- Step 1: local CW structure at u, w (F_R = F_{ab}^u = F_{ca}^w, etc.)
- Step 2 (new, complete): K_1 \ e leaves u into In(K_0) and arrives at w
from Out(K_0), so it crosses K_0 an odd number of times. Each crossing
uses a shared a-edge, so |E(K_0) ∩ E(K_1)| is even and ≥ 2. Closes
the case of a single shared edge.
- Step 3 (new, complete): Heawood's face-sum identity ∑ h_φ ≡ 0 (mod 3)
applied to every H-face inside a face Φ of K_0 ∪ K_1, with
multiplicity bookkeeping at degree-3 vs degree-2 boundary vertices,
yields ν_{2,Φ} ≡ -ℓ_Φ (mod 3) for every face Φ.
- Step 4 (open): use Lemma 5.2 alternation to force ν_{2,Φ} on some
face, and exhibit a violation.
Literature search via research-analyst confirms the theorem is novel
(Heawood face-sum and the +/-1 rotation sign are classical; no result
relating intersecting Kempe cycles via vertex signs found).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
If true, Theorem 5.5 + Lemma 5.3 → Conjecture 5.1: in the no-clause-3-witness
world, both V(K_b) and V(K_c) have constant Heawood (Lemma 5.3), but K_b
and K_c share the merged edge, contradicting Theorem 5.5.
Proof sketch sets up the two Lemma-5.2 applications (c-edges at u,w on
opposite sides of K_0; b-edges at u,w on opposite sides of K_1) and the
theta-curve K_0 ∪ K_1, but the planar-orientation contradiction is left
as "to be filled in".
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Standalone commentary document for readers of the paper:
- Headline table mapping each empirical / structural claim to its
proof status and the verification numbers we have.
- Statement of "what's actually open": the structural proof of
non-constancy of h_phi on V(K_b) (alone), which reduces to
Conjecture 5.1 via Corollary 5.4.
- Three reasons the proof appears to be hard:
(1) the obstruction has no slack (min flip count 2 -> 1 minority
vertex);
(2) the minority is not anchored to a structural vertex (~half
live on "other" non-named vertices);
(3) no single named-vertex-pair is always a mismatch (max 75%).
- List of candidate mechanisms ruled out by diagnostics:
- global sum identity, per-cycle sum identity,
- cycle-side balance |L| == |R|,
- specific-pair-always-mismatches.
- Index of diagnostic scripts in experiments/.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Two more diagnostics on chord-apex+Kempe colourings (n <= 18,
13,800 colourings) probing how thin the non-constancy obstacle on
V(K_b) is:
1. check_min_flip_structure.py
- Flip count on K_b drops as low as 2 (at n = 18, 12 colourings):
these have a single minority Heawood vertex on K_b. So the
structural obstacle has NO slack: proving "at least 1 minority
vertex on V(K_b)" is the bar.
- All n=14 colourings (216) have flip count = 8 exactly. At
larger n the distribution spreads.
2. check_minority_location.py
- For colourings with K_b flip count <= 4, identify the minority
Heawood vertices and tally where they sit:
v_n : 12.86%
A_{i+1} : 10.82%
A_{i+2} : 8.98%
A_i : 7.76%
A_{i+4} : 5.31%
A_{i+3} : 5.10%
"other" : 49.18%
- About half the minority vertices live on non-named vertices in
the rest of G'. No single named vertex is *always* the
minority. The obstruction is genuinely diffuse / global, not
anchored to a specific structural location.
These together imply that the structural proof of "h_phi non-constant
on V(K_b)" must be global (no local "this vertex must flip"
argument suffices) and handle the edge case where only one minority
vertex exists. Likely requires a topological / homological / global
counting argument.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Empirical refinement of Lemma 5.3: h_phi is non-constant on V(K_b)
alone (not just on the union) and likewise on V(K_c) alone, in every
one of 142,812 chord-apex+Kempe colourings tested (n in [12, 20]).
This is strictly stronger than what we previously reported.
The proof of Lemma 5.3 already constructs the (F, e_1, e_2) witness
from any consecutive same-Heawood failure on either Kempe cycle
through merged -- never needing the other cycle. Pull that out into
a separate Corollary 5.4 ("Per-cycle form"), which makes the
empirical-to-conjecture path more direct.
Update Remark 5.5 to:
- Cite Corollary 5.4 instead of the contrapositive of Lemma 5.3.
- Replace "non-constant on V(K_b) U V(K_c)" with the per-cycle form.
- Extend the empirical table with separate columns for K_b and K_c
non-constancy.
Also commit experiments/check_constancy_obstruction.py, the script
that produced these refined empirical findings. It additionally
records that no single named vertex (v_n, A_i, ..., A_{i+4}) is
structurally majority or minority -- the minority rates cluster in
31-39%, ruling out a single-vertex-mismatch identity.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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>
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