A Focal Inversion Area Ratio for Four-Periodic Elliptic Billiards

The experimental invariant list of Reznik, Garcia and Koiller prints the value 2 for the focal-inverse area of the outer tangent polygon divided by that of a four-periodic elliptic billiard orbit. We give a rational counterexample to this value and a self-contained area calculation showing that the correct constant is 1/2. The result concerns vertexwise inversion of both polygons about the same focus and signed shoelace area. It applies to primitive four-periodic orbits with a confocal elliptic caustic. This is a reciprocal correction to a tabulated constant, not a refutation of the underlying constancy or a claim about the corresponding general-period conjecture. Source record: AMR-050-0048 (raw ID 5100048, k806,b), Hugging Face dataset ulamai/UnsolvedMath. Self-audited, AI-assisted, unrefereed preprint; no independent human review or formalization is claimed. The source and novelty review credits known four-periodic product invariants and makes no absolute priority claim. The source ZIP includes the English LaTeX manuscript and portable standard-library exact rational checkers.