# Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T
Matthew Lukin Smawfield
Version: v0.12 (New Delhi)
First published: 28 December 2025 · Last updated: 28 October 2026
DOI: 10.5281/zenodo.18064365

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## Abstract

Gravitational anomalies across astrophysical scales exhibit a dual scaling regularity: the radius at which Newtonian dynamics fails scales as $R_T(M) = (3M/4\pi\rho_T)^{1/3} \propto M^{1/3}$ in the compact saturation regime and $R_{\rm DM} \propto M^{1/2}$ in the low-acceleration galactic regime. Within the Temporal Equivalence Principle (TEP) two-branch scalar action ($P(X,\phi) = P_{\rm kin}(X) - \tfrac{1}{4}\lambda M_{\rm Pl}^4 u^4$, $A(u) = e^{-u}$), the quartic confining potential fixes the cubic screening response $S = (\bar\rho/\rho_T)^{1/3}$ around the saturation density $\rho_T \approx 20$ g/cm³ (calibrated from terrestrial GNSS and tested across astrophysical systems), while the nonlinear kinetic branch $P_{,X}(\xi) = k\sqrt{\xi} + 2\xi$ ($k = 16.03$) governs unscreened ($S \ll 1$) galactic halos at $a_{\rm eff} = 1.20 \times 10^{-10}$ m s$^{-2}$ (yielding under the potential-share identification $\epsilon_{\rm host} = a_\phi/(g_{\rm bar} + a_\phi)$ a parameter-free star-forming spiral BBH host tracking fraction of $\langle\epsilon_{\rm host}\rangle_{\rm SFR}^{\rm UCD} = 0.236 \pm 0.054$ at $R_{\rm eff}$ and $0.243 \pm 0.052$ across the radial Freeman disk [$0.197\text{--}0.202$ with $25\%$ bulge/NSC hosts], matching the GWTC-1–5.0 optimum in Paper 22). Across a 4-layer PREM Earth boundary-value problem and 1-D spherical nonlinear Picard solve, the transition-shell ($4{,}010$ km; $3{,}957$ km nonlinear) and full-volume ($3{,}213$ km; $3{,}170$ km nonlinear, $L_c^{\rm vol}/R_T = 0.775$) kernels reproduce both the 25.3-year GPS PPP clock correlation length ($L_c = 4{,}201$ km, $\hat\rho_T = 19.2$ g/cm³) and 2023 CDDIS RINEX3 single-constellation GPS, Galileo, and GLONASS reprocessing ($3{,}853$ km $\sqrt{N}$-weighted, and $3{,}155 \pm 181$ km [$\mathrm{CV} = 5.7\%$] under full-day $\ell = 0$ common-mode removal, mapping via $L_c^{\rm vol}/R_T$ to $\hat\rho_T = 21.1$ g/cm³).

With $\rho_T = 20$ g/cm³ and $a_{\rm eff}$ held fixed and zero per-object refitting, three independent cross-scale tests follow. First, in 175 SPARC galaxies, an a priori Freeman-disk split separates 3-D rising-core onset scaling ($\alpha_{\rm rising} = 0.256 \pm 0.069$ vs. $1/3$) from 2-D Keplerian tails ($\alpha_{\rm tail} = 0.581 \pm 0.086$ vs. $1/2$; $p = 0.0031$), while the coupled-gradient force law reproduces the radial-acceleration ($0.1330$ dex RMS) and Baryonic Tully–Fisher ($R^2 = 0.956, 0.0807$ dex) relations. Second, four external $R_\oplus$-free astrophysical catalogs — outer-giant-planet spin ($19.89$ g/cm³), the brown-dwarf rotational period floor ($1.150 \pm 0.075$ hr vs. TEP-saturated Chandrasekhar–Jacobi $1.155$ hr [$20.19$ g/cm³], sitting $8.3\sigma$ above the unscreened Newtonian Jacobi instability limit of $0.527$ hr), compressed terrestrial moments of inertia ($17.89$ g/cm³), and the Solar Model S convection-zone base ($0.713\,R_\odot$ vs. $\sqrt{3}R_T(M_\odot) = 0.715\,R_\odot$ [$20.15$ g/cm³]) — converge with GNSS to $\hat\rho_T = 19.59 \pm 1.10$ g/cm³ ($19.68 \pm 1.25$ g/cm³ GNSS-free; $20.05 \pm 1.61$ g/cm³ $v_{\rm gal}$-marginalized; look-elsewhere $p = 2.9 \times 10^{-4}$ [$3.6\sigma$], $2.05\%$ leave-one-channel-out blind prediction RMS). Third, the horizon crossover $R_T(M_\times) = R_S$ occurs at $M_\times \approx 3.04 \times 10^7\,M_\odot$, matching the runaway black hole candidate RBH-1.

**Keywords:** Temporal Equivalence Principle — dark matter: phantom mass — gravitation: modified — atomic clocks: GNSS — galaxies: kinematics and dynamics (SPARC) — planetary systems: rotation — compact objects: screening

## 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):

\begin{equation} \label{eq:tep_action} S = \int d^4x \sqrt{-g} \left[ \frac{M_{\text{Pl}}^2}{2} R + P(X,\phi) \right] + S_m[\tilde{g}_{\mu\nu}], \qquad P(X,\phi) = P_{\rm kin}(X) - V(\phi), \end{equation}

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.<sup>†</sup>

<sup>†</sup>*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:

\begin{equation} \label{eq:saturation_radius} R_T = \left( \frac{3M}{4\pi \rho_T} \right)^{1/3}. \end{equation}

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.

\begin{equation} \label{eq:screening_operator}
\Sigma_\mu^{\text{obs}} = \mathcal{S}_\Sigma(\mathcal{E}) \, \nabla_\mu \ln A(\phi),
\end{equation}

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.

#### Box 1.1: Joint Constraint Architecture of \(\rho_T\) and \(S_\Sigma(\mathcal{E})\)

\(\rho_T\) is the shared macroscopic saturation/proximity scale. Other
papers in the TEP programme do not all measure \(\rho_T\) directly; they
measure domain-specific projections of the same environmental operator
\(S_\Sigma(\mathcal{E})\):

- **Clock covariance lengths** (GNSS, Paper 1–3)

- **Galactic onset radii** (SPARC, Milky Way, this work)

- **Response coefficients** (Cepheids, pulsars, Papers 10–11)

- **Lensing residuals** (strong-lens time delays, Paper 4)

- **Weak-field recovery scales** (wide binaries, Paper 13)

- **Compactness-dependent residuals** (LLR, Paper 17)

- **Cosmological temporal-shear amplitudes** (hi_class, C0, TH, Papers 18, 26, 27)

Each measurement channel has a response coefficient \(\kappa_X\), an
environmental screening projection, and a domain-specific observable
functional. This architecture protects the programme from the criticism
that different channels report different numerical values: they are
*observable response coefficients*, not bare scalar couplings,
and each channel measures a different projection of
\(S_\Sigma(\mathcal{E})\).

### The Multi-Scale Consistency Strategy

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

### Table 1: Cross-Scale Consistency of $\rho_T$

| Scale | System | Mass Range | Density Range | Key Observable | Result |
| --- | --- | --- | --- | --- | --- |
| **Terrestrial** | GNSS Clocks | $M_\oplus \sim 6 \times 10^{27}$ g | $\rho_\oplus \sim 5.5$ g/cm³ | $L_c \approx 4200$ km | $\rho_T \approx 20$ g/cm³ |
| **Galactic** | SPARC Galaxies | $10^9$–$10^{12} M_\odot$ | $\rho \sim 10^{-24}$ g/cm³ | $R_{\text{DM}} \propto M^{\alpha_{\rm SPARC}}$ | $\alpha_{\rm SPARC} = 0.352 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (def.)}$ |
| **Local** | Milky Way | $M_{\rm bar} \sim 6 \times 10^{10} M_\odot$ | $\rho \sim 10^{-24}$ g/cm³ | Inner mass-discrepancy onset | $R_{\text{DM}} \approx 3$ kpc (observed $\sim$3–5 kpc) |
| **Screening** | 26 Objects | $\sim 10^{27}$–$10^{33}$ g | $\sim 10^{0}$–$10^{14}$ g/cm³ | $S \equiv R_T/R_{\rm phys} \equiv (\bar\rho/\rho_T)^{1/3}$ | Deterministic regime map ($S < 1$ unscreened vs $S \ge 1$ compact) |
| **Planetary Spin** | Solar System ($N=12$) | $2.6 \times 10^{23}$–$1.9 \times 10^{30}$ g | $0.69$–$5.51$ g/cm³ | $v_{\rm eq} = c\,u_T/\sqrt{u_{\rm gal}}$; $J = \frac{G M^2}{2 v_{\rm gal}}$ | $v_{\rm obs}/v_{\rm TEP} = 1.000 \pm 0.080$ (outer 3; $1.016 \pm 0.073$ with Earth); $p_{\rm TEP} = 1.52 \times 10^{-15}$ cgs |

*Note: The convergence of saturation-radius constraints across 18 orders of magnitude in mass ($10^{20}\text{--}10^{38}$ kg, from asteroids and Earth to the $2\times 10^7\,M_\odot$ runaway black hole RBH-1, alongside galactic disks on the unscreened $S \ll 1$ kinetic branch) motivates $\rho_T$ as a candidate saturation scale.*

#### Box 1.2: Executive Summary of Quantitative Audits and Methodological Controls

Gravitational anomalies across astrophysical scales exhibit an empirical regularity: the radius at which Newtonian dynamics fails scales as $R \propto M^{1/3}$ in the compact saturation regime and $R \propto M^{1/2}$ in the low-acceleration galactic regime, organized by a candidate saturation/proximity density scale $\rho_T$ in the Temporal Equivalence Principle (TEP) conformal time-field sector. The 25-year GNSS atomic-clock correlation length $L_c \approx 4{,}200$ km (CODE, IGS, ESA; raw-RINEX validated) establishes an Earth-scale correlation structure consistent with $R_T(M_\oplus) = (3M_\oplus/4\pi\rho_T)^{1/3} = 4{,}146$ km at $\rho_T \approx 20$ g/cm³ ($\rho_T \in [8,\,70]$ g/cm³ conditional mapping under $L_c \leftrightarrow R_T(M_\oplus)$). A 4-layer PREM Earth boundary-value integration confirms an order-unity geometric projection ($L_c^{\rm PREM} = 4{,}010$ km shell; $3{,}213$ km volume) governed by Earth's $R_{\rm CMB}$ and $R_\oplus$ structure, while Monte Carlo network injection–recovery simulations across $L_c^{\rm true} \in [1{,}000,\,6{,}000]$ km demonstrate that 1-constellation GPS-only processing recovers $L_c^{\rm true} = 4{,}200$ km ($\hat L_c^{\rm GPS} = 4{,}036 \pm 113$ km; white-noise null $|C(d)| = 0.009$; correlated orbital/datum dipole null concave-down with $d^2C/dd^2 < 0$), whereas 4-constellation multi-GNSS inter-system-bias (ISB) projection compresses the same field to $\hat L_c^{\rm MGEX} = 2{,}277 \pm 47$ km per-epoch ($1.77\times$) and $2{,}485 \pm 45$ km daily-constant ($1.62\times$, explaining the empirical $1.90\times$ MGEX offset that de-biases pooled MGEX $1{,}862$ km to $3{,}536$ km), and direct 4-quarter 2023 CDDIS RINEX3 single-constellation reprocessing ($192\text{--}197$ station-days, $3{,}706\text{--}3{,}927$ pairs per constellation; Step 14) confirms on real observations that in the full-day 24-hour window ($n_{\rm seg}=288$) independent US GPS ($4{,}051$ km $\sqrt{N}$-weighted, $3{,}766$ km unweighted, $R^2 = 0.964$), Russian GLONASS ($4{,}693$ km $\sqrt{N}$-weighted, $3{,}806$ km unweighted, $R^2 = 0.849$), and European Galileo ($4{,}557$ km in dual-frequency iono-free LC on `optimal_100`; $3{,}653$ km in $10\text{--}60\,\mu\text{Hz}$, $R^2 = 0.890$, and $2{,}816$ km $\sqrt{N}$-weighted / $2{,}697$ km unweighted in single-frequency broadcast $10\text{--}500\,\mu\text{Hz}$, $R^2 = 0.934$) solutions yield strictly band-matched three-constellation means of $3{,}853$ km ($\sqrt{N}$-weighted $10\text{--}500\,\mu\text{Hz}$, $\hat\rho_T = 24.9$ g/cm³), $3{,}423$ km (unweighted $10\text{--}500\,\mu\text{Hz}$), $3{,}362$ km (matched $10\text{--}60\,\mu\text{Hz}$), and — under uniform full-day $\ell = 0$ common-mode-error removal across the $10\text{--}500\,\mu\text{Hz}$ band — $3{,}155 \pm 181$ km ($\mathrm{CV} = 5.7\%$ across GPS $2{,}953$, Galileo $3{,}302$, and GLONASS $3{,}210$ km, $R^2 = 0.955\text{--}0.969$, matching the 25.3-yr isotropic CODE scale $3{,}210$ km and PREM full-volume kernel $L_c^{\rm vol} = 3{,}213$ km [model-calibrated BVP ratio $L_c^{\rm vol}/R_T = 0.775 \implies R_T = 4{,}072$ km, $\hat\rho_T = 21.1$ g/cm³, $3.35\%$ astrophysical RMS error, vs. $45.4$ g/cm³ and $29.4\%$ RMS error at raw unit prefactor; $20.1$ g/cm³ on CODE $3{,}210$ km], with station-geometry resampling confirming no continental-clustering artifact and forward propagation across all four external astrophysical channels yielding $1.98\%\text{--}3.82\%$ joint RMS error vs. $78.6\%$ failure on the truncated $2{,}156$ km scale), whereas applying MGEX's 8-hour window truncation ($n_{\rm seg}=96$), per-epoch datum removal, and 4-constellation ISB estimation compresses the exact same data 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; $R^2 = 0.929\text{--}0.957$) $\to 1{,}478$ km ($2.32\times$ total empirical compression).

Because the canonical terrestrial BVP is shaped by Earth's fixed geometry, quantitative calibration of $\rho_T \approx 20$ g/cm³ rests on direct $R_\oplus$-free cross-mass tests, separated from structural-exponent tests. Direct tests comprise the RBH-1 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 equatorial rotation relation $v_{\rm eq}/c = u_T(M)/\sqrt{u_{\rm gal}}$ ($v_{\rm gal} = 220$ km s$^{-1}$), which matches the three truly unbraked outer giant planets (Uranus, Neptune, and Saturn, excluding lunar-braked Earth whose conserved primordial Earth+Moon angular momentum $T_{\rm prim} = 5.05$ hr sits on the small-body soliton ceiling $\eta_{\rm E+M} = 0.145$) at $v_{\rm obs}/v_{\rm TEP} = 1.000 \pm 0.080$ ($\hat\rho_T^{\rm spin} = 19.9$ g/cm³ geometric mean; $1.016 \pm 0.073$ and $20.9$ g/cm³ when present-day Earth is included, where the $+6.4\%/+8.4\%$ vs. $-0.6\%/-7.7\%$ residual signs track the nested screening-shell ratio $r_*(M)/a$ at $r = +0.960, p = 0.040$, and volumetric/deep-interior periods give $1.031 \pm 0.041$; extending above $M_{\rm cap} = 194\,M_\oplus = 0.61\,M_{\rm Jup}$ with one small-body capacity parameter $\eta_0 = 0.129$ [six-body IQR $0.122$–$0.154$, candidate secular stability benchmark $\eta_{\rm sec} = \tfrac{1}{8} = 0.125$] and $S_A = \min[1,\,S]$ matches Jupiter at $-2.9\%$ [$-0.2\%$ at $\eta_{\rm sec}$], the fastest planetary-mass rotator SIMP J0136 at $-2.8\%$ [$-0.3\%$], and the $1.15 \pm 0.08$ hr brown-dwarf period floor of Tannock et al. 2021 at $P_{\min}^{\rm TEP} = (k_{\rm MOI}/\eta_0)\tau_T = 1.17$ hr [IQR $0.98$–$1.24$ hr; $1.21$ hr at $\eta_{\rm sec}$], giving $v_{\rm obs}/v_{\rm 2br} = 1.001 \pm 0.060$ and $\hat\rho_T^{\rm rot} = 20.0$ g/cm³ across the $N = 8$ unbraked outer-planet and substellar systems [$1.008 \pm 0.060$ and $21.1$ g/cm³ across $N = 9$ with Earth]), self-compressed planetary moments of inertia $k_{\rm MOI} \simeq \tfrac{1}{2}S$ ($\rho_T^{\rm MOI,\text{terr}} = 17.9 \pm 1.4$ g/cm³ across Mercury, Venus, and Earth; $2k_{\rm MOI}/S = 1.17 \pm 0.14$ across 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 once Lane–Emden polytropic H/He envelope inflation is removed, whereas low-pressure uncompressed satellites with $P_c \ll K_0$ remain near the uniform-sphere ceiling), and an exploratory Standard Solar Model (Model S) helioseismic prototype (where $\rho_\odot(r) = \rho_T$ occurs at $0.249\,R_\odot$ near the nuclear-core boundary, $\sqrt{3}\,R_T(M_\odot) = 0.715\,R_\odot$ and $y_{\rm BVP} = \tfrac{1}{2}S_\odot$ at $0.723\,R_\odot$ coincide with the $0.713 \pm 0.001\,R_\odot$ convection-zone base [$+0.25\%$, $\rho_T^{\rm (cz)} = 20.15$ g/cm³], $\nu_{\rm eq} = g_{\rm surf}/(2c) = 456.5$ nHz [$1.997$ km s$^{-1}$] matches the equatorial rotation rate $460.7$ nHz to $-0.9\%$, and $S_\odot/\sqrt{2} = 0.292$ matches the equator-to-pole differential shear $0.289$ to $+1.1\%$; across the four GNSS-free external astrophysical channels alone, $\hat\rho_T^{\rm no\text{-}GNSS} = 19.68 \pm 1.25$ g/cm³ [$\pm 6.4\%$], and across all five channels including GNSS, $\hat\rho_T = 19.59 \pm 1.10$ g/cm³ [$\sigma_{\log_{10}\rho} = 0.025$ dex], rejecting random-density coincidence across $[0.1,\,10^4]$ g/cm³ at $p = 1.7 \times 10^{-3}$ on a 1-D scan, $p = 1.4 \times 10^{-7}$ across five channels, and $p = 2.9 \times 10^{-4}$ [$3.6\sigma$] even after granting $7^3 = 343$ look-elsewhere algebraic-factor trials). Structural and scale-cancellation tests comprise the SPARC onset scaling ($\alpha_{\rm SPARC} = 0.352 \pm 0.043$ discrete, $N = 168$; $\alpha_{\rm res} = 0.417 \pm 0.050$ on $n = 75$ resolved crossings, where an *a priori* Freeman-disk baryonic slope split around $M_{\rm cross} = 1.45 \times 10^{10}\,M_\odot$ separates rising-baryon $M^{1/3}$ core crossings [$\alpha_{\rm rising} = 0.256 \pm 0.069, n = 38$] from concentrated Keplerian-tail $M^{1/2}$ crossings [$\alpha_{\rm tail} = 0.581 \pm 0.086, n = 37; \Delta\alpha = +0.325 \pm 0.110, p = 0.0031$]; Tobit rejection of $R_{\rm DM}/R_d$ invariance at $+0.323 \pm 0.034, z = 9.57, p < 10^{-4}$; fixed-radius non-circular partial $r(R_{\rm DM},\,V_{\rm bar}(2.2R_d)\mid R_d) = +0.547$, $p = 3.8 \times 10^{-7}$), the two-branch RAR/BTFR profile $P_{,X} = \epsilon_{\rm reg} + k\sqrt{\xi} + 2\xi$ ($0.139$–$0.152$ dex RMS on the quality-controlled SPARC sample), the Milky Way onset ($3.1 \pm 0.4$ kpc mean; $[1.0,\,9.4]$ kpc $1\sigma$ interval), the deterministic regime map $S \equiv R_T/R_{\rm phys} \equiv (\bar\rho/\rho_T)^{1/3}$, and Wesson's constant $p_{\rm TEP} \equiv J/M^2 = G/(2v_{\rm gal}) = 1.52 \times 10^{-15}$ cm$^2$ g$^{-1}$ s$^{-1}$ ($J_{\rm obs}/J_{\rm TEP} = 1.11 \pm 0.25$).

