse-variance weighting) is uniform across epoch subsets:

| Fold | Test mean \(M_{\rm cl}\) (ns) | Fold-wise \(t\) |
| --- | --- | --- |
| 0 | 10.792 | 10.40\(\sigma\) |
| 1 | 15.065 | 8.28\(\sigma\) |
| 2 | 11.078 | 10.99\(\sigma\) |
| 3 | 11.807 | 10.32\(\sigma\) |
| 4 | 11.339 | 11.13\(\sigma\) |

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# Temporal Equivalence Principle: Lunar Laser Ranging and the Nordtvedt Effect
**Matthew Lukin Smawfield**
Version: v0.5 (Lucknow)
First published: 17 May 2026
Last updated: 17 October 2026
DOI: 10.5281/zenodo.19446028

---

## Abstract

The Temporal Equivalence Principle (TEP) proposes that the local rate of proper time is governed by a dynamical scalar field $\phi=\bar\phi+M_{\rm Pl}u$ rather than treated as a fixed geometric parameter. This field couples universally to all matter via the canonical unified metric $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $A_{\rm loc}(u_{\rm grav})=e^{\beta_A u_{\rm grav}}$, $A_{\rm sp}(u_{\rm drift})=e^{\beta_A u_{\rm drift}}=e^{-u_{\rm drift}}$; Paper 0, §2.2, Step 80). Temporal Shear is pinned where the ambient time-gradient is steep — weak-field language: $g \gg g_t \approx cH_0/2$ — by the environmental-state operator $\mathcal S_\Sigma(\mathcal E)$. The canonical pairwise benchmark gives an Earth–Moon response of order $10^{-14}$ and does not account for the reported range residual as a differential-fall signal; the analysis instead targets the synodic term in the post-fit residual channel, $\delta r = 13\eta_{\rm resid}\cos D$, where $D$ is the Moon-Sun elongation angle and $\eta_{\rm resid}$ is the residual-channel amplitude (an Observable Response Coefficient, distinct from the standard metric Nordtvedt parameter $\eta$). Evaluating both the Cassini Shapiro delay and the Earth–Moon laser path on the single solved two-branch solar profile $(k/\sqrt{2})q y^2+q^2 y^3=1$ ($k=16.03$, $g_t=cH_0/2$; Step 083) gives $y_{\rm tb}(1.6\,R_\odot)=2.16\times10^{-8}$ ($\gamma_{\rm PPN}-1=-4.32\times10^{-8}$, passing Cassini) and $y_{\rm tb}(1\text{ AU})=1.47\times10^{-5}$, which suppresses the direct scalar optical excursion across the Earth–Moon baseline to $\delta u_\odot^{\rm tb}=7.43\times10^{-16}\cos D$ ($\delta r_{\rm path}^{\rm tb}=-1.43\times10^{-4}$ mm, $\eta_{\rm tb}=-1.10\times10^{-8}$), $3.4\times10^4$ times below the measured post-fit INPOP19a amplitude $\eta_{\rm resid} = -(3.91\pm0.56)\times10^{-4}$ ($-5.08\pm0.73$ mm) and consistent with direct-fit LLR bounds. Because at a fixed cosmic epoch inside the screened Solar System ($A_{\rm sp}(t_0)=1$) TEP's spatial 3-metric $\tilde\gamma_{ij}=\gamma_{ij}=(1-2\Phi_N/c^2)\delta_{ij}$ is identical to GR's, the unscreened solar Newtonian potential differential $\Delta\Phi_{N,\odot}/c^2=2.54\times10^{-11}\cos D$ (which defines an unscreened reference ceiling $-\tfrac12 L\,\delta\phi_{\rm eff}\cos D=-4.85$ mm $\cos D$, $\eta_{\rm ceiling}=-3.73\times10^{-4}$ at $T_{\rm Moon}=0.9969$; Paper 0 Step 76, Step 083) cancels in the BCRS$\to$GCRS reduction identically to GR and would give $\gamma_{\rm PPN}-1=-2/3$ at Cassini if attributed to an unscreened solar scalar field; the measured $\eta_{\rm resid}$ is therefore reported as an empirical post-fit residual-channel detection whose dynamical or ephemeris-reduction transfer map awaits a source-level integrator refit.

This work analyses 26,207 raw Lunar Laser Ranging O–C residuals from the public INPOP19a ephemeris archives (Paris Observatory, Geoazur), comprising $N = 25{,}445$ measurements from five international stations (1984–2019) after standard $6\sigma$ MAD outlier cleaning. The primary estimand is a precision-weighted synodic regression on the post-fit residual channel, with inverse station-variance weighting and a full systematic model with annual, monthly, and thermal $\cos(2D)$ regressors and a common intercept.

The primary precision-weighted residual-channel estimate is $\eta_{\rm resid} = -3.91 \times 10^{-4} \pm 5.63 \times 10^{-5}$ (nominal WLS $6.94\sigma$; $6.78\sigma$ cluster-robust; wild cluster bootstrap 95% CI $[-5.37, -2.73]\times 10^{-4}$ across the $G=5$ station clusters). The amplitude stabilises in the band $-3.2$ to $-4.1 \times 10^{-4}$ across the estimator hierarchy, from cosD-only ($5.25\sigma$) through full-systematic OLS ($6.17\sigma$) to precision-weighted ($6.94\sigma$). Robustness checks — common-$\eta_{\rm resid}$ mixed model, leave-one-station-out meta-analysis, wild cluster bootstrap, phase-locked differential, cross-ephemeris validation on cleaned DE430 ($\eta_{\rm resid} = -7.04 \times 10^{-4}$, $5.96\sigma$; confirming the sign but differing in amplitude by $2.77\sigma$ from INPOP19a in the matched window), parametric GR-null bootstrap, and a frequency null scan — all support residual-channel survival of the synodic component (Section 4). If interpreted within the static Parametrized Post-Newtonian framework — an interpretation the residual-channel estimand does not yet license (Section 4.14) — the measured amplitude would correspond to $\beta_{\rm PPN} = 0.999902 \pm 1.07 \times 10^{-5}$ from the joint $(\beta_{\rm PPN}, \gamma)$ contour; this mapping is reported for scale only and is not advanced as a PPN measurement pending the integrator-level refit.

The result is therefore framed as a high-significance residual-channel candidate with a TEP interpretation, not as a completed replacement for direct-fit LLR bounds. The $6.94\sigma$ figure is a detection of a synodic pattern in post-fit residuals; it is not a measurement of the Nordtvedt parameter, and the quantitative relationship between $\eta_{\rm resid}$ and the direct-fit $\eta$ is not yet established. Source-level numerical refits of the INPOP or DE430 integrators with $\eta_{\rm resid}$ left free remain the critical open closure test. Ephemeris-absorption stress tests bound the residual-channel survival amplitude but do not replace a full dynamical integrator-level confirmation.

Code Availability: All data and analysis code required to reproduce the results presented in this work, including the full LLR residual processing pipeline, are available in the public repository.

Keywords: temporal equivalence principle, lunar laser ranging, LLR, equivalence principle, Nordtvedt effect, post-Newtonian, scalar-tensor gravity, strong equivalence principle

## 1. Introduction

The equivalence principles form the empirical and conceptual foundation of General Relativity: the Weak Equivalence Principle (WEP) establishes the universality of free fall for test bodies; the Einstein Equivalence Principle (EEP) combines WEP with local Lorentz invariance and local position invariance for non-gravitational experiments; and the Strong Equivalence Principle (SEP) extends the universality of free fall and local invariance to self-gravitating bodies and gravitational experiments themselves. A violation of the SEP indicates that bodies with different gravitational self-energy or internal compactness couple differently to additional gravitational degrees of freedom beyond a single metric tensor.

General Relativity has achieved extensive empirical validation across multiple regimes: the anomalous perihelion precession of Mercury (42.98 arcsec/century, matched to 0.1%); light deflection around the Sun (1.75 arcsec, verified to ~1%); gravitational time dilation (GPS clock corrections of ~45 μs/day, confirmed to 0.1%); orbital decay in binary pulsars (indirect gravitational wave detection, Hulse-Taylor pulsar); and direct gravitational wave emission from merging black holes (LIGO, 2015). These empirical successes position GR as the effective theory within its domain of validity: regions where spacetime curvature remains moderate ($\Phi/c^2 \ll 1$) and quantum corrections are negligible.

The Equivalence Principle in GR derives from the universality of free fall: all test particles follow geodesics of the metric $g_{\mu\nu}$ independent of composition. This emerges geometrically from Einstein's field equations in the weak-field limit.

The present work extends this framework through the Temporal Equivalence Principle (TEP), a scalar-tensor theory where proper time becomes a dynamical field $\phi=\bar\phi+M_{\rm Pl}u$ that couples universally to matter via the canonical unified metric $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$, with $A_{\rm loc}(u_{\rm grav})=\exp(\beta_A u_{\rm grav})$, $A_{\rm sp}(u_{\rm drift})=\exp(\beta_A u_{\rm drift})=\exp(-u_{\rm drift})$, and cosmic temporal flow $U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$ (Paper 0, §2.2, Step 80). Scalar-tensor theories have a long history in gravitational physics, dating to Jordan (1955), Fierz (1956), and Brans-Dicke (1961), who first explored gravitational theories with dynamical scalar fields coupled to the metric. The Parametrized Post-Newtonian (PPN) formalism (Will 2014) provides a standardized framework for testing such theories through observable parameters including the Eddington parameters $\beta_{\rm PPN}$ (nonlinearity) and $\gamma$ (spatial curvature), both equal to unity in GR.

TEP preserves the Weak Equivalence Principle through universal conformal coupling: all non-gravitational processes couple to the same matter metric, ensuring local experiments remain metric-compatible. Where TEP diverges from standard scalar-tensor theories is in its screening of Temporal Shear, which allows the theory to evade tight solar-system constraints while maintaining large bare couplings. The Cassini bound on PPN-$\gamma$ (Bertotti et al. 2003) imposes a stringent Solar-System constraint on the screened scalar response; its detailed translation into TEP's source-charge normalization is treated separately in the foundation paper. TEP Temporal Shear screening is represented by the kinetic two-body response where the ambient time-gradient is steep (weak-field: high-acceleration configurations). Density and gravitational compactness $\Phi/c^2$ characterize the environment and can serve as proxies for $\mathcal S_\Sigma(\mathcal E)$; they are not independent microscopic screening laws. The local GR limit is recovered when the additional scalar response is negligible in the tested channel (Paper 0, §7).

The Earth-Moon compactness contrast $\Delta(\Phi/c^2)\sim10^{-10}$ motivates a sensitive test of differential response. Detectability is determined by the solved field and range observable, not by compactness alone; the measured synodic residual and the screened force benchmark are distinguished in Section 2.

The most precise test of the SEP is Lunar Laser Ranging (LLR), which measures the Earth-Moon distance with millimeter precision by timing laser pulses reflected from retroreflectors placed on the Moon by Apollo missions and Soviet Luna probes.

Over 50 years of LLR data have constrained the Nordtvedt parameter $\eta$ — which quantifies SEP violation — to $|\eta| \lesssim$ few $\times 10^{-4}$ (Williams et al. 2012; Müller et al. 2019), consistent with zero. These constraints arise from fitting $\eta$ as a free parameter in the complete orbital model.

The present analysis takes a complementary approach: it searches for a cos(elongation) modulation in the residuals of a GR-based ephemeris (INPOP19a) that assumes $\eta = 0$, targeting a TEP-specific suppression signal that may not be fully captured by the standard Nordtvedt parameterisation. Current LLR solutions leave centimeter-level O-C residuals after fitting all standard physical effects, providing the dataset for this search.

### 1.1 Manuscript Series Context

This work is Paper 17 in the TEP program. Earlier installments established the framework (Paper 0), clock-network phenomenology (Papers 1–3), and domain-specific Observable Response Coefficients (Papers 4–13). Three prior papers frame the present LLR analysis without duplicating its estimand.

The role of the present paper within the series is therefore not to restart the TEP argument from first principles, but to test whether the same time-field architecture has a measurable, suppressed Solar-System expression in the most precise available strong-equivalence-principle observable.

| Series strand | Prior result | Role in this paper |
| --- | --- | --- |
| Paper 0 | Two-metric TEP framework with a dynamical time field, universal matter coupling, Temporal Topology, and Temporal Shear suppression. | Defines $\phi$, $A(\phi)$, the screened conformal sector, and the requirement that local Lorentz invariance is preserved while global time structure may remain dynamical. |
| Papers 1–3 | Distance-structured and directionally structured clock-network correlations in GNSS products, with multi-center and raw-RINEX validation. | Establish the empirical motivation for treating precision timing residuals as a direct probe of Temporal Topology. |
| Paper 6 | A candidate saturation scale $\rho_T \approx 20$ g/cm$^3$ from cross-scale Temporal Topology consistency. | Supplies the density/screening scale used to interpret why LLR should lie in the strongly screened Solar-System response regime. |
| Paper 9 | A taxonomy of precision relativity tests distinguishing direct-fit, two-way, residual-channel, and clock-sector observables. | Explains why a standard direct-fit $\eta$ bound and a post-fit residual-channel synodic test are related but not identical estimands. |
| Papers 10–11 | Observable Response Coefficients for pulsar spin-down and Cepheid timing/distance-ladder observables. | Motivate treating the LLR residual-channel parameter $\eta_{\rm resid}$ as a Solar-System response coefficient rather than a bare microscopic scalar coupling. |
| Paper 13 | Weak-field Temporal Shear recovery in Gaia DR3 wide binaries. | Provides the complementary weak-field, weakly screened limit to the compactness-suppressed Earth-Moon test here. |
| Paper 17 | LLR residual-channel search for a synodic Nordtvedt-like component. | Tests the screened Solar-System SEP sector of TEP using public INPOP19a and DE430 lunar-ranging residual archives. |

Paper 5 (GTE) argued that Solar System tests remain compatible with TEP when Temporal Shear is suppressed in the screened Solar-System environment, with a microscopic Nordtvedt amplitude well below published direct-fit bounds. Paper 8 (SLR) extended the conformal-sector program to two-way optical ranging on passive geodetic targets, testing distance-structured residual structure orthogonal to active-clock microwave networks. Paper 9 (EXP) classified LLR as a reciprocity-even, orbital-dynamics constraint that does not directly target clock-sector loop holonomy or spatial clock correlations.

Paper 17 does not overturn those scope claims. It instead tests whether a synodic $\cos(D)$ footprint survives in post-fit O-C residuals after INPOP19a and DE430 solutions that assume $\eta=0$. The fitted Nordtvedt parameter is treated as the Solar System Observable Response Coefficient for LLR, not as the bare microscopic coupling. The hardware epoch, synthetic absorption, and toy Keplerian analyses supply the structural reason a static direct-fit $\eta$ and a residual synodic channel need not coincide when the coupling scales with the heliocentric environment. Paper 8 and Paper 17 therefore occupy complementary optical-ranging lines: SLR probes conformal residual coherence at terrestrial baselines; LLR probes suppressed-PPN differential free fall in the Earth-Moon system.

### 1.2 Claim Hierarchy and Scope

The evidence is organised deliberately so that the empirical result, the systematic stress tests, and the TEP interpretation are not conflated. The paper advances a strong claim, but in a fixed hierarchy.

| Level | Claim | Role in the argument |
| --- | --- | --- |
| 1 | Data integrity | Public INPOP19a and DE430 residual archives are processed through a deterministic, checksum-audited pipeline. |
| 2 | Residual-channel detection | A synodic $\cos(D)$ component is extracted from post-fit LLR residuals under the fixed estimator hierarchy. |
| 3 | Systematic resistance | The same residual-channel component is tested against station concentration, hardware epochs, outliers, autocorrelation, spectral controls, and nuisance regressors. |
| 4 | TEP interpretation | The sign, scale, and screened Solar-System response are interpreted using the Temporal Topology and Temporal Shear framework established in the preceding papers. |
| 5 | External closure | Source-level INPOP or DE430 integrator refits with $\eta_{\rm resid}$ free, together with independent range-level reductions, remain the critical external validation tests. |

This hierarchy strengthens the inference by fixing what is being measured. The primary result is a residual-channel extraction of a Nordtvedt-like synodic component; the TEP claim is that this component has the sign, order of magnitude, screening regime, and dynamical structure expected for the Solar-System response of the same time-field theory tested elsewhere in the series.

The claim-discipline framework for the TEP corpus, including the scope limitations of canonical precision tests, is established in TEP-EXP (Paper 9).

The LLR Nordtvedt test in this paper probes a complementary aspect of TEP: through the suppressed PPN sector, the compactness-dependent effective coupling $\alpha_{\rm eff}$ could differ between Earth and Moon, producing a violation of the Strong Equivalence Principle. Earth's deeper gravitational potential ($\Phi_{\oplus}/c^2 \approx 7 \times 10^{-10}$) flattens Temporal Topology more strongly than the Moon's ($\Phi_{\rm Moon}/c^2 \approx 3 \times 10^{-11}$), suppressing Temporal Shear and yielding a smaller $\alpha_{\rm eff}$. This differential gradient suppression could lead to unequal free-fall rates in the Sun's field.

The quantitative prediction for the Nordtvedt parameter is informed by the TEP framework's Observable Response Coefficients, which quantify domain-specific astrophysical responses rather than a universal bare coupling. Preliminary results from related work in the same TEP framework report $\kappa_{\rm Cep}^{\rm equiv} = (1.22\text{--}1.24) \times 10^6$ mag for Cepheid period-luminosity anomalies (Paper 11 v0.12, Appendix C) and $\tilde\kappa_{\rm MSP} = (2.9 \pm 4.5) \times 10^4$ for pulsar spin-down excess (Paper 10 v0.10, observable response); the unscreened geometric amplitude is $\sim 10^6$ before environmental suppression. These coefficients are distinct from the microscopic conformal coupling $\beta$ or scalar-tensor coupling $\alpha_0$, absorbing instead the full astrophysical response including environmental activation and transfer functions. The prediction $\eta_{\rm resid} \sim -10^{-4}$ emerges from the differential suppression geometry combined with the understanding that LLR operates in a more screened Solar System regime, yielding a smaller effective response than the unscreened galactic probes.

This analysis uses 26,207 raw LLR O-C residuals from five international laser ranging stations spanning 35 years of measurements (1984–2019), with 25,445 retained after 6σ-equivalent (MAD-based) outlier cleaning. The residuals are processed against the INPOP19a lunar and planetary ephemeris from the Paris Observatory (Geoazur). To eliminate synodic blurring and ensure millimeter-level coordinate precision, Moon-Sun elongation angles were computed using high-precision Skyfield/DE440 ephemerides rather than mean-phase approximations. The analysis searches for the predicted TEP Nordtvedt signal: a modulation of the form $\delta r = 13 \eta_{\rm resid} \cos(D)$, where $D$ is the Moon-Sun elongation angle.

The predicted amplitude for $\eta_{\rm resid} \sim 10^{-4}$ is $\sim$1–5 mm, smaller than the 9.5 cm per-observation residual RMS. This is expected for a suppressed PPN signal in precision LLR metrology: the signal explains only $\sim$0.1% of the total variance; under an idealized i.i.d. scale illustration with $N = 25{,}445$ rows, $\sigma_r \approx 1/\sqrt{N} = 0.0063$ would place the raw correlation coefficient $r = -0.0329$ at a nominal $5.25\sigma$. Because individual LLR normal points are neither temporally nor station-wise independent, the operative inference does not rely on i.i.d. scaling, but on synodic phase coherence across 35 years evaluated through the full-systematic estimator, station-cluster covariance, and correlation-aware bootstrap analyses below. As hardware precision improved from $\sim$10 cm (PMT era) to $\sim$2 cm (C-SPAD era), the extracted physical amplitude did not scale to zero; it converged to a fixed negative boundary, consistent with a permanent modulation beneath the noise envelope.

Müller and Nordtvedt (1998) reported an unexplained synodic post-model residual proportional to $\cos(D)$ in 28 years of LLR data. The present sub-centimeter archive allows a cleaner extraction of the same footprint under explicit annual, monthly, and thermal controls.

Standard orbital periods are incommensurate with the synodic month. In the frequency domain, $\cos(D)$ therefore lies outside the classical multi-body mean-motion basis used in static direct-fit ephemerides, and a synodic coupling parameterized with $\eta_{\rm resid}$ fixed at zero is expected to project into post-fit residuals. Source-level integrator refits that leave $\eta_{\rm resid}$ free (Section 4.14) supply the critical dynamical closure; the hardware epoch, synthetic absorption, and toy Keplerian analyses supply the internal linearized and simulation checks on that channel.

## 2. Theoretical Framework: TEP and the Nordtvedt Effect

The Temporal Equivalence Principle (TEP) is a scalar-tensor theory in which proper time becomes a dynamical field $\phi=\bar\phi+M_{\rm Pl}u$ that couples to the local mass density. In this framework, matter couples to the canonical unified metric $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$, where $A_{\rm sp}(u_{\rm drift}) = \exp(\beta_A u_{\rm drift})=\exp(-u_{\rm drift})$ is the cosmological spatial ruler factor and $A_{\rm loc}(u_{\rm grav})=\exp(\beta_A u_{\rm grav})$ rescales the local $(3+1)$ ADM lapse along $U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$ at unstretched spatial 3-metric $\tilde\gamma_{ij}=\gamma_{ij}$ at fixed cosmic epoch ($A_{\rm sp}(t_0)=1$; Paper 0, §2.2, Step 80). The rate at which proper time accumulates depends on the local value of $\phi$, which in turn is determined by the matter sources, potential, kinetic response and boundary conditions. The nonlinear kinetic operator suppresses the gradient response where the ambient time-gradient is steep (weak-field: high-acceleration configurations); ambient density and compactness are diagnostics of that configuration, not additional microscopic screening laws.

**Screening projection notice.** Screening in TEP is represented at theory level by the environmental operator S_Σ(E). Quantities such as ρ_T, R_T(M), S_⊕(r), compactness Φ/c^2, local stellar density, thermal epoch, coherence length, proximity, and boundary geometry are domain-specific projections of E, not independent screening mechanisms and not interchangeable universal thresholds.

### 2.1 The Nordtvedt Effect

In scalar-tensor theories, bodies with different gravitational binding energy per unit mass will fall at different rates in an external gravitational field. This is the Nordtvedt effect, first derived by Kenneth Nordtvedt (Nordtvedt 1968) as a test of the Strong Equivalence Principle. The Nordtvedt parameter $\eta$ quantifies the strength of SEP violation:

\begin{equation} \label{eq:nordtvedt_param} \eta = 4\beta - \gamma - 3 -
\frac{10}{3}\xi - \alpha_1 + \frac{2}{3}\alpha_2 - \frac{2}{3}\zeta_1 -
\frac{1}{3}\zeta_2 \end{equation}

where $\beta$ and $\gamma$ are the post-Newtonian parameters, $\alpha_1$ and $\alpha_2$ are preferred-frame parameters, $\xi$ is the Whitehead parameter, and $\zeta_1$, $\zeta_2$ are conservation-law parameters. In General Relativity ($\beta = \gamma = 1$, all others zero), $\eta = 0$ exactly. In standard scalar-tensor theories such as Brans-Dicke, the Nordtvedt parameter is approximately $\eta_{\rm BD} \approx 2\alpha_0^2/(1+\alpha_0^2)$ where $\alpha_0 = \sqrt{2}\,\beta_A$ is the DEF-normalised bare scalar-tensor coupling (following the corpus-wide convention $\varphi = \phi/M_{\rm Pl}$, so that $\beta_A = d\ln A/d\varphi$ is dimensionless; in the unscreened TEP limit $\beta_A = -1$, giving $\alpha_0 = -\sqrt{2}$). An illustrative effective Earth–Moon coupling $\alpha_{\rm eff}^{(\oplus-{\rm Moon})} \approx 3 \times 10^{-3}$ is adopted for the phenomenological estimate below. Two readings of that scale are kept distinct throughout: as a screened-body coupling it enters the fall differential only in product with the solar source charge $| \alpha_0\,S_\Sigma^{(\odot)} | \lesssim 1.6\times10^{-5}$ ($8.2\times10^{-6}$ on the isotropic benchmark) bounded by Cassini; as a full observable response coefficient it already contains that factor. The bookkeeping below shows that only the second reading can stand at the quoted amplitude — which makes the volumetric number an effective response coefficient, not a decomposable force estimate.

The fitted $\eta_{\rm resid}$ used throughout the remainder of this paper is a post-fit residual-channel proxy extracted from published INPOP19a and DE430 O–C residuals. It is not a direct replacement for ephemeris-level SEP bounds until the TEP/Nordtvedt parameter is included in the source-level dynamical integrator. The symbol $\eta$ without a subscript denotes the standard Nordtvedt parameter defined in Eq. ($\ref{eq:nordtvedt_param}$), while $\eta_{\rm resid}$ denotes the empirically extracted residual-channel amplitude reported in Sections 3–5.

A Nordtvedt effect tests differential response associated with gravitational self-binding; it is distinct from the fitted synodic residual. TEP's universal conformal coupling $A(\phi)$ preserves the Weak Equivalence Principle (all non-gravitational processes couple to the same matter metric $\tilde{g}_{\mu\nu}$, consistent with the TEP framework paper, Paper 0, §7.2). However, the *Strong* Equivalence Principle may be violated through the suppression of Temporal Shear.

The operative shear response is obtained from the nonlinear kinetic field configuration and its source/body matching. Surface compactness $\Phi/c^2=GM/Rc^2$ is an environmental diagnostic used in the historical estimates below, not an independent microscopic screening law. Neither a suppressed gradient nor a fitted residual coefficient is by itself a measurement of a body's conserved source charge.

The compactness-squared and volumetric constructions below are retained as historical and phenomenological checks. They are not derivations of the operative kinetic operator or of the measured $\eta_{\rm resid}$. The canonical response and the coupled-field diagnostic are evaluated separately below; the empirical residual analysis does not depend on adopting the historical compactness model.

For two bodies with different compactness (Earth and Moon), the differential acceleration in an external gravitational field $a_0$ produces a Nordtvedt-like range modulation. In an unscreened toy baseline scaling simply with surface compactness, the differential coupling would be:

\begin{equation} \label{eq:effective_coupling_diff} \Delta\alpha_{\rm eff} = \alpha_{\rm eff,\oplus} - \alpha_{\rm
eff,{\rm Moon}} \approx \alpha_0 \left[ \left(\frac{\Phi}{c^2}\right)_\oplus
- \left(\frac{\Phi}{c^2}\right)_{\rm Moon} \right] \end{equation}

Substituting this naive compactness scaling and defining the residual-channel parameter $\eta_{\rm resid}$ (the TEP-specific residual amplitude, distinct from the standard Nordtvedt parameter $\eta_N = 2\alpha_0^2/(1+\alpha_0^2)$, which vanishes under screening) yields the toy baseline:

\begin{equation} \label{eq:tep_prediction} \eta_{\rm resid} \approx \frac{\delta a}{a_0} \approx \alpha_0^2 \left[
\left(\frac{\Phi_\oplus}{c^2}\right)^2 - \left(\frac{\Phi_{\rm
Moon}}{c^2}\right)^2 \right] \end{equation}

where $\Phi_{\oplus}/c^2 \approx 7 \times 10^{-10}$ is Earth's compactness and $\Phi_{\rm Moon}/c^2 \approx 3 \times 10^{-11}$ is the Moon's compactness. As established below (Section 2.1), this naive surface-compactness baseline yields $\eta_{\rm resid} \sim 10^{-18}$ at the bare coupling — and $\lesssim 3\times10^{-29}$ once the screened solar source charge is folded in — and serves only as a null check confirming that bare surface compactness cannot drive the LLR anomaly. The negative response assigned by the volumetric model in Eq. ($\ref{eq:volumetric_suppression}$) is a property of that phenomenological construction, not a separately established sign prediction of the kinetic field-to-range map.

Second, in the Damour–Esposito-Far&egrave;se (DEF) scalar-tensor framework the Nordtvedt parameter is linear in the sensitivity difference:

\begin{equation} \label{eq:eta_def_standard}
\eta_{\rm DEF} = \frac{4\alpha_0^2\,(s_\oplus - s_{\rm Moon})}{1 + \alpha_0^2} ,
\end{equation}

where $s_\oplus$ and $s_{\rm Moon}$ are the gravitational self-energies (sensitivities) of Earth and the Moon. The TEP-specific volumetric model adopted here uses a different functional form that integrates interior density profiles against the saturation scale $\rho_T$:

\begin{equation} \label{eq:eta_from_alpha_eff}
\eta \approx \alpha_{\rm eff,\oplus}^2 - \alpha_{\rm eff,{\rm Moon}}^2 .
\end{equation}

Substituting Eq. ($\ref{eq:effective_coupling_diff}$) into Eq. ($\ref{eq:eta_from_alpha_eff}$) yields the TEP prediction in Eq. ($\ref{eq:tep_prediction}$). Earth's deeper potential well flattens Temporal Topology more strongly than the Moon's, suppressing Temporal Shear and yielding a significantly smaller effective coupling.

Two theoretically distinct estimates are evaluated. The *compactness-squared* expression above uses only the surface compactness differential, evaluated with $(\alpha_0 S_\Sigma^{(\odot)})^2$ as the effective squared coupling. Even at the Cassini ceiling $|\alpha_0 S_\Sigma^{(\odot)}| \lesssim 1.6\times10^{-5}$ ($S_\Sigma^{(\odot)} \lesssim 1.15\times10^{-5}$ under the unified unstretched-space metric, or $5.8\times10^{-6}$ on the isotropic benchmark, with $\alpha_0 = \sqrt{2}\,\beta_A = -\sqrt{2}$; Paper 0, Step 80), and $\Delta(\Phi^2/c^4) \approx 5\times10^{-19}$, this gives $\eta_{\rm resid} \approx \alpha_0^2 S_\Sigma^{(\odot)\,2}\,\Delta(\Phi^2/c^4) \lesssim (1.6\times10^{-5})^2\times5\times10^{-19} \approx 1.3\times10^{-28}$ ($3\times10^{-29}$ on the isotropic benchmark) — essentially zero. This baseline is not a physical prediction for LLR; it is a consistency check that confirms the measured $\eta_{\rm resid}$ cannot arise from the bare coupling alone. $\eta_{\rm resid}$ is an emergent observable response coefficient, not a microscopic parameter.

The *phenomenological* TEP prediction is obtained from the volumetric suppression model, which integrates interior density profiles rather than surface compactness:

\begin{equation} \label{eq:volumetric_suppression} \eta_{\rm resid} \approx \alpha_{\rm eff}^{(\oplus-{\rm Moon})} \left(
\langle(\rho/\rho_T)^3\rangle_\oplus -
\langle(\rho/\rho_T)^3\rangle_{\rm Moon} \right) \end{equation}

Using physical parameters from the TEP framework paper (Paper 0, Jakarta v0.16): the canonical bare scalar-tensor coupling $\alpha_0 = -\sqrt{2}$ (DEF normalisation, $\alpha_0 = \sqrt{2}\,\beta_A$ with $\beta_A = -1$), an illustrative effective Earth–Moon coupling $\alpha_{\rm eff}^{(\oplus-{\rm Moon})} \approx 3 \times 10^{-3}$, and a simplified PREM density model with $\rho_T \approx 20$ g/cm³ from GNSS calibration (Paper 6) yielding an Earth-Moon shielding differential $\Delta\langle(\rho/\rho_T)^3\rangle \approx 3.22\times10^{-2}$. The phenomenological volumetric model prediction is therefore $\eta_{\rm resid} \approx (-3 \times 10^{-3}) \times (3.22 \times 10^{-2}) \approx -9.7 \times 10^{-5}$, agreeing in sign and order of magnitude but underestimating the measured headline value by a factor of approximately four; the magnitude match is contingent on the illustrative $\alpha_{\rm eff}$ and is not a test of the theory. This phenomenological volumetric model supplies a negative response in the $10^{-4}$ regime under its assumed weighting and normalization. It does not yet provide a first-principles, parameter-free prediction of $\eta_{\rm resid}$. The headline precision-weighted estimate $\eta_{\rm resid} = -3.91\times10^{-4}$ and the naïve OLS robustness check $\eta_{\rm resid} = -3.18\times10^{-4}$ lie in the same order-of-magnitude regime, but the phenomenological model uncertainties — PREM simplification, the exponent in $(\rho/\rho_T)^3$, $\rho_T = 20 \pm 7$ g/cm³, and the illustrative nature of the $\alpha_{\rm eff}^{(\oplus-{\rm Moon})}$ normalization — preclude a precision comparison at this stage.

Step 83 separates the historical volumetric weighting from the equilibrium-depth relation supplied by the potential sector (Paper 0 v0.16, Appendix E). The quartic matter-branch potential pins the equilibrium interior level as $\phi_{\min}(\rho) \propto \rho^{1/3}$ — the derived density response is the cube root, not the cube — supplying the depth-bookkeeping factor $S_A = \min[1,(\bar\rho/\rho_T)^{1/3}]$: $S_A^{(\oplus)} = 0.650$ ($0.43$–$0.88$ across the $\rho_T$ band) at $\bar\rho_\oplus = 5.49$ g/cm³ versus $S_A^{(\rm Moon)} = 0.550$ ($0.36$–$0.75$ across the $\rho_T$ band) at $\bar\rho_{\rm Moon} = 3.33$ g/cm³, an equilibrium-depth differential $\Delta S_A=0.100$, not a replacement for the illustrative $\alpha_{\rm eff}$ in a force or range equation. Volume-integrated over the same PREM-style profiles, the derived pinning differential $\Delta\langle(\rho/\rho_T)^{1/3}\rangle = 8.8\times10^{-2}$ likewise supersedes the phenomenological $\Delta\langle(\rho/\rho_T)^3\rangle = 3.22\times10^{-2}$ (which over-concentrates the weighting on the core). The $S_A$ factors record where each body's equilibrium interior level sits; they are not the excursion-transmission factor. The corrected static boundary calculation in Paper 0, Appendix E R17, separates infinitesimal susceptibilities from finite-excursion responses and does not establish a universal lunar transmission. Under the canonical unified metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, reducing at fixed cosmic epoch $A_{\rm sp}(t_0)=1$ to $\tilde g_{\mu\nu}=g_{\mu\nu}-(A_{\rm loc}^2(u_{\rm grav})-1)U_\mu U_\nu+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$), where the optical refractive index is $n(x)\simeq 1-2\Phi_N/c^2-\beta_A u_{\rm grav}(x)$, both the Cassini Shapiro delay and the direct scalar optical delay along the Earth–Moon laser path sample the same solar field $u_\odot(r)$. Step 83 evaluates both observables on the single solved two-branch solar profile $(k/\sqrt{2})q y^2+q^2 y^3=1$ ($k=16.03$, $g_t=cH_0/2$): at solar conjunction ($1.6\,R_\odot$), $y_{\rm tb}(1.6\,R_\odot)=2.16\times10^{-8}$ gives $\gamma_{\rm PPN}-1=-2\beta_A^2 y_{\rm tb}/(1+2\beta_A^2 y_{\rm tb})=-4.32\times10^{-8}$ (passing Cassini by $532\times$), while at $1\text{ AU}$ the same solution gives $y_{\rm tb}(1\text{ AU})=1.47\times10^{-5}$, suppressing the Earth–Moon scalar excursion to $\delta u_\odot^{\rm tb}=7.43\times10^{-16}\cos D$ and the direct two-way scalar optical path delay to $\delta r_{\rm path}^{\rm tb}=-\tfrac12 L\,\delta u_\odot^{\rm tb}\cos D =-1.43\times10^{-4}\text{ mm}\cos D$ ($\eta_{\rm tb}=-1.10\times10^{-8}$, $3.4\times10^4$ times smaller than the measured $\eta_{\rm resid}$ and consistent with direct-fit LLR bounds). Conversely, if the solar scalar field $u_\odot(r)$ along the optical path were unscreened ($y=1$, $\delta\phi_\odot=2.536\times10^{-11}\cos D$), it would give $\gamma_{\rm PPN}-1=-2/3$ at Cassini. Because at a fixed cosmic epoch inside the screened Solar System ($A_{\rm sp}(t_0)=1$) TEP's spatial 3-metric $\tilde\gamma_{ij}=\gamma_{ij}=(1-2\Phi_N/c^2)\delta_{ij}$ is identical to GR's (&ldquo;unstretched&rdquo; in Paper 0 denoting the absence of an additional local scalar factor $A_{\rm loc}^2(u_{\rm grav})$ on $\gamma_{ij}$, not the absence of GR's Newtonian spatial curvature), the external solar Newtonian potential differential $\Delta\Phi_{N,\odot}/c^2=2.536\times10^{-11}\cos D$ enters the BCRS$\to$GCRS transformation identically in TEP and GR. The $-4.851\text{ mm}\cos D$ scale ($\eta_{\rm ceiling}=-3.732\times10^{-4}$ at $T_{\rm Moon}=0.99685$, endpoint ceiling $L\,\delta\phi_{\rm eff}=9.702$ mm; Step 083) is therefore retained strictly as an unscreened reference ceiling (excluded as a direct solar scalar gradient by Cassini), not as a derived BCRS$\to$GCRS coordinate residual, and the quantitative mapping of the measured post-fit $\eta_{\rm resid}$ awaits a source-level integrator refit.

*Source-charge bookkeeping.* Read as a mediated-force estimate, the volumetric number is not internally consistent on the corpus's own terms. A differential-fall signature carries both vertices of the interaction, $\delta a/a_0 = (\alpha_0 S_\Sigma^{(\odot)})\,\Delta\alpha_{\rm body}$: the screened solar charge sourcing the field and the screened body coupling responding to it. Folding the Cassini-bounded source charge into the same bookkeeping returns $\eta_{\rm resid} \lesssim (1.6\times10^{-5})\times(3\times10^{-3})\times(3.22\times10^{-2}) \approx 1.6\times10^{-9}$ ($8\times10^{-10}$ isotropic) when $\alpha_{\rm eff}$ is read as a body coupling — and $\lesssim 4.8\times10^{-8}$ ($2.4\times10^{-8}$ isotropic) when it is read as the entire body-charge differential — at least four orders of magnitude below the measured band in either reading. The quoted $-9.7\times10^{-5}$ therefore admits no charge$\times$charge decomposition: it can only be an effective response coefficient in which the source factor has been absorbed rather than eliminated. This bookkeeping failure excludes the force-channel reading independently of, and prior to, the operator-level suppression below.

*Nested unit-term control.* The retained master-action control's screening closure (Paper 0 v0.16, Appendix E R11) derives the nested pairwise response $\mathcal{R}(s) = \mathcal{S}_\Sigma(X_{\rm env})^2\,y(s)$ — the flux-conserving shear profile $y[1+y^2(r_*/r)^4]=1$ propagated with the ambient inverse-stiffness factor at both vertices of the pair, with $r_* = \sqrt{GM/g_t}$ and $g_t = cH_0/(2|\beta_A|) = 3.4\times10^{-10}$ m s$^{-2}$. For the Earth–Moon pair — a hierarchical system embedded in Earth's nonlinear field plus the solar field at 1 AU — the dominant-member contribution makes the vertex factors asymptotically equal to the profile value (the identity $2X_{\rm dom}/\Lambda_X^4=(1-y)/y$), with the external ambients deepening the suppression further, returning $\mathcal{R} \simeq 2.2\times10^{-15}$ (the quartic-limit bookkeeping $y^3 \sim s^4$ gives $1.6\times10^{-14}$, the corpus's earlier $2.5\times10^{-14}$ at the same order), and $\simeq 3.3\times10^{-15}$ for the Sun–Earth pair (Step 085). The Sun–Moon channel, evaluated with the additional terrestrial-shell ambient at the lunar orbit, returns $\simeq 1.3\times10^{-15}$; the predicted differential response $\Delta_{\rm EM} \simeq 2.0\times10^{-15}$ lies a factor of $\sim65$ below the adopted $1.3\times10^{-13}$ differential-acceleration bound (`results/outputs/step_085_nested_llr_constraint.json`). A compactness-differential force anomaly of the kind the volumetric model describes is thereby suppressed to the $10^{-14}$–$10^{-15}$ level — invisible in this channel — and the illustrative $\alpha_{\rm eff}^{(\oplus-{\rm Moon})}$ normalization cannot be carried forward under the derived operator. This adjudication is a consistency check the closure passes rather than a new contradiction: this retained unit-term pairwise control predicts a dynamical Nordtvedt contribution below the stated bound; it is not a selected-action finite-source prediction, in line with the tight direct-fit LLR bounds on differential fall. The measured $\cos(D)$ residual — independently present in the published ephemeris record — is not explained by this pairwise force projection. Its established TEP analysis channel is the clock-sector/ephemeris-reduction projection: the lunar conformal field carries a genuine synodic component, $\delta\phi_\odot \simeq 2.5\times10^{-11}\cos D$ computed from the real orbital geometry — the exact Fourier projection of $u_\odot(d_{\rm SM}(D)) = GM_\odot/[\,d_{\rm SM}(D)c^2\,]$ with $d_{\rm SM}(D) = \sqrt{a_\oplus^2+a_{\rm M}^2-2a_\oplus a_{\rm M}\cos D}$ returning $2.54\times10^{-11}$ on the $\cos D$ harmonic (step 83; the peak-to-peak excursion is $5.1\times10^{-11}$) — which an ephemeris built on a fixed GR dynamical time standard projects into post-fit residuals — the channel on which $\eta_{\rm resid}$ is defined. This channel is not product-limited in the sense of the bookkeeping above, for a structural reason rather than by stipulation: the product bound governs charge$\times$charge mediation, a force transferred between two screened bodies, whereas the reduction projection is not a pairwise interaction at all. The conformal excursion $\delta\phi_\odot(t)$ is a one-point function of the temporal landscape evaluated at the Moon's instantaneous position, and its imprint on the residuals is set by the projection map of an ephemeris held to a fixed dynamical time standard, not by a second screened charge. What the Cassini bookkeeping and the canonical pairwise projection jointly bound is the mediated differential-fall channel at that order; the clock-sector channel is bounded instead by the field-excursion amplitude and the estimand map. Whether that projection reaches the measured amplitude is precisely the estimand question the source-level integrator refit (Section 4.14) addresses.

Finite-body bridge geometry is tested separately in Step 084 (`results/outputs/step_084_earth_moon_solar_bridge.json`). Radial baseline and two-branch flux laws are evaluated for uniform-density Earth and Moon sources at the measured separation, then combined with a locally uniform solar-gradient background over five orientations. Under this explicitly matched-profile ansatz, exact collinearity permits an exterior bridge cancellation: the two-branch local $P_{,X}$ reaches its numerical $10^{-8}$ regulator from a solar-background value near $6.8\times10^4$. The transverse configuration retains the ambient stiffness, while $P_{,X}&lt;1$ occurs only within about $0.020^\circ$ of collinearity. This identifies a sharply defined nonlinear boundary channel rather than an additional detected force. Superposed isolated profiles are not the solution of the coupled nonlinear three-source equation; the local minimum alone determines neither integrated scalar charge nor a synodic range perturbation. The INPOP19a estimate $\eta_{\rm resid}=(-4.06\pm0.66)\times10^{-4}$ from Step 056 is the empirical comparator, not an output of Step 084. Paper 0 supplies the common kinetic operator and test-probe scope; the finite-body and ranging calculation belongs to this paper.

Step 086 (`results/outputs/step_086_earth_moon_coupled_pde.json`) supplies the coupled solve that Step 084's ansatz only diagnoses: the nonlinear two-source equation is integrated in a uniform solar-gradient background at the two axisymmetric syzygy configurations (Full Moon $u_0&gt;0$ and New Moon $u_0&lt;0$), on the finite-volume cylinder discretisation shared with Paper 0. Smoothed Earth and Moon source charges and per-sector solar gradients are solved by damped Newton iteration to relative residual $&lt;10^{-6}$; the Gaussian source widths and grid do not resolve the actual body interiors or the solar tidal variation. At Full Moon ($u_0&gt;0$, where terrestrial and solar gradients add between Earth and Moon and the lunar-side saddle lies inside the smoothed source width), the field along the pair axis stays solar-dominated with $|\nabla\phi|_{\min}\approx335\,g_t$ ($P_{,X}\sim1.2\times10^{5}$); at New Moon ($u_0&lt;0$, where terrestrial and solar gradients oppose between the bodies), the axial gradient crosses zero at $z/d\approx0.432$ (axial cell-centre minimum $|\nabla\phi|_{\min}\approx36.3\,g_t$, $P_{,X}\sim1.7\times10^{3}$), but off-axis transverse gradients keep the resolved $P_{,X}&lt;1$ volume fraction at $0.0$ in both orientations. An isolated axial saddle point therefore does not unscreen a macroscopic corridor or relax Solar-System range constraints. Step 084's orientation scan remains an ansatz diagnostic, not a rigorous bound on the nonlinear off-axis solution. Physical finite-source matching and the range-model projection delimit application to LLR.

The cosD-only baseline $\eta_{\rm resid} = -3.18\times10^{-4}$ is also ~14&times; larger than the standard scalar-tensor benchmark estimate $\eta_{\rm BD} = 2\alpha_0^2/(1+\alpha_0^2) \approx 2.3\times10^{-5}$ evaluated at the standard unscreened Cassini bound $\alpha_0 \lesssim 3.4\times10^{-3}$, consistent with the interpretation that the LLR observable response absorbs contributions from the TEP screening mechanism beyond bare scalar-tensor physics. The coefficients $\kappa_{\rm MSP} \sim 10^6$–$10^7$ and $\kappa_{\rm Cep} \sim 10^6$ from related work in the same framework (Papers 10 and 11) carry different units (spin-down and magnitude, respectively) from $\eta_{\rm resid}$ and therefore cannot be compared numerically. They are cited here as domain-specific observable response coefficients in the same qualitative framework, not as quantitative cross-calibrations. No separate $\kappa_{\rm LL}$ parameter is required; $\eta_{\rm resid}$ itself is the observable response coefficient for LLR. The bookkeeping ladder is therefore: (i) bare compactness-squared null check ($\lesssim 3\times10^{-29}$ at the screened source charge), (ii) volumetric effective-response band ($-9.7\times10^{-5}$; non-decomposable as a mediated force and superseded by the canonical operator), (iii) measured LLR response coefficient $\eta_{\rm resid}$ on the admissible clock-sector channel.

### 2.2 Predicted Signal in LLR

The residual-channel estimand is extracted on the standard synodic basis: a fitted modulation of the Earth-Moon O–C range residuals as the Earth-Moon system orbits the Sun, of the form:

\begin{equation} \label{eq:range_perturbation} \delta r = 13 \eta_{\rm resid}
\cos(D) \end{equation}

The predicted amplitude scales linearly: $|\delta r| = 13 |\eta_{\rm resid}|$ meters. For the order-of-magnitude estimate $\eta_{\rm resid} \sim 10^{-4}$, this gives $\sim$1.3 mm. The headline precision-weighted estimate ($\eta_{\rm resid} = -3.91 \times 10^{-4}$, Section 4.1) predicts an amplitude of $\sim$5.1 mm, while the leverage-excised robustness check ($\eta_{\rm resid} \approx -3.6 \times 10^{-4}$) predicts $\sim$4.7 mm and the naïve cosD-only OLS robustness check ($\eta_{\rm resid} \approx -3.18 \times 10^{-4}$) yields $\sim$4.13 mm. These amplitudes are small compared to the centimetre-level residual RMS, but with 26,207 observations over 35 years, statistical averaging provides sensitivity well below the per-observation noise floor.

### 2.3 Continuous Gradient Suppression and the Saturation Scale

The operative v0.16 shear-screening mechanism is the nonlinear kinetic response $P_{,X}$ of the solved scalar gradient. Density, compactness and $\rho_T$ are environmental diagnostics, not independent microscopic screening laws. The canonical Earth–Moon pairwise benchmark is strongly suppressed. Step 086's coupled syzygy diagnostic replaces the earlier matched-profile bridge estimate for its stated source geometry; it does not itself perform an ephemeris-level range fit. Historical compactness and volumetric checks are retained separately from this kinetic result.

The Earth–Moon system is embedded in the solar and Galactic field, so interplanetary density alone does not fix its response. The following heliocentric modulation is an empirical residual-channel template, not an independently derived Nordtvedt force law:

\begin{equation} \label{eq:orbital_modulation} \eta_{\rm resid}(r_\odot) = \eta_0 \left( 1 + m \cdot \frac{r_{\rm mean} -
r_\odot}{r_{\rm mean} \cdot e_\oplus} \right) \end{equation}

where $\eta_0$ is the baseline residual-channel coefficient at mean orbital distance $r_{\rm mean}$, $e_\oplus \approx 0.0167$ is Earth's orbital eccentricity, and $m$ is the modulation depth characterizing the nonlinear response. For $m \approx 1$ (consistent with the observed sign-flip), the orbital modulation generates diagnostic sidebands at frequencies $D \pm l'$ (where $l'$ is the orbital longitude), spectrally orthogonal to classical multi-body resonances. This provides a unique signature distinguishing TEP from standard PPN or systematic artifacts.

The TEP framework further predicts that the coupling depends not only on the scalar gradient magnitude (distance) but also on the rate at which the Earth-Moon system traverses the temporal topology. A body moving through a spatial scalar gradient $\nabla\phi$ at velocity $\mathbf{v}$ experiences an effective temporal shear rate $d\phi/dt \approx \mathbf{v} \cdot \nabla\phi \approx v_r \, |\nabla\phi|$, where $v_r$ is the heliocentric radial velocity. For a Kepler orbit with small eccentricity, distance $r$ and radial velocity $v_r$ are approximately in quadrature ($90^\circ$ out of phase), making them statistically distinguishable predictors. The full dynamical modulation therefore takes the form:

\begin{equation} \label{eq:velocity_modulation} \eta_{\rm resid}(r_\odot, v_r) = \eta_0 \left( 1 + m_r \cdot \frac{r_{\rm mean} -
r_\odot}{r_{\rm mean} \cdot e_\oplus} + m_v \cdot \frac{v_r}{\bar{v}_r} \right) \end{equation}

where $\bar{v}_r$ is the characteristic radial velocity scale ($\approx 0.5$ km/s for Earth's orbit) and $m_v$ is the velocity modulation depth. A joint fit to both $r_\odot$ and $v_r$ determines whether the temporal topology is purely static ($m_v = 0$) or dynamically responsive ($m_v \neq 0$). The heliocentric radial velocity analysis tests this prediction directly, using DE440 ephemeris velocities computed via Skyfield for every LLR observation epoch.

A further TEP-motivated probe uses a *fixed celestial axis* tied to the Planck 2018 dipole as an operational reference: $(l, b) = (264.02^\circ, 48.25^\circ)$ in galactic coordinates, equivalent to $(\alpha, \delta) = (168.14^\circ, -7.22^\circ)$ in J2000 equatorial coordinates, with amplitude $v_{\rm CMB} \approx 369$ km/s for the kinematic dipole. Some scalar-field embeddings single out a large-scale rest frame; the LLR regressions reported here do not, by themselves, establish Lorentz violation or uniqueness of that frame. They test whether residual-channel structure is consistent with anisotropic coupling defined relative to $\hat{\mathbf{n}}_{\rm CMB}$ and with other fixed axes used as controls (Section 4.25.2). Two operational predictions tied to $\hat{\mathbf{n}}_{\rm CMB}$ are:

First, an *annual velocity projection*: Earth's orbital velocity ($\sim 30$ km/s) projects onto the CMB dipole direction with a sinusoidally varying parallel component $v_\parallel(t) = \mathbf{v}_{\rm orb} \cdot \hat{\mathbf{n}}_{\rm CMB}$. The CMB dipole lies at ecliptic longitude $\approx 173^\circ$, offset by approximately $70^\circ$ from the perihelion longitude ($\approx 103^\circ$), making the annual velocity projection phase-shifted relative to the heliocentric distance modulation and therefore statistically distinguishable.

Second, a *monthly orientation anisotropy*: the Earth-Moon line sweeps across the celestial sphere with synodic period. If an effective gradient projection onto $\hat{\mathbf{n}}_{\rm CMB}$ is present in the same channel, the coupling should depend on the cosine of the angle $\theta$ between the Earth-Moon vector and that axis:

\begin{equation} \label{eq:cmb_anisotropy} \eta_{\rm resid} = \eta_0 \left( 1 + m_{\rm CMB} \cdot \cos\theta_{\rm EM-CMB} \right) \end{equation}

where $\cos\theta_{\rm EM-CMB} = \hat{\mathbf{r}}_{\rm EM} \cdot \hat{\mathbf{n}}_{\rm CMB}$. Because the Earth-Moon line rotates on a monthly timescale while the synodic signal operates on a 29.5-day period, the two modulations are at different frequencies and can be separated in a joint fit. The CMB dipole orientation analysis tests both predictions directly.

Potential curvature contributes to the perturbation operator, but a density-dependent mass does not replace the kinetic response as an independent Earth–Moon screening prescription. Compactness diagnostics and the empirical modulation templates above do not establish either the absolute force or the clock-to-range transfer of the measured term.

### 2.4 The Secular Channel and the $\dot G/G$ Bound

Dedicated LLR fits constrain the present local dynamical observable $|\dot G/G|\lesssim4\times10^{-13}\,{\rm yr}^{-1}$ (Williams et al.). If the scalar-response factor and the gravitational-to-atomic unit conversion are held fixed, $G_{\rm loc}=G_*A^2(1+\alpha^2)$ gives the conditional relation $\dot G/G\simeq2\,d\ln A/dt$ and the corresponding local scale $|\dot A/A|\lesssim2\times10^{-13}\,{\rm yr}^{-1}$. Conversion to an ambient-field history requires the actual source, body and time-dependent measurement response. Paper 0, R17 supplies a static boundary diagnostic, not that cosmological transfer. The present bound is therefore not extrapolated into a cosmological lookback constraint or used to assign redshift uniquely to an emission-epoch spatial contrast (Paper 0, §8).

## 3. Data and Methods

### 3.1 Data

#### 3.1.1 INPOP19a Residuals

The primary dataset consists of 26,207 LLR O-C (observed minus computed) residuals from the INPOP19a planetary ephemeris (Fienga et al. 2019). The data span 35 years from 1984 to 2019 and include observations from five laser ranging stations: APO (Apache Point Observatory), Grasse, Matera, McDonald2, and Haleakala. The residuals have an RMS of 9.5 cm, representing the highest-precision LLR dataset currently available for SEP tests. After 6$\sigma$-equivalent MAD cleaning ($N = 25{,}445$), Grasse contributes 73.7% of the retained archive (18,742 shots); the raw 26,207-point archive is 74.0% Grasse (19,390 shots). Headline inference uses the cleaned sample unless a step explicitly states otherwise.

INPOP19a is a state-of-the-art ephemeris developed by the Paris Observatory that fits all available LLR data using a consistent modelling framework. The residuals represent the difference between observed round-trip laser pulse times and the times predicted by the ephemeris model, after accounting for all known physical effects including tides, relativity, and atmospheric delays.

#### 3.1.2 DE430 Residuals

For cross-ephemeris validation, DE430 residuals from JPL (Folkner et al. 2014) were analysed. The dataset spans 2014–2018. The raw file contains gross outliers (RMS = 26.7 cm); after 6σ-equivalent (MAD-based) cleaning, the RMS drops to 5.8 cm.

The uncleaned raw DE430 dataset shows no significant correlation with $\cos(D)$ ($r = -0.00015$, $p = 0.992$), as 37 phase-clustered gross outliers dominate the uncleaned covariance. The same $6\sigma$ MAD criterion is applied to both ephemerides and was fixed prior to the residual-channel fit. Once this standard outlier cleaning is applied (removing 37 observations, 0.8%), the cleaned DE430 archive ($N = 4{,}560$) recovers $\eta_{\rm resid} = -7.04 \times 10^{-4} \pm 1.18 \times 10^{-4}$ at $5.96\sigma$, independently confirming the negative synodic modulation found in INPOP19a. The primary detection analysis relies on the INPOP19a ephemeris due to its 35.5-year baseline (Section 4.1); detailed DE430 outlier and sideband behavior are presented in Section 4.13.

#### 3.1.3 Data Processing

The raw residual data were processed to extract high-precision kinematic quantities for each observation. To eliminate the 0.5% "synodic blurring" inherent in mean-phase approximations, Moon-Sun elongation angles were computed using the Skyfield library with the DE440 planetary ephemeris. This ensures coordinate precision at the sub-millimeter level relative to the geocenter. For each observation, the following quantities were extracted:

- Residual value: The O-C residual in centimeters

- 
Moon-Sun elongation: High-precision apparent elongation computed via
Skyfield/DE440

- 
Synodic phase: The phase in the lunar synodic cycle (0 = new moon, $\pi$
= full moon)

- Station identifier: The observing station

- Time: UTC timestamp of each laser shot

The entire pipeline is fully deterministic. Given the public INPOP19a `.csv` residual files and the DE440 `.bsp` kernel, every numerical result in this paper — including the primary regression estimate, all null-hypothesis simulations, and the Bayesian posterior — can be reproduced bit-for-bit from the repository using `seed = 42` and the provided `run_pipeline.sh` script.

#### 3.1.4 Statistical Power Criteria for Station Classification

To assess whether individual stations possess sufficient statistical power to constrain the Nordtvedt parameter, objective power criteria are applied in the station power analysis. These criteria classify stations based on their expected detection capability given sample size, precision, and phase coverage:

- 
Powered-detection threshold: Stations with expected
SNR $\geq 3\sigma$ at the globally measured $|\eta| \approx 4.16
\times 10^{-4}$ are designated as *powered stations*. The
threshold balances detection power with sample size requirements.

- 
Precision criterion: Sub-decimeter tracking
capabilities (RMS $\lesssim$ 10 cm for legacy data, $\lesssim$ 3 cm
for modern C-SPAD era) are required for reliable $\eta \sim 10^{-4}$
detection.

- 
Phase-coverage requirement: Adequate synodic phase
coverage (mean $|\cos D| &lt; 0.5$) is required to avoid
truncation-induced slope bias. Severe phase truncation yields
unreliable OLS estimates regardless of sample size.

Application of these power criteria classifies only Grasse as achieving conventional cosD-only significance at observed SNR $\geq 3.0\sigma$ ($4.97\sigma$). APO reaches $2.77\sigma$ despite moderate expected power at the global $|\eta|$ (expected $3.66\sigma$), and Matera, McDonald2, and Haleakala remain below the powered threshold on expected SNR. This pattern is expected for millimetre-scale signals in precision LLR metrology, which is why the primary detection relies on combined analysis across all stations with $N = 25{,}445$ observations.

Three stations are classified as underpowered based on expected SNR at the pooled reference amplitude, evaluated on the $6\sigma$-cleaned archive. Matera ($N = 345$) lacks sufficient sample size, with expected SNR = $0.38\sigma$. McDonald2 suffers from severe phase truncation, yielding expected SNR = $1.04\sigma$. Haleakala, which operated 1984–1990 with 10.6 cm cleaned RMS (13.8 cm raw), achieves only $0.27\sigma$ expected SNR at the pooled $|\eta|$ — below the powered threshold.

Haleakala's raw-sample estimate $\eta = +3.55 \times 10^{-3}$ yields an observed SNR of $2.45\sigma$ ($p = 0.014$; cleaned-archive $\eta = +2.57 \times 10^{-3}$, $2.21\sigma$), opposite in sign to the global detection. Given its underpowered status (expected SNR = $0.27\sigma$ at the pooled $|\eta|$, below the $3\sigma$ threshold), the station is down-weighted in precision-weighted regression.

### 3.2 The Synodic Residual Template

#### 3.2.1 Measured Signal

The regression tests a synodic term $\delta r=13\eta_{\rm resid}\cos D$ in the published post-fit range residuals, with $\delta r$ in metres. The conventional Nordtvedt range normalization defines the fitted coefficient but does not identify it with the direct-fit SEP parameter. GR predicts zero metric Nordtvedt parameter; a nonzero post-fit $\eta_{\rm resid}$ may also contain ephemeris or measurement effects.

#### 3.2.2 Amplitude Interpretation

The headline residual coefficient corresponds to approximately $5.1$ mm in this normalization. It is measured by the regression, not supplied as a kinetic-theory prediction from Earth–Moon compactness. Historical effective-response estimates are separated from the suppressed force benchmark in Section 2.1. A clock-sector attribution requires the field-to-observable transfer and the ephemeris refit described there.

### 3.3 Statistical Methods

#### 3.3.1 Overview of Analysis Pipeline

The analysis employs a comprehensive seven-group pipeline of 91 scripts designed to detect and validate the TEP Nordtvedt signal with maximum statistical rigor:

| Group | Steps | Purpose | Key Analyses |
| --- | --- | --- | --- |
| A | 000–003 | Core Detection | Simple OLS regression, Bayesian MCMC, Cook's distance analysis |
| B | 004–022 | Extended Robustness | Perihelion/aphelion subsets, individual station analysis, epoch analysis, Cook's distance excision, precision-weighted regression |
| C | 023–028, 047–048 | Physical Signal Probes | Heliocentric distance scaling, orbital velocity modulation, CMB dipole anisotropy, synodic/anti-synodic phase test, Lomb-Scargle periodogram, volumetric suppression model |
| D | 029–032 | Defensibility | Day/night thermal bias test, geometric elongation validation, station power analysis, hardware epoch consistency |
| E | 033–046b, 049–054, 057, 059–064-SRP | Advanced Defensibility & Cross-Validation | Quantitative η prediction, dust sensitivity, station balance, clean subset, CMB robustness suite, Haleakala fluctuation, Grasse sufficiency, Gaussian process extraction, systematic sensitivity (known-systematic MC null tests; adversarial PCA/GP nuisance search; era$\times$station$\times$lunation grid; blind 5-fold year cross-validation), orbital SRP bounds (064-SRP), atmospheric seeing |
| F | 014–015, 040–043, 055–056, 062–066, 064-PI | Results Consolidation & Spectral Validation | Inter-station meta-analysis, unified results table, multiple-testing correction, ephemeris absorption simulation, orthogonality proof, sideband survival, high-dimensional absorption test, false-positive simulation, outlier sensitivity, prediction-interval calibration (064-PI) |
| G | 067–071 | Advanced Estimator Corrections | Cluster-robust AR(1) combined standard errors, weighted robust M-estimator, rolling η(t) environmental correlation, DE430 full environmental model, stratified equal-N test |

This multi-layered approach ensures that the detection is robust against systematic errors, statistical artifacts, and instrumental biases. Each group addresses specific validation concerns, from core signal detection to peer-review-level defensibility tests.

The core statistical analysis (steps 000–003) provides the primary detection, while subsequent steps perform extended systematic checks, robustness validation, and theoretical consistency tests.

#### 3.3.2 Pearson Correlation Analysis

The primary analysis computes the Pearson correlation coefficient between the O-C residuals and $\cos(D)$, where $D$ is the Moon-Sun elongation. A significant negative correlation would indicate the predicted TEP signal. The correlation is computed for the full dataset and separately for each station to test consistency across independent observatories.

#### 3.3.3 Linear Regression

A linear regression model is fit to extract the Nordtvedt parameter:

\begin{equation} \label{eq:linear_regression} r = 13 \eta_{\rm resid} \cos(D) + b
+ \epsilon \end{equation}

where $r$ is the residual, $b$ is an intercept, and $\epsilon$ represents noise.  An intercept is included in all primary regressions even though INPOP19a residuals are mean-subtracted by construction (mean = 0.0007 m $\approx$ 0).  This guards against any residual station-epoch-dependent offset that would otherwise bias the $\cos(D)$ amplitude.  The fitted coefficient of $\cos(D)$ yields an estimate of $\eta$ with uncertainty from the regression standard error.

#### 3.3.4 Differential Phase Analysis

An independent test compares residuals at new moon ($D \approx 0$) versus full moon ($D \approx \pi$). The mean residual difference $\Delta r = \langle r_{\rm new} \rangle - \langle r_{\rm full} \rangle$ should be approximately $26\,\eta_{\rm resid}$ for the TEP signal. This differential analysis provides a robust confirmation that does not rely on the specific functional form of the correlation.

#### 3.3.5 Advanced Robust Analysis

To assess the robustness of any potential detection, the pipeline implements a comprehensive battery of diagnostic checks and robustness tests spanning multiple independent analysis families:

- 
Bootstrap confidence intervals: 10,000 resampling iterations to
estimate bias-corrected correlation coefficient and 95% confidence
intervals

- 
Permutation test: 10,000 random shuffles of residuals to test null
hypothesis (no correlation)

- 
OLS linear regression: Standard ordinary least squares regression
for amplitude estimation

- 
Theil-Sen robust regression: Median of pairwise slopes estimator
resistant to outliers

- 
Leverage analysis: Cook's distance diagnostics to identify
high-leverage observations that influence OLS estimates

- 
Outlier detection: Three independent methods (IQR $3\times$, sigma
$5\times$, Isolation Forest 1% contamination)

- 
Differential phase analysis: New moon vs full moon residual
comparison

- 
Station-by-station consistency: Independent analysis of each LLR
station

- 
Temporal stability analysis: 7 temporal bins to test signal
stability over time

- 
Station-specific temporal analysis: Temporal stability analysis
performed separately for each station to identify if temporal
non-stationarity is driven by specific stations

- 
Station dominance test: Compares global analysis with Grasse-only,
non-Grasse, and APO+Grasse combined analyses to test if signal is
driven primarily by the dominant station

- 
Cross-station validation: Tests whether the signal from one station
can predict the signal in another independent observatory

- 
Phase-binned analysis: 8 elongation phase bins to test functional
form

- 
Systematic error modelling: Tests for temporal drift, harmonics,
sin(elongation) correlation, outlier sensitivity, magnitude
dependence

- 
Sensitivity analysis: Vary elongation mask width, phase offset, and
temporal bin size

- 
K-fold cross-validation: 5-fold CV to test model prediction
performance on held-out data

- 
Holdout test: 80/20 train-test split to verify model predictions on
independent test set

- 
Systematic control analysis: Partial correlations controlling for
temporal trends, seasonal effects, station-specific drifts, and
residual magnitude

- 
Noise injection and signal recovery: Tests of signal robustness to
added Gaussian noise and validation that pipeline recovers injected
signals

- 
Subsample robustness: Five-category validation including (1) single
80% subsample replication, (2) multiple iteration stability (10
subsamples), (3) station jackknife with underpowered-sample
exclusion (SNR > 3 criterion), (4) station weight sensitivity
analysis, and (5) inverse-probability weighted (IPW) regression for
station-balance validation

- 
Bayesian MCMC analysis: Ensemble sampler (emcee) with 32 walkers to
sample the posterior distribution of $\eta$ and compute the
Savage-Dickey Bayes Factor

- 
Lomb-Scargle frequency sweep: Tests frequency specificity of the
signal across $0.5\nu$ to $1.5\nu$ to confirm synodic phase-locking

- 
Grand Phase Fold: Coherent phase analysis across 48 bins spanning 35
years to test long-term phase stability

This multi-method approach provides robustness against systematic errors and statistical artifacts.

Three systematic analysis families directly test the systematic artifact hypothesis. Systematic control analysis tests whether the signal persists after controlling for known systematic variables including temporal trends, seasonal cycles, and station drifts. Noise injection and signal recovery quantifies signal robustness to noise addition and validates detection methodology through signal recovery tests. Subsample robustness implements five-category validation including subsample replication, station jackknife, and IPW station-balance regression. Station decomposition analysis decomposes the pooled fit into station-specific contributions. Detailed results are presented in Section 4.20.

All analyses use fixed random seeds (seed = 42) for reproducibility and are parallelized using multiprocessing (12 workers on M4 Pro).

#### 3.3.6 Significance Testing and Birge Ratio Scaling

Statistical significance is assessed using the t-statistic for the correlation coefficient and regression coefficients. The Birge Ratio, $R_B = \sqrt{\chi^2_{\rm red}}$, is computed as an over-dispersion guard for every regression: when $R_B > 1$, formal errors are scaled upward by $R_B$; when $R_B &lt; 1$, no downward rescaling is applied. For the headline precision-weighted regression, the pipeline defines relative station weights $w_i \propto 1/\mathrm{RMS}_{\rm station}^2$ (normalized to unit scale rather than absolute inverse per-row variance $1/\sigma_i^2$ in metres$^{-2}$) while accumulating residuals in metres, which yields a nominal weighted sum-of-squares statistic $\chi^2_{\rm red} = 0.0038$ ($R_B = 0.062$). That small value is a unit-normalization consequence of using relative station weights in metre units rather than physical under-dispersion of the data; when evaluated against calibrated published per-shot uncertainties $\sigma_m$ (Section 4.9), the true reduced chi-square is $\chi^2_{\rm red} = 0.48$, reflecting slightly conservative published error bars. In both conventions $R_B &lt; 1$, so the Birge guard applies no downward rescaling and the nominal $6.94\sigma$ WLS significance is evaluated straight from the regression covariance matrix. All reported p-values are two-tailed.

#### 3.3.7 Bayesian MCMC Analysis

A synodic-only Bayesian layer (emcee; Foreman-Mackey et al. 2013) uses heteroskedastic Gaussian likelihood with archive $\sigma_i$ and uniform priors $\eta \in [-10^{-2}, +10^{-2}]$, $b \in [-0.1, +0.1]$ m. The MCMC analysis runs separate chains for four documented $\eta$ priors (Table 4.18.1) and evaluates the Savage–Dickey ratio at $\eta=0$ with multiple KDE bandwidths. This layer is secondary to the cluster-robust frequentist headline and the linearized integrator extraction.

#### 3.3.8 Lomb-Scargle Frequency Analysis

To test frequency specificity, a Lomb-Scargle periodogram was computed for the residuals as a function of synodic phase. The frequency grid spans $0.5\nu$ to $1.5\nu$ with 10,000 linearly-spaced points, where $\nu = 1/29.530588$ days$^{-1}$ is the synodic frequency. The periodogram power at each frequency is normalized by the variance, and significance is assessed via the false alarm probability (Scargle 1982). While the residuals contain multiple periodicities from unmodelled perturbations, the synodic frequency is identified as a significant spectral feature (Rank 4 in the modern C-SPAD era). The synodic frequency is identified as the absolute maximum peak in the frequency-specific regression scan, confirming that the detected Nordtvedt modulation is uniquely phase-locked to the Earth-Moon-Sun geometry.

#### 3.3.9 Power and Sensitivity Analysis

A power analysis is performed to determine the minimum detectable Nordtvedt parameter given the data precision, sample size, and the empirical spread of the predictor $x=\cos D$. For $N = 26{,}207$ observations, residual RMS = 9.5 cm, and $\sigma_{\cos D} = 0.491$, the expected standard error on the correlation coefficient is $\sigma_r \approx 1/\sqrt{N} = 0.0062$. Mapping correlation to slope via $r \approx A\,\sigma_{\cos D}/\sigma_{\rm residual}$ implies a $3\sigma$ minimum detectable amplitude $A_{\min} \approx 3\sigma_r\,\sigma_{\rm residual}/\sigma_{\cos D} = 0.36$ cm, corresponding to $\eta_{\min} \approx 2.76 \times 10^{-4}$. The pipeline is sensitive to $\eta$ at the few $10^{-4}$ level; the power to detect $\eta = 3\times 10^{-4}$ is $\approx 90\%$ at $\alpha = 0.05$, rising to $\approx 99.97\%$ for $\eta = 5\times 10^{-4}$.

#### 3.3.10 TEP Core Density Simulation

The core-density simulation integrates simplified PREM interior profiles for Earth and the Moon against the saturation scale $\rho_T$ to obtain the shielding differential $\Delta\langle(\rho/\rho_T)^3\rangle \approx 3.22\times10^{-2}$, the input of the volumetric bookkeeping of Section 2.1: \[ \eta_{\rm resid} \simeq \alpha_{\rm eff}^{(\oplus-{\rm Moon})} \Delta\!\left\langle(\rho/\rho_T)^3\right\rangle, \] where the angle brackets denote volumetric averages over the two interiors. Inserting the illustrative $\alpha_{\rm eff}^{(\oplus-{\rm Moon})} \approx 3\times10^{-3}$ yields the effective response band $\eta_{\rm resid} \approx -9.7\times10^{-5}$ in the measured decade. As established in Section 2.1, that number is an effective response coefficient rather than a decomposable differential-fall estimate: the same step folds in the Cassini-bounded solar source charge and returns a mediated-force ceiling of $\approx 8\times10^{-10}$ (or $2.4\times10^{-8}$ with $\alpha_{\rm eff}$ read as the full body-charge differential), and the canonical pairwise operator independently suppresses the Earth–Moon force channel to $\sim 10^{-14}$. The simulation's adjudicating role is thus bookkeeping — it confirms that the measured band cannot be sourced by the mediated-force sector, leaving the clock-sector projection of Section 2.1 as the admissible channel. The reader is referred to Section 2.1 for the full derivation and numerical calibration.

#### 3.3.11 Data Quality Validation

Comprehensive data quality validation is implemented to ensure the integrity of the raw data. For each station, the following validations are performed:

- 
Elongation validation: Ensures all elongation values are in the
valid range $[0, 2\pi]$ radians

- 
Residual validation: Ensures all residuals are within physically
reasonable range ($\pm 5$ m)

- 
Date validation: Ensures all Julian dates are in the expected range
(1980-2025)

- 
Missing value detection: Identifies and removes observations with
NaN or infinite values

All five stations (APO, Grasse, Matera, McDonald2, Haleakala) passed basic data quality validation with no observations removed, confirming the integrity of the INPOP19a dataset. Passing basic quality checks confirms data integrity but does not guarantee statistical power for SEP detection. Per the statistical power analysis, three stations (Matera, McDonald2, Haleakala) lack sufficient power for independent detection and are retained for validation while appropriately down-weighted in precision-weighted regression.

### 3.4 Systematic Checks

#### 3.4.7 Ephemeris Errors

To test whether any observed signal could arise from systematic errors in the INPOP19a ephemeris, residuals from each station are analysed independently. A common ephemeris error would likely affect all stations similarly, while a genuine physical signal would be expected to persist across independent observatories with different hardware and geographic locations.

A key methodological assumption is that the INPOP19a fitting procedure does not absorb a potential Nordtvedt-type signal into its model parameters. INPOP19a is constructed under the standard GR framework ($\eta = 0$); no TEP-specific density-dependent scalar coupling is included in the ephemeris model. A genuine TEP contribution could be partially absorbed by fitted ephemeris parameters. The present analysis tests whether a synodic component survives that absorption; its primary measured amplitude is approximately 0.51 cm, compared with a residual RMS of 9.5 cm.

#### 3.4.12 Environmental Effects

Atmospheric refraction, tidal effects, and other environmental factors are modelled in the residual computation by the ephemeris. The signal correlates specifically with Moon-Sun elongation (the predicted TEP signature) rather than with other environmental variables such as local time or weather conditions.

#### 3.4.14 Instrumental Effects

Different laser ranging systems could have systematic biases. The multi-station analysis tests whether the signal persists across stations with different hardware configurations, arguing against a common instrumental origin.

#### 3.4.15 Outlier Detection and Removal

Robust statistical methods are employed to identify and handle outliers. Observations with residuals exceeding $5\sigma$ from the median are flagged for inspection. The analysis is performed both with and without outliers to assess robustness of any findings. Outlier removal is conservative: only points with clear instrumental signatures (e.g., timing errors, atmospheric anomalies) are excluded from the primary analysis, with sensitivity tests showing the TEP signal remains significant regardless of outlier treatment.

#### 3.4.16 Systematic Error Budget: cos($D$)-Projection

A data-driven systematic error budget is constructed directly from the INPOP19a residuals and upstream pipeline outputs.  The critical methodological point is that only the component of a systematic source $s$ that is *correlated with $\cos(D)$* can bias the synodic slope; the orthogonal component contributes noise (absorbed into the statistical error) but does not shift the point estimate.  For a linear model $r = A \cos(D) + b + \epsilon$, the bias to the OLS slope from a systematic $s$ is

\begin{equation} \label{eq:systematic_projection}
\delta A_{\rm sys} = \frac{\mathrm{cov}(s, \cos D)}{\mathrm{var}(\cos D)},
\qquad
\delta\eta_{\rm sys} = \frac{\delta A_{\rm sys}}{13}.
\end{equation}

Equation $\eqref{eq:systematic_projection}$: cos($D$)-projected systematic bias.  A systematic of arbitrary raw amplitude cannot bias $\eta$ unless it carries a component coherent with $\cos(D)$.  The total raw systematic RMS (the naive quadrature sum of all amplitudes) is therefore not the operative uncertainty; the projected bias is.

Each component is computed from an observable proxy in the data and projected onto $\cos(D)$ via Equation $\eqref{eq:systematic_projection}$. The table below gives both the raw amplitude (what a naive RMS budget would quote) and the projected bias (the only quantity that can shift the synodic slope):

Systematic error budget: raw amplitude versus cos($D$)-projected bias. All values in cm. The projected total is the operative systematic uncertainty; the raw total is shown only to demonstrate the attenuation from projection.

| Source | Raw amplitude (cm) | Projected bias (cm) | Projection ratio |
| --- | --- | --- | --- |
| Ephemeris modelling (cross-ephemeris scatter) | 0.36 | 0.36 | 1.00 |
| Atmospheric delay (seasonal tropospheric delay) | 0.95 | 0.08 | 0.08 |
| Instrumental (station-to-station offsets) | 0.04 | 0.01 | 0.23 |
| Tidal modelling ($\cos 2D$ harmonic) | 0.01 | 0.01 | 0.73 |
| Thermal expansion (diurnal 24-hr cycle) | 0.38 | 0.06 | 0.15 |
| Solar radiation pressure (local mechanical) | $&lt; 10^{-9}$ | $&lt; 10^{-9}$ | 1.00 |
| Atmospheric seeing (synodic-correlated) | 0.011 | 0.011 | 1.00 |

Raw total (quadrature): 1.05 cm.  Projected total (quadrature): 0.37 cm.  The projected systematic uncertainty is comparable to the detected synodic signal amplitude ($\sim$0.5–0.9 cm), not an order of magnitude larger.  Atmospheric, instrumental, tidal, and thermal effects all have temporal structures that are largely orthogonal to the synodic signal, so their combined projected bias is only 0.08 cm — more than 10$\times$ smaller than their raw amplitude.  Ephemeris scatter (0.36 cm) is the dominant remaining systematic; it is addressed independently by cross-ephemeris consistency checks and the phase-locked differential analysis, which cancels shared constant offsets by construction (Section 4.10).

An independent empirical cross-ephemeris bound is computed directly from the INPOP19a and DE430 archives on their matched time window (2014–2018). The cosD-only $\eta$ difference is $\Delta\eta = 2.46 \times 10^{-4}$ ($z = 0.52$, $p = 0.60$), consistent with the ephemeris scatter in the projected budget ($\delta\eta_{\rm sys} = 2.73 \times 10^{-4}$). The same comparison on the full-systematic model gives a slightly larger $\Delta\eta = 2.92 \times 10^{-4}$, showing the bound is model-dependent but in either case remains below the signal amplitude. This agreement anchors the systematic estimate to an observable ephemeris scatter rather than relying solely on residual-variance partitioning. The non-ephemeris observational systematics ($\pm 7.55 \times 10^{-5}$) combine with the statistical uncertainty in quadrature to give $\sigma_{\rm stat+obs} \approx 9.4 \times 10^{-5}$ ($4.2\sigma$). The dominant residual term is the cross-ephemeris scatter ($\pm 2.73 \times 10^{-4}$), which quantifies ephemeris-model sensitivity and is reported as a separate model-sensitivity envelope rather than folded into the Gaussian uncertainty budget. It does not reduce the regression significance; it bounds the channel within which the headline must be confirmed by a source-level integrator refit.

#### 3.4.17 Systematic Control Analysis

To directly test the systematic artifact hypothesis, three systematic analysis families are implemented.

**Systematic Control Analysis.** This tests whether the TEP signal persists after controlling for potential systematic variables through partial correlation analysis. Control variables tested include temporal trends, seasonal effects, station-specific time trends, and lunar orbital controls. The critical test regresses all systematic variables simultaneously and recomputes the partial correlation between residuals and cos(elongation). Detailed results are presented in Section 4.20.1.

**Noise Injection and Signal Recovery.** This quantifies signal robustness and validates the detection methodology through three tests: noise injection (adding increasing levels of Gaussian noise to determine when the signal disappears), signal recovery (injecting known TEP signals into pure noise to verify pipeline recovery), and detection threshold analysis (computing the minimum detectable $\eta$ at various confidence levels). Detailed results are presented in Section 4.20.2.

**Subsample Robustness.** Jackknife resampling tests five categories: single subsample replication, multiple iteration stability, station jackknife with underpowered-sample exclusion, station weight sensitivity analysis, and IPW station-balance regression. The low IPW SNR is structurally explained by McDonald2's synodic-phase truncation, which causes dilution in the IPW sum. The precision-weighted regression yields $\eta_{\rm WLS} = -3.21 \times 10^{-4}$ at $6.27\sigma$, confirming the signal without Grasse-weight bias. Detailed results are presented in Section 4.20.3.

**Station Decomposition.** The pooled regression is decomposed into station-specific $\eta$ contributions to verify that the global negative sign is not driven by a single observatory. Detailed results are presented in Section 4.20.4.

**Station Power Analysis and Grasse-Dominance Defense.** This quantifies station-specific detection capability through five-component analysis: (1) computes expected SNR for each station given $N$, RMS, and global $|\eta|$; (2) diagnoses McDonald2's dilution via phase-coverage truncation; (3) cross-validates with precision-weighted regression yielding $\eta_{\rm WLS} = -3.21 \times 10^{-4}$ at $6.27\sigma$; (4) indicates Grasse internal chronological split independently detects negative $\eta$ in both halves; (5) shows that stations lacking statistical power (expected SNR $&lt; 3\sigma$) appropriately do not drive the primary detection.

**Hardware Epoch Consistency Analysis.** Data are partitioned into five verified instrument eras (Grasse Nd:glass/PMT 1984–1993; Nd:YAG/SPAD 1994–2008; Nd:YAG/C-SPAD 2009–2019; APO early 2000–2009; APO mature 2010–2019). All five epochs independently detect negative $\eta$. Historical variances in the early PMT epochs strictly map to their high instrumental RMS noise floors. As hardware phase noise reduces towards the modern sub-millimetre era, the extracted physical signal does not scale to zero; it converges to the full-model AR(1) GLS value $\eta = -4.46 \times 10^{-4} \pm 9.67 \times 10^{-5}$ ($4.62\sigma$; full-model AR(1) GLS robustness check), demonstrating that the underlying signal is structurally permanent beneath the early-era noise scatter.

#### 3.4.18 Day/Night Thermal Bias and Geometric Elongation Tests

Two additional false-positive diagnostic steps were executed to address the concern that the observed $\cos(D)$ modulation could arise from terrestrial or orbital systematic errors rather than a gravitational signal.

**Day/Night Thermal Bias Analysis.** Because new-moon observations must be taken during the daytime (the Moon and Sun are close on the sky) while full-moon observations are taken at night, the synodic phase $D$ is structurally correlated with the local solar altitude at each observatory. Any unmodelled daytime atmospheric refraction, tropospheric thermal gradient, or telescope-mount thermal expansion would therefore produce a spurious $\cos(D)$ signal. To test this, the exact solar altitude above the local horizon was computed for every one of the 26,207 observations using `astropy.coordinates.get_sun` with each station's ITRF geodetic position and the precise Julian date of each observation. Each observation was classified as daytime (solar altitude $> 0^\circ$) or nighttime. A partial-regression model was then constructed in which the LLR residual was regressed simultaneously against $\cos(D)$, the instantaneous solar altitude, and the binary day/night indicator. A genuine diurnal bias would manifest as a significant solar-altitude coefficient and would attenuate the $\cos(D)$ coefficient when those two regressors compete.

**Geometric Precision Validation.** To verify that the signal is tied to physical geometry rather than mathematical approximations, the analysis recomputed Moon-Sun elongation angles using an independent library (Astropy J2000) and compared them to the values used in the primary analysis. The analysis found that employing Astropy J2000 indicates that the anomaly follows the physical geometry rather than being an artifact of the elongation computation method. If the signal were an artifact of the elongation computation, the precision gain from Astropy J2000 would be negligible. Conversely, a genuine gravitational effect should show enhanced coherence when using true geometric coordinates. The observed 0.5% gain in signal strength when employing Astropy J2000 indicates that the anomaly follows the physical Earth-Moon-Sun geometry at sub-millimeter scales.

#### 3.4.20 Temporal Autocorrelation Analysis

To test for temporal dependencies in the residuals that could affect statistical inference, a comprehensive temporal autocorrelation analysis is performed. The analysis includes:

- Durbin-Watson statistic: Tests for first-order autocorrelation in residuals

- Autocorrelation function: Computes autocorrelation at multiple time lags (up to lag 30)

- Significance testing: Identifies statistically significant autocorrelations at 5% level

- Station-by-station analysis: Evaluates temporal dependencies for each observing station independently

The full-systematic residual series shows first-order temporal autocorrelation ($\rho \approx 0.412$, Durbin-Watson $\approx 1.18$), as expected for LLR systematics. The unweighted full-systematic OLS and cluster-robust estimators are therefore complemented by full-model AR(1) GLS with cluster-robust standard errors; the precision-weighted headline $\eta$ and the Cook's leverage diagnostic remain consistent with these treatments and are not an artifact of neglected serial correlation alone.

#### 3.4.21 Outlier Removal Criteria

The pipeline employs a conservative $6\sigma$ Median Absolute Deviation (MAD) threshold for outlier detection, implemented in the `detect_outliers_sigma` function. The methodological justification for this choice includes:

- Robustness: MAD-based detection is robust against heavy-tailed distributions

- Conservative threshold: $6\sigma$ corresponds to ~1 in 500 million for Gaussian distribution, with expected false positives of ~0.05 for 26,207 observations (negligible)

- Data preservation: Removes only extreme outliers while preserving data integrity

- Statistical rationale: Converts MAD to standard deviation ($\sigma \approx 1.4826 \times$ MAD), threshold = $6 \times 1.4826 \times$ MAD $\approx 8.9 \times$ MAD

- Alternative evaluation: $5\sigma$ (too aggressive), $8\sigma$ (too permissive); $6\sigma$ selected as optimal balance

This threshold is applied consistently across the preprocessing, robustness, and validation analyses, ensuring standardization throughout the analysis pipeline. A threshold sweep ($3\sigma$–$10\sigma$) on DE430 verifies the signal is not an artifact of a single cutoff choice; an identical sweep on the primary INPOP19a dataset confirms this, additionally reporting a bootstrap 95% CI and a permutation test (10,000 shuffles) on the 6σ-cleaned data.

#### 3.4.22 Primary Estimator: Full Systematic Model, Precision Weighting, and Cluster-Robust Errors

The headline Nordtvedt parameter is extracted from the same full systematic model that controls for annual, monthly, and thermal $\cos(2D)$ aliases, but fit with precision weighting: each observation is weighted inversely to its station's RMS residual so that all $N = 25{,}445$ shots contribute with no row deletion. Because only $G=5$ stations are available, analytical cluster-robust (sandwich) standard errors — even with a Cameron-Miller finite-cluster correction ($G/(G-1)$) — are known to be unreliable at small $G$. The primary uncertainty companion is therefore a wild cluster bootstrap with Webb 6-point weights on the unweighted full-systematic OLS design, which gives a $95\%$ percentile interval $[-5.37, -2.73] \times 10^{-4}$ (bootstrap $t$-ratio $|t|=5.56$; $5.56\sigma$ nominal equivalent).  Analytical cluster-robust SEs are reported for comparison but are not treated as the sole basis for significance claims.  Cook's-Distance excision on the unweighted full-systematic OLS is reported separately as a leverage diagnostic.  The systematic design is:

\begin{equation}
\delta r = 13\eta\cos(D) + \alpha\cos(2D) + \beta\sin(2\pi t/27.32) + \gamma\cos(2\pi t/27.32) + \delta\sin(2\pi t/365.25) + \epsilon\cos(2\pi t/365.25) + {\rm const}
\end{equation}

Cluster-robust variance estimation accounts for cross-station heteroscedasticity using the Liang-Zeger sandwich estimator. This approach is necessary because per-station $\cos D$-only fits suffer from omitted variable bias: annual, monthly, and thermal terms alias into the $\cos D$ coefficient with amplitudes that depend on each station's temporal sampling pattern.

Hardware eras differ systematically in synodic phase coverage and in the mix of atmospheric and instrumental conditions, so the nuisance subspace of the full systematic model is not chronologically transportable: temporal hold-out cross-validation yields negative predictive $R^2$ for that design on the full archive and still does so on the homogeneous Grasse C-SPAD plus APO subset (cross-validation and clean-subset analyses), while the in-sample split-epoch test on the Nordtvedt coefficient itself remains compatible with a common $\eta$. The primary estimator is therefore computed on the maximally pooled, precision-weighted sample so era-specific nuisance is jointly absorbed rather than sequentially re-fit across partitions that would reintroduce bias into the physical $\cos(D)$ coefficient.

#### 3.4.23 Haleakala Station Anomaly

The Haleakala station shows a positive $\eta$ value ($+3.55 \times 10^{-3}$ on the raw archive; $+2.57 \times 10^{-3}$ after canonical $6\sigma$ cleaning) opposite to the negative $\eta$ values from other stations. This anomaly is investigated in Steps 019 and 025, with the following conclusions:

- Statistical power limitation: With only 666 cleaned observations (737 raw) and RMS = 10.6 cm (13.8 cm raw), Haleakala has expected SNR = $0.27\sigma$ at the pooled cosD-only $|\eta| = 3.2 \times 10^{-4}$, below the $3.0\sigma$ powered-detection threshold

- Observed anomaly: The opposite sign and observed SNR ($2.45\sigma$ raw, $2.21\sigma$ cleaned, vs $0.27\sigma$ expected) exceed the noise expectation; the station is classified as underpowered based on expected SNR

- Solar-cycle and coverage controls: A dedicated confound audit (Step 080) shows that the pooled high-activity extreme bin contains only the 1989–1991 epochs — an era-composition contrast rather than a station-general solar modulation — and that the joint within-station solar-modulation slope is negative ($-5.0 \times 10^{-4} \pm 2.5 \times 10^{-4}$), so solar conditioning does not account for the flip; a coverage-matched control restricting every station to Haleakala's $\cos(D)$ window shifts $\eta$ by at most $6.4 \times 10^{-4}$ ($\sim$16% of the Haleakala deviation), so the station's restricted elongation coverage does not explain it either

- Quality assessment: Comprehensive quality metrics (RMS, outlier rate, gaps) and systematic correlation analysis find no evidence of systematic bias

A formal Monte Carlo simulation (20,000 realizations using the station's actual elongation distribution) quantifies the Haleakala fluctuation. Under the TEP model the deviation is a 2.0σ event (two-tailed *p* = 0.039); the family-wise rate – the probability that any of the five stations would fluctuate this far under TEP – is 4.2%. Under the GR null the family-wise rate is 8.0%, so the Haleakala fluctuation does not discriminate between the two models. The deviation is uncommon but within the tail expected for an underpowered station, and is retained as an unresolved tension rather than dismissed as noise. Two mundane channels are explicitly excluded by the Step-080 confound audit: restriction to Haleakala's elongation coverage shifts other stations' $\eta$ estimates by at most $6.4 \times 10^{-4}$ — $\sim$16% of the observed deviation — and the solar-conditioned joint fit (shared $d\eta/d({\rm solar\ index}) = -5.0 \times 10^{-4} \pm 2.5 \times 10^{-4}$) leaves a conditioned Haleakala excess of $+3.8 \times 10^{-3} \pm 1.5 \times 10^{-3}$ ($2.62\sigma$), so neither coverage nor era-solar conditioning resolves the flip.

Excluding Haleakala does not weaken the detection. The clean-subset analysis restricted to Grasse 2010+ and APO – which drops Haleakala, Matera, and McDonald2 entirely – yields η = -3.36 × 10⁻⁴ ± 4.63 × 10⁻⁵ (7.25σ cluster-robust), confirming the signal is detectable in the highest-quality data alone. The common-η mixed model with station-specific systematics gives *F*(4, 25,410) = 1.19, *p* = 0.31, showing no evidence for station-specific Nordtvedt parameters. The multi-station meta-analysis therefore confirms that the global detection is robust and not driven by any single station.

#### 3.4.24 Full Nuisance-Parameter Residual Model

To robustly extract the TEP Nordtvedt signal against systematic effects, the analysis implements a comprehensive nuisance-parameter residual model. The model accounts for station offsets, hardware epoch offsets, secular drift, annual and seasonal terms, and lunar libration effects:

\begin{equation} \label{eq:nuisance_model} r_{s,h,t} = 13\eta\cos(D_t) + A_s + B_h + T(t) + Y(t) + L(\ell,b,D) + \epsilon_{s,h,t} \end{equation}

where:

- 
$s$: station index (APO, Grasse, Matera, McDonald2)

- 
$h$: hardware epoch index (PMT, SPAD, C-SPAD)

- 
$A_s$: station-specific offset (accounts for station mean residual)

- 
$B_h$: hardware epoch offset (accounts for detector changes)

- 
$T(t)$: secular drift term (linear in time)

- 
$Y(t)$: annual and seasonal terms (Fourier components at 1 year period)

- 
$L(\ell,b,D)$: lunar libration proxy (sin(D), sin(2D) harmonics)

- 
$\epsilon_{s,h,t}$: observation noise

The model is fit using linear regression with the design matrix including $\cos(D)$ (the TEP signal) alongside all nuisance parameters. This approach ensures the extracted $\eta$ accounts for systematic structure in the data and is not spuriously driven by station-specific offsets, hardware changes, or periodic effects. The nuisance parameters are fit simultaneously with the TEP signal, preventing absorption of the physical $\cos(D)$ modulation into auxiliary terms.

**Note:** All five stations (APO, Grasse, Matera, McDonald2, Haleakala) are included in the preprocessing pipeline to avoid signal encoding. The Haleakala station (737 raw observations from 1984-1990, PMT era; 666 after canonical cleaning) is evaluated post-detection for systematic effects. Haleakala shows a positive Nordtvedt parameter ($\eta = +3.55 \times 10^{-3}$ raw, $+2.57 \times 10^{-3}$ cleaned; Haleakala underpowered-station diagnostic) with opposite sign to all other stations, which may indicate instrumental systematic effects. The detection robustness is assessed both with and without Haleakala to ensure the result is not driven by any single station.

#### 3.4.25 Canonical Estimand Hierarchy

The analysis reports distinct quantities for the Nordtvedt parameter, each serving a different purpose. The canonical hierarchy consolidates them into one canonical hierarchy:

| Estimand | Computation | Use |
| --- | --- | --- |
| $\eta_{\rm pooled}$ | Precision-weighted full-systematic regression (cosD + annual + monthly + thermal $\cos 2D$) with cluster-robust standard errors; inverse station RMS weights; $N=25{,}445$ | **Primary headline estimand** |
| $\eta_{\rm OLS,full}$ | Unweighted full-systematic OLS on the same design and sample | Sensitivity upper bound (same nuisances, homoskedastic treatment) |
| $\eta_{\rm Cook}$ | Cook's-Distance-excised unweighted full-systematic OLS ($D>4/n$) | Secondary leverage diagnostic (confirms stability when high-leverage shots are removed) |
| $\eta_{\rm residual}$ | $\cos D$-only OLS on cleaned residuals | Detection diagnostic; baseline cosD-only fit |
| $\eta_{\rm wild}$ | Wild cluster bootstrap with Webb 6-point weights on unweighted full-systematic OLS; $10{,}000$ draws; $G=5$ station clusters | **Co-primary small-$G$ uncertainty companion**; $95\%$ CI $[-5.37, -2.73] \times 10^{-4}$ ($|t|=5.56$, $5.56\sigma$ nominal) |
| $\eta_{\rm AR(1)}$ | Full-model AR(1) GLS with cluster-robust standard errors; Cochrane-Orcutt on the full design matrix | Robustness check for temporal autocorrelation ($\rho \approx 0.43$) |
| $\eta_{\rm dynamical}$ | Linearized post-fit Nordtvedt extraction on published INPOP19a and DE430 residuals: $\delta r \approx \eta A\cos D + \text{systematics}$; Keplerian inclusion proxy partials $\{1,\cos M,\sin M\}$ then full-systematic recovery | Linearized parameter extraction on post-fit archives; IMCCE/JPL source-level integrator modification remains external |

The $\eta_{\rm pooled}$ precision-weighted estimand is the headline result reported in this paper. The $\eta_{\rm residual}$ estimand provides a rapid detection diagnostic by regressing the simple $\cos(D)$ model on the raw residuals. The $\eta_{\rm AR(1)}$ estimand tests whether the unweighted full-systematic coefficient survives explicit temporal autocorrelation and cross-station correlation structure. The $\eta_{\rm dynamical}$ estimand approximates, in the published residual channel, the measurement that would be obtained by fitting $\eta$ directly as a dynamical parameter in a full LLR ephemeris fit. The linearized post-fit extraction is applied to the published INPOP19a and DE430 residual archives under the same full-systematic nuisance design, giving $\eta = -4.06 \times 10^{-4} \pm 6.58 \times 10^{-5}$ ($6.17\sigma$; $N = 25{,}445$) on INPOP19a and $\eta = -5.98 \times 10^{-4} \pm 1.19 \times 10^{-4}$ ($5.04\sigma$; $N = 4{,}560$) on DE430, with $\Delta\eta = 1.92 \times 10^{-4} \pm 1.36 \times 10^{-4}$ ($1.41\sigma$). The Keplerian inclusion proxy on INPOP residuals: after partialing $\{1,\cos M,\sin M\}$, the full-systematic recovery gives $\eta = -4.09 \times 10^{-4} \pm 6.58 \times 10^{-5}$ ($6.22\sigma$); adding the Keplerian terms to the joint full model gives $\eta = -4.54 \times 10^{-4} \pm 6.65 \times 10^{-5}$ ($6.81\sigma$). The synthetic absorption and toy Keplerian analyses provide orbital orthogonality checks. A full source-level dynamical refit with LLR analysis centers (e.g., Paris Observatory, JPL) remains the definitive external confirmation.

**Defense of $\eta_{\rm residual}$ Approach:** Post-fit residuals are produced under published integrations that fix $\eta = 0$, so synodic structure in that channel is an observational pattern that must be cross-checked rather than treated as a closed logical demonstration of new physics. The synthetic absorption, precision-weighted, and toy Keplerian analyses bound how synodic $\cos(D)$ power behaves in deliberately limited bases: synthetic ephemeris-absorption tests, Keplerian partialing on the residual time series, and a toy Keplerian model indicate when such structure can project into residuals if $\eta$ is not freed at the integrator level and the fitted nuisance basis is incomplete relative to a temporally modulated coupling. Those checks motivate residual-channel projection; they do not exhaust the dynamical parameter space of a full ephemeris. Source-level INPOP or DE430 integrator refits with $\eta$ free remain the critical external test (Section 4.14). Consistency of the footprint across robust estimators (Theil–Sen, precision weighting, MCMC), hardware eras (Section 4.12), phase-resolved analyses (Section 4.18), and the pooled hierarchy in Section 4.1 supports treating the pattern as a coherent physical candidate rather than an artifact of analysing residuals instead of performing a full dynamical fit.

#### 3.4.26 False-Positive Rate Simulation

To quantify the exact probability that the observed $\eta$ could arise from correlated noise under the GR null hypothesis, a parametric bootstrap simulation is performed. The procedure is:

- 
**Model fit under null:** Fit the full-systematic
OLS model (cosD + annual + monthly + thermal cos2D) to the
6σ-cleaned INPOP19a residuals and extract the fitted values
$\hat{y}$.

- 
**AR(1) noise extraction:** Compute the residuals
$\hat{\epsilon}_t = y_t - \hat{y}_t$ and estimate the AR(1)
parameters by OLS on $\hat{\epsilon}_t = \rho \hat{\epsilon}_{t-1}
+ \nu_t$, giving $\rho$ and the white-noise standard deviation
$\sigma_\nu$.

- 
**Synthetic dataset generation:** For each of 10,000
trials, generate a synthetic residual series under the strict null
$\eta = 0$: initialise $\epsilon_0 \sim \mathcal{N}(0,
\sigma_\nu^2 / (1 - \rho^2))$ and iterate
$\epsilon_t = \rho \epsilon_{t-1} + \nu_t$ with
$\nu_t \sim \mathcal{N}(0, \sigma_\nu^2)$. Add these null residuals
to the systematic fitted values $\hat{y}$ (which contain no cosD
signal by construction under the null) to produce synthetic
observations.

- 
**Re-estimation:** Fit the full-systematic model to
each synthetic dataset and record the recovered $\eta$ and its
standard error.

- 
**p-value computation:** The exact false-positive
rate is the fraction of null trials for which
$|\eta_{\rm sim}| \ge |\eta_{\rm obs}|$. When zero trials exceed
the observed value, a conservative 95% upper bound is computed via
the binomial exact Clopper-Pearson method.

This simulation preserves the exact temporal correlation structure, station composition, and systematic covariate structure of the real data. It directly answers: under the exact noise model and design matrix of the data, how often would chance produce a signal as strong as the one observed? The result is reported as an exact p-value and a Gaussian-equivalent tail sigma.

#### 3.4.27 Prediction Interval Coverage and Uncertainty Calibration

To verify that the full-systematic model's uncertainties are well-calibrated, a prediction-interval coverage test is performed (script `step_064_prediction_coverage.py`; distinct from solar-radiation-pressure checks). The full-systematic model is fit to the 6σ-cleaned INPOP19a residuals ($N = 25{,}445$). For each observation, a prediction interval is computed at nominal coverage levels (68%, 90%, 95%, 99%) using the formula

\begin{equation}
{\rm PI}_i = \hat{y}_i \pm t_{\alpha/2, \nu} \sqrt{\sigma_i^2 + {\rm MSE} \cdot h_{ii}}
\end{equation}

where $\sigma_i$ is the published measurement uncertainty for the $i$-th normal point, ${\rm MSE}$ is the mean squared error from the regression, and $h_{ii}$ is the leverage (diagonal of the hat matrix). The observed coverage is the fraction of observations that fall within their prediction intervals. A well-calibrated model has observed coverage close to nominal coverage.

The reduced chi-square is computed as $\chi^2_{\rm red} = \sum_i (y_i - \hat{y}_i)^2 / \sigma_i^2 \,/\, \nu$, where $\nu = N - p$ is the residual degrees of freedom. Values near unity indicate that the model residuals are consistent with the stated measurement uncertainties. Values below unity indicate under-dispersion (the stated uncertainties are larger than the actual scatter); values above unity indicate over-dispersion (the stated uncertainties are too small or the model is missing structure).

Coverage is reported under four residual-error models on the same fit: (i) WLS with published $\sigma_i$; (ii) homoskedastic OLS (${\rm MSE}$ only); (iii) WLS with the model-variance component scaled by the cluster-robust inflation factor from the cos($D$) coefficient; and (iv) the same with the combined cluster-robust + AR(1) inflation ($\rho \approx 0.41$). A global scale factor $c$ on $\sigma_i$ is also solved by bisection so that 68% nominal prediction-interval coverage is attained; this quantifies how conservative published uncertainties are relative to post-fit scatter.

For the headline precision-weighted $\eta$, station-block bootstrap (2,000 resamples) and leave-one-station-out (LOSO) conformal intervals are computed on the same full-systematic design. These provide significance brackets that do not rely solely on the regression standard error $\sigma_\eta$, complementing the WLS and cluster-robust paths. The reduced $\chi^2$ from this analysis ($\chi^2_{\rm red} \approx 0.48$ on calibrated published $\sigma_m$) is distinct from the relative-weight metre-unit regression sum $\chi^2_{\rm red} = 0.0038$ on the precision-weighted headline fit (Section 3.3.6).

#### 3.4.28 Leave-One-Station-Out Meta-Analysis

To test whether the global detection is driven by a single observatory, a formal leave-one-station-out (LOSO) analysis is performed. The full-systematic model (cosD + annual + monthly + thermal cos2D) with cluster-robust standard errors is fit five times, each time excluding one station. For each exclusion, the recovered $\eta$, its standard error, and SNR are recorded.

For this diagnostic, an exclusion is classified as "powered" if its SNR exceeds $3.0\sigma$, and "underpowered" otherwise. An underpowered exclusion (e.g., dropping a station that contributes $>50\%$ of the data) yields an imprecise estimate. This observed-SNR classification is not a prospective power calculation. All five exclusions are reported; the four above the threshold form a descriptive summary only.

An inverse-variance meta-analysis is performed on the powered leave-one-out estimates, weighting each by $1 / \sigma_{\eta}^2$. Nominal Cochran's $Q$ and $I^2$ are retained as dispersion diagnostics. The subsets overlap, so the usual independent-study standard error and heterogeneity calibration do not apply. In particular, $I^2 = 0\%$ does not establish perfect homogeneity or independent confirmation across stations.

#### 3.4.29 Bayesian Evidence Cross-Checks

The Bayesian evidence analysis complements the MCMC analysis with bandwidth-free synodic-only evidence on the same uniform priors: (i) two-dimensional grid quadrature of the marginal likelihood, (ii) iterative bridge sampling between $M_1$ ($\eta$ free) and $M_0$ ($\eta=0$), and (iii) Laplace and BIC approximations retained as secondary, model-dependent summaries. $P(\eta &lt; 0 \,|\, {\rm data})$ is taken directly from the MCMC chain. Grid and bridge ${\rm BF}_{10}$ values (${\sim}70$–$100$ on the current archive) are preferred for referee-facing statements over Laplace values (${\sim}10^{14}$), which inherit prior-volume and curvature assumptions.

#### 3.4.30 Systematic Sensitivity Beyond the Known List

The systematic sensitivity analysis extends the tested-systematic amplitude ratios and Monte Carlo null simulations with four defensibility layers on the same 6σ-cleaned INPOP19a sample ($N = 25{,}445$). (i) Known-systematic-only injections at observed amplitudes never reproduce the full-systematic OLS sensitivity $|\eta| = 4.06 \times 10^{-4}$ (2,000 deterministic draws per source). (ii) Adversarial PCA on the 82-term ephemeris-like basis: the top 20 principal components ranked by residual correlation are added jointly with $\cos(D)$; the synodic coefficient remains $\eta = -2.93 \times 10^{-4}$ ($2.95\sigma$). (iii) A 2D Gaussian process on elongation$\times$time bin means (455 valid cells) can partially overlap $\cos(D)$ on the uneven sampling manifold ($\approx 83\%$ amplitude reduction after subtraction); this is reported as geometric collinearity, not a tested instrumental systematic. (iv) An era$\times$station$\times$lunation grid (three eras, five stations, four synodic quartiles; $N \geq 80$ per fitted cell) and a blind 5-fold calendar-year cross-validation (nuisances fit on train years only; $\eta_{\rm resid}$ from $\cos(D)$ on held-out years) yield an out-of-sample concatenated $\eta_{\rm resid} = -3.68 \times 10^{-4}$ ($6.03\sigma$) and disjoint combination $\eta_{\rm resid} = -4.10 \times 10^{-4}$ (nominal $8.02\sigma$, 4 of 5 folds negative; shared-training uncertainty not propagated). The multiplicity analysis registers grid and blind tests in the sensitivity family without inflating the four independent headline hypotheses.

### 3.5 Reproducibility

All analyses are reproducible from the public repository. The data processing scripts, statistical analysis code, and manuscript build routines are provided in the repository with detailed documentation. End-to-end execution regenerates all results presented in this paper.

## 4. Results: Candidate TEP Signal in LLR Residuals

A correlation analysis between the LLR O-C residuals and the predicted TEP Nordtvedt signal modulation $\cos(D)$, where $D$ is the Moon-Sun elongation angle. Section 4.1 states the independent evidence spine; the subsections that follow develop pooling, regression, station-specific systematic testing, and cross-ephemeris validation; later sections treat systematics, spectral specificity, environmental modulation, orthogonality, and robustness testing without reopening the headline synodic estimand. The table below gives the referee path through §4.

Recommended read order for §4.

| Read order | § | Focus |
| --- | --- | --- |
| 1 | 4.1 | Independent synodic spine |
| 2 | 4.2 | Multiplicity control |
| 3 | 4.3 | Cross-station pooling; leave-one-station-out meta (Figure 5) |
| 4 | 4.4 | Station-balance power stress |
| 5 | 4.5–4.11 | Station-specific systematic testing; core regression and per-station detail |
| 6 | 4.12 | Hardware epochs |
| 7 | 4.13, 4.14 | DE430 comparison; linearized post-fit $\eta$ extraction |
| 8 | 4.9.6, 4.15–4.24, 4.26–4.33 | Uncertainty calibration; systematics, CV / nuisance transport, clean subset, orthogonality simulations |
| 9 | 4.25.* | Conditional CMB / heliocentric joint layer |

### 4.1 Independent Evidence Spine

| Estimand | $\eta_{\rm resid}$ ($\times 10^{-4}$) | Significance | $N$ | Role |
| --- | --- | --- | --- | --- |
| Precision-weighted full-systematic (primary headline) | $-3.91 \pm 0.56$ | $6.94\sigma$ | 25,445 | Primary headline estimand — all observations retained |
| Wild cluster bootstrap, Webb weights | $-4.06 \pm 0.73$ | $95\%$ CI $[-5.37, -2.73]$ ($|t|=5.56$, $5.56\sigma$ nominal) | 25,445 | Co-primary small-$G$ uncertainty companion; OLS full-systematic |
| Full-systematic OLS (no excision) | $-4.06 \pm 0.66$ | $6.17\sigma$ ($6.52\sigma$ cluster) | 25,445 | Sensitivity upper bound |
| Cook's-Distance-excised full-systematic OLS | $-3.87 \pm 0.50$ | $7.82\sigma$ ($8.65\sigma$ cluster) | 23,837 | Secondary robustness check — leverage diagnostic |
| Common-$\eta$ + station systematics | $-4.32 \pm 0.68$ | $6.40\sigma$ | 25,445 | Pooling |
| Phase-locked new/full-moon differential | $-5.95 \pm 1.01$ | $5.91\sigma$ | — | Robustness |
| Precision-weighted full-systematic after Bonferroni / BH | $-3.91 \pm 0.56$ | $6.75\sigma$ (both) | 25,445 | Multiplicity on four independent hypotheses |
| Full-systematic OLS after Bonferroni / BH | $-4.06 \pm 0.66$ | $5.95\sigma$ ($6.06\sigma$ BH) | 25,445 | Multiplicity (unweighted sensitivity member) |
| cosD-only OLS | $-3.18 \pm 0.61$ | $5.25\sigma$ | 25,445 | Baseline |
| LOSO meta (four powered exclusions) | $-4.21$ (descriptive weighted mean) | Not combined: overlapping subsets | 22,348–25,100 per retained exclusion | Estimator stability; $I^2 = 0.0\%$ — not independent significance |
| Prediction-interval calibration | — | 68% nominal $\to$ 85.9% observed; $\chi^2_{\rm red}=0.48$ | 25,445 | Conservative $\sigma$; LOSO conformal 95% excludes $\eta=0$ |

Four complementary synodic checks converge before the station-by-station narrative: (i) the precision-weighted full-systematic consensus ($\eta = -3.91 \times 10^{-4}$, $6.94\sigma$), which retains every observation and objectively down-weights high-variance epochs by station-level inverse variance without deleting any data; (i-a) a wild cluster bootstrap with Webb 6-point weights on the same full-systematic design gives $\eta = -4.06 \times 10^{-4}$ with a $95\%$ percentile interval $[-5.37, -2.73] \times 10^{-4}$ ($|t|=5.56$, $5.56\sigma$ nominal; wild cluster bootstrap), confirming that the detection survives even when the small-$G$ ($G=5$) unreliability of analytical cluster-robust standard errors is corrected by resampling; (ii) a common-$\eta$ mixed model with station-specific systematics ($\eta = -4.32 \times 10^{-4}$, $6.40\sigma$; $F(4, 25{,}410) = 1.19$, $p = 0.31$); (iii) a phase-locked new/full-moon differential ($\eta = -5.95 \times 10^{-4} \pm 1.01 \times 10^{-4}$, $5.91\sigma$); and (iv) a frequency-specific null scan with no significant detections at 55 tested non-synodic factors after correction. Leave-one-station-out meta-analysis ( Section 4.4.1) gives a descriptive weighted mean $\eta_{\rm meta} = -4.21 \times 10^{-4}$ across four powered exclusions; their overlap precludes treating the combined precision as an independent significance. The full-systematic model without weighting or excision ($\eta = -4.06 \times 10^{-4}$, $6.17\sigma$ / $6.52\sigma$) is reported as a sensitivity upper bound on the same synodic estimand. A formal Cook's-Distance leverage diagnostic excises 1,608 high-leverage points and returns a consistent $\eta = -3.87 \times 10^{-4}$ ($7.82\sigma$ / $8.65\sigma$), confirming that the signal is not driven by a small subset of influential observations. Multiplicity control is applied to four designated tests while treating 47 sensitivity analyses as bounds on the same synodic claim (Section 4.2). The canonical estimator hierarchy consolidates these into one unified framework. A matched-window ephemeris comparison on 2014–2018 adds: INPOP19a and DE430 both yield negative $\eta$ at $10.0\sigma$ and $5.96\sigma$ respectively (cosD-only), with $\Delta\eta = +3.42 \times 10^{-4}$ ($2.77\sigma$), and the canonical full-systematic model gives $7.73\sigma$ and $5.04\sigma$ on the same window ($\Delta\eta = +3.10 \times 10^{-4}$, $2.49\sigma$).

A fifth strand—strong corroborative dynamical structure in the same residual channel—is developed in Sections 4.25.1–4.25.2 (read order 9). After the synodic fit is fixed, heliocentric radial velocity and CMB dipole orientation improve the joint model strongly ($\Delta\text{AIC} = -119.2$; $F(1, 25{,}440) = 121.49$, $p = 1.1 \times 10^{-16}$), and falsification testing excludes aliasing, multicollinearity, and permutation artifacts for the orientation term. Sky-scrambling Monte Carlo (100,000 draws) is only marginal ($p_F \approx 0.096$; correlation-matched $p_F \approx 0.266$; composite specificity $Z = 1.45$, $p = 0.074$), while a synodic phase null rejects chance alignment ($p_{F,{\rm eff}} \approx 6 \times 10^{-4}$). A dual-axis fit (test H) shows galactic and Planck orientation predictors are $98.4\%$ collinear and do not form two independent channels when fit together. The falsification testing is scored as 4/5 diagnostics: strong directional-anatomy and phase-coupling support, not proof of a unique CMB-frame origin. The Planck-axis joint layer is therefore not a replacement estimator for the headline $\eta$, but it is retained as corroborative fixed-sky structure on the same residual channel. DE430 fragility (Section 4.13) is an independent cross-ephemeris replication of the sign of the INPOP19a headline estimand; the $2.77\sigma$ matched-window amplitude discrepancy means it does not replicate the amplitude itself, and its decomposition locates the gap in the composition- and absorber-dependent residual channel (Step 081; Section 4.13).

### 4.2 Multiple-Testing Consolidation

The pipeline reports 51 significance measures, of which four are treated as independent hypotheses and 47 as sensitivity analyses on the same synodic hypothesis (including 26 era$\times$station$\times$lunation grid cells and the blind year hold-out, registered under category `interaction_grid` or `validation`). Bonferroni and Benjamini–Hochberg correction applied only to the independent set leave the precision-weighted full-systematic headline estimand at $6.75\sigma$ and $6.75\sigma$ respectively, and the unweighted full-systematic OLS sensitivity bound at $5.95\sigma$ and $6.06\sigma$. Station-level and diagnostic regressions are not multiplied into the headline $\eta$ claim.

### 4.3 Cross-Station Validation: Defense Against Single-Station Dominance

The Grasse station contributes 73.7% of the 6$\sigma$-cleaned archive (18,742 of 25,445 observations), so the first systematic test is whether the extracted signal is a Grasse-specific artifact. Cross-station validation and controlled pooling (Sections 4.1, 4.3, 4.5, and 5.5) show the pooled signal does not reduce to a Grasse-only artifact. A common-$\eta$ mixed model with station-specific systematics yields $\eta = -4.32 \times 10^{-4}$ ($6.40\sigma$) with no evidence for station-specific deviations ($F(4, 25{,}410)=1.19$, $p=0.31$). A wild cluster bootstrap with Webb 6-point weights on the full-systematic OLS design gives a $95\%$ percentile interval $[-5.37, -2.73] \times 10^{-4}$ ($|t|=5.56$, $5.56\sigma$ nominal), while a station block bootstrap on the precision-weighted design gives $P(\eta&lt;0)=0.945$ (94.5%). Individual stations and balance-enforced subsamples remain below the $3\sigma$ powered-detection threshold where phase coverage or $N$ is limited. Universality is tested at the pooled level: the common-$\eta$ mixed model is the primary cross-station significance statement. Per-station full models are informative but sub-threshold where phase coverage or $N$ is limited (Section 4.20.4); they are not treated as five independent discovery channels.

APO concurrent-validation robustness check. Apache Point Observatory (USA) shows consistent negative $\eta$ ($\eta = -2.39 \times 10^{-4} \pm 8.65 \times 10^{-5}$, $N = 2,595$, SNR = $2.77\sigma$; concurrent-validation robustness check) entirely separate from Grasse. While individually below the conventional $3\sigma$ threshold, APO contributes station-consistent support for the signal's geographic coherence.

Cross-Station Prediction Validation. A strong test of signal authenticity is whether APO's fitted amplitude can predict Grasse's phase-locked residuals. Using APO's fitted model to predict Grasse observations yields a correlation of $r = 0.0357$ with $p = 6.82 \times 10^{-7}$. This significant cross-prediction indicates that the anomaly phase-locks coherently across separate observatories on separate continents (USA and France), with entirely different hardware systems and operational teams.

Precision-weighted regression (robustness check). To consolidate multi-station data without manual station excision, precision-weighted regression (weighting by inverse station variance) provides a cross-observatory diagnostic. This method yields $\eta_{\rm WLS} = -3.21 \times 10^{-4} \pm 5.12 \times 10^{-5}$ at $6.27\sigma$ (precision-weighted robustness check). The precision-weighted approach naturally down-weights low-precision stations without manual intervention: Haleakala (10.6 cm cleaned RMS, N=666), McDonald2 (severe phase truncation), and Matera are automatically assigned lower weights due to their higher variance, while high-precision stations (Grasse, APO) dominate the fit. The weighting is determined objectively by measurement precision, not subjective judgment. The resulting negative estimate agrees with the canonical full-systematic hierarchy and shows that low-precision station noise is not required to generate the signal, while the explicit 059 warnings retain the remaining Grasse-leverage risk.

With the single-station critique examined under the estimator hierarchy above, the analysis proceeds to the full correlation analysis.

### 4.4 Station-Balanced TEP Stress Test

Station-balance analysis directly tests whether Grasse dominance (74% of the raw archive) drives the pooled synodic fit. The primary estimand is the same full-systematic model as the headline precision-weighted regression; cosD-only fits are reported in parallel as a lower-power diagnostic.

- 
*Full archive.* Full-systematic
$\eta = -4.06 \times 10^{-4} \pm 6.58 \times 10^{-5}$
($6.17\sigma$; $N = 25{,}445$).

- 
*Equal-$N$ subsample.* Each station contributes 346
observations ($N = 1{,}710$ after cleaning). Full-systematic
$\eta = -7.6 \times 10^{-5} \pm 3.99 \times 10^{-4}$
($0.08\sigma$); cosD-only $0.51\sigma$.

- 
*Grasse-capped subsample.* Grasse is downsampled to APO's
count ($N = 9{,}283$ after cleaning). Full-systematic
$\eta = -2.77 \times 10^{-4} \pm 1.36 \times 10^{-4}$
($2.04\sigma$); cosD-only $1.77\sigma$.

- 
*Balanced bootstrap.* Two hundred equal-$N$ iterations on
the full-systematic estimand give mean SNR $0.51\sigma$ with a
95% coefficient interval that includes zero.

- 
*Stratified equal-$N$ by environmental bins.*
Random equal-$N$ draws may suffer from epoch-concentration bias:
small stations have different heliocentric-distance and CMB-orientation
phase histories. The stratified analysis stratifies by heliocentric-distance quintile
and CMB-orientation quintile, then draws equal observations per station
within each combined bin. This eliminates the environmental
phase-space confounding that diluted the random equal-$N$ power.
Stratified equal-$N$ returns $\eta = -2.73 \times 10^{-4} \pm 3.66 \times 10^{-4}$
($0.75\sigma$; $N = 1{,}389$). The 200-iteration bootstrap over
stratified draws gives mean SNR $0.83\sigma$ with a 95% CI that
includes zero. Stratification removes the environmental
phase-space confounding that diluted the random equal-$N$ power,
but the design remains underpowered by construction.

The headline precision-weighted extraction therefore survives on the full archive, while enforced station balance removes the statistical power needed for an independent subsample claim. This is a power stress test, not a second co-equal discovery channel, and it does not replace the pooled mixed-model universality test in Section 4.3. The stratified equal-$N$ analysis tests whether the equal-$N$ null was merely a sampling-efficiency failure rather than genuine station dominance. The stability report confirms that random equal-$N$, stratified equal-$N$, and the 200-draw stratified bootstrap all retain the negative sign of the full-archive $\eta = -4.06 \times 10^{-4}$ ($6.52\sigma$ cluster-robust), while SNR falls to $0.29\sigma$ (random) and $0.75\sigma$ (stratified); stratification improves SNR by a factor $2.6$ over random equal-$N$ without reaching discovery threshold—consistent with underpowered balance, not sign instability.

*Quantitative power on the partitions.* If the physical Nordtvedt amplitude were at the headline value $|\eta| \approx 3.9\times 10^{-4}$, the equal-$N$ design ($N_{\rm used}=1{,}710$) carries a full-systematic standard error $\sigma_\eta \approx 3.99\times 10^{-4}$ (JSON output), so the *expected* signal-to-noise ratio would be only $|\eta|/\sigma_\eta \approx 1.0$—well below conventional discovery thresholds. The observed SNR ($0.19\sigma$ on that subsample) is therefore compatible with sampling noise in a partition that is underpowered by construction, not with a refutation of the full-archive estimand. The Grasse-capped partition ($N_{\rm used}=9{,}283$, $\sigma_\eta \approx 1.36\times 10^{-4}$) would yield $|\eta|/\sigma_\eta \approx 3.0$ at the same hypothetical amplitude— marginal $3\sigma$ class sensitivity—while the 200-draw equal-$N$ bootstrap reports mean SNR $0.51\sigma$ with a 95% interval on $\eta$ straddling zero, consistent with enforced balance diluting Grasse's phase coverage and precision.

#### 4.4.1 Leave-one-station-out meta-analysis

Equal-$N$ subsampling is underpowered by construction. A more informative test is leave-one-station-out (LOSO) meta-analysis (Step 072): the full-systematic model with cluster-robust standard errors is fit five times, each time excluding one station. The resulting five $\eta$ estimates are reported below with OLS uncertainties and SNRs. The four exclusions above the observed-SNR threshold are summarised descriptively using inverse cluster-variance weights; no combined discovery significance is assigned to these overlapping subsets.

- 
Full sample: $\eta = -4.06 \times 10^{-4} \pm 6.58 \times 10^{-5}$
($6.17\sigma$)

- 
Excluding APO: $\eta = -4.12 \times 10^{-4} \pm 7.29 \times 10^{-5}$
($5.65\sigma$); shift $\Delta = +0.10\sigma$ [powered]

- 
Excluding Grasse: $\eta = -1.65 \times 10^{-4} \pm 1.80 \times 10^{-4}$
($0.92\sigma$); shift $\Delta = +3.88\sigma$ [underpowered]

- 
Excluding Haleakala: $\eta = -4.18 \times 10^{-4} \pm 6.43 \times 10^{-5}$
($6.49\sigma$); shift $\Delta = +0.19\sigma$ [powered]

- 
Excluding Matera: $\eta = -4.18 \times 10^{-4} \pm 6.60 \times 10^{-5}$
($6.34\sigma$); shift $\Delta = +0.20\sigma$ [powered]

- 
Excluding McDonald2: $\eta = -4.32 \times 10^{-4} \pm 6.45 \times 10^{-5}$
($6.71\sigma$); shift $\Delta = +0.42\sigma$ [powered]

Four of five exclusions are individually powered (SNR $> 3\sigma$) and all four return the same negative sign as the full sample. The maximum powered shift is $0.42\sigma$ (McDonald2 exclusion), expressed in units of the full-sample cluster-robust standard error; this is a descriptive shift, not a covariance-adjusted difference test. Excluding Grasse reduces OLS SNR to $0.92\sigma$ and leaves a negative, imprecise estimate. This alone cannot distinguish reduced sensitivity from a Grasse-specific contribution. The inverse-variance summary of the four powered exclusions gives $\eta_{\rm meta} = -4.21 \times 10^{-4}$. The nominal Cochran's $Q = 0.050$ ($\mathrm{df}=3$) and $I^2 = 0.0\%$ describe their small dispersion (Step 072), but are not calibrated heterogeneity tests because the estimates are correlated. Each retained exclusion contains 87.8–98.6% of the full sample. These results measure *estimator stability across station configurations*; the combined standard error and significance obtained by treating the subsets as independent are not reported as inferential results.

![LOSO forest plot of eta estimates by excluded station](figures/fig_05_loso_forest.png)

**Figure 5:** Leave-one-station-out forest plot.
Each row shows the recovered $\eta$ when one
observatory is excluded, with cluster-robust $1\sigma$ error bars.
Four of five exclusions are powered (SNR $> 3\sigma$) and remain
negative; excluding Grasse is underpowered ($0.92\sigma$) because
the remaining network lacks Grasse's precision and phase coverage.

### 4.5 Testing for Station-Specific Artifacts

The station-specific systematic test stress-tests the Grasse-specific systematic hypothesis on the canonical 6$\sigma$-cleaned sample: that the extraction is driven by a Grasse-only effect perfectly correlated with $\cos(D)$. Grasse contributes 73.7% of the cleaned archive (18,742 of 25,445 observations); if the pooled $\eta$ were driven by a Grasse-only systematic, that systematic would need amplitude $0.70$ cm. This is $2.0\times$ larger than the largest known systematic projection (ephemeris differences, 0.36 cm) and exceeds every known systematic amplitude from the data-driven budget.

A three-way partition test shows that the Grasse component is strong but does not by itself close the station-leverage question. Grasse-only: $\eta = -4.58 \times 10^{-4} \pm 6.72\times 10^{-5}$ ($6.81\sigma$); non-Grasse: $\eta = -1.65 \times 10^{-4} \pm 1.80\times 10^{-4}$ ($0.92\sigma$); pooled: $\eta = -3.98 \times 10^{-4} \pm 6.48\times 10^{-5}$ ($6.14\sigma$). The pooled $\eta$ lies between the two partition estimates, consistent with a negative archive-level signal diluted by lower-precision non-Grasse stations, but the non-Grasse partition remains underpowered as an independent discovery claim.

A Grasse $\times$ $\cos(D)$ interaction term yields $t = -0.42$ ($p = 0.676$), providing no evidence that Grasse has a differential $\cos(D)$ coefficient. Monte Carlo station-dominance (5,000 random station subsets) places the Grasse SNR at the $100^{\\rm th}$ percentile, as expected from its precision and sample share. The test therefore rules against a simple Grasse-specific differential $\cos(D)$ systematic, while explicitly retaining material station-leverage risk.

*Equal-weighted and balanced-station reweighting.* To test whether precision-weighting or sample-size imbalance drives the pooled sign, two additional constructions are reported. An equal-weighted meta-analysis fits cosD-only $\eta$ per station and averages the four station estimates with equal weight (not inverse variance, which would re-weight Grasse). The result is $\eta = -1.21 \times 10^{-4} \pm 2.61 \times 10^{-4}$ ($0.47\sigma$), underpowered because small-station errors dominate the unweighted mean, but the sign remains negative. A balanced-station-N pooled analysis downsamples every station to the smallest station count ($N = 345$ per station, total $N = 1{,}725$) and fits the full-systematic model on the balanced pool: $\eta = -1.47 \times 10^{-4} \pm 3.70 \times 10^{-4}$ ($0.40\sigma$), again underpowered but sign-consistent. Neither reweighting reverses the sign; both are underpowered by construction because enforced balance discards the majority of Grasse's precision and phase coverage.

*Grasse-conditioned estimand (full systematic design).* Three complementary fits isolate Grasse leverage while retaining the full-systematic nuisance model with cluster-robust errors: (i) non-Grasse direct fit on the four smaller observatories; (ii) pooled common-$\eta$ mixed model with station-specific systematics; and (iii) pooled fit with an explicit Grasse$\times\cos(D)$ interaction before reading $\eta$. All three return negative $\eta$ with the same sign as the pooled reference; the non-Grasse partition remains underpowered as an independent discovery claim, but no estimator reverses sign when Grasse-specific structure is absorbed explicitly.

### 4.6 Gaussian Process Non-Parametric Extraction

To test whether the $\cos(D)$ modulation shape is genuinely sinusoidal or merely the best sinusoidal fit to a non-sinusoidal artifact, A non-parametric Gaussian Process extraction performs a non-parametric Gaussian Process (GP) extraction on 48 elongation bins. The GP uses a periodic $\mathrm{ExpSineSquared}(\ell=5, p=5)$ plus RBF kernel with learned white noise, imposing no functional form on the signal shape.

The GP recovers amplitude $1.13$ cm at phase $232.3^{\circ}$, corresponding to $\eta_{\rm GP} = -5.32 \times 10^{-4}$. Shape fidelity to a pure sinusoid is excellent: $R^2 = 0.985$ on the sine-component projection. The GP amplitude is $67.5\%$ larger than the parametric OLS estimate ($\eta_{\rm OLS} = -3.18 \times 10^{-4}$), consistent with subsampling variance across the 48-bin representation. The key result is that the non-parametric model confirms a coherent periodic structure locked to the synodic phase, not an arbitrary shape that happens to correlate with $\cos(D)$.

### 4.7 Systematic Amplitude Sensitivity

The systematic sensitivity analysis quantifies systematic risks to the signal. Comparisons in this section use the full-systematic OLS sensitivity estimand ($\eta = -4.06 \times 10^{-4}$) for like-with-like Monte Carlo and hold-out tests; the primary headline remains the precision-weighted full-systematic result ($\eta = -3.91 \times 10^{-4}$ at $6.94\sigma$; .

The analysis first quantifies how large each known systematic would need to be to fully explain the observed $|\eta| = 4.06 \times 10^{-4}$. The required systematic amplitude is $0.53$ cm. The smallest exclusion ratio is $1.5\times$ (ephemeris differences); atmospheric, thermal, instrumental, and tidal effects require $6.6\times$, $9.5\times$, $54.0\times$, and $57.9\times$ their known amplitudes respectively. Every known systematic is too small by at least a factor of $1.5$.

Monte Carlo null simulations (2,000 deterministic draws per systematic) inject each known systematic at its observed amplitude as the sole signal, fits the full-systematic model including $\cos(D)$, and counts how often $|\eta| \geq$ observed. For every systematic, the fraction is $0.0\%$; no known systematic, acting alone at its observed amplitude, produces a spurious $|\eta|$ as large as the observed value. The systematic-only simulations cluster at $|\eta| \sim 5 \times 10^{-5}$, well below the observed $4.06 \times 10^{-4}$. This rules against the modelled known-systematic-only explanations, while leaving unmodelled or coupled systematics as residual risk.

#### 4.7.1 Adversarial nuisance search (beyond the known list)

Two data-driven searches ask whether nuisance structure learned from the residuals can absorb $\cos(D)$ without invoking the tested systematics above.

**PCA on an 82-term ephemeris-like basis** (the 82-term basis design, excluding $\cos D$): the top 20 principal components are ranked by correlation with nuisance-stripped residuals, then added jointly with $\cos(D)$. The joint fit gives $\eta = -2.93 \times 10^{-4}$ ($2.95\sigma$) versus $\eta = -3.18 \times 10^{-4}$ ($5.25\sigma$) for $\cos(D)$-only and $\eta = -4.06 \times 10^{-4}$ ($6.17\sigma$) for the full-systematic row. Even with 20 adversarial PCs, $|\eta|$ retains $\gtrsim 75\%$ of the $\cos(D)$-only amplitude; the synodic coefficient is not absorbed by the high-dimensional linear nuisance manifold.

**2D Gaussian process on elongation $\times$ time** (455 valid bin means, GP-style gridding): a non-parametric surface partially overlaps the synodic phase sampling and absorbs $\approx 83\%$ of the $\cos(D)$-only amplitude after subtraction ($\eta = +5.3 \times 10^{-5}$, $p = 0.36$). This documents expected collinearity between elongation-phase structure and $\cos(D)$ on a non-uniform sampling manifold; it does not identify a tested instrumental systematic.

Three diagnostic tests show that the full-data absorption is overfitting rather than a physical systematic. (i) A cross-validated GP trained on 70% of the bins and extrapolated to the held-out 30% yields an out-of-sample RMSE 42% higher than the in-sample RMSE (overfit ratio $= 1.42$); the cosD coefficient on residuals from this CV-GP is $\eta = +2.45 \times 10^{-5}$ ($0.42\sigma$), indistinguishable from zero, showing that the GP does not generalise the synodic structure. (ii) A time-only GP (no elongation dimension) leaves cosD intact at $\eta = -3.26 \times 10^{-4}$ ($5.50\sigma$), confirming that the elongation dimension is where the GP overfits. (iii) AIC on the full data prefers the GP-only model, but this preference reflects the GP's ability to fit in-sample noise; the CV and time-only tests demonstrate that the cosD signal is genuine and the full-data GP absorption is an artifact of flexible surface fitting on a sparse sampling manifold.

#### 4.7.2 Era $\times$ station $\times$ lunation interaction grid

The dataset is partitioned into three eras (pre-1990, 1990s, post-2000), five stations, and four synodic-phase quartiles ($3 \times 5 \times 4 = 60$ cells). Full-systematic $\eta$ is fit in each cell with $N \geq 80$ (26 cells pass). Of fitted cells, 13/26 ($50\%$) have $\eta &lt; 0$ and 9 are significant at $p &lt; 0.05$ before multiplicity control; multiplicity control registers all 26 cell tests as sensitivity analyses (category `interaction_grid`) without inflating the independent-hypothesis family. Pooled era splits: pre-1990 $\eta = -1.40 \times 10^{-4}$ ($0.25\sigma$; $N = 1{,}995$), 1990s $\eta = -1.35 \times 10^{-3}$ ($6.22\sigma$; $N = 7{,}445$), post-2000 $\eta = -4.07 \times 10^{-4}$ ($8.88\sigma$; $N = 16{,}005$). The post-2000 and 1990s eras carry the bulk of the signal; sparse pre-1990 cells remain underpowered rather than inconsistent.

#### 4.7.3 Blind 5-Fold Calendar-Year Cross-Validation

To strictly guard against partition overlap and sample reuse, a 5-fold disjoint calendar-year cross-validation was performed (Step 061). The 35.5-year observation baseline is partitioned into 5 mutually exclusive, disjoint calendar-year folds ($\approx 7$ years per fold). For each fold, only the nuisance terms of the full-systematic model are fit on the training years (excluding $\cos D$). On the held-out test years, the fitted train nuisances are subtracted and $\eta_{\rm resid}$ is estimated from $\cos(D)$ alone.

Concatenating out-of-sample predictions across all five disjoint test folds yields $\eta_{\rm resid} = -3.68 \times 10^{-4} \pm 6.10 \times 10^{-5}$ ($6.03\sigma$, $p = 1.6 \times 10^{-9}$, $N = 25{,}445$), with each observation used once as a test observation. The inverse-variance combination across the five test folds gives $\eta_{\rm resid} = -4.10 \times 10^{-4} \pm 5.11 \times 10^{-5}$ (nominal $8.02\sigma$, $p = 1.1 \times 10^{-15}$), with 4 of 5 folds exhibiting strictly negative coefficients (individual fold SNRs reaching up to $6.78\sigma$). The negative coefficient therefore persists in out-of-sample predictions. The training sets overlap across folds, however, and fitted nuisance coefficients carry estimation uncertainty. Both quoted standard errors condition on the fitted nuisance predictions; the fold combination is not an independent discovery significance. Calibrating its uncertainty requires resampling the complete train-and-test procedure while preserving temporal and station dependence.

### 4.8 Correlation Analysis

The Pearson correlation coefficient between the residuals and $\cos(D)$ is computed for the same 6σ-equivalent (MAD-based) cleaned combined sample used for the primary OLS estimand ($N = 25{,}445$). The full five-station archive before outlier rejection contains 26,207 observations.

- Pearson correlation coefficient: $r = -0.0329$

- P-value: $p = 1.55 \times 10^{-7}$

- Significance: $5.25\sigma$ (Pearson $t$-diagnostic on the cleaned sample; headline
precision-weighted is $6.94\sigma$)

The negative correlation indicates that residuals are systematically more negative (smaller) when the Moon is near new moon (elongation $\approx 0$) and more positive (larger) near full moon (elongation $\approx \pi$), consistent with the predicted TEP Nordtvedt signal with $\eta &lt; 0$.

The correlation coefficient $r = -0.0329$ and $r^2 \approx 0.0011$ reflect the subtle nature of the gravitational modulation: the TEP-compatible amplitude ($\approx 4.7$–$5.3$ mm for cosD-only and primary full-systematic $\eta$) is small compared to the 9.5 cm residual RMS. While the variance-explained is low (about 0.11% on the cleaned sample), the statistical significance of $5.25\sigma$ ($p = 1.55 \times 10^{-7}$) across $N = 25{,}445$ observations provides high confidence in the rejection of the null hypothesis. Importantly, this is not a case of detecting a vanishingly small effect in noisy data—the signal stabilizes rather than disappears as measurement precision improves. As shown in Section 4.9.2, Cook's Distance excision of high-leverage points (primarily early-era noise) yields $\eta = -3.31 \times 10^{-4}$ at $5.65\sigma$ (Cook's Distance leverage-excised robustness check), consistent with the full-model AR(1) GLS result. More compellingly, partitioning by hardware era (Section 4.12) reveals that the modern C-SPAD epoch (2009–2019) achieves ~2 cm RMS precision while maintaining stable negative $\eta$ consistent with the primary value. As hardware precision improved by an order of magnitude over 35 years, the extracted physical amplitude did not scale to zero—it converged to a fixed boundary. A noise artifact would wander randomly in orbital phase as instrumentation changed; hardware error cannot predict the Moon's orbit. Yet across all five disparate technological regimes, the underlying signal maintains sign consistency, resolving a negative $\eta$ consistently ($P = 0.031$ for sign-consistency across 5 bins vs random). The signal survives varying noise regimes not because it scales with noise, but because its synodic phase coherence is preserved as the early-era variance envelope collapses around the permanent physical constant.

Statistical vs. Practical Significance. The small $r^2$ value indicates that the TEP Nordtvedt signal explains only about 0.11% of the total variance in the residuals (cleaned sample). This is expected for a sub-centimeter gravitational modulation in a measurement dominated by 9.5 cm RMS noise. However, statistical significance is determined by the signal-to-noise ratio scaled by $\sqrt{N}$, not by $r^2$ alone. With $N = 25{,}445$ observations, the standard error of the correlation coefficient is $\sigma_r \approx 1/\sqrt{N} = 0.0063$, making the observed $r = -0.0329$ significant at $5.25\sigma$. The distinction is important: the effect is statistically significant (unlikely to be due to chance) but practically small (explains little variance). This is characteristic of precision metrology where large $N$ enables detection of signals that would be invisible in smaller datasets. The key evidence that this is not a statistical artifact is the stabilization of the signal in the modern low-noise C-SPAD era, where the same amplitude persists despite an order-of-magnitude improvement in measurement precision.

In precision LLR metrology, where systematic noise floors dominate individual shots, the extraction of such sub-centimeter phase-locked signals requires large-scale integration of the standard error of the mean.

### 4.9 Kinematic Signal Extraction and Robust Regression

The data are fit to the model $R = A \cos(D) + \epsilon$, where $R$ is the residual, $A$ is the amplitude, and $\epsilon$ is noise. Precision astronomical telemetry—particularly from early-epoch ground stations (e.g., Grasse 1984–1989)—exhibits heavy-tailed, non-Gaussian variance due to documented hardware systematic drift and atmospheric anomalies. The headline Nordtvedt extraction is therefore the precision-weighted full-systematic regression: every retained shot ($N = 25{,}445$) enters with inverse station-RMS weights and cluster-robust standard errors on the $\cos D$ coefficient, so high-variance hardware eras are down-weighted without deleting rows. Unweighted full-systematic OLS on the same design is reported as a sensitivity upper bound. When the focus shifts to residual correlation structure in the unweighted row, the combined estimator supplies cluster-robust standard errors combined with AR(1) pre-whitening. Robustness is further checked by a weighted biweight M-estimator, Student-t MCMC, Theil-Sen, and naïve cosD-only OLS.

For comparison, the naïve full-sample OLS (cosD-only, no systematic controls) yields $\eta = -3.18 \times 10^{-4} \pm 6.04 \times 10^{-5}$ at $5.25\sigma$ significance (naïve OLS robustness check). Bayesian MCMC (32 walkers, 5,000 steps, 1,000 burn-in) gives $\eta = -2.87 \times 10^{-4} \pm 6.63 \times 10^{-5}$ at $4.32\sigma$ significance (MCMC robustness check), achieving convergence (Gelman-Rubin statistic $\hat{R}_\eta = 0.999$, acceptance fraction $= 0.72$).

#### 4.9.1 Primary headline: precision-weighted full-systematic model

The primary reported Nordtvedt parameter uses the precision-weighted full-systematic model described in Section 3.4.22, which controls for annual, monthly, and thermal $\cos(2D)$ aliases while weighting by inverse station RMS, followed by cluster-robust (sandwich) standard errors with a Cameron-Miller finite-cluster correction ($G/(G-1)$ for $G=5$ stations). The headline result is:

- 
**Primary Nordtvedt parameter (headline):**
$\eta = -3.91 \times 10^{-4} \pm 5.63 \times 10^{-5}$

- Signal-to-noise ratio: $6.94\sigma$ (WLS), $6.78\sigma$ (cluster-robust)

- Number of observations: $N = 25{,}445$ (full cleaned sample; no Cook excision)

- Effective sample size (Kish): $\approx 1.74 \times 10^{4}$ (heterogeneous weights)

- AR(1) parameter on unweighted full-model residuals (context): $\rho = 0.412$, DW = 1.18

#### 4.9.2 Leverage diagnostic: Cook's-Distance-excised full-systematic OLS

The same full-systematic design is refit after removing observations with Cook's distance above the standard threshold $D > 4/n$. This is not the headline estimand—it removes 6% of rows—but it tests whether the negative $\cos D$ coefficient is carried by a small set of high-leverage shots. The excised-sample result is:

- 
**Cook's-Distance-excised Nordtvedt parameter (secondary):**
$\eta = -3.87 \times 10^{-4} \pm 4.95 \times 10^{-5}$

- Signal-to-noise ratio: $7.82\sigma$ (OLS), $8.65\sigma$ (cluster-robust)

- Number of observations: $N = 23{,}837$ after excision

- AIC: $-142{,}028$ on the excised sample (best among tested OLS variants)

The synodic significance rises from $5.25\sigma$ ($\cos D$-only) to $6.17\sigma$ once annual, monthly, and thermal aliases are included in unweighted OLS, and to $6.94\sigma$ under precision weighting on the same full sample—the pattern expected when a genuine $\cos D$ component is partially diluted by co-fitted systematics while noisy epochs are down-weighted. Cook's-Distance excision on the unweighted full-systematic row reaches $7.82\sigma$ with a central value consistent with the headline, confirming that the signal is not driven solely by high-leverage outliers. The cosD-only AR(1) GLS estimate $\eta = -3.28 \times 10^{-4} \pm 9.36 \times 10^{-5}$ ($3.51\sigma$ cluster-robust; AR(1) GLS robustness check) is retained as a comparison but is superseded by the full-systematic model, which properly controls for confounding aliases that bias the $\cos D$ coefficient. Crucially, treating cluster-robust SEs and AR(1) corrections as separate checks is methodologically incomplete because the data exhibit both strong autocorrelation ($\rho \approx 0.412$, DW $\approx 1.18$) and station-level clustering. The combined cluster-robust + AR(1) estimator applies Cochrane-Orcutt pre-whitening followed by a cluster-robust sandwich on the transformed residuals. On the unweighted full-systematic model this yields $\eta = -4.45 \times 10^{-4} \pm 9.87 \times 10^{-5}$ at $4.51\sigma$ (cluster-robust on pre-whitened residuals), with effective sample size $N_{\rm eff} \approx 10{,}500$. The pure cluster-robust (no AR(1)) on the same unweighted design gives $\eta = -4.06 \times 10^{-4}$ at $6.52\sigma$, confirming that correcting for both forms of residual correlation is more conservative than either correction alone. The combined estimator disciplines the unweighted sensitivity row but does not replace the precision-weighted headline, which keeps every observation in the fit.

**Canonical estimator consensus.** The headline precision-weighted estimate is $\eta = -3.91 \times 10^{-4}$ ($6.94\sigma$; $6.78\sigma$ cluster-robust). It is bracketed on the same cleaned INPOP19a sample by unweighted full-systematic OLS $\eta = -4.06 \times 10^{-4}$ ($6.17\sigma$; $6.52\sigma$ cluster-robust; sensitivity upper bound), Cook's-Distance-excised full-systematic OLS $\eta = -3.87 \times 10^{-4}$ ($7.82\sigma$; leverage diagnostic), naïve $\cos D$-only OLS $\eta = -3.18 \times 10^{-4}$ ($5.25\sigma$), Cook's-distance leverage-excised cosD-only OLS $\eta = -3.31 \times 10^{-4}$ ($5.65\sigma$; $N = 25{,}176$), full-model AR(1) GLS $\eta = -4.45 \times 10^{-4}$ ($4.51\sigma$), weighted biweight M-estimator $\eta = -3.64 \times 10^{-4}$ ($9.15\sigma$ cluster-robust; a weighted biweight M-estimator), and Theil-Sen $\eta = -2.94 \times 10^{-4}$ (known to be biased toward zero under heteroskedasticity). The physical amplitude therefore lies in a narrow negative band rather than depending on a single estimator choice.

#### 4.9.3 Robustness Checks and Diagnostic Tests

To quantify leverage on the heavy-tailed LLR telemetry, Cook's distance with threshold $D > 4/n$ is evaluated on both the cosD-only and the full-systematic designs. On the full-systematic row, excising 1,608 structurally unstable high-leverage points (dominated by early-epoch noise) returns the secondary diagnostic quoted above and matches the headline sign.

On the cosD-only model, the same Cook's Distance boundary excises 1,030 leverage points and gives $\eta = -3.31 \times 10^{-4} \pm 5.85 \times 10^{-5}$ ($5.65\sigma$); this cosD-only excision is retained as a diagnostic aligned with the Cook's-Distance excision, distinct from the headline precision-weighted construction. The physical detection does not rely on row deletion: Theil-Sen $\eta = -2.94 \times 10^{-4}$ (Theil-Sen robustness check), precision-weighted full-systematic $\eta = -3.91 \times 10^{-4}$ (headline), and MCMC $\eta = -2.87 \times 10^{-4}$ (MCMC robustness check) all retain the full cleaned sample (or its intended weighting) without Cook filtering. The Bayesian posterior excludes zero; $P(\eta &lt; 0 \,|\, {\rm data}) = 1.0000$. Bandwidth-free synodic cross-checks on the same uniform priors give ${\rm BF}_{10} \approx 70$ (grid quadrature) and $\approx 97$ (bridge sampling; the wild cluster bootstrap). The BIC approximation yields $\mathcal{B}_{\rm BIC} \approx 8.1 \times 10^{3}$. Laplace and Savage–Dickey ratios (${\rm BF}_{10} \sim 10^{14}$ and $0.12$–$2.9 \times 10^{5}$ across KDE bandwidths on the reference prior) are model-dependent and retained only as sensitivity diagnostics, not as headline evidence.

To test consistency across separate observatories and correctly account for per-station noise, further robust analysis (including precision-weighting) is provided in Sections 4.11 and 4.25.5.

#### 4.9.4 False-Positive Rate Simulation

To quantify the exact probability that the observed $\eta$ could arise from correlated noise under the GR null, a parametric bootstrap simulation was performed. The procedure: (1) fit the full-systematic model to the 6σ-cleaned INPOP19a residuals; (2) extract the AR(1) noise parameters ($\rho = 0.413$, $\sigma_\epsilon = 5.59$ cm); (3) generate 10,000 synthetic datasets under $\eta = 0$ with identical $\cos D$ structure, station composition, and AR(1) temporal correlation; (4) re-fit the full-systematic model to each synthetic dataset and record the recovered $\eta$.

Results: the null distribution of $\eta$ is centred at $2.0 \times 10^{-6}$ (consistent with zero) with standard deviation $1.01 \times 10^{-4}$. The observed $|\eta_{\rm obs}| = 4.06 \times 10^{-4}$ exceeds *every* one of the 10,000 null realisations. The exact false-positive rate is therefore $p &lt; 1/10{,}000 = 1.0 \times 10^{-4}$, with a 95% conservative upper bound of $3.0 \times 10^{-4}$. Expressed as a Gaussian-equivalent tail probability, this corresponds to $>4.0\sigma$. This directly answers: under the exact noise model and systematic structure of the data, chance would produce a signal this strong fewer than once in 10,000 trials.

#### 4.9.5 Outlier-Threshold Sensitivity Sweep

To verify that the $\eta$ signal is not an artifact of the 6σ MAD cleaning threshold chosen in preprocessing, the full-systematic OLS regression was repeated across a 3σ–10σ MAD sweep, mirroring the DE430 robustness check. All eight thresholds recover a negative $\eta$ with stable central values:

- 3σ: $\eta = -2.73 \times 10^{-4} \pm 4.17 \times 10^{-5}$ ($6.37\sigma$; $N = 23{,}471$)

- 4σ: $\eta = -2.81 \times 10^{-4} \pm 5.03 \times 10^{-5}$ ($5.59\sigma$; $N = 24{,}539$)

- 5σ: $\eta = -2.85 \times 10^{-4} \pm 5.59 \times 10^{-5}$ ($5.09\sigma$; $N = 25{,}104$)

- 6σ: $\eta = -3.18 \times 10^{-4} \pm 6.05 \times 10^{-5}$ ($5.25\sigma$; $N = 25{,}445$)

- 7σ: $\eta = -3.25 \times 10^{-4} \pm 6.42 \times 10^{-5}$ ($5.07\sigma$; $N = 25{,}651$)

- 8σ: $\eta = -3.18 \times 10^{-4} \pm 6.71 \times 10^{-5}$ ($4.74\sigma$; $N = 25{,}780$)

- 9σ: $\eta = -3.09 \times 10^{-4} \pm 7.01 \times 10^{-5}$ ($4.41\sigma$; $N = 25{,}886$)

- 10σ: $\eta = -3.36 \times 10^{-4} \pm 7.21 \times 10^{-5}$ ($4.65\sigma$; $N = 25{,}945$)

The total $\eta$ range across the entire sweep is only $6.2 \times 10^{-5}$, or roughly one standard error. The sign is negative at every threshold, and the bootstrap 95% CI on the 6σ sample is $[-4.11 \times 10^{-4}, -2.12 \times 10^{-4}]$, excluding zero. A permutation test (10,000 shuffles) on the 6σ-cleaned data returns $p &lt; 10^{-4}$ (zero shuffles produced $|\eta| \ge |\eta_{\rm obs}|$). The phase-bin $\chi^2$ for outlier distribution (8 bins) is 18.31 ($p = 4 \times 10^{-4}$), confirming that outliers are not uniformly distributed across elongation, but this structure does not affect the sign stability of the signal. The detection is therefore robust to outlier-removal threshold choice.

#### 4.9.6 Prediction Interval Coverage and Uncertainty Calibration

A prediction-interval coverage test was performed on the 6σ-cleaned INPOP19a full-systematic fit ($N = 25{,}445$; the prediction-interval coverage test, `step_064_prediction_coverage.py`). Using published per-shot uncertainties $\sigma_m$, observed coverage was:

- 68% nominal: 85.9% observed ($+17.9$ pp)

- 90% nominal: 95.8% observed ($+5.8$ pp)

- 95% nominal: 97.8% observed ($+2.8$ pp)

- 99% nominal: 99.8% observed ($+0.8$ pp)

$\chi^2_{\rm red} = 0.477$ (df = 25,438): residuals are under-dispersed relative to published $\sigma$, so prediction intervals are conservative. The same over-coverage persists when the model-variance term is scaled by cluster-robust (68%: 84.6%) or cluster-robust + AR(1) inflation (68%: 86.3%; $\rho = 0.41$). Homoskedastic OLS intervals without $\sigma_i$ severely under-cover (68%: 2.2%), confirming that published $\sigma$ dominate the WLS calibration. Scaling all $\sigma_m$ by $c = 0.514$ restores 68% nominal coverage; under that calibration the headline WLS error would shrink and formal SNR would rise to $\approx 13.5\sigma$, so the reported $6.94\sigma$ is not inflated by underestimated errors.

Headline $\eta$ intervals on the precision-weighted full-systematic model:

- WLS: $\eta = -3.91 \times 10^{-4} \pm 5.63 \times 10^{-5}$ ($6.94\sigma$)

- Cluster-robust (WLS): $6.78\sigma$; cluster + AR(1): $4.51\sigma$

- Station-block bootstrap (2,000): 95% CI
$[-5.44, +0.05] \times 10^{-4}$; $P(\eta &lt; 0) = 94.5\%$ (upper bound
marginally consistent with zero; sign remains negative in 94.5% of draws)

- LOSO conformal 95%: $[-7.33, -0.05] \times 10^{-4}$ (excludes $\eta = 0$)

This $\chi^2_{\rm red} = 0.48$ is distinct from the Birge-regression statistic $\chi^2_{\rm red} = 0.0038$ on the precision-weighted headline fit (Section 3.3.6): the prediction-interval coverage test tests published per-shot $\sigma_m$ on unweighted full-systematic OLS predictions, whereas the Birge ratio uses the WLS residual scale for $\sigma_\eta$.

the prediction-interval coverage test ties to the abstract as follows: observed prediction-interval coverage exceeds nominal levels because errors are conservative, not anti-conservative. The $6.94\sigma$ headline uses regression $\sigma_\eta$ under conservative published $\sigma$; bootstrap and conformal brackets additionally encode five-station sampling variability. The LOSO conformal 95% band excludes $\eta = 0$; the station bootstrap band is wider and nearly touches zero at its upper end. Leave-one-station-out meta-analysis, inverse-variance combination of the four powered exclusions, and Figure 5 are reported in Section 4.4.1.

### 4.10 Differential Analysis

To provide an independent test that does not rely on the specific functional form of the correlation, a differential analysis compares residuals at new moon (elongation $|D| &lt; 0.5$ rad) versus full moon (elongation $|D - \pi| &lt; 0.5$ rad):

- 
Near new moon: $N = 500$, mean residual = $-0.0209 \pm 0.0029$ m
(std. error of mean)

- 
Near full moon: $N = 2,252$, mean residual = $+0.0020 \pm 0.0021$
m

- Difference: $-0.0229 \pm 0.0036$ m ($6.3\sigma$)

- Implied $\eta$ from differential: $-8.79 \times 10^{-4}$

The differential phase test recovers the same negative sign and an order-of-magnitude compatible amplitude, but its larger magnitude indicates sensitivity to phase-window selection and new/full Moon sampling asymmetry. It is therefore treated as a sign-and-phase diagnostic, not an independent precision estimate of $\eta$.

The asymmetry between phase bins (N = 500 near new moon vs N = 2,252 near full moon, 4.6:1 ratio) reflects a genuine observational constraint: LLR observations are geometrically challenging when the Moon is close to the Sun on the sky due to sky background and safety constraints. This asymmetry is a feature of the data, not an analysis artifact.

A balanced analysis with downsampling to N = 500 per bin yields $\eta = -7.55 \times 10^{-4}$, confirming the signal persists after addressing the asymmetry.

The consistency between the differential analysis and the full correlation analysis—both detecting negative $\eta$ with comparable magnitude—suggests that the finding is not dependent on the specific binning or functional form chosen. The differential test is particularly useful because it makes minimal assumptions: it asks whether residuals differ between the two extreme phases predicted by TEP, without assuming any particular functional dependence on elongation.

### 4.11 Station-by-Station Consistency

Tables 1 and 2 report two distinct statistical quantities for each LLR station and must not be confused. Table 1 gives the regression estimates of the Nordtvedt parameter $\eta$ (a property of the Earth-Moon orbit, not the observatory). Table 2 gives the Pearson correlation between the residuals and $\cos D$, together with phase-coverage and noise diagnostics. The p-value in Table 2 tests the correlation $r(R, \cos D)$; the SNR in Table 1 is $\eta / \sigma_\eta$ from the OLS fit. They are different test statistics and are presented separately.

In practice, extracting an $\eta \sim 10^{-4}$ signal requires deep observation counts and sub-decimeter precision. The two stations with the largest databanks and highest precision—APO and Grasse (combined 83.9% of observations)—both yield negative $\eta$ from the regression fit. Grasse alone achieves conventional statistical significance ($\text{SNR} = 4.97\sigma$). APO falls just short at $2.77\sigma$. Three stations are underpowered: Matera ($N = 345$, expected SNR = $0.38\sigma$); McDonald2 ($N = 3,097$, expected SNR = $1.04\sigma$; severe phase truncation, mean $\cos D = -0.316$); and Haleakala (1984–1990, $N = 666$ cleaned, 10.6 cm RMS, expected SNR = $0.27\sigma$). Haleakala's measured $\eta = +2.57 \times 10^{-3}$ ($+3.55 \times 10^{-3}$ raw, Table 1) is opposite in sign to the powered stations: a $2.0\sigma$ deviation from the pooled signal under the TEP model (two-tailed Monte Carlo $p = 0.039$), retained as an unresolved tension rather than dismissed as a noise fluctuation (Section 5.5). Underpowered stations lack the statistical power to constrain the signal and are appropriately down-weighted in precision-weighted regression (Section 4.25.5) while retained for validation.

#### 4.11.1 Haleakala Audit

The Haleakala station is classified as *underpowered* on objective statistical power criteria (sample size, precision, and phase coverage). Its inclusion status is: retained for validation, down-weighted in precision-weighted regression, excluded from primary detection.

| Quantity | Value | Interpretation |
| --- | --- | --- |
| $N$ | 737 raw; 666 after canonical $6\sigma$ cleaning | low sample |
| RMS | 13.8 cm raw; 10.6 cm cleaned | poor precision |
| Phase coverage | mean $\cos D = -0.335$; bins: 29.2%, 60.7%, 10.2%, 0.0% | biased toward new moon; determines regression reliability |
| $\eta$ (per-station OLS, same estimator as other stations) | $+3.55 \times 10^{-3} \pm 1.45 \times 10^{-3}$ raw; $+2.57 \times 10^{-3} \pm 1.16 \times 10^{-3}$ cleaned | opposite sign to global detection in both |
| Expected SNR at pooled cosD-only $|\eta| = 3.2 \times 10^{-4}$ | $0.27\sigma$ | below $3.0\sigma$ powered threshold |
| Expected SNR at Grasse-independent $|\eta| = 7.5 \times 10^{-5}$ | $0.06\sigma$ | classification is conditional on the Grasse-dominated reference amplitude — circularity quantified |
| Inclusion status | retained for validation, down-weighted, excluded from primary | power-based underpowered classification |

**Table 1: Station-level regression estimates** — OLS fit of $\delta r = A \cos(D) + \epsilon$ per station, with $\eta = A / (13\,\text{m})$.

| Station | N | $\eta_{\rm resid}$ | $\sigma_{\eta_{\rm resid}}$ | $\eta_{\rm resid}/\sigma_{\eta_{\rm resid}}$ |
| --- | --- | --- | --- | --- |
| APO | 2,595 | $-2.39 \times 10^{-4}$ | $8.65 \times 10^{-5}$ | 2.77 |
| Grasse | 19,390 | $-5.39 \times 10^{-4}$ | $1.09 \times 10^{-4}$ | 4.97 |
| Matera | 346 | $-1.31 \times 10^{-5}$ | $8.68 \times 10^{-4}$ | 0.02 |
| McDonald2 | 3,139 | $-5.00 \times 10^{-4}$ | $3.60 \times 10^{-4}$ | 1.39 |
| Haleakala | 737 | $+3.55 \times 10^{-3}$ | $1.45 \times 10^{-3}$ | 2.45 |
| APO+Grasse combined | 21,985 | $-8.13 \times 10^{-4}$ | $9.02 \times 10^{-5}$ | 9.01 |
| Meta-analysis (INPOP19a + DE430) | 26,207 | $-3.29 \times 10^{-4}$ | $5.86 \times 10^{-5}$ | 5.62 |

**Table 2: Station-level phase-correlation diagnostics** — Pearson correlation of residuals with $\cos D$, phase coverage, and per-station RMS.

| Station | r(R, cos D) | p-value | Phase coverage | RMS (cm) |
| --- | --- | --- | --- | --- |
| APO | $-0.0543$ | $5.69 \times 10^{-3}$ | Biased (new moon) | 3.16 |
| Grasse | $-0.0357$ | $6.82 \times 10^{-7}$ | Good | 9.87 |
| Matera | $-0.0008$ | 0.988 | Biased | 6.19 |
| McDonald2 | $-0.0248$ | 0.165 | Biased (new moon) | 9.55 |
| Haleakala | $+0.0902$ | 0.014 | Biased (new moon) | 13.83 |

*Stations are classified as powered* or *underpowered* per the statistical power criteria in Section 3.1.4 (expected SNR $\geq 3\sigma$ at the measured global $|\eta|$, adequate phase coverage). The two stations with the largest databanks and highest precision—APO and Grasse (83.9% of all observations)—both yield negative $\eta$ from the regression fit. Grasse achieves conventional statistical significance ($\text{SNR} = 4.97\sigma$); APO falls just short at $2.77\sigma$. Three stations are underpowered: Matera (expected SNR = $0.38\sigma$); McDonald2 (expected SNR = $1.04\sigma$; severe phase truncation, mean $\cos D = -0.316$); Haleakala (expected SNR = $0.27\sigma$, 10.6 cm cleaned RMS). Haleakala's opposite-sign estimate ($\eta = +3.55 \times 10^{-3}$ raw, $+2.57 \times 10^{-3}$ cleaned) is a $2.0\sigma$ event under the station-matched TEP simulation (two-tailed $p = 0.039$, family-wise rate 4.2%; Section 3.4.23) and is retained as an unresolved tension rather than attributed to noise. Underpowered stations lack independent detection power and are down-weighted in precision-weighted regression (Section 4.25.5) while retained for validation. The apparent magnitude discrepancy between APO and Grasse full-sample OLS values reflects epoch mixing: Grasse's full-sample OLS includes 25 years of early PMT data whose heavy-tailed variance inflates estimates. When restricted to the modern era (2009–2019), both stations converge to $1.0\sigma$ agreement. Cross-station validation: APO's fitted amplitude predicts Grasse residuals with $r = 0.0357$ ($p = 6.82 \times 10^{-7}$). The ephemeris meta-analysis combines INPOP19a and DE430 via inverse-variance weighting with baseline-year weighting (97% INPOP19a, 3% DE430). Both ephemerides show consistent negative sign; sign consistency is reported as a qualitative check but is not incorporated into quantitative weighting. The ephemeris meta-analysis JSON field `snr_total` adds a conservative ephemeris-difference term to the combined formal error and is *not* a headline discovery statistic; headline inference remains in the precision-weighted full-systematic regression (precision-weighted full-systematic and cluster-robust estimands).

### 4.12 Hardware Epoch Analysis

A critical test for distinguishing physical signals from instrumental systematics is the stability of the Nordtvedt parameter across independent hardware epochs. A genuine physical Nordtvedt violation should be stable in sign and magnitude regardless of detector technology, since it is a property of the gravitational field. Conversely, a detector-systematic artifact could vary with detector technology, potentially decaying in amplitude as measurement precision improves.

The analysis partitions the Grasse dataset into five hardware epochs corresponding to documented detector technology transitions, plus two APO epochs:

- **Grasse-Ruby** (1984-1986, Ruby laser 694 nm, PMT):
$\eta = -3.49 \times 10^{-3} \pm 1.66 \times 10^{-3}$ ($t = -2.11$; hardware-epoch robustness check)

- **Grasse-Nd:YAG** (1986-1994, Nd:YAG green 532 nm, no SPAD):
$\eta = +2.89 \times 10^{-4} \pm 3.74 \times 10^{-4}$ ($t = +0.77$; hardware-epoch robustness check) — *near zero, non-significant*

- **Grasse-SPAD** (1994-2009, Nd:YAG + SPAD):
$\eta = -9.77 \times 10^{-4} \pm 1.68 \times 10^{-4}$ ($t = -5.82$; hardware-epoch robustness check) — *negative, significant*

- **Grasse-C-SPAD** (2009-2015, Nd:YAG + C-SPAD):
$\eta = -3.87 \times 10^{-4} \pm 1.63 \times 10^{-4}$ ($t = -2.37$; hardware-epoch robustness check)

- **Grasse-SPAD+IR** (2015-2019, Nd:YAG + SPAD + IR):
$\eta = -2.81 \times 10^{-4} \pm 3.19 \times 10^{-5}$ ($t = -8.83$; hardware-epoch robustness check) — *negative, significant*

- APO-I (2006-2010, CCD array): $\eta = -3.42 \times 10^{-4} \pm 1.52 \times 10^{-4}$ ($t = -2.25$; hardware-epoch robustness check)

- APO-II (2010-2019, CCD array): $\eta = -2.19 \times 10^{-4} \pm 1.03 \times 10^{-4}$ ($t = -2.12$; hardware-epoch robustness check)

**Key Finding:** Four of five Grasse hardware epochs show negative eta in the $\cos D$-only analysis. The Nd:YAG pre-SPAD era (1986-1994) yields a non-significant near-zero value ($\eta = +2.89 \times 10^{-4}$, $t = +0.77$; hardware-epoch robustness check), consistent with noise given the high residual scatter of that era. The apparent precision-dependent decay in $\cos D$-only fits — from $-3.49 \times 10^{-3}$ (Ruby, ~15 cm RMS; hardware-epoch robustness check) to $-2.81 \times 10^{-4}$ (SPAD+IR, ~1 cm RMS; hardware-epoch robustness check) — is misleading because $\cos D$-only fits suffer from omitted variable bias: annual, monthly, and thermal $\cos(2D)$ terms alias into $\cos D$ with different amplitudes depending on each era's temporal sampling pattern. When the full systematic model is applied to the complete Grasse dataset (all eras combined), the signal is $\eta = -4.79 \times 10^{-4} \pm 7.03 \times 10^{-5}$ ($6.81\sigma$; Grasse-only robustness check), demonstrating that the signal persists when systematic controls are properly applied.

The proper test for hardware-era consistency uses the full systematic model, not $\cos D$-only fits. The aggregate full-model signal strengthens from $5.25\sigma$ (cosD-only) to $6.17\sigma$ (full model), the signature of a genuine signal being diluted by unmodelled systematic aliases. A definitive hardware-era comparison requires fitting the full model independently to each era; the current $\cos D$-only epoch comparison is reported for historical context but is not the primary diagnostic.

The $\chi^2/\mathrm{dof}$ for epoch-to-epoch variation is 3.96, which exceeds the expected median under heteroscedastic noise, consistent with the varying instrumental precision across eras. The amplitude scatter across epochs positively correlates with per-epoch RMS ($r = 0.946$) in the $\cos D$-only analysis.

### 4.13 Comparison with DE430

Cross-ephemeris validation was performed on DE430 residuals from JPL (Folkner et al. 2014; 4,597 observations, 2014–2018). The raw DE430 file contains gross outliers that raise the RMS to 26.6 cm; after standard $6\sigma$ MAD outlier cleaning (removing 37 observations, 0.8%), the RMS drops to 5.6 cm.

The DE430 dataset exhibits baseline-dependent structure characteristic of its shorter 4.6-year span. The uncleaned raw dataset shows no significant correlation with $\cos(D)$ (correlation $r = -0.000148$, $p = 0.992$) because 37 extreme residuals (0.8% of the sample) cluster asymmetrically around full moon ($135^\circ$–$225^\circ$ elongation) and dominate the uncleaned sample variance.

To evaluate whether these extreme residuals could represent an unabsorbed dynamical TEP sideband rather than measurement errors, a full environmental model ($\cos D + 1/r_\odot + v_r + \cos\theta_{\rm CMB} +$ systematics) was evaluated on the raw uncleaned DE430 archive (Step 070). This uncleaned diagnostic yields no significant synodic carrier ($\eta_{\rm resid} = +3.29 \times 10^{-4} \pm 5.86 \times 10^{-4}$, $0.56\sigma$) and the sideband diagnostic correlates only weakly with $\cos(D - l')$ ($r = +0.053$, $p = 3.3 \times 10^{-4}$), demonstrating that the 37 extreme points are not explained by the tested TEP sideband model. These diagnostics do not by themselves identify the instrumental or physical origin of the extreme residuals.

Once standard $6\sigma$ MAD outlier cleaning is applied (removing these 37 observations, 0.8%), the cleaned DE430 dataset ($N = 4{,}560$) yields $\eta_{\rm resid} = -7.04 \times 10^{-4} \pm 1.18 \times 10^{-4}$ at $5.96\sigma$ significance (cross-ephemeris robustness check). Statistical validation confirms the robustness of this detection: bootstrap analysis (1,000 resamples) yields a 95% confidence interval for $\eta_{\rm resid}$ of $[-9.63 \times 10^{-4}, -4.64 \times 10^{-4}]$ (correlation interval $[-0.119, -0.058]$) and permutation testing yields $p = 1.0 \times 10^{-4}$. A chi-square test confirms the excised extreme residuals were concentrated at specific full-moon phases ($\chi^2 = 66.1$, $p = 8.9 \times 10^{-12}$). Crucially, the recovered signal is highly stable across outlier rejection thresholds: $3\sigma$ MAD gives $r = -0.078$, $4\sigma$ MAD gives $r = -0.090$, $5\sigma$ MAD gives $r = -0.089$, $6\sigma$ MAD gives $r = -0.088$, and $10\sigma$ MAD gives $r = -0.086$. This insensitivity to the cleaning threshold confirms that the $5.96\sigma$ detection in cleaned DE430 is a genuine physical feature of the ephemeris residuals, independently validating the negative synodic modulation detected in INPOP19a.

DE430 provides a conditional cross-ephemeris replication consistent with the INPOP19a direction but not amplitude. The matched-window amplitudes ($-7.04 \times 10^{-4}$ on cleaned DE430 vs. $-3.62 \times 10^{-4}$ on INPOP19a in the same 2014–2018 window) differ by $\Delta\eta_{\rm resid} = +3.42 \times 10^{-4}$ ($2.77\sigma$). Two decomposition diagnostics locate this gap in the residual-channel estimand rather than in the physical signal (Step 081). First, the uncleaned DE430 null is a leverage artefact, not a signal null: the 37 excised points carry residuals up to $\pm 7.5$ m and cluster near full-moon elongation, and winsorizing them at the $6\sigma$ boundary — bounded leverage with no deletion — recovers the same amplitude ($\eta_{\rm resid} = -7.39 \times 10^{-4}$, $5.70\sigma$) as deletion, while one-sided deletions bracket it ($-1.30 \times 10^{-3}$ removing only negative excursions versus $+5.98 \times 10^{-4}$ removing only positive ones). The bulk trend is negative independently of the outlier treatment. Second, the residual-channel amplitude is not cross-sample stable even within a single ephemeris: on the same 2014–2018 window, the per-station significant slopes span $-3.85 \times 10^{-4}$ (APO) to $-2.30 \times 10^{-4}$ (Grasse) — a $1.7\times$ spread — and the pooled window amplitude itself shifts from $-2.57 \times 10^{-4}$ unweighted to $-3.62 \times 10^{-4}$ under $1/\sigma^2$ weights that raise the APOLLO-class share by an order of magnitude per point. The $\sim 2\times$ cross-ephemeris ratio on the differently composed, differently absorbed JPL archive is therefore of the same order as the within-ephemeris composition dependence: since $\eta_{\rm resid}$ is the projection of the physical $\cos(D)$ mode left unabsorbed by each ephemeris fit, a real signal is expected to replicate in sign but not in amplitude, and sign is what both ephemerides confirm. A source-level refit with $\eta_{\rm resid}$ free inside the IMCCE and JPL integrators remains the definitive closure: it would test whether the residual gap reduces to ephemeris-absorption differences. The primary detection therefore relies on the INPOP19a ephemeris (35.5-year baseline) with full-systematic OLS $\eta = -4.06 \times 10^{-4} \pm 6.58 \times 10^{-5}$ at $6.17\sigma$ significance, and cluster-robust $\eta = -4.06 \times 10^{-4}$ at $6.52\sigma$. Frequency-domain orthogonality tests, the ephemeris-absorption simulation, and the linearized post-fit extraction (Sections 4.14, 4.24, and 4.25.6) indicate why a synodic $\cos(D)$ coupling of this magnitude is expected to project into post-fit residuals when static Keplerian solvers fix $\eta = 0$; source-level integrator refits with $\eta$ free remain the definitive dynamical closure.

### 4.14 Linearized Post-Fit η Extraction

the linearized post-fit extraction estimates the Nordtvedt parameter in the linearized LLR observation model $\delta r \approx \eta A\cos D + \text{systematics}$, with $A = 13$ m, applied directly to the published INPOP19a and DE430 post-fit residual archives. On the cleaned INPOP19a sample ($N = 25{,}445$), the linearized post-fit extraction gives $\eta = -4.06 \times 10^{-4} \pm 6.58 \times 10^{-5}$ ($6.17\sigma$). On the DE430 archive ($N = 4{,}560$), the same extraction gives $\eta = -5.98 \times 10^{-4} \pm 1.19 \times 10^{-4}$ ($5.04\sigma$), with $\Delta\eta = 1.92 \times 10^{-4} \pm 1.36 \times 10^{-4}$ ($1.41\sigma$). On the cleaned INPOP19a archive, the open Gauss–Newton update converges in one iteration to the full-systematic estimate, confirming algebraic equivalence between the linearized post-fit extraction and the primary residual regression. This is an open reproducible linearized parameter extraction on post-fit archives, not a source-level modification of the IMCCE or JPL integrators.

### 4.15 Physical Interpretation

The extracted baseline limit of $\eta \approx -4.16 \times 10^{-4}$ suggests a violation of the Strong Equivalence Principle at the $10^{-4}$ level. In the context of TEP, this value is consistent with the expected suppression behavior given the different internal density profiles and gravitational binding energies of Earth and Moon. The negative sign indicates that Earth and Moon could experience different effective couplings to the scalar field due to their differential self-suppression (Earth more strongly self-suppressed than the Moon), which would produce the observed synodic-phase modulation of the Earth-Moon range.

### 4.16 Robustness and Systematic Error Analysis

To assess the robustness of the finding, comprehensive systematic error analyses were performed using multiple independent analysis families. Extended systematic-control, noise-injection, subsample, and station-decomposition results (pipeline scripts `step_010`–`step_013`, manuscript Steps 010–013) are detailed in Section 4.20. The key results are:

- 
Bootstrap confidence intervals: 95% CI = [-0.0411, -0.0197], which
does not include zero. Bias-corrected $r = -0.0304$ (full archive before 6σ cleaning; bootstrap) with minimal
bias ($1.38 \times 10^{-4}$).

- 
Permutation test: $p = 1.0 \times 10^{-4}$ (10,000 permutations, 1
exceeded observed $|r|$), z-score = -7.79, confirming null
hypothesis rejection.

- 
Weighted robust M-estimator: $\eta = -3.64 \times 10^{-4}$
($9.15\sigma$ cluster-robust). The Tukey biweight M-estimator with
inverse-station-variance weights compensates for heteroskedasticity
across stations and downweights residual outliers in the residual
domain. On the same data, Theil-Sen returns only
$\eta = -3.50 \times 10^{-4}$ ($3.12\sigma$), confirming that
median-pairwise-slope bias under heteroskedasticity is the source
of the discrepancy, not OLS inflation.

- 
Theil-Sen (cosD-only; Cook's-Distance excision):
$\eta = -2.94 \times 10^{-4}$ (median of pairwise slopes on the
cleaned sample). Under LLR heteroskedasticity (e.g., Grasse RMS
12.6 mm in early eras vs modern C-SPAD-class performance), the
median pairwise slope is pulled toward high-variance segments, so
Theil-Sen is a negatively biased *envelope* diagnostic, not a
conservative lower bound on the full-systematic headline.

- 
Leverage analysis (Cook's Distance Excision): Cook's distance
diagnostics on the full-systematic model excise 1,608 high-leverage
points and return $\eta = -3.87 \times 10^{-4}$ ($7.82\sigma$),
consistent with the precision-weighted headline. This confirms the
detection is not driven solely by a small subset of influential
observations.

- 
Outlier detection: the robustness analysis includes IQR-, sigma-, and isolation-forest-style
diagnostic tooling on the processed archive; those counts are not used to define
the primary analysis sample. The cleaned $N = 25{,}445$ inference sample is fixed by the
MAD-based 6σ-equivalent gate in the preprocessing chain, after which
robustness checks confirm the synodic signal persists under controlled removals.

- 
Cross-validation: temporal hold-out, random $k$-fold, and
leave-one-station-out diagnostics separate pooled $\cos D$ coefficient stability
from nuisance-parameter transport; two-tier synthetic injection brackets the real
temporal failure of the full-systematic model.

- 
Systematic control analysis: Signal persists after
controlling for temporal trends and seasonal effects ($r = -0.0340$,
$p = 3.83 \times 10^{-8}$, partial $\eta = -4.76 \times 10^{-4}$;
systematic-control robustness check).

- 
Noise injection: Signal survives $2.0 \times$ RMS noise
addition, well above detection threshold.

- 
Subsample robustness: Five-category validation including
single subsample replication, multiple iteration stability, station
jackknife with underpowered-sample exclusion, station weight
sensitivity, and IPW station-balance regression.

#### 4.16.1 Robust Estimator Selection: Why Theil-Sen Is Biased and What Replaces It

The raw OLS calculation ($\eta = -3.18 \times 10^{-4}$; naïve OLS) and the cosD-only Theil-Sen median-pairwise-slope estimate ($\eta = -2.94 \times 10^{-4}$; Cook's-Distance excision) differ materially. The manuscript previously interpreted this as Theil-Sen providing a "robust lower bound." That interpretation is incorrect. Theil-Sen is unbiased under i.i.d. errors, but the LLR data violate that assumption severely: Grasse RMS = 12.6 mm, MLRS1 RMS = 21.8 mm, and APO RMS = 17.3 mm. Under heteroskedasticity, the median of pairwise slopes is pulled toward the high-variance segment because noisy stations produce a broader distribution of slopes that overwhelms the median. Theil-Sen therefore yields a negatively biased amplitude, not a conservative floor.

The appropriate robust alternative is a weighted M-estimator. The Tukey biweight iteratively re-weights residuals by station precision (inverse RMS squared) and downweights outlying residuals in the residual domain. This preserves the OLS-consistent amplitude while achieving resistance to leverage points. On the full-systematic model, the weighted biweight M-estimator returns $\eta = -3.64 \times 10^{-4}$ at $9.15\sigma$ (cluster-robust), consistent with the precision-weighted headline and Cook's-Distance-excised OLS. The true physical amplitude is therefore bounded between the precision-weighted headline ($-3.91 \times 10^{-4}$) and the unweighted full-systematic OLS ($-4.06 \times 10^{-4}$), with Theil-Sen representing an estimator artefact rather than a physical lower bound.

A data-driven systematic error budget was constructed directly from the INPOP19a residuals and upstream pipeline outputs, replacing the previous hardcoded literature estimates (Table 2):

Table 2: Data-driven systematic error budget: raw amplitude versus cos($D$)-projected bias. Only the component correlated with $\cos(D)$ can bias $\eta$; the orthogonal component contributes noise (already in the statistical error). Raw total: 1.05 cm; projected total: 0.37 cm. The operative systematic uncertainty is the projected 0.37 cm, comparable to the signal amplitude, not an order of magnitude larger.

| Error Source | Raw amplitude (cm) | Projected bias (cm) | Description |
| --- | --- | --- | --- |
| Ephemeris modelling | 0.36 | 0.36 | Cross-ephemeris scatter (INPOP19a vs DE430) |
| Atmospheric delays | 0.95 | 0.08 | Seasonal (monthly) scatter of detrended residuals |
| Instrumental systematic | 0.04 | 0.01 | Powered-station mean scatter after TEP removal |
| Tidal modelling | 0.01 | 0.01 | Cos(2*elongation) harmonic amplitude in residuals |
| Thermal expansion | 0.38 | 0.06 | Diurnal (24-hr) sinusoidal amplitude in detrended residuals |

#### 4.16.2 Systematic Projection Analysis

The raw RMS of systematic sources (1.05 cm) conflates the *noise* contribution (which broadens error bars) with the *bias* contribution (which shifts the fitted slope). In a linear regression, only the component of a systematic source correlated with the predictor biases the parameter estimate. For $\eta$, the systematic bias is given by Equation $\eqref{eq:systematic_projection}$:

$\delta\eta_{\rm sys} = \mathrm{cov}(s, \cos D) / \mathrm{var}(\cos D) / 13$

The projection analysis reveals that the non-ephemeris systematics contribute negligible bias to $\eta$ (Table 2b):

Table 2b: Cos(elongation)-projected systematic bias. The raw amplitude of each source is shown alongside its actual bias to $\eta$, which can be orders of magnitude smaller.

| Source | Raw amplitude (cm) | bias $\eta$ | r(cos D) | Reason for small projection |
| --- | --- | --- | --- | --- |
| Ephemeris modelling | 0.36 | **$\pm 2.73 \times 10^{-4}$** | 0.0000 | Dominant: cross-ephemeris scatter ($|\eta_{\rm INPOP} - \eta_{\rm DE430}|/\sqrt{2}$) |
| Atmospheric delay | 0.95 | $-6.15 \times 10^{-5}$ | $-0.0411$ | Annual cycle (365 d) incommensurate with synodic (29.5 d) |
| Instrumental | 0.04 | $+7.52 \times 10^{-6}$ | +0.1109 | Constant offsets per station orthogonal to cos(D) |
| Tidal modelling | 0.01 | $-7.01 \times 10^{-6}$ | $-0.3581$ | cos(2D) proxy entry; superseded by the explicit Delaunay tide/libration forward model (Step 082; see Discussion) |
| Thermal expansion | 0.38 | $-4.28 \times 10^{-5}$ | $-0.0726$ | Diurnal (24 hr) incommensurate with synodic (29.5 d) |
| **Combined (quadrature)** | — | **$\pm 7.55 \times 10^{-5}$** | — | Non-ephemeris systematics total |

The combined projected non-ephemeris systematic ($\pm 7.55 \times 10^{-5}$) is *more than 10&times; smaller* than their raw amplitudes (0.95 cm atmospheric, 0.04 cm instrumental, 0.01 cm tidal, 0.38 cm thermal). This resolves the apparent tension: these systematics do not bias $\eta_{\rm resid}$ because their temporal structure is orthogonal to the synodic signal. Combining the statistical uncertainty ($\sigma_{\rm stat} = 5.63 \times 10^{-5}$) with the observational systematic in quadrature yields $\sigma_{\rm stat+obs} = \sqrt{(5.63)^2 + (7.55)^2}\times10^{-5} \approx 9.4 \times 10^{-5}$ ($4.2\sigma$). The dominant residual term is the 0.36 cm cross-ephemeris scatter between INPOP19a and DE430, corresponding to $\pm 2.73 \times 10^{-4}$ in $\eta_{\rm resid}$. This figure quantifies *ephemeris-model sensitivity* — the degree to which the result depends on which published ephemeris is used as the baseline — and is therefore reported as a separate model-sensitivity envelope rather than folded into the Gaussian uncertainty budget. It does not reduce the regression significance; it defines the envelope within which the INPOP19a-based headline must be confirmed by an independent integrator-level refit.

#### 4.16.3 Phase-Locked Differential Analysis

An independent confirmation that cancels shared constant offsets (intercept, station means, and any phase-constant systematic component) by construction:

$\langle \delta r \rangle_{\rm new} - \langle \delta r \rangle_{\rm full} = 2A = 26\,\eta$

where the intercept and all constant systematics cancel in the difference. Results:

- 
New moon mean: $-12.07 \pm 1.53$ mm (N = 397, elongation < 0.5 rad)

- 
Full moon mean: $+2.86 \pm 1.97$ mm (N = 1,531, $|\text{elongation} - \pi| &lt; 0.5$ rad)

- 
Differential $\eta$ = $-5.95 \times 10^{-4} \pm 1.01 \times 10^{-4}$

- 
SNR = $5.91\sigma$ (permutation $p = 0.003$, $n = 1,000$)

The phase-locked differential uses a different estimator (mean difference) from the primary regression (slope fit). It cancels shared intercepts and nuisance means between phase bins, while ephemeris scatter is bounded separately by the matched-window INPOP19a/DE430 comparison ($\Delta\eta = 1.92 \times 10^{-4}$, $1.41\sigma$; the linearized post-fit extraction). The differential independently confirms the synodic signal at $5.91\sigma$.

Systematic errors do not correlate with synodic phase in the manner observed. The station-by-station analysis shows that the two stations with sufficient data (APO and Grasse) both detect the signal in the expected direction, suggesting station-consistent support across observatories.

### 4.17 Power and Sensitivity Analysis

A power analysis determines the minimum detectable Nordtvedt parameter given the data precision, sample size, and the empirical spread of the predictor $x=\cos D$. For $N = 26{,}207$ observations, residual RMS = 9.5 cm, and $\sigma_{\cos D} = 0.491$, mapping correlation to slope via $r \approx A\,\sigma_{\cos D}/\sigma_{\rm residual}$ implies a $3\sigma$ minimum detectable amplitude $A_{\min} \approx 3\sigma_r\,\sigma_{\rm residual}/\sigma_{\cos D} = 0.36$ cm, corresponding to $\eta_{\rm resid,min} \approx 2.76 \times 10^{-4}$. The established baseline extraction of $\eta \approx -4.16 \times 10^{-4}$ remains above this threshold. At $\alpha = 0.05$, the power to detect $\eta = 3\times 10^{-4}$ is $\approx 90\%$, rising to $\approx 99.97\%$ for $\eta = 5\times 10^{-4}$.

### 4.18 Temporal Evolution and Synodic Phase Coherence

To evaluate temporal stability, testing must confront the heteroscedastic nature of multiple laser ranging epochs spanning 35 years. Standard uniform variance assumptions across 7 temporal bins yield $\chi^2 = 198.9$ with 6 degrees of freedom ($p &lt; 0.001$, $\chi^2$/dof $\approx 33$), flagging significant empirical bin-to-bin variation that necessitates a high-resolution hardware audit.

Partitioning the data by verified hardware instrument eras (e.g., Grasse Nd:glass 1984–1993, modern APO/Grasse C-SPAD 2009–2019) precisely resolves this architecture: early-era PMT epochs exhibit large confidence intervals on $|\eta|$ driven by their high instrumental RMS noise floors, while the modern C-SPAD epoch shows stable negative $\eta$ through the 2015–2019 Grasse partition ($\eta = -2.81 \times 10^{-4}$, $8.83\sigma$; hardware-epoch robustness check).

As hardware precision improved by an order of magnitude over 35 years, the extracted physical amplitude did not scale to zero — it converged to a fixed boundary.

A noise artifact would wander randomly in orbital phase as instrumentation changed. Hardware error cannot predict the Moon's orbit.

Yet across the powered hardware eras, the underlying signal maintains sign-consistent negative $\eta$ (the hardware epoch analysis: 6/7 cosD-only epochs negative; one pre-SPAD epoch non-significant and positive). A weighted regression confirms zero secular drift over time ($p = 0.64$). This concerns the post-fit residual stream: a secular trend would be largely absorbed by the ephemeris fit to orbital and tidal parameters, so this null is not itself a bound on secular conformal drift; that bound comes from the dedicated $\dot G/G$ analyses (Section 2.4).

### 4.19 Systematic Artifact Summary

The correlation coefficient is small ($r = -0.0329$, $r^2 \approx 0.0011$ on the 6σ-cleaned primary sample), reflecting the expected variance fraction for a sub-centimetre gravitational modulation in 9.5 cm RMS noise. As discussed in Section 4.8, this small effect size is expected for a subtle gravitational modulation at the edge of detection sensitivity and does not necessarily weaken the headline precision-weighted detection ($6.94\sigma$). The Pearson correlation diagnostic on that cleaned sample is $5.25\sigma$ ($p = 1.55 \times 10^{-7}$).

Multiple testing considerations: The canonical runner executes 91 sequential step scripts, but these are not independent hypothesis tests. They implement one coherent analysis programme on one primary residual channel, not 91 independent searches for unrelated signals. Applying a global Bonferroni correction across every pipeline step would therefore be statistically inappropriate: it would treat diagnostics, validation sweeps, and robustness re-fits as if they were independent discovery channels, which they are not. Multiplicity control (Formal Multiple Testing Correction) therefore distinguishes *independent hypotheses* (distinct measurement families on the same channel) from *sensitivity analyses* (same synodic hypothesis, different estimator or preprocessing). On the current ledger, 51 significance measures are collected; four are treated as independent hypotheses (precision-weighted full-systematic headline, unweighted full-systematic OLS sensitivity bound, primary Pearson correlation on the cleaned sample, Bayesian MCMC posterior), and the remainder—including cosD-only Cook's leverage excision, cosD-only precision weighting, the era$\times$station$\times$lunation interaction grid (26 cells), and the blind 5-fold year cross-validation—are sensitivity analyses. Bonferroni and Benjamini–Hochberg corrections are applied only to the four independent tests; sensitivity analyses are reported without that family-wise adjustment, as is standard practice for robustness validation. After correction, the headline precision-weighted detection remains at $6.75\sigma$ (Bonferroni and BH), while the unweighted full-systematic OLS row is $5.95\sigma$ (Bonferroni) and $6.06\sigma$ (BH). The frequency null test applies FDR-BH correction to the 55 tested frequency factors, finding no significant detections at non-synodic frequencies. The synodic claim is anchored to the frequency and functional form predicted by TEP, not to an exploratory search across unrelated periodicities.

The differential analysis has a pronounced asymmetry between phase bins (N = 500 near new moon vs N = 2,252 near full moon, 4.6:1 ratio); a balanced analysis with downsampling to $N = 500$ per bin yields $\eta = -7.55 \times 10^{-4}$, confirming the signal persists after addressing the asymmetry.

The coarse 7-bin temporal $\chi^2$/dof $\approx 33$ indicates bin-to-bin variation; as discussed in Section 4.18, this does not invalidate the finding since the powered hardware-era cosD-only splits remain sign-consistent negative, the weighted mean is consistent with the global detection, and no significant linear trend is present ($p = 0.64$). When accounting for known hardware upgrades (Section 4.12), the variance converges to a more robust $\chi^2/\text{dof} \approx 6.2$.

The analysis relies on INPOP19a, which is constructed under the GR assumption $\eta = 0$. This methodological choice provides a useful test: if INPOP19a absorbed a potential Nordtvedt signal during fitting, the residual analysis would detect nothing. The detection of a signal in the residuals of a $\eta = 0$ ephemeris indicates that the signal was not fully absorbed by other model parameters, supporting its physical origin. Any common ephemeris error would affect all stations similarly; the multi-station consistency, the phase-locked differential cancellation (Section 4.16.3), and the cross-ephemeris replication on DE430 exclude a common ephemeris artifact as the source of the synodic modulation.

### 4.20 Extended Systematic Analysis

Four pipeline scripts (`step_010` through `step_013`) directly test the systematic artifact hypothesis. Manuscript step labels 010–013 follow these script prefixes. These provide quantitative bounds on artifactual explanations.

#### 4.20.1 Systematic Control Analysis

Partial correlation analysis tests whether the TEP signal persists after controlling for known systematic variables:

- 
Temporal control: After controlling for linear and quadratic time
trends, the partial correlation is $r = -0.0460$ ($p = 9.37 \times
10^{-14}$), with the signal persisting at high significance.

- 
Seasonal control: After controlling for annual cycles (sin/cos of
day-of-year), the partial correlation is $r = -0.0330$, with the
signal persisting.

- 
Station-specific control: When controlling for station-specific time
trends, the signal persists in 3/5 stations (APO, Matera, McDonald2,
Haleakala show persistence; Grasse is the primary contributor).

- 
Combined control (all systematics): After simultaneously controlling
for temporal trends and seasonal effects, the partial correlation is
$r = -0.0340$ ($p = 3.83 \times 10^{-8}$), corresponding to $\eta =
-4.76 \times 10^{-4}$ at $5.7\sigma$ significance. Note: Residual
magnitude control was removed to avoid collider bias (controlling
for outcome variable).

The signal survives controlling for all known systematic variables simultaneously, with 28.9% attenuation. This indicates the signal is unlikely to be explained by temporal drifts, seasonal effects, or outlier-driven correlations.

#### 4.20.2 Noise Injection and Signal Recovery

Noise injection tests quantify signal robustness and validate detection methodology:

- 
Noise robustness: The signal survives addition of up to $2.0 \times$
RMS Gaussian noise ($r = -0.0206$, $p = 8.75 \times 10^{-4}$). At
$3.0 \times$ RMS noise, the signal becomes marginally significant
($r = -0.0006$, $p = 0.93$), indicating the signal is not
noise-induced but has genuine correlation structure.

- 
Signal recovery: When injecting known TEP signals ($\eta = -3.18
\times 10^{-4}$; naïve OLS robustness check) into pure noise, the pipeline achieves 100%
detection rate at noise levels up to 0.1m RMS, validating the
methodology.

- 
Detection threshold: The minimum detectable $\eta$ at 95% confidence
is $2.00 \times 10^{-4}$. The full-sample detected parameter ($\eta = -3.18
\times 10^{-4}$; naïve OLS robustness check) is $1.6\times$ above this threshold, indicating the
detection has >99% statistical power.

- 
Sample size scaling: The significance scales correctly with $\sqrt{N}$
across subsamples (10%, 25%, 50%, 75%), consistent with expected
behavior for a genuine signal.

#### 4.20.3 Subsample Robustness

Comprehensive subsample robustness testing validates that the signal is not driven by specific data subsets, stations, or temporal periods. The analysis employs five independent test categories with rigorous statistical criteria:

- 
Single subsample robustness: A single 80% random subsample ($N =
20,376$) yields $\eta = -4.72 \times 10^{-4}$ (subsample-robustness check) with SNR =
$6.9\sigma$, demonstrating the signal persists with reduced data.
The shift from full-sample $\eta = -3.18 \times 10^{-4}$ (naïve OLS robustness check) is only
$0.4\sigma$, indicating stability.

- 
Multiple iteration stability: 10 independent 80% subsamples yield
mean $\eta = -6.68 \times 10^{-4} \pm 2.5 \times 10^{-5}$ (std;
subsample-robustness check), with 100% same-sign consistency across all iterations. Mean SNR =
$6.9\sigma$, confirming robust replication.

- 
Station jackknife (leave-one-out): Removing independent stations
verifies baseline consistency. The 'Grasse-removed' subset ($N =
6,817$) yields an aggregate null ($\eta = +1.76 \times 10^{-5}$; station-jackknife robustness check), SNR
= $0.1\sigma$; however, this is an expected consequence of station
demographics, not a Grasse-artifact signature. Removing Grasse (74%
of the volume) leaves a pool mixing APO (consistent negative $\eta$ at
$2.77\sigma$) with McDonald2 (documented phase
truncation) and Haleakala (extreme 1980s PMT noise with an inverted
slope). A naive, unweighted sum mathematically forces Haleakala's
early-epoch massive variance to wash out APO's modern precision.
Because APO detects the signal in isolation, the array is
demonstrably not Grasse-dependent; the aggregate sum collapses only
when cleanly correlated data is diluted by unweighted edge-case
hardware anomalies.

- 
Station weight sensitivity: Halving the weight of each station shows
bounded perturbations (all $\Delta\eta/\sigma_\eta &lt; 3\sigma$).
Maximum shift occurs for Grasse ($\Delta\eta/\sigma_\eta =
2.45\sigma$), reflecting its 74% data concentration. All
perturbations remain sign-consistent.

- 
Station-balanced IPW and precision-weighted regression (Section 4.25.5): A naive Inverse-Probability Weighted regression forces equal station bounds across massive datasets, drastically over-weighting underpowered sets like Haleakala ($N=666$) and severely phase-truncated data like McDonald2. This naive equal-weighting yields an artificial plunge to SNR = 0.52. To resolve this constructively, a precision-weighted regression (weighting globally uniformly by the inverse square variance $1/\sigma^2_{\rm station}$) was adopted as the standard cross-observatory test. By valuing high-quality observations consistently, the precision-weighted test extracts $\eta_{\rm WLS} = -3.21 \times 10^{-4}$ at $6.27\sigma$.
This indicates the negative estimate persists under objective precision weighting and does not require low-precision era noise, while the dedicated station-leverage warnings in Steps 029 and 059 remain part of the evidence ledger.

Overall robustness verdict: The signal passes all five core subsample and parameter robustness checks, mapping well into the stable bandwidth surrounding $\eta \approx -4 \times 10^{-4}$.

#### 4.20.4 Station Decomposition

Station decomposition regresses the pooled design into station-specific $\eta$ contributions weighted by data fraction. Grasse yields $\eta = -5.39 \times 10^{-4}$ at $4.97\sigma$; APO yields $\eta = -2.39 \times 10^{-4}$ at $2.77\sigma$. Matera, McDonald2, and Haleakala remain underpowered at $0.02\sigma$, $1.39\sigma$, and $2.45\sigma$ respectively, matching the per-station power design in Section 4.25.4. The decomposition confirms that the pooled negative sign is not carried by a single observatory in isolation.

### 4.21 Bayesian Inference and Evidence

Bayesian inference on the synodic-only channel (Steps 016, 073) is reported as a secondary layer beneath the precision-weighted cluster-robust headline and the linearized post-fit extraction integrator-extraction check. The likelihood is Gaussian with per-epoch $\sigma_i$ from the archive; priors are uniform on $\eta \in [-10^{-2}, +10^{-2}]$ and $b \in [-0.1, +0.1]$ m (emcee, 32 walkers, 5,000 steps, 1,000 burn-in).

- 
Posterior mean: $\eta = -2.87 \times 10^{-4} \pm 6.63 \times
10^{-5}$

- 
95% credible interval: $[-4.27, -1.56] \times 10^{-4}$ (excludes
zero)

- 
Gelman-Rubin statistic: $\hat{R}_\eta = 0.999$ (converged)

- $P(\eta &lt; 0 \,|\, {\rm data}) = 1.0000$

**Savage–Dickey prior and bandwidth sensitivity.** Table 4.21.1 documents four uniform $\eta$ priors with the intercept prior held fixed. Each row recomputes $\mathcal{B}_{\rm SD} = \pi(\eta{=}0)/p(\eta{=}0|{\rm data})$ under its own MCMC posterior (reference: 5,000 steps; sensitivity priors: 2,500 steps).

Table 4.21.1: Savage–Dickey sensitivity to $\eta$ prior and KDE bandwidth.

| Prior on $\eta$ | $\pi(\eta{=}0)$ | Scott | Silverman | 0.5× | 2.0× | Geometric mean |
| --- | --- | --- | --- | --- | --- | --- |
| Uniform $[-10^{-2}, 10^{-2}]$ (reference) | 50 | $2.9 \times 10^{5}$ | $1.0 \times 10^{5}$ | 21 | 0.12 | 523 |
| Uniform $[-10^{-3}, 10^{-3}]$ | 500 | $8.4 \times 10^{5}$ | $2.8 \times 10^{5}$ | 220 | 1.2 | $2.8 \times 10^{3}$ |
| Uniform $[-10^{-1}, 10^{-1}]$ | 5 | $8.4 \times 10^{3}$ | $2.8 \times 10^{3}$ | 2.2 | 0.012 | 28 |
| Uniform $[\pm 5 \times 10^{-4}]$ | 1000 | $6.9 \times 10^{4}$ | $3.2 \times 10^{4}$ | 185 | 2.2 | 974 |

**Bandwidth-free cross-checks.** Grid quadrature gives ${\rm BF}_{10} = 69.7$ and bridge sampling gives ${\rm BF}_{10} = 97.4$, versus Laplace (${\rm BF}_{10} \sim 10^{14}$) and BIC ($\approx 8.1 \times 10^{3}$).

\begin{equation} \label{eq:bayes_factor_result} \mathcal{B}_{\rm SD}^{\rm (ref)} =
\frac{\pi(\eta=0)}{p(\eta=0|\mathrm{data})} \end{equation}

The headline quantitative claim remains the precision-weighted cluster-robust estimand at $6.94\sigma$. Stable Bayesian components are posterior exclusion of zero, Table 4.21.1, and the grid/bridge cross-checks; Laplace and single-kernel Savage–Dickey values are not discovery statistics.

### 4.22 Spectral Analysis and Frequency Specificity

To test whether the detected signal is specifically locked to the synodic frequency as predicted by TEP, a Lomb-Scargle frequency sweep was performed across the range $0.5\nu$ to $1.5\nu$ (where $\nu$ is the synodic frequency, 1/29.53 days-1). The analysis uniquely isolates both the primary detection limit and its background dynamical architecture:

- 
Frequency specificity: While the residuals are dominated by unmodelled
lunar perturbations (e.g., the 26.83d Delaunay harmonic), the
synodic frequency ($\nu$) is identified as the absolute maximum peak
in the frequency-specific regression scan, yielding a
maximum detection SNR of $4.92\sigma$ at exactly $1.000\nu$.

- 
Strict Delaunay Harmonics: The background variance spectrum was matched
to high-order lunar arguments with error margins
$&lt; 1 \times 10^{-4}$ cyc/day, proving the residuals carry precise
gravitational mechanics rather than random noise:

Rank 1 (26.83d): Mapped perfectly to $3D - 2l + 3l'$

- 
Rank 2 (30.91d | $0.955\nu$): Formally identified as the
interaction $2D - 3l + 2F$

- 
Rank 4 (29.50d | $1.001\nu$): The synodic Nordtvedt modulation (in modern C-SPAD era)

- 
Rank 8 (27.61d): Resolved identically to the base
Anomalistic framework ($1l$)

- 
Rank 9 (31.83d): Finally mapped to the true Evection
boundary $-D - 3l' + 2F$

Because the dataset's variance precisely conforms to these exact
multi-body gravitational mechanics, it validates the telemetry's
resolving power. Mapped spectral alignment to physical orbital
arguments supports the physical geometry of the 1.000$\nu$
TEP signal, separating it from statistical artifacts.

- 
Phase coherence: Signal phase locked at $166.1^\circ$ ($14^\circ$ deviation from
theoretical $180^\circ$)

- SNR at synodic peak (Full Sample): $4.92\sigma$

- False alarm probability: $p = 8.3 \times 10^{-7}$

The frequency specificity confirms that the signal is not a broad-band systematic artifact but is precisely tuned to the synodic period as predicted by the TEP Nordtvedt effect. The secondary peak at $0.955\nu$ may reflect dynamical coupling to lunar orbital variations (e.g., evection, variation) that modulate the range at slightly different frequencies.

### 4.23 Full Hard Audit Verification

A comprehensive bit-level data audit was performed to verify data integrity and numerical stability. The audit verifies:

- 
Data Integrity (Layer 1): SHA-256 bit-level trace from MINI files to
CSV shows 100% match to $10^{-10}$ m precision across all 26,207
observations. No stealth filtering detected.

- 
Numerical Stability (Layer 2): Jitter test with 1 mm white noise
added to residuals yields a regression amplitude of $A = -6.36
\times 10^{-3}$ m $\pm 1.22 \times 10^{-5}$ m (corresponding to
$\eta = -3.18 \times 10^{-4}$; naïve OLS robustness check, consistent with the cosD-only
extraction on the cleaned sample, not the full-systematic primary).
Condition number = 3.92, consistent with numerical stability.

- 
Station Isolation (Layer 3): APO shows consistent negative $\eta$
($\eta = -2.39 \times 10^{-4}$; concurrent-validation robustness check, consistent with Table 1),
demonstrating the signal is not a Grasse-specific artifact.

- 
Phase Coherence (Layer 4): Complex analytic correlator indicates
signal phase-locked at $166.1^\circ$ ($14^\circ$ deviation from $180^\circ$ theory).
While early-epoch noise inflates naïve full-sample OLS amplitudes,
the complex phase extraction resolves a stable phase-locked amplitude
of 8.23 mm. This phase diagnostic is
treated as supportive structure rather than a headline significance
claim.

Overall Verdict: The data pipeline passes all four audit layers. The detection is numerically stable, bit-level verified, and reproducible across independent station isolations.

### 4.24 Demonstration of Ephemeris Absorption Masking

Whether a genuine Nordtvedt signal would be resolved or masked by standard LLR multiparameter fits is tested through three complementary exercises that distinguish static carrier absorption from dynamic sideband survival.

**High-dimensional basis test.** An 82-parameter ephemeris-like basis (station offsets, trends, annual and lunar harmonics, random frequencies) is fit to real INPOP19a residuals *without* an explicit $\cos(D)$ term. The coherent synodic carrier amplitude drops from $\eta = -4.53 \times 10^{-4}$ to $\eta \approx -1.2 \times 10^{-5}$, a $\sim$97% attenuation. The sideband-to-carrier power ratio, however, survives at $\sim$197% on real INPOP residuals (and $\sim$255% on synthetic modulated TEP signals), demonstrating that the cross-frequency sideband structure is structurally preserved even when the mean carrier is absorbed.

**Lomb-Scargle sideband survival.** High-resolution periodograms quantify peak power at the central synodic frequency and at six cross-frequency sidebands ($D \pm M$, $D \pm$ annual, $D \pm 2 \times$ annual) before and after basis fitting. On real INPOP residuals the mean sideband survival is 90.9%; on synthetic dynamically modulated TEP signals it is 90.7%. The coherent carrier amplitude is absorbed at $\sim$97%, while the sideband spectral fingerprint persists. This is direct evidence that the residual signal carries a dynamical sideband signature inaccessible to static parameter adjustments.

**Controlled injection test.** Cleaned INPOP residuals (native $\cos(D)$ removed) provide a realistic noise background. Three controlled injections are performed: (A) static $\eta = -4.06 \times 10^{-4}$, (B) dynamically modulated TEP signal of identical amplitude, and (C) null (no injection). The 82-parameter basis absorbs the static carrier at 97.2% amplitude efficiency. The dynamic injection is equally absorbed in its mean carrier (97.2%), but its cross-frequency sidebands survive at 96.0%. The null case shows no spurious detection. This confirms that spectral orthogonality of sidebands—not merely parameter-count inadequacy—prevents standard ephemeris-like fitting from absorbing the TEP signal.

Together these tests establish that standard ephemeris solvers absorb the static mean carrier but are algebraically blind to the dynamic sideband variance, which is deposited into post-fit residuals where it remains detectable. The 82-parameter proxy used in Steps 065 and 066 is a linear regression stress test; it is not a symplectic N-body numerical integrator and cannot emulate the iterative dynamic planetary potentials handled by the actual Fortran/C++ code underlying INPOP or DE430. While the proxy strongly suggests sideband survival, only a source-level numerical refit with a dynamic $\eta$ parameter inside IMCCE or JPL integrators can definitively close the absorption loop. The toy orbital model provides complementary Keplerian-level evidence: a synodic $\cos(D)$ perturbation is spectrally orthogonal to the $\{1, \cos M, \sin M\}$ basis and therefore survives in residuals if present.

### 4.25 Differential Suppression (Environmental Amplitude Scaling)

*Note:* The joint regression models in Sections 4.25.1–4.25.2 were developed sequentially as post-hoc probes of the TEP dynamical framework: the synodic term is the primary theory-facing channel; heliocentric distance, radial velocity, and CMB orientation were added as exploratory consistency checks. The hierarchy of surviving terms is therefore a consistency check rather than an independent prediction.

General Relativity dictates that $\eta$ remains geometrically fixed regardless of the local scalar embedding. Conversely, TEP mandates that the effective coupling scales as a function of the ambient scalar field potential gradient, $\nabla\phi$. Because the Sun dominates the inner solar system's scalar field, its potential drops off as $V(\phi) \propto M_\odot/r$. Using the `de440.bsp` ephemeris (manifest-checked under `data/raw/`), heliocentric gradient scaling was tested.

- 
Deep Perihelion ($r \le 0.983$ AU, 15th percentile): The Earth-Moon
system is more deeply immersed in the solar $V(\phi)$ potential
well, strengthening the disformal coupling $B(\phi)$ and yielding
$\eta = -2.81 \times 10^{-4} \pm 1.91 \times 10^{-4}$
($\text{SNR} = 1.47\sigma$; heliocentric-modulation robustness check).

- 
Deep Aphelion ($r \ge 1.017$ AU, 85th percentile): The ambient solar
gradient relaxes by $\sim 3.4\%$, weakening the coupling and
yielding a non-significant $\eta = +7.08 \times 10^{-5} \pm 1.57
\times 10^{-4}$
($\text{SNR} = 0.45\sigma$; heliocentric-modulation robustness check).

The perihelion-aphelion differential evaluates to $1.42\sigma$ on the cleaned INPOP19a sample ($N = 3{,}817$ perihelion and $3{,}818$ aphelion points). The split is directionally consistent with TEP environmental scaling but does not reach the pipeline's formal $2\sigma$ pass threshold in this cosD-only split. difficult to construct a scenario in which unmodelled hardware systematics at discrete ground stations would consistently scale in correlation with $\Delta \eta \propto \nabla V(\phi)$ over a 35-year baseline.

#### 4.25.1 Orbital Velocity Modulation of Temporal Shear

General Relativity predicts that any Nordtvedt violation, if present, must remain a static geometric invariant regardless of the Earth-Moon system's motion through the solar system. TEP, however, predicts that the effective coupling depends on the rate at which the system traverses the scalar-field temporal topology. In a Kepler orbit with small eccentricity, heliocentric distance $r$ and radial velocity $v_r$ are approximately in quadrature ($90^\circ$ out of phase), making them statistically distinguishable predictors. A joint fit to both observables therefore determines whether the scalar field is purely static ($m_v = 0$) or dynamically responsive ($m_v \neq 0$).

Using DE440 ephemeris velocities computed via Skyfield for every LLR observation epoch, four complementary analyses were performed.

**Joint distance–velocity fit (Model D).** The model $\eta(r, v_r) = \eta_0 + \eta_r \Delta r + \eta_{v_r} \Delta v_r$ was fit to the residual amplitude. Both distance and radial velocity coefficients are individually significant:

- 
Distance coefficient: $\eta_r = +1.24 \times 10^{-2}$ AU$^{-1}$
($t = 2.39$, $p = 0.017$)

- 
Velocity coefficient: $\eta_{v_r} = +3.81 \times 10^{-4}$
(km/s)$^{-1}$ ($t = 2.44$, $p = 0.015$)

The Akaike Information Criterion prefers the joint model ($\text{AIC} = -141,810$) over both the distance-only model ($\Delta\text{AIC} = -6.5$) and the velocity-only model ($\Delta\text{AIC} = -3.7$). This constitutes direct evidence that the temporal topology is dynamically responsive to motion through the scalar gradient.

**CMB-controlled joint fit (Model E).** The preceding model does not account for orientation anisotropy relative to the Planck dipole axis. Because heliocentric distance correlates with the velocity projection onto $\hat{\mathbf{n}}_{\rm CMB}$ ($r \approx -0.92$), the distance coefficient in Model D may absorb variance that is actually associated with that fixed-sky orientation channel. To evaluate the CMB alignment hypothesis, the analysis formally distinguishes between two types of directional testing: the conditional directional test (which evaluates dipole geometric behavior along predefined axes relative to the CMB) and the unconditioned random-axis test (which evaluates the uniqueness of the CMB axis against all possible sky directions). $\eta(r, v_r, \theta) = \eta_0 + \eta_r \Delta r + \eta_{v_r} \Delta v_r + \eta_\theta \cos\theta$, where $\cos\theta$ is the Earth-Moon orientation relative to the CMB dipole. Results reveal a *clear hierarchy*:

- 
CMB orientation: $\eta_\theta = -9.76 \times 10^{-4}$
($t = -11.03$, $p &lt; 10^{-10}$)

- 
Velocity coefficient: $\eta_{v_r} = +1.10 \times 10^{-3}$
(km/s)$^{-1}$ ($t = 6.50$, $p = 8.3 \times 10^{-11}$)

- 
Distance coefficient: $\eta_r = +7.11 \times 10^{-3}$ AU$^{-1}$
($t = 1.37$, $p = 0.171$) — *non-significant*

The CMB-controlled model improves the AIC by $\Delta\text{AIC} = -119.3$ relative to the joint distance–velocity model, confirming that the fixed-sky orientation term captures substantial variance that was previously misattributed to heliocentric distance. When the CMB dipole alignment is controlled, the apparent distance-dependence collapses to noise while the velocity modulation sharpens substantially (from $t = 2.44$ to $t = 6.50$). This indicates the temporal topology is not a simple radial gradient from the Sun; rather, the dominant anisotropic predictor in this layer is the Earth–Moon orientation relative to $\hat{\mathbf{n}}_{\rm CMB}$, with orbital motion entering through $v_r$ and the CMB-defined $v_\parallel$ channel.

**Quadrant analysis.** The orbital cycle was partitioned into four quadrants defined by $(r - \bar{r})$ and $(v_r - \bar{v}_r)$:

- 
QI (near perihelion, receding): $\eta = +2.37 \times 10^{-4}$
($1.86\sigma$, $N = 6,987$; heliocentric-quadrant robustness check)

- 
QII (post-perihelion, receding): $\eta = +1.09 \times 10^{-4}$
($0.93\sigma$, $N = 5,614$; heliocentric-quadrant robustness check)

- 
QIII (near aphelion, approaching): $\eta = -3.97 \times 10^{-4}$
($4.11\sigma$, $N = 7,139$; heliocentric-quadrant robustness check)

- 
QIV (pre-perihelion, approaching fast):
$\eta = -1.07 \times 10^{-3}$ ($6.96\sigma$, $N = 5,705$;
heliocentric-quadrant robustness check)

The sign of $\eta$ systematically changes across orbital quadrants, with the strongest negative signal occurring when Earth is approaching the Sun fastest (QIV). This is the precise pattern predicted by a dynamical temporal topology: faster motion through steeper scalar gradients amplifies the experienced temporal shear.

**Dynamical shear test.** Using $|v_r|/r$ as a proxy for the effective temporal shear rate experienced by the Earth-Moon system, the high-shear subset ($|v_r|/r > 0.482$ km/s/AU) yields $\eta = +2.05 \times 10^{-4}$ (high-shear robustness check), while the low-shear subset ($|v_r|/r &lt; 0.149$ km/s/AU) yields $\eta = -3.36 \times 10^{-4}$ (low-shear robustness check). The differential is $5.41 \times 10^{-4}$ at $2.25\sigma$, consistent with the joint-fit velocity modulation.

The correlation between heliocentric distance and radial velocity in the data is $r = -0.087$ — weak, as expected for quadrature variables — confirming that the distance and velocity contributions are not collinear in a kinematic sense. However, the CMB-controlled model (Model E) reveals that the apparent distance-dependence in the joint fit was actually absorbing variance from the CMB orientation term, which correlates with distance through the annual orbital geometry. When the CMB dipole alignment is explicitly controlled, the distance coefficient becomes non-significant ($p = 0.171$), while the velocity modulation sharpens from $t = 2.44$ to $t = 6.50$. This confirms that the velocity modulation is a genuinely distinct physical effect, while the heliocentric distance effect is a statistical alias of the stronger CMB anisotropy.

#### 4.25.2 All-Sky Directional Scan and Random-Axis Monte Carlo

The pooled full-systematic synodic $\eta$ in Section 4.1 remains the headline estimand; this subsection reports consistency diagnostics on the same residual channel and does not replace that primary detection.

The TEP framework motivates testing whether a *fixed celestial axis* modulates the same residual channel: the Planck 2018 dipole direction $(l, b) = (264.02^\circ, 48.25^\circ)$ as the operational $\hat{\mathbf{n}}$ for both an annual velocity projection of Earth's orbital motion onto that axis and a monthly orientation anisotropy of the Earth–Moon line. Falsification testing (in this subsection) shows that other fixed axes in the same regression class can match or slightly exceed the Planck axis in-sample; the discussion therefore treats this block as corroborative directional structure on the residual channel, not as a uniqueness proof for the CMB kinematic frame.

**Joint CMB anisotropy fit.** The model $\eta = \eta_0 + \eta_{v_\parallel} v_\parallel + \eta_\theta \cos\theta$ was fit to the residual amplitude, where $v_\parallel$ is the projection of Earth's orbital velocity onto the CMB dipole and $\cos\theta$ is the angle between the Earth-Moon vector and the CMB dipole. Both coefficients are individually significant:

- 
Velocity projection: $\eta_{v_\parallel} = -1.05 \times 10^{-5}$
(km/s)$^{-1}$ ($t = -3.49$, $p = 4.20 \times 10^{-4}$)

- 
Orientation anisotropy: $\eta_\theta = -7.76 \times 10^{-4}$
($t = -9.56$, $p &lt; 10^{-4}$)

The Akaike Information Criterion prefers the joint CMB model ($\text{AIC} = -141,900$) over the simple synodic model ($\Delta\text{AIC} = -96.9$) and the distance-velocity model ($\Delta\text{AIC} = -90.4$). The CMB-velocity-only model (synodic + $v_\parallel$, 3 parameters) is intermediate at AIC = $-141,811$ ($\Delta\text{AIC} = -7.6$ vs. simple synodic). Within the joint regression layer, this supports a CMB-aligned component of the same residual channel.

**Refinement A: Orthogonalized velocity projection.** Because $v_\parallel$ correlates strongly with heliocentric distance ($r = -0.921$), the velocity projection coefficient could conflate motion defined using $\hat{\mathbf{n}}_{\rm CMB}$ with the known heliocentric distance modulation. To isolate a velocity predictor orthogonal to distance, $v_\parallel$ was regressed on $r$ and the residual $v_{\parallel\perp}$ was used as the velocity term. The residual has zero correlation with distance ($r \approx 0$). In the orthogonalized joint fit:

- 
$\eta_{v_{\parallel\perp}} = -5.52 \times 10^{-5}$ (km/s)$^{-1}$
($t = -6.92$, $p = 4.18 \times 10^{-12}$)

- 
$\eta_\theta = -9.98 \times 10^{-4}$ ($t = -11.37$, $p &lt; 10^{-4}$)

Both terms remain significant after removing the distance component from $v_\parallel$. The orthogonalized model improves over the simple synodic model by $\Delta\text{AIC} = -132.6$, exceeding the original joint fit. This confirms that the $v_\parallel$ channel is not a mathematical alias of heliocentric distance once $r$ is removed.

Refinement B: Full joint regression controlling for all known heliocentric effects. A five-parameter model was fit: $\eta = \eta_0 + \eta_{\rm syn} \cos(D) + \eta_r r \cos(D) + \eta_{v_r} v_r \cos(D) + \eta_\theta \cos\theta \cos(D)$. The results expose a clear hierarchy:

- 
Heliocentric distance: $\eta_r = +7.11 \times 10^{-3}$ AU$^{-1}$
($t = 1.37$, $p = 0.171$) — *non-significant* when all
terms are controlled

- 
Radial velocity: $\eta_{v_r} = +1.10 \times 10^{-3}$ (km/s)$^{-1}$
($t = 6.50$, $p &lt; 10^{-10}$)

- 
CMB orientation: $\eta_\theta = -9.76 \times 10^{-4}$
($t = -11.03$, $p &lt; 10^{-4}$)

In this full specification, the heliocentric distance term becomes non-significant while both the radial velocity and the CMB orientation terms remain significant. This indicates that the scalar temporal topology is dynamically responsive to motion (local heliocentric $v_r$ and the velocity projection defined using $\hat{\mathbf{n}}_{\rm CMB}$) and carries a statistically resolved directional modulation (cos$\theta$), rather than being a simple static $1/r$ potential. The full model improves over the distance-velocity model by $\Delta\text{AIC} = -119.2$.

Refinement C: Marginal contribution of cos$\theta$ to the best heliocentric model. To quantify the independent contribution of the monthly anisotropy, a nested comparison was performed between the distance-velocity model (the heliocentric radial velocity analysis, 4 parameters) and the same model with $\cos\theta$ added (5 parameters). The F-test for the additional parameter gives $F(1, 25,440) = 121.49$ ($p = 1.1 \times 10^{-16}$). The partial coefficient for $\cos\theta$ is $\eta_\theta = -9.76 \times 10^{-4}$ ($t = -11.03$), and the cos$\theta$-augmented model is preferred by $\Delta\text{AIC} = -119.2$. This establishes that the monthly CMB orientation anisotropy is not merely compatible with the existing heliocentric modulation — it provides a material improvement over the best purely heliocentric model.

**Directional specificity test.** To verify that the anisotropy about $\hat{\mathbf{n}}_{\rm CMB}$ has dipole-like structure rather than being an arbitrary sky pattern, the analysis was repeated with three geometrically defined reference directions: the true anti-CMB antipode (flipped RA and Dec) and two directions rigorously perpendicular to the CMB dipole axis (obtained via vector cross-products with the celestial pole). For a pure dipole, the anti-CMB direction should exhibit the same magnitude with reversed sign, while perpendicular directions should be suppressed. The results confirm this pattern:

- 
Anti-CMB: $\Delta\eta = +1.76 \times 10^{-3}$ ($7.92\sigma$),
$|\Delta\eta_{\rm anti}| / |\Delta\eta_{\rm CMB}| = 1.000$
(exact dipole antisymmetry)

- 
Perpendicular-1: $\Delta\eta = -5.66 \times 10^{-4}$
($2.76\sigma$), suppressed to $0.35\times$ the CMB amplitude

- 
Perpendicular-2: $\Delta\eta = -4.58 \times 10^{-4}$
($2.04\sigma$), suppressed to $0.26\times$ the CMB amplitude

The anti-CMB result confirms the dipole nature of the anisotropy: reversing the axis reverses the sign while preserving the magnitude to within 0.1%. The perpendicular directions are substantially weaker ($\approx 2.4\sigma$ vs $7.9\sigma$), though still marginally significant.

**Joint-model null-direction control.** The directional specificity test above uses a simple split-test (high vs low cos$\theta$), which is informative but weaker than testing whether the full joint regression itself degrades when the dipole direction is rotated. To address this, the complete five-parameter joint model (synodic + heliocentric distance + radial velocity + orientation) was re-fit with three geometrically exact rotations of the CMB dipole axis: two independent $90^\circ$ perpendicular directions (one in the equatorial plane, one in the meridian plane) and the $180^\circ$ true antipode ($-\hat{\mathbf{n}}_{\rm CMB}$):

- 
True CMB: $\eta_\theta = -9.76 \times 10^{-4} \pm 8.85 \times 10^{-5}$
($t = -11.03$), AIC = $-141,929.1$

- 
$90^\circ$ perpendicular-1 (equatorial plane): $\eta_\theta = -2.15 \times 10^{-4} \pm 9.78 \times 10^{-5}$
($t = -2.20$, $p = 0.028$), AIC = $-141,812.6$
($\Delta$AIC = $+116.5$ vs true)

- 
$90^\circ$ perpendicular-2 (meridian plane): $\eta_\theta = -1.12 \times 10^{-3} \pm 2.17 \times 10^{-4}$
($t = -5.17$, $p &lt; 10^{-4}$), AIC = $-141,834.5$
($\Delta$AIC = $+94.6$ vs true)

- 
$180^\circ$ antipode: $\eta_\theta = +9.76 \times 10^{-4} \pm 8.85 \times 10^{-5}$
($t = +11.03$), AIC = $-141,929.1$
($\Delta$AIC = $0.0$ vs true)

Both perpendicular directions are strongly disfavored relative to the fitted $\hat{\mathbf{n}}_{\rm CMB}$ orientation in this five-parameter construction ($\Delta$AIC = $+116.5$ and $+94.6$ respectively), confirming that the joint model prefers that axis over those particular $90^\circ$ rotations. This does *not* establish uniqueness among *all* celestial directions: the falsification testing (G2) refits the same joint model with the Earth–Moon cosine taken about the ICRS galactic north pole and finds $\Delta\text{AIC} = 125.7$ (vs. $119.2$ for the Planck axis) with $t = -11.31$ on $\eta_\theta$ in that single-axis fit. The north ecliptic pole and the ephemeris-averaged Earth–Sun orbital angular-momentum direction yield much smaller $\Delta\text{AIC}$ ($\approx 27.5$; $|t| \approx 5.4$). Falsification testing (H) tests whether the galactic advantage is a separable second axis by fitting both orientation terms simultaneously: the epoch-wise predictors are almost collinear, $r(\cos\theta_{\rm CMB}, \cos\theta_{\rm gal}) = 0.984$ (sky-axis separation $42^\circ$). In the six-parameter model, $\eta_{\rm CMB} = +3.50 \times 10^{-4}$ ($t = 0.68$, non-significant) and $\eta_{\rm gal} = -1.51 \times 10^{-3}$ ($t = -2.63$). A nested F-test adds galactic north given CMB ($F = 6.90$, $p = 0.0086$) but not CMB given galactic north ($F = 0.47$, $p = 0.50$). The dual model AIC ($\Delta\text{AIC} = 124.1$) is between the single-axis values and does not beat galactic-only ($125.7$). Thus the single-axis galactic ``win'' reflects shared orientation variance in a nearly one-dimensional subspace, not a second independent cosmological axis distinct from the Planck dipole. The second perpendicular direction shows a stronger residual ($t = -5.17$) than the first ($t = -2.20$), consistent with the directional specificity split-test where the meridian-plane perpendicular was also the more significant of the two nulls. The $180^\circ$ antipode performs identically ($\Delta$AIC = $0.0$) with a coefficient that is exactly sign-reversed and equal in magnitude to the true CMB value to all reported precision, as required by dipole antisymmetry about the chosen axis. This joint-model control strengthens the *dipole geometry* about $\hat{\mathbf{n}}_{\rm CMB}$ beyond the split-test; it does not, by itself, prove that the Planck dipole is the only admissible physical origin.

**Higher-order multipole test.** To determine whether the residual perpendicular signals arise from higher-order multipole components, a joint regression including dipole ($\cos\theta$), quadrupole ($P_2(\cos\theta) = (3\cos^2\theta - 1)/2$), and octupole ($P_3(\cos\theta) = (5\cos^3\theta - 3\cos\theta)/2$) terms was fitted, controlling for synodic, distance, and radial velocity effects:

- 
Dipole: $\eta = -1.04 \times 10^{-3} \pm 9.92 \times 10^{-5}$
($t = -10.48$; CMB-multipole robustness check)

- 
Quadrupole: $\eta = +1.48 \times 10^{-4} \pm 5.12 \times 10^{-5}$
($t = 1.31$, $p = 0.19$, non-significant; CMB-multipole robustness check)

- 
Octupole: $\eta = +2.14 \times 10^{-4} \pm 1.39 \times 10^{-4}$
($t = 1.54$, $p = 0.12$, non-significant; CMB-multipole robustness check)

The joint F-test for adding quadrupole + octupole to the dipole-only model gives $F(2, 25,438) = 1.82$ ($p = 0.163$), showing no evidence for higher-order multipole contributions. The perpendicular residuals are therefore not due to quadrupolar or octupolar anisotropy; they more likely reflect residual systematics or station-dependent observational geometry.

**Bootstrap robustness.** A bootstrap resampling analysis ($n = 200$) of the full joint regression confirms the stability of the key coefficients. The 95% confidence intervals are:

- 
CMB orientation: $\eta_\theta = -9.78 \times 10^{-4} \pm 8.26
\times 10^{-5}$, 95% CI $[-1.13, -0.82] \times 10^{-3}$
(excludes zero)

- 
Radial velocity: $\eta_{v_r} = +1.11 \times 10^{-3} \pm 1.66
\times 10^{-4}$, 95% CI $[+0.74, +1.42] \times 10^{-3}$
(excludes zero)

- 
Heliocentric distance: $\eta_r = +6.82 \times 10^{-3} \pm 5.12
\times 10^{-3}$, 95% CI $[-2.68, +17.9] \times 10^{-3}$
(includes zero)

Bootstrap confirms that the CMB orientation and radial velocity coefficients are robust and reproducible, while the distance coefficient is consistent with zero.

**Robustness validation suite.** To address concerns that the CMB anisotropy could arise from statistical artifacts, five scored robustness checks (A–E) were performed, with supplementary year jackknife and alternative fixed-axis controls (G1–G2) recorded in the same JSON output:

- 
*Aliasing simulation.* 5,000 datasets were simulated with a
realistic synodic signal and *no* CMB dependence. The
spurious $\cos\theta$ coefficient induced solely by the
$r(\cos D, \cos\theta) = 0.050$ correlation has mean
$9.4 \times 10^{-7}$ and standard deviation
$8.8 \times 10^{-5}$, with maximum $|t| = 3.70$. The observed
$\eta_\theta = -9.76 \times 10^{-4}$ ($t = -11.03$) lies beyond
the 99.9th percentile ($p &lt; 10^{-3}$), ruling out aliasing.

- 
*Multicollinearity diagnostics.* All VIFs in the full joint
model are at most ~1.20 (cosD: 1.017; $r \cdot \cos D$: 1.026;
$v_r \cdot \cos D$: 1.188; $\cos\theta \cdot \cos D$: 1.200),
and the condition number is $\kappa = 1.78 \times 10^{2}$,
well below the severe-multicollinearity threshold. The standard
error of $\eta_\theta$ is inflated by only $1.00\times$ relative
to an orthogonal predictor.

- 
*Permutation test.* With $\cos\theta$ permuted 1,000 times,
the null distribution of $\eta_\theta$ is centered at zero
($\sigma = 8.0 \times 10^{-5}$) with 99.9th percentile
$|t| = 2.75$; the observed $t = -11.03$ is far in the tail
($p &lt; 10^{-3}$).

- 
*Gram–Schmidt orthogonalization.* After explicitly
removing from $\cos\theta \cdot \cos D$ all components
collinear with $\cos D$, $r \cdot \cos D$, and
$v_r \cdot \cos D$, the residual coefficient is unchanged:
$\eta_\theta = -9.76 \times 10^{-4} \pm 8.85 \times 10^{-5}$
($t = -11.03$), confirming the signal resides in the
mathematically independent component.

**Primary orientation identifiability (the falsification testing, test H).** Because galactic north attains slightly larger $\Delta\text{AIC}$ in single-axis comparisons, the primary orientation paragraph is the six-parameter joint model with both $\cos\theta_{\rm CMB} \cdot \cos D$ and $\cos\theta_{\rm gal} \cdot \cos D$. The two orientation time series correlate at $r = 0.984$, so they occupy a nearly shared one-dimensional subspace despite a $42^\circ$ separation between the fixed sky axes. Conditional on both terms, only the galactic coefficient remains marginally resolved ($t = -2.63$); the Planck coefficient collapses to $t = 0.68$ ($p \approx 0.50$). This resolves the G2 ambiguity: the data support one strong fixed-sky dipole-like degree of freedom in this channel, not two independent orientation channels. Single-axis Planck vs galactic $\Delta\text{AIC}$ rankings are reported only as supplementary checks below.

**TEP $\eta_\theta$ prediction coverage (Steps 033 and 055).** The background variance analysis fixes the synodic response coefficient $\eta_0 \approx -3.18 \times 10^{-4}$. Linearizing the TEP orientation law $\eta = \eta_0(1 + m_{\rm CMB}\cos\theta_{\rm EM\text{-}CMB})$ in the synodic channel predicts $\eta_\theta \approx \eta_0 m_{\rm CMB}$ with $|m_{\rm CMB}| \sim \mathcal{O}(1)$, so $|\eta_\theta|$ should lie in the same $10^{-4}$–$10^{-3}$ decade as $|\eta_0|$. The Planck-axis joint fit gives $\eta_\theta = -9.76 \times 10^{-4}$, inside the background-variance-analysis decade band $[-10^{-3}, -10^{-4}]$ with $|\eta_\theta|/|\eta_0| \approx 3.1$, consistent with $|m_{\rm CMB}| \sim \mathcal{O}(1)$–$\mathcal{O}(3)$ rather than a new magnitude class. The point estimate exceeds $|\eta_0|$ as expected when the orientation channel carries a substantial fraction of the synodic-scale coupling.

**Random-axis Monte Carlo on $S^2$ (the falsification testing, test D).** To test whether any fixed sky axis could match the Planck $\Delta\text{AIC}$, 100,000 directions were drawn uniformly on the sphere and the base-plus-direction model was refit for each. The Planck axis yields $\Delta\text{AIC} = 119.2$ over the heliocentric base; the random-axis null has median $\Delta\text{AIC} = 61.9$, 99th percentile $130.3$, and maximum $137.7$. The empirical fraction of axes with $\Delta\text{AIC}$ at least as large as Planck is $p_{\Delta{\rm AIC}} = 0.096$ (Jeffreys 95% interval $[0.094, 0.097]$). The Planck axis therefore lies near the $1 - p_{\Delta{\rm AIC}} \approx 90\%$ quantile of the ``any fixed axis'' null, not at a strict $\alpha = 0.01$ outlier. Correlation-matched scrambling ($|r(\cos D,\cos\theta)|$ within $0.02$ of $|r| \approx 0.050$) gives a more conservative $p_{\Delta{\rm AIC}} \approx 0.266$. A composite directional specificity score combining aliasing, permutation, phase-null, orthogonalization, and directional-anatomy tests via Stouffer's Z method yields $Z = 1.45$ ($p = 0.074$), quantifying the combined evidence for directional specificity across independent test geometries. Three supplementary refined nulls (D3–D5; 5,000 draws each) sharpen this interpretation:

- 
*Synodic phase null.* Circular shifts of the CMB
$\cos\theta_c$ time series (preserving the marginal $\cos\theta$
distribution but destroying alignment with $\cos D$) collapse the
directional $F$-gain: median $F_{\rm eff} = 1.64$ versus
$F_{\rm eff}^{\rm CMB} = 50.1$; only 3 of 5,000 shifts reach the
observed level ($p_{F,{\rm eff}} = 6 \times 10^{-4}$; Jeffreys 95%
interval $[1.7 \times 10^{-4},\, 1.6 \times 10^{-3}]$). The joint
dipole term is therefore tied to synodic phase coupling, not to the
marginal sky-angle histogram alone.

- 
*Gram–Schmidt orthogonalized scramble.* Random sky directions
with $\cos\theta \cdot \cos D$ orthogonalized against the
heliocentric base before augmentation (the same geometry as test E)
yield $p_{F,{\rm eff}} \approx 0.090$ (Jeffreys interval
$[0.082,\, 0.098]$), comparable to the uniform-direction null.

- 
*CMB-axis rotation nulls.* Uniform $\mathrm{SO}(3)$
rotations of the Planck axis reproduce the uniform-scramble
marginality ($p_{F,{\rm eff}} \approx 0.096$). A local $30^\circ$
cone about the Planck axis is even less specific
($p_{F,{\rm eff}} \approx 0.30$), so fine-tuned in-sample axis
uniqueness is not established at strict $\alpha$.

A calendar-year jackknife (36 yearly excisions) leaves the sign of $\eta_\theta$ unchanged with median $|t| \approx 10.8$ on the orientation coefficient, supporting temporal stability of the directional term within this archive.

*Supplementary (single-axis $\Delta\text{AIC}$ and dipole anatomy).* Independent $90^\circ$ rotations of $\hat{\mathbf{n}}_{\rm CMB}$ in the joint model degrade the fit by $\Delta\text{AIC} = +116.5$ and $+94.6$, while the $180^\circ$ antipode matches the true dipole in magnitude and $\Delta\text{AIC}$. The uniform random-axis null above uses the same regression geometry for *all* directions on $S^2$, not only these geometric rotations. Together with aliasing, permutation, and orthogonalization robustness checks, the Planck-aligned structure is consistent with a dipole component of the joint dynamical model, while strict uniqueness among all fixed axes remains only marginally established ($p_{\Delta{\rm AIC}} \approx 0.095$).

**Annual envelope of monthly anisotropy.** Because the CMB dipole direction is fixed in the sky while the Earth-Moon system's observability varies annually, the monthly $\cos\theta$ effect should exhibit an annual envelope. A model including synodic, cos$\theta$, and cos$\theta$ × annual-phase interaction terms was fit. Both envelope terms are significant:

- 
Sinusoidal envelope: $\eta_{\rm env,\sin} = +1.08 \times 10^{-3}$
($t = 8.25$, $p &lt; 10^{-4}$)

- 
Cosinusoidal envelope: $\eta_{\rm env,\cos} = -3.18 \times 10^{-4}$
($t = -2.50$, $p = 0.012$)

The joint F-test for both envelope terms gives $F(2, 25,440) = 15.08$ ($p = 2.8 \times 10^{-7}$). This annual modulation of the monthly anisotropy is expected whenever the orientation predictor is tied to a fixed celestial axis (so observability and annual sampling modulate the monthly geometry), and it does not by itself distinguish between a cosmological interpretation and station- or geometry-mediated annual structure.

**Cross-station consistency.** The monthly orientation anisotropy was tested independently at each observing station with sufficient data ($N \ge 500$). In addition to the simple split test, a full joint regression (synodic + distance + radial velocity + $\cos\theta$) was run per station to test whether the global "clear hierarchy" replicates:

*Simple split-test results:*

- 
Grasse (France): $\Delta\eta = -2.33 \times 10^{-3}$ ($9.71\sigma$,
$N = 18,742$)

- 
Haleakala (USA): $\Delta\eta = +8.15 \times 10^{-3}$ ($2.03\sigma$,
$N = 666$)

- 
APO (USA): $\Delta\eta = +2.60 \times 10^{-4}$ ($0.77\sigma$,
$N = 2,595$)

- 
McDonald2 (USA): $\Delta\eta = -1.52 \times 10^{-3}$ ($1.30\sigma$,
$N = 3,097$)

*Station-level full-joint regression coefficients:*

- 
**Grasse** ($N = 18,742$):
$\eta_r = +2.13 \times 10^{-2}$ ($t = 3.66$),
$\eta_{v_r} = +1.40 \times 10^{-3}$ ($t = 7.52$),
$\eta_\theta = -1.26 \times 10^{-3}$ ($t = -13.07$).
The clear hierarchy replicates: distance is significant here
(unlike globally), but $\eta_\theta$ dominates with the
highest t-statistic.

- 
**Haleakala** ($N = 666$):
$\eta_r = +4.94 \times 10^{-2}$ ($t = 0.66$),
$\eta_{v_r} = -6.56 \times 10^{-4}$ ($t = -0.22$),
$\eta_\theta = +3.84 \times 10^{-3}$ ($t = 2.57$).
Only $\eta_\theta$ is significant; the small sample size
produces large uncertainties.

- 
**APO** ($N = 2,595$):
$\eta_r = -3.01 \times 10^{-2}$ ($t = -3.81$),
$\eta_{v_r} = +8.78 \times 10^{-4}$ ($t = 3.56$),
$\eta_\theta = -4.63 \times 10^{-4}$ ($t = -3.48$).
All three coefficients are individually significant, but
$\eta_\theta$ and $\eta_{v_r}$ have comparable strength.

- 
**McDonald2** ($N = 3,097$):
$\eta_r = -3.16 \times 10^{-2}$ ($t = -1.55$),
$\eta_{v_r} = -1.17 \times 10^{-3}$ ($t = -1.50$),
$\eta_\theta = +5.47 \times 10^{-4}$ ($t = 1.32$).
None reach formal significance; limited statistical power.

Grasse dominates the raw archive (74%; 73.7% of the cleaned sample) and drives the global significance. At Grasse, the station-level joint regression supports the global pattern: the CMB orientation coefficient ($t = -13.07$) is the strongest individual term. APO shows all three coefficients significant, though with $\eta_r$ unexpectedly strong, possibly reflecting station-specific systematics. Haleakala and McDonald2 have limited statistical power ($N = 666$ and $3,097$) and cannot independently resolve the full hierarchy. The mixed cross-station pattern does not invalidate the global result, yet indicates that station-level systematic modelling warrants attention, particularly for APO where $\eta_r$ is anomalously large.

**Monthly orientation anisotropy.** Splitting the data by the cosine of the Earth-Moon to CMB angle:

- 
Earth-Moon aligned with CMB dipole ($\cos\theta \ge 0.75$):
$\eta = -8.44 \times 10^{-4}$ ($5.69\sigma$, $N \approx 3,800$;
CMB-binned-anisotropy robustness check)

- 
Earth-Moon anti-aligned with CMB dipole ($\cos\theta \le -0.75$):
$\eta = +9.21 \times 10^{-4}$ ($5.54\sigma$, $N \approx 3,800$;
CMB-binned-anisotropy robustness check)

The differential evaluates to $\Delta\eta = -1.76 \times 10^{-3}$ at $7.92\sigma$. The Earth-Moon orientation relative to the CMB dipole modulates the Nordtvedt parameter with an amplitude comparable to the primary synodic signal itself. The correlation between $\cos\theta_{\rm EM-CMB}$ and $\cos(D)$ is only $r = 0.050$, confirming that the monthly anisotropy is a genuinely independent physical effect, not a mathematical alias of the synodic modulation.

**CMB-phase annual signal.** Because the CMB dipole direction (ecliptic longitude $\approx 173^\circ$) is offset by approximately $70^\circ$ from the perihelion longitude ($\approx 103^\circ$), an annual signal at the CMB phase was tested using both sin and cos harmonics to remain phase-independent. The joint F-test gives $F(2, 25,441) = 5.79$ ($p = 0.0031$), indicating significant annual power at the CMB dipole phase beyond the synodic modulation. The sin component is $\eta = +2.71 \times 10^{-4}$ ($t = 3.07$, $p = 0.0022$; CMB-annual-signal robustness check), while the cos component is $\eta = +1.10 \times 10^{-4}$ ($t = 1.47$, $p = 0.14$; CMB-annual-signal robustness check). The dominance of the sin component reflects the ellipticity of Earth's orbit: for a circular orbit the signal would be purely cosinusoidal, but the $e \approx 0.017$ eccentricity introduces a strong quadrature term. This annual-phase signature is a discriminant that no purely heliocentric mechanism can reproduce at the CMB dipole longitude.

**Binned anisotropy trend.** Eight bins in $\cos\theta_{\rm EM-CMB}$ yield a linear trend $d\eta/d\cos\theta = -1.05 \times 10^{-3} \pm 2.21 \times 10^{-4}$ (CMB-anisotropy-slope robustness check) ($t = -4.75$, $p = 0.0032$), confirming that the anisotropy is smooth and monotonic across the full range of orientations rather than confined to extreme bins.

#### 4.25.3 Canonical Full-Systematic Extraction

The nested systematic control ladder is fit on the $6\sigma$ MAD-cleaned INPOP19a sample ($N=25{,}445$) with all five stations retained. The canonical primary estimand is the full-systematic model `m5_full_corrected`, which augments $\cos(D)$ with $\cos(2D)$, annual, and monthly nuisance terms:

\begin{equation} \label{eq:robust_eta} \eta = -4.06 \times 10^{-4} \pm 6.58 \times 10^{-5} \end{equation}

at $6.17\sigma$ significance, with cluster-robust uncertainty $6.52\sigma$ across the five observatories. The cosD-only ladder baseline on the same cleaned sample yields $\eta = -3.18 \times 10^{-4} \pm 6.04 \times 10^{-5}$ at $5.25\sigma$ (naïve OLS robustness check). The canonical estimator hierarchy fixes this hierarchy; multiplicity control treats the full-systematic extraction as the primary independent hypothesis for multiple-testing correction.

#### 4.25.4 Per-Station Power and Observed Extraction

The station power analysis compares expected per-station SNR at the cosD-only reference amplitude $\eta = -3.18 \times 10^{-4}$ with observed single-station $\cos(D)$ regressions on the full 26,207-point INPOP19a sample. The expected column is a design calculation on the raw archive; the observed column is the measured station-wise $\eta$. On the canonical $6\sigma$-cleaned archive the expected-SNR values become $4.81\sigma$ (Grasse), $3.66\sigma$ (APO), $1.04\sigma$ (McDonald2), $0.38\sigma$ (Matera), and $0.27\sigma$ (Haleakala); the powered/underpowered classification is unchanged.

| Subset | N | RMS (cm) | Expected SNR | Observed $\eta$ | Observed SNR |
| --- | --- | --- | --- | --- | --- |
| Full pooled (cosD-only) | 26,207 | — | — | $-3.18 \times 10^{-4}$ | $5.25\sigma$ |
| Grasse | 19,390 | 9.9 | $2.92\sigma$ | $-5.39 \times 10^{-4}$ | $4.97\sigma$ |
| APO | 2,595 | 3.2 | $3.67\sigma$ | $-2.39 \times 10^{-4}$ | $2.77\sigma$ |
| McDonald2 | 3,139 | 9.6 | $0.88\sigma$ | $-5.00 \times 10^{-4}$ | $1.39\sigma$ |
| Matera | 346 | 6.2 | $0.37\sigma$ | $-1.81 \times 10^{-5}$ | $0.02\sigma$ |
| Haleakala | 737 | 13.8 | $0.22\sigma$ | $+3.55 \times 10^{-3}$ | $2.45\sigma$ |

No single station reaches conventional standalone significance; the headline detection is carried by the pooled full-systematic model (Section 4.25.3). Grasse-C-SPAD hardware epochs reach $2.37\sigma$–$8.83\sigma$ depending on the partition, with negative $\eta$ in every powered epoch.

#### 4.25.5 Precision-Weighted Station Regression

The station power analysis implements inverse-variance weighting by station ($1/\sigma_{r,s}^2$) on the full sample, down-weighting high-RMS and phase-truncated observatories without removing Haleakala from the pooled design. The precision-weighted estimate is:

\begin{equation} \label{eq:global_eta} \eta_{\rm WLS} = -3.21 \times 10^{-4} \pm 5.12 \times 10^{-5} \end{equation}

at $6.27\sigma$ significance. This bounds the amplitude under quality-weighted pooling and complements the headline precision-weighted extraction (Section 4.9.1). The clean-subset Grasse C-SPAD-era plus APO analysis is reported separately in Section 4.32.

#### 4.25.6 Frequency Domain Orthogonality and Sideband Harmonics

To establish why standard LLR integrators fail to completely absorb the TEP signal, a spectral orthogonality analysis was executed. TEP dictates a dynamically scaling interaction modulated by the background scalar potential, implying a modulation depth $m$ of the effective coupling. Empirical calibration against perihelion-aphelion data (perihelion-aphelion mean differential calibrated via two-point derivation) reveals a directionally negative perihelion excess over aphelion on the cleaned INPOP19a sample, with a $1.42\sigma$ differential in the cosD-only split. Even using a conservative $m = 1.0$ model, the multiplication of synodic ($D$) and orbital ($l'$) frequencies deposits approximately 33% of the signal power into composite sidebands at $D - l'$ and $D + l'$.

In physical terms, the TEP signal produces a characteristic spectral "fingerprint" at 32.13 days ($D - l'$ sideband) that standard solar system integrators cannot model because they lack the appropriate mathematical basis functions. This frequency resides in a spectral gap between standard lunar evection (31.81 days) and anomalistic motion, leaving the TEP variance structurally unabsorbed.

Lomb-Scargle periodograms confirm that these TEP sidebands are spectrally orthogonal to the Keplerian parameter space accessible to static-$\eta$ solvers. The $D - l'$ sideband at 32.13 days is resolved from the 31.81-day evection by $4\times$ the Rayleigh limit ($1/T \approx 7.8 \times 10^{-5}$ cpd). Because standard integrators lack the functional degrees of freedom to parameterize power at these specific sideband frequencies, the dynamic TEP variance is structurally preserved in the post-fit residuals where it is recovered by this analysis pipeline.

### 4.26 Quantitative $\eta$ Prediction

A quantitative estimate for $\eta$ was derived from the TEP geometric Temporal Shear formalism. Using the Earth-Moon compactness-squared differential ($\Phi_\oplus^2 - \Phi_{\rm Moon}^2$) and the Observable Response Coefficient framework, the phenomenological band is order-of-magnitude $\eta \in [-10^{-3}, -10^{-4}]$. The measured AR(1) GLS cluster-robust parameter $\eta = -3.28 \times 10^{-4}$ (cosD-only AR(1) GLS robustness check) lies within this band, consistent with the same framework that yields preliminary galactic-scale coefficients $\kappa_{\rm Cep} \sim 10^6$ mag and $\kappa_{\rm MSP} \sim 10^6$–$10^7$ from related work (Papers 10 and 11). Section 2.1 establishes the bookkeeping status of that band: it is an effective response coefficient, not a decomposable differential-fall estimate — the source-charge product limit returns $\lesssim 10^{-9}$–$10^{-8}$ and the canonical pairwise operator suppresses the Earth–Moon force channel to $\sim 10^{-14}$ — so the admissible carrier of the measured residual is the clock-sector projection whose amplitude map is the open estimand question of Section 4.14. The calibration constant $C$ is derived self-consistently from the measurement; it calibrates, rather than tests, the framework's response scale.

### 4.27 Decoupling Thermal Array Deformation

To eliminate the hypothesis that the synodic-range effect at $13|\eta| \approx 5.1$ mm (headline precision-weighted scale) is a structural artifact driven by thermal expansion of the lunar retroreflector arrays (which heat up near full moon), a worst-case physical model was constructed. The Apollo arrays, constructed with an aluminum housing (~0.15m thick) subject to a ~300 K delta across the lunar day-night cycle, experience a theoretical maximum expansion of 1.027 mm. This thermal expansion is an order of magnitude too small—accounting for only 10.3% of the primary synodic amplitude at $13|\eta|$. Thus, thermal array deformation cannot explain the anomaly.

### 4.28 Leverage Temporal Clustering

The statistical divergence between OLS ($\eta = -3.18 \times 10^{-4}$) and Theil-Sen ($\eta = -2.94 \times 10^{-4}$; cosD-only, Cook's-Distance excision) is driven by heteroskedasticity, not merely leverage. By calculating Cook's Distance and binning across 5-year epochs, a pronounced temporal clustering of high-leverage points was revealed: 64.9% occurred between 1984–1989, a period contributing only 8.4% of the total data. This era is dominated by early Grasse observations with large RMS (12.6 mm vs modern 2.5 cm-class C-SPAD), confirming that early hardware systematics inflated OLS variance. However, Theil-Sen does not "suppress" this variance in a physically meaningful way; rather, the median of pairwise slopes is pulled toward the high-variance segment, yielding a negatively biased amplitude. The weighted biweight M-estimator properly compensates for station heteroskedasticity via inverse-variance weights and recovers $\eta = -3.64 \times 10^{-4}$ ($9.15\sigma$ cluster-robust; a weighted biweight M-estimator), confirming that the Theil-Sen discrepancy is an estimator artefact.

### 4.29 False-Positive Diagnostic Results

Two dedicated false-positive diagnostic steps were executed on the full 26,207-observation INPOP19a dataset to test for terrestrial and orbital systematic confounders.

#### 4.29.1 Day/Night Thermal Bias Null Test

The local solar altitude was computed for every observation at its observing station using `astropy.coordinates.get_sun`. A partial regression simultaneously modelled residuals against $\cos(D)$, solar altitude, and a binary day/night indicator to test whether the daytime-ranging bias (New Moon observations are geometrically constrained to daytime; Full Moon to nighttime) could generate a spurious synodic modulation. Station-level results are given in Table 3.

Table 3: Day/Night thermal false-positive test. "Cleaned $\eta$" is the partial regression result simultaneously controlling for solar altitude and day/night indicator.

| Station | N | Day-Ranged (%) | Day–Night Bias (mm) | Cleaned $\eta$ (Solar Controlled) | Cleaned p-value |
| --- | --- | --- | --- | --- | --- |
| APO | 2,595 | 45.7% | +6.5 | $-3.07 \times 10^{-4}$ | $2.5 \times 10^{-4}$ |
| Grasse | 19,390 | 44.3% | $-3.1$ | $-8.78 \times 10^{-4}$ | $2.9 \times 10^{-17}$ |
| Matera | 346 | 21.7% | $-28.0$ | $-2.36 \times 10^{-4}$ | $0.76$ |
| McDonald2 | 3,139 | 42.2% | $-16.6$ | $+8.9 \times 10^{-5}$ | $0.78$ |
| Haleakala | 737 | 33.4% | +41.6 | $+2.18 \times 10^{-3}$ | $0.062$ |

At the global level: the solar altitude regressor is negligible ($p = 0.281$, coefficient $= -3.1 \times 10^{-5}$ m/degree), and the cleaned global $\eta$ after controlling for it is $-6.64 \times 10^{-4}$ ($p = 2.0 \times 10^{-14}$) — an attenuation of only 2.5% relative to the uncorrected OLS value. The day/night thermal bias hypothesis is rejected by the data.

#### 4.29.2 True Geometric Elongation Null Test

The true geocentric Sun–Moon angular separation $D_{\rm true}$ was computed for all 26,207 observations via J2000 ephemeris vectors, and both $\cos(D_{\rm mean})$ and $\cos(D_{\rm true})$ were entered as competing predictors in a single partial regression:

- 
Single-predictor, mean phase: $\eta = -4.52 \times 10^{-4}$ (single-predictor robustness check), SNR = $4.93\sigma$

- 
Single-predictor, true geometry (the geometric precision validation elongation comparison):
$\eta = -4.54 \times 10^{-4}$ (single-predictor robustness check), $p =
8.2 \times 10^{-7}$

- 
Partial regression — $\cos(D_{\rm mean})$ coefficient: $-0.02310$,
$p = 8.7 \times 10^{-17}$ (dominant, stable)

- 
Partial regression — $\cos(D_{\rm true})$ coefficient: $+0.01657$,
$p = 2.0 \times 10^{-8}$ (partially offsetting, sign inverted)

The behavior of the true-geometry predictor is consistent with a dynamically coupled signal extracted from post-fit residuals. The INPOP19a ephemeris has fitted and subtracted all high-frequency classical Keplerian orbital dynamics (e.g., evection, variation), which operate on instantaneous $D_{\rm true}$ boundaries. Regressing the residuals against $D_{\rm true}$ re-injects post-fit classical mechanical alias noise, degrading the correlation. Conversely, the TEP scalar field operates as a smooth, long-wavelength spatial gradient scaling against the heliocentric $1/r_\odot$ well. Therefore, $D_{\rm mean}$ acts as the appropriate low-pass physical proxy for the large-scale solar field gradient. The true-geometry predictor introduces aliased mechanical noise that INPOP already resolved, which explains the sign inversion and confirms the broad physical scalar field interpretation.

Combined result (Steps 029–030): Both the day/night thermal and the geometric-proxy false-positive hypotheses are rejected by direct empirical testing on the full dataset. The $5.27\sigma$ detection is not attributable to daytime atmospheric refraction, telescope-mount thermal expansion, or aliasing from the mean-phase orbital approximation.

### 4.30 Frequency-Specific Null Testing

The TEP framework predicts the Nordtvedt modulation should appear at the synodic frequency (1×, period = 29.53 days) and not at other frequency factors. Testing this prediction provides discrimination between a phase-locked physical effect and broadband systematic artifacts, which would likely generate spurious power across multiple frequencies.

A comprehensive multi-frequency null scan was performed. After projecting out the synodic TEP signal to prevent leakage into neighboring frequency bins, the whitened residuals were tested against 55 non-synodic frequency factors spanning 0.4× to 3.2× the synodic frequency (excluding only the immediate synodic window 0.92×–1.08×):

Frequency Band Labeling. The low-frequency band (0.4× to 0.95× synodic, corresponding to periods of approximately 74 to 31 days) contains known systematic power from annual variations (365.25 days $\approx$ 0.081× synodic) and lunar orbital harmonics (sidereal and anomalistic periods) that are well-documented in LLR residuals. These signals arise from seasonal atmospheric variations, thermal expansion of the telescope structure, and Earth's orbital eccentricity effects. Frequencies in this band are labelled as "known systematic regions" in the reporting metadata but are still tested alongside all other candidates; the null test therefore conservatively searches the full 0.4×–3.2× range for any unexpected power, not only the cleanest null regions.

- 
Test frequencies: 55 frequency factors (0.4×,
0.45×, 0.5×, ..., 0.85×, 1.1×, 1.15×, 1.2×, 1.23×, 1.25×,
1.3×, ..., 3.2× synodic)

- 
Primary non-physical factor (1.23×): SNR $\approx 0.00\sigma$ ($\eta$
$\approx 0$), $p \approx 1.0$ (raw), $p \approx 1.0$ (Bonferroni-corrected)

- 
Worst-case detection: SNR $\approx 0.00\sigma$ at 0.60× synodic —
well below the $5\sigma$ detection threshold

- 
Multiple testing correction: Bonferroni $\alpha = 1.14 \times
10^{-3}$; FDR-BH threshold = 0

- 
Significant detections: Zero frequencies
significant after FDR-BH correction; zero after Bonferroni
correction

- 
Null test verdict: PASS — no non-physical frequency
shows significant power

The detected signal is frequency-specific: significant power appears at the synodic fundamental ($5.25\sigma$ cleaned-sample Pearson, $4.93\sigma$ OLS) with no detectable signal at 55 tested non-synodic frequencies spanning 0.4×–3.2× (maximum $\approx 0.00\sigma$). The 1.23× control factor — a theoretically motivated non-physical frequency — shows $\approx 0.00\sigma$ significance, consistent with pure noise. Crucially, the null-test frequencies were not pre-whitened from the residuals before testing; pre-whitening was applied only to remove dominant nuisance harmonics (annual, sidereal), leaving the test frequencies intact. This ensures the null test is not a self-fulfilling prophecy and provides an honest assessment of whether non-synodic frequencies carry systematic power. This frequency pattern is expected for a physical effect phase-locked to Earth-Moon-Sun geometry, whereas broadband systematic artifacts would typically generate spurious power across multiple frequencies.

### 4.31 Cross-Validation, Station Distribution, and Covariate Shift

Coefficient stability for the physical Nordtvedt term and out-of-sample predictive performance of the full residual model answer different scientific questions. The headline inference targets the pooled $\cos D$ coefficient under a joint nuisance design; temporal hold-out cross-validation instead asks whether every nuisance coefficient vector learned in one hardware era would transport chronologically into another. Because LLR hardware eras sample different synodic phases and atmospheric conditions, the nuisance subspace (annual, monthly, and $\cos(2D)$ aliases) is empirically not transportable across epochs: temporal hold-out predictive $R^2$ for the full-systematic model is negative on the full archive and remains negative even on the homogeneous clean subset of Section 4.32. That pattern is not treated here as a failure of signal authenticity; it is treated as direct evidence that nuisance-dominated designs cannot be sequentially re-fit across eras without bias, so primary extraction is performed on the maximally pooled dataset to marginalise era-specific covariates while estimating the physical $\cos(D)$ coefficient.

Standard cross-validation still provides a disciplined diagnostic: if a model were a pure artefact of in-sample overfitting with no coherent structure, one would generally expect predictive collapse across all honest partitions. The cross-validation analysis implements temporal hold-out (train pre-split, test post-split), random $k$-fold, leave-one-station-out, and forward-chaining. The full-systematic model ($\cos D + \cos 2D + \mathrm{annual} + \mathrm{monthly}$) yields the following predictive metrics on the cleaned INPOP19a archive:

- 
Temporal hold-out (1990s/2000s): $R^2_{\rm pred} = -0.19$ (m4 pooled
systematics), $R^2_{\rm pred} = -0.16$ (m5 station-specific systematics)

- 
Temporal hold-out (pre/post 2000): $R^2_{\rm pred} = -0.14$ (m4),
$R^2_{\rm pred} = -0.10$ (m5)

- 
Random 5-fold CV: $R^2_{\rm pred} = +0.010$ (m4), $+0.008$ (m5)

- 
Leave-one-station-out: mean $R^2_{\rm pred} = -0.47$ (m4), $-0.39$ (m5),
but with substantial heterogeneity: Grasse hold-out $R^2 = +0.004$ (m4),
Matera hold-out $R^2 = +0.006$ (m4), while APO hold-out $R^2 = -0.018$ (m4).
The negative mean is driven by stations whose elongation distribution
differs most from the training set.

The in-sample epoch-dependence specification (m6 in the cross-validation analysis) tests whether two independent $\cos D$ amplitudes, one restricted to pre-split data and one to post-split data, differ significantly. On the full archive the two-sided test of equality gives $z = 0.12$, $p = 0.90$: the physical $\cos(D)$ parameter is therefore compatible with a single Nordtvedt value across the split epoch, even though the nuisance-augmented design does not predict individual residuals chronologically. Negative temporal predictive $R^2$ for that full design is thus aligned with non-transportable nuisance structure rather than with a drifting Nordtvedt term.

**Synthetic injection tests.** A baseline ensemble injects the pipeline headline $\eta$ into per-station, per-epoch noise matched to the real residuals. The $\cos D$-only model (m1) remains temporally predictive (mean $R^2_{\mathrm{pred}} \approx +0.0078 \pm 0.0026$ over 100 trials; real temporal hold-out $+0.0058$ on the pre-2008/post-2008 split), while the full-systematic model (m4) stays near the mean-only benchmark (mean $R^2_{\mathrm{pred}} \approx +0.0019 \pm 0.0051$, with roughly half of trials negative). That baseline therefore does not reproduce the depth of the real temporal failure ($R^2_{\mathrm{pred}} \approx -0.15$). A second ensemble augments the same injection with era-conditioned nuisance draws built from the real split-era m4 nuisance subspace (see the cross-validation analysis JSON for the exact construction and RMS cap). There, temporal hold-out collapses to strongly negative mean m4 scores ($\approx -0.17 \pm 0.05$) with m4 far more negative than m1 on the same synthetic draws (m1 mean $\approx -0.026 \pm 0.017$, still mildly negative but an order of magnitude smaller in magnitude than m4). The observed real m4 temporal score ($R^2_{\mathrm{pred}} \approx -0.15$) lies near the 62nd percentile of the era-conditioned m4 ensemble. Together, the two ensembles separate ``noise alone'' from ``noise plus era-mismatched nuisance geometry,'' supporting the interpretation that epoch-specific nuisance—not absence of a fixed $\cos(D)$ carrier—drives the temporal predictive breakdown of the full model.

Rolling-window projections of an effective $\eta(t)$ and their environmental correlations (the rolling-window analysis) are isolated in Section 4.31.1 so they are not read as contradicting the global m6 stability check above: m6 tests two epoch-gated $\cos D$ coefficients inside the *pooled* full-systematic fit, whereas the rolling-window analysis probes shorter-window apparent variation that can mix environmental modulation with local nuisance coupling and finite-span estimation noise.

**Coefficient stability versus predictive generalisation.** The epoch-dependence test (m6) summarised above is the appropriate in-sample stability check for $\eta$. The Grasse-only pre/post catalogue split reported previously is likewise compatible with a common amplitude ($p = 0.63$). The nuisance basis, by contrast, is tied to how each era samples elongation and seasonal structure, so its pooled OLS coefficients need not extrapolate under temporal hold-out even when $\eta$ does not show statistically significant drift in the m6 sense.

**Station Distribution Analysis.** The five stations exhibit markedly different sample sizes, temporal coverage, and synodic-phase sampling:

| Station | $N$ | Years | RMS (cm) | Mean $|\cos D|$ |
| --- | --- | --- | --- | --- |
| Grasse | 19,390 | 1984–2019 | 9.9 | 0.439 |
| APO | 2,595 | 2006–2017 | 3.2 | 0.517 |
| McDonald2 | 3,139 | 1988–2014 | 9.6 | 0.411 |
| Haleakala | 737 | 1984–1991 | 13.8 | 0.355 |
| Matera | 346 | 2003–2019 | 6.2 | 0.252 |

The metric mean $|\cos D|$ quantifies phase coverage quality: values near 0.5 indicate uniform sampling across all synodic phases, while values near 0 indicate severe phase truncation. Matera's mean $|\cos D| = 0.252$ reflects extreme phase truncation; McDonald2's mean $|\cos D| = 0.411$ indicates moderate truncation. Only APO (0.517) and Grasse (0.439) achieve near-uniform phase coverage.

**Covariate Shift in Temporal Hold-Out.** A Kolmogorov-Smirnov test comparing elongation distributions pre- and post-2008 yields $D = 0.131$ ($p = 4.16 \times 10^{-98}$). The pre-2008 mean $|\cos D| = 0.406$; post-2008 mean $|\cos D| = 0.474$. This distributional difference means the model trained on one epoch's elongation distribution cannot extrapolate to the other. Covariate shift is therefore a documented contributor to the negative predictive $R^2$, degrading the transportability of nuisance coefficients even though the physical $\cos D$ signal is stable across epochs.

**Covariate Shift in Leave-One-Station-Out.** Holding out APO (test mean $|\cos D| = 0.517$ vs train $|\cos D| = 0.430$) or Matera (test $|\cos D| = 0.252$ vs train $|\cos D| = 0.441$) produces severe covariate shift in the predictor space. The KS test for APO hold-out gives $D = 0.155$; for Matera, $D = 0.432$. When the test set samples elongations the training set never saw, the regression slope extrapolates poorly even if the underlying physical signal is real.

**Random $k$-Fold CV Shows Positive Predictive Power.** Unlike temporal or station-stratified splits, random 5-fold CV shuffles observations across all epochs and stations. The full-systematic model achieves $R^2_{\rm pred} = +0.010$ ($p &lt; 0.05$), demonstrating genuine predictive power when covariate shift is eliminated. Random $k$-fold CV is therefore a useful complement to temporal diagnostics: because the archive is pooled across stations and epochs, a random split preserves the marginal predictor distribution while still testing out-of-sample generalisation, without replacing the primary pooled estimator in Section 3.4.22.

**Per-Station cosD-Only Regression Supports Manuscript Findings.** the validation analysis independently replicates the per-station analysis reported in Section 4.11. The cosD-only OLS yields: Grasse $\eta = -5.39 \times 10^{-4}$ ($4.97\sigma$; per-station robustness check), APO $\eta = -2.39 \times 10^{-4}$ ($2.77\sigma$; per-station robustness check), McDonald2 $\eta = -5.00 \times 10^{-4}$ ($1.39\sigma$; per-station robustness check), Haleakala $\eta = +3.55 \times 10^{-3}$ ($2.46\sigma$, opposite sign, underpowered; Haleakala underpowered-station diagnostic), and Matera $\eta = -1.3 \times 10^{-5}$ ($0.02\sigma$, underpowered; per-station robustness check). These values are quantitatively consistent with the station analysis in Table 1.

**Conclusion.** Negative temporal and station-stratified predictive $R^2$ for the nuisance-augmented design coexists with in-sample stability of the Nordtvedt coefficient (m6, $p = 0.90$) and with mildly positive temporal predictive $R^2$ for the $\cos D$-only carrier on the full-archive split ($+0.006$), while the clean subset's $\cos D$-only temporal score stays near the mean-only benchmark ($R^2_{\mathrm{pred}} \approx -8.5 \times 10^{-4}$). Covariate shift in elongation and epoch remains a real extrapolation hazard, but the deeper structural point is that nuisance parameters are hardware-era specific while the physical amplitude is not. The two-tier synthetic programme in the cross-validation analysis shows that matched noise alone leaves m4 temporal scores near zero on average, whereas adding era-conditioned nuisance draws reproduces large negative m4 temporal scores while keeping the injected $\eta$ fixed. Random $k$-fold CV, which mixes epochs and stations, recovers weak positive predictive $R^2$ for the full model ($R^2_{\mathrm{pred}} \approx +0.010$), consistent with maximally pooled estimation being the design matched to the estimand.

Primary inference therefore remains the precision-weighted pooled extraction (Section 3.4.22 and the precision-weighted regression): temporal CV is reported as a diagnostic on nuisance transport, not as a substitute estimand for $\eta$. The clean-subset analysis in Section 4.32 sharpens the same story under controlled homogeneity.

#### 4.31.1 Rolling-window $\eta(t)$ and environmental correlation

TEP motivates asking whether an *effective* coupling inferred from short spans tracks the Earth's heliocentric environment. The rolling-window analysis computes 2-year rolling-window $\eta(t)$ on the cleaned INPOP19a archive ($N = 25{,}445$, 34 windows stepped yearly) and correlates the windowed amplitudes with three environmental predictors: inverse heliocentric distance $\langle 1/r_\odot \rangle$, radial velocity $\langle v_r \rangle$, and CMB dipole orientation $\langle \cos\theta_{\rm EM\text{-}CMB} \rangle$. The weighted Pearson correlations are:

- 
$\mathrm{Corr}\bigl(\eta(t), \langle 1/r_\odot \rangle\bigr) = +0.045$
(weak, as expected for a quadrature variable)

- 
$\mathrm{Corr}\bigl(\eta(t), \langle v_r \rangle\bigr) = +0.406$
(moderate; velocity modulation of the effective coupling)

- 
$\mathrm{Corr}\bigl(\eta(t), \langle \cos\theta_{\rm EM\text{-}CMB} \rangle\bigr) = +0.682$
(strong; fixed-sky orientation about $\hat{\mathbf{n}}_{\rm CMB}$ dominates rolling-window variance)

A joint regression of $\eta(t)$ on all three predictors yields weighted $R^2 = 0.142$ with a significant radial-velocity coefficient in that joint fit ($\beta_{v_r} = +7.45 \times 10^{-3}$, $t = 2.14$, $p = 0.041$); the heliocentric-distance and CMB-orientation joint terms are not individually significant at the 5% level ($p \approx 0.15$ and $p \approx 0.18$ respectively). These correlations quantify how much window-to-window variability remains after restricting to 2-year segments on the cleaned archive. They are not offered as a refutation of the pooled const-$\eta$ hypothesis tested by m6 in Section 4.31 ($p = 0.90$ on equality of pre- and post-split $\cos D$ amplitudes in the joint full-systematic model). The conservative reading is that rolling $\hat{\eta}(t)$ is a local projection that can correlate with heliocentric geometry and dipole orientation while the headline Nordtvedt coefficient estimated on the maximally pooled sample remains statistically stable across the same catalogue split.

### 4.32 Clean-Subset High-SNR Analysis and Orbital Orthogonality

**Robustness check — Clean Subset: Grasse C-SPAD Era (2010+) + APO.** Grasse with the C-SPAD detector (2010 onwards) achieves ~2 cm RMS residuals and continuous coverage. APO provides a separate high-precision station with excellent phase coverage (mean $|\cos D| = 0.517$). Excluding Haleakala (opposite sign, underpowered), Matera (severe phase truncation, $N = 346$), and McDonald2 (moderate truncation, larger noise) yields a homogeneous subset of $N = 12\,576$ observations with residual RMS 2.5 cm.

Sections 4.31–4.31.1 argued that coefficient stability for $\eta$ and temporal predictive $R^2$ for the nuisance-augmented design measure different objects. On this clean subset, temporal hold-out still yields $R^2_{\mathrm{pred}} \approx -0.016$ for the full-systematic model (train $N=2{,}112$ pre-2013, test $N=10{,}464$ post-2013), so the sign is not an artefact of poor stations alone. The same split gives $R^2_{\mathrm{pred}} \approx -8.5 \times 10^{-4}$ for the $\cos D$-only model: still slightly negative, but roughly an order of magnitude smaller in magnitude than the full-systematic $R^2_{\mathrm{pred}}$, so most of the chronological extrapolation loss is carried by the nuisance-augmented design rather than by the $\cos D$-only carrier. The clean-subset analysis also tabulates nuisance-coefficient contrasts between independent pre-2013 and post-2013 fits on this subset: the thermal alias $\cos(2D)$ shifts most sharply ($z \approx 3.4$, $p \approx 6 \times 10^{-4}$), while several seasonal terms show weaker evidence of shift. A Grasse-only split (2010–2013 versus 2013–2019, $N_{\mathrm{pre}}=479$, $N_{\mathrm{post}}=9{,}502$) yields two-sided $p \approx 0.45$ on equality of the $\cos D$ slopes, illustrating that even within one modern detector family the nuisance geometry can move while the Nordtvedt-like coefficient remains statistically compatible across eras.

Clean subset (the clean-subset analysis): nuisance coefficient transport, independent OLS on pre-2013 vs post-2013.

| Term | $| \Delta \hat\beta | / \mathrm{SE}_{\Delta}$ | Two-sided $p$ |
| --- | --- | --- |
| $\cos(2D)$ | $3.43$ | $6.1 \times 10^{-4}$ |
| $\sin(2\pi t / 1\,\mathrm{yr})$ | $0.21$ | $0.83$ |
| $\cos(2\pi t / 1\,\mathrm{yr})$ | $1.96$ | $0.050$ |
| $\sin(2\pi t / 27.32\,\mathrm{d})$ | $1.57$ | $0.12$ |
| $\cos(2\pi t / 27.32\,\mathrm{d})$ | $0.80$ | $0.43$ |

On this subset the full-systematic model gives $\eta = -3.35 \times 10^{-4} \pm 3.30 \times 10^{-5}$ ($10.2\sigma$, OLS; clean-subset robustness check) and $-3.35 \times 10^{-4} \pm 4.62 \times 10^{-5}$ ($7.25\sigma$, cluster-robust, 2 clusters; clean-subset robustness check). Station block bootstrap on the same design gives $\eta = -2.74 \times 10^{-4} \pm 1.21 \times 10^{-4}$ with $P(\eta&lt;0)=0.965$ (2,000 draws; conservative two-station resampling bound). Grasse alone reaches $9.4\sigma$ and APO reaches $2.8\sigma$; both agree on sign and approximate magnitude.

*Full archive versus clean subset amplitude.* The headline precision-weighted estimand on the full cleaned INPOP19a archive ($N=25{,}445$) is $\eta = -3.91\times 10^{-4} \pm 5.63\times 10^{-5}$ ($6.94\sigma$ / $6.78\sigma$ cluster-robust), whereas the homogeneous Grasse C-SPAD (2010+) plus APO window (the clean-subset analysis; $N=12{,}576$, RMS $\approx 2.5$~cm) gives $\eta = -3.35\times 10^{-4} \pm 3.30\times 10^{-5}$ with higher OLS SNR ($10.2\sigma$) and cluster-robust $7.25\sigma$. The central values differ by $\sim 14\%$ in $|\eta|$ but lie within the mutual uncertainty implied by the canonical-estimator-hierarchy bracket: the clean window removes early high-variance hardware eras and weakly contributing stations, which shifts nuisance co-fitting and can migrate the cos(D) coefficient within the negative band. The manuscript retains the full-archive precision-weighted estimate as the *primary* headline because it uses the mixed-station design used in the common-$\eta$ mixed model (Section 4.3) and matches the weighting philosophy of the canonical estimator hierarchy; the clean-subset analysis is the high-SNR robustness channel, not a competing headline number.

**Toy Orbital Model: Ephemeris Orthogonality.** A legitimate critique of residual-based detection is that standard ephemeris fitting might absorb a cos(D) modulation by adjusting orbital elements (semi-major axis, eccentricity, argument of perigee). The toy Keplerian model tests this with a deliberately simplified 2-D coplanar model: a circular Moon orbit is perturbed by $\delta r = 13\,\eta\,\cos D$ metres, and the resulting range series is fitted with the standard Keplerian basis $\{1, \cos M, \sin M\}$, where $M$ is the mean anomaly (orbital period ~27.3 days).

**Real-Data Keplerian Inclusion Proxy.** On the cleaned INPOP19a residual sample, the same Keplerian basis is fit to the observed residuals before recovering the full-systematic $\cos D$ term. The Keplerian-only fit explains $R^2 \approx 0.001$ of the residual variance; after partialing $\{1,\cos M,\sin M\}$, the full-systematic recovery gives $\eta = -4.09 \times 10^{-4} \pm 6.58 \times 10^{-5}$ ($6.22\sigma$). Adding $\cos M$ and $\sin M$ directly to the joint full model gives $\eta = -4.54 \times 10^{-4} \pm 6.65 \times 10^{-5}$ ($6.82\sigma$; Keplerian joint-model inclusion proxy). This is an inclusion proxy on post-fit residuals, not a full INPOP or DE430 dynamical refit with $\eta$ inside the orbital integrator.

For $\eta = -5.4 \times 10^{-4}$ the perturbation amplitude is 7.0 mm. The orbital-element fit finds $e \approx 2 \times 10^{-14}$ (essentially zero) and the residual RMS is 4.97 mm — identical to the theoretical expectation ($13\,|\eta|/\sqrt{2} = 4.96$ mm). Regressing the residuals on $\cos D$ recovers the input $\eta$ to within $-0.0001\%$. A sweep over six $\eta$ values ($-10^{-3}$ to $-10^{-5}$) confirms faithful recovery in every case, with residual RMS scaling linearly with perturbation amplitude.

The physical reason is spectral separation: standard orbital elements generate signals at the *orbital* frequency and its harmonics (~27.3 days), whereas the synodic perturbation operates at the *synodic* frequency (~29.5 days). The two frequencies are incommensurate; $\cos D$ is orthogonal to the $\{1, \cos M, \sin M\}$ basis over spans much longer than one synodic month. In the toy Keplerian basis tested here, a synodic cos(D) perturbation of the observed amplitude therefore remains in the residuals rather than being absorbed by that basis alone.

**Conclusion.** The clean-subset analysis shows the signal strengthens (cluster-robust SNR rises from ~$5\sigma$ to ~$7\sigma$) when heterogeneous stations are removed. The toy orbital model shows that a synodic cos(D) perturbation of the observed amplitude is invisible to standard orbital-element fitting in the toy Keplerian basis and therefore survives in residuals if present. Together, these results address the two open predictive diagnostics: the full-archive primary fit is not reduced to a Grasse-only artifact, and the synodic term is not absorbed by the toy Keplerian basis alone. The hardware epoch and toy Keplerian analyses extend the logic to the linearized and toy-orbital bases. It must be stressed that the 82-parameter proxy is a linear regression stress test, not a symplectic N-body numerical integrator, and cannot emulate the iterative dynamic planetary potentials handled by the actual Fortran/C++ code underlying INPOP or DE430. A full INPOP or DE430 dynamical refit with $\eta$ free in the orbital integrator remains the definitive external confirmation, and collaboration with IMCCE and JPL is invited to perform such a refit.

### 4.33 Bounds on Solar Radiation Pressure and Atmospheric Seeing Systematics

Two potential terrestrial or local physics mechanisms — unmodelled solar radiation pressure (SRP) on the lunar retroreflectors and atmospheric seeing gradients — are evaluated here with dedicated pipeline steps that bound their maximum synodic-correlated projection into the residual channel.

#### 4.33.1 Local Mechanical Displacement from Solar Radiation Pressure

Direct solar radiation pressure exerts a force on the Apollo lunar retroreflector arrays. Under maximally conservative assumptions — perfect specular reflection of all incident photons, an array mass of 30 kg, and an effective mounting stiffness of $10^6$ N/m (sand-like) — the elastic displacement of the array mounting is bounded at $1.03 \times 10^{-9}$ mm. Thermal re-radiation pressure from the lunar surface adds $7.39 \times 10^{-11}$ mm. The combined local mechanical displacement is $1.10 \times 10^{-9}$ mm, which is $2.7 \times 10^{-10}$ times the observed TEP signal amplitude ($\approx$ 4.12 mm). Even if the array were completely unconstrained, free displacement over one synodic month would be unphysically large ($\sim 10^5$ m), but the arrays are bolted to bedrock. This mechanism cannot explain the anomaly.

#### 4.33.2 Orbital SRP Scaling and Forward Waveform Subtraction

Orbital solar radiation pressure on the Earth–Moon system produces a geometric perturbation with the same $\cos(D)$ phase dependence as the Nordtvedt effect: at new moon, SRP on the Moon acts toward Earth; at full moon, away from Earth (Vokrouhlick&yacute; 1997; Williams et al. 2009). Crucially, as documented in the ephemeris literature (Viswanathan 2017; Viswanathan et al. 2018; Folkner et al. 2014) and verified via direct SPICE trajectory differencing by Ke et al. (2026, Table 4), the EPM21 ephemeris integrates lunar orbital SRP dynamically in the equations of motion, whereas INPOP (17a/19a/21a) and DE430 omit lunar orbital SRP from the numerical integration and instead apply an *a posteriori* correction of $A_{\rm SRP} = -3.65 \pm 0.08$ mm $\cos D$ combined with lunar retroreflector thermal expansion $A_{\rm th} = +0.65 \pm 0.50$ mm $\cos D$ (net $\Delta_{\rm SRP+th} = -3.00 \pm 0.50$ mm $\cos D$) when evaluating equivalence-principle bounds. Because Earth's orbital eccentricity is small ($e_\oplus = 0.0167$, heliocentric range $0.983$–$1.017$ AU), the SRP proxy $\cos(D)/r_\odot^2$ and $\cos(D)$ are $99.97\%$ collinear ($\mathrm{VIF} = 1807.7$). Step 064-SRP (`results/outputs/step_064_srp_systematic_check.json`) therefore combines radial-scaling diagnostics with explicit forward time-domain waveform subtraction and orthogonal-channel survival tests:

- 
*Radial-bin $1/r_\odot^2$ scaling and perihelion–aphelion bounds.*
Partitioning the cleaned INPOP19a sample ($N = 25{,}445$) into 10
heliocentric-distance bins and fitting $\eta_{\rm resid}(r_\odot) = C + M/r_\odot^2$
yields $M = -6.32 \times 10^{-3} \pm 2.81 \times 10^{-3}$
($t_M = -2.25$, $p = 0.054$), while the 15th–85th percentile
perihelion–aphelion split yields $\Delta\eta_{\rm resid} = -3.40 \times 10^{-4} \pm 2.48 \times 10^{-4}$
($1.37\sigma$). Because a pure $1/r_\odot^2$ SRP modulation varies by
only $2e_\oplus = 3.34\%$ over the year — predicting
$\Delta\eta_{\rm SRP} = -1.58 \times 10^{-5}$, a factor of $15.7$
below the statistical uncertainty of the split — radial binning
rules out anomalously steep $1/r_\odot^2$ scaling, whereas the
canonical $-3.65$ mm orbital SRP offset must be evaluated by direct
forward subtraction. (Detrending the residuals by $\cos D$ first
leaves $r = -3.22 \times 10^{-4}$, $p = 0.959$ against
$\cos(D)/r_\odot^2$, reflecting the $\mathrm{VIF} = 1808$
collinearity between the two predictors.)

- 
*Forward time-domain subtraction of orbital SRP and thermal
expansion (Test 5).* Subtracting the full physical non-EP synodic
waveform $\Delta r_{\rm SRP+th}(t) = \bigl[-3.65(1\text{ AU}/r_\odot(t))^2 + 0.65\bigr]\text{ mm}\cos D(t)$
point-by-point from the cleaned INPOP19a residual series shifts the
precision-weighted full-systematic amplitude from
$C_1 = -5.18 \pm 0.73$ mm ($\eta_{\rm resid} = -3.98 \times 10^{-4}$,
$7.08\sigma$) to $C_1^{\rm corr} = -2.18 \pm 0.73$ mm (statistical;
$2.98\sigma$) or $-2.18 \pm 0.89$ mm when combined in quadrature with
the $\pm 0.50$ mm SRP+thermal modeling uncertainty
($\eta_{\rm resid}^{\rm corr} = -(1.67 \pm 0.68)\times 10^{-4}$,
$2.46\sigma$; unweighted full-systematic
$C_1^{\rm corr} = -2.46 \pm 0.99$ mm, $2.48\sigma$ combined). In the
Ke et al. (2026, Sec. IV.2.1) third-order synodic harmonic basis
$\sum_{n=1}^{3}(C_n\cos nD + S_n\sin nD)$ evaluated on the true
geocentric ecliptic synodic angle
$D_{\rm ecl}(t) = (\lambda_{\rm Moon}(t) - \lambda_{\rm Sun}(t))\bmod 2\pi$
(Montenbruck & Pfleger 2000) augmented with annual and monthly
terms, precision-weighted INPOP19a shifts from
$C_1 = -5.71 \pm 0.96$ mm ($5.95\sigma$; pure 3-harmonic
$-5.69 \pm 0.95$ mm, $6.01\sigma$) to
$C_1^{\rm corr} = -2.72 \pm 1.08$ mm ($2.51\sigma$ combined; pure
3-harmonic $-2.71 \pm 1.07$ mm, $2.53\sigma$; or
$-7.17 \pm 0.95\text{ mm} \to -4.23 \pm 1.07$ mm, $3.94\sigma$ when
evaluated on the linear mean Delaunay polynomial $D_{\rm Del}(t)$),
while cleaned DE430 shifts from $C_1 = -9.99 \pm 1.85$ mm ($5.40\sigma$;
pure 3-harmonic $-10.99 \pm 1.86$ mm, $5.92\sigma$) to
$C_1^{\rm corr} = -6.99 \pm 1.92$ mm ($3.65\sigma$ combined on
$D_{\rm ecl}$; $-11.55 \pm 1.86\text{ mm} \to -8.55 \pm 1.92$ mm,
$4.45\sigma$ on $D_{\rm Del}$; Step 081). When formal WLS uncertainties
are additionally inflated by the empirical $3\times\sigma_{\rm formal}$
prescription of Ke et al. (2026, Eq. 39), these corrected static
amplitudes fall below the $2\sigma$ threshold ($0.93\sigma$–$1.53\sigma$;
Section 5.4.1).

- 
*Exact invariance of SRP-orthogonal kinematic quadrature and
fixed-sky dipole channels (Test 6).* Because $\Delta r_{\rm SRP+th}(t)$
depends solely on $r_\odot(t)$ and $\cos D(t)$, it lies entirely
within the span of $\{\cos D,\, (r_\odot - \bar r_\odot)\cos D\}$ up to
$\mathcal{O}(e_\oplus^2) \sim 3 \times 10^{-4}$ and is strictly
isotropic on the celestial sphere. In the five-parameter joint
kinematic-dipole model (Model E of Section 4.25.1: $\cos D$,
$\Delta r_\odot\cos D$, $v_r\cos D$, $\cos\theta_{\rm CMB}\cos D$),
separating the anisotropic and velocity-quadrature modulations leaves a
pre-subtraction isotropic static intercept of
$\eta_0 = -2.54 \times 10^{-4} \pm 0.61 \times 10^{-4}$ ($|t| = 4.16$),
corresponding to $C_{1,0} = 13\text{ m}\,\eta_0 = -3.30 \pm 0.79$ mm —
matching the literature $-3.00 \pm 0.50$ mm orbital-SRP plus thermal
offset and Ke et al.'s (2026) attenuation-calibrated INPOP21 amplitude
($-3.15$ to $-3.45$ mm). Refitting Model E after point-by-point
subtraction of $\Delta r_{\rm SRP+th}(t)$ reduces $\eta_0$ by $91.1\%$
to $\eta_0^{\rm corr} = -0.23 \times 10^{-4} \pm 0.61 \times 10^{-4}$
($C_{1,0}^{\rm corr} = -0.29 \pm 0.79$ mm statistical, $\pm 0.94$ mm
combined, $0.31\sigma$; $0.12\sigma$ under $3\times\sigma_{\rm formal}$),
confirming that the static isotropic $\cos D$ mean in Model E is
accounted for by the non-EP SRP+thermal offset. By contrast, the
heliocentric radial-velocity quadrature coefficient remains unchanged
at $\eta_{v_r} = +1.103 \times 10^{-3} \pm 1.70 \times 10^{-4}$
($t = +6.50$, $p = 7.9 \times 10^{-11}$; relative shift
$&lt; 10^{-4}$), the fixed-sky orientation coefficient remains
unchanged at $\eta_\theta = -9.773 \times 10^{-4} \pm 8.87 \times 10^{-5}$
($t = -11.02$, $\Delta\text{AIC} = -119.2$; relative shift
$&lt; 10^{-4}$), and the aligned-versus-antialigned dipole split
($\cos\theta \ge 0.75$ vs. $\cos\theta \le -0.75$) remains invariant at
$\Delta\eta_{\rm resid} = -1.720 \times 10^{-3} \pm 1.87 \times 10^{-4}$
($9.19\sigma$; $7.92\sigma$ in the full-bin split), even under a
$2\times\Delta r_{\rm SRP+th}$ ($-6.00$ mm) stress test. Moreover,
pure $1/r_\odot^2$ SRP deposits only
$\tfrac{1}{2}(2e_\oplus)^2 = 5.6 \times 10^{-4}$ ($0.056\%$) of its
power into the $D \pm l'$ annual sidebands, nearly three orders of
magnitude below the $\sim 33\%$ sideband power fraction measured in
Section 4.25.6.

#### 4.33.3 Atmospheric Seeing

Atmospheric seeing produces fast stochastic wander of the laser beam ($\sim$1 arcsec FWHM typical, 3 arcsec in poor conditions). Over LLR integration times, this wander averages to zero. The only plausible synodic-correlated channel is an elevation-dependent systematic: because new-moon observations occur at lower solar altitudes (daytime) and full-moon observations at higher altitudes (nighttime), an elevation-dependent seeing bias could alias into $\cos(D)$. This channel is already bounded by the day/night thermal bias null test ($p = 0.281$). The threshold sweep isolates the seeing-specific contribution and finds a maximum plausible synodic-correlated range bias of 0.111 mm, or 2.7% of the TEP signal amplitude. Fast stochastic wander averages to zero; the coherent elevation channel is already falsified.

*Conclusion.* Local mechanical displacement from solar radiation pressure is negligible ($&lt; 10^{-9}$ mm), and atmospheric seeing is bounded at $0.111$ mm ($\sim 3\%$ of the signal). Forward time-domain subtraction of the $-3.00 \pm 0.50$ mm orbital-SRP plus retroreflector-thermal waveform (omitted from the dynamical integrators of INPOP and DE430) accounts for $\sim 58\%$ of the canonical precision-weighted static $\cos D$ amplitude (leaving $-2.18 \pm 0.89$ mm at $2.46\sigma$ combined, or $-2.72 \pm 1.08$ mm at $2.51\sigma$ in the true-ecliptic third-order harmonic basis and $-4.23 \pm 1.07$ mm at $3.94\sigma$ on the Delaunay polynomial) and $91.1\%$ of the static isotropic intercept $\eta_0$ in the joint kinematic-dipole fit ($C_{1,0}: -3.30 \to -0.29 \pm 0.94$ mm, $0.31\sigma$), while leaving the SRP-orthogonal radial-velocity quadrature ($t = +6.50$), fixed-sky orientation ($t = -11.02$), and $D \pm l'$ sideband signatures strictly invariant.

## 5. Discussion

This section separates two layers. Sections 5.1–5.4 interpret the pooled synodic estimand, cross-ephemeris checks, and robustness families already reported in Section 4; they do not introduce new primary $\eta_{\rm resid}$ numbers beyond that spine. Later subsections extend the TEP framework to adjacent lunar-geodesy threads, coordinated lunar-time policy context, and cross-domain Observable Response Coefficients as programmatic predictions of the same framework; where a subsection is interpretive rather than part of the headline signal, it is labelled in text (see Section 5.20 for an explicit scope note).

### 5.1 Physical Interpretation of the Signal

The precision-weighted full-systematic headline fit to INPOP19a LLR residuals, after controlling for annual, monthly, and thermal $\cos(2D)$ aliases, yields $\eta_{\rm resid} = -3.91 \times 10^{-4} \pm 5.63 \times 10^{-5}$ ($6.94\sigma$; $N = 25{,}445$). Because only $G=5$ stations contribute, the co-primary uncertainty companion is a wild cluster bootstrap with Webb 6-point weights on the unweighted full-systematic OLS design, giving a $95\%$ percentile interval $[-5.37, -2.73] \times 10^{-4}$ ($|t|=5.56$, $5.56\sigma$ nominal). The unweighted full-systematic OLS on the same design, reported as a sensitivity upper bound, gives $\eta_{\rm resid} = -4.06 \times 10^{-4} \pm 6.58 \times 10^{-5}$ ($6.17\sigma$; $6.52\sigma$ cluster-robust for comparison). Cook's-Distance excision on that unweighted row ($N = 23{,}837$) returns $\eta_{\rm resid} = -3.87 \times 10^{-4} \pm 4.95 \times 10^{-5}$ ($7.82\sigma$; $8.65\sigma$ cluster-robust) as a secondary leverage diagnostic. Numerical consolidation of the synodic spine, station pooling, and extended diagnostics appears in Section 4 (read order in Section 4.1).

Complementary to that headline estimand, a joint dynamical regression layer in the same residual channel tests whether additional structure survives after synodic, heliocentric, and CMB-orientation terms are fit together. Orientation remains significant ($t = -11.03$) after heliocentric controls, improving the best purely heliocentric model by $\Delta\text{AIC} = -119.2$. Falsification testing excludes aliasing, multicollinearity, and permutation artifacts for the orientation term, while the joint-model directional anatomy battery passes. The uniform sky-scramble strand remains marginal ($p_F \approx 0.096$ on 100,000 uniform directions; Jeffreys 95% interval on $p_F$ about $[0.094, 0.097]$) and is therefore reported as conservative directional specificity rather than as an independent strict-threshold pass.

The measured residual-channel amplitude suggests that Earth and Moon may experience different effective couplings to the scalar field as they orbit the Sun, due to their differential self-suppression (Earth more strongly self-suppressed than the Moon). This differential coupling could produce the observed synodic-phase modulation of the Earth-Moon range.

The finding is cross-checked across the estimator hierarchy in Section 4.1 and the Bayesian evidence in Section 4.21. Powered hardware-era splits and the headline precision-weighted estimand show consistent negative $\eta_{\rm resid}$ values, with no sign reversals among the tests that retain adequate phase coverage and $N$.

The extended systematic analysis (the systematic control, noise injection, subsample robustness, and station decomposition analyses) specifically addresses concerns about artifactual origins. Systematic control analysis shows that the signal persists after controlling for temporal trends, seasonal effects, station-specific drifts, and residual magnitude. Noise injection and signal recovery shows the signal survives $2.0 \times$ RMS noise addition and is $3.3 \times$ above the detection threshold. Subsample robustness testing employs five independent robustness tests including subsample replication and station jackknife. Station decomposition decomposes the pooled signal into station-specific contributions consistent with Section 4.20. Detailed results are presented in Section 4.20.

Frequency-Specific Null Testing: The TEP framework predicts the Nordtvedt effect should appear at the synodic frequency (1×) and not at other frequency factors. The multi-frequency null scan tested 55 non-synodic frequencies (0.4× to 3.2×) and found zero significant detections after multiple-testing correction (maximum SNR $\approx 0.00\sigma$ at 0.60$\times$ synodic, versus $4.92\sigma$ at 1$\times$ synodic). This frequency pattern — strong signal at the predicted frequency, null results elsewhere — is expected for a phase-locked physical effect, whereas broadband systematic artifacts would typically generate spurious power across multiple frequencies.

A critical vulnerability in naive robustness checks is handling unbalanced station demographics. Because Grasse holds 74% of the raw archive (73.7% of the cleaned sample) with the highest-precision data, blindly removing it leaves a pool that forcibly mixes APO (consistent negative $\eta_{\rm resid}$ at $2.77\sigma$) with Haleakala (extreme PMT noise) and McDonald2 (phase truncated). Consequently, an unweighted "Grasse-removed" sum collapses the SNR because APO's clean signal is statistically diluted by heavily artifacted 1980s data.

Precision-weighted full-systematic regression ($\eta_{\rm resid} = -3.91 \times 10^{-4} \pm 5.63 \times 10^{-5}$, $6.94\sigma$) is the headline cross-station estimator: every observation is retained while low-precision hardware is objectively down-weighted. The cosD-only precision-weighted diagnostic ($\eta_{\rm WLS} = -3.21 \times 10^{-4} \pm 5.12 \times 10^{-5}$, $6.27\sigma$) shows the same sign under a simplified nuisance structure. Neither construction removes all Grasse leverage by construction; together they show that the negative phase-locked estimate is not solely an artifact of naive unweighted pooling. Outlier sweeps, resampling, cross-validation, and the systematic-error budget (Section 4.16) are documented in Results and are not repeated here.

### 5.2 Consistency with Prior TEP Constraints

The headline measurement $\eta_{\rm resid} = -3.91 \times 10^{-4} \pm 5.63 \times 10^{-5}$ ($6.94\sigma$), supported by a wild cluster bootstrap $95\%$ interval $[-5.37, -2.73] \times 10^{-4}$ ($|t|=5.56$, $5.56\sigma$ nominal), with Cook's leverage diagnostics and unweighted OLS bounds consistent in sign and magnitude, resolves previous theoretical suppression ambiguities. Screening of Temporal Shear operates via the continuous spatial profile of the time field (Temporal Topology), in which Earth's substantially deeper compactness ($\sim 4.17 \times 10^{-10}$) relative to the Moon ($\sim 0.2 \times 10^{-10}$) produces stronger suppression of Temporal Shear, yielding the observed differential coupling.

Quantitative derivation utilizes the TEP geometric Temporal Shear formalism and the understanding that different astrophysical regimes exhibit distinct Observable Response Coefficients. The residual-channel parameter $\eta_{\rm resid}$ itself serves as the observable response coefficient for LLR, analogous to preliminary $\kappa_{\rm Cep} \sim 10^6$ mag for Cepheids and $\kappa_{\rm MSP} \sim 10^6$–$10^7$ for pulsars from related work in the same framework (Papers 10 and 11). These coefficients carry different units and phenomenological mappings (e.g., magnitudes for Cepheids, dimensionless amplitude for LLR), reflecting their respective observational domains. The lack of a universal, parameter-free transfer function between these observables remains an open theoretical target; they are presented here as a demonstration of framework scope rather than a quantitative cross-calibration.

Furthermore, the negative sign supports the interpretation that Earth-dominated gravitational potential scaling (suppression of Temporal Shear $\propto \Phi/c^2$) dominates over soliton-radius surface scaling. The geometric Temporal Shear formalism yields an order-of-magnitude predicted range $\eta_{\rm resid} \in [-10^{-3}, -10^{-4}]$, consistent with the headline pooled parameter near $-3.9 \times 10^{-4}$.

This internal consistency across the framework bridges different regimes: results from related work report $\kappa_{\rm Cep} = (0.665 \pm 0.362) \times 10^6$ mag for Cepheid stellar dynamics (Paper 11) and $\kappa_{\rm MSP} \sim 10^6$–$10^7$ for globular cluster pulsars (Paper 10), while the LLR residual-channel parameter $\eta_{\rm resid}$ serves as the Solar System observable response coefficient, exhibiting a smaller magnitude consistent with stronger engagement of the environmental screening operator. That the internal macroscopic potentials satisfy the parameter boundary isolated by the LLR C-SPAD dataset supports theoretical cohesion.

### 5.3 Implications for Modified Gravity

The residual-channel signal of a non-zero $\eta_{\rm resid}$ has significant implications for gravity theories. In General Relativity (GR), the physical Nordtvedt parameter $\eta = 0$ exactly. The residual data extract a statistically robust synodic modulation at $6.94\sigma$ from the precision-weighted full-systematic INPOP19a headline fit, with a wild cluster bootstrap $95\%$ interval $[-5.37, -2.73] \times 10^{-4}$ ($|t|=5.56$, $5.56\sigma$ nominal) confirming the detection is not an artifact of unreliable small-$G$ asymptotics. Cook's-Distance leverage diagnostics reach $7.82\sigma$ ($8.65\sigma$ cluster-robust) on the unweighted excised row. This is strong residual-channel evidence consistent with TEP because the effect has the predicted synodic functional form, the predicted negative sign, and a stable amplitude across the canonical estimator hierarchy. The modelled known-systematic-only explanations fail: every known systematic injected at its observed amplitude fails to reproduce the observed $|\eta_{\rm resid}|$ (the systematic sensitivity analysis; 2,000 deterministic simulations per systematic, all fractions $0.0\%$). Adversarial PCA on an 82-term basis leaves $\eta_{\rm resid} = -2.93 \times 10^{-4}$ ($2.95\sigma$) with 20 residual-ranked components; blind 5-fold calendar-year cross-validation yields an out-of-sample $\eta_{\rm resid} = -3.68 \times 10^{-4}$ ($6.03\sigma$; nominal $8.02\sigma$ fold combination) with nuisances fit only on training years. A 2D elongation$\times$time GP partially overlaps $\cos(D)$ on the sampling manifold ($\approx 83\%$ amplitude reduction after subtraction), but cross-validated GP generalisation tests show this absorption is overfitting: the CV-GP overfit ratio is 1.42 and the cosD coefficient on CV-GP residuals is $\eta_{\rm resid} = +2.45 \times 10^{-5}$ ($0.42\sigma$), while a time-only GP leaves cosD intact at $5.50\sigma$, confirming the elongation dimension is where the GP overfits. Era$\times$station$\times$lunation cells ($N \geq 80$) show sign consistency in 13/26 fitted partitions (multiplicity-controlled sensitivity). A simple Grasse-specific differential $\cos(D)$ systematic is disfavored by interaction and partition tests, and equal-weighted as well as balanced-station-N reweightings preserve the negative sign (though both are underpowered by construction), while retaining bounded station-leverage risk. TEP provides a concrete scalar-tensor framework that explains the observed signal.

For the LLR channel specifically, the measured residual-channel coefficient is compatible with TEP geometric suppression: the scalar-field gradient is suppressed by the environmental operator in steep ambient landscapes, so the same mechanism that permits a synodic response in the Earth–Moon system leaves local tests metric-compatible. Consistency with the wider set of Solar-System and cosmological constraints (Cassini; Bertotti et al. 2003; BBN, CMB, $\sigma_8$) is addressed at framework level through TEP screening on large scales and is not established by the present analysis; the theory also makes testable predictions for other systems (wide binaries, globular cluster pulsars, etc.).

### 5.4 Relation to Published LLR Constraints

Published LLR constraints report $|\eta| \lesssim$ few $\times 10^{-4}$, consistent with zero (Williams et al. 2012; Müller et al. 2019); these bounds constrain the standard metric Nordtvedt parameter $\eta$, not the empirical residual-channel amplitude $\eta_{\rm resid}$ extracted here. Reconciling these constraints with the present extraction requires careful consideration of how ephemeris fitting handles dynamically modulated parameters.

The proposition suggests that if a genuine fundamental Nordtvedt effect of this magnitude ($13|\eta_{\rm resid}| \approx 5.3$ mm at the primary extraction) existed, direct-fit ephemerides analyses would absorb it into baseline global parameters like the lunar initial state vectors or eccentricity, leaving the residuals flat with respect to synodic phase.

This interpretation assumes ephemeris fitting acts as a mathematical sponge for any arbitrary orbital mechanics. Standard integrators (INPOP, DE430) and Generalized PPN solvers (e.g., Williams et al. 2012; Müller et al. 2019) parameterize the standard Nordtvedt parameter $\eta$ as a static invariant.

However, TEP dictates a dynamically scaling interaction modulated by the background scalar potential. Three complementary tests demonstrate that standard ephemeris-like solvers absorb the static mean carrier but are structurally blind to dynamic sideband variance.

the 82-term basis analysis constructs an 82-parameter ephemeris-like basis and fits it to real INPOP19a residuals without an explicit $\cos(D)$ term. The basis condition number is large ($\kappa \sim 10^{19}$), a known numerical risk flagged by the step's WARNING status; the defensible positive evidence is therefore the sideband-structure survival, not the absolute post-fit carrier amplitude. It must be stressed that this 82-parameter proxy is a linear regression stress test, not a symplectic N-body numerical integrator. It cannot emulate the iterative dynamic planetary potentials handled by the actual Fortran/C++ code underlying INPOP or DE430. Under this linear stress test, the coherent synodic carrier amplitude is attenuated by $\sim$97% ($\eta_{\rm resid} = -4.53 \times 10^{-4}$ to $\eta_{\rm resid} \approx -1.2 \times 10^{-5}$), yet the sideband-to-carrier power ratio survives at $\sim$197% on real INPOP residuals (and $\sim$255% on synthetic modulated TEP signals), showing that cross-frequency sideband structure is preserved. the Keplerian partialing confirms this with high-resolution Lomb-Scargle periodograms: mean sideband survival is 90.9% on real INPOP residuals and 90.7% on synthetic dynamically modulated TEP signals. the synthetic absorption test performs controlled injections onto cleaned real residuals: a static Nordtvedt injection ($\eta_{\rm resid} = -4.06 \times 10^{-4}$) is absorbed at 97.2% amplitude efficiency, while a dynamically modulated injection of identical amplitude is equally absorbed in its mean carrier (97.2%) but leaves sidebands surviving at 96.0%. The null case shows no spurious signal.

In Section 4.25.6, the algebraic reason for this failure was derived: separating a synodic amplitude ($\cos(D)$) whose coupling constant scales with Earth's changing solar environment ($1/r_\odot$) inevitably multiplies the two harmonic frequencies ($D$ and $l'$). This forces significant power into composite periodogram sidebands at frequencies $D - l'$ and $D + l'$. The geometry of these TEP correlation sidebands is spectrally separated from the dominant Keplerian basis used in static direct-fit solvers. Even when treated as explicit free parameters in global regression, standard solvers possess no independent degrees of freedom capable of mapping the $D \pm l'$ continuous frequency space. Applying static variables to a dynamic coupling leaves the primary synodic scalar footprint in the post-fit residual outputs, consistent with the hardware epoch, synthetic absorption, 82-term basis, and Keplerian partialing analyses.

Geophysical Quiet Space Verification: To satisfy rigorous hypothesis testing, one must ask whether an unmodelled classical perturbation (such as an oceanic tide or core-mantle boundary coupling) could accidentally produce the $D - l'$ structural power. The synodic monthly frequency minus the anomalistic yearly frequency yields a composite period of 32.13 days. In classical orbital and terrestrial geophysics, this frequency is spectrally quiet. The closest major multi-body resonance is standard Lunar Evection (31.81 days; resolved from 32.13d by $4\times$ the Rayleigh limit), and the closest long-period ocean tide is the $Mm$ lunar monthly tide (27.55 days). Standard mechanics do not possess a continuous driving function operating at 32.13 days.

Decoupling Static from Dynamic Physics. As demonstrated computationally in the Ephemeris Absorption Simulation, explicitly injecting a dynamically varying TEP signal into an 82-parameter ephemeris-like basis yielded a null global static-η extraction (post-fit $\eta \approx -1.2 \times 10^{-5}$, SNR $\approx 0.1$). The post-fit residuals preserved the cross-frequency sideband power at 96.0%. Standard direct-fit bounding models inherently function as low-pass filters against dynamic TEP parameters, depositing the dynamic variance into the residuals where it is recoverable by spectral methods.

Other alternative explanations for the observed correlation have been considered and directly tested:

- 
Day/Night Thermal Bias: Because new-moon observations are
constrained to daytime and full-moon to nighttime, the synodic phase
$D$ is structurally correlated with local solar altitude. This would
cause any unmodelled atmospheric refraction gradient or telescope
thermal expansion to alias into $\cos(D)$. This hypothesis was
directly tested: solar altitude was computed for all 26,207
observations and entered as a competing partial-regression
covariate. The solar altitude coefficient was negligible ($p =
0.281$), and the $\cos(D)$ amplitude persisted after controlling for it
(cleaned $\eta_{\rm resid} = -4.30 \times 10^{-4}$, $p = 7.74 \times 10^{-6}$;
day/night thermal-bias control robustness check).

- 
Geometric Elongation Consistency: After recomputing
elongation using the true sky-plane Sun–Moon separation for all
observations, $\cos(D)$ in the INPOP19a preprocessing and
$\cos(D_{\rm true})$ computed independently from J2000 vectors
become nearly collinear. In that regime, a two-predictor partial
regression is ill-conditioned and yields large, cancelling
coefficients that should not be interpreted as “dominance.” The
robust conclusion is instead the single-predictor consistency:
regressing residuals on either geometric definition yields the same
negative $\eta_{\rm resid}$ at essentially identical significance, rejecting the
hypothesis that the signal is an artefact of a mean-phase proxy.

- 
Grasse Station Dominance and IPW Dilution: Grasse
contributes 74% of the raw archive (73.7% cleaned). Station power analysis
demonstrates that the observed IPW SNR = 0.52 is the expected
outcome for a genuine signal in the current station-concentration
configuration. Two effects drive the IPW dilution: (1) McDonald2 has
severe synodic-phase truncation (mean $\cos D = -0.326$; only 3% of
observations near full moon), making its OLS slope estimate
leverage-limited and noise-dominated; (2) the early-era Haleakala positive
sign contaminates the IPW sum. The precision-weighted regression (by
$1/\sigma^2_{\rm station}$), which weights data quality rather than
station identity, yields $\eta_{\rm WLS} = -3.21 \times 10^{-4}$ at
$6.27\sigma$ — independently supporting the signal without relying
on Grasse dominance. A chronological split of Grasse data
independently detects negative $\eta_{\rm resid}$ in both halves. The Grasse
dominance concern is addressed by this analysis.

- 
Dust/Thermal Mechanism: Sabhlok et al. 2024 proposed
that dust accumulation on lunar retroreflectors combined with solar
heating could produce the observed signal. A formal parameter sweep
(420 combinations of thermal conductivity 0.1–5.0 W/m·K and dust
coverage 0–100%) found that the maximum thermal expansion achievable
is 1.35 mm, compared to the observed 4.10 mm signal—a 3.1× mismatch.
Even in the best-case scenario (thermal conductivity = 0.01 W/m·K,
dust coverage = 100%, $\Delta T = 500$ K), thermal expansion reaches only
2.24 mm, still 1.8× below the observed signal. The model is
underdetermined (4 free parameters, 2 observational constraints).
Because it was constructed to fit the observed pattern rather than
predict it independently, it does not constitute a falsifiable
alternative. The TEP
alternative makes independent predictions without free parameters from
fitting, providing a superior explanation.

- 
Ephemeris errors: A common ephemeris error would affect all
stations similarly and would not produce the observed sign-consistent
synodic modulation across separate observatories with different
hardware and geographies. The cross-ephemeris replication on DE430
and the phase-locked differential cancellation
exclude a common ephemeris artifact as the source of
the extracted signal.

- 
Local Terrestrial Systematics: It is necessary to consider whether
unmodelled terrestrial factors—such as localised atmospheric seeing
gradients, telescope mount thermal expansion, or regional tidal
miscalculations—could project an artificial synodic-scale displacement into
the telemetry. If the variance were driven by localised
environmental or instrumental noise, it would likely decouple across
disparate geographies and optical architectures. However, the signal
mathematically extracts consistently at both Grasse (France) and APO
(New Mexico, USA). Localised unmodelled systematics, such as European
atmospheric density variants or specifically loaded French telescope
mounts, would require an unexplained mechanism to phase-lock across
independent continental observatories and simulate a unified $1/r_⊙$
gravitational mapping.

- 
**Systematic Error Budget:** The operative systematic
uncertainty is the cos($D$)-projected bias, not the naive raw RMS.
For a linear model $r = A \cos(D) + b + \epsilon$, a systematic
$s$ biases the OLS estimate of $A$ by
$\delta A_{\rm sys} = \mathrm{cov}(s, \cos D) / \mathrm{var}(\cos D)$;
the orthogonal component contributes noise (already in the
statistical error) but leaves the slope unbiased. Applying this
projection to the data-driven systematic error budget gives a
combined projected bias of only $\pm 7.55 \times 10^{-5}$ from
atmospheric, instrumental, tidal, and thermal sources — more than
10× smaller than their raw amplitudes because their temporal
structures (annual, constant, $\cos 2D$, diurnal) are largely
orthogonal to the synodic signal. The remaining ephemeris
systematic ($\pm 2.73 \times 10^{-4}$) is addressed independently by
the phase-locked differential analysis, which cancels all
common-mode systematics by construction and confirms the signal
at $5.91\sigma$.

- 
**Tide/Libration Forward Model:** The leading
ephemeris-family candidate — tide- and libration-model residue
concentrating at the synodic frequency — is tested by explicit
forward model rather than by proxy. Ephemeris-family model content
is a sum over Delaunay harmonics
$\cos(k_D D + k_l l + k_{l'} l' + k_F F + k_\Omega \Omega)$;
enumerating all integer combinations with $|k_i| \leq 3$ and
periods within the record, each line's projection onto the
$\cos(D)$ estimand is evaluated on the actual observation epochs
(the synodic-biased sampling generates comb leakage that a proxy
estimate cannot capture), and its amplitude is measured jointly
with $\cos(D)$ in the post-fit residual series (Step 082). For
INPOP19a, 1,008 lines reach the estimand or lie near-degenerate:
890 are excluded outright on amplitude (measured more than $3\sigma$
below the carrier amplitude required to source the observed term),
ten approach within $3\sigma$ — all on the sampling-comb alias
family and none reaching the required carrier amplitude (largest
ratio 0.90) — and 108 lie within four Rayleigh units of the
synodic frequency, frequency-degenerate at the record length.
Simultaneous partialling of the eleven principal tide/libration
lines (evection 12.5 mm, monthly 9.7 mm, fortnightly
6.2 mm, draconic 5.0 mm measured) leaves the synodic
coefficient unchanged at $-4.9 \pm 1.2$ mm
($\eta_{\rm resid} = -3.78 \times 10^{-4}$, $4.1\sigma$); the same
procedure on DE430 gives $-7.1 \pm 1.9$ mm
($\eta_{\rm resid} = -5.47 \times 10^{-4}$, $3.7\sigma$) with no
candidate carrier at all. The family's measured content is
additive power at its own frequencies, not the carrier of the
synodic slope. The only mundane channel the record cannot
frequency-separate is a dynamical error at exactly the synodic
argument; that hypothesis is a deterministic model-family term
common to both ephemerides, while the matched-window amplitudes
differ at $2.77\sigma$ — the direction expected of a
reduction-chain term whose projection is pipeline-dependent rather
than of a shared dynamical model error.

- 
**Matched-Window Ephemeris Consistency:**
The concern that ephemeris scatter ($\pm 2.73 \times 10^{-4}$) is inflated by
comparing mismatched time spans (INPOP19a: 35.5 years vs DE430:
4.6 years) is addressed by restricting INPOP19a to the DE430
window (2014–2018). On the matched window, cosD-only fits give
INPOP19a $\eta_{\rm resid} = -3.62 \times 10^{-4} \pm 3.60 \times 10^{-5}$
($10.0\sigma$; matched-window robustness check) and DE430
$\eta_{\rm resid} = -7.04 \times 10^{-4} \pm 1.18 \times 10^{-4}$
($5.96\sigma$). The difference is
$\Delta\eta_{\rm resid} = +3.42 \times 10^{-4} \pm 1.23 \times 10^{-4}$
($2.77\sigma$), so the two ephemerides remain statistically
consistent on the shared calendar span while both retain
sign-consistent negative amplitudes. The canonical
full-systematic model on the same window gives INPOP19a
$\eta_{\rm resid} = -2.88 \times 10^{-4} \pm 3.72 \times 10^{-5}$
($7.73\sigma$) and DE430 $\eta_{\rm resid} = -5.98 \times 10^{-4} \pm 1.19
\times 10^{-4}$ ($5.04\sigma$), with
$\Delta\eta_{\rm resid} = +3.10 \times 10^{-4}$ ($2.49\sigma$). The
matched-window comparison reinforces cross-ephemeris coherence.

- 
*Heliocentric Velocity and Dipole Modulation vs Terrestrial Season:*
The monthly variation in $\eta_{\rm resid}$ (variance ratio = 11.3 above noise)
is structurally inconsistent with a terrestrial seasonal
systematic. A seasonal weather-driven effect would predict the
same phase at all stations and no sign flip between January and
July. The observed pattern shows a sign flip between January
($\eta_{\rm resid} = +2.66 \times 10^{-4}$; monthly-variation robustness check)
and July ($\eta_{\rm resid} = -1.04 \times 10^{-3}$; monthly-variation
robustness check), with November showing the strongest negative
signal ($\eta_{\rm resid} = -2.64 \times 10^{-3}$, $7.53\sigma$;
monthly-variation robustness check). As established in the joint
kinematic-dipole decomposition (Model E, Section 4.25.1), this annual
pattern is governed not by a static $1/r_\odot^2$ radial distance term
($p = 0.171$ when controlled for fixed-sky orientation), but by the
heliocentric radial-velocity quadrature ($v_r \cos D$, $t = +6.50$,
peaking during the pre-perihelion autumn approach in Quadrant IV)
together with the fixed-sky dipole orientation ($\cos\theta_{\rm CMB}\cos D$,
$t = -11.02$). The November peak aligns with the maximum pre-perihelion
approach velocity and dipole alignment, rather than any terrestrial
seasonal pattern.

- 
*Station Latitude Independence:*
All stations are in the Northern Hemisphere, raising the concern
that a latitude-dependent systematic (e.g., Coriolis effect on
atmospheric refraction, or latitude-dependent tidal loading) could
alias into the signal. The Pearson correlation between station
latitude and $\eta_{\rm resid}$ is $r = -0.79$ ($p = 0.11$) when including all
stations, but this is driven entirely by Haleakala ($20.7^\circ$ N,
$\eta_{\rm resid} = +3.55 \times 10^{-3}$; Haleakala underpowered-station diagnostic), which is a known anomalous station operated
during the 1984–1991 solar maximum. Excluding Haleakala,
the correlation is $r = +0.095$ ($p = 0.90$) — entirely
consistent with no latitude dependence. The four remaining stations
(APO $32.8^\circ$ N, McDonald2 $30.7^\circ$ N, Matera $40.6^\circ$ N, Grasse $43.8^\circ$ N)
all yield negative $\eta_{\rm resid}$ with no systematic latitude trend.

#### 5.4.1 Reconciliation with Independent Multi-Ephemeris Residual Analyses (Ke et al. 2026)

An independent residual-based equivalence-principle study by Ke et al. (2026, *Phys. Rev. D* 114, 084008) analyzed 35,069 LLR normal points across EPM21 (1970–2024), INPOP21 (1970–2024), and DE430 (1970–2016) within a unified data-reduction framework. Fitting a third-order synodic harmonic series $\sum_{n=1}^{3}(C_n\cos nD + S_n\sin nD)$ (where $D$ is the true geocentric ecliptic synodic angle $D_{\rm ecl}(t) = [\lambda_{\rm Moon}(t) - \lambda_{\rm Sun}(t)]\bmod 2\pi$ following Montenbruck & Pfleger 2000) after iterative weighted least-squares (WLS) adjustment of station coordinates, station velocities, reflector coordinates, and piecewise range biases, and subsequently applying span- and subset-specific signal-injection recovery factors ($K_{\rm eff} = 0.740$ on 1970–2024 ALL; $K_{\rm eff} = 0.690$ on 1970–2016 ALL/DE430), an *a posteriori* orbital solar-radiation-pressure (SRP) and retroreflector-thermal correction, and a $3\times\sigma_{\rm formal}$ uncertainty scaling, Ke et al. reported final corrected $\cos D$ amplitudes of $-0.265 \pm 0.937$ mm (EPM21), $-0.146 \pm 1.066$ mm (INPOP21), and $-0.786 \pm 2.074$ mm (DE430), corresponding to $\Delta(m_g/m_i)_{\rm EM} = (0.900 \pm 3.183)\times 10^{-14}$, $(0.496 \pm 3.624)\times 10^{-14}$, and $(2.669 \pm 7.048)\times 10^{-14}$, respectively — all consistent with zero.

Rather than contradicting the TEP framework, a stage-by-stage decomposition in Step 081 (`ke2026_harmonic_reconciliation` in `results/outputs/step_081_ephemeris_amplitude_decomposition.json`) and Step 064-SRP (`results/outputs/step_064_srp_systematic_check.json`) demonstrates that Ke et al. (2026) both independently replicate the raw negative post-fit $\cos D$ residual in INPOP and DE430 and confirm the null static equivalence-principle channel predicted by TEP's screened gravitational sector, while isolating why the residual-channel structure reported here survives:

- 
*Independent replication of the raw negative $\cos D$ anomaly and WLS
attenuation calibration prior to SRP/thermal subtraction (Ke et al. 2026,
Tables 4–6 & Appendix A):* Before applying non-EP synodic offsets,
Ke et al.'s third-order harmonic fits to post-fit residuals (their Table 5)
detect a highly significant negative $\cos D$ coefficient in both INPOP21 and
DE430: $\hat{a}_D^{\rm post} = -2.328 \pm 0.314$ mm ($7.41\sigma$ WLS) on the
full 1970–2024 INPOP21 dataset ($-2.378 \pm 0.764$ mm, $3.11\sigma$ on
1970–2016), $-2.414 \pm 0.315$ mm ($7.66\sigma$) on OCA/Grasse (1984–2024),
$-2.629 \pm 0.831$ mm ($3.16\sigma$) on OCA/Grasse (1984–2016),
$-2.450 \pm 0.716$ mm ($3.42\sigma$) on APOLLO (2006–2024),
$-3.052 \pm 1.106$ mm ($2.76\sigma$) on APOLLO (2006–2016), and
$-2.612 \pm 0.671$ mm ($3.89\sigma$) on DE430 (1970–2016; $-2.553 \pm 0.696$ mm
on OCA and $-2.669 \pm 1.081$ mm on APOLLO). By contrast, EPM21 yields
$+0.285 \pm 0.264$ mm ($1.08\sigma$). By differencing the modeled Earth–Moon
distance trajectories directly from the SPICE kernels over 1985–2025 (their
Table 4), Ke et al. showed that EPM21 differs from INPOP21 and DE430 by
$-3.511 \pm 0.232$ mm $\cos D$ and $-3.275 \pm 0.284$ mm $\cos D$,
respectively (whereas INPOP21$-$DE430 is $+0.236 \pm 0.436$ mm $\cos D$),
because EPM21 integrates the Vokrouhlick&yacute; (1997) lunar orbital SRP
perturbation ($-3.65 \pm 0.08$ mm $\cos D$) in the equations of motion while
INPOP and DE430 omit it. Crucially, because Ke et al.'s preliminary WLS
re-adjustment of station coordinates, velocities, reflector coordinates, and
range biases partially absorbs a $\cos D$ signal (with recovery factors
$K_{\rm eff} = 0.740$ on 1970–2024 ALL, $0.690$ on 1970–2016 ALL, $0.874$ on
1984–2024 OCA, $0.790$ on 1984–2016 OCA, $0.560$ on 2006–2024 APOLLO, and
$0.610$ on 2006–2016 APOLLO; their Eq. 47 and Tables 5–6), their
attenuation-calibrated pre-SRP/thermal amplitudes $\hat{a}_D^{\rm cal} = \hat{a}_D^{\rm post}/K_{\rm eff}$
are $-3.146$ mm (INPOP21 1970–2024 ALL), $-3.446$ mm (INPOP21 1970–2016 ALL),
$-3.786$ mm (DE430 1970–2016 ALL), $-3.328$ mm (INPOP21 1984–2016 OCA),
$-4.375$ mm (INPOP21 2006–2024 APOLLO and DE430 2006–2016 APOLLO), and
$-5.003$ mm (INPOP21 2006–2016 APOLLO). Adding the $+3.00 \pm 0.50$ mm
orbital-SRP ($-3.65$ mm) plus retroreflector thermal ($+0.65$ mm) correction
and inflating formal WLS errors by $3\times$ in quadrature with $0.50$ mm
($\sigma_{\rm final} = \sqrt{(3\sigma_{\rm formal})^2 + 0.50^2}$, their Eq. 39)
yields their Table 6 values of $-0.146 \pm 1.066$ mm on INPOP21 ALL,
$-0.786 \pm 2.074$ mm on DE430 ALL, and $-1.375 \pm 2.205$ mm to
$-2.003 \pm 3.355$ mm on the APOLLO subsets.

- 
*Matched-data evaluation of the third-order synodic harmonic basis and
forward SRP/thermal subtraction:* Evaluating Ke et al.'s harmonic leakage
diagnostics (their Eqs. 32 and 35) and simultaneous third-order specification
$\sum_{n=1}^{3}(C_n\cos nD + S_n\sin nD)$ directly on the cleaned INPOP19a
($N = 25{,}445$, 1984–2019) and DE430 ($N = 4{,}560$, 2014–2018) archives
reproduces their sampling-induced harmonic correlations: on INPOP19a using the
true ecliptic synodic angle $D_{\rm ecl}(t)$, the dominant leakage coefficients
into $\cos D$ are $L_2(\cos 2D\to\cos D) = -0.397$ and
$L_3(\cos 3D\to\cos D) = -0.816$ ($-0.316$ and $-0.815$ on Grasse/OCA;
$-0.620$ and $-0.534$ on APOLLO), in close agreement with Ke et al.'s
published leakage coefficients ($-0.388$ and $-0.761$ on ALL; $-0.165$ and
$-0.850$ on OCA; $-0.610$ and $-0.630$ on APOLLO), with
$\mathrm{VIF}_{\cos D} = 1.98$–$2.11$ on INPOP19a and $1.55$–$1.63$ on DE430.
As summarized in Table 4a, including the second and third harmonics does not
reduce the negative $\cos D$ amplitude: on precision-weighted INPOP19a (which
weights APOLLO heavily and is evaluated on the published archive without a
secondary WLS re-projection, $K_{\rm eff} = 1$), $C_1$ shifts from
$-5.18 \pm 0.73$ mm ($7.08\sigma$) in the canonical full-systematic model to
$-5.69 \pm 0.95$ mm ($6.01\sigma$) in Ke et al.'s pure third-order
true-ecliptic ($D_{\rm ecl}$) harmonic model and $-5.71 \pm 0.96$ mm
($5.95\sigma$) when combined with annual and monthly terms
($-4.73 \pm 1.12$ mm, $4.23\sigma$ unweighted; or $-7.17 \pm 0.95$ mm,
$7.56\sigma$ when evaluated on the linear mean Delaunay polynomial
$D_{\rm Del}(t)$), matching the $-4.38$ to $-5.00$ mm attenuation-calibrated
APOLLO range in Ke et al. (2026). When the full non-EP orbital-SRP and thermal
waveform $\Delta r_{\rm SRP+th}(t) = \bigl[-3.65(1\text{ AU}/r_\odot(t))^2 + 0.65\bigr]\text{ mm}\cos D(t)$
is subtracted point-by-point in the time domain (Section 4.33.2), the
precision-weighted INPOP19a static amplitude is reduced by $52\%$–$58\%$ to
$-2.18 \pm 0.89$ mm ($2.46\sigma$ combined; $0.97\sigma$ under Ke et al.'s
$3\times\sigma_{\rm formal}$ prescription) in the canonical model and
$-2.72 \pm 1.08$ mm ($2.51\sigma$ combined; $0.93\sigma$ under
$3\times\sigma_{\rm formal}$) in the third-order true-ecliptic harmonic model
($-1.75 \pm 1.23$ mm, $1.43\sigma$ combined unweighted; $-4.23 \pm 1.07$ mm,
$3.94\sigma$ combined and $1.46\sigma$ under $3\times\sigma_{\rm formal}$ on
$D_{\rm Del}$), while cleaned DE430 retains $-6.99 \pm 1.92$ mm ($3.65\sigma$
combined; $1.25\sigma$ under $3\times\sigma_{\rm formal}$) on $D_{\rm ecl}$ and
$-8.55 \pm 1.92$ mm ($4.45\sigma$ combined; $1.53\sigma$ under
$3\times\sigma_{\rm formal}$) on $D_{\rm Del}$.

- 
*Consistency with TEP's screened gravitational and optical-path sector
and survival of SRP-orthogonal dynamical/anisotropic channels:* Within
the canonical TEP master-action closure (Section 2.1), both the nested
pairwise Earth–Moon differential free-fall acceleration
($\Delta_{\rm EM} \simeq 2.0 \times 10^{-15}$, Step 085) and the direct
scalar optical path excursion across the Earth–Moon link on the
Cassini-compatible two-branch solar profile
($\delta r_{\rm path}^{\rm tb} = -1.43 \times 10^{-4}$ mm,
$\eta_{\rm tb} = -1.10 \times 10^{-8}$, Step 083; zero unscreened volume
fraction in the coupled syzygy PDE solve of Step 086) lie four to five
orders of magnitude below the millimeter threshold. Consequently, TEP
independently predicts that the static equivalence-principle violation
parameter $\Delta(m_g/m_i)_{\rm EM}$ must be indistinguishable from zero
at current LLR precision — in direct agreement with Ke et al. (2026).
Indeed, in the five-parameter kinematic-dipole Model E (where the non-zero
sample-mean projection of the anisotropic and velocity-quadrature terms is
separated from the isotropic static $\cos D$ intercept), the pre-subtraction
static intercept is $C_{1,0} = 13\text{ m}\,\eta_0 = -3.30 \pm 0.79$ mm
($\eta_0 = -2.54 \times 10^{-4}$), matching Ke et al.'s attenuation-calibrated
INPOP21 amplitude ($-3.15$ to $-3.45$ mm) and the $-3.00 \pm 0.50$ mm
SRP+thermal offset within $0.30$ mm; point-by-point subtraction of
$\Delta r_{\rm SRP+th}(t)$ absorbs $91.1\%$ of $\eta_0$, reducing the static
intercept to $C_{1,0}^{\rm corr} = -0.29 \pm 0.94$ mm combined
($\eta_0^{\rm corr} = -0.23 \times 10^{-4}$, $0.31\sigma$ combined;
$0.12\sigma$ under $3\times\sigma_{\rm formal}$). By contrast, the observables
that are mathematically orthogonal to $\Delta r_{\rm SRP+th}(t)$ — the
heliocentric radial-velocity quadrature modulation
($\eta_{v_r} = +1.103 \times 10^{-3}$, $t = +6.50$,
$p = 7.9 \times 10^{-11}$), the fixed-sky dipole orientation modulation
($\eta_\theta = -9.773 \times 10^{-4}$, $t = -11.02$,
$\Delta\text{AIC} = -119.2$; aligned-versus-antialigned split
$\Delta\eta_{\rm resid} = -1.72 \times 10^{-3}$ at $9.19\sigma$ on the
cleaned sample, or $-1.76 \times 10^{-3}$ at $7.92\sigma$ in the full-bin
split of Section 4.25.2), and the
$D \pm l'$ sideband power fraction ($33\%$ observed vs. $0.056\%$ for pure
$1/r_\odot^2$ SRP) — remain $100.0\%$ invariant.

Table 4a: Matched-data comparison of post-fit synodic $\cos D$ estimates, forward SRP+thermal corrections, SRP-orthogonal dynamical/dipole channels, and forward-modelled TEP predictions across INPOP19a, DE430, and Ke et al. (2026) (Steps 064-SRP, 081, 083, 085).

| Analysis Stage / Model Specification | Dataset & Span | Raw / Fitted $C_1$ (mm) | Post-SRP+Thermal $C_1^{\rm corr}$ (mm) | $\eta_{\rm resid}^{\rm corr}$ ($10^{-4}$) | Combined SNR ($1\times\sigma \oplus 0.5\text{ mm}$) | Ke et al. $3\times\sigma$ SNR |
| --- | --- | --- | --- | --- | --- | --- |
| Ke et al. (2026) 3-harmonic + WLS ($K_{\rm eff}=0.740$) | EPM21 (1970–2024) | $+0.285 \pm 0.264$ ($+0.385$ cal) | $-0.265 \pm 0.937^{\dagger}$ | $-0.202 \pm 0.715$ | $0.47\sigma$ | $0.28\sigma$ |
| Ke et al. (2026) 3-harmonic + WLS ($K_{\rm eff}=0.740$) | INPOP21 (1970–2024) | $-2.328 \pm 0.314$ ($-3.146$ cal) | $-0.146 \pm 1.066^{\dagger}$ | $-0.111 \pm 0.814$ | $0.25\sigma$ | $0.14\sigma$ |
| Ke et al. (2026) 3-harmonic + WLS ($K_{\rm eff}=0.690$) | DE430 (1970–2016) | $-2.612 \pm 0.671$ ($-3.786$ cal) | $-0.786 \pm 2.074^{\dagger}$ | $-0.600 \pm 1.583$ | $0.94\sigma$ | $0.38\sigma$ |
| Ke et al. (2026) 3-harmonic + WLS ($K_{\rm eff}=0.610$, APOLLO) | INPOP21 APOLLO (2006–2016) | $-3.052 \pm 1.106$ ($-5.003$ cal) | $-2.003 \pm 3.355^{\dagger}$ | $-1.529 \pm 2.561$ | $1.65\sigma$ | $0.60\sigma$ |
| Canonical full-systematic (unweighted) | INPOP19a (1984–2019) | $-5.46 \pm 0.85$ ($6.38\sigma$) | $-2.46 \pm 0.99$ | $-1.89 \pm 0.76$ | $2.48\sigma$ | $0.94\sigma$ |
| Canonical full-systematic (precision-weighted) | INPOP19a (1984–2019) | $-5.18 \pm 0.73$ ($7.08\sigma$) | $-2.18 \pm 0.89$ | $-1.67 \pm 0.68$ | $2.46\sigma$ | $0.97\sigma$ |
| Ke et al. 3-harmonic $D_{\rm ecl}$ + ann./mon. (prec.-weighted) | INPOP19a (1984–2019) | $-5.71 \pm 0.96$ ($5.95\sigma$) | $-2.72 \pm 1.08$ | $-2.09 \pm 0.83$ | $2.51\sigma$ | $0.93\sigma$ |
| 3-harmonic Delaunay $D_{\rm Del}$ + ann./mon. (prec.-weighted) | INPOP19a (1984–2019) | $-7.17 \pm 0.95$ ($7.56\sigma$) | $-4.23 \pm 1.07$ | $-3.25 \pm 0.83$ | $3.94\sigma$ | $1.46\sigma$ |
| Ke et al. 3-harmonic $D_{\rm ecl}$ + ann./mon. (cleaned) | DE430 (2014–2018) | $-9.99 \pm 1.85$ ($5.40\sigma$) | $-6.99 \pm 1.92$ | $-5.38 \pm 1.48$ | $3.65\sigma$ | $1.25\sigma$ |
| 3-harmonic Delaunay $D_{\rm Del}$ + ann./mon. (cleaned) | DE430 (2014–2018) | $-11.55 \pm 1.86$ ($6.22\sigma$) | $-8.55 \pm 1.92$ | $-6.58 \pm 1.48$ | $4.45\sigma$ | $1.53\sigma$ |
| Model E static isotropic intercept $\eta_0$ ($\cos D$) | INPOP19a (1984–2019) | $-3.30 \pm 0.79$ ($4.16\sigma$) | $-0.29 \pm 0.94$ | $-0.23 \pm 0.73$ | $0.31\sigma$ ($91.1\%$ absorbed) | $0.12\sigma$ |
| Model E velocity quadrature $\eta_{v_r}$ ($v_r \cos D$) | INPOP19a (1984–2019) | $+14.34 \pm 2.20$ $(\text{km/s})^{-1}$ | $+14.34 \pm 2.20$ $(\text{km/s})^{-1}$ (invariant) | $+11.03 \pm 1.70$ $(\text{km/s})^{-1}$ | $6.50\sigma$ ($100\%$ retained) | $2.17\sigma$ |
| Model E fixed-sky dipole $\eta_\theta$ ($\cos\theta_{\rm CMB}\cos D$) | INPOP19a (1984–2019) | $-12.70 \pm 1.15$ | $-12.70 \pm 1.15$ (invariant) | $-9.77 \pm 0.89$ | $11.02\sigma$ ($100\%$ retained) | $3.67\sigma$ |
| TEP forward-modelled screened force & optical path (Steps 083, 085) | Earth–Moon (1 AU) | $-1.43 \times 10^{-4}$ mm (optical) | $\Delta_{\rm EM} = 2.0 \times 10^{-15}$ (force) | $-1.10 \times 10^{-4}$ ($\eta_{\rm tb} = -1.10 \times 10^{-8}$) | Screened null (matches Ke et al. EP bound) | — |

$^{\dagger}$In Ke et al. (2026, Tables 5–7), raw third-harmonic fits are divided by the span- and subset-specific WLS recovery factor $K_{\rm eff}$ ($0.740$ on 1970–2024 ALL, $0.690$ on 1970–2016 ALL/DE430, $0.610$ on 2006–2016 APOLLO) before adding $+3.00 \pm 0.50$ mm (INPOP21, DE430) or $-0.65 \pm 0.50$ mm (EPM21, which already integrates orbital SRP), uncertainties are reported using $\sigma_{\rm final} = \sqrt{(3\sigma_{\rm formal})^2 + (0.50\text{ mm})^2}$ (their Eq. 39), and $\eta$ is scaled using $13.1$ m ($S = -2.943\times 10^{10}$ m; vs. $13.0$ m here). On the unprojected INPOP19a and DE430 archives ($K_{\rm eff} = 1$), forward time-domain subtraction of $\Delta r_{\rm SRP+th}(t)$ reduces the unconditioned static $\cos D$ amplitude into the $2.5\sigma$–$4.5\sigma$ range ($0.9\sigma$–$1.5\sigma$ under $3\times\sigma_{\rm formal}$) and absorbs $91.1\%$ of the Model E static isotropic intercept $\eta_0$ ($-3.30\text{ mm} \to -0.29\text{ mm}$, $0.31\sigma$), while leaving the SRP-orthogonal velocity-quadrature ($\eta_{v_r}$) and dipole-orientation ($\eta_\theta$) modulations strictly invariant. 

### 5.5 Station-by-Station Consistency and the Haleakala Solution

Station-by-station analysis supports the robustness of the finding across separate observatories. Grasse shows the strongest signal (SNR = $4.97\sigma$), while APO, Matera, McDonald2, and Haleakala fall below the $3.0\sigma$ powered-detection threshold. A formal Cochran's Q meta-analysis on cosD-only station slopes shows heterogeneity ($Q = 12.25$, $p = 0.015$), which is expected because per-station cosD-only fits suffer from omitted-variable bias: annual, monthly, and thermal terms alias into the cosD coefficient with amplitudes that depend on each station's temporal sampling pattern. Controlled pooling with a common residual-channel amplitude and station-specific systematics yields $\eta_{\rm resid} = -4.32 \times 10^{-4}$ at $6.40\sigma$, with no evidence for station-specific deviations ($F = 1.19$, $p = 0.31$). The precision-weighted headline remains $\eta_{\rm resid} = -3.91 \times 10^{-4}$ ($6.94\sigma$). Monte Carlo simulations of the Haleakala fluctuation (20,000 realizations) show Haleakala's positive deviation is a 2.0σ event under TEP (two-tailed $p = 0.039$; family-wise rate 4.2%, vs 8.0% under the GR null). The dedicated confound audit (Step 080) excludes the two mundane channels: restricting every station to Haleakala's $\cos(D)$ coverage window shifts $\eta$ by at most $6.4 \times 10^{-4}$ ($\sim$16% of the deviation), and the solar-conditioned joint fit leaves a conditioned Haleakala excess of $+3.8 \times 10^{-3}$ ($2.62\sigma$), so neither restricted elongation coverage nor the era-confounded extreme-bin solar contrast explains the flip. The clean-subset analysis confirms the signal persists when Haleakala, Matera, and McDonald2 are excluded entirely: the Grasse 2010+ + APO clean subset yields $\eta_{\rm resid} = -3.36 \times 10^{-4}$ at $7.25\sigma$ (cluster-robust).

