A scalar-tensor framework for dynamical proper time, clock-network correlations, and synchronization holonomy
The Temporal Equivalence Principle proposes that proper time is a dynamical field. Its central claim is not that local relativity fails, but that proper-time accumulation and synchronization possess global structure that standard local, reciprocity-even tests are not designed to measure.
In TEP, matter and clocks couple to a causal metric governed by a scalar time field. The spatial and covariance structure of this field is called Temporal Topology; its locally active gradients are Temporal Shear. Local Lorentz invariance and locally measured c are preserved. The new observables are global: distance-structured clock correlations, environment-dependent rate effects, and residual synchronization holonomy around closed loops.
GR already predicts that clocks tick differently in different gravitational potentials. TEP asks whether, after standard GR redshift, Sagnac, Shapiro, Doppler, station-motion, and reference-frame effects are modeled and subtracted, reproducible residual structure remains in clock-rate covariance, Temporal Shear, or closed-loop synchronization.
The primary empirical programme tests this structure using global timing networks. GNSS analyses report distance-structured correlations in multi-center clock products, a 25-year CODE baseline, and raw RINEX-derived observables; satellite laser ranging provides an optical-domain consistency test. Astrophysical papers then test whether the same temporal-field structure organizes lensing, pulsars, Cepheids, wide binaries, and high-redshift galaxies.
All manuscripts, analysis code, derived outputs, and reproducibility instructions are released for open review. The next decisive steps are independent reproduction of the timing-network correlations, technology-independent clock and ranging tests, and direct closed-loop measurements of residual synchronization holonomy.
Start here: Key Concepts · Research Programme · Interactive Demo · Correlation Explorer · Replication Code · FAQ
These three concepts define the operational core of TEP. Temporal Topology describes the clock-rate landscape, Temporal Shear describes its local gradient / slope, and Synchronization Holonomy gives the cleanest closed-loop experimental discriminator.
Using the compact notation Θ = ln A:
Matter and clocks couple to the causal matter metric:
The landscape picture is only an analogy; the formal objects are ln A, CA, and g̃μν.
Using Θ ≡ ln A:
In screened or effectively saturated regimes, the observable shear/source-charge sector is suppressed:
with 𝒮Σ(ℰ) → 0 in locally screened regimes, where ℰ denotes environmental state.
Equivalently:
Foundations, timing-network tests, synchronization experiments, and astrophysical transfer tests.
Introduces TEP as a covariant framework in which proper time is dynamical while local Lorentz invariance and locally measured c are preserved. The paper develops the two-metric formulation, global synchronization structure, and first observational targets for clock-network and closed-loop tests.
Most high-precision tests of general relativity constrain reciprocity-even, largely local observables within single-metric frameworks. This leaves open a specific underdetermination between GR and two-metric scalar-tensor sectors involving conformal, disformal, gradient, covariance, and holonomy observables.
A phase-coherent spectral analysis of 62.7 million station-pair measurements from 364 GNSS stations (2023–2025) reports systematic distance-structured correlations in clock networks. These correlations follow an exponential decay with a median Temporal Topology correlation length λT = 3,330–4,549 km and show strong goodness-of-fit when evaluated...
A 25-year longitudinal analysis (1999–2024) of 165.2 million station-pair measurements from CODE clock products confirms the signal reported in Paper 1, tracking its evolution through multiple solar cycles and operational epochs.
A raw RINEX processing pipeline, from broadcast ephemeris-based SPP through DD ambiguity-fixing, confirms that the distance-structured correlations persist in unprocessed receiver observations and are not artifacts of precise clock-estimation pipelines.
An optical-domain consistency test using SLR residuals to LAGEOS-1/2 (2002–2025) reports distance-structured correlations comparable in magnitude and scale to the timing-network results. This independent-domain check supports the signal's robustness across measurement types.
Visualize the reported distance-dependent coherence pattern and compare it with the null expectation used in the analysis. Explore the correlation decay directly in the browser.
Integrates timing, geodetic, and astrophysical results into an evidence hierarchy, distinguishing primary clock-network tests from secondary transfer tests and speculative extensions.
Interprets ρT≈20 g/cm3 as a candidate Temporal Topology saturation scale and tests whether the associated M1/3 scaling organizes terrestrial, compact-object, galactic, and cosmological regimes.
Develops the Phantom Mass interpretation: temporal-composite image formation can project into static lens reconstructions as an apparent mass-like component normally modeled as dark matter.
Reports a spatially stratified spin-down anomaly in 197 globular-cluster millisecond pulsars relative to 346 field controls, with a 0.40 dex controlled residual and covariance-aware significance of 8.3σ under the stated model. The observed density scaling is suppressed relative to the Newtonian ensemble baseline.
Tests a proposed Cepheid Bias: an environment-dependent period contraction predicted by TEP, yielding a unified local value near H0≈68.17 km s−1 Mpc−1 under the model assumptions, with blind testing on independent distance-ladder samples still required.