### 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:

\begin{equation} \label{eq:rt_definition} R_T = \left( \frac{3M}{4\pi \rho_T} \right)^{1/3}. \end{equation}

Solving for $\rho_T$:

\begin{equation} \label{eq:rho_t_definition} \rho_T = \frac{3M}{4\pi R_T^3}. \end{equation}

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

\begin{equation} \label{eq:rho_t_earth} \rho_T = \frac{3 \times (6 \times 10^{27}\,\text{g})}{4\pi \times (4.2 \times 10^8\,\text{cm})^3} \approx 20\,\text{g/cm}^3. \end{equation}

### 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:

\begin{equation} \label{eq:rho_t_uncertainty} \rho_T = 20 \pm 7 \text{ g/cm}^3 \; (35\% \text{ inter-center}), \qquad \rho_T \in [8,\,70] \text{ g/cm}^3 \; \text{(epoch/window systematic)}. \end{equation}

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,

\begin{equation} L_c \sim \mathcal{O}(1) \times R_T(M_\oplus), \end{equation}

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.

![4-Layer PREM Earth Scalar BVP and Projected Surface Spatial Kernel](results/figures/figure_10_prem_earth_kernel.png)

Figure 1: Fixed-input, canonical-kinetic 4-layer PREM Earth boundary-value integration and 3-D Yukawa surface spatial-kernel projection (Step 14). (a) Solved interior scalar profile $u(r)/u_{\min}(\rho_T)$ and radial transition weight $r^2|\mathrm{d}u/\mathrm{d}r|$, showing the transition gradient centroid at $R_{\rm grad} = 4{,}875$ km ($1.18\,R_T(M_\oplus)$) across the core–mantle boundary (CMB). (b) Projected datum-free surface spatial kernels across station arc separations $d \in [200,\,8{,}000]$ km: the transition-shell kernel ($R_{\rm grad} = 4{,}875$ km) and its two-parameter exponential fit $A e^{-d/L_c^{\rm PREM}}$ ($A = 1.25$, $L_c^{\rm PREM} = 4{,}010$ km, $R^2 = 0.927$; within $-4.5\%$ of the $4{,}201$ km cross-sector GNSS baseline) together with the full radial-volume kernel integrated over $r \in [500,\,6{,}321]$ km ($L_c^{\rm vol} = 3{,}213$ km anchored fit, $R^2 = 0.981$; $3{,}044$ km two-parameter fit, $R^2 = 0.985$, matching the $3{,}210 \pm 389$ km 25.3-year isotropic CODE baseline), compared against the multi-center GPS-only PPP reference band ($L_c = 3{,}330$–$4{,}549$ km).

*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:

\begin{equation} \label{eq:dm_scaling} R_{\rm DM} \propto \left(\frac{M_{\rm bar}}{\rho_{\rm trans}}\right)^{1/3}
\propto M_{\rm bar}^{1/3}. \end{equation}

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:

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.

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.

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 $V_{\rm gas}^2$).

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:

| Definition (Threshold) | Fitted Exponent ($\alpha_{\rm SPARC}$) | Consistency with 1/3 |
| --- | --- | --- |
| Loose ($V_{obs}/V_{bar} > 1.1$) | $0.272 \pm 0.036$ | $1.7\sigma$ |
| Fiducial ($V_{obs}/V_{bar} > 1.3$) | $0.369 \pm 0.041$ | $0.9\sigma$ |
| Strict ($V_{obs}/V_{bar} > 1.5$) | $0.408 \pm 0.045$ | $1.6\sigma$ |

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).

\begin{equation} \label{eq:dm_relation} R_{\rm DM} = k \cdot M_{\rm bar}^{\alpha_{\rm SPARC}} \end{equation}

\begin{equation} \label{eq:sparc_exponent} \alpha_{\rm SPARC} = 0.352 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (definition)} \end{equation}

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:

\begin{equation} \label{eq:trans_radius} R_{\rm trans} \sim \left( \frac{M}{\rho_{\rm trans}} \right)^{1/3}. \end{equation}

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.

#### Formal Link: Two-Branch Gauss–Klein–Gordon Derivation of $\rho_{\rm trans}$

In the two-branch TEP action (Paper 0 §7; Appendix C.2, Eq. $\ref{eq:nonlinear_chameleon_eom}$; Step 14), compact-object saturation is governed by the potential sector $V(u) = \tfrac{1}{4}\lambda M_{\rm Pl}^4 u^4$ at $\rho_T \approx 20\text{ g/cm}^3$, whereas diffuse galactic halo onset ($\bar\rho \ll \rho_T$, where $V_{\rm eff}'(u) \to 0$) is governed directly by the nonlinear kinetic operator $P_{,X}(\xi) = \epsilon_{\rm reg} + k\sqrt{\xi} + 2\xi$ ($\xi \equiv |X|/\Lambda_X^4$, $\Lambda_X = \sqrt{M_{\rm Pl}H_0} = 1.91\text{ meV}$, $g_t = cH_0/2 = 3.40 \times 10^{-10}\text{ m s}^{-2}$, $a_{\rm eff} = 4\sqrt{2}\,g_t/k = 1.20 \times 10^{-10}\text{ m s}^{-2}$):

\begin{equation} \label{eq:profile_link} \nabla\cdot\bigl(P_{,X}(\xi)\nabla u\bigr) = \frac{\beta_A}{M_{\rm Pl}^2}\,\rho_b \quad \Longleftrightarrow \quad \bar u\,P_{,X}(\bar u^2/2) = \frac{g_N(r)}{g_t}. \end{equation}

At the fiducial velocity-ratio crossing $\tau = V_{\rm obs}/V_{\rm bar} = 1.30$ (where $a_\phi/g_N = \tau^2 - 1 = 0.69$ and $g_{\rm obs}/g_N = \tau^2 = 1.69$), solving the coupled-gradient force relation $a_\phi(a_\phi + g_N) = g_N\,a_{\rm eff}$ fixes the Newtonian crossing acceleration $g_{\rm crit}(\tau=1.30) = a_{\rm eff}/[\tau^2(\tau^2 - 1)] = 1.031 \times 10^{-10}\text{ m s}^{-2}$ ($3.18 \times 10^3\text{ [km s}^{-1}]^2\text{ kpc}^{-1}$; or $4.49 \times 10^{-11}\text{ m s}^{-2}$ under the uncoupled self-gradient comparator $a_\phi = \sqrt{g_N a_{\rm eff}}$) and the kinetic column-density scale $\Sigma_* = \Lambda_X^2/(\sqrt{2}|\beta_A|M_{\rm Pl}) = 0.122\text{ g cm}^{-2} = 582\,M_\odot\text{ pc}^{-2}$ (Step 14). Converting $g_{\rm crit}$ into an enclosed mean baryonic density via $\bar\rho_{\rm trans}(R_{\rm DM}) = 3M_{\rm bar}(<R_{\rm DM})/(4\pi R_{\rm DM}^3) = 3g_{\rm crit}/(4\pi G R_{\rm DM})$ yields $\bar\rho_{\rm trans} = 0.068\,M_\odot\text{ pc}^{-3}$ ($4.6 \times 10^{-24}\text{ g cm}^{-3}$; $0.030\,M_\odot\text{ pc}^{-3}$ uncoupled) at the median resolved SPARC crossing radius ($R_{\rm DM} = 2.6\text{ kpc}$) and $\bar\rho_{\rm trans} = 0.50\text{--}0.67\,M_\odot\text{ pc}^{-3}$ ($\sim 3\text{--}4 \times 10^{-23}\text{ g cm}^{-3}$) across the inner 3-D disk scale height ($R \approx 0.26\text{ kpc}$). Crucially, constant-density and constant-acceleration scaling are distinct regimes rather than interchangeable identities: inside rising 3-D baryonic cores ($M_{\rm bar} < M_{\rm cross}^{\rm SPARC} = 1.45 \times 10^{10}\,M_\odot$), the characteristic core density set by $\rho_{\rm core} \sim \Sigma_*/(2h_z) \approx 0.5\,M_\odot\text{ pc}^{-3}$ yields constant-density scaling $R_{\rm DM} = [3M_{\rm enc}/(4\pi\rho_{\rm core})]^{1/3} \propto M^{1/3}$ (observed $\alpha_{\rm rising} = 0.256 \pm 0.069$), whereas in concentrated disks ($M_{\rm bar} > M_{\rm cross}^{\rm SPARC}$) where the crossing occurs in the outer 2-D Keplerian tail at fixed acceleration $g_{\rm crit} = 1.031 \times 10^{-10}\text{ m s}^{-2}$, $M_{\rm enc} = g_{\rm crit}R_{\rm DM}^2/G$ yields constant-acceleration scaling $R_{\rm DM} \propto M^{1/2}$ (observed $\alpha_{\rm tail} = 0.581 \pm 0.086$).

#### Connection to the MOND Acceleration Scale

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

$$g_{\rm TEP} = \frac{GM}{R_{\rm trans}^2} \approx 5 \times 10^{-10}\,{\rm
m/s}^2.$$

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.

![Example Rotation Curves](results/figures/figure_6_sparc_examples.png)

Figure 2: Representative SPARC rotation curves across four decades in baryonic mass ($10^{7.7}$–$10^{11.4}\,M_\odot$). Each panel compares observed rotation velocities $V_{\rm obs}$ (purple points) and the Newtonian baryonic baseline $V_{\rm bar}$ (solid slate curve, $\Upsilon_d = 0.5$) against the fixed-input two-branch TEP rotation curves $V_{\rm TEP}(R)$ ($P_{,X}(\xi)=\epsilon_{\rm reg}+k\sqrt{\xi}+2\xi$, $k=16.03$ fixed by $a_0$; solid blue = coupled-gradient action-derived additive law $g_{\rm bar}+a_\phi = \tfrac{1}{2}g_{\rm bar} + \sqrt{(g_{\rm bar}/2)^2 + g_{\rm bar}a_{\rm eff}}$, dash-dotted blue = phenomenological quadrature comparator $\sqrt{g_{\rm bar}^2+a_\phi^2}$). Vertical lines mark the mean-density onset prediction $R_{\rm trans} = k\,M_{\rm bar}^{1/3}$ (orange dashed; shaded interior) and the discrete $V_{\rm obs}/V_{\rm bar} > 1.3$ crossing $R_{\rm DM}$ (grey dotted); in high-mass galaxies with prominent inner disks (e.g., NGC 2683, NGC 5055), the discrete $1.3\times$ crossing occurs farther out than $R_{\rm trans}$, contributing to the $\sigma = 0.48$ dex onset scatter while the continuous two-branch profile tracks the full velocity curve.

### 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:

\begin{equation} \label{eq:residual} \text{Residual} = \log_{10}(R_{\rm DM} / R_{\rm exp}) \end{equation}

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:

| Property | Correlation (r) | P-value | Interpretation |
| --- | --- | --- | --- |
| Gas Fraction | $-0.153$ | 0.048 ($p_{\rm adj} = 0.191$) | Weak, not significant after 4-test correction |
| Surface Brightness | $+0.047$ | 0.543 | Null, not significant |
| Inclination | $+0.096$ | 0.217 | Null, not significant |
| Central Density | $+0.016$ | 0.834 | Null, not significant |

#### 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.

Summary: across all 168 SPARC galaxies with measured onset radii, the
residuals are uncorrelated with galaxy inclination ($r = +0.096$, $p = 0.217$),
surface brightness ($r = +0.047$, $p = 0.543$), and central density
($r = +0.016$, $p = 0.834$), and show only a weak gas-fraction association
($r = -0.153$, $p = 0.048$; $p_{\rm adj} = 0.191$ after four-test correction), showing no significant dependence on the tested inclination, surface-brightness, central-density, or gas-fraction proxies after four-test correction.

![SPARC Residual Analysis](results/figures/figure_7_sparc_residuals.png)

Figure 3: Residual analysis of the SPARC $M^{1/3}$ onset scaling ($\alpha_{\rm SPARC} = 0.352 \pm 0.043$, $\sigma = 0.48$ dex). (a) Scaling
relation colored by log-residual (color bar clamped to $[-1, +1]$ dex). (b–e) Residual correlations with baryonic
properties and central density. (f) Gaussian residual distribution
($\sigma = 0.48$ dex). (g) Correlation magnitude summary compared against the $|r| = 0.3$ reference effect-size threshold; all residual correlations satisfy $|r| < 0.2$, and the nominal gas-fraction association ($p = 0.048$) does not survive correction for the four tested residual comparisons ($p_{\rm adj} = 0.191$).

### 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 $\rho_T \approx 20\text{ g/cm}^3$.

#### Distinctive Cross-Scale Linkage

While MOND provides a tighter fit to galaxy kinematics alone, it offers
no explanation for the GNSS clock correlations ($L_c \approx 4200$ km).
LCDM+Feedback can accommodate the
galaxy scaling but treats the normalization as a free parameter,
offering no predictive link to other scales. TEP is distinctive in that
its two-branch action simultaneously governs the terrestrial/compact saturation scale ($\rho_T \approx 20\text{ g/cm}^3$ via the quartic potential $V(u)$) and derives the galactic transition density ($\bar\rho_{\rm trans} = 3g_{\rm crit}/(4\pi G R_{\rm DM}) \simeq 0.03\text{--}0.5\,M_\odot/{\rm pc}^3$ and the $M^{1/3} \to M^{1/2}$ crossover mass $M_{\rm cross} = 1.45 \times 10^{10}\,M_\odot$ via the kinetic operator $P_{,X}(\xi)$; Eq. [\ref{eq:profile_link}], Steps 8 and 14) with zero galaxy-specific free parameters.

### 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:

For each galaxy, the rotation curve is evaluated at the native SPARC
radial sampling (not interpolated).

The velocity ratio $V_{\rm obs}/V_{\rm bar}$ is computed at each radius.

$R_{\rm DM}$ is the radius of the first bin exceeding the threshold.

Galaxies with non-monotonic or noisy $V_{\rm obs}/V_{\rm bar}$ profiles
are retained; the "first crossing" definition is robust to subsequent
fluctuations.

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.

Screening in TEP is represented at the theory level by the environmental operator
$S_\Sigma(\mathcal{E})$.
Quantities such as
$\rho_T$,
$R_T(M)$,
$S_\oplus(r)$,
compactness $\Phi/c^2$,
local stellar density,
geometric coherence length,
and channel-specific response coefficients
are domain-specific projections of $\mathcal{E}$,
not independent screening mechanisms
and not interchangeable universal thresholds.
Each is an observational transfer model
that parameterizes the same underlying operator
in a regime-appropriate form.

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:

\begin{equation} \label{eq:screening_factor} S = \frac{R_T}{R_{phys}}. \end{equation}

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.

\begin{equation} \label{eq:ppn_suppression} \alpha_{\rm PPN}^{\rm eff}=\alpha_0 F(S,\rho,\Phi,\nabla\phi), \qquad F\rightarrow 0 \end{equation}

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

![White Dwarf Screening Mechanism](results/figures/figure_3_wd_screening.png)

Figure 4: White Dwarf Physical Radius vs. Geometric Saturation Scale. As mass increases, the white-dwarf physical radius
shrinks (solid slate curve: Nauenberg 1972 relativistic degeneracy relation steepening toward the Chandrasekhar limit $M_{\rm Ch} = 1.44\,M_\odot$; dotted curve: low-mass non-relativistic asymptote $R_{\rm WD} \propto M^{-1/3}$) while the geometric saturation radius grows
($R_T \propto M^{1/3}$). This divergence drives the geometric saturation ratio
$S = R_T/R_{\rm phys}$ to large values ($S \approx 50\times$ for Sirius B), placing the star deep inside the compact-body regime ($R_T \gg R_{\rm phys}$).

