experiments/check_S8_hit8_pG.py finds that all 30 |S|=8 bad
colourings with hit = 8 have p_G = 11 EXACTLY. Not p_G ∈ {9, 10, 11}
as I'd expected, but always 11.
This means: when |S| = 8 and 8 G'-pentagons are hit, the parent
triangulation v has NO degree-5 neighbours (= all 5 neighbours have
degree ≥ 6), and hence the reduced dual has 12 - 1 - 0 = 11
G'-pentagons. Three G'-pentagons are uncovered, not merely one.
Updated Remark (gprime-pigeonhole-stop) in paper to reflect this
stronger regularity: the size of S = V \ (V(K_b) ∪ V(K_c)) is
structurally tied to the count of pentagonal F_k adjacent to F_v in
chord-apex+Kempe colourings. A non-empirical proof of this is open.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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