Rename the shared helper module to a number-resistant name. Update
all 26 dependent scripts via sed.
Add experiments/test_n_21_to_24.py — extends the empirical check
beyond |V(G)| ≤ 20 to n_G ∈ [21, 24]. Checks per chord-apex+Kempe
colouring:
(1) h_φ constant on V(K_b)? (counterexample to Corollary 5.4)
(2) h_φ constant on V(K_b) ∪ V(K_c)? (counterexample to Conj 5.1)
(3) Deciding face exists?
Writes results incrementally to test_n_21_to_24_results.jsonl (one
JSON line per triangulation, plus n-level and grand summaries).
Emits PROGRESS lines every 10 minutes (default) to stdout for live
monitoring.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW PROOF STRATEGY for Conjecture 5.1 (face-monochromatic-pair):
1. NEW Conjecture (Deciding face): For every chord-apex+Kempe
colouring φ of every reduced dual, the reduced dual has a face f
with ∂f ⊆ V(K_b) ∪ V(K_c) and |f| ≢ 0 (mod 3).
2. NEW Theorem: Deciding-face conjecture implies Conj 5.1.
Proof: contradiction. Assume no clauses-(1)-(3) witness for some
chord-apex+Kempe φ. By Lemma 5.3, h_φ ≡ ε ∈ {±1} on V(K_b) ∪ V(K_c).
By the deciding-face conjecture, ∃ face f with ∂f ⊆ V(K_b) ∪ V(K_c),
|f| ≢ 0 (mod 3). Heawood's face-sum identity (Heawood 1898) gives
Σ_{v ∈ ∂f} h_φ(v) = ε|f| ≡ 0 (mod 3). Since gcd(|f|, 3) = 1, we get
ε ≡ 0 (mod 3), but ε ∈ {±1} — contradiction.
3. EMPIRICAL: Conjecture (Deciding face) verified on 142,812 / 142,812
chord-apex+Kempe colourings of reduced duals up to |V(G)| ≤ 20 --
matching the full coverage of check_constancy_obstruction.py.
Face-length distribution:
|f| = 4: 13,074
|f| = 5: 102,498 (most common)
|f| = 7: 18,570
|f| = 8: 7,752
|f| = 10: 846
|f| = 11: 72
(All ≢ 0 mod 3.)
New scripts:
- check_kb_kc_coverage.py: |V(K_b) ∪ V(K_c)| / |V(Ĝ')| distribution.
73.87% of colourings have V(K_b) ∪ V(K_c) = V (full coverage); the
remaining 26% have coverage ≥ 70%, mostly ≥ 90%.
- check_deciding_face.py: existence of deciding face across all
colourings; 100.00% / 142,812.
Why this is the right reduction:
- It uses ALL THREE pieces of chord-apex+Kempe structure: Lemma 5.3
(constancy from no-witness), forced colour-equality at merged/spike,
and forced Kempe-cycle containment of merged + spike + side edges
(the latter two enter via V(K_b) ∪ V(K_c) covering specific
structural vertices).
- It uses Heawood's face-sum identity, which is the classical 3-fold
parity constraint on cubic plane 3-edge-colourings.
- The C28 counterexample to Conjecture 5.5 is not affected: it's not
a chord-apex+Kempe colouring of a reduced dual, so the deciding-face
structure doesn't apply.
Remaining work: prove the deciding-face conjecture structurally (likely
via the specific F_01 / F_12 "flank face" of the reduced dual, whose
length n_0 - 1 from the adjacent G'-face of length n_0 ≥ 5 is ≢ 0 mod 3
exactly when n_0 ≢ 1 mod 3, plus boundary-in-V(K_b) ∪ V(K_c) which
follows from Lemma 5.X kempe-spike + colour analysis at A_i).
Paper grows from 17 to 18 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis