Dimers, filters, and $q$-deformed real numbers

This article associates to each positive real number $x$ an inhomogeneous dimer model with an activity parameter $q>0$ and uses it to define a positive-real-valued function $q\mapsto [[x]{]}_q$ that is related to (and indeed was inspired by) the algebraic $q$-deformation $[x{]}_q$ introduced by Morier-Genoud and Ovsienko~\cite{MO22}. The technical details are most transparent when the dimer model is replaced by an equivalent model involving filters in partially ordered sets. When $x$ is rational we have $[[x]{]}_q=q[x{]}_q$, and it seems likely that more generally the two sides of the equation agree wherever both are defined.