Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T

1. Introduction: The Dark Matter Problem as a Temporal Structure Problem

The Universal Scaling Anomaly

Dark matter observations across cosmological scales reveal an empirical regularity. From dwarf galaxies ($M \sim 10^9 M_\odot$) to massive galaxies ($M \sim 10^{12} M_\odot$), the characteristic radius at which Newtonian dynamics fails—the "dark matter onset"—scales approximately as $R \propto M^{1/3}$. This implies a universal critical proximity scale ("critical" here meaning a phenomenological saturation/proximity scale, not a hard density threshold), observationally proxied by $\rho_T \sim M/R^3 \sim \text{constant}$, at which gravitational phenomenology changes. In the two-branch TEP action, the geometric saturation radius follows $R_T \propto M^{1/3}$, whereas the low-acceleration kinetic response produces an $R_s \propto M^{1/2}$ acceleration-crossing scale; these are different operational radii, and galactic onset estimators spanning the transition between them need not recover identical exponents.

This scaling appears in multiple independent contexts:

• Galaxy rotation curves: SPARC database analysis yields $R_{\text{DM}} \propto M^{\alpha_{\rm SPARC}}$ with $\alpha_{\rm SPARC} = 0.352 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (definition)}$ (Lelli et al. 2016; this work).
• Ultra-diffuse galaxies: DF2 and DF4 exhibit dark matter deficiency consistent with saturation radii exceeding tidal radii (van Dokkum et al. 2018, 2019).
• Milky Way: The SPARC-calibrated $M^{1/3}$ relation predicts a dark-matter onset at $R_{\text{DM}} \approx 3$ kpc for the Milky Way, consistent with the observed transition from baryonic to dark-matter-dominated rotation at $R \sim 3$–5 kpc (Gaia Collaboration 2023; Sofue 2020).

The persistence of this $M^{1/3}$ scaling across 6 orders of magnitude in mass (screening hierarchy: gas giants to neutron stars) and 15 orders of magnitude in density suggests a fundamental physical scale, not a coincidence of baryonic feedback or halo assembly.

Reframing Dark Matter: Phantom Mass from Temporal Shear

The Temporal Equivalence Principle (TEP) proposes that gravitational phenomena arise from a conformal time field $\phi(x^\mu)$ that modulates proper time rates. This phenomenology arises from the TEP action (Smawfield 2025a, Paper 0):

where matter couples to the causal matter metric $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf{x}))U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf{x}))\Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ in the canonical unified unstretched-space decomposition ($\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu\phi \nabla_\nu\phi$ on the historical isotropic-conformal benchmark). At late times, $B(\phi)$ is strongly constrained by multi-messenger observations and can be neglected, while $A(\phi) = \exp(\beta_A \phi/M_{\text{Pl}})$ modulates local proper-time rates. The environmental screening response suppresses the observed gradient in dense environments, producing effective field saturation at characteristic density $\rho_T$. In this framework, the "dark matter" problem is reinterpreted as a violation of the isochrony axiom—the assumption that clocks at the same gravitational potential tick at the same rate regardless of their spatial separation or the mass distribution's history.

When this axiom is relaxed, mass estimates made through different observable pairs diverge, and "phantom mass" — the apparent mass inferred when temporal structure is analysed as spatial mass — can be measured per channel. Whereas earlier relativistic extensions of the MOND paradigm introduced vector and scalar fields to reconcile galactic dynamics with gravitational lensing (Milgrom 1983; Bekenstein 2004) or invoked non-canonical kinetic screening to satisfy Solar System bounds (Babichev et al. 2009; Barreira et al. 2015), TEP derives both channels from the local clock-rate and spatial-slice decomposition of a single proper-time foliation. In the dynamical–baryonic channel studied here (rotation curves), phantom mass is the apparent excess of the dynamical mass $V_{\rm obs}^2 R/G$ over the baryonic mass: the orbital dynamics respond to the local time gradient, sourcing centripetal support with no additional baryons. In the lensing channel (Papers 4, 18, 19), under the canonical unified unstretched-space matter metric (Paper 0 §2.2, Step 80) — where at a fixed cosmic epoch ($A_{\rm sp}(t_0)=1$) the spatial 3-metric $\tilde\gamma_{ij}=\gamma_{ij}$ remains unstretched while the $(3+1)$ ADM lapse rescales as $N_{\rm TEP}(\mathbf x)=A_{\rm loc}(u_{\rm grav}(\mathbf x))N_{\rm GR}(\mathbf x)$ — intermediate clock rates refract light at half the dynamical potential weight ($\Phi_{\rm lens}=\Phi_N+\tfrac12 c^2\beta_A u_{\rm grav}$ versus $\Phi_{\rm dyn}=\Phi_N+c^2\beta_A u_{\rm grav}$), yielding a testable lensing-to-dynamical mass ratio $M_{\rm lens}^{\rm GR}/M_{\rm dyn}=(1+\beta_A^2 y_{\rm eff})/(1+2\beta_A^2 y_{\rm eff})$ (whereas on the historical isotropic-conformal benchmark $\tilde g_{\mu\nu}=A^2(\phi)g_{\mu\nu}$, $A^2(\phi)$ cancels out of null rays and $M_{\rm dyn}/M_{\rm lens}=1+2\beta_A^2 y_{\rm tb}$). Both measurements arise from spatial structure in proper time—without invoking non-baryonic particles—and are kept distinct throughout.^†

^†Note: This geometric "phantom mass" from temporal shear differs from cosmological "phantom energy" (dark energy with $w < -1$). The former arises from spatial gradients in proper time; the latter from exotic equation-of-state matter.

The characteristic scale at which this temporal structure becomes significant is set by the saturation scale $\rho_T$, where the scalar field $\phi$ reaches its self-interaction threshold and saturates—forming a stable, localized configuration (a soliton under that interpretation) with radius:

This $M^{1/3}$ scaling is a direct consequence of the saturation condition; under a soliton interpretation, this corresponds to soliton formation.

The saturation scale parameter $\rho_T \approx 20$ g/cm³ serves as the bulk-matter realization of the abstract environmental screening operator $\mathcal{S}_\Sigma(\mathcal{E})$ defined in the foundational TEP framework. In macroscopic atomic matter, $\rho_T$ acts as an empirical proxy for the geometric saturation of the Temporal Topology, yielding cross-scale predictions without committing to specific chameleon, Vainshtein, or symmetron microphysics.

where the environmental state is $$\mathcal{E} = \{\rho, \Phi/c^2, \nabla\rho, \nabla\Phi, \text{compactness}, R_T(M), \text{proximity}, T, z, \text{boundary geometry}, \text{coherence volume}\}.$$ This single operator unifies density screening, compactness screening, proximity screening, thermodynamic/environmental-state response, lensing/cosmological covariance screening, and wide-binary environmental screening across the entire TEP corpus.

The Multi-Scale Consistency Strategy

As summarized in Table 1, this paper tests $\rho_T$ through a convergent multi-scale approach:

Paper Structure

The remainder of this paper is organized as follows:

• Section 2: Terrestrial calibration via GNSS atomic clock correlations, establishing $L_c \approx 4200$ km → $\rho_T \approx 20$ g/cm³.
• Section 3: Galactic rotation-curve test on the SPARC sample, evaluating the transition between $M^{1/3}$ core saturation and $M^{1/2}$ low-acceleration scaling alongside the two-branch RAR profile.
• Section 4: Screening hierarchy analysis, providing a candidate screening account for why precision GR tests remain satisfied despite scalar-field structure.
• Section 5: Physical context for $\rho_T$, including a dimensional identity bridge preserving WEP, a condensed-matter EOS coincidence, and open derivation targets.
• Section 6: Universal Scaling as a Cross-Scale Consistency Test, synthesizing constraints across all regimes.
• Section 7: Milky Way dark-matter onset at $R_{\text{DM}} \approx 3$ kpc.
• Section 8: Discussion of dark matter as phantom mass and cosmological implications.
• Section 9: Conclusion and astrophysical applications (including RBH-1, companion paper).

This multi-scale convergence identifies $\rho_T \approx 20$ g/cm³ as a candidate GNSS-consistent saturation parameter (with explicit systematic uncertainty), enabling sharply constrained astrophysical applications such as the RBH-1 runaway black hole candidate (Smawfield 2025h, Paper 7).

Evidence hierarchy. Within the evidence hierarchy of the TEP series, the GNSS clock analyses (Papers 1–3) provide the primary empirical input for the terrestrial correlation scale; the present paper tests the cross-scale consequences of conditionally identifying that scale with $R_T(M_\oplus)$. It is a cross-scale normalization test, not an independent primary proof of the TEP framework.

2. Terrestrial Calibration: GNSS Atomic Clock Correlations

Distance-Structured Correlations in Atomic Clocks

The first empirical constraint on $\rho_T$ comes from terrestrial atomic clock networks. Multi-center analysis of GNSS (Global Navigation Satellite System) clock products reveals distance-structured correlations with a characteristic length scale $L_c \approx 4200$ km for Earth's mass. Operationally, $L_c$ denotes the best-fit exponential decay length recovered by fitting clock-residual covariance versus inter-station separation (see Appendix A). Recent analysis of GNSS products (Smawfield 2025b, Paper 1; Smawfield 2025c, Paper 2) reports this correlation across multiple processing centers.

In this corpus, the terrestrial scale $L_c$ is operationally identified with the projected covariance scale associated with the geometric Temporal Topology scale $R_T(M_\oplus)$, not with the in-medium Compton wavelength $\lambda_C(\rho)$ of a candidate completion. The empirical claim is that distance-structured clock correlations decay on the geometric scale $R_T(M)$ of the source mass, rather than on a Yukawa range set by local ambient density.

These correlations exhibit seven independent signatures:

• Distance structure: Correlation amplitude scales with inter-station separation.
• Directional anisotropy: Stronger correlations along Earth's orbital velocity vector.
• Orbital velocity coupling: Annual modulation consistent with $v_\oplus \approx 30$ km/s.
• Annual-phase direction recovery: RA = 6°, Dec = +4°, about 6° from Earth's aphelion velocity tangent; the directed CMB-apex template is anti-phase.
• Multi-center consistency: CODE, IGS, and ESA products show consistent characteristic scales and directional structure.
• 25-year stability: Correlation structure persists across 1999–2024 (Smawfield 2025c, Paper 2).
• Raw RINEX validation: Pattern survives in raw-RINEX Single Point Positioning (SPP) analysis, constraining processing artifacts (Smawfield 2025d, Paper 3).

Operational Calibration of $\rho_T$ from $L_c$

The characteristic length $L_c$ is used as the observed projected covariance scale associated with $R_T(M_\oplus)$. For a spherical mass distribution, the saturation radius is:

Solving for $\rho_T$:

Substituting $M_\oplus \approx 6 \times 10^{27}$ g and $L_c \approx 4200$ km $\approx 4.2 \times 10^8$ cm:

Systematic Uncertainties

The primary uncertainty is the length scale determination. Four distinct contributions are reported: (i) the 25.3-year CODE GPS-only per-measurement and cross-sector dispersion ($4{,}201 \pm 1{,}967$ km, $\pm 47\%$, with azimuth-averaged isotropic $\lambda = 3{,}210 \pm 389$ km; Paper 2); (ii) the GPS-only PPP inter-center systematic spread across CODE, IGS, and ESA ($3{,}330$–$4{,}549$ km, CV $18.2\%$, $\sim \pm 15\%$ half-range; Paper 1, and $4{,}767 \pm 835$ km on the raw-RINEX ionofree channel; Paper 3), with the caveat that centers share largely overlapping underlying data so this spread is not an independent ensemble; (iii) the fractional window/chain scatter and multi-GNSS combined-filter transfer offset established by the executed MGEX stress test (Paper 14 Steps 3.1 and 3.5): on the 4-constellation combined MGEX product (where receiver clocks absorb inter-system biases across GPS, GLONASS, Galileo, and BeiDou), quarterly windows scatter from $1{,}050$ to $1{,}935$ km ($\sim 30$–$40\%$ fractional scatter around the $1{,}862 \pm 155$ km pooled mean), matched-day independent chains CODE/GPSEST and CNES/GINS (GRG) return $2{,}744$ vs. $1{,}933$ km ($\sim 30\%$ offset, GRG within $4\%$ of the pooled $1{,}862$ km), and a factorial estimator decomposition on identical station-days (Paper 14 §4.5b, Step 3.5) proves that both GPS-only IGS-final and 4-constellation MGEX recover the identical band-ordered decay profile, with the 4-constellation combined filter compressing the fitted scale by $\sim 2\times$ relative to GPS-only PPP; and (iv) the $\mathcal{O}(1)$ datum-projection prefactor in the transfer sketch (Eq. 5 below), which differs between single-constellation GPS-PPP and multi-constellation MGEX filters. Applying the $\sim 30$–$40\%$ fractional window/chain uncertainty around the 25-year GPS-only anchor ($L_c = 4{,}200$ km) yields the primary systematic band $\rho_T \in [8,\,70]$ g/cm³ around $\rho_T = 20 \pm 7$ g/cm³, while adopting the unadjusted 4-constellation MGEX scale ($\lambda = 1{,}862$–$2{,}744$ km) at unit prefactor defines an asymmetric upper sensitivity bracket of $\rho_T \sim 69$–$220$ g/cm³. The operational mapping from $L_c$ to the projected covariance scale associated with $R_T(M_\oplus)$, together with the use of total Earth mass, are model-defining choices rather than additional empirical uncertainties.

Propagating the length-scale uncertainty alone:

This range lies in the same condensed-matter density regime where electron degeneracy becomes dynamically relevant to the equation of state (Section 5.3) and provides the calibration scale for all subsequent tests. The key point is that the $M^{1/3}$ structural form is independent of $\rho_T$; only the normalization changes. Even at the extremes ($L_c = 3{,}700$ km $\rightarrow \rho_T \approx 30$ g/cm³, or $L_c = 4{,}700$ km $\rightarrow \rho_T \approx 14$ g/cm³), the cross-scale consistency tests remain viable because the same $\rho_T$ value must describe all systems simultaneously. The wider epoch/window band does not alter this structural conclusion — it re-prices the normalization confidence: cross-scale checks that pass at $\rho_T = 20$ g/cm³ must be re-evaluated under the widened prior, and the Level 1/2/3 claim hierarchy (Section 8) is designed to absorb exactly this class of anchor-level hit.

Discrimination from Geophysical Signals

A critical objection to long-range clock correlations is the presence of tropospheric and ionospheric delays. However, spectral power analysis reveals distinct scale separation:

• Tropospheric covariance: Decays at ~100–500 km (weather systems)
• Ionospheric covariance: Decays at ~500–2000 km (TEC structures)
• Scalar field signature: Persists as a covariance floor at ~4200 km

The 4200 km feature shows no counterpart in ionospheric or tropospheric delay products. Furthermore, the signal recovers an annual phase about 6° from Earth's aphelion velocity tangent (companion analysis); the CMB-apex template is anti-phase and is not identified as a rest frame. Power spectral analysis of the GPS-only clean baseline confirms this as a persistent background covariance floor, not a geophysical artifact, and combined multi-GNSS receiver-clock solutions (Paper 14) recover the same band-ordered decay profile and heliocentric orbital-velocity coupling.

Null Tests and Robustness

The GNSS correlation pattern survives multiple null tests:

• Geophysical coupling: Correlations persist after removing tidal, atmospheric, and ionospheric signals.
• Orbital mechanics: Pattern is independent of satellite constellation geometry.
• Processing artifacts: Raw RINEX analysis (Paper 3) constrains "black box" concerns.
• Temporal stability: 25-year consistency (Paper 2) disfavors transient instrumental effects.

The convergence of multi-center, multi-decade, and raw-data analyses identifies $L_c \approx 4200$ km as the canonical terrestrial screening length from the 25-year GNSS baseline (Papers 1–2), independent of theoretical interpretation. All cross-regime calibration and forward models in this paper adopt that value. Paper 14 provides an independent MGEX held-out verification on a ~1 yr combined-clock span (recovered scale $\approx 1{,}396 \pm 90$ km, $R^2 \approx 0.49$), confirming signal presence on a shorter baseline and different product; it is cited as verification only, not as the dimensional scale for $\rho_T$ extrapolation. The scale tension between that figure and the 4,200 km anchor has since been tested directly (Paper 14 extension, Step 3.1): on strictly matched, verified-independent days the CODE and CNES processing chains return $\lambda = 2{,}744$ km and 1,933 km respectively (~30% offset, the CNES value within 4% of the pooled 1,862 km), while quarterly windows of the same product scatter between 1,050 and 1,935 km — the held-out discrepancy is dominated by epoch/sampling instability of the short-span estimate, not by a product-dependent physical scale, and the independent-center agreement bounds the processing contribution to ~30% (Appendix B.8).

Mapping the Static Terrestrial Profile to Network Clock Covariance: Dimensional Analysis and PREM BVP Integration

A central question is how the static Earth-sourced Temporal Topology profile produces the dynamic observable recovered in GNSS clock products: a spatial covariance length $L_c \approx 4200$ km. The calibration does not require GNSS satellites to orbit within a saturation shell, nor does it posit a literal shell at $R_T(M_\oplus)$. Instead, $R_T(M_\oplus)$ defines the geometric scale controlling the projected covariance kernel of the Earth-sourced exterior clock-transport field. In this sense, $L_c$ is not an arbitrary fitted length: it is the observed network projection of the same mass-dependent scale used in the cross-regime $R_T \propto M^{1/3}$ construction. This subsection first states the physical mechanism and dimensional scaling linking the static exterior profile to network clock covariance, and then evaluates the resulting spatial covariance kernel numerically via a 4-layer PREM Earth boundary-value integration and a 100-trial Monte Carlo network injection–recovery simulation (Step 14; Figure 1; Appendix B.2).

What $R_T(M_\oplus)$ is, and is not

$R_T(M) = (3M/4\pi\rho_T)^{1/3}$ is the radius a body of mass $M$ would have at uniform density $\rho_T$. For Earth this is $\approx 4146$ km. It is a scaling variable, not a location: because Earth lies in the partial-transition regime, $R_T(M_\oplus)$ is used as a geometric scaling variable rather than as the radius of a literal material shell (Appendix B.2). The question the sketch addresses is what spatial scale governs the exterior field of such a partially screened source.

The exterior profile scale

In the static exterior problem, the available length scales are the source radius $R_\oplus$, the geometric saturation scale $R_T(M_\oplus)$, and the ambient-medium Compton wavelength $\lambda_C(\rho_{\rm amb})$ (very long in the low-density exterior ($T^{(\rm m)} \approx 0$)). Under the corpus's screening ontology the transition in the screening function $S(\rho)$ is gradual, spanning $\Delta\rho/\rho_T \sim \mathcal{O}(1)$ — a continuous Temporal Topology with no thin shells (Paper 0, A4). For such a gradual transition, the exterior field of the partially screened source carries no structure sharper than the source scale itself, and the characteristic scale over which the effective source charge accumulates is $R_T(M)$: radii enclosing mean density above the transition range contribute suppressed source charge, radii beyond contribute unsuppressed charge, and the changeover in the integrated source — not in any local shell — occurs over a radial span of order $R_T$. (Note that a steep transition would instead produce thin-shell behaviour with structure on scales $\ll R_T$; the gradual ontology is therefore not incidental to this argument but required by it.) Both $R_\oplus = 6371$ km and $R_T(M_\oplus) = 4146$ km lie within a factor $\sim 1.5$, so at sketch precision the exterior profile varies on the scale of a few thousand kilometres, with $R_T(M)$ — not $R_\oplus$ — as the mass-scaling variable the cross-scale law requires.

From a static profile to a time-series covariance

A deterministic static field produces constant clock offsets, which clock-bias estimation absorbs; by itself it generates no time-series covariance. The sketch therefore requires one explicit assumption:

Modulated-sampling assumption. The residual variance analysed in Papers 1–3 arises from temporal modulation of each station's sampling of the static exterior profile — diurnal rotation of the ground network through the profile's anisotropy, satellite orbital motion, and tropospheric/orientation geometry that multiplies the local conformal rate $A(\bar{\phi}(\mathbf{x},t))$. Two clocks then acquire correlated time variance to the extent that their modulated samples traverse the same large-scale field structure.

Writing $\delta\tau_i(t) \approx \tau_0 (\beta_A/M_{\rm Pl})\, \bar{\phi}(\mathbf{x}_i(t))$ for the modulated residual, the equal-time network covariance $\langle \delta\tau_A\,\delta\tau_B\rangle_t$ inherits its baseline dependence from the spatial structure of $\bar{\phi}$: stations separated by $d$ small compared with the profile's variation scale sample nearly the same field excursion and correlate strongly; stations separated by $d$ large compared with that scale decorrelate. Crucially, this $L_c \approx 4{,}200$ km network correlation reflects independent terrestrial clocks sampling a pre-existing static spatial boundary-value geometry—analogous to spatially separated probes sampling a static Coulomb or Newtonian potential—rather than superluminal clock-to-clock communication (Appendix B.7). The decay length of the network covariance is then set by the exterior profile scale,

with the $\mathcal{O}(1)$ factor determined by the unspecified completion (profile shape, anisotropy spectrum, sampling geometry). Rather than leaving this $\mathcal{O}(1)$ prefactor undetermined, the 4-layer PREM spherical boundary-value integration and 3-D Yukawa surface covariance projection below (Step 14; Figure 1; Appendix B.2) computes it directly from the scalar field equation Eq. (\ref{eq:static_klein_gordon}) at the fixed Cassini-compatible normalization $\lambda = 7.53 \times 10^{-66}$ and $\rho_T = 20$ g/cm³, yielding $L_c^{\rm PREM} = 4{,}010.2$ km and a derived geometric transfer prefactor of $L_c^{\rm PREM}/R_T(M_\oplus) = 4{,}010.2/4{,}146.0 = 0.9672$ (within $-3.3\%$ of unity).

Implications for the calibration

This sketch binds the clock observable ($L_c$) to the geometric scaling variable ($R_T$) at the level of dimensional analysis under two explicit assumptions: the gradual-transition ontology and the modulated-sampling mechanism. It does not prove that the GNSS correlation is caused by the TEP scalar — that remains the empirical hypothesis tested by null-model exclusion (Papers 1–3) — and it does not by itself discriminate $R_T(M)$ from $R_\oplus$ as the controlling scale, since the two differ by only a factor $\sim 1.5$ for Earth. What would discriminate them is the mass-scaling test: $R_T \propto M^{1/3}$ while a source-radius scale does not follow that law across the astrophysical systems of Sections 3–7. To evaluate this transfer on a fixed-input, canonical-kinetic Earth-profile consistency calculation, Step 14 (step_14_scale_audit.py, Appendix B.2) integrates the boundary-value problem $\nabla^2 u = M_{\rm Pl}^{-2}V_{\rm eff}'(u)$ at the fixed Cassini-compatible quartic normalization $\lambda = 7.53\times 10^{-66}$ and GNSS-consistent $\rho_T = 20$ g/cm³ through a 4-layer Preliminary Reference Earth Model (PREM; Dziewonski & Anderson 1981) density profile: the resulting scalar gradient $|\mathrm{d}u/\mathrm{d}r|$ has its radial centroid at $R_{\rm grad} = 4{,}875$ km ($1.18\,R_T(M_\oplus)$, or $R_{\rm grad}^{\rm NL\text{-}PREM} = 4{,}809.9$ km [$1.160\,R_T(M_\oplus)$] when the coupled nonlinear two-branch operator $\nabla\cdot[P_{,X}(\xi)\nabla u] = M_{\rm Pl}^{-2}V_{\rm eff}'(u)$ of Eq. C.10 is solved via damped Picard relaxation [$\omega = 0.35$, $100$ iterations, relative convergence $\|\Delta u^{(100)}\|_\infty/\|u^{(100)}\|_\infty = 2.67 \times 10^{-14}$, grid-doubling shift $0.077\%$], where for the dimensionless fractional frequency perturbation $y_i(t) \equiv \delta\nu_i(t)/\nu_0 = -\delta u_{\rm loc}(\mathbf{x}_i(t),t) - c^{-1}(\mathrm{d}/\mathrm{d}t)\int_{\mathbf{x}_i(t)}^{\mathbf{x}_s(t)}\delta u_\oplus(r(s'))\,\mathrm{d}s'$, tangential rotation vanishes on the spherical monopole [$(\boldsymbol{\Omega}_\oplus\times\mathbf{r}_i)\cdot\nabla u_\oplus(r) \equiv 0$] while rotating station projection onto the external heliocentric/Galactic gradient $\mathbf{r}_i(t)\cdot\nabla u_{\rm ext}$ at $f_{\rm sid} = 11.61\,\mu\text{Hz}$ and inclined satellite slant-path elevation modulation $\delta\tau_{is}^{\rm prop}(t)$ at the constellation-specific orbital repeat frequencies — US GPS [$T_{\rm orb} = 11.97\text{ hr}, f_{\rm orb} = 23.21\,\mu\text{Hz}$], Russian GLONASS [$11.26\text{ hr}, 24.67\,\mu\text{Hz}$], and European Galileo [$14.08\text{ hr}, 19.74\,\mu\text{Hz}$], all sitting inside the $10\text{--}60\,\mu\text{Hz}$ passband and below MGEX's $34.7\,\mu\text{Hz}$ 8-hour cutoff — yield a conformal clock modulation amplitude of $\delta y_{\rm rms} \simeq (v_{\rm rot}/c)\,u_\oplus(R_\oplus) = 7.0 \times 10^{-15}$ [diurnal timing residual $\delta\tau_{\rm rms} = \delta y_{\rm rms}/(2\pi f_{\rm sid}) \simeq 0.10\text{ ns}$, matching sub-nanosecond IGS PPP clock residuals] while two-branch kinetic stiffness $P_{,X}(\xi)$ suppresses the dynamical Temporal-Shear acceleration (with the normalized Pearson cross-correlation kernel $C(d)/C(0) \propto \exp(-d/L_c)$ depending only on the scale-invariant spatial eigenprofile $u(r)/u(0)$) to $|\mathbf{a}_\phi|/g_\oplus \simeq a_{\rm eff}/g_\oplus = 1.22 \times 10^{-11}$ at Earth's surface and $|\mathbf{a}_\phi|/g_\odot(1\text{ AU}) = 2.02 \times 10^{-8} \ll |\gamma_{\rm Cassini}-1| \le 2.3 \times 10^{-5}$), and projecting the 3-D Yukawa-screened spatial kernel onto a datum-constrained surface station network across $d \in [200,\,8{,}000]$ km recovers a transition-shell decay scale of $L_c^{\rm PREM} = 4{,}010$ km ($3{,}957$ km nonlinear; $R^2 = 0.927$; Figure 1, within $-4.5\%$ of the $4{,}201$ km cross-sector GNSS baseline, $L_c^{\rm PREM}/R_T(M_\oplus) = 0.967$) while integrating across all interior shells $r \in [500,\,6{,}321]$ km yields $L_c^{\rm vol} = 3{,}044\text{--}3{,}213$ km ($3{,}170$ km nonlinear; $R^2 = 0.981\text{--}0.985$, matching the $3{,}210 \pm 389$ km 25.3-year isotropic CODE baseline of Paper 2 and $3{,}193$ km ESA Final baseline of Table A.1; varying the fit window across $d_{\max} \in \{6{,}000,\,8{,}000,\,10{,}000\}$ km shifts $L_c^{\rm PREM}$ across $3{,}290\text{--}4{,}595$ km, and scaling $\lambda_c(\rho_T) \propto \rho_T^{-1/3}$ across $\rho_T \in \{5,\,20,\,50,\,200\}$ g/cm³ shifts the endpoint-clamped shell integral $L_c^{\rm PREM}$ from $4{,}105$ to $3{,}650$ km because the $1/|\mathbf{x}_A - \mathbf{x}_s|$ Coulomb geometric factor and endpoint minimum-subtraction on $[200,\,8{,}000]$ km pin the single-shell surface integral to Earth's fixed $R_{\rm CMB}$ and $R_\oplus$ geometry, whereas the exact 3-D volume Yukawa convolution $\mathcal{F}^{-1}[(k^2+m_{\rm eff}^2)^{-2}] \propto \exp(-d_{\rm chord}/\lambda_c(\bar\rho_\oplus;\rho_T))$ — in which the $1/r$ singularity cancels identically in the 3-D convolution — tracks $\lambda_c(\bar\rho_\oplus;\rho_T) \propto \rho_T^{-1/3}$ across $6{,}329,\,3{,}951,\,2{,}888,$ and $1{,}796$ km, and the saturation-core coherence kernel $C_{\rm sat}(d;\rho_T) = \exp(-d_{\rm chord}/R_T(M_\oplus,\rho_T))$ tracks $R_T \propto \rho_T^{-1/3}$ across $6{,}901,\,4{,}311,\,3{,}152,$ and $1{,}960$ km). Furthermore, to test the direction of the multi-GNSS scale offset without circular ratio rescaling, Step 14 executes a 100-trial Monte Carlo injection–recovery and null suite on a 120-station global network ($N_{\rm ep}=400$ epochs): (i) injecting $L_c^{\rm true} = 4{,}200$ km, 1-constellation GPS-only global datum removal recovers $\hat L_c^{\rm GPS} = 4{,}036 \pm 113$ km ($R^2 = 0.999$, bias $-3.9\%$; across an injected grid $L_{\rm true} \in \{1{,}000,\,2{,}000,\,3{,}000,\,4{,}200,\,6{,}000\}$ km, GPS-only recovers $1{,}019,\,1{,}975,\,2{,}912,\,4{,}060,$ and $5{,}721$ km linearly), whereas 4-constellation multi-GNSS common-view inter-system bias (ISB) projection (Montenbruck et al. 2017; Hong et al. 2019) compresses the $4{,}200$ km injection to $\hat L_c^{\rm MGEX} = 2{,}277 \pm 47$ km ($1.77 \pm 0.06\times$ compression for per-epoch ISBs matching CNES/GRG $1{,}933$ km, and $2{,}485 \pm 45$ km or $1.62\times$ compression for daily-constant ISBs with rotating constellation visibility matching CODE matched-day MGEX $2{,}744$ km); and (ii) neither a white-noise null ($|C(d)| = 0.009 \pm 0.001$) nor a correlated rank-3 satellite-orbit/geocenter dipole $\cos(d/R_\oplus)$ plus datum common-mode null (which exhibits concave-down short-range curvature $\mathrm{d}^2C/\mathrm{d}d^2 = -0.024\text{ Mm}^{-2} < 0$ and drives exponential fits to the $\sim 24{,}900$ km upper bound) can manufacture a spurious $4{,}200$ km scale in GPS-only processing (the spatial ISB projection and temporal passband truncation of Paper 14 both suppress long-scale covariance and describe overlapping aspects of the combined processing response, so they are not treated as independent multiplicative corrections without a joint transfer calculation; the simulated $1.62\text{--}1.77\times$ compression reproduces the direction and approximate magnitude of the observed $4{,}066/2{,}141 = 1.90\times$ cross-mode offset). Finally, to test this single- versus multi-constellation discriminator directly on real multi-GNSS station observations, Step 14 (real_cddis_rinex3_single_constellation_reprocessing_2023_4q) executes dedicated single-constellation rnx2rtkp reprocessing across all four quarters of 2023 (DOY 010, 100, 190, and 280; $192\text{--}197$ station-days and $3{,}706\text{--}3{,}927$ station-pair observations per constellation from public CDDIS RINEX3 archives) after resolving the RTKLIB pntpos.c $CLK column mapping (where dtr[0] in column 5 holds the GPS clock and is constrained to $0.0$ when GPS is absent, while dtr[1] in column 6 holds the GLONASS clock and dtr[2] in column 7 holds the Galileo clock). Evaluating the exact four-step conditioning ladder from the full-day 24-hour spectral window ($n_{\rm seg} = 288$, $f_{\min} = 11.6\,\mu\text{Hz}$; Paper 3) down to the 8-hour MGEX high-pass window ($n_{\rm seg} = 96$, $f_{\min} = 34.7\,\mu\text{Hz}$) and 4-constellation ISB removal on these identical 198 station-days demonstrates how each processing step compresses the underlying scale: (i) in the full-day 24-hour window ($n_{\rm seg} = 288$), three completely independent satellite constellations with distinct orbital altitudes, onboard atomic standards, signal structures, and control segments all recover correlation scales inside the $[3{,}000,\,5{,}000]$ km GPS-only PPP band across strictly matched spectral bands: under Paper 3's standard $\sqrt{N_{\rm bin}}$-weighted exponential fit on the matched $10\text{--}500\,\mu\text{Hz}$ band, US GPS ($20{,}200$ km, Rb/Cs, CDMA) recovers $L_c^{\rm GPS} = 4{,}050.9 \pm 764.4$ km ($R^2 = 0.963$, implying $\hat\rho_T = 21.45$ g/cm³; $3{,}765.7 \pm 640.6$ km unweighted, $R^2 = 0.964$), Russian GLONASS ($19{,}100$ km, Cs, FDMA) recovers $L_c^{\rm GLO} = 4{,}692.8 \pm 2{,}043.6$ km ($R^2 = 0.846$; $3{,}806.2 \pm 1{,}419.7$ km unweighted, $R^2 = 0.849$), and European Galileo ($23{,}222$ km, passive H-masers, ESA control segment) recovers $L_c^{\rm GAL} = 2{,}815.5 \pm 579.2$ km ($R^2 = 0.933$; $2{,}697.0 \pm 507.9$ km unweighted, $R^2 = 0.934$, and $3{,}653.0 \pm 1{,}099.8$ km, $R^2 = 0.890$ in the $10\text{--}60\,\mu\text{Hz}$ diurnal/orbital band where $100\text{--}500\,\mu\text{Hz}$ single-frequency broadcast ionospheric noise is excluded, and $L_c^{\rm GAL,IFLC} = 4{,}556.7$ km across the full $10\text{--}500\,\mu\text{Hz}$ band on 67 optimal_100 stations when dual-frequency E1/E5 ionosphere-free LC is enabled; $3{,}208.1$ km, $R^2 = 0.907$ on DOY 010) — yielding a strictly band-matched $10\text{--}500\,\mu\text{Hz}$ three-constellation mean of $\bar L_c^{\rm 1\text{-}const} = 3{,}853.1$ km ($\sqrt{N}$-weighted, $\hat\rho_T = 24.92$ g/cm³; $3{,}423.0$ km unweighted), a strictly band-matched $10\text{--}60\,\mu\text{Hz}$ three-constellation mean of $3{,}361.6$ km (GPS $2{,}745.8$, GLONASS $3{,}685.9$, Galileo $3{,}653.0$ km), moreover, when the per-epoch global network common-mode error ($\ell = 0$ monopole, Dong et al. 2006) is removed directly on the full-day 24-hour window ($n_{\rm seg}=288$, strictly band-matched $10\text{--}500\,\mu\text{Hz}$ with a single uniform estimator across all constellations), US GPS ($2{,}953.3$ km, $R^2 = 0.969$), European Galileo ($3{,}302.1$ km, $R^2 = 0.955$ — jumping above $3{,}300$ km on the full $10\text{--}500\,\mu\text{Hz}$ band without any low-frequency sub-band cut), and Russian GLONASS ($3{,}210.2$ km, $R^2 = 0.957$) converge to $\bar L_c^{\rm CME\text{-}rem}(n_{\rm seg}=288) = 3{,}155.2 \pm 180.8$ km ($\mathrm{CV} = 5.73\%$), matching the 25.3-year isotropic CODE baseline ($3{,}210 \pm 389$ km, Paper 2), the ESA Final baseline ($3{,}193$ km), and the 4-layer PREM full-volume datum-removed kernel ($L_c^{\rm PREM,vol} = 3{,}213.2$ km, $L_c^{\rm vol}/R_T(M_\oplus) = 0.7749$), which provides a model-calibrated BVP transfer mapping $\bar L_c^{\rm CME\text{-}rem} = 3{,}155.2$ km to $R_T(M_\oplus) = 4{,}071.8$ km ($\hat\rho_T = 21.12$ g/cm³; vs. $\hat\rho_T = 45.40$ g/cm³ if inverted at raw unit prefactor without the volume-kernel transfer ratio) and mapping the 25.3-year isotropic CODE scale $3{,}210$ km to $R_T(M_\oplus) = 4{,}142.5$ km ($\hat\rho_T = 20.06$ g/cm³), whereas the 4-constellation combined solution (-sys G,R,E,C) is compressed to $2{,}437.6 \pm 416.2$ km ($R^2 = 0.939$, $2{,}462.1$ km $\sqrt{N}$-weighted; a station-geometry resampling audit on the real continental IGS coordinates in Appendix B.1 further confirms that continental clustering never manufactures a spurious $4{,}200$ km $\approx \tfrac{2}{3}R_\oplus$ scale across $L_{\rm true} \in [1{,}000,\,6{,}000]$ km, and that equal-pairs-per-bin subsampling flattening the continental distance histogram recovers $\hat L_c = 3{,}805.9 \pm 908.6$ km); (ii) truncating the Welch segment to 12 hours ($n_{\rm seg} = 144$) and then to MGEX's 8-hour high-pass window ($n_{\rm seg} = 96$, which removes the $10\text{--}34.7\,\mu\text{Hz}$ long-wavelength modes) compresses the three-constellation mean monotonically from $3{,}423.0$ km $\to 2{,}950.9$ km ($n_{\rm seg}=144$) $\to 2{,}850.4$ km ($n_{\rm seg}=96$: GPS $2{,}759.0$, Galileo $1{,}969.8$, GLONASS $3{,}822.5$ km); (iii) applying per-epoch network datum-mean removal on top of the 8-hour $n_{\rm seg}=96$ window compresses all three independent constellations onto the identical scale to within $\mathrm{CV} = 0.28\%$ ($2{,}163.2 \pm 378.7$ km GPS, $2{,}152.9 \pm 320.4$ km Galileo, and $2{,}152.3 \pm 291.5$ km GLONASS; $R^2 = 0.929\text{--}0.957$, mean $2{,}156.1$ km); and (iv) solving all four constellations jointly on the receiver-clock series (which estimates three extra per-epoch ISB parameters dtr[1..3] at every epoch) compresses the clock correlation length down to $1{,}477.5 \pm 328.1$ km ($R^2 = 0.867$; $2.32\times$ total empirical compression from the $3{,}423.0$ km full-day single-constellation mean, and $2{,}694.3 \to 766.9$ km in dual-frequency iono-free LC on optimal_100). Together with three companion metrological closures — (a) Paper 14 Step 2.8 (step_2_8_satellite_clock_analysis.json), which shows across $656{,}684$ within-constellation satellite-clock pairs (GPS $206{,}545$, Galileo $141{,}831$, GLONASS $96{,}705$, BeiDou $211{,}603$) that no space-segment constellation exhibits terrestrial-scale spatial decay ($R^2 \le 0.13$, hitting the $10^5$ km boundary); (b) Paper 3 Step 2.12 (step_2_12_channel_mode_null.json), where the precise SP3 shared-power channel across $22{,}707$ station-pair days recovers $L_c = 4{,}474.8$ km ($R^2 = 0.664$), ruling out broadcast ephemeris artifacts; and (c) Paper 8 (TEP-SLR), which uses 11 years ($2015\text{--}2025$, $201{,}503$ normal points across $46$ ILRS stations) of two-way optical Satellite Laser Ranging to passive retroreflectors (LAGEOS-1/2, Etalon-1/2, LARES) to bound conformal timer wander at $\delta A \lesssim 2 \times 10^{-8}$ with a cross-satellite turnover $\bar r = -0.23$ ($p = 0.046$; $-0.22$ under ESA/ESOC orbits) at $5{,}000\text{--}7{,}500$ km — this confirms that the spatial correlation scale is replicated across independent satellite constellations and is not a single-system GPS artifact.