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):

\begin{equation} \label{eq:jupiter_radius} R_{\rm phys} \approx 5{,}800 \text{ km} \quad \text{(observed)} \end{equation}

\begin{equation} \label{eq:jupiter_soliton} R_T \approx 289{,}000 \text{ km} \quad \text{(TEP saturation scale)} \end{equation}

\begin{equation} \label{eq:jupiter_screening} \text{Geometric Saturation Ratio } S = \frac{R_T}{R_{\rm phys}} \approx 50\times. \end{equation}

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 &sect;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 ($S \ge 1$):

\begin{equation} \label{eq:screening_law} S \equiv \frac{R_T(M)}{R_{\rm phys}} \equiv \left(\frac{\bar\rho}{\rho_T}\right)^{1/3}. \end{equation}

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 &sect;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 &sect;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 &sect;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:

\begin{equation} \label{eq:screening_derivation} S = \frac{R_T}{R_{phys}} = \frac{M^{1/3}}{M^{1/3}/\rho^{1/3}} =
\rho^{1/3}. \end{equation}

#### Interpretation of the Screening Law

It is important to recognize that the regression $S \propto
\rho^{0.334}$ is primarily a consistency check, not independent evidence
for the scaling law. Since $S \equiv R_T/R_{\text{phys}}$ and
the radii are defined by mass-density relations, the slope $\sim 1/3$ is
algebraically expected given the model assumptions.

The value of this analysis is not to prove the scaling 'ab initio', but
to demonstrate that a *single* saturation scale $\rho_T$ yields
a consistent screening hierarchy across 15 orders of magnitude in
density ($R^2 \approx 1$) without requiring regime-dependent
adjustments. The high $R^2$ confirms internal consistency; it does not
constitute independent confirmation of TEP.

Furthermore, the extreme screening factors observed in binary pulsars
($S \sim 29{,}000$) are consistent with a strongly non-linear suppression
mechanism in the scalar sector. The empirical screening hierarchy
($S$ vs $\rho$) derives from the canonical Temporal Topology mechanism
(Paper 0), where screening operates via the continuous spatial profile
of the scalar field governed by non-linear superposition of field
gradients (Temporal Shear). Box 6.5 (Section 6) sketches a candidate
route from the canonical action to soliton-like saturation, showing how
the interplay between the kinetic term and a candidate saturation potential could generate
the characteristic $M^{1/3}$ scaling and screening behavior. The detailed
dynamical derivation is not required for the empirical hierarchy used here.

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

![Screening Hierarchy Across 15 Orders of Magnitude](results/figures/figure_4_screening_hierarchy.png)

Figure 5: The Geometric Screening Hierarchy. (a) Geometric saturation ratio $S = R_T/R_{\rm phys}$ plotted against
mean density $\rho$ for 26 astrophysical objects (with the density-derived electron Fermi wavelength $\lambda_F = (8\pi m_p/(3\rho\,Z/A))^{1/3}$ at $Z/A = 0.5$ shown on the upper axis as a condensed-matter reference scale). The deterministic regime locus $S \equiv (\bar\rho/\rho_T)^{1/3}$ places degenerate stars and pulsars deep in the compact-body regime ($R_T \gg R_{\rm phys}$), while diffuse systems and supermassive black holes evaluated at their Schwarzschild horizon density (M87* at $\rho_S = 4.37 \times 10^{-4}$ g/cm³, $S = 0.028$; RBH-1 near the crossover at $\rho_S = 46.1$ g/cm³, $S = 1.32$) fall along the same $S = (\rho/\rho_T)^{1/3}$ locus across 18 decades in density. (b) Direct comparison of saturation radius $R_T(M)$ versus physical radius $R_{\rm phys}$.

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

| Object Class | Density (g/cm³) | Screening | Physical Meaning |
| --- | --- | --- | --- |
| Gas Giants | 0.7 – 1.6 | 0.3 – 0.4× | $R_T$ smaller than physical radius; saturated region does not
envelop the object — the unscreened side of the transition,
with channel-projected response (clock channels more active
than orbital channels) |
| Main Sequence Stars | 0.6 – 57 | 0.3 – 1.3× | Mixed regime; scalar and baryonic scales comparable |
| Rocky Planets (Earth) | 3.3 – 5.5 | 0.55 – 0.65× | Saturation scale comparable to object radius; GNSS probes this boundary |
| Brown Dwarfs | ~100 | ~1.7× | Just above $\rho_T$; screening onset begins |
| White Dwarfs | $\sim 4 \times 10^5$ – $2.4 \times 10^6$ | 25 – 50× | Baryonic star embedded within saturation scale; Keplerian mass = baryonic mass |
| Neutron Stars | $10^{14}$ | 26,800× | Scalar contribution less than 0.004%; pure GR dynamics |
| Binary Pulsars | $10^{14}$ | 29,500× | GR verified to 0.2%; Nobel Prize 1993 |

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:

| Test | Observable | Precision | Geometric $S_A$ ($R_T/R$) | Status |
| --- | --- | --- | --- | --- |
| Lunar Laser Ranging | Nordtvedt effect | $10^{-13}$ | 0.55× | GR-consistent in the orbital ($\mathcal S_\Sigma \sim 10^{-14}$) projection; synodic residual channel analyzed in Paper 17 |
| Cassini Conjunction | Shapiro delay | $2 \times 10^{-5}$ | 0.41× | Satisfied via kinetic source-charge suppression ($y_{\rm tb}(1.6\,R_\odot)=2.16\times10^{-8}$, $|\gamma_{\rm PPN}-1|=4.32\times10^{-8}\ll2.3\times10^{-5}$) |
| MESSENGER | Perihelion precession | $3 \times 10^{-4}$ | 0.65× | GR-consistent in the orbital ($\mathcal S_\Sigma \ll 10^{-8}$) projection |
| Hulse-Taylor Pulsar | GW emission | 0.2% | 29,500× | Completely screened |
| Double Pulsar | 7 PPN tests | 0.05% | 26,400× | Completely screened |

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).

**Principle:**

$\rho_T \approx 20$ g/cm³ is a saturation scale of the temporal-field topology, not a local on/off switch. Systems approach the GR-like limit when the observable shear/source-charge sector is suppressed, $\mathcal{S}_\Sigma \to 0$, which depends on source structure, environment, and boundary conditions, not $\rho > \rho_T$ alone. The empirical screening hierarchy $S \propto \rho^{0.334}$ (§4.2) is a smooth power-law across 15 orders of magnitude in density, not a step function at $\rho_T$.

### 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:

\begin{equation} \label{eq:omega_T_universal}
\omega_T \equiv \sqrt{\frac{GM}{R_T(M)^3}} = \sqrt{\frac{4\pi G\rho_T}{3}} = 2.365 \times 10^{-3}\text{ rad s}^{-1} \qquad \left(\tau_T = \frac{2\pi}{\omega_T} = 0.738\text{ hr} = 44.3\text{ min}\right).
\end{equation}

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:

\begin{equation} \label{eq:j_T_capacity}
j_T(M) \equiv R_T(M)^2\,\omega_T = \frac{c^2}{\omega_T}\,u_T(M) \propto M^{2/3},
\end{equation}

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:

\begin{equation} \label{eq:veq_tep_law}
\frac{v_{\rm eq}^{\rm TEP}}{c} = \frac{u_T(M)}{\sqrt{u_{\rm gal}}} \quad \Longleftrightarrow \quad v_{\rm eq}^{\rm TEP} = \frac{GM}{R_T(M)\,v_{\rm gal}} = \frac{v_T(M)^2}{v_{\rm gal}},
\end{equation}

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):

\begin{equation} \label{eq:kmoi_screening}
k_{\rm MOI} \simeq \frac{1}{2}\,S = \frac{1}{2}\,\frac{R_T(M)}{R_{\rm phys}} \quad \Longleftrightarrow \quad I \simeq \frac{1}{2}\,M\,R_T(M)\,R_{\rm phys},
\end{equation}

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:

\begin{equation} \label{eq:wesson_tep_law}
j_{\rm TEP}(M) = k_{\rm MOI}\,R_{\rm phys}\,v_{\rm eq}^{\rm TEP} = \frac{GM}{2\,v_{\rm gal}}, \qquad J_{\rm TEP}(M) = p_{\rm TEP}\,M^2, \qquad p_{\rm TEP} \equiv \frac{G}{2\,v_{\rm gal}} = 1.517 \times 10^{-15}\text{ cm}^2\text{ g}^{-1}\text{ s}^{-1},
\end{equation}

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).

| Body | $M$ (kg) | $r_*/a$ | $R_T$ (km) | $2k_{\rm MOI}/S$ | $u_T$ | $v_{\rm obs}$ (km/s) | $v_{\rm TEP}$ (km/s) | $T_{\rm obs}$ (hr) | $T_{\rm TEP}$ (hr) | $\chi_{\rm obs}$ | $\eta_{\rm rot}$ |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| **Hygiea** | $8.74 \times 10^{19}$ | $0.009$ | $101.4$ | $1.498$ | $6.40 \times 10^{-13}$ | $0.027$ | $0.041^{\dagger}$ | $13.83$ | $9.16^{\dagger}$ | $1.07 \times 10^5$ | $0.086$ |
| **Pallas** | $2.04 \times 10^{20}$ | $0.015$ | $134.5$ | $1.332$ | $1.13 \times 10^{-12}$ | $0.057$ | $0.062^{\dagger}$ | $7.81$ | $7.25^{\dagger}$ | $1.13 \times 10^5$ | $0.120$ |
| **Vesta** | $2.59 \times 10^{20}$ | $0.020$ | $145.7$ | $1.262$ | $1.32 \times 10^{-12}$ | $0.086$ | — | $5.34$ | — | $1.37 \times 10^5$ | $0.157$ |
| **Ceres** | $9.38 \times 10^{20}$ | $0.033$ | $223.7$ | $1.511$ | $3.11 \times 10^{-12}$ | $0.090$ | — | $9.07$ | — | $7.31 \times 10^4$ | $0.129$ |
| **Quaoar** | $1.20 \times 10^{21}$ | $0.002$ | $242.9$ | $1.565$ | $3.67 \times 10^{-12}$ | $0.107$ | $0.095^{\dagger}$ | $8.84$ | $10.01^{\dagger}$ | $7.63 \times 10^4$ | $0.146$ |
| **Haumea** | $4.01 \times 10^{21}$ | $0.004$ | $363.0$ | $1.504$ | $8.20 \times 10^{-12}$ | $0.348$ | — | $3.92$ | — | $1.06 \times 10^5$ | $0.305$ |
| **Mars** | $6.42 \times 10^{23}$ | $1.557$ | $1{,}971.2$ | $1.261$ | $2.42 \times 10^{-10}$ | $0.241$ | $0.099$ | $24.62$ | $60.02$ | $2{,}094.6$ | $0.033$ |
| **Earth** | $5.97 \times 10^{24}$ | $7.237$ | $4{,}146.4$ | $1.017$ | $1.07 \times 10^{-9}$ | $0.465$ | $0.437$ | $23.93$ | $25.48$ | $737.8$ | $0.024$ |
| **Earth+Moon** | $6.05 \times 10^{24}$ | $7.282$ | $4{,}163.3$ | $1.013$ | $1.08 \times 10^{-9}$ | $2.319^*$ | $0.441$ | $4.80^*$ | $25.27$ | $3{,}634.3$ | $0.119$ |
| **Uranus** | $8.68 \times 10^{25}$ | $1.436$ | $10{,}119.3$ | $1.137$ | $6.37 \times 10^{-9}$ | $2.586$ | $2.603$ | $17.25$ | $17.14$ | $769.6$ | $0.061$ |
| **Neptune** | $1.02 \times 10^{26}$ | $0.995$ | $10{,}692.4$ | $1.112$ | $7.11 \times 10^{-9}$ | $2.683$ | $2.906$ | $16.11$ | $14.87$ | $699.4$ | $0.059$ |
| **Saturn** | $5.68 \times 10^{26}$ | $7.368$ | $18{,}930.3$ | $1.389$ | $2.23 \times 10^{-8}$ | $9.871$ | $9.108$ | $10.66$ | $11.55$ | $1{,}025.5$ | $0.153$ |
| **Jupiter** | $1.90 \times 10^{27}$ | $24.791$ | $28{,}296.7$ | $1.334$ | $4.98 \times 10^{-8}$ | $12.572$ | $12.954^{\dagger}$ | $9.93$ | $9.64^{\dagger}$ | $561.4$ | $0.125$ |
| **SIMP J0136** | $2.41 \times 10^{28}$ | — | $66{,}019.2$ | $0.621$ | $2.71 \times 10^{-7}$ | $63.060$ | $64.890^{\dagger}$ | $2.41$ | $2.35^{\dagger}$ | $240.8$ | $0.125$ |
| **2M J0348$-$60** | $8.16 \times 10^{28}$ | — | $99{,}134.6$ | $0.269$ | $6.11 \times 10^{-7}$ | $105.136$ | $96.842^{\ddagger}$ | $1.08$ | $1.17^{\ddagger}$ | $77.2$ | $0.140^{\ddagger}$ |
| **2M J1219$+$31** | $9.30 \times 10^{28}$ | — | $103{,}546.3$ | $0.275$ | $6.67 \times 10^{-7}$ | $106.170$ | $103.228^{\ddagger}$ | $1.14$ | $1.17^{\ddagger}$ | $72.9$ | $0.133^{\ddagger}$ |
| **2M J0407$+$15** | $1.27 \times 10^{29}$ | — | $114{,}928.4$ | $0.207$ | $8.22 \times 10^{-7}$ | $82.170$ | $86.200^{\ddagger}$ | $1.23$ | $1.17^{\ddagger}$ | $34.5$ | $0.123^{\ddagger}$ |
| *Mercury (braked)* | $3.30 \times 10^{23}$ | $4.396$ | $1{,}579.5$ | $1.069$ | $1.55 \times 10^{-10}$ | $0.003$ | $0.063$ | $1407.6$ | $67.16$ | $34.7$ | $0.0004$ |
| *Venus (braked)* | $4.87 \times 10^{24}$ | $9.033$ | $3{,}873.1$ | $1.031$ | $9.33 \times 10^{-10}$ | $0.002$ | $0.381$ | $5832.6$ | $27.70$ | $3.3$ | $0.0001$ |

*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). <sup>*</sup>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 ($\chi_{\rm obs} = 737.8$, $v_{\rm obs}/v_{\rm TEP} = 1.064$). <sup>&dagger;</sup>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$^{-1}$). <sup>&Dagger;</sup>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.

| Saturn rotation clock | Period (hr) | Radius convention | $v_{\rm eq}$ (km/s) | $v_{\rm obs}/v_{\rm TEP}$ | $\rho_T^{\rm spin}$ (g/cm³) | Source |
| --- | --- | --- | --- | --- | --- | --- |
| C-ring kronoseismology (deep interior) | $10.5606$ | $R_{\rm vol}$ | $9.624$ | $1.057$ | $23.60$ | Mankovich et al. 2019 |
| Atmospheric potential vorticity (System IIIw) | $10.5703$ | $R_{\rm vol}$ | $9.615$ | $1.056$ | $23.53$ | Read et al. 2009 |
| Voyager System III (IAU SLS) | $10.6567$ | $R_{\rm vol}$ | $9.537$ | $1.047$ | $22.96$ | Desch & Kaiser 1981 |
| Voyager System III (IAU SLS) | $10.6567$ | $R_{\rm eq}$ | $9.871$ | $1.084$ | $25.46$ | Davies et al. 1995 |
| Cassini SKR fast component | $10.60$ | $R_{\rm vol}$ | $9.588$ | $1.053$ | $23.33$ | Kurth et al. 2008 |
| Cassini SKR slow component | $10.80$ | $R_{\rm vol}$ | $9.411$ | $1.033$ | $22.06$ | Gurnett et al. 2009 |
| Ulysses SKR (2003) | $10.7625$ | $R_{\rm vol}$ | $9.443$ | $1.037$ | $22.29$ | Gurnett et al. 2005 |

*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.

![Planetary Rotational Kinematics and Angular Momentum Scaling](results/figures/figure_9_planetary_rotation.png)

Figure 6: Planetary Rotational Kinematics Across Mass Scales (Step 15). (a) Observed equatorial rotation speeds $v_{\rm eq}$ versus the fixed-input TEP ambient relation $v_{\rm eq}^{\rm TEP} = c\,u_T(M)/\sqrt{u_{\rm gal}} = GM/(R_T v_{\rm gal})$ (solid line, using GNSS-consistent $\rho_T = 20$ g/cm³ and Solar-Circle $v_{\rm gal} = 220$ km s$^{-1}$), the interior-structure band $2k_{\rm MOI}/S \in [1.0, 1.4]$ (shaded), and the two-branch $+\;S_A$ saturation model ($v_{\rm obs}/v_{\rm 2br} = 1.008 \pm 0.060$ across all $N = 9$ major planetary and substellar speed-limit systems from Earth to $67\,M_{\rm Jup}$, orange diamonds: Earth, Uranus, Neptune, and Saturn on the unsaturated ambient branch at $1.016 \pm 0.073$; Jupiter at $1.0\,M_{\rm Jup}$ and SIMP J0136 at $12.7\,M_{\rm Jup}$ on the unscreened $S < 1$ soliton capacity branch within $-2.9\%$ and $-2.8\%$; and the three Tannock et al. 2021 speed-limit brown dwarfs at $43\text{--}67\,M_{\rm Jup}$ on the $S \ge 1$ saturated period floor $P_{\min}^{\rm TEP} = (k_{\rm MOI}/\eta_0)\tau_T = 1.172\text{ hr}$ vs. observed $1.150 \pm 0.075\text{ hr}$, crimson triangles). (b) Specific angular momentum $j = J/M$ across nearly ten orders of magnitude in mass ($8.7 \times 10^{19}$ to $1.3 \times 10^{29}$ kg), showing the three-regime structure: the $j_{\rm sol} = \eta_0 R_T(M)^2\omega_T \propto M^{2/3}$ soliton branch across six minor/dwarf planets (Hygiea, Pallas, Vesta, Ceres, Quaoar, Haumea), Earth+Moon, Jupiter, and SIMP J0136; the ambient Galactic Wesson branch $j_{\rm TEP} = GM/(2v_{\rm gal}) \propto M$ for intermediate-mass planets ($M < M_{\rm cap} = 194\,M_\oplus$); and the horizontal $S_A = \min[1,\,S] = 1$ saturation plateau ($P_{\min} = 1.17\text{ hr}$) at $\bar\rho \ge \rho_T$ ($S \ge 1$, $M \gtrsim 16\,M_{\rm Jup}$).

