Equal Focal Inverted Pedal Areas Can Vanish in Elliptic Billiards

For an elliptic billiard with a nondegenerate confocal elliptic caustic and effective even period, we prove that the pedals of the two unit-focus-inverted orbit polygons are congruent and have equal ordered signed areas. Equality does not imply nonvanishing. We construct a continuous family of genuine ten-periodic billiards of winding three and certify opposite signs of the common area at two rational major semiaxes. Exact rational interval arithmetic and the intermediate value theorem give a member for which both areas are zero. The construction satisfies the reflection law, has ten distinct vertices and a nondegenerate confocal elliptic caustic. As a contrast, we give an explicit positive formula for the counterclockwise four-periodic case. The focal area ratio remains one wherever its denominator is nonzero; the common-zero example obstructs an everywhere-defined literal quotient, not that ratio identity on its natural domain. Self-audited, unrefereed preprint prepared with AI assistance. Classical symmetry is credited; application novelty is undetermined after bounded literature searches. No independent human review, formal proof-assistant verification or absolute-priority claim is made. The source identity is AMR-050-0068 (raw ID 5100068), invariant k908,a in the frozen ulamai/UnsolvedMath v1.6.0 dataset. This is not a hyperbolic-caustic or whole-source-list closure. PDF, English LaTeX source and standard-library exact reproducibility checkers are included. Author: Alper Ferudun, Mercury Software GmbH.