Sloan's Analytical G\"omb\"oc Does Not Produce Mono-Monostatic Bodies: Computational Verification, Extended Construction, and a Thirteen-Member Verified Catalog

TL;DR

This paper computationally verifies Sloan's analytical bodies, extends the construction method, and publishes the first verified catalog of thirteen mono-monostatic geometries with detailed robustness and gentleness trade-offs.

Contribution

It introduces a new ECS oracle for stability measurement, extends Sloan's phase function with Fourier terms, and provides the first open catalog of verified mono-monostatic bodies.

Findings

No tested Sloan parameter produces a mono-monostatic body.

Thirteen verified mono-monostatic bodies are constructed and published.

A near-perfect trade-off between robustness and gentleness is observed.

Abstract

Varkonyi and Domokos (2006) proved mono-monostatic convex homogeneous bodies exist, and Sloan (2023) provided analytical equations. However, no verified open geometry exists and the relationship between Sloan's parameterization and actual mono-monostatic behavior has never been computationally tested. We introduce an ECS oracle measuring stable equilibria via drainage basin analysis on the COM height landscape. Applying it to Sloan's parameterization, we find no tested parameter value produces a mono-monostatic body. The surface function has two critical points as proven, but the COM height landscape exhibits 4-11 local minima. Surface critical points are necessary but not sufficient. We resolve this by extending Sloan's phase function with Fourier terms and adding radial perturbations, optimizing via differential evolution. Using both approaches, we construct thirteen verified mono-monostatic bodies (ECS=1 confirmed across merge thresholds 0.5-10%), the first openly published catalog of verified mono-monostatic geometries. The catalog reveals a near-perfect trade-off between self-righting robustness and gentleness (r=0.9993), spanning 7.2x asymmetry range from 0.4% to 3.0% deviation from spherical. Both perturbation families produce overlapping metric profiles, indicating construction method is a convenience not a constraint. All thirteen meshes and construction parameters are openly published.