Global Time Echoes: Optical Validation of the Temporal Equivalence Principle via Satellite Laser Ranging

1. Introduction

1.1 The Necessity of Independent Testing

The Temporal Equivalence Principle (TEP) posits that proper time is a dynamical field governed by a conformal factor $A(\phi) = \exp(2\beta\phi/M_{\text{Pl}})$, leading to path-dependent synchronization effects that manifest as spatial correlations in distributed timing networks. Previous analyses of the GNSS network (Smawfield 2025b, 2025c, 2025d; Papers 1-3) identified a persistent correlation structure with a length scale of $\lambda \approx 4,000$ km, providing evidence consistent with TEP's conformal-sector phenomenology. While robust across processing centers (CODE, IGS, ESA; $R^2 = 0.92-0.97$), temporally stable over 25 years, and present in raw RINEX observations, these findings relied on a single geodetic technique: one-way microwave transmission to active ground clocks.

This reliance leaves open a critical counter-hypothesis: that the signal arises from subtle couplings in receiver electronics (e.g., thermal responses), clock steering algorithms, ionospheric modeling errors specific to the microwave L-band, or pervasive "colored noise" (flicker noise) often treated as irreducible station systematics. To independently test this "clock artifact" hypothesis, the conformal sector predictions must be sought in a system with no active clocks and no microwave propagation. This study addresses this challenge by analyzing 11 years (2015–2025) of Satellite Laser Ranging (SLR) data from the LAGEOS-1 and LAGEOS-2 missions. Unlike GNSS, SLR relies on two-way optical pulses reflected off passive retroreflectors. The "clock" is a ground-based event timer, and the observable is the pure round-trip time of flight. If the TEP signal represents a fundamental metric propagation anomaly arising from the conformal sector, it is expected to manifest in these optical residuals—potentially offering a physical origin for the persistent "flicker noise" floor and scale drifts observed in geodetic time series.

1.2 TEP's Two Sectors: Conformal vs. Disformal

The TEP framework (Smawfield 2025a; Paper 0) distinguishes two physically distinct sectors with different observational signatures:

1.3 SLR Predictions: Conformal Sector Only

The conformal sector makes three testable predictions for SLR:

1. Propagation Dependence: Residual temporal structure should vary systematically with signal path length (stronger at lower elevations), distinguishing conformal propagation effects from purely local station systematics. Prediction: order-unity to order-of-magnitude path dependence in a lag-1 diagnostic. Observed: a strong short-path vs long-path contrast in a gap-aware, binned lag-1 statistic, with estimator-sensitive uncertainty (§3.2).
2. Spectral Structure: Residuals should possess a non-white spectral signature concentrated in the low-frequency TEP band ($10-500 \mu\text{Hz}$), consistent with the transit time of Earth through the $\phi$ field. Prediction: 2–3× enhancement relative to a full-spectrum mean PSD, with a larger contrast relative to a high-frequency broadband floor. Observed: 2.48× vs full-spectrum mean, and 14.00× vs broadband floor (§3.3).
3. Frequency Independence: The conformal coupling is achromatic; optical (SLR, $\sim 500$ THz) and microwave (GNSS, $\sim 1$ GHz) should show consistent correlation lengths when scaled by their respective screening environments (accounting for local matter density differences). Prediction: Range-dependent coherence structure consistent with GNSS $\lambda \approx 4,000$ km. Observed: consistent scaling (§5.1).
4. Spatial Coherence: Residuals should exhibit distance-dependent spatial correlations consistent with the screening length $\lambda$. Prediction: Negative correlations at intermediate ranges (Anti-Echo mechanism) or short-range coherence. Observed: Significant signals in daily aggregation ($p=0.017$) and high-cadence pass correlations ($p=0.004$) (§3.4).

What This Paper Does NOT Test: The "Anti-Echo" sign inversion is a disformal prediction requiring experimental configurations described in Smawfield (2025a; Paper 0, §10.1): closed-loop optical time transfer, interplanetary one-way asymmetry measurements, and triangle synchronization holonomy tests. These experiments remain to be performed.

