face_monochromatic_pairs: record graph6 + invariants of Conj-5.5 counterexample; drop partial proof attempt
- Disproof remark now records the canonical graph6 string (via G.canonical_label().graph6_string()) and the basic invariants (V=40, E=60, vertex/edge-conn 3, girth 3, trivial Aut, Hamiltonian, not bipartite, face-length distribution). - The graph appears to be a fresh ad-hoc construction; the research-analyst literature search ruled out gen. Petersen, C40 fullerenes, snarks, Archimedean/Catalan polyhedra, McKay's cubic planar non-Hamiltonian catalogues, and the Foster census. - counterexample_conj_5_5.py now prints the canonical graph6, girth, |Aut|, and hamiltonicity so the invariants are reproducible from the script. - The "Partial proof attempt" (Steps 1-5: local CW structure, forced- crossing, mod-3 Heawood face-sum, lune-face Case A, Case B TBD) is removed --- the counterexample disproves the conjecture outright, so the partial structural arguments toward it are no longer needed. Paper drops from 19 to 17 pages. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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@@ -314,6 +314,9 @@ def main():
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G = build_graph(EDGES)
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col = edge_colour_map(EDGES)
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print(f" |V(H)| = {G.order()}, |E(H)| = {G.size()}")
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print(f" canonical graph6 = {G.canonical_label().graph6_string()}")
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print(f" girth = {G.girth()}, |Aut| = {G.automorphism_group().order()}, "
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f"hamiltonian = {G.is_hamiltonian()}")
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degs = sorted(set(G.degree()))
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print(f" degree set = {degs}")
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@@ -844,20 +844,23 @@ $h_\varphi \equiv -1$ on the vertex set of each. Globally $h_\varphi$
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takes value $+1$ on $16$ vertices and $-1$ on $24$ vertices, with all
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of the $+1$-vertices concentrated in the inner ``tilted ladder''
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region, so that both Kempe cycles miss them entirely.
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The construction and verification (cubic, planar, proper
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$3$-edge-colouring, Kempe-cycle tracing, Heawood-number computation
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via the CW rotation at each vertex) are in
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\texttt{experiments/counterexample\_conj\_5\_5.py}, with the source
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drawing in \texttt{constant\_heawood\_counterexample.tikz}.
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The partial proof attempt below establishes
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$|E(K_0) \cap E(K_1)| \geq 2$ (Step~2) and closes the sub-case where
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two shared a-edges are consecutive on \emph{both} cycles (``Case~A''
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/ Step~4). The general claim fails because in the counterexample no
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pair of shared a-edges is consecutive on both cycles --- the
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$K_1$-arc between two $K_0$-consecutive shared a-edges itself passes
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through other shared a-edges, so the lune-face assumption of Step~4
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does not hold.
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\smallskip
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The underlying graph appears to be a fresh ad-hoc construction: it has
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$60$ edges, vertex- and edge-connectivity $3$, girth $3$ (two
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triangles), is Hamiltonian, is not bipartite, has trivial
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automorphism group, and its $22$ faces have lengths distributed
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as $\{3{:}2,\,4{:}4,\,5{:}10,\,6{:}1,\,7{:}1,\,8{:}2,\,9{:}1,\,10{:}1\}$.
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Its canonical \texttt{graph6} string (via
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\texttt{G.canonical\_label().graph6\_string()}) is
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\begin{center}
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\small\ttfamily
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ggE\_?C?O\_B?B?@C?P???B??\_?@???\_??Sa???W????@a??\_B????G\_???@?????O??@\_???@?\_???K?????\_????K?????o???B?????AG?????B?????G?????AG?????F
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\end{center}
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The construction, the proper $3$-edge-colouring, Kempe-cycle tracing,
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and the Heawood-number computation (via the CW rotation at each
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vertex) are all in \texttt{experiments/counterexample\_conj\_5\_5.py};
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the source drawing is in \texttt{constant\_heawood\_counterexample.tikz}.
