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face_monochromatic_pairs: reframe Lemma 5.2 as a non-existence result

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The previous statement "Heawood is constant on K through merged" was
strictly stronger than what the proof actually established without
Conjecture 5.3. Restate the lemma in the contrapositive direction:

  If h_phi is constant on V(K), then no edge e in E(K) admits a face
  F of G'^hat and edges e_1, e_2 on dF realising the clause-(3) arc
  of Conjecture 5.1 at the endpoints of e.

Proof structure is mostly preserved (same F_R/F_L geometry, same case
split on phi(e) in {a, b}, same reading-off of cyclic colour orders).
The hypothesis "h_phi(v_0) != h_phi(v_1)" becomes "h_phi(v_0) =
h_phi(v_1)", which flips the conclusion: the same-coloured non-e
edges at v_0, v_1 land on opposite faces of e instead of the same
face. No dependency on Conjecture 5.3 or Theorem 4.X.

Redraw the figure to match the new lemma: both vertices labelled
h_phi = +1, both showing CW order (a, b, c), and the same-colour pair
(b-edges in Case A, a-edges in Case B) drawn on opposite sides of e.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-24 22:31:10 -04:00
parent d99f8e23b3
commit 037d987c7d
6 changed files with 148 additions and 159 deletions
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papers/face_monochromatic_pairs/paper.aux
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@@ -38,9 +38,9 @@
\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}}
\@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 $h_\varphi (v_0) = +1$ (clockwise colour order $(a, b, c)$) and $h_\varphi (v_1) = -1$ (clockwise order $(a, c, b)$). \emph {Left (Case\nonbreakingspace A):} when $\varphi (e) = a$, the two $b$-edges at $v_0, v_1$ lie on the same face $F$, with $e$ as the unique $\partial F$-edge between them. \emph {Right (Case\nonbreakingspace B):} when $\varphi (e) = b$, the two $a$-edges at $v_0, v_1$ lie on the opposite face $F$ instead, again with $e$ between them on one arc. In either case $(F, e_1, e_2)$ witnesses clauses (1)--(3) of Conjecture\nonbreakingspace 5.1\hbox {}.}}{12}{}\protected@file@percent }
\@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}{13}}
\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}}
\bibcite{Heawood1898}{1}