diff --git a/papers/face_monochromatic_pairs/paper.pdf b/papers/face_monochromatic_pairs/paper.pdf
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--- a/papers/face_monochromatic_pairs/paper.tex
+++ b/papers/face_monochromatic_pairs/paper.tex
@@ -828,37 +828,49 @@ $V(K_0)$, then $h_\varphi$ is \emph{not} constant on $V(K_1)$.
 \begin{remark}[Disproof of
 Conjecture~\ref{conj:no-two-constant-kempe-cycles}]
 \label{rem:no-two-constant-kempe-cycles-counterexample}
-Conjecture~\ref{conj:no-two-constant-kempe-cycles} is \emph{false}:
-there exists a cubic plane graph $H$ with a proper $3$-edge-colouring
-$\varphi$ admitting an $\{a, b\}$-Kempe cycle $K_0$ and an
-$\{a, c\}$-Kempe cycle $K_1$ which share a colour-$a$ edge and on which
-$h_\varphi$ is simultaneously constant on $V(K_0)$ and on $V(K_1)$. A
-concrete counterexample is recorded in
-Figure~\ref{fig:no-two-constant-kempe-counterexample}.
+Conjecture~\ref{conj:no-two-constant-kempe-cycles} is \emph{false}.
+Figure~\ref{fig:no-two-constant-kempe-counterexample} exhibits a
+concrete counterexample: a cubic plane graph $H$ on $40$ vertices with
+a proper $3$-edge-colouring $\varphi$ (colours red/blue/green) in
+which both
+\[
+K_{\mathrm{red},\mathrm{blue}}
+\;=\; \text{the outer $8$-cycle}
+\qquad\text{and}\qquad
+K_{\mathrm{red},\mathrm{green}}
+\;=\; \text{the $12$-cycle (outer + upper-left ``ladder'' side)}
+\]
+share the colour-red edge $(0, 7)$ and satisfy
+$h_\varphi \equiv -1$ on the vertex set of each. Globally $h_\varphi$
+takes value $+1$ on $16$ vertices and $-1$ on $24$ vertices, with all
+of the $+1$-vertices concentrated in the inner ``tilted ladder''
+region, so that both Kempe cycles miss them entirely.
+The construction and verification (cubic, planar, proper
+$3$-edge-colouring, Kempe-cycle tracing, Heawood-number computation
+via the CW rotation at each vertex) are in
+\texttt{experiments/counterexample\_conj\_5\_5.py}, with the source
+drawing in \texttt{constant\_heawood\_counterexample.tikz}.
 
-The (partial) proof attempt below establishes the constraint
+The partial proof attempt below establishes
 $|E(K_0) \cap E(K_1)| \geq 2$ (Step~2) and closes the sub-case where
-two shared a-edges are consecutive on \emph{both} cycles
-(``Case~A''/Step~4), but the general claim fails because in the
-counterexample no pair of shared a-edges is consecutive on both
-cycles --- the $K_1$-arc between two $K_0$-consecutive shared a-edges
-itself passes through other shared a-edges, breaking the lune-face
-assumption.
+two shared a-edges are consecutive on \emph{both} cycles (``Case~A''
+/ Step~4). The general claim fails because in the counterexample no
+pair of shared a-edges is consecutive on both cycles --- the
+$K_1$-arc between two $K_0$-consecutive shared a-edges itself passes
+through other shared a-edges, so the lune-face assumption of Step~4
+does not hold.
 \end{remark}
 
 \begin{figure}[h]
 \centering
-% TODO: replace placeholder with the actual counterexample drawing
-% (e.g., the whiteboard photo or a TikZ rendering).
-\fbox{\parbox{0.7\textwidth}{\centering
-\emph{Placeholder for the counterexample figure.}\\[2pt]
-A cubic plane graph $H$ with three edge colours
-(red, blue, teal) showing two Kempe cycles sharing a colour edge,
-on which $h_\varphi$ is simultaneously constant on both cycles.\\
-See \texttt{figures/no-two-constant-kempe-counterexample.\{png,pdf\}}
-once the image is committed to the repo.}}
+\includegraphics[width=0.85\textwidth]{figures/no-two-constant-kempe-counterexample.png}
 \caption{Counterexample to
-Conjecture~\ref{conj:no-two-constant-kempe-cycles}.}
+Conjecture~\ref{conj:no-two-constant-kempe-cycles}: a cubic plane
+graph on $40$ vertices with a proper $3$-edge-colouring on which
+$h_\varphi$ is simultaneously constant ($\equiv -1$) on the outer
+red/blue $8$-cycle and on the red/green $12$-cycle (outer frame plus
+the upper-left ladder side), which share the colour-red edge
+$(0, 7)$.}
 \label{fig:no-two-constant-kempe-counterexample}
 \end{figure}