# Verification report

Title: Focal Inversion Area Ratios in Elliptic Billiards.

Author: Alper Ferudun, Mercury Software GmbH.

Problem identity: AMR-050-0046; frozen record 5100046; source invariant k805. Version 1.0, manuscript date 2 October 2026.

## Accepted explicit scope

For every nondegenerate confocal elliptic-caustic billiard of genuine least period n >= 6, n congruent to 2 modulo 4, the original ordered signed shoelace area divided by its unit focal-inverted ordered signed area is a positive phase-independent constant, identical at both foci. Every admitted coprime star winding is included. All real inverse vertices are finite, and the inverse signed area is nonzero. Repetitions of an admitted primitive orbit preserve this ratio. Circles are treated by a separate direct calculation.

The originating researcher accepted this theorem after a detailed self-audit on 2 October 2026. The proof is `problems/AMR-050-0046/proof.md`, SHA256 `4eac4b05b66a9f028c6006ee0b1b85621001e03831329a6009aaa9d84a659c57`. The actual records are `acceptance_20261002.json` and `verification.json` in that dossier. They record `proof_accepted=true`, `theorem_scope_resolved=true`, `original_problem_resolved=false`, `whole_source_record_resolved=false`, `novelty=UNDETERMINED` and `new_result=false`. The last flag means no novelty certification is made, not that the scoped proof was rejected.

This is an AI-assisted, self-audited, unrefereed preprint. Independent human review and proof-assistant verification have not been completed and are not implied by the finite exact controls.

## General analytic argument

The proof derives the actual confocal Jacobi billiard parameter from tangent intersections, rather than assuming a constant eccentric-anomaly step. Original vertices are inverted about an original boundary-ellipse focus, and the focal distance identity identifies the meromorphic inverse coordinates.

For n = 2m with m odd and coprime winding, cyclic relabeling reduces the common real period to 2K/m. The two area traces are periodic under 4iK' and antiperiodic under 2iK'. Their possible original and inverse poles reduce to the same two points on this genuine quotient torus. Original-area poles have order at most one. At a focal inverse pole, the negative Laurent coefficients of two adjacent singular vertices lie on one isotropic line; reversal symmetry then eliminates the potential second-order pole, leaving at most a simple pole.

Subtracting the appropriate residue multiple of the original area from the inverse area cancels both possible poles. The difference is entire on the compact torus and antiperiodic, so it is zero. This proves proportionality without presuming that a ratio has no poles at unknown denominator zeros. Positive directed-edge determinants, with the center and both foci inside the elliptic caustic, establish positive real areas and denominator nonvanishing. Central half-grid symmetry identifies the two focal constants. Reversal negates both areas, leaving the ratio unchanged.

This is a general all-admitted-period argument. Finite sampling does not execute the analytic pole, residue or compactness proof.

## Actual exact executions

The retained originating-researcher rerun is `problems/AMR-050-0046/attacks/algebra/executions/root-exact-run-002.json`, completed at 2026-10-02T03:03:18.897452+00:00. Its actual status is `PASS_EXACT_FOCAL_AREA_RATIO_CONTROLS`; it reports no floating-point use, no external requests and no canonical writes. The script SHA256 is `f058cdad79088592e00b1f48775665a6765700789a84c541f188c586a0be98e8`. The adjacent imported geometry helper is independently bound by SHA256 `afa502c78fd47fecd00376b8b5b4396d43297e937c5688ac95aa1fb855beef7c`.

For squared semiaxes 4 and 1 and caustic parameter 4/9, two primitive six-periodic phases give, at both foci:

- Horizontal phase: A = 20 sqrt(5)/9, inverted area B = 15 sqrt(5)/8, A/B = 32/27.
- Vertical phase: A = 32 sqrt(2)/9, inverted area B = 3 sqrt(2), A/B = 32/27.

The control verifies exact ellipse incidence, unit-velocity reflection, common confocal tangency, internal contact parameters, distinct primitive vertices, positive focal squared distances and finite inverse vertices. Two and three traversals preserve 32/27. The radius-two circle control gives ratio 16 exactly.