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\end{remark}
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\begin{figure}[h]
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@@ -873,185 +876,6 @@ $(0, 7)$.}
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\label{fig:no-two-constant-kempe-counterexample}
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\end{figure}
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\begin{proof}[Partial proof attempt (now superseded by
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Remark~\ref{rem:no-two-constant-kempe-cycles-counterexample})]
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\textbf{Note.}
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Conjecture~\ref{conj:no-two-constant-kempe-cycles} is false (see the
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counterexample in
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Remark~\ref{rem:no-two-constant-kempe-cycles-counterexample} /
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Figure~\ref{fig:no-two-constant-kempe-counterexample}). The argument
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below is preserved as partial progress: Steps~1--2 are unconditional
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and Step~4 closes the sub-case where some pair of shared a-edges is
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consecutive on both $K_0$ and $K_1$.
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Suppose for contradiction that $h_\varphi$ is constant on both
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$V(K_0)$ and $V(K_1)$, and that $K_0, K_1$ share a colour-$a$ edge
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$e = (u, w)$. Then $u, w \in V(K_0) \cap V(K_1)$ are consecutive on
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both cycles. Since the two constants agree at the shared vertex $u$,
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they agree everywhere on $V(K_0) \cup V(K_1)$; WLOG
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$h_\varphi \equiv +1$ on $V(K_0) \cup V(K_1)$, so the CW edge order at
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every such vertex is $(a, b, c)$.
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Orient both cycles so that they leave $u$ along $e$: write the $K_0$
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walk as $u = v_0 \xrightarrow{e} w = v_1 \to v_2 \to \cdots \to
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v_{L_0 - 1} \to v_0$, and the $K_1$ walk as $u = u_0 \xrightarrow{e}
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w = u_1 \to u_2 \to \cdots \to u_{L_1 - 1} \to u_0$. Let
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$F_R(e), F_L(e)$ be the two faces of $H$ incident to $e$, with $F_R$
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on the right and $F_L$ on the left of the $u \to w$ traversal.
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\textbf{Step 1 (local sides at $u, w$).} The CW-order $(a, b, c)$ at
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$u$ and $w$ partitions the faces of $H$ at each endpoint into three
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wedges. Direct inspection (cf.\ the proof of
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Lemma~\ref{lem:kempe-heawood-constant}) gives
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\[
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F_R(e) = F_{ab}^u = F_{ca}^w, \qquad F_L(e) = F_{ca}^u = F_{ab}^w,
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\]
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where $F_{\alpha\beta}^v$ is the face of $H$ at $v$ in the CW wedge
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between the colour-$\alpha$ and colour-$\beta$ edges. Consequently:
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\begin{itemize}
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\item $e_c^u$ lies between $F_{bc}^u$ and $F_{ca}^u = F_L(e)$, so $e_c^u$
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is on the side of $K_0$ containing $F_L(e)$ near $u$; call this side
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$\mathrm{In}(K_0)$.
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\item $e_c^w$ lies between $F_{bc}^w$ and $F_{ca}^w = F_R(e)$, so
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$e_c^w$ is on the side $\mathrm{Out}(K_0)$ containing $F_R(e)$ near $w$.
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\item Symmetrically, $e_b^u$ is on the side of $K_1$ containing
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$F_R(e)$, call this $\mathrm{Out}(K_1)$, and $e_b^w$ is on the side
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$\mathrm{In}(K_1)$ containing $F_L(e)$.
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\end{itemize}
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This recovers exactly the conclusion of
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Lemma~\ref{lem:kempe-heawood-constant} applied to $(u, w)$ on each
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cycle.
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\textbf{Step 2 (forced crossings).} Consider $K_1 \setminus e$, the
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path from $u$ to $w$ obtained by removing the open edge $e$ from
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$K_1$. By Step~1, this path leaves $u$ on the $\mathrm{In}(K_0)$ side
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and arrives at $w$ on the $\mathrm{Out}(K_0)$ side. Since $K_0$
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separates the plane into its two sides and $K_1 \setminus e$ is a
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continuous arc, the path must intersect $V(K_0)$ at an odd number of
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points strictly between $u$ and $w$ along the $K_1$-walk.
