Optimal nestings of all regular polygons

Starting from an equilateral triangle, place every other regular polygon once outside it, or independently once inside it, and optimize each containment over rotations and translations. We determine the infimum over all exhaustive orders. No ordinary permutation attains it: every optimal adjacency graph begins with the entire doubling ray 3, 6, 12, and so on. The infimum is attained by a countable linear order whose infinite blocks meet at limiting circles. Two exceptional blocks contain the square and the powers of two; all other blocks are given by an explicit matching of dyadic rays along powers of three. A summable arithmetic dual proves global optimality, and equality determines the unique unoriented adjacency graph. The infinite inequalities are analytic, with directed interval certificates for their finite remainder. In the normalization where the triangle has circumradius one, the optimal outer and inner circle radii are 2.3123644734... and 0.2162288885.... We compare these values with the Kepler-Bouwkamp construction and two direct adjacent-order models, and prove that the triangle is the unique best central side count for the two-sided radius ratio.

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