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dual_decomposition: move strengthened conjecture into Section 4 + 4CT implication

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- Cut Conjecture 3.8 + Remark 3.9 from Section 3 and move into a new
  Section 4 "The Four Colour Theorem from a strengthened conjecture".
- Add Remark 4.X spelling out the implication: clause (4)(i) forces the
  cyclic colour pattern (c,a,c,b) on the new 4-face f_n, two opposite
  edges of which satisfy the hypothesis of Theorem 3.9 verbatim; case
  (ii) is conjecturally reducible to case (i) via a Kempe swap on the
  {b,c}-cycle through X_1 X_2. Theorem 3.9 then produces the proper
  3-edge-colouring of the contraction, contradicting minimality of G.
- Rewrite the bridge prose into the cubic-contraction definition to
  reference Section 4 forward, rather than the conjecture directly.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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didericis
2026-05-24 13:43:09 -04:00
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papers/dual_decomposition_minimal_counterexamples/paper.log
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\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 4.2\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 T
he strength-ened con-jec-ture was tested on the same chord-
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