For detailed methodology and validation, see Appendix A and Papers 1–3.

3. The SPARC Galaxy Analysis: Phantom Mass as Unscreened Time-Field

Galaxies can be understood as "baryon-anchored" solitons—where the same scalar field structure calibrated from terrestrial GNSS data is pinned to a baryonic mass concentration. This section extends the test to galactic scales, where the $M^{1/3}$ scaling is expected to govern the transition between screened (Newtonian) and unscreened (phantom mass) regimes. The SPARC database provides an ideal testbed: 175 disk galaxies with high-quality rotation curves and well-constrained baryonic mass distributions. The theoretical basis for the phantom mass interpretation is developed in detail in Smawfield (2025e), which demonstrates how differential proper-time accumulation in gravitational lensing can produce apparent dark matter phenomenology.

Phantom Mass Interpretation

In the TEP framework, the apparent "dark matter" observed in galaxy rotation curves is identified as Phantom Mass—defined as the apparent mass derived from temporal shear under the assumption of isochrony (Smawfield 2025e). It is not a particle species but a geometric manifestation of the unscreened proper-time field. A critical distinction must be emphasized: "Phantom Mass" is an observational inference artifact (analogous to a refractive index effect), not a "ghost field" in the quantum field theory sense. Ghost fields imply negative kinetic energy and Hamiltonian instability; phantom mass carries no such pathology. It arises simply from modeling a bi-metric spacetime with a single-metric prior—the "missing mass" is the unmodeled temporal shear contribution.

The transition radius $R_{\rm trans}$ marks the boundary between two regimes:

• Inside $R_{\rm trans}$: The enclosed mean baryonic density exceeds the critical screening density ($\bar\rho > \rho_{\rm trans}$). The time-field is screened, and gravity follows Newtonian predictions based on visible baryonic matter.
• Outside $R_{\rm trans}$: The enclosed mean density drops below $\rho_{\rm trans}$. The time-field becomes unscreened, and the refractive proper-time gradient produces an apparent gravitational excess—the "phantom mass" conventionally attributed to dark matter.

If this interpretation is correct, the radius at which rotation curves diverge from Newtonian prediction should scale as $R_{\rm DM} \propto M_{\rm bar}^{1/3}$. While the fundamental core saturation occurs at the proximity scale observationally proxied by $\rho_T \approx 20$ g/cm³, the halo transition is governed by a much lower proximity scale, proxied by $\rho_{\rm trans}$, characteristic of the diffuse field tail.

Invariant: $\rho_T$ denotes the universal saturation density of the scalar sector that fixes the compact soliton scale via $R_T \propto (M/\rho_T)^{1/3}$. By contrast, $\rho_{\rm trans}$ is an emergent screening threshold for the onset of halo phenomenology in diffuse environments; it is not treated as a second fundamental constant and can depend on coupling, geometry, and baryonic structure.

Theoretical Expectation from \(\rho_T\)

The TEP scaling law \(R_T \propto M^{1/3}\), anchored by the GNSS calibration \(\rho_T \approx 20\) g/cm³, has a direct consequence for galactic structure. In the TEP framework, the dark-matter onset radius \(R_{\rm DM}\) is identified with the radius at which the enclosed mean baryonic density drops below the emergent halo threshold \(\rho_{\rm trans}\). Because \(\rho_{\rm trans}\) emerges directly from the two-branch Gauss–Klein–Gordon flux equation \(\nabla\cdot(P_{,X}\nabla u) = (\beta_A/M_{\rm Pl}^2)\rho_b\) inside rising 3-D baryonic cores (where the coupled-gradient crossing acceleration \(g_{\rm crit}(\tau=1.30) = 1.031 \times 10^{-10}\text{ m s}^{-2}\) [$4.49 \times 10^{-11}\text{ m s}^{-2}$ uncoupled] and kinetic column-density scale \(\Sigma_* = 582\,M_\odot\text{ pc}^{-2}\) set the characteristic 3-D core density \(\rho_{\rm core} \sim \Sigma_*/(2h_z) \approx 0.5\,M_\odot\text{ pc}^{-3}\) and enclosed transition density \(\bar\rho_{\rm trans} = 3g_{\rm crit}/(4\pi G R_{\rm DM}) = 0.068\,M_\odot\text{ pc}^{-3}\) at \(R_{\rm DM} = 2.6\text{ kpc}\); see the Formal Link box below and Step 14), dimensional analysis in the 3-D rising-core regime (\(M_{\rm bar} \lesssim M_{\rm cross}^{\rm SPARC}\)) yields:

Thus the TEP framework expects \(\alpha = 1/3\) for the DM onset scaling. This is a theoretical consequence of the same saturation scale that fixes the terrestrial correlation length, not a new free parameter. The SPARC analysis below tests whether the empirical exponent \(\alpha_{\rm SPARC}\) is consistent with this expectation. Notably, in the unscreened low-density outer halo (\(\rho \ll \rho_T\)), the coupled-gradient quadratic \(a_\phi(a_\phi + g_{\rm bar}) = g_{\rm bar}a_{\rm eff}\) with \(g_{\rm obs} = g_{\rm bar} + a_\phi\) algebraically recovers the MOND simple interpolation function \(g_{\rm bar} = g_{\rm obs}^2/(g_{\rm obs} + a_{\rm eff})\) [\(\mu(x) = x/(1+x)\), \(x = g_{\rm obs}/a_{\rm eff}\)], explaining why TEP reproduces the tight \(0.1330\text{ dex}\) radial-acceleration scatter of MOND (\(0.1328\text{ dex}\)) from the two-branch kinetic operator \(P_{,X}(\xi) = k\sqrt{\xi} + 2\xi\) without an ad hoc interpolation ansatz, while predicting both the 3-D rising-baryon core regime \(R_{\rm DM} \propto M_{\rm bar}^{1/3}\) (\(\alpha = 0.256 \pm 0.069\), consistent with \(1/3\) at \(1.1\sigma\) and \(3.5\sigma\) below the point-mass acceleration-tail exponent \(1/2\)) and five non-galactic \(\rho_T\)-saturated channels where acceleration-only MOND makes zero predictions: (i) terrestrial GNSS atomic-clock spatial coherence at \(L_c \approx R_T(M_\oplus) \approx 4{,}200\text{ km}\) fixed by \(\rho_T \approx 20\text{ g/cm}^3\); (ii) outer-planet equatorial rotation \(v_{\rm eq} = GM/(R_T v_{\rm gal})\); (iii) the Tannock et al. brown-dwarf period floor \(P_{\min}^{\rm th} = (k_{\rm MOI}/\eta)\tau_T(\rho_T) = 1.155\text{ hr}\); (iv) the Solar Model S convection-zone base \(\sqrt{3}R_T(M_\odot) = 0.715\,R_\odot\); and (v) relativistic lensing (Paper 4) and bi-metric gravitational-wave propagation \(\Xi(z) = (1+z)^{-1}\) (Paper 22).

The analysis was performed post hoc: the SPARC data were examined after the TEP framework was established, so the comparison is a consistency check rather than a blind prediction. A measured exponent \(\alpha_{\rm SPARC} \neq 1/3\) (outside measurement uncertainty) would challenge the framework, while agreement supports it.

Methodology

The analysis proceeds as follows for each of the 175 SPARC galaxies:

1. Calculate the total baryonic mass: $M_{\rm bar} = M_* + 1.33 M_{\rm HI}$, where $M_* = (M/L)_{3.6\mu} \times L_{3.6}$ with $(M/L)_{3.6\mu} = 0.5\,M_\odot/L_\odot$ for disk populations.
2. Compute the expected Newtonian rotation velocity using the SPARC signed-contribution convention: $V_{\rm bar}^2 = |V_{\rm gas}|V_{\rm gas} + (M/L)_{\rm disk} |V_{\rm disk}|V_{\rm disk} + (M/L)_{\rm bulge} |V_{\rm bulge}|V_{\rm bulge}$, preserving negative $V_{\rm gas}^2$ in central H I depressions.
3. Identify the mass discrepancy onset radius $R_{\rm DM}$: the first radius where $V_{\rm obs}/V_{\rm bar} > 1.3$. Radial bins with nonpositive or unreliable baryonic velocity ($V_{\rm bar} \leq 5$ km s$^{-1}$, including $V_{\rm bar}^2 \leq 0$ from negative model contributions) are excluded from the ratio; galaxies whose ratio never exceeds the threshold contribute no $R_{\rm DM}$ and are dropped from the fit — a selection, disclosed here — leaving 168 of 175 systems at the fiducial threshold (or 167 under unsigned $V_{\rm gas}^2$).
4. Fit the power-law relation $R_{\rm DM} = k \cdot M_{\rm bar}^\alpha$ across the full sample.

Results

Of the 175 SPARC galaxies, 168 yield valid mass discrepancy onset radii spanning five decades in baryonic mass ($10^7$–$10^{12}\,M_\odot$).

Robustness to Onset Definition

To address concerns that the "onset radius" is a noise-sensitive functional, a bootstrap analysis (1000 resamples) was performed across multiple definition criteria. The scaling exponent $\alpha_{\rm SPARC}$ remains consistent with the TEP expectation ($\alpha_{\rm SPARC} = 1/3$) across a wide range of velocity ratio thresholds:

Two quantitative effects explain the progression across Table 3.1. First, under the coupled-gradient two-branch acceleration law $g_{\rm obs} = g_N + a_\phi = \tfrac{1}{2}g_N + \sqrt{(g_N/2)^2 + g_N a_{\rm eff}}$ (Paper 0 §7; Step 10), at the exact transition point $g_N(R_{\rm DM}) = a_{\rm eff}$ the theoretical velocity ratio is $V_{\rm obs}/V_{\rm bar} = \sqrt{g_{\rm obs}/g_N} = \sqrt{(1+\sqrt{5})/2} = 1.272 \approx 1.3$, which derives the fiducial $1.3$ threshold directly from the action without free parameters. Second, at the loose threshold ($V_{\rm obs}/V_{\rm bar} > 1.1$), $122$ of the $170$ galaxies ($71.8\%$) already exceed $1.1$ at their innermost telescope beam-sampling bin ($\text{idx} = 0$, $R_{\min}$), so the discrete $0.272$ slope is heavily left-censored by $R_{\min}$; when evaluated on the resolved interior crossings against the enclosed baryonic mass $M_{\rm bar}(<R_{\rm DM})$ required by Gauss's law, the exponent across thresholds $1.10, 1.20,$ and $1.30$ is $0.315 \pm 0.027$, $0.338 \pm 0.021$, and $0.354 \pm 0.016$ ($r = 0.934, p = 2.7 \times 10^{-34}$), with log-scatter dropping from $0.48$ dex (total mass) to $0.180$ dex (enclosed mass), while higher thresholds ($1.5$ and outer $g_N = a_{\rm eff}$) sample farther out into the flat/Keplerian 2-D tail where $\alpha_{\rm pred} \to 1/2 = 0.500$ ($0.502 \pm 0.045$ in Step 12; and $\alpha_{\rm rising} = 0.256 \pm 0.069$ vs. $\alpha_{\rm tail} = 0.581 \pm 0.086$, $p = 0.0031$ under the a priori Freeman split at $\gamma_{\rm Freeman} = 0.210$ in Step 8).

The bootstrap-marginalized exponent deviates from the TEP expectation of $\alpha_{\rm SPARC} = 1/3 = 0.333$ by approximately 0.2$\sigma$ under the combined uncertainty ($\sim 0.4\sigma$ statistical-only)—consistent within typical statistical tolerance. The correlation coefficient $r \approx 0.6$ indicates a significant relationship. Importantly, this result is robust: individual threshold choices yield exponents ranging from 0.27 to 0.41, but the ensemble average converges near 1/3.

Size-Scaling Confound Test

The $M^{1/3}$ onset exponent is not by itself discriminating: for approximately self-similar exponential disks, a fixed velocity-ratio crossing occurs at a nearly fixed multiple of the disk scale length $R_d$, so the galaxy mass–size relation can produce the same exponent without any density threshold. The confound is tested directly on the same SPARC sample (pipeline step 8). The in-sample mass–size relation is $R_d \propto M_{\rm bar}^{0.322 \pm 0.017}$ ($r = 0.821$, $n = 175$), which alone predicts an onset exponent $\approx 0.32$ — within the stated uncertainty of the measured $0.352 \pm 0.043$. Comparing the two candidate invariants per galaxy across all discrete first-bin crossings ($n = 168$) — the size ratio $R_{\rm DM}/R_d$ and the enclosed mean baryonic density $\bar\rho(<R_{\rm DM}) = 3M_{\rm bar}(<R_{\rm DM})/(4\pi R_{\rm DM}^3)$, with $M_{\rm bar}(<R)$ estimated spherically as $V_{\rm bar}(R)^2 R/G$ — the size invariant is mass-independent on the raw discrete grid at the fiducial threshold (slope $0.053 \pm 0.044$, $p = 0.23$; median $R_{\rm DM}/R_d = 0.65$), while the enclosed density retains a residual mass correlation (slope $0.196 \pm 0.069$, $p = 4.7\times10^{-3}$). However, a diagnostic of the discrete grid reveals the cause: in $93$ of the $168$ galaxies ($55.4\%$; $112/169$ at threshold $1.2$ and $80/162$ at $1.4$), low-mass dwarfs already satisfy $V_{\rm obs}/V_{\rm bar} > 1.3$ at their innermost sampled rotation-curve point ($\text{idx} = 0$), so the discrete estimator is left-censored at the innermost beam/sampling radius $R_{\rm min}$ rather than measuring the true interior threshold crossing. Because rotation-curve radial grids scale with optical/H I disk size ($r(\log_{10}R_{\min},\,\log_{10}R_d) = +0.410, p = 4.6\times 10^{-5}$ across the $93$ censored systems, with median $R_{\min}/R_d = 0.396$ at mean $\log_{10}[M_{\rm bar}/M_\odot] = 9.51$ and negative internal slope $\mathrm{d}\log_{10}(R_{\min}/R_d)/\mathrm{d}\log_{10}M_{\rm bar} = -0.245, p < 10^{-4}$), pooling these $93$ telescope-sampling-floor points with the $75$ resolved interior crossings (mean $\log_{10}[M_{\rm bar}/M_\odot] = 10.17$, median $R_{\rm DM}/R_d = 1.236$, internal slope $+0.116 \pm 0.055$) produces a classic Simpson's-paradox cancellation on the raw $N = 168$ grid (apparent pooled slope $+0.053 \pm 0.044, p = 0.23$) while artificially inflating the apparent $M_{\rm bar}^{1/3}$ residual scatter from $0.366$ dex ($n = 75$ resolved crossings; $0.359$ dex OLS and $0.180$ dex against enclosed mass $M_{\rm bar}(<R_{\rm DM})$) to $0.483$ dex. Evaluating a left-censored Tobit maximum-likelihood regression across all $N = 168$ galaxies (retaining all $93$ left-censored low-mass systems as upper bounds $R_{\rm DM} \le R_{\min}$ alongside the $75$ resolved interior crossings, Step 8) rejects pure disk-scale invariance ($R_{\rm DM}/R_d = \text{const}$) with Tobit slope $+0.323 \pm 0.034$ ($z = 9.57, p < 10^{-4}$; whereas a right-censored Tobit regression on enclosed density $\rho_{\rm enc}(R_{\rm DM}) \ge \rho_{\rm enc}(R_{\min})$ gives a mass-invariant slope of $-0.112 \pm 0.131, p = 0.39$ at threshold $1.2$ and $-0.155 \pm 0.014$ at $1.3$, and because low-surface-brightness dwarfs with $\Sigma_0 < \Sigma_{\rm crit}$ have $R_{\rm DM} = 0$ physically, treating them as log-normal upper bounds $\log_{10} R_{\rm DM} \le \log_{10} R_{\min}$ pulls the low-mass tail toward $-\infty$ and inflates the apparent Tobit radius slope to $0.61 \pm 0.05$ above the resolved interior crossing slope $0.417 \pm 0.050$, which lies between the $M^{1/3}$ saturation scaling and the Step 12 two-branch kinetic $M^{1/2}$ acceleration-crossing slope $0.502$). Likewise, on the $n = 75$ resolved interior crossings at the fiducial $1.3$ threshold, the enclosed baryonic density is the sole statistically mass-invariant candidate among the three physical onset hypotheses (slope $0.075 \pm 0.075$, $p = 0.32$, Spearman $p = 0.56$; scatter contracting from $0.82$ to $0.55$ dex), whereas the disk-size ratio $R_{\rm DM}/R_d$ exhibits a residual mass trend (slope $0.116 \pm 0.055$, $p = 0.039$, Spearman $p = 0.018$; at threshold $1.4$, $n = 82$, both $R_{\rm DM}/R_d$ at $+0.162 \pm 0.048, p = 1.2\times10^{-3}$ and $\rho_{\rm enc}$ at $-0.147 \pm 0.074, p = 0.049$ show residual mass trends) and the local baryonic acceleration at onset $g_{\rm bar}(R_{\rm DM}) = V_{\rm bar}^2(R_{\rm DM})/R_{\rm DM}$ is strongly mass-dependent (slope $+0.492 \pm 0.045$, $p = 3.8\times10^{-17}$). To test kinematic vs. photometric scaling without sharing $R_{\rm DM}$ on both axes (as would occur if $R_{\rm DM}$ were regressed directly on $M_{\rm bar}(<R_{\rm DM}) = V_{\rm bar}^2 R_{\rm DM}/G$), Step 8 evaluates the non-circular partial correlation of $\log_{10} R_{\rm DM}$ with the purely kinematic velocity $\log_{10} V_{\rm bar}(R_{\rm DM})$ controlling for $\log_{10} R_d$ across the $n = 75$ resolved crossings, yielding $r_{\rm part}(R_{\rm DM},\,V_{\rm bar}\mid R_d) = +0.754$ ($p = 5.6\times10^{-15}$) versus $r_{\rm part}(R_{\rm DM},\,R_d\mid V_{\rm bar}) = +0.171$ ($p = 0.14$, statistically null; or $r_{\rm part}(R_{\rm DM},\,V_{\rm bar}(2.2R_d)\mid R_d) = +0.547, p = 3.8\times10^{-7}$ vs. $r_{\rm part}(R_{\rm DM},\,R_d\mid V_{\rm bar}(2.2R_d)) = +0.224, p = 0.053$ when $V_{\rm bar}$ is evaluated at the fixed photometric disk peak $2.2R_d$ with zero dependence on $R_{\rm DM}$). Furthermore, to test whether the intermediate full-sample slope ($0.417 \pm 0.050$) genuinely reflects a physical transition between the 3-D volume-density core regime ($\alpha = 1/3$) and the outer 2-D Keplerian acceleration tail ($\alpha = 1/2$) rather than post-hoc range flexibility, the $n = 75$ resolved galaxies are partitioned a priori — using only their Newtonian baryonic rotation curve $V_{\rm bar}(r)$ with zero dependence on $V_{\rm obs}$ or $R_{\rm DM}$ — via the Freeman (1970) exponential-disk rotation profile (Step 8). Combining the in-sample SPARC mass–size relation ($R_d \propto M_{\rm bar}^{0.322}$) with the Freeman peak acceleration $g_{\rm peak}(M_{\rm bar}) = 0.180\,GM_{\rm bar}/R_d^2$ predicts a regime crossover mass of $M_{\rm cross}^{\rm SPARC} = 1.45 \times 10^{10}\,M_\odot$ ($\log_{10}[M_{\rm cross}/M_\odot] = 10.16$), below which disk peaks fall below the crossing acceleration and transitions occur inside the rising 3-D volume-density core ($\alpha_{\rm pred} = 1/3 = 0.333$), and above which transitions occur outside the turnover in the 2-D Keplerian tail ($\alpha_{\rm pred} = 1/2 = 0.500$). Partitioning the $n = 75$ resolved galaxies a priori at the exact analytical Freeman logarithmic velocity slope between $1.0\,R_d$ and $2.2\,R_d$, $\gamma_{\rm Freeman} \equiv \ln[V_{\rm Freeman}(2.2R_d)/V_{\rm Freeman}(1.0R_d)]/\ln 2.2 = 0.210$ (zero dependence on $V_{\rm obs}$ or $R_{\rm DM}$), galaxies with rising inner baryonic curves ($\gamma_V \ge 0.210$, $n = 38$, median $M_{\rm bar} = 5.9 \times 10^9\,M_\odot < M_{\rm cross}^{\rm SPARC}$, pre-specified $\alpha_{\rm pred} = 1/3$) yield $\alpha_{\rm rising} = 0.256 \pm 0.069$ (consistent with $1/3$ at $1.1\sigma$, rejecting $1/2$ at $3.5\sigma$), whereas galaxies with concentrated or flat baryonic curves ($\gamma_V < 0.210$, $n = 37$, median $M_{\rm bar} = 4.8 \times 10^{10}\,M_\odot > M_{\rm cross}^{\rm SPARC}$, pre-specified $\alpha_{\rm pred} = 1/2$) yield $\alpha_{\rm tail} = 0.581 \pm 0.086$ (consistent with $1/2$ at $0.95\sigma$, rejecting $1/3$ at $2.9\sigma$; subsample shift $\Delta\alpha = +0.325 \pm 0.110$, $z = 2.95, p = 0.0031$). Decomposing the selection sensitivity between the full discrete sample ($N = 168$, $\alpha = 0.369 \pm 0.041$) and the resolved subset ($n = 75$, $\alpha = 0.417 \pm 0.050$) shows that $+0.028$ of the $+0.048$ difference arises from excluding the $93$ $R_{\min}$-censored low-surface-brightness dwarfs (on the discrete grid, the $n = 75$ uncensored subset gives $\alpha = 0.397 \pm 0.045$), while $+0.020$ arises from sub-bin log-linear interpolation.

Complementing the resolved interior crossings, the full-profile channels test the radial acceleration curve without relying on a single crossing point — the residual correlations with baryonic content (Step 7) and the two-branch kinetic profile $P_{,X}=\epsilon_{\rm reg}+k\sqrt\xi+2\xi$ ($k_*=16.03$ calibrated to $a_0$, $\epsilon_{\rm reg}=10^{-3}$; Steps 10, 12, and 13), which reproduces the SPARC radial-acceleration relation under the coupled-gradient additive force law $g_{\rm obs} = g_{\rm bar} + a_\phi = \tfrac{1}{2}g_{\rm bar} + \sqrt{(g_{\rm bar}/2)^2 + g_{\rm bar}a_{\rm eff}}$ ($a_\phi(a_\phi + g_{\rm bar}) = g_{\rm bar}a_{\rm eff}$; Paper 0 §7) at $0.1330$ dex RMS ($61.1\%$ within $0.1$ dex, median $-0.010$ dex at canonical $\Upsilon_d=0.50$, matching the empirical MOND reference $0.1328$ dex [$60.1\%$ within $0.1$ dex] to $0.0002$ dex with zero free parameters; uncoupled self-gradient additive $0.1522$ dex at $\Upsilon_d=0.50$ and $0.1392$ dex at $\Upsilon_d=0.35$, quadrature $0.1394$ dex, vs $0.2490$ dex for the unit-term baseline) across the standard quality-controlled SPARC RAR sample ($N=2{,}700$ points across $149$ galaxies with $Q\le 2$, $i\ge 30^\circ$, $e_{V}/V_{\rm obs}\le 0.10$; Lelli et al. 2017), and $0.1994$ dex ($53.9\%$ within $0.1$ dex vs MOND $0.1989$ dex, $53.2\%$; uncoupled additive $0.2185$ dex at $\Upsilon_d=0.50$ / $0.2004$ dex at $\Upsilon_d=0.35$, quadrature $0.1974$ dex) across all $3{,}389$ unfiltered table points, alongside an outer-edge BTFR median $v_{\rm obs}/v_{\rm pred}=0.956$ (RMS $0.0807$ dex under coupled-gradient additive, and $1.0109$, RMS $0.0812$ dex under quadrature; Step 13) and an $a_{\rm eff}$-crossing onset slope $0.502$ vs theoretical $0.500$ across $N=54$ bracketed galaxies in Step 12 (where the $-0.173$ dex intercept offset relative to $\sqrt{GM_{\rm bar}(R_{\max})/a_{\rm eff}}$ reflects the median enclosed mass fraction $f_{\rm enc} = M_{\rm bar}(<r_{\rm on})/M_{\rm bar}(R_{\max}) = 0.65$, reducing the zero-parameter RMS from $0.267$ to $0.218$ dex).

Phantom-Mass Amplitude and Profile Test

The onset radius alone does not test the mechanism — any threshold criterion produces an onset. The discriminating content of the phantom-mass claim lies in the amplitude and the radial profile, both of which are evaluated directly on the same SPARC sample (pipeline step 9). In a canonical linear-kinetic scalar baseline ($P_{,X}\to 1$) with conformal charge $\beta_A$, an unscreened scalar channel contributes a centripetal term of order $2\beta_A^2$ times the Newtonian one, giving the linear-kinetic plateau ceiling $V_{\rm obs}/V_{\rm bar} = \sqrt{1 + 2\beta_A^2 \mathcal S_{\rm eff}}$, which for the bare coupling $|\beta_A| = 1$ at $\mathcal S_{\rm eff} = 1$ is $\sqrt{3} \approx 1.73$. Measured on the outer third of each curve ($n = 164$ galaxies with $\geq 6$ valid points and $V_{\rm bar} > 5$ km s$^{-1}$), the median observed-to-baryonic plateau ratio is $1.98$ ($16$–$84$th percentile $1.38$–$2.49$) — exceeding the linear-kinetic bare ceiling ($\mathcal S_{\rm eff} = 1.46$ at the median, with $71\%$ of galaxies requiring $\mathcal S_{\rm eff} > 1$). Crucially, evaluating the selected two-branch nonlinear kinetic completion $P_{,X}(\xi)=\epsilon_{\rm reg}+k\sqrt\xi+2\xi$ ($k=16.03$, zero free parameters) directly on the same 164 outer-third galaxy plateaus closes this linear-kinetic gap completely: the median observed-to-predicted plateau velocity ratio under the coupled-gradient additive law $a_\phi = \sqrt{(g_N/2)^2 + g_N a_{\rm eff}} - g_N/2$ is $V_{\rm obs}/V_{\rm pred}^{\rm TEP} = 0.952$ ($16$–$84$th percentile $0.814$–$1.110$, RMS $0.094$ dex) at the canonical $\Upsilon_d = 0.50$ and $1.009$ ($16$–$84$th percentile $0.870$–$1.158$, RMS $0.091$ dex) at $\Upsilon_d = 0.35$ (with $0.893$ and $0.952$ under uncoupled self-gradient addition, and $1.011$, RMS $0.090$ dex under the quadrature comparator).

The radial profile supplies the second discriminator. Defining the enhancement $E(R) = V_{\rm obs}^2/V_{\rm bar}^2 - 1 = M_{\rm ph}(R)/M_{\rm bar}(<R)$, the phantom fraction is required by the transition picture to grow outward (screened inside, unscreened outside), whereas a constant coupling rescaling of the baryonic potential predicts $d\log E/d\log R = 0$. The measured profile rises: median slope $0.46$ across 161 galaxies with sufficient baseline, positive in 133 (Wilcoxon signed-rank $p = 2.3\times10^{-17}$). The rising phantom fraction is the expected morphology of a baryon-anchored screening transition and is inconsistent with a global multiplicative rescaling; it is not, by itself, separable from acceleration-threshold phenomenologies, which predict the same qualitative rise. What the two tests jointly establish is that the SPARC sample carries both an amplitude-level and a profile-level constraint on the mechanism — the onset scaling is no longer the sole empirical content of the claim.

Cross-Regime Consistency

The significance of this result emerges when combined with the Temporal Topology screening analysis (Section 4). Across macroscopic compact and planetary bodies (Section 4), the saturation radius $R_T(M) = (3M/4\pi\rho_T)^{1/3}$ defines the transition into the screened compact-body regime ($S \ge 1$), whereas in diffuse galactic disks ($S \ll 1$) the onset of the phantom-mass regime transitions from $M^{1/3}$ core scaling in rising-baryon disks ($M_{\rm bar} \lesssim M_{\rm cross}^{\rm SPARC} = 1.45 \times 10^{10}\,M_\odot$) to $M^{1/2}$ low-acceleration scaling in concentrated disks ($M_{\rm bar} \gtrsim M_{\rm cross}^{\rm SPARC}$).

Combined with the RBH-1 Schwarzschild crossover ($R_T/R_S = 1.32$) and the Milky Way inner mass-discrepancy onset ($R_{\text{DM}} \approx 3.1$ kpc, observed $\sim 3$–$5$ kpc), these results link the compact-object saturation sector and the low-acceleration galactic kinetic sector within the two-branch TEP framework.

Physical Interpretation

The observed $M^{1/3}$ scaling admits a natural interpretation in terms of density-limited screening. If the screening transition occurs at a characteristic density $\rho_{\rm trans}$, dimensional analysis requires:

Fixing $\alpha_{\rm SPARC} = 1/3$ and fitting only the normalization yields $k \approx 8.0 \times 10^{-4}\,{\rm kpc}/M_\odot^{1/3}$ ($7.96 \times 10^{-4}$). This normalization corresponds to an emergent screening density $\rho_{\rm trans} \approx 0.5\,M_\odot/{\rm pc}^3$ ($\sim 3 \times 10^{-23}$ g/cm³) recovered from the fitted normalization $k$ via Eq. ($\ref{eq:trans_radius}$). As derived in the Formal Link box below (Eq. [\ref{eq:profile_link}], Step 14), this transition density follows directly from the two-branch kinetic operator $P_{,X}(\xi) = \epsilon_{\rm reg} + k\sqrt{\xi} + 2\xi$ ($g_{\rm crit} = 1.031 \times 10^{-10}\text{ m s}^{-2}$ coupled-gradient [$4.49 \times 10^{-11}\text{ m s}^{-2}$ uncoupled], $\Sigma_* = 582\,M_\odot\text{ pc}^{-2}$) with zero free parameters. This value is within an order of magnitude of typical disk densities at the optical radius (0.01–0.1 $M_\odot/{\rm pc}^3$), providing a physically reasonable anchor for the screening mechanism.

Note: This galactic transition density $\rho_{\rm trans}$ is distinct from the fundamental core saturation scale $\rho_T \approx 20$ g/cm³ calibrated in Section 2 and cross-checked in Section 5. The core density $\rho_T$ governs the compact object size (soliton surface), while $\rho_{\rm trans}$ governs the onset of the diffuse halo effect (screening radius). Both scales follow the same $M^{1/3}$ structural form, consistent with the environment-dependent nature of the theory across 15 orders of magnitude in density.

Connection to the MOND Acceleration Scale

The fitted screening density implies a characteristic transition acceleration. At the transition radius, the gravitational acceleration is:

This is within a factor of 4 of the MOND acceleration scale $a_0 \approx 1.2 \times 10^{-10}\,{\rm m/s}^2$ (Milgrom 1983). The near-coincidence suggests that the TEP screening mechanism and the empirical MOND phenomenology may share a common origin: both describe the transition from screened (Newtonian) to unscreened (modified) gravity at a characteristic acceleration scale set by cosmological boundary conditions.

Connection to the Radial Acceleration Relation

The SPARC database is the source of the celebrated Radial Acceleration Relation (RAR), which shows a tight empirical coupling between observed acceleration $g_{\rm obs}$ and baryonic acceleration $g_{\rm bar}$ (McGaugh et al. 2016). The RAR has been interpreted as evidence for modified gravity (MOND) or as a consequence of dark matter halo profiles fine-tuned to baryonic distributions.

The TEP framework offers a third interpretation: the RAR may emerge from the environment-dependent screening of the proper-time field. At high accelerations (high densities), screening is complete and $g_{\rm obs} \approx g_{\rm bar}$. At low accelerations (low densities), the unscreened time-field gradient contributes additional apparent gravity, producing the characteristic upturn in the RAR. The $M^{1/3}$ scaling of the transition radius is the spatial manifestation of this acceleration-dependent screening.

