arXiv:2604.17095 · v2
Computational Construction and Engineering Evaluation of a Verified Mono-Monostatic Body
摘要
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 That claim is withdrawn.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.
作者备注：14 pages, 3 figures, 7 tables. Part III of the Substrate Geometry series. v2 amendment-of-record: withdraws the claim that Sloan's parameterization shows 4-11 local minima at published parameter values; narrows three verified bodies to one strictly convex body (two others are oracle-level only). Part I: arXiv:2604.12238. Part II: in preparation