Sloan's Analytical Gömböc at Published $β$: A Strict-Convexity-Constrained Reanalysis

Abstract

Varkonyi and Domokos (2006) proved that convex homogeneous bodies with exactly one stable and one unstable equilibrium point exist. Sloan (2023) gave the first analytical parameterization, with radial function $R(θ,ϕ)$ having exactly two critical points on $S^2$. This is the v2 amendment-of-record of arXiv:2604.17120. v1 claimed Sloan&#39;s parameterization does not produce mono-monostatic bodies and reported a 13-member catalog of Fourier/radial extensions certified at ECS=1 via mesh-vertex drainage-basin analysis. Following correspondence with P. L. Varkonyi (BME), an analytical verification suite was built around the Varkonyi-Gauss identity. Finding 1: Sloan&#39;s parameterization does produce mono-monostatic bodies in a strictly-convex sub-regime ($β\lesssim 0.036$), where $K_{\min} > 0$ and the identity certifies ECS=1. v1 missed this because its mesh-vertex oracle over-counted on shallow COM-height landscapes. At Sloan&#39;s published $β=0.05$, strict convexity is lost ($K_{\min}=-0.569$; $K<0$ over 4.01% of surface); the identity&#39;s precondition fails. v1&#39;s &#34;global surface information&#34; mechanism is replaced by the strict-convexity precondition. Finding 2: Of v1&#39;s 13 catalog instances only Phase-1 ($β=0.023149$, $a_1=0.234433$, $k=1$) survives identity-based verification; the remaining twelve were per-$k$ optimizer extrema overshooting the strict-convex boundary. Probing the regime interior verifies further mono-monostatic bodies in $k=2$ and $k=3$ sub-families: the verified set is an open regime in $(β, a_1, k)$, not a discrete list. Finding 3: v1&#39;s ECS=1 readings for the 9 radial-family members reflected drainage-basin merging; the $r=0.9993$ gentleness-robustness correlation is retracted.