Arranging circles of radii 1,2,...,n around a central circle: a Supnick TSP and certified finite optima

Abstract

We study a discrete-geometric optimization problem: circles of radii $1,2,\dots,n$ are all externally tangent to a central circle, and the central radius $R$ is minimized over cyclic orders of the surrounding circles. We prove that the chain-ordering component is governed by a fixed Supnick/anti-Monge traveling-salesman order. For every $R$, the angular-separation matrix is symmetric anti-Monge, so Supnick's theorem gives one minimizing cyclic order, independent of $R$. This proves the conjectured "pyramid" order optimal whenever the corresponding chain necklace is geometrically realizable, and gives an unconditional lower bound in all cases. Full geometric feasibility can fail because non-adjacent circle constraints are not captured by the chain equation; from $n=8$ the smallest circle can become a floating circle tangent only to the central circle. We formulate the full problem as a circular system of pairwise angular constraints, equivalently a simple temporal network, and certify global optima for $3\le n\le14$ using branch-and-bound plus an independent 50-digit verifier. We observe heuristically that the floating-circle cascade continues beyond the certified range, and we state the continuation and the asymptotic form $R^\ast(n)=n^2/8(1+o(1))$ as conjectures. The repository contains the saved certificate artifacts, verifier, and reproducibility commands.