Many engineering failures in orientation-dependent systems are geometric failure modes: changing the material delays the failure; changing the geometry can eliminate it. The mono-monostatic property (exactly one stable equilibrium under gravity) is mathematically proven to exist in convex homogeneous bodies, but, to our knowledge, no verified smooth geometry had been openly published when v1 appeared (April 2026). We introduce an Equilibrium Count Score (ECS) oracle that measures stable equilibria via drainage basin analysis on the center-of-mass height landscape. v2 correction: v1 reported that Sloan's (2023) analytical Gomboc parameterization shows 4-11 local minima "at all published parameter values." That claim is withdrawn. The beta values swept were this work's own grid points, not Sloan's (he states convexity bounds beta <= 0.15 and beta <= 0.17); they lie inside the strictly convex sub-regime beta < ~0.036, where a body at beta = 0.023 was later certified mono-monostatic by interval arithmetic; and the oracle's two stages are biased in opposite directions, so its count can be wrong either way. We construct one strictly convex mono-monostatic body by extending the Sloan phase function with a Fourier term and optimizing via differential evolution, with ECS=1 confirmed across merge thresholds from 0.5% to 10%; two more reach ECS=1 under the oracle but are not strictly convex. Conventional geometries cannot achieve ECS=1 through ballast alone: cylinders and cubes retain multiple stable equilibria even at 30% bottom-weighted mass. Applications: an IMU calibration housing (349x orientation precision improvement), aerial reforestation seed pods (20-67% germination loss from incorrect landing orientation), and marine buoy self-righting. Cross-layer scoring shows the Gomboc is 11.8x worse than the cylinder on contact distribution while optimal on equilibrium stability.
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 v3 amendment-of-record of arXiv:2604.17120 (v3 corrects an attribution in the v2 title and abstract; no finding changes). v1 claimed Sloan'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'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. Sloan states that $β\le 0.17$ should suffice for convexity, but at $β=0.05$, inside that range, strict convexity is lost ($K_{\min}=-0.569$; $K<0$ over 4.01% of surface) and the identity's precondition fails. v1's "global surface information" mechanism is replaced by the strict-convexity precondition. Finding 2: Of v1'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's ECS=1 readings for the 9 radial-family members reflected drainage-basin merging; the $r=0.9993$ gentleness-robustness correlation is retracted.