Computational checks of the necessity of Kempe-balance (Remark 5.8):
- check_medial_face_parity.py shows the naive "even P-coloured vertices
per medial face" claim is false (odd vertex-faces on the octahedron and
stacked triangulations), so the original face-parity justification was
wrong.
- check_remark58_bitefree.py builds genuine bite-free tire pieces (capped
triangulated annuli) and confirms every proper 3-colouring of M(G)
restricts to a Kempe-balanced colouring (|A(T)|=6,8,10,12, all
colourings, zero failures).
Rewrite Remark 5.8 to cite the correct mechanism: the up/down apexes lie
on level cycles, and a P-Kempe cycle meets each level cycle in an even
number of P-coloured incidences (Lemma 5.6). Note the bite case is not
yet checked end to end.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
didericis
5bed8b4dfb