Nonconstant Antipedal Area Ratios in Four-Periodic Elliptic Billiards

For a periodic elliptic billiard, let A' be the area of the polygon formed by consecutive tangents to the ellipse, and let A*_M be the antipedal area of the original orbit with respect to a fixed point M. We give an exact counterexample to the claimed constancy of A'/A*_M for periods divisible by four, listed as k402 in Table 5 of Reznik, Garcia and Koiller's 2021 article Fifty New Invariants of N-Periodics in the Elliptic Billiard. The denominator is unprimed and the printed point condition is All; taking M to be the center O is allowed. On a standard connected family of convex primitive four-periodic orbits in a fixed noncircular ellipse with one fixed confocal elliptic caustic, put s=a^2+b^2, D=a^2 b^2 and d(t)=(b^2-a^2) sin(t) cos(t). We derive A'/A*_O=D(D+d(t)^2)/(D^2+s^2 d(t)^2), with sharp range [D s^2/(s^2-2D)^2, 1] on the displayed family. For a=4,b=3, two orbits share the caustic with semiaxes 16/5 and 9/5 and give the ratios 1 and 90000/113569. The analytic continuous-family certificate proves reflection and strictly interior caustic tangency; all witness polygons are finite, convex and positively oriented. The four-periodic family and caustic are known geometry and are explicitly credited. This note refutes the literal k402 assertion and frozen dataset record AMR-050-0019 (raw ID 5100019), not every source invariant. A bounded primary-source search did not identify a correction of this exact row but does not certify novelty or absolute priority. Two standalone exact rational checkers accompany the analytic proof. This AI-assisted manuscript is unrefereed; no independent human review or proof-assistant verification is claimed. Alper Ferudun, Mercury Software GmbH. Contact: [email protected]. GitHub: https://github.com/AlperTheKing.