AI 中文总结
本文证明最小中心圆半径的渐近常数$C_*$存在，并通过有限线性规划有效刻画，给出显式区间，否定了先前提出的$1/8$系数。
AI 中文摘要
设${R^\ast}(n)$为半径为$1,\ldots,n$的不重叠圆外切于一个中心圆的最小半径。我们证明${R^\ast}(n)={C_\ast} n^2+o(n^2)$，并通过有限线性规划刻画${C_\ast}$，其显式误差趋于零。该归约保留了任意顺序和所有成对约束：极限问题将标记点置于一条直线上，成对间距至少为其标记的几何平均值。与有界边界代价的拼接证明了存在性，而平衡有限字程序提供了匹配的有效上界和下界。在定向算术误差之前，对于$k$种标记类型和长度为$r$的字，其认证间隙为$(1/k+1/r)/\pi$。一个定量的反射块恢复定理提供了真实的排列和完整的环几何，并具有可数扩展和严格四块改进。显式区间为$C_{\mathrm{term}}+\eta_{\mathrm{width}}\le{C_\ast}\le U_4$；两个端点均未断言为最优。特别地，先前有限研究中提出的系数$1/8$是错误的。${C_\ast}$的初等表达式、高效高精度评估和全局浮点圆结构仍然是开放问题。
英文摘要
Let ${R^\ast}(n)$ be the least radius of a central circle to which nonoverlapping circles of radii $1,\ldots,n$ are externally tangent. We prove that ${R^\ast}(n)={C_\ast} n^2+o(n^2)$ and characterize ${C_\ast}$ by finite linear programs with an explicit error tending to zero. The reduction preserves arbitrary orders and all pairwise constraints: the limiting problem places marked points on a line at pairwise separation at least the geometric mean of their marks. Concatenation with a bounded boundary cost proves existence, and balanced finite-word programs supply matching effective upper and lower bounds. Their certified gap is $(1/k+1/r)/π$, before directed arithmetic error, for $k$ mark types and words of length $r$. A quantitative reflected-block recovery theorem supplies genuine permutations and full ring geometry, with a countable extension and a strict four-block improvement. The explicit interval is $C_{\mathrm{term}}+η_{\mathrm{width}}\le{C_\ast}\le U_4$; neither endpoint is asserted sharp. In particular, the coefficient $1/8$ proposed in the preceding finite study is false. An elementary expression for ${C_\ast}$, efficient high-precision evaluation and global floating-circle structure remain open.
Comments8 pages, no figures. Asymptotic sequel to arXiv:2607.28654v1. Proves existence and an effective finite-program characterization of the global asymptotic constant. Proofs and code: https://github.com/falker47/ringmin/tree/3beb8d70c5b3748d370a92855847bdf574e5a14f
