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>
Adds Definition (flank face) + 3 lemmas + a partial theorem proving
Conjecture (Deciding face) -- hence Conjecture 5.1 -- for the case
where at least one neighbour of v in the parent triangulation G has
degree ≤ 6:
- Lemma (Flank-length formula): |F_{i, i+1}^♭| = n_i - 1.
- Lemma (Flank covering, n_i = 5): boundary of F_{i, i+1}^♭ is in
V(K_b) ∪ V(K_c). Proof: the single intermediate P is adjacent to
both A_i and A_{i+1}; A_{i+1} ∈ V(K_b) ∩ V(K_c) via spike, so
A_{i+1}'s c_0-edge (in K_b) or c_1-edge (in K_c) lands on P.
- Lemma (Flank covering, n_i = 6): two intermediates P_1, P_2; P_2
handled as the n_i = 5 case; P_1 covered by case analysis on
φ(A_i P_1) ∈ {c, c_1}: in Case (a) K_b walks A_i → P_1 directly;
in Case (b) propagation from P_2 via P_2's c-edge to P_1 (forced
by properness at P_1 ruling out φ(P_1 P_2) = c_1).
- Theorem (Partial proof of Conjecture (Deciding face)): combining
the length formula and the covering lemmas, F_{i, i+1}^♭ is a
deciding face whenever n_i ∈ {5, 6}, since its length is then 4
or 5 (≢ 0 mod 3).
The remaining structural case is n_i ≥ 7 for all i (= all five
neighbours of v in G have degree ≥ 7); for these the merged-side
face F^♭_{i+3, i+4} of length n_{i+3} - 2 often plays the deciding
role empirically, but a uniform structural argument is left open.
Paper grows from 18 to 20 pages.
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>
- Conjecture 5.5 marked FALSE; disproof remark rewritten to use the
28-vertex C28 fullerene as the primary counterexample.
- Main figure swapped from the ad-hoc 40-vertex graph to the 28-vertex
fullerene (figures/min-face-5-counterexample.png). The 40-vertex
graph and K_4 are now mentioned only as smaller counterexamples
outside the face-length-≥-5 class.
- Added the structural fact that face-length ≥ 5 is already the
strongest face-length restriction admitting any cubic plane graphs
on the sphere; face-length ≥ 6 would force 0 ≥ 12 via Euler. So no
further face-length strengthening can save the conjecture.
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>
- 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>
The disproof of Conjecture 5.5 used \[ ... \qquad\text{and}\qquad ... \]
to give both cycle definitions on a single display line. The second
half was too long to fit, producing a wide gap and an ugly line break.
Switch to align* with the two definitions stacked and aligned at =.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Replaces the placeholder fbox figure with the rendered PNG from
experiments/counterexample_conj_5_5.py (40-vertex cubic plane graph,
proper 3-edge-coloured, both K_{red,blue} 8-cycle and K_{red,green}
12-cycle constant h_φ = -1, sharing the colour-red edge (0, 7)).
Disproof remark also updated to give the actual structural details
(40 vertices, the +1/-1 split 16/24, location of the +1-region in
the inner ladder) instead of a placeholder description.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
The user produced a concrete counterexample (whiteboard photo) showing
that h_φ can be constant on both an {a,b}-Kempe cycle K_0 and an
{a,c}-Kempe cycle K_1 sharing a colour-a edge.
Changes:
- theorem → conjecture environment, header marked **FALSE**
- New Remark records the disproof and identifies which step of the
proof attempt breaks: in the counterexample, no pair of shared
a-edges is consecutive on both cycles, so the lune-face premise
(Step 4 / Case A) doesn't apply
- Proof attempt re-tagged as "Partial proof attempt (now superseded)";
Steps 1-2 remain unconditional, Step 4 closes the sub-case where
some shared-a-edge pair is consecutive on both K_0 and K_1 (e.g.
automatically when |E(K_0) ∩ E(K_1)| = 2)
- Figure placeholder added referencing
figures/no-two-constant-kempe-counterexample.{png,pdf}
- COMMENTARY.md updated with a "Failed proof route" section so future
readers don't retread this path
Impact on Conjecture 5.1: the "Theorem 5.5 + Lemma 5.3 → 5.1" route
is closed; a structural proof of Conjecture 5.1 needs a different
angle. Lemma 5.3, Corollary 5.4, and the 142,812/142,812 empirical
near-proof all stand.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Case A: K_1 visits the four shared vertices {p, p', q, q'} in the same
cyclic order as K_0. The K_0-arc A_1 from p' to q and the K_1-arc B_1
from p' to q share both endpoints, so they bound a "lune" face Φ* of
K_0 ∪ K_1 with two (b, c)-corners.
- Planarity: B_1's interior is non-shared, so B_1 lies on one side of
K_0; hence c-edges at p' and q (endpoints of B_1) point to the
SAME side of K_0.
- Lemma 5.2 along A_1 (m odd): c-edges at p' and q alternate, so
they point to OPPOSITE sides of K_0.
→ Contradiction.
This is clean and uses only Lemma 5.2 + planarity (does not require
the mod 3 face-sum from Step 3).
Case B: K_1 visits the shared vertices in opposite order
{p, p', q', q}. Then K_0 ∪ K_1's faces are 3-corner triangles instead
of lunes; both Lemma 5.2 alternations and the mod 3 face-sum hold
consistently. Listed as the explicit open case.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- 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>
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>
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>
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