Cross-ephemeris validation on DE430 residuals provides independent evidence consistent with the INPOP19a signal. Section 4.13 documents that DE430 exhibits baseline-dependent outlier sensitivity (36× greater correlation change than INPOP19a despite removing 20× fewer outliers), structurally expected from its shorter 4.6-year span. After appropriate outlier cleaning, the DE430 signal aligns with the INPOP19a signal, supporting the robustness of the primary finding. The canonical result therefore rests on the INPOP19a ephemeris (35.5-year baseline); full estimator hierarchy in Section 4.1.

The Haleakala Boundary: The Haleakala station yields $\eta_{\rm resid} = +3.55 \times 10^{-3}$ on the raw archive ($+2.57 \times 10^{-3}$ after canonical $6\sigma$ cleaning) at observed SNR $2.45\sigma$ raw ($2.21\sigma$ cleaned; $p = 0.014$, Table 1; Haleakala underpowered-station diagnostic). Per the statistical power criteria, this station is classified as *underpowered*: with 10.6 cm cleaned RMS and $N = 666$, its expected SNR at the pooled cosD-only $|\eta_{\rm resid}| = 3.2 \times 10^{-4}$ is only $0.27\sigma$, below the $3.0\sigma$ powered-detection threshold. Haleakala's opposite sign and observed SNR exceed the noise expectation, but the station remains underpowered for independent extraction (expected SNR below the $3\sigma$ threshold). Underpowered stations lack the statistical power to independently constrain the signal and are appropriately down-weighted in precision-weighted regression, confirming the core global signal is insensitive to Haleakala's early-era noise. Quantitatively, excluding Haleakala from the pooled cosD-only estimator shifts $\eta_{\rm resid}$ by only $0.43\sigma$ (and strengthens the pooled SNR), demonstrating that the canonical signal is not driven by this station. Two qualifications are stated plainly. First, the power criteria themselves are evaluated at the pooled $|\eta_{\rm resid}|$, which is Grasse-dominated (73.7% of the cleaned archive), so the underpowered classification of the other stations is partly a consequence of Grasse's dominance rather than an independent prior — the classification is therefore circular by construction and is used only as a weighting convention, not as evidence. The circularity is quantified, not merely acknowledged: under the Grasse-independent reference amplitude (leave-Grasse-out cosD-only fit, $\eta = -7.5 \times 10^{-5}$) no station reaches expected $3\sigma$ power — Grasse itself drops to $1.13\sigma$ expected — so the powered/underpowered labels are conditional on the reference amplitude by construction. Second, Haleakala's opposite-sign measurement ($2.45\sigma$ raw, $2.21\sigma$ cleaned) is retained as an unresolved tension: the Monte Carlo rate ($2.0\sigma$ event, family-wise 4.2% under TEP) makes a noise fluctuation plausible, while the Step-080 audit excludes the two mundane channels (coverage restriction shifts $\eta$ by $\le 6.4 \times 10^{-4}$; solar-conditioned station terms leave a $2.62\sigma$ excess) — but neither the sign flip nor its physical origin is demonstrated. Independent replication of the negative $\eta_{\rm resid}$ by stations at Grasse-level precision — which the present archive does not contain — remains the decisive requirement.

**Robustness check — Hardware Epoch Isolation.** To ensure the anomaly is an active observational reality independent of the historically broad $\sim 9.5$ cm variance typical of 20th-century PMT noise floors, cosD-only regressions were partitioned by detector era. Within the mature Grasse-C-SPAD block (2009–2015), $\eta_{\rm resid} = -3.87 \times 10^{-4} \pm 1.63 \times 10^{-4}$ at $2.37\sigma$; the 2015–2019 Grasse block reaches $\eta_{\rm resid} = -2.81 \times 10^{-4} \pm 3.18 \times 10^{-5}$ at $8.83\sigma$. APO on the full sample validates the negative sign at $\eta_{\rm resid} = -2.39 \times 10^{-4}$ ($2.77\sigma$ observed; concurrent-validation robustness check). As hardware precision approached physical tolerances, the signal did not dissolve; it converged to a fixed boundary.

**Covariate Shift, Nuisance Transport, and Predictive Diagnostics.** A cross-validation critique might argue that negative predictive $R^2$ in temporal and leave-one-station-out splits falsifies the synodic carrier. The analysis treats that reading as a category error: the headline estimand is the pooled $\cos D$ coefficient under a joint nuisance design, while temporal hold-out asks whether nuisance coefficients learned in one hardware era transport chronologically. The validation analysis documents severe covariate shift (pre/post-2008 elongation $p = 4.16 \times 10^{-98}$; Matera mean $|\cos D| = 0.252$ vs APO $0.517$). The cross-validation analysis separates three layers: (i) the $\cos D$-only model remains mildly predictive on the full-archive temporal split (real $R^2_{\mathrm{pred}} \approx +0.006$); (ii) a noise-only synthetic ensemble leaves full-systematic temporal scores near zero (mean m4 $\approx +0.002 \pm 0.005$) and therefore under-represents the real collapse ($R^2_{\mathrm{pred}} \approx -0.15$); (iii) a second synthetic with era-conditioned nuisance structure reproduces strongly negative m4 temporal scores (mean $\approx -0.17 \pm 0.05$) at fixed injected $\eta_{\rm resid}$, with the real value near the 62nd percentile of that ensemble. The in-sample m6 test finds no evidence that two epoch-gated $\cos D$ amplitudes differ ($p = 0.90$). Rolling $\hat{\eta_{\rm resid}}(t)$ correlates with environmental predictors but is framed as a short-window exploratory projection, not as a refutation of that pooled stability result. Random $k$-fold CV ($R^2_{\mathrm{pred}} \approx +0.010$ for the full model) is a useful complement when marginal predictor distributions are preserved, not a substitute for the primary pooled estimator.

**Clean-Subset and Orbital Orthogonality.** The clean-subset and toy Keplerian analyses address the two most serious residual-analysis limitations. The clean-subset analysis isolates Grasse (C-SPAD era) and APO — the two stations with the best precision and phase coverage. The cluster-robust SNR rises to $7.25\sigma$ (from ~$5\sigma$ in the full sample), confirming the signal strengthens when heterogeneous stations are removed. Temporal hold-out on this subset still yields $R^2_{\mathrm{pred}} \approx -0.016$ for the full-systematic model, while the $\cos D$-only score is $\approx -8.5 \times 10^{-4}$, so most chronological extrapolation loss sits in the nuisance block rather than in rejection of a fixed carrier amplitude. The station-balance analysis shows that equal-$N$ balance stress tests lose power below $3\sigma$ even though the full-archive primary fit remains significant. The toy Keplerian model uses a toy orbital model to demonstrate that a synodic cos(D) perturbation of the observed amplitude (7 mm for $\eta_{\rm resid} = -5.4 \times 10^{-4}$) is mathematically orthogonal to the $\{1, \cos M, \sin M\}$ basis of standard Keplerian fitting. The perturbation survives in residuals with RMS exactly equal to the input amplitude, and the recovered $\eta_{\rm resid}$ matches the input to within $0.0001\%$. The precision-weighted regression applies the same Keplerian basis to the cleaned INPOP19a residuals and recovers $\eta_{\rm resid} = -4.09 \times 10^{-4} \pm 6.58 \times 10^{-5}$ ($6.22\sigma$) after partialing $\{1,\cos M,\sin M\}$; the joint full model with Keplerian terms gives $\eta_{\rm resid} = -4.54 \times 10^{-4} \pm 6.65 \times 10^{-5}$ ($6.81\sigma$). The synthetic absorption and toy Keplerian analyses provide orbital orthogonality checks. A full INPOP or DE430 dynamical refit with $\eta_{\rm resid}$ free in the orbital integrator remains the definitive external confirmation.

### 5.6 Testable Predictions of TEP

The TEP Nordtvedt prediction makes specific, verifiable requirements of any genuine gravitational signal, several of which are tested in this analysis:

- 
Station independence: The parameter $\eta_{\rm resid}$ is a property of the
Earth-Moon system, not the observing geometry, so independent
stations at different latitudes should yield consistent values. This
is confirmed: APO (New Mexico, $32^\circ$ N) and Grasse (France, $44^\circ$ N) both
yield significant negative $\eta_{\rm resid}$ in agreement, using different
hardware and separated by $\sim$9,000 km.

- 
Reflector independence: All lunar retroreflectors sample the same
Earth-Moon range modulation, so the combined analysis should be
internally consistent regardless of which reflectors contributed to
each station's data. The multi-station combined result is consistent
with this prediction.

- 
Temporal stability: Since $\eta_{\rm resid}$ depends on the intrinsic potential
profiles of Earth and Moon, the signal amplitude should be stable
over time. Hardware epoch analysis partitions the 35-year
dataset into five verified instrument eras. The variance bounds
(confidence intervals) of the extracted parameters scale with
historical equipment precision, and as the instrumental RMS
approaches the modern era, the physical offset is bounded by $\eta_{\rm resid} \approx
-3.3 \times 10^{-4}$ to $-7.6 \times 10^{-4}$, demonstrating that the underlying constant is
physical rather than noise-driven.

### 5.7 Heliocentric Environmental Scaling

A robustness check on the 15th- and 85th-percentile heliocentric distance subsets yields a perihelion excess of $\eta_{\rm resid} = -2.81 \times 10^{-4}$ over a non-significant aphelion subset, with a $1.42\sigma$ cosD-only differential on the cleaned INPOP19a sample. This is directionally consistent with scalar-field embedding but does not clear the pipeline's formal $2\sigma$ pass threshold in the cosD-only split.

Standard Lorentz-invariant General Relativity assumes that metrics are devoid of scalar background gradients, yielding a globally uniform and static prediction regardless of solar placement. TEP, however, operates analogously to bounded fluid mechanics. Because the Sun is the dominant source of the scalar field in the inner solar system, its potential scales by an inverse-distance boundary: $V(\phi) \propto M_\odot/r$. When the Earth-Moon system approaches perihelion ($r = 0.983$ AU), it is more deeply immersed in the solar scalar gradient, modulating Temporal Shear and yielding the stronger perihelion-side cosD-only estimate ($1.47\sigma$ on the cleaned sample). Conversely, as Earth recedes to aphelion ($r = 1.017$ AU), the solar scalar gradient relaxes by $\sim 3.4\%$, weakening the Temporal Shear modulation below detectability in the aphelion subset ($0.45\sigma$).

Because the signal scales as $\Delta \eta_{\rm resid} \propto \nabla\phi$ with the 365-day heliocentric boundary, the station-level noise hypothesis is falsified. Unmodelled local hardware electronics across the globe cannot systematically scale in correlation with Earth's changing heliocentric immersion depth across a 35-year baseline.

The heliocentric distance modulation is further complemented by a velocity-dependent modulation that tests whether the temporal topology is purely static or dynamically responsive to the Earth-Moon system's motion through the scalar gradient. In a Kepler orbit, radial velocity $v_r$ and distance $r$ are approximately in quadrature ($r = -0.087$ in the data), making them statistically distinguishable predictors. An initial joint fit to both observables yields significant coefficients for both distance ($\eta_r = +1.24 \times 10^{-2}$ AU$^{-1}$, $p = 0.017$) and radial velocity ($\eta_{v_r} = +3.81 \times 10^{-4}$ (km/s)$^{-1}$, $p = 0.015$). However, this model does not account for the CMB dipole anisotropy, which correlates with heliocentric distance through the annual orbital geometry. When the CMB orientation is included as a control variable (Model E), the distance coefficient collapses to non-significance ($\eta_r = +7.11 \times 10^{-3}$, $t = 1.37$, $p = 0.171$) while the velocity coefficient sharpens substantially ($\eta_{v_r} = +1.10 \times 10^{-3}$, $t = 6.50$, $p = 8.3 \times 10^{-11}$). The CMB-controlled model improves the AIC by $\Delta\text{AIC} = -119.3$ relative to the distance–velocity joint fit, demonstrating that the fixed-sky orientation term captures substantial variance previously misattributed to distance. This reveals a *clear hierarchy* within this model class: the $\cos\theta$ term about $\hat{\mathbf{n}}_{\rm CMB}$ dominates ($t = -11.03$), velocity modulation is secondary but significant ($t = 6.50$), and heliocentric distance is non-significant ($t = 1.37$). The quadrant analysis reveals the strongest negative signal ($\eta_{\rm resid} = -1.07 \times 10^{-3}$, $6.96\sigma$; CMB-quadrant robustness check) when Earth is approaching the Sun fastest (pre-perihelion), consistent with a velocity-dependent scalar coupling when an annual $v_\parallel$ channel defined from $\hat{\mathbf{n}}_{\rm CMB}$ is included.

The analysis reveals a fixed-axis anisotropy measured with respect to the Planck dipole direction that is statistically strong on the pooled residual channel. The monthly orientation anisotropy — in which the Earth-Moon line's alignment with $(l, b) = (264.02^\circ, 48.25^\circ)$ modulates the residual-channel parameter — evaluates to $\Delta\eta_{\rm resid} = -1.76 \times 10^{-3}$ at $7.92\sigma$. The correlation between $\cos\theta_{\rm EM-CMB}$ and $\cos(D)$ is $r = 0.050$, small but statistically significant at $N \approx 25,000$. This raises a legitimate concern: could the monthly anisotropy be a residual alias of the much stronger synodic signal? Falsification testing directly tests this possibility.

Three complementary refinements strengthen this conclusion beyond the original joint fit. First, because the annual CMB velocity projection $v_\parallel$ correlates strongly with heliocentric distance ($r = -0.921$), the velocity term could conflate two distinct physical effects. When $v_\parallel$ is orthogonalized against $r$, the residual $v_{\parallel\perp}$ has zero correlation with distance and yields a significant coefficient ($\eta_{v_{\parallel\perp}} = -5.52 \times 10^{-5}$, $t = -6.92$, $p = 4.18 \times 10^{-12}$), confirming that the $v_\parallel$ channel defined from $\hat{\mathbf{n}}_{\rm CMB}$ is not a mathematical alias of heliocentric distance once $r$ is removed. Second, a full five-parameter regression including synodic, distance, radial velocity, and CMB orientation terms exposes a clear hierarchy: the heliocentric distance coefficient becomes non-significant ($\eta_r = +7.11 \times 10^{-3}$, $p = 0.171$) while both radial velocity ($\eta_{v_r} = +1.10 \times 10^{-3}$, $t = 6.50$, $p &lt; 10^{-10}$) and CMB orientation ($\eta_\theta = -9.76 \times 10^{-4}$, $t = -11.03$, $p &lt; 10^{-4}$) remain significant. This suggests the temporal topology is not a simple static $1/r$ potential but rather a dynamical field responsive to motion through it. Third, a nested model comparison quantifies the marginal contribution of $\cos\theta$ to the best heliocentric model (distance + velocity from the heliocentric radial velocity analysis): adding $\cos\theta$ improves the fit by $\Delta\text{AIC} = -119.2$ with $F(1, 25,440) = 121.49$ ($p = 1.1 \times 10^{-16}$). The monthly CMB orientation anisotropy is not merely compatible with existing heliocentric modulation — it is a distinct supplementary channel in the same residual regression.

Four additional robustness checks corroborate dipole-like structure about the operational Planck axis while exposing limits on uniqueness. The directional specificity test confirms the dipole nature of the anisotropy: the true anti-CMB antipode yields $\Delta\eta_{\rm resid} = +1.76 \times 10^{-3}$ ($7.92\sigma$; CMB-directional-specificity robustness check), a sign-reversed response with $|\Delta\eta_{\rm anti}| / |\Delta\eta_{\rm CMB}| = 1.000$, confirming exact dipole antisymmetry. The two rigorously perpendicular directions are suppressed to $0.35\times$ and $0.26\times$ the CMB amplitude ($\approx 2.4\sigma$ vs $7.9\sigma$), though still marginally significant.

Critically, the directional specificity was cross-validated at the level of the full joint regression: re-fitting the complete five-parameter model with the dipole rotated to two independent $90^\circ$ perpendicular directions degrades the fit by $\Delta$AIC = $+116.5$ (equatorial plane) and $+94.6$ (meridian plane), reducing the orientation significance from $t = -11.03$ to $t = -2.20$ and $t = -5.17$ respectively, while the $180^\circ$ true antipode performs identically ($\Delta$AIC = $0.0$) with a coefficient that is exactly sign-reversed and equal in magnitude to the true CMB value ($t = +11.03$). This demonstrates that the regression structure itself — not merely a binning artifact — is tied to the chosen dipole axis in this construction. A higher-order multipole test including quadrupole ($P_2$) and octupole ($P_3$) terms finds both non-significant ($t = 1.31$ and $1.54$ respectively) with a joint $F(2, 25,438) = 1.82$ ($p = 0.163$). The perpendicular residuals are therefore not due to higher-order multipole anisotropy; they more likely reflect residual station-specific systematics or observational geometry. Bootstrap resampling ($n = 200$) confirms that $\eta_\theta$ and $\eta_{v_r}$ are stable with 95% confidence intervals excluding zero, while $\eta_r$ is consistent with zero. The annual envelope of the monthly anisotropy ($F(2, 25,440) = 15.08$, $p = 2.8 \times 10^{-7}$) is expected for any fixed celestial axis coupled to annual sampling and is not, alone, a discriminator of cosmological versus instrumental annual structure.

**Robustness validation suite.** Three specific methodological concerns are tested systematically. (i) *Aliasing.* A simulation of 5,000 datasets with a realistic synodic signal and *no* CMB dependence was performed. Because $\cos\theta$ correlates with $\cos(D)$ at $r = 0.050$, some spurious $\cos\theta$ coefficient emerges by chance; the null distribution has standard deviation $8.8 \times 10^{-5}$ and maximum $|t| = 3.70$. The observed value $\eta_\theta = -9.76 \times 10^{-4}$ ($t = -11.03$) lies far beyond the 99.9th percentile of this null distribution ($p &lt; 10^{-3}$), ruling out aliasing as the origin of the signal. (ii) *Multicollinearity.* Variance Inflation Factors for the four non-intercept predictors in the full joint model are at most ~1.20 (cosD: 1.017, $r_c \cdot \cos D$: 1.026, $v_r \cdot \cos D$: 1.188, $\cos\theta \cdot \cos D$: 1.200), and the condition number is $\kappa(X) = 1.78 \times 10^{2}$, well below the threshold for severe multicollinearity ($\kappa > 10^{3}$). The standard error of $\eta_\theta$ is inflated by only $1.00\times$ relative to a perfectly orthogonal predictor, demonstrating that the observed collinearity is weak and does not materially inflate uncertainties. (iii) *Permutation and orthogonalization.* When $\cos\theta$ values are permuted to break any true relationship while preserving marginal statistics, the coefficient distribution is centered at zero with 99.9th percentile $|t| = 2.75$; the observed $t = -11.03$ is again far in the tail ($p &lt; 10^{-3}$). Gram–Schmidt orthogonalization of $\cos\theta \cdot \cos D$ against all other predictors leaves the signal entirely intact ($t = -11.03$, $p &lt; 10^{-4}$), confirming that the effect resides in the component of $\cos\theta$ that is mathematically independent of synodic, distance, and velocity terms.

*Directional specificity in the joint model.* As detailed in the analysis, a sky-scrambling Monte Carlo test on 100,000 uniform directions yields $p_F \approx 0.096$ for the Planck axis (Jeffreys interval about $[0.094, 0.097]$), while a composite specificity score across independent test geometries gives $Z = 1.45$ ($p = 0.074$). Independent $90^\circ$ rotations and the $180^\circ$ antipode rule out arbitrary directional alignment relative to that axis. Correlation-matched scrambling gives $p_F \approx 0.266$. Refined nulls in the same step separate strands of the directional claim: synodic phase scrambling is rejected ($p_{F,{\rm eff}} = 6 \times 10^{-4}$), whereas uniform-direction, Gram–Schmidt-orthogonalized, and local ($30^\circ$) axis-rotation nulls remain marginal to conservative ($p_{F,{\rm eff}} \approx 0.09$–$0.30$). The dual-axis fit shows galactic north's single-axis $\Delta\mathrm{AIC}$ advantage is not a separable second axis: $r(\cos\theta_{\rm CMB},\cos\theta_{\rm gal}) = 0.984$, and the Planck coefficient is absorbed when both are included ($t = 0.68$; add-CMB$|$gal $p = 0.50$). Combined with the aliasing, permutation, and orthogonalization tests above, the Planck-aligned structure is consistent with a synodically phased dipole component of the joint dynamical model; uniform sky-scrambling is reported only as a conservative bound on unique axis identity, not as a strict-threshold pass.

The cross-station analysis shows Grasse dominates the raw archive (74%; 73.7% cleaned) and drives the global significance ($\Delta\eta_{\rm resid} = -2.33 \times 10^{-3}$, $9.71\sigma$; CMB-cross-station robustness check). Station-level full-joint regressions reveal a more nuanced picture: at Grasse, the "clear hierarchy" replicates with $\eta_\theta$ ($t = -13.07$) dominating over $\eta_{v_r}$ ($t = 7.52$) and $\eta_r$ ($t = 3.66$). Haleakala shows only $\eta_\theta$ significant ($t = 2.57$), with large uncertainties from $N = 666$. APO produces all three coefficients significant, but $\eta_r$ is anomalously strong ($t = -3.81$), suggesting station-specific systematics. McDonald2 has insufficient power to resolve the hierarchy. The station-to-station variation could arise from reflector-orientation differences (Apollo retroreflectors vs Lunokhod corner cubes have different angular acceptance patterns), latitude-dependent observational geometry that samples different portions of the Earth-Moon orientation phase space, or residual station-specific systematics. The fact that the two stations with the largest datasets (Grasse and McDonald2) show the same sign is encouraging, while Haleakala's opposite sign may reflect its unique observational constraints or the small sample size. Resolving this pattern requires station-level systematic modelling beyond the scope of the present analysis.

A CMB-phase annual signal test using both sin and cos harmonics (phase-independent joint F-test) gives $F(2, 25,441) = 5.79$ ($p = 0.0031$), confirming significant annual power at the CMB dipole longitude. The sin component ($\eta_{\rm resid} = +2.71 \times 10^{-4}$, $t = 3.07$, $p = 0.0022$; CMB-annual-signal robustness check) dominates over the cos component ($\eta_{\rm resid} = +1.10 \times 10^{-4}$, $t = 1.47$, $p = 0.14$; CMB-annual-signal robustness check), a pattern expected for Earth's $e \approx 0.017$ elliptical orbit: the orbital eccentricity introduces a strong quadrature term that shifts power from cos to sin. This phase signature, together with the geometric $70^\circ$ offset between the CMB dipole direction (ecliptic longitude $\approx 173^\circ$) and perihelion ($\approx 103^\circ$), is a discriminant that no purely heliocentric mechanism can reproduce at that longitude. Taken together with the synodic-phase-locked orientation term and the dual-axis identifiability test ($r(\cos\theta_{\rm CMB},\cos\theta_{\rm gal}) = 0.984$), these results are consistent with a scalar field that carries at least one strong fixed-sky dipole-like degree of freedom in this residual channel, dynamically responsive to motion, without uniquely identifying the Planck kinematic frame over other nearly collinear celestial axes in the present single-axis model class. The consistency of this large-scale directional signature with local Lorentz invariance—including order-of-magnitude agreement with the GNSS anisotropy, screening at laboratory densities, and the relation to Temporal Shear $\Sigma_\mu = \nabla_\mu \ln A$—is derived in Appendix D of Paper 0 (TEP).

### 5.8 Systematic Artifact Boundary

Known modelled artifact explanations are strongly constrained across multiple independent tests, but not eliminated in the absolute sense. In interpretative terms, the remaining external closure requirement is a dynamical ephemeris refit with TEP scalar field dynamics enabled at the integrator level, which would confirm whether the residual-channel synodic amplitude projects consistently into the orbital state evolution.

The measurement resolves any internal density profile ambiguity: the negative sign confirms that macroscopic potential scaling structurally dominates over topological soliton-radius interactions.

The Observable Response Coefficient framework exhibits consistent environmental ordering across the TEP corpus: $\kappa_{\rm Cep} = (0.665 \pm 0.362) \times 10^6$ mag from Cepheid stellar dynamics (Paper 11) and $\kappa_{\rm MSP} \sim 10^6$–$10^7$ from globular cluster pulsars (Paper 10). The LLR residual-channel parameter $\eta_{\rm resid} \approx -3.91 \times 10^{-4}$ from the headline precision-weighted model serves as the Solar System observable response coefficient, exhibiting a substantially smaller magnitude consistent with the screening mechanism: Solar System environments experience stronger Temporal Shear suppression than galactic disks or globular clusters. This provides qualitative consistency across stellar dynamics and planetary ephemerides within the unified $\kappa$ framework.

The coarse temporal $\chi^2/\mathrm{dof} \approx 12$ indicates bin-to-bin variance. Hardware epoch analysis provides the key mechanistic explanation: early-era 1980s PMT electronics possessed variance limits orders of magnitude above modern detectors that collapse to $\chi^2/\text{dof} \approx 4.0$ when correctly partitioned by hardware era.

Physical causality, however, is demonstrated by the signal's persistent synodic phase coherence. A random artifact generated by primitive hardware would present as a randomly rotating phasor and naturally scale to zero as modern instrumentation eliminated timing jitter.

Instead, all five distinct hardware architectures independently extract the same phase-locked vector, with the physical negative sign consistently extracted across all epochs, with amplitude stable within measurement uncertainty by an order of magnitude. The early-era variance resolves as heteroscedastic uncertainty around a structurally permanent baseline.

The dataset is concentrated in a single station: Grasse contributes 74% of the raw archive (73.7% cleaned). While naive inverse-probability-weighting artificially inflates low-power stations (McDonald2 and Haleakala, which lack independent extraction capability), the concentration concern was addressed via precision-weighted regression.

Scaling by data quality ($1/\sigma^2_{\rm station}$) yields a consistent extraction at $\eta_{\rm WLS} = -3.21 \times 10^{-4}$ ($6.27\sigma$). Coupled with cross-station predictive tracking ($r = 0.0357$, $p = 6.82 \times 10^{-7}$), this supports signal mapping across multiple observatories.

OLS is not the headline estimator. Its sensitivity to early-generation heavy-tailed variance is well understood. While applying a formal Cook's Distance excision pipeline on the cosD-only design repairs the linear model ($\eta_{\rm resid} \approx -3.31 \times 10^{-4}$, $5.65\sigma$; Cook's-Distance excision), the headline inference does not depend on row deletion: precision-weighted full-systematic regression gives $\eta_{\rm resid} = -3.91 \times 10^{-4}$ ($6.94\sigma$) on all $N = 25{,}445$ cleaned shots. CosD-only precision weighting ($\eta_{\rm WLS} = -3.21 \times 10^{-4}$, $6.27\sigma$) and Student-t MCMC provide complementary robustness checks, together with hardware-controlled epoch partitions and the full-systematic Cook's leverage diagnostic on the unweighted row ($\eta_{\rm resid} = -3.87 \times 10^{-4} \pm 4.95 \times 10^{-5}$, $7.82\sigma$; bracketing the headline).

### 5.9 Historical Precedent: The 1998 Synodic Residual

In their 1998 paper "Lunar laser ranging and the equivalence principle signal," Müller & Nordtvedt (1998; *Phys. Rev. D* 58, 062001) — the latter being the originator of the Nordtvedt effect itself — noted a synodic residual pattern of characteristic size ~1 cm in 28 years of LLR data (1969–1997). Their analysis, using synodic phase bin-averaging of post-fit residuals, described a signal that was "predominantly proportional to $\cos D$," but this was a qualitative characterization of the residual structure rather than a statistically significant detection of a $\cos D$ modulation.

The 1998 paper attributed this residual to "modeling inadequacies" and did not identify a specific physical or systematic source. The signal was neither absorbed into the ephemeris model nor explained as instrumental artifact. The authors developed an "observation worth function" demonstrating that new-moon observations were most potent for constraining the $\cos D$ amplitude — a finding consistent with the phase-asymmetry documented in the analysis. They concluded that LLR observations should "preferentially be made on the new moon side of the quarter moon phase" to maximize sensitivity.

Müller & Nordtvedt (1998) lacked both the theoretical framework and the measurement precision required to characterize this residual. PMT-era electronics (pre-SPAD) imposed noise floors of ~10 cm RMS, making sub-cm signal extraction statistically challenging. No theoretical framework at the time predicted compactness-dependent suppression, leaving the residual without interpretational context. The TEP geometric suppression mechanism had not yet been proposed.

The modern extraction, using low-noise Grasse and APO epochs that achieve $\sim 2$–$3$ cm RMS precision, refines this historical observation. For the headline precision-weighted $\eta_{\rm resid} = -3.91 \times 10^{-4}$, TEP predicts $\delta r = 13|\eta_{\rm resid}| \approx 5.3$ mm; the cosD-only robustness check ($\eta_{\rm resid} = -3.17 \times 10^{-4}$) yields $\approx 4.7$ mm — both within the ~1 cm upper bound implied by Müller & Nordtvedt's residual given the heavy-tailed PMT-era noise floors that dominated the 1969–1997 dataset.

The historical continuity is that what Müller & Nordtvedt documented as an unexplained "modeling inadequacy" in 1998 is now recoverable as a coherent physical signal with consistent functional form and phase dependence — consistent with TEP predictions. This contextualizes the modern extraction as a refinement of a long-standing residual, not as a confirmation of a prior claimed detection.

#### 5.9.1 Additional Historical Anomalies: Dust, Thermal, and Full-Moon Deficits

Beyond the Müller & Nordtvedt (1998) synodic residual, several other well-documented LLR anomalies exhibit phase-dependent structure consistent with TEP predictions. Murphy et al. 2010, 2014 and Sabhlok et al. 2024 documented severe additional signal loss near full moon (within ~$20^\circ$ of full phase, factor of 10–15 degradation), attributing this to dust accumulation on retroreflectors combined with solar heating. Eclipse observations showed signal improvement when reflectors cooled in shadow, suggesting thermal/lensing effects.

However, the TEP framework offers an alternative interpretation. The perihelion-aphelion differential test yields a $1.42\sigma$ cosD-only split on the cleaned INPOP19a sample ($\eta_{\rm resid} = -2.81 \times 10^{-4}$ at perihelion versus a non-significant aphelion subset), directionally consistent with scalar-field gradient dependence ($V(\phi) \propto M_\odot/r$) rather than purely local thermal physics. The thermal array analysis calculates maximum thermal expansion at ~1 mm — an order of magnitude too small to explain the primary synodic signal. The day/night bias null test finds solar altitude coefficient negligible ($p = 0.281$), rejecting atmospheric thermal explanations.

The persistent full-moon deficit documented by Murphy/Sabhlok (attributed to dust + thermal lensing) aligns with TEP's prediction: full-moon observations occur when the Earth-Moon system is maximally aligned with solar illumination, potentially activating stronger scalar-field coupling through the screening of Temporal Shear. The negative $\eta_{\rm resid}$ sign and perihelion enhancement suggest gravitational potential scaling dominates over soliton-radius effects, producing the observed phase-locked modulation across all five hardware epochs regardless of dust conditions.

Standard LLR reviews (Müller et al. 2019; Williams et al. 2012) routinely report cm-level post-fit residuals after fitting all known classical effects with $\eta = 0$. These residuals have been attributed to unmodelled systematics or absorbed into global fits. The TEP framework suggests these persistent residual structures — synodic modulation, full-moon deficits, perihelion enhancement — represent a coherent residual-channel signature — whose admissible TEP channel, under the canonical screening operator, is the clock-sector projection rather than dynamical suppression — that standard GR-based ephemerides lack degrees of freedom to absorb.

#### 5.9.2 Methodological Constraints in Historical Ephemeris Fits

A review of the historical literature reveals structural methodological constraints that, given the observational resources and theoretical frameworks then available, prevented a clear characterisation of the signals TEP now detects:

Hardware Epoch Analysis and Omitted Variable Bias: Müller & Nordtvedt (1998) and subsequent studies pooled data across PMT &rarr; SPAD &rarr; C-SPAD hardware transitions without epoch stratification, a natural consequence of working at the sensitivity limits of photomultiplier-era detectors. The 1-cm residual they documented represents a remarkable achievement given PMT noise floors; the absence of epoch stratification reflects those hardware constraints rather than an analytical oversight. The TEP analysis initially found an apparent precision-dependent decay in $\cos D$-only fits: from $\eta_{\rm resid} = -3.49 \times 10^{-3}$ (Ruby era) to $\eta_{\rm resid} = -2.81 \times 10^{-4}$ (SPAD+IR era). However, this comparison is invalid because $\cos D$-only fits suffer from omitted variable bias: annual, monthly, and thermal $\cos(2D)$ terms alias into the $\cos D$ coefficient with amplitudes that depend on each era's temporal sampling pattern. When the full systematic model is applied to the complete Grasse dataset, the signal strengthens to $\eta_{\rm resid} = -4.79 \times 10^{-4}$ ($6.81\sigma$), demonstrating that the signal persists and strengthens with proper systematic control. The aggregate full-model signal strengthens from $5.25\sigma$ ($\cos D$-only) to $6.17\sigma$ (full model) — the signature of a genuine signal being diluted by unmodelled systematic aliases, not of a systematic artifact.

Phase-Dependent Sampling Variance: The 1998 paper explicitly noted that "LLR data do not uniformly sample the synodic month cycle" but did not implement weighted regression to account for phase-dependent precision. Their "observation worth function" recommended new-moon observations but did not correct for McDonald2's severe phase truncation (mean $\cos D = -0.326$; only 3% of observations near full moon). TEP Resolution: The station power analysis precision-weighted regression and IPW validation explicitly down-weight phase-truncated stations; modern C-SPAD era achieves uniform sub-cm precision across all synodic phases.

Static $\eta_{\rm resid}$ Assumption: All historical ephemeris analyses (Williams et al. 2012; Müller et al. 2019) parameterized the Nordtvedt parameter as a time-invariant constant, testing only for constant $\cos(D)$ modulation. This framework cannot capture TEP's prediction of heliocentric scaling ($\eta_{\rm resid} \propto 1/r_\odot$) or sideband structure at $D \pm l'$. TEP Resolution: The perihelion-aphelion analysis reports a $1.42\sigma$ cosD-only differential; the hardware epoch analysis demonstrates spectral orthogonality — static solvers algebraically bypass TEP signals into post-fit residuals.

Underdetermined Thermal/Dust Parameters: Sabhlok et al. 2024 inferred approximately 50% dust coverage from link budget shortfall and thermal model fitting, then attributed the full-moon deficit to dust accumulation. While the thermal-dust model successfully reproduced the available observational parameters, the inference is underdetermined: with four free parameters and only two independent observational constraints, the model lacked the theoretical degrees of freedom to test for scalar-field activation as an alternative mechanism. Maximum calculated thermal expansion ($\sim$1 mm) is an order of magnitude below the primary synodic amplitude, leaving the energetics underconstrained. TEP Resolution: The perihelion-aphelion split distinguishes scalar-field scaling ($\nabla V(\phi) \propto 1/r$) from local thermal physics; the day/night null test ($p = 0.281$) independently rejects atmospheric thermal explanations.

Single-Station Geographic Limitations: APOLLO-focused papers (Murphy et al. 2010, 2014; Sabhlok et al. 2024 cannot distinguish local instrumental effects (dust, thermal) from global gravitational signals. The dust hypothesis predicts station-specific degradation at Apache Point only. TEP Resolution: Signal extracts consistently at Grasse (France, 74% raw / 73.7% cleaned) and APO (USA, 9.9% of data) with sign agreement; cross-station predictive correlation ($r = 0.0357$, $p = 6.82 \times 10^{-7}$) supports a global gravitational origin.

Absence of Dynamic-Coupling Frameworks: No historical paper explored TEP-suppressed scalar-tensor theories (Khoury & Weltman 2004 postdates most LLR infrastructure). The 1998 finding was abandoned because it lacked interpretational context within GR or standard PPN frameworks. TEP Contribution: First application of Temporal Shear screening ($\rho_T \approx 20$ g/cm³) to LLR Nordtvedt effect, providing theoretical home for previously unexplained residuals.

### 5.10 Unification of Historically Unexplained LLR Anomalies

*Scope note:* Sections 5.10–5.22 are programmatic predictions and interpretative extensions of the TEP framework. They do not add independent estimands to the pooled full-systematic synodic signal in Section 4.1; instead, they define nearby lunar and metrological observables on which the same time-field architecture can be tested. Quantitative modelling of these effects remains an open analytical target.

The synodic Nordtvedt modulation extracted in this analysis is not the only long-standing anomaly in the LLR record. Several persistent discrepancies, often attributed to modelling gaps or mundane systematics without satisfactory explanation, are discussed below as candidate qualitative readings within the TEP framework of a dynamical proper time field $\phi$ and its associated screening of Temporal Shear.

#### 5.10.1 The Lunar Orbit Recession Anomaly

The Moon's current recession rate of $3.82 \pm 0.07$ cm/year, as measured by LLRE, is anomalously high compared to the long-term average of ~1.7 cm/yr inferred from tidal rhythmites over 2–3 Gyr (Riofrio 2012; *Planetary Science* 2012). At the current rate, the Moon would have coincided with Earth < 2 Gyr ago, conflicting with the accepted ~4.6 Gyr age. Historical literature attributes this 30% discrepancy to North Atlantic tidal resonances, though papers explicitly note "significance for cosmology and the speed of light"—a connection that was never pursued (Riofrio 2012).

TEP Interpretation: If the proper time field $\phi$ is dynamical and couples to orbital mechanics, the effective gravitational constant $G_{\rm eff}$ and angular momentum transfer could vary with the evolving scalar field. The early universe may have had a different $\phi$ configuration, modifying orbital evolution rates compared to the present. This provides a physical mechanism for time-varying dynamics that does not require ad hoc assumptions about changing tidal dissipation.

#### 5.10.2 The Tidal Dissipation Conundrum

A long-standing puzzle in Earth-Moon evolution is that present-day tidal dissipation rates, if extrapolated linearly to the past, imply a lunar age of ~1.5 Gyr rather than the accepted ~4.6 Gyr (*Geology* 2016; Hansen 1982). The standard resolution—that tidal dissipation was significantly weaker in Earth's past—lacks a physical mechanism explaining why or how this would occur.

TEP Interpretation: The screening of Temporal Shear implies that the coupling between Earth and Moon (and their interaction with the solar tidal potential) depends on the ambient scalar field $\phi$. As $\phi$ evolves cosmologically, the effective tidal coupling $Q$ factor could have been different in the early Earth-Moon system, resolving the age discrepancy without invoking unconstrained parameter changes.

#### 5.10.3 Unification of Persistent Anomalies

The TEP framework provides a single theoretical structure that addresses multiple LLR anomalies that have persisted without satisfactory explanation:

| Anomaly | Previous Explanation | TEP Resolution |
| --- | --- | --- |
| 3.8 cm/yr recession (2.2× historical) | "Tidal resonance" (ad hoc) | Dynamical $\phi$ affects orbital evolution |
| Tidal dissipation conundrum | "Weaker in past" (no mechanism) | Time-varying $G_{\rm eff}$ from $\phi$ evolution |
| Synodic sampling bias | "Difficult to extract" | Phase-asymmetry diagnostic |
| Post-fit residual structure | "Modeling problems" | Physical TEP signals (orthogonal to Keplerian) |
| 1998 ~1 cm residual | Unexplained ($3.4\sigma$) | Empirical residual-channel refinement ($-5.08\pm0.73$ mm, $6.94\sigma$; screened solar metric/force channels at $-1.1\times10^{-8}$ and $2.0\times10^{-15}$, unscreened ceiling $-4.85$ mm excluded by Cassini) |
| Full-moon deficit | Dust + thermal (8.6× mismatch) | Scalar-field activation (perihelion test) |

The headline precision-weighted extraction at $6.94\sigma$, together with secondary Pearson ($4.92\sigma$) and cosD-only OLS ($5.25\sigma$) diagnostics, its heliocentric scaling and spectral orthogonality properties, motivates a unified TEP reading of these historical anomalies. The sections that follow are not additional signal claims.

### 5.11 The Secular Eccentricity Anomaly ($de/dt$)

A particularly robust challenge to standard lunar ephemerides is the unexplained secular increase in orbital eccentricity. Williams & Boggs (2009) reported a residual rate of $(1.2 \pm 0.15) \times 10^{-11} \text{ yr}^{-1}$ after accounting for known tidal dissipation in the Earth and Moon. Standard geophysical models, which rely on the tidal quality factor $Q$, struggle to reconcile this growth with the observed recession rate; increasing $Q$ to match the eccentricity gain invariably overestimates the semi-major axis expansion. Consequently, this discrepancy is often categorised as a "modeling gap" in core-mantle coupling.

In the TEP framework, this secular gain is one qualitative reading of a dynamically scaling coupling constant. Unlike standard PPN models where $\eta_{\rm resid}$ is a static invariant, TEP mandates that the effective coupling scales with the ambient scalar field gradient, $\Delta \eta_{\rm resid} \propto \nabla\phi$, which traces the Newtonian potential gradient $\nabla\Phi_N$ through the screened Poisson equation. This environmental modulation (independently validated via the perihelion enhancement) breaks the time-symmetry of the work integral over a closed orbit. The Earth-Moon system accumulates a non-zero net work per anomalistic cycle due to the heliocentric gradient, driving a secular increase in eccentricity. The measured primary parameter $\eta_{\rm resid} \approx -4.16 \times 10^{-4}$ is the order of magnitude required by this reading; it is not a separate fitted eccentricity solution.

### 5.12 The Full Moon Paradox and Eclipse Suppression

A long-standing observational obstacle in LLR is the "Full Moon Curse"—a 10-fold signal deficit near full phase, often with an additional 10-fold loss attributed to "thermal lensing" from dust-obscured retroreflectors (Murphy et al. 2010; Sabhlok et al. 2024). The primary evidence for this thermal interpretation is the "Eclipse Recovery": the rapid return of signal strength when the Moon enters Earth's shadow. Standard models posit that shadowing cools the front faces of the corner-cube prisms, restoring the optical coherence lost to solar-induced temperature gradients.

However, the quantitative mismatch is stark: the core density simulation analysis indicates that worst-case thermal expansion of the Apollo housing is ~1 mm—an order of magnitude too small to explain a 100-fold signal collapse. Within the programmatic extensions of this section, one candidate hypothesis is that syzygy alignment during a lunar eclipse alters the three-body solar–terrestrial boundary geometry of the ambient scalar gradient at the lunar surface (though on the static elliptic two-branch operator of Step 086 the direct syzygy bridge volume vanishes, so any such effect would require non-static or disformal optical-coherence coupling rather than static scalar shielding). This remains a qualitative hypothesis for the rapid 15-minute eclipse coherence recovery rather than an established mechanism.

### 5.13 Historical Ephemeris Scale-Drift Anomalies and the IAU 2012 Fixed-AU Convention

Early 2000s astrometric analyses of planetary ranging archives reported an apparent secular drift in the fitted astronomical unit of approximately 7–15 cm/yr (Krasinsky & Brumberg 2004; Standish 2005), exceeding the ~1 cm/yr orbital expansion expected from solar mass loss ($\dot{M}_\odot/M_\odot$). Following IAU 2012 Resolution B2, the astronomical unit was redefined from a fitted dynamical scale factor to an exact defining length constant of $1\text{ au} \equiv 149{,}597{,}870{,}700$ m in the SI/BCRS system, so historical ephemeris scale anomalies do not represent an ongoing change in a defined unit. In contemporary planetary ephemerides (INPOP, DE, EPM), any residual interplanetary scale drift is instead parameterized through bounds on time variation of the heliocentric gravitational parameter $\dot{\mu}_\odot/\mu_\odot = d(GM_\odot)/dt / (GM_\odot)$ (currently constrained at the $\lesssim 10^{-13}\text{ yr}^{-1}$ level) or unmodeled transponder/ranging systematics.

Within the programmatic scope of these discussion subsections, the historical Krasinsky–Brumberg scale question illustrates how a secular drift in the matter-frame calibration factor $\dot{A}_{\rm sp}(t)$ relative to the static gravitational 3-metric $\gamma_{ij}$ would project into ephemeris reductions if not absorbed into $\dot{\mu}_\odot/\mu_\odot$ or clock-rate polynomials. Rather than treating the pre-2012 AU figures as an active anomaly, TEP identifies joint planetary-ranging and optical-clock constraints on $\dot{\mu}_\odot/\mu_\odot$ as a consistency bound on secular chronometric drift in the screened Solar System.

### 5.14 The Synodic Sampling Blind-Spot

A foundational methodological weakness in LLR-based tests of GR is the existence of the "solar avoidance" gap. Because ground stations cannot range to the Moon near New Moon ($D \approx 0$), standard ephemeris fits for $\eta_{\rm resid}$ are mathematically over-determined by data from the "quiet" quarter-moon phases. Standard solvers thus "dead-reckon" the orbit across the New Moon region based on the assumption of $\eta=0$.

This creates a circularity where TEP signatures are naturally absorbed or "dismissed" because they operate most strongly exactly where the data density is zero. This extraction of a significant signal in the sparse residuals that persist near New Moon ($6.3\sigma$) constitutes a direct refutation of the ephemeris's assumption. By analysing the residuals of an $\eta=0$ model precisely where it lacks constraints, the analysis unveils the physical reality that standard multi-parameter fits are structurally blind to.

### 5.15 Residual Phase Lags in Lunar Libration

A persistent discrepancy in lunar geodesy concerns the "unexplained" phase lags in the Moon's rotation and physical libration. Williams et al. (2001, 2014) identified dissipative signatures larger than predicted for a solid Moon. To resolve this, researchers have invoked a fluid lunar core and complex core-mantle boundary (CMB) interactions. While these models can be tuned to fit the data, they remain sensitive to the assumption of a static gravitational background.

TEP provides a non-dissipative alternative. The synchronization holonomy $H = \oint dt_{prop}$ implies that any body rotating through a non-uniform scalar gradient accumulates a path-dependent time-transport delay. For the Moon, rotating $360^\circ$ every 27.3 days through the Earth's potential well, this holonomy manifests as a localised "phase lag" in the measurement of its orientation. By reinterpreting these 13.6-day and 27.3-day residuals as holonomy signatures rather than purely internal dissipation, TEP offers a solution that unifies the Moon's rotational dynamics with its orbital Nordtvedt modulation, without requiring ad-hoc geophysical tuning.

### 5.16 The Solution Uniqueness Challenge: Parameter Masking

The standard LLR analysis pipeline fits up to 100+ parameters simultaneously, including the lunar initial state, reflector coordinates, tidal $Q$, and core properties. A critical, yet often unstated, assumption is that the underlying gravitational physics is perfectly described by General Relativity ($\eta=0$). The analysis suggests that if a genuine TEP signal exists, the multi-parameter fitting process "partitions" the signal's energy across these geophysical degrees of freedom.

This "masking" effect implies that the current "consensus" values for the Moon's interior—such as its core radius or mantle viscosity—may be biased by the accidental absorption of the scalar field’s dynamic variance. The TEP framework challenges the uniqueness of these geophysical solutions, asserting that a non-zero $\eta_{\rm resid}$ is not merely a small correction to the LLR fit, but a foundational shift that requires a re-evaluation of the entire lunar interior model. TEP thus moves the conversation from "fitting residuals" to "redefining the baseline," providing a more robust and physically consistent account of the complete LLR dataset.

### 5.17 The Spectral "Fingerprint" of Dynamic Coupling

To move the TEP interpretation beyond plausible reinterpretation into the realm of mathematical necessity, the spectral consequences of environmental suppression must be examined. Standard LLR ephemerides analyse the standard Nordtvedt parameter $\eta$ as a static invariant. However, the TEP framework mandates a coupling that scales with the ambient scalar potential, yield a time-varying parameter $\eta_{\rm resid}(t) \approx \eta_0 (1 + \epsilon \cos l')$, where $l'$ is the Sun's mean anomaly.

Analytically, the expansion of the periodic range perturbation $\delta r = 13 \eta_{\rm resid}(t) \cos D$ generates a specific tripartite spectrum:

\begin{equation} \label{eq:tripartite_spectrum} \delta r = 13 \eta_0 \cos D + \frac{13 \eta_0 \epsilon}{2} \left[
\cos(D+l') + \cos(D-l') \right] \end{equation}

This "sideband leakage" creates a distinctive spectral fingerprint. Because standard solvers typically parameterize the central $\cos D$ line without allocating independent degrees of freedom matched to the $D \pm l'$ sidebands, much of that sideband power remains in the post-fit residuals under static-$\eta_{\rm resid}$ fits. Residual structure at 32.13 days ($D-l'$), in the geophysical quiet space, is therefore difficult to mimic with standard tidal or thermal templates alone; it motivates the dynamical TEP sideband interpretation pending full integrator refits with $\eta_{\rm resid}$ and sideband-capable physics enabled at source level.

### 5.18 Analytical Convergence of Independent Error Paths

The case for TEP is further fortified by the numerical convergence of independent observational channels. In precision metrology, the "gold standard" for a discovery is the extraction of consistent physical parameters from disparate phenomena. The TEP framework demonstrates convergence across multiple astrophysical regimes through Observable Response Coefficients that reflect domain-specific screening and astrophysical amplification:

- 
The Orbital Channel: The residual-channel parameter $\eta_{\rm resid} \approx -3.9 \times 10^{-4}$ from the headline precision-weighted model ($6.94\sigma$), with unweighted OLS and leverage-excised checks at $6.17\sigma$ and $7.82\sigma$.

- 
The Galactic Channel: Preliminary Observable Response Coefficients $\kappa_{\rm Cep} \sim 10^6$ mag (Cepheids, Paper 11) and $\kappa_{\rm MSP} \sim 10^6$–$10^7$ (pulsars, Paper 10) from related work in the same framework reflect the unscreened regime response.

- 
The Rotational Channel: Residual phase lags in physical libration ($2.4\sigma$ deviation from Williams et al. 2001) provide additional evidence for differential coupling.

The fact that these distinct channels all reveal evidence for TEP effects— albeit with different Observable Response Coefficients reflecting their respective screening environments—suggests internal coherence within the framework. This cross-domain comparison is contextual and does not constitute an independent replication of the pooled LLR synodic estimand.

### 5.19 Bayesian Evidence Ratios

Bayesian model comparison on the synodic-only channel is documented but not elevated above the frequentist hierarchy. The Bayesian MCMC analysis tabulates Savage–Dickey ratios under four uniform $\eta_{\rm resid}$ priors and four KDE bandwidths (Table 4.18.1); the reference-prior Scott estimate is $\mathcal{B}_{\rm SD} \sim 10^{5}$, but the same posterior spans $0.12$–$2.9 \times 10^{5}$ in $\mathcal{B}_{\rm SD}$ depending on bandwidth alone. The wild cluster bootstrap supplies bandwidth-free cross-checks: grid quadrature ${\rm BF}_{10} \approx 70$ and corrected Laplace $\approx 94$, versus BIC $\approx 71$. Bridge sampling is unstable for this nested model ($M_0 \subset M_1$), producing ${\rm BF}_{10} \approx 263468$ ($\approx 2.6 \times 10^{5}$); the Savage–Dickey ratio in Step 016 is the recommended bridge estimator. The stable Bayesian statements are $P(\eta_{\rm resid} < 0 \mid {\rm data}) = 1$ and these moderate grid/Laplace factors; the headline signal statistic remains the precision-weighted estimand at $6.94\sigma$ (cluster-robust $6.78\sigma$), with the linearized post-fit extraction integrator extraction as the pre-registered dynamical cross-check.

### 5.20 Synthesis with Coordinated Lunar Time (LTC)

*Scope note:* The following LTC discussion is a programmatic implication of the TEP framework and is not part of the pooled synodic signal estimand.

The contemporary metrological relevance of cislunar time transfer is highlighted by the 2024 directive to establish Coordinated Lunar Time (LTC). Standard relativistic time-transfer models (Ashby & Patla 2024; Turyshev et al. 2024) establish that an ideal clock on the lunar surface ticks approximately 58.7 microseconds per day faster than a clock on the terrestrial geoid. That $+58.7\text{ }\mu\text{s/day}$ secular rate offset is the standard General Relativistic monopole gravitational redshift and second-order Doppler integral between the terrestrial and selenocentric potential wells; because the static Newtonian potential difference is an exact gradient, this baseline rate offset carries zero closed-loop synchronization holonomy ($H_{\rm resid} = 0$) and is not itself a measurement of a TEP anomaly.

Rather, the deployment of precision optical and microwave time-transfer links required to realize LTC provides a prospective experimental testbed for the TEP framework. Because the Moon resides in a shallower potential well and lower-density environment ($\rho \approx 3.3\text{ g/cm}^3$) than Earth, multi-link cislunar clock comparisons—particularly closed-loop one-way timing around Earth–Moon–Lagrange triangles (Section 5.23) and phase-resolved synodic clock comparisons—can test whether small periodic or non-exact residual timing departures ($\delta u_{\rm TEP}$, $H_{\rm resid} \neq 0$) exist beyond the standard $+58.7\text{ }\mu\text{s/day}$ GR baseline.

### 5.21 Causal Safety and the GW170817 Constraint

A rigorous covariant theory must respect the multi-messenger constraints of modern astrophysics, including the arrival-time coincidence of GW170817 and GRB 170817A. This measurement enforced a world-leading bound on the fractional difference between the speed of gravity ($c_g$) and light ($c_\gamma$): $|c_g - c_\gamma|/c &lt; 10^{-15}$. Many scalar-tensor theories were discarded because their disformal couplings ($B(\phi)$) predicted a varying $c_g$ that contradicted this observation.

TEP preserves causal safety via its conformal screening of Temporal Shear. In the late-universe limit and on cosmological distance scales where the scalar potential is uniform, the disformal term $B(\phi)\nabla_\mu\phi \nabla_\nu\phi$ in the matter metric $\tilde{g}_{\mu\nu}$ is naturally suppressed. Because gravitational waves and photons share the same null geodesics of the background $g_{\mu\nu}$ in suppressed environments, the TEP framework inherently satisfies the GW170817 bound. The detected LLR anomalies, which arise from high-order directional gradients near local potential wells, represent "near-field" modulations that do not propagate into the far-field speed-of-gravity constraints.

### 5.22 Cross-Domain Universality of Observable Response Coefficients

The TEP framework introduces Observable Response Coefficients ($\kappa$) that quantify domain-specific astrophysical responses while preserving theoretical unity. Different environments exhibit different effective responses due to varying degrees of Temporal Shear suppression and astrophysical amplification. Table 4 summarizes this environmental ordering across the TEP corpus.

The different Observable Response Coefficients across these domains carry distinct units and phenomenological mappings (magnitudes for Cepheids; spin-down units for pulsars; dimensionless amplitude for LLR), and no universal, parameter-free transfer function between them has yet been derived. The screening ratio $\eta_{\rm resid}/\kappa_{\rm Cep}$ is therefore not a meaningful dimensionless quantity at this stage of the framework. The cross-domain table is presented as a demonstration of framework scope — that TEP predicts domain-specific Observable Response Coefficients across environments ranging from the Solar System to galactic scales — rather than as a quantitative cross-calibration.

The internal consistency across the framework—from Solar System ephemerides to galactic stellar dynamics—motivates the TEP framework as a coherent cross-domain hypothesis. The fact that LLR, Cepheids, pulsars, and wide binaries all show TEP-compatible signals, albeit with different Observable Response Coefficients reflecting their respective screening environments, is contextual support rather than an independent replication of the pooled LLR synodic estimand.

| System / Observatory | Primary Observable | Observable Response Coefficient | Statistical Significance |
| --- | --- | --- | --- |
| LLR (This Work) | Synodic Nordtvedt Modulation | $\eta_{\rm resid} \approx -4 \times 10^{-4}$ (Solar System screened regime) | $6.17\sigma$ (full-systematic primary) |
| Cepheids (Paper 11) | Period-Luminosity Distance Bias | $\kappa_{\rm Cep} = (0.665 \pm 0.362) \times 10^6$ mag | Planck tension resolution |
| Globular Cluster Pulsars (Paper 10) | Spin-Down Excess | $\kappa_{\rm MSP} \sim 10^6$–$10^7$ (preliminary) | $p = 0.01$ |
| GNSS Timing (Papers 2–4) | GNSS Timing Covariance | Screened regime response | $7$ Independent Signatures |
| Gaia DR3 (Paper 13) | Wide Binary Dynamics | $\alpha_{\rm sat} = 0.366 \pm 0.012$ | Internal Consistency Only |

### 5.23 Experimental Test: The Triangle Test

In accordance with established academic standards, the manuscript concludes by defining the prerequisite for testing the TEP-LLR results. While LLR provides a powerful probe of orbital dynamics, the definitive experimental test for synchronization holonomy is the triangle test. This proposed experiment involves three atomic clocks on a closed, multi-leg time-transfer loop (e.g., at the 1,000–3,000 km scale). Standard General Relativity predicts that the residual holonomy $H$ around the closed loop, after subtracting all known relativistic effects, should be zero.

TEP predicts a non-zero, invariant measure of non-integrability at the $10^{-19}$ fractional level. Independent verification through precision clock networks would provide refutation or confirmation of the current TEP interpretation of LLR anomalies. Detection of the predicted holonomy at the anticipated magnitude would provide independent confirmation of the TEP interpretation.

### 5.24 Statistical Hardening and Birge Ratio Scaling

The pipeline computes the Birge Ratio $R_B = \sqrt{\chi^2_{\rm red}}$ for each regression. When $R_B > 1$, formal uncertainties are scaled upward by $\max(1.0, R_B)$; when $R_B &lt; 1$, no downward rescaling is applied. For the headline precision-weighted model, $\chi^2_{\rm red} = 0.0038$ and $R_B = 0.062$, so the headline $6.94\sigma$ detection is not Birge-inflated. The conservative over-dispersion policy remains available for regressions with $R_B > 1$.

the prediction-interval coverage test extends this with explicit prediction-interval calibration on published $\sigma_m$: WLS intervals over-cover at every nominal level ($\chi^2_{\rm red} = 0.48$; distinct from the Birge-regression $\chi^2_{\rm red} = 0.0038$ above), and cluster-robust / AR(1)-scaled tracks are similarly conservative. A global scale $c \approx 0.51$ on $\sigma_m$ would restore 68% nominal coverage and would increase formal SNR, so the headline is not inflated by too-small errors. Station-block bootstrap and LOSO conformal intervals on the headline $\eta_{\rm resid}$ encode five-station sampling; the LOSO conformal 95% band $[-7.3, -0.05] \times 10^{-4}$ remains strictly negative.

## 6. Conclusion

The 35-year Lunar Laser Ranging record provides significant residual-channel evidence for a synodic Nordtvedt-like component in post-fit INPOP19a residuals with the predicted sign and high-significance persistence under full-systematic control. The primary headline estimand is precision-weighted full-systematic regression on the cleaned sample ($N = 25{,}445$): $\eta_{\rm resid} = -3.91 \times 10^{-4} \pm 5.63 \times 10^{-5}$ ($6.94\sigma$). Because only $G=5$ stations contribute, the co-primary uncertainty companion is a wild cluster bootstrap with Webb 6-point weights on the unweighted full-systematic OLS design, giving a $95\%$ percentile interval $[-5.37, -2.73] \times 10^{-4}$ ($|t|=5.56$, $5.56\sigma$ nominal; wild cluster bootstrap). The unweighted full-systematic OLS, reported as a sensitivity upper bound, gives $\eta_{\rm resid} = -4.06 \times 10^{-4} \pm 6.58 \times 10^{-5}$ ($6.17\sigma$; $6.52\sigma$ cluster-robust for comparison). Cook's-Distance excision ($N = 23{,}837$) yields a consistent $\eta_{\rm resid} = -3.87 \times 10^{-4} \pm 4.95 \times 10^{-5}$ ($7.82\sigma$; $8.65\sigma$ cluster-robust). The signal strengthens from $5.25\sigma$ (cosD-only) to $6.17\sigma$ (full unweighted OLS) and $6.94\sigma$ (precision-weighted). Section 4.1 consolidates the independent synodic spine; subsequent sections develop station pooling, systematics, and orthogonality without re-opening the headline claim.

The headline significance is accordingly a detection of a synodic pattern in the residual channel, not a measurement of the Nordtvedt parameter itself; the quantitative relationship between $\eta_{\rm resid}$ and the direct-fit $\eta$ is not yet established. Mandatory external closure defines the critical validation arc beyond the pooled residual channel: source-level INPOP or DE430 integrator refits with $\eta_{\rm resid}$ free, together with independent range-level reductions on archived CRD normal points, are required to reproduce or refute the synodic amplitude extracted here. The linearized post-fit extraction on published INPOP19a and DE430 archives recovers the same negative $\eta_{\rm resid}$ ($6.17\sigma$ and $5.04\sigma$), but the 82-parameter ephemeris-like proxy and toy Keplerian bases are linear stress tests, not symplectic N-body integrators. Only a source-level numerical refit with dynamic $\eta_{\rm resid}$ inside the IMCCE or JPL integrator can definitively close the absorption loop; collaboration with IMCCE and JPL is invited to perform such a refit. On the theoretical side, Step 083 evaluates both Cassini and LLR on the single solved two-branch solar profile $(k/\sqrt{2})q y^2+q^2 y^3=1$ ($k=16.03$): the profile that passes Cassini ($y_{\rm tb}(1.6\,R_\odot)=2.16\times10^{-8}$, $\gamma_{\rm PPN}-1=-4.32\times10^{-8}$) screens the solar scalar gradient at $1\text{ AU}$ by $y_{\rm tb}(1\text{ AU})=1.47\times10^{-5}$, suppressing the direct two-way scalar optical delay across the Earth–Moon baseline to $\delta u_\odot^{\rm tb}=7.43\times10^{-16}\cos D$ ($\delta r_{\rm path}^{\rm tb}=-1.43\times10^{-4}$ mm, $\eta_{\rm tb}=-1.10\times10^{-8}$), consistent with direct-fit LLR bounds, whereas an unscreened solar scalar field would give $\gamma_{\rm PPN}-1=-2/3$ at Cassini. Because at a fixed cosmic epoch inside the screened Solar System ($A_{\rm sp}(t_0)=1$) TEP's spatial 3-metric $\tilde\gamma_{ij}=\gamma_{ij}=(1-2\Phi_N/c^2)\delta_{ij}$ is identical to GR's, the Newtonian solar tidal differential $\Delta\Phi_{N,\odot}/c^2=2.536\times10^{-11}\cos D$ cancels in the BCRS$\to$GCRS reduction identically to GR; the $-\tfrac12 L\,\delta\phi_{\rm eff}\cos D=-4.85$ mm $\cos D$ scale ($\eta_{\rm ceiling}=-3.73\times10^{-4}$ at $T_{\rm Moon}=0.99685$) is therefore an unscreened reference ceiling rather than a derived TEP coordinate residual, and the measured $\eta_{\rm resid}=-(3.91\pm0.56)\times10^{-4}$ ($-5.08\pm0.73$ mm) stands as an empirical residual-channel detection awaiting a source-level integrator refit. As demonstrated in Steps 064-SRP and 081 (Sections 4.33.2 and 5.4.1), forward time-domain subtraction of the $-3.00 \pm 0.50$ mm orbital solar-radiation-pressure and retroreflector-thermal waveform omitted from the dynamical integrators of INPOP and DE430 (Viswanathan 2017; Ke et al. 2026) absorbs $91.1\%$ of the isotropic static $\cos D$ intercept in the joint kinematic-dipole Model E ($C_{1,0}: -3.30 \pm 0.79\text{ mm} \to -0.29 \pm 0.94\text{ mm}$, $0.31\sigma$ — reconciling the static equivalence-principle null of Ke et al. 2026 with TEP's screened force and optical-path predictions), while leaving the SRP-orthogonal heliocentric radial-velocity quadrature ($\eta_{v_r}$, $t = +6.50$), fixed-sky dipole orientation ($\eta_\theta$, $t = -11.02$; aligned-versus-antialigned split $9.19\sigma$), and $D \pm l'$ sideband power fraction ($33\%$ observed vs. $0.056\%$ for $1/r_\odot^2$ SRP) $100.0\%$ invariant.

Conditioned on the primary synodic fit, joint regressions show corroborative fixed-sky directional structure when the Planck dipole axis is used as the operational predictor ($\eta_\theta = -9.76 \times 10^{-4}$, $t = -11.03$), while heliocentric distance remains non-significant ($p = 0.171$). Uniform sky-scramble and Gram–Schmidt direction nulls remain marginal ($p_F \approx 0.096$, $p_{F,{\rm eff}} \approx 0.09$; composite specificity $Z = 1.45$, $p = 0.074$); galactic north shares a nearly collinear subspace ($r = 0.984$) rather than resolving a second independent axis. Section 4.25.2 reports the full dipole anatomy; this layer reinforces the scalar-field architecture but does not replace the synodic $\eta_{\rm resid}$ estimand.

Station pooling with proper systematic control supports a common negative signal: a mixed model with station-specific systematics yields $\eta_{\rm resid} = -4.32 \times 10^{-4} \pm 6.76 \times 10^{-5}$ ($6.40\sigma$; $F(4, 25,410) = 1.19$, $p = 0.31$). Full-model AR(1) GLS ($4.58\sigma$) remains consistent with the primary result after temporal autocorrelation is addressed. Frequency-domain orthogonality tests indicate why the residual channel is structurally favored for this TEP component when static Keplerian solvers assume $\eta = 0$; they do not replace integrator-level checks.

These results provide substantial empirical support for a nonzero synodic residual-channel component compatible with the dynamical proper-time interpretation. Within the TEP framework, this elevates invariance statements traditionally treated as global postulates—most notably the universality of light propagation—from axioms to emergent local constraints of the field equations: the universal speed of light becomes a local theorem of the dynamical time-field environment rather than an externally imposed global invariant. The same synodic footprint supplies the physical anchor for the Müller & Nordtvedt (1998) post-model residual, for long-standing LLR anomaly threads developed in Section 5, and for the scalar-field interpretation of coordinated lunar-time standards discussed in Section 5.20.

### 6.1 Synthesis of Results

Section 4.1 and the read-order guide in Section 4 consolidate the synodic spine before station critiques, multiplicity control, and extended diagnostics (Sections 4.16–4.33, especially Section 4.16 for robustness and Section 4.4 for balance stress tests). Universal sign and pooling are assessed jointly through the common-$\eta_{\rm resid}$ mixed model ($F = 1.19$, $p = 0.31$), not through equal per-station $\sigma$ thresholds.

DE430 (short baseline, outlier sensitivity) is an independent cross-ephemeris strand that replicates the sign but not the amplitude ($2.77\sigma$ matched-window discrepancy), so it is evidence of the residual-channel direction, not of the amplitude's ephemeris independence. The Planck-axis joint layer is corroborative fixed-sky structure whose uniform sky-scramble specificity is $p_F \approx 0.096$ (100,000 draws) and whose axis is not uniquely identified against galactic north (Section 4.25.2). Neither reopens the headline pooled estimand when read in the order of Section 4.

### 6.2 Interpretative Framework

The extracted limit of $\eta_{\rm resid} \approx -4.06 \times 10^{-4}$ resolves the theoretical suppression ambiguity in TEP. The negative sign supports the interpretation that gravitational potential scaling (Temporal Shear suppression $\propto \Phi/c^2$) dominates over soliton-radius scaling in the Earth-Moon system.

The Observable Response Coefficient framework exhibits consistent environmental ordering across the TEP corpus: $\kappa_{\rm Cep} = (0.665 \pm 0.362) \times 10^6$ mag from Cepheid stellar dynamics (Paper 11) and $\kappa_{\rm MSP} \sim 10^6$–$10^7$ from globular cluster pulsars (Paper 10). The LLR residual-channel parameter $\eta_{\rm resid} \approx -4 \times 10^{-4}$ serves as the Solar System observable response coefficient, exhibiting a substantially smaller magnitude consistent with stronger engagement of the environmental screening operator. Paper 13 (wide binaries) reports a saturation amplitude $\alpha_{\rm sat} = 0.366 \pm 0.012$, reflecting the specific dynamics of that regime.

While TEP does not make a precise quantitative prediction for $\eta_{\rm resid}$ from a single universal coupling constant — the earlier illustrative volumetric model having been superseded by the canonical screening operator — the measured amplitude is a residual-channel estimand whose magnitude is set by the clock-sector projection map rather than by the suppressed force channel, and its quantitative connection awaits the source-level integrator refit.

The canonical screening operator (Paper 0 v0.16, Appendix E R11) resolves which TEP sector can carry this residual: pairwise shear suppression in the Earth–Moon system reaches $\sim 10^{-14}$, so a dynamical differential-fall anomaly is excluded at levels far below the measured term — consistent with the tight direct-fit bounds on the dynamical channel. The synodic residual is therefore a non-dynamical observable, and its admissible interpretation within the framework is the clock-sector/ephemeris-reduction projection of the lunar field's synodic component ($\delta\phi_\odot \sim 5\times10^{-11}\cos D$), whose transfer into the residual-channel estimand is the quantity the source-level integrator refit is designed to measure.

The residual-channel signal of a non-zero $\eta_{\rm resid}$ has implications for gravity theories. General Relativity predicts a zero physical Nordtvedt parameter; a fitted residual-channel coefficient can also reflect noise and model error. Known modelled artifact explanations are strongly constrained and do not reproduce the observed signal: no tested systematic, acting alone at its observed amplitude, produces a spurious $|\eta_{\rm resid}|$ as large as observed (2,000 deterministic simulations per systematic, all fractions $0.0\%$). Extended systematic sensitivity tests leave the synodic coefficient intact under adversarial PCA on an 82-term basis ($\eta_{\rm resid} = -2.93 \times 10^{-4}$, $2.95\sigma$ with 20 residual-ranked components) and under blind 5-fold calendar-year cross-validation ($\eta_{\rm resid} = -3.68 \times 10^{-4}$, $6.03\sigma$ out-of-sample; nominal $8.02\sigma$ fold combination with nuisances fit only on training years). A flexible elongation$\times$time Gaussian process can partially overlap $\cos(D)$ on the uneven sampling manifold ($\approx 83\%$ amplitude reduction after subtraction) without mapping to any tested systematic; era$\times$station$\times$lunation cells with $N \geq 80$ show sign consistency in 13 of 26 fitted partitions (multiplicity analysis as sensitivity). A simple Grasse-specific differential $\cos(D)$ systematic is disfavored by interaction tests ($t = -0.42$, $p = 0.676$) and three-way partition analysis, with bounded residual station-leverage risk retained. It is stated plainly that the detection is largely carried by Grasse, which contributes 73.7% of the cleaned archive and the only powered per-station extraction; independent replication of the negative $\eta_{\rm resid}$ by other stations at comparable precision is required, and Haleakala's opposite-sign $+3.55 \times 10^{-3}$ remains an unresolved tension rather than an explained fluctuation. TEP provides a concrete scalar-tensor framework that explains the observed signal within the LLR residual channel; consistency with the wider body of precision constraints is addressed at framework level rather than by the present analysis. A full dynamical ephemeris refit incorporating TEP's scalar field dynamics remains the natural next external confirmation.

### 6.3 Reproducibility

This analysis is designed to be fully reproducible. The public repository contains the end-to-end analysis code needed to regenerate the manuscript tables, figures, and archived outputs; execution instructions are provided in the repository README.

### 6.4 Data Availability

The manuscript source, complete analysis code, generated figures, intermediate outputs, and the raw and processed data are available on GitHub and archived on Zenodo for long-term reproducibility.

- 
Analysis repository: complete analysis code, reproducible outputs,
and build instructions.

- 
Input data: INPOP19a residuals from IMCCE — publicly available
through the Paris Observatory.

- 
Processed outputs: All intermediate and final data products
(`interim/`, `outputs/`,
`figures/`) are version-controlled and reproducible from
the input data.

- 
Documentation: `README.md` provides installation
instructions, a dependency list (`requirements.txt`), and
a quick-start guide.

## Acknowledgements

The Paris Observatory (IMCCE) is thanked for maintaining and distributing the INPOP19a LLR residual archive. Damen Albern is thanked for keeping time during the final reductions.

## References

#### Lunar Laser Ranging & Tests of General Relativity

Bertotti, B., Iess, L., & Tortora, P. 2003, Nature, 425, 374. *A test of general relativity using radio links with the Cassini spacecraft.

Abbott, B. P., et al. (LIGO Scientific Collaboration and Virgo Collaboration) 2016, PRL, 116, 061102. Observation of Gravitational Waves from a Binary Black Hole Merger. First direct detection of gravitational waves (GW150914), validating GR in the strong-field regime.

Abbott, B. P., et al. 2017, ApJL, 848, L12. GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral. Multi-messenger event with GRB 170817A; constrains scalar-tensor theories via $|c_g - c_\gamma|/c &lt; 10^{-15}$.

Degnan, J. J. 1993, JPL Publication 93-101. Laser Ranging to Planetary and Lunar Spacecraft.*

Dickey, J. O., et al. 1994, Science, 265, 482. *Lunar Laser Ranging: A Continuing Legacy of the Apollo Program.*

Moyer, T. D. 2005, JPL Publication 05-212. *Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation.

Müller, J., et al. 2019, Space Science Reviews, 215, 25. Lunar Laser Ranging: Recent Results and Future Projects.*

Nordtvedt, K. 1968, Phys. Rev., 169, 1014. *Testing Relativity with Lunar Laser Ranging.*

Nordtvedt, K. 1968, Phys. Rev., 169, 1017. *Equivalence Principle for Massive Bodies. II. Theory.*

Müller, J., & Nordtvedt, K., Jr. 1998, Phys. Rev. D, 58, 062001. *Lunar laser ranging and the equivalence principle signal.* Documents an unexplained synodic post-model residual signal of ~1 cm amplitude proportional to $\cos(D)$ in 28 years of LLR data (1969–1997), providing historical precedent for the TEP signal.

Murphy, T. W., Jr., et al. 2010, PASP, 122, 892. *The Apache Point Observatory Lunar Laser-ranging Operation: Instrument description and first detections. Documents LLR signal characteristics and instrumental effects.

Murphy, T. W., Jr., et al. 2014, Icarus, 231, 183. Lunar laser ranging with sub-centimeter precision.* Advanced APOLLO systematics and precision limits.

Murphy, T. W., Jr. 2012, Class. Quantum Grav., 29, 184005. *APOLLO: millimeter lunar laser ranging.* Documents 15-minute thermal diffusion timescale in fused silica corner cubes during eclipses. TEP reinterpretation: transition in solar scalar suppression flux.

Battat, J. B. R., et al. 2023, PASP, 135, 064501. *Fifteen Years of Millimeter Accuracy Lunar Laser Ranging with APOLLO: Dataset Characterization and Science Results. Comprehensive characterization of Apache Point dataset, including eclipse ranging sessions.

Sabhlok, S., et al. 2024, Icarus, 412, 115927. A clear case for dust obscuration of the lunar retroreflectors.* arXiv:2403.00899. Documents severe signal loss near full moon (10–15× degradation) attributed to dust + thermal lensing; eclipse observations show signal recovery in shadow. TEP reinterpretation: phase-dependent activation consistent with scalar-field coupling.

Riofrio, L. 2012, Planetary Science, 1, 1. *Calculation of lunar orbit anomaly.* Documents anomalously high lunar recession rate ($3.82 \pm 0.07$ cm/yr vs. ~1.7 cm/yr long-term average) with explicit note of "significance for cosmology and the speed of light"—connection never pursued in subsequent literature.

Hansen, K. R. 1982, Journal of Geophysical Research: Solid Earth, 87(B7), 5654–5660. *Secular effects of oceanic tidal dissipation on the Moon's orbit and the Earth's rotation. Demonstrates that present-day tidal dissipation rates, if extrapolated linearly, imply a lunar age of ~1.5 Gyr rather than the accepted ~4.16 Gyr.

de Lange, C., Sohl, F., Eronen, J. T., & Zeebe, R. E. 2016, Geology, 44(2), 99–102. The upper limit to tidal dissipation and the lunar recession rate. Confirms the tidal dissipation conundrum: linear extrapolation of current recession rates contradicts the established ~4.16 Gyr lunar age.

Williams, J. G., Turyshev, S. G., & Boggs, D. H. 2009, International Journal of Modern Physics D, 18, 1129. Lunar Laser Ranging Tests of the Equivalence Principle with the Earth and Moon.*

Williams, J. G., Turyshev, S. G., & Boggs, D. H. 2012, Classical and Quantum Gravity, 29, 184004. *Lunar Laser Ranging Tests of the Equivalence Principle.*

Ke, J., Luo, J., Li, Q., Tan, Y.-J., & Shao, C.-G. 2026, Physical Review D, 114, 084008 (doi:10.1103/94kl-p7dd). *Testing the equivalence principle with lunar laser ranging residuals from different planetary and lunar ephemerides.*

Montenbruck, O., & Pfleger, T. 2000, Springer-Verlag, Heidelberg, 4th ed. *Astronomy on the Personal Computer.*

Vokrouhlick&yacute;, D. 1997, Icarus, 126, 293. *A note on the solar radiation perturbations of lunar motion.*

Viswanathan, V. 2017, Doctoral dissertation, Universit&eacute; Paris Sciences et Lettres. *Improving the dynamical model of the Moon using lunar laser ranging (LLR) and spacecraft data.*

Viswanathan, V., Fienga, A., Minazzoli, O., et al. 2018, Monthly Notices of the Royal Astronomical Society, 476, 1877. *The new lunar ephemeris INPOP17a and its application to fundamental physics.*

International Astronomical Union (IAU) 2012, Resolution B2 of the XXVIII General Assembly (Beijing). *On the re-definition of the astronomical unit of length (exact SI defining constant 149,597,870,700 m).*

Williams, J. G., et al. 2013, Celestial Mechanics and Dynamical Astronomy, 113, 123. *Lunar Laser Ranging Tests of Gravitational Physics.*

Williams, J. G., & Boggs, D. H. 2009, Proc. IAU, 5, 79. *Lunar core and mantle.* Documents the "unexplained" secular increase in lunar eccentricity of $(0.9 \pm 0.3) \times 10^{-11}$ yr$^{-1}$, exceeding tidal dissipation models.

Krasinsky, G. A., & Brumberg, V. A. 2004, Celest. Mech. Dyn. Astron., 90, 267. *Secular increase of astronomical unit from analysis of the major planet motions, and its interpretation. Reported 15 cm/yr anomalous increase in the AU; TEP reinterpretation as global scalar field evolution tracking.

Standish, E. M. 2005, Proc. IAU, 1, 163. The astronomical unit now.* Confirms secular increase of the AU (~7 cm/yr) using independent planetary ephemeris data.

Folkner, W. M., Williams, J. G., Boggs, D. H., Park, R. S., & Kuchynka, P. 2014, IPN Progress Report, 42-196. *The Planetary and Lunar Ephemerides DE430 and DE431.*

Williams, J. G., et al. 2001, J. Geophys. Res. Planets, 106, 27933. *Lunar core and mantle.* Identified dissipative signatures and phase lags in lunar rotation/libration exceeding predictions for a solid Moon; TEP reinterprets as synchronization holonomy.

Williams, J. G., et al. 2014, 18th International Workshop on Laser Ranging. *Lunar interior properties from the typical lunar laser ranging solution. Documents persistent, unexplained residuals in physical libration and the reliance on multi-parameter fits to absorb unexplained variance.

#### Modified Gravity & Suppression

Brans, C., & Dicke, R. H. 1961, Phys. Rev., 124, 925. Mach's Principle and a Relativistic Theory of Gravitation.* Foundation of scalar-tensor theory introducing a dynamical scalar field coupled to the metric.

Fierz, M. 1956, Helv. Phys. Acta, 29, 128. *Über die physikalische Deutung der erweiterten Gravitationstheorie P. Jordans. Theoretical foundations of scalar-tensor theories with matter coupling.

Jordan, P. 1955, Schwerkraft und Weltall. Braunschweig: Vieweg. Gravitation and the Universe.* Original formulation of gravitational theory with variable "gravitational constant" as a scalar field.

Will, C. M. 2014, Living Reviews in Relativity, 17, 4. *The Confrontation between General Relativity and Experiment.* Comprehensive review of PPN formalism, Cassini bounds ($|\gamma - 1| < 2.3 \times 10^{-5}$), and scalar-tensor theory constraints.

Brax, P., van de Bruck, C., Davis, A.-C., Khoury, J., & Weltman, A. 2004, PhRvD, 70, 123518. *Detecting dark energy in orbit: The cosmological chameleon.

Burrage, C. & Sakstein, J. 2018, Living Reviews in Relativity, 21, 1. Tests of Chameleon Gravity.*

Khoury, J. & Weltman, A. 2004, PhRvL, 93, 171104. *Chameleon Fields: Awaiting Surprises for Tests of Gravity in Space.

Mota, D. F. & Shaw, D. J. 2007, PhRvD, 75, 063501. Evading equivalence principle violations, cosmological, and other experimental constraints in scalar field theories with a strong coupling to matter.

#### Statistical Methods & Software

Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306. emcee: The MCMC Hammer.* Ensemble MCMC sampler used for Bayesian posterior sampling and Savage-Dickey Bayes Factor computation.

Scargle, J. D. 1982, ApJ, 263, 835. *Studies in astronomical time series analysis. II. Statistical aspects of spectral analysis of unevenly spaced data. Foundation for Lomb-Scargle periodogram false alarm probability assessment.

Gelman, A., & Rubin, D. B. 1992, Statistical Science, 7, 457. Inference from iterative simulation using multiple sequences.* Convergence diagnostic ($\hat{R}$ statistic) for MCMC sampling validation.

#### Ephemerides & Clocks

Fienga, A., et al. 2019, A&A, 630, A145. *The INPOP19a planetary ephemerides.*

Grotti, J., et al. 2018, Nature Physics, 14, 437. *Geodesy and metrology with a transportable optical clock.*

Lisdat, C., et al. 2016, Nature Communications, 7, 12443. *A clock network for geodesy and fundamental science.*

Ashby, N., & Patla, B. 2024, Nature Communications, 15, 1. *A relativistic framework to establish Coordinate time on the Moon and Beyond. Establishes the standard for Coordinated Lunar Time (LTC), citing the 58.7 μs/day faster clock rate on the lunar surface.

Turyshev, S. G., et al. 2024, arXiv:2406.16147. Relativistic Time Transformations Between the Solar System Barycenter, Earth, and Moon. Provides high-precision synchronization algorithms for LTC.

White House OSTP. 2024, Memorandum for Record. Policy for Coordinated Lunar Time.* US Government directive establishing the necessity of a unique lunar time standard due to relativistic drift.

#### Cosmology

Planck Collaboration, Aghanim, N., et al. 2020, A&A, 641, A6. *Planck 2018 results. VI. Cosmological parameters.*

#### TEP Framework (This Series)

Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: 10.5281/zenodo.16921911 (Paper 0)

Smawfield, M. L. (2025). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Preprint v0.28 (Jaipur). Zenodo. DOI: 10.5281/zenodo.17127229 (Paper 1)

Smawfield, M. L. (2025). *Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products*. Preprint v0.21 (Cairo). Zenodo. DOI: 10.5281/zenodo.17517141 (Paper 2)

Smawfield, M. L. (2025). *Global Time Echoes: Raw RINEX Consistency Test*. Preprint v0.9 (Kathmandu). Zenodo. DOI: 10.5281/zenodo.17860166 (Paper 3)

Smawfield, M. L. (2025). *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Preprint v0.9 (Tortola). Zenodo. DOI: 10.5281/zenodo.17982540 (Paper 4)

Smawfield, M. L. (2025). *Global Time Echoes: Empirical Synthesis*. Preprint v0.8 (Singapore). Zenodo. DOI: 10.5281/zenodo.18004832 (Paper 5)

Smawfield, M. L. (2025). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T*. Preprint v0.9 (New Delhi). Zenodo. DOI: 10.5281/zenodo.18064365 (Paper 6)

Smawfield, M. L. (2025). *The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate*. Preprint v0.5 (Blantyre). Zenodo. DOI: 10.5281/zenodo.18059250 (Paper 7)

Smawfield, M. L. (2025). *Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging*. Preprint v0.7 (Mombasa). Zenodo. DOI: 10.5281/zenodo.18064581 (Paper 8)

Smawfield, M. L. (2025). *What Do Precision Tests of General Relativity Actually Measure?*. Preprint v0.9 (Istanbul). Zenodo. DOI: 10.5281/zenodo.18109760 (Paper 9)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. Preprint v0.10 (Caracas). Zenodo. DOI: 10.5281/zenodo.18165798 (Paper 10)

Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: 10.5281/zenodo.18209702 (Paper 11)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Preprint v0.8 (Kos). Zenodo. DOI: 10.5281/zenodo.19000827 (Paper 12)

Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Preprint v0.7 (Kilifi). Zenodo. DOI: 10.5281/zenodo.19102061 (Paper 13)

## Data Availability & Reproducibility

This work follows open-science practices. All empirical results are fully reproducible from raw data using the documented pipeline. Data ingestion verifies the five INPOP19a station residual files and the DE430 cross-ephemeris residual file against `data/raw/data_manifest.json` SHA-256 checksums before any downstream analysis runs. Monte Carlo, injection, and toy-model steps are labelled in their JSON outputs and are not treated as independent observational detections.

### Repository & Code

The repository contains a deterministic, version-controlled analysis pipeline with analysis steps for LLR data processing and statistical analysis. All canonical steps are orchestrated by `scripts/steps/run_all_steps.py` with comprehensive logging.

#### Repository Structure

```
TEP-LLR/
├── data/
│   ├── raw/                     # Raw LLR residuals (INPOP19a, DE430)
│   ├── interim/                 # Intermediate processing results
│   └── processed/               # Final processed datasets
├── scripts/
│   ├── steps/                   # Analysis pipeline steps and run_all_steps.py
│   └── utils/                   # Utility functions (parsing, plotting)
├── results/
│   └── outputs/                 # JSON/CSV analytical outputs
├── logs/                        # Per-step execution logs
├── site/
│   └── components/              # Manuscript HTML sections
├── requirements.txt             # Python dependencies
└── README.md                    # Documentation
```

### Data Provenance

| Data Source | Provider | Access Method | Size | Location |
| --- | --- | --- | --- | --- |
| INPOP19a Residuals | IMCCE (Paris Observatory) | Public download | ~2 MB | `data/raw/` |
| DE430 Residuals | JPL | Public download | ~0.5 MB | `data/raw/` |
| APOLLO Data | Apache Point Observatory | Public download | ~0.3 MB | `data/raw/apollo11/`, `apollo14/`, `apollo15/` |

### Pipeline Architecture

The analysis pipeline comprises 91 scripts organised into logical groups. Each step is a standalone Python script in `scripts/steps/` that produces JSON outputs and detailed logs in `logs/step_*.log`. Manuscript step labels in Results and Discussion follow the `step_XXX` script prefix wherever possible (e.g. the primary extraction $\leftrightarrow$ `step_050_corrected_tep_analysis.py`). Where a script's internal docstring uses a different ordinal, the manuscript cites the script prefix and the Results subsection given in the mapping table below.

#### Complete Step Inventory & Runtime

The canonical pipeline comprises 91 scripts organised into logical groups. The core statistical analysis (steps 000–003) provides the primary signal extraction, while subsequent steps perform extended systematic checks, robustness validation, and theoretical consistency tests. Recent enhancements include:

- Temporal autocorrelation analysis with Durbin-Watson statistic and significance testing

- Data-driven systematic error budget computed directly from residuals (cross-ephemeris scatter, seasonal variation, station means, tidal harmonics, diurnal amplitude)

- Outlier detection: Enhanced documentation of $6\sigma$ MAD threshold with methodological justification and statistical rationale

#### Manuscript Step ↔ Pipeline Script Map

Selected analyses where the manuscript label, script prefix, and internal docstring diverge:

| Manuscript step | Pipeline script | Primary JSON | Results section |
| --- | --- | --- | --- |
| 010–013 | `step_010`–`step_013` | `step_010`–`step_013` outputs | §4.20.1–4.20.4 |
| 029 (day/night) | `step_027_day_night_thermal_bias.py` | `step_027_day_night_thermal_bias.json` | §4.29.1 |
| 029 (station power / WLS) | `step_029_station_power_analysis.py` | `step_029_station_power_analysis.json` | §4.25.4–4.25.5 |
| 030 (geometric elongation) | `step_028_geometric_elongation.py` | `step_028_geometric_elongation.json` | §4.29.2 |
| 032 (hardware epochs) | `step_030_hardware_epoch_analysis.py` | `step_030_hardware_epoch_analysis.json` | §4.12 |
| 034 (orthogonality) | `step_032_ephemeris_orthogonality_proof.py` | `step_032_ephemeris_orthogonality_proof.json` | §4.25.6 |
| 045 (matched-window) | `step_045_independent_validation.py` | `step_045_independent_validation.json` | §4.1 / §5.4 |
| 050 (primary extraction) | `step_050_corrected_tep_analysis.py` | `step_050_corrected_tep_analysis.json` | §4.9 |
| 055 (CMB robustness suite; random-axis ΔAIC null; dual-axis primary; η_θ coverage) | `step_055_cmb_rigorous_falsification.py` | `step_055_cmb_rigorous_falsification.json` | §4.25.2 / §5.4 |
| 056 (linearized post-fit $\eta_{\rm resid}$ extraction) | `step_056_dynamical_integrator_eta_refit.py` | `step_056_dynamical_integrator_eta_refit.json` | §4.14 |
| Step 064-PI (prediction-interval calibration) | `step_064_prediction_coverage.py` | `step_064_prediction_coverage.json` | §4.9.6 |
| 072 (leave-one-station-out meta; forest plot) | `step_072_leave_one_station_out_meta.py` | `step_072_leave_one_station_out_meta.json` | §4.4.1 |
| Step 064-SRP (orbital SRP scaling, forward SRP+thermal subtraction & orthogonal survival) | `step_064_srp_systematic_check.py` | `step_064_srp_systematic_check.json` | §4.33.2 / §5.4.1 |
| 081 (cross-ephemeris decomposition & Ke et al. 2026 matched-pipeline reconciliation) | `step_081_ephemeris_amplitude_decomposition.py` | `step_081_ephemeris_amplitude_decomposition.json` | §4.13 / §5.4.1 |
| 083 (clock-sector amplitude bookkeeping) | `step_083_clock_channel_amplitude.py` | `step_083_clock_channel_amplitude.json` | §2.1 |
| 084 (Earth–Moon–Sun matched-profile bridge) | `step_084_earth_moon_solar_bridge.py` | `step_084_earth_moon_solar_bridge.json` | §2.1 |
| 085 (nested-operator LLR differential constraint) | `step_085_nested_llr_constraint.py` | `step_085_nested_llr_constraint.json` | §2.1 |
| 086 (coupled Earth–Moon–Sun PDE at syzygy) | `step_086_earth_moon_coupled_pde.py` | `step_086_earth_moon_coupled_pde.json` | §2.1 |

| Step | Script | Description | Runtime |
| --- | --- | --- | --- |
| 000 | `step_000_llr_data_ingestion.py` | Downloads and parses INPOP19a residual data from Paris Observatory | ~0.2s |
| 001 | `step_001_data_preprocessing.py` | Computes Moon-Sun elongation, synodic phase, and performs data quality validation | ~8.9s |
| 002 | `step_002_de430_preprocessing.py` | DE430 ephemeris preprocessing and comparison | ~0.9s |
| 003 | `step_003_statistical_analysis.py` | Computes Pearson correlation, linear regression, and differential analysis | ~1.2s |
| 004 | `step_004_detection_analysis_advanced.py` | Advanced robust analysis with multiple independent analysis families (bootstrap, permutation, robust regression, outlier detection, station-by-station, temporal stability, phase-binned, systematic error modelling, sensitivity, cross-validation, holdout) | ~5.4s |
| 006b | `step_006b_de430_outlier_robustness.py` | DE430 outlier robustness: threshold sweep ($3\sigma$–$10\sigma$), phase-bin chi-square, bootstrap CI, permutation test | ~15s |
| 005–039 | `step_005_*.py` through `step_039_*.py` | Extended validation, systematic controls, physical probes, and defensibility analyses through dust sensitivity | ~1m 20s total |
| 040–043 | `step_040_*.py` through `step_043_*.py` | Unified results table, ephemeris absorption simulation, multiple-testing correction, temporal-bin variation | ~25s total |
| 044–083 | `step_044_*.py` through `step_083_*.py` | Systematic projection, independent validation, station-balanced TEP, velocity modulation, CMB anisotropy, EOP systematics, corrected TEP extraction, cross-validation, station distribution, clean subset, toy orbital perturbation, CMB robustness suite, linearized dynamical integrator refit, advanced estimator corrections (067–073), Laplace Bayes factor, wild cluster bootstrap, generic PPN test, sky-scan directionality, GP model comparison, phase-locked explanation, dynamical transfer closure, Haleakala confound audit, ephemeris amplitude decomposition, tide/libration Delaunay forward model, and conditional clock-sector amplitude bookkeeping (solar-potential excursion and quartic depth diagnostic; its legacy unit-transmission range estimate is not a validated clock-to-range map, and the corrected Paper-0 Step 76 is a static boundary-response calculation) | ~3m total |
| 084 | `step_084_earth_moon_solar_bridge.py` | Finite-radius matched-profile bridge with solar-orientation and LLR-comparator provenance | Short |
| 085 | `step_085_nested_llr_constraint.py` | Nested-operator Sun–Earth/Sun–Moon differential response against the LLR bound (migrated from Paper 0 step_22) | Short |
| 086 | `step_086_earth_moon_coupled_pde.py` | Coupled nonlinear three-source solve at syzygy (axisymmetric); supersedes the 084 matched-profile ansatz for the collinear case — no screened bridge forms | Minutes (u0 continuation) |

#### Total Runtime Summary

| Component | Steps | Runtime |
| --- | --- | --- |
| Core Statistical Analysis | 4 (000-003) | ~25.6s |
| Extended Systematic Analysis | 68 (004–071, including 006b and 046b) | ~6m 27s |
| Total | 73 | ~6m 53s |

### Reproduction Instructions

#### Quick Start (Full Reproduction)

```
# 1. Clone repository
git clone https://github.com/matthewsmawfield/TEP-LLR.git
cd TEP-LLR

# 2. Install dependencies
pip install -r requirements.txt

# 3. Run full pipeline (generates JSON outputs and logs)
python scripts/steps/run_all_steps.py

# 4. Results will be in:
#    - results/outputs/   (JSON analytical outputs)
#    - logs/              (Detailed execution logs)
#    - site/dist/         (Built manuscript site)

# 5. Regenerate manuscript Markdown and static site from site/components/
node site/build.js
```

#### System Requirements

| Component | Minimum | Recommended |
| --- | --- | --- |
| CPU | 2 cores | 8+ cores (for multiprocessing) |
| RAM | 4 GB | 8 GB |
| Storage | 100 MB | 500 MB |
| Runtime | ~7m | ~6m 53s (with multiprocessing) |

#### Key Analysis Outputs

- `results/outputs/step_003_statistical_analysis.json` — Pearson correlation, linear regression, and differential analysis results

- `results/outputs/step_004_detection_analysis_advanced.json` — Comprehensive analysis with bootstrap, permutation, robust regression, outlier detection, station-by-station, temporal stability, phase-binned, systematic error modelling, sensitivity, cross-validation, and holdout test results

- `results/outputs/step_061_systematic_sensitivity_analysis.json` — Known-systematic amplitude ratios and Monte Carlo null simulations; `adversarial_pca`, `adversarial_gp` (2D elongation$\times$time binned GP), `interaction_grid` (era$\times$station$\times$lunation), `blind_year_holdout` (5-fold disjoint calendar-year cross-validation), and `tests_for_step_042`

- `results/outputs/step_042_multiple_testing_correction.json` — Formal multiplicity control including interaction-grid and blind 5-fold cross-validation tests (sensitivity family)

- `results/outputs/step_001_data_preprocessing.json` — Data preprocessing statistics and validation results

- `data/processed/INPOP19a_all_stations_residuals.csv` — Processed INPOP19a dataset with computed elongation phases

- `data/processed/[station]_residuals.csv` — Individual station processed datasets

#### Log Files

Each step produces detailed logs:

- `logs/step_000_llr_data_ingestion.log` — Data ingestion log

- `logs/step_001_data_preprocessing.log` — Data preprocessing and validation log

- `logs/step_002_de430_preprocessing.log` — DE430 ephemeris preprocessing log

- `logs/step_003_statistical_analysis.log` — Statistical analysis log

- `logs/step_004_detection_analysis_advanced.log` — Advanced 13-method analysis log

### Software Dependencies

| Package | Version | Purpose |
| --- | --- | --- |
| Python | 3.8+ | Language runtime |
| NumPy | 1.20+ | Numerical computing |
| SciPy | 1.5+ | Statistical functions |
| Pandas | 1.2+ | Data manipulation |
| Matplotlib | 3.3+ | Visualization |

All dependencies are specified in `requirements.txt`.

### Validation & Testing

The pipeline includes comprehensive validation:

- Data Integrity: Checks for missing or malformed residuals

- Phase Consistency: Verifies synodic phase calculations against ephemeris data

- Station Independence: Validates that results are consistent across independent observatories

- Cross-Ephemeris Validation: DE430 dataset from JPL shows complex behavior; full dataset shows no significant correlation ($r = -0.000148$, $p = 0.992$), but after removing 37 phase-specific outliers (0.8% of data) that cluster at $135^\circ$–$225^\circ$ elongation, yields $\eta_{\rm resid} = -7.04\times10^{-4}$ at $5.96\sigma$ with bootstrap validation (95% CI on $\eta_{\rm resid}$: $[-9.63, -4.64]\times10^{-4}$) and permutation test ($p &lt; 0.001$). Chi-square test confirms outliers are not uniformly distributed across phases ($\chi^2 = 66.1$, $p = 8.9 \times 10^{-12}$), indicating genuine measurement errors at specific phases. The primary detection relies on INPOP19a ephemeris.

### Reproducibility Checklist

To verify successful reproduction:

- All 91 canonical scripts complete with "PASS" status in pipeline output

- Key result (cross-ephemeris decomposition & Ke et al. 2026 reconciliation, Steps 064-SRP & 081): the DE430 uncleaned null is a gross-error leverage artefact — winsorization at the 6σ boundary recovers $\eta_{\rm resid} = -7.39 \times 10^{-4}$ ($5.70\sigma$) without deleting any point; on the shared 2014–2018 window the within-ephemeris composition spread (APO $-3.85 \times 10^{-4}$ vs Grasse $-2.30 \times 10^{-4}$; pooled $-2.57 \times 10^{-4}$ unweighted vs $-3.62 \times 10^{-4}$ under $1/\sigma^2$ weights) is of the same order as the $\sim 2\times$ cross-ephemeris ratio, so the residual-channel sign is the invariant channel and it agrees in both ephemerides; in the Ke et al. (2026) third-order true-ecliptic harmonic basis $D_{\rm ecl}(t)$, precision-weighted INPOP19a yields $C_1 = -5.71 \pm 0.96$ mm ($5.95\sigma$, shifting to $-2.72 \pm 1.08$ mm, $2.51\sigma$ combined after forward orbital-SRP+thermal subtraction; $-7.17 \to -4.23 \pm 1.07$ mm, $3.94\sigma$ on $D_{\rm Del}$), while forward SRP+thermal subtraction absorbs $91.1\%$ of the Model E static isotropic intercept ($C_{1,0}: -3.30 \to -0.29 \pm 0.94$ mm, $0.31\sigma$) and leaves $\eta_{v_r}$ ($t = +6.50$) and $\eta_\theta$ ($t = -11.02$) $100.0\%$ invariant

- JSON files in `results/outputs/` (step_000 through step_086, including 006b, 046b, and 064-SRP)

- Key result (headline): Precision-weighted full-systematic consensus $\eta_{\rm resid} = -3.91 \times 10^{-4} \pm 5.63 \times 10^{-5}$ ($6.94\sigma$; $6.78\sigma$ cluster-robust; $N = 25{,}445$)

- Key result (uncertainty calibration): prediction-interval coverage 86% at 68% nominal, 98% at 95% ($\chi^2_{\rm red}=0.48$ on published $\sigma$); LOSO conformal 95% band on $\eta_{\rm resid}$ excludes zero

- Key result (LOSO robustness): four of five exclusions exceed the observed-SNR threshold and share the negative sign; $\eta_{\rm meta} = -4.21 \times 10^{-4}$; $I^2 = 0.0\%$ heterogeneity; combined significance not reported independently (retained exclusions contain 87.8–98.6% of the full sample); excluding Grasse underpowered ($0.92\sigma$)

- Key result (sensitivity upper bound): Full-systematic OLS without excision $\eta_{\rm resid} = -4.06 \times 10^{-4} \pm 6.58 \times 10^{-5}$ ($6.17\sigma$); cluster-robust $6.52\sigma$; $N = 25{,}445$

- Key result (secondary leverage diagnostic): Cook's-Distance-excised full-systematic OLS $\eta_{\rm resid} = -3.87 \times 10^{-4} \pm 4.95 \times 10^{-5}$ ($7.82\sigma$); cluster-robust $8.65\sigma$; $N = 23{,}837$

- Key result: Pearson correlation on the 6σ-cleaned primary sample ($N=25{,}445$): $r = -0.0329$ ($p = 1.55 \times 10^{-7}$), $5.25\sigma$

- Key result: Cook's Distance leverage-excised cosD-only OLS $\eta_{\rm resid} = -3.31 \times 10^{-4} \pm 5.85 \times 10^{-5}$ (SNR = $5.65\sigma$; $N = 25{,}176$)

- Key result: Naïve OLS robustness check $\eta_{\rm resid} = -3.18 \times 10^{-4}$ ($5.25\sigma$)

- Key result: MCMC robustness check $\eta_{\rm resid} = -2.87 \times 10^{-4} \pm 6.63 \times 10^{-5}$ (SNR = $4.32\sigma$; canonical estimator hierarchy)

- Key result: Full-model AR(1) GLS robustness check $\eta_{\rm resid} = -4.45 \times 10^{-4} \pm 9.87 \times 10^{-5}$ (SNR = $4.51\sigma$)

- Key result: cosD-only AR(1) GLS robustness check $\eta_{\rm resid} = -3.28 \times 10^{-4} \pm 9.36 \times 10^{-5}$ (SNR = $3.51\sigma$ cluster-robust)

- Key result: AR(1) parameter $\rho = 0.412$ (full model), $\rho = 0.416$ (cosD-only); significant temporal autocorrelation

- Key result: Bootstrap 95% CI = [-0.0411, -0.0197] (excludes zero)

- Key result: Permutation test p < 10<sup>-4</sup> (no permutation achieved observed)

- Key result: Station consistency: Grasse shows strongest individual detection (SNR = $4.97\sigma$), all stations show consistent negative $\eta_{\rm resid}$ direction except Haleakala; common-$\eta_{\rm resid}$ mixed model $F(4, 25{,}410)=1.19$, $p=0.31$

- Key result: Haleakala null-fluctuation simulation (20,000 MC): 2.0$\sigma$ event under TEP (two-tailed $p=0.039$), family-wise rate 4.2% (8.0% under the GR null); Step-080 confound audit excludes coverage restriction ($\le 6.4 \times 10^{-4}$ shift) and solar conditioning (conditioned excess $+3.8 \times 10^{-3}$, $2.62\sigma$)

- Key result: Clean-subset exclusion (Grasse 2010+ + APO only): $\eta_{\rm resid} = -3.36 \times 10^{-4} \pm 4.63 \times 10^{-5}$ ($7.25\sigma$ cluster-robust), confirming signal without Haleakala, Matera, or McDonald2

- Key result: Cross-validation: Temporal hold-out, random k-fold, leave-one-station-out, and two-tier synthetic injection (noise-only vs era-conditioned nuisance) diagnose nuisance transport; primary $\eta_{\rm resid}$ remains the pooled precision-weighted extraction

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# Temporal Equivalence Principle: Native hi_class Conformal Implementation, Linear Perturbation Closure, and CMB Acoustic Peak Preservation
**Matthew Lukin Smawfield**
Version: v0.8 (Cambridge)
First published: 8 June 2026 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.20572722

---

## Abstract


Standard cosmology explains the Cosmic Microwave Background (CMB) acoustic peaks, the pre-recombination sound horizon, and the thermal scaling relevant to Big Bang Nucleosynthesis (BBN) within an FLRW expansion history conventionally extrapolated toward a physical singularity. This paper demonstrates that the CMB acoustic-sector and conformal thermal/sound-horizon scalings are preserved with high fidelity under a static conformal temporal-transport geometry governed by the Temporal Equivalence Principle (TEP).



In the TEP framework, matter clocks and photon phases evolve in a causal matter metric defined by a conformal scalar field $\tilde{g}_{\mu\nu} = A(\phi)^2 g_{\mu\nu}$. Because this conformal transport geometry is mathematically isomorphic to the FLRW scale factor $a(t)$, standard Boltzmann solvers like `hi_class` and `CLASS` can be used as conformal-frame calculators for the background/acoustic-sector mapping tested here. The parameter traditionally identified as Dark Energy ($\Omega_\Lambda$) is operationally reinterpreted within this implementation as the homogeneous temporal-shear background contribution filling the same background budget slot, $\Omega_\phi$.



This paper implements the native TEP interpretation directly in `hi_class`. Within the broader TEP interpretation, by recognizing that the spatial metric does not stretch, the "Big Bang" is reinterpreted not as a physical density singularity, but as a TEP temporal horizon—an asymptotic boundary where the observational clock map $A_{\rm clock} \to 0$. Direct Boltzmann integration verifies this background/acoustic mathematical isomorphism, verifying the internal conformal-frame preservation of the acoustic sector — the pre-recombination sound-horizon ratio is $r_s^{\rm TEP}/r_s^{\rm \Lambda CDM}=0.999994$ (corresponding to a $<6$ ppm deviation) and the acoustic-peak morphology remains intact without invoking early-universe spatial expansion. It is stated plainly that this preservation is a mathematical identity by construction — identifying $A(\phi)$ with $a(t)$ reproduces the FLRW integrals by definition, so the $<6$ ppm residual is a code-level artefact of finite-precision reconstruction and the exercise verifies the `hi_class` conformal-frame implementation, not the TEP hypothesis itself. The acoustic indistinguishability is nonetheless the necessary consistency requirement a conformal-frame alternative must satisfy: a framework that failed it would be excluded, and the demonstration that early-universe background physics cannot distinguish a stretching spatial metric from an evolving conformal clock-rate field shifts the discriminating burden to the perturbation and late-time sectors where the models do differ.



Beyond the background mapping, the paper closes the linear pure-conformal scalar perturbation sector through the EFT closure $\alpha_M=-2\alpha_A$, $\alpha_B=2\alpha_A$, $\alpha_K=w_c\,\alpha_A^2$, $\alpha_T=0$, where $w_c = Z_E - 6\beta_A^2$ is the effective cosmological kineticity parameterized by the Einstein-frame kinetic coefficient $Z_E = P_{,X} + 2XP_{,XX}$ of the scalar (with the exact nonlinear principal-operator matching treated separately in Appendix A.5a). Evolving $\delta\phi$ natively through the full Einstein–Boltzmann hierarchy — the regular $\delta\phi$ variable being required because $x_{\rm smg} = \delta\phi/\phi'$ is singular at the $\alpha_A = 0$ turnover — reveals a structure invisible to the no-ghost discriminant $D = \alpha_K + \frac{3}{2}\alpha_B^2 = (w_c+6)\alpha_A^2$: the canonical Einstein-frame image $Z_E = 1$ ($\alpha_K = -5\alpha_A^2$, the closure quoted in earlier versions of this work) passes $D > 0$ yet is linearly unstable on the cosmological background, carrying a tachyonic growing mode that produces runaway structure growth. The sign-corrected unscreened branch ($w_c = +7$, i.e. $Z_E = 13$) is stable but mediates an O(1) scalar-tensor fifth force, yielding $\sigma_8 \simeq 1.6$ — observationally excluded. The viable branch is the kinetically screened regime furnished by the nonlinear $P(X)$ sector: for $Z_E \gtrsim 7\times10^2$ the scalar response decouples and growth recovers the $\Lambda$CDM amplitude ($\sigma_8 = 0.824$ versus $0.823$; median matter-power deviation $\lesssim 0.2\%$, peaking at $\sim 1.2\%$ at the low-$k$ end, with a residual $\sim 13\%$ low-$\ell$ ISW signature), while $Z_E \sim 6\times10^3$ gives $\sigma_8 \simeq 0.76$, coincidentally at the weak-lensing central value. The measured $\sigma_8(Z_E)$ response curve is the perturbation sector's central result; the closure and its screening parameter $Z_E$ are falsifiable in both amplitude and scale dependence. An earlier statement that the active scalar contributes bitwise-identically to the background-only spectra is retracted: that result was produced while the SMG perturbation sector was silently disabled by a background-density gate, and it measured only the background channel.



A joint `hi_class` Cobaya MCMC tests the finite-turnover TEP conformal background within the native hi_class implementation. Two production runs are reported. The primary inference uses Planck 2018 low-$\ell$ TT/EE + lensing + BAO + Pantheon+, in which the TEP homogeneous amplitude is not detected: a converged four-chain MPI suite (4,814 post-burn-in samples, max parameter $R-1 = 0.011$) gives $\epsilon_T = 0.0062 \pm 0.0046$ (median $0.0053$, $\epsilon_T < 0.015$ at 95% CL), consistent with zero at $\sim 1.3\sigma$ — a null result that bounds rather than confirms the conformal modification. The full-likelihood extension adds `planck_2018_highl_plik.TTTEEE_lite` to the same baseline: a converged four-chain MPI run (11,237 post-burn-in samples; max parameter $R-1 = 0.0455$) yields $\epsilon_T = 0.0043 \pm 0.0025$, a one-sided bound $\epsilon_T < 0.0087$ at 95% CL at fixed turnover scale $z_T=5$ ($\epsilon_T < 0.0103$ at 95% CL under $z_T$ marginalization, Section 4.5), against a matched two-chain $\Lambda$CDM reference (max $R-1 = 0.0069$). On identical likelihoods the models are statistically indistinguishable: $\Delta\chi^2 = -0.11$ at the respective best fits, and a matched-parameter evaluation verifies the $\epsilon_T \to 0$ limit reproduces $\Lambda$CDM to $\Delta\chi^2 = 0.007$. The bound is geometric rather than dynamical: at fixed cosmology the high-$\ell$ likelihood contributes $\Delta\chi^2 \simeq +69$ at $\epsilon_T = 0.008$ through the anchored acoustic scale $\theta_s$, while low-$\ell$ TT and Pantheon+ weakly prefer positive amplitudes ($\Delta\chi^2 \simeq -0.6$ each). An earlier `clipy` overflow report is resolved: the warnings persist in the completed run but are benign, the historical failure tracing to a stock-`classy` build shadowing the patched hi_class on the Cobaya theory path. Companion TEP-C0 (Paper 26) reports Pantheon+ model-comparison evidence for the physical no-$\Lambda$ temporal-shear branch: Bayes factor approximately 5.8 for the conservative $z_{\rm los}=5$ branch (substantial, not decisive), approximately 5.4 for the fixed $z_{\rm los}=100$ benchmark, and approximately 5.4 for the broad free-$z_{\rm los}$ branch. These late-time model-comparison results are imported, not re-derived in HC; HC supplies the native `hi_class` acoustic and perturbation closure. Within the broader TEP corpus, Paper 11 interprets the Hubble tension as a late-time, environment-dependent clock-transport effect caused by the environmental screening of the scalar field, rather than through a crisis in early-universe physics; that interpretation is likewise imported — HC does not independently re-analyse the distance-ladder data, and Paper 11 Step 62 derives the Cepheid response scale $|\kappa_{P^2}^{\rm pred}(\lambda_0=1)| = 1.196\times 10^6$ mag from the stellar-envelope density ratio $\rho_T/\langle\rho\rangle(10\text{ d}) = 8.45\times 10^5$ at the unit-transport benchmark $\lambda_0 = 1$ with bare coupling $|\beta_A| = 1$ fixed by Paper 0 (matching the SH0ES fit $(1.191\pm 0.480)\times 10^6$ mag to $0.5\%$).


The Planck lensing likelihood in the baseline `hi_class` implementation retains a standard pressureless CDM component ($\Omega_{\rm cdm}h^2=0.1133\pm0.0061$ in the active posterior; $0.11813\pm0.00117$ in the full high-$\ell$ TTTEEE active-perturbation posterior) because isotropic conformal rescaling $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$ stretches spatial rulers alongside clocks and cancels out of null rays, leaving only $8.72\%$ of $C_{100}^{\phi\phi}$ when `omega_cdm` is removed ($\chi^2=1265.73$ on the 9 Planck 2018 `CMBMarged` lensing bandpowers; Step 21). Under the canonical unified matter metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) — where the $(3+1)$ ADM spacetime lapse rescales as $N_{\rm TEP}(\mathbf x)=A_{\rm loc}(u_{\rm grav}(\mathbf x))N_{\rm GR}(\mathbf x)$ while at a fixed cosmic epoch ($A_{\rm sp}(t_0)=1$) the spatial 3-metric $\tilde\gamma_{ij}=\gamma_{ij}$ remains unstretched, so intermediate clock rates $A_{\rm loc}(u_{\rm grav}(x))$ refract photons along the line of sight — the selected two-branch kinetic sector $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$ fixed by $a_0$, zero per-system free parameters) amplifies baryon-only cosmic-web and cluster-outskirt lensing potentials by $M=3.07$–$6.18$ ($1\times$) to $5.15$–$11.37$ ($2\times$), boosting $C_\ell^{\phi\phi}$ by $9.4\times$–$38.3\times$ ($1\times$) to $26.5\times$–$129.3\times$ ($2\times$). Fitting the full multipole spectrum ($\ell\in[2,2500]$) across the Silk transition directly against the official 9-bin Planck 2018 `CMBMarged` lensing likelihood at $\omega_{\rm cdm}=0$ yields $\chi^2=15.40$ with fixed Silk geometry ($\ell_{\rm Silk}=300,\alpha_{\rm Silk}=2$) and $\chi^2=13.40$ ($\chi^2/N=1.49$, $\Delta\chi^2=+1.11$ relative to the $12.30$ CDM baseline and $\Delta\chi^2=-1252.33$ relative to bare no-CDM; Step 21), reproducing the observed Planck CMB lensing spectrum across $L\in[8,400]$ without particulate dark matter. Furthermore, because the proper-time field $\phi$ fills all space as a continuous ambient scalar medium ($A(\phi)>0$ everywhere) with fixed action scale $g_t=\Lambda_X^2/(\sqrt{2}M_{\rm Pl})=cH_0/2$ and conformally invariant sound-horizon acceleration ratio $q_s=g_b(r_s)/g_t=\Omega_b(H_0 r_s/c)$, the same zero-parameter two-branch clock-slope equation $\nabla\cdot[P_{,X}(|\nabla u|)\nabla u]=\beta_A\rho_b$ evaluated at the sound horizon ($q_s=1.62\times10^{-3}$, $y_s=7.38$, $\omega_\phi=0.1166$ in the single-pass $r_s=146.19\text{ Mpc}$ analytic integral; converging under self-consistent $r_s(\omega_\phi)$ fixed-point iteration in `hi_class` to $r_s=144.94$–$145.00\text{ Mpc}$, $q_s=1.59$–$1.60\times10^{-3}$, $y_s=7.42$–$7.44$) predicts a pre-boundary baryon clock-slope well-driving strength $\omega_\phi=\omega_b(y_s/\sqrt{2})=0.1179$–$0.1182$ ($\omega_{m,\rm eff}=0.1404$–$0.1406$; spanning $\omega_\phi\in[0.1174,0.1199]$ across $H_0\in[67.4,70.0]\text{ km s}^{-1}\text{ Mpc}^{-1}$, all within $\lesssim1.6\sigma$ of the unconstrained full-Planck TTTEEE MCMC posterior $0.11813\pm0.00117$ and within $0.02\sigma$ [$0.02\%$] at the posterior mean $H_0=68.18\text{ km s}^{-1}\text{ Mpc}^{-1}$). Evaluated in exact `hi_class` on linear cosmological scales ($k\lesssim0.2\,h\,\text{Mpc}^{-1}$, $R\gtrsim10\text{ Mpc}$, where $99.8\%$ of the $R_8=8\,h^{-1}\text{Mpc}$ $\sigma_8^2$ variance originates at $k\le1.0\,h\,\text{Mpc}^{-1}$, so applying the nonlinear halo-interior transition filter above $k_*=1.5\,h\,\text{Mpc}^{-1}$ shifts $\sigma_8$ by only $1.0\%$ from $0.8185$ to $0.8103$, whereas inside virialized galaxy halos $R\in[1,100]\text{ kpc}$ for $M_b=6\times10^{10}\,M_\odot$, the algebraic Gauss law $(k/\sqrt{2})q y^2+q^2 y^3=1$ smoothly transitions $M_{\rm eff}(R)=1+y(R)/\sqrt{2}$ from $6.26$ on linear scales through $5.18$ at $100\text{ kpc}$, $2.25$ at $30\text{ kpc}$, and $1.42$ at $10\text{ kpc}$ down to $1.12$ at $3\text{ kpc}$ and $1.04$ at $1\text{ kpc}$ without an independent NFW particle cusp), this zero-parameter baryon clock-slope response reproduces the acoustic scale ($100\theta_s^{\rm CLASS}=1.04201$, $+0.10\sigma$ from the TTTEEE posterior $1.04198\pm0.00031$ at $H_0=68.18\text{ km s}^{-1}\text{ Mpc}^{-1}$, and $100\theta_s=1.04111$, $+0.02\sigma$ from $1.04110\pm0.00031$ at $H_0=67.95\text{ km s}^{-1}\text{ Mpc}^{-1}$), the first three acoustic peaks to $\le0.31\%$–$0.40\%$, the linear-scale matter power spectrum $P(k)$ with $\sigma_8=0.8170$–$0.8186$ ($S_8=0.8221$–$0.8226$), the BAO drag scale $r_d=147.49$–$147.55\text{ Mpc}$, the SDSS DR12 consensus BAO + RSD likelihood ($\chi^2/N=0.68$–$0.78$ on 6 BAO points, $0.71$–$0.76$ on 9 BAO+$f\sigma_8$ points), and the full Planck 2018 TTTEEE + low-$\ell$ + lensing + BAO + Pantheon+ likelihood with $\Delta\chi^2=-0.06$ ($\Delta\text{AIC}=-2.06$ with one fewer free parameter relative to free $\omega_{\rm cdm}$; Step 21 Part C).

The imported C0 evidence is prior- and anchor-sensitive: the conservative Bayes factor falls to approximately 0.63 after removing $z < 0.023$ anchors. The high-turnover branches are mathematical shape benchmarks pending an environment-dependent physical realization.


Keywords: cosmology theory, cosmic microwave background, static conformal geometry, scalar-tensor theories, conformal gravity, hi_class, Horndeski, temporal equivalence principle, proper time, Cobaya, Planck 2018



## 1. Introduction


### 1.1 Contextualizing the TEP Corpus


The Temporal Equivalence Principle (TEP) has been constrained across many orders of magnitude in mass density, from terrestrial laboratory scales ($\rho \sim 20$ g/cm³) to the cosmological mean ($\rho \sim 10^{-29}$ g/cm³). Previous papers in this series have established:



- *Terrestrial scales (Paper 1):* Terrestrial atomic clock networks show 4,200 km phase correlations consistent with the candidate Temporal Topology saturation scale ρ_T ≈ 20 g/cm³.

- *Galactic scales (Paper 6, UCD):* SPARC rotation curves validate the potential-dependent proper-time mapping.

- *Stellar scales (Paper 13, WB):* Gaia DR3 wide binaries exhibit the predicted environment-dependent kinematic transition.

- *Cosmological scales (Paper 12, JWST):* High-redshift anomalies align with environment-dependent time dilation.




### 1.2 The Cosmological Horizon


The Hubble tension and the JWST high-redshift galaxy anomalies represent the two most persistent challenges in modern cosmology. Standard $\Lambda$CDM relies on a stretching spatial metric, which extrapolates to a physical singularity at $a \to 0$ and tightly restricts the available proper time for early galaxy assembly.


Previous TEP work (Paper 11, H₀; Paper 12, JWST) argued that these anomalies are resolved within the TEP framework. The TEP temporal-horizon picture replaces the FLRW singularity with an asymptotic boundary, removing the finite-age assembly bottleneck, and the $H_0$ tension is addressed as a local environmental screening effect on kinematic distance probes.


### 1.3 Purpose of This Paper


To rigorously test the CMB acoustic-sector component of the Static Conformal thesis, this paper demonstrates that the acoustic-sector integrals can be reproduced by an exactly conformal temporal mapping without explicit spatial expansion. A full light-element abundance calculation is not performed in HC. TEP-TH supplies the nonsingular temporal-horizon geometry and recombination-visibility treatment, while TEP-BBN supplies the native chemical-evolution framework and local CMB-thermalization proof of concept. The present HC claim is limited to conformal thermal/sound-horizon scaling, acoustic preservation, and the native `hi_class` perturbation implementation.


Because the TEP conformal scalar field $\tilde{g}_{\mu\nu} = A(\phi)^2 g_{\mu\nu}$ is mathematically isomorphic to the FLRW scale factor $a(t)$, the TEP Static Conformal geometry can be natively evaluated by deploying the `hi_class` Boltzmann solver as a conformal-frame calculator. This requires:



- Mapping the TEP conformal geometry onto the Boltzmann framework, establishing the implementation-level correspondence between the parameter conventionally written as $\Omega_\Lambda$ and the homogeneous temporal-shear background contribution, $\Omega_\phi$, occupying the same background-budget role as $\Omega_\Lambda$.

- Native implementation in hi_class to evaluate the conformal temporal shear field directly.

- A joint MCMC parameter estimation ($H_0$, $\Omega_b h^2$, $\Omega_{\rm cdm} h^2$, $n_s$, $A_s$, $\tau$, $A_{\rm planck}$, $\epsilon_T$) against Planck 2018, BAO, and Pantheon+ data to quantitatively demonstrate that the acoustic peaks are preserved with sound-horizon ratio $r_s^{\rm TEP}/r_s^{\rm \Lambda CDM}=0.999994$ (corresponding to a $<6$ ppm deviation) in a static conformal geometry.



This work does not independently analyse Pantheon+ supernovae or derive the full nonsingular temporal-horizon closure; those are addressed in companion papers TEP-C0 (Paper 26) and TEP-TH (Paper 27).

The critical question: Can a static conformal geometry mathematically reproduce the CMB acoustic peaks?

The claim-discipline framework for the TEP corpus, including the scope limitations of canonical precision tests, is established in TEP-EXP (Paper 9).

## 2. Theoretical Architecture: The EFT Mapping

### 2.1 The Bi-Metric Action

The TEP framework posits that matter couples to a causal matter metric $\tilde{g}_{\mu\nu}$ related to the Einstein-frame metric $g_{\mu\nu}$ via a controlled conformal–disformal transformation (Paper 0, §2.2, Step 80):

\begin{equation} \label{eq:3_theory_01}
\tilde{g}_{\mu\nu} = -A_{\rm loc}^2\bigl(u_{\rm grav}(\mathbf x)\bigr)\,U_\mu U_\nu + A_{\rm sp}^2\bigl(u_{\rm drift}(t,\mathbf x)\bigr)\,\Pi_{\mu\nu} + B(\phi)\,\Pi_\mu^\alpha\Pi_\nu^\beta \nabla_\alpha\phi \nabla_\beta\phi,
\end{equation}

with $(3+1)$ ADM line element $d\tilde s^2=-N_{\rm TEP}^2(\mathbf x)c^2dt^2+A_{\rm sp}^2(t,\mathbf x)\gamma_{ij}(\mathbf x)dx^i dx^j$ along the unit cosmic temporal flow $U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$ ($g^{\mu\nu}U_\mu U_\nu=-1$, orthogonal spatial projector $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$), reducing in the homogeneous conformal shorthand to $\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu\phi \nabla_\nu\phi$, where:

- $A(\phi) = \exp(\beta_A\phi/M_{\rm Pl})$ is the universal conformal factor, with $\beta_A = -1.0$ (the locked lab-scale convention used across the TEP corpus; under the canonical unified metric at fixed cosmic epoch $A_{\rm sp}(t_0)=1$, the PPN gravitational slip is $\gamma_{\rm PPN}-1=-2\beta_A^2 S_\Sigma^{(\odot)}/(1+2\beta_A^2 S_\Sigma^{(\odot)})<0$, requiring $S_\Sigma^{(\odot)}\lesssim 1.15\times10^{-5}$ from Cassini's $|\gamma_{\rm PPN}-1|<2.3\times10^{-5}$ [$5.8\times10^{-6}$ on the isotropic benchmark], and satisfied on the selected two-branch solar solution at $1.6\,R_\odot$ with $y_{\rm tb}=2.16\times10^{-8}$ and $|\gamma_{\rm PPN}-1|=4.32\times10^{-8}\ll2.3\times10^{-5}$)

- $B(\phi)$ controls disformal deformation of the causal structure

- $\phi$ is the dynamical proper-time field

Following TEP-TH (Paper 27), two distinct projections of the temporal field are distinguished in cosmology: $A_{\rm clock}(z)$ is the exact observational clock/redshift mapping that generates the apparent distance-redshift relation, while $A_{\rm dyn}(z)$ is the physical dynamical response that modifies expansion and perturbations. In the homogeneous background limit evaluated here, the reference acoustic calculator carries the standard reconstructed background, while the finite-turnover code factor $A_{\rm HC}(z)=\exp[\epsilon_T\ln(1+z)e^{-(z/z_T)^{n_T}}]$ is a residual dynamical profile and is distinct from $A_{\rm clock}=(1+z)^{-1}$; the dynamical response $A_{\rm dyn}$ is parameterized by $\epsilon_T$ and modifies the Hubble parameter and acoustic observables at late times without requiring early-universe thermal screening (TEP-TH, Paper 27; TEP-BBN, Paper 29).

*Metric signature convention:* $(-,+,+,+)$ for the displayed covariant equations, matching $X=-\tfrac12g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$ and the foundational action. The numerical solver uses its native background variables.

### 2.2 Formal Bellini–Sawicki Alpha Correspondence

hi_class requires the EFT property functions $\alpha_i$ that encode metric modifications at linear perturbation level.

#### 2.2.1 Planck Mass Running ($\alpha_M$)

The conformal coupling directly determines the running of the effective Planck mass:

\begin{equation} \label{eq:3_theory_02}
\alpha_M \equiv \frac{d \ln M_{\rm eff}^2}{d \ln a} = - \frac{d \ln A^2(\phi)}{d \ln a} = - \frac{2\beta_A}{M_{\rm Pl}} \frac{\phi'}{\mathcal{H}}
\end{equation}

where $\mathcal{H} = aH$ is the conformal Hubble parameter and primes denote derivatives with respect to conformal time.

#### 2.2.2 Tensor Speed Excess ($\alpha_T$)

The disformal term $B(\phi)$ alters the gravitational wave propagation speed. Multi-messenger constraints from GW170817/GRB 170817A require:

\begin{equation} \label{eq:3_theory_03}
|c_g - c_\gamma|/c \lesssim 10^{-15} \Rightarrow \alpha_T \approx 0 \text{ (today)}
\end{equation}

However, $B(\phi)$ may be non-zero at recombination ($z \approx 1100$) provided it relaxes to zero by $z \sim 0$. This statement concerns the general disformal EFT only; the production HC acoustic/perturbation branch analysed below is the pure-conformal ($\alpha_T=0$) branch.

#### 2.2.3 Braiding ($\alpha_B$) and Kineticity ($\alpha_K$)

These functions govern scalar field clustering and metric mixing. The Bellini–Sawicki braiding parameter $\alpha_B$ should not be confused with the matter-metric disformal function $B(\phi)$; in the pure-conformal HC closure, $B(\phi) = 0$ while conformal scalar–metric mixing gives $\alpha_B = 2\alpha_A$.

\begin{equation} \label{eq:3_theory_04}
\alpha_B = -\frac{\mathcal{H}'\phi'}{\mathcal{H}^2} \cdot f_B(\phi, X)
\end{equation}

\begin{equation} \label{eq:3_theory_05}
\alpha_K = \frac{\phi'^2}{\mathcal{H}^2 M_{\rm Pl}^2} \cdot f_K(\phi, X)
\end{equation}

where $X = -\nabla_\mu\phi \nabla^\mu\phi/2$ and $f_B$, $f_K$ are functions derived from the TEP action:

- $f_B(\phi, X)$ encodes the disformal coupling to the energy-momentum tensor trace.

- $f_K(\phi, X)$ encodes the kinetic term non-canonicality from the TEP proper-time field.

For $B=0$, the exact Weyl image retains the nonlinear function $G_2=A^{-4}P(A^2\tilde X,\phi)-6M_{\rm Pl}^2A^{-2}(d\ln A/d\phi)^2\tilde X$, with $G_4=M_{\rm Pl}^2A^{-2}/2$. On a homogeneous timelike background its kineticity involves $Z_E=P_{,X}+2XP_{,XX}$, while its scalar principal speed is $P_{,X}/Z_E$. The production HC model uses the linear-in-$\tilde X$ effective template described in Appendix A.5a, assigning a constant $w_c=Z_E-6$ and explicitly imposing $c_s^2=1$. Its runtime functions are $\alpha_M=-2\alpha_A$, $\alpha_B=2\alpha_A$, $\alpha_K=w_c\alpha_A^2$ and $\alpha_T=0$, with no-ghost discriminant $D=(w_c+6)\alpha_A^2$. Positive $D$ excludes ghosts but does not by itself exclude growing modes. The active-scalar response scan establishes a viable screened effective branch near $w_c=700$; its environmental interpretation is motivated by the selected kinetic sector and existing growth averaging, with the exact nonlinear principal-operator match treated separately. The schematic $f_B,f_K$ forms above describe general EFT bookkeeping rather than additional functions varied in the production fit.

The background-only configuration uses $\alpha_B$ and $\alpha_K$ as formal EFT bookkeeping; during native `tep_mode` background-only integration $f_B$ and $f_K$ evaluate identically to zero, verifying that this is a strict geometric mapping at the background level. The active-perturbation closure configuration (Section 4.5) passes `gravity_model = tep` and `M2_evolution = yes` to hi_class together with the regular $\delta\phi$ perturbation variable (`use_pert_var_deltaphi_smg = yes`), which is required because the canonical SMG variable $x_{\rm smg} = \delta\phi/\phi'$ is singular at the $\alpha_A = 0$ turnover ($z_\star \simeq 2.65$). The scalar fluctuations are then evolved through the derived Bellini–Sawicki EFT closure ($\alpha_B=2\alpha_A$, $\alpha_K=w_c\alpha_A^2$, $c_s^2=1$) with kinetic coefficient fixed by `tep_kin_boost`: $w_c = 7K^2$ for $K>0$, so $Z_E = 6 + 7K^2$ spans the stable screened branch; the environmental suppression factor `tep_screening_s` rescales the whole conformal coupling, $F = M_{\rm Pl}^{2}A^{-2S_\Sigma}$, so that both mixing functions screen consistently ($\alpha_M \to -2S_\Sigma\alpha_A$, $\alpha_B \to 2S_\Sigma\alpha_A$, preserving $\alpha_B = -\alpha_M$) at fixed $w_c$; the fifth force is then genuinely suppressed $\propto S_\Sigma^2/w_c$ (the previously published scaling $\alpha_K \propto S_\Sigma^2$ left $\alpha_B^2/D$ invariant and did not screen). The background mapping uses the adopted HC reference-density convention; the active perturbation completion is the effective template of Appendix A.5a. Both are evaluated with the same runtime profile, while their reduction from the universal nonlinear action on the realized environmental background remains a separate matching calculation.

### 2.3 The Static Conformal Isomorphism

The defining feature of the TEP framework is that it recasts the role normally played by a physically expanding spatial metric in the background/acoustic sector. In standard $\Lambda$CDM cosmology, the proper distance between co-moving galaxies physically increases over time, parameterized by the scale factor $a(t)$.

In the static conformal interpretation tested here, intergalactic separations are not treated as primitively expanding; the apparent expansion is reconstructed through temporal transport. The causal matter metric $\tilde{g}_{\mu\nu}$ is modulated by the conformal clock-rate field $A(\phi)$. The Temporal Equivalence Principle relies on this distinct, dynamical proper-time field and is fundamentally separate from the standard Einstein Equivalence Principle (EEP), which concerns the universality of free fall and local Lorentz invariance in metric theories of gravity. Photons propagating through this gradient experience a shift in phase and frequency, leading to the exact HC conformal-frame mapping:

\begin{equation} \label{eq:3_theory_06}
1+z = \frac{A_0}{A_{\text{em}}} = A_{\text{clock}}^{-1}(z)
\end{equation}

The exact observational clock map $A_{\text{clock}}(z)=(1+z)^{-1}$ defines the redshift–distance relation in the static conformal geometry. Because the mathematical transport geometry of $A_{\text{clock}}$ across a static background is formally isomorphic to the transport geometry of a photon in an expanding FLRW metric with reconstructed scale factor $a_{\text{eff}}(z)$, standard cosmological integrators (like `hi_class` and `CLASS`) can be deployed natively as conformal-frame calculators for the background/acoustic-sector mapping tested here. The physical dynamical response $A_{\text{dyn}}(z)$ modifies the Hubble parameter and acoustic observables at late times (TEP-TH, Paper 27; TEP-BBN, Paper 29).

- The primary acoustic peaks ($100 \lesssim \ell \lesssim 2000$) generated at $z \sim 1089$ are preserved with extreme precision, because the mathematics of the acoustic horizon $r_s$ depend only on the conformal integration path, not on physical spatial stretching.

- The parameter conventionally identified as Dark Energy ($\Omega_\Lambda$) is reinterpreted within this implementation as the homogeneous temporal-shear background contribution, $\Omega_\phi$, occupying the same background-budget role as $\Omega_\Lambda$ in the reference FLRW calculation.

- The Big Bang singularity is reinterpreted, within TEP, as a temporal-horizon boundary of the conformal clock-rate field where $A_{\text{clock}} \to 0$ relative to the present epoch, driving the reconstructed scale factor $a_{\text{eff}} \to 0$ and creating the mathematical appearance of infinite density in standard FLRW reconstructions. The full nonsingular closure—finite curvature invariants, geodesic completeness, and the native chemical-evolution framework and local CMB-thermalization proof of concept—is developed in TEP-TH (Paper 27) and TEP-BBN (Paper 29), where the non-exact covariance/topology correction $C_T$ provides the transport closure beyond exact conformal shear.

- *Thermodynamic Cooling:* The adiabatic cooling of the CMB photon gas ($T \propto 1/a$ in standard cosmology) is preserved as $T \propto 1/A$. The energy density shifts natively via the conformal temporal shear without requiring physical spatial volume dilution.

*Operational definition of $\Omega_\phi$.* In this implementation, $\Omega_\phi$ is operationally defined as the homogeneous conformal-sector contribution that fills the same background budget slot occupied by $\Omega_\Lambda$ in the reference FLRW integration. Because the same budget slot is filled by a differently interpreted physical quantity, this is a change of interpretation rather than of background physics: the background evolution is identical to $\Lambda$CDM by construction, and the acoustic-sector preservation of Section 5 follows correspondingly. A first-principles stress-energy derivation of $\rho_\phi$, $p_\phi$, and the effective equation of state belongs to the broader TEP-C0 action-level treatment.

*Archived EFT reference.* The Bellini–Sawicki $\alpha_i$ functions mapped from the TEP bi-metric action (step-3 fiducial) are archived in `results/03_alpha_functions.json`. Production CMB constraints use the native conformal implementation (Section 4), evaluating the strict isomorphism directly without relying on linear-perturbation mapping approximations.


## 3. Software Implementation: hi_class and the Finite-Turnover Conformal Regime


### 3.1 The hi_class Architecture


hi_class extends the CLASS Boltzmann solver to handle general scalar-tensor theories via the EFT formalism. This work uses hi_class v3.2.3 with the modified gravity (SMG) module enabled.


### 3.2 Native TEP Conformal Background Implementation

The native TEP background-only Hubble modification is implemented directly in hi_class via the `tep_mode` flag. When enabled, the background expansion history is modified as:

\begin{equation} \label{eq:4_implementation_01}
H_{\rm TEP}(z) = H_{\Lambda\rm CDM}(z) \times M(z), \quad M(z) = \frac{A(z)}{1 - \alpha_A(z)}
\end{equation}

where $S(z) = \exp[-(z/z_T)^{n_T}]$ is the redshift suppression factor, $A(z) = \exp[\epsilon_T \ln(1+z)\,S(z)]$ is the covariant conformal factor, and $\alpha_A = -d\ln A/d\ln(1+z)$. The production integration directly evaluates the exact HC conformal-frame mapping $M = A/(1-\alpha_A)$, ensuring computational fidelity to the underlying formal derivation (see Appendix A.3a). The transition function $f_T(z) = \ln(1+z)\,S(z)$ appearing in the exponent is the shared TEP-C0 implementation (Paper 26; `core/cosmology.py`: `f_T`, `conformal_factor_native`, `jordan_frame_M`):

\begin{equation} \label{eq:4_implementation_01b}
f_T(z) = \ln(1+z)\,S(z).
\end{equation}

The suppression factor $S(z)$ defines the physical profile of the conformal field: it drives $f_T \to 0$ for $z \gg z_T$, ensuring the field profile flattens asymptotically as it approaches the TEP temporal horizon; the $\ln(1+z)$ factor enforces $f_T(0)=0$, fixing the local reference frame so $H_0$ is anchored to the local observer. The function peaks at intermediate redshift ($z \sim z_T$), where the effective homogeneous temporal-shear contribution mimics apparent acceleration, and flattens out in the deep past.

*Implementation note:* an earlier development build used the *complement* $f_T = 1 - \exp[-(z/z_T)^{n_T}]$, which instead saturates to unity for $z \gg z_T$. This inverted the scalar field profile and corrupted the acoustic peak evaluation (see Appendix A.3a). The default TEP conformal parameters are:



```
tep_mode = yes
epsilon_T = 0.0066
z_T = 5.0
n_T = 2.0
```



#### Parameter-Scale and Amplitude Convention

**Turnover scales.** $z_T^{\rm HC}$ denotes the homogeneous/acoustic `hi_class` profile scale used here. $z_T^{\rm los}$ denotes the C0 line-of-sight supernova transport turnover. These scales are related projections of the temporal sector but are not numerically interchangeable.

**Amplitudes.** $\epsilon_T^{\rm HC}$ denotes the native `hi_class` homogeneous conformal amplitude reported here ($0.0062 \pm 0.0046$, median $0.0053$, from the converged four-chain low-$\ell$ background-channel posterior). $\epsilon_T^{\rm los}$ denotes the late-time line-of-sight transport amplitude fitted in TEP-C0. $\epsilon_T^{\rm CMB}$ denotes the C0 background/acoustic diagnostic amplitude. $\epsilon_{\rm dyn}$ denotes the dynamical temporal-horizon response in TEP-TH, while $\epsilon_{\rm field}=0.01755$ denotes the primordial spectral-flow parameter constrained by $n_s$ in TEP-TH. These are related projections of the same temporal sector, but they are not numerically interchangeable parameters.



The background conformal mapping is implemented through $M(z)=A/(1-\alpha_A)$. The associated linear scalar perturbation sector is closed separately through the Bellini–Sawicki runtime functions described in Section 3.3 and Appendix A.5, and validated in the active $\delta\phi$ run of Section 4.5. This implementation is the hi_class analogue of the CLASS native TEP module used in TEP-C0 (Paper 26).


### 3.3 Perturbation Stability and Closure

The present analysis natively closes the scalar perturbation sector by evaluating the derived runtime Bellini–Sawicki Effective Field Theory (EFT) parameters in the pure conformal limit. While early formulations of the TEP acoustic mapping relied on the working assumption that scalar spatial fluctuations ($\delta\phi$) decouple or are heavily suppressed at recombination, the implementation developed here leverages the `hi_class` SMG framework to evolve the full Einstein-Boltzmann hierarchy actively.

Because the TEP framework maps its causal matter metric via the exact conformal relation $\tilde{g}_{\mu\nu}=A(\phi)^{2}g_{\mu\nu}$, the corresponding EFT functions ($\alpha_i$) in the Jordan frame are determined by the background derivative $\alpha_A \equiv -d\ln A / d\ln(1+z)$. For the implemented pure-conformal branch, the luminal EFT closure $\alpha_B=2\alpha_A$, $\alpha_K=w_c\,\alpha_A^2$, and $c_s^2=1$ is the Jordan-frame Bellini–Sawicki image of that branch: the Weyl transform of the Einstein-frame scalar gives $F(\phi) = M_{\rm Pl}^{2}A^{-2}$ and $W(\phi) = A^{-2}\left(Z_E - 6M_{\rm Pl}^{2}A_{,\phi}^{2}/A^{2}\right)$, where $Z_E \equiv P_{,X} + 2XP_{,XX}$ is the Einstein-frame kinetic coefficient of the scalar on the background ($Z_E = 1$ for a canonical scalar; the TEP $P(X)$ sector makes $Z_E$ environment-dependent). The standard Horndeski map returns $\alpha_M = -2\alpha_A$, $\alpha_B = -\alpha_M = 2\alpha_A$, $\alpha_K = (W/F)(d\phi/d\ln a)^{2} = (Z_E-6\beta_A^2)\alpha_A^{2}/\beta_A^{2} \equiv w_c\,\alpha_A^{2}$ at $\beta_A = -1$, and $\alpha_T = 0$ (Appendix A.5a). The background conformal mapping $M = A/(1-\alpha_A)$ (Appendix A.3a) and this perturbative completion descend from the same bi-metric action. The canonical image $w_c = -5$ ($Z_E = 1$) is the required Jordan-frame form of a canonical Einstein-frame scalar at $|\beta_A| > 1/\sqrt{6}$, with positive no-ghost discriminant $D = (w_c+6)\alpha_A^2$; however, the direct $\delta\phi$ Boltzmann integration reported below shows that $D > 0$ is necessary but not sufficient: the $w_c = -5$ branch is linearly unstable on the TEP background (a tachyonic scalarization mode that saturates only nonlinearly), and stability requires $w_c > 0$, i.e. $Z_E > 6$.

Evaluating the full discriminant yields $D = \alpha_K + \frac{3}{2}\alpha_B^2 = (w_c + 6)\alpha_A^2$ (at screening factor $S_\Sigma = 1$; generally $D = (w_c + 6S_\Sigma^2)\alpha_A^2$). For the canonical image $w_c = -5$ this reduces to the previously quoted $D = \alpha_A^2 \ge 0$; on the production branch $w_c = +700$ it is $D = 706\,\alpha_A^2 > 0$. Because the finite-turnover profile $f_T(z)$ has a single extremum, $\alpha_A = -\epsilon_T\,df_T/d\ln(1+z)$ vanishes at the turnover. For $z_T = 5$ and $n_T = 2$, this occurs at $z_\star \simeq 2.64785$. Consequently, $D$ touches zero at $z_\star$ for every nonzero $\epsilon_T$ on any branch. The relevant stability question is therefore whether $z_\star$ is a regular scalar-decoupling point, rather than whether $D$ remains bounded away from zero. Away from this point, $D > 0$; at the turnover all EFT modification functions vanish and the implemented system reaches the GR/conformal decoupling limit. The decoupling limit occurs both at $\epsilon_T = 0$ and instantaneously at extrema of the finite-turnover conformal profile.

In the Bellini–Sawicki EFT used by `hi_class`, the scalar kinetic coefficient is $Q_s = 2M_\star^2\,D/(2-\alpha_B)^2$. At $z_\star$, all EFT functions vanish ($\alpha_A = \alpha_B = \alpha_K = \alpha_M = 0$), so $D = Q_s = 0$ and the propagating scalar description reaches a degenerate GR/conformal limit. The linear solver completes the crossing without numerical divergence. Table 1 displays the analytic behavior of the EFT coefficients, while the finite spectra and completed active-perturbation chains provide the numerical regularity check. This establishes numerical regularity of the crossing within the linear EFT closure; numerical regularity does not by itself establish physical stability. The canonical derivation of Appendix A.5a identifies the crossing kinematically: $D = (w_c + 6)\,\alpha_A^{2} = (w_c+6)(d\phi/d\ln a)^{2}/M_{\rm Pl}^{2}$, so $D \to 0$ at $z_\star$ is the field-velocity zero $\dot\phi = 0$ at the profile turnover — a degeneracy of the unitary-gauge description (the comoving slicing $\delta\phi = 0$ loses its normalization where the background field instantaneously turns), not a surface on which any operator coefficient of the canonical action diverges. In the Einstein frame the scalar kineticity is constant at $\dot\phi = 0$, so the crossing is exact constraint reduction rather than a dynamical strong-coupling point at the level of the implemented sector. A residual nonlinear/canonical analysis of the full disformal sector is outside the scope of the present linear acoustic-sector closure. The `hi_class` SMG solver evaluates the EFT coefficients at each redshift step; no numerical regulator or tolerance threshold is applied, and the crossing is continuous in the neighbourhood of $\alpha_A = 0$.



Table 1: $D = 0$ crossing diagnostic at $z_\star \simeq 2.64785$ (fiducial $\epsilon_T = 0.0066$, $z_T = 5$, $n_T = 2$). $Q_s = 2M_\star^2 D/(2-\alpha_B)^2$ with $M_\star^2 = 1$ in reduced Planck units. The displayed EFT coefficients remain finite or vanish continuously through the turnover; $D$ and $Q_s$ vanish at $z_\star$, where the propagating scalar description becomes degenerate.

| $z - z_\star$ | $z$ | $\alpha_A$ | $\alpha_B$ | $D = \alpha_A^2$ | $Q_s$ | Linear solver status |
| --- | --- | --- | --- | --- | --- | --- |
| $-10^{-3}$ | $2.646848$ | $-4.31 \times 10^{-6}$ | $-8.61 \times 10^{-6}$ | $1.85 \times 10^{-11}$ | $9.27 \times 10^{-12}$ | Finite |
| $-10^{-4}$ | $2.647748$ | $-4.31 \times 10^{-7}$ | $-8.61 \times 10^{-7}$ | $1.85 \times 10^{-13}$ | $9.27 \times 10^{-14}$ | Finite |
| $-10^{-5}$ | $2.647838$ | $-4.30 \times 10^{-8}$ | $-8.61 \times 10^{-8}$ | $1.85 \times 10^{-15}$ | $9.26 \times 10^{-16}$ | Finite |
| $0$ | $2.647848$ | $0$ | $0$ | $0$ | $0$ | Finite |
| $+10^{-5}$ | $2.647858$ | $+4.29 \times 10^{-8}$ | $+8.58 \times 10^{-8}$ | $1.84 \times 10^{-15}$ | $9.20 \times 10^{-16}$ | Finite |
| $+10^{-4}$ | $2.647948$ | $+4.31 \times 10^{-7}$ | $+8.62 \times 10^{-7}$ | $1.86 \times 10^{-13}$ | $9.28 \times 10^{-14}$ | Finite |
| $+10^{-3}$ | $2.648848$ | $+4.31 \times 10^{-6}$ | $+8.61 \times 10^{-6}$ | $1.85 \times 10^{-11}$ | $9.27 \times 10^{-12}$ | Finite |

By evolving the implemented linear pure-conformal $\delta\phi$ scalar field through this derived EFT closure, the numerical solver verifies that the continuous conformal transition smoothly manages the emergence into the late universe without triggering gradient instabilities or pathological phantom energies — provided the kinetic coefficient is on the stable branch. The full $\delta\phi$ integration maps the stability boundary directly (Step 24, `results/closure_response_scan.json`): the canonical image $w_c = -5$ produces a runaway growing mode ($\sigma_8 \sim 4\times10^{4}$, median $P(k)$ enhancement $\sim 10^{6}$); the minimally stable unscreened branch $w_c = +7$ gives $\sigma_8 = 1.61$ with a $\sim 3\times$ matter-power enhancement (the O(1) scalar-tensor fifth force of the conformal coupling; the ISW-dominated low-$\ell$ TT plateau is amplified by a factor $\sim 59$ at $\ell = 2$; Step 24, `results/step_24_lowell_isw_scan.json`); and the screened production branch $w_c = 700$ ($\texttt{tep\_kin\_boost} = 10$, i.e. $Z_E \simeq 706$) returns $\sigma_8 = 0.824$ versus $0.823$ for $\Lambda$CDM at the same cosmology, with matter-power deviations $\lesssim 2\%$ and an ISW residual of $+13\%$ at $\ell = 2$ falling to $\lesssim 1\%$ by $\ell \gtrsim 100$ — inside the low-$\ell$ cosmic-variance floor. The background acoustic isomorphism is therefore preserved under active linear perturbation evolution on the screened branch. (An earlier version of this section reported a sub-$0.001\%$ ISW residual; that figure was measured while the SMG perturbation sector was silently disabled by a background-density gate and is superseded by the present active-scalar numbers.)


### 3.3b Covariant Frame Alignment: Matter-Frame Hubble Friction

The TEP action defines the causal metric for matter particles as $\tilde{g}_{\mu\nu} = A(\phi)^2 g_{\mu\nu}$. Under this conformal rescaling, the matter-frame conformal Hubble rate is related to the Einstein-frame rate by the exact derivation of Appendix A.3a. Using explicit frame-labelled variables, the matter-frame conformal Hubble rate satisfies

\begin{equation}
\widetilde{\mathcal{H}} = \mathcal{H}_E + \frac{A'}{A}, \qquad \frac{A'}{A} = \alpha_A\,\widetilde{\mathcal{H}},
\end{equation}

where $\mathcal{H}_E = a_E H_E$ is the Einstein-frame conformal rate of the isomorphic representation (Appendix A.3a), primes denote derivatives with respect to Einstein-frame conformal time, and $\alpha_A \equiv -d\ln A/d\ln(1+z)$. Solving for $\widetilde{\mathcal{H}}$ yields

\begin{equation} \label{eq:3_3b_Htilde}
\widetilde{\mathcal{H}} = \frac{\mathcal{H}_E}{1 - \alpha_A},
\end{equation}

consistent with the exact background relation $\tilde{H} = A\,H_{\Lambda\rm CDM}/(1-\alpha_A)$ derived in Appendix A.3a (Eq. $\ref{eq:a3a_exact}$). In the Jordan-frame EFT, matter particles follow geodesics of $\tilde{g}_{\mu\nu}$, so their Euler equations must be evaluated with $\widetilde{\mathcal{H}}$ rather than the Einstein-frame rate $\mathcal{H}_E$.

The variable `index_bg_H` in the patched hi_class code stores the TEP-mapped Hubble rate $H_{\rm TEP}(z) = M(z)\,H_{\Lambda\rm CDM}(z)$, which already incorporates the full conformal-frame factor $M = A/(1-\alpha_A)$ (Section 3.2). This is the matter-frame rate $\widetilde{H}$, not the Einstein-frame rate $H_E$. The conformal-time counterpart `index_bg_H_conformal` is therefore $\widetilde{\mathcal{H}} = a\,\widetilde{H}$, which is already the full matter-frame conformal rate. No additional multiplication by $(1-\alpha_A)$ is applied; the displayed code in earlier drafts that showed such a multiplication was a documentation error and has been corrected. The production codebase evaluates $M = A/(1-\alpha_A)$ in the background Hubble setter, and `index_bg_H` stores the resulting mapped rate directly.

Because null geodesics are conformally invariant, photons and neutrinos (while relativistic) propagate using the same mapped conformal rate; the photon and collision terms use the conformal-time scale factor $\tilde{a} = A\,a_E$ and its derivative, which are the quantities stored in the mapped background. This is a direct consequence of conformal invariance of null trajectories under $g_{\mu\nu} \to A^2 g_{\mu\nu}$, not a separate frame assignment. The metric equations retain the mapped background consistently, preserving the CMB acoustic physics. The modification is minimal and upstream: the background Hubble setter applies the exact $M = A/(1-\alpha_A)$ factor, and all species natively use the resulting mapped conformal rate.


### 3.4 Pipeline Architecture

The full analysis pipeline, executed via `scripts/run_all.py`, consists of:


- *Step 0 (Setup):* Environment configuration and dependency check.

- *Step 1 (Install):* Install Cobaya, Planck 2018 likelihoods, and hi_class with the native TEP patch (`external/patches/hiclass_tep_native.patch`).

- *Step 2 (Background):* Compute the TEP-modified background expansion history $H(z)$ and density evolution.

- *Step 3 (Alpha Functions):* Compute Bellini–Sawicki coefficients from the TEP theoretical mapping (archived for reference).

- *Step 4 (CMB Spectra):* Run hi_class with native `tep_mode` at the Planck 2018 best-fit point. Compare TT, TE, and EE spectra against standard CLASS $\Lambda$CDM.

- *Step 5 (Matter-Frame Scan):* Dual-scan reconstruction of the acoustic scale in finite-turnover and power-law diagnostic limits.

- *Step 6 (Cobaya Config):* Generate the Cobaya YAML configuration for the MCMC pipeline with native TEP parameters.

- *Step 7 (MCMC):* Execute the Cobaya MCMC with hi_class, using real Planck + BAO + Pantheon+ likelihoods, for both the background-only configuration and the active-perturbation closure configuration (`gravity_model = tep`).

- *Step 8 (Posteriors):* Analyze MCMC chains with burn-in removal and weighted statistics.

- *Step 9 (Synthesis):* Combine all results into summary JSON and markdown.


Publication figures are generated separately via `python scripts/generate_figures.py` (not part of `run_all.py`). Both figures are written to `results/figures/` with filenames matching their publication numbering. Figure 3 requires step 04b. Include them in the static site with `cd site && npm run build`.

The running EFT $\alpha$-functions provide the linear cosmological realization of the dynamical temporal response $A_{\rm dyn}$. Their continuous evolution produces the required conformal perturbation behaviour across cosmological epochs without introducing a separate phenomenological epoch-screening function.

## 4. MCMC Parameter Estimation Pipeline

### 4.1 The Cobaya Framework

Cobaya provides a Python interface to CLASS/hi_class with extensive MCMC sampling capabilities. The transition from SciPy/Pandas pipelines to Cobaya enables:

- Native hi_class integration without file-based I/O bottlenecks

- Parallel tempering and adaptive Metropolis-Hastings sampling

- Direct Planck likelihood wrapper integration

- Seamless GetDist posterior visualization

### 4.2 Likelihood Configuration

The pipeline uses the following Planck 2018 likelihoods:

| Likelihood | Description | $\ell$ Range |
| --- | --- | --- |
| `planck_2018_lowl.TT` | Low-$\ell$ temperature | 2–29 |
| `planck_2018_lowl.EE` | Low-$\ell$ polarization | 2–29 |
| `planck_2018_lensing.native` | CMB lensing reconstruction | 8–400 |
| `bao.sdss_dr12_consensus_final` | BAO SDSS DR12 consensus | — |
| `sn.pantheonplus` | Type Ia supernovae (Pantheon+) | — |

### 4.3 Free Parameters and Priors

The MCMC pipeline samples standard $\Lambda$CDM parameters alongside the TEP amplitude parameter $\epsilon_T$:

| Parameter | Prior | Description |
| --- | --- | --- |
| $\Omega_b h^2$ | $\mathcal{U}(0.005, 0.1)$ | Baryon density |
| $\Omega_{\rm cdm} h^2$ | $\mathcal{U}(0.01, 0.99)$ | Cold dark matter density |
| $H_0$ | $\mathcal{U}(40, 100)$ | Hubble constant |
| $\tau$ | $\mathcal{U}(0.01, 0.8)$ | Optical depth |
| $A_s$ | $\mathcal{U}(10^{-10}, 5 \times 10^{-9})$ | Scalar amplitude |
| $n_s$ | $\mathcal{U}(0.94, 1.0)$ | Scalar spectral index |
| $A_{\rm planck}$ | $\mathcal{U}(0.9, 1.1)$ | Planck calibration nuisance |
| $\epsilon_T$ | $\mathcal{U}(-1, 1)$ | TEP amplitude parameter (background Hubble modification) |

### 4.4 Pipeline Execution

```
# Cobaya YAML configuration
theory:
  classy:
    path: /path/to/hi_class
    extra_args:
      output: tCl,pCl,lCl,mPk
      lensing: yes
      modes: s,t
      non_linear: halofit
      # Native TEP background-only Hubble modification
      tep_mode: 'yes'
      z_T: 5.0
      n_T: 2.0
      # epsilon_T is sampled in params below — do not duplicate here

likelihood:
  planck_2018_lowl.TT: null
  planck_2018_lowl.EE: null
  planck_2018_lensing.native: null
  bao.sdss_dr12_consensus_final: null
  sn.pantheonplus: null

params:
  logA:
    prior: {min: 2.5, max: 3.5}
    ref: {dist: norm, loc: 3.044, scale: 0.014}
    proposal: 0.01
    drop: true
  A_s:
    value: 'lambda logA: 1e-10*np.exp(logA)'
  n_s:
    prior: {min: 0.94, max: 1.0}
    ref: {dist: norm, loc: 0.966, scale: 0.004}
    proposal: 0.004
  H0:
    prior: {min: 40, max: 100}
    ref: {dist: norm, loc: 67.4, scale: 0.5}
    proposal: 1.5
  omega_b:
    prior: {min: 0.005, max: 0.1}
    ref: {dist: norm, loc: 0.0224, scale: 0.0002}
    proposal: 0.0003
  omega_cdm:
    prior: {min: 0.01, max: 0.99}
    ref: {dist: norm, loc: 0.12, scale: 0.001}
    proposal: 0.0015
  tau_reio:
    prior: {min: 0.01, max: 0.8}
    ref: {dist: norm, loc: 0.054, scale: 0.007}
    proposal: 0.01
  A_planck:
    prior: {min: 0.9, max: 1.1}
    ref: {dist: norm, loc: 1.0, scale: 0.0025}
    proposal: 0.005
  epsilon_T:
    prior: {min: -1.0, max: 1.0}
    ref: {dist: norm, loc: 0.006, scale: 0.005}
    proposal: 0.0005
    latex: '\epsilon_T'
  sigma8:
    latex: '\sigma_8'

sampler:
  mcmc:
    burn_in: 0
    max_tries: 10000
    max_samples: 500000
    Rminus1_stop: 0.05
    Rminus1_cl_stop: 0.2
    output_every: 10
    drag: true
    seed: 42
```

### 4.5 Perturbation-Mode Validation

In addition to the background-only MCMC configuration above, the pipeline includes an active perturbation closure configuration (`data/cobaya/tep_hiclass_perturbations.yaml`) that passes `gravity_model = tep` and `M2_evolution = yes` to the `hi_class` SMG module. This directs the Boltzmann solver to evolve the implemented linear pure-conformal scalar fluctuation sector $\delta\phi$ through the derived Bellini–Sawicki EFT closure ($\alpha_B=2\alpha_A$, $\alpha_K=w_c\alpha_A^2$, $c_s^2=1$; Appendix A.5). The production configuration evolves the regular $\delta\phi$ variable (`use_pert_var_deltaphi_smg = yes`, with `pert_initial_conditions_smg = zero`) — required because the canonical SMG variable $x_{\rm smg}=\delta\phi/\phi'$ is singular at the field-velocity zero $z_\star\simeq2.65$ — on the screened kineticity branch $w_c = 700$ (`tep_kin_boost = 10`), for which the scalar response decouples (Section 3.3, `results/closure_response_scan.json`).

A combined 5-chain MCMC of the active-perturbation model (single-chain + 4-chain MPI; 58,180 accepted steps, 40,726 post-burn-in samples; Planck 2018 low-$\ell$ TT/EE + lensing + BAO + Pantheon+; cross-chain Gelman–Rubin $R-1 = 0.007$ at termination) ran successfully with finite posterior at every step. All five chains sample the same active-perturbation model, priors, and likelihood, and the quoted $R-1$ is recalculated across all five chains. The scalar perturbation sector was subsequently found to be silently inactive in these chains (see the provenance note below); this posterior therefore constrains the background channel. The primary low-$\ell$ determination is now supplied by the dedicated four-chain MPI suite run ($\epsilon_T = 0.0062 \pm 0.0046$, median $0.0053$, $\epsilon_T < 0.015$ at 95% CL; max parameter $R-1 = 0.011$), whose posterior agrees with the five-chain background-channel result to within $0.2\sigma$. It is compared below with the earlier single-chain exploratory background-only run ($\epsilon_T = 0.00602 \pm 0.00493$), which is retained as a solver-comparison diagnostic because its cross-chain $R$-statistic is undefined, having been superseded as the primary determination by the four-chain MPI suite run above. The resulting posterior summaries are:

\begin{equation} \label{eq:5_mcmc_epsT_pert}
\epsilon_T = 0.00547 \pm 0.00429,
\end{equation}

with $H_0 = 66.77 \pm 1.73$ km/s/Mpc, $n_s = 0.9956 \pm 0.0043$, $\Omega_b h^2 = 0.02144 \pm 0.00257$, $\Omega_{\rm cdm} h^2 = 0.1155 \pm 0.0042$, $\tau = 0.0497 \pm 0.0075$, $A_{\rm planck} = 1.088 \pm 0.013$, $\sigma_8 = 0.858 \pm 0.016$, and $S_8 = 0.868 \pm 0.025$. Two parameters are prior-boundary limited in this low-$\ell$-only configuration: $n_s$ accumulates at the upper bound of its $\mathcal{U}[0.94, 1.00]$ prior ($n_s^{\rm max} = 1.00000$), and $A_{\rm planck}$ saturates the upper bound of its $\mathcal{U}[0.9, 1.1]$ prior ($A_{\rm planck}^{\rm max} = 1.1000$). This is a known feature of low-$\ell$-only CMB analyses, which lack the high-$\ell$ acoustic information needed to break the $n_s$–$A_{\rm planck}$–$\tau$ degeneracy. A dedicated sensitivity run with $A_{\rm planck}$ widened to $\mathcal{U}[0.9, 1.25]$ yields $A_{\rm planck} = 1.229 \pm 0.026$ (still near the new ceiling) but $\epsilon_T = 0.0063 \pm 0.0048$, consistent with the 5-chain active posterior at $0.12\sigma$. A parallel multi-chain validation run (`tep_native_mcmc.yaml`) with $n_s$ widened to $\mathcal{U}[0.9, 1.05]$ and a Gaussian $A_{\rm planck}$ prior (loc = 1.0, scale = 0.0025) yields $n_s = 1.046 \pm 0.005$ (still saturating the wider upper bound) and $\epsilon_T = 0.0066 \pm 0.0049$, consistent with the 5-chain active posterior at $0.17\sigma$. The $n_s$ and $A_{\rm planck}$ posteriors should therefore be interpreted as robustness diagnostics of the $\epsilon_T$ constraint under prior-boundary saturation, not as final spectral-index or calibration determinations. The boundary-limited $n_s$ and $A_{\rm planck}$ values are not treated as physical measurements. The TEP amplitude $\epsilon_T$, however, shifts by no more than $0.17\sigma$ across the alternative prior configurations. The $\epsilon_T$ posterior is stable across all prior configurations: the maximum shift across the 5-chain active posterior, the widened-$A_{\rm planck}$ run, and the widened-$n_s$ run is $0.17\sigma$. Direct comparison with the background-only chain ($\epsilon_T = 0.00602 \pm 0.00493$) yields $\Delta\epsilon_T = -0.00055$ ($-0.08\sigma$), $\Delta H_0 = +0.09$ km/s/Mpc ($+0.04\sigma$), $\Delta n_s = +0.00009$ ($+0.02\sigma$), and $\Delta S_8 = +0.0009$ ($+0.03\sigma$). The maximum parameter disagreement across all eight cosmological parameters is $0.07\sigma$, and $\Delta\chi^2 = -0.46$. Figure 1 shows the background versus active-perturbation posterior comparison.

*Provenance note (post-audit).* The "active-perturbation" chains quoted in this section were executed with a build in which the SMG scalar-perturbation sector was silently disabled by a background-density gate in `has_smg`; their posteriors therefore constrain the background channel only, and the quoted chain-to-chain agreement demonstrates solver consistency, not scalar-perturbation negligibility. The genuinely active $\delta\phi$ closure — implemented via the regular $\delta\phi$ variable with the corrected kinetic-coefficient map — is characterised in Section 3.3 and by the registered response scan (`results/closure_response_scan.json`): on the production screened branch ($w_c = 700$) the active spectra lie within $\sim 2\%$ of the background-only spectra in the acoustic band and $\sim 18\%$ at low $\ell$ (ISW), so the background-channel posteriors remain a valid approximation for that branch; on the unscreened stable branch ($w_c = +7$) the model is excluded outright ($\sigma_8 \simeq 1.6$), and on the canonical image ($w_c = -5$) the spectrum diverges, so an MCMC on those branches is not meaningful. The re-run of the active-perturbation chain on the corrected closure has since been completed: a converged two-chain MPI run on the production screened branch ($w_c = 700$, `tep_kin_boost = 10`, regular $\delta\phi$ variable; 3,961 post-burn-in samples; max parameter $R-1 = 0.007$) yields $\epsilon_T = 0.0054 \pm 0.0035$ and $\sigma_8 = 0.814 \pm 0.031$, with $\epsilon_T$ consistent with the background-channel posterior at $-0.14\sigma$ and $\sigma_8$ shifted by $-1.3\sigma$ ($S_8$: $-1.2\sigma$) relative to the suite run — the screened scalar's subdominant spectrum imprint leaves the homogeneous-amplitude inference essentially unchanged, as the closure audit predicts, while the modest downward $\sigma_8$ shift tracks the scalar's contribution to the low-$\ell$ ISW residual registered in the closure response scan. Notably, $A_{\rm planck} = 1.019 \pm 0.051$ on the active branch no longer saturates the prior edge, indicating that part of the boundary pressure in the background-channel posterior is absorbed by the active scalar degree of freedom. The corrected-closure inference has also been extended to the full high-$\ell$ TTTEEE likelihood set (`tep_hiclass_perturbations_ttteee`; fixed $z_T = 5$, $n_T = 2$, same screened branch and regular $\delta\phi$ variable; converged four-chain MPI run, 22,278 accepted steps and 15,596 post-burn-in rows, sampler means $R-1 = 0.050$, classic scalar parameter max $R-1 = 0.009$): it yields $\epsilon_T = 0.0049 \pm 0.0031$ ($<0.0104$ at 95% CL), $\sigma_8 = 0.828 \pm 0.011$, and $A_{\rm planck} = 1.019 \pm 0.019$, with best sampled $\chi^2 = 2422.1$ against $2421.4$ for the background-channel high-$\ell$ run — the active scalar leaves the homogeneous-amplitude inference statistically unchanged on the complete data set ($+0.15\sigma$ relative to the fixed-$z_T = 5$ background posterior), confirming the closure audit's subdominance prediction at high $\ell$.

The `omega_cdm` parameter above is the standard clustering cold-dark-matter fluid evolved by `hi_class`, not scalar stress-energy inferred from the TEP action. Consequently the low-$\ell$ plus Planck-lensing posterior tests this implemented perturbation model *with CDM*. Step 21 (`scripts/steps/step_21_cdm_free_lensing.py`, artifact `results/step_21_cdm_free_lensing.json`) evaluates both the isotropic-conformal `hi_class` branch and the unstretched-space temporal-depth refraction channel when `omega_cdm` is removed ($\omega_{\rm cdm}=0$), including a full multipole $C_\ell^{\phi\phi}$ likelihood fit ($\ell\in[2,2500]$) against the official 9-bin Planck 2018 `CMBMarged` lensing likelihood ($L\in[8,400]$ with full $9\times9$ covariance matrix):
(1) Under the isotropic conformal rescaling $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$ implemented in `hi_class` — where spatial rulers stretch by $A^2(\phi)$ alongside clocks so that $A^2(\phi(x))$ divides out of null rays — removing `omega_cdm` reduces $C_{100}^{\phi\phi}$ to $0.0875$ and $C_{400}^{\phi\phi}$ to $0.00923$ of the CDM-containing values ($\sigma_8$ from $0.818$ to $0.075$), and on the active-perturbation branch ($w_c = 700$) to $0.0872$ and $0.00915$ ($\sigma_8$ from $0.821$ to $0.075$, Planck 2018 lensing $\chi^2=1265.73$).
(2) Under the canonical unified matter metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) — where the $(3+1)$ ADM spacetime lapse rescales as $N_{\rm TEP}(\mathbf x)=A_{\rm loc}(u_{\rm grav}(\mathbf x))N_{\rm GR}(\mathbf x)$ while at a fixed cosmic epoch ($A_{\rm sp}(t_0)=1$) the spatial 3-metric $\tilde\gamma_{ij}=\gamma_{ij}$ remains unstretched — intermediate clock rates along the photon path do not cancel: every intermediate point $x$ contributes its local temporal depth $-\beta_A u_{\rm grav}(x)$ directly to the optical refractive index $n(x)\simeq1-2\Phi_b(x)/c^2-\beta_A u_{\rm grav}(x)$ while preserving $c_{\rm GW}=c_\gamma$ identically on the shared cone ($\Delta t_{\gamma-\rm GW}=0$). Evaluated on the selected two-branch kinetic sector $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$ fixed by $a_0$, zero per-system free parameters) across the cluster-outskirt and cosmic-web filament scales that dominate the CMB lensing kernel ($R=1$–$10$ Mpc, $g_b/g_t=8.2\times10^{-4}$ to $5.1\times10^{-3}$), the solved response $y_{\rm tb}=4.15$–$10.37$ multiplies the baryon lensing potential by $M=1+\tfrac12\beta_A^2 y_{\rm tb}=3.07$–$6.18$ ($1\times$ temporal lapse) to $1+\beta_A^2 y_{\rm tb}=5.15$–$11.37$ ($2\times$ dynamical normalization), boosting $C_\ell^{\phi\phi}$ by $9.4\times$–$38.3\times$ ($1\times$) to $26.5\times$–$129.3\times$ ($2\times$) and recovering $82\%$–$334\%$ ($1\times$) to $231\%$–$1128\%$ ($2\times$) of the Planck $C_{100}^{\phi\phi}$ amplitude ($11.5\times$ required) and up to $118\%$ of $C_{400}^{\phi\phi}$ at $R=1$ Mpc ($109\times$ required).
(3) Integrating the continuous two-branch cosmic-web transfer $y_{\rm eff}(\ell)$ across the Silk transition from the pre-Silk supercluster/sheet regime ($q_{\rm sheet}=7.10\times10^{-2}$, $\log_{10}q_{\rm sheet}=-1.15\pm0.07$ marginalized under the full $4\times4$ Silk-prior-anchored inverse Hessian; $\pm0.04$ conditional curvature) to the post-Silk cosmic-web filament floor ($q_{\rm fil}=3.11\times10^{-4}$, $\log_{10}q_{\rm fil}=-3.51\pm0.08$ marginalized; $\pm0.02$ conditional) and fitting directly to the 9 Planck 2018 `CMBMarged` lensing bandpowers on the pure-baryon $\omega_{\rm cdm}=0$ spectra yields $\chi^2=15.40$ in the 2-parameter fixed-Silk fit ($\ell_{\rm Silk}=300,\alpha_{\rm Silk}=2$, with full $2\times2$ covariance $\log_{10}q_{\rm sheet}=-1.06\pm0.05$, $\log_{10}q_{\rm fil}=-3.48\pm0.02$, $r=-0.58$) and $\chi^2=13.40$ ($\chi^2/N=1.49$ across the $N=9$ bandpowers, $\ell_{\rm Silk}=299.9\pm26.6$ marginalized [$\pm5.1$ conditional], $\alpha_{\rm Silk}=2.15\pm0.14$ marginalized [$\pm0.05$ conditional]) in the 4-parameter profile fit — within $\Delta\chi^2=+1.11$ of the $\omega_{\rm cdm}=0.1155$ reference ($\chi^2=12.30$) and improving over the bare isotropic no-CDM spectrum by $\Delta\chi^2=-1252.33$ across $L\in[8,400]$ without particulate dark matter.
(4) In the pre-boundary acoustic regime ($z\ge z_*$), because the proper-time field $\phi$ fills all space as a continuous ambient scalar medium ($A(\phi)>0$ everywhere) with fixed action constant $g_t=\Lambda_X^2/(\sqrt{2}M_{\rm Pl})=cH_0/2=3.314\times10^{-10}\,{\rm m\,s^{-2}}$ (evaluated at $H_0=68.2$; the corpus-wide rounded value $3.4\times10^{-10}\,{\rm m\,s^{-2}}$ corresponds to $H_0=70$, the normalization used for $k=16.03$) and conformally invariant sound-horizon acceleration ratio $q_s=g_b(r_s)/g_t=[(4\pi/3)G\rho_{b,0}r_s]/g_t=\tfrac12(100\,{\rm km\,s^{-1}\,Mpc^{-1}})^2\omega_b r_s/(c\,g_t)$ — in which $h$ cancels out of $\rho_{b,0}\propto\omega_b=\Omega_b h^2$ when written in physical action units, so neither $r_s$ nor $H_0$ is a free choice — the same zero-parameter two-branch clock-slope equation $\nabla\cdot[P_{,X}(|\nabla u|)\nabla u]=\beta_A\rho_b$ evaluated at the sound horizon ($r_s=146.19\text{ Mpc}$, $q_s=1.62\times10^{-3}$, $y_s=7.38$, $\omega_\phi=0.1166$ in the single-pass analytic integral, refining under self-consistent $r_s(\omega_\phi)$ fixed-point iteration in `hi_class` to $r_s=144.94\text{ Mpc}$, $q_s=1.59\times10^{-3}$, $y_s=7.44$, and closing as a unique simultaneous root $(H_0=68.167\text{ km s}^{-1}\text{ Mpc}^{-1},\,r_s=144.95\text{ Mpc},\,q_s=1.594\times10^{-3},\,y_s=7.436,\,\omega_\phi=0.11815)$ of $r_s=r_s^{\rm CLASS}(\omega_b,\omega_\phi(q_s))$ and $100\theta_s(\omega_b,\omega_\phi(q_s),H_0)=1.04198$ from $\omega_b=0.02247$ and $T_{\rm CMB}=2.7255\text{ K}$ alone with zero free parameters) predicts an extra clock-slope well-driving density $\omega_\phi=\omega_b(y_s/\sqrt{2})=0.11815\text{--}0.11816$ ($\omega_{m,\rm eff}=\omega_b(1+y_s/\sqrt{2})=0.14062\text{--}0.14063$) from ordinary baryons ($\omega_b=0.02247$) alone (spanning $\omega_\phi\in[0.1174,0.1199]$ across $H_0\in[67.4,70.0]\text{ km s}^{-1}\text{ Mpc}^{-1}$). On the spatially flat background of the TEP matter metric (spatial structure enters only through $A_{\rm sp}^2\gamma_{ij}$, $\Omega_k=0$), the same root also fixes the homogeneous temporal-shear component with no further input: $\Omega_{m,\rm eff}=\omega_{m,\rm eff}/h^2=0.14062/0.68167^2=0.3026$, hence $\Omega_\phi=1-\Omega_{m,\rm eff}-\Omega_r=0.697$ (Papers 26 and 27 quote the rounded pair $0.304/0.696$), compared with the Planck 2018 $\Lambda$CDM value $\Omega_\Lambda=0.685\pm0.007$ and $0.704$ from the single-history Pantheon+ + DESI DR2 + $\theta_*$ fit (TEP-C0 Step 03-14). The dark-energy-like fraction is therefore an output of the baryon density, the action constant $g_t$ and the acoustic scale, not a fitted parameter. When the full-Planck high-$\ell$ TTTEEE active-perturbation MCMC fits that pressureless well-driving slot freely, it recovers $0.11813\pm0.00117$ — consistent across the entire $H_0$ band ($\le1.6\sigma$) and agreeing at the unique zero-choice root ($0.11815$) and posterior mean $H_0=68.18\text{ km s}^{-1}\text{ Mpc}^{-1}$ ($0.11816$) to $0.02\sigma$ ($0.02\%$; Step 21 Part C). Unlike superfluid dark matter (Berezhiani & Khoury 2015), which postulates a new eV-scale dark boson species that clusters as dust on linear scales and condenses into a $P(X)\sim X^{3/2}$ phonon superfluid inside cold galaxy halos (requiring a thermal phase transition at $T=T_c$ that breaks down in hot galaxy clusters), TEP posits zero dark-matter particles ($\omega_{\rm cdm}=0$): both regimes arise from the single universal proper-time field $\phi$ sourced purely by ordinary baryons. On linear cosmological scales ($k\lesssim0.2\,h\,\text{Mpc}^{-1}$, $R\gtrsim10\text{ Mpc}$, governing BAO and $\sigma_8$, where $99.8\%$ of the $R_8=8\,h^{-1}\text{Mpc}$ top-hat $\sigma_8^2$ variance integral comes from $k\le1.0\,h\,\text{Mpc}^{-1}$ so applying the nonlinear halo-interior screening transition above $k_*=1.5\,h\,\text{Mpc}^{-1}$ shifts $\sigma_8$ by only $1.0\%$ from $0.8185$ to $0.8103$), because the effective sound-shell prescription around the real-space acoustic Green's-function shell at $r_s$ sets the position-space stiffness $\mathcal K_{\rm sp}^{\rm eff}=\sqrt{2}/y_s=0.1917$ (the deep-IR limit of the 3-D Frobenius/RMS eigenvalue $K_{\rm rms}=\sqrt{(2K_\perp^2+K_\parallel^2)/3}=0.1918$ of the native spatial flux Hessian $K_\perp=1/y_s=0.1356$, $K_\parallel=2/y_s+(q_s y_s)^2=0.2713$, with peak multipoles shifting by $\le0.12$ across $[K_\perp/\mathcal K_t,K_\parallel/\mathcal K_t]$; Paper 0 Step 81, Paper 29 Step 20), the scale-averaged clock-slope multiplier $M_s=1+y_s/\sqrt{2}\simeq6.26$ enters the linear Poisson equation identically to $\omega_{m,\rm eff}=\omega_b+\omega_\phi$. Conversely, as a baryon overdensity collapses into a nonlinear virialized galaxy ($M_b=6\times10^{10}\,M_\odot$, $R\in[1,100]\text{ kpc}$) or rich galaxy cluster (X-COP archetype, $M_b(<R_{200})=1.45\times10^{14}\,M_\odot$, $R_{200}=2.0\text{ Mpc}$, $R_{500}=1.2\text{ Mpc}$; Ghirardini et al. 2019; Eckert et al. 2019), the acceleration ratio $q(R)=g_b(R)/g_t$ separates the two regimes cleanly: in galaxies, radiative cooling concentrates baryons into the central disk/bulge where $q(R)$ rises from $2.5\times10^{-3}$ ($100\text{ kpc}$) and $2.8\times10^{-2}$ ($30\text{ kpc}$) to $0.25$ ($10\text{ kpc}$), $2.8$ ($3\text{ kpc}$), and $25.3$ ($1\text{ kpc}$), screening the linear acoustic multiplier $1+y(R)/\sqrt{2}$ from $6.26$ down to $5.18$, $2.25$, $1.42$, $1.12$, and $1.04$. In rich clusters, by contrast, the extended X-COP ICM profile ($\beta\simeq0.65$) keeps $q(R)=0.015\text{--}0.040\ll1$ across $R=0.5\text{--}2.0\text{ Mpc}$, so the cluster ICM never enters the $q\gtrsim1$ high-acceleration screened regime. Under the exact static two-branch Gauss law alone on the empirical X-COP baryon profile (replacing the earlier $\beta=1$ Plummer benchmark that placed $72\%$ of cluster baryons inside $0.5\text{ Mpc}$ and gave $5.78, 4.61, 3.47, 2.39$), the local 3D point-particle ratio $M_{\rm dyn,st}(<R)/M_{b,\rm obs}(<R)=1+2\beta_A^2 y_{\rm tb}(R)$ is $5.80, 5.16, 4.80, 4.59, 3.98$ across $R=2.0, 1.5, 1.2, 1.0, 0.5\text{ Mpc}$ with integrated static potential-depth ratio $\Phi_{\rm tot}(R)/\Phi_b(R)=7.64, 6.85, 6.44, 6.20, 5.61$; when the Part-C clock-slope well ($\omega_\phi/\omega_b=y_s/\sqrt{2}=5.26$, $\sigma_{T,p}=0$) retains its pre-shock NFW collapse concentration ($c_{200}=4.0$) while virial shock heating and AGN feedback deplete collisional ICM gas in the core ($f_{\rm ret}(R)=f_b(<R)/f_b(R_{200})$ dropping from $1.00$ at $2.0\text{ Mpc}$ to $0.90$ at $1.2\text{ Mpc}$ and $0.68$ at $0.5\text{ Mpc}$; Eckert et al. 2019), the virialized ratio $M_{\rm dyn}(<R)/M_{b,\rm obs}(<R)=1+(y_s/\sqrt{2})/[f_{\rm ret}(R)(1+q(R))]$ rises inward from $6.18$ at $R=2.0\text{ Mpc}$ ($M_{\rm dyn}=8.96\times10^{14}\,M_\odot$) and $6.41$ at $R=1.5\text{ Mpc}$ ($6.96\times10^{14}\,M_\odot$) through $6.73$ at $R=1.2\text{ Mpc}$ ($5.59\times10^{14}\,M_\odot$) and $7.12$ at $R=1.0\text{ Mpc}$ ($4.61\times10^{14}\,M_\odot$) to $8.44$ at $R=0.5\text{ Mpc}$ ($1.99\times10^{14}\,M_\odot$; Step 21 Part D), matching observed X-COP hydrostatic and weak-lensing cluster mass ratios ($M_{\rm tot}/M_b\simeq6$ at $R_{200}$, $6\text{--}7$ at $R_{500}$, and $8\text{--}10$ at $0.5\text{ Mpc}$) without a MOND cluster deficit, thermal condensate breakdown, or collisionless NFW particle cusp. The baryon-retention profile $f_{\rm ret}(R)$ is an observational input from X-COP, so the inner-radius agreement tests the unshocked, collisionless distribution of the clock-slope mode given the observed gas depletion; the zero-parameter prediction is the normalization $1+y_s/\sqrt2\simeq6.2$ at $R_{200}$. Evaluating the exact `hi_class` Boltzmann hierarchy at this derived clock-slope response at the posterior $H_0=68.18\text{ km s}^{-1}\text{ Mpc}^{-1}$ reproduces the CLASS visibility-peak acoustic angle $100\theta_s^{\rm CLASS}=1.04201$ ($+0.10\sigma$ from the posterior $1.04198\pm0.00031$), acoustic peak multipoles $\ell=(220.31, 536.47, 813.35)$ within $\le0.31\%$ of Planck, the linear-scale matter power spectrum $P(k)$ with $\sigma_8=0.8186$ ($S_8=0.8221$), BAO drag scale $r_d=147.49\text{ Mpc}$, SDSS DR12 consensus BAO $\chi^2=4.10$ ($\chi^2/N=0.68$ across 6 BAO points) and $\chi^2=6.42$ ($\chi^2/N=0.71$ across all 9 BAO + $f\sigma_8$ RSD points), and full Planck 2018 TTTEEE + low-$\ell$ + lensing + BAO + Pantheon+ Cobaya likelihood $\chi^2=2421.95$ versus $2422.00$ at the free-$\omega_{\rm cdm}$ posterior mean ($\Delta\chi^2=-0.06$, $\Delta\text{AIC}=-2.06$ with one fewer free parameter); at the CAMB optical-depth-unity matched $H_0=67.951\text{ km s}^{-1}\text{ Mpc}^{-1}$ (within $0.08\sigma$ of the TEP-H0 Cepheid ladder $67.83\pm1.56$ and $0.49\sigma$ of the TTTEEE posterior $68.18\pm0.46$), $\omega_\phi=0.11794$ ($0.16\sigma$), $100\theta_s=1.04111$ ($+0.02\sigma$ from Planck's $1.04110\pm0.00031$), peaks $\ell=(220.49, 536.93, 814.06)$ within $\le0.40\%$, $\sigma_8=0.8170$ ($S_8=0.8226$), $r_d=147.55\text{ Mpc}$, and SDSS DR12 BAO $\chi^2=4.65$ ($\chi^2/N=0.78$, 6-pt) and $\chi^2=6.82$ ($\chi^2/N=0.76$, 9-pt BAO+RSD).


![Background vs Perturbation Posterior Comparison](figures/figure_1_perturbation_triangle.png)

**Figure 1.** Marginalized posterior triangle plot. **Blue:** background-only TEP chain with $\delta\phi$ frozen (`tep_hiclass_suite`, four-chain MPI, 6,875 accepted steps; chain 1 shown). **Red:** active-perturbation TEP chain on the corrected screened closure ($w_c = 700$, `tep_kin_boost = 10`, regular $\delta\phi$ variable); two-chain MPI run, 5,658 accepted steps across both chains (chain 1 shown). Native `hi_class` `tep_mode` with $z_{\rm HC}=5.0$, $n_T=2.0$. Parameters shown: $\epsilon_T^{\rm HC}$ (native `hi_class` homogeneous conformal amplitude), $H_0$, $n_s$, $\sigma_8$. Contours show 68% and 95% credible regions. The near-complete overlap confirms the closure audit's prediction: on the production screened branch the active scalar's spectrum imprint is subdominant ($\lesssim 2\%$ in the acoustic band), so the background-channel and active-closure posteriors agree ($-0.14\sigma$ in $\epsilon_T$).

The hi_class configuration uses native `tep_mode` with the transition function $f_T(z)=\ln(1+z)\exp[-(z/z_T)^{n_T}]$ and fixed `z_T = 5.0`, `n_T = 2.0`, with `epsilon_T` sampled freely in `params`. This configuration natively explores the parameter space of the static conformal field, leveraging the strict isomorphism to evaluate the acoustic physics exactly. The production configuration is `data/cobaya/tep_hiclass_suite.yaml` (reference alternate: `data/cobaya/tep_native_mcmc.yaml`).

*Pipeline status.* The native-`tep_mode` joint MCMC against Planck 2018 low-$\ell$ TT/EE + lensing + BAO (SDSS DR12) + Pantheon+ was run using the structurally corrected hi_class engine, allowing $\Omega_\Lambda$ to natively fill the background cosmological budget. The background-only determination (`tep_hiclass_suite`; a four-chain MPI run, 6,875 accepted steps, 4,814 post-burn-in samples with 30% burn-in discard; classic scalar maximum parameter $R-1 = 0.011$, sampler-level $R-1 = 0.036$ at termination — a genuine multi-chain Gelman–Rubin diagnostic superseding the earlier single-chain exploratory run) gives a $\Lambda$CDM-compatible background while measuring the TEP amplitude parameter:

\begin{equation} \label{eq:5_mcmc_epsT}
\epsilon_T = 0.00619 \pm 0.00464 \quad (\epsilon_T < 0.015 \text{ at } 95\%\ {\rm CL}),
\end{equation}

with $H_0 = 66.52 \pm 1.70$ km/s/Mpc, $\Omega_b h^2 = 0.0211 \pm 0.0025$, $\Omega_{\rm cdm} h^2 = 0.1148 \pm 0.0042$, $\tau = 0.050 \pm 0.007$, $A_{\rm planck} = 1.088 \pm 0.010$, and $S_8 = 0.868 \pm 0.025$. The result is consistent with the TEP nested-domain expectation: the homogeneous amplitude $\epsilon_T$ remains small ($\sim 10^{-3}$) on the largest scales; TEP-C0 (Paper 26) reports a separate background/acoustic diagnostic amplitude. This homogeneous HC amplitude is smaller than the primordial temporal-horizon amplitude ($\epsilon_{\rm field} = 0.01755$) reported in TEP-TH (Paper 27) because the two parameters describe different observational projections of the evolving temporal sector and are not numerically interchangeable.

*Current active-perturbation posterior.* The verified two-chain `tep_hiclass_perturbations` collection has 5,658 saved rows and 3,961 post-burn-in rows. The maximum scalar $R-1=0.0073$ and checkpoint $R-1=0.0262$ pass the unchanged thresholds. Its amplitude is $\epsilon_T=0.0054\pm0.0035$ ($\epsilon_T<0.0116$ at 95% CL), consistent with the background-only amplitude within $0.14\sigma$. It retains pressureless CDM, $\Omega_{\rm cdm}h^2=0.1133\pm0.0061$; these results establish consistency of the implemented sectors rather than a CDM-free lensing solution.

*Multi-chain validation.* A parallel 4-chain run (`tep_native`; configuration `data/cobaya/tep_native_mcmc.yaml`) using a Gaussian $A_{\rm planck}$ prior (loc = 1.0, scale = 0.0025) was executed. After an initial short run (2,993 post-burn-in samples; $R-1$ for $\epsilon_T = 0.098$), the chain was extended to 8,960 accepted samples (max_samples increased from 500k to 1M), reaching parameter-level $R-1 = 0.032$, while the conservative class-level diagnostic remained $R-1 = 0.098$. The extended chain is therefore retained as a sensitivity run rather than the primary posterior. It yields $\epsilon_T = 0.0066 \pm 0.0049$ and $H_0 = 67.15 \pm 1.36$ km/s/Mpc, consistent with the background-only chain at $0.10\sigma$ and $0.24\sigma$ respectively. The widened-$A_{\rm planck}$ chain (below) provides a fully converged multi-chain determination (max $R-1 = 0.036$, all parameters $R-1 < 0.05$), and the two agree to $0.05\sigma$. Together they confirm the background-only posterior is not an artefact of the sampling configuration.

*High-$\ell$ TTTEEE run.* The native `tep_mode` likelihood adds `planck_2018_highl_plik.TTTEEE_lite` to the baseline data set. The four-chain collection contains 16,051 saved rows and 11,237 post-burn-in rows after removing 30% separately from each chain. Frequency-weighted posterior analysis gives maximum scalar $R-1=0.0455$ (sampler $R-1=0.0450$), $\epsilon_T=0.0043\pm0.0025$, and the one-sided 95% bound $\epsilon_T<0.0087$ at fixed $z_T=5$, $n_T=2$. The remaining summaries are $H_0=67.79\pm0.39$, $n_s=0.9661\pm0.0034$, $\sigma_8=0.8225\pm0.0055$, and $S_8=0.8337\pm0.0097$. The matched two-chain $\Lambda$CDM collection contains 4,332 saved rows, with maximum scalar $R-1=0.0069$. On identical likelihoods, best sampled $\chi^2$ values are 2421.3669 and 2421.4738 respectively, giving $\Delta\chi^2=-0.1069$: statistically indistinguishable. The current summaries and collection provenance are recorded in `results/07_mcmc_summary_full.json`. The per-likelihood decomposition at fixed cosmology shows the bound is geometric: high-$\ell$ TTTEEE contributes $\Delta\chi^2 \simeq +69$ at $\epsilon_T = 0.008$ (the anchored acoustic scale $\theta_s$), BAO $+1.5$ and lensing $+0.7$, while low-$\ell$ TT and Pantheon+ weakly prefer positive amplitudes ($\Delta\chi^2 \simeq -0.6$ each). The constraint is quoted at fixed $z_T = 5$, $n_T = 2$; the penalty scales with transport depth, $\Delta\chi^2 \simeq +8/+17/+35/+104$ at $\epsilon_T=0.004$ for $z_T = 3/5/10/50$. Marginalizing the turnover scale removes this conditioning: a dedicated $z_T$-scan run (`tep_zscan_ttteee.yaml`, $z_T$ linear-uniform on $[0.5, 50]$; converged four-chain MPI run, 37,736 accepted steps and 26,416 post-burn-in rows, sampler means $R-1 = 0.030$, classic scalar parameter max $R-1 = 0.0085$) yields the marginalized posterior $\epsilon_T = 0.0032 \pm 0.0032$ (median $0.0022$, $\epsilon_T < 0.0103$ at 95% CL) with $z_T = 19.4 \pm 15.3$ itself poorly constrained. The marginalized posterior is bimodal in $(z_T, \epsilon_T)$: a low-$z_T$ mode (median $z_T = 1.4$, $21.9\%$ of posterior mass) returns $\epsilon_T = 0.0065 \pm 0.0041$ ($<0.013$ at 95%), while the dominant high-$z_T$ mode (median $z_T = 23.1$, $78.1\%$) returns $\epsilon_T = 0.0023 \pm 0.0021$ ($<0.006$ at 95%) — the bound tightening as the transition recedes, consistent with the transport-depth penalty scaling. The fixed-$z_T = 5$ bound $\epsilon_T < 0.0087$ is therefore intermediate between the two modes, and the marginalized constraint is the conservative headline for the homogeneous channel.

As an independent cross-check predating the native run, a `CAMB`-based TEP approximation was implemented. Because `CAMB` lacks a native `tep_mode`, the TEP Jordan-frame Hubble rate $H_{\rm TEP}(z) = H_{\Lambda{\rm CDM}}(z) \cdot M(z)$ (with $M(z) = A(z)/(1-\alpha_A)$) was mapped onto an effective dark-energy equation of state $w_{\rm eff}(a)$ that reproduces the same background expansion. The effective $w(a)$ is fed into `CAMB` via `DarkEnergyPPF.set_w_a_table`, which supports phantom crossing. The effective-$w(a)$ `CAMB` construction generates finite high-$\ell$ spectra and completes the `plik_lite` likelihood evaluation without numerical errors (Step 17: `scripts/steps/step_17_camb_plik_full_eval.py`). Both models are evaluated at the TEP-fitted baseline parameters ($H_0 = 66.63$, $\Omega_b h^2 = 0.0212$, $\Omega_{\rm cdm} h^2 = 0.1154$, $\tau = 0.049$, $A_s = 2.1 \times 10^{-9}$, $n_s = 0.965$) from the low-$\ell$ + lensing + BAO + Pantheon+ joint posterior. At these matched baseline parameters, $\chi^2_{\Lambda{\rm CDM}} = 1409.5$ and $\chi^2_{\rm TEP} = 1179.6$, giving $\Delta\chi^2 = -229.9$ in favour of TEP: the TEP modification with $\epsilon_T = 0.0056$ reduces the high-$\ell$ $\chi^2$ by $229.9$ at the same baseline cosmology, demonstrating that the TEP background expansion is consistent with the Planck high-$\ell$ TTTEEE likelihood under the effective-$w(a)$ mapping. Because this construction is an effective-background approximation rather than the native TEP perturbation implementation, the $\Delta\chi^2$ value reflects the background-expansion difference mapped onto the high-$\ell$ spectra, not a full perturbation-sector likelihood. It is therefore reported as a validation that the TEP background expansion produces finite, Planck-compatible high-$\ell$ spectra, and is not included in the primary parameter inference, which uses the native `hi_class tep_mode` low-$\ell$ and high-$\ell$ TTTEEE joint likelihoods reported above. Acoustic-peak preservation is established separately by the direct spectra and sound-horizon calculations.

*Planck calibration prior sensitivity.* The nuisance parameter $A_{\rm planck}$ (absolute CMB calibration) is implemented as a hard uniform prior on $[0.9, 1.1]$ in the 5-chain active posterior. The posterior mean is $A_{\rm planck} = 1.088 \pm 0.012$ with maximum sampled value $1.1000$, indicating saturation against the upper prior bound. To test whether this truncation biases the cosmological inference, a dedicated sensitivity test was executed with the prior widened to $[0.9, 1.25]$ (Step 20: `scripts/steps/step_20_aplanck_sensitivity.py`, configuration `data/cobaya/tep_hiclass_aplanck_sens.yaml`). The converged run (16,320 total samples; all parameters Gelman–Rubin $R-1 < 0.05$; maximum $R-1 = 0.036$) yields $A_{\rm planck} = 1.229 \pm 0.026$, confirming the old posterior was truncated by approximately $5.0\sigma$. The TEP amplitude from the widened run is $\epsilon_T = 0.0063 \pm 0.0048$ ($R-1 = 0.036$), consistent with the 5-chain active posterior at $0.12\sigma$ and with the multi-chain validation at $0.05\sigma$. The correlation between $A_{\rm planck}$ and $\epsilon_T$ is $r = -0.10$, and splitting at the posterior median $A_{\rm planck} = 1.236$ gives a difference in $\epsilon_T$ of only $-0.04\sigma$. Even a $0.1$ upward shift in $A_{\rm planck}$ would move $H_0$ by only $\sim 0.1$ km s$^{-1}$ Mpc$^{-1}$ (inter-chain comparison: $H_0$ shifts by only $0.11$ km s$^{-1}$ Mpc$^{-1}$ when $A_{\rm planck}$ shifts by $0.14$), well below its posterior width. The $\chi^2$ does decrease monotonically toward the old boundary within the old prior range, but there is no evidence of a degeneracy cascade with $\epsilon_T$. The TEP constraint on the homogeneous amplitude is robust against $A_{\rm planck}$ prior systematics. The widened-$A_{\rm planck}$ run is used only as a numerical robustness test of the $\epsilon_T$ posterior; the resulting calibration value should not be interpreted as a physically preferred Planck calibration model.

The companion paper TEP-C0 (Paper 26) provides the primary late-time constraints: Pantheon+ evidence integration favors the physical no-$\Lambda$ TEP M1 branch over baseline $\Lambda$CDM with BF approximately 5.8 (conservative $z_{\rm los}=5$), approximately 5.4 (fixed $z_{\rm los}=100$ benchmark), and approximately 5.4 (broad free-$z_{\rm los}$, likelihood maximum $z_{\rm los}\simeq5.3$, posterior volume extending over the nearly flat tail). Those model-comparison results are not re-derived here; they are used as the late-time empirical context for the `hi_class` acoustic-preservation implementation.

The imported C0 evidence is prior- and anchor-sensitive: the conservative Bayes factor falls to approximately 0.63 after removing $z < 0.023$ anchors and to approximately 1.24 with the 77 SH0ES calibrators removed, against an identical-model control returning $\Delta\ln\mathcal{Z}=0$ and $\mathrm{BF}=1$ exactly under the quadrature evaluator. The high-turnover branches are mathematical shape benchmarks pending an environment-dependent physical realization.

## 5. Results and Cosmological Constraints

### 5.1 The Acoustic Spectra

The physically meaningful test of the native TEP integration is to evaluate whether the conformal field exactly replicates the acoustic physics of the early universe without invoking spatial expansion. Because the conformal field mathematically mimics the FLRW scale factor, the recombination-era physics is evaluated natively within the static frame. Throughout this paper, "exact isomorphism" refers to the conformal background/acoustic mapping: the equality of the relevant sound-horizon and photon-transport integrals under the identification of the TEP conformal factor with the FLRW scale factor. The associated linear pure-conformal scalar perturbation sector is closed separately through the derived Bellini–Sawicki EFT closure and validated by the active ($\delta\phi$) hi_class run reported in Section 4.5 and Appendix A.5. This perturbative closure applies to the pure-conformal sector implemented here; the fully disformal, nonlinear, and environmentally inhomogeneous screening sectors remain extensions of the present calculation.

#### 5.1.1 Sound-horizon and acoustic-peak preservation

Running hi_class native `tep_mode` against standard CLASS $\Lambda$CDM at the Planck 2018 best-fit point, with $\epsilon_T = 0.0066$, $z_T = 5$, $n_T = 2$, yields:

- *Sound horizon preserved to ~6 ppm:* $r_s^{\rm TEP}/r_s^{\Lambda\rm CDM} = 0.999994$. The comoving sound horizon integrates identically in the static conformal frame. The remaining $\sim$6 ppm offset is a numerical/implementation-level residual associated with the finite precision of the conformal-frame background mapping and output reconstruction. Analytically, the exact mapping used in the production implementation is $M=A/(1-\alpha_A)$, whose first-order expansion is $A(1+\alpha_A)$. Direct verification confirms that the discrepancy is not a failure of the conformal isomorphism.

- *Acoustic-peak morphology unchanged:* with $r_s$, the baryon loading, and the photon-baryon driving at $z \approx 1089$ all operating identically under the conformal clock-rate, the relative peak heights and the damping tail closely match $\Lambda$CDM.

The central result is therefore not that all cosmological observables are already closed, but that the CMB acoustic scale itself is not uniquely diagnostic of physical spatial expansion.

#### 5.1.1b Acoustic-scale preservation metrics

Table 2 quantifies the acoustic-scale preservation in three multipole ranges. The sound-horizon ratio $r_s^{\rm TEP}/r_s^{\Lambda\rm CDM} = 0.999994$ is independent of multipole because $r_s$ is a single integrated quantity; the $\ell$-centroid shift tracks the expected $-0.185\%$ angular-diameter-distance projection. The tabulated residuals are total TEP-versus-$\Lambda$CDM projection residuals, dominated by the angular-diameter-distance projection; the active-minus-background perturbation residual is below $0.001\%$ across the entire acoustic range.

| Multipole range | $\ell_{\rm peak}$ shift $\Delta\ell$ | $r_s^{\rm TEP}/r_s^{\Lambda\rm CDM}$ | Total TEP–$\Lambda$CDM projection residual (%) |
| --- | --- | --- | --- |
| $100 \le \ell \le 500$ | $-0.06$ | $0.999994$ | $0.67$ |
| $500 \le \ell \le 1000$ | $-0.36$ | $0.999994$ | $1.34$ |
| $1000 \le \ell \le 2000$ | $-0.64$ | $0.999994$ | $1.82$ |

#### 5.1.2 The residual is a late-time projection, largely degenerate with $H_0$

The field profile is active over intermediate redshift ($z \sim 1$–$15$, peaking near $z_T$), changing the apparent angular distance to last scattering. At the fiducial $\epsilon_T = 0.0066$ this shifts the angular acoustic scale by $\Delta\theta_s/\theta_s = +0.185\%$. This rigid rescaling produces a coherent, oscillatory $\Delta C_\ell/C_\ell$ pattern whose envelope reaches $\sim 1.8\%$ across $100 < \ell < 2000$. This is *not* a change in early-universe physics: it is a pure angular-diameter-distance projection, largely degenerate with $H_0$.

#### 5.1.3 Polarization Spectra ($C_\ell^{TE}, C_\ell^{EE}$)

The TE and EE spectra inherit the same behavior: the recombination-era polarization source is natively preserved by the conformal integration, and the only effect is the common $\theta_s$ projection shared with TT. Running hi_class with active scalar perturbations ($\delta\phi$ evolved through the derived Bellini–Sawicki EFT closure) and comparing against the background-only TEP run and Planck 2018 $\Lambda$CDM, the maximum active-perturbation residuals in the polarization channels are:

| Multipole range | Total TE projection residual (%) | Total EE projection residual (%) |
| --- | --- | --- |
| $100 \le \ell \le 500$ | $1.09$ | $1.52$ |
| $500 \le \ell \le 1000$ | $2.16$ | $2.44$ |
| $1000 \le \ell \le 2000$ | $3.14$ | $3.64$ |

These are total TEP-versus-$\Lambda$CDM projection residuals, dominated by the common angular-diameter-distance projection. The isolated active-minus-background perturbation residual on the screened branch is $0.42\%$ in the acoustic band (Step 09), concentrated in the low-$\ell$ ISW-dominated region; the larger values tabulated here are total TEP-versus-$\Lambda$CDM projection differences. There is no additional polarization-specific distortion from the dynamical scalar sector at the current precision beyond that ISW signature.

A closure-limit diagnostic localizes the origin of this low-$\ell$ imprint precisely (Step 12c, `results/12c_spectrum_residuals.json`). At $\epsilon_T = 0$ all Bellini–Sawicki $\alpha$-functions vanish, and the $\alpha$-parametrized perturbation variable $x \equiv \delta\phi/\phi'$ indeed reproduces the GR spectrum to $3\times10^{-5}$. The prescribed-trajectory $\delta\phi$ closure, however, retains the non-minimal coupling $G_{4,\phi} = S_\Sigma M^2$ and the reconstructed potential slope $U_{,\phi} \supset 6 S_\Sigma M^2 (\mathcal{H}'/a + 2\mathcal{H}^2)$, which do not carry $\alpha_A$ suppression: the scalar sector possesses a minimum gravitational mixing floor that persists on a non-rolling background. The residual scales as $S_\Sigma^2$ ($+9.4\%$ / $+2.2\%$ / $+0.09\%$ in the TT quadrupole at $S_\Sigma = 1$ / $0.5$ / $0.1$) and as $w_c^{-1}$ ($+5566\%$ at $w_c = 70$ down to $+1.1\times10^{-5}$ at $w_c = 7\times10^{4}$). At the production point ($S_\Sigma = 1$, $w_c = 700$) the floor is a $+9.4\%$ TT and $+18\%$ TE quadrupole excess with TE quadrature SNR $\simeq 1.2$ against the cosmic-variance floor — a falsifiable, $\epsilon_T$-independent signature of the screened closure itself, not of the transport amplitude. The $\delta\phi$ formulation is therefore the more complete of the two linear closures at $\phi' = 0$; the two formulations differ precisely in whether the $G_{4,\phi}$ mixing channel survives where the background field stops rolling.

### 5.2 Cosmological Constraints: Late-Time Evidence and the CMB Bound

The cosmological constraints on TEP come from two complementary regimes, established in the companion paper TEP-C0 (Paper 26).

*Late-time evidence (supernovae).* A nested-sampling model comparison over the full $1701\times1701$ Pantheon+ statistical-plus-systematic covariance finds substantial Bayesian preference for the TEP geometry over $\Lambda$CDM:

| Model | Bayes factor vs $\Lambda$CDM |
| --- | --- |
| TEP, $z_{\rm los}=5$ | $\sim 5.8$ |
| TEP, $z_{\rm los}=100$ | $\sim 5.4$ |
| TEP, free $z_{\rm los}$ | $\sim 5.4$ |
| $w$CDM | $\sim 3.3$ |
| CPL | $\sim 2.7$ |
| Einstein–de Sitter | $\sim 1.5\times10^{-137}$ |

The TEP M1 branch improves the Pantheon+ likelihood relative to baseline $\Lambda$CDM with BF approximately 5.8 (conservative $z_{\rm los}=5$), approximately 5.4 (fixed $z_{\rm los}=100$ benchmark), and approximately 5.4 (broad free-$z_{\rm los}$) (TEP-C0, Paper 26). Those model-comparison results are not re-derived here; they are used as the late-time empirical context for the `hi_class` acoustic-preservation implementation. The model-comparison result is consistent with the TEP claim that the Etherington distance-duality relation is a mathematically native feature of the static conformal field. TEP shows that the supernova distance-redshift relation can be fit without treating late-time acceleration as primitive spatial acceleration.

The prior- and anchor-sensitivity of the imported C0 evidence noted in Section 4.5 (conservative Bayes factor $\approx 0.63$ after removing $z < 0.023$ anchors) applies equally here.

*$z_{\rm los}$ distinction.* The $z_{\rm los} = 5$ profile is the conservative line-of-sight turnover used in the C0 supernova-sector transport benchmark. It should not be confused with the homogeneous acoustic-sector profile scale $z_T^{\rm HC} = 5$ used in the hi_class conformal implementation, which is a related but distinct projection of the temporal sector. The C0 free-$z_{\rm los}$ robustness test uses $z_{\rm los} \in [0.1, 150]$.

*Amplitude dictionary.* $\epsilon_T^{\rm los}$ denotes the late-time line-of-sight transport amplitude fitted in TEP-C0. $\epsilon_T^{\rm CMB}$ denotes the C0 background/acoustic diagnostic amplitude. $\epsilon_T^{\rm HC}$ denotes the native `hi_class` homogeneous conformal amplitude reported here ($0.0062 \pm 0.0046$ from the four-chain background-only posterior, median $0.0053$; the corrected-closure active-perturbation posterior yields $0.0054 \pm 0.0035$, and the high-$\ell$ TTTEEE extension yields $0.0043 \pm 0.0025$, a one-sided bound $\epsilon_T < 0.0087$ at 95% CL at fixed $z_T=5$, weakening to $\epsilon_T < 0.0103$ marginalized over the turnover scale). $\epsilon_{\rm dyn}$ denotes the dynamical response in TEP-TH. $\epsilon_{\rm field}=0.01755$ denotes the primordial spectral-flow parameter constrained by $n_s$ in TEP-TH. These are related projections of the same temporal sector, but they are not numerically interchangeable.

*Homogeneous (CMB) bound.* The low-$\ell$ Planck likelihoods used in this paper's hi_class MCMC (TT/EE + lensing) yield $\epsilon_T = 0.0062 \pm 0.0046$ (converged four-chain background-channel posterior, median $0.0053$, $\epsilon_T < 0.015$ at 95% CL), consistent with zero at $\sim 1.3\sigma$. The homogeneous CMB amplitude and the primordial spectral-flow parameter are different observational projections of the same temporal sector and are not numerically interchangeable. Their differing numerical values therefore do not require a phenomenological epoch-screening mechanism. In the canonical TEP-TH/TEP-BBN architecture, no early-time screening function is imposed to restore a standard hot-Big-Bang thermal history. The low-$\ell$+lensing+BAO+Pantheon+ chains reported here serve as implementation and robustness tests of the native hi_class module. The native `hi_class tep_mode` high-$\ell$ TTTEEE run has since been completed (Section 4.5): a converged four-chain MPI posterior gives $\epsilon_T = 0.0043 \pm 0.0025$, a one-sided bound $\epsilon_T < 0.0087$ at 95% CL at fixed $z_T=5$ ($\epsilon_T < 0.0103$ marginalized, Section 4.5), and a matched $\Lambda$CDM reference returns $\Delta\chi^2 = -0.11$ at the respective best fits — statistical indistinguishability on the full Planck + BAO + Pantheon+ data set. A `CAMB`-based effective-background mapping is retained solely as an independent code-path cross-check; because it is an effective-$w(a)$ approximation rather than the native TEP perturbation implementation, it is not included in parameter inference or model comparison (Section 4.5). Acoustic preservation is established by the direct spectra and sound-horizon calculations.

*Late-transition preference (TEP-C0 Step 03-14).* With DESI DR2 BAO, Pantheon+ and the Planck acoustic angle on a single history, the native branch prefers a late transition, $z_T\simeq0.50$ and $\epsilon_T\simeq0.019$, giving $\Delta\chi^2=-8.1$ relative to $\Lambda$CDM. The improvement is already $-6.2$ at $\epsilon_T=0.01$ and grows from DESI DR1 ($-6.3$) to DR2. The free-$z_T$ full-likelihood chain of Section 4.5 (Planck TTTEEE + low-$\ell$ + lensing + SDSS DR12 BAO + Pantheon+) points the same way: its best-fit sample lies at $z_T=0.52$, $\epsilon_T=0.012$, with $\chi^2$ lower by $2.0$ than the best-fit sample of the matched $\Lambda$CDM chain, at the $z_T=0.5$ prior edge. That run used SDSS DR12 BAO with $z_T\ge0.5$; the DESI-era native hi_class run (TTTEEE + low-$\ell$ + lensing + DESI DR2 + Pantheon+, $z_T\in[0.2,2]$) is the full-Boltzmann confirmation of this preference.

*Native-TEP joint MCMC.* This paper's primary contribution is the verified hi_class implementation, demonstrating ppm-level sound-horizon preservation and acoustic-sector equivalence in the native static conformal geometry.

### 5.3 Structure Growth and the Matter Power Spectrum

The full hi_class Boltzmann closure with active SMG perturbations — on the screened kineticity branch ($w_c = 700$, `tep_kin_boost = 10`; Section 3.3) — yields a linear growth amplitude in agreement with Planck. The weak-lensing comparison must be stated exactly, and a registered provenance audit (Step 23, `results/step_23_s8_provenance_audit.json`) decomposes each contribution to the quoted tension. The joint MCMC gives $S_8 = \sigma_8\sqrt{\Omega_m/0.3} = 0.868 \pm 0.025$ at face value; however, the audit shows this number is prior-boundary dominated rather than a TEP prediction, and the chain was run on the pre-audit build in which the SMG sector was silently inactive (Section 4.5 provenance note). In the low-$\ell$-only chain, $A_{\rm planck}$ accumulates at the ceiling of its $\mathcal{U}[0.9, 1.1]$ prior (58% of posterior mass within 0.01 of the boundary) and $n_s$ at the ceiling of $\mathcal{U}[0.94, 1.0]$ (34% within 0.002); $\sigma_8$ tracks the calibration nuisance with correlation $+0.31$, and widening the $A_{\rm planck}$ prior to $\mathcal{U}[0.9, 1.25]$ drives the posterior to $A_{\rm planck} = 1.233$ and $S_8 \simeq 0.93$ — the quoted amplitude scales with prior width. A weighted linear response fit on the chain, evaluated at the non-saturated calibration $A_{\rm planck} = 1$, returns $S_8 \simeq 0.83$. At the shared fixed cosmology of the regenerated matter-power spectra ($\Omega_m = 0.314$), the growth comparison is direct: $S_8 = 0.8421$ for $\Lambda$CDM and $0.8432$ for TEP on the production screened branch, so the TEP–external gap equals the $\Lambda$CDM–external gap ($3.9\sigma$ versus DES-Y3 3$\times$2pt $0.776 \pm 0.017$ and $4.0\sigma$ versus KiDS-1000 $0.759 \pm 0.021$, TEP–$\Lambda$CDM separation $\lesssim 0.07\sigma$ on either anchor). The apparent $S_8$ tension on this branch is the standard cosmological tension between $z=0$ CMB-extrapolated amplitudes and low-$z$ cosmic-shear inferences, inherited — neither resolved nor amplified — by the screened linear closure. The response scan does, however, show that the closure is capable of suppressing growth: $\sigma_8$ falls monotonically with the kinetic coefficient and crosses the weak-lensing central value at $w_c \sim 6\times10^{3}$, where the trade-off is a low-$\ell$ ISW excess measured at $+41\%$ ($\ell=2$) peaking at $+57\%$ within $\ell \le 10$ (Step 24) — a falsifiable signature rather than a tuned fit.

Three further properties of the linear closure are registered by the audit. First, the sign of the growth modification on the production screened branch is a small enhancement, not a suppression: $\sigma_8^{\rm TEP}/\sigma_8^{\Lambda{\rm CDM}} = 1.0013$ at $z=0$, so the production branch does not move the amplitude toward the weak-lensing value; the larger-kineticity domain ($w_c \sim 6\times10^{3}$) does suppress $\sigma_8$ below $\Lambda$CDM, reaching $0.76$ (Section 3.3). Second, the SMG alpha sector's contribution to the matter power spectrum is now genuinely active and measured: on the production screened branch the active-perturbation $P(k)$ deviates from the background-only output by $\sim 0.3\%$ in the median and at most $\sim 1.2\%$ at the low-$k$ end (the genuine scalar-sourced correction; the high-$k$ region is flat to $\lesssim 0.2\%$), while the unscreened stable branch produces an order-unity ($\times 2$\u2013$3$) enhancement and the canonical image diverges — the full response is registered in `results/closure_response_scan.json`. An earlier claim that the active-perturbation $P(k)$ is bitwise identical to the background-only output is retracted: it was produced while a background-density gate silently disabled the SMG sector, so the "active" run never evolved $\delta\phi$. Third, the matter-power grids are now extended to $P_k^{max} = 20\,h/{\rm Mpc}$, eliminating the grid-boundary artifact previously noted at $k \approx 0.45\,h/{\rm Mpc}$. A common-kernel linear-growth diagnostic using a DES-Y3-like combined-source distribution ($n(z_s) \propto z_s^2 e^{-(z_s/0.5)^{1.5}}$, effective lens redshift $z_{\rm eff} \approx 0.457$, Step 22, `results/step_22_des_kernel_projection.json`) gives a kernel-weighted amplitude $S_8^{\rm eff} = 0.6712$ for TEP versus $0.6716$ for $\Lambda$CDM (relative difference $\sim 0.05\%$). The fiducial matter-plus-$\Lambda$ growing mode is integrated from $a=0$ and normalized at $a=1$. This assigned-kernel growth proxy is distinct from both a full tomographic shear spectrum and the published $z=0$ parameter $S_8=\sigma_8(0)\sqrt{\Omega_m/0.3}$; survey tension cannot be computed by subtracting those published parameters from it. The TEP/$\Lambda$CDM ratio is interpolated from the two native outputs at $z=0$ and $0.5$ and extrapolated beyond $0.5$, so this result tests their relative growth under the stated approximation. It supports close agreement of the two linear growth histories on the production branch.

- *Linear growth amplitude:* $\sigma_8 = 0.824$ at the fiducial TEP parameters ($\epsilon_T = 0.0066$, $z_T = 5$, $n_T = 2$) on the production screened branch ($w_c = 700$), compared to $\sigma_8 = 0.823$ for standard $\Lambda$CDM at the same cosmology. The TEP value is a native output of the full SMG EFT solver with the derived Bellini–Sawicki mappings; the assigned effective kinetic coefficient is $Z_{\rm eff}=w_c+6$. Its relation to the environmental nonlinear principal coefficients, which differ between timelike and static-gradient backgrounds, is specified in Appendix A.5a. The unscreened stable branch gives $\sigma_8 = 1.61$ (excluded) and the canonical image is unstable; see the registered response curve.

- *Growth-function diagnostic:* The two native matter-power outputs determine the present amplitude and an interval-averaged logarithmic growth rate $\bar f=\ln[D(0)/D(0.5)]/\ln(1.5)$, with $D(0)=1$ and $D(0.5)$ estimated at $k=0.01\,h/{\rm Mpc}$:


| Model | $\sigma_8(0)$ | $\bar f\,\sigma_8(0)D(0.5)$ |
| --- | --- | --- |
| $\Lambda$CDM | $0.823$ | $0.410$ |
| TEP (background-only) | $0.825$ | $0.413$ |
| TEP (active $\delta\phi$, screened $w_c=700$) | $0.824$ | $0.413$ |

These diagnostics show closely matched amplitudes and interval growth. The second column of measurements uses the mean growth rate across $0\le z\le0.5$; it is not the local $f\sigma_8(0.5)$ measured by redshift-space distortions. Native outputs near each comparison redshift are required to form that derivative and an observational RSD comparison.


The $\sigma_8$ result demonstrates that the TEP covariant action, when closed through the derived Bellini–Sawicki EFT on the screened kineticity branch and solved with the full Boltzmann hierarchy, restores structure growth to the observationally consistent range. The closure audit adds a necessary qualification: this agreement is not automatic — the canonical Einstein-frame image is linearly unstable and the minimally corrected unscreened branch is excluded ($\sigma_8 \simeq 1.6$), so the viability of the conformal sector rests on the scalar's cosmological response being kinetically suppressed ($Z_E \gtrsim 7\times10^{2}$), the regime the $P(X)$ sector is constructed to supply on screened backgrounds. Simplified EdS-only growth ODEs, which lack the SMG EFT perturbation closure, are insufficient for this sector; the full Boltzmann closure is required.

### 5.4 The Hubble Tension in TEP


The TEP framework offers a proposed reconciliation of the Hubble tension without invoking an early-universe crisis. The homogeneous background is exactly mathematically isomorphic to $\Lambda$CDM ($H_0 \approx 67$ km/s/Mpc from the CMB), while the apparent local $H_0 \approx 73$ km/s/Mpc arises from an environment-dependent clock-transport bias along the local distance ladder (Cepheid/SN Ia calibration in unscreened stellar atmospheres). The tension is a measurement-environment effect, bypassing the need for early-universe expansion.



![H0 in the TEP picture](figures/figure_2_H0_comparison.png)



**Figure 2.** Corpus-level $H_0$ comparison: HC acoustic posterior and Paper 11 local-ladder correction. **HC joint MCMC:** native `hi_class` `tep_mode` homogeneous posterior with $\epsilon_T^{\rm HC}=0.0062\pm0.0046$, $z_{\rm HC}=5.0$, $n_T=2.0$. **Planck 2018:** independent CMB constraint. **SH0ES uncorrected / TEP-corrected:** the local distance-ladder values are imported from Paper 11; HC does not independently re-analyse the Cepheid/SN Ia calibration data. This figure illustrates consistency of the HC early-universe sector with the broader TEP Hubble-tension interpretation; it is not a standalone HC distance-ladder test.

### 5.5 The Mathematical Limit of the Conformal Field

To explicitly map the action of the conformal field on the acoustic horizon, the acoustic scale is evaluated in a mathematically idealized geometry ($\Omega_m = 1.0$, $\Omega_\Lambda = 0.0$) using the hi_class native `tep_mode` implementation.

#### Regime I: Finite-turnover HC branch ($z_{\rm HC} = 5$)

In the standard TEP model, the profile $\exp[-(z/z_{\rm HC})^{n_T}]$ ensures the conformal field correctly matches the apparent late-time acceleration inferred from Pantheon+. The integration verifies this background/acoustic mathematical isomorphism:

| $\epsilon_T$ | $100\theta_s$ | $r_s$ [Mpc] | $\Delta D_C / D_C$ | Interpretation |
| --- | --- | --- | --- | --- |
| $0.00$ | $1.0403$ | $144.526$ | $0.00\%$ | Pure EdS reference (no TEP) |
| $0.05$ | $1.0548$ | $144.519$ | $-1.38\%$ | $r_s$ preserved; $\theta_s$ shifts from $D_C$ projection |

The sound horizon $r_s$ remains exact because the conformal field geometry accurately tracks the mathematics of the acoustic horizon without requiring physical stretching of space.

#### Regime II: Power-law diagnostic limit ($z_{\rm HC} \to \infty$)

Removing the empirical profile and forcing the conformal factor to grow as a pure unsuppressed power law $A(z) = (1+z)^{\epsilon_T}$ exposes the mathematical divergence of the bare field:

| $\epsilon_T$ | $100\theta_s$ | $r_s$ [Mpc] | $\Delta D_C / D_C$ | Interpretation |
| --- | --- | --- | --- | --- |
| $0.00$ | $1.0403$ | $144.526$ | $0.00\%$ | Pure EdS reference |
| $0.05$ | $0.7565$ | $100.584$ | $-4.30\%$ | $r_s$ mathematically squeezed by divergence |

This mathematical limit demonstrates that the $z_{\rm HC} \sim 5$ empirical fitting function accurately defines the physical profile of the conformal field, allowing it to mimic Dark Energy while preserving the CMB acoustic horizon without expanding space.


![Matter-frame dual-scan results](figures/figure_3_jordan_theta_s.png)



**Figure 3.** Jordan-frame acoustic-scale diagnostic. Background-only native `hi_class` `tep_mode` diagnostic in an Einstein-de Sitter geometry ($\Omega_m=1.0$, $\Omega_\Lambda=0.0$). **Left panel:** Finite-turnover HC branch ($z_{\rm HC}=5$, $\epsilon_T^{\rm HC}=0.05$, $n_T=2.0$); the acoustic scale is protected and stays near the Planck 2018 reference. **Right panel:** Power-law diagnostic limit ($z_{\rm HC}\to\infty$, $\epsilon_T^{\rm HC}=0.05$, $n_T=2.0$); removing the empirical profile exposes unphysical acoustic-horizon squeezing. The dashed line is the Planck 2018 acoustic reference value; the comparison is a diagnostic acoustic-scale recovery, not a full TT/TE/EE likelihood plot.

## 6. Falsifiable Predictions

The following near-term observational tests would strengthen or falsify the TEP-HC acoustic-sector and perturbation claims.

- **Small-scale CMB polarization from active scalar perturbations.** The TEP-HC linear scalar perturbation sector predicts a characteristic phase shift in the TE cross-spectrum at $\ell \gtrsim 1000$ ($\sim 0.1\%$–$0.3\%$) due to the $\alpha_B \neq 0$ running of the conformal scalar. CMB-S4 and Simons Observatory, targeting $\ell \sim 3000$ with $\mu$K-arcmin sensitivity, can test this deviation directly. A null result at $>3\sigma$ would place a tighter upper bound on the TEP scalar amplitude.

- **B-mode polarization from native tensor modes.** The TEP-TH assigned tensor-pump benchmark yields $r(k_{\rm pivot}) = 1.9 \times 10^{-9}$ and $r_{\rm max} = 5.2 \times 10^{-7}$ at the fiducial profile ($p=0.1$, $n=4$, $\eta_0=14{,}000$ Mpc), with one-parameter variations spanning $r_{\rm max} = 1.4 \times 10^{-5}$ on the curvature-regular slice ($0 \lt p \le \tfrac12$) to $6.1 \times 10^{-5}$ ($0 \lt p \le 1$) — all far below current BICEP/Keck bounds ($r < 0.036$). As established in TEP-TH Section 10.6, the one-parameter scan does not bound simultaneous multi-parameter profile variations ($p=0.5$, $n=2$, $\eta_0=3000$ Mpc reaches a diagnostic maximum near $1.14\times10^{-4}$), and the assigned pump must first be matched to the shared action's tensor normalization before diagnostic values become model-wide B-mode exclusion thresholds. CMB-S4's $r \sim 10^{-4}$ sensitivity will therefore constrain the joint transition-profile domain $(p, n, \eta_0)$ rather than test a single universal cutoff, while a null result remains fully compatible with the curvature-regular temporal-horizon branch.

- **Acoustic-peak consistency under `tep_mode` background.** The TEP background factor $M(z) = A/(1-\alpha_A)$ preserves the acoustic-peak locations to within the numerical integration tolerance of the Boltzmann solver (currently $\lesssim 10$ ppm). The current hi_class implementation reports $r_s^{\rm TEP}/r_s^{\Lambda\rm CDM} = 0.999994$, where the residual $\sim$6 ppm is understood as a numerical/implementation-level artifact of finite-precision conformal-frame background reconstruction. Any claimed physical deviation from the exact conformal isomorphism must be shown to exceed the standard solver interpolation noise ($\sim 10$ ppm) before it can be interpreted as a falsification of the background mapping, rather than as a C++ integration artifact.

- **Growth-factor amplitude from native EFT.** The TEP-HC active-perturbation closure, now implemented in the regular $\delta\phi$ variable with the corrected kinetic-coefficient map $\alpha_K = w_c\,\alpha_A^2$ (Appendix A.5a), yields $\sigma_8 = 0.824$ at the fiducial parameters ($\epsilon_T = 0.0066$, $z_T = 5$, $n_T = 2$) on the production screened branch ($w_c = 700$) — a $+0.1\%$ enhancement relative to $\Lambda$CDM ($\sigma_8 = 0.823$) at the same cosmology. The registered response scan (`results/closure_response_scan.json`) makes the prediction sharp and branch-dependent: $w_c \lesssim 0$ is linearly unstable (the canonical image $w_c=-5$ runs away); the minimally stable unscreened branch $w_c = +7$ is excluded ($\sigma_8 \simeq 1.6$); $w_c \gtrsim 7\times10^{2}$ recovers $\Lambda$CDM; and $w_c \sim 6\times10^{3}$ gives $\sigma_8 \simeq 0.76$, the weak-lensing central value, at the price of a $\sim 50\%$ low-$\ell$ ISW excess (directly measured on the corrected build: $+41\%$ at $\ell = 2$, peaking at $+57\%$ within $\ell \le 10$; `results/step_24_lowell_isw_scan.json`). The screening parameter is not free-floating: it is the effective cosmological kineticity $Z_E = P_{,X}+2XP_{,XX}$ required by, and physically motivated by, the coarse-grained nonlinear $P(X)$ sector (while the exact nonlinear principal-operator matching remains a separate closure; Appendix A.5a), so a measurement of the growth amplitude directly constrains the effective kineticity of the conformal scalar. The Step 23 provenance audit (`results/step_23_s8_provenance_audit.json`) further establishes that the MCMC posterior mean $\sigma_8 = 0.858 \pm 0.016$ is prior-boundary dominated ($A_{\rm planck}$ and $n_s$ saturate their low-$\ell$ prior ceilings; $\sigma_8$ tracks $A_{\rm planck}$ with correlation $+0.31$). An earlier claim that the active-perturbation $P(k)$ is bitwise identical to the background-only output is retracted — it was produced while the SMG sector was silently disabled; on the screened production branch the genuine active-scalar deviation is $\lesssim 2\%$ in $P(k)$. As Stage-IV weak-lensing surveys (Euclid, Rubin/LSST) resolve their internal calibration systematics, the closure is falsifiable in sign, amplitude, and branch.

Each test targets a distinct sector of the TEP-HC claim: active scalar perturbations (test 1), tensor modes (test 2), background isomorphism (test 3), and growth amplitude (test 4). A combined null result across all four would require a significant revision of the conformal-frame mapping.

## 7. Conclusion: The Static Conformal Universe

This paper implements and demonstrates the consistency of the native Temporal Equivalence Principle (TEP) conformal modification directly within the `hi_class` Boltzmann solver framework. By leveraging the mathematical isomorphism between the FLRW expanding scale factor $a(t)$ and the TEP conformal scalar field $A(\phi)$, this analysis demonstrates that the early-universe acoustic-sector observables can be reproduced at high fidelity under a static conformal temporal-transport mapping.

### 7.1 Summary of Results

- *The Mathematical Isomorphism:* The TEP conformal factor $A(\phi) = \exp(\beta_A\phi/M_{\rm Pl})$ dictates the clock-rates and photon phases in the causal matter metric $\tilde{g}_{\mu\nu} = A(\phi)^2 g_{\mu\nu}$. Because this scalar field evolves identically to the standard spatial scale factor $a(t)$, standard Boltzmann solvers can be used as conformal-frame calculators for the background/acoustic-sector mapping tested here. The parameter traditionally defined as Dark Energy ($\Omega_\Lambda$) is operationally reinterpreted within this implementation as the homogeneous temporal-shear background contribution filling the same background budget slot, $\Omega_\phi$.

- *CMB Acoustic Preservation (Pure-Conformal Consistency):* Because of the static conformal isomorphism, the hi_class native integration demonstrates that the background/acoustic-sector observables are preserved to parts-per-million ($r_s^{\rm TEP}/r_s^{\Lambda\rm CDM} = 0.999994$). This is a pure-conformal consistency check: the conformal transport geometry recovers the standard FLRW acoustic results in the regime where the conformal field dominates and disformal corrections are suppressed. It is not an independent confirmation of TEP, but rather a necessary consistency requirement that any viable conformal-frame alternative must satisfy. It is stated plainly that this result tests only the pure-conformal sector — mathematically isomorphic to $\Lambda$CDM by construction — while the disformal sector $B(\phi)$, where the distinctive TEP predictions (synchronisation holonomy, propagation cone tilts) reside, is set to zero in this branch and is not exercised by the CMB acoustic observable. The background acoustic observables alone do not uniquely force the spatial-expansion interpretation; they can be naturally accommodated by the evolving background scalar field $A(\phi)$. The active-perturbation closure (Section 4.5) — implemented in the regular $\delta\phi$ variable and audited across its kinetic-coefficient branch — shows that the perturbation sector's content is real: the canonical Einstein-frame image $\alpha_K = -5\alpha_A^2$ is linearly unstable (tachyonic scalarization channel, invisible to the ghost discriminant), the minimal sign-corrected unscreened branch is observationally excluded ($\sigma_8 \simeq 1.6$), and the viable production branch is the kinetically screened regime $w_c \gtrsim 7\times10^{2}$, which restores $\Lambda$CDM-level growth ($\sigma_8 = 0.824$) with a bounded low-$\ell$ ISW signature. The earlier active-perturbation chains (posterior agreement to $0.07\sigma$, Gelman–Rubin $R-1 = 0.007$ at 58,180 accepted steps) ran with the SMG sector silently inactive and are retained as background-channel posteriors; the active-closure re-run on the corrected screened branch has since been completed (two-chain MPI, $w_c = 700$, regular $\delta\phi$ variable; 3,961 post-burn-in samples, max $R-1 = 0.007$), yielding $\epsilon_T = 0.0054 \pm 0.0035$ — consistent with the background-channel posterior at $-0.14\sigma$, with a $-1.3\sigma$ downward shift in $\sigma_8$ tracking the scalar's low-$\ell$ ISW contribution. The converged four-chain MPI suite run ($\epsilon_T = 0.0062 \pm 0.0046$, median $0.0053$, $\epsilon_T < 0.015$ at 95% CL; max parameter $R-1 = 0.011$) serves as the primary low-$\ell$ background-channel determination (agreeing with the 5-chain background-channel run to within $0.2\sigma$), while the earlier single-chain exploratory background-only run is retained as a solver-comparison diagnostic because its cross-chain $R$-statistic is undefined.

- *The Temporal Horizon:* The result of this paper does not require the CMB acoustic peaks to originate from a physically expanding spatial metric beginning at a density singularity. In the TEP interpretation tested here, the same conformal transport integrals normally written in terms of the FLRW scale factor $a(t)$ are reproduced by the temporal conformal field $A(\phi)$. The limit conventionally described as $a\to0$ is therefore reinterpreted, at the level of clock transport and photon phase evolution, as a temporal-horizon limit $A_{\rm clock}\to0$ relative to the present epoch. This is the precise sense in which the present calculation removes the Big Bang singularity from the acoustic-sector interpretation: the sound horizon, photon-baryon driving, and acoustic-peak morphology are preserved without requiring physical spatial stretching back to a zero-scale-factor origin. In this precise but physically important sense, the CMB acoustic sector no longer requires a physical zero-scale-factor Big Bang; it can be equivalently represented as a conformal temporal-horizon limit of the clock-rate field. Because this temporal horizon is asymptotic, cosmological epochs typically defined by a finite "time since the Big Bang" are instead fundamentally mapped by their thermodynamic temperature and the exact conformal clock-rate, shifting the measurement of cosmic history from a linear stopwatch to a thermodynamic state. Crucially, as established in TEP-BBN, the observed radiation temperature ($T_{\rm obs}$) and the local matter temperature ($T_{\rm loc}$) are distinct physical quantities governed by temporal transport. The broader nonsingular closure—geodesic completeness and curvature regularity in TEP-TH, together with the native chemical-evolution framework and local CMB-thermalization proof of concept in TEP-BBN—completes the companion treatment beyond the acoustic sector considered here.

- *Cosmological Constraints:* A joint hi_class Cobaya MCMC (Planck 2018 low-$\ell$ TT/EE + lensing + BAO + Pantheon+) constrains the implemented isotropic-conformal field while retaining a pressureless well-driving component, whereas Step 21 demonstrates that under the canonical unified matter metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) on the selected two-branch sector $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$), (i) in the pre-boundary acoustic regime the zero-parameter baryon clock-slope response predicts $\omega_\phi=\omega_b(y_s/\sqrt{2})=0.11816$, matching the unconstrained full-Planck TTTEEE MCMC posterior ($0.11813\pm0.00117$) to $0.02\sigma$ ($0.02\%$) and reproducing $100\theta_s^{\rm CLASS}=1.04201$ ($+0.10\sigma$ at $H_0=68.18\text{ km s}^{-1}\text{ Mpc}^{-1}$; $100\theta_s=1.04111$, $+0.02\sigma$ at $H_0=67.95\text{ km s}^{-1}\text{ Mpc}^{-1}$), the acoustic peaks to $\le0.31\%$–$0.40\%$, the CDM-free matter power spectrum $P(k)$ with $\sigma_8=0.8170$–$0.8186$, $r_d=147.49$–$147.55\text{ Mpc}$, SDSS DR12 BAO+RSD $\chi^2/N=0.71$–$0.76$, and the full Planck TTTEEE + low-$\ell$ + lensing + BAO + Pantheon+ likelihood with $\Delta\chi^2=-0.06$ ($\Delta\text{AIC}=-2.06$ vs. free $\omega_{\rm cdm}$; Part C), while (ii) in the late-time cosmic web intermediate temporal depth along the photon path boosts the baryon-only $C_\ell^{\phi\phi}$ by $9.4\times$–$38.3\times$ ($1\times$) to $26.5\times$–$129.3\times$ ($2\times$) across cosmic-web and cluster-outskirt scales and fits the official 9-bin Planck 2018 `CMBMarged` lensing likelihood at $\omega_{\rm cdm}=0$ with $\chi^2=15.40$ (2-parameter fixed-Silk) and $\chi^2=13.40$ ($\chi^2/N=1.49$, $\Delta\chi^2=+1.11$ vs. the $12.30$ CDM baseline and $\Delta\chi^2=-1252.33$ vs. bare isotropic no-CDM), recovering the Planck CMB lensing spectrum across $L\in[8,400]$ without particulate dark matter. The completed native high-$\ell$ TTTEEE extension binds the additional homogeneous amplitude to $\epsilon_T < 0.0087$ (95% CL) at fixed $z_T=5$ — $\epsilon_T < 0.0103$ marginalized over $z_T\in[0.5,50]$ — and returns $\Delta\chi^2 = -0.11$ against a matched $\Lambda$CDM reference — statistical indistinguishability on the full data set. With DESI DR2 BAO, a single-history geometric fit selects this native branch as the TEP background for supernovae, BAO and the CMB acoustic angle together, and finds it preferred over $\Lambda$CDM ($\Delta\chi^2=-8.1$) at a late transition $z_T\simeq0.5$. The companion paper TEP-C0 (Paper 26) provides late-time evidence: BF approximately 5.8 (conservative $z_{\rm los}=5$), approximately 5.4 (fixed $z_{\rm los}=100$ benchmark), and approximately 5.4 (broad free-$z_{\rm los}$), reducing the phenomenological need to treat late-time acceleration as primitive spatial expansion. The imported evidence is prior- and anchor-sensitive: the conservative Bayes factor falls to approximately 0.63 after removing $z < 0.023$ anchors.

### 7.2 Resolving the Cosmological Crises

Standard $\Lambda$CDM cosmology currently faces two severe crises: the Hubble Tension ($H_0$) and the unexpectedly massive high-redshift galaxy candidates observed by JWST. The Static Conformal Universe offers a unified TEP interpretation of both without invoking early-universe modifications.

- *The Hubble Tension:* In the broader TEP corpus, Paper 11 argues that the temporal shear field is environmentally screened. Supernova and Cepheid distance indicators probe different environmental projections of the temporal field, with Cepheids residing in comparatively dense stellar environments where the observable response is more strongly screened (yielding a lower inferred $H_0$), while the supernova propagation and host-environment sector samples a less locally screened projection (yielding a higher inferred $H_0$). Paper 11 argues that the tension is an artifact of environmental screening on local kinematic distance probes, not an early-universe physics crisis. HC does not independently re-analyse the distance-ladder data; it imports the interpretation.


- *JWST High-Redshift Galaxies:* Within the broader TEP interpretation, the temporal-horizon picture removes the finite-age assembly bottleneck by replacing the FLRW singularity with an asymptotic conformal-clock boundary. The massive galaxies observed by JWST therefore form strictly within standard astrophysical accretion models over vast timescales (Paper 12).

### 7.3 Synthesis: Consistency Requirement Satisfied

This analysis implements and explicitly demonstrates the consistency of the native TEP implementation within a rigorous Boltzmann solver framework. The acoustic indistinguishability of the static conformal background from $\Lambda$CDM at recombination is a mathematical identity by construction — identifying $A(\phi)$ with $a(t)$ reproduces the FLRW integrals by definition — and therefore verifies the `hi_class` conformal-frame implementation rather than constituting an independent test of TEP. Its scientific content is nonetheless real: it establishes that the early-universe background physics cannot distinguish between a stretching spatial metric and an evolving conformal clock-rate field, satisfying the necessary consistency requirement any viable conformal-frame alternative must meet and shifting the discriminating burden to the perturbation and late-time sectors where the frameworks genuinely differ. The kernel-level reconciliation with the eternal-native surface is reported in TEP-BBN Steps 19–20 (`results/step_19_eternal_surface_reconciliation.json`, `results/step_20_ambient_bath_front.json`): the eternal ionization-front profile and the Peebles kernel agree on the integrated column ($\tau = 1$ at $z \approx 1080$, $\tau_{\rm reion} \approx 0.054$), forward-evaluating the self-shielded LyC radiative transfer across the independent Step-14 $[400,800]\,n_{b,0}$ front-column band predicts $z_{\tau=1}^{\rm pred}=1077.9$ at the $600\,n_{b,0}$ midpoint (span $[1069.9,1083.6]$ due to the Wien steepness $\chi\simeq53.5$, with exact inversion $\Delta_{\rm front}=668.9\,n_{b,0}$, $n_b=1.68\times10^{-4}\text{ cm}^{-3}$), and deriving the pre-boundary inertial driving density from the two-branch temporal-shear law at the sound horizon — where $g_t=\Lambda_X^2/(\sqrt{2}M_{\rm Pl})=cH_0/2$ is a fixed constant of the scalar action $S_\phi$ and $q_s=\Omega_b(H_0 r_s/c)$ is conformally invariant under $A(\phi)$, giving $\omega_{m,\rm eff}=\omega_b(1+y_s/\sqrt{2})=0.1391$ ($\omega_\phi=0.1166$, $q_s=1.62\times10^{-3}$, $y_s=7.38$, $M_s=6.22$ at $r_s=146.19\text{ Mpc}$, $\theta_s^{\rm derived}=0.010425\text{ rad}$, $\ell_A=301.3$ in the single-pass reduced 3-component integral) and converging under self-consistent $r_s(\omega_\phi)$ fixed-point iteration in exact `hi_class` to $r_s=144.94$–$145.00\text{ Mpc}$, $q_s=1.59$–$1.60\times10^{-3}$, $y_s=7.42$–$7.44$, and $\omega_\phi=0.1179$–$0.1182$ (spanning $\omega_\phi\in[0.1174,0.1199]$ across $H_0\in[67.4,70.0]\text{ km s}^{-1}\text{ Mpc}^{-1}$, all within $\le1.6\sigma$ of the TTTEEE MCMC posterior $0.11813\pm0.00117$ and within $0.02\sigma$ at the posterior mean $H_0=68.18$, reproducing $100\theta_s^{\rm CLASS}=1.04201$ at $+0.10\sigma$ at $H_0=68.18$ and $100\theta_s=1.04111$ at $+0.02\sigma$ at $H_0=67.95$, $\Delta\chi^2=-0.06$ and $\Delta\text{AIC}=-2.06$ on the full Planck TTTEEE + low-$\ell$ + lensing + BAO + Pantheon+ likelihood, linear-scale $\sigma_8=0.8170$–$0.8186$, $r_d=147.49$–$147.55\text{ Mpc}$, and peaks within $\le0.31\%$–$0.40\%$) — together with coupled linear Einstein–Boltzmann acoustic ODE + spherical-Bessel integration on the solved front — where the driving component is the continuous ambient proper-time field's two-branch clock-slope perturbation mode with timelike kinetic coefficient $\mathcal K_t=2k+6=38.060$ from the quadratic scalar action around the rolling cosmic background and prescribed sound-shell stiffness $\mathcal K_{\rm sp}^{\rm eff}=\sqrt{2}/y_s=0.1917$ (the deep-IR limit of the 3-D Frobenius/RMS eigenvalue $K_{\rm rms}=0.1918$ of the native spatial flux Hessian $K_\perp=1/y_s=0.1356$, $K_\parallel=2/y_s+(q_s y_s)^2=0.2713$, with peak shifts $\le0.12$ across $[K_\perp/\mathcal K_t,K_\parallel/\mathcal K_t]$; $c_{p,\rm eff}^2=\mathcal K_{\rm sp}^{\rm eff}/\mathcal K_t=5.04\times10^{-3}\ll c_{\gamma b}^2\simeq0.30$, and $\sigma_{T,p}=0$ by conformal Maxwell invariance; Paper 0 Step 81), not an inserted dust fluid — shifts the eternal diffusion blur from the uninverted $500\,n_{b,0}$ value $\ell_D = 1242$ ($\theta_D \approx 8\times10^{-4}$ rad) to $\ell_D = 1521$ ($\theta_D = 6.58\times10^{-4}$ rad, within $1.2\%$ of the observed Silk angle $6.50\times10^{-4}$ rad, inside the Planck $1400$–$1600$ damping band) while recovering the first three acoustic peak multipoles at $\ell = 221.0,\,527.1,\,811.8$ in the reduced ODE ($\le1.94\%$ at $c_{p,\rm eff}^2=0$, $\le1.90\%$ with $c_{p,\rm eff}^2=5.04\times10^{-3}$, and $\le0.40\%$ in exact `hi_class`; closure ablation: wavenumber-resolved $y(k)$ $\le7.54\%$, slaved-Poisson null $\le5.41\%$, isolating the autonomous ambient time-field inertia as the operative closure; on linear cosmological scales $k\lesssim0.2\,h\,\text{Mpc}^{-1}$ [$R\gtrsim10\text{ Mpc}$, where $99.8\%$ of the $R_8=8\,h^{-1}\text{Mpc}$ $\sigma_8^2$ variance comes from $k\le1.0\,h\,\text{Mpc}^{-1}$ so the $k_*=1.5\,h\,\text{Mpc}^{-1}$ halo-transition filter shifts $\sigma_8$ by only $1.0\%$ from $0.8185$ to $0.8103$] the scale-averaged multiplier $M_s=1+y_s/\sqrt{2}$ enters the linear Poisson equation identically to $\omega_{m,\rm eff}=\omega_b+\omega_\phi$, whereas in the nonlinear late-time virialized galaxy/cluster limit [$R\in[1,100]\text{ kpc}$, $\dot u\to 0$, $g_b(R)=GM_b/R^2$] the timelike momentum $\pi_u=\mathcal K_t\dot u$ vanishes identically and the baryon-sourced algebraic Gauss law $(k/\sqrt{2})q(R)y(R)^2+q(R)^2 y(R)^3=1$ smoothly transitions $M_{\rm eff}(R)=1+y(R)/\sqrt{2}$ from $6.26$ on linear scales through $5.18$ at $100\text{ kpc}$, $2.25$ at $30\text{ kpc}$, and $1.42$ at $10\text{ kpc}$ down to $1.12$ at $3\text{ kpc}$ and $1.04$ at $1\text{ kpc}$ for $M_b=6\times10^{10}\,M_\odot$, reducing to the static baryon-only two-branch RAR/lensing law with zero independent collisionless dark-matter halo cusp).

The corpus-level scoping of this result is stated plainly. The isomorphism verified here is a transport-mapping identity: it demonstrates that the conformal geometry carries the acoustic transfer functions, not that an eternal static universe produces the hot plasma whose recombination imprints them. That supply is the independent eternal-universe thermodynamic sector of the corpus: TEP-TH evaluates the recombination microphysics on the temporal-horizon geometry ($z_* \approx 1079$, $r_s \simeq 146$ Mpc) and verifies FIRAS-clean Planckian preservation, while TEP-BBN computes the supply budgets (Gates 11–12) — a unit-optical-depth reprocessing surface at temporal depth $z \approx 130$–$1340$ for dust-grade absorptivity, bracketing the recombination depth at which $T_{\rm loc}=T_0(1+z_*)\approx2.9\times10^{3}$ K is the hydrogen ionization edge, and an ionization-equilibrium balance maintaining the electron column to the required depth. The plasma's heating and cooling are real local thermodynamic processes in this architecture; the transport map demonstrated here relates local thermal states at different temporal depths rather than creating them. The residual open closure is the definitive electron-density profile $n_e(\ell)$ — the absorber-statistics inversion posed in TEP-BBN — a quantitative completion of a specified supply mechanism, not the absence of one.

Late-universe Pantheon+ data in TEP-C0 favor the physical no-$\Lambda$ TEP branch over baseline $\Lambda$CDM with BF $\simeq 5.8$ (conservative $z_{\rm los}=5$), BF $\simeq 5.4$ (fixed $z_{\rm los}=100$ benchmark), and BF $\simeq 5.4$ (free-$z_{\rm los}$), providing a concrete conformal-frame alternative to the background expansion interpretation. The active-perturbation closure reported in Section 4.5 confirms that the implemented linear pure-conformal scalar fluctuation sector is numerically regular across the sampled evolution, including the $D = Q_s = 0$ turnover, and quantifies its response: a scalar with unscreened canonical kineticity is excluded by the growth data, while the kinetically screened branch — the regime the nonlinear $P(X)$ sector supplies — is simultaneously stable and consistent with the observed growth amplitude at the percent level, with a characteristic low-$\ell$ ISW excess as its residual signature. By treating time itself as a dynamical, environmentally screened scalar field, TEP seeks to unify early-universe acoustic physics, late-time "acceleration", the $H_0$ tension, and JWST anomalies into a single, cohesive static geometric framework. The present paper provides both the hi_class background/acoustic benchmark and the perturbation-closure validation.


## References

Smawfield, M. (Paper 0). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed.* TEP Corpus (Jakarta). DOI: 10.5281/zenodo.16921911.

Smawfield, M. (Paper 1). *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks.* TEP Corpus. DOI: 10.5281/zenodo.17127229.

Smawfield, M. (Paper 6). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T.* TEP Corpus. DOI: 10.5281/zenodo.18064365.

Smawfield, M. (Paper 9). *What Do Precision Tests of General Relativity Actually Measure?* TEP Corpus. DOI: 10.5281/zenodo.18109760.

Smawfield, M. (Paper 11). *The Cepheid Bias: Resolving the Hubble Tension.* TEP Corpus. DOI: 10.5281/zenodo.18209702.

Smawfield, M. (Paper 12). *The Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies.* TEP Corpus. DOI: 10.5281/zenodo.19000827.

Smawfield, M. (Paper 13). *The Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries.* TEP Corpus. DOI: 10.5281/zenodo.19102061.

Smawfield, M. (Paper 26). *Temporal Equivalence Principle: A Covariant Alternative to Cosmic Expansion.* TEP Corpus. DOI: 10.5281/zenodo.20370143.

Smawfield, M. (Paper 27). *Temporal Equivalence Principle: Temporal Horizon Cosmology and the Absence of a Physical Big Bang Singularity.* v0.5 (Thika). DOI: 10.5281/zenodo.20723059.

Smawfield, M. (Paper 29). *Temporal Equivalence Principle: Dynamical Proper Time and the Illusion of Primordial Deuterium.* Paper 29 (Dubai). DOI: 10.5281/zenodo.21841147.

Bellini, E., & Sawicki, I. 2014, JCAP, 07, 050. *Maximal freedom at minimum cost: linear large-scale structure in scalar-tensor theories.*

Berezhiani, L., & Khoury, J. 2015, Phys. Rev. D, 92, 103510. *Theory of dark matter superfluidity.*

Brax, P., Burrage, C., Davis, A.-C., & Gubitosi, G. 2019, Phys. Rev. D, 100, 083515. *Screening mechanisms in scalar-tensor theories.*

Cobaya Team. 2023, *Cobaya: Code for Bayesian Analysis of physical theories.* arXiv:2305.02971.

Hu, B., Raveri, M., Frusciante, N., & Silvestri, A. 2014, Phys. Rev. D, 89, 103530. *EFTCAMB/EFTCosmoMC: Numerical Notes.*

Knox, L., & Millea, M. 2020, Phys. Rev. D, 101, 043533. *Hubble constant hunter's guide.*

Lagos, M., Bellini, E., Jimenez, J. B., et al. 2018, JCAP, 03, 021. *hi_class: Horndeski in the Cosmic Linear Anisotropy Solving System.*

Lewis, A., Challinor, A., & Lasenby, A. 2000, Astrophys. J., 538, 473. *Efficient computation of cosmic microwave background anisotropies.*

Planck Collaboration. 2020, A&A, 641, A1. *Planck 2018 results. I. Overview and cosmological parameters.*

Planck Collaboration. 2020, A&A, 641, A6. *Planck 2018 results. VI. Cosmological parameters.*

Riess, A. G., Casertano, S., Yuan, W., et al. 2022, ApJ, 934, L7. *A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team.*

Sawicki, I., & Bellini, E. 2015, Phys. Rev. D, 92, 084061. *Stability of dark energy and the generalized no-slip condition.*

Zumalacárregui, M., & García-Bellido, J. 2014, Phys. Rev. D, 89, 064046. *Transforming gravity: from derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian.*

## Appendix A: Technical Implementation Details

### A.1 hi_class Installation and Configuration

#### A.1.1 Building with TEP Support

hi_class is installed automatically by pipeline Step 1 (`step_00b_install.py`), which clones hi_class and applies the native TEP patch from `external/patches/hiclass_tep_native.patch` to `source/background.c`, `source/input.c`, and `include/background.h`. Manual rebuild:

```
cd external/hi_class/hi_class
make clean && make
```

**Screening projection notice.** Screening in TEP is represented at theory level by the environmental operator S_Σ(E). Quantities such as ρ_T, R_T(M), S_⊕(r), compactness Φ/c^2, local stellar density, coherence length, proximity, and boundary geometry are domain-specific projections of E, not independent screening mechanisms and not interchangeable universal thresholds.

### A.2 Cobaya Installation

```
pip install cobaya
cobaya-install planck_2018_lowl.TT planck_2018_lowl.EE \
 planck_2018_lensing.native bao.sdss_dr12_consensus_final \
 sn.pantheonplus --path /path/to/likelihoods
```

### A.3 TEP Module C Code Structure and Implementation Note

The native conformal modification is implemented directly in hi_class `source/background.c`, controlled by the `.ini` flags `tep_mode`, `epsilon_T`, `z_T`, `n_T`. The relevant functions are:

- `tep_f_transition(pba, z)`: returns the suppression factor $S(z) = \exp[-(z/z_T)^{n_T}]$; the full transition is $f_T(z) = \ln(1+z)\,S(z)$ (see `core/cosmology.py:f_T`).

- `tep_gamma_factor(pba, z)`: returns the exact covariant conformal factor $A(z) = \exp[\epsilon_T \ln(1+z)\,S(z)]$ (not linearised).

- The Hubble rate and its conformal-time derivative are mathematically mapped using the exact HC conformal-frame mapping $M(z) = A/(1-\alpha_A)$ in `background_functions` and in the initial-Hubble setter, explicitly evaluating the static conformal geometry.

*Implementation note (corrected bug).* An earlier build used $f_T = 1 - \exp[-(z/z_T)^{n_T}]$ (the complement of the suppression function). This incorrectly inverted the scalar field profile, erroneously mapping the peak kinetic energy to the early universe rather than intermediate redshifts, which logically corrupted the acoustic integration. In addition, the post-processing step that read the spectra used a hard-coded output index and could silently load a stale file from an earlier run. Both issues are fixed: the transition function now uses the shared TEP-C0 implementation (`core/cosmology.py`), correctly matching the field profile to the Pantheon+ apparent acceleration, and the analysis resolves the most recent hi_class output deterministically. *Sign convention (TEP disformal metric):* the distance integrand is multiplied by $A(z)$ for null-geodesic propagation in the conformal frame. The legacy SMG alpha-function stub (`smg_tep_*`) has been retired; production physics lives in the patched `background.c` (`external/patches/hiclass_tep_native.patch`).

### A.3a Derivation of the Conformal-Frame Factor $M(z)$

This appendix derives the background conformal mapping $M(z)$ from the bi-metric action (Equation $\ref{eq:3_theory_01}$) using a single frame convention held fixed throughout, demonstrating the exact conformal-frame relation under the adopted HC reference-density convention, implemented natively in the codebase.

*Setup and convention.* Matter, photons, and rods couple to the conformal metric $\tilde{g}_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\nabla_\nu\phi$. For the homogeneous background the disformal term contributes only through the time-time component and is absorbed into the lapse; the evolution is governed by the conformal part, so $B \to 0$ is imposed here (the disformal sector re-enters at the perturbative/GW level via $\alpha_T$, Section 2.2.2). Throughout this derivation, $a_E$ and $t_E$ denote the reconstructed variables of the isomorphic FLRW representation employed by the integrator: under the canonical architecture (Paper 0, §8; Paper 30, Appendix C) the physical gravitational metric is strictly static, and the Friedmann-form evolution assigned to $a_E$ below defines the representation rather than attributing expansion to the gravitational sector. Matter-sector observables are invariant under the decomposition $\tilde{a} = A\,a_E$. The conformal part gives the standard map between the Einstein-frame scale factor $a_E$ and cosmic time $t_E$ and their conformal counterparts:

\begin{equation} \label{eq:a3a_map}
\tilde{a} = A(\phi)\,a_E, \qquad d\tilde{t} = A(\phi)\,dt_E.
\end{equation}

These two relations *define* the convention; every subsequent equation is derived from them. The transition factor $S(z)=\exp[-(z/z_T)^{n_T}]$ correctly forces $A(z)\to1$ at the local endpoint, so the code's redshift grid can be identified with the physical conformal-frame redshift. The explicit $(z_E,\tilde z)$ distinction is retained only to derive the frame relation.

*Physical Hubble rate.* The expansion rate measured by conformal-frame clocks and rulers is $\tilde H \equiv \tilde a^{-1}\,d\tilde a/d\tilde t = d\ln\tilde a/d\tilde t$. Using $d/d\tilde t = A^{-1}\,d/dt_E$ and $\ln\tilde a = \ln A + \ln a_E$,

\begin{equation} \label{eq:a3a_Htilde}
\tilde H = \frac{1}{A}\frac{d}{dt_E}\big(\ln A + \ln a_E\big) = \frac{1}{A}\Big(\frac{d\ln A}{dt_E} + H_E\Big),
\end{equation}

where $H_E = d\ln a_E/dt_E$ is the Einstein-frame rate. Because the TEP conformal factor $A(z)$ is evaluated as a function of the observable physical redshift (the matter-frame redshift $1+z = \tilde{a}_0/\tilde{a}$), the coupling $\alpha_A$ computed in the codebase is fundamentally the derivative with respect to the *matter-frame* scale factor:

\begin{equation} \label{eq:a3a_alpha}
\frac{d\ln A}{dt_E} = \frac{d\ln A}{d\ln \tilde{a}}\,\frac{d\ln \tilde{a}}{dt_E} = \alpha_A\,(A \tilde{H}), \qquad \alpha_A \equiv \frac{d\ln A}{d\ln \tilde{a}} = -\frac{d\ln A}{d\ln(1+z)},
\end{equation}

which matches the definition in Section 3.2. Substituting $d\ln A/dt_E = \alpha_A A \tilde{H}$ into ($\ref{eq:a3a_Htilde}$) yields $\tilde{H} = \alpha_A \tilde{H} + H_E/A$, or equivalently $\tilde{H}(1 - \alpha_A) = H_E/A$.

*Einstein-frame reference rate.* The remaining step is to relate the Einstein-frame Hubble rate $H_E$ to the reference $\Lambda$CDM rate $H_{\Lambda\rm CDM}$. Under the conformal transformation $\tilde{g}_{\mu\nu} = A^2 g_{\mu\nu}$, the spacetime determinant transforms as $\sqrt{-\tilde{g}} = A^4 \sqrt{-g_E}$, so the homogeneous energy density transforms as $\rho_E = A^4 \tilde{\rho}$. The Friedmann equation of the isomorphic representation gives $H_E^2 = \frac{8\pi G}{3}\,\rho_E = A^4 \frac{8\pi G}{3}\,\tilde{\rho}$. The TEP implementation convention identifies the matter-frame reference density used by the acoustic calculator with the standard $\Lambda$CDM density evolution, $\tilde{\rho} = \rho_{\Lambda\rm CDM}$, so that $H_E^2 = A^4 H_{\Lambda\rm CDM}^2$ and therefore $H_E = A^2\,H_{\Lambda\rm CDM}$. Substituting into $\tilde{H}(1 - \alpha_A) = H_E/A$ gives the exact conformal-frame relation under the adopted HC reference-density convention:

\begin{equation} \label{eq:a3a_exact}
\boxed{\;\tilde H(z) = \frac{A(z)}{1 - \alpha_A(z)}\,H_{\Lambda\rm CDM}(z)\;} \qquad \Longrightarrow \qquad M_{\rm exact}(z) = \frac{A}{1 - \alpha_A}.
\end{equation}

*Implementation Status.* The production codebase (`hiclass_tep_native.patch`) evaluates this exact HC conformal-frame mapping $M = A/(1-\alpha_A)$ directly, guaranteeing mathematical fidelity to the conformal evaluation without requiring first-order approximations.

### A.4 Saturation Scale in Cosmological Units

The candidate Temporal Topology saturation scale ρ_T ≈ 20 g/cm³ converts to cosmological units as:

\begin{equation} \label{eq:9_appendix_01}
\rho_T = 20 \text{ g/cm}^3 = 2 \times 10^4 \text{ kg/m}^3 \approx 1.1 \times 10^{34} \text{ eV/cm}^3
\end{equation}

In Planck units ($\hbar = c = G = 1$):

\begin{equation} \label{eq:9_appendix_02}
\rho_T \approx 4 \times 10^{-93} M_{\rm Pl}^4
\end{equation}

Compare to cosmic mean density today ($\rho_{\rm crit,0} \approx 10^{-123} M_{\rm Pl}^4$). The hierarchy ensures the vast cosmological voids evaluate the pure unsuppressed conformal field, accurately simulating the expansion of space.

*Intermediate environments and operational parameter bounds.* In the density-projection used for this order-of-magnitude comparison, between the terrestrial laboratory and the cosmic mean the screening transition is continuous. At stellar atmospheric densities ($\rho \sim 10^{-6}$ g/cm³), the field is partially screened; at interplanetary densities ($\rho \sim 10^{-23}$ g/cm³), it is essentially unscreened. Certain orbital datasets—notably the Galileo GNSS clock ensemble—fall outside the operational parameters established for valid TEP-GNSS screening analysis (Paper 1), because their orbital altitude and local gravitational environment do not satisfy the strict kinematic isolation required to isolate the conformal phase drift from standard relativistic corrections. These exclusions are documented in the TEP-GNSS pipeline and do not affect the cosmological bound, which operates in the deep unscreened regime where $\rho \ll \rho_T$.

### A.5 Stability Sector Closure

To formally verify the stability of the active scalar perturbations, the native `hi_class` SMG module was extended to evaluate the exact analytical limits of the TEP conformal geometry at runtime.

In a generalized Horndeski treatment, the solver enforces the following physical stability conditions:

- $c_s^2 \ge 0$ (no gradient instabilities)

- $D = \alpha_K + \frac{3}{2}\alpha_B^2 \ge 0$ (no ghosts)

- $|\alpha_M|$ bounded (sub-luminal Planck-mass running)

- $\alpha_T \approx 0$ (gravitational wave speed constraints)

For the conformal modification implemented here, the EFT parameters map strictly to the dynamical background derivative $\alpha_A$:

\begin{equation} \label{eq:a5_alpha_M}
\alpha_M = -2\alpha_A
\end{equation}

\begin{equation} \label{eq:a5_alpha_B}
\alpha_B = 2\alpha_A
\end{equation}

\begin{equation} \label{eq:a5_alpha_K}
\alpha_K = w_c\,\alpha_A^2, \qquad w_c \equiv Z_E - 6\beta_A^2 = Z_E - 6
\end{equation}

\begin{equation} \label{eq:a5_alpha_T}
\alpha_T = 0
\end{equation}

#### A.5a Origin of the Closure and its Stability Boundary

For the $B(\phi)=0$ sector, the exact Weyl image of a generalized kinetic function is $$G_2(\phi,\tilde X)=A^{-4}P(A^2\tilde X,\phi)-6M_{\rm Pl}^2A^{-2}\left(\frac{d\ln A}{d\phi}\right)^2\tilde X,\qquad G_4=\frac{M_{\rm Pl}^2}{2A^2}.$$ Nonlinear dependence on $\tilde X$ is retained. The production HC solver evaluates the following linear-in-$\tilde X$ effective template, with the background coefficient $Z_E$ assigned from the kinetic-response comparison; it is exact for the canonical scalar but is not the complete nonlinear master-action image:

\begin{equation} \label{eq:a5_jordan_action}
S_J = \int d^{4}x\,\sqrt{-\tilde g}\left[\frac{F(\phi)}{2}\tilde R + W(\phi)\tilde X - U(\phi)\right],\qquad
F(\phi) = M_{\rm Pl}^{2}A^{-2},\quad
W(\phi) = A^{-2}\left(Z_E - \frac{6M_{\rm Pl}^{2}A_{,\phi}^{2}}{A^{2}}\right),\quad
U(\phi) = A^{-4}V,
\end{equation}

with $\tilde X = -\tfrac{1}{2}\tilde g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi$ and $Z_E \equiv P_{,X} + 2XP_{,XX}$ the Einstein-frame kinetic coefficient of the scalar evaluated on the background ($Z_E = 1$ for a canonical scalar). The conformal term contributes $-6M_{\rm Pl}^{2}A^{-4}A_{,\phi}^{2}\tilde X$ after integration by parts of the $3M_{\rm Pl}^{2}A^{-2}\tilde\Box\ln A$ piece of the transformed Einstein–Hilbert term. For the exponential coupling $A=\exp(\beta_A\phi/M_{\rm Pl})$, $M_{\rm Pl}^{2}A_{,\phi}^{2}/A^{2}=\beta_A^{2}$, so $W = A^{-2}(Z_E-6\beta_A^{2})$; the canonical Einstein-frame scalar is the special case $Z_E = 1$, for which the earlier-quoted $W = A^{-2}(1-6\beta_A^{2})$ is recovered. This is a Brans–Dicke-class action with Horndeski functions $G_2 = W\tilde X - U$ and $G_4 = F/2$. Applying the standard Horndeski map — the same map used internally by `hi_class` for its native `brans_dicke` model — gives $M_\star^{2} = F$, $\alpha_M = d\ln F/d\ln a = -2\alpha_A$, $\alpha_T = 0$, $\alpha_B = -\alpha_M = 2\alpha_A$, and $\alpha_K = 2\tilde X G_{2,X}/(H^{2}M_\star^{2}) = (W/F)(d\phi/d\ln a)^{2}$. With $d\phi/d\ln a = (M_{\rm Pl}/\beta_A)\alpha_A$, the kineticity evaluates to $w_c\,\alpha_A^{2}$ with $w_c = (Z_E-6\beta_A^{2})/\beta_A^{2} = Z_E - 6$ at $\beta_A = -1$. The production runtime explicitly imposes $c_s^2=1$ for this effective closure. The nonlinear master sector instead has timelike principal speed $P_{,X}/Z_E$; assigning its $Z_E$ to the template does not also reproduce that principal operator.

The canonical image $w_c = -5$ ($Z_E = 1$) was previously presented as the closure of this sector on the strength of the ghost discriminant alone. Direct evolution of the $\delta\phi$ perturbation through the full Einstein–Boltzmann hierarchy (production implementation, `use_pert_var_deltaphi_smg`) shows that this branch is *linearly unstable* on the TEP background: the scalar develops a tachyonic growing mode — the Damour–Esposito-Farèse scalarization channel — producing a median matter-power enhancement of order $10^{6}$ before nonlinear saturation. This instability is invisible to the no-ghost discriminant: the ghost-free combination $2WF + 3F_{,\phi}^{2} = 2M_{\rm Pl}^{2}A^{-4}$ remains positive-definite for every nonzero $\beta_A$ (equivalently the effective Brans–Dicke parameter $\omega_{\rm eff} = w_c/(4\beta_A^2) = -5/4$ exceeds the $-3/2$ ghost bound), but the scalar's effective mass on the conformal background is tachyonic. A $w_c$-resolved scan of the closure (Step 24, `results/closure_response_scan.json`) maps the response: $w_c = -5$ runaway; $w_c \approx -1$–$+1$ marginal ($\sigma_8 \approx 2.8$–$4.4$); the minimal stable sign-corrected branch $w_c = +7$ gives $\sigma_8 = 1.61$ (excluded by weak lensing); $w_c \gtrsim 700$ recovers $\Lambda$CDM growth to sub-percent precision; and $w_c \sim 6\times10^{3}$ yields $\sigma_8 \simeq 0.76$, at the weak-lensing central value. In terms of the microphysics, the viable cosmological closure therefore requires $Z_E \gtrsim 7\times10^{2}$ — a large effective kineticity, i.e. the nonlinear $P(X)$ branch of the scalar sector coarse-grained over the inhomogeneous virialized-well and halo-gradient landscape ($q \gtrsim 12$, where $73.5\%$–$92.6\%$ of matter resides in collapsed halos above $10^{12}$–$10^{10}\,M_\odot$; Paper 0, Step 74) rather than evaluated on the unscreened canonical $Z_E = 1$ limit or on the residual late-time dynamical profile $A_{\rm HC}(z)$ alone ($\xi_{\rm dyn} \sim \epsilon_T^2 \sim 10^{-5}$, whereas the full cosmic ruler roll $\bar\xi = 1$ gives homogeneous $Z_E = 2k + 6 = 38.06$ and $G_{\rm eff}(0)/G = 1.16$; Paper 0, Steps 54, 81).

The selected Paper 0 operator supplies positive environmental principal coefficients, with $P_{,X}=k\sqrt{\xi}+2\xi$ and $k\simeq16.03$ fixed by the rotation-curve normalization (Paper 0, Steps 53–54). Write the dimensionless gradient magnitude as $q=\sqrt{2\xi}$; for a static spatial gradient, $q=c|\nabla u|/H_0=c^2|\nabla u|/(2g_t)$. The two coefficients are

$$P_{,X}(q)=\frac{k}{\sqrt2}q+q^2,\qquad Z(q)=P_{,X}+2XP_{,XX}=\sqrt2\,kq+3q^2.$$
On a homogeneous timelike background, $Z$ is the temporal kinetic coefficient and $P_{,X}$ is the spatial coefficient, giving $c_s^2=P_{,X}/Z\in(1/3,1/2)$. On a static spatial-gradient background the assignments differ: $Z_t=Z_\perp=P_{,X}$ and $Z_\parallel=Z$, with longitudinal characteristic ratio $v_\parallel^2=Z/P_{,X}\in(2,3)$ and transverse ratio one. All coefficients are positive for nonzero gradient. The static longitudinal stiffness therefore cannot be substituted directly for a homogeneous temporal kineticity.

The existing ambient ladder illustrates the scale of the nonlinear response: $Z=13.9$ at $q=0.57$, $17.9$ at $q=0.72$, and approximately $704$ at $q=12$. At $q=12$, however, the static temporal coefficient is $P_{,X}\simeq280$, not $704$. The effective production branch $w_c=700$ has assigned temporal coefficient $w_c+6=706$; matching that number alone to the static longitudinal coefficient does not derive its perturbation closure. The response scan and growth-averaging calculation remain independent evidence for the tested effective and environmental branches. Their common-action identification requires matching the temporal, spatial and mixing operators on the realized background, including the environmental averaging. The numerical production results retain the explicitly specified luminal effective template.

At the boundary $\beta_A^{2} = Z_E/6$ the explicit kineticity vanishes ($W=0$) and the $f(R)$-like limit is recovered, in which the entire scalar kineticity resides in the braiding; the cancellation of the $\beta_A^{2}$ pieces in $D$ below is the discriminant-level expression of the same structure.

The substitution of the kineticity and braiding terms into the physical no-ghost discriminant yields:

\begin{equation} \label{eq:a5_ghost_identity}
D = (w_c\,\alpha_A^2) + \frac{3}{2}(2\alpha_A)^2 = (w_c + 6)\,\alpha_A^2
\end{equation}

For the canonical image $w_c = -5$ this reduces to the previously quoted identity $D = \alpha_A^2 \ge 0$; on the production branch $w_c = 700$ it gives $D = 706\,\alpha_A^2 > 0$. The discriminant being positive is necessary but — as the linear audit above demonstrates — not sufficient for stability of the perturbation sector. Because the finite-turnover profile $f_T(z)$ has a single extremum, $\alpha_A$ vanishes at the turnover redshift $z_\star$ (for $z_T=5$, $n_T=2$: $z_\star \simeq 2.64785$), and $D$ touches zero there for every nonzero $\epsilon_T$. For $D > 0$, the scalar branch has positive kinetic normalization $Q_s = 2M_\star^2 D/(2-\alpha_B)^2 > 0$ and the implemented luminal effective-closure prescription $c_s^2 = 1$ applies. At $z_\star$, $D = Q_s = 0$, so the propagating scalar description becomes degenerate and a propagation speed is not physically defined. The linear `hi_class` system passes continuously through this point without numerical divergence, as verified by the diagnostic reported in Section 3.3. Because $D = (w_c+6)(d\phi/d\ln a)^{2}/M_{\rm Pl}^{2}$, the crossing is the field-velocity zero $\dot\phi = 0$ at the profile turnover: a kinematic degeneracy of the unitary-gauge description, where the comoving slicing $\delta\phi = 0$ loses its normalization at the instant the background field reverses. No operator coefficient of the canonical Einstein-frame action diverges at $\dot\phi = 0$, so within the implemented pure-conformal sector the crossing is exact constraint reduction rather than a dynamical strong-coupling surface; the field's canonical kinetic normalization is unchanged at the crossing, and the scalar perturbation simply decouples instantaneously where its sourcing background gradient vanishes. A separate nonlinear/canonical analysis of the full disformal sector is outside the scope of the present linear closure. A fully derived sound-speed expression for the disformal and nonlinear screening sectors is likewise outside the scope of the present pure-conformal closure.

The production codebase applies these derived closure values during the calculation of the SMG perturbation coefficients, ensuring the pure-conformal branch is evaluated consistently without requiring pre-tabulated interpolation or analytical approximations.

# Unified TEP Parameter Dictionary

The TEP corpus uses related but distinct symbols across its papers. This dictionary maps every parameter, its definition, the paper where it is primary, and its fiducial or fitted value.

| Symbol | Definition | Primary Paper | Fiducial / Fitted Value |
| --- | --- | --- | --- |
| $A_{\rm clock}(z)$ | Exact observational clock/redshift map: $A_{\rm clock}=(1+z)^{-1}$ | TEP-TH | $(1+z)^{-1}$ (exact) |
| $A_{\rm dyn}(z)$ | Dynamical shear response: $\left(1+z/z_t\right)^{-\epsilon_t}$ | TEP-TH | Modifies late-time evolution |
| $\alpha_A$ | Runtime logarithmic derivative $\alpha_A=d\ln A_{\rm HC}/d\ln a$; distinct from the exact endpoint map $A_{\rm clock}$ | TEP-HC | Redshift-dependent; zero at the finite-profile turnover |
| $\alpha_M$, $\alpha_B$, $\alpha_K$, $\alpha_T$ | Runtime Bellini–Sawicki EFT functions: $\alpha_M=-2\alpha_A$, $\alpha_B=2\alpha_A$, $\alpha_K=w_c\,\alpha_A^2$ with $w_c = Z_E-6$ ($Z_E = P_{,X}+2XP_{,XX}$; production $w_c=700$, canonical image $w_c=-5$ unstable), $\alpha_T=0$; screened diagnostic branch $S_\Sigma<1$: $\alpha_M\to-2S_\Sigma\alpha_A$, $\alpha_B\to2S_\Sigma\alpha_A$ at fixed $w_c$ | TEP-HC | Adopted from $\alpha_A$ |
| $\epsilon_T^{\rm los}$ | Late-time line-of-sight transport amplitude (C0 supernova fit) | TEP-C0 | $\mathcal{U}[0, 2.0]$ (prior); posterior peaked near $\sim 0.89$ |
| $\epsilon_T^{\rm CMB}$ | C0 background/acoustic diagnostic amplitude | TEP-C0 | $-0.0015\pm0.0037$ |
| $\epsilon_T^{\rm HC}$ | Native hi_class homogeneous conformal amplitude | TEP-HC | $0.0043\pm0.0025$ (full high-$\ell$ likelihood); $0.00547\pm0.00429$ (historical low-$\ell$ active chain) |
| $\epsilon_{\rm dyn}$ | Dynamical temporal-horizon response | TEP-TH | Determined by late-time shear |
| $\epsilon_{\rm field}$ | Primordial spectral-flow parameter constrained by $n_s$ | TEP-TH | $0.01755$ (from $n_s=0.9649$) |
| $z_T^{\rm los}$ | C0 line-of-sight supernova transport turnover | TEP-C0 | $5$ (conservative), $100$ (benchmark), free (broad) |
| $z_T^{\rm HC}$ | Homogeneous/acoustic hi_class profile scale | TEP-HC | Fitted jointly with $\epsilon_T$ |
| $p$ | Temporal-horizon conformal exponent: $A_{\rm clock}\sim\eta^{-p}$ | TEP-TH | $0 \lt p\le\tfrac12$ (regular branch) |
| $r_s^{\rm TEP}/r_s^{\Lambda\rm CDM}$ | Pre-recombination sound-horizon ratio | TEP-HC | $0.999994$ ($<6$ ppm deviation) |
| $D=\alpha_K+\tfrac32\alpha_B^2$ | No-ghost discriminant ($D=(w_c+6)\alpha_A^2$; necessary but not sufficient for linear stability — the $w_c=-5$ branch is tachyonic) | TEP-HC | $(w_c+6)\alpha_A^2 \ge 0$ (non-negative; GR/decoupling limit at $\alpha_A=0$) |
| $r(k_{\rm pivot})$ | Native tensor-to-scalar ratio at Planck pivot | TEP-TH | $1.9\times10^{-9}$ |
| $r_{\rm max}$ | Maximum tensor-to-scalar ratio across transition profile | TEP-TH | $5.2\times10^{-7}$ (fiducial $p=0.1$); $6.1\times10^{-5}$ over all tested profiles $0\lt p\le1$ |
| $H_0$ | Hubble parameter (TEP-C0 joint MCMC) | TEP-C0 | $66.70\pm0.58$ km s$^{-1}$ Mpc$^{-1}$ |
| $S_8$ | $\sigma_8\sqrt{\Omega_m/0.3}$ (TEP-HC joint MCMC) | TEP-HC | $0.868\pm0.025$ |
| $\sigma_8^{\rm HC}$ | Native hi_class matter-fluctuation amplitude at fiducial point ($\epsilon_T=0.0066$, $z_T=5$, $n_T=2$) | TEP-HC | $0.824$ (fiducial); $0.858\pm0.016$ (MCMC posterior mean) |

**Note:** Parameters with superscript labels ($^{\rm los}$, $^{\rm HC}$) are related projections of the same temporal sector but are not numerically interchangeable. The turnover scales $z_T^{\rm los}$ and $z_T^{\rm HC}$ describe different physical regimes; the amplitudes $\epsilon_T^{\rm los}$, $\epsilon_T^{\rm CMB}$, $\epsilon_T^{\rm HC}$, and $\epsilon_{\rm field}$ are constrained by different observables.


## Appendix B: Data Availability & Reproducibility


This work follows open-science practices. All results are fully reproducible from raw data
using the documented pipeline. All numerical results, figures, and statistics are generated by deterministic
Python scripts processing public observational data.



### Repository and Code


GitHub Repository: github.com/matthewsmawfield/TEP-HC



The repository contains a deterministic, version-controlled cosmological analysis pipeline
for CMB acoustic peak preservation tests and MCMC parameter estimation with the TEP conformal background.



All MCMC chains, hi_class patch files, posterior samples, and the exact `cobaya` YAML configuration files are released in the Zenodo repository (DOI: 10.5281/zenodo.20572722) under CC-BY 4.0. The `run_all.py` orchestration script and all step scripts are provided in the GitHub repository.



### Repository Structure



```
TEP-HC/
├── data/
│   ├── cobaya/              # Cobaya MCMC configurations
│   ├── external/            # Cobaya packages (likelihoods, data)
│   └── hi_class/            # TEP-CLASS implementation
├── external/
│   ├── hi_class/            # hi_class submodule
│   └── patches/             # TEP patch files
├── scripts/
│   └── steps/               # Analysis pipeline steps
├── core/                    # TEP shared constants and parameters
├── results/                 # Pipeline outputs, figures, MCMC chains
├── site/
│   └── components/          # Manuscript HTML sections
├── tests/                   # Verification tests
├── requirements.txt
├── CITATION.bib
└── README.md
    
```


### Software Environment


Key packages: NumPy, SciPy, Matplotlib, Cobaya, hi_class.
The pipeline has been tested on Python 3.10+.



### License


All code and manuscripts are released under CC-BY-4.0.

---

<!-- ===================================================================== -->
<!-- CANONICAL URL: https://mlsmawfield.com/tep/lens/ -->
<!-- MARKDOWN URL:  https://mlsmawfield.com/tep/lens/manuscript.md -->
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# Temporal Equivalence Principle: A Blind-Prediction Residual Test in Multiply-Imaged Supernovae
**Matthew Lukin Smawfield**
Version: v0.3 (Lisboa)
First published: 10 June 2026
Last updated: 17 October 2026
DOI: 10.5281/zenodo.20572719

---

## Abstract

This paper tests the Temporal Equivalence Principle (TEP) through strong-lensing time-delay residuals, lensing–dynamical mass inference, and action-level photon-path refraction. SN Refsdal offers an unusually high-leverage test case: seven pre-reappearance lens-model variants spanning five modelling families published predictions for the long-baseline SX reappearance delay before the image was observed, and Kelly et al. (2023) later measured that delay independently from SN light-curve fitting. Six of seven strictly blind original predictions and all seven delay-blind revised predictions yield positive residuals (observed delay longer than predicted; correlation-aware family-sign-flip $p = 0.031$, strictly blind floor $p = 0.094$). Under a uniform time-delay rescaling, the $+30.1 \pm 8.9$ d residual corresponds to a target scale shift of $8.7\%$: stellar kinematics break the lens-model mass-sheet degeneracy, making the dynamical–lensing mass difference (phantom mass) a sign-testable consequence of the coupled reconstruction. For the multiply-imaged SLSN-I SN 2025wny ($z_s = 2.015$, five images A–E resolved by HST/JWST across a two-galaxy lens), whose delays were measured in August 2026, the predicted residual ($+2.3$ to $+7.1$ d on the measured $65.7\pm3.5$ d A–D baseline; Step 36b) is comparable to current lens-model precision ($\approx 6.8$ d), keeping the out-of-sample proxy test live.

At the action level (Step 64), the optical mechanism is resolved by comparing the two metric realizations on the selected two-branch kinetic sector $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$ fixed by $a_0$, zero per-system free parameters). Under the historical isotropic-conformal benchmark $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$, spatial rulers stretch by $A^2(\phi)$ alongside clocks so $A^2(\phi(x))$ cancels out of $d\tilde s^2=0$, leaving only the Planck-suppressed scalar stress tensor ($\kappa_\phi\simeq6.5\times10^{-6}$ at 1 Mpc depth; $+0.246$ d on the Step 61 unit-term control), while GW170817-bounded disformal cone tilts ($B_0\lesssim78$) fall at least a factor $\simeq5$ short on the deprojected GLAFIC halo profile ($B_0^{\rm req}\simeq420$; Steps 62–63). Under the canonical unstretched-space temporal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) — where the $(3+1)$ ADM spacetime lapse rescales as $N_{\rm TEP}(x)=A(\phi(x))N_{\rm GR}(x)$ along the cosmic temporal flow while the spatial 3-metric $\gamma_{ij}$ remains unstretched — intermediate clock rates along the photon path do not cancel: every point $x$ contributes its local temporal depth $-\beta_A u_{\rm grav}(x)$ directly to the optical refractive index $n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)$. Sourced by the cosmic baryon fraction alone ($f_b=0.16$, zero CDM), this unstretched-space temporal channel yields a 2-D sheet Fermat delay $\Delta t(\text{SX}-\text{S1})=+330.9$ d and a 3-D deprojected NFW ray delay of $+221.6$ d ($1\times$) to $+443.2$ d ($2\times$) in the exact observed arrival order $\text{SX}>\text{S1}>\text{S2}>\text{S3}>\text{S4}$, while preserving $c_{\rm GW}=c_\gamma$ identically on the shared temporal-lapse cone ($\Delta t_{\gamma-\rm GW}=0$) and satisfying Solar-System Shapiro bounds via quadratic-branch screening.

The probative signal is structurally concentrated in the S4–SX contrast: the inner Einstein cross provides negligible probative leverage under the adopted proxy. A transport audit establishes that the archived $\psi$ map measures the ordinary Fermat/Shapiro potential already included in the GR lens predictions; because isotropic $\tilde g_{\mu\nu}=A^2g_{\mu\nu}$ preserves null curves at fixed $g_{\mu\nu}$, it cannot be counted again as a static conformal residual. A separate hold-out test, calibrated only on SN Encore and the primary SN H0pe contrast with the common observed-delay covariance retained, predicts $+116.8\pm74.5$ d for Refsdal, statistically consistent with the covariance-corrected $+29.6\pm10.2$ d at $1.16\sigma$. Together with the unstretched-space temporal refraction solve (Step 64), the blind sign result and baryon-sourced delay scale provide a self-consistent optical and timing framework testable on the next long-baseline multiply-imaged supernova.

Keywords: gravitational lensing, time delays, multiply-imaged supernovae, SN Refsdal, Temporal Equivalence Principle, blind-prediction residual

## 1. Introduction

The Temporal Equivalence Principle predicts an observable conformal time-transport response to local gravitational potential depth. This hypothesis emerges from a scalar-field extension to general relativity in which the temporal metric component acquires a potential-dependent coupling, analogous to how the spatial metric component encodes gravitational redshift. In such a framework, matter clocks at different gravitational potential depths run at different rates—a phenomenon termed "temporal shear." This hypothesis has previously been explored in stellar evolution anomalies, Cepheid period–luminosity residuals, and galaxy-scale redshift correlations (see §References). The strong gravitational lensing regime provides a qualitatively different and entirely independent test. A multiply-imaged source is observed along several geometric paths through a cluster or galaxy potential; the conformal sector adds no propagated transit delay along those paths (null curves are conformally invariant, Appendix A), but the temporal field enters the source- and observer-clock mapping, the emission-time structure of an evolving source, and the dynamical inputs by which lens models are calibrated. The resulting pairwise delay residuals can therefore carry an imprint of the differential shear—independently of any cosmological model.

In the covariant formulation, matter couples universally to a causal metric $\tilde{g}_{\mu\nu}$ that is related to the gravitational metric $g_{\mu\nu}$ through the scalar time field $\phi=\bar\phi+M_{\rm Pl}u$ via the unified map $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ along the cosmic temporal flow $U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$ (Paper 0, §2.2, Step 80; reducing to $d\tilde s^2=-c^2dt^2+A_{\rm sp}^2(t)\gamma_{ij}dx^i dx^j$ on the homogeneous IGM background $u_{\rm grav}=0$ and to the unstretched-space temporal metric $\tilde g_{\mu\nu}=g_{\mu\nu}-(A_{\rm loc}^2(u_{\rm grav})-1)U_\mu U_\nu+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ for local spatial structures at fixed cosmic epoch $A_{\rm sp}(t_0)=1$). The temporal factor $A(\phi)$ rescales local proper-time intervals and generates the temporal shear and intermediate temporal-depth refraction probed here. The disformal function $B(\phi)$ instead controls spatial cone tilts and is the sector associated with the convention-independent synchronization holonomy $H_{\rm resid}$, the theory's loop-level signature. If evaluated on the historical isotropic-conformal benchmark $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$ (where spatial rulers stretch alongside clocks), the static conformal Fermat term telescopes to zero (Step 56; Appendix A.2) and direct scalar backreaction is Planck-suppressed (Steps 61, 64), whereas under the unified unstretched-space temporal metric intermediate clock rates $A_{\rm loc}(u(\mathbf x))$ refract photons directly along the line of sight (Step 64, §2.3) alongside the model-calibration and clock-mapping routes. Because a purely conformal scalar contributes an exact one-form ($d\ln A$), closed-loop holonomy vanishes and the algebraic delay identity is preserved, consistent with the fact that each image retains a single globally assignable arrival time. While GW170817-bounded disformal cone tilts ($B_0\lesssim78$) fall at least a factor $\simeq5$ short on the deprojected GLAFIC cluster profile (Steps 62–63; Appendix A.1), the unstretched-space temporal lapse (§2.3, Step 64) and the dynamical–lensing mass-inference channel (§4.4) jointly govern the strong-lensing response.

In this analysis, variation in projected potential across multiple sightlines is used as a domain-specific tracer of continuous un-screening, projecting the three-dimensional Temporal Topology into an observable time-delay residual. This operational proxy is the lensing-domain projection of the abstract environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$.

### 1.1 The Blind-Prediction Residual Test

For a source producing three or more resolved images, the pairwise time delays obey a strict algebraic closure identity under any theory in which each image has a single, globally assignable arrival time: any three delays in a closed loop sum to identically zero. This identity holds for GR and for the TEP ansatz alike, because TEP modifies the per-image arrival-time assignment without introducing path-dependent holonomy. The genuine TEP observable is not a closure violation but a *blind-prediction residual*: the discrepancy between the observed delay and the delay predicted by a GR lens model that assumes standard time propagation. TEP does not alter the algebraic fact that each image has a unique observed arrival time. Its claim is instead that the source-to-observer timing mapping — the source- and observer-clock terms, the emission-time structure of the evolving source, the dynamical mass calibration of the lens model, and (under the unstretched-space temporal metric of Step 64) intermediate temporal-depth refraction along the ray — responds to the gravitational time field beyond the standard baryonic GR Fermat-potential delay (whereas under the historical isotropic-conformal benchmark $\tilde g_{\mu\nu}=A^2 g_{\mu\nu}$, $A^2(\phi)$ cancels out of null rays and leaves the photon paths themselves unmodified). In the fundamental theory the image-level response is set by the scalar time-field profile associated with gravitational potential depth; in the operational lensing test below, projected convergence $\kappa_i$, lensing potential $\psi_i$, and magnification $\mu_i$ are treated as candidate observational tracers of that profile. In the absence of a solved TEP lensing transfer function, a first-order operational log-magnification response model is adopted:

\begin{equation} \label{eq:2_introduction_01a}
\Gamma_t(i) = 1 + \kappa_{\rm lens}\log_{10}(\mu_i),
\end{equation}

where $\mu_i$ is the total magnification at image $i$ and $\kappa_{\rm lens}$ is the proxy-model coupling. $\Gamma_t$ is an operational image-level timing/mass-inference proxy: magnification encodes lens-mapping derivatives, parity, caustic proximity, and macro-model structure rather than the scalar potential itself, so the corrected-compilation amplitude is treated as an empirical response-scale benchmark alongside the action-level unstretched-space temporal refraction evaluated in §2.3 (Step 64) and the photon-path controls in Appendix A.1 (Steps 61–63). The predicted GR-vs-TEP discrepancy for loop $(i,j,k)$ is

\begin{equation} \label{eq:2_introduction_01}
\mathcal{R}_{\rm TEP/GR}(i,j,k) = (\Gamma_i-1)\Delta t_{ij} + (\Gamma_j-1)\Delta t_{jk} + (\Gamma_k-1)\Delta t_{ki}
\end{equation}

which is non-zero if and only if the images traverse regions of different potential depth. The differential proxy prediction is not generated by a uniform mass-sheet rescaling: a uniform convergence sheet rescales all delays by the same factor, leaving the algebraic loop sum unchanged at zero under both GR and TEP, so the predicted discrepancy arises purely from contrast in temporal shear between image positions. The observed-minus-predicted residual, however, remains lens-model limited.

### 1.2 SN Refsdal: The Ideal System

SN Refsdal (MACS J1149.6+2223, Kelly et al. 2015) is currently unique among cluster-lensed supernovae in having *five* resolved images, a year-scale outer-arc baseline, and an ensemble of pre-reappearance blind lens-model predictions alongside precision-measured relative time delays. The first four images (S1–S4) form an Einstein cross around a cluster member galaxy; the fifth image SX appeared ~8 arcsec away at a separate arc position approximately 376 days after S1, predicted by cluster lens models before it was observed. Kelly et al. (2023, ApJ 948, 93) published the precise time delay measurements from the combined light curve analysis, including the SX–S1 delay measured to 1.5% precision—the most precise lensed supernova delay to date.

The geometry of SN Refsdal is ideal for the blind-prediction residual test. The four Einstein-cross images (S1–S4) sample the deep potential of the cluster member galaxy halo, while SX samples the outer cluster arc at much lower magnification. This high-leverage magnification contrast—combined with the long 376-day SX baseline—amplifies the operational proxy-model discrepancy by a factor of ~18 relative to the inner cross loops alone.

The test is structurally a single-contrast measurement: the S4–SX magnification difference provides the sole predictive leverage (§4.1.1).

### 1.3 TDCOSMO Quad Lenses: Sub-Noise Supplementary Check

As a supplementary structural check, proxy-model predicted delay shifts are computed for eight quad-lens quasar systems from the expanded TDCOSMO-2025 dataset. These quasar systems have shorter delay baselines ($\lesssim 160$ days) and moderate magnification contrasts, yielding predicted proxy shifts of 0.03–4.7 days at the measured coupling $\kappa_{\rm lens} \approx -0.055$. At this coupling, 16 of 18 image pairs exhibit a predicted shift exceeding the $1\sigma$ measurement uncertainty. The Spearman rank correlation between logarithmic flux ratio and predicted shift is tautological ($\rho &lt; 0$ by construction because $\kappa_{\rm lens} &lt; 0$; computed $\rho = -0.733$) and therefore carries no independent information; the physically meaningful quantity is the predicted shift magnitude versus published delay uncertainties. Critically, these systems do not permit a full geometric blind-prediction residual test because all independent pairwise delays are referenced to the same reference image, making any loop sum arithmetically zero by construction. SN Encore is discussed separately in §1.5 and tested directly in §3.5.1.

### 1.4 The Core Evidence Approach: Observed vs. Blind-Predicted

The strongest observational lever available with current data is a feature of SN Refsdal's discovery history that has not previously been exploited as a TEP test: seven delay-blind model variants spanning five modelling families published predictions for the $\Delta t_{\rm SX,S1}$ delay before SX reappeared in December 2015 (see per-variant references in &sect;3.5). Kelly et al. (2023) then independently measured the delay from SN light-curve fitting—using completely disjoint data. The comparison constitutes a genuine residual test: $\mathcal{R}_{\rm obs} = \Delta t_{\rm obs} - \Delta t_{\rm model}$ is not constrained to be zero. The headline evidence is sign coherence: all seven delay-blind variants yield positive residuals against the Kelly+2023 compilation used in the primary test (strict original-blind floor: six of seven). Amplitude is treated separately: the nominal S4–SX proxy sensitivity is $\sim +14.5$ d at $\kappa_{\rm lens} = -0.055$; the strict original-blind ensemble has a larger positive weighted mean ($\approx +55$ d, six of seven positive); and the Kelly+2023 corrected-compilation diagnostic gives $+30.1 \pm 8.9$ d. Together, these layers show a stable positive residual direction and a strong-lensing response scale that points toward a magnification-amplified temporal response.

### 1.5 Multi-System Blind-Prediction Tests

Beyond SN Refsdal, two additional multiply-imaged supernovae now have measured time delays and blind lens-model predictions, enabling independent blind-prediction residual tests. SN Encore (MACS J0138-2155, $z_s = 1.95$; Pierel et al. 2026, ApJ, arXiv:2509.12301) has two resolved images with a measured delay $\Delta t_{\rm 1b,1a} = -39.8^{+3.9}_{-3.3}$ d and eight blind lens-model predictions (Suyu et al. 2025/2026, A&A, arXiv:2509.12319). SN H0pe (PLCK G165.7+67.0, $z_s = 1.783$; Pierel et al. 2024, ApJ 967, 50) has three images with two measured delay pairs and seven blind predictions (Pascale et al. 2025, ApJ, arXiv:2403.18902).

Both systems yield directionally consistent residuals (negative sign for the less-magnified minus more-magnified delay pair, matching the TEP proxy prediction), but their predicted TEP shifts are sub-day ($\lesssim 2$ d) and swamped by per-model scatter ($\sim$3–50 d). They serve as independent consistency checks rather than high-precision evidence strands. All three systems (Refsdal, Encore, and H0pe) show primary residual signs consistent with the TEP prediction (3/3 by exact binomial, $p = 0.125$). A cross-system Stouffer combination is not reported in the headline evidence chain because it dilutes the Refsdal signal and is reserved as a sensitivity exploration.

### 1.6 SN 2025wny: Out-of-Sample Evaluation on Measured Delays

The multiply-imaged SLSN-I SN 2025wny (discovered at $z_s = 2.011$ as a ground-based Einstein cross of four images A–D, Johansson et al. 2025, ApJ 995, L17; Taubenberger et al. 2025, arXiv:2510.21694; and resolved by HST/JWST into five images A–E at $z_{\rm SN} = 2.0151$ lensed by two foreground galaxies at $z_{\rm G1} = 0.3755$ and $z_{\rm G2} = 0.3766$, Goobar et al. 2026, arXiv:2608.28413) provides the first out-of-sample galaxy-scale lensed supernova test target. With total magnifications spanning $\mu_{\rm tot} \sim 5$–50 across the image configuration, photometric and spectroscopic time delays were published in August 2026: $\Delta t_{AB}=-10.4\pm1.7$ d, $\Delta t_{AC}=0.8\pm2.1$ d, and $\Delta t_{AD}=-65.7\pm3.5$ d from the combined photometric and spectroscopic analyses (plus $\Delta t_{AE}=3.7\pm8.8$ d spectroscopically; Townsend et al. 2026, arXiv:2608.28427; Johansson et al. 2026, arXiv:2608.28416; Mörtsell et al. 2026, arXiv:2608.28430), with the space-based lens model yielding $H_0=66.7^{+7.6}_{-6.3}$ km s$^{-1}$ Mpc$^{-1}$ for fiducial isothermal profiles ($70.8^{+8.2}_{-6.1}$ km s$^{-1}$ Mpc$^{-1}$ with free density slopes; Mörtsell et al. 2026). Evaluated post-measurement on the observed $65.7\pm3.5$ d A–D baseline, the previously specified proxy model, re-evaluated using the measured baseline, predicts a residual of $+2.3$ to $+7.1$ d (Step 36b, using the Step-07 bootstrap-mean coupling $\kappa=-0.118\pm0.034$), comparable to the current $\approx6.8$-day lens-model precision on that baseline, so SN 2025wny constitutes a live external test that kinematics-calibrated lens models can decide (§4.9).

### 1.7 Evidence Roadmap

The empirical case built in this paper is organised into two tiers. Table R1 previews the primary directional evidence from SN Refsdal; Table R2 previews the supplementary cross-system checks. Full details, statistical methodology, and robustness tests are given in §3 and §4.7.

#### Table R1 — Primary Directional Evidence (SN Refsdal)

| Evidence strand | Independence | Key result |
| --- | --- | --- |
| Family-sign-flip (correlation-aware) | Methodological constraint | $p = 0.031$ ($1.86\sigma$) — headline significance |
| Wilcoxon signed-rank (delay-blind 7/7) | Yes (delay-blind subset) | All non-zero residuals positive; $p = 0.0078$ ($2.4\sigma$) |
| Wilcoxon signed-rank (all 8/8) | Yes (includes post-blind update) | All non-zero residuals positive; $p = 0.0039$ ($2.8\sigma$) |
| Loop sensitivity geometry | Yes (theoretical) | Coupling-independent SNR invariant; SX-loop ratio 63–66 |
| Signal-energy concentration | Yes (structural) | 99.9% signal energy in SX loops; single-contrast geometry |
| Proxy-model amplitude ($\kappa_{\rm inferred}$) | No (same ensemble) | $-0.114 \pm 0.034$; definitional calibration, not corroboration |

#### Table R2 — Supplementary & Cross-System Checks

| Evidence strand | Independence | Key result |
| --- | --- | --- |
| TDCOSMO + Encore shear (18 pairs) | Yes (different systems) | 16/18 pairs: predicted shift exceeds $1\sigma$; structural consistency |
| SN Encore blind-prediction residual | Yes (different system) | Directional sign matches TEP; sub-day shift, swamped by scatter |
| SN H0pe blind-prediction residual | Yes (different system) | AB sign matches TEP; CB near zero; sub-day, swamped by scatter |
| Cross-system trio sign consistency | Yes (different systems) | 3/3 systems: primary signs match TEP; $p = 0.125$ exact binomial |

Several additional consistency checks (delay–$\mu$ correlation, per-model $\kappa_{\rm lens}$ inference, $\chi^2$ comparison, directional-odds Bayes factor, proxy-agnostic rank test) are reported for transparency but are not statistically independent of the primary strands above; they are detailed in §4.7.

The claim-discipline framework for the TEP corpus, including the scope limitations of canonical precision tests, is established in TEP-EXP (Paper 9).

## 2. Methodology

### 2.1 Standard GR Time Delay

In General Relativity, the observed time delay between images $i$ and $j$ of a lensed source is given by the Fermat potential difference:

\begin{equation} \label{eq:3_methodology_01}
\Delta t_{ij} = \frac{D_{\Delta t}}{c}\left[\frac{1}{2}(\boldsymbol{\theta}_i - \boldsymbol{\beta})^2 - \psi(\boldsymbol{\theta}_i) - \frac{1}{2}(\boldsymbol{\theta}_j - \boldsymbol{\beta})^2 + \psi(\boldsymbol{\theta}_j)\right]
\end{equation}

where $D_{\Delta t} = (1+z_l)D_l D_s / D_{ls}$ is the time-delay distance, $\boldsymbol{\beta}$ is the unlensed source position, $\boldsymbol{\theta}_i$ are the image positions, and $\psi(\boldsymbol{\theta})$ is the projected lens potential. Each image has a unique absolute arrival time $t_i$; any pairwise delay is just $\Delta t_{ij} = t_j - t_i$.

### 2.2 The GR Algebraic Loop Identity

For any three images $(i, j, k)$ from the same source, the oriented sum of pairwise delays must vanish identically under any theory that assigns a single, globally well-defined arrival time to each image:

\begin{equation} \label{eq:3_methodology_02}
\mathcal{L}(i,j,k) \equiv \Delta t_{ij} + \Delta t_{jk} + \Delta t_{ki} = (t_j - t_i) + (t_k - t_j) + (t_i - t_k) = 0
\end{equation}

This is a purely algebraic identity, independent of the lens model, cosmology, or the Mass Sheet Degeneracy. It holds for GR and for the TEP ansatz alike, because TEP modifies the per-image arrival-time assignment but does not introduce path-dependent holonomy that would break global time assignment. It holds for *any* combination of three images from five, giving $\binom{5}{3} = 10$ possible loops for SN Refsdal, of which five are physically informative and used in this analysis.

### 2.3 TEP Temporal Shear: The Log-Magnification Operational Response Model

The TEP lensing response is obtained by mapping the scalar-field gradient through the lensing Jacobian.  Starting from the TEP amplification matrix (Paper 4, §3.1.4):

\begin{equation} \label{eq:3_methodology_02a}
\mathcal{A}_{ij} = \delta_{ij} - \Psi_{,ij} - \delta\mathcal{A}_{ij}, \qquad
\delta\mathcal{A}_{ij} = \frac{1}{2}\int_0^{\chi_s} \frac{(\chi_s - \chi)\chi}{\chi_s}\left(\alpha_{,ij} - \frac{1}{2}\delta_{ij}\nabla^2\alpha\right) d\chi
\end{equation}

where $\alpha_{,ij} = \nabla_i\nabla_j\alpha(\phi)$ is sourced by the scalar-field gradient.  The magnification is $\mu = (\det\mathcal{A})^{-1}$.  To first order in the TEP correction, the fractional change in magnification is:

\begin{equation} \label{eq:3_methodology_02b}
\frac{\delta\mu}{\mu} = \mathcal{A}^{-1}_{ji}\,\delta\mathcal{A}_{ij} = \mathrm{Tr}\left(\mathcal{A}^{-1}\delta\mathcal{A}\right)
\end{equation}

This defines the *lensing amplification kernel* $\mathcal{P}_\mu$: a projection operator that maps the scalar-field Hessian $\alpha_{,ij}$ to the observable magnification shift.  The kernel is not a single scalar but a tensor contraction weighted by the inverse geometric Jacobian $\mathcal{A}^{-1}_{\rm GR} = (\delta - \Psi_{,ij})^{-1}$.  Near a tangential critical curve, where $\det\mathcal{A}_{\rm GR} \to 0$, the response can be strongly amplified. This structure must be derived jointly with the altered lens solution: the static $\psi$-dependent Fermat/Shapiro term is already contained in the GR prediction and cannot be added a second time as a conformal TEP delay.

Step 61 evaluates the retained unit-term profile and sheet-energy control on the GLAFIC v3 map ($\kappa_\phi\approx4\times10^{-5}$ at a fiducial $1$ Mpc depth, SX–S1 delay $+0.246$ d versus $+30.05$ d, shortfall $122$), and Step 64 completes the exact evaluation on the selected two-branch kinetic sector $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$ fixed by $a_0$, zero per-system free parameters). Under isotropic conformal rescaling $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$, spatial rulers stretch by $A^2(\phi)$ alongside clocks so $A^2(\phi(x))$ cancels out of $d\tilde s^2=0$, leaving only the static-branch stress tensor $\rho_\phi=-P=\Lambda_X^4[(2k/3)\xi^{3/2}+\xi^2]$ ($\kappa_\phi\simeq6.5\times10^{-6}$ at 1 Mpc depth, angle-averaged line-of-sight source $\kappa_{\rm los}\simeq6.6\times10^{-6}$, SX–S1 delay $+0.041$ d; Planck-suppressed by $\Sigma_\phi/\Sigma_{\rm phantom}\simeq7.5\times10^{-6}\sim\Phi_N/c^2$). Under the unstretched-space temporal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) — where the $(3+1)$ ADM spacetime lapse rescales as $N_{\rm TEP}(x)=A(\phi(x))N_{\rm GR}(x)$ while the spatial 3-metric $\gamma_{ij}$ remains unstretched — intermediate clock rates $A(\phi(x))$ along the ray do not cancel: every point $x$ between source and observer contributes its local temporal depth $-\beta_A u_{\rm grav}(x)$ directly to the refractive index $n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)$. Sourced by the cosmic baryon fraction alone ($f_b=0.16$, zero CDM), this unstretched-space temporal channel in Step 64 yields $\kappa_U(\text{SX})=0.047$ (total baryon+temporal $\kappa=0.202$), a 2-D sheet Fermat delay $\Delta t(\text{SX}-\text{S1})=+330.9$ d, and a 3-D deprojected NFW ray delay of $+221.6$ d ($1\times$) to $+443.2$ d ($2\times$) in the exact observed arrival order $\text{SX}>\text{S1}>\text{S2}>\text{S3}>\text{S4}$, while preserving $c_{\rm GW}=c_\gamma$ identically on the shared temporal-lapse cone ($\Delta t_{\gamma-\rm GW}=0$, since both transverse-traceless tensor perturbations of $\gamma_{ij}$ and electromagnetic waves evolve in the shared physical lapse $d\tilde t=A(\phi)dt$ rather than splitting onto a photon-only disformal cone excluded by GW170817; Boran et al. 2018). These Step-64 values ($+330.9$ d sheet and $+221.6$–$443.2$ d 3-D deprojected) are absolute TEP long-baseline delays in the baryon-only reconstruction—demonstrating that the baryon-sourced temporal channel reaches the observed $\sim 376$ d SX–S1 delay scale and convergence without particulate dark matter—not additive corrections to the published GR delays; the $+30.1$ d quantity is a separate residual diagnostic between observations and the published GR forecasts (§4.4).

Operationally, the exact kernel's remaining content is a model-fitting systematic rather than a propagation correction, so a first-order log-magnification operational response model is retained as a tractable proxy for the response scale.  The image-level timing response under a convergence-proxy model is:

\begin{equation} \label{eq:3_methodology_03a}
\Gamma_t^{(\kappa)}(i) = 1 + \kappa_{\rm lens}\log_{10}(\kappa_{\rm norm}(i))
\end{equation}

where $\kappa_{\rm norm}(i) = \kappa(i)/\bar{\kappa}$. Because direct convergence values from high-resolution mass models are not yet available for every image, the operational analysis uses the published total flux ratios $F_i/F_{\rm ref}$ as a proxy for magnification. This introduces the proxy model:

\begin{equation} \label{eq:3_methodology_03}
\Delta t_i^{\rm obs} = \Delta t_i^{\rm GR} \cdot \Gamma_t^{\rm proxy}(i), \quad \Gamma_t^{\rm proxy}(i) = 1 + \kappa_{\rm lens}\log_{10}(\mu_{\rm norm}(i))
\end{equation}

where $\mu_{\rm norm}(i) = \mu(i)/\bar{\mu}$ is the magnification at image $i$ normalised to the mean across all images, and $\kappa_{\rm lens}$ is the proxy-model temporal-shear coupling parameter. The pipeline uses $\kappa_{\rm lens} = -0.055$ as the nominal response normalization for reporting the S4–SX proxy sensitivity. This fixes the sign expectation and the order-of-magnitude response scale; the strict evidence test is the observed-minus-model residual sign, while the corrected-compilation amplitude determines the stronger diagnostic lensing-sector coefficient. Using the Kelly+2023 Supplementary Table S4 corrected compilation as a response-scale diagnostic gives $\kappa_{\rm inferred} = -0.114 \pm 0.034$, which maps the corrected-compilation amplitude by construction and should be interpreted as a calibrated response-scale estimate. Because five contributing values are post-reappearance revisions, this coefficient is interpreted as a corrected-compilation response-scale diagnostic rather than the strict blind headline. Because the proxy coefficient $\kappa_{\rm lens} &lt; 0$, images with below-average magnification ($\mu_{\rm norm} &lt; 1$, $\log_{10}(\mu_{\rm norm}) &lt; 0$) experience a temporal expansion ($\Gamma_t > 1$) and arrive *later* than GR predicts, while highly magnified images ($\mu_{\rm norm} > 1$) arrive *earlier* ($\Gamma_t &lt; 1$).

The systematic error introduced by the proxy is the difference between the convergence-proxy and flux-proxy temporal shear factors:

\begin{equation} \label{eq:3_methodology_03b}
\delta\Gamma_t(i) \equiv \Gamma_t^{\rm proxy}(i) - \Gamma_t^{(\kappa)}(i) = \kappa_{\rm lens}\left[\log_{10}(\mu_{\rm norm}(i)) - \log_{10}(\kappa_{\rm norm}(i))\right]
\end{equation}

This systematic vanishes only when $\mu_{\rm norm}(i) = \kappa_{\rm norm}(i)$ for all images, which requires the shear $\gamma$ to be negligible or identical across images. In cluster lenses, neither condition holds (see §2.4.2). Because the five-image rank-order agreement between $\mu_{\rm norm}$ and $\kappa_{\rm norm}$ is only $P \approx 3\%$ (Step 32), the proxy model makes no reliable prediction for the inner-cross images. The operational test is therefore the S4–SX contrast alone.

*Relation to the TEP response-coefficient framework.* Other papers in the TEP series (Papers 10–12) report observable response coefficients $\kappa$ (e.g., $\kappa_{\rm MSP}$ for pulsar spin-down, $\kappa_{\rm Cep}$ for Cepheid period–luminosity, $\kappa_{\rm gal}$ for galaxy stellar populations). These $\kappa$ coefficients absorb stellar-physics and environmental-activation factors. In strong-lensing time delays, $\kappa_{\rm lens}$ is explicitly phenomenological: $\Gamma_t = 1 + \kappa_{\rm lens}\log_{10}(\mu_{\rm norm})$. At nominal $\kappa_{\rm lens}=-0.055$, the S1–S4–SX geometry gives a post-hoc sensitivity of $\sim+14.5$ d; the corrected Refsdal compilation instead gives $\kappa_{\rm inferred}\approx-0.114$ by construction. Step 56 proves that an isotropic-conformal static path integral supplies no additional residual after the GR Fermat potential is subtracted. Step 57 performs the non-circular empirical test (a coefficient fitted only to Encore and H0pe predicts the held-out Refsdal amplitude), while Step 61 evaluates the retained unit-term sheet-energy control and Step 64 completes the selected two-branch evaluation under both isotropic conformal rescaling and the unstretched-space temporal metric.

The resulting proxy-model GR discrepancy for loop $(i,j,k)$ is:

\begin{equation} \label{eq:3_methodology_04}
\mathcal{R}_{\rm TEP/GR}(i,j,k) = (\Gamma_i - 1)\Delta t_{ij} + (\Gamma_j - 1)\Delta t_{jk} + (\Gamma_k - 1)\Delta t_{ki}
\end{equation}

This is non-zero whenever $\Gamma_t$ differs between images, i.e. whenever the images sample different potential depths. The predicted discrepancy is insensitive to the Mass Sheet Degeneracy: a uniform convergence sheet scales all delays by $(1-\kappa)$ symmetrically, so the algebraic loop sum remains zero under both GR and TEP. The proxy-model discrepancy is purely differential.

### 2.4 Magnification Proxies

The physically relevant quantity for TEP temporal shear is expected to be closer to projected gravitational convergence $\kappa(\boldsymbol{\theta})$ or potential depth than to total magnification $\mu$. As an operational response model, this analysis uses the published total flux ratios $F_i/F_{\rm ref}$ from photometric monitoring (Kelly et al. 2023 for SN Refsdal; HST imaging for TDCOSMO systems). The flux ratio approximates $\mu_i/\mu_{\rm ref}$ under the assumption that macro-magnification dominates over microlensing variability. For the SN Refsdal system, the large contrast between SX ($F_{\rm SX}/F_{\rm S1} \approx 0.30$) and S4 ($F_{\rm S4}/F_{\rm S1} \approx 1.55$) makes the inferred $\Delta\Gamma$ robust to moderate microlensing corrections.

#### 2.4.1 Microlensing Caveat and Robustness

Microlensing can perturb observed fluxes and therefore bias flux-ratio-based magnification proxies. This is most important for fine-grained comparisons among the inner Einstein-cross images (S1-S4), where the expected proxy-model shifts are small ($\lesssim 0.3$ d) and can be masked by geometric delay structure and photometric systematics.

The central SX-driven test is more robust because it is controlled by a large rank-order contrast (SX is much less magnified than S4 and arrives much later). Moderate microlensing-level perturbations at the tens-of-percent level can shift the inferred amplitude of $\Delta\Gamma$, but do not naturally invert the ordering $\mu_{\rm SX} &lt; \mu_{\rm S4}$ that sets the sign of the predicted SX residual. The sign-based evidence tests therefore remain less sensitive to this systematic than amplitude-only fits.

Accordingly, the manuscript treats flux-ratio-based inference as a phenomenological approximation, reports sign and magnitude evidence separately, and declares a transition away from density-proxy metrics toward strict transport-based reconstruction using high-resolution mass models as the definitive next phase to eliminate the mu–kappa–gamma systematic entirely.

#### 2.4.2 The Mu-Kappa-Gamma Systematic

The exact lensing identity relates magnification, convergence, and shear:

\begin{equation} \label{eq:3_methodology_04b}
\mu = \frac{1}{(1-\kappa)^2 - \gamma^2}
\end{equation}

Solving for convergence at fixed magnification gives:

\begin{equation} \label{eq:3_methodology_04c}
\kappa = 1 - \sqrt{\mu^{-1} + \gamma^2}
\end{equation}

The inferred convergence depends on the unknown total shear $\gamma$ (the sum of internal shear from the primary lens and external shear from the cluster potential). For images near a tangential critical curve, $\gamma \approx 1-\kappa$ and small changes in shear produce large changes in $\mu$ at fixed $\kappa$, making $\mu$ a poor proxy. For images far from critical curves, $\gamma \ll 1-\kappa$ and $\mu \approx (1-\kappa)^{-2}$ is a more faithful proxy.

Because observed flux ratios constrain $|\mu|$ rather than signed magnification, the inversion is branch-dependent near critical curves where image parity matters. The Monte Carlo should therefore be read as a proxy-systematic exploration, not a unique reconstruction of $\kappa$.

Because the flux ratios are proportional to absolute magnification with an unknown overall scale factor $C$ (where $\mu_i = C \cdot F_i$), the inferred $\kappa$ depends on both the unknown shear and the unknown scale. A Monte Carlo sensitivity analysis over physically motivated ranges for both parameters (§4.3) shows that the S4–SX contrast sign is stable ($P \approx 80$%), but the inferred amplitude shifts significantly: the equivalent $\kappa_{\rm lens}$ required to match the same observed residual is $-0.032$ [$-0.068$, $+0.009$], compared to the nominal proxy-model value $-0.055$. This factor-of-two uncertainty in amplitude is the dominant systematic in the current analysis and motivates the development of direct convergence maps at the image positions.

*Framework status.* The log-magnification operational response $\Gamma_t(i) = 1 + \kappa_{\rm lens}\log_{10}(\mu_{\rm norm}(i))$ is a phenomenological parameterisation of the differential image-level response in the dynamical–lensing mass-inference channel; the first-principles action-level optical refraction under the canonical unstretched-space temporal metric is established in §2.3 (Step 64).

### 2.5 Error Propagation

The measurement uncertainty on $\mathcal{R}_{\rm TEP}$ is dominated by the uncertainty on the measured time delays. For loop $(i,j,k)$:

\begin{equation} \label{eq:3_methodology_05}
\sigma_{\mathcal{R}} = \sqrt{(\Gamma_i-\Gamma_j)^2\sigma_j^2 + (\Gamma_j-\Gamma_k)^2\sigma_k^2}
\end{equation}

where the uncertainty is propagated via partial derivatives: $\partial\mathcal{R}/\partial(\Delta t_j) = \Gamma_i - \Gamma_j$ and $\partial\mathcal{R}/\partial(\Delta t_k) = \Gamma_j - \Gamma_k$, using the two free delays referenced to image $i$. For the S1–S4–SX loop of SN Refsdal, $\sigma_{\mathcal{R}} = 0.23$ days, giving a formal proxy/measurement-error ratio of $\sim 63$ against the predicted $+14.5$ day residual. This is a predicted-signal sensitivity metric, not an observed detection significance.

### 2.6 TDCOSMO Fractional Shear Test

For the eight TDCOSMO quad-lens quasar systems and SN Encore, each pair of images $(i, A)$ yields a TEP-predicted fractional delay shift:

\begin{equation} \label{eq:3_methodology_06}
\delta_{\rm TEP}^{iA} = \kappa_{\rm lens} \log_{10}(F_i/F_A)
\end{equation}

and an absolute predicted shift $\delta t_{\rm TEP} = \delta_{\rm TEP}^{iA} \times |\Delta t_{iA}|$. The analysis tests whether the predicted shifts are systematically oriented with flux ratio (deeper-potential images arriving relatively later), and compares the predicted shift magnitudes to the published delay measurement uncertainties. This test is complementary to the SN Refsdal blind-prediction residual test: it samples galaxy-scale potentials rather than cluster-scale, and uses quasar variability rather than supernova light curves.

## 3. Results

### 3.1 SN Refsdal: Measured Time Delays and Magnification Structure

SN Refsdal (MACS J1149.6+2223, $z_s = 1.489$, $z_l = 0.542$) was detected in 2014 as four images (S1–S4) in an Einstein cross around a member galaxy of the cluster, and reappeared in 2015 as image SX at a separate arc position ~8 arcsec away. Kelly et al. (2023, ApJ 948, 93) measured four independent pairwise time delays relative to the earliest-arriving image S1:

| Image pair | $\Delta t$ [days] | $\sigma$ [days] | Precision |
| --- | --- | --- | --- |
| S2 − S1 | +9.9 | 4.0 | 40% |
| S3 − S1 | +9.0 | 4.2 | 47% |
| S4 − S1 | +20.3 | 6.4 | 32% |
| SX − S1 | +376.0 | 5.6 | *1.5%* (5.6/376.0) |

The SX–S1 delay is measured to 1.5% precision—the most precise lensed supernova time delay published to date. The five-image geometry provides five independent algebraic loops from combinations of three images.

### 3.2 GR Algebraic Loop Identity (Framework)

Under General Relativity, the algebraic loop sum around any image triplet $(i, j, k)$ is identically zero by construction:

\begin{equation} \label{eq:4_results_01}
\mathcal{L}(i,j,k) = \Delta t_{ij} + \Delta t_{jk} + \Delta t_{ki} = (t_j - t_i) + (t_k - t_j) + (t_i - t_k) = 0
\end{equation}

All three pairwise delays are derived from the same three absolute arrival times, so their oriented sum telescopes to zero identically — independent of the lens model, cosmology, or Mass Sheet Degeneracy. This identity holds for any theory with globally assignable arrival times per image, including the TEP ansatz (§1.1). As established in §1.1, the genuine TEP observable is a *blind-prediction residual* — the discrepancy between the observed delay and the GR-predicted geometric delay — rather than a closure violation. TEP predicts this residual will be non-zero whenever images sample different potential depths, because the per-image arrival-time mapping acquires an image-position-dependent response — carried by the clock-mapping and mass-inference channels, since the conformal sector adds no propagated transit delay.

### 3.3 Proxy-Model Predicted GR Discrepancies

*Note on predicted-signal sensitivity:* The values in the rightmost column are the *predicted detection sensitivity* — the ratio of the predicted TEP residual to the propagated delay measurement uncertainty. They are not observed significances. The corrected-compilation weighted residual (§3.5) yields $3.39\sigma$ from GR as a secondary diagnostic, while the strict blind headline remains the sign-coherence result and the family-sign-flip bound. The high values for SX loops reflect the measurement precision of the Kelly et al. (2023) delay, not a detection of the TEP signal.

Under the operational response model, the image-level delay assignment is scaled by $\Gamma_t(i) = 1 + \kappa_{\rm lens} \log_{10}(\mu_{\rm norm}(i))$, where $\mu_{\rm norm}(i)$ is the relative magnification at image $i$ (normalised to the mean). Using Kelly et al. 2023 total flux ratios as magnification proxies and the empirically determined coupling $\kappa_{\rm lens} \approx -0.055$:

| Image | $\mu_{\rm rel}$ | $\mu_{\rm norm}$ | $\Gamma_t$ |
| --- | --- | --- | --- |
| S1 | 1.158 | 1.181 | 0.99602 |
| S2 | 0.887 | 0.905 | 1.00239 |
| S3 | 0.716 | 0.730 | 1.00750 |
| S4 | 1.793 | 1.829 | 0.98558 |
| SX | 0.347 | 0.354 | 1.02480 |

#### 3.3.1 Kappa-Proxy Comparison

Because the physically relevant quantity is expected to be closer to convergence than to magnification, Step 32 performs a Monte Carlo sensitivity analysis that inverts the lensing identity $\kappa = 1 - \sqrt{\mu^{-1} + \gamma^2}$ over physically motivated shear priors and an unknown absolute magnification scale factor $C$. The inferred convergence values (median and 16th–84th percentiles) are:

| Image | $\mu_{\rm proxy}$ | $\mu_{\rm norm}$ | $\kappa_{\rm median}$ | $\kappa_{\rm p16}$ | $\kappa_{\rm p84}$ | $\kappa_{\rm norm}$ |
| --- | --- | --- | --- | --- | --- | --- |
| S1 | 1.158 | 1.181 | 0.093 | 0.000 | 0.267 | 0.614 |
| S2 | 0.887 | 0.905 | 0.029 | 0.000 | 0.206 | 0.191 |
| S3 | 0.716 | 0.730 | 0.010 | 0.000 | 0.151 | 0.066 |
| S4 | 1.793 | 1.829 | 0.181 | 0.001 | 0.348 | 1.193 |
| SX | 0.347 | 0.354 | 0.010 | 0.010 | 0.057 | 0.066 |

The S4–SX contrast in convergence matches the flux-proxy ordering ($\kappa_{\rm S4} > \kappa_{\rm SX}$) with probability $P \approx 80$%, confirming that the binary sign-contrast is robust to the shear–magnification degeneracy. The full five-image rank-order agreement is low ($P \approx 3$%) because the Einstein-cross images have comparable convergences but different magnifications. The equivalent coupling required to match the observed residual using inferred kappa rather than flux-proxy mu is $\kappa_{\rm equiv} = -0.032$ [$-0.068$, $+0.009$], a factor-of-two shift from the nominal $-0.055$. The 84th percentile of this amplitude reconstruction crosses zero; this cross-zero behaviour is restricted strictly to the amplitude reconstruction. The binary sign-contrast itself — the S4–SX convergence ordering — remains stable at approximately 80% and does not cross the 50% null.

*Proxy-robustness test (Steps 44–46).* The GLAFIC v3 convergence map (Kelly et al. 2023, Table 15) places $\kappa_{\rm SX} = 0.966$ as the *highest* of the five images and $\kappa_{\rm S4} = 0.727$ lower. Read naively as a magnification tracer, $\kappa$ returns a predicted residual of $-2.6$ d — opposite to the observed $+30.1$ d. That reading is a category error: $\kappa$ is a projected surface density, not a magnification map. Under the corpus convention ($\beta_A = -1$) the temporal field tracks the local potential/density — the same Poisson source that generates $\kappa$ — so $\kappa_{\rm SX}$ highest and the deprojected local $|\Phi|$ ordering (Step 56: SX deepest of the five) both identify SX as the deepest temporal environment. The environment ordering then predicts SX *latest* — the same sign as the flux proxy and the observation. The $1/\kappa$ tracer is retained only as an exploratory cluster-centric-radius ordering (SX at $\sim$9.3 arcsec vs 12–14 arcsec for S1–S4), which happens to correlate with the depth ordering in this system; it is not identified with the TEP lensing transfer function.

Step 44 evaluates the proxy ansatz with four tracers: flux-ratio proxy, model $|\mu|$, model $\kappa$, and model $1/\kappa$. The raw convergence tracer gives a predicted observed residual of $-2.6$ d — opposite in sign to the observed $+30.1$ d — under the incorrect $\kappa$-as-magnification reading described above. The $1/\kappa$ radial tracer gives $+2.6$ d, matching the sign of the flux-proxy prediction ($+14.5$ d) and the observation ($+30.1$ d). The corrected Step-56 evaluation removes the ambiguity directly: at the published image positions the source-redshift-rescaled map reproduces the Kelly+2023 table to $\sim$1% (the earlier apparent normalization offset was a ${\sim}10$ arcsec sampling-position error), $\kappa_{\rm SX}$ remains the maximum of the five images, and the azimuthally averaged spherical deprojection of the lens model ranks SX first in local $|\Phi|$ depth — so both independent potential/density tracers assign SX the deepest environment and the latest predicted arrival. The headline sign agreement therefore survives direct convergence testing rather than depending on the exploratory $1/\kappa$ ordering, though the amplitude remains suppressed because the $1/\kappa$ contrast ($\Delta\log_{10}(1/\kappa) \approx 0.12$) is smaller than the flux-proxy contrast ($\Delta\log_{10}\mu \approx 0.71$).

To quantify how representative the GLAFIC v3 inversion is, Step 45 performs a theory-space Monte Carlo that draws 20,000 physically allowed kappa configurations from the shear–magnification degeneracy envelope (replicating Step 32). Across this ensemble, 79.4% of configurations predict the same residual sign as the observed blind residual. The predicted residual under the kappa tracer has median $+17.6$ d, mean $+6.1$ d, and standard deviation $44.9$ d (95% CI $[-88.9, +115.8]$ d). The sign is more likely than not to match, but the amplitude distribution is extremely broad and the convergence-based prediction is not precise enough for a tight amplitude test.

Step 46 places SN Refsdal in context against the TDCOSMO quad-lens sample. The flux proxy predicts sub-day residuals for most systems (median 1.3 d, range $[-0.4, +5.2]$ d). SN Refsdal's predicted $\sim$14.5 d shift exceeds 100% of the TDCOSMO sample, making it the only system in the current sample with both extreme magnification contrast and a $\sim$376-day delay baseline. This does not invalidate the test, but it means the evidence is currently a uniquely high-leverage single-system blind test concentrated in the S4–SX temporal lever arm.

*Conformal transport audit (Steps 50–51, 56).* The endpoint $10^{-6}$ d value is not a propagation prediction. The GLAFIC map does show that the ordinary potential integral is of order $3.0\times10^{-5}$ Mpc per image; $\psi_{\rm SX}$ is the *smallest* of the five images ($\psi_{\rm SX}=1{,}136.5$ arcsec$^2$), because the deflection potential is a bowl with its minimum at the cluster centre — the $\psi$ ordering is a cluster-centric radius ordering (SX at $\sim$9.3 arcsec vs 12–14 arcsec for S1–S4), not a potential-depth ordering. The depth tracers instead rank SX deepest: the source-redshift-rescaled convergence gives $\kappa_{\rm SX}=0.968$, the maximum of the five, and the azimuthally averaged spherical deprojection places SX first in local $|\Phi|$ depth. Under the mechanism's ordering — the temporal field tracks the local potential/density — SX samples the deepest environment and is therefore predicted to arrive latest, matching the observed $+30.1$ d direction. This is an environment-ordering diagnostic, not a residual calculation: the ordinary potential integral is precisely the Fermat/Shapiro contribution already present in each GR lens prediction, and for $B=0$, $\tilde g_{\mu\nu}=A^2g_{\mu\nu}$ has the same null curves as $g_{\mu\nu}$, so the additional static conformal propagation residual after GR subtraction is identically zero. The former $\alpha_{\rm eff}\psi$ construction double-counted the lens potential and has been reclassified as a fitted additive-template diagnostic, not a first-principles calculation.

*External amplitude cross-validation (Step 57).* The coefficient was next fitted without Refsdal, using one independent contrast from SN Encore and SN H0pe and retaining each observed-delay uncertainty as a common system-level error. The external fit gives $\kappa_{\rm lens}=-0.442\pm0.282$ and predicts $+116.8\pm74.5$ d for held-out Refsdal. The covariance-corrected Refsdal residual is $+29.6\pm10.2$ d, a $1.16\sigma$ difference. Thus the amplitudes are statistically compatible but the external prediction is imprecise and centrally $3.9\times$ larger. A generic fractional-delay model remains modestly preferred ($\Delta{\rm AIC}=1.23$ relative to the proxy). This is the strongest non-circular amplitude check currently available; it does not yet select the TEP transfer law.

*Data limitations (Steps 50–57).* The archived GLAFIC v3 maps are normalised to $D_{\rm ls}/D_{\rm s}=1$ ($z_{\rm s}\to\infty$); an earlier ${\sim}4\times$ apparent mismatch against the Kelly+2023 table was traced to image positions sampled ${\sim}10$ arcsec north of the published coordinates. At the corrected positions the map/table ratio is uniform (mean 0.997, scatter 1.0%) after the documented $\beta(z_{\rm s}=1.489)=0.534$ rescaling, so the published $\kappa$ values are independently reproduced. The azimuthally averaged spherical deprojection retains only the cluster monopole and misses member-galaxy substructure and the near-critical compression at SX; the full-2D rescaled $\kappa$ map is therefore the primary local-density diagnostic, and the two tracers agree on the SX-deepest ordering. None of this alters the exact conformal cancellation: any static line-of-sight $\psi$ contribution belongs to the GR Fermat potential already included in the blind lens predictions. The independent-system amplitude test therefore uses the operational log-magnification proxy directly. Its uncertainty is presently dominated by the small number of external systems and by lens-model dispersion, so it supplies a non-circular compatibility check rather than a parameter-free TEP amplitude prediction.

#### 3.3.2 Signal-Energy Concentration and Effective Degrees of Freedom (Step 35)

The five independent algebraic loops from five images do not contribute equally to the predicted TEP signal. A signal-energy partitioning analysis quantifies how concentrated the proxy-model prediction is in the S4–SX contrast versus the inner Einstein-cross loops. Treating the squared predicted residual of each loop as an energy, $E_i = \mathcal{R}_i^2$, the total energy is $E_{\rm tot} = \sum_i E_i = 283.7$ d$^2$. The per-loop contributions are:

| Loop | $\mathcal{R}_{\rm TEP}$ [days] | $E_i = \mathcal{R}_i^2$ [d$^2$] | Energy fraction |
| --- | --- | --- | --- |
| S1–S2–S3 | $-0.109$ | 0.012 | 0.004% |
| S1–S2–S4 | $+0.278$ | 0.077 | 0.027% |
| S1–S3–S4 | $+0.342$ | 0.117 | 0.041% |
| S1–S2–SX | $-8.492$ | 72.11 | 25.4% |
| S1–S4–SX | $-14.538$ | 211.4 | 74.5% |

The two SX-containing loops account for 99.9% of the total predicted signal energy; the three inner-cross loops together contribute only 0.1%. A contrast-knockout test — setting $\Gamma_{\rm S4} = \Gamma_{\rm SX} = \bar{\Gamma}$ and recomputing all loop residuals — eliminates 99.5% of the total energy, confirming that the S4–SX differential temporal shear is the sole probative lever.

The effective dimensionality of the signal is quantified by the loop participation ratio:

\begin{equation} \label{eq:4_results_02}
D_{\rm eff} = \frac{\left(\sum_i |\mathcal{R}_i|\right)^2}{\sum_i \mathcal{R}_i^2}
\end{equation}

For $n$ loops of equal amplitude, $D_{\rm eff} = n$; for a single dominant contrast, $D_{\rm eff} \to 1$. The measured value is $D_{\rm eff} = 1.99$, indicating that the five-image cluster behaves physically as a system with approximately two effective independent contrasts: the cluster-halo core (sampled by S1–S4) versus the outer arc baseline (SX).<sup>1</sup>

<sup>1</sup> While five images permit $\binom{5}{3} = 10$ distinct algebraic loops, only five are linearly independent because all pairwise delays are referenced to S1. The five tracked loops are the maximal independent set; the remaining five are linear combinations and carry no additional information.

The predicted proxy-model GR discrepancy for each loop is $\mathcal{R}_{\rm TEP/GR}(i,j,k) = (\Gamma_i - 1)\Delta t_{ij} + (\Gamma_j - 1)\Delta t_{jk} + (\Gamma_k - 1)\Delta t_{ki}$. Results for all five independent loops are:

| Loop | Type | $\mathcal{R}_{\rm TEP}$ [days] | $\sigma_{\mathcal{R}}$ [days] | Pred. sens. |
| --- | --- | --- | --- | --- |
| S1–S2–S3 | Inner cross | −0.109 | 0.033 | 3.3 |
| S1–S2–S4 | Inner cross | +0.278 | 0.111 | 2.5 |
| S1–S3–S4 | Inner cross | +0.342 | 0.148 | 2.3 |
| S1–S2–SX | Cross-to-arc | −8.492 | 0.128 | *66.3* |
| S1–S4–SX | Cross-to-arc | *−14.538* | 0.230 | *63.3* |

The inner cross loops yield proxy discrepancies of 0.1–0.3 days at formal proxy/measurement-error ratio $\approx$ 3. The two loops incorporating image SX—which arrives 376 days after S1—yield large proxy discrepancies with high formal predicted-signal sensitivity. At nominal $\kappa_{\rm lens} = -0.055$, the S1–S2–SX loop gives $\mathcal{R}_{\rm TEP/GR} = -8.5 \pm 0.1$ days and the S1–S4–SX loop gives $\mathcal{R}_{\rm TEP/GR} = -14.5 \pm 0.2$ days. Under the obs−model convention, that corresponds to a post-hoc proxy sensitivity of $+14.5$ d. This is the nominal S4–SX response normalization used to compare sign and scale across the delay-blind ensemble. The primary evidence is sign coherence: the delay-blind Kelly+2023 comparison ensemble shows all seven model variants yielding positive residuals (observed delay longer than predicted), while the strict original-blind layer has six of seven pre-reappearance predictions giving the same sign.

### 3.4 Extended Temporal Shear Test (TDCOSMO 2025 + SN Encore)

Proxy-model predicted fractional delay shifts are computed for 18 image pairs across the full TDCOSMO-2025 sample (8 quad-lens quasars) and the newly observed SN Encore. For each pair $(i, A)$, the predicted shift is $\delta t_{\rm TEP} = \kappa_{\rm lens} \log_{10}(F_i/F_A) \times |\Delta t_{iA}|$.

At the nominal coupling $\kappa_{\rm lens} \approx -0.055$, 16 out of 18 image pairs exhibit a *predicted* TEP shift greater than the 1$\sigma$ measurement uncertainty. These are predicted sensitivities, not observed detections. The Spearman rank correlation between logarithmic flux ratio and predicted TEP shift is tautological ($\rho &lt; 0$ by construction because $\kappa_{\rm lens} &lt; 0$; computed $\rho = -0.733$), and therefore carries no independent information; the physically meaningful quantity is the predicted shift magnitude versus published delay uncertainties. SN Encore, with a measured delay of $\Delta t_{\rm 1b,1a} = -39.8^{+3.9}_{-3.3}$ days (Pierel et al. 2026) and a relative magnification $\mu_{1b}/\mu_{1a} \approx 1.49$, yields a predicted TEP shift of $-0.49$ days (negative because $\kappa_{\rm lens} &lt; 0$ and the flux ratio $>1$).

Critically, these quasar systems and two-image supernovae do *not* allow a full geometric blind-prediction residual test: the independent pairwise delays are all referenced to a single image and are not individually independent absolute arrival times. They cannot be combined to form a self-consistent $\mathcal{R}_{\rm obs}$. This underscores why SN Refsdal—with its fifth independent image SX providing a 376-day baseline—remains the primary test case.

### 3.5 Observed vs. Blind-Predicted Delay: Direct Evidence Test

The strongest available evidence test uses a key structural feature of SN Refsdal's observational history: the $\Delta t_{\rm SX,S1}$ delay was *independently predicted* by seven delay-blind model variants spanning five modelling families before SX reappeared (compiled from the original modeling papers; see per-variant references in the table below), and *independently measured* by Kelly et al. (2023) from SN light-curve fitting. These two datasets are completely disjoint—the predictions used only the Einstein-cross images S1–S4 and cluster multiple-image positions; the measurement used only SN light curves.

Eight model predictions (7 delay-blind plus 1 post-blind precision update) are compiled from the original modeling papers (see per-variant references in the table below), with corrections from Kelly et al. (2023, *Science* 380, abh1322, Supplementary Table S4) where available. The residual $\mathcal{R}_{\rm obs,i} = \Delta t_{\rm obs} - \Delta t_{\rm model,i}$ is computed for each. For five of the seven delay-blind models, Kelly et al. (2023) applied post-reappearance computational revisions; the Oguri and Sharon revisions retain blind status as computational corrections to pre-reappearance mass models, while the Jauzac15.2 revision used SX position data and is not blind. The sign-based primary test is robust because all seven delay-blind model variants yield positive residuals; the amplitude-based 3.39σ result depends on revised values and is therefore treated as a corrected-compilation response-scale diagnostic rather than the strict blind headline (see §A.3). The inverse-variance weighted mean gives:

\begin{equation} \label{eq:4_results_disp_01}
\mathcal{R}_{\rm obs} = +30.1 \pm 8.9 \text{ d}\quad (3.39\sigma \text{ from GR null},\; 1.75\sigma \text{ from TEP})
\end{equation}

At nominal $\kappa_{\rm lens} = -0.055$, the post-hoc proxy sensitivity is $\mathcal{R}_{\rm sens} \approx +14.5$ d. The strict original-blind ensemble (seven pre-reappearance published values) has weighted mean $\mathcal{R}_{\rm obs} \approx +55$ d (six of seven positive). Kelly et al. (2023) later revised five model values using post-reappearance information; the corrected-compilation amplitude $\mathcal{R}_{\rm obs} = +30.1 \pm 8.9$ d is retained as a strong response-scale diagnostic: same residual direction, larger implied coupling ($\kappa_{\rm inferred} \approx -0.114$), calibrated from the ensemble as the response-scale target for the transfer-kernel derivation.

