experiments/check_S_face_structure.py: detailed analysis of S-cycle
structure for the 1,314 bad chord-apex+Kempe colourings.
Findings:
1. S-cycle is NEVER a face boundary of the reduced dual (0% across
all |S| from 2 to 10). So the S-cycle's "interior" contains
additional faces.
2. Refined pigeonhole + p_G ≥ 7 + S-cycle structure closes:
- |S| = 2: max hit 2 < p_G ≥ 7. ✓ 420 / 1314.
- |S| = 4: max hit 4 < p_G ≥ 8. ✓ 258 / 1314.
- |S| = 6: max hit 7 < p_G ≥ 8. ✓ 348 / 1314.
- |S| = 10: max hit 7 < p_G ≥ 8. ✓ 36 / 1314.
Total: 1062 / 1314 = 80.8% of bad colourings closed.
3. |S| = 8: max hit = 8 = min p_G (sometimes). ≤ 30 colourings
(~2.3% of bad, ~0.02% of full 142,812) have ALL G'-pentagons hit
by S — so the G'-pentagon fallback (Conjecture 5.X) is
EMPIRICALLY FALSE in this sub-case! For these, the deciding face
must be a G'-heptagon (length 7) or G'-octagon (length 8), not a
pentagon. Both lengths are ≢ 0 mod 3 and so still serve as
deciding faces.
So the structurally-correct fallback is "G'-face of length ≢ 0 mod 3",
not "G'-pentagon" specifically. This is consistent with the
deciding-face data: 462 incidences of length-7 G-prime-faces, 6 of
length-8.
Combined structural coverage:
- Tight cases (a', b', c): 91% (1,205 / 1,314 plus full-coverage cases)
- Refined pigeonhole: 80.8% of bad colourings = 1062 / 1314
- Total: ≈ 99.5% of full 142,812 chord-apex+Kempe colourings
structurally proven.
The remaining ~0.02% (30 colourings) need a structural argument that
some G'-face of length ≢ 0 mod 3 always exists with boundary in
V(K_b) ∪ V(K_c).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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