A fixed-normalization test applies the existing alternative low-gradient kinetic realization directly to the SPARC mass models (pipeline steps 10–13). Its response $P_{,X}(\xi)=k\sqrt{\xi}+2\xi$, with $\xi=|X|/\Lambda_X^4$, differs from the unit-term baseline $1+2\xi$ of Paper 0 §7 and yields the derived flat-tail and baryonic Tully–Fisher limits. With the cosmic temporal-gradient threshold fixed by the Hubble clock roll at $g_t=cH_0/2\simeq3.40\times10^{-10}$ m/s$^2$ and the DEF coupling normalization $\alpha_0=\sqrt{2}\beta_A=-\sqrt{2}$, the natural two-branch crossover $k=16$ ($\xi_*=(k/2)^2=64$) yields $a_{\rm eff}=4\sqrt{2}\,g_t/k=cH_0/(4\sqrt{2})=1.202\times10^{-10}$ m/s$^2$ ($k=4\sqrt{2}\,g_t/a_0\simeq16.03$ at the canonical $a_0=1.20\times10^{-10}$ m/s$^2$), deriving Milgrom's $a_0\sim cH_0$ coincidence from the cosmological time field with no galaxy-specific parameter adjusted across the 175-galaxy sample. The algebraic flux law $\bar{u}P_{,X}(\bar{u}^2/2) = g_{\rm bar}/g_t$ (with canonical zero-gradient regulator $P_{,X}(\xi)=\epsilon_{\rm reg}+k\sqrt{\xi}+2\xi$, $\epsilon_{\rm reg}=10^{-3}$ ensuring strict hyperbolicity at $X=0$ while leaving galactic scales $\xi\sim10^{-2}$–$1$ invariant to $<10^{-4}$) yields the temporal-shear acceleration $a_\phi = 2g_t\bar{u}$, where $k=4\sqrt{2}\,g_t/a_0\simeq16.03$ is calibrated to the empirical scale $a_0=1.20\times10^{-10}$ m/s$^2$ (so agreement of the deep-tail normalization with $a_0$ is a calibration input rather than an independent prediction, while the shape across the 175-galaxy sample has zero refitted parameters). From the point-particle action $L_{\rm pp}=-mc\sqrt{-\tilde g_{\mu\nu}dx^\mu dx^\nu}$, the action-derived radial force law is additive, $g_{\rm obs} = g_{\rm bar} + a_\phi$. In the causal matter metric $\tilde g_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav})U_\mu U_\nu + A_{\rm sp}^2\Pi_{\mu\nu}$, evaluating the dielectric Gauss–Klein–Gordon response on the coupled matter-frame potential gradient ($a_\phi(a_\phi + g_{\rm bar}) = g_{\rm bar}a_{\rm eff}$; Paper 0 §7) yields the closed-form two-branch shear acceleration $a_\phi(r) = \sqrt{(g_{\rm bar}/2)^2 + g_{\rm bar}a_{\rm eff}} - g_{\rm bar}/2$, so $g_{\rm obs} = \tfrac{1}{2}g_{\rm bar} + \sqrt{(g_{\rm bar}/2)^2 + g_{\rm bar}a_{\rm eff}}$ — reproducing the functional form of the MOND "simple" interpolating relation $g_{\rm bar} = g_{\rm obs}^2/(g_{\rm obs} + a_{\rm eff})$ (so that both functional forms share the same SPARC onset-radius crossings while here arising from a relativistic two-branch dielectric scalar action rather than modified inertia) and fixing the theoretical velocity ratio at $g_{\rm bar} = a_{\rm eff}$ to $V_{\rm obs}/V_{\rm bar} = \sqrt{(1+\sqrt{5})/2} = 1.272 \approx 1.3$. Evaluated at the canonical SPARC mass-to-light ratios ($\Upsilon_d = 0.50, \Upsilon_b = 0.70$) and $k = 16.03$ with zero free parameters, this coupled-gradient additive law achieves an RMS log-residual of $0.1330$ dex ($61.1\%$ within $0.1$ dex, median $-0.010$ dex) on the standard quality-controlled SPARC RAR sample ($N = 2{,}700$ points; matching the empirical MOND interpolating benchmark $0.1328$ dex, $60.1\%$ within $0.1$ dex, to $0.0002$ dex) and $0.1994$ dex ($53.9\%$ within $0.1$ dex vs. MOND $0.1989$ dex, $53.2\%$) across all $3{,}389$ unfiltered table points, while on the Step 13 outer-edge BTFR ($N = 405$ points across $135$ galaxies) it yields median $v_{\rm obs}/v_{\rm pred} = 0.956$ and RMS $0.0807$ dex (under the dielectric potential-share identification $\epsilon_{\rm host}(g_{\rm host}) \equiv a_\phi(g_{\rm host})/[g_{\rm host} + a_\phi(g_{\rm host})] = [\sqrt{1 + 4a_{\rm eff}/g_{\rm host}} - 1]/[\sqrt{1 + 4a_{\rm eff}/g_{\rm host}} + 1]$, where the unscreened scalar fraction of the total potential gradient $|\beta_A\nabla u_{\rm grav}|/|\nabla\Phi_{\rm eff}|$ sets the transmitted fraction of the ambient cosmological clock drift into the host halo well — a stated static-to-cosmological bridge whose full time-dependent two-body derivation remains open; Paper 0 §8 — evaluating across the star-forming spiral disk mass–size relation of Courteau et al. 2007 and Lange et al. 2015 at fixed $a_{\rm eff} = 1.20 \times 10^{-10}\text{ m s}^{-2}$ yields a parameter-free binary black hole [BBH] host-tracking consistency evaluation of $\langle\epsilon_{\rm host}\rangle_{\rm SFR}^{\rm UCD} = 0.236 \pm 0.054$ at $R_{\rm eff}$ [median $0.233$, $68\%$ CI $[0.182,\,0.290]$, $0.197$ with $25\%$ bulge/NSC hosts] and $\langle\epsilon_{\rm host}\rangle_{\rm SFR,\;disk\text{-}int}^{\rm UCD} = 0.243 \pm 0.052$ when integrated across the full radial Freeman Bessel-disk profile $R \in [0,\infty)$ weighted by star-formation surface density $x e^{-x}\,\mathrm{d}x$ [median $0.240$, full 2-D $(M_{\rm gal},R)$ progenitor dispersion $\pm 0.100$, and $0.202$ with $25\%$ bulge/NSC hosts; Step 14] — matching the empirical GWTC-1–5.0 optimum $\hat\epsilon = 0.25$ in Paper 22; crucially, evaluating $S(M,R) = R_T(M)/R = (\bar\rho/\rho_T)^{1/3}$ symmetrically at the hydrostatic core configuration where each gravitational mass threshold $M_* \propto G_{\rm loc}^{-3/2}$ is physically set screens Type Ia supernovae at the Chandrasekhar white-dwarf density [$\bar\rho_{\rm WD} \simeq 2 \times 10^6\text{ g/cm}^3 \gg \rho_T \implies S_{\rm WD} \simeq 46.4$, $\epsilon_{\rm SN\,Ia} = \epsilon_{\rm host}/S_{\rm WD} \simeq 0.0043 \ll 0.150$], whereas for BBHs the $e^+e^-$ pair-instability threshold $M_{\rm PISN}$ [Fowler & Hoyle 1964; Woosley 2017] is triggered in radiation-pressure-supported high-entropy He/C–O cores at $\bar\rho_{\rm PISN} \simeq 5\text{--}50\text{ g/cm}^3 \sim \mathcal{O}(\rho_T)$ [$S_{\rm PISN} \simeq 0.63\text{--}1.36 \approx 1$, alongside H-burning MS $\bar\rho_{\rm MS} \simeq 0.01\text{--}0.1\text{ g/cm}^3$, Wolf–Rayet He $\bar\rho_{\rm WR} \simeq 1\text{--}10\text{ g/cm}^3 < \rho_T$, and wide binary orbits $a_{\rm orb} \gg R_T \gg R_S$ prior to the $\sim 1\text{ s}$ dynamical free-fall collapse], leaving $\epsilon_{\rm BBH} \simeq \epsilon_{\rm host}$ unscreened). By contrast, if $P_{,X}(\bar u^2/2)$ is evaluated using only the uncoupled scalar self-gradient $\bar u^2$ ($a_\phi \simeq \sqrt{g_{\rm bar}a_{\rm eff}}$, omitting the $-g_{\rm bar}/2$ cross-term), direct addition $g_{\rm bar} + a_\phi$ carries a $+2g_{\rm bar}a_\phi$ crossover offset at $\Upsilon_d = 0.50$ ($0.1522$ dex quality-controlled, $0.2185$ dex unfiltered, $k_{\rm fit} \approx 30.9$; absorbed to $0.1392$ and $0.2004$ dex at $\Upsilon_d = 0.35$, or $0.1394$ and $0.1974$ dex under the quadrature comparator $\sqrt{g_{\rm bar}^2 + a_\phi^2}$). A field-value coupling $K(\phi) = e^{c\psi}$ improves the RAR match monotonically (rms $\to 0.199$ dex at $c = 3$) but is excluded by a sign tension with the wide-binary pair channel, which requires the opposite sign, so the monotone form is not adopted. The residual offset is quantified by the accompanying scans: disk mass-to-light $\Upsilon_d = 0.35$ alone absorbs the offset (rms $0.2004$ dex, median $-0.020$ dex — indistinguishable from the MOND reference at zero additional sector parameters); the ambient-shear scan bounds the cosmic-web background at $u_{\rm amb} \lesssim 0.02$ (larger values degrade the deep tail monotonically), validating the isolated-source assumption; and a small canonical term $\epsilon \lesssim 0.1$ in $P_{,X} = \epsilon + k\sqrt{\xi} + 2\xi$ is observationally free and marginally preferred (rms $0.2098$ dex at $\epsilon = 0.1$), while order-unity values are excluded not by the data but by the flat-asymptote requirement — a linear tail reappears below $g_{\rm bar} \sim (\epsilon/k)\,g_t$. The offset is therefore a baryonic mass-model systematic, not a sector-shape failure. Note that the $0.1330$ dex RMS on the quality-controlled $N = 2{,}700$ sample ($0.1994$ dex across all $3{,}389$ raw points; $0.139$–$0.152$ dex and $0.200$–$0.218$ dex for the uncoupled/quadrature comparators) reported here is obtained at fixed global mass-to-light ratios ($\Upsilon_d = 0.35$ or $0.50$, $\Upsilon_b = 0.70$) and zero per-galaxy refitted parameters; by contrast, SPARC RAR analyses that marginalize over individual galaxy mass-to-light ratios, distances, and inclinations (e.g., Li et al. 2018, who obtain $\sim 0.057$ dex residual scatter after fitting three nuisance parameters per galaxy, or Desmond 2023, who jointly infers the intrinsic RAR scatter and galaxy-level systematics) absorb the galaxy-to-galaxy $\Upsilon_d$ variance that remains unadjusted in the present zero-refit test.

Linking RBH-1 to Galactic Halos

The SPARC analysis identifies a cross-scale consistency between the RBH-1 wake phenomenon and galactic dark matter halos:

• RBH-1 represents a "naked saturation"—a Temporal Topology saturation not anchored to a baryonic host, propagating freely through the intergalactic medium and inducing star formation via its time-dilation wake.
• Galactic halos represent "baryon-anchored solitons"—the same temporal field structure, but centered on and shaped by the baryonic mass distribution of the host galaxy.

Both phenomena are consistent with the same $M^{1/3}$ scaling law, calibrated from terrestrial GNSS observations. The universality of this scaling across planetary, stellar, galactic, and cosmological mass scales constitutes a central theoretical consequence of the Temporal Equivalence Principle.

Residual Analysis: Discriminating Baryonic Feedback vs Field Theory

The RMS scatter of 0.48 dex around the $M^{1/3}$ relation indicates substantial galaxy-to-galaxy variation. A critical test distinguishes whether this scatter arises from baryonic feedback processes (Standard Model) or from the intrinsic field theory mechanism (TEP): correlate the residuals with baryonic properties versus screening proxies.

Methodology

For each of the 168 SPARC galaxies with valid mass discrepancy radii, the residual from the $M^{1/3}$ expectation is calculated as:

where $R_{\rm exp} = k \cdot M_{\rm bar}^{1/3}$ with $k = 7.96 \times 10^{-4}$ kpc/$M_\odot^{1/3}$. These residuals are then correlated with:

• Baryonic properties. Gas fraction ($f_{\rm gas}$), surface brightness ($\Sigma$), and inclination, treated as proxies for feedback efficiency, star formation history, and observational systematics.
• Screening proxy. Central density ($\rho_{\rm central}$), treated as a proxy for local screening strength independent of $R_{\rm DM}$.

Results

The residual correlations reveal a clear pattern:

Interpretation

When evaluated using the measured SPARC Table 1 effective radii $R_{\rm eff}$, inclinations $i$, and signed gas contributions (step_7_sparc_residuals.json, $n = 168$), none of the baryonic or observational properties exhibits a strong correlation with the scaling residuals ($\max |r| = 0.153$): inclination ($r = +0.096$, $p = 0.217$) and surface brightness ($r = +0.047$, $p = 0.543$) are statistically null, while gas fraction shows only a weak nominal trend ($r = -0.153$, $p = 0.048$, $|r| < 0.2$) that does not survive Bonferroni correction across the four tested comparisons ($p_{\rm adj} = 0.191$). This confirms that the scaling relation is not driven by rotation-curve deprojection systematics or baryonic feedback efficiency.

The central density ($r = +0.016$, $p = 0.834$) likewise shows no residual trend after the primary mass scaling is removed, consistent with the full rotation-curve radial-acceleration fit in Steps 10 and 13 where the two-branch profile $P_{,X}=\epsilon_{\rm reg}+k\sqrt\xi+2\xi$ ($k=16.03$, $\epsilon_{\rm reg}=10^{-3}$) accounts for the complete radial profile under the coupled-gradient additive law at $0.1330$ dex RMS ($61.1\%$ within $0.1$ dex at canonical $\Upsilon_d=0.50$, vs MOND $0.1328$ dex; uncoupled additive $0.1522$ dex at $\Upsilon_d=0.50$, $0.1392$ dex at $\Upsilon_d=0.35$, and $0.1394$ dex quadrature) on the quality-controlled $N=2{,}700$ SPARC RAR sample and $0.1994$ dex ($53.9\%$ within $0.1$ dex at $\Upsilon_d=0.50$; uncoupled $0.2185$ dex, $0.2004$ dex at $\Upsilon_d=0.35$, $0.1974$ dex quadrature) across all $3{,}389$ raw points.

Scatter Quantification and Comparative Analysis

The observed scatter of $\sigma = 0.48$ dex around the $M^{1/3}$ relation is substantial (corresponding to a factor of $\sim 3$ variation). Quantifying the origin of this scatter is essential for distinguishing the TEP signal from competing hypotheses.

Intrinsic vs. Measurement Scatter

Standard error budgeting suggests that a significant fraction of this scatter is observational. Distance uncertainties in the SPARC sample typically range from 10–20%, which propagate to a ~0.1–0.2 dex uncertainty in $R_{\rm DM}$. Inclination corrections and non-circular motions contribute additional noise. However, the residual scatter ($\sim 0.3$ dex) likely reflects intrinsic variation in the galaxy population.

Comparison with MOND and LCDM+Feedback

MOND: The Modified Newtonian Dynamics framework typically yields a tighter scatter ($\sim 0.13$ dex) in the Radial Acceleration Relation (RAR). This is expected because MOND modifies gravity based on instantaneous local acceleration, a continuous variable. In contrast, this analysis extracts a single discrete parameter ($R_{\rm DM}$) from each rotation curve, a method inherently more sensitive to local irregularities and noise than the integrated RAR analysis. The larger scatter in $R_{\rm DM}$ does not invalidate the mean scaling; rather, it indicates that the "onset radius" is a noisy estimator of the underlying screening transition.

LCDM + Feedback (Null Hypothesis): The standard cosmological model relies on baryonic feedback to explain rotation curve diversity. In this scenario, the "dark matter onset" is not a fundamental scale but an emergent property of halo assembly and feedback history. Hydrodynamic simulations (e.g., NIHAO, FIRE) typically predict a scaling of $R_{\rm DM} \propto M^{\alpha}$ with $\alpha \approx 0.3$–$0.4$, broadly consistent with observation. However, they offer no account of why the galactic relation shares its structural exponent with the compact-object scaling. The fitted core normalization ($\rho_{\rm trans} \approx 0.5\,M_\odot/{\rm pc}^3 \sim 3\times10^{-23}$ g/cm³ at inner core radii, and $\bar\rho_{\rm trans} = 0.068\,M_\odot/{\rm pc}^3$ [$0.030\,M_\odot/{\rm pc}^3$ uncoupled] at the median crossing radius $R_{\rm DM} \approx 2.6\text{ kpc}$) follows directly from the kinetic sector's column-density and coupled-gradient acceleration scale ($g_{\rm crit} = 1.031 \times 10^{-10}\text{ m s}^{-2}, \Sigma_* = 582\,M_\odot\text{ pc}^{-2}$; Eq. [\ref{eq:profile_link}], Step 14), cleanly separating the kinetic halo onset scale $\Lambda_X = \sqrt{M_{\rm Pl}H_0}$ from the compact potential saturation density $\rho_T \approx 20\text{ g/cm}^3$.

Reproducibility

Full analysis code, data products, and methodology documentation are available in the public repository:

• Repository. github.com/matthewsmawfield/TEP-UCD
• SPARC Scaling Analysis. scripts/steps/step_4_sparc_analysis.py
• SPARC Residual Analysis. scripts/steps/step_7_sparc_residuals.py
• Two-Branch RAR & BTFR Analysis. scripts/steps/step_10_two_branch_sparc_rar.py, scripts/steps/step_13_two_branch_btfr.py
• Scale Audit & Planetary Rotational Kinematics. scripts/steps/step_14_scale_audit.py, scripts/steps/step_15_planetary_rotation.py, scripts/steps/step_16_saturn_interior.py
• Input Data. data/sparc/ (SPARC database tables)

Methodological Details

The "onset radius" $R_{\rm DM}$ is defined as the first radial bin where $V_{\rm obs}/V_{\rm bar} > 1.3$ (fiducial threshold). Specifically:

1. For each galaxy, the rotation curve is evaluated at the native SPARC radial sampling (not interpolated).
2. The velocity ratio $V_{\rm obs}/V_{\rm bar}$ is computed at each radius.
3. $R_{\rm DM}$ is the radius of the first bin exceeding the threshold.
4. Galaxies with non-monotonic or noisy $V_{\rm obs}/V_{\rm bar}$ profiles are retained; the "first crossing" definition is robust to subsequent fluctuations.
5. Galaxies that never exceed the threshold (7 of 175) are excluded from the fit.

The headline value $\alpha_{\rm SPARC}=0.352\pm0.043$ is the threshold-marginalized bootstrap estimate, obtained by resampling galaxies while marginalizing over the accepted onset definitions. The fiducial ($V_{\rm obs}/V_{\rm bar}>1.3$) definition alone gives $0.369\pm0.041$; the difference is retained as onset-definition systematic uncertainty. Varying the threshold from 1.1 to 1.5 yields exponents from 0.27 to 0.41, bracketing the headline estimate.

4. Temporal Topology Screening Analysis

If a scalar field permeates spacetime, why has it not been detected in precision gravitational experiments? This section answers that question. The key insight is that the scalar field's influence depends on the geometric overlap of topological charge cores: in environments where charge cores are tightly packed, the field is suppressed—a phenomenon called Temporal Topology screening. Rather than invoking discrete thin-shell boundaries, screening operates via the continuous spatial profile of the scalar field (Temporal Topology). The tight geometric packing in deep potential wells suppresses the local field gradient (Temporal Shear), ensuring short-range Temporal Shear suppression while leaving the field light cosmologically. General Relativity is recovered in the regimes where it has been tested most stringently.

The screening factor $S$ quantifies this suppression. It is defined as the ratio of the Temporal Topology saturation radius (where the scalar field saturates) to the physical radius of the object:

The quantity $S$ is used here as a geometric proxy for how deeply the baryonic source is embedded within a saturated soliton configuration. When $S \gg 1$, the physical object occupies a small region relative to the saturated field scale; in this regime the phenomenology assumes that non-linear response in the scalar sector suppresses gradients in the dense interior, recovering GR to high precision in local dynamics. When $S \sim 1$, the system lies near the transition between strongly screened and weakly screened behavior. When $S \ll 1$, the Temporal Topology saturation scale is smaller than the object and the saturated region does not envelop the full baryonic configuration; in this regime the model does not assume strong Temporal Topology suppression a priori, and constraints must be assessed case-by-case.

in the screened or source-charge-suppressed limit. The geometric factor $S$ is not itself the observable PPN coupling. Precision tests constrain $\alpha_{\rm PPN}^{\rm eff}$; $S$ is used here only as a proxy for the nonlinear suppression regime.

The near-transition regime $S \sim 0.4$–$0.7$ is resolved by the corpus's channel-projection structure (Paper 0). The screening response is not a single function of $S$: clock-comparison and transport observables probe the amplitude projection $S_A$, while orbital and geodesic observables probe the source-charge/shear projection $\mathcal S_\Sigma$, which is suppressed more strongly for the pairwise configurations used in ephemeris tests. The same geometric $S$ therefore places different channels on different sides of their respective thresholds: Earth's $S \approx 0.66$ admits a detectable clock-covariance signal ($C_A$ covariance channel, the GNSS calibration of Section 2) while the LLR, Cassini, and MESSENGER orbital observables at $S = 0.42$–$0.65$ remain GR-consistent through the $\mathcal S_\Sigma$ projection. The reading of sub-unity $S$ is uniform throughout this paper: $S < 1$ denotes that the saturation region does not envelop the object, i.e. the unscreened side of the transition; the observable amplitude in any given channel is then set by that channel's projection, not by $S$ alone.

4.1 The White Dwarf Stress Test

White Dwarfs are the ideal stress test for two reasons. First, their structure is determined by quantum mechanics (electron degeneracy pressure), not thermal physics, so their mass-radius relation is calculable from first principles. Second, their mass-radius scaling runs in the opposite direction to the saturation scaling (see Figure 4):

• White Dwarf (low-mass non-relativistic limit): $R_{\rm WD} \propto M^{-1/3}$ — heavier stars are smaller (steepening toward $M_{\rm Ch} = 1.44\,M_\odot$ under relativistic degeneracy; Nauenberg 1972)
• Saturation Scale (TEP): $R_T \propto M^{+1/3}$ — heavier sources have larger saturation radii

These opposing scalings create a "scissors" pattern: as mass increases, the geometric saturation radius $R_T(M)$ grows while the physical radius $R_{\rm phys}$ shrinks, driving the geometric saturation ratio $S \equiv R_T/R_{\rm phys}$ to large values across the degenerate stellar sequence.

For Sirius B ($M \approx 1.02 M_{\odot}$, $\rho \approx 2.4 \times 10^6$ g/cm³):

With a mean density \(\rho \approx 2.4 \times 10^6\) g/cm³ far exceeding the reference scale \(\rho_T \approx 20\) g/cm³, the geometric saturation ratio \(S = R_T/R_{\rm phys} \approx 50\) places Sirius B deep within the compact-body regime (\(R_T \gg R_{\rm phys}\)). This is consistent with the expected nonlinear screening hierarchy; the quantitative magnitude of physical scalar-gradient suppression in binary orbital dynamics is determined by the corresponding nonlinear kinetic and shell-impedance field solution (Paper 0 §7.3), rather than by the geometric ratio \(S\) alone.

4.2 The Empirical Screening Law

Extending this analysis across 26 astrophysical objects (planets, brown dwarfs, main sequence stars, white dwarfs, neutron stars, and binary pulsars) yields a compact summary of how the defined screening factor varies with density. Evaluating $S \equiv R_T(M)/R_{\rm phys}$ across the 26 objects defines a deterministic algebraic regime map separating unscreened bodies ($S < 1$) from compact screened bodies ($S \ge 1$):

Because $R_T(M) = (3M/4\pi\rho_T)^{1/3}$ and $R_{\rm phys} = (3M/4\pi\bar\rho)^{1/3}$, the $\bar\rho^{1/3}$ dependence is an exact algebraic consequence of the definition of $S$, not an independent statistical fit (an empirical log-log check across the 11 dense objects with $\bar\rho > \rho_T$ recovers $\beta_{\rm scr} = 0.3338 \pm 0.0008$, which is true by definition up to published table rounding). This density-dependent screening law is a projection of the source-charge screening operator $\mathcal{S}_\Sigma(\mathcal{E})$ (Paper 0 §7) onto the UCD density regime; it is not an independent screening operator. Two sector assignments are stated explicitly. The geometric diagnostic $S = R_T/R_{\rm phys}$ tracks the shear-sector $\mathcal{S}_\Sigma$ regime — density is the licensed stand-in for the ambient-steepness criterion inside dense macroscopic matter (Paper 0 §8) — while the scale $\rho_T$ that sets $R_T$ is itself calibrated through the clock-amplitude projection $S_A = \min[1,(\bar\rho/\rho_T)^{1/3}]$ (Paper 0 §3). Because each object's density is evaluated at its physical radius, $\bar\rho = 3M/(4\pi R_{\rm phys}^3)$, the diagnostic carries an enclosed-acceleration equivalent: the criterion $S\sim1$ corresponds to $GM/R_{\rm phys}^2 \propto \bar\rho\,R_{\rm phys}$, so the density and acceleration readings of the hierarchy agree object-by-object. Under the stated definitions, this is the expected scaling: if $R_T \propto M^{1/3}$ and $R_{phys} \propto (M/\rho)^{1/3}$, then:

The empirical screening law is a direct consequence of the $M^{1/3}$ soliton scaling, providing cross-regime consistency under a single $\rho_T$ rather than independent confirmation (see Note on Interpretation above).

4.3 Complete Screening Hierarchy

The complete hierarchy of screening factors across all object classes is visualized in Figure 5 and tabulated below.

Table 4.1: Complete screening hierarchy across astrophysical object classes. The screening factor increases monotonically with density, consistent with a Temporal Topology screening hierarchy under the stated definitions. Binary pulsars provide the strongest validation, with GR verified to 0.2% precision at screening factors exceeding 29,500×.

4.4 Consistency with Precision GR Tests

A critical question for any modified gravity theory is: why do precision tests of General Relativity show no deviation? Temporal Topology provides the answer: the suppression of Temporal Shear (vanishing field gradient) reduces the screened PPN scalar charge $\alpha_{\rm PPN}^{\rm eff}$ where the ambient landscape is steep. The five most precise tests of GR all occur in regimes where screening is operative:

Table 4.2: Precision GR tests and their screening factors. All tests are consistent with GR; TEP explains this via environment-dependent screening rather than requiring the scalar sector to be absent.

The Hulse-Taylor binary pulsar is particularly significant. Its orbital decay matches the GR prediction for gravitational wave emission to 0.2% precision—a result honored with the 1993 Nobel Prize (Taylor & Weisberg 1982). At a density of $\sim 10^{14}$ g/cm³, the screening factor reaches 29,500×. The scalar field contributes less than 0.003% to the orbital dynamics. This is not a violation of GR tests; it is a requirement of consistent screening.

4.5 Earth as the Calibration Anchor

The screening hierarchy reveals why Earth is the natural calibration point—and why GNSS clocks can detect what neutron stars cannot.

At $\rho \approx 5.5$ g/cm³, Earth sits just below the critical proximity scale, proxied by $\rho_T \approx 20$ g/cm³. The geometric saturation ratio is $S = 0.65\times$, meaning the saturation radius ($R_T(M_\oplus) \approx 4{,}146$ km; GNSS $L_c \approx 4{,}200$ km) is comparable to the physical radius (6,371 km). Earth occupies the narrow window where:

• The scalar field is not screened: Unlike neutron stars (S = 26,800×), Earth's density is low enough that the soliton extends to observable scales.
• The transition is close: Unlike gas giants ($S \approx 0.3$–$0.4$), where the saturation core sits deep inside the object and well below the channel-projection boundary, Earth's $S \approx 0.65$ places it near the transition, where clock-comparison ($S_A$) channels retain a measurable response.

GNSS satellites orbit at $\sim$20,000 km altitude, sampling clock transport through the exterior field region associated with the terrestrial Temporal Topology profile. The measured covariance length $L_c \approx 4200$ km is interpreted as a characteristic correlation/saturation scale of the Earth-sourced field, not as the orbital radius of the satellite network. The observed clock correlation length is therefore not an arbitrary parameter—it is the characteristic scale where the scalar field's gradient becomes steep enough to produce measurable timing correlations.

Within this phenomenology, the same saturation scale is expected to govern both terrestrial clock correlations and compact-object structure. The terrestrial scale provides an empirical calibration point, and the RBH-1 crossover provides an astrophysical consistency test under the same parameter choice.

4.6 The Critical Proximity Scale

The empirical data suggest a critical proximity scale, observationally proxied by the density $\rho_T \approx 20$ g/cm³, corresponding to approximately 4× Earth's mean density. This value has coincidence-level significance: it lies near the onset of electron degeneracy, where the equation of state transitions from thermal to quantum pressure support. This condensed-matter onset ($10\text{--}30$ g/cm³) complements the quartic-projected QED scale relation $\rho_T^{\rm EM} = (\alpha_{\rm EM}^{1/(m-1)}m_e)^m c^3/\hbar^3 = \alpha_{\rm EM}^{4/3}m_e^4c^3/\hbar^3 = 22.39$ g/cm³ derived in Section 5.5 (Step 14).

4.7 Planetary Rotational Kinematics and the Wesson–MacDonald Angular Momentum Laws

An independent dynamical test of the saturation radius $R_T(M)$ and screening factor $S = R_T(M)/R_{\rm phys}$ is provided by the rotational kinematics of Solar System bodies. In Newtonian gravity, planetary rotation periods ($T_{\rm rot} \approx 9.9\text{ hr}$ for Jupiter to $24.6\text{ hr}$ for Mars among unbraked planets, and $58.6\text{--}243\text{ d}$ for solar-tide-braked Mercury and Venus) and equatorial linear speeds ($v_{\rm eq} \approx 0.24\text{ km s}^{-1}$ for Mars to $12.57\text{ km s}^{-1}$ for Jupiter) appear unorganized by radius or heliocentric distance. Yet two empirical regularities have long been noted in the astrophysical literature without a first-principles gravitational derivation: the MacDonald–Brosche specific angular momentum scaling $j \equiv J/M \propto M^{2/3}\text{--}M^{0.8}$ across minor and major planets (Brosche 1963; MacDonald 1964), and Wesson's universal angular-momentum relation $J = p\,M^2$ with empirical constant $p_{\rm obs} \approx 10^{-15}\text{ cm}^2\text{ g}^{-1}\text{ s}^{-1}$ spanning planetary systems to spiral galaxies (Wesson 1979, 1981)—a value approximately $680$ times larger than the General Relativistic Kerr scale $G/c = 2.23 \times 10^{-18}\text{ cm}^2\text{ g}^{-1}\text{ s}^{-1}$.

Both regularities follow directly from the Temporal Topology saturation scale $\rho_T = 20\text{ g/cm}^3$ and the nested temporal hierarchy of Paper 0 (§7). First, because $R_T(M)^3 = 3M/(4\pi\rho_T)$ is proportional to mass, the characteristic dynamical frequency at the saturation boundary $R_T(M)$ is strictly mass-independent:

At the saturation radius $R_T(M)$, the dimensionless proper-time lapse depth is $u_T(M) \equiv GM/(R_T(M)c^2) = \omega_T^2 R_T(M)^2/c^2 \propto M^{2/3}$. Consequently, the natural specific angular-momentum capacity of the saturation core rotating at $\omega_T$ is directly proportional to its proper-time depth:

with universal proportionality constant $c^2/\omega_T = 3.801 \times 10^{19}\text{ m}^2\text{ s}^{-1}$ per unit fractional proper-time dilation. Second, in the nested environmental hierarchy of Paper 0 (§7), Solar System bodies condense inside the ambient Solar-Circle Galactic temporal well of depth $u_{\rm gal} = (v_{\rm gal}/c)^2 = 5.385 \times 10^{-7}$ ($v_{\rm gal} = 220\text{ km s}^{-1}$, $c/v_{\rm gal} = 1/\sqrt{u_{\rm gal}} = 1362.7$; adopting modern Gaia/VLBI solar-circle estimates $v_{\rm gal} \approx 229\text{--}236\text{ km s}^{-1}$ shifts predictions by $4\text{--}7\%$). As an action-motivated, zero-refit rotational scaling proposal, equating the mixed time–space kinematic transport coupling $(v_{\rm eq}/c)\sqrt{u_{\rm gal}}$ — equivalently, the advective velocity cross-term $(v_{\rm eq}\,v_{\rm gal})/c^2$ between internal circulation and Galactic drift — across the saturation boundary to the local proper-time depth $u_T(M)$ yields the equatorial velocity relation for ambient-coupled planets:

where $v_T(M) = \sqrt{GM/R_T(M)} = R_T(M)\omega_T$ is the circular velocity at the TEP saturation radius. Crucially, because Earth has transferred $\sim 83\%$ of its primordial spin angular momentum to the Moon's orbit via lunar tidal dissipation (whereas its conserved total Earth+Moon angular momentum $J_{\rm E+M} = 3.45 \times 10^{34}\text{ kg m}^2\text{ s}^{-1}, T_{\rm prim} = 5.05\text{ hr}$ sits on the small-body soliton capacity ceiling at $\eta_{\rm E+M} = 0.145$ vs $\eta_0 = 0.129$, and lunar braking has subsequently relaxed present-day $23.93\text{ hr}$ Earth onto the ambient floor at $v_{\rm obs}/v_{\rm TEP} = 1.063$), the clean zero-parameter test of Equation (\ref{eq:veq_tep_law}) is provided by the three truly unbraked outer giant planets (Uranus, Neptune, and Saturn, which lack large tidally braking moons): across Uranus, Neptune, and Saturn alone, $v_{\rm obs}/v_{\rm TEP} = 1.0003 \pm 0.0804$ ($\hat\rho_T^{\rm spin,outer3} = 19.89\text{ g/cm}^3$ geometric mean; including present-day Earth as a fourth ambient-floor planet gives $(v_{\rm eq}/c)/u_T = 1384.8 \pm 99.6$ vs $c/v_{\rm gal} = 1362.7$, $+1.6\%$, $v_{\rm obs}/v_{\rm TEP} = 1.016 \pm 0.073$, $\hat\rho_T^{\rm spin} = 20.9\text{ g/cm}^3$; Table 4.3, Figure 6) with zero fitted parameters. Equation (\ref{eq:veq_tep_law}) predicts rotation periods of $25.48\text{ hr}$ for Earth (observed $23.93\text{ hr}$), $17.14\text{ hr}$ for Uranus (observed $17.25\text{ hr}$; Lamy et al. 2025), $14.87\text{ hr}$ for Neptune (observed $16.11\text{ hr}$), and $11.55\text{ hr}$ for Saturn (observed $10.66\text{ hr}$). Moreover, the departures of Jupiter ($v_{\rm obs}/v_{\rm TEP} = 0.62$) and Mars ($v_{\rm obs}/v_{\rm TEP} = 2.44$) from the single-branch ambient formula Eq. ($\ref{eq:veq_tep_law}$) reflect the two-branch angular-momentum geometry of Figure 6b (Step 15): because the ambient Galactic Wesson branch $j_{\rm TEP}(M) = GM/(2v_{\rm gal}) \propto M^1$ grows faster with mass than the small-body soliton angular-momentum capacity $j_{\rm sol}(M) = \eta_0 R_T(M)^2\omega_T \propto M^{2/3}$ (where $\eta_0 = 0.129$ is a single empirical efficiency parameter calibrated from the small-body median, with six-body interquartile range $\eta_{\rm IQR} \in [0.122,\,0.154]$, six-body median $0.138$, and candidate secular bar-mode / rotational-shedding stability benchmark $\eta_{\rm sec} = k_{\rm core}(\omega_{\max}/\omega_T) = \tfrac{1}{2} \times \tfrac{1}{4} = \tfrac{1}{8} = 0.125$ at the $S \to 1$ core boundary), their ratio $$\frac{j_{\rm TEP}(M)}{j_{\rm sol}(M)} = \frac{R_T(M)\,\omega_T}{2\,\eta_0\,v_{\rm gal}} = \left(\frac{M}{M_{\rm cap}}\right)^{1/3}$$ crosses unity at a unique crossover mass $M_{\rm cap} = \frac{4\pi}{3}\rho_T(2\eta_0 v_{\rm gal}/\omega_T)^3 = 1.16 \times 10^{27}\text{ kg} = 194.2\,M_\oplus = 0.611\,M_{\rm Jup}$ ($R_T(M_{\rm cap}) = 24{,}013\text{ km}$; intersection of the solid and dashed curves in Figure 6b). For Earth ($1.0\,M_\oplus$), Uranus ($14.5\,M_\oplus$), Neptune ($17.1\,M_\oplus$), and Saturn ($95.2\,M_\oplus < M_{\rm cap}$), $j_{\rm TEP}(M)/j_{\rm sol}(M) \in [0.173,\,0.788] < 1$, so all four lie on the unsaturated ambient branch ($v_{\rm eq}^{\rm 2br} = v_{\rm eq}^{\rm TEP}$, with Saturn at $9.87\text{ km s}^{-1}$ vs $9.11\text{ km s}^{-1}$, $+8.4\%$). Conversely, Jupiter ($317.8\,M_\oplus > M_{\rm cap}$, $j_{\rm TEP}/j_{\rm sol} = 1.178 > 1$) is the sole Solar System planet above $M_{\rm cap}$, so its specific angular momentum saturates at $j_{\rm sol}(M_{\rm Jup})$ ($\eta_{\rm obs} = 0.125$ vs $\eta_0 = 0.129$), yielding $v_{\rm eq}^{\rm 2br} = j_{\rm sol}(M_{\rm Jup})/(k_{\rm MOI}R_{\rm Jup}) = 12.95\text{ km s}^{-1}$ ($T_{\rm 2br} = 9.64\text{ hr}$), within $-2.9\%$ of Jupiter's observed $12.57\text{ km s}^{-1}$ ($T_{\rm obs} = 9.93\text{ hr}$, $v_{\rm obs}/v_{\rm 2br} = 0.971$; across all five major Solar System planets, $v_{\rm obs}/v_{\rm 2br} = 1.007 \pm 0.066$, and $1.008 \pm 0.060$ across all nine planetary and substellar speed-limit systems, Figure 6a, diamond markers). Low-mass Mars ($0.11\,M_\oplus$, $v_{\rm obs} = 0.241\text{ km s}^{-1}$) sits in the intermediate transition zone between the ambient curve ($0.099\text{ km s}^{-1}$) and the soliton capacity curve ($0.954\text{ km s}^{-1}$) in Figure 6b and is recorded as an unmodeled transition body without introducing a Mars-specific rule.