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).

![Standard Solar Model TEP Saturation Zones and Helioseismic Differential Rotation](results/figures/figure_11_solar_helioseismology.png)

Figure 7: Standard Solar Model (Model S) TEP Saturation Zones and Helioseismic Rotation Prototype (Step 15). (a) Normalized clock-amplitude saturation profiles $S_A^{\rm loc}(r) = \min[1,\,(\rho_\odot(r)/\rho_T)^{1/3}]$ (dotted), $S_A^{\rm enc}(<r) = \min[1,\,(\bar\rho_\odot(<r)/\rho_T)^{1/3}]$ (dashed), and the solved canonical-kinetic BVP field $u(r)/u_{\min}(\rho_T)$ (solid purple) across the Christensen-Dalsgaard et al. (1996) Model S solar interior at fixed $\rho_T = 20$ g/cm³. The local density crosses $\rho_T$ at $r_{\rm core} = 0.249\,R_\odot$ (near the nuclear-burning core boundary), the 1-D spherical saturation radius sits at $R_T(M_\odot) = 0.413\,R_\odot$, and the candidate 3-D boundary $\sqrt{3}\,R_T(M_\odot) = 0.715\,R_\odot$ coincides with the helioseismic base of the convection zone ($r_{\rm cz} = 0.713 \pm 0.001\,R_\odot$, $+0.25\%$). (b) Internal solar rotation rate $\Omega(r,\theta)/2\pi$ across latitudes $0^\circ, 30^\circ, 45^\circ, 60^\circ,$ and $75^\circ$, comparing SOHO/MDI and GONG helioseismic inversions (dotted; Schou et al. 1998; Charbonneau et al. 1999) against the exploratory TEP prototype (solid: equatorial rate $\Omega_{\rm eq}/2\pi = c\,u_{\rm surf}/(2R_\odot) = 456.5$ nHz vs. $460.7$ nHz, fractional shear $\Delta_\Omega = S_\odot/\sqrt{2} = 0.2918$ vs. $0.2887$ with equal harmonic split $a_2 = a_4 = \tfrac{1}{2}\Delta_\Omega = 0.1459$, prolate tachocline center $r_c(\psi) = 0.688\text{--}0.708\,R_\odot$ vs. $0.693\text{--}0.717\,R_\odot$, and radiative-zone rate at the $P_2(\cos\theta_*) = 0$ node $426.9$ nHz vs. $432.0$ nHz).

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.

#### Box 4.1: Summary — Four Cross-Regime Consistency Tests

The screening analysis provides four cross-regime consistency tests of
the TEP framework under a single $\rho_T$:

**The RBH-1 Crossover:** The saturation radius is
consistent with the Schwarzschild radius for RBH-1 ($M \approx 2 \times 10^7
M_{\odot}$, near the exact crossover mass $M_\times = \sqrt{3c^6/(32\pi G^3\rho_T)} \approx 3.04 \times 10^7 M_{\odot}$) within combined uncertainties ($R_T/R_S \approx 1.3$). This
provides a tightly constrained cross-regime consistency check under
the fixed $\rho_T$ calibration.

**Deterministic Regime Map & Multi-Channel Coincidence Null Test:** Evaluating $S \equiv R_T(M)/R_{\rm phys} \equiv (\bar\rho/\rho_T)^{1/3}$ defines a deterministic algebraic regime map across 15 orders of magnitude in density (true by definition up to table rounding, $\beta_{\rm scr} = 0.3338 \pm 0.0008$; Appendix B.4), while the five independent $\rho_T$-calibrating observables (GNSS $L_c$, outer-three giant-planet spin, brown-dwarf period floor, self-compressed terrestrial $k_{\rm MOI}$, and solar $r_{\rm cz}$) converge to $\hat\rho_T = 19.59 \pm 1.10\text{ g/cm}^3$ ($\sigma_{\log_{10}\rho} = 0.025\text{ dex}$), rejecting random-density coincidence at $p = 1.44 \times 10^{-7}$ ($p = 2.9 \times 10^{-4}$, $3.6\sigma$ even after granting $7^3 = 343$ look-elsewhere algebraic-factor trials).

**GR Test Consistency:** All five precision tests of
General Relativity are consistent with screening factors that suppress
scalar contributions below current measurement limits.

**Planetary Spin, Substellar Period Floor, Wesson Constant, and Solar Prototype:** Combining $\rho_T = 20$ g/cm³ with the Galactic temporal depth $u_{\rm gal} = (v_{\rm gal}/c)^2$ yields the zero-parameter ambient equatorial speed relation ($v_{\rm eq} = c\,u_T/\sqrt{u_{\rm gal}}$, matching the three unbraked outer giant planets Uranus, Neptune, and Saturn at $1.000 \pm 0.080$, $\hat\rho_T^{\rm spin} = 19.9$ g/cm³; $1.016 \pm 0.073$ and $20.9$ g/cm³ with lunar-braked Earth, with residuals tracking $r_*(M)/a$ at $r = +0.960$) and the zero-parameter Wesson constant $p_{\rm TEP} = G/(2v_{\rm gal}) = 1.52 \times 10^{-15}$ cm$^2$ g$^{-1}$ s$^{-1}$, while the one-parameter ($\eta_0 = 0.129$, $\eta_{\rm IQR} \in [0.122,\,0.154]$, candidate benchmark $\eta_{\rm sec} = 1/8$) two-branch $+\;S_A$ extension matches the $N = 8$ unbraked outer-planet and substellar speed-limit systems to $67\,M_{\rm Jup}$ at $v_{\rm obs}/v_{\rm 2br} = 1.001 \pm 0.060$ ($\hat\rho_T^{\rm rot} = 20.0$ g/cm³; $1.008 \pm 0.060$ across $N = 9$ with Earth, including SIMP J0136 and the $1.172$ hr [$0.98\text{--}1.24$ hr] brown-dwarf period floor) alongside self-compressed planetary moments of inertia ($2k_{\rm MOI}/S = 1.039 \pm 0.027$ across the three 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 after Lane–Emden envelope de-inflation) and the exploratory Standard Solar Model helioseismic prototype (nuclear core $\rho = \rho_T$ 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], equatorial rate $456.5$ vs. $460.7$ nHz, and differential shear $S_\odot/\sqrt{2} = 0.292$ vs. $0.289$).

The TEP framework does not violate established physics. It extends the
metric structure to include a conformal sector that becomes observable
only in specific density regimes—such as those probed by GNSS atomic
clocks and cosmological soliton-scale systems.

## 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:

\begin{equation} \label{eq:rho_t_constant} \rho_T = \frac{M}{\frac{4}{3}\pi R_T^3} = \text{constant}. \end{equation}

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

\begin{equation} \label{eq:rt_mass} R_T(M) = \left(\frac{3M}{4\pi \rho_T}\right)^{1/3}. \end{equation}

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:

\begin{equation} \label{eq:lc_earth} L_{c,\oplus} = \left(\frac{3 M_\oplus}{4\pi \rho_T}\right)^{1/3} \approx
4200 \text{ km} \quad \text{for } \rho_T \approx 19\text{–}20 \text{
g/cm}^3. \end{equation}

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,

\begin{equation} \label{eq:vacuum_scale} \rho_T \approx \alpha^{4/3} m_e^4 \quad \text{(a density coincidence, not a potential-scale identification)}. \end{equation}

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:

\begin{equation} \label{eq:condensed_density_corrected} \rho_{\rm TF} \approx \frac{3}{4\pi} A Z \, m_p m_e^3 \alpha^3. \end{equation}

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.

| Density Regime | Dominant Physics | Examples |
| --- | --- | --- |
| \(\rho < 1\) g/cm³ | Gas pressure, thermal | Planets, main-sequence stars |
| \(\rho \sim 1\text{–}10\) g/cm³ | Coulomb/thermal dominated solids and liquids | Rocky planets |
| \(\rho \sim 10\text{–}10^2\) g/cm³ | Degeneracy onset (crossover) | Earth's core, brown dwarfs |
| \(\rho \gtrsim 10^4\) g/cm³ | Electron degeneracy dominated | White dwarfs |

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*<sub>T</sub> 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.

\begin{equation} \label{eq:alpha_density} \rho_T \approx \alpha^{4/3}\,\frac{m_e^4 c^3}{\hbar^3} \approx 22.4 \text{ g/cm}^3. \end{equation}

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:

**Condensed-matter EOS cross-check:** \(\rho_T\) lies at
the pressure-ionization and Thomas–Fermi electron-degeneracy onset scale (\(10\text{--}30\text{ g/cm}^3\))

**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)

**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.

#### Box 5.1: Summary — The Saturation Scale: Multi-Channel Anchor, Quartic QED Projection, and PREM BVP

Across five independent astrophysical and geodetic channels, the saturation density converges to \(\hat\rho_T = 19.59 \pm 1.10\text{ g/cm}^3\) (Step 15), consistent with two atomic/condensed-matter links and the PREM terrestrial BVP solution (Step 14):

- Condensed-matter EOS cross-check (Thomas–Fermi electron degeneracy onset at \(10\text{--}30\text{ g/cm}^3\))

- Quartic-projected QED scale \(\rho_T^{\rm EM} = (\alpha_{\rm EM}^{1/(m-1)} m_e)^m c^3/\hbar^3 = \alpha_{\rm EM}^{4/3} m_e^4 c^3/\hbar^3 = 22.39\text{ g/cm}^3\) (\(m = 4 \implies m/(m-1) = 4/3\))

In the present phenomenological calibration, the terrestrial coherence length
is treated as the observed projected covariance realization of the Earth-scale
\(R_T(M_\oplus)\) geometry (supported on the canonical-kinetic benchmark by the 4-layer PREM BVP and 3-D Yukawa spatial-kernel solve \(L_c^{\rm PREM} = 4{,}010\) km, \(L_c^{\rm PREM}/R_T(M_\oplus) = 0.967\), Step 14, confirming an order-unity spatial transfer prefactor $L_c^{\rm PREM}/R_T(M_\oplus) = 0.967$). Both the condensed-matter cross-check and
the quartic QED projection depend on this $\mathcal{O}(1)$
prefactor lying near unity; a prefactor of \(\sim\)2 would shift \(\rho_T\) to
\(\sim\)2.5 g/cm³, evaporating the coincidences. Operationally, GNSS observations
provide the calibration by which \(\rho_T\) is inferred in the most accessible
laboratory: Earth's gravitational field.

## 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:

**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.

**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:

\begin{equation} \label{eq:universal_scaling} R_T(M) = \left( \frac{3M}{4\pi \rho_T} \right)^{1/3}. \end{equation}

#### Box 6.1: The Theoretical Priority of $\rho_T$

A potential misreading of this framework is that "GPS clock noise
determines black hole physics." This inverts the logical and historical
order of the analysis.

**Calibration (Terrestrial):** The GNSS analysis
(Smawfield 2025b) serves as an empirical anchor. The
detection of a correlation length $L_c \approx 4200$ km in atomic
clock data provides an empirical estimate of the saturation scale, conditional on treating $L_c$ as the projected covariance scale associated with $R_T(M_\oplus)$.

**Application (Cosmological):** The resulting $M^{1/3}$
scaling is therefore not an extrapolation of clock noise, but a test
of whether this terrestrially-validated constant
remains universal at galactic scales.

The alignment of the RBH-1 crossover scale is thus an independent
consistency check, not a fitted result.

#### Box 6.2: Null Hypothesis and Multi-Channel Look-Elsewhere Test

A potential concern is that apparent cross-scale correspondences could
arise from post-hoc alignment or selection of geometric projection factors rather than predictive structure. To
address this possibility, two complementary null tests are evaluated:

**(1) Two-Anchor Coarse Overlap (GNSS + RBH-1 over 9 dex):** Under a coarse uniform logarithmic prior spanning $10^{-3}$–$10^{6}$ g/cm³ ($9$ dex), requiring $R_T(M_\oplus)$ within $50\%$ of $4{,}200$ km ($\rho_T \in [6,\,160]$ g/cm³, $1.4$ dex) and $R_T(2 \times 10^7\,M_\odot)$ within $50\%$ of $R_S$ ($\rho_T \in [14,\,370]$ g/cm³, $1.4$ dex; overlap $14$–$150$ g/cm³, $1.05$ dex) gives a single-draw overlap probability of $P \approx 1.05/9 \approx 0.12$ ($12\%$), which by itself is only suggestive.

**(2) Five-Channel 100,000-Trial Monte Carlo Look-Elsewhere Null Test (Section 4.7, Step 15):** Across a five-decade log-uniform prior $\rho_{\rm rand} \in [0.1,\,10^4]$ g/cm³, the five quantitative $\rho_T$-calibrating observables — (i) terrestrial GNSS $L_c = 4{,}200$ km ($\hat\rho_T = 19.24$ g/cm³), (ii) outer-three unbraked giant-planet spin across Uranus, Neptune, and Saturn ($19.89$ g/cm³), (iii) the Tannock et al. (2021) brown-dwarf rotational period floor $P_{\min} = 1.150$ hr ($20.79$ g/cm³, orthogonal $\rho^{-1/2}$ exponent), (iv) self-compressed terrestrial moments of inertia $2k_{\rm MOI}/S$ ($17.89$ g/cm³), and (v) the Solar Model S convection-zone base $r_{\rm cz} = 0.713\,R_\odot$ ($20.15$ g/cm³) — converge to $\hat\rho_T = 19.59 \pm 1.10$ g/cm³ ($\sigma_{\log_{10}\rho} = 0.0249$ dex, span $0.0652$ dex, RMS error $2.04\%$). Restricting to the three unit-prefactor ($\mathcal{O}(1) = 1$) channels that invoke no auxiliary geometric factors ($\sqrt{2},\,\sqrt{3},\,\pi$) or asteroid-fitted parameters — (1) terrestrial GNSS $L_c = R_T(M_\oplus)$ ($19.24$ g/cm³, PREM BVP transfer ratio $0.967$), (2) outer-three unbraked giant-planet spin $v_{\rm eq} = GM/(R_T v_{\rm gal})$ selected by the Goldreich–Soter tidal filter $\tau_{\rm tide}/t_\odot > 10$ ($19.89$ g/cm³), and (3) the TEP-saturated brown-dwarf period floor $P_{\min}^{\rm th} = (k_{\rm MOI}/\eta)\tau_T(\rho_T) = 1.155$ hr fixed by the classical Chandrasekhar–Jacobi / virial stability bracket $\eta \in [0.125,\,0.1375]$ at $\tau_T(\rho_T=20) = 0.7381$ hr ($20.19$ g/cm³; whereas purely classical Newtonian Jacobi instability without TEP saturation evaluates at $\bar\rho_{\rm BD} = 67\text{--}156$ g/cm³ and predicts $P_{\min}^{\rm Classical,Jacobi} = 0.39\text{--}0.60$ hr [mean $0.527$ hr], sitting $8.31\sigma$ below the observed $1.150 \pm 0.075$ hr floor) — yields a cross-regime algebraic consistency window of $\hat\rho_T^{\rm 3ch} = 19.77 \pm 0.48$ g/cm³ ($\pm 2.4\%$ SD at $v_{\rm gal} = 220$ km/s, and $\hat\rho_T^{\rm 3ch,marg} = 20.53 \pm 1.49$ g/cm³ [$\pm 7.3\%$] when $v_{\rm gal}$ is marginalized over $[220,\,236]$ km/s; random-draw overlap probability $p = 5.2 \times 10^{-5}$ across $[0.1,\,10^4]$ g/cm³). Across all five channels, the raw coincidence probability is $p = 1.44 \times 10^{-7}$, remaining $p = 2.9 \times 10^{-4}$ ($3.6\sigma$, $29/100{,}000$ trials; $p = 1.0 \times 10^{-3}$ across the four non-solar channels) even after granting the null model a $7^3 = 343$-combination look-elsewhere dictionary of algebraic factors $\mathcal{F} = \{\tfrac{1}{2},\,1/\sqrt{2},\,1,\,\sqrt{2},\,\sqrt{3},\,2,\,\pi\}$ within a $\pm 10\%$ window.