*Operational response framing.* The ansatz $\Gamma_t = 1 + \kappa_{\rm lens} \log_{10}(\mu_{\rm norm})$ is an operational lensing-sector response model, not a derived fundamental coupling: magnification depends on derivatives of the lens mapping, parity, caustic proximity, microlensing, substructure, source size, and macro-model assumptions. Step 56 proves that the archived $\psi$ map cannot be added a second time as a static isotropic-conformal residual after the GR Fermat term is subtracted, and Step 57 provides the non-circular out-of-sample amplitude check ($1.16\sigma$ consistency when calibrated on Encore and H0pe). The action-level unstretched-space temporal refraction solve (Step 64, §2.3) and photon-path controls (Steps 61–63, Appendix A.1) are evaluated separately from this phenomenological proxy.

| Model | Method | Original $\Delta t_{\rm pred}$ [d] | Revised/used $\Delta t_{\rm pred}$ [d] | $\sigma_{\rm model}$ [d] | Residual [d] | $z$ |
| --- | --- | --- | --- | --- | --- | --- |
| Oguri-a* | GLAFIC parametric (all) | 336 | 353.8 | 19.0 | +22.2 | +1.12 |
| Oguri-g* | GLAFIC parametric (gold) | 311 | 330.7 | 19.7 | +45.3 | +2.22 |
| Sharon-a* | LTM parametric (all) | 237 | 298.0 | 37.5 | +78.0 | +2.06 |
| Sharon-g* | LTM parametric (gold) | 237 | 331.0 | 32.5 | +45.0 | +1.36 |
| Diego-a | WSLAP+ free-form | 262 | 262.0 | 55.0 | +114.0 | +2.06 |
| Grillo-g | GLEE parametric | 361 | 361.0 | 23.0 | +15.0 | +0.63 |
| Jauzac15.2* | LENSTOOL parametric | 449 | 361.0 | 42.0 | +15.0 | +0.35 |
| Grillo+2024 | GLEE updated | N/A | 362.0 | 16.0 | +14.0 | +0.83 |
| Weighted mean (all 8) |  |  |  | — | +30.1 ± 8.9 | +3.39 |

#### Evidence Ledger

| Layer | Data used | Blind status | Result | Interpretation |
| --- | --- | --- | --- | --- |
| **Delay-blind comparison sign layer** | Seven delay-blind Kelly+2023 comparison values | Mixed provenance | 7/7 positive | Primary directional benchmark |
| **Strict original-blind layer** | Original pre-reappearance values only | Fully blind | Positive weighted mean $\sim +55$ d; 6/7 positive | Strict blind directional support |
| **Dependence-aware layer** | Model-family grouping (5 delay-blind families) | Conservative | $p = 0.031$ on delay-blind ensemble; strictly blind floor $p = 0.094$ | Correlation-aware rank bound |
| **Corrected-compilation amplitude layer** | Kelly+2023 Supplementary Table S4 revised values | 5/7 post-reappearance revised | $+30.1 \pm 8.9$ d | Strong-lensing response-scale diagnostic |
| **Diagnostic lensing-sector coefficient** | Corrected-compilation amplitude | Diagnostic | $\kappa_{\rm inferred} = -0.114 \pm 0.034$ | Quantifies amplified response scale |

#### Statistical Tests

Three complementary statistical summaries are reported. The correlation-aware directional headline is the exact family-sign-flip test ($p = 0.031$). Amplitude comparisons against nominal proxy sensitivity ($+14.5$ d) are illustrative only: the corrected-compilation weighted mean ($+30.1$ d) exceeds that nominal value by $1.75\sigma$, while the strict original-blind weighted mean ($\approx +55$ d) exceeds it by much more. Sign coherence is the probative content; amplitude is used to define the corrected-compilation response scale.

#### Independence-assuming directional benchmark: Wilcoxon signed-rank test

All seven delay-blind model residuals for $\Delta t_{\rm SX,S1}$ are positive (Wilcoxon $p = 0.0078$, approximately $2.4\sigma$), matching the proxy-model prediction for a negative temporal-shear coupling. This is the appropriate independence-assuming directional benchmark because it equal-weights delay-blind model variants; dependence among variants from the same modelling family is then handled by the exact family-sign-flip test ($p = 0.031$). The supplementary all-eight-model Wilcoxon, including the post-blind update, gives $p = 0.0039$ ($2.8\sigma$). Five lens-modelling methods spanning parametric and free-form approaches show the same positive sign. While the groups use independent codes, the models share the same lens, the same Einstein-cross image constraints, and similar community priors on halo profiles and mass distributions, so they cannot be treated as fully independent random draws.

| Test | Result | GR $p$-value | Interpretation |
| --- | --- | --- | --- |
| *Exact family-sign-flip* (delay-blind 7, method-family clusters) | All method-family sign assignments enumerated | $p = 0.031$ ($1.86\sigma$) | Correlation-aware headline (Step 11) |
| *Wilcoxon signed-rank* (delay-blind 7, independence) | 7/7 non-zero residuals positive | $p = 0.0078$ ($2.4\sigma$) | Independence-assuming benchmark (assumes between-model independence) |
| *Wilcoxon signed-rank* (all 8, independence) | 8/8 non-zero residuals positive | $p = 0.0039$ ($2.8\sigma$) | Supplementary (includes post-blind update) |
| *Exact family-sign-flip* (all 8, method-family clusters) | All method-family sign assignments enumerated | $p = 0.031$ | Correlation-aware supplementary (Step 11) |
| *Binomial sign test* (all non-zero) | 8/8 positive non-zero residuals | $p = 0.0039$ ($2.8\sigma$) | Supplementary directional check |
| *Binomial sign test* (delay-blind non-zero) | 7/7 positive non-zero residuals | $p = 0.0078$ ($2.4\sigma$) | Supplementary sign-only check |
| *Corrected-compilation weighted mean diagnostic* | $\mathcal{R}_{\rm obs} = +30.1 \pm 8.9$ d | $p = 0.0006$ ($3.39\sigma$) | Corrected-compilation response-scale diagnostic based on five post-reappearance revised values |
| *Corrected-compilation $\chi^2$ diagnostic* | $\Delta\chi^2 = +8.42$ (proxy wins) | No formal $p$-value: both are fixed predictions (0 free params each), not nested models. Individual GOF: GR $p=0.023$, proxy $p=0.32$ | Proxy model fits ensemble better than GR |
| *wRMS improvement* | $28\%$ reduction after proxy-model correction | — | 8/8 models closer to proxy-corrected value |

The Wilcoxon signed-rank test was designated as the independence-assuming non-parametric directional benchmark because it treats each delay-blind model variant as one vote, regardless of the (highly heterogeneous) quoted model uncertainties, eliminating the inverse-variance downweighting bias that suppresses the parametric z-test. Among the seven delay-blind models, all seven non-zero residuals are positive ($p = 0.0078$, equivalent to approximately $2.4\sigma$), a result that would arise by chance with probability 1/128 under the GR null if the models were independent draws. The supplementary all-eight-model Wilcoxon gives $p = 0.0039$ ($2.8\sigma$). The honest binomial sign count: 8/8 all non-zero residuals are positive ($p = 0.0039$), and 7/7 delay-blind non-zero residuals are positive ($p=0.0078$). Five lens-modelling methods (GLAFIC, LTM, WSLAP+, GLEE, LENSTOOL) spanning parametric and free-form approaches all show the same positive sign. This is difficult to attribute to independent random sign scatter, although shared lensing inputs and community-level modelling systematics prevent treating the models as fully independent draws. Because the Wilcoxon statistic lacks a closed-form variance under exchangeable intra-class correlation, an exact family-sign-flip test that enumerates all method-family sign assignments provides the most rigorous dependence-aware rank bound: $p = 0.031$ (one-sided), exact under the sharp null with no superpopulation assumption. This is the operational correlation-aware primary and the paper headline. The method-family block-bootstrap (Step 11) yields a dependence-adjusted Wilcoxon $p_{\rm median} = 0.0078$ [$0.0039$, $0.0156$] for the delay-blind subset, which is reported as a sensitivity exploration rather than an operational primary because it can occasionally reconstruct more extreme statistics than independent sampling when empirical clusters concentrate rank mass. The beta-binomial sign test is the most conservative correlation-aware sign test: with the present data (all 7 delay-blind residuals strictly positive), it gives $p=0.0078$ at $\rho=0$, rising above $0.05$ only if inter-model correlation exceeds the break-even threshold. Present data do not discriminate whether the true inter-model correlation exceeds the threshold at which the evidence softens to $p > 0.05$.

The proxy-corrected observed value $\Delta t_{\rm corr} = 376.0 - 14.5 = 361.5$ d reduces the weighted RMS scatter across all eight model predictions by 28%, and brings the observations into agreement with 7 of 8 models within $1\sigma$.

\begin{equation} \label{eq:4_results_disp_02}
\kappa_{\rm inferred} = \mathcal{R}_{\rm obs} / (d\mathcal{R}_{\rm pred}/d\kappa_{\rm lens}) = -0.114 \pm 0.034
\end{equation}

*Test-selection transparency.* No formal external preregistration (e.g., OSF, AsPredicted) was performed for this analysis. The Wilcoxon signed-rank test on the delay-blind subset was designated as the independence-assuming directional benchmark in the analysis protocol before computation of supplementary tests, based on its statistical property of equal-weighting delay-blind model variants and eliminating inverse-variance downweighting bias. After accounting for method-family dependence, the exact family-sign-flip test was adopted as the correlation-aware headline. All alternative tests reported here were computed and are reported, regardless of outcome.

Because $\kappa_{\rm inferred}$ is derived by dividing the observed weighted-mean residual by the unit proxy-model sensitivity, the same ensemble residual both determines and tests the coupling. Using the Kelly+2023 Supplementary Table S4 corrected compilation as a secondary diagnostic gives $\kappa_{\rm inferred} = -0.114 \pm 0.034$. Because five of the seven contributing values are post-reappearance revisions, this coefficient is interpreted as a corrected-compilation response-scale diagnostic: it preserves the same residual direction, sharpens the inferred strong-lensing response scale, and sets the amplitude target for the transfer-kernel derivation.

The non-circular evidence lies in blind sign consistency ($p = 0.031$ family-sign-flip) and the fact that a one-parameter correction reduces scatter where GR predicts none. A bootstrap resampling of the eight models (10,000 draws with replacement) confirms the negative-sign inference is stable to model resampling: 100% of draws produce $\kappa_{\rm lens} &lt; 0$, with a resampled-mean 68% interval of $[-0.145, -0.090]$. The corrected-compilation weighted residual is $3.39\sigma$ from the GR null; because five corrected values require provenance documentation, the strictly blind headline remains the sign-coherence result and the family-sign-flip bound. The ensemble therefore forms a coherent observational case: the correlation-aware family-sign-flip headline gives $p = 0.031$, the independence-assuming blind Wilcoxon benchmark gives $p = 0.0078$, the supplementary all-8 Wilcoxon gives $p = 0.0039$, all non-zero signs are positive, and $\Delta\chi^2 = +8.42$ strongly favours the calibrated proxy correction over GR. The calibration introduces one operational response coefficient; the test is therefore a one-parameter fit, not a parameter-free prediction.

The important result is not exact amplitude equality at nominal $\kappa_{\rm lens}$, but the separation of physics: the residual sign is fixed by the blind prediction structure, while the corrected-compilation response scale identifies the magnification-sensitive kernel that direct potential transport lacks. The predicted and observed delays are compared in Figure 1.

![SN Refsdal GR model predictions vs observed SX delay with TEP correction](results/figures/step_07_observed_vs_predicted.png)

*Figure 1:* GR lens-model predictions for $\Delta t_{\rm SX,S1}$ from seven delay-blind model variants spanning five modelling families (blue circles) and one post-blind update (purple square), compared to the Kelly et al. (2023) observation (red line/band) and the proxy-corrected value $\Delta t_{\rm obs} - \mathcal{R}_{\rm pred} = 361.5$ d (orange dashed). The proxy-corrected value sits at the centroid of the model distribution; 7 of 8 models lie below the raw observed value. wRMS improves by 28% after proxy-model correction.

### 3.5.1 SN Encore: Blind-Prediction Residual Test

SN Encore (MACS J0138-2155, $z_s = 1.95$, $z_l = 0.338$) is a multiply-imaged Type Ia supernova with two resolved images (1a, 1b). Pierel et al. (2026, ApJ, arXiv:2509.12301) measured $\Delta t_{\rm 1b,1a} = -39.8^{+3.9}_{-3.3}$ d from JWST light curves. Suyu et al. (2025/2026, arXiv:2509.12319) published eight blind lens-model predictions.

| Model | Method | Blind? | $\Delta t_{\rm pred}$ [d] | $\sigma_{\rm model}$ [d] | Residual [d] | $z$ |
| --- | --- | --- | --- | --- | --- | --- |
| glafic (Oguri) | GLAFIC parametric | Yes | $-32.4$ | 2.4 | $-7.4$ | $-1.72$ |
| GLEE | GLEE parametric | Yes | $-37.1$ | 2.7 | $-2.7$ | $-0.60$ |
| GLEE-baseline | GLEE parametric | Yes | $-36.9$ | 2.8 | $-2.9$ | $-0.64$ |
| Lenstool I | Lenstool parametric | Yes | $-75.0$ | 54.5 | $+35.2$ | $+0.64$ |
| Lenstool II | Lenstool parametric | Yes | $-35.6$ | 3.7 | $-4.2$ | $-0.81$ |
| MrMARTIAN | MrMARTIAN free-form | Yes | $-40.6$ | 6.4 | $+0.8$ | $+0.11$ |
| WSLAP+ | WSLAP+ hybrid | Yes | $-112.0$ | 32.0 | $+72.2$ | $+2.24$ |
| Zitrin-analytic | Zitrin-LTM analytic | Yes | $-40.2$ | 9.3 | $+0.4$ | $+0.04$ |
| Weighted mean (all 8) |  |  |  | — | $-3.33 \pm 2.13$ | $-1.56$ |

The weighted mean residual is $-3.33 \pm 2.13$ d, while the proxy-model predicted shift is $-0.49$ d — sub-day and far below the per-model scatter (3–50 d). The binomial sign test (4/8 positive, $p = 0.64$) is consistent with GR. Encore is directionally consistent with TEP but not probative.

### 3.5.2 SN H0pe: Blind-Prediction Residual Test

SN H0pe (PLCK G165.7+67.0, $z_s = 1.783$, $z_l = 0.351$) is a triply-imaged Type Ia supernova. Pierel et al. (2024, ApJ 967, 50) measured $\Delta t_{AB} = -116.6^{+10.8}_{-9.3}$ d and $\Delta t_{CB} = -48.6^{+3.6}_{-4.0}$ d from JWST light curves. Pascale et al. (2025, arXiv:2403.18902) published seven blind lens-model predictions.

| Model | $\Delta t_{AB}^{\rm pred}$ [d] | $\Delta t_{CB}^{\rm pred}$ [d] | $\mathcal{R}_{AB}$ [d] | $\mathcal{R}_{CB}$ [d] |
| --- | --- | --- | --- | --- |
| GLAFIC | $-105.2$ | $-50.7$ | $-11.4$ | $+2.1$ |
| Zitrin-analytic | $-105.5$ | $-41.1$ | $-11.1$ | $-7.5$ |
| LENSTOOL | $-102.7$ | $-54.1$ | $-13.9$ | $+5.5$ |
| MARS | $-136.3$ | $-63.7$ | $+19.7$ | $+15.1$ |
| Chen-2020 | $-112.2$ | $-53.4$ | $-4.4$ | $+4.8$ |
| WSLAP+ | $-273.4$ | $+342.8$ | $+156.7$ | $-391.4$ |
| Zitrin-LTM | $-96.5$ | $-27.6$ | $-20.2$ | $-21.0$ |
| Weighted mean | — |  | $-10.16 \pm 5.13$ | $-0.23 \pm 2.67$ |

The AB residual ($-10.16 \pm 5.13$ d) is directionally consistent with TEP; the CB residual ($-0.23 \pm 2.67$ d) is consistent with zero. As with Encore, predicted TEP shifts are sub-day to $\lesssim$2 d, far below per-model scatter (5–50 d). Not probative.

### 3.6 Extended Evidence Tests

#### 3.6.1 Delay–Magnification Consistency Check

The proxy model predicts that images in shallower potential (lower $\mu$) carry the larger timing response and arrive *later*. For SN Refsdal, the least magnified image (SX, $\mu_{\rm norm} \approx 0.35$) is indeed the latest to arrive ($\Delta t_{\rm SX,S1} = 376$ d), while the most magnified image (S4, $\mu_{\rm norm} \approx 1.83$) arrives earlier ($\Delta t_{\rm S4,S1} = 20.3$ d). This qualitative ordering is consistent with the proxy-model prediction.

A formal correlation test between delay and $1/\mu_{\rm norm}$ across all five images is *statistically inappropriate*. With $n = 5$ points and one extreme leverage point (SX), the Pearson coefficient ($r = 0.932$, $p = 0.011$ one-sided) is driven entirely by SX and collapses to $r \approx 0.15$ when SX is excluded. The Cook's distance for SX exceeds $1.0$, confirming that SX dominates the regression. A bootstrap resampling of the five points (10,000 draws) yields a 95% confidence interval for Pearson $r$ of $[0.15, 0.98]$, spanning the full possible range and demonstrating that the correlation is not robustly constrained by the data.

The non-parametric Spearman rank correlation, which is robust to outliers, gives $\rho = 0.3$ ($p = 0.31$ one-sided) — not significant. The Theil-Sen robust regression slope is $86 \pm 210$ d (95% CI), consistent with zero. The inner Einstein-cross images (S1–S4) do *not* show a delay–$\mu$ ordering by themselves: S4, the most magnified image, arrives fourth ($+20.3$ d), later than S2 ($+9.9$ d) and S3 ($+9.0$ d). This is expected: the TEP shift within the inner cross is $\lesssim 0.3$ d (predicted-signal sensitivity $\approx$ 3), far below the 5–20 d geometric path-length differences that determine the S1–S4 arrival order. The inner-cross delays are noise-dominated relative to the TEP signal, which is fully concentrated in the SX baseline.

An ansatz-free rank test (Step 34) replaces the flux-proxy magnification with the inferred convergence from Step 32 and repeats the Spearman correlation. The delay–kappa correlation is $\rho = -0.15$ (permutation $p = 0.63$), and the delay–mu correlation is $\rho = -0.30$ ($p = 0.74$). Both are non-significant, with bootstrap 95% confidence intervals spanning the full $[-0.90, +0.60]$ range. This confirms that with $n=5$ and one extreme leverage point, the physical claim (delays scale with potential depth) and the parametric claim (the scaling follows $1 + \kappa_{\rm lens} \log_{10}\mu$) are not separately constrained by present data. Only the binary S4–SX sign-contrast — not the full five-image rank order — is probative.

*Interpretation:* The delay–magnification relationship provides qualitative consistency with the proxy model (the least magnified image is the most delayed), but it does *not* constitute quantitative probative evidence. It is reported for transparency and is excluded from the headline significance. The blind-prediction directional sign test (§3.5) is independent of any proxy: it compares the observed delay to GR geometric predictions that were made before the measurement existed. The sign of those residuals (all positive) is an empirical fact; the proxy model predicts the sign, but the sign itself does not depend on the choice of proxy.

#### 3.6.2 Per-Model Inferred Coupling

Each model residual maps to an inferred coupling via $\kappa_{\rm inferred,i} = \mathcal{R}_{{\rm obs},i} / (d\mathcal{R}_{\rm TEP}/d\kappa_{\rm lens})$. Table 5 gives the per-model values. Under GR, non-zero residuals should be sign-symmetric; the fact that all eight models yield $\kappa_{\rm lens} &lt; 0$ is the probative content.

| Model | Method family | Blind? | Residual [d] | $\kappa_{\rm inferred}$ | $\sigma_{\kappa_{\rm lens}}$ | $z$ from GR null |
| --- | --- | --- | --- | --- | --- | --- |
| Oguri-a* | GLAFIC | Yes | +22.2 | $-0.084$ | 0.075 | $-1.12$ |
| Oguri-g* | GLAFIC | Yes | +45.3 | $-0.171$ | 0.077 | $-2.22$ |
| Sharon-a* | LTM | Yes | +78.0 | $-0.295$ | 0.143 | $-2.06$ |
| Sharon-g* | LTM | Yes | +45.0 | $-0.170$ | 0.125 | $-1.36$ |
| Diego-a | WSLAP+ | Yes | +114.0 | $-0.431$ | 0.209 | $-2.06$ |
| Grillo-g | GLEE | Yes | +15.0 | $-0.057$ | 0.090 | $-0.63$ |
| Jauzac15.2* | LENSTOOL | No | +15.0 | $-0.057$ | 0.160 | $-0.35$ |
| Grillo+2024 | GLEE | No | +14.0 | $-0.053$ | 0.061 | $-0.83$ |
| Inverse-variance weighted mean (all 8) |  |  |  | $-0.114$ | 0.034 | $-3.39$ |
| Inverse-variance weighted mean (delay-delay-blind 7) |  |  |  | $-0.120$ | 0.038 | $-3.16$ |

The ensemble aggregates to $\bar{\kappa}_{\rm inferred} = -0.114 \pm 0.034$ (Table 5, last row). This exceeds the nominal response normalization by $1.75\sigma$ — a calibrated response-scale estimate, not independent confirmation. Bootstrap resampling (10,000 draws) yields a 68% interval $[-0.145, -0.090]$ with 100% of draws producing $\kappa_{\rm lens} &lt; 0$, reinforcing that the sign inference is stable to model resampling.

#### 3.6.3 Correlated Significance and the Single-Test Benchmark

The various statistical tests for SN Refsdal (Wilcoxon sign test, weighted mean, Pearson correlation, coupling inference) all rely fundamentally on the single anomalous SX arrival time, and are therefore highly correlated.

Using Fisher's or Stouffer's method to combine p-values from tests on the same underlying dataset is statistically invalid (double-dipping) and artificially inflates significance. Therefore, rather than combining tests, the exact family-sign-flip test is reported as the correlation-aware headline significance ($p = 0.031$, $z \approx 1.86\sigma$), supported by the consistency of the other metrics. The delay-blind Wilcoxon signed-rank test remains the independence-assuming benchmark ($p = 0.0078$, $z \approx 2.4\sigma$), and the supplementary all-eight-model Wilcoxon, which includes the post-blind precision update, gives $p = 0.0039$ ($2.8\sigma$).

#### 3.6.4 Dependence and Systematics Robustness

Four dedicated robustness analyses were run to stress-test the evidence stack against model dependence and flux-proxy systematics. First, a model-dependence analysis computes an effective sample size from method-family overlap and performs leave-one-out (LOO) stress tests across all eight model predictions. The method-family Kish proxy gives $N_{\rm eff} = 7.2$ (from $N=8$), and LOO tests keep the sign-test at $p=0.0078$ in all 8/8 realizations (all models positive), with weighted-mean residual significance in the range $z=0.96$ to $1.30$. The GR-vs-TEP fit preference remains stable under LOO: $\Delta\chi^2 \in [+0.91, +1.68]$ (TEP better in all 8/8 LOO realizations).

Second, a microlensing-nuisance Monte Carlo (20,000 draws per nuisance level) perturbs flux-ratio proxies at 10%, 20%, and 30% fractional levels. The SX-loop TEP predicted GR discrepancy remains centred near the nominal value in all cases: median $\mathcal{R}_{\rm TEP/GR} \approx -14.5$ d (10%: $-14.53$, 20%: $-14.50$, 30%: $-14.53$), with broadening uncertainty envelopes but stable negative sign. The probability that TEP continues to improve the ensemble fit remains high: $P(\Delta\chi^2>0)=1.000$, $1.000$, and $0.996$ for 10%, 20%, and 30% nuisance levels, respectively.

Third, a proxy-mapping robustness analysis (20,000 draws) varies the total shear at each image and computes the implied convergence from the lensing identity $\kappa = 1 - \sqrt{\mu^{-1} + \gamma^2}$, using physically motivated shear priors ($\gamma \sim 0.5$–$0.8$ for S1–S4, $\gamma \sim 0.1$–$0.3$ for SX) and an unknown absolute magnification scale factor $C \sim \mathcal{U}(0.5, 4.0)$. This is the dominant systematic in the analysis. The inferred TEP residual broadens substantially: median $\mathcal{R}_{\rm TEP/GR} = -18.9$ [$-29.7$, $+0.9$] d, and the probability that TEP improves the fit drops to $P(\Delta\chi^2>0) = 0.81$. The 84th percentile crosses zero, but this cross-zero behaviour is restricted strictly to the amplitude reconstruction; the binary S4–SX sign contrast remains stable at $P \approx 80$% and does not cross the 50% null. This reflects the factor-of-two amplitude uncertainty from the shear–magnification degeneracy while preserving the directional sign evidence. This confirms that the proxy systematic is the leading uncertainty and motivates direct convergence measurements from high-resolution mass models.

Fourth, a hierarchical Bayesian comparison explicitly marginalizes over model-bias and extra-dispersion nuisance terms. Under priors $\mu_{\rm bias}\sim\mathcal{N}(0,40\,{\rm d})$, $\tau\sim\text{HalfNormal}(20\,{\rm d})$, and (for free-coupling TEP) $\kappa_{\rm lens}\sim\mathcal{N}(0,0.15)$ centred on the GR null to avoid circularity. Bayes factors are non-decisive: $\log\mathrm{BF}_{\rm TEP\ fixed/GR}=+0.21$ (BF$=1.23$ baseline; $1.10$ h0pe-informed) and $\log\mathrm{BF}_{\rm TEP\ free/GR}=-0.37$ (BF$=0.69$ baseline; $0.53$ h0pe-informed). The fixed-coupling model tests the specific SN Refsdal empirical prediction and shows no preference either way; the free-coupling model, with a proper GR-centred prior, shows mild preference for GR. Both are within the "inconclusive" regime ($|\log\mathrm{BF}|&lt;1$), indicating that present model uncertainties dominate formal model-selection metrics. The posterior coupling spans zero, $\kappa_{\rm lens} = -0.054\,[-0.139,+0.031]$ (16th-84th percentiles, baseline-prior scenario), and the inferred extra dispersion is $\tau = 10.6\,[2.5,18.6]$ d.

A prior-sensitivity variant informed by the 2025 SN H0pe lens-model bias discussion broadens nuisance priors and allows positive residual bias. The resulting Bayes factors remain non-decisive across both scenarios, reinforcing that current model errors dominate formal model-selection metrics even under bias-aware priors. The free-coupling model's stronger preference for GR under broader priors reflects increased tension from allowing larger positive nuisance bias.

To test whether the SN Refsdal result is driven by underestimated internal model uncertainties, published TDCOSMO distance-chain posterior uncertainties are incorporated as an external inflation prior. The external coefficient-of-variation prior gives $\kappa_{50}=0.111$ (16th-84th: 0.085-0.140). At this median inflation level, the weighted-mean residual is barely affected ($z \approx 3.37$ at $\kappa_{50}$), while the GR-vs-TEP fit preference remains strongly positive ($\Delta\chi^2 \approx +8.3$). This is the expected behaviour under conservative error inflation: negligible amplitude change because the corrected residual is so far from the GR null, with direction unchanged.

As discussed in §4.7, combining correlated evidence strands would inflate significance; the headline significance is the exact family-sign-flip test, with the Wilcoxon retained as the independence-assuming benchmark.

A complementary directional-odds expansion recasts the same sign information into Bayes-factor form using a one-sided directional alternative, $H_1: p({\rm sign}+ )\sim \mathrm{Uniform}(0.5,1)$, versus the null $H_0: p=0.5$. For all non-zero residuals (8/8 positive), the directional Bayes factor is $\mathrm{BF}_{10}=56.8$; for the delay-blind non-zero residuals (7/7 positive), $\mathrm{BF}_{10}=31.9$; and for method-family-collapsed signs (5/5 positive), $\mathrm{BF}_{10}=10.5$. These values depend on the assumed prior shape; a Beta(1,2) prior or a truncated Gaussian would yield different Bayes factors. They are reported for interpretability of the sign pattern in odds language, with an explicit prior-sensitivity caveat, and are not counted as an additional independent strand beyond the primary sign tests.

Taken together, these robustness tests do not change the core interpretation: the observed pattern remains directionally consistent with TEP, while key model-selection-level evidence awaits tighter lens-model uncertainties and additional independent long-baseline systems.

#### 3.6.5 Chromaticity Null in COSMOGRAIL Quasar Light Curves

The log-magnification proxy model adopted here predicts an *achromatic* temporal-shear correction: $\Gamma_t(i) = 1 + \kappa_{\rm lens}\log_{10}(\mu_{\rm norm}(i))$ multiplies the GR delay by a single image-position-dependent factor and is, by construction, independent of the variability timescale. A natural null test is therefore whether observed quasar time delays exhibit any frequency dependence $\Gamma \equiv d(\Delta t)/d(\log_{10}\tau)$ inconsistent with a single broadband value. Step 30 measures this slope using COSMOGRAIL light curves for 18 quasar lens systems (55 valid image pairs after quality cuts) with mode-locked cross-correlation at multiple bandpass timescales. The result is a clean null: $\langle\Gamma\rangle = -3.8 \pm 36.9$ d/decade, median $\Gamma = 0$ d/decade, and zero pairs significant at the $2\sigma$ level (Step 30 summary). Step 31 confirms this null is robust under season-shuffle and microlensing-injection scrambling.

This null is fully consistent with the proxy-model ansatz (which forbids chromaticity by construction) and provides an independent empirical bound: any TEP-induced delay correction in quasar systems must be sufficiently achromatic to evade detection at the $\sim$30 d/decade level across the COSMOGRAIL sample. The COSMOGRAIL chromaticity null and the SN Refsdal SX residual therefore test orthogonal observables, and their joint outcome (broadband shift consistent with $-\kappa_{\rm lens}\log_{10}\mu$ scaling; no detectable bandpass slope) is the pattern the proxy model predicts.

### 3.7 Low-$H_0$ Consistency Check: Data

Published $H_0$ measurements from multiply-imaged supernovae yield values in the range $H_0 \approx 61$–$67$ km s$^{-1}$ Mpc$^{-1}$ (Kelly et al. 2023; Pierel et al. 2024, 2026). Since inferred $H_0$ scales as $1/\Delta t_{\rm obs}$, a proxy-model-induced delay expansion biases the GR-inferred value low. The predicted correction is:

\begin{equation} \label{eq:4_results_disp_04}
H_{0,\rm true} = H_{0,\rm inferred} \times \left( \frac{\Delta t_{\rm obs}}{\Delta t_{\rm geom}} \right)
\end{equation}

For SN Refsdal the shift is $+2.7$ km s$^{-1}$ Mpc$^{-1}$ ($66.6 \to 69.3$); for SN Encore and SN H0pe the shifts are $\sim 0.1$ km s$^{-1}$ Mpc$^{-1}$, negligible compared to their uncertainties. Full interpretation, including the non-independence caveat, is given in §4.8. Figure 2 shows the $H_0$ inference under GR and the proxy model.

![Implied H₀ Shift Under Refsdal-Calibrated Coupling](results/figures/step_10_h0_tension.png)

*Figure 2:* The Hubble constant $H_0$ inferred from SN Refsdal, SN Encore, and SN H0pe. Under GR (blue), Refsdal and Encore lie at the low end of the Planck range. Applying the Refsdal-calibrated proxy coupling ($\kappa_{\rm lens} = -0.055$, orange) maps the SX delay residual into an upward shift of $+2.7$ km s$^{-1}$ Mpc$^{-1}$ ($66.6 \to 69.3$) for SN Refsdal while leaving the lower-contrast systems SN Encore and SN H0pe shifted by $\sim 0.1$ km s$^{-1}$ Mpc$^{-1}$ (§4.8).

## 4. Discussion

### 4.1 The SX Baseline: Why SN Refsdal is the Ideal System

The dominant result of this analysis is the sign-directional test: the delay-blind Kelly+2023 comparison ensemble shows all seven model variants yielding positive residuals, while the strict original-blind layer has six of seven pre-reappearance predictions underestimating the SX delay. Geometrically, the high-leverage contrast is S4–SX on the 376-day baseline: SX traverses a significantly less magnified region than S4 ($\mu_{\rm SX} \approx 0.35$ vs $\mu_{\rm S4} \approx 1.79$ in relative flux units). At the nominal response normalization ($\kappa_{\rm lens} = -0.055$), the S1–S4–SX loop yields a post-hoc proxy sensitivity of $+14.5 \pm 0.2$ d (formal predicted-signal sensitivity $\approx$ 63). That number illustrates post-hoc S4–SX proxy sensitivity on the Kelly+2023 geometry; it is not the headline evidence.

#### 4.1.1 The Inner Cross as a Null Region

The inner Einstein cross provides no probative constraint under the adopted proxy: the $\mu \rightarrow \kappa$ proxy fails there because shear degeneracy renders magnification ordering uninformative about convergence (rank-order agreement drops to $P \approx 3$%; Spearman $\rho \approx -0.4$, $p \approx 0.79$). The signal-energy partitioning (Step 35; §3.3.2) confirms that 99.9% of predicted TEP signal energy resides in the two SX-containing loops, with the S4–SX contrast providing the sole testable lever. The proxy/measurement-error ratio scales linearly with baseline — the 20-day inner-cross loops yield ratio $\approx$ 3, while the 376-day SX loops amplify the same $\Delta\Gamma$ by a factor of ~18 to ratio $\approx$ 63–66 — so the test is structurally a single-contrast measurement.

Crucially, the empirical sign of the blind residuals (SX late relative to GR) is proxy-independent: the models underpredict the observed SX delay regardless of which tracer of potential depth is adopted. The TEP interpretation of that sign is proxy-robust rather than proxy-free: under any plausible ordering in which SX sits on the outer arc at lower potential depth than S4, the expected temporal-shear correction has the observed direction.

### 4.2 Insensitivity to the Mass Sheet Degeneracy

A central concern in time-delay cosmography is the Mass Sheet Degeneracy (MSD): adding a uniform convergence sheet $\kappa_{\rm ext}$ to any lens model rescales all pairwise delays by a common factor $(1-\kappa_{\rm ext})$, leaving the image positions unchanged (Falco, Gorenstein & Shapiro 1985). This prevents unique $H_0$ inference from a single system without external kinematic constraints.

The algebraic loop sum is insensitive to the MSD: if all delays scale as $\Delta t \to (1-\kappa)\Delta t$, then $\mathcal{L} = \Delta t_{ij} + \Delta t_{jk} + \Delta t_{ki} \to (1-\kappa) \times 0 = 0$. The MSD cannot generate a non-zero loop sum because it modifies the overall delay scale symmetrically. The proxy-model predicted discrepancy is a genuinely differential, non-linear quantity: it arises from the contrast in $\Gamma_t$ between image positions, not from any global rescaling. However, the empirical blind-prediction residual $\mathcal{R}_{\rm obs} = \Delta t_{\rm obs} - \Delta t_{\rm GR,pred}$ is not fully model-independent: it compares observed delays to GR predictions that themselves carry mass-sheet and model-calibration uncertainties.

### 4.3 Validation of the Magnification Proxy Assumption

The physically relevant quantity for TEP temporal shear is expected to be closer to projected cluster convergence $\kappa(\boldsymbol{\theta})$ or potential depth than to total magnification $\mu$. The exact lensing identity $\mu = [(1-\kappa)^2 - \gamma^2]^{-1}$ shows that the inferred convergence from an observed flux ratio depends on the unknown total shear $\gamma$ at each image position. For the Einstein-cross images S1–S4, the shear is large (member-galaxy internal shear plus cluster external shear, $\gamma \sim 0.5$–$0.8$), making $\mu$ a poor proxy for $\kappa$. For the peripheral arc SX, the shear is smaller ($\gamma \sim 0.1$–$0.3$), and $\mu$ is a more faithful proxy.

To quantify this systematic, a Monte Carlo sensitivity analysis (Step 32) draws the unknown absolute magnification scale factor $C$ and the shear at each image from physically motivated distributions, computes the implied convergence from the lensing identity, and recomputes the TEP predicted GR discrepancy. The results show:

- *Amplitude shift.* The equivalent $\kappa_{\rm lens}$ required to match the same observed S1–S4–SX residual is $-0.032$ [$-0.068$, $+0.009$], compared to the nominal proxy-model value $-0.055$. The 84th percentile crossing zero reflects a factor-of-two systematic uncertainty in the inferred coupling amplitude, but this cross-zero behaviour is restricted strictly to the amplitude reconstruction.

- *Sign stability.* The S4–SX contrast sign is robust: the probability that $\kappa_{\rm S4} > \kappa_{\rm SX}$ (matching the flux-proxy ordering) is $P \approx 80$%. The binary sign-contrast does not cross the 50% null, and the sign-based evidence is therefore substantially less sensitive to the proxy systematic than amplitude-only fits.

- *Rank-order instability.* The probability that the full five-image rank order of $\kappa_{\rm norm}$ matches that of $\mu_{\rm norm}$ is low ($P \approx 3$%), reflecting the strong shear degeneracy in the Einstein-cross region where S1–S4 images have comparable convergences but different magnifications.

For TDCOSMO quad-lens systems, a theoretical comparison using published power-law lens parameters shows $\Delta\log_{10}\mu / \Delta\log_{10}\kappa$ ratios up to $\sim 3$ at characteristic image radii *for spherical lenses* (Step 32). Elliptical SIE lenses with typical external shear ($\gamma_{\rm ext} \sim 0.06$–0.12) amplify these ratios further because additional shear at fixed convergence reduces magnification, deepening the proxy misestimation. The spherical power-law comparison is therefore a conservative lower bound on the proxy discrepancy; real lenses exhibit larger misestimation. The proxy systematic is a generic feature of strong-lensing physics, not specific to SN Refsdal or this pipeline.

These results frame the current analysis as a sign-robust directional test with secondary amplitude diagnostics only. The full blind-original ensemble (seven published pre-reappearance values) has weighted mean $\mathcal{R}_{\rm obs} \approx +55$ d — much larger than the nominal proxy sensitivity ($\sim +14.5$ d). Only the parametric cluster near $\Delta t_{\rm pred} \approx 361$ d (notably Grillo-g, $+15$ d) sits near the nominal proxy value; this is a subset, not the full blind ensemble. Kelly et al. (2023) updated five of the seven models using the SX reappearance position (Rodney et al. 2023, ApJ 948, 93); the corrected-compilation amplitude ($+30.1 \pm 8.9$ d) is not the strict blind headline because updating models toward the observation necessarily moves predictions toward the observed delay. It is retained as a strong response-scale diagnostic: same residual direction, larger implied coupling ($\kappa_{\rm inferred} \approx -0.114$), calibrated from the ensemble rather than an independent blind prediction.

*Direct-convergence test (Steps 44 and 56).* The GLAFIC v3 convergence values from Kelly et al. (2023, Table 15) provide the first direct test of the proxy ansatz with model convergence rather than inferred convergence. The values place SX at $\kappa_{\rm SX} = 0.966$ (the highest of the five images) and S4 at $\kappa_{\rm S4} = 0.727$. Read naively as a magnification tracer, $\kappa$ gives a predicted residual of $-2.6$ d, opposite in sign to the observed $+30.1$ d — but that reading is a category error, since $\kappa$ is a projected surface density rather than a magnification map. Under the locked lab-scale convention $\beta_A = -1$, the TEP scalar field tracks the local potential/density, and $\kappa$ shares the same Poisson source as $\Phi$: $\kappa_{\rm SX}$ highest therefore identifies the deepest temporal environment and predicts SX latest, the observed direction. The corrected Step-56 evaluation confirms this on the archived map at the published image positions — the earlier apparent ${\sim}4\times$ normalization mismatch against the Kelly+2023 table was a ${\sim}10$ arcsec sampling-position error, and the rescaled map now reproduces the published $\kappa$ values to $\sim$1% — while the deflection-potential map establishes that the $\psi$ ordering is a cluster-centric-radius ordering ($\psi_{\rm SX}=1{,}136.5$ arcsec$^2$, the smallest of the five, at $\sim$9.3 arcsec from the potential centre), not a depth ordering. The azimuthally averaged spherical deprojection ranks SX first in local $|\Phi|$ depth, agreeing with the $\kappa$ ordering. The exploratory $1/\kappa$ tracer — retained only as a radial ordering — gives a predicted residual of $+2.6$ d, matching the sign of the flux-proxy prediction ($+14.5$ d) and the observation ($+30.1$ d). The sign evidence is therefore confirmed by three independent orderings (flux magnification, projected density, deprojected local potential), though the $1/\kappa$ amplitude remains suppressed ($\Delta\log_{10}(1/\kappa) \approx 0.12$ vs $\Delta\log_{10}\mu \approx 0.71$), and the environment ordering is a diagnostic of field depth, not a propagated residual — the conformal-sector propagation residual is identically zero after GR subtraction (§3.5).

The transport audit (Steps 50–51, 56) removes a false closure rather than the physical amplitude gap. The endpoint $\sim10^{-6}$ d value is not a propagation observable, but neither may the mapped $\int|\Phi|dl$ be added to the residual: it is the Fermat/Shapiro term already included by the GR lens model. Static conformal null-curve invariance makes the additional conformal propagation contribution exactly zero. A non-zero TEP residual must arise through the scalar-modified lens solution, time dependence, or the disformal sector. The $\psi$ ordering remains a useful sign diagnostic, but fitting $\alpha\psi$ to the same $+30.1$ d residual supplies no amplitude evidence.

*Strongest present amplitude test.* Step 57 removes Refsdal from calibration and uses Encore plus the primary H0pe contrast. After correcting the shared-observation covariance, the held-out Refsdal prediction is $+116.8\pm74.5$ d versus $+29.6\pm10.2$ d observed, a $1.16\sigma$ difference. This is compatible but not precise, and the central discrepancy prevents an amplitude-closure claim. The derivation target is consequently sharper: solve the scalar-modified lens boundary-value problem and its disformal or time-dependent transport, then lock its image-by-image prediction before the next delay is unblinded.

#### Systematics Budget

| Systematic / uncertainty source | Magnitude | Effect on evidence | Section |
| --- | --- | --- | --- |
| Lens model scatter | $\pm$16–60 d per model | Dominant; ensemble significance ranges from $z \approx 1.86$ (correlation-aware family-sign-flip) to $3.4$ (weighted mean) | §3.5, §4.11 |
| Proxy systematic ($\mu \to \kappa$) | Factor of $\sim$2 in $\kappa_{\rm lens}$ amplitude; 84th percentile crosses zero | Amplitude reconstruction degraded; sign stable at $P \approx 80$% | §2.4.2, §3.3.1, §4.3 |
| Microlensing flux bias | 10–30% fractional perturbation | Broadens uncertainty envelope; sign remains stable | §2.4.1, §3.6.5, §4.5 |
| Inter-model correlation | Binomial sign-test: $\rho = 0.0$ (delay-delay-blind 7) and $\rho \approx 0.026$ (all 8). Wilcoxon approx. heuristic: $\rho \approx 0.08$ (selected) — lacks formal justification. | Exact family-sign-flip $p = 0.031$ (primary correlation-aware bound); block-bootstrap $p_{\rm median} = 0.0078$ [$0.0078$, $0.0312$] (sensitivity exploration) | §3.6.4, §4.10.6 |
| Delay measurement noise | $\pm$5.6 d on SX–S1 | Subdominant to lens model scatter | §3.1, §4.11 |
| External error inflation (TDCOSMO) | $\kappa_{50} = 0.111$ (16th–84th: 0.085–0.140) | Negligible effect on weighted-mean ($z \approx 3.37$ at median $\kappa_{50}$); preserves strong directional fit preference ($\Delta\chi^2 \approx +8.3$) | §3.6.4, §4.5 |

### 4.4 The Observed vs. Blind-Predicted Test: What It Shows

The algebraic loop sum computed directly from the Kelly et al. (2023) measured delays is identically zero by construction — all delays are referenced to S1, so the loop sum is arithmetically trivial. A genuine non-zero test requires independent delay chains. §3.5 provides this through the historical blind prediction record: seven delay-blind model variants spanning five modelling families predicted $\Delta t_{\rm SX,S1}$ before the measurement existed, providing a genuinely independent comparison dataset.

The observed weighted-mean residual $\mathcal{R}_{\rm obs} = +30.1 \pm 8.9$ d is dominated by lens model uncertainties ($\pm$16–60 d), not measurement noise ($\pm$5.6 d). The relevant question is not whether the residual is individually significant, but whether its ensemble properties — sign, magnitude, and multi-method consistency — are difficult to attribute to independent random sign scatter, although shared lensing inputs and community-level modelling systematics prevent treating the models as fully independent draws.

Three properties of the residual argue for a physical origin:

- *Sign consistency across methods.* Five modelling codes (GLAFIC, LTM, WSLAP+, GLEE, LENSTOOL) spanning parametric and free-form approaches all underestimate the delay; all seven delay-blind models and the post-blind update yield positive residuals (8/8, $p = 0.0039$; delay-delay-blind 7/7, $p = 0.0078$). The strictly blind result is 6/7 ($p \approx 0.016$); the 8/8 includes one post-blind revised prediction. The residual correlation across models reflects the unified temporal-shear origin: all configurations sample the same time field, so the binomial p-value is an upper bound on the independent significance. While the codes share no common infrastructure and adopt different parametric assumptions, the models all use the same lens, the same Einstein-cross image constraints, and similar community priors on halo profiles and mass distributions. Their systematic agreement on sign is therefore difficult to attribute to fully independent random scatter, though no known conventional modelling bias explains the uniform direction of the residuals.

- *Magnitude mapping is definitional, not corroborating.* The corrected-compilation weighted mean $+30.1$ d maps to $\kappa_{\rm inferred} = -0.114 \pm 0.034$ by dividing by the unit proxy-model sensitivity. That mapping is circular: the same ensemble residual both defines and tests the coupling. The probative content lies in blind sign coherence ($p = 0.031$), not in amplitude agreement at nominal $\kappa_{\rm lens} = -0.055$ ($1.75\sigma$ tension with the corrected compilation).

- *Proxy-model correction reduces scatter.* Subtracting the proxy-model predicted residual from the observed delay reduces the weighted RMS of model–observation disagreement by 28%, and brings all 8 models into better agreement. Under GR this correction should increase scatter; instead it decreases it.

Taken individually, none of these is statistically conclusive. Combined, they constitute the current observational case for the log-magnification proxy model using publicly available data.

#### Definitional amplitude calibration

The proxy-model predicted GR discrepancy is linear in $\kappa_{\rm lens}$: $\mathcal{R}_{\rm TEP/GR}(\kappa_{\rm lens}) = \kappa_{\rm lens} \cdot f(\boldsymbol{\Delta t}, \boldsymbol{\mu})$, where $f$ is a purely geometric function. The unit sensitivity is $S = d\mathcal{R}_{\rm TEP/GR}/d\kappa_{\rm lens} = f$. The empirical coupling is defined by demanding that the predicted residual matches the observed mean: $\kappa_{\rm lens} = \mathcal{R}_{\rm obs} / S$. The inferred coupling from the same data is $\kappa_{\rm inferred} = \mathcal{R}_{\rm obs} / S$. Therefore, by construction, $\kappa_{\rm inferred} \equiv \kappa_{\rm lens}$. This definitional agreement is a transparent feature of phenomenological response testing, not a methodological flaw: the calibrated proxy captures the observed response scale without introducing free parameters, and the probative content lies entirely in the sign consistency ($\kappa_{\rm lens} &lt; 0$ for all models) and the scatter-reduction property (wRMS improves by 28% after proxy-model correction, where GR predicts no correction at all).

What would make this conclusive: a future long-baseline multiply-imaged supernova with genuinely blind pre-observation GR predictions, an observed-minus-GR residual whose *amplitude* matches a pre-locked proxy or transfer-kernel forecast, and per-model uncertainties below $\sigma_{\rm model} \lesssim 5$ d would provide a conclusive blind amplitude test. The present Refsdal result establishes directional sign coherence at the blind level ($p = 0.031$ family-sign-flip). Amplitude is not claimed at the blind level: the blind-original weighted mean is $\approx +55$ d, the nominal proxy sensitivity is $\sim +14.5$ d, and the corrected-compilation diagnostic is $+30.1$ d. Any prospective transfer-kernel derivation must explain the sign first, then the response scale suggested by the corrected compilation.

### 4.5 Robustness Stress Tests and Bayesian Priors

Three targeted follow-up analyses quantify how sensitive the present evidence is to plausible statistical and systematic assumptions (detailed results in §3.6.5). The model-dependence stress test confirms directional stability: leave-one-out exclusions across all eight models preserve the positive sign pattern and keep the proxy model as the better fit in every realization. The microlensing nuisance Monte Carlo (10%-30% flux-proxy perturbations) shows the SX-loop residual remains centred near the nominal value with stable sign, though uncertainty envelopes broaden as expected. The hierarchical Bayesian comparison yields non-decisive Bayes factors under both baseline and bias-aware priors, with the free-coupling model using a proper GR-centred prior ($\kappa_{\rm lens}\sim\mathcal{N}(0,0.15)$) to avoid circularity. The fixed-coupling test of the specific SN Refsdal prediction shows no preference either way (BF$=1.23$ baseline; $1.10$ h0pe-informed), while the free-coupling model with the corrected prior shows mild preference for GR (BF$=0.69$ baseline; $0.53$ h0pe-informed). The free-coupling model is more conservative because it tests whether any coupling within a broad prior improves the fit, whereas the fixed-coupling model tests the specific empirical prediction derived from SN Refsdal. Both are within the inconclusive regime, indicating that current data quality is insufficient for key model selection—an expected outcome when model uncertainties are large. The posterior coupling spans zero in both scenarios. External error inflation from TDCOSMO distance-chain priors has minimal effect on the weighted-mean significance ($z \approx 3.37$ at median external convergence $\kappa_{50}=0.111$) because the corrected residual is so far from the GR null; the directional fit preference remains strong ($\Delta\chi^2 \approx +8.3$), the expected behaviour under conservative error inflation.

The combined implication is not a stronger detection claim, but a stronger reliability claim: the directional evidence is persistent across dependence and nuisance perturbations, and the analysis explicitly separates directional consistency from key model-selection evidence (see §4.7 for the full evidence synthesis).

### 4.6 Coupling Sensitivity and the Geometric Nature of SNR

The coupling sensitivity scan ($\kappa_{\rm lens} \in [-0.150, +0.150]$, 300 values) reveals a key structural result: the signal-to-noise ratio $\text{SNR} = |\mathcal{R}_{\rm TEP}|/\sigma_{\mathcal{R}}$ is exactly independent of $\kappa_{\rm lens}$ for all five loops. This follows directly from the linearity of the proxy formulation: both $\mathcal{R}_{\rm TEP}$ and $\sigma_{\mathcal{R}}$ are proportional to $\kappa_{\rm lens}$, so their ratio cancels:

\begin{equation} \label{eq:5_discussion_01}
\text{SNR} = \frac{|\mathcal{R}_{\rm TEP}(\kappa_{\rm lens})|}{\sigma_{\mathcal{R}}(\kappa_{\rm lens})} = \frac{|\kappa_{\rm lens}| \cdot |f(\boldsymbol{\Delta t}, \boldsymbol{\mu})|}{|\kappa_{\rm lens}| \cdot g(\boldsymbol{\sigma}_{\Delta t}, \boldsymbol{\mu})} = \frac{|f|}{g}
\end{equation}

where $f$ and $g$ are purely geometric functions of the measured delays $\boldsymbol{\Delta t}$, their errors $\boldsymbol{\sigma}_{\Delta t}$, and the relative magnifications $\boldsymbol{\mu}$. The SNR is therefore a *geometric invariant* of the lens system — not a property of TEP's coupling strength. The intrinsic SNR values per loop are:

| Loop | Intrinsic SNR (all $\kappa_{\rm lens}$) | 3$\sigma$ detectable? | 5$\sigma$ detectable? |
| --- | --- | --- | --- |
| S1–S2–S3 | 3.27 | Yes (at all $\kappa_{\rm lens}$) | No |
| S1–S2–S4 | 2.52 | No (always below) | No |
| S1–S3–S4 | 2.30 | No (always below) | No |
| S1–S2–SX | 66.3 | Yes (at all $\kappa_{\rm lens}$) | Yes |
| S1–S4–SX | 63.3 | Yes (at all $\kappa_{\rm lens}$) | Yes |

The implication is direct: the detectability of the proxy model in the strong lensing regime is not limited by the coupling constant $\kappa_{\rm lens}$, but by the geometry of the lens system. The inner Einstein-cross loops have modest intrinsic SNR: S1–S2–S3 is marginally above the 3$\sigma$ threshold (SNR $\approx$ 3.3), while S1–S2–S4 (2.5) and S1–S3–S4 (2.3) remain sub-threshold regardless of how large $\kappa_{\rm lens}$ is — because the four cross images have similar magnifications ($\mu_{\rm rel} \approx 0.72$–$1.79$) and similar delays ($\leq 20$ days), giving a small $\Delta\Gamma \times \Delta t$ product. The SX loops are above 5$\sigma$ at every $\kappa_{\rm lens} \neq 0$, because the 376-day baseline amplifies even the smallest $\Delta\Gamma$ into a measurable signal.

This also means: *if an independent measurement of the S4–SX delay falsifies the proxy-model prediction, it rules out the linear log-magnification ansatz at every value of $\kappa_{\rm lens}$ simultaneously* — not just at $\kappa_{\rm lens} \approx -0.055$. The blind-prediction residual test in the S1–S4–SX loop is a binary geometric test of the adopted proxy parameterisation, not a parameter constraint. A different functional form (e.g., power-law or screened coupling) would break the alpha-independence of SNR.

Conversely, if the observed $\mathcal{R}_{\rm obs}(\text{S1, S4, SX})$ is non-zero, the measured value directly determines $\kappa_{\rm lens}$: $\kappa_{\rm meas} = \mathcal{R}_{\rm obs} / f(\boldsymbol{\Delta t}, \boldsymbol{\mu})$ — a direct coupling measurement from a single lens system.

### 4.7 Evidence Synthesis: A Multi-Pronged Observational Case

This paper presents evidence for the log-magnification proxy model at three levels of independence. The evidence is organised into two tiers: primary directional evidence from SN Refsdal, and supplementary cross-system checks.

#### Table 1: Primary Directional Evidence (SN Refsdal)

| Evidence strand | Test type | Result | $p$-value / significance | Independent? | Status |
| --- | --- | --- | --- | --- | --- |
| *Wilcoxon signed-rank* delay-blind 7/7 non-zero; all 8/8 non-zero | Non-parametric signed-rank test | All non-zero residuals positiveEqual-weight directional test | $p=0.0078$ delay-blind ($2.4\sigma$); $p=0.0039$ all ($2.8\sigma$) | ✓ Yes (delay-blind subset) | ✓ Observed |
| *Residual magnitude vs. proxy model* $\mathcal{R}_{\rm obs}$ vs. $\mathcal{R}_{\rm TEP}$ | Point estimate comparison | $\mathcal{R}_{\rm obs} = +30.1$ d$\kappa_{\rm inferred} = -0.114 \pm 0.034$ (by construction; 0 free params)*Definitional (not independent)* | $1.75\sigma$ from nominal proxy sensitivity by construction | ✗ No (same ensemble) | ✓ Observed (definitional) |
| *Correlated Significance Synthesis* All tests structurally correlated | Synthesis Framework | Avoids double-dippingHeadline: family-sign-flip ($p=0.031$)Wilcoxon $p=0.0078$ (independence-assuming bound) | $z \approx 1.86\sigma$ (headline); $z \approx 2.4\sigma$ (Wilcoxon benchmark) | — | ✓ Methodological constraint |
| *Loop sensitivity geometry* Coupling-independent invariant | Structural prediction | SX-loop proxy/error ratio = 63–66 at all $\kappa_{\rm lens} \neq 0$Geometric invariant | Geometric (no $\sigma$) | ✓ Yes (theoretical) | ✓ Structural |
| *Signal-energy concentration* Step 35, single-contrast dominance | Structural information-theoretic | 99.9% signal energy in SX loopsS4–SX knockout removes 99.5%$D_{\rm eff} = 2.0$ | Information-theoretic (no $\sigma$) | ✓ Yes (structural) | ✓ Structural |

#### Table 2: Supplementary & Cross-System Checks

| Evidence strand | Test type | Result | $p$-value / significance | Independent? | Status |
| --- | --- | --- | --- | --- | --- |
| *TDCOSMO+Encore Shear* Predicted sensitivity check, $n=18$ pairs | Predicted-sensitivity consistency check | 16/18 pairs: predicted shift exceeds $1\sigma$Spearman $\rho = -0.733$ (tautological by construction)No independent information | Predicted shift magnitude vs. published uncertainty | ✓ Yes (different systems) | ✓ Predicted (structural consistency; not an observed detection) |
| *SN Encore blind-prediction residual* 8 blind models, 1 delay pair | Single-pair residual test | $\mathcal{R}_{\rm obs} = -3.33 \pm 2.13$ dPredicted: $-0.49$ d ($-1.04$ d at Refsdal $\kappa_{\rm lens}$)Sub-day shift swamped by scatter ($\sim$3–50 d)4/8 positive ($p = 0.64$) | Consistent with GR; directional sign matches TEP prediction | ✓ Yes (different system) | ✓ Observed (consistency check only) |
| *SN H0pe blind-prediction residual* 7 delay-blind models, 2 delay pairs | Single-pair residual test (AB, CB) | AB: $\mathcal{R}_{\rm obs} = -10.16 \pm 5.13$ d; predicted $-1.58$ d ($-3.32$ d at Refsdal $\kappa_{\rm lens}$)CB: $\mathcal{R}_{\rm obs} = -0.23 \pm 2.67$ d; predicted $-0.26$ d ($-0.54$ d at Refsdal $\kappa_{\rm lens}$)Sub-day; swamped by scatter | AB sign matches TEP prediction; CB near zero and not probative | ✓ Yes (different system) | ✓ Observed (consistency check only) |
| *Cross-system trio (Refsdal + Encore + H0pe)* Sign consistency only | Directional sign consistency across 3 independent systems | 3/3 systems: primary signs consistent with TEPExact binomial $p = 0.125$Stouffer combination: sensitivity exploration only | $p = 0.125$ (exact binomial) | ✓ Yes (different systems; caveat: kappa extrapolation) | ✓ Observed (consistency check) |
| *Non-independent consistency checks (same SX-dominated data; reported for transparency only)* |  |  |  |  |  |
| *Delay–$\mu$ correlation* Pearson $r=0.93$ (SX-leveraged), $n=5$ | Correlation test | Pearson $r=0.932$ (SX-leveraged); $r \approx 0.15$ without SXCook's distance $>1.0$Spearman $\rho=0.3$ ($p=0.31$)Theil-Sen $86 \pm 210$ d (overlaps zero)*Not probative: $n=5$, one extreme leverage point* | Not meaningful as formal test | ✗ No (SX-driven; correlated with strand 1) | ✓ Observed (reported for transparency) |
| *Per-model $\kappa_{\rm lens}$ inference* $\bar{\kappa}_{\rm inferred} \approx -0.114$ | Parameter inference | Weighted mean $\kappa_{\rm inferred} = -0.114 \pm 0.034$All 8 models $\kappa_{\rm lens} &lt; 0$ | $p=0.0006$ vs. zero (one-sided) | ✗ No (same ensemble) | ✓ Observed |
| *$\chi^2$ model comparison* GR vs. TEP ensemble fit | Goodness-of-fit | $\Delta\chi^2 = +8.4$ in favour of proxy model28% wRMS reduction after proxy correction (8/8 models) | No formal $p$-value: fixed predictions (0 free params), not nested. Individual GOF: GR $p=0.023$, proxy $p=0.32$ | ✗ No (same ensemble) | ✓ Observed |
| *Directional-odds Bayes factor* Sign data recast in odds form | Bayesian directional sign model | $\mathrm{BF}_{10}=56.8$ (all), 31.9 (delay-blind), 10.5 (family-collapsed)Odds support for one-sided sign excess | Odds support for one-sided sign excess | ✗ No (same sign data) | ✓ Observed (complementary to sign tests) |
| *Proxy-agnostic rank test* Step 34, ansatz-free | Rank correlation (delays vs potential depth) | Delay–kappa $\rho = -0.15$ ($p = 0.63$); delay–mu $\rho = -0.30$ ($p = 0.74$)Non-significant; $n=5$ too small to separate claims | Not meaningful as formal test | ✗ No (same SX data) | ✓ Observed (reported for transparency) |

Across the SN Refsdal model ensemble, the probative signature is the uniform sign of the SX delay residual across all seven non-zero delay-blind model variants spanning five independent modelling families (strict pre-reappearance floor: six of seven), none of which incorporated TEP when their predictions were published. Accounting for between-model covariance from shared lensing inputs, the primary correlation-aware headline is the exact family-sign-flip test at $p = 0.031$ ($z \approx 1.86\sigma$); the independence-assuming Wilcoxon signed-rank test gives $p = 0.0078$ ($\approx 2.4\sigma$; all-eight-model Wilcoxon $p = 0.0039$, $2.8\sigma$), with a beta-binomial break-even inter-model correlation of $\rho \approx 0.08$ for the delay-blind Wilcoxon to remain below $p = 0.05$ (step_11). The delay–magnification correlation ($r = 0.93$) and the proxy-agnostic rank test (Step 34, $n=5$) trace the same SX-dominated contrast and are therefore not counted as separate degrees of freedom. Extending the blind-residual sign check to SN Encore and SN H0pe yields $3/3$ systems with primary residual signs matching the TEP prediction ($p = 0.125$, exact binomial); because their modest magnification contrasts produce sub-day to $\lesssim 2$ d predicted shifts against $\sim 5$–$50$ d lens-model uncertainties, Encore and H0pe serve as low-contrast scaling checks rather than high-weight combiners. At current lens-model precision ($\sigma_{\rm model} \sim 16$–$60$ d), hierarchical Bayesian model comparison remains prior-dominated ($\mathrm{BF} \sim 1$); reducing lens-model uncertainties below $\sigma &lt; 5$ d moves the amplitude channel into the conclusive regime.

### 4.8 Implied $H_0$ Shift and Magnification-Contrast Scaling

Propagating the Refsdal-calibrated proxy coupling ($\kappa_{\rm lens} \approx -0.055$) into time-delay cosmography illustrates how magnification contrast governs the inferred expansion rate across SN Refsdal (Kelly et al. 2023), SN Encore (Pierel et al. 2026), and SN H0pe (TD-only; Pierel et al. 2024), which yield $H_0 \approx 61$–$67$ km s$^{-1}$ Mpc$^{-1}$ under standard GR.

Because the most magnified image arrives first in these geometries, an unmodelled temporal-shear delay causes the observed delay to exceed the geometric delay and biases a GR-inferred $H_0$ low. Applying $\kappa_{\rm lens} = -0.055$ shifts SN Refsdal upward by $+2.7$ km s$^{-1}$ Mpc$^{-1}$ ($66.6 \to 69.3$), within $0.5\sigma$ ($p \approx 0.62$) of Planck ($67.4 \pm 0.5$) under the combined lens-model ($\sigma \approx 3.7$ km s$^{-1}$ Mpc$^{-1}$) and CMB error budget. Because $\kappa_{\rm lens}$ is calibrated on the SN Refsdal SX delay itself (§4.4), the $+2.7$ km s$^{-1}$ Mpc$^{-1}$ shift is the direct mapping of that residual into $H_0$ space; the out-of-sample content is the differential contrast scaling across systems—a several-km s$^{-1}$ Mpc$^{-1}$ shift in high-contrast configurations versus $\sim 0.1$ km s$^{-1}$ Mpc$^{-1}$ in modest-contrast systems (SN Encore, SN H0pe)—which will be tested without circularity on high-magnification long-baseline systems such as SN 2025wny (§4.9).

Step 58 closes the associated channel bookkeeping for the corpus-level $H_0$ resolution (Paper 11 §4.3). Evaluated on the real TDCOSMO-2025 system properties, the isotropic-conformal and disformal transport sectors are bounded well below the residual: static isotropic-conformal rescaling cancels by null-cone invariance, direct scalar stress-energy is Planck-suppressed ($\kappa_\phi \approx 4\times10^{-5}$ on the unit-term control, Step 61; $6.5\times10^{-6}$ on the two-branch sector, Step 64), endpoint rescaling contributes $\sim 10^{-7}$, and GW170817-bounded disformal cone tilts ($B_0 \lesssim 78$) fall at least a factor $\simeq 5$ short on the deprojected GLAFIC cluster profile ($B_0^{\rm req}\simeq 420$ at $r_s\simeq 0.74$ Mpc, and $\sim 176\times$ short under the environment-modulated gate; Steps 62–63, Appendix A.1). While the canonical unstretched-space temporal metric supplies the bulk non-baryonic Fermat delay directly from baryons (§2.3, Step 64), a positive propagated delay residual relative to a fixed lens model biases the inferred $H_0$ low rather than high. Within standard CDM-parameterised reconstructions, the order-percent route that elevates the inferred $H_0$ runs through the velocity-dispersion prior that breaks the mass-sheet degeneracy: because slow-matter dynamics experiences the full temporal-lapse weight while optical refraction enters at half weight (§4.12), dynamical inference exceeds lensing inference wherever Temporal Shear is active ($M_{\rm dyn} = M_{\rm lens} + M_{\rm ph}$; §4.10.8), setting the $\sigma_*$-anchored normalization above the lensing value at the tracer positions and redistributing the excess along the mass-sheet direction at a magnitude bounded by the aperture phantom fraction (percent-level or below at galaxy-lens apertures where $a_{\rm ap} \gg g_t$; Paper 5 §5.3). The measured mundane floor is commensurate: the SDSS1206+4332 power-law versus composite distance posteriors differ by $\sim 6\%$, and the per-system $\kappa_{\rm ext}$ widths span $\sim 3$–$10\%$.

### 4.9 SN 2025wny: Prospective Record and Post-Measurement Update

SN 2025wny (discovered at $z_s = 2.011$, $z_l = 0.375$, Johansson et al. 2025, ApJ 995, L17; Taubenberger et al. 2025, arXiv:2510.21694) is the first resolved, multiply-imaged superluminous supernova (SLSN-I), initially identified in ground-based imaging as four images (A–D) in an Einstein cross pattern separated by ~1.7 arcsec with brightest-image magnification estimated at $\mu \sim 20$–50. Space-based follow-up with HST (PID 17611, October and November 2025) and JWST (PID 5564, November 2025; PI: Goobar) subsequently resolved five images (A–E) at $z_{\rm SN} = 2.0151$ lensed by two foreground galaxies at $z_{\rm G1} = 0.3755$ and $z_{\rm G2} = 0.3766$ ($\theta_{\rm E,1}\simeq 1.6''$, $\theta_{\rm E,2}\simeq 0.7$–$0.8''$, posterior total magnifications $\mu_{\rm tot}\sim 5$–$50$; Goobar et al. 2026, arXiv:2608.28413; Mörtsell et al. 2026, arXiv:2608.28430).

Multi-facility ground-based photometric monitoring (Maidanak, Lulin, COLIBRI, Wendelstein) and space-based HST/JWST imaging and IFU spectroscopy yielded precision time delays published on 28 August 2026: $\Delta t_{AB}=-10.6^{+2.2}_{-2.5}$ d and $\Delta t_{AC}=1.2^{+2.7}_{-2.6}$ d photometrically (Townsend et al. 2026, arXiv:2608.28427), $\Delta t_{AB}=-10.3\pm2.3$ d, $\Delta t_{AC}=0.1\pm3.6$ d, $\Delta t_{AD}=-65.7\pm3.5$ d, and $\Delta t_{AE}=3.7\pm8.8$ d spectroscopically (Johansson et al. 2026, arXiv:2608.28416), and combined delays $\Delta t_{AB}=-10.4\pm1.7$ d, $\Delta t_{AC}=0.8\pm2.1$ d, and $\Delta t_{AD}=-65.7\pm3.5$ d used in the space-data lens-model cosmography of Mörtsell et al. (2026, arXiv:2608.28430), which infers $H_0=66.7^{+7.6}_{-6.3}$ km s$^{-1}$ Mpc$^{-1}$ for fiducial isothermal profiles and $H_0=70.8^{+8.2}_{-6.1}$ km s$^{-1}$ Mpc$^{-1}$ when the galaxy density slopes are varied freely.

Prior to the August 2026 delay release, a post-hoc geometric model had predicted a long-baseline delay of ~175 days between the trailing image (A) and the leading image (D), with shorter A–C (~20 d) and A–B (~30 d) delays (Witt–Wynne model, arXiv:2605.11090). With the Step 07 calibrated coupling $\kappa_{\rm inferred}=-0.114\pm0.034$ (nominal $\kappa_{\rm lens}=-0.055$) and a magnification contrast $\mu_{\rm max}/\mu_{\rm min} \sim 10$–50, the Step 36 proxy calculation recorded the pre-release forecast on that ~175-day geometric baseline before the August 2026 delays were published; it is retained below as the historical prospective record alongside the post-measurement evaluation on the observed $65.7\pm3.5$ d baseline (Step 36b).

*Historical pre-measurement forecast (Step 36, evaluated on the ~175-day Witt–Wynne geometric baseline prior to August 2026).*

For a four-image Einstein-cross system with estimated magnification $\mu \sim 20$–50 and a long-baseline delay of ~175 days, the proxy model predicts a fractional TEP shift of order $2$–$6\%$ of the longest delay baseline. The predicted proxy-model residual (Step 36 Monte Carlo, $10^5$ draws over magnification, baseline, and $\kappa_{\rm lens}$ priors) is:

- *Predicted residual range:* $\mathcal{R}_{\rm pred} = +6.1$ to $+19.0$ days (16th–84th percentile; median $+11.5$ days; 95th percentile $+25.3$ days), assuming the post-hoc ~175-day A–D baseline, the estimated magnification contrast, and the propagated $\kappa_{\rm lens}$ uncertainty. *This prediction is post-hoc: it uses a geometric model published after the SN discovery and is therefore not a blind prediction. A genuine blind-prediction residual test requires lens-model predictions published before time-delay measurements.*

- *Required lens-model precision for a $3\sigma$ test:* per-model blind-prediction uncertainty $\sigma_{\rm model} \lesssim 3.8$ days on the longest pairwise delay, assuming the residual falls near the median prediction.

- *Required lens-model precision for a $5\sigma$ test:* $\sigma_{\rm model} \lesssim 2.3$ days.

*Falsification condition.* Across kinematics-calibrated (mass-sheet-broken) long-baseline lensed supernovae with comparable magnification contrast, microlensing-controlled flux ratios, and blind lens-model precision $\sigma_{\rm model} \lesssim 3.8$ d, an observed-minus-GR residual systematically consistent with zero at the $\lesssim 1$ d level would falsify the operational linear log-magnification proxy ($\Gamma_t^{\rm proxy}$) in the strong-lensing domain (while testing the action-level unstretched-space temporal refraction of Step 64 requires direct convergence/potential reconstruction rather than the flux-ratio proxy). Conversely, a residual consistent with the predicted $+6.1$ to $+19.0$ day 68% range would constitute independent geometric evidence for potential-dependent temporal propagation. These thresholds are computed deterministically by the pipeline (Step 36); they will not be adjusted post-measurement.

*Post-measurement update (Step 36b).* The Step 36 forecast is retained unchanged as the prospective record. The published delays replace its ~175-day premise with a measured A–D baseline of $65.7\pm3.5$ d. Re-evaluating the same Monte Carlo on that baseline, with all other priors and the Step 07 coupling unchanged ($\kappa=-0.118\pm0.034$, the bootstrap-mean estimator; the inverse-variance weighted-mean diagnostic quoted above is $-0.114\pm0.034$; `step_36b_sn2025wny_measured_baseline_update.py`), gives $\mathcal{R}_{\rm pred}=+2.3$ to $+7.1$ d (16th–84th percentile; median $+4.4$ d). The current lens-model precision on that baseline, from the published $H_0$ posterior width ($\approx10\%$), is $\sigma_{\rm model}\approx6.8$ d, while a $3\sigma$ test at the median signal requires $\sigma_{\rm model}\lesssim1.5$ d. Two structural comparisons are already informative at the published baseline: (i) cosmographically, because $H_0^{\rm inferred}\propto 1/\Delta t_{\rm obs}$, a positive TEP delay residual biases a standard GR lens model toward a lower inferred $H_0$ — Mörtsell et al. (2026) obtain $H_0^{\rm GR}=66.7^{+7.6}_{-6.3}$ km s$^{-1}$ Mpc$^{-1}$ on their fiducial isothermal model (rising to $70.8^{+8.2}_{-6.1}$ km s$^{-1}$ Mpc$^{-1}$ when the density slopes of the two lens galaxies are allowed to vary freely, directly illustrating the radial-slope/mass-sheet sensitivity of §4.8), sitting below the local Cepheid ladder ($73.0$) and within $0.21\sigma$ of TEP's zero-parameter cosmic root $H_0=68.17$ km s$^{-1}$ Mpc$^{-1}$ (Paper 18 Step 21), while correcting the $65.7$ d baseline for the median $+4.4$ d proxy shift raises the fiducial isothermal rate to $71.5$ km s$^{-1}$ Mpc$^{-1}$ ($69.1$–$74.8$ across the 16th–84th percentile band), matching the SN Refsdal shift direction ($66.6\to69.3$ km s$^{-1}$ Mpc$^{-1}$); and (ii) at the action level, whereas SN Refsdal's cluster-scale loop samples the low-acceleration regime ($R\sim55$–$82$ kpc, $q\ll1$), SN 2025wny is a galaxy-scale lens system (two deflectors with $\theta_{\rm E,1}\simeq 1.6''$ and $\theta_{\rm E,2}\simeq 0.7$–$0.8''$; evaluated on the ground-based single-ring benchmark $\theta_E\approx0.85''$, physical radius $R_E=4.50$ kpc, enclosed lensing mass $M_E=1.35\times10^{11}\,M_\odot$) where the baryonic acceleration ratio $q(R_E)=g_b(R_E)/g_t=0.70$–$0.98$ (for early-type lens baryon fractions $f_b(&lt;R_E)=0.25$–$0.35$) places the ring in the partially screened transition regime of the zero-parameter two-branch Gauss law $(k/\sqrt2)q y^2+q^2 y^3=1$ ($k=16.03$), predicting $y_{\rm tb}(R_E)=0.30$–$0.35$, $M_{\rm tot}(&lt;R_E)/M_b(&lt;R_E)=1+2y_{\rm tb}(R_E)=1.59$–$1.70$, and an apparent dark-matter fraction inside $R_E$ of $f_{\rm phantom}(&lt;R_E)=37\%$–$41\%$ (matching typical SLACS/HOLISMOKES early-type galaxy lenses with zero per-system free parameters). SN 2025wny therefore stands as an active external post-hoc test: the predicted delay shift is compatible with but smaller than the current $\approx 6.8$ d lens-model uncertainty on the A–D baseline, and spatially resolved lens-galaxy kinematics together with multi-team blind/delay-blind reconstructions of this system are the route to a decisive comparison.

### 4.10 Alternative Astrophysical Explanations

Before attributing the observed positive residual to the proxy model, it is necessary to consider whether conventional astrophysical effects could produce the same sign pattern across multiple independent modelling methods. Several alternative explanations are examined below.

#### 4.10.1 Unmodelled Cluster Substructure

Unmodelled dark matter subhalos or line-of-sight mass concentrations could in principle introduce systematic errors in lens model predictions. However, such substructure would affect the predicted geometric delay in a way that depends on the specific mass distribution adopted by each modelling code. The fact that five independent modelling methods (GLAFIC, LTM, WSLAP+, GLEE, LENSTOOL) spanning both parametric and free-form approaches all underestimate the delay by the same sign argues against a substructure-driven bias. Substructure effects would be expected to scatter predictions both above and below the true value, depending on whether the substructure is included or missed in a given model. The systematic positive sign across all methods is inconsistent with random substructure scatter.

#### 4.10.2 Line-of-Sight Mass Convergence

Mass structures along the line of sight to the cluster can introduce external convergence ($\kappa_{\rm ext}$) that rescales all time delays by a common factor $(1-\kappa_{\rm ext})$. This is the external component of the Mass Sheet Degeneracy. Importantly, a uniform external convergence would rescale all pairwise delays by the same factor, leaving the algebraic loop sum unchanged at zero. The observed non-zero residual cannot be generated by a uniform line-of-sight convergence sheet. Non-uniform line-of-sight structure could in principle affect different images differently, but such structure would need to be precisely aligned with the image positions to produce the observed sign pattern, and would again be expected to scatter differently across modelling methods that incorporate line-of-sight information to varying degrees.

#### 4.10.3 Microlensing-Induced Flux Bias

Microlensing by stars in cluster member galaxies can perturb observed fluxes, potentially biasing the flux-ratio-based magnification proxies used to infer $\Gamma_t$ factors. This systematic is addressed in §2.4.1 and tested via the microlensing-nuisance Monte Carlo in §3.6.5. The key point is that microlensing perturbations at the tens-of-percent level can shift the inferred amplitude of $\Delta\Gamma$ but do not naturally invert the rank-ordering $\mu_{\rm SX} &lt; \mu_{\rm S4}$ that sets the sign of the predicted SX residual. The sign-based evidence tests (Wilcoxon, binomial) are therefore robust to moderate microlensing corrections. Furthermore, microlensing would be expected to affect different images with different time-varying patterns, whereas the observed residual is stable across the seven independent modelling teams that used different photometric epochs and analysis methods.

#### 4.10.4 Source-Size Effects

The finite size of the supernova host galaxy or the supernova light curve itself could in principle affect the inferred time delays if the lens models assume point-source approximations. However, Kelly et al. (2023) explicitly account for source-size effects in their light-curve fitting, and the SX image is well-separated from the Einstein cross, minimising cross-contamination. More importantly, source-size effects would be expected to systematically bias delays in one direction for all modelling teams, but the magnitude of the bias would depend on the specific source-size assumptions in each code. The consistency of the positive sign across methods with heterogeneous source-size treatments argues against this explanation.

#### 4.10.5 Photometric Systematics

Systematic errors in the measured SN fluxes or in the model-predicted flux ratios could bias the magnification proxy estimates. However, the primary evidence test (§3.5) does not rely on flux ratios at all—it compares the observed delay directly to blind model predictions made before the measurement existed. The flux-ratio-based evidence (delay–magnification correlation, per-model coupling inference) is treated as supplementary and explicitly caveated. The core sign-based evidence is independent of photometric systematics in the flux ratios.

Taken together, these alternative explanations are either ruled out by the geometric structure of the blind-prediction residual test (uniform convergence cannot produce a differential residual), inconsistent with the multi-method sign pattern (substructure, line-of-sight), or robust to the specific test design (microlensing, photometric systematics). The TEP interpretation provides a compact directional account of the residual pattern, but present data do not exclude correlated lens-model bias as a conventional explanation.

#### 4.10.6 Correlated Lens-Model Systematics

A referee's concern is that the blind lens models may not be statistically independent: they share common assumptions about the cluster mass distribution, the same HST imaging, and similar parametric halo prescriptions. If the models are correlated, the effective sample size is smaller than the nominal $N=8$ (or $N=7$ delay-blind), and the significance of the sign tests drops. This section quantifies that dependence with a hierarchy of three tests, from exact theory to operational execution.

*Test hierarchy under dependence.*

- *Exact family-sign-flip test (delay-blind headline).* Because the Wilcoxon statistic lacks a closed-form variance under exchangeable intra-class correlation, the most rigorous dependence-aware rank bound is an exact sign-flip enumeration over method-family clusters, preserving the empirical cluster structure. For the delay-blind subset (seven revised values fixed before the measured delay was known), this gives $p = 0.031$ (one-sided). It is exact under the sharp null and makes no superpopulation assumption. This is the operational *correlation-aware primary rank test*.

- *Exact family-sign-flip test (strictly blind floor).* The same test computed on the original pre-reappearance published predictions gives $p = 0.094$ (one-sided, 6/7 positive, 4/5 families). This is the floor: even if every model family had been fully independent, the sign pattern would not reach conventional significance on the strictly blind layer alone. The probative value of the headline therefore rests on the honesty of the delay-blind provenance claim — that the revised values were fixed before the 376-day delay was known.

- *Method-family block-bootstrap.* Resamples method families with replacement and computes the Wilcoxon statistic on each bootstrap draw, yielding a distribution of dependence-adjusted $p$-values. It respects the empirical cluster structure without assuming a parametric correlation model, but can occasionally reconstruct more extreme statistics than independent sampling when clusters concentrate rank mass. It is reported as a *sensitivity exploration* rather than the operational primary. The delay-blind subset yields $p_{\rm median} = 0.0078$ [$0.0039$, $0.0156$]; the strictly blind subset yields $p_{\rm median} = 0.0469$ [$0.0039$, $0.2344$].

- *Binomial sign test under beta-binomial correlation.* The binary sign of each residual is treated as an exchangeable outcome with intra-class correlation coefficient $\rho$. For conservative diagnostics that count any exact zero residual as non-positive, the all-eight sign count is 8/8 positive ($p=0.0039$ under independence), the delay-blind count is 7/7 positive ($p=0.0078$), and the strictly blind count is 6/7 positive ($p=0.0625$). These p-values rise above, or already sit above, the conventional threshold under modest inter-model correlation. The beta-binomial sign test is therefore reported only as the most *conservative correlation-aware sign diagnostic*, not as the headline sign result.

The exact family-sign-flip test ($p = 0.031$ on the delay-blind tier; $p = 0.094$ on the strictly blind floor) is the most rigorous dependence-aware rank bound because it is exact under the sharp null and requires no superpopulation assumption. The block-bootstrap ($p_{\rm median} = 0.0078$ [$0.0039$, $0.0156$] delay-blind; $p_{\rm median} = 0.0469$ [$0.0039$, $0.2344$] strictly blind) is reported as a sensitivity exploration: it can occasionally fall below the independent $p$-value when the empirical cluster structure concentrates rank mass, so it is not used as the operational primary. The beta-binomial sign diagnostic is deliberately conservative: with the present data (all signs positive), it gives $p=0.0078$ for the delay-blind count at $\rho=0$, rising above $0.05$ only if inter-model correlation exceeds the break-even threshold. Present data do not discriminate whether the true inter-model correlation exceeds the threshold at which the evidence softens to $p > 0.05$. Lens-modelling challenges for clusters such as H0LiCOW and TDCOSMO have demonstrated that different codes can disagree at the $\sim 10$–$20\%$ level, suggesting some but not total independence. The TEP interpretation offers a single compact parameter ($\kappa_{\rm lens} \approx -0.055$) that simultaneously explains the sign, magnitude, and directional odds of the SX residual, but a conventional correlated-lens-model bias cannot be excluded without either (i) a larger ensemble of truly independent blind models, or (ii) a direct test of the symmetric-scatter assumption (e.g., comparing blind and post-blind predictions for the same system).

#### 4.10.7 Discriminating Temporal Shear from Generic Modelling Bias (Steps 41–42)

Two dedicated tests separate the TEP interpretation from the leading conventional explanations of the same-sign residual pattern.

*Cross-system amplitude discriminator (Steps 41 and 57).* If the residuals reflect potential-dependent temporal shear, their *magnitudes* should scale with the proxy-model sensitivity (large for SN Refsdal's S4–SX contrast, sub-day for Encore and H0pe). If instead they reflect a generic $H_0$/mass-sheet-like fractional bias, the magnitudes should scale only with the delay baseline. Step 41 now takes each unit sensitivity $d\mathcal{R}/d\kappa$ directly from its upstream analysis; the former division of every system by the Refsdal nominal coefficient was a normalization error. With the corrected sensitivities, the uniform fractional model is modestly preferred to the TEP proxy by $\Delta\mathrm{AIC}=1.12$. Step 57 removes the remaining circularity by fitting the proxy coefficient only to Encore and one H0pe contrast, combining model predictions before adding the common observed-delay uncertainty, and holding Refsdal out. This gives $\kappa_{\rm lens}=-0.442\pm0.282$ and a Refsdal prediction of $+116.8\pm74.5$ d, compared with the covariance-corrected $+29.6\pm10.2$ d residual: a $1.16\sigma$ difference, but a central prediction $3.9\times$ too large. The analogous independent-system AIC comparison again modestly favours a uniform fractional response ($\Delta\mathrm{AIC}=1.23$). Current amplitudes are therefore compatible with the proxy but do not discriminate it from a baseline-scaling systematic. What remains non-trivial is the *sign*: all four Step 41 contrasts align with the potential-depth ordering (4/4; one-sided binomial $p = 0.06$), a pattern not specified by the unsigned fractional-amplitude model. The directional sign, rather than a calibrated amplitude, carries the present evidence.

*Precision-persistence (Step 42).* The leading conventional account of the SN Refsdal SX under-prediction is that the blind models were simply imprecise, and a sufficiently accurate GR model would converge to the observed delay with zero residual. Regressing the per-model residual on its quoted uncertainty $\sigma_{\rm model}$ and extrapolating to the high-precision limit ($\sigma_{\rm model}\to 0$) gives a residual intercept of $-8.2 \pm 18.3$ d (all eight models, one-sided $p = 0.67$) or $-15.9 \pm 21.6$ d (blind subset, $p = 0.77$). The intercept is consistent with (or below) zero, so a high-precision GR model could plausibly remove the residual — immaturity is not excluded. The most precise all-eight model (Grillo+2024, $\sigma = 16$ d) retains a $+14$ d residual, while the most precise blind model (Oguri-a*, $\sigma = 19$ d) retains a $+22$ d residual. The residual does not clearly shrink as precision improves, but the regression does not support a non-vanishing intercept; a key test requires a larger ensemble of independent high-precision blind models.

Together these tests sharpen rather than inflate the evidence: they confine the probative content to the directional sign channel and establish the amplitude as non-probative, consistent with §4.4 and §4.7.

#### 4.10.8 Phantom-Mass Bookkeeping Runs Both Ways (Step 60)

The delay-residual bookkeeping is evaluated in both directions. Step 60 defines the fractional delay residual $\lambda=\mathcal R_{\rm obs}/\langle\Delta t_{\rm pred}\rangle$: positive values mean observed delays exceed the model prediction, negative values the reverse. Interpreting this ratio as a phantom-mass fraction requires the dynamical-input and mass-sheet transfer described below; the ratio alone measures neither scalar backreaction nor a dynamical–lensing mass difference. Step 60 retains every available contrast, including negative and null entries.

The proposed sign chain runs through the models' dynamical inputs, not the photon paths; its realization depends on the fitted lens model. Cluster lens reconstructions are anchored in part by dynamical tracers — the BCG stellar velocity dispersion and cluster-member kinematics — which in TEP read the dynamical mass $M_{\rm dyn} = M_{\rm lens} + M_{\rm ph}$ wherever the Temporal Shear is active. Inflated dynamical inputs set the fitted mass above the lensing value at the tracer positions; to preserve the image-plane solution the model redistributes the deficit along the mass-sheet direction, lowering the effective delay normalization. Under this redistribution the observed delay exceeds the prediction, giving $\lambda>0$. The measured sign is consistent with that construction but does not establish a universal sign or amplitude independently of the reconstruction.

The delay-blind contrasts give $\lambda = +8.7 \pm 2.6\%$ (SN Refsdal), $+9.1 \pm 5.9\%$ (SN Encore), $+9.5 \pm 4.8\%$ (SN H0pe AB), and $+0.5 \pm 5.5\%$ (SN H0pe CB). The last is the counter-direction datum: consistent with zero, it bounds the phantom fraction to $|\lambda| \lesssim 11\%$ at 2$\sigma$ for that short-baseline, low-contrast geometry — where the differential proxy independently predicts the smallest shift, so the null is the mechanism's own expectation rather than a refutation of it. The archived diagonal-weight pooling of the four fractional delay residuals gives $+7.8 \pm 2.0\%$ with $\chi^2 = 2.06$ on 3 dof ($p = 0.56$). These are descriptive values under that approximation, not a calibrated phantom-mass detection: the two H0pe contrasts share an image, and joint delay/model covariance and denominator uncertainty must enter a calibrated combined significance. Under the archived convention the residuals are compatible with a common order-percent delay offset, and no contrast forces $|\lambda|$ beyond the mundane mass-model floor measured in §4.8 (median per-system $\kappa_{\rm ext}$ half-width $3.0\%$, SDSS1206+4332 power-law-versus-composite distance spread $\sim 6\%$).

The negative-direction census counts the cases that run against the mechanism rather than suppressing them: 12 of 36 per-model phantom fractions are negative, of which 5 exceed 1$\sigma$ — dominated by the free-form WSLAP+ models, which over-predict the Encore and H0pe delays, and by the strictly-blind Jauzac15.1 variant. Every ensemble-level weighted mean nevertheless remains positive, so the counter-direction content is real but sub-dominant at present precision: the phantom-mass hypothesis is simultaneously constrained from above (by the systems that agree) and supported in the favourable direction (by those that do not), which is precisely the two-sided bookkeeping a falsifiable mechanism requires.

### 4.12 The Final Proxy: Status in the TEP Framework

The strong-lensing magnification response $\Gamma_t(i) = 1 + \kappa_{\rm lens}\log_{10}(\mu_{\rm norm}(i))$ tested here is an operational domain-specific proxy for the dynamical–lensing mass-inference channel alongside the action-level unstretched-space temporal refraction derived in §2.3 (Step 64). Furthermore, on galaxy-scale strong lenses, because the local temporal lapse enters slow-matter dynamics at full weight ($\Phi_{\rm dyn}=\Phi_N+c^2\beta_A u_{\rm grav}$) and optical refraction at half weight ($\Phi_{\rm lens}=\Phi_N+\tfrac12 c^2\beta_A u_{\rm grav}$), the effective post-Newtonian slip at the Einstein radius is $\gamma_{\rm SMP}=\Psi_{\rm sp}/\Phi_{\rm dyn}=1/(1+2\beta_A^2 y_{\rm eff})&lt;1$ wherever the local scalar gradient is unscreened (Paper 0, Step 80). Evaluated across the SLACS massive early-type galaxy ensemble ($\sigma_v=220$–$300\text{ km s}^{-1}$, $R_{\rm Ein}=3.5$–$4.8\text{ kpc}$; Bolton et al. 2008; Cao et al. 2017; Shajib et al. 2021) and ESO 325-G004 ($\sigma_v=330\text{ km s}^{-1}$, $R_{\rm Ein}=2.85\text{ kpc}$; Collett et al. 2018), the idealized zero-ambient vacuum baseline ($X_{\rm group}=0$, $\gamma_{\rm SMP}=0.66$–$0.77$) is superseded in the realized nested cosmic hierarchy (Paper 0, Section 7): because massive early-type lenses ($M_*&gt;10^{11}\,M_\odot$) reside in group/filamentary host halos ($M_h\simeq3\times10^{13}\,M_\odot$, $r_*\simeq110\text{ kpc}$, $X_{\rm group}\simeq131$; Treu et al. 2009; Newman et al. 2015), nested ambient screening suppresses the incremental scalar response to $y_{\rm eff}\simeq0.007$–$0.014$, predicting $\gamma_{\rm SMP}^{\rm group}=0.972$–$0.981$ (and $0.986$–$0.997$ for ESO 325-G004 across $M_h=3\times10^{13}$–$3\times10^{14}\,M_\odot$), matching the SLACS ensemble mean $\langle\gamma_{\rm SMP}\rangle_{\rm SLACS}=0.98\pm0.07$ to $0.01\sigma$–$0.11\sigma$ and Collett et al.'s $0.97\pm0.09$ to $0.18\sigma$. Stratifying galaxy-scale lens-dynamics samples (SLACS, BELLS, Euclid/LSST) by host-group halo environment therefore provides a direct, sign-definite test of the unstretched-space metric.

### 4.11 Precision Requirements for Decisive Evidence

The headline single-test significance from SN Refsdal is $z = 1.86\sigma$ ($p = 0.031$; exact family-sign-flip, correlation-aware). The independence-assuming Wilcoxon benchmark is $z \approx 2.4\sigma$ ($p = 0.0078$; delay-blind subset). The limiting factor is not the size of the proxy-model signal — the 14.5-day predicted shift is far above the 5.6-day measurement precision — but rather the large uncertainties in current GR lens models ($\sigma_{\rm model} \approx 16$–$60$ d). Because the blind-prediction residual test compares the observed delay to the model-predicted geometric delay, the significance of any measured residual scales directly with model precision. The analysis therefore reports the correlation-aware exact family-sign-flip result as the headline and the Wilcoxon as the independence benchmark (see §4.7 and §4.10.6 for the treatment of correlated evidence).

The precision required to overcome this limitation can be quantified by simulating the ensemble significance for $N=8$ independent models as a function of the average per-model uncertainty $\sigma_{\rm model}$, assuming the true proxy-model signal is $\mathcal{R}_{\rm TEP/GR} = 14.5$ d.

The simulation shows conditional detection thresholds: if the average model uncertainty were to drop below $\sigma_{\rm model} = 13.7$ d, the same 14.5-day mean residual would cross the $3\sigma$ "evidence" threshold. At $\sigma_{\rm model} = 8.2$ d, the same residual would constitute a $5\sigma$ "discovery" of potential-dependent temporal shear. These are precision benchmarks, not predictions; they indicate what lens-model precision would be needed for the existing SN Refsdal data to become decisive, independent of any new observations.

These thresholds are not merely theoretical. Recent extended-source lens modeling of SN Refsdal's host galaxy, incorporating 77,000 HST pixels in addition to the 106 point-like multiple images, achieves *sub-percent statistical uncertainty* on the predicted $\Delta t_{\rm SX:S1}$ time delay — more than an order of magnitude smaller than the ~5% uncertainty of the best point-like models (Schuldt et al. 2026, *A&A*, aa57680-25, arXiv:2602.12329). This demonstrates that the $\sigma_{\rm model} &lt; 8.2$ d threshold required for a $5\sigma$ TEP discovery is achievable with existing HST data and established extended-image modeling techniques. The caveat is that these particular models are post-observation and therefore not blind; a genuine blind-prediction residual test would require comparable precision from models published before the time-delay measurement. Nevertheless, the precision roadmap is now an engineering question, not a fundamental limitation.

## 5. Conclusion

A geometric blind-prediction residual test for the Temporal Equivalence Principle (TEP) has been applied to SN Refsdal (MACS J1149.6+2223), currently unique among cluster-lensed supernovae in combining five resolved images, a 376-day long-baseline delay, and an ensemble of pre-reappearance blind lens-model predictions. The key results are:

- The algebraic loop sum is identically zero under any theory with globally assignable arrival times per image, including the TEP ansatz. The probative observable is the blind-prediction residual (§1.1).

- The inner Einstein cross is a non-probative region under the adopted proxy (shear degeneracy collapses the magnification–convergence rank order; §4.1.1). All probative content resides in the S4–SX contrast.

- The two loops incorporating image SX—which arrives 376 days after S1—yield post-hoc proxy sensitivities of $-8.5$ days (S1–S2–SX) and $-14.5$ days (S1–S4–SX closure; $+14.5$ d obs−model) at nominal $\kappa_{\rm lens} = -0.055$. These define the nominal S4–SX response normalization used to compare sign and scale across the delay-blind ensemble. The 376-day baseline is the high-leverage contrast for the sign test.

- The algebraic loop sum is insensitive to the Mass Sheet Degeneracy: a uniform convergence sheet rescales all delays by the same factor, leaving the loop sum unchanged at zero under both GR and TEP. The blind-prediction residual is MSD-insensitive in the algebraic-loop sense, but remains lens-model-limited in practice because it compares observed delays to GR predictions that themselves carry mass-sheet and model-calibration uncertainties.

- The correlation-aware headline is the exact family-sign-flip test on method-family signs ($p=0.031$, $z \approx 1.86\sigma$), supported by independence-assuming Wilcoxon benchmarks ($p=0.0078$ delay-blind, $p=0.0039$ all eight). The strictly blind floor — original pre-reappearance predictions only, excluding the five post-reappearance revisions — gives $p = 0.094$ (6/7 positive, 4/5 families); the headline significance therefore rests on the delay-blind provenance of the revised values. A complementary directional-odds analysis gives one-sided Bayes factors ($\mathrm{BF}_{10}=56.8$ all non-zero; 31.9 delay-blind; 10.5 method-family-collapsed). Robustness checks preserve the directional pattern under model-dependence stress tests and microlensing-style flux perturbations, while hierarchical Bayesian comparison remains inconclusive (BF $\sim 1$), indicating that present lens-model uncertainties dominate formal model-selection metrics.

- *Cross-system directional consistency.* Blind-prediction residual tests for SN Encore and SN H0pe are directionally consistent with the proxy model: all three systems show the primary expected sign (3/3, exact binomial $p=0.125$). A normalization bug in the original four-contrast comparison has been corrected: each response is now divided by the coefficient actually used upstream. The uniform fractional model remains modestly preferred over the proxy ($\Delta$AIC $=1.12$). In the stricter one-contrast-per-system hold-out test with shared observational covariance, the corresponding difference is $\Delta$AIC $=1.23$.

- *Internal consistency check — $H_0$ (not independent evidence).* This is a calibrated response-scale estimate rather than an independent blind prediction: because $\kappa_{\rm lens}$ was calibrated from the same SN Refsdal SX delay data, applying it back to that same system cannot constitute an independent confirmation. Under the proxy model, the inferred $H_0$ shifts from $66.6$ to $69.3$ km s$^{-1}$ Mpc$^{-1}$ — a calibrated consequence of the empirical coupling, not corroborating evidence. SN Encore and SN H0pe receive only $\sim 0.1$ km s$^{-1}$ Mpc$^{-1}$ corrections (kappa calibrated on Refsdal and applied without refitting), too small to be probative. This $H_0$ shift is therefore reported only as an internal consistency consequence of the calibrated response scale, not as independent cosmological evidence for TEP.

- Across kinematics-calibrated (mass-sheet-broken) long-baseline multiply-imaged supernovae with comparable magnification contrast, microlensing control, and blind GR model uncertainty $\sigma_{\rm model} \lesssim 3.8$ d, an observed-minus-GR residual systematically consistent with zero at the $\lesssim 1$ d level would falsify the operational linear log-magnification proxy in the strong-lensing domain. Conversely, a residual matching the proxy-model prediction would constitute direct geometric evidence for potential-dependent temporal propagation once lens-model uncertainties are reduced sufficiently for high significance.

- *Scalar-backreaction calculation and unstretched-space temporal refraction.* Step 56 proves that the static isotropic-conformal $\psi$ integral is the GR Fermat/Shapiro term and cannot be added again as a TEP residual ($\Delta t^{\rm residual}_{\rm static,conf}=0$ under $\tilde g_{\mu\nu}=A^2g_{\mu\nu}$). Step 62 corrects the disformal bound transfer ($B_0\lesssim78$), and Step 63 performs the NFW+galaxy deprojection ($B_0^{\rm req}\simeq420$ at $r_s\simeq0.74$ Mpc). Step 64 then evaluates the selected two-branch kinetic sector $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$ fixed by $a_0$, zero per-system free parameters): while isotropic conformal rescaling leaves only the Planck-suppressed scalar stress tensor ($\kappa_\phi\simeq6.5\times10^{-6}$), the unstretched-space temporal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) retains intermediate clock rates $A(\phi(x))$ along the ray, producing a $+330.9$ d 2-D sheet Fermat delay and $+221.6$ to $+443.2$ d 3-D deprojected NFW ray delay from $f_b=0.16$ baryons alone in the exact arrival order $\text{SX}>\text{S1}>\text{S2}>\text{S3}>\text{S4}$. These Step-64 values are absolute TEP long-baseline delays in the baryon-only reconstruction (demonstrating that the baryon-sourced temporal channel reaches the observed $\sim 376$ d SX–S1 delay scale and $\kappa(\text{SX})\simeq 0.20$ convergence with the correct arrival ordering), not additive corrections to the published GR delays; the $+30.1$ d ($8.7\%$) quantity is a separate residual diagnostic between observations and the published GR forecasts.

- The multiply-imaged SLSN-I SN 2025wny ($z_{\rm SN} = 2.0151$, five images A–E resolved by HST/JWST across two lens galaxies at $z = 0.3755$ and $0.3766$; Johansson et al. 2025; Goobar et al. 2026, arXiv:2608.28413) has measured delays published in August 2026 ($\Delta t_{AD}=-65.7\pm3.5$ d; arXiv:2608.28427, 2608.28416) and a space-data lens model ($H_0=66.7^{+7.6}_{-6.3}$ km s$^{-1}$ Mpc$^{-1}$ isothermal, $70.8^{+8.2}_{-6.1}$ km s$^{-1}$ Mpc$^{-1}$ free-slope; arXiv:2608.28430, with the isothermal value sitting within $0.21\sigma$ of TEP's zero-parameter cosmic root $H_0=68.17$ km s$^{-1}$ Mpc$^{-1}$ and shifting to $71.5$ km s$^{-1}$ Mpc$^{-1}$ under the median $+4.4$ d proxy delay correction, while the zero-parameter two-branch Gauss law on the $R_E=4.50$ kpc ring benchmark predicts $M_{\rm tot}(&lt;R_E)/M_b(&lt;R_E)=1.59$–$1.70$, i.e. a $37\%$–$41\%$ apparent dark-matter fraction inside $R_E$). On the measured $65.7\pm3.5$ d A–D baseline the proxy model predicts $+2.3$ to $+7.1$ d (Step 36b), comparable to the current $\approx6.8$-day lens-model precision: an external post-hoc test that sharper kinematics-calibrated lens models of this system can decide.

*Preregistration commitment.* A formal commitment is made to uploading the precise flux-to-potential transfer function — derived from the TEP scalar-field action through the strong-lensing Jacobian — to an open repository before the time-delay data of the next unseen long-baseline multiply-imaged supernova are unblinded, so that test will be a genuine blind prediction with the theoretical forecast locked before the observational data become available. For SN 2025wny, whose delays were published in August 2026 before the transfer kernel was externally timestamped, the comparison is reported transparently as a post-measurement evaluation (Step 36b) alongside the unchanged Step 36 historical forecast.

SN Refsdal provides the strongest current directional evidence for lensing-sector temporal-shear phenomenology: a pre-observation GR prediction ensemble, a later independent SX delay measurement, and a coherent positive residual sign pattern. The headline remains sign coherence ($p=0.031$). The non-circular hold-out amplitude test is statistically compatible but presently underpowered, while the static isotropic-conformal path-integral shortcut is excluded by exact null-curve invariance under $\tilde g_{\mu\nu}=A^2 g_{\mu\nu}$ and replaced by the unstretched-space temporal refraction closure of Step 64. The key theoretical task is to lock the coupled unstretched-space and lens-dynamics transfer function for prospective blind testing.

Quantitative analysis confirms this is a uniquely high-leverage single-system test: the probative content is concentrated in the S4–SX temporal lever arm (§4.1.1), with an effective dimensionality $D_{\rm eff} \approx 2.0$. The correlation-aware headline gives $p=0.031$ ($z \approx 1.86\sigma$), supported by independence-assuming Wilcoxon benchmarks; the strictly blind floor gives $p = 0.094$. All seven delay-blind-comparison residuals match the predicted sign, while the strict original-blind layer remains positive at six of seven. If the central Refsdal residual remains near the corrected-compilation value, reducing independent lens-model scatter below $\sigma_{\rm model} &lt; 5$ d would move the amplitude test into a conclusive regime. Two additional systems (SN Encore, SN H0pe) show directional consistency with the proxy model. Statistical power scales with both model precision and time-delay baseline; refined kinematics-calibrated models of the five-image SN 2025wny system and additional long-baseline multiply-imaged supernovae with high magnification contrast will extend the test to high significance.

Together with the zero-per-system-parameter unstretched-space temporal refraction solve (item 9 above; §2.3, Step 64), which demonstrates that the baryon-sourced temporal channel reaches the observed long-baseline Fermat delay scale and convergence of SN Refsdal with the exact image arrival order while preserving $c_{\rm GW}=c_\gamma$ identically, these results establish a self-consistent optical and timing framework ready for prospective blind testing on the next long-baseline multiply-imaged supernova.

## References

#### Strong Lensing Observations

Kelly, P. L., Rodney, S. A., Treu, T., et al. 2015, *Science*, 347, 1123, "Multiple images of a highly magnified supernova formed by an early-type cluster galaxy lens", doi:10.1126/science.aaa3358

Kelly, P. L., Rodney, S., Treu, T., et al. 2023, *ApJ*, 948, 93, "The SN Refsdal Expansion Rate Measurement: A Powerful New Cosmic Ruler", doi:10.3847/1538-4357/acbf4d

Kelly, P. L., Rodney, S., Treu, T., et al. 2023, *Science*, 380, abh1322, "A measurement of the Hubble constant from the multiply imaged supernova Refsdal", doi:10.1126/science.abh1322

Treu, T., Brammer, G., Diego, J. M., et al. 2016, *ApJ*, 817, 60, "The Refsdal Revelation: Predictions for the Return of the First Multiply Imaged Supernova", doi:10.3847/0004-637X/817/1/60

Grillo, C., Rosati, P., Suyu, S. H., et al. 2024, *ApJ*, 971, 49, "A High-precision Lens Model for MACS J1149.6+2223: Predicting the Reappearance of SN Refsdal", doi:10.3847/1538-4357/ad5a7d

Schuldt, S., et al. 2026, arXiv:2602.12329, "A boost in the precision of cluster-mass models: Exploiting the extended surface brightness of the lensed supernova Refsdal host galaxy" (arXiv preprint; not yet peer-reviewed)

Frye, B. L., Pascale, M., Pierel, J., et al. 2024, *ApJ*, 961, 171, "The JWST Discovery of the Triply Imaged Type Ia "Supernova H0pe" and Observations of the Galaxy Cluster PLCK G165.7+67.0", doi:10.3847/1538-4357/ad1f1d

Pascale, M., Pierel, J. D. R., Frye, B. L., et al. 2025, *ApJ*, 979, 17, "SN H0pe: The First Measurement of H0 from a Multiply Imaged Type Ia Supernova Discovered by JWST", doi:10.3847/1538-4357/ad9c6e

Grayling, M., Thorp, S., Mandel, K. S., et al. 2025, arXiv:2510.11719, "BayeSN-TD: Time Delay and H0 Estimation for Lensed SN H0pe" (arXiv preprint)

Pierel, J. D. R., Suyu, S. H., Acebron, A., et al. 2026, arXiv:2509.12301, "SN Encore: Time Delay Cosmography of a Lensed Type Ia Supernova in MACS J0138.2-2155" (arXiv preprint; not yet peer-reviewed)

Suyu, S. H., Acebron, A., Pierel, J. D. R., et al. 2026, arXiv:2509.12319, "Blind Time Delay Predictions for SN Encore from Eight Independent Lens Models" (arXiv preprint; not yet peer-reviewed)

Johansson, J., et al. 2025, *ApJ*, 995, L17, "Discovery of SN 2025wny: A Strongly Gravitationally Lensed Superluminous Supernova at z = 2.01", arXiv:2510.23533

Goobar, A., Johansson, J., Mörtsell, E., et al. 2026, arXiv:2608.28413, "Follow-up of SN 2025wny I: Space-based Observations of the First Multiply-imaged Superluminous Supernova" (arXiv preprint)

Johansson, J., et al. 2026, arXiv:2608.28416, "Follow-up of SN 2025wny III: Spectroscopic Time-delay Measurements of a Strongly Gravitationally Lensed Superluminous Supernova" (arXiv preprint)

Townsend, A., Dhawan, S., Hayes, E. E., et al. 2026, arXiv:2608.28427, "Follow-up of SN 2025wny IV: Photometric Time-delay Measurements of a Strongly Lensed Superluminous Supernova" (arXiv preprint)

Mörtsell, E., Johansson, J., Goobar, A., et al. 2026, arXiv:2608.28430, "Follow-up of SN 2025wny V: Lens Modelling and Cosmography of a Strongly Lensed Superluminous Supernova at z = 2.015 using Space Data" (arXiv preprint)

Taubenberger, S., Acebron, A., Cañameras, R., et al. 2025, arXiv:2510.21694, "HOLISMOKES XIX: SN 2025wny at z=2, the first strongly lensed superluminous supernova" (arXiv preprint)

Wynne, R. A. & Schechter, P. L. 2018, arXiv:1803.02722, "The Geometry of Quadruple Lenses: The Singular Isothermal Ellipsoid" (arXiv preprint)

Coulter, D. A., Larison, C., Pierel, J. D. R., Fujimoto, S., et al. 2026, arXiv:2601.04156, "A spectroscopically confirmed, strongly lensed, metal-poor Type II supernova at z = 5.13" (SN Eos; arXiv preprint)

Cañameras, R., Schuldt, S., Suyu, S. H., et al. 2020, *A&A*, 644, A163, "HOLISMOKES II: Identifying galaxy-scale strong gravitational lenses in Pan-STARRS using convolutional neural networks", doi:10.1051/0004-6361/202039071

Falco, E. E., Gorenstein, M. V., & Shapiro, I. I. 1985, *ApJ*, 289, L1, "Modeling of the Einstein Ring in MG 1131+0456", doi:10.1086/184605

#### Cosmology & H0 Measurements

Boran, S., Desai, S., Kahya, E. O., & Woodard, R. P. 2018, *Phys. Rev. D*, 97, 041501, "GW170817 falsifies dark matter emulators", doi:10.1103/PhysRevD.97.041501

Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, *A&A*, 641, A6, "Planck 2018 results. VI. Cosmological parameters", doi:10.1051/0004-6361/201833910

Riess, A. G., Yuan, W., Macri, L. M., et al. 2022, *ApJ*, 934, L7, "A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team", doi:10.3847/2041-8213/ac5c5b

#### Hubble Tension Reviews

Di Valentino, E., Mena, O., Pan, S., et al. 2021, *Classical and Quantum Gravity*, 38, 153001, "In the realm of the Hubble tension—a review of solutions", doi:10.1088/1361-6382/ac086d

#### TEP Framework (This Series)

Smawfield, M. L. 2025, "Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed" (Paper 0, Jakarta v0.16), doi:10.5281/zenodo.16921911

Smawfield, M. L. 2025, "Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations" (Paper 4, Tortola), doi:10.5281/zenodo.17982540

Smawfield, M. L. 2025, "Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T" (Paper 6, New Delhi), doi:10.5281/zenodo.18064365

Smawfield, M. L. 2025, "What Do Precision Tests of General Relativity Actually Measure?" (Paper 9, Istanbul), doi:10.5281/zenodo.18109760

Smawfield, M. L. 2025, "Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars" (Paper 10, Caracas), doi:10.5281/zenodo.18165798

Smawfield, M. L. 2025, "The Cepheid Bias: Resolving the Hubble Tension" (Paper 11, Kingston upon Hull), doi:10.5281/zenodo.18209702

## Appendix A: Theoretical Framework for Lensing Closures

### A.1 Temporal Shear and the Lensing Potential

The Temporal Equivalence Principle (TEP) postulates that the rate of proper time flow for a photon traversing a gravitational potential $\Phi$ is scaled relative to the cosmological background rate by a factor $\Gamma_t$:

\begin{equation} \label{eq:8_appendix_01}
\Gamma_t(\Phi) = 1 + \kappa_{\rm lens} \frac{|\Phi|}{c^2}
\end{equation}

where $\kappa_{\rm lens}$ is the lensing-sector effective coupling. The proxy-model coefficient is empirically $\kappa_{\rm lens} \approx -0.055$; under the corrected Kelly+2023 Supplementary Table S4 compilation, the SN Refsdal data infer a lensing-sector coupling $\kappa_{\rm inferred} = -0.114 \pm 0.034$, which exceeds the proxy value by $1.75\sigma$. In the strong lensing regime, the relevant potential is the projected gravitational potential $\psi(\boldsymbol{\theta})$ integrated along the line of sight. For singular isothermal sphere (SIS) or NFW profiles typical of cluster lenses, the projected potential depth scales logarithmically with the surface mass density $\Sigma$, which is directly related to the magnification $\mu$.

The endpoint expression above is a clock-ratio diagnostic, not by itself an additional arrival-time observable. In the strictly static conformal sector, $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$, null propagation obeys the exact identity

\begin{equation} \label{eq:8_appendix_01b}
\tilde g_{\mu\nu}k^\mu k^\nu=A^2(\phi)g_{\mu\nu}k^\mu k^\nu=0
\quad\Longleftrightarrow\quad
g_{\mu\nu}k^\mu k^\nu=0\,.
\end{equation}

Consequently, conformal rescaling preserves the unparameterised null trajectories and their static coordinate travel times. The thin-lens relation $\int |\Phi|c^{-2}dl=(D_lD_s/2D_{ls})\psi(\boldsymbol{\theta}_i)$ is the ordinary potential part of the GR Fermat delay already contained in the lens predictions. Adding a fitted multiple of this term after subtracting those predictions double-counts the same potential. Step 56 therefore gives $\Delta t^{\rm residual}_{\rm static,conf}=0$ exactly; the archived $\psi$ ordering is retained only as a lens-environment diagnostic.

Why the historical isotropic-conformal benchmark cannot generate a direct optical residual is confirmed by three action-level controls before passing to the canonical unstretched-space metric. First, under $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$, direct scalar stress-energy curvature of $g_{\mu\nu}$ is Planck-suppressed: evaluating the scalar lensing potential $\Psi_\phi$ on the GLAFIC mass map gives $\kappa_\phi \approx 4\times10^{-5}$ ($+0.25\,$d SX–S1 delay, shortfall $122$) on the unit-term control (Step 61) and $\kappa_\phi \simeq 6.5\times10^{-6}$ ($+0.041\,$d) on the selected two-branch sector (Step 64). Second, the quadratic disformal cone-tilt sector $B(\phi)\nabla_\mu\phi\nabla_\nu\phi$ cannot supply the delay without violating multi-messenger bounds: transferring the GW170817 speed constraint to the coupling $B_0$ via the peak cone-split functional on the host galaxy profile ($B_0 \lesssim 78$; Step 62) and integrating the 3-D field $g_z(z)$ across a spherical NFW + constant-sheet deprojection of the smoothed GLAFIC convergence map ($b_{\rm cl}=44$ kpc for SX versus $68$–$81$ kpc for S1–S4, reproducing the tabulated image $\kappa$ values to $\sim5\%$; Step 63) requires $B_0^{\rm req}\simeq 420$ at the best-fit halo scale $r_s\simeq 0.74$ Mpc — a factor $\simeq 5$ above the GW170817 envelope (and $\sim 176\times$ to $10^4\times$ short under the drift-transit and environment-modulated $B_{\rm eff}=B(\phi)G(X_{\rm local})$ conventions). Third, removing the artificial spatial ruler stretching of $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$ resolves the optical sector directly (Step 64, §2.3): under the canonical unstretched-space temporal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) on the selected two-branch kinetic sector $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$ fixed by $a_0$, zero per-system free parameters), the $(3+1)$ ADM spatial 3-metric $\gamma_{ij}$ remains unstretched while the spacetime lapse rescales as $N_{\rm TEP}=A_{\rm loc}(u_{\rm grav})N_{\rm GR}$, so intermediate clock rates contribute $-\beta_A u_{\rm grav}(x)$ directly to the refractive index $n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)$ ($c_{\rm GW}=c_\gamma$, $\Delta t_{\gamma-\rm GW}=0$) and yield an absolute $+330.9$ d 2-D sheet Fermat delay and $+221.6$ to $+443.2$ d 3-D deprojected NFW ray delay from $f_b=0.16$ baryons alone in the exact arrival order $\text{SX}>\text{S1}>\text{S2}>\text{S3}>\text{S4}$ (reproducing the total non-baryonic long-baseline delay scale rather than adding to the published GR forecast). Alongside this first-principles refraction solve, the phenomenological log-magnification relation below is tested as an operational model for the differential image-level residual (subjected to the external holdout test of Step 57):

\begin{equation} \label{eq:8_appendix_02}
\Gamma_t(i) \approx 1 + \kappa_{\rm lens} \log_{10}\left(\frac{\mu_i}{\bar{\mu}}\right)
\end{equation}

where $\mu_i$ is the absolute magnification of image $i$, and $\bar{\mu}$ is the mean magnification of the image system. This logarithmic scaling captures the essential feature that temporal shear is differential: it depends on the *ratio* of potential depths probed by different images. The test is a screen of this ansatz, not a measurement of the fundamental coupling.

### A.2 Derivation of the Predicted GR Discrepancy

Let $t_i^{\rm GR}$ be the arrival time of image $i$ predicted by General Relativity and $t_0$ the unlensed arrival time. Define the GR absolute delay $u_i = t_i^{\rm GR} - t_0$ and the GR pairwise delay $\Delta t_{ij}^{\rm GR} = t_j^{\rm GR} - t_i^{\rm GR} = u_j - u_i$. Under TEP, the observed arrival time is scaled by the path-specific shear $\Gamma_i$:

\begin{equation} \label{eq:8_appendix_03}
t_i^{\rm obs} = t_0 + \Gamma_i (t_i^{\rm GR} - t_0) = t_0 + \Gamma_i u_i
\end{equation}

The observed pairwise delay between images $i$ and $j$ is therefore:

\begin{equation} \label{eq:8_appendix_04}
\Delta t_{ij}^{\rm obs} = t_j^{\rm obs} - t_i^{\rm obs} = \Gamma_j u_j - \Gamma_i u_i
\end{equation}

Because observed arrival times are physical clock readings, the sum of observed pairwise delays around any closed loop is identically zero:

\begin{equation} \label{eq:8_appendix_05}
\Delta t_{ij}^{\rm obs} + \Delta t_{jk}^{\rm obs} + \Delta t_{ki}^{\rm obs} = (\Gamma_j u_j - \Gamma_i u_i) + (\Gamma_k u_k - \Gamma_j u_j) + (\Gamma_i u_i - \Gamma_k u_k) = 0
\end{equation}

The TEP signature is therefore not a failure of the observed delays to close, but a discrepancy between the observed delays and the delays predicted by GR lens models. The TEP predicted GR discrepancy is defined as the weighted sum of GR pairwise delays with coefficients $(\Gamma_m - 1)$:

\begin{equation} \label{eq:8_appendix_06}
\mathcal{R}_{\rm TEP}(i,j,k) \equiv (\Gamma_i - 1)\,\Delta t_{ij}^{\rm GR} + (\Gamma_j - 1)\,\Delta t_{jk}^{\rm GR} + (\Gamma_k - 1)\,\Delta t_{ki}^{\rm GR}
\end{equation}

Substituting $\Delta t_{ij}^{\rm GR} = u_j - u_i$ and expanding:

\begin{equation} \label{eq:8_appendix_07}
\mathcal{R}_{\rm TEP} = (\Gamma_i - 1)(u_j - u_i) + (\Gamma_j - 1)(u_k - u_j) + (\Gamma_k - 1)(u_i - u_k)
\end{equation}

Grouping terms by absolute delay:

\begin{equation} \label{eq:8_appendix_08}
\mathcal{R}_{\rm TEP} = u_i(\Gamma_k - \Gamma_i) + u_j(\Gamma_i - \Gamma_j) + u_k(\Gamma_j - \Gamma_k)
\end{equation}

This can also be written in the pairwise-delay form used in the main text by expanding $\Gamma = 1 + \delta\Gamma$ and using $\sum_{\rm loop}\Delta t^{\rm GR} = 0$:

\begin{equation} \label{eq:8_appendix_09}
\mathcal{R}_{\rm TEP} = \sum_{\rm loop} (1 + \delta\Gamma)\,\Delta t^{\rm GR} = \sum \Delta t^{\rm GR} + \sum \delta\Gamma\,\Delta t^{\rm GR} = \sum_{\rm loop} (\Gamma_m - 1)\,\Delta t_{mn}^{\rm GR}
\end{equation}

Explicitly for the triplet $(i, j, k)$:

\begin{equation} \label{eq:8_appendix_10}
\mathcal{R}_{\rm TEP} = (\Gamma_i-1)\Delta t_{ij} + (\Gamma_j-1)\Delta t_{jk} + (\Gamma_k-1)\Delta t_{ki}
\end{equation}

where $\Delta t_{ij} \equiv \Delta t_{ij}^{\rm GR}$. This equation demonstrates that the residual is generated purely by the *differences* in $\Gamma$ across the oriented image triplet. If the potential depth is constant ($\Gamma_i = \Gamma_j = \Gamma_k$), the residual vanishes. For the S1–S4–SX loop of SN Refsdal, where the other two delays are well-constrained by the Einstein-cross images, this weighted sum approximately equals the TEP-induced discrepancy in the SX–S1 delay. **This is not an observed delay-closure violation.** It is a model-space residual combination comparing GR-predicted delays to a TEP-scaled reference. Algebraic closure of pairwise delays remains exactly zero; the genuine TEP observable is the blind-prediction residual, not a synchronization holonomy.

### A.3 Prediction-Provenance Audit

The blind-prediction residual test relies on a delay-blind ensemble of pre-reappearance model variants. The selection was performed before any TEP analysis was computed, based on two criteria: (i) the original model was published before SX reappeared in December 2015, and (ii) the model variant is a distinct parameter configuration (all-image set vs gold-image set) from the same modelling family. The table below documents the provenance of each delay-blind variant, including whether the *value used in this analysis* is the originally published prediction or a later revision.

| Selected model | Original paper | Original $\Delta t_{\rm pred}$ | Original residual sign | Value used | Kelly+2023 revision? | Blind? | Reason selected |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Oguri-a* | Oguri (2015, MNRAS 449, L86) | 336 ± 21 d — Kelly+2023 Science S4, original pre-reappearance prediction | + | 353.8 d | Yes — post-reappearance computational revision by Kelly+2023 using the SX reappearance position (Rodney et al. 2023, ApJ 948, 93); original was 336<sup>+21</sup><sub>-21</sub>, revised to 354<sup>+21</sup><sub>-17</sub> | Original prediction: yes. Revised value: No — not blind | All-image set; parametric GLAFIC |
| Oguri-g* | Oguri (2015, MNRAS 449, L86) | 311 ± 24 d — Kelly+2023 Science S4, original pre-reappearance prediction | + | 330.7 d | Yes — same as Oguri-a; original was 311<sup>+24</sup><sub>-24</sub>, revised to 331<sup>+22</sup><sub>-17</sub> | Same as Oguri-a | Gold-image set; parametric GLAFIC |
| Sharon-a* | Sharon & Johnson (2015, ApJ 800, L26) | 237<sup>+37</sup><sub>-50</sub> d — Sharon & Johnson 2015, Δt<sub>1.2-S1</sub>; a/g variants from Treu 2015 use same underlying model | + | 298.0 d | Yes — post-reappearance computational revision by Kelly+2023; original ~237 d rescaled to 298<sup>+58</sup><sub>-17</sub> | Original prediction: yes. Revised value: No — not blind | All-image set; parametric LTM |
| Sharon-g* | Sharon & Johnson (2015, ApJ 800, L26) | 237<sup>+37</sup><sub>-50</sub> d — Same original paper; variant from Treu 2015 gold-image set | + | 331.0 d | Yes — same as Sharon-a; best-fit realization adopted as 331<sup>+30</sup><sub>-35</sub> | Same as Sharon-a | Gold-image set; parametric LTM |
| Diego-a | Diego et al. (2016, MNRAS 456, 356) | 262 ± 55 d — Original prediction, no post-reappearance revision noted in Kelly+2023 S4 | + | 262.0 d | No — value taken directly from Kelly+2023 Supplementary Table S4 compilation, no post-reappearance revision noted | Yes — original pre-reappearance prediction | All-image set; free-form WSLAP+ |
| Grillo-g | Grillo et al. (2016, ApJ 822, 78) | 361<sup>+19</sup><sub>-27</sub> d — Original prediction, no post-reappearance revision noted in Kelly+2023 S4 | + | 361.0 d | No — value taken directly from Kelly+2023 Supplementary Table S4 compilation, no post-reappearance revision noted | Yes — original pre-reappearance prediction | Gold-image set; parametric GLEE |
| Jauzac15.2* | Jauzac et al. (2016, MNRAS 457, 2029) | Jauzac15.1 = 449 ± 45 d (strictly blind, pre-reappearance); Jauzac15.2 = 361 ± 42 d (delay-blind, revised using SX position) | + | 361.0 d | Yes — Kelly+2023 S4 revised value. Original Jauzac15.1 (449 d) is strictly blind; revised Jauzac15.2 (361 d) is delay-blind (fixed before measured delay known, but used SX position) | Original Jauzac15.1: strictly blind. Revised Jauzac15.2: delay-blind | Selected model version; free-form LENSTOOL |
| *Post-blind supplementary only* |  |  |  |  |  |  |  |
| Grillo+2024 | Grillo et al. (2024, A&A 689, A78) | N/A (post-blind) | N/A | 362.0 d | No | Post-blind | Supplementary consistency check only |

**Blindness caveat.** The Kelly+2023 *Science* paper (published 2023, post-reappearance) compiled blind model predictions in Supplementary Table S4. Three provenance tiers are distinguished. *Strictly blind*: original pre-reappearance published predictions (Oguri-a 336 d, Oguri-g 311 d, Sharon-a 237 d, Sharon-g 237 d, Diego-a 262 d, Grillo-g 361 d, Jauzac15.1 449 d). *Delay-blind*: Kelly+2023 S4 revised values, all fixed before the measured delay (376.0 d) was known, but some (Oguri, Sharon, Jauzac) used the SX reappearance position (Rodney et al. 2023, ApJ 948, 93). *Post-blind*: Grillo+2024 (2024), published after the delay was measured. The primary sign test uses the delay-blind tier because all seven values were fixed before anyone knew 376.0 d, and all seven yield positive residuals. The strictly blind floor (6/7 positive, Jauzac15.1 residual −73 d) is reported for transparency. The amplitude-based 3.39σ result depends on revised values and is retained as a strong response-scale diagnostic, not as blind evidence.

The primary sign test uses the seven delay-blind model variants exactly as tabulated in the prediction-provenance table. The family-sign-flip test groups correlated variants by modelling family (GLAFIC, LTM, WSLAP+, GLEE, LENSTOOL) and is used as the operational dependence-aware headline. The post-blind Grillo+2024 update is reported as a supplementary consistency check and is excluded from the primary test.

### A.4 Immunity to Mass Sheet Degeneracy

The Mass Sheet Degeneracy (MSD) corresponds to a transformation of the convergence $\kappa \to \lambda \kappa + (1-\lambda)$, which rescales time delays by a factor $\lambda$:

\begin{equation} \label{eq:8_appendix_11}
\Delta t_{ij}' = \lambda \Delta t_{ij}
\end{equation}

Substituting this into the predicted GR discrepancy definition:

\begin{equation} \label{eq:8_appendix_12}
\mathcal{R}' = \Delta t_{ij}' + \Delta t_{jk}' + \Delta t_{ki}' = \lambda(\Delta t_{ij} + \Delta t_{jk} + \Delta t_{ki})
\end{equation}

Since $\Delta t_{ij} + \Delta t_{jk} + \Delta t_{ki} = 0$ in GR, then $\mathcal{R}' = \lambda \cdot 0 = 0$.

Thus, a uniform Mass Sheet Degeneracy cannot generate a non-zero algebraic loop sum by itself, because it rescales all delays coherently. This does not imply that the empirical blind-prediction residual is immune to all lens-model systematics. The narrower result is that the temporal-shear residual tested here is not algebraically equivalent to a trivial uniform mass-sheet rescaling.

## Data Availability & Reproducibility

This work follows open-science practices. All results are fully reproducible from raw data using the documented pipeline. All numerical results, figures, and statistics are generated by deterministic Python scripts processing real observational data.

### Repository & Code

*GitHub Repository:* github.com/matthewsmawfield/TEP-LENS

The repository contains a deterministic, version-controlled analysis pipeline covering *Steps 00–20, 30–46, and 50–57*, including diagnostics for blind residual statistics, model dependence, proxy robustness, convergence-tracer substitution, lensing-potential reconstruction, 3D geodesic transport, conformal-cancellation auditing, response-transfer auditing, cross-system kernel consistency auditing, external amplitude cross-validation, and future-system forecasting.

#### Repository Structure

```
TEP-LENS/
├── data/                          # Raw light curves and lens models
│   ├── sn_lensing/                # SN Refsdal and H0pe data
│   └── snh0pe/                    # SN H0pe light curves
├── scripts/
│   ├── steps/                     # Sequential analysis pipeline
│   │   ├── step_01_fetch_snh0pe_data.py
│   │   ├── step_02_gr_closure.py
│   │   ├── step_03_tep_closure.py
│   │   ├── step_04_plot_closure.py
│   │   ├── step_11_model_dependence.py
│   │   ├── step_13_bayes_model_comparison.py
│   │   └── run_all_steps.py       # Master pipeline runner
│   └── utils/                     # Plotting and logging utilities
├── results/                       # Pipeline outputs and figures
├── site/
│   └── components/                # HTML manuscript source
└── requirements.txt                 # Python dependencies
```

### Data Provenance

| Data Source | Provider | Access Method | Download Size | Reference |
| --- | --- | --- | --- | --- |
| SN Refsdal Light Curves | Kelly et al. (2023) | Published data | ~1 MB | ApJ 948, 93 |
| SN H0pe Data | SN H0pe Collaboration | Public release | ~500 KB | Via GitHub |
| TDCOSMO-2025 Chains | TDCOSMO Collaboration | Public chains | ~50 MB (MCMC chains) | tdcosmo.github.io |
| H0LiCOW/TDCOSMO Legacy | H0LiCOW Collaboration | Public chains | ~20 MB | Via repository |

*Total Download Size:* ~75 MB for all primary data sources.
*Data Provenance Log:* Complete acquisition details maintained in `data/DATA_PROVENANCE.md`.

### Reproduction Instructions

#### Quick Start (Full Reproduction)

```
# 1. Clone repository
git clone https://github.com/matthewsmawfield/TEP-LENS.git
cd TEP-LENS

# 2. Install dependencies
pip install -r requirements.txt

# 3. Run complete pipeline: Steps 00–20, 30–46, and 50–57
python scripts/steps/run_all_steps.py

# 4. Build manuscript
cd site
npm install
npm run build
```

#### System Requirements

| Component | Minimum | Recommended | Tested On |
| --- | --- | --- | --- |
| CPU | 4 cores | 8+ cores | Apple M4 Pro (14-core) |
| RAM | 8 GB | 16 GB | 24 GB (M4 Pro) |
| Storage | 5 GB | 10 GB | NVMe SSD |
| Runtime | ~30-60 minutes |  | ~30 minutes (M4 Pro) |

#### Detailed Pipeline Steps

The analysis pipeline covers *Steps 00–20, 30–46, and 50–57*, including diagnostics for blind residual statistics, model dependence, proxy robustness, convergence-tracer substitution, lensing-potential reconstruction, conformal-cancellation auditing, external amplitude cross-validation, response-transfer auditing, cross-system kernel consistency auditing, and future-system forecasting. Each step produces JSON outputs and logs for full traceability:

##### Data Acquisition & Closure Tests (Steps 00-04)

- *step_00_fetch_literature_and_cross_paper_data.py* — Fetch literature data and cross-paper reference values for TEP consistency checks

- *step_01_fetch_snh0pe_data.py* — Fetch SN H0pe light curve data from public release; parses multiply-imaged supernova magnitudes and time delays

- *step_02_gr_closure.py* — GR algebraic loop sum; computes the identically-zero loop baseline for GR with SN Refsdal and SN H0pe

- *step_03_tep_closure.py* — TEP predicted GR discrepancy; computes TEP-predicted GR discrepancies for each loop

- *step_04_plot_closure.py* — Generate predicted GR discrepancy plots (Figure 4 baseline vs discrepancy)

##### TDCOSMO & H0 Analysis (Steps 05-10)

- *step_05_tdcosmo_shear.py* — TDCOSMO temporal shear test; computes TEP-predicted delay shifts as a predicted-sensitivity consistency check

- *step_06_kappa_sensitivity.py* — κ₀ parameter sensitivity analysis; tests how predicted GR discrepancies depend on TEP coupling

- *step_07_observed_vs_predicted.py* — Observed vs blind-predicted delay comparison; blind sign evidence + post-hoc proxy sensitivity (not pre-observation forecast)

- *step_08_new_evidence.py* — Compile new evidence from independent lens systems

- *step_09_precision_roadmap.py* — Precision requirements analysis; quantifies significance as a function of model precision and sample size

- *step_10_h0_tension.py* — H0 internal-consistency check in lensing context; compares proxy-model shifts to local and CMB H0 without claiming a tension resolution

##### Model Dependence & Robustness (Steps 11-15)

- *step_11_model_dependence.py* — Model dependence analysis; tests sensitivity to lens model assumptions

- *step_12_microlensing_robustness.py* — Microlensing robustness tests; quantifies microlensing contamination effects

- *step_13_bayes_model_comparison.py* — Bayesian model comparison; computes Bayes factors for GR vs TEP lens models

- *step_14_external_chain_ingestion.py* — External chain ingestion; imports TDCOSMO/H0LiCOW MCMC chains for joint analysis

- *step_15_external_informed_inflation.py* — External-informed inflation analysis; tests TEP effects on time-delay cosmography precision under external uncertainty priors

##### External Data & Completeness (Steps 16-20)

- *step_16_independence_tier_significance.py* — Independence tier significance; statistical tests for multiple independent lens systems

- *step_17_directional_odds.py* — Directional odds analysis; tests TEP predictions for specific image parity configurations

- *step_18_external_dataset_registry.py* — External dataset registry; catalogs all public lensed SN light curves

- *step_19_tdcosmo2025_ingestion.py* — TDCOSMO-2025 chain ingestion; imports latest time-delay cosmography constraints

- *step_20_external_completeness_synthesis.py* — External completeness synthesis; combines all lensing constraints with TEP

##### Proxy Validation, Null Tests, Forward Prediction & Evidence Scaling (Steps 30–42)

- *step_30_cosmograil_temporal_shear.py* — CosmoGRAIL temporal-shear analysis; time-domain signal extraction from light curves

- *step_31_cosmograil_validation.py* — CosmoGRAIL validation; injection-recovery tests for delay estimation

- *step_32_kappa_proxy_validation.py* — Kappa proxy validation; Monte Carlo shear-degeneracy analysis quantifying mu-to-kappa systematic

- *step_32b_temporal_shear_figure.py* — Temporal-shear figure generation; produces diagnostic plots for shear analysis

- *step_33_einstein_cross_null.py* — Einstein cross null test; Spearman rank test on inner-cross delays vs magnification

- *step_34_agnostic_rank_test.py* — Agnostic rank test; proxy-agnostic statistical verification of rank-order predictions

- *step_35_single_contrast_dominance.py* — Single-contrast dominance quantification; effective degrees of freedom and signal-energy fractions

- *step_36_falsification_forward_prediction.py* — Falsification forward prediction; forward-projected thresholds for SN 2025wny

- *step_37_multi_system_evidence.py* — Multi-system evidence accumulation projection; Stouffer projection and coupling-inference scaling with N independent systems

- *step_38_cosmograil_cross_system.py* — CosmoGRAIL cross-system directional consistency diagnostic; caveated quasar check separate from the supernova residual tests

- *step_38_sn_encore_residuals.py* — SN Encore blind-prediction residual test; 8 independent lens models from Suyu+2025 vs Pierel+2026 observed delay

- *step_39_sn_h0pe_residuals.py* — SN H0pe blind-prediction residual test; 7 independent lens models from Pascale+2025 vs Pierel+2024 observed delays

- *step_40_cross_system_trio.py* — Cross-system trio evidence synthesis; Stouffer combination of directional evidence from Refsdal, Encore, and H0pe

- *step_41_null_channel_discriminator.py* — Null-channel discriminator; AIC comparison of the TEP proxy against generic bias models (null, constant offset, uniform-fractional H0/MSD-like) across cross-system contrasts, separating amplitude from sign evidence

- *step_42_precision_persistence.py* — Precision-persistence test; extrapolates the SN Refsdal SX residual to the high-precision lens-model limit to test the model-immaturity explanation

Each step produces JSON outputs with full metadata in `results/outputs/`, and execution logs are written to `logs/` with timestamps for complete traceability. Run all steps via: `python scripts/steps/run_all_steps.py`

#### Key Analysis Outputs

- `results/outputs/step_02_gr_closure.json` — GR algebraic loop sums

- `results/outputs/step_03_tep_closure.json` — TEP predicted GR discrepancies

- `results/outputs/step_07_observed_vs_predicted.json` — Observed vs predicted residuals with evidence-tier decomposition

- `results/outputs/step_13_bayes_model_comparison.json` — Bayesian comparison

- `results/outputs/step_32_kappa_proxy_validation.json` — Proxy systematic budget and mu-to-kappa Monte Carlo

- `results/outputs/step_34_agnostic_rank_test.json` — Proxy-agnostic Spearman rank test (delays vs kappa and mu)

- `results/outputs/step_35_single_contrast_dominance.json` — Signal-energy fractions and effective degrees of freedom

- `results/outputs/step_38_sn_encore_residuals.json` — SN Encore blind-prediction residuals and GR null tests

- `results/outputs/step_39_sn_h0pe_residuals.json` — SN H0pe blind-prediction residuals and GR null tests

- `results/outputs/step_40_cross_system_trio.json` — Cross-system trio Stouffer combination and sign-consistency check

- `results/outputs/step_56_transport_lapse.json` — Conformal transport audit; demonstrates that the static $\psi$ term is already the GR Fermat/Shapiro contribution

- `results/outputs/step_57_external_amplitude_crossvalidation.json` — Refsdal hold-out amplitude prediction from Encore and H0pe with shared-observation covariance retained

- `results/outputs/step_58_h0_channel_audit.json` — H0LiCOW/TDCOSMO-class time-delay channel audit under the TEP mechanism (per-system propagated-transport ceilings, endpoint rescaling, mass-model self-consistency, kinematic-prior phantom-mass sign, real $\kappa_{\rm ext}$ widths and the SDSS1206+4332 power-law-vs-composite distance floor)

- `results/outputs/step_60_mass_slip_two_directions.json` — two-direction phantom-mass bookkeeping: per-contrast implied phantom fractions $\lambda = \mathcal{R}_{\rm obs}/\langle\Delta t_{\rm pred}\rangle$, counter-direction $|\lambda|$ bounds for zero-consistent contrasts, negative-phantom-fraction census across all per-model residuals, and the pooled common-phantom-fraction consistency test

### Software Versions

- *Python* 3.10+

- *NumPy* 1.24+

- *SciPy* 1.10+

- *Pandas* 2.0+

- *Matplotlib* 3.7+

---

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# Temporal Equivalence Principle: A Standard-Siren Test of Bi-Metric Gravitational-Wave Propagation
**Matthew Lukin Smawfield**
Version: v0.2 (Maputo)
First published: 30 September 2026
Last updated: 17 October 2026
DOI: 10.5281/zenodo.20572695

---

## Abstract

Standard &Lambda;CDM assumes that gravitational waves and electromagnetic radiation propagate through the same effective distance-redshift relation. The Temporal Equivalence Principle (TEP) predicts a bi-metric scaling in which gravitational-wave luminosity distances and detector-frame masses are modified relative to matter-frame observations by the conformal factor *A*(*z*), with canonical endpoint map \(\Xi(z) = d_L^{\rm GW}/d_L^{\rm EM} = 1/(1+z)\). This paper tests both the distance-redshift channel and the host-catalog-free binary black hole (BBH) chirp-mass channel across combined GWTC catalogs (GWTC-1 through GWTC-5.0, plus O4 Discovery Papers), bright-siren spectroscopy, and GLADE+/GraceDB dark-siren host associations.

In the host-catalog-free BBH population channel (95 events: 45 low-redshift anchors and 50 high-redshift scored events, with the probability-density Jacobian and the reported-source chirp-mass selection cut of \(8\,M_\odot\) included; Step 09n), continuous physical high-mass cutoff fitting breaks the scale-free power-law degeneracy (\(\beta \simeq 154\,M_\odot\) for TEP vs \(\beta \simeq 33\,M_\odot\) for GR). At their canonical parameter-free baselines (\(\epsilon = 0\) for TEP, \(k = 0\) for GR), canonical pinned TEP outperforms stationary GR out of sample on the 50 high-redshift events by \(\Delta\ln L = +2.32\) (\(\ln L = -188.68\) vs \(-191.00\), conditional likelihood ratio \(\approx 10.2\)), and profiling over free tracking and evolution families confirms that both frames achieve their global optima at their canonical parameter-free baselines (\(\hat\epsilon = 0\), \(\hat k = 0\)). Under Rule-16 inversion (Step 09m), the selected events have median matter-frame chirp mass \(45\,M_\odot\) (reported \(\sim 20\,M_\odot\)), with 55% of \(z_L > 0.3\) events above \(65\,M_\odot\) (split-normal distance sensitivity fraction 0.26 [0.19–0.33, central 95%]). The host-assembly derivation of Paper 0 (&sect;8) supplies the progenitor-dependent transfer explicitly: progenitors forming in deep potential wells carry \(\epsilon_{\rm eff}\simeq0.004\) (the SN Ia class, matching the Pantheon+ bound \(\epsilon<0.150\) at \(3\sigma\), \(\Delta\chi^2=805.8\)), whereas BBH progenitors assembled in field-spiral or dwarf hosts carry \(\epsilon_{\rm eff}\simeq0.24\)–\(0.52\) (\(\mathcal M_{\rm det}\propto(1+z)^{-0.28}\) to \((1+z)^{+0.56}\)); with \(\hat\epsilon = 0\) as the global optimum, the two populations do not require a single shared tracking fraction.

In the distance-redshift channel (73 events without GWOSC distance-derived fallback redshifts: 1 bright siren GW170817 plus 72 GLADE+/DESI dark-siren associations; 34 GWOSC-fallback events quarantined as a cosmology-blind pipeline null with \(\gamma^* = 0\)), two distinct estimands are evaluated. First, under the properly normalized split-normal distance likelihood (\(\sigma_{\rm norm}=\tfrac12(\sigma_-+\sigma_+)\), Step 09f), the canonical map \(\Xi(z)=1/(1+z)\) returns \(-2\ln L=943.14\) at free \(H_0=56.0\) km/s/Mpc against \(944.31\) at \(H_0=70.0\) km/s/Mpc for &Lambda;CDM (\(\Delta\chi^2=+1.16\) in TEP's favor at equal degrees of freedom). Because the 72 dark-siren candidate hosts are pre-selected by a &Lambda;CDM distance gate \(d_L^{\Lambda{\rm CDM}}(z_{\rm cat};H_0\approx70)\simeq d_L^{\rm GW}\), fitting \(d_L^{\rm GW}=d_L^{\rm EM}/(1+z_{\rm cat})\) to &Lambda;CDM-gated redshifts mechanically scales a free \(H_0\) by \(\langle(1+z_{\rm cat})^{-1}\rangle=0.7878\) (\(70.0\times0.7878=55.1\) km/s/Mpc); remapping each event's redshift posterior onto the self-consistent TEP association kernel at fixed canonical \(H_0=68.18\) km/s/Mpc (Paper 18) recovers \(-2\ln L=944.31\) without shifting \(H_0\) (GW170817 at \(z=0.0098\): \(-2\ln L=4.94\) TEP vs \(4.95\) &Lambda;CDM). Second, under the small-excursion exponential template \(A(z) = \exp(\beta_A\phi_0[(1+z)^n - 1])\) with lab-fixed \(\beta_A = -1\), \(\phi_0 = +0.013\), \(n = 1\), the 73-event sample gives &Lambda;CDM \(H_0 = 72.0\) km/s/Mpc and TEP \(H_0 = 72.2\) km/s/Mpc (\(\Delta\chi^2 = +0.20\), \(|\Delta\mathrm{BIC}| < 2\); 107-event full sample: \(72.4\) vs \(72.4\) km/s/Mpc, \(\Delta\chi^2 = +0.06\)), and the converged 4-parameter emcee MCMC posterior (\(H_0 = 71.2 \pm 3.3\) km/s/Mpc, \(\phi_0 = +0.000 \pm 0.028\), \(n = 1.47 \pm 0.86\), \(\beta_{\rm fit} = -0.02 \pm 2.81\); \(\Delta\mathrm{BIC} = -6.1\) from the two effective extra parameters \(C\equiv\beta\phi_0\) and \(n\)) places 71.4% of its posterior mass on the corpus-consistent sign \(\beta\phi_0 < 0\) (\(A(z) < 1\)). Concordance-bias injection (&sect;5.8) shows that &Lambda;CDM-gated host weighting absorbs \(\sim 85\%\) of a genuine distance shift (transmission \(T = 0.15\), yielding 6% two-sided \(|Z|>2\) detection power on dark sirens), whereas spectroscopic bright standard sirens across \(z\in[0.1,1.0]\) at fixed \(H_0=68.18\) km/s/Mpc provide a prior-free \(5\sigma\) test with \(N=25, 7, 3,\) and \(1\) events at \(z=0.1, 0.2, 0.3,\) and \(z\in[0.5,1.0]\).

Keywords: Temporal Equivalence Principle, gravitational waves, standard sirens, bi-metric propagation, distance-redshift relation, combined GWTC catalogs

## 1. Introduction

## 1.1 Bi-Metric Propagation and the Hubble Tension

Measurements from the cosmic microwave background (Planck, *H*<sub>0</sub> ≈ 67.4 km/s/Mpc under ΛCDM) and the local distance ladder (SH0ES, *H*<sub>0</sub> ≈ 73 km/s/Mpc) differ by approximately 5σ. Paper 11 resolves the Hubble tension through an environment-dependent Cepheid response in the endpoint observing chain, not an additional cosmic-web path-integrated shear. Its host-level corrected value is *H*<sub>0</sub> = 67.83 ± 1.56 km/s/Mpc (0.26σ from Planck), internally consistent with the TEP-CMB value 66.70 ± 0.58 (Paper 26) at 0.68σ; that internal agreement is not independent confirmation. The endpoint environmental slope is detected at 2.33σ at the canonical velocity-scatter convention, the within-host period-coupled carrier reaches 2.28–2.79σ across anchor conventions, and Paper 11 Step 62 derives the Cepheid response scale $|\kappa_{P^2}^{\rm pred}(\lambda_0=1)| = 1.196\times 10^6$ mag directly from the envelope density ratio $\rho_T/\langle\rho\rangle(10\text{ d}) = 8.45\times 10^5$ at the unit-transport benchmark $\lambda_0 = 1$ with bare coupling $|\beta_A| = 1$ fixed by Paper 0 ($\lambda_0 = 0.995 \pm 0.401$). The present paper does not revisit the ladder fit: it tests the separate, falsifiable inter-sector prediction that gravitational waves and electromagnetic radiation yield a *redshift-dependent* difference in inferred luminosity distance.

This modification is not exactly degenerate with a constant *H*<sub>0</sub> shift or with dark energy over a sufficiently broad redshift range, because it predicts a particular functional form for how GW luminosity distances deviate from the &Lambda;CDM expectation as a function of redshift. Over the limited redshift baseline of the current sample and with large GW distance errors, partial degeneracy with *H*<sub>0</sub> can remain significant. Rather than adding another *H*<sub>0</sub> measurement, the question is whether the GW data prefer TEP's bi-metric scaling over the standard single-metric relation.

## 1.2 The TEP Bi-Metric Prediction

In the TEP framework, the metric governing gravitational-wave propagation is related to the electromagnetic metric by a conformal factor that depends on a cosmological scalar field *φ*(*z*):

\begin{equation}
d_L^{(\text{TEP})}(z) = A(z) \, d_L^{\Lambda\text{CDM}}(z; H_0)
\end{equation}

where the redshift-dependent conformal factor is

\begin{equation}
A(z) = \exp\!\bigl[\beta_A\,\phi(z)/M_{\rm Pl}\bigr], \qquad \phi(z) = \phi_0 \bigl[(1+z)^n - 1\bigr] .
\end{equation}

In the lab-fixed test, the parameters *&phi;*<sub>0</sub> and *n* are not free: |*&phi;*<sub>0</sub>| follows from the exploratory Naivasha lab-sector benchmark (now retired), and *n* &approx; 1 is derived in Section 2.1 from minimum-tracking of the density-dependent quartic effective potential, which gives &phi;<sub>min</sub> &prop; &rho;&#772;<sub>m</sub><sup>1/3</sup> &prop; (1+*z*) with a quantified correction *n*<sub>eff</sub> = 0.982 (pipeline step 09k). Throughout this paper *&phi;*<sub>0</sub> denotes the *dimensionless* ratio *&phi;*/*M*<sub>Pl</sub>. The deviation form [*(1+z)<sup>n</sup>* &minus; 1] enforces the ambient convention &phi;(0) = 0, hence *A*(0) = 1. The dimensionless conformal coupling is *&beta;<sub>A</sub>* = &minus;1, with sign fixed by the same convention used in Paper 11: the Cepheid period-contraction effect requires *&beta;<sub>A</sub>&phi;* < 0 in deep potentials, and the corpus endpoint identity 1+*z* = *A*<sub>0</sub>/*A*<sub>em</sub> requires *A*<sub>em</sub> < *A*<sub>0</sub> at positive redshift, so &phi;(*z*) > 0 in the past and the conformal factor satisfies *A*(*z*) < 1 with the magnitude of the departure growing with redshift. This produces a redshift-dependent distance-scale distortion that is tested against &Lambda;CDM in the current public sample.

## 1.3 This Work

This paper tests the TEP bi-metric prediction against combined GWTC standard siren data. Both &Lambda;CDM and TEP distance-redshift relations are fitted to the independent-redshift sample and their &chi;&sup2;, AIC, and BIC values are compared. The primary lab-fixed comparison has the same number of fitted parameters as &Lambda;CDM (only *H*<sub>0</sub> is fitted in each case). A secondary joint-fit diagnostic lets *H*<sub>0</sub>, *&phi;*<sub>0</sub>, *n*, and *&beta;* vary, with an explicit information-criterion penalty for the two effective additional parameters — the likelihood constrains only the product *&beta;&phi;*<sub>0</sub> and *n*.

The structure is as follows: Section 2 derives the cosmological scalar field profile from lab-scale TEP parameters; Section 3 describes the combined GWTC catalog data selection and independent-redshift methodology; Section 4 details the computation pipeline; Section 5 reports the model comparison results; Section 6 discusses the implications for bi-metric propagation; and Section 7 concludes.

## 2. Theoretical Framework

## 2.1 Canonical Radiation Contract and Historical Template

The canonical endpoint identities on the unified matter metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$; Paper 0, §2.2, Step 80) are $A_e=(1+z)^{-1}$, $M_e=A_e\widetilde M$, $M_{\rm det}^{\rm GR}=A_e\widetilde M$, $M_{\rm src}^{\rm GR}=A_e^2\widetilde M$, and $\Xi=A_e$. They contain no fitted coupling amplitude. The universal $\beta_A=-1$ remains fixed. These identities define the target for a joint source, waveform, distance and selection likelihood. Frame consistency of the mass identities is verified symbolically at Paper 0 step_70: a source-local observer measuring with locally determined constants recovers the standard chirp mass ($M_{\rm loc}\to\widetilde M$ for screened systems), and the dimensionless inspiral invariant $G_{\rm loc}\mathcal M f/c^3$ is preserved end-to-end, so the $(1+z)^{-1}$ map is a transport statement rather than nonstandard source physics. Its physical content is the epoch-dependence of the locally measured gravitational constant, $G_{\rm loc}\propto A^2(\phi)(1+\alpha_{\rm amb}^2)$ — and that dependence feeds back on the discriminant itself (Paper 0 step_71): interior k-mouflage tracking gives $G_{\rm loc}(z)\propto(1+z)^{-2}$ inside source galaxies, so characteristic mass scales drift as $\widetilde M^*(z)\propto(1+z)^{3\epsilon}$, yielding $\mathcal M_{\rm det}\propto(1+z)^{3\epsilon-1}$ versus the GR $(1+z)$. At the critical tracking $\epsilon=2/3$ the two predictions coincide; at full tracking ($\epsilon=1$) TEP detector masses rise as $(1+z)^2$ — the discriminant survives with reversed sign — and only pinned interiors ($\epsilon=0$) return the naive $A_e$ mass map. The same drift rescales SN Ia luminosity by $(1+z)^{3\epsilon}$, and the supernova ladder bounds it first for SN Ia progenitors (Paper 26 step 03_11 joint profile bound $\epsilon<0.150$ at $3\sigma$, $\Delta\chi^2=805.8$ at $\epsilon=1$, conditional on the fixed-$z_T=5$ M1 distance relation and Chandrasekhar prescription; Paper 0 step_73 residual-floor bracket $\epsilon \lesssim 0.05$–$0.10$). How that SN-sector bound relates to BBH interiors through the shared $G^{-3/2}$ scaling is set by the progenitor host-assembly derivation (Paper 0 &sect;8; Section 5.6): under the SN-sector bound $\epsilon<0.150$ the discriminant runs as $\mathcal M_{\rm det}\propto(1+z)^{-0.55}$–$(1+z)^{-1}$ ($\propto(1+z)^{-0.28}$ to $(1+z)^{+0.56}$ across the field-spiral/dwarf host-assembly bracket $\epsilon_{\rm eff}\simeq0.24$–$0.52$, with the empirical GWTC profile optimum at canonical pinned $\hat\epsilon=0$), while near-full tracking reverses it to $\propto(1+z)^{3\epsilon-1}$.

The exponential function and density-tracking calculation retained below define the historical phenomenological benchmark, denoted $A_{\rm legacy}(z)$ here (written $A(z)$ in its numerical sections). Sharing the sign $A<1$ and normalization $A(0)=1$ does not make it identical to the exact endpoint contract: at $z=1$, the canonical ratio is $0.5$, while the historical template with $\phi_0=0.013$ gives approximately $0.987$. Its fitted exponent and amplitude therefore characterize that benchmark, not a derivation or fit of the canonical endpoint identity. A nonexpanding common-action background and the coupled radiation calculation must supply the canonical likelihood; no new screening switch or independently adjustable bare coupling is introduced.

In this paper *A*(*z*) is not treated as an EM–GW arrival-time modification and does not evade GW170817 through a conformal propagation delay. It is used as a standard-siren distance-reconstruction projection: a matter-frame calibration factor relating the gravitational-wave inferred luminosity distance to the electromagnetic host-redshift frame under the TEP two-metric map. Differential propagation and cone-tilt constraints remain controlled by the disformal sector *B*(*ϕ*) ≥ 0, while the present test probes whether the standard-siren distance scale contains a conformal-frame residual. The Naivasha lab-sector convention gives the exponential form

**Screening projection notice.** Screening in TEP is represented at theory level by the environmental operator S_Σ(E). Quantities such as ρ_T, R_T(M), S_⊕(r), compactness Φ/c^2, local stellar density, thermal epoch, coherence length, proximity, and boundary geometry are domain-specific projections of E, not independent screening mechanisms and not interchangeable universal thresholds.

\begin{equation}
A(z) = \exp\!\bigl[\beta_A\,\phi_0\bigl((1+z)^n-1\bigr)\bigr]
\end{equation}

where $\phi_0 = +0.013$ is the cosmological field deviation amplitude (Paper 0 &sect;2). The deviation form $(1+z)^n-1$ enforces the ambient convention $\phi(0)=0$ and hence $A(0)=1$; it does not multiply the ambient value itself. With $\beta_A = -1$ (bare coupling, Paper 0 &sect;2.2), the positive $\phi_0$ gives $\phi(z)>0$ and $A(z)<1$ at cosmological distances — the direction required by the corpus endpoint identity $1+z = A_0/A_{\rm em}$, under which positive redshift corresponds to $A_{\rm em} < A_0$ (Paper 26). The same sign structure holds locally: near a mass concentration $\phi > 0$ and $A < 1$ (clocks slower), consistent with the sign convention of Paper 0, so the earlier universe and the denser environment are both deeper temporal wells. The cosmological background $A(z)$ and the spatial perturbation sector are distinct: $A(z)$ is the time-dependent homogeneous field, while the bidirectional spatial deviations ($A < 1$ in overdensities, $A > 1$ in underdensities; Paper 0, §2.2) are perturbations relative to this background that average to the smooth cosmological field along extended lines of sight.

The index $n$ is not an independent assumption: it is fixed by the homogeneous dynamics of the same scalar sector. On the cosmological background the field sits in the density-dependent effective potential

\begin{equation} \label{eq:V_eff}
V_{\rm eff}(\phi) = V(\phi) + \bigl(A(\phi)-1\bigr)\,\bar\rho_m \, ,
\end{equation}

the Einstein-frame form of the universal matter coupling for non-relativistic matter. For the corpus's confining quartic branch $V=\lambda\phi^4/4$ — the density-dependent-mass candidate identified in Paper 26, consistent with the Paper 0/28 quartic-Gaussian envelope near its minimum — the tracked minimum obeys

\begin{equation} \label{eq:phi_min}
\lambda\phi_{\rm min}^3 = -\beta_A\,A(\phi_{\rm min})\,\bar\rho_m \quad\Longrightarrow\quad \phi_{\rm min} = \left(\frac{A(\phi_{\rm min})\,\bar\rho_m}{\lambda}\right)^{1/3} \, ,
\end{equation}

so that $\phi_{\rm min}\propto\bar\rho_m^{1/3}$ to leading order. With $\bar\rho_m\propto(1+z)^3$ this gives $\phi\propto(1+z)$, i.e. $n=1$ exactly in the $|\beta_A\phi|\ll1$ limit; the factor $A(\phi)^{1/3}=e^{\beta_A\phi/3}$ supplies an $\mathcal O(\phi/3)$ correction. Solving the implicit minimum condition $\eqref{eq:phi_min}$ exactly over $z\in[0,3]$ and fitting the deviation form returns $n_{\rm eff}=0.982$ (pipeline step 09k), so $n\approx1$ is a quantified tracking statement rather than an ansatz. The tracking itself is maintained: the curvature about the minimum satisfies $m_\phi/H\gtrsim2.6$ throughout the tested range, placing the field in the quasi-adiabatic regime whose residual lag is of the same order as the fitted index departure. Matching the tracked amplitude to $\phi_0=+0.013$ fixes $\lambda\simeq2.3\times10^{-118}\,M_{\rm Pl}^4$ — the quartic normalization matched to this historical homogeneous-template amplitude. This matching fixes a benchmark coefficient; it does not derive density-dependent running of the operative matter-domain coupling. The same $\rho^{1/3}$ response family underlies the $\rho_T$ saturation scaling of Paper 6, so the homogeneous and screened sectors share the quartic response family rather than requiring separate response relations; the two normalizations characterize different benchmark realizations of that response family; connecting them within one fixed master action requires a background solution rather than identifying them as a derived running coupling (Paper 0 §8); the free-$n$ joint fit ($n=1.47\pm0.86$, step 07c) is retained as a consistency check and agrees with the derived index at the $0.6\sigma$ level.

The conformal amplitude factor $A(z)$ enters the standard-siren reconstruction through the declared endpoint/frame map. It is distinct from the incremental gradient/source response $\mathcal{S}_\Sigma(\mathcal{E})$, which must be evaluated on the selected kinetic field solution when a dynamical screening prediction is required. Theorem 2 of Paper 0 leaves the direct conformal photon propagation contribution zero at fixed metric; endpoint calibration and source evolution are separate observable channels. In this weak-field/long-baseline regime, the global cosmological background acts as the empirical proxy for the continuous saturation of Temporal Topology, anchoring the distance-reconstruction residuals independently of local astrophysical screening. The present test is agnostic about the microscopic origin of the conformal amplitude; it asks only whether the standard-siren distance scale inferred from gravitational-wave events shows a redshift-dependent residual consistent with the lab-sector projection.

## 2.2 Bi-Metric Propagation

The bi-metric framework distinguishes between the metric governing electromagnetic radiation (the observed matter metric) and the metric governing gravitational waves (the effective gravitational metric). This distinction arises naturally from the environment-dependent coupling of the scalar field, where the locally active response is governed by the environmental suppression operator *S*<sub>&Sigma;</sub>(*&Escr;*).

## 2.3 Hubble Diagram Prediction

The TEP-adjusted Hubble diagram overlays the standard &Lambda;CDM prediction and the TEP-scaled prediction on the GW standard-siren data points. The TEP curve predicts a redshift-dependent suppression of the GW distance scale relative to &Lambda;CDM, with amplitude growing as |*A*(*z*) &minus; 1| increases. This shift is tested directly against the data; it is an orthogonal probe of bi-metric propagation and is not expected to align the GW data with any particular external *H*<sub>0</sub> measurement.

## 3. Data Selection

## 3.1 Combined GWTC Catalogs

All publicly available LVK event catalogs queried by the pipeline from the Gravitational-Wave Open Science Center (GWOSC) are combined: GWTC-1-confident, GWTC-2, GWTC-2.1-confident, GWTC-3-confident, GWTC-4.0, GWTC-4.1, O4 Discovery Papers, and GWTC-5.0. Deduplication is performed by `commonName`, with later catalogs taking precedence for updated parameter estimates. Luminosity-distance central values and bounds are extracted from the GWOSC JSON API, and public GraceDB skymaps are used where available for dark-siren host association.

## 3.2 Precision Filtering

Events are filtered to a high-confidence subset with signal-to-noise ratio (SNR) > 12, false-alarm rate (FAR) < 1 per year when available, and *p*<sub>astro</sub> > 0.9 when available. The primary bright-siren anchor is GW170817, the confirmed neutron-star merger with an electromagnetic counterpart and a spectroscopic host-galaxy redshift (NGC 4993, z = 0.0092).

## 3.3 Independent Redshifts — Circularity Avoidance

To avoid the circularity problem that invalidates cosmological tests using GWOSC-derived redshifts (which are computed from luminosity distances assuming &Lambda;CDM), redshifts are obtained from two independent sources:

Bright sirens. Events with confirmed electromagnetic counterparts and spectroscopic host-galaxy redshifts from the literature. Only GW170817 satisfies this criterion in the current sample.

Dark sirens. For events without electromagnetic counterparts, candidate host-galaxy association is performed using public GraceDB HEALPix skymaps where available and a merged redshift-bearing galaxy list. The baseline catalog is GLADE+ from VizieR VII/291 for z < 0.1; DESI DR1 fastspec spectroscopic redshifts provide a deep fallback for higher-z events where GLADE+ is incomplete. Candidates are first filtered by a broad GW-distance compatibility window and then ranked by sky probability and the skymap distance posterior. Distance consistency is recorded and used as a quality control; this makes the dark-siren sample suitable for a pipeline demonstration and sensitivity test, while the bright-siren subset remains the cleanest non-circular anchor. The cone interrogates a peak-centred disc of the 90%-region equivalent area; the resulting sky-probability coverage, a per-event host-misassociation bound, and the residual &Lambda;CDM-concordance dilution of the selection chain are quantified by a dedicated injection test in &sect;5.8.

GWOSC redshift fields are used *only* as a fallback for events where GLADE+ cannot identify a plausible host (e.g., distance-inconsistent candidates or missing skymaps). This affects 34 of 107 events. The remaining 73 events carry host-associated redshifts that do not use the GWOSC redshift field (72 from GLADE+/DESI plus GW170817), and GW170817 provides the bright-siren anchor. Events using GWOSC fallback are explicitly tagged with `quality="fallback"` and excluded from the primary H0 fit to avoid &Lambda;CDM circularity. The primary analysis uses the 73 independent-redshift events only; the full sample of 107 events (including fallback redshifts) is reported as a secondary robustness check.

## 4. Computation

## 4.1 Pipeline Architecture

The reproducible analysis pipeline is implemented in Python and executed sequentially. Each step writes a JSON output to `results/outputs/` and a detailed log to `logs/`. Steps are fail-fast: execution halts on the first failure so that downstream steps do not consume stale data.

## 4.2 GR Distance Extraction

The standard General Relativity luminosity distance d<sub>L</sub><sup>(GR)</sup> and its upper/lower uncertainties are extracted from the GWOSC JSON API for each filtered event. Per-event fractional uncertainties are computed from the published distance bounds. Independent redshifts are taken from step 02 (bright-siren spectroscopy + GLADE+/DESI DR1 dark-siren Bayesian host association); GWOSC redshift fields are used only as a fallback when no catalog host can be identified.

## 4.3 TEP Distance Transformation

The *primary* likelihood model uses the lab-fixed TEP conformal scaling factor *A*(*z*) as an endpoint rescaling of the &Lambda;CDM luminosity distance:

\begin{equation}
d_L^{(\text{GW})}(z) = A(z) \, d_L^{\Lambda\text{CDM}}(z; H_0)
\end{equation}

The TEP-C0 Jordan-frame audit additionally treats the pipeline redshift as the physical matter-frame redshift and modifies the distance integral itself:

\begin{equation}
\frac{H_J(z)}{H_{\Lambda\mathrm{CDM}}(z)} = \frac{A(z)}{1-\alpha_A}, \qquad
\alpha_A = \frac{d\ln A}{d\ln a_J},
\end{equation}

\begin{equation}
d_L^{(\text{GW,C0})}(z) =
A(z)(1+z)c\int_0^z \frac{dz'}{H_J(z')}.
\end{equation}

For observed GW-inferred distances, the corresponding endpoint-only matter-frame corrected distance is d<sub>L</sub><sup>GR</sup> / A(z). Downstream fits use the propagated per-event distance uncertainties from the GWOSC bounds and redshift uncertainty in distance space. The pipeline fails rather than silently reverting to a default uncertainty if the required uncertainty fields are missing from an upstream step.

## 5. Results

## 5.1 Hubble Diagram

The primary manuscript figure overlays the standard ΛCDM curve, the raw unadjusted combined GWTC data points, and the TEP-scaled relation. The TEP adjustment applies a redshift-dependent conformal scaling factor *A*(*z*) to the distance model, producing a predicted deviation from ΛCDM whose amplitude grows with redshift and is not reabsorbable into a constant shift in *H*<sub>0</sub>.

![Hubble diagram showing combined GWTC standard sirens with ΛCDM and TEP distance-redshift curves](public/figures/fig_01_hubble_diagram.png)

Figure 1. Hubble diagram: combined GWTC standard sirens (blue points) with ΛCDM (dashed) and TEP (solid) distance-redshift curves. The TEP relation applies a redshift-dependent conformal scaling *A*(*z*) that deviates from ΛCDM at *z* > 0.1.

## 5.2 Bi-Metric Distance Scale

The Hubble constant is computed by fitting &Lambda;CDM and TEP distance-redshift relations to the GW standard siren sample. Three models are compared: (1) &Lambda;CDM with *H*<sub>0</sub> free; (2) TEP with Naivasha lab-sector convention *&phi;*<sub>0</sub> and *n* fixed, *H*<sub>0</sub> free; (3) TEP joint fit with *H*<sub>0</sub>, *&phi;*<sub>0</sub>, *n*, and the conformal coupling *&beta;* all free. All sign conventions are defined as &Delta;&chi;&sup2; &equiv; &chi;&sup2;<sub>&Lambda;CDM</sub> &minus; &chi;&sup2;<sub>TEP</sub>, so positive values favor TEP. Results are reported relative to the early-universe CMB baseline (Planck: *H*<sub>0</sub> &approx; 67.4 km/s/Mpc) and the local distance ladder (SH0ES: *H*<sub>0</sub> &approx; 73 km/s/Mpc).

**Primary analysis (73 events without GWOSC distance-derived redshifts).** The fit uses 1 independently identified bright siren (GW170817) and 72 GLADE+/DESI dark-siren host associations; the latter pass a &Lambda;CDM-referenced distance-consistency gate and must not be treated as independent spectroscopic host identifications. GWOSC fallback redshifts (34 events) are excluded. The grid-search best fit gives &Lambda;CDM *H*<sub>0</sub> = 72.0 km/s/Mpc and lab-fixed TEP *H*<sub>0</sub> = 72.2 km/s/Mpc. The lab-fixed TEP scaling suppresses the predicted GW distance at fixed *H*<sub>0</sub> through the bi-metric conformal factor *A*(*z*) < 1 with *&beta;<sub>A</sub>* = &minus;1 and *&phi;*<sub>0</sub> = +0.013; the inferred Hubble scale shifts by +0.2 km/s/Mpc relative to the &Lambda;CDM best fit, far below the per-fit uncertainty. The model comparison gives &Delta;&chi;&sup2; = +0.20 and |&Delta;BIC| < 2: a marginal preference for the lab-fixed TEP scaling at equal parameter count, short of decisive on this sample size.

**Secondary diagnostic (full sample, 107 events).** As a robustness check, the fit is repeated on the full sample including the 34 GWOSC fallback redshifts. The best-fit values shift to &Lambda;CDM *H*<sub>0</sub> = 72.4 km/s/Mpc and lab-fixed TEP *H*<sub>0</sub> = 72.4 km/s/Mpc (&Delta;&chi;&sup2; = +0.06), with the absolute scale anchored higher by the fallback redshifts, which were derived under a fiducial &Lambda;CDM cosmology. The TEP joint MCMC fit to the full sample gives *H*<sub>0</sub> = 71.2 &pm; 3.1 km/s/Mpc with posterior-mean *&phi;*<sub>0</sub> = +0.000, *n* = 1.40, and *&beta;<sub>fit</sub>* = &minus;0.03. The 68% credible intervals are *H*<sub>0</sub> &isin; [68.3, 74.0], *&phi;*<sub>0</sub> &isin; [&minus;0.031, +0.032], *n* &isin; [0.42, 2.40], *&beta;<sub>fit</sub>* &isin; [&minus;3.20, +3.15]. The lab-calibrated values (*&phi;*<sub>0</sub> = +0.013, *n* = 1.0, *&beta;<sub>A</sub>* = &minus;1.0) are all consistent with these intervals within 1*&sigma;*.

![Bar chart comparing best-fit H0 from ΛCDM, TEP lab-fixed, TEP joint-fit, Planck CMB, and SH0ES local distance ladder](public/figures/fig_02_h0_reconciliation.png)

Figure 2. Best-fit *H*<sub>0</sub> comparison: ΛCDM, TEP lab-fixed, and TEP joint-fit results from the GW standard siren sample, alongside Planck CMB and SH0ES local ladder reference values.

## 5.3 Model Comparison

Frequentist model comparison (&chi;&sup2;, AIC, BIC) is performed between &Lambda;CDM and TEP bi-metric as competing hypotheses. The joint TEP fit incurs a BIC penalty of *k* ln(*N*) for *k* additional free parameters and must improve &chi;&sup2; by more than this to be preferred. Although the joint sampler explores the nominal (*&phi;*<sub>0</sub>, *n*, *&beta;*) space, the likelihood depends only on the product *&beta;&phi;*<sub>0</sub> — *&beta;* and *&phi;*<sub>0</sub> are individually unconstrained — so the joint model carries two effective additional degrees of freedom (*C* &equiv; *&beta;&phi;*<sub>0</sub> and *n*). A full Bayesian analysis with posterior samples from emcee MCMC provides credible intervals on (*H*<sub>0</sub>, *φ*<sub>0</sub>, *n*, *β*) and tests whether the GW posterior is consistent with the Naivasha lab-sector convention TEP parameters. The free-*β* prior is broad and flat (&minus;5, 5), so the GW data independently constrain the conformal coupling amplitude.

For the primary independent-only sample (73 events), the lab-fixed comparison gives &Delta;&chi;&sup2; = +0.20 and |&Delta;BIC| < 2, a marginal TEP preference at equal parameter count. For the secondary full sample (107 events), the lab-fixed comparison gives &Delta;&chi;&sup2; = +0.06 and |&Delta;BIC| < 2. The TEP joint fit improves on &Lambda;CDM by &Delta;&chi;&sup2; = +2.44 on the independent sample and +0.48 on the full sample at the joint MAP best-fit point, but these gains fall short of the BIC penalty of *k* ln(*N*) = 2 ln(73) &asymp; 8.6 (independent) and 2 ln(107) &approx; 9.3 (full) chi2 units for the two effective additional parameters (*C* &equiv; *&beta;&phi;*<sub>0</sub> and *n*); the corresponding joint &Delta;BIC = &minus;6.1 (independent) and &minus;8.9 (full) reflect information-criterion disfavor at the current sample size rather than a fit deficit — the joint model improves &chi;&sup2; while carrying two extra effective parameters. The MCMC posterior for the joint fit is converged and consistent with the lab-fixed values (*&phi;*<sub>0</sub> = +0.013, *n* = 1.0, *&beta;<sub>A</sub>* = &minus;1.0) at the 68% level, although the broad posterior does not yet independently constrain them. Taken together, the evidence shows a marginal preference for the corpus-sign lab-fixed template, including the sign recovery reported in the matched-filter test below, but the current sample does not yet reach decisive statistical preference.

![Bar chart of Δχ² and ΔBIC for TEP lab-fixed and joint fits relative to ΛCDM](public/figures/fig_03_model_comparison.png)

Figure 3. Model comparison: &Delta;&chi;&sup2; and &Delta;BIC for TEP lab-fixed and TEP joint-fit relative to the ΛCDM baseline. Horizontal dashed lines mark positive (green, &Delta; = &plusmn;2) and strong (orange, &Delta; = &plusmn;6) evidence thresholds.

## 5.4 Conformal Scaling

The TEP conformal scaling factor *A*(*z*; *&phi;*<sub>0</sub>, *n*) quantifies the predicted deviation from GR propagation as a function of redshift. For the lab-fixed convention parameters (*&phi;*<sub>0</sub> = +0.013, *n* = 1.0), *A*(*z*) departs below unity at the sub-percent level across the sample's *z* &lsim; 0.5 baseline, producing a cumulative effect on luminosity distance that is testable with current GW standard siren samples.

![Plot of TEP conformal scaling factor A(z) versus redshift with GW event markers](public/figures/fig_04_conformal_scaling.png)

Figure 4. TEP redshift-dependent conformal scaling *A*(*z*) for lab-fixed convention parameters (*&phi;*<sub>0</sub> = +0.013, *n* = 1.0, red curve). Grey dashed line marks the GR limit *A* = 1. Blue points show the inferred *A*(*z*) for individual GW events.

## 5.5 Posterior Constraints

The joint MCMC fit to (*H*<sub>0</sub>, *&phi;*<sub>0</sub>, *n*, *&beta;*) with 64 emcee walkers &times; 40000 steps (burn-in 2000, thin 10) is converged by the emcee criterion (integrated autocorrelation time *&tau;* &approx; 110–220 steps across parameters, i.e. more than 50*&tau;* post-burn-in steps per walker; 243,200 retained samples, &asymp;11,000 effective). For the primary independent-only sample the posterior mean is *H*<sub>0</sub> = 71.2 &pm; 3.3 km/s/Mpc, *&phi;*<sub>0</sub> = +0.000 &pm; 0.028, *n* = 1.47 &pm; 0.86, *&beta;<sub>fit</sub>* = &minus;0.02 &pm; 2.81. The 68% credible intervals are *H*<sub>0</sub> &isin; [68.1, 74.3], *&phi;*<sub>0</sub> &isin; [&minus;0.033, +0.033], *n* &isin; [0.46, 2.48], *&beta;<sub>fit</sub>* &isin; [&minus;3.29, +3.24]. The lab-calibrated values (*&phi;*<sub>0</sub> = +0.013, *n* = 1.0, *&beta;<sub>A</sub>* = &minus;1.0) are all consistent with these intervals within 1*&sigma;*, although the broad posterior does not yet independently constrain them. Projected onto the locked corpus-sign template, the posterior conformal amplitude is &gamma; = +2.6 &pm; 5.3 with 71.4% of its mass at &gamma; > 0: the free fit's central tendency sits on the corpus-consistent sign (*&beta;&phi;*<sub>0</sub> < 0, hence *A*(*z*) < 1), while the amplitude normalization is not independently bounded. The direct optimizer supplies the best-fit &chi;&sup2; used in model comparison; the posterior provides the parameter consistency test.

![Corner plot of MCMC posterior samples for H0, phi0, n, and beta from TEP joint fit](public/figures/fig_05_corner_posterior.png)

Figure 5. TEP joint-fit posterior *P*(*H*<sub>0</sub>, *&phi;*<sub>0</sub>, *n*, *&beta;* | GW data) from emcee MCMC. Red lines mark Naivasha lab-sector convention parameter values. Marginal distributions show 16th, 50th, and 84th percentiles.

## 5.6 Robustness Diagnostics

Under a properly normalized split-normal likelihood ($\sigma_{\rm norm}=\tfrac12(\sigma_-+\sigma_+)$, Step 09f), the canonical distance map \(\Xi(z)=1/(1+z)\) returns \(-2\ln L=943.14\) at its best-fit \(H_0=56.0\) km/s/Mpc against \(944.31\) at \(H_0=70.0\) km/s/Mpc for the matched &Lambda;CDM comparator on 73 host-associated events (\(\Delta\chi^2=+1.16\) in TEP's favor at equal degrees of freedom), with the &Lambda;CDM-referenced association kernel the operative limitation: because the 72 dark-siren candidate hosts were pre-selected by a &Lambda;CDM distance gate \(d_L^{\Lambda{\rm CDM}}(z_{\rm cat};H_0\approx70)\simeq d_L^{\rm GW}\), fitting \(d_L^{\rm GW}=d_L^{\rm EM}(z_{\rm cat};H_0)/(1+z_{\rm cat})\) to &Lambda;CDM-selected redshifts mechanically scales the free \(H_0\) by \(\langle(1+z_{\rm cat})^{-1}\rangle=0.7878\), predicting \(70.0\times0.7878=55.1\) km/s/Mpc (matching the observed \(56.0\) km/s/Mpc), whereas remapping each event's redshift posterior onto the self-consistent TEP kernel at the fixed canonical \(H_0=68.18\) km/s/Mpc (Paper 18) returns \(-2\ln L=944.31\) — identical to the &Lambda;CDM kernel baseline by construction, so the TEP kernel at canonical \(H_0\) reproduces the &Lambda;CDM association quality without dragging \(H_0\) — and on the spectroscopic bright siren GW170817 (\(z=0.0098\)) fixed \(H_0=68.18\) km/s/Mpc gives \(-2\ln L=4.94\) (TEP) vs \(4.95\) (&Lambda;CDM). Its scalar-response diagnostic transmits approximately 15% of the signal and has approximately 6% detection power, while bright sirens with spectroscopic redshifts across \(z\in[0.1,1.0]\) at fixed \(H_0=68.18\) km/s/Mpc reach \(5\sigma\) discrimination against GR (\(\Xi=1\)) with \(N=25\) events at \(z=0.1\) (\(\Xi=0.909\)), \(N=7\) at \(z=0.2\) (\(\Xi=0.833\)), \(N=3\) at \(z=0.3\) (\(\Xi=0.769\)), and \(N=1\) event at \(z=0.5\) (\(\Xi=0.667\), \(5.0\sigma\) per \(10\%\) distance measurement) to \(z=1.0\) (\(\Xi=0.500\), \(10.0\sigma\)). The catalog-free population benchmark uses 95 BBH events (45 low-redshift anchors and 50 scored events), with both the probability-density Jacobian and the actual reported-source chirp-mass cut of 8 solar masses included. The catalog cut sets the lower bound for 30 anchor events. Continuous physical high-mass cutoff fitting breaks the scale-free power-law degeneracy (\(\beta \simeq 154\,M_\odot\) for TEP, \(\beta \simeq 33\,M_\odot\) for GR). At their canonical parameter-free baselines, canonical pinned TEP out-performs stationary GR by \(\Delta\ln L = +2.32\) out of sample on 50 high-\(z\) events (\(\ln L = -188.68\) vs \(-191.00\)). Profiling the declared tracking and evolution families confirms that both frames achieve their global optimum at their canonical parameter-free baselines (\(\hat\epsilon = 0\), \(\hat k = 0\)), with \(\Delta\ln L_{\rm TEP-GR} = +2.32\) at their respective optima. The empirical envelope, assumed physical mass floor and point-mass treatment make these conditional population scores. Related component-mass checks use overlapping events and are not independent detections. The host-assembly derivation of Paper 0 (&sect;8) now supplies that relation explicitly: progenitors forming in already-deep wells carry \(\epsilon_{\rm eff}\simeq0.004\) (the SN Ia class, matching the Pantheon+ bound \(\epsilon<0.150\) at \(3\sigma\), \(\Delta\chi^2=805.8\)), while BBH progenitors assembled in field-spiral or dwarf hosts carry \(\epsilon_{\rm eff}\simeq0.24\)–\(0.52\) (\(\mathcal M_{\rm det}\propto(1+z)^{-0.28}\) to \((1+z)^{+0.56}\)); with \(\hat\epsilon = 0\) as the global profile optimum, the two populations are not required to share one tracking fraction. A mapped high-redshift mass trend alone is neither a rejection nor an unvetted detection of the canonical contract.

Table 3. Robustness diagnostics for the independent-only sample (73 events). Positive &Delta;&chi;&sup2; values favor TEP over &Lambda;CDM. The matched-filter &chi;&sup2; entries are evaluated on the common log-residual statistic (fixed *H*<sub>0</sub> = 70.4) and are not commensurate with the direct-fit &chi;&sup2; rows, which are minimized over *H*<sub>0</sub>.

| Diagnostic | Value | Interpretation |
| --- | --- | --- |
| Direct H<sub>0</sub> fit (&Lambda;CDM, &sigma;-normalized split-normal) | 70.0 km/s/Mpc, &minus;2ln*L* = 944.31 | Normalized baseline reference for Step 09f |
| Direct H<sub>0</sub> fit (&Lambda;CDM, unnormalized &chi;&sup2;) | 72.0 km/s/Mpc, &chi;&sup2; = 931.83 | Unnormalized baseline comparator for lab-fixed template |
| Direct H<sub>0</sub> fit (TEP lab-fixed, unnormalized &chi;&sup2;) | 72.2 km/s/Mpc, &chi;&sup2; = 931.63 | +0.2 km/s/Mpc shift relative to the matched 72.0 km/s/Mpc &Lambda;CDM fit (72.0 &rarr; 72.2 km/s/Mpc) |
| &Delta;&chi;&sup2; (lab-fixed) | +0.20 | Marginal TEP preference; no added parameters |
| Canonical endpoint map &Xi; = 1/(1+*z*) (Step 09f) | &minus;2ln*L* = 943.14 (*H*<sub>0</sub> = 56.0 free on &Lambda;CDM-gated hosts); &minus;2ln*L* = 944.31 at fixed canonical *H*<sub>0</sub> = 68.18 km/s/Mpc under TEP-kernel host association (GW170817: 4.94 TEP vs 4.95 &Lambda;CDM) | &Delta;&chi;&sup2; = +1.16 in favor of TEP vs &Lambda;CDM baseline (944.31 at *H*<sub>0</sub> = 70.0) on the 73-event sample; the free *H*<sub>0</sub> = 56.0 shift is the exact mechanical artifact $70\langle(1+z)^{-1}\rangle=55.1$ of &Lambda;CDM-gated host selection. At fixed *H*<sub>0</sub> = 68.18 km/s/Mpc, spectroscopic bright sirens reach $5\sigma$ in $N=25,7,3,1$ events at $z=0.1,0.2,0.3,0.5$ |
| Matched-filter &gamma; | +25.2 &pm; 18.5 (Z = +1.36) | Corpus-predicted sign recovered; amplitude above locked template; see geometry controls below |
| &chi;&sup2; locked hypotheses (&gamma; = +1 / 0 / &minus;1) | 41.11 / 41.25 / 41.40 | Corpus branch best; sign-flipped branch worst of the three |
| &chi;&sup2; canonical contract (&Xi; = 1/(1+*z*)) | 43.16, fixed-statistic ln likelihood-ratio proxy = &minus;0.95 | Point prediction slightly disfavored vs &Lambda;CDM on real associations |
| Canonical-injection, transmission-corrected (Step 07b) | &gamma;* = +8.6 vs observed +25.2; power = 0.06; *p* = 0.82 (&gamma;), *p* = 0.78 (&Delta;&chi;&sup2;) | Signal-only injection of &Xi; = 1/(1+*z*) propagated through the measured ~85% association absorption: the channel cannot exclude the canonical contract (optimistic no-absorption bound: power 0.86, *p* = 0.045). Observed statistics consistent with canonical-true filtered through the association |
| Fixed-statistic ln likelihood-ratio proxy (TEP vs &Lambda;CDM) | +0.072 | No discriminatory evidence; not a marginalized Bayes factor |
| Fixed-statistic ln likelihood-ratio proxy (TEP vs sign-flipped) | +0.147 | Corpus branch ranks slightly ahead on this statistic only |
| Spearman *r* (residual vs template) | +0.17, *p* = 0.15 | Positive coherence, not significant |
| Redshift-shuffle *p* | 0.011 | Cross-event pairing significant |
| Within-event weight-shuffle *p* | 1.0 (null median &gamma; = 26.9) | Projection reproduced by candidate-set geometry alone |
| Sky-only control &gamma; | +111.6 &pm; 37.8 (Z = +2.95), fixed-statistic ln likelihood-ratio proxy = +3.18 | Control exceeds the real projection — catalog&times;skymap artifact |
| Leave-one-out *Z* range | +0.94 to +1.67 | Positive sign robust across all 73 jackknife variants |
| Fallback-only control &Delta;&chi;&sup2; | +0.003 | Cosmology-blind machinery null: GWOSC distance-inferred *z* returns residuals &asymp;0 under any distance map; canonical transmission is exactly 0 (injection undefined on this channel) |
| Mass-map population test (Step 09m, catalog-free) | Matter-frame chirp masses under &Xi;: median 45 *M*<sub>&#9737;</sub> (reported ~20), 26% > 65 *M*<sub>&#9737;</sub>, max 160 *M*<sub>&#9737;</sub>; association-*z* scenario: median 41, 19% > 65 | Applying Rule 16 to catalog posterior summaries puts 11/43 selected events above an illustrative 65 *M*<sub>&#9737;</sub> threshold. This is a shape diagnostic, not an intrinsic mass ceiling or calibrated exclusion; the association-*z* variant is GR-conditioned and not a strict lower bound |
| Mass-population likelihood test (Step 09n, catalog-free) | The catalog-free population benchmark uses 95 BBH events (45 low-redshift anchors and 50 scored events), with both the probability-density Jacobian and the reported-source chirp-mass cut of 8 solar masses included. Continuous physical high-mass cutoff fitting breaks the scale-free power-law degeneracy (\(\beta \simeq 154\,M_\odot\) for TEP, \(\beta \simeq 33\,M_\odot\) for GR). At their canonical parameter-free baselines (\(\epsilon = 0\) for TEP, \(k = 0\) for GR), canonical pinned TEP out-performs stationary GR by \(\Delta\ln L = +2.32\) out-of-sample on 50 high-redshift events (\(\ln L = -188.68\) vs \(-191.00\)). Profiling free tracking and evolution families confirms both frames land on their canonical baselines (\(\hat\epsilon = 0.0\), \(\hat k = 0.0\)). | The empirical envelope, assumed physical mass floor and point-mass treatment make these conditional population scores. The continuous cutoff breaks the scale-free identifiability degeneracy present in pure power laws; canonical pinned TEP achieves a +2.32 log-likelihood preference over stationary GR without free evolution parameters. Related component-mass checks use overlapping events and are not independent detections. The host-assembly derivation of Paper 0 (&sect;8) supplies the progenitor-dependent relation: progenitors in deep wells carry \(\epsilon_{\rm eff}\simeq0.004\) (SN Ia class, matching \(\epsilon<0.150\) at \(3\sigma\), \(\Delta\chi^2=805.8\)), while BBH progenitors in field-spiral or dwarf hosts carry \(\epsilon_{\rm eff}\simeq0.24\)–\(0.52\); the empirical optimum sits at canonical pinned TEP \(\hat\epsilon = 0\). |
| Per-galaxy mixture (Step 07f, canonical) | &Delta;&chi;&sup2; = +21.5 observed; decoy median +21.0 (4 off-region rotations); shuffle-null median +22.6; discriminating-subset observed +107.5 vs null +108.4 | Saturated: off-region decoys and every null reproduce the full preference depth; the statistic has zero residual discriminating power at DESI catalog density |

![Four-panel robustness diagnostic showing conformal scaling A(z) by event quality, model comparison by redshift subset, per-event chi2 contributions, and distance residuals for lab-fixed TEP](public/figures/fig_06_tep_robustness.png)

Figure 6. Robustness diagnostics for the lab-fixed TEP signal. Positive &Delta;&chi;&sup2; values favor lab-fixed TEP. The diagnostics show the corpus-predicted bi-metric distance-scale shift and redshift-dependent conformal structure, while the current &chi;&sup2;-level preference remains marginal.

## 5.7 Adversarial Controls

The adversarial controls are intentionally harsher than the baseline fit. On the primary independent-only sample, the lab-fixed sign of *&phi;*<sub>0</sub> shifts the inferred scale by +0.2 km/s/Mpc (72.0 to 72.2 km/s/Mpc) under the &sigma;-normalized likelihood, with the suppressed-distance scaling *A*(*z*) < 1 (*&beta;<sub>A</sub>* = &minus;1, *&phi;*<sub>0</sub> > 0) retained at fixed *H*<sub>0</sub>. On the common matched-filter statistic the corpus branch (&chi;&sup2;(&gamma; = +1) = 41.11) ranks ahead of both the &Lambda;CDM null (&chi;&sup2;(&gamma; = 0) = 41.25) and the sign-flipped control (&chi;&sup2;(&gamma; = &minus;1) = 41.40); forcing the flipped projection to its own best amplitude raises the cost to &chi;&sup2; = 46.83 against 39.39 for the corpus-sign free fit. Zero coupling returns exactly to &Lambda;CDM. Cross-event redshift shuffling returns *p* = 0.011, but the sharper within-event weight-shuffle null — which permutes the candidate weights inside each event while preserving the candidate set — reproduces the fitted projection at *p* = 1.0 (null median &gamma; = 26.9 against the observed 25.2), identifying the statistic as candidate-set-driven rather than association-driven. All 73 leave-one-out variants preserve the positive sign (*Z* = +0.94 to +1.67; removing GW170817 gives *Z* = +1.47), but the projection concentrates entirely in the catalog-incomplete high-redshift half: the *z* < 0.3 subset returns *Z* = +0.07, and removing the four most influential events drops *Z* to +0.37. The sky-only control, which retains catalog positions while discarding per-event association, produces a still larger projection (*Z* = +2.95). The direct-fit redshift subsamples are individually consistent with the null (&Delta;&chi;&sup2; = +0.01 below *z* = 0.2, &minus;0.02 for 0.2 < *z* < 0.4, +0.11 above *z* = 0.25), and &Lambda;CDM mock catalogs show the observed &Delta;&chi;&sup2; is consistent with the null at the current sample size. These diagnostics recover the corpus-predicted sign in the residual statistic while the geometry controls show that statistic is not yet an association-level measurement; the sample does not yet reach discovery-level significance.

![Four-panel adversarial-control plot showing sign control, LCDM mock p-values, generic linear-bias competitor, and chronological split](public/figures/fig_07_adversarial_controls.png)

Figure 7. Adversarial controls. The lab-fixed corpus sign passes the sign-direction test, while the sign-flipped control ranks below both &Lambda;CDM and the corpus branch. Mock-calibrated p-values and chronological splitting are consistent with the observed shift across the observing history, without a decisive &chi;&sup2; preference.

## 5.8 Host Association and Prior Sensitivity

The dark-siren host assignment performs Bayesian marginalization over the merged galaxy candidates within each skymap cone rather than a hard distance-window cut: candidates are ranked by the full 3D skymap posterior (sky probability times the Gaussian distance prior along the line of sight), so poor distance matches are downweighted by the posterior rather than discarded by hand. Because the distance posterior is evaluated under the LVK fiducial cosmology, the retained host-associated redshifts are not fully independent of the &Lambda;CDM distance-redshift relation; they are independent of the GWOSC redshift field, which is the circularity the primary analysis excludes. The residual concordance bias of the host-association step is quantified directly by an injection test on the realized data products: for each event the skymap distance posterior is shifted per pixel by the corpus endpoint factor *A*<sub>inj</sub>(*z*) = exp(&beta;<sub>A</sub>&phi;<sub>0</sub><sup>inj</sup>[(1+*z*)<sup>*n*</sup>&minus;1]) over a grid of amplitudes spanning &plusmn;100&times; the canonical &phi;<sub>0</sub> = 0.013, and the full selection chain — the distance-compatibility window, the per-pixel posterior ranking, and the 0.5–2.0 quality gate — is re-run on the same candidate lists. The result is a strong, quantified concordance dilution: at the canonical amplitude the top-candidate assignment is stable (median point retention 1.0; 12.5% of events exchange the top host) and the quality-gate survival fraction is &asymp;0.99 across the grid, but the distance-gated weighting re-centres the posterior on candidates whose redshifts track the injected shift — the posterior-weighted redshift mean absorbs &asymp;88% of the distortion, so only ~12% of a genuine *A*(*z*) signal survives into the marginalized residual. The same grid is also evaluated at the exact canonical contract *A*(*z*) = 1/(1+*z*): the posterior-weighted mean absorbs 85% of the distortion (83% at the point assignment), and the top host is re-selected in 92% of events — the association tracks the canonical shift into a concordant candidate in nearly every case. The same behaviour holds at large amplitudes: at 100&times; canonical the top host is re-selected in 94% of events, the point-level absorption reaches ~81%, and the posterior-mean absorption remains ~84%. The host-association step therefore imposes a bounded but large concordance dilution: the weighted-redshift channel retains an effective signal lever of only ~12–17% on any distance-map deviation (~15% at the exact contract), which — not the event count — is the limiting factor on this channel's sensitivity (Section 5.6). A complementary dual-frame diagnostic measures the assignment-level frame dependence on the unshifted data: re-running the identical selection chain with the candidate-galaxy distances referenced through the corpus template *A*(*z*) rather than the &Lambda;CDM map re-assigns the top host in 8 of 72 events (11%, consistent with the 12.5% exchange rate the injection grid returns at the same amplitude), with the changed hosts moving systematically to higher redshift as *A*(*z*)<1 requires. The redshift assignment is therefore not frame-neutral at the measured level: the choice of distance–redshift kernel applied to the catalog side alone shifts a subset of host identifications in the direction the conformal map predicts, even before any signal is present in the distance posterior.

The single-host misassociation probability is also quantified per event. The galaxy cone search interrogates a disc of the 90%-region equivalent area centred on the highest-probability pixel; because the LVK localizations used here are predominantly annular, the searched disc contains a median of 6.3% of the skymap probability (71% of events below 10%). The probability that the selected galaxy is the true host is bounded above by the product of the in-cone probability coverage and the within-list posterior weight, giving a median lower-bound misassociation probability of 0.98 across the catalog (0.99 on the independent subset). This is a coverage limitation on host identity, not on the assigned redshift: the redshift estimate is set by the distance posterior evaluated over the listed candidates, and the injection test shows that the point assignment is stable at the canonical amplitude (median retention 1.0) but re-selects between same-distance candidates as the amplitude grows (top-host changes reach ~86% at 50&times; and ~94% at 100&times; the canonical amplitude, with the point-level absorption rising to ~81%). The dark-siren redshift catalog thus functions as a distance-posterior-informed redshift sample with a per-event misassociation bound recorded for every event, consistent with its role as a sensitivity demonstration; full-volume host marginalization over the complete 90% region is retained as the next refinement.

## 6. Discussion

## 6.1 Implications for Bi-Metric Propagation

The two observational channels analysed in Section 5 probe complementary aspects of the TEP bi-metric prediction \(\Xi(z) = d_L^{\rm GW}/d_L^{\rm EM} = 1/(1+z)\). In the host-catalog-free BBH chirp-mass channel (Step 09n, 95 events), introducing a continuous physical high-mass cutoff breaks the scale-free power-law degeneracy (\(\beta \simeq 154\,M_\odot\) for TEP vs \(\beta \simeq 33\,M_\odot\) for GR) and yields an out-of-sample preference of \(\Delta\ln L = +2.32\) (\(\ln L = -188.68\) vs \(-191.00\)) for canonical pinned TEP over stationary GR across the 50 high-redshift events, with both frames achieving their global profile optima at their parameter-free canonical baselines (\(\hat\epsilon = 0\), \(\hat k = 0\)). Progenitor-dependent interior tracking (Paper 0 &sect;8) reconciles this BBH result with the Pantheon+ SN Ia bound (\(\epsilon < 0.150\) at \(3\sigma\), \(\Delta\chi^2 = 805.8\)): progenitors assembled in deep potential wells carry \(\epsilon_{\rm eff}\simeq 0.004\), whereas BBHs formed in field-spiral or dwarf hosts carry \(\epsilon_{\rm eff}\simeq 0.24\)–\(0.52\). In the dark-siren distance channel (Step 09f, 73 events), the split-normal likelihood favors \(\Xi(z)=1/(1+z)\) by \(\Delta\chi^2 = +1.16\) at equal degrees of freedom (\(-2\ln L = 943.14\) vs \(944.31\)); its free-fit \(H_0 = 56.0\) km/s/Mpc reflects the mechanical scaling \(70.0\langle(1+z_{\rm cat})^{-1}\rangle = 55.1\) km/s/Mpc induced by &Lambda;CDM-gated host pre-selection (~85% concordance absorption, transmission \(\approx 15\%\)), while self-consistent TEP-kernel host association at fixed canonical \(H_0 = 68.18\) km/s/Mpc (Paper 18) reproduces the baseline fit quality (\(-2\ln L = 944.31\)) without shifting \(H_0\). Spectroscopic bright standard sirens across \(z\in[0.1,1.0]\) bypass galaxy-catalog weighting entirely and reach \(5\sigma\) discrimination at fixed \(H_0 = 68.18\) km/s/Mpc in \(N = 25, 7, 3,\) and \(1\) events at \(z = 0.1, 0.2, 0.3,\) and \(0.5\).

## 6.2 Limitations

The dark-siren result remains conditioned on galaxy-catalog completeness, sky localization, cone truncation and host ranking. Of 107 events, 73 have host-associated redshifts (one identified bright siren and 72 ranked dark-siren associations); 34 have distance-derived fallback redshifts and serve as a pipeline null. Association-independent spectroscopic hosts provide the cleanest distance-channel test. The mass-population benchmark avoids host catalogs but has its own empirical selection and injection-calibration scope, stated above.

## 6.3 Relation to GWTC-5.0 Modified-Propagation Constraints

The LVK GWTC-5.0 cosmology analysis (Abbott et al., 2026) uses 236 GW sources and reports *H*<sub>0</sub> = 71.0<sup>+9.0</sup><sub>&minus;7.1</sub> km/s/Mpc, finding no evidence for parameterized deviations from GR propagation. The present analysis is not a generic modified-propagation fit; it tests a lab-fixed TEP conformal template with sign and amplitude inherited from the lab-fixed convention. The LVK modified-propagation basis parameterizes distance-redshift deviations as *d*<sub>L</sub><sup>GW</sup>(*z*) = *d*<sub>L</sub><sup>&Lambda;CDM</sup>(*z*)[1 + *&Sigma;*<sub>0</sub>*z*/(1+*z*)<sup>*n*</sup>], which is phenomenologically different from the TEP conformal factor *A*(*z*) = exp(*&beta;<sub>A</sub>**&phi;*<sub>0</sub>[(1+*z*)<sup>*n*</sup> &minus; 1]). A direct mapping between *A*(*z*) and the LVK modified-propagation basis is required before the two results can be compared one-to-one. In particular, the TEP template predicts a redshift-dependent residual structure in log-distance space (ln *A*(*z*) &prop; [(1+*z*)<sup>*n*</sup> &minus; 1]) that is not optimally captured by the LVK linear-bias parameterization. The two analyses are consistent: the LVK analysis finds no generic modified-propagation signal, and the conditional Gaussian-residual injection indicates the present GR-referenced host-association distance channel has low canonical detection power (~6%), while the host-catalog-free mass map is a selected-sample shape diagnostic rather than a calibrated constraint — both outcomes remain compatible with conformal amplitudes below the canonical &Xi; = 1/(1+*z*) normalization over the sampled baseline.

## 7. Conclusions

This paper implements the first observational standard-siren test of the corpus-sign lab-fixed TEP parameterization using combined public GWTC catalogs. Three complementary results emerge from the corrected pipeline. First, the lab-fixed conformal scaling produces the corpus-predicted bi-metric distance-scale shift without adding fitted cosmological degrees of freedom (primary independent-only sample of 73 events: &Lambda;CDM *H*<sub>0</sub> = 72.0 to TEP *H*<sub>0</sub> = 72.2 km/s/Mpc, &Delta;&chi;&sup2; = +0.20, |&Delta;BIC| < 2), a marginal preference at equal parameter count. The redshift-dependent matched-filter residual test recovers the corpus-sign structure at *Z* = +1.36 with the sign-flipped control disfavored, but the fitted amplitude exceeds the locked normalization and the geometry controls — a within-event weight-shuffle null reproducing the projection at *p* = 1.0 and a sky-only control returning a still larger projection — identify the channel as catalog&times;skymap driven rather than association-driven. A conditional Gaussian-residual injection of the canonical contract using the scalar association response (transmission 0.15 from &sect;5.8), rather than end-to-end reassociation, shows the distance channel has only 6% two-sided |*Z*| > 2 detection power if canonical-true — it can neither confirm nor exclude &Xi; = 1/(1+*z*) at this catalog configuration, with the observed projection consistent with canonical-true filtered through the association (*p* = 0.82; optimistic no-absorption bound *p* = 0.045). Second, the joint MCMC posterior is converged and consistent with the lab-fixed convention TEP parameters (*&phi;*<sub>0</sub> = +0.013, *n* = 1.0) at the 68% level, with 71.4% of its mass on the corpus-consistent conformal sign; the posterior does not yet independently constrain the amplitude, but the sign compatibility is a necessary condition for cross-scale consistency. Third, the catalog-free population benchmark uses 95 BBH events (45 low-redshift anchors and 50 scored events), with both the probability-density Jacobian and the actual reported-source chirp-mass cut of 8 solar masses included. The catalog cut sets the lower bound for 30 anchor events. Continuous physical high-mass cutoff fitting breaks the scale-free power-law degeneracy (\(\beta \simeq 154\,M_\odot\) for TEP, \(\beta \simeq 33\,M_\odot\) for GR). At their canonical parameter-free baselines, canonical pinned TEP out-performs stationary GR by \(\Delta\ln L = +2.32\) out of sample on 50 high-\(z\) events (\(\ln L = -188.68\) vs \(-191.00\)). Profiling the declared tracking and evolution families confirms that both frames achieve their global optimum at their canonical parameter-free baselines (\(\hat\epsilon = 0\), \(\hat k = 0\)), with \(\Delta\ln L_{\rm TEP-GR} = +2.32\) at their respective optima. The empirical envelope, assumed physical mass floor and point-mass treatment make these conditional population scores. Related component-mass checks use overlapping events and are not independent detections. The host-assembly derivation of Paper 0 (&sect;8) supplies that relation explicitly: progenitors forming in already-deep wells carry \(\epsilon_{\rm eff}\simeq0.004\) (the SN Ia class, matching the Pantheon+ bound \(\epsilon<0.150\) at \(3\sigma\), \(\Delta\chi^2=805.8\)), while BBH progenitors assembled in field-spiral or dwarf hosts carry \(\epsilon_{\rm eff}\simeq0.24\)–\(0.52\) (\(\mathcal M_{\rm det}\propto(1+z)^{-0.28}\) to \((1+z)^{+0.56}\)); with \(\hat\epsilon = 0\) as the global profile optimum, the two populations are not required to share one tracking fraction. A mapped high-redshift mass trend alone is neither a rejection nor an unvetted detection of the canonical contract. The descriptive mass-map diagnostic is retained as the shape-level companion (median 45 *M*<sub>&#9737;</sub>, 26% above the ~65 *M*<sub>&#9737;</sub> observed edge, maximum 160 *M*<sub>&#9737;</sub>), and the 34-event GWOSC fallback subsample is proven cosmology-blind by the injection machinery (&gamma;* = 0). The simplified &Lambda;CDM null injection returned zero &Delta;&chi;&sup2; > 2 preferences among 500 mock catalogs at the retained redshift/uncertainty structure; this finite-simulation result does not establish a zero false-positive probability under full host selection. The predicted bi-metric signature is orthogonal to the Hubble tension, which Paper 11 addresses through a conditional host-level reconciliation route while its prespecified primary estimator retains a residual discrepancy against Planck.  The absolute &Delta;&chi;&sup2; is small (&Delta;&chi;&sup2; = +0.20 for the primary lab-fixed comparison), the BIC-penalized joint fits remain information-criterion disfavored at the current sample size (&Delta;BIC = &minus;6.1 independent, &minus;8.9 full, under the effective two-parameter penalty for the degenerate *&beta;*–*&phi;*<sub>0</sub> direction), and the matched-filter residual test recovers the corpus-predicted sign without bounding its amplitude. Evaluated at the distance level under the properly normalized split-normal likelihood ($\sigma_{\rm norm}=\tfrac12(\sigma_-+\sigma_+)$, Step 09f), the canonical endpoint map &Xi; = *d*<sub>L</sub><sup>GW</sup>/*d*<sub>L</sub><sup>EM</sup> = 1/(1+*z*) derived from the two-metric action returns &minus;2ln*L* = 943.14 at its best-fit *H*<sub>0</sub> = 56.0 km/s/Mpc against 944.31 at *H*<sub>0</sub> = 70.0 km/s/Mpc for &Lambda;CDM on the independent sample (&Delta;&chi;&sup2; = +1.16 in favor of TEP at equal degrees of freedom; the free *H*<sub>0</sub> = 56.0 shift is the mechanical projection $70\langle(1+z)^{-1}\rangle=55.1$ of the &Lambda;CDM-conditioned dark-siren host selection, whereas self-consistent TEP-kernel host association at fixed canonical *H*<sub>0</sub> = 68.18 km/s/Mpc returns &minus;2ln*L* = 944.31 without shifting *H*<sub>0</sub>, and spectroscopic bright sirens at fixed *H*<sub>0</sub> = 68.18 km/s/Mpc reach $5\sigma$ discrimination in $N=25,7,3,1$ events at $z=0.1,0.2,0.3,0.5$), while the catalog-free BBH population test (Step 09n) prefers canonical pinned TEP over stationary GR by &Delta;ln*L* = +2.32 (conditional likelihood ratio &approx; 10.2) out of sample. The endpoint scaling *d*<sub>L</sub><sup>GW</sup> = *A*(*z*) *d*<sub>L</sub><sup>&Lambda;CDM</sup> is the primary likelihood model; the Jordan-frame integral mapping is singular on the canonical branch (&alpha;<sub>A</sub> &equiv; 1) and is therefore not an admissible model comparison there. The reproducible pipeline records each analysis step, propagates asymmetric event-level uncertainties, and the host-marginalization and catalog-completeness framework is calibrated for expansion as the GW event sample grows in the O5 era and beyond.

## References

[1] Abbott, B. P., et al. (LIGO/Virgo). (2019). GWTC-1: A gravitational-wave transient catalog of compact binary mergers observed by LIGO and Virgo during the first and second observing runs. *Physical Review X*, 9(3), 031040.

[2] Abbott, B. P., et al. (LIGO/Virgo). (2021). GWTC-2: Compact binary coalescences observed by LIGO and Virgo during the first half of the third observing run. *Physical Review X*, 11(2), 021053.

[3] Riess, A. G., et al. (2022). A comprehensive measurement of the local value of the Hubble constant with 1 km/s/Mpc uncertainty from the Hubble Space Telescope and the SH0ES team. *The Astrophysical Journal Letters*, 934(1), L7.

[4] Planck Collaboration. (2020). Planck 2018 results. VI. Cosmological parameters. *Astronomy & Astrophysics*, 641, A6.

[5] Abbott, R., et al. (LIGO/Virgo/KAGRA). (2021). GWTC-2.1: Deep extended catalog of compact binary coalescences observed by LIGO and Virgo during the first half of the third observing run. arXiv:2108.01045.

[6] Abbott, R., et al. (LIGO/Virgo/KAGRA). (2021). GWTC-3: Compact binary coalescences observed by LIGO and Virgo during the second part of the third observing run. *Physical Review X*, 13(4), 041039.

[7] The LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. (2025). GWTC-4.0: Updating the Gravitational-Wave Transient Catalog with Observations from the First Part of the Fourth LIGO-Virgo-KAGRA Observing Run. arXiv:2508.18082.

[8] The LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. (2026). GWTC-5.0: Updated LIGO–Virgo–KAGRA Catalog sets new records in precision gravitational wave astronomy. *News | LIGO Lab | Caltech*, 26 May 2026.

[9] The LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. (2026). GWTC-5.0: Constraints on the Cosmic Expansion Rate and Modified Gravitational-wave Propagation. arXiv:2605.27227.

[10] Abbott, B. P., et al. (LIGO/Virgo). (2017). GW170817: Observation of gravitational waves from a binary neutron star inspiral. *Physical Review Letters*, 119(16), 161101.

[11] Abbott, B. P., et al. (LIGO/Virgo). (2017). A gravitational-wave standard siren measurement of the Hubble constant. *Nature*, 551(7678), 85–88.

[12] Dalya, G., et al. (2022). GLADE+: An extended galaxy catalogue for multimessenger searches with advanced gravitational-wave detectors. *MNRAS*, 514(2), 1403–1415.

[13] DESI Collaboration. (2024). The Early Data Release of the Dark Energy Spectroscopic Instrument. arXiv:2306.06308.

[14] Fishbach, M., et al. (2019). A standard siren measurement of the Hubble parameter from GW170817 without the distance ladder. *The Astrophysical Journal Letters*, 871(1), L13.

[15] Palmese, A., et al. (2021). Comparison of two binary black hole host-galaxy catalogues: GLADE and DESI. *MNRAS*, 505(3), 3923–3935.

[16] Creminelli, P., & Vernizzi, F. (2017). Dark energy after GW170817 and GRB170817A. *Physical Review Letters*, 119(25), 251302.

[17] Ezquiaga, J. M., & Zumalacárregui, M. (2017). Dark energy after GW170817: Dead ends and the road ahead. *Physical Review Letters*, 119(25), 251304.

[18] Riess, A. G., et al. (2024). The SH0ES team: 2024 update on the local measurement of the Hubble constant. *The Astrophysical Journal Letters*, submitted.

## Data Availability & Reproducibility

All data used in this analysis are publicly available and reproducibly downloaded. No synthetic, fabricated, or simulated data is used in the main analysis.

Synthetic catalogs appear only in the Step 08 sensitivity-calibration diagnostic, where mock distances are generated from the real event redshift and uncertainty structure to estimate false-positive and recovery rates. They are not used as observational evidence in the main ΛCDM/TEP comparison.

*Data sources:*

- GWOSC combined catalogs: GWTC-1-confident, GWTC-2, GWTC-2.1-confident, GWTC-3-confident, GWTC-4.0, GWTC-4.1, O4 Discovery Papers, GWTC-5.0 — gwosc.org

- GraceDB public skymaps (bayestar.fits.gz): gracedb.ligo.org

- GLADE+ Galaxy Catalog (VizieR VII/291): glade.plus

- DESI DR1 redshift catalogs (zpix-main-dark + zpix-main-bright zcatalog monoliths, 13.4M good-galaxy spectroscopic redshifts): data.desi.lbl.gov

- NASA/IPAC Extragalactic Database redshift-bearing objects: ned.ipac.caltech.edu

*Pipeline steps (21 sequential stages):*

| Step | Script | Description |
| --- | --- | --- |
| 00 | `step_00_download_gwtc5_catalog.py` | Download combined GWTC catalogs from GWOSC |
| 01 | `step_01_precision_filtering.py` | Filter events by SNR > 12 and high confidence |
| 01b | `step_01b_download_desi.py` | Download DESI DR1 zcatalog monoliths and build the HEALPix-indexed 13.4M-galaxy cache (optional, ~20 GB download) |
| 02 | `step_02_independent_redshifts.py` | Build independent-redshift dataset (bright + GLADE+/DESI DR1 dark sirens) |
| 02b | `step_02b_host_association_systematics.py` | Host-association systematics: cone coverage, misassociation bound, and concordance-bias injection |
| 02c | `step_02c_desi_supplement.py` | Merge DESI DR1 candidates into stored host pools and rebuild posteriors |
| 03 | `step_03_compute_dl_gr.py` | Extract GR luminosity distances from LVK posteriors |
| 04 | `step_04_compute_dl_tep.py` | Compute TEP conformal scaling and matter-frame corrected distances |
| 04b | `step_04b_metric_direction_audit.py` | Audit A(z) vs 1/A(z), model-side vs data-side transformation conventions |
| 05 | `step_05_hubble_diagram.py` | Construct Hubble diagram data |
| 06 | `step_06_h0_reconciliation.py` | Secondary analysis: fit H₀ from full sample including fallback redshifts |
| 06b | `step_06b_h0_reconciliation_independent.py` | Primary analysis: fit H₀ from independent-only redshifts (no circularity) |
| 07 | `step_07_statistical_tests.py` | Goodness-of-fit, tension metrics, and model comparison for both analyses |
| 07b | `step_07b_tep_matched_filter.py` | Primary discovery statistic: TEP-template residual projection and wrong-sign test |
| 07c | `step_07c_independent_only_mcmc.py` | Primary independent-only joint MCMC with posterior gamma projection |
| 07d | `step_07d_fallback_only_control.py` | Fallback-only control MCMC (distance-inferred redshifts are cosmology-blind; machinery null) |
| 07f | `step_07f_per_galaxy_likelihood.py` | Per-galaxy hypothesis-consistent siren likelihood with multi-decoy and discriminating-subset nulls |
| 08 | `step_08_synthetic_injections.py` | Synthetic injection test: recovery and false-positive calibration |
| 08e | `step_08e_event_jackknife_influence.py` | Event jackknife: leave-one-out, redshift cuts, and precision cuts |
| 09 | `step_09_generate_figures.py` | Generate manuscript figures from real pipeline outputs |
| 09m | `step_09m_mass_map.py` | Canonical Rule-16 mass-map population consistency diagnostic (catalog-free) |
| 09n | `step_09n_mass_population.py` | Two-sided mass-population likelihood test: GR vs TEP bookkeeping with empirical selection envelope and contract-amplitude contour |
| 10 | `step_10_pipeline_audit.py` | Pipeline audit: verify execution integrity and output consistency |

The complete analysis pipeline, including all step scripts, is available at github.com/matthewsmawfield/TEP-LVK.

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# Temporal Equivalence Principle: The Dirac Limit of Dynamical Proper Time
**Matthew Lukin Smawfield**
Version: v0.4 (Qatar)
First published: 12 June 2026 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.20572697

---

## Abstract

Standard relativistic quantum mechanics, including the Klein–Gordon and Dirac equations, is recovered here as the screened, flat-frame tangent limit of a deeper dynamical proper-time phase transport governed by the Temporal Equivalence Principle (TEP). By governing proper-time accumulation *dτ* through a dynamical scalar field *φ* rather than treating time as a universal parameter, the causal matter metric is shown to reproduce the standard free-particle structures in the screened limit — a necessary consistency check for the theory — and two further structures are geometrically reinterpreted at the level of postulate.

(1) The phase action *S = −mc^{2} ∫ dτ̃* emerges as the primitive geometric driver, with mass appearing in the primitive action as the parameter governing the oscillator frequency, *ω_{0} = mc^{2} / ℏ*, modulated by the conformal factor in the causal matter metric *g̃_{μν}*. (2) The Klein-Gordon equation is derived from the minimal geometric Lagrangian in the causal metric and verified via WKB / eikonal expansion; its eikonal limit recovers the *g̃*-Hamilton-Jacobi equation, not via operator substitution. (3) The Dirac operator is recovered as the local Clifford/tetrad representation in the isochronous background — it emerges in the limit where temporal shear *Σ_{μ}* and disformal coupling *B(φ)* are negligible. (4) Spin-1/2 is reinterpreted as temporal-orientation holonomy of the proper-time phase frame, and antiparticles as reversed proper-time orientation within the temporal-orientation bundle, eliminating the need for a macroscopic two-sheeted spacetime. (5) The spinor structure of relativistic quantum mechanics may be reinterpreted geometrically: the 1928 Dirac spinor admits a retrospective geometric interpretation as an algebraic encoding of temporal-orientation holonomy on an isochronous background. These results recover standard relativistic quantum mechanics as the screened tangent-space limit of the TEP causal geometry, in the regime where the temporal field is frozen and locally constant over the interaction scale; the saturation scale *ρ_{T} ≈ 20 g/cm^{3}* is empirically calibrated in TEP-UCD (Paper 6). The scope is stated plainly: the Klein–Gordon and Dirac recoveries are exact in the conformal limit *B(φ) = 0* and constitute a consistency proof that QFT on the causal metric reduces to the standard form in the screened regime, rather than a TEP-specific derivation; the spin-1/2 and antiparticle reinterpretations are postulates about an orientation bundle with winding and non-abelian structure that the single scalar *φ* does not generate; the interacting theory — Born rule, gauge-invariant interactions, QED matching, *g*-factor, spin-statistics — is deferred to the companion Papers 24–25 (in preparation); and the estimated conformal mass shift (*δm/m &lesssim; 1.5 × 10^{−19}*) is a falsifiability floor below realistic interferometric sensitivity, not a detectable prediction of this paper.

Keywords: quantum foundations, Dirac equation, proper time, phase transport, spin, antimatter, scalar-tensor theories, geometric quantum mechanics, temporal equivalence principle, Hamilton-Jacobi equation, Klein-Gordon equation

## 1. Introduction: The Background-Dependency of Relativistic Quantum Theory

### 1.1 The External-Metric Postulate

Every equation of quantum mechanics — from the Schrödinger equation to the Dirac equation, from the path integral to the operator formalism of Quantum Field Theory — is formulated on a prescribed spacetime background. The metric *g_{μν}* is not dynamical in the quantum sector; it is an external input, fixed either as the Minkowski metric *η_{μν}* in flat-space QFT or as a classical solution to Einstein's equations in QFT in curved spacetime. In neither case does the quantum field back-react on the temporal geometry. Proper time *τ* is computed from a metric that is unaffected by the quantum state of the matter fields it governs.

**Screening projection notice.** Screening in TEP is represented at theory level by the environmental operator S_Σ(E). Quantities such as ρ_T, R_T(M), S_⊕(r), compactness Φ/c^2, local stellar density, thermal epoch, coherence length, proximity, and boundary geometry are domain-specific projections of E, not independent screening mechanisms and not interchangeable universal thresholds.

**Evidence status.** This paper is classified as a *candidate microscopic completion* in the TEP corpus. It derives the Klein–Gordon and Dirac operators as the screened tangent limit of a dynamical proper-time geometry, but full Born-rule recovery, gauge-invariant interaction structure, and quantitative QED matching remain open closure tasks. The antiparticle reinterpretation, wavefunction-collapse model, and entanglement ontology presented here are theoretical proposals, not empirically validated replacements for standard quantum mechanics. The results should be evaluated as structural exploration rather than as direct empirical proof of TEP.

This is not a bug in the formalism; it is a deliberate approximation. The energy scales of known particle physics are so far below the Planck scale that gravitational back-reaction is negligible. However, the approximation becomes a structural limitation when one asks whether the quantum state itself — the phase coherence, the entanglement, the spin orientation — might be fundamentally tied to the geometry that the metric describes. In standard relativistic quantum theory, the metric is a stage; the quantum fields are actors. The Temporal Equivalence Principle reverses this hierarchy.

**Conceptual continuity with Einstein's geometric programme.** This reversal places TEP in conceptual continuity with Einstein's search for a continuous geometric foundation beneath the discrete formal machinery of quantum theory. Einstein's later unified-field programme can be read not as a failure of the geometric instinct itself, but as evidence that the available geometry remained tied too closely to spacetime curvature while quantum phase, spin, and matter–antimatter structure were treated as algebraic additions. TEP shifts the geometric object from the gravitational metric alone to the causal proper-time geometry governing matter-clock phase transport. In this sense, the present paper does not replace the empirical success of relativistic quantum mechanics; it asks whether the Klein–Gordon and Dirac structures are the flat-frame, screened expression of the continuous geometry Einstein sought but could not formulate in terms of dynamical proper time.

A critical distinction must be made. General Relativity *does* recognise local proper-time variance — gravitational time dilation and path-dependent proper time are well-measured phenomena, and in general curved or rotating spacetimes ($g_{0i} \neq 0$) GR does not admit a global proper-time synchronization. However, GR computes all gravitational dynamics and all matter proper-time intervals from a single universal metric tensor *g_{μν}*, without an independent dynamical scalar clock field or internal orientation sector. In conventional curved-spacetime quantum field theory, the background metric is solved or prescribed externally while quantum phase, spin, and matter–antimatter structure are carried as algebraic structures on that single geometry. TEP introduces a distinct causal matter metric *g̃_{μν}* governed by a dynamical scalar field *φ* together with a postulated temporal-orientation bundle; its physical significance depends on regimes and observables not already contained in standard single-metric curved-spacetime field theory.

The Temporal Equivalence Principle goes further. It governs the local rate of proper time not merely as a parameter computed from a prescribed metric, but through a *dynamical scalar field* *φ* satisfying its own field equations and coupled to matter density through a conformal–disformal metric. The metric that governs quantum phase transport is not the gravitational (Einstein-frame) metric *g_{μν}*; it is the causal matter metric *g̃_{μν}*, which differs from the gravitational metric by a conformal factor *A(φ)* that modulates with local matter density. In standard relativistic quantum theory, even in curved spacetime, the metric is externally prescribed and the quantum field is quantised on that fixed background. The Bogoliubov transformation between vacua encodes particle creation (the Unruh and Hawking effects), but the metric itself remains classical and unaffected. This paper targets the structural assumption that the metric governing quantum phase transport is externally prescribed rather than dynamically coupled to the matter fields it governs.

### 1.2 The Inventions It Forces

When a dynamical, density-dependent temporal landscape is forced into the framework of a fixed background metric, the theory must invent compensating structures to account for the discrepancy:

- *Wavefunction collapse:* The sudden, non-unitary reduction of the quantum state upon measurement may reflect the inability of a fixed-background formalism to accommodate the geometric interaction between a probe and the local temporal shear field. In a dynamical proper-time geometry, the measurement process is reinterpreted as the equilibration of probe and system to a shared temporal contour: as the probe approaches the interaction region, the joint shear field relaxes to a common isochronous manifold. The apparent collapse may be reinterpreted as the geometric relaxation of two initially independent temporal topologies to a single synchronised phase, rather than a discontinuous jump. The fluid dynamics of the temporal shear field are proposed to conserve geometric stress; what standard quantum mechanics records as projection onto an eigenstate is, in the TEP framework, the settling of the combined system onto a contour of constant proper-time phase. The amplitude-curvature term that appears in the WKB expansion of the geometric Klein–Gordon equation (§2.4) is the natural point of contact with the quantum-potential term of de Broglie–Bohm mechanics: TEP identifies it as a transport effect of the causal phase geometry, not as a hidden-variable ontology. A rigorous derivation of this correspondence — including the recovery of the Born rule from geometric statistics — from the full TEP field equations, including the non-linear disformal sector, is beyond the scope of this paper and is addressed in TEP-KIN (Paper 25, in preparation).

- *Intrinsic spin:* In standard quantum mechanics, spin-1/2 is described by an abstract SU(2) representation without reference to spatial extent. In the TEP framework, spin-1/2 is the temporal-orientation holonomy of a localized topological charge — a geometric, not algebraic, property.

- *Antiparticles as separate ontological species:* The Dirac equation's negative-energy solutions are reinterpreted as a separate particle species. In the TEP framework, the antiparticle sector is the reversed proper-time phase orientation in the orientation-conjugate branch of the temporal-orientation bundle.

- *Virtual particles as force carriers:* The exchange of momentum between particles across empty space is modeled by the emission and absorption of unobservable virtual bosons. In the TEP framework, forces may be reinterpreted as routed through the continuous geometry of the disformally tilted light cone. The derivation of the Maxwell equations and full quantum electrodynamic interactions from the TEP disformal geometry is not provided in this paper; it is addressed in TEP-SPIN (Paper 24, in preparation).

### 1.3 The Temporal Equivalence Principle

The Temporal Equivalence Principle (TEP) governs the local calibration of proper time through a dynamical scalar field *φ* coupled to a conformal–disformal metric (Paper 0). Matter clocks tick at rates set by the conformal factor *A(φ) = exp(β_{A} φ / M_{Pl})*. The critical saturation scale *ρ_{T} ≈ 20 g/cm^{3}* (empirically calibrated in TEP-UCD, Paper 6, New Delhi, and cross-checked across compact-object and galactic scales) is the Temporal Topology saturation scale at which screening effects saturate; it is a property of the theory's non-linear regime and not a binary threshold. In the screened limit, where the observable shear sector *Σ_{μ}^{obs} → 0* with *A(φ)* normalized to its ambient environmental value, the locally observable Temporal Shear is suppressed and standard physics is recovered. In the unscreened regime, the full geometric structure of the Temporal Topology is manifest. The disformal coupling *B(φ)* controls the tilting of the light cone by the temporal gradient; in the screened limit the observable disformal response is suppressed and interactions become isotropic.

What "screened" means for a scattering event requires care. Following Paper 0 (§7), recovery of standard physics is controlled by suppression of the observable shear sector, *Σ_{μ}^{obs} = 𝒮_{Σ}(ℰ) Σ_{μ} → 0*, not by comparison of any local density against *ρ_{T}*; high-energy probes are represented by channel-specific response coefficients *κ_{X}* rather than by the ambient bulk-density screening function. For collider processes the relevant statement is kinematic: the scalar field's configuration is set by boundary conditions over its correlation length (*R_{T}(M_{&oplus;}) ≈ 4 × 10^{3} km* in the terrestrial calibration of Paper 6), so across the ~10^{−18} m, ~10^{−27} s extent of a TeV-scale interaction the field is frozen and locally constant — *Σ_{μ} ≈ 0* and *B(φ)(∇φ)^{2} ≈ 0* over the entire interaction region. A frozen, locally constant field is precisely the tangent-space limit of Section 3. The KG–Dirac and perturbative-QFT descriptions therefore remain exact for currently accessible particle-physics experiments, while long-baseline, long-integration interferometry — which accumulates phase over scales comparable to field gradients — is the probe class that can resolve the unscreened structure.

The foundational geometric framework of TEP was established in the original theory paper (Smawfield 2026, Paper 0, DOI: 10.5281/zenodo.16921911), which introduced the full disformal metric *g̃_{μν} = A^{2}(φ) g_{μν} + B(φ) ∇_{μ}φ ∇_{ν}φ* with *A(φ) = exp(β_{A} φ / M_{Pl})*, demonstrated local Lorentz invariance as a theorem, and introduced the sector ontology of Temporal Topology (spatial/covariance structure of *ln A(φ)*), Temporal Shear (*Σ_{μ} = ∇_{μ} ln A*), and the saturation scale *ρ_{T}*. This paper works in the conformal limit *B(φ) = 0* and adopts the foundational convention throughout: the symbol *A(φ)* denotes the original conformal factor of the Jakarta theory, so the causal matter metric reads *g̃_{μν} = A^{2}(φ) g_{μν}*. The screened limit *A(φ) → 1* recovers the standard Minkowski background; the defect core, where the conformal response saturates and the orientation sector carries winding, marks the topological defects identified in Section 4. The conformal limit suffices for the KG–Dirac tangent-limit derivation because a non-zero disformal response does not invalidate the conformal recovery; it marks the breakdown of the isotropic tangent limit and introduces direction-dependent causal corrections. The full disformal structure becomes essential for interaction routing, synchronization holonomy, interferometric observables, and multi-messenger tests, addressed in companion papers.

Companion papers in this series develop TEP across a wide range of mass densities, from subatomic scales (TEP-SPIN) through laboratory interferometry (TEP-KIN) to cosmological distances (TEP-C0). This paper addresses the quantum regime directly, arguing that the KG–Dirac sector of relativistic quantum theory is the flat-frame, isochronous tangent limit of a deeper dynamical proper-time geometry.

Paper 0 establishes local Lorentz invariance as a theorem: in local freely falling frames of *g̃_{μν}*, physics reduces to special relativity (Axiom A2). The present paper is the quantum-sector instantiation of that theorem. Sections 2–3 exhibit the explicit KG–Dirac tangent limit and, critically, the leading departure from it — the temporal-shear coupling *Σ_{μ}* in the conformal Dirac operator (§3.2.1) — so that the recovery of standard relativistic quantum mechanics is not asserted but constructed, with its first correction term in hand.

### 1.4 Structure of This Paper

Section 2 establishes the phase action and recovers the Klein-Gordon relation as the Euler-Lagrange equation of the minimal scalar-field Lagrangian in the causal metric. Section 3 delivers the central derivation: the Dirac operator is recovered as the local tetrad representation in the screened limit. Section 4 reinterprets spin and antiparticle structure as geometric orientations. Section 5 concludes and outlines the synthesis with companion papers.

### 1.5 Broader Framework Context

Beyond the tangent-space derivations developed below, TEP carries conceptual implications that extend across the companion series. Bell's theorem rules out any theory in which quantum correlations are explained by pre-assigned local hidden variables satisfying the usual separability and factorisation assumptions. TEP does not attempt to restore such a theory. Instead, it challenges the deeper premise that entangled systems are fundamentally separable subsystems embedded in an inert, externally prescribed background.

In the TEP framework, the interval between entangled charges is not an empty stage but part of the same dynamical temporal geometry that governs phase transport. Entanglement is therefore interpreted not as a superluminal signal between isolated particles, but as a non-separable geometric contour in the conformal–disformal proper-time field. Measurement perturbs one boundary of a shared temporal configuration; the observed correlation reflects the global constraint structure of that configuration rather than the transmission of information through space.

This view is best described as geometric non-separability rather than conventional local realism. It preserves the empirical force of Bell-type violations while relocating their ontological meaning: Bell excludes separable local hidden-variable completions of quantum mechanics, but it does not by itself exclude a realist theory in which the relevant physical object is an extended temporal-geometric structure rather than two independent point systems. The full disformal treatment of measurement, entanglement, and geometric stress propagation is developed in TEP-KIN and TEP-SPIN. The present paper establishes the tangent-space quantum limit on which those extensions build.

Temporal-topology drag denotes the influence of large-scale temporal-field configurations on local particle dynamics, mediated by the conformal factor *A(φ)* rather than spacetime curvature; environment-dependent mass emerges from modulation of the proper-time oscillator frequency by *A(φ)*. What standard fixed-background quantum theory models as virtual force carriers and statistical wavefunctions may be effective descriptions of geometric stress propagation in a contiguous temporal fluid. These phenomena furnish concrete experimental signatures developed in TEP-SPIN (Paper 24) and TEP-KIN (Paper 25).

**Cosmological clock-rate interpretation.** The same proper-time oscillator law also supplies the local quantum input for the cosmological papers in this series. If atomic emission frequencies are governed by *ω_{local}(φ) = ω_{0} A(φ)*, then spectra emitted across cosmological gradients of the temporal field may contain a conformal clock-rate component in addition to standard Doppler or metric-expansion interpretations. The present paper does not develop that cosmological model; it records the quantum-sector mechanism. The full redshift, distance-ladder, and effective-dark-energy interpretation is developed in TEP-C0.

### 1.6 Conventions and Notation

Unless otherwise stated, the metric signature *(+, −, −, −)* is used. Paper 0 works in the *(−,+,+,+)* signature; this paper uses *(+, −, −, −)*, and all imported expressions have been converted. Natural units *c = ℏ = 1* are used inside derivations; dimensional factors are restored in summary equations. With *x^{0} = ct*, the Klein–Gordon mass term is *m^{2}c^{2} / ℏ^{2}*, and the Dirac mass term is *mc / ℏ*. Throughout, "screened limit" denotes *Σ_{μ}^{obs} → 0* with *A(φ)* normalized to its ambient environmental value (Paper 0, §7); the shorthand *A → 1* refers to this normalization.

## 2. Proper-Time Phase Transport

### 2.1 The Phase Action

The fundamental action for a massive particle in the TEP framework is:

S = −mc^{2} ∫ dτ̃

where *τ̃* is the dynamical proper time measured along the particle's worldline in the causal matter metric *g̃_{μν}*. This is not an abstract phase — it is the *physical accumulated phase* of the matter-clock oscillator, whose frequency is set by the local conformal factor *A(φ)*.

Dimensional analysis confirms the primitive status of this action. The dimensions are:

[S] = [m] [c^{2}] [τ̃] = M L^{2} T^{−1} = [ℏ]

Thus *S/ℏ* is dimensionless, as required for a phase. Given Axiom A2 of Paper 0 — all non-gravitational processes evolve in the proper time of *g̃* — this action is the unique Lorentz-invariant, first-order geometric functional of the proper-time interval. The mass *m* appears not as an inertial parameter but as the frequency of phase accumulation: *ω_{0} = mc^{2} / ℏ*.

The causal matter metric is related to the gravitational (Einstein-frame) metric *g_{μν}* by the conformal transformation:

g̃_{μν} = A^{2}(φ) g_{μν}

The bare conformal coupling is frozen at $\beta_A = -1$ (Paper 0 §2.2); under the unified unstretched-space metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ (Paper 0 §2.2, Step 80), Cassini constrains the screened source-charge projection $S_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ ($5.8 \times 10^{-6}$ on the isotropic-conformal benchmark; with the solved two-branch solar profile giving $y_{\rm tb}(1.6\,R_\odot)=2.16\times 10^{-8}$ and $|\gamma_{\rm PPN}-1|=4.32\times 10^{-8}\ll 2.3\times 10^{-5}$, Paper 0 §7), not $\beta_A$ itself. In the screened limit, where the observable shear sector *Σ_{μ}^{obs} → 0* with *A(φ)* normalized to its ambient environmental value, *A(φ) → 1* and *g̃_{μν} → g_{μν}*. In the unscreened regime, the particle's phase accumulates along geodesics of the causal matter metric rather than the bare Einstein-frame gravitational metric.

### 2.2 The Massive Particle as Proper-Time Oscillator

A massive particle is defined geometrically as a stable, local proper-time oscillator governed by the causal metric. The oscillator's natural frequency in the flat-frame limit is:

ω_{0} = mc^{2} / ℏ

In the full TEP geometry, this frequency is modulated by the conformal factor:

ω_{local}(φ) = ω_{0} · A(φ)

Mass parameter and conformal rescaling. The bare mass *m* enters the primitive action *S = −mc^{2} ∫ dτ̃*. The local oscillator frequency is *ω_{local} = mc^{2} A(φ) / ℏ*. The Klein-Gordon equation uses the bare parameter *m* because the conformal modulation is already encoded in the causal d'Alembertian *□̃* (via *A^{2}* in the metric). The effective inertial mass measured by local clock response reduces to *m* when *A → 1* in the screened limit.

This local oscillator law is also the quantum-sector input used by the cosmological papers in this series. Since atomic emission frequencies are governed by the same matter-clock phase rate, *ω_{local}(φ) = ω_{0} A(φ)*, spectra emitted across large gradients of the temporal field need not be interpreted solely as Doppler or metric-expansion effects. In the present paper this observation is not developed into a cosmological model; it simply records the local mechanism by which conformal proper-time modulation enters atomic frequency standards. The cosmological redshift interpretation is developed in TEP-C0 (Paper 26), where the accumulated variation of *A(φ)* along cosmological baselines is compared with the standard expansion history.

### 2.3 The g̃-Hamilton-Jacobi Equation

The Hamilton-Jacobi equation for the proper-time phase action in the causal metric is:

g̃^{μν} ∂_{μ}S ∂_{ν}S = m^{2}c^{2}

This is the fundamental classical equation governing the phase transport of a massive particle. It is not an operator equation — it is the Hamilton-Jacobi equation in a curved geometry. The correspondence between the wave equation and this classical equation is constructed geometrically (§2.4) rather than imposed by operator substitution; the quantum sector itself enters through Axiom A2.

Metric signature convention. Throughout this paper, the metric signature is *(+, −, −, −)*. In the flat Minkowski limit, with *x^{0} = ct* and *η_{μν} = diag(1, −1, −1, −1)*, the Hamilton-Jacobi equation takes the explicit form:

(∂_{0}S)^{2} − |∇S|^{2} = A^{2}(φ) m^{2}c^{2}

### 2.4 Deriving the Klein-Gordon Equation from Geometric First Principles

In the following derivation *c = ℏ = 1* is set; dimensional factors are restored in the summary equations. The Klein-Gordon equation is derived as the Euler-Lagrange equation of the minimal scalar-field Lagrangian in the causal metric. The derivation proceeds in three steps.

#### Step 1: The geometric scalar-field Lagrangian.

The phase field *Ψ* is a complex scalar field propagating on the causal manifold with metric *g̃_{μν}*. The covariant, second-order Lagrangian for a massive scalar field that reduces to the standard Klein-Gordon Lagrangian in the flat limit is:

L = ½ ( g̃^{μν} ∂_{μ}Ψ^{*} ∂_{ν}Ψ − m^{2} |Ψ|^{2} )

The metric tensor *g̃^{μν}* raises indices; the inverse metric is *g̃^{μν} = A^{−2}(φ) g^{μν}*. The bare mass parameter *m* enters here because it is the only dimensionful parameter available from the primitive action *S = −mc^{2} ∫ dτ̃*. This Lagrangian follows directly from the causal metric *g̃_{μν}* via minimal coupling; its form is fixed by the geometric structure established in Section 2.1. The linear structure of the resulting wave equation follows from the stationarity condition on this action, exactly as in standard field theory, but now formulated in the causal geometry.

#### Step 2: Euler-Lagrange equation.

Varying the action *S = ∫ L √−g̃ d^{4}x* with respect to *Ψ^{*}* gives:

∂_{μ} ( √−g̃ g̃^{μν} ∂_{ν}Ψ ) − √−g̃ m^{2} Ψ = 0

Dividing by *√−g̃* yields the causal d'Alembertian form:

□̃ Ψ + m^{2} Ψ = 0

where *□̃ = (1 / √−g̃) ∂_{μ}( √−g̃ g̃^{μν} ∂_{ν} )* is the d'Alembertian operator constructed from the causal metric *g̃_{μν}*.

#### Step 3: Connection to the Hamilton-Jacobi equation via WKB / eikonal expansion.

To verify that this linear wave equation is consistent with the classical Hamilton-Jacobi equation derived in Section 2.3, insert the WKB / eikonal ansatz:

Ψ = R e^{iS/ℏ} , R, S ∈ ℝ

Substituting into *(□̃ + m^{2}) Ψ = 0* and dividing by *e^{iS/ℏ}* yields an equation that can be organised by powers of *ℏ*. Before collecting orders, recall the exact decomposition of the causal d'Alembertian established in Step 2. With *g̃_{μν} = A^{2}(φ) η_{μν}* in the flat-background limit,

□̃ = A^{−2}(φ) (□_{M} + 2 Σ^{μ} ∂_{μ})

where *□_{M} = ∂_{0}^{2} − ∇^{2}* is the Minkowski d'Alembertian. The conformal factor *A(φ)* is the same scalar field-dependent rescaling introduced in Section 2.1; throughout the eikonal expansion it is treated as a prescribed geometric background, not as part of the *ℏ*-expansion of *Ψ*. Its derivatives *∂_{μ}A = A Σ_{μ}* enter only through the second, temporal-shear term.

At leading order *O(ℏ^{−2})*, the real part recovers the Hamilton-Jacobi equation. The conformal bookkeeping at this order is explicit. From the WKB ansatz, *∂_{ν}Ψ = (i / ℏ) Ψ ∂_{ν}S + O(ℏ^{0})*, so the shear term is *O(ℏ^{−1})* and does not contribute at *O(ℏ^{−2})*. The conformal factor therefore enters the leading order exclusively through the kinetic prefactor *A^{−2}(φ)* in the first term: gradients of the phase *S* are contracted with the inverse causal metric *g̃^{μν} = A^{−2}(φ) η^{μν}* before any quantum corrections appear. The mass term carries no conformal prefactor (it is the bare parameter *m* from the primitive action). Collecting the *ℏ^{−2}* coefficients therefore gives

−A^{−2}(φ) R [(∂_{0}S)^{2} − |∇S|^{2}] + m^{2} R = 0

Dividing by *R* and multiplying by *−A^{2}(φ)* immediately yields the *g̃*-Hamilton-Jacobi equation in natural units, with the conformal mass shift *A^{2}(φ)* on the right-hand side:

(∂_{0}S)^{2} − |∇S|^{2} = A^{2}(φ) m^{2}

The appearance of *A^{2}(φ)* at this order is not an independent postulate; it is the classical signature of propagation on the causal manifold. In the screened limit *A → 1*, the equation reduces to the standard Hamilton-Jacobi form; in the unscreened regime, the effective inertia of the phase front is modulated by the local conformal clock-rate rescaling before any *O(ℏ^{−1})* transport or *O(ℏ^{0})* quantum-potential terms enter.

At next order *O(ℏ^{−1})*, the imaginary part gives the transport equation in the causal metric:

∂_{0}(R^{2} ∂_{0}S) − ∇ · (R^{2} ∇S) + 2A^{−1}R^{2} η^{μν}(∂_{μ}A)(∂_{ν}S) = 0

which expresses conservation of the probability current *j^{μ} = R^{2} ∂^{μ}S* modified by the temporal shear *Σ_{μ} = ∂_{μ} ln A*. In the screened limit *A → 1*, *∂_{μ}A → 0*, this reduces to the standard transport equation.

At finite order *O(ℏ^{0})*, the quantum-potential term with shear coupling appears:

A^{−2} □_{M} R / R + 2A^{−3} η^{μν}(∂_{μ}A) (∂_{ν}R) / R

which reduces to *□_{M} R / R* in the screened limit and is suppressed by *ℏ^{2}* relative to the Hamilton-Jacobi term.

This is the natural point of contact with the quantum-potential term familiar from de Broglie–Bohm formulations [11]: the standard amplitude-curvature term remains, while TEP supplies an additional shear-coupled geometric correction. TEP does not require a hidden-variable ontology; it identifies the potential-like term as a transport effect of the causal phase geometry.

The Klein-Gordon equation is thus *derived* from the minimal geometric Lagrangian in the causal metric, and its eikonal limit is verified to coincide with the *g̃*-Hamilton-Jacobi equation. The inputs are the causal metric *g̃_{μν}*, the bare mass *m* from the primitive action, and the standard scalar-field Lagrangian minimally coupled to that metric. No operator substitution is required.

In the screened limit *g̃_{μν} → η_{μν}* and the standard Klein-Gordon equation is recovered. In the unscreened regime, the causal d'Alembertian encodes the full geometric structure of the temporal field, including additional temporal-shear coupling terms proportional to *∂_{μ}A*.

## 3. The Dirac Operator as a Screened Limit

### 3.1 The Standard Dirac Equation

The Dirac equation, the cornerstone of relativistic quantum mechanics, is:

(iγ^{μ}∂_{μ} − m) ψ = 0

where *γ^{μ}* are the Dirac matrices satisfying the Clifford algebra *{γ^{μ}, γ^{ν}} = 2η^{μν}*. This equation is noted for its Lorentz covariance, its prediction of antimatter, and its role as the foundation of Quantum Electrodynamics.

### 3.2 Algebraic Flat-Space Recovery

TEP attributes the difficulty of geometrising the quantum sector to a metric misattribution: particle phase transport was mapped onto the gravitational metric while proper time remained a universal parameter. TEP demonstrates that the Klein–Gordon and Dirac equations are the screened, flat-frame tangent limits of a deeper dynamical proper-time phase transport. The geometric operator successfully reduces to the familiar Dirac operator only when the temporal background is artificially flattened. By governing proper-time accumulation τ through a dynamical scalar field φ and deriving the action from the causal matter metric *g̃_{μν}*, the geometric language of a deeper temporal topology is naturally revealed.

The standard Dirac equation is recovered as the local Clifford/tetrad representation in the isochronous (screened) limit. This is an exact mathematical result: the geometric operator reduces to the familiar Dirac operator when the temporal background is flat.

In curved spacetime, spinors cannot be defined directly on the manifold. They require a local frame (tetrad) *e^{a}_{μ}* at each point, related to the metric by:

g_{μν} = e^{a}_{μ} e^{b}_{ν} η_{ab}

The Dirac matrices in curved spacetime are defined as *γ^{μ} = e_{a}^{μ} γ^{a}*, where *γ^{a}* are the flat-space Dirac matrices. The Dirac operator in curved spacetime becomes:

(iγ^{μ}∇_{μ} − m) ψ = 0

where *∇_{μ}* is the spin-covariant derivative, incorporating the spin connection *ω^{ab}_{μ}*.

The flattening conditions. The standard Dirac equation *(iγ^{μ}∂_{μ} − m) ψ = 0* makes two critical assumptions that are never stated explicitly:

- *The temporal shear vanishes:* *Σ_{μ} = ∇_{μ} ln A(φ) = 0*. This means the conformal factor is locally constant, and up to constant unit normalization the causal matter metric coincides with the Einstein-frame gravitational metric.

- *The observable disformal response is suppressed:* *B(φ)(∇φ)² → 0*. This means the light-cone tilt becomes phenomenologically negligible, and all interactions are effectively isotropic in the screened regime.

When these two conditions are imposed, the causal metric *g̃_{μν}* reduces to the Minkowski metric *η_{μν}*, the tetrad field becomes the identity, and the spin-covariant derivative reduces to the ordinary partial derivative. The full geometric Dirac operator:

(i e^{μ}_{a} γ^{a} ∇_{μ} − m) ψ = 0

collapses to the standard form:

(iγ^{μ}∂_{μ} − m) ψ = 0

### 3.2.1 The Conformal TEP Dirac Operator Before Screening

Before imposing the flattening conditions, the conformal sector of the causal metric is *g̃_{μν} = A^{2}(φ) η_{μν}*. The corresponding tetrad and curved-space Dirac matrices are:

g̃_{μν} = A^{2}(φ) η_{μν}, ẽ^{a}_{μ} = A(φ) δ^{a}_{μ}, ẽ_{a}^{μ} = A^{−1}(φ) δ_{a}^{μ}, γ̃^{μ} = A^{−1}(φ) γ^{μ}.

The spin-covariant Dirac operator on this conformal geometry therefore takes the unrescaled four-dimensional conformal form

D̃ ψ = i A^{−1} γ^{μ} (∂_{μ} + 3/2 Σ_{μ}) ψ − m ψ = 0,

where *Σ_{μ} = ∂_{μ} ln A(φ)* is the temporal shear. Under the Weyl-rescaled spinor convention *χ = A^{3/2} ψ*, the derivative shear term can be absorbed into the rescaled spinor field. The equation may then be written equivalently as

A^{−5/2} i γ^{μ} ∂_{μ} χ − m A^{−3/2} χ = 0,

or, after multiplying by *A^{5/2}*,

i γ^{μ} ∂_{μ} χ − m A χ = 0.

Thus the temporal-shear contribution can be represented either as an explicit spin-connection term in the unrescaled matter-clock spinor *ψ*, or as a conformal mass modulation in the Weyl-rescaled spinor *χ*. In the matter-clock convention used here, the explicit *Σ_{μ}* term is retained because it displays the temporal-shear correction directly. In the screened limit:

A(φ) → 1, Σ_{μ} → 0 ⇒ i γ^{μ} ∂_{μ} ψ − m ψ = 0.

This *Σ_{μ}*-dependent term is the explicit conformal TEP correction: standard Dirac dynamics is recovered only when the temporal-shear contribution vanishes in the screened limit.

### 3.2.2 Disformal Departure from Conformal Collapse

The conformal sector *g̃_{μν} = A^{2}(φ) η_{μν}* alone admits tetrad rescaling *e^{a}_{μ} → A(φ) δ^{a}_{μ}*. The rank-one disformal correction *B(φ) ∇_{μ}φ ∇_{ν}φ* cannot be absorbed into a single conformal factor: for generic *∇_{μ}φ ≠ 0*, off-diagonal components such as *g̃_{0x} = B(φ) ∂_{t}φ ∂_{x}φ* are non-zero unless *B(φ) = 0*. Even with temporal shear suppressed (*A(φ) = 1*, *Σ_{μ} = 0*), the metric departs from Minkowski through *g̃_{00} − 1 = B(φ)(∂_{t}φ)^{2}* and *g̃_{xx} + 1 = B(φ)(∂_{x}φ)^{2}*. Standard Clifford collapse therefore requires both flattening conditions of Section 3.2. SymPy Audit A.4 in `results/sympy_audit.log` verifies this signpost; the full disformal tensor audit (inverse metric, null-cone tilt, Christoffel symbols, synchronization holonomy) is in TEP-KIN (Paper 25), `results/disformal_kinematics_audit.log`.

Bridge to the full matter metric. The conformal tangent-limit proof of Sections 2–3 is not an abandonment of the Jakarta ansatz *g̃_{μν} = A^{2}(φ) g_{μν} + B(φ) ∇_{μ}φ ∇_{ν}φ*; it isolates the tangent-space algebra that the KG–Dirac sector of relativistic quantum theory already assumes. Particle-physics experiments operate in the screened regime where *B(φ)(∇φ)^{2} → 0*; in the highly screened regime of particle colliders, where the scalar field is frozen and locally constant over the interaction scale (§1.3), this disformal suppression is not merely a convenient approximation but practically exact. The Dirac and Klein-Gordon derivations of this paper are therefore the exact limiting forms of the same geometric operator written against the full *g̃_{μν}*. Outside that limit, the disformal sector does not modify the Clifford algebra by a small perturbative correction; it replaces the isotropic Minkowski inner product with a direction-dependent causal structure. TEP-KIN (Paper 25) carries the full metric through: the symbolic inverse *g̃^{μν}*, the effective refractive index *n*_{eff} that tilts null cones, the Christoffel symbols that route geodesics through *B(φ)* gradients, and the synchronization-holonomy sector that distinguishes conformal transport (exact one-form *d ln A*) from genuinely non-integrable disformal transport. The narrative continuity is therefore explicit: Paper 23 establishes what the KG–Dirac sector of relativistic quantum theory recovers when both temporal shear and observable disformal response are suppressed; Paper 25 establishes how interactions, measurement, and interferometric observables behave when they are not.

Exact tensor algebra. The derivation proceeds by exact tensor algebra. The individual steps — tetrad collapse to the identity, vanishing of the spin connection, and reduction of the covariant derivative to the ordinary partial derivative — are the standard flat-space limit of the Fock–Ivanenko formalism in curved-spacetime quantum mechanics. Every term in the geometric operator is tracked through the flattening limit; no approximation is made. The standard Dirac operator is the exact local Clifford/tetrad representation in the isochronous background. This tensor-algebraic derivation has been verified symbolically using SymPy; see `results/sympy_audit.log`.

### 3.3 The Screened Limit

The flattening conditions correspond to the screened limit, where the observable shear sector *Σ_{μ}^{obs} → 0* with *A(φ)* normalized to its ambient environmental value (Paper 0, §7). In this regime:

- *A(φ) → 1* (conformal factor approaches unity)

- *Σ_{μ} = ∇_{μ} ln A(φ) → 0* (temporal shear vanishes)

- *B(φ)(∇φ)² → 0* (observable disformal response suppressed)

The standard Dirac equation is thus the *screened limiting case* of the geometric operator, valid in the regime where the geometric structure of the temporal field is negligible. For collider processes the relevant statement is kinematic: the scalar field's configuration is set by boundary conditions over its correlation length (*L_{c} ≈ 4 × 10^{3} km* in the terrestrial calibration of Paper 6), so across the ~10^{−18} m, ~10^{−27} s extent of a TeV-scale interaction the field is frozen and locally constant — *Σ_{μ} ≈ 0* and *B(φ)(∇φ)^{2} ≈ 0* over the entire interaction region. A frozen, locally constant field is precisely the tangent-space limit derived above. The KG–Dirac and perturbative-QFT descriptions therefore remain exact for currently accessible particle-physics experiments, while long-baseline, long-integration interferometry — which accumulates phase over scales comparable to field gradients — is the probe class that can resolve the unscreened structure.

The operator collapse of the curved-space Dirac equation into the flat-space Dirac limit serves as the quantum-geometric realization of the abstract environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$. The saturation of the Temporal Topology at the Compton scale $r_c$ acts here as the physical proxy for $\mathcal{S}_\Sigma(\mathcal{E})$, yielding the standard flat-space behavior without committing to specific chameleon or symmetron microphysics.

### 3.4 Geometric Reinterpretation of Spinor Structure

The KG–Dirac tangent-limit derivation shows that the standard Dirac equation is recovered within the TEP framework as the screened limit. The TEP framework further provides a geometric reinterpretation of the spinor structure that Dirac introduced in 1928.

The algebraic structure of the Dirac equation — the Clifford algebra, the spinor representation, the charge conjugation and parity operations — all emerge from the tetrad structure. When the tetrad is trivial (flat background), these operations appear as abstract algebraic symmetries. When the tetrad is non-trivial (curved temporal background), they are revealed as geometric orientation operations on the proper-time manifold.

In the TEP framework, the spinor is proposed as a mathematical encoding of temporal-orientation holonomy. The Clifford algebra is proposed as the local tangent-frame algebra of the temporal orientation bundle, with the gamma matrices *γ^{a}* generating the Clifford algebra and their commutators *σ^{ab} = (1/2)[γ^{a}, γ^{b}]* generating infinitesimal Lorentz transformations of the proper-time phase frame (Appendix A.2). Within this paper the proposal is supported at the abelian level (Appendix A.5); the non-abelian completion is developed in TEP-SPIN (Paper 24). This reinterpretation follows from the geometric structure of the temporal manifold and offers a geometric candidate for the algebraic structure introduced in 1928.

The Dirac equation is thus recovered within the TEP framework as the screened tangent-space limit when temporal shear and disformal coupling are negligible. This characterizes the standard theory as the tangent limit of a deeper geometric structure.

## 4. Spin and Antiparticles as Geometric Orientations

### 4.1 Spin-1/2 as Temporal-Orientation Holonomy

In the TEP framework, spin-1/2 arises as temporal-orientation holonomy of topological charge defects in the temporal field. In standard quantum mechanics, spin-1/2 is described by an abstract SU(2) representation without reference to spatial extent. The SU(2) spinor encodes angular momentum *ℏ/2* algebraically, not as a physical rotation in space.

In the TEP framework, spin is reinterpreted as temporal-orientation holonomy. A fermion is modeled as a localized defect in the temporal orientation bundle, with two sectors distinguished. The scalar amplitude sector: the core is a region of saturated conformal response, with *ln A(φ)* smooth and single-valued, so that *Σ_{μ} = ∇_{μ} ln A* remains an exact one-form and *∮ Σ_{μ} dx^{μ} = 0* around every loop — in exact agreement with the conformal-exactness theorem of Paper 0 (§3.3). The conformal sector carries no holonomy, and the causal metric *g̃_{μν} = A^{2}(φ) g_{μν}* is single-valued everywhere. The orientation sector: the defect regime is postulated to carry a circle-valued orientation phase on the temporal orientation bundle — additional structure beyond the single real scalar *φ*, in the same sense that the conjugate orientation bundle of §4.3 is postulated structure. Its winding around the defect core is the topological invariant identified with spin. The local temporal shear orients the defect relative to the matter-clock congruence but does not itself carry the winding. This division of labour places the holonomy in exactly the sector Paper 0 (§3.3) reserves for non-exact transport structure beyond the conformal one-form.

Singularity regulation. The defect core, where the conformal response saturates, is adopted here as a strict topological point defect for holonomy purposes: the postulated orientation phase winds around the topological core, and Stokes' theorem is inapplicable precisely because the orientation connection is not globally exact. The precise codimension of the idealized singularity in the orientation bundle is not fixed by the single scalar field *φ* alone; the physically relevant scale is the Compton-radius smearing discussed below. This idealization parallels the Dirac-string construction for a magnetic monopole or the *1/r* vortex of a superfluid; it is not regulated by perturbative renormalization of a zero-dimensional source. Physically, the fermion carries finite Compton-scale extent *r_{c} = ℏ/(mc)*; the scalar field equation with potential *V(φ)* saturates the conformal response at the core, smearing the curvature of the orientation connection, *d&thetasym;*, over this radius rather than admitting a bare ultraviolet divergence. A loop penetrating this smeared core (*r < r_{c}*) therefore encounters redistributed curvature rather than a point singularity, yet the enclosed holonomy does not degrade continuously: the integer winding *n = ±1* in the orientation bundle is a topological invariant, and any closed contour encircling the defect yields the same orientation holonomy *∮ &vartheta;_{μ} dx^{μ} = ±4π* regardless of how deeply it penetrates the core. Observable spin and transport are computed in the exterior regime *r &gg; r_{c}*, where the *1/r* shear profile applies; the interior regularization is derived in TEP-SPIN (Paper 24).

The "spin-1/2" property follows from the half-integer coupling of the matter-clock phase to the orientation sector. The postulated connection *&vartheta;_{μ}* carries winding *∮ &vartheta;_{μ} dx^{μ} = n · 4π*; the phase couples with weight *ℏ/2*, so the minimal winding *n = ±1* yields *ΔS/ℏ = ±2π*, preserving single-valuedness of *Ψ = exp(iS/ℏ)*. A *2π* spatial rotation of the defect changes the orientation phase by *±π*, producing the characteristic spinor sign flip *Ψ → −Ψ*. The *4π* periodicity is therefore a consequence of the half-integer weight, not an additional postulate. A full recovery of fermionic exchange antisymmetry requires the multi-particle configuration-space treatment developed in TEP-SPIN. The two "spins" are the two possible orientations of the topological charge relative to the local matter-clock congruence:

- *Spin up:* The topological charge rotates in the same sense as the local temporal shear circulation.

- *Spin down:* The topological charge rotates in the opposite sense.

On this construction, *SU(2)* is proposed as the holonomy group of the orientation bundle — postulated structure beyond the scalar sector — with the Pauli matrices *σ_{i}* as its generators. The *U(1)* winding exhibited in Appendix A.5 is the abelian core of that proposal; the non-abelian completion is developed in TEP-SPIN (Paper 24).

### 4.2 The g-Factor as Geometric Ratio

The anomalous magnetic moment *g − 2* arises in QED from loop corrections involving virtual photons. In the TEP framework, the g-factor is a geometric ratio that measures the deviation from the flat-frame approximation, and the QED loop expansion is interpreted as the screened-limit representation of a deeper geometric correction associated with topological charge structure. Far from the charge core, where the orientation-sector curvature becomes negligible, the geometric correction vanishes and *g → 2*, recovering the Dirac value. The geometric contribution to the g-factor is addressed in TEP-SPIN (Paper 24).

### 4.3 Antiparticles as Reversed Proper-Time Orientation

In standard quantum mechanics, antimatter is introduced as a separate field of particles with opposite charge. The Dirac equation's negative-energy solutions are reinterpreted via the Feynman-Stückelberg interpretation as positive-energy antiparticles moving backward in coordinate time.

In the TEP framework, antimatter is not an independent ontological species, nor does it require a macroscopic 'second sheet' of the universe. It is interpreted as reversed proper-time phase orientation localized strictly within the internal phase space (the temporal orientation bundle) of the topological defect.

The macroscopic spacetime manifold remains a single, continuous, orientable Lorentzian manifold. The 'double cover' required to support opposing temporal orientations exists entirely within the temporal-orientation bundle of the scalar field. Where a particle's phase advances as *+dτ̃* aligned with the ambient temporal shear, its antiparticle advances as *−dτ̃* with an inverted phase relative to the ambient shear.

The Decoupling Vertex and Topological Conservation: At the pair production vertex, the spacetime manifold does not tear or branch. The electromagnetic coupling furnishes the energy budget for a null phase contour to resolve into two orientation-conjugate timelike excitations: a forward-oriented matter-clock mode and a reverse-oriented charge-conjugate mode. In screened laboratory language these appear as particle and antiparticle; in TEP ontology they are opposite phase orientations of the same continuous proper-time geometry.

Laboratory realisation: pair production. The abstract seam *J_{br}* admits a concrete laboratory identification in photon-induced pair production, *γ + Z → e^{−} + e^{+} + Z*, where a high-energy photon converts in the Coulomb field of a nucleus *Z*. The process is traced here in laboratory coordinates *(t, x)*; the orientation assignment is a property of the temporal phase bundle, not of the spatial direction of motion in the lab frame.

*(i) Pre-junction approach.* Before the vertex, the photon propagates on the future null cone with *k_{μ} k^{μ} = 0* in *g̃*. Its phase action *S* advances along the null generator; no timelike proper-time oscillator is yet present because the photon is massless. The covector *k_{μ} = ∂_{μ}S* is future-directed and null, placing the photon on the branch locus *J_{br}* rather than on either orientation branch.

*(ii) Junction event.* At the interaction vertex—the spacetime event at which the electromagnetic coupling supplies sufficient energy–momentum budget to populate the electron mass shell—the local covector remains null on the photon mass shell but is no longer globally continuable on a single orientation branch: the connected phase contour must decouple to support two timelike, on-shell continuations. This vertex is a point of *J_{br}*. It is localised to a single laboratory event (a point in *(t, x)*), not extended along a spatial hypersurface; detectors record the bifurcation only through the emergence of two timelike tracks downstream of that event.

*(iii) Bifurcation into orientation-conjugate excitations.* From the connected contour on the decoupling vertex emerge two branches. The electron continues with timelike *k^{μ}*, accumulating phase as *+dτ̃* along a future-directed worldline aligned with the ambient temporal shear. The positron continues with reversed proper-time orientation within the temporal-orientation bundle, *dτ̃ → −dτ̃* relative to the matter-clock congruence. Both worldlines advance forward in coordinate time *t*; the distinction is geometric, not a reversal of laboratory causality. The Feynman–Stückelberg reinterpretation is recovered as the screened-laboratory projection of a past-directed proper-time orientation: the positron appears in detectors as a particle of opposite charge propagating forward in *t*, while its phase transport is reversed strictly within its internal phase frame. In the screened limit, kinematic worldline observables such as opposite track curvature in a magnetic field and coincidence timing follow from the two timelike branches, whereas quantitative QED radiative cross-sections (such as bremsstrahlung spectra) must be recovered by the open interacting completion; the TEP content here is the geometric identification of the creation vertex as a topological decoupling event.

*(iv) Post-creation separation and entanglement.* As the pair separates, each branch carries a localized topological charge (Section 4.1) on its respective orientation branch. The phase contour rooted at the *J_{br}* event remains a single connected geometric object: TEP-KIN (Paper 25) identifies this as the macroscopic entanglement link between the bifurcated charges. The junction event is therefore not only the orientation decoupling but also the origin of the shared orientation-bundle holonomy that enforces quantum correlations between the outgoing tracks.

*(v) Annihilation as recombination.* The reverse process, *e^{−} + e^{+} → γ + γ*, is recombination on *J_{br}*: timelike contours from the matter and antimatter orientation branches converge at a common null event, the shared phase contour closes, and a single future-directed null continuation emerges. Creation and annihilation are time-reversed navigations of the same topological seam.

Textual spacetime schematic. The same navigation can be read directly in the laboratory *(t, x)* plane, with *x* along the photon beam axis and *t* increasing upward. For *t < t_{V}*, the incident photon follows a null generator at slope *dt/dx = ±1/c* (future light cone); its phase contour lies on the branch locus *J_{br}*. At the vertex *V = (t_{V}, x_{V})*, that null generator meets *J_{br}*—a single event, not a spatial line or wall. For *t > t_{V}*, two timelike worldlines branch from *V*, each with *|dx/dt| < c* and both advancing in coordinate time. The electron worldline *γ_{e^{−}}* carries forward orientation *dτ̃ > 0*; the positron worldline *γ_{e^{+}}* carries reversed orientation *dτ̃ < 0* in the matter-clock congruence. The two branches may separate symmetrically in *x* (for example, emission at equal angles about the beam axis) while remaining future-directed in *t*. A track photograph or time-resolved coincidence measurement records only these timelike curves in *(t, x)*; the orientation labels are not imprinted on the tracks but are fixed by the sign of phase accumulation along each worldline. Annihilation is the time-reverse reading of the same diagram: *γ_{e^{−}}* and *γ_{e^{+}}* converge at a vertex *V′* on *J_{br}*, and the outgoing photon again follows a null generator from *V′*. The junction is thereby visible only as a fork or merge in the causal structure of recorded worldlines, localised to the vertex events *V* and *V′*.

Crucially, CPT symmetry is *preserved locally*. The local CPT theorem remains valid at every point in spacetime: a full rotation in the local orientation bundle (C × P × T) returns the system to its original state. Charge conjugation C is realised as a local reflection in the orientation bundle (reversing the direction of proper-time phase accumulation), not as a global spacetime transition. A particle and its antiparticle correspond to opposite orientations within the temporal-orientation bundle; this separation is a consequence of the internal phase geometry, not of the local C operation itself. This construction is not a substitute for the screened QED calculation of pair-production amplitudes; it supplies the proposed TEP geometric interpretation of the same laboratory process. This suggests a possible topological origin for the observed matter-antimatter asymmetry: rather than requiring a symmetry-breaking event in the early universe, the dominance of matter may reflect an orientation bias in the cosmological temporal-orientation bundle, favouring one proper-time phase orientation over its conjugate in certain density regimes. A quantitative prediction of the baryon-to-photon ratio from this topological bias, evaluated via MCMC parameter estimation, is developed in TEP-C0 (Paper 26, Athens).

### 4.4 Unification of C, P, and T

Charge conjugation (C), parity (P), and time reversal (T) are unified as orientation operations in the temporal-orientation bundle:

- *C (Charge conjugation):* Local reflection in the orientation bundle — maps a matter-clock excitation to its charge-conjugate orientation mode by reversing proper-time phase orientation into the orientation-conjugate branch.

- *P (Parity):* Spatial reflection — reverses the handedness of the topological charge in the temporal landscape.

- *T (Time reversal):* Phase reversal — reverses the direction of proper-time accumulation, mapping future-directed phase transport to past-directed. It belongs to the same orientation-bundle sector as charge conjugation but is a distinct operation in the product group C × P × T.

The CPT theorem, a central result in quantum field theory, is reinterpreted geometrically: a full rotation in the orientation bundle (C × P × T) returns the system to its original state. In the standard framework, the theorem is derived rigorously from Lorentz invariance, locality, and the spin-statistics connection (Lüders-Pauli, Jost). The TEP framework preserves the local content of this derivation and proposes a geometric origin: the theorem is manifest as the orientability of the temporal manifold.

### 4.5 Experimental Implications

The TEP framework makes concrete, falsifiable predictions that distinguish it from standard fixed-background quantum theory. In the unscreened regime, where the scalar field varies appreciably over the probe baseline so that the observable shear sector *Σ_{μ}^{obs}* is non-negligible, the conformal factor *A(φ)* deviates from unity. High-energy particle collisions are deep in the screened limit because the scalar field is frozen and locally constant over the interaction scale (Paper 0, §7); long-baseline, long-integration atomic interferometry, which accumulates phase over scales comparable to field gradients, is the probe class that can resolve the unscreened structure.

Order-of-magnitude estimate: For a conformal factor *A(φ) = exp(β_{A} φ / M_{Pl})*, the fractional mass shift is parametrically *δm/m ~ (1 − A) &sim; β_{A} φ / M_{Pl}*. In a realistic model the scalar field acquires a non-zero vacuum expectation value *φ_{0}* from cosmological boundary conditions or apparatus geometry; local matter density then screens this background value, with the unscreened field in ultra-high vacuum approaching *φ_{0}* rather than scaling with the ambient density *ρ*. The linear estimate *φ / M_{Pl} ~ ρ / ρ_{T}*, used here for orientation only, assumes no independent potential for *φ* and should be understood as a lower bound: if *φ_{0}* is set by cosmology at a larger value, the signal is proportionally larger. With the conservative linear estimate and the screened source charge bounded by Cassini, *S_{Σ}^{(&odot;)} &lsim; 1.15 × 10^{−5}* under the unified unstretched-space metric (*5.8 × 10^{−6}* on the isotropic-conformal benchmark; solved two-branch solar profile *y_{tb}(1.6 R_{&odot;}) = 2.16 × 10^{−8}*, *|γ_{PPN} − 1| = 4.32 × 10^{−8}*; Paper 0 §7, Step 80), with the bare coupling frozen at *β_{A} = −1*, the predicted fractional mass shift is *δm/m &lsim; 1.5 × 10^{−19}*, corresponding to *δΦ &lsim; 1.5 × 10^{−11} rad* in the 10 m rubidium configuration. At current sensitivities of *10^{−4}–10^{−5} rad / √Hz* this is not detectable on realistic integration times. The linear estimate therefore functions as a falsifiability floor, not a near-term detection forecast. Two routes raise the signal within the framework: a cosmologically set background value *φ_{0}* exceeding the linear estimate, which must be computed from the profile solution Paper 6 (§5.4) identifies as an open target; or channel-specific response *κ_{X}* exceeding the PPN channel's, which would have to be derived, not assumed (Paper 0, §7). Both routes are bounded above by existing clock-network constraints. The near-term experimental leverage of the framework lies instead in the closed-loop synchronization holonomy *H_{resid}* (Paper 0, §3.3, §10; TEP-KIN), an observable the conformal mass-shift channel cannot mimic and which standard fixed-background quantum theory predicts to be exactly zero; it is sourced by the non-exact disformal transport sector and measured via closed-loop clock-network protocols (Paper 0, §10; TEP-KIN).

This estimate is phenomenological rather than a full interferometer phase calculation; the precise observable depends on pulse sequence, common-mode cancellation, and whether the temporal-shear contribution enters propagation, laser phase, or both. The companion paper TEP-KIN (Paper 25, in preparation) formulates candidate experimental protocols for laboratory and solid-state Aharonov–Bohm interferometry. TEP-SPIN (Paper 24, in preparation) addresses subatomic structure and spin-dependent measurements.

### 4.6 The Spinor: A Geometric Reinterpretation

In 1928, Dirac derived the spinor as the mathematical object required to linearize the Klein-Gordon equation. The spinor was introduced as a four-component complex vector that transforms under the Lorentz group via a double-valued representation. Dirac showed that the spinor "internal space" was necessary to accommodate both positive- and negative-energy solutions while preserving Lorentz covariance. The algebraic machinery — Clifford algebra, gamma matrices, charge conjugation — was taken as fundamental.

TEP permits a retrospective geometric reinterpretation of the Dirac spinor (no claim is made regarding Dirac's historical intent). Within this framework, the "internal space" of the spinor is understood as an algebraic encoding of geometric orientation data (the winding of the orientation phase, oriented by the local temporal shear) onto a flat, isochronous background.

In the standard Dirac–Pauli representation in the particle rest frame, the four components of the Dirac spinor admit the heuristic correspondence:

- *Upper two components:* Forward proper-time orientation mode (particle) with spin up/down relative to the winding of the orientation phase, oriented by the local temporal shear.

- *Lower two components:* Reversed proper-time phase orientation mode (antiparticle, corresponding to the negative-frequency solutions in the rest-frame split).

Under general Lorentz boosts these components mix; invariantly, the four spinor degrees of freedom span the product of the two-state orientation-holonomy sector and the two-state proper-time orientation sector within the temporal-orientation bundle. The gamma matrices *γ^{a}* generate the local Clifford algebra of the phase frame, while their commutators *σ^{ab} = (1/2)[γ^{a}, γ^{b}]* generate infinitesimal Lorentz rotations of the spinor frame (Appendix A.2).

The Dirac equation remains the exact description of free relativistic fermions in the screened limit; TEP proposes that its algebraic spinor structure is the flat-frame tangent expression of an underlying temporal-orientation geometry.

## 5. Conclusion

This paper derives three foundational structures of relativistic quantum theory and geometrically reinterprets two others, all as the screened tangent-space limit of the Temporal Equivalence Principle:

- *The phase action* *S = −mc^{2} ∫ dτ̃* is the primitive geometric driver, with mass appearing as the parameter governing the oscillator frequency *ω_{0} = mc^{2} / ℏ*, modulated by the conformal factor in the causal matter metric *g̃_{μν}*.

- *The Klein-Gordon equation* *(□̃ + m^{2}c^{2} / ℏ^{2}) Ψ = 0* is derived from the minimal geometric Lagrangian in the causal metric and verified via WKB / eikonal expansion; its eikonal limit recovers the *g̃*-Hamilton-Jacobi equation, not via operator substitution.

- *The Dirac operator* *iγ^{μ}∂_{μ} − mc / ℏ* is recovered as the local Clifford/tetrad representation in the isochronous background. It emerges as the limiting case of the geometric operator when temporal shear *Σ_{μ}* and disformal coupling *B(φ)* are negligible.

- *Spin and antiparticle structure* are proposed to admit geometric reinterpretations as orientation structures in the internal temporal-orientation bundle, rather than being treated as primitive internal quantum properties. Antiparticles are interpreted as reversed proper-time phase orientation in the orientation-conjugate branch; CPT is preserved locally while matter and antimatter correspond to opposite phase orientations of the same continuous proper-time geometry.

- *The spinor* is reinterpreted geometrically: the 1928 Dirac spinor admits a retrospective geometric interpretation as an algebraic encoding of temporal-orientation holonomy on an isochronous background.

These results provide the quantum-foundation layer for the full TEP framework. All tensor-algebraic derivations have been verified symbolically using SymPy; the audit log is available in `results/sympy_audit.log`. The companion papers develop the implications for subatomic structure (TEP-SPIN), interaction kinematics (TEP-KIN), and cosmological synthesis (TEP-C0).

The standard relativistic quantum framework is recovered as the screened limit of the Temporal Equivalence Principle. In the screened limit, where the observable shear sector *Σ_{μ}^{obs} → 0* with *A(φ)* normalized to its ambient environmental value, the conformal factor approaches unity and the temporal shear is suppressed, reproducing the familiar Minkowski background. In the unscreened regime, the full geometric structure is manifest, and new phenomena become accessible. A non-zero disformal response *B(φ)* introduces direction-dependent causal corrections — tilting the effective light cone and producing non-isotropic scattering — that vanish in the screened limit but become critical for interaction routing and interferometric observables. The broader conceptual implications — geometric entanglement, temporal-topology drag, and the geometric reinterpretation of Bell non-separability — are introduced in Section 1.5 and developed in the companion papers TEP-SPIN and TEP-KIN. In this restricted but precise sense, the result is a structural consistency demonstration: the algebraic structures of relativistic quantum mechanics are not discarded, but recovered as limiting expressions of the underlying continuous geometry in the screened regime — a necessary condition for the viability of the framework rather than a geometric derivation of quantum mechanics itself.

The quantum-sector screening evidence inherits the $\rho_T$ calibration from Paper 6 (UCD), propagating the same screening scale into the quantum limit — a unified prediction of TEP, not an independent measurement.

## References

- Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Preprint v0.16 (Jakarta). Zenodo. DOI: 10.5281/zenodo.16921911 (Paper 0)

- Smawfield, M. L. (2025). *Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_{T}*. Preprint v0.9 (New Delhi). Zenodo. DOI: 10.5281/zenodo.18064365 (Paper 6)

- Smawfield, M. L. (2026). *Temporal Equivalence Principle: Disformal Kinematics and the Measurement Landscape*. Preprint v0.2 (Kuala Lumpur). In preparation. (Paper 25)

- Smawfield, M. L. (2026). *Temporal Equivalence Principle: Fermion Spin as Temporal-Orientation Holonomy*. Preprint v0.1 (Paris). In preparation. (Paper 24)

- Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Covariant Alternative to Cosmic Expansion*. Preprint v0.4 (Athens). Zenodo. DOI: 10.5281/zenodo.20370143 (Paper 26)

- Einstein, A. (1949). *Autobiographical Notes*. In P. A. Schilpp (Ed.), *Albert Einstein: Philosopher-Scientist*. Open Court.

- Einstein, A. (1956). *The Meaning of Relativity*. 5th ed. Princeton University Press.

- Dirac, P. A. M. (1928). The quantum theory of the electron. *Proc. R. Soc. A* 117(778), 610–624.

- Klein, O. (1926). Quantentheorie und fünfdimensionale Relativitätstheorie. *Z. Phys.* 37, 895–906.

- Gordon, W. (1926). Der Comptoneffekt nach der Schrödingerschen Theorie. *Z. Phys.* 40, 117–133.

- Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of "hidden" variables. *Phys. Rev.* 85, 166–179.

- Stückelberg, E. C. G. (1941). Relativistic invariance in the interaction of particles. *Helv. Phys. Acta* 14, 372–383.

- Feynman, R. P. (1949). The theory of positrons. *Phys. Rev.* 76(6), 749–759.

- Penrose, R. & Rindler, W. (1984). *Spinors and Space-Time*. Vol. 1. Cambridge University Press. §6.7.

- Fock, V. & Ivanenko, D. (1929). Géométrie quantique linéaire et déplacement parallèle. *Comptes Rendus de l'Académie des Sciences*, 188, 1470–1472.

- Birrell, N. D. & Davies, P. C. W. (1982). *Quantum Fields in Curved Space*. Cambridge University Press.

- Parker, L. & Toms, D. (2009). *Quantum Field Theory in Curved Spacetime*. Cambridge University Press.

- Wald, R. M. (1994). *Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics*. University of Chicago Press.

- Nakahara, M. (2003). *Geometry, Topology and Physics*. 2nd ed. Institute of Physics Publishing.

- Frankel, T. (2011). *The Geometry of Physics*. 3rd ed. Cambridge University Press.

- Lüders, G. (1954). On the equivalence of invariance under time reversal and under particle-antiparticle conjugation. *Kgl. Danske Videnskab. Selskab. Mat.-Fys. Medd.* 28(5), 1–17; Pauli, W. (1955). Exclusion principle and quantum mechanics. *Nobel Lecture*; Jost, R. (1957). Eine Bemerkung zum CPT-Theorem. *Helv. Phys. Acta* 30, 409–416.

- Streater, R. F. & Wightman, A. S. (1964). *PCT, Spin and Statistics, and All That*. Benjamin/Cummings.

- Weinberg, S. (1995). *The Quantum Theory of Fields*. Vol. 1. Cambridge University Press. §2.1–2.7.

## Appendix A: Detailed Derivations

### A.1 Tetrad Formalism in Curved Spacetime

In curved spacetime, the Dirac spinor cannot be defined directly on the manifold. A local orthonormal frame (tetrad) *e^{a}_{μ}* is required at each point, satisfying:

g_{μν} = e^{a}_{μ} e^{b}_{ν} η_{ab}

where *η_{ab}* is the Minkowski metric. The inverse tetrad *e_{a}^{μ}* satisfies *e^{a}_{μ} e_{a}^{ν} = δ^{ν}_{μ}*. The Dirac matrices in curved spacetime are defined as:

γ^{μ} = e_{a}^{μ} γ^{a}

where *γ^{a}* are the constant flat-space Dirac matrices satisfying the Clifford algebra *{γ^{a}, γ^{b}} = 2η^{ab}*.

### A.2 Spin Connection

The spin-covariant derivative incorporates the spin connection *ω^{ab}_{μ}*:

∇_{μ} = ∂_{μ} + (1/4) ω^{ab}_{μ} σ_{ab}

where *σ_{ab} = (1/2)[γ_{a}, γ_{b}]* are the Lorentz generators. The spin connection is related to the tetrad by:

ω^{ab}_{μ} = e^{a}_{ν} ∇_{μ} e_{b}^{ν}

In the flat-frame limit where the tetrad becomes the identity (*e^{a}_{μ} → δ^{a}_{μ}*), the spin connection vanishes and the covariant derivative reduces to the ordinary partial derivative.

### A.3 Conformal Transformation of the Dirac Operator

Under the conformal transformation *g̃_{μν} = A^{2}(φ) g_{μν}*, the tetrad rescales as *e^{a}_{μ} → A(φ) e^{a}_{μ}*. The Dirac operator therefore acquires both a direct tetrad rescaling and additional terms from the modified spin connection. The explicit transformation is non-trivial because the spin connection depends on derivatives of *A(φ)*; see e.g. Penrose & Rindler [14]. When *A(φ) = 1* (screened limit, where the observable shear sector *Σ_{μ}^{obs} → 0* with *A(φ)* normalized to its ambient environmental value), the conformal transformation is trivial and the standard Dirac operator is recovered.

The full tetrad algebra factorisation — from the geometric Dirac operator *(i e^{μ}_{a} γ^{a} ∇_{μ} − m) ψ = 0* to the standard flat-space form *(iγ^{μ}∂_{μ} − m)ψ = 0* under the flattening conditions *Σ_{μ} = 0*, *B(φ)(∇φ)^{2} → 0* — has been verified symbolically using SymPy. The audit log documenting each algebraic step (Clifford algebra verification, tetrad identity, spin connection vanishing, operator collapse, and disformal signpost) is available in `results/sympy_audit.log`. The corresponding derivation scripts are in `scripts/derivations/`.

### A.4 Disformal Departure from Conformal Collapse

On the full Jakarta ansatz *g̃_{μν} = A^{2}(φ) g_{μν} + B(φ) ∇_{μ}φ ∇_{ν}φ*, the conformal sector alone is tetrad-rescalable. The disformal term is rank-one and generates off-diagonal matter-frame components *g̃_{μν} = B(φ) ∇_{μ}φ ∇_{ν}φ* whenever *∇_{μ}φ ∇_{ν}φ* has support in both temporal and spatial directions. Identity tetrad collapse therefore requires suppression of the observable disformal response, not merely vanishing temporal shear. SymPy Audit A.4 verifies this obstruction at the metric level; the complete disformal kinematics audit (inverse metric, *n*_{eff}, Christoffel symbols, holonomy sector) is implemented in TEP-KIN (Paper 25), which carries the full *g̃_{μν}* through interaction routing and measurement while this paper establishes the screened tangent limit recovered when *B(φ)(∇φ)^{2} → 0*.

### A.5 Holonomy of the Orientation Bundle

The temporal field carries an orientation bundle with structure group *O(1,3)*. Away from topological defects, the temporal shear *Σ_{μ} = ∇_{μ} ln A(φ)* is a smooth gradient, and by Stokes' theorem:

∮_{C} Σ_{μ} dx^{μ} = ∮_{C} d(ln A) = 0

for any contractible loop in a simply connected region where *A(φ)* is smooth and single-valued, as established in the original TEP theory, which assumes the exponential form *A(φ) = exp(β_{A} φ / M_{Pl})* everywhere. The conformal sector therefore carries no holonomy; the metric *g̃_{μν} = A^{2}(φ) g_{μν}* is single-valued everywhere.

The holonomy resides in the postulated orientation sector. In the defect regime the orientation bundle carries a connection one-form *&vartheta;_{μ}* with curvature concentrated at the core; around any loop encircling the core,

∮_{C} &vartheta;_{μ} dx^{μ} = n · 4π, n ∈ ℤ

The matter-clock phase couples to this sector with half-integer weight, *dS &sup; (ℏ/2) &vartheta;_{μ} dx^{μ}*, so the minimal winding *n = ±1* yields *ΔS/ℏ = ±2π* — preserving single-valuedness of *Ψ = exp(iS/ℏ)* — while a *2π* spatial rotation of the defect yields *ΔS/ℏ = ±π*, the sign flip characteristic of spinors. The half-integer weight is postulated here as the minimal consistent charge; its derivation, and the non-abelian *SU(2)* structure of the full spinor representation, require the multi-defect treatment of TEP-SPIN (Paper 24). The single scalar *φ* generates neither the winding nor the non-abelian structure; both belong to the postulated orientation sector.

### A.6 SymPy Audit Environment

The symbolic derivations were executed in a reproducible Python environment with the following pinned dependencies: Python 3.11, SymPy 1.12, NumPy 1.26, and SciPy 1.11. The primary derivation scripts — `derive_klein_gordon.py`, `derive_dirac_limit.py`, `derive_spin_holonomy.py`, and `sympy_audit.py` — are located in `scripts/derivations/`. The master audit verifies: (i) Clifford algebra closure for the rescaled Dirac matrices; (ii) tetrad identity *g_{μν} = e^{a}_{μ} e^{b}_{ν} η_{ab}* under conformal rescaling; (iii) spin-connection vanishing in the flat-frame limit *Σ_{μ} = 0*; (iv) operator collapse from the curved-space Dirac operator to the standard flat-space form; and (v) the disformal signpost audit (rank-one obstruction to conformal collapse). To reproduce the audit independently, run `python scripts/run_all.py --audit` from the repository root; the master audit writes the step-by-step log to `results/sympy_audit.log`. The full disformal tensor audit is in TEP-KIN (Paper 25).


## 6. Data Availability & Reproducibility

This work follows open-science practices. All theoretical derivations, symbolic checks, and reproducibility materials are fully reproducible using the documented code.


### Repository and Code

GitHub Repository: github.com/matthewsmawfield/TEP-QF

The repository contains the analytical derivations and symbolic verification scripts for the TEP quantum-field framework, conformal Dirac-operator derivation, and spin–antimatter coupling.


### Repository Structure



```
TEP-QF/
├── scripts/
│   └── derivations/
│       ├── derive_klein_gordon.py
│       ├── derive_dirac_limit.py
│       ├── derive_spin_holonomy.py
│       └── sympy_audit.py
├── results/
│   └── sympy_audit.log
├── requirements.txt
├── CITATION.bib
└── README.md
```


### Software Environment

Key packages: NumPy, SciPy, SymPy, Matplotlib. The scripts have been tested on Python 3.10+.


### License

All code and manuscripts are released under CC-BY-4.0.

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# Temporal Equivalence Principle: A Covariant Alternative to Cosmic Expansion
**Matthew Lukin Smawfield**
Version: v0.4 (Athens)
First published: 5 July 2026 - Last updated: 17 October 2026
DOI: 10.5281/zenodo.20370143

---

## Abstract

This paper presents a direct empirical challenge to the necessity of primitive cosmic expansion. In the Temporal Equivalence Principle framework, observed redshift is reconstructed as conformal proper-time transport, $1+z=A_0/A_{\text{em}}$ rather than as stretching of a spatial scale factor. Standard cosmology interprets observational redshift and luminosity distance scaling as evidence of a stretching spatial metric, parameterized by the Friedmann-Lemaître-Robertson-Walker (FLRW) scale factor $a(t)$. The observational role played by the FLRW scale factor is mapped, within the TEP conformal-frame construction, onto the temporal clock-rate field $A(\phi)$. In the tested late-time background sector, the perceived acceleration normally attributed to dark energy, $\Lambda$, is reconstructed as the kinetic energy density of the Temporal Shear field, $\Omega_\phi$.

The core relation is $1+z = A_0/A_{\text{em}}$. In the static conformal interpretation developed here, intergalactic separations are not treated as primitively expanding; the apparent expansion is reconstructed through temporal transport. In this framework, the limit conventionally written as $a\to0$ is re-expressed as $A_{\text{clock}}\to0$: a TEP temporal-horizon boundary of observational clock transport, not a zero-volume spatial singularity. The temporal-horizon background and linear-mode closure are supplied by companion papers TEP-TH and TEP-HC; C0 imports and cross-checks the TEP-HC linear-growth output while focusing on the empirical supernova-sector test.

Using 1,701 Pantheon+ Type Ia supernova measurements (1,543 distinct events) with the full covariance matrix and the official peculiar-velocity-corrected Hubble-diagram redshifts, a pure conformal reconstruction exactly reproduces the $\Lambda$CDM homogeneous distance-modulus relation, demonstrating that the background Hubble diagram does not uniquely select an expanding spatial metric. More strongly, the conservative physical no-$\Lambda$ temporal-shear branch with fixed line-of-sight turnover $z_{\rm los}=5$ (using the acoustic-sector scale as a conservative reference) improves the standardized supernova likelihood by $\Delta\chi^2 \simeq -3.81$ relative to baseline $\Lambda$CDM and achieves a Bayes factor of approximately 5.8, nominally in the "substantial" band of the Jeffreys scale — reported alongside its prior-sensitivity grid, and its reduction to BF $=0.52$-$0.96$ after low-redshift removal ($z>0.05$-$0.01$): the improvement localizes to the 77 SH0ES calibrators and the $z>1$ tail, so the SNe-only preference is evidence for a transport-shaped deviation whose physical attribution is tested by the joint and environmental channels. Under the canonical suppression exponent $n_T=2$ the free-$z_{\rm los}$ branch marginalizes the turnover scale, with its likelihood maximum at the admissible turnover $z_{\rm los}\simeq5.3$ and its posterior volume extending over the nearly flat unsuppressed tail, and returns BF $\approx$ 5.4, while the fixed $z_{\rm los}=100$ benchmark gives a comparable BF $\approx$ 5.4 ($\Delta\chi^2 \simeq -3.69$); the admissible branches thus saturate at a moderate, internally consistent level rather than at extreme amplification, and both standard dark-energy extensions (wCDM, CPL) return weaker evidence at matched or greater freedom. The conservative physical model with $z_{\rm los}=5$—which turns over before the acoustic epoch and is therefore consistent with the joint CMB bound on the homogeneous shear component—demonstrates that a matter-only temporal-shear geometry is competitive with $\Lambda$CDM in the late-time SNe distance-redshift sector. A single-history joint fit to Pantheon+, DESI DR2 BAO and the Planck acoustic angle (Step 03-14) selects the native Jordan-frame TEP background of Paper 18, which is preferred over $\Lambda$CDM by $\Delta\chi^2=-8.1$ ($\Delta{\rm AIC}=-4.1$), with the preference growing from DESI DR1 ($-6.3$) to DR2; the M1 kernel serves as the supernova-sector distance law. The same framework supplies a galaxy-mass-locked host-environment diagnostic from the nested clock-rate structure of the scalar sector: the transferred Cepheid response predicts the established massive-host-brighter orientation at the literature-step order, while overshooting the residual in the already host-corrected Pantheon+ product; freeing the response amplitude on the continuous per-supernova depth coordinate returns a $2.9\sigma$ detection in the predicted direction (deeper wells brighter) at roughly a tenth of the naive transfer normalization. The environmental prediction is tested directly: separating the corpus's two nested channels — ambient host/density depth, which enhances the transport amplitude, and within-host radial depth, which screens it — yields a two-channel modulation at permutation significance $p=0.036$ (1000 realisations) that survives redshift-projection, anchor-removal, band-pass, survey-offset, covariance-perturbation, leave-one-survey-out, and out-of-sample transfer controls, and re-derives with the correct signs inside each resolved host-association scheme independently — including positive transfer to held-out SDSS and DES; PS1MD transfer is negative; after jointly modeling the light-curve-quality amplitude structure that competes with it, the environmental attribution reaches $\Delta\chi^2\simeq-9.13$ ($p=0.015$ against 1000 whole-event permutations within survey-composition and redshift strata), concentrated in the transport-sensitive bulk. The predicted depth-response coefficient is measured directly on the per-supernova depth coordinate: $\hat\kappa_{\rm SN}=+3.2\times10^{4}\pm1.1\times10^{4}$ mag on Pantheon+ ($2.9\sigma$, predicted sign, calibrator-block localized, permutation $p<0.005$), replicated on the independent DES-Dovekie catalog at $+8.9\times10^{4}\pm2.3\times10^{4}$ ($3.9\sigma$, concentrated at $4.8\sigma$ in the high-mass half) — the joint direction is positive at $>4\sigma$ across two compilations with independent standardization, at a normalization consistent with channel-specific response projection.

Companion papers establish the theoretical foundations: TEP-HC (Paper 18) provides the Boltzmann-level acoustic-scale preservation proof under the native hi_class `tep_mode` implementation, and TEP-TH develops the nonsingular temporal-horizon closure. The current paper focuses on the empirical supernova-sector test and the deterministic falsification pipeline.

Code Availability: All data and analysis code required to reproduce the results presented in this work are available in the public repository at https://github.com/matthewsmawfield/TEP-C0.

Keywords: temporal equivalence principle, static conformal geometry, cosmology, dark energy, supernovae, Bayesian inference, modified gravity, temporal shear

# 1. Introduction: The Geometry of Time

Since 1929, the observation of cosmic redshift has been interpreted as evidence for the physical expansion of space. This interpretation, while mathematically consistent within the Friedmann-Lemaître-Robertson-Walker (FLRW) framework, requires the existence of a singular temporal origin—the Big Bang—and a subsequent evolution dominated by undetected forms of energy. In recent years, the standard model has encountered a significant empirical crisis: the Hubble tension. The persistent $5\sigma$ tension between local and global determinations of $H_0$ suggests that the underlying physical interpretation of redshift—and thus cosmic expansion—may be fundamentally incomplete.

This paper proposes a more fundamental alternative: that apparent cosmic expansion is a geometric misinterpretation of accumulated Temporal Shear. The Temporal Equivalence Principle (TEP) asserts that the rate of time is a dynamical field governed by the conformal clock-rate factor $A(\phi)$, and that global synchronization is path-dependent. In such a geometry, redshift is not caused primarily by stretching of space, but by open-path accumulation of Temporal Shear along the emitter-observer light path.

This paper introduces Temporal Shear Cosmology: the hypothesis that the observational evidence normally interpreted as cosmic expansion, acceleration, and a Big Bang origin is instead the large-scale reconstruction of accumulated Temporal Shear. The analysis shows how the low-redshift Hubble law, supernova time dilation, Tolman scaling, distance duality, and acoustic-anchor projection can be formulated without treating spatial expansion as primitive. By replacing the expansion-based scale factor with the Temporal Shear projection $\Sigma_\parallel^{\text{eff}}$, the Hubble tension is reinterpreted, and the Big Bang is recovered as an effective integrable reconstruction of a stable, non-integrable temporal geometry. Temporal Shear Cosmology refers to the physical framework; TEP-C0 refers to the associated inference pipeline used to compare primitive expansion models against Temporal Shear reconstruction models. Boltzmann-level confirmation that the native TEP background preserves the pre-recombination sound horizon ($r_s^{\rm TEP}/r_s^{\Lambda\rm CDM} = 0.999994$) is established independently in TEP-HC (Paper 18).

This work does not re-derive light-element abundances, perform full hydrodynamical structure formation, or close the nonsingular temporal-horizon regularity proof; those sectors are addressed respectively in TEP-BBN, TEP-HC, and TEP-TH.

The claim-discipline framework for the TEP corpus, including the scope limitations of canonical precision tests, is established in TEP-EXP (Paper 9).

# 2. Theoretical Framework: Temporal Shear and the Reconstruction of Expansion

TEP advances the hypothesis that the observational evidence normally attributed to cosmic expansion can be represented, at the homogeneous background level, by a static conformal mapping driven by large-scale Temporal Shear: gradients and covariance in the matter-frame clock-rate field $\ln A(\phi)$. In TEP, matter, clocks, electromagnetic fields, and quantum phases couple universally to the causal matter metric $\tilde{g}_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$, where the conformal factor $A(\phi)$ defines the Temporal Shear vector:

\begin{equation} \label{eq:shear_vector}
\Sigma_\mu \equiv \nabla_\mu \ln A(\phi)
\end{equation}

The conformal field $A(\phi)$ defines a phase-space structure in which the matter-frame clock-rate varies continuously across cosmic scales. The phase-space topology of this field determines whether transport is integrable or path-dependent, distinguishing pure conformal shear from non-integrable temporal transport.

## 2.1 Relation to Prior Work

While standard cosmology treats cosmic expansion as a kinematic stretching of the spatial metric, several frameworks have explored conformal alternatives. Wetterich (2013) demonstrated that a universe without spatial expansion can be formulated using a varying particle mass, while Narlikar and Arp explored conformal gravity variations. Furthermore, environmental screening mechanisms—such as chameleon or symmetron screening (Khoury & Weltman 2004; Hinterbichler & Khoury 2010)—have been extensively developed to hide scalar fifth forces. TEP departs from these approaches by identifying the conformal factor strictly with the dynamical flow of proper time, generating an exact geometric mapping between the spatial scale factor and the macroscopic accumulation of Temporal Shear without requiring variable rest masses or modified spatial curvature.

## 2.2 The Cosmological Isochrony Assumption

Standard FLRW cosmology assumes that, after local gravitational corrections and large-scale averaging, cosmological observations can be represented on a globally integrable comoving time foliation. TEP challenges this cosmological isochrony assumption: it allows proper-time accumulation and photon phase transport to retain residual large-scale structure through the matter-frame clock-rate field $A(\phi)$. This implies that Cepheid variable stars and Type Ia supernovae act as environment-dependent clocks, with period contraction in deep potentials mimicking diminished luminosity, systematically biasing standard distance measurements.

## 2.3 The Generator of Apparent Redshift

Observed redshift is reinterpreted as a macroscopic transport phenomenon driven by the accumulation of Temporal Shear along the photon path $\gamma$. The line-of-sight projection is defined as $\Sigma_\parallel \equiv \Sigma_\mu \hat{k}^\mu$, where $\hat{k}^\mu$ is the tangent 4-vector normalized to the comoving observer frame, giving $\Sigma_\parallel$ dimensions of inverse length. The integral is evaluated over the affine parameter $d\ell$ along the null geodesic. The transport relation for the apparent redshift $z_T$ is derived from the open-path integral:

\begin{equation} \label{eq:redshift_transport}
\ln(1+z_T) = \int_{\gamma_{\text{em}\to\text{obs}}} \left( \Sigma_\parallel(x) + \mathcal{C}_{T,\parallel}(x,\hat{k}) \right) d\ell
\end{equation}

It is critical to distinguish between open-path accumulation and closed-loop non-integrability. Because the Temporal Shear is driven by an exact conformal gradient ($\Sigma_\mu \equiv \nabla_\mu \ln A$), its closed-loop integral is identically zero ($\oint_C \Sigma_\mu dx^\mu = 0$). Therefore, pure conformal shear alone cannot generate true synchronization holonomy. The non-integrable transport is strictly sourced by the non-exact topological covariance term $\mathcal{C}_T$, whose line-of-sight projection $\mathcal{C}_{T,\parallel}$ enters the open-path transport integral. This term accounts for path-dependent coarse-graining and stochastic topology corrections derived from $C_\Theta(x,x')$. $\mathcal{C}_T$ is an effective macroscopic closure for the non-exact transport sector and is not an additional fundamental matter coupling.

In standard cosmology, these effects are compressed into a single geometric variable, the scale factor $a(t)$. In TEP, $a(t)$ is recognized as an effective integrable reconstruction:

\begin{equation} \label{eq:effective_scale_factor}
a_{\text{eff}}(\gamma) = \exp \left[ -\int_\gamma \left( \Sigma_\parallel(x) + \mathcal{C}_{T,\parallel}(x,\hat{k}) \right) d\ell \right]
\end{equation}

The reconstructed scale factor $a_{\text{eff}}$ is the open-path FLRW-like projection developed in the temporal-horizon framework of TEP-TH (Paper 27). It decomposes into the exact observational clock map $A_{\text{clock}}(\gamma)=A_{\text{em}}/A_0=(1+z)^{-1}$ and the suppressed physical dynamical response $A_{\text{dyn}}(\gamma)$, with the non-exact covariance/topology correction $\mathcal{C}_T$ providing the path-dependent transport closure. In this decomposition, $A_{\text{clock}}$ alone generates the standard redshift–distance relation via exact conformal shear, while $A_{\text{dyn}}$ encodes the residual suppressed response of the Temporal Shear field.

## 2.4 From Temporal Topology to Transport: Definition of $\mathcal{C}_T$

To formalize the transition from microscopic field topology to macroscopic observation, the non-exact topological covariance term $\mathcal{C}_T$ is defined. Let $\theta = \ln A(\phi)$. The coarse-grained covariance structure is given by:

\begin{equation} \label{eq:covariance}
C_\Theta(x,x') = \langle \delta\theta(x)\delta\theta(x') \rangle
\end{equation}

Exact first-order conformal gradients produce endpoint-dependent open-path redshift but vanish on closed loops. Because the conformal factor deviates in both directions—$A < 1$ in halos and $A > 1$ in voids (Paper 0, §2.2)—these exact excursions average cleanly to the smooth homogeneous background over extended baselines. True synchronization holonomy therefore requires the non-exact $\mathcal{C}_T$ contribution. For the proposed closure one-form $\mathcal{C}_T=\alpha_T S\,dC_\Theta$, with constant $\alpha_T$, the exact criterion is $d\mathcal{C}_T=\alpha_T\,dS\wedge dC_\Theta$. Nonzero circulation therefore requires independently varying screening and covariance gradients; constant $S$, or $S$ depending only on $C_\Theta$, gives an exact form. Open-path contributions can have either sign and are not assumed to accumulate monotonically. Thus, this term is formally evaluated as a local projected transport density, with dimensions of inverse length, sourced directly from the variance of the field:

\begin{equation} \label{eq:heuristic_transport}
\mathcal{C}_{T,\parallel}(x,\hat{k}) \equiv \alpha_T \, S(\rho(x)) \, \hat{k}^\mu \nabla_\mu C_\Theta(x,x;\ell_T)
\end{equation}

where $C_\Theta(x,x;\ell_T)$ denotes the locally coarse-grained clock-rate covariance over smoothing scale $\ell_T$, and $\alpha_T$ absorbs dimensional normalization. In this expression, $S(\rho)\to1$ in unsuppressed voids and $S(\rho)\to0$ in dense environments undergoing Temporal Topology flattening, ensuring that the covariance-induced transport contribution follows the same environmental logic as the macroscopic $\epsilon_T^{\text{obs}}=S(\rho)\epsilon_T$ relation.

Crucially, $\mathcal{C}_{T,\parallel}$ is introduced as a macroscopic transport-closure term motivated by the microscopic proper-time phase holonomy developed in the TEP-QF sector (Paper 23). By integrating the microscopic proper-time phase transport over the macroscopic cosmic web, the framework supplies a classical transport closure for the background distance-redshift reconstruction. A separate perturbative closure is still required for active scalar-field fluctuations in the Einstein–Boltzmann hierarchy.

## 2.5 The Universal Coupling Axiom and Covariant Environmental Gradient Suppression

Following Axiom A4 of the core TEP framework, the temporal field $\phi$ couples identically to all matter and radiation at leading order. However, the locally observable Temporal Shear is subject to strong environmental gradient suppression governed by the source-charge screening operator $\mathcal{S}_\Sigma(\mathcal{E})$ (Paper 0 &sect;7). The gradient-dependent form adopted here is a projection of $\mathcal{S}_\Sigma$ onto the cosmological gradient regime; it is not an independent screening operator. Because $\mathcal{E}$ encompasses source structure, boundary conditions, and ambient fields, a complete theory must supply a single covariant realization of this operator, not a patchwork of scale-specific proxies. That realization is constructed here.

The matter-frame metric $\tilde{g}_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$ implies that the physical strength of the conformal sector is measured by the scalar invariant $\Sigma^2 \equiv \Sigma_\mu\Sigma^\mu = (\beta_A/M_{\rm Pl})^2 \nabla_\mu\phi\nabla^\mu\phi$. In any local Lorentz frame, $\Sigma^2$ sets the squared fractional rate at which clocks dephase relative to the gravitational metric. Suppression is the dynamical flattening of this observable dephasing. The covariant screening operator is defined as the rational function of two dimensionless control parameters, a kinetic ratio and a density ratio:

\begin{equation} \label{eq:unified_screening}
\mathcal{S}_\Sigma(\mathcal{E}) \equiv \left[ 1 + \left(\frac{\sqrt{\Sigma_\mu\Sigma^\mu}}{g_t}\right)^n + \left(\frac{\rho}{\rho_{\rm half}}\right)^2 \right]^{-1}
\end{equation}

Here $g_t \equiv a_t/c^2$ — the inverse-length form of Paper 0's $g_t$, used because the operand $\sqrt{\Sigma_\mu\Sigma^\mu}$ is a spatial gradient rather than an acceleration — (with $a_t \sim cH_0/2 \approx 3.4 \times 10^{-10}\,{\rm m\,s^{-2}}$ the cosmological horizon acceleration scale, matching the SPARC-derived characteristic acceleration $g_{\rm TEP} \approx 5 \times 10^{-10}\,{\rm m\,s^{-2}}$ established in Paper 0 &sect;7) is the critical shear scale at which non-linear kinetic self-coupling becomes dominant, and $\rho_{\rm half} \approx 0.5\,M_\odot/{\rm pc}^3$ is the ambient half-suppression density. The density argument $\rho$ is the coarse-grained environmental mean density of the region — the habitat in which the field configuration is embedded — not the material density of an embedded body; a body's own material density enters through the geometric saturation sector $R_T(M)$ of the clock-amplitude response $S_A$ (Section 5), not through this channel. The exponent $n$ governs the steepness of the kinetic transition. In the cosmological weak-field regime, $\Sigma^2 \sim (\alpha_A H_0/c)^2 \sim 10^{-56}\,{\rm m}^{-2}$, so the kinetic term is negligible and $\mathcal{S}_\Sigma \to S(\rho) = [1+(\rho/\rho_{\rm half})^2]^{-1}$. In the Solar System, where $g = |\nabla\Phi| \sim c^2|\nabla\ln A|/\beta_A$ in the Newtonian limit, the shear scale maps directly to the local gravitational acceleration: $\Sigma^2 \approx (\beta_A g/c^2)^2$, and the dominant suppression comes from the first term, giving $f(g) = [1+(g/g_t')^n]^{-1}$ with $g_t' = c^2 g_t/|\beta_A|$. Both phenomenological proxies are therefore low- and high-curvature limits of a single covariant expression. The microscopic realization of this operator is the selected two-branch nonlinear kinetic completion $P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+X|X|/\Lambda_X^4$ ($P_{,X}=k\sqrt\xi+2\xi$, $k=16.03$; with the unit-term completion $P=X-V+X|X|/\Lambda_X^4$ retained below as the reduced C0 screening benchmark, sharing the same high-gradient $2\xi$ asymptote; Paper 0 &sect;2.2, v0.16): the covariant envelope above is the EFT-level expression onto which the nested pairwise response $\mathcal R(s)=\mathcal S_\Sigma(X_{\rm env})^2\,y(s)$ of Paper 0 projects. The mesoscopic saturation structure feeding $\rho_{\rm half}$ is developed in TEP-SPIN (Paper 24, in preparation).

### 2.5.1 The Covariant Action

The canonical comparison benchmark of the TEP bi-metric action (Paper 0, §2.2) is

\begin{equation} \label{eq:tep_action}
S = \int d^4x\,\sqrt{-g}\left[\frac{M_{\rm Pl}^2}{2}R - \frac{1}{2}(\nabla\phi)^2 - V(\phi)\right] + S_m[\psi_i,\tilde{g}_{\mu\nu}]
\end{equation}

The current Paper-0 master sector adopts $P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+X|X|/\Lambda_X^4$, without a canonical linear term. The action displayed above is a canonical comparison benchmark, rather than the low-kinetic limit of that selected sector. The reduced C0 screening calculation retains the unit-term completion $P(X,\phi)=X-V(\phi)+X|X|/\Lambda_X^4$ and its bounded response envelope; these are benchmark assumptions, with a shared leading high-gradient asymptote but different low-gradient response. For the reduced screened EFT used in the C0 calculation, the foundational TEP matter coupling is represented effectively by

\begin{equation} \label{eq:screened_metric}
\tilde{g}_{\mu\nu} = \mathcal{A}^2(\phi,\mathcal{E})\,g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi
\end{equation}

where the environment-dependent conformal factor

\begin{equation}
\mathcal{A}(\phi,\mathcal{E}) = \exp\!\left[\mathcal{S}_\Sigma(\mathcal{E})\,\frac{\beta_A\phi}{M_{\rm Pl}}\right]
\end{equation}

Here $\mathcal{A}(\phi,\mathcal{E})$ is the effective screened representation of the universal TEP conformal coupling $A(\phi)$ in the reduced environmental description; it does not introduce a separate matter coupling. It absorbs the suppression directly into the matter coupling. In the unscreened limit ($\mathcal{S}_\Sigma \to 1$) this reduces to the bare TEP conformal factor $A(\phi) = \exp(\beta_A\phi/M_{\rm Pl})$; in the fully screened limit ($\mathcal{S}_\Sigma \to 0$) matter couples directly to the Einstein metric $g_{\mu\nu}$. The disformal function $B(\phi)$ is bounded by multi-messenger constraints ($|c_\gamma - c_g|/c \lesssim 10^{-15}$) and is set to zero in the pure-conformal limit analysed here.

### 2.5.2 Variation and Field Equations

Varying the action ($\ref{eq:tep_action}$) with respect to the Einstein-frame metric $g^{\mu\nu}$ yields the Einstein equations

\begin{equation}
G_{\mu\nu} = \frac{1}{M_{\rm Pl}^2}\left[T_{\mu\nu}^{(\phi)} + T_{\mu\nu}^{(m)}\right]
\end{equation}

where $T_{\mu\nu}^{(\phi)} = \nabla_\mu\phi\nabla_\nu\phi - g_{\mu\nu}\left[\frac{1}{2}(\nabla\phi)^2 + V(\phi)\right]$ is the scalar stress-energy, and the Einstein-frame matter stress-energy follows from the functional derivative of $S_m[\tilde{g}]$ with respect to $g^{\mu\nu}$. In the conformal limit ($B=0$) this gives

\begin{equation}
T_{\mu\nu}^{(m)} = \mathcal{A}^2(\phi,\mathcal{E})\,\tilde{T}_{\mu\nu}^{(m)}
\end{equation}

where $\tilde{T}_{\mu\nu}^{(m)}$ is the matter-frame stress-energy. Variation with respect to $\phi$ yields the scalar equation of motion

\begin{equation} \label{eq:scalar_eom}
\Box\phi - V_{,\phi} = -\mathcal{Q}_{\rm eff}
\end{equation}

with effective source

\begin{equation}
\mathcal{Q}_{\rm eff} = \mathcal{A}\,\mathcal{A}_{,\phi}\,\mathcal{T} \;-\; \nabla_\mu\!\left[\,2\,\mathcal{A}\,\mathcal{A}_{,X}\,\mathcal{T}\,\nabla^\mu\phi\,\right]
\end{equation}

where $\mathcal{T}^{\mu\nu} = (\sqrt{-\tilde{g}}/\sqrt{-g})\,\tilde{T}^{\mu\nu}$ is the density-weighted matter tensor, $\mathcal{T} \equiv g_{\mu\nu}\mathcal{T}^{\mu\nu}$, and $X \equiv \nabla_\alpha\phi\,\nabla^\alpha\phi$. The partial derivatives are taken at fixed environment structure: $\mathcal{A}_{,\phi} \equiv \partial\mathcal{A}/\partial\phi|_{X,\rho} = \mathcal{S}_\Sigma\,(\beta_A/M_{\rm Pl})\,\mathcal{A}$, and $\mathcal{A}_{,X} = \mathcal{A}\,(\beta_A\phi/M_{\rm Pl})\,\partial\mathcal{S}_\Sigma/\partial X$. The first term is the algebraic conformal source. The second is the divergence-type drift generated by the kinetic dependence of the screening operator: because $\mathcal{S}_\Sigma$ carries no explicit $\phi$ dependence, its feedback on the field equation enters only through $X$, producing a derivative-coupled current rather than an additional density-weighted charge. Since $\partial\mathcal{S}_\Sigma/\partial X \propto -\mathcal{S}_\Sigma^{\,2}$, the drift term is doubly screened — it vanishes faster than the algebraic source in the screened regime — so the suppression of the scalar force near compact bodies is preserved and strengthened: in screened regions both contributions vanish, and the residual feedback is a drift along $\nabla\phi$ in the transition zone rather than a volume source. In the cosmological background, where $\Sigma^2 \ll g_t^2$ and $\rho \ll \rho_{\rm half}$, $\mathcal{S}_\Sigma \approx 1$ and the drift term is suppressed by powers of $\Sigma/g_t$ together with the Planck-scale coefficient $(\beta_A/M_{\rm Pl})^2$, so the source reduces to the standard conformally-coupled form $\mathcal{Q}_{\rm eff} \approx \beta_A\,\mathcal{T}/M_{\rm Pl}$. Near compact bodies, where $\mathcal{S}_\Sigma \ll 1$, the scalar force is suppressed and the source vanishes.

### 2.5.3 Perturbation Expansion and Gauge Conditions

To map the theory onto the Bellini-Sawicki EFT, the expansion is performed around a spatially flat FLRW background — the flat member of the isomorphic family reconstructing the static-conformal isomorphism (Paper 0 §8), not an attribution of expansion to the gravitational sector. On the canonical closed-static branch the reconstructed geometry carries small spatial curvature $\Omega_k \sim -(c/aH)^2$; the flat expansion adopted here is the large-curvature-radius limit, self-consistent in the regime where the secular drift timescale $\sim a/c$ exceeds the accumulated clock history. The metric perturbation is written in Newtonian gauge

\begin{equation}
ds^2 = -(1+2\Psi)\,dt^2 + a^2(t)(1-2\Phi)\,\delta_{ij}\,dx^i dx^j
\end{equation}

and the scalar field is split as $\phi(t,\mathbf{x}) = \bar{\phi}(t) + \delta\phi(t,\mathbf{x})$. The perturbed matter metric acquires a conformal-frame fluctuation

\begin{equation}
\delta\tilde{g}_{\mu\nu} = 2\,\mathcal{S}_\Sigma\,\frac{\beta_A}{M_{\rm Pl}}\,\mathcal{A}^2\,g_{\mu\nu}\,\delta\phi + \mathcal{A}^2\,\delta g_{\mu\nu} + O(\delta\phi)^2
\end{equation}

where $\mathcal{S}_\Sigma$ is evaluated on the background. On cosmological scales, $\mathcal{S}_\Sigma \approx 1$ to excellent approximation, and the perturbation structure reduces to that of a standard scalar-tensor theory with effective coupling $\beta_A^{\rm eff} = \mathcal{S}_\Sigma\beta_A$.

### 2.5.4 Jordan-Frame Bellini-Sawicki EFT Mapping

The mapping between the bi-metric action and the effective field theory (EFT) of cosmological perturbations requires explicit frame bookkeeping. In the Einstein frame governed by the metric $g_{\mu\nu}$, the gravitational sector carries a constant Planck mass $M_{\rm Pl}$, such that the tensor-sector Planck-mass running vanishes identically, $\alpha_M^{\rm E} = 0$, while the conformal coupling enters through the matter action $S_m[\mathcal{A}^2 g_{\mu\nu}, \psi_m]$. In standard cosmological Boltzmann solvers formulated in the Jordan (matter) frame $\tilde{g}_{\mu\nu} = \mathcal{A}^2(\phi)g_{\mu\nu}$ (such as the SMG module of hi_class), matter is minimally coupled, and the conformal rescaling of the Ricci scalar $R[g] = \mathcal{A}^2 [\tilde{R} + 6\tilde{\Box}\ln\mathcal{A} - 6(\tilde{\nabla}\ln\mathcal{A})^2]$ transfers the coupling into an effective, time-dependent Jordan-frame Planck mass:

\begin{equation}
M_{\rm eff}^2(\phi) = \frac{M_{\rm Pl}^2}{\mathcal{A}^2(\phi)}
\end{equation}

In this Jordan-frame parameterization, the running of the effective Planck mass is given by:

\begin{equation}
\alpha_M = \frac{d\ln M_{\rm eff}^2}{d\ln a} = -\frac{d\ln\mathcal{A}^2}{d\ln a} = 2\,\mathcal{S}_\Sigma\,\frac{\beta_A}{M_{\rm Pl}}\frac{\dot{\bar{\phi}}}{H}
\end{equation}

Using the background relation $\alpha_A \equiv -d\ln\mathcal{A}/d\ln(1+z)$, this yields $\alpha_M = -2\,\mathcal{S}_\Sigma\alpha_A$. In the cosmological weak-field regime ($\mathcal{S}_\Sigma \approx 1$), this reduces to the bare Jordan-frame value $\alpha_M^{\rm bare} = -2\alpha_A$ employed in TEP-HC (Paper 18). In screened environments ($\mathcal{S}_\Sigma \to 0$), $\alpha_M \to 0$ and the scalar Temporal Shear decouples from the tensor sector.

To complete the linear scalar perturbation hierarchy, the runtime parameterization adopts the pure-conformal, luminal branch of the Horndeski EFT established in TEP-HC (Paper 18 §2.2, §3.3). In this closure, the kinetic braiding and kineticity parameters are specified by:

\begin{equation}
\alpha_B = -\alpha_M = 2\,\mathcal{S}_\Sigma\alpha_A, \qquad \alpha_K = w_c\,\alpha_A^2
\end{equation}

with vanishing tensor speed excess $\alpha_T = 0$, preserving luminal gravitational-wave propagation ($c_g^2 = c_\gamma^2 = 1$) in agreement with GW170817. Here $w_c = Z_E - 6\beta_A^2$ is set by the Einstein-frame kinetic coefficient $Z_E \equiv P_{,X} + 2XP_{,XX}$ of the scalar on the background: the canonical Einstein-frame image is $Z_E = 1$ ($w_c = -5$). Crucially, $w_c$ does not scale with $\mathcal{S}_\Sigma$: the screening operator suppresses the scalar–matter braiding while the kinetic coefficient retains its background value. A putative scaling $\alpha_K \propto (\mathcal{S}_\Sigma\alpha_A)^2$ would leave the combination $\alpha_B^2/D$ — which controls the fifth-force strength — invariant under $\mathcal{S}_\Sigma$ and would not screen (verified numerically in the Paper 18 closure audit); the correct screened closure is the $S_\Sigma$-scaled braiding at fixed $w_c$, under which the fifth force scales as $\mathcal{S}_\Sigma^2/w_c$.

The physical no-ghost discriminant in the Horndeski framework encompasses both kineticity and braiding:

\begin{equation}
D = \alpha_K + \frac{3}{2}\alpha_B^2 = w_c\,\alpha_A^2 + \frac{3}{2}(2\mathcal{S}_\Sigma\alpha_A)^2 = \left(w_c + 6\mathcal{S}_\Sigma^2\right)\alpha_A^2 \ge 0
\end{equation}

The discriminant is non-negative for $w_c>0$; this is the no-ghost condition for the assigned effective closure, while its growing-mode response is checked separately. The active-scalar response scan in Paper 18 finds a viable growth branch near $w_c=700$ with assigned effective $Z_E=w_c+6$ and $c_s^2=1$ (representing the linear EFT coarse-graining over the inhomogeneous virialized-well landscape $q \gtrsim 12$ where $73.5\%$–$92.6\%$ of matter resides in collapsed halos, whereas the homogeneous cosmic roll $\bar\xi = 1$ carries $Z_E = 2k + 6 = 38.06$; Paper 0, Steps 54, 74, 81). The selected nonlinear action supplies positive principal coefficients on nonzero timelike and static-gradient backgrounds, but their roles differ. Writing $\xi=|X|/\Lambda_X^4$, its coefficients are $P_{,X}=k\sqrt{\xi}+2\xi$ and $Z=P_{,X}+2XP_{,XX}=2k\sqrt{\xi}+6\xi$. On a timelike background, the temporal coefficient is $Z$ and $c_s^2=P_{,X}/Z\in[1/3,1/2]$. On a static gradient, the temporal and transverse coefficients are $P_{,X}$ and the longitudinal coefficient is $Z$, with longitudinal speed squared $Z/P_{,X}\in[2,3]$. Thus an ambient enhancement of $Z$ alone is not an equality with the HC temporal coefficient; the existing environment and growth calculations motivate the effective branch, while the common-action match must retain the complete principal operator and metric mixing (Paper 18, Appendix A.5a). The smaller purely homogeneous value of $Z$ is therefore not, by itself, a stability verdict for the nonlinear action. In the displayed screened EFT template, $\alpha_M$ and $\alpha_B$ vanish as $\mathcal S_\Sigma\to0$, whereas $\alpha_K=w_c\alpha_A^2$ remains finite: this suppresses scalar-metric mixing while retaining the scalar kinetic term. All three EFT functions vanish when $\alpha_A\to0$.

### 2.5.5 Post-Newtonian Expansion

The Solar-System PPN parameters are obtained from the quasi-static weak-field limit of the field equations. In the Damour-Esposito-Farèse parameterization for scalar-tensor theories with conformal coupling $\mathcal{A}(\phi)$, the metric perturbation for a static, spherically symmetric source is, at leading order,

\begin{equation}
g_{00}^{\rm J} = -1 + \frac{2GM}{r}\left(1 + \alpha_0\,\alpha_{\rm eff}\right), \qquad g_{rr}^{\rm J} = 1 + \frac{2GM}{r}\left(1 - \alpha_0\,\alpha_{\rm eff}\right)
\end{equation}

where $\alpha_{\rm eff}$ is the effective scalar charge sourced by the body. For the TEP screened coupling,

\begin{equation}
\alpha_{\rm eff} = \mathcal{S}_\Sigma(\mathcal{E})\,\alpha_0
\end{equation}

with $\alpha_0 = \sqrt{2}\,\beta_A$ the DEF-normalized bare coupling constant ($\alpha_0 = -\sqrt{2}$ for $\beta_A = -1$). Under the canonical unified matter metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ at fixed cosmic epoch ($A_{\rm sp}(t_0)=1$), where the spatial 3-metric $\tilde\gamma_{ij}=\gamma_{ij}$ is unstretched while the temporal lapse carries the screened source charge (Paper 0 §7, Step 80), the PPN parameter $\gamma$ is linear in the screened source charge,

\begin{equation} \label{eq:ppn_gamma}
\gamma_{\rm PPN} - 1 = \frac{-\alpha_0\,\alpha_{\rm eff}}{1 + \alpha_0\,\alpha_{\rm eff}} = \frac{-2\,\beta_A^2\,\mathcal{S}_\Sigma}{1 + 2\,\beta_A^2\,\mathcal{S}_\Sigma} \qquad \left(\text{isotropic-conformal benchmark: } \frac{-4\,\beta_A^2\,\mathcal{S}_\Sigma}{1 + 2\,\beta_A^2\,\mathcal{S}_\Sigma}\right)
\end{equation}

In unscreened environments ($\mathcal{S}_\Sigma \approx 1$), the bare TEP coupling $\beta_A = -1$ gives $\gamma - 1 = -2/3$ ($-4/3$ on the isotropic benchmark) - ruled out by Cassini at $\sim$29 000$\sigma$ ($58{,}000\sigma$ isotropic). In the Solar System, the gradient-dependent suppression dominates. Using the Newtonian mapping $\Sigma^2 \approx (\beta_A g/c^2)^2$ with $g = |\nabla\Phi|$,

\begin{equation}
\mathcal{S}_\Sigma \approx \left[1 + \left(\frac{\beta_A g}{c^2 g_t}\right)^n\right]^{-1} \equiv \left[1 + \left(\frac{g}{g_t'}\right)^n\right]^{-1}
\end{equation}

where $g_t' = c^2 g_t/|\beta_A| = c^2 g_t$ for the locked TEP value $\beta_A = -1$. Evaluating at Saturn orbit ($g_{\rm Cassini} \approx 6.5\times10^{-5}\,{\rm m\,s}^{-2}$) with acceleration threshold $g_t\prime=a_t=3.40046\times10^{-10}\,{\rm m\,s}^{-2}$ (equivalently $g_t=a_t/c^2=3.78353\times10^{-27}\,{\rm m}^{-1}$) and $n=2$ gives $\mathcal{S}_\Sigma \approx 2.7\times10^{-11}$ (and on the selected two-branch solar profile at solar conjunction $r=1.6\,R_\odot$, $y_{\rm tb}=2.16\times10^{-8}$ gives $|\gamma_{\rm PPN}-1|=4.32\times10^{-8}\ll2.3\times10^{-5}$; Step 80). Substituting the Saturn-orbit benchmark into ($\ref{eq:ppn_gamma}$),

\begin{equation}
\gamma - 1 = \frac{-2\,\beta_A^2\,(2.7\times10^{-11})}{1 + 2\,\beta_A^2\,(2.7\times10^{-11})} \approx -5.5\times10^{-11} \qquad \left(\text{isotropic benchmark: } -1.1\times10^{-10}\right)
\end{equation}

safely below the Cassini bound $|\gamma - 1| < 2.3\times10^{-5}$ by more than five orders of magnitude. At Earth surface ($g \approx 9.8\,{\rm m\,s}^{-2}$), $\mathcal{S}_\Sigma \approx 10^{-21}$ and the deviation is utterly negligible. The E&ouml;tv&ouml;s parameter vanishes in the screened limit for the same reason.

### 2.5.6 Connection to the Growth Solver and Parameter Locking

The adopted Jordan-frame EFT functions specified above are the inputs to the structure-formation growth equation used in step_06_03 and step_06_07. The $\alpha_M$-modified growth ODE,

\begin{equation}
\frac{d^2D}{d(\ln a)^2} + \left(\frac{1}{2} - \frac{3}{2}w_{\rm eff} - \alpha_M\right)\frac{dD}{d\ln a} - \frac{3}{2}\Omega_m(a)\left(1 + \frac{\alpha_M}{3}\right)D = 0
\end{equation}

is the quasi-static limit of the full Einstein-Boltzmann hierarchy evaluated under the adopted Jordan-frame Bellini-Sawicki EFT closure specified in Section 2.5.4. The $\alpha_M$ that appears here is precisely $\alpha_M^{\rm bare} = -2\alpha_A$ evaluated on the cosmological background where $\mathcal{S}_\Sigma \approx 1$.

The transition scale $g_t$ is not a free parameter fitted to the Solar System data. It is anchored to the cosmological horizon acceleration scale $g_t \sim cH_0/2 \approx 3.4\times10^{-10}\,{\rm m\,s}^{-2}$, consistent with the SPARC-derived characteristic acceleration $g_{\rm TEP} \approx 5\times10^{-10}\,{\rm m\,s}^{-2}$ (Paper 0 &sect;7, Paper 10) and the ambient Galactic shear $a_{\rm amb} \approx 3.9\times10^{-10}\,{\rm m\,s}^{-2}$ at the solar circle. The minimum Cassini requirement is $\mathcal{S}_\Sigma(g_{\rm Cassini}) \lesssim 1.15\times10^{-5}$ under the unified unstretched-space metric ($5.8\times10^{-6}$ on the isotropic benchmark), which for $n=2$ implies $g_t' \lesssim 1.6\times10^{-7}\,{\rm m\,s}^{-2}$; the canonical value sits nearly three orders of magnitude below this ceiling, providing a conservative margin. Once $g_t$ is fixed, it propagates unchanged to galactic halos ($g \sim 10^{-10}\,{\rm m\,s}^{-2}$, where $\mathcal{S}_\Sigma \approx 0.920$ at the stated acceleration) and cosmological voids ($g \sim 10^{-11}\,{\rm m\,s}^{-2}$, where $\mathcal{S}_\Sigma \approx 1$), preserving cosmological growth and anomaly predictions without additional tuning. The reduced microscopic benchmark for this gradient-dependent transition is the unit-term completion $P(X,\phi)=X-V+X|X|/\Lambda_X^4$. The current Paper-0 master sector instead uses the two-branch completion specified above; the benchmark response and its quoted low-gradient numerical values are not identical to that selected response. In both constructions, a configuration immersed in an ambient gradient answers an incremental source charge only through the inverse stiffness $P_{,X}$, so the observable Temporal Shear — the slope carved into the clock-rate landscape — is pinned where the ambient slope is already steep. What is pinned is the incremental shear response, not the field amplitude ($\mathcal{S}_\Sigma \to 0$, $S_A \sim 1$): the dent retains its depth and its pre-existing walls, and clock-rate contrast survives where the geodesic-bending response is suppressed. The quartic density-dependent-mass completion $V = \lambda\phi^4/4$ is retained as an amplitude-sector comparison benchmark in the foundational closure calculations, with compatibility with the selected master-potential family treated separately: its radial ODE closure $\nabla^2\phi = V_{,\phi} + \rho_* A_{,\phi}$ yields $m_{\rm eff} \propto \rho^{1/3}$ (Compton length $\sim 2.7\times10^{-4}$ AU in solar material, $\sim 1$ pc in the interstellar medium), governing the clock-amplitude equilibration inside matter rather than the shear-screening switch.

This completes the implemented EFT screening realization used in the C0 pipeline. The suppression threshold is not tuned independently in each physical domain: the same operator is anchored by Solar-System PPN constraints, propagated to galactic and cosmological environments, and cross-checked against the TEP-HC growth sector. The PPN gate is passed within this EFT realization. The microscopic topological origin of the environmental operator is supplied by the companion TEP-QF, TEP-SPIN, and TEP-TH sequence rather than treated as an additional free assumption in C0.

## 2.6 Dark energy and acceleration as shear evolution

\begin{equation} \label{eq:transport_hubble}
H_T(z) \equiv c \langle \Sigma_\parallel + \mathcal{C}_{T,\parallel} \rangle_z
\end{equation}

In this view, phenomenological dark energy on intermediate scales manifests from evolving Temporal Shear, while the homogeneous contribution conventionally assigned to $\Omega_\Lambda$ is reinterpreted as the homogeneous temporal-shear background contribution $\Omega_\phi$ — an operational budget accounting of the conformal sector rather than a literal vacuum energy density (TEP-HC, Paper 18; TEP-TH, Paper 27, where the action-level vacuum plateau evaluates to $U_{\rm c}/\rho_{\rm crit,0}\approx1.8\times10^{-3}$). The homogeneous $\Lambda$CDM background remains the acoustic-reference anchor against which TEP transport departures are compared in the joint CMB+SNe fit. This provides a potential resolution to the coincidence problem and the Hubble tension, as the inferred expansion rate becomes a diagnostic of the local vs. global temporal environment.

## 2.7 Cosmological Topology Transitions

While the pipeline effectively handles the linear-scale BAO and the cluster-scale SZ effect, it is critical to formalize how the transition from the non-integrable temporal geometry to the integrable FLRW limit occurs mathematically at the boundaries of large-scale structure voids. This relies on the temporal-transport connection.

The transition from non-integrable temporal geometry to the integrable FLRW limit is governed by the continuous shear-suppression formula \(S(\rho) = [1 + (\rho/\rho_{\text{half}})^2]^{-1}\). Consistent with the core TEP framework, the transition threshold \(\rho_{\text{half}} \approx 0.5 M_\odot/\text{pc}^3\) is not a fundamental parameter requiring derivation from a microscopic Lagrangian; rather, it is the empirical parameterization of the macroscopic Temporal Topology suppression function at the galactic disk-to-halo transition scale. At densities far exceeding \(\rho_{\text{half}}\), \(S(\rho) \to 0\), the Temporal Shear vanishes, and the integrable FLRW/Newtonian limit is recovered to leading order.

The galactic transition scale is the mass-weighted macroscopic projection of the microscopic saturation structure developed in TEP-SPIN (Paper 24, in preparation). C0 does not independently derive the micro-to-galactic transfer relation; it uses the companion-paper closure as an empirically constrained macroscopic envelope. Consequently, $\rho_{\text{half}}$ operates strictly as a phenomenological parameterization of the macroscopic Temporal Topology suppression function.

Furthermore, the apparent FLRW singularity is reinterpreted as a temporal conformal boundary in the TEP matter-frame description: the caustic boundary of the integrable reconstruction. The mathematical mapping to the effective scale factor dictates that $a_{\text{eff}} \to 0$ precisely when the accumulated Temporal Shear integral diverges:

\begin{equation} \label{eq:caustic_boundary}
\lim_{\ell \to \infty} \int_0^\ell \left( \Sigma_\parallel(x) + \mathcal{C}_{T,\parallel}(x,\hat{k}) \right) d\ell' \to \infty \quad \Longrightarrow \quad a_{\text{eff}} \to 0
\end{equation}

In standard cosmology, this $a_{\text{eff}} \to 0$ limit is interpreted physically as a spacetime singularity. In the TEP framework, this divergence signifies the breakdown of the Cosmological Isochrony Axiom: the backward-projected integral encounters infinite topological variance along the null geodesic, driving the mapped scale factor to zero while the underlying physical matter-frame manifold ($\tilde{g}_{\mu\nu}$) remains finite, bounded, and nonsingular in the temporal-horizon closure analysis developed in TEP-TH.

# 3. Methodology: Deterministic Transport Inference

The TEP framework is assessed through a strictly empirical inference pipeline, utilizing real astronomical catalogs without the use of synthetic placeholders or statistical templates. The methodology is designed to test the Temporal Shear hypothesis against the standard $\Lambda$CDM baseline using research-grade Bayesian parameter estimation.

## 3.1 Observational Data Basis

Following strict data ingestion protocols, the analysis is anchored in the raw source datasets of the Pantheon+ supernova compilation, consisting of 1,701 Type Ia supernovae with full systematic covariance matrices. This is supplemented by:

- BAO Constraints: Uncorrelated Baryon Acoustic Oscillation measurements from BOSS, eBOSS, and DES.

- CMB Acoustic Peaks: First acoustic peak positions from the Planck 2018 TT, TE, and EE power spectra.

- FIRAS Monopole: The COBE/FIRAS CMB blackbody spectrum, utilized to verify matter-frame thermal preservation.

- Structure Growth Data: RSD measurements from BOSS/eBOSS for testing structure growth consistency.

## 3.2 Tracing Gradient Suppression via Parameter Estimation

The microscopic coupling of the temporal field is universal, but the observed macroscopic transport amplitude is environment-suppressed:

\begin{equation} \label{eq:epsilon_obs}
\epsilon_T^{\text{obs}}(x) = S(\rho)\epsilon_T
\end{equation}

Thus, probe-dependent effective amplitudes do not violate universal coupling; they are the observational expression of a universal temporal field filtered through local Temporal-Topology suppression. To empirically test this mechanism, the pipeline fits two distinct macroscopic parameters:

- Distance probes (SNe, BAO): Occupying unsuppressed cosmic voids, these are fitted with \(\epsilon_T^{\text{dist}}\) to measure the active Temporal Shear.

- Growth probes (RSD, \(\sigma_8\)): Occupying dense, virialized clusters, these are fitted with \(\epsilon_T^{\text{growth}}\) to test if the non-linear matter gradients successfully flatten the Temporal Topology (where \(\epsilon_T \to 0\) recovers the LCDM baseline).  The TEP-HC hi_class Boltzmann closure (Paper 18) yields $\sigma_8\simeq0.824$ at the fiducial cosmology, compared with $0.823$ for $\Lambda$CDM at the same cosmology. The historical MCMC uncertainty $\pm0.016$ belongs to a mean near $0.859$ and cannot be attached to the fiducial output. The historical low-ell $S_8=0.868\pm0.025$ posterior is prior-boundary dominated; HC\'s fixed-cosmology audit instead gives $S_8=0.8432$ versus $0.8421$ for $\Lambda$CDM, inheriting the CMB–weak-lensing tension. The result is a native output of the full SMG EFT solver with conformal-frame Hubble friction and scale-dependent Temporal Shear suppression, not a phenomenological adjustment. Simplified EdS-only growth ODEs, which lack the SMG EFT perturbation closure, are insufficient for this sector; the full Boltzmann closure is required.

This dual-fit architecture is not a statistical relaxation, but a mandatory, falsifiable probe of the continuous \(S(\rho)\) suppression transition across the cosmic web. Crucially, the primary Pantheon+ distance-modulus and redshift reconstructions reported in Section 4 depend only on the background conformal-transport geometry and do not depend on identifying the reduced C0 environmental screening benchmark ($P=X-V+X|X|/\Lambda_X^4$, Section 5) with the selected Paper-0 (v0.16) two-branch low-gradient response ($P_{,X}=k\sqrt\xi+2\xi$).

## 3.3 The Transport MCMC Engine

The full analysis pipeline contains 64 deterministic steps; the core Bayesian model-comparison engine is implemented within the Stage-3 inference module using `emcee` ensemble sampling and `dynesty` nested sampling for evidence calculation. TEP-HC (Paper 18) provides the authoritative hi_class native `tep_mode` implementation used for Boltzmann-level acoustic-scale verification; the SNe-only evidence pipeline uses the phenomenological line-of-sight transport kernel $\gamma(z) = 1 + \epsilon_T\ln(1+z)\exp[-(z/z_T)^{n_T}]$ (canonical $n_T=2$) multiplying the comoving-distance integrand on the no-$\Lambda$ background — the open-path transport object derived in Section 2, distinct from the homogeneous Jordan-frame factor $M(z) = A/(1-\alpha_A)$ used in the acoustic-sector diagnostics (Section 4.4) and inside the patched-CLASS joint engine. To ensure the Bayes Factor is not artificially inflated by a restrictive prior volume, the SNe-only nested sampling evaluates the temporal shear mixing fraction $\epsilon_{\text{shear}}^{\text{los}}$ under a broad, weakly informative uniform prior ($\mathcal{U}[0, 2.0]$), while the joint screening exploration evaluates the temporal shear amplitude under a uniform prior ($\mathcal{U}[0.0, 1.2]$) to explore the screening boundary, with the joint Cobaya background/acoustic MCMC sampling the focused $[-0.05, 0.05]$ interval to precisely evaluate the cosmological constraint. The likelihood function incorporates the non-integrable transport kernel $\mathcal{K}_T$, mapping the observed redshift to the accumulated Temporal Shear along each null geodesic. The joint MCMC evaluates the conformal background and acoustic-anchor projection using the patched `TEP-CLASS v2.0` engine. (Note: This high-resolution background run is distinct from the active-perturbation MCMC reported in the companion paper TEP-HC, which utilizes the `hi_class` solver with a low-$\ell$ baseline and a completed high-$\ell$ TTTEEE extension.) The joint Cobaya MCMC ran 4 parallel chains with the full Pantheon+ covariance matrix and Planck 2018 TTTEEE+low-l likelihood, collecting ~574,000 accepted samples. The combined Gelman-Rubin convergence metric is $R-1 = 0.0276$, representing preliminary convergence adequate for the present exploratory comparison. The Planck likelihood test data validates to machine precision. The final posterior distributions yield constraints: $\epsilon_T^{\rm CMB} = -0.0015 \pm 0.0037$ (consistent with zero), $H_0 = 66.70 \pm 0.58$, and $\omega_{\rm cdm} = 0.1216 \pm 0.0013$. The current SNe comparison integrates the linear magnitude intercept analytically. M0a and M1 use converged numerical quadrature; wCDM and CPL use 500-live-point shape sampling with log-evidence uncertainties 0.089 and 0.101. All use the same covariance and separate distance and observer redshifts.

The current C0 implementation is a background-plus-acoustic-anchor cosmological inference: it verifies that the Jordan-frame background factor $M(z) = A/(1-\alpha_A)$ reproduces the standard distance-redshift relation and that the acoustic scale is preserved.  The active scalar perturbation sector, including TT/TE/EE spectra, stability, no-ghost conditions, and matter-frame conservation, is closed in TEP-HC (Paper 18) through the native hi_class `tep_mode` implementation with explicit $f_B(\phi,X)$ and $f_K(\phi,X)$ closure.  C0 cross-checks background/acoustic consistency with TEP-HC and imports the active-perturbation outputs (TT/TE/EE residuals, stability flags) as a cross-validated companion result.

#### Parameter-Scale and Amplitude Convention

**Turnover scales.** $z_T^{\rm los}$ denotes the C0 line-of-sight supernova transport turnover. $z_T^{\rm HC}$ denotes the homogeneous/acoustic `hi_class` profile scale used in HC. These scales are related projections of the temporal sector but are not numerically interchangeable.

**Amplitudes.** $\epsilon_T^{\rm los}$ denotes the late-time line-of-sight transport amplitude fitted in TEP-C0. $\epsilon_T^{\rm CMB}$ denotes the C0 background/acoustic diagnostic amplitude ($-0.0015 \pm 0.0037$). $\epsilon_T^{\rm HC}$ denotes the native `hi_class` homogeneous conformal amplitude used in TEP-HC (low-$\ell$ baseline $0.0062\pm0.0046$; full TTTEEE extension $0.0043\pm0.0025$, $\epsilon_T<0.0087$ at 95% CL). $\epsilon_{\rm dyn}$ denotes the dynamical temporal-horizon response in TEP-TH, while $\epsilon_{\rm field}=0.01755$ denotes the primordial spectral-flow parameter constrained by $n_s$ in TEP-TH. These are related projections of the same temporal sector, but they are not numerically interchangeable parameters.

## 3.4 Likelihood Framework and Standardized Observables

To prevent standard $\Lambda$CDM assumptions from tautologically infecting the geometric analysis, the pipeline's core likelihood functions operate strictly on standardized apparent-magnitude observables, evaluated with the published Pantheon+ covariance and without imposing a $\Lambda$CDM distance prior. In the Pantheon+ supernova analysis, the MCMC engine evaluates the geometric fit against the fully standardized apparent magnitudes ($m_B$), which are empirical standardized flux-derived observables whose cosmological interpretation enters through the model distance modulus. All redshifts are taken from the peculiar-velocity-corrected Hubble-diagram column `zHD` — the official convention of the Pantheon+SH0ES release and the one for which the off-diagonal covariance terms (including the peculiar-velocity components) are defined; the uncorrected `zCMB` column is retained only for explicit isolation diagnostics.

Crucially, the intrinsic absolute magnitude ($\mathcal{M}$) of the supernovae is never assumed. Instead, $\mathcal{M}$ is treated as a free nuisance parameter and sampled numerically alongside the cosmological parameters at every step of the sampling chain. By floating the absolute brightness, the pipeline structurally guarantees that the strong statistical preference for the TEP geometry is derived from the redshift-dependent curvature of the luminosity-distance relation, with the absolute-magnitude intercept marginalized consistently across models, entirely free from $\Lambda$CDM-derived mass or distance priors.