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At any such intersection $x \in V(K_0) \cap V(K_1) \setminus \{u, w\}$,
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$K_1$ uses the colour-$a$ edge at $x$ (since $K_1$ uses only colours
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$a, c$ and only colour-$a$ edges can lie on $K_0$). That colour-$a$
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edge is therefore a shared edge $e^* \in E(K_0) \cap E(K_1)$ with
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$e^* \neq e$; both endpoints of $e^*$ lie in $V(K_0) \cap V(K_1)$.
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So $|E(K_0) \cap E(K_1)| - 1$ equals the (odd) number of crossings,
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giving
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\[
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|E(K_0) \cap E(K_1)| \text{ is even, and } \geq 2.
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\]
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The symmetric argument applied to $K_0 \setminus e$ crossing $K_1$
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yields the same conclusion.
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\textbf{Step 3 (Heawood face-sum on each face of $K_0 \cup K_1$).}
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View $K_0 \cup K_1$ as a planar subgraph of $H$ and consider any face
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$\Phi$ of this subgraph. Let $F_\Phi$ be the set of $H$-faces lying
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inside (the closed region of) $\Phi$. Applying Heawood's classical
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face-sum identity
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$\sum_{v \in \partial f} h_\varphi(v) \equiv 0 \pmod 3$
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\cite{Heawood1898}
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to every $f \in F_\Phi$ and summing gives
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\[
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\sum_{f \in F_\Phi} \sum_{v \in \partial f} h_\varphi(v) \;=\;
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\sum_{v} \mathrm{mult}_\Phi(v)\, h_\varphi(v) \;\equiv\; 0 \pmod 3,
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\]
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where $\mathrm{mult}_\Phi(v)$ counts the number of $H$-faces in
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$F_\Phi$ whose boundary contains $v$.
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A direct case-check on the cubic vertex structure gives:
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\begin{itemize}
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\item $\mathrm{mult}_\Phi(v) = 3$ if $v$ is strictly interior to $\Phi$
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(all three $H$-faces at $v$ lie in $F_\Phi$);
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\item $\mathrm{mult}_\Phi(v) = 1$ if $v$ is a degree-$3$
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(shared, branching) vertex of $K_0 \cup K_1$ on $\partial\Phi$ (only
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one of $v$'s three wedges lies in $\Phi$);
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\item $\mathrm{mult}_\Phi(v) = 2$ if $v$ is a degree-$2$ (non-shared)
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vertex of $K_0 \cup K_1$ on $\partial\Phi$ \emph{and} $v$'s third edge
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points into $\Phi$ (the third edge subdivides $v$'s wedge in $\Phi$
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into two $H$-faces);
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\item $\mathrm{mult}_\Phi(v) = 1$ if $v$ is a degree-$2$ non-shared
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boundary vertex with its third edge pointing into the opposite face.
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\end{itemize}
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Under the contradiction hypothesis $h_\varphi \equiv +1$ on
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$V(K_0) \cup V(K_1) \supseteq \partial\Phi$, the boundary contribution
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collapses to
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\[
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\sigma_\Phi + \nu_{1,\Phi} + 2\,\nu_{2,\Phi}
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\;=\; \ell_\Phi + \nu_{2,\Phi}
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\;\equiv\; 0 \pmod 3,
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\]
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where $\sigma_\Phi$ counts shared boundary vertices of $\Phi$,
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$\nu_{1,\Phi}$ and $\nu_{2,\Phi}$ count non-shared boundary vertices
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with third edge pointing out of / into $\Phi$ respectively, and
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$\ell_\Phi$ is the boundary length of $\Phi$. (The interior
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contribution is a multiple of $3$ and drops out.) Hence
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\begin{equation}
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\label{eq:face-sum-mod3}
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\nu_{2,\Phi} \;\equiv\; -\ell_\Phi \pmod 3
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\quad\text{for every face } \Phi \text{ of } K_0 \cup K_1.
