AI 中文总结
该研究构造性证明单球谐扰动球体的平衡数公式，揭示单谐无法实现单单静态，提出多谐 regime 的已验证平衡计数方法，解决相关歧义问题。
AI 中文摘要
Varkonyi 和 Domokos 证明，存在齐次凸体，其稳定静态平衡数 S≥1 和不稳定静态平衡数 U≥1 可任意指定，其中 S=U=1 的情况为单单静态 Gomboc。他们的结果是一个存在性命题。我们对最简单的非退化形状给出完整构造性解答：边界为单位球面、经单个实球谐函数 Y_ℓ^m（ℓ≥2，1≤m≤ℓ）径向扰动的齐次体，在 m≥2 时的凸振幅范围内，以及 m=1 时的小振幅范围内，恰好具有 S = U = m(ℓ−m+1) 个稳定和不稳定平衡。对于 m≥2，向谐函数临界点的约化是恒等映射：单个田谐函数的对称性将质心精确固定在原点，因此质心到表面的距离是谐函数的严格递增函数。我们计数 Y_ℓ^m 的临界点，并以指数形式验证 Poincaré-Hopf 平衡，极猴鞍点保持不分裂，指数为 1−m。由此得出三个结论：单个谐函数仅填充 S=U 的对角线，因此次数≥2 的谐函数均非单单静态；任何中心对称（偶次）扰动的 S、U 均为偶数，存在奇偶性障碍；因此单单静态性本质上是多谐的。对八个物体的“先预测后验证”研究与公式完全匹配。对于单单静态体所在的多谐 regime，我们给出一个已验证的平衡计数反例（基于质心到表面距离的区间算术、Krawczyk 唯一性、区间 Hessian 分类、极射赤平极图），给定质心后，可保证既不遗漏也不多计。它验证了特定近球体为单单静态，包括已知的解析参数化，通过已验证计算解决了 drainage-basin 法和基于种子的方法无法明确的问题。
英文摘要
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.
Comments10 pages, 1 figure
