experiments/check_S_vs_pent_Fk.py: joint distribution of |S| and
# pentagonal F_k (= count of n_i, n_{i+1}, n_{i+3} = 5, i.e.
"visible" pent F_k via flank/merged faces).
Across all 142,812 chord-apex+Kempe colourings:
- |S| = 0 dominates: 73.9% have full coverage.
- For |S| = 2, 4, 6, 8: distribution of visible pent F_k spans 0-3
with no clean monotone trend.
- |S| = 12, 14 cases NEVER have visible = 0 (= 0 occurrences in
the 0-column for these |S| values).
- The 30 special "|S|=8 hit=8" cases all have full p_G = 11 (= all
5 of v's neighbours degree ≥ 6), not just visible = 0.
So the obvious |S| ↔ # pent F_k coupling doesn't hold uniformly.
The "|S|=8 hit=8 ⇒ p_G = 11" empirical fact is specific to the
conjunction (high |S| + high hit), not to |S| alone.
For a structural proof of "|S|=8 + hit=8 ⇒ p_G = 11", we'd need a
deep Kempe-cycle-structural argument that hitting 8 G'-pentagons
with an 8-vertex S-cycle requires specific local geometry around v.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
2d8c679691
