# Verification report Problem AMR-050-0044, frozen source record 5100044, invariant k804,a. Title: Focal Inversion Area Products in Elliptic Billiards. Author: Alper Ferudun, Mercury Software GmbH. ## Accepted theorem and repetition boundary For every nondegenerate confocal elliptic billiard of genuine least period divisible by four, the original ordered signed shoelace area times the signed area after unit inversion at either focus is a positive phase-independent constant. The inverse vertices are finite, and both foci give the same constant. All admitted coprime star windings are included. Repeating an admitted primitive orbit d times multiplies the product by d squared. The circle is treated separately; no hyperbolic or degenerate caustic extension is claimed. Two primitive convex six-periodic orbits in the ellipse with squared semiaxes 4 and 1, at caustic parameter 4/9, have products 125/6 and 64/3 at the same fixed focus. Two traversals give listed length twelve and products 250/3 and 256/3. A separate fourfold primitive-triangle control also gives unequal listed-twelve products. These refute only the unrestricted repeated-list extension, not the genuine-period theorem. Because the source does not explicitly settle that convention, `original_problem_resolved=false` and `whole_source_record_resolved=false` are preserved. The originating researcher accepted this scope after a detailed self-audit on 2 October 2026. The retained acceptance record binds the actual proof, source audit and executed controls. No independent human or proof-assistant verification is implied. ## General analytic argument The manuscript derives the actual confocal Jacobi parameter and the focal denominator identity. It works on the common torus with periods 4K and 4iK prime, rather than treating the imaginary half-period as a common period. All original and inverted vertex poles are enumerated, and the period-four parity calculation makes their traced singular sets disjoint. At a focal inverse pole, isotropic tangency forces both negative Laurent coefficients to lie on one line. Two adjacent singular vertices therefore contribute no fourth- or third-order pole. Reversal symmetry removes the second-order term, including for N4. The original area vanishes at these remaining possible simple poles. Conversely, central symmetry and focus exchange make the inverse area vanish at every possible original-area pole. The holomorphic product on the compact torus is constant. Real edge determinants prove positivity. This is a general argument for all admitted periods and windings, not an extrapolation from finite samples. ## Actual exact executions Three standard-library programs were actually executed and rerun by the originating researcher with exit code zero: - `check_exact_inverted_areas.py` verifies finite ellipse incidence, unit-velocity reflection, common confocal caustic tangency, internal contacts, focal denominators, both-focus N4 values, the N8 value 133/5, and repeated-triangle area arithmetic. - `check_repeated_six.py` independently assembles the two N6 phases and uses the hash-pinned adjacent geometry helper to verify their reflections, caustic contacts and double-traversal products. - `check_focal_inverted_area.py` verifies six coefficientwise rational identities, both-focus N4 products 4 and the independent inversion arithmetic for the earlier source-valid triangle. Its retained output truthfully says it does not rerun that triangle's earlier reflection certificate. The last control gives primitive triangle products `(294-2sqrt(105))/225` and `(294+2sqrt(105))/225`; four traversals multiply them by 16. These finite controls support the proof and scope distinctions but do not execute the complex torus argument. ## Numerical and document evidence A separate direct-geometry diagnostic was actually run with mpmath 1.3.0 at 85 decimal digits on 33 families and five phases per family. Of these, 27 have genuine least period divisible by four; six use excluded periods three or six to test the repetition boundary. It checks incidence, reflection, caustic tangency, closure and focal finiteness. This is supporting floating-point evidence, not exact certification or a proof of generality. Native LaTeX compilation and an actual cached Tectonic PDF build succeeded. The latter is a Tectonic build, not a native-editor PDF export. Final visual page review and the source-package extraction and replay are documented by their actual separate receipts when completed; this report does not preclaim those later operations. The package manifest binds the final payloads and relevant execution evidence after the build. ## Source and novelty limits The target is k804,a in arXiv:2004.12497v11 Table 9, page 11. The final journal version omits this area-product row and reuses k804 for a different cosine expression. Garcia and Reznik's 2022 companion renumbers the target k805,a and proves the known simple N4 value 4 in Proposition 4.9. That result is expressly credited. Stachel's canonical confocal parameter, NIST DLMF Jacobi identities and the classical meromorphic Poncelet method are prior tools. Chavez-Caliz's area-product theorem concerns the outer tangent polygon, not the focal-inverted polygon; no unproved transfer from it is used. Nine recorded targeted searches and inspected primary-source contexts did not identify a matched general target proof. This is a bounded finding, not a guarantee of novelty, current openness, exhaustive coverage or absolute priority. Novelty remains UNDETERMINED and `new_result=false` denotes absence of novelty certification. This is an English, AI-assisted, self-audited, unrefereed preprint. No independent human review, proof-assistant verification, guaranteed indexing, whole invariant-list resolution or absolute-priority claim is made.