arXiv:2607.28654 · v4

Minimum central circles: chain order, seam obstructions, and exact finite certification

首次提交 2026-06-25 16:45

版本更新 2026-11-12 02:18

雷达观测 2026-11-13 14:16

摘要

We minimize the radius of a central circle externally tangent to pairwise non-overlapping circles of radii $1,2,\ldots,n$. We first study the adjacent-chain relaxation, which imposes only the constraints between consecutive circles. Its angular kernel has strict anti-Monge structure, so Supnick's classical theorem supplies a fixed minimizing cyclic order at every central radius. A monotonicity argument transfers this order to the chain-radius problem. Full geometric feasibility requires every pairwise angular constraint. We give a self-contained criterion for feasibility of the canonical Supnick necklace on $\{k,\ldots,n\}$ at its chain root, expressed by one seam deficit. For every fixed $k$ the seam eventually fails and its failure persists. We prove the complete classifications for $k=1,2,3$, whose first failures occur at $n=8,13,17$, respectively. For the global infimum, we certify rational brackets $L_n<R^*(n)\le U_n$ of exact width $10^{-11}$ for every $3\le n\le14$. The lower proof combines independently enclosed angles, integer dynamic programming, and complete cyclic-order coverage; the upper proof uses exact rational Cartesian witnesses. The certificate does not depend on floating-point pruning in the numerical search. We distinguish these finite brackets and fixed-order theorems from numerical descriptions of floating circles and from unresolved claims about all optimal placements.

作者备注：18 pages. Streamlined the reproducibility statement, data availability and author declarations. Proofs and finite certificates unchanged. Standalone asymptotic sequel: arXiv:2609.13630. Source code and certificates: https://github.com/falker47/ringmin