n=16、32和64时的Reinhardt最大周长多边形问题：计算机辅助证明候选
Reinhardt's Maximum-Perimeter Polygon Problem for n=16, 32, and 64
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中文总结 AI 辅助
本文针对Reinhardt最大周长多边形问题的n=16、32、64开放情况，提出计算机辅助证明候选，采用通用证明架构完成符号编码筛选与唯一性论证，相关结果待独立评审。
中文摘要 AI 辅助
凸多边形若其直径不超过1则称为小凸多边形。Reinhardt证明了通用周长界perim(P) ≤ Uₙ:=2n sin(π/(2n))，且当n存在非平凡奇除数时该界可达。对于2的幂次n，除n=8外，其余情况均未得到精确解。本文针对前三个开放情况n=16、32、64，提出计算机辅助证明候选。每个情况的候选定理断言最大化同余类的唯一性。所有三个情况的证明架构相同：转换为差体P-P；通过符号编码对其重构进行编码；证明每个全局最大化元是饱和的，因此所有差体顶点均位于单位圆上；将每个竞争性构型局部化到正则角向量附近；使用精确算术穷尽筛选符号编码；消除所有非获胜二面体轨道；并通过强凸性和定量KKT论证证明获胜编码内部的唯一性。精确证书涵盖n=16的2¹⁵个归一化编码、n=32的2³¹个归一化编码，以及n=64的全部2⁶⁴个半编码，在轨道消除前分别剩余16、96和896个存活者。配套源代码包包含验证器、记录的输出及独立计算交叉核对结果。这些结果尚未获得独立人类专家评审，因此被刻意呈现为证明候选而非文献确立的定理。
英文摘要
A convex polygon is called small if its diameter is at most one. Reinhardt proved the universal perimeter bound $\operatorname{perim}(P)\le U_n:=2n\sin(π/(2n))$, and the bound is attained whenever $n$ has a nontrivial odd divisor. The remaining power-of-two cases have resisted exact solution beyond $n=8$. We give computer-assisted proofs of the first three cases, $n=16,32,64$, and in each case prove uniqueness of the maximizing congruence class. The proof architecture is common to all three cases: pass to the difference body $P-P$; encode its reconstruction by a sign code; prove that every global maximizer is saturated, so all difference-body vertices lie on the unit circle; localize every competitive configuration near the regular angle vector; exhaustively screen the sign codes using exact arithmetic; eliminate all nonwinning dihedral orbits; and prove uniqueness inside the winning code by strong convexity and a quantitative KKT argument. The exact certificates cover $2^{15}$ normalized codes for $n=16$, $2^{31}$ normalized codes for $n=32$, and all $2^{64}$ half-codes for $n=64$, leaving respectively $16$, $96$, and $896$ survivors before orbit elimination. The accompanying source package contains the verifiers, recorded outputs, hashes, and separate computational cross-checks.