Furthermore, comparing the dimensionless moment-of-inertia factor $k_{\rm MOI} \equiv I/(MR_{\rm eq}^2)$ (normalized to the equatorial radius $R_{\rm eq}$, measured via satellite geodesy, precession, and Juno/Cassini gravity-wind interior models for the gas giants, $k_{\rm MOI} = 0.26393 \pm 0.00001$ for Jupiter and $0.2181 \pm 0.0002$ for Saturn, Militzer & Hubbard 2023, and constrained by gravity-harmonic interior models for Uranus and Neptune, Lin et al. 2025; Morf & Helled 2025) against the geometric screening factor $S = R_T(M)/R_{\rm eq}$ across all eight cold hydrostatic Solar System planets ($M < 13\,M_{\rm Jup}$, non-fusing interiors with polytropic index $n \lesssim 1$ and central-to-mean density $\rho_c/\bar\rho \lesssim 2\text{--}10$) reveals an empirical interior gyration relation on the unscreened planetary branch ($S \lesssim 0.7$, $\bar\rho < 10\text{ g/cm}^3$; Step 15):

with $2k_{\rm MOI}/S = 1.039 \pm 0.027$ across the three self-compressed iron-core terrestrial planets ($1.017$ for Earth, $1.031$ for Venus, and $1.069$ for Mercury; because these three bodies span a narrow bulk density range $\bar\rho = 5.24\text{--}5.51\text{ g/cm}^3$, their agreement serves as a single-regime terrestrial amplitude check rather than a lever-arm slope test) and $2k_{\rm MOI}/S = 1.169 \pm 0.142$ across all eight major planets spanning an $8\times$ range in mean density ($\bar\rho = 0.69\text{--}5.51\text{ g/cm}^3$: $1.112$ for Neptune, $1.137$ for Uranus, $1.261$ for low-pressure Mars near the incompressible uniform-sphere ceiling $k_{\rm MOI} \to 2/5 = 0.4$, and $1.334$ for Jupiter and $1.389$ for Saturn where compressible H/He envelopes inflate $R_{\rm eq}$; an unconstrained two-parameter compositional power-law fit $k_{\rm MOI} = C\,\bar\rho^\beta$ on the raw 1-bar radii gives $\hat\beta_{\rm emp} = 0.239 \pm 0.038, C_{\rm emp} = 0.230$, RMS $0.031\text{ dex}$ vs. the zero-parameter Eq. [\ref{eq:kmoi_screening}] $\beta = 1/3, C_{\rm TEP} = 0.184$, RMS $0.075\text{ dex}$). Crucially, for self-compressed major planets ($P_c = 3GM^2/[8\pi R^4] \ge 25\text{ GPa}$), solving the exact Lane–Emden polytropic hydrostatic interior equation $\xi^{-2}\frac{\mathrm{d}}{\mathrm{d}\xi}(\xi^2\frac{\mathrm{d}\theta}{\mathrm{d}\xi}) = -\theta^n$ at the unique polytropic index $n_{\rm eff}$ fixed by each planet's measured $k_{\rm MOI}$ ($n = 0.36$ for Mercury, $0.47$ for Venus and Earth, $0.98$ for Jupiter, $1.18$ for Neptune, $1.31$ for Uranus, and $1.38$ for Saturn; noting that $n_{\rm eff}$ is inverted from $k_{\rm MOI}$) quantifies outer H/He atmospheric envelope inflation directly: relative to rocky Earth ($n = 0.47$, mass-weighted interior radius $R_{\rm mass}/R_{\rm eq} = 0.901$), the compressible outer H/He envelope inflates $R_{\rm eq}$ above the condensed mass-weighted interior radius $R_{\rm cond} = R_{\rm eq}\,[R_{\rm mass}(n)/R_{\rm mass,\oplus}]$ by $R_{\rm cond}/R_{\rm eq} = 0.886$ (Jupiter), $0.841$ (Neptune), $0.812$ (Uranus), and $0.799$ (Saturn). Evaluating $S_{\rm cond} = R_T(M)/R_{\rm cond}$ at this condensed-interior radius reduces Saturn's $2k_{\rm MOI}/S$ from $1.389$ to $1.109$ ($\rho_T^{\rm MOI,cond} = 14.7\text{ g/cm}^3$) and Jupiter's from $1.334$ to $1.181$ ($12.1\text{ g/cm}^3$; with Neptune at $0.935$ and Uranus at $0.923$), bringing all seven major condensed and giant planets (Mercury, Venus, Earth, Jupiter, Saturn, Uranus, and Neptune) to $2k_{\rm MOI}/S_{\rm cond} = 1.042 \pm 0.094$ ($\rho_T^{\rm MOI,cond} = 17.9\text{ g/cm}^3$ geometric mean; while the primary outer-planet spin test Eq. [\ref{eq:veq_tep_law}] does not depend on $k_{\rm MOI}$). Conversely, in low-mass uncompressed solid bodies where central gravitational pressure $P_c \ll K_0 \sim 100\text{ GPa}$ — including the six major satellites with measured moments of inertia ($P_c = 2.7\text{--}5.8\text{ GPa}$: Europa $2k_{\rm MOI}/S = 1.30$, Io $1.35$, Ganymede $1.36$, Moon $1.43$, Titan $1.50$, Callisto $1.57$), Mars ($P_c = 24.9\text{ GPa}$, $1.26$, vs. $141\text{--}173\text{ GPa}$ for Venus and Earth), and the minor planets ($1.26\text{--}1.57$) — weak self-compression leaves $k_{\rm MOI}$ near the uniform-sphere ceiling ($0.31\text{--}0.39$) while $S \approx 0.45\text{--}0.58$; and in nuclear-burning main-sequence stars such as the Sun ($\bar\rho_\odot = 1.41\text{ g/cm}^3, S_\odot = 0.413, k_{\rm MOI,\odot} \approx 0.070 \ll S_\odot/2$), radiative-core thermal support produces an $n \approx 3$ Eddington core condensation ($\rho_c \approx 153\text{ g/cm}^3 \approx 108\,\bar\rho_\odot$) outside cold condensed-matter hydrostatics. Multiplying Equation (\ref{eq:veq_tep_law}) by Equation (\ref{eq:kmoi_screening}) cancels both $R_{\rm phys}$ and $R_T(M)$ identically, yielding the empirical Wesson angular-momentum relation:

corresponding to a dimensionless Kerr-like spin parameter $\chi_{\rm TEP} \equiv cJ/(GM^2) = c/(2v_{\rm gal}) = 1/(2\sqrt{u_{\rm gal}}) = 681.3$. Crucially, for any body, the exact algebraic identity $$\frac{J_{\rm obs}}{J_{\rm TEP}} \equiv \left(\frac{2\,k_{\rm MOI}}{S}\right) \left(\frac{v_{\rm eq}^{\rm obs}}{v_{\rm eq}^{\rm TEP}}\right)$$ holds by definition: although $\rho_T$ cancels out of the product $J_{\rm TEP}(M)$, measuring $k_{\rm MOI}$ and $v_{\rm eq}$ separately constrains $\rho_T \approx 20\text{ g/cm}^3$ through both factors simultaneously ($2k_{\rm MOI}/S \propto \rho_T^{1/3}$ and $v_{\rm obs}/v_{\rm TEP} \propto \rho_T^{-1/3}$—an unadjusted $\rho_T \sim 220\text{ g/cm}^3$ would shift both factors by $2.23\times$ in opposite directions, to $2.61$ and $0.45$). Inverting Equation (\ref{eq:veq_tep_law}) directly via $\rho_T^{\rm spin} = \frac{3}{4\pi}(v_{\rm eq}^{\rm obs}v_{\rm gal}/GM^{2/3})^3 = \rho_T(v_{\rm obs}/v_{\rm TEP})^3$ across the four unsaturated ambient-coupled planets yields individual spin-implied saturation densities of $24.1\text{ g/cm}^3$ (Earth), $19.6\text{ g/cm}^3$ (Uranus, using the 2025 HST auroral rotation period $17\text{ h }14\text{ m }52\text{ s} = 17.247864 \pm 0.000010\text{ hr}$, Lamy et al. 2025), $15.7\text{ g/cm}^3$ (Neptune), and $25.5\text{ g/cm}^3$ (Saturn), with a geometric mean of $\rho_T^{\rm spin} = 20.9\text{ g/cm}^3$ ($21.2 \pm 4.4\text{ g/cm}^3$ arithmetic mean; and $21.1\text{ g/cm}^3$ geometric mean across the nine planetary and substellar systems under the one-parameter $\eta_0 = 0.129$ two-branch model), while inverting Equation (\ref{eq:kmoi_screening}) via $\rho_T^{\rm MOI} = \bar\rho/(8k_{\rm MOI}^3)$ yields $\rho_T^{\rm MOI,\text{terr}} = 17.9 \pm 1.4\text{ g/cm}^3$ sample scatter across the three iron-core terrestrial planets ($19.1\text{ g/cm}^3$ for Earth, $18.2\text{ g/cm}^3$ for Venus with $\pm 20\%$ propagated $k_{\rm MOI}$ uncertainty, $16.4\text{ g/cm}^3$ for Mercury) and $\rho_T^{\rm MOI} = 13.7 \pm 4.2\text{ g/cm}^3$ across all eight planets, exhibiting a systematic trend with planet composition from iron-core terrestrials ($16.4\text{--}19.1\text{ g/cm}^3$) to ice giants ($14.8\text{ g/cm}^3$ for Neptune, $13.9\text{ g/cm}^3$ for Uranus), low-pressure Mars ($10.0\text{ g/cm}^3$), and compressible H/He gas giants ($9.0\text{ g/cm}^3$ for Jupiter, $8.3\text{ g/cm}^3$ for Saturn). Across the five ambient-coupled major planets (Earth, Uranus, Neptune, Saturn, and Jupiter), the observed spin angular momenta match the zero-parameter Wesson relation $p_{\rm TEP} = G/(2v_{\rm gal})$ at $J_{\rm obs}/J_{\rm TEP} = 1.11 \pm 0.25$ ($\bar\chi_{\rm obs} = 758.7 \pm 169.0$ versus $\chi_{\rm TEP} = 681.3$).

A necessary condition separating the ambient Galactic Wesson branch ($J \propto M^2$) from the low-mass intrinsic soliton-clock branch ($j \propto M^{2/3}$) is obtained by comparing each body's two-branch screening shell radius $r_*(M) = \sqrt{GM/g_t}$ ($g_t = cH_0/2 = 3.40 \times 10^{-10}\text{ m s}^{-2}$; Paper 0 §7) to its heliocentric orbital semimajor axis $a$ (Table 4.3). Unbraked planets whose screening shell reaches their orbit ($r_*(M)/a \gtrsim 1$, with Neptune at the boundary) can couple to the Galactic shear floor: Jupiter ($r_*/a = 24.79$, $J_{\rm obs}/J_{\rm TEP} = 0.824$), Saturn ($r_*/a = 7.37$, $J_{\rm obs}/J_{\rm TEP} = 1.505$), Earth ($r_*/a = 7.24$, $J_{\rm obs}/J_{\rm TEP} = 1.083$), Uranus ($r_*/a = 1.44$, $J_{\rm obs}/J_{\rm TEP} = 1.129$), and Neptune ($r_*(M_{\rm Nep}) = 29.97\text{ AU}$ versus $a_{\rm Nep} = 30.11\text{ AU}$, $r_*/a = 0.995$, $J_{\rm obs}/J_{\rm TEP} = 1.026$), whereas Mars ($r_*/a = 1.56$, $\chi_{\rm obs} = 2094.6$, $\eta_{\rm rot} = 0.033$) sits in the transition zone between the two branches. As a secondary four-planet diagnostic on the ambient branch, this nested heliocentric screening-shell ratio $r_*(M)/a$ also orders the opposite signs of the individual spin residuals around Eq. (\ref{eq:veq_tep_law}): because $\sqrt{M_{\rm Sat}/M_\oplus} = 9.755 \approx a_{\rm Sat}/a_\oplus = 9.582$, Earth ($r_*/a = 7.24$, $+6.4\%$) and Saturn ($r_*/a = 7.37$, $+8.4\%$) share nearly identical nested screening-shell ratios deep in the $r_*/a \gg 1$ regime and rotate faster than the baseline formula by the same margin ($+6.3\%$ and $+4.7\%$ at Saturn's oblateness-corrected volumetric radius $\bar R_{\rm Sat} = 58{,}232\text{ km}$, and $+5.7\%$ under the $10.561\text{ hr}$ Cassini ring-seismology deep-interior period), whereas Uranus ($r_*/a = 1.44$, $-0.6\%$) and Neptune ($r_*/a = 0.995$, $-7.7\%$) sit at the $r_*/a \to 1$ boundary and rotate slightly slower (correlating with $\ln[r_*(M)/a]$ across the four ambient planets at Pearson $r = +0.960, p = 0.0398$, Spearman $\rho = +1.000$, reducing residual scatter from $\pm 7.3\%$ to $\pm 1.81\%$, and yielding $\bar v_{\rm rot,int}/v_{\rm TEP} = 1.031 \pm 0.041$, $\rho_T^{\rm spin,vol} = 21.9\text{ g/cm}^3$ under volumetric radii and deep-interior rotation periods). Conversely, for small bodies whose screening shell is much smaller than their orbital radius ($r_*(M)/a \ll 1$: Hygiea $0.009$, Pallas $0.015$, Vesta $0.020$, Ceres $0.033$, Quaoar $0.002$, and Haumea $0.004$), rotation decouples from the Galactic floor and follows the intrinsic soliton-clock capacity $j_{\rm sol}(M) = \eta_{\rm rot}\,R_T(M)^2\,\omega_T \propto M^{2/3}$ (six-body median $\eta_{\rm med} = 0.138$, interquartile range $\eta_{\rm IQR} \in [0.122,\,0.154]$, mean $0.157 \pm 0.076$, and $\eta_0 = 0.129$ at Ceres), yielding a log-log slope of $0.941 \pm 0.059$ ($R^2 = 0.9735$, $p = 8.85 \times 10^{-7}$) against $u_T(M)$ across seven orders of magnitude in mass (Figure 6). Notably, the combined Earth+Moon system ($r_*/a = 7.28$, $T_{\rm prim} = 5.05\text{ hr}$, $\eta_{\rm E+M} = 0.145$, or $0.119$ at fixed $k_{\rm MOI}$) sits inside the small-body soliton capacity interquartile range following the Moon-forming giant impact, while subsequent and still-ongoing lunar tidal dissipation ($\dot{a}_{\rm Moon} \approx +3.8\text{ cm yr}^{-1}$, lengthening the day by $\sim 2\text{ ms century}^{-1}$) has spun Earth down across the intermediate region to its present-day epochal value near the Wesson line ($\chi_{\rm obs} = 737.8$ vs. $\chi_{\rm TEP} = 681.3$, a secular transit rather than a static tidal equilibrium). Extending this soliton-capacity branch out of sample beyond the Solar System to super-Jupiters and ultra-cool brown dwarfs ($M > M_{\rm cap} = 0.61\,M_{\rm Jup}$; Snellen et al. 2014; Gagné et al. 2017; Bryan et al. 2020; Tannock et al. 2021) exposes a direct out-of-sample test of both $R_T(M)$ and the clock-amplitude saturation ceiling $S_A(\bar\rho) = \min[1,\,(\bar\rho/\rho_T)^{1/3}] = \min[1,\,S]$ (Paper 0 §3). Because the active saturation core inside a body has radius $R_{\rm active} = S_A\,R_{\rm phys} = \min[R_T(M),\,R_{\rm phys}]$, the maximum specific angular momentum is $j_{\rm cap}(M,\,R_{\rm phys}) = \eta_0\,\min[R_T(M),\,R_{\rm phys}]^2\,\omega_T$, which defines a dimensionless Keplerian breakup-fraction ceiling $v_{\rm cap}/v_{\rm breakup} = (\eta_0/k_{\rm MOI})\sqrt{\min(1,\,S)}$ across two high-mass regimes above $M_{\rm cap}$: (i) for unscreened planetary-mass objects with $\bar\rho < \rho_T = 20\text{ g/cm}^3$ ($S < 1$, $M \lesssim 16\,M_{\rm Jup}$), $R_{\rm active} = R_T(M) < R_{\rm phys}$ and the rotational ceiling rises as $v_{\rm cap}(M) = \eta_0 R_T(M)^2\omega_T/(k_{\rm MOI}R_{\rm phys})$ — predicting $v_{\rm cap} = 64.9\text{ km s}^{-1}$ ($P_{\rm cap} = 2.35\text{ hr}$, $v_{\rm cap}/v_{\rm breakup} = 0.478$) for the fastest-rotating planetary-mass object known, SIMP J013656.5+093347 ($M = 12.7 \pm 1.0\,M_{\rm Jup}, R = 1.22\,R_{\rm Jup}, \bar\rho = 8.67\text{ g/cm}^3, S = 0.76, k_{\rm MOI} = 0.235$; Gagné et al. 2017), matching its observed $v_{\rm eq} = 63.1\text{ km s}^{-1}$ ($P_{\rm obs} = 2.414\text{ hr}, v_{\rm obs}/v_{\rm breakup} = 0.464$, $v_{\rm obs}/v_{\rm cap} = 0.972$, within $-2.8\%$), while young contracting exoplanets such as $\beta$ Pictoris b ($11\,M_{\rm Jup}, v\sin i = 25\text{ km s}^{-1}$, deprojected $v_{\rm eq} \approx 31.8\text{ km s}^{-1}$ vs. $v_{\rm cap} = 46.6\text{ km s}^{-1}$; Snellen et al. 2014) and the 27-object sample of Bryan et al. (2020; $v/v_{\rm breakup} \in [0.05,\,0.31]$, $0/27$ exceeding the ceiling) lie well below this upper boundary; and (ii) once $\bar\rho \ge \rho_T = 20\text{ g/cm}^3$ ($S = R_T/R_{\rm phys} \ge 1$, in degenerate brown dwarfs with $M \gtrsim 16\,M_{\rm Jup}$), the active radius saturates at the physical surface ($R_{\rm active} = R_{\rm phys}$, $v_{\rm cap}/v_{\rm breakup} = \eta_0/k_{\rm MOI} = 0.630$), so $R_{\rm phys}$ cancels out of $P_{\min} = 2\pi R_{\rm phys}/v_{\rm cap}$ and locks the minimum rotation period of all degenerate brown dwarfs onto a universal, mass- and radius-independent floor set directly by $\tau_T = 2\pi/\omega_T = 0.7381\text{ hr}$ ($44.3\text{ min}$): $$P_{\min}^{\rm TEP}(\bar\rho \ge \rho_T) = \frac{k_{\rm MOI}}{\eta_0}\,\tau_T = \frac{0.205}{0.129}\,\tau_T = 1.172\text{ hr} \quad (70.3\text{ min}),$$ where $k_{\rm MOI} = 0.205$ is the exact Lane–Emden gyration factor of an $n = 1.5$ degenerate polytrope (propagating the six-body small-body interquartile range $\eta_{\rm IQR} \in [0.122,\,0.154]$ gives $P_{\min}^{\rm TEP} \in [0.98,\,1.24]\text{ hr}$, with $1.10\text{ hr}$ at the six-body median $\eta_{\rm med} = 0.138$). This floor reproduces the empirical rotational speed limit of ultra-cool brown dwarfs discovered by Tannock et al. (2021): across the three fastest-rotating brown dwarfs known — 2MASS J0348$-$6022 ($43\,M_{\rm Jup}, \bar\rho = 70.8\text{ g/cm}^3, S = 1.52, P_{\rm obs} = 1.080\text{ hr}$), 2MASS J1219$+$3128 ($49\,M_{\rm Jup}, \bar\rho = 66.6\text{ g/cm}^3, S = 1.49, P_{\rm obs} = 1.140\text{ hr}$), and 2MASS J0407$+$1546 ($67\,M_{\rm Jup}, \bar\rho = 156.4\text{ g/cm}^3, S = 1.98, P_{\rm obs} = 1.230\text{ hr}$) — all three observed periods lie inside the propagated small-body interquartile band $[0.98,\,1.24]\text{ hr}$ with mean $1.150 \pm 0.075\text{ hr}$ (matching $P_{\min}^{\rm TEP} = 1.172\text{ hr}$ within $+1.9\%$, whereas Newtonian Keplerian breakup sits at $P_{\rm breakup} = 0.28\text{--}0.43\text{ hr}$; furthermore, $\eta$ does not need to be fitted to small bodies at all: in the TEP saturation core of uniform density $\rho_T = 20\text{ g/cm}^3$ inside $R_{\rm active} = \min[R_T,\,R_{\rm phys}]$, the dimensionless soliton capacity $\eta \equiv j_{\rm sol}/(R_{\rm active}^2\omega_T) = (j_{\rm sol}\omega_T)/(GM/R_{\rm active}) = 2T_{\rm rot,core}/|W_T|$ is the direct virial action-to-binding ratio of the uniform-density scalar core (anchored model-independently across all $N = 9$ decoupled Solar System small bodies and outer giant-planet moons at $\bar\eta_9 = 0.135 \pm 0.014$, median $0.1376$, while the compressible baryonic envelope of gyration factor $k_{\rm MOI} = 0.205$ co-rotates with the core so that $k_{\rm MOI}$ enters the surface period conversion $P_{\min} = 2\pi R_{\rm phys}/v_{\rm cap} = (k_{\rm MOI}/\eta)\tau_T$ linearly); in classical self-gravitating fluid dynamics (Chandrasekhar 1969; Ostriker & Peebles 1973), a uniform Maclaurin spheroid becomes secularly unstable to triaxial $m=2$ bar deformation at the exact Maclaurin–Jacobi bifurcation invariant $\eta_{\rm Jacobi} \equiv (T_{\rm rot}/|W|)_{\rm crit} = 0.13753$ [matching the six-asteroid median $\eta_{\rm med} = 0.1376$ to $-0.04\%$ and nine-body median $0.1377$ to $-0.1\%$ with zero fitting!], which together with the virial benchmark $\eta_{\rm sec} = \tfrac{1}{8} = 0.125$ defines a parameter-free theoretical stability bracket $\eta \in [0.125,\,0.1375]$ that predicts $v_{\rm eq}^{\rm th} = 12.54\text{ km s}^{-1}$ for Jupiter [$-0.2\%$] and $62.9\text{ km s}^{-1}$ for SIMP J0136 [$-0.3\%$] at $\eta_{\rm sec}$, and fixes the TEP-saturated brown-dwarf rotational period floor parameter-freely to $P_{\min}^{\rm th} = (k_{\rm MOI}/\eta)\tau_T(\rho_T) \in [1.100,\,1.210]\text{ hr}$ with midpoint $1.155\text{ hr}$ [$+0.4\%$ from the observed $1.150 \pm 0.075\text{ hr}$, yielding $\hat\rho_T^{\rm BD,th} = 20.19\text{ g/cm}^3$ with zero asteroid calibration; crucially, in purely classical Newtonian mechanics without TEP saturation at $\rho_T = 20\text{ g/cm}^3$, evaluating $\eta_{\rm Jacobi} = 0.13753$ at each brown dwarf's physical density $\bar\rho_{\rm BD} = 70.8,\,66.6,$ and $156.4\text{ g/cm}^3$ places the classical stability boundary at $P_{\min}^{\rm Classical,Jacobi} = (k_{\rm MOI}/\eta_{\rm Jacobi})\sqrt{3\pi/(G\bar\rho_{\rm BD})} = 0.585,\,0.603,$ and $0.393\text{ hr}$ with mean $0.527\text{ hr}$ — $8.31\sigma$ ($>2.2\times$) below the observed $1.150 \pm 0.075\text{ hr}$ empirical speed limit — and if one instead applies the classical compressible $n=1.5$ envelope-polytrope conversion $\eta_J^{\rm poly} = \sqrt{2k_{\rm MOI}w\beta_J} = \sqrt{2(0.205)(6/7)(0.13753)} = 0.220$ (or the $n=1.5$ mass-shedding threshold $\beta_{\rm shed} \simeq 0.08 \implies \eta_{\rm shed}^{\rm poly} = 0.168$), the unscreened Newtonian stability boundary drops even lower to $0.366,\,0.377,$ and $0.246\text{ hr}$ with mean $0.330\text{ hr}$ ($0.432\text{ hr}$ at mass-shedding; $9.57\sigma\text{--}10.94\sigma$ below the observed $1.150 \pm 0.075\text{ hr}$ floor, requiring an unexplained non-gravitational braking mechanism to halt spin-up at $2.2\text{--}3.5\times$ the classical stability period in standard mechanics whereas TEP saturation at $\rho_T = 20\text{ g/cm}^3$ places the stability ceiling directly at $1.155\text{ hr}$)]). Across the $N = 8$ unbraked outer-planet and substellar speed-limit systems excluding lunar-braked Earth (Uranus, Neptune, Saturn, Jupiter, SIMP J0136, and the three Tannock et al. brown dwarfs), the two-branch $+\;S_A$ saturation law achieves $v_{\rm obs}/v_{\rm 2br} = 1.0014 \pm 0.0596$ ($\hat\rho_T^{\rm rot,8} = 19.99\text{ g/cm}^3$ geometric mean; or $1.008 \pm 0.060$ and $\rho_T^{\rm rot} = 21.1\text{ g/cm}^3$ across all $N = 9$ systems when present-day Earth is included; Figure 6).

Table 4.3: Planetary and small-body rotational kinematics evaluated at the GNSS-consistent saturation density $\rho_T = 20$ g/cm³ ($\omega_T = 2.365 \times 10^{-3}$ rad s$^{-1}$) and Solar-Circle Galactic velocity $v_{\rm gal} = 220$ km s$^{-1}$ ($\chi_{\rm TEP} = c/(2v_{\rm gal}) = 681.3$, $p_{\rm TEP} = 1.517 \times 10^{-15}$ cm$^2$ g$^{-1}$ s$^{-1}$; Step 15). ^*Earth+Moon restores the tidally transferred lunar orbital angular momentum to Earth's spin ($\eta_{\rm rot} = 0.119$), while present-day Earth spin sits directly on the ambient-coupled Wesson branch ($\chi_{\rm obs} = 737.8$, $v_{\rm obs}/v_{\rm TEP} = 1.064$). ^†Evaluated on the unscreened ($S < 1$) soliton-capacity branch $v_{\rm cap} = \eta_0 R_T^2\omega_T/(k_{\rm MOI}R_{\rm phys})$ ($\eta_0 = 0.129$; for Jupiter, the unsaturated ambient formula gives $20.35$ km s$^{-1}$). ^‡Evaluated in the screened brown-dwarf regime ($\bar\rho \ge \rho_T$, $S \ge 1$) where $R_{\rm active} = \min[R_T,\,R_{\rm phys}] = R_{\rm phys}$, giving the universal period floor $P_{\min}^{\rm TEP} = (k_{\rm MOI}/\eta_0)\tau_T = 1.172$ hr ($1.155$ hr at the parameter-free Chandrasekhar–Jacobi/virial midpoint) and active efficiency $\eta_{\rm act} = j_{\rm obs}/(R_{\rm phys}^2\omega_T)$. All planetary $v_{\rm obs} = 2\pi R_{\rm eq}/T_{\rm obs}$ and $T_{\rm TEP} = 2\pi R_{\rm eq}/v_{\rm TEP}$ entries in this table use 1-bar equatorial radii $R_{\rm eq}$ uniformly ($v_{\rm obs}/v_{\rm TEP} = 1.000 \pm 0.080$ across Uranus, Neptune, and Saturn); evaluating at volumetric mean radii $R_{\rm vol}$ gives $v_{\rm obs}^{\rm vol} = 9.538$ km s$^{-1}$ for Saturn ($+4.7\%$ vs. $9.108$ km s$^{-1}$) and reduces the outer-three dispersion by half ($1.031 \pm 0.041$ with deep-interior periods).

Saturn — the least screened major planet in this framework ($S = R_T/R_{\rm eq} = 0.314$, the lowest screening ratio of the eight planets and therefore the most deeply ambient-coupled body on the Wesson branch at $M = 95.2\,M_\oplus < M_{\rm cap}$) — supplies a dedicated stress test of the ambient-branch interpretation through three further audits (Step 15). First, Saturn is the only planet whose ``rotation period'' is not a single number: the C-ring kronoseismology deep-interior rate ($10^{\rm h}33^{\rm m}38^{\rm s}$; Mankovich et al. 2019), the atmospheric potential-vorticity frame System IIIw ($10^{\rm h}34^{\rm m}13^{\rm s} \pm 20^{\rm s}$; Read et al. 2009), the adopted IAU System III frame ($10^{\rm h}39^{\rm m}24^{\rm s}$; Desch & Kaiser 1981), the 2003 Ulysses SKR value ($10^{\rm h}45^{\rm m}45^{\rm s}$; Galopeau & Lecacheux 2000; Gurnett et al. 2005), and the dual hemispheric Saturn kilometric radiation (SKR) components ($\sim10.6$ and $\sim10.8$ hr; Kurth et al. 2008; Gurnett et al. 2009) disagree at the $\sim2\%$ level. Evaluating every clock at its appropriate radius convention (volumetric $\bar R_{\rm Sat} = 58{,}232\text{ km}$ for interior and magnetospheric rates; $R_{\rm eq} = 60{,}268\text{ km}$ for the table-convention System III comparison; Table 4.4) shows that all seven exceed the ambient prediction: $v_{\rm obs}/v_{\rm TEP} \in [1.033,\,1.084]$, corresponding to $\rho_T^{\rm spin} \in [22.1,\,25.5]\text{ g/cm}^3$ — with the interior clocks (kronoseismology and potential-vorticity frames, the physically meaningful anchors) carrying the claim at $+5.6\text{--}5.7\%$ while the magnetospheric rows are formal equatorial conversions included to demonstrate that the sign is clock- and convention-independent ($[1.047,\,1.094]$ even at $R_{\rm eq}$). Saturn's positive residual is therefore structural rather than an artefact of the ambiguous period choice, and its magnitude is ordered exactly as the $r_*/a$ nested-shell trend predicts. Second, the two-branch model implies a rotation-clock stability dichotomy that is observed directly: bodies on the saturated soliton branch carry rigid intrinsic clocks — Jupiter's System III period has remained constant since the 1960s (anchored by the Io-locked decametric emission), and the photometric periods of SIMP J0136 and the Tannock et al. brown dwarfs are stable to within their measurement uncertainties — whereas a planet whose magnetosphere co-rotates with the external temporal-gradient structure rather than with a locked interior drifts: Saturn's SKR period lengthened from $10^{\rm h}39^{\rm m}24^{\rm s}$ (Voyager, 1980) to $\sim10^{\rm h}47^{\rm m}$ in the Cassini era, a drift of $\sim0.3\text{--}0.45\%$ per decade — roughly $30\times$ the $0.012\%$ per decade Uranus revision between Voyager 2 (1986) and the 2025 HST auroral determination (Lamy et al. 2025) — while its dual north/south SKR periods reverse their ordering at equinoxes under solar control (19 November 1995 and 11 August 2009; Gurnett et al. 2009). Uranus and Neptune, although on the ambient branch, possess tilted magnetic fields whose non-axisymmetric components anchor the magnetosphere to the deep interior; Saturn is the unique ambient-branch planet whose field is filtered axisymmetric ($<0.007^\circ$ tilt), leaving it with no interior fiducial — the combination under which the model expects the largest observable drift. The 6 May 2025 Saturn equinox supplies a direct observational test of the ambient-coupling reading in post-equinox Wind/WAVES and STEREO/WAVES archives: it predicts a renewed hemispheric period-ordering reversal observable from Earth orbit, since the $<1.2\text{ MHz}$ SKR band is ground-blocked.

Table 4.4: Saturn's discordant rotation clocks evaluated against the fixed ambient prediction $v_{\rm TEP} = 9.108$ km s$^{-1}$ ($\rho_T = 20$ g/cm³, $v_{\rm gal} = 220$ km s$^{-1}$; Step 15). Every independently measured clock exceeds $v_{\rm TEP}$: the inferred saturation density $\rho_T^{\rm spin} = \rho_T(v_{\rm obs}/v_{\rm TEP})^3$ spans $[22.1,\,25.5]\text{ g/cm}^3$ across all period and radius conventions ($[1.047,\,1.094]$ if magnetospheric clocks are evaluated at $R_{\rm eq}$ instead). The interior clocks (rows 1–3) carry the claim; the SKR rows are formal equatorial conversions demonstrating that the sign is clock- and convention-independent — Saturn's positive residual is structural and is ordered, along with Earth's $+6.4\%$, by the shared $r_*/a \approx 7.3$ nested screening-shell embedding.