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\end{equation}
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\textbf{Step 4 (lune-face contradiction, ``Case A'' sub-case).}
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Suppose there exist two shared a-edges $e_1 = (p, p')$ and
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$e_2 = (q, q')$ which are consecutive on \emph{both} the $K_0$-walk
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\emph{and} the $K_1$-walk (so the $K_0$-arc from $p'$ to $q$ and the
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$K_1$-arc from $p'$ to $q$ both have all-non-shared interior). In
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particular, when $|E(K_0) \cap E(K_1)| = 2$ this is automatic, but in
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general it is an extra hypothesis. Let $A_1$ be the $K_0$-arc from
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$p'$ to $q$ of length $m$. The arc begins with the colour-$b$ edge at
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$p'$ and ends with the colour-$b$ edge at $q$, so $m$ is odd. Two
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cases for the cyclic order in which $K_1$ visits $\{p, p', q, q'\}$:
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\begin{description}
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\item[Case A.] $K_1$ visits them in the same cyclic order
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$p, p', q, q'$ as $K_0$. Then $K_1$ has a $K_1$-arc $B_1$ from $p'$
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to $q$ (between $e_1$ and $e_2$ on the $K_1$-walk), of odd length
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$n$, whose intermediate vertices are all non-shared.
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\item[Case B.] $K_1$ visits them in the opposite order
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$p, p', q', q$. Then $K_1$'s arcs between $e_1, e_2$ go from $p'$ to
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$q'$ and from $q$ to $p$; they do not share endpoints with $A_1$.
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\end{description}
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\emph{Case A is impossible.} The arcs $A_1$ and $B_1$ share both
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endpoints $p', q$ and meet only at those endpoints, so they bound a
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``lune'' face $\Phi^*$ of $K_0 \cup K_1$ whose boundary has exactly
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two corners (both of wedge type $(b, c)$) and length $m + n$. Now
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\begin{itemize}
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\item Every intermediate vertex of $B_1$ is a non-shared $K_1$-vertex,
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hence not on $K_0$, hence lies in one of the two open regions of
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$\mathbb{R}^2 \setminus K_0$. Since $B_1$ is a connected path joining
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$p', q \in V(K_0)$ and never visits $V(K_0)$ in its interior,
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$B_1 \setminus \{p', q\}$ lies entirely on \emph{one} side of $K_0$.
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In particular the colour-$c$ edges at $p'$ and at $q$ (the first
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and last edges of $B_1$) point into the \emph{same} side of $K_0$.
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\item By Lemma~\ref{lem:kempe-heawood-constant} applied along the
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$K_0$-arc from $p'$ to $q$, consecutive colour-$c$ non-cycle edges
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alternate sides. After $m$ steps with $m$ odd, the colour-$c$ edges
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at $p'$ and at $q$ lie on \emph{opposite} sides of $K_0$.
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\end{itemize}
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These two conclusions are incompatible, contradicting the assumption
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that both $K_0$ and $K_1$ have constant $h_\varphi$ in Case A.
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\textbf{Step 5 (Case B --- TBD).} In Case B, the four faces of
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$K_0 \cup K_1$ are each three-corner ``triangles'' bounded by one
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$K_0$-arc, one $K_1$-arc, and one shared a-edge (one corner each of
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types $(a,b)$, $(b,c)$, $(c,a)$). For such a face the
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Step~4 lune argument does not apply: $A_1$ and the corresponding
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$K_1$-arc $B_1$ no longer share both endpoints, and both
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Lemma~\ref{lem:kempe-heawood-constant} alternations and the mod-$3$
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constraint~\eqref{eq:face-sum-mod3} can be checked to be
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consistent on the parity counts. The contradiction in this case is
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\emph{open}.
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\emph{Empirical note.} The theorem's hypothesis is never observed:
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across the $142{,}812$ chord-apex+Kempe colourings of reduced duals
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with $|V(G)| \le 20$, ``$h_\varphi$ constant on $V(K_b)$'' fails on
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every colouring (see \texttt{experiments/check\_constancy\_obstruction.py}
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and Remark~\ref{rem:heawood-empirical}).
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\end{proof}
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\begin{remark}[Empirical near-proof of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} via Corollary~\ref{cor:single-cycle-non-constancy}]
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\label{rem:heawood-empirical}
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\sloppy
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