- 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- Proof attempt re-tagged as "Partial proof attempt (now superseded)";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