Tests whether high-redshift stellar-mass, age, dust, and compact-core anomalies are better organized by gravitational potential depth when the isochrony axiom is relaxed.
Finds evidence for a characteristic Temporal Shear recovery scale at Rs=2,646±182 AU statistical, with ±609 AU total uncertainty, and environmental ordering consistent with Temporal Shear recovery in weak-field environments.
Explores RBH-1 as a candidate Temporal Topology soliton/wake interpretation, using the terrestrial calibration as a consistency check rather than as part of the primary evidence chain.
This research programme develops the Temporal Equivalence Principle as a theory in which time is a dynamical field. More precisely, TEP proposes that proper-time accumulation is governed by a scalar field coupled to the matter/clock metric. The central question is whether proper time, after standard local relativistic corrections, can be transported globally without path dependence.
The framework is formulated as a covariant scalar-tensor theory with a gravitational metric and a matter/clock metric. Local physics remains Lorentz invariant. The new effects appear in global observables: distributed clock correlations, synchronization holonomy, Temporal Shear, and environment-dependent Temporal Shear recovery.
The primary empirical tests use GNSS because global navigation systems form a planetary-scale network of atomic clocks. The reported signal is a distance-structured correlation pattern with characteristic length λT≈4,200 km, seen in processed clock products, a 25-year longitudinal analysis, and raw RINEX-derived observables. Satellite laser ranging provides an optical-domain consistency check.
The astrophysical papers then test whether the same temporal-field structure organizes phenomena normally treated separately: lensing mass discrepancies, globular-cluster pulsar timing, Cepheid distance-ladder biases, JWST high-redshift anomalies, and Gaia wide-binary dynamics. These applications are not presented as independent proof of TEP, but as cross-regime tests of a common Temporal Topology / Temporal Shear framework.
All papers, code, and data products are released for open review. The clearest near-term tests are external reproduction of the timing-network correlations, multi-constellation checks, optical-clock or fiber-link experiments, non-GNSS timing systems, and closed-loop one-way synchronization holonomy.
This interactive visualization displays the reported exponential decay pattern across processed clock products and raw RINEX-derived observables. The dual axes compare phase alignment in processed clock products with magnitude-squared coherence in raw RINEX-derived observables. Although these quantities are measured on different scales, both exhibit distance-structured decay consistent with the Global Time Echoes timing-network result.
The dotted trend lines show averaged exponential fits with Temporal Topology correlation lengths λT ranging from 600–4,500 km and fit qualities R²=0.87–0.99. Shorter values arise in reduced or raw-observation demonstrations; the multi-center processed-clock analyses report longer several-thousand-kilometre scales. The recurrence across methods indicates that the effect is not confined to one tested processing pipeline. Whether the recurrence is physical rather than systematic is the central replication question.
All manuscripts, analysis code, processing pipelines, derived outputs, and reproducibility notes are publicly available under open-source licenses. The primary timing analyses include processed GNSS clock products, a 25-year CODE baseline, raw RINEX-derived observables, and optical SLR checks, with documented workflows from input data to final figures.
Independent replication by other research groups is the decisive next step. Researchers interested in replication may find Paper 1 (TEP-GNSS I) the most accessible entry point, using publicly available CODE/IGS/ESA clock products (compact .clk files). Paper 2 (TEP-GNSS II) extends this with 25 years of CODE clock data. Paper 3 (TEP-GNSS III) provides the most direct raw-data test via RINEX processing but requires more substantial computational resources. Feedback on methodology, interpretation, or potential collaboration is welcomed.
Core repository containing the theoretical framework, mathematical models, and LaTeX source. Includes derivation scripts, figure generation code, and the full manuscript source.
Complete analysis pipeline for Paper 1 (TEP-GNSS I). Features multi-center cross-validation scripts, processing logs, JSON statistical outputs, and generated figures for correlation decay quantification.
Research compendium for Paper 2 (TEP-GNSS II). Contains longitudinal analysis scripts, orbital coupling logs, CMB alignment data, and comprehensive JSON result files.
End-to-end pipeline for Paper 3 (TEP-GNSS III). Includes SPP processing scripts, raw RINEX analysis logs, consistency-test datasets, and resulting anisotropy figures from 1 billion samples.
Codebase for Paper 4 (TEP-GL). Contains phantom mass modeling scripts, rotation curve data, analysis logs, and the Python notebooks used to generate manuscript figures.
Synthesis framework for Paper 5 (TEP-GTE). Includes integration scripts, cross-study correlation logs, consolidated JSON datasets, and summary figures demonstrating signal convergence.
Scaling analysis codebase for Paper 6 (TEP-UCD). Features saturation-density calculation scripts, scaling law verification logs, and figure generation for the universal critical density ρT model.
Analysis codebase for Paper 7 (TEP-RBH). Contains JWST NIRSpec data processing scripts, soliton wake modeling, line-profile decomposition tools, and visualization for the RBH-1 runaway black hole candidate analysis.
Complete analysis pipeline for Paper 8 (TEP-SLR). Includes SLR data parsers, residual computation logs, correlation analysis scripts, and final result figures.
