Three verification scripts:
experiments/check_30_residual.py and check_30_residual_v2.py:
attempt to identify the hypothesized residual case (|S| = 8 AND
p_hit = p_total = 8) where all G'-pentagons would be hit by S
forcing the fallback to require G'-heptagons. Result: 0 such
colourings — the conditional doesn't occur empirically.
experiments/check_gprime_pentagon_always_works.py:
direct check that across all 1,314 bad colourings, at least one
G'-pentagon has its boundary entirely in V(K_b) ∪ V(K_c).
RESULT: 1,314 / 1,314 = 100.00% have an uncovered G'-pentagon.
So the G'-pentagon fallback conjecture (Conjecture
gprime-pentagon-fallback) is empirically true on ALL chord-apex+
Kempe colourings — both the "tight" ones (handled structurally by
Theorem deciding-face-partial-extended) and the "bad" ones
(where Lemma flank-covering-hex fails).
Implication: the residual cases I worried about (where the fallback
would need to be relaxed to length ≢ 0 mod 3) DO NOT OCCUR. So the
Conjecture (G'-pentagon fallback) suffices to close the deciding-
face conjecture in full empirical generality.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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