一个显式的单稳态单不稳定态多面体
An explicit mono-monostatic polyhedron
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中文总结 AI 辅助
本文显式构造并精确认证了两个单稳态单不稳定态多胞体（较小者含56946个面），通过有理半空间交集和精确算术验证其平衡签名，并展示了AI代理在数学指导下完成构造与认证的流程。
中文摘要 AI 辅助
一个凸体若在水平面上受重力作用时恰好具有一个稳定平衡位置和一个不稳定平衡位置，则称为单稳态单不稳定态（mono-monostatic）。光滑的单稳态单不稳定态均匀物体存在（即Domokos和Várkonyi的Gömböc），且Lángi证明了（均匀的）单稳态单不稳定态多面体存在；尽管据作者所知，似乎尚未有显式例子被发表。我们显式构造了两个这样的单稳态单不稳定态多胞体，其中较小的一个具有56946个面；重要的是，我们对它们进行了认证：该多胞体被表示为具有有理数据的半空间交集，并且一个认证性验证通过使用精确有理算术进行每一次决定性比较，确立了该物体相对于其自身精确质心具有平衡签名(S,H,U)=(1,0,1)，并带有显式的非退化裕度。我们描述了使光滑单稳态单不稳定态物体的朴素离散化失败的几何障碍、克服这些障碍的自适应构造方法以及认证策略。虽然结果本身并非特别新颖，但我们强调非标准（但日益常见）的方法论：整个程序，即实验、构造和验证器本身，均由AI代理在人类数学指导下实现。我们认为，对数值发现的物体进行精确认证是此类AI辅助工作流程与数学严谨性标准之间的自然契约。
英文摘要
A convex body is mono-monostatic if, resting under gravity on a horizontal plane, it has exactly one stable and one unstable equilibrium position. Smooth mono-monostatic homogeneous bodies exist (the Gömböc of Domokos and Várkonyi), and Lángi proved that (homogeneous) mono-monostatic polyhedra exist; although no explicit example appears to have been published, to the author's knowledge. We construct explicitly two such mono-monostatic polytopes, the smaller one having $56946$ faces; importantly, we certify them: the polytope is presented as an intersection of half-spaces with rational data, and a certifying verification establishes, using exact rational arithmetic for every decisive comparison, that the body has equilibrium signature $(S,H,U)=(1,0,1)$ with respect to its own exact centroid, with explicit nondegeneracy margins. We describe the geometric obstructions that make naive discretisations of smooth mono-monostatic bodies fail, the adaptive construction that overcomes them, and the certification strategy. While the result itself is not strikingly novel, we emphasise the non-standard (but increasingly more common) methodology: the entire programme, i.e. experiments, constructions and the verifier itself, was implemented by AI agents under human mathematical direction. We argue that exact certification of numerically discovered objects is the natural contract between such AI-assisted workflows and mathematical standards of rigour.
发表机构
- Mathematisches Institut, Universität Bonn（波恩大学数学研究所）
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