Codebase for Paper 9 (TEP-EXP). Methodological taxonomy of precision tests of general relativity, discriminating experiments, and analysis of which observables constrain conformal, disformal, local-gradient, clock-covariance, and synchronization-holonomy sectors.
Codebase for Paper 10, Suppressed Density Scaling in Globular Cluster Pulsars. Contains pulsar catalog compilation, field-control comparisons, cluster-density scaling tests, and adversarial dynamics checks.
Codebase for Paper 11 (TEP-H0). Includes SH0ES data processing, environment stratification, TEP correction optimization, and scripts testing whether an environment-dependent Cepheid correction can reduce the Hubble tension.
Codebase for Paper 12 (TEP-JWST). JWST high-redshift galaxy analysis, isochrony axiom tests, temporal shear calculations, SUSPENSE survey kinematic comparisons, and star formation efficiency modelling.
Codebase for Paper 13 (TEP-WB). Gaia DR3 wide-binary analysis, Temporal Shear recovery tests, disk/halo environmental ordering, and field-transition radius fitting.
TEP is a scalar-tensor framework proposing that time is a dynamical field. More precisely, proper-time accumulation is governed by a scalar field coupled to the matter/clock metric, while local Lorentz invariance and locally measured c are preserved.
TEP is proposed as a generalization of GR, not a simple replacement. GR is recovered to current experimental precision where the locally observable shear/source-charge sector is screened or saturated. Differences are expected in global or weakly screened observables: distributed clock correlations, closed-loop synchronization holonomy, and environment-dependent rate or screening transitions.
No. GR already predicts gravitational time dilation: clocks tick differently in different gravitational potentials. TEP accepts that and does not claim it as new. The proposed new physics is residual structure after the standard GR timing model is removed: distance-structured clock covariance, environment-dependent Temporal Shear, and possible closed-loop synchronization holonomy.
TEP is first a theoretical framework, but its strongest empirical motivation is the timing-network programme: distance-structured GNSS clock correlations across multi-center products, a 25-year CODE analysis, raw RINEX-derived observables, and an optical-domain SLR consistency test. Pulsars, Cepheids, wide binaries, lensing, and JWST systems are transfer tests of the same Temporal Topology and Temporal Shear framework.
TEP does not deny the lensing, timing, dynamical, or cosmological phenomena usually attributed to dark matter. It challenges the assumption that those phenomena uniquely require a new invisible particulate substance. Temporal-field gradients and nontrivial proper-time transport can project into lensing, timing, and dynamical inference as an apparent mass-like component, termed Phantom Mass.
In the conservative interpretation, this is a testable correction to time-domain and variability-dependent lensing observables. In the stronger interpretation, tested across the series, part of the dark-sector phenomenology may be temporal in origin: an effect of analyzing a universe with nontrivial time transport under the assumption of global isochrony.
TEP predicts that Cepheid periods may acquire an environment-dependent bias in deep gravitational potentials. In the current SH0ES-host analysis, correcting for this proposed bias shifts the inferred local value toward the Planck value. This should be read as a candidate distance-ladder systematic, not as a final resolution until tested blindly on independent Cepheid, TRGB, maser, and SN host samples.
Yes, in the intended parameter regime. GW170817 tightly constrains differential photon–graviton propagation: any disformal cone tilt must be extremely small. TEP's main conformal-sector effects are different. If electromagnetic and gravitational signals travel along the same path through the same conformal temporal landscape, the effect is common-mode rather than a photon–graviton speed split. Conformal sectors remain indirectly constrained by PPN, equivalence-principle, source-screening, redshift, and clock-comparison tests.
The framework is designed to be tested through several increasingly independent pathways:
No. GNSS works extraordinarily well. TEP does not claim navigation is failing. It asks whether residual clock-network covariance, after standard modeling and differencing, contains spatial structure normally treated as noise, covariance, or processing residual. The claim is about subdominant correlation structure, not operational GNSS accuracy.
Both TEP and MOND address phenomenology attributed to dark matter, but through distinct mechanisms. MOND proposes a universal acceleration threshold (a₀ ≈ 1.2×10⁻¹⁰ m/s²) below which gravitational behavior deviates from Newtonian predictions. TEP instead tests environment-dependent Temporal Shear recovery organized around the Temporal Topology saturation scale ρT ≈ 20 g/cm3, which produces environmental ordering that differs from the MOND/EFE parameterizations tested in the wide-binary analysis. The two frameworks make qualitatively different predictions for environmental stratification.
The full manuscript series is freely available at mlsmawfield.com with Zenodo DOIs. Analysis code and data pipelines are hosted on GitHub. All manuscripts, code, and data products are released under Creative Commons CC-BY-4.0 and MIT licenses.
Note on Disambiguation: The acronym TEP (Temporal Equivalence Principle) addresses the foundations of spacetime geometry. It is distinct from the thermoelectric power coefficient (Seebeck TEP) in condensed matter or the Total Extraperitoneal surgical procedure.