Phase + Width

Series I Test 1C

Introduction

Test 1C adds a minimal “width” coordinate to the twist model. The aim is to test whether allowing twist to depend on both phase and width improves geometric coherence beyond Test 1B.

Definitions

Phase coordinate: s = L₀ / 360 (mean longitude at J2000).

Width coordinate: w = signed obliquity-to-invariable / 90, derived from pole-vs-invariable geometry, with retrograde handled as underside.

Models

Plane baseline: best-fit plane to physical spin vectors.

Test 1B: φ(s)=φ₀+τ(s−0.5)

Test 1C: φ(s,w)=φ₀+τ(s−0.5)+ηw

De-twist is a rotation about the invariable pole axis k by −φ. Residuals are deviations to the best-fit plane after de-twist.

RMS Comparison

Plane only (Test 1A): 10.43°

Phase-only twist (Test 1B): 8.80°

Phase + width coupling (Test 1C): 2.38°

Best Parameters

(Coarse Search)

φ₀ ≈ 168.0°

τ ≈ 108.0° across full s-range

η ≈ -84.0° per unit width w

Per-Planet Deviations

(Degrees)

baseline = plane-only deviation.

twist = after phase-only model.

width-coupled = after phase+width model.

Planet	Baseline	Twist	Width-Coupled
Mercury	2.49°	8.90°	0.93°
Venus	7.01°	9.90°	1.97°
Earth	8.82°	8.75°	2.00°
Mars	18.63°	11.15°	3.43°
Jupiter	8.74°	7.84°	0.30°
Saturn	3.90°	4.99°	1.66°
Uranus	9.64°	0.17°	0.73°
Neptune	14.33°	12.44°	4.62°d

If the width-coupled RMS improves beyond the phase-only twist, it supports the working intuition that Möbius width affects the fit. Next refinements should replace the phase proxy with true orbital longitudes (Horizons) and replace the single coupling η with a smooth width-envelope α(s).

Produced by The Lilborn Equation Team:

Michael Lilborn-Williams

Daniel Thomas Rouse

Thomas Jackson Barnard

Audrey Williams