Dark matter observations across cosmological scales exhibit a regularity: the characteristic radius at which Newtonian dynamics fails scales as R ∝ M^ (1/3), implying a universal critical proximity scale, observationally proxied by ρT. This scaling appears in galaxy rotation curves (SPARC database), ultra-diffuse galaxies (DF2/DF4), and the Milky Way's dark-matter onset. Within TEP, this pattern is interpreted as evidence of a candidate saturation scale in the conformal time-field sector, where field saturation occurs at a characteristic proximity scale. Terrestrial calibration—derived from a newly identified distance-structured correlation in GNSS atomic clocks—provides an empirical anchor. A 25-year CODE analysis yields Lc ≈ 4, 200 km for Earth's mass (M_⊕ ≈ 6 × 10²7 g), consistent with multi-center results (CODE, IGS, ESA). The characteristic length Lc is operationally identified with the projected Temporal Topology covariance scale associated with RT (M_⊕), the geometric saturation scale for Earth's mass; a soliton interpretation is one candidate microscopic realization, not assumed in the calibration. This implies ρT ≈ 20 g/cm³. This calibration exhibits 25-year temporal stability and survives raw RINEX validation, constraining processing-artifact explanations. Galactic-scale validation comes from the SPARC rotation curve database (167 galaxies). The empirical dark matter onset scaling is αSPARC = 0. 355 ± 0. 043 (stat) ± 0. 07 (definition), consistent with the M^ (1/3) expectation within ~0. 3σ. For the Milky Way, the SPARC-calibrated M^ (1/3) relation predicts a dark-matter onset radius RDM ≈ 3 kpc, consistent with the observed transition from baryonic to dark-matter-dominated rotation at R ~ 3–5 kpc. For ultra-diffuse galaxies DF2 and DF4, the model predicts Temporal Topology saturation radii exceeding tidal radii, consistent with observed dark matter deficiency via tidal stripping of the scalar field envelope. Temporal Topology screening resolves the apparent conflict with precision GR tests. A hierarchy of 26 astrophysical objects spanning 15 orders of magnitude in density is assembled; regression on the 11 dense objects (ρ > ρT) yields S ∝ ρ⁰. 334 (R² = 0. 99995), algebraically expected from the RT (M) construction. This explains why GR tests pass (binary pulsars: S ~ 29, 000) while galactic dynamics (S ~ 10^-9 at ρ ~ 10^-24 g/cm³) are deeply unscreened, exhibiting strong scalar effects. The saturation density ρT ≈ 20 g/cm³ emerges as a candidate universal saturation scale of the temporal-field topology — not an ambient-density switch — supported by cross-scale consistency across 18 orders of magnitude in mass (Earth to galaxy), within stated uncertainties. This externally calibrated value enables tightly constrained astrophysical applications, including the RBH-1 runaway black hole candidate (Smawfield 2025h, Paper 7). The ρ^ (1/3) hierarchy is a consistency relation induced by the RT (M) construction; it is not, by itself, an independent discriminator of microscopic screening mechanism. 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 RT (M_⊕). Website: https: //matthewsmawfield. github. io/TEP-UCD/Code Availability: https: //github. com/matthewsmawfield/TEP-UCD DOI: 10. 5281/zenodo. 18064365 Keywords: dark matter – gravitation – scalar fields – galaxies: kinematics and dynamics – temporal equivalence principle Open Science Statement: This work is a preprint and is open to community review, ideas, and collaboration. All materials required for full reproducibility—including data downloads, analysis scripts, code, and manuscripts—are open-source. Feedback and contributions to further test these results are welcome.
Matthew Lukin Smawfield (Sun,) studied this question.