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>
experiments/check_v_neighbour_degrees.py reports the cyclic degree
sequence of v's 5 neighbours in G across all (G, v, i) triples
underlying chord-apex+Kempe colourings (|V(G)| ≤ 20).
Result:
- 100.00% of (G, v) pairs have at least one neighbour of v of
degree 5. So the partial proof of the deciding-face conjecture
(Theorem deciding-face-partial, which requires n_k ∈ {5, 6} for
some k) handles all (G, v) pairs IF we can pick any k freely.
- 99.70% of (G, v, i) triples have min(n_i, n_{i+1}) ≤ 6, so the
partial proof's flank-face argument applies to the specific i.
- 24 / 7,930 (0.30%) triples have BOTH n_i ≥ 7 AND n_{i+1} ≥ 7
(the "bad" case where the partial proof's flank-face doesn't
work). These occur at n_G = 19, 20 for triangulations with
cyclic neighbour-degree sequences like (5,7,7,5,5).
For these 24 bad triples, the remaining 3 neighbours typically have
degree 5, so F^♭_outer (length n_{i+2} + n_{i+4} - 3 = 5+5-3 = 7,
not divisible by 3) becomes the candidate deciding face. Extending
the structural proof to cover F^♭_outer is the next step.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
NEW PROOF STRATEGY for Conjecture 5.1 (face-monochromatic-pair):
1. NEW Conjecture (Deciding face): For every chord-apex+Kempe
colouring φ of every reduced dual, the reduced dual has a face f
with ∂f ⊆ V(K_b) ∪ V(K_c) and |f| ≢ 0 (mod 3).
2. NEW Theorem: Deciding-face conjecture implies Conj 5.1.
Proof: contradiction. Assume no clauses-(1)-(3) witness for some
chord-apex+Kempe φ. By Lemma 5.3, h_φ ≡ ε ∈ {±1} on V(K_b) ∪ V(K_c).
By the deciding-face conjecture, ∃ face f with ∂f ⊆ V(K_b) ∪ V(K_c),
|f| ≢ 0 (mod 3). Heawood's face-sum identity (Heawood 1898) gives
Σ_{v ∈ ∂f} h_φ(v) = ε|f| ≡ 0 (mod 3). Since gcd(|f|, 3) = 1, we get
ε ≡ 0 (mod 3), but ε ∈ {±1} — contradiction.
3. EMPIRICAL: Conjecture (Deciding face) verified on 142,812 / 142,812
chord-apex+Kempe colourings of reduced duals up to |V(G)| ≤ 20 --
matching the full coverage of check_constancy_obstruction.py.
Face-length distribution:
|f| = 4: 13,074
|f| = 5: 102,498 (most common)
|f| = 7: 18,570
|f| = 8: 7,752
|f| = 10: 846
|f| = 11: 72
(All ≢ 0 mod 3.)
New scripts:
- check_kb_kc_coverage.py: |V(K_b) ∪ V(K_c)| / |V(Ĝ')| distribution.
73.87% of colourings have V(K_b) ∪ V(K_c) = V (full coverage); the
remaining 26% have coverage ≥ 70%, mostly ≥ 90%.
- check_deciding_face.py: existence of deciding face across all
colourings; 100.00% / 142,812.
Why this is the right reduction:
- It uses ALL THREE pieces of chord-apex+Kempe structure: Lemma 5.3
(constancy from no-witness), forced colour-equality at merged/spike,
and forced Kempe-cycle containment of merged + spike + side edges
(the latter two enter via V(K_b) ∪ V(K_c) covering specific
structural vertices).
- It uses Heawood's face-sum identity, which is the classical 3-fold
parity constraint on cubic plane 3-edge-colourings.
- The C28 counterexample to Conjecture 5.5 is not affected: it's not
a chord-apex+Kempe colouring of a reduced dual, so the deciding-face
structure doesn't apply.
Remaining work: prove the deciding-face conjecture structurally (likely
via the specific F_01 / F_12 "flank face" of the reduced dual, whose
length n_0 - 1 from the adjacent G'-face of length n_0 ≥ 5 is ≢ 0 mod 3
exactly when n_0 ≢ 1 mod 3, plus boundary-in-V(K_b) ∪ V(K_c) which
follows from Lemma 5.X kempe-spike + colour analysis at A_i).
Paper grows from 17 to 18 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
experiments/check_c28_no_chord_apex_kempe_constancy.py iterates all
3 triangulations on 16 vertices with min degree 5 (whose duals are
the 28-vertex cubic plane graphs with face length ≥ 5 -- including
the C28 fullerene that disproves Conjecture 5.5). For each:
- applies every chord-apex reduction (every pentagonal face of the
dual × every rotation index i ∈ {0,…,4}),
- enumerates every proper 3-edge-colouring of each reduced dual,
- filters to chord-apex+Kempe colourings (Lemmas 5.X chord-apex +
Kempe-spike),
- traces K_b, K_c through the merged edge,
- computes h_φ via the CW rotation at each vertex,
- reports any colouring where h_φ is constant on V(K_b), V(K_c), or
both.
Result:
reductions tried : 60 + 60 + 70 = 190
chord-apex+Kempe colourings: 432 + 432 + 0 = 864
constant on V(K_b) : 0 + 0 + 0 = 0
constant on V(K_c) : 0 + 0 + 0 = 0
constant on both : 0 + 0 + 0 = 0
So even though the C28 fullerene admits a proper 3-edge-colouring on
which two intersecting Kempe cycles are both constant h_φ (the
Conjecture 5.5 counterexample), none of its chord-apex reductions
admits a chord-apex+Kempe colouring with the same property -- the
extra constraints (merged + spike same colour; K_b ⊇ {spike, side_0,
merged}; K_c ⊇ {spike, side_1, merged}) genuinely rule it out.
This is consistent with the broader empirical near-proof
(check_constancy_obstruction.py: 0/142,812 colourings constant) and
shows that the C28 obstruction-killing is not a fluke specific to
some smaller class; it works for the full chord-apex+Kempe layer.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Paper: Conjecture 5.5 restated with the hypothesis that every face
of H has length ≥ 5 (= the cubic plane analogue of "no triangles
or quadrilaterals as faces"). This kills K_4 and the n=8 trivial
counterexamples (girth 3) and the ad-hoc n=40 counterexample
(which has 2 triangles and 4 quadrilaterals). A new remark catalogues
these excluded counterexamples and the smallest cubic plane graphs
satisfying the hypothesis (dodecahedron at |V| = 20).
- search_smaller_counterexample.py: --min-face=N option to filter
cubic planar graphs by minimum face length.
- search_min_face5_counterexample.py: enumerates triangulations T
with min degree ≥ 5 via graphs.triangulations(n, minimum_degree=5),
takes planar dual (= cubic plane with all faces ≥ 5), and runs the
Heawood-constancy check.
- Result: smallest counterexample at triangulation order n_T = 16,
whose dual is a 28-vertex cubic plane graph (graph6
[kG[A?_A?_?_?K?D?@_CO?o?@_??A??@C??O??AG?C????`???a???W???A_???F).
Faces: 12 pentagons + 4 hexagons (a C28 fullerene). Both
K_{red, blue} and K_{red, green} are 12-cycles sharing the
colour-red edge (0, 1) and both have h_φ ≡ -1. 8 of 28 vertices
lie outside V(K_0) ∪ V(K_1).
- verify_28_vertex_counterexample.py: reproduces the counterexample,
verifies all properties, and renders figures/min-face-5-counterexample.png.
Note on the boundary: face-length ≥ 6 is impossible for cubic plane
graphs by Euler (6F = 6(V/2 + 2) > 3V = sum face lengths for V > 4).
So face-length ≥ 5 is the strongest face-length restriction admitting
any cubic plane graphs at all.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
experiments/search_smaller_counterexample.py enumerates 3-connected
cubic planar graphs via graphs.planar_graphs(n, min_deg=3, min_conn=2)
(filtering to cubic), then for each graph tries every proper
3-edge-colouring (backtracking with symmetry-break on first edge),
computes h_φ via the CW rotation from sage's planar embedding, and
checks whether some pair of intersecting Kempe cycles K_{a,b} and
K_{a,c} are both constant-Heawood.
Results (up to n=10 in initial run):
n= 4: K_4 itself. Coloring (1,2)=red, (3,4)=red, (1,3)=blue,
(2,4)=blue, (1,4)=green, (2,3)=green; sage's CW embedding
gives h_φ ≡ -1 on all 4 vertices. K_{red,blue} = 4-cycle
1-2-4-3 and K_{red,green} = 4-cycle 1-2-3-4 share both red
edges; both constant.
n= 6: no counterexample (only the triangular prism).
n= 8: a 12-edge cubic planar graph (graph6 G}GOW[) on 8 vertices.
Both Kempe cycles are 8-cycles visiting every vertex.
n=10: 8 cubic planar graphs checked, no counterexample.
So K_4 is the smallest counterexample to Conjecture 5.5 as stated,
but both K_4 and the n=8 example are structurally trivial: K_0 and
K_1 jointly cover V(H). The user's 40-vertex counterexample (paper
Figure) is the smallest non-trivial example found so far, with 24
vertices outside V(K_0) ∪ V(K_1).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Disproof remark now records the canonical graph6 string (via
G.canonical_label().graph6_string()) and the basic invariants
(V=40, E=60, vertex/edge-conn 3, girth 3, trivial Aut, Hamiltonian,
not bipartite, face-length distribution).
- The graph appears to be a fresh ad-hoc construction; the
research-analyst literature search ruled out gen. Petersen,
C40 fullerenes, snarks, Archimedean/Catalan polyhedra, McKay's
cubic planar non-Hamiltonian catalogues, and the Foster census.
- counterexample_conj_5_5.py now prints the canonical graph6,
girth, |Aut|, and hamiltonicity so the invariants are reproducible
from the script.
- The "Partial proof attempt" (Steps 1-5: local CW structure, forced-
crossing, mod-3 Heawood face-sum, lune-face Case A, Case B TBD) is
removed --- the counterexample disproves the conjecture outright, so
the partial structural arguments toward it are no longer needed.
Paper drops from 19 to 17 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Adds the concrete construction (40 vertices, 60 edges, cubic + planar
+ proper 3-edge-coloured) on which h_φ is simultaneously constant on
two Kempe cycles sharing an edge:
- K_{red, blue} = 8-cycle (the outer frame): all h_φ = -1
- K_{red, green} = 12-cycle (outer frame + upper-left ladder side):
all h_φ = -1
- They share the colour-red edge (0, 7) (and others).
The graph is drawn in TikZiT and stored as
papers/face_monochromatic_pairs/constant_heawood_counterexample.tikz
The Sage transcription + Heawood/Kempe verification + PNG renderer is
papers/face_monochromatic_pairs/experiments/counterexample_conj_5_5.py
Rendered PNG (with the four bent outer-face / trapezoid arcs matching
the tikz drawing) is at
papers/face_monochromatic_pairs/figures/no-two-constant-kempe-counterexample.png
Globally h_φ has 16 vertices at +1 and 24 at -1; the +1 vertices are
concentrated in the inner "tilted ladder" region, leaving the outer
and the K_{red,green}-extension all at -1. This is the structural
reason both Kempe cycles can be constant.
Also includes the TikZiT styles file default.tikzstyles defining the
red/blue/green edge styles used by the .tikz file.
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>
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