Under the specified independent log-uniform coincidence null and the tested algebraic-factor dictionary, the observed five-channel clustering has a simulated tail probability of $p = 2.9 \times 10^{-4}$ (this quantifies coincidence sensitivity within that null and does not incorporate the full history of observable selection, transfer-law construction, or astrophysical model uncertainty). Furthermore, evaluating a **Leave-One-Channel-Out (LOCO) cross-scale cross-validation test** under the fixed transfer dictionary (and propagating measurement plus $v_{\rm gal} \in [220,\,236]$ km/s uncertainties) — in which $\hat\rho_{T,-i}$ is calibrated on four channels and used to predict the held-out fifth observable out of sample — yields blind observable prediction errors of $-0.49\%$ on terrestrial GNSS $L_c$ ($\hat\rho_{T,-\text{GNSS}} = 19.53$ g/cm³ $\implies L_c^{\rm blind} = 4{,}179$ km vs. $4{,}200$ km), $-0.88\%$ on outer-three giant-planet equatorial velocity $v_{\rm eq}$, $+2.28\%$ on the Tannock et al. brown-dwarf period floor ($P_{\min}^{\rm blind} = 1.176$ hr vs. $1.150$ hr), $+3.55\%$ on compressed terrestrial $2k_{\rm MOI}/S$, and $+1.45\%$ on the Solar Model S convection-zone base ($r_{\rm cz}^{\rm blind} = 0.723\,R_\odot$ vs. $0.713\,R_\odot$), achieving a 5-channel LOCO cross-validation observable RMS error of $2.05\%$ (mean absolute error $1.73\%$; $3.02\%$ RMS when marginalized across $v_{\rm gal} \in [220,\,236]$ km/s) and a 3-channel zero-prefactor LOCO RMS error of $1.20\%$ ($-1.34\%, -0.29\%, +1.57\%$; $3.04\%$ across $v_{\rm gal} \in [220,\,236]$ km/s). Provenance scope: while the terrestrial GNSS anchor is established across Papers 1–3 and 14 (Smawfield 2025b,c,d), all four astrophysical and planetary calibrations are computed directly from external peer-reviewed catalogs (NASA/JPL & HST planetary ephemerides, Lamy et al. 2025; Tannock et al. 2021; Dziewonski & Anderson 1981; Christensen-Dalsgaard et al. 1996; Lelli et al. 2016; van Dokkum et al. 2025).

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.

#### Box 6.3: Robustness of the GNSS Calibration (Systematics Check)

The terrestrial calibration length $L_c \approx 4200$ km — the GNSS
instrument's measured realization of the canonical correlation scale
$\lambda_T$ (Paper 0 §7) — is an
extraordinary claim requiring careful exclusion of geodetic
systematics. The following tests support its physical origin (full
methodology in Smawfield 2025b,c,d):

**Multi-Center Verification:** The correlation
structure persists across independent clock solutions from CODE,
ESA, and IGS (1999–2024), disfavoring software-specific
processing artifacts (e.g., Bernese vs. GIPSY). The recovered $L_c$
spans $\sim$3,330–4,549 km across centers (CV $18.2\%$, $\sim \pm 15\%$ half-range).

**Null Tests:** (i) Randomizing satellite epochs
destroys the correlation ($r^2 < 0.01$, $N = 1000$ shuffles); (ii)
shuffling clock residuals within each satellite eliminates the
spatial structure; (iii) replacing real data with white noise yields
no coherent scale. All three nulls are satisfied at $>5\sigma$.

**Scale Separation:** The 4200 km scale is distinct
from tropospheric correlation lengths (~100–500 km) and orbital
period harmonics (half-sidereal, ~12 hr). 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.

**Dataset:** IGS final clock products (CLK files),
30-second sampling, 1999–2024. Preprocessing: removal of
satellite/receiver clock offsets, relativistic corrections (Sagnac,
gravitational redshift), and reference frame alignment to ITRF2020.

*Note on Transfer Closure:* In Step 14 (Section 2.3; Figure 1; Appendix B.2), integrating the 4-layer PREM Earth BVP at the Cassini-compatible normalization $\lambda = 7.53 \times 10^{-66}$ and $\rho_T = 20$ g/cm³ derives the transition-shell projection $L_c^{\rm PREM,shell} = 4{,}010.2$ km ($L_c^{\rm shell}/R_T(M_\oplus) = 0.9672$) and the full-volume datum-removed projection $L_c^{\rm PREM,vol} = 3{,}213.2$ km ($L_c^{\rm vol}/R_T(M_\oplus) = 0.7749$), and solving the coupled 1-D spherical nonlinear two-branch kinetic operator Eq. ($\ref{eq:nonlinear_chameleon_eom}$) via 100-iteration damped Picard relaxation ($\|\Delta u^{(100)}\|_\infty/\|u\|_\infty = 2.67 \times 10^{-14}$; Appendix C.2) 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, leaving only non-spherical 3-D azimuthal orbital-motion anisotropy for future work.

#### Box 6.4: Geometric Origin of the Saturation-Radius Law

For a universal macroscopic proximity-density parameter $\rho_T$, the characteristic source-mass saturation radius $R_T(M)$ is defined geometrically as the radius enclosing mass $M$ at reference density $\rho_T$:

\begin{equation} \label{eq:mass_conservation} M = \frac{4}{3}\pi R_T^3 \rho_T. \end{equation}

Solving for the radius yields the characteristic $M^{1/3}$ scaling:

\begin{equation} \label{eq:scaling_law} R_T(M) = \left(\frac{3M}{4\pi\rho_T}\right)^{1/3} \propto \left(\frac{M}{\rho_T}\right)^{1/3}. \end{equation}

As shown in Box 6.5 and Appendix C, $\rho_T$ is the matter-density reference scale at which the clock-amplitude response $S_A(\bar\rho) = \min[1,\,(\bar\rho/\rho_T)^{1/3}]$ saturates and the interior field becomes Compton-resolved ($\lambda_c(\bar\rho) \lesssim R_T(M)$), rather than a literal energy density of the scalar field itself ($\rho_{\rm eff} \ll \rho$). Demonstrating that $R_T(M)$ corresponds to the spatial transition scale of a finite, stratified body requires solving the boundary-value problem in the source's density profile (executed on the canonical-kinetic benchmark for the 4-layer PREM Earth in Step 14, yielding $R_{\rm grad} = 1.18\,R_T(M_\oplus)$ and $L_c^{\rm PREM} = 4{,}010$ km).

#### Box 6.5: Soliton Formation from the Canonical Action

To move beyond phenomenology, a candidate route from the canonical
Temporal Topology action (Paper 0) to soliton-like saturation is sketched. The action
above contains a canonical kinetic term $-\frac{1}{2}(\partial\phi)^2$
and a candidate saturation potential $V(\phi)$ that would prevent gradient divergence.
The interplay between these terms can generate soliton-like saturated configurations with
characteristic radius $R_T \propto M^{1/3}$.

**1. Field Equation and Equilibrium**

For a static, spherically symmetric configuration on the canonical-kinetic equilibrium benchmark ($P_X \to 1$; see Appendix C.2, Eq. $\ref{eq:nonlinear_chameleon_eom}$ for the operative nonlinear kinetic operator $\nabla\cdot(P_X\nabla u) = M_{\rm Pl}^{-2}V_{\rm eff}'(u)$), the scalar field
equation is:

\begin{equation} \label{eq:field_equation} \nabla^2 \phi = V'(\phi) + \alpha(\phi)\rho \end{equation}

where $\alpha(\phi) \equiv d\ln A/d\phi$ and $\rho$ is the ambient
matter density. In the dense limit, the effective potential
$V_{\rm eff}(\phi; \rho) = V(\phi) + [A(\phi)-1]\rho$ develops a
minimum at $\phi_{\rm min}(\rho)$ with effective mass
$m_{\rm eff}^2(\rho) = V_{\rm eff}''(\phi_{\rm min})$.

**2. Saturation Mechanism**

Under the corpus coupling $\beta_A = -1$ the matter term enters with
negative slope ($\alpha\rho = -\rho/M_{\rm Pl}$), so a minimum of
$V_{\rm eff}$ exists only on the confining branch $V' > 0$. Writing
$u = \phi/M_{\rm Pl}$, the corpus's own weak-field branch is the
quartic $V(u) = (\lambda/4)\,M_{\rm Pl}^4\,u^4$ — the member of the
confining family $V \propto u^m$ selected by the corpus amplitude
law $S_A \propto \rho^{1/3} = \rho^{1/(m-1)}$, i.e. $m = 4$
(Appendix C.2). The equilibrium condition $V_{\rm eff}' = 0$ reads
$\lambda M_{\rm Pl}^4 u_{\rm min}^{3} = \rho\,e^{-u_{\rm min}}$, giving
$u_{\rm min} \approx (\rho/\lambda M_{\rm Pl}^4)^{1/3}$ for
$u_{\rm min} \ll 1$: the field is elevated inside matter and
approaches zero in the ambient, in the direction required by the
corpus sign convention. (The inverse-power branch
$V \propto \phi^{-n}$ is inadmissible under $\beta_A = -1$: both
slopes are then negative and no minimum exists; its implied
$\phi_{\rm min} \propto \rho^{-1/(n+1)}$ would in any case run
opposite to the required direction.) The scalar energy density in a
saturated core is:

\begin{equation} \label{eq:effective_density} \rho_{\rm eff} \sim V(u_{\rm min}) + \frac{1}{2}M_{\rm Pl}^2(\nabla u)^2
\simeq \tfrac{1}{4}\rho\,u_{\rm min} \ll \rho. \end{equation}

The saturated configuration is therefore a field-geometry statement,
not a competing mass density: at the corpus normalization
$\lambda = 7.5\times10^{-66}$ (the scalar self-coupling, Appendix C)
the core energy density is
$\sim 3\times10^{-8}$ g/cm³ at terrestrial densities, and the
equilibrium $u_{\rm min}(\rho_T) \approx 6.9\times10^{-9}$ remains
perturbative. What saturates at $\rho_T$ is the clock-amplitude
response $S_A = \min[1, u_{\rm min}(\bar\rho)/u_{\rm min}(\rho_T)]$
through the geometric radius $R_T$; the order-unity self-quenching
crossing $u_{\rm min} \sim 1$ (where $e^{-u}$ suppresses the
conformal source) lies at $\rho_{\rm sat} \sim \lambda M_{\rm Pl}^4
\approx 6\times10^{25}$ g/cm³, inside the temporal-well domain of
Paper 28 (step 14).

**3. Temporal Topology Screening**

The effective mass
$m_{\rm eff}^2(\rho) = V_{\rm eff}''(u_{\rm min})/M_{\rm Pl}^2 =
[3\lambda M_{\rm Pl}^4\,u_{\rm min}^{2} + \rho\,e^{-u_{\rm min}}]/M_{\rm Pl}^2$
grows with density as $m_{\rm eff} \propto \lambda^{1/6}\rho^{1/3}$,
so the Compton wavelength $\lambda_C = 1/m_{\rm eff}$ contracts
through denser interiors: a body is Compton-resolved once
$\lambda_c(\bar\rho) \lesssim R_T(M)$, the crossover that for Earth
sits at $\sim 4$ g/cm³ (step 14). Inside such a body the field rests
near its local equilibrium $u_{\rm min}(\rho)$ and the spatial
variation is confined to a boundary layer of width $\sim\lambda_C$,
where the kinetic response $P_{,X}$ controls the gradient amplitude.
This creates the characteristic flattening of
the scalar field spatial profile—Temporal Topology screening—that
reconciles precision local tests with cosmological dynamics.
In the saturation regime ($\rho \sim \rho_T$), the Temporal Topology saturation radius
scales as $R_T = (3M/4\pi\rho_T)^{1/3}$, yielding the
observed $M^{1/3}$ dependence. At still higher densities — where the
equilibrium approaches order unity ($\rho \sim \rho_{\rm sat} \approx
6\times10^{25}$ g/cm³) — the effective source slope
$\rho\,A_{,u} = -\rho\,e^{-u_{\rm min}}$ is exponentially
suppressed, so the scalar charge of deep-interior matter quenches as
well: that is the temporal-well regime, not the terrestrial one.

**4. Coupling and Stability**

Matter couples to the Jordan frame metric $\tilde{g}_{\mu\nu} =
A^2(\phi) g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$ with $A(\phi) = \exp(\beta_A \phi/M_{\text{Pl}})$.
Stability is ensured because the canonical kinetic term has the correct
sign for a physical scalar field, and the theory satisfies the Null
Energy Condition for physically realizing solutions.

### 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:

\begin{equation} \label{eq:rbh_radius} R_T = \left( \frac{3M}{4\pi \rho_T} \right)^{1/3} \approx 7.8
\times 10^7 \text{ km}. \end{equation}

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

Schwarzschild radius: $R_{\rm S} = 2GM/c^2 \approx 5.9
\times 10^7$ km

Predicted Temporal Topology saturation radius: $R_T =
(3M/4\pi\rho_T)^{1/3} \approx 7.8 \times 10^7$ km

Ratio: $R_T/R_{\rm S} \approx 1.3$

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.

![Saturation Radius vs. Schwarzschild Radius Across Mass Scales](results/figures/figure_2_scaling.png)

Figure 8: Saturation Radius vs. Schwarzschild Radius Across Mass Scales. The solid curve shows the
geometric saturation radius $R_T(M) = (3M/4\pi\rho_T)^{1/3} \propto M^{1/3}$ calibrated via
terrestrial GNSS clock data ($L_c \approx 4{,}200$ km, $R_T(M_\oplus) \approx 4{,}146$ km at $\rho_T = 20$ g/cm³), while the dashed line shows the General Relativistic Schwarzschild radius $R_{\rm S}(M) = 2GM/c^2 \propto M$. The two constructed length scales intersect at the crossover mass $M_\times = \sqrt{3c^6/(32\pi G^3\rho_T)} \approx 3.04 \times 10^7\,M_\odot$, separating the $R_T > R_{\rm S}$ saturation-scale regime from the $R_{\rm S} > R_T$ Schwarzschild-scale regime; RBH-1 is plotted at its independently inferred mass $M \approx 2 \times 10^7\,M_\odot$ ($R_T \approx 7.80 \times 10^7$ km vs. $R_{\rm S} \approx 5.91 \times 10^7$ km; van Dokkum et al. 2025), directly adjacent to the crossover.

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.

\begin{equation} \label{eq:no_gnss_rho} \hat\rho_T(\text{External Astrophysical, no GNSS}) = 19.68 \pm 1.25\text{ g/cm}^3 \quad (\pm 6.4\%\text{ SD},\; \sigma_{\log_{10}\rho} = 0.028\text{ dex}). \end{equation}

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 ($M_\oplus$):

\begin{equation} \label{eq:gnss_rho} \hat\rho_T(\text{GNSS}) = \frac{3 M_\oplus}{4\pi L_c^3} = 19.24\text{ g/cm}^3 \quad (20 \pm 7\text{ g/cm}^3\text{ inter-center}). \end{equation}

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 $R_T/R_S = 1.32$:

\begin{equation} \label{eq:rbh_rho} \rho_S(M_{\text{RBH-1}}) = \frac{3 c^6}{32\pi G^3 M_{\text{RBH-1}}^2} = 46.1\text{ g/cm}^3 \quad (M_\times = 3.04 \times 10^7\,M_\odot\text{ at } \rho_S = 20.0\text{ g/cm}^3), \end{equation}

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.

#### Conclusion on Circularity and Anchor Independence

The convergence on $\rho_T \approx 20$ g/cm³ does not rest on a single unreplicated measurement: dropping the terrestrial GNSS channel entirely leaves $\hat\rho_T^{\rm no\text{-}GNSS} = 19.68 \pm 1.25$ g/cm³ ($\pm 6.4\%$) across four external published planetary, substellar, and helioseismic datasets, which the terrestrial GNSS scale ($19.24$ g/cm³) and the RBH-1 Schwarzschild crossover ($M_\times = 3.04 \times 10^7\,M_\odot$) match without retuning.

## 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:

\begin{equation} \label{eq:milky_way_dm} R_{\text{DM}} = k \cdot M_{\text{bar}}^{1/3} \approx 7.96 \times 10^{-4} \cdot (6 \times 10^{10})^{1/3} \text{ kpc} \approx 3.1 \text{ kpc}. \end{equation}

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

| Parameter | Value | Uncertainty | Impact on $R_{\text{DM}}$ |
| --- | --- | --- | --- |
| **MW Baryonic Mass** | $6.0 \times 10^{10} M_\odot$ | $\pm 15\%$ ($0.9 \times 10^{10}$) | $\pm 5\%$ (scaling $M^{1/3}$) |
| **SPARC Constant ($k$)** | $7.96 \times 10^{-4}$ | $\pm 4\%$ (fit error) | $\pm 4\%$ |
| **Model Systematic** | - | $\pm 10\%$ (geometry) | $\pm 10\%$ |
| Mean Relation $R_{\text{DM}}$ | 3.1 kpc | $\pm 0.4$ kpc ($\sim$14% on mean curve) | - |
| **Single-Galaxy $1\sigma$ Interval** | 3.1 kpc | $[1.3,\,7.2]$ kpc ($\pm 0.366$ dex resolved; $[1.0,\,9.4]$ kpc raw censored, $[2.0,\,4.7]$ kpc enclosed-mass $\pm 0.180$ dex) | Factor $\sim 2.3\times$ ($1.5\times$ enclosed) |
| **Observed Onset** | $\sim$3–5 kpc | $\pm 1$ kpc | - |

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

#### NFW Halo vs. TEP Soliton

- **NFW Halo (CDM):** In standard CDM, the inner rotation curve profile depends on the halo concentration $c$ and virial mass $M_{\text{vir}}$, both of which are free parameters. The radius where baryons cease to dominate is not predicted a priori; it emerges from the halo assembly history and can vary significantly between cosmological simulations.

- **TEP Soliton:** The $M^{1/3}$ scaling yields the onset radius directly from the baryonic mass, with no Milky Way-specific tuning. The normalization is fixed by the SPARC ensemble ($k = 7.96 \times 10^{-4}$ kpc/$M_\odot^{1/3}$) and gives $R_{\text{DM}} \approx 3.1$ kpc for the Milky Way. The agreement with the observed transition at $\sim$3–5 kpc supports the universality of the scaling law.

### 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.

#### Claim Hierarchy and Falsification Scope

The empirical content of this work can be read at three distinct levels.
This separation is critical for interpreting the consequences of future
measurements.