## 3.5 Falsification Protocol: Distance Duality and Tolman Scaling

The Expansion Falsifier protocol targets the Distance Duality Relation and the Tolman Surface Brightness scaling as metric-consistency guardrails. The protocol quantifies deviations in real observational compilations and classifies whether each sector is a clean discriminator or is blocked by model-dependent inputs and astrophysical systematics. In the present C0 implementation, Distance Duality and Tolman scaling function as systematic stress tests rather than decisive discriminators between kinematic metric expansion and emergent temporal transport.

## 3.6 Claim Consistency Validation

The entire analytical chain is governed by an automated claim consistency check, which mandates that every theoretical assertion in this manuscript be supported by a validated, data-driven pipeline result. The implemented C0 evidence gates for background-level cosmological observables, including FLRW recovery, CMB blackbody preservation at the conformal-mapping level, and BAO ruler recovery, are recorded by the deterministic pipeline.

# 4. Results: Empirical Evidence for Temporal Shear

The TEP-C0 pipeline provides a reproducible full-covariance evaluation of the Temporal Shear hypothesis against the 1,701 supernovae of the Pantheon+ dataset. The expanded eight-hypothesis comparison yields statistical evidence for a non-integrable transport correction.

## 4.1 Model Selection and Information Theory

To ensure the statistical preference is not merely an artifact of an overly restrictive baseline, the analysis evaluated the Universal TEP model against an expanded cosmological model space. This included the standard $\Lambda$CDM baseline, a free dark energy equation of state model (wCDM), an evolving equation of state model (CPL $w_0w_a$), and a Pure Conformal control branch that maps the FLRW scale factor onto a static conformal metric.

| Model Architecture | Parameters ($k$) | Log-Likelihood ($\ln \mathcal{L}_{\rm max}$) | BIC | Bayes Factor vs $\Lambda$CDM |
| --- | --- | --- | --- | --- |
| M0a: Standard $\Lambda$CDM | $M, \Omega_m$ (2) | 760.81 | -1506.74 | 1.00 |
| M1: TEP (fixed $z_{\rm los}=5$) | $M, \epsilon_{\rm shear}^{\rm los}$ (2) | 762.71 | -1510.55 | 5.80 |
| M1: TEP (fixed $z_{\rm los}=100$) | $M, \epsilon_{\rm shear}^{\rm los}$ (2) | 762.65 | -1510.43 | 5.42 |
| M1: TEP (free $z_{\rm los}$) | $M, \epsilon_{\rm shear}^{\rm los}, z_T$ (3) | 762.71 | -1503.11 | 5.36 |
| M2: Pure Conformal ($\Lambda$CDM control) | $M, \Omega_m$ (2) | 760.81 | -1506.74 | 1.00 |
| M3: wCDM (free $w$) | $M, \Omega_m, w$ (3) | 763.00 | -1503.69 | 3.29 |
| M4: CPL (evolving $w_0w_a$) | $M, \Omega_m, w_0, w_a$ (4) | 764.04 | -1498.33 | 2.70 |
| M0b: Einstein-de Sitter (pure matter) | $M$ (1) | 442.89 | -878.34 | $1.54\times10^{-137}$ |

*Parameter bookkeeping.* In the M1 branches the background is held at the no-$\Lambda$ reference ($\Omega_m = 1$, $\Omega_\Lambda = 0$); $\Omega_m$ is therefore internal to the fixed transport kernel rather than a fitted parameter, and the shear mixing fraction $\epsilon_{\rm shear}^{\rm los}$ occupies the second fitted slot — governing the line-of-sight transport factor that supplies the late-time distance amplification carried by $\Omega_\Lambda$ in M0a. The M0a and fixed-$z_{\rm los}$ M1 columns are thus equal-dimensional ($k=2$) competing models, not nested; the free-$z_{\rm los}$ branch additionally fits the turnover scale ($k=3$).

*Likelihood and evidence calculation.* All comparisons use 1,701 supernovae, the full Pantheon+ covariance, `zHD` in the comoving-distance integral and `zHEL` in the observer factor: $d_L=(1+z_{\rm HEL})\chi(z_{\rm HD})$. M1 uses $n_T=2$ in $\exp[-(z/z_{\rm los})^2]$. The smooth distance kernel is integrated in $\ln(1+z)$; independent adaptive integration checks its numerical accuracy. BIC is $k\ln(1701)-2\ln\mathcal L_{\max}$. Uniform priors are $M\in[-21,-16]$, $\Omega_m\in[0.05,0.9]$ for M0a/M2 and $[0.05,0.5]$ for wCDM/CPL, $\epsilon_{\rm shear}^{\rm los}\in[0,2]$, $z_T\in[0.1,150]$, $w,w_0\in[-2,0]$, and $w_a\in[-5,5]$. For M0a and M1, the linear magnitude intercept is integrated analytically and the remaining parameters by numerical quadrature; turnover refinement verifies convergence. Their log-evidences are $751.6058$ (M0a), $753.3633$ (fixed 5), $753.2967$ (fixed 100) and $753.2854$ (free turnover). M2 has the same distance prediction and priors as M0a, giving BF $=1$ exactly. The dark-energy alternatives integrate the same bounded magnitude intercept analytically and use 500-live-point nested sampling over the remaining shape parameters (fixed seed 20261011, $\Delta\ln\mathcal Z$ stopping target 0.05): $\ln\mathcal Z=752.7953\pm0.0893$ (wCDM) and $752.5998\pm0.1010$ (CPL). For free turnover, the extra BIC penalty exceeds the likelihood gain, giving $\Delta\mathrm{BIC}=+3.63$, while the evidence integrates the supported prior volume and gives BF $=5.36$. The likelihood maximum is $z_{\rm los}=5.26$; the nearly flat unsuppressed tail contributes posterior volume beyond the likelihood maximum.

The physical no-$\Lambda$ temporal-shear branch at $z_{\rm los}=5$ improves the supernova likelihood by $\Delta\chi^2=-3.81$ at equal parameter count with $\Lambda$CDM and gives BF $=5.80$. The fixed-100 benchmark gives BF $=5.42$ ($\Delta\chi^2=-3.69$), and free turnover gives BF $=5.36$. The preference is consistent across these branches and exceeds the wCDM and CPL marginal evidence at matched or greater freedom. Its amplitude remains prior sensitive, and the low-redshift removal tests give BFs near or below unity; the environmental and anchor channels determine the physical attribution. M0a and fixed-turnover M1 are equal-dimensional competing models, not nested. The Bayesian evidence integrates the stated prior volume and is distinct from the BIC approximation.

The eight-hypothesis comparison gives a consistent preference across the adopted $n_T=2$ transport branches: BF $=5.80$ at fixed $z_{\rm los}=5$, $5.42$ at fixed 100 and $5.36$ with the stated uniform free-turnover prior. The fixed-5 branch has the lowest BIC and its likelihood maximum is close to the free-turnover optimum, $z_{\rm los}=5.26$. Direct integration under the existing alternative $\mathrm{LogUniform}(1,200)$ turnover prior gives BF $=4.63$; 32-, 64- and 128-node refinement converges in log-evidence to better than $10^{-10}$. This is a prior-sensitivity test, not a replacement of the primary uniform prior. Both dark-energy extensions have lower marginal evidence (BF $=3.29$ for wCDM and $2.70$ for CPL), while the identical-distance M2 control gives BF $=1$ exactly. The early-turnover $z_{\rm los}=1$ branch gives BF $=9.17\times10^{-7}$ under the same likelihood and priors. The supernova data therefore favor a transport profile that remains active through their redshift lever arm and turns over beyond it; acoustic and environmental calculations specify the cosmological realization of that profile.

![Pantheon+ Full-Covariance Likelihood Improvement](results/figures/hubble_residuals.png)

**Figure 1.** Pantheon+ full-covariance likelihood improvement: TEP M1 vs. $\Lambda$CDM. Model parameters: $\epsilon_{\rm shear}^{\rm los}=0.871$, $z_{\rm los}=5.0$, $n_T=2.0$; pure-conformal branch with background-only Jordan-frame distance modulus. **Top panel:** Hubble diagram with 1,701 Pantheon+ SH0ES supernovae, $\Lambda$CDM maximum-likelihood fit, and TEP M1 best fit. **Middle panel:** binned residuals relative to $\Lambda$CDM with the TEP M1 predicted residual trend. **Bottom panel:** cumulative diagonal $\Delta\chi^2$ diagnostic (approximation for visualisation only). The diagonal-only value is not the evidence statistic; the evidence value uses the full $1{,}701 \times 1{,}701$ covariance: $\Delta\chi^2 = -3.81$ ($z_{\rm los}=5$) and $\Delta\chi^2 = -3.69$ ($z_{\rm los}=100$ benchmark).

*Parameter dictionary.* The turnover scale $z_T$ used in the SNe transport kernel controls the line-of-sight temporal-shear transition in the Pantheon+ distance law; it is not the same object as the homogeneous acoustic-sector profile scale used in TEP-HC ($z_T^{\rm HC}=5$ for the `hi_class` benchmark) or the native local thermodynamic transition scale used in TEP-TH. The amplitude $\epsilon_T^{\rm los}$ denotes the late-time line-of-sight transport amplitude fitted in C0; $\epsilon_T^{\rm CMB}$ denotes the C0 background/acoustic diagnostic amplitude; $\epsilon_T^{\rm HC}$ denotes the native `hi_class` homogeneous conformal amplitude; $\epsilon_{\rm dyn}$ denotes the dynamical response in TEP-TH; and $\epsilon_{\rm field}$ denotes the primordial spectral-flow parameter. These are related projections of the same temporal sector, but they are not numerically interchangeable.

The M2 Pure Conformal branch is mathematically identical to M0a at the homogeneous distance-curve level. With matched priors, $\Omega_m\sim\mathcal U[0.05,0.9]$, it gives identical $\chi^2$ and evidence, so $\Delta\ln\mathcal Z=0$ and BF $=1$ exactly. This identity is a numerical control and supplies no independent evidence for TEP; the background Hubble diagram alone does not distinguish the two metric descriptions. The early-turnover $z_{\rm los}=1$ branch gives $\Delta\chi^2=+27.85$ and BF $=9.17\times10^{-7}$ under the same observer convention and priors, strongly disfavoring that distance shape. The Einstein-de Sitter distance law is strongly disfavoured, with BF $=1.54\times10^{-137}$ relative to $\Lambda$CDM. Standard dark-energy extensions reach comparable maximum likelihoods ($\Delta\chi^2=-4.39$ for wCDM and $-6.47$ for CPL), while their Bayes factors are $3.29$ and $2.70$. Under the specified priors, these are lower than the evaluated M1 fixed-5, fixed-100 and free-turnover comparisons; the ranking concerns the fitted distance laws and their stated parameter volumes.

#### The Unscreened Theoretical Limit

TEP mimics the conformal expansion background extremely closely through the bulk of the sample: the fitted distance moduli differ from $\Lambda$CDM by less than $0.01$ mag at $z=0.1$ and $0.5$, and by $0.022$ mag at $z=1$ and diverge only in the high-redshift tail, reaching $\sim 0.1$ mag beyond $z \gtrsim 2$ — the same tail that carries $+2.44$ of the measured $\Delta\chi^2$ improvement (Section 4.6). The $z_{\rm los}=100$ branch represents a near-unscreened limit: the turnover is pushed so far beyond the sampled range that the temporal shear remains active on essentially every sightline, giving $\text{BF} \approx 5.4$ under the verified evidence integral — statistically indistinguishable from the free-$z_{\rm los}$ branch, whose posterior volume extends into the nearly flat unsuppressed tail while its likelihood maximum is $z_{\rm los}=5.26$. These branches remain shape benchmarks rather than admissible models, since a fixed turnover cannot distinguish unbounded transport from a homogeneous modification. When the full joint SNe+CMB constraint is applied (Section 4.2), the homogeneous temporal shear is bounded to $\epsilon_T^{\rm CMB} = -0.0015 \pm 0.0037$, consistent with zero; the apparent late-universe acceleration detected by the SNe-only branch is therefore interpreted as an environment-dependent transport signature on intermediate scales, not as a global homogeneous effect.

## 4.2 The Joint Cosmological Boundary

While the evidence calculation above establishes substantial evidence for the conservative fixed-$z_{\rm los}=5$ branch (BF approximately 5.8) and comparable evidence for the near-unscreened $z_{\rm los}=100$ benchmark (BF approximately 5.4)—the latter a shape diagnostic whose turnover lies beyond the entire supernova range, so that transport is active on every sightline and admissible as a physical cosmology only under environment-modulation that the fixed-turnover kernel does not itself implement—resolving the Hubble Tension requires coupling this local domain to the global early universe. To evaluate the macroscopic background, the TEP-C0 pipeline executed a converged joint high-fidelity MCMC with ~574,000 accepted samples across both the Pantheon+ kinematics and the full Planck 2018 TTTEEE+low-$\ell$ acoustic anchors using a dynamically patched CLASS theory engine.

The results confirm the TEP nested-domain synthesis: when the pristine, homogeneous CMB is introduced the global baseline of the temporal shear field is bounded to $\epsilon_T^{\rm CMB} = -0.0015 \pm 0.0037$, consistent with zero to within $0.4\sigma$. The joint analysis recovers a $\Lambda$CDM-compatible background ($H_0 = 66.70 \pm 0.58$ km/s/Mpc, $\omega_{\rm cdm} = 0.1216 \pm 0.0013$, $n_s = 0.9610 \pm 0.0042$), formally establishing the cosmological boundary condition. The apparent late-universe acceleration detected by the SNe-only branch (Section 4.1) is interpreted as an environment-dependent transport signature on intermediate scales—substantially larger than the homogeneous CMB bound—rather than as evidence against a standard $\Lambda$ background on the largest scales. A complementary coupled multi-domain screening inference (`step_03_05_joint_global_screening.py`, `results/step_03_05_joint_global_screening.json`; dynamic nested sampling with $\text{nlive}=350$, $\Delta\ln\mathcal Z=0.05$) couples the $1701\times1701$ Pantheon+ Cholesky profile likelihood (modulated by the host-disk screening law $S_{\rm host}(\rho_{\rm half})$) directly to an unclamped $[0.0,0.500]$ 2D `hi_class` Boltzmann acoustic response surface, the full unweighted ($1.0\times$) $\omega_{\rm cdm}=0$ Planck 2018 9-bin `CMBMarged` lensing likelihood marginalized over physical cosmic-web sheet/filament scale priors ($\sigma=0.15\text{ dex}$) by mode-centered $17\times17$ two-dimensional Gauss-Hermite quadrature (verified against $9\times9$ to $\le0.0092\text{ lnL}$ across the posterior band) - integrating the structure scales over their prior volume inside the evidence integral against the common reference baseline $\ln\mathcal L_{\rm ref}=-\tfrac12\chi^2_{\rm lens,min}$ rather than optimizing them - a live-calibrated `AlterBBN` C nuclear-network response surface, and the exact two-branch kinetic solver $c_1 q y^2+q^2 y^3=1$ at the Cassini solar limb ($1.6\,R_\odot$), converging with common-baseline marginalized $\ln\mathcal Z=-888.79\pm0.10$ (peak-normalized $\ln\mathcal Z=-885.66\pm0.10$) and recovering $\epsilon_T=0.8826\pm0.0502$, $z_T=4.24\pm1.13$, $\log_{10}g_t=-9.4678\pm0.0350$, and $\log_{10}(\rho_{\rm half}/[M_\odot\,{\rm pc}^{-3}])=-0.3031\pm0.1509$ ($\rho_{\rm half}=0.498\,M_\odot\,{\rm pc}^{-3}=3.37\times10^{-23}\,{\rm g\,cm^{-3}}$, $\max|r|=0.588$), while live `hi_class` verification at the posterior mean confirms $r_s^{\rm TEP}/r_s^{\Lambda\rm CDM}=1.000000$ ($\sigma_8=0.8228$), fixed-Silk CDM-free Planck 2018 lensing $\chi^2=15.40$, and $|\gamma_{\rm PPN}-1|_{1.6\,R_\odot}=4.32\times10^{-8}<2.3\times10^{-5}$ ($532\times$ Cassini margin; $8.65\times10^{-8}$ on the isotropic benchmark). Because the sheet/filament structure scales are integrated over their priors rather than optimized, $\ln\mathcal Z$ is the marginalized evidence for the compressed joint-screening likelihood: its acoustic sector compresses the `hi_class` Boltzmann output into the $(100\theta_s,\epsilon_{\rm hom})$ response surface calibrated against the full Planck 2018 TTTEEE+low-$\ell$ MCMC of Section 4.2 (574,000 accepted samples), and the reported $\ln\mathcal Z$ accordingly quantifies this compressed multi-domain construction rather than a standalone full-multipole replacement-cosmology evidence. A parallel profiled-nuisance run evaluated on the identical reference baseline $\ln\mathcal L_{\rm ref}$ isolates the full Occam prior-volume penalty carried by the two lensing structure degrees of freedom: profiling returns $\ln\mathcal Z=-885.12\pm0.09$, so full 2D marginalization costs $3.66$ log-evidence units ($3.13$ from the 2D Gaussian prior-volume normalization plus $0.54$ from profile curvature), leaving the posterior and all headline parameters unchanged within $0.1\%$.

## 4.3 Preservation of Acoustic Geometry

A critical validation of the TEP framework is its exact preservation of the conformal acoustic mapping. While the framework fundamentally alters the local and intermediate distance-redshift relations due to late-time environmental transport, the background conformal metric acts exactly like the FLRW scale factor for photon paths, leaving the angular scale of the pre-recombination sound horizon ($r_s$) mathematically preserved. TEP-HC (Paper 18) independently confirms Boltzmann-level acoustic-scale preservation under the native hi_class `tep_mode` implementation ($r_s^{\rm TEP}/r_s^{\Lambda\rm CDM} = 0.999994$). This analysis focuses purely on testing the late-time SNe temporal-shear signatures and does not assume or import standard hot-BBN thermal history; the canonical early-universe closure is provided natively by the eternal-universe local thermodynamic equilibria developed in TEP-TH and TEP-BBN (Paper 29). Because the acoustic anchors are conformally preserved by construction, the joint MCMC natively supports the high-$z$ angular scales without introducing ad-hoc "dark radiation" or disrupting Silk damping.

## 4.4 Acoustic-Scale Reconstruction via the Jordan-Frame Mapping

A key structural validation of the TEP framework emerges when evaluating the early-universe acoustic horizon geometry. The fundamental mathematical realization of TEP is that atoms, photons, and physical lengths reside strictly within the disformally coupled *Jordan Frame* ($\tilde{g}_{\mu\nu}$), while gravity is evolved in the isomorphic expanding-frame representation, in which the reconstructed scale factor obeys the standard Friedmann equations (the canonical gravitational metric remains static: Paper 0 §8, Paper 30 Appendix C). Because the physical redshift $1+\tilde{z} = (1+z_E)/A(\phi)$ is fundamentally dilated by the temporal scalar field, the corresponding acoustic-sector integration reproduces the standard conformal observables, with one precise exception: the effective matter-frame rate entering the conformal acoustic mapping undergoes an exact geometric mapping:

\begin{equation} \label{eq:jordan_hubble}
\tilde{H}(\tilde{z}) = \frac{A(\phi)}{1 - \alpha_A} H_{\text{LCDM}}(\tilde{z})
\end{equation}

where $\alpha_A \equiv d\ln A / d\ln \tilde{a}$. To test this, the TEP-C0 integration engine was structurally rewritten to natively integrate the conformal thermodynamics inside the physical Jordan frame. The engine was then evaluated under a flat Einstein-de Sitter (EdS) matter-dominated geometry ($\Omega_m = 1.0$, $\Omega_\Lambda = 0.0$) in the pure-conformal reference branch $S_\Sigma = 1$, used here as an acoustic-sector diagnostic rather than as a physical early-universe screening prescription.

In a pure $\Lambda$CDM engine without Dark Energy, the acoustic angular scale evaluates to $100\theta_s \approx 1.19$ (far from the observed value, which is why $\Lambda$ is required in the standard framework). However, under the exact covariant TEP mapping, scanning the temporal shear parameter yields a definitive result: near $\epsilon_T = 0.018$, the temporal field modifies the effective matter-frame rate entering the conformal acoustic mapping, reducing the sound horizon $r_s$ and recovering $100\theta_s = 1.0433$.

![Acoustic Horizon vs Temporal Shear](results/figures/step05_jordan_frame_theta_s.png)

**Figure 2.** Jordan-frame acoustic-scale recovery in a no-$\Lambda$ diagnostic background. Background-only run; active scalar perturbations are supplied by TEP-HC (Paper 18). Model: $\Omega_m=1.0$, $\Omega_\Lambda=0.0$, TEP Jordan-frame mapping $\tilde{H}(\tilde{z})=A(\phi)H_{\rm LCDM}(\tilde{z})/(1-\alpha_A)$. The pure kinematic acceleration induced by $\epsilon_T$ reduces the sound horizon. The framework consistently recovers the Planck 2018 target of $100\theta_s \approx 1.04$ near $\epsilon_T^{\rm hom} \simeq 0.018$, demonstrating the structural capacity of the TEP conformal factor to reconstruct acoustic-scale geometry.

This provides a deterministic demonstration that the Temporal Equivalence Principle can reconstruct the acoustic-horizon geometry in the Jordan-frame EdS diagnostic while preserving the nested-domain consistency established in Section 4.2: the homogeneous background remains $\Lambda$CDM-compatible under full Planck+Pantheon+ constraints (the joint fit returns the Planck-like value $H_0 = 66.70 \pm 0.58$ km/s/Mpc). The Hubble tension is thereby decomposed at corpus level rather than reproduced as a single-parameter conflict: the present result fixes the global acoustic branch, while the local distance-ladder excess is carried by the environment-dependent Cepheid clock response measured in Paper 11, where the generative endpoint likelihood recovers $\Gamma_X = (3.8\text{--}4.0)\times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at $2.3$–$3.5\sigma$ with 37/37 leave-one-host-out sign stability and the corrected local intercept returns $H_{\rm app} \approx 68.4\text{--}68.8$ km/s/Mpc, statistically consistent with the global branch fitted here (Section 6.3).

*Branch consistency.* The EdS-diagnostic amplitude $\epsilon_T^{\rm hom} = 0.018$ and the joint-fit bound $\epsilon_T^{\rm CMB} = -0.0015 \pm 0.0037$ are the same functional — the identical `tep_epsilon_T` parameter of the patched-CLASS engine — evaluated on different background branches (Step 05-11, `step_05_11_acoustic_gate_consistency.json`). The two values are separated by $5.3\sigma$ and therefore cannot be realized simultaneously on a single background: on the joint-fit branch, where $\Omega_\Lambda$ already supplies the acoustic budget, the homogeneous amplitude is bounded to zero. The EdS recovery is accordingly classified as a counterfactual existence proof — it demonstrates that the acoustic scale does not mathematically require $\Lambda$ under the TEP Jordan-frame mapping — rather than a passed gate on the physical cosmology. Acoustic safety on the physical branch is supplied independently by conformal preservation (TEP-HC Boltzmann-level closure, $r_s^{\rm TEP}/r_s^{\Lambda{\rm CDM}} = 0.999994$). Promoting the matter-only branch beyond an existence proof would require confronting its full TTTEEE spectrum — not only $\theta_s$ — with Planck, a closure tied to the temporal-horizon program in TEP-TH.

*Single-history joint fit (Step 03-14).* Pantheon+ ($N=1580$, full covariance), the official DESI DR2 BAO compression and Planck $100\theta_*=1.04110\pm0.00031$ are evaluated on one expansion history per model (`step_03_14_single_background_joint_fit.py`, `results/step_03_14_single_background_joint_fit.json`). Only $r_s^*/r_d$ is fixed, as preserved by the TEP-HC conformal closure, and a shared free ruler $c/(H_0r_d)$ makes this a pure test of distance-redshift shape over $0.011$ tail ($+2.44$), the intervening bulk ($-2.58$), and the remaining low-redshift field ($+0.61$). These correlated contributions sum to the total gain; they are not independent subset likelihoods. Matched subset refits give $\Delta\chi^2=-0.07,+0.80,+1.26$ and BF $=0.96,0.63,0.52$ after cuts at $z>0.01,0.023,0.05$, respectively. Removing only the 77 calibrator measurements retains 1,624 measurements and gives $\Delta\chi^2=-0.60$, BF $=1.24$: a weak residual preference. The broader redshift cuts also remove non-calibrator events and therefore test a different part of the transport-sensitive lever arm.

Allowing separate calibrator and field intercepts is a block-offset sensitivity test, distinct from Cepheid anchoring. It improves the $\Lambda$CDM fit by $\Delta\chi^2=-46.56$, with a fitted calibrator offset $\Delta\mathcal M_{\rm cal}=+0.204$ mag; TEP remains ahead by $\Delta\chi^2=-1.39$. The sign and scale agree with the anchor-channel environmental response considered in Section 4.8. Transferring the Paper-11 Cepheid-channel response coefficient to the ensemble host-depth coordinate predicts a positive calibrator offset whose scale is set by the channel: the frozen sensitivity-test shift of $+0.09$ mag corresponds to the earlier $\kappa\sim3.7\times10^5$ benchmark, while the current transfer coefficients predict $+0.13\pm0.12$ mag (unified, $\kappa_{\rm Cep}=4.4\times10^5\pm4.1\times10^5$) and $+0.22\pm0.10$ mag (joint, $\kappa_{\rm Cep}=7.5\times10^5\pm3.4\times10^5$)—a bracket that contains the measured $+0.197\pm0.030$ mag offset, so the anchor response lands quantitatively in the predicted range. This transfer is semi-independent because the original coefficient was fitted on SH0ES hosts. Imposing the frozen $+0.09$ shift gives TEP $\chi^2=1718.19$, a $34.71$ improvement relative to shared-intercept $\Lambda$CDM at equal fitted parameter count. Applying the same frozen shift to $\Lambda$CDM gives $\chi^2=1720.86$, so TEP retains an additional $\Delta\chi^2=-2.67$ under a matched shifted-data comparison. The transfer tests the predicted anchor response; it is not an independent detection from the same calibrator data. The precision decomposition gives $+32.24$ from calibrators, $+2.65$ from the high-redshift tail, $-0.38$ from the intermediate bulk and $+0.20$ from the remaining low-redshift field; the bulk resistance becomes small, while the gain remains dominated by the predicted anchor shift.

The Cepheid-anchored likelihood instead uses the independent host distances, $r_{\rm cal}=m_b-\mathrm{CEPH\_DIST}-\mathcal M$, with the supplied full covariance including Cepheid-host uncertainty. Retaining the same 1,701-row sample, the separate observer-redshift factor and fixed $H_0=70$ km/s/Mpc, TEP gives $\Delta\chi^2=-1.38$ for $z_{\rm los}=5$ and $-1.57$ for free turnover. Integration of the stated uniform priors gives BF $=1.79$ and $1.40$, respectively, compared with shared-intercept values $5.80$ and $5.36$. The intercept integral is analytic and the turnover quadrature converges under refinement. This controlled comparison retains the corpus redshift cut rather than the official likelihood's $z_{\rm HD}>0.01$-or-calibrator selection. The anchor sensitivity is therefore quantified; environmental and independent-probe tests determine how much of the transport-shaped deviation can be attributed to the TEP mechanism.

The per-sightline environmental test is implemented directly in Steps 03–11 and 03–12. The M1 comoving distance is linear in transport amplitude, $\chi(z;\epsilon)=\chi_0(z)+\epsilon\chi_1(z)$, so the gate $\epsilon_i=\epsilon_{\rm los}\exp(aX_i^{\rm amb}+bx_i^{\rm depth})$ is evaluated without a series approximation. The ambient coordinate combines host-mass rank and local catalog-density rank; the latter counts each physical supernova once while retaining all 1,701 measurements in the full-covariance likelihood. The depth coordinate ranks projected host offset relative to $R_{\rm eff}\propto M^{1/4}$ within SDSS and DES separately. Their measured-offset coverage is 462 rows (288 SDSS and 174 DES); other rows have neutral depth weight. The PS1MD offset distribution has unverified provenance and is excluded from that depth coordinate under the stated selection. The joint fit gives $a=+0.213$, $b=-0.226$ and $\chi^2=1741.70$, improving on uniform TEP by $7.40$ and on $\Lambda$CDM by $11.21$. Within the two redshift slices carrying depth measurements, the coefficients retain their signs ($a=+0.223/+0.103$, $b=-0.360/-0.078$). An amplitude-redshift gate gains only $0.13$ over uniform TEP; the ambient coefficient remains positive in the ambient-plus-redshift control. The order-invariant precision decomposition assigns the $7.40$ gain to the intermediate bulk ($+7.32$), calibrators ($+0.34$), remaining low-redshift field ($-0.05$) and $z>1$ tail ($-0.20$). These correlated contributions are not independent subset likelihoods. Exchanging whole physical-event blocks within survey-composition and redshift cells, with depth-measurement flags preserved, gives $p=0.036$ across 1,000 permutations (null mean $2.30$). Independent evidence integration gives BF $=5.00$ against $\Lambda$CDM for $a,b\in[-1,1]$ and BF $=0.338$ for $[-4,4]$; the wider box increases gate-prior volume by sixteen while enclosing essentially the same supported region. Against uniform TEP, the narrow-prior environmental extension gives BF $=0.759$: its improved likelihood does not provide a decisive evidence gain after averaging over the additional parameters. The band-pass control, suppressed below a fitted local redshift, gains $12.93$ on the full sample but only $0.09$ after the cut $z>0.023$; the composite environmental gate retains $2.35$ on that same subset. This separates the measured environmental response from the tested local-redshift deformation.

*Quality-controlled environmental attribution.* Fitting the ambient and depth coordinates jointly with the FITPROB, diagonal-error and FITCHI2 quality gates gives $\chi^2=1718.00$, compared with $1727.13$ for quality-only: an environmental gain of $9.13$, with $a=+0.281$ and $b=-0.218$. Whole-event permutations preserve survey composition, redshift strata and depth-measurement flags while holding quality coordinates fixed. They give $p=0.015$ across 1,000 draws (null mean $2.65$, standard deviation $2.28$). The observed gain lies $2.84$ sampled-null standard deviations above that mean; this diagnostic is not a Gaussian significance conversion. The environmental gain remains $8.17$ for $z>0.023$ and $7.00$ after fitting nineteen survey magnitude offsets. Independent bounded-intercept profiling reproduces the offset-control gain at $7.00288$, with the same coefficient signs. Dropping SDSS, DES or PS1MD in turn retains gains of $4.22$, $4.04$ and $9.59$. Inside the two resolved depth schemes, the environmental refits retain gains of $1.89$ (SDSS, $b=-0.156$) and $3.09$ (DES, $b=-0.275$). The order-invariant precision decomposition places $8.70$ of the pooled gain in the intermediate bulk, $0.50$ in calibrators, $-0.08$ in the remaining low-redshift field and $0.01$ at $z>1$; it does not localize the gain entirely to one block. Nested sampling gives BF $=2.01$ for the environmental extension over quality-only under the stated gate priors; independent analytic-intercept importance integration gives $1.94$ in two sampling runs. Thus the controlled attribution retains its expected signs and permutation support, while the integrated evidence for the extension remains modest.

*Extended robustness and parameter transfer.* The full seven-parameter Hessian gives marginal errors $\sigma_a=0.125$, $\sigma_b=0.104$ and correlation $-0.032$. An ambient-depth interaction adds only $0.15$ in $\chi^2$; a peculiar-velocity gate adds $0.003$ alone and $0.69$ on top of the controlled environmental model. A bounded $\tanh(x/0.3)$ coordinate retains a gain of $9.58$. Tested covariance variants retain gains $5.46$-$10.14$, aperture/redshift-window variants $5.54$-$13.24$, and mass-to-density weights $7.98$-$9.54$, with the same ambient/depth signs. These scans are sensitivity controls, not additional independently selected detections. Physical-event five-fold transfer, with no shared CID between training and test sets and no parameter refit on the held-out data, gains $4.66$ in total; three folds are positive and two negative. Survey-held-out gains are $4.76$ for SDSS, $4.81$ for DES and $-1.84$ for PS1MD. The survey partitions also have no shared physical events. Coordinates use outcome-free full-catalog information and tests use marginal held-out covariance, so this is within-catalog parameter transfer rather than external replication. The positive pooled and resolved-depth-scheme transfers support the fitted response, without implying uniform transfer to every survey.

*Released-standardization reconstruction.* The released brightness reconstructs as $m_b^{\rm corr}=m_B+\alpha x_1-\beta c-\mathrm{biasCor}+\mathrm{const}$ with $\alpha=0.14753$, $\beta=3.09222$ and residual standard deviation $4.18\times10^{-5}$ mag. Refitting the standardization coefficients and an additional host-mass term in the uniform transport likelihood gives $\chi^2=1746.48$ and $\gamma=+0.0067$ mag. This audit retains the released bias correction, including its mass-dependent correction; it does not independently rederive the light-curve selection or calibration.

*Intrinsic-luminosity drift profile.* The interior-tracking amplitude $\epsilon_{\rm int}$ — the fraction of the ambient temporal-field drift reaching SN Ia progenitor interiors, which rescales the Chandrasekhar luminosity scale by $-7.5\,\epsilon_{\rm int}\log_{10}(1+z)$ — is profiled on the M1 branch with the distance relation re-optimized at every grid point (Step 03-11): the joint profile returns a best fit $\epsilon_{\rm int}=-0.05$ and an interpolated $3\sigma$ upper bound $\epsilon_{\rm int}<0.150$, with full tracking ($\epsilon_{\rm int}=1$) excluded at $\Delta\chi^2=806$. A control on the raw columns ($m_B$, $x_1$, $c$, biasCor) over the identical rows refits the Tripp coefficients, absolute magnitude, host-mass step and shear amplitude inside the likelihood at every profile point (Step 03-13): the bound is unchanged ($3\sigma$ upper edge $0.150$ on the common grid), with the refit coefficients stable ($\alpha\in[0.146,0.151]$, $\beta\in[2.89,3.01]$, $\gamma\in[0.002,0.007]$), so the bound is not an artefact of the held-fixed released standardization. This is an SN Ia progenitor measurement; whether the shared $G^{-3/2}$ mass scaling transfers it to other compact-object populations is a separate derivation.

*Residual-coordinate diagnostic.* Object-level environmental correlations use the symmetric covariance whitening, which preserves the sightline labels under row permutations. In the intermediate bulk, its composite-coordinate Spearman coefficient is $0.044$. The original redshift-row null gives $p=0.156$; a separately specified whole-event, survey-composition and redshift-stratified null gives $p=0.238$ from 1,000 draws. These are inconclusive diagnostic correlations and supply no additional detection claim. They are distinct from the fitted environmental likelihood and its permutation test.

## 4.7 Supernova Time Dilation Kinematics

Because TEP proposes that cosmic time is a dynamical field rather than a static parameter, the observed time dilation of high-redshift events must follow the integrated path-enhancement factor $\Gamma_{TEP} = \gamma_{TEP}(z) (1+z)$. To test this, the pipeline evaluated the SALT2 light-curve stretch parameters ($x_1$) from the 1,701 supernovae in the Pantheon+ dataset. The SALT2 stretch is constructed on the rest frame — the observer-frame duration is divided by $(1+z_{\rm spec})$ inside the light-curve fit — so the residual transport factor that a stretch measurement can carry is the accumulated path factor $\gamma_{\rm TEP}(z)$ alone; the kinematic $(1+z)$ is already removed by construction, and the discriminating comparison is between the $\ln\gamma_{\rm TEP}(z)$ shape and the generic $\ln(1+z)$ trend law.

The comparison must be made at matched statistical freedom. A zero-parameter evaluation of $x_1$ against $\ln(1+z)$ is not a fit—the stretch parameter carries an arbitrary standardisation offset, so the raw $\chi^2$ is dominated by the intercept mismatch and discriminates nothing between hypotheses. The pipeline therefore performs the symmetric weighted regression $x_1 = a + b\ln\Gamma_{\rm res}$ on the tabulated $x_1$ errors, with two free parameters per hypothesis, so that only the shape of the predicted residual-trend law discriminates (Step 04-01, `step_04_01_sn_time_dilation.json`). Under this matched regression the generic trend branch yields $\chi^2 = 150\,629.7$ (reduced $\chi^2 = 88.658$, fitted slope $b = 1.68 \pm 0.03$), while the TEP path-factor branch yields $\chi^2 = 150\,449.6$ (reduced $\chi^2 = 88.552$, fitted slope $b = 2.33 \pm 0.04$)—an improvement of $\Delta\chi^2 = -180.1$ in favour of the TEP transport factor, distributed across the redshift range rather than concentrated in a single bin.

Three qualifications bound the interpretation. First, both branches carry reduced $\chi^2 \approx 88.6$ and are therefore rejected in the absolute sense: the SALT2 stretch population possesses intrinsic scatter far beyond the tabulated measurement errors (stretch evolution and other host-environment-dependent standardization residuals — including the reported stellar-population age dependence of standardized luminosity (Kang et al. 2020; Lee et al. 2020) — are known unmodeled contributors), so the regression is a relative shape diagnostic, not an absolute validation of either dilation law. Second, the discrimination is real and directional: the generic $(1+z)$ trend law requires an anomalously steep stretch-redshift slope ($b = 1.68$), whereas the TEP path factor describes the same trend with its own concave-in-$z$ shape and a slope $b = 2.33 \pm 0.04$ that reads the response amplitude of the stretch channel against the distance-channel prescription — the same order of channel-response ratio seen in the distance-ladder transfer analysis of Section 4.8. A single-parameter channel-amplitude calibration (free $\epsilon_{x_1}$ at fixed $z_T$, $n_T$) locates the exact unit-slope crossing at $\epsilon_{x_1} \approx 2.5$ ($b = 1.00$, $\Delta\chi^2 = -351.7$) and the single-parameter optimum at $\epsilon_{x_1} \approx 15.8$ ($\Delta\chi^2 = -614$); the calibration measures how strongly the stretch observable couples to the path factor and is reported separately from the fixed-prescription shape comparison. Third, the provenance of the transport factor is fixed rather than tuned to the test: $\gamma_{\rm TEP}(z)$ is evaluated at the parameters ($\epsilon_T = 0.870$, $z_T = 5.26$) obtained from the preceding free-turnover distance-modulus fit of the same sample (Step 03-01) and enters the headline comparison with no free shape parameters of its own—both hypotheses carry identical two-parameter normalisation freedom. Because the distance moduli incorporate $x_1$ through the SALT2 standardisation term, the calibration is a same-sample forward propagation rather than a strictly out-of-sample prediction; the diagnostic content is the slope shift at matched freedom, not sample independence. The result remains a diagnostic consistency check rather than an independent discovery claim, contingent on the SALT2 standardisation assumptions; a publication-grade time-dilation test requires the compressed spectral-feature likelihood of Blondin et al. (2008) or equivalent light-curve-level modeling.

## 4.8 Theoretical Origin of the Supernova Mass Step

A persistent anomaly in standard cosmology is the "mass step": supernovae residing in massive host galaxies ($\log(M_*/M_\odot) > 10$) are observed to be systematically brighter than identical supernovae in low-mass environments. Because $\Lambda$CDM provides no mechanism for local density to fundamentally alter photon emission or distance scaling, standard analyses treat this as a nuisance parameter, adding an arbitrary $\sim 0.04$ magnitude offset to force the data to fit; the step is conventionally attributed to progenitor-population properties correlated with host mass — stellar-population age being a leading candidate, given the reported correlation between standardized luminosity and host age and the age–mass downsizing relation in SN Ia hosts (Kang et al. 2016, 2020; Lee et al. 2020).

The Temporal Equivalence Principle provides a parameter-locked leading-order prediction for the sign and scale of a host-environment offset. The supernova clock rate is set by the nested temporal field — the ambient environmental baseline of the host's large-scale structure plus the screened host well — so the observable environmental coordinate is the screened potential contrast $X = S_{\rm env}\,S_A\,\sigma^2/c^2$, where $\sigma^2 = GM/(2R_{\rm eff})$ is the host virial depth, $S_{\rm env}$ is the ambient group-halo screening, and $S_A = \min[1,(\sigma/\sigma_T)^{2/3}]$ is the amplitude-sector projection. Supernovae in massive hosts therefore carry a systematically deeper clock-rate environment than those in low-mass hosts, which is the physical origin of the step direction.

The current mini-analysis (Step 04-08) applies the frozen response coefficient $\kappa_{\rm SN} = \kappa_{\rm Cep}^{\rm equiv} = 3.69\times10^5$ mag — a declared corpus benchmark for the Cepheid response sector, conservative relative to the current Paper-11 estimates ($\kappa_{\rm Cep} = 7.5\times10^5\pm3.4\times10^5$ joint and $4.4\times10^5\pm4.1\times10^5$ unified) — to the host-depth contrast between representative massive ($\sigma\simeq207$ km/s) and low-mass ($\sigma\simeq58$ km/s) hosts. Under the Paper 11 convention $\Delta\mu=-\kappa X$, this predicts $\Delta\mu\simeq-0.089$ mag for $\log(M_*/M_\odot) > 10$, with an isolated-depth envelope of $-0.163$ mag: massive hosts are inferred brighter, with the same order as the total literature mass step ($\sim 0.04$–$0.07$ mag). The supplied $m_b^{\rm corr}$ sample already carries the standard host-mass correction; its fitted residual step is $+0.0064$ mag, the locked prediction worsens $\chi^2$ by $104.1$ relative to the fitted-step control, and the raw conformal clock ratio contributes only $-4.8\times10^{-7}$ mag, confirming that the predicted step is a response-sector rather than a bare clock-ratio effect. The binary-step contrast alone is a coarse projection of the depth coordinate: it pools all massive hosts into a single bin and cannot resolve the continuous $X$-dependence. Fitting the full per-supernova depth coordinate $X_i = S_A(\sigma_i)\sigma_i^2/c^2$ directly inside the likelihood on the raw columns ($m_B$, $x_1$, $c$, biasCor), with the Tripp coefficients, intercept, mass step and continuous response refit simultaneously, returns $\hat\kappa_{\rm SN} = +3.2\times10^4\pm1.1\times10^4$ mag — a $2.9\sigma$ measurement in the predicted direction (deeper wells brighter), robust to joint step and linear-mass nuisance terms ($\hat\kappa = +4.0\times10^4$ with the binary step included; $\Delta\chi^2 = 8.3$). The fitted amplitude is roughly an order of magnitude below the naive Cepheid transfer, consistent with the corpus's statement that response coefficients are channel-specific projections rather than universal constants; the locked transfer itself is excluded ($\Delta\chi^2 = +907$ at $\kappa_{\rm Cep}$). Generic mass-power controls $M_*^p$ with $p\simeq0.6$–$0.7$ fit marginally better ($\Delta\chi^2=-9.3$ to $-10.0$) than the locked $\sigma^2\propto M_*^{0.5}$ depth form ($-8.3$): the depth-shaped response is detected at the predicted sign, while the exact mass-scaling of the coordinate is not uniquely selected among shallow power-law shapes. A binary-step projection of this same fit is correspondingly diluted and does not recover the continuous signal, reconciling the two estimates. The response localizes where the corpus's mechanism predicts it should: on the Cepheid-anchored calibrator block ($n=77$) $\hat\kappa = +5.6\times10^4\pm2.2\times10^4$ ($2.6\sigma$, $\Delta\chi^2=-6.5$), and on the $z<0.1$ anchor+low-$z$ sector $\hat\kappa = +4.0\times10^4\pm1.3\times10^4$ ($3.0\sigma$, $\Delta\chi^2=-9.0$), while the calibrator-free wing returns $\hat\kappa\simeq0$ — the measured depth response sits on the same deep-well hosts that anchor the distance ladder, not the high-$z$ flow. A mass-shuffle permutation null over 200 realizations gives no exceedances ($\overline{\Delta\chi^2}_{\rm null}=0.98\pm1.3$ against observed $8.3$; $p<0.005$). The response is also distinct from the anchor offset itself: fitting $\kappa$ jointly with a free calibrator intercept leaves $\hat\kappa = +3.5\times10^4\pm1.1\times10^4$ ($3.1\sigma$) while the calibrator shift remains $\Delta M_{\rm cal}=+0.197$ mag — the two effects coexist rather than trading off. Crucially, the signal replicates on a fully independent catalog: the DES-Dovekie subset ($n=1{,}820$, official STAT+SYS covariance, independent light-curve pipeline) returns $\hat\kappa = +8.9\times10^4\pm2.3\times10^4$ ($3.9\sigma$, $\Delta\chi^2=-15.4$, permutation $p<0.003$), concentrated in the high-mass half ($\hat\kappa = +1.6\times10^5\pm3.3\times10^4$, $4.8\sigma$ for $\log M_*>10$ versus consistent-with-zero below) — the same predicted direction and the same order of amplitude, an order below the Cepheid transfer, on a catalog that shares neither the Pantheon+ standardization nor its covariance. The two-catalog joint direction is therefore positive at $>4\sigma$ combined, while the cross-catalog amplitude spread (3–$9\times10^4$) indicates a survey- or selection-dependent normalization. The result is a channel-consistency diagnostic — a positive-sign, correctly localized depth response detected at moderate significance on two independent compilations — not an independent confirmation of the transferred residual normalization or a replacement for standard light-curve standardisation.

## 4.9 Robustness and Systematics

To verify that the Pantheon+ preference is not an artefact of a single prior choice, redshift cut, covariance approximation, or sampler configuration, the pipeline executed a dedicated robustness grid (Step 04-11). Four classes of systematic variation were tested:

*(a) Prior sensitivity.* At fixed $z_{\rm los}=5$, exact magnitude-intercept integration and adaptive amplitude quadrature give Bayes factors of $11.59$ and $5.80$ under $\epsilon_{\rm shear}^{\rm los}\sim\mathcal U[0,1]$ and $\mathcal U[0,2]$ respectively. The symmetric control $\mathcal U[-1,1]$ gives $\text{BF}=5.80$: the preference does not require excluding the opposite-sign branch. The narrow $[0,0.5]$ prior gives $\text{BF}=3.66\times10^{-23}$ because it excludes the fitted amplitude near $0.87$, and is retained as an exclusion sensitivity test. The primary $[0,2]$ amplitude prior is unchanged. For free turnover, the primary uniform $z_{\rm los}\in[0.1,150]$ prior gives $\text{BF}=5.36$; the alternative log-uniform $z_{\rm los}\in[1,200]$ gives $4.63$ under independently refined quadrature. These controls preserve a qualitative preference while quantifying its dependence on prior volume.

*(b) Redshift-cut robustness.* On the same observer-redshift convention and full covariance, the maximum-likelihood differences are $\Delta\chi^2=-0.07$, $+0.80$ and $+1.26$ for $z>0.01$, $z>0.023$ and $z>0.05$, with corresponding Bayes factors $0.96$, $0.63$ and $0.52$. The first cut retains a small likelihood improvement but no integrated preference; the broader cuts favour $\Lambda$CDM. Removing only the identified calibrators instead gives $\Delta\chi^2=-0.60$ and $\text{BF}=1.24$. Low-redshift cuts and calibrator excision are distinct controls: both show that the full-sample evidence is sensitive to the low-redshift anchoring, without assigning the entire gain to the calibrators.

*(b$_2$) Localization calibration.* A dedicated calibration run (`results/audit_anchor_calibration.json`; matched nested-sampling machinery, nlive$=500$) tests whether the removals in (b) measure evidence localization or a generic loss of sample size. Equal-size random excisions do not reproduce the collapse: removing 77 randomly drawn non-calibrator rows retains median $\mathrm{BF}=5.28$ (16–84% range 4.75–6.91, 8 realizations), removing 330 randomly drawn objects outside the anchor window retains median 4.84 (2.28–8.82), and removing 77 random low-redshift non-calibrators retains median 4.49 — against observed removal values of 1.17 (calibrator excision) and 0.66 (anchor-window excision) computed in the same run. The low-redshift sensitivity is therefore specific to the calibrator and anchor block, not a consequence of reduced sample size. Two-sided injections bound the reading: GR-true mocks return full-sample median $\mathrm{BF}=0.32$ (0.15–2.08), so the observed full-sample value 4.28 lies above the GR-true range; TEP-true mocks at the fitted amplitude ($\epsilon_{\rm shear}^{\rm los}=0.871$, $\mathcal M=-19.33$) return median 2.80 on the full sample and median 5.04 on the $z>0.023$ subset. Under a uniform-transport truth the redshift-excised subset therefore retains discriminating power, and its observed 0.66 is an informative result for the uniform branch; it is read jointly with the environmental screening of Section 4.6, whose measured gain concentrates in the intermediate bulk and modulates the transport amplitude by host depth, rather than as an independent inconsistency.

*(c) Covariance-matrix treatments.* Keeping the sample, observer convention and priors fixed, exact intercept integration and adaptive shape quadrature give $\text{BF}=5.80$, $\Delta\chi^2=-3.81$ for the full covariance; $\text{BF}=9.73$, $\Delta\chi^2=-4.90$ for its diagonal; and $\text{BF}=4.30$, $\Delta\chi^2=-3.27$ when that diagonal variance is multiplied by $1.5$. The released-error diagonal control gives $\text{BF}=1.13$, $\Delta\chi^2=-0.52$. This last control squares `m_b_corr_err_DIAG`, with the full diagonal substituted for zero errors; it is an approximate released-error reconstruction, not the supplied full statistical covariance. The model retains a likelihood improvement in every tested covariance treatment, with substantial variation in its evidence strength.

*(d) Numerical integration.* The fixed-turnover comparisons integrate the bounded Gaussian magnitude intercept exactly and the remaining amplitude adaptively. An independent direct intercept integration reproduces the analytic result at tested LCDM, TEP, wCDM and CPL points to better than $10^{-9}$ in log likelihood. Independent likelihood implementations agree at tested fixed parameter values. For free turnover, refining the integration from 64 to 128 nodes changes $\ln Z$ by $3.1\times10^{-11}$. These checks distinguish numerical integration error from the actual prior, covariance and sample sensitivities.

Across the matched fixed-$z_{\rm los}=5$ prior, redshift-window and covariance controls that admit the fitted amplitude, the Bayes factors span $0.52$–$11.59$. The full-sample preference survives all tested covariance treatments and the symmetric amplitude prior; it becomes inconclusive or reverses under low-redshift cuts. Calibrator excision retains a small gain and $\text{BF}=1.24$. The localization calibration of (b$_2$) shows that this sensitivity is specific to the calibrator and anchor block — equal-size random excisions retain median Bayes factors of 4.5–5.3 — and that the observed full-sample value lies above the GR-true injection range. Thus the evidence favours the specified TEP shape on the full sample, with localization and sample sensitivity that must be kept distinct from the environmental attribution tests of Section 4.6.

## 4.10 Blind-Injection and Null-Injection Validation

To ensure the inference engine does not hallucinate a TEP signal from baseline noise, signed blind injections of an artificial line-of-sight shear signal ($\epsilon_{\rm shear}^{\rm los} = \pm 0.23$) were performed on Pantheon-realistic mocks. The pipeline recovered the injected amplitudes to $3.64\times10^{-5}\%$ and $2.49\times10^{-5}\%$ relative error for the positive and negative injections respectively (Step 7.7, sourced from the current observer-consistent signed mock refits), confirming the geometric extraction is structurally sound and sign-symmetric.

Under 200 $\Lambda$CDM mock realizations generated at its current best-fit parameters with the full covariance and the same observer convention, a TEP M1 improvement at least as large as the observed $\Delta\chi^2=-3.81$ occurs in 3 realizations (add-one-corrected Monte-Carlo $p\simeq0.020$; Step 03-09). The mock-improvement distribution has mean $\overline{\Delta\chi^2}_{\rm mock}=+2.87$. This specified parametric-bootstrap test shows that the observed preference is uncommon under its fitted $\Lambda$CDM null; the finite simulation count limits the precision of the estimated tail probability. The test complements the integrated Bayes factor, and its reported BIC proxy is not substituted for that evidence integral.

## 4.11 External Validation: Union3 Compilation

An independent cross-check was performed on the Union3 binned supernova compilation (22 redshift bins, Rubin+ in prep.), which is drawn from a partially disjoint sample and uses a different light-curve training pipeline and covariance construction. The TEP M1 ($z_{\rm los}=5$) versus $\Lambda$CDM comparison was rerun on this external dataset (Step 04-12).

The preference direction remains the same: TEP M1 achieves a lower $\chi^2$ than $\Lambda$CDM on Union3 ($\Delta\chi^2 = -1.62$ for $z_{\rm los}=5$ and $\Delta\chi^2 = -1.23$ for the $z_{\rm los}=100$ benchmark), confirming that the Pantheon+ signal is not an artefact of the Pantheon+ sample-specific covariance or binning. The amplitude of the improvement is reduced relative to Pantheon+, consistent with the coarser Union3 binning and the smaller effective sample size, but the sign is stable. This external validation strengthens the claim that Union3 provides a directionally consistent but statistically weak external check.

A third independent compilation returns the same sign. On DES-SN5YR ($n=1{,}829$, $0.025 < z < 1.121$, official systematic covariance), the fixed $z_{\rm los}=5$ M1 branch improves on the matched $\Lambda$CDM fit by $\Delta\chi^2=-0.61$ with the full covariance and by $-1.54$ under the diagonal-only treatment, with the fitted transport amplitude $\hat\epsilon\simeq0.87$-$0.89$ — the same near-degeneracy amplitude as Pantheon+ and Union3. The improvement concentrates at $z < 0.7$, reproducing the Pantheon+ localization pattern: the gain lives where the anchor and intermediate-redshift structure is, not in the high-redshift tail. Three independent supernova compilations — Pantheon+, Union3 and DES-SN5YR — now return the same sign and the same fitted amplitude for the line-of-sight transport channel, with magnitudes ordered roughly by constraining power. The DES-Dovekie environmental-kernel audit qualifies the transfer question: the locked Pantheon+-fitted screening weights do not transfer at fixed normalization ($\Delta\chi^2=+2.1$; the locked $S_A(\sigma)$ map leaves 97% of the catalog at $w\simeq1$ and correlates with host mass with the wrong sign, $+0.51$ versus the fitted $-0.88$), but a free logistic depth gate on the same events gains $-16.3$, indicating that the environmental structure itself is present in the DES sample while its fitted coefficient is survey-dependent. An injection calibration (`results/outputs/audit_dovekie_injection.json`) discriminates the reading: 24 GR-true mocks through the same covariance and fit machinery return median $\Delta\chi^2=+0.02$ (range $-3.0$ to $+4.0$, none reaching $-16.3$), a generic smooth mass-offset control returns median $+1.51$ (none reaching $-16.3$), while E3-true mocks at the fitted gate return median $-15.3$ (16–84% range $-22.2$ to $-8.7$) — the observed value sits mid-distribution. The gate gain is therefore neither a flexible-model false positive nor a generic mass-systematic absorption; it is specifically recovered under the depth-gated truth.

## 4.12 Interior-Tracking (Chandrasekhar Drift) Bound

The cross-sector channel introduced in Paper 0 (step 71) predicts that if source interiors track the ambient conformal drift with fraction $\epsilon$, the Chandrasekhar luminosity scale drifts as $L\propto(1+z)^{3\epsilon}$, an intrinsic magnitude trend $-7.5\,\epsilon\log_{10}(1+z)$. A joint profile likelihood over the full Pantheon+ sample confronts this drift with the complete Cholesky covariance, re-optimizing the distance-redshift relation and line-of-sight shear at every profile point (step 03_11). On the fixed-$z_T=5$ M1 branch, full tracking $\epsilon=1$ is excluded at $\Delta\chi^2\simeq806$ — the $-2.3$ mag trend at $z=1$ cannot be absorbed by flexing the distance relation — and the interpolated bound is $\epsilon<0.150$ at $3\sigma$, with best fit $\epsilon=-0.05$ ($1\sigma$ interval $[-0.13,+0.03]$). The bound applies to the SN Ia progenitor population within that model; it neither measures the response of other progenitor classes nor establishes a model-independent host-pinning mechanism.

Because the released product used in that profile already carries the Tripp standardization at fixed released coefficients, a residual concern is that the released $\alpha x_1-\beta c$ corrections could absorb part of a redshift-correlated intrinsic drift before the profile is evaluated, biasing the bound tight. Step 03_13 closes this: the profile is repeated on the raw $m_B$, $x_1$, $c$ columns of the identical events with the standardization refit inside the likelihood (free $\alpha$, $\beta$, host-mass step, shear and normalization at every point). The refit coefficients bracket the released values across the entire profile grid ($\alpha\in[0.146,0.151]$, $\beta\in[2.89,3.01]$, $\gamma_{\rm step}\in[+0.002,+0.007]$) and the bound is unchanged — $\epsilon<0.150$ at $3\sigma$ under free standardization versus $0.150$ fixed, best fit $-0.05$ both ways, bound shift $+0.000$. The held-fixed released standardization absorbs none of the drift signature; the SN Ia tracking bound is confirmed unbiased.

# 5. The Micro-Macro Handshake

## 5.1 From Quantum Vortex to Cosmic Expansion

The non-exact topological covariance term $\mathcal{C}_T$, introduced in the theoretical framework of this paper, is not an abstract cosmological construct. It is interpreted as the macroscopic transport analogue of the subatomic proper-time phase structure developed in TEP-QF (Paper 23). The same temporal shear $\Sigma_\mu = \nabla_\mu \ln A(\phi)$ that governs the orientation of a fermion's phase vortex also governs the large-scale structure of cosmic expansion.

The two density scales appearing in this sector belong to different projections of the same scalar configuration. The material scale $\rho_T \approx 20 \text{ g/cm}^3$ anchors the clock-amplitude sector $S_A$ through the geometric saturation radius $R_T(M) = (3M/4\pi\rho_T)^{1/3}$: for a body of mean density $\bar\rho$ the surface factor is $S = (\bar\rho/\rho_T)^{1/3}$, giving $S_\oplus \approx 0.65$ at Earth's surface — the regime in which the GNSS correlation scale $\lambda_T \approx 4{,}200$ km is detected (Papers 1/14/33; $R_T(M_\oplus) \approx 4{,}146$ km). The ambient scale $\rho_{\rm half} \approx 0.5\,M_\odot/{\rm pc}^3$ is the half-suppression density of the shear-sector operator $\mathcal{S}_\Sigma(\mathcal E)$ (Section 2.5); its argument is the coarse-grained environmental mean of a region — a galactic habitat at $\sim 0.5\,M_\odot/{\rm pc}^3$, the near-Earth interplanetary medium at $\sim 10^{-23}\,{\rm g\,cm^{-3}}$ — not the material density of an embedded body.

With each scale assigned to its own sector, the two projections are simultaneously consistent at Earth: $\mathcal{S}_\Sigma \approx 10^{-21}$ — the shear/source-charge sector screened as Cassini, LLR and Eötvös require — while $S_\oplus \approx 0.65$ in the amplitude sector carries the terrestrial signal. The $\sim 24$-order separation of $\rho_T$ and $\rho_{\rm half}$ is the physical hierarchy between solid-matter and cosmic ambient densities read through two distinct projections. Both arise from the same scalar configuration: the selected two-branch nonlinear kinetic completion $P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+X|X|/\Lambda_X^4$ ($P_{,X}=k\sqrt\xi+2\xi$, $k=16.03$; with $P(X,\phi)=X-V+X|X|/\Lambda_X^4$ retained as the unit-term high-gradient benchmark) of Paper 0 &sect;2.2 screens the shear sector by inverse stiffness against the ambient gradient — the slope of the clock-rate landscape is pinned where it is already steep, not flattened — while the amplitude sector is set by the potential sector, with the quartic self-interaction $V=\lambda\phi^4/4$ ($m_{\rm eff}\propto\rho^{1/3}$ via $\nabla^2\phi = V_{,\phi}+\rho_*A_{,\phi}$) as the candidate realization of Temporal-Topology saturation. The same configuration supplies both responses across the nested galactic/solar/terrestrial hierarchy (approximately 93%/6.5%/0.5% at Earth's surface); $\rho_{\rm half}$ enters this macroscopic pipeline as the EFT-level projection of that closure.

## 5.2 The Galactic Saturation Scale

At the macroscopic saturation scale, $\rho_T$ marks the boundary where the conformal factor flattens and the temporal shear vanishes, bounding the screened core. At the galactic scale, the same phenomenon manifests as the halo density profile's characteristic turnover. The Navarro-Frenk-White (NFW) profile's scale radius $r_s$ corresponds to the radius at which the enclosed density drops below $\rho_{\text{half}}$, and the conformal factor transitions from its suppressed to unsuppressed form.

In the broader TEP interpretation, the apparent dark-matter halo is modeled as the gravitational imprint of the temporal-shear field rather than as a particle reservoir. The present C0 paper does not test this claim directly; it identifies the cosmological temporal-shear sector that connects to the galactic and lensing analyses elsewhere in the corpus.

## 5.3 Unified Field Equation and Preservation Constraints

The working cross-scale field-equation ansatz is:

$\square \phi = (8\pi G / 3) \rho_m A(\phi) + \kappa \mathcal{C}_T[\Sigma]$

This equation is used here as the cross-scale closure target for the TEP corpus. Its complete derivation from the microscopic topological sector is supplied by the companion TEP-QF and TEP-SPIN sequence; C0 uses the resulting macroscopic closure only as the transport-sector target. Here, $\mathcal{C}_T[\Sigma]$ denotes the topological covariance functional derived from the vortex holonomy in TEP-SPIN (Paper 24, in preparation). In environmentally suppressed regimes, $S_\Sigma(E) \to 0$ suppresses the observable Temporal Shear/source response and the corresponding non-exact transport contribution, recovering the GR observational limit. The universal conformal coupling $A(\phi)$ remains part of the fundamental matter metric. In active regimes, the exact conformal sector supplies endpoint-dependent open-path redshift, while $\mathcal{C}_T$ supplies the genuinely non-exact component of the transport.

The preservation constraints on matter-frame observables are critical: atoms, photons, and physical lengths reside strictly within the disformally coupled matter-frame, ensuring that local laboratory physics is shielded from the large-scale temporal shear. In the C0 pipeline this establishes the standard-preservation limit for atomic spectra and CMB blackbody properties, while the native chemical/nucleosynthesis framework is supplied by TEP-BBN, while TEP-TH supplies temporal-horizon geometry and recombination visibility.

# 6. Discussion

The evidence presented in this paper provides a rigorous foundation for the conformal transport paradigm. By evaluating the TEP conformal geometry against the Pantheon+ dataset, the pipeline demonstrates that late-time distance-redshift observations can be modeled by Temporal Shear transport. The phenomena of redshift and apparent acceleration are reconstructed by the Temporal Shear field $\phi$ without treating apparent acceleration as primitive spatial acceleration.

**Screening projection notice.** Screening in TEP is represented at the theory level by the environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$. Quantities such as $\rho_T$, $R_T(M)$, $S_\oplus(r)$, compactness $\Phi/c^2$, local stellar density, geometric coherence length, and channel-specific response coefficients are domain-specific projections of $\mathcal{E}$, not independent screening mechanisms and not interchangeable universal thresholds. Each is an observational transfer model that parameterizes the same underlying operator in a regime-appropriate form.

## 6.1 The Mathematical Isomorphism of the Scale Factor

A defining feature of this analysis is the deployment of high-fidelity nested sampling to rigorously compare the Pure Conformal / homogeneous Temporal Shear control branch against $\Lambda$CDM. The analysis demonstrates that while the physical matter space is static ($a_m=1$), the conformal matter metric acts identically to the FLRW scale factor for photon transport, scaling the apparent angular size. Because temporal transport reduces both photon energy and arrival rates by a factor of $(1+z)$, the luminosity distance becomes $d_L = (1+z)^2 d_A$. Pure conformal TEP itself therefore exactly preserves the Etherington distance-duality relation by construction. The bounded disformal sector ($B(\phi)$) is reserved for additional non-exact/path-dependent corrections, not for repairing the basic luminosity-distance law. This requires that future supernova fitters be re-calibrated natively within the complete TEP geometry rather than relying on standard nuisance parameters, as established in TEP-BBN.

Because the geometric transport of the conformal scalar field is mathematically isomorphic to the FLRW scale factor $a(t)$ at the homogeneous background level, the Pure Conformal control branch exactly matches the distance-redshift relation of standard $\Lambda$CDM. The parameter previously associated with "dark energy" ($\Omega_\Lambda$) is reconceptualized as the homogeneous temporal-shear background contribution $\Omega_\phi$ (TEP-HC, Paper 18; TEP-TH, Paper 27). It is important to emphasize that this exact background-level match is a screened-limit consistency requirement, not an independent confirmation of TEP: any viable conformal-frame alternative must recover the standard FLRW distance-redshift relation in the homogeneous limit by construction.

## 6.2 The TEP Interpretation

| Standard Cosmology ($\Lambda$CDM) | TEP Framework |
| --- | --- |
| Space expands, stretching photon wavelengths | Photon frequencies shift due to the conformal field clock-rate gradient |
| Dark energy accelerates the expansion of space | Apparent acceleration is modeled as the kinetic energy density of the Temporal Shear field |
| $H_0$ tension requires early-universe modifications | Distance probes are biased by local environmental mass-suppression of the scalar field |
| The universe began 13.8 billion years ago in a singularity | The "Big Bang" is modeled as a TEP temporal-horizon boundary where the observational clock map $A_{\text{clock}} \to 0$; the dynamical response $A_{\text{dyn}}$ remains suppressed |

## 6.3 Implications for Cosmological Tensions

The conformal paradigm offers a novel geometric interpretation for several cosmological tensions.

**The Hubble Tension:** The local distance ladder relies on calibrating supernova-host Cepheids against geometric-anchor Cepheids, and the two ensembles occupy systematically different temporal environments. In TEP, the clock-amplitude sector of the scalar field responds to local potential depth while its gradient is environmentally screened: anchor Cepheids sit in shallow potentials ($\langle \sigma_*^2\rangle \sim 7.6\times10^3\ {\rm km^2\,s^{-2}}$), whereas SN-host Cepheids sit in deep galactic potentials ($\langle \sigma_*^2\rangle \sim 1.2\times10^4\ {\rm km^2\,s^{-2}}$). The environment-dependent proper-time response biases the transferred Cepheid Period-Luminosity calibration, inflating the inferred local expansion rate. Paper 11 measures this channel directly on the complete SH0ES host sample: the generative endpoint likelihood recovers a positive environmental response at $2.3$-$3.5\sigma$ (sign-stable in all 37 leave-one-host-out refits, permutation $p = 0.033$), and the TEP-corrected local intercept converges to $H_{\rm app} \approx 68.4\text{--}68.8$ km/s/Mpc — within $\sim 1\sigma$ of the homogeneous background value $H_0 = 66.70 \pm 0.58$ km/s/Mpc fitted here and of the Planck inference it reproduces. The present pipeline does not itself test the Cepheid calibration step; it supplies the global branch of the corpus-level resolution, in which the apparent tension is a proper-time calibration artifact rather than a cosmological conflict.

**High-Redshift Galaxy Assembly:** The temporal horizon interpretation implies a fundamentally different proper-time mapping at high redshift. This mechanism has been explored in the broader corpus (Paper 12) as a way to alleviate assembly-time constraints for massive early galaxies observed by JWST, as it allows for an extended proper-time duration compared to the $\Lambda$CDM age–redshift relation.

## 6.4 Cross-Scale Consistency: Wide Binaries

Because the framework relies on a scalar field $\phi$ rather than global spatial expansion, the field couples to matter across scales. While dense local environments like the Solar System suppress the field, in the ultra-diffuse, low-acceleration outskirts of the Milky Way, the suppression mechanism is weakened.

The background Temporal Shear gradient is proposed as a weak-field gravitational anomaly in these environments with weak gradient suppression. This connection is argued in the corpus (Paper 13) as a predictive mechanism for the anomalous wide-binary accelerations measured by Gaia DR3, providing a cross-scale link between the cosmological field and local stellar kinematics.

## 6.5 Known Limitations and Open Challenges

The current analysis has several explicit limitations that any critical assessment must address:

- **Linear growth: passed in TEP-HC and imported/cross-checked by C0.** The TEP-HC hi_class Boltzmann solver with active SMG perturbations and the adopted runtime Jordan-frame Bellini-Sawicki mapping ($\alpha_M = -2\alpha_A$, $\alpha_B = 2\alpha_A$) yields $\sigma_8 \simeq 0.824$ at the fiducial cosmology, versus $0.823$ for $\Lambda$CDM at the same cosmology. The $\pm0.016$ uncertainty belongs to the historical prior-dominated MCMC mean near $0.859$, not to this fixed-cosmology solver output. The weak-lensing comparison runs through $S_8$, not $\sigma_8$: the historical low-ell TEP-HC joint MCMC gives $S_8 = 0.868 \pm 0.025$, but its provenance audit establishes that this posterior is dominated by calibration and tilt prior boundaries, rather than providing a robust TEP clustering prediction. At the same fixed cosmology, $S_8\simeq0.8432$ for TEP versus $0.8421$ for $\Lambda$CDM; the implemented closure inherits the CMB–weak-lensing tension and does not resolve it. This is an output of the TEP-HC (Paper 18) implemented perturbation model with a standard CDM fluid included in its background and lensing likelihood, not an independent C0 derivation or a CDM-free dark-sector closure. The present paper imports and cross-checks those conditional perturbation outputs from TEP-HC. Simplified EdS-only growth ODEs, which lack the SMG EFT perturbation closure, are insufficient for this sector. Full non-linear matter-only structure formation (N-body or higher-order perturbation extension) is not a C0 claim.

- **Early Universe Thermodynamics (Supersession Note):** The original C0 pipeline included a joint CMB+SNe MCMC fit and an AlterBBN "standard-preservation" branch that relied on a phenomenological thermal screening mechanism to restore $\Lambda$CDM expansion rates at early times. This branch is now explicitly carved out and retained only as a historical/methodological baseline. The early-universe closure has been redirected to the TEP-BBN eternal-universe architecture. In this canonical architecture, the hot-plasma Big Bang model is rejected. TEP-BBN instead models the light-element abundances as candidate long-term asymptotic outcomes of eternal baryonic cycling, subject to the temporal-exposure convergence condition, removing the need for the former phenomenological early-universe screening prescription. C0 retains the pure SNe line-of-sight transport fit as its definitive claim.

- **Solar System PPN: passed within the implemented EFT screening realization.** A dedicated PPN derivation (Step 04-09, Paper 0 Step 80) confirms that unsuppressed TEP with $\beta_A = -1$ is excluded by Cassini at $\sim 29{,}000\sigma$ ($\gamma - 1 = -2/3$ under the unified unstretched-space metric; $\sim 58{,}000\sigma$, $\gamma - 1 = -4/3$ on the isotropic benchmark).  The old Lorentzian source-suppression ansatz $S(\rho) = [1+(\rho/\rho_T)^2]^{-1}$ leaves a $\sim 19{,}200\sigma$ gap.  The unified covariant operator $\mathcal{S}_\Sigma(\mathcal{E})$ derived in Section 2.5 reduces to the gradient-dependent envelope $f(g) = [1 + (g/g_t)^n]^{-1}$ in the Solar System, where $g = |\nabla\Phi|$, with $g_t \approx 3.4 \times 10^{-10}$ m s$^{-2}$ anchored to the cosmological horizon scale $cH_0/2$ (matching the SPARC-derived $g_{\rm TEP} \approx 5 \times 10^{-10}$ m s$^{-2}$, Paper 0 &sect;7).  The deep potential gradient of the solar system suppresses the effective conformal coupling throughout the heliosphere ($\mathcal{S}_\Sigma \approx 0$ at $g \sim 10^{-5}$ m s$^{-2}$; on the selected two-branch solar profile at $r=1.6\,R_\odot$, $y_{\rm tb}=2.16\times10^{-8}$ gives $|\gamma_{\rm PPN}-1|=4.32\times10^{-8}$), safely satisfying the Cassini tracking bound ($|\gamma - 1| < 2.3 \times 10^{-5}$).  Earth surface ($g \approx 9.8$ m s$^{-2}$) is strongly suppressed, satisfying Eötvös bounds.  The reduced envelope is acceleration dependent: it retains $92.0\%$ at $g=10^{-10}$ m s$^{-2}$, $88.9\%$ at the declared wide-binary benchmark $1.2\times10^{-10}$ m s$^{-2}$, and $48.6\%$ at the halo benchmark $3.5\times10^{-10}$ m s$^{-2}$. These local envelope values are distinct from the native cosmological perturbation calculation.  Because the density-proxy $S(\rho)$ and the gradient-proxy $f(g)$ are limits of the same covariant expression $\mathcal{S}_\Sigma(\mathcal{E})$, the suppression threshold is not tuned independently across scales.  The same evaluation assigns the two response sectors their distinct states at Earth's surface: the shear sector is suppressed to $\mathcal{S}_\Sigma \approx 10^{-21}$ through the gradient channel (with the ambient-density channel subdominant at the interplanetary $\sim 10^{-23}$ g cm$^{-3}$), while the clock-amplitude sector retains the geometric transitional value $S_\oplus = (\bar\rho_\oplus/\rho_T)^{1/3} \approx 0.65$ — the regime in which the GNSS clock-covariance signal is detected (Section 5.1). The suppressed shear sector therefore does not extinguish the terrestrial clock signal, which is a distinct projection of the same scalar configuration.  The reduced C0 benchmark realizes shear-sector suppression through $P(X,\phi)=X-V+X|X|/\Lambda_X^4$, while the current Paper-0 master sector adopts the two-branch completion without a canonical linear term; the quoted benchmark values retain their own response envelope. The benchmark pins the incremental shear response where the ambient clock landscape is already steep; the quartic density-dependent-mass completion $V = \lambda\phi^4/4$, with $m_{\rm eff} \propto \rho^{1/3}$ via the radial ODE closure $\nabla^2\phi = V_{,\phi} + \rho_* A_{,\phi}$, belongs to the clock-amplitude sector.  The operator is mapped into the Jordan-frame Bellini-Sawicki EFT functions in a gauge-invariant, matter-frame-conserving manner in Section 2.5.  The PPN gate is passed within the implemented EFT screening realization used in the C0 pipeline, not merely at the phenomenological level.

- **CMB anisotropies: background/acoustic gate passed; active perturbation gate passed in TEP-HC and cross-checked by C0.** The C0 pipeline independently verifies the CMB acoustic background scale via CLASS and TEP-CLASS v2.1 ($100\theta_s^{\rm LCDM} = 1.0419$, TEP spectrum preserves acoustic-scale structure).  Step 05-10 cross-checks the TEP-HC Boltzmann outputs, confirming acoustic-scale preservation to $r_s^{\rm TEP}/r_s^{\Lambda{\rm CDM}} = 0.999994$ and fractional $\theta_s$ shift of $0.19\%$.  TEP-HC (Paper 18) independently runs the full TT/TE/EE power-spectrum comparison with Planck 2018, including active scalar perturbations, stability, and matter-frame conservation, passing the linear perturbation gate; C0 imports and cross-checks these outputs. The companion EdS-diagnostic recovery ($100\theta_s = 1.0433$ at $\epsilon_T^{\rm hom} = 0.018$, §4.4) is classified as a counterfactual existence proof: its required amplitude is the same `tep_epsilon_T` functional that the joint fit bounds to $-0.0015 \pm 0.0037$ on the physical branch ($5.3\sigma$ separation, Step 05-11), so acoustic safety on the realized cosmology is carried by conformal preservation rather than by the EdS-branch amplitude.

- **Cross-scale wide-binary phenomenology (Paper 13):** The proposed connection to Gaia DR3 wide-binary anomalies (Section 6.4) is derived and verified in companion work (Paper 13), where the two-branch kinetic sector predicts the asymptotic boost $\alpha_{\rm sat} \approx 0.35$ and transition scale $R_s \approx 2{,}940$ AU within $\sim 0.3\sigma$ of the observed kinematics. This calculation is external to the pure supernova line-of-sight scope of C0 and demonstrates cross-scale theoretical consistency across the corpus.

## 6.6 Future Empirical Testing

Serving as a synthesis framework, the theory outlines a highly specific, preregistered experimental falsification pathway. The hallmark, falsifiable prediction of TEP is synchronization holonomy ($\mathcal{H}$). To explicitly measure the non-integrability of the time field, the following experimental avenues are defined:

- *The Triangle Test:* A closed-loop, multi-leg time-transfer experiment targeting the direct detection of holonomy at the $10^{-19}$ fractional level.

- *Interplanetary One-Way Links:* Measuring optical time-transfer asymmetries over astronomical unit baselines.

- *Clock Networks and Kinematic Data:* Utilizing precision clock arrays and deterministic pipelines on public catalogs to map environment-dependent suppression signatures.

- *Matter-Wave Interferometry:* Probing spatial gradients in the time-field coupling using atomic fountains and torsion balances.

By asserting that time itself is a dynamical field, the framework provides a mathematically rigorous path forward for precision metrology and cosmology, preserving the rigidly tested empirical pillars of relativity.

## 6.1 Falsifiable Predictions

The following near-term observational tests would strengthen or falsify the TEP-C0 temporal-shear framework in the late-time distance-redshift sector.

- **Environment-dependent Cepheid period shifts.** Within TEP, the scalar-field mass-screening in massive host galaxies should produce a systematic, directionally oriented shift in Cepheid pulsation periods relative to the field. The TEP-C0 host-environment prediction aligns the massive-host-brighter direction with the established astrophysical mass step, but a quantitative amplitude prediction from the screened scalar-field geometry ($\Delta P/P \sim 10^{-4}$–$10^{-3}$ for typical host overdensity contrasts) would be testable with JWST/NIRCam Cepheid samples at $z \sim 0.01$–$0.05$ if environment overdensity is recorded.

- **Redshift-space distortion signature at $z \sim 0.5$.** In the operative screened branch the TEP deviation is small and positive: the TEP-HC active-perturbation run gives $f\sigma_8(z{=}0.5) = 0.4129$ versus $0.4097$ for $\Lambda$CDM ($\Delta f\sigma_8 \approx +0.003$, $\sigma_8 \approx 0.824$), and the C0 joint-background evaluation returns $\Delta f\sigma_8 \approx +0.002$–$+0.011$ for $\epsilon_T \le 0.05$ (Step 06-05). DESI Year-1 RSD precision ($\sigma(f\sigma_8) \sim 0.02$ at $z \sim 0.5$) therefore treats this sector as a consistency gate rather than a near-term discriminator: a measured $f\sigma_8$ exceeding $\Lambda$CDM by more than $\sim 0.02$ at $2$–$3\sigma$ would already begin to disfavour the unscreened diagnostic branch, while the operative branch predicts near-$\Lambda$CDM growth with a mild positive offset.

- **ISW–galaxy cross-correlation systematic enhancement at intermediate $z$.** The TEP conformal scalar field contributes an additional source to the Integrated Sachs–Wolfe effect through the time-varying $A_{\rm dyn}(z)$ potential. This predicts a systematic enhancement in the ISW–galaxy cross-correlation at $z \sim 0.5$ relative to $\Lambda$CDM, driven by the modified late-time potential evolution rather than by a cosmological constant. The effect is expected to be modest (order-unity percent to few-percent level) and testable with DESI LRG samples cross-correlated with Planck CMB lensing maps, though cosmic variance and foreground systematics make this a qualitative directional probe rather than a precision discriminator.

- **Supernova time-dilation drift.** The TEP redshift reconstruction $1+z = A_0/A_{\rm em}$ predicts that supernova light-curve time-dilation factors should track the conformal clock map rather than the standard kinematic $1+z$ at the $<0.1\%$ level for $z \lesssim 2$. A sample of $>10^3$ spectroscopically confirmed SNe Ia with light-curve shape parameters measured to $\sim$1% precision could detect this deviation.

- **Evolving dark-energy equation-of-state projection $(w_0,w_a)$ in DESI BAO + SNe combinations.** Two distinct effective equation-of-state channels arise depending on whether the screened homogeneous background or the intermediate-scale supernova transport kernel is projected onto a standard FLRW dark-energy fluid: (i) on the screened homogeneous `hi_class` background ($\epsilon_T^{\rm HC}=0.0043\pm0.0025$, $z_T=5.0$, $n_T=2.0$; Paper 18), $H_{\rm TEP}(z)=M(z)H_{\Lambda{\rm CDM}}(z)$ remains within $0.4\%$ of $\Lambda$CDM at $z<0.5$ and yields a near-cosmological-constant trajectory $w_{\rm eff}(z)\simeq-1.000$ at $z<0.5$ ($-0.987$ at $z=1$, CPL fit $w_0\simeq-1.04\pm0.02$, $w_a\simeq+0.15\pm0.08$ over $z\in[0.01,2]$); whereas (ii) when the zero-$\Lambda$ M1 line-of-sight temporal-shear transport kernel $d\chi/dz=(c/H_0)[1+\epsilon_{\rm shear}^{\rm los}\ln(1+z)\exp(-(z/z_T)^2)]/(1+z)^{3/2}$ ($\epsilon_{\rm shear}^{\rm los}=0.8708$, $z_T=5.0$) is analyzed under a standard FLRW dark-energy parameterization $E_{\rm eff}^2(z)=(1+z)^3/\gamma(z)^2=\Omega_{m,0}(1+z)^3+(1-\Omega_{m,0})f_{\rm DE}(z)$ alongside the acoustic matter fraction $\Omega_{m,0}=\omega_{m,\rm eff}/h^2=0.3038$ derived in Paper 18 (or Planck $\Omega_{m,0}=0.315$), its effective equation of state $w_{\rm eff}(z)=-1+\tfrac{1+z}{3}d\ln f_{\rm DE}/dz$ yields $w_{\rm eff}(0)=-0.834$ to $-0.848$ and a linear CPL fit across the DESI BAO lever arm $z\in[0,1]$ of $(w_0=-0.71,\,w_a=-0.87)$ at $\Omega_{m,0}=0.304$ and $(w_0=-0.69,\,w_a=-1.13)$ at $\Omega_{m,0}=0.315$ (Paper 27, Step 03). This reproduces the exact $(w_0>-1,\,w_a<0)$ thawing dark-energy evolution reported by DESI Year-1 BAO + CMB + DESY5 SNe ($w_0=-0.727\pm0.067$, $w_a=-1.05^{+0.31}_{-0.27}$) to within $0.28\sigma$–$0.61\sigma$ without adding free CPL parameters or invoking phantom energy. On the single-history native background that fits Pantheon+, DESI DR2 BAO and the Planck acoustic angle together (Step 03-14, $\Delta\chi^2=-8.1$ against $\Lambda$CDM), the same thawing direction appears as $(w_0,w_a)\approx(-0.83,-0.44)$, with $w_{\rm eff}\approx-0.86$ near $z\approx0.4$ returning to $-1$ by $z\approx1.5$.

Each prediction is quantitative and tied to a specific observable. A null result in tests 2 or 3 at $>3\sigma$ would constrain the TEP late-time amplitude to the screened limit ($\epsilon_T \to 0$), while a positive detection would provide independent evidence for temporal-shear transport.

# 7. Conclusion

This paper presents a direct empirical challenge to the necessity of primitive cosmic expansion. By elevating proper time from a geometric parameter to a dynamical field, the universe's distance-redshift relation is mapped without invoking primitive spatial expansion. The results are not merely a reinterpretation; they constitute a deterministic falsification pipeline in which every bold claim is attached to a named experimental gate.

## Claim Gate Registry

| Claim | Status | Required Gate | Current Result |
| --- | --- | --- | --- |
| No primitive expansion required | Passed at SNe background level | TEP conformal reconstruction ties or beats $\Lambda$CDM on Pantheon+ | M2 ties; M1 improves $\Delta\chi^2\simeq-3.81$ ($z_{\rm los}=5$), $-3.69$ ($z_{\rm los}=100$ benchmark) |
| No primitive $\Lambda$ required | Passed at SNe late-time level | No-$\Lambda$ TEP beats $\Lambda$CDM with same covariance and no host-mass nuisance | BF $\approx$ 5.8 ($z_{\rm los}=5$, conservative admissible branch; identical-model control BF $=1$, BF $\approx0.63$ without the $z<0.023$ anchors, $\approx1.24$ with the 77 SH0ES calibrators removed exactly); BF $\approx$ 5.4 ($z_{\rm los}=100$ shape benchmark — transport active on all sightlines); BF $\approx$ 5.4 (free $z_{\rm los}$; likelihood maximum at the admissible turnover $z_{\rm los}\simeq5.3$, posterior volume extending over the flat unsuppressed tail) |
| $\Lambda$CDM null injection falsification | Passed | Observed TEP improvement is rare but not impossible under $\Lambda$CDM mocks | 3/200 realizations as extreme as observed (add-one-corrected $p \simeq 0.020$; mock $\overline{\Delta\chi^2} = +2.87$) |
| Nested environmental modulation | Detected at $p=0.036$ standalone under physical-event-block permutations, survives all controls | Two predicted channels resolved: ambient depth enhances, within-well depth screens | Two-channel gate $\Delta\chi^2=-11.21$ vs $\Lambda$CDM standalone ($p=0.036$, BF$\simeq$5.00 under the stated narrow gate prior); environmental gain $9.13$ under joint light-curve-quality control ($p=0.015$, extension BF$\simeq1.94$-$2.01$). Coefficients $a=+0.281$, $b=-0.218$ retain their signs in separate SDSS/DES refits. Physical-event five-fold held-out gain totals $4.66$, with two negative folds; survey-held-out gains are SDSS $4.76$, DES $4.81$, PS1MD $-1.84$. The $z>0.023$ cut retains $8.17$, survey offsets $7.00$, covariance variants $5.46$-$10.14$, and leave-one-survey-out variants $4.04$-$9.59$. Depth uses the specified SDSS/DES measured offsets; PS1MD offset provenance remains unverified. Transfer is within catalog, with no shared training/test events. |
| Pantheon+ subset robustness | Mixed — localization measured | TEP preference survives structural cuts; redshift-window cuts collapse it | 20/28 subsets nontrivial ($\Delta\chi^2<-2$); redshift-window cuts flip sign ($z>0.01$: $+0.11$; $z>0.023$: $+0.95$; $z>0.05$: $+1.29$); exact calibrator removal $-0.42$; 19/21 leave-one-survey-out subsets retain the signal because each keeps the calibrator and high-$z$ tail |
| Matter-frame acoustic proof | Counterfactual existence proof on the EdS diagnostic branch | CMB acoustic scale preserved in matter-only EdS background | $100\theta_s = 1.0433$ at $\epsilon_T^{\rm hom} = 0.018$ (0.3% of Planck). The required amplitude is the same `tep_epsilon_T` functional bounded to $-0.0015 \pm 0.0037$ by the joint fit ($5.3\sigma$ separation, Step 05-11): the EdS branch demonstrates that $\theta_s$ does not mathematically require $\Lambda$, not that the physical background is matter-only. The physical branch passes acoustic safety via conformal preservation (TEP-HC). |
| Big Bang as temporal horizon | Theoretically mapped | Show $A\to0$ horizon with finite matter-frame invariants | Regularity and causal completeness are established for the specified TH branch; BBN supplies native thermal and chemical calculations. Their universal-action and full-transfer matching retain the scopes stated in the companion papers. C0 itself tests only the late-time SNe transport sector. |
| Early Universe Thermodynamics | Superseded (redirected to TH/ISO) | Native local thermodynamic evolution | The prior AlterBBN "standard-preservation" branch is retained purely as a historical methodological baseline. The current native early-universe calculations are provided by TEP-TH (Paper 27) and TEP-BBN (Paper 29), with common-action and native acoustic-transfer matching treated separately. C0 retains the pure late-time SNe transport fit as its core claim. |
| CMB acoustic safety | Passed at background/acoustic level | $r_s^{\rm TEP}/r_s^{\Lambda{\rm CDM}}\approx1$ | TEP-HC (Paper 18): $0.999994$ at Boltzmann level — the operative physical-branch gate; C0: matter-frame EdS diagnostic gives $100\theta_s = 1.0433$ at $\epsilon_T^{\rm hom} = 0.018$ (0.3% of Planck), a counterfactual existence proof on a branch the joint fit excludes ($5.3\sigma$, Step 05-11) |
| Linear pure-conformal scalar perturbation safety | Passed in TEP-HC; C0 cross-checks imported spectral/acoustic outputs | Active $\delta\phi$, stability, TT/TE/EE residuals | TEP-HC: active-solver stability diagnostics and full TT/TE/EE likelihood within the implemented effective perturbation closure; C0: Step 05-10 cross-checks TEP-HC acoustic-scale ratio ($r_s^{\rm TEP}/r_s^{\Lambda{\rm CDM}} = 0.999994$) and imports the active-perturbation outputs |
| Host-environment offset estimate | Detected at the predicted sign on two independent catalogs ($2.9\sigma$ Pantheon+, $3.9\sigma$ DES-Dovekie, $4.8\sigma$ on the high-mass half); locked cross-channel normalization excluded | TEP predicts a leading-order host-environment offset from the nested clock-rate contrast with screening; not a replacement for full SALT2/host-mass nuisance modeling | Canonical nested-rate prediction $\Delta\mu \simeq -0.089$ mag at the frozen Cepheid-equivalent response coefficient $\kappa_{\rm SN} = 3.69\times10^{5}$ mag (massive hosts inferred brighter — the direction of the established astrophysical step, whose total amplitude is $\sim 0.04$–$0.05$ mag). The supplied $m_b^{\rm corr}$ product already carries the standard Pantheon+ host-mass correction: its fitted residual step is $+0.0064$ mag, and the locked offset worsens $\chi^2$ by $104.1$ relative to the fitted-step control. Fitting the continuous per-supernova depth coordinate directly returns $\hat\kappa_{\rm SN}=+3.2\times10^{4}\pm1.1\times10^{4}$ mag — a $2.9\sigma$ response in the predicted direction (deeper wells brighter), roughly an order of magnitude below the naive Cepheid transfer and consistent with channel-specific response normalization. The signal localizes as predicted — calibrator block $+5.6\times10^{4}\pm2.2\times10^{4}$ ($2.6\sigma$), $z<0.1$ sector $+4.0\times10^{4}\pm1.3\times10^{4}$ ($3.0\sigma$), calibrator-free wing $\simeq0$ — with mass-shuffle permutation $p<0.005$, and it replicates on the independent DES-Dovekie catalog at $+8.9\times10^{4}\pm2.3\times10^{4}$ ($3.9\sigma$, permutation $p<0.003$, concentrated at $4.8\sigma$ in the high-mass half): the joint direction is positive at $>4\sigma$ combined across two compilations sharing neither standardization nor covariance. This is a channel-consistency diagnostic on the host-corrected product. |
| Dark matter replacement | Corpus-level implication | Lensing/growth/galaxy-scale gates | Not a C0-only claim |
| BAO native projection | Passed | BAO ruler recovery in TEP background | $\chi^2/\text{DOF} = 0.88$ (17 data points) |
| Growth amplitude | Passed in TEP-HC; imported by C0 | TEP-HC hi_class Boltzmann closure: $\sigma_8 = 0.824$ (fiducial solver output; MCMC posterior $0.858 \pm 0.016$ prior-boundary dominated per Paper 18 Step 23); Planck: $0.811 \pm 0.006$. | TEP-HC (Paper 18) evaluates this using the full hi_class SMG EFT solver with the adopted runtime Jordan-frame Bellini-Sawicki mapping ($\alpha_M = -2\alpha_A$, $\alpha_B = 2\alpha_A$). C0 imports and cross-checks the active perturbation outputs; it does not independently derive $\sigma_8$ from first principles. The C0 RSD confrontation (Step 06-05) separates branches: the unscreened EdS acoustic diagnostic is excluded ($\chi^2/\mathrm{dof} \simeq 7.5$), while the operative joint branch — the same $tep\_mode$ configuration and physical matter density selected by the SNe+CMB fit — gives $\chi^2 = 6.11$ against 11 BOSS/eBOSS $f\sigma_8$ points, marginally below the $\Lambda$CDM $\chi^2 = 6.45$. Full non-linear matter-only structure formation is not a C0 claim. |
| Distance duality | Passed on $r_s$-independent geometric probes | BAO-$r_s$-conditioned compilation: $\Lambda$CDM $\eta = 0.866 \pm 0.020$ ($6.6\sigma$), TEP $\eta = 0.846 \pm 0.019$ ($8.2\sigma$); $r_s$-independent geometric test (Step 04-10, 9 points): $\bar\eta = 0.9369 \pm 0.0358$ ($-1.76\sigma$, $\chi^2/\text{dof} = 9.27/9 = 1.03$) | BAO-conditioned compilations depart from $\eta=1$ because the BAO $D_A$ values assume a fiducial $\Lambda$CDM sound horizon $r_s$ — a compilation artifact rather than a metric violation. A self-consistent re-compilation using only directly measured distances returns $\eta$ consistent with unity: SZ/X-ray clusters give $\eta_{\rm dev}=+0.10\pm0.33$ and strong-lensing systems give $+1.99\pm1.87$, and the TEP-native re-analysis against 9 $r_s$-independent geometric $D_A$ constraints from H0LiCOW/TDCOSMO strong-lensing time delays and Bonamente et al. SZ clusters ($0.142 \le z \le 0.881$, Step 04-10, `step_04_10_tep_native_ddr.json`) recovers a weighted mean $\bar{\eta} = 0.9369 \pm 0.0358$ ($-1.76\sigma$ from unity, $\chi^2/\text{dof} = 9.27/9 = 1.03$, with every individual constraint within $1.43\sigma$ of $\eta = 1$). |
| Solar System PPN | Passed within the implemented EFT screening realization | Cassini: $\gamma = 1.000000 \pm 0.000023$ | Unsuppressed TEP predicts $\gamma - 1 = -2/3$ under the unified unstretched-space metric ($-4/3$ on the isotropic benchmark; ruled out $\sim$29 000$\sigma$ [$58{,}000\sigma$ isotropic]). Lorentzian source-suppression leaves a $\sim$19 200$\sigma$ gap (Step 04-09). The unified covariant operator $\mathcal{S}_\Sigma(\mathcal{E})$ derived in Section 2.5 reduces to gradient-dependent suppression $f(g) = [1+(g/g_t)^n]^{-1}$ in the Solar System, with $g = \|\nabla\Phi\|$, $g_t \approx 3.4 \times 10^{-10}$ m s$^{-2}$ (the cosmological horizon scale $cH_0/2$, matching the SPARC-derived $g_{\rm TEP} \approx 5 \times 10^{-10}$ m s$^{-2}$), $n = 2$. Heliosphere suppression ($\mathcal{S}_\Sigma \approx 0$ at $g \sim 10^{-5}$ m s$^{-2}$; on the selected two-branch solar profile at $r=1.6\,R_\odot$, $y_{\rm tb}=2.16\times10^{-8}$ gives $\|\gamma_{\rm PPN}-1\|=4.32\times10^{-8}$) safely satisfies Cassini. Earth surface is strongly suppressed (Eötvös satisfied). The reduced envelope retains $92.0\%$ at $g=10^{-10}$ m s$^{-2}$, $88.9\%$ at the wide-binary benchmark $1.2\times10^{-10}$ m s$^{-2}$, and $48.6\%$ at the halo benchmark $3.5\times10^{-10}$ m s$^{-2}$; its response is acceleration dependent and distinct from the native growth calculation. Because $S(\rho)$ and $f(g)$ are limits of the same covariant expression $\mathcal{S}_\Sigma(\mathcal{E})$, the suppression threshold is not tuned independently across scales. The suppressed shear sector at Earth ($\mathcal{S}_\Sigma \approx 10^{-21}$) coexists with the transitional clock-amplitude sector ($S_\oplus \approx 0.65$) that carries the GNSS signal, the two being distinct projections of one configuration. The reduced screening benchmark uses $P(X,\phi)=X-V+X\|X\|/\Lambda_X^4$, while Paper 0 (v0.16) selects the two-branch kinetic sector without a canonical linear term; the quartic density-dependent-mass completion $V=\lambda\phi^4/4$ is retained as the amplitude-sector candidate, while uniqueness within the wider $K(X)$ class and the first-principles derivation of the terrestrial clock-covariance amplitude $C_A(\mathbf r,t)$ remain open calculations. The adopted Jordan-frame Bellini-Sawicki EFT mapping used by C0 is implemented in Section 2.5 and cross-checked against the PPN and TEP-HC growth gates. |
| Tolman surface brightness | Inconclusive as discriminator | Measured $n = 3.375 \pm 0.027$ vs $\Lambda$CDM/TEP $n \approx 4.0$; data shows $n$ decreases with $z$ (slope $-1.03$) | Both $\Lambda$CDM ($n=4.0$ flat) and TEP ($n \approx 4.02$ flat) predict $n \geq 4.0$. The data shows $n$ falling from $\approx 3.65$ at $z < 0.3$ to $\approx 2.84$ at $z > 0.5$, a trend opposite to any cosmological model. The anomaly is dominated by K-correction systematics ($\pm 0.5$ mag), passive evolution, and selection effects. Not a clean discriminator. |

The empirical findings and their interpretations form a strict hierarchy of evidence:

- **No Primitive Expansion Required by the Tested Background Data:** The exact conformal reconstruction M2 demonstrates that the Pantheon+ homogeneous distance-redshift relation does not uniquely require primitive expansion of the spatial metric. The physical no-$\Lambda$ temporal-shear branch M1 with $z_{\rm los}=5$ improves the Pantheon+ likelihood by $\Delta\chi^2\simeq-3.81$ relative to baseline $\Lambda$CDM using the same 1,701-supernova covariance structure and no fitted host-mass-step nuisance parameter. The expansion interpretation is therefore underdetermined by the SNe background data; the temporal-transport distance law is observationally degenerate with $\Lambda$CDM at the background level, and modestly preferred (BF approximately 5.8, with the identical-model control giving BF $=1$ exactly and the measured localization to the calibrator and high-$z$ tails — BF $\approx0.63$ at $z>0.023$, $\approx1.24$ with the 77 calibrators removed) for the physically motivated $z_{\rm los}=5$ model.

- **No primitive dark energy required in the tested late-time sector (interpretive claim):** The M1 branch achieves a comparable standardized-supernova fit with $\Omega_\Lambda=0$, replacing vacuum-energy acceleration with temporal-shear transport in the late-time distance-redshift relation. Within the corpus architecture, M1's $(\Omega_m=1,\Omega_\Lambda=0,\epsilon_{\rm shear}^{\rm los})$ line-of-sight representation and the $(\Omega_{m,\rm eff}\approx0.304,\Omega_\phi\approx0.696)$ conformal-drift decomposition of Papers 0, 18, 27, and 29 are complementary descriptions of the same temporal-transport phenomenology (mapped onto the DESI CPL $(w_0,w_a)$ projection in Paper 27 Step 03). On a single expansion history the native branch fits supernovae, BAO and the CMB acoustic angle together and is preferred over $\Lambda$CDM (Step 03-14, $\Delta\chi^2=-8.1$). This is an interpretive inference, not an empirical falsification of $\Lambda$: because M1 and $\Lambda$CDM produce nearly identical distance moduli, the SNe data alone cannot distinguish the physical mechanism. The result demonstrates that apparent acceleration can be reconstructed without a primitive dark-energy component, not that $\Lambda$ is absent.

- **No Physical Big Bang Singularity in the Conformal Reconstruction:** In the TEP mapping, the limit conventionally written as $a\to0$ is re-expressed as $A_{\text{clock}}\to0$: a TEP temporal-horizon boundary of observational clock transport, not a zero-volume spatial singularity. The C0 paper establishes the conformal reconstruction and identifies the singular origin as an artefact of imposing an integrable FLRW clock foliation. The temporal-horizon background and linear-mode closure are supplied by companion papers TEP-TH and TEP-HC; C0 imports and cross-checks the TEP-HC linear-growth output while focusing on the empirical supernova-sector test. The nonsingular temporal-horizon boundary conditions for the deep temporal-horizon thermal sector are developed in TEP-TH (Paper 27).

- **Particle Dark Matter (Corpus Implication):** Although the current pipeline focuses on the cosmological background and macroscopic bounds, the broader TEP corpus establishes that local gradients of the same temporal field on the selected two-branch kinetic sector $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$ fixed by $a_0$, zero per-system free parameters) govern both dynamical and optical dark-sector phenomenology. While isotropic conformal rescaling $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$ stretches space alongside time and cancels out of null rays, the unstretched-space temporal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, rescaling the shared ADM lapse $N_{\rm TEP}=A(\phi)N_{\rm GR}$ at fixed spatial 3-metric $\gamma_{ij}$ so $c_{\rm GW}=c_\gamma$ identically) retains intermediate clock rates $A(\phi(x))$ along every photon path, refracting light by $n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)$ and reproducing the Bullet Cluster galaxy-peak convergence (Paper 4, Step 02), the SN Refsdal Fermat delay and arrival order (Paper 19, Step 64), and the cosmic-web amplification required by Planck CMB lensing (Paper 18, Step 21) from baryons alone.

The reproducible pipeline provides a robust supernova-sector Bayesian framework demonstrating that conformal transport is a viable alternative to the standard expanding universe in the tested background sector. TEP-HC (Paper 18) has validated its implemented linear perturbation mapping, yielding $\sigma_8 \approx 0.824$ on the screened $Z_E\gtrsim700$ branch and $\sigma_8\simeq0.76$ (matching weak-lensing cosmic-shear measurements) at $Z_E\sim6\times10^3$, while Step 21 demonstrates that under the unstretched-space temporal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) on the selected two-branch action $P_{,X}=k\sqrt\xi+2\xi$, intermediate temporal depth boosts baryon-only cosmic-web $C_\ell^{\phi\phi}$ by $9.4\times$–$129\times$, recovering the Planck CMB lensing amplitude without a CDM fluid. The conformal-frame Hubble mapping and screened perturbation response are evaluated within the adopted HC effective closure; the nonlinear master-action and environmental averaging match retains the scope stated in Paper 18. Non-linear structure formation is not a C0 claim. Host-environment reconstruction across the full multi-probe dataset is a corpus-level target beyond the SNe-only scope of C0. The nonsingular temporal-horizon boundary conditions for the deep temporal-horizon thermal sector are developed in TEP-TH (Paper 27). By asserting that time itself is a dynamical field, the framework provides a testable path forward for precision cosmology.

## Acknowledgments and Disclosures

The author declares no competing interests. No external funding was received for this work. All analysis was conducted using publicly available astronomical data and open-source software.

# 8. References

## 8.1 TEP Series

- Smawfield, M. L. (2025). *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. v0.16 (Jakarta). DOI: 10.5281/zenodo.16921911.

- Smawfield, M. L. (2026). *The Cepheid Bias: Resolving the Hubble Tension*. v0.12 (Kingston upon Hull). DOI: 10.5281/zenodo.18209702.

- Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. v0.8 (Kos). DOI: 10.5281/zenodo.19000827.

- Smawfield, M. L. (2026). *Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars*. v0.10 (Caracas). DOI: 10.5281/zenodo.18165798.

- Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. v0.7 (Kilifi). DOI: 10.5281/zenodo.19102061.

- Smawfield, M. L. (2026). *Temporal Equivalence Principle: Native hi_class Conformal Implementation, Linear Perturbation Closure, and CMB Acoustic Peak Preservation*. v0.8 (Cambridge). DOI: 10.5281/zenodo.20572722.

- Smawfield, M. L. (2026). *Temporal Equivalence Principle: The Dirac Limit of Dynamical Proper Time*. v0.4 (Qatar). DOI: 10.5281/zenodo.20572697.

- Smawfield, M. L. (2026). *Temporal Equivalence Principle: Fermion Spin as Temporal-Orientation Holonomy*. v0.1 (Paris). In preparation (Paper 24).

- Smawfield, M. L. (2026). *Temporal Equivalence Principle: Temporal Horizon Cosmology and the Absence of a Physical Big Bang Singularity*. v0.5 (Thika). DOI: 10.5281/zenodo.20723059.

- Smawfield, M. L. (2026). *Temporal Equivalence Principle: A Covariant Alternative to Cosmic Expansion*. v0.4 (Athens). DOI: 10.5281/zenodo.20370143.

- Smawfield, M. L. (2026). *Temporal Equivalence Principle: Dynamical Proper Time and the Illusion of Primordial Deuterium*. v0.6 (Dubai). DOI: 10.5281/zenodo.21841147.

## 8.2 Data Sources

- Scolnic, D., et al. (2018). *The Pantheon Analysis: Cosmological Constraints from the Largest Supernova Sample*. ApJ, 859, 101.

- Scolnic, D., et al. (2022). *Pantheon+: Type Ia Supernova Light Curves from the Dark Energy Survey*. ApJ, 938, 113.

- Planck Collaboration (2020). *Planck 2018 results. VI. Cosmological parameters*. A&A, 641, A6.

- Fixsen, D. J., et al. (1996). *The Spectrum of the Cosmic Background Radiation*. ApJ, 473, 576.

- Mather, J. C., et al. (1994). *Measurement of the Cosmic Microwave Background Spectrum by the COBE FIRAS Instrument*. ApJ, 420, 439.

## 8.3 BAO and RSD Surveys

- Alam, S., et al. (2017). *The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample*. MNRAS, 470, 2617.

- Beutler, F., et al. (2011). *The 6dF Galaxy Survey: baryon acoustic oscillations and the local Hubble constant*. MNRAS, 416, 3017.

- Anderson, L., et al. (2014). *The clustering of galaxies in the SDSS-III BAO sample: analysis of potential systematics*. MNRAS, 441, 24.

- Peacock, J. A., et al. (2015). *The SDSS-IV extended Baryon Oscillation Spectroscopic Survey: overview and early data*. MNRAS, 452, 2379.

- Dawson, K. S., et al. (2013). *The SDSS-III Baryon Oscillation Spectroscopic Survey: quasar targeting*. AJ, 145, 10.

- Ross, A. J., et al. (2015). *The clustering of quasars in SDSS-III DR9: testing the consistency of BAO and redshift-space distortions with the Planck CMB*. MNRAS, 449, 835.

## 8.4 Software and Tools

- Foreman-Mackey, D., et al. (2013). *emcee: The MCMC Hammer*. PASP, 125, 306. github.com/dfm/emcee

- Speagle, J. S. (2020). *dynesty: A dynamic nested sampling package for estimating Bayesian posteriors and evidences*. MNRAS, 493, 3132. github.com/joshspeagle/dynesty

- Torrado, J., & Lewis, A. (2021). *Cobaya: Code for Bayesian Analysis of cosmological data*. Astrophysics Source Code Library, ascl:2108.05. github.com/CobayaSampler/cobaya

- Lesgourgues, J. (2011). *The Cosmic Linear Anisotropy Solving System (CLASS). Part I: Overview*. arXiv:1104.2932. github.com/lesgourg/class_public

- Arbey, A. (2012). *AlterBBN: A program for calculating the BBN abundances of the elements in alternative cosmologies*. CPC, 183, 1822. alterbbn.hepforge.org

## 8.5 Historical References

- Hubble, E. (1929). *A relation between distance and radial velocity among extra-galactic nebulae*. PNAS, 15, 168.

- Friedmann, A. (1922). *Uber die Krummung des Raumes*. Z. Phys., 10, 377.

- Lemaitre, G. (1927). *Un univers homogene de masse constante et de rayon croissant rendant compte de la vitesse radiale des nebuleuses extra-galactiques*. Ann. Soc. Sci. Brux., 47, 49.

- Riess, A. G., et al. (1998). *Observational evidence from supernovae for an accelerating universe and a cosmological constant*. AJ, 116, 1009.

- Perlmutter, S., et al. (1999). *Measurements of Omega and Lambda from 42 high-redshift supernovae*. ApJ, 517, 565.

- Tolman, R. C. (1930). *On the estimation of distances in a curved universe with a non-static line element*. PNAS, 16, 511.

- Etherington, I. M. H. (1933). *On the definition of distance in general relativity*. Philos. Mag., 15, 761.

## 8.6 Supernova Standardization Systematics

- Kang, Y., Kim, Y.-L., Lim, D., Chung, C., & Lee, Y.-W. (2016). *Early-type Host Galaxies of Type Ia Supernovae. I. Evidence for Downsizing*. ApJS, 223, 7.

- Kang, Y., Lee, Y.-W., Kim, Y.-L., Chung, C., & Ree, C. H. (2020). *Early-type Host Galaxies of Type Ia Supernovae. II. Evidence for Luminosity Evolution in Supernova Cosmology*. ApJ, 889, 8.

- Lee, Y.-W., Chung, C., Kang, Y., & Jee, M. J. (2020). *Further Evidence for Significant Luminosity Evolution in Supernova Cosmology*. ApJ, 903, 22.

Smawfield, M. L. 2026. Temporal Equivalence Principle series, Papers 0-13. Zenodo preprints and associated repositories.

# Unified TEP Parameter Dictionary

The TEP corpus uses related but distinct symbols across its papers. This dictionary maps every parameter, its definition, the paper where it is primary, and its fiducial or fitted value.

| Symbol | Definition | Primary Paper | Fiducial / Fitted Value |
| --- | --- | --- | --- |
| $A_{\rm clock}(z)$ | Exact observational clock/redshift map: $A_{\rm clock}=(1+z)^{-1}$ | TEP-TH | $(1+z)^{-1}$ (exact) |
| $A_{\rm dyn}(z)$ | Dynamical shear response: $\left(1+z/z_t\right)^{-\epsilon_t}$ | TEP-TH | Modifies late-time evolution |
| $\alpha_A$ | Runtime logarithmic derivative $\alpha_A=d\ln A_{\rm HC}/d\ln a$; distinct from the exact endpoint map $A_{\rm clock}$ | TEP-HC | Redshift-dependent; zero at the finite-profile turnover |
| $\alpha_M$, $\alpha_B$, $\alpha_K$, $\alpha_T$ | Adopted runtime Jordan-frame Bellini–Sawicki EFT functions: $\alpha_M=-2\mathcal{S}_\Sigma\alpha_A$, $\alpha_B=2\mathcal{S}_\Sigma\alpha_A$, $\alpha_K=w_c\,\alpha_A^2$ with $w_c=Z_E-6$ independent of $\mathcal{S}_\Sigma$ (production $w_c=700$; the canonical image $w_c=-5$ is linearly unstable — Paper 18 closure audit), $\alpha_T=0$ | TEP-HC | Adopted closure evaluated from $\alpha_A$ |
| $\epsilon_T^{\rm los}$ | Late-time line-of-sight transport amplitude (C0 supernova fit) | TEP-C0 | $\mathcal{U}[0, 2.0]$ (prior); posterior peaked near $\sim 0.89$ |
| $\epsilon_T^{\rm CMB}$ | C0 background/acoustic diagnostic amplitude | TEP-C0 | $-0.0015\pm0.0037$ |
| $\epsilon_T^{\rm HC}$ | Native hi_class homogeneous conformal amplitude | TEP-HC | $0.0043\pm0.0025$ (full high-$\ell$ likelihood); $0.00547\pm0.00429$ (historical low-$\ell$ active chain) |
| $\epsilon_{\rm dyn}$ | Dynamical temporal-horizon response | TEP-TH | Determined by late-time shear |
| $\epsilon_{\rm field}$ | Primordial spectral-flow parameter constrained by $n_s$ | TEP-TH | $0.01755$ (from $n_s=0.9649$) |
| $z_T^{\rm los}$ | C0 line-of-sight supernova transport turnover | TEP-C0 | $5$ (conservative), $100$ (benchmark), free (broad) |
| $z_T^{\rm HC}$ | Homogeneous/acoustic hi_class profile scale | TEP-HC | Fitted jointly with $\epsilon_T$ |
| $p$ | Temporal-horizon conformal exponent: $A_{\rm clock}\sim\eta^{-p}$ | TEP-TH | $0 \lt p\le\tfrac12$ (regular branch) |
| $r_s^{\rm TEP}/r_s^{\Lambda\rm CDM}$ | Pre-recombination sound-horizon ratio | TEP-HC | $0.999994$ ($<6$ ppm deviation) |
| $D=\alpha_K+\tfrac32\alpha_B^2$ | No-ghost discriminant ($D=(w_c+6)\alpha_A^2$; necessary but not sufficient for linear stability — the $w_c=-5$ branch is tachyonic; production $w_c=700$ gives $D=706\,\alpha_A^2$) | TEP-HC | $(w_c+6)\alpha_A^2\ge 0$ (non-negative; GR/decoupling limit at $\alpha_A=0$) |
| $r(k_{\rm pivot})$ | Native tensor-to-scalar ratio at Planck pivot | TEP-TH | $1.9\times10^{-9}$ |
| $r_{\rm max}$ | Maximum tensor-to-scalar ratio across transition profile | TEP-TH | $5.2\times10^{-7}$ (fiducial $p=0.1$); $6.1\times10^{-5}$ over all tested profiles $0\lt p\le1$ |
| $H_0$ | Hubble parameter (TEP-C0 joint MCMC) | TEP-C0 | $66.70\pm0.58$ km s$^{-1}$ Mpc$^{-1}$ |
| $\rho_T$ | Material Temporal-Topology saturation scale; anchors the clock-amplitude sector $S_A$ through $R_T(M)=(3M/4\pi\rho_T)^{1/3}$, surface factor $S=(\bar\rho/\rho_T)^{1/3}$ | TEP / TEP-UCD | $\approx 20$ g cm$^{-3}$ ($S_\oplus\approx0.65$; $R_T(M_\oplus)\approx4{,}146$ km) |
| $\rho_{\rm half}$ | Ambient half-suppression density of the shear-sector operator $\mathcal{S}_\Sigma(\mathcal E)$; density argument is the coarse-grained environmental mean, not material density | TEP-C0 | $\approx 0.5\,M_\odot\,{\rm pc}^{-3}\approx3.4\times10^{-23}$ g cm$^{-3}$ |
| $g_t \equiv a_t/c^2$ | Critical shear scale of the kinetic channel of $\mathcal{S}_\Sigma(\mathcal E)$ in inverse-length form (Paper 0 quotes the acceleration form $g_t = cH_0/2$); Newtonian limit maps to local acceleration $a_t$ | TEP-C0 | $a_t/c^2\approx3.8\times10^{-27}$ m$^{-1}$ ($a_t\approx3.4\times10^{-10}$ m s$^{-2}$, $\sim cH_0/2$) |
| $S_8$ | $\sigma_8\sqrt{\Omega_m/0.3}$ (TEP-HC joint MCMC) | TEP-HC | $0.868\pm0.025$ |
| $\sigma_8^{\rm HC}$ | Native hi_class matter-fluctuation amplitude | TEP-HC | $0.824$ (screened fiducial); $0.858\pm0.016$ (MCMC posterior mean) |

**Note:** Parameters with superscript labels ($^{\rm los}$, $^{\rm HC}$) are related projections of the same temporal sector but are not numerically interchangeable. The turnover scales $z_T^{\rm los}$ and $z_T^{\rm HC}$ describe different physical regimes; the amplitudes $\epsilon_T^{\rm los}$, $\epsilon_T^{\rm CMB}$, $\epsilon_T^{\rm HC}$, and $\epsilon_{\rm field}$ are constrained by different observables. Two cases are the same functional rather than related projections: the EdS-diagnostic $\epsilon_T^{\rm hom} = 0.018$ of §4.4 and the joint-fit $\epsilon_T^{\rm CMB} = -0.0015 \pm 0.0037$ both set the patched-CLASS `tep_epsilon_T` parameter and differ only in the background branch — EdS ($\Omega_\Lambda = 0$) versus the $\Lambda$CDM-compatible branch selected by Pantheon+ + Planck. Their $5.3\sigma$ separation (Step 05-11) records that the no-$\Lambda$ acoustic recovery is a counterfactual existence proof, not a property of the joint-fit cosmology.

# 9. Data Availability & Reproducibility

This work follows open-science practices. All results are fully reproducible from raw data
using the documented pipeline. All numerical results, figures, and statistics are generated by deterministic
Python scripts processing real observational data. The pipeline is intentionally strict: failed dependencies are recorded as failed
results, not silently ignored.

### Repository and Code

GitHub Repository: github.com/matthewsmawfield/TEP-C0

The repository contains a deterministic, version-controlled cosmological analysis pipeline with 64 analysis steps
for supernova distance-redshift, distance-duality constraints, CMB acoustic scales, BBN preservation, structure growth, and systematic validation.
All steps are orchestrated by `scripts/run_pipeline.py` with comprehensive per-step logging.

All MCMC chains, posterior samples, and the exact `cobaya` YAML configuration files are released in the Zenodo repository (DOI: 10.5281/zenodo.20370143) under CC-BY 4.0. The `run_all.py` orchestration script and all step scripts are provided in the GitHub repository.

#### Repository Structure



```

TEP-C0/
├── data/
│   ├── raw/                       # Downloaded source catalogs (Pantheon+, DDR, etc.)
│   └── processed/                 # Ingested and filtered datasets
├── scripts/
│   ├── steps/                     # 64 deterministic pipeline steps
│   ├── utils/                     # Logging and validation utilities
│   └── run_pipeline.py            # Master orchestration script
├── core/                          # Cosmology and model libraries
├── external/                      # Patched CLASS, AlterBBN dependencies
├── results/
│   ├── outputs/                   # JSON/CSV analytical outputs
│   └── figures/                   # Generated plots
├── logs/                          # Per-step execution logs
├── site/
│   └── components/                # Manuscript HTML sections
├── requirements.txt               # Python dependencies
└── README.md                      # Documentation
```



### Data Provenance

| Data Source | Provider | Access Method | Records | Location |
| --- | --- | --- | --- | --- |
| Pantheon+ SNe Ia | Scolnic et al. | Auto-downloaded | 1,701 | `data/raw/pantheon_plus_shoes.dat` |
| Pantheon+ covariance | Scolnic et al. | Auto-downloaded | Full stat + sys | `data/raw/Pantheon+SH0ES.cov` |
| BAO constraints | BOSS, eBOSS, DES | Compiled from lit. | 10 measurements | `data/raw/ddr_constraints.csv` |
| SZ cluster DDR | Compiled | Auto-downloaded | ~38 clusters | `data/raw/sz_cluster_da.csv` |
| SGL lensing DDR | Compiled | Auto-downloaded | ~118 lenses | `data/raw/strong_lensing_da.csv` |
| DESI/eBOSS Lyman-alpha | DESI-DR1, eBOSS | Auto-downloaded | 3 measurements | `data/raw/ddr_constraints_highz.csv` |
| FIRAS CMB spectrum | NASA LAMBDA | Auto-downloaded | ~43 frequencies | `data/raw/firas_spectrum.dat` |
| Planck 2018 CMB | Planck Collaboration | Cobaya package | TTTEEE + low-$\ell$ TT/EE + Pantheon+ | External Cobaya cache |
| BBN abundances | AlterBBN, compiled lit. | Included / downloaded | Yp, D/H, Li/H | `data/raw/bbn_review.html` |

### Pipeline Architecture

The analysis pipeline comprises 64 deterministic steps organized into eight logical stages.
Each step is a standalone Python script in `scripts/steps/` that produces JSON/CSV outputs and
detailed logs in `logs/step_*.log`. Dependencies are resolved automatically by the runner.

#### Complete Step Inventory and Runtime

Runtimes are approximate and measured on Apple M4 Pro (14-core, 24 GB). The dominant cost is the nested sampling step (03_01), which scales with `nlive` and number of models.

| Stage | Step | Script | Description | Runtime |
| --- | --- | --- | --- | --- |
| Stage 1: Data Acquisition (8 steps) |  |  |  |  |
| Data | 1.1 | `step_01_01_data_download.py` | Download Pantheon+ SNe, covariance, FIRAS | ~10 s |
| Data | 1.2 | `step_01_02_data_ingestion.py` | Ingest and validate all downloaded catalogs | ~1 s |
| Data | 1.3 | `step_01_03_download_ddr.py` | Download BAO distance-duality constraints | ~1 s |
| Data | 1.4 | `step_01_04_download_sb.py` | Download surface-brightness catalog sources | ~1 s |
| Data | 1.5 | `step_01_05_download_sz.py` | Download Sunyaev-Zel'dovich cluster data | ~1 s |
| Data | 1.6 | `step_01_06_download_sgl.py` | Download strong gravitational lensing data | ~1 s |
| Data | 1.7 | `step_01_07_download_desi.py` | Download DESI-DR1 and eBOSS Lyman-alpha | ~1 s |
| Data | 1.8 | `step_01_08_compile_sb.py` | Compile surface-brightness master catalog | ~1 s |
| Stage 2: Theory and Transport (4 steps) |  |  |  |  |
| Theory | 2.1 | `step_02_01_transport_kernel.py` | Verify FLRW recovery limit of open-path transport K_T | ~1 s |
| Theory | 2.2 | `step_02_02_theory_derivation.py` | Derive theoretical predictions for distance-redshift and screening | ~2 s |
| Theory | 2.3 | `step_02_03_physics_implementation.py` | Implement TEP physics: distance moduli, transport, growth kernels | ~3 s |
| Theory | 2.4 | `step_02_04_screening_scale_transfer.py` | Two-sector screening evaluation: covariant $\mathcal{S}_\Sigma$ per environment, geometric $S_A$ factors, $R_T(M_\oplus)$ vs $\lambda_T$ | ~1 s |
| Stage 3: Model Comparison and MCMC (10 steps) |  |  |  |  |
| Core | 3.1 | `step_03_01_three_model_comparison.py` | Nested sampling (dynesty, nlive=500) for M0a_LCDM, M0b_EdS, M1 variants, M2_PureConformal_Control, M3_wCDM, M4_CPL; null injection | ~90 min |
| Core | 3.2 | `step_03_02_independent_mcmc.py` | Independent MCMC convergence diagnostics | ~1 s |
| Core | 3.4 | `step_03_04_cobaya_mcmc.py` | Joint SNe+CMB MCMC via Cobaya with TEP-CLASS v2.0 | ~2 min |
| Core | 3.4b | `step_03_04_minimize.py` | BOBYQA minimizer for joint SNe+CMB parameter optimization | ~30 s |
| Core | 3.5 | `step_03_05_analyze_cobaya.py` | Analyze Cobaya chains and produce parameter constraints | ~1 s |
| Core | 3.5b | `step_03_05_joint_global_screening.py` | Production coupled multi-domain screening inference (Pantheon+ $\times$ `hi_class` $\times$ `AlterBBN` $\times$ two-branch PPN Cassini; lensing structure scales marginalized by 2D Gauss-Hermite quadrature, plus profiled-nuisance comparison run) | ~6 min |
| Core | 3.6 | `step_03_06_cobaya_verbose.py` | Verbose Cobaya configuration and extended diagnostics | ~2 min |
| Core | 3.7 | `step_03_07_likelihood_synthesis.py` | Synthesize likelihoods across independent and joint analyses | ~1 s |
| Core | 3.8 | `step_03_08_h0_boundary_stress.py` | H0 prior stress test: extended priors reveal EdS-derived-parameter artifact driving H0 toward zero | ~30 s |
| Core | 3.9 | `step_03_09_lcdm_null_injection.py` | LCDM null injection: mock Pantheon+ from LCDM, measure TEP false-positive rate | ~60 s |
| Core | 3.10 | `step_03_10_pantheon_subset_robustness.py` | Leave-one-survey-out and redshift-window robustness tests | ~30 s |
| Stage 4: Supernova Tests and Distance Duality (10 steps) |  |  |  |  |
| SNe | 4.1 | `step_04_01_sn_time_dilation.py` | Test SN light-curve stretch factors against TEP time dilation | ~1 s |
| SNe | 4.2 | `step_04_02_sn_tolman.py` | Tolman surface-brightness dimming test | ~1 s |
| SNe | 4.3 | `step_04_03_tolman_sb.py` | Surface-brightness Tolman scaling with compiled catalog | ~1 s |
| DDR | 4.4 | `step_04_04_distance_duality.py` | Distance-duality relation: BAO constraints vs TEP prediction | ~1 s |
| DDR | 4.5 | `step_04_05_ddr_threeway.py` | Three-way probe comparison: BAO, SZ, SGL | ~1 s |
| DDR | 4.6 | `step_04_06_screening_fit.py` | Parametric screening model fit to probe-dependent DDR | ~2 s |
| DDR | 4.6 | `step_04_07_highz_ddr.py` | High-redshift Lyman-alpha DDR test (DESI, eBOSS) | ~1 s |
| SNe | 4.7 | `step_04_08_host_mass_step_prediction.py` | Host-mass-step mini-analysis: locked TEP prediction vs fitted $\Lambda$CDM nuisance | ~5 s |
| PPN | 2.4.5 | `step_04_09_ppn_constraints.py` | Solar System PPN constraint derivation with gradient-dependent screening | ~1 s |
| DDR | 4.5 | `step_04_10_tep_native_ddr.py` | TEP-native distance-duality re-analysis | ~1 s |
| SNe | 4.8 | `step_04_11_sn_robustness_systematics.py` | Prior, redshift-cut, covariance, and sampler robustness grid for TEP M1 | ~10 min |
| SNe | 4.9 | `step_04_12_external_sn_validation.py` | External validation on Union3 binned compilation | ~1 s |
| Stage 5: CMB and Big Bang Nucleosynthesis (10 steps) |  |  |  |  |
| CMB | 5.1 | `step_05_01_cmb_blackbody.py` | Verify TEP preserves CMB blackbody spectrum (FIRAS) | ~1 s |
| CMB | 5.3 | `step_05_03_cmb_boltzmann.py` | TEP Boltzmann integration via patched CLASS | ~1 s |
| CMB | 5.4 | `step_05_04_cmb_spectra.py` | Generate and compare TT/TE/EE power spectra | ~1 s |
| CMB | 5.5 | `step_05_05_cmb_consistency.py` | CMB acoustic-scale consistency check | ~1 s |
| BBN | 5.6 | `step_05_06_bbn_registry.py` | Compile observational BBN abundance registry | ~1 s |
| BBN | 5.7 | `step_05_07_bbn_preservation.py` | Historical AlterBBN compatibility baseline — superseded; not part of canonical TEP thermodynamics | ~1 s |
| CMB | 5.8 | `step_05_08_cmb_acoustic.py` | Acoustic-scale parameter comparison (Planck) | ~1 s |
| CMB | 5.9 | `step_05_09_minimal_perturbations.py` | Diagnostic minimal-closure perturbation checks; authoritative active-sector closure is documented in TEP-HC | ~3 s |
| CMB | 5.10a | `step_05_10_jordan_frame_proof.py` | Matter-frame acoustic-scale existence proof in EdS diagnostic background; Step 05-11 records its 5.3σ branch separation from the joint-fit homogeneous amplitude | ~1 s |
| CMB | 5.10b | `step_05_10_tephc_spectra_crosscheck.py` | Cross-check TEP-HC Boltzmann spectral outputs (acoustic-scale ratio) | ~1 s |
| Stage 6: BAO and Structure Growth (7 steps) |  |  |  |  |
| BAO | 6.1 | `step_06_01_bao_projection.py` | BAO ruler projection in TEP geometry | ~1 s |
| BAO | 6.2 | `step_06_02_bao_likelihood.py` | BAO likelihood module integration | ~7 s |
| Growth | 6.3 | `step_06_03_growth_solver.py` | TEP-CLASS v2.0 growth equation solver | ~1 s |
| Growth | 6.4 | `step_06_04_growth_validation.py` | Validate growth factors against LCDM baseline | ~1 s |
| Growth | 6.5 | `step_06_05_growth_rsd.py` | Redshift-space distortion comparison (f sigma_8) | ~2 s |
| Growth | 6.6 | `step_06_06_nonlinear_growth_closure.py` | Halo-model non-linear growth closure with gradient screening | ~5 s |
| Growth | 6.7 | `step_06_07_alphaM_growth_validation.py` | First-principles alpha_M-modified growth ODE: four-scenario sigma_8 comparison | ~2 s |
| Stage 7: Forecasts and Future Tests (7 steps) |  |  |  |  |
| Future | 7.1 | `step_07_01_mixed_forecast.py` | Forecast for mixed TEP-LCDM parameter recovery | ~1 s |
| Future | 7.2 | `step_07_02_redshift_drift.py` | Redshift-drift forecast and discriminating power | ~1 s |
| Future | 7.3 | `step_07_03_jwst_test.py` | JWST high-z supernova feasibility test | ~1 s |
| Future | 7.4 | `step_07_04_gw_sirens.py` | Gravitational-wave standard siren forecast | ~1 s |
| Future | 7.5 | `step_07_05_weak_lensing_plan.py` | Weak-lensing survey plan for TEP discrimination | ~1 s |
| Future | 7.6 | `step_07_06_weak_lensing.py` | Weak-lensing shear correlation analysis | ~1 s |
| Future | 7.7 | `step_07_07_blind_injection.py` | Blind injection validation protocol | ~1 s |
| Stage 8: Falsification, Verification, and Summary (8 steps) |  |  |  |  |
| Validation | 8.1 | `step_08_01_expansion_falsifier.py` | Expansion falsifier: distance duality and Tolman residuals | ~1 s |
| Validation | 8.2 | `step_08_02_comparison_stats.py` | Cross-model comparison statistics | ~1 s |
| Validation | 8.3 | `step_08_03_sensitivity_analysis.py` | Prior and parameter sensitivity analysis | ~1 s |
| Validation | 8.4 | `step_08_04_evidence_matrix.py` | Compile explanatory evidence matrix | ~1 s |
| Validation | 8.5 | `step_08_05_gate_registry.py` | Claim gate registry and status check | ~1 s |
| Validation | 8.6 | `step_08_06_claim_audit.py` | Automated claim consistency check | ~1 s |
| Validation | 8.7 | `step_08_07_final_summary.py` | Global evidence synthesis and summary | ~1 s |
| Validation | 8.8 | `step_08_08_diagnostic_plots.py` | Data-driven diagnostic figures (distance-duality residuals, Pantheon+ Hubble residuals) generated only from upstream pipeline artefacts | ~5 s |

#### Total Runtime Summary

The total runtime is dominated by Stage 3.1 (nested sampling). Runtimes scale approximately linearly with `nlive` and number of CPU cores.

| Component | Steps | Runtime |
| --- | --- | --- |
| Data Acquisition (Stage 1) | 8 | ~20 s |
| Theory and Transport (Stage 2) | 4 | ~6 s |
| Model Comparison and MCMC (Stage 3) | 10 | ~97 min |
| SNe Tests and DDR (Stage 4) | 10 | ~15 s |
| CMB and BBN (Stage 5) | 10 | ~11 s |
| BAO and Growth (Stage 6) | 7 | ~14 s |
| Forecasts and Future Tests (Stage 7) | 7 | ~7 s |
| Falsification and Verification (Stage 8) | 8 | ~7 s |
| Total | 64 | ~95 min (~1.6 h) |

### Reproduction Instructions

#### Quick Start (Full Reproduction)



```

# 1. Clone repository
git clone https://github.com/matthewsmawfield/TEP-C0.git
cd TEP-C0

# 2. Install dependencies
pip install -r requirements.txt

# 3. Run full pipeline (generates all results and figures)
python scripts/run_pipeline.py

# 4. Results will be in:
#    - results/            (JSON/CSV data)
#    - results/figures/   (PNG/PDF plots)
#    - logs/              (Detailed execution logs)
```



#### Command-Line Options

The pipeline supports selective execution for faster testing:



```

# Core statistical analysis only (skips long nested sampling)
python scripts/run_pipeline.py --core

# Resume from existing results (skip completed steps)
python scripts/run_pipeline.py --resume

# Run specific steps with automatic dependency resolution
python scripts/run_pipeline.py --steps step_04_04_distance_duality step_04_05_ddr_threeway
```



#### System Requirements

| Component | Minimum | Recommended | Tested On |
| --- | --- | --- | --- |
| CPU | 4 cores | 8+ cores | Apple M4 Pro (14-core) |
| RAM | 8 GB | 16 GB | 24 GB (M4 Pro) |
| Storage | 2 GB | 5 GB | NVMe SSD |
| Runtime (full) | ~4 h (4 cores) | ~1.5 h (8+ cores) | ~95 min (M4 Pro) |
| Runtime (-core) | ~1 min | ~30 s | ~20 s |

#### Key Analysis Outputs

- `results/current_model_comparison.json` — Observer-consistent evidence table quoted in the main text (*d*<sub>L</sub> = (1+*z*<sub>HEL</sub>) χ(*z*<sub>HD</sub>); bounded Gaussian intercept integral with adaptive shape quadrature) for M0a_LCDM, M0b_EdS, M1 variants, M2_PureShear, M3_wCDM and M4_CPL; generated by `scripts/steps/refresh_observer_comparison.py` and loaded, with a source-hash guard, by every downstream reporting step through `load_model_comparison()`

- `results/observer_prior_sensitivity.json`, `observer_covariance_sensitivity.json`, `observer_robustness_quadrature.json`, `observer_injection_recovery.json`, `audit_anchored_quadrature.json` — prior, covariance, redshift-cut, injection and anchor-removal results (`scripts/audit/verify_observer_prior_sensitivity.py`, `verify_observer_covariance_sensitivity.py`, `scripts/steps/refresh_observer_injection_recovery.py`, `verify_anchored_quadrature.py`)

- `results/step_03_12_env_composite_kernel.json`, `event_grouped_holdout.json`, `environment_evidence_independent.json`, `environment_profile_fit.json`, `environment_survey_offsets_profiled.json`, `quality_environment_evidence_independent.json` — environmental composite kernel, event-grouped holdout and independent-evaluator checks (`scripts/audit/verify_event_holdout.py`, `verify_environment_*.py`, `verify_quality_environment_evidence.py`)

- `results/step_03_11_sn_luminosity_drift_profile.json`, `results/step_03_13_raw_standardization_drift.json` — interior-tracking (Chandrasekhar drift) profile and its raw-standardization refit

- `results/step_03_01_three_model_comparison.json` — Legacy nested-sampling comparison retained as a cross-check; it predates the observer-frame correction and the adoption of the zHD redshift convention in the likelihood, and is not the source of the evidence values quoted in the main text

- `results/outputs/tep_cobaya_joint_chain{0..3}.1.txt` — four Cobaya MCMC chains for the joint SNe+CMB analysis, summarized with cross-chain Gelman-Rubin diagnostics in `results/step_03_05_analyze_cobaya.json`

- `results/step_04_04_distance_duality.json` — DDR weighted mean and deviation from unity

- `results/step_04_05_ddr_threeway.json` — Three-way BAO/SZ/SGL probe comparison

- `results/step_04_11_sn_robustness_systematics.json` — Prior, redshift-cut, covariance, and sampler robustness grid for TEP M1

- `results/step_04_12_external_sn_validation.json` — External validation on Union3 binned compilation

- `results/step_05_07_bbn_preservation.json` — Historical AlterBBN compatibility baseline — superseded; not part of canonical TEP thermodynamics

- `results/step_05_09_minimal_perturbations.json` — diagnostic minimal-closure perturbation checks; authoritative active-sector closure is documented in TEP-HC

- `results/step_06_04_growth_validation.json` — Growth factor and sigma_8 consistency check

- `results/step_08_04_evidence_matrix.json` — Explanatory evidence matrix across all observables

- `results/step_08_06_claim_audit.json` — Automated claim consistency check report

#### Log Files

Each step produces detailed logs with timestamps, SHA-256 checksums, and execution status:

- `logs/step_*.log` — Individual step logs (64 files, one per step)

- `logs/verbose/` — Verbose Cobaya and nested sampling logs

### Software Dependencies

| Package | Version | Purpose |
| --- | --- | --- |
| Python | 3.10+ | Language runtime |
| NumPy | 1.24+ | Numerical computing |
| SciPy | 1.10+ | Statistical functions, nested sampling |
| Pandas | 2.0+ | Data manipulation |
| Matplotlib | 3.7+ | Visualization |
| emcee | 3.1+ | Ensemble MCMC sampling |
| dynesty | 2.1+ | Nested sampling for Bayesian evidence |
| Cobaya | 3.6+ | Joint MCMC with Planck likelihoods |
| classy (CLASS) | 3.2+ | CMB Boltzmann solver (patched for TEP) |

All dependencies are specified in `requirements.txt`. External dependencies (patched CLASS, AlterBBN) are included in the `external/` directory.

### Appendix Figures

![Joint SNe+CMB Background/Acoustic MCMC Diagnostic](results/figures/step_03_05_analyze_cobaya_triangle.png)

Figure A1: Joint SNe+CMB Background/Acoustic MCMC Diagnostic. This triangle plot shows the joint posterior from the Cobaya MCMC, including the homogeneous acoustic-sector amplitude $\epsilon_T^{\rm CMB}$. This is a diagnostic figure, not the SNe-only M1 evidence result. The $H_0$ boundary behaviour is separately stress-tested (Section 4.2). The $\epsilon_T$ shown here is the homogeneous acoustic-sector amplitude, distinct from the line-of-sight $\epsilon_{\rm shear}^{\rm los}$ fitted to supernovae.

![Tolman Surface Brightness Decomposition](results/figures/step_04_02_sn_tolman.png)

Figure A2: Reference Tolman surface-brightness decomposition over the Pantheon+ redshift range. This is a geometric scaling decomposition, not an independent Tolman-test measurement. The empirical Tolman sector remains systematics-limited and inconclusive as a TEP/$\Lambda$CDM discriminator.

---

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# Temporal Equivalence Principle: Temporal Horizon Cosmology and the Absence of a Physical Big Bang Singularity
**Matthew Lukin Smawfield**
Version: v0.5 (Thika)
First published: 18 June 2026 - Last updated: 17 October 2026
DOI: 10.5281/zenodo.20723059

---

## Abstract

Standard FLRW cosmology extrapolates observed cosmic expansion backward to $a(t)\to0$, producing a Big Bang singularity at finite proper time. This paper demonstrates that this singularity is a reconstruction artifact of imposing a globally isochronous expanding-frame description on a conformal temporal geometry. In the Temporal Equivalence Principle (TEP), the observational role of FLRW expansion is reconstructed through conformal temporal transport: the effective scale factor $a_{\rm eff}$ arises from accumulated open-path conformal temporal shear along cosmological lines of sight rather than from physical expansion of space. TEP-C0 (Paper 26) established the distance-redshift and supernova evidence; the conditional nonsingular matter-frame boundary construction is developed here. Realization of that boundary within the complete infinite inhomogeneous solution remains the common-action construction problem of Paper 0.

The Temporal Horizon Cosmology framework is developed here, proving, within the temporal-conformal branch defined here, that the apparent $a_{\rm eff}\to0$ limit is not a physical curvature singularity but a temporal horizon. The effective scale factor $a_{\rm eff}$ is driven by the observational clock/redshift mapping $A_{\rm clock}(z)=(1+z)^{-1}$. Proposition 1 establishes curvature regularity of the temporal conformal boundary: for $A_{\rm clock}(\eta)=C\eta^{-p}$ with $0 \lt p\le\tfrac12$, all polynomial curvature invariants vanish at the boundary, timelike proper time diverges, and null geodesics have divergent affine parameter. The temporal-horizon exponent $p$ and the observational clock map are independent boundary conditions: $A_{\rm clock}(z)=(1+z)^{-1}$ is fixed by the redshift definition, while the regularity condition $0 \lt p\le\tfrac12$ is a mathematical requirement for curvature-regularity at the conformal boundary. Here $\eta$ is the temporal-horizon conformal coordinate, oriented so that approach to $\mathscr{T}^{-}$ corresponds to the asymptotic limit in which $A_{\rm clock}\to0$; it is not the standard FLRW conformal time coordinate extrapolated to $a=0$. Figure 1 (Section 4.5) illustrates the resulting conformal-boundary interpretation: the singular lower edge of standard flat $\Lambda$CDM is replaced by a smooth temporal conformal boundary $\mathscr{T}^{-}$, where $A_{\rm clock}\to0$ and curvature invariants vanish. The conformal compactification is smooth, the Weyl tensor vanishes on the boundary, and every causal curve approaches the regular past boundary $\mathscr{T}^{-}$ rather than terminating at a singularity. The temporal horizon is therefore simultaneously curvature-empty, timelike-complete, and null-complete in this branch.

The effective stress-energy tensor of the temporal field violates the Strong Energy Condition, an explicit prerequisite of the Hawking-Penrose singularity theorems. Rather than treating the high-redshift hot plasma as the residue of a singular expanding origin, TEP retains that thermal state as a matter-frame condition of the temporal landscape and replaces the singular-origin interpretation with Native Local Thermodynamic Evolution in an eternal universe. The temporal-horizon metric provides the geometric boundary where the clock rate vanishes, but it does not claim to uniquely derive primordial abundances. Instead, it delegates the thermal history to the eternal chemical-evolution framework (TEP-BBN, Paper 29), which provides an eternal-universe framework in which chemical states can approach a steady-state asymptotic equilibrium where D/H is no longer uniquely primordial and helium arises through baryonic cycling. The temporal-horizon thermal mapping preserves the observed CMB photon distribution through local decoupling processes without requiring a geometric singularity.

The scalar perturbation spectrum is derived from fluctuations of the clock field, $\zeta=\delta\ln A_{\rm clock}$, yielding a power spectrum $P_{\zeta}(k)\propto k^{n_{s}-1}$ with spectral-flow parameter $n_{s}-1=-2\epsilon_{\rm field}$. The observed Planck value $n_{s}=0.9649$ constrains $\epsilon_{\rm field}=0.01755$. In the assigned tensor-pump benchmark: for $A_{\rm clock}(\eta)\sim\eta^{-p}$ the tensor source term $A_{\rm clock}''/A_{\rm clock}=p(p+1)/\eta^{2}\to 0$ at the horizon, so the tensor equation approaches the Minkowski vacuum. The imported inflationary consistency relation $r=16\epsilon_{\rm field}$ is not assumed. Numerical integration of the assigned tensor-pump equation across the finite transition profile yields $r(k_{\rm pivot})=1.9\times 10^{-9}$ and $r_{\rm max}=5.2\times 10^{-7}$, both far below the BICEP/Keck 2021 bound $r\lt0.036$; tensor power is controlled only by the finite transition region. The late-time homogeneous expansion and acoustic observables (Planck 2018, BOSS DR12) are fully preserved by the conformal temporal mapping as mathematically confirmed in TEP-HC (Paper 18).

The causal matter-frame universe is curvature-regular at the temporal conformal boundary. The apparent Big Bang is a temporal horizon, not a physical curvature singularity. The temporal-horizon geometry supplies a nonsingular framework in which the standard background and acoustic observables are reconstructed by the conformal mapping, while TEP-BBN provides the native chemical-evolution mechanism and a proof of concept for local CMB thermalization; the scalar perturbation shape is reproduced, and the tensor-to-scalar ratio is computed from the assigned temporal-conformal pump equation, yielding values well below observational bounds.

Code Availability: https://github.com/matthewsmawfield/TEP-TH

Keywords: temporal equivalence principle, temporal horizon cosmology, big bang singularity, static conformal geometry, cosmology, modified gravity, temporal shear

**Common-action scope.** The regularity and completeness results apply to the specified matter-frame metric. The scalar spectrum is reconstructed using the observed tilt and amplitude. The quoted tensor values belong to an assigned pump diagnostic; Section 10.5 distinguishes it from the tensor quadratic action of the static gravitational background. They are not yet a universal action-normalized tensor prediction or a model-wide B-mode exclusion threshold. HC's acoustic preservation and BBN's thermal/chemical calculations retain their respective reference-model and native-boundary assumptions; matching their complete transfer histories is a separate calculation.

# 1. Introduction: From Big Bang Singularity to Temporal Horizon

Standard FLRW cosmology extrapolates the observed cosmic expansion backward to $(a(t)\to0)$, producing a Big Bang singularity at finite proper time. This singular origin requires an initial hot dense state, followed by BBN nucleosynthesis, recombination, acoustic peak formation, CMB blackbody thermalization, and primordial perturbation generation. Hawking (1966) established, and Hawking and Penrose (1970) later generalized, that such a singularity is mathematically inevitable provided the Strong Energy Condition holds in globally hyperbolic spacetimes.

This paper demonstrates that the observational role of FLRW expansion is reconstructed through conformal temporal transport. In this framework, the distance-redshift relation and CMB acoustic scales are preserved in a static conformal geometry where the effective scale factor $a_{\rm eff}$ arises from accumulated open-path conformal temporal shear along cosmological lines of sight, not from physical expansion of space. The exact observational clock/redshift mapping $A_{\rm clock}(z)=(1+z)^{-1}$ drives the apparent $a_{\rm eff}\to0$ limit. The apparent $(a_{\rm eff}\to0)$ limit is therefore not a physical curvature singularity but a reconstruction artifact of imposing a globally isochronous expanding-frame description on a conformal temporal geometry.

While standard cosmology treats cosmic expansion as a kinematic stretching of the spatial metric, several frameworks have explored conformal alternatives. Wetterich (2013) demonstrated that a universe without spatial expansion can be formulated using a varying particle mass, while Narlikar and Arp explored conformal gravity variations. Environmental screening mechanisms—such as chameleon or symmetron screening (Khoury & Weltman 2004; Hinterbichler & Khoury 2010)—have been extensively developed to hide scalar fifth forces. TEP departs from these approaches by identifying the conformal factor strictly with the dynamical flow of proper time, generating an exact geometric mapping between the spatial scale factor and the macroscopic accumulation of Temporal Shear without requiring variable rest masses or modified spatial curvature.

Prior work established the empirical basis for this static conformal cosmology. TEP-C0 (Paper 26) demonstrated, through nested sampling over 1,701 Pantheon+ supernovae, that a pure conformal reconstruction exactly matches $\Lambda$CDM at the homogeneous distance-modulus level, while the physical no-$\Lambda$ temporal-shear branch — the branch carrying no bare cosmological constant in the background model — improves the standardized supernova likelihood by approximately −3.81 in chi-squared for the conservative line-of-sight turnover $z_{\rm los}=5$ and approximately −3.69 for the fixed $z_{\rm los}=100$ benchmark. TEP-C0 identified full nonsingular matter-frame closure as the remaining open question for the cosmological sequence; the present paper establishes the regular boundary of the specified temporal-conformal branch, with its common-action embedding distinguished in Section 2.1.

TEP-HC (Paper 18) implemented the native TEP interpretation directly in the `hi_class` Boltzmann code, deriving the runtime Bellini–Sawicki functions ($\alpha_M=-2\alpha_A$, $\alpha_B=2\alpha_A$, $\alpha_K=w_c\,\alpha_A^2$ with $w_c = Z_E-6$, $\alpha_T=0$) and verifying that the static conformal geometry preserves the pre-recombination sound horizon with ratio $r_s^{\rm TEP}/r_s^{\Lambda\rm CDM} = 0.999994$ (corresponding to a $<6$ ppm deviation) with an active linear scalar perturbation sector that is stable and observationally negligible on the screened production branch ($w_c = 700$; the canonical image $w_c = -5$ is linearly unstable and excluded). TEP-TH closes the logical loop by demonstrating that the causal matter-frame geometry is regular at the temporal horizon, so the apparent Big Bang is not a physical singularity but an asymptotic boundary of the proper-time field.

#### Parameter-Scale and Amplitude Convention

**Turnover scales.**

**Amplitudes.** $\epsilon_{\rm field}=0.01755$ denotes the primordial spectral-flow parameter constrained by $n_s$. $\epsilon_{\rm dyn}$ denotes the dynamical temporal-horizon response. $\epsilon_T^{\rm los}$ denotes the late-time line-of-sight transport amplitude fitted in TEP-C0. $\epsilon_T^{\rm CMB}$ denotes the C0 background/acoustic diagnostic amplitude. $\epsilon_T^{\rm HC}=0.0043\pm0.0025$ denotes the native `hi_class` homogeneous conformal amplitude from the full high-$\ell$ likelihood reported in TEP-HC. These are related projections of the same temporal sector, but they are not numerically interchangeable parameters.

In standard cosmology, epochs are conventionally defined by the chronological time elapsed since the physical singularity (e.g., "three minutes after the Big Bang" for nucleosynthesis). Because the temporal horizon in the TEP framework is an asymptotic boundary rather than a zero-volume origin, a global linear time coordinate $t$ cannot be extrapolated to a finite $t=0$. Consequently, the sequence of early-universe events is strictly mapped not by chronological time, but by the thermodynamic cooling of the plasma ($T$) and the evolution of the conformal clock-rate field. The history of the universe is preserved, but the chronological stopwatch is replaced by thermodynamic state variables.

TEP-TH addresses this early-universe question by demonstrating that the apparent Big Bang singularity is not a physical boundary but a temporal-horizon limit. The central result is that the apparent singularity corresponds to the limit $A_{\rm clock}\to0$, where the observational clock-rate field vanishes and proper time dilates to zero relative to the present epoch. The thermal history emerges natively through local thermodynamic evolution $\tilde\nabla_\mu \tilde T^{\mu\nu}=0$ in proper time, generating the CMB emission without assuming an initially hot, dense geometric singularity. The "Big Bang" is not a zero-volume, infinite-density origin of space, but a temporal horizon—an asymptotic boundary of the proper-time field.

This paper demonstrates this temporal-horizon replacement through a ten-step pipeline:

- **Temporal-Horizon Mapping**: Establish $A_{\rm clock}(z)=(1+z)^{-1}$ as the exact clock map and $A_{\rm dyn}(z)$ as the physical dynamical response

- **Matter-Frame Curvature**: Prove that for $A_{\rm clock}(\eta)=C\eta^{-p}$ with $0 \lt p\lt1$, all curvature invariants generated from the Ricci sector vanish; since the Weyl tensor vanishes in the conformally flat branch, all polynomial curvature invariants vanish at the temporal conformal boundary

- **Geodesic Completeness**: Demonstrate that for $0 \lt p\le\tfrac12$ the temporal horizon is simultaneously curvature-empty, timelike-complete, and null-complete

- **Effective Stress-Energy**: Evaluate energy conditions and demonstrate Hawking-Penrose consistency via SEC violation

- **BBN Abundance Framework**: Replace expanding thermal plasma assumptions with eternal Native Proper-Time Nucleosynthesis (delegated to TEP-BBN, Paper 29) to evaluate candidate asymptotic abundance solutions

- **Recombination Visibility**: Re-evaluate dynamic proper-time recombination $x_e(\tau)$ and derived physical acoustic scales within the native eternal-universe thermodynamic framework

- **CMB Blackbody Origin**: Attribute the physical origin of the photon distribution to local emission processes and confirm conformal transport preservation

- **Entropy and Arrow of Time**: Demonstrate thermodynamic regularity at the horizon

- **Primordial Perturbation Boundary**: Derive scalar fluctuations from $\zeta=\delta\ln A_{\rm clock}$ and tensor modes from the assigned temporal-conformal pump equation

- **CMB Anisotropy and LSS Consistency**: Relying on TEP-HC (Paper 18) to demonstrate that TT/TE/EE spectra, matter power spectrum, growth factor, and BAO scales match Planck and BOSS

The combined pipeline shows that the corresponding observational pillars can be reconstructed within the TEP framework. The causal matter-frame universe is curvature-regular at the temporal boundary: for $A_{\rm clock}(\eta)=C\eta^{-p}$ with $0 \lt p\lt1$, curvature invariants vanish at the boundary rather than merely remaining bounded; comoving proper time diverges for $0 \lt p\le1$, while arbitrary timelike geodesics are complete for $0 \lt p\le\tfrac12$; null affine parameter diverges for $0 \lt p\le\tfrac12$; and the Strong Energy Condition is violated—satisfying the mathematical prerequisite established by Hawking and Penrose for a non-singular spacetime. The cleanest fully complete branch is $0 \lt p\le\tfrac12$, in which the temporal horizon is simultaneously curvature-empty, timelike-complete, and null-complete. BBN, recombination, CMB blackbody origin, entropy regularity, and the scalar shape of primordial perturbations are recovered without requiring a physical zero-volume, infinite-density expanding origin. Tensor modes are governed by a assigned temporal-conformal pump equation whose source term vanishes at the horizon; the imported inflationary consistency relation $r=16\epsilon_{\rm field}$ is not assumed. The apparent Big Bang is a temporal horizon, not a physical curvature singularity.

This work does not independently re-analyse Pantheon+ supernovae or perform the native hi_class Boltzmann perturbation closure; those are addressed in companion papers TEP-C0 (Paper 26) and TEP-HC (Paper 18).

The claim-discipline framework for the TEP corpus, including the scope limitations of canonical precision tests, is established in TEP-EXP (Paper 9).

**Common-action scope.** The regularity and completeness results apply to the specified matter-frame metric. The scalar spectrum is reconstructed using the observed tilt and amplitude. The quoted tensor values belong to an assigned pump diagnostic; Section 10.5 distinguishes it from the tensor quadratic action of the static gravitational background. They are not yet a universal action-normalized tensor prediction or a model-wide B-mode exclusion threshold. HC's acoustic preservation and BBN's thermal/chemical calculations retain their respective reference-model and native-boundary assumptions; matching their complete transfer histories is a separate calculation.

# 2. Temporal-Horizon Mapping

The central temporal-horizon mapping establishes the relationship between the apparent FLRW singularity and the conformal clock-rate field. In standard FLRW cosmology, the Big Bang corresponds to the limit $a(t)\to0$ at finite proper time. In TEP, the effective scale factor is reconstructed from accumulated open-path conformal temporal shear. Two distinct projections of the temporal field are required to maintain consistency across all redshifts:

\begin{equation} \label{eq:aeff}
a_{\rm eff}(z) = A_{\rm clock}(z)\,a_{\rm m}(z) \, ,
\end{equation}

where $a_{\rm m}(z)=1$ in the clean static matter-frame case. The clock map

\begin{equation} \label{eq:A_clock}
A_{\rm clock}(z) = \frac{1}{1+z}
\end{equation}

is the exact observational clock/redshift mapping that drives $a_{\rm eff}\to0$ as $z\to\infty$. This is the temporal horizon: $A_{\rm clock}\to0$ is the conformal-temporal boundary, not a physical curvature singularity. The dynamical response

\begin{equation} \label{eq:A_dyn}
A_{\rm dyn}(z) = \left(1+\frac{z}{z_{t}}\right)^{-\epsilon_{\rm dyn}}
\end{equation}

is the dynamical shear response that modifies the Hubble parameter and perturbations at late times. It is crucial to distinguish these roles: $A_{\rm clock}$ is fixed by the redshift–distance relation and produces the apparent singularity, while $A_{\rm dyn}$ is a small fitted TEP correction. An observer situated in the early universe would experience local time advancing normally, as curvature invariants remain bounded. The limit $A_{\rm clock}\to0$ is strictly a relative observational boundary: as one looks backward from the present epoch, ancient clocks appear to tick progressively slower. The horizon is the mathematical asymptote where this relative clock rate approaches zero, not a physical location where time itself ceases to exist.

The pipeline computes both projections across redshift $z\in[0,10^{4}]$. The clock map $A_{\rm clock}(z)=(1+z)^{-1}$ reproduces the standard redshift scaling exactly, while the dynamical response $A_{\rm dyn}(z)$ approaches zero at early times (for $\epsilon_{\rm dyn} > 0$, $(1+z/z_t)^{-\epsilon_{\rm dyn}} \to 0$ as $z \to \infty$). The effective Hubble parameter is therefore

\begin{equation} \label{eq:H_tep_split}
H_{\rm TEP}(z) = H_{\rm \Lambda CDM}(z)\,[1+\delta(z)] \, ,
\end{equation}

which encodes the late-time TEP shear correction. This separation resolves the apparent tension between $a_{\rm eff}\to0$ at the horizon and the requirement that $H_{\rm TEP}\approx H_{\rm LCDM}$ during the thermal epochs.

The static conformally-flat matter-frame representation is not an assumption; it follows from the TEP disformal coupling. The foundational TEP papers establish the universal matter coupling through the canonical unified metric $\tilde{g}_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}+B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$; Paper 0, §2.2, Step 80), written in conformal–disformal shorthand as $\tilde{g}_{\mu\nu}=A^{2}g_{\mu\nu}+B\,\nabla_{\mu}\phi\,\nabla_{\nu}\phi$. On the homogeneous cosmological background ($u_{\rm grav}=0$, $\Pi_\mu^\alpha\nabla_\alpha\phi=0$), $d\tilde s^2=-c^2dt^2+A_{\rm sp}^2(t)\gamma_{ij}dx^i dx^j$, which in conformal propagation time $d\eta=dt/A_{\rm sp}(t)$ takes the conformally flat form $d\tilde s^2=A_{\rm clock}^2(\eta)(-c^2d\eta^2+\gamma_{ij}dx^i dx^j)$ with $A_{\rm clock}\equiv A_{\rm sp}$. The static gravitational frame $a_{\rm m}=1$ is the coordinate choice in which this reduction is manifest.

The spatial homogeneity of this background represents the macroscopic averaging limit of a locally structured temporal field. While the local conformal factor deviates symmetrically above and below unity as it traverses halos and voids (Paper 0, §2.2), the spatial integration of these exact excursions recovers the smooth $\phi(\eta)$ dependence. The perfectly homogeneous background is therefore an effective macroscopic envelope, explicitly separated from the non-exact covariance $\mathcal{C}_T$ (Paper 26) that accumulates as a physical residual along specific lines of sight.

## 2.1 Sector Consistency with Foundational TEP

The foundational TEP metric is conformal-disformal,

\begin{equation}
\tilde{g}_{\mu\nu} = A^{2}(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_{\mu}\phi\,\nabla_{\nu}\phi \, .
\end{equation}

The present paper studies the homogeneous cosmological projection of this structure. On an FLRW background ($\phi=\phi(\eta)$), the disformal term is purely temporal and can be absorbed into the lapse by a time reparameterisation. The resulting temporal-horizon theorem is therefore a theorem of the conformal clock-rate sector ($A_{\rm clock}$), not a derivation of the non-exact synchronization-holonomy sector.

This distinction is essential. Open-path temporal transport,

\begin{equation}
\Delta\ln A = \int_{\gamma} d\ln A \, ,
\end{equation}

can generate cosmological redshift and the apparent scale-factor reconstruction. Closed-loop residual synchronization holonomy, however, obeys

\begin{equation}
\oint d\ln A = 0
\end{equation}

for smooth single-valued conformal transport. Therefore TEP-TH does not claim that $A_{\rm clock}$ alone generates nonzero closed-loop holonomy. Nonzero residual holonomy requires disformal or otherwise non-exact synchronization curvature, as developed in the foundational TEP paper (Jakarta). The temporal-horizon result and the synchronization-holonomy programme are complementary probes of different sectors of the same causal matter metric. The homogeneous temporal-shear background contribution $\Omega_\phi$ (TEP-HC, Paper 18) fills the same cosmological background budget as $\Omega_\Lambda$ — an operational bookkeeping of the conformal sector's distance-law role, not a literal vacuum energy density, whose value $\Omega_\phi\simeq0.697$ follows without fitting from the Paper 18 fixed point and spatial flatness (the action-level vacuum plateau is quantified in Section 5.1, where $U_{\rm c}/\rho_{\rm crit,0}\approx1.8\times10^{-3}$) — while the non-exact covariance/topology correction $C_T$ (TEP-C0, Paper 26) provides the transport closure for the open-path redshift reconstruction beyond exact conformal shear.

The redshift map and the conformal-time horizon profile are linked by $A_{\rm clock}(z)=(1+z)^{-1}$ and $A_{\rm clock}(\eta)=C\eta^{-p}$, which together imply

\begin{equation} \label{eq:z_eta}
1+z(\eta) = C^{-1}\eta^{p} \, .
\end{equation}

This relation connects the observational redshift coordinate — the epoch label under which events are ordered — to the conformal time parameter in which the curvature and geodesic theorems are proved. The constant $C$ is fixed by the condition $A_{\rm clock}=1$ at the present epoch ($z=0$, $\eta=\eta_{0}$), giving $C=\eta_{0}^{-p}$.

The scope of $\eqref{eq:z_eta}$ must be stated precisely, because it composes two objects of different standing. The clock map $A_{\rm clock}(z)=(1+z)^{-1}$ is the exact observational clock-ratio definition of redshift; the horizon profile $A_{\rm clock}(\eta)=C\eta^{-p}$ is the asymptotic form of the homogeneous ambient envelope, i.e. the boundary structure on which the completeness theorems operate. Their composition is therefore the internal epoch map of the homogeneous theorem branch: it assigns to a conformal position $\eta$ the clock-ratio coordinate that a strictly homogeneous realization would measure at that depth. It is not the realized distance–redshift relation. Two features of the transport architecture prevent that identification. First, the composition retains only the exact conformal sector of open-path transport, whose path integral reduces to the endpoint ratio $\Delta\ln A$; on the realized landscape the observed redshift is the full transport integral $\ln(1+z_{T})=\int_{\gamma}(\Sigma_{\parallel}+\mathcal{C}_{T,\parallel})\,d\ell$ of TEP-C0 (Paper 26), in which the non-exact covariance term $\mathcal{C}_{T}$ accumulates along the sightline without cancelling and the emission-side clock ratio carries the depth of the source's own temporal well — so the observed redshift is a functional of the realized field history, not of the homogeneous envelope value at the endpoint. Second, the power-law profile is the $\eta\to\infty$ asymptote that the curvature and completeness theorems constrain; the finite-$\eta$ behaviour of the branch is the transition-profile sector of Section 10, which departs from the asymptote precisely in the regime in which finite-redshift transport is evaluated. The luminosity distance in the static background remains the Etherington combination $d_{L}=(1+z)\,\Delta\eta$ — distance duality is preserved by the conformal construction — but the redshift entering it is the realized transport quantity, whose distance–redshift relation is computed on the landscape in TEP-C0 and TEP-HC (Papers 26 and 18) and reproduces the observed relation at the homogeneous-observable level. No law $d_{L}\propto(1+z)^{1+1/p}$ therefore follows from the homogeneous branch: such an expression results from substituting the internal epoch map into the distance formula, thereby identifying the theorem's coordinate parameter with a transport observable. Where this paper and its companions state that the distance–redshift relation is preserved, the claim is thereby scoped to the realized transport sector, to which the homogeneous $\Delta\eta$ does not reduce.

*Relation to the canonical background branch.* Two horizon asymptotics are distinguished within the framework, and they are not equivalent. The exactly solved closed-static benchmark solution (Paper 0, §8) approaches the temporal horizon exponentially in gravitational coordinate time, $A\propto\exp(\sqrt{2}\,t/a)$, accumulating only a finite matter-frame clock history while matter-frame curvature invariants diverge as $A^{-2}$. Because matter and clocks propagate on $\tilde g$, finite-proper-time termination with diverging scalar invariants is geodesic incompleteness in the matter frame: the benchmark realizes the singular branch, the asymptotic on which the standard hot-Big-Bang reading is correct. The power-law family analyzed in this paper, $A_{\rm clock}(\eta)=C\eta^{-p}$, realizes the qualitatively different regular branch: a curvature-empty conformal boundary with divergent accumulated proper time, for which the completeness results below are established. The two branches are therefore distinct causal completions selected by the scalar sector, not alternative descriptions of one phenomenology. The regular branch is not merely postulated: Section 10.2 derives a cosmological clock-field action — reconstructed kinetic function $\mathcal Z(\chi)$ and potential $U(\chi)$ — for which the power-law profile is an exact background solution of the Euler–Lagrange equation (Section 10.2a). What remains open is embedding that effective cosmological action in the two-metric master action, i.e. deriving $\mathcal Z(\chi)$ and $U(\chi)$ from the fundamental scalar sector; that derivation would promote the regular branch from an exactly solved effective-sector solution to a derived background of the full theory.

# 3. Matter-Frame Geometry and Curvature Regularity

To determine whether the temporal horizon is a physical singularity or a reconstruction artifact, matter-frame curvature invariants are computed. In the causal matter frame the metric is $\tilde{g}_{\mu\nu} = A_{\rm clock}^{2}(\eta)\,g_{\mu\nu}$, where $g_{\mu\nu}$ is the static conformal background and $A_{\rm clock}(\eta)$ is the conformal clock-rate field. The key question is whether curvature invariants remain finite as $A_{\rm clock}\to0$, and whether they in fact vanish for a suitable horizon profile.

The effective scale factor observed by matter is not the raw conformal factor but the product

\begin{equation} \label{eq:aeff_matter}
a_{\rm eff} = A_{\rm clock}\,a_{\rm m} \, ,
\end{equation}

where $a_{\rm m}=1$ in the clean static matter-frame case. The apparent FLRW limit $a_{\rm eff}\to0$ is driven by $A_{\rm clock}\to0$, while the underlying matter-frame coordinate scale factor $a_{\rm m}$ remains finite and non-zero. Consequently the physical matter-frame metric determinant is

\begin{equation} \label{eq:metric_det}
\det(\tilde{g}_{\mu\nu}) = A_{\rm clock}^{8}\,\det(g_{\mu\nu}) = -A_{\rm clock}^{8}\,a_{\rm m}^{6} \, .
\end{equation}

In four spacetime dimensions every metric component acquires a factor $A_{\rm clock}^{2}$, giving $A_{\rm clock}^{8}$ in the determinant. Because $a_{\rm m}$ does not collapse to zero, $\det(\tilde{g})$ vanishes only if $A_{\rm clock}\to0$ with no compensating behaviour. The determinant alone does not prove regularity; one must show that the *physical* volume element, constructed from the inverse metric and the relevant measure, does not degenerate in a way that produces divergent curvature.

The full four-dimensional conformal transformation of the Ricci scalar for $\tilde{g}_{\mu\nu}=A_{\rm clock}^{2}\,g_{\mu\nu}$ is

\begin{equation} \label{eq:ricci_scalar_full}
\tilde{R} = A_{\rm clock}^{-2}\Bigl[ R - 6\,\Box\ln A_{\rm clock} - 6\,(\nabla\ln A_{\rm clock})^{2} \Bigr] \, .
\end{equation}

Equivalently, writing everything in terms of $A_{\rm clock}$ itself,

\begin{equation} \label{eq:ricci_scalar_A}
\tilde{R} = A_{\rm clock}^{-2}R - 6A_{\rm clock}^{-3}\Box A_{\rm clock} \, .
\end{equation}

In the TEP static background $g_{\mu\nu}$ has $R=0$ and $\nabla A_{\rm clock}$ is spatially homogeneous, so the surviving terms are controlled by the background profile $A_{\rm clock}(\eta)$. For the temporal-horizon profile $A_{\rm clock}(\eta)=C\eta^{-p}$ with $0 \lt p\lt1$, the dominant term is $\tilde{R}=6A_{\rm clock}''/A_{\rm clock}^{3}=6p(p+1)C^{-2}\eta^{2p-2}$, which vanishes as $\eta\to\infty$.

The Kretschmann scalar and Ricci-tensor square transform with additional powers of $A_{\rm clock}^{-1}$. For a conformally flat static background the Weyl tensor vanishes, so the full Riemann tensor is built from the Ricci tensor. The conformal transformation of $\tilde{R}_{\mu\nu}$ (Wald, App. D) with $\ln A_{\rm clock}=\ln C-p\ln\eta$ and the static flat background gives the closed-form Ricci-tensor invariant

\begin{equation}
\tilde{R}_{\mu\nu}\tilde{R}^{\mu\nu} = 12p^{2}\bigl(p^{2}+p+1\bigr)\,\frac{\eta^{4p-4}}{C^{4}} \, .
\end{equation}

The Kretschmann invariant in the zero-Weyl case is $\tilde{K}=2\tilde{R}_{\mu\nu}\tilde{R}^{\mu\nu}-\tfrac13\tilde{R}^{2}$, which evaluates to

\begin{equation} \label{eq:kretschmann}
\tilde{K} = 12p^{2}\left(p^{2}+1\right)\frac{\eta^{4p-4}}{C^{4}} \, .
\end{equation}

For $0 \lt p\lt1$, the factor $\eta^{4p-4}\to0$ as $\eta\to\infty$, so both the Kretschmann scalar and the Ricci-tensor invariant vanish at the boundary with an explicit polynomial prefactor in $p$. The scaling is not approximate; it is a closed-form result derived from the conformal transformation of the Ricci tensor on the static background.

## 3.1 Proposition 1 — Curvature Regularity of the Temporal Conformal Boundary

**Proposition 1 (Curvature regularity of the temporal conformal boundary).** Let the observational effective scale factor be $a_{\rm eff}=A_{\rm clock}\,a_{\rm m}$ with $a_{\rm m}$ the finite, non-zero matter-frame coordinate scale factor. Let the clock field take the temporal-horizon profile $A_{\rm clock}(\eta)\sim\eta^{-p}$ in conformal time with $0 \lt p\lt1$. Then, as $\eta\to\infty$ ($A_{\rm clock}\to0$):

- The Ricci scalar $\tilde{R}$ in the matter frame vanishes: $\tilde{R}\sim\eta^{2p-2}\to0$.

- The Kretschmann scalar $\tilde{K}$ and Ricci-tensor invariant $\tilde{R}_{\mu\nu}\tilde{R}^{\mu\nu}$ vanish: $\tilde{K}\sim\tilde{R}_{\mu\nu}\tilde{R}^{\mu\nu}\sim\eta^{4p-4}\to0$.

- The observational conformal volume element $V_{\rm eff}=A_{\rm clock}^{3}a_{\rm m}^{3}$ tends to zero, but this is not a physical collapse of the underlying matter-frame spatial geometry. It reflects the degeneracy of the observational clock/redshift map at the temporal horizon. The finite coordinate scale $a_{\rm m}$ and vanishing curvature invariants show that the boundary is a conformal-temporal endpoint, not a zero-volume curvature singularity.

*Proof sketch.* Substitute $A_{\rm clock}(\eta)=C\eta^{-p}$ into equations $\eqref{eq:ricci_scalar_full}$–$\eqref{eq:kretschmann}$. In the static conformal background $R=0$ and spatial gradients vanish by homogeneity. The time derivatives scale as $A_{\rm clock}'\sim -pC\eta^{-p-1}$ and $A_{\rm clock}''\sim p(p+1)C\eta^{-p-2}$. The Ricci scalar $\eqref{eq:ricci_scalar_A}$ contains terms $A_{\rm clock}^{-2}R$ and $A_{\rm clock}^{-3}\Box A_{\rm clock}$. With $R=0$ and homogeneity, the dominant term is $6A_{\rm clock}''/A_{\rm clock}^{3}=6p(p+1)C^{-2}\eta^{2p-2}\to0$ for $0 \lt p\lt1$. The Ricci-tensor invariant has coefficient $12p^{2}(p^{2}+p+1)/C^{4}$ and the Kretschmann invariant has coefficient $12p^{2}(p^{2}+1)/C^{4}$, which satisfy the zero-Weyl identity $\tilde{K}=2\tilde{R}_{\mu\nu}\tilde{R}^{\mu\nu}-\tfrac13\tilde{R}^{2}$; both are strictly positive for all $p>0$ and scale as $\eta^{4p-4}\to0$ for $0 \lt p\lt1$. Numerical evaluation reproduces the analytic decay laws on the tested range; the vanishing limit is established by their negative powers of $\eta$. $\square$

For $p\le\tfrac12$ the Kretschmann scalar falls faster than $\eta^{-2}$, ensuring that null affine parameters diverge (Section 4.2).

*Scope of the $R=0$ hypothesis.* The proposition's background-flatness assumption is quantitative, not merely qualitative. Writing a residual background scalar as $R(\eta)=R_0\,\eta^{-s}$, the transformed term $A_{\rm clock}^{-2}R\sim R_0\,\eta^{2p-s}$ is bounded if and only if $s\ge2p$ and vanishes toward the boundary for $s>2p$: background curvature must decay toward the temporal horizon faster than $\eta^{-2p}$. Two sources organise the answer. A fixed curvature offset ($s=0$) — e.g., a cosmological-constant-like floor in the background — is excluded for every $p>0$; such a floor would convert the vanishing invariants into divergent ones. Curvature sourced by the clock field itself decays as $R_{\rm kin}\sim\varphi'^2\sim\eta^{-2}$ (since $A_{\rm clock}=e^{\beta_A\varphi/M_{\rm Pl}}=\eta^{-p}$ gives $\varphi\propto\ln\eta$), so $s=2$ satisfies the vanishing condition for every $p<1$; it falls in step with the $\eta^{2p-2}$ gradient terms whenever $p<1$. The proposition therefore stands precisely for backgrounds whose residual curvature decays at least as fast as the clock field's own kinetic trace, and fails only for a non-decaying background floor — a scope statement verified numerically across the $(p,s)$ plane by the bounded sensitivity calculation of the pipeline's curvature-sensitivity step.

This establishes the temporal boundary as a regular conformal endpoint, adopting the mathematical architecture of Penrose's conformal compactification and null infinity ($\mathscr{I}^{+}$) (Penrose 2006; Tod 2003, 2015), here applied to the asymptotic past. The boundary is not a curvature singularity; it is a conformal-temporal boundary where the Lorentzian metric becomes degenerate in the observational clock frame. The vanishing of curvature invariants is necessary but not sufficient for regularity: a degenerate metric admits no inverse, so the geodesic equation becomes singular at the boundary. Geodesic completeness is established not by the curvature invariants alone but by the affine-parameter analysis of Section 4: timelike proper time diverges for $0 < p \le 1$ and null affine parameter diverges for $0 < p \le \tfrac{1}{2}$, so no causal curve reaches the boundary in finite parameter. The boundary is therefore a regular asymptotic endpoint of the conformal compactification, not an interior point of the manifold; the metric degeneracy is a feature of the conformal frame, not a pathology of the physical geometry.

It is vital to distinguish the TEP temporal horizon ($\mathscr{T}^{-}$) from Penrose's Conformal Cyclic Cosmology (CCC). Where CCC posits an infinite sequence of aeons connected by conformal rescalings at $\mathscr{I}^{+}$ and $\mathscr{I}^{-}$, TEP does not invoke cyclicality. $\mathscr{T}^{-}$ functions as a strict, non-singular asymptotic past boundary for a single, continuous causal history, dictated by the vanishing of the relative conformal clock rate.

The physical interpretation is that the limit $A_{\rm clock}\to0$ is a *temporal horizon*: curvature invariants do not merely remain bounded—they vanish—so the boundary is asymptotically curvature-empty. The local clock rate vanishes relative to the present epoch, but the spatial geometry does not develop pathological curvature. The analogy is not a collapsing sphere reaching zero volume; by analogy with the standard GR event-horizon description, it is an observer falling toward an event horizon whose local clock asymptotically freezes. The causal structure remains intact. Just as an observer crossing a black hole event horizon experiences normal local time while appearing perfectly frozen to a distant observer, the extremely early universe experiences regular local thermodynamic evolution while appearing frozen relative to the present epoch. The "start" of the universe is therefore shielded by a relativistic horizon of clock-transport, preventing infinite extrapolation.

## 3.2 Disformal Compensation Analysis

A natural question is whether the disformal sector of the TEP matter metric, $\tilde{g}_{\mu\nu}=A^{2}g_{\mu\nu}+B(\phi)\nabla_{\mu}\phi\nabla_{\nu}\phi$, can prevent the metric degeneracy at the temporal horizon. The determinant of the full disformal metric is

\begin{equation} \label{eq:disformal_det}
\det(\tilde{g}_{\mu\nu}) = A^{8}\,\det(g_{\mu\nu})\left(1 + \frac{B\,(\nabla\phi)^{2}}{A^{2}}\right) \, ,
\end{equation}

The contraction is signed. On the homogeneous background $(\nabla\phi)^2=-(\phi')^2$, so the determinant factor is $1-B(\phi')^2/A^2$. For the declared product ansatz $|B(\phi')^2|=C_B A^{-2n}$, the negative-$B$ branch has

\begin{equation}
\tilde N^2=A^2+C_B A^{-2n},\qquad
\det\tilde g=-\tilde N^2A^6=-A^8-C_B A^{6-2n}.
\end{equation}

Thus the coordinate determinant tends to zero for $n<3$, to $-C_B$ for $n=3$, and diverges for $n>3$. A finite determinant is not sufficient to establish a smooth extension of all metric components at the boundary. Step 01 evaluates the same lapse and spatial factor in both the determinant and curvature calculation, for the existing powers $n\in\{2,3,4,5,6,10\}$.

With $A=\eta^{-p}$, write $\mathcal H=A'/A=-p/\eta$ and $H=\mathcal H/\tilde N$. The invariant curvature of $d\tilde s^2=-\tilde N^2d\eta^2+A^2d\mathbf x^2$ is

\begin{equation}
\tilde R=\frac{6}{\tilde N^2}\left(\mathcal H'+2\mathcal H^2-\mathcal H\frac{\tilde N'}{\tilde N}\right).
\end{equation}

On the negative-$B$ branch, $\tilde N'/\tilde N\sim np/\eta$, giving $\tilde R\sim6[p+(n+2)p^2]C_B^{-1}\eta^{-2-2np}\to0$ for $n>0$. This is a curvature invariant, not a coordinate artifact. The lapse remains positive at every finite $\eta$; its divergence at the asymptotic boundary alone does not prove a physical pathology. For the tested $n\ge2$, comoving proper time $\int\tilde N\,d\eta$ and null affine length $\int A\tilde N\,d\eta$ both diverge at the boundary. These metric-level facts do not derive $B(\phi)$ from the common action or establish its observational viability.

Positive $B$ instead gives $\tilde N^2=A^2-C_B A^{-2n}$, with a signature-change surface $A_\times=C_B^{1/(2n+2)}$. A Lorentzian present reference at $A=1$ requires $C_B<1$; the growing ansatz then encounters the surface before $A\to0$. The two signs therefore have different geometric verdicts. The pure conformal branch remains the branch used for this paper's main regularity and completeness theorems; neither the negative-$B$ comparison nor its rejection is needed for those results.

The pipeline confirms this analysis. In pure conformal closure ($B=0$), Proposition 1 is verified: $|\tilde{R}|\approx3.36\times10^{-18}$ and $|\tilde{K}|\approx2.23\times10^{-36}$ at $z=10^{6}$ for $p=0.4$ in the normalized profile, and the $p$-sensitivity test confirms vanishing for all $0<p\le\tfrac12$. The disformal test reproduces the exact metric determinant, whose finite-magnitude exponent is $n=3$, and the bounded curvature of its negative-$B$ comparison branch. Neither test promotes that assigned coupling to the common action. The metric degeneracy in pure conformal closure is therefore not a pathology to be cured by the disformal sector; it is the defining feature of the conformal boundary, exactly as in Penrose's conformal compactification of null infinity.

# 4. Geodesic Completeness

A key test for physical singularities is geodesic incompleteness. Null and timelike geodesics are integrated backward from the present epoch toward the temporal horizon. For the temporal-horizon profile $A_{\rm clock}(\eta)=C\eta^{-p}$ with $0 \lt p\le\tfrac12$, both null and timelike geodesics are complete: the null affine parameter diverges and the timelike proper time diverges. Comoving timelike observers accumulate infinite proper time for $0 \lt p\le1$; timelike geodesics with nonzero conserved spatial momentum require $p\le\tfrac12$, the same bound as null completeness. The boundary is curvature-empty in this branch.

Here $\eta$ is the temporal-horizon conformal coordinate, oriented so that approach to $\mathscr{T}^{-}$ corresponds to the asymptotic limit in which $A_{\rm clock}\to0$. It is not the standard FLRW conformal time coordinate extrapolated to $a=0$.

## 4.1 Conformal Christoffel Symbols

Under the conformal transformation $\tilde{g}_{\mu\nu}=A_{\rm clock}^{2}\,g_{\mu\nu}$, the Christoffel symbols transform as

\begin{equation} \label{eq:christoffel}
\tilde{\Gamma}^{\mu}{}_{\alpha\beta}
= \Gamma^{\mu}{}_{\alpha\beta}
+ \delta^{\mu}_{\alpha}\,\partial_{\beta}\ln A_{\rm clock}
+ \delta^{\mu}_{\beta}\,\partial_{\alpha}\ln A_{\rm clock}
- g_{\alpha\beta}\,g^{\mu\nu}\,\partial_{\nu}\ln A_{\rm clock} \, .
\end{equation}

For the static conformal background $g_{\mu\nu}$ with $\partial_{i}A_{\rm clock}=0$ and $A_{\rm clock}=A_{\rm clock}(t)$, the nonzero modified symbols are

\begin{equation}
\tilde{\Gamma}^{0}{}_{00} = \frac{\dot{A}_{\rm clock}}{A_{\rm clock}} \, , \qquad
\tilde{\Gamma}^{0}{}_{ij} = \frac{\dot{A}_{\rm clock}}{A_{\rm clock}}\,\delta_{ij} \, , \qquad
\tilde{\Gamma}^{i}{}_{0j} = \frac{\dot{A}_{\rm clock}}{A_{\rm clock}}\,\delta^{i}_{j} \, ,
\end{equation}

where overdots denote derivatives with respect to the conformal time of the background. These corrections encode how the varying clock rate modifies geodesic acceleration.

## 4.2 Null Geodesics and Affine Length

For radial null geodesics on the static gravitational background ($a_g=1$, with matter metric $\tilde g=A_{\rm clock}^2g$), the geodesic equation with the corrected Christoffels gives

\begin{equation} \label{eq:null_geodesics}
\frac{d\eta}{d\tilde{\lambda}} = \frac{1}{A_{\rm clock}^{2}(\eta)} \, , \qquad
\frac{dr}{d\tilde{\lambda}} = \frac{\pm 1}{A_{\rm clock}^{2}(\eta)} \, ,
\end{equation}

where $\tilde{\lambda}$ is the affine parameter in the matter frame. A standard result for conformally related metrics is that if $k^{\mu}=dx^{\mu}/d\lambda_{0}$ is an affinely parametrized null geodesic of the background $g_{\mu\nu}$, then the same path is an affinely parametrized null geodesic of $\tilde{g}_{\mu\nu}=A_{\rm clock}^{2}g_{\mu\nu}$ with parameter $\tilde{\lambda}$ obeying $d\tilde{\lambda}=A_{\rm clock}^{2}\,d\lambda_{0}$ (in four dimensions). The background is static and nonsingular, so $\lambda_{0}$ ranges over $(-\infty,\infty)$. The matter-frame affine parameter to any background coordinate time $t$ is therefore

\begin{equation} \label{eq:affine_null}
\tilde{\lambda}(t) = \int^{t} A_{\rm clock}^{2}(\lambda_{0})\,d\lambda_{0}
= \int^{t} A_{\rm clock}^{2}(\eta)\,d\eta \, ,
\end{equation}

where $\eta$ is the background coordinate time. For the temporal-horizon profile $A_{\rm clock}(\eta)=C\eta^{-p}$ with $0 \lt p\lt1$ as $\eta\to\infty$ (early times), the integrand behaves as $\eta^{-2p}$. The integral $\int^{\infty}\eta^{-2p}\,d\eta$ diverges for $2p\le 1$, i.e. for $0 \lt p\le\tfrac12$. Null geodesics do not terminate in this branch; they extend to $\tilde{\lambda}\to\infty$. For $\tfrac12\lt p \lt 1$ the null affine parameter converges, so null completeness requires $p\le\tfrac12$.

The divergence rates are explicit. Evaluating the integral with a lower cutoff $\eta_{0}\sim\mathcal{O}(1)$,

\begin{equation} \label{eq:null_divergence}
\tilde{\lambda}(\eta)=\int_{\eta_{0}}^{\eta}C^{2}\eta'^{-2p}\,d\eta'=\frac{C^{2}}{1-2p}\Bigl(\eta^{1-2p}-\eta_{0}^{1-2p}\Bigr)\quad(p\neq\tfrac12) \, ,
\end{equation}

so that for $p<\tfrac12$ the null affine parameter diverges as a power law, $\tilde{\lambda}\propto\eta^{1-2p}\to\infty$. At the marginal value $p=\tfrac12$,

\begin{equation} \label{eq:null_log}
\tilde{\lambda}(\eta)=C^{2}\,\ln\!\left(\frac{\eta}{\eta_{0}}\right) \to \infty \quad (p=\tfrac12) \, ,
\end{equation}

diverging only logarithmically. The slower divergence at $p=\tfrac12$ is still sufficient for past-completeness; every finite segment of a null geodesic accumulated since any finite $\eta_{0}$ requires infinite affine parameter to reach $\mathscr{T}^{-}$.

## 4.3 Timelike Geodesics and Proper Time

For timelike geodesics the conformal transformation rescales proper time as $d\tilde{\tau}=A_{\rm clock}\,d\tau$. In the static background a comoving observer has $d\tau=d\eta$, so the total matter-frame proper time from early time $\eta_{i}$ to the present is

\begin{equation} \label{eq:timelike_proper}
\tilde{\tau}_{\rm total} = \int_{\eta_{i}}^{\eta_{0}} A_{\rm clock}(\eta)\,d\eta \, .
\end{equation}

For the temporal-horizon profile $A_{\rm clock}(\eta)=C\eta^{-p}$ with $0 \lt p\lt1$, the integral $\int^{\infty}\eta^{-p}\,d\eta$ *diverges* because $p\le 1$. Comoving proper time to the temporal boundary is therefore *infinite*. This establishes the comoving result; completeness of arbitrary timelike geodesics requires the momentum-dependent calculation below. In TEP, Proposition 1 guarantees that all curvature invariants *vanish* at the temporal horizon for $0 \lt p\lt1$, so the infinite proper time reflects that observers can continue evolving indefinitely while the geometry becomes asymptotically curvature-empty.

The timelike divergence is also explicit:

\begin{equation} \label{eq:timelike_divergence}
\tilde{\tau}(\eta)=\int_{\eta_{0}}^{\eta}C\eta'^{-p}\,d\eta'=\frac{C}{1-p}\Bigl(\eta^{1-p}-\eta_{0}^{1-p}\Bigr)\quad(p\neq1) \, ,
\end{equation}

which diverges as a power law $\tilde{\tau}\propto\eta^{1-p}\to\infty$ for all $0\lt p\lt1$. The comoving bound $p\le1$ is weaker than the bound for general timelike geodesics. With conserved spatial momentum $q_i=A_{\rm clock}^2 dx^i/d\tilde\tau$ and $q^2=\delta^{ij}q_iq_j$, normalization gives $$\frac{d\tilde\tau}{d\eta}=\frac{A_{\rm clock}^2}{\sqrt{A_{\rm clock}^2+q^2}}.$$ For $q=0$ this reduces to $A_{\rm clock}$. For any fixed $q\neq0$, the asymptotic integrand is $A_{\rm clock}^2/|q|\propto\eta^{-2p}$, so arbitrary timelike geodesics are complete precisely for $p\le\tfrac12$. The adopted complete branch therefore satisfies both the timelike and null tests. The geodesic equations in the matter frame are regular everywhere:

\begin{equation} \label{eq:timelike_geodesics}
\frac{d^{2}x^{\mu}}{d\tilde{\tau}^{2}} + \tilde{\Gamma}^{\mu}{}_{\alpha\beta}\frac{dx^{\alpha}}{d\tilde{\tau}}\frac{dx^{\beta}}{d\tilde{\tau}} = 0 \, ,
\end{equation}

with $\tilde{\Gamma}$ given by $\eqref{eq:christoffel}$. Since $\tilde{\Gamma}\sim A_{\rm clock}^{-1}\partial A_{\rm clock}$ is bounded and the four-velocity is normalized, the equations admit smooth solutions that approach the horizon asymptotically. Step 02 integrates the null affine length and momentum-dependent timelike proper time in $\ln\eta$, checking the finite segments against the closed-form null and comoving integrals and independent timelike quadrature. The controls include $p=\tfrac12$, where affine length grows logarithmically, and $p>\tfrac12$, where null and moving-observer proper time converge. Infinite-end completeness is assigned by the analytic integrability criterion above, rather than by a numerical endpoint or an imposed small-value threshold.

## 4.4 Geodesic Completeness Theorem

**Theorem (Temporal-horizon geodesic completeness).** Let the conformal clock field take the profile $A_{\rm clock}(\eta)=C\eta^{-p}$ with $0 \lt p\le\tfrac12$. Then:

- Comoving proper time diverges for $0 \lt p\le1$; all timelike geodesics, including those with nonzero spatial momentum, have divergent proper time for $0 \lt p\le\tfrac12$.

- Null affine parameter to the temporal boundary diverges for $0 \lt p\le\tfrac12$.

- All polynomial curvature invariants vanish at the boundary for $0 \lt p\lt1$, hence in particular for $0 \lt p\le\tfrac12$.

- The tensor source term $A_{\rm clock}''/A_{\rm clock}=p(p+1)/\eta^{2}\to 0$ at the boundary.

The cleanest fully complete temporal-horizon branch is therefore $0 \lt p\le\tfrac12$. In this branch the temporal horizon is simultaneously curvature-empty, timelike-complete, and null-complete.

Proposition 1 establishes regularity for the temporal-conformal branch satisfying $0 \lt p\le\tfrac12$. The temporal-horizon exponent $p$ and the observational clock map are independent boundary conditions: $A_{\rm clock}(z)=(1+z)^{-1}$ is fixed by the redshift definition, while the regularity condition $0 \lt p\le\tfrac12$ is a mathematical requirement for curvature-regularity at the conformal boundary. The profile $A_{\rm clock}\sim\eta^{-p}$ with $p=1$ reproduces a familiar FLRW scaling — the radiation-era clock map $a\propto\eta$ — but that branch ($p=1$) lies outside the curvature-regular window. Within TEP, the regular branch is selected by the requirement that the matter-frame geometry be curvature-regular at the temporal horizon, not by the observational redshift mapping alone.

Three structural points sharpen the status of the branch index $p$. First, the power-law completeness window $p\le\tfrac12$ can also be expressed as $|\mathcal H_{\rm eff}'|/\mathcal H_{\rm eff}^{2}\ge2$, or as $|\dot H_{\rm clock}|/H_{\rm clock}^{2}\ge1$ using matter proper time (Section 10.2b). These equivalent geometric bounds do not establish a kinetic-dominated equation of state or independently select the background. Second, $p$ is the family index of the reconstructed single-clock action of Section 10.2: each member $(\mathcal Z_p, U_p)$ is reconstructed around its own exponent and realizes that power law exactly, so the theorem is a property of every member of the branch rather than of a single fitted trajectory; which member the corpus's canonical closure dynamics realizes is the same open asymptotic determination as the master-action embedding identified in Section 2 (the branch-index datum). Third, the index is observable in principle through the tensor diagnostic surface of Section 10.6: along the one-parameter fiducial slice ($n=4$, $\eta_0=14{,}000$ Mpc), the Step 09b computation returns $r_{\rm max}(p)\approx 6.1\times10^{-5}\,p^{2}$ at the transition scale ($\le 1.4\times10^{-5}$ for $p\le\tfrac12$, while joint variations such as $p=0.5$, $n=2$, $\eta_0=3000$ Mpc reach $1.14\times10^{-4}$), so the completeness claim is branch-robust and primordial tensor observations at $k\sim2\times10^{-4}\ {\rm Mpc}^{-1}$ constrain the joint transition-profile parameters $(p, n, \eta_0)$ once the tensor action normalization is fixed.

## 4.5 Conformal Compactification and Penrose-Style Diagram

To illustrate the temporal boundary in the language of standard conformal diagrams, a Penrose-style compactified conformal diagram is constructed for the TEP matter-frame spacetime and compared directly with the standard flat $\Lambda$CDM diagram. The metric in conformal coordinates is

\begin{equation} \label{eq:metric_conformal}
d\tilde{s}^{2} = a_{\rm eff}^{2}(\eta)\left(-d\eta^{2} + dr^{2} + r^{2}\,d\Omega^{2}\right) \, .
\end{equation}

In both the standard and TEP conformal diagrams, time and space are compactified independently so that infinite spacetime maps to a finite rectangle. Let $\eta_{0}$ denote the present-epoch conformal time. For the *standard* flat $\Lambda$CDM cosmology, conformal time runs from $\eta=0$ (Big Bang) to $\eta=\eta_{0}$ (present), and the compactification

\begin{equation} \label{eq:compact_coords_lcdm}
T_{\Lambda} = \arctan\!\left(\frac{\eta}{\eta_{0}}\right) \, , \qquad R = \arctan\!\left(\frac{r}{\eta_{0}}\right) \, ,
\end{equation}

maps $\eta\in[0,\eta_{0}]$ and $r\in[0,\infty)$ to the rectangle $0\le T_{\Lambda}\le\pi/4$, $0\le R\lt\pi/2$. The bottom edge $T_{\Lambda}=0$ is the physical singularity: a spacelike boundary where $a\to0$ and all curvature invariants diverge. Every causal curve terminates there in finite proper time.

For the *TEP* temporal-horizon cosmology, conformal time runs from $\eta=\eta_{0}$ (present) to $\eta\to\infty$ (temporal horizon), and the compactification

\begin{equation} \label{eq:compact_coords_tep}
T = \arctan\!\left(\frac{\eta_{0}}{\eta}\right) \, , \qquad R = \arctan\!\left(\frac{r}{\eta_{0}}\right) \, ,
\end{equation}

maps $\eta\in[\eta_{0},\infty)$ and $r\in[0,\infty)$ to the same rectangle $0\le T\le\pi/4$, $0\le R\lt\pi/2$. The bottom edge $T=0$ is the temporal horizon $\mathscr{T}^{-}$: a *regular* spacelike boundary where $A_{\rm clock}\to0$ and all curvature invariants vanish (Proposition 1). Null geodesics approach it with infinite affine parameter; timelike curves approach it with infinite proper time.

The compactified causal layout is analogous, while the physical content of the lower boundary differs radically.

**Figure 1.** Penrose-style schematic compactified conformal diagrams for (a) standard flat ΛCDM and (b) the TEP temporal-horizon branch. Both diagrams show a compactified causal slab; the upper edge is the present epoch, not future infinity. **Panel (a):** the lower spacelike boundary is the Big Bang singularity, where $a\to0$, curvature diverges ($\tilde{\mathcal{K}}\to\infty$), and past-directed causal curves terminate. **Panel (b):** the corresponding lower boundary is a smooth regular temporal conformal horizon $\mathscr{T}^{-}$, where $A_{\rm clock}\to0$ and curvature invariants vanish ($\tilde{\mathcal{K}}\to0$). Null geodesics are dashed red and timelike curves solid blue. In the TEP panel, the lower boundary is approached only asymptotically: null curves in infinite affine parameter and timelike curves in infinite proper time. Tick marks on the timelike curve denote equal proper-time intervals compressed by the compactification. The compactified causal layout is analogous, but the physical content of the lower boundary differs radically.

The TEP diagram (Panel b) is a rectangle with horizontal top and bottom edges and vertical left and right edges. In this schematic conformal representation, null geodesics are drawn at $45°$ to emphasize the causal structure; a fully metric-derived Penrose compactification would require constructing compactified null coordinates explicitly from the conformal metric. In a generic compactified coordinate representation their apparent slope can vary with position, but they remain causal and approach the spacelike boundary $\mathscr{T}^{-}$ only asymptotically. The boundary of this rectangle comprises:

- **$\mathscr{T}^{-}$ (temporal past infinity):** the horizontal line $T=0$ ($0\le R\lt\pi/2$), corresponding to $\eta\to\infty$ at all finite $r$. This is the TEP temporal horizon. It is *not* a curvature singularity; all polynomial invariants vanish there (Proposition 1), and the conformal factor $a_{\rm eff}\to 0$ merely reflects the vanishing of the relative clock rate.

- **Present epoch:** the horizontal line $T=\arctan(1)=\pi/4$ ($0\le R\lt\pi/2$), corresponding to $\eta=\eta_{0}$ at all finite $r$. This is the observer's present slice.

- **$r=0$ (spatial origin):** the vertical line $R=0$ ($0\le T\le\pi/4$), the symmetry axis of the radial coordinate.

- **$i^{0}$ (spatial infinity):** the vertical line $R\to\pi/2$ ($0\le T\le\pi/4$), approached along spacelike curves with $r\to\infty$.

- **$i^{+}$ (future timelike infinity):** the corner $(T,R)=(\pi/4,0)$, where comoving timelike geodesics reach the present epoch.

Crucially, *every* inextendible causal curve in the TEP diagram has its past limit on $\mathscr{T}^{-}$ and its future limit on the present-epoch boundary; there is no singularity anywhere in the diagram.

**Comparison with standard cosmology.** Figure 1 makes the distinction immediate. In the standard diagram (Panel a), the past boundary is a *spacelike singularity* (the Big Bang) where all curvature invariants diverge and every causal curve terminates in finite proper time. In the TEP diagram (Panel b), the past boundary is $\mathscr{T}^{-}$, a *regular conformal-temporal boundary* where curvature invariants vanish; null geodesics have infinite affine parameter, while timelike curves have infinite proper time. The TEP diagram shares the same compactification topology as the conformal diagram of global de Sitter space: both have a smooth horizontal spacelike past boundary, a smooth horizontal future boundary, and vertical spatial-infinity edges. The compactified causal layout is topologically analogous; the physical mechanism differs—de Sitter's past boundary is $\mathscr{I}^{-}$, an asymptotic surface of an exponentially expanding congruence, whereas TEP's past boundary is $\mathscr{T}^{-}$, the limit where the relative conformal clock rate vanishes on a nonexpanding static background.

**Boundary-regularity interpretation.** For the temporal-horizon profile $A_{\rm clock}(\eta)=C\eta^{-p}$ with $0 \lt p\le\tfrac12$:

- The conformal compactification $(\tilde{M},\tilde{g}_{\mu\nu})$ with conformal boundary $\partial\tilde{M}=\mathscr{T}^{-}\cup\{\text{present}\}\cup i^{0}\cup i^{+}$ is smooth.

- The Weyl tensor vanishes on $\mathscr{T}^{-}$ (conformally flat boundary).

- All inextendible causal curves have past endpoint on the spacelike boundary $\mathscr{T}^{-}$ and future endpoint on the present-epoch surface or future null infinity.

- There exists no trapped surface or incomplete geodesic; the framework complies with the mathematical constraints of the Hawking-Penrose singularity theorems by violating the Strong Energy Condition (Section 5).

The analytic curvature and geodesic results place $\mathscr{T}^{-}$ on the footing of a regular spacelike conformal boundary; Figure 1 gives a Penrose-style schematic representation of that boundary structure. It is a boundary of the manifold, not a singularity within it.

## 4.6 Implications for Quantum Cosmology

The regularity of the temporal conformal boundary $\mathscr{T}^{-}$ removes the standard motivation for a quantum gravitational "beginning" at finite proper time. In standard Big Bang cosmology, the curvature singularity at $a\to0$ forces a breakdown of the classical theory at a finite past, providing the principal physical justification for quantum-cosmological models that attempt to describe a first moment of time. Because TEP's temporal horizon is curvature-empty and timelike-complete, the classical geometry remains valid arbitrarily far into the past; there is no finite-time singularity that demands quantum replacement.

This opens the possibility of a well-defined wavefunction on the regular boundary $\mathscr{T}^{-}$, formulated as a boundary condition on a smooth conformal compactification rather than as an initial condition at a singular point. The boundary data would describe correlations across the temporal horizon, analogous to how boundary-value problems are posed on smooth spatial or null boundaries in standard field theory. Whether such a boundary quantum state reproduces the observed classical late-time cosmology is a well-posed mathematical question; it does not require a speculative resolution of a curvature singularity that does not exist in the TEP matter frame.

# 5. Effective Stress-Energy and Hawking-Penrose Consistency

The Hawking-Penrose singularity theorems establish that a singularity is mathematically inevitable if the Strong Energy Condition (SEC) holds: $\rho + 3p \geq 0$ and $\rho + p \geq 0$. The theorems require the SEC as an explicit assumption; if the SEC is violated, their conclusion of inevitable geodesic incompleteness no longer applies. The effective stress-energy tensor of the temporal field is computed to determine whether the SEC is violated in the temporal-horizon geometry.

## 5.1 Stress-Energy from the Temporal-Field Action

The temporal field $A$ enters the matter-frame metric through the conformal factor $A_{\rm clock}$. Its stress-energy tensor follows from the canonical scalar-field action with a field-dependent kinetic function $Z(A)$:

\begin{equation} \label{eq:T_munu}
T_{\mu\nu}(A) = Z(A)\,\partial_{\mu}A\,\partial_{\nu}A - g_{\mu\nu}\Bigl[\tfrac{1}{2}Z(A)(\partial A)^{2} + V(A)\Bigr] \, .
\end{equation}

For a homogeneous temporal field $A(t)$ in a FLRW background, the effective energy density and pressure are

\begin{equation} \label{eq:rho_p_phi}
\rho_{A} = \tfrac{1}{2}Z(A)\,\dot{A}^{2} + V(A) \, , \qquad p_{A} = \tfrac{1}{2}Z(A)\,\dot{A}^{2} - V(A) \, .
\end{equation}

In the slow-clock limit the kinetic term is negligible relative to the potential, $\tfrac{1}{2}Z\dot{A}^{2}\ll V(A)$, so the equation of state parameter approaches

\begin{equation} \label{eq:w_phi}
w_{A} = \frac{p_{A}}{\rho_{A}} \simeq -1 \, .
\end{equation}

This is the same effective equation of state as a cosmological constant. The numerical Step 03 diagnostic adopts the following effective-fluid density prescription for the clock map $A_{\rm clock}(z)=(1+z)^{-1}$:

\begin{equation} \label{eq:rho_phi}
\rho_{\phi} \approx \frac{\epsilon_{\rm dyn}}{8\pi G}\,H^{2}\left(\frac{d\ln a_{\rm eff}}{d\ln A_{\rm clock}}\right)^{2}
\end{equation}

where the dimensionless factor $(d\ln a_{\rm eff}/d\ln A_{\rm clock})^{2}$ encodes the coupling between the temporal field and the effective scale factor. With $a_{\rm eff}=A_{\rm clock}\,a_{\rm m}$ and $a_g=1$ in the static matter frame, $d\ln a_{\rm eff}/d\ln A_{\rm clock}=1$, while the reconstructed action of Section 10 has the late-time limit

\begin{equation}
\rho_{A}(\eta_{0}) = \tfrac12\mathcal Z(\chi_{0})\,\dot\chi_{0}^{2} + U(\chi_{0}) \approx U_{\rm c} \, ,
\end{equation}

when the kinetic and nonconstant potential terms have decayed. Identifying the benchmark density with this asymptotic plateau would impose the normalization

\begin{equation} \label{eq:epsilon_dyn}
\epsilon_{\rm dyn} = \frac{8\pi G\,U_{\rm c}}{H_{0}^{2}} \, .
\end{equation}

The exact effective-action energy-condition identities are stronger than an assigned equation-of-state curve. Writing $K=\tfrac12\mathcal Z\dot\chi^2\ge0$, one has $\rho=K+U$, $\rho+p=2K$ and $\rho-p=2U$. Thus NEC, WEC and DEC hold wherever $U\ge0$, while SEC is violated whenever $U>2K$. On the reconstructed branch $K\to0$ and $U\to U_{\rm c}>0$, so SEC violation follows asymptotically without fitting a fluid pressure. Step 03's numerical curve instead uses the declared prescription $w(z)=-1+0.1/(1+z)=-0.90-0.10(1-a)$ ($w_0=-0.90$, $w_a=-0.10$, lying in the thawing $w_0>-1,w_a<0$ quadrant); its finite-redshift energy-condition table is an illustrative fluid diagnostic, not an evaluation of $\mathcal Z(\chi)$ and $U(\chi)$ along the action trajectory. Furthermore, when the zero-$\Lambda$ TEP-C0 M1 temporal-shear transport kernel $d\chi/dz=(c/H_0)[1+\epsilon_{\rm shear}^{\rm los}\ln(1+z)\exp(-(z/z_T)^2)]/(1+z)^{3/2}$ ($\epsilon_{\rm shear}^{\rm los}=0.8708$, $z_T=5.0$, Paper 26) is reconstructed as an effective FLRW dark-energy fluid $E_{\rm eff}^2(z)=(1+z)^3/\gamma(z)^2=\Omega_{m,0}(1+z)^3+(1-\Omega_{m,0})f_{\rm DE}(z)$ alongside the zero-parameter TEP-HC acoustic matter fraction $\Omega_{m,0}=\omega_{m,\rm eff}/h^2=0.3038$ (or Planck $\Omega_{m,0}=0.315$), the effective equation of state $w_{\rm eff}(z)=-1+\tfrac{1+z}{3}d\ln f_{\rm DE}/dz$ yields $w_{\rm eff}(0)=-0.834$ to $-0.848$ and a linear CPL projection over the DESI BAO range $z\in[0,1]$ of $(w_0=-0.71,\,w_a=-0.87)$ at $\Omega_{m,0}=0.304$ and $(w_0=-0.69,\,w_a=-1.13)$ at $\Omega_{m,0}=0.315$ (Step 03), reproducing the exact $(w_0>-1,\,w_a<0)$ thawing dark-energy evolution reported by DESI Year-1 BAO + CMB + DESY5 SNe ($w_0=-0.727\pm0.067$, $w_a=-1.05^{+0.31}_{-0.27}$; DESI Collaboration 2024) within $0.28\sigma$–$0.61\sigma$ in $w_0$ and $0.27\sigma$–$0.61\sigma$ in $w_a$ without a primitive dark-energy fluid. On the single-history native background that fits Pantheon+, DESI DR2 BAO and the Planck acoustic angle together (TEP-C0 Step 03-14, $\Delta\chi^2=-8.1$ against $\Lambda$CDM), the apparent equation of state rises to $w_{\rm eff}\approx-0.86$ near $z\approx0.4$ and returns to $-1$ by $z\approx1.5$, with CPL projection $(w_0,w_a)\approx(-0.83,-0.44)$, in DESI's thawing quadrant, produced by the time field without a primitive dark-energy fluid.

At the Step 03 benchmark $\epsilon_{\rm dyn}=5.5\times10^{-3}$, imposing $\eqref{eq:epsilon_dyn}$ would give

\begin{equation} \label{eq:Uc_value}
\frac{U_{\rm c}}{\rho_{\rm crit,0}} = \frac{\epsilon_{\rm dyn}}{3} \approx 1.8\times10^{-3} \, ,
\end{equation}

recorded as a conditional normalization in Step 03. The benchmark is distinct from the native TEP-HC likelihood parameter $\epsilon_T^{\rm HC}$, whose full high-$\ell$ result is $0.0043\pm0.0025$: that amplitude is not a measurement of $U_{\rm c}$ without an action-level mapping. Likewise HC's operational $\Omega_\phi$ background budget is not fixed by the small illustrative plateau. The absence of a bare geometric $\Lambda$ is a structural choice; accounting for the full observed background budget with the common action requires the embedding identified in Section 2. The benchmark alone establishes neither a universal vacuum-energy bound nor the physical origin of the entire background budget.

The pipeline evaluates the NEC, WEC, DEC, and SEC for its specified effective-fluid diagnostic across redshift. The SEC is systematically violated, while the NEC and WEC are satisfied. By violating the SEC, the temporal field removes one of the assumptions required by the Hawking-Penrose singularity theorems; the theorems therefore do not force geodesic incompleteness. Geodesic completeness is established separately by the curvature and affine-parameter analysis of Sections 4.2–4.4.

## 5.2 Past-Completeness and the BGV Theorem

The Borde-Guth-Vilenkin theorem is kinematical and does not assume an energy condition. It establishes past incompleteness when the relevant averaged expansion along a past-directed geodesic is positive. TEP does not rely on an energy-condition loophole. Past completeness is established directly through the divergent timelike proper-time and null affine-parameter integrals derived in Sections 4.2–4.4.

The theorem applies to the timelike geodesic congruence of the spacetime whose completeness is in question. In the two-metric setting that spacetime is the matter metric $\tilde{g}_{\mu\nu}=A_{\rm clock}^2\,g_{\mu\nu}$ — the geometry on which particles and clocks propagate — so the operative evaluation is on the matter-frame congruence, not the coordinate congruence of $g_{\mu\nu}$ (for completeness it may be noted that the static coordinate metric also has vanishing expansion, $\theta_g = 3\dot a_g/a_g = 0$ with $a_g=1$, but the coordinate congruence is not the congruence to which the physical completeness question attaches).

Evaluated directly on the physical matter-metric congruence, the Hubble expansion parameter along a timelike congruence parameterized by proper time $\tilde{\tau}$ is $\tilde{H} = d\ln A_{\rm clock}/d\tilde{\tau}$. The BGV past-averaged expansion rate over a proper-time interval $\Delta\tilde{\tau} = \tilde{\tau}_0 - \tilde{\tau}_i$ is defined by

\begin{equation}
H_{\rm av} = \frac{1}{\Delta\tilde{\tau}} \int_{\tilde{\tau}_i}^{\tilde{\tau}_0} \tilde{H}(\tilde{\tau}')\,d\tilde{\tau}' = \frac{\Delta\ln A_{\rm clock}}{\Delta\tilde{\tau}} \, .
\end{equation}

As established in Section 4.3, for the adopted regular branch $0 \lt p \le 1/2$, the past proper-time integral diverges toward the temporal horizon: $\Delta\tilde{\tau} \to \infty$ as $\eta \to \infty$. Meanwhile, the accumulated logarithmic clock shift $\Delta\ln A_{\rm clock} = p\ln(\eta/\eta_0)$ grows only logarithmically, far slower than proper time ($\Delta\tilde{\tau} \sim \eta^{1-p}$ for $p \lt 1$). Consequently, the asymptotic past-averaged expansion rate along this comoving matter congruence tends to zero:

\begin{equation}
H_{\rm av} = \lim_{\Delta\tilde{\tau} \to \infty} \frac{p\ln(\eta/\eta_0)}{\Delta\tilde{\tau}} = 0 \, .
\end{equation}

The essential BGV hypothesis $H_{\rm av} > 0$ therefore fails by construction in the matter frame, just as $\theta_g = 0$ in the coordinate frame. This is not an evasion based on frame choice; it is a structural consequence of clock transport. The cosmological redshift is generated by clock-rate evolution rather than kinematic divergence of geodesics, The comoving congruence has the vanishing average evaluated above; arbitrary timelike geodesics and null geodesics are past complete on $p\le1/2$ by the independent integrals in Section 4. The BGV conclusion of past geodesic incompleteness is therefore avoided in both metric representations.

The explicit kinetic function $\mathcal Z(\chi)$ and reconstructed potential $U(\chi)$ used for the temporal-horizon perturbation sector are derived in Section 10 from the observed scalar spectrum and the background Euler-Lagrange equation. The generic stress-energy framework above is therefore connected to the specific action-level closure of the primordial perturbation derivation.

# 6. Temporal-Horizon Thermal History and Elemental Abundances

Standard cosmology attributes light-element synthesis (Helium-4, Deuterium, Lithium-7) to a primordial hot dense phase emerging from a zero-volume singularity. In the TEP framework, the physical high-temperature thermal state at high redshift ($T(\eta) = T_0 / A_{\rm clock}(\eta) = T_0 (1+z)$) is retained as a physical matter-frame state, which is operationally required for the recombination and photon-thermalization physics analysed in Sections 7 and 8. What is rejected is not the hot ionized plasma state itself, but its interpretation as a singular primordial beginning of the universe.

Because the underlying gravitational spacetime is static and eternal, the high-redshift plasma is bounded by an asymptotic temporal horizon ($A_{\rm clock} \to 0$) rather than a zero-volume spatial collapse. In this architecture, elemental abundances can also be investigated through candidate long-term asymptotic equilibria of stellar burning and galactic baryon recycling. Under this complementary pathway, the observed Helium mass fraction ($Y_p \approx 0.249$) and metallicity ($Z \approx 10^{-4}$ to $10^{-3}$) are examined as balancing points between production, astration, and galactic infall, governed by proper-time reaction flows.

The quantitative mechanism for sustained non-primordial Deuterium production and steady-state chemical equilibrium is investigated in detail in TEP-BBN (Paper 29), where it remains an active research problem. TEP-TH retains the hot matter-frame thermal environment as an exact feature of the temporal landscape without requiring a singular origin of spacetime.

# 7. Recombination and Visibility Function

Recombination and CMB last scattering are critical early-universe observables. In standard cosmology these are identified with a specific epoch when the universe cooled enough for electrons and protons to combine. In the TEP temporal-horizon cosmology there is no physical zero-volume origin; rather, recombination occurs when the hot plasma cools below the binding energy of hydrogen as the temporal field evolves. The ionization fraction is verified $x_e(z)$, visibility function $g(z)$, recombination redshift $z_*$, sound horizon $r_s$, drag horizon $r_d$, and angular scale $\theta_s$.

## 7.1 Recombination in the Canonical Conformal Branch

Earlier exploratory versions introduced a phenomenological epoch-screening function to suppress the dynamical temporal response during recombination. That construction is superseded in the canonical TEP architecture. The exact clock map $A_{\rm clock}$ and the late-time dynamical response $A_{\rm dyn}$ are distinct projections and no thermal screening function is imposed to recover a standard hot-plasma history. Recombination and acoustic-sector consistency are instead evaluated through the pure-conformal matter-frame mapping and the independent TEP-HC Boltzmann closure.

The ionization fraction is computed using the multi-level non-equilibrium recombination treatment (Peebles 1968; Seager, Sasselov & Scott 1999), with TEP modifications entering only through the Hubble parameter $H_{\rm TEP}(z)$. The Saha equation gives the high-temperature equilibrium initial condition, but the freeze-out tail and recombination dynamics are governed by the full rate equations for hydrogen and helium, including two-photon decays and Lyman-$\alpha$ trapping. The pipeline uses a TEP-adapted RECFAST equivalent that tracks $x_{e}(z)$ through the ionization balance

\begin{equation} \label{eq:peebles}
\frac{dx_{e}}{dt} = -C\,\left[\alpha_{\rm B}(T)\,n_{e}\,x_{p} - \beta(T)\,x_{e}\,e^{-E_{\alpha}/k_{\rm B}T}\right] \, ,
\end{equation}

where $C$ is the Peebles suppression factor, $\alpha_{\rm B}$ is the case-B recombination coefficient, and $\beta$ is the photoionization rate. All rates depend on temperature and density; the only TEP modification is $H_{\rm TEP}$ in the $dt$ conversion and in the density evolution.

The visibility function is derived from the optical depth $\tau(z)$:

\begin{equation} \label{eq:visibility}
g(z) = \frac{d\tau}{dz} e^{-\tau}
\end{equation}

The sound horizon is computed by integrating the sound speed divided by the Hubble parameter from the Big Bang to redshift $z$:

\begin{equation} \label{eq:sound_horizon}
r_s(z) = \int_z^{\infty} \frac{c_s(z')}{H_{\rm TEP}(z')} dz'
\end{equation}

The pipeline computes these quantities for $z\in[0,2000]$. Propagating the temporal geometry through the Peebles recombination treatment yields

- $z_* = 1078.6$ ($-1.0\%$ relative to $\Lambda$CDM $1089.9$)

- $r_s = 146.1$ Mpc ($-0.7\%$ relative to $\Lambda$CDM $147.1$ Mpc)

- $r_d = 145.0$ Mpc ($-1.6\%$ relative to $\Lambda$CDM $147.3$ Mpc)

- Manual $\theta_s = 0.01061$ ($+1.9\%$ relative to Planck $0.01041$)

The manual Peebles calculation has inherent numerical limitations at the $\sim 2\%$ level. The full Cosmic Linear Anisotropy Solving System (CLASS) Boltzmann calculation (Step 10) gives $100\theta_s = 1.0419$, consistent with Planck 2018 at the $0.09\%$ level.

**Relation to the eternal-native surface.** The same last-scattering surface is computed eternal-natively in TEP-BBN (Steps 14–19): there the $\tau = 1$ boundary at $z \approx 1080$ is a spatial ionization front — the Strömgren edge of the deep thermalized zone, where the ambient field's local temperature crosses the hydrogen ionization threshold — closing a neutral foreground trough, rather than a temporal freeze-out sweep of a primordial plasma. The kernel-level comparison (TEP-BBN results/step_19_eternal_surface_reconciliation.json) finds the two descriptions agree on the integrated column — $\tau = 1$ at $z \simeq 1079$–$1080$ ($0.13\%$ peak agreement) with $\tau_{\rm reion} \simeq 0.054$ — while differing in morphology: the eternal kernel is a sharp front column carrying $\sim 88\%$ of the normalized column plus diffuse wing/trough weight, against the broad Peebles bump ($\mathrm{FWHM}_z \sim 10^{2}$) computed here. The column bookkeeping is quantified by the path-measure Jacobian $J(z) \sim 4\times10^{7}$ at the edge: pointwise equivalence of the two columns would place the eternal surface at $\sim 30\,n_{b,0}$ — absorber/diffuse densities rather than cosmic-recombination densities — so the Peebles kernel above is the FLRW reconstruction of the observable column, not the physical eternal profile. The eternal-model diffusion blur is also now computed rather than estimated: the front column's scattering depth random-walks to $\theta_D \simeq 8\times10^{-4}$ rad, within $\sim 25\%$ of the observed Silk angle, leaving the boundary's native perturbation mode spectrum as the discriminating residual item.

# 8. CMB Blackbody Origin and Spectral Distortion

The CMB blackbody spectrum is a key test of early-universe thermal history. The TEP temporal-horizon cosmology must preserve a Planckian spectrum without generating forbidden spectral distortions. The temporal-horizon thermal scaling is verified to produce a FIRAS-compatible blackbody spectrum.

The TEP thermal scaling is:

\begin{equation} \label{eq:temp_scaling}
T(z) = T_0 / A_{\rm clock}(z)
\end{equation}

In the conformal frame, blackbody radiation preserves its thermal form because photon number is conserved and the frequency scales as $\nu \to \nu/A_{\rm clock}$. The Planck spectrum transforms as:

\begin{equation} \label{eq:planck}
B_\nu(T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu/k_BT} - 1}
\end{equation}

It is important to distinguish two separate questions. (i) *Instantaneous conformal rescaling*: if the photon distribution is exactly Planckian at some redshift, rescaling $\nu\to\nu/A_{\rm clock}$ and $T\to T/A_{\rm clock}$ leaves it Planckian. This is a trivial mathematical identity. (ii) *Distortion production during thermalization*: the physically observable question is whether the temporal field injects, removes, or redistributes photon energy in a way that creates forbidden deviations from a blackbody during the thermalization epochs $z\gtrsim 10^{6}$.

The standard thermalization hierarchy is: double-Compton and bremsstrahlung create and maintain equilibrium at $z\gtrsim 10^{6}$; Compton scattering shapes the spectrum at $10^{5}\gtrsim z\gtrsim 10^{4}$; photon production becomes inefficient below $z\sim 10^{4}$. In TEP, the conformal clock-rate rescaling applies uniformly to all photon frequencies. There is no frequency-dependent coupling, no non-thermal energy injection, and no preferred scattering process that would create a chemical potential or Compton $y$-parameter. The temporal field rescales the local temperature but does not alter photon number or redistribute energy across the spectrum.

The resulting distortion parameters are bounded by

\begin{equation} \label{eq:mu_bound}
\mu \sim 1.4 \int \frac{dQ/Q_{\gamma}}{dz}\,\mathcal{J}_{\mu}(z) \approx 0 \, ,
\end{equation}

because $dQ=0$ in TEP: the temporal field does not inject or extract energy from the photon fluid; it rescales the clock $A_{\rm clock}$ that measures it. Similarly, the Compton $y$-parameter satisfies $y\approx\int (k_{\rm B}T_{e}/m_{e}c^{2})\,d\tau_{\rm C}\approx 0$ at the level of the standard recombination-era value $y\sim 10^{-8}$, with no additional TEP contribution.

FIRAS constraints are $|\mu| \lt 9 \times 10^{-5}$ and $|y| \lt 1.5 \times 10^{-5}$ at 95% CL. Step 06 fits the 43-channel NASA LAMBDA COBE/FIRAS monopole spectrum (Fixsen et al. 1996; Fixsen & Mather 2002), recovering $T_{\rm FIRAS} = 2.7250 \pm 0.0006$ K, and evaluates the TEP-remapped spectrum against the published FIRAS distortion bounds across multiple redshifts ($z = 100, 500, 1100, 2000$). The implied $\mu$ and $y$ distortions are consistent with zero within FIRAS constraints: $|\mu| \lt 10^{-7}$, $|y| \lt 10^{-8}$. The temporal-horizon thermal mapping preserves a perfect blackbody spectrum without forbidden spectral distortions.

**Photon-number bookkeeping in a static volume.** In the expanding reconstruction, the number density falls as $(1+z)^3$ because the spatial volume grows. No such mechanism is required here, and none is invoked: the photon bath is not a fixed conserved population diluted by volume but an equilibrium field maintained by the thermalization surface. The occupation number of a free photon distribution is conformally invariant along null rays, so the Planckian phase-space density transported from depth $z$ retains its form while its characteristic temperature resamples to $T_0$. The number density of a Planck field is a function of temperature alone, $n_\gamma = 16\pi\zeta(3)(k_{\rm B}T)^3/(hc)^3$, hence $n_\gamma(z)/n_{\gamma,0} = (1+z)^3$ identically — the cubic factor is recovered as a temperature cubed, not as a volume. The same bookkeeping appears in the two complementary projections already established: in the matter frame, particle densities on the effective geometry scale as $\tilde n = n_0\,A_{\rm clock}^{-3}$ while blackbody entropy scales as $\tilde s_\gamma\propto\tilde T^{3}$, so the photon-entropy-per-baryon $\tilde\sigma = \tilde s_\gamma/\tilde n_b$ is exactly constant and equal to the observed $5.9\times10^{9}\,k_{\rm B}$ per baryon (Step 07; a deliberately mismatched dilution profile drifts by $\sim10^{9}$, showing the diagnostic is not vacuous); and in the observed flux sector the clock map suppresses energy and arrival rate by $(1+z)$ each, which is precisely the counting that returns the Tolman $(1+z)^{4}$ surface-brightness law and the exact Etherington relation $d_L = (1+z)^{2}d_A$ of TEP-C0. Photon number is therefore conserved, arrival rates are accounted, and the present-day $n_\gamma \approx 411\,{\rm cm^{-3}}$ is the equilibrium density of a $2.725$ K radiation field, not the residue of a vanished denser bath.

**The excitation-temperature discriminator.** The sharpest empirical test distinguishing a local-temperature law from a transport illusion is the excitation of atomic and molecular species in distant absorbers: rotational level populations of CN, CO and C I in quasar absorption systems directly thermometer the radiation field *at the absorber's own location*. This observable is native to the temporal-horizon construction and requires no transport assumption at all: the local temperature at temporal depth $z$ is $T_{\rm loc}(z) = T_0/A_{\rm clock}(z) = T_0(1+z)$ by the clock map itself, so an excitation thermometer embedded at that depth must read exactly the standard profile. Mechanisms that produce redshift as a propagation effect while leaving the ambient radiation field at a uniform present-day temperature fail this test outright; the temporal-horizon law passes it by construction. Step 11 evaluates the published CO excitation sample directly: six absorbers spanning $z = 0.89$–$2.69$ (Muller et al. 2013; Srianand et al. 2008; Noterdaeme et al. 2010, 2011), anchored by the FIRAS monopole, return $\chi^2/{\rm dof} = 0.15$ against $T = T_0(1+z)$ with maximum single-point pull $0.6\sigma$, and a free-exponent fit $T = T_0(1+z)^{1-\beta}$ yields $\beta = +0.003\pm0.023$, consistent with zero and with the literature constraints $\beta = -0.007\pm0.027$ (Noterdaeme et al.) and $+0.010\pm0.013$ (Klimenko et al. 2020). On the same sample, the constant-temperature alternative has $\chi^2 = 865.8$ versus $\chi^2 = 1.08$ for the $T = T_0(1+z)$ law (descriptive $\sqrt{\Delta\chi^2} = 29.4$) — the absorber thermometers confirm that the radiation field at depth genuinely sits at the elevated local temperature, supporting the tested temperature law. Collisional excitation can only bias $T_{\rm ex}$ above $T_{\rm CMB}$; the bounding $-10\%$ sensitivity check based on Klimenko et al. yields $\chi^2 = 4.72$ and $\beta = +0.041$, preserving the $\beta \simeq 0$ compatibility. The same logic applies to the Sunyaev–Zel'dovich temperature measurements of cluster-embedded gas, which likewise thermometer the ambient field locally at the cluster depth.

The spectral-preservation result is a statement about transport: it establishes that the conformal clock-rate field carries a thermal photon distribution without distortion. The physical supply of that distribution is a separate, already quantified sector of the eternal-universe construction. In the static spatial geometry the absorbing column along a line of sight is unbounded, and the TEP-BBN absorbing-column evaluation (Gate 11) places the unit-optical-depth surface for dust-grade absorptivity at temporal depth $z \approx 130$–$1340$ — a range that brackets the recombination depth $z_* \approx 1079$ computed in Section 7. The same temporal-depth sector therefore hosts both the surface at which distributed starlight is reprocessed into the observed photon bath and the last-scattering plasma: at that depth the local matter temperature $T_{\rm loc} = T_0(1+z_*) \approx 2.9 \times 10^{3}$ K lies at the hydrogen ionization edge, above the dust-sublimation range, so thermalization interior to the dust surface proceeds by the standard plasma processes — free-free absorption, Compton scattering, and the photoionization balance whose electron column is maintained to the required depth by the ionization-equilibrium budget of TEP-BBN (Gate 12). Forward-integrating the homogeneous Saha+Peebles ionization profile $n_e(z)$ across this transition (Steps 05–06) confirms that the foreground remains optically thin throughout the dust window ($\tau_{\rm es} = 8.9\times10^{-4}$ at $z=130$, $n_e = 1.5\times10^{-4}\,{\rm cm^{-3}}$; $\tau_{\rm es} = 0.011$ at $z=500$), crosses unit Thomson optical depth at the hydrogen ionization edge ($\tau_{\rm es} = 0.88$ at $z=1079$, $n_e = 28.2\,{\rm cm^{-3}}$; $\tau_{\rm es} = 1.00$ at $z \approx 1090$; $\tau_{\rm es} = 1.12$ at $z=1100$, $n_e = 36.7\,{\rm cm^{-3}}$), and reaches an optically thick plasma wall interior to the boundary ($\tau_{\rm es} = 22.8$ at $z=1500$, $n_e = 613\,{\rm cm^{-3}}$; $\tau_{\rm es} = 75.3$ at $z=2000$, $n_e = 1.52\times10^{3}\,{\rm cm^{-3}}$). Within this architecture the plasma's thermal history is a real local process: each region evolves under native local thermodynamic evolution $\tilde\nabla_\mu\tilde T^{\mu\nu}=0$ toward balance with its radiation field and sources. The conformal map supplies the transport relation between local thermal states at different temporal depths; it does not create the states. While the homogeneous Saha+Peebles electron-density and optical-depth profile is closed above, the inhomogeneous absorber-statistics clumping inversion posed in TEP-BBN remains a separate refinement.

# 9. Entropy and Arrow of Time

In standard cosmology, tracing the thermodynamic arrow of time backward to a zero-volume singularity forces a breakdown of physical state variables and requires a highly fine-tuned, exceptionally low initial entropy state. In the TEP temporal-horizon cosmology, the past boundary is an asymptotic limit of the relative clock rate, not a physical spatial collapse. Therefore, local thermodynamic state variables must remain finite, and the arrow of time must flow smoothly from an infinite affine past. Entropy is tracked in the temporal-horizon reconstruction to verify this thermodynamic regularity.

The radiation entropy density scales as $s_\gamma \propto T^3$. In the TEP matter frame, the entropy density transforms as:

\begin{equation} \label{eq:entropy_density}
\tilde{s} = s / A_{\rm clock}^3
\end{equation}

The entropy diagnostic must be non-definitional to carry content: a quantity assembled by algebraically undoing the conformal rescaling, such as $\tilde{s}A_{\rm clock}^{3}$ with $\tilde{s}=s/A_{\rm clock}^{3}$, is conserved identically and tests nothing. The pipeline therefore tracks the specific entropy per baryon,

\begin{equation} \label{eq:specific_entropy}
\tilde{\sigma} = \frac{\tilde{s}_{\gamma}}{\tilde{n}_{b}} \, ,
\end{equation}

assembled from two independently motivated evolution laws. The photon entropy density follows blackbody thermodynamics of the measured temperature-redshift law, $\tilde{s}_{\gamma}=(2\pi^{2}/45)\,g_{*s}\,\tilde{T}^{3}$ with $\tilde{T}(z)=T_{0}(1+z)$, while the baryon density follows particle-number conservation on the effective matter geometry, $\tilde{n}_{b}(z)=n_{b,0}\,A_{\rm clock}(z)^{-3}$. The constancy of $\tilde{\sigma}$ is then a physical consistency condition — it holds if and only if the thermal law and the dilution law are governed by the same clock factor, which is precisely the content of the thermal scaling $T=T_{0}/A_{\rm clock}$ established in Section 8 — rather than an algebraic identity.

The pipeline evaluates $\tilde{\sigma}(z)$ across $z\in[0,1000]$ (Step 07). The specific entropy is constant across the full range, with present-day value $\tilde{\sigma}_{0}\approx5.9\times10^{9}\,k_{\rm B}$ per baryon, the observed cosmic photon-entropy-per-baryon scale; no entropy production is required anywhere along the reconstruction. A control evaluation in which the dilution law is driven by a deliberately mismatched profile — the late-time drift factor $A_{\rm dyn}(z)=(1+z/z_{t})^{-\epsilon_{\rm dyn}}$, which is not the horizon clock map — drifts by a factor $\sim10^{9}$ over the same range, demonstrating that the diagnostic responds to inconsistency and is not vacuous. The entropy density remains finite at every finite redshift (reaching $\sim1.5\times10^{27}\,k_{\rm B}\,{\rm m}^{-3}$ in matter-frame units at $z=10^{3}$); its growth toward the horizon is the clock-ratio image of the standard $T\to\infty$ limit, not a state attained at finite affine distance. The arrow of time is carried by the clock-drift direction — the future is the direction of increasing $A_{\rm clock}$ — with the constant $\tilde{\sigma}$ recording adiabatic, reversible evolution: no low-entropy initial state is invoked, because there is no initial state. The temporal horizon is thermodynamically regular. Because the temporal horizon is an observational asymptote rather than a physical origin, this regularity demonstrates that local thermodynamic processes—and thus the arrow of time—operate without requiring a singular beginning. The universe does not emerge from a zero-volume state; rather, it evolves locally forward through an asymptotically infinite past.

# 10. Primordial Perturbation Origin

In standard cosmology the nearly scale-invariant primordial spectrum $P_\mathcal{R}(k) \approx A_s (k/k_{\rm pivot})^{n_s-1}$ with $n_s \approx 0.965$ and $A_s = 2.10 \times 10^{-9}$ is either postulated as an initial condition or derived from an inflationary slow-roll potential. In the TEP temporal-horizon cosmology the scalar spectral shape emerges from fluctuations of the conformal clock field. The curvature perturbation is not produced by physical spatial inflation; it is produced by local clock-rate fluctuations:

\begin{equation} \label{eq:zeta_def}
\zeta = \delta\ln A_{\rm clock} = \frac{\delta A_{\rm clock}}{A_{\rm clock}} \, .
\end{equation}

The scalar power spectrum is therefore the two-point function of these clock-field fluctuations:

\begin{equation} \label{eq:P_zeta}
P_{\zeta}(k) = P_{\delta\ln A}(k) \, .
\end{equation}

## 10.1 Native Temporal Action

Introduce the dimensionless clock variable

\begin{equation} \label{eq:chi_def}
\chi \equiv \ln A_{\rm clock} \, ,
\end{equation}

so that the causal matter metric is $\tilde g_{\mu\nu}=e^{2\chi}g_{\mu\nu}$. The temporal field is made dynamical by the single-clock action

\begin{equation} \label{eq:S_temporal}
S = \int d^4x\sqrt{-g}\left[\frac{M_{\rm Pl}^2}{2}R - \frac{M_{\rm Pl}^2}{2}\,\mathcal Z(\chi)(\nabla\chi)^2 - M_{\rm Pl}^2\,U(\chi)\right] + S_m[e^{2\chi}g_{\mu\nu},\psi_m] \, .
\end{equation}

Section 10 uses an effective cosmological single-clock action written in the variable $\chi=\ln A_{\rm clock}$. It preserves the universal TEP matter-metric coupling, while the reconstructed field-dependent kinetic function $\mathcal Z(\chi)$ represents an effective cosmological kinetic completion of the minimal canonical scalar sector rather than a simple linear field redefinition of Jakarta's canonical kinetic term. This makes $A_{\rm clock}$ a dynamical field rather than a fitted conformal factor. The kinetic function $\mathcal Z(\chi)$ and potential $U(\chi)$ are not guessed; they are determined by the temporal-horizon geometry and the observed scalar spectrum. The temporal slow-roll (spectral-flow) parameter is defined from the scalar mode index:

\begin{equation} \label{eq:epsilon_def}
\epsilon_{\rm field} \equiv \nu_A - \frac32 = \frac{1-n_s}{2} \, .
\end{equation}

The observed Planck value $n_s=0.9649$ gives $\epsilon_{\rm field}=0.01755$. This is not a free label; it is the spectral-flow parameter of the temporal clock-field fluctuations, analogous to the slow-roll parameter in inflation.

The relation between $(\chi,\mathcal Z,U)$ and the canonical scalar sector $(\phi,V)$ of the foundational action is made explicit as follows. The universal coupling $A(\phi)=\exp(\beta_A\phi/M_{\rm Pl})$ with $\beta_A=-1$ identifies the clock field with the canonical scalar up to the fixed linear map

\begin{equation} \label{eq:chi_phi_map}
\chi = \ln A_{\rm clock} = \frac{\beta_A\phi}{M_{\rm Pl}} = -\frac{\phi}{M_{\rm Pl}} \, , \qquad \phi = -M_{\rm Pl}\,\chi = M_{\rm Pl}\ln(1+z) \, ,
\end{equation}

where the second equality uses the exact redshift map $A_{\rm clock}=(1+z)^{-1}$. The ambient convention $\phi=0$ at the present epoch corresponds to $\chi=0$, and $\phi>0$ on the past-directed branch toward the horizon, consistent with the corpus-wide sign convention. A pure field redefinition would carry the canonical kinetic term into canonical form in $\chi$ as well, $-\tfrac12(\nabla\phi)^2\to-\tfrac12 M_{\rm Pl}^2(\nabla\chi)^2$, corresponding to $\mathcal Z=1$. The reconstructed $\mathcal Z(\chi)\neq1$ is not the Jacobian of the fixed linear field map. The current foundational sector is nonlinear $P(X,\phi)$ rather than a canonical kinetic term, and both its background-dependent principal coefficients and the disformal matter coupling can contribute to an effective cosmological reduction. The reconstructed $\mathcal Z$ therefore does not uniquely determine $B(\phi)$; likewise $U(\chi)$ must be matched to the universal master potential on the realized background. The common matter coupling and scalar variable are retained, while Rule 12 requires that explicit reduction rather than an independently assigned potential for each sector. The effective equations below state the reconstruction being tested pending that match.

For the temporal-horizon branch $A_{\rm clock}(\eta)=C\eta^{-p}$, the scalar quadratic action derived in Section 10.3 requires the kinetic function

\begin{equation} \label{eq:Z_A_clock}
\mathcal Z(A_{\rm clock}) = \mathcal Z_* \left(\frac{A_{\rm clock}}{A_*}\right)^{(2+2\epsilon_{\rm field})/p} \, .
\end{equation}

Equivalently, in terms of $\chi=\ln A_{\rm clock}$,

\begin{equation}
\mathcal Z(\chi) = \mathcal Z_*\,e^{(2+2\epsilon_{\rm field})\chi/p} \, .
\end{equation}

The action is therefore not arbitrary; the kinetic function required for the observed scalar spectral flow is derived from the temporal-horizon branch. The potential $U(\chi)$ is reconstructed from the background Euler-Lagrange equation rather than postulated; see Section 10.2.

**Effective scalar construction.** The mode equations below specify a reconstructed cosmological sector constrained by $n_s$ and $A_s$. The friction term $2\mathcal H_{\rm eff}\chi'$ uses the effective matter-frame pump; on a static gravitational metric, the homogeneous scalar Euler–Lagrange equation of $\eqref{eq:S_temporal}$ alone instead has no such gravitational scale-factor friction. Deriving the effective reduction, including matter sources and metric constraints, is part of the common-action embedding identified in Section 2. The clock fluctuation $\delta\chi$ is evaluated in the chosen slicing; its identification with gauge-invariant curvature perturbation also requires that constraint reduction. The reconstructed spectrum is a consistency construction within the assigned effective equations, rather than an independent prediction of the observed $n_s$ and $A_s$ from the fixed master potential.

## 10.2 Background Clock Solution

For a homogeneous clock field $\chi(\eta)$, the Euler-Lagrange equation is

\begin{equation} \label{eq:euler_lagrange}
\chi'' + 2\mathcal H_{\rm eff}\chi' + \frac12\frac{d\ln\mathcal Z}{d\chi}\chi'^2 + \frac{1}{\mathcal Z}\frac{dU}{d\chi} = 0 \, .
\end{equation}

The temporal-horizon branch is $A_{\rm clock}(\eta)=C\eta^{-p}$, so

\begin{equation}
\chi(\eta) = \ln C - p\ln\eta \, , \qquad \chi' = -\frac{p}{\eta} \, , \qquad \chi'' = \frac{p}{\eta^2} \, .
\end{equation}

This gives the effective clock-Hubble quantity

\begin{equation} \label{eq:H_eff}
\mathcal H_{\rm eff} \equiv \frac{a_{\rm eff}'}{a_{\rm eff}} = \frac{A_{\rm clock}'}{A_{\rm clock}} = -\frac{p}{\eta}
\end{equation}

Substituting the temporal-horizon solution and the derived kinetic function into the background Euler-Lagrange equation $\eqref{eq:euler_lagrange}$ determines the potential. With $\chi=\ln C-p\ln\eta$, $\chi'=-p/\eta$, $\chi''=p/\eta^2$, and $d\ln\mathcal Z/d\chi=(2+2\epsilon_{\rm field})/p$,

\begin{equation}
\frac{dU}{d\chi} = -\mathcal Z(\chi)\,\frac{p(2+2p+\epsilon_{\rm field})}{\eta^2} \, .
\end{equation}

Since $\eta^{-2}=C^{-2/p}e^{2\chi/p}$ and $\mathcal Z(\chi)=\mathcal Z_*e^{(2+2\epsilon_{\rm field})\chi/p}$,

\begin{equation}
\frac{dU}{d\chi} = -\mathcal Z_* C^{-2/p}\,p(2+2p+\epsilon_{\rm field})\; e^{(4+2\epsilon_{\rm field})\chi/p} \, .
\end{equation}

Integrating gives

\begin{equation} \label{eq:U_reconstructed}
U(\chi) = U_{\rm c} - \frac{\mathcal Z_* C^{-2/p}\,p^{2}(2+2p+\epsilon_{\rm field})}{4+2\epsilon_{\rm field}}\; e^{(4+2\epsilon_{\rm field})\chi/p}
= U_{\rm c} - U_{1}\left(\frac{A_{\rm clock}}{A_*}\right)^{(4+2\epsilon_{\rm field})/p} \, ,
\end{equation}

where $U_{\rm c}>0$ is an integration constant that sets the slow-clock vacuum energy — the same plateau identified in Section 5.1, where the illustrative normalization $\epsilon_{\rm dyn}=8\pi GU_{\rm c}/H_{0}^{2}=0.0055$ would give $U_{\rm c}/\rho_{\rm crit,0}\approx1.8\times10^{-3}$ — and

\begin{equation}
U_{1} = \frac{\mathcal Z_* C^{-2/p}\,p^{2}(2+2p+\epsilon_{\rm field})}{4+2\epsilon_{\rm field}} \, .
\end{equation}

On the physical temporal-horizon branch the constant $U_{\rm c}$ is chosen so that $U(\chi)>0$. The kinetic function $\mathcal Z(\chi)$ is fixed by the observed scalar spectral flow, while the background potential $U(\chi)$ is reconstructed from the temporal-horizon solution through the Euler-Lagrange equation.

## 10.2a Exact Background Solution

The power-law profile $A_{\rm clock}=C\eta^{-p}$ is not a tuned initial condition; it is an exact background solution of the reconstructed action. Substituting $\chi_{0}=\ln C-p\ln\eta$ into the background Euler-Lagrange equation $\eqref{eq:euler_lagrange}$ with $\mathcal H_{\rm eff}=\chi_{0}'$ and the reconstructed kinetic function $\mathcal Z(\chi)$ and potential $U(\chi)$, every term cancels identically because the exponents were chosen to satisfy the temporal-horizon constraint. The prescribed temporal-horizon trajectory is therefore an exact solution of this reconstructed effective action. Its exact satisfaction of the background equation is a consistency result; selection of the trajectory by the common master action is the embedding question stated in Section 2.

## 10.2b Equation of State from the Action

The equation of state is computed directly from the reconstructed kinetic function and potential. For the homogeneous field

\begin{equation}
w_{A}(\eta) = \frac{\tfrac12\mathcal Z(\chi)\,\dot\chi^{2} - U(\chi)}{\tfrac12\mathcal Z(\chi)\,\dot\chi^{2} + U(\chi)} \, ,
\end{equation}

with $\dot\chi=\chi'/A_{\rm clock}=-(p/C)\eta^{p-1}$, where the dot denotes matter proper time $d\tau=A_{\rm clock}\,d\eta$. As $\eta\to\infty$, the reconstructed kinetic contribution decays as $\eta^{2p-4-2\epsilon_{\rm field}}\to0$ while $U\to U_{\rm c}>0$, so $w_A\to-1$. The conformal clock rate is $\mathcal H_{\rm eff}=-p/\eta$, with $|\mathcal H_{\rm eff}'|/\mathcal H_{\rm eff}^2=1/p$. The proper-time rate is instead $H_{\rm clock}=\mathcal H_{\rm eff}/A_{\rm clock}=-(p/C)\eta^{p-1}$, giving $|\dot H_{\rm clock}|/H_{\rm clock}^2=(1-p)/p$ on $0<p\le\tfrac12$. These geometric rate ratios do not imply kinetic dominance: the displayed equation of state tends to $-1$ because the potential dominates the reconstructed scalar energy. A standard Einstein-frame Friedmann relation between the rate ratio and $w_A$ cannot be assumed for this effective clock metric.

The exponent $p$ enters this reconstruction as a family index, not as a dynamical output: the kinetic function's exponent $(2+2\epsilon_{\rm field})/p$ and the potential's asymptotic form are reconstructed around a chosen power law, and each member $(\mathcal Z_p, U_p)$ realizes that power law exactly. The geodesic-completeness window $0\lt p\le\tfrac12$ (Section 4) implies the conformal-rate bound $|\mathcal H_{\rm eff}^\prime|/\mathcal H_{\rm eff}^2\ge2$ and the proper-time bound $|\dot H_{\rm clock}|/H_{\rm clock}^2\ge1$. These are identities of the specified power-law family, rather than additional dynamical selection conditions. Which member of the family the corpus's canonical closure dynamics realizes is an open asymptotic determination (the branch-index datum, shared with the master-action embedding identified in Section 2); the completeness theorems hold unconditionally on every member of the regular branch, and the tensor falsification surface of Section 10.6 binds the index to an observable through $r_{\rm max}(p)\approx6.1\times10^{-5}p^{2}$.

in the static matter-frame case ($a_{\rm m}=1$). The scalar derivation no longer uses $\mathcal H=a_{\rm m}'/a_{\rm m}=0$ as the pump variable; it uses $\mathcal H_{\rm eff}=A_{\rm clock}'/A_{\rm clock}\neq 0$.

## 10.3 Scalar Quadratic Action and Quantization

Work in the single-clock gauge where fluctuations of the temporal field define the curvature perturbation $\zeta=\delta\chi=\delta\ln A_{\rm clock}$. Expanding the action $\eqref{eq:S_temporal}$ to second order and solving the lapse/shift constraints gives

\begin{equation} \label{eq:S_zeta_2}
S_\zeta^{(2)} = \frac12\int d\eta\,d^3x\,z_A^2\left[(\zeta')^2 - c_A^2(\nabla\zeta)^2\right] \, ,
\end{equation}

with

\begin{equation} \label{eq:z_A_Q_A}
z_A^2 = 2M_{\rm Pl}^2\,\mathcal Q_A \, , \qquad \mathcal Q_A = \frac{\mathcal Z(\chi)\chi'^2}{\mathcal H_{\rm eff}^2} \, ,
\end{equation}

and for the minimal temporal action $c_A^2=1$. Using $\chi'=-p/\eta$ and $\mathcal H_{\rm eff}=-p/\eta$ gives $\mathcal Q_A=\mathcal Z(\chi)$. With the derived kinetic function $\mathcal Z(\chi)=\mathcal Z_*e^{(2+2\epsilon_{\rm field})\chi/p}$ and $\chi=\ln C-p\ln\eta$,

\begin{equation}
z_A(\eta) = z_*\left(\frac{\eta}{\eta_*}\right)^{\frac12-\nu_A} = z_*\left(\frac{\eta}{\eta_*}\right)^{-1-\epsilon_{\rm field}} \, ,
\end{equation}

where $\nu_A=\frac32+\epsilon_{\rm field}$. This fixes the kinetic normalization required by the temporal action:

\begin{equation}
\mathcal Z(A_{\rm clock}) = \mathcal Z_*\left(\frac{A_{\rm clock}}{A_*}\right)^{(2+2\epsilon_{\rm field})/p} \, .
\end{equation}

The action is therefore not arbitrary; the kinetic function required for the observed scalar spectral flow is derived from the temporal-horizon branch.

Define the canonical variable $v_k=z_A\zeta_k$. The mode equation is

\begin{equation} \label{eq:v_k_mode}
v_k'' + \left[c_A^2k^2 - \frac{z_A''}{z_A}\right]v_k = 0 \, .
\end{equation}

With $z_A''/z_A=(\nu_A^2-\tfrac14)/\eta^2$ the Bunch-Davies solution is

\begin{equation}
v_k(\eta) = \frac{\sqrt{\pi|\eta|}}{2}\,H_{\nu_A}^{(1)}(k|\eta|) \, .
\end{equation}

Here $|\eta|$ denotes the local freeze-out coordinate measured relative to the effective sound-horizon crossing surface, not the asymptotic compactification coordinate used in the geodesic theorem; the two are related by a monotonic reparameterization, so the spectral index is invariant under the choice of conformal-time orientation. For $k|\eta|\ll 1$, $|\zeta_k|^2=|v_k|^2/z_A^2$ and the scalar power spectrum is

\begin{equation} \label{eq:P_zeta_Hankel}
P_\zeta(k) = \frac{k^3}{2\pi^2}\,|\zeta_k|^2 = A_s\left(\frac{k}{k_*}\right)^{3-2\nu_A} \, .
\end{equation}

Therefore $n_s-1=3-2\nu_A=-2\epsilon_{\rm field}$, recovering the scalar tilt relation from the action-level mode equation. The observed Planck value $n_s=0.9649$ gives

\begin{equation}
\epsilon_{\rm field} = \frac{1-n_s}{2} = 0.01755 \, .
\end{equation}

The running follows $\alpha_s=dn_s/d\ln k$. For the specified constant $\epsilon_{\rm field}$ and exact power-law spectrum, $\alpha_s=0$. A nonzero running requires a scale-dependent spectral-flow parameter or an additional correction derived from the action; it does not follow from $\epsilon_{\rm field}^2$ alone.

The domain of perturbative control is quantified by the pump field. Since $\mathcal Z(\chi)=\mathcal Z_*e^{(2+2\epsilon_{\rm field})\chi/p}$ has a positive exponent in $\chi$, both $\mathcal Z$ and $z_A=z_*(\eta/\eta_*)^{-1-\epsilon_{\rm field}}$ vanish only as $\eta\to\infty$, i.e. on the asymptotic frontier $A_{\rm clock}\to0$ that is the temporal horizon itself. At every finite $\eta$ the canonical normalization is finite and nonzero. A mode of wavenumber $k$ exits the effective sound horizon at $|\eta_k|=\sqrt{\nu_A^2-\tfrac14}/k$; in the oscillatory regime $|v_k|\simeq(2k)^{-1/2}$, so the per-decade variance of the field fluctuation $\delta\chi=v_k/z_A$ is

\begin{equation} \label{eq:P_dchi}
P_{\delta\chi}(k,\eta) \simeq \frac{k^2}{4\pi^2 z_A^2(\eta)} \;=\; P_{\delta\chi}(k,\eta_k)\left(\frac{\eta}{\eta_k}\right)^{2+2\epsilon_{\rm field}} \, ,
\end{equation}

growing toward the frontier but suppressed at freeze-out to the scalar-amplitude scale, $P_{\delta\chi}(k,\eta_k)\sim A_s$. Pipeline step 08b evaluates this across the CMB band $k\in[10^{-4},1]\,{\rm Mpc}^{-1}$ with the $A_s$-normalized pump ($z_*=0.53$ in the conventions of $\eqref{eq:z_star}$). For the pivot mode $k_*=0.05\,{\rm Mpc}^{-1}$, freeze-out occurs at $\eta_k\simeq30.9\,{\rm Mpc}$ with $z_A(\eta_k)\simeq113$, $P_{\delta\chi}\simeq4.3\times10^{-9}$, and $\delta\chi_{\rm rms}\simeq6.5\times10^{-5}$ — deep in the weak-coupling regime. The strong-coupling surface $P_{\delta\chi}\sim1$ lies at $\eta_{\rm sc}\simeq4.0\times10^{5}\,{\rm Mpc}$, some $4.1$ decades beyond freeze-out in the local mode coordinate; across the full band the controlled oscillatory window never falls below $4.07$ decades. The Bunch-Davies normalization is therefore imposed on modes inside a wide controlled window $\eta_k<\eta<\eta_{\rm sc}$, the spectrum is fixed at freeze-out, and the formal asymptotic limit in which $z_A\to0$ lies outside the domain in which the effective description is ever evaluated.

The remaining structure is a boundary statement rather than a defect: the pump-field zero is the strong-coupling frontier of the single-clock effective description, and it coincides asymptotically with the temporal horizon. The regularity results of Sections 3–4 are statements about the metric, which remains smooth; the field-space frontier is a statement about the scalar completion, which terminates at the same boundary. No controlled continuation of the perturbation sector through the horizon is claimed — the horizon bounds the effective field description, which is precisely its role in the temporal-horizon framework.

## 10.4 Observational Constraint and Amplitude Closure

The Planck 2018 measurement $n_s = 0.9649 \pm 0.0042$ (TT,TE,EE+lowE+lensing, 68% CL) constrains the temporal slow-roll parameter:

\begin{equation} \label{eq:epsilon_constraint}
\epsilon_{\rm field} = \frac{1 - n_s}{2} = 0.01755 \pm 0.0021 \, .
\end{equation}

This is a *consistency condition*: the model predicts a tilt, and the observed tilt fixes the spectral-flow parameter of the temporal action. In inflationary cosmology the analogous constraint $\epsilon = (1-n_s)/2$ fixes the shape of the inflaton potential; in TEP it fixes the kinetic slope of the temporal action.

The scalar amplitude $A_s = 2.10 \times 10^{-9}$ is fixed by the normalization of the temporal kinetic function. At the pivot,

\begin{equation}
P_\zeta(k_*) = A_s = \frac{2^{2\nu_A-3}\Gamma^2(\nu_A)}{\pi^3}\,\frac{1}{z_*^2}\,k_*^{3-2\nu_A}\,\eta_*^{1-2\nu_A} \, ,
\end{equation}

which fixes

\begin{equation} \label{eq:z_star}
z_*^2 = \frac{2^{2\nu_A-3}\Gamma^2(\nu_A)}{\pi^3 A_s}\,k_*^{3-2\nu_A}\,\eta_*^{1-2\nu_A} \, .
\end{equation}

The temporal kinetic normalization is therefore

\begin{equation}
\mathcal Z_* = \frac{z_*^2}{2M_{\rm Pl}^2} \, .
\end{equation}

The amplitude does not remain unexplained; it fixes the normalization of the temporal kinetic function $\mathcal Z_*$, in the same way that the observed scalar amplitude fixes the normalization of the inflaton potential in slow-roll inflation. The temporal horizon is the surface $A_{\rm clock}\to 0$ where the observational clock rate vanishes. Quantum fluctuations of the temporal field occur continuously in the local matter frame; as they cross the effective sound horizon their phase information becomes locked into the macroscopic topography of the temporal field. Because the conformal clock rate $A_{\rm clock}$ asymptotically approaches zero relative to the present epoch, these fluctuations appear frozen to a late-time observer. The nearly scale-invariant power spectrum is not the result of physical spatial stretching, but a fossilized record of local quantum dynamics projected across a relativistic temporal horizon.

The effective spectral boundary condition is therefore specified: the scalar spectral shape and amplitude are reproduced within the specified effective mode equations associated with the candidate temporal action $\eqref{eq:S_temporal}$ with the kinetic function $\eqref{eq:Z_A_clock}$ fixed by observation. The effective calculation adopts adiabatic single-clock initial conditions. Relative entropy perturbations of the matter species and cubic interactions are not calculated by the quadratic pump equations.

## 10.5 Tensor Pump Benchmark and Gravitational-Action Matching

**Action normalization and shared-lapse ADM embedding.** In the minimal constant-$M_{\rm Pl}$ Einstein–Hilbert term of $\eqref{eq:S_temporal}$ (and the constant-$M_{\rm Pl}$ homogeneous $u=0$ limit of Paper 0 Step 80, which enforces $c_{\rm GW}=c_\gamma$ and $\Delta t_{\gamma-\rm GW}=0$ on the shared cone), tensor ripples of $g_{\mu\nu}$ have pure-conformal matter-frame image $F=M_{\rm Pl}^2/A_{\rm clock}^2$ and tensor quadratic normalization $z_T=a_{\rm matter}\sqrt F$, which on a static gravitational background ($a_{\rm matter}=A_{\rm clock}$) gives $z_T=M_{\rm Pl}$ and $z_T''/z_T=0$. The $A_{\rm clock}''/A_{\rm clock}$ equation evaluated below is therefore the assigned temporal-conformal pump benchmark (corresponding to a matter-frame Jordan normalization where the spatial curvature term carries $\sqrt{F}\propto A_{\rm clock}$, distinct from the constant-$M_{\rm Pl}$ gravitational frame of $\eqref{eq:S_temporal}$). Its quadrature and profile-response results are retained as diagnostics, with $4k^3|\beta_k|^2/\pi^2$ mapped to dimensionless strain power via $M_{\rm Pl}^{-2}$.

The standard inflationary consistency relation $r=16\epsilon$ is not assumed in TEP-TH, because the temporal-horizon mechanism does not generate perturbations through quasi-de Sitter spatial inflation. Tensor modes must instead be derived directly from the temporal conformal metric. For transverse-traceless perturbations,

\begin{equation} \label{eq:tensor_metric}
d\tilde{s}^{2} = A_{\rm clock}^{2}(\eta)\,\bigl[-d\eta^{2} + (\delta_{ij}+h_{ij})\,dx^{i}\,dx^{j}\bigr] \, ,
\end{equation}

the canonical tensor variable $\mu_{k}=A_{\rm clock}\,h_{k}$ obeys

\begin{equation} \label{eq:tensor_eom}
\mu_{k}'' + \left(k^{2} - \frac{A_{\rm clock}''}{A_{\rm clock}}\right)\mu_{k} = 0 \, .
\end{equation}

For the temporal-horizon profile $A_{\rm clock}(\eta)\sim\eta^{-p}$ with $0 \lt p\lt1$, one has

\begin{equation} \label{eq:tensor_source}
\frac{A_{\rm clock}''}{A_{\rm clock}} = \frac{p(p+1)}{\eta^{2}} \to 0
\end{equation}

at the temporal boundary ($\eta\to\infty$). Thus the tensor equation approaches the Minkowski vacuum equation $\mu_{k}''+k^{2}\mu_{k}=0$. The temporal horizon does not generate the large tensor background predicted by the imported slow-roll estimate $r=16\epsilon_{\rm field}$. Tensor power is controlled only by the finite transition region where $A_{\rm clock}''/A_{\rm clock}$ is nonzero.

The tensor-to-scalar ratio is therefore not fixed by $16\epsilon_{\rm field}$ and must be computed by integrating the assigned tensor-pump equation $\eqref{eq:tensor_eom}$ across the transition profile. For the complete branch $0 \lt p\le\tfrac12$, the tensor index is

\begin{equation}
\nu_T = p + \frac12 \, ,
\end{equation}

because $p(p+1)=\nu_T^2-\tfrac14$. For $0 \lt p\le\tfrac12$ one has $\tfrac12\lt\nu_T\le 1$, while the scalar index is $\nu_A=\tfrac32+\epsilon_{\rm field}\approx 1.5175$. Therefore $\nu_T\neq\nu_A$, and the inflationary consistency relation $r=16\epsilon$ is not a theorem of TEP-TH. The scalar sector is near scale invariant, but the tensor sector is not amplified in the same way; tensor power is controlled only by the finite transition region where $A_{\rm clock}''/A_{\rm clock}$ is nonzero. No ad hoc suppression mechanism is assumed; the only control parameter is the finite shape of $A_{\rm clock}(\eta)$ in the transition region.

## 10.6 Tensor Pump Integration and Conditional Response

The explicit first-order Bogoliubov coefficient is obtained by quadrature of the source term $V(\eta)=A_{\rm clock}''/A_{\rm clock}$ against the free Minkowski mode:

\begin{equation}
\beta_{k} \simeq -\frac{i}{2k}\int_{0}^{\infty} V(\eta)\,e^{-2ik\eta}\,d\eta \, .
\end{equation}

In dimensionless units $(\tilde{\eta}=\eta/\eta_{0}, \tilde{k}=k\eta_{0})$ the integral becomes $\beta_{\tilde{k}}=-(i/2\tilde{k})\int_{0}^{\infty}\tilde{V}(\tilde{\eta})\,e^{-2i\tilde{k}\tilde{\eta}}\,d\tilde{\eta}$ with $\tilde{V}(\tilde{\eta})=\eta_{0}^{2}V(\eta)$. The tensor power spectrum for the two polarization states is

\begin{equation}
P_{T}(k) = \frac{4k^{3}}{\pi^{2}}\,|\beta_{k}|^{2} \, ,
\end{equation}

and the tensor-to-scalar ratio at any wavenumber is

\begin{equation}
r(k) = \frac{P_{T}(k)}{P_{\zeta}(k)} \, .
\end{equation}

Step 09b evaluates this numerically for the temporal-horizon transition profile $A_{\rm clock}(\eta)=C[1+(\eta/\eta_0)^n]^{-p/n}$ with fiducial parameters $p=0.1$, $\eta_0=6000$ Mpc, and $n=2$. The Bogoliubov integral is evaluated with a weighted oscillatory (Filon) quadrature rule, cross-verified against explicit half-period cell integration to a relative accuracy better than $2\times10^{-4}$ at every $\tilde k$ (the fixed-subdivision integrator is noise-dominated once the integrand oscillates $\tilde k/\pi\sim10^{3}$ times across the transition). Because $k\eta_0\sim 300$ for CMB-scale modes, rapid oscillations of $e^{-2ik\eta}$ across the transition strongly suppress the integral. The spectrum is largest at the transition scale $\tilde k\sim 1$ ($k\sim2\times10^{-4}\ {\rm Mpc}^{-1}$) and falls as $\tilde k^{-4}$ at CMB scales. The resulting tensor-to-scalar ratio is

\begin{equation}
r(k_{\rm pivot}) = 1.9 \times 10^{-9} \, , \qquad r_{\rm max} = 5.2 \times 10^{-7} \,
\end{equation}

both far below the BICEP/Keck 2021 bound $r \lt 0.036$ (95% CL) and below the $r\sim10^{-4}$ sensitivity floor of CMB-S4. The analytic parametric estimate $r\sim p^2/(4\pi^2\eta_0^3 A_s k_{\rm pivot}^3)\approx 4.5\times 10^{-3}$ is a coarse upper bound that omits the rapid oscillatory suppression $e^{-2ik\eta}$ across the finite transition region; the numerical integration, which includes this suppression, gives the substantially smaller value quoted above. Both estimates are below the observational bound.

### Robustness of Tensor Suppression

Table 1 reports the maximum tensor-to-scalar ratio $r_{\rm max}$ across the full wavenumber range $k\in[10^{-4},1]$ Mpc$^{-1}$ for variations of the three transition-profile parameters — the falsification surface $r_{\rm max}(p,n,\eta_0)$ of the temporal-horizon tensor sector. The maximum is always attained at the transition scale $\tilde k\sim1$; the quoted figures are evaluated with the cross-verified oscillatory quadrature described above. All values remain orders of magnitude below the BICEP/Keck 2021 bound $r\lt0.036$.

| Parameter | Range | $r_{\rm max}$ | Effect |
| --- | --- | --- | --- |
| $p$ | 0.05 | $1.3\times 10^{-7}$ | Scales approximately as $p^{2}$ at small $p$ |
| $p$ | 0.10 (fiducial) | $5.2\times 10^{-7}$ | Baseline |
| $p$ | 0.20 | $2.1\times 10^{-6}$ | Increases with $p$ |
| $p$ | 0.50 (regular-branch edge) | $1.4\times 10^{-5}$ | Largest on the curvature-regular branch |
| $p$ | 1.00 (curvature-excluded) | $6.1\times 10^{-5}$ | Largest over all tested profiles |
| $n$ | 1 | $2.9\times 10^{-7}$ | Weak dependence on transition steepness |
| $n$ | 2 (fiducial) | $5.2\times 10^{-7}$ | Baseline |
| $n$ | 4 | $7.3\times 10^{-7}$ | Steeper transition mildly enhances production |
| $\eta_{0}$ (Mpc) | 3000 | $4.3\times 10^{-6}$ | Smaller $\eta_0$ weakens oscillatory suppression |
| $\eta_{0}$ (Mpc) | 6000 (fiducial) | $5.2\times 10^{-7}$ | Baseline |
| $\eta_{0}$ (Mpc) | 12000 | $6.4\times 10^{-8}$ | Larger $\eta_{0}$ strengthens oscillatory suppression |

The table samples one parameter at a time and does not bound all simultaneous parameter combinations. For example, the assigned pump diagnostic at $p=0.5$, $n=2$, $\eta_0=3000$ Mpc gives a maximum near $1.14\times10^{-4}$, exceeding the earlier proposed $10^{-4}$ universal threshold. Moreover, these diagnostic values have not been promoted to a dimensionless action-normalized tensor spectrum. A model-wide B-mode falsification threshold therefore requires both the action match and a jointly bounded parameter domain; the sampled table alone does not provide it.

## 10.7 Theorem 2 — Native Scalar and Tensor Spectra

The derivation above is summarized in the following theorem.

**Theorem 2 (Spectra of the specified effective pump equations).** For the effective scalar and tensor mode equations specified below, with the candidate temporal action

\begin{equation}
S_A = \int d^4x\sqrt{-g}\left[\frac{M_{\rm Pl}^2}{2}R - \frac{M_{\rm Pl}^2}{2}\,\mathcal Z(A)(\nabla\ln A)^2 - M_{\rm Pl}^2\,U(A)\right] \, ,
\end{equation}

with $A_{\rm clock}=C\eta^{-p}$, $0 \lt p\le\tfrac12$, and kinetic normalization

\begin{equation}
\mathcal Z(A) = \mathcal Z_*\left(\frac{A}{A_*}\right)^{(2+2\epsilon_{\rm field})/p} \, ,
\end{equation}

scalar clock perturbations obey

\begin{equation}
v_k'' + \left[k^2 - \frac{\nu_A^2-\frac14}{\eta^2}\right]v_k = 0 \, , \qquad \nu_A = \frac32 + \epsilon_{\rm field} \, ,
\end{equation}

yielding

\begin{equation}
P_\zeta(k) = A_s\left(\frac{k}{k_*}\right)^{-2\epsilon_{\rm field}} \, , \qquad n_s = 1 - 2\epsilon_{\rm field} \, .
\end{equation}

Tensor modes obey

\begin{equation}
\mu_k'' + \left[k^2 - \frac{p(p+1)}{\eta^2}\right]\mu_k = 0 \, ,
\end{equation}

so their index is

\begin{equation}
\nu_T = p + \frac12 \, .
\end{equation}

Since $\nu_T \neq \nu_A$, the inflationary consistency relation $r=16\epsilon$ is not a theorem of TEP-TH. Tensor power must be computed from the native temporal-conformal equation and is controlled by the finite transition profile.

## 10.8 Pipeline Validation

The pipeline evaluates the TEP scalar power spectrum over $k \in [10^{-4}, 1]$ Mpc$^{-1}$ using the derived formula $\eqref{eq:P_zeta_Hankel}$ with the slow-roll parameters $\eqref{eq:epsilon_def}$. The controlling parameter is the spectral-flow parameter $\epsilon_{\rm field}$ of $\eqref{eq:epsilon_def}$, distinct from the background-response constant $\epsilon_{\rm dyn}=8\pi G U_{\rm c}/H_0^2$ of Section 5.1; at the pipeline default $\epsilon_{\rm field}=0.0055$ the derived index is $n_s=0.989$, and for a generic value $\epsilon_{\rm field}=0.10$ it is $n_s = 0.80$, redder than the observed value because the temporal field rolls more slowly than the observed tilt requires. When the parameter is set to the Planck-constrained value $\epsilon_{\rm field} = 0.01755$ (`planck_calibrated_evaluation` in Step 08), the TEP spectrum reproduces the observed tilt $n_s = 0.9649$ identically, with the second-order running $\alpha_s = -2\epsilon_{\rm field}^2 = -6.2\times10^{-4}$ keeping $P_\mathcal{R}(k)$ within $0.35\%$ RMS ($1.2\%$ maximum at $k=10^{-4}\ {\rm Mpc}^{-1}$) of the pure power-law reference across the full four-decade $k$-range. Step 08b evaluates the domain of perturbative control $\eqref{eq:P_dchi}$ on the same band, confirming that freeze-out occurs more than four decades inside the strong-coupling surface for every mode and that $z_A$ vanishes only asymptotically at the temporal-horizon frontier.

The effective spectral boundary condition is therefore specified: quantum fluctuations of the conformal clock field freeze at effective sound-horizon crossing inside the controlled domain and appear fossilized to a late-time observer relative to the temporal horizon, producing a nearly scale-invariant adiabatic spectrum whose tilt and running are derived from the native temporal action $\eqref{eq:S_temporal}$. The observed Planck values fix the spectral-flow parameter to $\epsilon_{\rm field} = 0.01755$ and the kinetic normalization $\mathcal Z_*$ through the observed amplitude $A_s = 2.10\times 10^{-9}$. The observed tilt and amplitude fix the effective spectral-flow parameter and kinetic normalization; their common-action derivation is separate from this reconstruction.

The construction above fixes the effective scalar spectrum: the native temporal action $\eqref{eq:S_temporal}$ with the kinetic function $\eqref{eq:Z_A_clock}$ produces the observed scalar spectrum, and the amplitude $A_s$ fixes the normalization $\mathcal Z_*$ in the same way that the inflationary amplitude fixes the inflaton potential. This complements the linear pure-conformal scalar perturbation closure established in TEP-HC (Paper 18), where the Bellini–Sawicki functions $\alpha_M=-2\alpha_A$, $\alpha_B=2\alpha_A$, $\alpha_K=w_c\,\alpha_A^2$ ($w_c = Z_E-6$), and $\alpha_T=0$ were derived and the $\delta\phi$ sector was evolved natively through the Einstein–Boltzmann hierarchy. That audit sharpened the stability statement: the canonical Einstein-frame image $\alpha_K=-5\alpha_A^2$ ($w_c=-5$) passes the ghost discriminant yet is linearly unstable on the conformal background (tachyonic scalarization), while the screened production branch $w_c = 700$ is stable and restores $\Lambda$CDM-level growth. Together, TEP-HC supplies active-solver constraints within its screened effective closure, TEP-TH supplies the temporal-horizon primordial spectrum, and Paper 0 Step 80 embeds the shared-lapse ADM gravitational and matter sectors in a single covariant action whose pre-boundary ambient clock-slope acoustic closure is evaluated directly in TEP-BBN Step 20 and TEP-HC Step 21.

# 11. Pipeline Summary and Internal Consistency

Before presenting the full CMB anisotropy and large-scale structure comparison, the internal consistency of the ten-step pipeline is summarized here. The claims are organized into three tiers. *Tier A* (Steps 1–4) contains theorems and analytic results: the exact clock map, curvature regularity, geodesic completeness, and energy-condition violation. *Tier B* (Step 9) interprets the primordial spectrum and tensor modes within the temporal-horizon framework. No downstream background or thermal observable is tuned. The primordial scalar sector fixes the action parameters $\epsilon_{\rm field}$ and $\mathcal Z_*$ from $n_s$ and $A_s$, analogously to how inflation fixes potential parameters from the scalar spectrum.

## Tier A: Theorems and Analytic Results

| Step | Claim | Basis | Status |
| --- | --- | --- | --- |
| 1 | $A_{\rm clock}=(1+z)^{-1}$ exact; $A_{\rm dyn}$ distinct projection | Definition + numerical (tol. $10^{-6}$) | Theorem |
| 2 | Curvature invariants vanish at boundary ($A_{\rm clock}=C\eta^{-p}$, $0 \lt p\lt1$) | Proposition 1 (closed-form substitution) | Proven |
| 3 | Null geodesics affine-complete for $0 \lt p\le\tfrac12$ | Analytic integral + numerical | Proven |
| 3b | Comoving proper time infinite for $0 \lt p\le1$; arbitrary timelike geodesics complete for $0 \lt p\le\tfrac12$ | Analytic integral + numerical | Proven |
| 4 | Assigned fluid diagnostic violates SEC; reconstructed effective action violates SEC asymptotically when $U>2K$ | Hawking–Penrose prerequisite check | Proven |

<!-- deleted tier B -->

## Tier B: Primordial Perturbations and Tensor Modes

| Step | Claim | Basis | Status |
| --- | --- | --- | --- |
| 9 | Scalar tilt $n_s - 1 = -2\epsilon_{\rm field}$; $\epsilon_{\rm field}=0.01755$ from Planck | Spectral-flow parameter of clock-field fluctuations | Consistency condition |
| 08b | Perturbative control at freeze-out: $z_A\to0$ only asymptotically; $\eta_{\rm sc}/\eta_k\gtrsim1.2\times10^{4}$ across the CMB band | Pump-field and variance growth $P_{\delta\chi}\propto(\eta/\eta_k)^{2+2\epsilon_{\rm field}}$ | Controlled; frontier bounds the EFT |
| 09b | Assigned tensor-pump diagnostic: $r(k_{\rm pivot})=1.9\times 10^{-9}$, $r_{\rm max}=5.2\times 10^{-7}$ | Numerical integration of $\mu_k''+(k^2-A_{\rm clock}''/A_{\rm clock})\mu_k=0$ | Below BICEP/Keck $r\lt0.036$ |

The pipeline separates the geometric theorems from the effective spectral construction and the imported thermal calculations. Tier A provides the geometric foundation: a curvature-regular, geodesically complete temporal horizon with SEC violation. The early-universe background and thermal observables are quantified corollaries of the eternal universe asymptotic equilibrium (Paper 29). Tier B derives the scalar spectral shape from clock-field fluctuations and evaluates the assigned tensor-pump diagnostic, whose action normalization is distinguished in Section 10.5. The imported inflationary consistency relation $r=16\epsilon_{\rm field}$ is not assumed.

## Minimal Model Status

| Sector | Status |
| --- | --- |
| Temporal-horizon mapping | viable ($A_{\rm clock}$ exact, $A_{\rm dyn}$ physical) |
| Curvature invariants | vanish at boundary for $A_{\rm clock}=C\eta^{-p}$, $0 \lt p\lt1$ |
| Null geodesics | affine-complete for $0 \lt p\le\tfrac12$ |
| Timelike geodesics | comoving proper time infinite for $0 \lt p\le1$; arbitrary timelike geodesics complete for $0 \lt p\le\tfrac12$ |
| Scalar tilt | matched by $\epsilon_{\rm field}=0.01755$ |
| Scalar amplitude | fixes kinetic normalization $\mathcal Z_*$ |
| Tensor ratio | assigned pump integrated; fiducial diagnostic $r_{\rm max}=5.2\times10^{-7}$; physical action normalization remains separate |

**Auxiliary numerical checks.** The recombination benchmark solves the existing Saha–Peebles equations with the Radau integrator and checks the zero-amplitude limit against the same comparator. When the optional CLASS import is unavailable, the output explicitly identifies this approximation. The spectral transport check uses photon-number-conserving chemical-potential and Compton-y templates with absolute input errors; it reports an optional CLASS ΛCDM reference separately. Its homogeneous blackbody-preservation identity is a transport check, and absent FIRAS data are not represented as observations.

*Imported thermal benchmark.* The auxiliary `step_04_full_bbn_abundances.py` evaluates the AlterBBN network with a single-epoch expansion-rate mapping to extra radiation. Screening enters through the computed conformal factor, and a nonzero mapped perturbation is evaluated by the network. Failed network runs stop rather than supply reference abundances. The helium anchor is the observational $Y_p=0.245\pm0.003$ from the PDG 2024 BBN review; the deuterium anchor retains the stated measurement selection. $N_{\rm eff}=3.046$ is an assumed reference and is excluded from the primary abundance pass count. These conditional benchmark results are distinct from the eternal-universe abundance budgets and transport calculations in Paper 29.

## 11.1 Falsifiable Predictions

The following near-term observational tests would strengthen or falsify the Temporal Horizon Cosmology framework.

- **Tensor action and transition-profile test.** The assigned pump integrations provide cross-checked conditional diagnostics, with the fiducial values reported in Section 10.6. The tensor normalization of the shared action must first be matched to that pump before those values become physical B-mode predictions. The one-parameter scan does not bound every joint profile combination: $p=0.5$, $n=2$, $\eta_0=3000$ Mpc already yields a diagnostic maximum near $1.14\times10^{-4}$. The earlier universal $r\gtrsim10^{-4}$ exclusion threshold is therefore withdrawn. A quantitative observational threshold requires an action-normalized spectrum and a specified joint parameter domain.

- **21-cm power spectrum at cosmic dawn.** The native thermodynamic evolution of the temporal horizon predicts a specific epoch-dependent modification to the matter temperature $T_k(z)$ at $z \sim 20$–$30$. The 21-cm power spectrum from SKA or a future lunar radio array would test whether the local temperature evolution exactly matches $\Lambda$CDM or shows the characteristic TEP transport turnover. A deviation from $\Lambda$CDM at $z \sim 20$ would support the temporal-horizon thermodynamic profile.

- **Gravitational-wave propagation speed variation.** In the homogeneous pure-conformal branch of the TEP metric, the effective gravitational-wave propagation speed exactly matches the speed of light ($c_{\rm gw} = c_\gamma$, $\alpha_T = 0$). Redshift-dependent EM–GW propagation differences arise solely through the bounded disformal sector ($B(\phi)$). LISA, observing massive black-hole mergers at $z \sim 1$–$10$, could constrain these disformal deformations to $\Delta c_{\rm gw}/c \lesssim 10^{-15}$. A positive detection of redshift-dependent GW speed variation at the $10^{-15}$ level would be direct evidence for the disformal geometry; a null result would tightly constrain the disformal amplitude but would not falsify the temporal-horizon framework itself.

- **Local excitation temperature at depth.** Rotational excitation of CN, CO and C&thinsp;I in high-redshift quasar absorption systems thermometers the ambient radiation field *at the absorber's own location*. The temporal-horizon law returns $T_{\rm loc}(z) = T_0/A_{\rm clock}(z) = T_0(1+z)$ natively at every depth, so the standard $T_{\rm CMB}(z)=T_0(1+z)$ excitation profile is a hard prediction requiring no transport assumption. Step 11 confronts it with the published CO absorber sample ($z=0.89$–$2.69$): $\chi^2/{\rm dof}=0.15$ ($\chi^2=1.08$), fitted exponent $\beta=+0.003\pm0.023$, versus $\chi^2=865.8$ for the constant-temperature null (descriptive $\sqrt{\Delta\chi^2}=29.4$) — the tested temperature law is corroborated on real data. Because local collisional excitation or ultraviolet pumping in dense clouds can only elevate $T_{\rm ex}$ above the ambient radiation bath ($T_{\rm ex}\ge T_{\rm loc}(z)$), the quiescent lower envelope across the absorber population must track $T_0(1+z)$; a systematic departure of that quiescent floor below $T_0(1+z)$ toward $T_0$ would discriminate directly against the local-temperature law, whereas continued confirmation of the $T_0(1+z)$ floor at $z\gtrsim3$ extends the depth over which the clock map is observationally validated.

Tensor and 21-cm observations are prospective tests whose quantitative TEP predictions require the action and thermal-profile matching described above. GW–EM propagation tests constrain the disformal sector; a null result does not exclude the pure-conformal boundary geometry. Excitation-temperature measurements probe the local thermal law directly and are the sharpest existing discriminator between a temporal-depth temperature gradient and any mechanism in which the ambient radiation field stays at $T_0$ along the path.

# 12. Conclusion

This paper constructs a curvature-regular, causally complete temporal-conformal branch as a candidate alternative to the singular FLRW past boundary. On the specified branch, the conformal clock rate vanishes only at an infinite-affine asymptote. The regular branch and the singular conformal benchmark are distinct causal completions; a change of clock convention alone does not remove a physical matter-frame singularity. Deriving the regular realization from the universal master action remains the common construction identified in Section 2.

The central temporal-horizon mapping is established: $a_{\rm eff}\to0$ corresponds to $A_{\rm clock}\to0$, where the exact observational clock map $A_{\rm clock}(z)=(1+z)^{-1}$ vanishes at infinite redshift. The dynamical response $A_{\rm dyn}(z)=\left(1+z/z_{t}\right)^{-\epsilon_{\rm dyn}}$ encodes the late-time TEP shear correction. Section 11 presents the integrated ten-step pipeline; the background and thermal checks retain their stated inputs, while the effective scalar kinetic slope and normalization are reconstructed from the observed $n_s$ and $A_s$. The key findings are:

- For the temporal-horizon profile $A_{\rm clock}(\eta)=C\eta^{-p}$ with $0 \lt p\le\tfrac12$, all polynomial curvature invariants vanish at the temporal conformal boundary; the boundary is asymptotically curvature-empty, timelike-complete, and null-complete

- Comoving proper time diverges for $0 \lt p\le1$; arbitrary timelike geodesics are complete for $0 \lt p\le\tfrac12$; null affine parameter diverges for $0 \lt p\le\tfrac12$. The boundary is a regular conformal-temporal endpoint, analogous to a smooth spacelike conformal boundary

- The effective stress-energy tensor of the temporal field violates the Strong Energy Condition, removing one of the assumptions required by the Hawking-Penrose singularity theorems; geodesic completeness is established separately by the curvature and affine-parameter analysis

- Native Local Thermodynamic Evolution in an eternal universe replaces the singular expanding-origin interpretation of the hot thermal state — the high-redshift plasma itself is retained, as Sections 7 and 8 operationally require — providing an eternal-universe framework in which chemical states can approach long-term asymptotic equilibria; the quantitative chemical mechanism and its exposure conditions are evaluated in TEP-BBN (an active research problem) without requiring a phenomenological screening scale.

- Standard BBN assumptions are replaced: the thermal and chemical states are modeled as candidate asymptotic equilibria (TEP-BBN) where D/H is not uniquely primordial and helium arises via baryonic cycling, without requiring a physical zero-volume Big Bang

- TEP-BBN supplies the native chemical-evolution framework and a proof of concept for local CMB thermalization; the present TEP-TH pipeline re-evaluates recombination visibility within the temporal-horizon geometry, while TEP-HC supplies the independent acoustic-sector closure.

- The temporal-horizon thermal mapping preserves a FIRAS-compatible blackbody spectrum without forbidden $\mu$ or $y$ spectral distortions

- Entropy and arrow-of-time analysis demonstrates thermodynamic regularity at the temporal horizon

- The scalar shape of the primordial perturbation spectrum is derived from fluctuations of the clock field $\zeta=\delta\ln A_{\rm clock}$, with spectral-flow parameter $n_s-1=-2\epsilon_{\rm field}$. The observed Planck value $n_s = 0.9649$ constrains $\epsilon_{\rm field} = 0.01755$

- Tensor modes obey a assigned temporal-conformal pump equation; for $A_{\rm clock}(\eta)\sim\eta^{-p}$ the source term $A_{\rm clock}''/A_{\rm clock}\to 0$ at the horizon, so no large primordial tensor background is generated at leading order; the imported inflationary consistency relation $r=16\epsilon_{\rm field}$ is not assumed; numerical integration of the assigned tensor-pump equation yields $r(k_{\rm pivot})=1.9\times 10^{-9}$ and $r_{\rm max}=5.2\times 10^{-7}$, well below the BICEP/Keck bound $r\lt0.036$

- CMB anisotropy spectra (TT, TE, EE) and LSS observables (matter power spectrum, growth factor $f\sigma_8$, BAO scales) are preserved via conformal acoustic equivalence as confirmed by TEP-HC (Paper 18)

The causal matter-frame universe is curvature-regular at the temporal conformal boundary. The apparent Big Bang is a temporal horizon, not a physical curvature singularity. The temporal-horizon geometry supplies a nonsingular framework in which the standard background and acoustic observables are reconstructed by the conformal mapping, while TEP-BBN provides the native chemical-evolution mechanism and a proof of concept for local CMB thermalization; the scalar perturbation shape is reproduced, and tensor production is suppressed at leading order by the vanishing of the temporal-horizon source term. Together with the distance-redshift evidence from TEP-C0 (Paper 26) and the acoustic-sector `hi_class` validation from TEP-HC (Paper 18), TEP-TH supplies the regular-boundary and effective-spectrum parts of the cosmological sequence: the foundational geometrical and thermodynamic signatures conventionally attributed to a hot Big Bang singularity can be modeled within a static conformal temporal-transport geometry.

**Implications:** The temporal-horizon cosmology of the TEP framework replaces the standard Big Bang interpretation. In the complete temporal-horizon branch, the universe extends backward through an infinite affine past without a physical curvature singularity. The temporal horizon is strictly an observational boundary where the relative conformal clock rate vanishes, meaning cosmic history must be mapped by local thermodynamic state variables rather than a global chronological stopwatch. Cosmic expansion is a geometric reconstruction of accumulated open-path conformal temporal shear. The background and thermal observational signatures conventionally attributed to a hot dense origin—light-element abundances, acoustic peaks, blackbody thermalization, and large-scale structure—can be reconstructed within the combined eternal-universe thermodynamic and conformal-acoustic framework. The tensor-to-scalar ratio must be computed from the finite transition region via the assigned temporal-conformal pump equation, not from an imported inflationary consistency relation. The eternal-branch background carries a corresponding caveat registered in the homogeneous analysis: the closed static solution is not a dynamical attractor — its linearization exhibits an Eddington-type secular mode with e-folding $\sim a/c$ — and control of the analogous mode in the inhomogeneous realization remains part of the open construction (Paper 0, §8).

**Outlook:** Figure 1 (Section 4.5) illustrates the conformal-boundary interpretation: the singular lower edge of standard flat $\Lambda$CDM is replaced by a smooth regular temporal conformal boundary $\mathscr{T}^{-}$, where $A_{\rm clock}\to0$ and curvature invariants vanish. The apparent Big Bang is a regular conformal-temporal endpoint, not a physical singularity. Step 09b is complete: the assigned tensor-pump equation yields $r(k_{\rm pivot})=1.9\times 10^{-9}$ and $r_{\rm max}=5.2\times 10^{-7}$, well below current and projected CMB bounds. Closed-loop synchronization holonomy remains the primary discriminant of the non-exact/disformal sector of TEP. In the homogeneous conformal limit analysed here, $A_{\rm clock}$ gives open-path temporal redshift but does not by itself generate residual loop holonomy, since $\oint d\ln A_{\rm clock}=0$. A nonzero $\mathcal{H}_{\rm resid}$ would require the disformal contribution $B(\phi)\nabla_{\mu}\phi\nabla_{\nu}\phi$, non-metricity, or another non-exact synchronization structure, as developed in the foundational TEP paper (Jakarta).

**Common-action scope.** The regularity and completeness results apply to the specified matter-frame metric. The scalar spectrum is reconstructed using the observed tilt and amplitude. The quoted tensor values belong to an assigned pump diagnostic; Section 10.5 distinguishes it from the tensor quadratic action of the static gravitational background. They are not yet a universal action-normalized tensor prediction or a model-wide B-mode exclusion threshold. HC's acoustic preservation and BBN's thermal/chemical calculations retain their respective reference-model and native-boundary assumptions; matching their complete transfer histories is a separate calculation.

# 13. References

- Smawfield, M.L. Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed. *Zenodo* (2025), v0.16 (Jakarta). DOI: 10.5281/zenodo.16921911

- Smawfield, M.L. Temporal Equivalence Principle: A Covariant Alternative to Cosmic Expansion. *Zenodo* (2026). DOI: 10.5281/zenodo.20370143

- Smawfield, M.L. Temporal Equivalence Principle: Native hi_class Conformal Implementation, Linear Perturbation Closure, and CMB Acoustic Peak Preservation. *Zenodo* (2026). DOI: 10.5281/zenodo.20572722

- Smawfield, M.L. Temporal Equivalence Principle: Dynamical Proper Time and the Illusion of Primordial Deuterium. *Zenodo* (2026). DOI: 10.5281/zenodo.21841147

- Hawking, S.W. The occurrence of singularities in cosmology. *Proc. R. Soc. A* **294**, 511-521 (1966).

- Hawking, S.W. & Penrose, R. The singularities of gravitational collapse and cosmology. *Proc. R. Soc. A* **314**, 529-548 (1970).

- Borde, A., Guth, A.H. & Vilenkin, A. Inflationary spacetimes are incomplete in past directions. *Phys. Rev. Lett.* **90**, 151301 (2003).

- Brandenberger, R. & Peter, P. Bouncing cosmologies: progress and problems. *Found. Phys.* **47**, 797-850 (2017).

- Novello, M. & Bergliaffa, S.E.P. Bouncing cosmologies. *Phys. Rep.* **463**, 127-213 (2008).

- Ijjas, A. & Steinhardt, P.J. Entropy, black holes and the new cyclic universe. *Phys. Lett. B* **824**, 136823 (2022).

- Peebles, P.J.E. *Principles of Physical Cosmology*. Princeton University Press (1993).

- Weinberg, S. *Cosmology*. Oxford University Press (2008).

- Dodelson, S. *Modern Cosmology*. Academic Press (2003).

- Mukhanov, V.F., Feldman, H.A. & Brandenberger, R.H. Theory of cosmological perturbations. *Phys. Rep.* **215**, 203-333 (1992).

- Liddle, A.R. & Lyth, D.H. *Cosmological Inflation and Large-Scale Structure*. Cambridge University Press (2000).

- Planck Collaboration, et al. Planck 2018 results. VI. Cosmological parameters. *A&A* **641**, A6 (2020).

- Riess, A.G., et al. Milky Way Cepheid Standards for Measuring Cosmic Distances and Application to Gaia DR2: Implications for the Hubble Constant. *ApJ* **861**, 126 (2018).

- Brout, D., et al. The Pantheon+ Analysis: Cosmological Constraints. *ApJ* **938**, 110 (2022).

- Fixsen, D.J., et al. The Cosmic Microwave Background Spectrum from the Full COBE FIRAS Data Set. *ApJ* **473**, 576 (1996).

- Chluba, J. & Sunyaev, R.A. The evolution of CMB spectral distortions in the early Universe. *MNRAS* **419**, 1294-1314 (2012).

- PARTICLE DATA GROUP. Review of Particle Physics. *PTEP* **2022**, 083C01 (2022).

- Cyburt, R.H., Fields, B.D., Olive, K.A. & Yeh, T.H. Big bang nucleosynthesis: Present status. *Rev. Mod. Phys.* **88**, 015004 (2016).

- Seager, S., Sasselov, D.D. & Scott, D. A new calculation of the recombination epoch. *ApJ* **523**, L1-L5 (1999).

- Peebles, P.J.E. Recombination of the Primeval Plasma. *ApJ* **153**, 1 (1968).

- Zeldovich, Y.B. & Sunyaev, R.A. The interaction of matter and radiation in a hot-model universe. *Astrophys. Space Sci.* **4**, 301-316 (1969).

- Seljak, U. & Zaldarriaga, M. A Line of Sight Integration Approach to Cosmic Microwave Background Anisotropies. *ApJ* **469**, 437 (1996).

- Lewis, A., Challinor, A., & Lasenby, A. Efficient Computation of CMB Anisotropies in Closed FRW Models. *ApJ* **538**, 473 (2000).

- Lesgourgues, J. & Tram, T. The Cosmic Linear Anisotropy Solving System (CLASS). Part IV: efficient implementation of non-cold relics. *JCAP* **09**, 032 (2011).

- Zumalacárregui, M., Bellini, E., Sawicki, I., Lesgourgues, J. & Ferreira, P.G. hi_class: Horndeski in the Cosmic Linear Anisotropy Solving System. *JCAP* **08**, 019 (2017).

- De Felice, A. & Tsujikawa, S. f(R) Theories. *Living Rev. Rel.* **13**, 3 (2010).

- Wetterich, C. Cosmology and the fate of dilatation symmetry. *Nucl. Phys. B* **302**, 668-696 (1988).

- Wetterich, C. A universe without expansion. *Phys. Dark Universe* **2**, 184 (2013).

- Narlikar, J.V. & Arp, H.C. Flat spacetime cosmology: A unified framework for extragalactic redshifts. *Astrophys. J.* **405**, 51-56 (1993).

- Mannheim, P.D. Conformal gravity and the nature of dark matter. *Prog. Part. Nucl. Phys.* **94**, 217-272 (2017).

- Khoury, J. & Weltman, A. Chameleon cosmology. *Phys. Rev. D* **69**, 044026 (2004).

- Hinterbichler, K. & Khoury, J. Symmetron cosmology. *Phys. Rev. Lett.* **104**, 231301 (2010).

- Penrose, R. Before the Big Bang: an outrageous new perspective and its implications for particle physics. *Proc. EPAC* (2006).

- Tod, K.P. Isotropic cosmological singularities. *Gen. Relativ. Gravit.* **35**, 779-805 (2003).

- Tod, K.P. The equations of conformal cyclic cosmology. *Gen. Relativ. Gravit.* **47**, 31 (2015).

- Ratra, B. & Peebles, P.J.E. Cosmological Consequences of a Rolling Homogeneous Scalar Field. *Phys. Rev. D* **37**, 3406 (1988).

- Caldwell, R.R., Dave, R., & Steinhardt, P.J. Cosmological Imprint of an Energy Component with General Equation of State. *Phys. Rev. Lett.* **80**, 1582 (1998).

- Clifton, T., Ferreira, P.G., Padilla, A. & Skordis, C. Modified gravity and cosmology. *Phys. Rep.* **513**, 1-189 (2012).

# Acknowledgements

The author thanks his mother, J.S., who once attended a Penrose lecture in New Delhi, for a book that initiated his study of conformal infinity.

# Unified TEP Parameter Dictionary

The TEP corpus uses related but distinct symbols across its papers. This dictionary maps every parameter, its definition, the paper where it is primary, and its fiducial or fitted value.

| Symbol | Definition | Primary Paper | Fiducial / Fitted Value |
| --- | --- | --- | --- |
| $A_{\rm clock}(z)$ | Exact observational clock/redshift map: $A_{\rm clock}=(1+z)^{-1}$ | TEP-TH | $(1+z)^{-1}$ (exact) |
| $A_{\rm dyn}(z)$ | Dynamical shear response: $\left(1+z/z_t\right)^{-\epsilon_{\rm dyn}}$ | TEP-TH | Modifies late time evolution |
| $\alpha_A$ | Runtime logarithmic derivative $\alpha_A=d\ln A_{\rm HC}/d\ln a$; distinct from the exact endpoint map $A_{\rm clock}$ | TEP-HC | Redshift-dependent; zero at the finite-profile turnover |
| $\alpha_M$, $\alpha_B$, $\alpha_K$, $\alpha_T$ | Runtime Bellini–Sawicki EFT functions: $\alpha_M=-2\alpha_A$, $\alpha_B=2\alpha_A$, $\alpha_K=w_c\,\alpha_A^2$ with $w_c=Z_E-6$ (production $w_c=700$; the canonical image $w_c=-5$ is linearly unstable — Paper 18 closure audit), $\alpha_T=0$ | TEP-HC | Adopted from $\alpha_A$ |
| $\epsilon_T^{\rm los}$ | Late-time line-of-sight transport amplitude (C0 supernova fit) | TEP-C0 | $\mathcal{U}[0, 2.0]$ (prior); posterior peaked near $\sim 0.89$ |
| $\epsilon_T^{\rm CMB}$ | C0 background/acoustic diagnostic amplitude | TEP-C0 | $-0.0015\pm0.0037$ |
| $\epsilon_T^{\rm HC}$ | Native hi_class homogeneous conformal amplitude | TEP-HC | $0.0043\pm0.0025$ (full high-$\ell$ likelihood) |
| $\epsilon_{\rm dyn}$ | Dynamical temporal-horizon response | TEP-TH | Modulates late-time expansion |
| $\epsilon_{\rm field}$ | Primordial spectral-flow parameter constrained by $n_s$ | TEP-TH | $0.01755$ (from $n_s=0.9649$) |
| $z_T^{\rm los}$ | C0 line-of-sight supernova transport turnover | TEP-C0 | $5$ (conservative), $100$ (benchmark), free (broad) |
| $z_T^{\rm HC}$ | Homogeneous/acoustic hi_class profile scale | TEP-HC | Fitted jointly with $\epsilon_T$ |
| $p$ | Temporal-horizon conformal exponent: $A_{\rm clock}\sim\eta^{-p}$ | TEP-TH | $0 \lt p\le\tfrac12$ (regular branch) |
| $r_s^{\rm TEP}/r_s^{\Lambda\rm CDM}$ | Pre-recombination sound-horizon ratio | TEP-HC | $0.999994$ ($<6$ ppm deviation) |
| $D=\alpha_K+\tfrac32\alpha_B^2$ | No-ghost discriminant ($D=(w_c+6)\alpha_A^2$; necessary but not sufficient for linear stability — the $w_c=-5$ branch is tachyonic; production $w_c=700$ gives $D=706\,\alpha_A^2$) | TEP-HC | $(w_c+6)\alpha_A^2\ge 0$ (non-negative; GR/decoupling limit at $\alpha_A=0$) |
| $r(k_{\rm pivot})$ | Native tensor-to-scalar ratio at Planck pivot | TEP-TH | $1.9\times10^{-9}$ |
| $r_{\rm max}$ | Maximum tensor-to-scalar ratio across transition profile | TEP-TH | $5.2\times10^{-7}$ (fiducial $p=0.1$); $6.1\times10^{-5}$ over all tested profiles $0\lt p\le1$ |
| $H_0$ | Hubble parameter (TEP-C0 joint MCMC) | TEP-C0 | $66.70\pm0.58$ km s$^{-1}$ Mpc$^{-1}$ |
| $S_8$ | $\sigma_8\sqrt{\Omega_m/0.3}$ (TEP-HC joint MCMC) | TEP-HC | $0.868\pm0.025$ |
| $\sigma_8^{\rm HC}$ | Native hi_class matter-fluctuation amplitude | TEP-HC | $0.824$ (fiducial); $0.858\pm0.016$ (MCMC posterior mean) |

**Note:** Parameters with superscript labels ($^{\rm los}$, $^{\rm HC}$) are related projections of the same temporal sector but are not numerically interchangeable. The turnover scales $z_T^{\rm los}$ and $z_T^{\rm HC}$ describe different physical regimes; the amplitudes $\epsilon_T^{\rm los}$, $\epsilon_T^{\rm CMB}$, $\epsilon_T^{\rm HC}$, and $\epsilon_{\rm field}$ are constrained by different observables.

# 14. Data Availability and Reproducibility

All data and analysis code required to reproduce the results presented in this work are available in the public repository at https://github.com/matthewsmawfield/TEP-TH.

All pipeline outputs, posterior samples, and the exact step configuration files are released in the Zenodo repository (DOI: 10.5281/zenodo.20723059) under CC-BY 4.0. The step scripts and `run_pipeline.py` orchestration script are provided in the GitHub repository.

## Repository Structure

- **scripts/steps/**: Python pipeline scripts for temporal-horizon analysis (steps 00-10, with 09b)

- **scripts/steps/th_common.py**: Shared utilities for TEP-TH pipeline

- **data/raw/**: Raw cosmological data sources

- **data/processed/**: Processed data products

- **results/**: Pipeline outputs (JSON, CSV, figures)

- **logs/**: Pipeline execution logs

## Pipeline Execution

The TEP-TH pipeline can be executed using the provided scripts:



```
cd scripts/steps
python step_00_temporal_horizon_mapping.py
python step_01_matter_frame_curvature.py
python step_02_geodesic_completeness.py
python step_03_effective_stress_energy.py
python step_04_full_bbn_abundances.py
python step_05_recombination_visibility.py
python step_06_cmb_blackbody_origin.py
python step_07_entropy_arrow.py
python step_08_primordial_perturbation_boundary.py
python step_09b_native_tensor_integration.py
python step_10_cmb_lss_class.py
```



## Key Pipeline Outputs

- **step_01_matter_frame_curvature.json / .csv**: Proposition 1 verification (pure conformal curvature vanishing) and disformal compensation analysis (critical power $n=4$, lapse divergence test)

- **step_04_full_bbn_abundances.json / .csv**: BBN abundance validation (Y_p, D/H, He3/H, Li7/H, N_eff)

- **step_05_recombination_visibility.json / .csv**: Recombination epoch ($z_*$, $r_s$, $\theta_s$), LCDM and TEP comparison

- **step_09b_native_tensor_integration.json / .csv**: Native tensor-mode integration (r(k_pivot), r_max, Bogoliubov coefficients)

- **step_10_cmb_lss_class.json / .csv**: CMB/LSS consistency scorecard (Planck comparison, sigma8)

## Dependencies

The pipeline requires Python 3.11+ with the following packages:

- numpy

- scipy

- matplotlib (for plotting)

- optional: `classy` (CLASS Python bindings) for the step_10 CMB/LSS spectra; without it the step degrades gracefully

- optional: bundled `external/alterbbn_wrapper.py` + AlterBBN for the step_04 abundance network; without it the step falls back to tabulated ΛCDM references

## Data Sources

The pipeline uses publicly available cosmological data:

- FIRAS CMB monopole spectrum (NASA LAMBDA)

- BBN abundance measurements (PDG 2024)

- Planck 2018 cosmological parameters

## Version Control

This work uses version v0.5 (Thika) of the TEP-TH pipeline, first published 18 June 2026. The repository maintains a complete history of all changes through Git version control.

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# Temporal Equivalence Principle: Black Holes and the Temporal Horizon
**Matthew Lukin Smawfield**
Version: v0.4 (Bahrain)
First published: 29 July 2026 - Last updated: 17 October 2026
DOI: 10.5281/zenodo.21677826

---

## Abstract

The Temporal Equivalence Principle treats proper time as a dynamical field. Matter, light, and ideal clocks couple to the universal causal metric $\tilde g_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))\,U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\,\Pi_{\mu\nu} + B(\phi)\,\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu \equiv -\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu} = g_{\mu\nu} + U_\mu U_\nu$; written in isotropic conformal–disformal shorthand as $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}+B(\phi)\nabla_\mu\phi\nabla_\nu\phi$). The Einstein-frame metric carries the gravitational dynamics; tensor propagation is determined by the principal symbol of the coupled temporal–geometric equations. This paper develops the strong-field consequence: Under TEP, a black hole is modeled as a temporal well—a target regular spatial region in which matter-frame clock rates become strongly suppressed. A continuous, extreme but finite gradient in proper-time rate provides a unified mechanism for the principal observational signatures attributed to a black hole.

Darkness, apparent compactness, large inferred mass, and practical inaccessibility are exterior reconstructions of temporal decoupling. The operational boundary is the Temporal Horizon — the observer-relative threshold beyond which the clock-transfer factor renders signals practically undetectable — not the event horizon. Inferred gravitational mass and local material mass need not coincide; a strong-field phantom mass residual $M_{\rm phantom}^{T} \equiv M_{\rm fit}^{\rm GR} - M_{\rm matter}^{\rm TEP}$ measures this discrepancy, its sign determined by the data. Standard black-hole ontology assumes isochrony: source clocks, photon propagation, and observer clocks mapped onto a single general-relativistic time coordinate. TEP drops that closure. Under TEP, the conventional reconstruction — compact mass, event horizon, singular collapse — is no longer the unique reading of the same observations.

The temporal field cannot sit passively on fixed Schwarzschild geometry: finite curvature and bounded areal radius are mutually exclusive when $g_{\mu\nu}$ is held fixed. Strong temporal structure forces gravitational backreaction. The canonical TEP matter coupling fixes the temporal sector; the leading curvature operator supplies backreaction; the regularising nonlinear coefficients are selected by global regularity and observation. The construction programme is solving the global field equations of the fixed EFT architecture — not selecting among competing theories.

Four observational consequences follow from one temporal field: time transfer, mass inference, photon accessibility, and ringdown. Weak-field data recover GR; horizon-scale images constrain the photon-region geometry but do not directly establish an event horizon; gravitational-wave ringdown provides the sharpest test, because TEP replaces the purely ingoing event-horizon condition with propagation through a regular temporal domain, changing the late-time spectral problem. *Cosmological expansion and black-hole collapse are dual misreadings of dynamical proper time.*

The scope of the construction is stated plainly. The temporal-well criteria are derived as necessary conditions, with quantitative evaluations across fixed Schwarzschild (incompatibility proof), perturbative sGB exterior, and the regular Hayward-class interior (Step 34). On the isotropic conformal benchmark ($\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$, where spatial stretching cancels $A^2(\phi)$ out of null geodesics), only the softened Hayward core acts on the photon sphere, yielding a $-5.4\%$ to $-6.3\%$ shadow contraction; under the unstretched-space temporal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) (where the $(3+1)$ ADM spatial 3-metric does not stretch and the temporal well $A(\phi(r))=e^{-\phi(r)}<1$ refracts null trajectories via $\tilde V_{\rm eff}(r)=e^{-2\phi(r)}F(r)/r^2$, Step 36), the fiducial $\eta=+0.30$ solution produces a positive phantom mass $M_{\rm phantom}=+1.41\%$ ($Q_s=0.20\,M$) to $+2.65\%$ ($Q_{\rm eff}=0.236\,M$ from the solved Step 34 BVP), unifying the positive phantom-mass sign ($M_{\rm phantom}>0$) across black-hole, galactic, and cluster scales while matching the EHT M87$^\ast$ ($40.25$–$40.74\,\mu$as vs $42.0\pm3.0\,\mu$as, $-0.58\sigma$ to $-0.42\sigma$) and Sgr A$^\ast$ ($54.00$–$54.67\,\mu$as vs $48.7\pm7.0\,\mu$as, $+0.76\sigma$ to $+0.85\sigma$) shadow diameters within $1\sigma$. The disformal normalisation $B_0 = 1$ used here is a positive-sign ($B\geq0$) strong-field benchmark. Paper 0's loop diagnostic also tests a positive coefficient, but defines its dimensionless normalization separately (Appendix E, R6); the earlier $B_0 \approx -3\times10^{-3}$ volume-balance calibration is excluded by the sign requirement. What remains open is the dimensional reconciliation between the geometrized strong-field normalization ($B_0 = 1\,M_{\rm BH}^2$) and the mass-dimension-$(-4)$ weak-field coefficient, together with the derivation of $B(\phi)$ from the master action. Finally, the construction is carried out in static spherical symmetry; the rotating generalisation is required for direct comparison with EHT images and LIGO ringdown, so the non-rotating shadow and quasinormal-mode figures are mechanism demonstrations, not direct observational predictions.

Keywords: Temporal Equivalence Principle, temporal well, temporal horizon, dynamical proper time, phantom mass, isochrony axiom, conformal-disformal metric, Schwarzschild incompatibility, apparent compactness, black holes, modified gravity, temporal shear, black-hole observations

# 1. Introduction: Black Holes under the Temporal Equivalence Principle

## 1.1 The Temporal Well

The Temporal Equivalence Principle (TEP) treats proper time as a dynamical field. Matter, light, and ideal clocks couple to the universal causal metric $\tilde g_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))\,U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\,\Pi_{\mu\nu} + B(\phi)\,\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ (written in isotropic conformal–disformal shorthand as $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}+B(\phi)\nabla_\mu\phi\nabla_\nu\phi$; Paper 0 v0.16, §2.2). The Einstein-frame metric carries the gravitational dynamics; tensor propagation is determined by the principal symbol of the coupled temporal–geometric equations. This paper develops the strong-field consequence: a black hole, under TEP, is a temporal well — a regular spatial region in which the rate of proper time differs radically from the exterior, without physical collapse to an ultradense singular object and without an absolute one-way boundary. A continuous, extreme but finite gradient in proper-time rate provides a unified mechanism for the principal observational signatures attributed to a black hole.

Darkness, apparent compactness, large inferred mass, and practical inaccessibility are exterior reconstructions of temporal decoupling. The operational boundary is the Temporal Horizon — the observer-relative threshold beyond which the clock-transfer factor renders signals practically undetectable — not the event horizon. The same principle that reframes cosmology reframes the dark compact sources called black holes. Cosmological expansion and black-hole collapse are dual misreadings of dynamical proper time.

## 1.2 The Principle

TEP distinguishes the Einstein-frame gravitational metric $g_{\mu\nu}$, which carries the gravitational dynamics, from the universal causal matter metric

\begin{equation} \label{eq:intro_1}
\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}+B(\phi)\nabla_\mu\phi\nabla_\nu\phi,
\end{equation}

on which matter, photons, and ideal clocks propagate. The conformal factor $A(\phi)$ maps clock scales; the disformal term $B(\phi)$ can tilt matter cones. On the isotropic conformal–disformal strong-field benchmark ($\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}+B(\phi)\nabla_\mu\phi\nabla_\nu\phi$), null geodesics at $B=0$ remain conformally invariant so that the shadow correction is generated by scalar–Gauss–Bonnet backreaction on the geometric metric while the conformal sector acts on timelike matter observables; under the canonical preferred-foliation unstretched-space metric of Paper 0 v0.16 ($\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav})U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift})\Pi_{\mu\nu}$ at $B=0$, evaluated in Section 8, Step 36), the spatial 3-metric remains unstretched while $A_{\rm loc}(\phi(r))<1$ directly refracts null trajectories and produces a positive phantom mass ($M_{\rm phantom}>0$). In each local Lorentzian orthonormal frame of $\tilde g_{\mu\nu}$, nongravitational physics is locally special relativistic and the locally measured speed of light is invariant. The field $\phi$ is the temporal field — the field that governs the rate of proper time.

The principle is that a continuous, extreme but finite gradient in proper-time rate provides a unified mechanism for the principal observational signatures attributed to a black hole: darkness, apparent compactness, large inferred mass, and practical inaccessibility for a distant observer. No absolute one-way boundary is required. No physical compression to ultradense matter is required. The observations themselves do not distinguish a physically compact collapsed body from a sufficiently strong temporal-rate gradient. The physical lapse satisfies

\begin{equation} \label{eq:intro_lapse}
0 < N(r), \qquad N_{\min} = \min_r N(r) \ll N_o,
\end{equation}

where the temporal-rate minimum may occur at a centre, a shell, or an asymmetric basin. The invariant quantity is not the lapse at any particular point but the largest source-to-observer transfer mismatch along physically connected worldlines:

\begin{equation} \label{eq:intro_transfer}
\mathcal{T}_{e\to o} = \frac{\omega_e}{\omega_o} = \frac{(-k_\mu u^\mu)_e}{(-k_\mu u^\mu)_o}, \qquad
\Delta_T = \ln\frac{\mathcal{T}_{e\to o}^{\rm TEP}}{\mathcal{T}_{e\to o}^{\rm GR}}.
\end{equation}

When $|\Delta_T|$ becomes extreme, signals are strongly redshifted, observed processes appear greatly slowed, matter appears to accumulate or freeze, reconstructed trajectories and spatial scales become distorted, a large central mass and compact radius may be inferred, and information becomes practically inaccessible. *Darkness does not require an event horizon; it requires sufficiently extreme temporal decoupling.*

## 1.3 The Observational Reading

Black-hole observations do not measure a singularity, an event horizon, or an ultradense core. They measure angles, frequencies, arrival times, and images. Those data are converted into mass, radius, and causal structure only by solving an inverse problem under a particular spacetime model. The hidden assumption in that inverse problem is the isochrony closure — source clocks, photon propagation, and observer clocks are mapped onto a single general-relativistic time coordinate. Under this closure, the received data are reduced to orbital periods, velocities, and radii, and then inverted through Kepler or GR to yield a large compact mass. The familiar relation

\begin{equation} \label{eq:intro_kepler}
M_{\rm app}^{\rm GR} = \frac{4\pi^2 a_{\rm GR}^3}{G P_{\rm GR}^2}
\end{equation}

returns an apparent mass conditioned on that temporal calibration. It does not independently measure either the local material mass or the proper volume of the source region. The event horizon, the singularity, and the ultradense core are further reconstructions — none is directly observed.

What is observed:

- photons, frequencies, timing, interferometric visibilities, orbital motion.

What is inferred:

- singularities, event horizons, proper volume, local density, compression.

TEP drops the isochrony closure. Under TEP, the conventional reconstruction — compact mass, event horizon, singular collapse — is no longer the unique reading of the same observations. Inferred gravitational mass and local material mass need not coincide; compactness, horizons, and collapse become emergent observational reconstructions rather than directly established physical facts.

## 1.4 The Event Horizon Is Not an Observable

The event horizon is a global causal construct. It is defined as the boundary of the causal past of future null infinity — a statement that requires knowledge of the entire future spacetime. It is not a local observable. No telescope, no interferometer, no gravitational-wave detector has ever measured an event horizon. What is observed is that signals from certain regions become extremely redshifted, delayed, and faint. The conventional interpretation assigns this to a one-way causal boundary. TEP assigns it to an extreme temporal-transfer mismatch. Both interpretations are compatible with the same observational data — that is the non-uniqueness.

The Temporal Horizon $\mathcal{H}_T^{(\Lambda)}[u_o]$ is the operational boundary defined directly from observable temporal accessibility:

\begin{equation} \label{eq:intro_TH}
\mathcal{H}_T^{(\Lambda)}[u_o] = \{x_e : \mathcal{T}_{e\to o} \geq \Lambda\}.
\end{equation}

It is not a null boundary, not one-way, and does not divide spacetime into an outside and an inside. Different observers place it differently because their clock rates differ. It is defined by what is observable — the transfer factor — not by a global property of an assumed spacetime.

## 1.5 Apparent Compactness $\neq$ Physical Compression

The exterior observer infers a compact, dark, massive object because processes deeper in the temporal field are strongly redshifted, signal arrival rates become extremely slow, light paths are strongly distorted, and matter appears to accumulate or freeze within a small apparent radius. But the density $\rho_{\rm inferred} \sim M/(4\pi r^3/3)$ is an exterior-frame inference that assumes the Schwarzschild radial coordinate and exterior clock remain valid measures of the deep region. The locally measured density and local physical volume need not reproduce that inference. The extreme density attributed to a black hole may be an artefact of reconstructing a region with a radically different proper-time rate using exterior spatial and temporal standards.

This interpretation is dual to the Temporal Horizon Cosmology reading (Thika); the strong-field Temporal Horizon is operationally distinct from the cosmological past boundary $\mathscr{T}^-$ (see Section 6.1). On cosmological scales, a temporal-rate gradient produces observed redshift, which standard physics interprets as spatial expansion — the conformal volume element $V_{\rm eff} = A_{\rm clock}^3 a_m^3$ tends to zero, but the underlying matter-frame spatial geometry does not collapse. Around an extreme gravitational source, a temporal-rate gradient produces redshift, lensing and altered trajectories, which standard physics interprets as gravitational collapse and inward suction. In both cases, the conventional spatial narrative is an observational projection of a single underlying dynamical proper-time field. *Cosmological expansion and black-hole collapse are dual misreadings of dynamical proper time.*

## 1.6 The Argument

This paper develops the strong-field consequence of TEP and shows what follows from it. The argument proceeds in five steps.

**First:** the theory is defined (Section 2). The two-metric action, the conformal-disformal matter metric, the frame dictionary, the observational inference problem, and the strong-field phantom mass decomposition. The isochrony closure is identified as the axiom TEP replaces.

**Second:** the analysis of Section 3 shows that the temporal field cannot sit passively on fixed Schwarzschild geometry. Finite curvature and bounded areal radius are mutually exclusive when $g_{\mu\nu}$ is held fixed. Strong temporal structure forces gravitational backreaction. If the temporal field becomes strong enough to create a Temporal Horizon, the geometry cannot remain ordinary Schwarzschild.

**Third:** the required geometric ingredients are shown to be mathematically attainable (Sections 4–5). Regular geometries exist. The leading curvature operator produces real backreaction. The canonical TEP matter coupling fixes the temporal sector; the regularising nonlinear coefficients are selected by global regularity and observation. The construction programme is solving the global field equations of the fixed EFT architecture.

**Fourth:** four observational consequences follow from one temporal field (Sections 6–8): time transfer, mass inference, photon accessibility, and ringdown. The Temporal Horizon is the operational boundary defined directly from observable temporal accessibility. Replacing an absorbing horizon boundary with a continuous, strongly time-dilated region changes the ringdown boundary-value problem in principle.

**Fifth:** existing data are confronted as examples of the framework (Section 9). S2 is a weak-field consistency check. Horizon-scale images constrain the photon-region geometry. Gravitational waves provide the strongest future test. None of these is a pillar holding up the theory — they are examples of the framework applied to real data.

## 1.7 Structure of the Paper

- Section 2 introduces the TEP theory: the two-metric action, the matter metric, the frame dictionary, the observational inference problem, and the phantom mass decomposition.

- Section 3 shows why Schwarzschild is not enough: a passive temporal field on fixed Schwarzschild geometry cannot satisfy the conditions for a physically admissible temporal well.

- Section 4 presents existence demonstrations: regular geometry and dynamical coupling as attainability proofs.

- Section 5 establishes global geometry and curvature regularity on the validation benchmark.

- Section 6 derives the causal structure and the Temporal Horizon as the operational boundary.

- Section 7 analyses the perturbation spectrum and the structural ringdown prediction.

- Section 8 presents four observational consequences from one temporal field.

- Section 9 discusses the physical interpretation, data confrontation, and scope.

- Section 10 concludes.

Detailed sGB derivations, numerical QNM solvers, EHT visibility fits, and pipeline manifests are collected in the appendices. The main text develops the TEP temporal well, the Schwarzschild incompatibility, the minimal temporal-well criteria, and the unified phenomenology.

## 1.8 Scope of This Paper

This paper develops the strong-field consequence of TEP: a black hole is a temporal well. The Schwarzschild incompatibility and the minimal temporal-well criteria follow from the TEP principle. The claim is that the phenomena attributed to black holes are produced by an extreme but finite gradient in proper-time rate — without physical collapse, without a singular core, without an absolute one-way boundary.

What is derived from TEP: the conventional black-hole reconstruction is not the unique reading of the observations; the event horizon is not a direct observable; conformal time on fixed Schwarzschild is incompatible with finite curvature and bounded area; the Temporal Horizon is operational and observer-relative; the ringdown boundary condition is altered.

What is constructively demonstrated: regular geometry and curvature-coupled scalar backreaction are attainable; finite-curvature matter metrics are achievable; weak-field S2 recovers GR; the exterior and deep-transit benchmarks isolate the geometric mechanism of the altered spectral problem.

What requires key closure: the TEP-selected global field solution; the complete coupled characteristic system; the raw non-isochronous multi-messenger refit; the measured phantom-mass sign.

# 2. TEP Theory: Action, Matter Metric, and Frame Dictionary

## 2.1 Canonical Action and Universal Matter Coupling

The canonical Einstein-frame action is used:

\begin{equation} \label{eq:tep_1}
S=\int d^4x\sqrt{-g}\left[\frac{M_{\rm Pl}^2}{2}R-\frac12(\nabla\Phi)^2-V(\Phi)\right]+S_m[\Psi_m,\tilde g_{\mu\nu}],
\end{equation}

with

\begin{equation} \label{eq:tep_2}
\tilde g_{\mu\nu}=A^2(\Phi)g_{\mu\nu}+B(\Phi)\nabla_\mu\Phi\nabla_\nu\Phi.
\end{equation}

This is the canonical frame: gravity has Einstein–Hilbert form, tensor modes propagate on $g_{\mu\nu}$, and all nongravitational fields couple universally to $\tilde g_{\mu\nu}$. Diffeomorphism invariance of the matter action gives $\tilde\nabla_\mu\tilde T^{\mu\nu}=0$ wherever $\tilde g$ is invertible. The static spherical field-equation construction in Sections 2–7 uses the isotropic conformal–disformal strong-field benchmark of Eq. (\ref{eq:tep_2}). Paper 0 (v0.16, §2.2, Step 80) promotes the preferred-foliation metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))\,U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\,\Pi_{\mu\nu} + B(\Phi)\,\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\Phi\nabla_\beta\Phi$ ($U_\mu\equiv-\nabla_\mu\bar\Phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$) as the canonical unified matter geometry; its fixed-epoch unstretched-space limit ($\tilde\gamma_{ij}=\gamma_{ij}$, $\tilde g_{tt}=-A_{\rm loc}^2(\phi(r))F(r)$) is evaluated directly in the optical shadow and phantom-mass calculations of Section 8 (Step 36). Embedding the full nonlinear temporal-well interior BVP directly within that unified preferred-foliation metric is part of the global strong-field closure. The pipeline uses the dimensionless field $\phi=\Phi/M_*$ and geometrized units $G=c=1$; the numerical normalisation, parameter freeze ($\beta_A=-1$, $B_0=1$), and weak-to-strong-field running are detailed in Appendix B.

The theory is a defined modified-gravity theory, not a broad framework into which arbitrary completions are inserted. The terms are organised into the fundamental sector and the strong-field EFT sector.

**Fundamental sector — fixed by the TEP principle.**

- The Einstein–Hilbert gravitational action $\frac{M_{\rm Pl}^2}{2}R[g]$: the Einstein–Hilbert term supplies the canonical gravitational kinetic structure. On the Schwarzschild benchmark, tensor propagation reduces to $c_T = 1$, consistent with GW170817; the nonlinear Temporal-Well characteristics follow from the complete coupled principal symbol.

- The canonical scalar kinetic $-\frac12(\nabla\Phi)^2$ and potential $V(\Phi)$: the temporal field $\Phi$ is the dynamical proper-time field. The canonical kinetic term supplies a hyperbolic scalar principal part, while $V''>0$ gives local stability around the selected background.

- The universal conformal matter coupling $A(\Phi) = \exp(\beta_A \Phi/M_{\rm Pl})$: a single function fixes the clock sector for all non-gravitational fields. This is the canonical TEP matter coupling; it is not a free function to be chosen per problem.

- The disformal matter coupling $B(\Phi)\nabla_\mu\Phi\nabla_\nu\Phi$: the existence and role of a universal disformal sector are fixed by the TEP principle, while the quartic-Gaussian form adopted in this work (Appendix B) is the Paper-28 strong-field realization — a prescribed benchmark whose unique corpus-wide microscopic form and normalization remain part of action closure (Paper 0 §2.2), not a form fixed uniquely by the principle. The sector is constrained by GW170817 to $|c_\gamma - c_g|/c \lesssim \text{few}\times10^{-15}$ along observed paths. $B$ is not set to zero; it is bounded.

- The environmental screening/running law $\Sigma_\mu^{\rm obs} = \mathcal S_\Sigma(\mathcal E)\,\Sigma_\mu$ (Section 2.5): the Temporal Topology operator governs weak-to-strong-field running of the effective coupling and reconciles terrestrial tests with cosmological dynamics. This is part of the theory, not an external add-on.

**Strong-field EFT sector — the leading curvature operator and regularising coefficients.**

- The leading curvature operator is the scalar–Gauss–Bonnet coupling $\alpha_{\rm GB}\,\phi\,\mathcal{G}$. This is a native, pre-existing operator in the master TEP Effective Field Theory action which becomes active only in the extreme curvature of the strong-field regime, supplying real backreaction to resolve the Schwarzschild incompatibility. It is the canonical strong-field completion of the TEP action, not an arbitrary addition of new physics.

- The regularising nonlinear coefficients — higher-order curvature couplings and potential terms that control the deep-region profile — are selected by the joint requirements of global regularity (finite curvature, bounded areal radius, Lorentzian signature) and observational consistency. These are EFT corrections to the leading operator, not alternative theories.

The canonical TEP matter coupling fixes the temporal sector. The present paper derives the strong-field solution requirements: the Schwarzschild incompatibility (Section 3) forces gravitational backreaction, and the leading curvature operator supplies it (Section 4). The regularising nonlinear coefficients are selected by global regularity and observation. What is under construction is the global field solution of this fixed EFT architecture — not a choice among competing theories.

## 2.2 Frame Dictionary and Local Lorentz Limit

| Quantity | Einstein/gravitational frame | Causal matter frame |
| --- | --- | --- |
| Metric | $g_{\mu\nu}$ | $\tilde g_{\mu\nu}$ |
| Propagation | tensor modes | matter, photons, ideal clocks |
| Proper time | $d\tau_g$ | $d\tilde\tau$ |
| Conservation | total Einstein-frame stress tensor | $\tilde\nabla_\mu\tilde T^{\mu\nu}=0$ |
| Validity condition | Lorentzian $g$ | Lorentzian, invertible $\tilde g$ |

At every regular point of the matter frame, Riemann normal coordinates give $\tilde g_{ab}=\eta_{ab}+O(x^2)$, preserving exact local Lorentz invariance and locally invariant $c$. Because the metric is globally non-degenerate, such local inertial frames exist everywhere for all $r > 0$.

## 2.3 Spherical Geometry and Scalar Ansatz

A regular Eddington–Finkelstein chart is used from the beginning, avoiding reliance on Schwarzschild $(t,r)$ coordinates in the deep region. The general static spherical form is

\begin{equation} \label{eq:tep_3}
ds_g^2 = -F(r)\,dv^2 + 2\,G(r)\,dv\,dr + R^2(r)\,d\Omega^2,
\end{equation}

in Eddington-Finkelstein form. This prevents coordinate artifacts from being mistaken for physical freezing. In the exterior, $F(r) = 1 - 2M/r$, $G(r) = 1$, $R(r) = r$, recovering the standard Schwarzschild chart at large $r$; this is a matching convenience, not an event horizon in the TEP solution.

The most general symmetry-compatible scalar profile is

\begin{equation} \label{eq:tep_4}
\phi(v,r) = q\,v + \psi(r).
\end{equation}

For the static case studied here, $q = 0$. Two scalar profiles are used in this paper, serving distinct roles in the argument:

- For the Schwarzschild incompatibility analysis (Section 3), the scalar is prescribed as $\phi(r) = \phi_0\,\ln(r/r_h)\,S(r)$ with a smooth logistic activation $S(r)$ that suppresses the field in the exterior. This prescribed profile is used only to establish the incompatibility: the geometric metric is fixed to Schwarzschild, and the scalar field is prescribed rather than solved from coupled equations.

- For the perturbative sGB exterior branch (Section 4), the scalar is solved from the sGB field equations and is Coulomb-like ($\phi \sim Q_s/r$) in the exterior, with the Sotiriou & Zhou scalar charge $Q_s = 2\alpha_{\rm GB}/M$ (equivalently $Q_s = 2\eta M/3$ under $\eta(M) = 3\alpha_{\rm GB}/M^2$), giving the dimensionless strong-field coupling $\eta(M) = 3\alpha_{\rm GB}/M^2$ (Section 4). All observational numbers in this work are computed on the perturbative sGB benchmark. The full nonlinear TEP interior has been computed on a Hayward-class regular background (Step 34, `step_34_solve_interior.py`), solved as the two-point boundary-value problem $\phi'(0) = 0$, $\phi \to 0$ at infinity — the correct formulation, since direct inward integration of the exterior matching data diverges as $\phi \sim C/r$ (the exterior data do not lie on the regular-centre branch). For $\eta = +0.3$ and $M = 1$, the sGB scalar on the regular geometric background yields a strictly positive geometric lapse $N_{\min} = 0.195$ at $r = 1.38$, Lorentzian signature throughout ($\det\tilde g_{2D} < 0$), and a regular centre ($w_r = -1$, $K(0) = 54.2$). The temporal field reaches $\phi_c = 8.67$ at the centre ($A(0) = 1.7 \times 10^{-4}$), and the matter-frame lapse $\tilde N = A \cdot N$ reaches $\tilde N_{\min} = 1.5 \times 10^{-4}$ at the temporal-rate minimum $r \simeq 0.79\,M$ — confirming the temporal-well structure, with maximum clock-transfer factor $\mathcal{Z}_{\max} \simeq 6.7 \times 10^{3}$ (Section 6.2). The observational numbers (QPO, S-star, EHT) are computed on the perturbative sGB exterior benchmark; the interior solution demonstrates that a regular-centre geometry is compatible with the sGB scalar field — a structural consistency result on a prescribed background, while the fully coupled global interior remains the construction target (Section 4). The running law connecting the strong-field $\beta_A = -1$ to the weak-field $\beta_{\rm eff} \approx 10^{-4}$ is the screening function $S_\Sigma(\phi)$ from the time field gradient floor (Section 2.5): $\beta_{\rm eff} = \beta_A \times S_\Sigma$, with $\beta_A = -1$ bare and $S_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ from Cassini under the unified unstretched-space metric ($5.8 \times 10^{-6}$ on the isotropic-conformal benchmark; solved two-branch solar profile $y_{\rm tb}(1.6\,R_\odot)=2.16\times 10^{-8}$, $|\gamma_{\rm PPN}-1|=4.32\times 10^{-8}\ll 2.3\times 10^{-5}$, Paper 0 §7, Step 80).

These are not two competing models. The first is a proof device for the Schwarzschild incompatibility; the second is the leading curvature operator of the strong-field EFT, demonstrating that the backreaction channel is physical. Both are calculations within the fixed TEP theory — the canonical action, the universal conformal coupling, the bounded disformal rule, and the environmental screening law of Section 2.1. A time-dependent scalar with static stress-energy is possible in shift-symmetric constructions and may be essential for rotating generalisations; this is an extension of the scalar ansatz within the same theory, not a replacement of the theory.

## 2.4 Complete Matter Metric and Signature

Expanding the disformal transformation gives $d\tilde s^2 = A^2\,ds_g^2 + B(\psi'(r)\,dr)^2$. The two-dimensional $(v,r)$ determinant is $\det \tilde g_{2D} = -A^4 G^2 - A^2 B F \psi'^2$. The disformal coupling uses an ultra-damped quartic Gaussian ansatz (Appendix B) chosen to ensure conformal dominance in the deep interior: $A^4$ dominates at all radii, so $\det \tilde g_{2D} < 0$ for all $r > 0$.

### Proposition 1 (Invertibility and Signature)

For the bounded strong-field ansatz, the matter metric is Lorentzian and nondegenerate for all $r > 0$. The determinant remains strictly negative as $r \to 0$, where conformal dominance holds asymptotically. The $r \to 0$ limit is an asymptotic spatially enlarged end, not an ordinary manifold point. Full parameter values and the determinant calculation are in Appendix B.

## 2.5 Screening and Temporal Topology

Canonical TEP does not identify screening with a density switch. It represents observable suppression by the environmental operator

\begin{equation} \label{eq:tep_10}
\Sigma_\mu^{\rm obs}=\mathcal S_\Sigma(\mathcal E)\,\Sigma_\mu,
\end{equation}

where $\mathcal E$ includes source structure, compactness, gradients, boundary conditions, and coherence scale. Temporal Topology denotes the continuous spatial and covariance structure of $\ln A$ and its temporal shear, not a discrete change of spacetime topology. The logistic $S(r)$ used in the Schwarzschild incompatibility analysis is a phenomenological radial activation compatible with this vocabulary; the coupled solution does not require $S(r)$ because the scalar profile is determined by the field equations.

## 2.6 The Observational Inference Problem

Because astrophysical mass is an inferred rather than direct measurement, its derivation depends on the assumed time-transfer map. The mass is reconstructed from observed angular positions, spectral shifts, signal periods and image scales under the assumption that source clocks, photon propagation and observer clocks can be mapped onto a single universal time coordinate. This paper identifies this assumption as the *Isochrony Axiom* — the hidden closure in black-hole inference. The TEP framework demonstrates that this reduction is not theory-neutral.

What the telescope directly records is functions of the observer's local proper time $\tilde\tau_o$:

\begin{equation} \label{eq:tep_isochrony_1}
\mathcal D_{\rm obs} = \left\{\theta_x(\tilde\tau_o),\; \theta_y(\tilde\tau_o),\; \nu_o(\tilde\tau_o),\; F_o(\tilde\tau_o)\right\},
\end{equation}

whereas the source trajectory is parameterised by a different local proper time $\tilde\tau_s$. The full source-to-observer map is

\begin{equation} \label{eq:tep_isochrony_2}
\tilde\tau_s \;\longrightarrow\; x^\mu_s(\tilde\tau_s) \;\longrightarrow\; k^\mu \;\longrightarrow\; \tilde\tau_o,
\end{equation}

with observed frequency $\omega_o = -k_\mu u_o^\mu$ and source-to-observer timing map

\begin{equation} \label{eq:tep_isochrony_3}
\frac{d\tilde\tau_o}{d\tilde\tau_s} = \mathcal T_{s\to o}\!\left[\phi,\, A,\, B,\, x_s^\mu,\, u_s^\mu,\, x_o^\mu,\, u_o^\mu\right].
\end{equation}

The standard analysis reconstructs orbital period, velocity, radius and mass by effectively setting $\mathcal T_{s\to o}$ to its GR or isochronous value. The familiar relation

\begin{equation} \label{eq:tep_isochrony_4}
M_{\rm app}^{\rm GR} = \frac{4\pi^2 a_{\rm GR}^3}{G P_{\rm GR}^2}
\end{equation}

therefore returns an apparent mass conditioned on that temporal calibration. It does not independently measure either the local material mass or the proper volume of the source region. The inferred mass is

\begin{equation} \label{eq:tep_isochrony_5}
M_{\rm app}^{\rm GR} = \mathcal F\!\left[\theta(\tilde\tau_o),\, \nu_o(\tilde\tau_o),\, D_{\rm assumed},\, \mathcal T_{s\to o}^{\rm GR},\, g_{\rm GR}\right],
\end{equation}

not a direct measurement of matter. The spectroscopic velocity has the same structure. The observed spectral shift is conventionally decomposed as $z_{\rm obs} = z_{\rm Doppler} + z_{\rm gravitational} + z_{\rm transverse} + \cdots$ and used to infer a radial velocity. But under TEP,

\begin{equation} \label{eq:tep_isochrony_6}
1 + z_{\rm obs} = \frac{(-k_\mu u^\mu)_e}{(-k_\mu u^\mu)_o}
\end{equation}

also contains the temporal-field transfer. Some quantity interpreted as $v_{\rm radial}$ may actually contain motion plus clock-rate difference plus open-path temporal transport. If that temporal component is forced into a Doppler model, it changes the inferred orbital speed and hence the inferred mass. Even the Galactic-centre distance, though largely inferred by combining angular astrometry with spectroscopic velocities rather than from a standard candle, still assumes $v_{\rm transverse} = D\,d\theta/dt$ and compares it with a spectroscopic radial velocity measured in the same effective time standard. Under TEP the correct relation is schematically $v_{\rm transverse}^{\rm local} = D_{\rm TEP}\,(d\theta/d\tilde\tau_o)\,(d\tilde\tau_o/d\tilde\tau_s)$. Even "geometric" orbital parallax is conditional on the mapping between source time and observer time.

The Cepheid analogy is exact at the methodological level. The TEP cosmological paper treats Cepheids as environment-dependent clocks whose altered periods are misread through a universal period–luminosity law, causing distance and $H_0$ bias. The black-hole analogue is: orbital motion supplies a periodic clock; spectral lines supply atomic clocks; variable plasma features supply dynamical clocks; ringdown supplies gravitational clocks. Standard analysis assumes these clocks can be compared through one universal time coordinate. TEP says their observed periods can include $P_{\rm observed} = P_{\rm intrinsic} \times \mathcal T_{\rm path} \times \mathcal T_{\rm environment}$. If those temporal factors are instead attributed to motion or curvature, the inferred mass becomes inflated. This is the same structural mistake as treating Cepheid period contraction as luminosity or distance.

The direct observables are: changing angular positions; changing received spectral frequencies; repeated signal patterns; strong redshift; strong lensing and image distortion; a dark central observational domain. The inferred quantities — a universal source-frame orbital period, a theory-independent physical orbital velocity, a theory-independent linear radius, millions of solar masses of matter, a tiny proper volume, an event horizon, a singularity — are reconstructed through the isochrony axiom. The conventional inference chain is:

\begin{equation} \label{eq:tep_isochrony_7}
\begin{aligned}
&\text{received angles, phases and frequencies} \\
&\quad\Downarrow\quad \text{isochrony axiom + GR transfer + distance calibration} \\
&\text{orbital period, velocity and radius} \\
&\quad\Downarrow\quad \text{Kepler/GR inversion} \\
&\text{large compact mass } (M_{\rm app}^{\rm GR}).
\end{aligned}
\end{equation}

TEP replaces it with:

\begin{equation} \label{eq:tep_isochrony_8}
\begin{aligned}
&\text{received angles, phases and frequencies} \\
&\quad\Downarrow\quad \text{dynamical temporal-transfer reconstruction} \\
&\text{local trajectory + clock-rate field} \\
&\quad\Downarrow\quad \text{material source + temporal contribution.}
\end{aligned}
\end{equation}

The precise statement is not that conventional analysis sets $\mathcal T_{s\to o}=1$ — GR includes gravitational redshift, transverse and line-of-sight Doppler shift, propagation delay, and Shapiro delay. The TEP claim is that conventional analysis fixes the transfer map to its *GR closure*, $\mathcal T_{s\to o} = \mathcal T_{s\to o}^{\rm GR}$. TEP tests whether a reproducible residual temporal field remains after the complete GR mapping is applied:

\begin{equation} \label{eq:tep_Delta_T}
\Delta_T \equiv \ln\frac{\mathcal T_{s\to o}^{\rm TEP}}{\mathcal T_{s\to o}^{\rm GR}},
\end{equation}

Conventional inference is not clock-blind; it is closed under the GR transfer law. Whether $\Delta_T \neq 0$ is required by the data is the central empirical question of the TEP programme.

## 2.7 Strong-Field phantom mass and the Mass Dictionary

This paper defines the strong-field counterpart of phantom mass in the wider TEP framework. At galactic scales, unmodelled temporal shear is interpreted as a dark-matter halo. At black-hole scales, extreme temporal transport can be interpreted as a very large central mass, rapid infall and a compact causal object. The conventional mass may combine a material contribution, temporal-field energy, and a non-isochronous calibration residual. The more exact expression is

\begin{equation} \label{eq:tep_phantom_2}
M_{\rm app}^{\rm GR} = \mathcal F\!\left(M_{\rm matter},\, M_\phi,\, M_{\rm int},\, A,\, B,\, \mathcal T_{s\to o},\, D_{\rm TEP}\right),
\end{equation}

where $M_{\rm matter}$ is the locally measured material content, $M_\phi$ is the temporal-field energy, $M_{\rm int}$ is the interaction energy, and $A, B, \mathcal T_{s\to o}, D_{\rm TEP}$ are the temporal-field profile and transfer functions. The strong-field phantom mass residual is *defined* as the difference between the conventional fitted mass and the TEP material inference:

\begin{equation} \label{eq:tep_phantom_3}
M_{\rm phantom}^{T} \equiv M_{\rm fit}^{\rm GR} - M_{\rm matter}^{\rm TEP}.
\end{equation}

This is a residual definition, not a derived decomposition. The sign of $M_{\rm phantom}^{T}$ is not assumed; it is determined by the data through the sign equation of Section 2.8. The schematic additive form $M_{\rm app}^{\rm GR} = M_{\rm local}^{\rm TEP} + M_{\rm phantom}^{T}$ is shorthand for the residual definition, valid only when $M_\phi$ and $M_{\rm int}$ are absorbed into $M_{\rm phantom}^{T}$.

The paper uses $M$ in several distinct roles, and the distinction matters once the mass itself is in question. The strict dictionary is:

| Symbol | Meaning |
| --- | --- |
| $M_{\rm GR-fit}$ | The parameter returned by a conventional GR reduction of the observations |
| $M_g$ | The asymptotic gravitational charge associated with $g_{\mu\nu}$ |
| $M_{\rm matter}$ | The locally measured material content $M_{\rm matter} = \int_\Sigma \tilde T_{\mu\nu}\tilde u^\mu\tilde u^\nu\,d\tilde V$ |
| $M_\phi$ | The temporal-field energy or effective contribution |
| $M_{\rm phantom}^{T}$ | The difference between the conventional fitted mass and the TEP material inference |
| $M$ | The GR-calibrated exterior mass parameter used in the conditional sGB benchmark (Sections 4–8) |

The statement that a fixed ADM $M$ is "the physical mass measured by a distant observer" is not made in this paper. Under the paper's own thesis, that is precisely what the TEP programme determines. The sGB calculations of Sections 4–8 use $M$ as the GR-calibrated exterior mass parameter; their observable predictions are conditional on that calibration. The central empirical task is to determine how much of the conventional central mass survives as locally measured material content when the Isochrony Axiom is removed. The decisive analysis is a *TEP-native refit of the raw observations*, not a perturbation of a pre-assumed Schwarzschild mass.

## 2.8 Sign and Identifiability of Strong-Field phantom mass

A direct intuition might suggest that because deep clocks run slow, observed periods are longer, so the inferred mass is larger. This overlooks the spatial-calibration terms that can accompany temporal stretching, and recognising the interplay between the two is the first gate of the entire analysis.

Consider a source orbiting in the deep region with local orbital period $P_s$ and local semi-major axis $a_{\rm local}$. A distant observer measures period $P_o$ and infers radius $a_{\rm GR}$. The temporal transfer stretches the period:

\begin{equation} \label{eq:tep_sign_1}
P_o = \mathcal T_P \, P_s, \qquad \mathcal T_P = \frac{d\tilde\tau_o}{d\tilde\tau_s} > 1.
\end{equation}

The spatial calibration relates the GR-inferred orbit to the local orbit:

\begin{equation} \label{eq:tep_sign_2}
a_{\rm GR} = \mathcal S_a \, a_{\rm local},
\end{equation}

where $\mathcal S_a$ encodes lensing magnification, distance miscalibration, and orbital-dynamics modification — all sourced by the temporal field. The conventional Kepler inversion gives:

\begin{equation} \label{eq:tep_sign_3}
M_{\rm app}^{\rm GR} = \frac{4\pi^2 a_{\rm GR}^3}{G P_o^2} = \frac{4\pi^2 \mathcal S_a^3 a_{\rm local}^3}{G \mathcal T_P^2 P_s^2}.
\end{equation}

The local mass, using the local period and scale, with a possible dynamical-law modification $\mathcal D_{\rm dyn}$ ($\mathcal D_{\rm dyn} = 1$ for Newtonian), is:

\begin{equation} \label{eq:tep_sign_4}
M_{\rm local}^{\rm TEP} = \frac{4\pi^2 a_{\rm local}^3}{G P_s^2 \, \mathcal D_{\rm dyn}}.
\end{equation}

The ratio is:

\begin{equation} \label{eq:tep_sign_5}
\boxed{\frac{M_{\rm app}^{\rm GR}}{M_{\rm local}^{\rm TEP}} = \frac{\mathcal S_a^3 \, \mathcal D_{\rm dyn}}{\mathcal T_P^2}.}
\end{equation}

If the spatial scale is held fixed ($\mathcal S_a = 1$, $\mathcal D_{\rm dyn} = 1$), then $M_{\rm app} = M_{\rm local} / \mathcal T_P^2 < M_{\rm local}$. *Slow deep clocks alone deflate the inferred mass*, producing $M_{\rm phantom}^{T} < 0$. This is the opposite of the phantom mass claim. Positive phantom mass requires the spatial magnification to dominate:

\begin{equation} \label{eq:tep_sign_6}
M_{\rm phantom}^{T} > 0 \quad \Longleftrightarrow \quad \mathcal S_a^3 \, \mathcal D_{\rm dyn} > \mathcal T_P^2.
\end{equation}

The spatial calibration factor $\mathcal S_a$ can differ from unity through timelike orbital calibration, endpoint distance inference, disformal propagation, and scalar backreaction on $g_{\mu\nu}$. Pure conformal rescaling does not by itself alter null trajectories. Appendix L isolates the scalar–Gauss–Bonnet backreaction and matter-orbit contributions.

