Static Equilibria of Perturbed Spheres: A Single-Harmonic Class Map, a Parity Obstruction, and a Certified Counter for the Mono-Monostatic Regime

Abstract

Varkonyi and Domokos proved that homogeneous convex bodies exist with any prescribed numbers $S\ge1$ of stable and $U\ge1$ of unstable static equilibria, the case $S=U=1$ being the mono-monostatic Gomboc. Their result is an existence statement. We give a complete constructive answer for the simplest nondegenerate shapes: a homogeneous body whose boundary is the unit sphere perturbed radially by a single real spherical harmonic $Y_\ell^m$ ($\ell\ge2$, $1\le m\le\ell$) has exactly $S = U = m(\ell-m+1)$ stable and unstable equilibria, for every amplitude in the convex range when $m\ge2$ and for small amplitude when $m=1$. For $m\ge2$ the reduction to the critical points of the harmonic is an identity: the symmetry of a single tesseral harmonic pins the centroid at the origin exactly, so the centroid-to-surface distance is a strictly increasing function of the harmonic. We count the critical points of $Y_\ell^m$ and verify the Poincare-Hopf balance in index form, the polar monkey-saddles persisting unsplit with index $1-m$. Three consequences follow: a single harmonic populates only the diagonal $S=U$, so none of degree $\ge2$ is mono-monostatic; any centrally symmetric (even-degree) perturbation has even $S,U$, a parity obstruction; hence mono-monostaticity is intrinsically multi-harmonic. A predict-then-confirm study on eight bodies matches the formula exactly. For the multi-harmonic regime, where mono-monostatic bodies live, we give a certified equilibrium counter (interval arithmetic on the centroid-to-surface distance, Krawczyk uniqueness, interval-Hessian classification, stereographic polar charts) that, given the centroid, provably neither under- nor over-counts. It certifies specific near-spherical bodies mono-monostatic, including a known analytic parameterization, settling by certified computation a question that drainage-basin and seed-based methods leave ambiguous.