A Telescoping Formula for Evolute Areas in Elliptic Billiards

For a noncircular elliptic billiard with a nondegenerate confocal elliptic caustic, we derive closed formulas for the signed area ratios of the caustic contact polygon and the boundary tangent polygon to their discrete evolutes. The formulas hold for every physical periodic orbit of least period greater than four, including star trajectories. Unit complex contact parameters satisfy a biquadratic relation. The consecutive circumcenters have a rational expression whose local area term differs from a constant multiple of the contact-area term by an explicit rational coboundary. A polynomial certificate and cyclic summation prove the inner formula. Conic polarity and a fixed-conic circumcenter transformation yield the outer formula. The exceptional coefficients force least periods three or four, which proves that the stated quotients have nonzero denominators. Exact examples also show why repeating primitive four-cycles does not extend the theorem to a list-length interpretation of period. The results address invariants k703 and k702 of Reznik, Garcia and Koiller in one manuscript, corresponding to AMR-050-0041 and AMR-050-0040 in the frozen ulamai/UnsolvedMath v1.6.0 dataset. The deposit includes the English manuscript, source archive and verification report. AI assistance was used for exploration, exact computation, literature searching and manuscript preparation. The author is responsible for the results. This is a self-audited, unrefereed preprint, without independent human review or proof-assistant formalization. The bounded literature review does not establish absolute priority. A DOI or public posting is not certification of correctness or novelty.