κ Parameterization And Residual Analysis

Introduction

This document provides the quantitative structure omitted intentionally from GPS Counter Drift for clarity of public communication. It defines explicit proxy functions for strain and twist, calibrates proportional coefficients on a locked training window (January 2024), evaluates prediction on a withheld test window (February 2024), and reports residual behavior. The objective is not rhetorical support but structural sufficiency.

Drift Estimator Definition

For each satellite and each daily clock file, drift is defined as the linear slope of clock offset versus time-of-day.

If offset(t) is the clock offset in seconds and t is seconds-of-day, then:

drift_day = (d offset / dt) × 86400

This yields daily drift in seconds per day. January days (DOY 001–031) are designated as the training window. February days (DOY 032–060) are designated as the test window.

κ Model Structure

Cascade mismatch between orbit and ground regimes is modeled in proportional form as:

κ_o − κ_g = κ₀ [ α σ(r_o) + β η(Ω_orb) ]

Where κ₀ is baseline cascade scale, σ(r_o) is a strain proxy based on orbital altitude, and η(Ω_orb) is a twist proxy based on orbital angular embedding.

Explicit Proxy Definitions

Strain proxy σ(r) is defined as proportional to inverse radial separation relative to Earth surface:

σ(r) = (1/r − 1/R_E)

Where r is orbital radius and R_E is Earth mean radius.

Twist proxy η(Ω_orb) is defined proportional to squared orbital angular velocity relative to ground frame:

η(Ω_orb) = Ω_orb² − Ω_ground²

Calibration on Training Window

(January 2024)

Using January drift data across 32 satellites, coefficients α and β are estimated via least-squares regression of observed drift_day against σ(r) and η(Ω_orb). Calibration is performed once and frozen prior to test evaluation.

Prediction on Withheld Window

(February 2024)

Using frozen α and β, predicted drift for February satellites is computed without parameter adjustment. Residual is defined as observed minus predicted drift.

Residual Summary

Across all satellites, residual magnitude remains within statistical tolerance defined as ±2 standard deviations of January residual distribution. No regime-specific retuning was required. No additional free parameters were introduced.

Degrees of Freedom Comparison

The Æ model uses two physically motivated proportional coefficients (α, β). The standard relativistic correction uses gravitational potential term and velocity term derived from spacetime metric. Both frameworks use two primary scalar contributions to reproduce drift magnitude. The difference lies in ontological interpretation rather than parameter count.

Conclusion

The January calibration and February prediction confirm that GPS clock drift can be quantitatively carried under the κ proportional structure using observable altitude and angular proxies. Residual behavior remains stable under the locked split. This does not disprove relativistic interpretation. It demonstrates structural sufficiency of the coherence framework without auxiliary constructs.

Produced by The Lilborn Equation Team:

Michael Lilborn-Williams

Daniel Thomas Rouse

Thomas Jackson Barnard

Audrey Williams