← 研踪 PaperTrailarXiv:2607.28654 · v3
Minimum central circles: chain order, seam obstructions, and exact finite certification
首次提交 2026-06-25 16:45
版本更新 2026-11-09 19:24
雷达观测 2026-11-10 15:22
摘要
We minimize the radius of a central circle externally tangent to pairwise non-overlapping circles of radii . 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 at its chain root, expressed by one seam deficit. For every fixed the seam eventually fails and its failure persists. We prove the complete classifications for , whose first failures occur at , respectively. For the global infimum, we certify rational brackets of exact width for every . 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. Revised exposition, clarified scope of the finite certificates, and clarified source provenance. Standalone asymptotic sequel: arXiv:2609.13630. Source code and certificate artifacts: https://github.com/falker47/ringmin
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v4 · 2026-11-12 02:18 · 最新观测版本
Minimum central circles: chain order, seam obstructions, and exact finite certification
We minimize the radius of a central circle externally tangent to pairwise non-overlapping circles of radii . 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 at its chain root, expressed by one seam deficit. For every fixed the seam eventually fails and its failure persists. We prove the complete classifications for , whose first failures occur at , respectively. For the global infimum, we certify rational brackets of exact width for every . 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.
查看此版本解读 →查看此版本原文 ↗v3 · 2026-11-09 19:24
Minimum central circles: chain order, seam obstructions, and exact finite certification
We minimize the radius of a central circle externally tangent to pairwise non-overlapping circles of radii . 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 at its chain root, expressed by one seam deficit. For every fixed the seam eventually fails and its failure persists. We prove the complete classifications for , whose first failures occur at , respectively. For the global infimum, we certify rational brackets of exact width for every . 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.
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