AI 中文总结
该研究针对半径1到n的圆围绕中心圆排列的优化问题，证明其最优循环顺序为Supnick顺序，验证3≤n≤14的全局最优解，提出R*(n)的渐近形式猜想。
AI 中文摘要
我们研究一个离散几何优化问题：半径为1,2,…,n的所有圆均与一个中心圆外切，且围绕这些圆的循环排列需使中心圆半径R最小。我们证明，链序分量由固定的Supnick/反Monge旅行商顺序决定。对于任意R，角分离矩阵为对称反Monge矩阵，因此Supnick定理给出一个与R无关的最小循环顺序。这证明了当对应的链项链几何可实现时，猜想的“金字塔”顺序是最优的，并在所有情况下给出无条件下界。由于非相邻圆约束未被链方程捕获，完整几何可行性可能失效；从n=8开始，最小的圆可能成为仅与中心圆相切的浮动圆。我们将完整问题表述为成对角约束的循环系统，等价于简单时间网络，并使用分支定界法加独立的50位验证器，为3≤n≤14的情况验证全局最优解。我们从启发式角度观察到，浮动圆级联会持续到已验证范围之外，并将其延续性和渐近形式R*(n)=n²/8(1+o(1))作为猜想。该代码库包含已保存的验证工件、验证器和可复现命令。
英文摘要
We study a discrete-geometric optimization problem: circles of radii $1,2,\dots,n$ are all externally tangent to a central circle, and the central radius $R$ is minimized over cyclic orders of the surrounding circles. We prove that the chain-ordering component is governed by a fixed Supnick/anti-Monge traveling-salesman order. For every $R$, the angular-separation matrix is symmetric anti-Monge, so Supnick's theorem gives one minimizing cyclic order, independent of $R$. This proves the conjectured "pyramid" order optimal whenever the corresponding chain necklace is geometrically realizable, and gives an unconditional lower bound in all cases. Full geometric feasibility can fail because non-adjacent circle constraints are not captured by the chain equation; from $n=8$ the smallest circle can become a floating circle tangent only to the central circle. We formulate the full problem as a circular system of pairwise angular constraints, equivalently a simple temporal network, and certify global optima for $3\le n\le14$ using branch-and-bound plus an independent 50-digit verifier. Continuation of the floating-circle pattern beyond that range remains conjectural. The v1 conjecture $R^\ast(n)=n^2/8(1+o(1))$ has been disproved by subsequent work (see the cited standalone sequel); this correction does not invalidate the finite results. The repository contains the saved certificate artifacts, verifier, and reproducibility commands.
Comments12 pages, 2 figures. Corrective v2: preserves the certified finite results; corrects superseded v1 asymptotic conjectures and clarifies floating-circle quantifiers. Standalone asymptotic sequel: arXiv:2609.13630. Source code and certificate artifacts: https://github.com/falker47/ringmin
