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

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-Mon