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face_monochromatic_pairs: add Lemma 5.3 (constancy on both Kempe cycles)

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Follow-up to Lemma 5.2. States that if Conjecture 5.1 has no
clauses-(1)-(3) witness for (G, G'^_{v,i}, phi), then h_phi is
constant on both Kempe cycles through merged, and the two constants
agree (since merged is on both cycles, so its endpoints force the
constants to match).

Proof is the V1-direction of the case analysis: differing h_phi on
either K_b or K_c reproduces a clause-(1)-(3) witness by the same
F_R/F_L geometry as Lemma 5.2's proof but with the hypothesis
"h_phi(v_0) != h_phi(v_1)", under which the matching-colour edges
land on the SAME face of e. Case B's merged-incidence corner is
handled by choosing a differing-Heawood pair away from merged's
endpoints.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-24 22:45:26 -04:00
parent 037d987c7d
commit cfc53dcbe2
4 changed files with 98 additions and 44 deletions
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papers/face_monochromatic_pairs/paper.aux
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@@ -37,13 +37,14 @@
\newlabel{sec:toward-4ct}{{5}{10}}
\newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{5.1}{10}}
\newlabel{lem:kempe-heawood-constant}{{5.2}{11}}
\newlabel{rem:conj-3-6-empirical}{{5.3}{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{conj:face-monochromatic-pair-strengthened}{{5.4}{12}}
\newlabel{rem:conj-3-8-empirical}{{5.5}{13}}
\newlabel{rem:implication-4ct}{{5.6}{13}}
\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}}
\bibcite{Heawood1898}{1}
\newlabel{rem:implication-4ct}{{5.7}{14}}
\bibcite{AH77a}{2}
\bibcite{AHK77}{3}
\bibcite{RSST97}{4}
@@ -53,5 +54,5 @@
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{14}{}\protected@file@percent }
\gdef \@abspage@last{14}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{15}{}\protected@file@percent }
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