didericis 037d987c7d face_monochromatic_pairs: reframe Lemma 5.2 as a non-existence result ...

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>

2026-05-24 22:31:10 -04:00

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