# Verification report

## The result and its limits

The manuscript gives an analytic proof of source invariant k501: for every odd-periodic elliptic billiard with a fixed nondegenerate confocal elliptic caustic, the original polygon's signed area divided by the signed area of its pedal polygon at its own Steiner curvature centroid is defined and constant. The centroid weight sum and the pedal area are proved nonzero. Coprime odd star windings and odd repeated traversals are included, under the explicitly stated internal-angle and signed-area conventions. The circle case has a separate elementary proof.

This is one source-record closure, not a resolution of adjacent k502/k503 or a whole conjecture table. Hyperbolic and degenerate caustics are excluded.

## Analytic structure

The proof derives the exact stationary pedal-area quadratic and the reflection trace identities. Complementary pole divisors give a constant area--trace product. Jacobi characters on the odd quotient torus identify both the area function and the centroid ellipse with nome Q raised to the odd primitive period. Residue ratios give exact axis coefficients. A strict folded-grid theta inequality proves the positive-quadratic case's axis ordering, closing the essential nonvanishing step before the area ratio is inverted. Odd repetitions preserve the centroid and multiply both signed areas equally.

## Reproducible exact controls

Run `python3 -I -B reproducibility/check_steiner_pedal.py` in the extracted source archive. It uses only standard-library rational and exact real quadratic-field arithmetic, with no network or repository imports. The checker verifies nine universal polynomial identities, the displayed grid-fold coefficient formula for all odd n from 3 through 201, the pedal-area quadratic on two generic rational polygons, and five genuine triangular billiards on one fixed confocal pair. Each triangle has verified reflection, distinct vertices and strictly internal caustic contact; its centroid is its circumcenter and its area ratio is exactly four. Threefold and fivefold repeated traversals are also checked exactly.

The coefficient regressions and low-period controls do not prove the all-odd theorem. The general proof is the analytic argument in `main.tex` and `reproducibility/proof.md`. The archive preserves original supporting derivations, a checker-provenance record and an actual exact execution receipt. The release's package manifest binds the final files to actual compilation, PDF export, all-page visual inspection, and an isolated extracted-archive checker run.

## Source and review status

The definition and signed-area convention were checked against the original arXiv v11 and final 2021 journal version. The classical fixed-polygon Steiner theorem and the triangle circumcenter case are credited as prior results. The bounded primary-literature audit did not identify an earlier general k501 closure, but does not certify novelty or priority.

This report records originating-researcher self-audit and reproducible exact checks with AI assistance. It does not claim independent human peer review, a proof-assistant formalization, or that successful code execution verifies every analytic inference.