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Verify Remark 5.8 mechanism; correct it to level-cycle conservation

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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>
This commit is contained in:
didericis
2026-06-11 16:33:00 -04:00
parent 79cbca8e00
commit 5bed8b4dfb
6 changed files with 354 additions and 10 deletions
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papers/medial_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk
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