Phase + Width Envelope
Series I Test 1D
Introduction
This test (Test 1D) replaces the single width-coupling term of the previous test with a smooth width-envelope α(s). This implements the intuition that the Möbius strip’s effective width can expand or compress as a function of recursion phase.
Definitions
Phase coordinate: s = L₀ / 360 (mean longitude at J2000; phase proxy).
Width coordinate: w = signed obliquity-to-invariable / 90.
Width envelope: α(s) = 1 + a cos(2πs) + b sin(2πs) (single smooth harmonic).
Model
φ(s,w) = φ₀ + τ(s−0.5) + η[α(s)·w]
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 re-fit): 8.80°
Phase + width coupling (Test 1C re-fit): 2.51°
Phase + width-envelope (Test 1D): 1.19°
Best-Fit Parameters (Test 1D; random search)
φ₀ ≈ 223.8°
τ ≈ 116.7° across full s-range
vη ≈ -88.4° per unit (α·w)
a ≈ -0.327
b ≈ -0.297
Per-Planet Deviations (degrees)
baseline = plane-only
twist = phase-only (Test 1A re-fit)
width-coupled = phase+width (Test 1B re-fit)
width-envelope = phase + α(s)·w (Test 1D)
| Planet | Baseline | Twist | Width-Coupled | Width-Envelope | α(s) |
| Mercury | 2.49° | 8.85° | 0.18° | 0.37° | 1.383 |
| Venus | 7.01° | 9.85° | 2.77° | 1.79° | 1.337 |
| Earth | 8.82° | 8.70° | 1.52° | 0.42° | 0.767 |
| Mars | 18.63° | 11.15° | 2.6° | 1.35° | 0.698 |
| Jupiter | 8.74° | 7.76° | 0.75° | 0.06° | 0.562 |
| Saturn | 3.90° | 5.20° | 2.93° | 1.17° | 0.562 |
| Uranus | 9.64° | 0.26° | 0.33° | 0.08° | 0.992 |
| Neptune | 14.33° | 12.51° | 4.94° | 2.16° | 1.057 |
Interpretation
This test implements a smooth, phase-dependent width mechanism rather than an arbitrary planet-by-planet adjustment. If the RMS improvement persists under this constrained envelope, it supports the claim that Möbius width is not constant and that phase and width interact in fitting planetary spin-axis geometry.
Next
Upgrade the phase proxy using Horizons true longitudes (or ephemeris vectors) to remove dependence on mean-longitude approximations.
Produced by The Lilborn Equation Team:
Michael Lilborn-Williams
Daniel Thomas Rouse
Thomas Jackson Barnard
Audrey Williams