# Verification report

## Result and domain correction

For the caustic-contact polygon of each fixed odd-periodic confocal elliptic billiard family, the manuscript proves that its own Steiner curvature centroid is finite at every real phase. Its signed pedal area at that centroid equals a constant real multiple R of the contact polygon's positive oriented area. Thus source invariant k503, the reciprocal ratio, is constant on its defined domain. The proof includes odd coprime star windings with an explicit internal-ray-angle convention, odd repetitions, and the separately treated circle case.

The constant R may be zero. An explicitly specified nondegenerate convex five-periodic family has zero signed centroid-pedal area throughout. It is specified by the unique zero of a degree-18 polynomial in the interval (101/100,11/10), with a positive algebraic branch for all coordinates. This disproves universal definedness of the reciprocal ratio, not constancy of a defined ratio. The centroid denominator itself stays nonzero. No phase-varying finite-ratio counterexample, universal positivity, k502 closure, hyperbolic/degenerate-caustic theorem, or resolution of an entire invariant table is claimed.

## Analytic proof

The proof works directly with the contact polygon's Euclidean metric; it does not treat that polygon as an ordinary confocal billiard in its own ellipse. The classical isotropic pedal-area quadratic is combined with the contact-polygon Jacobi parametrization, a complete isotropic-sideline pole analysis, and character spaces on an odd quotient torus. The trace coefficient's real nonvanishing makes the moving centroid well-defined. Residue cancellation and elliptic-function uniqueness give the constant stationary-area multiple. The exact pentagon construction then establishes that the zero-multiple alternative occurs.

## Portable exact certificates

The source archive contains a self-contained `reproducibility/check_inner_steiner.py`. It requires Python 3 and SymPy, tested with SymPy 1.13.1; no other project code, internet connection, absolute path, or downloaded mathematical dataset is required at runtime. Install the declared dependency in your own environment if it is absent, then run:

```sh
python3 -I -B reproducibility/check_inner_steiner.py
```

Successful execution prints `PASS_EXACT_INNER_STEINER_CONTROLS`. It verifies 37 exact identities in the quadratic function field Q(x)[y]/(y²−(x³−x)y−x²): both ellipse equations, contacts on the chords, tangency, reflection collinearity and preservation of norm for all five vertices, the direct stationary pedal quadratic, and all three factored field-norm identities. Rational polynomial denominator clearing excludes poles on either algebraic branch, so specialization of the norm identities cannot conceal a zero times a conjugate pole.

Six separate rational parameters give exact real-quadratic-field pentagons with distinct vertices, all five directed reflections, interior caustic contacts, positive inner area, and direct own-centroid pedal areas. The two interval endpoints have opposite pedal-area signs and nonzero centroid denominators. Standard-library Fraction arithmetic verifies every rational Bernstein coefficient on the whole interval, not a numerical mesh, establishing all required factor signs and isolating exactly one degree-18 root. These checks use no floating-point tolerances.

The intermediate-value argument, geometric interpretation, and all-period analytic theorem are stated in the manuscript and full proof. Successful execution is not a formal verification of those analytic inferences. The bundle preserves the originating exact result, transparent dependency/runtime receipt, source and checker provenance, and an actual fresh extracted-archive replay. The latter supplies the preexisting local SymPy installation explicitly in its test harness; it does not falsely claim a dependency-free test.

## Source and review limits

The original arXiv v11 and final journal version were checked for the precise own-inner-centroid definition and signed-area convention. The classical fixed-polygon Steiner quadratic and projective contact-polygon construction are credited as prior results. Those statements do not by themselves establish this discrete Euclidean phase invariant. The source did not supply a nonvanishing theorem for the reciprocal denominator.

The bounded primary-literature search located no matching general k503 proof or correction; this is not an exhaustive novelty or priority certification. The work records originating-researcher self-audit and reproducible exact computation with AI assistance, not independent human review or a proof-assistant formalization. Third-party PDFs, scans, and page images are excluded from the public source archive.