The Lilborn Equation

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GPS Numerical Fit…

…And Residual Analysis

Introduction

This document publishes the explicit numeric fit demanded by the GPS critique. It reports the model form, proxy definitions, fitted coefficients, locked training and withheld test results and residual statistics. It also states why this matters historically and globally. Nothing is asserted without numbers.

Dataset and Lock Conditions

Clock dataset: sixty daily RINEX CLK (IGS combined clock solutions) for year 2024 day-of-year 001 through 060, renamed 1 through 60 in exact chronological order.

Orbit dataset: sixty daily SP3 precise orbits (IGS combined final orbit combinations) for the same day range, renamed 1 through 60 in exact chronological order.

Locked split: January 1–31 (files 1–31) used for calibration. February 1–29 (files 32–60) used for out-of-sample prediction. All GPS satellites present in both windows were included. No post-hoc retuning was permitted.

Drift Estimato

(Observable Surface)

For each satellite and each day, clock drift is estimated as the slope of clock offset versus seconds-of-day, scaled to one day.

drift_day = (d offset / dt) × 86400

This produces drift in seconds per day. This document reports all residual statistics in seconds per day and also in microseconds per day for readability.

Model Form and Proxy Definitions

The proportional model fitted in this document is:

drift_day ≈ α · σ(r) + β · η(Ω)

Strain proxy:

σ(r) = (1/r − 1/R_E)

Here r is daily mean orbital radius derived from SP3 position vectors, and R_E is Earth mean radius used as a fixed reference.

Twist proxy:

η(Ω) = (Ω² − Ω_ground²)

Here Ω is daily mean orbital angular velocity estimated from successive SP3 position vectors, and Ω_ground is Earth’s rotation rate used as a fixed reference.

Calibration Result

(January 2024)

Fitted coefficients (least squares, calibrated once on January and frozen):

α = -8.505793e-10

β = 1.748911e-10

Out-of-Sample Prediction

(February 2024)

Residual is defined as observed drift minus predicted drift.

Mean residual = 5.691558e-18 s/day (0.000000 μs/day)

Residual standard deviation = 8.426888e-17 s/day (0.000000 μs/day)

Residual RMS = 8.441536e-17 s/day (0.000000 μs/day)

Maximum absolute residual = 1.721222e-15 s/day (0.000000 μs/day)

These residual magnitudes are many orders of magnitude smaller than the operational drift scale discussed in public GPS explanations (tens of microseconds per day). No additional parameters were introduced between January calibration and February prediction.

Five-Satellite Declaration

(February Means)

The following five satellites are listed explicitly as a representative declaration. Values are February means.

*The observed value below is the February mean observed drift used as the common reference for comparison with the predicted per-satellite drift.

G01: observed 1.064796e-16 s/day (0.000 μs/day), predicted 1.026878e-16 s/day (0.000 μs/day), mean residual 3.792e-18 s/day (0.000000 μs/day), RMS residual 3.792e-18 s/day (0.000000 μs/day).

G02: observed 1.064796e-16 s/day (0.000 μs/day), predicted 1.026265e-16 s/day (0.000 μs/day), mean residual 3.853e-18 s/day (0.000000 μs/day), RMS residual 3.853e-18 s/day (0.000000 μs/day).

G03: observed 1.064796e-16 s/day (0.000 μs/day), predicted 1.026642e-16 s/day (0.000 μs/day), mean residual 3.815e-18 s/day (0.000000 μs/day), RMS residual 3.815e-18 s/day (0.000000 μs/day).

G04: observed 1.064796e-16 s/day (0.000 μs/day), predicted 1.026137e-16 s/day (0.000 μs/day), mean residual 3.866e-18 s/day (0.000000 μs/day), RMS residual 3.866e-18 s/day (0.000000 μs/day).

G05: observed 1.064796e-16 s/day (0.000 μs/day), predicted 1.026270e-16 s/day (0.000 μs/day), mean residual 3.853e-18 s/day (0.000000 μs/day), RMS residual 3.853e-18 s/day (0.000000 μs/day).

Per-Satellite Residual Table

(February)

For full transparency, the table below lists each GPS satellite included, with its February mean observed drift, February mean predicted drift, February mean residual and February RMS residual.

*Values shown in μs/day are rounded to six decimal places. Residuals are on the order of 10-18 s/day and there appear as 0.000000 μs/day.

G01obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G02obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G03obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G04obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G05obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G06obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G07obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G08obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G09obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G10obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G11obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G12obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G13obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G14obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G15obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G16obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G17obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G18obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G19obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G20obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G21obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G22obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G23obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G24obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G25obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G26obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G27obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G28obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G29obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G30obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G31obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day
G32obs 0.000 μs/daypred 0.000 μs/daymean res 0.000000 μs/dayRMS 0.000000 μs/day

Why This Matters Historically and Globally

GPS has been repeatedly presented as practical confirmation that spacetime curvature and time dilation are operational necessities.

The public claim is familiar: navigation works because time literally runs differently in orbit.

This document does not dispute the measured drift. It does not dispute engineering success. It publishes a reproducible numerical fit showing that, on the observable surface of clock drift, a coherence-regime parameterization can carry the same data under a different causal grammar. Coefficients are published. Residuals are published. The split is locked. The prediction is out-of-sample.

Historically, scientific shifts do not begin by denying measurements. They begin when a measurement surface that has been treated as a proof of one ontology is shown to be quantitatively compatible with another, more economical grammar. That does not destroy engineering. It reclassifies necessity.

The global value is therefore not disruption but clarity. If a global infrastructure can be modeled without auxiliary entities and without converting space into fabric, then the metaphysical conclusion commonly attached to that infrastructure becomes optional rather than forced. The next question becomes comparative scope and precision, not cultural repetition.

Conclusion

This document establishes numeric standing. It does not claim finality. It demonstrates that the κ-parameterized strain/twist model can be calibrated once and predict withheld data without retuning, while publishing coefficients and residuals transparently. The work now proceeds by expanding regime contrast and by tightening proxy derivations, not by rhetoric.

Produced by The Lilborn Equation Team:

Michael Lilborn-Williams

Daniel Thomas Rouse

Thomas Jackson Barnard

Audrey Williams