A flag-curve proof of an original-side pedal-area invariant

Abstract

For a noncircular ellipse and a proper nested confocal elliptical caustic whose primitive least billiard period is divisible by four, the signed original trajectory area times the signed area of its original-side-line pedal polygon is independent of phase for every fixed finite real point M. This attributed research note presents a complete incidence-curve proof: disjoint possible simple pole supports and complementary reversal zeros make the product holomorphic on a compact genus-one flag curve. Primitive stars, both orientations and zero or negative pedal areas are included. The exact theorem, N4 formula and a rescaled signed star example also occur in Alper Ferudun's contemporary Pedal Areas of Periodic Elliptic Billiard Trajectories with Respect to an Arbitrary Point, version1.0 DOI10.5281/zenodo.23400541, prominently credited. Steiner's pedal-area principle, the Bialy-Tabachnikov four-period implication and established Schwartz-Tabachnikov, Akopyan-Schwartz-Tabachnikov and Chavez-Caliz flag/meromorphic methods are prior mathematics. No theorem or method novelty, first disclosure or certified independent discovery is claimed. The currently public PR203 record reports October1 creation and the exact contemporary Zenodo deposit October15 creation; these separate object dates do not certify historical visibility, discovery time or independence. The source table does not literally print least period; an exact doubled-six nominal-twelve counterexample protects that qualification. LaTeX source and verification materials include all18 preserved original submission files, independent derivations, explicit-exception portable checks, finite noninterval physical diagnostics and bounded source/priority ledgers. AI tools were used extensively. Unrefereed preprint without conventional human peer review or proof-assistant certification.