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>
didericis
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