Universal Critical Density: Unifying Atomic, Galactic, and Compact Object Scales

Dark matter observations across cosmological scales exhibit a striking regularity: the characteristic radius at which Newtonian dynamics fails scales as R ∝ M^ (1/3), implying a universal critical density ρc. This scaling appears in galaxy rotation curves (SPARC database), ultra-diffuse galaxies (DF2/DF4), the Milky Way's Keplerian transition, and compact object phenomena (magnetar anti-glitches). This pattern is shown to reflect a fundamental saturation scale in the conformal time-field sector of the Temporal Equivalence Principle (TEP), where gravitational solitons form at a characteristic density threshold. Terrestrial calibration—derived from a newly identified distance-structured correlation in GNSS atomic clocks—provides an independent measurement of this scale. Multi-center analysis (CODE, IGS, ESA) reveals correlations with characteristic length Lc ≈ 4200 km for Earth's mass (M_⊕ ≈ 6 × 10²7 g), implying ρc ≈ 20 g/cm³. This calibration exhibits 25-year temporal stability and survives raw RINEX validation, strongly constraining processing-artifact explanations. The derived density scale is independently constrained by atomic physics: requiring the soliton radius to reproduce the Bohr radius at the proton mass scale (Rₛol (mₚ) ~ a₀) yields ρc ~ 10–50 g/cm³, consistent with the GNSS measurement. Galactic-scale validation comes from the SPARC rotation curve database (175 galaxies). The empirical dark matter onset scaling is α = 0. 354 ± 0. 014, consistent with the M^ (1/3) prediction within 2σ. Gaia DR3 analyses report evidence consistent with a Keplerian-like decline near R ≈ 19 kpc in the Milky Way, broadly consistent with the predicted transition scale. For ultra-diffuse galaxies DF2 and DF4, the model predicts soliton 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. Analysis of 26 astrophysical objects spanning 15 orders of magnitude in density reveals an empirical scaling S ∝ ρ⁰. 334 (R² = 0. 9999), confirming the predicted ρ^ (1/3) dependence. At nuclear densities (binary pulsars: ρ ~ 10¹4 g/cm³), screening factors exceed S > 30, 000, suppressing scalar contributions to less than 0. 003% of orbital dynamics. This hierarchy explains why Solar System tests, binary pulsar timing, and gravitational wave observations show no deviation from GR, while galactic dynamics (ρ ~ 10^-24 g/cm³, S ~ 0. 01) exhibit strong scalar effects. Compact object consistency is assessed using magnetar anti-glitches. For a canonical neutron star mass (M ≈ 1. 4 M_⊙) and ρc ≈ 20 g/cm³, the model predicts a critical spin period Pcrit ≈ 6. 8 s, below which the soliton radius exceeds the stellar radius. The magnetar 1E 2259+586 (P = 6. 98 s) exhibits anti-glitch behavior, consistent with this threshold within 4%. The saturation density ρc ≈ 20 g/cm³ emerges as a candidate universal organizing parameter, supported by consistency across 40 orders of magnitude in mass (proton to galaxy cluster) and 15 orders of magnitude in density (cosmological voids to neutron stars), within stated uncertainties. This externally calibrated value enables tightly constrained astrophysical applications, including the RBH-1 runaway black hole candidate (companion paper). The convergence of terrestrial, galactic, and compact object constraints on a single density scale suggests a fundamental connection between quantum mechanics (Bohr radius), atomic timekeeping (GNSS), and cosmological structure formation (dark matter). Keywords: dark matter – gravitation – scalar fields – galaxies: kinematics and dynamics – pulsars: individual (1E 2259+586) – Galaxy: kinematics and dynamics – temporal equivalence principle