For each chord-apex+Kempe colouring, record:
- |V(K_b)|, |V(K_c)|, |V(K_b) cap V(K_c)|, |V(K_b) cup V(K_c)|
- "Flip count" on each cycle: #consecutive pairs whose third-colour
edges lie on opposite local sides (= #same-Heawood pairs by Lemma A).
Results (n in [12, 18], 13,800 colourings):
- |V(K_b) cap V(K_c)| is NEVER 2 -- always >= 6. The two Kempe cycles
through merged share many vertices.
- Distributions of flip_Kb and flip_Kc are identical multisets
(consistent with b <-> c symmetry of the construction).
- But per-colouring, flip_Kb == flip_Kc only 39.65% of the time --
the symmetry is statistical, not pointwise.
- Max observed flip count is 20, never the maximum possible |V(K)|.
Consistent with h_phi never being constant on V(K_b) U V(K_c).
The substantial overlap of K_b and K_c (>= 6 shared vertices) means
the constancy hypothesis would impose simultaneous alternation
constraints from both cycles at every shared vertex -- the topological
"trap" the proof needs to exploit.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Two diagnostic scripts probing the side-classification of c-edges at
K_b-vertices and their relationship to Heawood numbers:
1. check_heawood_side_correlation.py (first attempt)
- Defines "side" as connected component of H \ K_b.
- Result: K_b separates H into 2 components in 0% of cases, so
this notion doesn't capture the planar side. (Negative result --
kept for the record / so we don't redo it.)
2. check_heawood_local_side.py (correct version)
- Defines "side" locally via the planar CW embedding at v: c-edge
is on local RIGHT if, going CW from incoming K-neighbour at v,
we hit the c-neighbour before the outgoing K-neighbour; local
LEFT otherwise.
- Result on 625,200 consecutive K_b-pairs across 13,800
chord-apex+Kempe colourings (n in [12, 18]):
same h, same side: 0
same h, diff side: 372,456 (59.57%)
diff h, same side: 252,744 (40.43%)
diff h, diff side: 0
The empirical biconditional holds perfectly:
h_phi(v_0) == h_phi(v_1) <==> c-edges on opposite sides
This is "Lemma A" -- the corrected version of the proposed
orientation lemma. Equivalently: constant Heawood on a Kempe
cycle K forces the c-edges (off-K) to ALTERNATE inside/outside
of K along the cycle (not all on one side as I initially
conjectured).
This empirical result revises the spiral picture for Path 4: under
the Lemma 5.3 hypothesis of constant h on V(K_b) U V(K_c), the
c-edges alternate sides on K_b (and the b-edges alternate sides on
K_c). K_c must then cross K_b at every K_b-vertex it shares -- a
strong topological constraint we can now exploit.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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>
Follow-up to Lemma 5.2. States that if Conjecture 5.1 has no
clauses-(1)-(3) witness for (G, G'^_{v,i}, phi), then h_phi is
constant on both Kempe cycles through merged, and the two constants
agree (since merged is on both cycles, so its endpoints force the
constants to match).
Proof is the V1-direction of the case analysis: differing h_phi on
either K_b or K_c reproduces a clause-(1)-(3) witness by the same
F_R/F_L geometry as Lemma 5.2's proof but with the hypothesis
"h_phi(v_0) != h_phi(v_1)", under which the matching-colour edges
land on the SAME face of e. Case B's merged-incidence corner is
handled by choosing a differing-Heawood pair away from merged's
endpoints.
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