A cone-angle proof of Fuchs's labelled length-ratio conjecture

Abstract

This research note gives a cone-angle and classical Veech-cusp proof of Fuchs's Conjecture 2.7: the lengths of source-parallel first-vertex billiard trajectories in every regular n-gon, n≥3, have the canonical labelled sine ratio. The proof includes sides, odd and even double polygons, the physical folding-chart sign, segment-parity reversal, and separate triangular, square and hexagonal marked-lattice cases. Alper Ferudun's unrefereed preprint of 9 October 2026, DOI10.5281/zenodo.23267641, already states and proves the same nontrivial all-n theorem by angular indices, explicit uniform twists and descent. This note presents a different proof, credits that exact overlapping solution and classical cone-angle/hyperelliptic/Veech machinery, and claims neither a new invariant nor first or exclusive resolution. The repository candidate of 30 September 2026 predates that release, but its substantive folding-chart and physical-reversal correction was recorded on 11 October 2026 at23:57UTC and accepted on 12 October 2026 UTC. We describe the two proof routes as contemporaneous separate developments. The dates do not certify either author's awareness of the other's work or a literally simultaneous publication instant. The corrected proof is not backdated; historical novelty and first or exclusive priority are not certified. The deposit contains the manuscript PDF and a support archive with standalone LaTeX source, standard-library exact finite controls, an exact pentagon reversal counterexample, recorded outputs, metadata, source locators and priority-search limits. Finite controls support the analytic/geometric proof rather than replace it or prove its imported classical theorems. AI tools were used extensively in solving, drafting, source comparison, reproduction and independent adversarial checking. This preprint is unrefereed and has not undergone conventional human peer review. The author is responsible for its contents.