  We study a category $\mathcal{C}_2$ of $\mathbb{Z}$-graded MCM modules over
the $A_\infty$ curve singularity and demonstrate it has infinite type $A$
cluster combinatorics. In particular, we show that this Frobenius category (or
a suitable subcategory) is stably equivalent to the infinite type $A$ cluster
categories of Holm-Jorgensen, Fisher and Paquette-Yildirim. As a consequence,
$\mathcal{C}_2$ has cluster tilting subcategories modelled by certain
triangulations of the (completed) $\infty$-gon. We use the Frobenius structure
to extend this further to consider maximal almost rigid subcategories, and show
that these subcategories and their mutations exhibit the combinatorics of the
completed $\infty$-gon.
