constancy on V(K_b) U V(K_c))
Three empirical checks on all chord-apex+Kempe colourings up to
n = 20 (142,812 colourings):
1. check_heawood_on_kempe.py
- Sum_v h_phi(v): not zero in general; 17.6% of colourings have
sum 0, the rest range in {+-4, +-8, +-12, +-16, +-20, +-24}.
So the global "Heawood sum = 0" identity fails.
- h_phi constant on V(K_b) U V(K_c): NEVER (0/142,812). This is
the central empirical result -- by Lemma 5.3's contrapositive
it gives an empirical proof of Conjecture 5.1 on these
surrogates.
2. check_heawood_per_kempe_cycle.py
- Sum_{V(K_b)} h_phi and sum_{V(K_c)} h_phi range widely (-20 to
+20), with only ~23% zero. So the "Heawood sum on each Kempe
cycle = 0" identity also fails -- the per-cycle sum is not the
right invariant.
3. check_heawood_pair_mismatch.py
- For each of 16 named-vertex pairs (v_n with each A_j, A_j with
A_k for j, k in {i, ..., i+4}), counts how often h_phi differs.
No pair is *always* differing -- the closest are consecutive
pairs (A_j, A_{j+1}) at ~75% diff. So the Heawood mismatch
enforcing non-constancy on V(K_b) U V(K_c) is diffuse, not at
a fixed pair.
Together these results confirm Path 4 (Conjecture 5.1 reduces via
Lemma 5.3 to showing h_phi non-constant on V(K_b) U V(K_c)) but
rule out the simplest single-pair-identity proof; the structural
obstruction lives elsewhere (likely a topological/cycle-winding
argument or a chord-apex/Kempe-spike colour cascade).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
The previous statement "Heawood is constant on K through merged" was
strictly stronger than what the proof actually established without
Conjecture 5.3. Restate the lemma in the contrapositive direction:
If h_phi is constant on V(K), then no edge e in E(K) admits a face
F of G'^hat and edges e_1, e_2 on dF realising the clause-(3) arc
of Conjecture 5.1 at the endpoints of e.
Proof structure is mostly preserved (same F_R/F_L geometry, same case
split on phi(e) in {a, b}, same reading-off of cyclic colour orders).
The hypothesis "h_phi(v_0) != h_phi(v_1)" becomes "h_phi(v_0) =
h_phi(v_1)", which flips the conclusion: the same-coloured non-e
edges at v_0, v_1 land on opposite faces of e instead of the same
face. No dependency on Conjecture 5.3 or Theorem 4.X.
Redraw the figure to match the new lemma: both vertices labelled
h_phi = +1, both showing CW order (a, b, c), and the same-colour pair
(b-edges in Case A, a-edges in Case B) drawn on opposite sides of e.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Add Definition 3.1 "Heawood number of a vertex" (+1 if CW colour order
is (1,2,3), -1 if (1,3,2)) and cite Heawood 1898 in the bibliography.
- Add Lemma 5.2 "Heawood number is constant on the Kempe cycles through
the merged edge", positioned immediately after Conjecture 5.1. Its
proof exhibits a (F, e_1, e_2) witness for clauses (1)-(3) of the
conjecture from any pair (v_0, v_1) of consecutive K-vertices with
differing Heawood signs, by cases on whether phi(e) = a or b. The
proof does not invoke Conjecture 5.3 or Theorem 4.X.
- Add a two-panel figure illustrating Case A (b-edges on F_R when
phi(e) = a) and Case B (a-edges on F_L when phi(e) = b), with the
cyclic colour orders (a, b, c) at v_0 and (a, c, b) at v_1 visible
from the angular layout.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Main paper: dual_decomposition_minimal_counterexamples/ ->
face_monochromatic_pairs/. Title is now
"Face-Monochromatic Pairs and the Four Colour Theorem".
- Companion paper: dual_decomposition_iterated_reduction/ ->
iterated_reduction_in_reduced_dual/. Title is now
"An Iterated Reduction in the Reduced Dual". Its prose and bibliography
cite the parent under the new title.
- Update one absolute sys.path reference inside
check_conj_face_kempe_n15.py that pointed at the old folder.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis