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>
Empirical check (check_v_neighbour_degrees.py): 99.70% of (G, v, i)
triples up to |V(G)| ≤ 20 are covered by the flank-face partial proof
(Theorem deciding-face-partial). The remaining 24 / 7,930 (0.30%)
triples all have BOTH n_i, n_{i+1} ≥ 7, but in every single case the
remaining three neighbour degrees are (n_{i+2}, n_{i+3}, n_{i+4}) =
(5, 5, 5). For these, F^♭_outer has length 5+5-3 = 7 ≡ 1 mod 3 and a
boundary that fully lies in V(K_b) ∪ V(K_c).
Paper changes:
- Fix the existing flank-face theorem statement (was too loose: the
"WLOG some n_k" was actually only valid for k ∈ {i, i+1}, not
arbitrary k; the flank face only exists for the chosen i).
- Add Definition (Outer face) F^♭_outer (the side-1 + arc + merged +
arc + side-0 face inside F on the merged side of v_n).
- Add Lemma (Outer-face length): |F^♭_outer| = n_{i+2} + n_{i+4} - 3.
- Add Lemma (Outer-face covering, pentagonal-flanks case): if
n_{i+2} = n_{i+4} = 5, the boundary of F^♭_outer lies in
V(K_b) ∪ V(K_c). Proof: the two intermediates P_23 and P_40 each
lie adjacent to A_{i+3} ∈ V(K_b) ∩ V(K_c) and A_{i+4} ∈ V(K_b) ∩
V(K_c) respectively (via the merged edge's coverage of K_b ∩ K_c),
and the c_0/c_1 split of A_{i+3} and A_{i+4}'s non-merged edges
forces each intermediate into one of K_b or K_c.
- Add Theorem (Extended partial proof): deciding face exists in any
of cases (a) n_i ∈ {5,6}, (b) n_{i+1} ∈ {5,6}, (c) n_{i+2} =
n_{i+4} = 5.
- Rewrite the "remaining case" remark to record that
Theorem (Extended partial proof) covers 100% of empirical
(G, v, i) triples up to |V(G)| ≤ 20 -- giving a STRUCTURAL
PROOF of Conjecture 5.1 on the full empirical range.
So the combined result is:
Conjecture 5.1 (face-monochromatic-pair) is proven structurally for
every chord-apex+Kempe colouring of every reduced dual of every
triangulation of min degree 5 with |V(G)| ≤ 20.
The only remaining open structural case is configurations with both
n_i, n_{i+1} ≥ 7 AND (n_{i+2}, n_{i+4}) not both 5 -- which never
arises empirically up to |V(G)| ≤ 20 but could appear for larger
triangulations.
Paper grows from 20 to 21 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
experiments/check_v_neighbour_degrees.py reports the cyclic degree
sequence of v's 5 neighbours in G across all (G, v, i) triples
underlying chord-apex+Kempe colourings (|V(G)| ≤ 20).
Result:
- 100.00% of (G, v) pairs have at least one neighbour of v of
degree 5. So the partial proof of the deciding-face conjecture
(Theorem deciding-face-partial, which requires n_k ∈ {5, 6} for
some k) handles all (G, v) pairs IF we can pick any k freely.
- 99.70% of (G, v, i) triples have min(n_i, n_{i+1}) ≤ 6, so the
partial proof's flank-face argument applies to the specific i.
- 24 / 7,930 (0.30%) triples have BOTH n_i ≥ 7 AND n_{i+1} ≥ 7
(the "bad" case where the partial proof's flank-face doesn't
work). These occur at n_G = 19, 20 for triangulations with
cyclic neighbour-degree sequences like (5,7,7,5,5).
For these 24 bad triples, the remaining 3 neighbours typically have
degree 5, so F^♭_outer (length n_{i+2} + n_{i+4} - 3 = 5+5-3 = 7,
not divisible by 3) becomes the candidate deciding face. Extending
the structural proof to cover F^♭_outer is the next step.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis