A parity-selected endpoint theorem for the regular dodecahedron

Abstract

This note gives two precise answers associated with Fuchs Conjecture 3.2. The imported bare endpoint-angle condition is false: exactly the two proper face diagonals per initial face and vertex are additional paths and have graph distance two. An explicitly adopted parity-selected convention has a complete unbounded endpoint-exclusion proof through double-pentagon germ symmetry, a physical edge-midpoint half-turn, and a unique-common-neighbor argument. The convention is a supported disambiguation, not an authenticated author correction or uniquely compelled printed reading. The midpoint and refolding mechanisms are credited to Athreya–Aulicino–Hooper and Troubetzkoy. Alper Ferudun's current contemporaneous preprint, DOI 10.5281/zenodo.23267641, contains both the formulation distinction and endpoint theorem, and the public face-word type correction. GitHub records our draft PR creation on 1 October 2026 and DataCite records that related DOI registration on 9 October 2026. These dates do not establish historical exact-byte visibility, firstness, author awareness or independence. On 15 October 2026, the DOI-version files were retrieved successfully and authenticated against record checksums and the previously pinned website hashes; the inspected PDF and source ZIP are byte-identical to the DOI-version files. Uncompleted journal/body-version comparisons remain disclosed. No current openness or guaranteed novelty is claimed. The deposit includes a four-page research note, standalone LaTeX source and an exact verification package with current optimization-safe guards, meaningful wrong-science controls, distinct independent scientific audits and bounded priority ledgers. It independently validates the public face word while separating its terminal-angle error and additional displayed-coordinate norm inconsistency. Finite checks corroborate the data; they do not replace the universal proof. AI tools were used extensively in development, drafting, source checking, reproduction and adversarial verification. This preprint is unrefereed and has not undergone conventional human peer review.