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face_monochromatic_pairs: per-cycle refinement + Corollary 5.4

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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>
This commit is contained in:
didericis
2026-05-25 00:43:35 -04:00
parent a29d145cec
commit 33b51b675b
5 changed files with 295 additions and 44 deletions
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papers/face_monochromatic_pairs/paper.aux
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@@ -40,10 +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: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}}
\newlabel{cor:single-cycle-non-constancy}{{5.4}{13}}
\newlabel{rem:heawood-empirical}{{5.5}{13}}
\newlabel{rem:conj-3-6-empirical}{{5.6}{13}}
\newlabel{conj:face-monochromatic-pair-strengthened}{{5.7}{14}}
\newlabel{rem:conj-3-8-empirical}{{5.8}{14}}
\bibcite{Heawood1898}{1}
\bibcite{AH77a}{2}
\bibcite{AHK77}{3}
@@ -54,6 +55,6 @@
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\newlabel{rem:implication-4ct}{{5.8}{15}}
\newlabel{rem:implication-4ct}{{5.9}{15}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{15}{}\protected@file@percent }
\gdef \@abspage@last{15}