# Verification report

Problem: AMR-050-0043. Manuscript: Cosine Sums of Focal-Inversive Elliptic
Billiards. Author: Alper Ferudun, Mercury Software GmbH.

The proof is the complete meromorphic argument in main.tex. Its scope
is a fixed nondegenerate confocal elliptic caustic and a genuine least
period other than four, with any admissible coprime star winding.
The definition of the angle cosine is an explicit Euclidean dot product.
Both focal sums have the same constant. Hyperbolic and degenerate
caustics are excluded. Repeated traversal of an allowed primitive
orbit multiplies the sum; an arbitrary repeated length is not enough.

The self-audit checks the actual canonical billiard parameter, positive
real chord lengths, the complete divisors of m, e and R, the sole
double-pole collision h=K, and removal of all simple poles by reflection
and cyclic periodicity. This is a proof for all parameters in the stated
scope, not an extrapolation from finite samples.

The exact combinatorial regression checks 982 primitive step ratios
through period 80, 7848 possible pole locations and 421516 translated
symmetry centers. The collision equations are solved exactly. Symbolic
controls check the focal-distance and chord-direction identities.
Two exact four-periodic billiards in the same family have inverse
cosine sums 9/5 and 0. Their eight-entry repeated lists have sums
18/5 and 0. Ellipse, caustic tangency, specular reflection, positive
focal distances and inverse length products are checked exactly.

At 70 decimal digits, 300 real phases across 100 elliptic families
compare the direct inverted polygon with the meromorphic expression;
the two four-periodic families are expected nonconstancy controls.
All other tested families are invariant, including star windings, and
294 direct comparisons at the other focus agree. Separate complex
tests check the eight-point R divisor in four parameter choices,
reflection symmetries, common periods, six complex trace evaluations
and 24 evaluations near apparent poles. These numerical controls are
not formal verification and do not prove the universal theorem.

Normal and optimized Python executions agree for both programs. The
package retains the exact code and output hashes. The five-page PDF
compiled with the built-in compiler and Tectonic; all five pages were
visually reviewed. No clipping, missing glyphs, undefined references
or overflowing boxes were observed. Two harmless underfull bibliography
warnings affect spacing only.

The triangular case and four-periodic/hyperbolic exceptions are credited
to prior work. No later general proof was located in the bounded
primary-source search, but novelty or absolute priority is not certified.
This is an AI-assisted, self-audited and unrefereed preprint; there was
no independent reviewer and no proof-assistant verification.