Focal Distance Products of Outer Elliptic Billiard Polygons

We prove an explicit positive, phase-independent formula for the product of Euclidean distances from either focus of an ellipse to the consecutive boundary-tangent intersections of an elliptic billiard. The theorem assumes a nondegenerate confocal elliptic caustic and a least period divisible by four. It includes all admitted coprime star windings and repetitions of those genuine periods; concentric circular billiards are treated directly. A factorization of the actual squared focal distance, Jacobi quarter-period pairing and complete cyclic divisor cancellation give the general proof. Standard parametrization and Jacobi identities are credited, and the already known four-periodic case is not presented as new. An unrestricted divisible-by-four listed length is a different assertion: four traversals of an exact primitive triangle give two unequal distance products in the same fixed ellipse and caustic. Thus this is a complete proof of an explicit least-period theorem and a counterexample to its stronger repeated-list extension, not an unqualified closure of every interpretation of invariant k115 or frozen record AMR-050-0007 (5100007) in ulamai/UnsolvedMath v1.6.0. No hyperbolic or degenerate caustic is included. An unchanged standard-library exact checker and an optional, separately identified mpmath geometry diagnostic accompany the analytic proof. This English preprint is AI-assisted, self-audited and unrefereed. Novelty remains undetermined after a bounded primary-source review; no independent human review, proof-assistant verification or absolute-priority certification is asserted.