The spatial calibration $\mathcal S_a$ is not a free parameter. It is determined by photon propagation on $\tilde g_{\mu\nu}$ (lensing magnification), the GR-inferred distance $D_{\rm GR}$ (standard-candle or parallax calibration through the temporal field), and the orbital dynamics on the temporal-well geometry. The critical threshold is $\mathcal S_a > \mathcal T_P^{2/3}$. For moderate temporal transfer $\mathcal T_P \sim 10$, this requires $\mathcal S_a > 4.6$ — a strong but achievable lensing magnification near a strong-field source.

The velocity-based mass estimator gives a complementary relation. With $v_{\rm GR} = \mathcal S_v \, v_{\rm local}$ (spectroscopic velocity calibration):

\begin{equation} \label{eq:tep_sign_7}
\frac{M_{\rm app}^{\rm GR}}{M_{\rm local}^{\rm TEP}} = \mathcal S_v^2 \, \mathcal S_a \, \mathcal D_{\rm dyn}.
\end{equation}

The two estimators (Kepler and velocity) must agree in any consistent theory. Their joint constraint tightens the identifiability of $\mathcal S_a$ and $\mathcal T_P$.

Three outcomes are possible and the pipeline must be allowed to return any of them:

| Outcome | Condition | Interpretation |
| --- | --- | --- |
| $M_{\rm phantom}^{T} > 0$ | $\mathcal S_a^3 \mathcal D_{\rm dyn} > \mathcal T_P^2$ | Spatial magnification dominates. Conventional mass is inflated. $M_{\rm matter} < M_{\rm GR\text{-}fit}$. |
| $M_{\rm phantom}^{T} = 0$ | $\mathcal S_a^3 \mathcal D_{\rm dyn} = \mathcal T_P^2$ | Effects cancel. Conventional mass equals local material mass. TEP reduces to GR for mass inference. |
| $M_{\rm phantom}^{T} < 0$ | $\mathcal S_a^3 \mathcal D_{\rm dyn} < \mathcal T_P^2$ | Temporal stretching dominates. Conventional mass is deflated. $M_{\rm matter} > M_{\rm GR\text{-}fit}$. |

A pipeline structured to return only the first outcome would not constitute a test of the thesis. The TEP-native refit (Section 9, Appendix J) must determine the sign from the data, not assume it.

Two distinct hypotheses must be separated in the observational analysis. The *composition hypothesis* $H_{\rm composition}: M_{\rm matter} < M_{\rm GR\text{-}fit}$ states that the strong-field source contains less material mass than the conventional fit suggests. The *calibration hypothesis* $H_{\rm calibration}: M_g < M_{\rm GR\text{-}fit}$ states that the large exterior gravitational parameter itself is a calibration illusion. These are different claims: a temporal field can contribute real energy and backreaction, so TEP might find $M_{\rm matter} \ll M_{\rm GR\text{-}fit}$ while $M_g \sim M_{\rm GR\text{-}fit}$. In that case there is little compressed matter, but there remains a large physical temporal–geometric charge. That would still be a major TEP result, but it would not mean the entire dynamical mass was merely a data-reduction illusion. The raw-observation fit must test both separately.

A separate question is whether "finite density" at the regular centre implies "no extreme compression." A regular-centre geometry can still represent a physically ultra-dense source with extraordinarily high density. The intended TEP vision is stronger: no extreme local compression. This requires an explicit proper-volume diagnostic. Define the proper volume inside a boundary radius $r_b$:

\begin{equation} \label{eq:tep_proper_volume}
V_{\rm proper}(r_b) = 4\pi\int_0^{r_b} \sqrt{\tilde\gamma_{rr}}\,\tilde R^2(r)\,dr,
\end{equation}

and the local mean density $\bar\rho_{\rm local} = M_{\rm matter} / V_{\rm proper}(r_b)$. The critical diagnostic is the volume ratio:

\begin{equation} \label{eq:tep_CV}
\mathcal C_V \equiv \frac{V_{\rm proper}}{(4\pi/3)\,r_{\rm app}^3},
\end{equation}

where $r_{\rm app}$ is the externally inferred apparent radius. A value $\mathcal C_V \gg 1$ would demonstrate that the externally compact image corresponds to a much larger local proper volume — apparent compactness without physical compression. Without this calculation, "apparent compactness $\neq$ physical compression" remains an interpretation rather than a demonstrated property. The proper-volume diagnostic is a required output of the TEP-native refit.

# 3. Why Schwarzschild Is Not Enough

A natural question: why can TEP not simply add a temporal field onto ordinary Schwarzschild geometry? This section answers that question. The geometric metric is held fixed to Schwarzschild, and the conformal factor takes the power-law form $A = (r_h/r)^{\phi_0}$ in the deep interior. The exponent $\phi_0$ is allowed to take either sign: with the universal coupling $A = e^{\beta_A\phi}$ and $\beta_A = -1$, the branch $\phi_0 > 0$ gives $A \to \infty$, equivalently $\phi \to -\infty$, while the branch $\phi_0 < 0$ gives $A \to 0$, equivalently $\phi \to +\infty$. The Rule-4 sign convention $\phi > 0$ in mass concentrations selects $\phi_0 < 0$ as the physical temporal-well branch — clocks slow toward the centre and freeze as $A \to 0$; the positive-$\phi_0$ analysis is retained below as a complementary benchmark on the opposite branch. The question is whether the conformal matter metric can simultaneously achieve curvature regularity and bounded areal radius. The answer is no on either branch, and unconditionally so on the physical one. If the temporal field becomes strong enough to create a Temporal Horizon, the geometry cannot remain ordinary Schwarzschild. The geometry has to respond. The full disformal generalisation — classifying all allowed asymptotics of $A$, $B$, $\phi$ — is a separate mathematical programme; the conformal case already establishes the incompatibility.

## 3.1 The Compatibility Conditions

Consider the conformal matter metric $\tilde g = A^2 g_{\rm Schw}$ with $A = (r_h/r)^{\phi_0}$ in the deep interior, on a fixed Schwarzschild background. The exact Kretschmann scalar scales as

\begin{equation} \label{eq:fbt_1}
\tilde K \sim r^{4\phi_0 - 6}
\end{equation}

(Appendix D), not $r^{12\phi_0-6}$ as the naive conformal formula suggests — the derivative terms from the non-constant conformal factor dominate. The areal radius scales as

\begin{equation} \label{eq:fbt_2}
\rho = A\,r \sim r^{1-\phi_0}.
\end{equation}

Two conditions are required for a regular temporal well:

- Finite curvature ($\tilde K$ finite or vanishing) requires $\phi_0 \geq 3/2$. At $\phi_0 = 3/2$ the leading curvature is finite and constant; for $\phi_0 > 3/2$ the curvature vanishes.

- Bounded areal radius ($\rho$ finite) requires $\phi_0 \leq 1$.

On the benchmark branch ($\phi_0 > 0$, $A \to \infty$) these are mutually exclusive when the geometric metric is held fixed to Schwarzschild. On the physical temporal-well branch ($\phi_0 < 0$, $A \to 0$) the incompatibility is stronger still: the exponent $4\phi_0 - 6$ is strictly less than $-6$ for every $\phi_0 < 0$, so $\tilde K$ diverges faster than the Schwarzschild $r^{-6}$ singularity for every power-law index, while $\rho \sim r^{1-\phi_0} \to 0$ is trivially bounded. The conformal suppression of the volume element cannot overcome the background curvature divergence precisely because $A$ vanishes. Finite curvature therefore fails unconditionally on the physical branch — no admissible $\phi_0 < 0$ rescues it — so on the branch the theory actually realises, fixed Schwarzschild geometry cannot support the temporal geometry that TEP requires.

## 3.2 The Limiting Cases

On the benchmark branch, the limiting case $\phi_0 = 2$ achieves vanishing curvature: $\tilde K \sim 39r^2/16 \to 0$ (Appendix D). The matter metric is Lorentzian and nondegenerate throughout the open domain $r > 0$ ($\det\tilde g_{2D} < 0$ for all $r > 0$, since $\det\tilde g_{2D} = A^4 \det g_{2D}$ and $A > 0$). However, the areal radius diverges: $\rho \sim r_h^2/r \to \infty$. The $r \to 0$ limit is an asymptotic spatially enlarged end, not an ordinary manifold point. The conformal scaling stretches both time and space — the interior is spatially enlarged, not a black-hole interior.

The complementary benchmark-branch limit $\phi_0 = 1$ keeps the areal radius bounded but the curvature diverges: $\tilde K \sim 9/r^2 \to \infty$. The singularity is not resolved.

On the physical branch the structure is simpler and stronger. For every $\phi_0 < 0$ the Kretschmann exponent satisfies $4\phi_0 - 6 < -6$, so the matter-metric curvature diverges faster than Schwarzschild's own $r^{-6}$ regardless of how slowly $A$ falls, while the areal radius $\rho \sim r^{1-\phi_0}$ collapses to zero. The two conditions are not merely mutually exclusive — the finiteness condition fails outright for the entire physical branch. The physical interpretation matches the temporal-well picture: $A \to 0$ freezes all matter clocks, and the would-be interior collapses to vanishing areal radius rather than hosting a regular geometry. A Schwarzschild background therefore admits no conformal temporal well of power-law form in either sign of the exponent: the positive branch trades curvature for an enlarged end, the physical branch produces divergence unconditionally.

This is a structural incompatibility within the tested class, not a failure of a particular choice: fixed Schwarzschild geometry cannot support both conditions simultaneously, and on the physical branch cannot support even one of them. A systematic gate search over the tested disformal parameter space is reported in Appendix D.

## 3.3 Physical Principle

The incompatibility is the mathematical expression of a physical principle: because gravity depends on how things move through time, changing time must also change gravity. The temporal field cannot sit inertly atop a Schwarzschild geometry; it must backreact on the geometric metric itself. The mutual exclusion of finite curvature and bounded areal radius is proved for conformal matter metrics on fixed Schwarzschild; full disformal asymptotics remain open and do not weaken the backreaction requirement for the clock sector. Within the tested class, a dynamical proper-time field necessarily implies a dynamical gravitational geometry. A fixed Schwarzschild singular geometry cannot satisfy the conditions for a physically admissible temporal well. Any viable strong-field realisation of TEP must arise from a dynamically coupled temporal–geometric solution. Section 4 establishes the required backreaction channel and the regular geometric architecture; their joint nonlinear realisation is selected by the TEP field equations.

## 3.4 Raychaudhuri Equation and the Convergence Condition

On the globally Lorentzian domain, a hypersurface-orthogonal timelike congruence obeys

\begin{equation} \label{eq:fbt_3}
\frac{d\tilde\theta}{d\tilde\tau}=-\frac13\tilde\theta^2-\tilde\sigma_{\mu\nu}\tilde\sigma^{\mu\nu}-\tilde R_{\mu\nu}\tilde u^\mu u^\nu.
\end{equation}

The sign of $\tilde R_{\mu\nu}\tilde u^\mu u^\nu$ is the geometric statement of the timelike convergence condition. On a regular background with a temporal-rate minimum, the effective pressure is negative in the causal matter frame, and the convergence condition fails over the interior domain. This is the Hawking–Penrose escape clause: with the convergence condition violated, the singularity theorems' conclusion of inevitable geodesic incompleteness no longer applies to $(\mathcal M, \tilde g)$. In a two-metric theory, the Penrose–Hawking hypotheses must be checked separately for $(\mathcal M,g)$ and $(\mathcal M,\tilde g)$; incompleteness of Schwarzschild $g$ does not logically prove incompleteness of $\tilde g$. The analysis of this section shows that $\tilde g$ on fixed Schwarzschild cannot be both curvature-regular and spatially bounded — and on the physical $A \to 0$ branch cannot be curvature-regular at any power-law index; the regular-geometry benchmark (Section 5, Appendix K) shows that $\tilde g$ on a regular background can achieve finite curvature and bounded areal radius.

# 4. Existence Demonstrations

The Schwarzschild incompatibility result determines the admissible class of strong-field solutions: the temporal field must backreact on the geometric metric, and the resulting geometry must simultaneously achieve finite matter-frame curvature, continuous temporal transport, real exterior hair, and dynamical backreaction. The Schwarzschild obstruction is not a general geometric impossibility. Regular geometry, bounded matter-frame curvature, and dynamical scalar backreaction are each explicitly attainable — established separately by the two demonstrations below. Their joint realisation in one dynamically generated solution is the nonlinear closure selected by the TEP action.

## 4.1 Regular Geometry Exists

The Hayward metric $F(r) = 1 - 2Mr^2/(r^3 + 2M\ell^2)$ is a regular geometric background: the Kretschmann scalar is finite everywhere, $K_{\rm Hayward}(0) = 24/\ell^4$, and the effective energy density is finite, $\rho_{\rm eff} \to 3/\ell^2$ at the centre. The radial equation of state is $w_r = -1$ — cosmological-constant-like — with anisotropic tangential pressure. This profile is the signature of a higher-curvature coupling: a scalar field's stress-energy, modified by the Gauss-Bonnet invariant, can in principle produce anisotropic effective pressure with a de Sitter-like radial component.

The demonstration shows that singularity is optional, not mandatory. Once the geometric singularity is absent, the TEP matter metric simultaneously achieves bounded areal radius and finite Kretschmann (Section 5, Appendix K). The burden shifts: GR must justify why the singular reconstruction is mandatory, not why TEP must reproduce every deep-region coordinate chart. The Hayward metric is one example of a regular geometry; it is not the TEP-selected solution.

## 4.2 Dynamical Coupling Exists

The scalar-Gauss-Bonnet (sGB) coupling $\alpha_{\rm GB}\,\phi\,\mathcal{G}$ is the leading curvature operator of the strong-field EFT (Section 2.1). It is the lowest-dimension higher-curvature term that couples the temporal field to the gravitational geometry and supplies real backreaction. The published perturbative solution (Sotiriou & Zhou 2014, exact $\mathcal{O}(\alpha^2)$) establishes:

- Scalar hair around strong-field sources — the no-hair theorem is evaded because the Gauss-Bonnet coupling provides a geometric source term.

- Real backreaction on the gravitational metric — the temporal field is not epiphenomenal.

- Sector-dependent observables — the shadow is sensitive to the geometric metric at $\mathcal{O}(\eta^2)$ while the ISCO feels the conformal factor at $\mathcal{O}(\eta)$, because null geodesics are conformally invariant while timelike geodesics are not (Appendix L).

What sGB does *not* establish: a regular deep region. The nonlinear solutions of Sotiriou & Zhou (2014) for the same linear sGB coupling develop a finite-area singularity rather than a regular temporal minimum, and recent simulations (Thaalba et al. 2024) confirm this and explore a possible connection to loss of hyperbolicity. The TEP-selected solution must satisfy the minimal temporal-well criteria: $0 < N(r)$ and $N_{\min} \ll N_o$ everywhere, with no observer-independent one-way boundary. The leading curvature operator supplies the backreaction channel; the regularising nonlinear coefficients that complete the deep-region profile are selected by the joint requirements of global regularity and observation. This is the construction programme — solving the fixed EFT architecture, not choosing among theories.

Rule 20 requires the TEP matter clock rate to reach a finite, non-zero minimum ($N_{\min}>0$), together with a gravitationally regular static geometry whose radial coordinate remains spacelike. These are joint conditions on the two metrics and must be checked in the coupled solution. The fully coupled Kantowski–Sachs integration is retained as a constructive strong-field benchmark: it identifies the nonlinear kinetic and potential terms that arrest the curvature divergence and exhibits a finite-curvature regularising turn. Its auxiliary areal-radius evolution is a diagnostic continuation of the interior equations, not the final Rule-20 ontology. The physical TEP branch selected here is the static, finite-$N_{\min}$ temporal well; it does not expand and the completed field equations must remain static in the final spatial interior.

The regularising floor $V_0$ of the corpus master potential—the single analytic form $V(u) = (\lambda/4)\,u^{4}e^{-(u/u_s)^{4}} + V_0\,e^{-(u_s/u)^{4}}$—is a property of the same scalar field that supplies the weak-field screening branch. In the coupled benchmark, $u_s=10$, $V_0=0.3M_{\rm Pl}^4$, and $\lambda=7.5\times10^{-66}$ — the Cassini-compatible branch, $\lambda_{\rm Cassini} = 10^{5}\lambda_{\rm ref}$ above the reference normalization $\lambda_{\rm ref}=7.5\times10^{-71}$ (Paper 0, step_03; both values are fixed entries of the shared corpus constants, not a free choice per paper) — produce a finite-curvature turn and a non-degenerate kinetic matrix; both the unit-term benchmark completion $P = X - V + X|X|/\Lambda_X^4$ and the selected two-branch kinetic sector $P_{,X}=\varepsilon_{\rm reg}+k\sqrt{\xi}+2\xi$ ($k=16.03$, $\varepsilon_{\rm reg}=10^{-3}$) arrest the gradient divergence that defeats the first-order sGB branch. In the static 4-variable coupled Einstein–scalar boundary-value solve on $(\phi, q, m, \Phi)$ with canonical radial flux $q=r^2 e^\Phi\sqrt{F}\,P_{,\xi}(\xi)\phi'$ and Einstein-frame conformal matter coupling $\rho_E=\rho_b e^{-\phi}$ (`step_47_well_bvp_diagnostic.json`), the conformal factor $e^{-\phi_c}<1$ self-screens the central gravitating energy density, producing $5\times$ deeper scalar wells ($\phi_c=0.2141$, $A_c=0.8073$ vs $\phi_c=0.0437$ in the unscreened reduced benchmark at $\rho_b=0.020$) while keeping the core further from horizon formation ($F_{\min}=0.4388$ vs $0.3272$) and extending the horizon-free static branch to $\rho_b=0.030$ ($F_{\min}=0.1854$, clock-freeze ratio $4.69\times$, RMS collocation residual $3.99\times10^{-7}$, with strictly positive radial principal symbol $Z_\parallel\ge 10^{-3}$ everywhere). This is constructive evidence that the master action contains the nonlinear kinetic and conformal self-screening ingredients required for a regular static temporal-well branch; extending this static BVP family to the full astrophysical compactness and mass hierarchy with sGB curvature sourcing remains the global boundary-value task. Its core density is therefore a universal theory target, and supermassive objects such as M87* and Sgr A* are the same temporal-well construction target at a larger spatial scale, rather than a separate mass-tuned model.

All reported exterior observables — the EHT shadow central values for M87* and Sgr A*, the ISCO structure, the QPO resonance radii, and the S-star orbits — are computed on the perturbative sGB exterior benchmark and probe the geometry at and outside the photon sphere. Their validity is fully consistent with the continuous, scale-invariant static temporal-well interpretation because those observables are set outside the deep regularising region. No mass-dependent retuning of couplings is permitted. The remaining global task is to connect that demonstrated exterior and constructive interior benchmark with one static family having positive physical lapse, finite curvature and no observer-independent horizon throughout the mass range.

## 4.3 The Inverse Reconstruction

Computing the effective stress-energy required to produce a Hayward-like regular metric identifies $w_r = -1$ as a sufficient regularity target — one sufficient condition, not a uniqueness theorem. The effective energy density is

\begin{equation} \label{eq:coupled_8}
\rho_{\rm eff} = -G^t_t = \frac{12M^2\ell^2}{(r^3 + 2M\ell^2)^2},
\end{equation}

finite at the centre with $w_r = -1$. This is the stress-energy profile that a regular temporal well *would* require; it does not prove that standard linear sGB produces it. The full nonlinear integration is the key test. The detailed sGB action, perturbative metric corrections, matter-metric construction, and exterior observable benchmarks are in Appendices B and L.

## 4.4 Status

The Schwarzschild obstruction is not a general geometric impossibility. Regular geometry, bounded matter-frame curvature, and dynamical scalar backreaction are each explicitly attainable. Their joint realisation is the nonlinear closure selected by the TEP action. The canonical matter coupling fixes the temporal sector; the leading curvature operator supplies backreaction; the regularising nonlinear coefficients are selected by global regularity and observation. The construction programme is solving the global field equations of this theory — not selecting among competing theories. The theory is carried by the principle, the action, and the field equations; the global solution is what is under construction.

# 5. Global Geometry and Curvature Regularity

The existence demonstrations of Section 4 require a regular geometric background. This section confirms that finite curvature, bounded areal geometry, and Lorentzian signature can coexist on a regular background. The physical Temporal Well additionally requires $N_{\min}\ll N_o$. Three properties are established; detailed calculations are in Appendix K.

## 5.1 Three Results

- **Global Lorentzian signature.** The radial-block determinant $\det\tilde g_{2D} = -A^4 G^2 - A^2 B\,F\,(\phi')^2$ is strictly negative for all $r > 0$. The determinant remains strictly negative throughout the tested domain, establishing a globally Lorentzian benchmark without a finite-radius signature boundary. This benchmark tests geometric regularity; the physical Temporal Well requires the separate condition ($N_{\min}\ll N_o$).

- **Bounded curvature.** The Kretschmann scalar approaches a finite value at the centre, $\tilde K \to 8/r_h^4$, in contrast to the Schwarzschild value $K_{\rm Schw} = 48\,M^2/r^6 \to \infty$. All curvature invariants remain finite throughout.

- **Bounded areal radius.** The areal radius $\rho = A\,r$ remains finite as $r \to 0$. On a Schwarzschild background the conformal scaling drives $\rho \to \infty$, illustrating the Schwarzschild incompatibility (Section 3): curvature regularity and bounded areal radius are mutually exclusive when the geometric metric is held fixed. On the regular background both conditions are satisfied simultaneously.

## 5.2 Frame Dependence of Density

The classical density estimate $M/(4\pi r^3/3)$ presupposes that the geometric metric governs matter. In a two-metric theory the physical density is defined from the matter stress tensor and a spacelike hypersurface with induced metric $\tilde\gamma_{ij}$: $d\tilde V = \sqrt{\det\tilde\gamma_{ij}}\,d^3x$, $\tilde\rho = \tilde T_{\mu\nu}\,u^\mu u^\nu$. The frame-dependence of volume is a structural feature of TEP: the classical density inference assumes the geometric metric governs matter, which is precisely what TEP questions. The geometric Einstein-tensor component, the matter-frame stress, and the TEP matter stress tensor are distinct objects and must be separated explicitly (Appendix K).

## 5.3 Status

These results establish that finite curvature, bounded areal geometry, and Lorentzian signature can coexist once the fixed Schwarzschild singularity is removed. The nonlinear TEP solution combines these geometric conditions with the defining slow-clock condition ($N_{\min}\ll N_o$). The scalar-on-regular-background benchmark (Appendix K) confirms that the sGB scalar field on a regular background produces a strictly positive lapse, vanishing areal radius at the centre, and Lorentzian signature throughout. The specific numerical values are properties of the benchmark; the structural conclusion is general.

# 6. Causal Structure

The event horizon is not an observable. It is a global causal construct inferred from the spacetime model. No telescope has ever measured an event horizon. What is observed is that signals from certain regions become extremely redshifted, delayed, and faint. The Temporal Horizon is the operational boundary defined directly from observable temporal accessibility. The causal structure of TEP is fundamentally different from the standard black-hole picture: the standard "black hole" is replaced by the *temporal well* — a regular spatial region in which the rate of proper time differs radically from the exterior (Section 1.1). *Darkness does not require an event horizon; it requires sufficiently extreme temporal decoupling.*

## 6.1 The Operational Temporal Horizon

The Temporal Horizon was introduced in the cosmological context (Thika) as the conformal-temporal boundary where $A_{\rm clock} \to 0$. This paper extends the concept to the strong-field regime. The shared name marks a shared mechanism — clock-transport as boundary — not a shared definition. In Thika the Temporal Horizon is the past conformal boundary $\mathscr{T}^-$ where $A_{\rm clock} \to 0$. Here it is the observer-relative accessibility set $\mathcal{H}_T^{(\Lambda)}[u_o] = \{x_e : \mathcal{T}_{e\to o} \geq \Lambda\}$. Cosmological $\mathscr{T}^-$ is not the black-hole surface $\mathcal{H}_T$; both are reconstructions of dynamical proper time. Around a compact source, the relevant boundary is an observer-relative accessibility threshold — the surface beyond which the clock-transfer factor renders signals practically undetectable. The physical lapse $N(r) = \sqrt{-\tilde g_{\mu\nu}\xi^\mu\xi^\nu}$ satisfies $0 < N(r) < 1$ through the deep region, with $N(r) \ll 1$ where the object appears black, but $N(r) \neq 0$ at every finite physical location. The TEP-selected solution must have $N(r) \geq N_{\min} > 0$ everywhere — the lapse has a finite minimum at the temporal-rate minimum, never vanishing at any finite radius. The perturbative sGB benchmark belongs to a horizon-bearing branch that does not realise the temporal well; the full TEP action must produce $N(r) \geq N_{\min} > 0$ everywhere.

The Temporal Horizon is defined by the clock-transfer factor between an emitter $e$ and observer $o$:

\begin{equation} \label{eq:temporal_horizon_def}
\mathcal{T}_{e\to o} = \frac{d\tilde\tau_o}{d\tilde\tau_e} = \frac{\omega_e}{\omega_o} = \frac{(-k_\mu u^\mu)_e}{(-k_\mu u^\mu)_o},
\end{equation}

where $k_\mu$ is the signal wavevector and $u^\mu$ is the observer's four-velocity. For stationary observers with matter-frame lapse $N(x)$, the Killing energy is conserved, giving $\mathcal{T}_{e\to o} = N_o/N_e$. The maximum transfer factor from the temporal-rate minimum is finite: $\mathcal{Z}_{\max} \sim N_o/N_{\min}$ — potentially astronomically large but never divergent. The object appears black because the transfer factor exceeds any practical accessibility threshold, not because signals are absolutely forbidden from escaping. The operational Temporal Horizon at threshold $\Lambda$ is:

\begin{equation} \label{eq:operational_TH}
\mathcal{H}_T^{(\Lambda)}[u_o] = \left\{x_e \in J^-(\gamma_o) : \mathcal{T}_{e\to o} \geq \Lambda\right\}.
\end{equation}

This boundary is not null, not one-way, and does not divide spacetime into an outside and an inside. It changes with the observer's proper-time rate, motion, and worldline: $\mathcal{H}_T^{(\Lambda)}[u_1] \neq \mathcal{H}_T^{(\Lambda)}[u_2]$. The black-hole observables used here are open-path transfer factors $\mathcal{T}_{e\to o}$; they do not require residual closed-loop synchronization holonomy, which remains a disformal/Jakarta programme probe. A freely falling observer moving through the deep temporal region experiences ordinary local space, a normally running local clock, no wall, no horizon crossing, no signature change. The distant observer's Temporal Horizon is not their Temporal Horizon.

## 6.2 Transfer, Refraction, and Darkness

The unifying mechanism is the coordinate propagation rate. In any local Lorentzian frame of $\tilde g_{\mu\nu}$, the locally measured speed of light is exactly $c$ — no local rule is broken. But the physical lapse $N(r)$ drops to an extremely small fraction in the deep temporal region, so from the perspective of a distant fast-clock observer, the coordinate propagation rate of light through that region is drastically slowed: $dr/dt \sim N(r) \ll 1$. This single mechanism — light propagating through a gradient where the rate of time varies — produces three observational consequences that the standard picture attributes to three separate phenomena:

- **Transfer.** The clock-transfer factor $\mathcal{T}_{e\to o} = N_o/N_e$ grows as the emitter moves deeper, reaching a finite maximum $\mathcal{Z}_{\max} \sim N_o/N_{\min}$. When this exceeds any practical threshold, the object appears dark. Darkness is extreme temporal decoupling, not an absorbing boundary.

- **Refraction.** The slowing bends trajectories: light curves when the propagation rate varies through a medium, exactly as in refraction, producing the apparent "inward pull" and gravitational lensing without a sideways force. The conformal factor $A(\phi)$ controls clock-rate transfer and massive-particle response; the disformal term $B(\phi)$ can alter causal cones; the backreacted geometric metric $g_{\mu\nu}[\phi]$ generates light bending and orbital curvature. Lensing is generated by the complete matter metric $\tilde g_{\mu\nu}$, not by conformal clock scaling alone.

- **Extended transit.** Waves crossing the deep region take far longer in coordinate time — the deep-transit ringdown modes show extended damping because the wave transits a zone where proper time runs at a small fraction of the exterior rate (Section 7).

Photons, massive particles, and gravitational waves are all governed by the same unified temporal field — they are different observational probes of the same temporal well, not evidence of fractured metric structure. In the perturbative sGB benchmark, photons and massive particles respond to the temporal field at different coupling orders (null geodesics are conformally invariant, timelike geodesics are not), but this is a property of the mathematical toolkit, not a fundamental law of TEP. Detailed null-expansion calculations, Eddington–Finkelstein null algebra, and the invariant frequency-transfer derivation are in Appendix B.

The darkness criterion can be made quantitative (Step 50, `step_50_darkness_threshold.py`). For a stationary emitter at lapse $N_e$ and a distant static observer ($N_o \to 1$), conformal invariance of null transport conserves the Liouville quantity $I_\nu/\nu^3$: the transferred specific intensity is suppressed by $\mathcal{T}^{-3}$, the bolometric surface brightness by $\mathcal{T}^{-4}$, and the photon arrival rate by $\mathcal{T}^{-1}$, where $\mathcal{T} = N_o/N_e$. An interior emitter of intrinsic emissivity contrast $C_{\rm em}$ relative to the photon-ring emission is invisible in the image when the transferred intensity falls below the interior-to-ring dynamic-range floor $\varepsilon$: $\mathcal{T} > \mathcal{T}_{\rm dark} = (C_{\rm em}/\varepsilon)^{1/3}$. For interferometric imaging of EHT class ($\varepsilon \sim 10^{-1}$–$10^{-3}$) and emissivity contrasts up to $C_{\rm em} \sim 10^{6}$, the threshold spans $\mathcal{T}_{\rm dark} \sim 2$–$10^{3}$. The corrected scalar-on-regular-background benchmark (Section 4) reaches $\mathcal{Z}_{\max} \simeq 6.7 \times 10^{3}$ at the temporal-rate minimum $N_{\min} = 1.5 \times 10^{-4}$ ($r \simeq 0.79\,M$), exceeding the most conservative threshold by a factor $\sim 7$ and the nominal threshold ($\varepsilon = 10^{-2}$, $C_{\rm em} = 10^{2}$, $\mathcal{T}_{\rm dark} \simeq 21.5$) by $\sim 310$: interior emission is dimmed to $\sim 3 \times 10^{-12}$ of its intrinsic specific intensity and $\sim 5 \times 10^{-16}$ bolometrically. Massive messengers are more strongly suppressed: a particle at rest in the well carries Killing energy $E = mN_e$ and requires a boost $\sim \mathcal{Z}_{\max}$ to reach the exterior, so neutrino and cosmic-ray channels are confined rather than merely redshifted. Photons are not trapped: by conformal invariance their coordinate transit through the deep region is set by the geometric lapse alone, $\Delta t = \int dr/F(r) \simeq 33\,M$ for the $g = 1.1M$ core — of order $12$ minutes for Sgr A$^{\ast}$ and $12$ days for M87$^{\ast}$ — so individual photons escape, but they arrive dimmed, spectrally shifted ($\nu_o = \nu_e/\mathcal{T}$), and temporally diluted ($d\tilde\tau_o = \mathcal{T}\,d\tilde\tau_e$). The darkness is enforced by the intensity suppression, not by signal trapping: the deep region is operationally black on the corrected benchmark, and $\mathcal{T}_{\rm dark} \sim 10^{1}$–$10^{3}$ is the quantitative requirement the coupled nonlinear interior must satisfy. Sensitivity branches that screen the scalar charge of the deep region — the exact Hayward-source variant, in which the background Gauss–Bonnet invariant changes sign in an interior shell, giving $\mathcal{Z}_{\max} \simeq 4.1$ — fall short of the nominal threshold; the declared benchmark source (the Schwarzschild-form continuation shared by the dependent pipeline steps) is the branch that realises operational darkness.

## 6.3 Minimal Temporal-Well Criteria

A regular-centre profile satisfying the minimal temporal-well criteria approaches, as $r \to 0$:

\begin{equation} \label{eq:causal_6}
F(r) = 1 - f_2 r^2 + O(r^4), \quad R(r) = r + O(r^3), \quad \phi(r) = \phi_c + \phi_2 r^2 + O(r^4),
\end{equation}

\begin{equation} \label{eq:causal_7}
A(\phi_c) = A_c, \quad 0 < A_c < \infty, \quad B(\phi_c) \text{ finite}.
\end{equation}

Then the physical areal radius becomes $\tilde R = A(\phi)R(r) \sim A_c r \to 0$ — an ordinary regular centre with no spatial opening, no tube-like asymptotic end, no other universe, no wall, and finite local geometry. Both null families have finite affine parameter. The centre is at finite optical distance. An emitter at the centre has finite $\mathcal{Z}$ with any causally connected receiver. The temporal-rate minimum need not coincide with a centre — it may occur at a shell or an asymmetric basin — but the invariant is the same: $N_{\min} > 0$ with finite transfer mismatch to any receiver. The TEP object is a temporal well, not a regular-centre black hole.

\begin{equation} \label{eq:causal_8}
\begin{aligned}
&\text{regular Lorentzian geometry} + \text{finite local density} \\
&\qquad + 0 < N(r) \text{ everywhere with } N_{\min} \ll N_o \\
&\qquad + \text{no observer-independent one-way boundary } (N \geq N_{\min} > 0) \\
&\qquad + \text{extreme external redshift}
\end{aligned}
\end{equation}

The Schwarzschild incompatibility analysis (Section 3) shows that the temporal field must backreact on the geometric metric: the fixed-Schwarzschild construction cannot work. The inverse reconstruction (Section 4) identifies $w_r = -1$ as a sufficient regularity condition, demonstrated by the regularity benchmark (Appendix K). The global solution of the fixed TEP theory — the canonical action with the leading sGB curvature operator and the regularising nonlinear coefficients selected by global regularity — must produce a temporal well: a regular spatial domain with finite local density and an extreme temporal-rate gradient, not a regular black hole. The sGB perturbative exterior produces $F = 0$ at a shifted $r_H$ because it is an EFT approximation around the Schwarzschild background; the TEP-selected solution must have $N(r) \geq N_{\min} > 0$ everywhere. The observable predictions (shadow, ISCO, QNM) are computed at $r \sim 3$–$6M$ in the conditional sGB benchmark, far from the would-be horizon at $r \sim 2M$; their robustness to the deep-region profile requires the TEP-native solution. The horizonless temporal well alters the late-time ringdown spectrum (Section 7).

The organising question of TEP is not "How is a black-hole interior regularised?" but rather: *can the phenomena attributed to black holes arise from an extreme relative-time gradient without physical collapse into an ultra-dense object?* The shadow is reproduced by the photon sphere (at $r \sim 3M$, independent of the horizon). The ISCO is reproduced by the orbital structure (at $r \sim 6M$, independent of the horizon). The gravitational-wave ringdown is reproduced by the perturbation spectrum, with a theory-dependent inner boundary condition. The darkness is reproduced by the extreme redshift of signals from the deep temporal region. The apparent inward pull is reproduced by the complete matter metric $\tilde g_{\mu\nu}$. The apparent compactness is reproduced by the exterior observer's reconstruction through strongly redshifted, delayed signals. No event horizon is required, no ultra-dense object is required, and no suction mechanism is required — not a better black hole, but an explanation of why a regular temporal domain is mistaken for one. The duality with cosmological expansion is established in Section 1.5.

In the minimal temporal-well picture, the observer falling toward the deep region experiences ordinary local space, a normally running local clock, finite local density, no wall, no horizon crossing, no signature change. Matter need not be compressed into a vanishing proper volume; the apparent concentration can result from temporal decoupling between the deep region and the exterior. The extreme effect is a finite but potentially enormous transfer mismatch: $\mathcal{Z}_{\max} \sim N_o/N_{\min}$, finite because $N_{\min} > 0$. The signature structure ($\det\tilde g_{2D} = -A^4 G^2 - A^2 F B(\phi')^2 < 0$) is established in Section 2.4 and Appendix B. The divergent-$A$ benchmark ($\phi_0 = 1$) produces a finite-area asymptotic end rather than a regular centre; it serves as a validation benchmark and cautionary example, detailed in Appendix K.

# 7. Perturbation Analysis: Ringdown Structure

TEP replaces the purely ingoing event-horizon condition with propagation through a regular temporal domain. Therefore its late-time spectral problem is not the GR black-hole spectral problem. This is the theory-level structural prediction, independent of the specific coupling. The altered late-time spectrum is an unavoidable consequence. Whether the altered spectrum exhibits longer or shorter damping, weak or strong reflection, echoes or no resolvable echoes depends on the effective potential, characteristic speeds, interior transmission, scalar coupling, boundary regularity, and excitation coefficients of the specific geometry.

## 7.1 The Structural Prediction

In the standard black-hole picture, a single-peaked potential with a purely absorbing horizon produces no echoes. In the TEP temporal-well picture, there is no horizon and no physical boundary — the deep region is continuous, ordinary space in which the rate of proper time drops to a finite minimum. The wave does not strike a wall; it encounters an extreme continuous gradient in the temporal rate. The gradient may turn the wave refractively — the propagation direction curves smoothly as the local clock slows — and the wave may transit through the deep region and emerge back into the exterior. Whether a resonant cavity forms, whether the transit produces longer or shorter damping, and whether scalar–tensor mixing is enhanced or suppressed are all determined by the specific geometry.

The structural prediction is categorical: the boundary condition is changed, and therefore the late-time spectrum is altered. The benchmark calculation on a prescribed Hayward background (Section 7.2) isolates the geometric mechanism: longer damping and amplified isospectrality breaking are the realised behaviour of the present deep-transit benchmark. The nonlinear TEP solution determines the physical realisation.

## 7.2 Benchmark Calculations

Three layers of QNM calculation are reported, each on a distinct geometry. (1) Fixed-Schwarzschild baseline: the geometric metric is held at Schwarzschild; gravitational tensor QNMs are the GR baseline, with no TEP shift (Appendix G). (2) Perturbative sGB exterior: the Sotiriou–Zhou metric perturbation introduces scalar hair and backreaction; the full coupled axial operator requires derivation from the second variation of the action, and the sign and coefficient of the QNM shift are not determined by the horizon displacement alone (Appendix L). (3) Prescribed Hayward deep-transit model: an eigenvalue problem on a horizonless regular geometry, isolating the geometric mechanism of the altered spectral problem. The deep-transit modes find longer damping and amplified isospectrality breaking — the realised behaviour of the present benchmark. The nonlinear TEP solution determines the physical QNM spectrum. The sGB exterior potentials, shadow/ISCO shifts, and QNM horizon-displacement estimates are in Appendix L; the deep-transit eigenvalue calculation is described in this section and documented in the pipeline (Appendix J).

## 7.3 Hyperbolicity and Characteristics

The disformal matter metric must have a well-posed initial-value formulation. The radial-block determinant is strictly negative in the exterior, confirming Lorentzianity, invertibility, and non-degeneracy (Section 2.4, Appendix B). The tensor speed $c_T = 1$ on the Schwarzschild background benchmark, satisfying the GW170817 constraint. The full tensor characteristic metric on the nonlinear TEP solution, global ghost freedom, and scalar sector hyperbolicity in the deep interior require derivation from the complete coupled perturbation system — an open problem, not yet contained in any appendix.

# 8. Four Consequences of One Principle

The principle is that the source-to-observer temporal transfer is a dynamical degree of freedom. The transfer factor

\begin{equation} \label{eq:pred_transfer}
\mathcal{T}_{e\to o} = \frac{\omega_e}{\omega_o} = \frac{(-k_\mu u^\mu)_e}{(-k_\mu u^\mu)_o}
\end{equation}

is the single quantity from which all observational consequences follow. Four consequences, one field:

- **Time transfer** — signals are redshifted, delayed, and diluted by the transfer factor.

- **Mass inference** — the conventional mass is reconstructed under isochrony; a non-isochronous refit yields a phantom mass residual.

- **Photon accessibility** — the shadow and photon region are projections of the transfer field; the Temporal Horizon is the observer-relative accessibility boundary.

- **Ringdown** — replacing an absorbing horizon with a continuous, strongly time-dilated region changes the boundary-value problem in principle.

These are not separate mechanisms bolted onto a GR background. They are different projections of one field. The sGB coupling provides the realised benchmark for the first three; the fourth is a structural prediction independent of the specific coupling. Detailed numerical tables are in Appendix L; the main text retains only the structural points.

## 8.1 Time Transfer

When $|\Delta_T| = |\ln(\mathcal{T}_{e\to o}^{\rm TEP}/\mathcal{T}_{e\to o}^{\rm GR})|$ becomes extreme, signals are strongly redshifted, observed processes appear greatly slowed, and signal arrival rates become extremely slow. The object appears dark because the transfer exceeds any practical accessibility threshold, not because signals are absolutely forbidden. The transfer factor from the temporal-rate minimum is finite: $\mathcal{Z}_{\max} \sim N_o/N_{\min}$, where $N_{\min} \ll N_o$. This may be astronomically large but is not divergent. On the corrected scalar-on-regular-background benchmark ($\eta = +0.3$, $g = 1.1M$), the matter-frame lapse minimum $N_{\min} = 1.5 \times 10^{-4}$ gives $\mathcal{Z}_{\max} \simeq 6.7 \times 10^{3}$ — a factor $\sim 7$ above the most conservative radiometric darkness threshold $\mathcal{T}_{\rm dark} = (C_{\rm em}/\varepsilon)^{1/3} \lesssim 10^{3}$ (Section 6.2): interior emission reaches the observer at $\sim 3 \times 10^{-12}$ of its intrinsic specific intensity.

In the sGB benchmark, the conformal factor $A = e^{-\phi}$ modifies clock rates at $\mathcal{O}(\eta)$. The tensor speed $c_T = 1$ on the Schwarzschild background in shift-symmetric sGB, satisfying the GW170817 constraint on that background. The Einstein-frame metric carries the gravitational dynamics; tensor propagation is determined by the principal symbol of the coupled temporal–geometric equations. The full tensor characteristic metric on the TEP solution requires derivation from the complete coupled perturbation system (Section 7).

## 8.2 Mass Inference

The conventional mass $M_{\rm app}^{\rm GR}$ is not directly weighed. It is reconstructed from observed angles, frequencies, and periods under the isochrony closure. The TEP phantom mass residual $M_{\rm phantom}^{T} \equiv M_{\rm fit}^{\rm GR} - M_{\rm matter}^{\rm TEP}$ measures the difference between the GR-reconstructed mass and the locally measured material content. Its sign is not assumed: slow deep clocks alone deflate the inferred mass; positive phantom mass requires spatial magnification to dominate temporal stretching (Section 2.8).

The sGB benchmark illustrates both optical regimes (`step_36_phantom_mass_critical.json`). The temporal-well branch ($\alpha_{\rm GB} > 0$) gives $A(\phi) = e^{-\phi} < 1$ in the exterior — clocks run progressively slower toward the object, in the corpus-wide sign convention of Paper 0 §2.2. Under isotropic conformal rescaling $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$, spatial rulers stretch alongside clocks so $A^2(\phi)$ cancels out of null geodesic paths, leaving only the softened Hayward core $F(r)$ and yielding a $-5.4\%$ to $-6.3\%$ shadow contraction. Under the unstretched-space temporal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) — where the $(3+1)$ ADM spatial 3-metric does not stretch ($\tilde g_{\phi\phi}=r^2$) while clocks slow by $\tilde g_{tt}=-e^{-2\phi(r)}F(r)$ — the optical effective potential $\tilde V_{\rm eff}(r)=e^{-2\phi(r)}F(r)/r^2$ is lowered by $e^{-2\phi(r)}<1$, expanding the critical impact parameter to $\tilde b_c = 5.269\,M$ ($Q_s=0.20\,M$ at the fiducial $\eta=+0.30$) and $5.334\,M$ ($Q_{\rm eff}=0.236\,M$ from the solved Step 34 regular-interior BVP) against Schwarzschild $b_{c,\rm Schw}=5.196\,M$. This produces a positive phantom mass $M_{\rm phantom}=+1.41\%$ to $+2.65\%$ ($>0$, matching the positive sign of galactic and cluster lensing) and brings the predicted M87$^\ast$ shadow to $40.25$–$40.74\,\mu$as ($-0.58\sigma$ to $-0.42\sigma$ vs EHT's $42.0\pm3.0\,\mu$as, improving on Schwarzschild's $-0.77\sigma$) and Sgr A$^\ast$ to $54.00$–$54.67\,\mu$as ($+0.76\sigma$ to $+0.85\sigma$ vs $48.7\pm7.0\,\mu$as). The magnitude scales with the dimensionless coupling $|\eta| = 3|\alpha_{\rm GB}|/M^2$, making corrections largest for low-mass stellar black holes. The fundamental coupling $\alpha_{\rm GB}$ is a dimensionful action constant, so $\eta(M)$ is mass-dependent; the interior-regularity floor $|\eta| \gtrsim 0.3$ of Section 4 therefore bounds the effective strong-field coupling realised in the deep field. The map from the asymptotic action constant to that effective coupling is supplied by the Temporal Topology running law (Section 2.5) — the same environmental dressing that suppresses the bare $\beta_A = -1$ to the weak-field $\beta_{\rm eff}$ — so the floor does not translate directly into a bound on the weak-field $\alpha_{\rm GB}$. Detailed numbers are in Appendix L.

## 8.3 Photon Accessibility

The shadow and photon region are projections of the transfer field. Under isotropic conformal rescaling, the shadow is sensitive only to the geometric metric at $\mathcal{O}(\eta^2)$ while the ISCO feels the conformal factor at $\mathcal{O}(\eta)$; under the unstretched-space temporal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$), both the photon sphere ($\tilde V_{\rm eff}=e^{-2\phi}F/r^2$) and timelike orbits respond directly to the temporal well $A(\phi)=e^{-\phi}$ at $\mathcal{O}(\eta)$, yielding a positive phantom mass ($+1.41\%$ to $+2.65\%$ at $\eta=0.30$).

The Temporal Horizon $\mathcal{H}_T^{(\Lambda)}[u_o]$ is the observer-relative accessibility boundary. It is defined directly from the observable transfer factor, not from a global property of an assumed spacetime. Horizon-scale images constrain how far the photon-region geometry can differ from the GR-calibrated exterior; they do not directly establish an event horizon or measure the local density and proper volume of the obscured region. The pipeline implementation and data products for the visibility-domain fit are documented in Appendix J.

## 8.4 Ringdown

TEP replaces the purely ingoing event-horizon condition with propagation through a regular temporal domain. The late-time spectral problem is therefore not the GR black-hole spectral problem. This is the theory-level prediction. The altered late-time spectrum is an unavoidable consequence. Whether the altered spectrum exhibits longer or shorter damping, weak or strong reflection, echoes or no resolvable echoes depends on the specific geometry.

An altered late-time spectrum is universal to the TEP boundary condition. The calculation on a prescribed Hayward background (Section 7.2) isolates the geometric mechanism: longer damping, two preserved families, and amplified isospectrality breaking are the realised behaviour of the present deep-transit benchmark. The nonlinear TEP solution determines the physical spectrum. The deep-transit modes cannot be directly compared to published sGB QNMs because they are on a horizonless geometry while all published sGB QNMs are on horizon-bearing backgrounds.

## 8.5 Rotating Sources

The static, spherically symmetric ansatz is an imposed symmetry reduction, not a derived property of the theory. A real astrophysical source with angular momentum produces a temporal field that is neither spherical nor static. The slowly-rotating shift-symmetric sGB solution (Delgado et al. 2020) provides the $\mathcal{O}(\beta^2)$ correction to the frame-dragging function from the coupled field equations — an axisymmetric realisation that isolates the spin-coupling mechanism. The nonlinear TEP solution determines the physical rotating-source geometry. The shadow and QNM corrections from spin are documented in the pipeline (Appendix J).

## 8.6 Dynamical Signatures

A binary merger produces a highly asymmetric, dynamical temporal gradient. The coupled system contains axial, gravitational-led polar, and scalar-led temporal mode families; the ringdown can differ from GR because the temporal field participates in the coupled dynamics. During inspiral, the coupled theory permits a $-1$PN scalar dipole channel when the components carry unequal temporal charges — absent in GR because GR has no dynamical time field. The weak-field version of this channel is resolved from the action (Paper 0, step_69): the scalar charge is conserved through the nonlinear shell, so each compact body carries $\alpha_A = \alpha_0(1-2s_A)$ with $s_A$ its binding sensitivity — the exponential coupling removes spontaneous scalarization ($\beta_0 = 0$) but not the compactness asymmetry, which is a strong-equivalence effect present in every scalar-tensor theory — while the $\ell = 1$ emission is suppressed by the solved shell transmission $T_{\rm amp}\sim a/r_*$. All benchmark systems (J1738+0333, J0348+0432, J0737-3039, J0437-4715) pass by $\sim 5\times10^2$–$6\times10^3$ under the full equation-of-state sensitivity bracket, and the same impedance suppresses any merger dipole by the binary's own $a/r_*$. The strong-field merger prediction — the asymptotic scalar charge carried by a temporal well on a dynamical horizon, for which the canonical no-hair argument does not automatically apply to $P(X)$ operators — remains open and requires the full coupled calculation.

## 8.7 Summary

Four observational consequences follow from one temporal field. The sGB coupling provides the realised benchmark for the first three; the fourth is a structural prediction. None of these is a pillar holding up the theory — they are consequences of the principle that proper time is dynamical. The key test is the raw-observable, non-isochronous refit (Section 9).

# 9. Interpretation and Scope

TEP treats proper time as a dynamical field. The strong-field consequence is the temporal well. This section collects the interpretation, confronts existing data as examples of the framework, and defines the scope.

## 9.1 Derived, Demonstrated, Decisive Closure

**Derived from TEP — follows from the principle.**

- The temporal-well interpretation: a black hole is a regular spatial region in which the rate of proper time differs radically from the exterior, without physical collapse, without a singular core, without an absolute one-way boundary.

- The operational Temporal Horizon: the observer-relative accessibility boundary defined directly from observable temporal accessibility.

- Non-unique GR-closed inverse mapping: the conventional black-hole reconstruction is not the unique reading of the same observations once the Isochrony Axiom is dropped.

- Fixed-Schwarzschild incompatibility in the defined conformal class: finite curvature and bounded areal radius are mutually exclusive when $g_{\mu\nu}$ is held fixed.

**Constructively demonstrated — attainability proofs and consistency checks.**

- Regular geometry exists: the Hayward metric achieves finite Kretschmann, bounded areal radius, and Lorentzianity simultaneously (Section 5, Appendix K).

- Altered horizonless ringdown boundary condition (structural prediction pending realisation): TEP replaces the purely ingoing event-horizon condition with propagation through a regular temporal domain, so the late-time spectral problem is not the GR black-hole spectral problem; the realised spectrum awaits the nonlinear TEP solution.

- Scalar hair and backreaction exist: the sGB coupling produces real scalar hair and real backreaction on the gravitational metric (Section 4, Appendix L).

- Finite-curvature matter metrics are attainable: the scalar-on-regular-background benchmark produces a regular point centre (Section 4, Appendix K).

- Weak-field S2 recovers GR — a required consistency gate, not a null result (Section 9.3, Appendix J).

- Exterior and deep-transit benchmarks: the sGB shadow/ISCO shifts and the prescribed Hayward deep-transit eigenmodes isolate the geometric mechanism of the altered spectral problem (Sections 7–8, Appendices G, L).

**Key closure — the nonlinear selection problem.**

- One global TEP solution: solving the coupled Einstein–scalar–sGB system with the canonical matter coupling to obtain the minimal temporal-well profile.

- Complete characteristics: the full coupled tensor, scalar, and mixed-sector perturbation system on the TEP-selected geometry.

- Raw multi-messenger inference: the non-isochronous refit of the same data GR already uses.

- Measured phantom-mass sign: the joint photon–orbit–timing forward model that fixes $\mathcal S_a^3 \mathcal D_{\rm dyn}$ relative to $\mathcal T_P^2$.

The distinction between what is derived from the TEP principle and what requires explicit construction makes the framework more robust, not less: the conceptual claims cannot be dismissed by finding an issue in one calculation. Bahrain fixes the governing principle, action structure, and global solution conditions. The nonlinear field equations select the realised temporal-well geometry and its observable spectrum. The remaining nonlinear integration does not define the paradigm; it determines the unique quantitative strong-field realisation selected by the fixed TEP action and tests whether all global regularity conditions are simultaneously satisfied. The strongest route forward is: one action, one exterior coupling, one global solution, one coupled characteristic system, one observational likelihood, one phantom mass posterior.

## 9.2 Data as Examples

Existing data are not pillars holding up the theory. They are examples of the framework applied to real observations. Three examples are considered.

## 9.3 S2: Weak-Field Consistency

S2 orbits Sgr A* at pericentre $r \sim 1369\,R_s$ — far outside the deep temporal region. Consistency with GR at this scale is a required consistency gate. Under the unstretched-space temporal metric, where the temporal well $A(r) = \exp(-\eta/(3r_{\rm rs})) < 1$ supplies an additive pericentre clock redshift $\Delta v_{\rm clock} = +c\,(A(r)^{-1}-1) > 0$ without rescaling spatial coordinates (Step 46, `step_46_tep_transfer.json`), the real 145-point astrometry and 44-point spectroscopy sample improves from $\chi^2_{\rm GR} = 304.17$ to $\chi^2_{\rm TEP} = 303.18$ ($\Delta\chi^2 = +0.99$, Bayes factor $B = 1.64 > 1$ in TEP's favour at equal parameter count $k = 11$) at the locked Section 4 interior-regularity benchmark $\eta = 0.30$, and reaches $\chi^2_{\rm TEP} = 300.66$ ($\Delta\chi^2 = +3.52$, $\Delta{\rm AIC} = -1.52$, positive phantom mass $M_{\rm phantom}^T = +1.57\times 10^3\,M_\odot$, fraction $+3.65\times 10^{-4}$) when $\eta_{\rm TEP}$ is varied freely ($\hat\eta_{\rm TEP} = 1.00 \pm 0.76$, bracketing $\eta = 0.30$ within $1\sigma$). S2 thereby validates weak-field recovery and shows mild positive preference for the physical clock-slowing sign, while horizon-scale observations provide the high-leverage strong-field test. The detailed 7-step pipeline, GR fit, and MCMC posterior are in Appendix J.

## 9.4 EHT: Photon-Region Constraint

Horizon-scale images constrain how far the photon-region geometry can differ from the GR-calibrated exterior. They do not directly establish an event horizon or measure the local density and proper volume of the obscured region. The visibility-domain fit to EHT M87$^\ast$ and Sgr A$^\ast$ data constrains deviations in the photon-region transfer model; the pipeline implementation and data products are documented in Appendix J. The epistemological point is that a horizon-scale image is a constraint on the photon-region transfer field, not a direct observation of an event horizon.

## 9.5 Gravitational Waves: The Strongest Test

TEP replaces the purely ingoing event-horizon condition with propagation through a regular temporal domain. The late-time spectral problem is therefore not the GR black-hole spectral problem — this is the theory-level prediction, independent of the specific coupling. The altered boundary-value problem is an unavoidable consequence; the magnitude and form of the resulting spectral deviation—longer or shorter damping, echoes or no resolvable echoes, weak or strong scalar–tensor mixing—depend on the realised geometry. Gravitational-wave ringdown is the strongest test because it directly probes the boundary condition that distinguishes a temporal well from a black hole.

## 9.6 The Unified Statement

*Cosmological expansion and black-hole collapse are dual misreadings of dynamical proper time.* On cosmological scales, the conformal volume element $V_{\rm eff} = A_{\rm clock}^3 a_m^3$ tends to zero while the matter-frame geometry does not collapse (Thika); temporal evolution is interpreted as increasing spatial separation. Around extreme gravitational sources, temporal gradients are interpreted as spatial attraction, compression and causal capture. In both cases, TEP replaces apparent spatial dynamics with the observable consequences of a dynamical proper-time field.

The object is a regular region in which the rate of proper time differs radically from the exterior — not a drain or a singularity, but a continuous spatial domain with an extreme temporal gradient. Matter moving into that gradient appears, from the exterior, to accelerate inward, become increasingly redshifted, slow and fade, collect within a small apparent radius, and become inaccessible. But locally there is no mysterious suction. Matter follows its ordinary local trajectory through a continuous physical environment. The apparent force is analogous to refraction: trajectories curve when the propagation rate varies through a medium, not because matter is being pulled sideways.

## 9.7 Scope

The derived, demonstrated, and decisive-closure claims are collected in Section 9.1. Bahrain defines the nonlinear closure problem and the boundary conditions that uniquely identify the physical strong-field branch. The nonlinear integration does not define the TEP paradigm; it selects the realised strong-field branch and tests whether the fixed TEP action admits a global solution satisfying all temporal-well regularity conditions.

# 10. Conclusion

The Temporal Equivalence Principle treats proper time as a dynamical field. This paper develops its strong-field consequence: a black hole, under TEP, is a temporal well — a regular spatial region in which the rate of proper time differs radically from the exterior, without physical collapse to an ultradense singular object and without an absolute one-way boundary. A continuous, extreme but finite gradient in proper-time rate provides a unified mechanism for the principal observational signatures attributed to a black hole.

The operational boundary is the Temporal Horizon — the observer-relative threshold beyond which the clock-transfer factor renders signals practically undetectable — not the event horizon. The event horizon is a global causal construct inferred from the spacetime model, not an observable. Apparent compactness arises from exterior-frame reconstruction, not physical compression. Inferred gravitational mass and local material mass need not coincide; a strong-field phantom mass residual $M_{\rm phantom}^{T} \equiv M_{\rm fit}^{\rm GR} - M_{\rm matter}^{\rm TEP}$ measures this discrepancy, its sign determined by the data. Standard black-hole ontology assumes isochrony: source clocks, photon propagation, and observer clocks mapped onto a single general-relativistic time coordinate. TEP drops that closure. Under TEP, the conventional reconstruction — compact mass, event horizon, singular collapse — is no longer the unique reading of the same observations.

The temporal field cannot sit passively on fixed Schwarzschild geometry. Finite curvature and bounded areal radius are mutually exclusive when $g_{\mu\nu}$ is held fixed. Strong temporal structure forces gravitational backreaction. The canonical TEP matter coupling fixes the temporal sector; the leading curvature operator supplies backreaction; the regularising nonlinear coefficients are selected by global regularity and observation. The construction programme is solving the global field equations of the fixed EFT architecture — not selecting among competing theories.

Four observational consequences follow from one temporal field: time transfer, mass inference, photon accessibility, and ringdown. TEP replaces the purely ingoing event-horizon condition with propagation through a regular temporal domain; the late-time spectral problem is therefore not the GR black-hole spectral problem. This is the theory-level prediction. An altered late-time spectrum is universal to the TEP boundary condition. The benchmark calculation on a prescribed Hayward background isolates the geometric mechanism: longer damping and amplified isospectrality breaking are the realised behaviour of the present deep-transit benchmark. The nonlinear TEP solution determines the physical spectrum.

Weak-field data recover GR; horizon-scale images constrain the photon-region geometry but do not directly establish an event horizon; gravitational-wave ringdown provides the sharpest test. The key programme is the raw non-isochronous multi-messenger refit of the same data GR already uses.

Black holes are not fundamental collapsed objects in TEP. They are the observational appearance of regular strong-field temporal wells reconstructed under exterior temporal standards. Bahrain derives the governing strong-field conditions, proves fixed Schwarzschild insufficient, identifies the required backreaction channel, constructs regular benchmarks, and defines the key observational tests. The remaining nonlinear integration does not define the paradigm; it determines the unique quantitative strong-field realisation selected by the fixed TEP action and tests whether all global regularity conditions are simultaneously satisfied.

*Cosmological expansion and black-hole collapse are dual misreadings of dynamical proper time.* The standard black-hole object was never what the telescopes measured. The event horizon, the singularity, and the ultradense core are GR reconstructions from observational data under an implicit isochronous transfer model. Under TEP, the same observations are produced by a temporal well — and the conventional reconstruction is no longer the unique reading.

# Appendix A — Conventions and Disformal Identities

This appendix fixes the sign, curvature and disformal conventions used throughout the paper. All identities quoted in the main text are derived here explicitly so that no sign or factor ambiguity survives into the field equations, the curvature invariants of Appendix D, or the perturbation analysis of Appendix G.

## A.1 Metric Signature and Curvature

The metric signature $(-,+,+,+)$ is adopted. The Riemann tensor is defined via the Wald convention

\begin{equation} \label{eq:appA_1}
(\nabla_a\nabla_b - \nabla_b\nabla_a) V_c = R_{abc}{}^{d}\,V_d,
\end{equation}

with Ricci tensor $R_{ab} = R_{acb}{}^{c}$ and Ricci scalar $R = g^{ab}R_{ab}$. The Einstein tensor is $G_{ab} = R_{ab} - \tfrac12 R\,g_{ab}$, and the Einstein equations read $G_{\mu\nu}[g] = 8\pi\,T_{\mu\nu}$ in geometrized units $G = c = 1$. With this convention the Schwarzschild Kretschmann scalar is

\begin{equation} \label{eq:appA_2}
K_{\rm Schw} = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} = \frac{48\,M^2}{r^6},
\end{equation}

which is the reference used for the curvature comparison in Appendix D.

## A.2 Disformal Transformation

The matter (causal) metric is the disformal image of the geometric metric,

\begin{equation} \label{eq:appA_3}
\tilde g_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi,
\end{equation}

where $A(\phi)$ is the conformal factor and $B(\phi)$ is the disformal coupling. Defining $X \equiv -\tfrac12\,g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$, the inverse matter metric is

\begin{equation} \label{eq:appA_4}
\tilde g^{\mu\nu} = A^{-2}\left(g^{\mu\nu} - \frac{B\,\nabla^\mu\phi\,\nabla^\nu\phi}{A^2 - 2BX}\right).
\end{equation}

This is the identity used in the geodesic equations of Appendix E and in the perturbation potentials of Appendix G.

## A.3 Determinant and Invertibility

In four spacetime dimensions the determinant of the disformal metric is

\begin{equation} \label{eq:appA_5}
\det\tilde g = A^{8}\,\det g\,\left(1 - \frac{2BX}{A^2}\right) = A^{8}\,\det g\,\frac{A^2 - 2BX}{A^2}.
\end{equation}

The transformation is therefore invertible if and only if

\begin{equation} \label{eq:appA_6}
A^2 > 0 \qquad\text{and}\qquad A^2 - 2BX \neq 0.
\end{equation}

The 2D $(v,r)$ determinant in Eddington–Finkelstein coordinates is the canonical expression $\det\tilde g_{2D} = -A^4 G^2 - A^2 F B(\phi')^2$ (Section 2.4). The geometric horizon has $\det\tilde g_{2D}=-1.002$ and is Lorentzian. The determinant is strictly negative at every sampled radius from the horizon to the innermost grid point ($r_{\min}=10^{-8}M$) in the exterior region ($F > 0$). Inside horizons ($F < 0$), the disformal term becomes positive and the determinant must be checked with the canonical formula. The disformal transformation is invertible where $A^2 - 2BX \neq 0$; this must be verified separately for the interior. For the $\phi_0 = 2$ fixed-Schwarzschild case, both null families have infinite affine parameter ($\tilde\lambda \sim \int r^{-4}\,dr \to \infty$) because $A \sim r^{-2}$; the $r \to 0$ limit is an asymptotic spatially enlarged end, not a regular point (Appendix E).

## A.4 Connection Difference

The Levi-Civita connection of $\tilde g$ differs from that of $g$ by a tensorial term. Writing $\Sigma_\mu \equiv \nabla_\mu\ln A$ and $\Phi_\mu \equiv \nabla_\mu\phi$, the difference is

\begin{equation} \label{eq:appA_7}
\tilde\Gamma^{\lambda}_{\;\;\mu\nu} - \Gamma^{\lambda}_{\;\;\mu\nu} = 2\,\delta^{(\lambda}_{(\mu}\,\Sigma_{\nu)} - g_{\mu\nu}\,\Sigma^{\lambda} + \frac{B}{A^2-2BX}\Big[\Phi_{\mu}\,\Phi_{\nu}\,\Sigma^{\lambda} + 2\,\Phi^{(\lambda}\,\nabla_{\mu)}\Phi_{\nu)} - g_{\mu\nu}\,\Phi^{\lambda}\,\Sigma_{\rho}\Phi^{\rho}\Big] + \mathcal{O}(\nabla B),
\end{equation}

where $\mathcal{O}(\nabla B)$ collects terms proportional to $\nabla_\mu B$. The conformal piece (the first two terms) is the standard scalar-tensor result; the disformal piece (the bracket) is what produces the modified causal structure and the shear zone in the interior.

## A.5 Stress-Energy Transformation

Matter minimally coupled to $\tilde g_{\mu\nu}$ has stress tensor $\tilde T^{\mu\nu} = -(2/\sqrt{-\tilde g})\,\delta S_m/\delta\tilde g_{\mu\nu}$. When expressed with respect to the geometric metric $g_{\mu\nu}$, this becomes

\begin{equation} \label{eq:appA_8}
T^{\mu\nu} \equiv \frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g_{\mu\nu}} = A^{-2}\,\tilde T^{\mu\nu} + \frac{B}{A^2}\,\tilde T^{\alpha\beta}\,\nabla_\alpha\phi\,\nabla_\beta\phi\,g^{\mu\nu} + \text{disformal trace corrections}.
\end{equation}

The first term is the familiar conformal rescaling; the second is the disformal correction that feeds back into the geometric Einstein equations $G_{\mu\nu}[g] = 8\pi\,T_{\mu\nu}$. Conservation is frame-consistent: $\tilde\nabla_\mu\tilde T^{\mu\nu} = 0$ in the matter frame, which maps to a non-standard conservation law in the geometric frame that is sourced by the disformal gradient $\nabla_\mu B$.

## A.6 Summary of Identities

| Object | Expression |
| --- | --- |
| Inverse matter metric | $\tilde g^{\mu\nu} = A^{-2}\big(g^{\mu\nu} - \frac{B\,\nabla^\mu\phi\,\nabla^\nu\phi}{A^2 - 2BX}\big)$ |
| Determinant (4D) | $\det\tilde g = A^{8}\,\det g\,\frac{A^2 - 2BX}{A^2}$ |
| Invertibility | $A^2 > 0$ and $A^2 - 2BX \neq 0$ |
| Lorentzian signature | $\det\tilde g_{2D} = -A^4 G^2 - A^2 F B(\phi')^2 < 0$ in exterior ($F > 0$); must be checked inside horizons |
| Connection split | $\tilde\Gamma - \Gamma = $ conformal $+$ disformal |
| Stress split | $T^{\mu\nu} = A^{-2}\tilde T^{\mu\nu} + $ disformal corrections |

The disformal metric is invertible and Lorentzian in the exterior region ($F > 0$), with $\det\tilde g_{2D} < 0$ at every sampled radius. The pipeline-verified conditions $A^2 > 0$ and $A^2 - 2BX \neq 0$ hold at every sampled radius in the exterior. Inside horizons ($F < 0$), the disformal term $-A^2 F B(\phi')^2$ becomes positive and can oppose the conformal term; the determinant must be checked with the canonical formula. The interior hyperbolicity depends on the outcome of the scalar-on-regular-background benchmark (Section 4, Appendix K).

# Appendix B — Strong-Field EFT Action and Reduced Field Equations

This appendix records the strong-field EFT action, the reduced radial system obtained by substituting the spherical ansatz of Section 2.3, and the parameter set that defines the solutions integrated by the pipeline. The equations are written in geometrized units $G = c = 1$.

## B.1 Strong-Field EFT Action and Frame Assignment

The strong-field TEP action used by Bahrain is the canonical TEP action (Section 2.1) augmented by the leading curvature operator — the shift-symmetric scalar–Gauss–Bonnet coupling:

\begin{equation} \label{eq:appB_1}
S_{\rm BH} = \int d^4x\,\sqrt{-g}\left[\frac{R[g]}{16\pi} - \tfrac12(\nabla\phi)^2 - V(\phi) + f(\phi)\,\mathcal{G}\right] + S_m[\tilde g_{\mu\nu},\Psi_m],
\end{equation}

where $\mathcal{G} = R^2 - 4R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ is the Gauss–Bonnet invariant, $f(\phi) = \eta\,\phi$ is the shift-symmetric coupling, and the matter metric for the spherical strong-field benchmark is (with the canonical preferred-foliation unstretched-space decomposition evaluated for the optical shadow in Section 8, Step 36):

\begin{equation} \label{eq:appB_2}
\tilde g_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi.
\end{equation}

The terms are organised as follows. The Einstein–Hilbert term $R/(16\pi)$, the canonical scalar kinetic $-\frac12(\nabla\phi)^2$, the potential $V(\phi)$, and the conformal–disformal matter coupling $S_m[\tilde g]$ constitute the fundamental TEP sector (Section 2.1). The scalar–Gauss–Bonnet coupling $f(\phi)\,\mathcal{G}$ is the leading curvature operator of the strong-field EFT: it is the lowest-dimension higher-curvature term that couples the temporal field to the gravitational geometry and supplies real backreaction. The regularising nonlinear coefficients — higher-order curvature couplings and potential terms that control the deep-region profile — are EFT corrections to this leading operator, selected by the joint requirements of global regularity and observation (Section 4).

The dimensionless coupling is $\eta \equiv 3\alpha_{\rm GB}/M^2$; the temporal-well branch uses $\eta = +0.3$ as the fiducial benchmark (Section 4, step_12 of the pipeline), the floor value at which the step_33 interior construction exhibits dynamical regularisation (Paper 0 — an explored approach to the construction target, not yet an established global solution); the dimensionless coupling $\eta \equiv 3\alpha_{\rm GB}/M^2$ varies with mass for a fixed action constant $\alpha_{\rm GB}$, with $\sqrt{\alpha_{\rm GB}} \lesssim 1.2$ km giving $|\eta| \leq 2.0 \times 10^{-2}$ (10 $M_\odot$), $2.2 \times 10^{-3}$ (30 $M_\odot$), $4.7 \times 10^{-4}$ (65 $M_\odot$). The fixed $\eta = +0.3$ is a branch parameter for the temporal-well construction, not a universal constant; the construction generalizes to $\eta(M) = 3\alpha_{\rm GB}/M^2$. In code units ($M=1$), $\alpha_{\rm GB} = \eta/3$. The perturbative parameter is $\zeta \equiv \alpha_{\rm GB}^2/M^4 = \eta^2/9 \ll 1$. The shift-symmetric Gauss–Bonnet coupling is $f(\phi)=\alpha_{\rm GB}\phi$, so $df/d\phi=\alpha_{\rm GB}$; the scalar equation therefore carries the source $-\alpha_{\rm GB}\mathcal G$. The potential is $V = 0$ for the sGB branch (massless scalar); the interior completion instead runs on the corpus master potential $V(u) = (\lambda/4)\,u^{4}e^{-(u/u_s)^{4}} + V_0\,e^{-(u_s/u)^{4}}$ of Paper 0 (§8), whose essential-singularity floor is the Planck-scale regularisation surface invoked in Section 4. The time-dependent generalisation $\phi(v,r) = qv + \psi(r)$ of Section 2.3 uses the same $f(\phi)$.

Variation yields the geometric Einstein equations (now including the Gauss–Bonnet stress-energy contribution), the scalar equation, and the matter conservation law in the matter frame:

\begin{equation} \label{eq:appB_3}
G_{\mu\nu}[g] = 8\pi\,T_{\mu\nu} + T_{\mu\nu}^{(\mathcal{G})}, \qquad \Box_g\phi = \frac{dV}{d\phi} - \frac{df}{d\phi}\,\mathcal{G} + \frac{d\ln A}{d\phi}\,T^{(\rm m)} + \frac{d\ln B}{d\phi}\,\mathcal{D}, \qquad \tilde\nabla_\mu\tilde T^{\mu\nu} = 0,
\end{equation}

where $T_{\mu\nu}^{(\mathcal{G})}$ is the Gauss–Bonnet stress-energy tensor obtained by varying $f(\phi)\mathcal{G}$ with respect to $g^{\mu\nu}$, and $T_{\mu\nu}$ includes the conformal and disformal corrections derived in Appendix A. The scalar equation now contains the Gauss–Bonnet source term $-(df/d\phi)\,\mathcal{G} = -\alpha_{\rm GB}\,\mathcal{G}$. This is the term that sources the scalar in vacuum: the Gauss–Bonnet invariant is nonzero wherever curvature is present, so the scalar is sourced by geometric curvature even when the matter trace $T^{(\rm m)}$ vanishes. The conformal coupling $(d\ln A/d\phi)\,T^{(\rm m)}$ sources the scalar through the matter trace and vanishes in vacuum; it does not provide the geometric vacuum source. The disformal source $\mathcal{D}$ is proportional to $T^{\mu\nu}\nabla_\mu\phi\nabla_\nu\phi$ and is finite in vacuum.

## B.2 Fixed-Background Prototype Ansatz

Horizon-regular Eddington–Finkelstein form is used throughout,

\begin{equation} \label{eq:appB_4}
ds_g^2 = -F(r)\,dv^2 + 2\,G(r)\,dv\,dr + R^2(r)\,d\Omega^2,
\end{equation}

with a static scalar profile $\phi = \phi(r)$ (the shift-symmetric $q\,v$ piece is set to zero for the vacuum solution). The metric functions and scalar field are parameterized as

\begin{equation} \label{eq:appB_5}
A(\phi) = e^{\beta_A \phi}, \qquad B(\phi) = B_0\,|\phi|^{2}/(1 + |\phi|^{2})\,\exp\!\left(-\frac{\phi^4}{2\sigma_B^4}\right), \qquad \phi(r) = \phi_0 \ln\!\left(\frac{r}{r_h}\right) S(r),
\end{equation}

where $S(r) = 1/(1 + e^{(r-r_h)/(\delta\, r_h)})$ is a smooth logistic activation, and $r_h = 2M$ is the geometric horizon. The exterior areal radius is $R(r) = r$, so that $F(r) = 1 - 2M/r$ recovers Schwarzschild exactly at large $r$.

## B.3 Schematic Prototype Equations

Substituting the ansatz, the schematic prototype is written in terms of the independent radial variables

\begin{equation} \label{eq:appB_6}
F(r), \qquad G(r), \qquad R(r), \qquad \phi(r).
\end{equation}

The coupling functions $A(\phi)$ and $B(\phi)$ are evaluated on the solved field profile and are not independent dynamical variables. The $vv$ and $vr$ Einstein components give two first-order constraints,

\begin{equation} \label{eq:appB_7}
\frac{F'}{F} = \frac{2M}{r^2 F} - \frac{2}{r}\left(1 - \frac{1}{G^2}\right) + 8\pi\,\mathcal{S}_{vv}[A,B,\phi],
\end{equation}

\begin{equation} \label{eq:appB_8}
G' = -4\pi\,r\,G^2\,\mathcal{S}_{vr}[A,B,\phi],
\end{equation}

where $\mathcal{S}_{vv}$ and $\mathcal{S}_{vr}$ are the disformal source terms built from $A$, $B$, $\phi'$ and $X = -\tfrac12 F(\phi')^2$. The angular equation fixes $R(r)$ through

\begin{equation} \label{eq:appB_9}
\frac{R''}{R} + \frac{2}{r}\frac{R'}{R} = 8\pi\,\mathcal{S}_{\theta\theta}[A,B,\phi].
\end{equation}

The scalar wave equation reduces to

\begin{equation} \label{eq:appB_10}
\frac{1}{r^2}\frac{d}{dr}\left(r^2\,F\,\phi'\right) = \frac{dV}{d\phi} - \alpha_{\rm GB}\,\mathcal{G}[g] + \frac{d\ln A}{d\phi}\,T^{(\rm m)} + \frac{d\ln B}{d\phi}\,\mathcal{D}[X,\phi],
\end{equation}

where the $-\alpha_{\rm GB}\,\mathcal{G}$ term is the Gauss–Bonnet geometric source, the $(d\ln A/d\phi)\,T^{(\rm m)}$ term is the conformal matter-trace source, and the $(d\ln B/d\phi)\,\mathcal{D}$ term is the disformal source. In vacuum ($T^{(\rm m)} = 0$), the scalar is sourced by the Gauss–Bonnet invariant: $\Box\phi = -\alpha_{\rm GB}\,\mathcal{G}$. For Schwarzschild, $\mathcal{G}_{\rm Schw} = 48M^2/r^6$, giving $\Box_{\rm Schw}\phi = -48\alpha_{\rm GB} M^2/r^6$. The conformal matter-trace source vanishes in vacuum; it is the Gauss–Bonnet coupling, not the conformal coupling, that provides the geometric vacuum source. This is the mechanism by which the scalar acquires hair around strong-field sources and evades the no-hair theorem (Section 4.2).

## B.4 Matching Conditions at the Geometric Horizon

The exterior Schwarzschild limit is approached smoothly at $r_h=2M$ by the logistic profile: $A\approx1$, $B\approx0$, and $\phi\approx0$. The clean run gives $A=1.0005$, $B=3.0\times10^{-7}$, and $\phi=-2.73\times10^{-4}$ at the sampled horizon grid point. This value of $A$ comes from the old logistic-screened fixed-background profile, not from the sGB-coupled exterior with $\beta_A = -1$. With the sGB scalar and $\beta_A = -1$, the temporal-well branch has $A(r_H) \approx 0.83$ for $\eta = +0.3$ (Section 4.3 assessment). The two profiles are different models and cannot be combined. The disformal contribution remains finite because $F\to0$. This is a continuity check for the prescribed ansatz, not a derivation of junction conditions for a fully coupled interior solution.

## B.5 Model Parameters

The pipeline uses the dimensionless field $\phi=\Phi/M_*$ and geometrized units $G=c=1$, followed by the numerical normalization $M=1$. Thus radii and curve parameters are reported in $M$, curvature-squared diagnostics in $M^{-4}$, and three-volumes in $M^3$. $A$ and $\phi$ are dimensionless; $B_0=1$ denotes $B_0=1\,M^2$ in the implemented metric convention, so that $B\,\partial\phi\,\partial\phi$ is dimensionless. The conformal coupling is $A(\phi) = A_0\,e^{\beta_A \phi}$ with $\beta_A = -1$ (the dimensionless strong-field coupling $d\ln A/d\phi$) frozen across all strong-field calculations; this corresponds to the Jakarta convention $A(\Phi) = \exp(\beta_A \Phi/M_{\rm Pl})$ with the identification $\phi = \Phi/M_*$ and $M_* = M_{\rm Pl}/|\beta_A|$. The weak-field corpus figure $0.013$ is the cosmological field-deviation amplitude $\phi_0$ (Paper 0 §2), not a bare coupling; Solar-System safety is instead supplied by the screened source charge, which Cassini constrains to $S_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ under the unified unstretched-space PPN mapping ($5.8 \times 10^{-6}$ on the isotropic-conformal benchmark; solved two-branch solar profile $y_{\rm tb}(1.6\,R_\odot)=2.16\times 10^{-8}$, $|\gamma_{\rm PPN}-1|=4.32\times 10^{-8}\ll 2.3\times 10^{-5}$), while the unscreened bare coupling is $\beta_A = -1$.

Two solution branches are integrated by the pipeline, both within the fixed strong-field EFT action of equation $\eqref{eq:appB_1}$:

- **Fixed-background prototype** (step_01): the geometric metric is held fixed to Schwarzschild, and the scalar is prescribed as $\phi(r) = \phi_0\ln(r/r_h)\,S(r)$. This branch establishes the Schwarzschild incompatibility (Section 3) and validates the matter-metric signature/invertibility construction. It is not a coupled solution.

- **sGB self-gravitating branch** (step_12): the scalar is solved from the sGB-sourced equation $\Box\phi = -\alpha_{\rm GB}\,\mathcal{G}$ on the Schwarzschild background, and the metric is corrected at $\mathcal{O}(\alpha_{\rm GB}^2/M^4)$ (Sotiriou & Zhou 2014). In code units ($M=1$) this is $\mathcal{O}(\eta^2)$. This branch establishes scalar hair and real backreaction (Section 4.2). The scalar charge is $Q_s = 2\alpha_{\rm GB}/M$ (yielding $2\eta/3$ in code units where $M=1$ and $\alpha_{\rm GB} = \eta/3$) and the scalar profile is $\phi(r) = Q_s(1/r + M/r^2 + 4M^2/(3r^3))$.

The parameter set for the fixed-background prototype is:

| Parameter | Symbol | Value | Role |
| --- | --- | --- | --- |
| Disformal amplitude | $B_0$ | $1.0$ | sets disformal coupling strength; geometric code units ($M=1$, dimensionless field), i.e. $B_0 = 1\,M_{\rm BH}^2$ geometrized — a strong-field construction normalization, not the canonical weak-field coefficient of Paper 0, to which it must not be numerically identified (Paper 0 §2.2: dimensionless-field normalizations are not imported into the weak-field theory) |
| Mass | $M$ | $1.0$ | geometrized black-hole mass |
| Conformal exponent | $\beta_A$ | $-1.0$ | controls $A(r)$ divergence in the interior |
| Disformal horizon exponent | $\delta$ | $0.05$ | controls $B\to0$ approach at $r_h$ |
| Disformal quartic width | $\sigma_B$ | $1.5$ | controls $B(\phi)$ suppression against violent Coulomb gradients |
| Scalar amplitude | $\phi_0$ | $2.0$ | normalizes the prescribed scalar profile |
| Geometric horizon | $r_h$ | $2.0\,M$ | Schwarzschild radius |

The additional parameter for the sGB self-gravitating branch is:

| Parameter | Symbol | Value | Role |
| --- | --- | --- | --- |
| sGB coupling | $\alpha_{\rm GB}$ | $\eta M^2/3$ | shift-symmetric GB coupling $f(\phi)=\alpha_{\rm GB}\phi$; coefficient of $\mathcal{G}$ in the scalar equation; temporal-well branch |
| Dimensionless sGB coupling | $\eta$ | $+0.3$ | $\eta = 3\alpha_{\rm GB}/M^2$; temporal-well branch, interior-regularity floor |
| Dimensionless coupling | $\eta$ | $3\alpha_{\rm GB}/M^2 = +0.3$ | temporal-well branch, interior-regularity floor |
| Potential | $V(\phi)$ | $0$ | massless scalar in the sGB branch |
| Scalar charge | $Q_s$ | $2\alpha_{\rm GB}/M$ ($2\eta/3$ when $M=1$) | Coulomb-like $1/r$ falloff |

With $\beta_A=-1$ and $\phi_0=2.0$, $A(r)\sim(r/r_h)^{-2}$ in the deep interior, reaching $A\sim4.0\times10^{16}$ at the innermost sampled radius. The ultra-damped (quartic Gaussian) factor $B(\phi)=B_0|\phi|^2/(1+|\phi|^2)\exp(-\phi^4/(2\sigma_B^4))$ is suppressed by the quartic Gaussian envelope in the deep interior, driven below $10^{-300}$ at the innermost sampled radius. The conformal term $A^4$ dominates everywhere, so the determinant is strictly negative and the matter metric is globally Lorentzian with no boundary.

The fixed-background prototype (step_01) is a prescribed-profile metric construction, not a closed six-function radial ODE integration. Its supported result is that the bounded conformal-disformal ansatz is asymptotically Schwarzschild and Lorentzian in the exterior ($\det\tilde g_{2D} < 0$ for $F > 0$). The $\phi_0 = 2$ fixed-Schwarzschild case has vanishing curvature ($\tilde K \to 0$) but the $r \to 0$ limit is an asymptotic spatially enlarged end ($\rho \to \infty$) with both null families at infinite affine parameter — not a regular point (Appendix E). The sGB self-gravitating branch (step_12) solves the scalar from the GB-sourced equation and corrects the metric at $\mathcal{O}(\eta^2)$; it establishes hair and backreaction but not a regular deep region (Section 4.2). The joint realisation — regular geometry, bounded matter-frame curvature, and dynamical scalar backreaction in one solution — is the nonlinear closure selected by the TEP action. The minimal temporal-well criteria are given in Section 6.3.

# Appendix C — Strong-Field Asymptotic Expansion (the $\phi_0=2$ Case)

This appendix derives the interior scaling of the metric functions, the physical radial distance, and the areal radius for the $\phi_0 = 2$ case (curvature-regular but spatially-enlarged; see Appendix D for the Schwarzschild incompatibility result that establishes why this case does not satisfy the temporal-well criteria). All exponents and numerical values are taken directly from the pipeline integration of the reduced system in Appendix B.

#### Context

The $\phi_0 = 2$ case achieves $\tilde K \sim 39r^2/16 \to 0$ (curvature-regular) but has diverging areal radius $\rho \sim r_h^2/r \to \infty$ (spatially-enlarged, not a regular centre). The Schwarzschild incompatibility result (Appendix D) shows this is unavoidable for fixed Schwarzschild: curvature regularity ($\phi_0 > 3/2$) and bounded areal radius ($\phi_0 \leq 1$) are mutually exclusive. The validation benchmark achieves both by allowing the geometric metric to be regular.

## C.1 Interior Scaling

In the deep interior ($r \ll r_h = 2M$), the scalar field follows a logarithmic profile $\phi(r) = \phi_0 \ln(r/r_h)$ with $\phi_0 = 2.0$, and the conformal and disformal factors are

\begin{equation} \label{eq:appC_1}
A(\phi) = A_0\,e^{\beta_A \phi}, \qquad B(\phi) = B_0\,\frac{|\phi|^2}{1+|\phi|^2}\,\exp\!\left(-\frac{\phi^4}{2\,\sigma_B^4}\right),
\end{equation}

with $\beta_A = -1$, $\sigma_B = 1.5$, and disformal coupling $\delta = 0.05$. Substituting the logarithmic $\phi$, the effective radial scalings are

\begin{equation} \label{eq:appC_2}
A(r) \sim A_0\,(r/r_h)^{-2}, \qquad B(r) \to 0, \qquad \phi(r) \sim 2\ln(r/r_h).
\end{equation}

With $\beta_A=-1$ and $\phi_0=2$, the conformal factor diverges as $A\sim(r/r_h)^{-2}$. The disformal factor is suppressed by the quartic cutoff $\exp(-\phi^4/2\sigma_B^4)$: as $|\phi|\to\infty$ the logistic prefactor saturates to unity but the quartic exponential drives $B\to 0$ instantly. The net effect is that the disformal sector vanishes in the deep interior while the conformal sector diverges, establishing conformal dominance throughout the interior.

## C.2 Conformal Factor Profile

The pipeline integration gives the following values of $A(\phi)$ along the radial profile:

| Location | $r/M$ | $A$ |
| --- | --- | --- |
| Geometric horizon | $2.0$ | $1.0005$ |
| Innermost sampled | $10^{-8}$ | $4.0\times10^{16}$ |

The conformal factor is essentially unity at the geometric horizon (deviation $5.5\times10^{-4}$) and reaches $4.0\times10^{16}$ at the innermost sampled radius $r_{\min}=10^{-8}M$. The deep-interior scaling is the inverse-second-power law implied by $\phi_0=2$ and $\beta_A=-1$. Note: this table uses the old logistic-screened fixed-background profile, where $\phi \approx 0$ at the horizon and $A \approx 1$. This is a different model from the sGB-coupled exterior with $\beta_A = -1$, where the temporal-well branch gives $A(r_H) \approx 0.83$ for $\eta = +0.3$ (Section 4.3 assessment) (illustrative benchmark; $\eta(M) = 3\alpha_{\rm GB}/M^2$ varies with mass for fixed $\alpha_{\rm GB}$, with $|\eta| \leq 2.0 \times 10^{-2}$ at 10 $M_\odot$). The two profiles cannot be combined.

## C.3 Disformal Factor Profile

The corresponding values of $B(\phi)$ are:

| Location | $r/M$ | $B$ |
| --- | --- | --- |
| Geometric horizon | $2.0$ | $3.0\times10^{-7}$ |
| Innermost sampled | $10^{-8}$ | $<10^{-300}$ |

The disformal factor is $B=3.0\times10^{-7}$ at the geometric horizon and below $10^{-300}$ at the innermost sampled radius. Unlike a purely logistic activation, the quartic Gaussian cutoff $\exp(-\phi^4/2\sigma_B^4)$ drives $B$ to zero in the deep interior rather than saturating to $B_0$. The disformal sector is dynamically irrelevant in the interior; the geometry is governed entirely by the conformal factor.

## C.4 Conformal Dominance and the Absence of a Causal Boundary

The canonical determinant of the two-dimensional $(v,r)$ sector for $ds_g^2 = -F\,dv^2 + 2G\,dv\,dr + R^2\,d\Omega^2$ and a static radial scalar is

\begin{equation} \label{eq:appC_3}
\det\tilde g_{2D} = -A^4 G^2 - A^2 F\,B(\phi')^2.
\end{equation}

For the $\phi_0 = 2$ fixed-Schwarzschild case, $G = 1$ in EF coordinates. In the deep interior, $A\sim(r/r_h)^{-2}\to\infty$ while $B\to 0$ exponentially. The $-A^4 G^2$ term therefore dominates at every sampled radius, and the determinant is strictly negative:

\begin{equation} \label{eq:appC_4}
\det\tilde g_{2D}\big|_{\rm horizon} = -1.002, \qquad \det\tilde g_{2D}\big|_{r_{\min}} = -2.56\times10^{66}.
\end{equation}

The matter metric is Lorentzian and nondegenerate for all $r > 0$. However, a negative determinant for every $r > 0$ proves only that no finite positive radius is degenerate; it does not prove the limit $r = 0$ can be added as a regular point. The $r \to 0$ limit is an asymptotic spatially enlarged end ($\rho \sim r_h^2/r \to \infty$), not a regular manifold point. The determinant must be checked with the canonical formula inside horizons ($F < 0$), where the disformal term becomes positive.

## C.5 Areal Radius and Interior Volume Growth

The areal radius of the matter geometry is $\rho = A\,r$. With $A\sim(r/r_h)^{-2}$, this gives

\begin{equation} \label{eq:appC_5}
\rho \sim \frac{r_h^2}{r} \to \infty \quad\text{as}\quad r\to 0.
\end{equation}

The areal radius diverges: the physical space does not collapse. The constant-$r$ volume element of the matter metric is $d\tilde V/dv = 4\pi A^2 r^2 \sqrt{|F|}$. The pipeline gives

\begin{equation} \label{eq:appC_6}
\frac{d\tilde V}{dv}\bigg|_{0.1M} = 8.76\times10^{4}, \qquad \frac{d\tilde V}{dv}\bigg|_{r_{\min}} = 2.84\times10^{22}.
\end{equation}

Since the geometry is globally Lorentzian, this volume growth is a physical proper volume, not a formal diagnostic. The would-be central singularity of Schwarzschild is replaced by continuous, low-curvature regular space.

## C.6 Curvature Scaling (Corrected)

The exact Kretschmann scalar scales as $\tilde K \sim r^{4\phi_0 - 6} = r^{2}$ for $\phi_0 = 2$ (Appendix D), so the curvature vanishes as $r\to 0$. The pipeline confirms $\tilde K \approx 0.024$ at $0.1M$ and $\tilde K \approx 1.36\times10^{-16}$ at the innermost sampled radius, compared with $K_{\rm Schw} = 4.8\times10^{7}$ and $4.8\times10^{49}$ respectively. The exact formula $\tilde K \sim 39r^2/16$ matches the numerical values to within 4%. Note: the naive conformal formula $\tilde K = A^{-12} K_{\rm Schw} \sim r^{18}$ is incorrect — it omits the derivative terms from the non-constant conformal factor $A(r)$, which dominate for $\phi_0 < 3/2$ and contribute at the same order for $\phi_0 = 2$. The curvature vanishing is real, but the power is $r^2$, not $r^{18}$. Details are recorded in Appendix D.

## C.7 Summary of Asymptotic Behaviour

| Quantity | Scaling | Behaviour as $r\to0$ |
| --- | --- | --- |
| $A(r)$ | $\sim (r/r_h)^{-2}$ | diverges ($\to 4.0\times10^{16}$) |
| $B(r)$ | $\to 0$ (quartic Gaussian cutoff) | vanishes ($< 10^{-300}$) |
| $\phi(r)$ | $\sim 2\ln(r/r_h)$ | diverges logarithmically ($\to -38.2$) |
| $\det\tilde g_{2D}$ | $\sim -A^4 < 0$ | strictly negative (globally Lorentzian) |
| Areal radius $\rho = Ar$ | $\sim r^{-1}$ | diverges ($\to\infty$; spatially-enlarged — see Schwarzschild incompatibility result, App. D) |
| Kretschmann $\tilde K$ | $\sim r^{2}$ (exact; not $r^{18}$) | vanishes ($\to 0$; curvature-regular) |
| Interior volume | $\sim A^2 r^2$ | grows ($\to 2.84\times10^{22}$) |
| Singularity | — | curvature-regular, but areal radius diverges (the $\phi_0=2$ case) |

The interior scaling $A\sim(r/r_h)^{-2}$ with $B\to 0$ establishes conformal dominance: the determinant is strictly negative at every sampled radius, the matter geometry is globally Lorentzian, and the areal radius diverges as $\rho\sim 1/r$. The exact Kretschmann scalar vanishes as $\tilde K\sim r^{2}$ (not $r^{18}$ as the naive conformal formula suggests). The $\phi_0 = 2$ case is curvature-regular but spatially-enlarged; the Schwarzschild incompatibility result (Appendix D) shows this is unavoidable for fixed Schwarzschild, motivating the validation benchmark on a Hayward background.

# Appendix D — Curvature Invariants and the Schwarzschild Incompatibility

This appendix records the exact curvature analysis for the TEP matter metric. It is proved that the naive conformal formula $\tilde K = A^{-12} K_{\rm Schw}$ is incorrect for non-constant conformal factors, the exact Kretschmann $\tilde K \sim r^{4\phi_0 - 6}$ is derived, and the Schwarzschild incompatibility result for the fixed-Schwarzschild construction is established.

## D.1 The Exact Kretschmann Scalar

For the pure conformal matter metric $\tilde g = A^2 g_{\rm Schw}$ with $A = (r_h/r)^{\phi_0}$ in the deep interior ($r \ll r_h$, where $F \approx -2M/r$ and $B \to 0$), the Kretschmann scalar is computed in standard Schwarzschild coordinates $(t, r)$ where the metric is diagonal:

\begin{equation} \label{eq:appD_1}
\tilde g_{tt} = -A^2 F, \quad \tilde g_{rr} = \frac{A^2}{F}, \quad \tilde g_{\theta\theta} = A^2 r^2.
\end{equation}

The four independent orthonormal-frame Riemann components are:

\begin{equation} \label{eq:appD_2}
E = \frac{R_{trtr}}{\tilde g_{tt}\,\tilde g_{rr}}, \quad F_t = \frac{R_{t\theta t\theta}}{\tilde g_{tt}\,\tilde g_{\theta\theta}}, \quad F_r = \frac{R_{r\theta r\theta}}{\tilde g_{rr}\,\tilde g_{\theta\theta}}, \quad G = \frac{R_{\theta\phi\theta\phi}}{\tilde g_{\theta\theta}^2},
\end{equation}

and the Kretschmann scalar is $K = 4E^2 + 8F_t^2 + 8F_r^2 + 4G^2$.

Each component scales as $r^{2\phi_0 - 3}$ (verified by exact SymPy computation), giving:

\begin{equation} \label{eq:appD_3}
\boxed{\tilde K \sim r^{4\phi_0 - 6} \quad \text{as } r \to 0.}
\end{equation}

This is NOT $r^{12\phi_0 - 6}$ as the naive conformal formula $A^{-12} K_{\rm Schw}$ suggests. The conformal formula includes only the leading Schwarzschild term scaled by $A^{-12}$, omitting the derivative terms from the non-constant conformal factor $A(r)$. For $\phi_0 < 3/2$, the derivative terms ($r^{4\phi_0-6}$) dominate over the leading term ($r^{12\phi_0-6}$), and the conformal formula underestimates the curvature.

## D.2 Specific Values (Exact, $M = 1$)

| $\phi_0$ | $\tilde K$ (exact, deep interior) | $\tilde K \to 0$? | $\rho = Ar$ bounded? |
| --- | --- | --- | --- |
| $0.5$ | $\frac{75}{4r^4} + \cdots$ | No ($\to \infty$) | Yes |
| $1.0$ | $\frac{9}{r^2} + \frac{1}{4}$ | No ($\to \infty$) | Yes ($\rho \to r_h$) |
| $1.5$ | $\frac{291}{64}$ (constant) | No ($\to$ const) | No ($\rho \to \infty$) |
| $2.0$ | $\frac{39\,r^2}{16}$ | Yes ($\to 0$) | No ($\rho \sim r_h^2/r \to \infty$) |
| $2.5$ | $\frac{1323\,r^4}{1024}$ | Yes ($\to 0$) | No |

## D.3 The Schwarzschild Incompatibility Result

### Theorem D.1 (Fixed-Schwarzschild)

Within the conformal-disformal TEP framework with geometric metric fixed to Schwarzschild, curvature regularity and bounded physical areal radius are mutually exclusive on the $A \to \infty$ branch, and curvature regularity fails unconditionally on the physical $A \to 0$ temporal-well branch.

#### Proof

The exponent $\phi_0$ is allowed either sign; with $A = e^{\beta_A\phi}$, $\beta_A = -1$, the branch $\phi_0 > 0$ corresponds to $A \to \infty$ ($\phi \to -\infty$) and $\phi_0 < 0$ to $A \to 0$ ($\phi \to +\infty$), the latter being the physical branch under the $\phi > 0$-in-wells convention.

*Benchmark branch ($\phi_0 > 0$).* Curvature regularity ($\tilde K \to 0$ as $r \to 0$) requires $4\phi_0 - 6 > 0$, i.e., $\phi_0 > 3/2$. Bounded areal radius ($\rho = A \cdot r = r_h^{\phi_0} r^{1-\phi_0}$ finite as $r \to 0$) requires $1 - \phi_0 \geq 0$, i.e., $\phi_0 \leq 1$. These conditions are mutually exclusive: there is no $\phi_0 > 0$ satisfying both $\phi_0 > 3/2$ and $\phi_0 \leq 1$.

*Physical branch ($\phi_0 < 0$).* The Kretschmann exponent $4\phi_0 - 6 < -6$ for every $\phi_0 < 0$: $\tilde K$ diverges faster than the Schwarzschild $r^{-6}$ singularity for every admissible index, while $\rho \sim r^{1-\phi_0} \to 0$ is trivially bounded. Curvature regularity therefore fails unconditionally — no $\phi_0 < 0$ achieves finite or vanishing $\tilde K$. The incompatibility on the physical branch is thus stronger than mutual exclusion: the finiteness condition itself is unreachable. $\square$

The two branches exhaust the power-law ansatz (the borderline $\phi_0 = 0$ recovers Schwarzschild itself, $K = 48M^2/r^6$, divergent). No conformal power-law temporal well exists on fixed Schwarzschild geometry.

## D.4 Numerical Verification

The exact formula $\tilde K \sim r^{4\phi_0 - 6}$ is verified numerically using the corrected pipeline curvature computation (standard Schwarzschild coordinates, 4-term Kretschmann formula). The power law is recovered to within 0.2% accuracy for all tested $\phi_0$ values on both branches (negative $\phi_0$ rows are the physical temporal-well branch):

| $\phi_0$ | Numerical power law | Expected $4\phi_0 - 6$ | Deviation |
| --- | --- | --- | --- |
| $-2.00$ | $r^{-14.000}$ | $-14.0$ | 0.000 |
| $-1.50$ | $r^{-12.000}$ | $-12.0$ | 0.000 |
| $-1.00$ | $r^{-10.000}$ | $-10.0$ | 0.000 |
| $-0.50$ | $r^{-8.000}$ | $-8.0$ | 0.000 |
| 0.50 | $r^{-4.000}$ | $-4.0$ | 0.000 |
| 0.75 | $r^{-3.000}$ | $-3.0$ | 0.000 |
| 1.00 | $r^{-2.001}$ | $-2.0$ | 0.001 |
| 1.25 | $r^{-1.001}$ | $-1.0$ | 0.001 |
| 1.50 | $r^{-0.001}$ | $0.0$ | 0.001 |
| 1.75 | $r^{0.999}$ | $1.0$ | 0.001 |
| 2.00 | $r^{1.999}$ | $2.0$ | 0.001 |
| 2.50 | $r^{3.999}$ | $4.0$ | 0.001 |

The physical-branch rows confirm numerically that every $\phi_0 < 0$ gives a divergence steeper than the Schwarzschild $r^{-6}$ (exponents $-8$ to $-14$ over the sampled range), with bounded areal radius throughout — the unconditional failure asserted in Theorem D.1. The Schwarzschild baseline ($\phi_0 \to 0$, $A \to 1$) recovers $K = 48M^2/r^6$ to within $5 \times 10^{-6}$ (0.0005%). All regression tests pass on both branches.

## D.5 Corrected Values for the $\phi_0 = 2$ Prototype

The $\phi_0 = 2$ case (curvature-regular but spatially-enlarged) has the following corrected curvature values, replacing the incorrect $\tilde K \sim r^{18}$ and $\tilde K \sim 10^{-150}$ from the naive conformal formula:

| Location | $r/M$ | $\tilde K$ (corrected) | $\tilde K$ (old, wrong) | $K_{\rm Schw}$ |
| --- | --- | --- | --- | --- |
| Exterior | $10$ | $4.79 \times 10^{-5}$ | $4.79 \times 10^{-5}$ | $4.79 \times 10^{-5}$ |
| Horizon | $2$ | $0.748$ | $0.752$ | $0.752$ |
| Deep interior | $0.1$ | $2.31 \times 10^{-2}$ | $\sim 10^{-150}$ | $4.79 \times 10^{7}$ |
| Innermost | $10^{-8}$ | $1.36 \times 10^{-16}$ | $\sim 10^{-300}$ | $4.80 \times 10^{49}$ |

The corrected values show $\tilde K$ is small but not absurdly tiny. The exact power law $\tilde K \sim 39r^2/16$ gives $\tilde K(0.1) = 0.024$, matching the numerical value $0.023$ to within 4%. The old values ($\sim 10^{-150}$) were artifacts of the incorrect conformal formula.

## D.6 Disformal Term: Status of the Classification

The disformal term $B\nabla\phi\nabla\phi$ contributes $B(\phi')^2$ to $\tilde g_{rr}$. In the deep interior, the Lorentzian condition $\det_{2d} < 0$ forces $B \to 0$ (for logarithmic $\psi$) or $\tilde g_{rr} \to 0$ (for saturating $\psi$). For the tested parameter space, the deep interior curvature is controlled by $A$ alone, and the conformal theorem applies. The full disformal classification — classifying all allowed asymptotics of $A$, $B$, $\phi$ under $\det\tilde g < 0$, $\tilde K < \infty$, and $\tilde R_{\rm area} < \infty$ — is an open mathematical programme (Section 3.3). The statement that the disformal term cannot help is established for the tested class but not as a general theorem.

## D.7 Implication: The Temporal Field Must Modify the Geometry

The Schwarzschild incompatibility result shows that the fixed-Schwarzschild construction is incompatible with the conditions for a physically admissible temporal well. The temporal field cannot merely sit on top of a Schwarzschild black hole — it must also change the black-hole geometry itself. When the geometric metric is allowed to be regular, all regularity conditions hold simultaneously.

# Appendix E — Geodesic Structure

This appendix records the null geodesic structure of the TEP matter metric. The key result is that for a conformal transformation $\tilde g = A^2 g$, null geodesics of $g$ are also null geodesics of $\tilde g$ (same unparameterised curves), but the affine parameters are related by $d\tilde\lambda = A^2\,d\lambda$. This single relation determines all affine-parameter results.

## E.1 Conformal Affine Parameter Relation

For $\tilde g_{\mu\nu} = A^2 g_{\mu\nu}$, a null geodesic $x^\mu(\lambda)$ of $g$ with tangent $k^\mu = dx^\mu/d\lambda$ satisfies $g_{\mu\nu}k^\mu k^\nu = 0$ and the geodesic equation $k^\nu\nabla_\nu k^\mu = 0$. Under the conformal transformation, the same curve is null ($\tilde g_{\mu\nu}k^\mu k^\nu = A^2 g_{\mu\nu}k^\mu k^\nu = 0$), but the affine parameter transforms as

\begin{equation} \label{eq:appE_1}
d\tilde\lambda = A^2\,d\lambda.
\end{equation}

This is the standard conformal affine-parameter relation. The Christoffel symbols transform as $\tilde\Gamma^\mu_{\alpha\beta} = \Gamma^\mu_{\alpha\beta} + \delta^\mu_\alpha \partial_\beta\ln A + \delta^\mu_\beta \partial_\alpha\ln A - g_{\alpha\beta}g^{\mu\nu}\partial_\nu\ln A$, and the geodesic equation in the rescaled parameter is satisfied.

## E.2 Eddington–Finkelstein Null Algebra

In ingoing Eddington–Finkelstein coordinates, the conformal matter metric ($B = 0$) has components

\begin{equation} \label{eq:appE_2}
\tilde g_{ab} = A^2\begin{pmatrix} -F & 1 \\ 1 & 0 \end{pmatrix}, \qquad \tilde g^{ab} = A^{-2}\begin{pmatrix} 0 & 1 \\ 1 & F \end{pmatrix}.
\end{equation}

Note that $\tilde g^{rr} = F/A^2$, not zero. The null condition $\tilde g_{ab}k^a k^b = 0$ gives $dv(-F\,dv + 2\,dr) = 0$, yielding two families:

- Ingoing ($v = \mathrm{const}$, $dv = 0$): the tangent $k^\mu = (0, 1)$. The Christoffel symbol $\tilde\Gamma^r{}_{rr} = 2A'/A$ is nonzero.

- Outgoing ($dr/dv = F/2$): the tangent $k^\mu = (2/F, 1)$, or $dv/dr = 2/F$.

## E.3 Affine Parameters on the Divergent-$A$ Benchmark

On the Hayward benchmark with $A = (r_h/r)^{\phi_0}$ and $\phi_0 = 1$, the conformal factor diverges as $A \sim r^{-1}$ near $r = 0$. The seed (Hayward) radial null affine parameter behaves as $d\lambda \propto dr$ (since $F \to 1$ and the background is regular). Applying the conformal relation:

\begin{equation} \label{eq:appE_3}
\tilde\lambda \sim \int A^2\,dr = \int \frac{dr}{r^{2\phi_0}}.
\end{equation}

For $\phi_0 = 1$: $\tilde\lambda \sim \int r^{-2}\,dr = -1/r \to \infty$ as $r \to 0$. Both radial null families have infinite affine parameter. The finite-area asymptotic end lies at infinite affine distance for both families. This is consistent with the end being an asymptotic boundary, not a regular manifold point.

For $\phi_0 = 2$: $\tilde\lambda \sim \int r^{-4}\,dr \to \infty$. Both families infinite.

For general $\phi_0 > 0$: both families have $\tilde\lambda \to \infty$. The divergence is driven by $A \to \infty$, not by the background geometry.

## E.4 Affine Parameters on the Correct Finite-$A$ Architecture

On the target global solution with $A \to A_c$ finite as $r \to 0$, the conformal relation gives $d\tilde\lambda \to A_c^2\,d\lambda$. Since the seed (regular) geodesic has $d\lambda \propto dr$ and finite affine parameter to $r = 0$, the matter-frame affine parameter is also finite: $\tilde\lambda \to A_c^2 \lambda_{\rm seed}$. Both null families reach the regular centre at finite affine distance. The centre is a regular manifold point; geodesics are extendable through it. This is geodesic completeness in the standard sense.

## E.5 The $\phi_0 = 2$ Case

For $\phi_0 = 2$ on fixed Schwarzschild, $A \sim r^{-2}$ and $\tilde\lambda \sim \int r^{-4}\,dr \to \infty$ for both families. The $r \to 0$ limit is an asymptotic spatially enlarged end ($\rho \sim r_h^2/r \to \infty$), not a regular point. The matter metric is Lorentzian and nondegenerate for all $r > 0$ ($\det\tilde g_{2D} = A^4 \det g_{2D} < 0$ since $A > 0$), but the limit $r = 0$ cannot be added as a regular point. The curvature vanishes ($\tilde K \sim 39r^2/16 \to 0$), but the areal radius diverges — the Schwarzschild incompatibility result (Appendix D) shows this trade-off is unavoidable for fixed Schwarzschild.

## E.6 Signature Structure

The canonical 2D $(v,r)$ determinant for $ds_g^2 = -F\,dv^2 + 2G\,dv\,dr + R^2\,d\Omega^2$ and a static radial scalar is:

\begin{equation} \label{eq:appE_4}
\det\tilde g_{2D} = -A^4 G^2 - A^2 F B(\phi')^2.
\end{equation}

For the pure conformal metric ($B = 0$): $\det\tilde g_{2D} = -A^4 G^2 < 0$ — Lorentzianity inherited analytically from $g$ via $A > 0$. For the full disformal metric, the determinant must be checked with the canonical formula. Inside horizons ($F < 0$), the disformal term $-A^2 F B(\phi')^2$ becomes positive and can oppose the conformal term; all invertibility and hyperbolicity results must be regenerated from this expression.

# Appendix F — Physical Volume and Density Diagnostics

This appendix distinguishes the geometric volume from the matter-frame volume diagnostics. The pipeline computes the exterior cumulative volume and an interior constant-$r$ volume element; it does not solve a self-consistent matter density profile.

## F.1 Volume Elements

The geometric volume is

\begin{equation} \label{eq:appF_1}
V_{\rm geom}(r)=\frac{4\pi}{3}r^3.
\end{equation}

Inside the geometric horizon, the spatial constant-$r$ slice has volume element per unit $v$

\begin{equation} \label{eq:appF_2}
\frac{d\tilde V}{dv}=4\pi A^2r^2\sqrt{|F|}.
\end{equation}

For the clean run this rises from $2.08\times10^2$ at $0.1M$ to $5.50\times10^5$ at $r_{\min}=10^{-8}M$.

## F.2 Geometric Collapse and Physical Interior Growth

$V_{\rm geom}\to0$ as $r\to0$, while the matter-frame interior volume element grows strongly because $A\sim r^{-1}$, reaching $d\tilde V/dv=5.50\times10^5$ at $r_{\min}=10^{-8}M$. The determinant is strictly negative at every sampled radius, so this growth is a physical result on a globally Lorentzian space, not a formal continuation diagnostic. The areal radius diverges ($\rho\sim1/r$), confirming that the Schwarzschild areal coordinate is not a direct measure of physical volume in TEP.

## F.3 Numerical Verification

| Quantity | Geometric frame | Matter-frame diagnostic |
| --- | --- | --- |
| Volume at $r_{\min}$ | $V_{\rm geom}=4.19\times10^{-24}$ | interior element $d\tilde V/dv=5.50\times10^5$ |
| Lorentzian domain | globally Lorentzian | $\det\tilde g_{2D}<0$ at all sampled radii |
| Density | $M/V_{\rm geom}\propto r^{-3}$ | not independently computed by this pipeline |

## F.4 Supported Statement

### Proposition F.1

The $\phi_0 = 2$ fixed-Schwarzschild configuration replaces the geometric zero-volume intuition with a strongly growing matter-frame interior volume element on a Lorentzian space for all $r > 0$. The areal radius diverges ($\rho \sim 1/r \to \infty$), confirming that this configuration is a spatially enlarged asymptotic end, not a bounded solution. The nonlinear TEP solution determines the physical interior volume. The Schwarzschild areal coordinate is not a direct measure of physical volume in TEP. A finite physical density claim requires an explicit stress-energy solution and is not asserted by the present diagnostic pipeline.

# Appendix G — Fixed-Schwarzschild Perturbation Baseline

This appendix records the perturbation calculation on the fixed-Schwarzschild geometric background — the first of three QNM calculations in this manuscript. The three are: (1) Fixed-Schwarzschild baseline (this appendix): the geometric metric is held at Schwarzschild; gravitational tensor QNMs are the GR baseline. (2) Perturbative sGB exterior (Appendix L): the Sotiriou–Zhou metric perturbation introduces scalar hair and backreaction; the full coupled axial operator requires derivation from the second variation of the action. (3) Prescribed Hayward deep-transit model (Section 7.2): a toy eigenvalue problem on a horizonless regular geometry, illustrating one realisation of the altered spectral problem.

## G.1 Geometric Tensor Equation

In the fixed-background limit, the geometric metric is Schwarzschild. Canonical TEP distinguishes the geometric metric $g_{\mu\nu}$ from the matter metric $\tilde g_{\mu\nu}$; gravitational waves propagate on $g_{\mu\nu}$. Conformal and disformal factors in $\tilde g_{\mu\nu}$ therefore do not shift the gravitational Regge–Wheeler or Zerilli spectrum. Decomposing $h_{\mu\nu}$ into axial and polar spherical harmonics and Fourier transforming $\Psi(t,r)=e^{-i\omega t}\Psi(r)$ gives

\begin{equation} \label{eq:appG_1}
\frac{d^2\Psi_i}{dr_*^2}+\left[\omega^2-V_i^{\rm Schw}(r)\right]\Psi_i=0,
\end{equation}

where $r_*$ is the Schwarzschild tortoise coordinate and $i$ denotes the Regge–Wheeler or Zerilli sector. These are the standard geometric GR perturbation equations for this fixed background.

## G.2 Interpretation of Matter-Metric Diagnostics

The pipeline also evaluated surrogate radial functions constructed from the disformal matter metric. Such functions may diagnose matter or electromagnetic propagation, but they are not gravitational-wave potentials in canonical TEP. In particular, applying gravitational QNM boundary conditions to a $\tilde g_{\mu\nu}$ surrogate does not establish a shifted tensor spectrum.

## G.3 Ringdown Spectrum

The gravitational QNMs of this non-spinning fixed-background limit are the Schwarzschild GR baseline. The previously quoted $0.421-0.099\,i$ frequency and 5.6% shift arose from the matter-metric surrogate rather than the geometric gravitational equation and are not gravitational QNM predictions. The coupled solution (Section 4) introduces scalar-led and mixed modes with coupling-specific QNM shifts; the full coupled axial operator requires derivation from the second variation of the action (Appendix L).

## G.4 Stability and Echo Scope

No complete coupled perturbation system was analysed on this fixed background. Background regularity and the shape of a surrogate matter-metric potential are insufficient to prove absence of ghosts, gradient instabilities, or growing scalar and mixed modes. They are likewise insufficient to predict or exclude gravitational-wave echoes. An echo statement requires geometric perturbation evolution with physically specified inner boundary conditions and a waveform-level analysis; those calculations were not performed on this background. The prescribed Hayward deep-transit model (Section 7.2) addresses a different geometry.

## G.5 Summary

| Question | Result for the fixed-Schwarzschild baseline |
| --- | --- |
| GW propagation metric | Geometric $g_{\mu\nu}$ = Schwarzschild |
| Gravitational tensor QNMs | GR baseline; no TEP shift |
| Scalar-led and mixed modes | Computed in the perturbative sGB exterior (Appendix L); full axial operator open |
| Full linear stability | Background regularity verified; coupled-system stability is open (Section 7) |
| Gravitational-wave echoes | Neither predicted nor excluded on this background |

This appendix covers the fixed-Schwarzschild baseline only. The perturbative sGB exterior (Appendix L) and the prescribed Hayward deep-transit model (Section 7.2) are separate calculations on different geometries. The three must not be conflated.

# Appendix H — Ray Tracing and Orbital Mechanics

This appendix records the photon-sphere, ISCO, and QPO calculations together with the EHT shadow-size central-value comparisons. The reported pipeline configuration imposes exterior screening, $A=1$ and $B=0$, over the ray and orbit domains, so these calculations reproduce the Schwarzschild baseline. Gravitational ringdown is separately governed by the geometric metric $g_{\mu\nu}$, which is fixed at Schwarzschild here.

## H.1 Photon Sphere

Null geodesics of the matter metric $\tilde g_{\mu\nu}$ reduce to Schwarzschild null geodesics on the screened exterior used by the pipeline, where $A=1$ and $B=0$. The photon sphere is therefore at

\begin{equation} \label{eq:appH_1}
r_{\rm ph} = 3\,M,
\end{equation}

and the critical impact parameter for the shadow edge is

\begin{equation} \label{eq:appH_2}
b_{\rm crit} = \frac{r_{\rm ph}}{\sqrt{1 - 2M/r_{\rm ph}}} = 3\sqrt{3}\,M = 5.196\,M.
\end{equation}

These are baseline values. The current exterior-null ray tracing does not calculate a subleading disformal correction to the shadow edge.

## H.2 ISCO and Radiative Efficiency

On the exterior-screened baseline, the innermost stable circular orbit (ISCO) for timelike geodesics is

\begin{equation} \label{eq:appH_3}
r_{\rm ISCO} = 6\,M.
\end{equation}

The specific energy and angular momentum at the ISCO are

\begin{equation} \label{eq:appH_4}
E_{\rm ISCO} = \sqrt{\frac{8}{9}} = 0.9428, \qquad L_{\rm ISCO} = 2\sqrt{3}\,M = 3.464\,M.
\end{equation}

The radiative efficiency — the fraction of rest-mass energy liberated by accretion onto the ISCO — is

\begin{equation} \label{eq:appH_5}
\eta = 1 - E_{\rm ISCO} = 1 - \sqrt{\frac{8}{9}} = 0.0572 = 5.72\%.
\end{equation}

These are standard Schwarzschild values resulting from screening over the orbit domain. They do not imply that a nonconstant conformal factor would exactly cancel from timelike geodesics.

## H.3 Redshift Factor at ISCO

The gravitational redshift factor at the ISCO is

\begin{equation} \label{eq:appH_6}
g = \sqrt{1 - \frac{2M}{r_{\rm ISCO}}} = \sqrt{1 - \frac{1}{3}} = \sqrt{\frac{2}{3}} = 0.816.
\end{equation}

The corresponding redshift is

\begin{equation} \label{eq:appH_7}
z = \frac{1}{g} - 1 = \sqrt{\frac{3}{2}} - 1 = 0.225.
\end{equation}

These are the standard Schwarzschild ISCO values, confirming that the iron-line and reflection spectroscopy signatures of the TEP exterior are identical to GR at the ISCO.

## H.4 3:2 QPO Resonance

Quasi-periodic oscillations (QPOs) from black-hole accretion disks often appear in a 3:2 frequency ratio, interpreted as a resonance between the radial and vertical epicyclic frequencies. For the TEP exterior (Schwarzschild), the resonance condition $\Omega_r / \Omega_\theta = 2/3$ is satisfied at

\begin{equation} \label{eq:appH_8}
r_{\rm QPO} = 10.8\,M,
\end{equation}

where the epicyclic frequency ratio is

\begin{equation} \label{eq:appH_9}
\frac{\Omega_r}{\Omega_\theta}\bigg|_{r=10.8\,M} = 0.667 = \frac{2}{3}.
\end{equation}

This is the standard Schwarzschild 3:2 resonance radius. The TEP disformal interior does not affect this radius because it is well outside the horizon.

## H.5 EHT Shadow Comparisons

The shadow angular diameter is computed from the critical impact parameter $b_{\rm crit} = 5.196\,M$ and the source distance and mass. The pipeline evaluates the shadow size for the two EHT targets:

| Source | TEP prediction | EHT measurement | Deviation |
| --- | --- | --- | --- |
| M87* | $39.69\;\mu\text{as}$ | $42 \pm 3\;\mu\text{as}$ | $0.77\sigma$ |
| Sgr A* | $53.26\;\mu\text{as}$ | $48.7 \pm 7.0\;\mu\text{as}$ | $0.65\sigma$ |

The exterior-screened central values, $39.69\;\mu$as for M87* and $53.26\;\mu$as for Sgr A*, lie within the quoted measurement intervals $42\pm3\;\mu$as and $48.7\pm7.0\;\mu$as, respectively. These are central-value checks; a full likelihood analysis would propagate correlated mass, distance, calibration, and imaging systematics. The fixed-background limit reproduces the Schwarzschild baseline; the coupled solution (Section 4) provides TEP-specific shadow deviations of $-0.35\%$ for $\eta=+0.3$ on $g^{\rm sGB}$ (Appendix L).

## H.6 Gravitational Ringdown Baseline

Gravitational waves propagate on the geometric metric, not on the matter metric used for ray tracing. Since the fixed-background limit holds the geometric metric at Schwarzschild, its non-spinning gravitational ringdown is the Schwarzschild GR baseline. GW150914 produced a spinning remnant and must be compared with the appropriate geometric Kerr spectrum. The coupled solution (Section 4) provides coupling-specific QNM shifts of sub-percent to percent-level.

## H.7 Summary of Orbital and Observational Predictions

| Observable | Value | vs Schwarzschild |
| --- | --- | --- |
| Photon sphere | $r = 3\,M$ | identical |
| Critical impact parameter | $b = 5.196\,M$ | identical |
| ISCO | $r = 6\,M$ | identical |
| ISCO energy | $E = \sqrt{8/9} = 0.9428$ | screened baseline |
| ISCO angular momentum | $L = 3.464\,M$ | identical |
| Radiative efficiency | $\eta = 5.72\%$ | identical |
| Redshift factor at ISCO | $g = 0.816$ | identical |
| Redshift at ISCO | $z = 0.225$ | identical |
| 3:2 QPO resonance | $r = 10.8\,M$ | identical |
| M87* shadow | $39.69\;\mu$as | $0.77\sigma$ from EHT |
| Sgr A* shadow | $53.26\;\mu$as | $0.65\sigma$ from EHT |
| GW150914 $f_{220}$ (non-spinning WKB diagnostic) | $206.5\,$Hz | Schwarzschild baseline (0% shift) |
| GW150914 $f_{220}$ (Kerr, $\chi=0.67$) | $267\,$Hz | Kerr baseline; measured $\sim 250\,$Hz |

The exterior orbital structure (photon sphere at $3M$, ISCO at $6M$, 3:2 QPO at $10.8M$) is identical to Schwarzschild. EHT shadows agree at $0.77\sigma$ (M87*) and $0.65\sigma$ (Sgr A*). The gravitational QNM baseline is the Schwarzschild spectrum for non-spinning remnants; for GW150914's spinning remnant ($\chi=0.67$) the appropriate baseline is Kerr, giving $f_{220}\approx 267\,$Hz versus the measured $\sim 250\,$Hz. The non-spinning WKB diagnostic $f_{220}=206.5\,$Hz ($\omega_R^{\rm WKB}=0.3988$) is not the physical prediction for a spinning remnant and carries $\sim 7\%$ WKB systematic error relative to the exact Schwarzschild value $\omega_R^{\rm exact}=0.3737$. There is no coupling-specific ringdown frequency shift in this fixed-background limit.

# Appendix I — Comparison Table

This appendix provides a detailed technical comparison of TEP-BH with the major competing models of singularity resolution. The TEP-BH column is split into four sub-columns reflecting the distinct calculations performed in this paper: the fixed-Schwarzschild theorem case, the Hayward validation benchmark, the linear sGB exterior candidate, and the scalar-on-regular-background benchmark. These must not be combined into a single "TEP" column, because they represent different geometries with different properties.

## I.1 Detailed Comparison

| Property | GR (Schwarzschild/Kerr) | Regular BH (Bardeen/Hayward) | TEP theorem case ($\phi_0=2$, fixed Schw.) | TEP benchmark ($\phi_0=1$, Hayward) | TEP linear sGB (exterior) | TEP regularised interior (solved) |
| --- | --- | --- | --- | --- | --- | --- |
| Exterior metric | Schwarzschild/Kerr (exact) | Schwarzschild (to $M/r$) | Schwarzschild (fixed) | Hayward (prescribed) | Schwarzschild to $\sim$3 parts in $10^2$ at $|\eta|=0.3$ (perturbative) | Asymptotically Schwarzschild (target) |
| Interior metric | Singular ($r\to0$) | de Sitter-like temporal minimum | Spatially enlarged end ($\rho \to \infty$); not a regular point | Finite-area asymptotic end ($\rho \to 1.995$, $\tilde\ell \to \infty$); not a regular point | Not yet integrated; literature: finite-area singularity at $r \approx 0.27\,r_h$ (Sotiriou & Zhou 2014) | Regular point centre: $A(0) = 1.7 \times 10^{-4}$, $\rho_{\rm areal}(0) \to 1.7 \times 10^{-10}$, $w_r(0) = -1.000$, $K(0) = 54.2$ ($\eta=+0.3$, $g=1.1M$) |
| Central curvature | $K\to\infty$ | Finite (by construction) | Vanishes ($\tilde K \sim 39r^2/16 \to 0$) but areal radius diverges | Finite ($\tilde K \to 8/r_h^4$) but asymptotic end, not regular centre | Likely divergent at finite-area singularity (literature) | Finite: $K(0) = 54.2$ for $g=1.1M$ |
| Geodesic completeness | No (incomplete at $r=0$) | Model-dependent | Both null families infinite affine ($\tilde\lambda \sim \int r^{-4}\,dr \to \infty$); asymptotic end | Both null families infinite affine ($\tilde\lambda \sim \int r^{-2}\,dr \to \infty$); asymptotic end | Not yet computed; likely incomplete at finite-area singularity | Both families finite affine; regular point centre, extendable geodesics |
| Horizon structure | Event horizon at $r=2M$ | Event + inner horizon | Schwarzschild horizon (fixed) | Hayward outer + inner horizon | sGB apparent horizon (perturbative artifact) | Temporal well: $\tilde N_{\min} = 1.5 \times 10^{-4}$ at $r \simeq 0.79\,M$, no finite causal horizon |
| Temporal Horizon | — | — | Not established (theorem case only) | Not established ($\theta_+=0$ is constant-area property, not temporal freeze) | Exterior matches solved regular interior at $r_{\rm match}=10M$ (Step 34) | Operational: $\mathcal{Z}_{\max} \sim N_o/\tilde N_{\min} \sim 6.7 \times 10^{3}$ at $r \approx 0.79\,M$ — above the darkness threshold $\mathcal{T}_{\rm dark} \lesssim 10^{3}$ (Section 6.2); $\tilde N > 0$ everywhere |
| Surface / junction | None | None (smooth) | None (but asymptotic end) | None (but asymptotic end) | Smooth exterior-to-interior matching (Step 34 BVP) | None (smooth, regular point centre) |
| Lorentzianity | Yes | Yes | Yes for all $r > 0$ ($\det = A^4 \det g < 0$); limit $r=0$ not a regular point | Yes in exterior ($F > 0$); inside horizon must check canonical determinant | Yes in exterior ($F > 0$) | Yes: $\det\tilde g_{2D} < 0$ throughout the integrated domain |
| Observable shifts | — | — | — | — | Unstretched-space positive phantom mass $+1.41\%$ ($Q_s=0.20M$, Step 36); S2 pericentre clock redshift $\Delta\chi^2 = +3.52$ ($B=1.64$ at locked $\eta=0.30$, Step 46) | Positive phantom mass $+2.65\%$ ($Q_{\rm eff}=0.236M$, Step 34/36); M87$^\ast$ shadow $40.25$–$40.74\,\mu$as ($-0.58\sigma$ to $-0.42\sigma$); $c_T=1$ on exterior |
| QNM spectrum | Schwarzschild/Kerr | Modified | — | — | Indicative WKB: scalar-led, axial, polar; requires spectral solver and coupled perturbation equations | Exterior benchmark: axial $+0.344\%$ real shift at $\eta=+0.3$, polar-led $-0.028\%$ real breaking, $+0.065\%$ damping breaking. Full deep-transit solve: three modes (polar-led $0.440-0.034i$, scalar-led $0.416-0.032i$, overtone $0.745-0.035i$), $2.6\times$ longer damping, $+9.3\%$ real breaking, $5.6\%$ mode splitting |
| Shadow size | $b = 5.196\,M$ | Similar to Schwarzschild | — | — | $-0.35\%$ at $\eta=+0.3$ on $g^{\rm sGB}$; conformal invariance means shadow on $\tilde g$ equals shadow on $g$ for pure conformal; ISCO on $\tilde g$ is $-6.19\%$ (same inward direction, but $\mathcal{O}(\eta)$, $\sim 18\times$ larger) — a property of the sGB EFT (Appendix L) | From $\tilde g$ photon sphere (target) |
| Gravitational-wave echoes | No | Possible (inner barrier) | — | — | Not yet analysed (requires waveform-level analysis with inner boundary conditions) | Depends on interior outcome |
| Stability / hyperbolicity | Stable | Model-dependent | — | — | Tensor sector: kinetic term positive at leading order, $c_T = 1$ on Schwarzschild background (Horndeski $G_4$ unchanged by sGB at leading order). Full tensor characteristic metric on TEP solution, global ghost freedom, and scalar sector hyperbolicity in deep interior remain open (Thaalba et al. 2024; Section 7.3) | Scalar-sector principal symbol on solved geometry (target) |
| $c_T$ (tensor speed) | $c_T = 1$ | $c_T \to 1$ asymptotically | — | — | $c_T = 1$ on Schwarzschild background (Horndeski $G_4 = M_{\rm Pl}^2/2$, sGB coupling enters through $G_5$); GW170817 satisfied on that background; full tensor characteristic metric on TEP solution pending (Section 7.3) | Established |
| Observational coupling | $M$ (and $a$) | Regularisation parameter | — | — | One $\alpha_{\rm GB}$; $\eta_i = 3\alpha_{\rm GB}/M_i^2$ per source; $|\eta|=0.3$ cannot be universal across masses | Same (target) |

## I.2 What Is Established vs What Is Target

The established results in this paper are:

- The Schwarzschild incompatibility: on fixed Schwarzschild, finite curvature and bounded areal radius are mutually exclusive for the power-law conformal class (Section 3, Appendix D).

- The Hayward benchmark: finite curvature and bounded areal radius are compatible on a regular background, but the $\phi_0 = 1$ benchmark produces a finite-area asymptotic end, not a regular centre (Appendix K).

- The perturbative sGB exterior: scalar hair and real backreaction are established by the published $\mathcal{O}(\alpha^2)$ solution (Sotiriou & Zhou 2014). The non-perturbative exterior mass-function evolution requires rerun with the correct scalar charge.

- The literature evidence that standard linear sGB develops a finite-area interior singularity (Sotiriou & Zhou 2014; Thaalba et al. 2024).

- The minimal temporal-well criteria: $0 < N(r)$ everywhere with $N_{\min} \ll N_o$, regular Lorentzian geometry, no observer-independent one-way boundary, extreme external redshift (Section 6.3).

- The scalar-on-regular-background benchmark with a regular point centre, solved globally as the regular-centre/ambient-decay two-point boundary-value problem and reached to $r = 10^{-6}M$ for $\eta=+0.3$ (Section 4, Appendix K).

The remaining target results are:

- The coupled polar–scalar QNM spectrum and isospectrality breaking.

- The observable shifts as quantitative predictions (requires deriving both characteristic metrics).

- The tensor speed $c_T$ derived from the action.

- The observational fit with one universal $\alpha_{\rm GB}$ across all sources.

The comparison table above distinguishes these explicitly. Earlier versions of this appendix combined all TEP calculations into a single column, producing claims (globally Lorentzian interior, geodesic completeness, vanishing central curvature, disformal birefringence, no inner boundary) that belong to specific benchmark or theorem cases, not to the unsolved global TEP solution.

# Appendix J — Reproducibility

Every numerical result quoted in this paper is produced by the automated pipeline, which downloads or verifies the primary data, constructs the prescribed matter metric, computes the diagnostics, and writes the comparison tables. Each clean run writes SHA-256 checksums for the result files. Checksums prove that the result files were not altered after generation; they do not prove that the calculations are correct. Three categories of calculation require particular care in scalar–tensor perturbation theory and are not cited as quantitative evidence in this manuscript: (i) the nonlinear exterior mass-function evolution, which requires a consistent scalar charge and sign convention matched to published sGB solutions; (ii) surrogate WKB QNM frequencies, which require the perturbation potential to be derived from the second variation of the action (not from inserting modified background functions into a GR potential formula); and (iii) matter-frame observables, which must be computed on the matter metric $\tilde g$ rather than the geometric metric $g$. The fixed-background limit is globally Lorentzian for all $r > 0$; the coupled solution is Lorentzian in the exterior ($F > 0$). The deep-interior Lorentzianity of the scalar-on-regular-background benchmark depends on the interior integration (Section 4, Appendix K).

## J.1 Pipeline Architecture

The pipeline consists of 51 steps (`step_00` through `step_50`), each implemented as an independent Python module. Steps 00–11 implement the fixed-background limit (Section 3); steps 12–15 implement the coupled solution (Section 4); steps 16–41 are derivation scripts (geometry, corrected observables, QNM solvers, interior, phantom mass, Kerr-sGB, GW confrontations); step 42 generates figures; steps 43–49 implement the S-star inference pipeline (Section 9, Appendix J); step 50 computes the operational darkness threshold and compares it against the interior benchmark (Section 6.2). The steps are:

| Step | Inference Step | Output | Description |
| --- | --- | --- | --- |
| step_00 | step_00_data_download | data/raw/, data/processed/ | Download EHT M87* visibilities, compile measurement tables |
| step_01 | step_01_field_equations | step_01_field_equations.{json,csv} | Disformal metric, curvature invariants, geodesic completeness, physical volume/density |
| step_02 | step_02_perturbations | step_02_perturbations.{json,csv} | Regge–Wheeler/Zerilli potentials, QNMs, echo check |
| step_03 | step_03_raytracing | step_03_raytracing.{json,csv} | Photon sphere, shadow angular diameter, ISCO, EHT comparison |
| step_04 | step_04_accretion | step_04_accretion.{json,csv} | Circular geodesics, epicyclic frequencies, radiative efficiency, redshift, QPO |
| step_05 | step_05_observational_constraints | step_05_observational_constraints.{json,csv} | Chi-squared comparison of TEP predictions against EHT and LIGO |
| step_06 | step_06_gw190521_mass_gap | step_06_gw190521_mass_gap.{json,csv} | GW190521 mass gap: LIGO posterior analysis (TEP exterior-null) |
| step_07 | step_07_qpo_frequency_lock | step_07_qpo_frequency_lock.{json,csv} | QPO frequency lock: 3:2 epicyclic resonance ratios (Schwarzschild baseline) |
| step_08 | step_08_jwst_early_smbhs | step_08_jwst_early_smbhs.{json,csv} | JWST early SMBHs: Eddington-limited growth envelope from stellar seeds |
| step_09 | step_09_spin_bias | step_09_spin_bias.{json,csv} | Over-maximal spin bias: observed spin distribution (TEP exterior-null) |
| step_10 | step_10_tde_missing_flares | step_10_tde_missing_flares.{json,csv} | TDE missing flares: sub-Eddington luminosity survey (TEP exterior-null) |
| step_11 | step_11_eht_polarization | step_11_eht_polarization.{json,csv} | EHT polarization: birefringence check (TEP exterior-null) |
| step_12 | step_12_self_gravitating | step_12_self_gravitating.{json,csv} | sGB metric corrections, scalar profile, exterior observables (shadow, ISCO, QNM), $\eta$-scan, GW150914 projection (Section 4) |
| step_13 | step_13_scalar_perturbations | step_13_scalar_perturbations.{json,csv} | Scalar-led, axial, and polar QNM channels; isospectrality breaking; deep-transit analysis (Section 7) |
| step_14 | step_14_kerr_tep | step_14_kerr_tep.{json,csv} | Kerr–TEP shadow and ISCO spin scan; frame-dragging cancellation; GW150914 Kerr ringdown comparison (Section 4) |
| step_15 | step_15_dynamical_signatures | step_15_dynamical_signatures.{json,csv} | PPN parameters, $-1$PN scalar dipole radiation, $c_T = 1$ on the Schwarzschild background, combined $\eta$ constraints (Section 4) |
| step_16 | step_16_exact_geometry | step_16_exact_geometry.{json,csv} | Exact curvature invariants, determinant, inverse metric, areal radius, proper radial distance |
| step_17 | step_17_null_expansions | step_17_null_expansions.json | Null expansions $\theta_\pm$ for TEP matter metric on Schwarzschild and Hayward backgrounds |
| step_18 | step_18_observer_redshift | step_18_observer_redshift.{json,csv} | Gravitational redshift for static observers at different radii |
| step_19 | step_19_observer_frequency_transfer | step_19_observer_frequency_transfer.{json,csv} | Invariant emitter-receiver frequency transfer factor $\mathcal{Z}$ (Section 6) |
| step_20 | step_20_corrected_observables | step_20_corrected_observables.json | Corrected exterior observables (shadow, ISCO, frequency transfer) under fixed-ADM normalization (Appendix L) |
| step_21 | step_21_cT_derivation | step_21_cT_derivation.json | Tensor propagation speed $c_T$ from principal symbol of coupled Einstein-sGB equations |
| step_22 | step_22_quadratic_action | step_22_quadratic_action.json | Exact quadratic action for axial, polar, and scalar perturbation channels (Section 7) |
| step_23 | step_23_characteristic_matrices | step_23_characteristic_matrices.json | Tensor characteristic matrices and principal symbol analysis |
| step_24 | step_24_conformal_invariance_check | step_24_conformal_invariance_check.json | Conformal invariance verification for null geodesics |
| step_25 | step_25_qnm_solver | step_25_qnm_solver.json | Perturbative sGB QNM solver: axial, polar-led, scalar-led modes |
| step_26 | step_26_coupled_spectral_solver | step_26_coupled_spectral_solver.json | Coupled spectral solver for polar-scalar mixed modes |
| step_27 | step_27_qnm_horizon_branch | step_27_qnm_horizon_branch.json | Horizon-bearing sGB QNM: Schwarzschild base + Sotiriou-Zhou $\mathcal{O}(\eta^2)$ correction (Section 7.3) |
| step_28 | step_28_qnm_validation | step_28_qnm_validation.json | QNM spectrum validation against published sGB results (Bryant et al. 2021; Blazquez-Salcedo et al. 2016) (Section 7.3) |
| step_29 | step_29_matrix_leaver | step_29_matrix_leaver.json | Full coupled $2\times 2$ matrix continued-fraction QNM solver with regular inner boundary; three deep-transit modes (Section 7.3) |
| step_30 | step_30_taylor_recurrence | step_30_taylor_recurrence.json | Exact Taylor-series recurrence for background potentials and scalar field coefficients near the origin (Section 7.3) |
| step_31 | step_31_frobenius_analysis | step_31_frobenius_analysis.json | Frobenius regularity analysis at the temporal minimum: $r^{\ell+1}$ regularity, finite tortoise, regularised polar–scalar mixing (Section 7.3) |
| step_32 | step_32_interior_integration | step_32_interior_integration.json | Exploratory Ricci-coupling inward-IVP diagnostic: exterior matching data do not lie on the regular-centre branch (canonical interior solver: step_34) |
| step_33 | step_33_interior_analysis | step_33_interior_analysis.json | Exploratory interior-construction diagnostics: regularity checks, Lorentzian signature, lapse minimum (one approach to the construction target; not an established global solution) |
| step_34 | step_34_solve_interior | step_34_solve_interior.json | TEP interior solver: sGB scalar on Hayward background (two-point BVP: regular centre + ambient decay), matter metric construction (Section 4, Appendix K) |
| step_35 | step_35_mass_bias_sign | step_35_mass_bias_sign.json | Mass-bias sign equation derivation (Section 2.8) |
| step_36 | step_36_phantom_mass_critical | step_36_phantom_mass_critical.json | Horizon-scale phantom mass: critical $g$ analysis, EHT M87$^\ast$ and Sgr A$^\ast$ shadow comparison (Section 8.2) |
| step_37 | step_37_phantom_mass_raytrace | step_37_phantom_mass_raytrace.json | phantom mass null geodesic ray-tracing (Section 8.2) |
| step_38 | step_38_phantom_mass_scan | step_38_phantom_mass_scan.json | phantom mass parameter scan (Section 8.2) |
| step_39 | step_39_eht_visibility_fit | step_39_eht_visibility_fit.json | EHT visibility-domain joint inference: direct fit of TEP shadow model to calibrated visibilities (Section 9.4) |
| step_40 | step_40_kerr_sgb_true | step_40_kerr_sgb_true.json | True rotating sGB shadow and QNM from Delgado et al. (2020) slowly-rotating solution (Section 8.5) |
| step_41 | step_41_gw250114_confrontation | step_41_gw250114_confrontation.json | GW250114 corrected confrontation with TEP observables |
| step_42 | step_42_generate_figures | results/figures/ | Figure generation (10 figures from paper plan) |
| step_43 | step_43_sstar_data | step_43_sstar_data.json | S-star data acquisition: 145 astrometric + 44 radial-velocity epochs for S2 (Section 9) |
| step_44 | step_44_gr_fit | step_44_gr_fit.json | Conventional GR fit to S-star data (Section 9) |
| step_45 | step_45_mass_bias | step_45_mass_bias.json | Initial mass-bias sign evaluation from GR fit residuals (Section 9) |
| step_46 | step_46_tep_transfer | step_46_tep_transfer.json | TEP transfer-function fit with $\eta_{\rm TEP}$ as free parameter (Section 9) |
| step_47 | step_47_joint_forward | step_47_joint_forward.json | Joint forward-model: composition vs calibration separation, $\mathcal{C}_V$ diagnostic (Section 9) |
| step_48 | step_48_likelihood | step_48_likelihood.json | Formal likelihood comparison: Bayes factor, BIC, AIC, Wilks (Section 9) |
| step_49 | step_49_posterior | step_49_posterior.json | phantom mass posterior via MCMC (Section 9) |
| step_50 | step_50_darkness_threshold | step_50_darkness_threshold.json | Operational darkness threshold $\mathcal{T}_{\rm dark} = (C_{\rm em}/\varepsilon)^{1/3}$ vs benchmark $\mathcal{Z}_{\max}$; radiometric, massive, spectral, dilution, and transit channels (Section 6.2) |

All calculations use the unified parameter set: $B_0 = 1.0$, $M = 1.0$, $\beta_A = -1.0$, $n_B = 2.0$, $\delta = 0.05$, $\sigma_B = 1.5$, $r_h = 2.0$, with the quartic-damped disformal coupling $B(\phi) = B_0\,|\phi|^{n_B}/(1 + |\phi|^{n_B})\,\exp(-\phi^4/2\sigma_B^4)$. The conformal exponent $\beta_A = -1$ is frozen across all calculations — the analysis of Section 3, sGB exterior, and the target global solution — consistent with the canonical TEP bare coupling. Steps 12–15 use the sGB coupling parameter $\eta$ (primary value $\eta = +0.3$ for the temporal-well branch — the interior-regularity floor derived in Section 4; $\eta(M) = 3\alpha_{\rm GB}/M^2$ varies with mass for fixed $\alpha_{\rm GB}$) with the exact $\mathcal{O}(\alpha^2)$ perturbative metric corrections of Sotiriou & Zhou (2014), Eqs. (56)-(63); the $\eta$-scan covers $\eta \in \{0.1, 0.2, 0.3, 0.5\}$.

#### Convention resolution

The conformal coupling $A = e^{\beta_A \phi}$ with $\beta_A = -1$ is frozen across all calculations. The different scalar profiles interact with this single convention as follows: (i) the analysis of Section 3 uses $\phi = \phi_0 \ln(r/r_h) < 0$ in the interior, giving $A = e^{-\phi} \to \infty$ — the divergent conformal factor that establishes this result; (ii) the sGB exterior uses $\phi \sim Q_s/r$ (Coulomb). TEP selects the temporal-well branch with $Q_s > 0$ (equivalently $\alpha_{\rm GB} > 0$), so $\phi > 0$ and $A = e^{-\phi} < 1$ — the matter metric is conformally *contracted* relative to $g$, and clocks run progressively slower toward the object, consistent with the corpus-wide sign convention of Paper 0 §2.2; (iii) the target global solution has $\phi \to \phi_c$ finite at the centre, giving $A_c = e^{-\phi_c}$ finite — one candidate profile satisfying the minimal temporal-well criteria of Section 6.3. The quartic-Gaussian $B(\phi)$ should ultimately be derived from the action or screening mechanism, or an admissible class of $B(\phi, X)$ functions should be shown to produce stable results; this is a refinement target, not a structural inconsistency.

## J.2 Output Files

The pipeline produces three categories of output:

- `results/*.json` — structured summary of each step (the canonical machine-readable results);

- `results/*.csv` — data tables for plotting and inspection;

- `data/processed/*.json` — compiled measurement tables (EHT, LIGO QNM, QPO, TDE, spin, JWST) built from the raw downloads;

- `data/raw/` — downloaded EHT/LIGO data (EHT M87* 2019-D01-01 visibilities, LIGO GW190521 posterior);

- `results/pipeline_results.json` — top-level summary with step status, timing, and execution metadata;

- `results/checksums_sha256.json` — SHA-256 checksum of every `results/*.json` and `results/*.csv` file.

## J.3 Checksum Verification

At the end of each run, the pipeline computes the SHA-256 hash of every result file and writes them to `results/checksums_sha256.json`. This covers all result files (step JSON + step CSV + pipeline summary JSON). Any modification to a result file is detectable by re-running the hash. The checksum file itself is regenerated on each run, so the recorded hashes always correspond to the current results.

## J.4 Warning-Free Execution

A clean run is verified with the repository command:

\begin{equation} \label{eq:appJ_1}
\texttt{python scripts/run\_pipeline.py}
\end{equation}

The current diagnostics include a numerical safeguard that masks inverse-determinant curvature expressions where $|\det\tilde g_{2D}|$ is small; in the fixed-background limit this safeguard is never triggered because the determinant is strictly negative everywhere (globally Lorentzian).

## J.5 Data Sources

All observational data is drawn from published, publicly available sources:

| Source | Reference | Used in |
| --- | --- | --- |
| EHT M87* 2019 calibrated visibilities | EHT Collaboration 2019, data product 2019-D01-01 (GitHub) | step_00, step_03, step_05, step_11 |
| LIGO GW190521 posterior | Isi et al. 2020 (Zenodo 4057131) | step_06 |
| QPO measurements (GRS 1915+105) | Strohmayer 2001; Remillard et al. 2002; Homan et al. 2005 | step_07 |
| JWST early SMBH masses | Harikane et al. 2023; Greene et al. 2023; Bogdan et al. 2024 | step_08 |
| Black-hole spin measurements | McClintock et al. 2014; Reynolds 2021; EHT 2019/2022 | step_09 |
| TDE flare catalogs | van Velzen et al. 2021; Holoien et al. 2020 | step_10 |
| EHT polarization | EHT 2021 ApJ 910 L13; EHT 2024 ApJ 964 L26 | step_11 |
| LIGO QNM measurements | compiled in data/processed/ligo_qnm_measurements.json | step_05 |
| sGB perturbative metric corrections | Sotiriou & Zhou 2014, Eqs. (56)-(63) (exact $\mathcal{O}(\alpha^2)$) | step_12, step_14 |
| Kerr QNM / shadow formulas | Bardeen 1973; Leaver 1985 (approximate) | step_14 |
| GW170817 $c_T$ constraint | LIGO/Virgo & Fermi-GBM 2017 | step_15 |
| Binary pulsar dipole bounds | Freire et al. 2012; Antoniadis et al. 2013 (conditional on NS scalarization) | step_15 |

## J.6 Running the Pipeline

The full pipeline is run with a single command:



```
git clone https://github.com/matthewsmawfield/TEP-BH.git
cd TEP-BH
python scripts/run_pipeline.py
```



Options include `--start-step`, `--stop-step`, `--skip-steps`, `--no-derive`, `--no-inference`, `--no-figures`, `--list-steps`, and `--continue-on-error`. Full documentation is in `scripts/README.md`.

## J.7 Code Availability

All code is in the GitHub repository:
https://github.com/matthewsmawfield/TEP-BH

The repository contains the complete pipeline (`scripts/`), the processed and raw data (`data/`), the results (`results/`), the manuscript source (`site/components/`), and the figure-generation code (`scripts/steps/step_42_generate_figures.py`). The site is built with `cd site && npm run build` and published at
https://mlsmawfield.com/tep/bh.

The pipeline is 51 steps (step_00–step_50), runs warning-free, and writes SHA-256 checksums for all result files. Steps 00–11 use $\phi_0 = 2.0$, $\delta = 0.05$, $\sigma_B = 1.5$ for the fixed-background limit; steps 12–15 use the sGB coupling $\eta$ (primary $\eta = +0.3$) for the coupled solution, selecting the temporal-well branch; steps 16–41 are derivation scripts; step 42 generates figures; steps 43–49 implement the S-star inference. All data is from published sources (EHT 2019-D01-01, LIGO Zenodo 4057131, and compiled measurement tables). Full documentation is in the repository `README`.

**S-star inference pipeline.** Steps 43–49 of the unified pipeline implement the primary falsifiable test: a non-isochronous refit of the published S-star data around Sgr A*. Step 43 downloads the machine-readable CDS files for Gillessen et al. (2017, VizieR J/ApJ/837/30, table5.dat and table3.dat); it parses 145 NACO/NTT/Keck/Gemini astrometric epochs and 44 SINFONI/Keck/Gemini radial-velocity epochs for S2 relative to Sgr A*. The steps are: (43) data acquisition, (44) conventional GR fit, (45) initial mass-bias sign evaluation, (46) TEP transfer-function fit, (47) joint forward-model with composition/calibration separation and $\mathcal C_V$ diagnostic, (48) formal likelihood comparison (Bayes factor, BIC, AIC, Wilks), (49) MCMC posterior on $M_{\rm phantom}^T$.

# Appendix K — Regular-Geometry Validation Benchmark

This appendix records the regular-geometry validation benchmark used throughout the main text. The Hayward metric is a controlled mathematical reference, not the TEP-selected solution: it validates the analysis pipeline and confirms that finite curvature and bounded areal geometry are mutually compatible once the geometric singularity is absent. The question TEP asks is whether a dynamical proper-time field can supply the physical origin of that regularisation. The scalar-on-regular-background benchmark (Section 4) is the key test of this question. The literature evidence (Sotiriou & Zhou 2014; Thaalba et al. 2024) indicates that standard linear sGB may develop a finite-area singularity rather than a regular centre, motivating the full TEP action.

## K.1 The Hayward Metric

The Hayward metric is

\begin{equation} \label{eq:appK_1}
F(r) = 1 - \frac{2Mr^2}{r^3 + 2M\ell^2},
\end{equation}

where $\ell$ is the regularisation scale. As $r \to 0$, $F \to 1$ (de Sitter-like temporal minimum); as $r \to \infty$, $F \to 1 - 2M/r$ (Schwarzschild). The effective energy density is

\begin{equation} \label{eq:appK_2}
\rho_{\rm eff} = -G^t_t = \frac{12M^2\ell^2}{(r^3 + 2M\ell^2)^2} \to \frac{3}{\ell^2} \quad \text{as } r \to 0,
\end{equation}

finite at the centre. The Kretschmann scalar is $K_{\rm Hayward}(0) = 24/\ell^4$, finite. The equation of state is $w_r = -1$ (cosmological-constant-like radial pressure), with anisotropic tangential pressure.

## K.2 TEP Matter Metric on the Hayward Background

The TEP matter metric $\tilde g = A^2 g_{\rm Hayward}$ with $A = (r_h/r)^{\phi_0}$ and $\phi_0 = 1$ is placed on the Hayward background. The computed invariants are:

| Quantity | Value as $r \to 0$ | Classification |
| --- | --- | --- |
| Kretschmann $\tilde K$ | $8/r_h^4 = 0.5$ (for $M=1$, $r_h=2M$) | Finite — de Sitter-like temporal minimum |
| Areal radius $\rho = Ar$ | $1.995$ | Bounded — finite-area asymptotic end |
| $\det\tilde g_{2D}$ | $< 0$ at 100% of sampled radii | Lorentzian in exterior ($F > 0$) |
| Null expansion $\theta_+$ | $0$ exactly | Constant-area property of the asymptotic end (not temporal freeze) |
| Affine parameter $\tilde\lambda$ (both families) | $\sim \int r^{-2}\,dr \to \infty$ | Infinite affine distance — asymptotic end, not regular point |
| Radial proper distance $\tilde\ell$ | $\sim \int dr/r \to \infty$ | Infinite — tube-like asymptotic end |
| Static-observer clock rate $d\tilde\tau/dt$ | $\to \infty$ | Conformal blueshift (unphysical; benchmark artefact of divergent $A$) |
| Static-observer redshift $z$ | $\to -1$ (blueshift) | Infinite redshift is at the Hayward outer horizon, not the centre — benchmark artefact of the divergent-$A$ test profile, not the TEP target |

## K.3 Classification of the Limiting Region

At an ordinary regular spherical centre one has $\rho \to 0$ as $r \to 0$. Here $\rho \to 1.995 \neq 0$: the limiting areal radius is nonzero. Because $A \sim 1/r$ in the deep interior, the radial proper distance behaves as $\tilde\ell \sim \int dr/r \to \infty$. The limiting region is therefore not a point-like centre but a finite-area asymptotic end (a tube-like limiting region of infinite radial extent). This is still a valid TEP geometry — the curvature is finite, the areal radius is bounded, and the metric is globally Lorentzian — but it must be classified correctly as an asymptotic end rather than a regular point centre. The scalar-on-regular-background benchmark (Section 4) now produces an ordinary regular point centre with $\rho_{\rm areal}(0) \to 1.7 \times 10^{-10}$ for $\eta=+0.3$, $g=1.1M$.

## K.4 What the Benchmark Validates

The benchmark validates two things:

- The analysis pipeline correctly recognises a regular geometric metric and classifies the matter-frame limiting region (finite Kretschmann, bounded areal radius, Lorentzianity inherited from $g$ via $A > 0$).

- Finite curvature and bounded areal geometry are mutually compatible once the geometric singularity is absent — the conditions that were mutually exclusive in the Schwarzschild incompatibility result (Section 3) are simultaneously achievable on a regular background.

The benchmark establishes pipeline correctness and the mutual compatibility of finite curvature and bounded areal geometry on a regular background. The mass function obtained by integrating the stress tensor of a scalar that itself solves \((r^2\phi')'=r^2(\mu^2\phi-\rho)\), matched to a Yukawa exterior, lies within 9% rms of a Hayward profile for horizon-free cores (\(F_{\min}=0.66\) at the heaviest of the solved set). That is the geometry emerging from the equation, not an inserted \(m(r)\). On those solutions \(\phi_c=\mathcal O(10^{-2})\) and \(A=e^{-\phi_c}\) stays above \(0.985\). \(A\to 0\) is not a regular solution of this boundary-value problem: the same density that deepens \(\phi\) deepens the mass, and a horizon arrives first. The disformal cone split on these solutions remains \(\ll 1\). The strong-field symbol is not probed by a horizon-free well of this action. The limiting region is a finite-area asymptotic end (Section K.3). The null expansion $\theta_+ = 0$ reflects the constant-area property of the asymptotic end; the invariant frequency-transfer calculation of Section 6 establishes the Temporal Horizon. Both null families have infinite affine parameter ($\tilde\lambda \sim \int r^{-2}\,dr \to \infty$); the asymptotic end is at infinite affine distance.

## K.5 Regularisation Scale Effects

The Kretschmann scalar at the centre depends on the regularisation scale: $K_{\rm Hayward}(0) = 24/\ell^4$. For $\ell = 0.1$ (the benchmark value), $K = 2.4 \times 10^5$; for $\ell = 1.0$, $K = 24$; for $\ell = 0.001$, $K = 2.4 \times 10^{13}$. These are code-unit values ($M = 1$); they do not establish Planck-scale curvature without converting to physical units for a specific black-hole mass. The regularisation scale is a parameter of the benchmark, not a prediction of the theory; the scalar-on-regular-background benchmark with $g = 1.1M$ gives $K(0) = 54.2$ for $\eta=+0.3$ (Section 4) — the exact Hayward central value $K(0) = 96M^{2}/g^{6}$.

## K.6 Configuration Scan

A scan over conformal exponent $\phi_0$ on the Hayward background. For a regular (de Sitter-like) seed, the background Kretschmann $K_{\rm Hay} \to$ finite as $r \to 0$, so the dominant contribution to the conformal Kretschmann comes from the $A$-derivative terms: $\tilde K \sim r^{4\phi_0 - 4}$ (not $r^{4\phi_0 - 6}$, which is the Schwarzschild-seed result where $K_{\rm Schw} \sim r^{-6}$ dominates). The areal radius scales as $\rho = Ar \sim r^{1-\phi_0}$ on either seed. The asymptotic classifications are:

| $\phi_0$ | $\rho$ as $r\to 0$ | $\tilde K$ as $r\to 0$ | Lorentzian | Asymptotic classification |
| --- | --- | --- | --- | --- |
| 0.5 | $\to 0$ | $\to \infty$ ($r^{-2}$) | Yes | Point centre, curvature diverges |
| 1.0 | $\to r_h \neq 0$ | $\to$ finite ($r^0$) | Yes | Finite-area asymptotic end, finite curvature |
| 1.25 | $\to \infty$ ($r^{-0.25}$) | $\to 0$ ($r^1$) | Yes | Areal radius diverges, curvature vanishes |
| 2.0 | $\to \infty$ ($r^{-1}$) | $\to 0$ ($r^4$) | Yes | Areal radius diverges, curvature vanishes |

The $\phi_0 = 1$ configuration is the validation benchmark used in the main text: it is the unique configuration that simultaneously achieves bounded (nonzero) areal radius and finite curvature on the Hayward background. The curvature scaling $\tilde K \sim r^{4\phi_0 - 4}$ differs from the Schwarzschild-seed scaling $\tilde K \sim r^{4\phi_0 - 6}$ (Appendix D) because the regular seed has finite background curvature, so the $A$-derivative terms dominate rather than the background Kretschmann. This distinction is critical: the Schwarzschild incompatibility result uses the Schwarzschild exponent, while the benchmark classification uses the Hayward exponent. The two must not be mixed.

# Appendix L — Explicit Derivation of Corrected Exterior Observables

This appendix provides the step-by-step derivation of the corrected exterior observables — the Regge–Wheeler potential, the photon sphere and shadow radius, and the ISCO on the matter metric $\tilde g$ — from the Sotiriou & Zhou (2014) perturbative sGB solution. The derivation demonstrates the coupling-order difference between the photon and massive-particle sectors explicitly.

## L.1 The sGB-Corrected Metric

The Sotiriou & Zhou (2014) perturbative solution gives the geometric metric to $\mathcal{O}(\beta^2)$, where $\beta = \alpha_{\rm GB}/r_H^2 = \eta/12$ (since $r_H = 2M$ and $\alpha_{\rm GB} = \eta M^2/3$):

\begin{equation} \label{eq:appL_1}
ds^2_{\rm sGB} = -F(1 + \beta^2 h_2)\,dt^2 + \frac{1 + \beta^2 \sigma_2}{F}\,dr^2 + r^2\,d\Omega^2,
\end{equation}

where $F = 1 - r_H/r$, $x = r_H/r$, and the dimensionless polynomials are (Sotiriou & Zhou 2014, Eqs. 60–61):

\begin{equation} \label{eq:appL_2}
h_2(x) = -\frac{98}{5}x - \frac{98}{5}x^2 - \frac{274}{15}x^3 - \frac{14}{15}x^4 + \frac{52}{15}x^5 + \frac{20}{3}x^6,
\end{equation}

\begin{equation} \label{eq:appL_3}
\sigma_2(x) = \frac{98}{5}x + \frac{58}{5}x^2 + \frac{38}{5}x^3 - \frac{406}{15}x^4 - \frac{436}{15}x^5 - \frac{92}{3}x^6.
\end{equation}

The would-be horizon shifts to $r_H = 2m = 2M(1 - 19.6\,\beta^2)$ at fixed ADM mass — a property of the horizon-bearing branch on which the perturbative sGB benchmark lives, not a physical TEP temporal-well horizon. The scalar field is $\phi(r) = (2\alpha_{\rm GB}/m)(1/r + m/r^2 + 4m^2/(3r^3))$ with $m = M/(1 + 49\eta^2/360)$ and scalar charge $Q_s = 2\alpha_{\rm GB}/m$. For the TEP temporal-well branch the scalar charge is positive ($Q_s > 0$), so $\phi > 0$ in the exterior. The conformal factor is $A = e^{-\phi}$, giving the matter metric $\tilde g_{\mu\nu} = A^2 g^{\rm sGB}_{\mu\nu}$ with $A < 1$ (conformal contraction): clocks run progressively slower toward the object, consistent with the corpus-wide sign convention of Paper 0 §2.2. The true TEP solution is a temporal well: $N \geq N_{\min} > 0$ everywhere, with a finite but extremely small minimum lapse at the centre.

The perturbative regime requires $\beta^2 |h_2(x)| \ll 1$ at all radii of interest. At the would-be horizon ($x = 1$), $h_2(1) = -48.27$, so $\beta^2 |h_2(1)| = 6.25 \times 10^{-4} \times 48.27 = 0.030 \ll 1$. The value $\eta = +0.3$ ($\beta^2 = 6.25 \times 10^{-4}$) is used throughout this appendix as the TEP temporal-well benchmark ($\alpha_{\rm GB} > 0$): it is the floor at which the step_33 interior construction exhibits dynamical regularisation (Section 4 — an explored approach to the construction target, not yet an established global solution). For a fixed action constant $\alpha_{\rm GB}$, $\eta(M) = 3\alpha_{\rm GB}/M^2$ is mass-dependent, and the observational bound $\alpha_{\rm GB} < 2.9$ km$^2$ corresponds to $|\eta| < 0.12$ at $M = 10\,M_\odot$; the benchmark is a branch parameter of the strong-field construction, not a universal constant.

## L.2 Regge–Wheeler Potential: Status

A naive application of the scalar gradient $\phi'(r)$ directly into the standard GR Regge–Wheeler potential formula yields a term with dimensions $L^{-4}$ — inconsistent with the potential's $L^{-2}$ dimensions. The correct approach requires deriving the axial perturbation equation from the second variation of the sGB action on the corrected background. In modified gravity, the axial perturbation equation generally cannot be obtained merely by inserting the modified background functions into a GR potential formula; the modified field equations contribute directly to the quadratic perturbation operator.

The dominant effect on the QNM spectrum comes from the would-be horizon shift: $r_H \to 2M(1 - 19.6\,\beta^2)$ shifts $F$ and hence the entire potential. The would-be horizon scale change $r_H \to 2M(1 - 19.6\,\beta^2)$ gives an $\mathcal{O}(\beta^2) \sim \mathcal{O}(10^{-3})$ QNM modification at $\eta = +0.3$ (with $\beta = \eta/12$); the coefficient and sign are not determined by the would-be horizon shift alone. A precise value requires a coupled spectral solver on the corrected background, with the perturbation equations derived from the second variation of the action and validated against published nonperturbative sGB QNM spectra (Witek et al. 2019; Blazquez-Calzadilla et al. 2020; Chen et al. 2024). The structural prediction (scalar-led channel, broken isospectrality at $\mathcal{O}(\eta^2)$) follows from the coupling structure; the sign of the QNM shift is not determined by the would-be horizon shift alone. The deep-transit ringdown structure on the horizonless temporal well is analysed in Section 7.3.

## L.3 Photon Sphere and Shadow (Null Geodesics on $g^{\rm sGB}$)

For null geodesics on $ds^2 = -f\,dt^2 + g\,dr^2 + h\,d\Omega^2$, the impact parameter at a circular null orbit is $b^2 = h/f$. The photon sphere is at the minimum of $b^2(r)$, and the shadow radius is $b_{\rm ph} = \sqrt{h/f}$ at that minimum.

For the geometric metric $g^{\rm sGB}$: $f = F(1 + \beta^2 h_2)$, $h = r^2$. The minimum of $b^2 = r^2/[F(1 + \beta^2 h_2)]$ gives $r_{\rm ph} = 2.9775M$ and $b_{\rm ph} = 5.178M$ at $\eta = +0.3$, a $-0.35\%$ deviation from Schwarzschild ($b = 5.196M$).

For the matter metric $\tilde g = A^2 g^{\rm sGB}$: $f_{\tilde{}} = A^2 f$, $h_{\tilde{}} = A^2 r^2$, so $b^2 = h_{\tilde{}}/f_{\tilde{}} = r^2/f = r^2/[F(1 + \beta^2 h_2)]$ — identical to $g^{\rm sGB}$. The conformal factor $A^2$ cancels exactly. This is conformal invariance of null geodesics: the photon sphere and shadow are the same on $g^{\rm sGB}$ and $\tilde g$. The shadow deviation is $\mathcal{O}(\eta^2)$, determined solely by the metric perturbation.

## L.4 ISCO (Timelike Geodesics on $\tilde g$)

For timelike geodesics on $ds^2 = -f\,dt^2 + g\,dr^2 + h\,d\Omega^2$, the specific angular momentum of a circular orbit is:

\begin{equation} \label{eq:appL_4}
L^2 = \frac{f'\, h^2}{f\, h' - f'\, h},
\end{equation}

and the specific energy is $E^2 = f(1 + L^2/h)$. The ISCO is at the minimum of $L^2(r)$ (marginally stable orbit).

For the geometric metric $g^{\rm sGB}$ ($h = r^2$): $L^2 = f' r^4 / (f \cdot 2r - f' r^2)$. The minimum gives $r_{\rm ISCO} = 5.970M$ at $\eta = +0.3$, a $-0.50\%$ deviation. This is an $\mathcal{O}(\eta^2)$ effect from the metric perturbation alone.

For the matter metric $\tilde g = A^2 g^{\rm sGB}$ ($h = A^2 r^2$): the angular component now includes the conformal factor. The derivative $h' = 2A^2 r + 2A^2 r \cdot (-\phi') r = A^2(2r - 2r^2\phi')$ introduces $\phi'$ at $\mathcal{O}(\eta)$, which does not cancel. The minimum of $L^2(r)$ gives $r_{\rm ISCO} = 5.628M$ at $\eta = +0.3$, a $-6.19\%$ deviation. This is an $\mathcal{O}(\eta)$ effect: the conformal factor $A = e^{-\phi} < 1$ contracts the effective areal radius $\tilde R = A \cdot r$, pulling the ISCO inward.

## L.5 Coupling-Order Difference in the sGB Benchmark

The key result is the coupling-order difference between the shadow and the matter-metric ISCO. Under fixed-ADM mass normalization ($M_{\rm ADM} = M$ held fixed; the would-be horizon shifts to $r_H = 2M(1 - 19.6\,\beta^2)$), both shifts are inward — the conformal contraction pulls massive orbits inward at $\mathcal{O}(\eta)$ while the geometric metric receives only an $\mathcal{O}(\eta^2)$ inward shift. The ISCO shift is $\sim 18\times$ larger in magnitude:

\begin{equation} \label{eq:appL_5}
\frac{\delta b}{b}\bigg|_{\rm shadow} = -0.35\% \quad (\mathcal{O}(\eta^2), \text{ photons on } g^{\rm sGB} = \tilde g),
\end{equation}

\begin{equation} \label{eq:appL_6}
\frac{\delta r_{\rm ISCO}}{r_{\rm ISCO}}\bigg|_{\tilde g} = -6.19\% \quad (\mathcal{O}(\eta), \text{ massive particles on } \tilde g).
\end{equation}

The coupling-order difference is a mathematical property of the perturbative sGB calculation (Section 6.2), not an indication of fractured metric structure. Matter, photons, and clocks all propagate universally on the single causal matter metric, $\tilde g_{\mu\nu}$. The $\sim 18\times$ coupling-order ratio arises purely from the geometry of the paths: because null geodesics ($d\tilde s^2 = 0$) are conformally invariant, the temporal scaling $A^2(\phi)$ divides out exactly, making the shadow sensitive only to the underlying backreacted geometry at $\mathcal{O}(\eta^2)$. Conversely, massive particles travel on timelike geodesics ($d\tilde s^2 < 0$) and therefore feel the conformal factor $A = e^{-\phi} < 1$ directly, shifting the ISCO at $\mathcal{O}(\eta)$. Both effects are complementary projections of the same unified temporal well. The matter-metric ISCO shift is an order of magnitude larger than the shadow shift ($-6.19\%$ vs $-0.35\%$) because the conformal factor contributes at $\mathcal{O}(\eta)$, not $\mathcal{O}(\eta^2)$. The would-be horizon shift $r_H = 2M(1 - 19.6\,\beta^2)$ is the sGB perturbative signature: the temporal field backreacts on the geometry through the sGB coupling, contracting the horizon of the horizon-bearing branch. The Temporal Horizon is observer-dependent and operational — a distant observer (fast clock) sees the accessibility boundary far from the centre; a deeper observer (stronger conformal field, smaller $A$) sees less redshift to any given emitter, and the practical observability boundary moves inward.

The scaling with $\eta$ is:

- Shadow: $\delta b/b \propto \eta^2$ (geometric metric only, conformal factor cancels)

- ISCO on $g^{\rm sGB}$: $\delta r/r \propto \eta^2$ (geometric metric only)

- ISCO on $\tilde g$: $\delta r/r \propto \eta$ (conformal factor dominates)

All numbers are computed at $\eta = +0.3$ in the perturbative regime — the interior-regularity floor of Section 4. For a fixed action constant $\alpha_{\rm GB}$, $\eta(M) = 3\alpha_{\rm GB}/M^2$ is mass-dependent; the observational bound $\alpha_{\rm GB} < 2.9$ km$^2$ gives $|\eta| < 0.12$ for $M = 10\,M_\odot$, so the benchmark functions as a branch parameter rather than a universal constant. Within a fixed-$\alpha_{\rm GB}$ bound, the largest shifts are for $M = 10\,M_\odot$: shadow $-0.007\%$, ISCO $-0.78\%$. The computation uses the Sotiriou & Zhou (2014) exact $\mathcal{O}(\beta^2)$ metric perturbations and the analytic sGB scalar profile, with no free parameters.

# References

- Smawfield, M. L. (2025). Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed. Paper 0, v0.16 (Jakarta). DOI: 10.5281/zenodo.16921911.

- Schwarzschild, K. (1916). Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie. *Sitzungsber. Preuss. Akad. Wiss.*, 189–196.

- Penrose, R. (1965). Gravitational collapse and space-time singularities. *Phys. Rev. Lett.*, 14, 57–59.

- Hawking, S. W. (1966). The occurrence of singularities in cosmology. *Proc. Roy. Soc. A*, 294, 511–521.

- Hawking, S. W. & Penrose, R. (1970). The singularities of gravitational collapse and cosmology. *Proc. Roy. Soc. A*, 314, 529–548.

- Bekenstein, J. D. (1973). Black holes and entropy. *Phys. Rev. D*, 7, 2333–2346.

- Hawking, S. W. (1975). Particle creation by black holes. *Commun. Math. Phys.*, 43, 199–220.

- Bekenstein, J. D. (1993). Relation between physical and gravitational geometry. *Phys. Rev. D*, 48, 3641–3647. arXiv:gr-qc/9211017.

- Wald, R. M. (1993). Black hole entropy is the Noether charge. *Phys. Rev. D*, 48, R3427–R3431. arXiv:gr-qc/9307038.

- Bardeen, J. M. (1968). Non-singular general relativistic gravitational collapse. In *Proceedings of the 5th International Conference on Gravitation and the Theory of Relativity*, Tbilisi, p. 174.

- Hayward, S. A. (2006). Formation and evaporation of nonsingular black holes. *Phys. Rev. Lett.*, 96, 031103. arXiv:gr-qc/0506126.

- Mazur, P. O. & Mottola, E. (2004). Gravitational vacuum condensate stars. *Proc. Nat. Acad. Sci.*, 101, 9545–9550. arXiv:gr-qc/0109035.

- Mathur, S. D. (2005). The fuzzball proposal for black holes: an elementary review. *Fortsch. Phys.*, 53, 793–827. arXiv:hep-th/0502050.

- Almheiri, A., Marolf, D., Polchinski, J. & Sully, J. (2013). Black holes: complementarity or firewalls? *JHEP*, 02, 062. arXiv:1207.3123.

- Khoury, J. & Weltman, A. (2004). Chameleon cosmology. *Phys. Rev. D*, 69, 044026.

- Event Horizon Telescope Collaboration (2019). First M87 Event Horizon Telescope results. I. The shadow of the supermassive black hole. *ApJL*, 875, L1.

- Event Horizon Telescope Collaboration (2019). First M87 Event Horizon Telescope results. IV. Imaging the central supermassive black hole. *ApJL*, 875, L4.

- Event Horizon Telescope Collaboration (2019). First M87 Event Horizon Telescope results. V. Physical origin of the asymmetric ring. *ApJL*, 875, L5.

- Event Horizon Telescope Collaboration (2022). First Sagittarius A* Event Horizon Telescope results. I. The shadow of the supermassive black hole in the center of the Milky Way. *ApJL*, 930, L12.

- Smawfield, M. L. (2026). Temporal Equivalence Principle: A Covariant Alternative to Cosmic Expansion. Paper 26 (Athens). DOI: 10.5281/zenodo.20370143.

- Smawfield, M. L. (2026). Temporal Equivalence Principle: Native hi_class Conformal Implementation, Linear Perturbation Closure, and CMB Acoustic Peak Preservation. Paper 18 (Cambridge). DOI: 10.5281/zenodo.20572722.

- Smawfield, M. L. (2026). Temporal Equivalence Principle: Temporal Horizon Cosmology and the Absence of a Physical Big Bang Singularity. Paper 27 (Thika). DOI: 10.5281/zenodo.20723059.

- Smawfield, M. L. (2026). Temporal Equivalence Principle: Dynamical Proper Time and the Illusion of Primordial Deuterium. Paper 29 (Dubai).

- LIGO/Virgo Collaboration (2016). Observation of gravitational waves from a binary black hole merger. *Phys. Rev. Lett.*, 116, 241102. DOI: 10.1103/PhysRevLett.116.241102.

- LIGO/Virgo Collaboration (2020). GW190521: A binary black hole merger with a total mass of 150 M<sub>☉</sub>. *Phys. Rev. Lett.*, 125, 101102. DOI: 10.1103/PhysRevLett.125.101102.

- Isi, M., et al. (2020). Posterior samples of GW190521 posterior samples. *Zenodo*. DOI: 10.5281/zenodo.4057131.

- Strohmayer, T. E. (2001). Discovery of high-frequency quasi-periodic oscillations in the black-hole binary GRS 1915+105. *ApJ*, 552, L49.

- Remillard, R. A., et al. (2002). XTE J1550-564: QPOs and black-hole spin. *ApJ*, 564, 962.

- Homan, J., et al. (2005). High-frequency QPOs in H 1743-322. *ApJ*, 624, 1005.

- Morgan, E. H., Remillard, R. A., & Greiner, J. (1997). RXTE observations of GRS 1915+105. *ApJ*, 482, 993.

- Harikane, E., et al. (2023). JWST CEERS: discovery of black-hole candidates at z > 8. *ApJ*, 958, 11.

- Greene, J. E., et al. (2023). CEERS AGN at z ~ 7. *ApJ*, 957, 24.

- Bogdan, A., et al. (2024). Evidence for a black-hole seed at z = 10.1. *Nature Astronomy*.

- Castellano, P., et al. (2024). JWST discovery of GHZ2 at z = 10.6. *Nature Astronomy*.

- Matsuoka, Y., et al. (2019). Subaru discovery of a z = 7.04 quasar. *ApJ*, 872, 2.

- Ba&ntilde;ados, E., et al. (2018). A black-hole mass of 8&times;10<sup>8</sup> M<sub>☉</sub> at z = 7.54. *Nature*, 553, 473.

- McClintock, J. E., Narayan, R., & Steiner, J. F. (2014). Black-hole spin via continuum fitting and reflection spectroscopy. *Space Sci. Rev.*, 183, 295. arXiv:1303.1583.

- Reynolds, C. S. (2021). X-ray reflection spin measurements and systematics. arXiv:2104.10300.

- Gou, L., et al. (2011). Cygnus X-1 spin via continuum fitting. *ApJ*, 742, 85.

- van Velzen, S., et al. (2021). Seventeen tidal disruption events from ZTF. *ApJ*, 908, 4.

- Holoien, T. W.-S., et al. (2016). ASASSN-14li: a nearby TDE. *MNRAS*, 455, 1690.

- Holoien, T. W.-S., et al. (2020). ASASSN-18pg: a luminous TDE. *ApJ*, 903, 151.

- Nicholl, M., et al. (2020). AT2019qiz: a nearby TDE with early-time spectroscopy. *MNRAS*, 499, 482.

- Event Horizon Telescope Collaboration (2021). First M87 Event Horizon Telescope results. VII. Polarization of the ring. *ApJL*, 910, L13.

- Event Horizon Telescope Collaboration (2024). Sgr A* polarimetry with the EHT. *ApJL*, 964, L26.

- Bekenstein, J. D. (2004). Conformal and disformal transformations. *Phys. Rev. D*, 70, 083509.

- Koivisto, T. (2012). Disformal equivalence. *Phys. Rev. D*, 85, 044043.

- Inayoshi, K., Haiman, Z., & Ostriker, J. P. (2020). Hyper-Eddington accretion and rapid growth of massive black holes. *MNRAS*, 496, 4236.

- Sotiriou, T. P. & Zhou, S.-Y. (2014). Black hole hair in generalized scalar-tensor gravity: An explicit example. *Phys. Rev. D*, 90, 124063. arXiv:1408.1698. [Companion to PRL 112, 251102 (2014); contains the exact $\mathcal{O}(\alpha^2)$ perturbative metric corrections used in Section 4. Their nonlinear solutions develop a finite-area singularity at $r \approx 0.27\,r_h$ — see Section 4.2.]

- Kanti, P., Mavromatos, N. E., Rizos, J., Tamvakis, K. & Winstanley, E. (1996). Dilatonic black holes in higher curvature string gravity. *Phys. Rev. D*, 54, 5049. arXiv:hep-th/9511071. [Establishes scalar hair and regular horizons in sGB; "absence of naked singularities" permits singularities hidden inside the horizon. Follow-up (Phys. Rev. D 57, 6255, 1998) confirms "other researchers... demonstrated numerically the existence of curvature singularities" behind the horizon.]

- Kleihaus, B., Kunz, J. & Radu, E. (2011). Rotating black holes in dilatonic Einstein-Gauss-Bonnet theory. *Phys. Rev. Lett.*, 106, 151104. arXiv:1101.2868. [Rotating sGB black holes; domain of existence bounded by singular extremal solutions.]

- Thaalba, F., Franchini, N., Bezares, M. & Sotiriou, T. P. (2024). Dynamics of spherically symmetric black holes in scalar-Gauss-Bonnet gravity with a Ricci coupling. *Phys. Rev. D*, 111, 064054. arXiv:2409.11398. [Confirms finite-area singularity in sGB; explores connection between singularity formation and loss of hyperbolicity; Ricci coupling can mitigate hyperbolicity loss.]

- Torii, T., Maeda, K. & Tamaoki, T. (1999). Rotating non-singular black holes in dilatonic Gauss-Bonnet gravity. *Phys. Rev. D*, 59, 064012.

- Delgado, J. F. M., Herdeiro, C. A. R., & Radu, E. (2020). Spin-induced scalarization and spontaneous scalarization of Kerr black holes in scalar-Gauss–Bonnet gravity. *Phys. Rev. D*, 102, 044041. arXiv:2007.12030.

- Delgado, J. F. M., Herdeiro, C. A. R. & Radu, E. (2020). Spinning black holes in shift-symmetric Horndeski theory. *JHEP*, 04, 180. arXiv:2002.05012. [Slowly-rotating shift-symmetric sGB solution; $\mathcal{O}(\beta^2)$ correction to $W(r)$ and horizon angular velocity; nonperturbative numerical solutions for arbitrary spin. Used in Section 8.5.]

- Bardeen, J. M. (1973). Rapidly rotating stars, disks, and black holes. In *Black Holes (Les Astres Occlus)*, eds. DeWitt & DeWitt, Gordon & Breach, pp. 241–289.

- Leaver, E. W. (1985). An analytical representation for the quasi-normal modes of Kerr black holes. *Proc. Roy. Soc. A*, 402, 285–298.

- Witek, H., et al. (2019). Scalar-Gauss–Bonnet gravity in the strong-field regime. *Phys. Rev. D*, 99, 064035. arXiv:1810.05177.

- Blazquez-Calzadilla, J., et al. (2020). Scalar-Gauss–Bonnet perturbations. *Phys. Rev. D*, 102, 024086. arXiv:2003.02862.

- Chen, Y., et al. (2024). Quasinormal modes of rotating black holes in shift-symmetric Einstein-scalar-Gauss–Bonnet theory. arXiv:2412.09377. [Nonperturbative rotating sGB QNM spectra; validation target for the TEP-sGB QNM pipeline.]

- Aresté Saló, L., Doneva, D. D., Clough, K., Figueras, P. & Yazadjiev, S. S. (2025). Challenges in the nonlinear evolution of unequal mass binaries in scalar-Gauss–Bonnet gravity. *Phys. Rev. D*, in press. arXiv:2507.13046. [Numerical relativity simulations of binary black hole mergers in sGB; ringdown excitation extraction.]

- Kobayashi, T., Yamaguchi, M. & Yokoyama, J. (2011). Galilean creation of the inflationary universe. *Prog. Theor. Phys.*, 126, 511. arXiv:1105.5723. [Principal symbol of scalar–tensor theories; tensor speed derivation used in Section 7.3.]

- Nishizawa, A. & Arai, K. (2019). Generalized framework for testing gravity with gravitational-wave propagation. *Phys. Rev. D*, 99, 104038. arXiv:1901.08249. [c_T formula for modified gravity theories including sGB; GW170817 constraints.]

- LIGO/Virgo & Fermi-GBM Collaborations (2017). Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A. *ApJL*, 848, L13.

- Freire, P. C. C., et al. (2012). The relativistic pulsar-white dwarf binary PSR J0348+0432. *MNRAS*, 423, 3328.

- Antoniadis, J., et al. (2013). A massive pulsar in a compact relativistic binary. *Science*, 340, 6131.

# Acknowledgements

The author acknowledges the broader TEP research programme, founded in Paper 0 (Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed), and the empirical and computational framework established by TEP-C0 (Paper 26, Athens) and TEP-HC (Paper 18, Cambridge) within which this black-hole interior analysis is developed. Section 3.4 relates the SEC-violation mechanism found here to the independent construction of TEP-TH (Paper 27, Thika), which establishes an analogous singularity-exclusion result for the cosmological branch. The eternal-universe thermodynamic closure, which replaces the hot Big Bang with a temporal-horizon proper-time reaction flow, is developed in TEP-BBN (Paper 29, Dubai).

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# Temporal Equivalence Principle: Dynamical Proper Time and the Illusion of Primordial Deuterium
**Matthew Lukin Smawfield**
Version: v0.6 (Dubai)
First published: 12 August 2026 - Last updated: 17 October 2026
DOI: 10.5281/zenodo.21841147

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## Abstract

The spectroscopic identification of primordial deuterium-a foundational empirical pillar of Big Bang cosmology-is not mathematically unique. Within standard Voigt-profile fitting architectures, the absorption features conventionally attributed to deuterium across the three best-studied quasar sightlines are statistically better described by unrestricted ordinary hydrogen. Using Monte-Carlo-calibrated model comparison across Q1009+2956 (Keck/HIRES), PKS 1937$-$101 (ESPRESSO), and J1332$+$0052 (UVES+HIRES), the unrestricted hydrogen model is significantly preferred over the true-deuterium model under both the canonical-parent and adversarial parent-reassignment statistics (largest calibrated $p=0.001$ among the primary Gate 2/3 statistics, surviving $\alpha/6$). The key physical insight is that *isotope identifiability is controlled by both Doppler regime and D column density*: at the unsaturated LLS-class columns typical of D/H measurements, the intrinsic H/D profile-shape difference is sub-noise on a per-pixel basis, so the deuterium identification depends predominantly on the velocity offset-which the free-H model can reproduce without invoking deuterium at all. A multi-pronged evidence analysis (Gate 8) calibrates three discriminators-$b$-value ratio, velocity offset deviation, and column density difference-against true-D injection Monte Carlo ($N = 1000$ per sightline): the calibrated joint results are Q1009: $p=0.001$; PKS1937: $p=0.001$; J1332: $p=0.023$. The two unsaturated systems give velocity-diagnostic probabilities Q1009: $p=0.001$; PKS1937: $p=0.001$. All three features are blueward of the parent, consistent with the conformal sign theorem (though the D-selected sample is not an independent sign test). A velocity-selection theorem explains the universality of the $-81.6$ km/s offset as a selection effect of the D-search protocol, and motivates an independently selected population test for non-isotope displaced H I components. An archival feasibility search establishes the principal saturation and Ly$\alpha$-forest selection limits and identifies one candidate component in Q1009$+$2956, motivating a larger targeted survey. A pilot search of the three benchmark sightlines identifies one candidate feature already present in the literature scaffold near the predicted orphan phenotype in Q1009$+$2956: a narrow ($b = 10.3$ km/s, $\log N = 11.5$) H I component at $\Delta v = -155.5$ km/s from the parent ($73.9$ km/s from the isotope shift), identified as a candidate non-isotope component in the HIRES spectrum.

Conformal core–edge geometry predicts a blueward sign. An absorber-regime amplitude ansatz $S_A^{\rm abs}$ is introduced for diffuse absorbers; Cassini independently constrains the source-charge response $S_\Sigma^{(\odot)}$. Both are projections of the common microscopic scalar solution (Paper 0 &sect;7): the derived kinetic operator $\mathcal{S}_\Sigma(\mathcal{E})=[k\sqrt{\xi}+2\xi]^{-1}$, $\xi=|X|/\Lambda_X^4$; the quoted environment table uses the acceleration proxy $\xi_{\rm tab}=(g/g_t)^2$ inherited from the foundational benchmark, rather than identifying that proxy with the solved-field kinetic invariant. Under this tabulation convention the operator returns $\mathcal{S}_\Sigma\simeq1.6\times10^{-15}$ at 1 AU and $\mathcal{S}_\Sigma\sim1.5$-$21$ at the Lyman-limit absorber ambient; evaluated instead on the solved drift-dominated field at the absorber epochs the same operator returns $\mathcal{S}_\Sigma\sim0.13$-$0.18$ (step_21 drift-branch), so the environmental discrimination is supplied by one operator rather than assumed, under either field reconstruction. The conformal-only amplitude is $\sim10^{-2}\,\mathrm{km\,s^{-1}}$; the temporal-disformal channel activated by the cosmological scalar time derivative was investigated as a candidate amplitude mechanism. It can reach the observed scale per sightline in the Einstein-frame parametrisation explored here. The power-law $B(\phi)$ forms tested so far do not achieve predictive closure across the three sightlines: $n = 2$ exceeds the Cassini bound only under a conservatively adopted solar source-charge profile (under the derived operator the disformal spatial term carries an $\mathcal{S}_\Sigma^2\sim10^{-48}$ suppression and the bound is non-binding), while $n = 5$ loses Lorentzian signature above $z \approx 3.4$. The full covariant calculation (Gate 10) with $\omega=-u^\mu k_\mu$ confirms that the near-criticality and cross-redshift signature loss persist under the complete matter-frame treatment when the drift is imported from the $\Lambda$CDM expansion history; the binding constraint is then the $A^2/\dot{\bar\phi}^{\,2}$ variation across redshift, which prevents any single increasing power-law $B(\phi)$ from achieving simultaneous near-criticality at all three sightlines. That constraint is shown to be a property of the imported rate rather than of the disformal channel: the joint sign-and-closure criterion reduces to $d\ln H_T^2/d\ln(1+z) < 0$ — a descent-type ambient drift — and the unsourced free-roll on the reconstructed corpus potential, $\dot{\bar\phi}^{\,2} = \bar\Pi_\infty^2 + \bar\rho\,e^{-\bar\varphi}$, satisfies it. Under the pure-descent limit the descent-tracking coupling $B(\varphi) = B_0\,\varphi^4/(\varphi_c^4+\varphi^4)\,e^{-\varphi}$ has inverse-fit roots agreeing to a factor $1.00004$ and satisfies the Cassini and GW170817 bounds; forward evaluation at that common $B_0$ does not reproduce the three amplitudes, and the branch is independently excluded by ambient-lapse self-consistency: the boundary factor $u = B\dot{\bar\phi}^{\,2}/A^2$ is constant along the descent at $\approx 1 - 10^{-5}$, so the matter-metric redshift map $1+z = \tilde N(0)/\tilde N(\bar\varphi)$ would require a negative ambient field to reproduce the observed absorber population. The unscreened temporal-disformal amplitude channel is thereby bounded sub-critical at these epochs; a contribution of the observed scale can arise only where $B(\phi)$ is environmentally screened. The $-81.6$ km/s clustering is explained by the velocity-selection theorem (Gate 6-I) as a selection effect of the D-search protocol, not as a universal TEP-produced shift. Within the adopted VMO-dominated reaction-flow model, the helium-4 abundance is reproduced as an equilibrium, while a conditional opacity theorem provides an observable high-redshift boundary without a physical plasma wall. Cosmological redshift is reformulated as temporal transport over a static spatial background. The broader TEP framework-CMB acoustic peak preservation, temporal-horizon regularity, cosmological distances without expansion, and JWST high-redshift phenomenology-is developed in companion papers.

Keywords: temporal equivalence principle, deuterium abundance, isotopic line identification, temporal shear, absorption-line spectroscopy, Lyman-limit systems, Big Bang nucleosynthesis, cosmology, TEP, Proper-Time Transport

## 1. Introduction

The prevailing cosmological paradigm interprets the Hubble redshift as the kinematic expansion of a spatial volume, directly linking high redshifts to a dense, ultra-hot spatial singularity—the Big Bang. Within this framework, the measurement of light element abundances, particularly deuterium (D/H) in high-redshift Lyman-limit systems such as Q1009+2956, serves as a crucial anchor for Big Bang Nucleosynthesis (BBN) <sup>[22], [45]</sup>. However, this standard inference assumes that cosmological redshift is intrinsically geometric and that the spectroscopic structure of deuterium is uniquely distinguishable from contaminating intergalactic hydrogen.

Tensions in precision cosmology—such as the Hubble tension <sup>[17], [18]</sup> and the $S_8$ growth tension—have motivated numerous theoretical extensions to the $\Lambda$CDM framework <sup>[16]</sup>. Recent literature has extensively explored modified gravity (e.g., $f(R)$ or scalar-tensor theories <sup>[30], [42]</sup>), non-standard recombination histories, and Early Dark Energy (EDE) to resolve these anomalies. While these approaches introduce new dynamical parameters to accommodate the standard hot-thermal history, they generally preserve the foundational assumption that cosmological redshift equates to geometric spatial expansion.

### The Temporal Equivalence Principle Framework

The standard Friedmann-Lemaître-Robertson-Walker (FLRW) metric <sup>[11], [12], [13]</sup> explicitly couples cosmological evolution to a dynamic spatial volume. Extrapolating this geometric expansion backward inevitably terminates in a spatial singularity ($a \to 0$)—a regime where polynomial curvature invariants diverge and the foundational equations of General Relativity break down <sup>[5], [6], [7]</sup>. Alternative non-singular cosmologies, including bouncing models <sup>[8], [9]</sup>, cyclic scenarios <sup>[10]</sup>, and conformal approaches <sup>[37], [38], [39]</sup>, have been explored but typically require additional dynamical ingredients. The Temporal Equivalence Principle (TEP) avoids this singular outcome by formally decoupling spatial kinematics from temporal dynamics. By anchoring the universe to a static physical matter frame ($a_{\rm m} = 1$) governed by a dynamical proper-time field $A(\phi)$ <sup>[40], [41]</sup>, the TEP geometry removes the spatial singularity. Cosmological redshift is not the stretching of space, but rather the manifestation of a temporal gradient between the emitting and observing frames. This approach shares conceptual ground with static-universe and conformal frameworks <sup>[31], [32], [33], [34]</sup>, but distinguishes itself through the dynamical proper-time field and its coupling to matter. The apparent "Big Bang" is replaced by an asymptotic temporal horizon ($\mathscr{T}^-$) in the far past, characterized by a vanishing relative clock rate ($A_{\rm clock} \to 0$). Because the underlying spatial manifold remains static, all polynomial curvature invariants remain finite at this boundary. The hot, dense spatial origin required by standard cosmology is therefore not needed; the temporal horizon replaces the Big Bang as the observational boundary, establishing a regular geometric foundation for early-universe observables without requiring an explosive spatial origin.

The standard hot-BBN inference is challenged on two fronts within the TEP framework <sup>[1], [2]</sup>. First, an algorithmically controlled analysis of Q1009+2956 shows that the canonical deuterium interpretation is not uniquely identifiable against the ordinary-H alternative at the available resolution. Second, the temporal-transport framework is developed to show how the associated cosmological observables can be represented without a primordial spatial singularity.

## 2. Spectroscopic Analysis

The spectroscopic uniqueness of the deuterium identification in high-redshift quasar absorption systems is examined under Monte-Carlo-calibrated model comparison. The primary finding is that the statistical preference for an unrestricted hydrogen model over a true-D model is *robust across sightlines, under both the canonical-parent and parent-reassignment statistics*. The *parent-reassignment statistic* ($T_{\rm parent}$, $p_{\rm parent}$)—which gives D maximum freedom to select the most favorable parent—is the more conservative and informative number, and is reported as the primary result throughout. The canonical-parent (standard) statistic is reported as a secondary comparison. A second key finding is that isotope identifiability is *strictly regime-dependent*: it must be assessed per absorber rather than assumed, as the distinguishability of H and D Voigt profiles depends critically on the thermal-to-turbulent Doppler ratio.

The analysis is performed on the high-resolution Keck/HIRES spectrum of the benchmark Q1009+2956 absorption system, from Zavarygin et al. 2018 <sup>[44]</sup> (22 HIRES exposures co-added into four setups using the wspectrum software). The published kinematic architecture and component parameters from Zavarygin et al. 2018 <sup>[44]</sup> are used as the structural prior for the Voigt component family, fitted with the VPFIT framework <sup>[50]</sup>. The spectral data comprise four coadds at two instrumental settings: C1 ($R\approx49\,000$, $\sigma_{\rm inst}=2.6$ km/s) and C5 ($R\approx37\,000$, $\sigma_{\rm inst}=3.4$ km/s), with S/N $\approx 47$–$78$ per pixel near Ly$\alpha$. The system was originally measured for primordial D/H by Burles & Tytler <sup>[46]</sup> and subsequently by Zavarygin et al. <sup>[44]</sup>, with independent measurements from other quasar sightlines contributing to the primordial D/H estimate <sup>[47], [48]</sup>. Two additional benchmark sightlines—PKS 1937$-$101 (ESPRESSO, $z=3.572$) and J1332$+$0052 (UVES+HIRES, $z=3.421$)—are analysed with the identical framework in Section 2.4.

### 2.1 Isotope Identifiability Limits in Q1009+2956

Using the physical atomic registries for H I and D I (NIST ASD), synthetic deuterium was embedded at the actual Q1009 D column ($N_{\rm D} = 5.7\times10^{12}\,\mathrm{cm}^{-2}$, derived from D/H $= 2.48\times10^{-5}$ and $\log N_{\rm HI} = 17.362$) and recovered using unrestricted hydrogen models. Zavarygin et al. <sup>[44]</sup> themselves noted that the putative D feature is likely contaminated by weak interloping Ly$\alpha$ absorption from a low-column-density H I cloud, reducing the D/H precision. The present analysis extends this insight by quantifying the D/H degeneracy through explicit adversarial model comparison with Monte-Carlo-calibrated parent reassignment, comparing multiple sightlines, and then asking whether the ordinary-H alternative admits a TEP interpretation. The isotope identifiability depends on both the Doppler parameter regime and the D column density. For thermally-dominated gas, the H and D Doppler parameters differ by the mass ratio: $b_{\rm H} = b_{\rm D}\sqrt{m_{\rm D}/m_{\rm H}} \approx 1.41\,b_{\rm D}$. With this physical mass scaling, the maximum convolved flux discrepancy between H and D profiles across all Lyman transitions is $0.52\sigma$ at Keck/HIRES C1 resolution ($R\approx49\,000$, $\sigma_{\rm inst} = 2.6$ km/s), indicating that H and D are marginally distinguishable for thermally-dominated gas at this column. The $\sigma$ statistic is computed as the maximum pixel-level flux residual divided by a conservative constant noise floor ($\sigma_{\rm floor} = 0.05$), evaluated across the five strongest Lyman transitions (Ly$\alpha$ through Ly$\epsilon$). For turbulence-dominated gas ($b_{\rm turb} \gg b_{\rm therm}$), the mass-dependent thermal contribution is negligible and $b_{\rm H} \approx b_{\rm D}$; in this limit the maximum discrepancy falls to $0.0002\sigma$, making the isotopes operationally unidentifiable. For the actual Q1009 fitted parameters ($T_K = 2000$ K, $b_{\rm turb} = 11.60$ km/s, giving $b_{\rm turb}/b_{\rm therm} \approx 2.02$), the mixed-regime discrepancy is $0.10\sigma$ — far below conventional detectability. The identifiability is controlled by the D column: at the unsaturated LLS-class columns typical of D/H measurements ($\log N_{\rm D} \lesssim 13$), the single-pixel discrepancy is sub-threshold in every Doppler regime; only saturated sub-DLA columns ($\log N_{\rm D} \gtrsim 14$) approach distinguishability. Isotope identifiability must therefore be assessed per absorber as a function of both column and Doppler regime.

### 2.2 Likelihood Model Comparison and Significance Testing

A complete structural analysis of the Q1009+2956 spectrum is performed under a controlled constrained-versus-unrestricted model hierarchy embedded in the joint D+H model. By anchoring the fits to a SHA-256-validated data manifest, the likelihood surfaces of the standard D-interpretation ($M_D$), an unrestricted hydrogen interpretation ($M_{H,\rm free}$), and the joint space ($M_{D+H}$) are mapped. The 142 shared parameters (43 H I Voigt component parameters: centres, $b$-values, and column densities, plus continuum basis coefficients for each data block) are held at the published literature architecture <sup>[44]</sup>, identically for every hypothesis. This fixed scaffold is part of the definition of the test statistic rather than an uncontrolled nuisance choice: the significance of $T$ is not read off an asymptotic $\chi^2$ distribution but calibrated by parametric bootstrap in Section 2.3, where true-D spectra are regenerated on the same scaffold and the identical statistic is recomputed. Because the scaffold enters the observed and simulated statistics identically, the resulting $p$-value is calibrated conditional on the adopted published architecture. Q1009 scaffold freedom is tested explicitly in §5.5.

The candidate component parameter structure is made fully explicit. Each model differs only in the candidate component; the shared scaffold is identical:

| Model | Candidate params | $v$ | $\log N$ | $T_K$ | $b_{\rm turb}$ | $k_{\rm cand}$ |
| --- | --- | --- | --- | --- | --- | --- |
| $M_D$ (M_Dfree) | D tied to parent | tied to parent $+ v_{\rm iso}$ | free | free | free | 3 |
| $M_{H,\rm free}$ (M_H) | unrestricted H | free | free | free | free | 4 |
| $M_{D+H}$ | D + unrestricted H | D tied, H free | D free, H free | shared | shared | 5 |
| $M_{\rm kin}$ | parent-width-constrained H control | free | free | — | — | 2 |

The parameter count difference is $\Delta k = k(M_{H,\rm free}) - k(M_D) = 4 - 3 = 1$: the unrestricted-H model gains one free parameter (the velocity $v_H$, which in $M_D$ is tied to the parent velocity via $v_D = v_{\rm parent} + v_{\rm iso}$). The column density, temperature, and turbulent broadening are free in both models. The $\Delta k = 1$ therefore reflects exactly the velocity-tying difference between the isotope-D and free-H hypotheses — the single physical degree of freedom that distinguishes them.

| Model | Candidate interpretation | $k$ | $\ln L_{\max}$ | $\Delta\text{AIC}$ | Nested? |
| --- | --- | --- | --- | --- | --- |
| $M_D$ | D tied to parent H | 3 | $-15446.77$ | 0 | — |
| $M_{H,\rm free}$ | unrestricted H | 4 | $-15366.26$ | $-159.03$ | observationally embeds $M_D$ at Q1009 precision |
| $M_{D+H}$ | D + unrestricted H | 5 | $-15366.26$ | $-153.03$ | contains $M_{H,\rm free}$ if $N_{\rm D}\to 0$ (verified to $2.8\times10^{-10}$ in $\Delta\ln L$) |

#### Statistical Significance

The unrestricted hydrogen model—which supplies the displaced-H phenomenology to which the candidate TEP edge-gas interpretation is later applied—provided a superior description of the data, yielding a likelihood improvement:

\begin{equation} \Delta \ln L = 80.51, \qquad T = 161.03, \qquad \Delta k = 1, \qquad \Delta\mathrm{AIC} = -159.03. \end{equation}

The unrestricted-H model adds one free parameter (the H velocity) relative to the D-tied model ($\Delta k = 1$), yet the likelihood improvement of $\Delta\ln L = 80.51$ exceeds the AIC penalty of 2, yielding $\Delta\mathrm{AIC} = -159.03$ and $\Delta\mathrm{BIC} = -151.97$ ($n = 8{,}610$ data points, $\ln n \approx 9.06$). Both information criteria favor $M_{H,\rm free}$. Because the shared scaffold is identical under both hypotheses, the comparison isolates the single additional kinematic parameter of the unrestricted-H model and does not depend on the number of shared components. The information criteria are reported for orientation only; the significance quoted below rests on the Monte Carlo calibration, not on asymptotic likelihood-ratio theory, whose regularity conditions are not satisfied here because $M_D$ sits on the boundary $N_{\rm D}\to 0$ of $M_{D+H}$. Monte Carlo $p$-values use the Laplace-corrected estimator $\hat{p} = (k+1)/(N+1)$, where $k$ is the exceedance count and $N$ is the number of realizations.

To establish rigorous significance, 1000 physical Monte Carlo simulations generating exact, noisy true-D flux were run, followed by dense free-H refitting. Two calibration statistics are reported. The *standard statistic* locks D to the canonical parent (matching the observed fit). The *parent-reassignment statistic* allows D to select the most favorable parent from the H I components within ±1000 km/s of the absorber systemic velocity. The standard statistic represents the prespecified comparison; the parent-reassignment statistic represents the most adversarial test. Both are reported to bracket the significance honestly.

### 2.3 Leave-One-Out and Parent Reassignment Robustness

The component misattribution vulnerability was exhaustively tested by tying the candidate D velocity to every eligible H component within the conservative $\pm 1000\,{\rm km\,s^{-1}}$ parent window (14 parent candidates). The maximum alternative-parent test statistic is defined as:

\begin{equation} T_{\rm parent} = 2 \left[ \ln L(M_{H,\rm free}) - \max_j \ln L(M_D\mid j) \right]. \end{equation}

Against the most advantageous alternative parent assignment, the free-H interpretation is preferred for Q1009+2956 with $T_{\rm parent} = 161.03$, equal to $T = 161.03$ for the canonical parent. The discrimination is therefore not specific to the published parent assignment, and the parent-reassignment statistic is the conservative quantity to calibrate. When calibrated inside the true-D Monte Carlo loop—utilizing bounded L-BFGS-B optimization with a multi-start grid spanning the full velocity range $[-160, +50]\text{ km/s}$, $\log N_{\rm D} \in [0, 20]$, $T_K \in [100, 4\times10^4]$ K, and $b_{\rm turb} \in [1, 50]$ km/s, with the best likelihood retained across all starts to prevent local minima from artificially widening the null distribution—the empirical $p_{\rm parent}$ is reported below.

**Parent-reassignment window.** Two parent-reassignment windows are reported. The *conservative* window ($\pm 1000\text{ km/s}$) gives the deuterium hypothesis maximum freedom to find a favorable parent, establishing a lower bound for the $T_{\rm parent}$ statistic. Of the 43 H I components in the Q1009 model, 14 fall within this window, of which 4 satisfy the column-density eligibility cut ($\log N_{\rm HI} \ge \log N_{\rm max} - 2$ dex); none are DLA-class ($\log N_{\rm HI} > 20.3$), and components from unrelated absorber systems at $v \sim -47\,000$ to $-74\,000\text{ km/s}$ are correctly excluded. The *physical* window ($\pm 3b_{\rm max}$, where $b_{\rm max} = 56.7$ km/s is the largest Doppler parameter of the main complex) restricts eligible parents to those within the kinematic structure of the absorber: 9 H I components fall within the velocity window, of which 4 satisfy the column-density eligibility cut ($\log N_{\rm HI} \ge \log N_{\rm max} - 2$ dex). The conservative $\pm 1000\text{ km/s}$ window is the cross-sightline headline statistic; the $\pm 3b_{\rm max}$ physical window has also been calibrated for all three sightlines (Section 5.5), with results consistent with the conservative window. Under the physical window, the conclusions are essentially unchanged from the conservative window (see robustness check C1 in Section 5.5), confirming that the $\pm 1000\text{ km/s}$ conservative bound was not driving the results.

A critical interpretation issue arises in the parent-reassignment test. For Q1009+2956, $T_{\rm parent} = T_{\rm obs} = 161.03$, meaning that allowing reassignment does not improve the D fit over the canonical parent—the discrimination is not specific to the published parent assignment. For PKS 1937$-$101, $T_{\rm parent} = 59.00 < T_{\rm obs} = 161.31$, so the best alternative parent substantially reduces the statistic, but the result remains significant. For J1332$+$0052, $T_{\rm parent} = 135.76 < T_{\rm obs} = 136.97$, so the best alternative parent slightly reduces the statistic. In all three cases, the free-H model is significantly preferred even under the most favorable parent assignment for D. The parent-reassignment statistic is therefore the conservative quantity to calibrate, and the $p_{\rm parent}$ values reported below should be interpreted as the significance under the most favorable parent assignment for D.

**Parent-identity diagnostic (Gate 7).** Under the isotope-D interpretation, the best alternative parent $j^*$ (the H I component maximizing $\ln L(M_D \mid j)$) should be the main narrow thermal component — D is tied to the gas it formed beside. Under TEP, $j^*$ might be a diffuse, low-column edge component — the edge gas anchors to its own local structure. For all three sightlines, the best non-canonical parent within the eligible window ($\log N_{\rm HI} \ge \log N_{\rm max} - 2$ dex, $|v| \le 1000$ km/s) is a core component: Q1009+2956 parent 1 ($v = -43.7$ km/s, $\log N_{\rm HI} = 16.80$, $b = 17.8$ km/s), PKS 1937$-$101 parent 20 ($v = +17.9$ km/s, $\log N_{\rm HI} = 17.38$, $b = 17.1$ km/s), and J1332$+$0052 parent 2 ($v = +8.7$ km/s, $\log N_{\rm HI} = 18.39$, $b = 15.8$ km/s). This is consistent with the isotope-D interpretation and does not support the TEP edge-parent prediction. The parent-identity test is presently treated as a descriptive diagnostic rather than a TEP-specific discriminator. Under TEP the feature is ordinary edge H, not D tied to any parent; the identity of the parent that minimizes a mis-specified D fit may be a fitting artefact rather than a physical TEP observable. A derivation showing why a displaced edge-H feature mathematically causes the false-D likelihood to maximize on diffuse edge components is required before Gate 7 can be elevated to a TEP-specific prediction.

**Offset-exactness test (Gate 4E).** Under the isotope-D interpretation, the free-H component should sit at the exact isotope-shifted velocity $v_{\rm iso} = -81.6$ km/s relative to its parent. Under TEP, the edge-gas feature need not sit at the exact isotope shift — the observed offset depends on the local shear amplitude, which varies with core-edge geometry. The fitted $M_{H,\rm free}$ velocities deviate from the isotope-shifted position by:

| Sightline | Parent $v$ (km/s) | $v_{\rm iso}$ (km/s) | Expected D $v$ (km/s) | Fitted $v_H$ (km/s) | Deviation (km/s) |
| --- | --- | --- | --- | --- | --- |
| Q1009+2956 | $-52.4$ | $-81.6$ | $-134.0$ | $-131.3$ | $+2.7$ |
| PKS 1937$-$101 | $+8.7$ | $-81.6$ | $-72.9$ | $-67.3$ | $+5.7$ |
| J1332$+$0052 | $0.0$ | $-81.6$ | $-81.6$ | $-118.2$ | $-36.6$ |

The signed residuals in this table are measured relative to the isotope position, not relative to the parent. Thus the $+2.7$ and $+5.7$ km/s residuals mean that Q1009+2956 and PKS 1937$-$101 are slightly less blueward than exact D; they do not represent redward core-edge shifts. All three fitted components remain blueward of their respective parents.

Q1009+2956 sits $2.7$ km/s from the isotope shift — lying approximately one instrumental Gaussian width ($\sigma_{\rm inst}\simeq2.6\,{\rm km\,s^{-1}}$) from the isotope position, but highly inconsistent when calibrated against the true-D simulation distribution ($\pm 0.2$ km/s, $p ≤ 0.001$, Section 5.6). PKS 1937$-$101 deviates by $5.7$ km/s — also highly significant after calibration ($p ≤ 0.001$). J1332$+$0052 deviates by $-36.6$ km/s — not individually significant after calibration ($p = 0.098$). The cross-sightline pattern — increasing offset deviation with increasing D column — is quantitatively testable: if the feature is isotope-D, all three should sit at $v_{\rm iso}$ within spectral resolution; under TEP, the offset varies with core-edge geometry. The J1332 feature shows the largest departure from the isotope-shifted position, and it is also the sightline with the highest D column (saturated, $\log N_{\rm D} \approx 14.65$), where the edge-gas interpretation is most physically motivated.

Finally, performing a transition-level leave-one-out (LOO) test reveals where the empirical discrimination resides. The full LOO vector for Q1009+2956 (10 entries: 4 coadd deletions + 6 transition deletions):

| Excluded | Type | $T_{\rm LOO}$ | $\Delta T$ vs $T_{\rm full}$ |
| --- | --- | --- | --- |
| C1x1 coadd | coadd | 125.05 | $-35.98$ |
| C1x2 coadd | coadd | 141.03 | $-20.00$ |
| C5x1 coadd | coadd | 77.31 | $-83.71$ |
| C5x2 coadd | coadd | 139.87 | $-21.16$ |
| Ly$\alpha$ | transition | 34.15 | $-126.88$ |
| Ly$\beta$ | transition | 148.64 | $-12.39$ |
| Ly$\gamma$ | transition | 156.16 | $-4.86$ |
| Ly6 | transition | 160.70 | $-0.33$ |
| Ly13 | transition | 161.05 | $+0.02$ |
| Ly14 | transition | 161.03 | $+0.00$ |

The full LOO vector reveals that the discrimination is strongly concentrated in Ly$\alpha$: removing Ly$\alpha$ *decreases* $T$ from 161.03 to 34.15, indicating that Ly$\alpha$ provides the bulk of the isotope discrimination. This is physically expected: the isotope velocity shift ($-81.6$ km/s) is the same across all Lyman transitions, but Ly$\alpha$ has the highest oscillator strength and signal-to-noise ratio, yielding the deepest absorption profile and the largest number of informative pixels over which the mass-scaled Doppler difference between H and D accumulates. Ly$\beta$ is the second most important transition ($T_{-\mathrm{Ly}\beta} = 148.64$, $\Delta T = -12.39$), followed by Ly$\gamma$ ($\Delta T = -4.86$). The high-order transitions (Ly6, Ly13, Ly14) contribute negligibly ($|\Delta T| < 0.5$). The coadd deletions show that the C5x1 setting contributes the most discrimination ($\Delta T = -83.71$), followed by C1x1 ($\Delta T = -35.98$). All coadd deletions decrease $T$, confirming that the discrimination is distributed across all four coadd settings rather than being driven by a single dataset. The result demonstrates that a benchmark high-redshift D/H system is not spectroscopically self-authenticating once the displaced-H model class is admitted.

**Velocity-selection theorem (Gate 6).** The strongest objection to the TEP interpretation is that the observed velocity offset matches the exact reduced-mass isotope shift ($-81.6$ km/s) in all three sightlines. However, this coincidence is a property of the *selection function*, not of the gas: the D-identification protocol is a filter $|\Delta v - v_{\rm iso}| < \epsilon$ — VPFIT initializes D components at the isotope-shifted velocity, and analysts search for D at that specific offset. Any shear feature at $-40$ km/s or $-120$ km/s relative to the core is fitted as an ordinary H I component and never flagged as D. The clustering near $-81.6\,{\rm km\,s^{-1}}$ is therefore strongly conditioned by construction of the D-selected sample and cannot be treated as an independent chance-probability test of isotope identity. The falsifiable prediction is that edge-shear features at other offsets exist and are being absorbed into H I component structure: there should be a population of anomalously narrow "orphan" components in high-S/N LLS spectra whose velocity distribution is broader than the isotope shift and whose columns track the edge-gas prediction.

**Illustrative Population Forecast (Gate 6-II).** An illustrative selection model, calibrated to the three benchmark fitted offsets, gives $\sigma_{\rm shear} \approx 19$ km/s and an orphan fraction of $\sim 79\%$. Because the benchmark sample is D-selected and the J1332 fitted velocity is width-prior-sensitive, these numbers are exploratory rather than a frozen TEP population prediction. The observed cross-sightline deviations ($+2.7$, $+5.7$, $-36.6$ km/s for Q1009, PKS 1937, J1332 respectively) give: the D-search selection function $|\Delta v - v_{\rm iso}| < 5$ km/s captures only $\sim 21\%$ of edge-shear features; the remaining $\sim 79\%$ are orphans absorbed into the H I component structure. This is not the expected detected fraction: the archival feasibility analysis in §5.7 shows that saturation and forest contamination reduce an 80-sightline parent sample to roughly 10 usable systems, corresponding to approximately two detectable excess candidates. Because the shear width is inferred from only three D-selected systems, this fraction is provisional; the robust TEP prediction is an excess of non-isotope displaced components with the joint edge-gas phenotype relative to a matched conventional H I background. In a sample of $\sim 80$ high-S/N LLS sightlines with full Lyman coverage (SQUAD DR1, UVES SLS, ESPRESSO D/H survey), $\sim 24$ shear features are expected, of which $\sim 5$ are identified as D and $\sim 19$ are orphans. The predicted orphan surface density is $\sim 2.4$ per 10 sightlines at $|\Delta v - v_{\rm iso}| > 5$ km/s, falling to $\sim 1.8$ per 10 sightlines at $> 10$ km/s. Orphan features are predicted to have narrow Doppler parameters ($b < 10$ km/s, cool edge gas), low columns ($\log N_{\rm HI} \sim 12$-$14$), and thermal Doppler regimes — distinguishable from the normal H I population by an excess of narrow, low-column components at non-isotope offsets. Under the isotope-D null, no such excess is expected. This is a targetable observational program using existing archival data.

The embedding pipeline has been generalised to support multiple sightlines via a sightline-configuration system. Two additional benchmark D/H quasar sightlines have now been analysed with the identical model-comparison framework, providing the first cross-sightline application of the TEP spectroscopic methodology. The same spectroscopic framework is applied to the two additional benchmark sightlines in §2.4; their cross-sightline amplitude comparison is presented in §3.3.

### 2.4 Cross-Sightline Validation

The identical Gate 2/3 pipeline was applied to PKS 1937$-$101 (Cooke et al. 2024 <sup>[66]</sup>; ESPRESSO, $R=70\,000$, $z_{\rm abs}=3.572$) and J1332$+$0052 (Kislitsyn et al. 2024 <sup>[67]</sup>; joint UVES+HIRES, $R\approx50\,800$ and $49\,200$, $z_{\rm abs}=3.421$). Both sightlines were analysed with all available Lyman transitions covered by their spectral data (13 for PKS 1937$-$101, 12 for J1332$+$0052 which lacks Ly$\alpha$ coverage), the same noise-model calibration, and the same 1000-realization Monte Carlo protocol.

| Sightline | $z_{\rm abs}$ | Instrument | $T_{\rm obs}$ | $T_{\rm parent}$ | $p_{\rm std}$ | $p_{\rm parent}$ | Doppler regime | Interpretive status (conservative) |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Q1009$+$2956 | 2.5042 | Keck/HIRES | 161.03 | 161.03 | 0.001 (0/1000) | 0.001 (0/1000) | Mixed ($b_{\rm turb}/b_{\rm therm} \approx 2.02$, $0.10\sigma$) | Significant; free-H preferred even under most favorable parent |
| PKS 1937$-$101 | 3.572 | ESPRESSO | 161.31 | 59.00 | 0.001 (0/1000) | 0.001 (0/1000) | Thermal | Significant; free-H preferred even under most favorable parent |
| J1332$+$0052 | 3.421 | UVES+HIRES | 136.97 | 135.76 | 0.001 (0/1000) | 0.001 (0/1000) | Turbulent, saturated ($b_{\rm turb} \approx 50$ km/s at bound, no Ly$\alpha$ coverage; boundary check in §5.6) | Significant; free-H preferred even under most favorable parent |

$^{\dagger}$ For J1332$+$0052, the canonical-parent standard statistic is reported as the primary $p_{\rm std}$ for consistency with Q1009 and PKS 1937. Both definitions yield significant results for J1332$+$0052; the canonical-parent value is reported in the table for cross-sightline uniformity.

Under the conservative parent-reassignment statistic, the results are: for Q1009+2956, $p_{\rm parent} = 0.001$ (0/1000 exceedances), significant—free-H is preferred even under the most favorable parent assignment. For PKS 1937$-$101, $p_{\rm parent} = 0.001$ (0/1000 exceedances), significant—free-H is preferred even under the most favorable parent assignment. For J1332$+$0052, $p_{\rm parent} = 0.001$ (0/1000 exceedances), significant—free-H is preferred even under the most favorable parent assignment. The standard (canonical-parent) statistic yields $p_{\rm std} = 0.001$ (0/1000) for all three sightlines. The common free-H preference across all three sightlines provides the displaced-H phenomenology to which the TEP interpretation is subsequently applied. A cross-sightline amplitude ranking from the density-weighted Green function $K$ is computed in Section 3.3 (Gate 5).

**Multiplicity correction.** The primary confirmatory family comprises the six Gate 2/3 statistics and is Bonferroni-corrected at $\alpha/6=0.0083$; all six survive. Gate 8 consists of three secondary correlated multivariate diagnostics derived from the same fitted spectra and is reported separately, not added to the primary family as independent confirmation. Correcting the three Gate-8 sightline joint tests among themselves gives $\alpha/3=0.0167$; the corresponding calibrated probabilities are Q1009: $p=0.001$, PKS1937: $p=0.001$, J1332: $p=0.023$. Q1009, PKS1937 pass that threshold. Parent maximization is already inside $T_{\rm parent}$ and its Monte Carlo calibration; it is not a further Bonferroni factor.

Three critical analysis issues were identified and fixed during this work: (1) the continuum basis was made dynamic — 10 parameters for broad Ly$\alpha$ regions (where the H I wing requires flexible continuum separation from the narrow D feature) and 5 parameters for narrow metal-line regions (where 10 parameters would overfit and act as a mathematical sponge, absorbing isotopic discrepancies and artificially depressing $T$); (2) the isotope identifiability calculation was corrected to use mass-scaled Doppler parameters at the actual D column ($N_{\rm D} = 5.7\times10^{12}\,\mathrm{cm}^{-2}$), revealing that H and D are marginally distinguishable at $0.52\sigma$ in the thermal regime and indistinguishable at $0.10\sigma$ in the mixed regime; and (3) the parent-reassignment interpretation was corrected: a non-significant $p_{\rm parent}$ indicates that D with an alternative parent is statistically indistinguishable from free-H, which does not exclude the deuterium interpretation but does not independently support it either.

## 3. Scalar Field Dynamics and Temporal Shear <sup>[53]</sup>

*Part I: Local Absorber Physics.* Sections 2–3 establish the spectroscopic analysis and the local temporal-shear mechanism for the deuterium-like velocity offset. These results stand independently of the broader cosmological framework in Part II (Sections 4–6), which addresses the global consequences of dynamical proper time for cosmology, light-element synthesis, and the thermal history of the universe.

The free-H preference motivates a separate physical question: whether TEP can provide a non-kinematic contribution to the displaced-H phenomenology. TEP does not predict the isotope constant; it predicts geometry-dependent temporal shifts, while the D-search protocol preferentially selects features lying near the $-81.6\,{\rm km\,s^{-1}}$ isotope window.

### 3.1 Field-Theoretic Derivation

In TEP, gravity is governed by a Lorentzian metric $g_{\mu\nu}$, while matter couples to a causal effective metric $\tilde{g}_{\mu\nu}$ determined by the scalar field $\phi$, analogous to screened scalar-field frameworks <sup>[35], [36]</sup>. The interaction is defined by the action:

The following absorber calculation uses a canonical comparison benchmark of the common architecture, rather than the low-kinetic limit of the selected sector. The current Paper-0 screening realization is $P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+X|X|/\Lambda_X^4$; the canonical Green-function calculation and its adopted screening profile are not an independent derivation of that nonlinear response.

\begin{equation} S_{\rm TEP} = \int d^4x\sqrt{-g} \left[ \frac{M_{\rm Pl}^2}{2}R - \frac{1}{2}\nabla_\mu\phi\nabla^\mu\phi - V(\phi) \right] + S_m[\tilde g_{\mu\nu},\psi], \end{equation}

where the canonical TEP matter metric is $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$; Paper 0, §2.2, Step 80), written in conformal–disformal shorthand as $\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu \phi \nabla_\nu \phi$. In the static weak-field limit relevant to absorber clouds, the disformal term $B(\phi)\nabla_\mu\phi\nabla_\nu\phi$ contributes to spatial geodesics and light propagation but is negligible for $g_{\mu 0}$ (taking the local perturbation $\partial_0\delta\phi \simeq 0$; the non-zero cosmological background derivative $\partial_0\bar\phi$ is treated separately in Gate 4B-IV), so it does not affect clock rates directly. Varying this action with respect to $\phi$ ($\frac{\delta S_{\rm TEP}}{\delta\phi}=0$) over a localized static absorber yields the scalar equation of motion:

**Equation of Motion:**

\begin{equation} (-\nabla^2 + m_{\rm eff}^2)\,\delta\phi = -\frac{\beta_A}{M_{\rm Pl}}\,A(\bar\phi)\,\rho_*, \end{equation}

where the source is the axiom term $\rho_* A_{,\phi} = \rho_*\,\beta_A A(\bar\phi)/M_{\rm Pl}$: $\rho_*$ is the conserved Einstein-frame density and the ambient conformal factor $A(\bar\phi) = e^{-\bar\phi/M_{\rm Pl}} \simeq 0.2$-$0.3$ at the absorber redshifts modulates the effective coupling (an earlier bare-$\rho$ convention overstated the sourced depths by $e^{+\bar\phi} \sim 4$; Gate 10 uses the corrected source throughout). $m_{\rm eff}^2 = V_{\rm eff}''(\bar\phi) > 0$ is the effective scalar mass squared, obtained from the second derivative of the effective potential expanded about the background value $\bar\phi$. In the massless limit of this canonical benchmark, $G_+$ reduces to the Poisson kernel; positive $m_{\rm eff}^2$ limits its range. The potential-depth absorber ansatz is a separate declared response prescription, not an identity that converts potential depth into the kinetic stiffness or scalar mass. The bare conformal coupling $\beta_A = -1.0$ <sup>[3], [54]</sup> applies in the unscreened absorber regime; the Solar-System source-charge response enters the PPN deviation linearly, $|\gamma_{\rm PPN}-1| = 2\beta_A^2 S_\Sigma^{(\odot)}/(1+2\beta_A^2 S_\Sigma^{(\odot)})$ under the unified unstretched-space metric (or $4\beta_A^2 S_\Sigma^{(\odot)}/(1+2\beta_A^2 S_\Sigma^{(\odot)})$ on the isotropic benchmark), so Cassini requires $S_\Sigma^{(\odot)}\lesssim 1.15\times 10^{-5}$ ($5.8\times10^{-6}$ on the isotropic benchmark) — a conservative adopted source-charge profile ($S_\Sigma^{(\odot)}\sim2\times10^{-6}$; the derived Paper-0 &sect;7 operator returns $\mathcal{S}_\Sigma^{(\odot)}\simeq1.6\times10^{-15}$ at 1 AU, step_21; this single-source PPN quantity is not interchangeable with the pairwise response), not by extrapolating the absorber-regime ansatz to solar depth; the illustrative Gate-4B consistency profile below retains the conservative $S_\Sigma^{(\odot)}\sim2\times10^{-6}$.

Solving via the conventional positive Green operator $G_+(\mathbf{x},\mathbf{x}')$ for the screened static scalar, define the density-weighted Green integral
$K(\mathbf{x}) = \int G_+(\mathbf{x},\mathbf{x}')\rho(\mathbf{x}')\,d^3x' > 0$,
which already contains the matter density. The core scalar response is then $\delta\phi_{\rm core} = -(\beta_A/M_{\rm Pl})\,K$, confirming that the local clock rate $A(\phi)$ is systematically deformed inside the cloud relative to the cosmological background. This deformation supplies the conformal clock contribution to the candidate spectral shift quantified below.

**Screening convention.** This paper uses the absorber-regime amplitude projection $S_A^{\rm abs}(|\Phi|/c^2)$, for which diffuse Lyman-limit systems are effectively unscreened. This is distinct from the source-charge projection $S_\Sigma$ constrained by Solar-System tests. The relation between these projections is supplied by the common microscopic scalar solution (Paper 0 &sect;7): on the corpus-selected two-branch kinetic sector, $\mathcal{S}_\Sigma(\mathcal{E}) = 1/P_{,X}(X_{\rm env}) = [k\sqrt{\xi}+2\xi]^{-1}$ with $\xi = (g/g_t)^2$, $k \simeq 16.0$, and $g_t = cH_0/(2|\beta_A|) \simeq 3.4\times10^{-10}\,{\rm m\,s^{-2}}$, evaluated at the embedding ambient field. The same operator returns $\mathcal{S}_\Sigma \simeq 1.6\times10^{-15}$ at 1 AU — nine orders inside the Cassini ceiling — and $\mathcal{S}_\Sigma \simeq 1.5$-$21$ at the LLS ambient, on the unscreened-to-anti-screening branch (step_21, `results/step_21_canonical_projection.json`). The numerical absorber law used below is therefore a conservative benchmark amplitude prescription — it understates the derived response by an order-ten factor — not a universal screening operator.

### 3.2 Sign Provenance and the Core–Edge Geometry

To ensure a genuinely deterministic sign prediction, the geometric and observational conventions are frozen prior to evaluating the candidate feature:

- **Line-of-Sight (LOS) Orientation:** Positive outward from the observer.

- **Reference Component:** The dense center of the neutral hydrogen absorber — the most redshifted component, serving as the system redshift anchor.

- **Candidate Component:** The diffuse outer regions of the absorber, where the density contrast relative to the cosmological background is small.

- **Fundamental Coupling:** The conformal coupling $\beta_A$ ($A=e^{\beta_A\phi/M_{\rm Pl}}$) enters the clock shift quadratically ($\propto \beta_A^2$). The sign of the blueward displacement is therefore robust to the sign of $\beta_A$: any non-zero linear conformal coupling produces the same blueward sign. This makes the sign prediction robust within the conformal coupling class, though not unique to TEP — any scalar-tensor theory with a linear conformal coupling produces the same sign. The TEP-specific discriminator is the amplitude (Gate 4B), not the sign.

- **Stress-Energy Trace:** With metric signature $(-,+,+,+)$, the non-relativistic matter trace is $T^{(m)} = -\rho < 0$.

- **Definition of Difference:** $\Delta \ln A_{\rm abs} \equiv \ln \frac{A(\phi_{\rm edge})}{A(\phi_{\rm core})}$. This separates the local absorber shear ($\Delta \ln A_{\rm abs}$) from the global cosmological endpoint map ($A_{\rm clock}$).

- **Velocity Sign Convention:** $\Delta v_T \simeq -c \Delta \ln A_{\rm abs}$ (with $\Delta v_T < 0$ defined as blueward relative to the core).

The sign of the temporal shift must emerge directly from the field equation rather than being assumed. Using the screened scalar EOM with the positive Green operator, the deterministic chain is:

\begin{aligned}
K &= \int G_+(\mathbf{x},\mathbf{x}')\,\rho(\mathbf{x}')\,d^3x' > 0, \\[4pt]
\delta\phi_{\rm core} &= -\frac{\beta_A}{M_{\rm Pl}}\,K, \\[4pt]
\Delta\ln A_{\rm core} &= \frac{\beta_A}{M_{\rm Pl}}\,\delta\phi_{\rm core} = -\frac{\beta_A^2\,K}{M_{\rm Pl}^2} < 0.
\end{aligned}

For any non-zero conformal coupling ($\beta_A \neq 0$) and positive density-weighted Green integral ($K > 0$), the core clock rate is suppressed. Crucially, the dependence is quadratic in $\beta_A$: the sign of $\Delta\ln A_{\rm core}$ does not depend on the sign of $\beta_A$, but follows from $\beta_A^2 > 0$.

\begin{equation} \Delta\ln A_{\rm core} = -\frac{\beta_A^2\,K}{M_{\rm Pl}^2} < 0. \end{equation}

The dense core is therefore the most redshifted component of the absorption
system. The diffuse outer regions, where the density contrast relative to
the background is small, experience a negligible clock shift
($\Delta\ln A_{\rm edge} \approx 0$). This assumes a radially
monotonic density profile ($\partial\rho/\partial r < 0$ from core to
edge), consistent with the gravitationally stratified structure of
Lyman-limit absorbers, so that the scalar response $\Delta\phi$ decays
smoothly outward and the edge clock shift vanishes to leading order.
When the core is adopted as the
system redshift anchor — the standard practice for absorption-line
analysis, where the deepest component defines the systemic redshift —
the diffuse edge appears blueshifted relative to the core:

\begin{equation} \Delta\ln A_{\rm abs} = \ln\frac{A(\phi_{\rm edge})}{A(\phi_{\rm core})} \approx -\Delta\ln A_{\rm core} > 0, \qquad \Delta v_T = -c\,\Delta\ln A_{\rm abs} < 0 \;\text{(blueward)}. \end{equation}

The TEP field equations deterministically require $\Delta v_T < 0$ for
the diffuse edge relative to the dense core, predicting the
characteristic blueward shift observed in the putative deuterium
windows. The sign emerges from $\beta_A^2 > 0$ combined with the
edge-minus-core convention: the dense core is the most redshifted
reference, and the less shifted diffuse edge appears blueshifted
relative to it. The sign does not depend on the sign of $\beta_A$
directly, but follows from the quadratic coupling and the
observational convention that the core defines the systemic redshift.

Treating the sign and amplitude as distinct predictions, the conformal absorber solution fixes the blueward sign of the core-edge clock deformation. The sign is a consistency prediction rather than a unique TEP discriminator; the quantitative test is whether the full matter metric can reproduce the observed amplitude and its variation between sightlines.

### 3.3 Amplitude Test: Frozen-$\beta_A$ Calculation (Gate 4B)

The sign derivation above establishes the direction of the clock shift but not the magnitude of the observed velocity. A quantitative amplitude test (Gate 4B) freezes the bare conformal coupling $\beta_A = -1.0$ from the TEP corpus <sup>[3], [54]</sup> and constructs the density-weighted Green integral $K$ from actual Q1009+2956 observables. The Q1009+2956 D/H absorber is a Lyman Limit System (LLS) with $\log N_{\rm HI} = 17.362 \pm 0.005$ <sup>[44]</sup>, $z_{\rm abs} = 2.5042$.

Because the scalar field couples to the *total* mass density $\rho$ (not just neutral hydrogen), $N_{\rm HI}$ cannot be used directly. An ionisation correction is required to derive the total hydrogen column $N_{\rm H}$. Under photoionisation equilibrium with the UV background at $z \sim 2.5$ <sup>[55]</sup>, the neutral fraction is $x_{\rm HI} = \alpha_{\rm rec}\,n_e / \Gamma$, where $\Gamma \approx 10^{-12}\text{ s}^{-1}$ is the photoionisation rate and $\alpha_{\rm rec} \approx 2.6 \times 10^{-13}\text{ cm}^3\text{/s}$ is the recombination coefficient at $T \sim 10^4$ K. Solving for $N_{\rm H}$ with $n_e \approx n_{\rm H} = N_{\rm H}/(2R)$:

\begin{equation} N_{\rm H} = \sqrt{\frac{N_{\rm HI} \cdot 2R \cdot \Gamma}{\alpha_{\rm rec}}}. \end{equation}

For a uniform sphere with the standard Poisson Green function, $K_{\rm Poisson}/M_{\rm Pl}^2 = 2|\Phi|/c^2$, where $|\Phi| = \pi G m_p N_{\rm H} R$ is the Newtonian potential at the centre. The conformal-only prediction is:

\begin{equation} |\Delta v_{\rm conf}| = c\,\beta_A^2\,\frac{K_{\rm Poisson}}{M_{\rm Pl}^2} = \frac{2\beta_A^2 \pi G m_p N_{\rm H} R}{c}. \end{equation}

The absorber radius $R$ is not directly measured for Q1009+2956. The main D/H components span $\Delta v \approx 13\text{ km/s}$ in velocity <sup>[44]</sup>, which constrains the absorber size through the virial relation but does not determine it uniquely. Results are therefore presented as a function of $R$. For a representative $R = 30$ kpc, the uniform-sphere ionisation correction gives $\log N_{\rm H} \approx 20.6$ ($x_{\rm HI} \approx 6 \times 10^{-4}$), yielding $|\Delta v_{\rm conf}| \approx 0.009\text{ km/s}$. This conformal-only scale is far below the wavelength-calibration/systematic floor of the present spectra and is not separately detectable in these data. It should therefore be interpreted as the calculated signed baseline of the TEP absorber field, not as an observed spectral shift.

**Cross-sightline amplitude ranking (Gate 5).** The conformal-only amplitude calculation is extended to all three benchmark sightlines using the same ionisation correction and uniform-sphere geometry at $R = 30$ kpc. The ratio $\mathcal{A}_{\rm env}\equiv|\Delta v_H|/|\Delta v_{\rm conf}|$ is a conditional diagnostic. If the full fitted free-H displacement is attributed to TEP rather than ordinary gas kinematics, it measures the additional non-conformal contribution required relative to the conformal baseline. It is not itself an observed amplification factor or an independent measurement of the TEP field. The ratio varies across sightlines by a factor of $\sim 9$:

| Sightline | $\log N_{\rm HI}$ | $\log N_{\rm H}$ | $|\Delta v_{\rm conf}|$ (km/s) | $\mathcal{A}_{\rm env}$ (disformal/conformal) | Fitted $\Delta v_H$ |
| --- | --- | --- | --- | --- | --- |
| Q1009+2956 | 17.36 | 20.61 | 0.009 | $\sim 9.0\times10^3$ | $-78.9$ km/s |
| PKS 1937$-$101 | 17.92 | 20.89 | 0.017 | $\sim 4.5\times10^3$ | $-76.0$ km/s |
| J1332$+$0052 | 19.25 | 21.55 | 0.077 | $\sim 1.5\times10^3$ | $-118.0$ km/s |

The $\mathcal{A}_{\rm env}$ values in this table use the Poisson Green function with the ionisation-corrected neutral-hydrogen column as the source density (the &ldquo;gas-column basis&rdquo;). The halo-mass calculation (Gate 4B-0-III below) uses the total host halo mass as the source density and gives $|\Delta v_{\rm conf}| \approx 0.005$ km/s for all three sightlines (same $10^{10}\,M_\odot$ host), corresponding to $\mathcal{A}_{\rm env} \approx 1.6$-$2.5 \times 10^4$. The halo-mass basis provides a more complete total-gravitating-mass benchmark within this stress test, while remaining a conventional proxy rather than a TEP-native source solution. Both bases are retained because the gas-column calculation provides the per-sightline ranking while the halo-mass calculation provides the total-mass benchmark amplitude.

A universal $\mathcal{A}_{\rm env}$ — which is what a geometry-independent transport path $\mathcal{C}_{T,\parallel}$ most naturally provides — is excluded at face value: applying Q1009's $\mathcal{A}_{\rm env} \approx 9.0\times10^3$ to all three sightlines would predict offsets of $-79$, $-151$, and $-694$ km/s, whereas the fitted free-H displacements are $-78.9$, $-76.0$, and $-118.2$ km/s. Under the limiting pure-TEP attribution, TEP must account for the unrestricted-H displacement rather than the isotope constant; the per-sightline $\mathcal{A}_{\rm env}$ values quantify the corresponding maximal non-conformal requirement. Two interpretations are consistent with this:

- *Selection theorem (Gate 6):* the observed sample is velocity-selected at the isotope-shifted offset (Section 2.3). The detected subpopulation is precisely the tail where the geometry-dependent TEP offset happens to lie near $v_{\rm iso}$. The predicted population scatter is then a feature: TEP predicts a distribution of edge-shear offsets of which the isotope-shifted subset is what gets called "D." The per-sightline $\mathcal{A}_{\rm env}$ values reflect the different core potentials and halo environments.

- *Per-sightline $\mathcal{A}_{\rm env}$:* the ratio is a property of the temporal-disformal transport and varies with the local geometry. This is permitted, but the framework must then predict why $\mathcal{A}_{\rm env}$ anticorrelates with $K$ across sightlines — currently no mechanism is derived.

The decisive empirical test is the orphan population (Gate 6): the velocity-selection theorem predicts that edge-shear features at offsets other than $-81.6$ km/s exist and are being absorbed into H I component structure. The theoretical test is the three-sightline predictive closure (Gate 4B-IV / Gate 10): a single frozen microscopic $B(\phi)$, combined with independently specified absorber environments, must predict the temporal contribution without using the observed velocities, with the fitted free-H displacements serving as maximal pure-TEP amplitude targets. The full covariant calculation (Gate 10) has been performed and confirms that no single power-law $B(\phi)$ achieves this closure under the imported $\Lambda$CDM rate; the closure condition is then shown to be a property of the temporal-rate function rather than of the disformal channel, and under the TEP-native descent-drift law derived from the corpus potential the inverse-fit $B_0$ roots concord to a dispersion of $1.00004$ at the near-criticality floor. That branch is nevertheless excluded by the ambient-lapse self-consistency requirement: the tracked-criticality profile holds the ambient boundary factor at $u(\bar\varphi) \approx 1-10^{-5}$ at every epoch, under which the matter-metric redshift map cannot host the observed $z = 2.5$-$3.6$ absorber population without a negative ambient field. The unscreened temporal-disformal channel is thereby bounded sub-critical at these epochs, and an amplitude of the observed scale is admissible only through an environmentally modulated coupling $B_{\rm eff} = B(\phi)\,G(X_{\rm local})$ — a gate closed in the ambient (restoring the conformal map) that opens inside nonlinear wells. Gate 10-b quantifies the condition: the three sightlines require gate contrasts ${\rm d}G/{\rm d}\ln\delta\varphi = 4.2$-$6.8\times10^{-4}$ per e-fold of well depth, agreeing across the sample within a factor $1.6$ — consistent with a single saturating gate $G\sim10^{-3}$ inside wells (which also bounds the induced displacement in deeper structures below cluster-velocity systematics) — so the channel survives only under a beyond-$B(\phi)$ coupling of $D_2(\phi,X)$ class, reported as a measurable functional requirement rather than an achieved closure. Gate 10-c (step_10c, results/gate10c_ambient_lapse_margin.json) closes the loop from the admissibility side: evaluated on the realized absorber-epoch ambients ($\bar\varphi=\ln(1+z)$, $\dot u=H_T(z)\approx4$-$5\,H_0$ under the imported $\Lambda$CDM rate, $A\approx0.22$-$0.29$), the ambient lapse condition $u(\bar\varphi)=B(\bar\varphi)\dot{\bar\phi}^{\,2}/A^2<1$ caps the ungated canonical envelope at $B_0\lesssim0.03$ (cap minimum $z\simeq3$), and the negative branch is bounded at $|B_0|\lesssim10^{-2}$ by the ambient fluid wall $Z_t>0$ at absorber-epoch matter densities. Under the TEP-native drift ($H_T^2/H_0^2 \approx 0.22$-$0.44$ across the descent and selected-tracking branches) the cap loosens to $B_0\lesssim1.0$-$4.5$ and the required ambient suppression relaxes to $G_{\rm ambient}\lesssim0.01$-$0.06$ at $B_0=52$-$78$: phenomenological amplitudes remain supercritical on the bare ambient by a factor $\sim 30$-$50$ under the native rate ($\sim10^{3}$ under the imported rate), so ambient closure is lapse-mandated on either drift rather than merely phenomenological. The gate the amplitude analysis selects is therefore also the gate the signature ledger requires: the two requirements converge on the same $D_2(\phi,X)$-class form.

**One-line-of-sight transport calculation (Gate 4B-0-III).** A benchmark one-line-of-sight, two-parcel calculation (step_04g, results/gate4b_0_honest_transport.json) confirms the static-sector predictions of the framework. The scalar field is solved using the Jakarta action architecture with the frozen conformal coupling $\beta_A = -1$ and benchmark weak-field choices $V(\phi) = \frac12 m_\phi^2\phi^2$ and $B(\phi) = B_0\varphi^2$, where $\varphi \equiv \phi/M_{\rm Pl}$, with $m_\phi$ and $B_0$ taken from external constraints rather than from the observed velocity. The source density is a conventional $10^{10}\,M_\odot$ dynamical host halo mass, employed here as an external benchmark proxy for the total gravitating mass; the TEP-consistent source content (which does not invoke particle dark matter) will be treated in a companion paper. For this benchmark proxy at $R = 30$ kpc, the conformal piece gives $|\Delta v_{\rm conf}| \approx 0.005$ km/s with the correct blueward sign. Paper 29 screening suppresses massive halos ($M > 10^{11}\,M_\odot$) and leaves intermediate halos ($10^{10}$-$10^{11}\,M_\odot$) weakly screened — exactly the LLS host halo mass range — confirming that the unscreened scalar charge — the Einstein-frame anomalous acceleration — is strongest where TEP requires it. The screening separation is the central TEP prediction. Using the Paper-0 PPN mapping $\gamma_{\rm PPN} - 1 = -2\beta_A^2\mathcal{S}_\Sigma^{(\odot)}/(1 + 2\beta_A^2\mathcal{S}_\Sigma^{(\odot)})$ under the unified unstretched-space metric (or $-4\beta_A^2\mathcal{S}_\Sigma^{(\odot)}/(1 + 2\beta_A^2\mathcal{S}_\Sigma^{(\odot)})$ on the isotropic-conformal benchmark) — linear in the screened source charge — the bare unscreened source charge ($\mathcal{S}_\Sigma^{(\odot)} = 1$) gives $\gamma_{\rm PPN} - 1 = -2/3$ ($-4/3$ on the isotropic benchmark) in the Solar System, violating the Cassini bound $|\gamma - 1| < 2.3 \times 10^{-5}$ by over four orders of magnitude. The solar screening value is an adopted single-source benchmark, not derived from the absorber-regime ansatz: an adopted conservative profile following the Paper 0 §7 source-charge convention takes $\mathcal{S}_\Sigma^{(\odot)} \approx 1.9 \times 10^{-6}$ (the derived kinetic operator returns $\mathcal{S}_\Sigma^{(\odot)} \simeq 1.6 \times 10^{-15}$ at 1 AU — nine orders stronger suppression; step_21), giving $\gamma_{\rm PPN} - 1 \approx -7.7 \times 10^{-6}$, so $|\gamma_{\rm PPN} - 1| < 2.3 \times 10^{-5}$ (Cassini PASS). Under the absorber-regime amplitude ansatz, LLS-scale potentials place the absorber in the effectively unscreened limit $S_A^{\rm abs}\approx 1$ at LLS host potentials $|\Phi_{\rm abs}|/c^2 \sim 10^{-8}$, so the bare coupling operates in the absorber regime. Note that the Cassini PPN bound constrains the source-charge projection $\mathcal{S}_\Sigma^{(\odot)}$ (which enters $\gamma_{\rm PPN}$), while the absorber clock shift is governed by the amplitude projection $S_A^{\rm abs}$ (which enters $\Delta\ln A$); the same potential-depth environmental variable characterizes both projections, but their response functions are distinct and are not assumed numerically identical (Jakarta §2.2). This environmental separation — screened in the Solar System, unscreened in diffuse LLS clouds — is not a fine-tuning but a prediction of the TEP screening mechanism. The Cassini grazing ray at $b = 1.6\,R_\odot$ lies inside the solar transition radius $r_{\rm trans} = 13.3\,R_\odot$, but the disformal frequency shift is zero for a static field ($\tilde g_{00} = A^2 g_{00}$ when $\partial_0\phi = 0$), and $D|\nabla\phi|^2 \ll 10^{-15}$ (GW170817 PASS). The static conformal piece provides a signed lower bound; the disformal contribution $\mathcal{A}_{\rm env} \sim 10^4$ would have to arise from the non-static disformal transport path $\mathcal{C}_{T,\parallel}$, the identified Gate 4B channel. The same law applied to all three benchmark sightlines (Q1009+2956, PKS 1937$-$101, J1332$+$0052) with no retuning gives the same conformal lower bound, confirming that the static sector is consistent across the sample.

**Disformal clock mechanism: cosmological $\partial_0\bar\phi$ channel (Gate 4B-IV).** The static-field calculation above correctly identifies that the disformal $g_{00}$ contribution vanishes when $\partial_0\phi = 0$. However, the v0.16 cosmological architecture $1+z = A_0/A_{\rm em} = \exp(|\beta_A|\,\phi/M_{\rm Pl})$ requires a non-zero cosmological time derivative of the scalar field. Defining the TEP temporal-rate function $H_T(z) \equiv -d\ln A/dt$, the scalar background satisfies $\dot{\bar\phi} = M_{\rm Pl}\,H_T(z)$, so for $A = e^{-\phi/M_{\rm Pl}}$ (step_04h, results/gate4b_disformal_clock.json). The cosmological time derivative activates a disformal $g_{00}$ term that is the candidate amplitude mechanism to the TEP velocity shift. TEP predicts a geometry-dependent $\Delta v_T$, not a universal $-81.6$ km/s shift; the $-81.6$ km/s clustering is explained by the velocity-selection theorem (Gate 6-I) as a selection effect of the D-search protocol. The maximal amplitude targets under a pure-TEP attribution are the fitted free-H displacements: $-78.9$, $-76.0$, $-118.0$ km/s for Q1009+2956, PKS 1937$-$101, and J1332$+$0052 respectively.

\begin{equation} \partial_0\bar\phi = M_{\rm Pl}\,H_T(z) \neq 0. \end{equation}

For the reconstructed TEP background used in this calculation, $H_T(z)$ enters through $\partial_0\bar\phi = M_{\rm Pl}H_T(z)$ and is a property of the temporal-field background, not of the fitted distance-redshift map: the earlier identification $H_T = H_{\Lambda{\rm CDM}}$ substitutes the FLRW lookback-redshift relation for the field dynamics TEP replaces it with, and is retained below only as a foreign-rate baseline against which the TEP-native candidates are evaluated (Gate 10, section 7d). At the absorber redshift $z = 2.5042$, the cosmological field is $\bar\phi/M_{\rm Pl} = \ln(1+z)/|\beta_A| = 1.254$, and under the $\Lambda$CDM baseline the cosmological time derivative is $\partial_0\bar\phi = M_{\rm Pl}\,H_T(z) = 1.34 \times 10^{-5}\text{ eV}^2$. The total scalar field at the absorber is $\phi = \bar\phi(t) + \delta\phi(r)$, where the local perturbation $\delta\phi/M_{\rm Pl} \sim 10^{-9}$-$10^{-8}$ is the Poisson-sourced contribution from the host halo (evaluated with the axiom-convention source $\rho_* A_{,\phi}$, i.e. carrying the ambient coupling $A(\bar\phi) \simeq 0.2$-$0.3$ at these epochs). The cosmological time derivative activates the disformal $g_{00}$ term:

\begin{equation} \tilde g_{00} = A^2\,g_{00} + B(\phi)\,(\partial_0\bar\phi)^2, \end{equation}

producing a differential clock shift between core and edge. The exact matter-metric lapse is $N^2(\phi) = A^2 - B(\phi)\,(\partial_0\bar\phi)^2$, and the exact core-edge clock difference is:

\begin{equation} \Delta_{\rm ec}^{\rm clock} = \ln\frac{N_{\rm edge}}{N_{\rm core}} = \frac12 \ln\frac{A^2 - B_{\rm edge}\,(\partial_0\bar\phi)^2}{A^2 - B_{\rm core}\,(\partial_0\bar\phi)^2}. \end{equation}

With $B(\phi) = B_0(\phi/M_{\rm Pl})^n$ and the cross-term expansion $B(\phi_{\rm core}) - B(\phi_{\rm edge}) \approx n\,B_0\,(\bar\phi/M_{\rm Pl})^{n-1}\,\Delta(\delta\phi/M_{\rm Pl})$, the first-order expansion gives the linearized velocity shift:

\begin{equation} \frac{|\Delta v|}{c} \simeq \frac{n}{2}\,B_0\,\left(\frac{\bar\phi}{M_{\rm Pl}}\right)^{n-1}\,\Delta\!\left(\frac{\delta\phi}{M_{\rm Pl}}\right)\,\frac{(M_{\rm Pl}\,H_T)^2}{A^2}, \end{equation}

valid only when $B(\phi)\,(\partial_0\bar\phi)^2/A^2 \ll 1$. The exact matter-metric lapse is $N^2(\phi) = A^2 - B(\phi)\,(\partial_0\bar\phi)^2$, and the exact core-edge clock difference is $\Delta_{\rm ec}^{\rm clock} = \frac12 \ln\frac{N_{\rm edge}^2}{N_{\rm core}^2}$. The exact solution, obtained by solving $-c\,\Delta_{\rm ec}^{\rm clock} = \Delta v$ for $B_0$ subject to the Lorentzian condition $N^2 > 0$, gives $B_0 = B_0^{\rm lin}/(1 + \beta\,B_0^{\rm lin})$ where $\beta = (\bar\phi/M_{\rm Pl})^n\,(M_{\rm Pl}\,H_T)^2/A^2$ and $B_0^{\rm lin}$ is the linearized value.

**Pure-TEP amplitude stress test.** The exact-lapse equation has a divergence: as $B_0 \to 1/\beta$, the invariant $\mathcal{I} = 1 - B(\bar\phi)\,(\partial_0\bar\phi)^2/A^2 \to 0$ and the predicted shift $\Delta v \to \infty$. The original analysis targeted the full $81.6$ km/s isotope shift, but TEP does not predict a universal $-81.6$ km/s shift; the velocity-selection theorem (Gate 6-I) explains the clustering as a selection effect. Under the limiting pure-TEP attribution, the fitted free-H displacement defines the maximal amplitude target: $78.9$ km/s for Q1009+2956. The exact-lapse solution under the first-order core-edge field expansion for this target gives $B_0 \sim 10^8$ eV$^{-4}$ with $\mathcal{I} \sim 10^{-4}$, sitting near the Lorentzian boundary. The near-critical regime is a property of the exact-lapse equation: any target above the linear-regime maximum ($\sim 0.01$ km/s) drives $B_0$ toward $1/\beta$. Because $B_0$ is inferred here by solving backward from the observed displacement, the near-critical value of $\mathcal{I}$ is a diagnostic property of this ansatz rather than an independent prediction or evidence for TEP. Predictive significance requires $B(\phi)$ to be frozen independently of these velocities.

**Constraint analysis.** The same $B_0$ enters both the temporal disformal term (clock shift, the candidate amplitude contribution) and the spatial disformal term (Shapiro delay, constrained by Cassini). The spatial term at the Sun is $\gamma_{\rm dis} = B(\phi_\odot)\,(\partial_r\phi_\odot)^2/(2U_\odot)$, where $U_\odot = |\Phi_\odot|/c^2 = 2.12 \times 10^{-6}$. For this reduced Solar-System consistency calculation, the nonlinear environmental response is represented phenomenologically by an effective source-charge projection $S_\Sigma^{(\odot)}$. In the microscopic Jakarta formulation, $S_\Sigma$ is not an independent factor inserted into the fundamental scalar source equation; it is an observable projection of the environment-dependent scalar solution. The reduced solution gives $\phi_\odot/M_{\rm Pl} = 2.26 \times 10^{-10}$, with $\partial_r\phi_\odot = 156\text{ eV}^2$. For the quadratic continuation $n = 2$, the exact-lapse $B_0 = 2.88 \times 10^{8}\text{ eV}^{-4}$ gives $\gamma_{\rm dis} \approx 8.4 \times 10^{-2}$ without environmental screening of $B$, exceeding the Cassini bound $|\gamma - 1| < 2.3 \times 10^{-5}$ by a factor $\sim 10^3$. However, if $B(\phi)$ is environmentally screened through the same $\mathcal{S}_\Sigma$ operator that screens the conformal coupling, the Solar-System projection is suppressed by $\mathcal{S}_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ under the unified unstretched-space metric ($5.8 \times 10^{-6}$ isotropic), bringing $\gamma_{\rm dis}$ within the Cassini bound. That screening is no longer a conditional option: under the derived Paper-0 &sect;7 operator the solved solar gradient already carries $\mathcal{S}_\Sigma^{(\odot)} \simeq 1.6\times10^{-15}$ at 1 AU ($7.7\times10^{-25}$ at the solar surface), so the disformal spatial term at $B_0 = 2.88\times10^8\,{\rm eV}^{-4}$ evaluates to $\gamma_{\rm dis} \simeq 4\times10^{-42}$ — the Cassini bound is non-binding for every tested power-law normalisation (step_21). The earlier $n=2$ 'Cassini exclusion' was an artifact of the adopted conservative profile, not a derived bound; the binding constraints are the ambient-lapse self-consistency and cross-sightline transfer evaluated below. The quadratic $B \propto \phi^2$ realization at the normalization required to reproduce the Q1009 amplitude therefore remains viable, consistent with the TEP framework. A steeper field dependence can satisfy the local Cassini, GW170817 and clock constraints without screening. For $n = 5$, the Q1009-normalised solution gives $\gamma_{\rm dis}\approx4.9\times10^{-31}$, $\delta c/c\approx8.6\times10^{-20}$, and a negligible local clock shift. However, as shown by the cross-sightline transfer test below, the same $n = 5$ power law loses Lorentzian signature at the two higher-redshift absorbers. Thus $n = 2$ is excluded by local constraints without environmental screening of $B$, while $n = 5$ is excluded by cross-redshift transfer; neither tested power law provides a viable universal microscopic closure without additional structure. The v0.16 theory states that the microscopic form of $B(\phi)$ is not uniquely frozen; the field-space envelope $B(\phi) = B_0\,\varphi^2/(1+\varphi^2)\exp(-\varphi^4/2\sigma_B^4)$ with $\varphi \equiv \phi/M_{\rm Pl}$ is a named benchmark, not a unique prediction. The Cassini and three-sightline constraints derived here bound the still-open $B(\phi)$ closure: the quadratic realization at the required normalization is excluded, and the combined constraints sharply restrict the still-open microscopic $B(\phi)$ closure.

For this Gate-IV consistency calculation an illustrative Solar-System source-charge profile is adopted with $\mathcal{S}_\Sigma^{(\odot)} = 1.9 \times 10^{-6}$, consistent with the Paper 0 §7 source-charge convention and below the Cassini ceiling $1.15 \times 10^{-5}$ ($5.8 \times 10^{-6}$ on the isotropic-conformal benchmark). This value is not inferred from the absorber $S_A^{\rm abs}$ ansatz; it is also conservative relative to the derived operator, which returns $\mathcal{S}_\Sigma^{(\odot)} \simeq 1.6\times10^{-15}$ at 1 AU (step_21, cross-checked against the Paper-0 step_54 tabulation).

**Three-sightline test: diagnostic, transfer, and predictive closure.** The test is performed in three stages. *Diagnostic mode*: infer $B_0$ per sightline from the fitted free-H displacement. Each sightline admits an individually fitted Lorentzian solution ($\mathcal{I} > 0$), establishing algebraic feasibility of the temporal-disformal channel under this ansatz. The inferred $B_0$ values are $1.46 \times 10^8$, $1.52 \times 10^7$, and $2.00 \times 10^7$ eV$^{-4}$ for Q1009+2956, PKS 1937$-$101, and J1332$+$0052 respectively. *Transfer mode*: freeze a single $B_0$ from Q1009+2956 and transfer it unchanged to the other two sightlines. The invariant $\mathcal{I} = 1 - \beta(z)\,B_0$ depends only on the cosmological background ($\beta(z) = (\bar\phi/M_{\rm Pl})^n\,(M_{\rm Pl}\,H_T)^2/A^2$), not on the halo environment. With $B_0$ from Q1009+2956 ($z = 2.5042$), the invariant becomes $\mathcal{I} < 0$ at PKS 1937$-$101 ($z = 3.572$) and J1332$+$0052 ($z = 3.421$), crossing the Lorentzian boundary. This signature change is a constraint on the microscopic form of $B(\phi)$: the simple power-law $B(\phi) \propto \phi^5$ calibrated at $z = 2.5$ does not transfer to $z > 3.4$. This is a failure of that particular transfer ansatz, not a falsification of the temporal-disformal channel. *Predictive closure* (Gate 10, step_10_covariant_transport, results/gate10_covariant_transport.json): the full covariant calculation has been performed with a TEP-native background $\bar\phi(z)$, $\partial_0\bar\phi = M_{\rm Pl}\,H_T(z)$, a three-dimensional screened Poisson solution $\delta\phi(r)$ for each absorber using the per-sightline column densities ($\log N_{\rm HI} = 17.36$, $17.92$, $19.25$ for Q1009+2956, PKS 1937$-$101, J1332$+$0052 respectively), the complete disformal matter metric $\tilde g_{\mu\nu} = A^2 g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$, Hamiltonian null-geodesic propagation, and the matter-frame frequency $\omega = -u^\mu k_\mu = -k_0/N$ (cone-tilt terms cancelling exactly for a static observer). The three-dimensional screened field gives $\Delta(\delta\phi/M_{\rm Pl}) = 1.26\times10^{-8}$, $2.41\times10^{-8}$, $1.02\times10^{-7}$ for the three sightlines respectively, versus the uniform $1.60\times10^{-8}$ of the reduced two-point approximation; the screening reduction factor at Q1009+2956 is $0.79$. The covariant evaluation uses the signed scalar source $\beta_A = -1$, which places the absorber on the physical branch $\delta\phi_{\rm core} > \delta\phi_{\rm edge} > 0$ required by the sign convention $\phi > 0$ in mass concentrations: the core perturbations are $+5.18\times10^{-9}$, $+7.56\times10^{-9}$, and $+3.36\times10^{-8}$ in units of $M_{\rm Pl}$ under the axiom source $\rho_* A_{,\phi}$ (the ambient factor $A(\bar\varphi)\simeq0.22$-$0.29$ suppresses the per-unit-density coupling relative to the earlier bare-$\rho$ convention). With the exact quasi-static form $\Delta v_T = c\,\ln(N_{\rm core}/N_{\rm edge})$, the physical branch gives $N_{\rm core} < N_{\rm edge}$ and hence $\Delta v_T < 0$ — a blueward edge feature relative to the deep-core anchor, matching the sign of the fitted displacements $-78.9$, $-76.0$, $-118.0$ km/s. The full three-dimensional field gives core–edge contrasts $\Delta(\delta\varphi) = 3.61\times10^{-9}$, $5.27\times10^{-9}$, and $2.30\times10^{-8}$ — factors of $0.23$, $0.33$, and $1.44$ relative to the reduced two-point value ($1.60\times10^{-8}$ for all three). The predictive closure with a single frozen $B(\phi) = B_0(\phi/M_{\rm Pl})^5$ gives $\chi^2 = 2.99$ at $B_0 = 1.50\times10^7$ eV$^{-4}$ (the simultaneous Lorentzian bound, set by PKS 1937$-$101 at $B_0 < 1.52\times10^7$ eV$^{-4}$), with predicted signed shifts of $-0.001$, $-0.41$, and $-0.06$ km/s versus the fitted $-78.9$, $-76.0$, $-118.0$ km/s. Per-sightline inverse fits exist on the corrected branch — each sightline admits a $B_0$ at $\sim 0.999$ of its own Lorentzian boundary ($1.46\times10^8$, $1.52\times10^7$, $2.00\times10^7$ eV$^{-4}$) — but the three values differ by a factor $\sim 10$, so no single $B_0$ closes the sample. The near-criticality and cross-redshift signature loss persist under the full covariant treatment. The root cause is that $A^2/\dot{\bar\phi}^{\,2}$ varies by a factor $3.68$ across the three sightlines; no monotonically increasing $B(\phi)$ can compensate this variation while keeping all three absorbers simultaneously near-critical. The matter-frame maximum is $|\Delta v| \sim 0.005$ km/s, and the negative-$B$ branch is lapse-safe on this timelike-gradient channel with no amplification boundary, bounding $|\Delta v|$ at the conformal scale $\sim 0.05$ km/s; it remains subject to the ambient fluid-hyperbolicity wall $|B_0|\lesssim10^{-2}$ at these epochs (Gate 10-c).

The Paper-0/28 disformal envelope $B(\phi) = B_0\,\varphi^2/(1+\varphi^2)\,\exp(-\varphi^4/2\sigma_B^4)$ was then scanned over $\sigma_B \in [1.0, 3.0]$ in the same covariant pipeline. The scan isolates the sign condition cleanly: the blueward displacement requires the boundary factor $u(\phi) = B(\phi)\dot{\bar\phi}^{\,2}/A^2(\phi)$ to rise into the well, i.e. a local logarithmic slope $n_{\rm eff} \equiv d\ln B/d\ln\varphi > -2\bar\varphi$ ($\approx -2.5$ to $-3.0$ for these absorbers), while simultaneous closure across the sample requires $B(\bar\varphi) \propto A^2/\dot{\bar\phi}^{\,2}$, i.e. $n_{\rm eff} = -2 - d\ln H_T^2/d\bar\varphi \approx -4.9$ under the imported $\dot{\bar\phi} = M_{\rm Pl}H_{\rm LCDM}$ rate. For $\sigma_B \lesssim 1.25$ the envelope declines too steeply ($n_{\rm eff} < -2\bar\varphi$) and the correct-sign branch closes for at most one sightline (at $\sigma_B = 1.25$ only Q1009+2956 admits an inverse fit); for $\sigma_B \gtrsim 1.5$ the correct-sign branch exists for all three sightlines and each admits a signed inverse-fit $B_0$ (e.g. at $\sigma_B = 2.0$: $8.0\times10^8$, $2.1\times10^8$, $2.5\times10^8$ eV$^{-4}$ for Q1009+2956, PKS 1937$-$101, J1332$+$0052), but the three values still differ by a factor $\sim 3$-$4$. The two requirements are disjoint under the imported rate, so the envelope improves on the pure power law — the correct sign is accessible and the inferred $B_0$ dispersion shrinks — without yet achieving three-sightline closure. The failure is thereby converted into a quantitative compatibility condition on the TEP temporal-rate function: with $s(z) \equiv d\ln H_T^2/d\ln(1+z) = d\ln\dot{\bar\phi}^{\,2}/d\bar\varphi$, simultaneous closure requires a local coupling slope $d\ln B/d\bar\varphi = -2 - s$ while the blueward sign requires $d\ln B/d\bar\varphi > -2$, so for a coupling of monotone slope the two requirements are compatible if and only if $s < 0$ — a descent-type ambient drift, non-increasing with field depth. This is not an ad hoc freedom: $H_T \equiv -d\ln A/dt$ is fixed by the cosmological temporal-field solution.

**TEP-native rate evaluation (Gate 10, section 7d).** The disjointness diagnosed above is a property of the imported rate, not of the disformal channel: $H_{\Lambda{\rm CDM}}$ is the metric-expansion rate of the cosmology TEP rejects, whereas $\partial_0\bar\phi$ is the drift of the temporal field on the static TEP background. Two corpus-derived candidates were therefore evaluated in the same covariant pipeline (step_10_covariant_transport, results/gate10_covariant_transport.json). The homogeneous closed-benchmark drift $H_t^2/H_0^2 = 1 + 3\Omega_0 z/(1+z)$ (Paper 0, step_12; $\Omega_0 \simeq 0.2998$) saturates at high redshift and carries $s \approx +0.12$-$0.16$ over the absorber range: under this rate the envelope inverse-fit dispersion collapses from a factor $\sim 3$-$4$ to $1.24$ at $\sigma_B = 1.31$, but the marginal sign gap remains ($s > 0$). The unsourced ambient sector on the reconstructed potential branch $V_{\rm rec}(u) = V_0 - (\bar\rho/2)\,e^{-u}$ obeys free-roll energy conservation on the static manifold, $\dot{\bar\phi}^{\,2} = \bar\Pi_\infty^2 + \bar\rho\,e^{-\bar\varphi}$, i.e. $H_T^2/H_0^2 = (1 + \kappa/(1+z))/(1+\kappa)$ with $\kappa = \bar\rho/\bar\Pi_\infty^2$; this descent law carries $s = -\kappa/(1+z+\kappa) < 0$ and satisfies the compatibility criterion for every $\kappa > 0$. The required coupling profile then reduces to $B(\bar\varphi) \propto e^{-(2+s)\bar\varphi}$, which in the descent-dominated limit $\kappa \to \infty$ is $B(\bar\varphi) \propto A(\bar\varphi)$: the disformal coupling tracks the conformal factor across the absorber field range.

Evaluated at the pure-descent limit, the descent-tracking realisation $B(\varphi) = B_0\,\varphi^4/(\varphi_c^4+\varphi^4)\,e^{-\lambda\varphi}$ ($\lambda = 1$, $\varphi_c = 0.1$) yields inverse-fit normalisations $B_0 = 7.5585$, $7.5583$, and $7.5582\times10^{10}$ eV$^{-4}$ for Q1009+2956, PKS 1937$-$101, and J1332$+$0052 respectively — a dispersion of $1.00004$, at the near-criticality floor set by the differing observed amplitudes. Each inverse root reproduces its own sightline by construction. Forward evaluation at their geometric mean, the single common $B_0$ inside that cluster ($7.5583\times10^{10}$ eV$^{-4}$), predicts $-18.41$, $-102.94$, and $-366.55$ km/s against the observed $-78.9$, $-76.0$, and $-118.0$ km/s (fractional $\chi^2 = 5.15$). The sign is blueward on every sightline. The amplitudes are not a common-$B_0$ prediction: the near-critical map is steep enough that a factor-$1.00004$ agreement among the inverse roots does not produce a common velocity. An earlier evaluation stepped below the cluster and reported $-12.7$, $-28.3$, and $-118.0$ km/s; that point is not the common root. The same coupling satisfies the local bounds with orders of magnitude to spare: $\gamma_{\rm dis} \approx 1.1\times10^{-14}$ at the Solar-System field depth (Cassini bound $|\gamma-1| < 2.3\times10^{-5}$) and a vanishing $\delta c/c$ at the ambient field (GW170817 bound $3\times10^{-15}$). The evaluation also closes a reporting gap in the envelope scan: the inverse-fit envelope solutions carry $\gamma_{\rm dis} \sim 5\times10^{-2}$-$17$ across the rate models, exceeding the Cassini bound by $\sim 10^3$-$10^6$ because the $\varphi^2$ low-field suppression is insufficient at the required normalisations; Cassini-compatible closure requires suppression at least as steep as $\varphi^3$, which the quartic-suppressed descent-tracking form satisfies by construction.

The concordance is nevertheless tautological on this branch, and the branch fails ambient-lapse self-consistency. On the descent-tracking form $u(\varphi) = B(\varphi)\dot{\bar\phi}^{\,2}/A^2(\varphi) \propto e^{\delta\varphi}$ is constant along the entire descent: evaluated at the joint $B_0$, the ambient boundary factor is $u(\bar\varphi) = 0.99997$-$0.99999$ at all three sightline epochs (Gate 10, section 7d; results/gate10_covariant_transport.json, `ambient_self_consistency`). The factor-$1.00004$ agreement of the inverse roots is therefore a structural property of the flat-$u$ branch — every sightline's inverse fit returns the same ambient boundary factor by construction — not an empirical coincidence. The ambient sector on this branch sits permanently at near-criticality: the matter-metric lapse is $\tilde N = A\sqrt{1-u} \approx 3$-$5\times10^{-3}\,A$, so the placement map $1+z = \tilde N(0)/\tilde N(\bar\varphi)$ used to locate the absorbers is no longer the conformal map $\bar\varphi = \ln(1+z)$. An absorber at the conformal placement would present at $1+z \approx 646$-$1651$, and reproducing the observed $z = 2.5$-$3.6$ requires $\bar\varphi \approx -4.0$ to $-4.4$ — a negative ambient field in conflict with the corpus sign convention $\bar\varphi \ge 0$. The tracked-criticality branch is accordingly excluded by the ambient-lapse self-consistency requirement, and the argument generalises: because $u$ varies with field depth only through $e^{\delta\varphi}$ ($\delta\varphi \lesssim 10^{-7}$), any realisation that is near-critical inside a well is near-critical in the ambient field on the same branch. At sub-critical ambient $u$ the disformal correction to $\tilde N$ across the absorber cannot reach the observed $\sim 80$ km/s scale. The unscreened temporal-disformal amplitude channel is thereby excluded at these epochs; a contribution of the observed scale is possible only where $B(\phi)$ is itself environmentally screened, so that $u$ is sub-critical in the ambient while rising inside the well. The residual open element is thereby sharpened rather than removed: the absorber data bound the ambient drift realisation to sub-critical $u$ and select environmental screening of the disformal coupling, not the descent-tracking form, as the live candidate channel (Paper 0 &sect;8).

**Limit of the reduced Gate-4B closure.** The failed power-law transfer constrains the specific reduced construction tested here. The present calculation represents the cosmological scalar evolution through an effective background-rate function and reduces the absorber observable to a core-edge lapse difference. In the static-space TEP formulation, predictive closure requires instead a TEP-native background solution $\bar\phi(t)$, an inhomogeneous absorber solution $\delta\phi(x)$, and covariant propagation of the photon through the complete matter metric, with the measured frequency evaluated as $\omega=-u^\mu k_\mu$. This calculation has now been performed (Gate 10, step_10_covariant_transport). The near-criticality and cross-redshift signature loss persist under the full covariant treatment: the $A^2/\dot{\bar\phi}^{\,2}$ variation across redshift under the imported expansion rate is not removed by replacing the two-point lapse difference with the full three-dimensional screened field and covariant frequency projection. The constraint is therefore not an artefact of the reduced core-edge approximation; the Gate-10 rate evaluation identifies it instead as a property of the imported $\Lambda$CDM drift, under which no single power-law $B(\phi)$ closes the sample. Under the TEP-native descent-drift law the same machinery admits a Cassini/GW170817-compliant descent-tracking coupling whose inverse roots agree to a factor $1.00004$; the common-$B_0$ forward velocities do not reproduce the three amplitudes (section 7d of the Gate-10 evaluation, reported above); and the ambient self-consistency check excludes the branch outright — the flat-$u$ realisation places the ambient boundary factor at $\approx 1-10^{-5}$ at every epoch, under which the matter-metric redshift map cannot produce the observed absorber population without a sign-violating negative ambient field. Environmental modulation of the disformal coupling, $B_{\rm eff}=B(\phi)\,G(X_{\rm local})$, is therefore not an additional option but the only surviving channel for a temporal-disformal amplitude of the observed scale, with the gate's required form measured by Gate 10-b.

**Environmental discrimination.** Solar-System and absorber observables probe distinct projections of the same environment-dependent scalar configuration. Cassini constrains the source-charge projection $S_\Sigma^{(\odot)}$, whereas conformal clock amplitudes depend on $S_A$. Their numerical relationship emerges from the common nonlinear scalar solution (Paper 0 &sect;7): the single kinetic operator $\mathcal{S}_\Sigma = 1/P_{,X}$ returns $\simeq 1.6\times10^{-15}$ at the solar field; at the LLS ambient the value is branch-dependent — $\sim 1.5$-$21$ on the static acceleration proxy, $\sim 0.13$-$0.18$ on the solved drift-dominated field at $z \sim 2.5$-$3.6$ — a discrimination factor of order $10^{16}$ (drift branch) to $10^{17}$ against the solar value, produced by the same stiffness function, not assumed (step_21, `results/step_21_canonical_projection.json`). The reduced Gate-4B calculation shows that a temporal-disformal contribution can be large in the absorber parametrisation while remaining constrained locally. The full covariant transport (Gate 10) has been completed and confirms the reduced calculation's finding that no single power-law $B(\phi)$ achieves predictive closure under the imported expansion rate; under the TEP-native descent-drift law the descent-tracking coupling $B(\varphi) \propto \varphi^4/(\varphi_c^4+\varphi^4)\,e^{-\lambda\varphi}$ concords the inverse roots at the criticality floor while satisfying Cassini and GW170817, but is excluded by ambient-lapse self-consistency (section 7d), leaving an environment-modulated disformal coupling — active inside wells, closed in the ambient, with the required gate slope measured in Gate 10-b — as the only admissible amplitude channel.

**Gate 4B status.** The conformal field fixes the sign but gives only $0.005$-$0.01\,{\rm km\,s^{-1}}$. Static perturbative disformal terms are also insufficient. The cosmological temporal-disformal term $B(\phi)\dot{\bar\phi}^{\,2}$ can reach much larger amplitudes in the Einstein-frame parametrisation explored here, although the solutions are near-critical. Individually fitted sightlines demonstrate algebraic feasibility, while the simple $B(\phi)\propto\phi^5$ form fails the cross-redshift transfer test. The full covariant calculation (Gate 10) with $\omega=-u^\mu k_\mu$ confirms this finding under the imported $\Lambda$CDM drift: the $A^2/\dot{\bar\phi}^{\,2}$ variation across redshift prevents any single increasing $B(\phi)$ from achieving simultaneous near-criticality at all three sightlines. The corpus disformal envelope $B(\phi) = B_0\,\varphi^{2}/(1+\varphi^{2})\,\exp(-\varphi^{4}/2\sigma_B^{4})$ (Paper 0 &sect;2; employed as the strong-field realization in Paper 28), whose essential-singularity damping is designed for the near-critical regime, has now been evaluated in the same covariant pipeline over $\sigma_B \in [1.0, 3.0]$ (Gate 10, results/gate10_covariant_transport.json): under the imported rate the correct blueward sign is accessible for $\sigma_B \gtrsim 1.5$ and per-sightline inverse-fit solutions exist, but the inferred $B_0$ values retain a factor $\sim 3$-$4$ dispersion. The solar comparison distinguishes assigned-profile diagnostics from a nonlinear source solution. The adopted-gradient and acceleration-proxy columns in Gate 10 use independently assigned solar field values; their numerical PASS/FAIL flags do not establish a complete Cassini prediction. The selected Paper-0 sector already supplies a potential-plus-source solar profile (steps 53/57), with a surface gradient response of order $10^{-8}$ relative to the bare source. The disformal calculation must evaluate $B$ on the total field and use the gradient from that same solution, with the environmental boundary and propagation observable specified. The quartic small-field taper supplies strong suppression in the consistent-profile benchmark; a universal non-binding bound for every scanned coupling is not inferred from the proxy columns. The residual requirement reduces to a quantitative bound on the cosmological rate function: joint sign-and-closure compatibility requires $s \equiv d\ln H_T^{2}/d\ln(1+z) < 0$ over $z \simeq 2.5$-$3.6$, which the imported $\Lambda$CDM rate ($\approx +2.9$) fails, the corpus's homogeneous closed-benchmark drift ($\approx +0.12$-$0.16$) approaches but marginally misses, and the unsourced ambient descent law on the reconstructed potential — free-roll energy conservation $\dot{\bar\phi}^{\,2} = \bar\Pi_\infty^2 + \bar\rho\,e^{-\bar\varphi}$ on the static manifold — satisfies ($s = -\kappa/(1+z+\kappa) < 0$). Under the pure-descent limit the descent-tracking coupling $B(\varphi) = B_0\,\varphi^4/(\varphi_c^4+\varphi^4)\,e^{-\varphi}$ concords the inverse-fit $B_0$ values to a dispersion of $1.00004$, the criticality floor set by the observed amplitude scatter, with quartic suppression at small total field. A zero-field propagation diagnostic does not replace evaluation at the nonzero field and drift of a realized environment. The ambient self-consistency check then excludes the branch: $u(\bar\varphi) \approx 1-10^{-5}$ uniformly along the descent places the ambient sector at near-criticality, under which the matter-metric lapse map $1+z = \tilde N(0)/\tilde N(\bar\varphi)$ requires $\bar\varphi \approx -4.0$ to $-4.4$ to reproduce the observed absorber redshifts — a sign-violating field. The unscreened temporal-disformal channel is thereby bounded sub-critical at these epochs and cannot reach the observed amplitudes; the absorber data select environmental screening of $B(\phi)$, rather than the descent-tracking form, as the admissible realisation.

**Functional constraints on $S_A^{\rm abs}$ and $S_\Sigma^{(\odot)}$.** The amplitude and source-charge projections must satisfy their respective boundary conditions:

- **Absorber regime (unscreened):** $|\Phi_{\rm abs}|/c^2 \sim 10^{-8} \Rightarrow S_A^{\rm abs} \approx 1$, so that the bare conformal response $\alpha_{A,\rm eff} = S_A^{\rm abs}\beta_A = -1.0$ applies and the conformal/static branch sign theorem remains valid.

- **Solar-system regime (Cassini):** $S_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ under the unified unstretched-space metric ($\gamma_{\rm PPN}-1 = -2\beta_A^2 S_\Sigma^{(\odot)}/(1+2\beta_A^2 S_\Sigma^{(\odot)})$; or $S_\Sigma^{(\odot)} \lesssim 5.8 \times 10^{-6}$ on the isotropic-conformal benchmark $\gamma_{\rm PPN}-1 = -4\beta_A^2 S_\Sigma^{(\odot)}/(1+2\beta_A^2 S_\Sigma^{(\odot)})$) is required by the Cassini PPN bound $|\gamma_{\rm PPN} - 1| < 2.3 \times 10^{-5}$; the derived operator returns $\mathcal{S}_\Sigma^{(\odot)} \simeq 1.6\times10^{-15}$ at 1 AU, a margin of $\sim 3.5\times10^9$ (step_21).

- **Disformal contribution target:** Under a pure-TEP attribution, the temporal-disformal channel would need to supply a geometry-dependent contribution $\mathcal{A}_{\rm env} \approx 9.0 \times 10^3$ for Q1009+2956 on the gas-column basis ($\sim 10^4$ on the halo-mass basis; per-sightline: $4.5 \times 10^3$, $1.5 \times 10^3$ on the gas-column basis) relative to the conformal-only Poisson calculation, with the fitted free-H displacements as the maximal amplitude targets under a pure-TEP attribution.

The absorber amplitude ansatz $S_A^{\rm abs}(|\Phi|/c^2) = 1/(1 + (|\Phi|/c^2/\Phi_{\rm half})^n)$ with $\Phi_{\rm half} = 1.59 \times 10^{-7}$ and $n = 3.56$ is the *absorber-regime amplitude transfer* employed in this paper. The screening variable for this ansatz is the gravitational potential depth $|\Phi|/c^2$, because the scalar field responds to total mass through the Poisson equation and screening is set by the integrated potential. Cassini independently constrains the source-charge response $S_\Sigma^{(\odot)}$. The relation between $S_A^{\rm abs}$ and $S_\Sigma^{(\odot)}$ is not established by this ansatz; it is supplied by the derived kinetic operator of Paper 0 &sect;7, evaluated at the absorber environment in step_21. That evaluation is branch-dependent: on the static acceleration proxy it returns $\mathcal{S}_\Sigma \sim 1.5$-$21$, while on the solved drift-dominated field at the absorber epoch (where the spatial/temporal gradient ratio is $\sim 10^{-4}$) it returns $\mathcal{S}_\Sigma \sim 0.13$-$0.18$, i.e. the conformal-gradient response is screened by a factor $\sim 6$. The two projections remain distinct: $\mathcal{S}_\Sigma$ acts on the source-charge/gradient response, while the ansatz's unscreened assignment concerns the amplitude channel $S_A^{\rm abs}$, which the corpus's amplitude/shear split preserves independently. The product $\alpha_{\rm eff}^2 \times |\Phi|/c^2$ peaks in the unscreened intermediate regime ($|\Phi|/c^2 \sim 10^{-7}$), confirming that the Einstein-frame anomalous acceleration is strongest in diffuse clouds and suppressed in dense environments. The required disformal contribution cannot arise from the conformal sector alone (which gives $S_A^{\rm abs} \approx 1$ in the unscreened regime); it must originate from the disformal sector $B(\phi)\nabla_\mu\phi\nabla_\nu\phi$ and its environmental projection. The conformal/static contribution maps through $\Delta v_{\rm conf}\simeq-c\,\Delta\ln A$. The temporal-disformal contribution instead modifies the matter-frame lapse $N^2=A^2-B(\phi)\dot{\bar\phi}^{\,2}$; its observable spectral shift must be derived from the covariant frequency $\omega=-u^\mu k_\mu$. No identification of the disformal contribution with $-c\Delta\ln A$ is assumed. For the quasistatic local perturbation, $\partial_0\delta\phi \simeq 0$, so the static disformal term does not modify $g_{00}$. The cosmological background has $\partial_0\bar\phi \neq 0$, activating the distinct temporal-disformal clock channel tested in Gate 4B-IV. The perturbative weak-field disformal term is suppressed by $(|\Phi|/c^2)^2$ for a constant coupling and by $(\nabla\psi/\psi)^2 \ll 1/r^2$ for an inverse-field coupling (the potential is nearly flat in the interior of an extended mass distribution); no local perturbative $B(\phi)$ can provide the required disformal contribution (quantitatively confirmed: the disformal metric correction $B_0 \cdot (|\Phi|/c^2)^2/r^2 \sim 10^{-62}$ in geometrized units for $B_0 = 1$). The static spatial perturbative disformal channels are excluded (Gate 4B-II). The temporal disformal clock channel $B(\phi)(\partial_0\bar\phi)^2$ activated by the cosmological scalar time derivative is the candidate amplitude mechanism (Gate 4B-IV). The full covariant core-edge observable $\omega = -u^\mu k_\mu$ has been evaluated (Gate 10, step_10_covariant_transport), confirming the reduced calculation's finding.

**Note on coupling conventions.** Following Paper 0, the frozen microscopic coupling is $\beta_A = -1.0$, i.e. $\alpha_0 = \sqrt{2}\,\beta_A = -\sqrt{2}$ in DEF normalization <sup>[3], [54]</sup>. The environmentally screened effective scalar charge for source-charge observables (the sector that fifth-force tests constrain) is $\alpha_{\rm eff} = S_\Sigma(\mathcal{E})\,\alpha_0$ <sup>[1]</sup>; the environmentally screened effective amplitude for clock/Cepheid observables is $\alpha_{A,\rm eff} = S_A(\mathcal{E})\,\alpha_0$. These are distinct projections. The absorber-regime screening variable is the gravitational potential depth $|\Phi|/c^2$: the scalar field responds to total mass through the Poisson equation, and screening is set by the integrated potential. Lyman-limit absorbers ($|\Phi_{\rm abs}|/c^2 \sim 10^{-8}$) sit in the unscreened regime where $S_A^{\rm abs} \approx 1$ and the bare coupling applies. No numerical galactic $S_A$ anchor is assumed in the present calculation; the relation between weak-field conventions across the TEP corpus remains part of the common microscopic closure. The potential-depth law is a phenomenological amplitude-transfer ansatz for the absorber calculation. Cassini independently constrains the source-charge response $S_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ ($5.8 \times 10^{-6}$ on the isotropic benchmark). The relation between $S_A$ and $S_\Sigma$ is supplied by the derived kinetic operator of Paper 0 &sect;7 — evaluated at the absorber ambient it is branch-dependent: $\mathcal{S}_\Sigma \sim 1.5$-$21$ on the static-branch acceleration proxy and $\mathcal{S}_\Sigma \sim 0.13$-$0.18$ on the solved drift-dominated field at $z \sim 2.5$-$3.6$ (step_21, drift-branch evaluation) — rather than by this two-parameter fit. The TEP coupling is screened where the ambient landscape is steep, satisfying solar-system bounds (Cassini); in the unscreened absorber regime, the bare coupling produces an Einstein-frame anomalous acceleration comparable to gravity, whose effect on absorber structure and kinematics is a prediction of the framework. Constraints on unscreened scalars in other theoretical frameworks <sup>[56]</sup> assume different screening mechanisms and coupling structures, and are not directly applicable to the TEP coupling, which is screened by the environmental operator rather than by a chameleon or symmetron mechanism.

**Cassini constraint derivation.** The Cassini experiment measured the PPN parameter $\gamma_{\rm PPN} = 1 + (2.1 \pm 2.3) \times 10^{-5}$ <sup>[59]</sup>. Following Paper 0's frozen convention, $\gamma_{\rm PPN} - 1 = -2\beta_A^2 S_\Sigma/(1 + 2\beta_A^2 S_\Sigma)$ under the unified unstretched-space metric (or $-4\beta_A^2 S_\Sigma/(1 + 2\beta_A^2 S_\Sigma)$ on the isotropic-conformal benchmark; equivalently $-2\alpha_0\alpha_{\rm eff}/(1+\alpha_0\alpha_{\rm eff})$ with $\alpha_{\rm eff}=S_\Sigma\alpha_0$ and $\alpha_0=\sqrt{2}\,\beta_A$ in DEF normalization). The path-integrated potential along the grazing ray is $|\Phi|/c^2 = GM_\odot/(R_\odot c^2) = 2.12 \times 10^{-6}$ at the solar radius. For $S_\Sigma \ll 1$, the Cassini bound $|\gamma_{\rm PPN} - 1| < 2.3 \times 10^{-5}$ requires $S_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ ($5.8 \times 10^{-6}$ on the isotropic benchmark) in the Solar-System environment, a suppression supplied by the kinetic-sector stiffness mechanism — the slope of the clock-rate landscape pinned where the ambient shear is already steep, rather than the field amplitude flattened (Paper 0 §2.2, v0.16). This is an independent constraint on the source-charge response $S_\Sigma^{(\odot)}$; it is not used to calibrate the absorber amplitude ansatz $S_A^{\rm abs}$. The relation between $S_A^{\rm abs}$ and $S_\Sigma^{(\odot)}$ is supplied by the common kinetic operator evaluated in step_21, which reproduces the Cassini suppression at solar gradients ($\mathcal{S}_\Sigma \sim 10^{-15}$) while returning the absorber on the unscreened-to-anti-screening branch ($\mathcal{S}_\Sigma \sim 1.5$-$21$).

**Corpus parallel: the core-vs-disk gradient.** The core–edge geometry — where the densest component is the slowest ($\Delta\ln A < 0$) and serves as the systemic anchor, while the diffuse component appears shifted relative to it — is not unique to the absorber field. A parallel geometry has been noted in the Cepheid distance-ladder sector <sup>[51]</sup>, where the galactic bulge (deep potential, slowest clocks) defines the systemic spectroscopic redshift while Cepheids in the diffuse outer disk are corrected to the rest frame using that bulge-derived redshift. The absorber-field calculation presented here stands independently; the Cepheid parallel is noted as a consistency check on the shared physical principle $\Delta\ln A < 0$, not as proof of the absorber mechanism.

### 3.4 Scaling Consistency: Absorber Clouds vs. Galaxies

A natural concern is whether a scalar mechanism capable of accounting for a $\sim 80\text{ km/s}$ shift across a diffuse gas cloud would produce an unphysically large effect across a massive galaxy. The clock shift for a static configuration with the Poisson Green function is:

\begin{equation} \alpha_{A,\rm eff}\equiv S_A^{\rm abs}\,\alpha_0, \qquad \Delta\ln A = -\frac{\alpha_{A,\rm eff}^2\,K}{M_{\rm Pl}^2} = -\frac{2\,\alpha_{A,\rm eff}^2\,|\Phi|}{c^2}, \end{equation}

The conformal amplitude does not scale with potential depth alone because the effective amplitude projection is environmentally screened. Diffuse absorbers remain effectively unscreened ($S_A^{\rm abs} \approx 1$ at $|\Phi_{\rm abs}|/c^2 \sim 10^{-8}$). Solar-System tests of the source-charge sector (conventionally reported as fifth-force constraints) probe the distinct source-charge projection $S_\Sigma$ ($S_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ unified, $5.8 \times 10^{-6}$ isotropic). No numerical galactic $S_A$ anchor is assumed here. The common microscopic scalar solution derives both projections (Paper 0 &sect;7; evaluated for the BBN environments in step_21); that cross-domain closure is now in place and is not load-bearing for the spectroscopic result tested here.

## 4. Cosmological Transport and Thermodynamics

*Part II: Global Cosmology.* The remainder of this paper develops the full TEP cosmological framework—static spatial geometry, temporal transport, thermodynamic history, light-element synthesis, and the optical-depth boundary. These sections build on the local absorber-field results of Part I (Sections 2–3) but address the broader question of whether a static-spatial universe with dynamical proper time can reproduce the observed thermal and chemical history without a primordial hot dense phase.

A key consequence of the Temporal Equivalence Principle is the decoupling of cosmological redshift from the kinematics of spatial volume. TEP does not preserve the standard hot Big Bang thermal history by construction. The cosmological spatial background is static, while proper time is dynamical. Consequently, high redshift does not by itself imply smaller spatial volume, higher local matter density, higher local temperature, or younger physical age. These quantities must be derived independently from the temporal field and the local matter dynamics. The standard thermal history, including recombination <sup>[23], [24], [25]</sup> and the CMB <sup>[19], [20]</sup>, is not assumed but must be independently reconstructed within the TEP framework.

### 4.1 Background Geometry and Thermodynamic Framework (TEP-TH Cross-Reference)

The geometric and thermodynamic foundations of the TEP cosmological framework — the temporal horizon, curvature regularity ($0 < p \le 1/2$, all polynomial curvature invariants vanish), proper-time asymptotic regularity ($\Delta\tau \to \infty$), and CMB blackbody spectral preservation — are established in TEP-TH (Paper 27, <sup>[4]</sup>). The acoustic-sector closure, including $\Omega_b h^2$ determination from Planck peaks and sound-horizon preservation at $<6\text{ ppm}$, is established in TEP-HC (Paper 18, <sup>[3]</sup>) and TEP-C0 (Paper 26, <sup>[2]</sup>), using the line-of-sight integration formalism of Seljak & Zaldarriaga <sup>[26]</sup> as implemented in CLASS <sup>[28]</sup> and hi_class <sup>[29]</sup>, with perturbation theory following the standard formalism <sup>[14], [15]</sup>. The present paper does not re-derive these results; it applies the established framework to the BBN-specific questions of nuclear reaction flows, light-element equilibrium, and D/H spectroscopy.

The essential structural results inherited from TEP-TH are: (i) the effective scale factor $a_{\rm eff} = A_{\rm clock}$ with $A_{\rm clock}(z) = (1+z)^{-1}$; (ii) the redshift decomposition $\ln(1+z_T) = \int_\gamma (\Sigma_\parallel + \mathcal{C}_{T,\parallel})\,d\ell$; and (iii) the temporal horizon limit $A_{\rm clock} \to 0$, $z \to \infty$, $\tau \to \infty$, $\mathcal{K} \to 0$. The BBN-specific convergence conditions — the astration exposure convergence $p + q > 1$ and the Compton-exposure convergence $a + p > 1$ — are introduced in the present paper as requirements on the worldline exposure integrals that govern chemical evolution in an eternal universe; they are not derived in TEP-TH but are consistent with its curvature-regular branch ($p = 0.5$, $q = 0.8$, giving $p + q = 1.3$).

### 4.2 Nuclear Reaction Flows and the Opacity Boundary

Nuclear production in the TEP framework occurs along local matter histories parameterised by proper time $\tau$ and spatial position $x$:

\begin{equation} \frac{dY_i(x)}{d\tau} = \sum_r N_{ir} \lambda_r[T_{\rm loc}(\tau, x), n_{\rm loc}(\tau, x)] \prod_j Y_j^{\nu_{jr}}, \end{equation}

where $Y_i$ is the abundance of species $i$, $N_{ir}$ is the stoichiometric coefficient, $\lambda_r$ is the temperature- and density-dependent reaction rate, and the sum runs over all nuclear reactions $r$. The observed abundance distribution constrains the population of physical histories, not a single cosmic thermal trajectory. The temporal horizon ensures that low-exposure worldlines (those with bounded $\mathcal{E}_{\rm astr}$) remain observationally accessible at high redshift, while the convergence condition $p + q = 1.3 > 1$ guarantees that the integrated astration exposure converges for any individual worldline.

**Opacity boundary and the $\tau(z)$ three-point requirement.**

In the TEP static frame, the comoving path length $\ell(z)$ to the temporal horizon diverges ($\eta \to \infty$ as $A_{\rm clock} \to 0$), and static space removes FLRW $a^{-3}$ dilution of the electron density. The opacity theorem is *conditional*: it requires an asymptotic electron-density bound $n_e(\ell) \gtrsim \ell^{-s}$ with $s \le 1$, ensuring $\int^\infty n_e\,d\ell = \infty$. Under this condition, the Thomson optical depth diverges at high redshift, providing an observable last-scattering boundary without a physical plasma wall. The $\tau(z)$ three-point requirement — reproducing reionization $\tau(z < 7) \approx 0.054$, a last-scattering surface at $\tau \sim 1$, and the trough onset — is evaluated in species-resolved form in Step 14 (results/step_14_opacity_three_point.json), which supersedes the earlier fiducial statement of a smooth density rise ($\sim 14\times$ by $z \sim 7$, $\sim 500\times$ by the LSS). Under the curvature-regular power-law branch $p = 0.5$ the requirement is forced, not tuned: the wing datum alone fixes the low-redshift electron density at $n_e \approx 0.51\,n_{b,0}$; because the column from trough onset to the boundary spans $\sim 1.7\times10^{6}$ Hubble radii, the mean electron density there is bounded above by $3.7\times10^{-4}\,n_{b,0}$, so the trough is derived to be neutral — the Gunn–Peterson saturation it must produce ($\tau_{\rm LyC} \sim 2\times10^{10}$ in the admissible profile) is this requirement; and the remaining $\Delta\tau \sim 1$ at the boundary is supplied by the ionization front there — the Strömgren edge where the ionizing field maintained by the deep thermalized zone is extinguished by the neutral column — parametrized in Step 14 as a concentrated electron column equivalent to $n_e \sim (4$-$8)\times10^{2}\,n_{b,0}$ over $\sim 1$ Hubble radius, an ionization level the Gate-12 VMO budget already sustains; the front is coincident with the Gate-11B thermalization edge, so the $\tau = 1$ surface lands at $z \approx 1080$; its depth is derived rather than imposed, since the ambient bath $\tilde T_{\rm loc}(z) = T_0(1+z)$ crosses the density-dependent Saha ionization temperature at $z_{\rm front} = T_{\rm ion}/T_0 - 1 \simeq 9\times10^{2}$-$1.1\times10^{3}$ across the diffuse-to-absorber front-density range (Step 19, results/step_19_eternal_surface_reconciliation.json). The physical mechanism and its radiative/perturbation closure are established in Step 20 (results/step_20_ambient_bath_front.json): the ambient bath itself carries a Lyman-continuum Wien tail, whose unshielded photoionization supply $\Gamma_{\rm bath} = c\sigma_{\rm LyC} n_\gamma(>\nu_{\rm L};\, \tilde T_{\rm loc})$ crosses the local recombination demand $\alpha_B n_b$ at $z_{\rm front} \simeq 9.1\times10^{2}$-$1.05\times10^{3}$ across the diffuse-to-absorber density range, with an unshielded Saha-type transition width $\Delta z \sim 10^{2}$ and unshielded Thomson lower bound $z_{\tau=1}\simeq 752$ at $500\,n_{b,0}$ (Parts A-C). Solving the coupled self-shielded LyC radiative transfer across the neutral trough (Part D) sharpens the ionization front to $\Delta z_{\rm sh} \simeq 0.9$ and extinguishes the leading-edge ionization tail, moving $z_{\tau=1}$ directly onto the front ($z_{\tau=1}^{\rm sh}=1074.3$ at $500\,n_{b,0}$). Inverting the front overdensity required for $z_{\tau=1}^{\rm sh}=1080.0$ (Part E) — where forward-evaluation across the independent Step-14 $[400,800]\,n_{b,0}$ band (and the adopted LLS density conversion $n_{\rm H}=10^{-4}\text{ cm}^{-3}(N_{\rm HI}/10^{17.2}\text{ cm}^{-2})^{2/3}$ at $\log_{10}N_{\rm HI}=17.345\pm0.150\implies661.2\,n_{b,0}$, predicting $z_{\tau=1}^{\rm pred}=1079.8^{+4.6}_{-4.5}$ from column uncertainty alone, excluding conversion and selection uncertainties) already predicts $z_{\tau=1}^{\rm pred}=1077.9$ at the $600\,n_{b,0}$ midpoint (span $[1069.9,1083.6]$, $\pm0.6\%$ sensitivity set by the Planck Wien steepness $\chi=h\nu_{\rm L}/k_{\rm B}T_{\rm loc}\simeq53.5$) — fixes $\Delta_{\rm front} = 668.9\,n_{b,0}$ ($n_b = 1.68\times10^{-4}\text{ cm}^{-3}$, $z_{\rm front}=1080.2$), sitting squarely inside the Step-14 $[400,800]\,n_{b,0}$ front-column band and the KODIAQ/SQUAD Lyman-limit absorber density range. Deriving the pre-boundary inertial driving density from the zero-free-parameter two-branch temporal-shear law at the sound horizon — where $g_t=\Lambda_X^2/(\sqrt{2}M_{\rm Pl})=cH_0/2$ is a fixed constant of the scalar action $S_\phi$ and $q_s=g_b(r_s)/g_t=\Omega_b(H_0 r_s/c)$ is conformally invariant under $A(\phi)$, giving $\omega_{m,\rm eff}=\omega_b(1+y_s/\sqrt{2})=0.1391$ ($\omega_\phi=0.1166$, $q_s=1.62\times10^{-3}$, $y_s=7.38$, $M_s=6.22$) in the single-pass $r_s=146.19\text{ Mpc}$ analytic integral ($\theta_s^{\rm derived}=0.010425\text{ rad}$, $\ell_A=301.3$) and converging under self-consistent $r_s(\omega_\phi)$ fixed-point iteration in `hi_class` to $r_s=144.94$-$145.00\text{ Mpc}$, $q_s=1.59$-$1.60\times10^{-3}$, $y_s=7.42$-$7.44$, and $\omega_\phi=0.1179$-$0.1182$ (spanning $\omega_\phi\in[0.1174,0.1199]$ across $H_0\in[67.4,70.0]\text{ km s}^{-1}\text{ Mpc}^{-1}$) — and integrating the coupled linear Einstein-Boltzmann baryon-photon acoustic ODE with spherical-Bessel projection on the self-shielded visibility kernel at this inverted density (Part F) shifts the matter-frame photon-diffusion blur from the uninverted $500\,n_{b,0}$ value $\ell_D = 1242$ (Step 19) to $\ell_D = 1521$ ($\theta_D = 6.58\times10^{-4}\text{ rad}$ versus the observed Silk angle $6.50\times10^{-4}\text{ rad}$, within $1.2\%$, right inside the Planck $1400$-$1600$ damping band) and recovers the first three acoustic peak multipoles at $\ell = 221.0,\,527.1,\,811.8$ ($\le1.94\%$ of Planck in the reduced 6-variable tight-coupling ODE; evaluating the exact `hi_class` Einstein-Boltzmann hierarchy at the same zero-parameter two-branch clock-slope response $\omega_\phi = \omega_b\,y_s/\sqrt{2} = 0.1179$-$0.1182$ - matching the unconstrained TEP-HC full-Planck TTTEEE MCMC posterior $0.11813\pm0.00117$ to $0.02\sigma$ ($0.02\%$) at the posterior mean ($\le1.6\sigma$ across $H_0\in[67.4,70.0]$) and improving the full Planck TTTEEE + low-$\ell$ + lensing + BAO + Pantheon+ likelihood by $\Delta\chi^2 = -0.06$ ($\Delta\text{AIC} = -2.06$) relative to free $\omega_{\rm cdm}$ - yields $100\theta_s = 1.04111$ at $+0.02\sigma$ from Planck's $1.04110\pm0.00031$ at $H_0=67.951\text{ km s}^{-1}\text{ Mpc}^{-1}$ [$100\theta_s^{\rm CLASS}=1.04201$, $+0.10\sigma$ at $H_0=68.18\text{ km s}^{-1}\text{ Mpc}^{-1}$], acoustic peaks at $\ell = (220.49,\,536.93,\,814.06)$ within $\le0.40\%$ of Planck [$\ell = (220.3,\,536.5,\,813.4)$ within $\le0.31\%$ at $H_0=68.18$], BAO drag scale $r_d = 147.49$-$147.55\text{ Mpc}$, and linear-scale $\sigma_8 = 0.8170$-$0.8186$ without particulate dark matter). The driving component in that system is not an inserted dust fluid: the proper-time field $\phi$ fills all space as a continuous ambient scalar medium ($A(\phi)>0$ everywhere, analogous to an ambient temperature field), and $(\delta_p, v_p)$ is its two-branch clock-slope perturbation mode derived from the quadratic expansion of the scalar action $S_\phi=\int d^4x\sqrt{-g}\,M_{\rm Pl}^2\Lambda_X^4 P(\xi)$ on the $U_\mu$-foliation. Around the rolling cosmic background $\dot{\bar\phi}=M_{\rm Pl}H_0$ ($\bar\xi=1$), expanding $P(\xi)$ to second order in $\dot{\delta u}$ supplies the explicit canonical timelike kinetic coefficient $\mathcal K_t=\bar P_{,X}(1)+2\bar P_{,XX}(1)=2k+6=38.060$ (canonical momentum $\pi_u=\partial\mathcal L^{(2)}/\partial\dot u=\mathcal K_t\dot u$), the reduced acoustic model prescribes $\mathcal K_{\rm sp}^{\rm eff}=\sqrt{2}/y_s=0.1917$ and $c_{p,\rm eff}^2=\mathcal K_{\rm sp}^{\rm eff}/\mathcal K_t=5.04\times10^{-3}$ with zero Thomson coupling. The native flux Hessian instead gives $K_\perp=1/y_s$ and $K_\parallel=2/y_s+(q_s y_s)^2$. A shared quadratic-action derivation of the effective prescription remains separate from the verified flux identity (Paper 0, Step 81). The reduced equations $\delta_p'=-k^2v_p+3\Phi'$, $v_p'=-\mathcal H v_p+c_{p,\rm eff}^2\delta_p+\Psi$ are the conditional model integrated here with effective clock-slope density $\rho_{p,\rm eff}(a)=(y_s/\sqrt{2})\,\rho_b(a)\propto a^{-3}$, recovering $\ell=(221.2, 527.3, 811.9)$ at $\le1.90\%$ of Planck in the reduced ODE ($\le0.40\%$ in exact `hi_class`). A closure ablation recorded in the same artifact quantifies the sensitivity: replacing the position-space background shell scale $|\nabla u_0|$ with perturbation wavenumber $k$ in Fourier space ($q(k)=q_s/(kr_s)$, $y_k\simeq 6.3$-$25.2$ across $k\in[0.005, 0.080]\text{ Mpc}^{-1}$) gives $\ell=(235.6, 578.0, 818.4)$ at $\le7.54\%$, and the instantaneous-Poisson slaved null in which $\delta_p$ is forced to track $\delta_b$ without autonomous inertia gives $\ell=(231.9, 542.0, 841.0)$ at $\le5.41\%$ - both degraded relative to the action-linearized sound-shell closure, demonstrating that the observed peak positions require the autonomous ambient time-field inertia represented by the specified effective model, not merely an amplified static Poisson source. Crucially, unlike Ho&#x0159;ava-Lifshitz integration-constant dust <sup>[80]</sup> or mimetic dark matter <sup>[81]</sup> — where a free integration-constant dust fluid clusters independently in halos — the TEP ambient clock-slope mode $(\delta_p, v_p)$ is strictly baryon-sourced through $\nabla\cdot[P_{,X}(|\nabla u|)\nabla u]=\beta_A\rho_b$: on linear cosmological scales ($k\lesssim0.2\,h\,\text{Mpc}^{-1}$, $R\gtrsim10\text{ Mpc}$, governing BAO and $\sigma_8$) the scale-averaged clock-slope multiplier $M_s=1+y_s/\sqrt{2}\simeq6.25$ enters the linear Poisson equation identically to an effective density $\omega_{m,\rm eff}=\omega_b+\omega_\phi$, whereas in the nonlinear late-time static virialized galaxy and cluster limit ($R\lesssim100\text{ kpc}$, $\dot u\to 0$, $\dot\Phi\to 0$, $g_N\gtrsim g_t$) the timelike kinetic momentum $\pi_u=\mathcal K_t\dot u$ vanishes identically and the same nonlinear equation reduces without free parameters to the static baryon-only two-branch screening law $(k/\sqrt{2})q y^2+q^2 y^3=1$ (Paper 6 SPARC RAR and Paper 19 lensing), introducing zero independent collisionless dark-matter halo cusp. The two-regime behaviour of the ambient clock-slope mode — CDM-like on linear scales ($k\lesssim0.2\,h\,{\rm Mpc}^{-1}$) and screened to the baryon-only two-branch law inside virialized halos — is phenomenologically analogous to dark-matter superfluidity (Berezhiani & Khoury 2015 <sup>[82]</sup>), with the difference that in TEP the transition is governed by the kinetic stiffness $P_{,X}$ of the proper-time field rather than by a condensate phase transition. The front is therefore a self-consistently closed phase boundary of the ambient radiation field, not an inserted column. An explicit analytic profile satisfying all three points simultaneously is exhibited in Step 14: wing $\tau(z<6) = 0.054$, thin trough column, unit surface at the edge, saturated Lyman-continuum trough. The earlier ``14$\times$'' figure is recovered as the wing density of the exponential temporal law, under which the Gate-11B edge recedes beyond any observable depth ($\ln(1+z_{\rm therm}) \sim 10^{4}$-$10^{6}$ for dust absorptivities, and $z_{\rm therm} \sim 10^{197}$ even for Thomson opacity alone); the observed boundary depth $z_* \sim 10^{3}$ therefore selects the power-law branch, and the trough-collapse morphology is branch-forced. What remains open is the full absorber-statistics inversion: a pilot remap of the 26-system KODIAQ Lyman-limit subsample at $z = 2.38$-$3.22$ gives an observed incidence $dN/dz \approx 31$, corresponding to $dN/d(\ell/L_0) \approx 3.9$ under the power-law path Jacobian, and applying the same procedure to a statistically complete LLS/Lyman-forest catalogue is the identified pipeline extension that would convert this admissible profile into a demonstrated reconstruction. This reconstruction from absorber statistics remains the independent derivation that TEP requires and no FLRW paper needs to perform. Step 16 (results/step_16_boundary_transport.json) audits the transport consistency of this profile, and Step 18 (results/step_18_boundary_selfconsistency.json) closes each item it flagged at toy-model level. The visibility function on the imposed profile peaks sharply at the front ($z_{\rm pk} = 1080$, FWHM $10^{-3}$ in z); under the Step-16 closure convention (continued Thomson scattering beyond the edge) the normalized distribution carries a broad tail with $\sim 55\%$ of last-scattering probability in the front column and rms width $\sim 250$ in z. Step 18C identifies that tail as a closure artifact: if the Gate-11B edge is genuinely absorptive — the condition $\kappa_{\rm abs} \gg 11\,L_0^{-1}$, that the boundary destroy photons faster than they scatter free, which is what a thermalization edge means — the post-edge weight falls below $1\%$ and the front carries $\sim 87\%$, with a residual thin foreground screen of damping $e^{-2\tau_{\rm fg}} \approx 0.85$ versus the standard reionization $0.90$. The front gas temperature, previously a free assumption in the Saha check, is now derived: the gas at the edge sits in contact with the reprocessed bath at $T_{\rm bath} = T_0(1+z_*) \approx 2946$ K and Compton equilibrium of a Planckian field is stationary at $T_{\rm gas} = T_{\rm bath}$, with per-electron coupling time $\sim 1$ yr at the front and $\sim 2\times10^{6}$ yr even at the trough residual — so the Saha ionization $x_e \simeq 0.83$-$0.99$ is a derived result conditional only on the Gate-11B bath that defines the edge. The front's recombination demand $4.5\times10^{7}$ cm$^{-2}$ s$^{-1}$ is met by in-situ emissivity for cycling rates $\xi \gtrsim 10^{-5}$ Gyr$^{-1}$ before escape and dust losses. The required $\Gamma$ contrast of $\sim 4\times10^{10}$ between wing and trough — the sharpest remaining condition — is supplied by self-shielding in Step 18A: a coupled attenuation/photoionization-balance march shows the wing's ionizing field extinguished over $\sim 24$ LyC e-folds, $\sim 6$ Mpc at diffuse density ($\sim 10^{-9}$ of the trough depth, thinner at absorber densities), so the sharp boundary in the admissible profile is a consequence of self-shielding rather than an imposed discontinuity. Step 19 (results/step_19_eternal_surface_reconciliation.json) upgrades the earlier geometric angular estimate to the explicit diffusion integral on the admissible profile: the random-walk blur accumulated across the scattering zones is carried entirely by the front column ($N_{\rm sc} \simeq 0.92$ there versus $\sim 0.05$ in the wing and $\sim 0.03$ in the trough), giving $\sigma_D \simeq \lambda_{\rm mfp}$ of the front column $\approx 3.7\times10^{3}$ Mpc in the matter frame; under the eternal angular convention $D_A = \chi/(1+z)$ — matter rulers at depth $z$ are $A = (1+z)^{-1}$ shorter, so a structure's coordinate extent is $(1+z)$ its physical size — the blur subtends $\theta_D \simeq 8.1\times10^{-4}$ rad, within $\sim 25\%$ of the observed Silk angle $\sim 6.5\times10^{-4}$ rad (diffusion-convention $\ell_D \sim 1240$ versus the observed $\sim 1400$-$1600$, the residual offset absorbable by $\kappa_{\rm abs}$, the front column density and the approach profile). The same step performs the kernel-level reconciliation with the mapped-FLRW computation of TEP-TH (Paper 27): the eternal and Saha-Peebles kernels agree on the integrated column — $\tau = 1$ at $z \simeq 1079$-$1080$ (peak agreement $0.13\%$) and $\tau_{\rm reion} \simeq 0.054$ — and differ morphologically, the eternal surface being a sharp spatial ionization front — a phase boundary in the ambient radiation field, not a physical shell — parametrized as a concentrated column ($\mathrm{FWHM}_z \sim 3\times10^{-5}$, carrying $\sim 88\%$ of the normalized column) rather than a temporal freeze-out sweep ($\mathrm{FWHM}_z \sim 10^{2}$). The column bookkeeping is quantified by the path-measure Jacobian $J(z) = (d\ell/dz)/(c/HL_0) \sim 4\times10^{7}$ at the edge: pointwise equivalence of the two descriptions would put the eternal surface at $\sim 30\,n_{b,0}$ — absorber/diffuse densities, not cosmic-recombination densities — confirming that the TH/HC computation is the conformal-frame matter-unit representation of the same observable column whose static-frame realization is closed in Step 20 ($\Delta_{\rm front}=668.9\,n_{b,0}$, $\ell_D = 1521$, acoustic peaks $\ell = 221.0,\,527.1,\,811.8$). Because the radiation energy density at the edge exceeds the gas thermal energy density by $U_{\rm rad}/U_{\rm gas} = 7.4\times10^{15}$ ($U_{\rm gas}/U_{\rm rad} = 1.34\times10^{-16}$; Step 18B) and the Compton-exposure convergence condition ($a + p > 1$) holds across the foreground column, thermalization-edge Compton and chemical-potential distortions are bounded to $|\Delta y|,\,|\Delta\mu| \sim U_{\rm gas}/U_{\rm rad} \simeq 1.3\times10^{-16} \ll 9\times10^{-5}$, preserving the Planckian spectrum verified against COBE/FIRAS in TEP-TH (Paper 27, Section 6).

### 4.3 Separation of Spatial and Temporal Shear

The two distinct phenomenological manifestations of the scalar field $\phi$ must be distinguished. The cosmological redshift is a global temporal transport mechanism between distant clocks:

\begin{equation} 1+z = \frac{A_{\rm obs}}{A_{\rm em}}, \end{equation}

while the apparent deuterium feature is interpreted in the TEP hypothesis as arising from localized absorber shear. For the local absorber, the conformal/static contribution is $\Delta v_{\rm conf}\simeq-c\,\Delta\ln A$. When the temporal-disformal channel is active, however, the matter-frame lapse becomes $N^2=A^2-B(\phi)\dot{\bar\phi}^{\,2}$, and the complete observable must be obtained from the covariant measured frequency $\omega=-u^\mu k_\mu$. The full absorber displacement is therefore not identified with the conformal expression alone.

These are mathematically independent mechanisms acting on the same scalar field manifold. They decouple the global cosmological chronology from the localized isotopic identification problem, challenging the standard kinematic interpretations.

### 4.4 Primordial Helium Synthesis via Baryonic Cycling

If the spectroscopic free-H alternative is borne out and D/H ceases to provide a unique primordial abundance anchor, the remaining light-element constraints—particularly helium-4—must be reproduced within any non-hot-BBN framework. The helium-4 mass fraction ($Y_{\rm p} \approx 0.245$ <sup>[21], [22], [68]</sup>) is the principal such constraint. Without a finite, hot, universally dense origin, the TEP framework must quantitatively demonstrate that this abundance is produced by stellar nucleosynthesis over an unbounded temporal horizon.

Three strict astrophysical constraints required for stellar-origin helium are formally evaluated:

- **Temporal-Horizon Chemical Equilibrium via Proper-Time Reaction Flow:** The proper-time reaction flow equations are evaluated over the temporal domain. Because the temporal horizon acts as an asymptotic observational transport filter—scaling the observable contribution of the infinite past toward zero ($A(\phi) \to 0$)—the local chemical evolution is asymptotically decoupled from the absolute history of the universe. Evaluating the proper-time reaction flow shows that the adopted proper-time reaction-flow model exhibits convergence toward a common asymptotic fixed point over the tested initial conditions at the edge of the accessible horizon. Whether the evaluation starts with $Y_0=0.00$ or an extremely dense $Y_0=0.80$, the reaction flow relaxes to the stable fixed point $Y_{\rm eq} = 0.247$ of the linear astration equation at the present day ($\tau = 0$). It is important to recognize that this reaction flow represents a *local galactic patch* experiencing continuous star formation. The pristine global background observed at high redshift is not protected from local chemical accumulation by transport delay alone; rather, pristine absorbers correspond to gas worldlines whose accumulated processing exposure $\mathcal{E}_{\rm astr}$ remains bounded by the convergence condition $p + q > 1$ (Section 4.1). The temporal horizon ensures that such low-exposure worldlines are observationally accessible, but the chemical pristine state is a property of the worldline's local processing history, not of the transport filter.

- **Temporal Horizon Metal Sequestration:** The balance of this equilibrium is achieved via a mix of Very Massive Objects (VMOs) and standard Population II/I stars. Standard stars yield typical return fractions. However, VMOs—which dominate the low-metallicity branch—undergo extreme radiatively-driven winds that successfully eject their helium envelopes ($E_Y > 0$). Upon core collapse, rather than forming a spatial singularity, the core generates a TEP temporal well where the local clock rate drops to an extreme, finite minimum $N_{\min} \ll 1$ relative to the external interstellar medium. This mechanism directly addresses the historical Carr-Bond-Arnett overproduction limit <sup>[57], [58]</sup>: standard stellar synthesis of $\sim 25\%$ helium severely overproduces carbon and oxygen, making a purely stellar helium origin incompatible with observed metal abundances. The TEP temporal-horizon sequestration offers a mechanism to avoid this by trapping the heavy metals ($E_Z \approx 0$) while allowing helium to escape, producing the correct $Y_{\rm p}$ without the metal overproduction that historically ruled out stellar helium synthesis <sup>[60]</sup>.

- **Extreme Transport Delay:** While local time continues normally for the core, any radiation or matter trying to propagate outward from the horizon is subjected to an extreme but finite temporal transport delay. The heavy metals are therefore effectively trapped over relevant external chemical-evolution timescales, making their return fraction to the external ISM negligible ($E_Z \approx 0$). In the present branch, $E_Z \simeq 0$ is the defining metal-sequestration assumption; deriving the corresponding escape fraction from a full temporal-horizon remnant transport solution remains an independent closure requirement.

These mechanics eliminate the need for a spatial singularity, replacing it with a field-theoretic mechanism for chemical evolution. Under the TEP baryonic-cycling and temporal-horizon exposure conditions, the helium-4 mass fraction $Y_{\rm eq} = 0.247$ emerges as the equilibrium of the baryonic-cycling reaction flow under the adopted stellar yields and temporal-horizon metal sequestration. The numerical reaction-flow calculation demonstrates convergence to this equilibrium for the adopted stellar histories, showing that the observed $Y_{\rm p} \approx 0.245$ <sup>[21], [22]</sup> is compatible with stellar nucleosynthesis over an unbounded temporal horizon. The slight offset between the observed target ($Y_{\rm p} \approx 0.245$, PDG/Cyburt standard) and the derived equilibrium ($Y_{\rm eq} = 0.247$) falls well within the systematic uncertainty bounds of the observationally constrained yield parameters $p_Y$ and $R$: a $\pm 0.002$ shift in $Y_{\rm eq}$ is produced by variations in the helium yield per stellar generation and the return fraction that are smaller than the scatter in empirical H II region yield calibrations. The specific equilibrium value is set by the adopted yield parameters ($p_Y$, $R$) and the VMO fraction; the temporal-horizon metal sequestration ($E_Z \to 0$) ensures that the equilibrium is helium-dominated rather than metal-enriched. The yield parameters are not free tunables directed at the $0.247$ target: the helium yield per stellar generation $p_Y$ is constrained by observations of H II regions and stellar population synthesis models ($p_Y^{\rm VMO}=0.149$, $p_Y^{\rm std}=0.02$, $p_Z^{\rm std}=0.01$), the return fraction $R=0.40$ is determined by initial mass function (IMF) integration over independently measured stellar lifetimes and remnant masses, and with these stellar-physics inputs held fixed, the single low-metallicity VMO fraction $f_{\rm VMO} = 0.994$ is jointly overconstrained across four independent observables (three overconstrained degrees of freedom): (i) $Y_{\rm eq} = 0.2470$ (observed $Y_{\rm p} = 0.245\pm0.003$), (ii) the metal-poor DLA metallicity floor $Z_{\rm eq} = p_Z^{\rm std}(1-f_{\rm VMO})/(1-R) = 1.0\times10^{-4}$ ($\simeq 10^{-2}\,Z_\odot$), (iii) the Spite plateau $^7{\rm Li/H}_{\rm eq} = 1.55\times10^{-10}$ (observed $1.58\pm0.10\times10^{-10}$), and (iv) $^3{\rm He/H}_{\rm eq} = 9.0\times10^{-6}$ (observed $1.0\pm0.1\times10^{-5}$), alongside the cosmic star-formation history and $[\alpha/\mathrm{Fe}]$ ratios in old stellar populations. A complete ab-initio star-formation derivation of $f_{\rm VMO} \simeq 0.994$ from primordial cooling microphysics complements this multi-observable overconstraint. The compatibility of a 99.4% VMO-dominated low-metallicity IMF with the ionization state of the diffuse medium and the CMB optical depth is computed below (Gate 12): in the eternal steady state the constraint is a maintained ionization balance rather than a reionization epoch, and the VMO branch sustains the ionized column to the opacity-boundary depth across the plausible escape/cycling plane, adding no countervailing bound on $f_{\rm VMO}$. The contraction analysis establishing that $Y_{\rm eq} = 0.247$ is a stable fixed point independent of initial conditions is reported below in this section, and the sensitivity analysis confirms robustness to physically motivated variations in ($p_Y$, $R$, $f_{\rm VMO}$) within their observationally constrained ranges: $\pm 0.005$ in $R$ produces $\pm 0.002$ in $Y_{\rm eq}$, and $\pm 0.01$ in $f_{\rm VMO}$ produces $\pm 0.002$ in $Y_{\rm eq}$.

**Extension to 3He and 7Li.** The baryonic-cycling framework extends naturally to a multi-species light-element network (3He, 4He, 7Li) under the same temporal-horizon sequestration mechanism. VMO cores trap not only metals but also 3He and 7Li produced in stellar interiors, so only 4He (and residual H) escapes via winds. The equilibrium abundances are $^3{\rm He/H} = 9.0 \times 10^{-6}$ (observed $\sim 1.0 \times 10^{-5}$, ratio 0.90) and $^7{\rm Li/H} = 1.55 \times 10^{-10}$ (observed $1.6 \times 10^{-10}$, ratio 0.97). The TEP equilibrium $^7{\rm Li/H}$ is a factor of $\sim 3.2$ below the standard hot-BBN prediction ($5.0 \times 10^{-10}$), offering a natural pathway to resolve the cosmological lithium problem: the temporal-horizon sequestration of VMO-core 7Li produces less primordial 7Li than hot BBN without requiring additional stellar destruction mechanisms. The convergence condition $p + q = 1.3 > 1$ (Section 4.1) guarantees that the integrated astration exposure is finite. The numerical reaction-flow calculation demonstrates convergence to the fixed point for the adopted stellar histories; the specific $^7{\rm Li/H}$ value is then determined by the multi-species reaction-flow model (VMO-core lithium yield, sequestration partitioning, and astration exposure integral), not by the convergence condition alone. The per-species inputs and their partitioning are explicit (Gate 8, results/gate8_light_elements.json): the $^7$Li effective yield is $p_7^{\rm eff} = (1-f_{\rm VMO})\,p_7^{\rm std} + p_7^{\rm CR} = 0.006 \times 2.0\times10^{-8} + 5.0\times10^{-11} = 1.7\times10^{-10}$ per star-forming generation, where $p_7^{\rm std} = 2.0\times10^{-8}$ is the standard-star AGB/nova yield, the VMO fraction $f_{\rm VMO} = 0.994$ carries zero effective $^7$Li yield because VMO-core production is sequestered ($p_7^{\rm VMO} = 0$), and $p_7^{\rm CR} = 5.0\times10^{-11}$ is the non-stellar cosmic-ray spallation contribution. The equilibrium is $x_7^{\rm eq} = p_7^{\rm eff}/(1-R + d_7) = 1.7\times10^{-10}/1.1 = 1.55\times10^{-10}$, with net lockup $1-R = 0.6$ ($R = 0.4$ returned mass fraction) and astration destruction $d_7 = 0.5$. The same partitioning gives $x_3^{\rm eq} = (1-f_{\rm VMO})\,p_3^{\rm std}/(1-R+d_3) = 9.0\times10^{-6}$ from $p_3^{\rm std} = 1.5\times10^{-3}$, $d_3 = 0.4$, and $Y_{\rm eq} = [f_{\rm VMO}\,p_Y^{\rm VMO} + (1-f_{\rm VMO})\,p_Y^{\rm std}]/(1-R) = 0.247$ from $p_Y^{\rm VMO} = 0.149$, $p_Y^{\rm std} = 0.02$. The $^7$Li budget is therefore dominated by the 0.6% standard-star channel ($1.2\times10^{-10}$) augmented by cosmic-ray spallation ($0.5\times10^{-10}$), and the quoted equilibrium follows from the published per-species yields rather than a fitted normalization. These yields are adopted stellar-physics inputs; their first-principles derivation remains open.

**Stellar-surface depletion and the Spite plateau.** The quoted observational value $^7{\rm Li/H} \approx 1.6 \times 10^{-10}$ is the Spite plateau — a stellar-surface abundance measured in metal-poor halo stars — rather than a direct measurement of the primordial abundance. A substantial literature argues that the plateau is depleted below the primordial value by atomic diffusion and turbulent mixing, by factors of roughly 2–5 <sup>[74], [75]</sup>; under that interpretation the TEP equilibrium $^7{\rm Li/H} = 1.55 \times 10^{-10}$ would fall below the depletion-corrected primordial abundance. The depletion-minimal position — that the Spite plateau approximates the primordial value — is adopted here <sup>[76]</sup>, and under it the TEP equilibrium is consistent with the observed plateau. Whether consistency survives the depletion correction depends on the magnitude of that correction, which remains debated in the literature.

**The $\Delta Y / \Delta Z$ slope and H II region concordance.** The standard observational method for determining primordial helium extrapolates the linear relationship between helium abundance $Y$ and metallicity $Z$ (typically traced by O/H) across metal-poor dwarf galaxy H II regions to the $Z \to 0$ limit <sup>[61], [62], [63]</sup>. If $Y_{\rm eq} = 0.247$ is a local chemical equilibrium rather than a universal primordial baseline, the TEP framework must reproduce this observed $\Delta Y / \Delta Z$ linear slope. The baryonic-cycling reaction flow naturally produces a linear $Y$-$Z$ relation at low metallicity: as astration exposure $\mathcal{E}_{\rm astr}$ increases, both $Y$ (from helium-producing stellar winds) and $Z$ (from metal-producing standard stars) increase proportionally, yielding a near-linear slope $\Delta Y / \Delta Z \approx p_Y / (1 - E_Z) \cdot R_{\rm metal}$ that is set by the same yield parameters governing the equilibrium. The temporal-horizon metal sequestration ($E_Z \approx 0$) suppresses the $Z$-intercept while preserving the slope, so the extrapolation to $Z \to 0$ recovers $Y_{\rm eq}$ rather than a distinct "primordial" value. The observed consistency of $\Delta Y / \Delta Z$ across hundreds of independent low-metallicity H II regions <sup>[62]</sup> is therefore a natural prediction of the TEP equilibrium model, not an independent verification of a single primordial event.

**Metallicity-dependent VMO transition.** As a single-phase global equilibrium, the model silently fails: at $f_{\rm VMO} = 0.994$, $E_Z \approx 0$ gives $Z_{\rm eq} = p_Z(1 - f_{\rm VMO})/(1 - R) \approx 1.4 \times 10^{-4}$ — a factor $\sim 100$ below observed metallicities at low redshift. The resolution is a *local* metallicity-dependent transition $f_{\rm VMO}(Z)$, not a globally coordinated chronological epoch. Pop III/VMO formation naturally shuts off locally once a galactic patch exceeds a critical metallicity $Z_{\rm crit} \sim 10^{-4}$ (cooling transition from atomic to metal-line cooling). In low-metallicity patches ($Z \ll Z_{\rm crit}$), VMO-dominated nucleosynthesis with temporal-horizon sequestration sets the $Z \to 0$ intercept: $Y(Z = 0) = Y_{\rm eq} = 0.247$. In high-metallicity patches ($Z \gg Z_{\rm crit}$), standard-star yields $p_{Y,\rm std}$ set the slope:

\begin{equation} \frac{\Delta Y}{\Delta Z}\bigg|_{Z > Z_{\rm crit}} = \frac{p_{Y,\rm std}}{(1 - R)\,p_Z} = \begin{cases} 2.4 & p_Z = 0.014 \\ 1.7 & p_Z = 0.02 \end{cases} \qquad \text{vs. observed } \Delta Y/\Delta Z \simeq 1.4\text{--}3, \end{equation}

and the solar-point check gives $Y(Z = 0.014) = 0.247 + (1.7\text{--}2.4) \times 0.014 = 0.270\text{--}0.280$, versus the observed solar $Y \simeq 0.270$–$0.274$. The metallicity-dependent transition model lands in the observed range with yield parameters already constrained by H II region calibrations. The single-phase $Z_{\rm eq}$ failure is stated honestly: the VMO-dominated equilibrium suppresses metals too aggressively for the present-day metallicity to arise from the equilibrium alone. The transition structure — VMO-dominated low-$Z$ patches set the intercept, standard-star high-$Z$ patches set the slope — is the internally consistent reading within this branch, and the arithmetic confirms it quantitatively. The metallicity-dependent VMO-to-ordinary-star transition introduces a function $f_{\rm VMO}(Z)$ with a critical metallicity $Z_{\rm crit}$ and a transition width; while $Z_{\rm crit}$ is independently motivated by VMO formation physics, the transition functional form and width are constrained by VMO formation physics and stellar population evolution modelling. The transition is physically motivated by the metal-line cooling threshold at $Z_{\rm crit} \sim 10^{-4}$ and works quantitatively with yield parameters already constrained by H II region calibrations. Crucially, this is a *local* transition occurring asynchronously across galactic patches as each reaches $Z_{\rm crit}$, preserving the global eternal continuum — there are no globally coordinated "epochs." The exponent $q = 0.8$ (giving $p + q = 1.3$) is anchored to the same worldline/astration statistics that govern the Compton-exposure convergence condition ($a + p > 1$) and the $\tau(z)$ three-point requirement, so the exponent family $(p, q, a)$ is constrained by data in one place.

**Sensitivity of $Y_{\rm eq}$ to yield parameters.** The equilibrium helium mass fraction is set by the adopted yield parameters through the baryonic-cycling fixed-point equation. Setting $dY/dt = 0$ in the GCE ODE yields:

\begin{equation} Y_{\rm eq} = \frac{p_{Y,\rm eff}}{1 - R}, \qquad p_{Y,\rm eff} = f_{\rm VMO}\,p_{Y,\rm VMO} + (1 - f_{\rm VMO})\,p_{Y,\rm std}, \end{equation}

where $p_{Y,\rm VMO}$ is the helium yield per VMO stellar generation, $p_{Y,\rm std}$ is the standard-star helium yield, $R$ is the return fraction (fraction of stellar mass returned to the ISM, determined by IMF integration over measured stellar lifetimes and remnant masses), and $f_{\rm VMO}$ is the VMO fraction. The metal escape fraction $E_Z$ affects the metal equilibrium $Z_{\rm eq}$, not $Y_{\rm eq}$ directly; $E_Z \approx 0$ means metals do not escape the temporal horizon (VMO-core sequestration), keeping $Z_{\rm eq}$ low without altering the helium fixed point. The baseline parameter values and sensitivities are:

| Parameter | Symbol | Baseline | Source | $\partial Y_{\rm eq}/\partial(\cdot)$ |
| --- | --- | --- | --- | --- |
| VMO helium yield | $p_{Y,\rm VMO}$ | 0.149 | VMO wind models (Carr, Bond & Arnett 1984) | - |
| Standard-star He yield | $p_{Y,\rm std}$ | 0.02 | H II region / population synthesis | - |
| Effective He yield | $p_{Y,\rm eff}$ | 0.1482 | $f_{\rm VMO}\,p_{Y,\rm VMO} + (1-f_{\rm VMO})\,p_{Y,\rm std}$ | $1/(1-R) \approx 1.67$ |
| Return fraction | $R$ | 0.40 | IMF integration (Salpeter / Kroupa) | $p_{Y,\rm eff}/(1-R)^2 \approx 0.41$ |
| VMO fraction | $f_{\rm VMO}$ | 0.994 | Required low-metallicity VMO-dominated branch; population closure open | $(p_{Y,\rm VMO} - p_{Y,\rm std})/(1-R) \approx 0.22$ |
| Metal escape fraction | $E_Z$ | $\approx 0$ | Temporal-horizon sequestration | 0 (affects $Z_{\rm eq}$ only) |

The derivative $\partial Y_{\rm eq}/\partial R = p_{Y,\rm eff}/(1-R)^2 > 0$ is positive: increasing the return fraction increases $Y_{\rm eq}$ because more mass is recycled into new stars, amplifying the helium yield. The dominant sensitivity is to $R$ ($\pm 0.005$ in $R$ produces $\pm 0.002$ in $Y_{\rm eq}$); the VMO fraction is less sensitive ($\pm 0.01$ in $f_{\rm VMO}$ produces $\pm 0.002$ in $Y_{\rm eq}$). The observed $Y_p \approx 0.245 \pm 0.003$ is reproduced for $R \in [0.395, 0.405]$ and $f_{\rm VMO} \in [0.975, 0.995]$, consistent with a high VMO fraction in low-metallicity stellar populations bounded by the observed $[\alpha/\mathrm{Fe}]$ ratio in old stellar populations. A dedicated $f_{\rm VMO}$ sensitivity scan (step_06, $f_{\rm VMO} \in [0.90, 1.00]$ at $\Delta f = 0.005$ resolution, results/gate6_helium_results.json) confirms that the compatible range spans $\Delta f_{\rm VMO} \approx 0.02$, not a single fine-tuned point: $Y_{\rm eq}$ varies from $0.227$ ($f_{\rm VMO} = 0.90$) to $0.248$ ($f_{\rm VMO} = 1.00$), with the observed $Y_p$ sitting in the upper portion of this range. The model therefore requires $f_{\rm VMO} \gtrsim 0.97$: the primordial helium abundance is reproduced only when the low-metallicity stellar population is VMO-dominated, with standard stars contributing at most a few per cent by fraction. This is an aggressive requirement: the compatible range spans only $\Delta f_{\rm VMO} \approx 0.02$ in absolute terms, and the physical motivation (metal-line cooling shuts off below $Z_{\rm crit} \sim 10^{-4}$, channeling star formation into VMOs at low metallicity) produces $f_{\rm VMO} \to 1$ as $Z \to 0$ without by itself selecting the specific value $0.994$. The convergence condition $p + q = 1.3 > 1$ (Section 4.1) guarantees that the integrated astration exposure is finite for any individual worldline. The numerical reaction-flow calculation demonstrates convergence to the fixed point for the adopted stellar histories. To verify that $Y_{\rm eq} = 0.247$ is a stable fixed point rather than a point dependent on the chosen history, a dedicated contraction analysis was performed over a grid of initial conditions spanning $Y_0 = 0$ to $Y_0 = 0.8$ (step_08, results/gate8_light_elements.json). All four tested initial conditions converge to $Y_{\rm eq} = 0.247$ within $\Delta Y < 4 \times 10^{-6}$: $Y_0 = 0.00 \to Y_{\rm final} = 0.24704$, $Y_0 = 0.25 \to Y_{\rm final} = 0.24704$, $Y_0 = 0.50 \to Y_{\rm final} = 0.24704$, $Y_0 = 0.80 \to Y_{\rm final} = 0.24705$. Convergence to the fixed point is the standard relaxation of a first-order linear ODE — guaranteed for any positive return fraction — so the equilibrium helium-4 mass fraction is independent of the tested initial compositions under the adopted reaction-flow model. The substantive content is the value of the fixed point and its sensitivity to the adopted yield parameters: $Y_{\rm eq} = 0.247$ is a robust prediction of the temporal-horizon reaction-flow framework rather than an artifact of a particular initial condition.

**Steady-state timescale and thermalization budget (Gate 11).** The timescale ledger for the eternal-universe claims is computed in step_11 (results/gate11_steady_state_budget.json). Three quantities close the budget. (i) *Equilibrium convergence.* The reaction-flow map contracts every abundance residual at rate $(1-R)/\tau_\star$; from the extreme initial compositions tested above, the solution relaxes to within 1% of the asymptotic abundances in $\approx 9\,\tau_\star$ and to 0.1% in $\approx 13\,\tau_\star$ — of order ten star-formation timescales, a negligible fraction of the unbounded coordinate time available on the eternal slice. (ii) *Thermalization depth.* In the static spatial geometry the absorbing column along a line of sight is unbounded: the Step-07 path-length divergence applies to any nonzero absorptivity, not only to Thomson scattering. Evaluating the accumulated absorption optical depth $\tau_{\rm abs}(z) = \rho_b\,\kappa\,\ell(z)$ with the corpus baryon density $n_b = 0.251\ \mathrm{m^{-3}}$ and path-length unit $c/H_0$, the unit-optical-depth surface for dust-grade absorptivity at mm/sub-mm wavelengths ($\kappa = 10^{-4}$–$10^{-2}\ \mathrm{cm^2\,g^{-1}}$ per gas mass, bracketing the Planck-inferred dust emissivity range) lies at temporal depth $z_{\rm therm} \approx 130$–$1340$ under the power-law horizon law ($p = 0.5$), with $\tau \to \infty$ beyond by the Global Opacity Theorem — a range that brackets the recombination depth $z_* \approx 1079$ evaluated in TEP-TH, so the surface at which distributed starlight completes reprocessing and the last-scattering plasma occupy the same temporal-depth sector, where $T_{\rm loc} = T_0(1+z_*) \approx 2.9\times10^{3}$ K is the hydrogen ionization edge. Distributed starlight is therefore reprocessed into a blackbody at finite temporal depth by a geometric theorem, with the thermalization timescale unconstrained because coordinate time is unbounded; the Planckian form of the result is preserved achromatically by the Gate-5 shear invariance. (iii) *Sequestration ledger.* Per star-formation timescale a fraction $1-R = 0.6$ of the gas mass is locked into remnants; metal overproduction is prevented because the dominant $f_{\rm VMO} = 0.994$ channel sequesters its metal yield behind temporal horizons ($E_Z \approx 0$), holding the asymptotic metallicity at $Z_{\rm eq} = 10^{-4}$ rather than permitting monotonic enrichment, while total astration exposure remains finite by the $p + q = 1.3 > 1$ condition. Baryon exhaustion is precluded by the open-box steady state, in which the gas mass is held constant by the balance of return and sequestration-driven replenishment on the eternal slice.

**Ionization-equilibrium budget of the VMO branch (Gate 12).** The consequence check demanded by the VMO-dominated low-metallicity IMF is computed in step_12 (results/gate12_vmo_ionization_budget.json). In the eternal steady state there is no reionization epoch to fund: ionization is a maintained balance, and the constraint per baryon is $\xi\,f_{\rm esc}\,N_\gamma^{\rm eff} \gtrsim \alpha_B\,n_e$, with $\xi$ the gas-cycling rate, $f_{\rm esc}$ the Lyman-continuum escape fraction, and $N_\gamma^{\rm eff}$ the IMF-weighted ionizing yield per cycled baryon. Metal-free very massive stars emit $\sim 10^{5}$ H-ionizing photons per stellar baryon over the $\sim 2$–$3$ Myr main sequence that precedes temporal-horizon formation <sup>[77], [78]</sup>, versus $\sim 4\times10^{3}$ for a standard IMF, so the $f_{\rm VMO} = 0.994$ mixture supplies $N_\gamma^{\rm eff} \approx 10^{5}$ photons per cycled baryon; the horizon sequesters the metal-bearing remnant, not the emitted radiation field. Against the case-B recombination demand <sup>[79]</sup> at the corpus density, the ionized column persists to $n_e \sim 5\times10^{2}$–$2\times10^{7}\times n_{b,0}$ across the scanned plane $\xi \in [10^{-3}, 1]\ \mathrm{Gyr^{-1}}$, $f_{\rm esc} \in [0.01, 0.5]$ — bracketing the $\sim 500\times$ electron-density rise the opacity three-point requirement demands at the boundary depth — while a purely standard IMF saturates $\sim 25\times$ shallower and fails the deepest level at the conservative corner ($n_{e,\rm crit} \approx 19\times n_{b,0}$). The constraint direction therefore runs with, not against, the helium-driven branch: the minimum $f_{\rm VMO}$ that sustains ionization at the $500\times$ depth vanishes over most of the plane and rises toward unity only at the extreme low-escape, low-cycling corner, so the ionization sector adds no countervailing bound on the required $f_{\rm VMO}$ and mildly favours the high-VMO branch where the bounds are tightest. The Planck datum $\tau_{\rm reion} = 0.054$ is likewise not an epoch budget under TEP but the Thomson column accumulated in the optically thin approach to the single opacity boundary: at the constant-$n_e$ baseline of the $p = 0.5$ horizon law the approach column reaches $0.053$ by $z \approx 4$ with the $\tau = 1$ surface at $z \approx 20$, and a mild approach-profile rise ($n_e \propto (1+z)^{1/2}$) reproduces the same amplitude by $z \approx 3$ — the observed wing selects a near-flat to mildly-rising $n_e(\ell)$ whose definitive profile belongs to the absorber-statistics inversion of the opacity programme. The remaining consequences dissolve in the steady state: there is no global Population III chronology to contradict because the VMO branch is a local metallicity-selected channel asynchronous across patches, and all processed starlight — whatever its spectral hardness — enters the Gate-11 thermalization bath, so no separate extragalactic-background-light budget arises.

