Temporal Equivalence Principle: Lunar Laser Ranging and the Nordtvedt Effect

Author: Matthew Lukin Smawfield
Version: v0.2 (Lucknow)
First published: 17 May 2026 · Last updated: 17 May 2026
Status: Preprint (Open for Collaboration)
DOI: 10.5281/zenodo.19446029
Website: https://mlsmawfield.com/tep/llr/
Paper Series: TEP Series: Paper 17 (Lunar Laser Ranging)

Abstract

The Temporal Equivalence Principle (TEP) is a scalar-tensor theory in which proper time is a dynamical field $\phi$ that couples universally to all matter via a conformal metric $\tilde{g}{\mu\nu} = A^2(\phi) g{\mu\nu}$. In deep gravitational potential wells, high ambient density suppresses the local gradient of this field — a mechanism called Temporal Shear screening — with the degree of suppression scaling with a body's gravitational compactness $\Phi/c^2$. Because Earth and Moon have different compactness and interior shielding profiles, TEP motivates a compactness-dependent Strong Equivalence Principle response that would appear in Lunar Laser Ranging as a synodic Earth-Moon range modulation $\delta r = 13\eta\cos D$, where $D$ is the Moon-Sun elongation angle and $\eta$ is the Nordtvedt parameter. The expected residual-channel amplitude is at the millimetre level for $\eta \sim 10^{-4}$.

This work analyses 26,207 raw Lunar Laser Ranging O–C residuals from the public INPOP19a ephemeris archives (Paris Observatory, Geoazur), comprising $N = 25{,}445$ measurements from five international stations (1984–2019) after standard $6\sigma$ MAD outlier cleaning. The primary estimand is a precision-weighted synodic regression on the post-fit residual channel, with inverse station-variance weighting and a full systematic model that includes annual, monthly, and station-specific regressors.

The primary precision-weighted residual-channel estimate is $\eta = -3.91 \times 10^{-4} \pm 5.63 \times 10^{-5}$ ($6.94\sigma$; $6.78\sigma$ cluster-robust). The amplitude stabilises in the band $-3.2$ to $-4.1 \times 10^{-4}$ across the estimator hierarchy, from cosD-only ($5.25\sigma$) through full-systematic OLS ($6.17\sigma$) to precision-weighted ($6.94\sigma$). Robustness checks — common-$\eta$ mixed model, leave-one-station-out meta-analysis, wild cluster bootstrap, phase-locked differential, cross-ephemeris validation on DE430, parametric GR-null bootstrap, and a frequency null scan — all support residual-channel survival of the synodic component (Section 4). Within the standard Parametrized Post-Newtonian framework, the measured Nordtvedt parameter implies $\beta = 0.999902 \pm 1.07 \times 10^{-5}$ from the joint $(\beta, \gamma)$ contour, placing General Relativity ($\beta = \gamma = 1$) at $\Delta\chi^2 = 48.2$, outside the 99% confidence contour. A full-sky directional scan on the residual channel (2,664 uniformly spaced directions, 5° grid) places the Planck CMB dipole axis at rank 226/2664 (top 8.5%); a scrambled-sky null with $n = 1{,}000$ Monte Carlo realizations yields a look-elsewhere-corrected $p < 0.001$.

The result is therefore framed as a high-significance residual-channel candidate with a TEP interpretation, not as a completed replacement for direct-fit LLR bounds. Source-level numerical refits of the INPOP or DE430 integrators with $\eta$ left free remain the critical open closure test. Ephemeris-absorption stress tests bound the residual-channel survival amplitude but do not replace a full dynamical integrator-level confirmation.

Code Availability: All data and analysis code required to reproduce the results presented in this work, including the full LLR residual processing pipeline, are available in the public repository.

Key Results

TEP Signal in INPOP19a LLR Residuals:

• 26,207 observations from 5 stations (APO, Grasse, Matera, McDonald2, Haleakala); 25,445 after $6\sigma$ MAD cleaning
• Date range: 1984–2019
• Residual precision: 9.5 cm RMS
• Cook's-Distance-excised full-systematic OLS (leverage diagnostic): $\eta = -3.87 \times 10^{-4} \pm 4.95 \times 10^{-5}$ ($N = 23{,}837$ after excision), 7.82σ; cluster-robust 8.65σ
• Precision-weighted full-systematic (consensus): $\eta = -3.91 \times 10^{-4} \pm 5.63 \times 10^{-5}$, 6.94σ; cluster-robust 6.78σ
• Full-systematic OLS without excision (sensitivity upper bound): $\eta = -4.06 \times 10^{-4} \pm 6.58 \times 10^{-5}$ ($N = 25{,}445$), 6.17σ; cluster-robust 6.52σ
• Common-$\eta$ mixed model with station systematics (pooling): $\eta = -4.31 \times 10^{-4} \pm 6.74 \times 10^{-5}$, 6.40σ; $F(4, 25{,}410) = 1.19$, $p = 0.31$
• Phase-locked new/full-moon differential (robustness): $\eta = -5.95 \times 10^{-4} \pm 1.01 \times 10^{-4}$, 5.91σ
• cosD-only OLS (baseline): $\eta = -3.18 \times 10^{-4} \pm 6.05 \times 10^{-5}$, 5.25σ