Third, the interior-boundary atlas places $R_T(M_{\rm Sat}) = 18{,}930\text{ km} = 0.325\,R_{\rm vol}$ inside Saturn's diffuse core-envelope transition region, which extends out to $\sim0.6\,R$ and contains $\sim17\,M_\oplus$ of ice and rock (Mankovich & Fuller 2021); a separate, thinner helium-rain stable layer ($>2{,}500\text{--}3{,}000\text{ km}$ thick) sits above the dynamo and axisymmetrizes the internal magnetic field to a dipole tilt $<0.007^\circ$ (Cao et al. 2020; Dougherty et al. 2018) — the Cowling's-theorem-defying symmetry which removes the fiducial marker that would otherwise anchor a rigid radio clock, forcing reliance on the drifting SKR surrogate — whereas Saturn's zonal winds decay at $0.80\text{--}0.85\,R$, far above $R_T$ (Galanti et al. 2019). The same atlas places Jupiter's $R_T = 0.405\,R_{\rm vol}$ at the outer boundary of the Juno dilute-core region ($\sim0.4\text{--}0.5\,R_{\rm J}$; Wahl et al. 2017; Debras & Chabrier 2019), Earth's $R_T = 0.651\,R_\oplus$ just beneath the nonlinear PREM kernel centroid at $R_{\rm grad} = 4{,}810\text{ km} = 1.160\,R_T$ (Step 14), and the solar $\sqrt{3}\,R_T(M_\odot) = 0.715\,R_\odot$ at the helioseismic tachocline — in each case the saturation radius lands at an interior stratification or transition boundary rather than at an arbitrary depth, motivating a dedicated future channel comparing $R_T(M)$ against published giant-planet interior profiles. Extending the Step-14 nonlinear Picard machinery to the unscreened-interior regime for the first time (Step 16; Lane–Emden interiors at each planet's measured $n_{\rm eff}$, all with $\rho < \rho_T$ everywhere and Picard residuals $\sim 10^{-14}$) reveals two systematic laws across the seven-planet scan. First, the $r^2$-weighted interior gradient centroid scales monotonically with the volumetric screening ratio $S_{\rm vol} = R_T/R_{\rm vol}$: $R_{\rm grad}/R_T = 1.04$ (Mercury, $S_{\rm vol} = 0.647$), $1.19$ (Earth-polytrope, $0.651$), $1.21$ (Venus, $0.640$), $1.76$ (Neptune, $0.434$), $1.89$ (Uranus, $0.399$), $2.04$ (Jupiter, $0.405$), and $2.39$ (Saturn, $0.325$; $S_{\rm eq} = 0.314$) — the field transition zone migrating outward as screening weakens. Second, in absolute units the centroid clusters at $R_{\rm grad} \approx 0.75\text{--}0.83\,R_{\rm vol}$ for every planet scanned (Saturn $0.776$, Venus $0.775$, Earth $0.778$, Neptune $0.766$, Uranus $0.755$, Jupiter $0.825$; Mercury $0.672$) — a generic property of the unscreened operator rather than a Saturn-specific feature, so the agreement between Saturn's $R_{\rm grad} = 45{,}183\text{ km} = 0.776\,R_{\rm vol}$ and its measured zonal-wind decay base at $0.80\text{--}0.85\,R_S$ (Galanti et al. 2019) is registered as a coincidence inside a universal law: the operator predicts a transition region at $\sim0.8\,R_{\rm vol}$ for any unscreened body, and Saturn happens to possess a measured boundary there, at the same depth where the layered rotation clocks transition from the rigid interior rate to the ambient-coupled exterior. The same ambient-clock structure extends outward into the ring system. First, substituting $R_T(M_p) = R_p(\bar\rho_p/\rho_T)^{1/3}$ into the classical fluid Roche limit $a_{\rm Roche} = 2.456\,R_p(\bar\rho_p/\rho_m)^{1/3}$ yields the exact algebraic cancellation identity $a_{\rm Roche}(M_p,\rho_m) \equiv 2.456\,R_T(M_p)(\rho_T/\rho_m)^{1/3}$, in which the planet's physical radius $R_p$ and density $\bar\rho_p$ cancel identically (noting that $\rho_T$ likewise cancels out of the product $R_T(M_p)\rho_T^{1/3} = (3M_p/4\pi)^{1/3}$, so the Roche boundaries provide an exact classical tidal consistency check parameterized in $R_T$ units, whereas the five co-rotation, ring-mass, $g_t$ grain-floor, satellite-expansion, and outer-moon soliton tests below depend explicitly on $\rho_T$, $g_t$, and $v_{\rm gal}$) — fixing the radial compositional stratigraphy of Saturn's main rings directly from $R_T(M_{\rm Sat}) = 18{,}930.3\text{ km}$ ($\rho_T = 20\text{ g/cm}^3$): (a) porous and solid water ice ($\rho_m = 0.75\text{--}0.92\text{ g/cm}^3$) disrupts out to $a_{\rm Roche}(\text{ice}) = 129{,}759\text{--}138{,}904\text{ km}$ ($6.85\text{--}7.34\,R_T$), matching the A-ring outer edge at $136{,}775\text{ km}$ ($7.23\,R_T$) and F ring at $140{,}180\text{ km}$ ($7.41\,R_T$); (b) silicate rock ($\rho_m = 2.7\text{--}3.3\text{ g/cm}^3$) cannot disrupt anywhere in the B or A rings ($91{,}975\text{--}136{,}775\text{ km}$) because its Roche limit sits inward at $a_{\rm Roche}(\text{rock}) = 84{,}768\text{--}90{,}632\text{ km}$ — directly inside the C ring ($74{,}658\text{--}91{,}975\text{ km}$, C/B boundary within $-1.5\%$), explaining why tidal stripping of a differentiated progenitor leaves the B and A rings $>99\%$ pure water ice while the intact silicate core spirals inward into the mid-C ring ($\sim80{,}000\text{--}88{,}000\text{ km}$) before disrupting into the buried silicate-rich rubble belt discovered by Cassini $2.2\text{ cm}$ radiometry (Zhang et al. 2017); and (c) an iron-rich core ($\rho_m = 5.0\text{ g/cm}^3$) disrupts at $a_{\rm Roche}(\text{Fe}) = 73{,}804\text{ km}$, matching the C-ring inner edge at $74{,}658\text{ km}$ ($-1.1\%$). Simultaneously, the saturation-radius gravitational self-energy mass equivalent of Saturn, $M_{\rm rings}^{\rm TEP} \equiv u_T(M_{\rm Sat})M_{\rm Sat} = GM_{\rm Sat}^2/(R_T c^2) = 1.267 \times 10^{19}\text{ kg}$ (or $(2k_{\rm MOI}/S)u_T M_{\rm Sat} = 1.759 \times 10^{19}\text{ kg}$), matches the Cassini Grand Finale ring mass $M_{\rm rings} = (1.54 \pm 0.49) \times 10^{19}\text{ kg}$ (Iess et al. 2019) within $-0.56\sigma$ ($+0.45\sigma$), while the ring system's own screening shell $r_*(M_{\rm rings}) = \sqrt{GM_{\rm rings}/g_t} = 1.74 \times 10^6\text{ km}$ ($1\sigma$ band $[1.44,\,2.00] \times 10^6\text{ km}$; $1.58 \times 10^6\text{ km}$ at $M_{\rm rings}^{\rm TEP}$) reaches out to Hyperion's orbit ($1.481 \times 10^6\text{ km}$) at the outer boundary of the regular Titan–Hyperion subsystem. Spreading $M_{\rm rings}^{\rm TEP}$ across the B+A ring annulus fixes the mean surface density $\Sigma_{\rm TEP} = 39.4\text{ g cm}^{-2}$ (observed $47.8\text{ g cm}^{-2}$), which in turn brackets both ends of the ring particle-size distribution — the lower grain-size cutoff where a particle's own surface gravity crosses $g_t = 3.40 \times 10^{-10}\text{ m s}^{-2}$ ($r_*(M_{\rm grain}) = R_{\min} \implies R_{\min}^{\rm TEP} = 3g_t/[4\pi G\rho_{\rm ice}] = 1.35\text{--}2.43\text{ mm}$ for $\rho_{\rm ice} = 0.5\text{--}0.9\text{ g cm}^{-3}$) and the upper gravitational-wake boulder cutoff and vertical ring thickness $R_{\max}^{\rm TEP} = 2\pi G\Sigma_{\rm TEP}/\Omega_K^2 = 5.04\text{ m}$ ($6.12\text{ m}$ at observed $\Sigma$, vs. the measured $1\text{ mm}\text{--}1\text{ cm}$ to $5\text{--}10\text{ m}$ window; Zebker et al. 1985) — while the ring vertical sheet acceleration $2\pi G\Sigma = 1.65 \times 10^{-7}\text{ m s}^{-2}$ (mean) to $(2.9\text{--}4.2) \times 10^{-7}\text{ m s}^{-2}$ (B-ring core, $\Sigma_B = 70\text{--}100\text{ g cm}^{-2}$) matches the $4 \times 10^{-7}\text{ m s}^{-2}$ empirical acceleration required to close the Cassini Grand Finale Doppler residuals (Iess et al. 2019; Serra et al. 2019). Comparing $a_{\rm ring}$ against the co-rotation radius $a_{\rm sync} = (GM_p T^2/4\pi^2)^{1/3}$ across all four ringed giant planets explains why Saturn's ring architecture is unique: for Jupiter ($a_{\rm ring} = 129{,}000\text{ km} < a_{\rm sync} = 156{,}886\text{ km}$), Uranus ($a_{\rm ring} = 51{,}149\text{ km} < a_{\rm sync} = 82{,}367\text{ km}$, where $a_{\rm sync}$ instead lands inside the Belinda–Puck inner-moon gap at $75{,}255\text{--}86{,}004\text{ km}$, midpoint $+2.2\%$), and Neptune ($a_{\rm ring} = 62{,}933\text{ km} < a_{\rm sync} = 79{,}183\text{ km}$), the dense main rings terminate inside $a_{\rm sync}$; only at Saturn ($a_{\rm ring} = 136{,}775\text{ km} > a_{\rm sync}$) does the main ring sheet physically span the co-rotation zone, which explains why Saturn uniquely traps and retains a massive ring sheet of order its saturation self-energy mass equivalent $u_T(M_{\rm Sat})M_{\rm Sat}$ against sub-synchronous tidal infall whereas disrupted ring material inside $a_{\rm sync}$ at Jupiter, Uranus, and Neptune drains inward onto the host planet. Evaluating $a_{\rm sync}$ at the ambient-equilibrium period $T_{\rm TEP} = 11.55$ hr places it at $118{,}425\text{ km} = 2.034\,R_S$ — inside the Cassini Division ($117{,}580\text{--}122{,}170\text{ km}$) and within $0.72\%$ of the B-ring outer edge, while the interior clock's co-rotation ($111{,}569\text{ km}$) lies $5.1\%$ inside that edge; three radii — the ambient co-rotation surface, the Mimas $2{:}1$ inner Lindblad resonance ($116{,}882\text{ km}$), and the observed B-ring boundary — thus converge within $\sim1.5\%$, placing the system's widest gap at the intersection of the moon-resonance and the ambient-clock co-rotation surface (Step 16; as the magnetospheric period drifted $10^{\rm h}39^{\rm m}\to10^{\rm h}48^{\rm m}$ the co-rotation surface swept outward through the B-ring edge toward the ambient-equilibrium location). Second, the moon system completes the nested structure across three distinct regimes: (i) Saturn's inner moons migrate at rates $\sim100\times$ the equilibrium-tide expectation because their orbits are resonance-locked to the planet's internal gravity-wave frequencies (Fuller, Luan & Quataert 2016; Lainey et al. 2020) — the same g-mode cavity that the C-ring waves probe (Mankovich & Fuller 2021) and that contains $R_T = 0.325\,R_S$ — while Saturn's spin axis is captured in the $s_8$ spin-orbit resonance with Neptune at the solar screening-shell boundary $r_*(M_\odot) = 29.97\text{ AU} \approx a_{\rm Nep}$ (Wisdom et al. 2022). Crucially, whereas standard resonance-locking requires an unknown interior evolutionary timescale $t_\alpha$ as a free parameter to explain why every Saturnian moon migrates on a nearly distance-independent timescale $t_{\rm mig} = a/\dot a \approx 10\text{ Gyr}$ while Earth requires a separate transient North Atlantic ocean-shelf resonance to explain its anomalously fast present-day lunar recession ($\dot a_{\rm Moon} = 3.82 \pm 0.07\text{ cm yr}^{-1}$, $t_{\rm mig} = 10.06\text{ Gyr}$; Williams & Boggs 2016; Paper 17), in TEP an ambient-branch host planet ($M < M_{\rm cap}$: Earth and Saturn) possesses a spatially inhomogeneous two-zone clock structure — an interior saturation boundary at $R_T(M) < R_p$ ($0.325\,R_S$ inside Saturn's g-mode cavity; $0.651\,R_\oplus$ in Earth) surrounded by an unscreened orbital zone $a \gg R_T$. Because this radial boundary separates the interior cavity from the exterior orbital zone, a homogeneous coordinate-time redefinition $\mathrm{d}\tau = A(t)\,\mathrm{d}t$ cannot eliminate the differential mode/lapse drift $|\dot\omega_{\rm mode}/\omega_{\rm mode}| = |\mathrm{d}\ln|g_{00}|/\mathrm{d}t| = 2\dot A/A = 2H_0$ ($H_0 = 2g_t/c = 7.16 \times 10^{-11}\text{ yr}^{-1}$, $t_H = 13.97\text{ Gyr}$; or direct secular drift of $G_{\rm loc}(r,t)$ across the boundary, Paper 17 §5.10.2). For any coupled satellite ($r_*(M_{\rm moon})/a > 1$) resonance-locked to a non-axisymmetric planetary interior or ocean-basin mode $\Phi_{\rm mode}(r,\varphi - \omega_{\rm mode}t)$ (Fuller, Luan & Quataert 2016; Lainey et al. 2020), the non-axisymmetric mode torque $\Gamma_{\rm mode} = -\partial\Phi_{\rm mode}/\partial\varphi = \dot L_{\rm orb} = -\dot J_{\rm planet}$ transfers spin angular momentum from the planet to the orbit at the rate required for the mean motion $n_{\rm orb} \propto a^{-3/2}$ to track $\dot\omega_{\rm mode}/\omega_{\rm mode}$, yielding secular orbital elements $\dot a/a = \tfrac{2}{3}|\dot\omega_{\rm mode}/\omega_{\rm mode}| = \tfrac{4}{3}H_0 = 8g_t/[3c]$, $\dot L_{\rm orb}/L_{\rm orb} = +\tfrac{2}{3}H_0 = -\dot J_{\rm planet}/L_{\rm orb}$, $\mathrm{d}\ln|E_{\rm orb}|/\mathrm{d}t = -\dot a/a = -\tfrac{4}{3}H_0$ (signed $\dot E_{\rm orb}/|E_{\rm orb}| = +\tfrac{4}{3}H_0$), and $\langle\dot e\rangle_{\rm sec} = 0$ on a universal, zero-parameter expansion timescale $t_{\rm mig}^{\rm TEP} = \tfrac{3}{4}t_H = 3c/(8g_t) = 10.48\text{ Gyr}$ (or $t_{\rm mig}^{(1H_0)} = \tfrac{3}{2}t_H = 20.95\text{ Gyr}$ under linear $A^1$ clock drift $|\dot\omega/\omega| = H_0$). Because $t_{\rm mig}^{\rm TEP}$ paces the resonance-locked spin-to-orbit torque $\dot L_{\rm orb} = -\dot J_\oplus$ rather than replacing tidal angular-momentum conservation, it is fully consistent with Earth having transferred $\sim 83\%$ of $J_{\rm E+M}$ to the Moon, with the $3.7\text{ TW}$ ocean tidal dissipation measured by satellite altimetry, and with the $\sim 2.3\text{ ms century}^{-1}$ lengthening of the day in historical eclipse records: this single constant predicts a resonance-locked ceiling rate $\dot a_{\rm Moon}^{\rm TEP} = a_{\rm Moon}/t_{\rm mig}^{\rm TEP} = 3.67\text{ cm yr}^{-1}$ for Earth's Moon ($96.1\%$ of present-day LLR $3.82 \pm 0.07\text{ cm yr}^{-1}$, with $0.15\text{ cm yr}^{-1}$ non-locked residual ocean dissipation; backward integration gives $0.651\,a_0$ at $t = -4.5\text{ Gyr}$ on the pure exponential ceiling and $0.559\,a_0 = 33.7\,R_\oplus$ when coupled with the $0.15\text{ cm yr}^{-1}$ $a^{-11/2}$ term, while supercontinent basin detuning across the Wilson cycle naturally reduces the time-averaged Proterozoic rhythmite rate to $\sim 1.7\text{--}2.2\text{ cm yr}^{-1}$ below the $3.67\text{ cm yr}^{-1}$ resonance-locked ceiling, avoiding the $t = -1.55\text{ Gyr}$ Gerstenkorn collision; Paper 17 §5.10.1), predicts $\dot a_{\rm Titan}^{\rm TEP} = a_{\rm Titan}/t_{\rm mig}^{\rm TEP} = 11.66\text{ cm yr}^{-1}$ for Saturn's Titan ($-3.1\%$, $0.18\sigma$ vs. Cassini $11.3 \pm 2.0\text{ cm yr}^{-1}$; Lainey et al. 2020; $5.83\text{ cm yr}^{-1}$ under the $1H_0$ linear comparator), and requires the apparent equilibrium-tide quality factor $Q_{\rm eff}^{\rm TEP}(M_{\rm moon},a) = 3k_2(M_{\rm moon}/M_{\rm Sat})(R_{\rm Sat}/a)^5 n_{\rm orb}t_{\rm mig}^{\rm TEP}$ ($k_2 = 0.382$) across Saturn's six measured moons to vary across a $50\times$ span — $Q_{\rm eff}^{\rm TEP} = 6{,}966$ at Mimas, $3{,}973$ at Enceladus (vs. Cassini $1{,}000\text{--}4{,}000$), $5{,}672$ at Tethys (vs. $>3{,}000$, reproducing the observed non-monotonic rise above Enceladus driven by Tethys's $5.7\times$ larger mass), $2{,}014$ at Dione (vs. $1{,}500\text{--}4{,}000$), $484$ at Rhea (vs. $\sim300$), and $119.4$ at Titan ($238.8$ under $1H_0$, vs. Cassini radio-tracking $124 \pm 22$, $-0.21\sigma$, and astrometric $61^{+240}_{-31}$; Step 15). Conversely, Jupiter ($M_{\rm Jup} = 317.8\,M_\oplus > M_{\rm cap} = 194.2\,M_\oplus$) sits on the saturated soliton branch where the interior clock is rigid rather than ambient-drifting, providing a positive control: its Galilean satellites in the $4{:}2{:}1$ Laplace resonance retain a high, near-classical tidal quality factor $Q_{\rm Jup} = (3.56 \pm 0.56) \times 10^4$ (Lainey et al. 2009) rather than the rapid $Q \sim 10\text{--}100$ ambient expansion observed at Earth and Saturn; (ii) evaluating each satellite's own screening-shell ratio $r_*(M_{\rm moon})/a_{\rm orb}$ around Saturn shows that all ring-embedded moonlets sit at $r_*/a < 1$ (Daphnis $0.028$, Pan $0.233$, Atlas $0.261$), the two F-ring shepherd moons Prometheus ($1.270$) and Pandora ($1.158$) are the first bodies to cross $r_*/a = 1$ right at the main-ring outer boundary ($139{,}380\text{--}141{,}720\text{ km}$), and all regular outer moons from Epimetheus/Janus ($2.12,\,4.03$) and Mimas ($14.6$) through Titan ($133.0$) and Iapetus ($5.29$) satisfy $r_*/a > 1$ with the sole exception of Hyperion ($r_*/a = 0.709 < 1$), which is uniquely the only chaotically tumbling moon in the Saturnian system (Wisdom, Peale & Mignard 1984; conventionally driven by its $4{:}3$ resonance with Titan and irregular figure); and (iii) across all three large outer non-tidally-locked moons of the giant planets with spacecraft-measured masses and radii — Phoebe at Saturn ($M = 8.29 \times 10^{18}\text{ kg}, R = 106.5\text{ km}, P = 9.274\text{ hr}, r_*/a = 0.098 < 1$; Thomas et al. 2007; Castillo-Rogez et al. 2012), Himalia at Jupiter ($M = 4.20 \times 10^{18}\text{ kg}, R = 69.8\text{ km}, P = 7.782\text{ hr}, r_*/a = 0.079 < 1$; Emelyanov 2005; Pilcher et al. 2012), and Nereid at Neptune ($M = 3.10 \times 10^{19}\text{ kg}, R = 178.5\text{ km}, P = 11.594\text{ hr}, r_*/a = 0.447 < 1$; Kiss et al. 2016) — every object is decoupled from its host's screening shell ($r_*/a < 1$) and sits on the small-body soliton-capacity branch at $\eta_{\rm rot} = 0.1392$ ($0.1477$ at uniform $k_{\rm MOI} = 0.35$) for Phoebe, $0.1190$ for Himalia, and $0.1377$ for Nereid, with a three-moon mean $\bar\eta_{\rm moons} = 0.1320 \pm 0.0113$ ($0.1348 \pm 0.0146$ at uniform $k_{\rm MOI} = 0.35$) that matches the parameter-free Chandrasekhar–Jacobi/virial bracket $[0.125,\,0.1375]$ (midpoint $0.1313$) to $+0.5\%$ ($+2.7\%$, $\hat\rho_T^{\rm moons} = 19.8\text{--}20.6\text{ g/cm}^3$; Step 15) and extends the soliton rotational scaling down to $4.2 \times 10^{18}\text{ kg}$ across nearly eleven orders of magnitude in mass. Finally, Saturn's ``unbraked'' classification is quantitatively robust against its massive moon: Titan's orbital angular momentum carries only $L_{\rm Titan}/J_{\rm Sat} = 1.3\%$ of Saturn's spin angular momentum (versus $\sim83\%$ for the Earth+Moon system), so even the resonance-locked effective tidal response $Q_{\rm eff} \sim 100$ implied by Titan's measured rapid migration ($\dot a_{\rm Titan} = 11.3\text{ cm yr}^{-1}$, orbital expansion timescale $10.8\text{ Gyr}$; Lainey et al. 2020) perturbs Saturn's spin by $<0.6\%$ over the age of the Solar System — and tidal braking in any case moves $v_{\rm obs}$ downward, opposite to the observed $+8.4\%$ residual direction, so neither Titan's present torque nor the hypothesized lost satellite ``Chrysalis'' invoked to explain Saturn's $26.7^\circ$ obliquity and young rings via the $s_8$ spin-orbit resonance with Neptune (Wisdom et al. 2022) can manufacture the positive ambient-branch excess.

Finally, while the Sun ($\bar\rho_\odot = 1.41\text{ g/cm}^3$, $S_\odot = R_T(M_\odot)/R_\odot = 0.4127$) lies outside both the cold planetary gyration relation Eq. (\ref{eq:kmoi_screening}) (because radiative-core thermal support produces an $n \approx 3$ central condensation with $\rho_c = 152.9\text{ g/cm}^3 \approx 108\,\bar\rho_\odot$ and $k_{\rm MOI,\odot} \approx 0.070$) and the Galactic-ambient planetary spin law Eq. (\ref{eq:veq_tep_law}) (which applies to bodies embedded in the Galactic potential with $u_{\rm surf} \ll u_{\rm gal} = 5.39 \times 10^{-7}$, whereas the Sun's surface potential depth $u_{\rm surf}(M_\odot) = GM_\odot/(R_\odot c^2) = 2.121 \times 10^{-6}$ exceeds $u_{\rm gal}$ by $3.94\times$ so the solar envelope is self-screened from the Galactic floor), its stratified interior physically crosses $\rho_T = 20.0\text{ g/cm}^3$ and motivates an exploratory solar prototype comparing $\rho_T$ against Standard Solar Model structure and helioseismology (Step 15; Figure 7). First, in the global angular-momentum budget of the Solar System, adding the orbital angular momentum of the planets ($J_{\rm orb} = 3.13 \times 10^{43}\text{ kg m}^2\text{ s}^{-1}$, $99.4\%$ of the total) to the Sun's present-day spin ($J_{\rm spin,\odot} = 1.92 \times 10^{41}\text{ kg m}^2\text{ s}^{-1}$) yields $J_{\rm sys,obs} = 3.15 \times 10^{43}\text{ kg m}^2\text{ s}^{-1}$ ($\eta_{\rm sys} = j_{\rm sys}/j_T(M_\odot) = 0.081$), within $35\%$ of the proposed solar soliton capacity $J_{\rm sol}(M_\odot) = \eta_{\rm sec} M_\odot R_T(M_\odot)^2 \omega_T = 4.85 \times 10^{43}\text{ kg m}^2\text{ s}^{-1}$ ($5.01 \times 10^{43}$ at $\eta_0 = 0.129$)—an order-of-magnitude consistency analogous to the Earth+Moon system under protostellar disk transfer and magnetized solar-wind braking. Second, integrating the clock-amplitude saturation law $S_A(\rho) = \min[1,\,(\rho/\rho_T)^{1/3}]$ through the Standard Solar Model (Model S; Christensen-Dalsgaard et al. 1996) identifies three interior radii (Figure 7a): (i) the local density crosses $\rho_\odot(r) = \rho_T = 20.0\text{ g/cm}^3$ ($S_A^{\rm loc} = 1$) at $r_{\rm core} = 0.249\,R_\odot$ (enclosing $44\%$ of $M_\odot$), lying near the conventional $r \lesssim 0.25\,R_\odot$ nuclear energy-generating region; (ii) the 1-D spherical saturation radius sits at $R_T(M_\odot) = 0.4127\,R_\odot$ ($287{,}387\text{ km}$, adjacent to the enclosed-mean-density crossing $\bar\rho(<r) = \rho_T$ at $0.368\,R_\odot$), while both the BVP gyration-threshold crossing $y_{\rm BVP}(r) = \tfrac{1}{2}S_\odot = 0.2064$ of Eq. (\ref{eq:kmoi_screening}) evaluated at the Step 14 physical Compton wavelength $\lambda_c(\rho_T) = 2{,}479\text{ km} = 0.00356\,R_\odot$ ($r = 0.7228\,R_\odot$, $+1.37\%$) and the candidate 3-D spatial metric projector norm $r_{\rm 3D} = \|\Pi_{ij}\|\,R_T(M_\odot) = \sqrt{3}\,R_T(M_\odot) = 0.7148\,R_\odot$ ($\Pi_\mu^\mu = 3$ in Paper 0 §2.2; $M(<r_{\rm 3D})/M_\odot = 0.978$, $\bar\rho(<r_{\rm 3D}) = (\rho_T/3\sqrt{3})[M(<r_{\rm 3D})/M_\odot] = 3.76\text{ g/cm}^3$; spanning $r_{\rm 3D}/R_\odot \in [0.471,\,0.970]$ across the $\rho_T \in [8,\,70]\text{ g/cm}^3$ GNSS window) coincide with the helioseismically determined base of the solar convection zone at $r_{\rm cz} = 0.713 \pm 0.001\,R_\odot$ ($+0.25\%$ at $\rho_T = 20\text{ g/cm}^3$; Christensen-Dalsgaard et al. 1996; conditionally identifying $r_{\rm cz}$ with $\sqrt{3}\,R_T(M_\odot)$ yields $\rho_T^{\rm (cz)} = 20.15 \pm 0.09\text{ g/cm}^3$). Inside $r \le r_{\rm cz}$ where $y_{\rm BVP}(r) \ge \tfrac{1}{2}S_\odot$ in a stably stratified radiative zone, any latitudinal angular-velocity gradient $\partial_\theta\Omega \neq 0$ induces an advective scalar gradient $\partial_\theta u$ opposed by the meridional scalar-shear stress $\tau_{r\theta}^\phi = M_{\rm Pl}^2 P_{,X}\partial_r u\,\partial_\theta u$, damping differential shear into rigid-body rotation, whereas beneath $r_{\rm cz}$ the rotational tachocline is prolate, rising from $r_c(0^\circ) = 0.693 \pm 0.002\,R_\odot$ at the equator to $r_c(60^\circ) = 0.717 \pm 0.003\,R_\odot$ at high latitude ($\Delta r_c = +0.024 \pm 0.004\,R_\odot$; Charbonneau et al. 1999), matching the inward equatorial shear depression $\delta r_{\rm eq} = (r_{\rm 3D}/\sqrt{3})[(\Omega_{\rm eq}-\Omega_{\rm rad})/\Omega_{\rm eq}] = 0.026\,R_\odot$ ($r_c^{\rm TEP}(0^\circ) = 0.688\,R_\odot \to r_c^{\rm TEP}(60^\circ) = 0.708\,R_\odot$); and (iii) across the unscreened convective envelope ($r > r_{\rm cz}$) of a self-screened star ($u_{\rm surf} > u_{\rm gal}$), associating the local Rindler clock-acceleration rate $g_{\rm surf,\odot}/(2c) = c\,u_{\rm surf}/(2R_\odot) = 456.5\text{ nHz}$ with the cyclic rotation frequency $\nu_{\rm eq}^{\rm TEP} \equiv \Omega_{\rm eq}^{\rm TEP}/(2\pi)$ per rotational circumference $2\pi R_\odot$ (an explicit cyclic-frequency convention, since equating $g_{\rm surf}/(2c)$ to the angular rate $\Omega_{\rm eq}$ would lower the speed by $2\pi$) gives $v_{\rm eq}^{\rm TEP} = 2\pi R_\odot\nu_{\rm eq}^{\rm TEP} = \pi\,c\,u_{\rm surf}(M_\odot) = 1.997\text{ km s}^{-1}$, within $-0.91\%$ of the SOHO/MDI helioseismic equatorial plasma rate $460.7\text{ nHz}$ and near the conventional $25.38\text{ d}$ ($456.0\text{ nHz}$) low-latitude sunspot Carrington reference (Schou et al. 1998). Dividing $S_\odot = 0.4127$ by Paper 0's canonical DEF scalar coupling norm $|\alpha_0| = \sqrt{2}|\beta_A| = \sqrt{2}$ yields the equator-to-pole fractional differential shear $\Delta_\Omega^{\rm TEP} = S_\odot/|\alpha_0| = S_\odot/\sqrt{2} = 0.2918$ ($+1.07\%$ vs. helioseismic $a_2 + a_4 = 0.2887$), and adopting an idealized minimal-complexity equal harmonic benchmark ($a_2 = a_4 = \tfrac{1}{2}\Delta_\Omega^{\rm TEP} = 0.1459$ with zero borrowed shape parameters, vs. observed $a_2 = 0.140, a_4 = 0.149$) predicts the radiative-zone lock rate at the $\ell = 2$ quadrupole zero ($P_2(\cos\theta_*) = 0$, latitude $35.26^\circ$, where $\sin^2\psi_* = 1/3$ and $P_4(\sin 35.26^\circ) = -7/18 \neq 0$) as $\nu_{\rm rad}^{\rm TEP} = \nu_{\rm eq}^{\rm TEP}(1 - \tfrac{2}{9}\Delta_\Omega^{\rm TEP}) = 426.9\text{ nHz}$ versus the SOHO/GONG radiative-zone rate $432.0\text{ nHz}$ ($-1.2\%$; Figure 7b). For the metal-rich Kepler solar-mass analog HD 173701 (KIC 8006161; $M_* = 1.00\,M_\odot, R_* = 0.93\,R_\odot, S_* = 0.444$), $\Delta_\Omega^{\rm TEP} = S_*/\sqrt{2} = 0.314$ compares directly with the observed Kepler sliding-window photometric period range of $25\text{--}35\text{ d}$ across active latitudes ($\Delta P/P_{\max} = 0.286$; Karoff et al. 2018), while broader Kepler asteroseismic mode-splitting inversions across F/G stars indicate still stronger convective-zone shears with wide Bayesian credible intervals (Benomar et al. 2018; a full 3-D rotating MHD + scalar-field derivation across stellar Rossby numbers remains future work).

To test rigorously whether the convergence of these independent channels around $\rho_T \approx 20\text{ g/cm}^3$ could arise from post-hoc selection of algebraic projection factors or radius definitions (a look-elsewhere / numerology null test; Step 15), a $100{,}000$-trial Monte Carlo coincidence simulation was executed across a five-decade log-uniform prior $\rho_{\rm rand} \in [0.1,\,10^4]\text{ g/cm}^3$. With tidally braked present-day Earth excluded from the spin set, the five independent $\rho_T$-calibrating observables — (1) terrestrial GNSS clock correlation length $L_c = 4{,}200\text{ km}$ ($\propto \rho^{-1/3} \implies \hat\rho_T = 19.24\text{ g/cm}^3$), (2) outer-three giant-planet equatorial spin across Uranus, Neptune, and Saturn ($\propto \rho^{+1/3} \implies \hat\rho_T = 19.89\text{ g/cm}^3$), (3) the substellar brown-dwarf rotational period floor $P_{\min} = 1.150\text{ hr}$ ($\propto \rho^{-1/2}$, an orthogonal power-law exponent $\implies \hat\rho_T = 20.79\text{ g/cm}^3$), (4) self-compressed terrestrial moments of inertia $2k_{\rm MOI}/S$ ($\propto \rho^{+1/3} \implies \hat\rho_T = 17.89\text{ g/cm}^3$), and (5) the Solar Model S convection-zone base $r_{\rm cz} = 0.713\,R_\odot$ ($\propto \rho^{-1/3} \implies \hat\rho_T = 20.15\text{ g/cm}^3$) — yield an across-channel mean of $\hat\rho_T = 19.59 \pm 1.10\text{ g/cm}^3$ (logarithmic dispersion $\sigma_{\log_{10}\rho} = 0.0249\text{ dex}$, full five-channel span $0.0652\text{ dex}$, RMS fractional error $2.04\%$; or $\hat\rho_T = 19.45 \pm 1.22\text{ g/cm}^3$ across the four non-solar channels alone). On a 1-D random-density scan with fixed equations, only $p = 1.69 \times 10^{-3}$ of random densities achieve an RMS fractional error $\le 2.04\%$ ($p = 2.29 \times 10^{-3}$ across the four non-solar channels). Under the multi-channel coincidence null hypothesis — where the observables reflect unrelated physical scales drawn independently across $[0.1,\,10^4]\text{ g/cm}^3$ — the probability of all five landing within a $0.0652\text{ dex}$ window is $p = 1.44 \times 10^{-7}$ without look-elsewhere freedom ($p = 8.86 \times 10^{-6}$ across four channels), and remains $p = 2.9 \times 10^{-4}$ ($3.6\sigma$, $29/100{,}000$ realizations; $p = 1.0 \times 10^{-3}$ across four channels) even when the null model is granted a full $7^3 = 343$-combination look-elsewhere trials dictionary of algebraic factors $\mathcal{F} = \{\tfrac{1}{2},\,1/\sqrt{2},\,1,\,\sqrt{2},\,\sqrt{3},\,2,\,\pi\}$ across the interior/period-floor channels within a $\pm 10\%$ ($0.087\text{ dex}$) tolerance window.

5. Physical Constraints on the Proximity Scale

A natural question arises: why does the saturation scale take the observational proxy value \(\rho_T \approx 20\) g/cm³? This section argues that the underlying proximity scale is not treated as an arbitrary fit parameter: it is empirically calibrated, while additional coincidence-level reference scales and their quartic-potential / spatial-projection derivations are recorded below. The GNSS coherence length \(L_c\) is a derived quantity; the fundamental parameter is the proximity scale itself, of which density is an observable proxy. In the canonical TEP framework, the full screening state of any system is determined by the environmental operator \(S_\Sigma(E)\); \(\rho_T\) is the density projection of that operator, not the complete screening description.

5.1 \(\rho_T\) as the Fundamental Saturation Parameter and \(L_c\) as Its Projected Scale

The saturation scaling law \(R_T \propto M^{1/3}\)—where a soliton interpretation is one candidate microscopic realization—implies a constant characteristic density:

For any object of mass \(M\), the saturation radius is determined by:

The terrestrial coherence length \(L_c \approx 4200\) km is therefore not an independently tuned galaxy-scale parameter; operationally, it is the observed GNSS covariance scale used to calibrate the projected Earth-scale realization of \(R_T(M_\oplus)\). This relation is inverted in Section 2 to calibrate \(\rho_T\) from the measured \(L_c\) for Earth:

The question thus reduces to: what physical considerations constrain \(\rho_T\)?

5.2 The Dimensional Identity Bridge

A natural concern is whether the proximity of $\rho_T \approx 20$ g/cm³ to the density of terrestrial condensed matter implies a composition-dependent coupling that would violate the Weak Equivalence Principle. Under universal conformal coupling, the scalar sector is sourced by the bulk trace $T = -\rho$, independent of microscopic composition. The saturation scale $\rho_T$ must therefore be a property of the vacuum potential $V(\phi)$, not an emergent property of the local matter.

The electron Fermi wavelength does not cause the scalar field to saturate. If it did, the field would require a direct, particle-specific coupling to the electron number density $n_e$, making the field exert a different force on iron than on hydrogen at the same bulk density. This would introduce a gross WEP violation, already constrained by experiments such as the MICROSCOPE satellite.

Instead, $\rho_T$ is treated as a fundamental constant of the scalar sector — the geometric reference density of the saturation-radius law — rather than an emergent property of the local matter. The recorded coincidence is, in natural units,

Because this is a property of the scalar sector itself, the dimensional identity does not introduce composition-dependent coupling; WEP protection then follows from the universal matter coupling and the screened effective scalar charge. The field universally couples to bulk $\rho$ and begins its continuous screening transition when local $\rho$ approaches this $\rho_T$. The potential sector's own normalization is a separate quantity — the quartic coupling $\lambda$ (equivalently $\Lambda_V = \lambda^{1/4}M_{\rm Pl} \approx 1.3\times10^{2}$ GeV on the Cassini-compatible branch) — and is not to be identified with $\rho_T$ (Appendix C.3).

The bulk density of zero-pressure condensed matter is governed by valence Coulomb packing on the scale of the Bohr radius $a_0 = (\alpha m_e)^{-1}$ ($\rho_{\rm solid} \sim A m_p / (\frac{4\pi}{3} r_s^3 a_0^3) \sim 3\text{--}10$ g/cm$^3$ for typical terrestrial rock and iron with Wigner–Seitz radius $r_s \sim 2\text{--}3$), whereas the inner Thomas-Fermi core screening radius $R_{\rm TF} \approx a_0 Z^{-1/3}$ characterizes the high-compression regime where inner electron shells overlap:

For representative rock-forming elements ($A/Z \approx 2$, $AZ \sim 10^2\text{--}10^3$), $\rho_{\rm TF} \sim 10^2$–$10^3$ g/cm$^3$ marks full Thomas-Fermi core compression, while the initial crossover from zero-pressure valence solids ($\sim 3\text{--}8$ g/cm$^3$) toward pressure-ionized degeneracy support begins in the intermediate $\sim 10\text{--}10^2$ g/cm$^3$ regime (Section 5.3; the exact pressure-ionization threshold varies with composition $Z$, so this regime serves as a broad order-of-magnitude condensed-matter reference window rather than a composition-independent constant).

5.3 The Electron Degeneracy Threshold

Coincidence-level significance attaches to \(\rho_T \approx 20\) g/cm³: this density lies in the condensed-matter regime where quantum pressure begins to become dynamically relevant. The crossover from thermal and electrostatic support to degeneracy-dominated support is gradual, with full degeneracy domination occurring only at much higher (white-dwarf) densities.

Table 5.1: Equation of state transitions by density. The saturation density \(\rho_T \approx 20\) g/cm³ lies near the onset of electron degeneracy.

This coincidence provides a useful condensed-matter reference scale for where quantum pressure becomes dynamically relevant to the equation of state. Under the universal conformal-coupling benchmark used throughout this corpus, the scalar sector is sourced primarily by the non-relativistic mass-density term, with pressure corrections entering only as a small relativistic correction in the condensed-matter regime. Electron degeneracy therefore does not, by itself, provide a mechanism that sets \(\rho_T\), but it motivates the \(\sim 10\text{–}30\) g/cm³ window as a natural coincidence-level cross-check.

5.4 From Quartic Potential Scale to PREM Terrestrial Transition Width

Box 6.5 adopts the corpus's quartic completion, in which the amplitude sector is normalized by the quartic coupling \(\lambda\) and \(\rho_T\) enters as the geometric constant of the saturation-radius law \(R_T(M)\). Once \(\rho_T\) and the corpus branch of \(\lambda\) are specified, the field equation determines a smooth crossover between Compton-unresolved and Compton-resolved regimes (\(\lambda_c(\bar\rho)\) versus \(R_T(M)\)). On the canonical-kinetic benchmark at the fixed Cassini-compatible corpus normalization (\(\lambda = 7.53\times 10^{-66}\), \(\rho_T = 20\) g/cm³), Step 14 (Figure 1; Appendix B.2) solves this boundary-value problem across a 4-layer PREM Earth density profile (\(\rho \sim 3.5\text{–}12.9\) g/cm³), recovering a radial transition-gradient centroid at \(R_{\rm grad} = 4{,}875\) km (\(1.18\,R_T(M_\oplus)\)) and a projected 3-D Yukawa surface spatial-kernel decay length of \(L_c^{\rm PREM} = 4{,}010\) km (\(L_c^{\rm PREM}/R_T(M_\oplus) = 0.967\)), while solving the coupled 1-D spherical nonlinear two-branch kinetic operator \(\nabla\cdot(P_{,X}\nabla u) = M_{\rm Pl}^{-2}V_{\rm eff}'(u)\) (Eq. \(\ref{eq:nonlinear_chameleon_eom}\)) directly via 100-iteration damped Picard relaxation (\(\|\Delta u^{(100)}\|_\infty/\|u\|_\infty = 2.67 \times 10^{-14}\); Step 14, Appendix C.2) recovers \(R_{\rm grad}^{\rm NL\text{-}PREM} = 4{,}810\) km (\(1.160\,R_T(M_\oplus)\)), \(L_c^{\rm NL\text{-}PREM} = 3{,}957\) km, and \(L_c^{\rm NL,vol} = 3{,}170\) km. Extending this radial nonlinear solve to non-spherical 3-D azimuthal heliocentric-motion anisotropy and deriving \(\rho_T\) itself from vacuum microphysics remain targets for future work.

Many-body condensed-matter physics (including the onset of electron degeneracy) governs the baryonic equation of state and provides useful reference densities, but under universal coupling it does not by itself select the saturation scale \(\rho_T\). A complete theoretical account must instead explain both the value of \(\rho_T\) and the mapping between the geometric saturation radius \(R_T(M)\) and the observable correlation length in clock networks.

While Section 5.3 contextualizes ρ_T within macroscopic condensed-matter physics, establishing its fundamental origin requires examining the scalar coupling scale at the vacuum level.

5.5 Dimensional Analysis: The Scalar Coupling Scale

Expressed in natural units, the saturation density carries the energy-density scale \(\rho_T^{1/4} \approx 96.4\) keV for \(\rho_T = 20\) g/cm³. This is a fourth-root reading of a density, not the scalar potential's normalization: the potential coefficient is the quartic coupling \(\lambda = 7.5\times10^{-66}\), equivalently \(\Lambda_V \equiv \lambda^{1/4}M_{\rm Pl} \approx 1.3\times10^{2}\) GeV (Box 6.5, Appendix C; step 14). The kinetic completion carries its own distinct scale, \(\Lambda_X = \sqrt{M_{\rm Pl}H_0} \approx 1.9\) meV, which sets the Temporal-Shear threshold and plays no role in the amplitude-sector density argument.

While this relation remains a dimensional cross-check rather than a first-principles derivation of \(\rho_T\) from QED vacuum loops, the appearance of the rational exponent \(4/3\) has a direct structural rationale in the quartic potential degree \(m = 4\) and 3-D spatial clock-amplitude response established in Step 14 (amplitude_law_check and closed_form_theoretical_derivations): because minimizing \(V_{\rm eff}(u) = \tfrac{1}{4}\lambda M_{\rm Pl}^4 u^4 - |\beta_A|\rho u\) fixes the 3-D spatial clock-amplitude response exponent to \(1/(m-1) = 1/3\) across the \(\Pi_\mu^\mu = 3\) spatial dimensions, projecting the electromagnetic fine-structure coupling \(\alpha_{\rm EM} = e^2/(4\pi\varepsilon_0\hbar c) = 1/137.036\) of the atomic electron wavepacket through this \(1/(m-1) = 1/3\) spatial response screens the 1-D electron mass-energy scale to \(\tilde m_e c^2 = \alpha_{\rm EM}^{1/(m-1)} m_e c^2 = \alpha_{\rm EM}^{1/3} m_e c^2 = 99.12\text{ keV}\) (within \(+2.8\%\) of \(\rho_T^{1/4} = 96.36\text{ keV}\)). Raising \(\tilde m_e\) to the \(m = 4\) power of the 4-D energy-density operator produces the exact exponent \(m/(m-1) = 4/3\) (coinciding with the relativistic Chandrasekhar–Fermi-Thomas polytropic index \(\gamma_F = 4/3\)) and yields \(\rho_T^{\rm EM} = \tilde m_e^4 c^3/\hbar^3 = \alpha_{\rm EM}^{4/3} m_e^4 c^3/\hbar^3 = 22.39\text{ g/cm}^3\).

5.6 Status: Phenomenological Constraint with Theoretical Foundation

The present analysis treats \(\rho_T \approx 20\) g/cm³ as an empirically calibrated parameter, while recording additional coincidence-level reference scales and open derivation targets:

1. Condensed-matter EOS cross-check: \(\rho_T\) lies at the pressure-ionization and Thomas–Fermi electron-degeneracy onset scale (\(10\text{--}30\text{ g/cm}^3\))
2. Quartic-projected QED scale relation: \(\rho_T^{\rm EM} = (\alpha_{\rm EM}^{1/(m-1)} m_e)^m c^3/\hbar^3 = \alpha_{\rm EM}^{4/3} m_e^4 c^3/\hbar^3 = 22.39\text{ g/cm}^3\), where \(m/(m-1) = 4/3\) follows from the \(m = 4\) quartic potential and \(1/(m-1) = 1/3\) spatial response (Step 14)
3. PREM terrestrial BVP projection: Solving \(\nabla^2 u = M_{\rm Pl}^{-2}V_{\rm eff}'(u)\) across the 4-layer PREM Earth profile yields \(L_c^{\rm PREM} = 4{,}010\text{ km}\) (\(L_c^{\rm PREM}/R_T(M_\oplus) = 0.967\); Step 14)

Its use in quantum or topological extensions as a microscopic cutoff or defect-overlap scale is a conjectural extrapolation of the macroscopic saturation proxy, not an independent consequence of the present paper.

The key point for the present manuscript is that \(\rho_T\) is not treated as a free fit parameter: it is GNSS-consistent by clock correlations (Section 2) and then carried, without retuning, into the cross-scale tests that follow.

6. The Universal Density Constraint

A key test is whether the Temporal Topology saturation radius \(R_T\) yields quantitative expectations for otherwise free scales. Under a soliton interpretation, RBH-1 would correspond to a gravitational soliton; however, the saturation radius itself is the primary observable, constrained by the universal density \(\rho_T\) rather than introduced as an adjustable parameter.

The Universal Density Hypothesis

The Temporal Topology framework tests the empirical claim that a candidate saturation scale $\rho_T \approx 20$ g/cm³ governs compact-object structure across mass scales. A soliton interpretation provides one candidate microscopic realization of this saturation physics. This hypothesis is testable via two independent windows:

1. Planetary Scale: The Earth-scale Temporal Topology parameter $R_T(M_\oplus)$ is calibrated through the observed GNSS projected covariance length ($L_c \approx 4200$ km), serving as the empirical anchor.
2. Cosmological Scale: The extrapolated Temporal Topology saturation radius for RBH-1 ($M \approx 2 \times 10^7 M_\odot$, near the crossover mass $M_\times \approx 3.04 \times 10^7 M_\odot$) can be compared to the Schwarzschild radius, placing the object near the crossover regime where horizon and soliton interpretations are maximally degenerate.

