  Recently an exact duality between topological string and the spectral theory
of operators constructed from mirror curves to toric Calabi-Yau threefolds has
been proposed. At the same time an exact quantization condition for the cluster
integrable systems associated to these geometries has been conjectured. The
consistency between the two approaches leads to an infinite set of constraints
for the refined BPS invariants of the toric Calabi-Yau threefolds. We prove
these constraints for the $Y^{N,m}$ geometries using the $K$-theoretic blowup
equations for $SU(N)$ SYM with generic Chern-Simons invariant $m$.
