Test 1K

Introduction

This test (Test 1K) examines whether the Möbius surface identified in Series I has predictive power rather than merely descriptive power. Earlier tests established that the planetary spin axes are far better organized by a twisted Möbius surface than by planar, spherical or other orientable alternatives. The next question is whether that structure can successfully predict the position of a planetary spin axis that was not included in the fit.

Method

The leave-one-out procedure is simple and rigorous.

For each planet in the dataset:
• remove that planet from the fit

• optimize the Möbius surface using the remaining seven planets

• do not use the removed planet during optimization

• measure the angular residual between the removed planet’s spin axis and the predicted surface

This process is repeated for all eight planets.

The result is a set of out-of-fit prediction residuals rather than a single descriptive fit to all planets at once.

Why This Test Matters

A model that fits the same points used to build it can always be accused of overfitting. This test removes that objection by forcing the Möbius surface to predict the held-out planet.

If the held-out planets continue to fall close to the predicted twisted surface, then the Möbius organization is not simply a way of describing the existing dataset. It becomes a predictive structural model.

Evaluation Metrics

The following quantities are to be recorded for each leave-one-out realization:
• held-out planet name

• RMS angular error of the seven-planet fit

• angular residual of the held-out planet to the predicted surface

• optimized strip parameters (major radius, width, rotation), when needed

The key value is the residual of the held-out planet. Low residuals indicate predictive continuity of the twisted surface.

Interpretation

If the held-out planets continue to land near the predicted surface, then the Möbius structure is not merely fitting the known set; it is capturing an underlying organization of the planetary spin axes.

If the held-out residuals grow large, then the Möbius surface may be overfitting the full eight-planet dataset and the structural claim would weaken accordingly.

Expected Outcome

Given the low residuals already observed in the continuous eight-planet optimization, the expectation is that most held-out planets will still land near the predicted strip. The most informative cases will likely be Venus and Uranus, since they occupy the inversion region of the twisted surface.

A strong outcome in this test would show that the Möbius topology possesses predictive power within the planetary dataset itself.

Conclusion

Test 1K is the natural completion of the Series I testing ladder. Earlier tests established that twist is required and that the Möbius surface remains the best-performing continuous surface among the models examined. This test now asks whether that same twisted topology can predict withheld planetary orientations.

A strong result here would mark the transition from descriptive fit to predictive structure.

Produced by The Lilborn Equation Team:

Michael Lilborn-Williams

Daniel Thomas Rouse

Thomas Jackson Barnard

Audrey Williams