Focal Inversion Area Ratios in Elliptic Billiards

For every nondegenerate confocal elliptic-caustic billiard of genuine least period n >= 6 with n congruent to 2 modulo 4, we prove that the original ordered signed area divided by the signed area after unit inversion about either boundary-ellipse focus is a positive phase-independent constant, identical at the two foci. The proof includes all admitted coprime star windings and establishes finite inverse vertices and a nonzero denominator. In confocal Jacobi coordinates, the two area traces have the same two possible simple poles on a reduced quotient torus. Residue cancellation and imaginary antiperiodicity force proportionality; real geometry proves positivity. Repetitions of admitted primitive cycles preserve the ratio, and the circular case is treated directly. The target is k805 in arXiv:2004.12497v11, corresponding to frozen record AMR-050-0046 / 5100046. Garcia and Reznik's 2022 Proposition 4.16 already gives the simple-six-periodic formula, which is expressly credited. Exact controls reproduce the ratio 32/27 at two six-periodic phases. Two primitive triangles in one fixed family, each traversed twice to give listed length six, instead have ratios (10584 + 72 sqrt(105))/25 and (10584 - 72 sqrt(105))/25; this excludes only the stronger unrestricted listed-length interpretation. A separately executed 85-decimal diagnostic covers 27 target families and 18 odd-period controls, five phases each; these finite checks are not the general proof. No hyperbolic or degenerate caustic extension or unqualified whole-source closure is claimed. This English preprint is AI-assisted, originating-researcher self-audited and unrefereed. Novelty remains undetermined; no independent human review, proof-assistant verification, guaranteed indexing or absolute-priority claim is made.