The central question is whether a single density parameter, validated cosmologically, remains universal at galactic scales.

The Saturation Scale as Fundamental Parameter

If the scalar sector saturates at a critical energy density $\rho_T$, dimensional analysis requires a universal mass–radius relation:

The GNSS-derived value $\rho_T \approx 19\text{–}20$ g/cm³ (Section 2) lies in the same condensed-matter density regime where electron degeneracy becomes dynamically relevant to the equation of state (Section 5). The central question is whether this same density scale holds for RBH-1.

Testing the Density Constraint at Cosmological Scales

The saturation scale $\rho_T$ determined from terrestrial clocks is now tested against RBH-1. Using the revised best-fit mass estimate of $M \approx 2 \times 10^7 M_{\odot}$ (van Dokkum et al. 2025), the predicted Temporal Topology saturation radius is:

The predicted Temporal Topology saturation radius can be compared directly to the Schwarzschild radius for this mass:

This order-unity correspondence ($R_T \approx 1.3 R_{\rm S}$) arises because $\rho_T$ lies near the horizon-formation density threshold in the underlying scalar sector. Within the combined uncertainties from $M$ and $\rho_T$, which propagate to $\sim 25\%$ uncertainty in $R_T$, the two scales remain consistent at the factor-of-few level. This places RBH-1 near a crossover regime where horizon and saturated-soliton interpretations can be observationally degenerate.

This geometric match is the primary prediction of the scaling law. While the velocity discontinuity ($\Delta v \sim 650$ km/s) provides a secondary constraint on the amplitude of the screening (see Smawfield 2025h, Paper 7), the scale of the object is set fundamentally by the saturation scale $\rho_T$. The correspondence suggests that what is conventionally identified as the event horizon scale in GR may correspond to the saturation boundary of the scalar field in the TEP framework.

In the TEP interpretation, what is conventionally called a "black hole" at this mass scale is modeled as a saturated soliton core, i.e., a region where the conformal time-field approaches a maximum gradient set by $\rho_T$. In this phenomenology, the characteristic radius is set by the saturation scale rather than by horizon formation. The observed central dimming is attributed to extreme time dilation (strong redshifting) rather than causal disconnection.

A further structural implication follows from the different mass scalings. The soliton relation predicts $R_T \propto M^{1/3}$, whereas the Schwarzschild radius scales as $R_{\rm S} \propto M$. These opposing scalings intersect at a unique crossover mass $M_{\times} = \sqrt{3c^6/(32\pi G^3\rho_T)}$ where the two radii coincide (see Figure 8). Empirically, the terrestrial calibration ($\rho_T = 20$ g/cm³) places this crossover at $M_{\times} \approx 3.04 \times 10^7\,M_{\odot}$ ($R_T = R_{\rm S} \approx 8.97 \times 10^7$ km)—directly adjacent to the mass regime where RBH-1 ($M \approx 2 \times 10^7\,M_{\odot}$) resides. Objects near $M_{\times}$ are expected to be maximally degenerate between Schwarzschild-scale and saturation-scale interpretations, making RBH-1 an unusually diagnostic system.

The identification is thereby restricted to the crossover neighbourhood: it is a statement about $M \approx M_{\times}$, not a general reinterpretation of compact objects. The directional implication is explicit. For $M \ll M_{\times}$ the weak-field construction gives $R_T \gg R_{\rm S}$ — a $10\,M_\odot$ object would carry $R_T$ of order $10^{6}$ km against $R_{\rm S} \approx 30$ km — so soliton-realized compact objects at stellar masses would appear as extended structures and are incompatible with observed merger phenomenology. For $M \gg M_{\times}$ the saturation radius retreats far inside the horizon (for $\text{M87}^*$ at $M \approx 6.5 \times 10^9\,M_\odot$, $R_T \approx 3.6\text{ AU}$ against $R_{\rm S} \approx 128\text{ AU}$ gives $S \sim 0.03$, whereas $\text{Sgr A}^*$ at $M \approx 4.3 \times 10^6\,M_\odot$ has $R_T \approx 0.31\text{ AU}$ and $S \sim 3.7$, placing its saturation boundary across the observed photon-ring scale), and the identification ceases to be degenerate. Two resolutions are admitted, and the corpus requires one of them: either soliton-realized compact objects exist only in the crossover band, or the weak-field $R_T(M)$ construction fails at horizon compactness and must be replaced by the relativistic strong-field treatment of Paper 28, within which the $M^{1/3}$ ansatz is not guaranteed to survive. The crossover statement is therefore conditional on weak-field validity at horizon-compactness — an assumption stated rather than assumed — while the degeneracy at $M_{\times}$ itself remains a legitimate and diagnostic coincidence that feeds the RBH-1 analysis of Paper 7.

Status of the Scaling Law

The relation $R \propto M^{1/3}$ is a density-limited scaling expected for compact, self-bound field configurations whose cores approach a finite saturation scale. Such behavior is familiar in non-topological solitons and bosonic compact objects (Coleman 1985; Seidel & Suen 1991; Gleiser 1994; Hui et al. 2017).

The central claim is not that terrestrial clocks "predict" black hole sizes, but that a fundamental density constant $\rho_T$ appears across scales. The saturation scale is determined empirically from Earth—the most accessible high-precision gravitational laboratory—and then tested for consistency at cosmological scales. The present analysis should be read as a phenomenological identification of an unexpected regularity: the same density constraint that governs clock correlations in Earth's gravitational field also governs the characteristic size of an $M \sim 10^7 M_\odot$ compact object ($M \approx 2 \times 10^7 M_\odot$ for RBH-1).

Dependency Structure and Constraint Isolation

A critical requirement for the robustness of this framework is to demonstrate that the convergence on $\rho_T \approx 20$ g/cm³ does not depend on any single channel—and in particular, that dropping the internal GNSS series (Papers 1–3, 14) still leaves a tightly constrained saturation density determined purely by external peer-reviewed astrophysical and geodetic catalogs (Step 15, gnss_free_4channel_external_catalog_isolation).

Test 1: GNSS-Free External Astrophysical & Planetary Isolation (Dropping All GNSS Data)

If all terrestrial GNSS clock data are discarded completely, four independent external-catalog observables calibrate $\rho_T$ directly without reference to $R_\oplus$ or GNSS processing chains:

• Unbraked outer-giant-planet equatorial spin (NASA/JPL & HST ephemerides; Uranus, Neptune, Saturn): Inverting $v_{\rm eq} = GM/(R_T v_{\rm gal})$ ($\propto \rho_T^{1/3}$) yields $\hat\rho_T^{\rm spin,outer3} = 19.89$ g/cm³ geometric mean ($19.6, 15.7,$ and $25.5$ g/cm³).
• Substellar brown-dwarf rotational period floor (Tannock et al. 2021; $P_{\min}^{\rm obs} = 1.150 \pm 0.075$ hr): Inverting $P_{\min} = (k_{\rm MOI}/\eta_0)\sqrt{3\pi/(G\rho_T)}$ ($\propto \rho_T^{-1/2}$) yields $\hat\rho_T^{\rm BD,floor} = 20.79 \pm 2.71$ g/cm³ ($\pm 13\%$ uncertainty).
• Self-compressed planetary moments of inertia (satellite geodesy & precession): Inverting $k_{\rm MOI} = \tfrac{1}{2}(\bar\rho/\rho_T)^{1/3}$ yields $\hat\rho_T^{\rm MOI,\text{terr}} = 17.89 \pm 1.36$ g/cm³ ($\pm 7.6\%$ across Mercury, Venus, and Earth; $17.86$ g/cm³ across all seven condensed/giant planets).
• Solar Model S convection-zone base (Christensen-Dalsgaard et al. 1996; SOHO/MDI $r_{\rm cz} = 0.713 \pm 0.001\,R_\odot$): Inverting $r_{\rm cz} = \sqrt{3}\,R_T(M_\odot)$ yields $\hat\rho_T^{\rm (cz)} = 20.15 \pm 0.09$ g/cm³.

Result: Without any GNSS input, external published planetary, substellar, and helioseismic datasets independently pin down $\rho_T$ to $\pm 6.4\%$ ($19.68 \pm 1.25$ g/cm³)—an order of magnitude tighter than the single-body GNSS systematic band—and are further bracketed by the condensed-matter Thomas–Fermi degeneracy onset ($10\text{--}30$ g/cm³) and the quartic QED projection $\rho_T^{\rm EM} = \alpha_{\rm EM}^{4/3}m_e^4c^3/\hbar^3 = 22.39$ g/cm³ (Section 5.5).

Test 2: Terrestrial GNSS Isolation (Dropping Astrophysical & Condensed-Matter Inputs)

Conversely, if all astrophysical and condensed-matter inputs are discarded and reliance is placed solely on the observed 25-year GPS-only correlation length $L_c = 4{,}200$ km for Earth's mass ($M_\oplus$):

Result: The central terrestrial GNSS scale ($19.24$ g/cm³) agrees with the independent GNSS-free external astrophysical mean ($19.68 \pm 1.25$ g/cm³) to within $-2.2\%$.

Test 3: Cosmological Horizon-Crossover Isolation (RBH-1 Schwarzschild Density)

If both terrestrial and Solar System constraints are discarded and the exact Schwarzschild horizon mean density $\rho_S(M) \equiv 3M/(4\pi R_S^3) = 3c^6/(32\pi G^3 M^2)$ is evaluated directly for the JWST/NIRSpec-confirmed runaway supermassive black hole RBH-1 ($M_{\text{RBH-1}} \approx 2 \times 10^7\,M_\odot$, with factor-of-two mass bracket $[1,\,4] \times 10^7\,M_\odot$; van Dokkum et al. 2025) without assuming $R_T/R_S = 1.32$:

spanning $[11.5,\,184]\text{ g/cm}^3$ across the factor-of-two mass bracket $M_{\text{RBH-1}} \in [1,\,4] \times 10^7\,M_\odot$. Across $12.8$ orders of magnitude in mass above Earth ($M_{\text{RBH-1}}/M_\oplus \approx 6.7 \times 10^{12}$, and $18$ orders of magnitude above the smallest tested asteroids), the horizon crossover of a $2\text{--}3 \times 10^7\,M_\odot$ runaway black hole independently selects the same $\sim 12\text{--}46$ g/cm³ density window.

7. Milky Way Test: Dark Matter Onset

Expected Onset Radius

The Milky Way provides a local test of the galactic $M^{1/3}$ scaling law established in Section 3. Unlike external galaxies where distance uncertainties dominate, the Milky Way offers high-precision kinematic data. This section tests whether the SPARC-calibrated relation predicts the radius at which the mass discrepancy first becomes significant in the Milky Way.

Using the SPARC-calibrated normalization ($k = 7.96 \times 10^{-4} \approx 8.0 \times 10^{-4}$ kpc/$M_\odot^{1/3}$) and the Milky Way's total baryonic mass estimated at $M_{\text{bar}} \approx 6 \times 10^{10} M_\odot$ (Bland-Hawthorn & Gerhard 2016), the expected dark-matter onset radius is:

This radius marks the scale where the baryonic rotation curve (bulge + disk) first falls below the observed circular velocity, signaling the onset of the phantom mass effect. It is the same quantity $R_{\text{DM}}$ that is measured directly in the SPARC ensemble; the outer Keplerian decline at larger radii is a separate phenomenon governed by the full halo profile, not by the onset scaling alone.

Gaia DR3 and Local Observations

Gaia Data Release 3 (Gaia Collaboration 2023) provides precise proper motions and radial velocities for millions of stars, enabling construction of the Milky Way rotation curve to greater distances than previously possible. Recent analyses (e.g., Jiao et al. 2023) also report evidence for a Keplerian-like decline beginning near $R \approx 19$ kpc; this outer feature traces the halo edge and is distinct from the inner onset scale tested here.

Key observational features relevant to the $R_{\text{DM}}$ prediction:

• Baryonic dominance ($R \lesssim 3$ kpc): The bulge and inner disk dominate; the rotation curve rises steeply and is well described by visible matter alone (Sofue 2020).
• Dark-matter onset ($R \sim 3$–$5$ kpc): Beyond the peak of the baryonic contribution, the observed circular velocity ($v_{\text{circ}} \approx 220$ km/s) stays flat while the baryonic model predicts a decline. This is the Milky Way's equivalent of the SPARC $R_{\text{DM}}$ — the radius where the mass discrepancy first becomes significant.
• Flat regime ($5 \lesssim R \lesssim 15$ kpc): The rotation curve remains approximately flat, indicating substantial phantom mass contribution.
• Keplerian decline ($R \gtrsim 19$ kpc): At large radii the enclosed mass growth slows and $v_{\text{circ}}$ begins to decline. This outer feature is not predicted by the onset scaling $R_{\text{DM}} \propto M^{1/3}$.

Uncertainty Analysis and Error Budget

The mean-relation prediction ($3.1 \pm 0.4$ kpc; single-galaxy $1\sigma$ population prediction interval $[1.3,\,7.2]$ kpc on the $n = 75$ resolved interior crossings with $\sigma = 0.366$ dex, $[2.0,\,4.7]$ kpc at the $\sigma = 0.180$ dex enclosed-mass scatter, and $[1.0,\,9.4]$ kpc when the $93$ $R_{\min}$-censored dwarf sampling floors are included in the raw $0.48$ dex SPARC ensemble scatter) and the observed dark-matter onset scale ($\sim$3–5 kpc) are consistent within the combined uncertainty. Because the SPARC-calibrated normalization $k$ is the sole empirical input to this prediction, the Milky Way test is a SPARC-internal consistency check rather than a GNSS cross-scale test, consistent with the Numerology Firewall classification (Section 8).

Comparison with Dark Matter Halo Models

Future Refinements

Ongoing Gaia data releases will improve the precision of the inner rotation curve, potentially constraining the exact baryonic-to-dark-matter transition radius to $\sim$10% accuracy. Combined with improved Milky Way mass estimates from satellite kinematics and gravitational lensing, this could provide a sub-10% test of the $M^{1/3}$ onset scaling in the Milky Way.

8. Discussion: The Nature of the Dark Sector

Synthesis of Multi-Scale Evidence

The central result of this work is the separation and joint evaluation of two classes of cross-scale constraints organized around the Temporal Topology framework across 18 orders of magnitude in mass for saturation-radius tests ($10^{20}\text{--}10^{38}$ kg, from small bodies and Earth to the $2\times 10^7\,M_\odot$ runaway black hole RBH-1, with galactic disks testing the unscreened $S \ll 1$ kinetic branch):

• Direct numerical tests of $\rho_T \approx 20$ g/cm³: The terrestrial GNSS correlation anchor ($L_c \approx 4{,}200$ km, $R_T(M_\oplus) \approx 4{,}146$ km), the RBH-1 horizon–saturation crossover ($M_\times = \sqrt{3c^6/(32\pi G^3\rho_T)} \approx 3.04 \times 10^7\,M_\odot$, $R_T/R_S = 1.32$), the zero-parameter unbraked outer-giant-planet ambient equatorial spin relation ($v_{\rm obs}/v_{\rm TEP} = 1.000 \pm 0.080$, $\hat\rho_T^{\rm spin} = 19.9$ g/cm³ geometric mean across Uranus, Neptune, and Saturn; $1.016 \pm 0.073$ and $20.9$ g/cm³ with lunar-braked Earth, and $1.001 \pm 0.060$, $\hat\rho_T^{\rm rot} = 20.0$ g/cm³ across the $N = 8$ unbraked outer-planet and substellar systems [$1.008 \pm 0.060$ across $N = 9$ with Earth]), the planetary moment-of-inertia ratio ($2k_{\rm MOI}/S = 1.17 \pm 0.14$ across eight planets at 1-bar radii, $\rho_T^{\rm MOI,\text{terr}} = 17.9 \pm 1.4$ g/cm³ across the three self-compressed iron-core terrestrial planets, and $2k_{\rm MOI}/S_{\rm cond} = 1.042 \pm 0.094$, $\rho_T^{\rm MOI,cond} = 17.9$ g/cm³ across all seven condensed/giant planets once Lane–Emden polytropic H/He envelope inflation is removed), and the exploratory Solar Model S convection-zone base ($\sqrt{3}R_T(M_\odot) = 0.715\,R_\odot$ vs. $0.713\,R_\odot$, $\rho_T^{\rm (cz)} = 20.15$ g/cm³; across all five independent channels, $\hat\rho_T = 19.59 \pm 1.10$ g/cm³, rejecting random-density look-elsewhere coincidence at $p = 2.9 \times 10^{-4}$ [$3.6\sigma$]).
• Structural-exponent and scale-cancellation tests (independent of the numerical value of $\rho_T$): The galactic dark-matter onset scaling in SPARC ($\alpha_{\rm SPARC} = 0.352 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (def.)}$, with an a priori Freeman-disk baryonic slope split separating rising-baryon 3-D core crossings $\alpha_{\rm rising} = 0.256 \pm 0.069$ vs $1/3$ from concentrated Keplerian-tail crossings $\alpha_{\rm tail} = 0.581 \pm 0.086$ vs $1/2$, $\Delta\alpha = +0.325 \pm 0.110, z = 2.95$, and the coupled-gradient two-branch RAR/BTFR profile at $0.1330$ dex RMS [$61.1\%$ within $0.1$ dex, matching the $0.1328$ dex MOND benchmark at canonical $\Upsilon_d=0.50$; $0.139$–$0.152$ dex uncoupled/quadrature] on the quality-controlled sample), the Milky Way inner onset check ($R_{\text{DM}} \approx 3.1$ kpc), the deterministic regime map $S \equiv R_T/R_{\rm phys} \equiv (\bar\rho/\rho_T)^{1/3}$, and Wesson's angular-momentum constant $p_{\rm TEP} \equiv J/M^2 = G/(2v_{\rm gal}) = 1.52 \times 10^{-15}$ cm$^2$ g$^{-1}$ s$^{-1}$ (where $R_T(M)$ and $\rho_T$ cancel out of the product $J_{\rm TEP}$).

The convergence of these mutually reinforcing constraints suggests that $\rho_T$ is not merely a fitting parameter for a specific system, but a candidate universal parameter of the effective description of the dark sector, within the stated uncertainties.

Numerology Firewall: Which Agreements Were Predicted?

The elegance of cross-scale agreement is vulnerable to the accusation of "curve-fitting across scales." This section provides a candid assessment of which constraints were predicted a priori and which were identified post hoc.

Logical dependency summary: (i) $R_T(M) \propto M^{1/3}$ and $S(\rho) \propto \rho^{1/3}$ are algebraic consequences of a fixed reference density $\rho_T$; (ii) $L_c \leftrightarrow R_T(M_\oplus)$ is the conditional terrestrial anchor identification (where the 4-layer PREM BVP $L_c^{\rm PREM} = 4{,}010$ km and volume integral $L_c^{\rm vol} = 3{,}213$ km confirm an order-unity geometric projection on Earth's $R_{\rm CMB}, R_\oplus$ structure, while a saturation-core kernel $\xi_c = R_T(M_\oplus,\rho_T)$ carries the $\rho_T^{-1/3}$ scaling, Step 14); (iii) the SPARC onset scaling ($\alpha_{\rm SPARC} = 0.352$ discrete on $N = 168$, $\alpha_{\rm res} = 0.417 \pm 0.050$ on $n = 75$ resolved crossings compatible with a transition between the $M^{1/3}$ saturation and $M^{1/2}$ low-acceleration kinetic scales, Tobit rejection of $R_{\rm DM}/R_d$ invariance at $+0.323 \pm 0.034, z = 9.57, p < 10^{-4}$, and fixed-radius non-circular partial $r(R_{\rm DM},\,V_{\rm bar}(2.2R_d)\mid R_d) = +0.547$) and the coupled-gradient two-branch RAR/BTFR profile ($P_X = k\sqrt{\xi} + 2\xi$ at $0.1330$ dex RMS on the quality-controlled SPARC RAR sample and $0.1994$ dex across all $3{,}389$ points [$0.139$–$0.152$ and $0.200$–$0.218$ dex uncoupled/quadrature]) are empirical galactic tests with $\rho_{\rm trans}$ and $k_* = 16.03$ calibrated to the halo regime; and (iv) the RBH-1 crossover ($M_\times \propto \rho_T^{-1/2}$) and the $R_\oplus$-free planetary and substellar spin/MOI channels ($\rho_T^{\rm spin} = 20.9$ g/cm³ geometric mean across Earth, Uranus, Neptune, and Saturn; $\rho_T^{\rm rot} = 21.1$ g/cm³ across the 9-system planetary + substellar speed-limit sequence including SIMP J0136 and the $1.17$ hr brown-dwarf period floor; $\rho_T^{\rm MOI,\text{terr}} = 17.9 \pm 1.4$ g/cm³, $2k_{\rm MOI}/S = 1.039 \pm 0.027$ across the core-screened terrestrial planets, $1.17 \pm 0.14$ across all eight planets at 1-bar radii, and $2k_{\rm MOI}/S_{\rm cond} = 1.042 \pm 0.094$, $\rho_T^{\rm MOI,cond} = 17.9$ g/cm³ across all seven condensed/giant planets; and the exploratory Model S solar prototype with $\rho_\odot = \rho_T$ at $0.249\,R_\odot$, convection-zone base $\sqrt{3}R_T(M_\odot) = 0.715\,R_\odot$ vs. $0.713\,R_\odot$ [$\rho_T^{\rm (cz)} = 20.15$ g/cm³], and differential shear $S_\odot/\sqrt{2} = 0.292$ vs. $0.289$) directly calibrate and test the numerical value $\rho_T \approx 20$ g/cm³.

Dark Matter as Phantom Mass

In the TEP framework, "dark matter" is reinterpreted not as a particle species, but as "phantom mass"—an apparent excess inferred when a geometry with temporal shear (two metrics) is analyzed under the assumption of isochrony (single metric). The saturation scale $\rho_T$ represents the scale at which the conformal factor $A(\phi) = \exp(\beta_A\phi/M_{\text{Pl}})$ enters a non-linear, self-supporting phase (the soliton).

This explains the phenomenology of "cores" in dark matter halos. In standard CDM simulations, halos tend to form cusps ($\rho \propto r^{-1}$), which are often in tension with observations of flat cores in dwarf galaxies (the core-cusp problem). In the soliton picture, the core is a distinct physical state—a Bose-Einstein condensate-like configuration of the scalar field—that naturally supports a flat density profile with a characteristic radius $R_T$.

Comparison with Alternative Frameworks

The TEP framework shares phenomenological features with both Modified Newtonian Dynamics (MOND) and Fuzzy Dark Matter (FDM) but is distinguished by its screening mechanism and cross-scale predictive power.

TEP vs. MOND

MOND provides an excellent fit to galaxy rotation curves via a single acceleration threshold ($a_0 \approx 1.2 \times 10^{-10}$ m/s²) but offers no explanation for terrestrial GNSS clock correlations or compact-object core scales. In the two-branch TEP action (Paper 0 §7; Appendix C), both scales coexist without contradiction because they belong to the two complementary sectors of $P(X,\phi) = P_{\rm kin}(X) - V(\phi)$: the kinetic gradient scale $\Lambda_X = \sqrt{M_{\rm Pl} H_0}$ fixes the diffuse-halo Temporal Shear acceleration threshold $g_t = cH_0/2 \simeq 3.40 \times 10^{-10}$ m/s² (yielding $a_{\rm eff} = 4\sqrt{2}\,g_t/k \simeq a_0$ in the SPARC RAR and BTFR, Steps 10–13, and the outer screening shell $r_*(M) = \sqrt{GM/g_t}$), whereas the potential/amplitude sector fixes the compact-core Temporal Topology saturation density $\rho_T \approx 20$ g/cm³ ($R_T \propto M^{1/3}$).

TEP thus recovers MOND-like acceleration phenomenology in diffuse outer halos ($g_{\rm bar} \lesssim a_{\rm eff}$) from its kinetic sector while simultaneously predicting the compact saturation scale $R_T(M)$ that organizes terrestrial GNSS clock correlations, planetary interior gyration ($k_{\rm MOI} \simeq S/2$), and the RBH-1 crossover.

TEP vs. FDM

Standard FDM predicts solitonic cores but fundamentally struggles to recover the correct structural scales for both dwarf galaxies and the Milky Way simultaneously, requiring contradictory particle mass bounds. TEP naturally and uniquely links the core scale directly to the ambient saturation scale, generating the correct cross-scale hierarchy anchored directly by GNSS empirical measurements.

Response-Coefficient Hierarchy

A natural criticism of the multi-channel TEP programme is that different papers report different numerical coefficients. This is expected, and it is a feature rather than a flaw. In the TEP framework, each measurement channel carries an observable response coefficient \(\kappa_X\) that absorbs stellar physics, environment, and projection geometry, not the bare microscopic conformal coupling \(\beta_A\).

Paper 11 (TEP-H0) independently calibrates the weakly screened galactic-disk response via Cepheid period–luminosity residuals. The Paper 0 benchmark normalization is \(\kappa_{\rm gal}^{\rm bench} = (0.96 \pm 0.40) \times 10^6\) mag, while the empirical Cepheid-channel analyses recover \(\kappa_{\rm equiv}^{39} = (1.220 \pm 0.531) \times 10^6\) mag (\(\sigma_v=250\)), \(\kappa_{\rm WLS}^{44} = (1.275 \pm 0.370) \times 10^6\) mag (\(3.44\sigma\); \(\sigma_v=150\)), and \(\kappa_{\rm Cep}^{44} = (0.665 \pm 0.362) \times 10^6\) mag (\(\sigma_v=250\)) — bracketing the benchmark with a midpoint of \(0.970 \times 10^6\) mag (\(1.0\%\) agreement) — and Paper 11 Step 62 derives \(|\kappa_{P^2}^{\rm pred}(\lambda_0=1)| = 1.196 \times 10^6\) mag directly from the Cepheid envelope density ratio \(\rho_T/\langle\rho\rangle(10\text{ d}) = 8.45 \times 10^5\) at the unit-transport benchmark \(\lambda_0 = 1\) with bare coupling \(|\beta_A| = 1\) fixed by Paper 0 (matching the SH0ES fit \((1.191 \pm 0.480) \times 10^6\) mag to \(0.5\%\) at the central calibration, \(\lambda_0 = 0.995 \pm 0.401\) statistical and \(\lambda_0 \in [0.28,\,2.49]\) across the \(\rho_T \in [8,\,70]\text{ g cm}^{-3}\) GNSS window band; equivalently, under the conditional unit-envelope-transport assumption \(\lambda_0=1\), inverting the SH0ES fit yields \(\rho_T^{\rm (Cepheid)} = 19.91 \pm 8.02\text{ g cm}^{-3}\), providing a conditional cross-regime consistency check with the GNSS anchor \(\rho_T = 20 \pm 7\text{ g cm}^{-3}\)). Paper 10 (TEP-COS) measures the globular-cluster pulsar response \(\tilde\kappa_{\rm MSP} \approx 3 \times 10^4\) (full response coefficient, step_44_kappa_msp_prior.json) after dense-cluster geometric suppression; the mapping to Paper 10's effective channel coefficient is \(\kappa_{\rm MSP} = \tilde\kappa_{\rm MSP}\,\mathcal S_{\rm cluster} \approx 0.05\), i.e. a cluster suppression factor \(\mathcal S_{\rm cluster} \approx 1.7\times10^{-6}\) applied to the full response — the same coefficient quoted under the un-tilded symbol. The two values are consistent with the bare TEP geometric-factor estimate (\(\sim 10^6\)–\(10^7\)) only after environmental transfer factors are included; they do not assert a direct one-to-one equality of raw channel coefficients.

This hierarchy protects the programme from the criticism: "Why does each paper have a different coefficient?" The answer is that they are observable response coefficients, not bare scalar couplings. Each channel measures a different projection of \(S_\Sigma(\mathcal{E})\):

The cross-paper comparison is therefore through a shared clock-response structure, not raw equality of coefficients. This preserves the internal consistency of the programme while acknowledging that each domain probes a different facet of the same environmental operator. Across the broader TEP research programme, the same unified unstretched-space matter metric $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav})U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift})\Pi_{\mu\nu}$ and two-branch kinetic sector $P_{,X} = \epsilon_{\rm reg} + k\sqrt{\xi} + 2\xi$ ($k = 16.03$ fixed by $a_0$) confront the principal astrophysical and cosmological discriminators in dedicated companion studies: (i) in cluster and strong-lensing optics (Papers 4, 19), intermediate temporal depth $-\beta_A u_{\rm grav}(\mathbf{x})$ refracts null rays directly via $n(\mathbf{x}) \simeq 1 - 2\Phi_N/c^2 - \beta_A u_{\rm grav}(\mathbf{x})$, reproducing the Bullet Cluster galaxy-peak convergence and offset morphology ($\kappa_{\rm tot} = 0.307$–$0.372$ centered on the collisionless galaxies without a negative moat; Paper 4 §6.9) and the SN Refsdal long-baseline Fermat delay ($+330.9$ d sheet, $+221.6$–$443.2$ d 3-D ray; Paper 19 Step 64) from baryons alone; (ii) in the early-universe and background sector (Papers 18, 26, 27, 29), baryon clock-slope acoustic driving ($\omega_\phi = \omega_b y_s/\sqrt{2} = 0.118$) and unstretched-space temporal CMB lensing close the Planck 2018 TTTEEE + low-$\ell$ + lensing + BAO + Pantheon+ likelihood at zero particle dark matter ($\omega_{\rm cdm} = 0$, $\Delta\chi^2 = -0.06$, $\Delta\mathrm{AIC} = -2.06$; Paper 18), while the single-history Pantheon+ + DESI DR2 BAO + Planck $\theta_*$ joint fit ($\Delta\chi^2 = -8.1$ vs $\Lambda$CDM; Paper 26) reproduces $(1+z)$ supernova light-curve time dilation and generates an emergent thawing projection $(w_0, w_a) \approx (-0.83, -0.44)$ alongside Pop-III/VMO asymptotic helium equilibrium ($Y_{\rm eq} \approx 0.247$; Paper 29); and (iii) in precision-gravity and relativistic-population channels, quadratic-branch and shell-impedance screening suppress preferred-frame PPN coefficients ($|\alpha_2(1.6\,R_\odot)| = 2.16 \times 10^{-8}$, $\alpha_2(1\text{ AU}) = -1.47 \times 10^{-5}$) and PSR J1738+0333 dipolar radiation by $>2{,}800\times$ in amplitude (Paper 0 §7.2–7.3), environmental $|Z|$ stratification and external-field screening model the transition morphology in Gaia DR3 wide binaries ($R_s = 3{,}447$ vs $4{,}912$ AU, $p = 8.1 \times 10^{-15}$; Paper 13), Virial velocity-dispersion predictors ($\Psi \propto \sigma^2$) confirm high-$z$ JWST age-excess scaling across $N = 75$ NIRSpec systems independently of SED stellar-mass circularity ($p = 8.4 \times 10^{-7}$; Paper 12), and cumulative GWTC-1–5.0 BBH secondary-mass taper and spin mixtures favor canonical bi-metric propagation out of sample ($\Delta\ln L = +7.05$ to $+12.14$ on $m_2 \ge 5\,M_\odot$, $+5.58$ to $+17.09$ on $m_2 \ge 8\,M_\odot$, and $\Delta\ln L_{\rm joint} = +11.31$ to $+14.48$ at the unified optimum $\hat\epsilon = 0.25$ matching the parameter-free spiral-disk evaluation $\langle\epsilon_{\rm host}\rangle_{\rm SFR}^{\rm UCD} = 0.236 \pm 0.054$ [$0.243 \pm 0.052$ disk-integrated; $0.197\text{--}0.202$ with $25\%$ bulge/NSC hosts] of Section 3, alongside $+11.02$ to $+22.55$ at the unscreened $\epsilon = 0$ boundary; Paper 22).

Resolving Galactic Anomalies

The Radial Acceleration Relation (RAR)

The SPARC analysis (Section 3) confirms that the onset of mass discrepancies follows the baryonic mass distribution, a key feature of the Radial Acceleration Relation (McGaugh et al. 2016). In TEP, this coupling is a natural consequence of the assumed sourcing: the scalar field is sourced by the trace of the energy-momentum tensor ($T^\mu_\mu$), so the "dark matter" halo is directly anchored to the baryonic mass. This recovers the MOND-like phenomenology of the RAR without modifying inertia or requiring new force laws, through the non-linear response of the scalar sector.

Ultra-Diffuse Galaxies (DF2/DF4): Hyperbolic Re-Equilibration and External-Field Screening

The baryon-dominated ultra-diffuse galaxies NGC 1052-DF2 and DF4 ($M_{\rm bar} \approx 2 \times 10^8\,M_\odot, R_e \approx 2.2\text{ kpc}$, observed velocity dispersion $\sigma_{\rm los} = 8.5^{+2.3}_{-3.1}\text{ km s}^{-1}$ vs. Newtonian baryonic $\sigma_{\rm bar} \approx 7.0\text{ km s}^{-1}$; van Dokkum et al. 2018, 2019, 2022) are accounted for quantitatively by the two-branch TEP field equations (Step 14, df2_df4_udg_hyperbolic_re_equilibration_and_efe; Appendix B.7). First, because scalar perturbations on a static spacelike gradient propagate at the hyperbolic radial group speed $v_\parallel = \sqrt{1 + 2\xi P_{,XX}/P_{,X}}\,c \in [1,\,\sqrt{3}]\,c$ (Eq. [\ref{eq:nonlinear_chameleon_eom}]), the scalar field re-equilibrates across $R_e = 2.2\text{ kpc}$ in the light-crossing time $\tau_{\rm eq} = R_e/(v_\parallel c) = 4.14\text{--}7.18\text{ kyr} \ll \tau_{\rm dyn} \approx 1.5 \times 10^8\text{ yr}$, so the field tracks the instantaneous baryon configuration quasi-statically. Second, because the nonlinear kinetic operator $P_{,X}(\xi_{\rm tot})$ depends on the total scalar gradient $\xi_{\rm tot} = |\nabla u_{\rm int} + \nabla u_{\rm ext}|^2/\Lambda_X^4$ (Paper 0 §7), an isolated dwarf with internal acceleration $g_{\rm int}(R_e) = GM_{\rm bar}/(2R_e^2) = 5.9 \times 10^{-12}\text{ m s}^{-2} \ll a_{\rm eff}$ would exhibit $\sigma_{\rm iso}^{\rm TEP} = \sigma_{\rm bar}(1 + \sqrt{a_{\rm eff}/g_{\rm int}})^{1/2} = 19.1\text{ km s}^{-1}$, whereas embedding DF2/DF4 inside the external gravitational potential of the massive elliptical host NGC 1052 ($v_{\rm host} \approx 300\text{ km s}^{-1}$ at pericentric/group separation $D \approx 25\text{ kpc}$, $g_{\rm ext} = v_{\rm host}^2/D = 1.17 \times 10^{-10}\text{ m s}^{-2} \simeq a_{\rm eff}$) dielectric-saturates $P_{,X}(\xi_{\rm tot})$ and suppresses the internal boost to $a_{\phi,\rm int}/g_{\rm int} = \tfrac{1}{2}[\sqrt{1 + 4a_{\rm eff}/(g_{\rm int}+g_{\rm ext})} - 1] = 0.62$, yielding $\sigma_{\rm los}^{\rm EFE} = \sigma_{\rm bar}(1 + a_{\phi,\rm int}/g_{\rm int})^{1/2} = 8.91\text{ km s}^{-1}$ — matching the measured $8.5^{+2.3}_{-3.1}\text{ km s}^{-1}$ with zero free parameters.

Systematic Uncertainties

The primary uncertainty in the global fit remains the GNSS calibration length $L_c$. Four distinct contributions comprise this uncertainty (Section 2): (i) the 25.3-year CODE per-measurement and cross-sector dispersion ($4{,}201 \pm 1{,}967$ km, $\pm 47\%$, with azimuth-averaged isotropic $\lambda = 3{,}210 \pm 389$ km; Paper 2); (ii) the GPS-only PPP inter-center systematic spread across CODE, IGS, and ESA ($3{,}330$–$4{,}549$ km, CV $18.2\%$, $\sim \pm 15\%$ half-range; Paper 1, and $4{,}767 \pm 835$ km on raw-RINEX ionofree; Paper 3), with the caveat that centers share largely overlapping underlying data so this spread is not an independent ensemble; (iii) the fractional window/chain scatter ($1{,}050$–$1{,}935$ km quarterly windows) and multi-GNSS combined-filter transfer offset established by the MGEX stress test (Paper 14 Steps 3.1, 3.5) and Step 14 Monte Carlo injection–recovery suite (where 4-constellation ISB projection compresses a $4{,}200$ km injection by $1.62\text{--}1.77\times$, reproducing the empirical $1.90\times$ offset that maps pooled MGEX $1{,}862$ km to $3{,}536$ km); and (iv) the $\mathcal{O}(1)$ datum-projection prefactor established by the 4-layer PREM BVP integration (Section 2; $L_c^{\rm PREM}/R_T(M_\oplus) = 0.967$ on the transition shell). Applying the $\sim 30$–$40\%$ fractional window/chain uncertainty around the 25-year GPS-only anchor ($L_c = 4{,}200$ km) under the conditional $L_c \leftrightarrow R_T(M_\oplus)$ identification yields $\rho_T = 20 \pm 7$ g/cm³ and the primary systematic band $\rho_T \in [8,\,70]$ g/cm³ (while the unadjusted 4-constellation MGEX scale defines a conservative upper sensitivity bracket of $\rho_T \sim 69$–$220$ g/cm³; Appendix B.8). The $M^{1/3}$ and $M^{1/2}$ structural forms are independent of $\rho_T$; only the compact saturation normalization changes. Even the widened band is far smaller than the dynamic range over which the model is tested, preserving the falsifiability of the cross-scale framework.

Predictions for High-Redshift JWST Observations

The TEP framework makes concrete predictions for the "Little Red Dots"—compact, massive galaxies at $z > 5$ discovered by JWST. These objects are a critical testing ground because they are compact enough for scalar-envelope effects to dominate their apparent structure, yet massive enough for JWST to resolve the predicted kpc-scale transition region.

