Cosine Sums of Focal-Inversive Elliptic Billiards

Fix an ellipse and a nondegenerate confocal elliptic caustic admitting a periodic billiard family. We prove that inversion of the vertices about either focus preserves a phase-independent sum of internal-angle cosines, provided the genuine least period is not four. The proof includes every admissible coprime star winding. A meromorphic continuation of the inverted-edge unit direction has an explicit eight-point divisor; the only possible double-pole collision occurs at genuine period four. Reflection symmetry removes all remaining simple poles of the cyclic cosine trace. The two focal constants coincide. An exact four-periodic pair also explains why the hypothesis cannot be replaced by the length of an arbitrary repeated vertex list. The theorem addresses the elliptic-caustic, genuine-period formulation of the focal-inversive cosine conjecture, recorded as AMR-050-0043 in UnsolvedMath. Hyperbolic caustics are not asserted, and known special cases and exceptions are credited. This AI-assisted preprint is self-audited and unrefereed; no independent review, formal verification or absolute-priority certification is claimed.