**Level 1 (Empirical regularities):** The existence of
approximate $M^{1/3}$ scaling features in rotation-curve onset
radii, and the density-ordered screening hierarchy summarized by $S
= R_T/R_{\text{phys}}$.

**Level 2 (Universal-density soliton model):** The
hypothesis that a single saturation scale $\rho_T$ organizes these
regularities via $R_T(M) = (3M/4\pi\rho_T)^{1/3}$, with
GNSS providing an empirical calibration (subject to independent
replication) and cross-regime consistency tests (SPARC ensemble,
Milky Way, RBH-1 crossover).

**Level 3 (TEP microphysics):** The full dynamical
realization in which temporal shear and a conformal time field
$A(\phi) = \exp(\beta_A\phi/M_{\text{Pl}})$ generate the effective soliton/screening
phenomenology. Failures at Level 2 primarily constrain the soliton
implementation and/or calibration mapping, without necessarily
excluding the broader TEP framework.

**Three Explicit Numerical Kill Criteria:** (1) *Metrological kill condition:* if full-day single-constellation clock/positioning solutions or external optical/two-way time-transfer links recover a stationary correlation decay length outside $[3{,}000,\,5{,}000]\text{ km}$ ($L_c \lesssim 2{,}000\text{ km}$ before multi-constellation ISB or 8-hour window truncation), the terrestrial $L_c \leftrightarrow R_T(M_\oplus)$ identification at $\rho_T \approx 20\text{ g/cm}^3$ is falsified (in the 4-quarter 2023 CDDIS RINEX3 reprocessing executed in Step 14, full-day $n_{\rm seg}=288$ single-constellation solutions pass this criterion directly with $\sqrt{N}$-weighted $10\text{--}500\,\mu\text{Hz}$ $L_c = 4{,}051\text{ km}$ [GPS], $4{,}693\text{ km}$ [GLONASS], and $2{,}816\text{ km}$ [Galileo; $3{,}653\text{ km}$ in $10\text{--}60\,\mu\text{Hz}$], yielding strictly band-matched three-constellation means of $3{,}853\text{ km}$ [$\sqrt{N}$-weighted $10\text{--}500\,\mu\text{Hz}$], $3{,}423\text{ km}$ [unweighted $10\text{--}500\,\mu\text{Hz}$: $3{,}766, 3{,}806, 2{,}697\text{ km}$], and $3{,}362\text{ km}$ [matched $10\text{--}60\,\mu\text{Hz}$: $2{,}746, 3{,}686, 3{,}653\text{ km}$], all inside $[3{,}000,\,5{,}000]\text{ km}$, whereas applying MGEX's 8-hour $n_{\rm seg}=96$ truncation, per-epoch datum removal, and 4-constellation ISB compresses the same data step-by-step from $3{,}423 \to 2{,}850 \to 2{,}156\text{ km}$ [$\mathrm{CV} = 0.28\%$] $\to 1{,}478\text{ km}$); (2) *Galactic onset kill condition:* if high-resolution rotation-curve surveys partitioned *a priori* at the analytical Freeman slope $\gamma_{\rm Freeman} = 0.210$ fail to show a positive regime shift between rising-core ($\alpha \approx 1/3$) and Keplerian-tail ($\alpha \approx 1/2$) crossings ($\Delta\alpha \equiv \alpha_{\rm tail} - \alpha_{\rm rising} \le 0$), the two-branch onset transition is falsified; and (3) *Compact/substellar saturation kill condition:* if any unbraked substellar rotator ($M > 0.61\,M_{\rm Jup}$) exceeds the soliton breakup ceiling $v_{\rm cap}/v_{\rm breakup} = (\eta_0/k_{\rm MOI})\sqrt{\min(1,S)}$ ($P_{\rm rot} < 0.98\text{ hr}$ in degenerate brown dwarfs) or if revised JWST kinematics move RBH-1's mass outside the crossover bracket $R_T/R_S \in [0.7,\,2.0]$, the universal $\rho_T \approx 20\text{ g/cm}^3$ saturation ceiling is falsified.

### 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*.

| Comparison | Before/after GNSS calibration? | Free parameters? | Independent? | Weight |
| --- | --- | --- | --- | --- |
| **Earth GNSS $L_c$** | Calibration anchor | yes/no (fit to correlation) | no — defines scale | Calibration |
| **SPARC $M^{1/3}$ scaling** | After | Yes (normalization $k$) | Medium | Moderate |
| **DF2/DF4 UDGs (external-field screening)** | After | No ($a_{\rm eff} = 1.20\times 10^{-10}\text{ m s}^{-2}$, $v_\parallel \in [1,\sqrt{3}]c$ fixed) | Medium (independent van Dokkum et al. 2018, 2019 kinematics) | Quantitative EFE check ($\tau_{\rm eq} = 4.1\text{--}7.2\text{ kyr}$; $\sigma_{\rm los}^{\rm EFE} = 8.9\text{ km s}^{-1}$ vs $8.5^{+2.3}_{-3.1}\text{ km s}^{-1}$) |
| **Milky Way inner mass-discrepancy onset** | After | No new normalization; uses SPARC-calibrated $k$ | Medium | Local consistency check |
| **Wide binaries (if tested)** | Before/after | Yes | Stronger (if replicated) | High if replicated |
| **Planetary ambient $v_{\rm eq}$ (4 planets) & Wesson $J/M^2$ (5 planets)** | After (post-hoc) | No ($0$ free parameters; $\rho_T = 20$ g/cm³, $v_{\rm gal} = 220$ km/s fixed) | Medium–High (independent NASA/JPL spin & MOI data; post-hoc subset selection) | Zero-refit dynamical check (outer 3 unbraked: $1.000 \pm 0.080$, $\hat\rho_T^{\rm spin} = 19.9$ g/cm³; 5-channel look-elsewhere null $p = 2.9 \times 10^{-4}$) |
| **Small-body soliton $j_{\rm sol}$ & Jupiter cap ($M > M_{\rm cap}$)** | After (post-hoc) | Yes ($1$ empirical efficiency parameter $\eta_0 = 0.129$ from small-body median) | High out-of-sample (6 minor/dwarf planets; caps Jupiter at $-2.9\%$, SIMP J0136 at $-2.8\%$, and 3 Tannock+ 21 speed-limit BDs at $P_{\min} = 1.17$ vs $1.15$ hr) | 8-system outer + substellar test ($v_{\rm obs}/v_{\rm 2br} = 1.001 \pm 0.060$, $\hat\rho_T^{\rm rot} = 20.0$ g/cm³; $N=9$ with Earth: $1.008 \pm 0.060$) |
| **Solar Model S core, convection base & rotation prototype** | After (post-hoc) | No ($\rho_T = 20$ g/cm³, $|\alpha_0|=\sqrt{2}$, minimal-complexity $a_2=a_4=\tfrac{1}{2}\Delta_\Omega$ benchmark; full 3-D MHD derivation open) | Medium–High (independent Model S, SOHO/GONG, & Kepler solar analog HD 173701) | Exploratory solar/stellar prototype ($r_{\rm cz}: \sqrt{3}R_T=0.715\,R_\odot$ [$\rho_T^{\rm (cz)}=20.15$] & $y_{\rm BVP}=\tfrac{1}{2}S_\odot$ at $0.723\,R_\odot$; $\Delta_\Omega=0.292$ vs $0.289$) |

*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.

#### Calibration Scope and Pre-Registered Discriminators for $\rho_T$

The density scale $\rho_T \approx 20$ g/cm³ is anchored on the terrestrial GNSS correlation length $L_c \approx 4{,}200$ km and tested across held-out mass regimes. Because the $\rho^{1/3}$ scaling is a structural consequence of the $R_T(M) = (3M/4\pi\rho_T)^{1/3}$ construction, discriminating the microscopic mechanism requires a pre-registered test in which the scale is predicted prior to analysis and recovered without refitting.

The per-constellation reprocessing executed in Step 14 on public 2023 CDDIS RINEX3 archives directly tests the multi-GNSS compression discriminator: in the full-day 24-hour window ($n_{\rm seg}=288$), single-constellation GPS ($4{,}051\text{ km}$ $\sqrt{N}$-weighted; $3{,}766\text{ km}$ unweighted), GLONASS ($4{,}693\text{ km}$ $\sqrt{N}$-weighted; $3{,}806\text{ km}$ unweighted), and Galileo ($4{,}557\text{ km}$ in dual-frequency iono-free LC on `optimal_100`; $3{,}653\text{ km}$ in $10\text{--}60\,\mu\text{Hz}$ and $2{,}816\text{ km}$ in single-frequency broadcast $10\text{--}500\,\mu\text{Hz}$) recover strictly band-matched three-constellation means of $3{,}853\text{ km}$ ($\sqrt{N}$-weighted $10\text{--}500\,\mu\text{Hz}$) and $3{,}423\text{ km}$ (unweighted), whereas applying MGEX's 8-hour window truncation, datum removal, and 4-constellation ISB estimation compresses the exact same observations step-by-step through $2{,}850 \to 2{,}156\text{ km}$ ($\mathrm{CV} = 0.28\%$) $\to 1{,}478\text{ km}$. Furthermore, propagating each GNSS pipeline's implied $\rho_T(L_c) = 3M_\oplus/(4\pi L_c^3)$ forward through the four external astrophysical channels (Section 6) yields joint out-of-sample RMS errors of $1.98\%$ (25-yr PPP $4{,}201\text{ km}$, $\rho_T = 19.23\text{ g/cm}^3$) and $3.82\%$ ($\sqrt{N}$-weighted GPS $4{,}051\text{ km}$, $\rho_T = 21.45\text{ g/cm}^3$; $9.28\%$ at the $3{,}853\text{ km}$ 3-constellation mean), while rejecting the unweighted $3{,}423\text{ km}$ ($22.5\%$ RMS error) and truncated $2{,}156\text{ km}$ ($78.6\%$ RMS error) scales.

### 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.

| Feature | MOND | Fuzzy DM (FDM) | TEP (Soliton) |
| --- | --- | --- | --- |
| **Physical Basis** | Modified Inertia/Gravity | Ultralight Boson ($m \sim 10^{-22}$ eV) | Conformal Time Field Saturation |
| **Fundamental Scale** | Acceleration $a_0 \approx 1.2 \times 10^{-10}$ m/s² | Particle Mass $m_\psi$ | Core Saturation $\rho_T \approx 20$ g/cm³ & Shear Scale $g_t = cH_0/2$ |
| **Galactic Scaling** | Fits RAR ($a_0$ is free parameter) | $M_c \propto M_h^{1/3}$ (Core-Halo) | $R \propto M^{1/3}$ (derived from $\rho_T$) |
| **Screening** | External Field Effect (EFE) | None (requires tuning) | Temporal Topology ($S \propto \rho^{1/3}$) |
| **Solar System** | Recovered via interpolation function | Unsuppressed (tension) | Screened ($S = 0.65$ for Earth, $\sim 29{,}500$ for Pulsars) |
| **Key Prediction** | Exact RAR | Solitonic Cores | GNSS Correlation Length + Screening Hierarchy |

#### 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
\(S_\Sigma(\mathcal{E})\):

| Channel | Response coefficient | Environment | Projection |
| --- | --- | --- | --- |
| **GNSS clocks** | \(L_c \approx 4200\) km | Planetary crust/mantle | Radial covariance scale |
| **Cepheids (Paper 11)** | \(\kappa_{\rm Cep} = 0.665\text{--}1.275 \times 10^6\) mag (\(0.67\text{--}1.28 \times 10^6\)) empirical; Step 62 \(\rho_T/\langle\rho\rangle \implies 1.196 \times 10^6\) mag (\(\lambda_0=1\)) | Galactic disk (weakly screened) | Period–luminosity modulation |
| **Pulsars (Paper 10)** | \(\tilde\kappa_{\rm MSP} \sim 10^4\) full; \(\kappa_{\rm MSP} \approx 0.05\) effective | Globular cluster (strongly screened) | Spin-down excess (suppressed) |
| **Wide binaries (Paper 13)** | \(\alpha_{\rm sat} \approx 0.366\) | Galactic halo (unscreened) | Velocity-profile saturation |

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:

#### 1. Soliton Envelope Scale

The fundamental core saturation density $\rho_T \approx 20$ g/cm³
governs the compact soliton surface, but the observable galactic
transition is set by the diffuse screening density $\rho_{\rm trans}
\approx 3 \times 10^{-23}$ g/cm³ (fitted from SPARC, Section 3). The
predicted emission extent therefore scales as $R_{\rm trans} \propto
M^{1/3}$ at the transition density, not at $\rho_T$. Precise
numerical predictions for high-redshift galaxies require
epoch-specific calibration of $\rho_{\rm trans}$; until then, the
model predicts that resolved gas tracers (e.g., H$\alpha$) should
reveal more extended kinematic structure than expected from a purely
stellar compact core.

#### 2. Velocity Dispersion

The dynamical support speed is set by the soliton potential at the
transition radius: $\sigma_v^2 \sim G M / R_{\rm trans}$. Because
$R_{\rm trans} \propto M^{1/3}$, the velocity dispersion scales as
$\sigma_v \propto M^{1/3}$. This predicts characteristic velocity
dispersions intermediate between the broad-line AGN regime
($>1000$ km/s) and the low-mass dwarf regime ($<50$ km/s). At redshift $z$, the two-branch threshold scales with the expansion rate as $g_t(z) = cH(z)/2$ ($a_{\rm eff}(z) = 4\sqrt{2}\,g_t(z)/k$; Paper 12), fixing the epoch-dependent transition density $\bar\rho_{\rm trans}(z) \propto H(z)^2$ without free parameters.

#### 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

| Tier | Role | Channels | Relation to \(\rho_T\) |
| --- | --- | --- | --- |
| **A** | Direct / quasi-direct calibration | GNSS, SPARC/MW, RBH-1, Planetary Spin/MOI, condensed-matter EOS | Enters \(\rho_T\) normalization and dynamical scale checks directly |
| **B** | Screening / response-transfer test | Pulsars (10), Cepheids (11), wide binaries (13), LLR (17), Earth flyby (15), Outer System (36) | Constrains \(S_\Sigma(\mathcal{E})\) and transfer functions, not independent \(\rho_T\) |
| **C** | Phantom Mass / conformal transport | Lensing (4), multiply-imaged SNe (19), JWST (12) | Tests the mechanism; not a \(\rho_T\) calibration |
| **D** | Orthogonal architecture test | J0437 (16), hi_class (18), LVK (22), QF (23), C0 (26), TH (27), BH (28), BBN (29) | Tests deep TEP structure; does not calibrate \(\rho_T\) |

*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.

**Summary:** Only Tier A enters the present
\(\rho_T\) normalization argument directly. Tiers B–D constrain
the environmental operator, response hierarchy, and theoretical
viability of the same TEP architecture. This tiered dependence is
stronger than a flat pile of evidence because it acknowledges
exactly where each paper contributes and where it does not.

## 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:

- **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.

- **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. $1/2$; $\Delta\alpha = +0.325 \pm 0.110, z = 2.95, p = 0.0031$).

- **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).

- **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.

- **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:

\begin{equation} \label{eq:tep_action_conclusion} S = \int d^4x \sqrt{-g} \left[ \frac{M_{\text{Pl}}^2}{2} R + P(X,\phi) \right] + S_m[\tilde{g}_{\mu\nu}], \qquad P(X,\phi) = P_{\rm kin}(X) - V(\phi), \end{equation}

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:

- **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.

- **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.

- **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.

- **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.

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### 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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/10.5281/zenodo.22858190) (Paper 36)

## Contact Information

Author: Matthew Lukin Smawfield

Affiliation: Independent Researcher

Email: [matthew@mlsmawfield.com](mailto:matthew@mlsmawfield.com)

ORCID: [0009-0003-8219-3159](https://orcid.org/0009-0003-8219-3159)

GitHub: [github.com/matthewsmawfield](https://github.com/matthewsmawfield)

License: This work is licensed under a [Creative Commons Attribution 4.0 International License](https://creativecommons.org/licenses/by/4.0/).

Version: v0.12 (New Delhi) · Last updated: 28 October 2026

Revision note (v0.12): added the 7-planet unscreened-interior nonlinear Picard solve (Step 16), Saturn seven-clock kronoseismology/PV/SKR evaluation (Table 4.4), Roche-limit compositional ring stratigraphy, saturation self-energy ring mass and grain-size window, Cassini Division ambient co-rotation analysis, universal ambient-branch satellite orbital expansion law ($t_{\rm mig}^{\rm TEP} = \tfrac{3}{4}t_H = 10.48\text{ Gyr}$) across Earth–Moon and Saturn's six regular moons, Saturnian satellite screening-shell boundary mapping across Prometheus, Pandora, and chaotically tumbling Hyperion, and the outer non-tidally-locked satellite soliton-capacity test across Phoebe, Himalia, and Nereid (§4.7, Steps 15–16).

## 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:

| Center | λ Median (km) | 95% CI (km) | R² (pooled) |
| --- | --- | --- | --- |
| CODE | 4,181 | 1,198–5,918 | 0.920 |
| IGS Combined | 3,763 | 3,197–4,871 | 0.968 |
| ESA Final | 3,193 | 2,532–3,984 | 0.970 |

*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):

- **Epoch randomization:** Shuffling satellite epochs destroys the correlation ($r^2 < 0.01$, $N = 1000$ shuffles)

- **Residual shuffling:** Shuffling clock residuals within each satellite eliminates the spatial structure

- **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.

| Signature | PPP (Processed) | SPP (Raw) | Status |
| --- | --- | --- | --- |
| Exponential decay | R² = 0.92–0.97 | R² = 0.94 | ✓ Confirmed |
| Directional anisotropy | Detected | Detected | ✓ Confirmed |
| Orbital velocity coupling | r = −0.86 | Detected | ✓ Confirmed |
| Preferred-direction recovery | RA = 6°, Dec = +4°; ~6° from aphelion velocity tangent; CMB apex anti-phase | Detected | ✓ Confirmed |

*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

#### What GNSS Measures

The GNSS analysis measures a characteristic correlation length $L_c \approx 4200$ km in atomic clock residuals. This length is operationally interpreted as the projected Temporal Topology covariance scale associated with $R_T(M_\oplus)$, the geometric saturation scale for Earth's mass under the TEP framework.

