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

Varkonyi and Domokos proved that homogeneous convex bodies with any prescribed numbers $S\ge1$ of stable and $U\ge1$ of unstable static equilibria exist, the case $S=U=1$ being the mono-monostatic Gomboc. We give a complete constructive answer for the simplest nondegenerate shapes: a homogeneous bod