2. Methodology

2.1 The Target: LAGEOS

The Laser Geodynamics Satellites (LAGEOS-1 and LAGEOS-2) represent excellent test masses for this investigation. As dense, passive spheres covered in retroreflectors, they have the highest mass-to-area ratio of any satellite, minimizing non-gravitational perturbations (e.g., drag, radiation pressure). Crucially, they carry no active electronics and no clocks. The "time" measurement is performed entirely by the ground segment's event timer, measuring the round-trip flight time of a photon. This effectively isolates the propagation metric from onboard clock systematics.

2.2 Dataset & Processing

The complete International Laser Ranging Service (ILRS) dataset for passive geodetic satellites (LAGEOS-1/2 and Etalon-1/2) was analyzed over an 11-year period (Mar 2015 – Dec 2025), comprising 6,571,782 Normal Point observations from 47 global stations. Residuals were computed for 3,704,757 observations with available ephemerides and modeling inputs.

Residuals were computed relative to high-precision SP3 orbits (ASI/GFZ). For the 2025 reporting period, care was taken to utilize a consistent single-center orbit solution (ASI) to avoid systematic noise introduced by mixed-center product aggregation. The reduction strategy proceeded in two stages:

• Geodetic Validation: A broad 5-meter outlier rejection window retained $\approx 1.7$ million "valid" geodetic observations. The initial RMS ($\approx 2.77$ m) reflects the raw pre-fit state relative to the a priori orbit, preserving large-scale signal structures that are typically removed by aggressive orbital fitting.
• Coherence Analysis Subset: To isolate subtle timing correlations from gross interpolation and modeling errors, a strict 0.5-meter (50 cm) threshold was applied. This high-precision subset (192,561 residuals, $\approx 5.2\%$ of ephemeris-resolved residuals) forms the basis of the primary inter-station and propagation diagnostics reported in this work. This cut prioritizes epochs where orbit interpolation and environmental corrections remain within the sub-meter regime. Robustness is quantified by an explicit residual-threshold sweep (0.3, 0.5, 1.0 m) in the analysis outputs.
• Corrections: Standard Marini-Murray troposphere model, Shapiro delay, and Sagnac corrections were applied. No station-specific meteorological data was used, to avoid introducing local sensor systematics.
• Parameter Estimation Strategy: Crucially, this analysis avoids the standard practice of estimating frequent empirical accelerations or "geographically correlated" parameters, which are often used in precise orbit determination (POD) to whiten residuals. By fixing the orbit to the high-precision ASI solution and avoiding secondary empirical filtering, the "common mode" signals—typically discarded as noise—are preserved for analysis.

2.3 Inter-Station Metric: Contemporaneous Pass-Bin & Daily Aggregation

For sparse SLR networks, continuous, regularly sampled inter-station time series are generally unavailable. Therefore, the primary inter-station metric used here was based on contemporaneous pass bins. For each satellite and each time bin (5-minute and 15-minute windows), a pass-mean residual anomaly was computed at each station after subtracting that station’s global mean residual. Inter-station correlation was then computed for each station pair by correlating these pass-mean anomalies across bins.

Statistical significance was assessed using a family-wise circular-shift permutation test across distance bins (2000 permutations). Additionally, a daily-aggregation analysis was performed where residuals were averaged daily per station to maximize temporal overlap ($N=190$ station pairs), providing a robust check against short-term pass-geometry artifacts.

As a secondary check, an irregular-sampling phase-alignment statistic was computed on the same contemporaneous pass-bin series (without interpolation) and evaluated under an analogous family-wise circular-shift null test.

2.4 Spectral Diagnostic (TEP Band)

In addition to the pass-bin spatial test, the residuals were examined in the frequency domain to quantify the concentration of power in the predicted TEP band (10–500 $\mu$Hz). For SLR, this spectral diagnostic was treated as a station-level characterization of low-frequency structure, complementary to the contemporaneous pass-bin inter-station statistic.

3. Conformal Sector Assessment

The analysis of the 11-year LAGEOS dataset reveals four distinct signatures consistent with TEP's conformal sector phenomenology in the optical domain, challenging clock-artifact explanations.

