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

Let $R^{\ast }(n)$ be the least radius of a central circle to which nonoverlapping circles of radii $1,\dots ,n$ 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 $(1\mathrm{/}k+1\mathrm{/}r)\mathrm{/}\pi$, 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 $C_{\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}}+{\eta }_{\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}}\le C_{\ast }\le U_4$; neither endpoint is asserted sharp. In particular, the coefficient $1\mathrm{/}8$ 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.