For each of the 1,314 chord-apex+Kempe colourings on which Lemma
flank-covering-hex's conclusion empirically fails (the audit-revealed
sub-case (b)(ii) bad cases), classify the actual deciding face.
experiments/check_bad_subcase_deciding_face.py findings:
Deciding-face TYPE distribution (per colouring; multiple deciding
faces possible per colouring):
G-prime-face (= face of G' not modified by reduction): 7,872
outer (F_outer^♭): 1,236
flank-upper: 1,188
merged: 516
Per-colouring coverage:
G-prime-face available: 1,314 / 1,314 = 100.00% ← always
outer: 1,236 / 1,314 = 94.06%
flank-upper: 1,188 / 1,314 = 90.41%
merged: 516 / 1,314 = 39.27%
100% of bad colourings have at least one G'-pentagon (length 5) as a
deciding face -- i.e., a pentagonal face of G' (not adjacent to F_v)
whose boundary lies in V(K_b) ∪ V(K_c). This suggests the missing
piece is a "G'-pentagon fallback" lemma.
Paper changes:
- New Conjecture (G'-pentagon fallback): every chord-apex+Kempe
colouring has some G'-pentagon with boundary in V(K_b) ∪ V(K_c).
- Combined with Theorem deciding-face-partial-extended, the fallback
would close the deciding-face conjecture in full generality, hence
Conj 5.1 (face-monochromatic-pair). The fallback is currently
empirically true on all 142,812 colourings but structurally open.
- Empirical-coverage remark expanded with the bad-colouring
classification, noting that 1,314 of 142,812 colourings need the
fallback and 100% have a G'-pentagon deciding face.
Paper grows from 21 to 22 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
2f82f6e0bc
