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face_monochromatic_pairs: empirical near-proof of Conjecture 5.1 via Lemma 5.3

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Add Remark 5.5 immediately after Lemma 5.3's proof, recording the
empirical reduction of Conjecture 5.1 via the contrapositive of
Lemma 5.3: the conjecture follows from "h_phi is not constant on
V(K_b) U V(K_c)", and we have verified that non-constancy holds on
every one of 142,812 chord-apex+Kempe colourings up to n <= 20
(including the six Holton-McKay duals as a special case).

This is an independent empirical near-proof of Conjecture 5.1,
complementary to the direct (1)-(3) witness check in
Remark 5.6 / rem:conj-3-6-empirical. A structural proof of the
non-constancy claim would upgrade this to a proof of the
conjecture.

Also include two diagnostic scripts that informed the remark:
  - check_shared_parity.py: parity-bucket symmetry n_{0,0} = n_{1,1},
    n_{0,1} = n_{1,0} at vertices in V(K_b) cap V(K_c). 100%.
  - check_cw_parity_prediction.py: structural identity
    s_b XOR s_c = i_b XOR i_c XOR 1 holds at every shared vertex
    (263,004 / 263,004), and the simple constancy prediction matches
    exactly 50% of shared vertices per colouring with 0 perfectly
    matching colourings.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-25 00:21:52 -04:00
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commit 703b523161
6 changed files with 476 additions and 40 deletions
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papers/face_monochromatic_pairs/paper.log
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