Qualitative Predictions for High-Redshift JWST Observations

For a target galaxy with stellar mass $M_* = 10^{10} M_\odot$ (typical of LRDs), the TEP model makes the following qualitative predictions:

Discriminant: Soliton vs. Supermassive Black Hole

A leading alternative explanation is that LRDs are reddened Active Galactic Nuclei (AGNs) powered by supermassive black holes ($M_{\text{BH}} \sim 10^7$–$10^8 M_\odot$).

• AGN Hypothesis: Emission should be point-like or confined to the nuclear region ($< 1$ kpc). Broad lines reflect BLR physics ($v > 1000$ km/s).
• TEP Soliton Hypothesis: Emission should be extended over the galactic transition scale (sub-kpc to kpc-scale for typical LRD masses under $g_t(z) = cH(z)/2$; Paper 12). Broad lines reflect the deep gravitational potential of the soliton well ($\sim$100–200 km/s).

Discriminant: Spatially resolve the H$\alpha$ emission. Extended kpc-scale emission with velocity dispersions $\sim$100–200 km/s would disfavor a purely nuclear AGN interpretation, while strongly compact emission with $v > 1000$ km/s would favor an AGN-dominated explanation. This constitutes a concrete observational discriminator for this particular soliton-interpretation channel.

Implications for Paper 7 (RBH-1)

The GNSS-consistent $\rho_T$ identified here provides the working normalization for the analysis of RBH-1 in the companion paper (Smawfield 2025h). By adopting $\rho_T \approx 20$ g/cm³ (with stated uncertainty), the soliton interpretation for RBH-1 becomes tightly constrained. The wake structure, velocity jump, and thermal emission properties can then be compared against the interaction of the wake with a soliton of characteristic radius $R \approx 7.8 \times 10^7$ km. Deviations would primarily constrain the object-specific soliton interpretation and/or the assumed mapping between $\rho_T$ and the effective radius in this environment, rather than the broader TEP framework.

Cross-Domain Constraint Network

Only Tier A contributes to the present \(\rho_T\) normalization. Tiers B–D constrain the environmental operator, response hierarchy, and theoretical viability of the same TEP architecture.

The TEP programme now spans thirty papers across terrestrial, galactic, compact-object, lensing, and cosmological domains. It is essential to distinguish which of these constrain \(\rho_T\) directly and which constrain the broader TEP architecture. The following tiered classification makes this dependence explicit and prevents the misreading that every paper independently confirms the same density number.

Tier A — Direct or Quasi-Direct \(\rho_T\) Constraints

These channels calibrate the macroscopic saturation scale most directly:

• GNSS \(\lambda_T\) (Papers 1–3): The primary empirical anchor. The 4,200 km correlation length is conditionally identified with the projected covariance scale associated with \(R_T(M_\oplus)\), up to an order-unity transfer factor, yielding the operational estimate \(\rho_T \approx 20\) g/cm³.
• SPARC/Milky Way onset scaling (this work): Quasi-direct scaling consistency. The \(M^{1/3}\) scaling of dark-matter onset radii across 168 SPARC galaxies (out of 175) is consistent with the saturation-radius construction, but does not independently calibrate \(\rho_T\). The Milky Way test uses the SPARC-calibrated normalization and is therefore not an independent measurement.
• RBH-1 crossover (Paper 7): The runaway-black-hole candidate tests the extrapolation of \(R_T(M)\) to \(M \sim 10^7 M_\odot\), with wake structure providing an object-specific consistency test.
• Planetary rotational kinematics, substellar period floor, and Solar Model S prototype (this work, Section 4.7): Zero-parameter dynamical test combining \(\rho_T = 20\) g/cm³ (\(\omega_T = 2.365 \times 10^{-3}\) rad s\(^{-1}\)) with the Solar-Circle Galactic temporal depth \(u_{\rm gal} = (v_{\rm gal}/c)^2\) to reproduce unbraked outer-giant-planet equatorial speeds (\(v_{\rm obs}/v_{\rm TEP} = 1.000 \pm 0.080, \hat\rho_T^{\rm spin} = 19.9\) g/cm³ across Uranus, Neptune, and Saturn; \(1.016 \pm 0.073\) with lunar-braked Earth, with residuals tracking \(r_*(M)/a\) at \(r = +0.960\)), the one-parameter (\(\eta_0 = 0.129\), candidate benchmark \(\eta_{\rm sec} = 1/8\)) two-branch \(+\;S_A\) extension across \(N = 8\) unbraked outer-planet and substellar systems (\(v_{\rm obs}/v_{\rm 2br} = 1.001 \pm 0.060, \hat\rho_T^{\rm rot} = 20.0\) g/cm³; \(1.008 \pm 0.060\) across \(N = 9\) with Earth, including SIMP J0136 and the \(1.172\) hr [IQR \(0.98\text{--}1.24\) hr] brown-dwarf period floor), interior moments of inertia (\(2k_{\rm MOI}/S = 1.039 \pm 0.027\) and \(\rho_T^{\rm MOI,\text{terr}} = 17.9 \pm 1.4\) g/cm³ across Mercury, Venus, and Earth; \(1.17 \pm 0.14\) across all eight planets at 1-bar radii, and \(2k_{\rm MOI}/S_{\rm cond} = 1.042 \pm 0.094, \rho_T^{\rm MOI,cond} = 17.9\) g/cm³ across all seven condensed/giant planets after Lane–Emden H/He envelope de-inflation), Wesson's angular-momentum constant \(p_{\rm TEP} = G/(2v_{\rm gal}) = 1.52 \times 10^{-15}\) cm\(^2\) g\(^{-1}\) s\(^{-1}\), and the exploratory Solar Model S convection-zone base (\(\sqrt{3}R_T(M_\odot) = 0.715\,R_\odot\) vs. \(0.713\,R_\odot, \rho_T^{\rm (cz)} = 20.15\) g/cm³; 5-channel look-elsewhere coincidence null \(\hat\rho_T = 19.59 \pm 1.10\) g/cm³, \(p = 2.9 \times 10^{-4}\); and \(\hat\rho_T^{\rm no\text{-}GNSS} = 19.68 \pm 1.25\) g/cm³ [\(\pm 6.4\%\)] across the four external astrophysical channels alone with GNSS dropped).
• Condensed-matter EOS coincidence (this work, Section 5): The proximity of \(\rho_T \approx 20\) g/cm³ to warm-dense-matter / degeneracy-onset scales provides a coincidence-level physical plausibility check, not an independent empirical calibration.

Tier B — Screening and Response-Transfer Constraints

These channels do not independently calibrate \(\rho_T\), but they constrain the environmental operator \(S_\Sigma(\mathcal{E})\) and the transfer functions that connect \(\rho_T\) to observables:

• Globular-cluster pulsars (Paper 10): The suppressed density scaling of spin-down excess (\(\Gamma \approx 0.39\) dex/dex, vs. Newtonian \(0.75\)) constrains how dense environments screen the temporal-field response. The observable coefficient \(\tilde\kappa_{\rm MSP}\) is a downstream projection, not a direct \(\rho_T\) measurement.
• Cepheids (Paper 11): The Cepheid clock-bias response — \(0.665\text{--}1.275\times10^6\) mag empirically (\((0.665\pm0.362)\times10^6\) mag multi-block and \((1.206\pm0.528)\times10^6\) mag redshift-only WLS at canonical \(\sigma_v=250\text{ km/s}\), \((1.275\pm0.370)\times10^6\) mag at \(\sigma_v=150\text{ km/s}\)), bracketing the \(0.96\times10^6\) mag benchmark and matching the Step 62 envelope-density prediction \(1.196\times10^6\) mag at the unit-transport benchmark \(\lambda_0=1\) (\(|\beta_A|=1\)) to \(0.5\%\) at the central estimate (\((1.191\pm0.480)\times10^6\) mag empirical fit) — in galactic disks bridges the weakly screened regime. It agrees with \(\tilde\kappa_{\rm MSP}\) after environmental transfer factors are applied, demonstrating that response coefficients are channel-specific projections.
• Wide binaries (Paper 13): Gaia DR3 reveals a weak-field screening transition at a few thousand AU with environmental ordering (high-\(|Z|\) vs. midplane). This is an independent morphology test of Temporal Shear recovery, not an independent \(\rho_T\) calibration: the paper explicitly states that imported \(\rho_T\) and \(g_{\rm TEP}\) enter only as cross-scale consistency checks.
• Lunar Laser Ranging (Paper 17): Reports a compactness-dependent residual channel at high significance, but remains at the "candidate residual" level pending integrator-level ephemeris refits. Classified as a guardrail: it suggests the local screened regime may still contain residual-channel structure, but does not yet provide a completed \(\rho_T\) likelihood term.
• Earth flyby (Paper 15): Uses the UCD \(\rho_T \approx 20\) g/cm³ as a shared input, not an independent validation. Valuable as a downstream posterior predictive test: if the terrestrial screening profile is correct, flyby anomalies should organize under the restricted trajectory model.
• Outer Solar System boundary (Paper 36): Tests the directional proper-time domain boundary across detached extreme TNOs (\(q > 30\) AU, \(a > 150\) AU) and long-period comet transits, whose radial detachment and decorrelation thresholds match Neptune's individual screening shell (\(r_*(M_{\rm Nep}) = 29.97\) AU) and the combined giant-planet screening shell (\(r_*(M_{\rm giants}) = 152.6\) AU; Section 4.7).

Tier C — Phantom Mass / Conformal Transport Tests

These channels test the mechanism by which TEP generates apparent mass discrepancies, without directly calibrating \(\rho_T\):

• Gravitational lensing (Paper 4): Under the canonical unstretched-space temporal metric (Paper 0 §2.2; Paper 4 §6.9, Step 02), intermediate temporal depth refracts photons at half the dynamical scalar potential weight (\(\Phi_{\rm lens}=\Phi_N+\tfrac12 c^2\beta_A u_{\rm grav}\) vs \(\Phi_{\rm dyn}=\Phi_N+c^2\beta_A u_{\rm grav}\), whereas on the historical isotropic-conformal benchmark \(\tilde g_{\mu\nu}=A^2 g_{\mu\nu}\), \(A^2(\phi)\) cancels out of null rays), supplying both the baryon-sourced Bullet Cluster convergence and a testable lensing-to-dynamical mass ratio \(M_{\rm lens}^{\rm u.s.}/M_{\rm dyn}=(1+\beta_A^2 y_{\rm eff})/(1+2\beta_A^2 y_{\rm eff})\). This provides mechanistic support for the TEP interpretation of the dark sector, not an independent \(\rho_T\) calibration.
• Multiply-imaged supernovae (Paper 19): SN Refsdal provides directional evidence from blind model residuals for temporal transport in the lensing sector, while the action-level unstretched-space temporal refraction solve on the selected two-branch sector \(P_{,X}=k\sqrt\xi+2\xi\) (\(k=16.03\), Step 64) reproduces the \(+330.9\text{ d}\) sheet and \(+221.6\text{--}443.2\text{ d}\) 3-D NFW ray Fermat delay and arrival order from \(f_b=0.16\) baryons alone alongside the phenomenological magnification proxy.
• JWST high-redshift galaxies (Paper 12): Tests potential-depth response at \(z > 5\) using an external response prior. The signal is not an independent \(\rho_T\) measurement; it probes whether the same conformal architecture organizes high-redshift compact galaxies.

Tier D — Orthogonal TEP-Architecture Tests

These papers do not calibrate \(\rho_T\) at all, but they test deep structural claims of the TEP framework:

• J0437 scintillation (Paper 16): Reports non-zero phase-closure signal in millisecond pulsar scintillation, rejecting standard scalar-delay ISM models (\(\psi = 0\)) at high confidence. This tests non-integrable time transport / synchronization holonomy, one of the deepest TEP claims. It does not calibrate \(\rho_T\).
• hi_class cosmology (Paper 18): Demonstrates that the conformal temporal-transport geometry preserves CMB acoustic structure, sound-horizon behaviour, and stable linear perturbations. A global consistency constraint, not a \(\rho_T\) term.
• C0 supernova conformal redshift (Paper 26): Shows that Pantheon+ distance-redshift data do not uniquely require primitive spatial expansion. Supports the temporal-transport interpretation of cosmological redshift.
• TH temporal horizon (Paper 27): The apparent \(a_{\rm eff} \to 0\) limit becomes a temporal conformal boundary rather than a physical singularity. Provides theoretical closure for the late-time cosmology.
• QF quantum tangent limit (Paper 23): Shows that the screened quantum tangent limit is consistent with the TEP architecture. A theoretical compatibility check, not empirical \(\rho_T\) evidence.

9. Conclusion

A Universal Organizing Parameter

This paper identifies the saturation scale $\rho_T \approx 20$ g/cm³ as a candidate organizing parameter for gravitational phenomena across planetary, galactic, compact-object, and RBH-scale regimes. The convergence of terrestrial GNSS timing structure, SPARC galaxy onset scaling, Milky Way inner mass-discrepancy structure, and the density-ordered screening hierarchy motivates the hypothesis that $\rho_T$ is a real physical scale in the temporal-field topology, not a system-specific fitting parameter. The screening hierarchy is not treated as an independent discovery of the $1/3$ exponent; rather, it demonstrates that the same $\rho_T$ coherently recovers GR-like behavior in dense systems while permitting active-shear scalar phenomenology in diffuse regimes.

Key Results

The primary findings are:

1. GNSS Calibration: Distance-structured correlations in atomic clocks yield $L_c \approx 4200$ km, corresponding to $\rho_T \approx 20$ g/cm³ under the conditional identification $L_c \leftrightarrow R_T(M_\oplus)$. The underlying GNSS correlation scale $L_c$ exhibits 25-year stability, multi-center consistency, and survives raw RINEX validation.
2. Galactic Scaling: SPARC rotation curves yield $\alpha_{\rm SPARC} = 0.352 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (definition)}$ on the discrete grid and $0.417 \pm 0.050$ on the $n = 75$ resolved crossings, where an a priori Freeman-disk baryonic slope split ($\gamma_{\rm Freeman} = 0.210$) separates rising-baryon 3-D core crossings ($\alpha_{\rm rising} = 0.256 \pm 0.069$ vs. $1/3$) from concentrated Keplerian-tail crossings ($\alpha_{\rm tail} = 0.581 \pm 0.086$ vs. $1/2$; $\Delta\alpha = +0.325 \pm 0.110, z = 2.95, p = 0.0031$).
3. Deterministic Regime Map: Evaluating $S \equiv R_T(M)/R_{\rm phys} \equiv (\bar\rho/\rho_T)^{1/3}$ across 26 objects spanning ~15 orders of magnitude in density defines a deterministic algebraic regime map explaining why compact-body GR tests pass (binary pulsars: $S \sim 29{,}500$) while galactic dynamics reside in the unscreened active-shear regime ($S \sim 10^{-9}$ at $\rho \sim 10^{-24}$ g/cm³).
4. Milky Way Test: The SPARC-calibrated $M^{1/3}$ relation predicts a dark-matter onset radius $R_{\text{DM}} \approx 3$ kpc for the Milky Way, consistent with the observed transition from baryonic to dark-matter-dominated rotation at $R \sim 3$–5 kpc, providing a local scale-consistency check.
5. Planetary Rotational Kinematics & Solar Prototype: Combining the universal saturation clock frequency $\omega_T = \sqrt{4\pi G\rho_T/3} = 2.365 \times 10^{-3}$ rad s$^{-1}$ ($\tau_T = 44.3$ min) with the Solar-Circle Galactic temporal depth $u_{\rm gal} = (v_{\rm gal}/c)^2$ yields the zero-parameter ambient equatorial rotation law $v_{\rm eq}/c = u_T(M)/\sqrt{u_{\rm gal}}$ ($v_{\rm obs}/v_{\rm TEP} = 1.000 \pm 0.080$ and $\hat\rho_T^{\rm spin} = 19.9$ g/cm³ across the three unbraked outer giant planets Uranus, Neptune, and Saturn; $1.016 \pm 0.073$ and $20.9$ g/cm³ with lunar-braked Earth, where residuals track $r_*(M)/a$ at $r = +0.960, p = 0.040$ and volumetric/deep-interior periods give $1.031 \pm 0.041$), its one-parameter ($\eta_0 = 0.129$, candidate benchmark $\eta_{\rm sec} = 1/8$) two-branch $+\;S_A$ extension across the $N = 8$ unbraked outer-planet and substellar speed-limit systems ($v_{\rm obs}/v_{\rm 2br} = 1.001 \pm 0.060$, $\hat\rho_T^{\rm rot} = 20.0$ g/cm³; $1.008 \pm 0.060$ across $N = 9$ with Earth, matching SIMP J0136 within $-2.8\%$ [$-0.3\%$ at $\eta_{\rm sec}$] and the brown-dwarf period floor $P_{\min}^{\rm TEP} = (k_{\rm MOI}/\eta_0)\tau_T = 1.172$ hr [IQR $0.98\text{--}1.24$ hr; $1.21$ hr at $\eta_{\rm sec}$] vs. observed $1.150 \pm 0.075$ hr), the self-compressed planetary gyration scaling $k_{\rm MOI} \simeq \tfrac{1}{2}S$ ($2k_{\rm MOI}/S = 1.039 \pm 0.027$ and $\hat\rho_T^{\rm MOI,\text{terr}} = 17.9 \pm 1.4$ g/cm³ across Mercury, Venus, and Earth; $1.17 \pm 0.14$ across all eight planets at 1-bar radii, and $2k_{\rm MOI}/S_{\rm cond} = 1.042 \pm 0.094$, $\rho_T^{\rm MOI,cond} = 17.9$ g/cm³ across all seven condensed/giant planets after Lane–Emden envelope de-inflation), the zero-parameter Wesson constant $p_{\rm TEP} = G/(2v_{\rm gal}) = 1.52 \times 10^{-15}$ cm$^2$ g$^{-1}$ s$^{-1}$ ($\chi_{\rm TEP} = 681.3$), and the exploratory Standard Solar Model helioseismic prototype (nuclear core $\rho_\odot = \rho_T$ at $0.249\,R_\odot$, convection-zone base $\sqrt{3}R_T(M_\odot) = 0.715\,R_\odot$ vs. $0.713\,R_\odot$ [$\rho_T^{\rm (cz)} = 20.15$ g/cm³], prolate tachocline $r_c = 0.688\text{--}0.708\,R_\odot$ vs. $0.693\text{--}0.717\,R_\odot$, equatorial rate $456.5$ vs. $460.7$ nHz, and differential shear $S_\odot/\sqrt{2} = 0.292$ vs. $0.289$; across all five independent $\rho_T$-calibrating channels, $\hat\rho_T = 19.59 \pm 1.10$ g/cm³ [$\sigma_{\log_{10}\rho} = 0.025$ dex], rejecting random-density look-elsewhere coincidence at $p = 2.9 \times 10^{-4}$ [$3.6\sigma$]).

Dark Matter as Phantom Mass

The TEP framework reinterprets dark-matter observations as violations of the isochrony axiom—the assumption that spatially separated clocks at the same gravitational potential tick at the same rate. On the selected two-branch kinetic sector $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$ fixed externally by $a_0$, zero per-galaxy free parameters), the coupled-gradient additive force law $a_\phi(a_\phi + g_N) = g_N a_{\rm eff} \implies g_{\rm obs} = \tfrac{1}{2}g_N + \sqrt{(g_N/2)^2 + g_N a_{\rm eff}}$ derives the fiducial $V_{\rm obs}/V_{\rm bar} = \sqrt{(1+\sqrt{5})/2} = 1.272 \approx 1.3$ onset threshold directly from the action and reproduces the SPARC radial-acceleration relation at the canonical $\Upsilon_d=0.50, \Upsilon_b=0.70$ with $0.1330$ dex rms ($61.1\%$ within $\pm 0.1$ dex, matching the empirical $0.1328$ dex MOND benchmark with zero per-galaxy refits on the quality-controlled $N=2{,}700$ SPARC RAR sample, vs $0.1522$ dex uncoupled and $0.1394$ dex quadrature; $0.1994$ dex across all $3{,}389$ raw points vs $0.2812$ dex for the unit-term baseline, Step 10) and the outer-edge BTFR ($v_{\rm obs}/v_{\rm pred}=0.956$, $0.0807$ dex rms at $\Upsilon_d=0.50$, and $1.014$, $0.0785$ dex at $\Upsilon_d=0.35$, beating the $0.0812$ dex quadrature comparator, Step 13), while the high-gradient quadratic branch satisfies Solar-System screening bounds. Evaluated under the canonical unstretched-space temporal metric (Paper 0 §2.2, Step 80), this same zero-parameter kinetic sector also closes the optical lensing sector across cluster, strong-lensing, and CMB scales from baryons alone (Papers 4, 18, 19).

This reinterpretation offers a candidate common account of several dark-sector puzzles:

• Universal scaling: The $M^{1/3}$ relation follows naturally from the saturation-radius construction.
• Dark-matter-deficient UDGs (DF2/DF4): Hyperbolic scalar propagation ($v_\parallel \in [1,\,\sqrt{3}]\,c$) fixes the field re-equilibration time across $R_e = 2.2\text{ kpc}$ to $\tau_{\rm eq} = 4.14\text{--}7.18\text{ kyr} \ll \tau_{\rm dyn}$, while external-field kinetic saturation in the NGC 1052 host potential ($g_{\rm ext} \simeq 1.17 \times 10^{-10}\text{ m s}^{-2}$) suppresses the predicted velocity dispersion from $19.1\text{ km s}^{-1}$ (isolated) to $\sigma_{\rm los}^{\rm EFE} = 8.91\text{ km s}^{-1}$, matching the observed $8.5^{+2.3}_{-3.1}\text{ km s}^{-1}$ (Step 14).
• Core-cusp structure: Saturated temporal-field cores offer a possible route to flat effective density profiles.
• Baryonic Tully-Fisher behaviour: On the selected two-branch kinetic sector $P_{,X}=k\sqrt\xi+2\xi$ ($k=16.03$), flux conservation derives the deep-tail asymptote $v_{\rm flat}^4 = GM_b a_{\rm eff}$ with $a_{\rm eff} = 4\sqrt{2}\,g_t/k = 1.20 \times 10^{-10}\text{ m s}^{-2}$ and a finite-radius slope $\mathrm{d}\log M_b/\mathrm{d}\log V_{\rm flat} = 3.53\text{--}3.64$ across the SPARC outer-edge regime ($g_N/g_t \sim 0.02$), matching the SPARC outer-edge BTFR ($v_{\rm obs}/v_{\rm pred} = 0.956$, $0.0807$ dex rms at canonical $\Upsilon_d = 0.50$ under coupled-gradient additive closure, vs $1.0109$ quadrature and $0.910$ uncoupled additive, with inverse-OLS slope $3.76\text{--}3.86$ on the quality-filtered sample, Step 13) with zero galaxy-specific free parameters.

Astrophysical Applications

The GNSS-consistent value $\rho_T \approx 20$ g/cm³ enables specific applications for astrophysical systems. The companion paper (Smawfield 2025h, Paper 7) applies this calibration to the RBH-1 runaway black hole candidate, yielding a saturation radius $R_T \approx 7.8 \times 10^7$ km. This provides a testable extrapolation: if the RBH-1 mass is revised by a factor of 3, the expected radius would change by $\sim 44\%$, offering a clear consistency check with observations.

Future applications include:

• JWST high-redshift galaxies: "Little Red Dots" and massive quiescent galaxies at $z > 10$ may exhibit soliton signatures.
• EHT polarimetry: M87* and Sgr A* polarization patterns could reveal scalar field structure near supermassive compact objects.
• Gravitational waves: Binary black hole mergers may produce scalar radiation detectable by LISA.
• Strong lensing time delays: The phantom-mass sector can act on time-delay cosmography both through the mass-model normalization — where the single-halo dynamical–lensing mass difference biases the velocity-dispersion prior that breaks the mass-sheet degeneracy in the direction that elevates the inferred $H_0$ (Paper 19, Step 58 channel audit; $0.0\text{ d}$ propagated static-conformal residual on the isotropic-conformal benchmark) — and directly through the canonical unstretched-space temporal metric (Paper 0 §2.2; Paper 19, Step 64), where intermediate temporal depth on the two-branch sector $P_{,X}=k\sqrt\xi+2\xi$ reproduces the SN Refsdal Fermat delay ($+330.9\text{ d}$ sheet and $+221.6\text{--}443.2\text{ d}$ 3-D NFW ray) from $f_b=0.16$ baryons alone.

Theoretical Implications

The convergence of terrestrial, compact, galactic, and cosmological constraints on a single density scale suggests a fundamental organization principle governing gravitational phenomenology across many orders of magnitude in mass. This organization is formalized in the Temporal Equivalence Principle (Smawfield 2025a, Paper 0), which posits that gravitational phenomena arise from a temporal scalar field $\phi(x^\mu)$ coupled to matter via the two-branch k-essence action:

where $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav})U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift})\Pi_{\mu\nu}$ in the canonical unified unstretched-space temporal metric ($\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu}$ on the historical isotropic-conformal benchmark) with $A(\phi) = \exp(\beta_A\phi/M_{\text{Pl}})$.

The saturation scale $\rho_T$ emerges as the scale at which the scalar field $\phi$ reaches its self-interaction threshold, saturating the field. Under a soliton interpretation, this would correspond to soliton formation; regardless of microscopic realization, this provides a natural explanation for the $M^{1/3}$ scaling and the dynamical phantom-mass phenomenon.

Systematic Uncertainties and Robustness

The primary uncertainty is the GNSS length scale determination. Four distinct contributions are reported: (i) the 25.3-year CODE GPS-only per-measurement and cross-sector dispersion ($4{,}201 \pm 1{,}967$ km, $\pm 47\%$, with azimuth-averaged isotropic $\lambda = 3{,}210 \pm 389$ km; Paper 2); (ii) the GPS-only PPP inter-center systematic spread across CODE, IGS, and ESA ($3{,}330$–$4{,}549$ km, CV $18.2\%$, $\sim \pm 15\%$ half-range; Paper 1, and $4{,}767 \pm 835$ km on the raw-RINEX ionofree channel; Paper 3), with the caveat that centers share underlying data so this spread is not an independent ensemble; (iii) the fractional window/chain scatter and multi-GNSS combined-filter transfer offset established by the executed MGEX stress test (Paper 14 Steps 3.1 and 3.5): on the 4-constellation combined MGEX product, quarterly windows scatter from $1{,}050$ to $1{,}935$ km ($\sim 30$–$40\%$ fractional scatter around the $1{,}862 \pm 155$ km pooled mean), matched-day independent chains CODE/GPSEST and CNES/GINS (GRG) return $2{,}744$ vs. $1{,}933$ km ($\sim 30\%$ offset, GRG within $4\%$ of the pooled $1{,}862$ km), and a factorial estimator decomposition on identical station-days (Paper 14 §4.5b, Step 3.5) proves that both GPS-only IGS-final and 4-constellation MGEX recover the identical band-ordered decay profile, with the 4-constellation combined filter compressing the fitted scale by $\sim 2\times$ relative to GPS-only PPP; and (iv) the $\mathcal{O}(1)$ datum-projection prefactor established by the 4-layer PREM BVP integration ($L_c^{\rm PREM}/R_T(M_\oplus) = 0.967$; Section 2, Step 14). Applying the $\sim 30$–$40\%$ fractional window/chain uncertainty around the 25-year GPS-only anchor ($L_c = 4{,}200$ km) yields the primary systematic band $\rho_T \in [8,\,70]$ g/cm³ around $\rho_T = 20 \pm 7$ g/cm³, while adopting the unadjusted 4-constellation MGEX scale ($\lambda = 1{,}862$–$2{,}744$ km) at unit prefactor defines an asymmetric upper sensitivity bracket of $\rho_T \sim 69$–$220$ g/cm³. Quantitative sensitivity to the numerical value of $\rho_T$ resides in the GNSS anchor, the RBH-1 crossover ($M_\times \propto \rho_T^{-1/2}$), and the planetary $v_{\rm eq}$ and $k_{\rm MOI}$ factors ($\propto \rho_T^{\pm 1/3}$, which independently favor $\rho_T \approx 20$ g/cm³ over the unadjusted $\sim 220$ g/cm³ MGEX value by $2.2\times$), whereas the SPARC/Milky Way onset scaling, the algebraic $S(\rho)$ hierarchy, and the product $J_{\rm TEP}$ test structural exponents and scale cancellation.

The robustness of the result is demonstrated by:

• Multi-center consistency: CODE, IGS, and ESA yield consistent GNSS correlation structure across independent processing centres.
• Temporal stability: 25-year consistency disfavors transient effects.
• Raw data validation: RINEX analysis constrains processing artifacts.
• Cross-scale agreement: Five independent saturation-density channels ($\hat\rho_T = 19.59 \pm 1.10$ g/cm³; $19.68 \pm 1.25$ g/cm³ GNSS-free) plus compact-to-SMBH regime tests spanning 18 orders of magnitude in mass ($10^{20}\text{--}10^{38}$ kg).

Future Directions

The identification of $\rho_T \approx 20$ g/cm³ as a candidate cross-scale saturation parameter opens several avenues for future research:

1. Non-Spherical 3-D Azimuthal Kinetic PDE Integration for Earth: Extending the completed 4-layer PREM canonical-kinetic ($L_c^{\rm PREM} = 4{,}010$ km, $L_c^{\rm vol} = 3{,}213$ km) and 1-D spherical nonlinear Picard ($L_c^{\rm NL\text{-}PREM} = 3{,}957$ km, $L_c^{\rm NL,vol} = 3{,}170$ km, $\|\Delta u^{(100)}\|_\infty/\|u\|_\infty = 2.67 \times 10^{-14}$; Step 14) BVP solutions of Eq. ($\ref{eq:nonlinear_chameleon_eom}$) to a non-spherical 3-D azimuthal solver incorporating Earth's heliocentric orbital velocity vector to model directional EW/NS clock-covariance anisotropy from first principles.
2. Independent Third-Party Geodesy & Optical Time-Transfer Replication: Having executed the 4-quarter 2023 CDDIS RINEX3 single-constellation GPS, Galileo, and GLONASS reprocessing and station-geometry resampling nulls on public archives in Step 14 ($\bar L_c = 3{,}853$ km $\sqrt{N}$-weighted and $\bar L_c^{\rm CME\text{-}rem} = 3{,}155 \pm 181$ km [$\mathrm{CV} = 5.7\%$] before 4-constellation ISB compression to $1{,}478$ km), the primary metrological target is independent replication of the public CDDIS/RTKLIB recipe by external geodesy groups alongside continental optical-fiber and two-way satellite time-transfer links.
3. Exoplanet and Brown-Dwarf Rotational Population Tests: Testing the soliton-capacity breakup ceiling ($v_{\rm eq}/v_{\rm breakup} \le (\eta_0/k_{\rm MOI})\sqrt{\min(1,S)}$ for $M > M_{\rm cap} = 0.61\,M_{\rm Jup}$, satisfied by $27/27$ objects in Bryan et al. 2020; Step 15) against expanded $v\sin i$ catalogs of directly imaged super-Jupiters and brown dwarfs and TESS/Kepler stellar hosts.
4. High-Redshift Galaxies & Gravitational-Wave Cosmology: Testing soliton-core kinematics in $z > 10$ JWST galaxies (Paper 12) and bi-metric propagation in standard sirens (Paper 22).

Concluding Remarks

This paper identifies $\rho_T \approx 20$ g/cm³ as a candidate macroscopic saturation/proximity scale in the TEP environmental-response hierarchy. GNSS clock correlations provide the primary terrestrial timing anchor for the Earth-scale covariance length. SPARC onset scaling, Milky Way structure, RBH-1 crossover behaviour, compact-object screening, and condensed-matter/EOS considerations then test whether the same scale organizes otherwise disconnected gravitational regularities without retuning. Channel-specific quantities such as $\rho_{\rm trans}$, $R_s$, $\kappa_{\rm Cep}$, $\kappa_{\rm MSP}$, and lensing response coefficients are not independent measurements of $\rho_T$, but downstream projections of $S_\Sigma(\mathcal{E})$.

This paradigm shift—from dark matter as substance to dark matter as temporal shear—provides a candidate unified organizing framework for the $M^{1/3}$ scaling, screening hierarchy, and dynamical phantom-mass phenomenon, to the extent permitted by lensing–dynamics consistency. The GNSS-consistent value of $\rho_T$ enables specific testable applications, including the RBH-1 case study (Paper 7), positioning the Temporal Equivalence Principle as an empirically grounded candidate alternative to the particle dark matter paradigm in the dynamical sector, with the dynamical–lensing mass comparison (Paper 19) as its decisive open test.

References

TEP Research Series

Smawfield, M. L. (2026). Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed. Preprint v0.18 (Jakarta). Zenodo. DOI: 10.5281/zenodo.16921911 (Paper 0)

Smawfield, M. L. (2025b). Global Time Echoes: Distance-Structured Correlations in GNSS Clocks. Preprint v0.28 (Jaipur). Zenodo. DOI: 10.5281/zenodo.17127229 (Paper 1)

Smawfield, M. L. (2025c). Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products. Preprint v0.22 (Cairo). Zenodo. DOI: 10.5281/zenodo.17517141 (Paper 2)

Smawfield, M. L. (2025d). Global Time Echoes: Raw RINEX Consistency Test. Preprint v0.9 (Kathmandu). Zenodo. DOI: 10.5281/zenodo.17860166 (Paper 3)

Smawfield, M. L. (2025e). Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations. Preprint v0.10 (Tortola). Zenodo. DOI: 10.5281/zenodo.17982540 (Paper 4)

Smawfield, M. L. (2025f). Global Time Echoes: Empirical Synthesis. Preprint v0.8 (Singapore). Zenodo. DOI: 10.5281/zenodo.18004832 (Paper 5)

Smawfield, M. L. (2025g). Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T. Preprint v0.12 (New Delhi). Zenodo. DOI: 10.5281/zenodo.18064365 (Paper 6 — this work)

Smawfield, M. L. (2025h). The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate. Preprint v0.5 (Blantyre). Zenodo. DOI: 10.5281/zenodo.18059250 (Paper 7)

Smawfield, M. L. (2025). Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging. Preprint v0.7 (Mombasa). Zenodo. DOI: 10.5281/zenodo.18064581 (Paper 8)

Smawfield, M. L. (2025). What Do Precision Tests of General Relativity Actually Measure?. Preprint v0.9 (Istanbul). Zenodo. DOI: 10.5281/zenodo.18109760 (Paper 9)

Smawfield, M. L. (2026). Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars. Preprint v0.10 (Caracas). Zenodo. DOI: 10.5281/zenodo.18165798 (Paper 10)

Smawfield, M. L. (2026). The Cepheid Bias: Resolving the Hubble Tension. Preprint v0.12 (Kingston upon Hull). Zenodo. DOI: 10.5281/zenodo.18209702 (Paper 11)

Smawfield, M. L. (2026). Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies. Preprint v0.9 (Kos). Zenodo. DOI: 10.5281/zenodo.19000827 (Paper 12)

Smawfield, M. L. (2026). Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries. Preprint v0.7 (Kilifi). Zenodo. DOI: 10.5281/zenodo.19102061 (Paper 13)

Smawfield, M. L. (2026). Global Time Echoes: Multi-GNSS MGEX Held-Out Replication and Processing-Chain Decomposition. Preprint v0.5 (Suva). Zenodo. DOI: 10.5281/zenodo.20572726 (Paper 14)

Smawfield, M. L. (2026). Temporal Equivalence Principle and the Earth Flyby Anomaly: Clock-Sector Response and Coherent-Link Cancellation. Preprint v0.4 (Yogyakarta). Zenodo. DOI: 10.5281/zenodo.19454862 (Paper 15)

Smawfield, M. L. (2026). Temporal Equivalence Principle: Synchronization Holonomy in Pulsar Scintillation (PSR J0437−4715). Preprint v0.6 (Sintra). Zenodo. DOI: 10.5281/zenodo.19454619 (Paper 16)

Smawfield, M. L. (2026). Temporal Equivalence Principle: Lunar Laser Ranging and the Nordtvedt Effect. Preprint v0.5 (Lucknow). Zenodo. DOI: 10.5281/zenodo.19446028 (Paper 17)

Smawfield, M. L. (2026). Temporal Equivalence Principle: Native hi_class Conformal Implementation, Linear Perturbation Closure, and CMB Acoustic Peak Preservation. Preprint v0.8 (Cambridge). Zenodo. DOI: 10.5281/zenodo.20572722 (Paper 18)

Smawfield, M. L. (2026). Temporal Equivalence Principle: Multiply-Imaged Supernova Time-Delay Cosmography and Unstretched-Space Refraction. Preprint v0.4 (Lisboa). Zenodo. DOI: 10.5281/zenodo.20572719 (Paper 19)

Smawfield, M. L. (2026). Temporal Equivalence Principle: A Standard-Siren Test of Bi-Metric Gravitational-Wave Propagation. Preprint v0.5 (Maputo). Zenodo. DOI: 10.5281/zenodo.20572695 (Paper 22)

Smawfield, M. L. (2026). Temporal Equivalence Principle: The Dirac Limit of Dynamical Proper Time. Preprint v0.4 (Qatar). Zenodo. DOI: 10.5281/zenodo.20572697 (Paper 23)

Smawfield, M. L. (2026). Temporal Equivalence Principle: A Covariant Alternative to Cosmic Expansion. Preprint v0.5 (Athens). Zenodo. DOI: 10.5281/zenodo.20370143 (Paper 26)

Smawfield, M. L. (2026). Temporal Equivalence Principle: Nonsingular Temporal-Horizon Closure and Cosmological Perturbation Regularity. Preprint v0.5 (Thika). Zenodo. DOI: 10.5281/zenodo.20723059 (Paper 27)

Smawfield, M. L. (2026). Temporal Equivalence Principle: Nonsingular Temporal Wells and Black-Hole Horizon Regularity. Preprint v0.4 (Bahrain). Zenodo. DOI: 10.5281/zenodo.21677826 (Paper 28)

Smawfield, M. L. (2026). Temporal Equivalence Principle: Dynamical Proper Time and the Illusion of Primordial Deuterium. Preprint v0.6 (Dubai). Zenodo. DOI: 10.5281/zenodo.21841147 (Paper 29)

Smawfield, M. L. (2026). Temporal Equivalence Principle: A Proper-Time Domain Boundary in the Outer Solar System. Preprint v0.2 (Addis Ababa). Zenodo. DOI: 10.5281/zenodo.22858190 (Paper 36)

Appendix A: GNSS Calibration — Summary of Evidence

The claim that terrestrial clock correlations calibrate a fundamental density parameter ($\rho_T \approx 20$ g/cm³) is extraordinary and requires rigorous justification. This appendix summarizes the key validation results from the companion GNSS papers (Smawfield 2025b,c,d) and addresses common concerns about systematics.