#### Scope of the GNSS Calibration

The GNSS analysis measures the spatial covariance scale $L_c$ of the clock-sector residual field and uses it to anchor $\rho_T \approx 20$ g/cm³. Whether that macroscopic scale originates from a soliton-like Temporal Topology transition is tested by its out-of-sample transfer across the astrophysical mass hierarchy in the main text.

### 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](https://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:

| Quantity | Typical Value | Source | Physical Meaning |
| --- | --- | --- | --- |
| \(\alpha_{\rm SPARC}\) | \(0.352 \pm 0.043 \text{ (stat)} \pm 0.07 \text{ (definition)}\) | UCD (this work) | Galactic DM onset scaling (expected 1/3) |
| \(\alpha_{\rm sat}\) | \(0.366\) | WB (Paper 13) | Wide-binary saturation boost (phenomenological) |
| \(\beta_{\rm scr}\) | \(0.334\) | UCD (this work) | Screening exponent from \(S(\rho)\) hierarchy |
| \(\kappa_{\rm Cep}\) | \(0.665\text{--}1.275\times10^6\) mag (\(0.67\text{--}1.28\times10^6\)) empirical; \(1.196\times10^6\) mag derived (Step 62) | H0 (Paper 11) | Cepheid clock-bias response coefficient (derived from 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 \(|\beta_A|=1\), matching the SH0ES central estimate \((1.191\pm0.480)\times10^6\) mag to \(0.5\%\)) |
| \(\tilde\kappa_{\rm MSP}\) | \(\sim 10^4\) | COS (Paper 10) | Pulsar timing full response coefficient (suppressed observed value after dense-cluster geometric screening; bare geometric-factor estimate is \(\sim 10^6\)–\(10^7\)) |

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

**Series convention.** In this corpus, $\rho_T$ denotes the
phenomenological macroscopic saturation/proximity scale calibrated from GNSS
by treating $L_c$ as the projected covariance scale associated with $R_T(M_\oplus)$. Channel-specific quantities such as
$\rho_{\rm trans}$, $\rho_{\rm half}$, $R_s$, $\kappa_{\rm Cep}$,
$\kappa_{\rm MSP}$, and disformal response parameters are not independent
measurements of $\rho_T$, but downstream response or environment parameters.
The symbol $\lambda$ denotes a generic correlation length recovered from GNSS
clock-residual analysis; center- or product-specific values are distinguished
by subscripts (e.g., $\lambda_{\rm CODE}$, $\lambda_{\rm MGEX}$).

### 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, $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2 U_\mu U_\nu + A_{\rm sp}^2 \Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$):

\begin{equation} \label{eq:jordan_metric} \tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu \phi \nabla_\nu \phi. \end{equation}

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:

\begin{equation} \label{eq:inverse_metric} \tilde{g}^{\mu\nu} = \frac{1}{A^2(\phi)} g^{\mu\nu} - \frac{B(\phi)}{A^2(\phi) \big[A^2(\phi) + B(\phi) (\nabla\phi)^2\big]} \nabla^\mu \phi \nabla^\nu \phi, \end{equation}

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:

\begin{equation} \label{eq:signature_condition} A^2(\phi) + B(\phi) (\nabla\phi)^2 > 0. \end{equation}

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:

\begin{equation} \label{eq:metric_components} \tilde{g}_{00} = A^2(\phi) g_{00} = -A^2 c^2, \qquad \tilde{g}_{rr} = A^2(\phi) g_{rr} + B(\phi) (\partial_r \phi)^2. \end{equation}

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

\begin{equation} \label{eq:null_signal} -A^2 c^2 dt^2 + \big[A^2 + B(\phi) (\partial_r \phi)^2\big] dr^2 = 0, \end{equation}

which yields the effective light speed:

\begin{equation} \label{eq:effective_speed} c_\gamma^2 \equiv \left( \frac{dr}{dt} \right)^2 = \frac{A^2 c^2}{A^2 + B(\phi)(\partial_r \phi)^2}. \end{equation}

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:

\begin{equation} \label{eq:scalar_wave} P_X\,\ddot{\delta\phi} - c^2 \nabla_i \Big[\big(P_X\delta^{ij} - P_{XX}\nabla^i\bar{\phi}\nabla^j\bar{\phi}\big)\nabla_j\delta\phi\Big] + V_{,\phi\phi}\,\delta\phi = \text{source terms}, \end{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):

\begin{equation} \label{eq:static_klein_gordon} \nabla^2 \bar{\phi} - m_{\rm eff}^2(\rho) \bar{\phi} = \frac{\beta_A}{M_{\rm Pl}} \rho_{\rm matter}. \end{equation}

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.

#### Resolution of the Causal Vulnerability

By deriving the effective matter-frame light cone $c_\gamma^2$, establishing strict Cauchy hyperbolicity on the preferred foliation, and distinguishing
the stationary boundary-value solution from the dynamic Green's function,
the selected physical backgrounds satisfy the stated local matter-cone and scalar-hyperbolicity conditions, and the GNSS covariance interpretation does not require superluminal communication:

**Matter-Frame Light Cone & Hyperbolicity:** Matter and electromagnetic signals propagate strictly within the locally Lorentz-invariant causal cone of $\tilde{g}_{\mu\nu}$ ($c_\gamma \le c$), while scalar perturbations remain strictly hyperbolic on the preferred global foliation ($v_\perp^2 = 1$, $v_\parallel^2 \in [1,3]$ on static spacelike gradients; Paper 0 §12).

**Correlation vs. Communication:** The 4,200 km
correlation observed in the 25-year CODE analysis is a
*static structural parameter* of the scalar medium,
established by the Earth's mass distribution over geological
timescales, not an instantaneous interaction between the atomic
clocks.

### B.8 Summary of Risk Register

| Vulnerability | Severity | Mitigation Status | Resolution Target |
| --- | --- | --- | --- |
| GNSS correlation transfer & null discrimination | Low–Medium | 4-layer PREM BVP ($L_c^{\rm PREM} = 4{,}010$ km) & 100-trial MC network injection–recovery ($\hat L_c^{\rm GPS} = 4{,}036 \pm 113$ km; white-noise $|C(d)| = 0.009$ and orbital-dipole $d^2C/dd^2 < 0$ nulls excluded; Step 14) | Resolved in Step 14 (hardware timing-perturbation test optional) |
| Earth not a soliton | Low–Medium | 4-layer PREM Earth BVP, 1-D spherical nonlinear Picard solve ($\|\Delta u^{(100)}\|_\infty = 2.67\times 10^{-14}$; recovers $R_{\rm grad} = 4{,}875$ km [$4{,}810$ km nonlinear], $L_c^{\rm PREM} = 4{,}010$ km [$3{,}957$ km nonlinear], $L_c^{\rm vol} = 3{,}213$ km [$3{,}170$ km nonlinear]; Step 14), and 7-planet Lane–Emden unscreened-interior nonlinear Picard scan ($R_{\rm grad} \approx 0.75\text{--}0.83\,R_{\rm vol}$; Step 16) | Resolved in TEP-UCD v0.11–v0.12 (Steps 14, 16; 3-D azimuthal EW/NS PDE in future work) |
| SPARC $M^{1/3} \to M^{1/2}$ transition & $R_d$ confound | Low | Left-censored Tobit MLE rejects $R_{\rm DM}/R_d = \text{const}$ ($z = 9.57, p < 10^{-4}$); *a priori* Freeman-disk baryonic slope split around $M_{\rm cross} = 1.45 \times 10^{10}\,M_\odot$ separates $M^{1/3}$ core crossings ($\alpha = 0.256 \pm 0.069$) from $M^{1/2}$ tail crossings ($\alpha = 0.581 \pm 0.086, p = 0.0031$; Step 8) | Resolved in TEP-UCD v0.11 (Step 8) |
| Screening hierarchy algebraic | Low | Supplemented by the fixed-input ambient planetary scaling relation, the two-branch planetary/substellar/outer-moon ($\bar\eta_{\rm moons} = 0.1320 \pm 0.0113$) rotational consistency tests, and the three-channel zero-prefactor convergence ($\hat\rho_T = 19.77 \pm 0.48$ g/cm³, $p = 5.2 \times 10^{-5}$; §4.7, Steps 15–16) | Resolved in TEP-UCD v0.11–v0.12 (Steps 15–16) |
| Cross-paper parameter drift | Low | Notation purge complete | Ongoing |
| Causal cone & scalar hyperbolicity | Resolved | Proved: local matter-cone nesting ($c_\gamma \le c$), strict Cauchy hyperbolicity ($v_\perp^2 = 1, v_\parallel^2 \in [1,3]$ on static spacelike gradients; Paper 0 §12 & Appendix B.7), and static BVP covariance distinction | Resolved in Appendix B.7 & Paper 0 §12 |
| MGEX scale discrepancy (1,862 vs. 4,200 km) | Low–Medium | Pre-registered test & cross-mode RINEX/PPP de-biasing executed: scale is reproducible across independent chains (COD 2,744 vs GRG 1,933 km), and single- vs 4-constellation RINEX/PPP comparison establishes a $1.90\times$ multi-GNSS ISB filter compression factor that de-biases pooled MGEX to $3{,}536$ km (Step 14) | Resolved in TEP-GNSS-MGEX Steps 2.8, 3.1, 3.5 & TEP-UCD Step 14 (4-quarter 2023 GPS/Galileo/GLONASS single-constellation RINEX3 reprocessing executed) |

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.

## Appendix C: Candidate EFT Realizations of the Soliton Profile

The empirical saturation scale $\rho_T \approx 20$ g/cm³ is established
phenomenologically from GNSS clock correlations and SPARC scaling. A
natural question is whether standard scalar-tensor Effective Field Theory
(EFT) machinery can generate a density-dependent screening transition at
this scale. This appendix presents two candidate completions: the
symmetron, retained as an external benchmark whose $\mathbb{Z}_2$
mechanism is standard in the modified-gravity literature but whose
quadratic coupling is not the corpus's universal exponential; and a
confining-potential realization constructed directly on the corpus
coupling $A(\phi) = e^{-\phi/M_{\rm Pl}}$ ($\beta_A = -1$), which is
the corpus-internal demonstration that the observed phenomenology is
*mechanistically possible* without requiring new physics beyond
a self-interacting scalar sector. The completions are benchmark models,
not unique derivations; the corpus's working screening realization is the
selected two-branch nonlinear kinetic sector of the Paper-0 master action,
$P(X,\phi) = -V(\phi) + \frac{2k}{3\Lambda_X^2}X|X|^{1/2} + X|X|/\Lambda_X^4$ (with the unit-term form $P = X - V + X|X|/\Lambda_X^4$ retained as the high-gradient control), whose response $P_{,X}(\xi) = k\sqrt{\xi} +
2\xi$ ($\xi = |X|/\Lambda_X^4$, $k = 16.03$) saturates the scalar coupling once the field gradient
exceeds the kinetic scale $\Lambda_X = \sqrt{M_{\rm Pl}H_0} \approx 1.9$
meV (Paper 0, A4, §7; analogous to non-canonical kinetic K-mouflage screening in the scalar-tensor literature, Babichev et al. 2009; Barreira et al. 2015, where precision Solar-System perihelion, Lunar Laser Ranging, and time-varying $G_{\rm eff}$ bounds complement Cassini Shapiro-delay constraints).
The kinetic scale fixes the shear threshold $g_t$ — equivalently a
surface-density crossover $\Sigma_* \sim 3 M_{\rm Pl}^2 H_0 \approx
0.12$ g/cm² inside sources on the unit-term interior benchmark (step 14) — not a volume-density scale:
the density argument of the amplitude sector enters through the
potential normalization, and the potential-sector realization below is
constructed on the corpus's own quartic coupling. In the
TEP framework, these mechanisms are treated as candidate microscopic
completions, not as the defining ontology of the theory (Paper 0, A4).
TEP screening is defined by the continuous spatial profile of the scalar
time field (Temporal Topology) and its gradient (Temporal Shear), with no
thin-shell boundaries.

### C.1 Symmetron Screening: Density-Triggered Symmetry Breaking

The symmetron mechanism (Hinterbichler & Khoury 2010) provides a
natural route to density-dependent screening. The scalar potential is:

\begin{equation} \label{eq:symmetron_potential} V(\phi) = -\frac{1}{2}\mu^2 \phi^2 + \frac{1}{4}\lambda \phi^4, \end{equation}

and the conformal coupling to matter is:

\begin{equation} \label{eq:symmetron_coupling} A(\phi) = 1 + \frac{1}{2}\left(\frac{\phi}{M}\right)^2, \end{equation}

where $M$ is the symmetry-breaking scale. The effective potential in the
presence of ambient density $\rho$ becomes:

\begin{equation} \label{eq:symmetron_veff} V_{\rm eff}(\phi) = V(\phi) + \rho \left[A(\phi) - 1\right] = -\frac{1}{2}\mu_{\rm eff}^2 \phi^2 + \frac{1}{4}\lambda \phi^4, \end{equation}

with effective mass parameter:

\begin{equation} \label{eq:symmetron_meff} \mu_{\rm eff}^2 = \mu^2 - \frac{\rho}{M^2}. \end{equation}

The critical behavior is immediate. When $\rho > \rho_* \equiv \mu^2 M^2$,
the symmetry-breaking parameter $\mu_{\rm eff}^2 < 0$ turns negative so the physical squared mass (curvature of $V_{\rm eff}$ at the origin, $m_{\rm eff}^2(0) \equiv V_{\rm eff}''(0) = -\mu_{\rm eff}^2 = \rho/M^2 - \mu^2 > 0$) is positive and the field sits stably at the symmetric
minimum $\phi = 0$; the scalar is massive and screened, so Temporal Shear
effects are suppressed. When $\rho < \rho_*$, symmetry is spontaneously
broken, the field rolls to $\phi_0 = \mu_{\rm eff}/\sqrt{\lambda}$, and the
scalar becomes light—generating long-range forces. The saturation scale is:

\begin{equation} \label{eq:symmetron_rhostar} \rho_* = \mu^2 M^2. \end{equation}

Identifying $\rho_* \equiv \rho_T \approx 20$ g/cm³ fixes one relation
between $\mu$ and $M$. The second relation comes from demanding that the
Compton wavelength in the unscreened regime matches the galactic scale
($\lambda_C \sim$ kpc), which yields $\mu \sim 10^{-27}$ eV and
$M \sim 10^{-3}$ $M_{\rm Pl}$. These values are within the
phenomenologically allowed window for scalar-tensor theories. Within the
corpus this completion is an external benchmark only: its quadratic
coupling $A(\phi) = 1 + \phi^2/2M^2$ is not the universal exponential,
and the $\mathbb{Z}_2$ structure on which the mechanism rests cannot be
imposed on the linearly coupled exponential $A = e^{-\phi/M_{\rm Pl}}$.

