Semiprime Rigidity and Growth along Powers in Integer-Sided Cyclic Polygons

For a positive integer n, let a(n) be the largest number of sides of a simple cyclic polygon of circumradius n whose side lengths are positive integers. We classify semiprime radii exactly: a(pq) can only be 6, 8, or 18, with the exceptional values characterized by Gaussian and Eisenstein angle conditions. In sharp contrast, for every fixed integer N >= 2, the values along its powers satisfy a(N^e) -> infinity and limsup a(N^e)/e >= 2. A quadratic chord tower and Farey neighbors give explicit constructions, including a(8192) >= 38. We also prove that a(n) is even whenever at most one distinct prime divisor of n is congruent to 1 modulo 4. The proofs use Conway--Radin--Sadun angle splitting, arithmetic in quadratic fields, an exact finite verification for even semiprime radii, and a one-dimensional Farey bound.