# Verification report

Problem: AMR-050-0007; frozen source record 5100007; invariant k115.

Title: Focal Distance Products of Outer Elliptic Billiard Polygons.

Author: Alper Ferudun, Mercury Software GmbH.

## Accepted theorem and excluded assertion

For a nondegenerate confocal elliptic-caustic billiard of least period divisible by four, the product of Euclidean distances from either original ellipse focus to consecutive boundary-tangent intersections is independent of phase. The proof includes admitted coprime star windings and repetitions of those genuine periods. Concentric circles are handled separately. No hyperbolic or degenerate caustic is claimed.

Four repetitions of an exact primitive triangle give listed length twelve but least period three, with two distinct positive focal distance products in the same fixed ellipse and caustic. This refutes only the unrestricted listed-length extension. The source does not explicitly settle that convention, so `original_problem_resolved=false` and `whole_source_record_resolved=false`.

Root accepted this precise theorem after its detailed originating-researcher self-audit on 2 October 2026. The immutable decision is `reproducibility/acceptance_20261002.json`, with actual evidence hashes. No independent human or proof-assistant verification is implied.

## Analytic proof

The manuscript derives the boundary and actual outer-tangent parametrizations, factors the squared Euclidean distance, proves positivity of both factors and uses the genuine quarter shift. It then accounts for all zeros and poles, with multiplicities, of the cyclic denominator product on the compact Jacobi torus. Complete cancellation, rather than divisor translation-invariance alone, proves constancy. The coincident-zero case and positive real square root are retained.

The argument establishes all admitted periods and windings. Finite computational tests are not substituted for it.

## Actual originating controls

The unchanged standalone exact checker was actually rerun by root in isolated mode, with exit code zero and output identical to its originating run. The retained result verifies nine coefficientwise rational polynomial identities, an exact primitive N8 billiard with focal product `2000000/3`, actual boundary/outer-tangent incidence, caustic contact and reflection, and the repeated-triangle squared-product arithmetic. It uses Python standard-library `Fraction` and no floating-point arithmetic.

The repeated-triangle checker output truthfully says it does not rerun the older triangle reflection certificate. The manuscript independently includes reflection and support-line calculations; no unavailable project file is needed to run the distributed program.

A distinct direct-geometry implementation was actually run with mpmath 1.3.0 at 85 decimal digits on 39 families and four phases per family. It constructs and intersects actual tangent lines without importing the algebra-lane formula. This is optional numerical evidence, not an exact or general proof.

The source-package builder, when explicitly invoked after final PDF and page QA, extracts its newly produced ZIP into a fresh temporary directory and executes only the required exact checker. Its actual portable receipt and manifest then bind the program, exact output, payloads and originating receipt hashes. This report does not preclaim that a package build or replay has already occurred.

## Sources and novelty

Reznik, Garcia and Koiller list k115 in arXiv:2004.12497v11, Table 2, printed page 5, and in the final Arnold Mathematical Journal article, Table 2, page 345. Stachel's confocal Jacobi parametrization and quarter-period technique, and standard NIST DLMF identities, are credited. The N4 product `4a^2 b^2` is already implied by the earlier AMR-050-0048 calculation and is not claimed as new here.

Thirteen recorded targeted searches did not identify a direct all-period k115 proof in the inspected primary sources. This bounded nonfinding cannot certify exhaustive coverage or priority. Novelty remains UNDETERMINED; `new_result=false` denotes absence of novelty certification.

This is an AI-assisted, self-audited, unrefereed preprint. No proof-assistant verification, independent human review, absolute-priority certification, guaranteed indexing or full-source-record closure is asserted.