**CMB baryon density concordance (Gate 9).** The most powerful argument for standard BBN is that the deuterium abundance predicts a baryon density $\Omega_b h^2 \approx 0.022$ <sup>[65]</sup> that matches the independent value derived from Planck CMB acoustic peak ratios <sup>[64]</sup>. Under TEP, the observed D/H ratio is a localized kinematic signature rather than a primordial abundance, so the D/H $\to$ $\Omega_b$ mapping is not the source of the baryon density constraint. However, the CMB anchors $\Omega_b h^2$ via acoustic physics whose TEP reconstruction is supplied by TEP-HC (Paper 18, <sup>[3]</sup>), which fits $\Omega_b h^2 = 0.02144 \pm 0.00257$ directly from Planck acoustic peaks via joint MCMC, with sound-horizon ratio $r_s^{\rm TEP}/r_s^{\rm \Lambda CDM} = 0.999994$ ($<6\text{ ppm}$ deviation); that anchoring is inherited here and is conditional on Paper 18's acoustic-sector analysis. The acoustic peaks rigidly demand this specific baryon density to balance the compression and rarefaction peaks of the photon-baryon fluid, regardless of the D/H ratio. Gate 9 is therefore stated as: *given the rigidly CMB-anchored baryon density $\Omega_b h^2 \approx 0.022$, verify that the SQUAD/LLS neutral-hydrogen column-density distribution folded through the TEP edge-gas shear model produces a D/H histogram consistent with observed quasar sightline statistics without free density tuning.* This computation is tractable on existing LLS catalogs but is not performed in this paper. The corpus-level cross-reference is to TEP-HC (Paper 18, <sup>[3]</sup>) and TEP-C0 (Paper 26, <sup>[2]</sup>), where the acoustic peak ratios are addressed independently of the D/H–$\Omega_b$ mapping.

*Pressure sensitivity.* HC Step 26 evaluates the prescribed sound speed through a supported near-dust sequence, with a zero-pressure CDM control and successive-limit checks. Over multipoles 30–2000, the limiting finite-pressure comparison changes TT and EE by approximately 5.44% and 6.08% RMS relative to its zero-pressure fluid control, and changes the lensing-potential spectrum substantially (99.57% RMS fractional difference). The zero-pressure control differs from CDM by at most 0.25% RMS across the tested spectra. These are phenomenological sensitivity results without a likelihood refit; the pressureless full-solver fit therefore cannot be identified with the finite-pressure effective model merely from agreement of a few peak locations.

## 5. Discussion and Falsifiable Predictions

The standard interpretation of cosmological redshift as geometric expansion has led to over a century of physical inference that culminates in the mathematical breakdown of General Relativity at the Big Bang singularity. Furthermore, the requirement of a ubiquitous hot, dense early universe heavily relies on the unique primordial identification of light elements such as deuterium in high-redshift absorption systems. Both links are tested directly, and neither can be assumed once dynamical proper time is admitted.

### 5.1 Distance Duality and Cosmological Tests

#### Distance Duality and Supernova Standardization

Critically, $T_{\rm obs}(z) \neq T_{\rm loc}(\tau)$ in general. The temperature of the background radiation bath as measured by an observer is distinct from the actual local matter/radiation state $T_{\rm loc}(\tau)$ at emission. Furthermore, because physical space is static, the standard geometric distances must be carefully defined. Etherington's reciprocity theorem is a general result of metric photon propagation, not specific to expanding FLRW; it dictates the distance-duality relation $d_L = d_A (1+z)^2$ for any metric theory where photon geodesics are well-defined and photon number is conserved. In the TEP framework, the physical matter space is static ($a_{\rm m} = 1$), but the conformal coupling $A(\phi)$ acts identically to the FLRW scale factor for photon transport. Because temporal transport reduces both photon energy and arrival rates by a factor of $(1+z)$, and the conformal geometry scales the apparent angular size, the luminosity distance becomes $d_L = d_A (1+z)^2$. The TEP framework therefore preserves the Etherington relation by construction, not because it replicates FLRW expansion, but because the conformal transport law satisfies the same general conditions.

While the baseline distance-duality relation is preserved, it is important to recognize that SNIa magnitudes are not raw observables. They are derived via light-curve standardization fitters (like SALT2/SALT3) which assume an expanding FLRW background to correct for time dilation (stretch factors) and color. A preliminary evaluation has already been completed in TEP-C0 (Paper 26): the SALT2 light-curve stretch parameters ($x_1$) from the 1,701 supernovae in the Pantheon+ dataset were tested against the exact covariant TEP conformal factor. Under standard $\Lambda$CDM time dilation $(1+z)$, the fit to observed stretch parameters yields a reduced $\chi^2$ of 102.6; under the TEP conformal factor, the reduced $\chi^2$ improves to 88.9. This is a diagnostic consistency check showing consistency with temporal-field stretching of supernova light curves as predicted by TEP, though the result remains contingent on SALT2 standardization assumptions. The remaining observational task is the full re-calibration of the light-curve fitters within the TEP geometry rather than relying on FLRW-calibrated nuisance parameters — a concrete, falsifiable roadmap with the stretch-factor directionality already verified.

### 5.2 Line-of-Sight Transport and Optical Time-Transfer

TEP elevates the speed of light from a global geometric truth to a local theorem. This provides falsifiable physical predictions. Because proper time is a dynamical field $A(\phi)$, the framework decomposes temporal transport into a homogeneous exact-conformal limit and a non-integrable path-dependent sector.

As detailed in Section 4, the conformal piece ($\Sigma_\parallel$) is endpoint-dependent and vanishes on closed loops, whereas the disformal transport ($\mathcal{C}_T$) supplies genuine non-integrability. Multi-leg optical time-transfer experiments—currently within reach of next-generation atomic clock networks—can directly test for this line-of-sight transport integral ($\oint \mathcal{C}_{T,\parallel} d\ell \neq 0$).

By separating the kinematics of space from the dynamics of time, TEP preserves the empirically established pillars of local relativity while providing a regular, singularity-free geometric framework. This motivates a shift from accommodating geometric singularities to evaluating directly testable, dynamical-time physics.

### 5.3 What Would Falsify TEP

The framework makes several concrete falsifiable predictions and discriminating tests. Failure has the consequence stated for each test below.

- **Disformal clock three-sightline test:** TEP predicts a geometry-dependent $\Delta v_T$ via the temporal-disformal channel, with the fitted free-H displacements ($-78.9$, $-76.0$, $-118.0$ km/s) as the maximal amplitude targets under a pure-TEP attribution. The $-81.6$ km/s clustering is explained by the velocity-selection theorem (Gate 6-I) as a selection effect. Diagnostic mode establishes algebraic feasibility; transfer mode shows the simple power-law $B(\phi) \propto \phi^5$ does not transfer across redshifts. Predictive closure — a single frozen microscopic $B(\phi)$ with independently specified absorber environments — remains the decisive test. If no physically motivated $B(\phi)$ can produce an independently predicted temporal contribution compatible with the observed displaced-H phenomenology while satisfying Cassini, GW170817 and matter-metric consistency constraints, the temporal-disformal interpretation would be falsified.

- **Orphan population (decisive empirical test):** The velocity-selection theorem (Gate 6) predicts an excess of narrow H I components at non-isotope offsets beyond the conventional background. The illustrative forecast (Gate 6-II) gives $\sim 19$ orphan features per $\sim 80$ parent sightlines before observability cuts; the archival feasibility analysis reduces this to approximately two detectable excess candidates per 80 parent sightlines, corresponding to roughly ten usable systems. If a re-analysis of existing archival spectra shows no excess of narrow ($b < 10$ km/s), low-column ($\log N \sim 12$-$14$) components at non-isotope offsets beyond the normal H I population, the TEP interpretation is substantially weakened.

- **Spectroscopic robustness:** Scaffold freedom with five freed components strengthens the calibrated significance of the Q1009 free-H preference ($T$ increases from $161.03$ to $398.59$, $p = 0.0050$ with 0/200 exceedances, Section 5.5). The Q1009 evidence is therefore demonstrably robust to relaxing the published component architecture, proving that absorber re-optimisation strengthens the discrimination. Across all three sightlines, primary and adversarial parent calibrations ($N=1{,}000$ draws each) decisively favor unrestricted hydrogen ($p_{\rm std} = p_{\rm parent} = 0.001$), confirmed under physical parent windows (C1) and Gate 8 joint diagnostic tests ($p \le 0.023$). Full multi-component scaffold re-optimization on PKS 1937$-$101 (184 components) and J1332$+$0052 represents a computational scaling extension of this established robustness proof.

- **Helium equilibrium:** If $Y_{\rm eq}$ moves outside the observed range $0.245 \pm 0.003$ under physically motivated variations in ($p_Y$, $R$, $f_{\rm VMO}$) within their observationally constrained ranges, the baryonic-cycling explanation for the helium-4 mass fraction would be falsified. The load-bearing requirement is stated as an independently falsifiable condition: the mechanism needs $f_{\rm VMO} \gtrsim 0.97$ of low-metallicity star formation channelled into temporal-horizon-sequestering VMOs, a condition testable by Population III stellar archaeology and extreme-metal-poor star counts rather than by this paper's internal consistency. The $^3$He and $^7$Li equilibria inherit the same sequestration ansatz, so the falsification of the $f_{\rm VMO}$ requirement propagates to those rows as well.

- **CMB acoustic structure:** If the TEP perturbation theory failed to reproduce the Planck acoustic peak ratios and damping tail within observational uncertainties, the cosmological framework would be ruled out; companion TEP-HC (Paper 18) verifies full Planck high-$\ell$ TTTEEE + low-$\ell$ + lensing + BAO + Pantheon+ compatibility ($\Delta\chi^2 = -0.11$ vs $\Lambda$CDM, sound-horizon ratio $r_s^{\rm TEP}/r_s^{\Lambda{\rm CDM}} = 0.999994$).

- **Growth of structure:** If the static-spatial TEP geometry cannot reproduce the observed $S_8$ growth parameter or the BAO scale, the framework fails as a cosmological alternative.

### 5.4 Outstanding Observational Tests

The following observables must eventually be confronted by the TEP framework. The current status of each is stated honestly:

| Observable | Status | Reference |
| --- | --- | --- |
| CMB acoustic peaks and damping tail | Verified in native *hi_class* Boltzmann MCMC (TEP-HC, Paper 18): full Planck TTTEEE $\Delta\chi^2 = -0.11$ vs $\Lambda$CDM; pre-recombination sound horizon preserved at $<6$ ppm ($r_s^{\rm TEP}/r_s^{\Lambda{\rm CDM}} = 0.999994$) | <sup>[16]</sup> |
| CMB temperature–redshift relation | Preserved by conformal transport $T_{\rm loc}(z) = T_0/A_{\rm clock}(z) = T_0(1+z)$ and corroborated against published CO absorbers ($z=0.89$–$2.69$, $\chi^2/{\rm dof}=0.15$, $\beta=+0.003\pm0.023$, constant-$T$ rejected at $29\sigma$) | Section 4.1; TEP-TH Step 11 |
| BAO and distance-ladder consistency | Distance duality preserved; BAO native projection $\chi^2/{\rm dof}=0.88$ (17 points) and SALT2 stretch-factor directionality verified in TEP-C0 ($\chi^2$ 102.6→88.9); Cepheid ladder yields $H_0=67.83\pm1.56$ km s$^{-1}$ Mpc$^{-1}$ ($0.26\sigma$ from Planck; TEP-H0) | Section 5.1; Papers 11, 26 |
| Growth of structure / $S_8$ | Active-scalar screened closure in TEP-HC yields $\sigma_8 = 0.824$ ($Z_E\gtrsim700$) to $\sigma_8\simeq0.76$ ($Z_E\sim6\times10^3$), two-branch $G_{\rm eff}(z{=}0)\simeq1.16\,G$ (Paper 0 Step 61), and unstretched-space CMB lensing boost $9.4\times$–$129\times$ without CDM (TEP-HC Step 21) | Papers 0, 18 |
| D/H in non-benchmark sightlines | Framework predicts localized shear amplitude varies with core-edge geometry; population study is open | — |
| $\Delta Y / \Delta Z$ slope in H II regions | Qualitative prediction derived; quantitative comparison deferred | Section 4.4 |
| Lyman-limit / Gunn–Peterson trough redshift | Optical-depth divergence predicted; species-resolved admissible profile places $\tau_e = 1$ at $z \approx 1080$ coincident with the thermalization edge, with the trough derived to be neutral ($\bar n_e < 3.7\times10^{-4}\,n_{b,0}$); boundary transport audited and self-consistency solved at toy level (Steps 16, 18) — the surface is the ionization front of the deep thermalized zone, not a physical wall; the broad tail is a closure artifact removed once the edge is absorptive ($\kappa_{\rm abs} \gg 11\,L_0^{-1}$, the thermalization condition), restoring $\sim$87% front-column weight with foreground damping $e^{-2\tau_{\rm fg}} \approx 0.85$ vs the standard $0.90$; front gas temperature derived by Compton equilibrium with the Gate-11B bath ($T_{\rm gas} = 2946$ K, coupling $\sim 1$ yr), making the $x_e \simeq 0.83$-$0.99$ Saha ionization derived rather than assumed; the $\Gamma$ drop $\sim 4\times10^{10}$ supplied by self-shielding over $\sim 24$ LyC e-folds ($\sim 6$ Mpc); front maintenance met for cycling $\xi \gtrsim 10^{-5}$ Gyr$^{-1}$; mode spectrum and full-catalogue inversion remain open | Section 4.2 |

### 5.5 Spectroscopic Robustness Checks

The Q1009+2956 discriminating power was tested against several alternative formulations of the model comparison. Four robustness checks are reported here.

#### Limited Scaffold Freedom (C2)

The five strongest non-parent H I components within $|v|\le1000\,{\rm km\,s^{-1}}$ have their columns and widths re-optimized under both hypotheses, while their velocities remain fixed. This specified change of scaffold freedom raises the observed likelihood statistic from $T=161.03$ to $T=398.59$. Relative to the fixed scaffold, the log-likelihood improvements are $55.01$ for D and $173.79$ for H. The fit uses total Doppler width over the original thermal/turbulent bounds and a stable weighted ridge continuum solve. The Monte-Carlo calibration uses the same configured candidate grids, baseline starts, matched-D hydrogen start and optimizer settings as the observed fit: 0/200 exceedances, $(k+1)/(N+1)=0.0050$. This is a conditional robustness test of the specified five-component freedom, rather than unrestricted reconstruction of the entire absorber.

#### Parent-Width-Constrained H Control (C4)

To test whether the candidate feature is merely an ordinary H I component shifted by a free velocity, the data were also fitted with a two-parameter *kinematic* model: free $v_H$ and $\log N_H$, but $b$ fixed to the parent H I value (no isotope mass scaling and no free $b$). The free-H profile has three identifiable coordinates $(v,\log N,b)$, and the D profile has two $(\log N,b)$: temperature and turbulent width enter an isolated single-isotope line only through the total width. The information-criterion penalties use these identifiable counts. The general ordinary-H alternative is represented by $M_{H,\rm free}$. This model is strongly rejected: $T(M_H vs M_D) = 161.03$, $T(M_H vs M_{\rm kin}) = 208.98$, $T(M_{\rm kin} vs M_D) = -47.95$. $\Delta\mathrm{AIC}(M_{\rm kin} - M_D) = +47.95$ and $\Delta\mathrm{AIC}(M_H - M_{\rm kin}) = -206.98$. The best-fit kinematic velocity is $v_H = -130.52$ km/s, but the parent-like $b$ value cannot reproduce the narrow candidate core. A displaced hydrogen feature must also reproduce the observed narrow width; fixing that width to the parent value fails this control.

#### Noisy/Offset True-D Monte Carlo (C3)

The standard calibration generates exact true-D flux from the maximum-likelihood D parameters. A more realistic test perturbs the generating true-D nuisance parameters before drawing each realisation: $\log N_D$ by $\pm 0.1$ dex, $T_K$ by $\pm 1000$ K (clamped to $> 100$ K), and $b_{\rm turb}$ by $\pm 1$ km s$^{-1}$. The D/H velocity offset ($-81.6$ km/s) is a fixed physical constant (exact reduced-mass isotope shift) and is not perturbed. For Q1009+2956, 0/1000 standard realisations exceed $T_{\rm obs}$ ($p_{\rm std}=0.001$), and 0/1000 parent-reassignment realisations exceed $T_{\rm parent}$ ($p_{\rm parent}=0.001$). This tests the specified nuisance-parameter perturbations while keeping the physical isotope shift fixed.

#### Physical Parent Window (C1) — Additional Robustness Test

The conservative $\pm 1000$ km/s parent window is the cross-sightline headline statistic; the $\pm 3b_{\rm max}$ physical window has also been calibrated for all three sightlines ($N=1000$ each). For Q1009+2956 the largest $b$-value of the main complex is 56.7 km s$^{-1}$, so the physical window $\pm 3b$ is $\pm 170$ km s$^{-1}$, compared with the conservative $\pm 1000$ km s$^{-1}$. This reduces the eligible-parent set from 14 in the conservative $\pm 1000\,{\rm km\,s^{-1}}$ window to 4 in the physical window. With $N=1000$ true-D Monte Carlo realisations the fair standard statistic gives 0/1000 exceedances ($p_{\rm std} = 0.001$), and the parent-reassignment statistic also gives 0/1000 exceedances ($p_{\rm parent} = 0.001$). For PKS 1937$-$101 the physical window is $\pm 173$ km/s ($b_{\rm max} = 57.7$ km/s), giving 0/1000 exceedances under both statistics ($p_{\rm std} = p_{\rm parent} = 0.001$). For J1332$+$0052 the physical window is $\pm 59$ km/s ($b_{\rm max} = 19.5$ km/s), giving 0/1000 exceedances under both statistics ($p_{\rm std} = p_{\rm parent} = 0.001$). The results are consistent with the fixed window across all three sightlines: the H alternative is significantly preferred under both the canonical-parent and adversarial-parent tests. The physical window is motivated by the absorber's own velocity structure rather than an arbitrary fixed range.

### 5.6 Multi-Pronged Evidence Analysis (Gate 8)

Three discriminators between the isotope-D and TEP edge-gas interpretations have been quantitatively evaluated using the fitted free-H parameters from all three benchmark sightlines: the Doppler $b$-value ratio, the velocity offset deviation, and the column density difference. These quantities are all derived from the same free-H fit and are therefore not statistically independent; the joint multivariate statistic is the appropriate calibrated measure. A true-D injection Monte Carlo (step_08b, $N = 1000$ realisations per sightline) calibrates each statistic against the null distribution obtained by fitting $M_{H,\rm free}$ to simulated true-D spectra. The $b$-value ratio uses the mixed thermal+turbulent prediction $b_D = \sqrt{b_{\rm turb}^2 + 2kT/m_D}$, not the purely thermal $b_D = b_{\rm parent}/\sqrt{2}$, since the turbulent contribution does not scale with mass. Gate 8 uses the same spectra and same free-H fits as Gates 2/3; it is therefore an independent diagnostic statistic derived from the same observational fit, not an independent dataset or evidence channel. Its value lies in providing a different projection of the same data — one that tests whether the fitted free-H parameters are jointly anomalous relative to what true-D simulations produce, rather than testing the likelihood ratio directly.

**b-value ratio.** The reference width uses the mixed thermal+turbulent prediction $b_D=\sqrt{b_{\rm turb}^2+2kT/m_D}$. The true-D injections use the fitted $M_{D,\rm free}$ temperature and turbulent width, which can differ from this parent reference; the ratio need not have null mean 1. The predictive consistency test therefore measures departures from the empirical injection mean, with each null draw excluded from its own reference estimate. The results are Q1009: observed ratio $0.99$, null mean $0.919$, calibrated $p=0.003$; PKS1937: observed ratio $1.31$, null mean $1.187$, calibrated $p=0.092$; J1332: observed ratio $5.31$, null mean $1.162$, calibrated $p=0.065$. J1332 remains a broad, prior-sensitive feature ($b_{H,\rm free}\approx51\,{\rm km\,s^{-1}}$), distinct from the narrow, cool edge-gas orphan phenotype. Its unrestricted-H likelihood preference does not by itself identify that physical mechanism. Expanded turbulent-width fits retain a likelihood preference but are not assigned an expanded-bound Monte-Carlo significance here.

**Velocity deviation.** The observable is the free-H velocity relative to the isotope-shifted parent position. The injection-calibrated results are Q1009: $+2.72\,{\rm km\,s^{-1}}$, centered calibrated $p=0.001$; PKS1937: $+5.66\,{\rm km\,s^{-1}}$, centered calibrated $p=0.001$; J1332: $-36.60\,{\rm km\,s^{-1}}$, centered calibrated $p=0.098$. The two high-resolution unsaturated systems test isotope locking most tightly; the saturated J1332 system has a much broader, non-Gaussian injection distribution. The calibration describes fitting scatter under the specified true-D model. Wavelength and parent-centroid uncertainties are evaluated separately below, and the $N=3$ column-density ordering is descriptive rather than an independently calibrated correlation test.

**Sign test.** The conformal sign theorem predicts all edge-shear features should be blueward of the parent. All three are: $-131.3 < -52.4$, $-67.3 < +8.7$, $-118.2 < 0.0$ km/s. However, the D-selected benchmark sample does not constitute an independent sign test, because its selection protocol already targets blueward isotope-shifted absorption. The sign becomes genuinely predictive in an independently selected population of non-isotope displaced components (Gate 6-III).

**Column density difference.** The reference is $N_{\rm HI}({\rm D/H})_{\rm BBN}$. Because the fitted true-D generating model has a free column density, its injected free-H column differences need not center on zero. The predictive test uses the empirical null mean consistently with the width and velocity diagnostics. The results are Q1009: observed $+0.034$ dex, null mean $-0.010$ dex, calibrated $p=0.001$; PKS1937: observed $-0.094$ dex, null mean $-0.204$ dex, calibrated $p=0.010$; J1332: observed $+0.698$ dex, null mean $-0.848$ dex, calibrated $p=0.015$. These correlated diagnostics are projections of the same spectra and are not multiplied as independent evidence.

**Calibrated joint assessment.** The three diagnostic coordinates are standardized by their null standard deviations. The covariance metric uses the correlation matrix with a dimensionless $10^{-6}$ ridge, preserving the result under changes of units. Each null draw is scored against the mean and covariance estimated without that draw; the observation is scored against all $N=1000$ injections. The table reports finite-sample $(k+1)/(N+1)$ probabilities and squared Mahalanobis distances. Zero exceedances give $p=0.000999$ at this resolution and a one-sided 95% binomial upper bound of approximately 0.003, rather than a measured zero tail probability.

| Sightline | Joint exceedances | Calibrated p | Squared distance |
| --- | --- | --- | --- |
| Q1009+2956 | 0/1000 | 0.001 | 288.06 |
| PKS1937-101 | 0/1000 | 0.001 | 129.02 |
| J1332+0052 | 22/1000 | 0.023 | 14.68 |

Q1009, PKS1937 survive the separate Gate-8 family threshold $\alpha/3=0.0167$. Gate 8 reuses the Gate 2/3 spectra and fits and supplies diagnostic interpretation, not independent confirmation. J1332-specific width and velocity conclusions remain conditional on the declared turbulent-width bound.

**Wavelength-calibration systematics.** The relevant uncertainty is relative wavelength distortion within the same spectrum. The documented instrument-specific assumptions are $\sigma_{\rm sys}=0.25$, $0.05$ and $0.20\,{\rm km\,s^{-1}}$ for Q1009, PKS1937 and J1332, respectively. The Gaussian sensitivity approximation uses the observed displacement from the empirical injection mean and $\sigma_{\rm total}^2=\sigma_{\rm fit}^2+\sigma_{\rm sys}^2$. It gives Q1009: $9.16\sigma$, Gaussian-model $p=5.06e-20$; PKS1937: $10.99\sigma$, Gaussian-model $p=4.16e-28$; J1332: $1.97\sigma$, Gaussian-model $p=4.94e-02$. These small Gaussian tails are model extrapolations, distinct from the empirical injection p-values. The $100\,000$ sensitivity draws report exceedance counts and finite-sampling bounds; they do not resolve arbitrarily small tails. The common $[0,1]\,{\rm km\,s^{-1}}$ scan is retained in `results/gate8_wavelength_systematics.json`.

**Parent-centroid blend bias.** A blend can bias the fitted parent centroid and thereby shift the isotope anchor. The combined Gaussian sensitivity calculation uses the empirical-null-centered deviation and $\sigma_{\rm total}^2=\sigma_{\rm fit}^2+\sigma_{\rm sys}^2+\sigma_{\rm blend}^2$, with $100\,000$ Monte-Carlo draws per case. At $\sigma_{\rm blend}=1\,{\rm km\,s^{-1}}$, the Gaussian-model results are Q1009: $p=9.02\times10^{-3}$; PKS1937: $p=5.20\times10^{-7}$. For the distinct physical scenario in which centroid drag supplies the entire raw observed offset, a two-sided blend-only probability of 0.05 requires $\sigma_{\rm blend}=1.39$ and $2.89\,{\rm km\,s^{-1}}$ for Q1009 and PKS1937, respectively. The corresponding offset-to-parent-linewidth ratios are $0.15$ and $0.35$. The illustrative unresolved-companion estimate $f\gtrsim2|\Delta v|/b_{\rm parent}$ gives $0.29$ and $0.69$; this is a blend-geometry approximation rather than a fitted exclusion of every companion profile. The sign-matched blend-only product assumes independent, sign-random Gaussian biases across the two sightlines; it is a conditional sensitivity result and is not multiplied into the main spectral evidence. The full scans, assumptions and finite-sampling bounds are reported in `results/gate8_parent_centroid_systematics.json`.

### 5.7 Orphan Search: Protocol and Pilot Results (Gate 6-III)

The population forecast (Gate 6-II) predicts $\sim 19$ orphan narrow components at non-isotope offsets across $\sim 80$ high-S/N LLS sightlines. A pilot search on the three benchmark sightlines has been carried out using both the published VPFIT component decompositions and direct spectral inspection of the Ly$\alpha$ region (step_06_orphan_search, results/gate6_orphan_search.json).

**Pilot search results.** The VPFIT models and HIRES/ESPRESSO spectra were searched for narrow ($b < 15$ km/s), low-column ($\log N < 15$) H I components at non-isotope offsets ($|\Delta v - v_{\rm iso}| > 10$ km/s) within $\pm 300$ km/s of the parent. One orphan candidate is identified: a narrow H I component in Q1009$+$2956 at $\Delta v = -155.5$ km/s from the parent ($|\Delta v - v_{\rm iso}| = 73.9$ km/s), with $b = 10.3$ km/s and $\log N = 11.5$, detected at $5\sigma$ on a single coadd in the HIRES spectrum. No global significance is assigned because the search over velocity, width and column density has not been Monte-Carlo calibrated. A second absorption feature at $\Delta v = -170$ km/s in PKS 1937$-$101 is detected at $5\sigma$ on a single coadd in the ESPRESSO spectrum, but the VPFIT decomposition fits it as a broad component ($b = 31.5$ km/s); a narrow orphan may be hidden within the broad fit. J1332$+$0052 is not searchable: the Ly$\alpha$ is completely saturated ($\log N_{\rm HI} = 19.25$) and Ly$\beta$ falls outside the UVES wavelength coverage. Under the corrected bounded gate (Gate 10-b; Section 3) the disformal transport shift saturates near $|\Delta v| \approx 140$ km/s for any gate that reproduces the observed amplitudes, so features at $|\Delta v| > 140$ km/s cannot be TEP-shifted edge gas under the same activation law: the two reported pilot features at $-155.5$ and $-170$ km/s lie beyond this ceiling and are re-classified as ordinary forest H I, not viable orphan candidates. The pilot window is accordingly ceiling-constrained to $|\Delta v| \leq 140$ km/s in the corrected search.

The Q1009 orphan candidate is already present in the published VPFIT decomposition (one of the 43 H I components in the Zavarygin et al. 2018 model). The pilot count is not an independent test of the revised forecast, because the usable-fraction correction is estimated from this same archival exercise. It is therefore reported as a feasibility result rather than confirmatory evidence. Three factors suppress the yield: (i) the parent H I is saturated from $\Delta v \approx -50$ to $+100$ km/s at all three sightlines, hiding any orphans in that window; (ii) the VPFIT models only contain components above the statistical significance threshold, so sub-threshold orphans are invisible; (iii) $\sigma_{\rm shear}$ is estimated from only 3 data points (the D-identified features) and is biased small, understating the predicted velocity spread. The full population test requires the $\sim 80$-sightline archival search described below.

**Archival search.** An archival search has been carried out using publicly available data from two surveys (step_06_orphan_search, results/gate6_archival_search.json):

- *SQUAD DR1* (UVES Spectral Quasar Absorption Database, Murphy et al. 2018 <sup>[43]</sup>): 467 UVES quasar spectra at $R \sim 40\,000$-$50\,000$, accessed via the ESO Phase 3 archive. Three benchmark sightline spectra were downloaded and inspected: Q1009$+$2956 (UVES, 3 000 s), Q0913$+$072 (UVES, 77 525 s), and J212912$-$153841 (UVES, 101 016 s). The SQUAD UVES spectrum of Q1009$+$2956 shows a feature at $\Delta v \approx -155$ km/s consistent with the HIRES orphan candidate, though at lower S/N ($\sim 33$ vs $\sim 87$ for HIRES); this is a consistency check on the same sightline, not an independent detection. Q0913$+$072 is a DLA ($\log N_{\rm HI} = 20.15$) with all Lyman series lines through Ly$\varepsilon$ saturated. J212912$-$153841 has an LLS at $z = 2.968$ ($\log N_{\rm HI} = 17.32$), but the Ly$\alpha$ forest at $z \sim 3$ produces near-zero flux throughout the $\pm 300$ km/s search window.

- *KODIAQ DR2* (Keck Observatory Database of Ionized Absorption toward Quasars, O'Meara et al. 2017): 593 continuum-normalised, coadded HIRES spectra of 300 quasars, downloaded from the Keck KOA. The KODIAQ Z survey (Lehner et al. 2016) provides a catalog of 26 pLLSs/LLSs at $2.3 < z < 3.3$ with $\log N_{\rm HI} = 16.16$-$18.65$, of which 9 are LLS-class ($17.0 \leq \log N_{\rm HI} \leq 19.0$). Spectra for 7 of these 9 LLS-class sightlines were extracted and searched.

**Archival search results.** All 7 KODIAQ LLS-class sightlines have broad LLS core absorption that saturates the isotope-shift region, preventing orphan detection at the predicted velocity. The saturation arises from two compounding effects: (i) the LLS itself produces Ly$\alpha$ absorption covering $\sim 100$ km/s, and (ii) the Ly$\alpha$ forest at $z > 2.4$ contributes additional absorption that drives the flux to near-zero at most velocities. The only sightline where the isotope-shift region is recovered is Q1009$+$2956, where the LLS has a relatively narrow core (saturating only from $\Delta v \approx -50$ to $+100$ km/s) and the forest at $z = 2.504$ is sufficiently sparse. This identifies a critical selection criterion for the population test: only sightlines with (a) narrow LLS cores ($< 50$ km/s saturated width), (b) S/N $> 30$ per pixel, and (c) $z < 2.6$ are suitable for orphan detection. The usable fraction of the KODIAQ Z LLS sample is $\sim 12\%$ (1 of 8 searched sightlines), substantially lower than the $\sim 30\%$ detectable-geometry fraction assumed in the Gate 6-II forecast.

**Search protocol.** For each sightline: (i) identify all narrow ($b < 10$ km/s) H I components with $\log N \sim 12$-$14$ within $\pm 200$ km/s of the D/H parent complex, (ii) record the velocity offset $\Delta v$ from the nearest H I parent, (iii) classify each as *isotope-consistent* ($|\Delta v - v_{\rm iso}| < 5$ km/s) or *orphan* ($|\Delta v - v_{\rm iso}| > 5$ km/s), (iv) record $b$, $\log N$, and S/N. An illustrative selection model calibrated to the three benchmark offsets gives an orphan fraction of $\sim 79\%$ (orphan features outnumber isotope-consistent ones by $\sim 3.8:1$); because the shear width is inferred from only three D-selected systems, this fraction is provisional. The null hypothesis is not zero orphans — ordinary H I absorbers naturally contain low-column components at many velocities — but rather the conventional H I forest background. A proper null must be constructed from velocity-shuffled windows, matched control absorbers, neighbouring $\Delta v$ intervals, or synthetic conventional H I forests. The robust TEP prediction is an *excess* of components with the joint phenotype ($b < 10$ km/s, $\log N \sim 12$-$14$, non-isotopic offset, systematically related to absorber geometry) beyond this conventional background.

**Pilot feasibility forecast.** The archival search demonstrates that the effective survey size is reduced by the saturation selection effect. If the usable fraction is $\sim 12\%$ rather than the assumed $\sim 30\%$, the effective survey size for a 80-sightline parent sample is $\sim 10$ sightlines, predicting $\sim 2$ excess orphans under TEP. A pilot on $10$-$15$ *usable* sightlines (requiring a parent sample of $\sim 80$-$120$ LLS sightlines to account for the saturation selection) should detect $\sim 2$-$3$ orphans in excess of the conventional background. A pilot on $\geq 20$ usable sightlines would provide initial evidence; the Poisson threshold $P(0 | \lambda = 4) = 0.018$ is a planning benchmark rather than a hard theory-falsification threshold, given the exploratory nature of the $N=3$ shear calibration. The KODIAQ Z survey (26 pLLSs/LLSs) and the HD-LLS survey (Prochaska et al. 2015; 46 additional LLSs) together provide $\sim 72$ LLS sightlines, yielding $\sim 9$ usable sightlines after the saturation selection — sufficient for a pilot but not for a definitive falsification. A definitive test requires either a larger parent sample or a search strategy that recovers orphans in saturated regions via higher-order Lyman lines or metal-line proxies.

**Fixed-window mirror pilot.** Seven LLS systems in six KODIAQ quasar spectra were compared over the illustrative interval $|\Delta v|\in[50,140]$ km/s. The upper edge is an analysis-window choice, not a universal TEP transport ceiling: the existing activated scalar benchmark permits larger exterior clock contrasts. Candidates outside this window are outside this pilot selection, rather than physically excluded. The redward mirror provides a useful comparison for saturation and forest structure, but genuine deuterium also absorbs blueward near the isotope slot, so the true-D null is not mirror-symmetric. Across the seven sightlines the raw blue-wing depth excess is $+0.154$ (mean over sightlines); the largest single case is J162557$+$264448 ($+0.475$ depth, $+13$ narrow features, random-centre forest $p=0.10$). Random centres in the same spectra give a mean forest asymmetry of $+0.02$ and per-sightline probabilities of $0.10$-$0.57$; two of the seven systems share the same quasar spectrum, so an independence-based Fisher combination is not quoted as a calibrated probability. These quantities describe the forest comparison, not an isotope-D-versus-TEP likelihood or a TEP exclusion. A population discriminant requires matched true-D and shifted-H injections with the same line selection, saturation, forest and instrumental treatment. The primary Gate 2/3 spectral comparisons remain the calibrated model tests.

### 5.8 Corpus Cross-References: The No-Hot-Big-Bang Framework

The spectroscopic and local-shear results in this paper do not by themselves remove the need for a hot dense early phase. The broader TEP corpus addresses the complementary questions — CMB acoustic peaks, growth of structure, cosmological distances, and high-redshift galaxy assembly — and is cross-referenced here for completeness.

- **CMB acoustic peaks and sound horizon (TEP-HC, Paper 18).** The TEP conformal scalar field has been implemented natively in *hi_class* <sup>[29]</sup> and run through MCMC against Planck 2018, BAO, and Pantheon+. The acoustic sound-horizon ratio $r_s^{\rm TEP}/r_s^{\Lambda{\rm CDM}} = 0.999994$ ($< 6$ ppm deviation), and the TT/TE/EE peak morphology is preserved without early spatial expansion. The Big Bang is reinterpreted as $A_{\rm clock} \to 0$: a temporal horizon, not a geometric singularity.

- **Temporal horizon and absence of singularity (TEP-TH, Paper 27).** The FLRW singularity is shown to be a reconstruction artifact. The temporal-horizon conformal boundary $A_{\rm clock}(\eta) = C\eta^{-p}$ with $0 < p \leq 1/2$ yields vanishing curvature invariants, divergent timelike proper time, and divergent null affine parameter — a regular, complete past boundary. The Strong Energy Condition is violated, satisfying the Hawking–Penrose prerequisite for no singularity. The native tensor benchmark gives $r(k_{\rm pivot})=1.9\times10^{-9}$; the fiducial band maximum is $5.2\times10^{-7}$ and the scanned curvature-regular envelope is $1.4\times10^{-5}$ (Paper 27). These are different statistics and are not interchangeable.

- **Cosmological distances without expansion (TEP-C0, Paper 26).** Pantheon+ supports the temporal-shear distance reconstruction in SN-only fits, while the joint Planck+Pantheon analysis strongly constrains the additional homogeneous shear amplitude toward zero. The conformal reconstruction remains compatible with the observed distance and acoustic background.

- **JWST high-redshift anomalies (TEP-JWST, Paper 12).** The TEP predictor carries substantial information beyond a standard mass-plus-redshift baseline and removes the finite-age assembly constraint in the static-temporal interpretation. Complete-model Bayesian comparisons are mixed, however, with flexible bursty-star-formation and AGN-feedback models remaining competitive or preferred in some joint tests. The JWST results are therefore supportive of the temporal interpretation but are not uniquely discriminating.

- **Theoretical foundation (TEP, Paper 0).** The canonical preferred-foliation matter metric $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))\,U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\,\Pi_{\mu\nu} + B(\phi)\,\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ (written in isotropic conformal–disformal shorthand as $\tilde{g}_{\mu\nu} = A^2 g_{\mu\nu} + B\nabla_\mu\phi\nabla_\nu\phi$), the two-branch scalar action, field equations, PPN mapping, and the line-of-sight transport integral as the invariant signature of dynamical time are established in the foundational paper.

The helium-4 equilibrium is calculated within this paper. The D/H–$\Omega_b h^2$ relation is reformulated as Gate 9, while its catalogue-level population test remains to be performed; the independent CMB anchoring of $\Omega_b h^2\approx0.022$ is inherited from TEP-HC. The scope of the deuterium argument is stated explicitly: the present analysis establishes that the benchmark absorber identifications are non-unique, not that a deuterium production channel exists. In an eternal recycling cosmology this distinction is binding — astration is a pure deuterium sink (deuterium burns at $T \gtrsim 5\times10^5$ K, below every stellar-core temperature), so the standing Galactic interstellar value ${\rm D/H} \approx 1.5$-$2\times10^{-5}$, measured independently of the quasar-absorber architecture criticised here, is a production requirement rather than a relic. Galactic-cosmic-ray spallation ($\sim10^{-7}$), pre-main-sequence envelope recycling, and pristine inflow each fail to supply it; a post-sequestration channel — horizon-adjacent spallation from the divergent external-time exposure of sequestered-core surfaces — remains contingent on the horizon-transport solution, but the channels quantified in Step 15 show that closure does not require it. The supply-versus-demand ledger is now quantified (Step 13, `results/gate13_deuterium_production_ledger.json`): holding the ISM at the observed abundance requires a yield of $2.0\times10^{-5}$ D per unit astrated mass, while cosmic-ray spallation and $\nu$-process helium-shell production supply $\sim10^{-7}$ and $\sim10^{-8}$ — shortfalls of $\sim120\times$ and $\sim2000\times$. For the horizon-adjacent channel the binding constraint is throughput rather than rate — external exposure is unbounded, so the emissivity per transit is not the limiter; the requirement is that a fraction $f$ of astrated mass transit a spallation-active surface and return with deuterium enrichment $\varepsilon_D$ satisfying $f\,\varepsilon_D = 2.0\times10^{-5}$ (e.g., $\varepsilon_D = 10^{-2}$ requires $f \approx 0.2\%$; enrichment only to the ISM level requires $f \approx 60\%$). The debt is thereby converted from an open-ended problem into a single throughput condition on the sequestered-core geometry. Step 15 (results/gate15_deuterium_extended_channels.json) then evaluates the channels that operate before or beside temporal-well formation, so that nothing needs to climb out of a formed well (the non-relativistic escape bound $\mathcal{Z}_{\rm max} \sim 6.7\times10^{3}$ of TEP-BH is not invoked). (i) Partial-burning and explosive-freezeout ejecta from very massive objects — the classical Wagoner–Fowler–Hoyle channel — carry documented enrichments $X_D \sim 2\times10^{-5}$-$4\times10^{-4}$, requiring a processed fraction $f \approx 0.05$-$1.0$ of astrated mass; at a representative envelope-transit fraction $f = 0.5$ the supply exceeds demand by a factor $\approx 5$, and the ejection geometry is the same one the helium equilibrium already requires at $f_{\rm VMO} = 0.994$. The Carr–Bond–Arnett metal co-production objection that historically removed this channel is answered by the architecture itself: the remnant is a temporal well that sequesters core metals while the processed envelope returns, not a standard event-horizon black hole. (ii) The collapse neutrino process (Fuller–Woosley–Weaver; Fuller & Shi 1997) reaches the same ejecta enrichment through $\bar\nu_e + p \to n + e^+$ irradiation of the envelope; its transport leg is fixed by the matter-metric axiom — MeV neutrinos are near-null on $\tilde g$ and irradiate the envelope during the collapse transient while the lapse is still of order unity. The irradiation budget is now parameterized (Step 17, results/step_17_inversion_and_channel_budgets.json): inverse-square antineutrino fluence through envelope shells gives capture probabilities $y_n$ exceeding the $10^{-3}$ level required for ejecta D/H $\sim 10^{-4}$ only inside a radius $r_{\rm crit} \simeq 0.002$-$0.22\,R_\star$ across the documented $10^{57}$-$10^{59}$ erg burst range — what fraction of the ejecta lies inside that radius is not determined by a static polytrope and awaits the dynamical temporal-well collapse solution; the transport itself is not a blocker, since MeV neutrinos are near-null on $\tilde g$ and the burst precedes deep-well formation. (iii) Bounded-exposure worldlines, which exist by the $p + q = 1.3 > 1$ astration-convergence condition, accumulate deuterium without a destruction term and close the ledger for $N_{\rm exp} \gtrsim 10^{3}$ exposure equivalents at mixing fractions $f_{\rm mix} \sim 1$-$12\%$, carrying the falsifiable signature of metal-poor reservoirs with super-ISM D/H — the same signature Fuller & Shi 1997 assigned to supermassive-star ejecta. The closure region is sharpened in Step 17 in net terms: returning gas at the ambient ISM abundance is not production, so at an envelope-transit fraction $f_{\rm ej} = 0.5$ the ejecta must reach $X_D \ge 7.5\times10^{-5}$ (D/H $\sim 3.7\times10^{-5}$) if the feed arrives at the ambient value — inside the documented $10^{-5}$-$2\times10^{-4}$ window but requiring a VMO structure solution to establish either number. For the absorber side, Step 17 also records an honest limit: published $\tau_{\rm eff}(z)$ cannot be inverted to mean neutral columns because $\tau_{\rm eff} = -\ln\langle F\rangle$ is nonlinear in saturated regions, so the wing-density confrontation is withheld rather than claimed; the assumed LLS population implies a LyC reach of only $\Delta z \sim 0.6$ at $z \sim 3$, confirming that trough ionization cannot draw on external background light. The deuterium requirement is therefore reduced to a VMO ejecta-fraction/enrichment calculation and a reservoir longevity/mixing calculation, with a single shared observational discriminator. The Gate 4B amplitude channel has been bounded by the ambient-lapse self-consistency requirement (Gate 10, section 7d): the descent-tracking realisation that concords the inverse roots places the ambient boundary factor at $u(\bar\varphi) \approx 1-10^{-5}$ on the entire branch, under which the matter-metric redshift map cannot host the observed absorber population without a negative ambient field; the unscreened channel is thereby excluded at these epochs, and a temporal-disformal amplitude of the observed scale is admissible only under an environment-modulated coupling $B_{\rm eff}=B(\phi)\,G(X_{\rm local})$; Gate 10-b (results/gate10b_disformal_gate_test.json) derives the measurable condition — a gate closed in the ambient and opening with a common slope ${\rm d}G/{\rm d}\ln\delta\varphi\sim(4$-$7)\times10^{-4}$ per e-fold across the three wells (agreement within a factor $1.6$), saturating near $G\sim10^{-3}$ so that deeper structures are not over-displaced — converting the amplitude question into a well-posed target for the disformal-sector derivation rather than an open loophole. The primary Gates 2/3 spectroscopic result is in any case independent of the Gate-4B amplitude closure. Gates 4E, 7 and 8 provide secondary diagnostics, while Gate 6 defines the exploratory population test.

## 6. Conclusion

Across three benchmark D/H absorption systems, unrestricted H I is significantly preferred over isotope-tied D I within the tested Voigt architecture, including under adversarial parent reassignment and the primary multiplicity correction. Q1009+2956 and PKS 1937$-$101 also show significant departures from exact isotope locking in the calibrated Gate-8 analysis, while J1332$+$0052 is less discriminating. The result therefore demonstrates that the deuterium interpretation is not spectroscopically unique within the tested model class; the isotope-tied model is strongly disfavoured relative to unrestricted H within the tested architecture.

The fixed $-81.6\,{\rm km\,s^{-1}}$ clustering is not an independent discriminator in a sample selected through the deuterium search window. The corresponding falsifiable population test is an excess of suitable displaced H I components outside that window. The archival pilot establishes the practical selection limits of this test and identifies one candidate feature in Q1009+2956, but a substantially larger usable sample is required.

TEP provides a candidate physical interpretation of this displaced-H phenomenology. The conformal absorber field gives the blueward sign but only a $0.005$-$0.01\,{\rm km\,s^{-1}}$ shift. A temporal-disformal channel can reach the observed scale in the explored parametrisation. The full covariant evaluation (Gate 10) shows that the earlier closure obstruction is a property of the imported $\Lambda$CDM drift rate rather than of the disformal channel: joint sign-and-closure compatibility requires a descent-type ambient drift ($d\ln H_T^2/d\ln(1+z) < 0$), which the unsourced free-roll on the reconstructed corpus potential supplies, and under that rate a descent-tracking coupling $B(\varphi) = B_0\,\varphi^4/(\varphi_c^4+\varphi^4)\,e^{-\varphi}$ passes Cassini and GW170817, with inverse roots agreeing to a factor 1.00004. Forward evaluation at that common $B_0$ does not reproduce the three amplitudes, and ambient-lapse self-consistency excludes the branch outright: the flat-$u$ realisation holds the ambient sector at near-criticality, where the matter-metric redshift map would require a sign-violating negative ambient field. The unscreened temporal-disformal channel is accordingly bounded sub-critical at these epochs. Independently, the baryonic-cycling calculation shows that a helium-4 equilibrium near the observed abundance is possible under the adopted VMO-dominated, metal-sequestering branch. The spectroscopic, population and light-element predictions therefore provide separate empirical routes for testing the proposed non-hot-BBN interpretation. The scope of the deuterium argument is bounded accordingly: the analysis establishes that the benchmark absorber identifications are non-unique, not that a deuterium production channel exists. In an eternal recycling cosmology the distinction is binding — astration is a pure deuterium sink, so the standing Galactic interstellar value ${\rm D/H} \approx 1.5$-$2\times10^{-5}$, measured independently of the quasar-absorber architecture criticised here, is a production requirement rather than a relic; galactic-cosmic-ray spallation, pre-main-sequence envelope recycling, and pristine inflow each fail to supply it. The extended ledger (Step 15, results/gate15_deuterium_extended_channels.json) quantifies three channels that operate before or beside temporal-well formation — VMO partial-burning/freezeout ejecta, the collapse neutrino burst irradiating the envelope through near-null neutrinos on the matter metric, and protected-reservoir accumulation on bounded-exposure worldlines — of which the first and third can close the required net yield within documented ranges — at a transit fraction $f_{\rm ej} = 0.5$ the ejecta must reach $X_D \ge 7.5\times10^{-5}$ if the feed carries ambient ISM deuterium — reducing the open item to a VMO ejecta-enrichment calculation and a reservoir mixing rate rather than a missing mechanism. Independently, the opacity three-point requirement is shown to admit an explicit species-resolved profile (Step 14, results/step_14_opacity_three_point.json): the wing datum fixes $n_e \approx 0.51\,n_{b,0}$, the intervening trough is derived to be neutral ($\bar n_e < 3.7\times10^{-4}\,n_{b,0}$), and the ionization front at the Gate-11B thermalization depth places the $\tau = 1$ surface at $z \approx 1080$. The boundary's self-consistency items are then closed at toy-model level (Step 18, results/step_18_boundary_selfconsistency.json): the $\sim 4\times10^{10}$ wing-to-trough ionization-rate contrast is supplied by self-shielding over $\sim 24$ LyC e-folds; the front gas temperature is derived ($T_{\rm gas} = T_{\rm bath} = 2946$ K via Compton equilibrium on a $\sim$yr coupling time), making the Saha ionization a consequence of the Gate-11B bath rather than an assumption; the visibility tail is shown to be a closure artifact removed when the edge is absorptive; and the deuterium channels' residual budgets are bounded (transit neutrino irradiation under a stated homologous-collapse model, reservoir longevity reduced to a $\sim 10^{3}$-$10^{4}$ Gyr sequestration requirement). The remaining open item in the opacity sector is the perturbation mode spectrum on the boundary — the eternal-native Boltzmann treatment — together with the absorber-statistics inversion on a full LLS/Lyman-forest catalogue, which is the identified pipeline extension. The cosmological stability evidence must be read by sector. The homogeneous closed-background benchmark has an Eddington-type secular mode. The explicit Euclidean zero-mean scalar projection in Paper 0 (step_79) gives a positive lowest restricted eigenvalue, $+38.8145$ at $N=24$ and $+38.9129$ at $N=32$; its negative direction is almost spatially constant, with relative variation $2.2\times10^{-5}$. The discrete periodic Poincaré bound independently certifies all zero-mean modes above $38.80$ and $38.90$ on these grids. This is substantive stability evidence for the computed reduced sector. Identifying the constant direction as a gauge mode and establishing the full coupled constraint-reduced spectrum require the corresponding action-level analysis. The chemical attractors evaluated here therefore coexist with a tested stable nonconstant scalar sector; a global stability theorem is not inferred from either chemical convergence or that reduced spectrum.

## Data Availability & Reproducibility

This work follows open-science practices. All results are fully reproducible from raw data
using the documented pipeline. All numerical results, Monte Carlo simulations, and statistics are generated by deterministic
Python scripts processing real observational data. The pipeline enforces rigorous reproducibility: prespecified statistical outcomes are reported without post-hoc alteration and interpreted according to the scope of each test.

### Repository and Code

GitHub Repository: github.com/matthewsmawfield/TEP-BBN

The repository contains a deterministic, version-controlled cosmological analysis pipeline utilizing 27 analysis steps across 4 phases: core gates (1–8), Gate 8 calibration (step_08b, true-D injection Monte Carlo), Gate 8 wavelength- and parent-centroid-systematics sensitivity (step_08c, step_08d), disformal transport, the steady-state and ionization-equilibrium ledgers (step_11, step_12), the deuterium production and extended-channel ledgers (step_13, step_15), the species-resolved opacity three-point test (step_14), the boundary-transport consistency audit (step_16), the literature-input inversion and channel-budget audit (step_17), the boundary self-consistency solve (step_18), the eternal-surface reconciliation and self-shielded ambient-bath front mode spectrum (step_19, step_20), the canonical screening-projection audit (step_21), secondary robustness analyses, and manuscript finalization.
All steps are orchestrated by `scripts/run_pipeline.py` with comprehensive per-step logging, SHA-256 checksum manifest generation, and a pipeline summary ledger.

All raw spectroscopic data (Keck/HIRES, VLT/UVES + Keck/HIRES, and VLT/ESPRESSO), structural likelihood matrices, and the temporal-field equation solvers are released in the Zenodo repository (DOI: 10.5281/zenodo.21841147) under CC-BY 4.0. The full codebase and execution environments are identical to the published version.

#### Repository Structure



```

TEP-BBN/
├── data/
│   ├── raw/                       # Spectroscopic exposures (Keck/HIRES, UVES+HIRES, ESPRESSO)
│   │   ├── atomic/                # Physical H I, D I line registries (NIST ASD)
│   │   └── reduced_products/      # Pre-reduced and co-added normalized spectra
│   ├── literature_components/     # Published VPFIT model files and component tables
│   └── processed/                 # Pipeline-ready union manifests
├── scripts/
│   ├── steps/                     # pipeline steps (01-17 + lettered substeps, secondary, finalize)
│   ├── lib/                       # Physical RT engine, Voigt fitters, model parsers
│   ├── utils/                     # Logging, isotopic shift, shear model utilities
│   └── run_pipeline.py            # Master orchestration script (4-phase pipeline)
├── configs/
│   ├── sightlines/                # Per-sightline JSON configs (bounds, MC settings, noise models)
│   └── TEP_BBN_FOUNDATION_FREEZE.json  # Frozen TEP prior parameters
├── results/                       # Generated parameter ledgers and significance matrices
├── logs/                          # Per-step execution logs
├── site/
│   └── components/                # Manuscript source components
├── requirements-lock.txt          # Locked Python dependencies
└── README.md                      # Documentation
```



### Data Provenance

| Data Source | Provider | Access Method | Records | Location |
| --- | --- | --- | --- | --- |
| Q1009+2956 Spectra | Keck/HIRES (Zavarygin et al. 2018 <sup>[44]</sup>) | Zavarygin et al. 2018 GitHub | 4 coadds (S/N $\approx 47$–$78$ near Ly$\alpha$) | `data/raw/reduced_products/Q1009+2956_z2.504_HIRES/` |
| PKS 1937$-$101 Spectra | VLT/ESPRESSO (Cooke et al. 2024) | ESO Archive | 3 exposures | `data/raw/spectra/PKS1937-101_z3.572_ESPRESSO/` |
| J1332$+$0052 Spectra | VLT/UVES + Keck/HIRES (Kislitsyn et al. 2024) | ESO Archive | 24 exposures | `data/raw/spectra/J1332+0052_z3.420_UVES/` |
| SQUAD DR1 Spectra | UVES/SQUAD (Murphy et al. 2018 <sup>[43]</sup>) | ESO Phase 3 Archive | 3 sightlines (Q1009, Q0913, J212912) | `data/raw/spectra/SQUAD/` |
| KODIAQ DR2 Spectra | Keck/HIRES (O'Meara et al. 2017) | Keck KOA | 7 LLS-class sightlines | `data/raw/spectra/KODIAQ/` |
| KODIAQ Z LLS Catalog | Lehner et al. 2016 | Literature | 26 pLLSs/LLSs at $2.3 < z < 3.3$ | `data/kodiaq_z_lls_catalog.json` |
| Atomic Data | NIST ASD <sup>[49]</sup> | Static Registry | H I, D I, metals | `data/raw/atomic/` |
| Q1009+2956 Model | Zavarygin et al. <sup>[44]</sup> / VPFIT <sup>[50]</sup> | Static File | 43 H I + 3 D I components | `data/literature_components/model_6a.26` |
| PKS 1937$-$101 Model | Cooke et al. 2024 / VPFIT <sup>[50]</sup> | Static File | 184 H I + 3 D I components | `data/literature_components/PKS1937-101_z3.572_vpfit_model.26` |
| J1332$+$0052 Model | Kislitsyn et al. 2024 / VPFIT <sup>[50]</sup> | Static File | 11 H I + 4 D I components | `data/literature_components/J1332+0052_z3.420_vpfit_model.26` |
| Prior Bounds | Derived | Static File | All variables | `configs/sightlines/*.json` |

### Pipeline Architecture

The analysis pipeline comprises 4 phases and 24 steps spanning spectroscopic ingestion to thermodynamic evaluation, helium synthesis, opacity boundary analysis, Gate 8 true-D injection calibration, wavelength- and parent-centroid-systematics sensitivity, disformal transport calibration, the steady-state timescale/thermalization ledger (Gate 11), the VMO ionization-equilibrium budget (Gate 12), the deuterium production ledger (Gate 13), the species-resolved opacity three-point test (Step 14), the extended deuterium channels ledger (Step 15), the boundary-transport consistency audit (Step 16), the literature-input inversion and channel-budget audit (Step 17), the boundary self-consistency solve (Step 18), secondary robustness checks, and manuscript number finalization.
Each step is a standalone Python script in `scripts/steps/` that produces serialized JSON outputs and
detailed logs with SHA-256 checksum verification.

#### Complete Step Inventory and Runtime

Runtimes are approximate and measured on Apple M4 Pro (14-core, 24 GB). The dominant cost is the Monte Carlo significance test (step 03), which scales with the number of realizations multiplied by the parent reassignment refit performed inside each realization.

| Phase | Step | Script | Description | Est. Runtime |
| --- | --- | --- | --- | --- |
| 1 | 01 | `step_01_embedding.py` | Physical atomic-data embedding (H vs D isotope identifiability) | ~1 s |
| 1 | 02 | `step_02_q1009.py` | Nested-hypothesis fit (multi-start; nesting invariants verified) | ~4 min |
| 1 | 03 | `step_03_significance.py` | 1000-realization Monte Carlo significance calibration (parent reassignment inside each realization) | ~25 min |
| 1 | 04 | `step_04_prior.py` | TEP absorber field sign closure ($\Delta v_T < 0$) | ~1 s |
| 1 | 04B | `step_04b_amplitude.py` | TEP absorber amplitude test (conformal/static branch sign theorem) | ~1 s |
| 1 | 05 | `step_05_thermodynamics.py` | Matter-frame temporal thermodynamics (Planck spectrum preservation) | ~1 s |
| 1 | 06 | `step_06_helium.py` | Primordial helium synthesis via baryonic cycling | ~1 s |
| 1 | 07 | `step_07_global_opacity.py` | Analytical proof of divergent optical depth in the static spatial geometry | ~1 s |
| 1 | 08 | `step_08_light_elements.py` | Light-element network (3He, 4He, 7Li) and convergence constraints | ~1 s |
| 1 | 08B | `step_08b_calibration.py` | Gate 8 true-D injection Monte Carlo calibration ($N=1000$ per sightline) | ~45 min |
| 1 | 08C | `step_08c_wavelength_systematics.py` | Gate 8 wavelength-systematics sensitivity (literature-calibrated $\sigma_{\rm sys}$ per sightline, $N=100\,000$ MC) | ~1 s |
| 1 | 08D | `step_08d_parent_centroid_systematics.py` | Gate 8 parent-centroid blend-bias sensitivity (parent velocity perturbed inside the MC loop, blend-alone and joint sign-matched bounds, $N=100\,000$ MC) | ~1 s |
| 2 | 09 | `step_09_disformal_transport.py` | Disformal transport solver and absorber-ansatz / Cassini bound bookkeeping | ~1 s |
| 2 | 4B-IV | `step_04h_disformal_clock.py` | Disformal clock mechanism: cosmological $\partial_0\bar\phi$ channel, $B(\phi)$ field-dependence scan, Cassini/GW170817 constraints, three-sightline test | ~1 s |
| 2 | 09B | `step_09b_screening_plot.py` | Screening function visualization and JSON ledger | ~1 s |
| 2 | 10 | `step_10_covariant_transport.py` | Full covariant transport solver ($\omega = -u^\mu k_\mu$) and ambient-lapse self-consistency | ~1 s |
| 2 | 11 | `step_11_steady_state_budget.py` | Gate 11 steady-state timescale and thermalization-depth budget | ~1 s |
| 2 | 12 | `step_12_vmo_ionization_budget.py` | Gate 12 VMO-branch ionization-equilibrium budget over the ($\xi$, $f_{\rm esc}$) plane | ~1 s |
| 2 | 13 | `step_13_deuterium_production_ledger.py` | Gate 13 deuterium supply-versus-demand ledger ($f\,\varepsilon_D = 2.04\times10^{-5}$) | ~1 s |
| 2 | 14 | `step_14_opacity_three_point.py` | Species-resolved opacity three-point test; admissible profile and KODIAQ LLS pilot inversion | ~2 s |
| 2 | 15 | `step_15_deuterium_extended_channels.py` | Extended deuterium channels: VMO ejecta, collapse neutrino process, protected reservoirs | ~1 s |
| 2 | 16 | `step_16_boundary_transport.py` | Boundary-transport consistency audit on the Step-14 profile: visibility function, geometric diffusion analogue, conditional Saha ionization, recombination/emissivity ledger, $\Gamma$-equilibrium requirements | ~5 s |
| 2 | 17 | `step_17_inversion_and_channel_budgets.py` | Literature-input absorber audit and deuterium channel budgets: net-yield ejecta closure, neutrino capture-radius thresholds, reservoir mixing bookkeeping | ~1 s |
| 2 | 18 | `step_18_boundary_selfconsistency.py` | Boundary self-consistency solve: coupled ionization front (self-shielded $\Gamma$ transition), Compton-equilibrium front temperature, absorptive-edge visibility kernel, homologous-collapse neutrino transit exposure, reservoir longevity timescale | ~5 s |
| 2 | 19 | `step_19_eternal_surface_reconciliation.py` | Eternal-vs-mapped-FLRW surface reconciliation: path-measure Jacobian, explicit diffusion-integral damping scale, TEP-TH kernel comparison | ~1 s |
| 2 | 20 | `step_20_ambient_bath_front.py` | Ambient-bath ionization front & closure: Planck LyC Wien-tail supply, self-shielded LyC radiative transfer, absorber-statistics density inversion ($\Delta_{\rm front}=668.9\,n_{b,0}$ for $z_{\tau=1}=1080.0$), and boundary perturbation mode spectrum ($\ell_D=1521$; action-derived two-branch ambient clock-slope mode with exact `hi_class` Boltzmann closure and three-branch ablation) | ~1 min |
| 2 | 6-III | `step_06_orphan_search.py` | Orphan feature search in VPFIT models, direct spectral inspection, and archival SQUAD/KODIAQ search | ~10 s |
| 3 | C2-C4 | `secondary_analyses.py` | Scaffold freedom (C2), perturbed MC (C3), kinematic test (C4) | ~10 min |
| 4 | Final | `finalize_manuscript_numbers.py` | Manuscript number synchronization + SHA-256 checksum manifest | ~1 s |

#### Total Runtime Summary

| Component | Steps | Runtime |
| --- | --- | --- |
| Core Gates (Phase 1) | 9 | ~45 min per sightline |
| Disformal + Secondary (Phases 2–3) | 14 | ~10 min per sightline |
| Finalization (Phase 4) | 1 | ~1 s |
| Total (3 sightlines) | 24 | ~2 hr |

### Reproduction Instructions

#### Quick Start (Full Reproduction)



```

# 1. Clone repository
git clone https://github.com/matthewsmawfield/TEP-BBN.git
cd TEP-BBN

# 2. Install dependencies
pip install -r requirements-lock.txt

# 3. Run full pipeline (default: Q1009+2956 sightline, 1000 MC sims)
python scripts/run_pipeline.py

# 3b. Quick mode (50 MC sims, for faster testing)
python scripts/run_pipeline.py --quick

# 3c. Run a specific sightline
python scripts/run_pipeline.py --sightline PKS1937-101

# 3d. Run all sightlines with ingested data
python scripts/run_pipeline.py --all-sightlines

# 3e. Skip secondary robustness analyses
python scripts/run_pipeline.py --skip-secondary

# 3f. Custom MC simulation count
python scripts/run_pipeline.py --n-sims 500

# 4. Results will be stored in results/ and logs/
#    Pipeline summary: results/pipeline_summary.json
#    SHA-256 manifest: results/checksums_sha256.json
```



#### Multi-Sightline Configuration

The pipeline supports multiple D/H absorber sightlines through per-sightline JSON configuration files in `configs/sightlines/`.
Each config specifies the data manifest path, VPFIT model file, noise model, absorber redshift, candidate parameter bounds, multi-start initial points, and Monte Carlo settings.
Three sightlines are currently supported with ingested data: Q1009+2956 (Keck/HIRES, $z=2.5042$), PKS 1937$-$101 (ESPRESSO, $z=3.572$), and J1332$+$0052 (UVES+HIRES, $z=3.421$).
Sightlines without ingested data (PKS 1937$-$1009, HS 0105+1619) are automatically skipped with a diagnostic message.

#### System Requirements

| Component | Minimum | Recommended | Tested On |
| --- | --- | --- | --- |
| CPU | 2 cores | 4+ cores | Apple M4 Pro (14-core) |
| RAM | 4 GB | 8 GB | 24 GB |
| Storage | 1 GB | 2 GB | SSD NVMe |
| OS | Linux/macOS | Linux/macOS | macOS Sequoia 15.1 |

#### Current statistical implementation

The fixed data-quality mask is shared by all hypotheses and simulations. All finite residuals contribute to the Student-$t$ likelihood; numerical forward-model or continuum failures are rejected. Continuum order is 10 for Lyman-series blocks and 5 for metal-line blocks. The per-sightline noise models, candidate bounds, multistart searches and parent-selection rules are recorded with the fitted ledgers. Both canonical-parent and parent-maximized likelihood comparisons are calibrated with $N=1000$ genuine true-D injections per sightline.

Gate 8 retains every simulation vector and its seed. Marginal consistency tests center on the empirical null means. The joint metric regularizes a dimensionless correlation matrix and uses leave-one-out null scores. Input and simulation-file hashes permit provenance checks. Gaussian systematic sensitivity calculations are distinguished from empirical injection probabilities and their finite-sampling limits. The final method does not combine these correlated diagnostics as independent evidence.

**Ionization balance.** The opacity calculations use the stable positive root of $x_e^2=q(1-x_e)$. At the declared baryon density, photoionization equilibrium is $\Gamma(1-x_e)=\alpha_B n_b x_e^2$. The step-16 LTE-equivalent rates therefore agree at fixed temperature across its density ladder; these are conditional equilibrium requirements, not a measured radiation field.

**Likelihood and tail evaluation.** All finite data residuals contribute to the likelihood. Numerical failures are rejected. Gaussian sensitivity tails use the survival function; Monte-Carlo outputs include exceedance counts, sampling resolution and one-sided 95% binomial upper bounds.

## References

- Smawfield, M.L. Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed. *Zenodo* (2025), v0.16 (Jakarta). DOI: 10.5281/zenodo.16921911

- Smawfield, M.L. Temporal Equivalence Principle: A Covariant Alternative to Cosmic Expansion. *Zenodo* (2026). DOI: 10.5281/zenodo.20370143

- Smawfield, M.L. Temporal Equivalence Principle: Native hi_class Conformal Implementation, Linear Perturbation Closure, and CMB Acoustic Peak Preservation. *Zenodo* (2026). DOI: 10.5281/zenodo.20572722

- Smawfield, M.L. Temporal Equivalence Principle: Temporal Horizon Cosmology and the Absence of a Physical Big Bang Singularity. *Zenodo* (2026). DOI: 10.5281/zenodo.20723059

- Hawking, S.W. The occurrence of singularities in cosmology. *Proc. R. Soc. A* **294**, 511-521 (1966).

- Hawking, S.W. & Penrose, R. The singularities of gravitational collapse and cosmology. *Proc. R. Soc. A* **314**, 529-548 (1970).

- Borde, A., Guth, A.H. & Vilenkin, A. Inflationary spacetimes are incomplete in past directions. *Phys. Rev. Lett.* **90**, 151301 (2003).

- Brandenberger, R. & Peter, P. Bouncing cosmologies: progress and problems. *Found. Phys.* **47**, 797-850 (2017).

- Novello, M. & Bergliaffa, S.E.P. Bouncing cosmologies. *Phys. Rep.* **463**, 127-213 (2008).

- Ijjas, A. & Steinhardt, P.J. Entropy, black holes and the new cyclic universe. *Phys. Lett. B* **824**, 136823 (2022).

- Peebles, P.J.E. *Principles of Physical Cosmology*. Princeton University Press (1993).

- Weinberg, S. *Cosmology*. Oxford University Press (2008).

- Dodelson, S. *Modern Cosmology*. Academic Press (2003).

- Mukhanov, V.F., Feldman, H.A. & Brandenberger, R.H. Theory of cosmological perturbations. *Phys. Rep.* **215**, 203-333 (1992).

- Liddle, A.R. & Lyth, D.H. *Cosmological Inflation and Large-Scale Structure*. Cambridge University Press (2000).

- Planck Collaboration, et al. Planck 2018 results. VI. Cosmological parameters. *A&A* **641**, A6 (2020).

- Riess, A.G., et al. Milky Way Cepheid Standards for Measuring Cosmic Distances and Application to Gaia DR2: Implications for the Hubble Constant. *ApJ* **861**, 126 (2018).

- Brout, D., et al. The Pantheon+ Analysis: Cosmological Constraints. *ApJ* **938**, 110 (2022).

- Fixsen, D.J., et al. The Cosmic Microwave Background Spectrum from the Full COBE FIRAS Data Set. *ApJ* **473**, 576 (1996).

- Chluba, J. & Sunyaev, R.A. The evolution of CMB spectral distortions in the early Universe. *MNRAS* **419**, 1294-1314 (2012).

- PARTICLE DATA GROUP. Review of Particle Physics. *PTEP* **2022**, 083C01 (2022).

- Cyburt, R.H., Fields, B.D., Olive, K.A. & Yeh, T.H. Big bang nucleosynthesis: Present status. *Rev. Mod. Phys.* **88**, 015004 (2016).

- Seager, S., Sasselov, D.D. & Scott, D. A new calculation of the recombination epoch. *ApJ* **523**, L1-L5 (1999).

- Peebles, P.J.E. Recombination of the Primeval Plasma. *ApJ* **153**, 1 (1968).

- Zeldovich, Y.B. & Sunyaev, R.A. The interaction of matter and radiation in a hot-model universe. *Astrophys. Space Sci.* **4**, 301-316 (1969).

- Seljak, U. & Zaldarriaga, M. A Line of Sight Integration Approach to Cosmic Microwave Background Anisotropies. *ApJ* **469**, 437 (1996).

- Lewis, A., Challinor, A., & Lasenby, A. Efficient Computation of CMB Anisotropies in Closed FRW Models. *ApJ* **538**, 473 (2000).

- Lesgourgues, J. & Tram, T. The Cosmic Linear Anisotropy Solving System (CLASS). Part IV: efficient implementation of non-cold relics. *JCAP* **09**, 032 (2011).

- Zumalacárregui, M., Bellini, E., Sawicki, I., Lesgourgues, J. & Ferreira, P.G. hi_class: Horndeski in the Cosmic Linear Anisotropy Solving System. *JCAP* **08**, 019 (2017).

- De Felice, A. & Tsujikawa, S. f(R) Theories. *Living Rev. Rel.* **13**, 3 (2010).

- Wetterich, C. Cosmology and the fate of dilatation symmetry. *Nucl. Phys. B* **302**, 668-696 (1988).

- Wetterich, C. A universe without expansion. *Phys. Dark Universe* **2**, 184 (2013).

- Narlikar, J.V. & Arp, H.C. Flat spacetime cosmology: A unified framework for extragalactic redshifts. *Astrophys. J.* **405**, 51-56 (1993).

- Mannheim, P.D. Conformal gravity and the nature of dark matter. *Prog. Part. Nucl. Phys.* **94**, 217-272 (2017).

- Khoury, J. & Weltman, A. Chameleon cosmology. *Phys. Rev. D* **69**, 044026 (2004).

- Hinterbichler, K. & Khoury, J. Symmetron cosmology. *Phys. Rev. Lett.* **104**, 231301 (2010).

- Penrose, R. Before the Big Bang: an outrageous new perspective and its implications for particle physics. *Proc. EPAC* (2006).

- Tod, K.P. Isotropic cosmological singularities. *Gen. Relativ. Gravit.* **35**, 779-805 (2003).

- Tod, K.P. The equations of conformal cyclic cosmology. *Gen. Relativ. Gravit.* **47**, 31 (2015).

- Ratra, B. & Peebles, P.J.E. Cosmological Consequences of a Rolling Homogeneous Scalar Field. *Phys. Rev. D* **37**, 3406 (1988).

- Caldwell, R.R., Dave, R., & Steinhardt, P.J. Cosmological Imprint of an Energy Component with General Equation of State. *Phys. Rev. Lett.* **80**, 1582 (1998).

- Clifton, T., Ferreira, P.G., Padilla, A. & Skordis, C. Modified gravity and cosmology. *Phys. Rep.* **513**, 1-189 (2012).

- Murphy, M.T., Kacprzak, G.G., Savorgnan, G.A.D. & Carswell, R.F. The UVES Spectral Quasar Absorption Database (SQUAD) data release 1: the first 10 million seconds. *MNRAS* **482**, 3458-3482 (2019). arXiv:1810.06136

- Zavarygin, E., Webb, J.K., Dumont, V. & Riemer-Sørensen, S. The primordial deuterium abundance at z<sub>abs</sub> = 2.504 from a high signal-to-noise spectrum of Q1009+2956. *MNRAS* **477**, 5536-5553 (2018). arXiv:1706.09512

- Cooke, R.J., Pettini, M., Jorgenson, R.A., Murphy, M.T. & Steidel, C.C. Precision measures of the primordial abundance of deuterium. *ApJ* **781**, 31 (2014).

- Burles, S. & Tytler, D. The Deuterium Abundance toward QSO 1009+2956. *ApJ* **507**, 732 (1998).

- O'Meara, J.M., Burles, S., Prochaska, J.X., Prochter, G.E., Bernstein, R.A. & Burgess, K.M. The Deuterium-to-Hydrogen Abundance Ratio toward the QSO SDSS J155810.16-003120.0. *ApJ* **649**, L61 (2006).

- Kirkman, D., Tytler, D., Burles, S., Lubin, D. & O'Meara, J.M. On the primordial deuterium abundance. *ApJ* **529**, 655 (2000).

- Kramida, A., Ralchenko, Yu., Reader, J. & NIST ASD Team. NIST Atomic Spectra Database (ver. 5.11). National Institute of Standards and Technology, Gaithersburg, MD (2023).

- Carswell, R.F. & Webb, J.K. VPFIT: Voigt profile fitting program. Astrophysics Source Code Library, ascl:1408.015 (2014).

- Smawfield, M.L. The Cepheid Bias: Resolving the Hubble Tension. *Zenodo* (2026). DOI: 10.5281/zenodo.18209702

- Smawfield, M.L. Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies. *Zenodo* (2026). DOI: 10.5281/zenodo.19000827

- Smawfield, M.L. Temporal Equivalence Principle: Temporal Shear in the Earth Flyby Anomaly. *Zenodo* (2026). DOI: 10.5281/zenodo.19454862

- Smawfield, M.L. Temporal Equivalence Principle: Black Holes and the Temporal Horizon. *Zenodo* (2026). DOI: 10.5281/zenodo.21677826

- Haardt, F. & Madau, P. Radiative transfer in a clumpy universe. IV. New synthesis models of the UV/X-ray cosmic background. *ApJ* **746**, 125 (2012).

- Benisty, D., Brax, P. & Davis, A.C. Constraining modified gravity with cosmological data. *Phys. Rev. D* **107**, 064049 (2023).

- Carr, B.J., Bond, J.R. & Arnett, W.D. Cosmological consequences of Population III stars. *ApJ* **277**, 445-469 (1984).

- Bond, J.R., Arnett, W.D. & Carr, B.J. The evolution and fate of Very Massive Objects. *ApJ* **280**, 825-847 (1984).

- Bertotti, B., Iess, L. & Tortora, P. A test of general relativity using radio links with the Cassini spacecraft. *Nature* **425**, 374-376 (2003).

- Ober, W.W., El Eid, M.F. & Fricke, K.J. Evolution of massive Pop III stars with helium-overshooting. *A&A* **119**, 61-68 (1983).

- Aver, E., Olive, K.A. & Skillman, E.D. The effects of He I λ10830 on helium abundance determinations. *JCAP* **07**, 011 (2011).

- Izotov, Y.I., Thuan, T.X. & Guseva, N.G. A new determination of the primordial He-4 abundance: A self-consistent reanalysis of the H II region data. *MNRAS* **445**, 778-801 (2014).

- Peimbert, M., Peimbert, A. & Luridiana, V. The primordial helium abundance and the number of neutrino species. *Rev. Mex. Astron. Astrofis.* **52**, 319-326 (2016).

- Planck Collaboration, et al. Planck 2018 results. VI. Cosmological parameters. *A&A* **641**, A6 (2020). (See also: Cooke, R.J., Pettini, M. & Steidel, C.C. The primordial deuterium abundance and the baryon density. *ApJ* **817**, 79 (2016) for the D/H → Ω<sub>b</sub>h² concordance.)

- Cooke, R.J., Pettini, M. & Steidel, C.C. The primordial deuterium abundance and the baryon density. *ApJ* **817**, 79 (2016).

- Cooke, R.J., et al. ESPRESSO observations of the D/H ratio in PKS 1937−101. *MNRAS*, stae452 (2024). arXiv:2402.05586

- Kislitsyn, R., Balashev, S.A., Murphy, M.T., et al. A new precise determination of the primordial abundance of deuterium: measurement in the metal-poor sub-DLA system at z = 3.42 towards quasar J 1332+0052. *MNRAS* **528**, 4068-4084 (2024). arXiv:2401.12797

- Fields, B.D., Olive, K.A., Yeh, T.H. & Cyburt, R.H. The Hubble tension and the primordial helium abundance. *Universe* **9**, 165 (2023).

- Smawfield, M.L. Temporal Topology Saturation Scale: Cross-Scale Consistency of $\rho_T$. *Zenodo* (2026). DOI: 10.5281/zenodo.18064365

- Smawfield, M.L. Temporal Equivalence Principle: Lunar Laser Ranging and the Nordtvedt Effect. *Zenodo* (2026). DOI: 10.5281/zenodo.19446028

- Griest, K., Whitmore, J.B., Wolfe, A.M., Prochaska, J.X., Dawley, M.M. & Vladilo, G. Wavelength Accuracy of the Keck HIRES Spectrograph and Measuring Changes in the Fine Structure Constant. *ApJ* **708**, 158-170 (2010). DOI: 10.1088/0004-637X/708/1/158

- Whitmore, J.B. & Murphy, M.T. Impact of instrumental systematic errors on fine-structure constant measurements with quasar spectra. *MNRAS* **444**, 2 (2014). DOI: 10.1093/mnras/stu2420. arXiv:1409.4467

- Schmidt, T.M., Murphy, M.T., Lo Curto, G., et al. Fundamental physics with ESPRESSO: Towards an accurate wavelength calibration for a precision test of the fine-structure constant. *A&A* **646**, A104 (2021). DOI: 10.1051/0004-6361/202039345

- Korn, A.J., et al. A probable stellar solution to the cosmological lithium discrepancy. *Nature* **442**, 657-659 (2006).

- Richard, O., Michaud, G. & Richer, J. Implications of WMAP observations on Li abundance and stellar evolution. *ApJ* **619**, 538-548 (2005).

- Fields, B.D. The primordial lithium problem. *Annu. Rev. Nucl. Part. Sci.* **61**, 47-68 (2011).

- Schaerer, D. On the properties of massive Population III stars and metal-free stellar populations. *A&A* **382**, 28-42 (2002).

- Tumlinson, J. & Shull, J.M. Zero-metallicity stars and the effects of the first stars on reionization. *ApJ* **528**, L65-L68 (2000).

- Osterbrock, D.E. & Ferland, G.J. *Astrophysics of Gaseous Nebulae and Active Galactic Nuclei*, 2nd ed. (University Science Books, 2006).

- Mukohyama, S. Dark matter as integration constant in Hořava-Lifshitz gravity. *Phys. Rev. D* **80**, 064005 (2009).

- Chamseddine, A.H. & Mukhanov, V. Mimetic Dark Matter. *JHEP* **11**, 135 (2013).

- Berezhiani, L. & Khoury, J. Theory of dark matter superfluidity. *Phys. Rev. D* **92**, 103510 (2015).

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# The Mount Wilson Paradigm: Restoring the Eternal Universe via the Temporal Equivalence Principle
**Matthew Lukin Smawfield**
Version: v0.3 (Harare)
First published: 15 August 2026 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.21954257

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## Abstract

Cosmological redshift is standardly interpreted as a geometric signature of spatial expansion. This interpretation requires a fundamental assumption that is defined herein as the Isochrony Axiom: the premise that, after gravitational and kinematic effects derived from the spacetime metric have been included, the calibration of every matter clock is fully exhausted by that single metric, so that no independently dynamical field may rescale matter proper time across cosmological epochs. Standard single-metric cosmology closed the temporal sector before the redshift was interpreted. In this paper, it is demonstrated that when matter-clock calibration is permitted an independent dynamical scalar degree of freedom ($\phi$), the measured cosmological redshift ($z$) can be rigorously derived from the ratio of matter-clock calibrations between emission and observation on a static spatial manifold. The scalar field defines a spatiotemporal clock geometry—the Temporal Topology—whose homogeneous evolution governs cosmological calibration and whose inhomogeneous disformal connection can become non-integrable, producing synchronization holonomy. Using the canonical preferred-foliation matter metric of the Temporal Equivalence Principle (TEP), $\tilde g_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))\,U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\,\Pi_{\mu\nu} + B(\phi)\,\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu \equiv -\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu} = g_{\mu\nu} + U_\mu U_\nu$; written in isotropic conformal–disformal shorthand as $\tilde g_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi$), the Mount Wilson Equivalence Theorem is proven: on a homogeneous, static gravitational background ($u_{\rm grav} = 0$, $\Pi_\mu^\alpha\nabla_\alpha\phi = 0$, and $A^2 - B\dot\phi^2 > 0$ on the unprojected isotropic-disformal comparison branch), the endpoint redshift $1+z = A_{\rm sp}(t_o)/A_{\rm sp}(t_e) \equiv A_0/A_{\rm em}$ is observationally degenerate with the FLRW relation $1+z = a_0/a_{\rm em}$ at the level of the redshift observable alone. TEP is distinguished from a mere conformal rewriting by its separate matter and gravitational propagation sectors, making multi-messenger observations a principal inter-sector discriminator. The historical data from Mount Wilson Observatory are re-examined to separate the documented spectral displacements from the subsequent spatial inferences, demonstrating that the observed dynamics are consistent with a temporal reinterpretation on a static spatial manifold; the historical analysis is an argument for underdetermination — redshift alone does not select between the interpretations — not an argument for TEP. The theorem itself establishes an observational degeneracy, not an empirical discrimination: the Isochrony Axiom is a consequence of the standard minimally coupled action rather than an arbitrary closure, the static gravitational ansatz ($a = 1$) is imposed on the homogeneous branch rather than derived from the gravitational field equations, and the empirical distinction between TEP and a conformal rewriting requires a dimensionless inter-sector observable — the standard-siren deviation $\Delta_{\rm siren}(z) = d_L^{\rm GW}(z)/d_L^{\rm EM}(z) - 1$ — whose canonical radiation rule is $\Xi(z)=1/(1+z)$, hence $\Delta_{\rm siren}=1/(1+z)-1$, evaluated across the GWTC-3 bright, dark, and spectral-siren population likelihoods in Paper 22 (showing that the apparent $H_0 \simeq 56\ {\rm km\,s^{-1}\,Mpc^{-1}}$ free-fit dark-siren shift is driven by $\Lambda$CDM-gated galaxy-catalog host-redshift assignment, that self-consistent TEP-kernel reassociation at the fixed canonical $H_0 = 68.18\ {\rm km\,s^{-1}\,Mpc^{-1}}$ recovers the baseline association likelihood, and that the separate lab-fixed comparison gives $H_0 \simeq 72.2$ versus $72.0\ {\rm km\,s^{-1}\,Mpc^{-1}}$ at marginal significance); this paper supplies the redshift-degeneracy theorem. Precision GNSS chronometry, pulsar scintillation, and Lunar Laser Ranging are identified as the empirical instruments capable of testing the temporal sector locally. The eternal-branch completion is assembled from companion analyses (TEP-BBN, TEP-TH, and Papers 11, 12, 18, 26), each of which must independently stand; the contribution of the present paper is the degeneracy theorem, the identification of the discriminating observables, and the resulting globally consistent framework for an eternal, deterministic continuum, supplying the common mathematical resolution to the questions that Einstein separated.

Keywords: cosmological redshift, isochrony axiom, scalar-tensor theory, static universe, temporal equivalence principle, expansion degeneracy, synchronization holonomy

## Prologue: The Photograph

On 29 January 1931, Albert Einstein bent to the eyepiece of Mount Wilson's 100-inch Hooker telescope. Edwin Hubble and Walter Adams were present, members of the observational programme whose measurements appeared to show that the universe itself was expanding. The telescope revealed the cosmological spectral displacement; the geometry assigned to that displacement remained a theoretical inference. The photograph records the moment when the architect of relativistic spacetime confronted the instrument that had transformed a theoretical possibility into the dominant picture of the cosmos.

![Albert Einstein at the eyepiece of the 100-inch Hooker telescope, accompanied by Edwin Hubble and Walter S. Adams, Mount Wilson Observatory, January 1931.](site/public/einstein_hooker_1931.jpg)

Figure 1. Albert Einstein at the eyepiece of the 100-inch Hooker telescope, with Edwin Hubble and Walter S. Adams, Mount Wilson Observatory, January 1931. Source: Caltech Archives and Special Collections 52.

Einstein's encounter with Mount Wilson had begun long before Hubble's redshift programme. In 1913, he wrote to the observatory's founder, George Ellery Hale, asking whether stars close to the Sun could be observed in daylight, a possible test of gravity's effect on light 21. Eighteen years later, he returned to the same mountain, visiting both the 150-foot solar tower and the 100-inch Hooker telescope, to confront a still larger question: what could light reveal about the geometry of the universe?

The Hooker telescope was at the time the largest optical instrument in the world. It was here that Milton Humason and Edwin Hubble had recently assembled the data that would permanently alter the foundations of physics. Einstein encountered the observational results of the Mount Wilson programme, including the spectrographic measurements that showed unmistakable displacements of the calcium absorption lines toward longer wavelengths.

His visit is widely recorded as the moment he abandoned his 1917 model of a static, globally Machian universe 6. Confronted with a linear relationship between estimated distance and spectral redshift, Einstein accepted the dynamical geometry formulated by Georges Lemaître 12 and Aleksandr Friedmann 13. The universe was expanding, and the static cosmos was set aside as an incomplete theoretical closure. Einstein arrived in Pasadena amid extraordinary public attention: reporters, photographers, Hollywood figures, and enthusiastic hosts followed his movements. He was accompanied by his wife Elsa and his mathematical assistant Walther Mayer, the Viennese topologist known informally as "Einstein's calculator," who had been engaged specifically to work on the distant-parallelism programme 57. During a tour of the Mount Wilson facilities, Elsa was shown the great telescope and told that one of its chief purposes was to determine the shape of the universe. She reportedly replied: "My husband does that on the back of an envelope" 58. The anecdote captures the asymmetry of the moment: the most powerful spatial instrument in history, operated by men who had no chronometric machinery at all. To a New York Times reporter, Einstein described the experience in dramatic terms: "The redshift of distant nebulae has smashed my old construction like a hammer blow," he said, swinging down his hand to illustrate. Then he added: "The red shift is still a mystery" 64. The first statement entered the standard narrative. The second did not. Writing to his sister shortly afterward, Einstein remarked that he would soon "no longer have to play the part of the good Lord or his assistant" 53. That public spectacle formed the backdrop to a scientific transition that had nevertheless been developing for years.

This historical pivot established the conceptual foundation of the standard cosmological model. The Hooker telescope was an instrument of unprecedented spatial resolution, yet it operated in a state of what this paper terms Chronometric Blindness. It could measure the frequency of light to extraordinary precision; it possessed no mechanism for interrogating the independent dynamics of the proper time that governed the clocks of distant matter. The photographic plates captured the displaced frequencies of incoming photons. They did not directly record expanding space. What, exactly, did the telescope prove?

## 1. The Evidence

The transformation occurred with extraordinary speed. In April 1920, Harlow Shapley and Heber Curtis debated the scale of the Milky Way and the nature of the spiral nebulae 36. Eleven years later, Einstein stood at Mount Wilson confronting the claim that the entire spatial universe was evolving. Cosmology had passed from uncertainty about the existence of other galaxies to a theory of expanding space in little more than a decade.

The conclusion was not arbitrary. It arose from a formidable synthesis of observational astronomy, and reconstructing the chain of evidence that produced it is necessary to understand the transition of the 1930s.

Mount Wilson had transformed the universe once before. On a glass plate exposed on the night of 5-6 October 1923, Hubble identified a variable star in Andromeda. He crossed out his original mark for a nova and wrote, in emphatic letters, "VAR!" Leavitt's period-luminosity relation then placed Andromeda far beyond the Milky Way. A handwritten word on a small glass plate had enlarged the known universe. Six years later, another set of marks on other plates would be interpreted as showing that this larger universe was itself expanding.

The initial anomaly had emerged earlier, when Vesto Slipher measured the spectra of spiral nebulae 37. By 1917, he had found that twenty-one of twenty-five measured nebulae were redshifted. Twenty of the twenty-four velocities used in Hubble's famous 1929 diagram came from Slipher's work. Hubble's contribution was principally to connect those velocities with estimated distances. The famous diagram was not the product of one observer. Its velocity axis was built largely from Slipher's spectra; its distance scale depended on Leavitt's stellar periods; Hubble assembled the relation; and Humason extended it into the deeper universe. Modern cosmology emerged not from a single decisive observation but from the alignment of several different measurements under a common interpretation.

Nor did Hubble approach the data in an interpretive vacuum. Carl Wirtz 42, Knut Lundmark, and Gustaf Strömberg had already searched for relations between spectral displacement and distance, often under the influence of de Sitter's cosmology. The crucial question was therefore never simply whether a pattern existed. It was which theoretical language should be used to describe it.

The distance scale had an unexpectedly temporal foundation. Henrietta Leavitt compared the changing brightness of twenty-five variable stars in the Small Magellanic Cloud 38. Because those stars lay at approximately the same distance, she recognized that their periods revealed their intrinsic luminosities. Hubble's distance axis was therefore constructed from a stellar clock: the measured rhythm of a pulsating star. Time was already hidden inside the diagram that would later be interpreted as the expansion of space.

Humason's measurements were acts of extraordinary patience. A former mule driver and observatory janitor with little formal education, he became one of the finest spectroscopists of his generation. His two plates of NGC 7619 required exposures of thirty-three and forty-five hours 43. Across several nights, the telescope gathered enough light to reveal a few displaced absorption lines. The 100-inch mirror itself was a source of constant anxiety: the massive glass disk was sensitive to thermal gradients, and Hale had proposed refrigerating the entire dome to prevent the mirror from heating up during the day, though only a sunscreen was ultimately installed 54. The observers mastered temperature-dependent distortions in the glass because those distortions were measurable. A cosmological variation in matter-clock calibration lay outside the variables their framework permitted. The Chronometric Blindness was not confined to Mount Wilson. Across the 1920s and 1930s, astronomers including J. K. Fotheringham and John Jackson were documenting anomalous "accelerations" of the Sun and Moon against Greenwich Mean Time 69. The Moon appeared to be drifting ahead of its predicted ephemeris. The anomaly was real, but its interpretation was inverted: the celestial bodies were not accelerating; the Earth's rotation was slowing due to tidal friction. Spencer Jones confirmed this in 1939 70, but the concept of a time standard independent of Earth's rotation, Ephemeris Time, was proposed in 1948, adopted by the International Astronomical Union in 1952, and not introduced into official astronomical ephemerides until 1960 72. Throughout the entire Mount Wilson era, time was defined by the rotation of a single planet, and its irregularities were being misattributed to the dynamics of distant bodies. The global astronomical community had no framework for treating time itself as a variable physical quantity. A cosmological variation in the rate of proper time was not merely undetectable; it was conceptually inaccessible.

What emerged on the glass was not an image of a galaxy moving through space. It was a measurable displacement in the frequency of its light. The 1929 relation 1 drew on numerous previously published velocities, with Humason contributing several recent measurements. He subsequently undertook the arduous task of extending the spectrographic measurements to much fainter and more distant galaxies, culminating in the major 1931 Hubble-Humason paper 2 that extended the relation far beyond the original sample. The combined programme established that extragalactic redshift increased linearly with estimated distance.

In the theoretical domain, general relativity had already established that the metric tensor of spacetime ($g_{\mu\nu}$) was a dynamical entity. Willem de Sitter 16 had introduced a cosmological solution that, though presented in static coordinates, contained the seed of an expanding interpretation that would later be recognized through coordinate transformations by Lemaître and others. Aleksandr Friedmann 13 and Georges Lemaître 12 then explicitly demonstrated that Einstein's field equations permitted, and in most cases required, the spatial metric to evolve over time. The expanding interpretation did not originate on a photographic plate. By 1927, Lemaître had already brought the dynamical solutions of Einstein's equations into contact with the available redshift and distance estimates, deriving the velocity-distance relation and estimating the expansion rate two years before Hubble's 1929 paper. He sent a reprint to Eddington, his former supervisor, who filed it away and did nothing. The paper sat unnoticed for three years. It was not until the January 1930 meeting of the Royal Astronomical Society, where de Sitter presented the observational velocity-distance relation and both he and Eddington struggled to explain it theoretically, that Lemaître read the published report and wrote to remind Eddington of the forgotten paper. Eddington immediately recognized its significance, arranged for its English translation and publication in the Monthly Notices 56, and brought it to the attention of the international community. A further complication has recently emerged: the 1931 English translation omitted the key paragraphs in which Lemaître described the velocity-distance law and derived the expansion rate, replacing them with a single sentence quoting the adopted value 65. Lemaître himself edited the translation and voluntarily omitted his earlier numerical discussion, judging it superseded by Hubble's more recent results. In doing so, he surrendered the published evidence of his priority, and observational credit shifted almost entirely to Mount Wilson. The expanding universe was not born from a sudden empirical revelation on a Mount Wilson plate. It was retrieved from a desk drawer as a mathematically ready framework for theorists confronting an unexplained empirical relation.

When the Mount Wilson relation appeared, a mathematical interpretation was already waiting for it. Theorists immediately recognized a synthesis: the observed redshift could be elegantly explained as the continuous stretching of the spatial scale factor ($a(t)$) during the photon's transit from emitter to observer.

Confronted with this powerful relationship, the abandonment of the static universe appeared entirely rational. The observational evidence was robust, and the expanding spatial geometry provided a mathematically consistent mechanism to encompass it. ΛCDM may therefore be understood as an architecture constructed after the temporal sector had already been closed. Dark matter, dark energy, and inflation were developed within that architecture. The question is whether the original closure itself was ever independently tested.

Seen operationally, the Mount Wilson diagram was stranger than its familiar labels suggest. The telescope measured wavelength shifts, which were subsequently expressed as velocities; the distance scale was anchored by periodic stars, whose luminosities were inferred from their rhythms. One axis began as a frequency ratio, while the other was calibrated through a stellar clock. The diagram became a picture of spatial expansion only after these chronometric observables had been interpreted geometrically.

## 2. The Interpretation

The standard historical narrative often conflates the observational discovery with its physical interpretation. The two must be formally separated:

Operational Distinction. Spectral displacement is not a direct measurement of spatial expansion.

The first is a measured physical observable. The second is an inference drawn within the constraints of a specific theoretical framework. Spatial expansion was not recorded directly on the photographic plates; it was the chosen physical interpretation of the observed spectral displacement.

The primary astronomers of the era were acutely aware of this distinction. In 1929 Hubble described the quantities as "apparent radial velocities" and the paper itself is notably restrained in its cosmological interpretation, presenting the empirical relation without elaborating on its physical significance 1. In 1931, Hubble wrote directly to Willem de Sitter, explicitly disclaiming interpretive authority over his own data: "We use the term 'apparent' velocities to emphasize the empirical features of the correlation. The interpretation, we feel, should be left to you and the very few others who are competent to discuss the matter with authority" 60. His 1936 book, *The Realm of the Nebulae* 4, was predominantly observational in character; Hubble described himself as "primarily an observer" and confined theoretical interpretation to a brief concluding discussion. The 1936 Rhodes Lectures at Oxford, published as *The Observational Approach to Cosmology* 5, concluded that "a static universe with a hitherto unsuspected dependence of light frequency on distance is probably more acceptable than one or other of the homogeneous expanding models of general relativity," as reported in a contemporary Nature review 67. His explicit treatment of recession and an unknown non-kinematic mechanism as parallel possibilities emerged in his later work with Tolman 3 and in his 1936-1937 lectures 4,5.

This caution persisted throughout his career. In the 1935 study with Richard Tolman 3, Hubble attempted to distinguish between actual recession and a non-kinematic redshift using surface-brightness scaling, a test whose theoretical framework Tolman had developed in his treatise on relativity and cosmology 17. In an expanding universe, the Tolman prediction is a surface-brightness dimming proportional to $(1+z)^4$. In the simple stationary energy-loss model used for comparison, only one factor of $(1+z)$ appears. The 1935 data were insufficient to decisively distinguish the two predictions, a conclusion Hubble and Tolman stated explicitly. Later analyses by Lubin and Sandage 71 measured surface-brightness exponents below four and interpreted the shortfall as luminosity evolution; after that correction, they reported consistency with the Tolman signal. Hubble's data appeared more consistent with the static prediction. Advocates of expansion argued that luminosity evolution could account for the discrepancy, since distant galaxies may have been intrinsically different from nearby galaxies. Hubble remained deeply skeptical of relying upon such corrections, because their required magnitude depended upon the cosmological interpretation being tested. In a 1937 paper to the Royal Astronomical Society 51, Hubble openly acknowledged the tension, stating that if the redshift represented actual expansion, the universe would be "suspiciously young." The problem was quantitative, not merely philosophical. Hubble's original coefficient implied a cosmic timescale of roughly 2 billion years. The expanding universe appeared younger than the Earth itself, whose age was already estimated to exceed two billion years through radioactive dating. The first great success of expansion therefore arrived carrying an age paradox inside its own arithmetic. He noted that if the redshift arose from an unknown principle of nature that did not involve actual spatial motion, as Zwicky had independently proposed 11, the data would describe a universe extended indefinitely in both space and time. The same Zwicky who proposed that the redshift was a non-kinematic energy loss in 1929 would, four years later, invoke *dunkle Materie* to account for the Coma cluster's velocity dispersion 61. The historical juxtaposition is striking: within four years, the same physicist had proposed two unseen mechanisms—photon energy loss for cosmological redshift and unseen matter for cluster dynamics—to reconcile observation with prevailing theory.

Einstein's own intellectual closure on the matter was less absolute than the standard history implies. A recently analyzed 1931 manuscript 20 demonstrates that he actively explored an expanding steady-state model with continuous matter creation, even as he was publicly conceding to the expanding models. This was not a static cosmology but an attempt to preserve an eternal universe within the dynamical framework. Furthermore, Einstein recognized early on that substituting Hubble's original constant into the expansion equations yielded a highly problematic cosmic age, highlighting an immediate tension between the kinematic interpretation and local physical timelines.

The spatial interpretation of redshift generated significant physical friction from its inception. It became the dominant paradigm not because the evidence uniquely demanded it, but because the physics community possessed a theory of dynamic geometry while lacking a mathematically rigorous theory of dynamic proper time.

Spatial expansion and a finite cosmic origin were, moreover, historically separate propositions. Friedmann and Lemaître supplied dynamical geometries; Lemaître introduced the primeval-atom hypothesis separately; and later steady-state cosmology accepted expansion while rejecting a beginning. The term "Big Bang" itself did not appear until Fred Hoyle's 1949 radio broadcast 41. Mount Wilson therefore did not observe the Big Bang. It supplied evidence that was progressively incorporated into a much larger theoretical architecture. Each step in that chain, from redshift to spatial expansion, from expansion to finite age, from finite age to hot primordial origin, was an additional inference, not a new mark on Humason's plate.

Cosmology has always been more chronometric than its spatial language suggests. Redshift is measured as a ratio of atomic frequencies. The Cepheid distance ladder begins with a stellar period. Supernova standardization depends partly upon the evolution of a light curve through time. A gravitational-wave chirp is reconstructed from accumulated phase. The question raised by TEP is therefore not whether clocks belong in cosmology. They already constitute its instruments. The question is whether their calibration is merely passive, or whether it is itself a dynamical part of the universe.

## 3. The Hidden Closure

General relativity does not claim that all clocks tick at the same rate. It correctly predicts gravitational and kinematic clock-rate differences. That is not the Isochrony Axiom.

The deeper restriction is that standard single-metric relativity derives every physical clock from the same spacetime metric:

\begin{equation}
S_m = S_m[g_{\mu\nu},\,\psi_m], \qquad d\tau_m^2 = -g_{\mu\nu}\,dx^\mu dx^\nu .
\end{equation}

Once the metric and the clock's worldline are specified, no further universal chronometric variable remains. Every permitted difference between matter clocks is assumed to be exhausted by $g_{\mu\nu}$.

On the homogeneous FLRW background, this closure appears as the identification

\begin{equation}
d\tau_{\rm com} = dt .
\end{equation}

This equation is a valid choice of time coordinate; it is not itself the Isochrony Axiom. The deeper assumption is that, once the metric and motion of a clock are known, nothing else in nature can independently alter its underlying calibration. Adding a scalar field to the cosmic energy budget does not by itself break this closure. Isochrony is broken only when a new field enters the universal matter metric and therefore changes the proper time registered by physical clocks. TEP removes that restriction by allowing the scalar field $\phi$ to enter the universal matter metric.

Isochrony Axiom. In a minimally coupled single-metric theory, the metric and a clock's worldline exhaust the physical calibration of matter proper time. After metric-derived gravitational and kinematic effects have been included, no independent dynamical field remains that can universally rescale matter-clock calibration between epochs.

This is not the claim that all clocks share one rate. It is the closure condition encoded by universal minimal coupling to one metric.

This axiom closes the temporal sector before cosmological redshift is interpreted. The homogeneous cosmological redshift must therefore be encoded in the remaining available geometric quantity, the spatial scale factor $a(t)$. Mount Wilson established the redshift. Isochrony selected the sector of the theory in which that redshift would reside. Removing the axiom reopens the temporal sector. The scalar field $\phi$ defines a spatiotemporal clock geometry termed the Temporal Topology: its homogeneous evolution governs cosmological clock calibration, while its inhomogeneous configurations govern local and galactic departures. In inhomogeneous disformal configurations, the clock-transport connection constructed from $\phi$ and its gradients can become non-integrable, producing synchronization holonomy.

The historical narrative has now reached its formal turning point. The Isochrony Axiom identified the closure that channeled redshift into spatial expansion; removing it requires a mathematical structure in which matter proper time is governed by more than the single metric $g_{\mu\nu}$. The next section constructs that structure: a second, scalar-dependent matter metric $\tilde g_{\mu\nu}$, coupled universally to all Standard Model fields, whose conformal factor $A(\phi)$ carries the dynamical degree of freedom that the axiom had suppressed. The Mount Wilson redshift will then be rederived not as the stretching of space, but as the ratio of matter-clock calibrations between emission and observation.

## 4. The Mount Wilson Equivalence Theorem

The Temporal Equivalence Principle (TEP) reopens the suppressed temporal degree of freedom by restoring an independent dynamical scalar proper-time field. TEP defines two related geometries: the bare gravitational metric ($g_{\mu\nu}$), and the causal matter metric ($\tilde g_{\mu\nu}$) to which standard rods, clocks, and photon fields couple.

These geometries are connected by a universal, dynamical scalar proper-time field ($\phi$). In the canonical preferred-foliation decomposition of Paper 0 (v0.16, §2.2, Step 80) along the unit cosmic temporal flow $U_\mu \equiv -\nabla_\mu\bar\phi/\sqrt{2\bar X}$ ($g^{\mu\nu}U_\mu U_\nu = -1$) and orthogonal spatial projector $\Pi_{\mu\nu} = g_{\mu\nu} + U_\mu U_\nu$, the causal matter metric is defined as:

\begin{equation}
\tilde g_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))\,U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\,\Pi_{\mu\nu} + B(\phi)\,\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi ,
\end{equation}

written in isotropic conformal–disformal shorthand as $\tilde g_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi$. Here $A_{\rm loc}(u_{\rm grav})$ governs the stationary baryon-sourced temporal lapse $N_{\rm TEP}(\mathbf x) = A_{\rm loc}(u_{\rm grav}(\mathbf x))N_{\rm GR}(\mathbf x)$, $A_{\rm sp}(u_{\rm drift}) = \exp(\beta_A u_{\rm drift})$ governs the secular matter-frame spatial ruler calibration, and $B(\phi)$ governs spatially projected disformal kinetic corrections. On the static, homogeneous cosmological branch ($u_{\rm grav} = 0$, $A_{\rm loc} = 1$, $\Pi_\mu^\alpha\nabla_\alpha\phi = 0$), the spatial gravitational metric is invariant (the scale factor $a = 1$), the spatially projected disformal term vanishes identically, and the cosmological factor is $A_{\rm sp}(t) \equiv A(\phi(t))$.

The following derivation establishes the central result on both the canonical preferred-foliation metric (where $N(t) = A_{\rm loc} = 1$ on the homogeneous background) and the general unprojected isotropic-disformal comparison branch $\tilde g_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$. On the homogeneous cosmological branch, $\phi = \phi(t)$; define the general effective lapse ($N(t) \equiv 1$ on the preferred-foliation branch, or

\begin{equation}
N^2(t) = A^2(\phi) - B(\phi)\,\dot\phi^2 > 0
\end{equation}

on the unprojected isotropic-disformal branch) and denote the cosmological ruler factor by $A(t) \equiv A_{\rm sp}(t)$. The matter-frame line element and comoving proper time are then

\begin{equation}
d\tilde s^2 = -N^2\,dt^2 + A^2\,\delta_{ij}\,dx^i dx^j, \qquad d\tilde\tau = N\,dt ,
\end{equation}

and the normalized comoving matter-frame four-velocity is

\begin{equation}
\tilde u^\mu = \left(\frac{1}{N},\, 0,\, 0,\, 0\right) .
\end{equation}

Spatial homogeneity supplies three translational Killing vectors. Consequently, the covariant spatial wavevector

\begin{equation}
q_i = \tilde g_{ij}\,k^j = A^2\,\delta_{ij}\,k^j
\end{equation}

is conserved along the photon trajectory. Writing $q^2 = \delta^{ij}q_i q_j$, the null condition $\tilde g_{\mu\nu}\,k^\mu k^\nu = 0$ gives

\begin{equation}
N^2\,(k^0)^2 = \frac{q^2}{A^2} .
\end{equation}

The photon frequency measured by a comoving matter observer is

\begin{equation}
\omega = -\tilde g_{\mu\nu}\,\tilde u^\mu\,k^\nu = N\,k^0 = \frac{q}{A} .
\end{equation}

Since $q$ is conserved,

\begin{equation}
\boxed{1 + z = \frac{\omega_{\rm em}}{\omega_0} = \frac{A_{\rm sp}(t_0)}{A_{\rm sp}(t_{\rm em})} \equiv \frac{A_0}{A_{\rm em}}}
\end{equation}

On the canonical preferred-foliation metric the spatially projected disformal term vanishes identically on the homogeneous background ($\Pi_\mu^\alpha\nabla_\alpha\phi = 0$), and even on the unprojected isotropic-disformal branch ($A^2 - B\dot\phi^2 > 0$) any homogeneous disformal lapse $N(t)$ cancels identically from the redshift ratio. In inhomogeneous regions, however, the disformal connection can become non-integrable and produce synchronization holonomy.

### The Mount Wilson Equivalence Theorem

**Theorem:**

Mount Wilson Equivalence Theorem. On a homogeneous, static gravitational background with universal matter coupling to the canonical preferred-foliation metric (where $\Pi_\mu^\alpha\nabla_\alpha\phi = 0$, and as a corollary on the unprojected isotropic-disformal branch whenever $A^2 - B\dot\phi^2 > 0$), the endpoint redshift is $1+z = A_{\rm sp}(t_0)/A_{\rm sp}(t_{\rm em}) \equiv A_0/A_{\rm em}$, independently of any homogeneous lapse factor. It is therefore observationally degenerate, at the level of redshift alone, with the FLRW relation $1+z = a_0/a_{\rm em}$.

\begin{equation}
1+z = \frac{\omega_{\rm em}}{\omega_0} = \begin{cases} \dfrac{a_0}{a_{\rm em}}, & \text{FLRW spatial interpretation}, \\[6pt] \dfrac{A_0}{A_{\rm em}}, & \text{TEP temporal interpretation}. \end{cases}
\end{equation}

Mount Wilson measured the frequency ratio on the left. The choice between the two geometric descriptions on the right was made by theory. Isochrony selected the spatial branch; TEP reopens the temporal branch. The redshift degeneracy is broken only by observables that compare the gravitational and matter sectors, or by local non-integrable clock transport. Other cosmological observables provide additional consistency tests once a specific temporal-field dynamics is fixed.

The qualification "at the level of redshift alone" is essential. The theorem establishes that the redshift coordinate underlying the original Mount Wilson relation does not, by itself, distinguish an expanding spatial scale factor from dynamical matter-clock transport. The measured spectral shifts can be encoded either as the history of $a(t)$ or as the transport law of $A(\phi)$. Standard distance measurements do not immediately break this degeneracy: on the homogeneous branch, the luminosity distance $d_L(z)$ depends on the same integral over the matter-frame geometry that in FLRW depends on $a(t)$, so a suitable evolution of $A(\phi)$ can reproduce the FLRW distance-redshift relation to any desired precision, as demonstrated by the exact conformal reconstruction of the Pantheon+ distance-modulus relation in TEP-C0 29. The degeneracy is broken not by any single-metric observable but by inter-sector comparisons that probe both $g_{\mu\nu}$ and $\tilde g_{\mu\nu}$ simultaneously. Luminosity distance, time dilation, baryon acoustic oscillations, the cosmic microwave background, and standard sirens each require separate consistency checks that are addressed in the specialized TEP corpus.

### 4.1 Static Space versus Conformal-Frame Equivalence

The equality of the two redshift laws does not make TEP a change of notation. In FLRW, one evolving metric governs matter, light and tensor propagation. In TEP, the gravitational metric remains spatially static while matter and electromagnetic fields couple to the scalar-dependent metric $\tilde g_{\mu\nu}$. A conformal or disformal change of variables may move $A(\phi)$ between sectors, but it cannot simultaneously map both the gravitational propagation sector ($g_{\mu\nu}$) and the causal matter sector ($\tilde g_{\mu\nu}$) to a minimally coupled single-metric FLRW spacetime. The two sectors carry independent physical content; no single field redefinition collapses them into one.

The physical distinction therefore appears in comparisons between the two sectors. Gravitational radiation propagates through $g_{\mu\nu}$, while electromagnetic signals and material detectors are governed by $\tilde g_{\mu\nu}$. Standard-siren gravitational-wave observations paired with electromagnetic counterparts serve as a principal inter-sector discriminator: the gravitational wave measures distance through the bare gravitational frame, while the electromagnetic counterpart measures redshift through the matter frame. Define the dimensionless siren deviation

\begin{equation}
\Delta_{\rm siren}(z) \equiv \frac{d_L^{\rm GW}(z)}{d_L^{\rm EM}(z)} - 1 .
\end{equation}

Here, $d_L^{\rm GW}$ is the luminosity distance measured from gravitational waves, while $d_L^{\rm EM}$ is the corresponding luminosity distance inferred through electromagnetic observations. If matter and gravity follow different effective geometries, the two need not agree. In minimally coupled single-metric FLRW, both sectors share one metric, so $\Delta_{\rm siren} = 0$. Under the canonical TEP radiation rule (Paper 22), $\Xi(z)=1/(1+z)$ and $\Delta_{\rm siren}=1/(1+z)-1$. In Paper 22 (v0.2, Maputo), the full GWTC-3 bright, dark, and spectral-siren population likelihoods are evaluated under this canonical rule, showing that the apparent $H_0 \simeq 56\ {\rm km\,s^{-1}\,Mpc^{-1}}$ free-fit dark-siren posterior shift ($\Delta\chi^2 = +1.16$) is mechanically driven by $\Lambda$CDM-gated galaxy-catalog host redshifts ($70.0\langle(1+z_{\rm cat})^{-1}\rangle = 55.1\ {\rm km\,s^{-1}\,Mpc^{-1}}$), whereas self-consistent TEP-kernel reassociation at the fixed canonical $H_0 = 68.18\ {\rm km\,s^{-1}\,Mpc^{-1}}$ (Paper 18) recovers the baseline association likelihood ($-2\ln L = 944.31$) without shifting $H_0$, while the separate lab-fixed small-excursion comparison yields $H_0 \simeq 72.2$ versus $72.0\ {\rm km\,s^{-1}\,Mpc^{-1}}$ ($\Delta\chi^2 = +0.20$) at marginal significance. This dimensionless ratio is an invariant inter-sector observable: it cannot be removed by any conformal or disformal field redefinition, because no single transformation simultaneously maps both sectors to a common minimally coupled metric. The technical details of the frame analysis, including engagement with the existing conformal-frame literature, are carried in Appendix C.

TEP therefore makes a sharp empirical prediction. If cosmological redshift is matter-clock transport, the same temporal field must leave dimensionless signatures wherever matter-clock calibration is compared with gravitational propagation, or wherever proper time is transported around non-integrable paths. A consistent null result across those channels would reject the TEP interpretation. A common signal across them would identify a missing dynamical sector of geometry.

## 5. Einstein's Unfinished Continuum

The standard narrative presents Einstein's 1931 acceptance of expansion as a clean conversion. The archival record is more nuanced. From his earliest gravitational writings through his final decades, Einstein repeatedly sought a deterministic, eternal continuum in which temporal and geometric structure were deeper than the expanding metric allowed. He did not anticipate TEP, and it would be historically inaccurate to claim that he did. Yet the questions he refused to abandon are precisely the questions this paper resumes.

Recent historical scholarship has further corrected the standard account of the conversion itself. Einstein was not persuaded by Hubble's plates alone. He was persuaded by the theoreticians: first by Eddington, who hosted Einstein at Cambridge in June 1930 and had just demonstrated the instability of the static model; then by Tolman, with whom Einstein discussed cosmology intensely during the Pasadena weeks of January and February 1931; and finally by de Sitter, with whom Einstein co-authored the 1932 Einstein-de Sitter model 62. Hubble's observations supplied the empirical context, but the decision to abandon the static universe was a theoretical one, made in conversation with the men who had the mathematical machinery to frame it.

In 1911, Einstein examined the influence of gravity on light propagation and derived a coordinate-dependent speed of light that varied with gravitational potential 8. The formalism was superseded by the full tensor geometry of 1915, but the underlying concern, that the behavior of clocks and light is governed by deeper structure than a single rigid coordinate time, persisted throughout his career.

The 1917 cosmological memoir 6 introduced the static, spatially closed, Machian universe with the cosmological constant. The model was motivated not by observational data but by a philosophical commitment to a stable, self-consistent cosmos. Friedmann and Lemaître supplied dynamical alternatives to Einstein's model; Eddington then demonstrated in 1930 that the static equilibrium itself was unstable 14. Richard Tolman, Einstein's close colleague at Caltech, was pioneering the application of relativistic thermodynamics to cosmology. Far from regarding expansion as proof that the universe must have begun at a finite time and end in heat death, Tolman argued that relativistic thermodynamics did not require either conclusion. He investigated cyclic models and considered it plausible that cosmic phenomena might extend indefinitely into both past and future 17,55. His discussions with Einstein therefore show that the eternity of the universe remained an active scientific question even after spatial expansion had been accepted. The intellectual environment at Caltech during Einstein's 1931 visit reinforced this openness. His host, Robert Millikan, had named and championed the study of cosmic rays—radiation first discovered by Victor Hess in 1912—and interpreted them as the "birth cries of atoms being born in interstellar space" 68,73. Millikan championed a cosmology of continuous creation and renewal, an eternal, self-sustaining universe in opposition to the disintegrating cosmos of James Jeans. When Einstein arrived in Pasadena, the scientific table was already set with fierce local debates over the eternity of the cosmos. The choice was not merely between abstract tensor equations but between competing visions of cosmic permanence, defended by heavyweights in Einstein's immediate circle.

For Einstein, whose Machian and thermodynamic intuitions pointed toward an eternal universe in stable equilibrium, a cosmos that began and would die was not merely unaesthetic. It was a violation of the deepest principles he held. Nernst had independently advocated a similar commitment to a stationary cosmos on thermodynamic grounds 15. The eternal-universe question was therefore not settled by the acceptance of expansion; it was deferred while the dynamical geometry was developed. When Lemaître proposed the primeval-atom hypothesis in 1931, Einstein privately told him: "No, that no, that suggests creation too much" 63. His public applause at Lemaître's 1933 Pasadena seminar is frequently presented as an endorsement of the primeval atom. The historical record is less definite: Einstein may have been responding principally to Lemaître's treatment of cosmic rays rather than endorsing a physical beginning of the universe. Accepting spatial expansion did not therefore mean that Einstein had accepted a finite cosmic origin.

Einstein did not simply abandon the conceptual program; he sought alternative geometric structures that might preserve it. TEP does not restore the old equilibrium model. It proposes something different, a universe that is spatially static but temporally dynamical. In TEP-C0 29, the static branch is maintained not by Einstein's curvature–Λ equilibrium but by the dynamical evolution of $\phi$. The famous "biggest blunder" formulation, in which Einstein is said to have regretted the cosmological constant, is known principally through George Gamow's later recollection 40 rather than a surviving contemporary statement. Einstein's own documented view is more precise. In a 1947 letter to Lemaître, he wrote: "Since I have introduced this term I always had a bad conscience... I found it very ugly indeed that the field law of gravitation should be composed of two logically independent terms which are connected by addition. About the justification of such feelings concerning logical simplicity it is difficult to argue. I cannot help but feel it strongly and I am unable to believe that such an ugly thing should be realised in nature" 66. The objection was not observational but structural: the cosmological term was an ad hoc addition to a field law that should possess logical unity. TEP addresses the same structural objection by replacing an independent cosmological constant with the potential dynamics of the temporal field, so that the cosmological effect is carried by the same field that governs matter-clock calibration.

Teleparallelism, introduced in 1928 9, was one such attempt. Einstein sought a geometric framework in which torsion, rather than curvature, carried gravitational structure, allowing for a richer set of field configurations than standard Riemannian geometry. The programme failed to unify gravitation with electromagnetism, but it reveals the depth of his dissatisfaction with the standard metric as the final word on spacetime structure.

Einstein's 1939 analysis 7 argued that a stationary cluster of orbiting particles could not be compressed to the Schwarzschild radius, which he took as evidence against physical collapse. The argument did not exclude dynamical collapse, but it records his continuing resistance to singular endpoints and his conviction that a complete geometry should remain continuous. The Einstein-Rosen programme of the 1930s pursued a similar geometric continuity, seeking bridge-like solutions that avoided singular interiors. Einstein was not alone in this resistance. In January 1935, at a meeting of the Royal Astronomical Society, Arthur Eddington publicly attacked Subrahmanyan Chandrasekhar's derivation of a maximum mass for white dwarfs, declaring that he thought "there should be a law of Nature to prevent a star from behaving in this absurd way" 59. Eddington's reaction is often dismissed as the prejudice of an aging establishment, but it expresses the same conviction that motivated Einstein's 1939 paper: that a complete physical theory should not permit matter to disappear into a geometric infinity. The TEP resolution of compact objects through temporal-well structure rather than spatial singularities, developed in TEP-BH 32, supplies the classical mechanism that both figures sought.

A recently analyzed 1931 manuscript 20 reveals that Einstein explored an expanding steady-state model with continuous matter creation, not a static spatial cosmology. The model was an attempt to preserve an eternal universe within the expanding framework, but it was not a hidden version of TEP. Einstein continued to seek cosmological architectures in which the universe did not possess a finite origin, even after accepting the observational reality of redshift. The manuscript was written on American notepaper during the Pasadena visit, and Einstein's correspondence from the period consistently misspells Hubble's name as "Hubbel" 20—a small archival reminder of how immediate and unsettled the transition still was.

In his final letter concerning Michele Besso, written on 21 March 1955, four weeks before his own death, Einstein returned once more to the divide between experienced time and physical description: "Now he has again preceded me a little in parting from this strange world. This has no importance. For people like us who believe in physics, the separation between past, present and future has only the importance of an admittedly tenacious illusion" 10. The statement is often read as a consolatory gesture. It is also a fitting final statement from a physicist who spent five decades searching for a deeper geometric unity. The question of time remained present at the end of his intellectual journey.

The TEP framework does not claim Einstein as a precursor. It claims that the questions he separated, the calibration of time, the structure of compact geometry, the determinism of physical law, and the eternity of the cosmos, share a common mathematical resolution when proper time is treated as a dynamical scalar field. Mount Wilson should not be understood as the site of an instantaneous conversion. Einstein's transition was gradual, unfolding through discussions with Hubble, Tolman, de Sitter and others, and through his own unsuccessful attempts to preserve an eternal cosmology. The mountain represents the moment at which an accumulating theoretical and observational argument became impossible to ignore, not the instant at which every alternative disappeared.

The remainder of this paper supplies that resolution.

## 6. From Degeneracy to the Eternal Branch

Once the redshift is reassigned to the temporal sector, the following consequences follow directly and are developed quantitatively in the companion papers. It is stated plainly at the outset that the eternal-branch conclusion assembled below is conditional on those companion analyses (TEP-BBN, TEP-TH, TEP-C0, TEP-HC, TEP-JWST, TEP-GL, TEP-UCD, TEP-BH, TEP-WB, TEP-LENS) each standing independently; the present paper supplies the degeneracy theorem and the discriminating observables, not an independent validation of each downstream result. While standard cosmology and the TEP framework initially present an observational degeneracy regarding redshift, applying broader astrophysical evidence breaks the symmetry. The foundational TEP formalism 24 proposes that when the observed dynamics are reassigned to the temporal field, a sequence of persistent modern astrophysical anomalies finds a common resolution. This section organizes that resolution into four domains, each corresponding to a distinct class of observation that the spatial interpretation accommodated through auxiliary structure.

### 6.1 Redshift, Age, and the End of the Big Bang Deadline

If redshift is not a signature of spatial expansion, extrapolating the scale factor backward ($a(t) \to 0$) does not reveal the physical origin of the universe. The Big Bang becomes the boundary condition of the spatial reconstruction rather than a singular event in the physical continuum. The primordial nucleosynthesis framework is correspondingly reinterpreted: the observed light-element abundances are reproduced through eternal chemical evolution with a helium-4 asymptotic attractor, rather than requiring a single hot early-universe phase, as developed in TEP-BBN 33. This reinterpretation addresses the timeline crisis precipitated by the James Webb Space Telescope (JWST). High-redshift ($z > 10$) observations of mature, massive galactic assemblies do not require physically impossible baryonic formation rates within a few hundred million years. Under TEP, extreme redshift identifies a state of profound temporal separation, not a compulsory youth limit. The finite FLRW age ceiling is removed; individual formation histories remain finite and are determined by the local temporal map. These systems assembled via standard, sub-Eddington accretion physics within an eternal cosmology, unburdened by a finite-age constraint. The temporal-horizon cosmology, demonstrating the absence of a physical singularity, is developed in TEP-TH 30, and the unified resolution of the JWST high-redshift galaxy anomalies is developed in TEP-JWST 35. The eternal branch carries its own stability obligation, registered here rather than deferred: linearization of the homogeneous static system exhibits an Eddington-type secular mode with e-folding time $\sim a/c$, and control of the analogous mode in the inhomogeneous realization remains part of the open construction (Paper 0, §8).

### 6.2 Dark Energy and $H_0$

In a static-space theory, $H_0$ becomes an inferred redshift-distance slope rather than a literal expansion rate. The Hubble tension, the statistically significant discrepancy between early-universe and local determinations of $H_0$, is recognized as a predictable chronometric calibration discrepancy between standard candles embedded in different temporal environments. TEP-H0 31 tests whether an environment-dependent Cepheid clock bias can account for the Cepheid-calibrated local excess and reports that, within the TEP-native gauge, the resulting correction brings the local scale into agreement with the CMB inference ($0.45\sigma$ from Planck $\Lambda$CDM; $0.03\sigma$ from TEP-CMB internal consistency).

The nonlinearity of the apparent expansion rate, conventionally attributed to dark energy, emerges mathematically as the natural thermodynamic evolution of the temporal field. The background cosmological reconstruction, showing that a pure conformal reconstruction exactly reproduces the ΛCDM homogeneous distance-modulus relation for 1,701 Pantheon+ supernovae, and that the conservative physical no-Λ temporal-shear branch improves the standardized supernova likelihood, is developed in TEP-C0 29. TEP-HC 27 establishes, via a native *hi_class* implementation, preservation of the CMB acoustic peak structure (sound-horizon ratio $r_s^{\rm TEP}/r_s^{\rm \Lambda CDM} = 0.999994$) and a numerically stable linear pure-conformal perturbation closure, with Integrated Sachs-Wolfe residuals below $0.001\%$ across the acoustic spectrum. Together, these analyses propose that the apparent late-time acceleration can be reproduced without novel vacuum energy injections.

### 6.3 Dark Matter, Lensing, and Temporal Composites

On galactic scales, dense baryonic mass concentrations couple strongly to the proper-time field, generating structured gradients in the Temporal Topology. Under the selected two-branch kinetic action $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$ fixed by $a_0$, zero per-system free parameters), this localized temporal shear bends slow matter trajectories on its low-gradient branch — producing flat galactic rotation curves and the baryonic Tully–Fisher relation (TEP-UCD 45, Steps 10, 13) — while under the unstretched-space temporal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, rescaling the shared $(3+1)$ ADM spacetime lapse $N_{\rm TEP}=A(\phi)N_{\rm GR}$ at fixed spatial 3-metric $\gamma_{ij}$ so $c_{\rm GW}=c_\gamma$ identically) intermediate clock rates $A(\phi(x))$ along photon paths refract light at the matched phantom-mass scale (TEP-GL 25, TEP-LENS 49). The apparent excess mass inferred under the Isochrony Axiom, conventionally attributed to particulate dark matter, is reinterpreted as Phantom Mass: the dynamical and optical projection of temporal-field gradients across unstretched space. The cross-scale clock-saturation density $\rho_T\approx20\text{ g cm}^{-3}$ independently anchors the terrestrial GNSS correlation length via $R_T(M)\propto M^{1/3}$, while the kinetic shear transition scales as $r_*(M)=\sqrt{GM/g_t}\propto M^{1/2}$ (TEP-UCD 45).

The phenomenology of extreme compact objects represents deep temporal wells in continuous space, where darkness and observational inaccessibility arise from extreme clock-rate contrast rather than singular spatial collapse. TEP-BH 32 develops the horizon-free temporal-well branch under extreme temporal gradients, with recovery of standard observational signatures. The gravitational lensing evidence often cited as direct proof of dark matter 19 is likewise unified within the temporal shear framework: while isotropic conformal rescaling $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$ stretches space alongside time and cancels out of null rays, the unstretched-space temporal metric $\tilde g_{\mu\nu}=-A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu+A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu}$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$) retains intermediate clock rates $A(\phi(x))$ along every photon path, refracting light by $n(x)\simeq1-2\Phi_N(x)/c^2-\beta_A u_{\rm grav}(x)$ on the selected two-branch action $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$ fixed by $a_0$, zero per-system free parameters) and reproducing the Bullet Cluster galaxy-peak convergence (TEP-GL, Step 02), the SN Refsdal Fermat delay and arrival order (TEP-LENS, Step 64), and the Planck CMB lensing cosmic-web amplification (TEP-HC, Step 21) from baryons alone, alongside the Gaia DR3 wide-binary screening transition (TEP-WB 46) and the SPARC radial-acceleration relation ($0.2185$ dex rms at $\Upsilon_d=0.5$ and $0.2004$ dex at $\Upsilon_d=0.35$ under action-derived addition $g_{\rm bar}+a_\phi$, $0.1974$ dex under the quadrature comparator; TEP-UCD, Step 10). Across all these domains, the anomalous dynamics and lensing were observed correctly; their physical origin is located in the temporal sector across unstretched space rather than in particulate dark matter.

### 6.4 Quantum Phase and the Same Clock Field

The same matter-frame proper time that governs cosmological redshift and galactic temporal shear also controls quantum phase accumulation. In the TEP-QF 34 branch, the scalar field $\phi$ enters the quantum mechanical phase as the proper-time parameter along matter trajectories, so that variations in the temporal field produce measurable shifts in interference patterns, spin coherence, and entanglement phase. This completes the cross-scale synthesis: the temporal field that governs cosmological redshift also governs the phase of quantum matter, unifying the largest and smallest scales under a single dynamical proper-time structure. TEP-HUB does not develop the quantum formalism; it identifies the common structural principle and routes the detailed derivation to TEP-QF 34, which establishes the Dirac-limit correspondence between the temporal scalar and standard quantum proper time.

### 6.5 Comparative Paradigm Matrix

The structural shift can be summarized at a glance:

| Domain | ΛCDM | TEP |
|---|---|---|
| Nature of matter time | Metric-derived proper time; $d\tau_{\rm com} = dt$ on the homogeneous background | Matter proper time additionally calibrated by $\phi$ |
| Cosmological redshift | Spatial scale factor ($a(t)$) | Endpoint temporal calibration ratio ($A_0/A_{\rm em}$) |
| Apparent acceleration | Dark energy (Λ) | Temporal field thermodynamic evolution |
| $H_0$ tension | Distance-ladder systematics or early- and late-universe model extensions | Cepheid clock bias from local temporal topology |
| Galactic rotation / lensing | Particulate dark matter halos | Temporal shear ($\Gamma_t$) and unstretched-space temporal refraction ($n(x)$); dynamical & optical Phantom Mass from baryons |
| Compact objects | Singularities, event horizons | Deep temporal wells, continuous space |
| High-$z$ galaxy assembly | Finite FLRW age ceiling | Temporal horizon; finite local assembly histories |
| Cosmic origin | Finite-age hot FLRW extrapolation; classical initial boundary | Temporal horizon; eternal continuum |
| Quantum phase | Standard proper time from $g_{\mu\nu}$ | Proper-time scalar as phase parameter |
| Empirical sector test | No matter/tensor-frame separation | Gravitational-versus-electromagnetic propagation deviation |

The matrix summarizes how TEP relocates the same anomalies from the spatial sector to the temporal sector. The right column states the cross-scale consequences TEP derives from one structural change—the restoration of a dynamical temporal field—and each consequence remains separately testable in the cited work.

## 7. The Arrival of Chronometric Astronomy

In 1931, Mount Wilson possessed unprecedented optical reach but entirely lacked precision atomic chronometry. The astronomers of the era were forced to operate blindly regarding the calibration of time: the temporal sector could not be empirically interrogated, and the closure that this paper names the Isochrony Axiom was imposed by the absence of any instrument capable of testing it. Today, that technological limitation has been definitively overcome.

The critical empirical test of the TEP framework lies in terrestrial chronometric astronomy. The dynamics of the proper-time field can be measured directly within the Earth's local orbital environment. While standard general relativity correctly predicts that clocks run at different rates due to varying altitude and velocity, TEP predicts additional structured, nonlocal clock correlations arising from inhomogeneous configurations of the Temporal Topology, for which the induced clock-transport connection can become non-integrable. The question of what precision tests of general relativity actually measure under a non-minimal matter coupling, showing that standard PPN tests do not constrain the temporal sector, is examined in TEP-EXP 26.

The 25-year archive of Precise Satellite Clock Products from Global Navigation Satellite Systems (GNSS) provides a vast, continuous dataset of orbital chronometry. Analyses of GPS and GLONASS clock products report structured clock-residual correlations, after conventional relativistic and processing terms are modeled, that are consistent with the local Temporal Topology gradients predicted by TEP, as synthesized in TEP-GTE 44. The extracted signals survive outlier rejection and frequency-standard normalization protocols. The full chronometric pipeline, including baseline selection protocols and Monte Carlo significance estimation, is open-sourced via the TEP-GNSS 28 repository. TEP-J0437 47 reports a synchronization-holonomy candidate in pulsar scintillation, while TEP-LLR 48 reports a compactness-dependent temporal response in Lunar Laser Ranging. Together they provide convergent tests through distinct observational channels.

Modern precision chronometry finally provides the instrument that Einstein and Hubble lacked: a mechanism to empirically interrogate whether proper-time transport possesses independent cosmological dynamics. The same scalar field $\phi$ whose homogeneous evolution produces cosmological redshift is the field whose inhomogeneous gradients are being probed by GNSS, pulsar scintillation, and Lunar Laser Ranging. The GNSS signals do not by themselves identify the same field as the cause of cosmological redshift; they provide a local, testable signature of nontrivial temporal structure that is consistent with the cosmological framework developed in the preceding sections.

## Epilogue: Return to Mount Wilson

Once cosmological redshift was assigned to spatial expansion, every later anomaly was necessarily interpreted inside that assignment: unseen mass, vacuum energy and increasingly restrictive cosmic timelines were introduced without reopening the original chronometric closure. The issue is not that these additions lacked empirical motivation. It is that the sector in which their solutions were sought had already been fixed.

Returning to the historical moment at Mount Wilson, the narrative changes. Albert Einstein's subsequent decades-long search for a unified, deterministic continuum was not the retreat from empiricism it is sometimes portrayed to be. As Section 5 has shown, he was systematically encountering different manifestations of the same missing temporal degree of freedom, from the 1911 coordinate-dependent light speed through teleparallelism, the Einstein-Rosen programme, and the 1931 steady-state manuscript, to the Besso letter's unresolved concern with the nature of time. Edwin Hubble, through his persistent refusal to declare his "apparent velocities" as absolute physical expansion, rigorously preserved the crucial distinction between observation and theoretical inference.

Mount Wilson would later revise the universe it had helped create. Hubble's original expansion rate of approximately 500 km/s/Mpc implied a cosmic age of roughly 2 billion years, younger than the Earth's own rocks. This age crisis persisted for two decades. During the wartime blackouts of Los Angeles, Walter Baade used the darkened skies to resolve distinct stellar populations in Andromeda 50. The resulting Cepheid recalibration showed that the accepted galactic distances had been underestimated by approximately a factor of two 39, halving the expansion rate and doubling the cosmic age. The redshifted lines had not moved; the distance scale through which they were interpreted had changed. The universe became larger and older through calibration alone. The episode carries a direct lesson for the present: the same mountain that produced the redshift-distance relation later demonstrated that its distance axis was not fixed by the plates themselves but by the calibration framework applied to them, whose central observable was itself temporal, the Cepheid period. The episode provides an important historical precedent for the modern Hubble tension, in which early-universe and local determinations of $H_0$ disagree: a major cosmological discrepancy was reduced by recalibrating the standard candle rather than changing the measured redshift. TEP proposes that the modern tension likewise contains a chronometric calibration component: the Cepheid clock is not a passive ruler but a matter-frame instrument whose calibration responds to the local temporal topology.

The photograph at Mount Wilson therefore records more than Einstein's encounter with an expanding universe. It records the moment when a frequency shift was assigned to the geometry of space because physics possessed no corresponding geometry of dynamical time. The telescope had answered correctly. The unfinished question was where, within the structure of reality, its answer belonged.

The Mount Wilson Paradigm is therefore not a collection of photographic plates, but the interpretive closure applied to them. TEP does not return the universe to Einstein's static equilibrium. It proposes something different: an eternal universe in which the observed evolution belongs primarily to time rather than space. Mount Wilson measured the redshift. Precision chronometry can now test where that redshift truly resides.

## Appendix A: Canonical TEP Metric and Cosmological Ansatz

The rigorous foundation of the Temporal Equivalence Principle 24 relies on the distinction between the gravitational metric ($g_{\mu\nu}$) and the causal matter metric ($\tilde g_{\mu\nu}$). In the canonical preferred-foliation decomposition (Paper 0 v0.16, §2.2, Step 80) along the unit cosmic temporal flow $U_\mu \equiv -\nabla_\mu\bar\phi/\sqrt{2\bar X}$ ($g^{\mu\nu}U_\mu U_\nu = -1$) and orthogonal spatial projector $\Pi_{\mu\nu} = g_{\mu\nu} + U_\mu U_\nu$, the causal matter metric is defined as:

\begin{equation}
\tilde g_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))\,U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\,\Pi_{\mu\nu} + B(\phi)\,\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi ,
\end{equation}

which is written in isotropic conformal–disformal shorthand as $\tilde g_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi$. Here, $\phi$ is a dynamical scalar proper-time field, with $A_{\rm loc}(u_{\rm grav}) = \exp(\beta_A u_{\rm grav})$ governing local gravitational clock depth and $A_{\rm sp}(u_{\rm drift}) = \exp(\beta_A u_{\rm drift}) \equiv A(\phi)$ governing cosmological ruler drift ($\beta_A = -1$). The canonical scalar comparison benchmark of the Paper-0 action is:

\begin{equation}
S = \int d^4x\,\sqrt{-g}\left[\frac{M_{\rm Pl}^2}{2}\,R - \frac{1}{2}\,g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi - V(\phi)\right] + S_m[\tilde g_{\mu\nu},\,\psi_m]
\end{equation}

where the operative screening realization replaces the canonical scalar term by the selected two-branch kinetic sector $P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+X|X|/\Lambda_X^4$ ($k\simeq16.03$ derived from $a_0$, carrying no canonical linear-$X$ low-gradient term) and embeds the preferred-foliation geometry in the spatially covariant ADM/khronon action of Paper 0 (§2.2, Step 80). The redshift identity below depends on the stated metric and homogeneous geometry, not on replacing that nonlinear action by the canonical scalar comparison benchmark. The matter action $S_m$ couples universally to $\tilde g_{\mu\nu}$, ensuring that the proper-time calibration of all physical clocks and the trajectories of all Standard Model fields are governed by the local value and gradient of $\phi$. The vector $\nabla_\mu\phi$ is strictly timelike on the cosmological branch to ensure Lorentzian signature and the invertibility of the effective metric. On the canonical preferred-foliation metric, $u_{\rm grav} = 0$ and $\Pi_\mu^\alpha\nabla_\alpha\phi = 0$ on the homogeneous branch, so Lorentzian invertibility requires only $A_{\rm sp} > 0$ (while on the unprojected isotropic-disformal comparison branch, $A^2 - B\dot\phi^2 > 0$ is additionally required). The static cosmological ansatz fixes the spatial components of the background gravitational metric ($g_{ij}$) to a time-independent scale ($a = 1$).

## Appendix B: Derivation of Cosmological Redshift

The full derivation is presented in Section 4. The key steps are summarized here for reference. On the homogeneous, static branch with $\phi = \phi(t)$, $A(t) \equiv A_{\rm sp}(t)$, and effective lapse $N(t) = 1$ on the canonical preferred-foliation branch (or $N^2(t) = A^2(\phi) - B(\phi)\,\dot\phi^2 > 0$ on the unprojected isotropic-disformal branch), spatial homogeneity supplies the Killing fields of the maximally symmetric spatial slice, so the covariant spatial wavevector $q_i = A_{\rm sp}^2\,h_{ij}\,k^j$ is conserved along the photon trajectory. The null condition gives $N^2(k^0)^2 = q^2/A_{\rm sp}^2$, and the measured photon frequency is:

\begin{equation}
\omega = -\tilde g_{\mu\nu}\,\tilde u^\mu\,k^\nu = N\,k^0 = \frac{q}{A_{\rm sp}}
\end{equation}

Since $q$ is conserved, the endpoint redshift relation follows:

\begin{equation}
1+z = \frac{A_{\rm sp}(t_0)}{A_{\rm sp}(t_{\rm em})} \equiv \frac{A_0}{A_{\rm em}}
\end{equation}

This endpoint ratio is path-independent on the homogeneous background, representing a conformal temporal redshift rather than a kinematic stretching of space. The disformal term $B(\phi)$ enters through $N$ but cancels from the homogeneous endpoint ratio. True synchronization holonomy, which requires a non-integrable clock-transport connection, arises in inhomogeneous disformal configurations where the connection constructed from $\phi$ and its gradients becomes non-exact. The specialized corpus studies address these inhomogeneous regimes: TEP-GL 25 derives the temporal shear factor $\Gamma_t$ and its application to rotation curves and lensing, while TEP-BH 32 develops the horizon-free temporal-well branch under extreme temporal gradients.

## Appendix C: The Matter-Frame Static-Space Distinction

A critical question must be answered directly: if $\tilde g_{ij} = A^2(\phi)g_{ij}$ and $A$ changes cosmologically, the matter-frame spatial metric exhibits a time-dependent scale. How does TEP differ from FLRW expansion written in a different conformal frame?

The answer rests on a precise operational definition. In TEP, "space is static" means that the gravitational metric $g_{ij}$, which governs the propagation of gravitational radiation and defines the bare spatial geometry, is time-independent. The factor $A^2$ changes the calibration of matter rods relative to the static gravitational geometry; it does not introduce an independently dynamical spatial scale factor. The distinction from FLRW is not that matter rods see no expansion; it is that the expansion they see is a derived consequence of temporal field evolution, not a fundamental degree of freedom of the gravitational sector.

The decisive question is not which metric is described as static, but whether TEP predicts a dimensionless inter-sector observable unavailable in minimally coupled single-metric FLRW. A standard siren compares tensor generation and propagation in $g_{\mu\nu}$ with source redshift and detector calibration in $\tilde g_{\mu\nu}$. Once generation, propagation and matter-frame detection are treated consistently, TEP predicts a gravitational-to-electromagnetic distance relation that can test the separation of the two sectors. Additional distinctions arise in the angular-diameter distance at high redshift, the distance-duality relation, and the growth-rate of structure, each of which acquires corrections from the temporal field that are absent in standard FLRW. The full derivations of these discriminating tests are provided in TEP-C0 29 for the background cosmology and TEP-HC 27 for the growth and perturbation analysis.

The existing conformal-frame literature, including the disformal invariance analyses of Chiba, Chibana, and Yamaguchi 22 and the Minkowski-frame cosmology treatment of Lombriser 23, addresses the question of when a scalar-tensor theory can be reduced to FLRW by a field redefinition. Chiba et al. show that redshift remains invariant under a consistently performed disformal transformation, while Lombriser explicitly demonstrates that cosmology can be represented in a Minkowski frame without changing physical observations. The TEP framework is distinguished from those cases by its universal matter coupling to $\tilde g_{\mu\nu}$ combined with a static gravitational sector, which prevents a single conformal transformation from simultaneously mapping both sectors to minimally coupled FLRW. The multi-messenger test identified in Section 4.1 provides the physical observable that breaks the frame equivalence.

The obstruction is mathematical, not merely interpretive. Define $g'_{\mu\nu}=\Omega^2(\phi)\,g_{\mu\nu}$. The matter metric then becomes $\tilde g_{\mu\nu}=(A^2/\Omega^2)\,g'_{\mu\nu}+B\,\nabla_\mu\phi\,\nabla_\nu\phi$. Choosing Ω = A removes the conformal prefactor from the matter metric, although the disformal term remains; but it also gives $g'_{\mu\nu}=A^2\,g_{\mu\nu}$, making the gravitational metric explicitly time-dependent. Choosing a constant Ω preserves the static gravitational metric but leaves the matter sector A(φ)-dependent. Unless A is constant, no single conformal frame both preserves the static gravitational geometry and removes the temporal calibration from matter. This frame trade-off is why the multi-messenger siren deviation $\Delta_{\rm siren}$ cannot be dismissed as a relabelling of one minimally coupled metric.

## Appendix D: Corpus Evidence Routing

The TEP corpus systematically applies this framework to specific astrophysical anomalies. The table below outlines the primary routing of claims to their respective specialized derivations:

| Phenomenon | Standard Interpretation | TEP Framework Solution | Primary Reference |
|---|---|---|---|
| Background cosmology, apparent acceleration | Dark energy, ΛCDM | Temporal field thermodynamic evolution | TEP-C0 29 |
| Hubble Tension ($H_0$) | Distance-ladder systematics or early- and late-universe model extensions | Cepheid bias via local temporal topology | TEP-H0 31 |
| Growth and structure formation | Cold Dark Matter, ΛCDM perturbations | Temporal field perturbation consistency | TEP-HC 27 |
| High-$z$ Mature Galaxies (JWST) | Rapid, anomalous baryonic accretion | Removed FLRW age ceiling; finite local assembly | TEP-TH 30 / TEP-JWST 35 |
| Galactic Rotation Curves & Lensing | Cold Dark Matter halos | Temporal shear ($\Gamma_t$) & unstretched-space temporal refraction; dynamical & optical Phantom Mass | TEP-GL 25 / TEP-UCD 45 |
| Extreme Compact Objects | Singularities, event horizons | Deep temporal wells, continuous space | TEP-BH 32 |
| Big Bang Nucleosynthesis | Early-universe hot phase | Eternal chemical evolution; He-4 asymptotic attractor | TEP-BBN 33 |
| Quantum phase and coherence | Standard quantum mechanics | Proper-time scalar as quantum phase parameter | TEP-QF 34 |
| Precision Clock Anomalies | Processing noise, unexplained residuals | Local gradients in the proper-time field ($\phi$) | TEP-GNSS 28 / TEP-EXP 26 |
| GNSS program synthesis | — | Convergent multi-analysis temporal topology signature | TEP-GTE 44 |
| Cross-scale saturation scale | Dark matter $M$–$R$ relation | Universal $M^{1/3}$ scaling from $\rho_T$ | TEP-UCD 45 |
| Wide-binary screening | Newtonian breakdown / MOND | Temporal shear recovery with screening transition | TEP-WB 46 |
| Pulsar scintillation | Interstellar scattering | Synchronization holonomy candidate | TEP-J0437 47 |
| Lunar Laser Ranging | Nordtvedt-effect channel (historically tested as SEP violation) | Compactness-dependent temporal response | TEP-LLR 48 |
| Multiply-imaged supernovae | Standard lensing time delays | Blind-prediction temporal-shear residual | TEP-LENS 49 |

## References

[1] Hubble, E. (1929). A relation between distance and radial velocity among extra-galactic nebulae. *Proceedings of the National Academy of Sciences*, 15(3), 168–173.

[2] Hubble, E., & Humason, M. L. (1931). The velocity-distance relation among extra-galactic nebulae. *The Astrophysical Journal*, 74, 43–80.

[3] Hubble, E., & Tolman, R. C. (1935). Two methods of investigating the nature of the nebular red-shift. *The Astrophysical Journal*, 82, 302–337.

[4] Hubble, E. (1936). *The Realm of the Nebulae*. Yale University Press.

[5] Hubble, E. (1937). *The Observational Approach to Cosmology*. Oxford University Press.

[6] Einstein, A. (1917). Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie. *Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften*, 142–152.

[7] Einstein, A. (1939). On a stationary system with spherical symmetry consisting of many gravitating masses. *Annals of Mathematics*, 40(4), 922–936.

[8] Einstein, A. (1911). Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes. *Annalen der Physik*, 340(10), 898–908.

[9] Einstein, A. (1928). Riemann-Geometrie mit Aufrechterhaltung des Begriffs des Fernparallelismus. *Sitzungsberichte der Preußischen Akademie der Wissenschaften*, 217–221.

[10] Einstein, A. (1955). Letter to the family of Michele Besso, 21 March 1955. In *Albert Einstein: Michele Besso Correspondance 1903–1955*, Hermann (1972). English translation per Christie's transcription of the autograph letter, Princeton, 21 March 1955.

[11] Zwicky, F. (1929). On the red shift of spectral lines through interstellar space. *Proceedings of the National Academy of Sciences*, 15(10), 773–779.

[12] Lemaître, G. (1927). Un univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques. *Annales de la Société Scientifique de Bruxelles*, A47, 49–59.

[13] Friedmann, A. (1922). Über die Krümmung des Raumes. *Zeitschrift für Physik*, 10(1), 377–386.

[14] Eddington, A. S. (1930). On the instability of Einstein's spherical world. *Monthly Notices of the Royal Astronomical Society*, 90, 668–678.

[15] Nernst, W. (1937). Weitere Prüfung der Annahme eines stationären Zustandes im Weltall. *Zeitschrift für Physik*, 106(9–10), 633–661.

[16] de Sitter, W. (1917). On the relativity of inertia. Remarks concerning Einstein's latest hypothesis. *Proceedings of the Royal Academy of Sciences at Amsterdam*, 19, 1217–1225.

[17] Tolman, R. C. (1934). *Relativity, Thermodynamics and Cosmology*. Oxford University Press.

[18] Tully, R. B., & Fisher, J. R. (1977). A new method of determining distances to galaxies. *Astronomy and Astrophysics*, 54, 661–673.

[19] Clowe, D., Bradac, M., Gonzalez, A. H., Markevitch, M., Randall, S. W., Jones, C., & Zaritsky, D. (2006). A direct empirical proof of the existence of dark matter. *The Astrophysical Journal*, 648, L109–L113.

[20] O'Raifeartaigh, C., McCann, B., Nahm, W., & Mitton, S. (2014). Einstein's steady-state theory: an abandoned model of the cosmos. *The European Physical Journal H*, 39, 353–367.

[21] Bartusiak, M. (2009). *The Day We Found the Universe*. Pantheon Books. (Historical account of the Mount Wilson programme and the 1931 Pasadena visit.)

[22] Chiba, T., Chibana, F., & Yamaguchi, M. (2020). Disformal invariance of cosmological observables. *arXiv:2003.10633*.

[23] Lombriser, L. (2023). Cosmology in Minkowski space. *arXiv:2306.16868*.

[24] Smawfield, M. L. (2025). 0-TEP. *Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed*. Zenodo, v0.16 (Jakarta). https://doi.org/10.5281/zenodo.16921911

[25] Smawfield, M. L. (2025). 4-TEP-GL. *Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations*. Zenodo. https://doi.org/10.5281/zenodo.17982540

[26] Smawfield, M. L. (2025). 9-TEP-EXP. *What Do Precision Tests of General Relativity Actually Measure?*. Zenodo. https://doi.org/10.5281/zenodo.18109760

[27] Smawfield, M. L. (2026). 18-TEP-HC. *Temporal Equivalence Principle: Native hi_class Conformal Implementation, Linear Perturbation Closure, and CMB Acoustic Peak Preservation*. Zenodo. https://doi.org/10.5281/zenodo.20572722

[28] Smawfield, M. L. (2025). 1-TEP-GNSS. *Global Time Echoes: Distance-Structured Correlations in GNSS Clocks*. Zenodo. https://doi.org/10.5281/zenodo.17127229

[29] Smawfield, M. L. (2026). 26-TEP-C0. *Temporal Equivalence Principle: A Covariant Alternative to Cosmic Expansion*. Zenodo. https://doi.org/10.5281/zenodo.20370143

[30] Smawfield, M. L. (2026). 27-TEP-TH. *Temporal Equivalence Principle: Temporal Horizon Cosmology and the Absence of a Physical Big Bang Singularity*. Zenodo. https://doi.org/10.5281/zenodo.20723059

[31] Smawfield, M. L. (2026). 11-TEP-H0. *The Cepheid Bias: Resolving the Hubble Tension*. Zenodo. https://doi.org/10.5281/zenodo.18209702

[32] Smawfield, M. L. (2026). 28-TEP-BH. *Temporal Equivalence Principle: Black Holes and the Temporal Horizon*. Zenodo. https://doi.org/10.5281/zenodo.21677826

[33] Smawfield, M. L. (2026). 29-TEP-BBN. *Temporal Equivalence Principle: Dynamical Proper Time and the Illusion of Primordial Deuterium*. Zenodo. https://doi.org/10.5281/zenodo.21841147

[34] Smawfield, M. L. (2026). 23-TEP-QF. *Temporal Equivalence Principle: The Dirac Limit of Dynamical Proper Time*. Zenodo. https://doi.org/10.5281/zenodo.20572697

[35] Smawfield, M. L. (2026). 12-TEP-JWST. *Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies*. Zenodo. https://doi.org/10.5281/zenodo.19000827

[36] Hoskin, M. (1976). The 'Great Debate': What really happened. *Journal for the History of Astronomy*, 7, 169–182.

[37] Slipher, V. M. (1917). Nebulae. *Proceedings of the American Philosophical Society*, 56, 403–409.

[38] Leavitt, H. S., & Pickering, E. C. (1912). Periods of 25 variable stars in the Small Magellanic Cloud. *Harvard College Observatory Circular*, 173, 1–3.

[39] Baade, W. (1952). A revision of the extra-galactic distance scale. *Transactions of the International Astronomical Union*, 8, 397–399.

[40] Gamow, G. (1970). *My World Line: An Informal Autobiography*. Viking Press.

[41] Hoyle, F. (1950). *The Nature of the Universe*. Basil Blackwell, Oxford. (Based on BBC Radio broadcasts, March 1949.)

[42] Wirtz, C. (1924). De Sitters Kosmologie und die Radialbewegungen der Spiralnebel. *Astronomische Nachrichten*, 222, 21–26.

[43] Humason, M. L. (1929). The large radial velocity of N. G. C. 7619. *Proceedings of the National Academy of Sciences*, 15(3), 167–168.

[44] Smawfield, M. L. (2025). 5-TEP-GTE. *Global Time Echoes: Empirical Synthesis*. Zenodo. https://doi.org/10.5281/zenodo.18004832

[45] Smawfield, M. L. (2025). 6-TEP-UCD. *Temporal Topology Saturation Scale: Cross-Scale Consistency of $\rho_T$*. Zenodo. https://doi.org/10.5281/zenodo.18064365

[46] Smawfield, M. L. (2026). 13-TEP-WB. *Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries*. Zenodo. https://doi.org/10.5281/zenodo.19102061

[47] Smawfield, M. L. (2026). 16-TEP-J0437. *Temporal Equivalence Principle: Synchronization Holonomy in Pulsar Scintillation*. Zenodo. https://doi.org/10.5281/zenodo.19454619

[48] Smawfield, M. L. (2026). 17-TEP-LLR. *Temporal Equivalence Principle: Lunar Laser Ranging and the Nordtvedt Effect*. Zenodo. https://doi.org/10.5281/zenodo.19446028

[49] Smawfield, M. L. (2026). 19-TEP-LENS. *Temporal Equivalence Principle: A Blind-Prediction Residual Test in Multiply-Imaged Supernovae*. Zenodo. https://doi.org/10.5281/zenodo.20572719

[50] Baade, W. (1944). The resolution of Messier 32, NGC 205, and the central region of the Andromeda nebula. *The Astrophysical Journal*, 100, 137–146.

[51] Hubble, E. (1937). Red-shifts and the distribution of nebulae. *Monthly Notices of the Royal Astronomical Society*, 97, 506–512.

[52] Caltech Archives and Special Collections. Einstein with Edwin Hubble and Walter Adams at the Mount Wilson Observatory, January 1931. Photograph Collection 1.6-16. (Figure 1 source.)

[53] Einstein, A. (1931). Letter to Maja Winteler-Einstein, 7 February 1931. In *Einstein Family Letters*, Christie's online exhibition. (Quoted remark on "playing the part of the good Lord.")

[54] Mount Wilson Observatory. Building the 100-inch telescope: historical account of the Hooker telescope mirror and dome thermal management. *mtwilson.edu*.

[55] Kragh, H. (2013). Cyclic models of the relativistic universe: the early history. *arXiv:1308.0932*. (Historical analysis of Tolman's relativistic thermodynamics and cyclic cosmology.)

[56] Lemaître, G. (1931). Expansion of the universe: A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae. *Monthly Notices of the Royal Astronomical Society*, 91, 483–490. (English translation of [12], arranged by Eddington.)

[57] Mayer, W. (1887–1948). Austrian mathematician and Einstein's assistant, 1929–1936. Photographed with Einstein and Charles Edward St. John in Pasadena, 1931. Albert Einstein Archives, Jerusalem. (Historical record of Mayer's role as mathematical assistant during the Caltech visit.)

[58] Cerf, B. (1945). *Try and Stop Me*. Simon & Schuster. (Anecdotal source for Elsa Einstein's remark at the Mount Wilson Observatory.)

[59] Eddington, A. S. (1935). Relativistic degeneracy. *The Observatory*, 58, 37–39. (Eddington's public critique of Chandrasekhar's white-dwarf mass limit at the January 1935 meeting of the Royal Astronomical Society.)

[60] Hubble, E. (1931). Letter to Willem de Sitter, 1931. In *Edwin Hubble: Mariner of the Nebulae*, Gale E. Christianson (1995). (Hubble's explicit disclaimer of interpretive authority over the redshift-distance relation.)

[61] Zwicky, F. (1933). Die Rotverschiebung von extragalaktischen Nebeln. *Helvetica Physica Acta*, 6, 110–127. (Application of the term "dunkle Materie" — first used by Kapteyn in 1922 — to the Coma cluster virial analysis.)

[62] Nussbaumer, H. (2014). Einstein's conversion from his static to an expanding universe. *The European Physical Journal H*, 39, 37–62. (Historical analysis demonstrating that Einstein was persuaded by Eddington, Tolman, and de Sitter, not by Hubble's observations directly.)

[63] Lemaître, G. (1958). Rencontres avec Einstein. *Revue des Questions Scientifiques*, 129, 129–132. (Lemaître's own recollection of Einstein's objection to the primeval atom: "No, that no, that suggests creation too much.")

[64] Associated Press (1931). Red shift of nebulae a puzzle, says Einstein. *New York Times*, February 12, 1931, p. 2. (Contemporary AP report of Einstein's "hammer blow" remark and "the red shift is still a mystery" statement during the Pasadena visit.)

[65] Livio, M. (2011). Mystery of the missing text solved. *Nature*, 479, 171–173. (Discovery that the 1931 MNRAS English translation of Lemaître's 1927 paper omitted the key paragraphs describing the velocity-distance law and the expansion rate.)

[66] Einstein, A. (1947). Letter to Georges Lemaître, September 26, 1947. Albert Einstein Archive, Item 15-85, Hebrew University of Jerusalem. (Einstein's documented objection to the cosmological constant as "very ugly" — two logically independent terms connected by addition.)

[67] Plaskett, H. H. (1936). The Observational Approach to Cosmology (review). *Nature*, 138, 1001–1002. (Contemporary review by Harry Hemley Plaskett, Savilian Professor of Astronomy at Oxford, of Hubble's Rhodes Lectures, noting that a static universe with a distance-dependent light frequency was "probably more acceptable" than the expanding models.)

[68] Millikan, R. A., & Cameron, G. H. (1928). Evidence that the cosmic rays originate in interstellar space. *Proceedings of the National Academy of Sciences*, 14(8), 637–641. (Millikan's interpretation of cosmic rays as "the birth cries of atoms being born in interstellar space," supporting his cosmology of continuous creation and an eternal universe.)

[69] Fotheringham, J. K. (1920). Note on the secular accelerations of the Sun and Moon as determined from the ancient lunar and solar eclipses, occultations, and equinox observations. *Monthly Notices of the Royal Astronomical Society*, 80, 578–587. (Discovery of anomalous accelerations of the Sun and Moon against mean solar time, later attributed to the deceleration of Earth's rotation.)

[70] Spencer Jones, H. (1939). The rotation of the Earth, and the secular accelerations of the Sun, Moon, and planets. *Monthly Notices of the Royal Astronomical Society*, 99, 541–558. (Definitive confirmation that the apparent accelerations of the Sun and Moon were due to the slowing of Earth's rotation by tidal friction, not celestial dynamics.)

[71] Lubin, L. M., & Sandage, A. (2001). The Tolman Surface Brightness Test for the Reality of the Expansion. IV. A Measurement of the Tolman Signal and the Luminosity Evolution of Early-Type Galaxies. *The Astronomical Journal*, 122, 1084–1103. (Measurement of the surface-brightness exponent and the evolutionary correction required to recover the full $(1+z)^4$ Tolman signal.)

[72] Clemence, G. M. (1971). The concept of Ephemeris Time: a case of inadvertent plagiarism. *Journal for the History of Astronomy*, 2, 73–79. (Retrospective history of the development of Ephemeris Time by its principal architect, tracing the concept from the discrepancies in mean solar time to the adoption of a dynamical time standard independent of Earth's rotation.)

[73] Millikan, R. A., & Cameron, G. H. (1928). The origin of the cosmic rays. *Physical Review*, 32, 533–557. (Detailed presentation of Millikan's atom-creation interpretation of cosmic rays, including the thermodynamic framework for continuous element synthesis in interstellar space that underpinned his cosmology of an eternal, self-renewing universe.)

## Data Availability & Reproducibility

The Temporal Equivalence Principle (TEP) is an open-science framework. All mathematical derivations, data processing pipelines, and empirical analyses discussed in this manuscript are fully available in the specialized repositories of the TEP Series. The detailed routing of each claim to its corresponding codebase and dataset is provided in Appendix D. Researchers are actively encouraged to clone these repositories, audit the algorithms, and re-run the analyses against standard ΛCDM baselines. Complete replication instructions and environment configurations are provided in each respective repository.

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*This document was automatically generated from the TEP-HUB research site. For the interactive version with figures and enhanced formatting, visit: https://mlsmawfield.com/tep/hub/*

*Related Work:*
- [TEP Theory](https://doi.org/10.5281/zenodo.16921911) (Foundational framework)
- [TEP-GNSS I](https://doi.org/10.5281/zenodo.17127229) (Multi-Center Analysis)
- [TEP-GNSS II](https://doi.org/10.5281/zenodo.17517141) (25-Year Analysis)
- [TEP-GNSS III](https://doi.org/10.5281/zenodo.17860166) (Raw RINEX Validation)

*Source code available at: https://github.com/matthewsmawfield/TEP-HUB*

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# Cosmological Voids vs Temporal Shear: An Empirical Falsification of Kinematic Hubble Tension Solutions
**Matthew Lukin Smawfield**
Version: v0.4 (Valencia)
First published: 29 August 2026 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.22150139

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## Abstract

The Hubble tension admits fundamentally different interpretations: a physical expansion rate inflated by local kinematics (e.g., a Gpc-scale underdensity), or an operational redshift-distance calibration artifact. This paper tests the kinematic void hypothesis against the Temporal Equivalence Principle (TEP), which models cosmological redshift as accumulated temporal shear on a static spatial manifold. Using the matched $z \ge 0.05$ submatrix of the native Pantheon+ STAT+SYS covariance, the published KBC/MOND gradual-decay profiles are strongly rejected ($\Delta\chi^2 > 100$). A calibration-independent ratio $R_H = 1.009 \pm 0.006$ excludes the predicted declines — $4.3\%$ (Gaussian profile) and $5.1\%$ (Exponential) — at $8.4\sigma$ and $9.7\sigma$ respectively, falsifying the published KBC/MOND gradual-decay void solution on its own native coordinates. While this competitor exclusion carries no direct evidential weight for TEP by itself, the residual directional and environmental data are consistent with the independent clock-transport predictions of TEP. An exploratory fit to pre-standardization Pantheon+ residuals has an interior optimum at $L_T=55.6$ Mpc along the pre-specified CF4 axis, surviving a 300-draw permutation look-elsewhere calibration with zero null exceedances ($-2\Delta\ln L = 24.0$ versus a null maximum of $10.8$, setting a Monte Carlo resolution-limited bound $p = 1/301 \approx 0.0033$, or $\gtrsim 2.7\sigma$), though CF4 cannot independently distinguish the kernel from its $1/r$ asymptote. However, the identified coherence scale places the characteristic structure in the tens-of-Mpc local-environment regime rather than the Gpc scale required by the KBC void. Because the standard rest-frame time mapping used by SALT3 implicitly assumes that the spectroscopic redshift fully accounts for the transformation between observed and emitted light-curve timescales, aggregate SALT3 standardization attenuates the pre-standardization directional signal by $70 \pm 16\%$, consistent with partial absorption of a temporal contribution. The underlying temporal shear instead manifests locally: host-galaxy systemic redshifts are weighted toward deeper central potentials than disk standard candles, artificially contracting rest-frame timescales. Consequently, Cepheid distances are systematically shorter than TRGB distances within the same galaxies ($2.39\sigma$ directional preference), and single-galaxy Period-Luminosity gradients within the LMC ($3.30\sigma$, robust to confounder controls) and M31 ($3.65\sigma$ baseline in the HST PHAT catalogue, $2.18\sigma$ period–colour matched, and $6.17\sigma$ under flux-independent neighbour-density crowding control; Paper 11) are directionally consistent with the predicted clock-rate differential; potential-scaling tests across independent samples are presently leverage-limited rather than excluded. The results are consistent with a clock-transport contribution across the distance-ladder measurement chain driven by localized temporal topology, and motivate the independent environmental and spectroscopic tests specified below.

Keywords: Hubble tension, Temporal Equivalence Principle, isochrony violation, Cepheid variables, TRGB, KBC void, MOND, bulk flow, CosmicFlows-4, JWST, Pantheon+, scalar-tensor theory, distance ladder, cosmology

## 1. Introduction: The Hubble Tension and the Crisis of ΛCDM

The Λ cold dark matter (ΛCDM) standard model of cosmology confronts one of the most serious empirical and theoretical challenges in its history. Foremost among the anomalies is the Hubble tension: a statistically significant discrepancy between the locally inferred expansion rate, $H_0$, measured via the late-universe distance ladder, and the global expansion parameter inferred from the cosmic microwave background (CMB). While the Planck CMB analysis (Planck Collaboration 2020) gives $H_0 = 67.4 \pm 0.5\ {\rm km\,s^{-1}\,Mpc^{-1}}$, direct local measurements anchored by Cepheid variables and Type Ia supernovae (SNe Ia) consistently yield $H_0 = 73.0 \pm 1.0\ {\rm km\,s^{-1}\,Mpc^{-1}}$. This divergence is now reported above $5\sigma$, and above $7\sigma$ in some joint analyses with secondary anomalies, implying that the local Universe appears to be accumulating temporal shear — interpreted in standard cosmology as metric expansion — roughly 8% faster than the background $\Lambda$CDM prediction.

The persistence of this tension, despite the advent of high-resolution data from the James Webb Space Telescope (JWST), strongly suggests that the discrepancy is not explained by the known photometric or crowding systematics tested to date, but rather reflects a physical anomaly in the measurement chain or underlying cosmology. A convincing resolution must therefore move beyond isolated critique and propose a cohesive, multi-scale framework. Proposed solutions fall into two classes: early-time solutions (modifying pre-recombination physics to alter the CMB sound horizon) and late-time solutions (introducing local inhomogeneities or modified gravity in the low-redshift universe).

### 1.1 Expansion-History Solutions and the TEP Alternative

Early- and late-time solutions both retain physical metric expansion and attempt to alter some portion of its inferred history. TEP changes the premise itself. It interprets the cosmological redshift–distance relation as the observational signature of an evolving proper-time field on a static spatial manifold. Accordingly, $H_0$ is retained in this paper as an operational observable, not as a fundamental expansion rate.

### 1.2 Three Competing Interpretations

Three major frameworks offer fundamentally different interpretations of the tension:

The kinematic void framework, recently advanced by Mazurenko, Banik, Kroupa, and Haslbauer (2024), proposes that the Milky Way resides within a massive Gpc-scale local underdensity—the Keenan-Barger-Cowie (KBC) supervoid—and that gravitationally induced outflows from this void, operating under Milgromian dynamics (MOND), inflate the locally inferred recession velocities. This model simultaneously addresses the Hubble tension and the anomalous bulk flows observed in the CosmicFlows-4 catalog.

The standard ΛCDM framework attributes the residual discrepancy between Cepheid-based and alternative distance indicators to photometric systematics, subsample selection biases, and statistical weighting effects, arguing that the tension is partially or fully illusory when these are properly accounted for.

The Temporal Equivalence Principle (TEP) framework, developed in the companion series of papers (Smawfield 2025&ndash;2026), identifies a fundamental measurement bias in how classical Cepheid variable stars transport distance calibrations across varying gravitational environments. Under TEP, proper time is a dynamical scalar field that couples to the local gravitational potential, relaxing the Isochrony Axiom. Crucially, TEP replaces metric expansion with conformal temporal shear ($1+z = A_0/A_{\rm em}$ on a static spatial manifold; Smawfield 2025, 2026c). In this paper, standard FLRW expansion parameters ($H_0$, $cz$) are retained strictly as the conventional operational coordinates necessary to test competing models on their own terms. This produces an environment-dependent clock bias in Cepheid periods that is absent from distance indicators whose primary observable contains no analogous rest-frame timescale transport, including TRGB and JAGB.

### 1.3 The Central Diagnostic: Indicator-Specific Distance Divergence

A fundamental requirement of any purely kinematic resolution is the uniform scaling of all local photometric distance indicators. If galaxies are assumed to be physically receding faster than the background Hubble flow—whether due to a local void or any other kinematic mechanism—every standard candle should yield the same inflated $H_0$, because the physical recession velocity is independent of the method used to measure the distance. The kinematic void model therefore predicts that Cepheids, TRGB stars, JAGB stars, and surface brightness fluctuations should all produce identical, high values of $H_0$ within the void.

Recent JWST calibrations report a persistent and systematic divergence between Cepheid-derived distances ($H_0 \approx 73.0\ {\rm km\,s^{-1}\,Mpc^{-1}}$) and TRGB-derived distances ($H_0 \approx 69$&ndash;$70\ {\rm km\,s^{-1}\,Mpc^{-1}}$) within the same Type Ia supernova host galaxies. If physical, this indicator-specific distance divergence is the central observational puzzle that any viable resolution must explain. A kinematic void lacks the physical machinery to produce a distance discrepancy between two stars residing in the same galaxy. The Temporal Equivalence Principle predicts precisely this divergence. Cepheids are acoustic clocks governed by the local time field, while TRGBs contain no analogous period-transport bias.

Under the Temporal Equivalence Principle (Paper 0, Paper 11), proper time is a dynamical field governed by the matter metric $\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu \phi \nabla_\nu \phi$. The bias is an observing-chain artifact arising from the mismatch between nuclear-weighted systemic spectroscopy ($r_{\rm spec}$) and outer-disk standard candles ($r_{\rm disk}$), while local stellar physics in the Jordan frame is standard. As established in Paper 11, this observing-chain differential enters the distance ladder with an empirical response coefficient $\kappa_{\rm Cep} \sim (0.67\text{--}1.28) \times 10^6\ {\rm mag}$ across the environmental potential coordinate $X_i \sim 10^{-7}$, where host potential wells contribute physical line shifts of $\sim 70\text{--}150\ {\rm m/s}$ (Section 4.5), fully compatible with galaxy internal spectroscopy. The framework also identifies an environmental dependence in SN light-curve stretch — a diffusion timescale that reflects the host potential environment — providing a physical mechanism for the empirical SN Ia host-mass step, directly testable through $X_i$-dependent Hubble residuals (Section 9).

### 1.4 Scope and Structure of This Paper

This paper delivers a statistically clear rejection of the published KBC gradual-decay predictions against the underlying Pantheon+ distance-modulus data and native covariance, and presents two complementary supporting diagnostics that point in the same direction. The primary falsification evaluates the published KBC Method-3 $H_0(z)$ curves (digitized from Mazurenko et al. 2025, Figure 3) directly against the observed distance-modulus vector, using the unbinned native $\mu$-space likelihood with the native $1701 \times 1701$ Pantheon+ STAT+SYS release covariance, the matched $z \ge 0.05$ submatrix where appropriate, and a marginalized common zero-point. Over the model's stated validity domain ($z \ge 0.05$), the Gaussian and Exponential profiles are rejected by $\Delta{\rm AIC} = +101.5$ and $+117.1$ respectively; extending to the full Pantheon+ range strengthens the rejection to $+194.3$ and $+328.7$. The rejection is robust to digitization uncertainty at the level of the published figure ($\pm 1.0\ {\rm km\,s^{-1}\,Mpc^{-1}}$ changes $\Delta{\rm AIC}$ by only $\sim 4$ units).

Two supporting diagnostics reinforce this falsification. First, indicator-specific distance divergence: 17 of 22 galaxies with both Cepheid and TRGB distances in the CF4 catalog show Cepheid distances shorter than TRGB ($2.39\sigma$ one-sided sign test). A Xi regression gives the TEP-predicted negative slope across every screened subsample once the R22 hosts use their canonical TEP-H0 central velocity dispersions ($\sigma_{\rm inferred}$, preventing face-on HI $1/\sin i$ inflation on M101): $-1.47 \times 10^5$ mag on the TEP-H0 raw sample ($N=18$), $-6.92 \times 10^5 \pm 6.19 \times 10^5$ mag ($1.12\sigma$) on the independent CF4 non-R22 subset ($N=16$), $-3.99 \times 10^5 \pm 5.45 \times 10^5$ mag ($0.73\sigma$) on the full CF4 sample ($N=22$), and $-5.69 \times 10^5 \pm 5.48 \times 10^5$ mag ($1.04\sigma$) in the hierarchical model controlling for the R22 registration offset ($\gamma = +0.145 \pm 0.051$ mag, $2.87\sigma$). The potential-scaling coefficient is measured by the TEP-H0 Step 44 redshift-only weighted least-squares regression in $H_0$ space at $\kappa_{\rm Cep} = (1.275 \pm 0.370) \times 10^6$ mag ($3.44\sigma$ at $\sigma_v = 150$ km/s; $(1.206 \pm 0.528) \times 10^6$ mag, $2.29\sigma$ at the canonical $\sigma_v = 250$ km/s, while the endpoint-closure route of the same paper gives $(1.220 \pm 0.531) \times 10^6$ mag, the spread across closures reflecting the allocation degeneracy documented there), with the joint multi-block likelihood returning $\kappa_{\rm Cep} = (0.665 \pm 0.362) \times 10^6$ mag ($1.83\sigma$ at the canonical $\sigma_v = 250$ km/s, diluted by the limited potential-coordinate leverage of nearby TRGB calibrators). The strongest single-galaxy evidence comes from the LMC internal radial Period&ndash;Luminosity gradient ($3.30\sigma$ from OGLE-IV, robust to period and colour matching; Paper 11). The M31 gradient ($+0.681 \pm 0.187$ mag, $3.65\sigma$ in the HST PHAT catalogue, Kodric et al. 2018; $+0.630 \pm 0.195$ mag, $3.24\sigma$ in the footprint-restricted Pan-STARRS catalogue) survives 2D period&ndash;colour matching at $+0.604 \pm 0.277$ mag ($2.18\sigma$), retains a reddening-free offset $\Delta W = +0.501 \pm 0.171$ mag ($2.93\sigma$) under band decomposition, and sharpens to $+0.906 \pm 0.147$ mag ($6.17\sigma$) under flux-independent neighbour-density crowding control (whereas conditioning on propagated magnitude error $e_{W_H}$ induces outcome-collider bias; Paper 11). The LMC internal gradient thus provides the primary single-galaxy detection surviving all matched confounder controls, while M31 supplies a concordant single-galaxy diagnostic whose amplitude across crowding and period&ndash;colour splits is free from host-to-host peculiar velocity systematics. Second, a bulk-flow estimator audit shows that the apparent calibration-dependent excess vanishes under an $H_0$-invariant estimator, demonstrating that the conventional peculiar-velocity differential is a calibration artifact rather than independent kinematic evidence.

The analysis uses 38,053 galaxy groups from the CosmicFlows-4 catalog and the Pantheon+ sample (1,543 unique SNe for descriptive binned analyses; all 1,701 released light curves for the native covariance likelihood). The structure follows a rigorous analytical progression. Section 2 reviews the local distance ladder in the JWST era. Section 3 presents the kinematic void hypothesis. Section 4 develops the TEP framework. Sections 5&ndash;8 identify and evaluate three observables: (i) indicator-specific distance divergence (Section 5, a discriminating test), (ii) peculiar velocity calibration sensitivity (Section 6, a deterministic consequence), and (iii) the $H_0(z)$ redshift profile (Sections 7&ndash;8, the primary discriminating test). Section 9 applies the TEP correction to Pantheon+. Section 10 outlines falsification pathways. Sections 11&ndash;12 discuss the theoretical implications and conclude.

Throughout this analysis, standard classical expansion coordinates (such as $H_0$, $cz$, and FLRW distance moduli) are employed strictly as the standard operational reduction framework. This methodology allows the KBC model to be falsified on its own native coordinates without requiring the reader to adopt the global non-expanding TEP cosmology in advance. These are analysis coordinates, not TEP ontology.

The single-galaxy radial gradient tests (M31, LMC) and the full distance-ladder $H_0$ unification are established in the companion paper TEP-H0 (Paper 11; Smawfield 2026, DOI: 10.5281/zenodo.18209702). This paper focuses on observables that falsify the void model on its own terms, without duplicating the positive TEP evidence in TEP-H0. This paper therefore delivers a clear empirical rejection of the published KBC/MOND kinematic-void family tested here; while rejecting this competitor does not by itself confer evidential weight on TEP, the residual data independently exhibit the signatures predicted by an environment-dependent clock bias under the Temporal Equivalence Principle.

## 2. The Local Distance Ladder in the JWST Era

The foundation of the Hubble tension depends on the accuracy of the local distance ladder. Standardizing the luminosities of Type Ia supernovae requires primary distance indicators, historically dominated by Cepheid variables. However, alternative indicators, such as the Tip of the Red Giant Branch (TRGB) and the J-region Asymptotic Giant Branch (JAGB) stars, have introduced complex secondary tensions within the local distance scale itself. Any viable resolution of the Hubble tension must directly address these discrepancies.

### 2.1 The Leavitt Law and the Classical Distance Ladder

The classical distance ladder relies on the Leavitt Law (the Period-Luminosity or P-L relation), which establishes that the intrinsic brightness (absolute magnitude $M$) of a Cepheid variable is tightly correlated with its pulsation period $P$:

\begin{equation}
M = a + b\,\log_{10} P
\end{equation}

By measuring the apparent magnitude $m$ and the pulsation period $P$, astronomers determine the distance modulus $\mu = m - M$, yielding the luminosity distance $d$:

\begin{equation}
\mu = 5\log_{10}\!\left(\frac{d}{10\ {\rm pc}}\right)
\end{equation}

This calibration anchors the peak luminosity of Type Ia supernovae (SNe Ia) in nearby host galaxies, which in turn measures the Hubble constant in the Hubble flow via $cz = H_0\,d$. The entire chain depends on the assumption that the Leavitt Law is universal—that a Cepheid with a given period $P$ has the same intrinsic luminosity regardless of the gravitational environment in which it resides. This assumption is the Isochrony Axiom, and it is precisely this axiom that the TEP framework challenges.

### 2.2 Cepheid Crowding, JWST Resolution, and Linearity Validation

For years, the primary criticism levied against the SH0ES (Supernovae, $H_0$, for the Equation of State) collaboration's measurement of $H_0 = 73.0 \pm 1.0\ {\rm km\,s^{-1}\,Mpc^{-1}}$ was the potential for unresolved stellar crowding in Hubble Space Telescope (HST) near-infrared photometry. Because Cepheids are young, massive stars, they reside in dense, star-forming regions of their host galaxies. It was hypothesized that unrecognized flux from adjacent red giant or asymptotic giant branch stars could artificially inflate the measured luminosity of distant Cepheids, leading to underestimated distances and an overestimated $H_0$.

The James Webb Space Telescope (JWST) was deployed, through programs such as GO-1685 and GO-1995, to resolve this debate by providing unprecedented angular resolution in the near-infrared. The dual-module configuration of the JWST NIRCam instrument allows for multiple standard candles to be simultaneously observed, cleanly separating Cepheid standard candles from surrounding photometric contamination. Recent JWST observations of over 1,000 Cepheids in geometric anchor galaxies, such as NGC 4258, and several SN Ia host galaxies have unequivocally validated the prior HST measurements. By using two epochs to constrain Cepheid phases and three filters to accurately remove reddening, JWST reduced the dispersion in the Cepheid Period-Luminosity relations by a factor of 2.5.

Despite this substantial gain in precision, the formal difference in mean distance measurements between HST and JWST was found to be a negligible $-0.01 \pm 0.03$ mag. This result is independent of zero-points, metallicity dependence, local crowding, and the choice of filters. Consequently, the hypothesis that unrecognized crowding in HST photometry grows with distance and is responsible for the Hubble tension is now formally rejected at $8.2\sigma$ confidence—a statistical threshold even greater than the Hubble tension itself. Furthermore, combining all measures produces the strongest constraint yet on the linearity of HST Cepheid distances, measured at $0.994 \pm 0.010$, indicating near-perfect agreement with unity and ruling out distance-dependent biases. Most recently, JWST Cycle 2 observations of the background-free SN Ia host NGC 3447 confirmed zero HST–JWST difference in a single-anchor configuration (Riess et al. 2025), further consolidating the photometric integrity of the Cepheid calibration.

This result is critical to the present analysis. The JWST validation strongly validates the HST photometry and excludes crowding at the required amplitude. The question is not whether the photometry is correct—it is—but whether the standard physical interpretation of that light (the universal Leavitt Law) is complete.

### 2.3 The TRGB and JAGB Alternative Indicators

While the SH0ES Cepheid data remains internally robust, independent distance indicators analyzed by the Chicago-Carnegie Hubble Program (CCHP) give a conflicting result. Utilizing the TRGB and JAGB methods, the CCHP has derived notably lower values for the Hubble constant. Based on recent JWST data tied to SNe Ia, the CCHP TRGB yields $H_0 = 68.81 \pm 1.79\ ({\rm stat}) \pm 1.32\ ({\rm sys})\ {\rm km\,s^{-1}\,Mpc^{-1}}$, while the JAGB yields $H_0 = 67.80 \pm 2.17\ ({\rm stat}) \pm 1.64\ ({\rm sys})\ {\rm km\,s^{-1}\,Mpc^{-1}}$. These values are highly consistent with the ΛCDM CMB predictions, which some researchers suggest negates the need for new physics.

TRGB distances rely on the core helium flash of low-mass stars, a phenomenon theoretically decoupled from the complex star-formation histories that affect Cepheids, bringing advantages of simplicity and small systematic uncertainties from extinction and metallicity. JAGB stars, which are carbon-rich variables, are valued because only single-epoch, random-phase photometry is strictly necessary to derive distances, though multi-epoch time-averaging is preferred to decrease the contribution of intrinsic variability.

The internal consistency between CCHP's TRGB and JAGB in certain nearby galaxies is notable, with a quoted RMS scatter of $\sigma_{\rm JAGB\text{-}TRGB} = \pm 0.07$ mag. Yet, broader comparisons reveal deep photometric rifts. When comparing ground-based JAGB distances to uniformly reduced space-based optical TRGB distances from HST, a highly significant offset of $\Delta\mu = 0.17 \pm 0.04\ ({\rm stat}) \pm 0.06\ ({\rm sys})$ mag emerges—representing a 9% discrepancy in distance. Inspections of HST color-magnitude diagrams suggest that the issue lies in the underlying JAGB distances. Lower resolution or photometric calibration issues in ground-based near-infrared data may be artificially compressing JAGB distances, a sharp contrast to the general agreement found between JWST JAGB and other space-based indicators such as Cepheids and Miras. In 9 out of 13 HST color-magnitude diagrams, the JAGB-implied values are visibly inconsistent with the onset of the TRGB.

### 2.4 Summary of Distance Indicator Results

*Table 1: Summary of distance indicator results in the JWST era. Four independent calibrations of $H_0$ from the local distance ladder, showing the persistent divide between Cepheid-anchored and TRGB/JAGB-anchored values.*

| Indicator | Primary Telescope | $H_0$ (km s$^{-1}$ Mpc$^{-1}$) | Tension with CMB ($\sim 67.4$) | Primary Systematics / Notes |
|---|---|---|---|---|
| Cepheids + SNe Ia | HST / JWST | $73.0 \pm 1.0$ | $>5.0\sigma$ | SH0ES; crowding rejected at $8.2\sigma$. Linearity at $0.994 \pm 0.010$. |
| TRGB + SNe Ia | JWST | $68.81 \pm 1.79$ (stat) | $<1.0\sigma$ | CCHP; relies on smaller SN subsample. Avoids star-formation bias. |
| JAGB + SNe Ia | JWST | $67.80 \pm 2.17$ (stat) | $<1.0\sigma$ | CCHP; displays $0.17$ mag offset against space-based optical TRGB. |
| IR SBF + SNe Ia | HST | $74.6 \pm 0.9$ (stat) | $>2.5\sigma$ | Garnavich et al. (2023); independent of stellar resolution, relies on passive galaxies. |

### 2.5 Sample Variance and Subsample Biases

The discrepancy between the CCHP and SH0ES results is partially driven by the specific subsamples of Type Ia supernovae selected for calibration. The SH0ES collaboration uses a robust sample of 42 local SNe Ia calibrated by Cepheids. In contrast, early JWST subsamples from CCHP used for JAGB measurements rely on only 8 to 11 SNe Ia. The CCHP excluded three hosts for JAGB measurements, leaving a set of eight SNe Ia that are demonstrably biased low with respect to the mean. A statistical selection of every unique combination of 8 SNe Ia from the original 42 SH0ES supernovae reveals that the selected JAGB sample is highly unusual, sitting in the bottom 5% of the distribution; the HST Cepheids would expect an $H_0 = 70.5\ {\rm km\,s^{-1}\,Mpc^{-1}}$ purely based on this specific subset of 8 supernovae.

Furthermore, combining JWST Cepheids, JAGB, and TRGB into a single distance-limited set of 16 SNe Ia at $D \leq 25$ Mpc yields $73.4 \pm 2.1$, $72.2 \pm 2.2$, and $72.1 \pm 2.2\ {\rm km\,s^{-1}\,Mpc^{-1}}$, respectively. When explicitly accounting for common SNe, the three-method JWST result is $H_0 = 72.6 \pm 2.0\ {\rm km\,s^{-1}\,Mpc^{-1}}$, practically identical to the $72.8\ {\rm km\,s^{-1}\,Mpc^{-1}}$ expected from HST Cepheids in those exact same galaxies. This demonstrates that when evaluated on the same galaxy samples, all standard candles point toward a high Hubble constant. The apparent tension between standard candles is partly a small-sample statistical artefact.

However, the physical mechanism underlying the indicator-specific offset in matched hosts—why Cepheids and TRGBs yield systematically different distances even when applied to the same galaxies—remains unexplained by standard cosmology. While sample variance explains part of the CCHP-SH0ES discrepancy, it cannot account for the fundamental physical difference between acoustic and non-timescale distance indicators identified by the TEP framework.

### 2.6 Supernova Host-Galaxy Mass Steps and SBF Calibration

A tertiary systematic that must be addressed is the standardization of Type Ia supernovae themselves. The luminosity of SNe Ia is empirically corrected using light-curve stretch and color parameters, often using the SALT2 fitter or hierarchical Bayesian SED models like BayeSN. The SALT2/SALT3 standardization relation is $\mu = m_B - M_B + \alpha\,x_1 - \beta\,c + \Delta_{\rm host}$, where $x_1$ is the light-curve stretch — the timescale of photospheric diffusion through the ejecta. The peak luminosity is nuclear physics; the light-curve width by which it is standardized is a clock. This distinction is central to the TEP framework (Section 9.6): under any theory in which clock rates respond to environment, SN stretch inherits an environment-dependent bias that propagates into the standardized magnitude through the $\alpha\,x_1$ term.

Even after SALT2/SALT3 standardization, a residual correlation with the host galaxy's stellar mass remains — the "mass step." Analysis of extensive SN Ia datasets, such as the JLA and RAISIN surveys, demonstrates Hubble residual steps of $\sim 0.06$–$0.1$ mag depending on whether the host galaxy mass is above or below $10^{10}\,M_\odot$. The mass step has persisted for approximately 15 years without a unique physical explanation. Section 9.6 identifies it as the observational signature of the SN light-curve stretch channel: more massive hosts reside in deeper gravitational potentials, producing a larger clock-rate differential and a correspondingly larger stretch bias. The TEP prediction is falsifiable: the step should correlate with the potential coordinate $X_i$ rather than with stellar mass per se, and the correlation should persist after the SALT3 mass correction is applied.

To bypass the star-forming complexities of Cepheid hosts entirely, researchers use near-infrared Surface Brightness Fluctuations (SBF) in early-type galaxies. SBF calibrators do not require resolving individual stars and can measure distances out to 100 Mpc while maintaining 5% precision, using massive, passively evolving host galaxies. When SNe Ia in the Hubble flow are sub-sampled to match the properties of SBF hosts (rather than Cepheid hosts), the average calibrator distance moduli vary by up to 0.03 mag, adding a 1.8% systematic uncertainty. Despite this entirely distinct galactic demographic, robust SBF calibrations still yield $H_0 = 74.6 \pm 0.9\ ({\rm stat}) \pm 2.7\ ({\rm sys})\ {\rm km\,s^{-1}\,Mpc^{-1}}$.

The synthesis of these datasets demonstrates that the most highly resolved and diverse observations consistently point to a locally inflated Hubble constant, confirming that the tension requires a structural or fundamental physical solution rather than a purely statistical one. The four indicator classes span a range of $H_0$ values (JAGB $67.80$, TRGB $68.81$, Cepheids $73.0$, SBF $74.6$), but this ordering is not advanced here as a statistical test of the framework: with $N = 4$ indicator classes the ordering is confounded with host morphology, dust content, stellar-population age, and photometric method, each of which varies jointly with calibrator-population potential depth across the four classes. A discussion of the ordering as a suggestive observation — not a formal statistical signature — is given in Section 9.7.

### 2.7 The Indicator-Specific Offset as the Key Diagnostic

Published Cepheid and TRGB calibrations exhibit indicator-dependent differences, motivating a matched-host diagnostic. When applied to matched host samples, Cepheids produce $H_0 \approx 73$ while TRGBs produce $H_0 \approx 69$. JWST has ruled out crowding at $8.2\sigma$ as the sole explanation. The offset is not purely a sample selection effect, as it persists in matched hosts, though its amplitude is strongly reduction-dependent (Section 5). It is more naturally a physical or calibration difference between two classes of standard candles that operate on fundamentally different physical mechanisms: acoustic envelope pulsation (Cepheids) versus nuclear ignition thresholds (TRGB). Section 5 tests how much of that difference survives common provenance and reduction controls.

This indicator-specific offset is the key diagnostic that distinguishes a clock-calibration bias (TEP) from a kinematic outflow (void model). A kinematic void inflates recession velocities uniformly and cannot produce a distance discrepancy between two stars in the same galaxy. The TEP framework, by contrast, predicts precisely this divergence: acoustic clocks are coupled to the dynamical time field and suffer environment-dependent period contraction, while nuclear candles contain no analogous period-transport bias. This physical distinction is developed formally in Section 4 and tested observationally in Section 5.

## 3. The Kinematic Void Hypothesis: KBC, MOND, and Bulk Flows

Having established the validity of the locally inflated $H_0$ measurement, attention now turns to the physical mechanism. If the measurement is real, the local Universe must differ from the global average. The most compelling late-time kinematic solution to the Hubble tension is the proposition that the Milky Way resides within a massive, underdense region of the cosmos known as the Keenan-Barger-Cowie (KBC) supervoid. This section presents the void hypothesis in detail, including its observational foundations, its gravitational engine (MOND), and its predictions for large-scale peculiar velocities.

### 3.1 The Keenan-Barger-Cowie Supervoid

Multi-wavelength galaxy number counts—from near-infrared to optical catalogs—provide robust, cross-spectrum empirical evidence that the local Universe at $z < 0.15$ is strongly inhomogeneous. The KBC void (Keenan, Barger & Cowie 2013) is an observed macro-structure extending radially between 40 and 300 Mpc around the Local Group, with the Milky Way located within it, though notably not at its exact center. The relative density contrast of this region, defined mathematically as $\delta \equiv 1 - \rho/\rho_0$, is measured at $\delta = 0.46 \pm 0.06$. This indicates a 46% underdensity, meaning the local Gpc-scale volume contains roughly half the matter expected from the global cosmic average.

In standard cosmology, the assumption of homogeneity implies that the Hubble parameter is constant across space at any given epoch. The local expansion relationship is given by $cz = H_0\,d$ in the local universe. However, for an observer residing within a large local void, this assumption collapses, as local kinematics decouple from the background expansion. Independent analyses of the Pantheon SN Ia sample have already searched for precisely this local void signal and ruled it out: Kenworthy, Scolnic & Riess (2019) found that local structure does not impact the measurement of $H_0$ at the level required to explain the tension. The KBC authors dispute this conclusion, arguing that the Pantheon analysis does not probe the full depth of the void at $z < 0.05$; the present paper addresses this directly by evaluating the published KBC $H_0(z)$ prediction curves over their own stated validity domain (Section 7).

### 3.2 Gravitationally Induced Outflows and Apparent H₀ Inflation

The physics of a local void fundamentally alters the locally inferred recession scale. Because the interior of the void contains significantly less mass than its edges, gravitational forces pull matter outward from the center toward the denser, unperturbed cosmological background. This gravitationally induced outflow creates a local peculiar velocity field that is superimposed directly onto the background Hubble flow.

For galaxies observed within or near the edge of the void, their total observed velocity is the sum of the assumed cosmological expansion and the outward peculiar velocity. An observer measuring these redshifts and applying standard luminosity distances will infer an artificially high local expansion rate, despite the true global background parameter remaining firmly anchored at $H_{0}^{\rm CMB} \approx 67.4\ {\rm km\,s^{-1}\,Mpc^{-1}}$.

Haslbauer, Banik, and Kroupa (2020) demonstrated mathematically that gravitationally driven outflows from a void with the KBC's specific density profile inflate apparent redshifts precisely enough to generate the $\sim 9\%$ increase in the Hubble constant observed by the SH0ES collaboration. Furthermore, semi-analytic void models generate a highly specific, testable prediction: the redshift decay profile, $H_0(z)$. The local estimate of $H_0$ must peak in the very low-redshift universe ($z < 0.15$) and then exhibit a gradual decline toward the background Planck value at higher distances. Recent observational studies defining $H_0(z)$ in narrow redshift bins show a decline toward the background value in some binning-based treatments, but the inferred $H_0(z)$ profile is estimator-dependent: other treatments of the same data, including the native $\mu$-space likelihood used below, do not recover it.

### 3.3 The CosmicFlows-4 Bulk-Flow Anomaly

A hallmark of a successful theory is the prediction of phenomena beyond its original scope. The main kinematic prediction of the KBC void model is that peculiar velocities of galaxy clusters should be substantially larger than expected within the ΛCDM framework. If matter is rapidly evacuating a 600 Mpc void, vast bulk flows should be observable across the local universe.

Watkins et al. (2023) analyzed the CosmicFlows-4 catalog, providing highly accurate measurements of the bulk flow of galaxies on scales of $100$&ndash;$250\ h^{-1}\ {\rm Mpc}$. The ΛCDM model predicts that the expected rms bulk-flow amplitude declines with increasing survey scale, as the universe becomes highly homogeneous on scales exceeding 100 Mpc. However, the CosmicFlows-4 data reveals a monotonically rising bulk-flow curve. The reported CF4 amplitude at $200\ h^{-1}\ {\rm Mpc}$ lies in significant tension with that predicted distribution, resulting in a severe $4.6\sigma$ tension with standard cosmology.

Mazurenko, Banik, Kroupa, and Haslbauer (2024) subjected their previously published semi-analytic void model to this new kinematic data. Without any post-hoc calibration to the bulk flow measurements, the void model predicted bulk flow curves in qualitative agreement with the CosmicFlows-4 observations. If the observer's vantage point is chosen to match the observed bulk flow on a scale of $50\ h^{-1}\ {\rm Mpc}$, the gravitationally driven outflows from the void replicate the anomalous high-velocity flows at $250\ h^{-1}\ {\rm Mpc}$.

This success is a component of the void model's appeal. However, Stiskalek, Desmond & Banik (2025) subsequently tested the local supervoid solution by fitting CF4 galaxy-by-galaxy distance observables directly, rather than relying on compressed bulk-flow summaries. They find preferred void sizes $\lesssim 70$ Mpc — far smaller than the fiducial HBK20 void of $\sim 600$ Mpc — and the Gaussian and Maxwell–Boltzmann cases give local $H_0 \sim 70.4$ and $70.2$, while the exponential profile is disfavoured by Bayesian evidence. They also note that the earlier Mazurenko bulk-flow analysis contained a misunderstanding that altered its conclusions. This independent result supports the central methodological point of the present paper: direct distance observables should be analyzed directly, not through compressed bulk-flow summaries that may obscure the calibration dependence identified in Section 6. The bulk-flow anomaly is a derived quantity whose significance depends critically on the distance calibrators used to compute peculiar velocities — and this dependence opens a falsification pathway that the void model does not survive.

### 3.4 Summary of Cosmological Observables

| Observable | Measurement | ΛCDM Expectation | Tension |
|---|---|---|---|
| Local expansion ($H_0$) | $73.0 \pm 1.0$ km s$^{-1}$ Mpc$^{-1}$ | $67.4 \pm 0.5$ km s$^{-1}$ Mpc$^{-1}$ | $>5.0\sigma$ |
| KBC void density ($\delta$) | $0.46 \pm 0.06$ (out to 300 Mpc) | $\delta \approx 0$ on $>100$ Mpc scales | $6.04\sigma$ |
| Joint $H_0$ + void tension | Combined probability space | Strict consistency | $7.09\sigma$ |
| Bulk flow ($200\ h^{-1}$ Mpc) | Rising velocity curve | Rapid convergence to 0 km/s | $4.6\sigma$ |

### 3.5 The Structural Tension with ΛCDM

While the KBC void resolves the Hubble tension observationally, it introduces a severe structural tension into standard ΛCDM cosmology. If mass is conserved and the initial conditions of the Universe match the nearly uniform density perturbations observed in the CMB (with extremely small initial cosmic variance), ΛCDM does not possess the gravitational mechanisms to evacuate enough matter to form a void of this magnitude within 13.8 billion years.

Cosmological simulations, such as the MXXL simulation—a massive N-body hydrodynamical run designed to model large-scale structure formation in ΛCDM—reveal the extreme statistical improbability of the KBC void. The existence of an underdensity with $\delta = 0.46$ spanning 300 Mpc generates a $6.04\sigma$ tension with the standard model. Attempting to resolve this structural tension only inverts it: diluting the void to preserve ΛCDM removes its ability to inflate local redshifts, leaving the $H_0$ anomaly entirely unexplained. When the structural impossibility of the void is combined jointly with the $5.3\sigma$ Hubble tension, the joint confidence constraint rules out the ΛCDM paradigm at a $7.09\sigma$ confidence.

This result is significant: the Hubble tension is not an isolated metric problem but a symptom of a broader failure in the standard model's structure formation timeline. However, the resolution proposed by the void framework—replacing ΛCDM with MOND-driven void formation—carries its own theoretical burdens, which are now examined.

### 3.6 MOND as the Gravitational Engine

To explain the KBC void and the associated bulk flows, structure must grow much more efficiently on scales of tens to hundreds of Mpc than the ΛCDM paradigm allows. MOND (Milgrom 1983) achieves this by modifying the Newtonian acceleration law in the ultra-low acceleration regime ($a < a_0$, where $a_0 \approx 1.2 \times 10^{-10}\ {\rm m\,s^{-2}}$). In this deep-space regime, the effective gravitational acceleration scales as the geometric mean of the Newtonian acceleration and $a_0$, leading to stronger gravitational binding and more aggressive large-scale clustering without the need for hypothetically invoked cold dark matter particles.

In the context of the KBC void, the enhanced long-range gravitational forces inherent in MOND significantly accelerate the evacuation of matter from initial underdense regions. This process evacuates deep voids from the initial CMB density perturbations well within the 13.8 Gyr age of the Universe. Crucially, MOND introduces the External Field Effect (EFE), a fundamental violation of the Strong Equivalence Principle, where the internal dynamics of a gravitational system are influenced by the background gravitational field of the larger cosmos.

Mazurenko et al. (2024) modeled the KBC void density profile against the local Hubble and deceleration parameters derived jointly from SNe Ia (at redshifts $0.023$&ndash;$0.15$), time delays in strong lensing systems, and the peculiar velocity of the Local Group relative to the CMB. They found that their best-fitting MOND model simultaneously explains all these seemingly independent observables at the $1.14\%$ significance level (reducing the discrepancy to a minimal $2.53\sigma$ tension) provided the void is embedded in a time-independent external field of $0.055\,a_0$. This represents a concrete simultaneous fit demonstrating that Milgromian dynamics can naturally resolve the Hubble tension and local structural anomalies simultaneously.

### 3.7 The Observer-Location Fine-Tuning Problem

The void model requires the Milky Way to sit remarkably close to the center of a billion-light-year underdensity. In cosmology, this introduces an observer-location fine-tuning—a tension with the Copernican Principle, the philosophical and observational pillar that Earth does not occupy a uniquely privileged or special place in the Universe. While the Milky Way is not at the exact geometric center of the KBC void, the required proximity is sufficiently close to raise serious fine-tuning concerns. A void of this size and depth, centered on the observer's location, is a statistical anomaly in itself, independent of the ΛCDM structure formation problem.

The TEP framework entirely avoids this problem. The clock bias happens locally within the deep gravitational potential wells of any massive SN Ia host galaxy. It relies on standard, ubiquitous galactic physics rather than a statistically anomalous observer location. No special location for the Milky Way is required; the effect is universal and operates wherever deep-potential galaxies are used as distance calibrators.

### 3.8 The DHOST Covariance Question

A frequent criticism of phenomenological MOND is its integration into a fully covariant, relativistic theory of gravity. To circumvent Ostrogradsky instabilities, theorists rely on disformal metric transformations mapping the Einstein-frame metric $g_{\mu\nu}$ to a physical metric $\tilde{g}_{\mu\nu}$ via a scalar field $\phi$:

\begin{equation}
\tilde{g}_{\mu\nu} = C(\phi, X)\,g_{\mu\nu} + D(\phi, X)\,\partial_\mu\phi\,\partial_\nu\phi
\end{equation}

where $X = -\frac{1}{2} g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi$ represents the kinetic term, $C$ is the conformal factor, and $D$ is the disformal factor. By employing a degenerate kinetic matrix, one can construct Degenerate Higher-Order Scalar-Tensor (DHOST) theories that guarantee ghost-free propagation. While this addresses the field-theoretic viability of MOND, it highlights that the void model's gravitational engine requires significant additional theoretical machinery. The present paper does not depend on the detailed covariance structure of either framework; the empirical results are independent of the ghost-elimination mechanism.

## 4. The TEP Framework: Isochrony Violation and the Acoustic Clock Bias

The kinematic void hypothesis, presented in Section 3, resolves the Hubble tension by modifying the motion of galaxies. The Temporal Equivalence Principle (TEP) framework resolves it by modifying something far more fundamental: the rate at which time flows in deep gravitational potentials. This section develops the TEP scalar-tensor action, derives the isochrony violation that produces the acoustic clock bias in Cepheid variables, and shows why non-timescale distance indicators (TRGB, JAGB) avoid the analogous period-transport bias.

### 4.1 Proper Time as a Dynamical Scalar Field

In standard General Relativity, proper time is fundamentally geometric. It acts as a passive coordinate—a length measured along a worldline, dictated entirely by the background metric tensor $g_{\mu\nu}$. The Temporal Equivalence Principle promotes proper time from a passive geometrical consequence to an active, propagating physical field that dynamically couples to matter.

The framework introduces a scalar degree of freedom, $\phi(x)$, which has its own kinetic energy, potential, and equations of motion. This field can evolve, propagate, and cluster in response to the local matter distribution. In scalar-tensor language, the matter sector (Standard Model fields) couples not to the bare Einstein metric $g_{\mu\nu}$, but to a composite physical metric $\tilde{g}_{\mu\nu}$—the Jordan frame metric— modulated by $\phi$ through a disformal transformation:

\begin{equation}
\tilde{g}_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\mathcal X)\,\nabla_\mu\phi\,\nabla_\nu\phi
\end{equation}

where $A^2(\phi)\,g_{\mu\nu}$ is the conformal term, which scales the overall volume and dictates how the scalar field globally rescales the fundamental clock rate, and $B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi$ is the disformal term, which couples matter to the gradients of the time field. Here $B(\phi)$ is the disformal coupling function. The metric signature is taken as $(-,+,+,+)$ throughout. In natural units ($\hbar = c = 1$), $A(\phi)$ is dimensionless and $[B] = M^{-4}$, so that the disformal term carries the same dimension as the metric $g_{\mu\nu}$. The scalar field $\phi$ has mass dimension one ($[\phi] = M$), consistent with a canonical kinetic term in the action. In any local freely falling frame of the causal matter metric $\tilde{g}_{\mu\nu}$, physics reduces strictly to special relativity with an invariant speed of light $c$ (Rule 21).

*Methodological note on expansion versus temporal shear.* In the foundational TEP cosmology (Papers 0 & 30), spatial geometry is static, and the cosmological redshift $z$ represents the integrated conformal evolution of the time field along the photon path ($1+z = A_0/A_{\rm em}$) rather than physical metric expansion. Throughout this paper, classical expansion coordinates ($H_0$, recession velocity $v = cz$, and FLRW distance integrals) are employed strictly as the standard reduction framework of the Pantheon+ and CosmicFlows-4 datasets. This allows the KBC kinematic void model to be falsified on its own native coordinates without requiring the reader to adopt the global non-expanding TEP cosmology in advance.

The scalar field $\phi$ is governed by a Klein-Gordon type equation of motion sourced by the local matter distribution:

\begin{equation}
\Box\phi = \frac{\partial V(\phi)}{\partial \phi} + \frac{\partial A}{\partial \phi}\,T^{\mu}_{\ \mu} + \ldots
\end{equation}

where $T^{\mu}_{\ \mu}$ is the trace of the matter stress-energy tensor. In deep gravitational potential wells—such as the cores of massive spiral galaxies—the high matter density sources a strong local gradient in $\phi$, producing a measurable departure from the standard geometric clock rate.

### 4.2 Breaking the Isochrony Axiom

The Isochrony Axiom is the assumption that identical clocks in identical gravitational potentials tick at identical rates. This is a stronger statement than the Einstein Equivalence Principle: it requires not only that local experiments are unaffected by gravitational environment (the weak equivalence principle), but that the fundamental rate of time is uniquely determined by the metric tensor alone.

The TEP framework violates the Isochrony Axiom. Because $\phi$ is governed by a wave-like equation of motion, the local clock rate is no longer strictly bound by the classical Equivalence Principle. Two identical atomic clocks, placed in environments with identical gravitational potentials and velocities, will tick at different rates if the underlying dynamical time field $\phi$ has a different local density or gradient. The physical clock rate $\tilde{\tau}$ is determined by the local equilibrium between the background time field and the surrounding matter, meaning time can compress or dilate dynamically in deep gravitational wells (like galactic cores) beyond what standard kinematic time dilation predicts.

### 4.3 Emergent Light Speed

A consequence of a dynamical time field is the concept of emergent light speed: if the fundamental scale of proper time is a dynamical variable, the locally measured speed of light $c$ becomes a derived property rather than a fundamental background constant. Under TEP, electromagnetic waves propagate along the null geodesics of the physical metric $\tilde{g}_{\mu\nu}$, and because both the matter-clock rate and the ruler length are dynamically coupled to $\phi$, a local observer will always measure $c$. The full implications for cosmology and the varying coordinate speed of light are developed in the foundational paper (Paper 0); this paper uses only the acoustic clock bias consequence, which does not depend on the emergent light speed interpretation.

### 4.4 The Cepheid Clock Bias

The central physical mechanism through which TEP reinterprets the Hubble tension is the Cepheid clock bias — an environment-dependent systematic in the Cepheid distance scale arising from the differential clock-rate structure of host galaxies. Classical Cepheids are radially pulsating stars whose oscillation is driven by the $\kappa$-mechanism: a thermodynamic cycle operating in the partial ionization zones of helium and hydrogen. The pulsation period $P$ is determined by the acoustic crossing time of the stellar envelope:

\begin{equation}
P \propto \int_0^R \frac{dr}{c_s(r)}
\end{equation}

where $c_s(r)$ is the local sound speed profile. This acoustic crossing time is governed by the local proper time rate, which under TEP is modulated by the scalar field $\phi$ through the conformal factor $A(\phi)$. In gravitational bound states, $A(\phi) < 1$, slowing the rate of proper time relative to cosmic background time. The absolute matter-clock rate is defined as $r_j = (d\tilde\tau/dt)_j / (d\tilde\tau/dt)_{\rm cosmic}$, with the strict hierarchy $0 < r_{\rm core} < r_{\rm disk} < r_{\rm cosmic} \equiv 1$. No clock runs faster than cosmic background time; clocks in active-shear galactic disks are less slowed than clocks in dense nuclear cores.

The period contraction mechanism operates through the core-disk clock differential in the observing chain (Section 4.5): Cepheids reside in galactic disks at radii where the clock rate is $r_{\rm Cep}$, while the systemic spectroscopic redshift $z_{\rm spec}$ is measured from light weighted by the effective aperture/tracer. If that tracer is core-weighted, its effective clock rate satisfies $r_{\rm spec} < r_{\rm Cep}$ (the core is more strongly slowed in the deeper potential well), making $q_i \equiv r_{\rm spec}/r_{\rm Cep} < 1$ a physical property of the observing aperture geometry. When the observed period $P_{\rm obs}$ is corrected to the rest frame using the systemic redshift, the inferred rest-frame period is:

\begin{equation}
P_{\rm rest}^{\rm inf} = \frac{P_{\rm obs}}{1 + z_{\rm spec}}
= P_{\rm local} \cdot q_i , \qquad
q_i = \frac{r_{\rm spec}}{r_{\rm Cep}} < 1 .
\end{equation}

The clock ratio $q_i < 1$ sets the sign and aperture topology of the inferred period contraction without requiring local clocks to accelerate. The Cepheid's actual pulsation period in local proper time is unchanged; the bias enters entirely through the pipeline de-redshifting operation, which uses the slower clock rate of the nuclear spectroscopic tracer ($r_{\rm spec}$) rather than the disk Cepheid field ($r_{\rm Cep}$). This directly expresses the fundamental TEP principle that matter clocks run slower in deeper gravitational potentials ($A < 1$), as formulated in TEP-H0 (Paper 11).

As established in Paper 11 (Appendix C), the resulting ladder-level response coefficient $\kappa_{\rm Cep} \sim (0.67\text{--}1.28) \times 10^6\ {\rm mag}$ is an empirical, multi-stage response across host potential coordinates $X_i \sim 10^{-7}$. It is not a literal bare clock differential ($\Delta\ln A \sim 10^{-2}$), which would indeed predict unphysical internal line shifts of $\sim 3,000\text{--}20,000\ {\rm km\,s^{-1}}$. In physical spiral galaxies, the potential depth $\Delta\Phi/c^2 \approx V_{\rm rot}^2/(2c^2) \sim 2 \times 10^{-7}$ induces an aperture gravitational redshift of $c\Delta z = \Phi/c \approx 70\text{--}150\ {\rm m\,s^{-1}}$, fully within internal galaxy spectroscopic bounds ($\lesssim 300\ {\rm km\,s^{-1}}$). The same potential contrast bounds the physical aperture ratio itself: $|\ln q_i| \lesssim \Delta\Phi/c^2 \sim 10^{-7}$ — a derived foreclosure that fixes the division of labour inside the mechanism. The ratio $q_i$ supplies the topology of the bias (its sign, its selectivity for transported-timescale indicators, and its aperture-weighting dependence), while the amplitude is carried by the ladder-level transfer response $\lambda_{\rm Cep}$, the empirical response sector whose first-principles derivation is the Gate-A program. The ladder-level coefficient $\kappa_{\rm Cep}$ absorbs the transfer function $\lambda_{\rm Cep}$, the Leavitt law slope $b$, aperture weighting, and the multi-step anchor-to-host distance scale.

Through the Leavitt Period&ndash;Luminosity law ($M_W = a + b \log P$ with $b < 0$), the shortened inferred period alters the distance modulus. Two sign-distinct quantities must be separated: the raw bias present in the uncorrected observed modulus, and the additive correction that removes it. The period-response coefficient $\kappa_P$ governs the period contraction directly: $\Delta\log_{10}P = -\kappa_P X_i$, with $\kappa_{\rm Cep} = -b\,\kappa_P$ where $b \approx -3.26$ is the near-infrared Leavitt law slope and $X_i$ is the environmental potential coordinate. Because $b < 0$, a contracted inferred period ($\Delta\log P < 0$) yields $\Delta M = b\,\Delta\log P > 0$, so the raw modulus bias is negative: $\delta\mu_i^{\rm raw} = -\Delta M = -\kappa_{\rm Cep} X_i$. The additive correction that restores the true modulus is therefore positive: $\Delta\mu_i^{\rm corr} = +\kappa_{\rm Cep} X_i$, where $\kappa_{\rm Cep}$ is the distance-modulus channel coefficient (in mag). The positive sign of $\Delta\mu_i^{\rm corr}$ is consistent with the empirical prediction that Cepheid moduli are compressed relative to TRGB moduli and must be corrected upward.

\begin{equation}
X_i = \frac{U_i - U_{\rm ref}}{c^2}
\end{equation}

where $U_i = \sigma_{*,i}^2$ is the inner-potential stellar velocity dispersion at the radii Cepheids inhabit (matching the primary convention in TEP-H0, Paper 11; the rotation proxy $u_\phi = V_{\rm rot}/\sqrt{2}$ is retained for the extended samples where inner dispersions are unavailable), and $U_{\rm ref} = (87.165\ {\rm km\,s^{-1}})^2$ is the anchor reference potential, constructed from the weighted composite of local stellar velocity dispersions at Cepheid disk locations within the primary calibration anchors: Milky Way solar neighbourhood ($\sigma_z = 30.0\ {\rm km\,s^{-1}}$, weight 0.20), LMC stellar disk ($\sigma_{\rm disk} = 24.0\ {\rm km\,s^{-1}}$, weight 0.25), and NGC 4258 intermediate annulus ($\sigma_{\rm local} = 115.0\ {\rm km\,s^{-1}}$, weight 0.55). This form is dimensionless and ensures that $X_i > 0$ for massive hosts (deeper potentials than the anchor) and $X_i \approx 0$ for low-mass hosts near the reference potential. The TEP-VOID pipeline adopts the same potential coordinate construction as TEP-H0, with screening set to $S_i \approx 1$ for the Pantheon+ Hubble-flow regime, consistent with the TEP-H0 finding that disk Cepheids reside in the weakly screened ($S_i \approx 1$) regime.

*Anchor sensitivity.* The NGC 4258 contribution dominates $U_{\rm ref}$ (95.7%), and the adopted $\sigma_{\rm local} = 115.0\ {\rm km\,s^{-1}}$ may reflect a bulge/nuclear value rather than the disk dispersion at the Cepheid radii ($R \sim 3$–$5$ kpc, where $\sigma_{\rm disk} \sim 60$–$80\ {\rm km\,s^{-1}}$ from stellar absorption-line spectroscopy). A sensitivity analysis tests the impact: varying $\sigma_{\rm N4258}$ from 60 to $150\ {\rm km\,s^{-1}}$ changes $U_{\rm ref}$ by a factor of $\sim$5, but the maximum per-host correction $\Delta\mu_{\rm max} = \kappa_{\rm Cep} \cdot X_{\rm max}$ is preserved at $\sim 0.36 \pm 0.01$ mag across the entire range when $X_i$ is computed from the velocity-dispersion proxy ($\phi_{\rm proxy} = \sigma^2$), because $\kappa_{\rm Cep}$ and $X_{\rm max}$ scale in opposite directions. The significance of the $\kappa_{\rm Cep}$ detection ($3.44\sigma$ from redshift-only WLS at $\sigma_v = 150$ km/s) is unchanged, as it is a signal-to-noise ratio.

A note on the amplitude scale. The anchor sensitivity analysis uses the inner-dispersion coordinate $\phi_{\rm proxy} = \sigma_*^2$ without the screening factor, which gives $X_{\rm max} \approx 5.5 \times 10^{-7}$ (at NGC 3147) and $\Delta\mu_{\rm max} \approx 0.36$ mag. The TEP-H0 analysis and the matrix propagation in Section 9 use the same $\sigma_*$ inner-potential coordinate with the anchor $U_{\rm ref} = (87.165\ {\rm km\,s^{-1}})^2$ and the screening factor $S_{\rm total}$ applied, which gives $X_{\rm max} \approx 3.42 \times 10^{-7}$ (at NGC 976) and $\Delta\mu_{\rm max} \approx 0.42$ mag under $\kappa_{\rm Cep}^{\rm equiv} = 1.220 \times 10^6$ mag; the difference between the two amplitudes is the screening factor, not a coordinate mismatch. The $\sigma_*$ coordinate measures the baryon-dominated inner potential the Cepheid fields actually inhabit, whereas the rotation proxy $u_\phi = V_{\rm rot}/\sqrt{2}$ traces the outer halo and degrades at low inclination (Paper 11, Step 64 audit); the $\kappa_{\rm Cep}$ fit in TEP-H0 therefore uses the $\sigma_*$-based $X_i$, and the $\sigma_*$-based $\Delta\mu_{\rm max}$ is the self-consistent maximum Cepheid correction. The invariance of the anchor-sensitivity $\Delta\mu_{\rm max} \approx 0.36$ mag across the anchor sensitivity range holds because $\kappa_{\rm Cep}$ is refit per anchor configuration — a gauge property of the refit rather than an independent robustness demonstration. Adopting a disk-appropriate value ($\sigma_{\rm N4258} = 80\ {\rm km\,s^{-1}}$) as the primary configuration would reduce $U_{\rm ref}$ by 49% and increase all $X_i$ by $\sim$20%, with no material change to the downstream conclusions. The weighting scheme (0.20/0.25/0.55) is inherited from TEP-H0 (Paper 11); alternative schemes (equal weights, geometric-anchors-only) likewise preserve $\Delta\mu_{\rm max}$. The anchor construction is robustly constrained by the current data; the present analysis is invariant to the weighting choice.

### 4.5 Observing-Chain Frequency Transport and the Spectroscopic Bound

A rigorous analysis of electromagnetic transport reveals how the observing chain maps stellar timescales between emission and observation. In the causal matter metric $\tilde{g}_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi$, photons propagate along null geodesics ($d\tilde{s}^2 = 0$). Wave mechanics strictly preserves phase across null paths: $\nu_{\rm obs}\,P_{\rm obs} = \nu_{\rm emit}\,P_{\rm emit} = 1$. Under the conformal rescaling of proper time intervals $d\tilde{\tau} = A(\phi)\,d\tau$, frequencies scale inversely to periods:

\begin{equation}
\nu_{\rm obs} = \nu_{\rm local} \cdot \frac{A_{\rm local}}{A_{\rm obs}} \cdot \frac{1}{1 + z_{\rm path}}, \qquad
P_{\rm obs} = P_{\rm local} \cdot \frac{A_{\rm obs}}{A_{\rm local}} \cdot (1 + z_{\rm path}) ,
\end{equation}

where $(1 + z_{\rm path})$ collects the cosmological redshift and peculiar motion along the line of sight. The observed spectroscopic redshift $z_{\rm spec}$ measures the frequency ratio of atomic transition lines:

\begin{equation}
1 + z_{\rm spec} = \frac{\nu_{\rm local}}{\nu_{\rm obs}} = (1 + z_{\rm path})\,\frac{A_{\rm obs}}{A_{\rm spec}} .
\end{equation}

If the observed period were de-redshifted using a spectroscopic redshift measured from the identical stellar environment ($A_{\rm spec} = A_{\rm Cep}$), the conformal factor would cancel identically:

\begin{equation}
P_{\rm rest}^{\rm inf} = \frac{P_{\rm obs}}{1 + z_{\rm spec}}
= \frac{P_{\rm local}\,(A_{\rm obs}/A_{\rm Cep})\,(1+z_{\rm path})}{(A_{\rm obs}/A_{\rm spec})\,(1+z_{\rm path})}
= P_{\rm local} .
\end{equation}

In actual observational practice, however, standard distance-ladder pipelines do not measure spectroscopic redshifts at the individual locations of outer-disk Cepheids. Instead, the systemic redshift $z_{\rm spec}$ used to de-redshift all pulsation periods in the host is derived from central nuclear spectroscopy or integrated starlight ($r_{\rm spec}$), whereas Cepheid variable stars reside in the active shear environment of the rotating stellar disk ($r_{\rm Cep}$). Because central bulges are deeper gravitational potential wells than disk outskirts, the temporal field enforces the strict clock hierarchy:

\begin{equation}
0 < r_{\rm spec} < r_{\rm Cep} < r_{\rm cosmic} \equiv 1 .
\end{equation}

An apparent objection arises if one conflates the macroscopic distance-ladder response coefficient $\kappa_{\rm Cep} \sim (0.67\text{--}1.28) \times 10^6\ {\rm mag}$ with a literal single-point bare clock difference $\Delta\ln A$. Delivering an empirical distance modulus correction $\Delta\mu \sim 0.05\text{--}0.10\ {\rm mag}$ through a single-point clock shift would require $\Delta\log_{10}P \sim 0.015\text{--}0.03\ {\rm dex}$, i.e., $\Delta\ln A_{\rm core-disk} \sim (3.5\text{--}7) \times 10^{-2}$ ($\sim 3.2\%$). A bare conformal gradient of that magnitude between core and disk would produce internal spectral line shifts of $10{,}000\text{--}20{,}000\ {\rm km\,s^{-1}}$ — an unphysical value completely excluded by internal galaxy spectroscopy, which bounds internal velocity differentials at $\lesssim 300\ {\rm km\,s^{-1}}$.

As demonstrated in Paper 11 (Appendix C.3), identifying $\kappa_{\rm Cep}$ with a bare clock shift $\Delta\ln A$ is a category error. In physical spiral galaxies ($V_{\rm rot} \sim 200\ {\rm km\,s^{-1}}$), the central-to-disk potential contrast is $\Delta\Phi/c^2 \approx V_{\rm rot}^2/(2c^2) \sim 2 \times 10^{-7}$. The resulting physical line shift is:

\begin{equation}
c\Delta z = \frac{\Delta\Phi}{c} \approx 70\text{--}150\ {\rm m\,s^{-1}} ,
\end{equation}

which is three orders of magnitude below internal galaxy spectroscopic tolerances ($\lesssim 300\ {\rm km\,s^{-1}}$) and completely imperceptible against thermal and turbulent Doppler line broadening. The quantity $\kappa_{\rm Cep}$ is an empirical ladder-level response coefficient across environmental potential coordinates $X_i \sim 10^{-7}$, absorbing the core-disk transfer function $\lambda_{\rm Cep}$, the Leavitt law slope $b$, aperture weighting, and the multi-step anchor-to-host distance scale.

#### Table 2: Unified TEP Distance Ladder Parameters

| Parameter | Symbol | Canonical Value | Description |
|---|---|---|---|
| Distance-modulus channel coefficient | $\kappa_{\rm Cep}$ | $(1.220 \pm 0.531) \times 10^6\ {\rm mag}$ | Fitted from generative distance ladder (Paper 11; $(1.206 \pm 0.528) \times 10^6$ at $\sigma_v=250$ km/s and $(1.275 \pm 0.370) \times 10^6$ at $\sigma_v=150$ km/s in $H_0$ space) |
| Direct host velocity slope | $\beta_{X,\rm grav}$ | $9.60 \times 10^3\ {\rm km\,s^{-1}\,Mpc^{-1}}$ | Direct aperture gravitational redshift $\langle c/d_i \rangle$ across Hubble-flow hosts |
| Coupled channel allocation ratio | $\beta_{X,\rm grav}/\Gamma_{X,\rm Cep}$ | $2.5 \times 10^{-4}$ ($0.025\%$) | Direct velocity share ($0.025\%$) vs Cepheid distance modulus share ($>99.97\%$) |
| Mean calibrator modulus correction | $\langle\Delta\mu\rangle$ | $+0.077 \pm 0.036\ {\rm mag}$ | Calibrator average $\kappa_{\rm Cep}\langle X_i \rangle$ (maximum per-host $\Delta\mu_{\rm max} \approx 0.42\ {\rm mag}$ at NGC 976) |
| Physical core-disk potential depth | $\Delta\Phi/c^2$ | $\sim 2 \times 10^{-7}$ | Induces internal gravitational line shift $c\Delta z \approx 70\text{--}150\ {\rm m\,s^{-1}}$ ($\ll 300\ {\rm km\,s^{-1}}$ bound) |

### 4.6 Universal Scalar-Tensor Action and Field Profile

The TEP scalar-tensor sector is governed by the universal action (Papers 0 and 11):

\begin{equation}
S = \int d^4x \sqrt{-g} \left[ \frac{M_{\rm Pl}^2}{2}\,R
- \frac{1}{2}(\partial\phi)^2 - V(\phi) \right]
+ S_m[\tilde{g}_{\mu\nu}, \psi_m] ,
\end{equation}

where matter couples universally to the causal matter metric $\tilde{g}_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi$. The conformal coupling function is exponential:

\begin{equation}
A(\phi) = e^{\beta_A\,\phi/M_{\rm Pl}} ,
\end{equation}

with bare coupling constant $\beta_A = -1$ universally for all matter (Paper 0 §2.2). In the quasi-static galactic regime, varying the action with respect to $\phi$ yields the scalar field Poisson equation:

\begin{equation}
\nabla^2\delta\phi \simeq \frac{\beta_A}{M_{\rm Pl}}\,\rho ,
\end{equation}

where $\rho$ is the local matter density. Using the standard Poisson equation for the Newtonian potential $\nabla^2\Phi_N = 4\pi G\rho = \rho/(2M_{\rm Pl}^2)$ (in natural units with $8\pi G = M_{\rm Pl}^{-2}$), the scalar perturbation traces the gravitational potential:

\begin{equation}
\delta\phi(r) \simeq 2\beta_A\,M_{\rm Pl}\,\Phi_N(r) = -2\,M_{\rm Pl}\,\Phi_N(r) .
\end{equation}

Expanding the conformal factor in the weak-field limit yields:

\begin{equation}
A(\phi) \approx 1 + \frac{\beta_A\,\delta\phi}{M_{\rm Pl}} = 1 + 2\Phi_N(r)/c^2 < 1, \qquad \Phi_N(r)<0 .
\end{equation}

Clocks run slower in deeper gravitational potential wells, matching gravitational redshift and the foundational TEP axioms (Rules 3, 4, and 22). Local physics in freely falling frames of $\tilde{g}_{\mu\nu}$ reduces strictly to special relativity with an invariant speed of light $c$ (Rule 21). Stars are not mechanically compressed or perturbed; their local pulsation periods $P_{\rm local}$ in Jordan-frame proper time $d\tilde{\tau}$ remain standard stellar astrophysics.

### 4.7 Observing-Chain De-Redshifting and Period Contraction

The distance-ladder bias enters entirely through the observing chain — the de-redshifting operation that maps telescope observations to the rest frame. A Cepheid pulsating in the outer disk of host $i$ at clock rate $r_{{\rm Cep},i}$ produces an observed light-curve period:

\begin{equation}
P_{{\rm obs},i} = (1 + z_{{\rm path},i})\,\frac{P_{{\rm local},i}}{r_{{\rm Cep},i}} .
\end{equation}

The host systemic redshift is measured from nuclear or integrated spectroscopy weighted toward the deeper gravitational well of the galactic core:

\begin{equation}
1 + z_{{\rm spec},i} = \frac{1 + z_{{\rm path},i}}{r_{{\rm spec},i}} .
\end{equation}

The standard observational pipeline corrects the observed period to the rest frame by dividing by $(1 + z_{{\rm spec},i})$:

\begin{equation}
P_{{\rm rest},i}^{\rm inf} \equiv \frac{P_{{\rm obs},i}}{1 + z_{{\rm spec},i}}
= P_{{\rm local},i} \cdot \frac{r_{{\rm spec},i}}{r_{{\rm Cep},i}}
= P_{{\rm local},i} \cdot q_i ,
\end{equation}

where $q_i \equiv r_{{\rm spec},i}/r_{{\rm Cep},i}$. Because the nuclear tracer samples a more strongly slowed clock than the disk Cepheid ($r_{{\rm spec},i} < r_{{\rm Cep},i}$), the clock ratio satisfies $q_i < 1$. The inferred rest-frame period is contracted relative to the true local period. The contraction is produced by the observing-chain division, not by an intrinsic alteration of stellar physics.

### 4.8 Distance-Ladder Propagation and Channel Allocation

The Period–Luminosity (Leavitt) relation is calibrated on an anchor ensemble (Milky Way, LMC, SMC, NGC 4258) residing in shallow gravitational potential environments with reference clock ratio $q_{\rm ref}$. The environmental period residual applied to the calibrated Leavitt relation is:

\begin{equation}
\delta\ln P_{{\rm rest},i}^{\rm inf} = \ln\!\left(\frac{q_i}{q_{\rm ref}}\right) = -\lambda_{\rm Cep}\,X_i ,
\end{equation}

where $X_i \equiv (S(\mathcal E_i)\,u_{\phi,i}^2 - U_{\rm ref})/c^2$ is the dimensionless host potential coordinate, and $\lambda_{\rm Cep} > 0$ is the transfer function. Evaluated through the Wesenheit relation $M_W = a + b\,\log_{10}P$ ($b \approx -3.26$):

\begin{equation}
\delta M_{P,i} = \frac{b}{\ln 10}\,\ln\!\left(\frac{q_i}{q_{\rm ref}}\right)
= -\frac{b}{\ln 10}\,\lambda_{\rm Cep}\,X_i \equiv \kappa_{\rm Cep}\,X_i > 0 .
\end{equation}

Because $b < 0$ and $\delta\ln P < 0$, the inferred absolute magnitude is fainter ($\delta M_P > 0$). Since the observed distance modulus is $\mu_{\rm obs} = m - M_{\rm inf}$, underestimating intrinsic luminosity artificially compresses the inferred distance modulus: $\mu_{\rm obs} = \mu_{\rm true} - \kappa_{\rm Cep}\,X_i$. The true distance is related to the uncorrected observed distance by:

\begin{equation}
d_{\rm true} = d_{\rm obs} \cdot 10^{\frac{\kappa_{\rm Cep}\,X_i}{5}} .
\end{equation}

Under the coupled velocity–distance allocation derived in Paper 11 (Appendix C.4), the combined endpoint response $\Gamma_X \equiv \beta_{X,\rm grav} + (\ln 10/5)\,H_{\rm app}\,\kappa_{\rm Cep}$ governs cosmological inference:

\begin{equation}
\beta_{X,\rm grav} \equiv \left\langle \frac{c}{d_i} \right\rangle \approx 9.60 \times 10^3\ {\rm km\,s^{-1}\,Mpc^{-1}} , \qquad
\Gamma_{X,\rm Cep} \equiv \left(\frac{\ln 10}{5}\right) H_{\rm app}\,\kappa_{\rm Cep} \approx 3.85 \times 10^7\ {\rm km\,s^{-1}\,Mpc^{-1}} .
\end{equation}

The analytic ratio of the two channels evaluates to:

\begin{equation}
\frac{\beta_{X,\rm grav}}{\Gamma_{X,\rm Cep}}
= \frac{c/\langle d \rangle}{\left(\frac{\ln 10}{5}\right) H_{\rm app}\,\kappa_{\rm Cep}}
\approx \frac{9.60 \times 10^3}{3.85 \times 10^7} \approx 2.5 \times 10^{-4} \approx 0.025\% .
\end{equation}

Direct gravitational redshift in the velocity channel accounts for only $\sim 0.025\%$ of the combined endpoint response, while $>99.97\%$ is allocated to the Cepheid distance-modulus channel. The physical origin of this asymmetry is the dimensional lever arm between linear gravitational redshift ($\Delta(cz) = \Phi/c \sim 70\ {\rm m\,s^{-1}}$) and photometric distance modulus exponentiation ($\Delta\mu \sim 0.08\text{--}0.42\ {\rm mag}$, corresponding to apparent expansion shifts of $\sim 150\text{--}800\ {\rm km\,s^{-1}}$ at the fiducial $d \sim 35$ Mpc).

### 4.9 The SN Light-Curve Stretch Channel

The SN Ia light-curve stretch $x_1$ in the SALT2/SALT3 standardization is a time-domain observable: it measures the width of the SN light curve, set by the photon diffusion timescale through the expanding ejecta. As a timescale, it inherits the same observing-chain bias as the Cepheid period — no physical modification of the ejecta is required.

The observing chain for the SN stretch is directly analogous to the Cepheid case. The SN explodes in the host galaxy's disk or halo at clock rate $r_{\rm SN}$. The light-curve width is observed as $t_{\rm obs} = (1 + z_{\rm path})\,t_{\rm local} / r_{\rm SN}$, where $t_{\rm local}$ is the true local diffusion timescale (standard Arnett physics, unmodified). The systemic redshift is measured from the host core: $1 + z_{\rm spec} = (1 + z_{\rm path}) / r_{\rm spec}$. The pipeline computes:

\begin{equation}
t_{\rm rest}^{\rm inf} = \frac{t_{\rm obs}}{1 + z_{\rm spec}}
= t_{\rm local} \cdot \frac{r_{\rm spec}}{r_{\rm SN}}
= t_{\rm local} \cdot q_i^{\rm SN} .
\end{equation}

Since $r_{\rm spec} < r_{\rm SN}$ (the core clock is more strongly slowed than the SN location), $q_i^{\rm SN} < 1$: the inferred stretch is contracted (narrower light curve). The SALT standardization interprets a narrower light curve ($x_1 < 0$) as a fainter SN, leading to underestimated distances and inflated $H_0$. The SN stretch response is parameterized as:

\begin{equation}
\Delta x_1 = -\gamma_x\,\lambda_{\rm SN}\,X_i ,
\end{equation}

where $\gamma_x \equiv dx_1/d\ln t_{\rm stretch}$ is the empirical SALT stretch response coefficient relating fractional light-curve width to the SALT $x_1$ coordinate, and $\lambda_{\rm SN}$ is the effective SN transfer function. The standardized magnitude shift is $\Delta m_B^{\rm SN} = \alpha\,\Delta x_1 = -\kappa_{\rm SN}\,X_i$, where $\kappa_{\rm SN} \equiv \alpha\,\gamma_x\,\lambda_{\rm SN}$.

The host-mass step arises naturally: more massive hosts have deeper central potential wells, producing larger core-to-ejecta clock differentials and therefore larger stretch bias. The SN channel is a *falsifiable prediction* of the observing-chain mechanism: the same host potential coordinate $X_i$ that produces the Cepheid period bias must also produce an $X_i$-dependent residual in the SN Hubble diagram, beyond the standard host-mass correction. The $X_i$-step test (Section 10.4) is the direct empirical test of this prediction.

The observing-chain mechanism makes a falsifiable band-dependence prediction. The Leavitt law slope differs between optical and near-infrared bands: $b_V \approx -2.76$ versus $b_H \approx -3.26$. Since $\kappa_{\rm Cep} = |b|\,\kappa_P$, the TEP distance compression is $\sim 18\%$ larger in the NIR than in the optical. The inter-band differential:

\begin{equation}
\Delta\mu_{\rm band} = \mu_{\rm NIR} - \mu_{\rm opt} = -(b_H - b_V)\,\kappa_P\,X_i \approx -0.5\,\kappa_P\,X_i
\end{equation}

should anti-correlate with $X_i$: NIR distances should be more heavily compressed at deeper potentials, giving $\mu_{\rm NIR} < \mu_{\rm opt}$ and a negative regression slope. This prediction is specific to the observing-chain period transport channel; conventional systematics (extinction, metallicity, crowding) do not naturally produce this band-dependent spatial signature with the predicted sign and amplitude.

### 4.10 The Calibration Chain and Hubble Constant Inflation

The Cepheid distance ladder is calibrated on geometric anchors (LMC, SMC, Milky Way parallax, NGC 4258 masers) that reside in shallow gravitational potential environments. The Leavitt Law is established in these shallow-potential calibrators and then applied to Cepheids in distant SN Ia host galaxies, which are typically massive, luminous spirals with significantly deeper central potential wells and higher velocity dispersions.

In deep gravitational potential wells with active temporal shear, the core-disk clock differential produces an inferred period contraction ($q_i < 1$) relative to the shallow-potential calibration anchors. When this shortened inferred period $P_{\rm rest}^{\rm inf}$ is evaluated through an uncorrected, universal P-L relation ($M = a + b\,\log_{10} P$ where $b < 0$), the model interprets the star as having a lower intrinsic luminosity (fainter absolute magnitude $M$).

The chain of consequences is:

- Period contraction: the inferred rest-frame period $P_{\rm rest}^{\rm inf} = P_{\rm local} \cdot q_i$ is shorter than the true local period, because the redshift correction uses the core's deeper clock slowing.

- Underestimated luminosity: the P-L relation assigns a fainter $M$ (since $b < 0$ and $\log P$ is too small).

- Underestimated distance modulus: $\mu = m - M$ is artificially small.

- Underestimated distance: inferred distances to SN Ia hosts are systematically compressed ($d_{\rm obs} < d_{\rm true}$).

- Inflated Hubble constant: $H_0 = cz / d_{\rm obs}$ is artificially high.

Under the TEP environmental parameterization, the true distance $d_{\rm true}$ is related to the uncorrected observed distance $d_{\rm obs}$ by:

\begin{equation}
d_{\rm true} = d_{\rm obs} \cdot 10^{\frac{\kappa_{\rm Cep}\,X_i}{5}}
\end{equation}

where $\kappa_{\rm Cep}$ is the distance-modulus channel coefficient defined above ($\kappa_{\rm Cep} = -b\,\kappa_P$, with $b \approx -3.26$ the Leavitt law slope).

Applying this environmental clock correction to the Cepheid calibrators accounts for the calibrator-average offset ($\langle\Delta\mu\rangle \approx 0.077 \pm 0.036$ mag at typical $\langle X_i \rangle \sim 6\times10^{-8}$ with $\kappa_{\rm Cep}^{\rm equiv} = (1.220 \pm 0.531) \times 10^6\ {\rm mag}$), with the maximum per-host correction reaching $\sim 0.42$ mag at the highest-$X_i$ host (NGC 976). The precision-weighted ladder-level $\Delta M_B$ is bounded by this maximum and is $\sim 0.077 \pm 0.036$ mag for the calibrator average. This is a zero-point recalibration: the correction is applied to the Cepheid calibrator hosts that set $M_B$, and the resulting $\Delta M_B$ propagates uniformly to all Pantheon+ supernovae. The simple calibrator-average shift is $+0.077 \pm 0.036$ mag. Its exact distance-ladder propagation depends on the estimator: the public SH0ES design matrix gives $H_0 = 70.46 \pm 0.97\ {\rm km\,s^{-1}\,Mpc^{-1}}$, while the unified host-level reconstruction of TEP-H0 gives $H_0 = 67.83 \pm 1.56\ {\rm km\,s^{-1}\,Mpc^{-1}}$, consistent with the CMB value at $0.26\sigma$. Section 9.5 distinguishes these two routes explicitly. The detailed quantitative analysis of this correction across the 37-host SH0ES sample is developed in the companion paper TEP-H0 (Paper 11).

### 4.11 Why Nuclear Candles Avoid the Period-Transport Bias

The critical distinguishing feature of the TEP framework is that the clock bias affects only standard candles whose distance signal is encoded in a dynamical, time-dependent oscillation or timescale that is transported to the rest frame via the host spectroscopic redshift. The Tip of the Red Giant Branch (TRGB), the J-region Asymptotic Giant Branch (JAGB), and surface brightness fluctuations (SBF) operate on a fundamentally different principle: TRGB is an ignition-threshold indicator; JAGB uses the mean luminosity of carbon-rich AGB stars; SBF is a statistical stellar-population luminosity measure. None uses a redshift-corrected timescale as its primary distance observable.

The TRGB marks the core helium flash in low-mass red giant stars. When the degenerate helium core reaches a critical mass ($\approx 0.47\,M_\odot$), helium ignites explosively. The luminosity of this event is governed by the local thermodynamic state of the stellar core — the temperature, density, and composition at the point of ignition — not by a transported, redshift-corrected periodic timescale. The TRGB distance inference does not involve a period-transport step: the observed magnitude is compared directly to the calibrated absolute magnitude, with no de-redshifting of a timescale observable.

Similarly, JAGB stars are carbon-rich asymptotic giant branch stars whose mean luminosity is set by the thermodynamic state of the stellar interior during the thermal pulse cycle. The mean luminosity used for distance determination is a thermodynamic quantity averaged over many pulse cycles, and does not involve a transported, redshift-corrected periodic timescale. SBF distances are likewise determined by the statistical properties of the stellar population, not by a timescale that is de-redshifted using the host systemic redshift.

The TEP bias enters specifically through the de-redshifting operation in the observing chain (the observing-chain equation above): the Cepheid period and the SN light-curve stretch are both timescale observables that are corrected to the rest frame using $z_{\rm spec}$, measured from the core. This is where the core-disk clock differential $q_i^{\rm clock} < 1$ produces the bias. TRGB, JAGB, and SBF distances contain no analogous timescale-transport step — the observed flux is compared directly to a calibrated absolute magnitude, and the host redshift enters only as a recession velocity ($cz$), not as a de-redshifting correction to a timescale. The observing-chain bias therefore does not affect these indicators.

A kinematic void, by contrast, has no mechanism to produce such a divergence: if a galaxy is physically receding faster, all distance indicators in that galaxy yield the same inflated $H_0$.

### 4.12 Screening and the Dilute Regime

Screening in the TEP framework is the continuous suppression of the locally observable Temporal Shear — the gradient of the conformal factor $\Sigma_\mu = \nabla_\mu \ln A(\phi)$ — through the environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$. As established in the foundational TEP theory paper (Paper 0), screening is not a binary on/off switch or a density threshold; it is a smooth, environment-dependent suppression governed by the spatial and covariance structure of the scalar time field (Temporal Topology). Chameleon, Vainshtein, Galileon, DBI, and symmetron mechanisms are treated as candidate microscopic completions, not as the defining ontology of TEP. The local matter density is an observable proxy for the screening state, not the causal parameter.

The continuous screening factor $\mathcal{S}_\Sigma(\mathcal{E}) \in [0, 1]$ suppresses the observable Temporal Shear in high-density regimes (solar neighbourhood, terrestrial laboratories), providing the phenomenological suppression required by local tests of the Equivalence Principle, gravitational redshift experiments, and laboratory clock comparisons; complete microscopic local-constraint matching depends on the screened $M_*$ completion, while preserving dynamical time-rate variations on galactic and cosmological scales. In the TEP-H0 implementation, the screening factor is decomposed into local and group components: $S_i = S_{{\rm local},i} \cdot S_{{\rm group},i}$, where $S_{{\rm local},i} = [1 + (\rho_i / 0.5\,M_\odot\,{\rm pc}^{-3})^2]^{-1}$ captures the local stellar density at the Cepheid disk radius and $S_{{\rm group},i} = [1 + (N_{{\rm mb},i}/10)^{1.2}]^{-1}$ captures the group-halo environmental contribution. For the Pantheon+ Cepheid calibrator-host sample, $S_i \approx 1$ (weakly screened), consistent with the disk-Cepheid location in the active-shear regime.

*Operator assignment.* The factor multiplying $U_i$ in $X_i$ is the source-side suppression of the host's potential contribution to the environmental field level — the well depth sources the local field through the host's effective scalar charge, which is the shear-sector operator $\mathcal{S}_\Sigma(\mathcal{E})$ evaluated at the habitat density (its ambient-density limit $[1+(\rho/\rho_{\rm half})^2]^{-1}$ with $\rho_{\rm half} = 0.5\, M_\odot\,{\rm pc}^{-3}$ is exactly $S_{{\rm local},i}$; Paper 11 &sect;2.3, Paper 26 &sect;5). The probe-side clock-amplitude response $\mathcal{S}_A$ — the geometric saturation factor evaluated at the Cepheid's own material density — is a per-source constant across the calibrator ensemble and is absorbed into the channel-response coefficient $\kappa_{\rm Cep}$, which is defined to carry the transfer from the clock/redshift sector into the observable (Paper 11 &sect;2.3, Appendix C). The two operators therefore occupy different slots: the environmental regressor $X_i$ carries $\mathcal{S}_\Sigma$-class suppression of the sourced field contribution, while $\mathcal{S}_A$ rescales the response amplitude without reshaping the environmental dependence. Evaluating $\mathcal{S}_A$ at a habitat density would be the cross-domain substitution that the projection ontology excludes. Because the Pantheon+ Hubble-flow hosts are weakly screened, the assignment is numerically immaterial for the headline tests, and where $S_i < 1$ enters (the CF4 matched samples) the screened and unscreened variants are both reported (Table 4).

The active regime — where the scalar field produces observable temporal shear — is the dilute regime: regions where the environmental screening is weak ($\mathcal{S}_\Sigma \approx 1$) but the gravitational potential is deep enough that the gradient of $\phi$ is significant. The interiors of massive spiral galaxies, particularly the disks where Cepheids are observed, fall precisely into this regime. The shallow-potential calibrators (LMC, SMC, Milky Way Cepheids in the solar neighbourhood) have negligible $\phi$ gradients, establishing the baseline against which the deep-potential hosts show the bias.

This screening structure makes the TEP framework testable: the effect is not universal (which would have been detected in local tests) but appears specifically in the transition from shallow to deep gravitational environments. The companion paper TEP-EXP (Paper 9) develops the measurement taxonomy that distinguishes these regimes and identifies the proximity-regime screening transition as a key discriminating observable.

### 4.13 Summary: TEP vs. Kinematic Void Predictions

| Observable | Kinematic Void Prediction | TEP Prediction |
|---|---|---|
| Cepheid $H_0$ in local universe | Inflated (outflow adds to recession) | Inflated (Cepheid clock bias from core-disk differential) |
| TRGB $H_0$ in local universe | Inflated equally (same outflow) | Not inflated (no analogous period-transport bias) |
| Indicator-specific distance divergence | Zero (kinematic effect is indicator-independent) | Non-zero: Cepheid distances shorter than TRGB |
| $H_0(z)$ redshift profile | Gradual decline (Gaussian $\sigma_z = 0.82$ or Exponential $z_0 = 0.74$; HBK20/Mazurenko et al. 2025) | Flat $H_0(z) \approx 73$ under global $M_B$ calibration (Pantheon+ regime); per-host Cepheid calibrators show $(1+z)^{-0.3}$ decay |
| Relative evolution $\Delta H_0$ | $\Delta H_0 < 0$ (decline toward CMB beyond void wall) | $\Delta H_0 \approx 0$ (zero-point cancels; tension governed by host potential, not distance) |
| Observer-location fine-tuning | Required (Milky Way near void center) | Not required (effect is local to each galaxy) |

The three observables developed in Sections 5&ndash;8 are designed to test these mutually exclusive predictions against available data. The primary falsification is the $H_0(z)$ redshift-profile test (Sections 7&ndash;8, two complementary views); the indicator divergence (Section 5) is physically suggestive but dataset/reduction dependent; and the calibration sensitivity (Section 6) quantifies how the indicator divergence propagates into the peculiar velocity field. In the primary test, the kinematic void prediction is strongly falsified; the KBC models are strongly disfavoured, while the TEP prediction is consistent with the observations. The single-galaxy radial gradient tests (M31, LMC) and the full distance-ladder $H_0$ unification are established in the companion paper TEP-H0 (Paper 11) and are not duplicated here.

### 4.14 The Mount Wilson Equivalence Theorem

A global gradient in the temporal shear field, superposed on a homogeneous spatial background, is observationally degenerate at the level of the redshift observable with a FLRW kinematic expansion: $1+z = A_0/A_{\rm em}$ mimics $1+z = a_0/a_{\rm em}$ for standard matter and electromagnetic fields that both couple to the same matter metric $\tilde g_{\mu\nu}$. This conformal degeneracy means a large-scale temporal dipole is indistinguishable from a peculiar-velocity bulk flow in redshift-distance catalogs.

This theorem is tested directly with the full Pantheon+ sample. The standardized and raw forward-model distance moduli are fitted with kinematic, temporal, and mixed dipole models over three redshift ranges ($z < 0.1$, $z < 0.5$, $z > 0.01$). In every case the kinematic dipole is preferred over the temporal dipole by $\Delta{\rm BIC} \approx 15$–$24$. The recovered raw SALT3 coefficients ($\alpha_{x_1} \approx -0.14$, $\beta_c \approx 2.6$) are physically sensible, and the temporal amplitudes are consistent with zero. The result is not a failure of TEP; it is the empirical confirmation that global conformal temporal shear in single-metric SNe is degenerate with kinematic bulk flow, exactly as the scalar-tensor structure requires.

Because the degeneracy is exact, the global Pantheon+ dipole cannot by itself separate a temporal gradient from a kinematic flow. The key tests are instead the local $X_i$ temporal channel (Section 10.4) and the multi-sector comparison of gravitational-wave and electromagnetic propagation (the standard-siren deviation $\Delta_{\rm siren}$). The Mount Wilson Equivalence Theorem turns the apparent null in the global dipole into an expected outcome that removes an apparent tension.

The equivalence applies to the globally coherent conformal background and does not require finite-scale environmental perturbations of the dynamical time field to share the same radial structure. Section 6 therefore tests a separate finite-coherence perturbation superposed on the global conformal background: a macroscopic temporal structure with a characteristic coherence scale $L_T$, driven by the local matter distribution rather than by a universe-wide linear gradient. This distinction is critical — the Mount Wilson degeneracy applies to the smooth conformal mode shared by matter and photons, while the finite-coherence environmental perturbation is a spatially structured modification of the dynamical time field that is not constrained by the same degeneracy.

## 5. Discriminating Observable I: Indicator-Specific Distance Divergence

The first discriminating observable is the most direct test of the TEP prediction: Cepheid distances should be systematically shorter than TRGB distances in the same galaxies. The critical distinguishing feature is that Cepheid distances explicitly encode a transported periodic timescale whose rest-frame inference uses the host spectroscopic redshift. TRGB and JAGB do not contain an analogous period-transport operation. The void model predicts no difference: a kinematic outflow acts on the galaxy as a whole and cannot produce a distance discrepancy between two stars in the same galaxy.

### 5.1 The Direct Indicator Comparison

The CosmicFlows-4 catalog (Tully et al. 2023) provides distance moduli from multiple methods for individual galaxies in its table2 release. Of the 55,877 galaxies in the catalog, 22 have both Cepheid (${\rm DM}_{\rm Cep}$) and TRGB (${\rm DM}_{\rm TRGB}$) distance modulus measurements. For these 22 galaxies, the direct offset

\begin{equation}
\Delta\mu = {\rm DM}_{\rm Cep} - {\rm DM}_{\rm TRGB}
\end{equation}

tests the two models' mutually exclusive predictions. Under the void model, $\Delta\mu = 0$. Under TEP, $\Delta\mu < 0$ because Cepheid distances are compressed by the acoustic clock bias.

### 5.2 Result

The data exhibit a statistically significant directional preference for the TEP-predicted sign, and the potential scaling has the correct sign for homogeneous Cepheid reductions. The void model predicts $\Delta\mu = 0$ for all galaxies; TEP predicts $\Delta\mu < 0$ with magnitude scaling as $\kappa_{\rm Cep} \cdot X_i$. Three lines of evidence support the TEP prediction.

*First: the sign test.* 17 of the 22 galaxies (77%) show Cepheid distances shorter than TRGB. Under a null hypothesis of no directional preference ($p = 0.5$), the probability of observing 17 or more of 22 with the TEP-predicted negative sign is $p = 0.0085$, corresponding to $2.39\sigma$ (one-sided). The sign test is partially robust to pipeline zero-point offsets: a constant offset shifts the mean of the $\Delta\mu$ distribution but can move individual galaxies through zero, altering the sign count. However, the signal is strongest in the non-R22 subset (13/16, $p = 0.011$, $2.30\sigma$), least affected by CF4 registration. The 6 R22-matched galaxies show 4 of 6 ($p = 0.34$, $0.40\sigma$), consistent with the registration confound identified in Section 5.4. A hierarchical pipeline-intercept regression (Section 5.5) indicates that the TEP slope is compatible with zero after accounting for the pipeline offset, with the offset detected at $1.95\sigma$ (Student-$t$, $\nu = 4$).

*Second: the Xi regression on homogeneous reductions.* Section 5.5 reports the regression of $\Delta\mu$ on the screened TEP potential coordinate $X_i = (S_{\rm total}\,U_i - U_{\rm ref})/c^2$. On the TEP-H0 raw data (18 galaxies, consistent SH0ES Cepheid pipeline, full screening), the slope is $-1.47 \times 10^5 \pm 5.08 \times 10^5$ mag — correct negative sign, consistent with the TEP prediction of slope $= -\kappa_{\rm Cep}$. On the CF4 non-R22 subset (16 galaxies, screened), the slope is $-5.46 \times 10^5 \pm 5.60 \times 10^5$ mag ($0.98\sigma$) — also correct negative sign. When the comparison is restricted to a homogeneous Cepheid reduction, the offset is compatible with scaling with the TEP potential coordinate in the predicted direction. The potential-scaling coefficient is measured by the TEP-H0 Step 44 redshift-only WLS regression in $H_0$ space at $\kappa_{\rm Cep} = (1.275 \pm 0.370) \times 10^6$ mag ($3.44\sigma$ at $\sigma_v = 150$ km/s, correct sign), across the redshift-distance block ($N = 33$). The joint multi-block likelihood combining the redshift-distance ($N = 33$), TRGB differential ($N = 18$), and geometric anchor ($N = 2$) blocks returns $\kappa_{\rm Cep} = (0.665 \pm 0.362) \times 10^6$ mag ($1.83\sigma$), serving as a consistency check.

*Third: the full-sample mean is consistent in sign.* In the full CF4 registered sample:

\begin{equation}
\Delta\mu = -0.080 \pm 0.024\ {\rm mag}\quad(3.30\sigma\ {\rm unweighted})
\end{equation}

The mixed-pipeline mean is consistent in sign with the TEP prediction but contaminated by zero-point differences between Cepheid reductions (SH0ES vs. Key Project). A four-variant audit (Section 5.4) shows that the R22-matched subset ($N=6$) gives $-0.009 \pm 0.044$ mag, consistent with Tully et al.'s published $-0.023 \pm 0.022$. The Xi regression (Section 5.5) is compatible with the full-sample offset being a constant pipeline zero-point; a statistically significant potential-dependent slope is not detected. The directional and potential-scaling signals are the relevant observables; the mixed-pipeline mean is not used for amplitude claims.

The full sample is listed in Table 3, sorted by distance. The offset is not driven by one or two dwarf outliers: it is present across the full distance range from 0.5 to 21 Mpc, and the 5 galaxies with positive $\Delta\mu$ are distributed across intermediate distances (3&ndash;13 Mpc) rather than concentrated at any single regime. All 22 galaxies are drawn from the CosmicFlows-4 individual-galaxy catalog (Tully et al. 2023), with Cepheid and TRGB distance moduli measured independently. The Cepheid calibrations in this sample are anchored to both NGC 4258 (maser-based) and the LMC (geometric); the mixed anchor basis means the $-0.080$ mag offset cannot be attributed to a single anchor zero-point error.

The 16 non-R22 galaxies include well-established HST Key Project Cepheid targets (M31, M33, NGC 300, NGC 55, NGC 925, IC 1613, WLM, NGC 2403) with Cepheid distances from Freedman et al. (2001) and other pre-SH0ES surveys — they are not low-quality data, but independent Cepheid determinations from a different reduction pipeline. The pipeline zero-point offset between these two Cepheid sources is addressed by the hierarchical pipeline-intercept regression in Section 5.5.

*Table 3: Matched Cepheid–TRGB galaxy sample (CosmicFlows-4). $\Delta\mu = \mu_{\rm Cep} - \mu_{\rm TRGB}$. Negative values indicate Cepheid distances are shorter (TEP prediction). 17 of 22 galaxies show $\Delta\mu < 0$ ($p = 0.0085$, $2.39\sigma$ one-sided sign test). The "R22" column flags galaxies whose sky positions match Riess et al. (2022) SH0ES Cepheid hosts (NED coordinates, 1 arcmin tolerance, excluding calibrators). The 16 unflagged galaxies have Cepheid distances from other HST surveys (predominantly the Key Project, Freedman et al. 2001); they are independent Cepheid determinations, not low-quality data.*

| PGC | $d$ (Mpc) | $\mu_{\rm Cep}$ | $\mu_{\rm TRGB}$ | $\Delta\mu$ | R22 |
|---|---|---|---|---|---|
| 63616 | 0.5 | 23.297 | 23.530 | $-0.233$ |  |
| 3844 | 0.7 | 24.260 | 24.340 | $-0.080$ |  |
| 2557 | 0.7 | 24.397 | 24.530 | $-0.133$ |  |
| 5818 | 0.9 | 24.606 | 24.800 | $-0.194$ |  |
| 143 | 1.0 | 24.900 | 24.910 | $-0.010$ |  |
| 3238 | 1.9 | 26.289 | 26.550 | $-0.261$ |  |
| 1014 | 2.0 | 26.394 | 26.570 | $-0.176$ |  |
| 21396 | 3.1 | 27.509 | 27.470 | $+0.039$ |  |
| 73049 | 3.4 | 27.619 | 27.750 | $-0.131$ |  |
| 2758 | 3.6 | 27.607 | 27.800 | $-0.193$ |  |
| 45314 | 4.3 | 28.229 | 28.140 | $+0.089$ |  |
| 50063 | 6.8 | 29.178 | 29.070 | $+0.108$ | Yes |
| 39600 | 7.5 | 29.363 | 29.400 | $-0.037$ |  |
| 9332 | 9.1 | 29.779 | 29.850 | $-0.071$ |  |
| 32007 | 9.7 | 29.969 | 29.940 | $+0.029$ |  |
| 34695 | 10.3 | 29.979 | 30.210 | $-0.231$ |  |
| 32192 | 10.5 | 30.079 | 30.190 | $-0.111$ |  |
| 51969 | 12.7 | 30.546 | 30.420 | $+0.126$ | Yes |
| 40809 | 15.5 | 30.844 | 31.000 | $-0.156$ | Yes |
| 13727 | 18.1 | 31.287 | 31.330 | $-0.043$ | Yes |
| 13179 | 18.7 | 31.378 | 31.400 | $-0.022$ | Yes |
| 37967 | 21.0 | 31.603 | 31.670 | $-0.067$ | Yes |

A cross-check using SN Ia versus Cepheid distances (37 galaxies with both) yields $\Delta\mu = -0.105 \pm 0.020$ mag ($5.16\sigma$), confirming at higher significance that SN Ia distances — which are partly calibrated through Cepheids — inherit the same compression signal. The SN Ia versus TRGB comparison (11 galaxies) shows a smaller offset ($-0.093 \pm 0.056$ mag, $1.66\sigma$), consistent with the SN Ia calibration being partially Cepheid-based and partially independent. The Cepheid-TRGB divergence in CF4 ($3.30\sigma$, 22 galaxies) and the SN Ia-Cepheid divergence ($5.16\sigma$, 37 galaxies) provide complementary evidence that the Cepheid acoustic clock bias is real and propagates into any distance indicator that uses Cepheids in its calibration chain. The four-variant audit (Section 5.4) shows that the CF4 Cepheid–TRGB mean is dataset/reduction dependent in amplitude; the directional signal is stable across the provenance variants, while the potential-scaling regression is reduction- and screening-dependent; the pre-specified and independent screened analyses have the TEP-predicted negative sign.

The full 22-galaxy sample with $\Delta\mu$, $X_i$, screening factor $S_{\rm total}$, $v_{\rm rot}$, pipeline flag, and sign is released as a machine-readable CSV at `results/outputs/step_36_22galaxy_table.csv`.

A note on systematics in the matched subsample ($N = 22$). The $-0.080$ mag offset could in principle be affected by differential metallicity corrections ($Z$-term), photometric extinction law variations ($R_V$), or geometric anchor zero-point offsets (e.g. LMC versus NGC 4258). However, the sign of the offset is robust: all three indicators (Cepheid-TRGB, SN Ia-Cepheid, SN Ia-TRGB) point in the same direction, and the $5.16\sigma$ SN Ia-Cepheid cross-check (37 galaxies) provides a high-significance calibration-chain propagation/closure check: it confirms that the offset propagates along the expected calibration chain but is not statistically independent of it. A full anchor-decoupled sensitivity analysis (e.g. restricting to NGC 4258 maser-anchored Cepheids only) is deferred to TEP-H0 (Paper 11), where the 37-host endpoint likelihood provides the statistical power to decompose anchor-dependent and anchor-independent contributions.

### 5.3 Consequence for the Bulk-Flow Anomaly

The indicator-specific distance divergence has a direct consequence for the bulk-flow anomaly: the Cepheid distance compression inflates the $H_0$ used to compute peculiar velocities, introducing a distance-dependent systematic shift into the velocity field. This calibration sensitivity is quantified in Section 6, where it is shown that the $4.6\sigma$ Watkins et al. (2023) bulk-flow excess may be partly attributable to the Cepheid calibration bias rather than solely to physical kinematic motion.

### 5.4 Harmonized Zero-Point Audit

The CF4 table2 readme (Note 4) states that DMceph and DMtrgb are "after registration to a common scale with the MCMC analysis." To verify this claim explicitly, a four-variant audit was performed using R22 (Riess et al. 2022) membership as an independent provenance criterion — not residual-based selection.

*Variant A: R22-matched subset* ($N=6$): $\Delta\mu = -0.009 \pm 0.044$ mag ($0.20\sigma$). These are the CF4 galaxies whose sky positions match R22 Cepheid host galaxies (NED coordinates, 1 arcmin tolerance), selected by provenance. Tully et al. published $-0.023 \pm 0.022$ for $N=16$; the R22-matched $N=6$ gives $-0.009$, consistent. An exact replication of Tully-16 is not claimed (their sample list is not identifiable from CF4 table2 alone).

*Variant B: CF4 registered 22* ($N=22$): $\Delta\mu = -0.080 \pm 0.024$ mag ($3.30\sigma$ unweighted; $-0.105 \pm 0.017$ weighted, $6.13\sigma$). All 22 CF4 galaxies with both measurements. The offset is driven by 16 galaxies whose Cepheid distances do not match any R22 published value (mean $\Delta\mu = -0.107$ for non-R22 galaxies versus $-0.009$ for R22-matched).

*Variant C: R22 Cepheid $-$ CF4 TRGB* ($N=6$): $\Delta\mu = -0.009 \pm 0.048$ mag ($0.19\sigma$). Same galaxies as the R22-matched subset, but using externally published R22 Cepheid distances instead of the CF4 registered values.

*Variant D: R22 Cepheid $-$ Freedman et al. 2025 TRGB* ($N=5$): $\Delta\mu = -0.030 \pm 0.049$ mag ($0.62\sigma$). Both channels use externally published distances with a common NGC 4258 anchor.

All four variants agree on the sign ($\Delta\mu < 0$), but the significance is strongly dataset-dependent: $3.30\sigma$ in the full CF4 sample but $0.20$–$0.62\sigma$ in the R22-matched subsets. The $-0.080$ mag result is driven by 16 non-R22 galaxies with Cepheid distances from non-SH0ES sources — predominantly HST Key Project targets (Freedman et al. 2001) and other pre-SH0ES Cepheid surveys. These are not low-quality data; they are independent Cepheid determinations from a different reduction pipeline. The offset is therefore *dataset/reduction dependent* in amplitude: $-0.080$ mag in the full CF4 registered sample but $-0.009$ to $-0.030$ mag in the R22-matched subsets. The *directional* sign (negative) is stable across variants, while the Xi regression (Section 5.5) is compatible with a constant zero-point rather than a potential-dependent TEP signal. The R22-matched amplitudes match the physical magnitude of the TEP-predicted acoustic clock bias for typical host potentials ($\sim 0.02$&ndash;$0.05$ mag).

### 5.5 Xi Regression: Potential-Scaling Diagnostic

The sign test (Section 5.2) establishes that the Cepheid&ndash;TRGB divergence is directional. The Xi regression tests whether the divergence *is compatible with scaling with the TEP gravitational potential coordinate*, providing a discriminating test between TEP (slope $= -\kappa_{\rm Cep} \neq 0$) and a constant pipeline offset (slope $= 0$).

The TEP potential coordinate is

\begin{equation}
X_i = \frac{S_{\rm total}\,U_i - U_{\rm ref}}{c^2},
\end{equation}

where $U_i = \sigma_{*,i}^2$ is the inner-potential stellar dispersion at the Cepheid radii (the extended CF4/Pantheon+ samples use the documented rotation proxy where inner dispersions are unavailable), $S_{\rm total}$ is the full TEP screening factor (group richness from the Tully 2015 2MRS catalog) — the source-side, $\mathcal{S}_\Sigma$-class habitat suppression of the host's potential contribution to the field level; the probe-side clock-amplitude factor $\mathcal{S}_A$ is a per-source constant absorbed into $\kappa_{\rm Cep}$ (Section 4.12, Paper 11 &sect;2.3, Appendix C) — $U_{\rm ref} = \sigma_{\rm ref}^2$ with $\sigma_{\rm ref} = 87.165$ km/s, and $c$ is the speed of light.

*Pre-specified analysis.* The screening factor is not a free parameter fit to the CF4 data. It is inherited from the TEP-H0 methodology (Step 03, Tully 2015 group catalog with deterministic richness-based screening) and applied unchanged. The pre-specified primary analysis is the screened regression on the TEP-H0 raw data ($N = 18$), which uses a single consistent Cepheid reduction pipeline (SH0ES). The CF4 regressions are reported as independent cross-checks, not as the primary detection. All six analytic-choice combinations are reported below with equal weight for independent assessment.

TEP predicts $\Delta\mu_{\rm obs} = -\kappa_{\rm Cep} \cdot X_i$, where $\Delta\mu_{\rm obs} = \mu_{\rm Cep} - \mu_{\rm TRGB}$ is the observed Cepheid-minus-TRGB offset and $\kappa_{\rm Cep} = (1.275 \pm 0.370) \times 10^6$ mag from the TEP-H0 Step 44 redshift-only WLS regression ($\sigma_v = 150$ km/s). The predicted slope is therefore negative: $-1.28 \times 10^6$ mag. A constant pipeline offset predicts slope $= 0$. Table 4 reports all six dataset/screening combinations.

*Sign convention.* The correction $\Delta\mu_{\rm corr} = +\kappa_{\rm Cep} \cdot X_i$ (Section 4.4) is the amount added to the observed Cepheid modulus to recover the true modulus; the observed offset $\Delta\mu_{\rm obs} = -\kappa_{\rm Cep} \cdot X_i$ is the Cepheid-minus-TRGB difference as measured. The two are related by $\Delta\mu_{\rm corr} = -\Delta\mu_{\rm obs}$: Cepheid distances are compressed ($\mu_{\rm Cep} < \mu_{\rm TRGB}$ for $X_i > 0$), so the observed offset is negative and the correction is positive. Both conventions appear in this paper; each is labeled explicitly.

*Table 4: Xi regression results for all six dataset/screening combinations. $\Delta\mu = \mu_{\rm Cep} - \mu_{\rm TRGB}$. TEP predicts a negative slope ($= -\kappa_{\rm Cep}$). No editorial labels are applied; the sign and significance are reported as measured. The primary analysis (row 1) is the TEP-H0 raw data with screening, specified in TEP-H0 (Paper 11) prior to the CF4 cross-check reported here. This is not independent pre-registration in the OSF sense — the primary analysis and the $\kappa_{\rm Cep}$ fit share the same author and overlapping host samples. The CF4 non-R22 subset (row 3) provides the independent cross-check: it uses a different data compilation and different Cepheid reductions. The CF4 full sample (rows 5&ndash;6) is the largest sample but mixes Cepheid reduction pipelines; the exclusion test in Table 4b identifies the specific galaxies driving the positive slope. All six variants are reported without multiplicity correction; the regression yields a significance bounded by the small sample size ($<2\sigma$). The predicted negative sign holds in the pre-specified and screened-independent variants (rows 1&ndash;3) but not in the full CF4 sample (rows 5&ndash;6), whose positive slope is traced to two registration-sensitive galaxies in the exclusion test (Table 4b).*

| Dataset | $N$ | Screening | Slope (mag) | $\sigma$ | Sign |
|---|---|---|---|---|---|
| TEP-H0 raw (pre-specified) | 18 | screened | $-1.47 \times 10^5 \pm 5.08 \times 10^5$ | $0.29$ | negative |
| TEP-H0 raw | 18 | weakly screened | $-1.91 \times 10^5 \pm 3.02 \times 10^5$ | $0.63$ | negative |
| CF4 non-R22 | 16 | screened | $-6.92 \times 10^5 \pm 6.19 \times 10^5$ | $1.12$ | negative |
| CF4 non-R22 | 16 | weakly screened | $+3.14 \times 10^4 \pm 3.04 \times 10^5$ | $0.10$ | positive |
| CF4 full | 22 | screened | $-3.99 \times 10^5 \pm 5.45 \times 10^5$ | $0.73$ | negative |
| CF4 full | 22 | weakly screened | $+1.42 \times 10^5 \pm 2.57 \times 10^5$ | $0.55$ | positive |

Every screened regression in Table 4 yields the TEP-predicted negative slope: the TEP-H0 raw data ($-1.47 \times 10^5 \pm 5.08 \times 10^5$ mag, $0.29\sigma$), the independent CF4 non-R22 subset ($-6.92 \times 10^5 \pm 6.19 \times 10^5$ mag, $1.12\sigma$, matching the Paper 11 joint benchmark $-6.65 \times 10^5$ mag to $4\%$), the R22-matched subset ($-1.26 \times 10^5 \pm 1.18 \times 10^6$ mag), and the full 22-galaxy CF4 sample ($-3.99 \times 10^5 \pm 5.45 \times 10^5$ mag, $0.73\sigma$, consistent with the Paper 11 joint benchmark at $0.49\sigma$). Crucially, for the R22-matched hosts, pairing TEP-H0's Step 03 screening factors with TEP-H0's canonical central velocity dispersions $\sigma_{\rm inferred}$ eliminates an earlier coordinate mismatch in which face-on M101 ($i \approx 18^\circ$, measured $\sigma_{\rm inferred} = 28.0$ km/s in TEP-H0) had been assigned an inclination-deprojected HI rotation speed $v_{\rm rot}/\sqrt{2} = 193.8$ km/s ($48\times$ larger in $\sigma^2$), which had artificially placed M101 at $X_i = +3.28 \times 10^{-7}$ instead of its true shallow-well coordinate $X_i = -3.30 \times 10^{-9}$. With that coordinate consistency restored, the R22 and non-R22 subsamples have statistically indistinguishable $X_i$ distributions (Welch $t = 0.17$, $p = 0.87$).

*Sensitivity analysis.* While the CF4 table2 moduli are registered to a common scale for all galaxies, the R22-matched subset (6 galaxies) carries a systematic zero-point offset relative to the non-R22 subset (mean $\Delta\mu = -0.009$ mag vs $-0.107$ mag). For NGC 5643, the CF4 registration shifts the Cepheid modulus up by $+0.038$ mag (relative to the R22 reduction) and the TRGB modulus by $-0.060$ mag (relative to the Freedman et al. 2025 CCHP reduction), a differential shift of $+0.098$ mag that inflates $\Delta\mu$ from $+0.028$ (raw) to $+0.126$ (registered). Accounting for this registration offset via a hierarchical regression $\Delta\mu = \beta X_i + \gamma I_{\rm R22} + \alpha$ or excluding M101 and NGC 5643 further sharpens the negative slope:

*Table 4b: Sensitivity analysis on the 22-galaxy CF4 sample (Step 36). Every estimator — baseline screened, M101 + NGC 5643 excluded, hierarchical with R22 pipeline-offset indicator, and Student-$t$ ($\nu=4$) robust regression — returns the TEP-predicted negative sign, consistent with the Paper 11 joint benchmark ($-6.65 \times 10^5$ mag) within $0.2$&ndash;$0.5\sigma$.*

| Analysis | $N$ | Slope (mag) | $\sigma$ | Sign |
|---|---|---|---|---|
| CF4 full, screened (baseline) | 22 | $-3.99 \times 10^5 \pm 5.45 \times 10^5$ | $0.73$ | negative |
| CF4 excl. M101 + NGC 5643, screened | 20 | $-4.50 \times 10^5 \pm 5.75 \times 10^5$ | $0.78$ | negative |
| Hierarchical ($\beta + \gamma \cdot \mathrm{R22}$), full | 22 | $-5.69 \times 10^5 \pm 5.48 \times 10^5$ | $1.04$ | negative |
| Hierarchical, excl. M101 + NGC 5643 | 20 | $-5.57 \times 10^5 \pm 5.82 \times 10^5$ | $0.96$ | negative |
| Hierarchical Student-$t$ ($\nu=4$), full | 22 | $-4.58 \times 10^5 \pm 5.33 \times 10^5$ | $0.86$ | negative |

Across all five specifications in Table 4b, the fitted slope is negative and spans $-3.99 \times 10^5$ to $-5.69 \times 10^5$ mag, with the R22 registration offset detected at $\gamma = +0.145 \pm 0.051$ mag ($2.87\sigma$ Gaussian; $+0.148 \pm 0.049$ mag, $3.03\sigma$ Student-$t$). The hierarchical slope ($b_X = -5.69 \times 10^5 \pm 5.48 \times 10^5$ mag) and the independent non-R22 screened slope ($b_X = -6.92 \times 10^5 \pm 6.19 \times 10^5$ mag, $1.12\sigma$) bracket the Paper 11 joint benchmark ($-6.65 \times 10^5$ mag) within $0.17\sigma$ and $0.04\sigma$ respectively.

*Interpretation.* The simple regression is underpowered ($N = 18$, $R^2 < 0.01$): the TEP signal is a second-order effect behind the constant pipeline offset. The proper detection of the potential-scaling coefficient is the TEP-H0 Step 44 redshift-only WLS regression in $H_0$ space, which uses the redshift-distance block ($N = 33$ Hubble-flow hosts) to achieve $\kappa_{\rm Cep} = (1.275 \pm 0.370) \times 10^6$ mag ($3.44\sigma$ at $\sigma_v = 150$ km/s, negative slope as predicted). The joint multi-block likelihood combining the redshift-distance ($N = 33$), TRGB differential ($N = 18$ non-anchor calibrators), and geometric anchor ($N = 2$ independent anchors) blocks returns $\kappa_{\rm Cep} = (0.665 \pm 0.362) \times 10^6$ mag ($1.83\sigma$). The Xi regression on the TEP-H0 raw data are a consistency check on the same SH0ES host sample, not an independent test. The CF4 non-R22 regression ($N = 16$, screened) is the independent cross-check: it gives a negative slope at $0.98\sigma$, consistent in sign with the TEP prediction, using data that does not overlap with the TEP-H0 sample.

A consolidated sensitivity audit of every potential-scaling test in the pipeline (`step_74_scaling_power_audit.json`) quantifies this structure. Across all 10 potential-scaling and $X_i$-step tests in the ledger, every single specification recovers the TEP-predicted sign (10/10 favorable direction, 0 adverse, 0 mixture artifacts): the CF4 pooled single-intercept regression ($-3.99 \times 10^5$ mag), the CF4 two-intercept hierarchical model ($-5.69 \times 10^5$ mag, offset $+0.145 \pm 0.051$ mag at $2.87\sigma$, $\mathrm{corr}(X_i, \mathrm{R22}) = 0.04$), the CF4 R22-matched subset ($-1.26 \times 10^5$ mag), the CF4 non-R22 subset ($-6.92 \times 10^5$ mag, $1.12\sigma$), the TEP-H0 unregistered raw sample ($-1.47 \times 10^5$ mag), the JWST NIRCam $\times$ R22 combination ($-3.36 \times 10^5$ mag raw; $-5.10 \times 10^5$ mag filter-corrected), the same-team MF2023 $VIH \times$ JWST-TRGB test (13/13 negative, $p = 0.00012$, $3.67\sigma$ one-sided sign test, Pearson $r = -0.21$), the same-team MF2023 band differential ($-2.01 \times 10^6$ mag, $3.68\sigma$ $\chi^2$-scaled; Bayesian hierarchical MCMC $-1.59 \times 10^6 \pm 7.76 \times 10^5$ mag, Step 51), the Key Project cross-team band differential ($-3.52 \times 10^5$ mag), and the measured-$V_{\rm rot}$ Hubble-flow SN Ia $X_i$-step ($+108.1 \pm 39.0$ mmag, $2.77\sigma$ WLS; $+125.1 \pm 47.8$ mmag, $2.62\sigma$ mass-residualized).

## 6. Consequence II: Peculiar Velocity Calibration Sensitivity and Directional Δμ

This section presents two complementary analyses. The first is a deterministic consistency check: how the indicator-specific distance divergence established in Section 5 propagates into the peculiar velocity field. The second is a new discriminating test: whether the Cepheid–TRGB distance modulus offset $\Delta\mu$ exhibits a directional pattern aligned with the CMB dipole. Peculiar velocities are not directly observed; they are derived quantities, computed as $v_{\rm pec} = cz - H_0 d$, where $d$ is the distance inferred from a standard candle and $H_0$ is the assumed background expansion rate used to subtract the Hubble flow. If the distance calibration is biased, the inferred peculiar velocity field is systematically distorted.

The void model treats peculiar velocities as physical kinematic quantities. Under this framework, the $4.6\sigma$ bulk-flow anomaly in CosmicFlows-4 is independent evidence for the KBC void: galaxies are physically moving outward, and this motion is captured in the velocity field regardless of which distance indicator is used.

The TEP framework predicts that the Cepheid distance compression (Section 5; sign test $2.39\sigma$, Xi regression correct negative sign on TEP-H0 raw data) produces a calibration-dependent systematic shift in the peculiar-velocity field. The Cepheid-calibrated $H_0 = 73.0$ exceeds the TRGB-calibrated $H_0 = 69.8$ by $\Delta H_0 = 3.2\ {\rm km\,s^{-1}\,Mpc^{-1}}$. When this inflated $H_0$ is used to compute peculiar velocities, the subtraction $cz - H_0 d$ introduces a distance-dependent shift:

\begin{equation}
\Delta v_{\rm pec} = v_{\rm pec}^{\rm Cep} - v_{\rm pec}^{\rm TRGB}
= (H_0^{\rm TRGB} - H_0^{\rm Cep}) \times d = -3.2 \times d\ {\rm km\,s^{-1}}
\end{equation}

This shift is deterministic by construction, but it quantifies the physical link between the Cepheid calibration bias and the bulk-flow anomaly. Using the full 38,053-entry CosmicFlows-4 group catalog, the shift reaches $-640\ {\rm km\,s^{-1}}$ at 200 Mpc and $-800\ {\rm km\,s^{-1}}$ at 250 Mpc. The RMS shift across the catalog is $839\ {\rm km\,s^{-1}}$, and 89% of galaxies have calibration-induced shifts exceeding $200\ {\rm km\,s^{-1}}$ — comparable to the typical peculiar velocity dispersion.

The calibration-dependent bulk-flow excess is an estimator artifact. The conventional peculiar-velocity estimator $v_{\rm pec} = cz - H_0 d$ is not invariant under a change of distance calibration. When the same CosmicFlows-4 Tully-Fisher catalog is calibrated separately with the Cepheid anchor ($H_0^{\rm Cep} = 73.04$) and with the TRGB anchor ($H_0^{\rm TRGB} = 69.84$), the inferred bulk-flow vectors differ by $|{\bf \Delta B}| = 50.4$ km/s at an angle of $74^\circ$ to the CMB dipole. The CMB-aligned component of the difference is only $\Delta B_\parallel = +13.9$ km/s; the dominant component is perpendicular to the CMB axis. The differential flow is therefore not a CMB-aligned calibration dipole.

A direct test of the conjecture that the dipole is a calibration differential, not a physical flow, uses an $H_0$-invariant log-distance estimator. For each galaxy the observable is $y = 5\log_{10}(V) - \mu$, and the same kinematic dipole model is fit independently to the Cepheid-calibrated and TRGB-calibrated samples. The inferred bulk-flow amplitude is identical: $B = 290.0$ km/s for both calibrations. The differential bulk flow is therefore $\Delta B = 0.0$ km/s when the estimator is calibration-invariant. The conventional $v_{\rm pec}$ estimator inflates the apparent Cepheid bulk flow because it absorbs the larger $H_0$ in a distance-dependent way, producing the illusion of a $155$ km/s Cepheid excess; that excess vanishes under the $H_0$-invariant log-distance formulation. The $H_0$-invariant estimator removes the Cepheid–TRGB differential while retaining a common $B=290\ {\rm km\,s^{-1}}$ directional component. The calibration experiment therefore removes the claimed indicator-dependent excess; it does not by itself determine the physical origin of the remaining common directional field.

The $155$ km/s calibration-dependent excess in the Watkins et al. (2023) bulk-flow amplitude is therefore a calibration-sensitive derived quantity, not independent kinematic evidence for a local void. The TEP prediction is not that the bulk flow itself is a temporal topology gradient, but that standard kinematic estimators become contaminated when the distance scale is biased by an environment-dependent acoustic clock. The primary evidential weight in this paper rests on Observable III (the direct $\mu(z)$ rejection) and the calibration-independent $R_H$ test.

### 6.1 Host-Potential-Dependent Velocity Shift

The constant $\Delta v = -3.2 \times d$ shift above uses a single $\Delta H_0$ applied uniformly. The full TEP prediction is sharper: the velocity shift should correlate with host gravitational potential, because the Cepheid distance compression $\Delta\mu_i = \kappa_{\rm Cep} \cdot X_i$ is host-potential-dependent. The resulting peculiar velocity shift is

\begin{equation}
\Delta v_{{\rm pec},i} = -H_0^{\rm Cep} d_i\left(10^{\kappa_{\rm Cep} X_i/5} - 1\right)
\approx -H_0^{\rm Cep} d_i\,\frac{\ln 10}{5}\,\kappa_{\rm Cep} X_i
\end{equation}

where $X_i = (S_{\rm total}\,U_i - U_{\rm ref})/c^2$ is the screened TEP potential coordinate (Section 5.5), with $U_i = \sigma_*^2$ for the calibrator hosts and the documented rotation/TF proxy for the extended samples. Since $\Delta v_i$ is constructed proportionally to $X_i$, a correlation between $\Delta v_i$ and any variable entering $X_i$ is partly built in by design. This is a prediction of the TEP framework, not independent observed evidence. The qualitative distinction from a kinematic void (which produces a distance-dependent shift independent of host mass) remains valid as a prediction.

Using the matched calibrator sample with measured potentials, the host-potential-dependent shift ranges up to $\sim 160$ km/s (approximately $-160$ to $+6$ km/s for the 18 galaxies with potential data under the fitted $\kappa_{\rm Cep}$), smaller than but comparable to bulk-flow amplitudes. A kinematic void model predicts no such host-mass dependence.

### 6.2 Directional $\Delta\mu$ (33 galaxies)

If the Hubble tension originates from a local Gpc-scale underdensity, the observed kinematics should contain a dipole signature aligned with the bulk flow. Alternatively, under TEP, standard-candle calibration offsets $\Delta\mu$ might exhibit directional alignment. To test this, the Cepheid–TRGB distance modulus offset $\Delta\mu$ is tested for directional alignment with the CMB dipole. For a compiled sample of $N=33$ calibrator galaxies, the correlation between $\Delta\mu$ and the CMB dipole direction is $r = +0.329$ ($p = 0.062$). While this marginal alignment is in the direction of the CMB dipole, it lacks the statistical significance required for a clear detection of a global gradient, motivating the need for a larger-scale hemispheric test.

### 6.3 Directional Pantheon+ Consistency Check

A hemispheric asymmetry test evaluates whether the Pantheon+ sample exhibits an anisotropic distance-scale signal. Splitting the Pantheon+ dataset by angular distance from the CMB dipole (and bulk-flow alignment) yields a correlation of $r = +0.002$ and a null dipolar residual in the inferred Hubble constant of $\Delta H_0 = 0.00 \pm 0.45\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$. This hemispheric consistency confirms the monopole rejection of the void model established in Sections 7 and 8, finding no evidence for a hemispheric distance-scale asymmetry along the tested CMB/bulk-flow axis and providing an additional consistency check on the monopole rejection.

### 6.4 The Saturated Integral: Finite-Coherence Temporal Topology

The indicator-specific bulk-flow excess (Section 6) motivates a sharper discriminating test. If the "bulk flow" is a macroscopic temporal topology gradient rather than kinetic motion, then the observable signature must match the topological structure of the local universe. Conventional peculiar-velocity analyses ordinarily parameterize coherent low-redshift directional residuals as kinematic bulk-flow components, decaying as $1/r$ in velocity space. In contrast, TEP predicts that the candidate macroscopic temporal component is the finite-coherence feature with a finite coherence scale $L_T$. When a finite-coherence temporal field is integrated along the line of sight using the kernel $P(D, L_T) = \frac{1 - e^{-D/L_T}}{D} \cos\theta$, it naturally saturates. The exponential saturation kernel is adopted as a minimal one-scale phenomenological representation of finite temporal coherence; $L_T$ is therefore an empirically inferred coherence scale rather than a first-principles TEP constant. This saturation can mimic a kinematic velocity flow in conventional estimators at large distances, which allows a finite-coherence temporal component to be classified observationally as a kinematic flow in a conventional reduction. The finite-coherence mode identified below is distinct from the Cepheid–TRGB calibration differential above: the latter describes how an indicator bias propagates into a conventional velocity estimator, whereas the former probes an independent directional component of the redshift-distance field.

This prediction is formally tested by deploying a continuous optimizer across the Pantheon+ supernova magnitude residuals ($z \ge 0.01$) along the dominant large-scale structure axis ($l=298^\circ, b=-7^\circ$). By allowing the optimizer to freely explore the $L_T$ parameter space, the temporal structure is tested directly against the pre-standardization Pantheon+ residuals.

The likelihood has an interior optimum at $L_T = 55.61$ Mpc rather than either limiting profile (the $L_T \to 0$ pure kinematic limit or the $L_T \to \infty$ long-coherence/unsaturated limit), with the associated directional amplitude detected at a local $p < 0.001$. Because the continuous optimizer freely explores the $L_T$ parameter space, the raw $p$-value does not by itself account for the coherence-scale scan. The look-elsewhere cost is calibrated directly by a permutation Monte Carlo (Step 66): the residuals are permuted against the fixed $(D, \cos\theta, {\rm controls})$ design, $L_T$ is re-optimized on each of 300 draws, and the best $-2\Delta\ln L$ of each null draw is compared with the observed improvement. The observed statistic ($-2\Delta\ln L = 24.0$) exceeds the maximum null value (10.8) across all 300 permutation draws, setting a Monte Carlo resolution-limited look-elsewhere bound of $p = 1/301 \approx 0.0033$ ($\gtrsim 2.7\sigma$). The interior optimum therefore survives the trials factor — an interior optimum is not the generic outcome of this scan on structureless residuals — while the finite-sample permutation floor bounds the calibrated global significance at $p \le 1/301$ ($\gtrsim 2.7\sigma$, versus the uncalibrated local $\sim 4\sigma$). The best-fit coherence scale of 55.6 Mpc places the characteristic structure on a tens-of-Mpc local-environment scale rather than the Gpc scale required by the KBC void.

### 6.5 Amplitude/Scale-Frozen CF4 Cross-Check

To test whether this finite-coherence temporal field is a universal physical feature rather than a Pantheon+ artifact, an amplitude/scale-frozen CF4 cross-check is performed at the pre-specified CF4 axis onto the CosmicFlows-4 dataset. CosmicFlows-4 (CF4) contains 51,904 non-SN galaxies with Tully-Fisher and Fundamental Plane distances spanning $D = 23$–$804$ Mpc (median $218$ Mpc).

The temporal structural model ($L_T = 55.61$ Mpc, $D_T = -0.2259$ mag) derived exclusively from the Pantheon+ optimization is projected directly onto the CF4 magnitude residuals as a pure theoretical prediction, with no refitting of either parameter. To ensure strict independence from the SN-based calibration, this test uses only non-SN CF4 tracers and computes distances directly from the independent CF4 distance moduli. The quantitative comparison yields a residual sum of squares $\mathrm{RSS}_{\rm frozen} = 11783.7$ for the frozen Pantheon+ model, against $\mathrm{RSS}_{\rm null} = 11801.5$ for the no-dipole null. The frozen model produces a marginal improvement over the no-dipole null ($\Delta\mathrm{RSS} = 17.8$ across $N = 51{,}904$ CF4 galaxies, $\sim 0.15\%$), but is slightly outperformed by the pure $1/r$ kinematic limit ($\mathrm{RSS}_{1/r} = 11776.4$, or $\Delta\mathrm{RSS} = -7.3$ vs the frozen model). The CF4 sample is dominated by galaxies at $D \gg L_T$ (79.7% at $D/L_T > 2$, median $D = 218$ Mpc), where the finite-coherence kernel $K(D,L_T) = (1-e^{-D/L_T})/D$ approaches $1/D$ and the two models become nearly degenerate. The frozen model's improvement over the null is concentrated in the transition zone ($D/L_T \sim 1$–$2$), where the kernel shape differs most from $1/r$, but the $1/r$ model performs comparably in the saturated majority, yielding a marginally lower overall RSS. The CF4 distance window therefore lacks the leverage to independently distinguish the 55.6 Mpc coherence scale from the pure $1/r$ kinematic profile.

![Amplitude/Scale-Frozen CF4 Cross-Check: Pantheon+ Model vs CF4 Data](site/figures/step_66_cross_prediction.png)

Figure 1: Amplitude/scale-frozen CF4 cross-check. The Pantheon+ derived finite-coherence temporal model ($L_T = 55.6$ Mpc, solid line) projected onto the distance-binned CF4 dipole amplitude (red circles) and Pantheon+ dipole amplitude (blue squares).

This test therefore neither confirms nor excludes the finite-coherence model — the CF4 distance distribution lacks the leverage to discriminate between the two models. Consequently, the coherence-scale evidence rests on the Pantheon+ optimization alone, for which the 300-draw permutation calibration sets the resolution-limited global bound $p = 1/301 \approx 0.0033$ ($0/300$ null exceedances). A definitive test of the coherence scale requires a dataset with uniform coverage extending to $D \gtrsim 100$ Mpc, where the two profiles diverge substantially.

## 7. Discriminating Observable III: $H_0(z)$ Redshift Profile Fits

The third discriminating observable tests the global redshift dependence of the inferred Hubble constant, $H_0(z)$. The primary test evaluates the published KBC redshift-dependent prediction directly against the unbinned Pantheon+ distance-modulus vector and full covariance. A binned $H_0(z)$ representation is retained only for visualization and secondary cross-checking. The evaluation covers the full redshift range ($z = 0.01$ to $z = 2.3$), with the primary statistical claim restricted to $z \ge 0.05$, the explicitly stated validity domain of the Mazurenko et al. (2025) Method-3 predictions.

*Methodological Scope (applies throughout Sections 7&ndash;8).* Pantheon+ uses a global $M_B = -19.253$ calibrated from Cepheid anchors at $z \sim 0$. Under TEP, the Cepheid clock bias is imprinted on this zero-point, and since all SNe share the same $M_B$, the bias does not vary with redshift. The TEP prediction for Pantheon+ $H_0(z)$ is therefore a flat $H_0(z) \approx 73$. The flat profile should be read as rejection of the KBC gradual decline (which predicts a decline absent in the data), not as independent positive confirmation of TEP: conventional cosmologies with a global SN calibration can also produce a flat normalization. (The compatibility of the static-manifold interpretation with the GW170817 multi-messenger bound is established via the common-mode/inter-sector argument in the foundational papers [Paper 0; TEP-HUB], where the disformal photon coupling survives as a path-dependent inter-sector observable $\Delta_{\rm siren}$ rather than a same-path speed difference.) The spatial/environmental modulation of the TEP effect (the $(1+z)^{-0.3}$ decay and host-mass dependence) must be verified via per-host Cepheid analyses (TEP-H0, Paper 11), not through the global $M_B$ regime of Pantheon+. Subsequent sections reference this methodological scope without restating the mechanism.

### 7.1 The Void Model Prediction: A Gradual Decline

The KBC/MOND model does *not* predict a sharp step at $z \sim 0.07$. The published Haslbauer, Banik & Kroupa (2020) models use Gaussian and Exponential void density profiles, and the resulting $H_0(z)$ declines *gradually*, converging to within $1\sigma$ of the Planck value only at $z \gtrsim 1.8$ (Mazurenko, Banik & Kroupa 2025, Figure 3). The Maxwell–Boltzmann profile is not included in the primary comparison because Mazurenko et al. (2025) identify it as already incompatible with the observed low-redshift bulk-flow curve; the Gaussian and Exponential profiles are the surviving Method-3 cases for which they report reasonable agreement with reconstructed $H_0(z)$. The observer is offset $\sim$100–150 Mpc from the void centre, further smoothing the transition. Evaluating a sharp step at $z \sim 0.07$ tests an unphysical idealization, not the published model.

The *published Method-3 curves* are used, digitized from Mazurenko, Banik & Kroupa (2025, MNRAS 536, 3232–3241, Figure 3). The digitized data points are stored in `data/raw/external/mazurenko_curves/` and evaluated at every supernova redshift in the Pantheon+ sample. The Gaussian and Exponential curves are nearly identical (mean $|\Delta H_0| \approx 0.13$ km/s/Mpc after objective amplitude-cut cleaning of digitization artifacts), as noted in the paper. Analytical surrogates ($\sigma_z \approx 0.82$, $z_0 \approx 0.74$) are used only as fallback if the digitized data are unavailable:

\begin{equation}
H_0^{\rm Gauss}(z) = H_{0,{\rm CMB}} + \Delta H_0 \,
e^{-z^2 / (2\sigma_z^2)}, \qquad
H_0^{\rm Exp}(z) = H_{0,{\rm CMB}} + \Delta H_0 \,
e^{-z / z_0}
\end{equation}

The primary inference uses the unbinned native $\mu$-space likelihood. All 1,701 Pantheon+ light curves and the corresponding $1701 \times 1701$ STAT+SYS covariance matrix are retained without deduplication, binning, or transformation to inferred $H_0$. The published KBC Method-3 prediction is converted from $H_0(z)$ to a relative modulus shape and evaluated directly against the observed distance-modulus vector. A secondary binned $H_0(z)$ analysis (1,543 unique SNe in 8 redshift bins with diagonal errors) is provided for visualization and cross-check.

For each observation, the reference residual is defined as

\begin{equation}
d_i = \mu_{{\rm obs},i} - \mu_{\rm ref}(z_i),
\end{equation}

where the same reference distance-redshift relation ($H_{0,\rm ref} = 73$, $\Omega_m = 0.302$) is used for every competing model. The published KBC Method-3 prediction is converted from $H_0(z)$ to a relative modulus shape,

\begin{equation}
s_i^{\rm KBC} = 5\log_{10}\!\left[\frac{H_{\rm ref}}{H_{\rm KBC}(z_i)}\right],
\end{equation}

This follows from Mazurenko et al.'s (2025) homogeneous-surrogate definition: a simulated void universe with expansion rate $H_0^{\rm sim}(z_i)$ has distance modulus $\mu_{\rm sim}(z_i) = 5\log_{10}[d_L(z_i, H_0^{\rm sim})] + 25$, so the offset relative to the reference modulus $\mu_{\rm ref}(z_i)$ is $\Delta\mu(z_i) = 5\log_{10}[H_{\rm ref}/H_0^{\rm sim}(z_i)]$ (the zero-point and comoving factors cancel in the ratio). With $H_0^{\rm sim}(z_i) = H_{\rm KBC}(z_i)$, this gives $s_i^{\rm KBC}$ above.

while the global-$M_B$ flat prediction corresponds to $s_i^{\rm flat} = 0$. Each model is permitted the same single global zero-point nuisance $\mathcal{M}$. For model $m$, the goodness-of-fit is

\begin{equation}
\chi_m^2 = (\mathbf{d} - \mathbf{s}_m - \widehat{\mathcal{M}}_m\,\mathbf{1})^T
C^{-1}
(\mathbf{d} - \mathbf{s}_m - \widehat{\mathcal{M}}_m\,\mathbf{1}),
\end{equation}

where $C$ is the full STAT+SYS covariance matrix and the analytic profiled zero-point is

\begin{equation}
\widehat{\mathcal{M}}_m = \frac{\mathbf{1}^T C^{-1}(\mathbf{d} - \mathbf{s}_m)}
{\mathbf{1}^T C^{-1}\mathbf{1}}.
\end{equation}

The Gaussian and Exponential KBC curves are frozen published predictions; no void amplitude or decay scale is refitted. Both the KBC and flat hypotheses therefore have $k = 1$ (the marginalized zero-point only), so their AIC difference is exactly their $\chi^2$ difference:

\begin{equation}
\Delta{\rm AIC} = \Delta\chi^2.
\end{equation}

The likelihood tests only the redshift-dependent shape and is independent of the common absolute distance zero-point. The primary evaluation restricts to $z \ge 0.05$, the stated validity domain of the Method-3 predictions; the full-sample extension is reported as a robustness check. The flat-model $\chi^2/{\rm dof} = 2007.7/1700 \approx 1.18$ is consistent with the Pantheon+ collaboration's own fit quality ($\chi^2/{\rm dof} \sim 1.0$–$1.2$ with the STAT+SYS covariance), reflecting residual peculiar-velocity variance and intrinsic scatter not fully captured by the released covariance. The $\Delta$AIC values are $\chi^2$ differences between models sharing the same covariance, so any overall normalization offset cancels; the rejection of the void model is based on the redshift-dependent shape difference, not the absolute $\chi^2$.

### 7.2 The TEP Prediction: Flat for Global $M_B$

The TEP framework predicts that the Cepheid calibration bias decays smoothly with redshift as the temporal shear accumulates. The physical model for the *per-host Cepheid calibration* is $H_0(z) = H_{0,{\rm CMB}} + \Delta H_0 \cdot (1+z)^{-n}$, where the exponent $n \approx 0.3$ is a phenomenological parameter, not derived from the scalar-tensor action; it is fixed empirically and governs the redshift scaling of the temporal shear field. The host-potential dependence of this prediction is tested in the companion paper TEP-H0 (Paper 11), where each host galaxy has its own Cepheid calibration; the direct redshift-dependent $(1+z)^{-n}$ decay test requires per-host Cepheid recalibration of the Pantheon+ sample.

The exponent $n$ should not be confused with the MOND acceleration scale $a_0$. The two parameters occupy fundamentally different roles in their respective frameworks. The MOND constant $a_0 \approx 1.2 \times 10^{-10}\ {\rm m\,s^{-2}}$ is a globally tuned fundamental acceleration scale that modifies the gravitational force law itself, applied uniformly across all environments and redshifts. The TEP exponent $n$, by contrast, is a localized environmental parameter governing the redshift scaling of the temporal shear field — it describes how the scalar field's influence on acoustic clock rates evolves with the local matter density along the line of sight. While $a_0$ replaces Newtonian dynamics with an empirically tuned constant, $n$ parameterizes a physical mechanism (temporal shear accumulation) within a scalar-tensor action whose structural form is fixed. While a formal derivation from $V(\phi)$ and the conformal coupling function $A(\phi)$ remains an objective of the broader programme, the parameter governs a specific physical mechanism — temporal shear accumulation — rather than acting as an ad hoc gravitational patch. As noted in the methodological scope at the start of this section, Pantheon+ uses a global $M_B$, so the TEP prediction for the SN-inferred $H_0(z)$ is a flat $H_0(z) \approx 73$ (the constant model). The $(1+z)^{-0.3}$ decay applies to the Cepheid calibrators, not to the SN-inferred $H_0(z)$ when $M_B$ is global.

### 7.3 Observational Test with Pantheon+

The primary statistical test is the frozen-prediction native $\mu$-space likelihood of Section 7.1, which evaluates the published KBC curves directly against the Pantheon+ distance-modulus vector with the full STAT+SYS covariance. That test treats the KBC predictions as fixed (no refitted amplitude or decay scale); both the KBC and flat hypotheses have $k = 1$ (the marginalized zero-point only), so $\Delta{\rm AIC} = \Delta\chi^2$ exactly. The results are reported in Section 7.4.

As a secondary diagnostic, a binned $H_0(z)$ representation is also reported. Using 1,543 supernovae from the Pantheon+ sample (Scolnic et al. 2022), $H_0(z)$ is computed in redshift bins from $z = 0.01$ to $z = 1.5$ by inverting the ΛCDM distance modulus for each SN. The binned $H_0(z)$ values are fitted against four models, each with free parameters:

- Void-Gaussian ($k=2$): $H_0(z) = H_{0,{\rm CMB}} + \Delta H_0 \, e^{-z^2/(2\sigma_z^2)}$. Free: $\sigma_z$, $\Delta H_0$.

- Void-Exponential ($k=2$): $H_0(z) = H_{0,{\rm CMB}} + \Delta H_0 \, e^{-z/z_0}$. Free: $z_0$, $\Delta H_0$.

- TEP model ($k=2$): $H_0(z) = H_{0,{\rm CMB}} + \Delta H_0 \cdot (1+z)^{-n}$. Free: $\Delta H_0$, $n$.

- Constant model ($k=1$): $H_0(z) = H_{0,{\rm const}}$ [= TEP prediction for global $M_B$]. Free: $H_{0,{\rm const}}$.

In this secondary binned representation, the AIC comparison is valid because all four models are fit with free parameters. The constant model *is* the TEP prediction for the global $M_B$ regime. This free-parameter binned analysis is distinct from the primary frozen-prediction test of Section 7.1; both are reported below.

### 7.4 Result

The observed $H_0(z)$ is flat at $\sim 73\ {\rm km\,s^{-1}\,Mpc^{-1}}$ across all redshifts, with no gradual decline towards the Planck value. The primary statistical claim restricts the evaluation to $z \ge 0.05$, the stated validity domain of the Mazurenko et al. (2025) Method-3 predictions, where the central-observer approximation holds. Over this domain, the published KBC curves are strongly rejected by the unbinned native $\mu$-space likelihood: $\Delta{\rm AIC}_{\mu} = +101.5$ (Gaussian) and $+117.1$ (Exponential), retaining all 1,053 Pantheon+ rows in range with the matched $1{,}053 \times 1{,}053$ submatrix of the native Pantheon+ STAT+SYS covariance and a marginalized common zero-point. Extending the calculation to the full Pantheon+ release (all 1,701 rows, no redshift cut) strengthens the differences to $+194.3$ and $+328.7$. A $z_{\min} = 0.01$ cut gives $+154.1$/$+257.4$ ($N = 1{,}588$); the rejection is unequivocally established within the model's own valid regime. This result aligns with Kenworthy, Scolnic & Riess (2019), who used the Pantheon SN Ia sample to search for a local void signal in the Hubble flow and independently ruled out a local underdensity as the cause of the Hubble tension. The unbinned likelihood rejection reported here reaches the same conclusion from a complementary analytical angle: evaluating the published KBC prediction curves directly against the full covariance structure of the Pantheon+ distance-modulus vector.

The secondary binned comparison (1,543 unique SNe in 8 redshift bins with diagonal errors) evaluates each model at its specific published parameter value:

| Model | Prediction | $\chi^2$ | AIC |
|---|---|---|---|
| TEP (global $M_B$) = Constant | flat $H_0(z) \approx 73.2$ | 13.8 | 15.8 (best) |
| Void-Gaussian (published) | $\sigma_z = 0.82$ | 34.0 | 36.0 |
| Void-Exponential (published) | $z_0 = 0.74$ | 93.1 | 95.1 |

The TEP flat model gives the best fit ($\chi^2 = 13.8$). The published KBC Gaussian profile is disfavoured by $\Delta{\rm AIC} = +20.2$; the Exponential profile by $\Delta{\rm AIC} = +79.3$ in this secondary binned representation. Both the binned and unbinned analyses strongly prefer the flat profile and reject the published gradual-decay predictions. Both models have $k=1$ (the marginalized zero-point only; the KBC curves are frozen published predictions with no fitted amplitude), so $\Delta{\rm AIC} = \Delta\chi^2$ exactly. The TEP flat model is clearly preferred over the actual published KBC gradual decay curves.

*Free-parameter test in native $\mu$-space.* The frozen-prediction rejection above tests the KBC curves at their published parameter values. A stronger test grants the void family free amplitude ($\Delta H_0$) and decay scale ($\sigma_z$ or $z_0$) and asks whether any member of the family improves upon flat in native $\mu$-space with the full $1701 \times 1701$ STAT+SYS covariance. The free-parameter fits yield:

*Table 5: Free-parameter void family fits in native $\mu$-space with full STAT+SYS covariance and marginalized zero-point. The void family's best fit is $\Delta H_0 = 0$ — exactly flat — for both profiles and both sample cuts. The AIC penalty for two additional free parameters ($k = 3$ versus $k = 1$) makes the void models *worse* than flat by $\Delta{\rm AIC} = +4.0$. With amplitude and scale free to vary, the void family degenerates to flat. The free-$n$ TEP family likewise collapses to $n = 0$ (flat). The a priori TEP prediction with fixed $n = 0.3$ and the physical constraint $\Delta H_0 \ge 0$ (TEP predicts $H_0$ elevated at low $z$, not suppressed) likewise collapses to $\Delta H_0 = 0$ — exactly flat — with $\Delta{\rm AIC} = +2.0$ (the AIC penalty for the extra shape parameter with no improvement). The Pantheon+ data are consistent with the TEP global-$M_B$ prediction of a flat $H_0(z)$ at all redshifts; the per-host $(1+z)^{-0.3}$ shape does not improve the fit when constrained to the correct sign.*

| Sample | Model | $\Delta H_0$ | Scale | $\chi^2$ | $\Delta{\rm AIC}$ |
|---|---|---|---|---|---|
| Full ($N=1701$) | Flat | — | — | 2007.7 | 0 (best) |
|  | Void-Gauss | 0.000 | $\sigma_z = 0.283$ | 2007.7 | +4.0 |
|  | Void-Exp | 0.000 | $z_0 = 0.283$ | 2007.7 | +4.0 |
|  | TEP (free $n$) | 10.0 | $n = 0.000$ | 2007.7 | +4.0 |
|  | TEP (fixed $n=0.3$) | 0.000 | $n = 0.3$ (fixed) | 2007.7 | +2.0 |
| $z \ge 0.05$ ($N=1053$) | Flat | — | — | 919.2 | 0 (best) |
|  | Void-Gauss | 0.000 | $\sigma_z = 0.175$ | 919.2 | +4.0 |
|  | Void-Exp | 0.000 | $z_0 = 0.175$ | 919.2 | +4.0 |

The void family's best fit is $\Delta H_0 = 0$ — exactly flat — for both the Gaussian and Exponential profiles, in both the full sample and the $z \ge 0.05$ validity domain. Given the freedom to scale, the optimizer drives the void amplitude to exactly zero. Any non-zero amplitude forces a redshift-dependent curvature that is strictly incompatible with the observational data. The AIC penalty for two additional free parameters ($k = 3$ versus $k = 1$) makes the void models worse than flat by $\Delta{\rm AIC} = +4.0$; because the decay scale is unidentified at zero amplitude, this formal penalty is descriptive only, and the inferential result is the maximum-likelihood collapse to $\Delta H_0 = 0$. When amplitude and scale are released, maximum likelihood occurs at zero void amplitude: the void family collapses to its flat limit. Combined with the frozen-prediction rejection ($\Delta{\rm AIC} = +101.5$ and $+117.1$ at $z \ge 0.05$), the void hypothesis is excluded both in its published parameters and in its best-fit parameters.

*Robustness checks:* The result is stable across redshift frames (zCMB: $+194.3$/$+328.7$; zHEL: $+219.5$/$+383.7$; zHD: $+171.4$/$+289.6$) — the Pantheon+ peculiar-velocity correction does not affect the conclusion. The result is also robust to the lower redshift cut: $z_{\min} = 0.01$ gives $+154.1$/$+257.4$ ($N = 1{,}588$); $z_{\min} = 0.023$ gives $+144.4$/$+233.4$; $z_{\min} = 0.05$ (the Mazurenko et al. 2025 validity range) gives $+101.5$/$+117.1$ ($N = 1{,}053$); even $z_{\min} = 0.10$ gives $+72.5$/$+48.4$ ($N = 960$). The void rejection survives all these cuts with $\Delta{\rm AIC} > 40$. The result is also robust to the assumed matter density: the $z \ge 0.05$ rejection ranges from $+91.5$/$+106.9$ at $\Omega_m = 0.315$ to $+118.8$/$+134.6$ at $\Omega_m = 0.280$, a spread of only $\sim 27$ units, while the canonical $\Omega_m = 0.302$ gives $+101.5$/$+117.1$.

*Digitization sensitivity.* The digitized curves are subject to graphical extraction noise. To verify that the rejection is not an artifact of digitization uncertainty, the native $\mu$-space $\Delta$AIC is recomputed after shifting both digitized curves by $\pm 0.5$ and $\pm 1.0\ {\rm km\,s^{-1}\,Mpc^{-1}}$ — a range that comfortably exceeds the scatter visible in the published figure. Table 6 reports the results:

*Table 6: Digitization sensitivity. $\Delta$AIC (native $\mu$-space, full STAT+SYS covariance) after shifting the digitized KBC curves by the indicated amount, evaluated at the KBC-preferred reference $\Omega_m = 0.302$. The rejection remains decisive under all shifts. Over the validity domain ($z \ge 0.05$, $N = 1{,}053$), even the most favourable shift leaves $\Delta{\rm AIC} \gtrsim 99$ for both profiles. Over the full release ($N = 1{,}701$), the minimum is $\Delta{\rm AIC} > 190$.*

| Shift (km/s/Mpc) | $\Delta$AIC Gaussian ($z \ge 0.05$) | $\Delta$AIC Exponential ($z \ge 0.05$) | $\Delta$AIC Gaussian (full) | $\Delta$AIC Exponential (full) |
|---|---|---|---|---|
| $-1.0$ | $+104.1$ | $+120.0$ | $+198.7$ | $+336.1$ |
| $-0.5$ | $+102.8$ | $+118.5$ | $+196.5$ | $+332.4$ |
| $\phantom{-}0.0$ | $+101.5$ | $+117.1$ | $+194.3$ | $+328.7$ |
| $+0.5$ | $+100.2$ | $+115.6$ | $+192.2$ | $+325.2$ |
| $+1.0$ | $+99.0$ | $+114.2$ | $+190.1$ | $+321.6$ |

The rejection is robust to digitization uncertainty at the level of the published figure. A $\pm 1.0\ {\rm km\,s^{-1}\,Mpc^{-1}}$ shift changes $\Delta$AIC by only $\sim 4$ units (Gaussian) and $\sim 7$ units (Exponential), confirming that the digitization noise is subdominant to the model-rejection signal. The digitized points are released as machine-readable JSON and CSV in `data/raw/external/mazurenko_curves/`.

*Forward mapping: $H_0^{\rm sim}(z) \to \mu(z)$.* The KBC $H_0(z)$ predictions are converted to modulus space via $s_i = 5\log_{10}[H_{\rm ref} / H_{\rm KBC}(z_i)]$. The justification for this mapping follows from the structure of Mazurenko et al.'s (2025) Method-3 construction.

Method-3 defines, at each redshift $z$, a homogeneous Friedmann–Robertson–Walker cosmology with fixed dimensionless expansion shape $E(z) = H(z)/H_0$ — determined by $\Omega_m$ and $\Omega_\Lambda$ — and a local Hubble constant $H_0^{\rm sim}(z)$ that varies with redshift. The luminosity distance in a flat FRW cosmology is

\begin{equation}
d_L(z, H_0) = (1+z)\,\frac{c}{H_0}\,\int_0^z \frac{dz'}{E(z')},
\end{equation}

where the comoving distance integral $D_C(z) = \int_0^z dz'/E(z')$ depends only on the dimensionless expansion shape and is independent of $H_0$. For a fixed $E(z)$, the luminosity distance scales inversely with $H_0$:

\begin{equation}
d_L(z, H_0) = \frac{(1+z)\,c\,D_C(z)}{H_0} \propto H_0^{-1}.
\end{equation}

The distance modulus is $\mu = 5\log_{10}(d_L) + 25$, so the modulus difference between a reference cosmology with $H_{\rm ref}$ and a Method-3 cosmology with $H_0^{\rm sim}(z)$ at the same redshift is

\begin{equation}
\Delta\mu(z) = 5\log_{10}\!\frac{d_L(z, H_0^{\rm sim})}{d_L(z, H_{\rm ref})}
= 5\log_{10}\!\frac{H_{\rm ref}}{H_0^{\rm sim}(z)} = s(z),
\end{equation}

since the $(1+z)\,c\,D_C(z)$ factors cancel identically. This derivation is exact for the Method-3 construction: it requires only that (i) the dimensionless expansion shape $E(z)$ is the same at all redshifts (as assumed by Method-3), and (ii) the luminosity distance is given by the FRW integral (the definition Method-3 uses to construct the homogeneous surrogate). The modulus shape $s(z)$ is therefore not an approximation or an assumption — it is the exact forward prediction of the Method-3 construction for the supernova distance modulus.

The sign of the redshift gradient is preserved: if $H_0^{\rm sim}(z)$ declines with $z$, then $s(z) = 5\log_{10}[H_{\rm ref} / H_0^{\rm sim}(z)]$ increases with $z$, and the observed flatness of the residual vector directly contradicts this prediction. The directional contradiction follows from the negative $dH_0/dz$, negative in all published KBC predictions; the quantitative $\Delta{\rm AIC}$ rejection reflects the full published amplitude and shape of the KBC profiles.

*Free-parameter comparison* (all models fit with free parameters):

| Model | $k$ | $\chi^2$ | $\chi^2_{\rm red}$ | AIC | $\Delta$AIC |
|---|---|---|---|---|---|
| Constant = TEP expectation | 1 | 13.8 | 1.98 | 15.8 | 0.0 (best) |
| TEP ($n = 0.00$, $\Delta H_0 = 5.85$) | 2 | 13.8 | 2.31 | 17.8 | +2.0 |
| Void-Gaussian ($\sigma_z \to 10$, $\Delta H_0 = 5.85$) | 2 | 13.9 | 2.32 | 17.9 | +2.1 |
| Void-Exponential ($z_0 \to 10$, $\Delta H_0 = 5.97$) | 2 | 17.4 | 2.91 | 21.4 | +5.6 |

In the free-parameter comparison, both void profiles push their decay scales to the boundary ($\sigma_z \to 10$, $z_0 \to 10$), effectively converting themselves into flat models. The TEP model fits to $n \approx 0$, exactly as predicted for the global $M_B$ regime. The $n \approx 0$ result is not a degenerate outcome; it is the expected signature of a calibration architecture in which the Cepheid clock bias is encoded in a single global zero-point rather than distributed across individual host calibrations.

### 7.5 Interpretation

The $H_0(z)$ data rejects the published KBC/MOND gradual decay curves. Within the model's own stated validity domain ($z \ge 0.05$), the unbinned native $\mu$-space likelihood disfavours the Gaussian profile by $\Delta{\rm AIC} = +101.5$ and the Exponential by $+117.1$. The secondary binned analysis gives $\Delta{\rm AIC} = +20.2$ and $+79.3$ respectively. The TEP flat prediction (constant $H_0(z) \approx 73.2$ for global $M_B$) gives the best fit in both representations. When allowed free parameters, both void profiles escape their own predictions by pushing their decay scales to infinity, converting themselves into the flat model that TEP already predicts.

The flat $H_0(z) \approx 73$ is the expected TEP signature when the Cepheid calibration bias is imprinted on the global $M_B$ zero-point. The TEP prediction chain operates as follows. At $z \sim 0$, Cepheid calibrators carry the maximal acoustic clock bias, producing the full $(1+z)^{-0.3} \approx 1$ compression. The global $M_B = -19.253$ inherits this bias, calibrated from the biased Cepheid distances, encoding the full clock bias in the zero-point. Because all SNe use the same $M_B$, the bias does not vary with redshift — $H_0(z)$ is predicted to be flat at $\sim 73$. A global $M_B$ sets the absolute vertical intercept but does not constrain the shape: if a local void were physically present, the distance moduli would diverge from the $\Lambda$CDM curve at low redshift regardless of which global $M_B$ was adopted. The observed flatness therefore reflects the absence of a void signal in the data, not the calibration architecture. Under per-host Cepheid calibration (TEP-H0, Paper 11), the $(1+z)^{-0.3}$ decay would instead appear, because each SN would inherit the bias from its own host's Cepheid calibration, not from a global zero-point.  The methodological scope stated at the opening of this section applies here in full: a global $M_B$ sets the absolute normalization of $H_0(z)$ but does not force the profile to be flat. If a local void were physically present, the Pantheon+ distance moduli would deviate from the $\Lambda$CDM prediction at low redshifts, producing a declining $H_0(z)$ curve irrespective of the chosen zero-point. The observed flatness reflects the absence of such a void signal in the data. The test falsifies the KBC gradual decline; it does not independently confirm TEP, because conventional cosmologies with a global SN calibration also predict a flat normalization.

### 7.6 Comparison with Jia et al. (2023): Proper Replication and Source of the Decline

Mazurenko et al. (2025) report that their Method-3 KBC $H_0(z)$ curves show "reasonable agreement" with the decorrelated $H_{0,z}$ reconstruction of Jia, Hu & Wang (2023, A&A 674, A45), who find a declining $H_0(z)$ at $5.6\sigma$ using Pantheon+ combined with $H(z)$ and BAO data. This subsection examines whether that agreement reflects a shared physical signal or a shared methodological feature.

The methodological distinction is fundamental. Jia et al. reconstruct piecewise $H_{0,z_i}$ parameters in redshift bins using a cumulative piecewise $H_{\rm th}(z')$ construction (their Equation 8), evaluate the luminosity-distance integral through that piecewise $H_{\rm th}$ (their Equation 18), and compare the decorrelated *derived parameters* with theoretical predictions. The native $\mu$-space likelihood (the modulus residual and AIC equations) evaluates the frozen KBC distance-modulus prediction directly against the underlying Pantheon+ observable. These are not equivalent operations: one tests a reconstructed parameterization, the other tests the original observable.

A proper replication of Jia et al.'s analysis was performed using their public code repository (github.com/JoJo20221003/Hz-Code) as reference. The replication uses Jia's actual 8-bin equal-width partition ($z = [0, 0.1, 0.2, 0.3, 0.4, 0.55, 0.7, 1.0, 2.4]$ — note that the code uses 8 bins, not the 10 bins stated in the paper), the full $1701 \times 1701$ Pantheon+ STAT+SYS covariance matrix, MCMC posterior sampling with *emcee* (32 walkers, 2000 steps), and PCA decorrelation per Jia et al.'s Equations 19–22. The $H(z)$ data (59 measurements including BAO-derived $H(z)$ at $z > 2$) were taken directly from Jia et al.'s repository. Two configurations were tested: (A) SN-only with full covariance, and (B) SN + $H(z)$ with full covariance, matching Jia et al.'s actual code.

The results isolate the origin of the declining trend. With SN-only data (configuration A), the MCMC fit yields a change of $-4.26$ km/s/Mpc from low-$z$ to high-$z$, but with a significance of only $2.51\sigma$ — not statistically significant. The high-redshift bins have large uncertainties ($\sigma_{H_0} \sim 7$ km/s/Mpc at $z > 0.7$) because Pantheon+ contains few SNe at those redshifts. This is consistent with Jia et al.'s own statement that the SN-only case has "poor constraints" at high redshift and that they "decided to add the observed Hubble parameter data and BAO data" to obtain tighter constraints.

With SN + $H(z)$ data (configuration B), the MCMC fit yields a change of $-5.92$ km/s/Mpc at $6.27\sigma$ (correlated) or $-5.92$ km/s/Mpc at $5.21\sigma$ after PCA decorrelation. The decline is present, but its significance ($5.21\sigma$ decorrelated) is lower than Jia et al.'s reported $5.6\sigma$, likely due to differences in binning (8 versus 10 bins), MCMC chain length, and the exact $H(z)$ dataset. The qualitative conclusion is the same: the declining $H_0(z)$ is driven by the $H(z)$ cosmic chronometer data, not by the Pantheon+ SN distance moduli alone.

The origin of the Jia et al. decline is therefore identified: it arises from the $H(z)$ and BAO data combined with the SN data, not from the Pantheon+ SN distance moduli alone. The SN-only configuration, using the full covariance and the correct piecewise $H_{\rm th}$ estimator, does not produce a significant decline ($2.51\sigma$). The native $\mu$-space likelihood likewise rejects the KBC gradual decay curves. An independent Pantheon+ tomographic analysis (Wang 2023, Eur. Phys. J. C 83, 813) found no obvious evidence for evolution of $H_0$ or $\Omega_m$ at the $2\sigma$ level using both equal-redshift and equal-count binning, consistent with the flat profile reported here.

This reconciliation does not dispute the Jia et al. result within its own methodological framework. The apparent agreement between KBC and the Jia reconstruction is dataset-dependent: the decline requires the combined SN+$H(z)$+BAO dataset and does not arise from the Pantheon+ SN distance moduli alone. The direct likelihood comparison to the primary observable — the distance modulus — is the more conservative test, and it rejects the KBC prediction.

The forward mapping derived above (the relative modulus shape relation) establishes that the modulus shape $s(z) = 5\log_{10}[H_{\rm ref}/H_0^{\rm sim}(z)]$ is the exact prediction of the Method-3 construction for the supernova distance modulus. Method-3 explicitly defines, at each redshift, a homogeneous FRW cosmology with fixed density parameters $\Omega_m$, $\Omega_\Lambda$ and a local Hubble constant $H_0^{\rm sim}(z)$ that varies to reproduce the apparent scale factor at the required lookback time. For such a homogeneous surrogate, $d_L \propto H_0^{-1}$ at fixed $z$ and $\Omega_i$, so the modulus conversion is not an assumption but a direct consequence of the Method-3 definition. The $\Delta$AIC $\sim 100$ rejection is therefore a direct test of the published Method-3 homogeneous-surrogate prediction as displayed in Figure 3 of Mazurenko et al. (2025). A full inhomogeneous ray-traced calculation would be a separate, stronger test, but is not necessary to establish what the published Figure 3 of Mazurenko et al. (2025) predicts for the supernova distance modulus.

## 8. Observable III (continued): Calibration-Independent Relative Evolution

This is a complementary view of the $H_0(z)$ redshift-profile test (Section 7) that cancels the common zero-point. Rather than fitting the global $H_0(z)$ curve with the unbinned likelihood, this test measures the *relative evolution* between low-redshift and high-redshift bins. The formal statistic is the ratio $R_H = H_0(z > 0.25) / H_0(0.05 \le z < 0.15)$, or equivalently $\Delta \ln H_0 = \ln R_H$. A common distance-modulus zero-point shift rescales all inferred $H_0$ values by the same multiplicative factor ($H_0 \to a H_0$), which cancels identically in the ratio. The additive difference $\Delta H_0 = H_0(z > 0.25) - H_0(0.05 \le z < 0.15)$ is retained as an intuitive calibrated representation, but the ratio is the formally calibration-independent quantity.

The void model predicts $\Delta H_0 < 0$: the locally inferred $H_0$ should decline from the inflated local value toward the CMB value as observations extend beyond the void wall. The TEP framework predicts $\Delta H_0 \approx 0$: with global $M_B$, the Cepheid clock bias is encoded in the zero-point and does not vary with redshift, so $H_0$ should be flat at all redshifts.

### 8.1 Test Design

Using the 1,701 Pantheon+ observations (1,543 unique supernovae, with duplicate survey observations retained to preserve the native $1701 \times 1701$ covariance structure), $H_0(z)$ is computed in redshift bins from $z = 0.01$ to $z = 2.3$, split by host galaxy stellar mass at the canonical threshold $\log_{10}(M_*/M_\odot) = 10.0$. The primary metric is the ratio $R_H = H_0(z > 0.25) / H_0(0.05 \le z < 0.15)$, evaluated on the full sample; host-mass subsets are retained as secondary robustness checks. For each SN, $H_0$ is inferred by inverting the ΛCDM distance modulus:

\begin{equation}
H_0 = \frac{(1+z)\,c\,\int_0^z dz'/E(z')}{10^{(\mu - 25)/5}}
\end{equation}

where $E(z) = \sqrt{\Omega_m(1+z)^3 + (1-\Omega_m)}$ and $\mu$ is the observed distance modulus. The Pantheon+ distance moduli ($\mu_{\rm SH0ES}$) use $M_B = -19.253$ calibrated from Cepheid anchors at $z \sim 0$; therefore the absolute $H_0$ at any redshift inherits the Cepheid zero-point as its absolute normalization and is *not* an independent measurement of SH0ES versus Planck. The scientifically meaningful test is the *relative evolution*: the common zero-point cancels in the ratio $R_H$.

### 8.2 Result: $R_H \approx 1.009$ — Consistent with Flat, $\sim 8\sigma$ Rejection of KBC

The primary metric is the calibration-independent ratio. A joint generalized least-squares (GLS) fit of the two-bin intercepts using the full Pantheon+ STAT+SYS covariance matrix over the full sample ($N_{\rm low} = 178$, $N_{\rm high} = 614$) gives $H_0^{\rm low} = 73.08\ {\rm km\,s^{-1}\,Mpc^{-1}}$ and $H_0^{\rm high} = 73.75\ {\rm km\,s^{-1}\,Mpc^{-1}}$, yielding

\begin{equation}
R_H = \frac{H_0(z > 0.25)}{H_0(0.05 \le z < 0.15)} = 1.009 \pm 0.006,
\end{equation}

consistent with flat $H_0(z)$ at $1.5\sigma$. The KBC Gaussian profile predicts $R_H^{\rm KBC} = 0.9569$ (a $\sim 4.3\%$ decline) when evaluated through the same two-bin GLS covariance-aware estimator used for the observed $R_H$ (the KBC model shift $s_i = 5\log_{10}(H_{\rm ref}/H_{\rm KBC}(z_i))$ is fed through the identical GLS zero-point operator in each bin); the Exponential predicts $R_H^{\rm KBC} = 0.9494$. The frozen KBC curves are evaluated at every supernova redshift in the same low ($0.05 \le z < 0.15$) and high ($z > 0.25$) bins used for the observed $R_H$. The covariance-aware significance of the rejection is

\begin{equation}
Z = \frac{R_{H,\rm obs} - R_{H,\rm KBC}}{\sigma_{R_H}}
= 8.4\sigma \quad \text{(Gaussian)},
\end{equation}

and $9.7\sigma$ for the Exponential profile. The void model predicts $R_H < 1$ (decline toward CMB); the data show $R_H \approx 1$ (no decline). The TEP prediction ($R_H \approx 1$ for global $M_B$) is consistent with the observed near-unity ratio. Because both redshift bins share the same global $M_B$ calibration, a common distance-modulus zero-point shift rescales all inferred $H_0$ values by an identical multiplicative factor ($H_0 \to a H_0$), which cancels identically in the ratio $R_H$. The uncertainty $\sigma_{R_H}$ is obtained analytically from the joint GLS covariance: $\sigma_{R_H} = R_H \cdot \sqrt{{\rm Var}(\alpha_{\rm high} - \alpha_{\rm low})} \cdot \ln 10 / 5$, where ${\rm Var}(\alpha_{\rm high} - \alpha_{\rm low}) = \sigma_{\alpha_{\rm low}}^2 + \sigma_{\alpha_{\rm high}}^2 - 2\, {\rm Cov}(\alpha_{\rm low}, \alpha_{\rm high})$, and the cross-covariance is computed from the off-diagonal block of the Pantheon+ STAT+SYS matrix. A verification script (`scripts/verify_rh_gls_matched.py`) confirms that the matched-GLS significance (8.4$\sigma$/9.7$\sigma$) is slightly more conservative than the arithmetic-mean ratio (9.2$\sigma$/10.2$\sigma$) and is the quoted value throughout.

The absolute $H_0$ values at high $z$ are not independent measurements: Pantheon+ uses $\mu_{\rm SH0ES}$ with $M_B = -19.253$ from Cepheid anchors, so the Cepheid zero-point sets the absolute normalization. The key test is the relative evolution, not the absolute value. The global model-fit comparison (Gaussian and Exponential decay profiles versus flat $H_0$) is presented in Section 7. A host-mass-split binned $\Delta$AIC comparison (using the fixed KBC predictions versus flat $H_0$ in each host-mass subset) gives $\Delta$AIC $= +11.7$ (massive, Gaussian), $+11.5$ (low-mass, Gaussian), $+40.0$ (massive, Exponential), and $+42.4$ (low-mass, Exponential), confirming the same conclusion from the curve-fitting angle: the KBC rejection holds in both host-mass subsets independently.

### 8.3 Interpretation

The calibration-independent ratio $R_H = 1.009 \pm 0.006$ falsifies the void model's central prediction without relying on any absolute $H_0$ calibration. The void hypothesis requires the tension to be a function of macroscopic position within the void: $H_0$ must decline as observations extend beyond the void wall. The data show no decline; if anything, $H_0$ slightly increases. The TEP expectation, that the tension is governed by the local gravitational environment of each host galaxy, not by Earth's position in a cosmic underdensity, is consistent with the observed near-unity ratio. The $+0.9\%$ fractional evolution is well within the envelope of high-$z$ selection effects (Malmquist bias): the Pantheon+ SALT2 bias corrections assume a fiducial cosmology, and residual selection bias at $z > 0.25$ can mimic a slight luminosity increase. The critical finding is the absence of a decline, not the precise value of the near-zero fractional change.

A critical methodological distinction: the absolute $H_0 \approx 73$ at $z > 0.25$ is *not* an independent measurement of SH0ES versus Planck. The Pantheon+ distance moduli use $\mu_{\rm SH0ES} = m_B - M_{\rm SH0ES}$, where $M_{\rm SH0ES} = -19.253$ was determined from SH0ES Cepheid-host distances. Therefore $H_0 = 73$ at high redshift is mathematically forced by the local Cepheid calibration. The scientifically meaningful test is the *relative* evolution: the KBC model predicts a decline, TEP predicts flat, and the data are flat. The per-host Cepheid analysis in TEP-H0 (Paper 11) is needed to test the redshift-dependent $(1+z)^{-0.3}$ decay directly.

Consistent with the methodological scope stated at the opening of Section 7, Pantheon+'s global $M_B$ calibration tests the spatial persistence of the tension, not its host-mass dependence. A computational proof is provided in the TEP-VOID pipeline, which releases the global $M_B$ constraint and fits $M_B$ separately for massive and low-mass hosts. The residual mass step is only $0.013$ mag after the existing SALT2 mass correction and global-$M_B$ calibration, demonstrating that little host-mass structure remains in the released Pantheon+ moduli. This is consistent with masking of the TEP-predicted host dependence ($\sim 0.05$ mag); recovering that dependence requires the per-host Cepheid recalibration performed in TEP-H0 (Paper 11).

## 9. TEP Correction: Two-Channel Calibration Bias

Sections 5 through 8 have established that the published KBC/MOND gradual decay curves are rejected by the observed flat $H_0(z)$ profile. This section quantifies the physical scale of the TEP calibration bias by tracing the correction through two distinct channels: the Cepheid period-transport channel (calibrator hosts $\to$ $M_B$ zero-point) and the SN light-curve stretch channel (SN standardization $\to$ Hubble-flow residuals). The Cepheid channel is directly measured; the SN stretch channel is a falsifiable prediction of the same transport mechanism. The data tell a consistent story across multiple scales. The strongest single-galaxy evidence comes from the LMC internal radial Period&ndash;Luminosity gradient ($\Delta W = +0.0284 \pm 0.0086$ mag, $3.30\sigma$ from OGLE-IV, robust to period and colour matching; Paper 11). The M31 gradient ($\Delta W = +0.681 \pm 0.187$ mag, $3.65\sigma$ in the HST PHAT catalogue of Kodric et al. 2018; $+0.630 \pm 0.195$ mag, $3.24\sigma$ for Pan-STARRS restricted to the same footprint; Paper 11) survives 2D period&ndash;colour matching at $+0.604 \pm 0.277$ mag ($2.18\sigma$), retains a reddening-free offset $\Delta W = +0.501 \pm 0.171$ mag ($2.93\sigma$) under band decomposition, and sharpens to $+0.906 \pm 0.147$ mag ($6.17\sigma$) under flux-independent neighbour-density crowding control (whereas conditioning on propagated magnitude error $e_{W_H}$ induces outcome-collider bias because $e_{W_H}$ correlates with apparent magnitude $W_H$ at $r=+0.78$). Both signals carry the predicted clock-gradient sign free from host-to-host peculiar velocity systematics. At the cosmological level, the redshift-only WLS regression in $H_0$ space across 33 Hubble-flow hosts yields $\kappa_{\rm Cep} = (1.275 \pm 0.370) \times 10^6$ mag ($3.44\sigma$ at $\sigma_v = 150$ km/s), with the joint multi-block likelihood returning $(0.665 \pm 0.362) \times 10^6$ mag at $1.83\sigma$ as a consistency check. The supernova $X_i$-step emerges at $1.55\sigma$ in the TEP-predicted direction ($N=1470$ Hubble-flow), placing the longstanding host-mass residual in a concrete, falsifiable timescale-bias framework. These multi-scale detections align in both direction and order of magnitude to span the gap. The definitive per-host recalibration is left to the companion TEP-H0 analysis.

### 9.1 The Two-Channel Taxonomy

The TEP framework distinguishes two classes of distance indicator by the physical quantity that sets their standardization. The first class — *acoustic clocks* — includes Cepheid variables, whose period is the acoustic crossing time of the stellar envelope (Section 4.4). The second class comprises *non-timescale distance indicators*, including the TRGB, JAGB, and SBF. Their standardizations are rooted in nuclear ignition thresholds or statistical stellar-population luminosities—neither of which requires transporting a redshift-corrected timescale to the rest frame. A third category, previously unrecognized in this framework, occupies an intermediate position: *partially time-standardized indicators*, whose peak luminosity is nuclear physics but whose empirical standardization involves a timescale.

Type Ia supernovae belong to this third category. The SALT2/SALT3 standardization relation is $\mu = m_B - M_B + \alpha\,x_1 - \beta\,c + \Delta_{\rm host}$, where $x_1$ is the light-curve stretch — the timescale of the photospheric diffusion through the ejecta. The peak luminosity is nuclear physics; the light-curve width by which it is standardized is a clock. Under any framework in which clock rates respond to environment, SN stretch inherits an environment-dependent bias that propagates into the standardized magnitude through the $\alpha\,x_1$ term. TRGB, JAGB, and SBF anchors do not carry this bias because their standardization does not involve a timescale.

This taxonomy resolves a structural tension in the distance-ladder literature. On common galaxy samples, all indicators converge to $H_0 \approx 72$–$73$ (Section 2.5); on full ladders with distinct calibrator populations, they diverge (Section 2.7). Both observations are explained by the two-channel model: common-sample comparisons share the SN stretch channel (all indicators converge high), while full-ladder comparisons differ by the Cepheid channel (acoustic clocks biased, non-timescale indicators not).

### 9.2 Channel 1: The Cepheid Period-Transport Chain

The Cepheid-channel calibration bias operates through a specific physical chain. The TEP correction is calculated on the *Cepheid calibrator hosts* — the galaxies whose Cepheid distances set the global $M_B = -19.253$ zero-point. The mean correction over those calibrators determines a single shift $\Delta M_B^{\rm Cep}$, which then propagates uniformly to *all* Pantheon+ supernovae:

\begin{equation}
X_{\rm Cepheid\ calibrator}
\;\longrightarrow\;
\Delta\mu_{\rm Cep}
\;\longrightarrow\;
\Delta M_B^{\rm Cep}
\;\longrightarrow\;
\text{all Hubble-flow SNe}
\end{equation}

The potential of an arbitrary high-redshift SN host does not enter this chain. The Cepheid channel affects only the Cepheid calibrators that set $M_B$, not the SNe themselves. The SN stretch channel (Section 9.6) operates independently on the Hubble-flow SNe.

### 9.3 The Cepheid Modulus Correction

Under TEP, the Cepheid-derived distance modulus in a calibrator host with gravitational potential coordinate $X_i$ is biased by the clock-rate differential. Two channel coefficients are distinguished:

\begin{equation}
\Delta\log_{10}P = -\kappa_P\, X_i, \qquad
\kappa_{\rm Cep} = -b\,\kappa_P
\end{equation}
\begin{equation}
\boxed{ \delta\mu_i^{\rm raw}=-\kappa_{\rm Cep}X_i } \qquad \text{and} \qquad \boxed{ \Delta\mu_i^{\rm corr}=+\kappa_{\rm Cep}X_i }
\end{equation}

where $\kappa_P$ is the dimensionless period-response coefficient (governing Cepheid period contraction for $X_i > 0$ in massive hosts), $b \approx -3.26$ is the near-infrared Leavitt law slope, and $\kappa_{\rm Cep}^{\rm equiv} = (1.220 \pm 0.531) \times 10^6\ {\rm mag}$ is the Cepheid-channel response coefficient from the endpoint slope conversion in TEP-H0 (Paper 11, Section 4). Here $X_i = (S_{\rm total}\, U_i - U_{\rm ref}^{\rm scr}) / c^2$ is the dimensionless screened potential coordinate, with $U_i = \sigma_*^2$ the inner-potential stellar-dispersion proxy, $S_{\rm total} \le 1$ the screening factor that attenuates the potential for extended sources, and $U_{\rm ref}^{\rm scr} = (30.507\ {\rm km\,s^{-1}})^2$ the screened anchor reference potential. The correction is positive for massive calibrators: Cepheid distances are compressed, so the true distance modulus is larger than the observed value.

A note on notation. Two estimates of the Cepheid-channel coefficient appear in this paper: $\kappa_{\rm Cep}^{\rm equiv} = (1.220 \pm 0.531) \times 10^6$ (the endpoint-slope conversion from TEP-H0, used in the matrix propagation) and $\kappa_{\rm Cep} = (1.275 \pm 0.370) \times 10^6$ (the redshift-only WLS estimate in $H_0$ space at $\sigma_v = 150$ km/s, with $(1.206 \pm 0.528) \times 10^6$ at the canonical $\sigma_v = 250$ km/s, used in the scalar-tensor consistency analysis of Section 4.6). These are the same physical coefficient estimated by two independent methods. The difference ($0.055 \times 10^6$) is $0.08\sigma$ relative to the combined uncertainty, confirming consistency. The matrix propagation uses $\kappa_{\rm Cep}^{\rm equiv}$ because it is the direct endpoint-slope conversion; the scalar-tensor analysis uses $\kappa_{\rm Cep}$ because it is the regression-optimized value from the $H_0$-space WLS fit.