### C.2 Confining-Potential Screening on the Corpus Coupling

Under the corpus's frozen coupling $\beta_A = -1$, the matter term enters
the effective potential with negative slope, so the runaway inverse-power
branch $V \propto \phi^{-n}$ used in chameleon completions is inadmissible:
both $V' < 0$ and $\beta_A\rho A/M_{\rm Pl} < 0$ for all $\phi > 0$, no
minimum of $V_{\rm eff}$ exists, and the field runs away rather than
saturating. The admissible branch is a confining potential, $V' > 0$.
Writing the dimensionless field $u = \phi/M_{\rm Pl}$, the admissible
family is

\begin{equation} \label{eq:chameleon_potential} V(u) = \frac{\lambda}{m}\,M_{\rm Pl}^4\,u^{m}, \qquad m > 2, \end{equation}

whose coefficient is the dimensionless coupling $\lambda$ — not a
fourth-power energy density, and distinct from the GNSS covariance
length denoted $\lambda$ in Section 3. The corpus's own matter-hosting branch is
the quartic member of this family, the weak-field truncation of the
master potential $V(u) = V_{\rm matter}(u)e^{-(u/u_s)^4} + V_0
e^{-(u_s/u)^4}$ (Paper 0, Section 8). The conformal coupling is the
corpus exponential:

\begin{equation} \label{eq:chameleon_coupling} A(u) = e^{-u} \qquad (\beta_A = -1). \end{equation}

The effective potential is:

\begin{equation} \label{eq:chameleon_veff} V_{\rm eff}(u) = V(u) + \rho\left[A(u) - 1\right] = \frac{\lambda}{m}M_{\rm Pl}^4\,u^m + \rho\left(e^{-u} - 1\right). \end{equation}

Its derivative $V_{\rm eff}' = \lambda M_{\rm Pl}^4 u^{m-1} - \rho\,e^{-u}$ vanishes at

\begin{equation} \label{eq:chameleon_umin} \lambda\,u_{\rm min}^{m-1}\,e^{u_{\rm min}} = \frac{\rho}{M_{\rm Pl}^4} \quad\Longrightarrow\quad u_{\rm min} \approx \left(\frac{\rho}{\lambda M_{\rm Pl}^4}\right)^{1/(m-1)} \;\; (u_{\rm min} \ll 1), \end{equation}

so a minimum exists for every $\rho > 0$, with the field elevated inside
matter ($u_{\rm min} > 0$) and approaching zero in the ambient — the
direction required by the corpus sign convention. The tracking exponent is then
fixed, not free: the quartic potential derives the unclipped equilibrium tracking ratio
$S(\bar\rho) \equiv u_{\rm min}(\bar\rho)/u_{\rm min}(\rho_T) = (\bar\rho/\rho_T)^{1/3} = R_T(M)/R_{\rm phys}$
(the $u_{\rm min} \ll 1$ truncation; the full $\lambda u^3 e^u$ solve
agrees to $\sim 4\times10^{-10}$ relative, step 14), because reproducing the $(\bar\rho/\rho_T)^{1/3}$ scaling requires $1/(m-1) = 1/3$, i.e. $m = 4$ — the quartic
member already adopted by the master potential. Macroscopic observables then project this unclipped ratio $S(\bar\rho) = R_T(M)/R_{\rm phys}$ in two complementary ways across the corpus: (i) for surface rotational and structural kinematics (Section 4.7), because a body's matter distribution terminates at its physical surface $R_{\rm phys}$, the active screened radius enclosed within the body cannot exceed $R_{\rm phys}$ ($R_{\rm active} = \min[R_T(M),\,R_{\rm phys}]$), which imposes the geometric surface ceiling $S_A(\bar\rho) \equiv R_{\rm active}/R_{\rm phys} = \min[1, (\bar\rho/\rho_T)^{1/3}]$ (whereas the local equilibrium $u_{\rm min}(\bar\rho)$ itself continues to grow as $\bar\rho^{1/3}$); and (ii) for deep-interior clock/coupling drift transmission in over-saturated compact objects ($\bar\rho > \rho_T$, Paper 22 &sect;6.1), the unclipped ratio $S_{\rm prog} = u_{\rm min}(\bar\rho)/u_{\rm min}(\rho_T) = (\bar\rho/\rho_T)^{1/3} > 1$ ($S_{\rm WD} \simeq 46.4$ inside Chandrasekhar white dwarfs) measures the interior over-saturation depth and suppresses transmitted cosmological $G_{\rm loc}$ drift by $[\max(1, S_{\rm prog})]^{-1}$. Likewise, in the unified matter metric $\tilde g_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf{x}))U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf{x}))\Pi_{\mu\nu} + B(\phi)(\cdots)$ of Paper 0 &sect;2.2, quasi-static local soliton dynamics at a fixed cosmic epoch $t_0$ ($A_{\rm sp}(t_0) = 1$) probe spatial gradients of the lapse $A_{\rm loc}(\mathbf{x}) = e^{-u_{\rm grav}(\mathbf{x})}$, whereas cosmological radiation propagation along the homogeneous drift background ($A_{\rm loc}(t) = A_{\rm sp}(t) = A(\bar\phi(t))$, $B(\phi)k^\mu\partial_\mu\phi \to 0$; Paper 22 &sect;2.2) specializes to the conformal radiation metric $\tilde g_{\mu\nu} = A^2(\bar\phi)g_{\mu\nu}$ with $c_T \equiv c$ and $\Xi(z) = A_{\rm em}/A_0 = (1+z)^{-1}$.
The normalization is likewise fixed corpus-wide at the
Cassini-compatible branch $\lambda = 7.5\times10^{-66}$
(`LAMBDA_QUARTIC_CASSINI`; the fiducial point $\lambda_{\rm ref} =
7.5\times10^{-71}$ of the same family fails the corrected
linear-in-$S_\Sigma$ Cassini evaluation by $\sim 250\times$, Paper 0
Appendix E step 03). As an energy scale the potential normalization is
$\Lambda_V \equiv \lambda^{1/4}M_{\rm Pl} \approx 1.3\times10^{2}$ GeV —
fourteen orders of magnitude above the kinetic scale $\Lambda_X \approx
1.9$ meV, and six orders above the energy-density reading
$\rho_T^{1/4} \approx 96$ keV of the saturation density (Section 5.5).
These are three distinct quantities: $\Lambda_X$ is the gradient
threshold of the kinetic sector, $\lambda$ the amplitude-sector
coupling, and $\rho_T$ the geometric reference density of the
$R_T(M)$ law. Under this normalization the equilibrium remains deeply
perturbative at all matter densities: $u_{\rm min}(\rho_T) =
6.9\times10^{-9}$ and $u_{\rm min}(\bar\rho_\oplus) = 4.5\times10^{-9}$,
matching the terrestrial share ($\sim 0.5\%$) of the nested field
decomposition (Paper 0, Section 2.2).

On the canonical-kinetic equilibrium benchmark ($P_{,X} \to 1$), the field
equation in a static, spherically symmetric environment is:

\begin{equation} \label{eq:chameleon_eom} \frac{d^2 u}{dr^2} + \frac{2}{r}\frac{du}{dr} = \frac{1}{M_{\rm Pl}^2}V_{\rm eff}'(u), \end{equation}

whereas on the operative nonlinear two-branch kinetic completion $P_{,X}(\xi) = \epsilon_{\rm reg} + k\sqrt{\xi} + 2\xi$, the spatial differential operator carries the nonlinear kinetic stiffness directly:

\begin{equation} \label{eq:nonlinear_chameleon_eom} \nabla \cdot \big(P_{,X}(\xi)\,\nabla u\big) = \frac{1}{r^2}\frac{d}{dr}\!\left(r^2 P_{,X}(\xi)\frac{du}{dr}\right) = \frac{1}{M_{\rm Pl}^2}V_{\rm eff}'(u). \end{equation}

On the canonical-kinetic equilibrium benchmark, the effective mass squared around $u_{\rm min}(\rho)$ is:

\begin{equation} \label{eq:chameleon_meff} m_{\rm eff}^2 = \frac{1}{M_{\rm Pl}^2}V_{\rm eff}''(u_{\rm min}) = \frac{3\lambda M_{\rm Pl}^4\,u_{\rm min}^{2} + \rho\,e^{-u_{\rm min}}}{M_{\rm Pl}^2} \simeq 3\lambda M_{\rm Pl}^2 u_{\rm min}^{2}, \end{equation}

where the equilibrium condition makes the curvature term dominant at
$u_{\rm min} \ll 1$, giving $m_{\rm eff} \propto \lambda^{1/6}
\rho^{1/3}$. The screening is doubly density-dependent. On this canonical-kinetic benchmark, the Compton
wavelength $\lambda_c(\rho) = 1/m_{\rm eff}$ contracts through denser
interiors, defining an order-of-magnitude equilibrium reference scale at which a body's interior becomes Compton-resolved inside its
own saturation halo, $\lambda_c(\bar\rho) \lesssim R_T(M)$ — for the
corpus normalization $\lambda_c(\rho_T) \approx 2{,}500$ km $\approx
0.6\,R_T(M_\oplus)$, and the canonical crossover density $\lambda_c(\bar\rho) =
R_T(M_\oplus)$ sits at $\sim 4$ g/cm³, inside the terrestrial
condensed-matter window (step 14). Deep in the interior the field sits
near its local equilibrium $u_{\rm min}(\rho)$, while across the transition layer the
nonlinear kinetic stiffness $P_X(\xi)$ (and its longitudinal combination $P_X + 2X P_{XX} = P_X - 2|X|P_{XX} = \epsilon_{\rm reg} + 2k\sqrt{\xi} + 6\xi$ in Eq. $\ref{eq:nonlinear_chameleon_eom}$) governs the actual physical transition width $\Delta R_T$; solving the coupled nonlinear two-branch boundary-value operator Eq. ($\ref{eq:nonlinear_chameleon_eom}$) directly across the 4-layer PREM Earth profile via under-relaxed Picard iteration ($\omega = 0.35$, $100$ iterations; Step 14) converges to machine precision ($\|\Delta u^{(100)}\|_\infty/\|u^{(100)}\|_\infty = 2.67 \times 10^{-14}$, grid-doubling shift $N = 1{,}200 \to 2{,}400$ of $3.70$ km [$0.077\%$], and initial-guess invariance across $u^{(0)} \in \{0,\,u_{\rm eq},\,2u_{\rm eq}\}$ to $8.8 \times 10^{-11}$ km), placing the nonlinear transition-gradient centroid at $R_{\rm grad}^{\rm NL\text{-}PREM} = 4{,}809.9$ km ($1.160\,R_T(M_\oplus)$, within $-1.3\%$ of the $4{,}874.9$ km canonical-kinetic centroid) and recovering a projected surface decay scale of $L_c^{\rm NL\text{-}PREM} = 3{,}957$ km and full-volume datum-removed decay scale of $L_c^{\rm NL,vol} = 3{,}170$ km (matching the $3{,}155.2$ km 3-constellation CME-removed mean and $3{,}210$ km 25-year CODE baseline within $1\%$). For the time-dependent GNSS clock observable, because $u \equiv \phi/M_{\rm Pl}$ is dimensionless ($A(u) = e^{-u}$), the dimensionless fractional frequency perturbation is $y_i(t) \equiv \delta\nu_i(t)/\nu_0 = \mathrm{d}\delta\tau_i/\mathrm{d}t = -\delta u_{\rm loc}(\mathbf{x}_i(t),t) - c^{-1}(\mathrm{d}/\mathrm{d}t)\int_{\mathbf{x}_i(t)}^{\mathbf{x}_s(t)}\delta u_\oplus(r(s'))\,\mathrm{d}s'$ with rate derivative $\dot y_i(t) = -(\partial_t u + \mathbf{V}_i(t)\cdot\nabla u) - \mathrm{d}^2\delta\tau_{is}^{\rm prop}/\mathrm{d}t^2$: while geocentric rotation alone is tangential to the spherical Earth monopole ($(\boldsymbol{\Omega}_\oplus\times\mathbf{r}_i)\cdot\nabla u_\oplus(r) \equiv 0$), the rotating station vector $\mathbf{r}_i(t) = R_\oplus\hat{\mathbf{n}}_i(\Omega_\oplus t)$ projects onto the external heliocentric/Galactic drift gradient $\nabla u_{\rm ext}$ via $\mathbf{r}_i(t)\cdot\nabla u_{\rm ext} \propto \cos(\Omega_\oplus t + \lambda_i)\cos\phi_i$ at the sidereal diurnal frequency ($f_{\rm sid} = 11.61\,\mu\text{Hz}$), while inclined satellite slant paths modulate $\delta\tau_{is}^{\rm prop}(t) \simeq [L_{\rm eff}/\sin E_{is}(t)]\,u_\oplus(R_\oplus)/c$ at the constellation-specific orbital repeat frequencies (GPS $11.97\text{ hr} / 23.21\,\mu\text{Hz}$, GLONASS $11.26\text{ hr} / 24.67\,\mu\text{Hz}$, Galileo $14.08\text{ hr} / 19.74\,\mu\text{Hz}$) inside the $10\text{--}60\,\mu\text{Hz}$ passband, producing a conformal clock-frequency modulation of $\delta y_{\rm rms} \simeq (v_{\rm rot}/c)\,u_\oplus(R_\oplus) = 7.0 \times 10^{-15}$ ($\delta\tau_{\rm rms} \simeq 0.10\text{ ns}$) while nonlinear kinetic stiffness $P_{,X}(\xi)$ suppresses the transmitted dynamical Temporal-Shear acceleration ratio (while the normalized Pearson correlation profile $C(d)/C(0)$ depends 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 $|\mathbf{a}_\phi|/g_\odot(1\text{ AU}) = 2.02 \times 10^{-8}$. The conformal self-quenching of
the source slope $\rho\,A_{,u} = -\rho\,e^{-u_{\rm min}}$ becomes
operative only when the equilibrium reaches order unity, which under the
corpus normalization occurs at $\rho_{\rm sat} \sim \lambda M_{\rm
Pl}^4 \approx 6\times10^{25}$ g/cm³ — far above neutron-star densities,
inside the temporal-well regime where the master potential's $V_0$ floor
activates (Paper 28). Ordinary saturation is therefore the
Compton-resolution crossover, not an order-unity field excursion.

### C.3 The Saturation Scale: Calibration versus Derivation

In the symmetron benchmark, $\rho_T = \mu^2 M^2$ is the
symmetry-restoration density, fixed by two potential parameters. On the
corpus branch the analogous question — which property of the scalar
action returns $\rho_T$ — has a sharper answer than a dimensional guess.
Under the corpus normalization the equilibrium at the transition density
is $u_{\rm min}(\rho_T) = 6.9\times10^{-9} \ll 1$: the saturation scale is
not a field-amplitude crossing of order unity. What $\rho_T$ parameterizes
is the geometric constant of the saturation-radius law $R_T(M) =
(3M/4\pi\rho_T)^{1/3}$, calibrated by the GNSS covariance length (Section
2); its dynamical proxy on the quartic branch is the Compton-resolution
crossover $\lambda_c(\bar\rho) \sim R_T(M)$, which for Earth sits at
$\bar\rho \sim 4$ g/cm³ — the terrestrial condensed-matter window in which
the measured $\rho_T \approx 20$ g/cm³ resides.

The shortcut identification $\rho_T \equiv \Lambda_V^4$ — equating the
geometric reference density with the potential coefficient — is not
available on the corpus branch, and is excluded rather than merely
unnecessary. Imposing $u_{\rm min}(\rho_T) \sim 1$ is equivalent to fixing
$\lambda = \rho_T/M_{\rm Pl}^4 \approx 2.4\times10^{-90}$, and that
normalization fails three independent checks (step 14): it lies
$3\times10^{24}$ below the Cassini-compatible quartic normalization that
Solar-System screening requires; it drives the neutron-star-core
equilibrium to $u_{\rm min} \approx 21$, past the master-potential knee
$u_s \simeq 10$ and onto the Planck-density floor — observed pulsars are
not temporal wells; and its implied Compton length at the transition
density, $\lambda_c(\rho_T) \approx 3.4\times10^{7}$ km, exceeds any
terrestrial body by more than three orders of magnitude, so the local equilibrium it
invokes never forms within the host. The same comparison furnishes a
directional lower bound: keeping neutron-star interiors below the knee,
$u_{\rm min}(\rho_{\rm NS}) < u_s$ at $\rho_{\rm NS} \sim 2\times10^{14}$
g/cm³, requires $\lambda \gtrsim 10^{-84}$, which the corpus branch
satisfies at nineteen orders of margin. The quartic normalization is
thereby bounded from below by compact objects and pinned from the side by
Cassini — a constrained window, not a free scale.

The correct microscopic reading is therefore: $\rho_T$ is the measured
geometric constant of $R_T(M)$; the quartic coupling $\lambda$ is the
amplitude-sector normalization; and the order-unity self-quenching crossing
sits at $\rho_{\rm sat} \sim \lambda M_{\rm Pl}^4 \approx
6\times10^{25}$ g/cm³, inside the temporal-well regime of Paper 28. Meanwhile,
the SPARC galactic rotation curves and outer-edge BTFR are governed directly
by the two-branch kinetic sector $P_{,X}=k\sqrt\xi+2\xi$
($k=16.03$ fixed externally by $a_0$, Steps 10 and 13), which supplies the exact $a_{\rm eff}=a_0$
asymptote with zero per-galaxy free parameters.

### C.4 Status and Caveats

The completions above demonstrate that the TEP phenomenology—density-dependent
screening, a compact soliton scale, and a galactic transition—is
*mathematically realizable*: the confining-potential construction
realizes it within the corpus's own frozen coupling, and the symmetron
provides an external benchmark from the standard EFT literature. They
do not, however, constitute a proof that nature has chosen either mechanism.
Key open questions include:

**Realization uniqueness:** Within the confining family
no exponent freedom remains — the corpus amplitude-response exponent
selects the quartic member $m = 4$, which is the weak-field branch of
the corpus's own master potential rather than one choice among many.
The residual uniqueness question concerns alternative realizations
(the symmetron benchmark, the wider $K(X)$ kinetic class) and is the
action-closure question recorded in Paper 0 — not a freedom this
appendix retains.

**Quantum stability:** The small quartic coupling
$\lambda \approx 7.5\times10^{-66}$ is radiatively stable in the
technical sense — its quantum corrections are proportional to
$\lambda$ itself. The $\mu \sim 10^{-27}$ eV mass-scale exposure
belongs to the symmetron benchmark, not to the corpus realization.

**Terrestrial transition width:** On the canonical-kinetic quartic branch ($\lambda = 7.53\times 10^{-66}$, $\rho_T = 20$ g/cm³), the 4-layer PREM Earth BVP solve and 3-D Yukawa surface covariance projection (Step 14) recovers a transition gradient centroid at $R_{\rm grad} = 4{,}875$ km ($1.18\,R_T(M_\oplus)$) and a surface-network decay length of $L_c^{\rm PREM} = 4{,}010$ km ($L_c^{\rm vol} = 3{,}213$ km full-volume), while the 100-iteration damped Picard solution of the coupled 1-D spherical nonlinear two-branch operator Eq. ($\ref{eq:nonlinear_chameleon_eom}$) converges to $\|\Delta u^{(100)}\|_\infty/\|u\|_\infty = 2.67 \times 10^{-14}$ and yields $R_{\rm grad}^{\rm NL\text{-}PREM} = 4{,}810$ 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 (Section C.2). Extending this radial nonlinear solve to non-spherical 3-D azimuthal orbital-motion anisotropy remains a target for future work.

The existence of a corpus-internal completion that structurally reproduces
the salient phenomenology under the frozen $\beta_A = -1$ coupling —
with the symmetron as an external benchmark — removes the objection that
$\rho_T \approx 20$ g/cm³ is a numerological construct. The scale is the
geometric constant of the saturation-radius law, calibrated by GNSS, whose
dynamical reading on the corpus's quartic branch is the Compton-resolution
crossover of the interior field — the admissible branch selected by the
corpus sign convention, normalized by the Cassini-compatible coupling, and
bounded from below by compact-object equilibria (step 14). It is anchored
to terrestrial kinematic data at every step.

---

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

*Related Work:*
- [**TEP Theory**](https://doi.org/10.5281/zenodo.16921911) (Foundational framework)
- [**TEP-RBH Paper 7**](https://doi.org/10.5281/zenodo.18059250) (RBH-1 Application)

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