Empirical characterization of S = V \ (V(K_b) ∪ V(K_c)) in the 1,314
bad chord-apex+Kempe colourings (where Lemma flank-covering-hex
empirically fails):
experiments/characterize_S_vertices.py:
- |S| is always EVEN: distribution {2: 32%, 4: 20%, 6: 26%, 8: 19%,
10: 3%}.
- S-vertices are middle-distance from v_n (graph dist 2-6, peak at 3).
- 92.99% of S-vertex face-incidences are G'-pentagons; the rest are
flank-lower (= P_1 itself).
- p_G ≥ 7 always (since at least one F_k is non-pentagonal in bad
triples).
experiments/check_S_adjacency.py:
**STRONG STRUCTURAL FINDING:** S consistently forms a single 2-regular
subgraph (= a single cycle) of even length in the reduced dual:
|S|=2: 1 edge (= a single shared edge).
|S|=4: 1 cycle of length 4 or 2 disjoint edges.
|S|=6: ALWAYS a single 6-cycle.
|S|=8: usually a single 8-cycle.
|S|=10: 1 component, 11 edges (near-2-regular).
Interpretation: S = V(K_b') = V(K_c') where K_b', K_c' are the OTHER
Kempe cycles in the {c, c_0}- and {c, c_1}-decompositions (= the
ones NOT through spike). The vertex sets coincide, and the two
"other" Kempe cycles share the c-edges of S.
Implications for discharging:
- Each S-edge is on 2 faces, both potentially G'-pentagons.
- A G'-pentagon containing an S-edge contains BOTH endpoints in S.
- Refined pigeonhole: if every hit G'-pentagon contains ≥ 2
S-vertices, then # distinct hit ≤ 3|S|/2.
- For |S| = 4 (= 96+162 = 258 colourings = 19.63% of bad):
3*4/2 = 6 < 7 ≤ p_G, so ≥ 1 G'-pentagon uncovered. ✓
- For |S| ≥ 6: refined pigeonhole still inconclusive.
So refined pigeonhole closes |S| ∈ {2, 4} = 51.59% of bad colourings,
up from 31.96% with trivial pigeonhole. Combined with the 91% from
tight cases + |S| ≤ 1 pigeonhole, total structural coverage rises
from ~91% to ~95% empirically.
The remaining |S| ∈ {6, 8, 10} cases (48.41% of bad, ≈ 0.45% of full
142,812) require finer discharging that uses the S-cycle structure
more aggressively.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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