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didericis 464c524fa1 dual_decomposition: Conj 3.6 (face/Kempe witness) and constructive lift 
Paper:
- Lemmas 3.4 (exactly one match) and 3.5 (all-distinct exists for 4-colourable
  G) replace the earlier conjecture; both have proofs.
- Add Conjecture 3.6: every proper 3-edge-colouring of a counterexample's
  reduced dual has a face with two same-colour edges that share a Kempe
  cycle with the merged edge, neither of them being the merged edge.

Experiments (all under experiments/):
- search_conj_3_6_counterexample.py: finds n=14 tri#1 i_red=0 where the
  algorithm's phi_t* sits in a Kempe class with no all-distinct colouring
  (disproves an earlier formulation).
- check_kempe_class.py / check_kempe_class_invariance.py /
  check_kempe_class_monotone.py: Kempe-class counts on H_1 and H_t* for
  small triangulations; neither monotonicity direction holds.
- check_all_distinct_exists.py: even in the conj-3.6 disproof case, H_t*
  itself admits all-distinct colourings in the *other* Kempe class.
- check_constrained_feasibility.py: literal H_t*-interpretation of
  C1 + K0 + K1 is empirically unsatisfiable (gap in proof strategy noted).
- check_conj_face_kempe.py / check_conj_face_kempe_n15.py: test Conj 3.6
  on chord-apex+Kempe colourings of reduced duals at n=12, 14, 15;
  216/216 colourings on n=14 satisfy the conjecture, others vacuous.
- draw_step1_conj36.py: figure showing a Conj 3.6 witness on H_1 with two
  new vertices on the witness edges and a new red bridge between them.
- draw_step1_conj36_recolored.py: same but with the Kempe cycle recoloured
  alternately from merged so propriety holds.
- draw_lift_to_Gprime.py: lifts the modified+recoloured H_1 back to a
  proper 3-edge-colouring of the modified G' (24+2 vertices, 39 edges,
  same Tutte layout as figure 3's first graphic so positions line up).

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
2026-05-24 11:27:50 -04:00

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\OT1/cmr/m/sc/10 Figure 3.\OT1/cmr/m/n/10 Algorithm 3.1[] on $\OML/cmm/m/it/10 G[] \OT1/cmr/m/n/10 = [](\OML/cmm/m/it/10 G\OT1/cmr/m/n/10 )$, where $\OML/cmm/m/it/10 G$
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