# Verification report

Manuscript: *Center-Pedal Area Products in Elliptic Billiards*.

Author: Alper Ferudun, Mercury Software GmbH. Date: 1 October 2026.

The theorem concerns the original orbit's signed shoelace area A and the signed shoelace area A_O of the actual perpendicular feet from the common center to its original sidelines. For a fixed noncircular boundary ellipse and nondegenerate confocal elliptic caustic, A A_O is constant for genuine billiard families whose least period is odd or divisible by four, including all admitted coprime star windings. The source row k203,b uses N not congruent to 2 modulo 4; its least-period convention is stated as this manuscript's qualification, not quoted as an explicit source definition. Hyperbolic or degenerate caustics and arbitrary padded lists are excluded.

The all-period argument is analytic. A reflection-coordinate telescope gives 2D S_2=s S_1. Direct ordered perpendicular-foot determinants then give A_O=sqrt(lambda)[sD-lambda(alpha^2+beta^2)]S_1/(2D). In the genuine Jacobi parameter, the original area and the corresponding trace have complementary pole sets precisely for the claimed primitive period classes. Odd trace pairing removes area poles; reversed-edge pairing removes trace poles. Every possible pole of the elliptic product is covered, including simultaneous nonadjacent antipodal vertex poles in the even case. No division by a signed area or extrapolation from finite samples is used.

check_low_period.py uses exact sparse rational polynomial rings and real quadratic fields. It verifies the universal triangle conic/pedal identities and source-valid same-caustic N=3 controls; the elementary N=3 formula is 27 delta D^2/(s+2delta)^3. The exact N=4 diamond and rectangle at a=4,b=3 both yield A A_O=165888/625, in agreement with 8D^2/s^2. Two primitive N=6 trajectories in the fixed a=5,b=3,lambda=225/64 family yield 72665775/65536 and 122650422111/110166016, differing by 7800849/1721344>0. Doubling each traversal multiplies the product by four but preserves least period six, documenting why the unrestricted padded-list extension is false.

check_centered_pedal_prefactor.py certifies 11 coefficientwise zero-remainder identities, without floating-point sampling, for the actual incoming/outgoing feet, sign convention, local area bracket, reflection tensor and final prefactor. Its output explicitly does not certify the geometric hypotheses or the all-period pole-constancy lemma by itself. Both scripts and their originating results/execution receipts are preserved unchanged. The package's portable receipt is created only after actual fresh isolated extraction and successful execution of both scripts, with exact parsed-output comparison to those retained results.

The originating researcher's proof and self-audit establish the general result. Numerical scouting over 63 accepted families is corroboration, not a proof, an interval certificate or a certified counterexample. The package builder waits for activated acceptance, verification, actual final native compilation, actual PDF export and complete final page-by-page visual QA, and pins those records and the unchanged manuscript bytes. It does not itself mark a paper ready or published.

The preprint is AI-assisted and unrefereed; no independent human review, proof-assistant verification or absolute priority certification is claimed. The centered-pedal trace factor and the complete pole-class cancellation are presented with credited prior geometry, parametrization and analytic methods. A bounded primary-literature search did not locate the exact general conclusion; novelty remains undetermined.