For squared semiaxes 21 and 16 and caustic parameter 336/25, the two primitive triangles T and -T are certified to lie in one fixed family. At the fixed focus (sqrt(5), 0), their ratios are (10584 + 72 sqrt(105))/25 and (10584 - 72 sqrt(105))/25, with positive denominators. Repeating each triangle twice produces listed length six and preserves the unequal ratios. This refutes the unrestricted listed-length extension, not the genuine least-period-six theorem. It is the reason no unqualified original-record closure is asserted.

## Portable layout and comparison policy

The package configuration places `main.tex` at the ZIP root and the required exact entry point at `reproducibility/check_focal_ratio.py`, with the unchanged shared helper at `reproducibility/check_exact_inverted_areas.py`. Run `python3 -I -B reproducibility/check_focal_ratio.py` from the extraction root. This requires only the Python standard library, not a project checkout.

The reviewed adaptation makes exactly two path-bootstrap changes: remove the unused `ROOT=HERE.parents[3]` assignment so shallow extraction paths cannot raise an IndexError, and replace the `HELPER` assignment to find the adjacent shared helper. Mathematical logic and shared-helper bytes are unchanged. The unchanged original checker is retained at `reproducibility/original/check_focal_ratio.py`. The original result and explicit adaptation record are included for provenance. Because the path-bootstrap adaptation changes the checker's own SHA256, the replay must verify both original and current embedded source hashes separately. It compares all mathematical output fields exactly after omitting only `actual_execution_completed_at` and `script_sha256`; it must not claim raw-output or full parsed-result equality. These are configured checks, not a claim that package execution has already completed. A fresh isolated extraction/replay requires its actual receipt.

## Actual numerical diagnostic

The separate receipt is `problems/AMR-050-0046/attacks/geometry/run-001.json`, completed at 2026-10-02T02:58:20.593795+00:00, status `PASS_NUMERICAL_DIAGNOSTIC_NOT_PROOF`. The script SHA256 is `03f9fa35b683b5fc3cb22e3d2be86756ea79bd92511c0f74f1a75fbe1212307f`; the executed dependency was mpmath 1.3.0.

The run used 85 decimal digits on 45 families, five phases each. Of these, 27 target families have genuine least periods 6, 10, 14 or 18, covering the admitted coprime windings and three moduli. The other 18 are odd-period controls, not instances of the theorem. It checks closure, ellipse incidence, unit-velocity reflection and caustic tangency. All target ratio spreads and focus differences satisfy the script's 10^(-60) threshold. This is finite floating-point evidence without certified error bounds, not formal verification or proof of all periods.

The configured archive's optional entry point is `reproducibility/diagnostic.py`, with its unchanged original retained under `reproducibility/original/diagnostic.py`. Its two bootstrap-only removals are the unused `ROOT=Path(__file__).resolve().parents[4]` assignment and the project-specific dependency `sys.path` insertion; mathematical logic is unchanged. The declared optional dependency is mpmath 1.3.0. The package harness does not execute this optional diagnostic or install dependencies, and no adapted diagnostic execution is asserted here.

## Source, novelty and document limits

Garcia and Reznik's 2022 Proposition 4.16 already supplies the simple-six value 4 a^3 b^4 / ((2a-b)(a+b)^2) for unit inversion. The exact 32/27 controls reproduce that published special case. The source's original k805 was removed from the final 2021 journal table and renumbered k806 in the 2022 companion. Deletion or renumbering is not itself a general proof or refutation.

Eight targeted primary-source searches and retained source-context inspections did not identify a matched general exact theorem. This bounded result does not certify novelty, priority, exhaustive coverage or current openness. Stachel's Jacobi parameter, NIST DLMF identities and meromorphic residue cancellation are standard methods, not claimed inventions.

This report records proof acceptance, already executed originating checks and the reviewed package configuration. Successful native compilation, a built PDF, visual page review, source-package extraction/replay, paper readiness and external publication require their own actual receipts; the configuration is not evidence of those outcomes. No DOI, hyperbolic or degenerate caustic extension, guaranteed indexing, arXiv submission or whole invariant-list resolution is claimed here.