Additional Null Models
Test 1J (b)
Introduction
This test (Test 1J (b)) expands the comparison set of orientable alternatives used in Series I. Earlier tests established that the optimized Möbius surface fits the planetary spin axes far more tightly than either a planar band or a spherical band. The purpose of This test is to determine whether any additional orientable null models can approach the performance of the Möbius twisted surface.
Models Evaluated
The following orientable null models were evaluated against the planetary spin axes:
• Planar band
• Spherical band
• Thick ecliptic disk
• Spherical cap
• Clustered random band baseline
Each model is evaluated using the same angular-residual framework used in earlier tests so that comparisons remain consistent across the entire Series I testing sequence.
Results
The quantitative comparison of models is:
Planar band RMS: ~10.43°
Spherical band RMS: ~9.31°
Thick ecliptic disk RMS: ~69.50°
Spherical cap RMS: ~66.01°
Möbius surface RMS: ~1.34°
The thick ecliptic disk and spherical cap models perform dramatically worse than even the planar and spherical band models. Neither model captures the observed directional structure of the planetary spin axes.
The Möbius surface reduces the RMS angular error by approximately an order of magnitude relative to the planar baseline and by a factor of roughly seven relative to the best spherical band.
Clustered Random Baseline
A Monte Carlo baseline of random axis clusters was generated to estimate how frequently random directional sets produce similar band structures.
Mean clustered-band RMS: ~20.97°
Standard deviation: ~5.16°
Fraction of random clusters fitting as well or better than the observed spherical band: ~0.003
This result confirms that the planetary spin axes do not resemble a random directional cluster.
Interpretation
The additional orientable null models do not close the residual gap. Curvature alone is insufficient to organize the planetary spin axes into a tight continuous surface.
The decisive improvement appears only when twist is introduced through the Möbius topology. This means the observed planetary orientation structure behaves like a twisted directional band rather than a planar or purely curved one.
Conclusion
This test strengthens the Series I result by demonstrating that the planetary spin axes are not merely planar, curved or randomly clustered. The Möbius twisted surface remains the strongest structural candidate presently identified for organizing the planetary spin axes of the solar system.
Among all continuous surfaces tested so far, the Möbius surface provides the best-performing structural description by a large margin.
Produced by The Lilborn Equation Team:
Michael Lilborn-Williams
Daniel Thomas Rouse
Thomas Jackson Barnard
Audrey Williams