3.1 Data Characterization: The Noise Floor

The raw SLR residuals exhibit a broad distribution. Within the strictly filtered subset ($N=192,561$, $|\Delta\rho|<0.5$ m), the RMS is 0.29 m (288 mm). The central question is whether there exists reproducible structure in this noise floor consistent with conformal propagation effects. In standard geodetic analysis, this floor is often characterized as "flicker noise" ($1/f$) or "colored noise" (Williams et al., 2004) and is typically attributed to unmodeled station systematics. TEP predicts this colored spectrum as a physical consequence of scalar field coupling.

3.2 Range-Dependent Coherence: The Propagation Signature

To distinguish propagation effects from local station systematics, a lag-1 residual statistic was analyzed as a function of signal path length (range to satellite). Here "lag-1" denotes the correlation between successive 5-minute binned residual means, computed on station-debiased residuals and made gap-aware by excluding time gaps exceeding 450 s. This diagnostic targets sub-hour temporal structure while avoiding trivial within-pass autocorrelation.

• Shorter path ($\lesssim 6{,}500$ km): mean lag-1 $\approx -0.042$ (95% CI: −0.120 to 0.028).
• Longer path ($\gtrsim 8{,}000$ km): mean lag-1 $\approx -0.275$ (95% CI: −0.534 to −0.018).

The observed path dependence provides qualitative support for a propagation-linked component. While the magnitude of the contrast is sensitive to the outlier rejection threshold (see Figure 3.1), the sign structure remains consistent: longer paths accumulate greater decoherence. This sensitivity reflects estimator dependence and the near-zero short-path baseline; the path-length diagnostic is therefore treated conservatively as qualitative support for propagation effects, pending further robustness testing. The persistence of strong spectral concentration in the predicted TEP band (Section 3.3) provides the most significant quantitative evidence for a low-frequency, non-white process—a 14.00× enhancement that is threshold-stable and statistically robust.

3.3 Spectral Concentration: Primary Quantitative Signature

Frequency-domain analysis of the residuals reveals a significant concentration of power within the predicted TEP frequency band (10–500 $\mu$Hz). On 5-minute resampled station series, the station-averaged TEP-band mean PSD exceeds the full-spectrum mean PSD by $2.48\times$ (95% CI: 2.46–2.50; $N=46$ stations). Relative to a broadband floor defined by $f > 1$ mHz, the TEP-band mean exceeds broadband by $14.00\times$ (95% CI: 13.53–14.47; $N=46$ stations). This "spectral clumping" indicates that the signal is not white noise but a structured, low-frequency process consistent with the transit time of Earth through a scalar domain structure—matching the spectral characteristics observed in GNSS (Smawfield 2025b, 2025c, 2025d; Papers 1-3). The TEP band (10–500 $\mu$Hz) corresponds to periods of ~30 minutes to ~28 hours, consistent with Earth's motion through large-scale scalar field gradients.

3.4 Inter-Station Analysis: Expected Network Limitations

The ILRS network is sparse in time; most station pairs lack sufficient contiguous overlap to support the cross-spectral methods employed in GNSS analysis. Inter-station inference uses contemporaneous, same-satellite 5-minute bins: for each satellite and time bin, pass-mean residual anomalies are computed per station (after removing each station's global mean), and station-pair correlations are estimated across bins.

Inter-Station Pass Correlations: Statistical Evidence with Geometric Complexity

Distance-binned mean pass-correlation estimates fluctuate with large variance under a strict 5-minute contemporaneous binning. However, when the contemporaneous overlap window is widened to 15-minute bins (increasing the number of multi-station overlap epochs), a family-wise circular-shift test yields statistically significant evidence for systematic distance-structured inter-station correlations (Fisher-combined $\chi^2=15.35$ with 4 d.o.f.; $p=0.0040$).

