Four-Periodic Counterexamples to Two Elliptic Billiard Invariants

We give two exact counterexamples using one connected family of convex, primitive four-periodic billiards in a noncircular ellipse with a fixed confocal elliptic caustic. Write A for the orbit's ordered signed area, A' for the area of its consecutive tangent-intersection polygon, A'' for the area of its caustic-contact polygon, and theta_i for the internal angles of the original orbit. The quantity (A'/A) product_i sin(theta_i/2), listed as k107 for N congruent to 0 modulo 4, varies on this family. So does the product A'A'', listed as old k111 for even N in arXiv:2004.12497v11. The first assertion survives in the final 2021 journal table; the old second row does not. The renumbered final-journal k111 is a different, odd-period identity and is not contradicted. With s=a^2+b^2, D=a^2 b^2 and d(t)=(b^2-a^2) sin(t) cos(t), the displayed family has A'A''=16D(D+d(t)^2)/s^2 and (A'/A) product_i sin(theta_i/2)=2(D+d(t)^2)^2/(D s^2). Both vary whenever a>b>0. For a=4, b=3, an axis diamond and an axis-aligned rectangle share the caustic with semiaxes 16/5 and 9/5. Their area products are 331776/625 and 576, and their angle-area quantities are 288/625 and 625/1152. Boundary membership, reflection, interior segment tangency, primitive period and positive signed areas are established explicitly. The contradiction does not rely on self-intersection, hyperbolic caustics or numerical inference. The standard four-periodic geometry and Garcia and Reznik's Proposition 4.9 area identities are credited. The old product row's omission is documented without speculating about its reason. This is one version-qualified corrective note covering frozen dataset records AMR-050-0001 and AMR-050-0005 (raw IDs 5100001 and 5100005), not two deposits or a solution of the full source list. Bounded literature checks do not establish absolute novelty or priority. The manuscript is AI-assisted and unrefereed; no independent human review or formal proof-assistant verification is claimed. Author affiliation and contact: Mercury Software GmbH; [email protected]; https://github.com/AlperTheKing.