diff --git a/papers/face_monochromatic_pairs/paper.pdf b/papers/face_monochromatic_pairs/paper.pdf
index 37bb4de..7069c7c 100644
Binary files a/papers/face_monochromatic_pairs/paper.pdf and b/papers/face_monochromatic_pairs/paper.pdf differ
diff --git a/papers/face_monochromatic_pairs/paper.tex b/papers/face_monochromatic_pairs/paper.tex
index ff7ed24..b6f36b2 100644
--- a/papers/face_monochromatic_pairs/paper.tex
+++ b/papers/face_monochromatic_pairs/paper.tex
@@ -833,13 +833,12 @@ 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
-\[
+\begin{align*}
 K_{\mathrm{red},\mathrm{blue}}
-\;=\; \text{the outer $8$-cycle}
-\qquad\text{and}\qquad
+  &= \text{the outer $8$-cycle, and} \\
 K_{\mathrm{red},\mathrm{green}}
-\;=\; \text{the $12$-cycle (outer + upper-left ``ladder'' side)}
-\]
+  &= \text{the $12$-cycle (outer frame $+$ upper-left ``ladder'' side)}
+\end{align*}
 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