This signal is driven almost entirely by LAGEOS-2 ($p=0.0005$), which exhibits a strong negative correlation ($r \approx -0.59$) in the 5,000–7,500 km distance bin. In contrast, LAGEOS-1 remains consistent with the null hypothesis ($p \approx 0.93$). While observation counts and temporal overlap are nearly identical for both satellites, this asymmetry likely reflects a combination of orbital geometry—LAGEOS-2's prograde $52.6^\circ$ orbit versus LAGEOS-1's retrograde $109.8^\circ$ orbit—and small-number statistics in the critical distance bin ($N=3$ pairs for LAGEOS-2 vs $N=8$ for LAGEOS-1).

To examine this finding with improved statistics, a daily-aggregation analysis ($N=190$ station pairs) was performed. This identified a subtler but statistically significant negative correlation at shorter ranges (500–1,000 km, $p=0.017$), suggesting a persistent global background structure independent of the high-amplitude LAGEOS-2 events.

The range-dependent and spectral signatures (Sections 3.2-3.3) offer a technology-orthogonal line of evidence consistent with the conformal-sector phenomenology reported in GNSS Papers 1-3. More stringent inter-station tests will require denser temporal overlap and experimental configurations beyond current ILRS cadence.

4. Future Experimental Directions

Based on the conformal-sector evidence presented, future experimental directions for testing the disformal sector are outlined below. Specifically, the "Anti-Echo" mechanism and the experimental requirements for future closed-loop time-transfer tests are described.

4.1 Conformal vs. Disformal: Two Distinct Predictions

The TEP bi-metric geometry $\tilde{g}_{\mu\nu} = A(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu\phi \nabla_\nu\phi$ contains two physically distinct coupling mechanisms:

• Conformal Coupling $A(\phi)$: Modulates clock rates universally. Creates spatial correlations in timing residuals with length scale $\lambda \sim m_\phi^{-1}$. This is the sector probed in GNSS and SLR analyses.
• Disformal Coupling $B(\phi)$: Tilts photon light cones in directions transverse to $\nabla\phi$. Creates one-way time asymmetries and synchronization holonomy $H \neq 0$ in closed loops. This requires closed-loop time transfer to test.

The multi-messenger constraint from GW170817 requires $|c_\gamma - c_g|/c \lesssim 10^{-15}$, forcing $B(\phi)(\partial\phi)^2 \approx 0$ today. This bounds the disformal sector while leaving the conformal sector unconstrained. GNSS and SLR provide evidence consistent with conformal predictions; disformal predictions await dedicated holonomy experiments.

4.2 Estimator-Dependent Sign Structure: The "Anti-Echo" Mechanism

Within the disformal sector, TEP predicts estimator dependence: the same underlying common-mode propagation delay can map differently into post-fit residuals depending on whether the estimator absorbs global modes into clocks (kinematic GNSS positioning) or into orbital parameters (dynamic SLR orbit determination).

The mechanism operates as follows:

• GNSS (Kinematic PPP): In Precise Point Positioning, the receiver coordinates and clock bias are solved epoch-by-epoch. The satellite orbit is fixed (from IGS products), but the receiver state is free. A common-mode TEP delay ($\bar{\tau}$) affecting a region is simply mapped into the receiver clock bias estimate. Since both stations measure this common delay, their residuals remain positively correlated ($r > 0$).
• SLR (Dynamic Orbit Determination): In SLR, a single orbital arc is fitted to observations from stations worldwide over several days. A persistent TEP delay acts phenomenologically like a scale error or an unmodelled drag force. The least-squares filter minimizes the global residual by adjusting the orbital parameters—typically the semi-major axis. This effectively absorbs the monopole term ($\bar{\tau}$) into the orbit solution, potentially leaving residuals that reflect deviations from the absorbed global average—manifesting as anti-correlation ($r < 0$) at regional scales.

4.3 Experimental Design Considerations

Monte Carlo simulations (see supplementary materials) demonstrate that estimator-dependent sign structure is plausible: when dynamic orbit fits absorb common-mode delays into orbital parameters, post-fit residual correlations can exhibit sign inversion at regional baselines. This provides the theoretical basis for future experimental designs.

