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didericis 33b51b675b face_monochromatic_pairs: per-cycle refinement + Corollary 5.4 
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>
2026-05-25 00:43:35 -04:00

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\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 5.8\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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