For each chord-apex+Kempe colouring (n in [12, 18]), record:
(1) (#L, #R) split of c-edge sides along K_b and K_c. #L == #R only
in 35.43% of colourings (the rest have unbalanced sides --
consistent with the empirical Heawood non-constancy).
(2) Ordered sequence of (i_b mod 2, i_c mod 2) parity pairs at
shared K_b cap K_c vertices in K_b walk order, plus a tally of
transitions in the 4-state space.
Two clean structural observations on the transition matrix:
(A) i_b parity strictly alternates between consecutive shared
K_b-vertices. Every transition goes (0, *) -> (1, *) or
(1, *) -> (0, *); transitions within (0, *) or within (1, *) are
never observed. So shared positions on K_b alternate even/odd in
walk order -- the gap on K_b between consecutive shared vertices
is always odd.
(B) From odd-i_b states, i_c parity must flip too: (1, 0) only
transitions to (0, 1) and (1, 1) only to (0, 0). From even-i_b
states, both i_c outcomes occur.
(B) is explained structurally: at an odd-i_b shared vertex K_b leaves
via the a-edge (which is also on K_c), so K_b and K_c traverse the
same edge and K_c advances exactly one step, flipping i_c. At an
even-i_b shared vertex K_b leaves via the b-edge (off K_c), so K_c
advances at its own pace and i_c can be either parity at the next
shared vertex.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Add Remark 5.5 immediately after Lemma 5.3's proof, recording the
empirical reduction of Conjecture 5.1 via the contrapositive of
Lemma 5.3: the conjecture follows from "h_phi is not constant on
V(K_b) U V(K_c)", and we have verified that non-constancy holds on
every one of 142,812 chord-apex+Kempe colourings up to n <= 20
(including the six Holton-McKay duals as a special case).
This is an independent empirical near-proof of Conjecture 5.1,
complementary to the direct (1)-(3) witness check in
Remark 5.6 / rem:conj-3-6-empirical. A structural proof of the
non-constancy claim would upgrade this to a proof of the
conjecture.
Also include two diagnostic scripts that informed the remark:
- check_shared_parity.py: parity-bucket symmetry n_{0,0} = n_{1,1},
n_{0,1} = n_{1,0} at vertices in V(K_b) cap V(K_c). 100%.
- check_cw_parity_prediction.py: structural identity
s_b XOR s_c = i_b XOR i_c XOR 1 holds at every shared vertex
(263,004 / 263,004), and the simple constancy prediction matches
exactly 50% of shared vertices per colouring with 0 perfectly
matching colourings.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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>
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