didericis 873c2ccdbd face_monochromatic_pairs: retract n_i = 6 lemma as empirically false ...

CRITICAL AUDIT FINDING: experiments/check_subcase_iib.py shows that
Lemma (Flank covering, n_i = 6) is empirically FALSE in full
generality, not just unproven:

  Across 142,812 chord-apex+Kempe colourings up to |V(G)| ≤ 20:
    -   9,228 (6.46%) reach sub-case (ii.B) of Case (b)
        (φ(A_i P_1) = c_1 AND φ(P_1 P_2) = c_0);
    -   1,314 (0.92%) of those have P_1 ∉ V(K_b) ∪ V(K_c),
        falsifying the lemma's conclusion ∂F_flank^♭ ⊆ V(K_b) ∪ V(K_c).

So the original n_i = 6 lemma cannot be saved by patching the proof;
the conclusion itself is wrong.

Paper changes:
  - Lemma (Flank covering, n_i = 6): retracted in full generality.
    Restated with a weakened conclusion (true only for Case (a) and
    Case (b) sub-case (i)), with explicit acknowledgement that the
    sub-case (b)(ii) configuration falsifies the lemma on 1,314
    colourings.
  - Proof of the lemma: rewritten to honestly stop at the proven sub-
    cases; sub-case (b)(ii) is identified as unprovable by local
    argument (and now demonstrated empirically false).
  - Theorem (Partial proof via flank): restricted from n_i ∈ {5, 6}
    to n_i = 5 only.
  - Theorem (Extended partial proof): cases relabelled (a'), (b'), (c)
    with a' = (n_i = 5), b' = (n_{i+1} = 5), c = (n_{i+2} = n_{i+4} = 5).
  - Empirical coverage remark: structural proof covers
    7,531 / 7,930 (94.97%) of (G, v, i) configurations up to
    |V(G)| ≤ 20. The other 399 (5.03%) have at least one n_k = 6 but
    no n_k = 5 in the right position; the flank face on the n_k = 6
    side is the natural candidate but is no longer a tight covering.
  - Deciding-face conjecture itself remains empirically true on all
    142,812 colourings; the proof's structural step is what's open
    on the 399 triples.

Lessons from the audit:
  - The "+P_2 ∈ V(K_b) ∪ V(K_c) implies P_1 ∈ V(K_b) ∪ V(K_c)"
    propagation in the original n_i = 6 proof was wrong: the cycle
    type (K_b vs K_c) matters in a way the proof glossed over, and
    specifically when φ(P_1 P_2) = c_0 the K_c cycle through P_2
    doesn't use that edge.
  - A correct n_i = 6 lemma would require a global K_b-walk argument
    showing the {c, c_0}-cycle through P_2 coincides with K_b in the
    bad sub-case. Empirically this is FALSE in 0.92% of colourings,
    so no such argument exists; the n_i = 6 covering must instead come
    from a different face entirely for those colourings.

Paper stays at 21 pages.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

2026-05-25 05:10:48 -04:00

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