Phase Proxy:
Mean Longitude at J2000
Series I Test 1B
Introduction
Test 1B extends Test 1A by introducing a minimal twist model. The intent is to determine whether a single smooth twist applied to the planetary spin-axis vectors produces a tighter geometric coherence than a simple planar fit.
Data Sources
Planetary north-pole directions (RA/Dec, J2000) are taken from the IAU WGCCRE standard reference table. The solar-system invariable-plane pole is used as the reference axis for the twist.
A first-pass phase coordinate s is supplied by the JPL SSD Table 1 mean longitude L₀ (J2000) for each planet.
Model
1. Convert each planetary pole to a 3D unit vector.
2. Convert to the physical spin direction by flipping retrograde bodies (Venus, Uranus).
3. Apply a de-twist rotation about the invariable-plane pole axis k, using:
φ(s) = φ₀ + τ (s − 0.5)
V’ = Rot(k, −φ(s)) · V
4. Fit the best plane to the de-twisted vectors V’ and compute the angular deviation of each vector from that plane.
This is the simplest possible “twisted manifold” stress test: one axis, one smooth twist function, no ad hoc exceptions.
Baseline
(From Test 1A)
RMS deviation from best-fit plane (no twist): 10.43°
Best-Fit Twist Parameters
φ₀ ≈ 149.6°
τ ≈ -243.9° across the full s-range (linear twist rate)
RMS deviation after de-twist: 8.80°
Per-Planet Deviations
(Degrees)
Deviations are the angles of each de-twisted spin-axis vector from the best-fit plane computed after the de-twist.
| Planet | Baseline | After Twist |
| Mercury | 2.49° | 8.86° |
| Venus | 7.01° | 9.85° |
| Earth | 8.82° | 8.70° |
| Mars | 18.63° | 11.15 |
| Jupiter | 8.74° | 7.76° |
| Saturn | 3.90° | 5.19° |
| Uranus | 9.64° | 0.26° |
| Neptune | 14.33° | 12.51° |
Interpretation
The single-axis twist model improves global coherence relative to the planar baseline, reducing RMS deviation. This does not constitute proof of a Möbius topology; it establishes that a minimal smooth twist is a better fit to the planetary spin-axis geometry than a flat plane alone.
Conclusion
Next tests (1C and 1D) will introduce a width-envelope term and alternative recursion-phase coordinates to determine whether coherence strengthens under a more realistic two-coordinate topology.
Produced by The Lilborn Equation Team:
Michael Lilborn-Williams
Daniel Thomas Rouse
Thomas Jackson Barnard
Audrey Williams