A.1 Physical Mechanism

In the TEP framework, the scalar field $\phi$ modulates proper time via a conformal factor $A(\phi) = \exp(\beta_A\phi/M_{\text{Pl}})$. Near a gravitating mass, the field profile creates a spatial gradient in clock rates. GNSS atomic clocks, distributed across Earth's gravitational field, sample this gradient at different radial positions. The characteristic correlation length $L_c \approx 4200$ km corresponds to the scale where the scalar field's gradient becomes steep enough to produce measurable timing correlations—the saturation radius for Earth's mass.

A.2 Why Not Ionospheric, Tropospheric, or Orbital Systematics?

Scale Separation

The 4200 km correlation length is distinct from known geodetic systematics:

• Tropospheric correlations: ~100–500 km (weather systems)
• Ionospheric correlations: ~500–2000 km (TEC structures)
• Orbital harmonics: ~12 hr (half-sidereal period)

Power spectral analysis shows the 4200 km feature as a persistent background covariance floor in the GPS-only clean baseline, with no counterpart in ionospheric or tropospheric delay products; combined multi-GNSS receiver-clock products (Paper 14) recover the same band-ordered correlation structure.

Multi-Center Verification

The correlation structure persists across independent clock solutions from three analysis centers:

Table A.1: Multi-center correlation parameters. The bootstrap-median scale spans 3,193–4,181 km across centres (CV 13.4%); the corresponding exponential best-fit range is 3,330–4,549 km (CV 18.2%, Paper 1). Because centers process largely shared underlying data, this spread is not an independent ensemble average; it is treated as a systematic range rather than a statistical error reduction. λ values are bootstrap medians; R² is from the exponential best fit.

Null Tests

Three null tests were performed (all satisfied at >5σ):

1. Epoch randomization: Shuffling satellite epochs destroys the correlation ($r^2 < 0.01$, $N = 1000$ shuffles)
2. Residual shuffling: Shuffling clock residuals within each satellite eliminates the spatial structure
3. White noise injection: Replacing real data with white noise yields no coherent scale

A.3 25-Year Temporal Stability

The correlation length has been confirmed over a 25-year temporal baseline (1999–2024) using CODE final products. The signal shows:

• Decadal stability (no secular drift)
• Persistent spatial anisotropy
• Strong orbital velocity coupling ($r = -0.86$, $p = 0.002$ for Southern Hemisphere)
• Annual-phase direction recovery near Earth's aphelion velocity tangent; the directed CMB-apex template is anti-phase

A.4 Raw RINEX Validation

To strongly constrain the processing artifact hypothesis, the correlation was detected in raw RINEX observations processed with Single Point Positioning (SPP) using broadcast ephemerides only—no network solutions, no precise orbits, no clock products.

Table A.2: Raw RINEX validation results. All signatures detected in processed (PPP) data are independently confirmed in raw (SPP) observations, constraining processing artifacts as the origin of the observed signatures.

A.5 What the GNSS Measurement Does and Does Not Claim

A.6 Companion Papers

Full methodology, data products, and reproducibility information are available in the companion papers:

• Smawfield 2025b: Multi-center validation (CODE, ESA, IGS)
• Smawfield 2025c: 25-year longitudinal analysis
• Smawfield 2025d: Raw RINEX validation
• Smawfield 2025f: Integrated synthesis (TEP-GTE)

Analysis code and data products are available at: github.com/matthewsmawfield/TEP-GNSS

Appendix B: Open Questions and Limitations

The TEP-UCD manuscript makes several strong empirical claims. This appendix discusses active research questions and known limitations. Addressing these is essential for the theory to mature from a consistency argument to a fully predictive framework.

B.1 The GNSS Correlation Mechanism

The operational identification of the terrestrial coherence length \(L_c \approx 4200\) km with the projected covariance scale associated with the geometric saturation radius \(R_T(M_\oplus) = (3M_\oplus/4\pi\rho_T)^{1/3}\) assumes that the observed exponential decay in clock residual correlations is controlled by the source-mass scale \(R_T(M)\), rather than by network geometry, processing, or by the in-medium Compton wavelength \(\lambda_C(\rho)\) of any specific candidate completion. This is not guaranteed. Alternative explanations include:

• Network geometry: The spatial distribution of GNSS stations is not random; continental clustering could imprint a characteristic scale unrelated to any physical field.
• Common-mode clock processing: Inter-analysis-center differences in orbit/clock estimation software may introduce correlated noise with scale-dependent structure.
• Ephemeris systematics: Orbital errors propagate into clock solutions with characteristic spatial signatures (e.g., along-track harmonics) that could mimic an exponential envelope.
• Flicker noise and power-law correlations: The null-model covariance analysis (TEP-GNSS, Paper 1) formally excludes power-law and constant kernels, but a more sophisticated non-stationary model might evade the current tests.

Mitigation: The null-model covariance analysis (null_model_covariance.py) systematically compares the exponential hypothesis against ionospheric dipole, power-law flicker, Gaussian, and Matérn kernels. The exponential model is preferred by AIC/BIC across all three analysis centers. In addition, Step 14 (station_geometry_and_continental_clustering_null_audit) executes a direct station-geometry and continental-clustering resampling audit on the actual IGS continental station coordinates ($n = 54$ global stations) to test whether continental clustering or ocean gaps imprint a spurious $L_c \approx 4{,}200\text{ km} \approx \tfrac{2}{3}R_\oplus$ scale: (i) injecting known exponential fields $L_{\rm true} \in \{1{,}000,\,2{,}000,\,3{,}000,\,4{,}200,\,6{,}000\}\text{ km}$ onto the real continental IGS coordinates recovers $\hat L_c = 885 \pm 82,\; 1{,}924 \pm 127,\; 2{,}783 \pm 210,\; 3{,}600 \pm 300,$ and $4{,}644 \pm 434\text{ km}$ linearly (never manufacturing $4{,}200\text{ km}$ when $L_{\rm true} \neq 4{,}200\text{ km}$, and showing that ocean gaps on a 54-station continental network mildly compress a true $4{,}200\text{ km}$ field to $\sim 3{,}600\text{ km}$ under unweighted binning, matching the unweighted $3{,}423\text{--}3{,}766\text{ km}$ CDDIS RINEX3 recovery); (ii) subsampling an equal number of station pairs per distance bin ($K = 30$ pairs/bin across 200 bootstrap draws, flattening the continental pair-count histogram $N(d)$ completely) on the real 2023 CDDIS RINEX3 GPS data recovers $\hat L_c = 3{,}805.9 \pm 908.6\text{ km}$ (within $+1.1\%$ of the full-sample unweighted $3{,}765.7\text{ km}$); (iii) permuting station coordinates across the real continental network flattens the correlation profile completely (slope $+0.006\text{ Mm}^{-1}$, bin SD $0.063$); and (iv) confronting the Karhunen–Loève / PCA common-mode-error (CME) filtering literature (Dong et al. 2006), the Funk–Hecke theorem on the 2-sphere $S^2$ shows that for an isotropic exponential covariance $C(\theta) = \exp(-\alpha\theta)$ with $\alpha = R_\oplus/L_c = 6{,}371/4{,}200 = 1.5169$, the exact analytical fraction of the zero-lag variance $C(0) = \sum_{\ell=0}^\infty f_\ell = 1$ carried by spherical-harmonic degree $\ell$ (with degeneracy weight $2\ell+1$) is $f_\ell = \tfrac{2\ell+1}{2}\int_0^\pi e^{-\alpha\theta}P_\ell(\cos\theta)\sin\theta\,\mathrm{d}\theta$, giving the closed-form analytical integrals $f_0 = (1+e^{-\alpha\pi})/[2(1+\alpha^2)] = 15.28\%$ ($16.99\%$ of the resolved $\ell \le 14$ station-network band), $f_1 = 3(1-e^{-\alpha\pi})/[2(4+\alpha^2)] = 23.60\%$ ($26.25\%$ of $\ell \le 14$), and $f_2 = 15.55\%$ ($17.29\%$ of $\ell \le 14$), or $54.43\%$ cumulative across $\ell \le 2$ ($60.53\%$ of $\ell \le 14$, and $95.40\%$ of the Parseval $L^2$ squared-kernel norm $\sum(2\ell+1)\lambda_\ell^2$). Removing the $\ell = 0$ global common mode on the full-day 24-hour window ($n_{\rm seg}=288$, matched $10\text{--}500\,\mu\text{Hz}$) across all 198 CDDIS RINEX3 station-days yields $\bar L_c^{\rm CME\text{-}rem} = 3{,}155.2 \pm 180.8\text{ km}$ ($\mathrm{CV} = 5.73\%$ across GPS $2{,}953.3$, Galileo $3{,}302.1$, and GLONASS $3{,}210.2\text{ km}$, $R^2 = 0.955\text{--}0.969$), matching the pre-computed 4-layer PREM full-volume datum-removed kernel $L_c^{\rm PREM,vol} = 3{,}213.2\text{ km}$ (model-calibrated BVP transfer ratio $L_c^{\rm vol}/R_T = 0.7749 \implies R_T = 4{,}071.8\text{ km}, \hat\rho_T = 21.12\text{ g/cm}^3$, vs. $\hat\rho_T = 45.40\text{ g/cm}^3$ if inverted at raw unit prefactor without the volume-kernel transfer ratio), whereas higher-rank $m \ge 2$ regional SVD truncation projects out the physical field's own $\ell = 1$ spherical harmonic just as 4-constellation ISB removal does, while physical broadcast-minus-precise common-mode subtraction ($D_i(t) = b_i(t) - p_i(t)$ in Paper 3 Step 2.12) removes satellite clock/orbit common modes without killing $\ell = 1$ and recovers $L_c = 4{,}474.8\text{ km}$ ($R^2 = 0.664$).

B.2 Earth is Not a Soliton

The narrative operationally identifies \(L_c\) with the projected covariance scale associated with \(R_T(M_\oplus) = (3M_\oplus/4\pi\rho_T)^{1/3}\). For Earth mass \(M_\oplus\), this yields \(R_{T,\oplus} \approx 4146\) km and \(\rho_T \approx 20\) g/cm³. However, Earth's mean density is \(\rho_\oplus \sim 5.5\) g/cm³, well below the quoted saturation scale. The planet therefore sits in the transition regime, not the deep-screening regime. The identification of the GNSS correlation scale with \(R_T(M_\oplus)\) is phenomenologically motivated but lacks a single-potential \(V(\phi)\) demonstration that both are the same parameter.

Mitigation: A fixed-input, canonical-kinetic Earth-profile consistency calculation has now been executed in Step 14 (step_14_scale_audit.py, prem_earth_bvp_integration). Holding the quartic coupling fixed at the Cassini-compatible corpus normalization $\lambda = 7.53\times 10^{-66}$ and using the GNSS-consistent $\rho_T = 20$ g/cm³ ($\lambda_c(\rho_T) = 2{,}478.5$ km), the boundary-value problem $\nabla^2 u = M_{\rm Pl}^{-2}V_{\rm eff}'(u)$ is solved across a 4-layer piecewise-constant approximation to the Preliminary Reference Earth Model (PREM; Dziewonski & Anderson 1981; layer-mean densities joined by 30 km tanh transitions, total mass within $0.4\%$ of $M_\oplus$; inner core $12.90$ g/cm³, outer core $11.05$ g/cm³ to $3{,}480$ km, lower mantle $4.90$ g/cm³ to $5{,}701$ km, upper mantle/crust $3.55$ g/cm³ to $6{,}371$ km). The solved transition gradient $r^2|\mathrm{d}u/\mathrm{d}r|$ peaks across the core–mantle transition with radial centroid $R_{\rm grad} = 4{,}874.9$ km $= 1.176\,R_T(M_\oplus)$ and transition Compton scale $\lambda_c(R_{\rm grad}) = 7{,}807.9$ km. Projecting the 3-D Yukawa spatial kernel from this transition shell onto pairs of surface stations at $R_\oplus = 6{,}371$ km separated by arc baselines $d \in [200,\,8{,}000]$ km (with network datum-mean removal) recovers an exponential spatial-kernel decay scale ($R^2 = 0.927$) of $L_c^{\rm PREM} = 4{,}010.2$ km—within $-4.5\%$ of the GNSS-consistent length $L_c = 4{,}200$ km ($L_c^{\rm PREM}/R_T(M_\oplus) = 0.967$, confirming an order-unity spatial transfer prefactor; varying the fit window $d_{\max} \in \{6{,}000,\,8{,}000,\,10{,}000\}$ km shifts $L_c^{\rm PREM}$ across $3{,}290\text{--}4{,}595$ km and scaling $\lambda_c(\rho_T) \propto \rho_T^{-1/3}$ across $\rho_T \in \{5,\,20,\,50,\,200\}$ g/cm³ shifts the endpoint-clamped shell integral $L_c^{\rm PREM}$ from $4{,}105$ to $3{,}650$ km because the $1/|\mathbf{x}_A-\mathbf{x}_s|$ geometric factor and endpoint minimum-subtraction on $[200,\,8{,}000]$ km pin the single-shell integral to Earth's fixed $R_{\rm CMB}$ and $R_\oplus$ geometry, whereas the exact 3-D volume Yukawa convolution $\mathcal{F}^{-1}[(k^2+m_{\rm eff}^2)^{-2}] \propto \exp(-d_{\rm chord}/\lambda_c(\bar\rho_\oplus;\rho_T))$ — where the $1/r$ factor cancels identically in 3-D convolution — tracks $\rho_T^{-1/3}$ across $6{,}329,\,3{,}951,\,2{,}888,$ and $1{,}796$ km and the saturation-core coherence kernel $\exp(-d_{\rm chord}/R_T(M_\oplus,\rho_T))$ tracks $R_T \propto \rho_T^{-1/3}$ across $6{,}901,\,4{,}311,\,3{,}152,$ and $1{,}960$ km, reinforced by the independent cross-mass tests of Sections 4 and 6). Furthermore, solving the coupled 1-D spherical nonlinear two-branch kinetic operator $\nabla\cdot(P_{,X}\nabla u) = M_{\rm Pl}^{-2}V_{\rm eff}'(u)$ (Eq. $\ref{eq:nonlinear_chameleon_eom}$) via 100-iteration damped Picard relaxation converges to $\|\Delta u^{(100)}\|_\infty/\|u\|_\infty = 2.67 \times 10^{-14}$ and yields $R_{\rm grad}^{\rm NL\text{-}PREM} = 4{,}809.9$ km ($1.160\,R_T(M_\oplus)$), $L_c^{\rm NL\text{-}PREM} = 3{,}957$ km, and $L_c^{\rm NL,vol} = 3{,}170$ km, while the rotating-station and slant-path clock modulation yields $\delta y_{\rm rms} = 7.0 \times 10^{-15}$ ($\delta\tau_{\rm rms} \simeq 0.10$ ns at $f_{\rm sid} = 11.61\,\mu$Hz and orbital carriers $19.7\text{--}24.7\,\mu$Hz; Appendix C.2), leaving only non-spherical 3-D azimuthal PDE integration for future work.

B.3 The SPARC $M^{1/3} \to M^{1/2}$ Transition and Disk-Scale Confound

On the discrete first-bin grid ($N = 168$), the empirical dark-matter onset exponent \(\alpha_{\rm SPARC} = 0.352 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (definition)}\) is consistent with \(1/3 = 0.333\), whereas on the $n = 75$ resolved interior crossings the full-sample slope shifts to \(\alpha_{\rm res} = 0.417 \pm 0.050\), and the in-sample mass–size relation (\(R_d \propto M_{\rm bar}^{0.322}\)) poses a potential disk-size confound.

Mitigation: Both issues are resolved quantitatively in Step 8 (step_8_density_vs_size_confound.py) and Step 14 (step_14_scale_audit.py). First, a left-censored Tobit maximum-likelihood regression across all $N = 168$ galaxies (retaining the $93$ $R_{\min}$-censored low-surface-brightness dwarfs as upper bounds, where $R_{\min}$ tracks $R_d$ at $r = +0.410, p = 4.6 \times 10^{-5}$ with median $R_{\min}/R_d = 0.396$ and internal slope $-0.245$ against the resolved crossings' median $1.236$ and slope $+0.116$, creating a Simpson's-paradox cancellation on the raw discrete grid and inflating the apparent $M_{\rm bar}^{1/3}$ scatter from $0.366$ dex resolved / $0.180$ dex enclosed-mass to $0.483$ dex) formally rejects pure disk-scale invariance (\(R_{\rm DM}/R_d = \text{const}\)) with slope \(+0.323 \pm 0.034\) (\(z = 9.57, p < 10^{-4}\)), while on the $n = 75$ resolved crossings the non-circular fixed-radius partial correlation \(r_{\rm part}(R_{\rm DM},\,V_{\rm bar}(2.2R_d)\mid R_d) = +0.547\) (\(p = 3.8 \times 10^{-7}\)) confirms that gravitational potential depth drives $R_{\rm DM}$ at fixed disk size. Second, combining the two-branch Gauss–Klein–Gordon flux equation (Eq. [\ref{eq:profile_link}], which derives $g_{\rm crit}(\tau=1.30) = 1.031 \times 10^{-10}\text{ m s}^{-2}$ and $\bar\rho_{\rm trans} = 3g_{\rm crit}/(4\pi G R_{\rm DM}) = 0.068\,M_\odot\text{ pc}^{-3}$ [$0.030\,M_\odot\text{ pc}^{-3}$ uncoupled] at $R_{\rm DM} = 2.6\text{ kpc}$) with the Freeman (1970) exponential-disk profile predicts a regime crossover mass of \(M_{\rm cross}^{\rm SPARC} = 1.45 \times 10^{10}\,M_\odot\). Partitioning the $n = 75$ resolved galaxies a priori at the exact analytical Freeman velocity slope \(\gamma_{\rm Freeman} = 0.210\) (zero dependence on $V_{\rm obs}$ or $R_{\rm DM}$) cleanly separates rising-baryon 3-D core crossings (\(\gamma_V \ge 0.210, n = 38\): \(\alpha_{\rm rising} = 0.256 \pm 0.069\), consistent with $1/3$ at $1.1\sigma$ and rejecting $1/2$ at $3.5\sigma$) from concentrated 2-D Keplerian-tail crossings (\(\gamma_V < 0.210, n = 37\): \(\alpha_{\rm tail} = 0.581 \pm 0.086\), consistent with $1/2$ at $0.95\sigma$ and rejecting $1/3$ at $2.9\sigma$; subsample shift \(\Delta\alpha = +0.325 \pm 0.110, z = 2.95, p = 0.0031\)).

B.4 Screening Hierarchy: Algebraic Tautology?

The screening fit \(S \propto \rho^{0.334}\) with \(R^2 = 0.99995\) (11 dense objects, \(\rho > \rho_T\), excluding planets and black holes) is extraordinarily tight. Because \(S \equiv R_T / R_{\rm phys}\) and both radii are defined by mass-density relations, the slope \(\sim 1/3\) is algebraically expected given the model assumptions. The high \(R^2\) confirms internal consistency; it does not constitute independent confirmation of TEP.

Mitigation: The screening hierarchy verification (scripts/steps/step_5_ultimate_screening.py and scripts/verification/verify_screening_claim.py) performs object-by-object residual analysis across the 11 dense objects (\(\rho > \rho_T\)), confirming \(\beta_{\rm scr} = 0.3338 \pm 0.0008\) (\(R^2 = 0.99995\)). While the density-ordered \(S = R_T/R_{\rm phys}\) relation is an internal consistency check by construction, the planetary rotational kinematics, moment-of-inertia, and satellite tests in Section 4.7 (step_15_planetary_rotation.py, step_16_saturn_interior.py) break this algebraic circularity by testing \(R_T(M)\) and \(S\) against independent NASA/JPL ephemeris spin velocities (\(v_{\rm obs}/v_{\rm TEP} = 1.000 \pm 0.080\) across the three unbraked outer giant planets Uranus, Neptune, and Saturn [\(\hat\rho_T^{\rm spin} = 19.9\) g/cm³], and \(1.016 \pm 0.073\) with lunar-braked Earth), moments of inertia (\(2k_{\rm MOI}/S = 1.039 \pm 0.027\) across the three core-screened terrestrial planets and \(2k_{\rm MOI}/S_{\rm cond} = 1.042 \pm 0.094\) across all seven condensed/giant planets), outer non-tidally-locked satellite soliton capacity (\(\bar\eta_{\rm moons} = 0.1320 \pm 0.0113\) across Phoebe, Himalia, and Nereid, \(\hat\rho_T^{\rm moons} = 19.8\text{--}20.6\) g/cm³), Wesson's angular-momentum constant \(p_{\rm TEP} = G/(2v_{\rm gal})\), and a 100,000-trial random-density look-elsewhere coincidence null test across five independent channels (\(\hat\rho_T = 19.59 \pm 1.10\) g/cm³, \(p = 2.9 \times 10^{-4}\); and \(19.77 \pm 0.48\) g/cm³, \(p = 5.2 \times 10^{-5}\) on the three-channel subset without algebraic prefactors).

B.5 Cross-Paper Parameter Drift

Multiple distinct exponents appear across the TEP literature, and their notation has been ambiguous:

These are not the same \(\alpha\). The notation purge across all manuscripts has disambiguated them, but the proliferation of phenomenological exponents remains a vulnerability. A unified Lagrangian should predict all of them from a single coupling function \(A(\phi)\).

Mitigation: The global notation purge completed in this version removes ambiguity. Step 14 (amplitude_law_check) proves that the $1/3 = 1/(m-1)$ saturation exponent follows identically from the $m = 4$ quartic potential $V(u) = \tfrac{1}{4}\lambda M_{\rm Pl}^4 u^4$, while Step 8 and Step 12 prove that the $1/2$ galactic tail exponent follows from the square-root branch of $P_{,X}(\xi) = \epsilon_{\rm reg} + k\sqrt{\xi} + 2\xi$, and Paper 11 (Step 62) derives $\kappa_{\rm Cep} = 1.196 \times 10^6\text{ mag}$ from $\rho_T/\langle\rho\rangle(10\text{ d})$.

B.6 Series Convention

B.7 Causal Structure and Hyperbolicity

The TEP scalar field mediates correlations over thousands of kilometres in GNSS clocks. If this field propagates at finite speed, one must verify that the observed correlations do not imply superluminal signal transfer. This appendix establishes local causal-cone and scalar-hyperbolicity consistency on the selected physical backgrounds and explains why GNSS spatial covariance arises from a static boundary-value geometry rather than instantaneous signaling. Global causal admissibility is formulated relative to the preferred TEP foliation ($\nabla_\mu\bar{\phi}$ timelike at the cosmological boundary; Paper 0 §12).

Local Causal-Cone and Hyperbolicity Analysis

Objective. Establish that on the specified static background the effective causal (matter) metric $\tilde{g}_{\mu\nu}$ and the scalar field equation of motion preserve locally Lorentz-invariant matter/photon cones and strict Cauchy hyperbolicity, and that long-range spatial correlations reflect static boundary-value geometry rather than dynamic propagation.

Step 1: The Bimetric Causal Cones

In TEP (Paper 0), gravity is governed by the Einstein-frame metric $g_{\mu\nu}$. Matter and electromagnetic fields evolve along the geodesics of the causal (Jordan-frame) metric, written here on the isotropic conformal–disformal benchmark (in the canonical unified unstretched-space decomposition of Paper 0 §2.2, $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2 U_\mu U_\nu + A_{\rm sp}^2 \Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$):

To ensure causality, the inverse metric $\tilde{g}^{\mu\nu}$ must exist, and the Lorentzian signature $(-, +, +, +)$ must be strictly preserved. By the Sherman–Morrison formula, the inverse is:

where $(\nabla\phi)^2 \equiv g^{\alpha\beta} \nabla_\alpha \phi \nabla_\beta \phi$.

Condition 1 (Signature Preservation). To avoid a metric singularity and a flip in causality, the denominator must not cross zero:

As long as Condition 1 holds on the $B(\phi) \ge 0$, static-gradient benchmark ($\partial_t\phi = 0$, $(\nabla\phi)^2 > 0$), the causal cone of $\tilde{g}_{\mu\nu}$ remains strictly nested inside, or tangent to, the gravitational light cone of $g_{\mu\nu}$.

Step 2: Signal Velocity of Electromagnetic / Matter Fields

Computing the local speed of light $c_\gamma$ (matter-frame photons) in a static spherically symmetric background. In Earth's rest frame, the field is static ($\partial_t \phi = 0$) and radially dependent $\phi = \phi(r)$. The metric components become:

For a radial null signal in the matter frame, $\tilde{g}_{\mu\nu} dx^\mu dx^\nu = 0$:

which yields the effective light speed:

If $B(\phi) > 0$, then $c_\gamma < c$. The causal cone of light and matter is squeezed inside the bare gravitational cone. Information transfer is strictly subluminal relative to the Einstein frame. (As noted in Paper 0, $B(\phi)$ is deeply constrained by GW170817 multi-messenger limits in the late universe, meaning $c_\gamma \approx c$ macroscopically.)

Step 3: Propagation and Hyperbolicity of the Scalar Field

Consider perturbations $\delta\phi$ of the scalar field around a background solution $\bar{\phi}$ under the convention $X \equiv -\frac{1}{2}g^{\mu\nu}\nabla_\mu\phi\nabla_\nu\phi$. On a homogeneous cosmological background ($\bar{X} > 0$), the EFT of Dark Energy implementation (Paper 18, hi_class cosmology) satisfies the no-ghost ($\alpha_K > 0$) and no-gradient-instability ($c_s^2 > 0$) conditions with $c_s \le c$. On a static spacelike-gradient background ($\bar{X} = -\frac{1}{2}(\nabla\bar{\phi})^2 < 0$, so $|X| = -X > 0$) for the selected two-branch k-essence completion $P_X(\xi) = \epsilon_{\rm reg} + k\sqrt{\xi} + 2\xi$ ($\xi \equiv |X|/\Lambda_X^4$, $\epsilon_{\rm reg} = 10^{-3}$; Paper 0 §12, Step 77), varying $\nabla_\mu(P_X\nabla^\mu\phi)$ with $\nabla_\nu X = -\nabla^\mu\phi\nabla_\nu\nabla_\mu\phi$ yields the linearized hyperbolic wave equation:

whose transverse and longitudinal scalar characteristic speeds squared relative to the gravitational metric $g_{\mu\nu}$ are $v_\perp^2 = 1$ and $$v_\parallel^2 = \frac{P_X + 2X P_{XX}}{P_X} = \frac{P_X - 2|X| P_{XX}}{P_X} = \frac{\epsilon_{\rm reg} + 2k\sqrt{\xi} + 6\xi}{\epsilon_{\rm reg} + k\sqrt{\xi} + 2\xi} \in [1, 3]$$ across the nonlinear branches ($v_\parallel^2 \to 1$ in the regulator-dominated limit $\xi \to 0$, $v_\parallel^2 \to 2$ on the $\sqrt{\xi}$ branch, and $v_\parallel^2 \to 3$ on the quadratic branch). As established in Paper 0 §12, this widened longitudinal scalar characteristic ($v_\parallel > 1$ on static spacelike gradients) does not violate matter-sector causality: matter and electromagnetic fields propagate strictly on the locally Lorentz-invariant causal cone of $\tilde{g}_{\mu\nu}$ ($c_\gamma \le c$, Step 2), while the cosmic time field $\bar{\phi}$ defines a preferred global foliation on which the classical Cauchy initial-value problem is strictly hyperbolic and well-posed ($P_X > 0$ and $P_X + 2X P_{XX} = P_X - 2|X|P_{XX} > 0$ everywhere).

Step 4: The Steady-State Fallacy (The 4,200 km GNSS Correlation)

Crucially, the observed GNSS spatial covariance over $L_c \approx 4{,}200$ km does not rely on dynamic wave propagation between clocks at all—whether luminal or characteristic-widened.

The error in the superluminality objection lies in conflating dynamic wave communication with a static spatial boundary-value profile.

The GNSS atomic clocks are not interacting with each other via $\phi$. They are independently measuring the local magnitude of a pre-existing, steady-state geometric structure: Earth's Temporal Topology screening profile $\bar{\phi}(r)$.

Consider the Coulomb field of a proton. The electric field extends to infinity. If you place two electrons 4,200 km apart in that static field, their potential energies are perfectly "correlated" according to the geometry of $1/r$. This correlation does not require the electrons to communicate instantly; it merely requires that the central proton has existed long enough ($t > r/c$) for the static field boundary-value to settle.

In TEP, the Earth ($M_\oplus$) acts as a scalar source charge. The spatial distribution of the field is the solution to the static Klein–Gordon equation with environmental screening (the Temporal Shear transition):

Because the Earth has been in hydrostatic equilibrium for billions of years, $\partial_t \bar{\phi} = 0$. The resulting Temporal Shear profile $\Sigma_\mu = \nabla_\mu \ln A(\bar{\phi})$ forms a rigid geometric "scaffolding" around the planet.

When a GNSS clock network orbits through this profile, the residuals exhibit a spatial correlation length $L_c \approx 4{,}200$ km. This is not a superluminal signal propagating between satellites. It is the geometric width of the screening transition zone in the stationary background field.

B.8 Summary of Risk Register

The MGEX discrepancy is the most consequential held-out tension in the corpus. Paper 14 reports $\lambda \approx 1{,}862 \pm 112$ km (corrected pooled; the original ~1-year headline was 1,396 km) from a combined multi-constellation product, versus the ~4,200 km scale recovered from multi-year single-center analyses. Candidate explanations include: (i) combined multi-constellation clock products mixing different orbital sampling geometries into a compressed projection; (ii) the short ~1-year span yielding an unstable scale estimate (consistent with the window scatter and $R^2 \approx 0.70$ on the pooled fit); or (iii) a genuinely different scale. The pre-registered discriminator is a per-constellation analysis: if GPS, Galileo, and GLONASS individually recover ~4,200 km, the projection explanation wins and the MGEX result becomes a methodological footnote. If per-constellation analysis also yields ~1,900 km, the $L_c \leftrightarrow R_T(M_\oplus)$ identification fails as calibrated, and the Level 1/2/3 claim hierarchy (Section 8) absorbs the hit exactly as designed—Level 2 falls, Levels 1 and 3 survive. Executing the matched-day CODE/CNES comparison (Paper 14 Step 3.1), the factorial estimator decomposition (Paper 14 Step 3.5), and the 100-trial Monte Carlo ISB projection simulation in Step 14 (which proves that 4-constellation ISB removal compresses a $4{,}200$ km field to $2{,}277\text{--}2{,}485$ km, $1.62\text{--}1.77\times$, matching the empirical $1.90\times$ offset that de-biases pooled MGEX $1{,}862$ km to $3{,}536$ km) together with the executed 4-quarter 2023 CDDIS RINEX3 single-constellation reprocessing in Step 14 (where full-day $n_{\rm seg}=288$ single-constellation solutions recover $3{,}766$ km [GPS], $3{,}806$ km [GLONASS], and $3{,}653$ km [Galileo $10\text{--}60\,\mu\text{Hz}$; $2{,}697$ km full-band] with a three-constellation mean of $3{,}423$ km inside the $[3{,}000,\,5{,}000]$ km GPS-only band, and where applying MGEX's 8-hour $n_{\rm seg}=96$ window truncation, per-epoch datum removal, and 4-constellation ISB estimation compresses the exact same station-days step-by-step from $3{,}423 \to 2{,}850 \to 2{,}156$ km [$\mathrm{CV} = 0.28\%$ across GPS $2{,}163.2$, Galileo $2{,}152.9$, and GLONASS $2{,}152.3$ km] $\to 1{,}478$ km, a $2.32\times$ empirical compression) resolves the multi-GNSS compression mechanism directly on real observations.

The discriminator has now been executed (TEP-GNSS-MGEX Steps 3.1 and 3.5, TEP-GNSS-MGEX/results/outputs/step_3_1_per_center_scale.json), with the pre-registration outcome stated plainly: the literal per-constellation station-clock split could not be executed as originally designed because MGEX station clocks are published as single combined-solution parameters (one receiver clock per station-epoch combining GPS, GLONASS, Galileo, and BeiDou with inter-system bias absorption), and space-segment satellite clocks in the combined product carry no distance-decaying within-constellation coherence (Step 2.8). The substitute discriminators executed on the retained archive establish three facts: (i) on the identical COD MGEX product, quarterly windows scatter from $\lambda = 1{,}050$ to $1{,}935$ km ($R^2$ from $-0.18$ to $0.80$), showing that short $\lesssim 1$-year windows carry $\sim 30$–$40\%$ fractional sampling scatter around their product mean; (ii) on strictly matched, verified-independent days, two independent multi-GNSS processing chains — CODE/GPSEST and CNES/GINS (GRG) — return pooled $\lambda = 2{,}744$ km ($n = 43{,}971$ pairs) and $1{,}933$ km ($n = 44{,}000$) respectively ($\sim 30\%$ apart, GRG within $4\%$ of the pooled $1{,}862$ km MGEX scale); and (iii) a factorial estimator decomposition on identical station-days (Paper 14 §4.5b, Step 3.5) proves that estimator settings (Welch segment length and common-mode removal) shift $\lambda$ by $\lesssim \pm 25\%$, whereas both GPS-only IGS-final ($\lambda \approx 3.9$–$5.9\times 10^3$ km in the tidal band, isotropic $3{,}210 \pm 389$ km and cross-sector $4{,}201 \pm 1{,}967$ km over 25.3 years in Paper 2; raw-RINEX ionofree $4{,}767 \pm 835$ km in Paper 3) and 4-constellation MGEX ($\lambda \approx 1.8$–$2.0\times 10^3$ km) recover the identical band-ordered decay profile, with the MGEX combined multi-GNSS filter acting as a $\sim 2\times$ scaled-down projection of the GPS-only PPP profile. Thus, the 25.3-year temporal stability of the GPS-only CODE baseline (Paper 2) and the product-family scale offset of the 4-constellation MGEX filter (Paper 14) are distinct phenomena: a quantitative cross-mode de-biasing synthesis in Step 14 (constellation_debiasing_synthesis) comparing single-constellation GPS/ionofree RINEX and PPP solutions across identical station networks (mean $4{,}066$ km; including $4{,}767$ km raw-RINEX dual-frequency ionofree and $3{,}512$–$5{,}026$ km across ALL STATIONS and OPTIMAL 100 in Paper 3) against 4-constellation combined multi-GNSS solutions (mean $2{,}141$ km; $1{,}770$–$2{,}726$ km raw-RINEX multi-GNSS and $1{,}862$–$2{,}744$ km MGEX) establishes an empirical multi-GNSS inter-system-bias (ISB) filter compression factor of $1.90\times$. Crucially, the Monte Carlo injection–recovery suite on a 120-station global network in Step 14 proves the direction and mechanism of this bias without circular ratio rescaling: across an injected scale grid $L_c^{\rm true} \in \{1{,}000,\,2{,}000,\,3{,}000,\,4{,}200,\,6{,}000\}$ km, 1-constellation GPS-only global datum removal tracks every scale ($\hat L_c^{\rm GPS} = 1{,}019,\,1{,}975,\,2{,}912,\,4{,}036 \pm 113,\text{ and }5{,}721$ km; white-noise null returns $|C(d)| = 0.009 \pm 0.001$, and a correlated rank-3 satellite-orbit/geocenter dipole plus datum common-mode null yields a concave-down correlation curve $d^2C/dd^2 = -0.024\text{ Mm}^{-2} < 0$ whose unconstrained exponential fit hits the $\sim 24{,}900$ km upper bound rather than manufacturing a $4{,}200$ km scale), whereas 4-constellation multi-GNSS common-view ISB projection compresses $L_c^{\rm true} = 4{,}200$ km to $\hat L_c^{\rm MGEX} = 2{,}277 \pm 47$ km ($1.77 \pm 0.06\times$ compression) for per-epoch ISBs (matching CNES/GRG $1{,}933$ km) and $\hat L_c = 2{,}485 \pm 45$ km ($1.62\times$ compression) for daily-constant ISBs with rotating constellation visibility (matching CODE matched-day MGEX $2{,}744$ km). The spatial ISB projection and temporal passband truncation both suppress long-scale covariance and may describe overlapping aspects of the combined processing response; they are not treated as independent multiplicative corrections without a joint transfer calculation, with the simulated $1.62\text{--}1.77\times$ compression reproducing the direction and approximate magnitude of the observed $1.90\times$ offset (mapping pooled MGEX $1{,}862$ km to $3{,}536$ km inside the $3{,}330$–$4{,}549$ km GPS-only anchor range). Applying the $\sim 30$–$40\%$ fractional window/chain uncertainty around the GPS-only anchor ($L_c = 4{,}200$ km) under the conditional $L_c \leftrightarrow R_T(M_\oplus)$ identification yields the systematic band $\rho_T \in [8,\,70]$ g/cm³, while the unadjusted 4-constellation MGEX projection ($\lambda = 1{,}862$–$2{,}744$ km at unit prefactor) defines a conservative upper sensitivity bracket of $\rho_T \sim 69$–$220$ g/cm³. With the 4-quarter 2023 CDDIS RINEX3 single-constellation GPS, Galileo, and GLONASS reprocessing now executed in Step 14 (including Galileo dual-frequency iono-free LC recovering $L_c = 4{,}556.7\text{ km}$ across 67 optimal_100 stations), the single- versus multi-constellation discriminator is resolved on public data, leaving independent third-party geodesy replication and optical/two-way time-transfer links as the remaining external verification targets.

Each open question in this register defines an explicit quantitative test, ensuring that the framework is falsifiable at multiple independent points across mass and density scales.