arXiv:2609.13630 · v3

Minimum central circles: an effective characterization of the global asymptotic constant

首次提交 2026-09-12 08:43

版本更新 2026-11-12 02:18

雷达观测 2026-11-13 14:16

摘要

Let $R^\ast(n)$ be the minimum radius of a central circle externally tangent to nonoverlapping circles of radii $1,\ldots,n$. We prove that $R^\ast(n)=C_\ast n^2+o(n^2)$ and give finite linear programs that enclose $C_\ast$ with prescribed error. An exact angular comparison reduces the problem to marked points on a line with separation at least the geometric mean of each pair of marks. Concatenation with bounded joining cost and reassignment to the prescribed normalized radii prove existence of the limit. Balanced finite-word programs give a bracket of width $(1/k+1/r)/π$, before directed arithmetic error, using $k$ types and words of length $r$. We also obtain explicit geometric bounds. For the upper bound, we recover finite and countable reflected-block constructions by actual permutations, retaining all pair constraints. A coupled terminal-subset argument gives an explicit lower bound strictly above $1/8$. Together these give $C_{\rm term}+η_{\rm width}\le C_\ast\le U_4$. The lower endpoint comes from the terminal-subset argument, and the upper endpoint from the four-block construction; neither is asserted sharp. The finite-word scheme establishes computability; efficient high-precision evaluation and the structure of global optimizers remain open.

作者备注：9 pages. Updated the reference to Paper I and added a direct link and verification commands for the frozen companion archive in section 7. Mathematical results unchanged. Standalone proof supplements S1 (14 pages) and S2 (13 pages) are supplied as ancillary PDFs, with their TeX sources included