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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
parent d659bf40d5
commit 703b523161
6 changed files with 476 additions and 40 deletions
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papers/face_monochromatic_pairs/paper.aux
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@@ -40,11 +40,11 @@
\newlabel{lem:both-kempe-constant}{{5.3}{11}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The two cases in the proof of Lemma\nonbreakingspace 5.2\hbox {}. Vertices $v_0, v_1$ are consecutive on the $\{a, b\}$-Kempe cycle $K$, joined by an edge $e$, with the lemma's hypothesis $h_\varphi (v_0) = h_\varphi (v_1) = +1$ --- so both vertices share the clockwise colour order $(a, b, c)$. \emph {Left (Case\nonbreakingspace A):} when $\varphi (e) = a$, the colour-$b$ edge at $v_0$ lies south of $e$ (on $\partial F_R$) and the colour-$b$ edge at $v_1$ lies north of $e$ (on $\partial F_L$); the two would-be witness edges are on opposite faces, so no face of $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,i}$ contains both. \emph {Right (Case\nonbreakingspace B):} when $\varphi (e) = b$, the colour-$a$ edges at $v_0, v_1$ are likewise on opposite sides of $e$. In either case the clause-$(3)$ arc of Conjecture\nonbreakingspace 5.1\hbox {} cannot be realised at $e$.}}{12}{}\protected@file@percent }
\newlabel{fig:lemma-kempe-heawood}{{5}{12}}
\newlabel{rem:conj-3-6-empirical}{{5.4}{13}}
\newlabel{conj:face-monochromatic-pair-strengthened}{{5.5}{13}}
\newlabel{rem:conj-3-8-empirical}{{5.6}{13}}
\newlabel{rem:heawood-empirical}{{5.4}{13}}
\newlabel{rem:conj-3-6-empirical}{{5.5}{13}}
\newlabel{conj:face-monochromatic-pair-strengthened}{{5.6}{14}}
\newlabel{rem:conj-3-8-empirical}{{5.7}{14}}
\bibcite{Heawood1898}{1}
\newlabel{rem:implication-4ct}{{5.7}{14}}
\bibcite{AH77a}{2}
\bibcite{AHK77}{3}
\bibcite{RSST97}{4}
@@ -54,5 +54,6 @@
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\newlabel{rem:implication-4ct}{{5.8}{15}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{15}{}\protected@file@percent }
\gdef \@abspage@last{15}