# Verification report

Title: Products of the Two Focal Inversion Areas in Elliptic Billiards.

Author: Alper Ferudun[^author]

[^author]: Mercury Software GmbH. alper@mercurycodelab.com; https://github.com/AlperTheKing.

Problem identity: AMR-050-0062; frozen record 5100062; source invariant k903,a. English preprint, version 1.0, manuscript date 2 October 2026.

This work is AI-assisted, originating-researcher self-audited and unrefereed. No independent human review, formal proof-assistant verification or absolute-priority certification is claimed. The general theorem rests on the analytic argument, not numerical constancy.

## Accepted scope and provenance

For every fixed odd real-periodic billiard family in a noncircular ellipse, the two original-boundary-focus unit inversions of the original vertices have a positive phase-independent product of their signed shoelace areas. The result includes every primitive odd period n >= 3, every admitted coprime star winding, orientation reversal and every fixed odd repeated traversal. Circular boundaries are proved separately. A real-caustic parity lemma excludes an omitted odd hyperbolic or degenerate trajectory.

The originating researcher accepted the proof after a detailed self-audit on 2 October 2026, recorded in `problems/AMR-050-0062/acceptance_20261002.json` and `verification.json`. The complete general proof is `proof.md`, SHA256 `9a83ef210e4e61b1a3ee8da4ffa3170ea90db03d8f3c7f07b8982319fba84593`, in that dossier. Its acceptance records `proof_accepted=true`, `theorem_scope_resolved=true`, `original_problem_resolved=true`, `whole_source_record_resolved=true`, `novelty=UNDETERMINED` and `new_result=false`. The whole-record flag refers only to AMR-050-0062, not all source invariants.

The polygons preserve trajectory order and use original billiard vertices and original boundary foci. For stars, the quantity is signed shoelace area, not unsigned face-union area. The inversion radius is one. Translation back by the inversion center does not affect area.

## General argument actually accepted

The confocal Jacobi parametrization is derived from tangent intersections and agrees with Stachel's billiard map. It is not equal eccentric-anomaly spacing. Write the original vertices as P(u)=(-a sn(u),b cn(u)), let h be the actual billiard shift, and set g_e(u)=a+e c sn(u) for the focus F_e=(e c,0). The bilinear identity (P-F_e) dot (P-F_e)=g_e^2 determines the meromorphic inverse coordinates Q_e=(P-F_e)/g_e^2.

Every ordinary Jacobi pole is removable for Q_e, where it vanishes. The proof exhausts the remaining four simple roots of g_e on the common 4K,4iK' torus. At a possible area pole, exactly two consecutive inverse vertices are singular. Their two negative Laurent coefficients lie on the same nonzero isotropic line. Hence their determinant has neither a fourth- nor a third-order pole; all other contributions have order at most two. Exact cyclic index reversal makes each focal area odd about that phase, removing the second-order term.

Coprime cyclic relabeling reduces the real period to 4K/n. For odd n, the two focal pole sets differ by a half real period and are disjoint. At each possible simple pole of one focal area, the opposite focal area is holomorphic and odd about that point, hence zero. Their product has no poles on the compact quotient torus and is therefore constant. This removes principal parts, rather than falsely inferring constancy from a fixed pole set.

For forward real elliptic-caustic trajectories, both foci lie strictly inside the caustic. Each inverse-edge determinant is positive, including admitted coprime stars; all inverse vertices are finite. This proves positivity and nonvanishing of the two real areas. Reversal negates both areas, preserving their product. An r-fold traversal multiplies each area by r and their product by r^2. Odd listed length has odd primitive divisor and odd repetition count, so repeated odd lists are included.

For a circular boundary of radius a, the formula is n^2 sin^2(2 pi tau/n)/(4 a^4) for a primitive star of winding tau. The supplementary normalized chord-line tangency equation proves that hyperbolic-caustic chords cross the interior focal segment and alternate nonzero y-signs. At the degenerate endpoints the remaining exceptions are axial two-cycles or zero-length grazing. Thus no nondegenerate odd real family lies outside the elliptic-caustic proof. This is not an extension to arbitrary complex billiards.