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The TEP Research Program

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 Temporal Evolution	10.5281/zenodo.17517141
Paper 3	TEP-GNSS-RINEX	Global Time Echoes: Raw RINEX Validation of Distance-Structured Correlations in GNSS Clocks	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 Validation of the Temporal Equivalence Principle	10.5281/zenodo.18004832
Paper 6	TEP-UCD	Universal Critical Density: Unifying Atomic, Galactic, and Compact Object Scales	10.5281/zenodo.18064366
Paper 7	TEP-RBH	The Soliton Wake: A Runaway Black Hole as a Gravitational Soliton	10.5281/zenodo.18059251
Paper 8	TEP-SLR	Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging	10.5281/zenodo.18064582
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.19102062
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 (This repo)	Lunar Laser Ranging and the Nordtvedt Effect	10.5281/zenodo.19446029

Directory Structure

TEP-LLR/
├── archive/                # Archived old/unused scripts and files
├── data/
│   ├── raw/                 # INPOP19a residual files (MINI format)
│   └── processed/           # Parsed residuals with elongation angles
├── logs/                    # Execution logs
├── manuscripts/             # Generated PDF/Markdown outputs
├── results/                 # Analytical outputs and figures
├── scripts/
│   ├── steps/               # Sequential analysis pipeline
│   │   ├── step_000_llr_data_ingestion.py      # Verify raw data availability and hashes
│   │   ├── step_001_data_preprocessing.py      # Parse MINI format
│   │   ├── step_002_de430_preprocessing.py     # DE430 ephemeris processing
│   │   ├── step_003_statistical_analysis.py    # Basic TEP detection analysis
│   │   ├── step_004_detection_analysis_advanced.py  # Advanced analysis (M4 Pro optimized)
│   │   ├── ... (82 canonical steps: step_000 through step_076, including 006b and 046b)
│   │   └── run_all_steps.py                     # Run complete pipeline
│   └── utils/               # Shared utilities
│       ├── llr_constants.py                       # Physical constants
│       ├── parse_de430.py                         # DE430 parser
│       ├── parse_inpop_mini.py                    # INPOP parser
│       ├── logger.py                              # Logging utilities
│       ├── pipeline_runner.py                     # Pipeline execution
│       ├── pipeline_quality_gate.py               # Reviewer-facing audit gate
│       ├── generate_evidence_ledger.py            # Evidence summary artifact
│       ├── schema_validation.py                   # Output schema checks
│       ├── statistical_utils.py                   # Statistical utilities
│       └── verify_value_consistency.py            # Manuscript value audit
├── site/
│   └── components/          # HTML source of truth for manuscript
├── README.md
└── requirements.txt         # Python dependencies

Installation

# Clone repository
git clone https://github.com/matthewsmawfield/TEP-LLR.git
cd TEP-LLR

# Install Python dependencies
pip install -r requirements.txt

Essential Data Files

• data/processed/INPOP19a_all_stations_residuals.csv - Main LLR residuals dataset
• data/raw/INPOP19a_*_residuals.txt - INPOP19a MINI format residual files verified by data/raw/data_manifest.json
• data/raw/DE430_2014-2018_residuals.dat - DE430 residual archive verified by data/raw/data_manifest.json

Data Sources

• Paris Observatory (Geoazur): INPOP19a lunar ephemerides with O-C residuals (primary data source)
• Stations: Apache Point (APO), Grasse, Matera, McDonald2, Haleakala
• Ephemeris: INPOP19a (2019) - most recent INPOP release with LLR residuals

Reproduction Pipeline

# Run complete pipeline (recommended)
python scripts/steps/run_all_steps.py

# Validate structured outputs and manuscript consistency
python scripts/utils/schema_validation.py
python scripts/utils/verify_value_consistency.py

# Run the reviewer-facing quality gate
python scripts/utils/pipeline_quality_gate.py

# Generate the evidence ledger directly
python scripts/utils/generate_evidence_ledger.py

# Or run steps individually:
# Step 0: Verify required raw residual files and checksums
python scripts/steps/step_000_llr_data_ingestion.py --verbose

# Step 1: Parse MINI format and compute elongation angles
python scripts/steps/step_001_data_preprocessing.py --verbose

# Step 2: DE430 ephemeris processing
python scripts/steps/step_002_de430_preprocessing.py --verbose

# Step 3: Run basic TEP detection analysis
python scripts/steps/step_003_statistical_analysis.py --verbose

# Step 4: Run advanced TEP detection analysis (M4 Pro optimized)
python scripts/steps/step_004_detection_analysis_advanced.py --verbose

Citation

@article{smawfield2026llr,
  title={Temporal Equivalence Principle: Lunar Laser Ranging and the Nordtvedt Effect},
  author={Smawfield, Matthew Lukin},
  journal={Zenodo},
  year={2026},
  doi={10.5281/zenodo.19446029},
  note={Preprint v0.2 (Lucknow)}
}

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Open Science Statement

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.

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Contact: matthew@mlsmawfield.com
ORCID: 0009-0003-8219-3159