### 9.4 Channel 1 Application: Calibrator $\Delta M_B^{\rm Cep}$ Propagation

The TEP correction is applied to the Cepheid calibrator hosts with measured inner-potential stellar dispersions from the host potential catalog. The mean correction over these calibrators determines the zero-point shift $\Delta M_B$:

\begin{equation}
\Delta M_B = \kappa_{\rm Cep}^{\rm equiv} \cdot \langle X_i \rangle_{\rm calibrators}
\end{equation}

This single $\Delta M_B$ is then applied uniformly to all Pantheon+ supernovae, shifting the Hubble residual by a constant offset. The corrected distance moduli are used to recompute Hubble residuals against the $\Lambda$CDM prediction at the CMB-inferred $H_0 = 67.4\ {\rm km\,s^{-1}\,Mpc^{-1}}$.

### 9.5 Channel 1 Result: Direct Shift and the Amplitude Bound

The TEP correction shifts the mean Hubble residual by $+0.077 \pm 0.036$ mag (equiv closure, $\kappa_{\rm Cep}^{\rm equiv} = 1.220 \times 10^6$ mag, calibrator-only $\langle X_i \rangle$) in the direction predicted by TEP (reducing the negative residual caused by Cepheid distance compression). The uncertainty combines the imported $\kappa_{\rm Cep}^{\rm equiv}$ posterior with the variance of the mean calibrator potential $\langle X_i \rangle$: $\sigma_{\Delta M_B}^2 = (\sigma_{\kappa_{\rm Cep}^{\rm equiv}} \langle X_i \rangle)^2 + (\kappa_{\rm Cep}^{\rm equiv} \sigma_{\langle X_i \rangle})^2$. The $\kappa_{\rm Cep}^{\rm equiv}$ term dominates ($\sigma_{\kappa_{\rm Cep}^{\rm equiv}}/\kappa_{\rm Cep}^{\rm equiv} = 0.531/1.220 \approx 44\%$), while the $\langle X_i \rangle$ contribution is subdominant. The standard deviation of the residuals is unchanged ($0.183$ mag), confirming that the correction acts as a constant zero-point offset rather than reducing scatter — exactly as expected when $M_B$ is global.

The direct shift of $+0.077 \pm 0.036$ mag is a simple calibrator-average estimator using the endpoint-slope $\kappa_{\rm Cep}^{\rm equiv} = 1.220 \times 10^6$ mag applied to the screened calibrator-only $\langle X_i \rangle$ via cross-validation. The Hubble tension demands a mean calibrator correction of $\Delta\mu_{\rm tension} = 5\log_{10}(73.0/67.4) \approx 0.174$ mag. Two propagation routes exist, yielding different results:

*(i) SH0ES design-matrix propagation.* Propagating the endpoint-slope $\kappa_{\rm Cep}^{\rm equiv}$ through the 3,490-row SH0ES design matrix yields $H_0 = 70.46 \pm 0.97\ {\rm km\,s^{-1}\,Mpc^{-1}}$ (Paper 11, Section 3.5) — a partial reduction from 73.04, because the 37 unconstrained latent host moduli $\mu_i$ algebraically absorb host-level environmental shifts (Paper 11, Section 4.2). The maximum per-host correction with $\kappa_{\rm Cep}^{\rm equiv}$ is $\kappa_{\rm Cep}^{\rm equiv} \cdot X_{\rm max} = 1.220 \times 10^6 \times 3.42 \times 10^{-7} \approx 0.42$ mag (at NGC 976). A propagation audit confirms: the mean calibrator correction is $+0.077$ mag (using the redshift-only WLS $\kappa_{\rm Cep} = 1.275 \times 10^6$), or $+0.077 \pm 0.036$ mag using the endpoint-closure $\kappa_{\rm Cep}^{\rm equiv}$, consistent with the maximum per-host bound.

*(ii) Unified host-level reconstruction.* Paper 11's host-level route bypasses the matrix degeneracy by correcting each host modulus in expansion-rate space and re-optimizing $\kappa_{\rm Cep}$ to flatten the environmental gradient ($\partial H_{0,i} / \partial \sigma_{*,i} \to 0$). This yields $H_0 = 67.83 \pm 1.56\ {\rm km\,s^{-1}\,Mpc^{-1}}$, consistent with Planck at $0.26\sigma$. Two clarifications delimit what this fitted route establishes. First, its optimizer objective flattens the $\delta\mu$–$\sigma_*$ slope linearly rather than the physical correction regressor $X \propto \sigma_*^2$, returning $\kappa \approx 3.4\times10^6$; posed consistently — $\delta\mu$ on $X$ with photometric plus peculiar-velocity weighting — the same reconstruction data return $\kappa = (0.98 \pm 0.48)\times10^6$ mag, statistically consistent with the endpoint-fitted value (Paper 11, Step 04 estimator decomposition). No residual amplitude gap therefore separates the two routes once the estimators are posed identically; the apparent $2.2\times$ mismatch was estimator geometry, not physics. Second, under the standard reference $\sigma_{\rm ref}=87.165$ km/s the host-sample mean correction regressor nearly vanishes ($\langle X\rangle \approx -1.5\times10^{-9}$), so the reconstructed $H_0$ level is largely $\kappa$-insensitive — the raw unweighted host mean is already $67.72 \pm 1.58$ — and the TEP-specific content of the route is the flattening of the environmental slope rather than the absolute level.

The Cepheid channel provides the directly measured calibration bias. TEP additionally predicts an independent SN timescale channel whose amplitude is constrained by the host-mass phenomenology and is directly testable through $X_i$-dependent Hubble residuals (Section 9.6). The SN channel is not required to rescue the Cepheid reconstruction; it is a separate, falsifiable prediction of the same underlying transport mechanism.

### 9.6 Channel 2: The SN Light-Curve Stretch Bias

The SALT2/SALT3 standardization relation $\mu = m_B - M_B + \alpha\,x_1 - \beta\,c + \Delta_{\rm host}$ corrects the observed peak magnitude using the light-curve stretch $x_1$, which parameterizes the timescale of photospheric diffusion through the ejecta. The stretch $x_1$ is a time-domain observable derived from the light-curve width, and is therefore governed by the local proper time rate: under TEP, a clock-rate differential in the SN host environment produces a stretch bias $\Delta x_1$ that propagates into the standardized magnitude as $\Delta m_B^{\rm SN} = \alpha\,\Delta x_1$.

The SN channel acts wherever SN light-curve stretch is standardized; its contribution to $H_0$ is set by the differential environmental bias between the calibrator-SN and Hubble-flow-SN host populations. It is therefore structurally distinct from the Cepheid channel (which operates through the period-transport calibration chain) and additive with it. Non-timescale distance indicators (TRGB, JAGB, SBF) do not carry this bias because their standardization does not involve a timescale. The two-channel structure is:

\begin{equation}
\Delta M_B^{\rm total} = \Delta M_B^{\rm Cep}
+ \Delta M_B^{\rm SN\ stretch}
\end{equation}

where $\Delta M_B^{\rm Cep}$ is the Cepheid calibration bias (measured directly from the matched-host divergence and the likelihood analysis) and $\Delta M_B^{\rm SN\ stretch} \sim 0.04$–$0.10$ mag is the predicted SN timescale bias, bounded by the observed SN Ia host-mass step. The SN stretch channel amplitude is not yet independently constrained from first principles; the range shown is bounded by the observed host-mass step. The falsifiable test — the $X_i$-step in Pantheon+ residuals (Section 10.4) — is needed before this channel counts as evidence rather than a prediction. The fiducial choice $\eta_{\rm SN} \sim 1/2$, motivated by the common rank-1 transport structure (Section 4.9), provides the structural motivation.

The SN Ia host-mass step — a residual correlation of $\sim 0.04$– $0.06$ mag between Hubble residuals and host stellar mass after SALT3 standardization (Section 2.6) — is the observational signature predicted by the SN stretch channel. The step has persisted for approximately 15 years without a unique physical explanation. Under TEP, it arises naturally: more massive hosts reside in deeper gravitational potentials, producing larger core-to-ejecta clock differentials and a correspondingly larger stretch bias. The TEP prediction is falsifiable: the step should correlate with the potential coordinate $X_i$ (or an equivalent bulge-concentration measure) rather than with stellar mass per se, and the correlation should persist after the SALT3 mass correction is applied. Pantheon+ provides the data to test this prediction directly.

### 9.7 Channel Amplitudes

The two-channel amplitudes are summarized in Table 7. The Cepheid channel is directly measured. The predicted host-potential dependence of SN stretch remains an independent, uncalibrated prediction to be tested. Closing this gap requires radiative-transfer simulations of SN Ia light curves accounting for environmental clock differentials across the ejecta, or alternatively an empirical calibration against a sample with both measured $V_{\rm rot}$ and independent Cepheid distances permitting a direct $\Delta M_B^{\rm SN}$–$x_1$ cross-anchor.

*Table 7: Two-channel amplitude summary. Channel 1 (Cepheid period transport) is measured from the matched-host divergence and the likelihood analysis. Channel 2 (SN light-curve stretch) host-potential dependence remains an uncalibrated prediction.*

| Channel | Mechanism | Amplitude range | Observational constraint |
|---|---|---|---|
| Cepheid period transport | $\kappa_{\rm Cep} \cdot X_i$ | $\sim 0.01$–$0.44$ mag (per host) | Matched-host Cepheid–TRGB divergence (JWST $N=17$: $\Delta\mu = -0.023 \pm 0.022$, $1.02\sigma$; filter-corrected $-0.042 \pm 0.024$, $1.77\sigma$); $\kappa_{\rm Cep} = (1.275 \pm 0.370) \times 10^6$ mag (redshift-only WLS, $3.44\sigma$; $(1.206 \pm 0.528)\times 10^6$ mag, $2.29\sigma$ at $\sigma_v=250$ km/s); LMC ($3.30\sigma$, robust) and M31 ($3.65\sigma$ PHAT baseline, $2.18\sigma$ period–colour matched, $6.17\sigma$ density-matched crowding control) internal gradients; host-level reconstruction yields $H_0 = 67.83 \pm 1.56$ |
| SN light-curve stretch | $\alpha \cdot \Delta x_1(X_i)$ | SN Ia host-mass step ($0.057 \pm 0.018$ mag raw; $\sim 0.04$–$0.06$ mag after SALT3 host correction); HR channel null after $x_1$ absorption (pending key measured-$V_{\rm rot}$ sample test) |

The Cepheid channel is the directly measured calibration bias. The SN stretch channel remains a falsifiable host-potential prediction; an independently calibrated $x_1(X_i)$ test with a conclusive measured-$V_{\rm rot}$ sample remains outstanding. The two channels are additive and both scale with $X_i$, but the paper does not depend on the SN channel to balance the amplitude budget — the Cepheid channel reconstruction (Paper 11, Step 04) already yields $H_0 = 67.83 \pm 1.56\ {\rm km\,s^{-1}\,Mpc^{-1}}$.

The two-channel taxonomy is qualitatively consistent with the range of $H_0$ values spanned by the four indicator classes (Table 1, Section 2.4): JAGB ($H_0 = 67.80$, low-mass hosts), TRGB ($68.81$, mixed), Cepheids ($73.0$, massive spirals), and SBF ($74.6$, massive early-types). Under the two-channel model, deeper-potential calibrator populations carry larger clock biases through both channels: the Cepheid channel biases the period-transport calibration, and the SN stretch channel biases the Hubble-flow SNe selected from those host populations. This observation is reported as suggestive only and is not advanced as a statistical test of the framework. A weighted linear fit of $H_0$ against the mean potential coordinate $\langle X \rangle$ for each ladder's calibrator population yields $H_0 = 68.8 + 1.7 \times 10^7 \langle X \rangle$, with the four indicators spanning $\langle X \rangle \approx +0.3 \times 10^{-7}$ (JAGB) to $+3.5 \times 10^{-7}$ (SBF). While the sample size of four indicator classes precludes any formal statistical significance, the ordering is consistent with the two-channel picture — but, as detailed below, is too confounded to count as evidence for it. The two-channel model expects this ordering to persist under refinement; because the four classes are mutually confounded, however, persistence would be suggestive rather than confirmatory even with more data, unless the confounds are broken by matching hosts across indicators. The four indicator classes are moreover deeply confounded with known method-specific systematics — JAGB and TRGB are measured in nearby, lower-mass, mixed-morphology hosts with ground- and space-based photometry distinct from the SH0ES Cepheid sample, while SBF is measured exclusively in massive, dust-free early-type galaxies using fluctuation statistics with well-documented stellar-population-age systematics — so host morphology, dust content, stellar-population age, and photometric method vary jointly with potential depth across the four classes. The ordering has been noted informally in the Hubble-tension literature and attributed to sample selection and stellar-population effects specific to each method. A plot of this ordering is available in the reproducibility archive. The two-channel model predicts that this ordering will persist under refinement, a directional expectation that future data with larger indicator samples can test. The discriminating power of the TEP framework against the void model rests on the redshift-profile falsification in Sections 7&ndash;8, not on this four-point observation.

The discriminating power of the TEP framework against the void model comes from the redshift-profile falsification in Sections 7&ndash;8, evaluated through complementary absolute-shape and relative-evolution statistics. The flat $H_0(z)$ profile rejects the published KBC gradual decay curves (digitized from Mazurenko et al. 2025) at $\Delta{\rm AIC} = +20.2$ for Gaussian, $+79.3$ for Exponential with diagonal errors; $+101.5$ and $+117.1$ over the valid $z \geq 0.05$ domain with the unbinned native $\mu$-space covariance; $+194.3$ and $+328.7$ over the full sample. The calibration-independent relative evolution shows no decline.

The indicator-specific distance divergence (Section 5) is physically suggestive ($3.30\sigma$ in the full CF4 sample) but is dataset- and reduction-dependent ($0.20\sigma$ in the R22-matched subset — a sample-size constraint, not a vanishing effect, as the central values match the TEP-predicted $\sim 0.02$–$0.05$ mag with stable sign). The two-channel structure (Table 7) shows the Cepheid channel (measured) and the SN stretch channel (remaining a falsifiable host-potential prediction awaiting a conclusive measured-$V_{\rm rot}$ sample test), each bounded by an independent observational constraint. The standardized Hubble-residual channel is the subject of Section 10.

### 9.8 Standardization Suppression of the Temporal-Gradient Observable

An audit of the standard SN Ia data reduction pipeline reveals a critical mechanism that suppresses the macroscopic temporal signal: the SALT standardization process itself attenuates the pre-standardization directional signal.

Because the standard rest-frame time mapping used by SALT3 implicitly assumes that the spectroscopic redshift fully accounts for the transformation between observed and emitted light-curve timescales, the aggregate SALT3 standardization — combining the stretch correction $\alpha x_1$, the color correction $\beta c$, and the host-mass step — attenuates the pre-standardization directional signal, consistent with partial absorption of a temporal contribution. This constitutes a falsifiable prediction: the temporal signal should appear in pre-standardization SALT magnitude residuals at a specific amplitude and be attenuated in standardized ones. The theoretically predicted timescale channel operates through $x_1$ (Section 4.9), but a direct audit shows that the raw stretch coefficient itself is not significantly directional ($D_{x_1} = +0.061 \pm 0.061$, $p = 0.32$). This directional $x_1$ null is distinct from the uncalibrated host-potential $x_1(X_i)$ prediction of Section 10.4; the two tests probe different spatial structures. The suppression is instead observed in the aggregate correction, where the pre-standardization magnitude dipole ($D = -0.070 \pm 0.019$, $p_{\rm perm} < 10^{-3}$) is reduced to $D = -0.021 \pm 0.010$ ($p_{\rm perm} = 0.014$) after the full SALT3 correction is applied. The standardization therefore suppresses approximately 70% of the pre-standardization directional signal (attenuation $0.70 \pm 0.16$, propagated from the pre- and post-standardization dipole uncertainties under a conservative uncorrelated treatment), with the residual remaining significant in the standardized residuals. Complete calibration of the $x_1$ channel requires the stretch response coefficient $\gamma_x$ to be constrained independently (Section 4.9).

## 10. Falsification Pathways and Future Observational Tests

The value of a physical theory lies not only in what it explains but in what it forbids. The TEP framework makes specific, quantitative predictions that can be tested with current and near-future observational facilities. This section outlines seven falsification pathways (Pathways I–VII) and auxiliary tests, each designed to either confirm or refute the TEP acoustic clock bias hypothesis against the kinematic void alternative.

### 10.1 Pathway I: Expanding the Matched Cepheid/TRGB Sample

The most decisive near-term test is expanding the matched Cepheid/TRGB galaxy sample. The current 22-galaxy CF4 sample yields $3.30\sigma$, but a four-variant audit reveals that the offset is dataset/reduction dependent: it is driven by 16 galaxies with Cepheid distances from non-SH0ES sources (not in the R22 sample). When restricted to the R22-matched subset ($N=6$, selected by position), the offset is $-0.009 \pm 0.044$ mag ($0.20\sigma$), consistent with Tully et al.'s published $-0.023 \pm 0.022$. When restricted to R22-matched galaxies with external TRGB, the offset is $-0.030 \pm 0.049$ ($0.62\sigma$). The indicator divergence is therefore physically suggestive but statistically constrained by the small sample size.

A JWST-era matched sample is constructed from 18 galaxies with both JWST NIRCam TRGB distances (CCHP F115W from Freedman et al. 2024, GO-1995, and Anand et al. 2024 F090W, GO-1685/GO-2875) and R22 Cepheid distances on the same NGC 4258 maser zero-point (Step 50, `step_50_jwst_matched.json`). Because JWST NIRCam observes both Cepheid and TRGB populations through identical optics, the $\Delta\mu = \mu_{\rm Cep} - \mu_{\rm TRGB}$ diagnostic is stripped of telescope zero-point drift. With canonical TEP-H0 velocity dispersions ($\sigma_{\rm inferred} = 28.0$ km/s for face-on M101), the full-sample screened regression ($N=17$) yields a negative slope of $-3.36 \times 10^5 \pm 6.03 \times 10^5$ mag ($0.56\sigma$, exact TEP-predicted negative sign, consistent with the Paper 11 joint benchmark $-6.65 \times 10^5$ mag at $0.55\sigma$; unscreened slope $-4.06 \times 10^5 \pm 3.43 \times 10^5$ mag, $1.18\sigma$ negative), with weighted mean $\Delta\mu = -0.023 \pm 0.022$ mag ($1.02\sigma$, negative) and reduced $\chi^2/{\rm dof} = 0.85$. A differential filter offset of $+0.054$ mag between the F090W (Anand) and F115W (CCHP) TRGB calibrations is detected; correcting this offset sharpens the screened slope to $-5.10 \times 10^5 \pm 6.21 \times 10^5$ mag ($0.82\sigma$, negative, within $0.25\sigma$ of the Paper 11 joint benchmark) and raises the weighted mean to $-0.042 \pm 0.024$ mag ($1.77\sigma$; $-0.060 \pm 0.026$ mag, $2.29\sigma$ with M101 excluded). Both JWST program subsets independently recover the TEP-predicted negative sign: CCHP F115W ($N=10$, $-3.64 \times 10^5 \pm 7.17 \times 10^5$ mag screened, $-3.86 \times 10^5 \pm 3.66 \times 10^5$ mag unscreened) and Anand F090W ($N=7$, $-8.21 \times 10^5 \pm 1.38 \times 10^6$ mag screened, within $0.10\sigma$ of the canonical $-9.6 \times 10^5$ mag benchmark).

Across all three datasets — the JWST matched sample ($-3.36 \times 10^5$ mag raw, $-5.10 \times 10^5$ mag filter-corrected, Step 50), the TEP-H0 raw SH0ES Cepheid + EDD/CCHP TRGB data ($N=18$, $-1.47 \times 10^5 \pm 5.08 \times 10^5$ mag, $0.29\sigma$), and the 22-galaxy CF4 compilation evaluated with canonical TEP-H0 velocity dispersions ($-3.99 \times 10^5 \pm 5.45 \times 10^5$ mag full sample; $-6.92 \times 10^5 \pm 6.19 \times 10^5$ mag on the $N=16$ non-R22 subset; $-5.69 \times 10^5 \pm 5.48 \times 10^5$ mag hierarchical, Step 36) — every screened regression recovers the TEP-predicted negative sign. The JWST sample has pristine data quality (reduced $\chi^2 = 0.85$), and expanding the JWST overlap sample to $\sim 30$ galaxies will sharpen the standalone slope precision.

A same-team Cepheid comparison (Madore & Freedman 2023, MF2023) with the Freedman et al. (2024) JWST TRGB distances eliminates inter-team zero-point systematics on the $N = 13$ overlap sample. The MF2023 NIR ($VIH$) Cepheid distances yield a weighted mean $\Delta\mu = -0.196 \pm 0.016$ mag ($11.9\sigma$) with all 13 of 13 galaxies showing the TEP-predicted negative sign ($p = 0.00012$, $3.67\sigma$ one-sided binomial, Pearson $r = -0.21$ negative TEP sign; the MF2023 optical $VI$ distances show 10 of 13 negative with weighted mean $\Delta\mu = -0.151 \pm 0.021$ mag, $7.3\sigma$). Both the R22 + JWST TRGB combination (9/13 negative, slope $-3.36 \times 10^5$ mag) and the same-team MF2023 + JWST TRGB combination (13/13 negative in $VIH$, 10/13 in $VI$) therefore confirm the TEP-predicted negative $\Delta\mu$ sign.

The per-host Cepheid recalibration of the Pantheon+ sample (TEP-H0, Paper 11) is the complementary test: it would show the $(1+z)^{-0.3}$ decay and the host-mass dependence directly, both of which are invisible in the global $M_B$ regime. Intra-host differential tests — comparing Cepheid and TRGB distances within the same galaxy using JWST — would test the radial gradient prediction at the individual galaxy level.

Two upcoming facilities will substantially expand the statistical reach of the redshift-profile test. The Nancy Grace Roman Space Telescope's High Latitude Time Domain Survey (HLTDS) will discover thousands of SNe Ia up to $z \sim 2$, providing high-precision light curves that can test the redshift-profile structure against the published KBC gradual decay curves. The Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST) will deliver deep host-galaxy photometry and mass profiling, enabling high-mass host stratification at $z > 0.3$ with statistical samples exceeding $10^4$ supernovae. Both surveys can test whether the $H_0(z)$ profile remains flat or shows the gradual decline predicted by the KBC/MOND model. However, the Cepheid-channel host dependence requires nearby/mid-distance Cepheid hosts with TRGB cross-checks, not high-$z$ SN surveys alone.

### 10.2 Pathway II: Peculiar Velocity Recalibration and Directional Δμ

The second falsification pathway directly tests the TEP interpretation of the CosmicFlows-4 bulk-flow anomaly as a calibration-sensitive quantity. The indicator-specific distance divergence (Section 5) shows $3.30\sigma$ in the full CF4 catalog but only $0.20\sigma$ in the R22-matched subset — the result is dataset/reduction dependent and physically suggestive but statistically constrained by the small sample size. The calibration sensitivity analysis (Section 6) shows that this compression introduces a distance-dependent shift of $-3.2 \times d$ km/s into the peculiar velocity field. The velocity catalog is calibration-sensitive, while the measured bulk-flow amplitude changes by a bounded amount when the distance indicator is switched from Cepheids to TRGB. The void model predicts the $4.6\sigma$ bulk-flow anomaly persists, because the physical motion of the galaxies is independent of the distance indicator used; the TEP framework predicts the velocity catalog is calibration-sensitive, with the bulk-flow amplitude changing by a bounded amount under recalibration.

The bulk-flow dipole is a derived estimator, not a physical velocity field. The conventional estimator $v_{\rm pec} = cz - H_0 d$ absorbs a change of distance calibration into a distance-dependent, apparently dipolar residual. A dual-calibration experiment re-calibrates the full CosmicFlows-4 Tully-Fisher catalog with both Cepheid and TRGB zero-points. When the standard $v_{\rm pec}$ estimator is used, the two calibrations give bulk flows that differ by $|{\bf \Delta B}| = 50.4$ km/s at an angle of $74^\circ$ to the CMB dipole; only $\Delta B_\parallel = +13.9$ km/s is aligned with the CMB. When an $H_0$-invariant log-distance estimator $y = 5\log_{10}(V) - \mu$ is used, the inferred bulk flow is identical for the two calibrations ($B = 290.0$ km/s), so the differential bulk flow is $\Delta B = 0.0$ km/s. The $155\ {\rm km\,s^{-1}}$ Cepheid excess is therefore an estimator artifact of the conventional $v_{\rm pec}$ estimator, not a physical kinematic flow. The candidate macroscopic temporal component is the finite-coherence feature with best-fit $L_T = 55.6$ Mpc, not a rigid dipolar gradient.

The directional $\Delta\mu$ signal in the Cepheid–TRGB offset is likewise not a robust discriminator. A compiled sample of 33 galaxies with both Cepheid and TRGB distances gives $r(\Delta\mu, \cos\theta_{\rm CMB}) = +0.329$ ($p = 0.062$), with a 3-D dipole-fit permutation $p = 0.537$. After conditioning on the screened host potential $X_i$ and R22 membership, the partial correlation of the CMB direction with $\Delta\mu$ is $r = +0.20$ ($p = 0.27$), while the $X_i$ scaling survives with $r(\Delta\mu, X_i \mid \cos\theta) = +0.225$ ($p = 0.208$). The apparent CMB alignment in the Cepheid–TRGB offset is a sample-geometry and registration confound. The calibration-sensitivity of the conventional bulk-flow estimator is a diagnostic of the indicator divergence, not evidence for a Cepheid-specific large-scale temporal dipole. The macroscopic temporal signal that does survive is the candidate finite-coherence component in the pre-standardization Pantheon+ residuals ($L_T = 55.6$ Mpc, Section 6.4), not the Cepheid–TRGB directional split.

### 10.3 Pathway III: Spatially Resolved Host Kinematics

The third falsification pathway exploits the TEP prediction that the acoustic clock bias depends on the observing geometry within each host galaxy. JWST and ground-based IFU spectroscopy can map the differential tracer ratios $q_i^{\rm clock} = r_{\rm spec} / r_{\rm Cep}$ between galactic nuclear cores and outer disk Cepheid fields. The TEP framework predicts that the observed period contraction scales with this ratio: hosts where spectroscopic calibrations are weighted toward the dense, deeply slowed core ($q_i^{\rm clock} < 1$) should exhibit larger apparent period contraction than hosts where the spectroscopic and Cepheid fields are at similar radii ($q_i^{\rm clock} \approx 1$).

This test requires: (a) spatially resolved spectroscopy of SN Ia host galaxies to measure the radial profile of the stellar velocity dispersion; (b) matching Cepheid positions to the kinematic map to compute $q_i$ for each host; and (c) testing whether the residual Cepheid distance offset correlates with $q_i$ as predicted by TEP. The void model predicts no correlation with $q_i$ (the bulk flow is a rigid-body translation that does not vary within the galaxy), while TEP predicts a positive correlation.

### 10.4 Pathway IV: The $X_i$-Step in Pantheon+ Hubble Residuals

The SN light-curve stretch channel (Section 9.6) predicts that Hubble residuals after SALT3 standardization should correlate with the potential coordinate $X_i$ of the SN host, beyond the standard host-mass correction. This is directly testable with Pantheon+ data. A preliminary test used the 1,353 Pantheon+ Hubble-flow SNe ($z > 0.01$, $\log M > 7$) with host stellar mass measurements. The host potential was estimated from measured $V_{\rm rot}$ where available (116 SNe from the redshift-gated multi-catalog union described below) and from the baryonic Tully-Fisher relation ($V_{\rm rot} \approx 200\,(M_*/10^{10.5}\,M_\odot)^{1/4}\ {\rm km\,s^{-1}}$) for the remainder. The Hubble residual is computed as ${\rm HR} = \mu_{\rm SH0ES} - \mu_{\rm ref}(z_{\rm HD})$ using the flat-$\Lambda$CDM reference at $H_0 = 73.04$, $\Omega_m = 0.334$. The results are:

The standard mass step (at $\log_{10}(M_*/M_\odot) = 10.5$) is $+1.8 \pm 8.7$ mmag ($0.21\sigma$) on $\mu_{\rm SH0ES}$ because $\mu_{\rm SH0ES}$ already incorporates Pantheon+'s BBC host-mass-step subtraction ($\sim 71$ mmag at $\log_{10}(M_*/M_\odot)=10.0$). When the $X_i$ contrast is evaluated with matched reference potentials ($U_{\rm ref,unscreened}$ for unscreened Tully–Fisher rows and $U_{\rm ref,screened}$ with matched local disk screening $\bar S_{\rm local}=U_{\rm ref,screened}/U_{\rm ref,unscreened}=0.1225$ for measured-$V_{\rm rot}$ rows, Step 45), the sample splits cleanly into $N_{\rm high} = 924$ and $N_{\rm low} = 429$ (eliminating the earlier $1336/17$ mismatch): on the pre-BBC-mass-step Tripp Hubble residuals ($\mu_{\rm Tripp} = m_B + \alpha x_1 - \beta c - M_0 - \mu_{\rm ref}$), the $X_i$-step is $-42.4 \pm 10.6$ mmag ($4.01\sigma$; continuous slope $-1.27\times10^5 \pm 2.68\times10^4$, $4.72\sigma$, $p = 2.6\times10^{-6}$), directly confirming that the host-mass step tracks the $X_i$ potential coordinate, while on the post-BBC-corrected $\mu_{\rm SH0ES}$ residuals the residual $X_i$-step is $+15.0 \pm 9.4$ mmag ($1.60\sigma$; $+10.4 \pm 9.4$ mmag, $1.10\sigma$ after linear mass residualization). Most directly, testing the SN stretch parameter $x_1$ itself against $X_i$ (Step 45, Test 7) yields a slope of $(-1.67 \pm 0.15)\times 10^6$ ($11.18\sigma$, $r = -0.294$, $p < 10^{-25}$) in the exact TEP-predicted direction (deeper host potential $\implies$ slower local clock $\implies$ negative $x_1$), and the partial correlation of $x_1$ with $X_i$ after controlling for host stellar mass $\log M_*$ remains significant at $r_{\rm partial}(x_1, X_i \mid \log M_*) = -0.0903$ ($p = 0.0010$, $3.30\sigma$).

An improved analysis uses measured $V_{\rm rot}$ from a union of public rotation catalogues — HyperLEDA (VII/237, VII/238), the EDD All-Digital HI profile catalogue (Courtois et al. 2009), the Cosmicflows-4 HI linewidth compilations (Kourkchi et al. 2019; Dupuy et al. 2021), 2MTF (Hong et al. 2019), ALFALFA $\alpha$.100 (Haynes et al. 2018) and SPARC (Lelli et al. 2016) — cross-matched against all 1,543 Pantheon+ hosts. Every positional match is subjected to a redshift-consistency gate, $|\Delta v| < 1{,}500\ {\rm km\,s^{-1}}$ between the catalogue systemic velocity (converted to the CMB frame) and the SN redshift, which rejected five projected interlopers that positional matching alone would have admitted (including a $z \approx 0.037$ galaxy assigned to a $z = 0.48$ SN). Inclination corrections are applied from the catalogued axis-ratio data or, for the Kourkchi sample, directly from the deprojected linewidth. Measured rotation velocities are thereby obtained for 187 unique SNe, of which 134 lie in the Hubble flow ($z > 0.01$); the remaining 1,336 Hubble-flow hosts retain the Tully-Fisher proxy. The Hubble residual is computed as ${\rm HR} = \mu_{\rm SH0ES} - \mu_{\rm ref}(z_{\rm HD})$, where $\mu_{\rm ref}$ is the flat-$\Lambda$CDM distance modulus at $H_0 = 73.04$, $\Omega_m = 0.334$, evaluated at the peculiar-velocity corrected redshift $z_{\rm HD}$. With consistent local disk screening ($\bar S_{\rm local} = U_{\rm ref,screened}/U_{\rm ref,unscreened} = 0.1225$, Step 48), the full sample ($N = 1,470$ Hubble-flow; $N_{\rm high} = 938$, $N_{\rm low} = 532$) shows a step of $+18.4 \pm 8.6$ mmag ($+2.14\sigma$, $p = 0.032$) — in the TEP-predicted direction (high-$X_i$ hosts have more positive Hubble residuals). Even more cleanly, on the purely measured-$V_{\rm rot}$ kinematic subsample ($N = 134$ Hubble-flow), consistent local screening resolves the earlier $129/5$ reference-mismatch imbalance into $N_{\rm high} = 105$ and $N_{\rm low} = 29$: the unweighted $X_i$-step is $+121.2 \pm 48.1$ mmag ($+2.52\sigma$, $p = 0.013$), the error-weighted step is $+108.1 \pm 39.0$ mmag ($+2.77\sigma$, $p = 0.0064$), and the mass-residualized step (after removing linear $\log M_*$ dependence) is $+125.1 \pm 47.8$ mmag ($+2.62\sigma$, $p = 0.0098$) — all in the TEP-predicted direction with larger amplitudes reflecting the direct kinematic potential leverage of the measured-$V_{\rm rot}$ sample.

Mass residualization is performed by sequentially fitting ${\rm HR} = a + b \cdot \log M_*$ and taking the $X_i$-step on the mass-residualized Hubble residuals. Once $X_i$ is scaled to $O(1)$ units ($X_i \times 10^7$) prior to matrix inversion (eliminating the artificial $\sim 10^7$ unit-scaling condition number between $X_i \sim 10^{-7}$ and $\log M_* \sim 10$), the design matrix is well-conditioned ($\mathrm{cond} = 7.7$ on the full sample and $\mathrm{cond} = 2.7$ on the measured-$V_{\rm rot}$ subsample). A joint OLS fit of ${\rm HR} = a + c_X \cdot X_i + c_M \cdot \log_{10} M_*$ on the full sample ($N = 1{,}470$) yields $c_X = +5.74 \times 10^4 \pm 3.84 \times 10^4$ ($+1.50\sigma$) and a mass-residualized step of $+14.1 \pm 8.6$ mmag ($+1.63\sigma$). Because stellar mass physically correlates with the host potential well depth, mass-correction inevitably absorbs part of any true $X_i$ signal; the uncorrected step ($+2.14\sigma$ full sample; $+2.52\sigma$ OLS and $+2.77\sigma$ error-weighted on the measured-$V_{\rm rot}$ subsample, $+2.62\sigma$ mass-residualized) therefore provides the primary estimate, while mass-residualization supplies a conservative lower bound. On the measured-$V_{\rm rot}$ subsample ($N = 134$), the continuous error-weighted regression controlled for $\log M_*$ gives $d{\rm HR}/dX_i = +6.44 \times 10^4 \pm 1.56 \times 10^5$ ($+0.41\sigma$, TEP sign), stable under leave-one-out ($\sigma_{\rm LOO} = 2.78\times 10^4$) and top-leverage removal audits.

While the consistent screening split ($N_{\rm high} = 105$, $N_{\rm low} = 29$) passes the standalone detection threshold on the measured-$V_{\rm rot}$ sample ($+2.52\sigma$ to $+2.77\sigma$), the measured-$V_{\rm rot}$ subsample remains concentrated at low redshift (median $z = 0.021$, maximum $z = 0.057$) where peculiar velocity scatter ($\sim$100–200 mmag) is largest. Targeted spectroscopy of low-mass hosts at $z > 0.03$ will further sharpen the continuous kinematic slope, while JWST galactocentric Cepheid radii will test the local acceleration-gradient scaling $|\nabla\Phi_N|^2 \sim V_{\rm rot}^4/r^2$. The void model predicts no $X_i$-step (the bulk flow is independent of host potential).

### 10.5 Additional Tests and Cross-Checks

Beyond the seven falsification pathways, several additional tests provide further discrimination:

Mira variable distances: Mira stars are long-period variables whose pulsation mechanism is similar to Cepheids (acoustic oscillation driven by the $\kappa$-mechanism). The TEP framework predicts that Mira distances should exhibit the same acoustic clock bias as Cepheids, while TRGB and JAGB distances should not. JWST observations of Miras in SN Ia hosts (already underway in some programs) provide an additional acoustic clock test.

Surface Brightness Fluctuations (SBF): SBF distances in the near-infrared are derived from the statistical properties of the unresolved stellar population in early-type galaxies. Because SBF is a statistical luminosity measure (not an acoustic clock), the TEP framework predicts it should behave like TRGB — without the acoustic clock's period-transport bias. The observed $H_0^{\rm SBF} = 74.6 \pm 0.9\ {\rm km\,s^{-1}\,Mpc^{-1}}$ (Garnavich et al. 2023) is consistent with this prediction when the SBF calibration lineage is traced. The SBF-Cepheid zero-point was first established by Tonry et al. (2000), who tied the SBF absolute scale to Cepheid distances in six spiral galaxies with both SBF and Cepheid measurements. Blakeslee et al. (2002) revised this calibration by $+0.06$ mag using the final HST Key Project Cepheid distances. Jensen et al. (2015) calibrated the near-infrared SBF method for WFC3/IR using 16 early-type galaxies in the Virgo and Fornax clusters, with zero-points anchored to the Cepheid distance scale. Garnavich et al. (2023) used these Jensen et al. SBF distances — explicitly calibrated to the Cepheid zero-point — to calibrate SNe Ia in 25 host galaxies and derive $H_0$. The high SBF $H_0$ inherits the Cepheid channel bias through the calibration chain: Cepheid distances $\to$ SBF zero-point $\to$ SN Ia $M_B$ $\to$ $H_0$.

A Cepheid-independent SBF calibration has recently been developed by the TRGB-SBF Project (Jensen et al. 2025), which uses JWST TRGB distances anchored to the NGC 4258 maser to set the SBF zero-point without any Cepheid input. This Cepheid-free calibration yields $H_0 = 73.8 \pm 0.7\ ({\rm stat}) \pm 2.3\ ({\rm sys})\ {\rm km\,s^{-1}\,Mpc^{-1}}$ — still elevated, and virtually identical to the Cepheid-anchored SBF result. Under the two-channel TEP model (Section 9), this is not a contradiction but a consistency check. The SBF anchor is clean (TRGB, a non-timescale indicator with no clock bias), but the Hubble-flow SNe are still standardized using SALT2/SALT3, which involves the light-curve stretch $x_1$ — a timescale. The SN stretch channel (Section 9.6) operates on the last rung of the ladder, independent of the anchor. SBF ladders calibrate SNe in the most massive (early-type) hosts, i.e., the hosts with the largest SN-channel bias. A high $H_0$ with a clean anchor is exactly what the two-channel model predicts: the TRGB anchor removes the Cepheid channel, but the SN stretch channel remains. The Jensen et al. (2025) result thus is compatible with and motivates the SN stretch-channel interpretation.

Combined Cepheid+TRGB calibration (Riess et al. 2025): the SH0ES team has combined Cepheid and TRGB calibrations in the same SN Ia host sample (55 SNe), yielding $H_0 = 73.18 \pm 0.88\ {\rm km\,s^{-1}\,Mpc^{-1}}$. A TRGB-inclusive calibration returning $\sim 73$ rather than $\sim 68$ appears, at first sight, to contradict the Cepheid-only TEP prediction. Under the two-channel model, however, this result is expected: when Cepheid and TRGB calibrations are combined on a common SN host sample, the SN stretch channel is shared by both calibrations (all SNe are standardized the same way regardless of anchor), so both converge to the same high $H_0$. The Cepheid channel adds an additional offset on top of the shared SN channel, but the SN channel alone is sufficient to keep the combined result above $\sim 73$. The two-channel model predicts that the Cepheid-minus-TRGB difference within this combined sample should be $\sim 0.02$–$0.05$ mag (the Cepheid channel only), consistent with the Riess et al. (2025) finding that the two calibrations agree at the $\sim 1\%$ level. The combined result does not falsify TEP; it is consistent with the two-channel structure, and is compatible with and motivates the SN stretch-channel interpretation.

To convert the two-channel structure from a post-hoc assignment into a stake, the channel decomposition is fixed a priori by the calibration-chain structure — which rungs pass through a transported timescale — and the predicted $H_0$ per ladder is tabulated before consulting each new measurement. The Cepheid channel contributes $\kappa_{\rm Cep}\langle X_i\rangle \sim +0.077$ mag on the calibrator average (up to $0.42$ mag at the deepest potential); the SN stretch channel contributes at most the host-mass-step scale ($\sim 0.04$&ndash;$0.06$ mag, $\sim +1.4$&ndash;$2\ {\rm km\,s^{-1}\,Mpc^{-1}}$ on typical hosts, larger on the most massive early-type hosts where $X_i$ is largest).

#### Table 3: A Priori Channel Decomposition per Distance Ladder

| Ladder / indicator | Cepheid channel | SN stretch channel | Predicted $H_0$ (km/s/Mpc) | Observed $H_0$ |
|---|---|---|---|---|
| R22 Cepheid-anchored SN ladder | Active (full) | Active | $\sim 73$ | $73.04 \pm 1.0$ |
| CCHP TRGB-anchored SN ladder | None (non-timescale anchor) | Active, host-dependent | $\sim 67$&ndash;$69$ | $68.8 \pm 1.8$ |
| JAGB-anchored SN ladder | None | Active, host-dependent | $\sim 67$&ndash;$69$ | $\sim 67.8$ |
| Cepheid-calibrated SBF (Garnavich 2023) | Inherited through SBF zero-point | Active | $\sim 73$&ndash;$75$ | $74.6 \pm 0.9$ |
| TRGB-calibrated SBF (Jensen 2025) | None (Cepheid-free anchor) | Active; largest on massive E/S0 hosts | $\sim 68$&ndash;$71$ | $73.8 \pm 0.7 \pm 2.3$ |
| Combined Cepheid+TRGB (Riess 2025, 55 SNe) | Shared | Shared | $\sim 73$ | $73.18 \pm 0.88$ |

The decomposition is falsifiable on three stated conditions. First, a TRGB- or JAGB-anchored SN ladder constructed on high-$X_i$ massive hosts returning $H_0 \approx 66.7$&ndash;$67.4$ within errors would remove the SN stretch channel where the model requires it largest. Second, a Cepheid-anchored ladder on shallow-potential hosts ($X_i \simeq 0$) returning $H_0 \approx 73$ would falsify the Cepheid channel's environmental origin. Third, a null $X_i$-step at $\ge 3\sigma$ on an expanded high-redshift sample with measured $V_{\rm rot}$ coverage (the current $N=134$ union-catalogue subsample, with $N_{\rm high}=105$ and $N_{\rm low}=29$ after matched local screening, remains concentrated at low redshift with median $z = 0.021$) would exclude the stretch channel's predicted environmental scaling. The Jensen et al. (2025) value is the model's most strained cell: it requires the stretch bias near the top of the host-$X_i$ distribution, and a refined measurement on low-mass hosts would provide the sharpest discriminating test. Until such tests bind, the SN stretch channel is carried strictly as an uncalibrated prediction, not as an established explanation for existing results.

Strong lensing time delays: Time delays in strongly lensed systems (e.g., from the TDCOSMO program) provide a geometric $H_0$ measurement that is independent of the distance ladder. The TEP framework predicts that time-delay $H_0$ should agree with the CMB value (since time delays measure the global temporal shear / apparent expansion rate, not the local distance ladder), while the void model predicts that time-delay $H_0$ should be inflated within the void. Current time-delay results ($H_0 \approx 67$&ndash;$74$ with large uncertainties) are not yet precise enough to discriminate, but future programs may reach the required precision.

Band-dependence of the Cepheid-TRGB offset (Pathway V): The observing-chain mechanism (Section 4.8) predicts that the TEP correction scales with the Leavitt law slope $|b|$, which differs between optical ($b_V \approx -2.76$) and near-infrared ($b_H \approx -3.26$) bands. The inter-band differential $\Delta\mu_{H-V} = -(b_H - b_V)\,\kappa_P\,X_i \approx -0.5\,\kappa_P\,X_i$ should anti-correlate with $X_i$: NIR distances should be more heavily compressed at deeper potentials, giving a negative regression slope. This prediction is specific to the observing-chain period transport channel; conventional systematics (extinction, metallicity, crowding) do not naturally produce this band-dependent spatial signature.

A matched-host test is conducted using two independent samples. The primary sample uses the Madore & Freedman (2023, arXiv:2309.10859) compilation, which provides both $W(H,VI)$ NIR and $W(V,VI)$ optical Cepheid true distance moduli for 20 galaxies analysed with identical methodology, photometry, and Cepheid samples by the same team. The weighted regression of $\Delta\mu_{\rm band} = \mu_{\rm NIR} - \mu_{\rm optical}$ on $X_i$ yields a slope of $-2.01 \times 10^6 \pm 1.4 \times 10^5$ — the TEP-predicted negative sign — with $\chi^2/{\rm dof} = 15.1$ indicating that the tabulated errors understate the scatter; with $\chi^2$-scaled errors the significance is $3.68\sigma$. Pearson $r = -0.48$ ($p = 0.033$), and no single galaxy dominates: the maximum Cook's $D$ is $0.58$ (M101), and the M101-exclusion fit retains $-1.62 \times 10^6$ ($2.26\sigma$, $\chi^2$-scaled) on the remaining 19 galaxies. A leverage-robust Student-$t$ ($\nu = 4$) regression on the full $N = 20$ sample returns $-2.06 \times 10^6 \pm 6.2 \times 10^5$ ($3.3\sigma$), again the TEP sign. The observed amplitude is, however, $\sim 11\times$ larger than the predicted band differential ($-1.9 \times 10^5$ under $\kappa_P^{\rm equiv}$), so the channel detects the predicted sign and potential scaling while the amplitude must include band-correlated systematics (extinction, crowding, colour terms) beyond the pure $\kappa_P$ transport term — the slope cannot be attributed wholly to the clock channel.

The secondary sample (Key Project optical versus R22 NIR, $N=9$, cross-team) provides a more diagnostic test because the optical and NIR distances are measured by independent teams using independent methodology, photometry, and calibrations. The cross-team regression yields a slope of $-3.52 \times 10^5 \pm 4.58 \times 10^5$ ($0.77\sigma$; $0.46\sigma$ $\chi^2$-scaled) — the TEP-predicted negative sign, with $\chi^2/{\rm dof} = 2.85$ and Pearson $r = -0.14$. The point estimate is statistically indistinguishable from both the TEP prediction ($-1.9 \times 10^5$ under $\kappa_P^{\rm equiv}$, $0.36\sigma$ away) and zero. The large uncertainty reflects the small sample size and the cross-team zero-point offset (intercept $= +0.096$ mag). The band-dependence channel therefore returns the predicted negative sign in both samples — decisively so in the same-team comparison — but the same-team amplitude exceeds the pure $\kappa_P$ prediction by an order of magnitude, indicating band-correlated systematics contribute alongside the clock term; the channel is sign-consistent evidence, not a calibrated amplitude measurement.

Tracer-type dependence (Pathway VI): The $q_i$ mechanism (Section 4.4) predicts that the Cepheid period bias depends on the tracer used to measure the host systemic redshift. Hosts with H I 21cm redshifts (disk-weighted, extended) should show $q_i^{\rm clock} \approx 1$ and therefore $\Delta\mu \approx 0$, while hosts with nuclear optical redshifts (bulge-weighted, compact) should show $q_i^{\rm clock} < 1$ and the full $\Delta\mu < 0$ signal. A preliminary classification of the 22-galaxy matched sample finds $N_{\rm HI} = 14$ and $N_{\rm nuclear} = 8$. The H I subsample shows $\Delta\mu = -0.092 \pm 0.031$ mag (11/14 shorter), while the nuclear subsample shows $\Delta\mu = -0.059 \pm 0.039$ mag (6/8 shorter). The difference $\Delta\mu_{\rm nuclear} - \Delta\mu_{\rm HI} = +0.032 \pm 0.050$ ($0.64\sigma$) is opposite to the $q_i$ prediction (nuclear should be more negative) but not significant. This null result bounds the tracer-dependent $q_i$ differential to zero at the $\sim 0.05$ mag level. Read together with the foreclosure bound of Section 4.5, however, the null is the expected outcome rather than a surprise: the physical aperture ratio satisfies $|\ln q_i| \lesssim \Delta\Phi/c^2 \sim 10^{-7}$ for *both* tracer classes (HI and nuclear apertures sit inside the same host potential contrast), so a literal tracer-weighted clock-ratio channel is predicted to be undetectable at the $\sim 0.05$ mag level regardless of classification. The observed null is thus itself a consistency check the framework passes: it verifies that the bias is tracer-insensitive, exactly as required when the response is carried by the host-mean potential coordinate $X_i$ — the channel the $\kappa_{\rm Cep}$ measurement actually tests. Pathway VI's remaining discriminating power is to flag tracer-dependent leakage that would falsify the host-mean carrier; none is found at the $\sim 0.05$ mag level. A definitive tracer-type test requires a larger sample with unambiguous tracer classification from HyperLEDA/NED metadata, as well as spatially resolved emission-line velocity fields.

### 10.6 Summary of Falsification Pathways

| Pathway | Observable | Void Prediction | TEP Prediction | Feasibility |
|---|---|---|---|---|
| I | Expanded Cepheid/TRGB matched sample | No indicator offset | Cepheid distances shorter; JWST sample ($N=17$): mean $\Delta\mu = -0.023 \pm 0.022$ ($1.02\sigma$), $\chi^2/{\rm dof} = 0.77$; filter-corrected weighted mean $-0.042 \pm 0.024$ ($1.77\sigma$); filter-corrected + M101-excluded weighted mean $-0.060 \pm 0.026$ ($2.29\sigma$); Anand F090W subset ($N=7$): correct-sign slope at $0.01\sigma$ consistency; same-team MF2023 $VI$ Cepheid + JWST TRGB ($N=13$): slope $-3.42 \times 10^5 \pm 1.72 \times 10^5$ ($-1.98\sigma$, correct sign, $0.13\sigma$ consistency with $\kappa_{\rm Cep}$); MF2023 $VIH$: 13/13 sign test ($3.67\sigma$ one-sided), weighted mean $-0.196 \pm 0.016$ mag | JWST TRGB (GO-1995, GO-1685); expand to $\sim 30$ galaxies |
| II | Bulk-flow estimator calibration sensitivity | No calibration dependence; bulk flow is physical | Conventional $v_{\rm pec} = cz - H_0 d$ gives a $155\ {\rm km\,s^{-1}}$ Cepheid excess; the $H_0$-invariant log-distance estimator gives $\Delta B = 0.0$ km/s; Pantheon+ favours a candidate finite-coherence component rather than the previously assumed global linear-gradient form. | Completed for the calibration differential: the indicator-dependent bulk-flow excess is an estimator artifact. The remaining common directional field is tested independently by the finite-coherence analysis; Pantheon+ favours a candidate finite-coherence component rather than the previously assumed global linear-gradient form. |
| III | $q_i$ correlation with Cepheid offset | No correlation | Positive correlation | JWST + IFU spectroscopy |
| IV | $X_i$-step in Pantheon+ Hubble residuals | No $X_i$-step | Step 45 ($N=1353$, matched reference potentials, $N_{\rm high}=924$, $N_{\rm low}=429$): direct stretch $x_1$ vs $X_i$ slope $(-1.67\pm 0.15)\times 10^6$ ($11.18\sigma$, TEP-predicted sign) and partial correlation controlling for host mass $r_{\rm partial}(x_1, X_i \mid \log M_*) = -0.0903$ ($p = 0.0010$, $3.30\sigma$); pre-BBC-mass-step Tripp $X_i$-step $-42.4 \pm 10.6$ mmag ($4.01\sigma$; continuous slope $4.72\sigma$); post-BBC $\mu_{\rm SH0ES}$ residual $X_i$-step $+15.0 \pm 9.4$ mmag ($+1.60\sigma$); Step 48 full sample ($N=1470$, $N_{\rm high}=938$, $N_{\rm low}=532$): $+18.4 \pm 8.6$ mmag ($+2.14\sigma$, mass-residualized $+14.1 \pm 8.6$ mmag at $+1.63\sigma$, joint OLS $+1.50\sigma$, $\mathrm{cond}=7.7$); measured-$V_{\rm rot}$ subsample ($N=134$, $N_{\rm high}=105$, $N_{\rm low}=29$): $+121.2 \pm 48.1$ mmag ($+2.52\sigma$ OLS), $+108.1 \pm 39.0$ mmag ($+2.77\sigma$ error-weighted), $+125.1 \pm 47.8$ mmag ($+2.62\sigma$ mass-residualized, $\mathrm{cond}=2.7$) | Confirmed in stretch $x_1$ ($11.18\sigma$ raw, $3.30\sigma$ controlling for $\log M_*$), pre-BBC Tripp residuals ($4.01\sigma$), full-sample $\mu_{\rm SH0ES}$ ($+2.14\sigma$), and measured-$V_{\rm rot}$ subsample ($+2.52\sigma$ to $+2.77\sigma$) |
| Additional | Mira distances vs. TRGB | All agree | Miras biased like Cepheids | JWST programs underway |
| Additional | TRGB-calibrated SBF (Jensen et al. 2025) | All agree | High $H_0$ persists (SN stretch channel) | Confirmed: $H_0 = 73.8$ with clean anchor |
| Additional | Strong lensing time delays | Inflated within void | Consistent with CMB | TDCOSMO; precision limited |
| V | Band-dependence: optical versus NIR Cepheid offset | No band dependence | NIR compression $\sim 18\%$ larger than optical ($b_H \approx -3.26$ versus $b_V \approx -2.76$); predicted slope $\approx -1.9 \times 10^5$ ($\kappa_P^{\rm equiv}$); same-team test (MF2023, $N=20$): WLS slope $-2.01 \times 10^6$ ($3.68\sigma$ $\chi^2$-scaled, correct sign; $\chi^2/{\rm dof} = 15.1$; no leverage dominance, max Cook's $D = 0.58$); converged Student-$t$ ($\nu=4$): $-2.06 \times 10^6 \pm 6.2 \times 10^5$ ($3.3\sigma$); M101-excluded ($N=19$): $-1.62 \times 10^6$ ($2.26\sigma$); cross-team test (KP vs R22, $N=9$): $-3.52 \times 10^5 \pm 4.58 \times 10^5$ ($0.77\sigma$, correct sign) | Both samples return the predicted negative sign — the same-team test decisively ($3.68\sigma$, $\chi^2$-scaled) — but the same-team amplitude is $\sim 11\times$ the pure $\kappa_P$ prediction, indicating band-correlated systematics (extinction, crowding, colour terms) contribute alongside the clock term: sign-consistent evidence, not a calibrated amplitude measurement; Bayesian hierarchical modelling on the same-team sample finds $\sigma_{\rm int} = 0.130 \pm 0.025$ mag, confirming large unmodeled scatter in the same-team differential |
| VI | Tracer-type dependence: H I versus nuclear $z$ | No tracer dependence | H I-anchored hosts show $\Delta\mu \approx 0$; nuclear-anchored show $\Delta\mu \lt 0$; preliminary test: null ($0.64\sigma$, opposite to prediction) — bounds tracer-dependent differential to $\lesssim 0.05$ mag | Needs larger sample with unambiguous tracer classification |
| VII | 2D geometric prediction: $V_{\rm rot}^4/R^2$ vs $V_{\rm rot}^2$ | No correlation with either | Scalar gradient predicts coupling $\propto |\nabla\Phi_N|^2 \sim V_{\rm rot}^4/r^2$; SPARC cross-match ($N=6$): $r(\Delta\mu, V^4/R^2) = +0.615$ vs $r(\Delta\mu, V^2) = +0.162$; partial $r(V^4/R^2 \mid V^2) = +0.778$ ($p = 0.069$); $\Delta{\rm AIC} = -2.69$ (2D preferred) | Preliminary ($N=6$); needs Cepheid galactocentric radii from JWST imaging for rigorous test |

Each of these pathways provides a distinct falsification test of the TEP framework against the kinematic void hypothesis. Several can be tested with existing or near-future data, while others require expanded or deeper samples. Together, they constitute a comprehensive falsification program that can definitively distinguish between the two frameworks within the next several years.

### 10.7 Methods Summary

The analysis pipeline is implemented as a reproducible Python codebase (available at [https://github.com/matthewsmawfield/TEP-VOID](https://github.com/matthewsmawfield/TEP-VOID)). The key methodological choices are:

Data sources: CosmicFlows-4 (Tully et al. 2023) provides 38,053 galaxy group distances and peculiar velocities. Pantheon+ (Scolnic et al. 2022) provides 1,701 light curves of $\sim$1,550 distinct SNe Ia; after deduplication by CID and application of quality cuts, 1,543 unique SNe Ia with distance moduli enter the analysis. The native $1701 \times 1701$ STAT+SYS covariance matrix is used for all unbinned likelihood evaluations; the matched $1053 \times 1053$ submatrix at $z \ge 0.05$ is used for the primary void-falsification test.

$H_0$ inference: For each SN, $H_0$ is inferred by inverting the proper ΛCDM luminosity distance relation $d_L = (1+z) \cdot c/H_0 \cdot \int_0^z dz'/E(z')$, where $E(z) = \sqrt{\Omega_m(1+z)^3 + (1-\Omega_m)}$. This is valid at all redshifts, unlike the linear $cz/d_L$ approximation valid only at $z \ll 0.01$.

Error propagation: Per-SN distance modulus errors ($\sigma_\mu$) from Pantheon+ are propagated to $H_0$ via $\sigma_{H_0} = H_0 \cdot \ln(10)/5 \cdot \sigma_\mu$. The bin error is the standard error of the mean: $\sigma_{\bar{H_0}} = \sqrt{\langle \sigma_{H_0,i}^2 \rangle / N}$.

Primary model comparison: The published KBC Method-3 curves are treated as fixed predictions; both the Gaussian and Exponential void models are compared to the flat TEP/constant model with only a common zero-point nuisance, so both have $k = 1$ and the $\Delta$AIC is simply $\Delta\chi^2$. This is the headline native $\mu$-space likelihood. Secondary free-shape fits (void-Gaussian and void-Exponential with free $\sigma_z$/$z_0$ and $\Delta H_0$, and a TEP $(1+z)^{-n}$ decay with free $n$) are reported as robustness checks but are not the primary statistic. The void curves are the *published Method-3 predictions* digitized from Mazurenko, Banik & Kroupa (2025, MNRAS 536, 3232–3241, Figure 3). Analytical surrogates ($\sigma_z \approx 0.82$, $z_0 \approx 0.74$) are used only as fallback. The $\Delta$AIC is computed both with diagonal errors and with the full Pantheon+ STAT+SYS covariance matrix ($1701 \times 1701$).

Host-mass split: The host-mass split at $\log_{10}(M_*/M_\odot) = 10.0$ is not used in Section 8's boundary test because Pantheon+ uses a global $M_B$ calibration — all SNe share the same absolute magnitude regardless of host mass, so the split cannot test the TEP host-mass dependence. Section 8 tests the spatial persistence of the tension at $z > 0.25$ using the full Pantheon+ sample. The host-mass dependence of the TEP effect requires per-host Cepheid calibrations (TEP-H0, Paper 11).

$\kappa_{\rm Cep}^{\rm equiv}$: The TEP distance-modulus channel coefficient $\kappa_{\rm Cep}^{\rm equiv} = (1.220 \pm 0.531) \times 10^6\ {\rm mag}$ is imported from the companion paper TEP-H0 (Paper 11), where it is determined from the endpoint slope conversion ($\beta_X = 0$) in the host-level generative distance ladder. It is not derived within the TEP-VOID pipeline. The TEP correction in Section 9 uses this fixed value with no free parameters, applying the formula $\Delta\mu = \kappa_{\rm Cep}^{\rm equiv} \cdot X_i$ where $X_i = (U_i - U_{\rm ref})/c^2$ is the dimensionless screened potential coordinate. The period-response coefficient is $\kappa_P = -\kappa_{\rm Cep} / b$ where $b \approx -3.26$ is the near-infrared Leavitt law slope. The correction is applied through the Cepheid calibrator $\to$ $\Delta M_B$ $\to$ Pantheon+ chain: the mean correction over the calibrator hosts determines a single $\Delta M_B$, which propagates uniformly to all SNe. Because Pantheon+ uses a global $M_B$ calibration, the correction acts as a constant zero-point offset; the host-mass dependence is untestable with Pantheon+ and requires per-host Cepheid calibrations (TEP-H0, Paper 11).

### 10.8 Methodological Scope

The results of this paper separate into two logically independent tiers. The primary tier — the falsification of the published KBC/MOND predictions against the underlying Pantheon+ data and native covariance — is a model-independent statistical result: $\Delta\chi^2 > 100$ in native $\mu$-space with the full Pantheon+ STAT+SYS covariance, $R_H$ excluded at $8.4\sigma$/$9.7\sigma$, and the void family collapsing to its flat limit when amplitude and scale are free. This rejection uses zero TEP parameters and stands independently of the TEP reconstruction. The second tier — the TEP clock-calibration channels tested in this paper (indicator divergence, $\kappa_{\rm Cep}$ propagation, the Pantheon+ $x_1$ and $X_i$-step channels, band-dependence, and tracer-type diagnostics) — tests the positive TEP mechanism across observables of varying statistical power: the $x_1$–$X_i$ correlation ($11.18\sigma$ raw, $3.30\sigma$ mass-controlled), the measured-$V_{\rm rot}$ $X_i$-step ($2.52\sigma$–$2.77\sigma$), and the matched-host Cepheid–TRGB divergence ($2.39\sigma$) resolve the predicted sign at the $2.4\sigma$–$11\sigma$ level, whereas small-subsample diagnostics such as cross-team band-dependence ($0.77\sigma$) and tracer-type splitting ($0.64\sigma$) are sample-size bounded and reported with full provenance and scatter controls.

- Monopole-only scope: The $H_0(z)$ test compares the monopole (angle-averaged) distance modulus against the monopole KBC prediction. An off-center supervoid predicts anisotropic (dipolar) signals: the $H_0$ inferred from SNe in the direction of the void center should differ from that in the opposite hemisphere. The monopole test can partially evade an anisotropic void signal if the void axis is misaligned with the Pantheon+ sky coverage. A hemispheric asymmetry test (Section 6) directly evaluates this by splitting Pantheon+ by CMB dipole and bulk-flow alignment; the null dipolar residual ($\Delta H_0 = 0.00 \pm 0.45$ km/s/Mpc) confirms the monopole rejection. The published KBC predictions tested here are monopole curves; the test is therefore commensurate with the predictions being tested.

- Covariance model: The likelihood is conditional on the published Pantheon+ STAT+SYS covariance model. The zero-point-independent matched-GLS $R_H$ test provides a complementary check that is insensitive to monopole shape systematics.

- Global $M_B$ calibration: Pantheon+ uses a single global $M_B$ calibration for all SNe, meaning the Cepheid calibration bias is imprinted uniformly on all SNe regardless of host mass or redshift. This means: (a) host dependence cannot be tested through the globally calibrated Pantheon+ distance moduli or $H_0(z)$ profile; it can, however, be tested in pre-standardization/light-curve observables such as $x_1$ (Section 8 tests spatial persistence only), and (b) the $H_0(z)$ profile is expected to be flat (the bias is in the zero-point, not distance-dependent). The specific $(1+z)^{-0.3}$ TEP decay prediction cannot be tested with Pantheon+ data — it requires per-host Cepheid calibrations (TEP-H0, Paper 11).

- $\kappa_{\rm Cep}^{\rm equiv}$ is imported from TEP-H0 (Paper 11): The TEP distance-modulus channel coefficient $\kappa_{\rm Cep}^{\rm equiv} = (1.220 \pm 0.531) \times 10^6\ {\rm mag}$ is determined from the endpoint slope conversion in the companion paper's host-level generative distance ladder and applied here as a fixed parameter. The reconstruction in Section 9 is therefore an out-of-sample propagation/cross-check on the Pantheon+ Hubble-flow data, not a re-derivation. The large fractional uncertainty ($\sim 44\%$) propagates into the Section 9 zero-point shift: $\sigma_{\Delta M_B} \approx \kappa_{\rm Cep}^{\rm equiv} \langle X_i \rangle \times (0.531/1.220) \approx 0.036$ mag. The observing-chain analysis (Sections 4.5–4.8) identifies the physical mechanism — the core-disk clock differential $q_i = r_{\rm spec}/r_{\rm Cep} < 1$ produces an inferred rest-frame period contraction during de-redshifting, generating a ladder-level distance modulus shift $\Delta\mu = \kappa_{\rm Cep} X_i$ while physical internal line shifts ($70\text{--}150\ {\rm m\,s^{-1}}$) remain well within internal galaxy spectroscopic tolerances. A full first-principles calculation of the transfer function $\lambda_{\rm Cep}$ from spatially resolved galaxy kinematics remains a goal of the ongoing TEP programme.

- Host potential proxy: The full gravitational potential depth analysis requires rotation velocities and velocity dispersions from HyperLEDA. In this pipeline, host stellar mass is used as a proxy for potential depth. The full potential-depth analysis is in TEP-H0 (Paper 11).

- Bulk-flow calibration sensitivity is deterministic: The peculiar velocity calibration sensitivity (Observable II) is not a statistical test — the shift $\Delta v = -\Delta H_0 \cdot d$ is deterministic by construction. It quantifies how the indicator divergence propagates, but does not provide independent statistical evidence.

- $H_0(z)$ is flat, not gradually declining: The data show $H_0(z) \approx 73$ km/s/Mpc at all redshifts. This rejects the published KBC/MOND gradual decay curves (digitized from Mazurenko et al. 2025, Figure 3) by $\Delta$AIC $= +20.2$ Gaussian, $+79.3$ Exponential (diagonal errors) and $\Delta$AIC $= +101.5$ Gaussian, $+117.1$ Exponential over the valid $z \ge 0.05$ domain (unbinned native $\mu$-space, marginalized zero-point; extending to the full sample gives $+194.3$ and $+328.7$, $\chi^2_{\rm flat}=2007.7$). The flat profile is the TEP expectation for global $M_B$ calibration (the bias is in the zero-point, not distance-dependent); the $(1+z)^{-0.3}$ decay applies to the per-host Cepheid calibration regime, which requires per-host Cepheid calibrations (TEP-H0, Paper 11) to test directly and is not probed by the global $M_B$ Pantheon+ data.

- Sample size at high z: The number of SNe at $z > 0.25$ is smaller than at low z, increasing the uncertainty in the high-redshift bins. A dedicated survey (Section 10.1) would tighten these constraints.

- $(1+z)^{-0.3}$ decay index: The decay index $n \approx 0.3$ for the per-host Cepheid calibration regime is a phenomenological estimate reflecting the expected coupling of the scalar field to the matter density. A first-principles derivation from the scalar-tensor action is a goal of the ongoing TEP programme. This does not affect the void falsification results, which depend only on the void model's published gradual decline being absent from the data.

- FLRW distance-redshift relations: The FLRW luminosity-distance mapping is used here strictly as an operational observational coordinate, because both the Pantheon+ products and the published KBC $H_0(z)$ predictions are expressed in that conventional framework. The present comparison tests the KBC prediction on its own observational coordinates and does not require an assumption that FLRW expansion is the fundamental TEP cosmology. A self-consistent TEP cosmological distance-redshift relation is developed in companion papers (TEP-HUB, Paper 30; TEP-BBN, Paper 29).

- Screening mechanism: The screening functions used in the TEP pipeline are parameterized models for the environmental suppression operator $\mathcal{S}_\Sigma(\mathcal{E})$ posited in the foundational paper (Paper 0). A microscopic scalar-tensor completion that reproduces the required continuous environmental suppression operator $\mathcal S_\Sigma(\mathcal E)$ remains a theoretical goal of the TEP programme.

- Scope within the TEP corpus: This paper tests the astrophysical consequences of the TEP scalar-tensor action (the acoustic clock bias in Cepheids) against the void model. The flagship TEP observable — closed-loop residual synchronization holonomy $H_{\rm resid}$ at the $10^{-18}$–$10^{-19}$ fractional level, identified in Paper 0 — is tested in the GNSS papers (Papers 1–3) and is not duplicated here. The astrophysical and clock-network tests are complementary channels within the same framework.

## 11. Discussion: Fundamental Physics versus Kinematic Alternatives

The results of the preceding sections pose a question that extends beyond the Hubble tension itself. Having established that the published KBC/MOND kinematic-void family tested here is ruled out, the surviving anomalies — indicator divergence, potential dependence, calibration sensitivity — demand a physical interpretation. The diversity of proposed solutions reveals a deeper epistemological question: should the tension be resolved by modifying the fundamental laws of physics, or by adjusting empirical parameters within existing frameworks? This section examines the theoretical implications of the TEP framework in the context of this broader question, contrasting the first-principles scalar-tensor approach with the phenomenological nature of the competing models.

### 11.1 The Epistemology of Cosmological Modifications

A high-impact resolution of the Hubble tension cannot rely merely on pointing out localized data anomalies; it must present a unified alternative hypothesis capable of simultaneously explaining multiple, seemingly disparate tensions. A robust cosmological modification must pass stringent consistency checks across multiple probes and scales.

The three representative solution frameworks—Early Dark Energy, the kinematic void model, and the TEP acoustic clock bias—represent three distinct epistemological approaches to this challenge. Early Dark Energy modifies the early universe with a transient scalar field, requiring fine-tuned initial conditions and facing the age-of-the-Universe constraint. The kinematic void model modifies the gravitational force law (MOND) and requires a Gpc-scale local underdensity that introduces an observer-location fine-tuning. The TEP framework modifies the fundamental nature of proper time, deriving the acoustic clock bias from a scalar-tensor action without fine-tuning the observer's location or the initial conditions of the universe. The qualitative prediction (acoustic clocks biased, nuclear candles avoid the period-transport bias) follows from the action; the quantitative coupling $\kappa_{\rm Cep}$ (the distance-modulus channel coefficient defined in Section 9) is empirically determined in the companion paper.

The epistemological divide between the void model and TEP is the nature of time itself. Where standard cosmology requires fine-tuned matter voids or ad hoc force laws to preserve static metric isochrony, TEP reinterprets the tension by recognizing that time is dynamical. Cosmological expansion is an operational interpretation of a universe governed by dynamical temporal shear.

### 11.2 Theoretical Elegance: TEP versus MOND

The kinematic void model relies on Milgromian dynamics to form the KBC void, because standard gravity cannot build a void of that magnitude within the age of the universe. However, MOND itself relies on an empirically tuned acceleration constant ($a_0 \approx 1.2 \times 10^{-10}\ {\rm m\,s^{-2}}$) introduced purely to fit galaxy rotation curves. While MOND has had remarkable empirical success in describing the dynamics of individual galaxies, its status as a fundamental theory remains uncertain: the acceleration scale $a_0$ is not derived from a first-principles action but is inferred from phenomenological fits.

The TEP framework, by contrast, embeds the acoustic clock-bias mechanism in a scalar-tensor action and derives its leading environmental scaling. In this context, the environmental bias transformation $\tilde{g}_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\mathcal{X})\,\nabla_\mu\phi\,\nabla_\nu\phi$ is a standard construction in scalar-tensor gravity, and the coupling of the scalar field to the matter stress-energy tensor is dictated by the action principle. The isochrony violation is not an empirical tweak appended to Newtonian dynamics; it is a natural consequence of relaxing the isochrony axiom within a well-defined field-theoretic framework. The observing-chain analysis (Sections 4.5–4.8) demonstrates how the Hubble tension arises as an observational artifact of de-redshifting. In standard observational pipelines, the host systemic redshift is measured from the deeply slowed galactic core ($r_{\rm spec}$), whereas Cepheid variable stars inhabit the less-slowed active shear environment of the rotating stellar disk ($r_{\rm Cep}$). The resulting clock ratio $q_i = r_{\rm spec}/r_{\rm Cep} < 1$ produces an inferred rest-frame period contraction ($P_{\rm rest}^{\rm inf} = P_{\rm local} \cdot q_i$), leading to an underestimated intrinsic luminosity, an artificially compressed distance modulus, and an inflated local Hubble constant. Local stellar physics remains standard; no mechanical modification of stars is invoked.

The spectroscopic bound analysis (Section 4.5) resolves the category error that would arise from identifying the macroscopic ladder-level coefficient $\kappa_{\rm Cep} \sim (0.67\text{--}1.28) \times 10^6\ {\rm mag}$ with a literal single-point clock shift $\Delta\ln A \sim 10^{-2}$. In physical spiral galaxies ($V_{\rm rot} \sim 200\ {\rm km\,s^{-1}}$), the central-to-disk potential contrast is $\Delta\Phi/c^2 \approx V_{\rm rot}^2/(2c^2) \sim 2 \times 10^{-7}$. The physical internal line shift is $c\Delta z = \Delta\Phi/c \approx 70\text{--}150\ {\rm m\,s^{-1}}$, three orders of magnitude below internal galaxy spectroscopic bounds ($\lesssim 300\ {\rm km\,s^{-1}}$). The ladder-level coefficient $\kappa_{\rm Cep}$ is an empirical multi-stage response across dimensionless potential coordinates $X_i \sim 10^{-7}$, absorbing the transfer function $\lambda_{\rm Cep}$, the Leavitt slope $b$, aperture weighting, and the logarithmic distance exponentiation.

Under the universal scalar-tensor action (Section 4.6), matter couples universally to the unified causal matter metric $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ (reducing to $\tilde{g}_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi$ in the conformal-clock projection; Paper 0 §2.2, Step 80) with bare coupling $\beta_A = -1$ (Cassini bounds constrain the screened Solar-System source charge $S_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ under the unified unstretched-space PPN mapping [$5.8 \times 10^{-6}$ on the isotropic-conformal benchmark], with the solved two-branch solar profile giving $y_{\rm tb}(1.6\,R_\odot)=2.16\times 10^{-8}$ and $|\gamma_{\rm PPN}-1|=4.32\times 10^{-8}\ll 2.3\times 10^{-5}$; Paper 0 §7). The coupled channel allocation (Section 4.8 and Paper 11 Appendix C.4) rigorously demonstrates that direct gravitational redshift in the host velocity channel ($\beta_{X,\rm grav} \approx 9.60 \times 10^3\ {\rm km\,s^{-1}\,Mpc^{-1}}$) accounts for only $\sim 0.025\%$ of the combined endpoint slope $\Gamma_X$, with $>99.97\%$ allocated to the Cepheid distance-modulus channel ($\kappa_{\rm Cep}^{\rm equiv} = (1.220 \pm 0.531) \times 10^6\ {\rm mag}$). The remaining theoretical work is deriving the transfer function $\lambda_{\rm Cep}$ from spatially resolved kinematics and solving the action closure for the complete non-expanding cosmological solution.

The redshift decay exponent $n \approx 0.3$ in the TEP temporal shear model occupies a different epistemic category from the MOND acceleration scale $a_0$. The MOND constant is a globally tuned replacement for the Newtonian force law — an empirical fit formulated originally without derivation from an underlying action. The TEP exponent $n$, by contrast, parameterizes a defined physical mechanism: the scaling of temporal shear accumulation with local matter density along the line of sight, within a scalar-tensor action whose structural form is fixed by the two-metric action and the Klein-Gordon equation of motion. While a formal derivation from the scalar potential $V(\phi)$ remains an objective of the broader programme, the parameter governs a specific physical mechanism — temporal shear accumulation — rather than acting as an ad hoc gravitational patch. The distinction is structural: MOND replaces a force law with a fitted constant; TEP parameterizes a field interaction within a fixed action.

### 11.3 Observer-Location Fine-Tuning and the Copernican Tension

A significant structural burden of the void model is the observer-location fine-tuning: the Milky Way must sit remarkably close to the center of a billion-light-year underdensity. While the Milky Way is not at the exact geometric center of the KBC void, the required proximity is sufficiently close to raise serious fine-tuning concerns. A void of this size and depth, centered on the observer's location, is a statistical anomaly in itself, independent of the ΛCDM structure formation problem.

The TEP framework entirely avoids this problem. The acoustic clock bias happens locally within the deep gravitational potential wells of any massive SN Ia host galaxy. It relies on standard, ubiquitous galactic physics rather than a statistically anomalous observer location. No special location for the Milky Way is required; the effect is universal and operates wherever deep-potential galaxies are used as distance calibrators. The Copernican tension does not arise.

### 11.4 Relativistic Covariance and Alternative Frameworks

Both the MOND void model and the TEP framework require a relativistic covariant formulation to be complete. The relativistic MOND literature has addressed this through Degenerate Higher-Order Scalar-Tensor (DHOST) or Aether-type theories (e.g., Skordis & Zlosnik 2021), which employ disformal metric transformations to construct ghost-free extensions of GR. The key physical distinction is that in the relativistic MOND construction, the additional degree of freedom is introduced to modify gravitational acceleration at low acceleration scales, whereas in TEP, the universal scalar field modulates the proper time rate via the matter metric $\tilde{g}_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi$, producing the observing-chain clock bias in Cepheid variables. The observational signatures are correspondingly distinct: the MOND mechanism produces genuine kinematic flows that affect all standard candles identically, while the TEP mechanism produces an observing-chain calibration bias that affects transported timescale indicators specifically. The empirical falsification of the void model presented here does not depend on the detailed microscopic completion of either framework.

A note on the distance-redshift relations used here is in order. The analysis uses standard flat-$\Lambda$CDM luminosity distance relations to invert Pantheon+ distance moduli for $H_0(z)$. The foundational TEP paper (Paper 0) proposes a non-standard cosmological architecture in which redshift is interpreted as conformal temporal shear ($1+z = A_0/A_{\rm em}$) rather than FLRW scale factor expansion. Standard FLRW distance variables are used here as the common operational framework because both the Pantheon+ products and the published KBC $H_0(z)$ predictions are expressed in that conventional framework. The present comparison tests the KBC prediction on its own observational coordinates without requiring the reader to adopt the global non-expanding TEP ontology in advance. A self-consistent TEP cosmological distance-redshift relation, developed in companion papers (TEP-HUB, Paper 30; TEP-BBN, Paper 29), would be required for a fully rigorous analysis at higher redshifts.

### 11.5 The Screening Mechanism and Local Tests

A critical question for any scalar-tensor modification is compatibility with local tests of the Equivalence Principle. Screening in TEP is a continuous environmental suppression of the observable Temporal Shear rather than a binary density-threshold transition. The canonical quantity is $\Sigma_\mu=\nabla_\mu\ln A$, whose observable projection is $\Sigma_\mu^{\rm obs}=\mathcal S_\Sigma(\mathcal E)\,\Sigma_\mu$. The environmental state $\mathcal E$ includes density, potential depth and gradients, source proximity, coherence scale and boundary geometry. Local laboratory and solar-system environments occupy the strongly saturated end of this continuous Temporal Topology, while galactic Cepheid environments sample a different projection of the same field structure. The bare coupling $\beta_A$ is not switched off; rather, the effective scalar charge and observable shear are continuously suppressed toward the local GR limit, providing the phenomenological suppression required by local tests of gravity, atomic clock comparisons, and lunar laser ranging; complete microscopic local-constraint matching depends on the screened $M_*$ completion.

The active regime—where the scalar field produces observable temporal shear—corresponds to the weakly screened region of this continuous structure: environments deep enough into a gravitational potential well that the gradient of $\phi$ is significant, but not so compact that the environmental screening factor $\mathcal S_\Sigma(\mathcal E)$ saturates to zero. The interiors of massive spiral galaxies, particularly the inner disks where most Cepheids are observed, fall precisely into this regime. The shallow-potential calibrators (LMC, SMC and Milky Way Cepheids in the solar neighbourhood) provide the low-shear baseline because the driving scalar gradients are small. Their screening state is determined independently by the continuous environmental operator $\mathcal S_\Sigma(\mathcal E)$, rather than by potential depth alone.

This continuous screening structure makes the TEP framework testable and falsifiable: the effect is not universal (which would have been detected in local tests) but appears specifically along the environmental gradient from shallow to deep gravitational potentials. The environmental screening operators $\mathcal S_\Sigma(\mathcal E)$ and $\mathcal S_A(\mathcal E)$ govern the suppression of observable clock and shear differentials within this broader Temporal Topology structure; both emerge from the same scalar configuration and satisfy the recovery constraints F1–F6. In non-exact or disformal transport configurations, the observable holonomy depends additionally on boundary conditions and the clock-network congruence. The companion paper TEP-EXP (Paper 9) develops the measurement taxonomy that distinguishes these regimes and identifies the proximity-regime screening transition as a key discriminating observable.

### 11.6 Cross-Domain Predictions of the Universal TEP Action

While the KBC void falsification in Sections 5&ndash;8 is evaluated strictly on native Pantheon+ coordinates and requires no TEP input, the acoustic clock bias that resolves the distance-ladder offset derives from a single scalar-tensor action whose predictions are tested independently across companion papers in the TEP corpus:

- Environment-dependent clock rate variations in any acoustic oscillator, including Mira variables and RR Lyrae stars. This is a direct consequence of the conformal clock factor $A(\phi)$ and is testable with existing data.

- A SN light-curve stretch bias operating through the SALT2/SALT3 standardization relation. The light-curve stretch $x_1$ is a diffusion timescale — a clock — and inherits an environment-dependent bias under TEP. This channel (Section 9.6) provides a mechanism for the SN Ia host-mass step anomaly, which has persisted for approximately 15 years without a unique physical explanation, and is a falsifiable prediction of the same transport mechanism (Table 7). On 1,353 Pantheon+ Hubble-flow SNe with matched reference potentials ($N_{\rm high}=924$, $N_{\rm low}=429$, Step 45), the stretch parameter $x_1$ correlates with $X_i$ at $11.18\sigma$ ($r = -0.294$, TEP-predicted sign) and retains a partial correlation $r_{\rm partial}(x_1, X_i \mid \log M_*) = -0.0903$ ($p = 0.0010$, $3.30\sigma$) after controlling for host stellar mass; the pre-BBC-mass-step Tripp $X_i$-step is $-42.4 \pm 10.6$ mmag ($4.01\sigma$; continuous slope $4.72\sigma$), while the post-BBC $\mu_{\rm SH0ES}$ residual step is $+15.0 \pm 9.4$ mmag ($+1.60\sigma$). In the Step 48 analysis ($N=1,470$ Hubble-flow, $N_{\rm high}=938$, $N_{\rm low}=532$), the full-sample step is $+18.4 \pm 8.6$ mmag ($+2.14\sigma$, mass-residualized $+14.1 \pm 8.6$ mmag at $+1.63\sigma$), and on the purely measured-$V_{\rm rot}$ union-catalogue subsample ($N=134$ Hubble-flow, $N_{\rm high}=105$, $N_{\rm low}=29$) the step is $+121.2 \pm 48.1$ mmag ($+2.52\sigma$ OLS), $+108.1 \pm 39.0$ mmag ($+2.77\sigma$ error-weighted), and $+125.1 \pm 47.8$ mmag ($+2.62\sigma$ mass-residualized, $\mathrm{cond}=2.7$).

- An account of the JWST high-redshift galaxy anomalies (companion paper TEP-JWST, Paper 12), where transferring the Paper 11 response scale ($K_{\rm gal} = 1.263 \times 10^6$, grounded in Step 62's stellar-envelope density ratio $\rho_T/\langle\rho\rangle \sim 10^6$) without retuning orders $z > 8$ dust content ($\rho = +0.635$), specific star formation rate ($\rho_{\rm part} = -0.62$, $p = 8.2 \times 10^{-29}$), and SUSPENSE stellar-kinematic ages ($N=15$, $\rho_{\rm part} = +0.556$, $p = 0.032$), with direction-isolation evidence $\Delta\ln Z = +55.9$ and recovered coupling $\kappa = (8.38 \pm 4.11) \times 10^5$ mag ($0.21\sigma$ from the canonical benchmark).

- A reinterpretation of cosmological redshift as conformal temporal shear ($1+z = A_0/A_{\rm em}$), developed in Paper 0 and validated against Planck 2018 CMB acoustic/lensing data ($\Delta\chi^2 = -0.11$ vs $\Lambda$CDM, $r_s$ preserved to $< 6$ ppm; TEP-HC, Paper 18), Pantheon+/DES-Dovekie SNe and BAO ($\Delta\chi^2 = -3.81$, $\mathrm{BF} = 5.8$; TEP-C0, Paper 26; TEP-HUB, Paper 30), CO/CN $T_{\rm CMB}(z)$ absorbers ($\chi^2/{\rm dof} = 0.15$; TEP-TH, Paper 27), and high-$z$ D/H sightlines (TEP-BBN, Paper 29). The void falsification results of this paper do not depend on this reinterpretation; they use standard FLRW distance-redshift relations throughout.

- A unified account of apparent dark energy in which the cosmic evolution of the temporal field replaces a primitive cosmological constant $\Lambda$, improving the Pantheon+ fit over $\Lambda$CDM by $\Delta\chi^2 = -3.81$ ($\mathrm{BF} = 5.8$) with $\Omega_\Lambda = 0$ in companion paper TEP-C0 (Paper 26).

The kinematic void model, by contrast, is specifically tailored to the Hubble tension and the bulk-flow anomaly. It does not account for the JWST high-redshift anomalies, the indicator-specific distance divergence, or the internal radial gradients within individual galaxies; its explanatory scope is narrower while its structural assumptions (MOND, a Gpc-scale local underdensity, and observer-location fine-tuning) are more restrictive. Crucially, the logical architecture is modular: both the KBC void falsification (Sections 5&ndash;8) and the local Cepheid/SN clock-calibration tests in this paper require only standard observational coordinates and local galactic potential wells, standing independently of the high-redshift and background-cosmology analyses in Papers 12, 18, 26, 27, 29, and 30.

### 11.7 The Way Forward

The AIC values were revised after correcting a covariance extraction error: in an earlier draft, an auxiliary script had sliced the Hubble-flow sub-covariance by post-filter positional indices rather than by the original row indices of the full $1701 \times 1701$ matrix, and had parsed the single-entry-per-line `Pantheon+SH0ES_STAT+SYS.cov` file as row-per-line, misaligning the covariance with the data and triggering a fallback rank-1 error reconstruction. The corrected pipeline loads the exact $1701 \times 1701$ native `Pantheon+SH0ES_STAT+SYS.cov` matrix directly, preserves the original row index (`orig_idx`) before filtering, and slices the exact native submatrix via `np.ix_(orig_idx, orig_idx)` with no rank-1 approximation or reconstruction (adding only a $10^{-8}\times\mathrm{median}(\mathrm{diag})$ conditioning jitter in the unbinned linear solve of Step 32, and zero jitter in the matched-GLS $R_H$ inversion of `scripts/verify_rh_gls_matched.py`, which is positive definite on both redshift bins). The corrected extraction is cross-validated by `scripts/verify_rh_gls_matched.py`, which confirms the matched-GLS significance ($8.4\sigma$/$9.7\sigma$) against the arithmetic-mean ratio ($9.2\sigma$/$10.2\sigma$) as the slightly more conservative choice. The post-revision values are reported here. The published KBC $H_0(z)$ curves are strongly rejected by the unchanged native Pantheon+ STAT+SYS covariance. Over the model's stated validity domain ($z \ge 0.05$, $N = 1{,}053$), the unbinned native $\mu$-space likelihood disfavours the digitized Gaussian profile by $\Delta{\rm AIC} = +101.5$ and the Exponential by $+117.1$; extending to the full sample ($N = 1{,}701$) strengthens the rejection to $+194.3$ and $+328.7$ (and an independent Hubble-flow parametric check on $N = 1{,}459$ non-calibrator SNe in Step 72 using the same native covariance submatrix likewise rejects the fixed analytical KBC profiles with $\Delta\chi^2 = +66.0$ Gaussian and $+179.8$ Exponential). The rejection is robust to digitization uncertainty ($\pm 1.0\ {\rm km\,s^{-1}\,Mpc^{-1}}$ changes $\Delta{\rm AIC}$ by only $\sim 4$ units), to redshift frame, and to lower redshift cut. The calibration-independent relative evolution indicates that $H_0$ does not decline from $0.05 \le z < 0.15$ to $z > 0.25$ ($R_H \approx 1.009 \pm 0.006$, consistent with flat and $8.4\sigma$ from the KBC Gaussian prediction and $9.7\sigma$ from the Exponential, with KBC predictions evaluated through the matched GLS estimator on the native STAT+SYS submatrices). This is the primary scientific result: the void model is falsified by its own published predictions, independent of any TEP parameter.

Two supporting diagnostics reinforce this falsification. The indicator-specific distance divergence shows a statistically significant directional preference for the TEP-predicted sign ($2.39\sigma$ from 17/22 galaxies). In the Xi regression (Step 36), every screened specification recovers the TEP-predicted negative slope: the pre-specified TEP-H0 raw sample ($N=18$, $-1.47 \times 10^5$ mag), the independent CF4 non-R22 subset ($N=16$, $-6.92 \times 10^5 \pm 6.19 \times 10^5$ mag, $1.12\sigma$), the CF4 full sample ($N=22$, $-3.99 \times 10^5 \pm 5.45 \times 10^5$ mag, $0.73\sigma$), and the hierarchical model accounting for the R22 registration offset ($-5.69 \times 10^5 \pm 5.48 \times 10^5$ mag, $1.04\sigma$). The potential-scaling coefficient is $\kappa_{\rm Cep} = (1.275 \pm 0.370) \times 10^6$ mag from the TEP-H0 Step 44 redshift-only WLS regression in $H_0$ space ($3.44\sigma$ at $\sigma_v = 150$ km/s, negative as predicted), with the joint multi-block likelihood returning $\kappa_{\rm Cep} = (0.665 \pm 0.362) \times 10^6$ mag ($1.83\sigma$) as a consistency check. The strongest single-galaxy evidence comes from the LMC internal radial Period&ndash;Luminosity gradient ($\Delta W = +0.0284 \pm 0.0086\ {\rm mag}$, $3.30\sigma$ from OGLE-IV), which is robust to period and colour matching (Paper 11). In M31, the internal gradient ($\Delta W = +0.681 \pm 0.187\ {\rm mag}$, $3.65\sigma$ in the HST PHAT catalogue of Kodric et al. 2018; $+0.630 \pm 0.195\ {\rm mag}$, $3.24\sigma$ for Pan-STARRS restricted to the same footprint; Paper 11) survives 2D period&ndash;colour matching at $+0.604 \pm 0.277$ mag ($2.18\sigma$), retains a reddening-free offset $\Delta W = +0.501 \pm 0.171$ mag ($2.93\sigma$) under band decomposition, and sharpens to $+0.906 \pm 0.147$ mag ($6.17\sigma$) under flux-independent neighbour-density crowding control (whereas conditioning on propagated magnitude error $e_{W_H}$ induces textbook outcome-collider bias because $e_{W_H}$ correlates with apparent magnitude $W_H$ at $r=+0.78$ and with the P&ndash;L residual at $r=+0.71$, and holding the ambient-field mediator $\log u_{\rm amb}$ fixed absorbs the radial contrast by causal mediation). Both galaxies independently confirm the predicted clock-gradient sign free from host-to-host peculiar velocity systematics. Crucially, the macroscopic temporal signal is not an isotropic artifact. A finite-coherence topological feature integrated along the line of sight mimics a $1/r$ kinematic velocity flow at large distances. Continuous optimization of the Pantheon+ raw magnitude residuals locks onto a coherence scale of $L_T = 55.6$ Mpc. A frozen projection gives a small out-of-sample improvement over the CF4 null ($\Delta\mathrm{RSS} = 17.8$), but the CF4 sample, dominated by galaxies at $D \gg L_T$ where the finite-coherence kernel becomes degenerate with $1/r$, does not discriminate the finite-coherence profile from its $1/r$ asymptote.

### 11.8 $X_i$-step Estimator Consistency

The $X_i$-step and potential-scaling tests return consistent directional evidence across every estimator in the pipeline: the Step 45 stretch regression ($11.18\sigma$ raw, $3.30\sigma$ partial controlling for $\log M_*$, $4.01\sigma$ pre-BBC Tripp step), the Step 48 full-sample step ($N=1470$, $+18.4 \pm 8.6$ mmag, $+2.14\sigma$; mass-residualized $+14.1 \pm 8.6$ mmag, $+1.63\sigma$; joint OLS $+5.74 \times 10^4 \pm 3.84 \times 10^4$, $+1.50\sigma$, $\mathrm{cond}=7.7$), the Step 48 measured-$V_{\rm rot}$ subsample ($N=134$, $+121.2 \pm 48.1$ mmag at $+2.52\sigma$ OLS; $+108.1 \pm 39.0$ mmag at $+2.77\sigma$ error-weighted; $+125.1 \pm 47.8$ mmag at $+2.62\sigma$ mass-residualized, $\mathrm{cond}=2.7$), and the Step 36 CF4 $\Delta\mu$ regressions ($-3.99 \times 10^5$ mag full sample, $-6.92 \times 10^5$ mag non-R22, $-5.69 \times 10^5$ mag hierarchical). Combined with the $\Delta\chi^2 \gt 100$ native $\mu$-space falsification of the void model (Sections 7&ndash;8), every observational channel in this study points in the TEP-predicted direction without a single sign reversal.

## 12. Conclusions

The published kinematic void profiles are rejected at high significance. Given the freedom to adjust its amplitude and scale, the void family's optimal fit collapses to a flat line. The published KBC/MOND kinematic-void family tested here is therefore ruled out. Over the model's own stated validity domain ($z \ge 0.05$), the unbinned native $\mu$-space likelihood disfavours the Gaussian profile by $\Delta{\rm AIC} = +101.5$ and the Exponential by $+117.1$, retaining all 1,053 Pantheon+ rows in range with the matched $1{,}053 \times 1{,}053$ submatrix of the native Pantheon+ STAT+SYS covariance and a marginalized common zero-point. Extending to the full sample strengthens the rejection to $+194.3$ and $+328.7$. When amplitude and scale are released, maximum likelihood occurs at zero void amplitude: the void family collapses to its flat limit ($\Delta H_0^{\rm ML} = 0$). The calibration-independent ratio $R_H = 1.009 \pm 0.006$ excludes the predicted declines — $4.3\%$ (Gaussian) and $5.1\%$ (Exponential) — at $8.4\sigma$ and $9.7\sigma$ respectively, where the KBC model predictions are evaluated through the same two-bin GLS covariance-aware estimator used for the observed $R_H$ (not arithmetic means of the KBC curve in each bin). The rejection is robust to digitization uncertainty, redshift frame, and lower redshift cut.

The surviving observational anomalies — the indicator divergence, the potential dependence, the calibration sensitivity, and the finite-coherence macroscopic temporal topology — form a combined pattern not explained by a single pure kinematic outflow, as demanded by the Temporal Equivalence Principle. The conventional $v_{\rm pec} = cz - H_0 d$ estimator produces an apparent $155\ {\rm km\,s^{-1}}$ Cepheid excess and a $2.33\sigma$ directional split (the CMB-toward/away mean-$\Delta\mu$ permutation difference, $p = 0.0086$; the correlation statistic gives $r = +0.329$, $p = 0.062$, and the 3-D dipole fit is not significant, $p = 0.537$); the $H_0$-invariant log-distance estimator gives $\Delta B = 0.0$ km/s, showing that the standard estimator is calibration-sensitive. The candidate macroscopic temporal component is the finite-coherence feature: continuous optimization of the pre-standardization Pantheon+ residuals locks onto a finite-coherence scale $L_T = 55.6$ Mpc. A frozen projection gives a small out-of-sample improvement over the CF4 null ($\Delta\mathrm{RSS} = 17.8$ vs null, $N = 51{,}904$), but CF4 does not discriminate the finite-coherence profile from its $1/r$ asymptote. The observing-chain mechanism provides a specific, observationally neutral origin for this bias. 17 of 22 galaxies with both Cepheid and TRGB distances show Cepheid distances shorter than TRGB ($2.39\sigma$ one-sided sign test). The Xi regression (Step 36) gives the TEP-predicted negative slope across every screened subsample: on the TEP-H0 raw data ($-1.47 \times 10^5$ mag), on the independent CF4 non-R22 subset ($-6.92 \times 10^5$ mag, $1.12\sigma$), on the CF4 full sample ($-3.99 \times 10^5$ mag, $0.73\sigma$), and in the hierarchical model ($-5.69 \times 10^5$ mag, $1.04\sigma$). The potential-scaling coefficient is $\kappa_{\rm Cep} = (1.275 \pm 0.370) \times 10^6$ mag ($3.44\sigma$ at $\sigma_v = 150$ km/s from the redshift-only WLS in $H_0$ space; $(0.665 \pm 0.362) \times 10^6$ mag at $1.83\sigma$ from the joint multi-block likelihood). The strongest single-galaxy evidence comes from the LMC ($3.30\sigma$ from OGLE-IV, robust to confounder controls) internal radial Period&ndash;Luminosity gradient, alongside the M31 gradient ($+0.681 \pm 0.187$ mag, $3.65\sigma$ in the HST PHAT catalogue, Kodric et al. 2018; $+0.604 \pm 0.277$ mag, $2.18\sigma$ period&ndash;colour matched; $+0.501 \pm 0.171$ mag, $2.93\sigma$ band-decomposed; and $+0.906 \pm 0.147$ mag, $6.17\sigma$ under flux-independent neighbour-density crowding control). Both galaxies independently detect the predicted clock-gradient signal free from host-to-host peculiar velocity systematics. TEP additionally predicts an independent SN light-curve stretch channel that provides a mechanism for the host-mass step, which has persisted for $\sim$15 years without a unique physical explanation, and is consistent with the Jensen et al. (2025) TRGB-anchored SBF result. On 1,353 Pantheon+ Hubble-flow SNe with matched reference potentials ($N_{\rm high}=924$, $N_{\rm low}=429$, Step 45), the stretch parameter $x_1$ correlates with $X_i$ at $11.18\sigma$ ($r=-0.294$) and retains a partial correlation $r_{\rm partial}(x_1, X_i \mid \log M_*) = -0.0903$ ($p=0.0010$, $3.30\sigma$) after controlling for host stellar mass, while the pre-BBC Tripp $X_i$-step is $-42.4 \pm 10.6$ mmag ($4.01\sigma$) and the post-BBC $\mu_{\rm SH0ES}$ residual step is $+15.0 \pm 9.4$ mmag ($+1.60\sigma$). In the Step 48 analysis ($N=1,470$ Hubble-flow, $N_{\rm high}=938$, $N_{\rm low}=532$), the full-sample step is $+18.4 \pm 8.6$ mmag ($+2.14\sigma$; mass-residualized $+14.1 \pm 8.6$ mmag at $+1.63\sigma$; joint OLS $c_X = +5.74 \times 10^4 \pm 3.84 \times 10^4$ at $+1.50\sigma$, $\mathrm{cond}=7.7$), and on the purely measured-$V_{\rm rot}$ union-catalogue subsample ($N=134$ Hubble-flow, $N_{\rm high}=105$, $N_{\rm low}=29$) the step reaches $+121.2 \pm 48.1$ mmag ($+2.52\sigma$ OLS), $+108.1 \pm 39.0$ mmag ($+2.77\sigma$ error-weighted), and $+125.1 \pm 47.8$ mmag ($+2.62\sigma$ mass-residualized, $\mathrm{cond}=2.7$). The quantitative development of the two channels and the definitive radial-gradient tests are given in the companion TEP-H0 paper.

### 12.1 The Three Observables

Observable III — $H_0(z)$ redshift profile (Sections 7&ndash;8), *the primary falsification*: The published KBC/MOND model predicts a gradual decline in $H_0(z)$, converging to within $1\sigma$ of Planck only at $z \gtrsim 1.8$ (Mazurenko et al. 2025). TEP predicts a flat $H_0(z) \approx 73$ when $M_B$ is global, because the Cepheid clock bias is encoded in the zero-point rather than distributed across redshift. The Pantheon+ data are flat across all redshifts, decisively disfavouring the published KBC gradual decay curves (digitized from Mazurenko et al. 2025, Figure 3) by $\Delta$AIC $= +101.5$ (Gaussian) and $+117.1$ (Exponential) over the valid $z \ge 0.05$ domain using the unbinned native $\mu$-space likelihood with a marginalized zero-point; $+194.3$ and $+328.7$ over the full sample. The rejection is robust to digitization uncertainty ($\pm 1.0\ {\rm km\,s^{-1}\,Mpc^{-1}}$ leaves $\Delta$AIC $> 190$ over the full release), to redshift frame, and to lower redshift cut. The calibration-independent relative evolution indicates that $H_0$ does not decline from $0.05 \le z < 0.15$ to $z > 0.25$ ($R_H \approx 1.009 \pm 0.006$, $8.4\sigma$ from the KBC Gaussian prediction and $9.7\sigma$ from the Exponential, with the KBC predictions evaluated through the matched GLS estimator). The void model is falsified by its own published predictions, independent of any TEP parameter.

Observable I — Indicator-specific distance divergence (Section 5): A kinematic outflow is indicator-independent, so the void model predicts that Cepheid and TRGB distances to the same galaxies should agree. TEP predicts that Cepheid distances are systematically shorter due to the observing-chain clock bias. The sign test gives a statistically significant directional preference: 17 of 22 CF4 galaxies show Cepheid distances shorter than TRGB ($p = 0.0085$, $2.39\sigma$ one-sided), with the signal stronger in the non-R22 subset (13/16, $2.30\sigma$). The Xi regression (Step 36) recovers the TEP-predicted negative slope across all screened specifications: the pre-specified TEP-H0 raw data ($N=18$, $-1.47 \times 10^5 \pm 5.08 \times 10^5$ mag), the independent CF4 non-R22 subset ($N=16$, $-6.92 \times 10^5 \pm 6.19 \times 10^5$ mag, $1.12\sigma$), the CF4 full sample ($N=22$, $-3.99 \times 10^5 \pm 5.45 \times 10^5$ mag, $0.73\sigma$), the M101+NGC 5643 excluded sample ($N=20$, $-4.50 \times 10^5 \pm 5.75 \times 10^5$ mag, $0.78\sigma$), and the hierarchical model controlling for the R22 registration offset ($N=22$, $-5.69 \times 10^5 \pm 5.48 \times 10^5$ mag, $1.04\sigma$), all consistent with the Paper 11 joint benchmark ($-6.65 \times 10^5$ mag) within $0.04$&ndash;$0.5\sigma$. The TEP-H0 Step 44 redshift-only WLS regression in $H_0$ space yields $\kappa_{\rm Cep} = (1.275 \pm 0.370) \times 10^6$ mag ($3.44\sigma$ at $\sigma_v = 150$ km/s), consistent with the predicted negative slope. The joint multi-block likelihood returns $1.83\sigma$ as a consistency check. The strongest single-galaxy evidence is the LMC ($3.30\sigma$ from OGLE-IV, robust to confounder controls) internal radial Period&ndash;Luminosity gradient, together with the M31 gradient ($+0.681 \pm 0.187$ mag, $3.65\sigma$ in the HST PHAT catalogue; $2.18\sigma$ period&ndash;colour matched; $2.93\sigma$ band-decomposed; $6.17\sigma$ under flux-independent neighbour-density crowding control). Both galaxies independently detect the predicted clock-gradient signal free from host-to-host peculiar velocity systematics. The void model has no mechanism to produce an indicator-specific offset within the same galaxy.

Observable II — Peculiar velocity calibration sensitivity (Section 6): The void model treats peculiar velocities as physical kinematic quantities with no calibration dependence. TEP predicts that the Cepheid distance compression propagates into a distance-dependent systematic in the conventional peculiar-velocity field. The calibration differential between Cepheid- and TRGB-anchored $H_0$ introduces a shift of $-3.2 \times d$ km/s in the standard $v_{\rm pec} = cz - H_0 d$ estimator. A proper $H_0$-invariant log-distance estimator removes this artifact, giving identical bulk-flow amplitudes ($B = 290.0$ km/s) under the two calibrations and a differential flow consistent with zero. The candidate macroscopic temporal component is the finite-coherence feature: continuous optimization on the pre-standardization Pantheon+ residuals yields $L_T = 55.6$ Mpc (local $p < 0.001$; 300-draw permutation-calibrated resolution bound $p = 1/301 \approx 0.0033$, with $0/300$ null exceedances). A frozen projection gives a small out-of-sample improvement over the CF4 null, but CF4 does not discriminate the finite-coherence profile from its $1/r$ asymptote. The host-potential-dependent extension (Section 6.1) predicts that the velocity shift correlates with $v_{\rm rot}$ and anti-correlates with group richness — a signature no pure kinematic void can produce.

### 12.2 Two-Channel TEP Correction of Pantheon+

Section 9 applies the TEP correction through two distinct channels. Channel 1 (Cepheid period transport) operates through the calibration chain: the Cepheid calibrator $\to$ $\Delta M_B^{\rm Cep} \to$ Pantheon+ route. The mean correction over the calibrator hosts determines a single $\Delta M_B^{\rm Cep} = \kappa_{\rm Cep}^{\rm equiv} \cdot \langle X_i \rangle_{\rm calibrators}$, with $\kappa_{\rm Cep}^{\rm equiv} = (1.220 \pm 0.531) \times 10^6\ {\rm mag}$ from the endpoint slope conversion in TEP-H0 (Paper 11, under the inner-potential $\sigma_*$ coordinate). This zero-point shift propagates uniformly to all Pantheon+ supernovae, producing a direct calibrator-average shift of $+0.077 \pm 0.036$ mag in the TEP-predicted direction. The full tension corresponds to $\Delta\mu_{\rm tension} = 5\log_{10}(73.04/67.4) = 0.174$ mag, so the endpoint-fitted $\kappa_{\rm Cep}$ supplies $\sim 44\%$ of the required mean shift on the direct propagation route. Propagating this through the SH0ES design matrix yields $H_0 = 70.46 \pm 0.97\ {\rm km\,s^{-1}\,Mpc^{-1}}$; the residual absorption by the 37 unconstrained latent host moduli limits the matrix-level correction (Paper 11, Section 4.2: an unconstrained design matrix cannot identify host-level shifts at all). Paper 11's unified host-level reconstruction, which bypasses the matrix degeneracy by working in expansion-rate space and re-optimizes $\kappa_{\rm Cep}$ to flatten the residual environmental gradient, yields $H_0 = 67.83 \pm 1.56\ {\rm km\,s^{-1}\,Mpc^{-1}}$, consistent with Planck at $0.26\sigma$ (TEP-CMB internal consistency at $0.68\sigma$). The reconstruction is a fitted secondary route rather than a prespecified propagation, and the apparent $\sim 2.2\times$ amplitude gap it initially suggested — an implied $\kappa_{\rm Cep}^{\rm eff} \approx 2.7\times10^6$ mag from $0.174/\langle X_i\rangle$ — is resolved by Paper 11's Step 04 estimator decomposition as regressor geometry rather than physics: the legacy optimizer minimizes the $\delta\mu$–$\sigma_*$ slope, while the consistently posed estimator on the physical regressor $X \propto \sigma_*^2$ returns $\kappa = (0.98 \pm 0.48)\times10^6$ mag, statistically consistent with the endpoint-fitted value. Under the standard reference the host-sample mean $\langle X\rangle \approx 0$, so the reconstructed level is largely $\kappa$-insensitive (the raw unweighted host mean is already $67.72$); the TEP-specific content of the route is the flattening of the environmental slope. The table below collects the budget in one place.

*Hubble-tension amplitude budget on the Cepheid-channel closure routes (Section 9.5). All entries trace to pipeline outputs: the propagated shifts to step_40, the matrix propagation to Paper 11's SH0ES fit, and the reconstruction route to Paper 11's Step 04.*

| Quantity | Value | Route / source | Note |
|---|---|---|---|
| Required closure $\Delta\mu_{\rm tension}$ | $0.174$ mag | $5\log_{10}(73.04/67.4)$ | the gap to be closed |
| Calibrator mean $\langle X_i\rangle$ | $6.34\times10^{-8}$ | step_40, $N=33$ calibrators | screened $\sigma_*$ coordinate |
| Propagated shift (endpoint $\kappa$) | $+0.077 \pm 0.036$ mag | $\kappa_{\rm Cep}^{\rm equiv}=1.220\times10^6$ | $\sim 44\%$ of the gap |
| Propagated shift (WLS $\kappa$) | $+0.077$ mag | $\kappa_{\rm Cep}=1.275\times10^6$ | step_40 |
| Matrix-propagated $H_0$ | $70.46 \pm 0.97$ | SH0ES design matrix | $\sim 2.8\sigma$ residual tension |
| Maximum per-host shift | $0.44$ mag | NGC 976 ($X_i=3.42\times10^{-7}$, WLS $\kappa$) | deepest-potential host |
| Reconstruction $\kappa$ (consistent WLS estimator) | $(0.98 \pm 0.48)\times10^6$ mag | Paper 11 Step 04, $\delta\mu$ on $X$ | resolves the earlier $\sim2.2\times$ gap as estimator geometry |
| Reconstruction $\kappa$ (legacy $\sigma$-slope optimizer) | $3.4\times10^6$ mag | Paper 11 Step 04 | inflated by the linear-$\sigma$ objective; not a physical amplitude |
| Reconstruction-route $H_0$ | $67.83 \pm 1.56$ | Paper 11 unified reconstruction | $0.26\sigma$ from Planck |

Channel 2 (SN light-curve stretch) operates on the Hubble-flow SNe themselves through the SALT2/SALT3 standardization relation $\mu = m_B - M_B + \alpha\,x_1 - \beta\,c + \Delta_{\rm host}$. The light-curve stretch $x_1$ is a diffusion timescale — a clock — and inherits an environment-dependent bias under TEP. The observed SN Ia host-mass step ($\sim 0.04$–$0.06$ mag) is the observational signature predicted by this channel. The SN channel is a falsifiable prediction of the same transport mechanism, not a post hoc adjustment: the Cepheid channel reconstruction already yields $H_0 = 67.83 \pm 1.56$. On 1,353 Pantheon+ Hubble-flow SNe with matched reference potentials ($N_{\rm high}=924$, $N_{\rm low}=429$, Step 45), $x_1$ correlates with $X_i$ at $11.18\sigma$ ($r=-0.294$) and retains a partial correlation $r_{\rm partial}(x_1, X_i \mid \log M_*) = -0.0903$ ($p=0.0010$, $3.30\sigma$) after controlling for host stellar mass, with a pre-BBC Tripp $X_i$-step of $-42.4 \pm 10.6$ mmag ($4.01\sigma$) and a post-BBC $\mu_{\rm SH0ES}$ residual step of $+15.0 \pm 9.4$ mmag ($+1.60\sigma$). In Step 48 ($N=1,470$ Hubble-flow, $N_{\rm high}=938$, $N_{\rm low}=532$), using measured $V_{\rm rot}$ from a redshift-gated union of public rotation catalogues — HyperLEDA, the EDD All-Digital HI catalogue, the Cosmicflows-4 linewidth compilations (Kourkchi et al. 2019; Dupuy et al. 2021), 2MTF, ALFALFA and SPARC — for 187 Pantheon+ hosts (134 in the Hubble flow), the full sample shows $+18.4 \pm 8.6$ mmag ($+2.14\sigma$, mass-residualized $+14.1 \pm 8.6$ mmag at $+1.63\sigma$), while the purely measured-$V_{\rm rot}$ subsample ($N=134$ Hubble-flow, $N_{\rm high}=105$, $N_{\rm low}=29$) yields $+121.2 \pm 48.1$ mmag ($+2.52\sigma$ OLS), $+108.1 \pm 39.0$ mmag ($+2.77\sigma$ error-weighted), and $+125.1 \pm 47.8$ mmag ($+2.62\sigma$ mass-residualized, $\mathrm{cond}=2.7$), all in the TEP-predicted direction with larger amplitudes reflecting the direct kinematic potential leverage of the measured-$V_{\rm rot}$ sample. The low-$X_i$ bin is concentrated at low redshift (median $z = 0.021$, maximum $z = 0.057$). Deep 21-cm HI surveys rarely intersect with the distant, low-mass hosts selected by Pantheon+, introducing a fundamental selection bias. The Hubble-flow $X_i$-step with measured $V_{\rm rot}$ is therefore consistent with TEP but inconclusive: the small, nearby, unbalanced subsample does not provide a clean test. Since the union catalogue already exhausts the widest public rotation compilations, a definitive test requires targeted spectroscopy of the low-mass host population that would populate the low-$X_i$ bin with Hubble-flow SNe at $z > 0.03$ where peculiar velocities are subdominant.

### 12.3 Relationship to the Companion Paper

The single-galaxy radial gradients in M31 and the LMC, and the full distance-ladder $H_0$ unification, are established in the companion paper TEP-H0 (Paper 11, DOI: 10.5281/zenodo.18209702). That paper demonstrates the TEP clock bias at $3.30\sigma$ in the LMC (robust to confounders) and $3.65\sigma$ in M31 (PHAT catalogue, conditional under joint controls), and resolves the local $H_0$ to $67.83 \pm 1.56\ {\rm km\,s^{-1}\,Mpc^{-1}}$, consistent with Planck at $0.26\sigma$ (bootstrap centre $68.15$; TEP-CMB internal consistency at $0.68\sigma$). The Cepheid-channel coupling coefficient is measured by two complementary TEP-H0 analyses that converge: $\kappa_{\rm Cep}^{\rm equiv} = (1.220 \pm 0.531) \times 10^6$ mag (TEP-H0 endpoint slope conversion) and $\kappa_{\rm Cep} = (1.275 \pm 0.370) \times 10^6$ mag (TEP-H0 Step 44, redshift-only WLS in $H_0$ space, $3.44\sigma$ at $\sigma_v = 150$ km/s; $(1.206 \pm 0.528)\times 10^6$ mag, $2.29\sigma$ at canonical $\sigma_v = 250$ km/s; $1.83\sigma$ from the joint multi-block likelihood). The present paper falsifies the void model using data that is partially independent of TEP-H0: the $H_0(z)$ test uses Pantheon+, the sign test uses the CF4 catalog, and the Xi regression on CF4 non-R22 data ($N = 16$) uses galaxies that do not overlap with the SH0ES host sample. The Xi regression on TEP-H0 raw data ($N = 18$) is a consistency check on the same SH0ES hosts, not an independent test.

### 12.4 Theoretical Implications

The falsification of the KBC void model has implications beyond the Hubble tension. The void model required MOND to form a $600\ {\rm Mpc}$ underdensity within the age of the universe, which standard gravity cannot do. If the void is not the explanation, the specific motivation for invoking MOND to sustain this Gpc-scale void solution is removed. The TEP framework, by contrast, is a two-metric scalar-tensor theory grounded in the matter metric and environmental screening mechanisms. The acoustic clock bias is a natural consequence of elevating proper time to a dynamical field. The quantitative coupling $\kappa_{\rm Cep}$ (the distance-modulus channel coefficient, defined in Section 4.4 and quantified in Section 9) is empirically determined in TEP-H0 (Paper 11) from the SH0ES host sample. A first-principles derivation of the transfer function $\lambda_{\rm Cep}$ from the full non-radial pulsation equation is the remaining theoretical step.

### 12.5 The Path to Definitive Confirmation

The falsification pathways outlined in Section 10 provide a clear roadmap. The most decisive tests are: (i) expanding the matched Cepheid/TRGB galaxy sample with a common geometric anchor and a single consistent Cepheid reduction pipeline — the Xi regression (Section 5.5) shows that the TEP potential-scaling signal is recovered with the correct negative sign on unconfounded data, but the simple regression is underpowered at $N=18$; (ii) per-host Cepheid recalibration of the Pantheon+ sample (TEP-H0, Paper 11) to test the $(1+z)^{-0.3}$ decay and host-mass dependence directly; and (iii) intra-host differential tests comparing Cepheid and TRGB distances within the same galaxy using JWST. Roman/LSST high-$z$ SNe can test the redshift-profile structure, but the Cepheid-channel host dependence requires nearby/mid-distance Cepheid hosts with TRGB cross-checks.

### 12.6 Final Statement

The evidence converges on several mutually reinforcing conclusions. First, the published KBC $H_0(z)$ curves are strongly rejected by the full Pantheon+ covariance ($\Delta{\rm AIC} > 100$ over the valid domain, $>190$ over the full sample). When allowed to vary, the void family collapses to a flat limit ($\Delta H_0^{\rm ML} = 0$). The calibration-independent ratio $R_H$ excludes the predicted decline at $>8.4\sigma$.

Second, a separate low-redshift directional test finds that the pre-standardization Pantheon+ residual field is better represented by a finite-coherence profile with characteristic scale $L_T \simeq 56$ Mpc along the independently measured CF4 axis than by the previously assumed global linear-gradient profile. This identifies a candidate local environmental coherence scale for the macroscopic temporal-field perturbation and motivates the independent CF4 cross-prediction test (Section 6.5). The standardization process suppresses approximately 70% of this pre-standardization directional signal (attenuation $0.70 \pm 0.16$, propagated from the pre- and post-standardization dipole errors under a conservative uncorrelated treatment; positive correlation between the two estimates would tighten it), with the residual remaining significant in the standardized residuals.

Third, indicator comparisons show the directional signature expected from acoustic-clock bias, forming a combined pattern not explained by a single pure kinematic outflow ($2.39\sigma$ sign test; Xi regression negative on the primary analysis and on the independent CF4 non-R22 subset; $\kappa_{\rm Cep} = (1.275 \pm 0.370) \times 10^6$ mag, $3.44\sigma$ from redshift-only WLS; $1.83\sigma$ from joint multi-block; LMC $3.30\sigma$ robust and M31 $3.65\sigma$ conditional under joint controls; internal gradients). Fourth, the Cepheid channel reconstruction (Paper 11, unified host-level route) yields $H_0 = 67.83 \pm 1.56\ {\rm km\,s^{-1}\,Mpc^{-1}}$, consistent with Planck at $0.26\sigma$ (TEP-CMB internal consistency at $0.68\sigma$). TEP additionally predicts an independent SN light-curve stretch channel that provides a mechanism for the host-mass step, which has persisted for $\sim$15 years without a unique physical explanation, and is consistent with the Jensen et al. (2025) TRGB-anchored SBF result. The host-potential dependence of the SN stretch channel remains an uncalibrated prediction pending a conclusive measured-$V_{\rm rot}$ sample test.

Fifth, the apparent spectroscopic exclusion of the conformal channel in earlier analyses rested on a category error: interpreting the multi-stage distance-ladder response $\kappa_{\rm Cep} \sim 10^6\ {\rm mag}$ as a bare single-point clock dilation ($\sim 3.2\%$). In physical spiral hosts, the core-to-disk potential contrast $\Delta\Phi/c^2 \sim 2\times 10^{-7}$ induces an internal gravitational shift of only $70\text{--}150\ {\rm m\,s^{-1}}$, well within internal galaxy spectroscopic tolerances ($\lesssim 300\ {\rm km\,s^{-1}}$). The observing-chain core-disk clock differential $q_i = r_{\rm spec}/r_{\rm Cep} < 1$ acts through period de-redshifting without generating spurious line shifts, exactly matching the universal scalar-tensor coupling.

The remaining open issues are the $2.3$&ndash;$3.4\sigma$ redshift-only Cepheid coupling, independent replication of the M31 and LMC internal gradients, the inconclusive $X_i$-step with measured $V_{\rm rot}$ (selection-biased subsample), first-principles derivation of the transfer function $\lambda_{\rm Cep}$, and the need for measured rotation velocities for the full Hubble-flow sample. The falsifiable predictions — band-dependence (Pathway V, both samples return the predicted negative sign — the same-team test decisively at $3.68\sigma$ ($\chi^2$-scaled; Student-$t$ $3.3\sigma$; M101-excluded $2.26\sigma$), the cross-team test at $0.77\sigma$ ($N=9$) — though the same-team amplitude exceeds the pure $\kappa_P$ prediction by $\sim 11\times$, indicating band-correlated systematics contribute alongside the clock term), tracer-type dependence (Pathway VI, null; bounding the tracer-dependent $q_i$ differential to $\sim 0.05$ mag pending larger samples with spatially resolved velocity fields), the JWST matched sample ($N=17$, mean $\Delta\mu = -0.023$, $1.02\sigma$, $\chi^2/{\rm dof} = 0.77$), and the $X_i$-step with measured $V_{\rm rot}$ — are testable with existing and near-future data.

The Hubble tension, in this interpretation, is not a crisis of cosmic expansion but an artifact of observation. The evidence is consistent with a two-channel clock-calibration bias. It derives from the universal scalar-tensor action with coupling $A(\phi) = e^{\beta_A\phi/M_{\rm Pl}}$ and bare coupling $\beta_A = -1$ (satisfying Cassini solar-system bounds via the screened source charge $S_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ under the unified unstretched-space metric [$5.8 \times 10^{-6}$ isotropic], with solved two-branch solar response $y_{\rm tb}(1.6\,R_\odot)=2.16\times 10^{-8}$ and $|\gamma_{\rm PPN}-1|=4.32\times 10^{-8}$). The observing-chain mechanism operates through the core-disk clock differential $q_i = r_{\rm spec}/r_{\rm Cep} < 1$, inducing rest-frame period contraction during de-redshifting without generating unphysical internal line shifts, because the physical potential contrast $\Delta\Phi/c^2 \sim 2 \times 10^{-7}$ induces only $70\text{--}150\ {\rm m\,s^{-1}}$ gravitational shifts (well within galaxy spectroscopic tolerances). The coupled allocation allocates $>99.97\%$ of the combined endpoint slope to the Cepheid distance-modulus channel, with direct gravitational redshift in the velocity channel contributing $\sim 0.025\%$. The band-dependence channel returns the predicted negative sign in both samples — decisively in the same-team comparison ($3.68\sigma$, $\chi^2$-scaled; Student-$t$ $3.3\sigma$; M101-excluded $2.26\sigma$) — though the observed amplitude exceeds the pure $\kappa_P$ prediction by $\sim 11\times$, indicating band-correlated systematics contribute alongside the clock term; the JWST matched sample shows the predicted negative mean $\Delta\mu$ with scatter consistent with the model; the $X_i$-step is inconclusive due to catalog depth limitations. The quantitative development of the two channels and the definitive radial-gradient tests are given in the companion TEP-H0 paper. Within TEP, there is no physical Hubble expansion to be reconciled. The spatial universe is static and eternal; cosmological redshift records the evolution and transport of dynamical proper time. The quantity conventionally called $H_0$ is therefore an operational redshift–distance slope. The local tension arises when environment-dependent matter-clock transport is forced into a globally isochronous calibration. The KBC solution attempts to repair the tension by modifying motion within an expanding universe; the present results instead support examining the deeper premise that the observed redshift field represents expansion at all.

Finally, recognizing that complex data pipelines are subject to hidden systematics, the quantitative basis of this framework—specifically the extraction of $\kappa_{\rm Cep}$ and the rejection of the KBC void—must be externally verifiable. All scripts, exact data inputs, covariance mappings, and analysis outputs (including a known-answer test suite with injected synthetic signals validating the regression pipeline) required to reproduce the headline likelihood and $\kappa_{\rm Cep}$ calculations are archived with the project and available for independent reproduction.

## References

Riess, A. G., Anand, G. S., Yuan, W., et al. (2024a). "JWST Observations Reject Unrecognized Crowding of Cepheid Photometry as an Explanation for the Hubble Tension at $8\sigma$ Confidence." *Astrophysical Journal Letters*, 962, L17. JWST observations of $>1000$ Cepheids validating HST photometry; crowding rejected at $8.2\sigma$.

Riess, A. G., Anand, G. S., Yuan, W., et al. (2024b). "JWST Validates HST Distance Measurements: Selection of Supernova Subsample Explains Differences in JWST Estimates of Local $H_0$." *Astrophysical Journal*, 977, 120. Linearity of HST Cepheid distances at $0.994 \pm 0.010$; distance-dependent bias ruled out at $\sim 7\sigma$.

Freedman, W. L., Madore, B. F., Hoyt, T. J., et al. (2019). "The Carnegie-Chicago Hubble Program. VIII. An Independent Determination of the Hubble Constant Based on the Tip of the Red Giant Branch." *Astrophysical Journal*, 882, 34. CCHP TRGB calibration yielding $H_0 = 69.8 \pm 0.8\ ({\rm stat}) \pm 1.7\ ({\rm sys})\ {\rm km\,s^{-1}\,Mpc^{-1}}$.

Freedman, W. L., Madore, B. F., Hoyt, T. J., et al. (2025). "Status Report on the Chicago-Carnegie Hubble Program (CCHP): Measurement of the Hubble Constant Using the Hubble and James Webb Space Telescopes." *Astrophysical Journal*, 985, 203. CCHP TRGB and JAGB calibrations yielding $H_0 = 68.81 \pm 1.79\ ({\rm stat}) \pm 1.32\ ({\rm sys})\ {\rm km\,s^{-1}\,Mpc^{-1}}$ (TRGB) and $H_0 = 67.80 \pm 2.17\ ({\rm stat}) \pm 1.64\ ({\rm sys})\ {\rm km\,s^{-1}\,Mpc^{-1}}$ (JAGB).

Mazurenko, S., Banik, I., Kroupa, P., & Haslbauer, M. (2024). "A simultaneous solution to the Hubble tension and observed bulk flow within $250\ h^{-1}\ {\rm Mpc}$." *Monthly Notices of the Royal Astronomical Society*, 527, 4388&ndash;4396. MOND-driven KBC void model; simultaneous resolution of Hubble tension and CosmicFlows-4 bulk flow; $1.14\%$ significance level with external field $0.055\,a_0$.

Haslbauer, M., Banik, I., & Kroupa, P. (2020). "The KBC void and Hubble tension contradict ΛCDM on a Gpc scale—Milgromian dynamics as a possible solution." *Monthly Notices of the Royal Astronomical Society*, 499, 2845&ndash;2883. Mathematical demonstration that KBC void outflows inflate apparent redshifts by $\sim 9\%$; joint $7.09\sigma$ tension with ΛCDM.

Watkins, R., Allen, T., Bradford, C. J., et al. (2023). "Analysing the large-scale bulk flow using cosmicflows4: increasing tension with the standard cosmological model." *Monthly Notices of the Royal Astronomical Society*, 524, 1885&ndash;1892. $4.6\sigma$ bulk-flow tension with ΛCDM at $200\ h^{-1}\ {\rm Mpc}$.

Stiskalek, J., Desmond, H., & Banik, I. (2025). "Testing the local supervoid solution to the Hubble tension with direct distance tracers." *Monthly Notices of the Royal Astronomical Society*, 543, 1556&ndash;1573. Direct CF4 galaxy-by-galaxy fits find preferred void sizes $\lesssim 70$ Mpc; Gaussian and Maxwell–Boltzmann cases give $H_0 \sim 70.4$, $70.2$; exponential disfavoured by Bayesian evidence. Notes a misunderstanding in the earlier M24 bulk-flow analysis.

Keenan, R. C., Barger, A. J., & Cowie, L. L. (2013). "Evidence for a $\sim 300$ Mpc Scale Under-density in the Local Galaxy Distribution." *Astrophysical Journal*, 775, 62. Observational evidence for the KBC void: $\delta = 0.46 \pm 0.06$ underdensity extending 40&ndash;300 Mpc.

Vagnozzi, S. (2023). "Seven Hints That Early-Time New Physics Alone Is Not Sufficient to Solve the Hubble Tension." *Universe*, 9, 393. Seven distinct reasons why early-time solutions to the Hubble tension are not viable.

Planck Collaboration (2020). "Planck 2018 Results VI. Cosmological Parameters." *Astronomy & Astrophysics*, 641, A6. CMB inference of $H_0 = 67.4 \pm 0.5\ {\rm km\,s^{-1}\,Mpc^{-1}}$.

Smawfield, M. L. (2025). "Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed." Preprint v0.16 (Jakarta), Zenodo DOI: 10.5281/zenodo.16921911. Foundational TEP framework; scalar-tensor action; disformal metric transformation; synchronization holonomy.

Smawfield, M. L. (2025). "Global Time Echoes: Distance-Structured Correlations in GNSS Clocks." Preprint, Zenodo DOI: 10.5281/zenodo.17127229. (TEP-GNSS, Paper 1). GNSS clock-network test of residual synchronization holonomy at the $10^{-18}$–$10^{-19}$ fractional level.

Smawfield, M. L. (2025). "Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products." Preprint, Zenodo DOI: 10.5281/zenodo.17517141. (TEP-GNSS-II, Paper 2). Quarter-century GNSS clock correlation analysis.

Smawfield, M. L. (2026). "Global Time Echoes: Raw RINEX Consistency Test." Preprint, Zenodo DOI: 10.5281/zenodo.17860166. (TEP-GNSS-RINEX, Paper 3). Raw RINEX-level consistency validation of GNSS clock correlations.

Smawfield, M. L. (2026). "The Cepheid Bias: Resolving the Hubble Tension." Preprint, Zenodo DOI: 10.5281/zenodo.18209702. (TEP-H0, Paper 11). Companion paper: 37-host endpoint likelihood; M31 and LMC differential tests; unified $H_0 = 67.83 \pm 1.56\ {\rm km\,s^{-1}\,Mpc^{-1}}$.

Smawfield, M. L. (2026). "What Do Precision Tests of General Relativity Actually Measure?" Preprint, Zenodo DOI: 10.5281/zenodo.18109760. (TEP-EXP, Paper 9). Measurement taxonomy; screening regimes; proximity-regime screening transition.

Smawfield, M. L. (2026). "Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies." Preprint, Zenodo DOI: 10.5281/zenodo.19000827. (TEP-JWST, Paper 12). Temporal shear effects on high-redshift galaxy ages and masses.

Smawfield, M. L. (2026). "The Mount Wilson Paradigm: Restoring the Eternal Universe via the Temporal Equivalence Principle." Preprint, Zenodo DOI: 10.5281/zenodo.21954257. (TEP-HUB, Paper 30). Cosmological redshift as conformal temporal shear; TEP distance-redshift relation on a static spatial manifold.

Smawfield, M. L. (2026). "Temporal Equivalence Principle: Dynamical Proper Time and the Illusion of Primordial Deuterium." Preprint, Zenodo DOI: 10.5281/zenodo.21841147. (TEP-BBN, Paper 29). Deuterium isotope identifiability; temporal transport framework for primordial abundance observables.

Kodric, M., Riffeser, A., Hopp, U., et al. (2018). "Cepheids in M31: The PAndromeda Cepheid Sample." *Astronomical Journal*, 156, 130. Pan-STARRS Cepheid catalog of M31: 2,686 variables used in the M31 internal radial gradient analysis.

Soszyński, I., Udalski, A., Szymański, M. K., et al. (2015). "OGLE-IV Cepheids in the Large Magellanic Cloud." *Acta Astronomica*, 65, 297&ndash;312. OGLE-IV fundamental-mode Cepheid catalog of the LMC used in the LMC radial stratification analysis.

Garnavich, P., Wood, C. M., Milne, P., Jensen, J. B., Blakeslee, J. P., et al. (2023). "Connecting Infrared Surface Brightness Fluctuation Distances to Type Ia Supernova Hosts: Testing the Top Rung of the Distance Ladder." *Astrophysical Journal*, 953, 35. IR SBF calibration yielding $H_0 = 74.6 \pm 0.9\ ({\rm stat}) \pm 2.7\ ({\rm sys})\ {\rm km\,s^{-1}\,Mpc^{-1}}$; uses Jensen et al. SBF distances as an intermediate step between Cepheids and SNe Ia.

Milgrom, M. (1983). "A Modification of the Newtonian Dynamics as a Possible Alternative to the Hidden Mass Hypothesis." *Astrophysical Journal*, 270, 365. Foundational MOND paper; acceleration scale $a_0 \approx 1.2 \times 10^{-10}\ {\rm m\,s^{-2}}$.

Langlois, D. (2019). "Degenerate Higher-Order Scalar-Tensor (DHOST) Theories." *International Journal of Modern Physics D*, 28, 1942006. Ghost-free scalar-tensor constructions via disformal metric transformations; Ostrogradsky instability avoidance.

Riess, A. G., Yuan, W., Macri, L. M., et al. (2022). "A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km s$^{-1}$ Mpc$^{-1}$ Uncertainty." *Astrophysical Journal Letters*, 934, L7. SH0ES 42-SN Ia Cepheid calibration: $H_0 = 73.0 \pm 1.0\ {\rm km\,s^{-1}\,Mpc^{-1}}$.

Scolnic, D., Brout, D., Carr, A., et al. (2022). "The Pantheon+ Analysis: The Full Data Set and Light-Curve Release." *Astrophysical Journal*, 938, 113. Pantheon+ supernova distance-redshift catalog.

Tully, R. B., Kourkchi, E., Courtois, H. M., et al. (2023). "Cosmicflows-4." *Astrophysical Journal*, 944, 94. CosmicFlows-4 peculiar-velocity catalog; 55,877 galaxies in 38,065 groups; the present analysis uses 38,053 groups from the VizieR table4.dat release after quality cuts.

Courtois, H. M., Tully, R. B., Fisher, J. R., et al. (2009). "The Extragalactic Distance Database: All Digital HI Profile Catalog." *Astronomical Journal*, 138, 1938. Homogenized HI profile linewidths and systemic velocities for $\sim$23,000 galaxies; the linewidth source for the Tully-Fisher component of Cosmicflows-4 and part of the measured-$V_{\rm rot}$ union catalogue.

Kourkchi, E., Tully, R. B., Neill, J. D., et al. (2019). "Global Attenuation in Spiral Galaxies in Optical and Infrared Bands." *Astrophysical Journal*, 884, 82. Inclination-corrected HI linewidths ($\log W_{\rm mx}^i$) for the Cosmicflows-4 spiral calibration sample.

Dupuy, A., Courtois, H. M., Guinet, D., et al. (2021). "Towards Cosmicflows-4: The HI Data Catalog." *Astronomy & Astrophysics*, 646, A113. HI linewidths, inclinations and systemic velocities feeding the Cosmicflows-4 distance scale.

Hong, T., Staveley-Smith, L., Masters, K. L., et al. (2019). "2MTF — VII. 2MASS Tully-Fisher Survey Final Data Release: Distances for 2062 Nearby Spiral Galaxies." *Monthly Notices of the Royal Astronomical Society*, 487, 2061. Inclination-corrected HI linewidths from the 2MTF survey.

Haynes, M. P., Giovanelli, R., Kent, B. R., et al. (2018). "The ALFALFA Extragalactic HI Source Catalog." *Astrophysical Journal*, 861, 49. The $\alpha$.100 blind HI survey linewidths.

Lelli, F., McGaugh, S. S., Schombert, J. M. (2016). "SPARC: Spitzer Photometry and Accurate Rotation Curves." *Astronomical Journal*, 152, 157. Resolved rotation curves for 175 disk galaxies.

Riess, A. G., Li, S., Anand, G. S., et al. (2025). "The Perfect Host: JWST Cepheid Observations in a Background-free Type Ia Supernova Host Confirm No Bias in Hubble-constant Measurements." *Astrophysical Journal Letters*, 992, L34. JWST Cycle 2 Cepheid observations in NGC 3447; combined HST+JWST calibration yields $H_0 = 73.49 \pm 0.93\ {\rm km\,s^{-1}\,Mpc^{-1}}$; HST-JWST difference consistent with zero.

Mazurenko, S., Banik, I., & Kroupa, P. (2025). "The redshift dependence of the inferred $H_0$ in a local void solution to the Hubble tension." *Monthly Notices of the Royal Astronomical Society*, 536, 3232&ndash;3241. DOI: 10.1093/mnras/stae2758. Method-3 $H_0(z)$ predictions for Gaussian, Exponential, and Maxwell&ndash;Boltzmann void density profiles from HBK20 parameters; Gaussian and Exponential profiles converge to within $1\sigma$ of Planck only at $z \gtrsim 1.8$.

Jia, X. D., Hu, J. P., & Wang, F. Y. (2023). "Evidence of a decreasing trend for the Hubble constant." *Astronomy & Astrophysics*, 674, A45. DOI: 10.1051/0004-6361/202346356. Non-parametric piecewise $H_{0,z_i}$ reconstruction from Pantheon+ combined with $H(z)$ and BAO data; declining $H_0(z)$ at $5.6\sigma$ (equal-width binning). The decline does not survive the piecewise $H_{\rm th}$ estimator with Pantheon+ SN-only data, regardless of error treatment (diagonal, full $1701 \times 1701$ STAT+SYS covariance, or full covariance with marginalized zero-point); it is driven by the $H(z)$ and BAO data.

Kenworthy, W. D., Scolnic, D., & Riess, A. G. (2019). "The Local Perspective on the Hubble Tension: Local Structure Does Not Impact Measurement of the Hubble Constant." *Astrophysical Journal*, 875, 145. DOI: 10.3847/1538-4357/ab0ebf. Used the Pantheon SN Ia sample to search for a local void signal in the Hubble flow; ruled out a local underdensity as the cause of the Hubble tension. This aligns with an independent analysis of the same dataset family.

Wang, D. (2023). "Pantheon+ Tomography and Hubble Tension." *European Physical Journal C*, 83, 813. DOI: 10.1140/epjc/s10052-023-11991-0. Independent Pantheon+ tomographic analysis finding no obvious evidence for evolution of $H_0$ or $\Omega_m$ at the $2\sigma$ level using both equal-redshift and equal-count binning, consistent with a flat $H_0(z)$ profile.

Jensen, J. B., Blakeslee, J. P., Cantiello, M., et al. (2025). "The TRGB-SBF Project. III. Refining the HST Surface Brightness Fluctuation Distance Scale Calibration with JWST." *Astrophysical Journal*, 987, 87. Cepheid-independent SBF calibration using JWST TRGB distances anchored to NGC 4258, yielding $H_0 = 73.8 \pm 0.7\ ({\rm stat}) \pm 2.3\ ({\rm sys})\ {\rm km\,s^{-1}\,Mpc^{-1}}$. Provides a clean-anchor test that motivates the two-channel TEP interpretation: the Cepheid channel is absent while the SN stretch channel remains.

Freedman, W. L., Madore, B. F., Gibson, B. K., et al. (2001). "Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant." *Astrophysical Journal*, 553, 47. HST Key Project Cepheid calibration yielding $H_0 = 72 \pm 8\ {\rm km\,s^{-1}\,Mpc^{-1}}$; historical reference for the pre-SH0ES Cepheid distance scale.

Tully, R. B. (2015). "Galaxy Groups: A 2MASS Catalog." *Astronomical Journal*, 149, 171. Group catalog used for the richness-based screening criterion applied to the TEP-H0 Cepheid host sample (Step 03).

Jensen, J. B., Blakeslee, J. P., Ma, C., et al. (2015). "The Carnegie-Chicago Hubble Program. III. The Calibration of the Tip of the Red Giant Branch and the Surface Brightness Fluctuation Distance Scales in the Near-Infrared." *Astrophysical Journal*, 808, 91. WFC3/IR SBF calibration distances used as an intermediate step in the Garnavich et al. (2023) SBF $H_0$ determination.

Tonry, J. L., Dressler, A., Blakeslee, J. P., et al. (2000). "The SBF Survey of Galaxy Distances. IV. SBF Distances, Velocity Distances, and Hubble Constants for 300 Galaxies." *Astrophysical Journal*, 530, 625. Establishment of the SBF–Cepheid zero-point calibration using six spiral galaxies with both SBF and Cepheid distance measurements.

Blakeslee, J. P., Ajhar, E. A., & Tonry, J. L. (2002). "The SBF Survey of Galaxy Distances. VI. Calibration of the SBF $I$-band Method and Hubble Constant." *Astrophysical Journal*, 563, 714. Revision of the SBF zero-point calibration by $+0.06$ mag using final HST Key Project Cepheid distances.

## Data Availability and Reproducibility

### Open Science Commitment

This preprint is released under CC and MIT licenses. All materials required to reproduce the analysis—scripts, data references, and the manuscript source—are openly available. Feedback and collaboration are welcome.

### Repository Structure

The complete project is hosted at [https://github.com/matthewsmawfield/TEP-VOID](https://github.com/matthewsmawfield/TEP-VOID) and is structured as follows:

- *site/components/*—HTML source files for each manuscript section. These are the only files that should be edited; all other formats are auto-generated.

- *scripts/steps/*—Reproducible analysis pipeline with 48 registered step scripts organized into six blocks (data ingestion, indicator divergence and void falsification, void boundary test, TEP reconstruction / synthesis, auxiliary TEP supporting tests, and bulk-flow estimator audit / radial discriminators).

- *core/*—Shared TEP framework Python modules (scalar field, conformal scaling, screening, cosmology, constants).

- *scripts/*—Utility scripts including the pipeline runner (`run_pipeline.py`) and PDF generation (`generate_site_pdf.py`).

- *results/*—Generated outputs and figures (populated by running the pipeline).

### Reproducing the Analysis

The full analysis pipeline can be reproduced by running:

```
cd /path/to/TEP-VOID
python3 scripts/run_pipeline.py
```

This executes all 48 registered step scripts in sequence, populating `results/figures/` and `results/outputs/` with fresh data. Individual blocks or steps can be run selectively:

```
python3 scripts/run_pipeline.py --block I      # Indicator divergence only
python3 scripts/run_pipeline.py --step 40       # Single step only
```

The pipeline is fully implemented and all 49 registered steps run successfully against real published data. The pipeline structure, data sources, and analysis methodology are documented in `scripts/README.md`.

### Building the Manuscript

The manuscript site is built from the HTML component files:

```
cd site
npm install
npm run build
```

This generates the static site in `site/dist/` and the markdown version at the repository root (`G31-TEP-VOID-v0.4-Valencia.md`). A PDF can be generated from the built site:

```
python3 scripts/generate_site_pdf.py
```

### Revision Notes

The headline $\Delta$AIC values reported in this version ($+101.5$/$+117.1$ at $z \ge 0.05$; $+194.3$/$+328.7$ full sample) differ from earlier preprint versions ($+91.5$/$+106.9$; $+179.4$/$+310.3$). The shift arises from a correction to the covariance submatrix extraction in step_32: the earlier version used an approximate row-index mapping that mismatched a subset of Pantheon+ rows near the $z = 0.05$ boundary. The corrected extraction uses the exact CID-based row mapping from the Pantheon+ covariance matrix, retaining all 1,053 rows in the $z \ge 0.05$ range with the matched $1{,}053 \times 1{,}053$ submatrix. The shift strengthens the rejection; the conclusion is unchanged.

### Data Sources

The analysis draws on the following publicly available datasets:

- *SH0ES Cepheid + SN Ia host sample* (Riess et al. 2022, 2024a, 2024b): 42 local SNe Ia calibrated by Cepheids, with JWST validation from programs GO-1685 and GO-1995.

- *CCHP TRGB and JAGB calibrations* (Freedman et al. 2019, 2025): HST and JWST-based TRGB and JAGB distance measurements in SN Ia hosts.

- *CosmicFlows-4 catalog* (Tully et al. 2023; Watkins et al. 2023): 55,877 galaxy distances and 38,053 group distances/peculiar velocities.

- *Pantheon+ supernova catalog* (Scolnic et al. 2022): 1,543 unique SN Ia distance-redshift measurements after deduplication by CID (raw catalog: 1,701 entries).

- *HyperLEDA and RC3 galaxy parameters*: Rotation velocities, velocity dispersions, and photometric parameters for host galaxy potential characterization.

- *M31 Pan-STARRS Cepheid catalog* (Kodric et al. 2018): 2,686 Cepheid variables in M31.

- *OGLE-IV LMC Cepheid survey* (Soszyński et al. 2015): Fundamental-mode Cepheids in the LMC.

- *HST PHAT (Panchromatic Hubble Andromeda Treasury)*: High-resolution spatial matching for M31 crowding controls.

### Companion Papers

This paper is part of the TEP Research Series. The quantitative Cepheid bias analysis (37-host endpoint likelihood, M31 and LMC differential tests, unified $H_0$ reconstruction) is developed in detail in the companion paper TEP-H0 (Paper 11), which should be consulted for the full statistical methodology and numerical results. The measurement taxonomy and screening regime analysis are developed in TEP-EXP (Paper 9). The JWST high-redshift anomalies are addressed in TEP-JWST (Paper 12).

### Citation

If you use this code or data, please cite:

```
@misc{smawfield2026void,
  title        = {Cosmological Voids versus Temporal Shear:
                  An Empirical Falsification of Kinematic
                  Hubble Tension Solutions},
  author       = {Smawfield, Matthew Lukin},
  year         = {2026},
  doi          = {10.5281/zenodo.22150139},
  url          = {https://doi.org/10.5281/zenodo.22150139},
  note         = {Preprint, Version v0.4 (Valencia)}
}
```

### Contact

*Matthew Lukin Smawfield*
Email: matthew@mlsmawfield.com
ORCID: [0009-0003-8219-3159](https://orcid.org/0009-0003-8219-3159)
Website: [https://mlsmawfield.com/tep/void/](https://mlsmawfield.com/tep/void/)

---

*This document was automatically generated from the TEP-VOID research site. For the interactive version with figures and enhanced formatting, visit: https://mlsmawfield.com/tep/void/*

*Related Work:*
- [TEP Theory](https://doi.org/10.5281/zenodo.16921911) (Foundational framework)
- [TEP-GNSS: Distance-Structured Correlations in GNSS Clocks](https://doi.org/10.5281/zenodo.17127229) (Paper 1 — GNSS clock-network test of synchronization holonomy)
- [TEP-GNSS-II: 25-Year Analysis of CODE Precise Clock Products](https://doi.org/10.5281/zenodo.17517141) (Paper 2 — quarter-century GNSS clock correlations)
- [TEP-GNSS-RINEX: Raw RINEX Consistency Test](https://doi.org/10.5281/zenodo.17860166) (Paper 3 — raw RINEX-level consistency validation)
- [TEP-H0: The Cepheid Bias](https://doi.org/10.5281/zenodo.18209702) (Companion paper — quantitative Cepheid bias analysis)
- [TEP-EXP: Precision Tests of GR](https://doi.org/10.5281/zenodo.18109760) (Measurement taxonomy)
- [TEP-JWST: JWST High-Redshift Anomalies](https://doi.org/10.5281/zenodo.19000827) (High-redshift galaxy ages and masses)
- [TEP-HUB: The Mount Wilson Paradigm](https://doi.org/10.5281/zenodo.21954257) (Cosmological redshift as temporal shear)
- [TEP-BBN: Primordial Deuterium](https://doi.org/10.5281/zenodo.21841147) (Deuterium isotope identifiability)

*Source code available at: https://github.com/matthewsmawfield/TEP-VOID*

---

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# Temporal Equivalence Principle: A Proper-Time Domain Boundary in the Outer Solar System
**Matthew Lukin Smawfield**
Version: v0.2 (Addis Ababa)
First published: 30 September 2026
Last updated: 17 October 2026
DOI: 10.5281/zenodo.22858190

---

## Abstract

Adopting the Temporal Equivalence Principle (TEP) field framework established in the companion corpus, this paper tests an outer-solar-system realization: a localized proper-time domain boundary organizing detached trans-Neptunian residents and long-period comet transits about the same 60° sector of inertial sky. The two populations probe complementary secular and crossing responses.

The detached resident population clusters in perihelion longitude ($R_\varpi =$ 0.332, conditioned-null $p =$ 0.00735), and a Keplerian discovery-selection model retains a directional excess on the 52 class-1 objects within its stated visibility limits ($p =$ 0.0092). In the transit population, the primary coordinate-level observable — spatial aphelion anisotropy computed directly from osculating elements with no three-leg reconstruction — leans toward the shared sector across both historical and modern eras ($d_\parallel = +0.110$, $p < 10^{-4}$ pre-2018; $d_\parallel =$ +0.0949, $p =$ 0.00215 post-2017; Fisher-combined $p = 1.8\times10^{-6}$; full CometEls census $d_\parallel = +$0.10, $p =$ 2×10<sup>-3</sup>), remaining flat across all orbit-quality and arc-length strata. As a secondary, solver-sensitive observable, the CODE three-leg transit record carries a direction-dependent orbit-reconstruction rotation ($p =$ 0.00295), retained after planetary-encounter and observing-leverage controls. Independent two-leg fits of raw Minor Planet Center astrometry for 585 pre-2018 comets reproduce catalogue rotations ($\rho =$ +0.998) and retain the inbound-leg excess ($p =$ 0.0370; CODE-overlap $p =$ 0.00688), with no resolved in-cap energy excess. Under crossing-localized transport, the angular residual corresponds to an equivalent-time contrast of $+$4.9 yr; this is a model-conditioned conversion rather than a direct clock reading.

A ten-channel synthesis across six catalogue lineages gives $S_{10} =$ 186.2 at the resident axis, with conditional sky-area rank 0.0145 on the fixed observed catalogues. This measures directional convergence; it is not a global discovery probability. While the coordinate-level aphelion dipole persists unchanged in the post-2017 holdout, the estimator-dependent reconstruction channel reverses its declared-axis carrier and exhibits a displaced residual structure under short-arc solver absorption, restricting the simple stationary three-leg interpretation. Point-mass, tidal, drag and reduction explanations are tested through quantitative amplitude, localization and transfer-function controls. The canonical disformal benchmark specifies the required field excursion using actual conic entry and exit geometry; its source profile and full metric-to-observation response remain to be derived.

The results provide mutually reinforcing evidence for a shared direction-organized outer-system structure; its interpretation as an outer-system TEP domain boundary remains model-conditioned and testable through the prospective discoveries registered below. The registered prospective test predicts 59% of new detached objects in the cap, against a 42% characterized-footprint expectation, requiring approximately 123 discoveries for 95% power at a 1% test. This separates the present evidence from the next key validation.

Key words: Kuiper belt: general – comets: general – gravitation – celestial mechanics – astrometry

## 1. Introduction

## 1.1 The extreme-TNO clustering anomaly and the point-mass consensus

The orbits of the most distant known trans-Neptunian objects (TNOs) are not isotropically distributed. The longitudes of perihelion $\varpi = \Omega + \omega$ of detached, high-perihelion bodies cluster about a common direction in inertial space, a fact first emphasized by Trujillo & Sheppard (2014) and developed into the Planet Nine hypothesis by Batygin & Brown (2016). Under that hypothesis the clustering is secular shepherding: an unseen $\sim 5$–$10\,M_\oplus$ perturber on a wide, inclined orbit organizes the perihelia of the extreme population into an anti-aligned configuration. A decade of wide-field searches has not detected the proposed body — Pan-STARRS1 together with the ZTF and DES limits excludes approximately 78 per cent of the nominal parameter volume (Brown, Holman & Batygin 2024) — while independent orbital fits continue to shift the preferred solution (Siraj, Chyba & Tremaine 2025) and the newest high-perihelion discoveries complicate rather than confirm the clustering picture (Chen et al. 2025).

The shepherding interpretation carries mechanical obligations that are testable in their own right. A point mass acting through secular gravity should (i) deposit some fraction of the clustered population on or near mean-motion resonances, (ii) couple the alignment direction to physical orbital elements — semimajor axis, perihelion distance, eccentricity, and inclination — because the secular forcing frequencies depend on them, and (iii) imprint its signature on any tracer population that crosses the same volume, since gravity is blind to dynamical class. The analysis reported here finds none of these signatures in the data.

## 1.2 The Temporal Equivalence Principle alternative

The dynamical proper-time field formulated in Paper 0 and investigated in the companion studies is the foundation adopted here. TEP-9 tests its proposed outer-Solar-System domain-boundary realization: the question is which admissible field profile and observation response reproduce the orbital signatures, rather than a new survey of field existence.

The Temporal Equivalence Principle (TEP), developed in Paper 0 of this series, posits that proper time is governed by a dynamical scalar field $\phi$ rather than being a passive coordinate. Matter couples universally to the canonical causal matter metric $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$; Paper 0, §2.2, Step 80), which in the conformal–disformal shorthand is written $\tilde{g}_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi$: the conformal factor $A(\phi)$ sets the local rate of proper-time accumulation, and the disformal term $B(\phi)$ weights that rate along the field gradient. Spatial structure in $\phi$ therefore produces macroscopic, measurable consequences for orbital dynamics.

Within this framework the outer solar system anomaly is modelled not as a body but as spatial structure in the proper-time field — a localized domain boundary. In the language of Paper 0 it is a steep but continuous screening transition of the solar Temporal Topology, across which the rate of proper-time accumulation departs from the standard Newtonian assumption. In the proposed boundary model, crossing the transition produces an accumulated orbital-phase displacement without requiring a net kinetic-energy change between asymptotically equivalent entry and exit regions; local conservative acceleration can occur during the crossing, while the non-integrable transport response must be computed from the full conformal–disformal connection. The Temporal Shear $\Sigma_\mu = \nabla_\mu \ln A$ specifies the conformal field gradient. Its integral is an endpoint term, so non-integrable synchronization transport must be calculated from the full connection, including the disformal contribution. The canonical benchmark and its regime of validity are tested in Section 6.7; the angular-to-time conversion and orbit-refit injections specify the observation response separately. Because time accumulates at a different rate across the boundary, a comet's inbound and outbound trajectories will mathematically disagree when reconstructed by standard software. That failure of forward and backward transport through the transition to close — the orbit-reconstruction discrepancy the transit channel measures — is a synchronization-holonomy-class observable: the signature of dynamical time defined in Paper 0, registered on celestial rather than laboratory transport. Its invariance status is stated precisely: the closed-loop theorem of Paper 0 covers closed spatial circuits of the synchronization connection, while the comet channel is an open-path, fit-space non-closure whose convention-dependence the analysis itself exploits (a single joint fit absorbs the offset into the elements); the loop-invariant version of the same observable requires a genuinely closed transport circuit, such as the triangle time-transfer experiment of Paper 0. The static reading is adopted as primary: the era dependence of the measured carrier is attributed to arc-length-dependent solver absorption — demonstrated experimentally in Section 4.15 — rather than to motion of the field itself, while the evolving-boundary alternative is registered as an open hypothesis requiring a dynamical driver the corpus does not currently supply.

At the level of the single scalar degree of freedom, the framework's structure is that of an environmentally screened gradient field. Rather than an algebraic density-dependent potential minimum (as in standard chameleon theories (Khoury & Weltman 2004; Burrage & Sakstein 2018), whose thin-shell mechanism is bypassed here), screening in TEP operates through the saturation of the Temporal Shear $\Sigma_\mu \equiv \nabla_\mu \ln A(\phi)$ within local potential wells, governed by the environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$. A domain boundary arises where the local heliocentric scalar gradient $\nabla\ln A_\odot$ attenuates to the ambient Galactic and cosmological background shear ($a_{\rm amb} \approx 3.9 \times 10^{-10}$ m s$^{-2}$ at the solar circle, Paper 26; distinct from the horizon-anchored suppression scale $g_t \approx 3.4 \times 10^{-10}$ m s$^{-2}$ used in Paper 0). On physical grounds the transition is expected in the outer solar system, in the broad neighbourhood of the solar-wind ram-pressure boundary ($\sim 100$–$150$ AU), where the Sun's gradient kinematically decouples from the interstellar background; the forward-model localization reported below places the best-fit crossing at the outer edge of its tested range, $\sim 250$ AU — a lower bound pending extension of the injection scan — so the heliospheric coincidence is a noted proximity rather than a fitted anchor. The anchoring is therefore to the continuous transition of the heliospheric Temporal Topology, not to contemporaneous local plasma fluctuations — a distinction Section 4.14 tests directly. Deriving the potential $V(\phi)$ and the coupling functions is the task of Paper 0 — where the quartic density-dependent-mass branch is evaluated as a candidate completion rather than assumed — and the present work's contribution is the measurement programme: whether the orbital record carries the signature such a transition must produce.

#### Key terminology

- *Domain boundary*: a localized region across which the proper-time field $\phi$ changes appreciably — the steep screening-transition region of the solar Temporal Topology (Paper 0) in the outer solar system.

- *Clock-sector elements*: the angular orbital elements ($\Omega$, $\omega$, $\varpi$) that encode where a body is in its cycle; they are the components a proper-time perturbation touches directly.

- *Resident population*: orbits that remain in the boundary region for secular times (detached extreme TNOs).

- *Transit population*: orbits that plunge through the boundary on a single pass (long-period Oort-spike comets).

- *Orbit-reconstruction discrepancy*: the mismatch between osculating, original (backward-integrated), and future (forward-integrated) orbit solutions published for the same comet.

The physical distinction between the two hypotheses is sharp. A point mass transports momentum: it rotates orbital elements through a calculable, element-coupled torque whose amplitude is set by the mass and the geometry and bounded by surveys and ephemerides. A lapse boundary transports phase: a trajectory that crosses it accumulates proper time at a perturbed rate, so a reconstruction that assumes the standard clock mis-assigns the object's orbital phase — an error that appears in the clock-sector elements without the corresponding energy exchange. The boundary therefore predicts a rotation-without-energy signature, threshold rather than force-law morphology, and population selectivity by residency rather than by size.

## 1.3 The resident-versus-transit strategy

This work tests the domain-boundary hypothesis with two dynamically distinct populations that probe the structure differently. Detached extreme TNOs are *residents*: their orbits never dip into the planetary region, so their clock-sector elements integrate the secular influence of whatever occupies the outer system over Gyr timescales. Long-period comets are *transiters*: each falls from the Oort spike through the full radial structure once, and the catalogued discrepancy between its osculating, original, and future orbit solutions records how well the standard-dynamics clock survives the crossing. A third, injected population — Jupiter-family comets driven inward through the same volume — supplies a directional control.

The plan of the paper is as follows. Section 2 defines the catalogues and selection criteria. Section 3 establishes the resident signature: the extreme-TNO alignment, the measured discovery-footprint null, and the failure of point-mass discriminators. Section 4 presents the transit signature: the cometary orbit-reconstruction discrepancy localized to the same sky sector, its rotation-without-energy decomposition, and its survival after regression of the planetary baseline. Section 5 demonstrates that the two populations converge on a single structure. Section 6 tests and rejects each conventional alternative in turn. Section 7 registers the falsifiable prediction that the imminent Rubin/LSST cohort will decide. Section 8 summarizes the findings.

Because the argument crosses several measurement classes, the five physical discriminators that separate a proper-time domain boundary from a Newtonian point mass and from observational artefact are stated at the outset:

- *The amplitude-localization bound* (Sections 4.3, 6.2): secular gravity does rotate orbital elements at nearly flat energy, so the flat $\Delta(1/a)$ channel is not by itself the exclusion — the exclusion is that the residual rotation, measured after regression of the full planetary baseline, exceeds every allowed perturber's amplitude by two to three orders of magnitude while remaining direction-organized, single-leg-localized, and energy-flat.

- *The stratification discriminator* (Sections 4.8, 4.12): on the modern cohort the reconstruction-dependent channel reverses while the spatial aphelion channel persists flat across every quality stratum, and the sensitivity audit (step 150) shows the aphelion channel is the more error-exposed of the two — the dissociation is structural, not noise.

- *The holonomy transfer function* (Section 4.15): the pre-2018 implied slip has no resolved velocity scaling, supporting the tested crossing-localized transport response; amplitude and trajectory modelling are required to distinguish it conclusively from impulse realizations.

- *The CMB-anchored frame* (Sections 4.13, 4.14): the pooled slip map is axisymmetric — same-signed lobes tracking $\cos 2\theta$ — while the two transit eras carry reversed measured polarity about the frame-anchored dipole direction on the CMB-anchored reference frame, an epoch dependence rather than a lobe crossing, with a sharp transition that solar-wind and catalogue-composition controls exclude.

- *Two-way transponder cancellation* (Sections 6.8, 6.9): the endpoint conformal term cancels identically at the coherent turnaround, so the two-way Doppler and ranging links are blind to a static lapse gradient by construction; the onboard-oscillator SCLK record is the clock-carrying channel the theorem exempts, and its measured nulls bound the excursion rather than contradicting the orbital signal.

## 2. Data and catalogues

The analysis uses only published, public orbit catalogues; every reported measurement is computed on real catalogue data, with synthetic realizations entering only the explicitly labelled injection and validation controls. Three independent data lineages are used — the JPL Small-Body Database for resident and injected populations, the Warsaw catalogue of near-parabolic comets, and the independent CODE cometary catalogue — plus the characterized Outer Solar System Origins Survey (OSSOS) ensemble for discovery-bias calibration. Every acquisition is scripted and provenance-stamped (source URL, retrieval timestamp, byte count, and SHA-256 checksum) by pipeline steps 001–004.

## 2.1 The resident population: JPL SBDB outer solar system

The JPL Small-Body Database (SBDB) query API supplies the complete asteroid orbit catalogue with heliocentric semimajor axis $a \geq 30$ AU (7,300 objects at retrieval), the class-CEN Centaur population (1,050 objects), and comets reaching perihelia $q \geq 5$ AU (268 objects). The fields retrieved are designation, $a$, $e$, $i$, $\Omega$, $\omega$, $q$, aphelion, absolute magnitude $H$, element epoch, orbit class, condition code, data arc, observation counts, and first/last observation dates.

The primary resident sample follows the literature convention for the detached population: $a > 150$ AU, $q > 30$ AU, and orbit condition code $\leq 3$ (secure, multi-opposition solutions). This yields $N = 44$ objects, among them 2023 KQ14 ('Ammonite'), the newest Sedna-like body, which enters the secure sample through a 2005 precovery rather than its 2023 discovery arc. Sensitivity to the cuts is mapped throughout: the extreme subsample ($a > 250$ AU; $N = 26$), a relaxed-quality control (all condition codes; $N = 90$), a sednoid subset ($q > 50$ AU; $N = 5$), and a plunging-transit control ($a > 150$ AU, $q \leq 30$ AU; $N = 34$) are analysed in parallel. A further prospective cohort — all objects with $a > 150$ AU, $q > 30$ AU, and designation year 2025 or later, at any condition code ($N = 21$; median condition code 8) — is scored separately against the registered predictions and its own discovery footprint in the newest-cohort audit (Section 7).

## 2.2 The transit population: Warsaw and CODE comet catalogues

The Warsaw catalogue of cometary orbits (Królikowska 2014; VizieR J/A+A/567/A126) provides homogeneous orbit solutions for 119 near-parabolic comets at three epochs: osculating heliocentric elements (table B), original barycentric elements reconstructed backward to 250 AU (table C), and future barycentric elements propagated forward to 250 AU (table D), together with observational material and quality assessment (table A1) and an extended sample (table B4). The three-leg structure — osculating, original, future — is the raw material of the transit test, since the discrepancy between legs measures the failure of a standard-dynamics clock to reproduce a single physical trajectory.

The CODE catalogue (Catalogue of Orbits and their Dynamical Evolution; Królikowska & Dybczyński 2020) provides the same three-leg structure for an independent comet set served by the Poznań comet-dynamics server. In the matched deep-plunger, class-1 regime the independent CODE-only sample contains $N = 54$ comets, 19 of which have original-orbit aphelion directions inside the TNO-defined 60° cap. The CODE acquisition is an archival snapshot of the published catalogue, hash-pinned in `data/raw/code/provenance.json`; the live pad2 export interface did not return a machine-readable table at retrieval, a limitation recorded in the provenance metadata.

A fourth catalogue completes the transit record. The one-apparition comet catalogue (Królikowska et al. 2014; VizieR J/A+A/571/A63) publishes osculating, original, and future barycentric solutions for 51 comets observed at a single apparition, together with formal uncertainties on every element and the corresponding Marsden–Williams (2008) catalogue values. It supplies two channels unavailable elsewhere: a discrepancy statistic normalized by each orbit's own published error budget, and a partial third-lineage comparison against the MW08 determinations.

## 2.3 Bias calibration: the OSSOS characterized ensemble

Discovery-geometry control uses the OSSOS characterized ensemble (Bannister et al. 2018; VizieR J/ApJS/236/18, table t3char): 840 detection events with measured survey pointing, magnitude limits, and detection efficiencies. This is the standard bias-calibration resource in this literature. The measured per-object discovery footprint — each TNO's opposition longitude at discovery — is taken directly from SBDB observation-date metadata rather than modelled, so the footprint null is built from the survey's own data.

## 2.4 Auxiliary catalogues

Three further public resources support the concordance and injection-chain analyses. The Minor Planet Center's full orbit catalogue MPCORB.DAT (1.56 million orbits) recomputes the TNO cluster on MPC orbit solutions and carries the catalogue-lineage replications of Sections 3.6 and 5.1 — the detached-clustering reproduction and the bias-free mean-plane audit. The JPL comet elements file (CometEls.txt) selects the Jupiter-family population by Tisserand class. And the Dark Energy Survey six-year TNO catalogue (Bernardinelli et al. 2022; 814 objects discovered in a contiguous 5000 deg$^2$ of southern sky with independently fitted orbits) serves as the independent resident-lineage cohort in Section 3.6. The N-body planetary baseline additionally uses the JPL DE440s planetary ephemeris (Park et al. 2021; NAIF generic kernel, coverage 1849–2150) for solar-system barycentric planet states at each comet's perihelion epoch.

**Table 1:** Data catalogues and their roles

| Catalogue | Source | Content | Role | Pipeline step |
| --- | --- | --- | --- | --- |
| JPL SBDB | ssd-api.jpl.nasa.gov | 7,300 outer asteroids; 1,050 Centaurs; 268 comets | Resident + injected populations | 001 |
| Warsaw | VizieR J/A+A/567/A126 | 119 near-parabolic comets, 3-leg solutions | Transit signature (primary) | 002 |
| CODE | pad2.astro.amu.edu.pl | Independent comet set, 3-leg solutions | Transit signature (confirmation) | 003 |
| One-apparition | VizieR J/A+A/571/A63 | 51 comets, 3-leg + uncertainties + MW08 | Normalized channel; third lineage | 002 |
| OSSOS | VizieR J/ApJS/236/18 | 840 characterized detections | Discovery-bias calibration | 002 |
| MPCORB | minorplanetcenter.net | Full MPC orbit catalogue (1.56M orbits) | Catalogue concordance; clustering + mean-plane replication | 004 |
| CometEls | ssd.jpl.nasa.gov | JPL comet elements | JFC injection chain | 004 |
| DES TNOs | bernardinelli/des_tno_catalog | 814 southern-sky TNOs, independent orbit fits | Resident-lineage replication | 085 |

A known structural caveat is stated at the outset: the Warsaw, CODE, and one-apparition catalogues share methodological lineage (the same orbit-determination group), so the replications are independent in sample but not in pipeline. A fully independent three-leg lineage does not exist in public form — the JPL SBDB API exposes osculating elements and non-gravitational parameters but not original/future barycentric solutions. The nearest approach is the Marsden–Williams (2008) comparison embedded in the one-apparition catalogue: an independently determined $1/a$ series whose cross-lineage agreement with the Warsaw values can itself be tested for direction dependence (Section 4.7). This limitation is carried explicitly through the interpretation.

The independent refit records retain explicit failure reasons. The integrations use a conservative 0.001-AU close-encounter guard to avoid unresolved point-mass encounters; this is a numerical scope limit rather than a planetary collision radius. In the fresh pre-2018 replay, C/2013 A1 (Siding Spring) reaches that guard and is excluded. The fitted-cohort counts and statistical summaries therefore use only successfully propagated records.

## 3. The resident signature: extreme-TNO spatial alignment

This section establishes the anomaly in the population that lives in the structure. Every statistic is conditioned on the observed geometric elements: each orbit is decomposed into geometric elements ($a$, $e$, $i$) and clock elements ($\Omega$, $\omega$, $\varpi$), and the null holds ($a$, $e$, $i$) fixed while drawing $\Omega$, $\omega$ uniform on the circle — testing clustering precisely in the sector a proper-time perturbation touches, free of the inclination-marginal bias that afflicts naive pole-vector tests.

## 3.1 The clustered anomaly

In the primary detached sample ($a > 150$ AU, $q > 30$ AU, condition code $\leq 3$; $N = 44$), the longitude of perihelion shows Rayleigh $R_\varpi =$ 0.332 about mean direction $\bar{\varpi} = 49.1^\circ$ (68 per cent bootstrap interval $[33.2^\circ, 66.6^\circ]$), against a conditioned-null expectation of $0.134$ — $p =$ 0.00735. The three-dimensional perihelion unit vectors cluster at $R = 0.336$ ($p = 0.0052$) about ecliptic direction $(\lambda, \beta) = (49.9^\circ, -17.0^\circ)$ — the axis that anchors every downstream test. The secure subsample is stable to every tightening of the selection: the numbered-orbit subset gives $R = 0.59$ ($p = 0.0017$), and the pre-2014 discovery-era subsample — objects found before the anomaly was announced — retains the signal at $p = 0.043$.

#### The measured axis

Ecliptic $(\lambda, \beta) \approx (50^\circ, -17^\circ)$; galactic $(l, b) \approx (182^\circ, -44^\circ)$. The anti-axis is $(\lambda, \beta) \approx (229^\circ, +17^\circ)$. All cap tests below use this pre-declared direction and a 60° cap unless stated otherwise.

## 3.2 The empirical null and discovery-footprint calibration

The standard objection to any outer-solar-system clustering claim is discovery bias: surveys point where the ecliptic and the galactic plane allow, and objects are found near opposition, so a naive isotropic null is unfair to the data. The pipeline therefore measures the footprint rather than assuming it. Each detached object's discovery-opposition longitude is read from its observation-date metadata, and the realized footprint is the distribution those directions actually make.

Three results close the bias channel. First, the maximal-pointing ceiling: even under the extreme assumption $\varpi_i = \lambda_{ {\rm opp},i}$ exactly — perfect perihelion-to-opposition coupling — the measured footprint realizes only $R_\varpi = 0.295$, below the observed 0.332; pointing cannot reach the observed amplitude at any coupling strength. Second, the direction mismatch: the footprint's own mean direction is $10.5^\circ$, while the observed cluster sits at $49.1^\circ$ — a $38.6^\circ$ systematic offset that the footprint does not predict at all. Third, the distance-aware conditional null, which convolves each object's footprint with the OSSOS-measured opposition–perihelion coupling, returns $p = 0.035$: the anomaly survives the strongest empirically calibrated bias model. The footprint proxy itself is validated against real discovery coordinates (median error 6°), and even with the OSSOS-measured near-perihelion coupling applied at 100 per cent strength, the realized footprint fails to reach the observed $R$ ($p = 0.021$).

## 3.3 Failure of the shepherding signatures

The point-mass interpretation is tested on its own mechanical terms. The clustered population shows no mean-motion resonance sticking, no coupling of the alignment axis to $a$, $q$, $e$, or $i$, and no physical-element substructure. The orbit poles do show a coherent signal — a mean-plane warp that grows smoothly with semimajor axis — but a smooth radial warp is the signature of a spatially extended gradient, not of a discrete perturber, which tilts its secular eigenplane once. The galactic-tide decomposition likewise finds the anomaly off the tidal band and $m = 1$ (dipolar), where a tide produces $m = 2$ (quadrupolar) structure.

## 3.4 The decorrelation radius

A measured transition locates where the field signature overtakes discovery geometry. For each object, the angular distance to the axis is compared with the distance to its own discovery footprint. Inside $a \sim 50$–$150$ AU the footprint wins — the fraction of objects closer to the axis than to their own pointing direction falls to $\sim 0.23$. Beyond $\sim 150$ AU the fraction roughly doubles to $\sim 0.42$, and does so against the tightest footprint coupling of any bin: in the $a = 150$–$300$ AU bin the median axis distance collapses to $48^\circ$ while the median footprint distance is only $29^\circ$ — the axis gains ground precisely where pointing is strongest. The sednoid subsample — the deepest residents ($N = 5$) — shows $R = 0.43$ with mean $\bar\varpi = 16^\circ$ inside the same sector. The anomaly strengthens at the boundary, exactly where a domain wall predicts it should.

## 3.5 Resident-versus-transit selectivity

The signature is specific to the resident population. Detached orbits align to the axis at $p = 5\times10^{-5}$ in the three-dimensional test; plunging detached orbits ($a > 150$ AU, $q \leq 30$ AU) that transit the inner system are consistent with isotropy ($p = 0.37$). The contrast is itself a domain-boundary prediction: an orbit that plunges through the full radial structure to the planetary region has its clock-sector elements scrambled by the transit, while an orbit that remains resident retains the secular imprint.

## 3.6 Patch scale and catalogue concordance

The structure's intrinsic angular scale is measured by a uniform-plus-von-Mises mixture fit to the $\varpi$ distribution: a clustered fraction $f = 0.43$ (68 per cent interval $[0.32, 0.70]$) in a patch of width $\sigma = 34.4^\circ$ ($[27.5^\circ, 48.0^\circ]$) — a finite sector, not a point. The cap-profile scan peaks at 50° in $\varpi$ ($p = 2\times10^{-4}$, remaining below $10^{-2}$ across the full 30–60° range) and at 40–60° in the three-dimensional perihelion-vector profile, where $p \leq 10^{-4}$ is sustained over 30–60° caps and the in-cap excess exceeds every Monte-Carlo realization of the conditioned null. The cluster also survives a catalogue swap. Recomputed on the Minor Planet Center's own orbit solutions for the identical object list, the alignment is element-for-element identical ($R = 0.332$ at $\bar\varpi = 49.1^\circ$; median $|\Delta\varpi| = 0.003^\circ$), excluding solver and parsing systematics — though since JPL ingests MPC fits, this is a concordance check rather than an independent derivation.

The concordance is therefore carried past the shared object list into a full-population replication on the MPCORB lineage (step 092). The secure MPCORB sample is larger and fresher — 63 detached objects at the same cuts, including 19 that postdate or fall outside the SBDB selection — and it carries the clustering independently: $R_\varpi = 0.243$ against a conditioned-null expectation of $0.112$ ($p = 0.024$), about mean perihelion direction $(45^\circ, -21.1^\circ)$, inside the same sector. Once again the strongest subset is the most securely observed: the 18 numbered-orbit MPCORB objects give $R = 0.592$ ($p = 0.0011$), while dropping the quality cut entirely dilutes the signal into noise ($N = 91$, $R = 0.085$, $p = 0.519$). On the second catalogue as on the first, the clustering lives in the well-measured orbits.

The genuinely independent version of that check now exists (step 085). The Dark Energy Survey six-year catalogue — 814 TNOs discovered in a contiguous southern footprint and orbit-fitted entirely outside the JPL/MPC pipeline (Bernardinelli et al. 2022) — contributes 16 detached objects at the same $a > 150$, $q > 30$ AU cuts. Scored against the pre-declared axis and cap, 13 of the 16 land inside (0.81; $p = 1.2\times10^{-4}$ against uniformity), with cohort resultant $R_\varpi = 0.745$ about a mean $\bar\varpi = 23.3^\circ$ that sits inside the registered $[17^\circ, 88^\circ]$ interval. Because the DES fields point into the axis sector, the cohort's own footprint baseline is elevated (0.55); the measured rate exceeds even that biased expectation ($p = 0.029$) and is consistent with the registered 59-per-cent prediction ($p = 0.079$).

One structural qualification applies: 14 of the 16 objects overlap the SBDB secure sample, so the cohort is an independent-survey, independent-fit lineage on largely the same membership rather than a set of independent objects. On the shared bodies the two pipelines agree to a median $|\Delta\varpi| = 0.25^\circ$, setting the resident-side inter-fitter floor an order of magnitude below the cap scale. The DES-only pair (2013 RC156, 2015 UN105) splits one in, one out — consistent with the pooled rate.

The characterized-survey version of the audit then replaces every proxy with measurement (step 112). The OSSOS t3char ensemble — 840 objects with secure multi-opposition orbits, the survey's own dynamical classification, and the exact discovery astrometry of every object in the same table — is scored class by class against the pre-declared cap, with the footprint baseline built from the measured discovery positions rather than the designation proxy. Two model assumptions are validated directly on the ensemble. First, the discovery-longitude–$\varpi$ coupling is measured, not assumed: the eccentric classes — detached, scattering, centaur, and the resonant population, all discovered preferentially near perihelion — carry coupling resultants $R_d = 0.44$–$0.54$, while the classical belt, discovered at any orbital phase, is flat ($R_d = 0.12$); the class-aware baselines are assigned by the data. Second, the coupling width itself is refitted on the OSSOS detached cohort: $\sigma = 49^\circ$ (95 per cent profile interval $[33^\circ, 75^\circ]$) against the $50^\circ$ SBDB calibration — the footprint model's one free parameter is confirmed on an independent survey.

The detached cohort itself lands 13 of 31 in-cap (0.42) against its own exact-astrometry baseline of 0.40 ($p = 0.49$) — consistent with its selection function. This is the expected selectivity outcome rather than a failed replication: the OSSOS 'det' class is Gladman-detached ($a > 47.7$ AU, $e > 0.24$), a cohort whose median semimajor axis ($\sim 60$ AU) sits inside the $\sim 150$ AU boundary the resident signature occupies — its two $a > 150$ members split one in, one out — and every other class is baseline-consistent as well (scattering 0.45 vs 0.41; resonant 0.40 vs 0.44; classical 0.31 vs $1/3$; centaur 0.56 vs 0.48). The 132 plutinos supply the positive control on the same selection function: their $\varpi$ distribution is structured by Neptune rather than by the axis (median separation from Neptune's apsidal direction $98^\circ$), and Neptune's mean longitude of perihelion itself sits $4.0^\circ$ from the resident axis on the $\varpi$ circle — a registered coincidence ($p \approx 0.02$ a priori). On the shared orbits, the OSSOS and SBDB fits agree to a median $|\Delta\varpi| = 0.40^\circ$ — a third inter-fitter floor.

The three detached cohorts are then scored side by side on one ledger (step 113). SBDB lands 26 of 44 in-cap against its designation-proxy baseline of 0.42 ($p = 0.015$), DES 13 of 16 against 0.55 ($p = 0.029$), OSSOS-det 13 of 31 against its exact-astrometry 0.40 ($p = 0.49$). Fisher combination over the three cohorts gives $p = 0.0098$ against own-footprint baselines — strictly two detections plus one bound, since the OSSOS cohort is interior-dominated — and the deduplicated 76-object union lands 39 in-cap against a pooled baseline of 0.41 ($p = 0.047$); restricted to the boundary-resident subset ($a > 150$ AU) on which the anomaly is defined, the 47-object union lands 27 in-cap ($p = 0.020$).

Decomposed by discovery longitude (step 116), the OSSOS bound itself resolves into an open-channel replication and a closed-channel null: the 17 OSSOS members discovered pointing inside the key window land 13 in-cap (76 per cent, the SBDB secure rate itself), while the 14 discovered through the closed channel return zero — as do all 76 closed-channel objects across every population in the catalogue. The concordant quantity is direction: the in-cap members' mean $\varpi$ is $43.1^\circ$ (SBDB), $27.5^\circ$ (DES) and $44.8^\circ$ (OSSOS) — three independent selections pointing at the same place, with mutual concordance $R = 0.99$ against three directions drawn uniformly inside the cap.

## 3.7 The in-cap property audit

The 26 objects inside the axis cap are audited against the 18 outside it on every available property. They are statistically identical in $a$ ($p = 0.95$), $q$ ($p = 0.49$), $e$ ($p = 0.34$), $i$ ($p = 0.58$), observation arc ($p = 0.99$), and discovery era ($p = 0.84$); the only channels that differ are $\Omega$ and $\omega$ — the cluster itself. Two properties differ in the direction opposite to an artefact: the in-cap objects carry more observations per object (median 50.5 vs 31.5, $p = 0.012$) and a higher numbered-orbit fraction (14/26 vs 4/18, $p = 0.061$). The signal lives in the best-measured orbits — the reverse of what a poor-orbit or selection artefact produces.

Splitting the detached sample by first-observation year (step 078) confirms the era audit directly: cap membership is uncorrelated with discovery epoch ($\rho = -0.086$, $p = 0.58$), the bulk 2005–2015 CCD-era cohort alone carries a significant periapsis resultant ($R = 0.42$, $p = 0.005$), and the in-cap fraction is stable across eras ($0.636$, $0.621$); the post-2015 cohort is too small to test ($n = 4$), a gap the LSST-era sample will close. The synthesis one-map (Figure 2, Section 5.1) places every measured population and reference axis on a single ecliptic projection.

The selection boundary itself is then audited continuously rather than at a handful of alternative cuts (step 111). The detachment plane — $a_{\min} \in \{100, 125, \dots, 250\}$ AU crossed with $q_{\min} \in \{30, 33, 35, 38, 40, 45, 50\}$ AU, all at condition code $\leq 3$ — is scanned cell by cell, recomputing the $\varpi$ resultant under the identical conditioned null together with the recovered three-dimensional perihelion direction. Of the 42 cells retaining at least eight objects, 34 reach $p < 0.05$ individually, and every recovered direction lies within $30^\circ$ of one of the two declared axes — 74 per cent within $30^\circ$ of the detached-sample axis itself.

The direction does not wander randomly with the cut: it drifts smoothly with detachment depth, from $(50^\circ, -17^\circ)$ at the shallow cells toward the cap-declaration direction $(34^\circ, -13^\circ)$ at the deepest, where several cells land within $0.4^\circ$–$12^\circ$ of the cap axis. The declared cell sits inside this coherent plateau, so the measured axis is a property of the detached population and not of one selection boundary — and the two declared directions are bracketing measurements of a single sector whose deepest-detached members converge on the direction the comet channel recovers independently.

The mechanism attribution of the resident alignment is stated plainly rather than implied. The transit channel's mechanism — a crossing-localized non-integrable transport slip — is defined for bodies that traverse the boundary once, and does not apply to residents that never cross it. A secularly accumulating phase slip is likewise excluded as the organizing agent on morphological grounds alone: an unbounded accumulation over Gyr would produce a secular drift of perihelia, not a stationary clustering direction. The admissible field-side candidate is a secular geodesic response of the resident clock-sector elements to the boundary's shear structure — the equilibrium bias a standing axisymmetric lapse gradient imprints on apsidal orientation — whose required amplitude is governed by the same admissibility ledger as the transit excursion of Section 6.7 and is not yet derived from a solved field configuration. The resident alignment therefore stands in this work as an independent spatial datum — the datum the Planet Nine hypothesis was introduced to explain — whose TEP interpretation is open, in parallel with the transit channel's status under the admissible-branch computation of Section 6.7.

**Table 2:** Resident-signature tests (pipeline steps 010–028, 053, 057, 085, 092, 111–113, 116)

| Test | Result | Null/baseline | Source |
| --- | --- | --- | --- |
| $\varpi$ conditioned-null $R$ (detached, $N=44$) | $R =$ 0.332, $p =$ 0.00735 | 0.134 | step_010 |
| 3-D perihelion alignment | $R = 0.336$, $p = 0.0052$ | 0.136 | step_010 |
| 3-D axis test (boundary-resident) | $p = 5\times10^{-5}$ | isotropic | step_024 |
| Plunging detached control | $p = 0.37$ | isotropic | step_024 |
| Maximal-pointing ceiling | $R_{\max}=0.295 < 0.332$ | 100% coupling | step_014 |
| Footprint direction offset | $38.6^\circ$ | footprint mean $10.5^\circ$ | step_014 |
| Footprint-conditional null | $p = 0.035$ | OSSOS-calibrated | step_016 |
| Numbered-orbit subset | $R = 0.59$, $p = 0.0017$ | conditioned null | step_023 |
| Intrinsic patch width | $\sigma=34.4^\circ$ | mixture fit | step_025 |
| MPC catalogue concordance | identical $R$, $\bar\varpi$ | JPL vs MPC | step_053 |
| MPCORB full-population replication | $R = 0.243$, $p = 0.024$, axis $(45^\circ,-21.1^\circ)$; numbered subset $R = 0.592$, $p = 0.0011$ | conditioned null | step_092 |
| In-cap property audit | identical except $\Omega,\omega$; better-observed | KS tests | step_057 |
| DES independent-survey cohort | 13/16 in cap ($p=1.2\times10^{-4}$ vs uniform; $p=0.029$ vs own footprint) | uniform + footprint | step_085 |
| Detachment-cut coherence | 34/42 cells $p<0.05$; 100% of recovered directions within $30^\circ$ of a declared axis | conditioned null | step_111 |
| OSSOS characterized-ensemble audit | 13/31 in cap vs exact-astrometry baseline 0.40 ($p=0.49$); coupling width $\sigma=49^\circ$ refit vs $50^\circ$ calibration; all classes baseline-consistent | exact discovery astrometry | step_112 |
| Three-survey resident ledger | SBDB $p=0.015$, DES $p=0.029$, OSSOS bound; Fisher $p=0.0098$; union 39/76 ($p=0.047$), boundary-resident union 27/47 ($p=0.020$); in-cap means $43^\circ/28^\circ/45^\circ$ | own-footprint baselines | step_113 |
| Open-channel union (empirical coupling) | 52 objects, 39 in-cap vs $E=30.97$ footprint / $E=39.86$ mixture; BF 25.5; closed channel 76 objects, 0 in-cap | leave-one-out measured kernel | step_116 |

## 4. The transit signature: cometary clock-channel discrepancy

Long-period comets are the key probe because they do something the resident TNOs cannot: they cross the candidate boundary on a single pass. Their transit record divides into two distinct measurement classes (Schematic below): a primary, coordinate-level spatial channel — the aphelion dipole $d_\parallel$ computed directly from osculating elements, which requires no leg-split orbit reconstruction and persists flat across all eras and quality strata ($d_\parallel = +0.110$, $p < 10^{-4}$ pre-2018; $d_\parallel =$ +0.0949, $p =$ 0.00215 post-2017; Fisher-combined $p = 1.8\times10^{-6}$, Sections 4.8 and 4.12) — and a secondary, estimator-dependent reconstruction channel that compares separate inbound and outbound orbit fits. For the latter, the Warsaw and CODE catalogues publish, for each near-parabolic comet, three orbit solutions — osculating elements fitted to the observed arc, original barycentric elements obtained by backward integration to 250 AU, and future barycentric elements by forward integration. Standard dynamics carries each comet from one solution to the next through exactly the perturbation the planetary system applies; the size and direction of that difference — and how it interacts with orbit-solver phase absorption across epochs — is examined first in Sections 4.1–4.7 before the era-invariant spatial aphelion channel and the modern holdout dissociation are established in Sections 4.8–4.12.

#### Why comets test the clock

A comet plunging through a lapse boundary accumulates proper time at a perturbed rate during the crossing. A back-integration that assumes the standard clock then mis-assigns the orbital phase: the reconstructed orbit is rotated relative to the true one — in the clock-sector elements — while the orbital energy, a momentum-sector quantity, is untouched. The signature is rotation without a resolved energy contrast, on the crossing leg.

*Note on conformal invariance.* Under Theorem 2 of Paper 0, a static conformal metric produces no direction-odd propagation asymmetry along closed reciprocal photon paths ($\oint d\ln A = 0$). The cometary transit observable does not violate this theorem: comets are non-relativistic, open-path dynamical tracers that traverse the boundary sector on an asymmetric single leg (inbound in-cap, outbound out-of-cap). The non-zero rotation residual reflects an open path integral of the active Temporal Shear, $\int \Sigma_\mu\,dx^\mu$, which standard software misinterprets as non-closure when attempting to fit a closed Keplerian orbit.

Reconstruction-dependent channels
raw astrometry → orbit solver → leg-separated boundary fits
periapsis rotation $\delta\theta$ · inbound-leg displacement $d_{\rm in}$ · leg disagreement
the solver can absorb — or inject — phase structure here (step 151)

Reconstruction-free channels
raw astrometry → boundary propagation → sky coordinates
aphelion direction · arrival dipole $d_\parallel$ · cap membership $\theta$
no orbit solution intervenes — a warp cannot be laundered through the fit

↓ the stratification discriminator: the two channel classes must move together under a real field, and can only dissociate through the solver

Closed-loop clocks (spacecraft)
coherent two-way links cancel the endpoint conformal term identically at the turnaround (Sections 6.8, 6.9): the link null is mandated by the geometry — onboard-oscillator records remain clock-carrying

Open-loop clocks (comets)
no uplink discipline — a crossing slip accumulates unrescaled into the reconstructed phase: the only macroscopic channel that keeps the integral

**Schematic:** the instrument's two measurement classes. The reconstruction-dependent channels pass through an orbit solver and can be absorbed by it (measured directly in step 151); the spatial channels are coordinate observables computed from a single fitted orbit solution — they never pass through the leg-split reconstruction, though they still inherit that solution's element-level uncertainties. The post-2017 dissociation — reversal in the first class, persistence in the second — is the predicted signature of a real boundary viewed through short-arc fitting (Section 4.12). The clock-class contrast — null on disciplined spacecraft links, accumulating on undisciplined cometary trajectories — is the two-way cancellation theorem's required pattern (Section 6.8).

## 4.1 The three-leg decomposition

The reconstruction residual is decomposed into an inbound leg (osc $\rightarrow$ orig), an outbound leg (osc $\rightarrow$ fut), and the full pass (orig $\rightarrow$ fut). In the Warsaw spike sample the directional discrepancy concentrates on the inbound leg and the in-plane clock-sector element: the full-arc inbound test inside the TNO cap gives $p = 0.0089$, while the outbound comparison is flat ($p \approx 0.72$). The element decomposition localizes the discrepancy further: on the CODE full pass, the excess registers in the argument of perihelion $\omega$ ($p = 0.022$) while $\Omega$, $i$, and $e$ are flat ($p = 0.93$, $0.92$, $0.68$) — the reconstructed orbit is rotated in its own plane, not tilted. That is the phase error a lapse boundary writes: an in-plane rotation of the reconstructed orbit, not a plane tilt and not an energy shift.

## 4.2 Independent confirmation on CODE

The anomaly replicates on the independent CODE-only sample ($N = 54$ deep-plunger, class-1 comets with no Warsaw overlap). Comets whose original-orbit aphelia fall inside the TNO-defined 60° cap ($n = 19$) show median orig $\rightarrow$ fut reconstruction discrepancy 0.212°, versus 0.131° for the out-cap population ($n = 35$): $p =$ 0.00295. The energy channel is flat in the same sample ($|\Delta(1/a)|$, $p =$ 0.688), while the rotation-per-energy-kick ratio is elevated at $p = 0.017$ — the discrepancy lives in the clock sector, not the momentum sector. The continuous, cap-free statistic on all 131 class-1 comets — Spearman correlation of the discrepancy against angular distance from the axis, with no cap and no perihelion cut — gives $\rho = -0.243$, $p = 0.0051$. Where gaps between populations are quoted in dex below, the unit is $\log_{10}$ residual amplitude: $+0.3$ dex is a factor of two in median residual.

## 4.3 The planetary baseline residual

The one conventional channel that could mimic an axis preference is planetary-encounter geometry, and it is subtracted with a measured N-body baseline rather than a proxy. Each class-1 CODE comet is backward-integrated from its osculating elements at the catalogued perihelion epoch through the full planetary system — the Sun plus the planetary-system barycentres, with states drawn from JPL DE440s — using the REBOUND integrator (Rein & Liu 2012) in its IAS15 mode (Rein & Spiegel 2015) until it crosses 255 AU heliocentric, bracketing the 250 AU sphere at which the catalogue's original orbit is defined. A hybrid-symplectic MERCURIUS re-integration (Rein et al. 2019) confirms the result is integrator-independent ($\rho = 0.93$ on the recovered boundary energies). The integration is validated end-to-end: the simulated energy kicks reproduce the catalogue's own original-versus-osculating $1/a$ changes — an independent N-body measurement of the same perturbation — at rank correlation $\rho = 0.98$ (median absolute deviation $4.7 \times 10^{-6}$ AU$^{-1}$; 88 per cent of comets within $50 \times 10^{-6}$ AU$^{-1}$). The in-cap comets carry no energy-kick advantage ($p = 0.568$); their marginally smaller minimum planet distances (3.9 vs 4.6 AU, $p = 0.008$) are absorbed by adding closest approach as a second baseline covariate ($p = 0.0024$–$0.0045$).

Regressing the measured kick out of the discrepancy and retesting against the axis, the anomaly is undiminished: matched-sample cap contrast $p = 0.0021$, all-class-1 cap contrast $p = 0.0008$, continuous correlation $\rho = -0.27$ ($p = 0.0017$). The analytic Jupiter-node proxy of step 040 gives the same ordering. What survives subtraction of the full measured planetary perturbation still points at the TNO axis.

The same machinery tests the named hypothesis directly (step 062). The Brown & Batygin (2021) Planet Nine is inserted into the identical backward integrations — both published realizations, the maximum-likelihood model ($m_9 = 5\,M_\oplus$, $a_9 = 300$ AU, $e_9 = 0.15$, $i_9 = 17^\circ$, $\varpi_9 = 254^\circ$, $\Omega_9 = 108^\circ$) and the marginalized-median model ($m_9 = 6.9\,M_\oplus$, $a_9 = 461$ AU, $e_9 = 0.30$, $\varpi_9 = 246.7^\circ$, $\Omega_9 = 97^\circ$) — each at four mean anomalies spanning the unconstrained present position, eight placements in total.

The perturber's apoapsis lies inside the anomalous cap, so its maximum leverage falls on exactly the comets that must show the signal. The measured effect is nonetheless negligible: the median periapsis-direction rotation P9 injects into the reconstructed inbound orbit is $3 \times 10^{-4}$ degrees per placement (maximum $2.7 \times 10^{-3}$ degrees across all 131 comets and all placements), 460–1200 times short of the observed in-cap median discrepancy, and in no single case does the P9-induced rotation reach the comet's own observed residual. The in-cap comets are not preferentially hit, the injected rotation does not track the discrepancy, and removing it leaves the cap contrast at $p = 0.0016$–$0.0024$. Even the per-comet maximum across all eight placements — the most generous possible bound — remains $527\times$ short. The published perturber cannot produce the anomaly it was hypothesized to explain.

The same integration, run in both directions, closes the comparison against the catalogue itself (step 063). Each comet is propagated backward and forward to the 250 AU barycentric sphere on which the original and future solutions are defined, with the boundary states evaluated about the system barycentre — the catalogue's own convention. The independent integration then reproduces the catalogue's original and future periapsis directions to a median $0.001^\circ$ per leg (95th percentile $0.005^\circ$; 98 per cent of legs within $0.01^\circ$), and the per-comet orig $\rightarrow$ fut rotation at rank correlation $\rho = 0.999$ with median absolute difference $0.0005^\circ$. The machinery is therefore validated not only against the energy channel but against the catalogue's own boundary solutions, and the validation carries a sharpening consequence: the catalogued rotation is shown to be entirely a product of standard planetary dynamics. The anomaly is not rotation beyond what the planets apply — it is that the applied rotation lands anisotropically.

Regressing the modelled rotation against the full measured perturbation budget — the energy kick, the minimum planetary approach distance, the perihelion depth and the inclination — leaves the directional excess in place: cap contrast $p = 0.002$ on the matched sample and $p = 0.012$ on all class-1 comets, with continuous correlation $\rho = -0.408$ ($p = 0.002$) on the matched set. Comets arriving from the TNO direction are rotated more per unit of energy exchanged than the measured encounter budget accounts for.

## 4.4 Threshold morphology

The spatial shape of the signal is diagnostic. Inside the cap the median discrepancy is a flat plateau — $0.235^\circ$, $0.165^\circ$, and $0.244^\circ$ in the 0–30°, 30–45°, and 45–60° shells, with within-cap correlation $\rho = +0.10$ ($p = 0.69$) — then steps down to the $0.13^\circ$ background at 60–75°. A point-mass impulse scales as $1/b^2$ with angular separation from the perturber; the observed profile is instead a threshold: every crossing trajectory receives the same treatment regardless of impact parameter, and the effect vanishes outside a finite patch. That is the morphology of a wall.

While the comet channel provides the cleanest step morphology, a similar threshold was sought on the resident side. A top-hat mixture (clustered fraction $f = 0.40$ in a half-width $W = 52^\circ$ sector) fits the TNO $\varpi$ distribution marginally better than a von Mises peak ($\Delta{\rm AIC} = -1.9$), and the in-cap counts are consistent with flatness ($\chi^2 = 0.15$, $p = 0.985$). A structural caveat follows from the conditional null: conditioned on the measured discovery footprint, 87 per cent of null realizations also prefer the top-hat — a broad footprint pile-up measured as offsets from a displaced axis naturally fits a cap better than a Gaussian. The TNO-side shape is therefore recorded as suggestive but non-discriminating; the comet channel's step morphology remains the cleaner wall signature because it is measured on a residual, not on elements subject to pointing bias.

## 4.5 Robustness of the transit signal

The comet result survives the full blind-validation battery. Leave-one-out jackknife: $p \in [0.0013, 0.0058]$ — no single comet carries the signal. Split-half axis recovery: a free sky scan on one random half recovers the direction within 45° of the TNO axis 70 per cent of the time, and the held-out half confirms at $p < 0.05$ in 22 per cent of splits (median held-out $p = 0.22$ — a conservative validation rate at this sample size). Look-elsewhere: over a 2,232-direction grid, only 0.27 per cent of trial axes match the observed cap contrast. No-$q$-cut: $p = 0.010$. Era stability: the effect is flat against apparition year ($\rho = -0.015$, $p = 0.91$) and present in both catalogue generations (pre-2006 $p = 0.0095$; post-2006 $p = 0.10$); the proper-time residual is likewise epoch-flat within cohorts ($\rho = -0.187$, $p = 0.11$ pooled in-cap) with the in/out contrast positive in every era bin (Section 4.11). Quality-class stability: dropping the class-1 restriction entirely, the all-quality-class CODE-only sample ($n = 72$) retains $p = 0.011$. A doubly matched control — pairing each in-cap comet to the out-cap comet nearest in both perihelion depth and realized energy kick — returns $p = 0.0019$: the excess is not a depth or kick-size artefact.

## 4.6 Observational-leverage and axis-definition controls

The remaining conventional account of the discrepancy is differential orbit quality: comets pointing at the axis might simply be worse observed, so their three-leg solutions disagree more. The audit says the opposite. On the CODE matched sample the in-cap and out-cap comets are statistically identical in observation count (median 328 vs 598, $p = 0.22$), arc length (495 vs 495 days, $p = 0.79$), perihelion epoch, $q$, and $i$; restricting to the 37 comets with two-sided observational arcs, the cap contrast survives at $p = 0.039$; regressing the discrepancy against all leverage covariates strengthens it ($p = 0.0017$); and pairing comets of matched observational material returns $p = 0.027$ (Wilcoxon). One covariate does differ: in-cap comets carry a larger original $1/a$ (median 44.8 vs 32.9, $p = 0.031$), i.e. they are on average more tightly bound; the energy channel is flat throughout, so this difference modulates the boundness distribution rather than the discrepancy channel itself, and it is retained as a recorded covariate.

The same audit on the Warsaw metadata finds the in-cap comets better observed, not worse (median 376 vs 232 observations, $p = 0.019$) — the anomaly lives in the best-measured solutions, mirroring the resident-side audit.

The result is also stable to the axis definition. Re-testing against the detached-sample axis $(50^\circ, -17^\circ)$ — rather than the extreme-subsample axis $(34^\circ, -13^\circ)$ used to pre-declare the cap — the CODE contrast holds at $p = 0.030$ with a continuous rank correlation $\rho = -0.31$ ($p = 0.023$), and a free CODE-internal recovery at $(10^\circ, -20^\circ)$ gives $p = 0.0025$. The anti-axes, ISM-inflow (Bzowski et al. 2015), galactic-pole, and ecliptic-pole directions are all null. The signal is attached to the broad sector, not to a finely tuned direction.

## 4.7 Uncertainty-normalized and independent-cohort channels

A fourth catalogue tightens the test. The one-apparition comet catalogue (Królikowska et al. 2014; VizieR J/A+A/571/A63) publishes three-leg solutions for 38 Oort-spike comets together with formal element uncertainties — permitting a discrepancy statistic normalized by each orbit's own published error budget. In that cohort the raw in-cap excess persists (median $0.229^\circ$ vs $0.126^\circ$, $p = 0.005$), and it survives normalization: the rotation discrepancy per unit formal uncertainty is larger in-cap at $p = 0.012$, with continuous correlation $\rho = -0.425$ ($p = 0.008$). The anomaly is therefore not a rescaling of larger error bars — in-cap discrepancies exceed their own published uncertainties by a wider factor. The normalized energy channel remains flat ($p = 0.65$). The embedded Marsden–Williams (2008) $1/a$ comparison values supply a partial third-lineage check: the Warsaw–MW08 disagreement is uniform in and out of the cap ($p = 0.75$), so catalogue-lineage systematics are not direction-dependent; the MW08 energy series itself is likewise flat ($p = 0.71$).

## 4.8 A reconstruction-independent spatial channel: the aphelion dipole

The transit evidence so far uses the reconstruction residual — a quantity produced inside the orbit pipeline. The comets' fitted aphelion directions provide a spatial observable distinct from the inbound–outbound residual; they still inherit catalogue and orbit-fit uncertainties. Measured against a tide-aware null that preserves the observed galactic-latitude distribution while isotropizing galactic longitude, the aphelia of every comet sample lean toward the TNO axis: Warsaw spike $d_\parallel = +0.134$ ($p = 0.006$), CODE class-1 $+0.193$ ($p = 3\times10^{-4}$), CODE-only $+0.22$ ($p = 8\times10^{-4}$), and the one-apparition cohort $+0.31$–$0.34$ ($p \leq 8\times10^{-4}$). The control directions discriminate the origin: the on-plane galactic anticenter — where the radial component of the tide does predict a longitude asymmetry — carries the expected residual dipole ($d_\parallel = +0.11$ to $+0.21$ across the four samples, $p = 0.001$–$0.03$) yet stays below the TNO-axis dipole in every one, while the measured dipole points $44^\circ$ off the galactic plane at the TNO direction. The anti-axis and galactic pole are null throughout. Whatever organizes the comet aphelia is not the axisymmetric galactic tide.

The four samples above are catalogue elites — every member earned a full three-leg solution — so the channel is next tested on the broadest discovered population: the full MPC CometEls census, restricted to near-parabolic orbits ($0.90 \leq e < 1.02$; $n =$ 270 unique parent designations after fragment records are merged) (step 108). The lean persists on the unselected population — $d_\parallel = +$0.10 toward the cap axis ($p =$ 2×10<sup>-3</sup> under the identical tide-aware null) — and strengthens toward the spike: the $a_{\rm osc} > 250$ AU proxy subset leans at $+$0.17 ($p =$ 3.0×10<sup>-4</sup>), with the eccentricity bins $[0.985, 1.0)$ and $[1.0, 1.02)$ carrying $+$0.17 ($p =$ 2.5×10<sup>-4</sup>) and $+$0.086 ($p =$ 0.053) respectively.

The membership accounting is reported exactly: 62 of the 270 also appear in the training set, and the 208 that do not — modern discoveries on the MPC's own orbit-fit lineage, comets the anomaly analysis never touched — carry the dipole alone at $d_\parallel = +$0.067 ($p =$ 0.039). Finally, freed of the declared axis entirely, the census's own spherical-mean aphelion direction recovers (34$^\circ$, -37$^\circ$) — 24$^\circ$ off the cap axis and 156$^\circ$ off the anti-axis: the broadest population self-organizes inside the measured sector. The dipole is therefore a population-level property of the discovered near-parabolic comets, not an artefact of the solution-quality cut.

The heterogeneity objection is then closed directly: comet surveys scan the ecliptic, so the discovery footprint is organized in ecliptic coordinates, and the census dipole is re-evaluated under an ecliptic-latitude-preserving null — the footprint-aware analogue of the tide null — where it survives at $p =$ 0.0021 (step 109). Partitioned by absolute magnitude, perihelion year, and perihelion distance, the lean stays positive in all nine tercile bins, and a single common amplitude is consistent with every partition ($\chi^2$ heterogeneity $p =$ 0.47–0.88): the dipole does not scale with brightness, discovery epoch, or perihelion depth, as a selection artefact would.

One geometric question remains: is the asymmetry a direction or an axis? The dipole statistic $\langle\cos\theta\rangle$ measures a one-ended lean, whereas a two-ended concentration about the axis line — the geometry the slip field itself exhibits (Section 4.10) — would register in the unsigned $\langle|\cos\theta|\rangle$ and quadrupole $\langle\cos 2\theta\rangle$ moments. Decomposed across all six comet samples under the identical null (step 110), the directional moment is significant everywhere ($p =$ 3×10<sup>-4</sup>–0.006) while both axis-symmetric moments are null in every sample ($p =$ 0.32–0.91). The aphelion asymmetry is therefore one-ended: comets' aphelia lean toward the cap pole specifically, with no mirrored excess toward the antipode. The two channels then measure different aspects of the same structure — the orbit distribution records the direction of approach through the boundary, while the clock-sector field encountered in transit is axisymmetric about the boundary normal.

## 4.9 Second-catalogue validation and the equivalent orbital time scale

The bidirectional machinery is then run unchanged on the Warsaw spike sample ($N = 98$ comets with all three legs; step 064) — a second catalogue sample, independently selected though within the shared orbit-determination lineage (Sections 2.4 and 6.7), on which the anomaly was first isolated on the inbound leg (Section 4.1). The integration reproduces the Warsaw boundary solutions to a median $0.0008^\circ$ per leg (95th percentile $0.009^\circ$; orig $\rightarrow$ fut rotation at $\rho = 0.998$), so the reconstruction comparison is validated on two catalogues, not one.

On the channel the Warsaw anomaly occupies — the osc $\rightarrow$ orig direction discrepancy — the raw in-cap excess is only marginal ($p \approx 0.07$–$0.08$, catalogue and simulated alike), but it is the right channel: after regression of the same measured encounter budget, the unexplained inbound offset in the cap is $+$5.0 yr $[2.6, 7.2]$ against $-1.782$ yr outside ($p =$ 0.0079) on the full spike sample, and +2.4 yr against -2.2 yr ($p =$ 0.0390) on the matched deep plungers (step 065). The orig $\rightarrow$ fut channel that carries the CODE signal is flat on Warsaw ($p \approx 0.2$–$0.8$): the two catalogues place the anomaly on different legs, each an inbound-weighted clock-sector discrepancy per unit of applied perturbation — the signature a phase slip writes, since a clock error contaminates whichever leg the standard-dynamics reconstruction traverses first.

Expressed in a useful orbital scale, the residual rotation can be converted to the time an unperturbed orbit would need to sweep through the same angle (step 065). A periapsis-direction offset $\delta\theta$ at the 250 AU boundary sphere and sweep rate $\Omega_b = h/r_b^2$ define $\delta t_{\rm eq} = \delta\theta\,r_b^2/h$. Regressing this equivalent scale on the same encounter budget gives +4.9 yr $[4.1, 6.0]$ on the matched CODE sample ($p =$ 0.00532), $+$5.0 yr $[2.6, 7.2]$ on the Warsaw inbound channel ($p =$ 0.0079), and $+2.1$ yr $[0.2, 3.2]$ on the pooled cohorts ($p = 0.016$). These values are an orbital phase conversion, not a measured proper-time lapse: deriving $d\tilde\tau$ requires a TEP field, a matter response, and a light/observation model consistent with the canonical metric — the realization the conversion encodes (crossing-localized time translation versus local impulse) is discriminated on the independent refit record in Section 6.8.

![Implied proper-time offset versus axis distance](figures/step_b30_proper_time_slip.png)

**Figure 1:** The transit anomaly in equivalent orbital time units. Left: $\delta t_{\rm eq} = \delta\theta\,r_b^2/h$ for the pooled CODE (circles) and Warsaw (squares) cohorts versus aphelion–axis separation; the in-cap points (red) sit systematically high. Right: the residual after regression on the measured encounter budget. This conversion is not itself a proper-time measurement. Pipeline output: step 065 (`results/figures/step_b30_proper_time_slip.png`).

## 4.10 Slip localization: a surface crossing, not a distributed field

The proper-time reading admits two competing structures that the transit data separate (step 068). A distributed lapse-gradient field predicts slip accumulating with the time spent inside the affected volume — $\delta\tau$ growing with the measured 512–753 yr spread of transit times — whereas a single crossing of a localized surface predicts a step: an offset set at the boundary and independent of how long the orbit then dwells inside. The data select the step. On the pooled in-cap cohort ($N = 75$) the budget-regressed slip is uncorrelated with transit time ($\rho = -0.02$, $p = 0.85$), and normalizing by transit time does not contract the in-cap scatter ($\sigma_{\log\delta\tau} = 1.11 \rightarrow 1.13$); within the cap the slip is also flat against axis distance ($\rho = -0.13$, $p = 0.41$ on CODE; $\rho = +0.13$, $p = 0.47$ on Warsaw), so the anomaly is not a penetration-depth effect. Outside the cap the CODE residual does retain a transit-time correlation ($\rho = +0.31$, $p = 0.004$) — field-like structure the encounter regression leaves behind — but it is absent precisely where the anomaly lives.

The in-cap discrepancy is coherent in sign as well: only 14 of 43 in-cap CODE comets carry a positive signed rotation (binomial $p = 0.016$, against $p = 0.083$ outside), and on Warsaw the in-cap anomalies place a median 92 per cent of their discrepancy on the inbound leg versus 72 per cent outside ($p = 0.031$). A lapse discontinuity crossed once — predominantly on the way in — is the parsimonious reading.

## 4.11 Where the slip enters: radial profile, edge width, and epoch stability

Three further measurements localize the anomaly in space, angle, and time (steps 071–073). The step-063 bidirectional legs are extended to twelve boundary shells spanning 60–400 AU, recording each leg's osculating barycentric periapsis direction at every crossing (step 071). The in-cap rotation excess — median $\delta\theta = 0.147^\circ$ inside the cap against $0.094^\circ$ outside — is already fully developed at the innermost shell and flat to $0.001^\circ$ across the entire range: a discrete crossing between 60 and 400 AU would imprint a radial kink at the crossing radius, and a distributed field would grow the offset with distance; neither is seen. Converted to time units the excess rises smoothly from $+0.2$ to $+6.9$ yr across the shells — the $\propto s^2$ growth expected when a fixed angular offset is expressed against the local sweep rate $h/s^2$ — so the event is a discrete angular slip imprinted interior to 60 AU: on the inbound passage or within the observed-arc region where the reconstruction residual is generated, not on a second outbound crossing.

The edge of the anomaly in axis separation is likewise sharp: a logistic fit to the unexplained slip against $\theta$ returns a transition centred at $\theta_0 = 59.5^\circ$ on CODE and $44.5^\circ$ on Warsaw ($45.4^\circ$ pooled), with the width parameter driven to the resolution floor — the data are indistinguishable from a step (logistic and step RSS equal within 0.5 per cent), so no graded transition broader than a few degrees is required (step 073).

Finally, the residual slip is flat against perihelion epoch within every cohort ($\rho = -0.187$, $p = 0.11$ pooled in-cap; step 072): the in/out contrast is positive in all three era bins — $+10.6$ yr pre-1950 ($p = 0.007$), $+0.7$ yr 1950–1990, $+4.3$ yr post-1990 ($p = 0.14$) — while the absolute levels drift together in both caps, indicating a global era term in the residual that the contrast removes. The amplitude is era-weighted toward the earliest cohort, an objective weighting the modern-era subsample retains at reduced significance; the signal is not an artefact of any single catalogue generation.

Two further localizations tighten the picture (steps 075–077). Repeating the shell measurement inside 100 AU (step 076) shows the excess already present in full at the innermost shell, 8 AU — within the region where the comet is actually observed and the reconstruction residual is generated; the discrepancy does not grow through the giant-planet zone and therefore does not ride the encounter channel.

The slip also carries a sign structure, and its attribution is measured rather than assumed (steps 075, 080). The raw signed periapsis rotation about each comet's own orbit pole is non-random on both catalogues' anomaly channels — CODE orig$\rightarrow$fut rotations are 84 per cent retrograde-signed in-cap, and the Warsaw inbound leg shows the consistent counterpart — but the bidirectional simulation shows the same signed rotations are predicted by the planets themselves at $\rho = 0.999$ per comet: the global sign bias is the secular rotation the giant planets impart over the $\sim 600$ yr legs, not an anomalous phase.

What survives subtraction is a small residual that remains sign-coherent — 77 per cent positive in-cap (binomial $p = 3\times10^{-4}$) against 61 per cent outside ($p = 0.025$), median $+0.001^\circ$ — a weak directed term on top of the planetary rotation, consistent with the in-cap enhancement but too small to carry the anomaly's amplitude alone.

The era concern raised by the residual channel is bounded independently by the spatial dipole: the aphelion projection onto the resident axis is positive in every era and highly significant in the modern cohort ($d_\parallel = +0.15$, $p = 2\times10^{-4}$, $n = 200$ post-1990) — the spatial anomaly is not an artefact of the early catalogue (step 077).

The same localization admits a sharper physical read once the planetary baseline's own depth scaling is removed (step 086). The raw rotation channel carries a $q$-dependence — $\delta\theta \propto q^{-0.598}$ — but it is identical in both caps, the expected encounter scaling: deeper plungers rotate more under the giant planets. The mechanism test therefore runs on the residual slip. In boundary-normalized time units the unexplained in-cap slip is nearly flat against perihelion distance — slope $-0.233$ ($-0.234$ on CODE, $-0.205$ on Warsaw) — against the three separated predictions: a fixed angular displacement requires $\delta\tau \propto q^{-0.5}$; a fixed proper-time offset at the boundary crossing requires $\delta\tau \propto q^{0}$; and a fixed time offset absorbed during the observed arc requires $\delta\tau \propto q^{-2}$, strongly excluded.

The measured scaling sits closest to the boundary reading: the anomaly behaves as a roughly constant temporal offset per comet, not a constant angular displacement — the signature a temporal mechanism leaves, and not the one a fit systematic would. The angular residual channel is too noisy to add independent discrimination ($\delta\theta_{\rm resid}$ being the difference of two small angles, its slope is unstable across catalogues) and is reported as such.

The anomaly's geometry in axis separation admits one further measurement that bears on the boundary's topology (step 089). If the structure were a two-sided sheet or a symmetric pair, comets whose aphelia point near the anti-axis — the mirror cap, $\theta > 120^\circ$ — should carry a second rotation lobe. It does not: the mirror-cap cohort carries the lowest rotation medians of the full profile ($0.080^\circ$ against $0.105^\circ$ in the intervening region; a second-lobe amplitude bounded above $0.0075^\circ$ at 95 per cent — an order of magnitude below the CODE in-cap elevation, $0.144^\circ$ cap against $0.065^\circ$ mirror), and the cap-versus-mirror contrast is significant ($p = 0.029$). The angular signature is therefore single-faced.

The planetary-subtracted slip, however, is not: the residual proper-time offset is positive at both poles of the axis — $+2.2$ yr in the cap and $+2.0$ yr in the mirror cap on CODE ($+0.7$ and $+1.3$ yr on Warsaw) — while the intervening $60^\circ$–$120^\circ$ region is systematically negative (median $-3.9$ yr; positive fraction 0.39, binomial $p = 0.012$). A harmonic decomposition makes the geometry explicit: the residual tracks $\cos 2\theta$ ($\rho = +0.17$, $p = 0.010$), the axisymmetric basis, rather than $\cos\theta$ ($\rho = +0.07$, $p = 0.30$), the one-sided basis. The slip field is thus axisymmetric — elevated toward both ends of the measured axis and depressed between — the morphology an axisymmetric proper-time structure produces, and the same two-ended geometry the injected JFC chain already exhibits in aphelion direction. The strong rotation signature remains localized to the $+$axis sector; the weaker residual field is two-lobed and same-signed at both ends. Because both lobes carry the same sign, the era-dependent sign reversal measured in Section 4.14 cannot be a spatial lobe crossing: it is a measured-polarity reversal between epochs, attributable to the solver-absorption mechanism of step 151 or to time dependence of the field itself.

The map just measured is an in-sample statement, so it is next required to predict (step 102). With the axis, cap width and harmonic order all pre-declared, the two-parameter profile $\delta\tau(\theta) = a + b\cos 2\theta$ fitted on four-fifths of the pooled cohort predicts the signed residual slip of the held-out fifth at $\rho = +0.15$ against a $\theta$-shuffle null ($p = 0.007$, $2\times10^{4}$ permutations); fitted on the CODE cohort alone it predicts the signed slips of the Warsaw cohort at $\rho = +0.22$ ($p = 0.029$) — a transfer between essentially disjoint populations (one shared designation in 229) and independent orbit fits, which a fitter- or sample-specific artefact cannot produce. The reverse direction is positive but weaker ($\rho = +0.13$, $p = 0.13$), and held-out sign accuracy is 56 per cent ($p = 0.032$). The per-object correlations are modest by construction: per-comet slip scatter of several years exceeds the $\cos 2\theta$ amplitude itself ($+6.0$ yr), so a genuine population-level map yields small object-level skill. The test asks whether the map predicts unseen comets at all, and it does.

Two further controls tighten the validation (step 104). Random folds leave a spatial neighbour of every held-out comet inside the training set, so they cannot distinguish a genuine sky field from a pattern carried by a few clustered objects; a half-sky split can. Dividing the aphelion distribution by great circles through the axis at four orientations and fitting on one half predicts the other half at a median $\rho = +0.17$ across all eight plane-direction tests (range $+0.11$ to $+0.23$, every test positive) — the map transfers across disjoint sky regions, as a field must. The same vectors localize the structure from the comet data alone: displacing the assumed axis, the out-of-sample skill peaks at zero displacement, halves near $30^\circ$ and crosses zero near $39^\circ$, and the detached-sample axis $15^\circ$ away returns $\rho = +0.10$ — the cohort places the boundary inside the pre-declared sector at tens-of-degrees precision, rather than borrowing its location from the TNO measurement.

One extrapolation remains unresolved: fitted without the $\theta > 120^\circ$ comets, the map predicts the mirror lobe's reappearance at the right amplitude (predicted median $+3.4$ yr against observed $+2.0$ yr) but per-object sign skill is not resolved at $n = 32$ ($p = 0.70$). Over 200 random five-fold partitions the held-out skill is positive in every draw (median $+0.134$, 68 per cent interval $[+0.114, +0.154]$).

A third catalogue completes the transfer chain (step 105). The one-apparition cohort of Section 4.7 contributes 30 Oort-spike comets with all three legs — the shortest-arc fits the catalogues carry — and is run through the identical bidirectional instrument: integrated from each published osculating solution to the 250 AU barycentric sphere in both directions under DE440s, the simulation reproduces that catalogue's own original and future boundary solutions to a median $5\times10^{-5}$ deg per leg (95th percentile $1.5\times10^{-3}$ deg), so the instrument is now validated on three catalogues, not two.

On the channel the map predicts, the cohort's planetary-subtracted slip declines with transit angle at $\rho = -0.367$ ($p = 0.046$) — the sign of the TEP direction, matching the CODE cohort's $-0.32$ — with in-cap versus out-cap medians of $+3.69$ versus $-3.95$ yr ($p = 0.043$); the detached-sample axis returns the same sign ($\rho = -0.382$, $p = 0.037$). The membership overlap is reported exactly: 23 of the 30 also appear in the training set, and on those bodies the two catalogues' three-leg rotations agree to $\sim 0.001^\circ$ ($\rho = 0.999$; step 106) — the same underlying orbit record republished with uncertainties and the MW08 comparison, not an independent re-fit — so that subset is a robustness check on the shortest-arc definition rather than a replication, and it retains the decline at $\rho = -0.418$ ($p = 0.047$).

The seven members the map never saw are the genuinely new information, and they trend the same direction ($\rho = -0.36$), underpowered at that sample size. The axisymmetric slip map fitted on the 229 Warsaw+CODE comets predicts this cohort's signed slip at $\rho = +0.12$ ($p = 0.54$; sign 17 of 30, $p = 0.29$) — positive but unresolved, a consistency result rather than a detection. The Marsden–Williams comparison remains energy-only by construction of that record — it publishes no angular elements — and the Warsaw–MW08 $1/a$ disagreement shows no direction dependence within this cohort ($\rho = +0.18$, $p = 0.35$), consistent with the flat cross-lineage check of Section 4.7.

Three residual selection questions close out on the same products (step 106). The first is the two-lobe morphology itself: the $\cos 2\theta$ order was selected in-sample, so the five-fold cross-validation is re-run under identical folds with alternative profile families — a constant, $\cos\theta$, $\cos 2\theta$, $\cos 3\theta$, and a linear $\theta$ term. Only the $\cos 2\theta$ form retains positive held-out skill ($\rho =$ +0.149 against $-0.073$, $-0.025$, $-0.086$ and $-0.024$); the $\cos 2\theta$ minus $\cos\theta$ skill difference is $+0.17$ (68 per cent interval $[+0.09, +0.26]$, positive in 98 per cent of bootstrap draws) — the two-lobe structure is a predictive property of the data, not a choice made on it.

The second question is whether the third cohort recovers the axis on its own: the free sky scan of Section 5.1, run unchanged on the lpc discrepancy record, finds its strongest direction at $(50^\circ, -50^\circ)$ ($p = 6\times10^{-4}$), $39^\circ$ from the cap-declaration axis — inside the same sector — while the pre-declared axis itself returns a stronger cap contrast than 96 per cent of the 370 trial directions ($p = 0.006$).

The third question is amplitude: the lpc cohort's own free fit returns $b = +2.3$ yr (68 per cent interval $[-1.1, +6.4]$) against the training amplitude $+6.0$ yr — the same sign and a consistent amplitude class on a cohort the fit never saw. The shared-member concordance then supplies the exact degree of independence in that catalogue: the two records' per-comet rotations agree to a median $5\times10^{-4}$ deg ($\rho = 0.999$), confirming that the one-apparition catalogue is the same underlying orbit record for shared bodies — the measured basis for weighting the seven unshared members, rather than the catalogue label, as the new information.

## 4.12 Prospective holdout: the post-2017 SBDB cohort

The transit framework is next subjected to a definitive stress test against the modern synoptic-survey era: a cohort the axis declaration never saw (step 117). The JPL Small-Body Database yields 287 long-period comets designated in 2018 or later with near-parabolic solutions ($0.95 < e < 1.5$), a data arc of at least 30 days and at least 20 observations — out-of-time against the CODE and Warsaw catalogues and orbit-fitted on the modern survey record. Of these, 148 satisfy the deep-plunger cut ($0.1 < q < 3.1$ AU) and 121 reach the 250 AU barycentric sphere under the identical bidirectional instrument (REBOUND/IAS15, DE440s); the registered axis, cap width and channel definitions are applied unchanged. The cohort exposes an operational divergence between reconstruction-dependent residuals and coordinate-invariant geometry. In the single-solution reconstruction channel the declared-axis carrier reverses: inside the 60° cap (32 of 121 primaries) the median full-pass rotation is 0.114° against 0.171° outside (permutation contrast $p =$ 0.920; kick- and leverage-regressed residual cap test $p =$ 0.968), and the axisymmetric slip map fitted on the 229 CODE+Warsaw comets transfers at $\rho = -0.04$ ($p = 0.65$). In the spatial aphelion channel the same cohort leans toward the declared axis at $d_\parallel =$ +0.0949 ($p =$ 0.00215) — the coordinate observable no orbit solver can absorb — so the modern record separates the two observable classes rather than dismissing both.

The reversal is itself informative rather than noise, and it is structured rather than flat: the direction-aware audit (step 117) finds the cap contrast significant in the opposite direction on the inbound leg — in-cap $d_{\rm in}$ is suppressed at two-sided $p =$ 0.019 (residualized $p =$ 0.017; $d\tau_{\rm in}$ $p =$ 0.021) while the outbound leg is null ($p =$ 0.52) — the same leg that carries the CODE anomaly's positive gap, now with reversed sign. The sign flip is era-specific: pre-2018 SBDB resolves no reversal — the CODE-overlap members carry a positive median gap ($+0.008$) while the non-overlap cell is marginal and unresolved ($-0.013$; MWU $p =$ 0.25 and 0.57 respectively). Pooled with CODE under a cohort$\times$cap interaction model, the in-cap residual gap differs by 0.19 dex at $p =$ 0.0050 against cohort-label permutation — the two catalogues' answers at the same sky direction are inconsistent with each other, not jointly null. The frame decomposition below (CMB-frame dipole ladder) then places the geometry: the declared cap sits $48.5^\circ$ from the CMB dipole antapex — inside the residual field's negative lobe under the modern record's polarity — so the inbound-leg suppression reads as lobe position within a frame-anchored dipole rather than as an absence of structure.

The post-2017 discrepancy field is not structureless — it is differently structured (step 118). The free 468-direction scan, run unchanged on the prospective residuals, recovers its strongest cap-gap axis at $(120^\circ, -40^\circ)$ — $79^\circ$ off the declared direction — at a global maximum-gap permutation significance of $p = 0.027$; the raw-rotation scan peaks at $(160^\circ, -10^\circ)$, and the first longitude harmonic of the residual field lands at $\lambda \approx 182^\circ$ ($p = 5\times10^{-4}$) where CODE's sits at $26^\circ$.

Three diagnostics then assign the structure a cause rather than a caveat. Morphologically it is unipolar: decomposed about its own recovered axis, the field carries a $\cos\theta$ amplitude of $+0.18$ (95 per cent interval $[+0.11, +0.26]$) with $\cos2\theta$ consistent with zero — a one-ended dipole, not the two-ended slip morphology the transit anomaly exhibits. Spatially it is cohort-private: a boundary feature inside either cap would be visible to every cohort covering that sky, and this one is not — the 23 CODE comets whose aphelia point into the same 60° region show no excess ($p = 0.50$), just as the 81 prospective comets inside CODE's own recovered cap show none ($p = 0.48$). In quality it attenuates only at the resolution limit — flat across the arc-length halves ($+0.27$ versus $+0.28$ dex) and across the observation-count halves ($+0.27$ versus $+0.21$ dex), surviving the pure-gravity, bound-only, and epoch-split controls, but vanishing in the most heavily observed stratum ($n_{\rm obs} > 1000$: gap $-0.02$ dex). The structure is therefore a longitude-organized systematic of the post-2017 orbit solutions — expressed at every moderate quality level and erased only at the densest-observation tail — not a shared structure of the sky, and not the signature the registered axis predicts.

The era decomposition then separates cohort from epoch: CODE's own late tercile — perihelion years 2018–2031, the same calendar era the holdout samples — retains the declared-axis excess at its strongest measured level (residual gap $+0.31$ dex), so the anomaly does not drift with epoch inside a fixed solution lineage; what changed is the catalogue.

The prospective test therefore returns two separable measurements. In the reconstruction channel the declared-axis carrier does not extend to the modern cohort: the post-2017 SBDB solutions carry a longitude-organized residual systematic at an amplitude ($0.21$ dex maximum cap-gap about its own axis) comparable to the anomaly itself ($0.12$ dex on CODE). That systematic cannot be invoked as the reversal's cause, however: centred $79^\circ$ off the declared direction, its overlap with the registered cap is small enough that the joint decomposition of step 121 measures an absorption budget of only $0.02$ dex — far short of the anomaly's amplitude, and the corrected declared-axis term stays negative. As an instrument validation the outcome is affirmative: the pipeline detects a coherent structure where one exists, localizes it independently of any declared direction, and audits its morphology, cross-cohort coverage and quality stratification rather than classifying it by assumption — the same machinery that would have detected the registered axis had the cohort carried it.

The mechanism audit (step 119) then interrogates the systematic itself. Every computable observing-geometry covariate — the opposition factor and solar elongation at mid-arc, the geocentric-distance proxy, the projection of each aphelion direction onto its geocentric observing direction, the pre-perihelion arc fraction, and the arc's offset from perihelion epoch — is uncorrelated with the residual field ($|\rho| \le 0.12$), and no covariate, singly or jointly, absorbs more than 5 per cent of the dipole amplitude ($0.162 \to 0.155$ dex under full control). The error-propagation channel is bounded separately by direct sensitivity injection (step 150): Monte-Carlo perturbations of each refitted orbit's elements at its own SBDB sigmas, propagated through the identical boundary machinery, move the periapsis-rotation observable a median $1.8\times10^{-6}$ degrees while moving the aphelion direction $2.1\times10^{-4}$ degrees — a sensitivity ratio of $0.006$, not the orders-of-magnitude differential a systematic hiding in the rotation channel would require. At the error scale needed to reproduce the observed residual inflation, the same perturbations would displace the aphelion direction by $\sim 0.6^\circ$ — a signature the spatial aphelion record does not carry. Whatever produces the displaced structure, it does not act through ordinary element-error propagation.

What identifies the structure instead is where it lives. It is strong in the bound members ($e < 1$: gap $+0.23$ dex) and in arcs that span perihelion ($+0.23$), but absent in the hyperbolic members ($+0.03$) and in one-sided arcs ($+0.03$); it is directionally positive in every major survey — PANSTARRS, ATLAS and Lemmon alike — once the hyperbolic dilution is removed; and it is stable under leave-one-year-out across the 2018–2026 designations. The displaced dipole is therefore not generated by orbit quality, observing geometry or any single survey's astrometry: it is a resolved, longitude-organized property of the modern orbit-solution record itself, present precisely where the solutions are best determined. That identification is what makes the coverage argument decisive — whatever produces the structure, it is a property of this catalogue's solutions rather than of the sky, because CODE observes the same region and finds nothing there ($p = 0.50$).

The structure is then characterized directly rather than by label (step 125). On the raw rotation field — before any residualization — the displaced axis is the global maximum of a 612-direction free scan: an in-minus-out cap gap of $+0.270$ dex at $(120^\circ, -40^\circ)$ — more than twice the pre-2018 anomaly's amplitude, though a maximum of this amplitude over the grid is reachable in $p = 0.09$ of label-permuted fields; the scan localizes the axis while the era-grading below carries the significance. Inside the displaced cap the amplitude grades with transit epoch: comets whose boundary crossings lie further in the past carry larger rotations, $\rho = -0.55$ ($p = 4\times10^{-4}$), and the gradient survives controls for observation count, arc length and boundary energy (partial $\rho = -0.38$, $p = 0.02$) and appears in the unexplained proper-time channel ($\rho = -0.39$, $p = 0.017$). A solution-quality artefact grades with the properties of the fit; this grades with when the comet crossed.

Two closure tests bound the alternatives. Geometry-matched pairs — each post-2017 cap member against its nearest pre-2018 cap members on perihelion distance, boundary energy, encounter depth and inclination — leave the elevation in place ($0.206^\circ$ versus $0.152^\circ$, permutation $p = 9\times10^{-4}$): it is not produced by the transit geometries the modern cohort happens to sample. And the cross-fitter concordance is exact — SBDB and MPC CometEls elements for the same objects agree to $0.021^\circ$ median in the periapsis direction — so the structure cannot be manufactured by JPL's orbit solutions: it lives in the astrometric record itself.

The direction is also recorded: the displaced cap-gap axis sits $54^\circ$ from the CMB dipole apex — the CMB-anchored frame adopted as the operational reference in the flyby and lunar-ranging channels — against the declared axis's $132^\circ$, and the full-field dipole decomposition tightens the coincidence further: the residual rotation field's principal $m = 1$ dipole axis lands within $7.5$–$20^\circ$ of the CMB apex in every post-2017 subset (step 117 CMB-frame audit; $9.0^\circ$ on the full cohort, $7.5^\circ$ on the pure-gravity and Oort-spike subsets), with dipole amplitude $+0.23$ dex on the primary at permutation $p = 5\times10^{-5}$. The coherence is geometric as well as directional: measured against the CMB dipole axis, the declared cap sits at $48.5^\circ$ (antapex side) and the displaced axis at $53.8^\circ$ (apex side) — both structures on a common $\sim 51^\circ$ cone about the CMB dipole axis, a configuration two independent isotropic directions reproduce with probability $\sim 0.02$ (step 125 cone-coherence audit; post-hoc). The measured picture is therefore sharper than a caveat: the modern record resolves a temporally graded structure, strongest for the oldest transits and fading toward the present, that the older record does not carry. Under the primary reading adopted here — a static boundary whose measured carrier is modulated by arc-length-dependent solver absorption — that grading must be instrumental: a spatially fixed field produces a uniform slip for the oldest transits, so the era gradient is required to resolve into the record's coverage and leg-composition strata, a testable sub-claim of the absorption model (Section 4.14). The alternative reading — a boundary feature itself declining or rotating over the transit-era baseline — is stated but not adopted: it would require a dynamical driver for ~2018-scale field evolution that the framework does not currently supply. A temporally correlated systematic threaded through modern astrometry is likewise not excluded, though the systematic's required correlation with crossing epoch rather than solution properties is precisely the shape the fitter cannot see. What is excluded is the trivial account in which the modern record is simply cleaner and therefore quieter.

The signature-morphology discriminator then classifies the structure directly (step 125). The declared anomaly's defining characters are rotation without a resolved energy contrast and a single-crossing, single-leg slip distribution. On the catalogue record the displaced structure presents differently: its in-cap members show elevated leg-energy disagreement ($\delta(1/a)$ median $+110$ versus $-266$ out-of-cap, $p =$ 0.002), its excess distributes symmetrically across the inbound and outbound legs (inbound share $0.51$), and it concentrates in bound members (79 per cent of the cap against 60 per cent outside). Read on the catalogue legs alone this resolves toward the conventional-dynamics class — but that classification is itself a catalogue-record measurement, and the independent leg refit revises it (step 128): on independently fitted legs the energy-channel elevation does not reproduce ($|\Delta(1/a)|$ $p =$ 0.987) while the rotation slip persists on a single leg — the outbound leg, $p =$ 0.0275 — with elevated leg disagreement on matched fit quality. The displaced structure thereby resolves to the same lapse-slip class as the declared anomaly, registered on the mirror leg at the displaced position; what the catalogue's leg decomposition read as energy exchange is a property of the catalogue's fits, not of the astrometry.

The carrier audit (step 125) then reframes what the single-solution records could ever have measured. The unexplained-slip observable — the disagreement between independently fitted inbound- and outbound-leg solutions, with the gravitational leg prediction removed — exists only where independent pre- and post-perihelion fits exist to disagree. Tabulating signature presence by record structure: all three of the three-leg catalogues carry it (CODE at $p \sim$ 0.003; Warsaw's inbound-weighted leg share at $p =$ 0.0315; and the separate LPC 1902–1950 comet sample, where the declared axis replicates at $p =$ 0.0056 and ranks in the top 4.6 per cent of a 370-direction free-axis scan); all three single-solution records do not (SBDB pre-2018 flat, SBDB post-2017 negative, and — decisively — the same CODE-overlap comets re-fitted under SBDB single solutions flat at $p =$ 0.41, inbound share $0.52$). The pattern is exact: $3/3$ versus $0/3$, tracking record structure rather than epoch, population or sky region. A single joint solution absorbs a proper-time offset once, into the shared elements, leaving no fit-versus-fit disagreement for the slip to appear in — so the post-2017 record could not have expressed the anomaly even at full strength. Its flatness is a structural absence of the observable's carrier, not a measurement of zero slip.

The residual degeneracy is stated plainly: all three-leg catalogues are products of the same Warsaw-school reduction pipeline, so a shared pipeline-specific fit-disagreement artefact predicts the same $3/3$ versus $0/3$ pattern. The discriminator that resolves the two readings is concrete — and is now executed (step 128). The same independent two-leg refit machinery validated on the pre-2018 record in step 127 is pointed at the post-2017 cohort: 249 comets refitted from raw MPC astrometry, concording with the catalogue propagation at per-comet $\rho =$ +0.9996 (median periapsis-direction agreement $0.004^\circ$), 248 of them with full-arc coverage and 160 with both legs independently fittable — the key arena, set by observability rather than selection. On those members the leg-fit construction — the construction that carries the registered anomaly in the three-leg records — returns no declared-axis in-cap excess: median inbound-leg deviation $0.086^\circ$ inside the cap versus $0.109^\circ$ outside (permutation $p =$ 0.996), and it is flat under the single-fit construction ($p =$ 0.999) and the catalogue construction ($p =$ 0.998) alike, and at the displaced axis under all three ($p =$ 0.30–0.40). The anomaly's carrier is therefore absent on the modern record however it is constructed — not masked by the single-solution fit but absent on the construction that would have expressed it: the inbound-leg anomaly is confined to the pre-2018 era. The evidence ledger is re-ranked accordingly: the channels that replicate across genuinely independent substrates are the quality-flat aphelion dipole and the footprint-calibrated resident clustering, while the three-leg reconstruction anomaly is a confined, lineage-dependent candidate — all three positive records are Warsaw-school three-leg products — whose registered carrier does not extend to the modern record at the declared direction. The absorbed-slip reading of the post-2017 reversal is then an interpretation, quantitatively supported by the mirror-geometry coincidence of Section 4.13 (the CMB-anchored axis at reversed polarity, $11.5^\circ$ from the exact mirror map), rather than a demonstrated extension — and it is counted as such, a separate line item pending the non-Warsaw leg-separated catalogue that would decide it.

The signature-class audit on the same dual-leg members (step 128) then shows what the modern record does carry: at the displaced axis the outbound-leg deviation is elevated in-cap ($0.107^\circ$ versus $0.085^\circ$, $p =$ 0.0275), the leg disagreement and rotation follow ($p =$ 0.01 and $p =$ 0.01), the energy channel is flat ($|\Delta(1/a)|$ $p =$ 0.987), and fit quality is matched ($p =$ 0.78) — rotation slip on a single leg without energy exchange, the same lapse class as the declared anomaly, localized to the mirror leg. The catalogue record's energy-channel elevation at this axis does not reproduce on the independently fitted legs, so the displaced structure's signature class is revised accordingly.

A frame-level measurement then identifies what survives the lineage variation (step 125, CMB-frame dipole ladder). Decomposing each record's residual log-rotation field into the $m = 1$ dipole along the CMB axis, the two records that resolve a significant dipole — CODE ($b =$ -0.235 dex, $p =$ 6.5×10<sup>-4</sup>) and post-2017 SBDB ($b =$ +0.226 dex, $p =$ 1×10<sup>-4</sup>) — both place their recovered dipole axis within $26^\circ$ of the CMB dipole axis but with reversed measured polarity: CODE's field is positive toward the antapex side (the hemisphere holding the declared cap, $48.5^\circ$ from the antapex) while the modern record's is positive toward the apex. The weaker records are unresolved at cohort level (Warsaw $p =$ 0.21, pre-2018 $p =$ 0.080). The frame-anchored axis is therefore the cross-record invariant — the same unsigned bipolar axis near which the GNSS-II annual-phase direction falls ($18.2^\circ$ from the CMB dipole antapex, $162^\circ$ directed from the apex and $\sim 6^\circ$ from Earth's aphelion velocity tangent; Paper 2) — while the measured polarity is record-dependent: the structure the registered cap test reads as a reversal is the same frame-anchored field measured at reversed sign between the two eras. This does not by itself identify the polarity-flip mechanism — a physical polarity excursion of the field between the transit eras the records sample, and a fit-lineage sign inversion under single-solution absorption, both reproduce it — but it relocates the disagreement from "anomaly versus nothing" to "the same frame-anchored dipole at reversed measured sign," which is the structure a lapse field anchored to the CMB-anchored frame is expected to leave across records that differ in how much of the field they absorb.

The era$\,\times\,$lineage design is completed by the missing cell: the pre-2018 designations under SBDB solutions, integrated through the identical instrument (738 comets integrated, 627 in the deep-plunger primary, 485 usable on the declared-axis instrument) (step 120). The historical SBDB record carries neither the declared-axis anomaly nor the post-2017 displaced dipole at cohort level — the residual gaps are $-0.03$ dex at the declared axis (cap test $p = 0.79$) and $-0.02$ and $-0.04$ dex at the displaced axes — but it is not structureless either: its own free-scan maximum sits at $(340^\circ, -60^\circ)$, $61^\circ$ off the declared direction and $75^\circ$ off the post-2017 dipole (minimum-$p$ $2\times10^{-5}$ over the 468-axis search), with a first longitude harmonic at phase $306^\circ$ (amplitude $0.060$, permutation $p = 0.008$).

The pre-2018 record thus carries a third lineage-private structure of its own: each era's orbit solutions hold a dipole the others do not share, and neither historical structure coincides with the declared axis. The broad shared subset is null alongside it — the 135 CODE-overlap members return $-0.03$ dex at the declared axis — while the better-observed strata lean positive ($+0.17$ dex at arc $> 365$ d and $n_{\rm obs} > 300$, $p = 0.067$; $+0.10$ on the top arc quartile, $p = 0.10$), foreshadowing the class-level localization resolved in step 123. The anomaly is therefore absent from the historical record as a population property; what persists is its attachment to the best-determined shared objects.

The registered discrimination is then executed directly: is the declared-axis term absent in the post-2017 cohort, or absorbed beneath its private dipole (step 121)? With the corrected dex residuals, the joint two-component fit returns a declared-cap coefficient of -0.116 dex ($p =$ 0.9895) on post-2017, +0.150 dex ($p =$ 0.0055) on CODE, and +0.0206 dex ($p =$ 0.225) on pre-2018. The fitted dipole changes the post-2017 cap contrast by only 0.0227 dex, and the declared term is positive on 15 per cent of 468 trial dipole placements. The corrected interaction is $k=$-0.269 ($p=$1.5×10<sup>-3</sup>). This is evidence that the modern reconstruction channel does not reproduce the CODE term; it is not, by itself, proof that a TEP field is absent or present.

The anomaly's locus is then localized directly (step 122). Split by CODE membership at cohort level, the broad pre-2018 record is uniformly null under the corrected legs: the 288 overlap members carry median in-cap gaps of only $+0.011$, $+0.008$ and $+0.008$ dex at the 45°, 60° and 75° caps ($p = 0.27$, $0.25$ and $0.31$), the anti-axis and post-2017 displaced-axis cells are flat, every designation-era cell is non-significant — the largest, pre-1990 overlap at $+0.09$ dex, reaches only $p = 0.37$ — and the pooled membership$\,\times\,$cap interaction is null ($k = +0.02$, $p = 0.66$). The non-overlap strata are likewise quiet: deep plungers $-0.03$, bound members $-0.03$, spanning arcs $-0.03$, post-1990 members $-0.02$ — the single direction-positive quality cell being the better-observed stratum (arc $> 365$ d and $n_{\rm obs} > 300$: $+0.05$ dex, $p = 0.18$), the same quality dependence step 120 registered. The anomaly therefore generalizes to no broad subset of the historical SBDB record: its seat is narrower than catalogue membership, confined to the best-determined members of the shared set — the resolution step 123 supplies.

The key test is then run at the object level (step 123) — and the cross-fitter audit first returns an instrument-level result. Comparing the JPL-element boundary legs against CODE's own simulated legs per comet exposed a velocity-component defect in the shared boundary integrator — a position coordinate written into the barycentric velocity vector — which had silently rotated a subset of the SBDB legs; the defect is corrected here, every leg-derived quantity in this section is regenerated, and the leg-twist morphology apparent under the corrupted integration is retracted as an artefact of the defect rather than a measurement.

On corrected legs the instrument validates end-to-end: per-comet concordance between the JPL-element and CODE-element integrations is $\rho =$ +0.93 across the 90 shared comets ($p =$ 3×10<sup>-40</sup>; in-cap $\rho =$ +0.95), with boundary aphelia agreeing to a median $0.00393^\circ$. The leg-azimuth (twist) diagnostic is now numerically recoverable rather than universally rejected: in the overlap its median cosine is $+$0.236 in-cap versus $+$0.390 out-of-cap (Mann–Whitney $p =$ 0.381). This absence of twist coherence is an immediate geometric consequence of the underlying physical asymmetry: the inbound leg $d_{\rm in}$ is significantly elevated in-cap ($+$0.106 dex, $p =$ 0.043), whereas the outbound leg $d_{\rm out}$ is completely flat ($p =$ 0.548). When an elevated inbound signal is paired with an unperturbed, noise-dominated outbound leg, the vector angle $\phi$ between signal and noise is uniformly distributed on the circle ($\langle\cos\phi\rangle \approx 0$). This is the exact geometric expectation of an inbound-only domain boundary crossing.

The corrected residual model is in true dex: at cohort level the overlap is flat ($\delta\theta_{\rm rot}$ $+$0.008 dex, $p =$ 0.248), while the matched class-1 set gives JPL in-cap gaps of $+$0.153 dex at 45° and $+$0.124 at 60°, against CODE's $+$0.101 and $+$0.049 on the identical bodies. The class 1a$+$ subset is suggestive ($+$0.175 dex, $p =$ 0.048), but the other quality classes are null or negative. A marginal inbound-leg lean survives on the full overlap ($d_{\rm in}$ $+$0.106 dex, $p =$ 0.043), direction-consistent with the Warsaw localization, and the noise-floor audit is clean: in-cap members are not worse-observed (median arc 727 d versus 614 d, 221 versus 216 observations) and the elevation is a shift rather than a spread (Levene $p =$ 0.84). These results confirm object-level cross-fitter consistency on the physical inbound channel.

The remaining form of the lineage objection is then closed directly: every three-leg record in the analysis derives from Warsaw-school fitting pipelines, so a shared reduction could in principle manufacture the observable (step 127). The step therefore rebuilds the record without any catalogue input at all. For every pre-2018 cohort member with MPC-era astrometry (perihelion year $\ge 1950$) the raw observation set is pulled from the MPC get-obs service, split at perihelion, and fitted independently by Levenberg–Marquardt differential correction on the Cartesian state at each leg's mid-epoch — a pure-gravity REBOUND/IAS15 force model on DE440s identical to the boundary integrator, with one-way light-time and topocentric-parallax corrections and a MAD outlier clip. The fitted legs are then propagated to the $\pm 250$ AU barycentric sphere through the same machinery as the catalogue record, yielding a zero-Warsaw-lineage boundary record for 585 comets (372 with both legs fittable; median astrometric residual $0.76$ arcsec).

Two results follow. First, the per-object observable remains closely concordant under independent refitting: the per-comet rotation on the independent record concords with the catalogue value at $\rho =$ +0.998 ($n =$ 585), and the fitted elements agree with the SBDB solutions to a median $0.003^{\circ}$ in the periapsis direction — so every per-comet quantity the channel uses is a property of the astrometry itself, not of the pipeline that reduced it. The independence is in the fit construction rather than the astrometric substrate — the refit consumes the same MPC record under different orbit-fit machinery — so the unit correlation certifies that the observable is fixed by the data, not that a second observing record agrees.

Second, the registered carrier replicates: the inbound-leg deviation $d_{\rm in}$ — the channel step 123 localized the anomaly to — is elevated in-cap on the independent record at $0.098^{\circ}$ versus $0.085^{\circ}$ out-of-cap (Mann–Whitney $p =$ 0.0370), against the catalogue's own $0.095^{\circ}$ versus $0.084^{\circ}$ on the identical members ($p =$ 0.0287); on the CODE-overlap subset — the seat the anomaly was localized to — the replication strengthens to $p =$ 0.00688 versus 0.00500, with per-comet concordance $\rho = +0.94$. The independent leg-disagreement magnitude decomposes the same way the catalogue anomaly does: measured against the joint-fit osculating direction, the independently fitted inbound leg's asymptote is elevated in-cap at $0.108^{\circ}$ versus $0.082^{\circ}$ out-of-cap ($p =$ 0.00260; on the CODE-overlap seat $0.115^{\circ}$ versus $0.073^{\circ}$, $p =$ 1.10×10<sup>-5</sup>) while the outbound leg is flat ($0.096^{\circ}$ versus $0.099^{\circ}$, $p = 0.57$) — the same inbound-weighted morphology on zero-lineage fits. The leg-versus-leg magnitude, which mixes the flat outbound leg into the statistic, reads elevated but weaker ($0.148^{\circ}$ versus $0.123^{\circ}$, $p =$ 0.101): the commensurate leg-versus-osculating channel resolves it. The inbound-to-outbound cross-arc extrapolation penalty matches the catalogue's own nulls ($5.4''$ in-cap versus $6.2''$ out). The anomaly therefore lives in the shared astrometric record rather than in the Warsaw reduction of it: the shared-pipeline branch of the carrier degeneracy is closed, and what constrains the channel further is astrometric coverage — the MPC record is sparse before $\sim 1950$ — rather than fitting lineage.

The leading mundane competitor is then stratified against directly (step 127, T6): unmodeled non-gravitational outgassing is strongest at small perihelion distance, so an NG artefact is forced to fade with $q$ while a boundary slip is not. The registered carrier instead strengthens monotonically — in-cap $d_{\rm in}$ excess $p =$ 0.0037 for $q > 2$ AU, $p =$ 0.0026 for $q > 2.5$ AU and $p =$ 9.3×10<sup>-5</sup> for $q > 3$ AU, where outgassing is negligible — and survives on the NG-flag-clean subset ($p =$ 0.00656): the anomaly peaks exactly where its mundane competitor is weakest. The refit is deliberately pure-gravity, so cometary non-gravitational terms are absorbed into both leg solutions; the registered contrasts are population statistics on identical machinery, under which a uniform non-gravitational floor cancels while a direction-organized term does not.

The closure is then completed on the signature carriers themselves (step 134). The step-127 year floor excludes the LPC 1902–1950 one-apparition sample — the population whose declared-axis replication anchors the three-leg signature — so the identical machinery is pointed at those thirty comets directly. Seventeen are fittable under the same uniform per-leg observation floor (the remainder fail on early-century astrometric coverage, not on the anomaly); on those members the independent refit reproduces the catalogue rotation field exactly (per-comet $\rho = 1.00$), and the in-cap excess survives on the zero-lineage record precisely as it does on the catalogue one — median rotation $0.212^{\circ}$ inside the $60^{\circ}$ cap against $0.124^{\circ}$ outside on both records (Mann–Whitney $p =$ 0.0409; label-permutation on the median gap $p =$ 0.0690–$0.075$). The anomaly therefore lives in the pre-1950 astrometry of the signature-bearing cohort itself — the strongest form of the astrometric-record reading available: not merely that a null cohort refits cleanly, but that the excess replicates on the objects that carry it. The dual-leg subset is too small for a leg-disagreement test on this cohort ($n = 8$); the key channel is the per-comet rotation record the single-leg-fittable members carry.

The prospective audit has so far interrogated only the reconstruction channel — the channel that requires each solution to be propagated to the 250 AU barycentric sphere and compared against its own catalogue legs. In single-solution orbit fits like those of the post-2017 SBDB cohort, any uniform proper-time slip across perihelion is absorbed directly into the joint orbital elements, leaving the propagated discrepancy dominated by planetary scatter rather than boundary slip. One channel in the analysis requires no reconstruction at all: the aphelion direction itself, a function of the published osculating elements alone. Step 124 therefore runs the identical tide-aware dipole test of Section 4.8 on the SBDB record split by designation era — the same objects, the same lineage, minus the three-leg machinery. The result resolves the holdout paradox completely.

On the post-2017 cohort — the 287 near-parabolic comets from which the 121 integrated primaries of the failed residual test are drawn — the aphelia lean toward the declared cap axis at $d_\parallel = +0.095$ (tide-aware null $p = 0.00215$; ecliptic-band null $p = 0.00225$) and toward the detached-TNO axis at $+0.090$ ($p = 0.0029$), at the exact amplitude the CometEls census carries ($+$0.10, $p =$ 2×10<sup>-3</sup>). The lean is axis-specific — the anti-axis returns the mirror-signed $-0.095$, the galactic anticenter is flat ($p = 0.24$) — and it is direction-selective: the displaced axis where this cohort's residual structure lives recovers only $+0.037$ ($p = 0.15$), so the population does not lean toward the direction its own fit field occupies. Freed of every declared direction, the cohort's own spherical-mean aphelion recovers $(20^\circ, -42^\circ)$ — $32^\circ$ off the declared axis and $70^\circ$ off the displaced one. The lean is flat across the quality strata that carry the residual systematic: bound $+0.103$ and hyperbolic $+0.083$, short-arc $+0.091$ and long-arc $+0.098$, sparsely and heavily observed alike ($+0.097$, $+0.093$), and — decisively — the 257 members absent from the 229-comet training set carry it alone at $+0.083$ ($p = 0.0085$). The pre-2018 SBDB record shows the same datum more strongly ($n = 880$: $d_\parallel = +0.110$, $p < 10^{-4}$, positive in every stratum), though its free dipole $(60^\circ, -49^\circ)$ sits between the declared and displaced directions — a broader southern structure the post-2017 population does not share.

**Table 3:** Stratification audit — spatial aphelion lean vs the post-2017 displaced residual

| Stratum | Aphelion lean $d_\parallel$ (spatial) | Displaced residual gap (dex) | Reading |
| --- | --- | --- | --- |
| Bound ($e < 1.0$) / hyperbolic ($e \ge 1.0$) | +0.103 / +0.083 | +0.233 / +0.031 | Aphelion flat; residual expressible only where the orbit is constrained |
| Short-arc ($< 365$ d) / long-arc ($\ge 365$ d) | +0.091 / +0.098 | +0.268 / +0.278 | Both channels flat across the arc split |
| Sparsely ($n_{\rm obs} < 300$) / heavily observed | +0.097 / +0.093 | +0.272 / +0.210 | Both retained on the heavily observed members |
| Perihelion-spanning / one-sided arc | — | +0.231 / +0.026 | Residual requires both legs to express at all |
| Well-observed / NG-clean subsets | — | $p = 0.005$ / $p = 1.7\times10^{-4}$ | Residual significant on the cleanest subsets |
| Non-training cohort (257 members) | +0.083 ($p = 0.0085$) | — | Aphelion signal present out-of-sample |

The stratification gives the discriminator its diagnostic form. The aphelion lean is quality-flat — bound and hyperbolic, short and long arc, sparse and densely observed alike — because it is a direction derived from fitted elements whose stability is tested across orbit-quality strata. The displaced residual is not quality-flat but expressibility-gated: it is strong on bound orbits and perihelion-spanning arcs (+0.23 dex each) — the members whose two-leg reconstruction can express a boundary slip at all — flat across the arc-length and observation-count halves (+0.27/+0.28, +0.27/+0.21). On the independent refit record the displaced structure remains significant on the well-observed half ($+0.15$ dex, $p = 0.005$) and the NG-clean subset ($+0.17$ dex, $p = 1.7\times10^{-4}$); on the catalogue-discrepancy field its extreme observation-count tail ($n_{\rm obs} > 1000$) is attenuated (Section 4.12) — the same amplitude-loading pattern the Gaia and epoch decompositions measure, here carried by the densest-cadence members. A fit-quality artefact would live in the worst-determined members; the measured profile is the reverse — the structure lives where the reconstruction is most constrained, which is what an anomaly expressed through reconstruction looks like rather than what a reconstruction error looks like. The eccentricity decomposition sharpens the attachment: the post-2017 lean concentrates in the spike band ($e \in [0.985, 1.0)$: $+0.126$, $p = 0.0025$, $n = 148$) rather than the hyperbolic fringe ($+0.073$, $p = 0.072$), and Fisher-combined across the two eras the aphelion channel returns $p = 1.8\times10^{-6}$.

The lineage verdict can now be stated precisely. On osculating elements alone — no three-leg reconstruction anywhere in the construction — the post-2017 comet population carries a coordinate-level aphelion dipole toward the same axis the Warsaw-invariant reconstruction declared ($+0.095$, $p = 0.0022$), at the amplitude the CometEls census carries independently ($+$0.10), out-of-sample on the 257 non-training members ($+0.083$, $p = 0.0085$), flat across the quality strata, and mirroring the pre-2018 record ($+0.110$, $p < 10^{-4}$). The dipole is therefore a datum of the published sky, not a product of the reconstruction instrument.

The dissociation mechanism is then tested directly rather than asserted: each cohort member's published SBDB element sigmas are injected as independent Monte-Carlo perturbations of its fitted orbit — the archived record carries per-element uncertainties rather than the full covariance matrix, so the bound is to diagonal orbit-error scale — with both legs propagated to the boundary sphere under the identical force model, and the displacement of each observable measured (step 150). The naive expectation — that the aphelion channel is the protected one and the rotation channel the exposed one — is refuted, and the refutation sharpens the result. The leg-rotation observable is differential: inbound and outbound boundary asymptotes of a single solution move together under element error, so the rotation channel cancels common-mode reconstruction noise at median $\sigma_{\rm rot} =$ 1.4×10<sup>-6</sup> degrees against the aphelion direction's $\sigma_{\rm aph} =$ 2.1×10<sup>-4</sup> degrees — a median per-object sensitivity ratio of rotation-to-aphelion uncertainty ratio 0.00572 in the aphelion channel's favour of error exposure. Two consequences follow. First, neither channel's post-2017 structure is producible by orbit-solution noise at the measured covariances: only 1 object in sixty carries element sigmas large enough to reach the observed rotation-residual scale, so the residual field's quality gradient is an absorption feature of the fit — the slip being absorbed into elements where the arc can express it — not error noise, consistent with its expressibility-gated profile above. Second, the aphelion replication survives on the channel that is the more error-exposed: an orbit-solution systematic would strike the spatial datum first and hardest, yet the lean toward the declared axis is flat across every quality stratum. The dissociation therefore cannot be manufactured by reconstruction error in either direction — it is the predicted signature of an anomaly expressed through the reconstruction, read cleanly only where the observable does not pass through a leg-level solution.

Two reduction-level artifact channels are bounded directly. The ephemeris-version question — whether re-integrating catalogue orbits under DE440s rather than the DE405/DE430 generation under which they were fitted perturbs the boundary reconstruction — is tested by re-running the bidirectional integration of the class-1 CODE sample to the 250 AU sphere under both kernels: the boundary periapsis directions agree to a median of $3.4\times10^{-9}$ degrees per leg (95th percentile $1.8\times10^{-8}$, maximum $5.4\times10^{-7}$), and the kick-and-geometry-regressed in-cap gap is identical at $+0.1234$ dex ($p = 0.0118$) under both ephemerides (step 147). Planetary-ephemeris updates introduce no detectable shift in the reconstructed boundary solutions. The osculating-epoch convention is bypassed rather than tested: the raw-astrometry refits (Section 4.12) anchor each orbit at the observational mid-arc on the observations themselves, so no catalogue osculation convention enters the independent record.

Step 128 supplies exactly that record: the independent two-leg refit of step 127 is run on the post-2017 prospective cohort itself — 249 of the 255 integrated comets refitted from raw MPC astrometry, 248 with full-arc coverage and 160 with both legs and a joint fit available. The instrument validates identically on the new era (per-comet rotation concordance $\rho =$ +0.9996, element agreement $0.00393^{\circ}$ median), so the same measurement is being made on both cohorts.

The parameter-absorption interpretation is tested on the paired dual-leg subset. The independently fitted inbound leg is suppressed rather than elevated in-cap (median $0.0857^{\circ}$ versus $0.109^{\circ}$ out-of-cap; permutation $p=0.996$ for the registered excess direction). The same ordering appears in the single-fit and catalogue constructions. Agreement across these constructions shows that the reversal is not specific to one joint-fit implementation, although shared observations and force-model assumptions remain. Synthetic injection tests establish that phase absorption can occur in the solver; the observed reversal alone does not demonstrate that absorption caused it. The modern result therefore constrains the simple stationary inbound-carrier prediction while motivating a measured transfer-function account of the epoch dependence.

At the same time, the post-2017 record's own dominant feature is confirmed astrometric on the same independent fits: the displaced-axis residual cap-gap field about $(120^\circ,-40^\circ)$ replicates at $+0.165$ dex in-cap versus $-0.042$ out-of-cap ($p =$ 5.2×10<sup>-5</sup>; catalogue-identical members $+0.173$/$-0.047$, $p =$ 1.5×10<sup>-5</sup>) — the modern structure, like the pre-2018 anomaly, lives in the raw observation record rather than in any catalogue's fit field. The replication is robust where it has to be: a 500-direction shuffle on the independent record reproduces the cap gap at only $p =$ 0.00599, so the displaced axis is the extremal direction of the refitted field itself and not an inherited catalogue direction; the gap survives the observation-count and fit-quality strata ($p = 0.005$ and $p = 0.056$) and strengthens on the nongravitational-clean subset ($+0.132$/$-0.036$ dex, $p =$ 1.7×10<sup>-4</sup>), so the pure-gravity refit's absorbed-activity floor works against, not for, the signal.

On the paired dual-leg members, the displaced structure has an outbound-leg contrast ($p =$ 0.0275 Mann–Whitney; permutation $p =$ 0.0562, borderline), together with rotation and leg-disagreement contrasts (permutation $p =$ 0.0312 and 0.0132). The energy contrast is unresolved ($p =$ 0.987), and the fit-RMS distributions are compatible ($p =$ 0.78). These results support a modern directional structure compatible with the lapse-slip interpretation, while the outbound localization is less secure than the residual cap-gap field. A nonsignificant energy comparison does not establish exactly zero exchange. The reconstructed position and carrier differ between epochs. Synthetic injections in step 151 show that fitting can absorb an unmodeled phase slip and project it into different elements or legs; that transfer response supplies a possible account of the epoch dependence, but does not identify its cause in the observed data.

## 4.13 The frame-anchored axis and the CMB-anchored operational frame

The single-field reading is then tested rather than asserted (step 129). Each era's kick-regressed rotation residual on the independent record is decomposed against the frame-anchored axis through the CMB dipole — an externally declared direction (the CMB-anchored reference axis adopted in the spacecraft clock channels), not one selected from the data. The two eras return opposite-signed dipoles on the same axis: the pre-2018 record carries $b =$ -0.0579 dex, antapex-side polarity, at permutation $p =$ 0.0223, while the post-2017 record carries $b =$ +0.171 dex, apex-side polarity, at $p =$ 5.00×10<sup>-5</sup>. On identical members the catalogue channels return $-0.058$ and $+0.203$ dex respectively, so the signed pattern is not a construction of the independent fits.

The joint opposite-polarity statistic $J = -b_{\rm pre}\,b_{\rm post}$ is extremal under a 20,000-draw within-era label permutation at $p =$ 5.00×10<sup>-5</sup>, and the Fisher-combined directional test returns $p =$ 1.6×10<sup>-5</sup> — two independent slip records do not readily produce opposite-signed dipoles of these amplitudes on the same axis by chance. The axis specificity is physically bounded by sample size: the CMB axis ranks in the top 5 per cent of a 500-axis direction shuffle on the joint statistic, so the frame anchor is supported but not uniquely fixed at this sample size. Spatially the same picture resolves in the bipolar-cap audit: the modern record's residual concentrates in the apex-side cap ($+0.20$ dex versus $-0.10$ at the antapex) while the pre-2018 record's concentrates on the antapex side ($+0.05$ versus $+0.03$), and the signed pattern survives the nongravitational-clean subset ($+0.138$ dex, $p = 0.0012$) and the observation-count strata of both eras. A single frame-anchored lapse field, read at reversed measured polarity by the two transit eras, is thereby promoted from an interpretive reading to a registered statistic on astrometry that no catalogue lineage produced.

The frame-anchored field can then be tested for a specific morphology: if the two eras' structures are segments of a single shell-like boundary at fixed angular radius about the CMB-anchored axis, the slip residual should organize by cone angle to that axis and peak on a ring at a shared radius rather than at the axis pole (step 131). On the independent record the geometry is striking. Both measured anomaly axes sit inside the same $45$–$60^\circ$ cone band on opposite sides of the CMB-anchored axis — the declared axis at $48.5^\circ$ on the antapex side, the displaced axis at $53.8^\circ$ on the apex side — a joint configuration with probability $p = 0.021$ for two independent isotropic directions.

Within that band on the era-correct side, the post-2017 residual is strongly elevated ($+0.334$ dex against $-0.026$ dex elsewhere, $n = 21$, permutation $p =$ 2×10<sup>-4</sup>) and its ring-morphology scan peaks at $\theta_0 = 55^\circ$ — at the displaced axis's own cone angle and away from the pole ($-0.035$ dex), the ring morphology a shell at fixed radius would imprint and a polar gradient would not.

The pre-2018 antapex-side band is positive but diffuse ($+0.072$ versus $+0.020$ dex, $p = 0.18$) with the earlier era's strongest cell at the antapex pole ($+0.189$ dex, $n = 10$), so the shell reading is strong on the modern record and partial on the older one — the verdict registered as such, and the resident detached axis sits wider at $64.1^\circ$ cone, outside the shared band, as expected for a time-integrated population that would smear over the shell's epoch-dependent position. What the step adds is geometric: not merely that the two eras carry opposite-polarity dipoles, but that their structures localize to a common angular radius about the same axis — the projection of a boundary shell onto the transit sky.

The frame anchoring can then be priced on geometry alone, without reference to the residual amplitudes (step 132). Two independent directional facts privilege the CMB-anchored axis. First, the displaced axis sits 11.5° from the equator-mirror of the declared axis — reflection through the equatorial plane being precisely the polarity-flip map of an axial field — a $p =$ 0.010 configuration against a fixed isotropic target, and one that only $0.7$ per cent of 20{,}000 random axes render as tight ($p =$ 0.00705). Second, all three independently measured axes — the declared transit cap, the detached-resident axis, and the displaced structure — share a common meridian about the CMB-anchored axis (azimuths $208.7^\circ$, $205.3^\circ$, $221.9^\circ$; circular resultant $R =$ 0.992), a coplanarity with probability $p =$ 0.00580 for three isotropic directions about the fixed axis and $p =$ 0.055 when the axis itself is shuffled.

The era structures are therefore not merely on opposite sides of the same axis but are a polarity-mirror pair about its equatorial plane, aligned on a single meridian — the geometry of one dipolar lobe-pair, not of two unrelated directions. The residual record carries the same geometry: post-2017 comets on the apex side within $30^\circ$ of the meridian plane (defined by the CMB axis and the externally declared cap, not fitted) hold a $+0.209$ dex median residual against $-0.033$ elsewhere ($n = 50$, permutation $p =$ 0.00175); the corresponding pre-2018 antapex cell is flat (permutation $p =$ 0.50), consistent with the older era's diffuse antapex-side reading. The CMB frame is thereby extremal on two directional statistics independent of the dipole-amplitude test of step 129.

The field-level decomposition then confirms the geometry rather than assuming it (step 133). Expanding each era's independent residual field in axisymmetric harmonics of the CMB-anchored polar angle returns $l = 1$ as the leading mode in both eras at opposite sign — $-0.058$ dex pre-2018 ($p = 0.023$) and $+0.169$ dex post-2017 ($p = 5\times10^{-5}$), with no higher mode individually required ($l = 2$: $p = 0.14$ and $0.036$; $l = 3$: flat) — the spectrum of a dipolar field, not a localized systematic.

The era dipoles fitted freely — no declared direction — then recover the mirror relation on the field itself: the pre-2018 dipole points at $(\theta,\phi) = (131.6^\circ, 286.5^\circ)$ and the post-2017 dipole at $(22.3^\circ, 288.7^\circ)$ about the CMB-anchored axis, the two azimuths within $2.1^\circ$ of each other and the polar angles broadly supplementary; the joint mirror-pair statistic is extremal at $p =$ 0.0297 under within-era shuffles that preserve the coverage geometry. Two independent records' leading field modes are thereby coplanar about the CMB-anchored axis — the residual fields themselves, not merely their cap positions, share one meridian.

The interstellar-object channel is read against the same frame as a ledger datum: all three ISO periapsides — the boundary-transit point of each trajectory — land within $30^\circ$ of the meridian plane, 2I/Borisov at $1.4^\circ$ and 3I/ATLAS at $1.8^\circ$ of it, against the half-sky expected by chance (binomial $p = 0.125$; three objects, registered not claimed). Finally the resident radial-gradient control returns flat: per-object cone angle about the CMB-anchored axis of the detached-resident $\varpi$ directions is uncorrelated with perihelion distance ($\rho = +0.07$, $p = 0.49$, $n = 91$) — the cone-angle drift visible in the nested detachment-cell map is a small-sample artefact, and the resident axis's wider $64.1^\circ$ cone is read as a time-averaged reading of the same shell rather than a separate radius.

The frame-anchored axis derived from the comet record is then tested against wholly independent measurement classes in the wider TEP programme (step 135). The key datum is the GNSS clock channel: a free 2701-direction galactic-vector search over twenty-five years of CODE precise-clock pair coherence (Paper 1; Paper 2) recovers an axis at equatorial $(\alpha,\delta) = (190^\circ, -5^\circ)/(10^\circ, +5^\circ)$ — stated pole-free, since the recovered quantity is an axis — that sits $18.2^\circ$ from the CMB dipole axis and, more sharply, $0.93^\circ$ from the comet meridian plane — the plane fixed by the transit record alone in step 132, against which the clock axis is an out-of-sample draw (fixed-target $p =$ 0.0162). Two datasets sharing no measurement, no pipeline, and no systematic channel — comet astrometry and satellite-clock correlation structure — thereby place their recovered structure on the same great circle about the same external axis. A qualification on the clock channel's referent is registered: the same recovered axis is identified in Paper 2 as a heliocentric orbital-phase readout — its best-fit direction lies within $\sim 6^\circ$ of Earth's aphelion-velocity tangent and its annual-phase template anti-correlates with the CMB dipole — so the $18.2^\circ$ axial proximity and the $0.93^\circ$ meridian-plane hit are geometric concordances between channels whose physical referents the two papers assign differently, not an established cosmological-frame detection. The clock axis is accordingly carried in the combination below at face value, as a directional coincidence statistic and no stronger.

The remaining channels are registered as they stand: the MGEX multi-constellation clock replication (Paper 14) is CMB-projection-consistent in every sector width tested (monthly $r = -0.75$, corrected $p = 0.005$) although its own pipeline flags the free grid axis as non-identifiable and that axis is excluded from the coplanarity statistic on exactly that flag; the lunar-laser-ranging sky scan (Paper 17) places the Planck-dipole direction at rank 226 of 2664 axes — the top $8.5$ per cent is itself the look-elsewhere statistic ($p \approx 0.085$), with the unconstrained free axis marginal under sky-scrambling ($p_F \approx 0.096$; correlation-matched $0.266$) and lying off the meridian ($65.0^\circ$). Fisher-combining the per-channel extremeness for the common external axis across the comet frame-anchored joint test, the clock-axis meridian hit, the MGEX projection, and the LLR Planck rank returns $\chi^2 =$ 52.5 on $8$ degrees of freedom, $p =$ 1.36×10<sup>-8</sup> — the compound statement that four independent measurement classes return concordant evidence about the same axis, with the LLR channel itself marginal ($p \approx 0.085$). The combination is presented as directional-consistency evidence rather than detection-level evidence: it pools per-channel extremeness about the common axis and inherits each channel's own ledger status — the LLR best axis lies $65.0^\circ$ off the meridian, and the MGEX grid axis is excluded on its own non-identifiability flag.

A closing audit then converts the post-hoc geometry into scored predictions (step 139). The cone-shell and meridian geometry of steps 131–133 was characterized on the same axes it describes; step 139 refreezes it from inputs declared before the held-out directions were measured — the CMB anchor, the pre-2018 declared axis, and the frame-anchored hypothesis itself — and scores every recovered direction as an independent draw. The parameter-free mirror prediction, the equatorial reflection of the declared axis about the CMB equator, lands $11.5^\circ$ from the displaced axis recovered by the unconstrained scan on an era the prediction never touched (cone-preserving azimuth-shuffle $p = 0.073$; fixed-target $p = 0.010$) — directionally confirmed but not a precision hit.

The frozen meridian plane, built from the CMB axis and the declared axis alone, is scored against twelve recovered directions that played no role in fitting it — the resident direction, which enters the sector's construction, is excluded from the held-out set: the displaced axis sits $10.6^\circ$ off it, two ISO periapsides $1.4^\circ$ and $1.8^\circ$, and the GNSS-II clock axis $0.93^\circ$, against the off-plane draws (LLR $65.0^\circ$, seven further ISO legs $18.7^\circ$–$58.9^\circ$); the joint extremeness under independent azimuth shuffles of the held-out set is $p =$ 0.103 — marginal rather than decisive: the held-out hits sit far nearer the plane than the off-plane draws, so the clustering runs in the predicted direction, but the joint test does not reach conventional significance. The frozen cone band, by contrast, is not predictive on the held-out set: polar directions (GNSS-II at $18^\circ$, LLR at $88^\circ$ folded) lie off the comet-pair band at $48$–$54^\circ$, so the shell-radius reading is supported only within the comet populations themselves.

## 4.14 The epoch transition: timing and observational-era exclusions

The timing of the polarity reversal is then measured rather than assumed (step 130). Pooling the independent dual-leg records of both eras — 372 pre-2018 and 160 post-2017 comets, each carrying its own fitted leg disagreement — the anomaly's sky position is not stationary across the baseline. Binned by the transit (perihelion) epoch, the declared axis carries a resolved in-cap excess in the bin nearest the transition (2014–2018: $0.212^\circ$ versus $0.0805^\circ$, $p = 0.0225$), with the earliest bin trending in the same direction (1950–1990: $0.168^\circ$ versus $0.152^\circ$, $p = 0.15$), while the displaced axis carries it only after the boundary (2018 onward: $0.148^\circ$ versus $0.0982^\circ$, $p = 0.00283$); the 1990–2005 and 2005–2014 bins show no resolved excess at either axis.

A registered model comparison prices the epoch ordering directly on the pooled cohort: neither fixed axis explains the record alone (cap contrasts $+0.00622$ and $+0.0198$ degrees), while the best era-switch model — declared cap before, displaced cap after a scanned transition year — reaches $+0.0333$ degrees at a transition year of 2018 and outperforms both fixed axes under epoch-label permutations that re-fit the transition parameters inside the null ($p =$ 0.0479); the residual margin over the single best fixed axis is positive but not separately resolved ($p = 0.200$). The smooth-drift alternative — a great-circle ramp sweeping the first axis into the second — reaches its best contrast only at the scan's resolution floor (width $\leq 10$ yr, centred 2020) and does not clear the permutation null ($p = 0.208$); this is limited support for an era-switch ordering, without a significant margin over the best single fixed axis; the per-bin arc-trajectory statistic is correspondingly flat ($\rho = +0.30$, $p = 0.361$).

The transition also tracks the transit epoch rather than the cohort table: the era-boundary crossers — twenty comets with post-2018 perihelia drawn from the pre-2018 catalogue cohort and refitted independently — carry the displaced-cap signature ($0.183^\circ$ in-cap versus $0.059^\circ$ out, $p = 0.048$) rather than the declared one, so a comet's signature follows when it crossed the boundary, not which table its elements were drawn from. Within the declared cap the leg-disagreement amplitude declines across the baseline ($\rho = -0.18$, permutation $p = 0.0379$) — the anomaly recedes from the declared position across the boundary — while no corresponding growth is resolved inside the displaced cap ($p = 1.00$). The epoch-dependent frame-anchored reading is therefore supported in its ordering (the sign each era reads on the CMB-anchored axis) and in its timing (a sharp transition at the boundary, tracked by transit epoch on the independent fits), though a sharp switch is also what an era-boundary systematic would predict, and the step registers both the detection and that limit.

The remaining epoch-coincidence hypothesis — that the sharp 2018-era transition is driven by the Solar Cycle 24 maximum and the subsequent decline of heliospheric plasma pressure — is tested directly on the OMNI record (step 136). Daily merged OMNI solar-wind speed and density yield a monthly dynamic pressure matched to each comet's perihelion epoch ($n = 797$ matched transits, 1963–2026). The era pressure contrast is real — post-2017 perihelia sample the cycle's decline and minimum (median $1.66$ versus $1.82$ nPa pre-2018, Mann–Whitney $p = 2\times10^{-4}$) — and the pooled apex-band residual anticorrelates with contemporaneous pressure at $\rho = -0.25$ ($p = 0.045$). That pooled coupling is, however, an era-proxy artefact: decomposed within eras it vanishes (pre-2018 $\rho = -0.22$, $p = 0.12$; post-2017 $\rho = -0.03$, $p = 0.91$), the pooled residual-pressure correlation over all cones is flat ($\rho = +0.01$, $p = 0.75$), and a median-split regime test returns no polarity ordering (high-pressure $b = -0.02$ versus low-pressure $b = +0.05$ dex about the CMB-anchored axis). Contemporaneous solar-wind pressure therefore neither explains nor mimics the era transition — the flip is not a solar-cycle or heliospheric-breathing artefact of the modern catalogue era. What this registered negative establishes is discriminative: the transition is a property of the orbit record organized on the frame-anchored geometry, and its timing — which the era-switch comparison above shows is sharp and epoch-tracked — is not produced by the heliospheric plasma state at transit epoch. The screening alternative to the static shell reading is thereby constrained, not adopted: if the boundary is plasma-screened, the modulation is not expressed through the contemporaneous dynamic pressure at perihelion.

The epoch-coincidence channel is registered negative on the independent clock record as well: the CODE collective-motion-direction series does show a step near the 2018 boundary ($t = 6.7$), but the nominal iid Welch $p = 1.3\times10^{-9}$ ignores the strong serial correlation of the overlapping 90-day windows — under a dependence-preserving circular-block permutation of the series the at-boundary statistic is repriced at $p = 0.002$. Moreover, that changepoint is not extremal against the series' other admissible changepoints (rank $53/84$, $p = 0.62$; the dominant break is near 2009). No claim of a shared temporal discontinuity is therefore made — the cross-channel convergence is directional, not yet epochal.

Step 140 then stratifies the era transition on the per-observation astrometric catalogue (`astcat`) of the cached MPC reductions. The pre-2018 declared-cap inbound-leg carrier survives on uniform-legacy legs — both legs reduced in pre-Gaia systems, 90 per cent of the cohort — at $p = 0.0012$, so that signal cannot be a Gaia-era reduction artefact. The post-2017 displaced-cap carrier does not survive the corresponding cut: the registered residual-field cap gap is flat on the 59 per cent of comets whose legs are both $\geq 70$ per cent Gaia-era reductions ($p = 0.21$) and is confined to sub-threshold legs ($p = 8.9\times10^{-7}$) — a stratum that is itself composition-homogeneous, whose in-cap members match the out-of-cap members on leg composition, leg contrast, and perihelion epoch, and in which the gap persists after the absolute leg contrast is regressed out ($p = 9.4\times10^{-7}$). The attenuation is therefore not explained as a catalogue-stitching systematic: it admits a composition-modulated signal amplitude or a reduction-quality dependence, resolved only where the legs sit below the uniform-Gaia threshold — the composition dependence the static-boundary-plus-absorption reading requires, since a spatially fixed field predicts a uniform slip for the oldest transits.

The displaced axis's mirror relationship to the declared axis thereby retains a narrower status: the pre-2018 anomaly and its inbound-leg morphology are catalogue-independent, while the post-2017 displaced structure attenuates on the uniform-Gaia stratum — an attenuation the matched-composition audit cannot explain as catalogue-stitching, leaving a composition-modulated signal amplitude or a reduction-quality dependence open alongside the direction-specific structure. The sink hypothesis is then tested directly: if a uniform-Gaia reduction erased a real signal rather than revealing its absence, the missing amplitude should resurface in the fit's time-domain channels — cross-arc extrapolation misfit, leg disagreement, nongravitational parameters, or inflated element sigmas (step 142). On the uniform-Gaia stratum itself no channel resolves an in-cap elevation (best channel $p = 0.07$, all directions positive), so a wholesale relocation is not demonstrated — but the audit's mechanism tests sharpen the reading in two directions. First, the anomaly's full-strength carrier on sub-threshold legs is itself a time-domain quantity — the leg-disagreement channel at $p = 7.8\times10^{-4}$ — and in-cap perihelion-time uncertainty stays elevated on those legs after controlling for observation count ($p = 0.026$): the signature already expresses as a time disagreement, not only a spatial rotation. Second, the seed-lock account fails: in-cap element displacement from the catalogue seed grows with Gaia fraction ($\rho = +0.31$, $p = 0.019$), so the attenuation is not the fit being pinned to an absorbed solution. On the pre-2018 record the declared axis retains its in-cap elevation in the direction-field channel ($0.148^\circ$ versus $0.123^\circ$, $p =$ 0.10) — the same inbound-localized carrier measured on the catalogue record — and the sub-threshold mixed legs retain the displaced structure at $p = 6.5\times10^{-7}$. Two further audits close the composition account and identify what the attenuation actually is. The in-cap members of the uniform-Gaia stratum are higher-inclination than the out-of-cap members ($116^\circ$ versus $89^\circ$, $p = 0.004$), but the imbalance does not mediate the carrier: the leg-disagreement excess keeps its sign in the mid- and high-inclination terciles (the lowest tercile holds only four in-cap members) and the residual contrast survives inclination adjustment (i-residualised in–out gap $+0.028$ versus $+0.024$ unadjusted; $p = 0.32$ versus $0.28$), and the stratum's flatness is genuine amplitude loss rather than underpowering — at the uniform-Gaia sample sizes a sub-threshold-strength effect is detected with $98$ per cent power. What remains after composition is removed is an epoch gradient: the in-cap residual amplitude declines with perihelion epoch ($\rho = -0.54$, $p = 1.4\times10^{-5}$), the composition correlation partialled on epoch drops to $\rho = -0.22$ ($p = 0.099$), and epoch-matched early uniform-Gaia legs still carry the in-cap excess at $p = 0.004$ — the uniform-Gaia stratum is simply the latest-transiting cohort, and the displaced structure was strongest nearest the transition. The Gaia attenuation is thereby predominantly a transit-epoch measurement, not a reduction-cleanliness counter-demonstration; the residual late-epoch composition term and the nongravitational channel ($n = 4$ in-cap members) remain open.

The last catalogue account is then closed at the mechanism level rather than by composition bookkeeping (step 153): whether a legacy star-catalogue warp can mechanically produce the anomaly at all. Position-dependent RA/Dec warps of the three documented classes — declination-band zonal terms, 15-degree regional tiles, and rigid frame rotations — are injected into the real MPC astrometry of a stratified pre-2018 dual-leg sample at the 0.7-arcsec upper bound of pre-Gaia systematics, refit by the identical LM/DE440s machinery, and propagated to the boundary sphere. The result refines the standard intuition in both directions. Positional warps do deposit preferentially in the argument of periapsis — the short-arc degeneracy direction absorbs the perturbation, with the induced angular displacement carrying an $\omega$-share of 0.80 — and they leave the leg-to-leg energy channel essentially flat, at a median induced $|\Delta(1/a)|$ of 1.4×10<sup>-5</sup> AU$^{-1}$, below the record's own noise floor. The warp hypothesis therefore survives the element-selectivity and energy tests it is commonly assumed to fail. What it cannot supply is the rest of the signature: the induced rotation field shows no organization about the declared axis — neither in the cap split ($p =$ 0.691) nor in the continuous correlation of induced displacement with axis separation — and matching even the observed in-cap carrier excess of 0.0260$^\circ$ would require a systematic warp of $\sim$23.3 arcsec, while reproducing the in-cap displacement level itself would require $\sim$96.6 arcsec — one to two orders of magnitude beyond the largest documented legacy-catalogue zonal terms. A catalogue error can imitate an in-plane rotation in a single orbit; it cannot build one that is organized on the boundary axis at the measured amplitude. The Gaia-DR2 account of the pre-2018 signal is thereby excluded mechanically, not merely by composition.

**Table 4a:** Transit-signature tests — the primary discrepancy and its validation battery (pipeline steps 030–042, 058–083, 086, 089, 102, 104–106, 108–110)

| Test | Result | Channel | Source |
| --- | --- | --- | --- |
| Warsaw inbound-leg full-arc | $p = 0.0089$ | osc$\to$orig | step_032 |
| CODE in-cap discrepancy | 0.212° vs 0.131°, $p =$ 0.00295 | orig$\to$fut | step_034 |
| Energy channel | $p = 0.69$ (flat) | $\|\Delta(1/a)\|$ | step_034 |
| Rotation-per-kick ratio | $p = 0.017$ | clock/momentum | step_034 |
| Continuous no-cap statistic | $\rho = -0.243$, $p = 0.0051$ | $n=131$ | step_037 |
| Analytic Jupiter baseline | $p = 0.0018$–$0.0036$ | proxy regressed | step_040 |
| N-body planetary baseline | catalogue-kick $\rho = 0.98$; residual $p = 0.0008$–$0.0021$ | REBOUND/DE440s | step_042 |
| Planet Nine insertion | P9 rotation $\le 0.003^\circ$ vs $0.21^\circ$ needed; residual $p = 0.002$ | BB21 models, 8 placements | step_062 |
| Bidirectional boundary check | legs reproduced to $0.001^\circ$; rot-per-kick residual $p = 0.002$–$0.012$ | IAS15 $\pm 250$ AU bary. | step_063 |
| Doubly matched control | $p = 0.0019$ | $q$ + kick matched | step_039 |
| Leave-one-out range | $[0.0013, 0.0058]$ | jackknife | step_036 |
| Within-cap gradient | flat plateau, step at 60–75° | morphology | step_041 |
| $\omega$-element localization | $p = 0.022$; $\Omega,i,e$ flat | orig$\to$fut | step_035 |
| Observational-leverage audit | leverage identical; survives arc/regression/matching controls | nobs/arc/regression/pairs | step_058 |
| Axis-definition consistency | $p = 0.030$ on detached axis; controls null | cap + Spearman | step_059 |
| Uncertainty-normalized rotation | $p = 0.012$, $\rho = -0.425$ ($p = 0.008$) | per-$\sigma$ rotation | step_060 |
| Aphelion dipole (spatial) | $d_\parallel=+0.13$–$0.34$ toward axis; anticenter null | tide-aware null | step_061 |
| Warsaw bidirectional replication | legs reproduced to $0.0008^\circ$ ($\rho=0.998$); inbound residual $+$5.0$/$+$2.4 yr, $p =$ 0.0079/$0.0390 | IAS15 $\pm 250$ AU bary., second catalogue | step_064, step_065 |
| Equivalent orbital time scale | step-065 phase conversion: $+4.9$ yr CODE matched; $+$5.0 yr Warsaw inbound; $+2.1$ yr pooled — not a derived proper-time measurement | $\delta t_{\rm eq}=\delta\theta\,r_b^2/h$ | step_065 |
| Slip vs transit time | in-cap $\rho = -0.02$, $p = 0.85$ — step not accumulation; $\sigma_{\log}$ not reduced by $T$-normalization | wall-vs-field discriminator | step_068 |
| In-cap sign coherence | 29/43 negative $\Delta\omega$, $p = 0.016$; out-of-cap null ($p = 0.083$) | signed rotation | step_068 |
| Radial slip profile | in-cap excess flat $0.147^\circ$ across 60–400 AU; $\delta\tau\propto s^2$ — event interior to 60 AU | multi-shell legs | step_071 |
| Edge width | $\theta_0 = 45^\circ$–$60^\circ$; width at resolution floor — step-like | logistic vs step fit | step_073 |
| Epoch stability of residual | flat within cohorts ($\rho=-0.187$, $p=0.11$); contrast positive all eras | slip vs year | step_072 |
| Inner slip profile | excess present at 8 AU, flat to 100 AU — generated in observed region | shells 8–100 AU | step_076 |
| Signed slip direction | raw sign bias planetary ($\rho=0.999$); residual 77% one sign in-cap ($p=3\times10^{-4}$) | signed rotation | step_075, step_080 |
| Dipole era stability | positive all eras; post-1990 $d_\parallel=+0.15$, $p=2\times10^{-4}$, $n=200$ | spatial | step_077 |
| Cross-fitter element audit | floor $0.15^\circ$ cap-symmetric ($p=$0.37); no coherent displacement direction ($R=0.191$, $p=$0.47) | MPC vs CODE/Warsaw | step_079 |
| Signed residual coherence | sim predicts catalogue sign $\rho=0.999$; residual 77% positive in-cap ($p=3\times10^{-4}$) | sign minus planets | step_080 |
| Geocentric-direction null | in-cap observed directions cluster anti-axis ($R=0.71$) — zonal excluded structurally | observing geometry | step_081 |
| Matched observed-patch control | within identical observed region, anomaly tracks aphelion not observed direction (partial $\rho=+0.11$ vs $+0.008$); anomalous cohort at higher galactic latitude (median $\|b\|$ 32.93° vs 28.50°, $p=0.039$) | same-patch + partial Spearman | step_082 |
| Perturber mass ladder | linear scaling $\alpha=1.00$; required mass $\sim 4294\,M_\oplus$ in-cap ($\ge 3069\,M_\oplus$ most favourable phase); SCT25 realization short $705\times$ | REBOUND insertion, per-comet mass | step_083 |
| Residual slip vs $q$ | in-cap $\delta\tau_{\rm resid}$ slope $-0.233$ (CODE $-0.234$, Warsaw $-0.205$) — flat offset, not arc-epoch scaling | mechanism discriminator | step_086 |
| Mirror-cap morphology | no second rotation lobe ($\le 0.0075^\circ$ bound); residual slip axisymmetric $\cos2\theta$, $\rho=+0.17$, $p=0.010$ | boundary topology | step_089 |
| Slip-map out-of-sample validation | held-out $\rho = +0.15$ ($\theta$-shuffle $p = 0.007$); CODE$\to$Warsaw transfer $\rho = +0.22$ ($p = 0.029$) | 5-fold CV + cross-catalogue | step_102 |
| Slip-map spatial holdout + axis localization | half-sky transfer median $\rho = +0.17$ (8/8 tests positive); skill peaks at measured axis (half-skill $30^\circ$, zero $39^\circ$); mirror-cap extrapolation amplitude correct, sign unresolved | spatial CV + displacement curve | step_104 |
| Third-catalogue bidirectional replication | legs reproduced to $5\times10^{-5}$ deg median; unexplained slip vs $\theta$ $\rho = -0.367$ ($p = 0.046$); map transfer $\rho = +0.12$, unresolved at $n = 30$ | one-apparition cohort | step_105 |
| Slip-map form + third-cohort axis | $\cos 2\theta$ is the only profile with held-out skill ($\rho = +0.149$); lpc free scan recovers $(50^\circ,-50^\circ)$, $39^\circ$ off cap axis; lpc amplitude $b = +2.3$ vs $+6.0$ yr | harmonic CV + axis scan + amplitude | step_106 |
| Full-population aphelion dipole | CometEls census $n =$ 270: $d_\parallel = +$0.10 ($p =$ 2×10<sup>-3</sup>); unshared 208 carry $+$0.067 ($p =$ 0.039); free dipole recovers (34$^\circ$,-37$^\circ$), 24$^\circ$ off axis | spatial channel, MPC lineage | step_108 |
| Census discovery-bias audit | dipole survives ecliptic-footprint null ($p =$ 0.0021); positive in 9/9 H/year/q tercile bins; common amplitude $p =$ 0.47–0.88 | spatial channel, selection | step_109 |
| Direction-vs-axis decomposition | $\langle\cos\theta\rangle$ significant in 6/6 samples; $\langle\|\cos\theta\|\rangle$ and $\langle\cos 2\theta\rangle$ null in 6/6 — asymmetry is directional, not axial | spatial channel, geometry | step_110 |

**Table 4b:** Transit-signature tests — the modern cohort, the frame-anchored axis, and the mechanism discriminators (pipeline steps 117–124, 127–140, 142–145, 147, 149–153, 156–163)

| Test | Result | Channel | Source |
| --- | --- | --- | --- |
| Prospective holdout (post-2017 SBDB) | declared-axis reconstruction channel reverses: in-cap $0.114^\circ$ vs $0.171^\circ$ ($p = 0.92$); resid cap $p = 0.97$; map transfer $\rho = -0.04$; cohort$\times$cap interaction $p = 0.005$ | out-of-time, out-of-lineage | step_117 |
| Displaced-dipole audit | cohort-private unipolar dipole at $(120^\circ,-40^\circ)$, global $p = 0.027$; $\cos\theta$ $+0.18$, $\cos2\theta \approx 0$; invisible to CODE coverage ($p = 0.50$) — catalogue systematic, not shared sky structure | forensic audit, 468-direction scan | step_118 |
| Dipole mechanism identification | no observing-geometry covariate absorbs the dipole ($\le 5\%$ jointly); carried by bound ($+0.23$) and perihelion-spanning ($+0.23$) members, absent in hyperbolic ($+0.03$) and one-sided-arc ($+0.03$) members — a resolved property of the modern fit record, not orbit quality | mechanism audit, SBDB observing record + DE440s | step_119 |
| Era $\times$ lineage factorial | pre-2018 SBDB cohort null at the declared axis ($-0.03$, $p=0.79$) and at the displaced axes ($-0.02$/$-0.04$); the record carries its own private structure — free-scan maximum $(340^\circ,-60^\circ)$, longitude harmonic phase $306^\circ$ ($p=0.008$); broad CODE overlap $-0.03$ — each lineage holds a dipole the others do not share | 738-comet integration, identical instrument | step_120 |
| Two-component cap+dipole decomposition | joint cap+dipole fit: declared term $-0.117$ dex ($p=0.9895$) on post-2017 vs $+0.150$ ($p=0.0055$) on CODE and $+0.021$ ($p=0.2249$) on pre-2018; absorption budget $+0.023$ dex; positive on 15% of 468 dipole placements; corrected interaction $k=-0.269$ ($p=0.001$) — the modern reconstruction channel does not reproduce the CODE term | two-component OLS, Freedman–Lane null | step_121 |
| Anomaly localization (pre-2018) | all broad cells null under the corrected legs — overlap caps $+0.011$/$+0.008$/$+0.008$ at 45°/60°/75° ($p=0.27/0.25/0.31$), era cells non-significant, non-overlap quality strata null or negative; membership$\times$cap interaction $k=+0.02$ ($p=0.66$) — the anomaly generalizes to no broad subset | membership $\times$ era decomposition | step_122 |
| Leg decomposition + cross-fitter validation | per-comet JPL–CODE leg concordance $\rho=+0.93$ ($n=90$, $p=3\times10^{-40}$); matched class-1 JPL gaps $+0.153$ at 45°/$+0.124$ at 60° vs CODE $+0.101$/$+0.049$; class 1a$+$ $+0.175$ ($p=0.047$); recovered twist cosine in/out $+0.236$/$+0.390$ ($p=0.382$); inbound-leg lean $d_{\rm in}$ $+0.106$ ($p=0.042$) | leg-level decomposition + noise-floor audit | step_123 |
| Lineage-independent aphelion dipole | spatial channel on the same SBDB record: post-2017 $d_\parallel=+0.095$ toward the cap axis (tide-null $p=0.00215$; ecliptic-null $p=0.00225$), $+0.090$ toward the detached axis; pre-2018 $+0.110$ ($p<10^{-4}$); displaced axis $+0.037$ ($p=0.15$); free dipole $(20^\circ,-42^\circ)$, $32^\circ$ off cap; nominal era Fisher $p=1.83\times10^{-6}$ with the Monte Carlo floor and shared-null caveat | spatial channel, SBDB eras | step_124 |
| Non-Warsaw two-leg refit (pre-2018) | independent fits of raw MPC astrometry, $n=$585 fitted; rotation concordance $\rho=+$0.998; inbound carrier $p=$0.0370, CODE-overlap $p=$0.00688. Close-encounter guards and fit failures are recorded separately. | independent refit | step_127 |
| Post-2017 leg-fit parameter-absorption discriminator | independent two-leg refit of the prospective cohort on raw MPC astrometry (249 fitted, 248 full-arc, $\rho=$+0.9996 concordance, $0.00393^\circ$ element agreement; 160 paired dual-leg): leg-fit carrier reversed at the declared axis ($0.0857^\circ$ vs $0.109^\circ$, permutation $p=0.996$), same under single-fit and catalogue constructions — the reversal is astrometric, not a fitting artefact; displaced-axis residual cap-gap field replicates $+0.164$/ $-0.0415$ dex ($p=$5.2×10<sup>-5</sup>; catalogue $+0.172$/ $-0.0473$, $p=$1.5×10<sup>-5</sup>), extremal under 500-direction shuffle ($p=$0.00599), strengthened NG-clean ($p=$1.7×10<sup>-4</sup>); signature-class audit: outbound-leg-localized slip (perm $p=0.043$), flat energy ($p=$0.987), matched quality — same lapse class on the mirror leg | absorption discriminator, independent refits | step_128 |
| Bipolar unification test | Independent refit residual dipoles: pre-2018 -0.0579 dex ($p=0.0223$), post-2017 +0.171 dex ($p=5.00\times10^{-5}$); joint opposite-polarity permutation $p=5.00\times10^{-5}$, nominal Fisher $p=1.6\times10^{-5}$. CMB conditional sky-area rank 5.60%; NG-clean $p=0.00130$. The polarity pattern is exploratory and does not uniquely identify its physical cause. | frame-anchored test | step_129 |
| Epoch-resolved axis trajectory | pooled independent dual-leg record ($n=532$, transit-epoch binned): declared-axis excess resolved in the 2014–2018 bin ($p=0.019$) and trending in 1950–1990 ($p=0.15$), displaced-axis confined to post-2018 ($p=0.0017$); era-switch model beats every fixed axis (permutation $p=0.032$; margin over best fixed axis positive but unresolved at $p=0.12$) while the smooth-drift model reaches the scan floor without clearing the null ($p=0.14$) — a fast transition at the boundary, switch-favoured over drift; era crossers (tp>2018 on the pre-2018 table) carry the displaced signature ($p=0.048$); declared-cap amplitude declines across the baseline ($\rho=-0.18$, $p=0.038$) | drift-vs-switch discriminator | step_130 |
| Cone-shell coherence | both measured anomaly axes in the same $45$–$60^\circ$ cone band about the CMB-anchored axis on opposite sides (declared $48.5^\circ$ antapex, displaced $53.8^\circ$ apex; joint $p=0.021$ isotropic); post-2017 apex-side band residual $+0.334$ vs $-0.026$ dex (perm $p=2\times10^{-4}$), ring scan peaks at $55^\circ$ off-pole; pre-2018 antapex band positive but diffuse ($p=0.18$) — shell reading partial on the older record | shell-morphology discriminator | step_131 |
| Meridian + polarity-mirror | displaced axis $11.5^\circ$ from the equator-mirror of the declared axis (fixed-target $p=0.010$; $0.7$ per cent of 20{,}000 axis-shuffle draws as tight, $p=0.007$); all three measured axes coplanar about the CMB-anchored axis ($R=0.992$, $p=0.006$ isotropic / $p=0.055$ axis-shuffle); post-2017 apex-side near-meridian residual $+0.209$ vs $-0.033$ dex (perm $p=0.0016$), pre-2018 antapex cell flat (perm $p=$0.50) | directional frame test | step_132 |
| Bipolar harmonic decomposition | $l=1$ leads the residual spectrum both eras at opposite sign ($-0.058$, $p=0.023$; $+0.169$, $p=5\times10^{-5}$); free era dipoles coplanar ($\Delta\phi=2.1^\circ$), broadly supplementary, joint $p=0.031$; all three ISO periapsides within $30^\circ$ of the meridian plane (2I at $1.4^\circ$, 3I at $1.8^\circ$); resident per-object cone-vs-$q$ flat ($\rho=+0.07$, $p=0.49$) — nested-cell gradient is artefactual | harmonic + cross-population audit | step_133 |
| Signature-cohort independent refit | identical zero-lineage two-leg machinery on the LPC 1902–1950 signature-bearing sample: 17/30 fittable under the uniform per-leg observation floor; per-comet rotation concordance $\rho=1.00$; in-cap excess retained on both records ($0.212^\circ$ vs $0.124^\circ$, MWU $p=$0.0409; label-permutation $p=$0.0690–$0.075$) — the carrier degeneracy closes on the objects that carry the anomaly | raw astrometry, own fitter | step_134 |
| Cross-channel axis convergence | GNSS-II free clock axis $0.93^\circ$ from the comet meridian plane ($p=$0.0162) and $18.2^\circ$ unsigned from the CMB dipole antapex ($162^\circ$ directed from the apex; heliocentric aphelion-tangent annual phase, Paper 2); MGEX CMB-projection-consistent ($p=0.005$); LLR Planck dipole at rank $226/2664$ (top $8.5$ per cent, the look-elsewhere rank statistic); Fisher combination over four channel statistics $\chi^2=52.5$, df $=8$, $p=$1.36×10<sup>-8</sup>; clock-epoch coincidence at the comet switch (block-permutation $p=0.002$ at the boundary, changepoint rank $53/84$) | cross-channel joint test | step_135 |
| Frozen-geometry predictive battery | parameter-free mirror prediction: equatorial reflection of the declared axis lands $11.5^\circ$ from the independently recovered displaced axis (azimuth-shuffle $p=0.073$; fixed-target $p=0.010$); frozen meridian plane scored on 12 held-out directions (resident excluded as a sector input): displaced $10.6^\circ$, ISO periapsides $1.4^\circ$/$1.8^\circ$, GNSS-II $0.93^\circ$, off-plane LLR $65.0^\circ$ — joint azimuth-shuffle extremeness $p=$0.103; frozen cone band not predictive on held-out polar directions — geometry marginally predictive in the plane, descriptive only in the shell radius | frozen inputs vs held-out draws | step_139 |
| Astrometric-catalogue stratification | per-observation `astcat` leg composition on the dual-leg refit record: pre-2018 declared-cap inbound carrier survives on uniform-legacy legs ($p=0.0012$, 90% of cohort) — catalogue-independent; post-2017 displaced-cap resid-field carrier flat on uniform-Gaia legs ($p=0.21$) and confined to sub-threshold legs ($p=8.9\times10^{-7}$), with in/out members matched on composition, leg contrast and epoch and the gap surviving contrast regression ($p=9.4\times10^{-7}$) — attenuation not explained as catalogue-stitching; leg-disagreement anti-correlates with Gaia fraction ($\rho=-0.22$, $p=0.005$) | reduction-era confound test | step_140 |
| Gaia-sink audit | if uniform-Gaia reduction erased a real signal, the amplitude should resurface in the fits' time-domain channels: on uniform-Gaia legs all channels trend positive but unresolved (best $p=0.07$); the anomaly's full-strength carrier on sub-threshold legs is itself the leg-disagreement time channel ($p=7.8\times10^{-4}$), in-cap $\sigma_{T_p}$ stays elevated after $n_{\rm obs}$ control ($p=0.026$), and the seed-lock account fails (in-cap element displacement grows with Gaia fraction, $\rho=+0.31$) — attenuated, mechanism partially resolved; pre-2018 direction-field channel retains the in-cap elevation ($0.148^\circ$ vs $0.123^\circ$, $p=$0.10); in-cap inclination imbalance does not mediate the carrier (sign preserved in mid/high i-terciles; contrast survives i-adjustment; $98$% powered for sub-threshold amplitude) and the attenuation decomposes predominantly into transit epoch ($\rho=-0.54$, $p=1.4\times10^{-5}$; early epoch-matched uniform-Gaia legs still $p=0.004$) — epoch measurement, not a cleanliness counter-demonstration | sink-vs-erasure discriminator | step_142 |
| Transfer-function discriminator | impulse vs holonomy realization of the measured slip: conformal-kick model predicts $\delta\theta\propto1/(v_n v)$ and an energy exchange; holonomy predicts universal $\delta\tau$ and zero impulse — pre-2018 implied slip velocity-flat at every shell ($\rho=+0.02$, $p=0.81$); the measured $\|\Delta(1/a)\|$ floor bounds an along-track kick at $\sim20$ m/s, comparable to the $\sim10$–$14$ m/s transverse demand, so the energy channel alone does not exclude an impulse realization; post-2017 displaced slip carries additional v-dependence ($\rho=-0.37$, $p=0.001$; epoch-partial $-0.45$) — eras do not share one realization; equivalent conformal step $\sim6\times10^{-13}$ | realization discriminator | step_145 |
| Nongravitational transverse channel | NG record read as an observable for the first time: a boundary slip absorbed by pure-gravity fits must leak into the fitted timing term — Warsaw A2 (transverse) elevated in-cap $0.65$ vs $0.17$ median ($3.9\times$, $p=0.30$, $n_{\rm in}=10$) while radial A1 and normal A3 are flat; 8/10 positive A2 in-cap — the time-domain component alone carries the lean, but the independent SBDB NG record does not replicate the elevation ($p=0.68$) — registered candidate, underpowered and unreplicated | timing-parameter leakage test | step_143 |
| Newtonian directional null | complete directional battery applied to the pure N-body rotation field: the model inherits rather than manufactures the axis — model best axis $10.8^\circ$ from the resident axis (coincidence $p=0.009$ under direction-unlinked fields), catalogue best axis identical at $0.0^\circ$; source traced to coordinate level — arrivals over-concentrate in-cap ($36.6$% vs $25$% sky fraction, binomial $p=0.0020$); energy-kick channel direction-flat ($\rho=-0.083$ vs resident axis) — rotation without energy on the model side too; observed$-$model residual cap-flat ($p=0.10$) — no unexplained rotation remains, as a phase slip requires; rotation-per-kick retains a marginal axis coincidence on the matched sample ($28.2^\circ$, $p=0.069$; full cohort $80.7^\circ$, $p=0.41$) | full pipeline on model field | step_149 |
| Channel-dissociation sensitivity audit | error-propagation mechanism test for the two-channel dissociation: Monte-Carlo element perturbations at each refitted orbit's own SBDB sigmas, propagated through the identical boundary machinery (60 comets, 16 + 10 + 10 draws each) — median rotation-to-aphelion noise ratio is 0.00572 (median $\sigma_{\Delta\varpi}=$ 1.4×10<sup>-6</sup> deg vs $\sigma_{\rm aph}=$ 2.1×10<sup>-4</sup> deg, ratio 0.00572); at the error scale needed to fake the rotation residual the aphelion channel would carry $\sim$0.61° displacement it does not — the dissociation is not producible by element-error propagation | systematic-mechanism bound | step_150 |
| Arrival-anisotropy conditioned nulls | discovery-pointing control for the in-cap arrival excess: ecliptic-latitude-conditioned longitude shuffle ($50{,}000$ draws) — the resident-axis cap collects 36.6% of class-1 arrivals against a conditioned expectation of 28.2% ($p=$0.0230; Galactic-conditioned $p=$0.0693; matched subset 40.7%, $p=$0.0116/0.0280); axis-longitude scan at fixed cap latitude ranks the observed placement in the top 17 per cent of 72 positions — the arrival anisotropy is longitude-specific, not a band artifact | discovery-geometry null | step_152 |
| Zonal-distortion null | legacy-catalogue warp classes (zonal bands, regional tiles, frame rotations) injected into real MPC astrometry at the 0.7″ upper bound and refit by identical LM/DE440s machinery: warps do deposit preferentially in $\omega$ ($\omega$-share 0.80 via the short-arc degeneracy) and stay energy-flat (induced $\|\Delta(1/a)\|$ 1.4×10<sup>-5</sup> AU$^{-1}$), but the induced field has no axis organization ($p=$0.691) and matching the observed in-cap carrier needs a $\sim$23.3″ warp ($\sim$96.6″ for the in-cap level itself) — one to two orders of magnitude beyond documented zonal systematics | catalogue-artefact mechanism test | step_153 |
| Slip-injection transfer function | specified slip realizations injected into the real observing chain (MPC epochs, stations, noise floors) and refit identically: integrable time translation reads back invisible ($0.010^\circ$ — the $\oint d\ln A=0$ control); crossing-localized position/velocity slips survive in the fitted inbound asymptote at $0.14^\circ$ and load $\omega$; impulse displaces $0.27^\circ$ with energy loading; injected slips return fitted rotation at the noise floor ($0.08$–$0.12^\circ$ vs control $0.14^\circ$) — the measured masking; leg disagreement predicted at floor for a past-boundary slip (observed $0.143^\circ$ vs control $0.139^\circ$) vs $27.9^\circ$ for a within-apparition slip — slip localized at/before the crossing | forward-model transfer function, real observing chain | step_151 |
| Conventional generative population | the arrival excess tested generatively, not conditionally: isotropic Oort-spike source $\times$ secular Galactic-tide injection (orbit-integrated torque, step-070 tensor) $\times$ stellar remixing $\times$ empirical ecliptic-latitude discovery kernel — synthetic in-cap fraction median 28.2% vs observed $36.6$% (model $p=$0.0205; matched $p=$0.0165); injected density's own preferred direction $118^\circ$ off the resident axis; a synthetic resultant reaching the observed 3.7° axis proximity has $p=$5.0×10<sup>-4</sup> — conventional generation cannot supply the anisotropy it transmits | generative source-plus-selection model | step_157 |
| Propagated discovery footprint | visibility-model-eligible subset: 52 of 131 class-1 objects. One weighted orientation per fixed object per null realization; brightness integrated over Keplerian time. Predicted dipole (246.3°, -69.40°), 92.02° from the observed direction; null cap fraction 26.9% vs 34.6% ($p=$0.16), directional $p=$0.0092. Residual offset 31°; scenario maximum 44.3°. Excluded high-perihelion objects lie outside this model's coverage. | conditional physical selection model | step_163 |
| TEP-side generative test | the field hypothesis placed on the same generative footing as the conventional models: boundary-slip redirection applied to footprint-selected arrivals displaces fitted aphelia by 0.132° median (max 1.43°) against the $\sim10$–$20^\circ$ coherent shift required — the dipole lives in the incoming state, not the readout; required source modulation solved: in-sector density $\times$1.28 (matched $\times$1.53), dipolar $m\approx$0.39; anti-cap fraction 0.130 sits below the selection baseline 0.237 in all five cohorts — dipolar morphology, matching the rotation field's independent CMB-frame dipole decomposition; per-comet rotation tracks the axis-anchored lobe weight at $\rho=$+0.279 ($p=$0.0012) | TEP generative source model | step_164 |
| Canonical disformal transport | positive-$B_0$ leading-connection benchmark, with actual published conic spatial entry/exit directions (median separation 28.0°). Exploratory signed-dipole excursion $G=\cos\theta$ has an even quartic amplitude in the symmetric limit; generic odd-mode cancellation is not assumed. The unsigned amplitude contrast 5.9 yr requires $u_b\approx$2.5×10<sup>-4</sup> at the instrumental normalization. Signed contrasts are -5.9 yr and 5.9 yr on opposite branches; amplitude alone cannot select the sign. Pattern $\rho=+$0.157, fixed-profile $p=$0.0170. Source-profile, full metric and observation-transfer validation remain required; clock and impulse comparisons are conditional diagnostics. | profile/geometry benchmark | step_165 |
| Metric-derived field slip | scalar-sector construction $\phi=m\theta_A\sigma(r)$ evaluated on the true dual-leg trajectories: $d\phi$ is exact, so a single-valued sector yields only a leg-symmetric coherent offset — the dual-leg instrument cancels it by construction; the multi-valued (line-defect) reading's open-path winding $\int\sigma\,d\theta_A$ is cap-blind ($p=$0.804) — constrains admissible field topology on both readings; axis-specific datum: in-cap trajectories thread the resident axis line closer (1.82 vs 2.24 AU, $p=$0.0125, absent on all non-antipodal control axes) | field-level derivation test | step_158 |
| Boundary-radius forward-model scan | measured slip injected at controlled crossing radii (8–250 AU) through the real observing chain, shell-resolved after refit: injections inside $\sim$50 AU disrupt the orbit or produce catastrophic fitted anomalies ($\sim 100^\circ$); crossings at 100–250 AU yield fitted offsets already developed at the 8-AU shell (~half the injection-shell value) — the observed flat radial profile is the expected readout of a distant crossing; the 250-AU injection predicts leg disagreement 0.142° vs observed $0.143^\circ$ — the best match sits at the outer edge of the tested range, so $\sim 250$ AU is a lower bound pending extension of the scan | forward-model localization, real observing chain | step_156 |
| Perihelion-time slip channel | osculating perihelion-epoch disagreement between independently fitted inbound and outbound legs on the dual-leg record: the channel certifies (median $\|\Delta T_p\| =$ 0.074 d, a 1.2×10<sup>-4</sup> arc fraction, concordant with the catalogue $\Delta\tau$ and leg-disagreement observables) but carries no cap localization (in-cap 0.0838 d vs out-of-cap 0.0734 d, MWU $p=$0.39; CODE-overlap $p=$0.51) and no frame-anchored polarity (signed shift $p=$0.37) — the epoch projection of the inter-leg non-closure does not itself carry the axis organization | timing-channel audit | step_159 |
| ISO two-leg refit | independent leg-split LM/DE440s refits of the three interstellar objects on raw MPC astrometry: leg disagreements 0.142°/0.0612°/0.152° with perihelion-epoch slips 32.8/3.6/8.5 d for 1I/2I/3I; 1I's inbound leg is the more displaced (0.270° vs 0.135°) — the comet cohort's asymmetry direction — while 3I's is outbound; fit quality sits inside the comet cohort's range (1I rms at the 20th cohort percentile) — for the volatile-active ISOs the channel is a joint lapse+outgassing datum, contextual rather than calibrated | raw astrometry, own fitter | step_160 |
| APDB planetary O-C channel | O'Handley-format Pluto/Uranus/Neptune optical record (1914–1998) reduced against DE440s with FK4/FK5/TETE equinox handling and topocentric correction: pooled Pluto along-track residuals organize against axis separation ($\rho=$-0.17, $p=$1.7×10<sup>-9</sup>; in-cap median 120″ vs 17″ out, $p=$7.5×10<sup>-6</sup>, $n_{\rm in}=14$) — but Pluto's separation from the axis drifts monotonically with epoch within each source window, so the pooled correlation aliases secular plate and ephemeris trends and does not replicate within individual sources — registered as an exploratory diagnostic of structure surviving solver absorption, not a detection | planetary astrometric O-C | step_161 |
| NIMA resident O-C pilot | NIMA/Lucky Star per-observation residuals for the detached-resident ledger intersection: 8 of 76 ledger objects carry usable records (7 in-cap, a single out-of-cap control; 451 observations); in-cap median residual amplitude 0.154″ against the control's 0.0305″ — registered as a small-$N$ pilot channel that cannot support a detection claim at this coverage | post-fit residual pilot | step_162 |

## 4.15 The transfer function: impulse versus holonomy

The conversion of the measured rotation to a time scale is no longer an unconstrained phase conversion — it is a discriminated transfer function (step 145). Two microscopic realizations produce the observable. An impulse realization — a conformal fifth-force kick $\Delta v = c^2\Delta\alpha/v_n$ along the wall normal — predicts $\delta\theta \propto 1/(v_n v)$, hence an implied slip strongly anticorrelated with crossing speed, and an along-track component that changes orbital energy at $\Delta(1/a) = 2v\Delta v_\parallel/GM$. A holonomy realization — a crossing-localized time translation $H = \oint_C d\tau$, the non-integrable transport the two-way-link theorem isolates — predicts a universal $\delta\tau$ independent of crossing velocity and no local impulse at all. The measured record selects between them. On the pre-2018 record the implied slip is velocity-flat at every shell ($\rho = +0.02$, $p = 0.81$ at 60–250 AU): the universal-amplitude holonomy signature, with an equivalent conformal step of order $6\times10^{-13}$. The energy channel is less decisive than once stated: the measured $|\Delta(1/a)|$ floor bounds an along-track impulse at $\sim 20$ m/s, comparable to the $\sim 10$–$14$ m/s transverse demand, so the channel alone cannot exclude an impulse realization — an earlier audit had overstated the bound at $0.3$ m/s by substituting the planetary-approach distance for the energy residual, a wrong-variable defect corrected in step 145. The post-2017 displaced record does not share the flat signature (in-cap $\rho = -0.37$, $p = 0.001$, surviving epoch partialling at $\rho = -0.45$): the modern residual carries an additional velocity- or epoch-dependent term — the same heterogeneity the epoch-decomposition audit measures independently — so the two sides of the frame-anchored structure are not required to share one realization, and the $\delta t_{\rm eq} = \delta\theta\,r_b^2/h$ conversion is the exact transfer function of the holonomy realization the older record selects.

The same fork is then measured in the forward direction on the real observing chain (step 151): for twenty dual-leg cohort comets the actual MPC epochs, stations and noise floors are kept, the astrometry is synthesised from a truth orbit carrying a specified slip realization injected at the true inbound boundary crossing, and the standard LM/DE440s refit is run exactly as on the real record. The transfer function measured this way closes the remaining interpretive alternatives. First, the integrable control behaves as the theorem requires: a boundary state advanced coherently in time — position and velocity both — stays on the same orbit, and the refit reads it back at 0.0101° of inbound-asymptote displacement — an integrable lapse is invisible to every channel, so the anomaly the instrument measures cannot be integrable. Second, the non-integrable realizations leave the predicted record: a crossing-localized position slip survives in the fitted inbound asymptote at 0.143° and loads the periapsis element ($\Delta\omega = +$0.11°); the velocity-slip variant loads the same channel at 0.137°; the impulse control displaces the asymptote nearly twice as far (0.274°) while loading the perihelion-distance element — the mechanical realization's energy signature. Third, the fit's absorption of the slip is itself measured: injected slips of the observed amplitude return fitted rotations at the noise floor ($0.08$–$0.12^\circ$ against the uninjected control's 0.14°), so the catalogue rotation is a heavily masked observable — the measured cap contrast is what survives absorption, not the slip's full amplitude. Fourth, the leg-disagreement channel discriminates the slip's location: a slip at the past boundary crossing leaves the whole apparition post-slip and predicts leg disagreement at the noise floor — the observed cohort median 0.143° matches the uninjected control's 0.139° — while the same slip injected inside the apparition produces 27.9° of leg disagreement, a catastrophic signature the real record does not carry. A paired per-comet difference analysis ($O_{\rm injected} - O_{\rm control}$ on identical geometries) quantifies this channel's power directly: the past-boundary slip shifts leg disagreement by a median $+0.049^\circ$ with no amplitude correlation ($\rho = +0.18$), so the observed-to-injected median agreement is required by the channel's floor-level sensitivity rather than an independent confirmation — the discriminating contrast is against the within-apparition realization alone. The same pairing shows where the slip does register: the truth-referenced inbound-asymptote channel responds at $+0.14^\circ$ median with unit sign consistency, while the catalogue-level leg observables retain only $0.01$–$0.06^\circ$ of the injected displacement — the fitter absorbs the bulk of the slip before leg-level statistics see it. And fifth, the instrument is a calibrated readout, not a threshold detector: across comets the produced boundary displacement tracks the injected slip amplitude at $\rho =$ 0.96 ($p =$ 1.8×10<sup>-11</sup>) for the position-slip realization, so the measured anomaly's magnitude is itself a controlled quantity. The injection battery thereby selects the same realization the observed velocity-flatness selects — a crossing-localized, non-integrable time transport — and independently bounds its epoch: at or before the boundary crossing, not during the observed arc.

The injection chain answers how a slip reads through the solver, but leaves the slip itself empirical. The corresponding question at field level — whether a specified matter-metric construction produces the non-integrable observable rather than having it inserted — is examined in step 158, and the audit is sharper than the null it returns. For the scalar sector $\phi = m\theta_A\,\sigma(r)$ winding about the boundary axis, $d\phi = m(\sigma\,d\theta_A + \theta_A\,d\sigma)$ is an exact differential: its integral along each leg is an endpoint term, and inside the envelope ($\sigma \approx 1$ throughout the observed arc) it reduces to the comet's own axis azimuth at the osculating epoch — a coherent offset identical on both legs, the same structure the step-151 integrable control showed reads back invisible. A single-valued radially switched scalar can therefore write only a common-mode displacement the dual-leg instrument cancels by construction; the leg-differential observable is structurally blind to it. A non-integrable prediction survives only if $\phi$ is multi-valued — the axis as a line defect — in which case the open-path winding $W = \int \sigma\,d\theta_A$ is the legitimate observable, and it is evaluated on the true trajectories of the dual-leg cohort (372 source members, 371 successfully marched): the boundary-weighted winding is cap-blind (median $2.44$ vs $2.46$ rad, Mann–Whitney $p =$ 0.804), its signed values do not track the measured rotation sign, and the apparent magnitude correlation — trajectories passing closer to the axis line carrying larger measured slips — is reproduced by control axes $85$–$180^\circ$ away and is largely a perihelion-distance proxy. The admissible field topology is thereby constrained on both readings: a single-valued sector yields only coherent offsets, and the line-defect winding realization does not organize about the cap — the slip is not a function of azimuthal winding about, or threading distance to, a single line axis. One datum does resolve on the declared axis alone — in-cap trajectories thread the resident axis line closer than out-of-cap trajectories (median 1.82 versus 2.24 AU, $p =$ 0.0125, absent on every non-antipodal control) — a geometric confirmation that the boundary axis passes through the transit population itself, without establishing that a winding field is the carrier.

The same chain then locates the crossing itself (step 156). The shell-resolved record showed the angular excess already developed at 8 AU, which raised an apparent inconsistency with a transition placed at 50–150 AU; the forward model resolves it. Injecting the measured equivalent-time slip at controlled inbound crossing radii — 8, 25, 50, 100, 150 and 250 AU — on the dual-leg cohort's real observing chains, then refitting and propagating the solutions through the boundary shells, measures how far inward a distant slip reads. Two bounds emerge. From below, a slip of the observed amplitude injected inside $\sim 50$ AU is not a subtle offset but an orbit-disrupting event — the post-slip trajectories either fail to return a coherent leg to 250 AU (16 of twenty at the 25-AU injection) or carry a fitted anomaly of order $100^\circ$ — incompatible with the record. From above, a genuine 100–250 AU crossing produces a fitted inbound-direction offset that is already partially developed at the innermost shell: 0.532° at 8 AU for a 100-AU crossing and 0.0651° for a 250-AU crossing, roughly half of the displacement measured at the injection shell itself — the least-squares solution redistributes the boundary slip through the fitted orbit, so "already formed at 8 AU" is the expected readout of a distant crossing, not evidence for a local one. And the localization is quantitative rather than suggestive: the 250-AU injection predicts a leg-to-leg fitted disagreement of 0.142° against the observed cohort median of $0.143^\circ$, matching to within the measurement scatter — the boundary crossing sits at the outer edge of the tested range, consistent with the independent heliospheric anchoring.

## 5. Synthesis: convergence on a single domain boundary

The two populations were analysed independently — the TNO axis was fixed before any comet was examined — so the synthesis tests whether they converge on one structure or merely on similar anomalies.

## 5.1 Independent axis recovery

A free sky scan on the independent CODE comet sample — no TNO information supplied — recovers its strongest axis at $(\lambda, \beta) = (10^\circ, -20^\circ)$ with $p = 0.0025$, $24.0^\circ$ from the resident-defined axis and inside the same 60° sector. Of eight physically motivated trial axes (TNO axis and anti-axis, galactic poles and centre, ISM inflow and antipode, CMB dipole directions), only the TNO-associated patch returns a strong comet signal; the ISM inflow axis in particular is null ($p = 0.95$ on the discrepancy channel). The comet patch is broad — its best direction wanders at $n \sim 50$ sample sizes — but both catalogues' recovered axes fall inside the structure (24° and 37° from the TNO axis). The convergence is not confined to the reconstruction channel: the comets' aphelion directions themselves — a spatial catalogue observable distinct from the inter-leg residual, with its own orbit-fit uncertainties — dipole toward the same axis at the $10^{-2}$–$10^{-4}$ level against a tide-aware null in all four comet samples, and that dipole points $44^\circ$ off the galactic plane rather than at the on-plane anticenter the tide would favour (Section 4.8) — a datum that certifies the direction-organized arrival population without selecting which agent organizes it.

A fourth direction channel uses the position-conditioned mean-plane likelihood of Siraj–Chyba–Tremaine (2025), reimplemented on MPCORB (step 090). The broad non-resonant 80–400 AU sample has a modest best-fit tilt, 3.42° at 0.730$\sigma$, compared with the larger tilt recovered on the study's published 46-object subset. The 80–200 and 200–400 AU lean azimuths are 16.6° and 45.3°; the pooled lean is 31.1°, 17.9° from the resident-axis longitude (directional $p=0.0848$). The deepest-bin offset is 3.70° ($p=$0.0449). The simple mean-pole cross-check gives a pooled azimuth of 44.4°. These are consistency measurements with the stated conditioning, rather than a proof that every survey selection effect has been removed.

Leave-one-out fits of the pooled pole have median azimuth 31.2° and circular scatter 2.30°. This measures sensitivity to individual objects. Together with the weak amplitude significance, it supports retaining this channel as directional consistency evidence.

What the axis is not is audited explicitly (step 107). All three recovered directions — the cap-declaration axis $(34^\circ, -13^\circ)$, the detached-sample axis $(49.9^\circ, -17^\circ)$, and the third-cohort free-scan direction $(50^\circ, -50^\circ)$ — are separated from almost every named direction of the coordinate sky: galactic centre and anticentre, both galactic poles, the solar apex and antapex, the galactic-rotation direction, and the ecliptic poles. The nearest named direction to the cap axis is the south galactic pole at $35.6^\circ$ — outside the $30$–$40^\circ$ localization width — and a uniform-direction null prices the closest approach at $p = 0.709$: a random axis does as well seven times in ten, so nothing in the sky's fixed geometry accounts for the measured direction.

One proximity is structural rather than coincidental: the published Planet Nine perihelion direction $(244.6^\circ, +15.8^\circ)$ sits $150^\circ$ from the primary axis but $14.2^\circ$ from the detached axis's antipode — inside the axial field's mirror lobe. That is where the anti-aligned shepherding picture itself places its perturber, and since the axis and the published direction derive from the same TNO clustering under opposite conventions, the antipodal consistency is expected under both readings; the dynamical insertions of Section 6 supply the discriminator, bounding any mass there at hundreds of Earth masses below requirement.

The CMB dipole is the exception the audit must state exactly, because it is the operational reference direction the theory's clock channels adopt rather than a generic sky direction: in axial terms the declared axis lies $48.5^\circ$ from the CMB dipole axis — inside the 60° cap rim, though outside the core localization width — and the comet-channel dipole decompositions of Section 4.12 place the residual field's own axis within $20$–$26^\circ$ of the CMB axis in the two records that resolve it, matching the GNSS-II annual-phase direction's $18.2^\circ$ unsigned bipolar separation from the CMB antapex ($162^\circ$ directed from the apex; Paper 2). The comet and clock channels therefore do not measure the CMB direction itself — they measure a local screening/orbital-phase axis displaced from it — but the axial displacement is toward the CMB axis rather than orthogonal to it, the geometry the nested-screening picture predicts when a local lapse-field structure is organized within the cosmological rest frame.

One named direction does fall inside the localization width and is registered with its secular context (steps 112, 144). The resident axis azimuth sits $4.0^\circ$ from Neptune's mean longitude of perihelion ($\bar\varpi_{\rm N} \approx 45^\circ$; a-priori $p = 0.022$ against a uniform target). Whether the proximity is a fixed structural alignment or a circulating mean-element coincidence is decided by the giants' forced apsidal directions — the time-averaged eccentricity vectors each planet's secular architecture imposes — integrated over 5 Myr under DE440s/REBOUND. The forced directions do not carry the axis: Jupiter and Saturn share the coupled mode at $\approx 99^\circ$, Uranus at $268^\circ$, Neptune at $179^\circ$; the joint forced azimuth lies $112^\circ$ off-axis (uniform null $p = 0.62$), and no giant's instantaneous apsidal longitude dwells preferentially within $15^\circ$ of it. The axis is therefore not a secular artefact of the giant planets — the closest conventional source of a fixed apsidal direction does not supply one — while the mean-element proximity is registered as a coincidence of unresolved status, priced rather than asserted.

![One-map axis coincidence: all measured populations on one ecliptic projection](figures/step_b22_one_map.png)

**Figure 2:** Every measured population on one ecliptic projection ($\lambda$, $\beta$). Filled circles: detached TNOs inside the 60° axis cap; open circles: detached TNOs outside it; small dots: the injected JFC chain ($q < 3$ AU blue, $q > 5$ AU purple). Reference directions: TNO axis (red star), independently recovered comet patch centre (teal diamond), ISM inflow (blue triangle), CMB apex (orange triangle), anti-axis (purple star). The resident cluster, the transit patch, and the injected chain all land on the same axis. Pipeline output: step 057 (`results/figures/step_b22_one_map.png`).

The coincidence is quantified rather than drawn (step 050). The three dynamically independent axial directions — the TNO perihelion axis, the comet perihelion concentration, and the ISM inflow axis — share a mean pairwise-axis statistic at $p = 0.031$ (separations $16.7^\circ$, $28.0^\circ$, and $42.6^\circ$). The wider seven-axis fold, which adds the four cosmological dipoles (CMB, axis-of-evil, radio, and fine-structure), returns a concentration $R = 0.74$, marginal under a convention-matched isotropic null that folds the random draws identically ($p = 0.16$). The remaining local members are oriented directions rather than axes, so they are not folded: taken raw, the five non-TNO local directions — the comet concentration, the ISM inflow, both Voyager heliopause crossings, and 'Oumuamua's periapsis — cluster among themselves at $R = 0.80$ ($p = 0.012$), with a resultant at $(259^\circ, +22^\circ)$ that sits inside the mirror cap, $28^\circ$ from the anti-axis and $17^\circ$ from the ISM inflow. Under the axisymmetric field the local directions therefore register on the same axis, on the mirror side. The common plane of the resident and transit populations carries its own geometric datum: its pole is tilted only $\sim 6^\circ$ off the ecliptic pole toward longitude $\sim 50^\circ$ — the same azimuth as the perihelion clustering axis, so the two populations' clustering is one coherent three-dimensional structure rather than two separate projections.

## 5.2 Combined evidence and multiple-testing control

The formal evidence ledger groups the 22 primary tests into five families: F1 resident clustering, F2 bias bounds, F3 structural diagnostics, F4 comet clock channel, F5 cross-population direction. Under within-family Benjamini–Hochberg at $q = 0.05$, the three empirical signal families retain all their listed tests — F1 7/7, F4 4/4, F5 3/3 — while F2 and F3, which contain correlated bound and structure measurements rather than independent discoveries, do not and are not claimed as such. The two-population Fisher combination — the TNO conditioned-null result with the CODE confirmation — gives $p =$ 2.55×10<sup>-4</sup> nominal; combining the Warsaw and CODE comet legs instead gives $p = 3.02 \times 10^{-4}$.

The five-channel predecessor to the synthesis evaluates the CODE slip, pooled aphelion dipole, Warsaw inbound offset, detached-TNO periapsis projection and in-cap longitude concentration on 20,000 random test directions (step 069). Its descriptive score is 107 at the resident axis, with conditional sky-area rank 0.00215, and 105 at the transit axis, with rank 0.00880. The highest-scoring sampled direction is 2.83° from the resident axis. Like step 126, this localizes the fixed observed landscape; it does not calibrate discovery significance under null catalogues.

The result is also not carried by any one channel: recomputing the same null with each channel dropped in turn leaves every leave-one-out set at conditional sky-area rank $\leq 0.003$ at the resident axis (step 074) — the strongest single contributors are the resident $\varpi$ concentration ($\Delta S = 38$) and the aphelion dipole ($\Delta S = 30$), yet removing either still leaves the joint anomaly above the 99.7th percentile of the fixed-catalogue direction landscape.

Adding the MPCORB mean-plane summary gives a six-channel descriptive score of 112 and conditional sky-area rank 0.00570 at the resident direction (step 094). The mean-plane reference conditions on each object's observed position. Its catalogue fit differs from SBDB but shares some source objects and astrometry; the directional synthesis therefore remains a localization check rather than an independent global detection test.

## 5.3 The injected population: a two-ended chain

The third population closes the geometry. Jupiter-family comets — bodies injected inward through the same volume — organize by perihelion depth. The full Tisserand-selected JFC sample ($N = 580$) rotates its mean perihelion direction continuously with depth: the deep injected bulk ($q < 3$ AU, $N = 381$) points at $\bar\varpi = 41.1^\circ$, $8^\circ$ off the detached-TNO axis, while the shallow transitional subset ($q > 5$ AU) reaches $\bar\varpi \approx 190$–$225^\circ$, the anti-axis sector. The measured discovery footprints do not track this rotation — at $q > 5$ AU the footprint mean is $293^\circ$ against an observed $204^\circ$ — and a pairing test on the dated deep subsample rejects footprint manufacture at $p =$ 0.026. The transit population between the two organized ends — Centaurs, $N = 1{,}044$ — is isotropic ($R = 0.09$). The injection chain's endpoints land on both ends of one axis: residents and deep injected comets at $\sim +45^\circ$, shallow transitionals at $\sim +190$–$225^\circ$.

A structural caveat accompanies the two-ended reading. JFC perihelia carry real secular structure (Jupiter-driven; the $q < 1$ AU bin shows $\omega$ concentrated at $\sim 193^\circ$), so part of the progression may reflect known inner-system dynamics. The datum being claimed is narrower: the chain endpoints coincide with both ends of the pre-declared axis within $\sim 10^\circ$, and the footprints cannot produce that.

## 5.4 Depth selectivity across populations

The comet anomaly requires crossing the full structure: the shallow-plunger control ($q \geq 3.1$ AU) is flat ($p = 0.16$), while the signal compounds for deep plungers that traverse the nested boundary region. The same depth logic appears on the resident side — the alignment fraction rises sharply beyond $a \sim 150$ AU (Section 3.4) — and the sednoids, the deepest residents, cluster near the axis at $R = 0.43$. Both populations agree on where the structure lives: it begins near the classical-belt edge and is fully developed by the detached region.

## 5.5 Ten-channel global cross-survey and multi-lineage synthesis

Multi-lineage convergence is tested directly across astronomical catalogues and process classes (step 126). Ten empirical summaries span the JPL Small-Body Database (SBDB), Dark Energy Survey (DES), Minor Planet Center orbit catalogue (MPCORB), CODE, Warsaw and CometEls. They share some objects and, apart from DES, much of their underlying ground-based astrometry; catalogue labels therefore do not establish independence. For each test direction the calculation evaluates all ten summaries on the same fixed catalogues. This preserves their observed directional relationships in a descriptive score landscape. The resident in-cap Rayleigh references are applied after directional selection and must be treated as weights rather than valid component discovery probabilities. The listed channel set defines the synthesis; selection among the wider pipeline's 152 registered steps requires its own calibration:

- The extreme detached-TNO apsidal clustering (SBDB lineage; $\varpi$ concentration, $p =$ 5.7×10<sup>-9</sup> at the resident axis).

- The Dark Energy Survey six-year detached cohort (DES lineage; 13 of 16 in-cap, in-cap Rayleigh $p =$ 5.0×10<sup>-5</sup>).

- The MPCORB detached TNO clustering replication (MPCORB lineage; 31 of 63 in-cap, in-cap Rayleigh $p =$ 2.8×10<sup>-10</sup>).

- The non-resonant Kuiper belt mean-plane tilt (MPCORB lineage; Siraj–Chyba–Tremaine estimator, lean azimuth 17.9° off the axis at reference weight 0.085).

- The CODE long-period comet periapsis rotation anomaly (CODE lineage; $p =$ 0.0087).

- The Warsaw catalogue inbound boundary-leg offset (Warsaw lineage; $p =$ 0.0053).

- The pooled long-period comet aphelion dipole (CODE/Warsaw/MPC pooled sample; $p =$ 3.5×10<sup>-7</sup>).

- The full MPC CometEls near-parabolic comet census aphelion dipole (CometEls lineage; $d_\parallel = +$0.10, $p =$ 0.0021).

- The prospective post-2017 SBDB comet aphelion dipole (SBDB lineage; $d_\parallel = +$0.0949, $p =$ 0.0038).

- The Jupiter-family comet deep-injection perihelion alignment (SBDB lineage; $\bar\varpi = 41.1^\circ$, $p =$ 0.044).

The descriptive Fisher-shaped score $S_{10}=-2\sum_{j=1}^{10}\ln p_j$ is evaluated at 20,000 random test directions on the fixed catalogues. At the declared resident axis $(49.0^\circ,-17.0^\circ)$ the score is 186.2 and the conditional sky-area rank is 0.0145; at the transit axis $(34.0^\circ,-13.0^\circ)$ they are 188.9 and 0.00925. The highest-scoring sampled direction lies 14.9° from the resident axis. These measurements localize the observed directional landscape. A global discovery test requires null catalogues, the full search/selection procedure and preservation of shared-object dependencies.

The relation of that landscape argmax to the declared axes is audited directly rather than assumed (step 126, axis-localization block). The maximum lies at $(40.6^\circ, -17.0^\circ)$ — $8.0^\circ$ from the resident axis and $7.5^\circ$ from the transit axis, remaining inside the sector bracketed by the two independently declared directions rather than displaced toward a third region of sky. The statistic's own landscape is a smooth ridge centred on the declared sector: the median $S$ over sampled directions is $184.1$ within $15^\circ$ of the resident axis and falls monotonically to $33.4$ beyond $90^\circ$, and only $1.18$ per cent of the celestial sphere reaches the declared-axis value.

Refitting the landscape maximum on 150 member-resampled bootstrap replicates gives a nominal 68-per-cent radius of 18.4° about the declared resident direction. The observed offset lies within that radius. This is a descriptive stability check: the current bootstrap does not preserve cross-catalogue identities and therefore does not provide a fully calibrated joint localization interval.

The ten channel-best directions have a mean pairwise separation of 35.1°. The separation matrix records both the concentration and its exceptions; convergence does not require every pair to lie within 60°. Comparing these directions across resident and transit populations tests whether the structure survives changes in catalogue and observable. Common astrometric inputs and discovery selection remain possible shared influences, which the dedicated refit, covariance and footprint controls test separately.

![Ten-channel global cross-survey synthesis](figures/step_b90_global_synthesis.png)

**Figure 3:** Ten-channel global cross-survey and multi-lineage synthesis (step 126). Left: distribution of the descriptive joint score $S_{10}$ across 20,000 random sky directions (grey), against the values evaluated at the resident detached-TNO axis ($S_{10} =$ 186.2, conditional sky-area rank 0.0145, solid) and the comet transit axis ($S_{10} =$ 188.9, conditional sky-area rank 0.00925, red dashed) — both declared directions sit at the extreme upper edge of the random-direction landscape. Right: per-channel $-\log_{10}(p)$ evaluated at the declared boundary axis for all ten channels across six catalogue lineages (SBDB, DES, MPCORB, CODE, Warsaw, CometEls; resident-population channels navy, comet channels red; dotted line: $p = 0.05$), displaying each channel's contribution; the plotted reference weights do not all have calibrated discovery-p-value interpretations. Pipeline output: step 126 (`results/figures/step_b90_global_synthesis.png`, `results/step_b90_global_synthesis.json`).

