Audit of the structural proof of Conjecture 5.1 (via deciding-face
conjecture) identifies one real proof gap:
Lemma (Flank covering, n_i = 6), Case (b) sub-case (ii) -- when
φ(A_i P_1) = c_1 AND φ(P_1 P_2) = c_0 -- the propagation argument
"the cycle at P_2 passes from P_2 to P_1" requires the {c, c_0}-Kempe
cycle through P_2 to be K_b, which forces P_1 onto K_b via the c_0
edge P_1 P_2. Properness at P_2 only forces P_2 ∈ V(K_c) (via
φ(A_{i+1} P_2) = c_1), not P_2 ∈ V(K_b). The further step requires
controlling the {c, c_0}-walk through the rest of the graph, which
the local argument doesn't do.
experiments/audit_tight_coverage.py quantifies the impact across
empirical data:
- 7,930 / 7,930 (G, v, i) triples up to |V(G)| ≤ 20 are covered
by the FULL partial proof (including the n_i = 6 lemma);
- 7,531 / 7,930 (94.97%) are covered by the TIGHT subset
(n_i = 5 OR n_{i+1} = 5 OR (n_{i+2}, n_{i+4}) = (5, 5)) which
has no proof gap;
- 399 (5.03%) genuinely require the n_i = 6 lemma.
So the gap matters: empirical coverage of the tight subset alone is
~95%, not 100%.
Paper changes:
- Lemma (Flank covering, n_i = 6) marked as "partial" with a status
note in the statement itself.
- Proof of Lemma includes an "Audit note" identifying the open
sub-case explicitly, after establishing the parts that ARE proven.
- Empirical coverage remark softened: the 100% claim is restated
as "modulo the open sub-case", with the 94.97% tight figure
given separately.
Empirically the n_i = 6 lemma is robust (all 142,812 colourings have
a deciding face), so the gap is probably patchable — likely either
via a structural argument that rules out the bad sub-case in
chord-apex+Kempe colourings, or via a global K_b-walk argument
showing P_2 ∈ V(K_b) anyway. But this is open.
Paper stays at 21 pages (only added text within existing lemma + remark).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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