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Important illustration to the principle ``partition functions in stringtheory are $\tau$-functions of integrable equations'' is the fact that the(dual) partition functions of $4d$ $\mathcal{N}=2$ gauge theories solvePainlev\'e equations. In this paper we show a road to self-consistent proof ofthe recently suggested generalization of this correspondence: partitionfunctions of topological string on local Calabi-Yau manifolds solve$q$-difference equations of non-autonomous dynamics of the``cluster-algebraic'' integrable systems.  We explain in details the ``solutions'' side of the proposal. In the simplestnon-trivial example we show how $3d$ box-counting of topological stringpartition function appears from the counting of dimers on bipartite graph withthe discrete gauge field of ``flux'' $q$. This is a new form of topologicalstring/spectral theory type correspondence, since the partition function ofdimers can be computed as determinant of the linear $q$-difference Kasteleynoperator. Using WKB method in the ``melting'' $q\to 1$ limit we get a closedintegral formula for Seiberg-Witten prepotential of the corresponding $5d$gauge theory. The ``equations'' side of the correspondence remains theintriguing topic for the further studies.

Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux