Author: Matthew Lukin Smawfield
Version: v0.6 (New Delhi)
Date: First published: 28 December 2025 · Last updated: 12 June 2026
Status: Preprint
DOI: 10.5281/zenodo.18064365
Website: https://matthewsmawfield.github.io/TEP-UCD/
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 ρ_T. 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 L_c ≈ 4,200 km for Earth's mass (M_⊕ ≈ 6 × 10^27 g), consistent with multi-center results (CODE, IGS, ESA). The characteristic length L_c is operationally identified with the projected Temporal Topology covariance scale associated with R_T(M_⊕), the geometric saturation scale for Earth's mass; a soliton interpretation is one candidate microscopic realization, not assumed in the calibration. This implies ρ_T ≈ 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 α_SPARC = 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 R_DM ≈ 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 > ρ_T) yields S ∝ ρ^0.334 (R^2 = 0.99995), algebraically expected from the R_T(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 ρ_T ≈ 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 R_T(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 R_T(M_⊕).
A universal critical density ρ_T ≈ 20 g/cm³ emerges as a candidate saturation scale of the temporal-field topology, supported by cross-scale consistency across 18 orders of magnitude in mass (Earth to galaxy). Terrestrial calibration from GNSS atomic clocks (L_c ≈ 4,200 km) provides an independent anchor consistent with galactic-scale observations (SPARC rotation curves: α_SPARC = 0.355 ± 0.043 (stat) ± 0.07 (definition), consistent with the M^(1/3) expectation within ~0.3σ). Analysis of 26 astrophysical objects yields a consistency relation S ∝ ρ^0.334 (R^2 = 0.99995), algebraically expected from the R_T(M) construction.
| Paper | Repository | Title | DOI |
|---|---|---|---|
| Paper 0 | TEP | Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed | 10.5281/zenodo.16921911 |
| Paper 1 | TEP-GNSS | Global Time Echoes: Distance-Structured Correlations in GNSS Clocks | 10.5281/zenodo.17127229 |
| Paper 2 | TEP-GNSS-II | Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products | 10.5281/zenodo.17517141 |
| Paper 3 | TEP-GNSS-RINEX | Global Time Echoes: Raw RINEX Consistency Test | 10.5281/zenodo.17860166 |
| Paper 4 | TEP-GL | Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations | 10.5281/zenodo.17982540 |
| Paper 5 | TEP-GTE | Global Time Echoes: Empirical Synthesis | 10.5281/zenodo.18004832 |
| Paper 6 | TEP-UCD (This repo) | Universal Critical Density: Cross-Scale Consistency of ρ_T | 10.5281/zenodo.18064365 |
| Paper 7 | TEP-RBH | The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate | 10.5281/zenodo.18059250 |
| Paper 8 | TEP-SLR | Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging | 10.5281/zenodo.18064581 |
| Paper 9 | TEP-EXP | What Do Precision Tests of General Relativity Actually Measure? | 10.5281/zenodo.18109760 |
| Paper 10 | TEP-COS | The Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars | 10.5281/zenodo.18165798 |
| Paper 11 | TEP-H0 | The Cepheid Bias: Resolving the Hubble Tension | 10.5281/zenodo.18209702 |
| Paper 12 | TEP-JWST | The Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies | 10.5281/zenodo.19000827 |
| Paper 13 | TEP-WB | The Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries | 10.5281/zenodo.19102061 |
| Paper 15 | TEP-EFA | Temporal Equivalence Principle: Temporal Shear in the Earth Flyby Anomaly | 10.5281/zenodo.19454863 |
| Paper 16 | TEP-J0437 | Synchronization Holonomy in Pulsar Scintillation | 10.5281/zenodo.19454620 |
| Paper 17 | TEP-LLR | Lunar Laser Ranging and the Nordtvedt Effect | 10.5281/zenodo.19446029 |
| Paper 18 | TEP-HC | Temporal Equivalence Principle: hi_class Background Implementation and CMB Acoustic Peak Preservation | — |
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site/components/: HTML components comprising the manuscript sections.-
1_abstract.html: Paper abstract. -
2_introduction.html: The Dark Matter problem as a temporal structure problem. -
3_gnss_calibration.html: Derivation of$\rho_T$ from atomic clocks. -
4_sparc_validation.html: Galactic rotation curve analysis. -
5_screening_hierarchy.html: Temporal Topology screening mechanism. -
6_atomic_boundary.html: Physical constraints on ρ_T (electron degeneracy, dimensional analysis). -
7_universal_scaling.html: The unified scaling law. -
8_milky_way_test.html: Local Milky Way dark-matter onset. -
10_discussion.html: Theoretical implications (Phantom Mass). -
11_conclusion.html: Summary of findings. -
12_visual_evidence.html: Key figures. -
13_references.html: Bibliography. -
appendix_a_gnss.html: GNSS methodology appendix.
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- Preceded by: Papers 0–4 and the Synthesis (TEP, TEP-GNSS series, TEP-GL, TEP-GTE), which establish the empirical reality of the clock correlations and the theoretical framework.
- Companion to: Paper 7 (TEP-RBH), which applies the ρ_T value derived here to test the soliton hypothesis for the runaway black hole candidate RBH-1.
@article{smawfield2025ucd,
title={Universal Critical Density: Cross-Scale Consistency of ρ_T},
author={Smawfield, Matthew Lukin},
journal={Zenodo},
year={2025},
doi={10.5281/zenodo.18064365},
note={Preprint v0.6 (New Delhi)}
}This project is licensed under Creative Commons Attribution 4.0 International (CC-BY-4.0). See LICENSE for details.
These are working preprints shared in the spirit of open science—all manuscripts, analysis code, and data products are openly available under Creative Commons and MIT licenses to encourage and facilitate replication. Feedback and collaboration are warmly invited and welcome.
Contact: matthew@mlsmawfield.com
ORCID: 0009-0003-8219-3159
The manuscript is assembled from the HTML components in site/components/.