## Actual originating exact execution

The source checker `problems/AMR-050-0062/attacks/algebra/check_focal_pair.py` has SHA256 `4d231ca868f086eb99da882b9c90b3ef391bb99f33af29c3324819a0ac9170b1`. The actual originating rerun receipt is `root-exact-run-002.json` in the same directory, completed at 2026-10-02T05:32:19.612894+00:00, SHA256 `1fc20c7bcef8ac0efdbd72f5ce330e6b92aa50bc618645da509907fef6053935`. Its status is `PASS_EXACT_FOCAL_PAIR_PRODUCT_CONTROLS`.

Arithmetic uses the Python standard library and Fraction quadratic fields; the run reports no floating-point use, repository dependencies, external requests or canonical writes. Physical coordinate/area scaling is explicitly tracked rather than confusing logical quadratic-field coordinates with physical area.

For a^2=21, b^2=16 and lambda=336/25, the two physical triangular phases are

    H=((sqrt(21),0),(-3sqrt(21)/5,16/5),(-3sqrt(21)/5,-16/5)),
    V=((0,4),(-21/5,-8/5),(21/5,-8/5)).

They are not cyclic relabelings or central negations of one another. Exact incidence, unit-velocity reflection, internal common-confocal-caustic contact, primitive period and finite focal inversion pass. Both focal-area products are 1/324. At V both individual areas are 1/18; at H the individual areas are different. Odd repeat counts 3,5,7 give respectively 1/36,25/324,49/324. Reversal, cyclic reindexing and central negation preserve the product. The geometric Poncelet porism is a theorem dependency; connected-family membership is not inferred from two samples alone.

In the period-four family a^2=4, b^2=1, lambda=4/5, the axial and rectangular phases have products 1 and 25/16. Both are genuine primitive four-cycles with internal contacts. This refutes only removal of the odd-period hypothesis. The radius-two circular triangle control gives 27/256. No exact noncircular primitive star control is claimed by this checker; finite controls are not the all-period proof.

## Configured portable execution

The archive is `arxiv_source.zip` locally and on Zenodo, with website download name `source.zip`. It contains root `main.tex` and the exact entry point `reproducibility/check_focal_pair.py`. From a fresh extraction root, run `python3 -I -B reproducibility/check_focal_pair.py`. It needs only the standard library and no repository imports. The checker is copied without mathematical or path-bootstrap changes. The package replay verifies its source hash and compares every output field to the retained originating result except the explicitly identified execution timestamp. Raw-output equality is not claimed when timestamps differ.

The separate isolated-execution receipt and package manifest record the package result. The command and originating checks described here do not replace those execution records. No dependency installation is required.

## Actual numerical diagnostic

The separate originating receipt `problems/AMR-050-0062/attacks/algebra/numerical-run-001.json` completed at 2026-10-02T05:30:16.509883+00:00. Its status is `PASS_NUMERICAL_FOCAL_PAIR_DIAGNOSTICS_NOT_PROOF`; SHA256 is `9999eadfae136dabbc1574949a746be1b773eea3e478a1d1ef2128e9f7c643f0`. The executed script SHA256 is `635f48daa3d36528500e8534279874952dc98b72bc209c86502a0a8007deb438`.

The actual mpmath 1.3.0 run uses 85 decimal digits on 42 odd-period families, including 27 stars, five phases each, with primitive periods 3,5,7,9,11. It checks closure and geometric residuals, positive focal distances, area-product spreads and threefold repetition scaling. This is finite floating-point evidence without certified error bounds. It neither formally verifies the analytic proof nor supplies its arbitrary-period conclusion. mpmath is optional diagnostic provenance, not a dependency of the exact required replay.

## Prior art and document boundaries

Reznik, Garcia and Helman's arXiv:2012.03020v2 Proposition 7 already supplies the N=3 area-product formula. The two-phase exact controls reproduce a known special case; no new triangular theorem is claimed. The original source k903,a is the same mathematical quantity as k807 in Garcia and Reznik's 2022 companion. The final 2021 source journal omits the older pair-area table, which is neither proof nor correction. Stachel's parametrization, standard DLMF identities, and compact-torus/pole-cancellation methods are credited.

The bounded primary-source audit does not certify novelty, absolute priority or exhaustive coverage. PDF compilation/export, visual page QA, extracted source replay, paper readiness and external publication are distinct operations requiring actual receipts. These documents claim none of those not-yet-recorded outcomes. No DOI, guaranteed indexing or arXiv action is asserted.