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face_monochromatic_pairs: reframe Lemma 5.2 as a non-existence result

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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>
This commit is contained in:
didericis
2026-05-24 22:31:10 -04:00
parent d99f8e23b3
commit 037d987c7d
6 changed files with 148 additions and 159 deletions
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papers/face_monochromatic_pairs/paper.log
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\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 5.5\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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\OT1/cmr/m/n/10 apex+Kempe colour-ings as Re-mark 5.3[]; for each colour-ing we
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Output written on paper.pdf (14 pages, 1081744 bytes).
Output written on paper.pdf (14 pages, 1077738 bytes).
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