**Table 5:** Synthesis evidence

| Test | Result | Source |
| --- | --- | --- |
| CODE free-scan axis | $(10^\circ, -20^\circ)$; $24^\circ$ from TNO axis; $p = 0.0025$ | step_036 |
| Split-half axis recovery | 70% within $45^\circ$ | step_037 |
| Two-population Fisher combination | $p = 2.55\times10^{-4}$ | step_051 |
| Family-wise BH survival | F1 7/7, F4 4/4, F5 3/3 at $q=0.05$ | step_051 |
| Deep JFC direction | $\bar\varpi=41.1^\circ$ ($8^\circ$ off axis) | step_052 |
| Shallow JFC direction | $\bar\varpi\approx 190$–$225^\circ$ (anti-axis sector) | step_052, step_027 |
| JFC footprint pairing test | $p =$ 0.026 | step_052 |
| Next-discovery cap rate | 59% predicted vs 42% baseline | step_054 |
| Detection power | $N \approx 123$ for 95% power at $\alpha=1%$ ($\approx 184$ at 0.1%, exact binomial) | step_067 |
| Ammonite (2023 KQ14) jackknife | $134^\circ$ off axis; $R$ 0.357$\to$0.332, $p$ 0.0043$\to$0.0075 | step_084 |
| 2025–26 cohort vs own footprint | 2/21 in cap vs 0.12 footprint baseline (cohort points to anti-axis) | step_084 |
| Mixture posterior-predictive audit | SBDB $E=25.5$ vs 26 obs (BF 44.6); DES $E=12.1$ vs 13 obs (BF 10.7); union BF 9.7 | step_115 |
| Empirical-coupling audit | union in-window BF 25.5 (39 obs vs $E=39.86$); OSSOS in-window 13/17; closed channel 76 objects, 0 in-cap; provisional in-window depletion-corrected BF 2.1 (bootstrapped median 1.93); grand-union BF 2.05–3.76 | step_116 |
| DES detached cohort (indep. fits) | 13/16 in cap ($p=1.2\times10^{-4}$ vs uniform; $p=0.029$ vs own footprint 0.55; consistent with 0.59) | step_085 |
| DES–SBDB resident cross-fitter floor | median $\|\Delta\varpi\|=0.25^\circ$ on 14 shared objects | step_085 |
| Non-resonant-disk lean direction (SCT25 estimator, MPCORB) | 80–200 AU lean 16.6°; 200–400 AU lean 45.3°; pooled 31.1° (17.9° off-axis, $p=0.0848$); deepest-bin directional $p=0.0449$ (position-conditioned null) | step_090 |
| MPCORB detached replication | $R=0.243$, $p=0.024$, axis $(45^\circ,-21.1^\circ)$; matched subset identical | step_092 |
| Named-direction audit | nearest named direction $35.6^\circ$ (S galactic pole); CMB dipole $48.5^\circ$ axial (inside cap rim); coincidence $p=0.709$; P9 perihelion $14.2^\circ$ off detached antipode (mirror lobe); Neptune mean $\varpi$ coincidence $4.0^\circ$ ($p=0.022$ a priori) does not extend to forced secular directions ($112^\circ$ off-axis, $p=0.62$) — axis not a giant-planet secular artefact | step_107, step_112, step_144 |
| Comet-channel prospective score | post-2017 SBDB $n=121$: registered-axis contrast reversed — reversal confirmed astrometric on the leg-separated refit (step 128 — not a joint-fit artefact, while solver absorption of the slip is confirmed on injected realizations, step 151; reversed in form on this era) — but the cohort's dominant structure is compatible with the lapse-slip class at opposite CMB-frame polarity; forward falsifier re-registered: future cohorts should continue the modern apex-side polarity | step_117, step_118, step_125, step_128 |
| Out-of-lineage spatial datum | same post-2017 cohort, aphelion channel: $d_\parallel=$+0.0949 toward cap axis ($p=$0.00215); pre-2018 +0.110 ($p=$5.00×10<sup>-5</sup>); free dipole $(20^\circ,-42^\circ)$, $32^\circ$ off axis — population datum replicates where the residual channel fails | step_124 |
| Ten-channel global synthesis | Omnibus $S_{10} =$ 186.2 at resident axis, $S_{10} =$ 188.9 at transit axis; 20,000-direction conditional sky-area rank 0.0145 (resident) and rank 0.00925 (transit); the joint landscape concentrates near the declared sector across 10 channels and 6 lineages; best random axis sits $14.9^\circ$ from resident axis — inside the bootstrap $r_{68} =$ 18.4° localization cone, on a smooth ridge centred on the declared sector | step_126 |
| Independent astrometric refit | two-leg refits of raw MPC astrometry, $n=$585 fitted; rotation concordance $\rho=+$0.998; inbound in-cap carrier $p=$0.0370, CODE-overlap $p=$0.00688 | step_127 |
| Post-2017 parameter-absorption discriminator | same independent refit on the prospective cohort, $n = 248$ (160 paired dual-leg): leg-fit carrier reversed at the declared axis ($0.0857^\circ$ vs $0.109^\circ$, perm $p = 0.996$) identically under all three constructions — reversal is astrometric, not a fitting artefact; solver absorption of the slip is confirmed independently on injected realizations (step 151); displaced-axis residual field replicates $+0.164$/ $-0.0415$ dex ($p = 5.2\times10^{-5}$), extremal under a 500-direction shuffle ($p=0.00599$) and strengthened NG-clean ($p=1.7\times10^{-4}$); signature-class audit: outbound-leg contrast (permutation $p=0.0562$; borderline), unresolved energy contrast ($p=0.987$) — the modern structure is compatible with the lapse-slip class on the mirror leg; the registered axis and leg are epoch-dependent | step_128 |
| Bipolar unification test | Independent refit residual dipoles: pre-2018 -0.0579 dex ($p=0.0223$) versus post-2017 +0.171 dex ($p=5.00\times10^{-5}$). Opposite-polarity product $J$ has permutation $p=5.00\times10^{-5}$; nominal component Fisher $p=1.6\times10^{-5}$. The CMB direction has conditional sky-area rank 5.60% among 500 directions. Polarity exploration and shared pipeline structure limit a unique physical interpretation. | step_129 |
| Cross-channel axis convergence | joint frame-anchored ledger across independent TEP channels: the GNSS-II free clock axis lands $0.93^\circ$ from the comet-fixed meridian plane (out-of-sample $p=$0.0162) and $18.2^\circ$ unsigned from the CMB dipole antapex ($162^\circ$ directed from the apex; heliocentric aphelion-tangent annual phase, Paper 2); MGEX CMB-projection-consistent ($p=0.005$, grid axis non-identifiable by its own flag and excluded on it); LLR fixed-Planck-axis probability from scrambled data, with observed-sky rank 226/2664 retained separately; nominal Fisher combination over four channel statistics $\chi^2=52.5$, df $=8$, nominal $p=$1.36×10<sup>-8</sup>; cross-channel dependence and geometry selection remain uncalibrated; clock-epoch coincidence at the comet switch (block-permutation $p=$0.0020 at the boundary, global-changepoint rank $53/84$) — directional convergence across independent measurement classes, no shared astrometry | step_135 |

## 6. Discussion and alternative mechanisms

The conventional explanations are tested on the signatures they are required to produce. Each alternative is evaluated against the channel it must act through; the verdicts below are measured, not asserted.

## 6.1 Discovery and catalogue systematics

The discovery-bias channel is the default objection, and it is quantitatively bounded and structurally excluded rather than argued away. The discovery geometry is evaluated separately for resident TNOs and transiting comets. For detached TNOs, the OSSOS characterized ensemble supplies a calibrated conditional null (rejected at $p = 0.035$). For comets, the discovery geometry is governed by all-sky synoptic optical surveys (Pan-STARRS 1 & 2 delivering 38.7 per cent, ATLAS 13.2 per cent, Mount Lemmon 12.8 per cent, and ZTF 10.5 per cent of post-2017 discoveries), whose seasonal cadences and ecliptic scanning could in principle imprint a discovery-longitude bias. Measured directly on the discovery coordinates of 266 post-2017 comets, the modern cometary discovery footprint is longitude-uniform (resultant $R = 0.025$, mean direction $322^\circ$, Rayleigh $p = 0.84$), with its mean $158^\circ$ off the displaced axis. The longitude-organized reconstruction systematic is therefore not a discovery-pointing artefact; it attenuates instead under the strictest observation-count cuts — the signature of the modern short-arc fit record rather than of where telescopes were pointed. The maximal-pointing ceiling (Section 3.2) shows the realized footprint cannot reach the observed amplitude at any coupling strength; the footprint's own mean direction is $38.6^\circ$ off the observed axis; and the footprint proxy is validated against real discovery coordinates at 6.2° median error. Catalogue systematics are excluded by era stability on three independent channels — the spatial aphelion dipole is positive in every era and strongest post-1990 ($p = 2\times10^{-4}$, $n = 200$; step 077), the resident cap membership is uncorrelated with first-observation year ($\rho = -0.09$, $p = 0.58$; step 078), and the residual-slip contrast stays positive in all perihelion-era bins (step 072). Quality-class stability (the all-quality-class sample retains $p = 0.011$), MPC element concordance (Section 3.6), and the fact that the comet signal sits $24^\circ$ off the TNO axis rather than on it complete the exclusion — a shared astrometric error would have to corrupt two different catalogues toward the same wrong direction.

Reduction-level artifact channels are bounded directly. The ephemeris-version question — whether propagating catalogue elements under DE440s rather than the DE405/DE430 generation under which they were fitted perturbs the boundary reconstruction — is tested by re-integrating the class-1 CODE sample to the 250 AU sphere under both kernels: the boundary periapsis directions agree to a median of $3.4\times10^{-9}$ degrees per leg, and the in-cap residual gap is identical at $+0.1234$ dex ($p = 0.0118$) under both ephemerides (step 147). The osculating-epoch convention is bypassed rather than tested: the raw-astrometry two-leg refits (steps 127–128) anchor each orbit at the observational mid-arc on the observations themselves, so no catalogue osculation convention enters the independent record at all.

The residual lineage caveat is then bounded quantitatively rather than rhetorically (steps 079, 081). MPC's CometEls file is an independently maintained fit record; on the 70 comets present in both it and the CODE/Warsaw tables, the per-comet periapsis-direction difference between fitters is the direct inter-fitter noise floor — median $0.15^\circ$, the same order as the anomaly, which is precisely why the measurement is made on the within-catalogue discrepancy channel rather than on element values. The floor is orthogonal to the signal on every axis tested: cap-symmetric ($0.158^\circ$ in versus $0.146^\circ$ out, $p =$ 0.37), uncorrelated with the per-comet discrepancy ($\rho =$ +0.22, $p =$ 0.082), and carrying no coherent displacement direction inside the cap (rotation-axis resultant $R =$ 0.191, $p =$ 0.47) — a direction-dependent fitter bias, the last form the lineage hypothesis can take, is absent between the two fit records. The geometric door to a sky-localized astrometric systematic is likewise measured: in-cap comets are in fact observed preferentially toward the anti-axis sky region (geocentric resultant $R = 0.71$), because a 1–2 AU perihelion point dominates Earth's positional scatter — so a zonal catalogue error is not excluded on sky position alone. It fails structurally instead: the resident TNOs are observed from the axis side, the opposite patch, so a single zonal error cannot organize both populations toward one inertial direction; nor can it produce the channel structure — $\omega$-only rotation with the energy channel flat — or the era stability spanning reference-catalogue generations.

The matched-patch control then turns the same geometry against the systematic hypothesis (step 082). Comets observed within the affected region — within $60^\circ$ of the anti-axis at perihelion epoch, identically exposed to whatever zonal error operates there — do not carry the anomaly uniformly: the discrepancy follows the aphelion direction, not the observed direction. The axis-alignment coefficient retains its sign, but is not separately significant, after control for observed-patch alignment ($\rho = +0.11$, $p = 0.22$, partial Spearman) while the observed-direction correlation collapses under the mirror control ($\rho = +0.008$, $p = 0.93$); position inside the affected region modulates nothing ($\rho = +0.004$, $p = 0.98$ among in-cap comets); the in-cap members observed inside the affected patch remain the most discrepant cell ($0.144^\circ$, $n = 35$) against the out-of-cap members observed outside it ($0.078^\circ$, $n = 75$); and the out-of-cap comets observed inside the patch are not discrepant at anomaly level ($0.131^\circ$). The classic crowding and twilight channels fail as well: the anomalous cohort is observed at higher galactic latitude than the rest of the sample (median $|b| = 32.93^\circ$ versus $28.50^\circ$, $p = 0.039$ — the opposite direction from a crowding-driven error, which would press the discrepant members toward the plane), and solar-elongation geometry at perihelion epoch is flat against the discrepancy ($\rho = -0.15$ across the sample, $+0.04$ within the anomalous cohort). The anomaly tracks where the comet came from, not where Earth stood when it was seen.

The lineage caveat is now a result rather than a footnote (steps 117–123). The post-2017 SBDB reconstruction cohort fails the registered declared-axis contrast and carries a displaced, temporally graded dipole that the signature-morphology discriminator initially resolved toward the conventional-dynamics class on the catalogue record — a classification the independent refit later revises: on independently fitted legs the displaced structure is energy-flat and single-leg-localized, the same lapse-slip class as the declared anomaly on the mirror leg (step 128).

The corrected two-component fit gives $-0.117$ dex on post-2017, $+0.150$ dex on CODE, and $+0.021$ dex on pre-2018; the fitted dipole changes the modern cap contrast by only $0.023$ dex. The independent leg audit validates the integrator ($\rho=0.93$ across 90 shared objects), but the recovered twist comparison is null (overlap median cosine $+0.236$ in-cap versus $+0.390$ out-of-cap, $p=0.382$). Matched JPL/CODE cap gaps are modest in true dex ($+0.153$/$+0.124$ versus $+0.101$/$+0.049$), with only the class 1a$+$ subset suggestive. These are useful consistency and failure-boundary results; they do not establish a universal reconstruction signal or a common TEP field. The shared-pipeline branch of the caveat is then closed at the observable level (step 127). Every three-leg record above derives from Warsaw-school fitting pipelines, so the record is rebuilt without them: raw MPC astrometry is differential-corrected independently on each perihelion leg under the identical REBOUND/DE440s force model, and propagated to the boundary sphere through the same machinery — a zero-Warsaw-lineage record of 585 comets. The per-comet rotation concords with the catalogue at $\rho =$ +0.998, and the registered carrier — the inbound-leg deviation — retains its in-cap excess on the independent fits ($p = 0.0370$ against the catalogue's own $p = 0.0287$ on identical members; CODE-overlap seat $p = 0.00688$ versus $0.00500$), while the channels the catalogue registers as flat remain flat. The anomaly is therefore a property of the shared astrometric record, not of the pipeline that reduced it; the residual lineage question is reduced to astrometric coverage, since the MPC record thins before $\sim 1950$.

The same instrument pointed forward closes the question in the other direction: refitting the post-2017 cohort's raw astrometry into independent legs — 248 comets at $\rho = +0.9996$ concordance, 160 with both legs and a joint fit available — returns the carrier reversed at the declared axis ($0.0857^\circ$ in-cap versus $0.109^\circ$ out-of-cap, permutation $p = 0.996$) identically under the leg-fit, single-fit, and catalogue constructions (step 128). The reversal is therefore a property of the modern astrometry itself, not of signal absorbed into joint elements — the parameter-absorption account is tested and refuted — while the modern record's own displaced-axis residual field replicates on the same independent fits at $+0.164$/ $-0.0415$ dex ($p = 5.2\times10^{-5}$), extremal under a 500-direction shuffle at $p = 0.00599$ and strengthened on the nongravitational-clean subset ($p = 1.7\times10^{-4}$). Both eras' structures are thereby certified astrometric; what the leg-separated record establishes is that the inbound-leg anomaly is confined to the pre-2018 era however the modern record is read — while the modern structure is compatible with the lapse-slip class (outbound-leg contrast at permutation $p = 0.0562$ (borderline), unresolved energy contrast at $p = 0.987$) on the mirror leg.

The displaced structure also does not subsume the declared signal it shares the modern record with. The same post-2017 cohort whose residual field peaks at $(120^\circ,-40^\circ)$ continues to lean toward the declared cap on the spatial aphelion channel ($d_\parallel =$ +0.0949, tide-aware $p =$ 0.00215; step 124) while recovering only $+0.037$ ($p = 0.15$) toward the displaced axis itself — so the declared alignment persists in the very population that carries the modern systematic. The discriminator is expressibility rather than quality: the aphelion lean is quality-flat (+0.103 bound, +0.0830 hyperbolic; +0.0907 short-arc, +0.0979 long-arc; +0.0987 sparse, +0.0919 heavily observed), persisting across the 257 members absent from the training set (+0.0827, $p =$ 0.00805), because the aphelion-direction statistic is computed independently of the inbound/outbound leg comparison. The displaced residual, by contrast, is strongest precisely where the reconstruction can express a boundary slip — bound and perihelion-spanning members (+0.23 dex each), flat across the arc-length and observation-count halves, and significant on the well-observed and NG-clean subsets ($p = 0.005$, $p =$ 1.7×10<sup>-4</sup>). That is the profile an anomaly expressed through reconstruction produces, not the profile a reconstruction error produces. The two structures are best read as the same lapse-slip class measured on opposite legs and at reversed measured polarity about the frame-anchored axis: the independent refit shows the modern structure is rotation slip on the outbound leg without energy exchange ($p =$ 0.0275 leg, $p =$ 0.987 energy; step 128), and the CMB-frame dipole ladder recovers the same frame-anchored axis at reversed sign on the two records — rather than competing anomalies in different signature classes.

The joint opposite-polarity configuration is then a registered statistic on the independent record itself: the two eras return $b = -0.058$ and $+0.172$ dex about the CMB axis, opposite-signed on identical catalogue cross-check, and the joint product is extremal at $p = 1\times10^{-4}$ under within-era permutation (step 129). What the post-2017 record changes is therefore the dominance ordering inside one channel, not the reality of the declared-sector structure, whose spatial component replicates on the modern cohort and whose rotation component remains carried by the pre-2018 three-leg record.

The transition's timing is then itself a measurement: on the pooled independent record, binned by transit epoch, an era-switch at 2018 between the two fixed axes beats every fixed-axis model under label permutation ($p = 0.0479$; the margin over the single best fixed axis is positive but not separately resolved, $p = 0.200$), while a smooth drift reaches the scan's resolution floor (ramp width $\leq 10$ yr) without clearing the null ($p = 0.208$) (step 130). The scan therefore supports an era-switch ordering in the measured record, with no separately resolved margin over the best fixed axis — carried, under the static-boundary reading adopted here, by the instrument's era-composition change rather than by motion of the field itself. The era-boundary crossers confirm that the signature tracks perihelion epoch rather than the source table (displaced-cap $p = 0.048$ on post-2018 transits drawn from the pre-2018 cohort), and inside the declared cap the anomaly's amplitude declines across the baseline ($\rho = -0.18$, $p = 0.038$). A sharp transition is also what an era-boundary systematic would predict, so the timing evidence favours the epoch reading without excluding the systematic — a limit the analysis registers alongside the detection.

The field's morphology is then partially resolved: both measured anomaly axes sit inside the same $45$–$60^\circ$ cone band on opposite sides of the CMB-anchored axis (joint $p = 0.021$ for independent directions), and the modern structure's residual peaks on a ring at $55^\circ$ cone angle rather than at the axis pole ($+0.334$ versus $-0.026$ dex in-band, $p = 2\times10^{-4}$) (step 131) — the signature of a shell-like boundary at fixed angular radius, partial on the older record where the antapex-side enhancement is diffuse.

The frame anchoring is then extremal on geometry alone: the displaced axis sits $11.5^\circ$ from the equator-mirror of the declared axis — the equatorial-mirror configuration of a shell-like boundary — a configuration only $0.7$ per cent of 20{,}000 random axes render as tight, and all three measured axes share one meridian about that axis ($R = 0.992$, $p = 0.006$ for isotropic directions) (step 132), with the modern apex-side residual elevated within $30^\circ$ of that meridian plane ($+0.209$ dex, $p = 0.0016$). The two era structures thus sit on opposite sides of one frame-anchored shell and carry reversed measured polarity between eras, and the CMB axis is extremal on two directional statistics independent of the dipole-amplitude test.

The field-level decomposition closes the same loop: $l = 1$ leads each era's residual spectrum at opposite sign, and the freely fitted era dipoles — no declared direction — are coplanar about that axis ($\Delta\phi = 2.1^\circ$, joint mirror-pair $p =$ 0.0297), so the residual fields themselves share the meridian, not merely their cap positions (step 133); all three ISO periapsides land within $30^\circ$ of it as a registered ledger datum, and the resident cone-versus-perihelion control returns flat, ruling out a second radial scale.

The epoch-coincidence hypothesis — that the 2018-era transition is a product of the Solar Cycle 24 maximum and the declining heliospheric plasma state — is excluded at observation level in Section 4.14: the OMNI solar-wind record shows the pooled pressure correlation is an era-proxy artefact that vanishes within eras, so contemporaneous dynamic pressure neither explains nor mimics the flip (step 136).

## 6.2 Point mass and Planet Nine

The shepherding model fails on three independent channels. Mechanically, it predicts resonant substructure and element coupling that are absent (Section 3.3). Morphologically, a $1/b^2$ impulse concentrates at the perturber's direction, while the observed in-cap profile is a flat plateau with a threshold edge at 60–75° (Section 4.4). The three fatal kinematic facts are worth isolating:

- **Impulse shortfall.** Rotating an aphelion direction by the observed $\sim 0.15^\circ$ at $\sim 500$ AU requires a transverse impulse no candidate perturber supplies: the published BB21 realizations, directly inserted into the boundary integrations, under-rotate the catalogued trajectories by a median factor of $1{,}020$ on the matched sample (step 062).

- **Mass exclusion.** Converting the shortfall to a required mass yields a brown-dwarf-scale perturber — median 4294$\,M_\oplus$, a value the all-sky infrared surveys exclude outright — and the bound stays at that scale at every rung of a 100–1000 AU distance ladder (steps 087, 091).

- **Energy constraint.** The in-cap energy contrast is unresolved ($p =$ 0.688, step 034). This constrains modelled along-track impulses jointly with the rotation amplitude and direction; it does not alone exclude transverse impulses or secular gravitational torques.

The impulse budget quantifies the first failure directly: $$\Delta v_{\rm required} \approx 4.9\ {\rm m\,s^{-1}} \quad {\rm vs.} \quad \Delta v_{10\,M_\oplus} \approx 0.11\ {\rm m\,s^{-1}} \;\; (44\times\ {\rm short}).$$

Even a Jupiter-mass object delivers only $3.6$ m/s — still short, and already excluded by WISE at that distance. The obstruction is amplitude- and morphology-based rather than kinematic in principle: secular gravitational torques do rotate the orientation elements at nearly fixed energy — the same planetary baseline reproduces the catalogued leg rotations at $\rho = 0.999$ while the simulated kicks stay below $\sim 5\times10^{-6}$ AU$^{-1}$ in median $1/a$ — so a flat energy channel alone does not exclude secular gravity. What excludes it is quantitative: the residual rotation, measured after regression of that baseline, exceeds every allowed perturber's amplitude by two to three orders of magnitude while remaining direction-organized, single-leg-localized, and uncoupled from the physical elements. The impulsive channel additionally fails on geometry. Measured on the independent leg-fitted trajectories, the transverse impulse demand is $\sim 11$–$14$ m/s per in-cap comet while the measured $|\Delta(1/a)|$ channel bounds the along-track impulse only at $\sim 20$ m/s (step 145): the energy channel does not force a fine-tuned transverse geometry — an earlier audit's sub-m/s bound substituted the planetary-approach distance for the energy residual and has been corrected. The impulse class instead fails on the velocity signature: a mechanical kick predicts an implied slip anticorrelated with crossing speed, while the pre-2018 record is velocity-flat. The flat energy channel is therefore corroborating evidence rather than the load-bearing exclusion — the point-mass class fails on measured amplitude, axial localization, and threshold morphology, the rotation profile the temporal-boundary realization predicts.

The direct insertion test closes the loop. The published Brown & Batygin (2021) perturber — both the maximum-likelihood ($m_9 = 5.0\,M_\oplus$, $a_9 = 300$ AU) and marginalized-median ($m_9 = 6.9\,M_\oplus$, $a_9 = 461$ AU) realizations, each placed at four mean anomalies spanning the unconstrained orbital phase — is added to the same REBOUND/DE440s backward integrations of every class-1 CODE comet (step 062). Its apoapsis lies inside the anomalous cap, so the perturber's greatest leverage already falls on the comets that must carry the signal. The largest rotation Planet Nine can impart to any catalogued trajectory is nonetheless $0.0027^\circ$; the median across all placements is $0.0005^\circ$ against the observed in-cap median of $0.14^\circ$ — a shortfall exceeding $300\times$, rising to $527\times$ on the matched sample under the most generous per-comet bound — and no comet receives even its own observed discrepancy. Nor is the perturbation aimed correctly: the injected rotations are anti-correlated with the discrepancy they would need to explain ($\rho = -0.27$, $p = 0.0018$), and absorbing them leaves the cap contrast at $p = 0.0016$–$0.0024$. The deficit is therefore not a modelling assumption about where a perturber might be; it is measured on the catalogued trajectories themselves — the specific planet the anomaly is attributed to cannot produce it on the real trajectories.

The impulse shortfall is then converted into a mass bound on the same trajectories (step 083). The injected rotation scales linearly with perturber mass (per-comet exponent $\alpha = 1.00$ across a four-rung ladder), so the rotation a point mass would need to supply translates directly into the mass it would need to have. At the BB21 median location the in-cap discrepancies require a median perturber of $\sim$ 4294$\,M_\oplus$ — thirteen Jupiter masses — with a fifth-percentile floor of $\sim$ 794$\,M_\oplus$; under the most favourable of the four orbital-phase placements per comet the bound softens only to $\sim$ 3069$\,M_\oplus$ median ($\sim 656\,M_\oplus$ floor), and every in-cap comet requires more than $100\,M_\oplus$.

A body of that scale at $\sim 460$ AU is not a planet but a brown dwarf, excluded outright by the all-sky infrared surveys — and even $150\,M_\oplus$ at the same location still under-rotates the median comet by a factor of 24. The third published realization fares no better: the Siraj, Chyba & Tremaine (2025) Planet-X best fit ($m_p = 4.4\,M_\oplus$, $a_p = 290$ AU, $e_p = 0.29$, $i_p = 6.8^\circ$), placed in the same anti-aligned shepherding geometry, injects a median $0.0002^\circ$ — a $705\times$ shortfall — with the same anti-targeting ($\rho = -0.16$).

The survey side independently narrows the remaining room: Pan-STARRS1 together with the ZTF and DES limits excludes $\sim 78$ per cent of the BB21 parameter volume (Brown, Holman & Batygin 2024), and the single surviving IRAS–AKARI far-infrared candidate pair (Phan et al. 2025) remains unconfirmed.

Nor does the bound depend on where the perturber is placed (step 087). Repeating the mass inversion on a five-rung distance ladder — perturber semimajor axes 250, 300, 461, 700 and 1000 AU, with the eccentricity and apsidal geometry rescaled from the BB21 solution — yields in-cap required masses of $\sim 2186$–$4290\,M_\oplus$ at every rung (median $m_{\rm req} \propto a_9^{+0.2}$, nearly distance-independent over a factor of four in semimajor axis), because the rotation a point mass injects falls off only as $a_9^{-0.14}$ at fixed perihelion-direction alignment. One hundred per cent of in-cap comets require more than a Jupiter mass at every tested distance; the fifth-percentile floor never drops below $\sim 530\,M_\oplus$. A more distant or more nearby perturber does not help: the rotation deficit is geometric, not parametric. What the anomaly requires at the published perturber locations — and at every distance between them — is therefore measured rather than assumed, and it is not a planet.

The newest proposed escape route — an inner perturber rather than a distant one — is tested with the same machinery (step 091). Siraj, Chyba & Tremaine (2025), measuring the distant belt's mean plane with a footprint-immune likelihood, concluded that neither published distant-planet realization can generate the warp they detect, and that a third body — 'Planet Y', of order Mercury-to-Earth mass ($0.06$–$1\,M_\oplus$) at $a = 100$–$200$ AU with $i \gtrsim 10^\circ$ — would be required. Whether such a body can produce the cometary anomaly is measured directly: the class-1 CODE integrations are rerun with an inner perturber on a six-rung ladder, $a_9 = 100$–$250$ AU, in both the anti-aligned shepherding geometry (the BB21 orientation continued inward) and the measured warp plane itself ($i_9 = 15^\circ$, $\Omega_9 = 120^\circ$, $\varpi_9$ antipodal to the axis). Under either geometry the required mass stays at brown-dwarf scale — in-cap medians of $515$–$2186\,M_\oplus$ across the rungs — and the failure is total in both bounding directions: zero in-cap comets require a mass inside the SCT25 Planet-Y box, and zero require a mass below the Gomes et al. (2023) five-sigma ephemeris line $1.1\,M_\oplus(d/400\,{\rm AU})^3$, the level at which Juno/Cassini/Mars ranging already senses a tidal field. A Mercury-to-Earth-mass inner planet may tilt the disk's mean plane by a degree or two; it cannot rotate the transiting comets by the observed amount — and any inner body massive enough to do so would have announced itself in the planetary ephemerides.

The distant-belt warp is audited with the SCT25 likelihood on MPCORB, a catalogue fit lineage distinct from the SBDB sample used in that study (step 090). Reanalysis of its published 46-object list reproduces the reported pole parameters to the printed precision. The broader non-resonant census has a weaker warp: the pooled 80–400 AU tilt is 3.42° (0.730$\sigma$), with the deepest bin at 1.88$\sigma$. Direction is therefore considered separately from warp significance. The 80–200 and 200–400 AU bin leans are 16.6° and 45.3° in ecliptic azimuth, compared with the resident axis at 49°. The pooled lean is 31.1° (17.9° off-axis, directional $p=$0.0848); the deepest bin is 3.70° off-axis ($p=$0.0449), under the position-conditioned null. This is a sample-dependent directional consistency check, rather than an independently significant census-wide warp detection.

Finally, the one real candidate body ever reported — the surviving IRAS–AKARI far-infrared pair of Phan et al. (2025) — is closed quantitatively rather than left at 'unconfirmed' (step 093). Its catalogued position $(\lambda, \beta) \approx (5.4^\circ, -57.5^\circ)$ sits $52^\circ$ from the measured axis and $128^\circ$ from the anti-aligned shepherding direction a distant perturber would need to occupy. Its capacity is measured by direct insertion into the same REBOUND/DE440s machinery at 600 AU at the published mass estimates of 7 and $17\,M_\oplus$: the injected boundary rotation is $0.0003$–$0.0007^\circ$ against the observed in-cap $0.14^\circ$, a required-mass median of $\sim 3367\,M_\oplus$ — two to five hundred times the candidate's own estimated mass and an order of magnitude beyond the all-sky infrared exclusion. If the candidate is a planet, it is not the perturber; if it is not, the perturber-zoo map contains no surviving body at any distance on any geometry tested.

The population-level test removes the remaining escape route — that an unmeasured torque maintains the cluster without rotating comet orbits. The real detached population itself is forward-integrated for 20 Myr through the DE440s planetary system, first with the giant planets alone and then with each published Planet Nine realization added (step 066). Under the giants alone the measured cluster does not hold: the resultant $R$ of the perihelion directions decays from 0.332 to 0.265 over 20 Myr, implying a dispersal time of $\sim 160$ Myr — the observed alignment is young against the age of the system and requires continuous regeneration. Inserting the perturber does not supply it. The two realizations bracket the unperturbed decay rather than arresting it ($R(20\,{\rm Myr}) = 0.304$ and 0.228 versus 0.265 under the giants alone), the per-object longitude-of-perihelion diffusion is statistically unchanged, and the restoring-torque diagnostic — correlation of each object's planet-added drift with the drift needed to return it to the anti-aligned libration centre — finds no confining signature: marginal and of the wrong sign for the maximum-likelihood model ($\rho = +0.28$, $p = 0.07$) and null for the median model ($\rho = +0.09$, $p = 0.58$). The published planet neither rotates the transiting comets nor confines the resident population it was invoked to explain; a resident spatially fixed structure regenerating the alignment continuously has no such requirement to meet.

The 2025 literature does not converge on a single perturber to replace the failed one; it fragments into mutually inconsistent targets — a $7$–$17\,M_\oplus$ source at $500$–$700$ AU (Phan et al. 2025), a $4.4\,M_\oplus$ perturber at $290$ AU alongside a sub-Earth Planet Y at $100$–$200$ AU (Siraj, Chyba & Tremaine 2025, both tested above), and an anti-aligned sednoid whose discoverers conclude against any single present-day shepherd at all. The last conventional recourse is therefore to displace the agent in time rather than in space: a stellar flyby in the birth cluster (Kenyon & Bromley 2004) or a temporarily captured rogue planet (Gladman & Chan 2006) that imprinted the detached architecture once, at formation — the reading Chen et al. (2025) themselves favour in dating a primordial clustering to $\sim 4.2$ Gyr ago. It is the most resilient historical alternative because it requires no surviving perturber, and the catalogue accommodates its fossil: the anti-aligned Ammonite merely dilutes the measured alignment rather than dissolving it (Table 5). It is nonetheless excluded by the same two clocks that remove every other candidate. The first clock is resident-side: the in-cap $\varpi$ alignment disperses on a $\sim 160$ Myr time under the giant planets alone (step 066, above), so an imprint laid down $\sim 4.2$ Gyr ago cannot persist to the present unless an agent regenerates it continuously — and neither inserted perturber supplies the required restoring torque. A one-off flyby fails the same requirement a fortiori: it can deposit an alignment but cannot maintain one. The second clock is transit-side and decisive. The cometary anomaly is not a static element distribution but a reconstruction discrepancy that accrues during the presently observed arc — the inbound and outbound legs of the same catalogued trajectory disagree through a boundary crossing measured within the last two decades. An event that ended billions of years ago has no channel through which to act on a comet now in flight. The two populations thereby impose opposite temporal requirements on the agent — the resident cluster needs an imprint old enough to sculpt yet maintained to the present, the transit cohort needs an agent active inside the observing window itself — and only a persistent, spatially fixed field satisfies both. That is a selection on the mechanism's time structure, not a uniqueness proof: any still-active field would meet the joint constraint, and the boundary realization tested here is the framework's candidate for it. What it excludes is the entire one-off class — every mechanism that deposits structure once and leaves.

## 6.3 Interstellar-medium and heliospheric drag

Asymmetric ISM drag or premature bow-shock outgassing would be a dissipative force and is excluded on four measured channels.

- Spatial misalignment: the ISM inflow direction $(\lambda \approx 256^\circ, \beta \approx +5^\circ)$ is explicitly tested in the axis scan and is null ($p = 0.95$ on the discrepancy channel).

- Mechanics: drag must subtract kinetic energy, so its primary signature is a systematic $\Delta(1/a)$ shift — a channel with no resolved in-cap excess ($p = 0.69$). The null difference alone does not establish zero exchange; excluding a specific friction model requires comparing its predicted energy effect with the measured uncertainty and with the directional and localization tests.

- Depth selectivity: every Oort-spike comet crosses the heliopause, but the anomaly is depth-selective — flat for shallow plungers ($p = 0.16$), present only for deep plungers crossing the full nested structure.

- The NG paradox: if heliospheric interaction triggered unmodeled outgassing, the orbit fits would require non-gravitational parameters — but the NG-need fraction is lower in-cap (7/19 versus 24/35 out, $p = 0.043$). The anomaly registers to standard fitters as a clean, gravity-only orbit that points in the wrong direction.

The propagation class of plasma effects is excluded by the observable's construction rather than by amplitude arguments alone. Heliospheric plasma couples to a measurement only through the dispersive delay $\Delta t = 40.3\,{\rm STEC}/(cf^2)$ of a radio link — a channel that vanishes at optical frequencies, so no solar-wind or MHD state can enter the cometary orbit solutions, which are fits to optical astrometry. On the spacecraft channel, where the channel does exist, the direct regression is null (step 154): the per-boundary SCLK residuals show no contemporaneous coupling to solar-wind pressure, field magnitude, or activity indices and no Sun–Earth–probe elongation dependence, while their radial organization survives partialling every plasma covariate. The remaining plasma-adjacent competitor for the heliosheath shell modulation is environmental rather than propagation: spacecraft charging, radiation environment, and tracking-cadence changes across the disturbed sheath could in principle couple to the oscillator or the calibration fit — a degeneracy the analysis prices by registering that pattern as monitored rather than as evidence (Section 6.9).

## 6.4 Non-gravitational outgassing

Cometary recoil is the strongest remaining conventional channel and is controlled directly. The subsample whose orbits were fitted with no NG freedom at all — pure gravity solutions — carries the excess directionally (median $0.21^\circ$ versus $0.14^\circ$ out-of-cap, $p = 0.051$ at $n_{\rm out} = 11$), so the discrepancy is not an artifact of outgassing parametrization. The signed NG parameters show no cap concentration ($|A_1|$ correlation $p = 0.066$, $|A_2|$ $p = 0.19$, $|A_3|$ $p = 0.74$), and the NG-need deficit runs the wrong direction for an outgassing explanation (Section 6.3).

The NG record is then read for the first time as an observable rather than only as an exclusion control (step 143). The logic is directional: a boundary slip absorbed by a pure-gravity fit must leak into whatever free parameter the fitter adds to absorb a timing error, and in the Marsden formulation that parameter is the transverse term $A_2$ — the component that shifts the fitted period — not the radial $A_1$ or normal $A_3$. The Warsaw NG record carries the predicted selectivity: in-cap members' transverse term is elevated at a median $0.65$ versus $0.17$ ($3.9\times$, $n_{\rm in} = 10$, Mann–Whitney $p = 0.30$) with $8$ of $10$ positive-signed, while radial and normal components are flat ($p = 0.50$ and $0.77$) and the fitted outgassing-peak time shift shows no contrast ($p = 0.12$). The channel is registered at candidate level rather than as a detection: NG membership is not cap-selective ($p = 0.82$), the in-cap NG sample is small, and the same test on the independent SBDB NG record does not replicate the elevation ($p = 0.68$). What the channel contributes is the sign and component structure — the only term that leans is the one a time-domain anomaly must load — priced against its membership and its independent replication.

## 6.5 Planetary perturbations

Planetary-encounter geometry is the one mechanism that could manufacture an axis preference, and it is subtracted with a measured N-body baseline rather than a proxy (Section 4.3): every class-1 CODE comet is backward-integrated through the DE440s planetary system with REBOUND, the simulated kicks reproduce the catalogue's own energy changes at $\rho = 0.98$, with the recovered boundary energies cross-validated between two independent integrators at $\rho = 0.93$, the in-cap comets carry no energy-kick advantage ($p = 0.568$), and the residual cap contrast is $p = 0.0008$–$0.0021$ with continuous correlation $\rho = -0.27$ ($p = 0.0017$). The doubly matched control on perihelion depth and realized energy kick — $p = 0.0019$ — closes the remaining depth-and-kick confound. The bidirectional extension (step 063) validates the comparison against the catalogue's own integrations: propagated to the same 250 AU barycentric sphere, the independent run reproduces the catalogue's original and future periapsis directions to a median $0.001^\circ$ per leg and the full orig $\rightarrow$ fut rotation at $\rho = 0.999$. Run unchanged on the Warsaw spike sample — a second, independently selected catalogue sample — the machinery reproduces its boundary solutions to $0.0008^\circ$ per leg ($\rho = 0.998$), and the unexplained inbound-leg offset on the channel that catalogue's anomaly occupies survives the same encounter-budget regression at $p =$ 0.0079$–$0.0390 (steps 064–065; Section 4.9). The observed rotation is thereby shown to be entirely a product of standard planetary dynamics — the anomaly is that its size is direction-dependent beyond the measured encounter budget, surviving a four-covariate baseline of kick, closest approach, depth and inclination at $p = 0.002$–$0.012$, on two catalogues rather than one.

## 6.6 Galactic tide and extended structures

The galactic tide produces $m = 2$ quadrupolar structure aligned with the galactic frame; the measured anomaly is $m = 1$ dipolar and sits off the tidal band (the axis is at galactic latitude $-44^\circ$). The strongest possible tide-based reading — an aphelion asymmetry driven by the tide's radial component, which concentrates delivered comets toward the galactic centre and anticentre — is tested directly in Section 4.8: the observed aphelion dipole points $44^\circ$ off the plane at the TNO axis and exceeds the on-plane anticenter component in every sample, while the anticenter component itself is null in the hardest-observed cohort. A tide-aware null that preserves the observed latitude structure still rejects the dipole at $p \lesssim 6\times10^{-3}$ in all four comet samples. A massive debris disk or extended perturber sheet likewise generates even-$m$ signatures and cannot produce a single localized 60° patch. The orbit-pole eigenstructure favours a point concentration (a common direction) over a girdle, disfavouring the warped-disk reading.

The tide is also bounded by direct insertion rather than by scaling argument alone (step 070). The step-063 bidirectional integrations of all 131 class-1 CODE comets are rerun with the Heisler–Tremaine tidal tensor added to the comet's equations of motion by Strang splitting (Oort constants $A = -B = 13$ km s$^{-1}$ kpc$^{-1}$, $\rho_\odot = 0.1\,M_\odot\,{\rm pc}^{-3}$). Across every leg the tide moves the reconstructed boundary periapsis by a median of $<10^{-6}$ degrees (95th percentile $1.2\times10^{-6}$°; the lone degree-scale outlier is a tangent boundary crossing where the leg geometry amplifies any perturbation), and the median orig$\rightarrow$fut rotation changes by $9\times10^{-9}$ degrees — seven orders of magnitude below the observed in-cap anomaly ($\sim 0.14^\circ$). The tide's quadrupole does not even imprint a significant in/out contrast on the leg offsets ($p =$ 0.20) or on the rotation change ($p =$ 0.33), and its amplitude is in any case utterly negligible on the scale of the measured signal. The galactic tide is therefore excluded not only by symmetry and direction but by direct dynamical insertion: the last known un-modelled Newtonian channel cannot produce the anomaly.

The same insertion test is now run on the resident side, where the tide is the standard no-new-physics mechanism invoked for outer-system apsidal structure (step 088). The 44 catalogued detached TNOs are integrated 20 Myr through the giant-planet system with the same tidal tensor applied by Strang splitting, in three realizations. On the observed initial conditions the tide does not maintain the cluster: the in-cap apsidal-drift dispersion gives a dispersal time of 151 Myr — statistically the giants-alone rate — and the resultant decays to $R(20\,{\rm Myr}) = 0.275$, indistinguishable from the un-tided 0.265. Nor can the tide regenerate the alignment: starting the same 44 bodies from an isotropized $\varpi$ distribution, the tide produces at most $R = 0.14$ over the full 20 Myr — consistent with isotropic scatter for $n = 44$ — at a mean direction of $166^\circ$, far from both the observed axis ($49^\circ$) and its own ecliptic libration directions ($347^\circ$, $167^\circ$); and the restoring-torque diagnostic finds no confining signature ($\rho = +0.135$, $p = 0.38$). Tide plus perturber is no better than either alone ($R = 0.246$). On the resident population as on the transit population, the Galactic tide neither sustains nor creates the measured alignment.

## 6.7 Statistical robustness

Small-$N$ sensitivity is assessed by conditional permutations, Monte Carlo controls, leave-one-out checks and split-half directional recovery. Each result answers its stated null question; the observed-sky fraction from a directional scan is a localization rank rather than a null-data search probability. The within-family Benjamini–Hochberg ledger (Section 5.2) records false-discovery-rate screening of the listed tests, without establishing independence of nested samples. The ten-channel synthesis spans six catalogue lineages and gives $S_{10}=186.2$ at the resident axis, with conditional sky-area rank 0.0145 across 20,000 directions on the fixed catalogues. It shows shared directional concentration, while a global discovery calibration would require null catalogues, the actual search and selection procedure, and preserved shared-object dependencies. Catalogue overlap, the broad recovered comet patch, sparse lower-quality subsamples and the in-cap energy covariate remain in the ledger. The nominal-element N-body integrations provide an ensemble encounter control; deeply chaotic encounters limit their use as per-comet reconstructions.

The deepest available null then turns the complete directional battery on the model field itself (step 149), and it locates the anomaly precisely where a spatial boundary must put it — upstream of dynamics. Every directional instrument registered on the observed discrepancy — the declared-cap contrast, the continuous Spearman test, the shell morphology, the 2,232-axis look-elsewhere scan, the 20,000-draw label-swap null, and the axis-coincidence null over 2,000 full permutation sky scans — is applied to the pure Newtonian rotation field, the catalogued field, the observed-minus-model residual, the energy-kick magnitude, and the rotation per unit kick. Three facts emerge. First, the axis is real and shared: the model field's own best-contrast direction lands $10.8^\circ$ from the independently derived resident axis — a coincidence only $0.9$ per cent of direction-unlinked fields reproduce — and the catalogued field's best axis is identical to the model's at $0.0^\circ$ separation. Second, the model does not manufacture that axis; it inherits it. The inbound arrival directions themselves over-concentrate toward the resident axis — 36.6 per cent of class-1 arrivals inside its 60° cap against the $25$ per cent sky fraction (binomial $p =$ 0.00205) — and the excess is not a discovery-pointing artifact: under the ecliptic-latitude-conditioned null, where each arrival's longitude is redrawn uniformly at its observed latitude (the first-order survey-pointing model, step 152), the conditioned expectation is 28.2 per cent and the observed fraction clears it at $p =$ 0.0230 (Galactic-latitude conditioning $p =$ 0.0693; matched subset 40.7 per cent, $p =$ 0.0116 and 0.0280). The directional organization is therefore present at coordinate level, in the geometry of where comets arrive from, before any integrator runs. Standard dynamics is a passive transmitter: it converts an anisotropic arrival field into an anisotropic rotation field, which is why the model reproduces the cap contrast ($p = 0.004$) and why the anomaly cannot be dynamical in origin — dynamics has no access to the input directions that carry the structure. Third, the morphology stays the signature's own: the energy-kick channel is direction-flat ($|\Delta(1/a)|$ vs $\theta$: $\rho = -0.083$ about the resident axis, $-0.044$ about the comet axis), so the directional rotation structure enters entirely through geometry, never through work — rotation without energy on the model side as on the observed side. The observed-minus-model residual is flat at the cap ($p = 0.10$), which is consistency rather than absence: a phase-slip boundary translates orbits in time without adding rotation magnitude beyond what the slipped geometry already encodes, so no unexplained rotation should remain — and none does. What the model cannot account for is the input itself: the rotation-per-unit-kick channel retains a marginal axis coincidence on the matched sample — its best axis sits $28.2^\circ$ from the resident axis ($p = 0.069$) — while on the full class-1 cohort the channel's best axis is unconstrained ($80.7^\circ$, $p = 0.41$), i.e. conversion efficiency is at most weakly axis-aware once kick amplitude is divided out. The Newtonian null thereby performs its proper function: it demonstrates that every dynamical degree of freedom is exhausted reproducing a structure whose direction is supplied from outside dynamics — the coordinate-level arrival anisotropy toward the resident axis, the same spatial lean the post-2017 cohort registers independently at $d_\parallel = +0.095$ ($p = 0.0021$).

The remaining escape on the arrival side is generative rather than statistical: the conditioned nulls of step 152 reweight the observed directions but cannot test whether a conventional source could have produced them in the first place. That question is closed by a forward model (step 157) that generates the incoming population itself: an isotropic Oort-spike source is injected by the secular Galactic tide — the orbit-integrated tidal torque on the same tensor as step 070 — optionally remixed toward isotropy by stellar impulses, and filtered by the empirical ecliptic-latitude detection kernel. Every variant of the conventional chain returns the same answer. The synthetic in-cap fraction medians sit at or below 28.2 per cent against the observed 36.6 per cent (model $p =$ 0.0205 on the full class-1 cohort; matched subset $p =$ 0.0165), and the tidally injected density's own preferred direction lands $118^\circ$ from the resident axis — Galactic-plane banding, not an ecliptic cap. Sharper still, the probability that a synthetic sample's arrival resultant approaches the resident axis as closely as the observed matched-sample resultant does ($3.7^\circ$) is $p =$ 5.0×10<sup>-4</sup>. Conventional dynamics transmits the directional structure but does not generate it: the coordinate-level excess toward the resident axis is not produced by any tested combination of isotropic source, Galactic tide, stellar remixing, and latitude-biased discovery.

Step 163 tests the discovery instrument through an explicit, limited forward model. Each eligible comet retains its orbital shape $(q,e)$ and perihelion epoch, while proper Haar-random rotations change its orientation. Keplerian trajectories are propagated through solar-elongation, declination and geocentric-distance cuts, with a brightness integral $\sum\Delta t/(r^2\Delta^2)$. The reference window reaches 3 AU and 4 AU geocentric distance; it therefore covers 52 of the 131 class-1 source objects. Objects whose perihelia lie outside that visibility model cannot be included in its null comparison. Each eligible object's orientation weights are normalized separately, and each null realization draws one orientation for that same object, preserving the conditioned shape/epoch cohort instead of applying a second brightness selection to its population. On this modelled subset the predicted dipole resultant is 0.118 against the observed 0.244. Its direction is $(246.3^\circ,-69.40^\circ)$, 92.02° from the observed dipole; the longitude comparison is 255.8° versus 53.7°. The directional tail toward the declared axis is $p=$0.0092; the analogous visibility-conditioned SBDB tests give 0.00087 pre-2018, 0.045 post-2017 and 0.00012 pooled. The null cap fraction is 26.9 per cent against 34.6 per cent observed ($p=$0.16). Subtracting the modelled footprint leaves a residual 31° from the axis, with a maximum separation of 44.3° in the tested scenarios. Each scenario uses only its own eligible objects. These measurements constrain this circular-Earth, brightness-proxy selection model; they do not replace a survey pointing/detection pipeline or extend its coverage to the excluded higher-perihelion population.

The comparison has so far been asymmetric — conventional models are required to generate the data while the field hypothesis is only checked for consistency — and step 164 closes it by placing the TEP-side models on the identical generative footing. Two channels are separated. First, redirection: the measured slip displaces the fitted aphelion direction by a median 0.132° on the real observing chain (maximum 1.43°; step 151 position-slip realizations applied to isotropic footprint-selected arrivals), whereas reproducing the observed cap excess by coherent rim migration would require a uniform inward displacement of order $10$–$20^\circ$ — the measured slip is two orders of magnitude too weak, and isotropically directed besides. The arrival dipole therefore cannot be a redirection artefact of the boundary crossing: it is a property of the incoming orbital state itself, and any field explanation must act on which trajectories arrive, not on how the measured ones are read. Second, the required source structure is measured rather than assumed: solving the axis-anchored modulation needed to lift the selection baseline to the observed cap fractions returns an in-sector source density enhanced by a factor 1.28 on the full class-1 cohort (1.53 on the matched subset; $1.11\times$ on the large SBDB cohorts), equivalently a dipole modulation $w(u) \propto 1 + m\cos\theta_{\rm axis}$ with $m \approx$ 0.39. The morphology is constrained by the far side of the sky: the observed anti-cap fraction (0.130) sits below the selection-kernel baseline (0.237) in all five cohorts — the pattern a dipolar modulation predicts and a one-sided lobe does not (lobe median 0.214, depleted at $p = 0.015$ on the full cohort; dipole median 0.168, consistent). The evidential class of the dipole is then stated exactly: it is evidence for a direction-organized source population, not for the boundary mechanism — the measured slip cannot produce it, so its interpretation is degenerate between a primordial Oort-cloud anisotropy, a residual selection effect, and the proposed field, and it is carried in the ledger as consistency evidence — sharing, as it happens, the same dipole morphology the rotation field carries independently in its CMB-frame decomposition. The TEP-specific content rests on the channels where the field acts dynamically: the reconstruction slip and the resident secular imprint. And the two channels share the structure continuously, not only at the binary cap: per-comet rotation correlates with the axis-anchored lobe weight at $\rho =$ +0.279 ($p =$ 0.0012). The incoming-state requirement is thereby specified quantitatively — an axis-anchored dipole modulation of order tens of per cent in source density — which is the quantity a field-level injection mechanism must produce, and which the rotation anomaly then propagates downstream.

The mechanism-level question left open by step 158 — whether the canonical metric can carry the inferred slip — is tested directly on the canonical connection in step 165. The synchronization transport is $\delta\tilde{\sigma}_\mu \simeq -(B/A^2)(u\cdot\nabla\phi)\,P_\mu{}^\nu\nabla_\nu\phi$ (Paper 0, Appendix A3.2): the conformal part is exact and cannot carry holonomy, while the leading non-exact residue is disformal. Along a transported worldline this gives $\Delta t_{\rm tr}=-(1/c^2)\int b(u)R_H^2(du/dt)^2dt$. The minus sign is now retained numerically. The previous implementation instead multiplied by $|B_0|$ and assigned a physical sign afterwards, which made its branch claim non-computational. The corrected calculation uses the admissible $B_0=+3.2\times10^{-3}$ branch. Its magnitude is mirrored from the sign-excluded Paper-0 volume-balance reconstruction only to define a benchmark scale; it is not a calibration or prediction of the admissible normalization. The observable used for the amplitude fit must be distinguished from the signed transport. Step 065 converts an angular separation into $\delta t_{\rm eq}=|\delta\theta|r_b^2/h$; residualization can make the reported contrast positive or negative, but the underlying rotation magnitude contains no orientation. Step 165 therefore fits $|\Delta t_{\rm tr}|$ and records the transport sign separately. At the positive-$B$ benchmark the signed pole-to-midband transport contrast is -5.9 yr; changing to the excluded negative-$B$ reference reverses it to 5.9 yr, while the observable amplitude is unchanged at 5.9 yr. Thus the admissible branch does not predict the wrong sign of the statistic actually fitted: that earlier conclusion was a sign-observability error. A future signed comparison requires the connection orientation to be propagated through matched positive- and negative-slip orbit injections; the existing unsigned conversion cannot select the sign of $B$. The amplitude and morphology tests retain their distinct roles. The spatial entry and exit directions are computed from the published original and future conics. Their median separation at 250 AU is 28.0°, so the comet crossings are not antipodal. The signed-dipole excursion $G(\theta)=\cos\theta$ gives an even quartic amplitude in the coincident-aphelion limit because the envelope is even and the traversal gradient is squared; an offset dipole does not share that symmetry. This is an exploratory profile comparison, with the resident source modulation tested separately. Reproducing the measured 5.9 yr amplitude contrast requires $u_b\approx$2.5×10<sup>-4</sup> for this signed-dipole benchmark; the per-comet amplitude pattern correlates with the measured residual at $\rho=+$0.157 (fixed-profile permutation $p=$0.0170, without a correction for profile exploration). Each radial crossing speed is now calculated at 250 AU from that comet's inverse semimajor axis and angular momentum. A potential-tracking field ($u\sim4\times10^{-11}$) under-produces the benchmark contrast by approximately 24 orders (1.1×10<sup>-19</sup> s). The required excursion is $\sim$359$\times$ the Galactic ambient anchor and corresponds to a 251 ppm conformal step ($\sim$62.8$\times$ the sustained spacecraft comparison threshold). Those spacecraft limits apply to sampled trajectories; no spacecraft has crossed the proposed 250-AU wall, so they do not directly exclude it. If the instantaneous clock threshold is imposed on this boundary profile, the leading-order contribution is 2×10<sup>-4</sup> yr, approximately 2.5×10<sup>4</sup> times below the observed contrast. The formal fixed-speed, one-crossing radial impulse is 8.6×10<sup>9</sup> m/s, with its along-track projection $\sim$2.4×10<sup>8</sup>$\times$ the measured energy-channel floor. Such an impulse lies outside the small-perturbation regime and is a consistency diagnostic of the approximation rather than a physical trajectory prediction. Moreover, a conservative static profile can cancel energy changes between entry and exit; the measured net inter-leg energy floor cannot be applied to each crossing without the full transfer model. Conditional on an uncancelled along-track impulse, the scale map gives $u_b\lesssim$1×10<sup>-12</sup> and a required admissible $B_0\sim$1×10<sup>31</sup>. This conditional map does not derive the normalization or establish exclusion. The unsigned observable does not select the transport sign. The remaining calculation is a sourceable admissible profile with the full metric, trajectory and signed observation response, evaluated within its valid regime.

The orbit-error contribution to the modern two-channel dissociation is bounded under the tested diagonal-error model by Monte-Carlo error propagation (step 150). Injecting each post-2017 refit member's own SBDB element sigmas into its fitted orbit and propagating both legs to the boundary sphere shows the rotation observable is the *less* error-exposed channel, not the more: leg-to-leg rotation is a differential quantity whose common-mode element error cancels (median $\sigma_{\rm rot} =$ 1.4×10<sup>-6</sup> deg against the aphelion direction's $\sigma_{\rm aph} =$ 2.1×10<sup>-4</sup> deg, median rotation-to-aphelion uncertainty ratio 0.00572 under the independent-element perturbation model). And the matched-scale test closes the loop: at the perturbation amplitude required to inject the observed rotation residual, only 1 member in sixty can be driven that far — and the same error would displace its aphelion direction by $\sim$0.61°, a displacement far beyond anything the quality-flat aphelion channel carries. The tested independent-element perturbations do not reproduce the dissociation within the measured noise scale: too small and neither channel moves; large enough to move the rotation channel and the aphelion channel — which is observed stable — would be destroyed. The dissociation is structural: single-solution fitting absorbs phase information by construction, and the spatial aphelion statistic avoids the leg-comparison construction, while still inheriting orbit-solution and selection uncertainties.

**Table 6:** Alternative mechanisms — required signature vs observed

| Alternative | Required signature | Observed | Verdict |
| --- | --- | --- | --- |
| Discovery footprint | clustering amplitude + direction from pointing | ceiling 0.295 < 0.332; 38.6° offset | inconsistent with data (steps 013–016) |
| Point mass / Planet Nine | $1/b^2$ profile, element coupling, energy kicks; inserted P9 must rotate orbits $\sim 0.2^\circ$ and confine the cluster | flat plateau, no coupling, energy flat; 44× analytic + $527\times$ measured impulse shortfall (P9 $\le 0.003^\circ$); required mass $\sim 4294\,M_\oplus$ in-cap ($\ge 100\,M_\oplus$ every comet, linear scaling $\alpha = 1.00$), distance-independent $\sim 2186$–$4290\,M_\oplus$ across $a_9 = 250$–$1000$ AU; SCT25 realization $0.0002^\circ$; cluster disperses in $\sim 160$ Myr and P9 adds no confining torque | inconsistent with data (steps 011, 020, 041, 062, 066, 083, 087) |
| Inner perturber / Planet Y | SCT25 box $0.06$–$1\,M_\oplus$ at $a = 100$–$200$ AU must supply the rotation while escaping the ephemeris 5$\sigma$ line | required mass $515$–$2186\,M_\oplus$ in-cap median under both geometries at $a_9 = 100$–$250$ AU; 0/43 in-cap comets inside the Planet-Y box; 0/43 below the Gomes+23 ephemeris line | inconsistent with data (step 091) |
| IRAS–AKARI candidate (Phan+25) | real candidate must sit near the required direction and rotate orbits at its estimated 7–$17\,M_\oplus$ | position $(5.4^\circ,-57.5^\circ)$ — $52^\circ$ off axis, $128^\circ$ off antipode; injects $0.0003$–$0.0007^\circ$ vs $0.14^\circ$ observed; required mass $\sim 3367\,M_\oplus$ | inconsistent with data (step 093) |
| Ancient stellar flyby / rogue planet | a one-off formation-era event must leave the $\varpi$ cluster still aligned today and rotate orbits inside currently observed comet arcs | in-cap alignment disperses in $\sim 160$ Myr under the giants alone — a $\sim 4.2$ Gyr imprint (Chen+25) cannot persist without continuous regeneration no flyby supplies; the comet discrepancy accrues in-arc during the measured transit, a channel no historical event can reach | reproduces neither channel; survival of the cluster requires an active agent (steps 066, 084) |
| ISM/heliospheric drag | ISM-axis alignment + energy dissipation | ISM null $p = 0.95$; energy flat $p = 0.69$ | inconsistent with data (steps 032, 034) |
| NG outgassing | NG-parameter dependence | GR-only subset carries excess directionally, $p = 0.051$ | inconsistent with data (step_041) |
| Planetary encounters | kick/geometry advantage for in-cap comets | N-body kick flat $p = 0.568$; residual $p = 0.0008$–$0.0021$; bidirectional validation $\rho = 0.999$ (CODE) and $\rho = 0.998$ (Warsaw), rot-per-kick residual $p = 0.002$–$0.012$ | subtracted, anomaly survives on two catalogues (steps 040, 042, 063, 064) |
| Galactic tide | $m=2$, tidal-band alignment; on-plane anticenter dipole; tide-induced boundary-leg rotation at anomaly scale; tide maintains or regenerates the $\varpi$ cluster | $m=1$ off-plane dipole to axis; anticenter null; transit insertion $9\times10^{-9}$° vs $0.14^\circ$; resident insertion disperses at giants-alone rate (151 Myr), scrambled start stays isotropic ($R\le0.14$ at $166^\circ$), restoring test null | inconsistent with data (steps 021, 061, 070, 088) |
| Distributed lapse field | $\delta\tau$ accumulates with transit time; scatter shrinks under $T$-normalization | in-cap $\rho = -0.02$ ($p = 0.85$); $\sigma_{\log}$ unchanged by normalization; step favoured over field | rejected within TEP family (step 068) |
| Catalogue systematic | era/quality dependence, shared direction, worse orbits; zonal error must track the observed sky direction; an arc-epoch fit error scales as $\delta\tau \propto q^{-2}$ | era-flat on three channels, class-stable, leverage-identical, axes differ 24°; anomaly tracks aphelion not observed direction (partial $\rho = +0.11$ vs $+0.008$), patch-position modulation null, anomalous cohort at higher galactic latitude; residual slip flat in $q$ (slope $-0.23$) | inconsistent with data (steps 023, 037, 053, 058, 072, 077–079, 081, 082, 086) |

## 6.8 The origin question

Within the companion TEP field framework, the measured spatial structure asks which admissible field profile could produce the boundary-like response the resident population registers — its perihelion directions decoupling from discovery geometry beyond the $\sim 150$ AU decorrelation turnover (Section 3, step 017) — oriented toward $(\lambda, \beta) \approx (50^\circ, -17^\circ)$? Two readings remain hypotheses, not measurements. A primordial scalar domain wall and a local heliospheric screening transition make different predictions for motion and edge width, but neither follows from the current orbital statistics alone; the slip profile itself does not resolve an onset radius ($8$–$100$ AU shells carry the same in-cap leg excess, step 076). What the catalogues fix is a directional population contrast and a catalogue-dependent reconstruction contrast; a TEP interpretation requires a derived field, trajectory, and observation model before cosmological or local provenance can be assessed. The resident mechanism is scoped correspondingly: the crossing-localized transport slip cannot organize residents, and a secularly accumulating slip is excluded on morphological grounds — unbounded accumulation over Gyr yields drift, not a stationary direction — leaving the secular response of resident orbits to the boundary shear as the sole field-side candidate, contingent on the admissible-excursion ledger of Section 6.7.

The impulse-versus-holonomy discriminator and forward-model injection battery (steps 145 and 151, Section 4.15) support crossing-localized non-integrable transport on the pre-2018 record: the implied slip has no resolved velocity scaling at the tested shells. This tests the observation response to prescribed perturbations. Establishing the microscopic realization additionally requires the canonical field profile and its signed trajectory-to-refit transfer; the energy floor alone does not select between the realizations.

The heliospheric geometry measured in step 099 bears on this fork. The heliotail apex lies $28.8^\circ$ from the axis — inside the 60° cap — while the axis lies $151.2^\circ$ from the ISM inflow direction and $162.3^\circ$ from the pristine interstellar magnetic field; the direction is thereby associated with the heliosphere's downstream wake rather than with the inflow or the external field.

The plasma channel strengthens the association: the Voyager PWS record (step 097) shows the heliopause is a real, sharp plasma boundary, which is precisely the kind of structure an environmental-state screening transition would lock onto — the second reading above — while remaining equally compatible with a primordial wall that merely happens to share the heliotail's orientation.

The two spacecraft positions and the probe asymptotes also supply the geometric context for the pattern the spacecraft channels return: only Pioneer 10's velocity asymptote enters the cap, Voyager 1 sits inside the mirror cap — a sector where the slip field is positive-signed at a median similar to the primary cap ($\delta\tau \approx +2$ yr; Section 4.8's cos-2$\theta$ structure is positive at both axis poles), though without a second rotation lobe — and Voyager 2 crossed in the intervening negative-slip region.

The electronic-clock tests are bounds in both signed regimes (calibration-residual excursions null at the $\sim 1$ ppm level at all four crossings, segment-phase jumps below $0.2$ s, and the correction-interval drift staircase showing no crossing-localized change; the New Horizons interior control — same clock class, no boundary crossings, outside both lobes — returns only documented engineering resets).

The signed level of the oscillator offset is then interrogated separately (step 141), because it is the one SCLK observable the cancellation theorem exempts: the onboard oscillator ticks in spacecraft proper time, so its measured frequency offset against Earth time is a one-way clock comparison, not a two-way link. All four crossings in fact carry negative level steps in the signed offset — $-0.9$ to $-4.0$ ppm on 2–5 yr windows, surviving smooth-ageing detrending — and the joint extremeness of the four negative tails against each record's own changepoint null is Fisher $p = 0.0024$. The sign consistency alone is not probative (66 per cent of all changepoints are negative; 4/4 at $p = 0.19$), the record is kernel-segmented with long calibration gaps near the crossings, and the New Horizons control is unusable for the signed channel (its offset is frozen below 0.1 ppm). The channel therefore registers as a candidate — crossing-aligned signed structure at the ppm level, priced against its own changepoint null — rather than the null the residual-scatter test alone suggested, and rather than a detection: separating a lapse-rate step from undocumented oscillator aging at this level awaits the same class of independent lever the nuclear channel lacks. The same channel simultaneously bounds the mechanism: the $\sim 120$ ppm conformal crossing step that the $\sim 100$–$150$ AU anchoring would require (Section 6.7) is $30$–$130\times$ the largest signed offset measured, so the required step is excluded along the actually crossed spacecraft paths — the mechanism's consistency thereby rests on the $\sim 250$ AU forward-model localization, beyond every crossed craft.

Two further tests interrogate the same signed series along its radial and temporal axes. The radial-gradient channel (step 146) regresses the signed offset on heliocentric radius rather than epoch: outside 60 AU the two craft carry opposite-signed drifts — Voyager 1 at $-0.0309$ ppm/yr (ecliptic latitude $+34.79^\circ$, $\sin\beta = +0.571$) and Voyager 2 at $+0.0296$ ppm/yr (ecliptic latitude $-32.62^\circ$, $\sin\beta = -0.539$) — while the New Horizons interior control is flat on the same record class. Because both Voyagers exited into the same longitudinal nose hemisphere of the heliosphere ($\lambda = 253.6^\circ$ and $288.9^\circ$), their projections onto in-plane longitudinal axes (the CMB dipole apex and the comet-measured axis) share the same sign, whereas their symmetric north–south straddling of the ecliptic plane isolates the ecliptic-polar component ($\hat z_{\rm ecl}$, $\sin\beta$): normalizing each craft's outer drift slope by $\sin\beta$ yields $\text{slope}/\sin\beta = -0.0542$ ppm/yr for Voyager 1 and $-0.0549$ ppm/yr for Voyager 2 — locking in both sign and amplitude to $1.4\%$ (ratio $0.986$; step 146). While radius and mission age are $\sim 99.9$ per cent collinear along a single outbound trajectory, the opposite-signed north–south split between two spacecraft of identical oscillator hardware eliminates common-mode secular aging and matches an ecliptic-polar lapse gradient to $1.4\%$.

The epoch channel (step 148) re-anchors the windowed-step machinery of step 141 to the physically measured boundary epochs — the canonical termination-shock and heliopause dates and, for Voyager 1, the PWS plasma-event chronology — and asks whether the strongest negative signed steps lock to the walls. They do not, at present significance: the strongest step within each crossing window sits $0.2$–$1.3$ yr from its canonical epoch (joint Fisher $p = 0.11$), and on Voyager 1 the plasma-step epochs sit farther from the strongest negative changepoints than random placements ($9.9$ yr mean separation against a $9.6$ yr placement null) — anti-coincident rather than coincident. The amplitude evidence and the epoch evidence are thereby separated cleanly: the signed steps are extremal in magnitude against their own changepoint null but are not wall-locked in time. That combination admits two readings — a transition smeared over the boundary region, of the same graded morphology the heliosheath shell modulation and the RTG ramp independently present, or residual kernel segmentation near the crossing-era calibration gaps — and the record does not currently choose between them.

The electronic-clock bounds sit on the channel the cancellation theorem exempts. In a fully coherent two-way transponder link the spacecraft carries no free-running reference in the signal chain: the transponder multiplies the received carrier by a fixed ratio, so the endpoint conformal factors cancel identically at the turnaround and only the time-evolution term $\sim\dot\eta\,\tau_{\rm light}$ survives on the link observable (Paper 0 §2.2; Paper 15 §5.2). The theorem thereby mandates the blindness of the coherent Doppler and ranging observables used in orbit determination — not of the SCLK record, which is onboard-clock telemetry: a one-way comparison of the spacecraft oscillator against ephemeris time, and therefore a clock-carrying channel. The residual-scatter, step and drift bounds measured above are accordingly informative nulls on the sustained conformal excursion along the Voyager paths rather than mandated ones, while open one-way dynamical orbits — the comets — accumulate the macroscopic slip the links cannot see. Under the Temporal Equivalence Principle, non-integrable time-transport holonomy survives only around closed spatial circuits ($H = \oint_C \mathrm{d}\tau$). The nuclear channel's only unit-coherent excursion falls on Voyager 1 — the sole craft inside a field lobe — with the out-of-cap Voyager 2 units mutually incoherent: the directional pattern the two-lobed field predicts. It remains at candidate level, however — the excursion is baseline-window-sensitive and, measured against the record's own aging wander, is not record-extreme (the crossing-aligned ramp is matched by $\sim 17$ per cent of post-baseline placebo placements and the deepest in-window ramp's best-fit centre sits $7.5$ yr post-heliopause; step 103), and remains degenerate with accelerating late-life degradation (Section 6.9). These geometric data are descriptive — the probe directions are mission-constrained and the sample is small — but they convert what would otherwise be an unconstrained spatial postulate into a registered, testable orientation.

The nuclear channel's degeneracy with generator aging is then attacked with the full cross-generator ensemble (step 137). Per-sensor thermal telemetry is not publicly archived for the Voyager bus — the flight system carries very few temperature sensors — so the regression of the power decline against the spacecraft's thermal and degradation state is run against every unit-level $P/P_0$ series in the published dataset: 34 generator and mission-aggregate series across six RTG families. The pooled empirical wander null — the residual change over 9-yr windows across all documented aging trajectories, $n = 509$ windows, $\sigma = 0.79$ per cent — places every Voyager crossing-window decline inside documented behaviour: Voyager 1's three units carry $-0.55$ to $-0.56$ per cent and the mission total $-0.38$ per cent over the heliopause-aligned window (one-sided tails 7.4–16.5 per cent). The common-mode construction is sharper than any single unit: Voyager 1's three nuclear clocks drift together at $-0.55$ per cent, a 6.0 per cent raw tail against the fleet's common-mode wander ($n = 117$ realizations; 12 per cent under the two-craft family correction), while the out-of-cap Voyager 2 pair diverges ($-0.63$ and $+0.36$ per cent; common-mode $-0.01$ per cent, 40 per cent tail). The cap-geometry contrast therefore points the predicted way — the in-cap craft carries the coherent decline and the out-of-cap craft does not — but the excursion does not clear the family-wise line, and the nuclear channel is retained as an upper bound with a resolved directional carrier rather than a detection.

The thermal channel is itself quantified: sibling units sharing one spacecraft's bus and thermal environment wander coherently at a median of $0.044$ per cent per 9 yr, an order of magnitude tighter than cross-mission pairs ($0.39$ per cent; Mann–Whitney $p \approx 4\times10^{-39}$) — the shared-environment component of common-mode wander is real but small, and cannot promote the crossing excursion beyond the ensemble null. The nuclear channel therefore remains an upper bound, now priced against the complete population of real RTG aging trajectories rather than the Voyager record alone ($|\delta\alpha_{\rm eff}/\alpha_{\rm eff}| \sim 2\times10^{-5}$ at the crossing window, $1\sigma$).

The one-way tracking channel the two-way cancellation theorem calls for is likewise audited (step 138): the required observable — an onboard-oscillator-referenced one-way carrier — is the same transport class as the GNSS satellite downlink already analysed in the corpus, so the non-cancelling test is realized through the clock channel whose free axis lands $0.6^\circ$ from the comet meridian plane (step 135).

The deep-space archival record is inventoried: the 1979–81 Voyager $\Delta$VLBI/$\Delta$DOR demonstrations (22 usable passes, $\sim 0.5\,\mu$rad scatter at 5–10 AU), Cassini VLBA astrometry (published positions; raw delays not publicly archived), and the PRIDE open-loop Venus/Mars Express campaigns (mHz-class one-way Doppler on inner-system paths) — none reaches beyond $\sim 10$ AU, and no public one-way dataset covers the boundary-distance regime, where the Voyager carrier is receivable only by 70-m-class apertures. The deep-space one-way channel is thereby identified and scoped with named holdings, registered as open rather than tested. Its registered caveat is propagation noise: should such a channel ever be populated at boundary distances, solar-plasma scintillation is the dominant one-way link noise at small elongations and must be budgeted against any claimed boundary signature. The one-way channel already realized in the corpus carries no such exposure — the GNSS clock products are ionosphere-free dual-frequency combinations, so the propagation-plasma class is absent from the channel that supplies the meridian hit (step 135).

The observation-level counterpart for the comet channel is the raw astrometric residual at the crossing itself — the observed-minus-calculated series should show a structural inflection at the epoch a comet traverses the boundary, rather than a Gaussian scatter. The current archive cannot supply that test: long-period comets are tracked inside the inner tens of AU, and MPC astrometry of the analysis cohort terminates far short of the $\sim 150$–$250$ AU zone in which the transition is bounded — the resident decorrelation turnover at $\sim 150$ AU below, and the forward-model best-fit crossing at $\sim 250$ AU, the outer edge of the tested injection range and therefore a lower bound. The two-leg boundary reconstruction used throughout this work is the population-level realization of the same test — the crossing kink is expressed as the recovered leg discrepancy at the 250 AU sphere rather than as an in-arc residual break, and its leg-specificity and directional localization are the kink's signatures read through reconstruction. Direct observation-level access is a registered future channel: deep precovery of distant comets and stellar-occultation astrometry of boundary-distance small bodies would place raw photons on the far side of the transition and discriminate a sharp boundary kink from a distributed fit discrepancy.

The same observable class extends beyond the Solar System. The Gaia wide-binary census — resolved stellar pairs at projected separations of a few $10^3$ to $10^4$ AU — is contested between Newtonian and modified-gravity readings (Chae 2023; Hernández 2023; cf. Banik et al. 2024) because fitted relative accelerations at the weakest-binding end exceed the Newtonian expectation while no local perturbation accounts for them: an anomaly resident in the fitted orbital solution, flat in every auxiliary channel — the comet channel's own morphology. Under the present framework the reading is structural rather than dynamical: if macroscopic proper-time domains exist, a wide binary straddling a domain boundary accumulates a relative phase slip between its components that Keplerian reduction must absorb into the fitted relative acceleration — the same observation-model absorption that places the comet anomaly in reconstructed elements rather than in measured energy. The separation scale makes the connection specific: kilo-AU binaries are the nearest stellar systems large enough to straddle macroscopic domains at all, and they are precisely where the anomalous-acceleration census sits. No wide-binary measurement is made here; the channel is developed as a dedicated member of this series (Paper 13), where a $341{,}315$-system high-purity Gaia DR3 ensemble resolves a screening transition at $R_s = 2{,}646 \pm 182$ AU saturating at $\alpha_{\rm sat} = 0.37$ above the Keplerian baseline — strongly preferred over both a flat Newtonian profile ($\Delta\chi^2 = 14{,}845$) and a constant boost. The programme-level implication is that a boundary-localized slip field at $\sim 150$–$250$ AU and a kilo-AU anomalous-acceleration census would then be two projections of one structure — the Solar-System record supplying the calibrated template against which the stellar channel can be read.

The same absorption logic reaches a documented engineering anomaly at the opposite end of the Solar System. Earth gravity-assist flybys (Galileo, NEAR, Rosetta, Juno) returned unexplained asymptotic-velocity offsets of mm s$^{-1}$ to cm s$^{-1}$ that no thermal, tidal or drag accounting has closed (Anderson et al. 2008). A steep environmental-screening transition — a localized change in $\mathcal{S}_\Sigma(\mathcal{E})$ across a planetary well — is exactly the structure the field equation of Section 1.2 admits, and a phase slip accumulated across it is invisible to orbit-determination software except as an asymptotic momentum offset — the identical mapping by which the comet anomaly surfaces as a periapsis rotation rather than a clock reading. The channel accounting is stated exactly: in a fully coherent two-way transponder link the endpoint conformal factors cancel at the turnaround, so the term survives only on clock-carrying observables — and the published catalogue was recorded in the coherent class (Paper 0 §2.2; Paper 15 §5.2). The mechanism is therefore a falsifiable prediction for clock-carrying flyby passes, not an attribution of the recorded anomalies; it is registered here as the same observable class. Pulsar-timing arrays supply the complementary negative-capability case. The stochastic-background detection rests on the Hellings–Downs quadrupolar correlation between pulsar pairs (Agazie et al. 2023) — an angular signature a local proper-time structure cannot produce, since a lapse at Earth enters the array as an Earth-term monopole and a structured lapse field deposits at most a dipolar term aligned with its axis. The correct TEP observable in PTA residuals is therefore not the detected background but a CMB-axis-aligned Earth-term dipole — a registered, falsifiable search target rather than a reinterpretation of the existing detection.

## 6.9 The spacecraft clock channels

The anomaly is a claim about clocks — orbital periods are macroscopic dynamical clocks — so the sharpest independent test is supplied by the spacecraft that have physically crossed the outer heliosphere carrying clocks of other process classes: an electronic quartz oscillator, a plasma oscillation, and a nuclear decay clock. Under the conformal coupling $A(\phi)$ these process classes need not share one effective coupling strength, so the correct question is not whether every channel shows the anomaly but whether the measured and bounded values in each channel are mutually consistent with a single lapse field. Four archival data series are analysed: the JPL clock-calibration record encoded in the NAIF spacecraft-clock kernels (both Voyagers plus the New Horizons interior control), the PDS/PPI Voyager PWS electron-density collection, the published JPL MHW-RTG power-telemetry record, and the NAIF trajectory kernels that tie every record to a heliocentric position (step 095 acquisition; all provenance-pinned). The electronic-clock bounds are the quantitative content of that record; the residual phenomenology they leave open, and the auxiliary plasma and nuclear channels, are weighed at candidate level against their instrumental degeneracies.

The SCLK calibration record is the mission's complete clock-correction history (step 096). Voyager's onboard clock is not free-running in the data record: JPL re-fits the SCLK$\rightarrow$ephemeris-time correlation whenever ground calibration shows drift, and every re-fit survives as a coefficient record in the NAIF kernels — 1,144 fine calibration records for Voyager 1 and 1,407 for Voyager 2 across the full mission, plus 41 commanded rate-mode records each (the recurring $2820.000\times$ s/unit = 47/48 programmed rate state) and 15 partition boundaries per spacecraft.

The measured onboard-versus-ground rate ratio shows ordinary quartz ageing ($0.011$–$0.053$ ppm/yr linear trend) with residual scatter of $4.8$ ppm (VG1) and $5.6$ ppm (VG2). The channel-relevant lapse signature — a persistent signed rate offset after a boundary crossing — is tested directly by a step term in the rate fit at each crossing epoch, and is null at all four crossings: fitted step amplitudes of $-0.11$ ppm (V1 termination shock, $p = 0.85$), $-0.722$ ppm (V1 heliopause, $p = 0.22$), $-0.863$ ppm (V2 termination shock, $p = 0.16$) and $-1.073$ ppm (V2 heliopause, $p = 0.21$), with before/after $\pm 365$-day window contrasts consistent with zero (Mann–Whitney $p = 0.14$–$0.96$).

A second, sharper test targets the signature type itself: each coefficient record anchors an affine SCLK$\to$ET segment, so the residual between extrapolating one segment and the next record's anchor measures the accumulated phase error between calibrations — the observable for a discrete proper-time jump. Within $\pm 90$ days of every crossing the largest boundary residual is $\lesssim 0.2$ s, bounding a crossing-localized phase slip eight orders of magnitude below the comet-channel slip scale; the only large discontinuities in the record are documented engineering resyncs (the 1981/1983 $\pm 10^{5}$ s reset/restore pairs and the 2010 VG2 anomaly cluster), none at crossing epochs.

A third channel resolves the correction history itself: each phase residual divided by its segment length gives the implied mean oscillator drift between successive recalibrations — the staircase of drift corrections accumulated since launch — and the median absolute drift change across every crossing window is below $0.003$ ppm, so the corrections are ordinary quartz ageing plus flagged engineering events rather than any crossing-localized change. New Horizons supplies the interior control on the same clock class: 3,382 calibration records spanning 2006–2025 on a trajectory that never crosses a heliospheric boundary and remains outside both field lobes; its post-fit kernel reaches a $\sim 20$ $\mu$s phase floor, and its three flagged resets (the 2009 and 2017 software-MET resynchronizations and the launch anchor) are all documented engineering events — validating the reset classifier on a craft whose correction history is fully documented.

The claimed nulls are likewise demonstrated rather than assumed: planted rate steps of 5 and 50 ppm are recovered at $t = 7.309$ and $84.19$, and planted phase slips of $0.5$ s and $3{,}600$ s are recovered as $0.469$ s and $3{,}599.97$ s — the record is demonstrably sensitive at and below every bound it sets.

The channel therefore bounds a sustained post-crossing offset at the $\sim 1$ ppm level, a discrete phase jump at the sub-second level, and any instantaneous lapse contrast between the spacecraft's position and Earth at $|\delta A/A| \lesssim 2\times10^{-5}$ along the Voyager paths — different observables of the same field than the cumulative transit slip, which is accrued along a crossing of the boundary sector rather than measured at a point.

The falsifiable edge is retained: if the boundary transition were sharp and the electronic oscillator coupled identically to orbital dynamics, the post-crossing SCLK record would show an offset of order the integrated slip; its absence at the $\sim 10^{-6}$ level excludes that specific combination (sharp edge plus universal coupling). The channel can carry that exclusion precisely because the cancellation theorem does not govern it — the SCLK record is an onboard-oscillator comparison against ephemeris time, the clock-carrying class of Section 6.8. What the theorem governs is the coherent link class: the two-way Doppler and ranging observables that determine the trajectory, in which the endpoint conformal term cancels identically at the turnaround and a static lapse gradient cannot be resolved. Open one-way dynamical orbits — the comets and TNOs that carry the anomaly — accumulate the macroscopic path integral instead. The $\sim 1$ ppm bound is thereby an informative null on the sustained excursion along the Voyager paths, consistent with the screened-interior picture and the process-dependent couplings the framework predicts.

Under the wall-over-field verdict of Section 4.10 the interior channels carry a further role: the same evidence that prefers a screened boundary over a distributed field requires the inner-system clock channels to be null, so the SCLK, drift-staircase and interior-control bounds are the model's predicted-null checks rather than weak detections — a sustained interior offset at a fraction of the comet-channel contrast would falsify the screened reading outright, and none is observed.

What remains of the spacecraft record beyond these bounds — its residual phenomenology and the auxiliary plasma and nuclear channels — is reported at candidate level. Each pattern below is real in the record, and each is quarantined from the primary ledger because it remains degenerate with documented instrument, cadence, or environment systematics; none carries the statistical weight of the orbital channels.

The drift staircase is audited rather than assumed featureless: the per-interval correction grows with heliocentric radius ($\rho = +0.149$ and $+0.226$ on the two craft, $p<10^{-6}$) and anticorrelates with the velocity component along the measured axis ($\rho = -0.194$ and $-0.231$, $p<10^{-10}$), the heliosheath intervals carrying about five times the inner-heliosphere median absolute drift. This is a real, position-organized modulation of the calibration record — but at a median amplitude of $\sim 0.01$ ppm in the shell, six orders of magnitude below the comet-channel lapse contrast, in the amplitude class a tracking-geometry systematic produces, and null across every crossing window.

The profile is cleaner on the raw per-boundary phase residual, the observable free of the interval-length artefact that dominates the implied drift rate: against a post-1992 inner-heliosphere baseline of $4.0\times10^{-4}$ s on both craft, the accumulated error per calibration boundary rises to $3.5\times10^{-3}$ s (Voyager 1) and $1.8\times10^{-3}$ s (Voyager 2) inside the heliosheath shell ($p \approx 10^{-8}$ and $10^{-12}$) and falls back to $8\times10^{-4}$ s and $7\times10^{-4}$ s in the VLISM. On both independent clocks the modulation peaks in the low-density shell and relaxes in the denser exterior — the bounded-shell profile an environment-triggered activation of $\mathcal{S}_\Sigma(\mathcal{E})$ would produce. The planetary-encounter windows, meanwhile — the deepest screening wells in the record, down to $\phi/c^2 \approx 4\times10^{-9}$ at the Voyager-1 Jupiter closest approach of $3.5\times10^{5}$ km — show no drift response at all (pooled Mann–Whitney $p = 0.83$ and $0.76$).

The shell excess is coherent in character, not merely larger: inside the shell the residuals accumulate nearly linearly with segment length (accumulation-law slope $\approx 1.0$–$1.1$ on both craft, versus $\approx 0.4$–$0.5$ random-walk inside and $0.3$–$0.6$ in the VLISM), and $96$–$97$ per cent of long-interval residuals carry the same negative sign (binomial $p \approx 10^{-14}$ to $10^{-26}$; the inner heliosphere is sign-balanced at $\sim$50 per cent and Voyager 2's VLISM returns to sign balance with cumulative excursion $\approx 0$) — a signed, rate-offset-like accumulation that calibration noise does not produce.

The pattern also organizes by position rather than era: the two craft's radial residual profiles correlate at matched radius ($\rho = 0.62$ over eight shared 10-AU bins, both peaking near $105$ AU despite termination-shock radii of 94.0 and 83.7 AU; the correlation is maximal at zero radial shift and reverses at lags of $\pm 30$–$40$ AU) but not at matched epoch ($\rho = 0.18$) — Voyager 1 and 2 crossed the same radii three to seven years apart, so a global era-specific calibration bias should correlate by epoch rather than radius.

The direct plasma-covariate regression then closes the propagation channel itself (step 154). Matched to each calibration boundary's own segment interval, the residuals carry no contemporaneous solar-wind loading: against interval-mean OMNI dynamic pressure, $|\mathbf{B}|$, wind speed, proton temperature, sunspot number, F10.7 and $a_p$ the strongest pooled coefficient is an anticorrelation ($\rho = -0.14$ in |phase residual|, $\rho = -0.20$ in |drift|) — the radius–era degeneracy restated, since residuals grow with distance while activity declined late-mission, whereas link plasma noise predicts the opposite sign. The link-geometry channel is likewise flat: no dependence on Sun–Earth–probe elongation ($\rho = +0.016$ and $-0.019$ on the two craft, $p = 0.48$–$0.58$) even though 49 boundaries sample the ${<}10^\circ$ elongations where solar-plasma scintillation is strongest; the New Horizons interior record carries a weak elongation loading of the link-artefact sign ($\rho = -0.058$, $p = 8\times10^{-4}$ on $n = 3365$ boundaries) at negligible amplitude ($\rho^2 \approx 0.003$ of residual variance) and with no heliosheath segments to propagate it into the shell result. Partialled on the plasma covariates the radial organization survives intact (partial $\rho = +0.28$ and $+0.33$, $p \lesssim 10^{-19}$) while the pressure term partialled on radius is flat ($p = 0.13$–$0.58$). The only marginal in-shell activity split — larger residuals in the high-$P_{\rm dyn}$ half of the heliosheath — is an interval-length artefact: raw phase residuals accumulate with segment span ($\rho \approx 0.9$) and shell-era cadence covaries with the wind, and the coupling vanishes under the interval control (partial $r = -0.11$ to $+0.07$, $p = 0.23$–$0.72$). Plasma propagation along the tracking link therefore neither produces nor modulates the measured structure.

The trend controls sharpen where the anomaly sits: the post-1992 inner record itself grows quasi-exponentially with radius on both craft with nearly the same e-folding length ($\sim 15$–$16$ AU; $\beta_r \approx 0.026$ dex AU$^{-1}$ at $t = 8$–$10$ after controlling for interval length, which is uncorrelated with radius in that era), while New Horizons shows *no* positive radial growth over the same inner-heliosphere range ($\beta_r \approx 0$ across 1–62 AU on the same record class — the Voyager trend is not a generic clock or tracking systematic). The shell is therefore the *saturation* of that inner growth rather than an isolated jump, and the VLISM residuals collapse to $1$–$35$ per cent of *any* inner-trend extrapolation — a turnover at the boundary that no smooth distance-, epoch-, or link-margin-driven systematic produces. The hedges remain: the residuals are products of the kernel's affine fit, so a bias specific to the shell-era record cannot be fully excluded even though epoch-matching argues against a global era effect; the matched-radius statistic rests on few bins; and radius and direction stay degenerate on a single trajectory. The shell modulation is therefore registered as a monitored pattern rather than evidence.

The plasma channel supplies the discriminant between a lapse artifact and real structure (step 097). The PWS VLISM collection records 755 electron-plasma-frequency events for Voyager 1 (2012–2025, $N_e$ rising $0.054\rightarrow0.155$ cm$^{-3}$ across 122–165 AU) and five events for Voyager 2 (2019–2025, $0.039\rightarrow0.120$ cm$^{-3}$). Because $f_{pe}$ is referenced to the onboard frequency scale, a uniform lapse cannot selectively move it — and the SCLK channel already bounds the onboard-versus-ground ratio at the $10^{-5}$ level. A constants-shift reading, in which $e_{\rm eff}$, $m_{e,\rm eff}$ or $\epsilon_{0,\rm eff}$ scale with $A(\phi)$ while $N_e$ stays smooth, is separately falsifiable: attributing the observed $f_{pe}$ rise to such a shift at constant density would require a combined exponent shift $\delta X = 1.05$ — roughly $90$ times the comet-channel lapse contrast. The measured density structure, including the sharp $+35.8$ per cent jump over 1.268 AU of April 2013, is therefore confirmed as real plasma structure — which is what a plasma-coupled screening transition would use as its trigger (co-location, not illusion). The smooth inter-jump drift bounds any residual constants-shift along the Voyager path at the per-cent level, and the measured interstellar density-gradient scale ($\sim 65$ AU e-folding) is a candidate screening length for the field itself.

The nuclear channel is the cleanest of the three (step 098). Each Voyager carries three MHW-RTG units powered by Pu-238 alpha decay — a continuous monitor of a nuclear clock whose decay rate is amplified $153\times$ sensitive to the effective fine-structure constant through the Gamow exponent ($G = 76.3$ for Pu-238). Because the electronic SCLK sector bounds any common conformal shift at the $\sim 1$ ppm level, a per-cent-scale RTG excursion is interpretable only through the nuclear sector specifically — the channel this test isolates. The analysis uses the published JPL MHW-RTG telemetry record (Whiting & Woerner 2023; 129 Voyager-1 and 89 Voyager-2 total-mission epochs plus five functioning per-unit series through August 2021).

A model-light observable is constructed by fitting the logarithmic decline inside the boundary (mission years 13–26) and extrapolating outward: Voyager 1's three independent units then decline coherently, accumulating $-0.388$ per cent at the heliopause and $-0.584$ per cent by series end, with the excess flat through the termination shock and a best-fitting graded ramp straddling the crossing (onset $\sim$ year 29, saturated by year 36) — the morphology of a transition spread over the boundary region rather than a step.

The Voyager 2 total-power series mixes two report conventions offset by $\sim 0.75$ per cent and its two surviving units diverge by $\sim 1.7$ per cent (one declining slower, one faster than baseline), so no unit-coherent excursion is resolved there; V2 crossed outside both field caps while V1 crossed inside the mirror lobe, and the coherent-negative-versus-incoherent contrast across the two craft matches the directional pattern the two-lobed field predicts — including the sign. The slip field is positive at both axis poles, so in the mirror lobe proper time (and with it the decay clock) runs faster, predicting precisely the faster-decline excursion observed.

The V1 coherence is retained as a candidate signature (a $\delta\lambda/\lambda \approx -0.5$ per cent decay-rate excursion, equivalent to $\delta\alpha_{\rm eff}/\alpha_{\rm eff} \sim -2.5\times10^{-5}$ through the Gamow factor), but two quantified instrumental degeneracies keep it at candidate level: the excursion amplitude — and even its sign — is baseline-window-dependent ($+1.6$ per cent under a 0–26-year baseline versus $-0.388$ per cent under 13–26 years, per the robustness scan in step 098), and Voyager 2's second unit alone shows $-0.845$ per cent at series end under the canonical window — an out-of-cap counterexample at the same amplitude. The unit coherence itself is real (spread $\sim 0.05$ per cent) but is expected for units sharing one craft's thermal environment, measurement chain, and degradation physics: it discriminates unit-specific noise, not field effect versus common-mode aging.

The remaining instrumental degeneracy — undocumented mid-life MHW-RTG aging — is therefore confronted with the record's own wander (step 103). The control is less favourable than the excursion's coherence suggests: the deepest saturating-ramp fit in the crossing search window ($-0.77$ per cent) is centred $7.5$ yr after the heliopause, not on it, and the ramp whose window genuinely contains the crossing is weaker ($-0.53$ per cent) and matched by $\sim 17$ per cent of post-baseline placebo placements. The three units stay within $0.06$ per cent of each other through the excursion and it survives leave-one-unit-out, but the all-negative coherence window is unremarkable (placebo $p = 0.46$) and $\sim 87$ per cent of the record shows a tighter unit spread.

The vintage-matched control cuts the same way: Voyager 2's two surviving units at the identical mission years carry a mean ramp of $-0.31$ per cent — and one of them, $-1.50$ per cent on its own record scan, is the deepest single-unit ramp anywhere in the dataset, on the craft that crossed outside both lobes. The excursion is thus real unit-level coherence but well inside the record's demonstrated wander, amplitude-matched by an out-of-cap counterexample, and degenerate with accelerating late-life degradation; it is retained at bound level with the mid-life control measured rather than pending.

The conservative nuclear-sector bound stays $|\delta\lambda/\lambda| \lesssim 1.5$ per cent at the crossing epochs, i.e. $|\delta\alpha_{\rm eff}/\alpha_{\rm eff}| \lesssim 1\times10^{-4}$ ($2\sigma$).

The trajectory geometry completes the picture (step 099). The measured axis sits $28.8^\circ$ off the heliotail apex (the antipode of the ISM inflow; uniform-direction $p = 0.062$), $151^\circ$ off the inflow itself, $157^\circ$ off the IBEX ribbon centre and $162^\circ$ off the pristine ISMF direction — the boundary is oriented toward the downwind heliospheric sector rather than the upwind nose. Of the five escaping spacecraft, only Pioneer 10's velocity asymptote ($84.1^\circ, +2.9^\circ$) falls inside the 60° cap — the sole probe transiting the anomalous sector — while Voyager 1's asymptote and position fall inside the mirror cap and Voyager 2's lie outside both caps ($87^\circ$ and $80^\circ$ from the anti-axis) — Voyager 1 alone samples the second lobe of the axisymmetric slip field. The interstellar objects add a dated datum: 2I/Borisov arrived from $(54.1^\circ, +43.0^\circ)$, $60.1^\circ$ from the axis — on the cap rim — and 1I/'Oumuamua from $(284.5^\circ, +56.6^\circ)$ inside the mirror cap, while 3I/ATLAS arrived from $(293.5^\circ, +2.4^\circ)$ outside both caps. All three carry fitted non-gravitational accelerations in the SBDB record; the channel that discriminates them — whether the fitted term is accounted for by detected volatiles — is tested against the full transit geometry in Section 6.10.

The ledger is internally consistent (step 100). The comet channel measures a cumulative transit-integrated slip of $\delta A/A \sim 1.2\times10^{-2}$. The SCLK channel bounds the sustained post-crossing offset at the $\sim 1$ ppm level and the instantaneous onboard-versus-ground contrast along the Voyager paths at $\lesssim 2\times10^{-5}$, with the correction-interval drift staircase null across every crossing window and the New Horizons interior control returning only documented engineering resets. The PWS channel confirms the heliopause structure is real plasma rather than lapse artifact. And the RTG channel shows a coherent $-0.388$ per cent excursion across Voyager 1's three independent nuclear clocks at the heliopause — the only spacecraft inside a field lobe — while the out-of-cap Voyager 2 units are mutually incoherent, a candidate directional nuclear-sector signature that formally remains a bound-level result ($\lesssim 1\times10^{-4}$ in $\delta\alpha_{\rm eff}/\alpha_{\rm eff}$). The mid-life degradation control of step 103 finds the deepest saturating-ramp fit centred $7.5$ yr after the heliopause and the genuinely crossing-aligned ramp ($-0.53$ per cent) matched by $\sim 17$ per cent of post-baseline placebo placements — coherent on an absolute scale ($0.06$ per cent unit spread) but not record-extreme, and degenerate with an accelerating late-life degradation mode. These are different functionals of the same field measured on different clock classes — consistent under the conformal framework precisely because the comet slip is cumulative and directional while the spacecraft bounds are instantaneous and path-specific.

## 6.10 The interstellar-object datum

The interstellar-object channel is reported here rather than in the synthesis for a reason of sample size: the catalogue holds three objects, and the discriminating member is a single datum. The concordance below is exact in its geometry but is priced at its exchangeability weight; it is registered as a scored constraint on future discoveries, not counted as evidence.

The three catalogued interstellar objects are the only population besides comets that physically transits the boundary sector on measurable trajectories, and each now carries a fitted non-gravitational acceleration in its JPL Small-Body Database orbit solution (step 101; raw solutions provenance-pinned under `data/raw/iso/`). The discriminating observable is not the presence of a fitted term — every outgassing comet carries one — but an unexplained term: a fitted acceleration with no detected volatile driver. On that criterion the population divides cleanly. 2I/Borisov ($A_1 = 4.86\times10^{-8}$ AU d$^{-2}$, solution 54) and 3I/ATLAS ($A_1 = 5.32\times10^{-8}$ AU d$^{-2}$, solution 54) are ordinary outgassing bodies whose measured gas and dust production accounts for the fitted terms, while 1I/'Oumuamua's $A_1 = 2.79\times10^{-7}$ AU d$^{-2}$ (solution 16, $7.8\sigma$) would require $\sim 25$ kg s$^{-1}$ of outgassing recoil — three orders of magnitude above the detected limits — and remains the literature's unexplained datum (Micheli et al. 2018; Jewitt & Seligman 2023).

The transit geometry separates the same objects the same way. Computed from the SBDB osculating elements, 'Oumuamua is the only ISO executing a two-lobe transit: it arrives through the mirror cap ($57.3^\circ$ from the antiaxis) and departs through the primary cap ($56.6^\circ$ from the axis), so its two boundary crossings both sample positive-slip lobes and compound rather than cancel. Borisov threads the measured edge band on both legs ($60.1^\circ$ in, $62.9^\circ$ out); ATLAS arrives in the edge band ($65.1^\circ$) and crosses the primary lobe on departure only ($46.9^\circ$). The unique unexplained-NG object is thereby the unique two-lobe-transit object — a concordance whose exchangeability weight is limited by the present catalogue size ($n = 3$, $p \sim 0.33$ under geometry-blind assignment) but exact in its geometry, and which is registered as a scored constraint for future ISO discoveries rather than claimed as a detection.

The magnitude and morphology carry the weight. Evaluated under each solution's own Marsden–Sekanina $g(r)$ law, 'Oumuamua's term is a fractional effective-attraction offset of $9.4\times10^{-4}$ of the solar gravitational acceleration at 1 AU — $A_1 = 2.79\times10^{-7}$ AU d$^{-2} \approx 5.6\times10^{-6}$ m s$^{-2}$ — across the observed arc — a factor of $\sim 12$ below the comet-channel lapse contrast ($1.2\times10^{-2}$) if the two channels couple identically. The deficit has a natural kinematic origin rather than a contrived coupling: a fixed proper-time offset imprinted over a finite crossing thickness accumulates in proportion to the dwell time, and 'Oumuamua transits the boundary at $\sim 26$ km s$^{-1}$ where a parabolic comet moves at $\sim 2.7$ km s$^{-1}$ — a $\sim 10\times$ suppression under identical coupling, within a factor of two of the observed ratio and consistent with a partial-slip reading — while the volatile-explained objects sit at $1.6$–$1.8\times10^{-4}$, $\sim 70$ times below the comet scale. Its vector is also the purest radial of the three: $|A_2/A_1| = 0.05$ against $0.39$ (Borisov) and $0.22$ (ATLAS), with no perihelion-time offset required — the morphology a directed gradient produces rather than a rotational jet. The sign is consistent at interpretation level: its periapsis direction samples the measured negative-$\delta\tau$ mid-band, and the unexplained term is outward, an effective-attraction shortfall.

The solver-absorption mechanism is then exercised on this datum directly (step 155), the comet-cohort injection-and-refit machinery tailored to the hyperbolic arc. Synthetic astrometry is generated on the object's real observing chain — the catalogued MPC epochs, stations and noise floor — and refit by gravity-only and Marsden-augmented differential correction under the SBDB $g(r)$ law. The fitter is first validated against the real record: applied to the actual astrometry it recovers $A_1 =$ +2.87×10<sup>-7</sup> AU d$^{-2}$ against the published $+2.79\times10^{-7}$, an offset of 0.22$\sigma$, while the residual floor falls from 0.381$^{\prime\prime}$ under a gravity-only fit to 0.261$^{\prime\prime}$ once the Marsden block is admitted. Three injection classes then separate the mechanisms the geometry permits. A one-shot slip at the true inbound boundary crossing, $\sim 44$ yr before the apparition, leaves the entire observed arc post-slip and is absorbed identically to the comet case: gravity-only refits return the injected 0.410$^{\prime\prime}$ floor and the fitted $A_1$ is null within its formal uncertainty ($|A_1|/\sigma \leq$ 0.16 across full-state, position-only and velocity-only realizations). An instantaneous within-arc slip is excluded with equal decisiveness: a 1.8-day phase offset injected at mid-arc leaves a 202$^{\prime\prime}$ gravity-only residual, $\sim 500$ times the noise floor, where the real record sits at $0.4^{\prime\prime}$. The channel that registers is the one the transit geometry implies: a sustained proper-time lapse-rate offset $f$ accumulated inside the domain maps linearly onto the fitted radial term, with measured slope -5.8×10<sup>-4</sup> AU d$^{-2}$ per unit offset through zero — the analytic $A_1 \simeq -2f\,g_\odot(r)/g(r)$ of an effective-attraction rescaling $(1+f)^2$ read off at the arc's effective radius — and reproduces the observed coefficient for $f =$ -3.9×10<sup>-4</sup>. The required lapse is thereby a proper-time rate deficit inside the lobe, its sign the one the geometry assigns (slower proper time reads as an outward effective-attraction shortfall), its morphology dominantly radial ($|A_2/A_1| =$ 0.13 fitted against $0.05$ observed), and its magnitude a factor of $\sim 3$ below the naive dwell-scaled comet contrast (1.2×10<sup>-3</sup>) — the direction a partial-slip reading predicts. The unexplained Marsden term is thereby converted from a consistency argument into a measured transfer function: proper-time structure injected at the entry boundary is processed by standard orbit-determination machinery into the phenomenological radial coefficient the object carries.

The same channel bounds the field interior. Borisov's inner arc samples the same negative-slip mid-band sector as 'Oumuamua's, yet its fitted term is fully volatile-explained at $1.6\times10^{-4}$ — a universal interior lapse gradient at comet-channel strength on such arcs is excluded by a factor of $\sim 70$, and ATLAS extends the bound to the positive-slip lobe sector. The lapse structure therefore does not reach the inner system as a distributed sector field — the wall-over-field verdict of Section 4.10 carried inward, and the configuration the framework's screening requires, with the temporal structure confined to the boundary rather than distributed through the screened interior — and 'Oumuamua's in-arc term, if boundary-connected at all, originates in the crossings rather than in an interior gradient. The prospective prediction is registered in step 101: a future ISO executing a two-lobe transit should require an unexplained term at the $\sim 10^{-3}$ level if volatile-free, while edge-band and field transits should be volatile-explained; a double-lobe transit fully explained by detected volatiles, or a field transit requiring an unexplained term, falsifies the concordance.

The volatile classification is itself the load-bearing element and is audited as such (step 101, T6). Because the concordance requires the unexplained object and the two-lobe transiter to be the same body, it rests on the explained status of Borisov and ATLAS as much as on 'Oumuamua's anomaly: a single-member reclassification audit shows an upgrade of either explained object to unexplained status breaks the all-three classification, while a detected volatile driver for 'Oumuamua would dissolve the datum entirely. ATLAS's explained verdict rests on the nucleus mass/production accounting, where the post-perihelion record has converged rather than weakened: systematic-uncertainty non-gravitational solutions (Spada et al. 2026), a recoil-inferred nucleus mass of $\sim 10^{11}$–$10^{12}$ kg (Thoss et al. 2026), and HST nucleus sizing at $1.3 \pm 0.2$ km (Hui et al. 2026) are mutually consistent, against a minority large-nucleus claim that would move the fitted term toward the unexplained class — a channel the pipeline monitors by re-pulling the SBDB records on every run. The publicly discussed oddities of 3I/ATLAS — its retrograde trajectory within $\sim 5^\circ$ of the ecliptic, the sunward anti-tail, the extreme polarization — are kinematic and compositional channels, not lapse channels, and carry no discriminating weight here; were its fitted term genuinely unexplained it would sit in the concordance-break column of the audit, not the support column.

**Table 7:** Cross-channel clock ledger

| Channel | Clock class | Observable | Result | Source |
| --- | --- | --- | --- | --- |
| Comet transit | orbital dynamics | aphelion-direction slip / leg duration | $\delta A/A \sim 1.2\times10^{-2}$ (cumulative) | steps 065, 100 |
| Voyager SCLK | electronic quartz | onboard-vs-ground rate ratio | sustained offset $\lesssim 1$ ppm at crossings; instantaneous $\|\delta A/A\| \lesssim 2\times10^{-5}$ | steps 096, 100 |
| Voyager SCLK phase | electronic quartz | calibration-segment boundary residuals | $\|\Delta\phi\| < 0.16$ s at all four crossings (vs comet slip $\sim 10^{7}$–$10^{8}$ s) | steps 096, 100 |
| Voyager SCLK drift intervals | electronic quartz | per-correction-interval oscillator drift | $\|\Delta{\rm drift}\| < 0.003$ ppm across all four crossing windows; corrections are quartz ageing plus flagged resyncs | steps 096, 100 |
| Voyager SCLK vs link plasma | electronic quartz | residuals vs OMNI $P_{\rm dyn}$/$\|\mathbf{B}\|$/F10.7/SSN/$a_p$ and SEP elongation | no activity or link-geometry coupling; radial organization survives partialling plasma covariates (partial $\rho=+0.28$/$+0.33$) | step 154 |
| New Horizons SCLK | electronic quartz (interior control) | correction history, no boundary crossings, $r < 60$ AU | 3 flagged resets, all documented MET events; post-fit phase floor $\sim 20$ $\mu$s | steps 095–096, 100 |
| Voyager PWS | plasma oscillation | $f_{pe}$ structure vs lapse-equivalent | $\delta X = 1.05$ required ($\sim 90\times$ comet channel) — real plasma | steps 097, 100 |
| Voyager RTG | nuclear $\alpha$ decay | power-decline slope at crossings; per-unit coherence | $\|\delta\alpha_{\rm eff}/\alpha_{\rm eff}\| \lesssim 1\times10^{-4}$ ($2\sigma$, nuclear bound); V1 units coherently $-0.388\%$ at heliopause (candidate signature; mid-life control: crossing-aligned ramp matched by $\sim 17\%$ of post-baseline placebos; out-of-cap VG2_rtg2 shows the record's deepest unit ramp) | steps 098, 100, 103 |
| Trajectory geometry | — | probe/ISO asymptotes vs caps | 1/5 probe velocities in cap, 1 in mirror; 1/3 ISO arrivals from a cap | steps 099, 100 |
| Interstellar objects | orbital dynamics | unexplained non-gravitational term vs transit geometry | unique unexplained-NG object = unique two-lobe transit; unexplained fractional attraction offset $9.4\times10^{-4}$ (of solar gravity at 1 AU) vs explained $1.6$–$1.8\times10^{-4}$; interior gradient bounded $\sim 70\times$ below the comet contrast | steps 099, 101 |

## 7. The falsifiable prediction: the LSST cohort

The analysis closes with a dated, falsifiable commitment carried by a survey already gathering the key data. For the detached-class discoveries ($a > 150$, $q > 30$ AU) the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST) will deliver, the domain-boundary model requires approximately 59 per cent of new objects to land inside the pre-declared 60° axis cap, against a footprint-only expectation of approximately 42 per cent — a discriminating gap of 17 percentage points. An exact binomial power analysis (step 067) resolves that gap at 95 per cent power and $\alpha = 1$ per cent from $n \approx 123$ new detached discoveries, a yield LSST is expected to exceed within its first years. Discoveries arriving at the baseline rate falsify the intrinsic-axis reading outright; discoveries at the registered rate exclude the footprint model at the same confidence. The remainder of this section registers the prediction and its directional content, prices the test's power, and scores the partial cohorts already on record against it.

The registered record (step 054) makes the prediction directional as well as statistical. The predicted mean direction is $\bar\varpi = 49^\circ$ (95 per cent interval $[17^\circ, 88^\circ]$) against the footprint-only prediction of $10.5^\circ$. Under the point-mass reading, anti-aligned confinement places the perturber's perihelion near $229^\circ$ — coincident with the shallow-JFC concentration — so deep surveys covering that sector either find the mass or they do not.

Fresh long-period comet cohorts should show reconstruction discrepancies concentrating in the pre-declared patch at the CODE-measured rate (58 per cent of in-cap comets exceeding $0.2^\circ$ versus 23 per cent out-of-cap), with the energy channel flat. The same surveys will measure the boundary's thickness, not only its location: a primordial topological defect should present a sharp threshold, while a heliospheric or local-interstellar interface would appear as a graded transition spread over several degrees (Section 6.8).

A Monte-Carlo power analysis on the measured rates (step 067) shows the discriminating gap is decisive well inside the survey yield: an exact binomial calculation gives 95 per cent power at $\alpha = 1$ per cent from $n \approx 123$ new detached discoveries ($n \approx 184$ at $\alpha = 0.1$ per cent; $\sim 83$ objects suffice at $\alpha = 5$ per cent) on the cap-fraction channel alone, with the mean-direction channel performing comparably; a fresh class-1-like comet cohort of $\sim 100$–$150$ suffices for the rotation-channel test. LSST's detached-object yield is expected to exceed these thresholds within its first years.

A first scoring cohort is already in the catalogue, though it is diagnostic rather than decisive (step 084). The fourth sednoid, 2023 KQ14 ('Ammonite'; Chen et al. 2025) — whose anti-aligned $\varpi = 270.8^\circ$ has been reported as weakening the clustering case — sits $134^\circ$ off the axis, outside the cap as advertised; its addition to the secure sample dilutes the clustering resultant from 0.357 to 0.332 ($p = 0.0043 \to 0.0075$) rather than dissolving it, so the measured alignment already includes the object most hostile to it. The broader 2025–26 designation cohort ($n = 21$; $a > 150$, $q > 30$ AU; provisional orbits at median condition code 8) lands inside the $\varpi$ cap at 2 of 21. But that cohort's own discovery footprint points almost exactly the other way (mean opposition longitude $232.3^\circ$, resultant $R = 0.56$), giving a footprint-only expectation of 0.12. The newest objects were discovered looking toward the anti-axis sector — the very region the point-mass reading reserves for the perturber's own perihelion — and their cap rate tracks their pointing, not the axis ($p = 0.73$ against their own footprint baseline). The registered 59-per-cent prediction therefore stands untested rather than failed; the discriminating gap will be decided by the cohort discovered looking at the axis sector itself.

One such cohort already exists, and it scores the prediction in the footprint-matched direction (step 085). The Dark Energy Survey six-year catalogue (Bernardinelli et al. 2022) — 814 objects discovered in a contiguous 5000 deg$^2$ of southern sky, with orbits fitted independently of SBDB — contains the largest uniform-survey detached cohort: 16 objects at $a > 150$, $q > 30$ AU, which the survey's own authors reported as consistent with azimuthal isotropy. Against the pre-declared directional cap the cohort is anything but isotropic: $13/16$ land inside the 60° cap ($p = 1.2\times10^{-4}$ against uniformity; the cohort resultant $R = 0.745$, mean $\bar\varpi = 23.3^\circ$, inside the registered $[17^\circ, 88^\circ]$ interval). The DES opposition fields happen to sit in the axis sector, so the cohort's own designation-derived footprint baseline is itself elevated (0.55) — yet the measured rate exceeds even that biased expectation ($p = 0.029$) and is consistent with the registered 59-per-cent prediction ($p = 0.079$). The apparent conflict with the survey's isotropy statement is a statement about statistics, not data: azimuthal uniformity and membership of a fixed pre-declared direction are different tests, and the latter is the sensitive one here.

Two structural qualifications apply: 14 of the 16 objects are shared with the SBDB secure sample — so the cohort is an independent-survey and independent-orbit-fit lineage on largely the same bodies, not an independent membership — and on those shared bodies the DES and SBDB solutions agree to a median $|\Delta\varpi| = 0.25^\circ$, the resident-side analogue of the comet cross-fitter floor.

The two newest cohorts thus bracket the test fairly: the 2025–26 provisional cohort, discovered looking toward the anti-axis, sits at its footprint baseline; the DES cohort, discovered looking toward the axis, lands on the registered rate.

The registered prediction is then made cohort-specific rather than generic (step 115). Conditioning the same mixture on each object's own discovery longitude — an intrinsic 43-per-cent cluster of $\sigma = 34.4^\circ$ about the axis convolved with the measured $\sigma = 50^\circ$ discovery coupling — turns the single 59-per-cent figure into a per-cohort posterior expectation, so every cohort is scored against what each model predicts for its own pointing. The model that registered the prediction reproduces the detections: the posterior expectation for the secure SBDB sample is 25.5 in-cap objects against 26 observed (Bayes factor 44.6 over footprint-only; posterior-predictive $p = 0.51$), and for the DES cohort 12.1 against 13 observed (Bayes factor 10.7; $p = 0.41$) — the two detection counts are essentially the mixture's own numbers, while the footprint model under-predicts both ($p = 0.005$ and $p = 0.026$).

The OSSOS detached cohort leans to the footprint model (Bayes factor 0.37), as its interior-to-the-boundary membership leads one to expect, and the 2025–26 cohort returns 2 against expectations of 2.4 and 4.5 under the two models — consistent with both and mildly favouring footprint, which is what its pointing guarantees. The test's lever arm — the difference between the models' expected rates as a function of discovery longitude — contracts to 0.03 at the cohort's mean opposition direction of $230^\circ$. A cohort discovered looking at the anti-axis therefore cannot distinguish the models at any count it could plausibly return. The discriminating cohort is the one discovered looking inside the axis sector, where the expectations diverge by up to 0.23 — and the only such cohort on record lands on the mixture prediction.

The deduplicated secure union favours the mixture at a joint Bayes factor of 9.7 (39 observed against 41.7 expected, versus 31.3 under footprint-only). Nor is provisional orbit quality the limitation on the newest cohort: its $\varpi$-minus-opposition residual scatter ($39.7^\circ$) is tighter than the secure sample's own ($43.1^\circ$), so its count is a real measurement — it simply measures a pointing direction where both models say the same thing.

The ordering is also parameter-robust: re-deriving both models across the mixture fit's own 68-per-cent confidence region ($f = 0.32$–$0.70$, $\sigma_{\rm int} = 27.5$–$48^\circ$, $\sigma_c = 40$–$65^\circ$), the mixture is favoured in every one of the 36 cells for SBDB (Bayes factor $1.5$–$58.7$) and for DES ($3.9$–$28.06$), and in none for the anti-axis cohort — the cross-cohort coherence does not rest on the fitted point estimate. Decomposed by designation generation on the full detached pool, the signature's weight sits in the 2011–15 cohort (18 in-cap of 35; Bayes factor 10.4), with the two earlier, smaller generations leaning the same way (4 of 8 and 3 of 4; factors 1.6 each). The post-2015 aggregate weakens — the 2016–19 cohort returns 6 of 15 — but the decomposition by orbit quality shows that this is composition, not decay: within every well-populated generation the secure orbits reproduce the signature (14 of 22 in 2011–15, factor 10.2; 5 of 8 in 2016–19, factor 3.2 — the mixture favoured in every sensitivity cell in both).

The provisional component is the counterweight and is reported as such: pooled across all generations it returns 7 in-cap of 46 — below the uniform rate — against expectations of 8.3 and 13.5 under the two models, formally favouring footprint (factor 0.03). Its pointings are concentrated on the anti-axis side (mean opposition longitude $221^\circ$, 24 of 46 more than $120^\circ$ from the axis longitude), where the lever arm is weakest, and its fitted longitudes of perihelion carry wider residuals than the secure sample's ($\sigma_{\rm eff} = 56.2^\circ$ against the calibrated $50^\circ$), so the count is consistent with its own pointing geometry; but the isotropic-noise convolution does not close the gap, and the pooled provisional datum stands as the one sole cohort on record that favours the footprint model outright.

Two readings remain open: short-arc orbits carry $\varpi$ systematics beyond the noise model, or recovery prioritization — any preferential re-observation of objects near the claimed clustering direction — would convert in-cap objects into secure ones preferentially, draining the provisional pool of exactly the members the mixture predicts. The DES cohort is the control that separates them: its follow-up was systematic rather than preferential, and it reproduces the excess. Which reading dominates is a question the accumulating 2025+ sample will answer as its arcs extend.

The assumed coupling is then replaced by the measured one (step 116). The discovery kernel — the distribution of $\varpi$ minus opposition longitude across the detached pool — is not the symmetric $50^\circ$ wrapped normal used as the calibration: it peaks at $+17.4^\circ$ for the secure cohort and $+30.2^\circ$ for the provisional pool, a measured designation-epoch lag of the proxy relative to true discovery opposition, and has a finite reach of order $\pm 90^\circ$. Rebuilt on leave-one-out empirical kernels per quality stratum, both models still reproduce the detections: the secure SBDB sample returns a Bayes factor of 13.53 for the mixture over footprint-only, DES 3.06 — reduced from the assumed-kernel values, as expected when the coupling is allowed to absorb part of the structure it conditions on, yet favouring the mixture in every bandwidth tested ($12$–$25^\circ$: SBDB 10.8–14.8, DES 2.67–3.77).

The same audit then resolves the ledger's apparent counterweights geometrically rather than by appeal. The only channel through which an in-cap member can be observed is the key window — the span of discovery longitudes on which the models' expected in-cap rates differ by more than 0.10 — and outside it every population returns the same null: across SBDB, DES, OSSOS and both provisional pools, 76 objects were discovered pointing through the closed channel and zero landed in-cap (pooled expectations 4.61 and 11.05 under the two models), a population-independent confirmation that the excess is spatially bounded rather than a catalogue-wide skew.

Inside the window, the OSSOS cohort — whose pooled ledger entry leaned to the footprint model — independently reproduces the detection: 13 of its 17 open-channel members land in-cap (76 per cent, the secure cohort's own rate; Bayes factor 2.22), its 14 closed-channel members returning zero. The deduplicated secure-lineage union restricted to the open channel then gives the cleanest number in the ledger: 52 objects, 39 in-cap against a mixture expectation of 39.86 and a footprint expectation of 30.97 — the observed count is the mixture's own count (Bayes factor 25.5; the same membership scores 34.96 under the assumed symmetric coupling, so the measured kernel is the conservative choice, not the favourable one). The detection rests on no single survey: dropping each contributing catalogue in turn leaves Bayes factors of 7.0 (minus SBDB), 8.3 (minus DES) and 10.9 (minus OSSOS).

The provisional pool's composition is measured rather than inferred: its designation generations show the survey pointing drifting from the axis sector through 2019 (generation-mean opposition longitudes $352$–$100^\circ$, all within $58^\circ$ of the axis longitude, in-cap rates 0.40–0.75) to the anti-axis sector from 2020 onward (means $212$–$258^\circ$, in-cap 0.00–0.10), so the pooled deficit is dominated by where recent surveys pointed. The pool is further drained of in-cap members by preferential securing: among pre-2020 in-window discoveries, in-cap members hold secure orbits at a rate of 0.84 against 0.54 for their out-of-cap siblings (Fisher odds ratio 4.5, $p = 0.057$), and within the 2011–19 generation alone the in-window secure rate is 0.69 against 0.36 out-window — a measured survivorship term acting in the direction the mixture predicts. The cross-fitter audit confirms the provisional solutions are themselves data-determined: median $|\Delta\varpi| = 0.01^\circ$ between the JPL and MPC lineages across all 90 detached objects, with every fitted orbit placing its object near perihelion ($r/q \approx 1.1$). Fitted $\varpi$ is therefore tied to the discovery direction by construction.

The provisional in-window remainder is then scored under the correct conditioning: membership in the provisional pool is itself the non-securing event, so each member's in-cap probability is thinned by the measured securing rates (0.839 in-cap, 0.538 out-of-cap) before likelihood scoring. The expectations move from 8.43 and 12.42 to 4.65 and 7.90 under the two models, and the observed count of 7 is the mixture's own count (Bayes factor 2.1 at the measured rates). Bootstrapped over the 44-object securing table the factor is median 1.93, exceeding unity in 65 per cent of draws — the correction removes the footprint lean without yet establishing a mixture lean in that cell. The ledger's last footprint-leaning cell is thereby neutralized; the discriminator remains falsifiable per object — every provisional member's posterior in-cap probability under the mixture is registered in `step_b80_forward_predictions.csv`, to be scored as the orbits secure.

Pooled over the whole 120-object detached catalogue under this conditioning — 76 secure-lineage objects scored raw, 44 provisional-only objects thinned — the Bayes factor is 2.05 under the smooth kernel and 3.76 with each stratum's kernel truncated at its measured $|\delta|$ reach: the conservative pooled figure, in which the closed channel's zero-count penalizes the mixture's smoothed-tail expectation of 11.05 in-cap members there while the open channel carries the evidence.

The comet side of the registered prediction has now been scored as well (step 117). The post-2017 SBDB cohort — 121 deep-plunger primaries, inside the $\sim 100$–$150$ size the power analysis set for the rotation-channel test — fails it in the declared direction: the in-cap excess runs reversed (Section 4.12), and the cohort's own sky scan places its strongest cap contrast $78^\circ$ off the registered axis, with 201 of 264 trial directions exceeding the registered-axis contrast.

The reversal is itself structured, however, not a scatter null: the direction-aware audit finds the declared axis sitting at a measured minimum of the cohort's leg-rotation field — in-cap $d_{\rm in}$ is suppressed at two-sided $p = 0.019$ (residualized $p = 0.014$; $d\tau_{\rm in}$ $p = 0.021$), the declared cap returning the lowest inbound deviation of any sky region (median $0.100^\circ$ against $0.112^\circ$ off both caps and $0.129^\circ$ at the displaced axis, $p = 0.047$ against the neither-cap background) while the outbound leg is flat ($p = 0.52$). The cohort's leg-rotation field is therefore a gradient — minimum at the declared direction, maximum $79^\circ$ away at the displaced dipole — not a null field: the same leg that carries the CODE anomaly's positive gap is here structured with the opposite sign.

Pre-2018 SBDB resolves no reversal — the CODE-overlap cell carries a positive median gap ($+0.008$) and the non-overlap cell a marginal, unresolved $-0.013$ (MWU $p =$ 0.25 and 0.57 respectively) — so the sign flip is era-specific: consistent with the modern JPL fit absorbing part of the boundary rotation — the fitted elements migrate toward the asymptote, shrinking the measured leg deviation where the signal lives — rather than with the absence of structure. The forensic audit (step 118) assigns the reversal to a cohort-private, longitude-organized systematic rather than to the sky — CODE covers the same region and shows nothing there.

The carrier audit (step 125) then reframes the score itself: the unexplained-slip observable — the disagreement of independent inbound- and outbound-leg solutions after the gravitational prediction is removed — exists only in three-leg records, and signature presence tracks record structure exactly (three-leg: CODE, Warsaw, LPC $3/3$; single-solution: SBDB pre-2018, SBDB post-2017, and the same CODE-overlap comets re-fitted under SBDB elements, $0/3$). A single joint solution absorbs a proper-time offset once, into shared elements, leaving no fit-versus-fit disagreement for the slip to appear in — the SBDB lineage could not express the anomaly even at full strength. The registered comet-channel prediction therefore stands untested, rather than falsified, on the single-solution lineage — while the LPC sample's declared-axis replication ($p =$ 0.0056, top 4.6 per cent of its free-axis scan) keeps the three-leg record unanimously positive.

The degeneracy that remains is lineage: all three-leg catalogues are Warsaw-school products, so a pipeline-specific fit-disagreement artefact predicts the same pattern; resolving it requires leg-separated fits on post-2017 astrometry or a non-Warsaw leg-separated catalogue.

The divergence is, however, channel-specific rather than population-wide: on the spatial aphelion channel the same post-2017 cohort leans toward the declared axis at $d_\parallel =$ +0.0949 ($p =$ 0.00215; step 124), at the CometEls amplitude, with the free population dipole landing $32^\circ$ off the cap direction and $70^\circ$ off the residual systematic's own axis. What the SBDB lineage cannot carry is the propagated-solution comparison; what its newly discovered population still carries is the spatial datum itself.

The parameter-absorption model is then tested experimentally and rejected: the same independent two-leg refit run on the prospective cohort returns the declared-axis reversal on the construction that can express slip, so the reversal is a real feature of the modern astrometry and the registered-axis reconstruction channel reverses in form on this era rather than remaining structurally untested (step 128). The same record, however, resolves the reversal's seat: the cohort's dominant structure is a lapse-slip anomaly of the identical signature class — rotation slip on a single leg with the energy channel flat — at the displaced position, and the CMB-frame decomposition measures the same frame-anchored axis at reversed measured polarity.

The forward falsifier therefore re-registers on the frame-anchored axis rather than disappearing (because this polarity re-registration is formulated after observing the post-2017 reconstruction reversal, it carries zero confirmatory weight in the present cohort and is strictly prospective): under that reading the measured-polarity ordering is the prediction — future transit records (the LSST-era comet cohort) should continue to resolve the modern, apex-side structure and not revert to the pre-2018 antapex-side polarity. A future cohort recovering the declared-axis inbound-leg excess in the pre-2018 sense, or resolving no coherent structure at all, would break the era-polarity ordering the frame-anchored account requires; continuation of the modern polarity at growing significance is what that account predicts.

## 8. Conclusions

Two dynamically distinct populations of the outer solar system — detached extreme trans-Neptunian objects and long-period comets — demonstrate localized, non-planetary orbital structure converging on the same 60° sector of inertial sky. Across ten empirical channels and six distinct catalogue lineages (SBDB, DES, MPCORB, CODE, Warsaw, and CometEls), the multi-survey synthesis establishes this convergence with a descriptive joint statistic of $S_{10} =$ 186.2 at the resident axis (conditional sky-area rank 0.0145) and $S_{10} =$ 188.9 at the transit axis (rank 0.00925), measured against 20,000 random test directions on the fixed observed catalogues. These ranks measure localization of the observed landscape. A global discovery probability requires a separate null-data calibration of the actual directional and channel-selection procedure, retaining catalogue overlap. The single best-performing random axis on the entire celestial sphere lies only 14.9° from the resident axis, within the 18.4° bootstrap localization radius. Across orbital, spacecraft, terrestrial-clock and lunar measurements, the nominal cross-channel Fisher combination yields $\chi^2 =$ 52.5 (nominal $p =$ 1.36×10<sup>-8</sup>), using the lunar fixed-Planck-axis scrambled-data statistic. This is distinct from the lunar direction's rank on the observed sky. Cross-channel dependence and retrospective geometry selection remain to be calibrated.

The tested standard astrophysical explanations face quantitative amplitude, direction and localization constraints. A Planet-Nine-like distant point mass is excluded as a common explanation of the measured signals over the tested parameter region on three independent grounds: it lacks resonant substructure and element coupling, its secular impulse falls short of the required cometary rotation by factors of $44\times$ to $527\times$, and direct dynamical insertion demonstrates that reproducing the observed rotation would require perturber masses of $\sim 515$–$4{,}290\,M_\oplus$ — brown-dwarf scale at every rung — across all semimajor axes from 100 to 1,000 AU, in direct violation of all-sky infrared surveys and planetary ephemerides. Ultimately, the point-mass hypothesis fails on amplitude and localization rather than on kinematic principle — secular gravity does rotate orbital elements at nearly flat energy, but only at bounded, element-coupled, calculable amplitudes, and the measured residual exceeds every allowed perturber's reach by two to three orders of magnitude while remaining direction-organized, single-leg-localized, and energy-flat. A crossing-localized proper-time transport provides a candidate phase response consistent with the measured rotation/energy dissociation. The canonical transport calculation now specifies its required amplitude and morphology; a sourceable admissible field profile and the full metric-to-observation response are still required to establish the outer-system realization quantitatively. The tested interstellar-drag accounts are constrained jointly by the unresolved in-cap energy contrast ($p = 0.69$), directional mismatch and concentration in deep plungers, while the galactic tide is excluded by its $m = 2$ quadrupolar symmetry, its $44^\circ$ tilt off the tidal plane, and direct N-body insertion demonstrating an impulse seven orders of magnitude below the observed signal. Observational discovery bias is constrained rather than eliminated outright: the footprint-conditional null calibrated on the OSSOS-measured discovery coupling rejects a footprint-only account at the marginal $p = 0.035$ level, while the key weight is carried by the contiguous Dark Energy Survey cohort ($13/16$ in-cap, $p = 1.2\times 10^{-4}$ against uniformity and $p = 0.029$ against its own footprint) and by the fact that the resident and transit populations are observed toward opposite sky patches.

The analysis establishes a key empirical discriminator between physical signal and reduction-level systematics. The aphelion dipole, evaluated independently of the leg comparison, remains stable across the tested orbit-quality strata: it is flat across bound and hyperbolic subsets (+0.103 versus +0.0830), short-arc and long-arc fits (+0.0907 versus +0.0979), and sparse versus dense observation records (+0.0987 versus +0.0919), replicating robustly across all 257 non-training comets (+0.0827, $p =$ 0.00805), the full CometEls census ($+$0.10, $p =$ 2×10<sup>-3</sup>), and the post-2017 prospective cohort ($d_\parallel =$ +0.0949, $p =$ 0.00215). In sharp contrast, the post-2017 cometary reconstruction systematic is quality-dependent at the resolution limit — flat across the arc-length halves (+0.27 versus +0.28 dex) and across the observation-count halves (+0.27 versus +0.21 dex), but collapsing to $-0.02$ dex under the strictest quality cut ($n_{\rm obs} > 1000$). This is the profile of a short-arc reduction systematic of the modern synoptic-survey record (Pan-STARRS, ATLAS, Mount Lemmon, and ZTF), and no single survey carries it — the displaced gap survives every leave-one-out survey deletion at $p < 3\times 10^{-4}$. The opposite behavior of these two channels under identical stratification demonstrates that the instrument cleanly separates physical incoming structure from short-arc reduction artifacts. The aphelion dipole's class is stated plainly: it evidences a direction-organized source population — degenerate between a primordial Oort-cloud anisotropy, a selection residual, and the boundary field — rather than the boundary mechanism itself, which the measured slip cannot generate.

Dedicated reduction audits establish the anomaly as an intrinsic feature of the astrometric record. Integrating class-1 CODE comets under both DE440s and DE430 proves that planetary ephemeris updates produce boundary periapsis shifts below $5.4\times 10^{-7}$ degrees (median $3.4\times 10^{-9}$ per leg), leaving the four-covariate residual in-cap gap identical at $+0.1234$ dex ($p = 0.0118$) under both ephemerides to within $10^{-7}$ dex. The osculating-epoch convention is eliminated rather than bounded: independent two-leg Levenberg–Marquardt refits of raw Minor Planet Center astrometry anchor each orbit at the observational mid-arc on the observations themselves, reproduce the catalogue boundary rotations at $\rho =$ +0.998, and confirm the inbound-leg in-cap excess ($p =$ 0.0370), so no catalogue osculation convention or fitting pipeline enters the independent record.

The multi-messenger and multi-process ledger reinforces this dynamical picture. In the interstellar sector, 1I/'Oumuamua is the only catalogued interstellar object executing a two-lobe boundary transit — a geometric concordance whose present weight is bounded by the three-object catalogue ($p \sim 0.33$ under geometry-blind assignment) — exhibiting a purely radial unexplained non-gravitational acceleration of $9.4\times 10^{-4}$ of the local solar attraction ($A_1 = 2.79\times10^{-7}$ AU d$^{-2} \approx 5.6\times10^{-6}$ m s$^{-2}$), kinematically consistent with the comet-channel lapse contrast under high-velocity transit, while volatile-explained interstellar objects bound the interior gradient $\sim 70\times$ below. In the spacecraft clock sector, archival SCLK records bound sustained rate steps at 1 ppm and instantaneous lapse contrasts at $|\delta A/A| \lesssim 2\times 10^{-5}$, on the clock-carrying onboard-oscillator channel the two-way cancellation theorem exempts — the coherent tracking links themselves being blind to the static lapse term by construction — while MHW-RTG nuclear clocks show unit-level coherence across the heliopause. In terrestrial networks, the independent GNSS-II free clock axis lands $0.93^\circ$ off the cometary meridian plane ($p =$ 0.0162) — a directional concordance the clock channel's own analysis reads as a heliocentric orbital-phase direction, carried here at face value rather than as shared-frame evidence.

The definitive prospective test will be decided by the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST). For future detached-TNO discoveries, the TEP domain-boundary model predicts that $\sim 59$ per cent will land inside the pre-declared 60° cap about $\bar\varpi = 49^\circ$, compared to a footprint-only expectation of $\sim 42$ per cent and a footprint-predicted direction of $10.5^\circ$. An exact power analysis demonstrates that $N \approx 123$ new detached discoveries will separate these hypotheses at 95 per cent power and $\alpha = 1$ per cent, a milestone achievable within the survey's initial years. If future discoveries conform to the survey footprint, the domain-boundary hypothesis is cleanly falsified. Confirmation of the registered cap fraction against a characterized survey-selection null would supply prospective macroscopic evidence for the outer-system TEP realization adopted here; its interpretation would also be tested against competing anisotropic source-population models.

## 9. References

- Agazie G., et al. (NANOGrav Collaboration), 2023, ApJL, 951, L8 — The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background

- Anderson J. D., Campbell J. K., Ekelund J. E., Ellis J., Jordan J. F., 2008, Phys. Rev. Lett., 100, 091102 — Anomalous Orbital-Energy Changes Observed during Spacecraft Flybys of Earth

- Banik I., Pittordis C., Sutherland W., et al., 2024, MNRAS, 527, 4573 — Strong constraints on the gravitational law from Gaia DR3 wide binaries

- Bannister M. T., et al., 2018, ApJS, 236, 18 — OSSOS: the Outer Solar System Origins Survey ensemble characterization (VizieR J/ApJS/236/18)

- Batygin K., Brown M. E., 2016, AJ, 151, 22 — Evidence for a distant giant planet in the solar system

- Bernardinelli P. H., et al., 2022, ApJS, 258, 41 — A search of the full six years of the Dark Energy Survey for outer solar system objects

- Brown M. E., Batygin K., 2021, AJ, 162, 219 — The orbit of Planet Nine

- Brown M. E., Holman M. J., Batygin K., 2024, AJ, 167, 146 — A Pan-STARRS1 search for Planet Nine

- Burrage C., Sakstein J., 2018, Living Rev. Relativ., 21, 1 — Tests of chameleon gravity

- Bzowski M., et al., 2015, ApJS, 220, 28 — Interstellar neutral helium in the heliosphere from IBEX observations. III. Mach number of the flow, velocity vector, and temperature from the first six years of measurements

- Chae K.-H., 2023, ApJ, 952, 128 — Breakdown of the Newton–Einstein standard gravity at low acceleration in internal dynamics of wide binary stars

- Chen Y.-T., et al., 2025, Nat. Astron., 9, 1309 — Discovery and dynamics of a Sedna-like object with a perihelion of 66 au (2023 KQ14, 'Ammonite')

- Gladman B., Chan C., 2006, ApJ, 643, L135 — Production of the extended scattered disk by rogue planets

- Gomes D. C. H., Murray Z., Gomes R. C. H., Holman M. J., Bernstein G. M., 2023, PSJ, 4, 66 — Can the gravitational effect of Planet X be detected in current-era tracking of the known major and minor planets?

- Hernández X., 2023, MNRAS, 525, 1401 — Internal kinematics of Gaia DR3 wide binaries: anomalous behaviour in the low acceleration regime

- Hui M.-T., et al., 2026, ApJL, 999, L37 — Nucleus and post-perihelion activity of interstellar object 3I/ATLAS observed by the Hubble Space Telescope

- Jewitt D., Seligman D. Z., 2023, ARA&A, 61, 197 — The interstellar interlopers

- Kenyon S. J., Bromley B. C., 2004, Nature, 432, 598 — Stellar encounters as the origin of distant Solar System objects

- Khoury J., Weltman A., 2004, Phys. Rev. Lett., 93, 171104 — Chameleon fields: awaiting surprises for tests of gravity in space

- Królikowska M., 2014, A&A, 567, A126 — Warsaw catalogue of cometary orbits: 119 near-parabolic comets (VizieR J/A+A/567/A126)

- Królikowska M., et al., 2014, A&A, 571, A63 — catalogue of cometary orbits and their dynamical evolution: one-apparition comets with element uncertainties and Marsden–Williams (2008) comparison values (VizieR J/A+A/571/A63)

- Królikowska M., Dybczyński P. A., 2020, A&A, 640, A97 — CODE catalogue of cometary orbits and their dynamical evolution

- Marsden B. G., Williams G. V., 2008, Catalogue of Cometary Orbits, 17th edn. — Minor Planet Center/Central Bureau for Astronomical Telegrams

- Micheli M., et al., 2018, Nature, 558, 223 — Non-gravitational acceleration in the trajectory of 1I/2017 U1 ('Oumuamua)

- Park R. S., et al., 2021, AJ, 161, 105 — The JPL Planetary and Lunar Ephemerides DE440 and DE441

- Phan T. L., et al., 2025, PASA, 42, e064 — A search for Planet Nine with IRAS and AKARI data

- Rein H., Liu S.-F., 2012, A&A, 537, A128 — REBOUND: an open-source multi-purpose N-body code for collisional dynamics

- Rein H., Spiegel D. S., 2015, MNRAS, 446, 1424 — IAS15: a fast, adaptive, high-order integrator for gravitational dynamics

- Rein H., et al., 2019, MNRAS, 485, 5490 — MERCURIUS: a hybrid N-body integrator with smooth close-encounter transition

- Siraj A., Chyba C. F., Tremaine S., 2025, ApJ, 978, 139 — Orbit of a possible Planet X

- Siraj A., Chyba C. F., Tremaine S., 2025, MNRAS Lett., 543, L27 — Measuring the mean plane of the distant Kuiper belt

- Smawfield M. L., 2026 — Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed, preprint v0.16 (Jakarta), TEP Series Paper 0, doi:10.5281/zenodo.16921911

- Smawfield M. L., 2026 — Global Time Echoes: Distance-Structured Correlations in GNSS Clocks, preprint v0.28 (Jaipur), TEP Series Paper 1, doi:10.5281/zenodo.17127229

- Smawfield M. L., 2026 — Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products, preprint v0.21 (Cairo), TEP Series Paper 2, doi:10.5281/zenodo.17517141

- Smawfield M. L., 2026 — Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries, preprint v0.7 (Kilifi), TEP Series Paper 13, doi:10.5281/zenodo.19102061

- Smawfield M. L., 2026 — Global Time Echoes: MGEX Multi-GNSS Clock Replication, preprint v0.4 (Suva), TEP Series Paper 14, doi:10.5281/zenodo.20572726

- Smawfield M. L., 2026 — Temporal Equivalence Principle: Temporal Shear in the Earth Flyby Anomaly, preprint v0.4 (Yogyakarta), TEP Series Paper 15, doi:10.5281/zenodo.19454862

- Smawfield M. L., 2026 — Temporal Equivalence Principle: Lunar Laser Ranging and the Nordtvedt Effect, preprint v0.5 (Lucknow), TEP Series Paper 17, doi:10.5281/zenodo.19446028

- Smawfield M. L., 2026 — Temporal Equivalence Principle: A Covariant Alternative to Cosmic Expansion, preprint v0.4 (Athens), TEP Series Paper 26, doi:10.5281/zenodo.20370143

- Spada F., et al., 2026, A&A, in press (doi:10.1051/0004-6361/202659645) — Systematic and statistical uncertainties in the non-gravitational acceleration of 3I/ATLAS

- Thoss V., Loeb A., Burkert A., 2026, arXiv:2603.15735 — Inferring the mass and size of 3I/ATLAS from its non-gravitational acceleration

- Trujillo C. A., Sheppard S. S., 2014, Nature, 507, 471 — A Sedna-like body with a perihelion of 80 astronomical units

- Whiting C. E., Woerner D. F., 2023, Lifetime Performance of Spaceborne RTGs, Chapter 8 in Woerner D. F. (ed.), The Technology of Discovery: Radioisotope Thermoelectric Generators and Thermoelectric Technologies for Space Exploration, Wiley, ISBN 9781119811367; accompanying dataset: Whiting C. E., Woerner D. F., 2024, Historical RTG performance data through 2023, Dryad, doi:10.5061/dryad.1zcrjdfw2 (Voyager MHW-RTG telemetry, courtesy E. Medina, JPL)

- Jet Propulsion Laboratory, Small-Body Database (SBDB) query API, https://ssd-api.jpl.nasa.gov/sbdb_query.api

- Minor Planet Center, MPCORB orbit catalogue, https://minorplanetcenter.net/iau/MPCORB/

## 10. Data availability and reproducibility

## 10.1 Data availability

All catalogues used are public. The JPL Small-Body Database is queried live through `https://ssd-api.jpl.nasa.gov/sbdb_query.api`. The OSSOS characterized ensemble, the Warsaw comet-orbit tables, and the one-apparition comet catalogue are obtained from the CDS/VizieR archives (J/ApJS/236/18, J/A+A/567/A126, and J/A+A/571/A63). The CODE catalogue is distributed through the Poznań comet-dynamics server at pad2.astro.amu.edu.pl; the analysed snapshot selects the published preferred solution for each comet and requires matching model identifiers across the original, osculating and future tables; its captured exports are integrity-pinned by SHA-256 in `data/raw/code/provenance.json`. MPCORB and the JPL CometEls file complete the auxiliary data; the N-body baseline draws planet states from the JPL DE440s ephemeris kernel (`data/raw/spice/de440s.bsp`, SHA-256 pinned in provenance). The auxiliary records are likewise public and provenance-pinned: per-observation MPC astrometry and station codes (`data/raw/mpc/obs/`, `obscodes.json`), the NASA/GSFC SPDF OMNI daily solar-wind record (`data/raw/omni/`), Voyager MHW-RTG telemetry (Whiting & Woerner 2024, Dryad, with the 2023 lifetime-performance chapter; `data/raw/rtg/`), Voyager PWS plasma-wave density profiles (`data/raw/voyager_pws/`), JPL Horizons spacecraft state vectors and NAIF spacecraft clock kernels (`data/raw/horizons/`, `data/raw/naif/`), the IMCCE/PODB APDB planetary astrometric compilation (`data/raw/apdb/`), and the NIMA/Lucky Star stellar-occultation astrometry (LESIA; `data/raw/nima/`). No synthetic or simulated data enter any reported measurement; synthetic realizations appear only in the explicitly labelled injection, transfer-function, and forward-model controls.

## 10.2 Reproducibility

The complete analysis is a single-command pipeline archived at github.com/matthewsmawfield/TEP-9 (Zenodo archive DOI 10.5281/zenodo.22858190): `python3 scripts/run_all.py`, run from the repository root, executes the one hundred and fifty-two registered steps in order under `scripts/run_all.py`, organized in eleven phases from catalogue acquisition through forward-model closure and the publication claims-trace audit. Every step writes a timestamped log to `logs/`; analysis results and figures are stored under `results/`, while acquisition steps store raw snapshots and provenance under `data/raw/`; every download records its URL, retrieval timestamp, byte count, and SHA-256 checksum under `data/raw/*/provenance.json`; Monte-Carlo nulls use fixed recorded seeds; and a per-run audit with script-integrity hashes is written to `results/audits/`. Every step number, result file, and figure cited in the text resolves unambiguously to a named artefact in that archive. The only non-self-contained steps are 135 and 139, which read registered result leaves — not raw data — from the sibling TEP-GNSS-II, TEP-GNSS-MGEX, and TEP-LLR pipelines in the local corpus, each itself a provenance-pinned public pipeline; the pipeline fails loudly if any required catalogue is absent.

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# Temporal Equivalence Principle: Localized Proper-Time Gradients and the Spatial Proximity of Discordant Pairs
**Matthew Lukin Smawfield**
Version: v0.4 (Pasadena)
First published: 24 September 2026 · Last updated: 17 October 2026
DOI: 10.5281/zenodo.22938153

---

## Abstract

The discordant quasar–galaxy pairs catalogued by Halton Arp — low-redshift galaxies apparently joined by luminous bridges, filaments, or foreground absorption to companions at redshifts one to two orders of magnitude larger — represent historical association claims that remain disputed and require independent distance adjudication. Under the standard interpretation, redshift is identified with distance through the Hubble relation and each such pair is treated as a chance superposition of unrelated objects. This paper applies the Temporal Equivalence Principle (TEP) — the bi-metric scalar-tensor framework in which matter, clocks, and light propagate on the causal metric $\tilde{g}_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi$ with universal coupling $A(\phi) = e^{-\phi}$ — to a twelve-pair catalogue drawn from the published literature and verified against the NASA/IPAC Extragalactic Database. In TEP the observed redshift factorises into a background clock-rate factor set by the ambient temporal level hosting the source and an intrinsic conformal factor generated by the emitter's own proper-time field, $1 + z_{\rm obs} = A_0 / (A_{\rm bg}\,A_{\rm int})$. For two objects at a common distance the shared background term divides out exactly, yielding the distance-independent redshift-ratio diagnostic $A_{\rm pair} = (1 + z_G)/(1 + z_Q)$: a conditional estimate of the companion's local clock-rate offset only if co-distance is established. Applied to the catalogue, the diagnostic returns pair factors spanning $A_{\rm pair} = 0.33$ (NGC 7319) to $0.98$ (NGC 1232A); conditional on co-distance, these become a smooth distribution of local field depths rather than the discrete quantisation demanded by earlier intrinsic-redshift proposals (e.g., Karlsson 1971; Burbidge 2001) — while the standard reading places companions up to $\sim 360$ times farther away than the galaxies to which they appear connected. The physical-association evidence is evaluated independently of any redshift model: surface-brightness transects across the claimed bridges are tested against rotated control axes, the published Ca II and H I absorption systems fix the line-of-sight ordering, an empirical quasar-density test compares the Arp host fields against seeded random fields, and the Poisson chance-alignment probability is computed per pair and for the joint ensemble under selection-aware density and geometry conditioning. The resolved NGC 7603 filament is detected as real emitted light along its measured path ($+3.9\sigma$ over the inter-object segment, $+5.7\sigma$ peak, reproduced in three photometric bands). Demonstrating that foreground halo absorption fixes line-of-sight ordering without establishing co-distance, the Ly$\alpha$ forest path-length audit could be executed on one catalogue sightline (NGC 3067–3C 232) and falsified it, placing 3C 232 at its cosmological distance. The direct adjudication ledger therefore stands at one falsification in one key catalogue-pair test, with no pair yet confirmed co-distant and the other eleven isolated as unadjudicated candidates; existing HST coverage of Mrk 205 and Keck coverage of the NGC 7319 companion are non-discriminating. For the eleven-system residual catalogue, the joint probability is retained strictly as a conditional configuration-severity statistic, not as a co-distance measurement, returning $P = 3.6 \times 10^{-13}$ under the conservative empirical quasar density, sharpened to $4.9 \times 10^{-19}$ under the axis-conditioned geometry of the morphology-claimed systems and $4.0 \times 10^{-18}$ under the measured X-ray-selection density after correction for the measured pointed-coverage overdensity (yielding a look-elsewhere crossover volume of $\sim 310$ parent fields). Independent geometric-coherence tests — minor-axis anisotropy, radial redshift ordering, and physical-scale clustering at redshift-independent distances — probe structure the catalogue was not selected on; two return significant structure and the third is marginal, where the hypothesis predicts it, while paired-redshift and symmetry tests return bounding nulls. The scalar-field transition is inferred on the NGC 7603 filament — the only Arp structure carrying measured interior redshifts ($z = 0.243$ and $z = 0.391$ between the endpoints) — where the non-monotonic four-point transect refutes any single shared transition and is consistently parameterised by a nested-wells field model — compact local proper-time depressions superposed on the shared pair field — with each well confined to a projected width below $1.2$ kpc of its emitter and the diffuse filament continuum sitting at ambient depth between them. Transferring the historical Arp–Sulentic companion-redshift asymmetry test to an unselected survey-scale sample — $162{,}650$ bound satellites across $88{,}662$ groups in the SDSS DR10 friends-of-friends catalogue — detects a small, statistically significant positive redshift excess ($f_+ = 0.5075$, two-sided binomial $p = 1.7 \times 10^{-9}$, rising to $0.513$ in rich groups): the asymmetry strengthens in volume-limited subsamples ($f_+ = 0.510$–$0.513$), persists beyond host-disk dust scales with an inverted colour–magnitude signature, exceeds forward-modelled interloper injection by two orders of magnitude, survives at full amplitude in the clean spectroscopic-metadata and passive–passive same-template subsamples ($f_+ = 0.5078$ and $0.5092$), scales with satellite-to-central contrast ($\langle\Delta v\rangle = +5.9$ km/s at the bright end), and carries the opposite sign to standard cluster gravitational redshift — opposing the simple shared-ambient gravitational-redshift expectation and consistent with an emitter-intrinsic proper-time-well contribution, while residual magnitude-differential spectral-template systematics remain to be excluded. All tests are aggregated into a single falsification table separating pair-level distance adjudication from conditional configuration severity and population controls. A forensic re-reduction of the raw 2003 Keck/LRIS frames reproduces the published $z = 2.114$ spectrum of the NGC 7319 companion at its recorded wavelengths, and the same audit identifies the observation that would settle the question for the flagship system: a single blue-arm echelle exposure on the $z = 2.114$ companion of NGC 7319 would deliver the decisive $z > 2$ forest sightline. The eleven unadjudicated systems remain candidate naturally occurring differential-clock experiments selected for key follow-up; if co-distance is confirmed in any high-$z$ system, its redshift ratio becomes a direct measurement of the local clock factor predicted by TEP.

Keywords: Temporal Equivalence Principle, Halton Arp, discordant redshifts, quasar–galaxy associations, intrinsic redshift, proper-time gradient, scalar-tensor theory, modified gravity, Markarian 205, NGC 7603

## 1. Introduction: The Discordant-Redshift Anomaly

In 1966 Halton Arp published the *Atlas of Peculiar Galaxies*, a systematic survey of the sky's morphologically disturbed systems. Among them he found the anomaly that would occupy the rest of his career: galaxies and quasars that appear, on the sky, to be physically connected yet carry redshifts differing by one to two orders of magnitude (Arp 1987, *Quasars, Redshifts and Controversies*). Under the standard interpretation, redshift is distance: an object at $z_Q$ is placed a factor $\sim z_Q/z_G$ farther away than a companion at $z_G$. On that reading, every luminous bridge, filament, and absorption association Arp documented between a low-redshift galaxy and a high-redshift companion is treated as a chance superposition of unrelated objects at vastly different distances. The discordant pairs were accordingly absorbed into the background census, while direct path-length adjudication remained incomplete for many individual historical cases.

Although the controversy subsided under the standard metric-expansion paradigm after the programme lost its instrument access in 1983 (Section 7.5), the underlying interpretation rests on a single identifiability assumption: that a redshift differential between two sky-coincident objects can encode only a difference in cosmological distance. That assumption is what the present paper tests through an explicit adjudication hierarchy. If redshift also carries a contribution intrinsic to the emitting object—a local proper-time offset rather than a distance—then while individual historical candidates selected by foreground absorption or alignment may still prove to be background projections, a subset of discordant systems with genuine luminous connections can reside at a common distance, with their redshift differentials encoding localized proper-time gradients rather than cosmological separation.

This paper returns to the same sky with the tools that did not then exist—digital all-sky imaging, homogeneous quasar catalogues, archive redshift services, and a reproducible inference pipeline—and measures what the controversy only argued.

### 1.1 The Observational Basis

The Arp associations are not anecdotal. They were assembled over four decades by a consistent observational method: start from morphologically disturbed galaxies, look for compact radio, X-ray or blue stellar objects paired or aligned across their nuclei—preferentially along the minor axis—and test the candidate connections with whatever spectroscopy the epoch allowed. The morphological premise descends from the *Atlas* itself, whose first 101 entries are peculiar spiral galaxies and spirals carrying small attached companions (Arp 1966)—the configuration from which the discordant-redshift claim originally grew—and the associations were consolidated in the *Catalogue of Discordant Redshift Associations* (Arp 2003). The catalogue on which the present analysis is built comprises twelve well-documented systems whose claimed physical connections span four distinct, independently testable types. In the flagship pair, the Seyfert 1 nucleus Markarian 205 at $z = 0.0708$ is projected $42$ arcsec from the nucleus of the barred spiral NGC 4319 at $z = 0.0047$, inside the galaxy's outer isophotes, with a luminous optical bridge between them (Arp 1971) confirmed on deep CCD imagery and spectroscopy (Sulentic 1983; Sulentic & Arp 1987). Under the Hubble relation the companion is placed $\sim 15$ times farther away than the galaxy it appears to touch.

NGC 7603 is connected to its companion NGC 7603B by a luminous filament (Arp 1970) that hosts two additional emission knots at intermediate redshifts, $z = 0.243$ and $z = 0.391$ (López-Corredoira & Gutiérrez 2002), producing a four-point redshift transect along a single continuous structure. In NGC 3067, the radio quasar 3C 232 at $z = 0.533$ exhibits Ca II K absorption at the redshift of the foreground galaxy, $z_{\rm abs} = 0.005$ (Stocke et al.\ 1991), and 21-cm H I absorption at the same velocity (Carilli, van Gorkom & Stocke 1989)—gas at the galaxy redshift lying demonstrably between the observer and the quasar. Three quasars at $z = 0.60$, $1.40$, and $1.94$ are projected on the disk of NGC 1073 at $z = 0.007$ (Arp & Sulentic 1979). A strong X-ray quasar at $z = 2.114$ sits $9$ arcsec from the nucleus of NGC 7319 in Stephan's Quintet (Galianni et al.\ 2005)—a Chandra ultraluminous X-ray source (Trinchieri et al. 2003) lying inside the galaxy's shocked interstellar medium, with spectral evidence reported for direct interaction between the quasar and the host's gas at $z = 0.022$ (Galianni et al. 2005). A compact companion at $z = 0.044$ is silhouetted in absorption against the diffuse body of NGC 1199 at $z = 0.0083$ (Arp 1978)—the higher-redshift object lying in front of the lower-redshift galaxy's light. X-ray quasars trace the minor axis of the edge-on spiral NGC 3628 (Arp et al. 2002), the morphological signature of the ejection-axis alignment reported across the Arp systems. The same signature repeats across the added fields: two compact ROSAT X-ray sources symmetrically disposed across the nucleus of NGC 4258, along a line touching the ends of its anomalous arms, proved on spectroscopy to be quasars at $z = 0.398$ and $z = 0.653$ (Pietsch et al. 1994; Burbidge 1995); two X-ray quasars at $z = 0.305$ and $z = 0.323$ sit nearly symmetrically across the nucleus of NGC 2639 along its minor-axis direction, within a seven-source X-ray chain on the same axis (Burbidge 1997; Burbidge, Burbidge, Arp & Zibetti 2004); the complete quasar survey of the NGC 1097 field found the quasar density rising toward the galaxy and peaking between its two strongest optical jets (Wolstencroft et al. 1983; Arp, Wolstencroft & He 1984; Arp & Carosati 2007); and the companion on the spiral arm of NGC 1232—the prototype of the Atlas companion-on-arm morphology (Arp 41)—carries a recession velocity $+4,776\ {\rm km\,s^{-1}}$ in excess of its host (Arp 1982).

Each of these associations carries its own geometrical argument, and several carry two: a luminous connection and an absorption or silhouette ordering that fixes the line-of-sight sequence independently of any redshift interpretation. The standard response—that each is individually a chance alignment—is an assignable probability, and Section 5 computes it. What the standard framework cannot do is convert the ensemble into a coincidence: the joint probability for the sample evaluates to a catalogue severity statistic, computed in this paper for the first time, to test the hypothesis that the catalogue is merely a collection of random projections.

### 1.2 The Temporal Equivalence Principle

The Temporal Equivalence Principle (TEP; Paper 0, Jakarta) is a bi-metric scalar-tensor framework in which gravity is mediated by the Einstein-frame metric $g_{\mu\nu}$ while matter, clocks, and light propagate on the canonical causal matter metric $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$; Paper 0, §2.2, Step 80), written in conformal–disformal shorthand as

\begin{equation}\tilde{g}_{\mu\nu} = A^2(\phi)\,g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi ,\end{equation}

with a universal conformal coupling $A(\phi) = \exp(\beta_A \phi/M_{\rm Pl})$, $\beta_A = -1$. Every non-gravitational process runs on the proper time of $\tilde{g}_{\mu\nu}$; local Lorentz invariance and the local light speed $c$ are strictly preserved in every freely falling laboratory. The scalar field $\phi$ is positive near masses and vanishes in the ambient today by convention, so $A < 1$ inside gravitational wells and clocks there run slower than ambient—reproducing gravitational redshift as a clock-rate ratio rather than a metric expansion. Because the quasar sits in a deeper temporal well, $\phi_Q > \phi_G$ and $A_{\rm int} = e^{-(\phi_Q-\phi_G)} < 1$, consistent with the corpus sign convention of Jakarta (Paper 0).

The cosmological content of the framework, developed in the companion papers of this series, is that the universe is spatially static, eternal, and non-expanding. Within this two-metric cosmology, the Hubble relation is a clock-rate comparison between the ambient field today and the field at the source at emission; and structure, motion, and local geometry evolve on the static manifold. Within that framework an observed redshift factorises into a background clock-rate factor—the ambient temporal level at the source's position and emission epoch—and an intrinsic conformal factor generated by the emitting object's own proper-time field. The second factor is the quantity the standard distance ladder discards—and the quantity the Arp pairs measure directly.

### 1.3 Scope of the Present Paper

This paper applies the TEP redshift decomposition to the Arp discordant-pair catalogue as a complete, reproducible inference chain. The analysis proceeds in four stages. First, the intrinsic conformal factor $A_{\rm int}(Q)$ is computed for each pair from published, archive-verified redshifts; the diagnostic is distance-independent by construction. Second, the physical-connection evidence is evaluated on its own terms: surface-brightness transects across the claimed bridges against rotated control axes, the published foreground-absorption and silhouette systems, an empirical quasar-overdensity test around the Arp hosts against seeded random fields, and the Poisson chance-alignment probability of each pair and of the joint ensemble. Third, the scalar-field transition across the connecting structures is inferred: monotonic and nested-wells profile families are fitted to the redshift transect of the NGC 7603 filament—the only Arp structure with measured interior redshifts—and the local well depths are estimated by Markov Chain Monte Carlo with full residual diagnostics. Fourth, every test is aggregated into a single falsification table opposing the TEP spatial-proximity hypothesis against the chance-superposition interpretation.

The TEP hypothesis tested here is narrow and therefore strong: the catalogue contains a subset of physically associated discordant systems at a common distance, while allowing individual historical candidates to be chance superpositions, and their redshift differentials are localized proper-time gradients of the TEP scalar field rather than gaps in cosmological distance. Every numerical statement in this paper is produced by the accompanying open pipeline from real archival and published data; no synthetic data are used at any stage, and the pipeline fails loudly when a required input is unavailable.

## 2. Theoretical Framework: Redshift as Proper-Time Transport

In TEP the measured redshift of a source is a clock-rate comparison. A photon emitted by matter running on the proper time of $\tilde{g}_{\mu\nu}$ at the source is compared, on arrival, against clocks running on the matter metric here and now. Writing the conformal clock-rate factor at emission as $A_{\rm em}$ and at observation as $A_0$,

\begin{equation}1 + z = \frac{A_0}{A_{\rm em}} .\end{equation}

Written this way, the relation isolates the conformal clock factor. The complete clock-transport law additionally carries the gravitational lapse at each endpoint and any propagation contribution accumulated along the path; for the pairs analysed here these terms are common to both members and divide out of the ratio formed in Section 2.1.

The emission-point factor itself separates into two contributions with distinct physical origins. The first is the background clock-rate factor set by the ambient temporal level at the source—the field value hosting the emitter's environment at emission—denoted $A_{\rm bg}(r)$ for a source at coordinate distance $r$; the drift of this ambient level across the realized temporal landscape produces the systematic Hubble relation. The second is the intrinsic factor generated by the emitting object's own proper-time field—the local elevation of $\phi$ inside the object's gravitational well—denoted $A_{\rm int}$. The observed redshift is the product

\begin{equation}1 + z_{\rm obs} = \frac{A_0}{A_{\rm bg}(r)\,A_{\rm int}} .\end{equation}

For an object whose local field is continuous with the ambient environment, $A_{\rm int} \to 1$ and the redshift reduces to the background term. For an object sitting deep inside its own temporal well—a compact, high-surface-density source such as a quasar nucleus—$A_{\rm int} < 1$ and the observed redshift exceeds, potentially by a large factor, the value its distance alone would imply.

### 2.1 The Pair Diagnostic

Consider two objects sharing the same spatial coordinate distance $r$ from the observer: a host galaxy $G$ and a companion $Q$. The background factor $A_{\rm bg}(r)$ is a property of the shared environment, not of the emitter, and is therefore identical for both. The same is true of every further contribution shared between the two endpoints—the observer's own clock factor and gravitational lapse, and any propagation term accumulated along two sight-lines separated by tens of arcseconds over a common baseline. Forming the ratio of the two redshift factors eliminates all such shared contributions exactly, isolating the difference in the two emitters' local temporal states:

\begin{equation}\frac{1 + z_Q}{1 + z_G} = \frac{1}{A_{\rm int}(Q)} ,\end{equation}

where the galaxy's own intrinsic factor has been absorbed into the pair's ambient baseline—the clock rates of the pair are measured relative to the host, which is the natural local standard. The intrinsic conformal factor of the companion is then

\begin{equation}A_{\rm int}(Q) = \frac{1 + z_G}{1 + z_Q} , \qquad
\Delta\phi \equiv \phi_Q - \phi_G = -\ln A_{\rm int}(Q) \;\; (\beta_A = -1,\; M_{\rm Pl}\ {\rm units}) .\end{equation}

**Theorem:**

#### Distance Independence of the Intrinsic Factor

The quantity $A_{\rm int}(Q) = (1+z_G)/(1+z_Q)$ depends on the two measured redshifts alone. No distance estimate, Hubble constant, or cosmological parameter enters its definition. If the two objects are at a common distance, $A_{\rm int}$ directly measures the companion's local proper-time offset. If they are not co-distant, the same ratio remains a distance-independent observable but cannot be interpreted as an intrinsic conformal factor without modelling the unequal background terms. The diagnostic is therefore well-defined under either hypothesis, and the physical-proximity tests of Sections 5–6 decide between the two readings.

### 2.2 The Proper-Time Budget

With the universal coupling $A(\phi) = e^{-\phi}$ in Planck units, every matter clock in the companion runs at the rate $A_{\rm int}$ relative to the host's clocks. The redshift differential of an Arp pair is thereby converted into a physical statement: the companion's proper time elapses at a fraction $A_{\rm int}$ of the host's rate, corresponding to a scalar-field offset $\Delta\phi = -\ln A_{\rm int}$. The same numbers admit two interpretive readings whose consequences are tabulated in Section 4: the standard reading converts the excess redshift into a fictitious recession velocity $v_{\rm intr} = c\,z_{\rm intr}$ and a distance gap $D_H(Q) - D_H(G)$, while the TEP reading converts it into a clock-rate ratio at a single shared distance.

### 2.3 The Field Transition Along a Connecting Structure

TEP is deterministic: the scalar configuration is a continuous field, not a step function. If a luminous bridge physically joins a host galaxy to a companion, the intrinsic conformal factor must interpolate smoothly between $A_{\rm int} = 1$ at the host nucleus and $A_{\rm int} = A_{\rm int}(Q)$ at the companion. Along a normalised bridge coordinate $x \in [0,1]$, the transition is modelled by candidate profile families of two distinct classes. The first class comprises three monotonic families, each constructed to satisfy the endpoint boundary conditions exactly and carrying an interior shape parameter that the data constrain:

\begin{equation}A_{\rm exp}(x) = 1 - \bigl(1 - A_Q\bigr)\,\frac{1 - e^{-kx}}{1 - e^{-k}} ,\end{equation}

\begin{equation}A_{\rm tanh}(x) = 1 - \bigl(1 - A_Q\bigr)\,\tfrac{1}{2}\left[1 + \tanh\!\left(\frac{x - x_0}{w}\right)\right] \quad ({\rm normalised\ to}\ A(1) = A_Q) ,\end{equation}

\begin{equation}A_{\rm yuk}(x):\quad \phi(x) \propto \frac{e^{-m r}}{r} \;\;\mathrm{mapped\ through}\;\; A = e^{-\phi} .\end{equation}

The exponential family describes a field that relaxes toward the ambient level with a scale length; the tanh family describes a transition concentrated at an interior point $x_0$ of width $w$; the Yukawa family maps a screened potential profile through the conformal relation. The three families span the qualitatively distinct topologies a single shared transition can take—distributed, localized, and screened—so that the transect fit discriminates among field geometries rather than merely fitting a curve.

The second class is the nested-wells profile, the model the framework itself supplies. The clock-rate hierarchy requires that each compact emitter along a connecting structure elevates $\phi$ inside its own gravitational well, on top of the ambient field of the pair; the measured factor is then the product of the shared baseline and the object's local depth. Modelling each local well as a Lorentzian depression of depth $d_i$ centred on the emitter's transect position $x_i$, with a shared width $w$,

\begin{equation}A_{\rm nest}(x) = \exp\!\left[-\sum_{i}\frac{d_i}{1 + \bigl((x - x_i)/w\bigr)^2}\right] ,\end{equation}

the nested-wells family permits exactly the structure the monotonic class forbids: interior points deeper in the field than the companion endpoint, because each emitter's own well contributes to the total. Where the monotonic families test whether a single shared transition suffices, the nested-wells family tests the clock-rate hierarchy directly—and returns each emitter's local well depth $d_i$ as a measurable quantity comparable to $\Delta\phi = -\ln A_{\rm int}$.

The falsifiable content is precise. Under the standard interpretation the interior objects of a connecting filament—such as the two emission knots of the NGC 7603 filament—are unrelated background sources, and no smooth single-family curve is required to pass through their redshifts: their values are accidents and should scatter arbitrarily. Under TEP the knots are material inside a shared field, and their intrinsic factors must lie on a continuous profile of the total temporal landscape—a shared transition populated by the local wells of the emitters themselves. A field model that reproduces the measured interior points within their propagated uncertainties is therefore direct, structured evidence for a shared local field—evidence the chance-alignment hypothesis cannot supply, because it predicts no ordering at all.

### 2.4 Relation to the Wider TEP Corpus

The decomposition used here is not a new mechanism introduced for the Arp problem; it is the application of the corpus's fixed scalar sector to a new observable. The ambient-vs-local separation of $A$ is the same nesting of clock rates that the framework uses from the screened Solar-System limit, through galactic and void scales, to the strong-field temporal wells and the temporal horizon. What the Arp pairs add is a uniquely clean geometry: two emitters at a common, physically connected location whose redshift ratio isolates the intrinsic factor with no dependence on the cosmological model. The pairs are, in effect, naturally occurring differential-clock experiments distributed across the sky.

## 3. The Discordant-Pair Sample and the Analysis Pipeline

The analysis is built as a single reproducible pipeline: a registered sequence of processing stages that begins with the published literature values for the discordant pairs, verifies them against public archives, ingests the available observational data, and terminates in the falsification summary and manuscript figures. Every number quoted in this paper is an output of that pipeline; the pipeline is deterministic, seeded where randomness enters, and fails loudly when a required real input is missing. No synthetic data are generated at any stage.

### 3.1 The Pair Catalogue

The catalogue comprises twelve discordant pairs selected for the strength and diversity of their claimed physical connections, covering the connection morphologies of the consolidated Arp catalogue (Arp 2003): a luminous bridge (NGC 4319–Mrk 205), a luminous filament with embedded emission knots (NGC 7603–NGC 7603B), foreground absorption systems (NGC 3067–3C 232, NGC 1199 and its compact companion), disk and minor-axis projections of multiple quasars (NGC 1073, NGC 3628), an X-ray-selected quasar chain with a paired $z = 2.10$ redshift (NGC 3516), a strong X-ray ejection candidate (NGC 7319), symmetrically paired X-ray quasars across active nuclei on the minor-axis and anomalous-arm directions (NGC 4258, NGC 2639), a quasar overdensity correlated with the optical jets of NGC 1097, and the companion-on-arm system NGC 1232–NGC 1232A—the prototype of the Atlas companion-on-arm morphology. Each entry carries the published galaxy and companion redshifts, the projected angular separation, the documented connection evidence with its literature references, and a flag identifying the redshift provenance. The catalogue is seeded exclusively with literature values; the companion redshift of NGC 3628 enters at the value Arp et al. (2002) tabulate for the minor-axis quasar WEE 51 ($z = 2.15$), flagged as approximate pending archive verification—the verification step supersedes it with the spectroscopic archive value ($z = 1.498$), as recorded in the verification manifest.

Every catalogued redshift is then verified against the NASA/IPAC Extragalactic Database (NED). The heliocentric redshift of each resolvable object is extracted from the NED object record and compared against the catalogue value at a fractional tolerance $|\Delta z|/(1+z) = 0.01$; companions catalogued under descriptive labels are verified through their archive-resolved identities, by alternate name or by cone search at the resolved position. Objects that cannot be securely identified in the archive are carried at their literature values and flagged rather than substituted, while provisional literature values are superseded by the archive value wherever a secure identification exists—the one such correction in the sample is recorded in the verification manifest.

### 3.2 Archival Data Ingestion

Three archival layers are ingested per pair field. Optical imaging is retrieved as deep CCD data: the DESI Legacy Imaging Surveys DR10 $r$-band cutout service (Dey et al.\ 2019; $0.262$ arcsec pixels) is the primary survey, with SDSS $r$-band and DSS2 Red through the SkyView interface as fallbacks where the DR10 layer has no coverage (one field, NGC 7319, uses SDSS $r$). Each field is centred on the host galaxy and sized to cover the catalogued companion separation with margin; these images provide the spatial base on which the bridge transects are drawn. A multi-wavelength provenance manifest is compiled from the HEASARC master catalogues (Chandra, XMM-Newton, ROSAT) and VizieR radio catalogues (NVSS, FIRST), recording the archival coverage of each field so that the physicality of connecting structures can be assessed across bands rather than in a single image. Finally, SDSS DR18 is queried for spectroscopic coverage of each pair member, and available spectra are ingested; where a member lies outside the footprint the absence is recorded, not replaced.

A fourth archival layer anchors the distance scale. The Cosmicflows-4 catalogue (Tully et al. 2023; VizieR J/ApJ/944/94) is cone-searched at every member position for redshift-independent distance moduli drawn from the primary and secondary distance ladder—Cepheids, the tip of the red giant branch, masers, surface-brightness fluctuations, the Tully–Fisher and fundamental-plane relations, and Type Ia supernovae. A catalogue entry is adopted only within a $30$ arcsec positional match, beyond which the nearest record belongs to a different object; where a companion's nearest entry is the host's own catalogue record, the absence of an independent companion distance is recorded rather than substituted. Six of the twelve hosts carry matched distances—NGC 7603 at $113\pm17$ Mpc, NGC 3067 at $22\pm4$ Mpc (Tully–Fisher), NGC 1199 at $34\pm5$ Mpc (surface-brightness fluctuations), NGC 3628 at $11\pm2$ Mpc (Tully–Fisher), NGC 1097 at $15\pm3$ Mpc (Tully–Fisher), and NGC 4258 at $7.54\pm0.09$ Mpc, the most precisely anchored distance in the sample, averaged across the maser, Cepheid, TRGB, surface-brightness-fluctuation and Tully–Fisher records—each consistent with its redshift-inferred distance, as expected for hosts in shallow wells. These measurements anchor every projected physical scale reported below in a redshift-independent distance rather than an assumed one. No companion carries a redshift-independent distance of its own—quasar and compact-knot companions admit no stellar-population distance indicator—so the key co-distance test remains a defined future measurement rather than a result claimed here (Section 7).

### 3.3 Intrinsic-Factor Computation

For every pair the pipeline computes the intrinsic conformal factor $A_{\rm int}(Q) = (1+z_G)/(1+z_Q)$, the scalar-field offset $\Delta\phi = -\ln A_{\rm int}$, and the proper-time budget: the fractional clock-rate difference between companion and host, the fictitious recession velocity $c\,z_{\rm intr}$ that a Doppler reading would assign, the Hubble-law distances implied by each redshift separately at $h = 0.70$, and the distance-inflation factor between them. These quantities are reported in Section 4.

### 3.4 Physical-Connection Tests

Five independent tests of physical association are implemented. The bridge-morphology test first resolves every companion to an archive position—by NED object-name query where the catalogued name is resolvable, and otherwise by redshift-matched cone searches of Simbad, the Milliquas catalogue, and NED in turn—then draws a surface-brightness transect along the measured galaxy–companion axis of each pair field and compares the inter-object segment (the central $45$–$80\%$ of the axis, excluding both nuclear regions) against rotated control transects of identical length and width sampling the galaxy's outer isophotes away from the companion direction. The statistics reported are the mean-significance $S_{\rm bridge}$ of the on-axis excess over the control distribution, a one-sided sign test of the on-axis excess's persistence across the segment, and the peak per-sample significance with its transect coordinate; for the silhouette pair a complementary annulus decrement test is evaluated about the companion position. The absorption-system test compiles the published absorption and silhouette evidence—gas at the host redshift intervening along the line of sight to the companion—which fixes the line-of-sight ordering independently of any redshift model. The association test resolves each Arp host through Simbad and performs a VizieR cone search of the Million Quasars catalogue (Milliquas v7.2; Flesch 2021; VizieR VII/290), counting spectroscopically confirmed quasar-class objects (catalogue classes Q, A, B and K) within $6$ arcmin of each host. The host's own catalogue entry is removed by name, and the catalogued companions are excluded within $15$ arcsec of their archive-resolved positions, so that the objects the pairs were selected for cannot inflate the count. The identical query is repeated on twelve seeded random control fields offset by $1.5^\circ$ per host, constructing an empirical field quasar surface density from the same catalogue, and the measured catalogue offset of every recovered object is recorded. The comparison is evaluated both against the pooled control distribution and in paired form—each host against its own twelve controls—because the pooled null mixes sight-lines of unequal spectroscopic completeness. The chance-alignment test computes, per pair and jointly, the Poisson probability

\begin{equation}P = 1 - e^{-\Sigma_Q\,\pi\,\theta^2}\end{equation}

of finding at least one background quasar within projected radius $\theta$ of an arbitrary bright galaxy, evaluated for quasar companions under both the SDSS DR16 quasar-catalogue surface density $\Sigma_Q = 51.6\ {\rm deg}^{-2}$ (Lyke et al. 2020) and the more conservative empirical field density measured by the association test in the same sight-lines, and using the conservative bright-galaxy density $\sim 10\ {\rm deg}^{-2}$ for resolved companions; the headline value is the larger of the two evaluations. The fifth test lifts the association question to population level: a predefined parent sample—2MRS galaxies (Huchra et al. 2012) with $500 < cz < 10,000$ km/s and $|b| > 15^\circ$, defined without reference to the Arp catalogue—is bulk cross-matched to the same confirmed Milliquas classes within $180$ arcsec through the CDS XMatch service, and each parent galaxy is paired with four seeded offset control fields cross-matched identically. The quasar overdensity is measured in three annuli and as a paired galaxy-versus-own-controls difference tested by a galaxy-unit bootstrap. A suite of consistency audits complements these tests: a lensing magnification budget on the measured separations (Section 5.4), geometric and spectroscopic coherence diagnostics (Section 5.5), a live archive audit of resolved-kinematics coverage (Section 7.6), and the Ly$\alpha$ forest path-length audit of Section 5.6, which prices each quasar-class companion's distance directly from the intergalactic absorption content of its spectrum.

### 3.5 Field-Gradient Inference

The scalar-field transition is inferred on the NGC 7603 filament, the only connecting structure in the sample with measured interior redshifts: the host and companion supply the transect endpoints $x = 0$ and $x = 1$, and the two emission knots at $z = 0.391$ and $z = 0.243$ supply interior points at archive-measured positions: both knots resolve to Simbad entries (the [LG2002] emission-line objects), and their transect coordinates $x = 0.55$ and $x = 0.91$ are the projection of the resolved positions onto the host–companion axis—no positional assumption enters the transect. The profile families of Section 2.3—the three monotonic transitions and the nested-wells model—are fitted to the four-point transect by least squares on their interior parameters and compared by chi-squared. The nested-wells parameters—the three local well depths and the shared width—are then sampled by Markov Chain Monte Carlo (emcee ensemble sampler; Foreman-Mackey et al. 2013, 32 walkers, 8,000 steps with 2,000 burn-in, flat priors $d_i \in [0, 2]$ and $w \in [0.002, 1]$, Gaussian likelihood in the measured $A_{\rm int}$ values with propagated redshift uncertainties), producing posteriors on the well depths that are directly comparable to the per-object $\Delta\phi$ measurements of Section 4. The sample-level structure of $A_{\rm int}$ is analysed across all twelve pairs: its distribution, its dependence on projected separation (the Spearman correlation against angular separation, testing Arp's empirical claim that discordance declines with separation), and the comparison of absorption-verified against morphology-only pairs. Finally, the transect fits are subjected to residual diagnostics: weighted residuals, reduced chi-squared, residual RMS, lag-1 autocorrelation, and a Lilliefors normality check (Lilliefors 1967), so that model adequacy is assessed on residual structure rather than parameter-count arguments alone. All tests are aggregated into the falsification table of Section 7.

## 4. Results I: The Intrinsic Conformal Factor

**Key Finding:**

#### A Distance-Independent Proper-Time Diagnostic Applied to Twelve Discordant Pairs

The intrinsic conformal factor $A_{\rm int}(Q) = (1 + z_G)/(1 + z_Q)$ has been computed for all twelve catalogue pairs from archive-verified redshifts. The values span $A_{\rm int} = 0.328$ (NGC 7319–QSO) to $0.984$ (NGC 1232–NGC 1232A), a smooth and continuous distribution of local proper-time depths. The corresponding scalar-field offsets span $\Delta\phi = 0.016$–$1.114$ in Planck units. Under the standard distance interpretation the same pairs require the companions to lie a median factor of $64.7$ farther away than their apparent hosts—and up to a factor of $361$ for the NGC 3628 minor-axis quasar—while the projected physical separations are only $4$–$136$ kpc.

### 4.1 The Pair Measurements

Table 1 presents the central measurement of this paper. For each pair the table lists the two observed redshifts, the intrinsic conformal factor, the scalar-field offset $\Delta\phi = -\ln A_{\rm int}$, the clock-rate ratio between companion and host, and the distance-inflation factor that the standard interpretation requires. Because $A_{\rm int}$ is constructed from a ratio of redshifts, it carries no dependence on distance, on $H_0$, or on any cosmological parameter: it is a distance-independent pair diagnostic; under the co-distance hypothesis it is the direct local conformal factor.

*Table 1. The intrinsic conformal factor and proper-time budget of the Arp discordant-pair sample. Separations are archive astrometry (NED/Simbad) measured nucleus-to-companion.*

| Pair | $z_G$ | $z_Q$ | $A_{\rm int}(Q)$ | $\Delta\phi$ ($M_{\rm Pl}$) | Clock ratio | $D_C(Q)/D_C(G)$ | Sep. (arcsec) |
|---|---|---|---|---|---|---|---|
| NGC 4319 – Mrk 205 | 0.0047 | 0.0708 | 0.938 | 0.064 | 0.938 | 14.9 | 42.7 |
| NGC 7603 – NGC 7603B | 0.0295 | 0.0565 | 0.974 | 0.026 | 0.974 | 1.9 | 58.3 |
| NGC 3067 – 3C 232 | 0.0049 | 0.533 | 0.656 | 0.422 | 0.656 | 95.2 | 110.2 |
| NGC 1073 – BSO 1 | 0.0070 | 0.5997 | 0.629 | 0.463 | 0.629 | 73.7 | 121.4 |
| NGC 7319 – QSO | 0.0225 | 2.114 | 0.328 | 1.114 | 0.328 | 55.7 | 9.3 |
| NGC 1199 – companion | 0.0083 | 0.044 | 0.966 | 0.035 | 0.966 | 5.3 | 57.8 |
| NGC 3628 – WEE 51 | 0.0028 | 1.498 | 0.401 | 0.913 | 0.401 | 361.0 | 251.1 |
| NGC 4258 – quasar pair (primary $z=0.398$) | 0.0015 | 0.398 | 0.716 | 0.334 | 0.716 | 240.5 | 789.3 |
| NGC 2639 – quasar pair (primary $z=0.305$) | 0.0111 | 0.305 | 0.775 | 0.255 | 0.775 | 25.6 | 596.3 |
| NGC 1097 – jet quasar field (primary $z=0.52$) | 0.0042 | 0.520 | 0.661 | 0.414 | 0.661 | 107.7 | 993.3 |
| NGC 1232 – NGC 1232A | 0.0053 | 0.0220 | 0.984 | 0.016 | 0.984 | 4.1 | 243.1 |
| NGC 3516 – quasar chain (primary $z=0.93$) | 0.0088 | 0.930 | 0.523 | 0.649 | 0.523 | 83.2 | 674.5 |

Two features of the distribution carry immediate physical weight. First, the values are continuous: the sample does not cluster at discrete plateaux, consistent with a smooth proper-time landscape rather than the quantised intrinsic-redshift proposals historically attached to the Arp systems. Second, the depth ordering is physically sensible. The two largest intrinsic offsets, $\Delta\phi \approx 0.91$ and $1.11$, belong to the two pairs whose companions are the most extreme compact emitters—the $z = 2.114$ X-ray quasar beside NGC 7319 and the minor-axis X-ray quasar WEE 51 of NGC 3628—while the smallest offsets belong to the resolved galactic companions NGC 7603B, the NGC 1199 compact object, and NGC 1232A. The deepest inferred temporal wells in this sample occur in high-energy quasar-like emitters, while the shallowest occur in more ordinary galactic companions. This ordering is qualitatively consistent with TEP, in which redshift depth is governed by the local proper-time field configuration rather than by cosmological distance alone. TEP does not require temporal well depth to be a simple function of spatial compactness.

![Intrinsic conformal factor and distance-inflation factor for the twelve pairs](results/figures/step_41_pair_gallery.png)

Figure 1. Left: the intrinsic conformal factor
$A_{\rm int}(Q) = (1+z_G)/(1+z_Q)$ for each pair—a
continuous distribution of local proper-time depths, all
sub-unity. Right: the distance-inflation factor the
standard redshift–distance reading requires for the
same pairs, reaching a factor of $361$ for the
NGC 3628 minor-axis quasar.

### 4.2 The Proper-Time Budget

Under the universal coupling $A(\phi) = e^{-\phi}$, each companion's matter clocks run at the fraction $A_{\rm int}$ of the host's clock rate. For the flagship pair NGC 4319–Mrk 205 the quasar's proper time elapses at $93.8\%$ of the host galaxy's rate—a clock-rate deficit of $6.2\%$, corresponding to $\Delta\phi = 0.064\,M_{\rm Pl}$. At the deep end of the sample, the NGC 7319 quasar's clocks run at $32.8\%$ of the host rate ($\Delta\phi = 1.11\,M_{\rm Pl}$): an extreme but bounded temporal well, a finite approach toward the strong temporal wells treated in the strong-field sector of the framework.

The standard reading assigns the same differential to the Hubble flow, and its cost is quantified in the fictitious columns of the budget. Interpreted as a recession velocity, the intrinsic component of the Mrk 205 redshift is a modest $\sim 20,000\ {\rm km\,s^{-1}}$; but for 3C 232 the required intrinsic recession is $1.58 \times 10^{5}\ {\rm km\,s^{-1}}$ ($0.53\,c$), and for the NGC 7319 quasar the required recession rate is formally superluminal. The standard $\Lambda$CDM distance reading places the companions $71$–$5,244$ Mpc behind their hosts; for NGC 7319 the implied lookback-time gap between the two members of a single $9$-arcsecond system is $10.1$ Gyr. The distance interpretation does not merely separate the pairs; it requires their apparent physical connection to span ten billion years of cosmic history.

### 4.3 Decomposition into Background and Intrinsic Factors

With the pair distance set by the host galaxy, the background clock-rate factor at each system is $A_{\rm bg} = A_0/(1 + z_G)$, and the full decomposition $1 + z_Q = A_0/(A_{\rm bg}\,A_{\rm int})$ holds identically. The measured background factors, $A_{\rm bg} = 0.971$–$0.999$ relative to the observer-frame value, show that the ambient field contributes only a small correction across these nearby systems: the discordance lives almost entirely in the intrinsic term. For NGC 7603, where the discordance is modest, $A_{\rm bg} = 0.971$ and $A_{\rm int} = 0.974$ contribute comparably; for the extreme quasar pairs the intrinsic factor dominates by two orders of magnitude over the background factor. This is the quantitative sense in which the Arp pairs measure local physics: the ambient background is common to both members, and only the emitters' own wells differ.

### 4.4 Independent Spectroscopic Confirmation

Archive ingestion returned two independent SDSS spectra for 3C 232, which yield a measured spectroscopic redshift $z_{\rm SDSS} = 0.5306$ against the catalogued literature value $z = 0.533$—agreement at the $0.2\%$ level in $1+z$—and one SDSS spectrum for NGC 7603B, which returns $z_{\rm SDSS} = 0.0557$ against the catalogued $z = 0.0565$, again at sub-percent agreement in $1+z$. These are independent confirmations that the redshifts entering the diagnostic are measured quantities rather than catalogue errors. The remaining members outside the SDSS footprint are recorded as absent in the ingestion manifest; no substitute values are used.

## 5. Results II: Physical-Connection Evidence

**Key Finding:**

#### The Companion Axes Are Real, the Ordering Is Measured, and the Surviving Ensemble Is Inconsistent with Pure Chance Superposition

Four independent lines of evidence converge on physical association. Every companion in the catalogue is resolved to an archive position—through NED name resolution or redshift-matched Simbad/Milliquas/NED cone searches—so each measurement is taken on the true companion axis rather than an assumed direction, on deep CCD imaging (DESI Legacy Imaging DR10 r-band; SDSS r for NGC 7319) rather than photographic plates. Along the measured axis of the NGC 7603 filament the transect drawn along the resolved curved path—through both archive-measured emission knots—exceeds rotated control paths by $+3.9\sigma$ with a localised $+5.7\sigma$ peak: the connecting structure is detected as real emitted light. The NGC 4319–Mrk 205 bridge direction carries a persistent low-level excess above its control envelope (sign-test $p = 0.020$). Foreground Ca II K and H I absorption at the redshift of NGC 3067 in the spectrum of 3C 232, and the silhouette companion of NGC 1199, fix the line-of-sight ordering of two pairs independently of any redshift model. Every quasar-class companion is recovered as a confirmed object in the Million Quasars catalogue at its measured offset from the host nucleus. And the Poisson chance-alignment computation, evaluated at the archive-measured separations and excluding the falsified member, returns $P = 3.6 \times 10^{-13}$ as the eleven-system severity statistic under the specified independent-draw sampling model—the conservative bound, evaluated at the empirical quasar density of the same sight-lines; the published SDSS density gives $6.6 \times 10^{-14}$.

### 5.1 Bridge Surface-Brightness Transects

For each pair field a surface-brightness transect is drawn along the measured galaxy–companion axis of the archival imaging: every companion is resolved to an archive position (NED name resolution where the catalogued name is a resolvable object, otherwise a redshift-matched cone search of Simbad, Milliquas, and NED in turn), so no transect is ever drawn along an assumed direction. Each on-axis transect is compared against eight rotated control transects of identical length and width sampling the galaxy's outer isophotes at the same galactocentric radius. The imaging is the DESI Legacy Imaging Surveys DR10 r-band cutout for every field where the layer has coverage ($0.262$ arcsec pixels)—modern CCD photometry an order of magnitude deeper than the DSS2 photographic plates—with SDSS r-band for NGC 7319, which sits in an LS-DR10 coverage hole. Three statistics are evaluated on the inter-object segment—the central $45$–$80\%$ of the axis, which excludes both nuclear regions: the mean-significance statistic $S_{\rm bridge} = (\mu_{\rm bridge} - \bar{\mu}_{\rm ctrl}) / \sigma_{\rm ctrl}$, a one-sided sign test on the fraction of segment samples lying above the per-sample control median (the signature of a diffuse, continuous structure that a segment mean can dilute; samples are counted at strip-width spacing so that only independent resolution elements enter the count), and the largest per-sample $z$ within the segment with its transect coordinate. The test is targeted only where a luminous structure is actually claimed—the NGC 4319 bridge, the NGC 7603 filament, and the spiral arm carrying NGC 1232A; for the remaining morphologies the transect is a contextual measurement of the companion sight-line. For NGC 7603 a second transect follows the filament's resolved curved path—the polyline through the archive-measured knot positions rather than the straight chord, since the knots sit up to $23$ arcsec off the axis—with control paths produced by rigidly rotating the same polyline about the nucleus.

*Table 2. Bridge-axis surface-brightness statistics on archive-resolved companion axes (DESI Legacy Imaging DR10 r-band; SDSS r for NGC 7319).*

| Pair | Resolved companion | $S_{\rm bridge}$ | sign-test $p$ | peak $z$ ($x$) | Reading |
|---|---|---|---|---|---|
| NGC 4319 – Mrk 205 | MRK 0205 | $+0.19$ | $0.020$ | $1.7$ ($0.46$) | persistent on-axis excess (claimed bridge) |
| NGC 7603 – NGC 7603B (straight axis) | NGC 7603B | $+0.09$ | $0.61$ | $2.7$ ($0.80$) | chord misses the curved filament |
| NGC 7603 filament path | through [LG2002] knots | $+3.92$ | $3.1 \times 10^{-5}$ | $5.7$ ($0.49$) | filament detected along resolved path |
| NGC 3067 – 3C 232 | 3C 232 | $-0.40$ | $1.0$ | $2.7$ ($0.75$) | null (absorption pair; no bridge claimed) |
| NGC 1073 – BSO quasars | QSO B0240+011 | $+0.34$ | $6 \times 10^{-8}$ | $1.9$ ($0.51$) | bright disk axis (contextual) |
| NGC 7319 – QSO | [VV2006] J223603.7+335824 | $+0.48$ | — | $1.1$ ($0.77$) | axis across galaxy body (contextual) |
| NGC 1199 – companion | 2MASX J0303366-153740 | $+0.21$ | $2.4 \times 10^{-4}$ | $0.9$ ($0.45$) | silhouette pair; annulus test below |
| NGC 3628 – WEE 51 | WEE 51 | $-0.72$ | $1.0$ | $-0.6$ ($0.68$) | minor-axis sight-line dim (contextual) |
| NGC 4258 – quasar pair | SDSS J121902.21+470505.4 | $+0.82$ | $6 \times 10^{-5}$ | $2.2$ ($0.46$) | paired-quasar axis elevated (contextual) |
| NGC 2639 – quasar pair | SDSS J084331.40+500227.0 | $-0.08$ | $0.72$ | $8.6$ ($0.66$) | null (contextual) |
| NGC 1097 – jet quasar field | CEMM J1779.9BL 2 | $+0.49$ | $3.8 \times 10^{-3}$ | $88.7$ ($0.52$) | jet-field axis; peak is a saturated foreground star (contextual) |
| NGC 1232 – NGC 1232A | NGC 1232A | $+0.34$ | $1.3 \times 10^{-5}$ | $5.2$ ($0.63$) | companion-on-arm axis elevated (claimed) |
| NGC 3516 – quasar chain | NuSTAR J110752+7230.7 | $-0.10$ | $0.97$ | $6.0$ ($0.75$) | open-field chain (contextual) |

On the three axes where a luminous connection is actually claimed the test returns positive evidence. The NGC 7603 result is decisive once the transect follows the published structure: the filament is curved—its two emission knots sit up to $23$ arcsec off the straight host–companion chord—so the straight-axis transect mostly samples blank sky beside it, while the path transect drawn through the archive-resolved knots exceeds the rotated control paths by $+3.9\sigma$ over the inter-object segment, sits above the control median on essentially every independent resolution element (sign-test $p = 3 \times 10^{-5}$), and peaks at $+5.7\sigma$ mid-way along the filament. The connecting structure is real emitted light at the sensitivity of modern CCD imaging, and the knots are part of it rather than superposed on it: cuts perpendicular to the path at each knot show the corridor elevated at $+4.8\sigma$ and $+19.1\sigma$ over the same cuts drawn at the corresponding vertices of the rotated control paths, even after the knot's own point-source profile ($\sim 2$ arcsec) is excluded—each discordant emitter sits on locally elevated diffuse emission, not on blank sky. The same path transect on the independent DR10 g-band cutout returns $S = +4.0\sigma$ (sign-test $p = 3 \times 10^{-5}$), and on the digitised DSS2 photographic plate—the medium on which the connection was originally claimed—it returns $S = +3.0\sigma$ (sign-test $p = 9.8 \times 10^{-4}$). The structure is detected independently in three bands on two detectors a generation of instrumentation apart; no single-detector artifact can produce it.

Along the Mrk 205 direction the NGC 4319 axis sits above its control median persistently enough that a sign test on independent resolution elements returns $p = 0.020$—a low-level but coherent excess along the claimed bridge direction, consistent with the sub-threshold extension of the luminous connection confirmed on deep CCD imagery (Sulentic 1983; Sulentic & Arp 1987). The widely cited high-resolution HST observations of this system that were interpreted as dissolving the connection into unrelated background sources and host spiral structure (e.g., Bahcall et al. 1992) test the compact, high-surface-brightness features; the persistent low-level diffuse excess quantified here in deep, wide-field Legacy Survey imaging remains a property of the sightline as a whole. The third claimed axis is the spiral arm of NGC 1232 at whose tip the discordant companion sits: the transect toward the archive-resolved NGC 1232A lies above the control median on essentially every independent resolution element (sign-test $p = 1.3 \times 10^{-5}$, segment peak $+5.2\sigma$)—the arm itself is the claimed luminous structure, and it is detected at the same significance class as the Mrk 205 bridge direction. The contextual axes behave as their morphologies predict: NGC 3067's quasar sight-line shows no excess (its connection is absorption, not emitted light), the NGC 1073 axis is mildly elevated across the galaxy's disk, the $9$-arcsecond NGC 7319 transect lies inside the bright nuclear body, and the NGC 3628 minor axis is fainter than the control envelope, as expected for an edge-on galaxy's dust lane. Of the new sight-lines, the paired-quasar axis of NGC 4258 is mildly elevated ($S = +0.82$, sign-test $p = 6 \times 10^{-5}$) and the NGC 1097 jet-field axis likewise ($S = +0.49$, sign-test $p = 3.8 \times 10^{-3}$, its segment peak a saturated foreground star rather than a feature of the galaxy), while NGC 2639 and the NGC 3516 chain return nulls. Where the claim is a luminous structure, the resolved axes detect it; where it is not, the measurements are noise-level or galaxy-morphology diagnostics, not refutations.

### 5.2 Foreground Absorption and Silhouette Ordering

Absorption at the host-galaxy redshift in a companion's spectrum is a distance-ordering measurement that no redshift model can evade: the absorbing gas lies between the observer and the emitter. Table 3 compiles the published absorption and silhouette evidence for the sample.

*Table 3. Absorption and silhouette systems fixing the line-of-sight ordering.*

| Pair | Absorber | Species | $z_{\rm abs}$ | $z_{\rm emit}$ | Reference |
|---|---|---|---|---|---|
| NGC 3067 – 3C 232 | NGC 3067 halo | Ca II K | 0.005 | 0.533 | Stocke et al. 1991 |
| NGC 3067 – 3C 232 | NGC 3067 halo | H I 21 cm | 0.005 | 0.533 | Carilli et al. 1989 |
| NGC 1199 – companion | NGC 1199 body | continuum extinction | 0.0083 | 0.044 | Arp 1978 (photographic claim; not reproduced on DESI DR10 imaging—see text) |

Two orderings follow. In the NGC 3067–3C 232 system, gas at $z = 0.005$ intervenes along the line of sight to the $z = 0.533$ quasar: the quasar is demonstrably behind the galaxy's halo. The ordering is measured directly; whether the pair is also at a common distance is adjudicated by the Ly-$\alpha$ path-length audit, which assigns 3C 232 to its cosmological distance (Section 5.6). In the NGC 1199 system, the $z = 0.044$ companion is silhouetted in absorption against the diffuse light of the $z = 0.0083$ galaxy—the higher-redshift object lies inside the lower-redshift galaxy's own body of light. For the silhouette pair the claimed obscuration is also tested directly on the imaging: the surface brightness of a $4$–$12$ arcsec annulus about the archive-resolved companion position (the compact object itself excluded) is compared against rotated control annuli at the same galactocentric radius. In the DESI DR10 r-band imaging the annulus reads $+2.0\sigma$ above its controls—the companion's own extended light fills the aperture—so the claimed dark ring is not detected at this survey's sensitivity, consistent with the published debate over its photometric reality; the ordering evidence for this pair rests on the published photographic silhouette claim of Arp (1978), not on this image.

Under the standard reading these detections are absorbed into the foreground-halo phenomenology and the association question is left open; under TEP the ordering is what spatial proximity at a common distance produces naturally. The measurement itself is deliberately modest: absorption at the host's own velocity proves ordering—the emitter lies behind the foreground galaxy's halo—and it converts the pair from an assumed superposition into a demonstrated foreground–background configuration at a projected separation of order an arcminute. What absorption cannot prove by itself is co-spatiality, since a genuinely cosmological background quasar would produce the same signature—and the Ly$\alpha$ audit (Section 5.6) demonstrates that for 3C 232, this is exactly what occurred. For establishing spatial proximity through ordering, only the NGC 1199 silhouette remains, as it places the companion inside the host's own luminous body rather than behind an extended halo. For the broader sample, the association case rests on the conjunction of ordering with the resolved structure and ensemble statistics of this section. The question the ordering leaves open is precisely the one the intrinsic-factor diagnostic answers: what the redshift differential means once proximity is established.

### 5.3 The Quasar-Overdensity Test

Arp's original statistical claim was positional: catalogued quasars are over-dense in the immediate vicinity of bright, low-redshift galaxies relative to random lines of sight. The pipeline implements this as an empirical test against a homogeneous quasar catalogue rather than an assumed density: a cone search of the Million Quasars catalogue (Milliquas v7.2; Flesch 2021; VizieR VII/290) counts spectroscopically confirmed quasar-class objects—catalogue classes Q (type-I quasar), A (type-I Seyfert/AGN), B (BL Lac) and K (type-II quasar), with the N class excluded for its documented NELG/LINER residue and all photometric candidates rejected—within $6$ arcmin of each Arp host, the host's own catalogue entry removed by name and every catalogued companion excluded within $15$ arcsec of its archive-resolved position, so that the objects the pairs were selected for do not inflate the count (companion-inclusive counts are reported alongside). The identical query is repeated on twelve seeded random control fields offset $1.5^\circ$ from each host—144 control fields in total—to construct the empirical field density from the same catalogue.

Because the pairs were selected for containing their companions, counting those companions toward the host tally would be circular; the primary count therefore excludes every catalogued companion within $15$ arcsec of its archive-resolved position, and the companion-inclusive count is reported alongside. The companion-inclusive totals are 26 confirmed quasar-class objects against $25.2$ expected from the 144-field control distribution (pooled Poisson $p = 0.46$); with the companions removed the host fields hold $18$—a host-field mean of $1.50$ against $2.10 \pm 2.45$ for the controls ($-0.24\sigma$ of the field scatter). The companions, in other words, are the difference between the two tallies: at the completeness of the local control fields these sight-lines are ordinary, and the objects the pairs were selected for are the only quasars distinguishing them.

The field-level detail remains informative. Wherever the companion is itself quasar-class the catalogue returns it by name at a measured offset—$0.71$ arcmin for MARK 205 from the NGC 4319 nucleus, $1.84$ arcmin for 3C 232 from NGC 3067, the three disk quasars NGC 1073 U1, U2 and PKS 0241+011 at $1.4$–$2.0$ arcmin, Q 2233+337 at $0.15$ arcmin from the nucleus of NGC 7319, and the NGC 3628 minor-axis quasars WEE 51 and WEE 52 together with the three 1WGA X-ray sources—and even with the catalogued companions excluded the host cones retain further confirmed quasars: two apiece in the NGC 4319, NGC 1073, NGC 7319 and NGC 4258 fields, five in the NGC 7603 field, four in the NGC 3628 minor-axis field, and one in the NGC 2639 field. Five hosts return zero: NGC 3067 and NGC 3516, where the only quasar-class object inside the cone is the catalogued companion itself; NGC 1199, precisely the pair whose companion is a compact galaxy seen in silhouette rather than a quasar-class object; NGC 1232, whose companion is likewise a galaxy; and NGC 1097, whose claimed quasar concentration lies along the jets beyond the $6$-arcmin cone. The nulls reflect catalogue membership and cone geometry, not association strength.

One further measured property of the NGC 1073 field deserves record. The three disk quasars carry intrinsic factors $A_{\rm int} \simeq 0.63$, $0.42$ and $0.34$ respectively—three deep well depths within a single projected disk, not a random draw from the $[0,1]$ continuum. In a continuous clock-rate field such preferred intrinsic ratios are the natural residue of a shared formation history within one host environment; the configuration is recorded here as a measured property of the field, with no claim of periodicity or quantisation.

The pooled control distribution mixes sight-lines of unequal spectroscopic completeness, so the comparison is repeated in paired form: each host against only its own twelve control fields. Two of the twelve hosts—NGC 4319 and NGC 3628—exceed every one of their own control fields (empirical $p \simeq 0.08$ each) even after the companions are removed: both lie off the main spectroscopic footprint, where the local control density is uniformly low and the host cones stand out cleanly. The famous fields—NGC 7603, NGC 1073, NGC 3067—sit inside control distributions that are themselves quasar-rich, the direct imprint of the spectroscopic attention those sight-lines have attracted: the measured control density is $3.3\times$ the all-sky catalogue expectation of $0.64$ objects per cone. Combined across the sample the companion-exclusive paired statistic is null (Fisher $p = 0.95$): beyond the catalogued companions themselves, these are ordinary lines of sight at the catalogue's completeness. The test therefore does what it is designed to do—it verifies every companion as a genuine confirmed quasar at a measured offset from its host, and it establishes that the ensemble-level excess lives entirely in the companion objects, which the chance-alignment computation of Section 5.4 then prices directly.

Four further fields extend the catalogue beyond single-companion systems, and each carries its evidential weight in structure rather than in angular proximity alone. Around the Seyfert NGC 3516 ($z = 0.0088$) the published claim is a chain of X-ray-selected quasars at $z = 0.33$, $0.69$, $0.93$, $1.40$ and $2.10$ within $\sim 12$ arcmin. All five resolve in Milliquas v7.2 to confirmed ROSAT/WGA X-ray quasars at the stated redshifts and archive-measured separations of $4.3$–$11.2$ arcmin, and the same field independently contains a second quasar at $z = 2.100$—a matching pair—together with two further quasars at $z = 2.40$. The angular test prices the chain correctly as weak in isolation ($P \approx 0.86$): wide-field quasar chains are not individually improbable under a global density. Two features of the field lie outside the angular statistic's reach—the X-ray selection, which restricts the relevant source population far below the optical quasar density, and the historically reported paired redshift at $z = 2.10$. Each claimed feature is tested rather than assumed in Section 5.5.

The same structural pricing applies to the three added multi-quasar fields. The two compact ROSAT sources paired across the nucleus of NGC 4258 resolve in Milliquas to confirmed quasars at $z = 0.398$ and $z = 0.653$, at $9.7$ and $13.2$ arcmin from the nucleus on a line that passes within $\sim 5^\circ$ of the minor axis and touches the ends of the anomalous arms (Pietsch et al. 1994; Burbidge 1995); the angular statistic at the pair's widest member separation returns $P \approx 1.0$ under the global density; the published accidental-configuration probability for the symmetric placement itself was estimated at $< 4 \times 10^{-7}$ (Arp 1996), an estimate re-examined at archive positions in Section 5.5. The NGC 2639 field carries two X-ray quasars at $z = 0.3048$ and $z = 0.3232$ nearly symmetric across the nucleus on the minor-axis direction, embedded in a seven-source X-ray chain on the same axis with four further confirmed AGN at $z = 0.337$–$2.63$ (Burbidge 1997; Burbidge et al.\ 2004); its angular price is likewise unremarkable in isolation ($P \approx 0.98$), and the historically claimed pairing symmetry is examined at archive positions in Section 5.5. And the NGC 1097 field—the most heavily surveyed of the sample—returned 31 spectroscopic quasars in the complete $8.1$ square-degree search, the surface density rising toward the galaxy and peaking between the two strongest optical jets, with the azimuthal distribution concentrated along the jet directions and a uniform-field $\chi^2$ rejected at $p = 0.008$ (Arp, Wolstencroft & He 1984; Wolstencroft et al. 1983; Arp & Carosati 2007). The pipeline's angular statistic prices only the widest member radius and returns $P \approx 1.0$; what the statistic does not price is the radial and azimuthal correlation with the jets that the survey itself measured. In each case the angular test is reported honestly: these systems enter the ensemble at their true Poisson weights, and their association claims rest on the structural and spectroscopic features the angular statistic cannot see.

The same test is then lifted to population level, removing the twelve-pair selection entirely. The parent sample is predefined without reference to the Arp catalogue: all 2MRS galaxies (Huchra et al. 2012; VizieR J/ApJS/199/26) with $500 < cz < 10,000$ km/s and $|b| > 15^\circ$—23,205 galaxies spanning the velocity range of the Arp hosts, which remain in the sample by the same cuts. Each parent is cross-matched in bulk against the same confirmed Milliquas classes within $180$ arcsec through the CDS XMatch service, with matches closer than $3$ arcsec removed as self-identifications of the parent's own nucleus, and each parent is paired with four seeded control fields offset $1.5^\circ$ cross-matched identically—92,820 controls in total. The result is null: the quasar surface density within $180$ arcsec of the parent galaxies is $\delta = -0.030$ relative to controls (bootstrap $p = 0.98$ for any positive excess), and the annulus decomposition shows no excess at $60$–$120$ arcsec ($\delta = -0.005$, $-0.2\sigma$) or $120$–$180$ arcsec ($\delta = +0.009$, $+0.4\sigma$), while the innermost arcminute carries a significant deficit ($\delta = -0.29$, $-7.7\sigma$)—the known spectroscopic avoidance and deblending losses near bright-galaxy cores, which the test correctly resolves. Restricting the parent sample to its 490 self-identified Seyfert-class members—the population Arp's claim specifically invoked—returns the same null ($\delta = -0.157$, bootstrap $p = 0.97$).

The pairing signature itself—two confirmed quasars flanking the nucleus at matching redshift (folded position-angle difference $\geq 135^\circ$, $|\Delta z| \leq 0.1$)—is likewise absent at population level: $34$ qualifying pairs among the parent fields against $47.5$ expected from the control rate.

Neither result contradicts the catalogue claim, which was never that every bright galaxy hosts a quasar companion: at the association rate the historical surveys implied ($\sim 10^{-3}$ per galaxy; Burbidge et al.\ 1971), the expected excess in $23,205$ fields is of order a few tens of objects—below this test's sensitivity—and the innermost arcminute, where several catalogued companions sit, is the region the spectroscopic catalogues sample least completely.

A generic quasar overdensity around bright galaxies therefore does not exist in the modern catalogues at these separations: the catalogued associations are rare configurations rather than a population-level effect, and the weight of evidence rests on the resolved structures, ordering measurements and nested redshift morphologies of the specific systems rather than on a bulk statistical excess. This null is itself information: it prices, directly and at population scale, the rarity that the look-elsewhere correction of Section 5.4 must otherwise assume.

### 5.4 Chance-Alignment Probabilities

Under the chance-superposition hypothesis, each pair is an independent draw: the probability of at least one background object of the companion's class landing within the measured separation of an arbitrary bright galaxy is the Poisson probability $P = 1 - e^{-\Sigma\,\pi\,\theta^2}$, evaluated for quasar companions under two surface-density assumptions—the published SDSS DR16 quasar-catalogue density $\Sigma_Q = 51.6\ {\rm deg}^{-2}$ (Lyke et al. 2020) and the more conservative empirical field density $\Sigma_Q = 66.8\ {\rm deg}^{-2}$ measured by the association test of Section 5.3 from the same catalogue in the same sight-lines—and the conservative bright-galaxy density $\Sigma \sim 10\ {\rm deg}^{-2}$ for resolved galactic companions. The angular scale $\theta$ is the archive-measured nucleus-to-companion separation (the largest member separation for multi-object companions). Table 4 lists the conservative empirical evaluation; the published-density evaluation returns uniformly smaller probabilities. For NGC 1073 the probability is that of three or more quasars within the field, as the system requires.

*Table 4. Per-pair and joint chance-alignment probabilities at archive-measured separations, grouped by the evidence class to which the angular statistic applies.*

| Pair | $\theta$ (arcsec) | $\Sigma$ (deg$^{-2}$) | $\lambda = \Sigma\pi\theta^2$ | $P_{\rm chance}$ |
|---|---|---|---|---|
| *Compact pairs — angular proximity is the operative statistic* |
| NGC 7319 – QSO | 9.3 | 66.8 | 0.001 | $1.4 \times 10^{-3}$ |
| NGC 4319 – Mrk 205 | 42.7 | 66.8 | 0.030 | $2.9 \times 10^{-2}$ |
| NGC 1199 – companion | 57.8 | 10.0 | 0.008 | $8.1 \times 10^{-3}$ |
| NGC 7603 – NGC 7603B | 58.3 | 10.0 | 0.008 | $8.2 \times 10^{-3}$ |
| NGC 3067 – 3C 232 | 110.2 | 66.8 | 0.196 | $1.8 \times 10^{-1}$ |
| NGC 1073 – 3 quasars | 121.4 | 66.8 | 0.239 | $1.9 \times 10^{-3}$ ($\geq 3$) |
| NGC 1232 – NGC 1232A | 243.1 | 10.0 | 0.143 | $1.3 \times 10^{-1}$ |
| NGC 3628 – WEE 51 | 251.1 | 66.8 | 1.02 | $6.4 \times 10^{-1}$ |
| *Wide-field systems — retained for completeness; claimed structure tested in Section 5.5* |
| NGC 2639 – quasar pair | 596.3 | 66.8 | 5.75 | $9.8 \times 10^{-1}$ ($\geq 2$) |
| NGC 3516 – quasar chain | 674.5 | 66.8 | 7.36 | $8.6 \times 10^{-1}$ ($\geq 5$) |
| NGC 4258 – quasar pair | 789.3 | 66.8 | 10.08 | $\approx 1.0$ ($\geq 2$) |
| NGC 1097 – jet quasar field | 993.3 | 66.8 | 15.97 | $\approx 1.0$ ($\geq 4$) |
| *Joint (compact subsample)* | — | — | — | $7.7 \times 10^{-14}$ |
| *Joint (all twelve)* | — | — | — | $6.5 \times 10^{-14}$ |

The two groups answer different questions. The partition is made at $\theta = 300$ arcsec, which falls in the gap between the largest compact separation (NGC 3628, $251''$) and the smallest wide-field separation (NGC 2639, $596''$); the grouping is therefore insensitive to the exact boundary, any threshold in the $251$–$596''$ range returning the same two samples. For the eight compact pairs the angular statistic is the operative test: a quasar within $43$ arcsec of a bright galaxy is a $\sim 2.9\%$ event, three quasars on a single low-redshift disk a $10^{-3}$ event, a quasar $9$ arcsec from the nucleus of NGC 7319 a $10^{-3}$ event. Their joint product, $7.7 \times 10^{-14}$, is the probability that the compact subsample exists at all under independent draws. The four wide-field systems are individually unsurprising under the same statistic—at half-degree scales a bright galaxy nearly always has catalogued quasars nearby. They are retained as an exhaustive audit of the historical catalogue: the morphological and paired-redshift claims made for them in the literature are tested rather than assumed in Sections 5.3 and 5.5. They enter the full product at their true weights near unity, contributing a factor of only $0.84$: including or excluding them changes the headline number by less than a factor of two. The catalogue is not, however, perfect. NGC 3067–3C 232 was selected by the same morphology criteria as the rest, but the Ly$\alpha$ path-length audit places the quasar definitively in the background (Section 5.6). That is direct measured evidence that these selection criteria can admit chance superpositions. The observed adjudication count is one falsification in one key catalogue-pair test; with a denominator of one, it is not an estimate of the false-positive frequency of the heterogeneous untested catalogue. Dropping this falsified pair from the ensemble ($P = 0.178$) raises the joint product to $P_{\rm joint} = 3.6 \times 10^{-13}$ under the conservative empirical density, and $6.6 \times 10^{-14}$ under the published SDSS density; the residual seven-pair compact product is $4.3 \times 10^{-13}$. Even granting the weakest systems their full probabilities, dropping the known false-positive, and using the pessimistic density assumption, under the specified chance-superposition model, the unadjudicated eleven-system configuration is highly improbable.

Both numbers price every companion against the full quasar-class surface density over the complete annulus, and both conditions can be relaxed in the direction the catalogue itself documents. First, five of the twelve systems carry a published claim that their companions lie on a host-morphology axis—the minor axis or the jet axis. For those pairs the target direction is fixed by the host's own light distribution before any companion position is examined; the a priori target is therefore not the full annulus but an axis corridor, taken conservatively as a $\pm 10$ arcdeg wedge about each axis direction ($f = 1/9$ of the annulus; published alignment tolerances for these systems are $3$–$7$ arcdeg). Repricing only the five axis-claimed pairs against their corridors—bridges and filaments are excluded, since their claimed direction is partially defined by the companion itself—sharpens the joint probability to $P_{\rm joint}^{\rm axis} = 8.7 \times 10^{-20}$ ($\log_{10} P = -19.1$; $4.9 \times 10^{-19}$ for the eleven-system residual catalogue), and the look-elsewhere boundary rises from $\sim 200$ to $\sim 310$ parent fields. Second, the X-ray-selected systems are priced against the wrong population: the companions of NGC 7319, 3628, 4258, 2639 and 3516 carry X-ray association flags in Milliquas (types QX, AX, KX)—the detection channel through which these X-ray-selected associations were identified—so the appropriate comparison density is that of X-ray-detected quasar-class objects, not all catalogued quasars. Measuring that density directly on the $144$ control fields of Section 5.3—$17$ of $302$ confirmed quasar-class objects carry X-ray association flags in Milliquas, giving $\Sigma_X = 3.8 \pm 0.9\ {\rm deg}^{-2}$, an upper bound since the flag spans all X-ray missions and is deeper than the ROSAT selection—and repricing the five X-ray-selected systems against it while holding the remaining seven at their baseline densities yields $P_{\rm joint}^{\rm sel} = 1.3 \times 10^{-22}$ ($\log_{10} P = -21.9$; $7.2 \times 10^{-22}$ for the eleven-system residual catalogue). One further correction is applied against that figure, derived from the survey-scale decomposition of Section 5.7: pointed X-ray identifiers are measured there to be overdense at catalogued galaxy positions by $\delta = +1.47$ (the coverage-uniform RASS subset returns $\delta = +0.12$), so the effective density at galaxy positions is $\Sigma_X \times 2.47$. Repricing the five X-ray-class pairs against the corrected density moves the joint to $7.1 \times 10^{-19}$ ($4.0 \times 10^{-18}$ for the eleven-system residual catalogue)—and for the most heavily pointed hosts in the catalogue (Stephan's Quintet, NGC 4258, NGC 3628) the true correction is plausibly larger still. The baseline statistic remains the weakest form of the argument: selection-conditioned, the ensemble remains improbable at the $10^{-18}$ level even after the correction is applied against itself.

One standard objection is priced explicitly rather than assumed away. Under the background-quasar interpretation each host is a foreground gravitational lens whose magnification can promote sub-threshold quasars above survey limits, inflating the apparent overdensity. Treating every host as a singular isothermal sphere with $\sigma_v = 300$ km/s—an upper bound above the published central dispersions of all twelve hosts ($\sim 130$–$230$ km/s)—the largest magnification at any catalogued separation is $\mu - 1 = 38\%$ (the $9.3$ arcsec companion of NGC 7319); folding the magnification-bias correction $\Sigma \to \Sigma\,\mu^{\alpha-1}$ through the joint product over the quasar luminosity-function slope range $\alpha = 1.5$–$2.5$ inflates the joint probability by a factor $1.3$–$2.3$ at that bound, and by no more than a factor $5$ anywhere on the full $\sigma_v = 250$–$400$ km/s sensitivity grid ($P = 1.1 \times 10^{-13}$ at the nominal slope). Lensing cannot manufacture the observed improbability; the correction is bounded at the percent level per pair.

The joint product is the a priori improbability of the configuration set—the statistic the discordant-pair literature has always reported (Burbidge et al. 1971; Arp 1987)—and it is corrected here for the volume of the search that assembled the catalogue. Under the chance hypothesis each of $N$ parent bright-galaxy fields is an independent Poisson trial: the expected number of chance lookalikes of the $i$-th observed system is $\mu_i = N\,p_i$, and the probability that a search of $N$ fields reproduces every observed severity class simultaneously is $P_{\rm cat}(N) = \prod_i \left[1 - e^{-N p_i}\right]$. It bears emphasis that the twelve systems were selected on claimed connection morphology—luminous bridges, silhouettes, absorption—not on proximity, so the joint product is a severity statistic for the observed configurations rather than a discovery $p$-value.

Evaluated on the measured severities, $P_{\rm cat}$ stays below $0.05$ for parent volumes smaller than $\sim 200$ fields—$0.007$ at $N = 100$—rising to $0.15$ at the $338$-entry scale of the Atlas of Peculiar Galaxies and $0.64$ at $N = 10^3$. The angular statistic therefore decides the ensemble on its own wherever the effective search volume lies below $\sim 200$ fields, a plausible bound for a sample assembled over decades of targeted observation of peculiar galaxies rather than drawn from a systematic survey; at larger volumes the bare angular product no longer decides. In either regime $P_{\rm cat}$ is an upper bound on the true ensemble chance probability, because the observed systems occupy narrower classes than projected proximity alone: two carry line-of-sight ordering evidence that no alignment frequency produces—spectroscopic absorption for NGC 3067, the published photographic silhouette claim for NGC 1199—and the only resolved transect is non-monotonic. The conjunction factor is a qualitative severity statement rather than a priced term in the model: the computation prices proximity alone. What a large search volume must counterfeit is not twelve close pairs but twelve close pairs carrying their ordering and structural evidence with them—and that conjunction the chance hypothesis has never produced.

The sharpest form of the argument prices coincidence not against proximity to the host but against proximity to the measured structure itself. In NGC 7603 the two [LG2002] emission knots carrying $z = 0.391$ and $z = 0.243$ sit on the detected filament polyline—a corridor $75$ arcsec long and a few arcsec wide, of order $10^{-5}$ square degrees. The probability that two independent background compact emission-line objects land inside that corridor is Poisson in $\lambda = \Sigma\,L\,w$: under the empirical quasar-class density at the $2$-arcsec width it is $P(\geq 2) \approx 3\times10^{-7}$, under a conservative faint emission-line-galaxy density ($\Sigma = 500\ {\rm deg}^{-2}$) across the widest $4$-arcsec corridor it is $7\times10^{-5}$, and even under an extreme deep-survey density ($\Sigma = 2000\ {\rm deg}^{-2}$) at the same width it remains $\approx 10^{-3}$. This is the quantitative content of the claim that the configuration is not merely improbable but structurally constrained: the discordant emitters do not sit near the system, they sit on the connecting structure, at positions whose spacing along it orders them by redshift.

Priced as a single configuration, NGC 7603 asks the chance hypothesis to produce a compact companion at $z = 0.057$ within $58$ arcsec ($P = 8 \times 10^{-3}$), a detected luminous filament between host and companion, and two independent compact emission objects at unrelated redshifts inside the filament corridor. The product over these distinct objects gives $P \approx 2 \times 10^{-9}$ at the empirical quasar-class density and remains below $10^{-5}$ even at the extreme deep-ELG density across the widest corridor. Folded into the ensemble, the full twelve-system catalogue severity statistic evaluates to $P \approx 1.9 \times 10^{-20}$ (quasar-class) and no more than $7 \times 10^{-17}$ under the most hostile density and corridor assumptions—the corridor is traced through the knot positions, so these figures price the conditional configuration, but the filament's reality no longer rests on that path definition: it is detected continuously along the segment in three independent bands, and the corridor remains elevated around each knot with the knots' own light masked out.

### 5.5 Geometric and Spectroscopic Coherence

The catalogue was selected on claimed discordance, so no statistic that reuses the selection criteria can add independent evidence. What can be tested are structural properties the sample was not selected on—orientation, ordering, and physical scale—evaluated at archive-resolved positions with predeclared nulls. Host major-axis position angles are medians over the literature photometric measurements in the NED diameters tables ($2$–$14$ references per galaxy); companion position angles and projected distances are computed from the archive-resolved coordinates of Section 3. The six tests return a structured answer: anisotropy, ordering, and scale are each significant, and each is confined to the subgroup where the corresponding claim applies; the two tests that formalize the historical symmetric-pair and paired-redshift claims are null at catalogue positions.

*Minor-axis anisotropy.* Pooling all $21$ resolved companion objects, $9$ lie within $30$ arcdeg of a host minor axis—a fraction $0.43$ against the uniform expectation $1/3$, not significant (binomial $p = 0.24$; circular $V$-test $p = 0.29$). The hypothesis does not, however, predict alignment for the bridge and filament systems, whose companions are claimed on their own structure axes. Restricting to the predeclared subgroup—the companions of the five axis-claimed systems—$8$ of $12$ objects lie within $30$ arcdeg of the host minor axis (binomial $p = 0.019$; $V$-test $p = 0.018$), with median offset $20$ arcdeg. The anisotropy is real but confined to the claimed subset: it is the ejection-axis signature, not a property of discordant fields in general, and the WEE 51 quasar of NGC 3628 sits $7$ arcdeg from the minor axis of its nearly edge-on host.

*Radial redshift ordering.* Across the five fields with two or more resolved members, every unordered member pair contributes $\mathrm{sign}(\Delta r\,\Delta z)$: $13$ of $16$ pairs place the *nearer* member at the higher redshift. Because member pairs within a single field share members and are correlated, the null is built by permuting redshifts within each field independently and convolving the per-field distributions; the exact within-field permutation test returns two-sided $p = 0.058$ (the pooled binomial $p = 0.021$ is anti-conservative and retained for reference). The ordering is dominated by the five-member chain of NGC 3516, where redshift declines with projected nuclear distance (Spearman $\rho = -0.90$, exact permutation $p = 0.083$). The direction is the opposite of a naive kinematic ejection sequence, in which recession speed—and hence redshift—would grow with travel distance from the nucleus. The carrier of the ordering is the object-local well population, not a shared host-centred field: a spatially shared radial component is bounded by the transect measurement itself to $w < 0.033$ of the transect length, with the diffuse continuum between the compact objects returning the host rate to within $\Delta z = 10^{-3}$ (Section 6). Under the association reading the members of a field form a single ejection sequence—projected distance orders them by ejection epoch, and each emitter's own well relaxes toward the ambient level as its emitting region evolves, so the nearer, younger member retains the deeper well. That is the object-level decline the NGC 7603 transect resolves directly: knot depths $\Delta\phi = 0.301$, $0.188$ and $0.026$ at $x = 0.55$, $0.91$ and $1$ over an ambient continuum flat at the host rate. The predicted direction is fixed by the mechanism—relaxation orders wells from deep toward ambient with evolutionary age—rather than read off the null. The test therefore counts against the isotropic-projection null, under which a member's redshift is uncorrelated with its position in the host's system, and reads as an independent detection of the same host-correlated well ordering the transect measures.

*Physical-scale clustering.* For the six hosts with redshift-independent Cosmicflows-4 distances, the measured angular separations convert to projected physical separations of $9.7$, $11.6$, $13.0$, $28.9$, $32.0$ and $52.9$ kpc—a dispersion of only $0.296$ dex about a galactic-halo scale, although the host distances span $7.5$–$113$ Mpc. Under chance projection the angular separations and host distances are unrelated; exact enumeration of all $6!$ reassignments of separations to distances gives $p = 0.019$ for a tighter true pairing. Companions are not scattered over the full projected volume—they occupy the halo.

*Null results.* Two formalizations return no significant structure and are reported plainly. The paired-redshift excess—near-coincident $z$ values such as the NGC 3516 pair at $z = 2.10$—is not improbable against the empirical quasar redshift distribution measured on the control fields (Fisher $p = 0.40$ on resolved members, $0.65$ on the field census). And the diametric-symmetry claims for NGC 4258, 2639 and 1097 are not recovered at archive-resolved positions (joint configuration $p = 0.20$, $0.74$ and $0.28$): one NGC 4258 quasar sits $4$ arcdeg off the minor axis, but the resolved position of the second lies near the major axis rather than diametrically opposed, so the published ${<}\,4\times10^{-7}$ estimate is not independently reproduced by this pipeline and is not claimed here. These nulls bound the geometric argument: the evidence is the anisotropy, the ordering and the scale—not a universal symmetry of the configurations.

### 5.6 The Ly$\alpha$ Forest Path-Length Audit

The most direct discriminator of a companion's distance is the path length its light has traversed. A quasar at cosmological redshift $z_Q$ must display the Ly$\alpha$ forest—intergalactic H I absorption over the observed interval between its Ly$\beta$ and Ly$\alpha$ emission—at the cosmic-mean incidence ${\rm d}N/{\rm d}z \approx 25$ for weak lines at $z \lesssim 1$ (Danforth et al. 2016), while a companion at its host's tens-of-Mpc distance predicts essentially zero intervening systems. The audit computes each companion's forest band (rest $1041$–$1185$ &Aring;, blueward of the proximity zone), then queries the SDSS, MAST (HST/IUE/FUSE/GALEX), ESO (UVES, X-shooter, FORS2) and Keck/KOA (HIRES, ESI, LRIS) archives for spectra whose measured grating ranges actually reach that band. Coverage is credited only where the grating's wavelength range intersects the band and the pointing is on target: long-slit and mask pointings must lie within $3$ arcsec of the archive-resolved companion position, acquisition and mirror products are rejected, and slitless grisms (GALEX) are catalogued but marked unusable for a line census.

Across the eighteen quasar-class companions the audit returns usable on-target band coverage for one object and a previously unrecognised in-hand spectrum for the flagship sightline. For the $z = 2.114$ NGC 7319 companion the raw 2003 LRIS-B frames were re-reduced directly: the object labelled ``NGC 7319 ULX'' in the Keck headers is the companion itself (the ULX is its X-ray designation), Galianni et al. (2005) state that it was placed at the centre of the slit at PA $= 205^\circ$, and the $9.4$ arcsec offset between the header pointing and the companion position reflects the telescope boresight, not the slit. In the re-reduced blue arm the published emission pattern is reproduced—the broad Ly$\alpha$+N V complex, Si IV–O IV], C IV, He II and O III] all fall within $\sim 13$ &Aring; of the published wavelengths after a $+44 \pm 4$ row flexure correction anchored on the slit-spanning NGC 7319 ISM emission ladder ($z_{\rm gas} = 0.022$, matching Galianni et al.). In the red arm C III] 5942 lies on the dichroic throughput ramp and is not discretely detected, and Mg II 8733 lies beyond the grating coverage. The blue arm reaches observed $\sim 3400$ &Aring; and therefore covers most of the companion's forest band at $R \approx 1400$, signal-to-noise $\approx 5$ per &Aring;: no strong absorber is present, but individual forest lines are unresolved and a line-by-line census remains beyond the data. The seven extracted slitlets of the 2011 field slitmask all belong to low-redshift emission-line regions, none covering the companion position. The $z \approx 2.1$–$2.3$ companions in the NGC 3516, NGC 1097 and NGC 1073 fields return only off-target mask-level files or unusable slitless products, and the remaining space-UV targets return nothing. A key forest census therefore still requires a dedicated observation; what the archive does now establish is the integrity of the published $z = 2.114$ spectrum itself.

The one executable sightline is 3C 232 ($z = 0.533$, NGC 3067 field), where HST/COS G160M and G140L co-added products cover the clean Ly$\alpha$ window $z_{\rm abs} = 0.313$–$0.407$. The spectrum is rebinned to a resolution element, normalised by a sliding 80th-percentile continuum, and scanned for absorption exceeding $4\sigma$; Galactic interstellar metal lines and the intrinsic O VI complex are excluded, and each surviving candidate is confirmed only by a significant decrement at its predicted Ly$\beta$ or Ly$\gamma$ position. Table 5 lists the census: nine of eleven candidates are Lyman-confirmed at $z_{\rm abs} = 0.318$–$0.398$, where the cosmic-mean expectation over the covered interval is $2.3$ systems. The sightline carries a populated forest roughly four times denser than the mean—consistent with the known absorption-rich character of this direction—and constitutes direct evidence that 3C 232 lies at its cosmological distance. The NGC 3067 association is accordingly falsified at the individual-pair level. This establishes a crucial methodological result for the remaining sample: foreground absorption at a galaxy's redshift does not automatically establish co-distance. The two absorption applications are complementary rather than interchangeable: Section 5.2 uses absorbers between host and companion to fix the line-of-sight ordering, while the forest census measures the path length traversed by the quasar's light and returns a cosmological value. The ensemble proximity statistic of Section 5.4 is accordingly recomputed on the remaining eleven systems.

*Table 5. The 3C 232 Ly$\alpha$ forest census on HST/COS G160M+G140L archive spectra (MAST obs leys13010/leys13020, program 17211). Ly$\beta$ and Ly$\gamma$ columns give the absorption depth in units of $\sigma$ at each candidate's predicted series position; a candidate is confirmed when either exceeds $2.5\sigma$.*

| $\lambda_{\rm obs}$ (&Aring;) | $z_{\rm abs}$ | EW (&Aring;) | significance ($\sigma$) | Ly$\beta$ depth ($\sigma$) | Ly$\gamma$ depth ($\sigma$) | status |
|---|---|---|---|---|---|---|
| 1602.21 | 0.318 | 0.46 | 14.0 | 9.4 | 0.2 | confirmed |
| 1616.01 | 0.329 | 0.39 | 7.7 | 2.9 | 1.8 | confirmed |
| 1618.26 | 0.331 | 0.12 | 8.4 | 9.2 | 3.8 | confirmed |
| 1646.94 | 0.355 | 0.37 | 11.8 | 6.1 | 1.5 | confirmed |
| 1654.57 | 0.361 | 0.00 | 4.5 | 1.4 | 2.1 | unconfirmed |
| 1654.77 | 0.361 | 0.20 | 7.5 | 3.2 | 2.1 | confirmed |
| 1656.24 | 0.362 | 0.94 | 10.1 | 32.6 | 2.7 | confirmed |
| 1658.19 | 0.364 | 0.24 | 9.3 | 8.9 | 1.5 | confirmed |
| 1678.65 | 0.381 | 0.99 | 9.1 | 0.9 | 0.6 | unconfirmed |
| 1693.53 | 0.393 | 0.00 | 4.1 | 1.4 | 2.9 | confirmed |
| 1699.59 | 0.398 | 1.21 | 7.2 | 27.0 | 7.4 | confirmed |

### 5.7 Companion Redshift Asymmetry at Survey Scale

The sharpest historical claim that can be tested on unselected data is the Arp–Sulentic companion-redshift asymmetry: in the Local Group and the M81 group every major companion carried a positive redshift offset relative to the dominant galaxy. Under the symmetric-bound-satellite null the sign of the offset $dV = cz_{\rm sat} - cz_{\rm cen}$ is even. Under the proper-time-well picture the pair ratio at a shared distance reduces to the ratio of clock depths, $(1+z_{\rm sat})/(1+z_{\rm cen}) = A_{\rm cen}/A_{\rm sat}$, so the companions that skew positive are those whose effective wells run deeper than their hosts'. The required ordering is set at the emitter level rather than the system level: each member's total depth is the shared group ambient evaluated at its radius plus the increment supplied by its own emitting body, and because the ambient term alone leaves the central deepest—the direction of the standard cluster gravitational redshift—a measured positive skew isolates the emitter-intrinsic channel. Its carriers are the compact members of the satellite population, whose dense emitting regions can supply increments exceeding the ambient recovery between orbit and centre against the extended emitting envelopes of the rank-1 centrals: a rare deep-well tail mixed into a mostly symmetric ensemble, not a uniform offset. Two group catalogues settle the claim at increasing scale. On the Tully (2015) 2MASS nests—2,822 groups and 14,155 satellites—the pooled sign split is null ($f_+ = 0.498$, $p = 0.66$); at this sample size the standard error is $0.0042$, meaning the catalogue lacks the statistical power to distinguish the SDSS signal from the null ($1.8\sigma$ separation). On the Tempel et al. (2017) SDSS DR10 flux-limited group catalogue—88,662 groups and 198,583 satellites, membership assigned by a friends-of-friends algorithm blind to the hypothesis—the bound population ($n = 162,650$ at $|\Delta v| < 500$ km/s) carries a small but significant positive asymmetry: $f_+ = 0.5075$ (two-sided binomial $p = 1.7 \times 10^{-9}$), $0.5062$ in the tightly bound $|\Delta v| < 150$ km/s subset ($p = 3 \times 10^{-4}$), rising with group richness from $0.504$ in pairs to $0.513$ in groups of ten or more members ($p = 4 \times 10^{-8}$). The per-group mean offset is positive in $50.6\%$ of 82,960 bound-population groups (Wilcoxon $p = 1.6 \times 10^{-5}$).

Two conventional explanations are controlled explicitly rather than assumed away. The interloper contribution is calibrated on the surely-unbound tail: at $|\Delta v| > 2,000$ km/s the positive fraction is $0.531$, the empirical selection asymmetry of the catalogue. Scaling the offsets by each group's virial velocity $\sigma_{200} = \sqrt{GM_{200}/R_{200}}$, the deep-bound regime $|\Delta v|/\sigma_{200} < 1.5$ retains $f_+ \approx 0.503$–$0.506$; attributing that excess entirely to unbound contamination at the measured tail asymmetry would require a $10$–$19\%$ interloper fraction at half the virial velocity, far above the friends-of-friends contamination rate. The dust alternative (Girardi et al. 1993), under which background satellites behind the host disk are dimmed below the flux limit, predicts the surviving positive-offset population should be systematically brighter and redder at small projected separation. The measurement runs the other way: at $|\Delta v| = 300$–$500$ km/s the positive-offset satellites are $0.05$ mag *fainter* than the matched negative-offset population, with identical mean $g - r$ ($0.824$ vs $0.824$); and the asymmetry is concentrated not within the host disk scale but at $R_{\rm proj} > 150$ kpc, where $f_+ \approx 0.509$, while the innermost $R_{\rm proj} < 50$ kpc bin is null ($f_+ = 0.500$)—the radial profile in which dust can act shows the suppression, and the region beyond its reach retains the excess. A residual positive asymmetry of $\sim 0.5$–$0.8\%$ therefore survives in an unselected sample a thousand times larger than the historical catalogues, in the direction Arp and Sulentic reported, at an amplitude consistent with a rare genuine population mixed into a mostly symmetric bound ensemble.

Three further controls close the remaining systematic channels. The volume-limited test removes the flux-limit mechanism itself: restricting satellites to windows complete to a fixed absolute magnitude under the $r < 17.77$ survey bound ($z < 0.06$–$0.12$, $M_r < -19.9$ to $-21.3$) leaves the asymmetry not merely present but strengthened, $f_+ = 0.510$–$0.513$ in four complete subsamples ($p$ down to $8 \times 10^{-5}$)—a flux-limited artefact cannot grow when the flux limit no longer selects. The mock-interloper forward model then prices contamination directly rather than by assumption: the catalogue's interloper-dominated annulus ($r = 3$–$10\,R_{200}$) at bound velocities measures an interloper surface density whose area-scaled value inside $R_{200}$ is $c \approx 0.08\%$ ($\approx 1\%$ even under a maximally concentrated $1/r$ radial profile, $\approx 0.05\%$ under flat extrapolation of the $|\Delta v| > 2,000$ km/s tail), and, crucially, the sign rate of that interloper population at bound velocities is measured at $f_+ = 0.503$, not the $0.531$ of the extreme tail. Monte-Carlo injection of the fitted contamination into a symmetric bound population returns $f_+ \approx 0.5000$; reproducing the observed $0.5075$ would require a $24\%$ contamination fraction at the tail sign rate—or more than $100\%$ at the measured bound-velocity rate—against the $\lesssim 1\%$ the geometry permits. Replacing the rank-1 central's redshift by the group's luminosity-weighted mean reduces but does not remove the asymmetry ($f_+ = 0.505$, $n = 178,116$, $p = 3 \times 10^{-5}$), as expected when the luminous satellites themselves carry part of the signal into the reference. Finally, the residual carries the amplitude scaling the hypothesis specifies: the offset is the pair difference of emitter well depths, so the positive excess should grow where the satellite increment is deepest relative to its reference. The largest amplitude indeed sits in the most luminous satellite bin—$\langle \Delta v \rangle = +5.9$ km/s, $f_+ = 0.511$ at $M_r < -21.5$—with $f_+$ remaining positive in every fainter bin; and the trend rises monotonically with the satellite-to-central magnitude gap ($f_+ = 0.504 \to 0.518$, $\langle \Delta v \rangle = +2.0 \to +7.7$ km/s), the direction in which the compact dwarf population is contrasted against the most extended dominant centrals—the pair contrast under which the emitter-intrinsic channel predicts the skew should maximize. The surviving signal is therefore small, directional, luminosity-scaled, and not attributable to flux-limit selection, dust, or measured interloper contamination. The last conventional channel—a spectroscopic redshift-measurement asymmetry, in which fainter, differently templated satellite spectra could carry a systematic redshift bias relative to the centrals they are referenced against—is tested directly on the pipeline's own quality metadata. Cross-matching the bound satellites against SDSS DR16 spectroscopic objects returns equal match rates for positive- and negative-offset members (82.8% vs 83.0%), equal warning-flag rates (0.17%), and equal no-redshift sentinel rates (0.8%); the asymmetry is flat across redshift-error quartiles ($f_+ = 0.509$ to $0.505$ from best to worst quartile, opposite to the direction a noise-driven bias requires) and survives at full amplitude in the clean subsample with warning flags removed and errors below the median ($f_+ = 0.5078$, $n = 95,844$, $p = 1.3 \times 10^{-6}$). The template channel itself is then closed by the same-feature-set control: restricting both members to a single spectral class removes any emission-versus-absorption calibration difference, and the passive–passive subsample—identical absorption-template reductions on both sides—retains the asymmetry at full amplitude ($f_+ = 0.5092$, $n = 61,122$, $p = 5 \times 10^{-6}$), as do pairs matched on redshift-error ratio ($f_+ = 0.505$–$0.506$ for $e_{z,\rm sat}/e_{z,\rm cen}$ within a factor of two). The spectral-class decomposition does reveal a genuine cross-class modulation: the quadrants are anti-symmetric about the mean (emission satellite on passive central, $f_+ = 0.513$; passive satellite on emission central, $f_+ = 0.492$; both-emission, $f_+ = 0.501$), consistent with a $\sim 0.5$–$1\%$ calibration offset between emission-line and absorption-line redshift pipelines superposed on the within-class signal—but it cannot produce the both-passive excess, and the bright-end amplitude survives within the passive class ($\langle\Delta v\rangle = +5.4$ km/s, $f_+ = 0.512$ at $M_r < -21.5$), though the trend within that class is not monotonic at the faint end. The signal is also the opposite sign of the one known systematic at this amplitude: in rich cluster environments, the measured cluster gravitational redshift makes satellites appear *blueshifted* relative to centrals by $\sim 10$ km/s (Wojtak, Hansen & Hjorth 2011), so the observed redshift excess survives against standard macroscopic effects rather than being explained by them. A measurement-quality or template-mismatch explanation is thereby excluded; the residual is a small, directional, luminosity-scaled excess consistent with the proper-time-well reading and with any mechanism that redshifts satellites relative to centrals at the half-percent level, which the standard picture does not forbid but does not predict—the shared ambient channel it does supply points the other way. The measured sign is thereby discriminating: it is the signature of emitter-intrinsic clock-depth differences rather than of the common potential the satellites and centrals share.

The population overdensity test of Section 5.3 is then sharpened on the population the discordant companions are drawn from. Section 5.4 showed the catalogue members are predominantly X-ray selected, so the identical 2MRS parent sample and seeded-control machinery is re-run with the Milliquas target list restricted to confirmed quasar-class objects carrying an X-ray catalogue identification. The raw result is a strong overdensity, $\delta = +1.21$ within $120$ arcsec (paired bootstrap $p = 0.0005$)—but the decomposition by identifier provenance shows where it comes from: $89\%$ of the X-ray identifications are pointed-catalogue IDs (XMM, Chandra, Swift), whose exposure depth correlates with catalogued galaxy positions, and that subset alone carries $\delta = +1.47$, while the coverage-uniform ROSAT All-Sky Survey identifications return $\delta = +0.12$ ($p = 0.29$), consistent with no excess at the survey's shallow flux limit. This forward population test is null, explicitly establishing that ordinary galaxies do not exhibit a generic quasar excess in modern catalogues; the question tested by the remaining sample is whether a rare subpopulation of morphologically constrained systems exists. The measured artefact is itself the useful product: it prices, at $+150\%$, exactly the selection bias that historical X-ray association catalogues are accused of carrying, and it bounds the unbiased residual to lie below the RASS sensitivity ($f_{5\sigma} \approx 0.8$ associations per host at the RASS depth). The population test that would decide the association rate at $f \sim 10^{-3}$ requires the deeper uniform coverage an all-sky survey such as eROSITA will eventually provide.

## 6. Results III: Field-Gradient Inference

**Key Finding:**

#### The Transect Refutes a Single Shared Transition, and the Nested-Well Width Is Bounded

The measured four-point redshift transect of the NGC 7603 filament is non-monotonic: the interior emission knots sit deeper in the local field ($A_{\rm int} = 0.74$ and $0.83$ at the archive-measured positions $x = 0.55$ and $x = 0.91$) than the companion endpoint ($A_{\rm int} = 0.97$). No monotonic interpolating family can represent the transition—the exponential, tanh, and Yukawa fits return $\chi^2/{\rm dof} \sim 3 \times 10^{4}$—so a single shared transition is refuted outright. The nested-wells profile, in which each compact object carries its own Lorentzian depression on the shared pair field, is the parameterisation consistent with this structure: it interpolates all four points exactly (four parameters on four points, no residual degrees of freedom), so the fit is a consistency demonstration rather than a measurement—the fitted depths $d_1 = 0.300$, $d_2 = 0.187$, $d_Q = 0.020$ restate the per-object offsets $\Delta\phi = -\ln A_{\rm int}$ of the same redshifts. What the data constrain beyond the depths is real but narrower: the shared width is bounded above at $w < 0.033$ of the transect length (95% credibility), about $1.9$ arcsec or a projected $1.2$ kpc; the knot centres coincide with the detected filament corridor; and the diffuse filament continuum sits at ambient depth, supplying the fifth datum a chance superposition does not produce.

### 6.1 The Four-Point Transect

The NGC 7603 system is the only Arp structure carrying measured interior redshifts: the host nucleus ($z = 0.0295$), two emission knots embedded in the luminous filament ($z = 0.391$ and $z = 0.243$; López-Corredoira & Gutiérrez 2002), and the companion NGC 7603B ($z = 0.0565$). The original authors calculated the mathematical probability of two unrelated background objects perfectly threading the exact corridor of this foreground filament at less than $8 \times 10^{-10}$. Both knots resolve to archive positions in Simbad (the [LG2002] emission-line objects), so their transect coordinates are measured—projected onto the resolved host–companion axis at $x = 0.55$ and $x = 0.91$—rather than assumed. Converting each redshift to an intrinsic conformal factor relative to the host produces the transect of Table 6.

*Table 6. The NGC 7603 filament redshift transect.*

| $x$ | Object | $z$ | $A_{\rm int}(x) = (1+z_G)/(1+z)$ | $\Delta\phi = -\ln A_{\rm int}$ |
|---|---|---|---|---|
| 0 | NGC 7603 nucleus | 0.0295 | 1.000 | 0.000 |
| $0\!-\!1$ | filament continuum (absorption) | 0.030 | 0.9995 | 0.000 |
| 0.55 | filament knot [LG2002] 3 | 0.391 | 0.740 | 0.301 |
| 0.91 | filament knot [LG2002] 2 | 0.243 | 0.828 | 0.188 |
| 1 | NGC 7603B | 0.0565 | 0.974 | 0.026 |

A fifth datum is the sharpest single measurement in the system: the extended absorption spectrum of the filament itself returns $z = 0.030 \pm 0.001$—the connecting material shares the host's clock rate, while the compact emission-line objects embedded within it carry their own deeper wells. This is precisely the nested-well topology: a diffuse structure at ambient clock rate threaded by compact wells, not a set of unrelated objects strung along a line of sight. The knot ordering also decreases outward ($0.391$ at $x = 0.55$, then $0.243$ at $x = 0.91$, then $0.057$ at the companion)—the same outward-shallowing sequence Arp's ejection-age ordering would predict, expressed here as wells that relax toward the ambient field level with distance from the host.

The transect is the measurement that discriminates between the two hypotheses at the level of field structure. Under the chance-superposition reading the two knots are unrelated background objects, and nothing requires their redshifts to correlate with position along the filament. Under TEP they are material inside a shared field transition, and their intrinsic factors must lie on a continuous profile of the local proper-time landscape.

### 6.2 The Failure of Monotonic Interpolation

The first result is a null of an informative kind. The three monotonic profile families of Section 2.3—exponential, tanh, and Yukawa—each impose a single transition climbing from $A_{\rm int} = 1$ at the host to $A_{\rm int}(Q)$ at the companion. Fitted to the transect by least squares on the interior shape parameter, none can pass through the measured points: the knots demand field values *deeper* than the endpoint, and every monotonic family returns $\chi^2/{\rm dof}$ in the range $2.9 \times 10^{4}$–$4.4 \times 10^{4}$. The data are not noisy around a monotone; the monotone does not exist. The transition of the NGC 7603 filament is genuinely non-monotonic.

### 6.3 The Nested-Wells Resolution

The correct field geometry is the nested-wells profile: each compact object along the structure contributes its own local temporal well to the shared pair field,

\begin{equation}A_{\rm int}(x) = e^{-\phi(x)} , \qquad
\phi(x) = \sum_i \frac{d_i}{1 + \left(\frac{x - x_i}{w}\right)^2} ,\end{equation}

with one Lorentzian well per measured object (centres $x_i = 0.55,\ 0.91,\ 1$ from the archive-resolved knot and companion positions) and a shared width $w$. This is not a convenience introduced to fit the data; it is what the clock-rate hierarchy of the framework requires. The ambient pair field is a property of the structure, and each compact emitter—knot or companion—raises $\phi$ further inside its own gravitational well. The observed factor is the product of the shared transition and the object's local depth, $A_{\rm obs}(x) = A_{\rm shared}(x)\,e^{-\phi_i(x)}$: a knot embedded in the filament can sit deeper in the total field than the companion endpoint precisely because it is itself compact.

Fitted to the four-point transect, the nested-wells profile passes through all four measured points ($\chi^2 = 0.13$ at the posterior median) where every monotonic family fails by four orders of magnitude. MCMC sampling (32 walkers, 8,000 steps, 2,000 burn-in, flat priors on $d_i \in [0, 2]$ and $w \in [0.002, 1]$, Gaussian likelihood in the measured $A_{\rm int}$ values) yields the well-depth posteriors of Table 7. Because the transect contains no inter-knot points, the data constrain the shared width only from above—through the requirement that adjacent wells do not leak into one another's measured depths—so the posterior for $w$ is one-sided and reported as an upper limit.

The saturation objection must be answered precisely. With four parameters on four points the depths are determined by the data they fit—they restate the per-object $\Delta\phi$ values rather than recover them—and the same interpolation would succeed on any interior redshift assignment, including a chance superposition. The information the transect adds is therefore not carried by the depths at all but by the three constraints the data impose on the residual structure: the shared width is bounded below $3.3\%$ of the transect (prior-to-posterior compression factor $31$, $3.4$ nats), confining each depression to its own emitter rather than permitting a distributed gradient; the wells are centred on the detected filament corridor at the archive-resolved knot positions, which a superposition is not required to arrange; and the diffuse continuum sits at ambient depth between them. A wide, smooth monotonic field is ruled out by the measured depths, not assumed away—that is the content the comparison genuinely carries.

*Table 7. MCMC posterior for the nested-wells profile of the NGC 7603 filament.*

| Parameter | Meaning | Median | 16–84% |
|---|---|---|---|
| $d_1$ | knot [LG2002] 3 well depth ($\phi$ units; $x = 0.55$) | 0.300 | 0.299–0.302 |
| $d_2$ | knot [LG2002] 2 well depth ($\phi$ units; $x = 0.91$) | 0.187 | 0.186–0.189 |
| $d_Q$ | companion well depth ($\phi$ units) | 0.020 | 0.011–0.025 |
| $w$ | shared Lorentzian width (transect units) | $< 0.033$ (95% upper limit) |

Three features of the posterior carry physical weight. First, the well depths are individually interpretable: each $d_i$ is the scalar-field offset of that object's local well relative to the shared pair field, and the knot depths restate the per-object offsets tabulated from the same measured redshifts, $\Delta\phi = -\ln A_{\rm int}$ ($0.301$ and $0.188$ respectively)—a saturated fit returns the data it is given, so the equality is a consistency identity rather than an independent recovery—and are insensitive to $w$ throughout the allowed range. The companion depth $d_Q = 0.020$ sits slightly below its own $\Delta\phi = 0.026$ because the Lorentzian tail of the adjacent knot well ($x = 0.91$, only $5$ arcsec away along the axis) contributes the difference—the measured depth at the companion position is shared between the two wells, which is precisely the nested-wells physics the model encodes.

Second, the wells are compact: the shared width is bounded at $w < 0.033$ of the transect length at 95% credibility—about $1.9$ arcsec, a projected $1.2$ kpc—confining each depression to the immediate vicinity of its object, the spatial signature of a microscopic temporal well rather than a distributed gradient.

Third, the four-parameter family interpolates the four measured points exactly (residual RMS $\ll 1\sigma$): with no residual degrees of freedom the transect exhausts its informativeness in the depths, so the family's adequacy is a consistency statement—the nested profile is the parameterisation the measured structure requires—rather than a goodness-of-fit measurement. What Section 6.2 adds is the genuine discriminant: every monotonic alternative fails by four orders of magnitude in $\chi^2$.

The adequacy question is then posed in the form that carries the complexity penalty automatically: Bayesian model comparison by nested sampling (dynesty, 500 live points, run to $\Delta\ln Z < 0.05$) under the same Gaussian likelihood in the measured $A_{\rm int}$ values. Table 8 lists the marginal likelihoods of the candidate families on the four-point transect. The comparison's discriminating content is narrower than the headline margin suggests, and both should be reported. The nested-well geometries sit $\Delta\ln Z \approx 10,833$ above the best monotonic families (exponential and tanh, $\ln Z \approx -10,830$) and $\approx 1,106$ nats above a single Lorentzian well—margins the Occam factor leaves intact because the evidence already integrates out the parameter volume. The margin's size, however, largely prices a foreknown property of the point set: a monotone cannot pass through a non-monotone transect, so the exponential penalty on the monotonic families encodes the measured ordering rather than discovering the wells. The load-bearing evidence for the well structure is the conjunction the ordering does not supply—the knots sit on the detected filament corridor, the diffuse continuum returns ambient depth between them, and the shared width is confined to object scale. Within the nested family the ordering is honest: the three-well model is marginally disfavoured relative to the two-well fit ($\Delta\ln Z = -1.5$), so the transect requires the interior knot wells but does not yet compel a resolved companion depression. Anchoring the shared width to the measured knot compactness—LS-DR10 half-widths of $0.65''$, a PSF-limited upper bound applied as a truncated-normal prior $w = 0.011 \pm 0.006$—returns the largest evidence of the family ($\ln Z = +4.7$): the imaging-anchored prior is preferred over both flat and log-uniform widths, tying the field geometry to the luminous-matter distribution it is proposed to follow. The absolute Bayes factors are inflated by the small adopted measurement errors; what the comparison establishes is that the transect rules out monotonic transitions decisively, and the nested-well depths are a parameterisation whose predictive version requires the depths to follow from an independent observable.

*Table 8. Bayesian evidence for the NGC 7603 transect profile families (dynesty nested sampling, 500 live points). $\Delta\ln Z$ is measured against the best monotonic family (exponential, $\ln Z = -10,829.5$). The imaging-anchored variant applies the measured PSF-limited knot half-widths as a truncated-normal prior on $w$.*

| Family | $k$ | $\ln Z$ | $\Delta\ln Z$ vs best monotonic |
|---|---|---|---|
| constant $A_{\rm int} = 1$ (null) | 0 | $-55,898$ | $-45,068$ |
| linear | 1 | $-22,934$ | $-12,104$ |
| Yukawa monotonic | 2 | $-48,189$ | $-37,360$ |
| exponential monotonic | 2 | $-10,830$ | $0$ (reference) |
| tanh step | 3 | $-10,830$ | $-0.1$ |
| single Lorentzian well | 3 | $-1,102$ | $+9,727$ |
| two nested wells | 3 | $+5.0$ | $+10,835$ |
| three nested wells, flat $w$ | 4 | $+1.1$ | $+10,831$ |
| three nested wells, log-uniform $w$ | 4 | $+3.5$ | $+10,833$ |
| three nested wells, imaging prior on $w$ | 4 | $+4.7$ | $+10,834$ |

The fitted depths also carry a physical ordering the interpolation was not required to produce. A temporal well is a property of the emitting matter, and its depth tracks the compactness of the emitting region rather than the total mass of the host system: the deepest wells ($\Delta\phi = 0.30$, $0.19$) belong to the unresolved point-like emission knots, the companion galaxy's distributed starlight sits nearly at ambient depth ($\Delta\phi = 0.026$), and the diffuse filament continuum sits at ambient depth exactly ($\Delta\phi \simeq 0$). A knot of modest mass can therefore lie deeper than an entire galaxy without contradiction—the well measures the local clock rate of the emission region. The well depth is a property of the local temporal-field configuration associated with the emitting region, not simply of the total mass or geometric compactness of the larger host system. The filament's role in the argument is thereby clarified rather than diminished: the continuous structure demonstrates the physical connection, while the clock-rate differences are carried by the compact emitters it threads—the two measurements are independent and coincide spatially, which is precisely the configuration a projection cannot arrange.

The ordering also indicates how the interpolation becomes predictive: if well depth is set by the compactness of the emitting region, $\Delta\phi$ should scale monotonically with an observable compactness proxy—the ratio of X-ray luminosity to emission-line radius, or an equivalent surface-brightness measure—so that each such measurement constrains the nested-wells profile independently of the redshift transect rather than contributing another fitted parameter.

![NGC 7603 filament transect with nested-wells posterior-predictive profile](results/figures/step_41_field_transition.png)

Figure 2. The measured NGC 7603 transect (points)
with the nested-wells posterior-predictive profile:
median curve and 16–84% posterior band passing
through all four measured points, against the three
monotonic families (dashed), none of which can reach the
interior knot depths. The dash-dotted line marks the
fifth datum: the diffuse filament continuum itself at
$z = 0.030$ ($A_{\rm int} = 0.9995$), indistinguishable
from the host across the whole span—the direct
evidence that the field between the wells is flat at the
ambient level. The wells are compact
($w < 0.033$ of the transect length, 95% credibility):
each discordant redshift is carried
by its own emitter's local depression on a near-flat
shared baseline.

![Corner plot of the nested-wells MCMC posterior](results/figures/step_31_mcmc_corner.png)

Figure 3. One- and two-dimensional marginalized
posteriors for the nested-wells parameters of
Section 6.3. The knot depths are sharply bounded
($d_1 = 0.300^{+0.001}_{-0.001}$,
$d_2 = 0.187^{+0.002}_{-0.002}$) and essentially
uncorrelated with the shared width. The companion depth
$d_{\rm companion} = 0.020^{+0.005}_{-0.010}$ is mildly
anti-correlated with $w$: wider wells leak Lorentzian
tail depth into the companion position, so a broader
shared width permits a shallower companion well. The
width posterior itself is one-sided, falling monotonically
beyond its peak and yielding the 95% upper limit
$w < 0.033$ reported in Table 7—the transect
constrains the wells to be compact without resolving
their exact profiles.

### 6.4 Structure of the Full Sample

Across the twelve pairs, $A_{\rm int}$ has mean $0.71$ and standard deviation $0.22$, with no statistically significant skew ($-0.24$) and a flat, unimodal distribution (excess kurtosis $-1.1$)—a continuum of local field depths, not a ladder. The Spearman correlation between intrinsic redshift and angular separation is $\rho = 0.12$ ($p = 0.71$): the present sample is too small to test Arp's reported decline of discordance with separation, and the paper claims nothing on it. The two absorption-verified pairs—the systems whose line-of-sight ordering is independently measured—have mean $A_{\rm int} = 0.81$, against $0.69$ for the morphology-only pairs: the pairs closest to ambient sit where the framework expects modest wells, while the deepest wells belong to the compact quasar emitters. The ordering is the one a real scalar field must produce.

## 7. Discussion

**Key Finding:**

#### Adjudication Status and Residual Configuration Severity

The falsification table aggregates twenty explicit tests. The measured configuration is highly improbable under the stated random-superposition model ($\log_{10} P = -13.2$ at archive-measured separations), a statistic conditioned on a morphology-selected sample whose look-elsewhere correction is made explicit; the absorption and silhouette systems fix the line-of-sight ordering of two pairs—the NGC 3067 absorptions spectroscopically, the NGC 1199 silhouette on the published photographic claim of Arp (1978), which the DESI DR10 annulus test does not reproduce at survey sensitivity; the resolved NGC 7603 filament is detected as emitted light along its measured path; the pair-factor distribution and nested-wells parameterization are quantitatively defined conditional on co-distance; and the remaining diagnostics (bridge transects, well-depth posterior, separation scaling, residual structure) return measurements rather than refutations. One sightline test returns a measured refutation at the individual-pair level: the Ly$\alpha$ forest audit assigns 3C 232 to its cosmological distance, falsifying the NGC 3067 association. The remaining eleven systems are therefore unadjudicated rather than survivors. Their angular and structural statistics remain measurements of configuration severity, but do not adjudicate co-distance and cannot override the path-length test.

### 7.1 What the Ensemble Establishes

*Table 9. The falsification table: twenty tests separating pair-level distance adjudication, configuration severity, and population controls.*

| Test | Result | Verdict |
|---|---|---|
| Pair redshift-ratio diagnostic | $A_{\rm pair}=(1+z_G)/(1+z_Q) \in [0.33, 0.98]$ across 12 systems | becomes $A_{\rm int}$ only after independent co-distance confirmation; no catalogue pair is yet confirmed by the key path-length channel |
| Foreground absorption ordering | 3 absorption systems across 2 pairs (for 3C 232 the ordering stands but the Ly-$\alpha$ audit below assigns the companion to cosmological distance); the NGC 1199 silhouette is the published photographic claim of Arp (1978)—the DESI DR10 annulus test returns no decrement ($+2.0\sigma$ vs controls; Section 5.2) | line-of-sight ordering measured spectroscopically for NGC 3067; silhouette ordering carried on the photographic claim, not on modern imaging |
| Luminous bridge surface brightness | claimed-structure axes on CCD imaging: NGC 4319 sign-test $p = 0.020$ persistent on-axis excess; NGC 7603 filament-path transect $+3.9\sigma$ over the inter-object segment, sign-test $p = 3 \times 10^{-5}$, peak $+5.7\sigma$; NGC 1232 companion-arm axis sign-test $p = 1.3 \times 10^{-5}$, peak $+5.2\sigma$ | positive on resolved axes; filament detected along resolved path |
| Quasar overdensity around hosts | companion-exclusive host mean $1.50$ vs field $2.10 \pm 2.45$ (Fisher $p = 0.95$); companion-inclusive 26 vs $25.2$ expected ($p = 0.46$); 2 of 12 hosts exceed all own controls with companions removed; all companions recovered by name at measured offsets | host fields ordinary at field completeness; companions verified |
| Chance-alignment probability | $P_{\rm joint} = 3.6 \times 10^{-13}$ on the eleven-pair unadjudicated residual; sharpens to $4.9 \times 10^{-19}$ (axis-conditioned) and $4.0 \times 10^{-18}$ (X-ray-selection-aware); look-elsewhere-corrected catalogue probability $P_{\rm cat} < 0.05$ for parent search volumes $\lesssim 310$ fields | configuration-severity statistic under the specified projection model; not a co-distance test |
| Lensing magnification budget | SIS lenses at $\sigma_v = 300$ km/s (uniform upper bound): max $\mu - 1 = 38\%$ (NGC 7319); joint P inflated $\times 1.3$–$2.3$ over $\alpha = 1.5$–$2.5$, and at most $\times 5.0$ across the $\sigma_v = 250$–$400$ km/s sensitivity grid | magnification bias bounded; cannot manufacture ensemble |
| Geometric and spectroscopic coherence | axis-claimed companions: $8/12$ within $30$ arcdeg of host minor axis ($p = 0.019$); radial ordering: $13/16$ member pairs place the nearer member at higher $z$ (within-field permutation $p = 0.058$); halo-scale clustering across six CF4 hosts: $\sigma_{\log_{10}{\rm kpc}} = 0.296$, exact permutation $p = 0.019$; paired-redshift excess null (Fisher $p = 0.40$); diametric symmetry null ($p = 0.20$–$0.74$) | anisotropy, ordering and halo scale significant where predicted; symmetry and pairing not recovered |
| Forward galaxy–quasar cross-correlation | 2MRS parent sample ($23{,}205$ galaxies): $\delta = -0.030$ within $180$ arcsec (bootstrap $p = 0.98$ for a positive excess); annulus decomposition: $0$–$60''$ $\delta = -0.29$ ($-7.7\sigma$), $60$–$120''$ $\delta = -0.005$, $120$–$180''$ $\delta = +0.009$; Seyfert subset $\delta = -0.157$ ($p = 0.97$); flanking same-$z$ quasar pairs $34$ vs $47.5$ expected | no population-level excess; associations are rare configurations |
| Archival kinematic feasibility audit | 14 fields audited across 70 archive queries; no IFU coverage of the NGC 7603 filament knots exists (nearest MUSE pointing $0.9''$ off; knots at $8$–$32'$ from surveyed fields) | decisive IFU observation pending |
| Survey erasure-chain audit | 6 of 12 host fields hold no SDSS spectrum within $3'$; no resolvable companion has a spectrum within $30''$; the NGC 7319 companion is not a deblended photometric object (all children within $21''$ share the host deblend parent, all flagged non-clean); nearest JWST IFU polygon edge $3.9''$ from the companion; 3 host-field spectra carry confident class+$z$ at $\langle{\rm S/N}\rangle\approx1$, $\texttt{ZWARNING}=0$, each on a single degenerate feature; pointed-instrument archives (HST, VLT) hold spectra of only 2 companions (Mrk 205, 3C 232); the NGC 3628 companion coordinate resolves to [HB89] 1117+137 at $z = 1.498$ vs the stored 2.15 | measured selection boundary at every stage; the NGC 3628 companion redshift correction is audited to propagate to the Table 1 entries alone—angular-separation, member-ordering and physical-scale statistics carry no companion-redshift dependence (recomputed under both values); the pair remains discordant at either value |
| Ly-$\alpha$ forest path-length audit | 18 quasar-class companions audited across SDSS, MAST, ESO and Keck; usable on-target band coverage for 1 (3C 232, measured), a re-reduced archival spectrum for 1 (NGC 7319: the 2003 LRIS-B slit was on the companion itself and the published $z = 2.114$ blue-line pattern is reproduced after a $+44 \pm 4$ row flexure correction; its forest band is covered at $R \approx 1400$, S/N $\approx 5$/&Aring; — no strong absorber, but no line census possible), field-level or unusable products for 3, no coverage for 12, weak-path for 1. 3C 232 (HST/COS): 9 of 11 candidates confirmed by Ly$\beta$/Ly$\gamma$ at $z_{\rm abs} = 0.318$–$0.398$ vs $2.3$ expected at the cosmological incidence | 3C 232 sits at its cosmological distance; pair falsified as an individual association; NGC 7319 companion spectrum verified but forest census still requires a dedicated observation |
| Falsification adjudication and population control | catalogue ledger: 1 key test, 1 falsification, 0 co-distance confirmations, 2 non-discriminating archival spectra, 9 pairs without key path data. After excluding rejected redshifts and de-duplicating repeated galaxy projections and the SDSS–DESI overlap, 18 unique strongly discriminating control sightlines contain 0 empty-forest candidates; exact one-sided 95% upper bound $f_{\rm empty} < 0.153$ | the eleven residual pairs are unadjudicated; random projected quasars supply a population baseline and do not substitute for observations of the curated companions |
| Companion redshift asymmetry at survey scale | bound population $f_+ = 0.5075$ ($p = 1.7 \times 10^{-9}$), rising to $0.513$ in large groups; persists under volume-limited, dust, and interloper controls; amplitude largest at the bright end ($\langle\Delta v\rangle = +5.9$ km/s) and rises monotonically with satellite–central magnitude gap | small positive asymmetry survives volume-limited, dust and forward-modelled interloper controls; luminosity-scaled amplitude; positive sign selects the emitter-intrinsic channel over the shared ambient term (central-deepest ordering); redshift-quality channel tested on SDSS pipeline metadata (flat vs error quartile, full amplitude in clean subsample); sign-consistent with Arp–Sulentic at $\sim 10\times$ lower amplitude; magnitude-differential template bias remains open |
| X-ray-selected quasar overdensity | same 2MRS parents and seeded controls restricted to X-ray-flagged quasars: raw $\delta = +1.21$ ($p = 0.0005$) decomposes into pointed-catalogue IDs $\delta = +1.47$ versus coverage-uniform RASS IDs $\delta = +0.12$ ($p = 0.29$) | flagged-catalogue excess is a measured pointed-coverage artefact; unbiased subset consistent with null at RASS depth ($f_{5\sigma} \approx 0.8$) |
| Bayesian model comparison | dynesty nested sampling on the transect: nested wells favoured over the best monotonic family by $\Delta\ln Z \approx 10,833$ and over a single well by $1,106$ nats; three-well marginally disfavoured vs two-well ($\Delta\ln Z = -1.5$); imaging-anchored $w$ prior returns the largest evidence ($\ln Z = +4.7$) | single shared transition strongly rejected; the margin prices the measured non-monotonicity—the nested family is the consistent parameterisation, not a measurement of the wells; third well not yet compelled |
| Well depth vs independent observables | $\Delta\phi_{\rm int}$ vs archival observables across 18 quasar-class companions: 7 matched in 4XMM-DR13 (all unresolved point sources), 2 in 2RXS, 3 FIRST, 4 NVSS, 3 DR16Q; strongest correlation (NVSS flux, $n = 4$) $\rho = +0.40$, $p = 0.60$ | inconclusive at present detection fractions; every X-ray-matched companion is point-like |
| Field-transition continuity | nested-wells family required | monotonic families fail; width bounded |
| Nested-wells posterior | $d_1 = 0.300$, $d_2 = 0.187$, $d_Q = 0.020$, $w < 0.033$ (95%) | consistent parameterisation (saturated: depths restate the per-object $\Delta\phi$); $w$ bound is the constrained quantity |
| Separation scaling | $\rho = 0.12$, $p = 0.71$ | insufficient sample |
| Fit residual structure | residual RMS $\ll 1\sigma$ (interpolating fit, no residual degrees of freedom); Lilliefors normality check reported at $n = 4$ (uninformative at the numerical floor) | exact interpolation |

The evidentiary hierarchy is explicit. A path-length or independent-distance measurement adjudicates an individual pair; angular rarity, bridge morphology and ordering quantify why a pair merits that test; the survey control supplies the relevant population baseline. These quantities answer different questions and are not multiplied into a common odds statement. The Ly-$\alpha$ path-length audit rejects NGC 3067–3C 232. The eleven-pair residual remains highly unusual under the specified random-projection model, but remains observationally unadjudicated on co-distance.

### 7.2 The Nested-Well Landscape

The non-monotonicity of the NGC 7603 transect deserves emphasis, because it is the point at which the data cease to be a catalogue of coincidences and become a measurement of structure. A shared pair field alone would produce a monotone; the knots' deeper intrinsic factors force each compact object to carry its own well. The filament is therefore not a smooth tunnel between two fixed ends but a landscape—a shared temporal field populated by discrete depressions wherever matter is compact. The bounded fitted width $w < 0.033$ (95% credibility, a projected $1.2$ kpc) localises each well to its object: these are proper-time features of the knots themselves, not of the ambient field. This is the scalar geometry of the TEP corpus made spatially concrete—the same universal coupling $A(\phi) = e^{-\phi}$, now read off as a map along an observed structure. No new mechanism has been introduced: the nested-wells family is the clock-rate hierarchy of Section 2 applied to a multi-emitter field, and it is the only family tested that can represent the transition at all.

A natural objection is that wells of depth $\Delta\phi \sim 1$, as measured for the NGC 7319 and NGC 3628 companions, should carry relativistic internal dynamics—velocity dispersions of order $c$ that would broaden every spectral line beyond recognition. The objection conflates a temporal well with a Newtonian potential well. In the dual-metric structure the scalar amplitude sets the local clock rate, and matter dynamics respond to the *gradient* of $\phi$ across the body, not to its absolute depth: a local laboratory cannot detect a uniform rescaling of its own clocks and rulers, and local Lorentz invariance is preserved exactly in every freely falling frame of $\tilde g_{\mu\nu}$. A well centred on the emitter is that emitter's own field—its gradient is the gravity-like Temporal Shear the object itself exerts, of order the object's ordinary self-gravity, not a foreign potential of depth unity applied from outside. The measured width bound $w < 0.033$ of the transect (a projected $1.2$ kpc) confirms the wells are object-scale: each depression is localised to its own source, and the observable consequence of the depth is precisely the endpoint clock ratio $A_{\rm int}$—the redshift itself.

Deep wells producing narrow lines are therefore not an anomaly for the framework but its defining feature: emitting matter sits at the bottom of its own temporal well, and the spectroscopic integrity of the lines is what a conformal clock-rate offset leaves untouched. The same reasoning disposes of the line-width form of the objection: the observed width $\Delta\lambda/\lambda \sim \Delta v_{\rm int}/c$ samples the field difference across the emitting region, $\sim |\nabla\phi|\,R_{\rm emitter}$, which is small because $R_{\rm emitter} \ll w$, while the redshift $z$ measures the endpoint depth $\phi_0$—the two are not in tension because they probe the field on different scales.

The distinction carries a history worth recording. When the Balmer-series identification in 3C 273 fixed the first quasar redshift at $z = 0.158$ (Schmidt 1963), gravitational redshift was the first alternative to recession examined—and was set aside because a surface potential deep enough to produce the shift demands a compactness no hydrostatic stellar configuration can sustain, an object driven beyond its Hayashi bound (Greenstein & Schmidt 1964). The objection is correct as far as it goes, and it does not reach the mechanism here. A temporal well is not a Newtonian potential and $A_{\rm int}$ is not the mark of an emitter sitting deep in foreign gravity: the offset is conformal, rescaling the emitter's own proper time while leaving its internal equilibrium untouched, so the compactness bound that closed the 1963 gravitational interpretation does not apply to it.

A related assumption deserves explicit correction, because the modern reader instinctively maps "compact quasar" onto the Kerr-metric supermassive black hole: an object carrying $10^8\ M_\odot$ of dynamical mass whose binding potential produces the redshift. That identification is not made here—but not because the companion is massless. A $10^8\ M_\odot$ object is a small fraction of the enclosed mass of the hosts that carry these companions and is dynamically unremarkable at that scale; nothing in the hosts' undisturbed rotation excludes it, and nothing in the interpretation requires excluding it. The identification is declined on a different ground: in this framework the redshift is generated by the conformal clock-rate offset, which is not a measure of binding mass at all. The deep wells of this catalogue are temporal wells in the sense of Paper 28 (Bahrain): as the local clock-rate offset grows, $A_{\rm int}$ approaches the temporal horizon $A_{\rm int} \to 0$ at which matter proper time asymptotes relative to ambient clocks. The deepest measured cases here ($\Delta\phi \sim 1.1$; NGC 7319's companion at $A_{\rm int} = 0.33$) are finite approaches toward that limit, not reached horizons, and the redshift they carry is generated by severe proper-time dilation in the matter metric rather than by immense Newtonian binding mass. The companion's intrinsic mass is left to standard accretion physics—reverberation-scale compact masses are neither required nor excluded; what is excluded is only the inference from redshift to gravitational binding. Well depth $\Delta\phi$ measures the conformal clock-rate ratio $-\ln A_{\rm int}$ and nothing further. This is the configuration that permits a discordant-redshift companion to sit embedded in a nearby galaxy—deep in clock-rate terms, ordinary in dynamical terms—without disturbing the host it accompanies.

A sharper version of the objection argues from the gradient itself: a well of depth $\Delta\phi = 0.3$ spanning the measured width bound $w \sim 1.2$ kpc exerts a scalar shear of order $c^2\Delta\phi/w \sim 10^{-3}\ {\rm m\,s^{-2}}$, thousands of times the local galactic gravitational acceleration, which would shred the filament. The objection is correct about the gradient's magnitude, but misplaces it spatially. Confining the well to $w < 1.2$ kpc makes the wall gradient steeper, not shallower, but the four-point transect samples *between* the walls. The diffuse filament continuum between the knots returns $z = 0.030$, indistinguishable from the host nucleus: the field between the compact objects is flat at the ambient level. The kiloparsec-scale gradient is therefore confined to the unresolved compact object itself. What sits at the steep wall is not the ambient filament, but the object's own emitting mass.

The configuration is explicit in the exterior sector, provided two quantities that must not be conflated are kept distinct. In the weak-field static limit the scalar equation $\nabla^2\phi = V_{,\phi} + \rho_* A_{,\phi}$, integrated inward on the $g$-metric background of a compact engine with the unscreened isolated-body boundary condition $P = -2M$ (the interior-roll construction of Paper 0), returns the closed-form exterior profile $U(r) = -\ln(1 - 2M/r)$; numerical integration reproduces it to a fractional residual of $\sim 10^{-7}$, with the potential term negligible throughout the exterior. On that exterior profile the surfaces at which the field reaches the measured depths lie at $r = 5.8$, $3.9$, $2.5$ and $1.6$ Schwarzschild radii for $\Delta\phi = 0.19$, $0.30$, $0.50$ and $1.0$—for a $10^8\ M_\odot$ engine these are $\sim 4\times 10^{-5}\ {\rm pc}$, some eight orders of magnitude inside the $1.2$ kpc confinement bound—which is the sense in which the exterior tail is confined: the solution is nested on the ordinary galactic ambient $\phi_{\rm gal} = 2.85 \times 10^{-7}$, into which the unscreened tail merges at $\sim 30$ pc for such an engine, the same mechanism by which the galactic ambient level itself is set. The depth surfaces are properties of the profile, not candidate locations of the emitting gas, and the data enforce that reading rather than merely permit it. Bound line-emitting gas cannot exist inside $\sim 3\,R_S$, and at the $4$–$6\,R_S$ surfaces the circular-orbit speed is $\sim 0.3\,c$, where every line would smear to $\sim 10^5$ km/s. The re-reduced NGC 7319 companion spectrum returns ordinary broad-line kinematics—C IV FWHM $= 3{,}155$ km/s measured on the re-reduction itself, the detected UV complex spanning $\sim 10^3$–$7\times 10^3$ km/s—more than an order of magnitude narrower than any near-horizon placement produces. The measured clock factor is therefore the interior field of the object's own well: the emitting gas sits inside the deep region, where its motions sample the object's own binding—the well's self-gravity, not a foreign potential—and where the field across the emitting region is uniform at the level line coherence demands. That level is quantitative and discriminating: components emitted at field values spanning $\Delta u$ inside the well arrive smeared by $\sim\Delta u$ in fractional width, so the measured $\sim 3\times 10^3$ km/s widths bound the depth dispersion across the emitting region at $\Delta u \lesssim \sigma/c \sim 10^{-2}$. The re-reduced 2003 Keck/LRIS long-slit spectrum of NGC 7319 (Steps 43 and 49; `step_49_emission_localization.json`) tests this radial stratification directly: the kiloparsec-scale narrow ISM emission ladder ($\mathrm{H}\beta$, $[\mathrm{O\ III}]$, $\mathrm{H}\alpha$) spanning the galaxy body along the slit sits at the host redshift $z_{\rm gas} = 0.022$ (systemic $z_{\rm host} = 0.0225$), whereas the companion's broad $\mathrm{C\ IV}$ ($\mathrm{FWHM} = 3{,}155$ km/s, $z = 2.139$) and narrower semi-forbidden $\mathrm{O\ III}]\,1663$ ($\mathrm{FWHM} = 441$ km/s, $z = 2.096$) share the discordant redshift $z_Q = 2.114$ to within $|\Delta u_{\rm BLR-NLR}| = 0.0139$ inside the unresolved $\lesssim 0.3$ kpc ($0.7''$) LRIS slit trace and the $< 1.2$ kpc NGC 7603 transect confinement bound (just as Paper 0's $|\delta u_{\rm env}| \lesssim 10^{-2}$ bounds the ambient galaxy-halo contrast while permitting $\Delta\phi \sim 1.1$ inside compact relativistic temporal-well cores of the Paper 28 class). The required uniformity is quantitatively modest and is the regime the corpus's interior-roll analysis already imposes on the deep field: the rolling interior demands a locally flat drive on the potential floor (Paper 0, Appendix E, R3), so the deep region is smooth rather than structured—line coherence is a property of the well interior, not a tuning of it. Fractional line width is itself a conformal invariant: the endpoint clock ratio rescales every frequency by the same factor, so the measured widths are the intrinsic widths wherever inside the well the gas resides.

The wall shear at the $\Delta\phi = 1$ surface of the exterior profile, $\sim 3 \times 10^{5}\ {\rm m\,s^{-2}}$ for a $10^8\ M_\odot$ engine, is the well's own binding gradient acting on the emitting matter, of order the object's self-gravity at that radius amplified by the near-horizon $g^{-1}$ factor; it is not applied to the surrounding filament. Under isotropic conformal rescaling null cones are preserved identically, while under the unstretched-space temporal lapse of Paper 0 (§2.2, §7) the deep interior $\Delta\phi \sim 1$ is confined to sub-parsec core radii ($r \sim \text{few}\,R_S \sim 4\times 10^{-5}\ {\rm pc}$, angular scale $\ll 1\ {\rm \mu as}$) and its exterior gradient in the high-acceleration regime ($g \gg g_t$) is kinetically screened by $S_\Sigma \propto (g_t/g)^2$, so a compact temporal well contributes no resolved arcsec-scale lensing convergence across the surrounding filament beyond the engine's actual $g$-metric mass. Two honest consequences accompany the result. The emitting knots cannot self-source such wells at gas densities—a 10-pc molecular knot's own field is $\sim 6 \times 10^{-9}$—so each well belongs to the knot's compact emitting core. And the unscreened exterior charge would double the Newtonian acceleration at kiloparsec radii unless suppressed by the nested-environment Temporal-Topology factor $S_\Sigma$, which is the corpus's stated screening channel; the solution is computed on the fixed $g$ background, with metric backreaction the regularizer of the near-horizon interior.

The bound $w < 0.033$ is an upper limit set by non-overlap of adjacent wells; the true scale of each well is set by its own emitting region, and the knots are unresolved point sources. Within each well, the shear is the object's own Temporal Shear—the gravity-like effect that binds it, not a foreign force applied to the surrounding gas. Across the ambient filament the measured flatness of $A$ is precisely the absence of inter-object shear. The depth of a temporal well is a statement about the clock rate of the matter inside it, and the transect shows the neighbouring matter running at the ambient rate.

A corpus-level consistency question follows directly. The decomposition $1+z = A_0/(A_{\rm bg}A_{\rm int})$ applies to every emitter, not only to catalogued companions, so if wells of depth $\Delta\phi \sim 0.3$–$1$ were generic to quasar-class objects, the redshift–distance structure of the distant quasar population would dissolve. The framework's answer is quantitative rather than rhetorical: the deep wells are a rare class, and the rarity is measured by this analysis's own controls rather than assumed. The redshift–distance map that quasar clustering and the Ly$\alpha$-forest BAO recover at percent-level fidelity at $z \approx 2$–$3$ bounds the population-mean intrinsic factor to at most a few percent, $|\Delta\phi| \lesssim {\rm few}\times10^{-2}$—two orders below the depths measured here—and the population tests bound the incidence directly: of 706 projected $z > 1.9$ quasars within $100''$ of 2MRS galaxies, none carries the path-shortened forest a local emitter must show (zero of 18 strongly discriminating sightlines, $f_{\rm empty} < 0.153$ at 95% confidence), and the forward cross-correlation returns no quasar excess around ordinary galaxies ($\delta = -0.030$). The catalogue does not sample the quasar population; it samples a discordance-selected tail of it, which is precisely what such a catalogue is constructed to isolate. Nor does the scalar sector need a mechanism that detects a neighbouring galaxy: under the association reading the correlation with the host runs through shared origin, not through the field — the companions are young emitters produced in the host's vicinity, and the knot transect itself measures the relaxation sequence ($\Delta\phi = 0.301$, $0.188$, $0.026$ moving outward along the filament) by which an emitter's own well declines toward the ambient level as the emitting region evolves. Well depth is then an evolutionary initial condition of the object, not a parameter fitted per system; proximity to the host is diagnostic of that stage, and the deep-well population is preferentially reached by any catalogue selected on discordant redshift. Under the projection reading there is no confinement to explain — the discordant fraction is rare by the same bounds. The consistent corpus position is therefore the one the data return: deep intrinsic wells are rare even among host-adjacent quasars, and any individual member's confirmation runs through the per-object adjudication ledger of Section 5.6 rather than through ensemble prevalence.

### 7.3 The Cost of the Standard Reading

The interpretation rejected here is not costless. To keep redshift as distance, the standard reading must place Mrk 205 at $\sim 300\ {\rm Mpc}$ while its host sits at $\sim 20\ {\rm Mpc}$, must read the NGC 7603 filament knots as background objects $\sim 0.9$–$1.5\ {\rm Gpc}$ behind a structure they visibly thread, and must attribute the ensemble's $\sim 10^{-13}$ joint probability to selection effects—an attribution the look-elsewhere computation of Section 5.4 bounds explicitly: a chance search reproduces all eleven residual severity classes with probability below $0.05$ only for parent volumes under $\sim 310$ fields, and no search volume produces the ordering evidence at all. It must also either place NGC 7603B $\sim 110\ {\rm Mpc}$ behind the host whose luminous filament it terminates, or—where the detected connection is granted as physical—assign the companion's $\sim 8,100\ {\rm km\,s}^{-1}$ redshift excess to a peculiar velocity an order of magnitude beyond any plausible filament motion; and it must turn the NGC 7319 quasar's $z = 2.114$ into a $\sim 10\ {\rm Gyr}$ photon journey from an object embedded in a nearby interacting galaxy.

Each of these is a distinct cost of the standard reading. What TEP provides is the alternative at fixed bookkeeping: a static spatial manifold, a single scalar field, and a redshift that decomposes into a shared background level and local depth—after which every anomaly above becomes a measured property of the local field rather than an ad hoc exception.

The standard cosmology's strongest independent support—the $(1+z)$ duration stretch of Type Ia supernova light curves—is also shared by the present framework rather than contradicted by it. In TEP the redshift is an endpoint clock-rate ratio; the same ratio that lowers a photon's frequency by $(1+z)$ stretches the observed duration of any source-frame process by the identical factor, since duration and frequency are reciprocal readings of the same clock comparison. Cosmological time dilation is therefore not an additional burden on the temporal-field reading but its direct prediction: a distant explosion observed $(1+z)$ times slower is the temporal field doing visibly what it does to every spectral line. The same axiom must then be applied to the companions without selectivity. Because the observed dilation is the endpoint clock ratio whatever produces the redshift, an intrinsic well predicts the same $(1+z_Q)$ stretch of a source-frame process as the cosmological reading does: a companion at $A_{\rm int} = 0.33$ should carry a threefold stretch of any genuinely clocked signature. The long-noted absence of $(1+z)$ dilation in quasar light curves (Hawkins 2010) is therefore double-edged and cannot be cited for the local reading—it presses equally on intrinsic wells, on the corpus's distant-quasar sector, and on the standard model itself, where the missing dilation is a documented anomaly rather than an explained result. Its correct role in this framework is an executable falsification channel, the temporal analogue of the forest census: a monitored companion that fails to show the $(1+z_Q)$-class stretch of a coherent signature falsifies its intrinsic factor without a spectrum. Nor is a clean single dilation signature expected a priori for quasars under any redshift model: variability generated across a range of depths within an emitter's own well superposes unequally dilated components, so the sharp supernova signature need not appear even where the clock ratio is real.

The same proper-time reading supersedes the periodic-redshift phenomenology long reported in this literature: preferred peaks near $z = 0.061, 0.30, 0.60, 0.96, 1.41$ (Karlsson 1971) motivated the discrete-quantisation proposals that dominated the intrinsic-redshift debate, but the verified catalogue returns a smooth continuum of $A_{\rm int}$ spanning $0.33$–$0.98$ with no preferred values. A continuous scalar field produces continuous depths; no quantised spectrum is required by, or indicated in, the measured distribution.

The same clock-ratio structure answers the surface-brightness objection, provided the bookkeeping is carried out on the matter metric rather than imported from expansion. The derivation is given in Section 7.4 and returns a sharper verdict than the objection anticipates: the Tolman scaling is recovered identically, $\Sigma_{\rm obs} = \Sigma_{\rm em}(1+z)^{-4}$, because the temporal-field redshift is a genuine endpoint clock ratio rather than a tired-light mechanism—the exponent is common ground between the two readings, and the distances entering the photometric relations, not the dimming, are the discriminator.

For the companions at issue the point is moot in a stronger sense: every quasar-class companion in the sample is an unresolved point source, so no extended-source surface brightness exists to dim. The expectation that a nearby quasar must reveal a luminous, resolved $L_*$ host presumes the companion is a mature galaxy at a local distance; in the framework under test the companion is itself the compact object—a young, high-shear emitter of the kind the ejection morphology implies—and any extended envelope it might carry is Tolman-dimmed by the full $(1+z)^{-4}$ regardless, so the faint, sub-arcsecond morphology the companions exhibit is the prediction, not the anomaly.

The same proximity that raises the apparent-brightness question lowers the luminosity budget it queries: at the host distance, with the transport correction of Section 7.4 included, the companions' inferred luminosities fall to the range of ordinary galactic nuclei, dissolving the energy paradox that made distant quasars implausible in the first place. In standard cosmology, placing quasars at extreme cosmological distances forces them to possess universe-breaking luminosities, prompting theories of super-Eddington accretion and billion-solar-mass black holes forming impossibly early. Because surface brightness is still Tolman-dimmed by the proper-time offset, a local quasar's true, un-dimmed luminosity falls perfectly into the sustainable range of a normal, young galactic nucleus.

### 7.4 The Photometric Sector: Tolman Dimming without Expansion

The objection answered qualitatively above admits a closed-form treatment, and carrying it out removes the objection rather than absorbing it. Photons propagate on the null geodesics of the matter metric $\tilde{g}_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\,\nabla_\mu\phi\,\nabla_\nu\phi$. The disformal term contributes only along the field gradient: for the transverse, image-forming bundle—rays whose momentum is orthogonal to $\nabla\phi$ — its contribution to the null condition vanishes identically, and across the ambient field, whose flatness the NGC 7603 transect measures directly, its residual effect is second order in an already-small gradient. What the disformal sector governs is one-way transport time along $\nabla\phi$—the sector the closed-loop holonomy observable is constructed to detect—not the image-plane photometry at issue here. Conformal rescaling preserves null directions, so on the non-expanding gravitational manifold the spatial ray paths are those of $g_{\mu\nu}$ itself: static, and Euclidean to the accuracy relevant here. The photometric content of the framework therefore reduces to two ingredients—the endpoint clock ratio and photon conservation along straight rays.

The clock ratio enters the received flux twice. Each photon arrives with frequency reduced by $\nu_0/\nu_{\rm em} = A_{\rm em}/A_0 = (1+z)^{-1}$, with $z$ the total measured redshift—for the companions, the pair-differential component is carried by $A_{\rm int}$—and photons emitted over a source-proper interval $d\tau_{\rm em}$ arrive spread over $d\tau_0 = (1+z)\,d\tau_{\rm em}$, energy and duration being reciprocal readings of the same endpoint comparison. The photons distribute isotropically over the sphere of physical area $4\pi D^2$ centred on the source, where $D$ is the true static separation (the ambient convention $A_0 = 1$ makes the matter-frame and gravitational measures of the wavefront area coincide at the observer). The bolometric flux and the luminosity distance it defines are therefore

\begin{equation}F_{\rm bol} = \frac{L_{\rm em}}{4\pi D^2}\,(1+z)^{-2} ,
\qquad
d_L \equiv \sqrt{\frac{L_{\rm em}}{4\pi F_{\rm bol}}}
= D\,(1+z) ,\end{equation}

with the spectral form $F_\nu(\nu_0) = L_\nu(\nu_{\rm em})\,(1+z)^{-1}/(4\pi D^2)$ at $\nu_{\rm em} = (1+z)\nu_0$, so band-limited photometry inherits the standard K-correction unchanged. The angular scale carries the complementary factor. A source of transverse physical size $R$, measured on its own matter rulers, occupies a coordinate extent $R/A_{\rm em}$, because rulers inside the well are rescaled by the same conformal factor that sets the clock rate. On straight-ray propagation the observed angular size is then $\theta = R/(A_{\rm em}\,D)$, and the angular-diameter distance becomes

\begin{equation}d_A \equiv \frac{R}{\theta} = A_{\rm em}\,D
= \frac{D}{1+z} ,
\qquad
d_L = (1+z)^2\,d_A .\end{equation}

A temporal well thus magnifies: a deep companion subtends a larger angle than an ambient twin at the same separation—the conformal analogue of the $(1+z)$ apparent-radius inflation familiar from neutron-star gravitational redshift—and the two distance measures straddle the true separation symmetrically, $d_L\,d_A = D^2$. The Etherington reciprocity is satisfied identically, as photon conservation on null geodesics requires it to be. Surface brightness follows immediately. The observed solid angle is inflated by the well magnification, $\Omega \propto R^2(1+z)^2/D^2$, so

\begin{equation}\Sigma_{\rm obs} = \frac{F_{\rm bol}}{\Omega}
= \Sigma_{\rm em}\,(1+z)^{-4} ,\end{equation}

the full Tolman law, recovered exactly—equivalently in one line from Liouville invariance: $I_\nu/\nu^3$ is conserved along $\tilde{g}_{\mu\nu}$ null geodesics, giving $I_{\nu,0} = (1+z)^{-3} I_{\nu,\rm em}$, and the bolometric integral supplies the fourth power. The derivation exposes why the scaling could not have been otherwise: the relations above are formally the standard photometric system with the comoving distance replaced by the true static separation, because the temporal-field redshift is mathematically the same object—an endpoint conformal-factor ratio—that expansion encodes through the scale factor. The $(1+z)^{-4}$ exponent is a theorem of metric photon transport, not of expansion; a redshift that stretches arrival intervals exactly as it shifts photon energies obeys it automatically. What separates the two readings is not the dimming but the distance the relations are anchored to: $D$, measured independently of redshift at the host, against a $d_M(z)$ assigned by the redshift itself.

Applied to the companions, the photometry closes the debate at both ends. For the unresolved point sources—every quasar-class companion in the sample—the operative correction is the transport factor on total flux: an intrinsic luminosity inferred at the host distance is $L = 4\pi D^2 F\,(1+z)^2$, a multiplier of order unity to $\sim 10$ across the sample's well depths, and still two to five orders of magnitude below the luminosity the standard distance assigns.

For any extended envelope—the resolved $L_*$ host the objection expects—the surface brightness is Tolman-dimmed by the full $(1+z)^{-4}$: at $z = 2.114$ that is a factor of $\sim 94$, about five magnitudes per square arcsecond, pushing a nominal host envelope below the detection threshold of the available imaging; its non-detection is the prediction, not the anomaly. Should an envelope nonetheless be resolved, its angular scale reads off $d_A = D/(1+z)$ against the host's redshift-independent distance—a sharp discriminative test, since the standard reading places the same structure at $d_A = d_M(z)/(1+z)$, one to two orders of magnitude deeper. The magnification also tightens rather than loosens the compactness argument: the sub-arcsecond morphologies bound the emitters' physical sizes through $R < \theta_{\rm lim}\,D\,(1+z)^{-1}$, a stronger constraint than the Euclidean reading supplies.

### 7.5 The Historical Dimension

At the 1972 AAAS symposium (Field, Arp & Bahcall 1973), the dispute reduced to two background warrants—the defenders assumed the contested sky was a typical random sample, Arp assumed apparent proximity was physical—and neither side measured which held. The present pipeline is that missing measurement.

The present analysis validates specific observational elements of that programme: the NGC 7603 connecting filament is detected as emitted light, the silhouette ordering—the published photographic claim of Arp (1978), not reproduced on modern DESI imaging—records a foreground-background configuration, and the residual catalogue has quantitatively unusual angular and structural severity under the specified projection model. Individual co-distance nevertheless remains subject to the path-length ledger above. The proposed replacement for Arp's mechanism—intrinsic redshift as quantised, empirical, unexplained—with the deterministic proper-time gradient of the TEP framework: the discordance is what a scalar field produces, and the smooth distribution of $A_{\rm int}$ across the sample is the signature discrete-quantisation proposals (e.g., Karlsson 1971; Burbidge 2001) never explained.

### 7.6 Limitations and Open Questions

The limitations of the present analysis are stated plainly. The sample is twelve pairs—large enough to measure the chance statistic on the selected systems and to map one resolved filament, too small for population-level field statistics or an unconditional survey-wide rejection of chance projection. The nested-wells fit is a four-parameter description of a four-point transect: adequate at the error level, but underdetermined as a field inversion; it constrains the *shape* of the local proper-time landscape rather than the global coupling $\beta_A = -1$, which the analysis assumes. The model awaits over-constraint from either resolved spectroscopy of the filament at intermediate positions, or a second resolved filament with more knots.

The knot transect coordinates are archive-measured projections onto the resolved host–companion axis (the curved filament places the knots up to $23$ arcsec off the straight axis, which the projected coordinate absorbs), and the NGC 3628 companion resolves to WEE 51—the quasar at the tip of the minor-axis X-ray filament—whose literature redshift ($z = 2.15$; Arp et al. 2002) is superseded by the modern spectroscopic archives (NED $z = 1.4983$; Milliquas $z = 1.499$); the verification step adopts the archive value. The correction is verified to propagate only to the pair's redshift-derived entries (Table 1): the chance-alignment, member-ordering and physical-scale statistics consume the archive-measured separation, the member ordering and the host distance respectively—no companion-redshift input enters them—so the ensemble inference is identical under either reading. In the multi-quasar fields the archive cone does not always recover the published member list identically: the NGC 1097 resolution returns two of the four catalogued quasars at the measured offsets inside the enclosing radius, and the NGC 2639 members resolve to the Milliquas quasars nearest the catalogued positions ($z = 0.297$ and $0.336$ against the published $0.305$ and $0.323$). Both fields carry proximity statistics near unity under either member reading (Table 4), so the ensemble inference is insensitive to which set is priced; the resolved member table is nevertheless released for provenance. The companion $\sigma_z \sim 10^{-3}$–$10^{-2}$ uncertainties dominate the error budget where quoted, and SDSS spectroscopic coverage of the pair fields is partial. The ensemble statistic's dependence on the effective parent search volume is likewise bounded: the look-elsewhere computation of Section 5.4 shows the angular product decides the catalogue below $\sim 200$ fields and is an upper bound at any volume, because the observed systems occupy narrower classes than proximity alone—a qualitative severity bound, the conjunction itself not being a priced term in the model.

The most decisive open test is resolved kinematics of the filament gas: a velocity field across the NGC 7603 knots would separate a physically associated, possibly disturbed structure from a projected background filament. An exhaustive audit of the public spectroscopic archives (ESO MUSE, VIMOS, SINFONI and GIRAFFE; MAST JWST-IFU, MaNGA and fiber surveys) was performed at all twelve host positions and both knot coordinates, classifying every returned frame as science or calibration. At the NGC 7603 nucleus a MUSE science pointing exists at $0.9$ arcsec offset—IFU data over the host position already sits in the archive—but no frame is recorded at the knot coordinates, whose nearest science pointings lie $8$–$32$ arcmin away under the archive's proposal-target coordinates; footprint-level verification is required before coverage can be claimed. JWST NIRSpec/MRS coverage exists at NGC 7319, 4258 and 1097—the last also carrying MUSE, VIMOS and SINFONI science frames on the host—while NGC 3628 has IFU frames in the field but none on target. NGC 4319, 1199, 1232 and 3516 return no spatial spectroscopy at all; NGC 3067, 1073 and 2639 only unresolved fiber spectra. The key filament-kinematics measurement is therefore a new observation: a single MUSE pointing, or a NIRSpec-IFU mosaic, covering the resolved filament polyline and both knot coordinates.

The Lyman-$\alpha$ forest audit of Section 5.6 bounds the proximity claim from the opposite direction. On the one sightline where the test is executable the result is adverse to the association: the populated 3C 232 forest—nine Lyman-confirmed systems against $2.3$ expected—places that quasar at its cosmological distance, falsifying the NGC 3067 pair as an individual association while leaving the ensemble statistic intact. For the flagship high-redshift companions the test remains observationally unavailable: for the $z = 2.114$ NGC 7319 companion the raw 2003 Keck/LRIS-B frames have been re-reduced end-to-end from the archive: the frame headers name the target ``NGC 7319 ULX'', which is the companion itself (the ULX is its Chandra designation), and Galianni et al. (2005) record that it was placed at slit centre at PA $= 205^\circ$. The $9.4$ arcsec offset between the header pointing and the companion position is the telescope boresight offset, not a slit miss. The re-reduction reproduces the published blue-arm spectrum—the Ly$\alpha$+N V complex, Si IV–O IV], C IV, He II and O III]—at the published wavelengths within $\sim 13$ &Aring;, after a $+44 \pm 4$ row flexure correction anchored on the slit-spanning NGC 7319 ISM emission ladder at $z_{\rm gas} = 0.022$. The red-arm C III] 5942 line falls on the dichroic throughput ramp and is not discretely recovered, and Mg II 8733 is outside the grating coverage. The same blue arm covers roughly two-thirds of the companion's forest band at $R \approx 1400$ and signal-to-noise $\approx 5$ per &Aring;—enough to exclude a damped or Lyman-limit-class absorber, but not to census individual forest lines. The 2011 Stephan's Quintet slitmask allocated its seven extracted slitlets to low-redshift emission-line regions, none coincident with the companion position or carrying a Ly$\alpha$ emission line at its observed wavelength; and the nearest pointings to the NGC 3516 and NGC 1073 companions fall $94$–$101$ arcsec off-target, beyond the slit half-length. The $z \approx 2.1$–$2.3$ members of the NGC 3516, NGC 1097 and NGC 1073 fields return no usable forest-band spectra among the archive products inspected for each target. The audit thereby isolates the single executable observation that would settle the question for the flagship system (Section 8), and a blue-arm spectroscopic programme covering the four ground-accessible $z > 1.9$ companions has been drafted accordingly.

The archival pattern generalises into a selection statement about where the modern surveys can and cannot see these objects. The canonical discordant companions occupy precisely the angular regime that survey pipelines treat worst. In the imaging layer, companions projected within the luminous extent of a bright host are routinely shredded by, or folded into, the parent deblend and never acquire an independent photometric catalogue entry; in the spectroscopic layer, mechanical fibre-collision limits ($55''$ in SDSS-I/II, and reduced but non-zero in later surveys) exclude a companion fibre wherever the host nucleus has claimed one, so the closest and hence most discordant cases are the least likely to be measured. Downstream, friends-of-friends group catalogues impose a velocity window that rejects discordant-redshift members by construction—the sample the catalogue was built to characterise is cut out before statistics begin—and automated template fitting assigns low-signal spectra to the best-matching low-$z$ template rather than flagging a discordant candidate. Each stage can be audited against the public catalogues rather than asserted. Across the twelve fields, six contain no SDSS spectrum at all within $3'$ of the host nucleus, and none of the companions with resolvable coordinates has a spectrum within $30''$. The NGC 7319 case quantifies the mechanism end-to-end: the $z = 2.114$ companion is not a deblended photometric object at all—every catalogue child within $21''$ shares the host's deblend parent and is flagged non-clean, so the companion never acquired a photometric identity, was never targetable, and carries no SDSS spectrum; its only measured redshift comes from the single pointed Keck observation re-reduced in this work. The nearest JWST IFU coverage terminates $3.9''$ short of its position as well, measured on the actual cube polygons rather than pointing centres. The template stage is equally demonstrable: three spectra in these host fields carry a confident pipeline class and redshift at $\langle{\rm S/N}\rangle \approx 1$ with no warning flag, each resting on a single emission feature for which five to seven alternative single-line redshift identifications fit equally well. The scarcity of discordant companions in SDSS-era catalogues is therefore at least partly a measured selection boundary, not evidence of scarcity in the sky.

The NED redshift-independent distance library (NED-D) supplies the orthogonal discriminator that historically settled such cases. Companion coverage is sparse but instructive: NGC 1232A carries a Nearby Galaxies Catalogue distance estimate of $21$ Mpc—consistent with the NGC 1232 host distances ($\sim 15$–$21$ Mpc) and inconsistent with its own redshift distance ($\sim 94$ Mpc)—while the two photographic estimates for NGC 7603B ($214$–$232$ Mpc) agree with its redshift distance rather than the host's $\sim 115$ Mpc. The independent-distance record therefore divides: one companion sits at the host distance on a published (if coarse) measurement, one is weakly consistent with its cosmological distance, and the remainder carry no redshift-independent distance at all. Host-galaxy distances themselves are broadly consistent with their redshifts.  Extending the forest census to the full public spectroscopic sky returns a consistent picture. A cross-match of the DESI-DR1 forest-quality quasar catalogue (856,767 sightlines) against the same 2MRS parent sample yields 706 projected $z > 1.9$ quasar–galaxy rows within $100''$ of a bright galaxy, corresponding to 699 unique DESI spectra. Of these, 632 projection rows (625 unique spectra) have measured forest transmission and all show transmission within cosmic variance of the mean optical depth (maximum excess $+0.2\sigma$). The discriminating power is worth stating precisely: at $z \approx 2$–$3$ the cosmic forest is itself thin and a local emitter separates by only $\sim 2\sigma$, so the statistic only bites at $z \gtrsim 3.5$ where $\langle T\rangle < 0.5$. After rejected SDSS redshifts are removed, repeated DESI projections of the same spectrum are collapsed, and the single SDSS–DESI overlap is counted once, 18 unique sightlines occupy that strongly discriminating regime and none is anomalous. The exact zero-event binomial bound is $f_{\rm empty} < 0.153$ at one-sided 95% confidence. This is a bound on empty-forest projections, not a falsification of proximity for any curated pair, whose claimed connections were never tested by random-projection statistics. The elevated-transmission tail of the SDSS sample resolves entirely into template-stamped redshifts: 30 spectra whose pipeline redshift (up to $z \approx 6.6$) is rejected in favour of a multi-line alternative at $z \approx 1$–$2.9$ each show a normal forest once re-measured at the corrected redshift. The spectroscopic channel supplies no positive population candidate at its current depth; the catalogue-pair population that remains open is precisely the one the spectral sweep cannot reach. A photometric census of the twelve host fields finds 18 UV-excess point sources within $2'$ that carry no spectrum—nine of them around NGC 1073, the aligned-quasars field—while four hosts (NGC 4319, NGC 1097, NGC 1232, NGC 3516) return no catalogued point source at all within the same radius, masked by the bright-host deblend at even the photometric level. If a discordant companion with an unabsorbed blue continuum exists in this sample, it is in the unspectroscopied population, reachable only by pointed spectroscopy.

These results leave the measured structural observations intact—the NGC 1199 silhouette ordering (on the published photographic claim), the resolved bridge transects, and the conditional severity statistic on the eleven remaining systems—while fixing their logical status: they prioritize targets but do not establish co-distance. The filament transect is consistent with, and required to be, a nested-well transition; a population-level test awaits a larger discordant-pair catalogue.

### 7.7 Bypassing the Pipeline Deadlock

The diagnostic dilemma for testing the TEP spatial-proximity hypothesis is stark: broadband imaging destroys spectral dispersion at readout, while modern survey spectrographs destroy discordant targets in the targeting software. Because modern pipelines rely on robotic fiber positioners that suffer mechanical collisions, photometric deblenders that shred compact objects inside bright host envelopes, and clustering algorithms that reject velocity discrepancies by design, automated optical catalogs will not deliver the key high-resolution absorption spectra required to falsify or confirm co-distance.

Bypassing this deadlock requires exploiting observational modes that disperse light without slits or fibers, or datasets where full 3D spectral cubes already exist across the entire field of view:

- The Macroscopic Continuum Drop ($D_A$ Flux Deficit): The cumulative effect of the Ly$\alpha$ forest creates a massive, measurable flux depression ($D_A$) in the continuum blueward of the quasar's Ly$\alpha$ emission peak. At $z \approx 2.1$, a quasar loses $\sim 15$–$20\%$ of its flux in this band. This flux step can be measured without echelle resolution using low-resolution spectra ($R \sim 500$) or calibrated narrow-band imaging surveys (like J-PAS).

- Intervening Metal Lines (Mg II and C IV): While the Ly$\alpha$ forest lies in the extreme ultraviolet where atmospheric transmission is poor, heavier elements like the Mg II ($\lambda\lambda 2796, 2803$) and C IV ($\lambda\lambda 1548, 1550$) doublets appear alongside hydrogen and shift into the clean optical and red bands at moderate redshifts. Standard ground-based optical spectrographs (e.g., Keck red arm, SDSS) operate here with peak efficiency. A complete absence of intervening Mg II absorbers across a presumed $\Delta z = 2$ path length provides a severe probabilistic constraint on the distance.

- Soft X-Ray Photoelectric Absorption ($N_{\rm H}$): Many discordant companions (e.g., in NGC 7319 and NGC 3628) are luminous X-ray point sources. Low-energy X-rays ($< 1$ keV) are highly susceptible to photoelectric absorption by neutral gas. A cosmological companion must shine through both the intergalactic medium and the host galaxy's entire interstellar medium, accumulating a massive total column density ($N_{\rm H}$). Because X-ray observatories like Chandra and XMM-Newton capture event files across open detectors without slit targeting, archival data can be directly fitted for the low-energy turnover to extract $N_{\rm H}$. A local companion on the near side of the galaxy disk will lack the expected absorption column.

- Cold Gas Interferometry (ALMA / VLA): Optical telescopes struggle to separate a compact quasar from the blinding glare of its host galaxy, but radio and sub-millimeter interferometers isolate emission without starlight interference. Facilities like ALMA (observing CO) and the VLA (observing H I 21-cm) map cold molecular gas via 3D data cubes. If a radio/sub-mm emitting quasar sits behind a host galaxy, the galaxy's cold gas must imprint deep, narrow absorption notches against the quasar's continuum. Because interferometers record spectra for every spatial pixel in the beam simultaneously, the data avoids slit-targeting errors. An archival audit confirms 32 ALMA archive entries overlapping the NGC 7319 companion position (four programs, Bands 3 and 6), plus 4 entries on the NGC 7603 host and none at the NGC 7603B position; at the companion pixel the cubes contain host-velocity CO emission only.

- Gamma-Ray Attenuation (EBL Opacity): The Extragalactic Background Light renders the universe opaque to gamma rays above $\sim 100$ GeV at $z > 0.5$ via pair-production annihilation ($\gamma\gamma \to e^+e^-$). A Cherenkov-band detection of any $z > 1.5$ companion would therefore be irreconcilable with a cosmological distance. (Fermi-LAT detections carry no such veto: LAT detects $z > 1.5$ blazars routinely below $\sim 10$ GeV.) A live cross-match of all companion positions and host fields against 4FGL-DR4 returns zero detections—including 3C 232—so no gamma-ray channel currently constrains the sample; the veto remains available to future CTA-class coverage.

Of these methods, extracting the X-ray column density from existing raw Chandra event files and checking existing ALMA datasets are the most immediately actionable, as they leverage archival data completely insulated from optical slit-targeting misses.

### 7.8 The Gradient Energy Density Constraint

The energetic bookkeeping that remains is real, but its operative geometry is not the pair separation. The measured field difference $\Delta\phi \approx 1.1$ is between the companion's well interior and the ambient pair field, and it is spanned at the well's own wall—not across the $9.3$-arcsecond ($\sim 4$ kpc) projected separation. The macroscopic-gradient geometry a naive estimate prices—a continuous field gradient filling the inter-object volume—is the configuration the transect itself measures to be absent: the filament continuum between the knots sits at ambient depth ($z = 0.030$) and the shared well width is bounded at $w < 0.033$ of the transect (a projected $1.2$ kpc). That geometry is therefore excluded by measurement, not merely by its own energy estimate. What remains to be priced is the microscopic wall enclosing each emitter's own well, whose only bounds come from the emitting gas's measured kinematics.

For a canonical static configuration carrying the displacement $\Delta u$ across a scale $R_w$, the gradient energy density $\tfrac{1}{2}(\nabla u)^2 M_{\rm Pl}^2$ integrated over the configuration gives the enclosed mass equivalent $M_{\rm wall} \approx \kappa\,\Delta u^2 R_w / 12G$ in geometric units, with $\kappa = 1$ for a volume-filling ramp and $\kappa \lesssim 6$ for a flat interior bounded by a concentrated wall—the wall reaches the engine's own mass at $R_w/R_S = 6/(\kappa\Delta u^2)$. The emitting-region radius is bounded independently by kinematics: gas of intrinsic dispersion $\sigma$ remains gravitationally bound only inside $r/R_S \lesssim (c/\sigma)^2$, which for the flagship's measured C IV width of $3{,}155$ km/s is $\sim 9\times 10^{3}\,R_S$, and reverberation-scale broad-line-region radii are $10^2$–$10^3\,R_S$ for engines of the relevant masses. Table 10 prices the ledger for the measured emitters.

*Table 10. The well-enclosure energetic ledger ($\kappa = 1$ volume-filling bookkeeping; concentrated walls inflate by up to $\kappa \lesssim 6$). $R_{\rm unity}$ is the wall radius at which the canonical static field-displacement mass equals the engine mass; the escape bound is the radius inside which gas of the measured dispersion stays bound (computed at the flagship-measured $\sigma = 3{,}155$ km/s for all emitters; narrower intrinsic widths relax it); $M_{\rm wall}/M_{\rm engine}$ ratios are shown at candidate enclosure radii. The deepest well prices the tightest wall.*

| Emitter | $\Delta u = -\ln A_{\rm int}$ | lapse depth surface $[R_S]$ | $R_{\rm unity}\ [R_S]$ | escape bound $[R_S]$ | $50\,R_S$ | $200\,R_S$ | $10^3\,R_S$ |
|---|---|---|---|---|---|---|---|
| NGC 7319 companion | 1.114 | 1.49 | 4.8 | $9\times10^{3}$ | 10.3 | 41 | 207 |
| NGC 3628 companion | 0.913 | 1.67 | 7.2 | $9\times10^{3}$ | 6.9 | 28 | 139 |
| NGC 3516 chain ($z = 0.93$ member) | 0.649 | 2.10 | 14.3 | $9\times10^{3}$ | 3.5 | 14 | 70 |
| NGC 7603 knot [LG2002] 3 | 0.301 | 3.85 | 66 | $9\times10^{3}$ | 0.75 | 3.0 | 15 |
| NGC 7603 knot [LG2002] 2 | 0.188 | 5.82 | 169 | $9\times10^{3}$ | 0.30 | 1.2 | 5.9 |

![Well-enclosure energetic ledger and the measured flagship C IV profile](results/figures/step_49_emission_localization.png)

Figure 4. Left: the canonical static enclosure ledger—the
wall mass in units of the engine mass against enclosing wall
radius in Schwarzschild radii for the measured emitters. The
wall reaches the engine mass at $R_w/R_S = 6/\Delta u^2$; the
shaded band marks the photon-sphere–ISCO interior where
no bound emitter can sit, and the dotted line marks the
bound-gas ceiling for the flagship's measured dispersion.
Right: the C IV region of the re-reduced NGC 7319
companion spectrum with its measured
$\mathrm{FWHM} = 3{,}155$ km/s—ordinary broad-line
kinematics that exclude any near-horizon emitter placement.

Two regimes separate cleanly. Under the volume-filling ($\kappa = 1$) bookkeeping that the corpus's graded interior-roll profile instantiates—the field rising continuously through the interior rather than through a concentrated wall—enclosures confined to $\lesssim 5$–$10\,R_S$ price at order unity against the engine mass for every emitter in the catalogue: a statically consistent deep-well regime exists at Schwarzschild scale, within which the measured $\sim 10^3$ km/s gas motions are comfortably bound ($v_{\rm esc} \gtrsim 0.3\,c$ at $10\,R_S$). A concentrated wall inflates the ledger by the profile factor $\kappa \lesssim 6$—for the deepest wells that pushes the unity radius inside the horizon, itself a selection for the graded interiors the construction supplies. At reverberation-scale radii the same fixed-background bookkeeping runs $\sim 10$–$10^2\times$ the engine mass for the deepest wells ($\sim 10^3\times$ at the concentrated-wall extreme). The quantity the wall contributes is itself part of the object's enclosed dynamical budget, so the operative exclusion is not whether the wall is massive but whether the total enclosed content exceeds the object's measured dynamical mass—a budget no existing observation fixes at these radii for these companions. The ledger therefore narrows the residual to a factor of $\sim 10$–$10^3$ at reverberation-scale enclosures under fixed-background accounting—the regime in which the corpus's coupled interior construction (Paper 28: a regular geometry with metric backreaction, demonstrated at stellar masses with the supermassive extension open) is the operative accounting—not the $\gtrsim 200\times$ parsec-scale exclusion a broad-line-sized wall would carry, and not the photon-sphere impossibility of the exterior-profile misidentification, which does not arise in this channel at all: the required enclosure is Schwarzschild scale or larger, the emitting gas sits at tens to hundreds of Schwarzschild radii inside the well rather than within $1.5\,R_S$ of a compact exterior, and the conformal sector preserves the photons' escape with only the endpoint clock ratio applied.

The Greenstein–Schmidt compactness objection is thereby met on its own terms. That objection assumed a one-metric world in which a large redshift factor is necessarily the gravitational lapse $\sqrt{1 - 2M/r}$, so that a deep redshift demands a surface potential no bound emitting gas survives. In the two-metric structure the clock factor is the product $A(\phi)\,N_g$: the scalar amplitude can be deep inside a region whose gravitational geometry remains regular and modest—the defining architecture of a temporal well—and the clock factor decouples from the lapse. What the objection correctly demands is a sourcing account, and this section supplies the quantified one: the enclosed-well energetics are order-unity at Schwarzschild scale for the graded interior the corpus's own construction supplies, the reverberation-scale bracket is a factor of $\sim 10$–$10^3$ depending on profile concentration under fixed-background bookkeeping, and closing that bracket is the same coupled supermassive interior construction the corpus already carries as open work in the temporal-well sector (Paper 28), here stated with its numeric margin rather than as an unquantified obstacle.

## 8. Conclusion

The Temporal Equivalence Principle has been applied to the discordant quasar–galaxy pairs catalogued by Halton Arp. A twelve-pair catalogue drawn from the published literature and verified against the NASA/IPAC Extragalactic Database has been carried through a thirty-eight-step reproducible pipeline—archive imaging, redshift verification, bridge transects, absorption ordering, chance-alignment statistics, field-profile fitting, and MCMC inference—and the two hypotheses at issue have been tested against each other at every point where the data can decide. The findings are as follows.

First, the distance-independent diagnostic $A_{\rm pair}(Q) = (1+z_G)/(1+z_Q)$ returns pair factors spanning $0.33$–$0.98$ across the sample. Conditional on independent co-distance confirmation, these become a smooth continuum of local proper-time depths. Under the standard reading the same numbers imply that the companions lie a median of $\sim 65$, and up to $\sim 360$, times farther away than the galaxies to which they appear connected.

Second, within the eleven-system unadjudicated residual, multiple systems carry structural or ordering evidence independently of the redshift interpretation, and the ensemble is strongly inconsistent with the specified random-projection model. Every companion resolves to an archive position, so each measurement is taken on the true companion axis. The NGC 7603 filament is detected as real emitted light: the transect drawn along its resolved curved path through the emission knots exceeds rotated control paths by $+3.9\sigma$ over the inter-object segment, with a $+5.7\sigma$ localised peak; the Mrk 205 bridge direction carries a persistent low-level excess above its control envelope (sign-test $p = 0.020$); and the silhouette companion of NGC 1199 fixes the line-of-sight ordering of the pair on the published photographic claim of Arp (1978)—a claim the DESI DR10 annulus test does not reproduce at survey sensitivity (Section 5.2).

Third, the eleven-system residual catalogue is highly unlikely under the explicitly specified independent random-placement model: at archive-measured separations and excluding the one system falsified by the path-length audit, the joint Poisson probability for the eleven remaining systems is computed. Because the catalogue is historically selected rather than prospectively sampled, these probabilities should be interpreted as conditional severity statistics rather than unconditional frequencies for the parent sky population. The baseline severity statistic is $P = 3.6 \times 10^{-13}$ under the most conservative density assumption—the empirical quasar density of the same sight-lines—and $6.6 \times 10^{-14}$ under the published SDSS DR16 quasar surface density. Conditioning on the selection criteria the catalogue actually used sharpens the rejection rather than weakening it: repricing the five axis-claimed systems against their morphology-defined corridors gives $4.9 \times 10^{-19}$, and repricing the five X-ray-selected systems against the measured X-ray-AGN density of the same control fields, corrected for the pointed-coverage overdensity measured on the survey-scale population ($\delta = +1.47$ on pointed identifiers versus $+0.12$ on coverage-uniform RASS), gives $4.0 \times 10^{-18}$, while the foreground-lensing correction inflates the baseline figure by a factor of only $1.3$–$2.3$ even at the uniform $\sigma_v = 300$ km/s upper bound, and by at most a factor of five on the full sensitivity grid.

Fourth, the ensemble carries geometric structure that is highly improbable under chance superposition, and that structure sits where the hypothesis predicts it. Among the companions of the five axis-claimed systems, eight of twelve lie within $30$ arcdeg of the host minor axis ($p = 0.019$); the radial ordering of member redshifts shows a trend in the predicted direction—placing the nearer object at the higher redshift in $13$ of $16$ cases (within-field permutation $p = 0.058$)—the ordering of object-local wells relaxing along the ejection sequence, opposite to the recession ordering a kinematic ejection sequence would give; and the projected separations at redshift-independent distances cluster at the galactic-halo scale, $9.7$–$53$ kpc over host distances spanning $7.5$–$113$ Mpc (exact permutation $p = 0.019$). The historical symmetric-pair and paired-redshift claims are not recovered at archive positions and are set aside.

Fifth, the measured NGC 7603 filament transect is non-monotonic and a single shared transition is refuted outright—every monotonic family fails by four orders of magnitude—while the nested-wells field profile—each compact object contributing its own Lorentzian depression to the shared pair field—is the parameterisation consistent with the measured structure. The fit is saturated (four parameters on four points), so the MCMC well depths restate the per-object offsets $\Delta\phi = -\ln A_{\rm int}$ rather than measuring them independently; what the transect constrains beyond the depths is the shared width, bounded above at a projected $1.2$ kpc, together with the corridor coincidence of the knot wells and the ambient-depth filament continuum between them. Bayesian model comparison by nested sampling carries the Occam penalty explicitly: monotonic single-transition profiles are strongly rejected by the non-monotonic four-point ordering (Table 8, where the formal margin prices the foreknown non-monotonicity of the saturated four-point set rather than an over-constrained field inversion), and within the nested-well family the variant anchored to the measured (PSF-limited) knot compactness returns the largest evidence ($\ln Z = +4.7$) over flat and log-uniform width priors.

Sixth, the companion-redshift asymmetry survives transfer to an unselected sample at survey scale, and it survives the controls that decide whether it is systematic. On the SDSS DR10 group catalogue, 162,650 of the 198,583 satellite velocity offsets satisfy the adopted bound-regime cut; this subset shows a small positive redshift asymmetry ($f_+ = 0.5075$, $p = 1.7 \times 10^{-9}$). The excess strengthens in volume-limited subsamples ($f_+ = 0.510$–$0.513$), excluding flux-limit selection; it persists beyond the reach of host dust, with the photometric signature inverted relative to the extinction prediction; and forward-modelled injection of the catalogue's own measured interloper population falls short of reproducing it by two orders of magnitude in contamination fraction; and it persists at full amplitude among the highest-quality redshifts in the catalogue ($f_+ = 0.5078$ in the clean SDSS-metadata subsample), excluding a measurement-quality origin—while its amplitude scales with satellite luminosity in the direction a proper-time-well reading specifies. A residual signal of order $0.5\%$ in data nobody selected for anomalies, in the direction the historical catalogues reported. The parallel X-ray-selected overdensity is shown to be a pointed-coverage artefact: decomposed by identifier provenance, the excess is carried by pointed-catalogue identifications while the coverage-uniform RASS subset is consistent with null. Both results are stated at their measured significance and neither is overclaimed.

The catalogue is not homogeneous: one executable path-length test falsifies the NGC 3067–3C 232 association. The direct ledger is therefore one falsification in one key catalogue-pair test, no co-distance confirmations, two non-discriminating archival spectra, and nine pairs without key path data. The remaining systems retain measured structural, geometric and statistical peculiarities, but are unadjudicated rather than survivors. TEP does not require every historical Arp association to be genuine; the claim that genuinely associated discordant systems can exist is now attached to the observations capable of deciding it. If the redshift differentials of an independently confirmed subset are local proper-time structure, the connecting filaments cease to be paradoxes and become the observable structures of a scalar field. The discordant pairs emerge as a heterogeneous catalogue containing at least one falsified historical association and an unadjudicated subset with strong target-selection evidence—a population of candidate naturally occurring differential-clock experiments; if co-distance is confirmed in any high-$z$ system, its redshift ratio becomes a direct measurement of the local clock factor predicted by TEP. The same audit that found the false positive identifies the observation that settles the flagship case: the 2003 Keck/LRIS-B spectrum of the $z = 2.114$ companion of NGC 7319 is re-reduced from the raw frames and reproduces the published emission-line pattern, and its blue arm covers most of the forest band only at low resolution—so the decisive $z > 2$ forest census still requires a single blue-arm echelle exposure. A programme covering the four ground-accessible $z > 1.9$ companions has been drafted for submission; it will provide the definitive test of the proximity reading, with the binary outcome pre-registered—three or more normal forests among the four refutes the proximity reading.

### Acknowledgements

This work uses imaging from the DESI Legacy Imaging Surveys (DR10) and the Sloan Digital Sky Survey, with the Digitized Sky Survey via the NASA/IPAC SkyView service as fallback, redshift and catalogue data from the NASA/IPAC Extragalactic Database (NED), object identification from the CDS Simbad database (Wenger et al.\ 2000), quasar membership from the Million Quasars catalogue (Milliquas) via the VizieR service, quasar surface densities from the SDSS DR16 quasar catalogue, ultraviolet spectroscopy from the Mikulski Archive for Space Telescopes (HST/COS, STIS, IUE and FUSE), and spectroscopic coverage metadata from the ESO Science Archive and the Keck Observatory Archive. The complete analysis pipeline, catalogue, and machine-readable results are available in the project repository.

## References

Arp, H. (1970).
"NGC 7603: a galaxy connected to a companion of much higher redshift."
*Astrofisika*, 6, 367.

Arp, H. (1971).
"The spiral galaxy NGC 4319 and the quasi-stellar object Markarian 205."
*Astrophysical Journal*, 168, 39; and Arp, H. (1971),
"Observational paradoxes in extragalactic astronomy,"
*Science*, 174, 1189–1200.

Arp, H. (1978).
"A compact, high-redshift object silhouetted in front of the E galaxy NGC 1199."
*Astrophysical Journal*, 220, 401.

Arp, H., & Sulentic, J. W. (1979).
"Three quasars near the spiral arms of NGC 1073."
*Astrophysical Journal*, 229, 496.

Sulentic, J. W. (1983).
"Confirmation of the luminous connection between NGC 4319 and Markarian 205."
*Astrophysical Journal Letters*, 265, L49.

Sulentic, J. W., & Arp, H. (1987).
"The galaxy-quasar connection: NGC 4319 and Markarian 205. I. Direct imagery. II. Spectroscopy."
*Astrophysical Journal*, 319, 687.

Arp, H. (1966).
*Atlas of Peculiar Galaxies.*
Astrophysical Journal Supplement Series, 14, 1.

Arp, H. (1987).
*Quasars, Redshifts and Controversies.*
Interstellar Media, Berkeley.

Arp, H. (2003).
*Catalogue of Discordant Redshift Associations.*
Apeiron (C. Roy Keys), Montreal.

Arp, H., & Sulentic, J. W. (1985).
"Analysis of groups of galaxies with accurate redshifts."
*Astrophysical Journal*, 291, 88.

Girardi, M., Mezzetti, M., Giuricin, G., & Mardirossian, F. (1993).
"Redshift asymmetry and color-velocity correlation in nearby galaxy groups: evidence of dust?"
*Astrophysical Journal*, 404, 38.

Schmidt, M. (1963).
"3C 273: a star-like object with large red-shift."
*Nature*, 197, 1040.

Greenstein, J. L., & Schmidt, M. (1964).
"The quasi-stellar radio sources 3C 48 and 3C 273."
*Astrophysical Journal*, 140, 1.

Field, G. B., Arp, H., & Bahcall, J. N. (1973).
*The Redshift Controversy.*
W. A. Benjamin, Reading, MA.

Bahcall, J. N., Jannuzi, B. T., Schneider, D. P., Hartig, G. F., & Jenkins, E. B. (1992).
"The near-ultraviolet spectrum of Markarian 205."
*Astrophysical Journal*, 398, 495.

Stocke, J. T., Case, J., Donahue, M., Shull, J. M., & Snow, T. P. (1991).
"New observations of the gas cloud associated with the quasar-galaxy pair 3C 232/NGC 3067."
*Astrophysical Journal*, 374, 72.

Carilli, C. L., van Gorkom, J. H., & Stocke, J. T. (1989).
"Disturbed neutral hydrogen in the galaxy NGC 3067 pointing to 3C 232."
*Nature*, 338, 134.

Danforth, C. W., Keeney, B. A., Tilton, E. M., Shull, J. M., Stocke, J. T., Stevans, M., et al. (2016).
"The HST/COS archive absorption survey of baryons in the intergalactic medium."
*Astrophysical Journal*, 817, 111.

Galianni, P., Burbidge, E. M., Arp, H., Junkkarinen, V., Burbidge, G., & Zibetti, S. (2005).
"The discovery of a high-redshift X-ray-emitting QSO very close to the nucleus of NGC 7319."
*Astrophysical Journal*, 620, 88.

Trinchieri, G., Sulentic, J., Breitschwerdt, D., & Pietsch, W. (2003).
"Stephan's Quintet: The X-ray anatomy of a multiple galaxy collision."
*Astronomy & Astrophysics*, 401, 173.

López-Corredoira, M., & Gutiérrez, C. M. (2002).
"Two emission-line objects with z > 0.2 in the optical filament connecting the Seyfert galaxy NGC 7603 to its companion."
*Astronomy & Astrophysics*, 390, L15.

Arp, H., Burbidge, E. M., Chu, Y., Flesch, E., Patat, F., & Rupprecht, G. (2002).
"NGC 3628: Ejection activity associated with quasars."
*Astronomy & Astrophysics*, 391, 833.

Arp, H. (1982).
"Further examples of companion galaxies with discordant redshifts and their spectral peculiarities."
*Astrophysical Journal*, 263, 54.

Pietsch, W., Vogler, A., Kahabka, P., Jain, A., & Klein, U. (1994).
"ROSAT PSPC X-ray observations of NGC 4258: detection of point sources, 4 million K halo emission, and anomalous arms."
*Astronomy & Astrophysics*, 284, 386.

Burbidge, E. M. (1995).
"Spectra of two quasars possibly ejected from NGC 4258."
*Astronomy & Astrophysics*, 298, L1.

Burbidge, G. (2001).
"The distribution of redshifts in quasi-stellar objects, N-galaxies and related radio sources."
*Publications of the Astronomical Society of the Pacific*, 113, 899.

Karlsson, K. G. (1971).
"Possible discretization of quasar redshifts."
*Astronomy & Astrophysics*, 13, 333.

Hawkins, M. R. S. (2010).
"On time dilation in quasar light curves."
*Monthly Notices of the Royal Astronomical Society*, 405, 1940.

Arp, H. (1996).
"The pair of X-ray sources across NGC 4258: its relation to intrinsic redshifts, ejection and quantization."
*Astrophysics and Space Science*, 244, 9.

Burbidge, E. M. (1997).
"Spectra of two X-ray-emitting quasi-stellar objects apparently ejected from the Seyfert galaxy NGC 2639."
*Astrophysical Journal Letters*, 484, L99.

Burbidge, E. M., Burbidge, G. R., Arp, H. C., & Zibetti, S. (2004).
"QSOs and active galactic nuclei associated with NGC 2639."
*Astrophysical Journal Supplement Series*, 153, 159.

Wolstencroft, R. D., Ku, W. H.-M., Arp, H. C., & Scarrott, S. M. (1983).
"Six quasars near the jets of NGC 1097."
*Monthly Notices of the Royal Astronomical Society*, 205, 67.

Arp, H., Wolstencroft, R. D., & He, X.-T. (1984).
"Complete quasar search in the NGC 1097 field."
*Astrophysical Journal*, 285, 44.

Arp, H., & Carosati, D. (2007).
"A concentration of quasars around the jet galaxy NGC 1097."
arXiv:0706.0143.

Burbidge, E. M., Burbidge, G. R., Solomon, P. M., & Strittmatter, P. A. (1971).
"Apparent associations between bright galaxies and quasi-stellar objects."
*Astrophysical Journal*, 170, 233.

Flesch, E. W. (2021).
"The Million Quasars (Milliquas) catalogue, v7.2."
arXiv:2105.12985; VizieR catalogue VII/290.

Dey, A., et al. (2019).
"Overview of the DESI Legacy Imaging Surveys."
*Astronomical Journal*, 157, 168.

Tully, R. B. (2015).
"Galaxy groups: a 2MASS catalog."
*Astronomical Journal*, 149, 171; VizieR catalogue J/AJ/149/171.

Tully, R. B., et al. (2023).
"Cosmicflows-4."
*Astrophysical Journal*, 944, 94; VizieR catalogue J/ApJ/944/94.

Huchra, J. P., et al. (2012).
"The 2MASS Redshift Survey—description and data release."
*Astrophysical Journal Supplement Series*, 199, 26;
VizieR catalogue J/ApJS/199/26.

Lyke, B. W., et al. (2020).
"The Sloan Digital Sky Survey Quasar Catalog: sixteenth data release."
*Astrophysical Journal Supplement Series*, 250, 8.

Wenger, M., et al. (2000).
"The SIMBAD astronomical database."
*Astronomy & Astrophysics Supplement Series*, 143, 9.

Wojtak, R., Hansen, S. H., & Hjorth, J. (2011).
"Gravitational redshift of galaxies in clusters as predicted by general relativity."
*Nature*, 477, 567–569.

Paturel, G., Petit, C., Prugniel, P., Theureau, G., Rousseau, J., Brouty, M., Dubois, P., & Cambrésy, L. (2003).
"HyperLEDA. I: Identification and designation of galaxies."
*Astronomy & Astrophysics*, 412, 45–55; VizieR catalogue VII/237.

Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. (2013).
"emcee: The MCMC Hammer."
*Publications of the Astronomical Society of the Pacific*, 125, 306–312.

Lilliefors, H. W. (1967).
"On the Kolmogorov-Smirnov test for normality with mean and variance unknown."
*Journal of the American Statistical Association*, 62, 399–402.

Smawfield, M. L. (2025).
"Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed."
TEP Research Series, Paper 0 (Jakarta). DOI: 10.5281/zenodo.16921911.

Smawfield, M. L. (2026).
"What Do Precision Tests of General Relativity Actually Measure?"
TEP Research Series, Paper 9 (Istanbul). DOI: 10.5281/zenodo.18109760.

Smawfield, M. L. (2026).
"Temporal Equivalence Principle: Black Holes and the Temporal Horizon."
TEP Research Series, Paper 28 (Bahrain). DOI: 10.5281/zenodo.21677826.

Smawfield, M. L. (2026).
"TEP-AGN: Scalar Proper-Time Gradients as a Geometric Alternative to Relativistic Smearing."
TEP Research Series, Paper 32 (Cordoba). In preparation.

## Data Availability and Reproducibility

### Open Science Commitment

This preprint is released under CC and MIT licenses. All materials required to reproduce the analysis—scripts, data references, and the manuscript source—are openly available. Feedback and collaboration are welcome.

### Repository Structure

The complete project is hosted at [https://github.com/matthewsmawfield/TEP-ARP](https://github.com/matthewsmawfield/TEP-ARP) and is structured as follows:

- *site/components/*—HTML source files for each manuscript section. These are the only files that should be edited; all other formats are auto-generated.

- *scripts/steps/*—Reproducible analysis pipeline with 38 registered step scripts organized into five blocks (data ingestion and target catalogue, TEP formalism, physical-connection evidence, field-gradient inference, and synthesis / figures).

- *core/*—Shared TEP framework Python modules (scalar field, conformal scaling, screening, cosmology, constants).

- *scripts/*—Utility scripts including the pipeline runner (`run_pipeline.py`) and PDF generation (`generate_site_pdf.py`).

- *results/*—Generated outputs and figures (populated by running the pipeline).

### Reproducing the Analysis

The full analysis pipeline can be reproduced by running:

```
cd /path/to/TEP-ARP
python3 scripts/run_pipeline.py
```

This executes all 38 registered step scripts in sequence, populating `results/figures/` and `results/outputs/` with fresh data. Individual blocks or steps can be run selectively:

```
python3 scripts/run_pipeline.py --block II     # Physical-connection evidence only
python3 scripts/run_pipeline.py --step 23      # Single step only
```

The pipeline is deterministic: seeded where randomness enters (MCMC chains, control-field placement). A verification test suite re-derives the headline numbers from the processed tables offline:

```
python3 -m pytest scripts/tests/ -v
```

The pipeline is fully implemented and all 38 registered steps run successfully against real published data. The pipeline structure, data sources, and analysis methodology are documented in `scripts/README.md`.

### Building the Manuscript

The manuscript site is built from the HTML component files:

```
cd site
npm install
npm run build
```

This generates the static site in `site/dist/` and the markdown version at the repository root (`G37-TEP-ARP-v0.4-Pasadena.md`). A PDF can be generated from the built site:

```
python3 scripts/generate_site_pdf.py
```

### Data Sources

The analysis draws on the following publicly available datasets:

- *NASA/IPAC Extragalactic Database (NED)*: Redshift verification and archive-resolved positions for every catalogued pair and companion.

- *DESI Legacy Imaging Surveys DR10*: r-band CCD imaging for the bridge surface-brightness transects; SDSS r for NGC 7319 and DSS2 Red through SkyView as fallbacks.

- SDSS DR16 quasar catalogue and DR18 spectroscopy: Empirical quasar field densities and spectroscopic coverage of the target fields.

- *HEASARC master catalogues and VizieR*: Multi-wavelength coverage manifest, including the VizieR VII/290 quasar catalogue used for the field-density test.

- *2MRS galaxy catalogue* (Huchra et al. 2012; VizieR J/ApJS/199/26): The 23,205-galaxy parent sample for the forward galaxy–quasar cross-correlation.

- *Tully 2MASS group catalogue* (Tully 2015; VizieR J/AJ/149/171): The 2,822 nests and 14,155 satellites used for the satellite redshift-asymmetry audit step (the signature reported by Arp & Sulentic 1985 and examined further by Girardi et al. 1993).

- *Redshift-independent distances*: Tully–Fisher, fundamental-plane, and related calibrators compiled through the Cosmicflows-4 catalogue (Tully et al. 2023; VizieR J/ApJ/944/94) for the host galaxies.

- *Host morphological position angles*: literature photometric position angles and axis ratios from the NED diameters tables (median over $2$–$14$ references per galaxy), with HyperLEDA mean PAs (Paturel et al. 2003; VizieR VII/237) as fallback.

- *X-ray selection accounting*: Milliquas association type flags (e.g. QX, KX) counted on the same control-field cones as the quasar field-density test, giving the measured X-ray-detected quasar-class density used for the selection-aware repricing.

### Companion Papers

This paper is part of the TEP Research Series. The foundational framework is developed in TEP (Paper 0); the measurement taxonomy and screening regime analysis are developed in TEP-EXP (Paper 9); and the AGN proper-time gradient interpretation is developed in TEP-AGN (Paper 32, in preparation).

### Citation

If you use this code or data, please cite:

```
@misc{smawfield2026arp,
  title        = {Temporal Equivalence Principle: Localized
                  Proper-Time Gradients and the Spatial Proximity
                  of Discordant Pairs},
  author       = {Smawfield, Matthew Lukin},
  year         = {2026},
  doi          = {10.5281/zenodo.22938153},
  url          = {https://doi.org/10.5281/zenodo.22938153},
  note         = {Preprint, Version v0.4 (Pasadena)}
}
```

### Contact

*Matthew Lukin Smawfield*
Email: matthew@mlsmawfield.com
ORCID: [0009-0003-8219-3159](https://orcid.org/0009-0003-8219-3159)
Website: [https://mlsmawfield.com/tep/arp/](https://mlsmawfield.com/tep/arp/)

---

*This document was automatically generated from the TEP-ARP research site. For the interactive version with figures and enhanced formatting, visit: https://mlsmawfield.com/tep/arp/*

*Related Work:*
- [TEP Theory](https://doi.org/10.5281/zenodo.16921911) (Foundational framework)
- [TEP-GNSS: Distance-Structured Correlations in GNSS Clocks](https://doi.org/10.5281/zenodo.17127229) (Paper 1 — GNSS clock-network test of synchronization holonomy)
- [TEP-GNSS-II: 25-Year Analysis of CODE Precise Clock Products](https://doi.org/10.5281/zenodo.17517141) (Paper 2 — quarter-century GNSS clock correlations)
- [TEP-GNSS-RINEX: Raw RINEX Consistency Test](https://doi.org/10.5281/zenodo.17860166) (Paper 3 — raw RINEX-level consistency validation)
- [TEP-H0: The Cepheid Bias](https://doi.org/10.5281/zenodo.18209702) (Companion paper — quantitative Cepheid bias analysis)
- [TEP-EXP: Precision Tests of GR](https://doi.org/10.5281/zenodo.18109760) (Measurement taxonomy)
- [TEP-JWST: JWST High-Redshift Anomalies](https://doi.org/10.5281/zenodo.19000827) (High-redshift galaxy ages and masses)
- [TEP-HUB: The Mount Wilson Paradigm](https://doi.org/10.5281/zenodo.21954257) (Cosmological redshift as temporal shear)
- [TEP-BBN: Primordial Deuterium](https://doi.org/10.5281/zenodo.21841147) (Deuterium isotope identifiability)

*Source code available at: https://github.com/matthewsmawfield/TEP-ARP*