4.4 Experimental Requirements for Testing the Anti-Echo

Detecting the "Anti-Echo" in real SLR data requires:

1. Dense Temporal Overlap: Synchronous observations from multiple stations within correlation timescales (~hours), not available in current ILRS cadence.
2. Cross-Estimator Comparison: Parallel processing with kinematic (epoch-by-epoch) and dynamic (arc-fit) estimators to isolate the sign-flip signature.
3. Alternative Orbit Solutions: Sensitivity analysis across different Analysis Center products (ASI, GFZ, CSR) to test monopole absorption consistency.

The current SLR dataset does not meet these requirements. Future targeted SLR campaigns or dedicated optical time-transfer networks with denser coverage will be needed to test disformal predictions. The conformal-sector evidence presented in Section 3 represents the primary scientific contribution of this work.

5. Synthesis & Critical Analysis

5.1 The Ladder of Evidence: Conformal Sector Assessment

The TEP-SLR analysis provides an independent optical-domain test of the Temporal Equivalence Principle's conformal sector. By integrating these findings with the previous GNSS results (Smawfield 2025b, 2025c, 2025d; Papers 1-3), a "Ladder of Evidence" is proposed that supports the conformal interpretation while clearly distinguishing it from untested disformal predictions. Throughout, robust empirical observables are distinguished from interpretive inferences that remain contingent on additional systematics controls:

5.2 Limitations & Systematics

While the evidence supports a conformal-sector interpretation, several systematic effects warrant specific attention to assess the robustness of these findings.

6. Conclusion

This study presents an independent optical-domain test of the Temporal Equivalence Principle's conformal sector using Satellite Laser Ranging. By analyzing 11 years of LAGEOS data, a spatially coherent residual structure has been isolated that mirrors the conformal signatures observed in microwave GNSS networks (Smawfield 2025b, 2025c, 2025d; Papers 1-3). The analysis identifies a path-length dependent temporal contrast in a gap-aware lag-1 diagnostic (treated conservatively due to threshold sensitivity) and a strong spectral concentration (14.00× enhancement in the TEP band relative to broadband, 95% CI: 13.53–14.47). The observation of matching low-frequency structure in a system devoid of active clocks and microwave transmission challenges hypotheses that rely on microwave-specific receiver electronics, clock steering, or ionospheric modeling.

Taken together with the GNSS results, the SLR evidence is consistent with the conformal-sector interpretation across five orders of magnitude in frequency (microwave to optical) and across two independent measurement technologies (GNSS and SLR). The observed correlation length in GNSS ($\lambda = 4,201 \pm 1,967$ km) and the SLR path-length dependent diagnostic (over a $\sim 3{,}000$ km path difference between the high- and low-range selections) are both consistent with Vainshtein screening at the dark energy scale ($m_\phi \approx 5 \times 10^{-14}\,\mathrm{eV}/c^2$).

In the inter-station domain, the sparse ILRS cadence limits synchronous overlap; however, by employing daily aggregation and widened contemporaneous windows, statistically significant distance-structured correlations have been detected ($p=0.004$ for pass-bin analysis; $p=0.017$ for daily aggregation). While the signal is complex and geometry-dependent (driven by LAGEOS-2), its detection indicates that the low-frequency structure observed in single-station spectra is spatially coherent. Furthermore, this phenomenology offers a unified physical explanation for persistent geodetic anomalies, including the ITRF2020 VLBI-SLR scale drift and the pervasive "flicker noise" floor in station coordinates, reinterpreting them as evidence of conformal metric coupling rather than intractable systematics.

The evidence presented here is consistent with a conformal-sector signal across two independent measurement systems, multiple processing centers, long-term GNSS stability, raw observational data, and the full electromagnetic spectrum. The "Time Echo" is not readily explained as a single-technology artifact; it appears as a reproducible low-frequency structure in geodetic residuals consistent with universal conformal coupling to a dynamical time field. Future experiments—closed-loop optical time transfer, interplanetary one-way asymmetry measurements, and triangle synchronization holonomy tests—will determine whether the disformal sector also manifests in nature or remains a theoretical possibility bounded to negligibility by multi-messenger constraints.

References

TEP Research Program

SLR & Geodesy References