# Verification report

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

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

The theorem concerns the signed area A' of consecutive boundary-tangent intersections and the signed area B_M of perpendicular feet of a fixed point M on those same boundary tangents. For a fixed noncircular ellipse and nondegenerate confocal elliptic caustic, A' B_M is constant for genuine billiard families whose least period is congruent to two modulo four, including every admitted coprime star winding. This is invariant k303,a with an explicit least-period convention, not a theorem for arbitrary repeated lists or unsigned face areas.

The general proof is analytic. The derived tangent-intersection map gives A'(u)=dn(Delta)/cn(Delta)^2 times A(u+Delta), with the indispensable half-step shift. The exact pedal map splits into U-R_M; opposite-index pairing cancels mixed signed-area terms for arbitrary M. Its two denominator roots and the outer-area poles lie in disjoint lattice classes under the primitive parity condition. The foot map is regular at ordinary Jacobi poles, and adjacent foot residues are parallel at its other poles, so every possible area pole is at most simple. The reversal j to 1-j gives zero pedal area at every outer-area pole; the reversal j to -j gives zero original area at the shifted pedal poles. The elliptic product is consequently entire and constant. All common-lattice translates and the otherwise possible N=4 adjacent coincidence are accounted for. No area division or extrapolation from samples is used.

check_outer_pedal.py uses exact rational polynomial rings and real quadratic fields. Two polynomial identities prove isotropy of the homogeneous quadratic pedal-area term; the Gram identity and four axis identities supplement the explicit geometry. Sixteen six-periodic orbit instances are checked, with closure, six distinct vertices, all reflection laws, strict internal caustic contacts, tangent intersections, all six area-product coefficients in M, and direct feet for three fixed points. These are not asserted to be sixteen distinct geometric orbits. Their values agree with ab(a+2b)(2a+b)(1+|M|^2/(a+b)^2), derived for an axis member and transferred by the general theorem.

For a^2=1, b^2=3/8, lambda=9/25 and M=(1,0), two centrally reflected least-period-three trajectories have products 6561/1600 and 1701/1600. Doubling their lists gives products 6561/400 and 1701/400, differing by 243/20. Exact geometry and projection checks establish that the unrestricted six-entry-list extension is false. This does not refute the genuine least-period-six theorem.

The unchanged checker, original JSON result, derivation note and actual execution receipt are included. Package portability is recorded only after fresh isolated archive extraction, actual execution and exact parsed-output comparison with the retained originating result. The written analytic argument, not the finite checker or binary64 scouting, establishes the general result. The package builder pins actual root acceptance, final native compilation, PDF export and all-page visual QA. The harmless XeTeX inputenc compatibility warning is not a mathematical or layout defect; the source is UTF-8. The builder itself does not mark a paper ready or publish it.

The argument is originating-researcher self-audited, AI-assisted and unrefereed. No independent human review, proof-assistant verification or absolute priority certification is claimed. Source geometry, canonical Jacobi parametrization and the established pole-cancellation method are credited. A bounded primary-literature search did not locate the precise conclusion; novelty remains undetermined.

## Version 1.1 centered theorem

The k303,a arbitrary-fixed-point theorem retains its explicit genuine least-period condition N congruent to two modulo four. The additional k303,b center-pedal theorem covers every closed traversal length not divisible by four, including repeated traversals and all admitted signed star windings. Both use a fixed noncircular ellipse and fixed nondegenerate confocal elliptic caustic, boundary-tangent feet and signed shoelace areas. The centered N=4 products 8*a^2*b^2 and 2*(a^2+b^2)^2 differ by 2*(a^2-b^2)^2; this refutes an all-period extension, not k303,b. No hyperbolic, degenerate or whole invariant-list extension is asserted.

For the center the pedal map is U=n/(n.n). Its area has possible simple poles at Z-Delta-j*h, while the outer area has possible simple poles at z_p-Delta-j*h. These classes are disjoint for odd and two-modulo-four least periods. The reversal j to 1-j forces the centered pedal area to vanish at every outer-area pole, and j to -j forces the shifted original area to vanish at every pedal pole. All lattice translates, removable Jacobi-pole feet, and excluded N=4 adjacent coincidences are explicitly handled in the analytic proof. Repetition multiplies each signed area by r and their product by r^2; if four does not divide the traversal length, four cannot divide its least period. No division by an area is used.

The exact new controls retain their originating and root-replay receipts. For a=5,b=3 and lambda=225/34 the axis diamond and axis-aligned rectangle give 1800 and 2312, a difference of 512. These are genuine same-caustic N=4 orbits, illustrating precisely the excluded extension. The archive also retains every original k303,a reproducibility member unchanged. Fresh isolated execution of both scripts is separately recorded by the revision builder; only actual compile, PDF export and complete visual QA receipts can authorize assembly.

This is a self-audited, AI-assisted, unrefereed preprint. No independent human review, formal proof-assistant verification, first-proof or absolute priority certification is claimed. Novelty remains undetermined after bounded primary literature searches. Credited confocal geometry, Jacobi parametrization and the established meromorphic pole-cancellation method are prior mathematics.