Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the general $q$-Painlev\'e III$_3$ tau functions

We reformulate the $q$-difference linear system corresponding to the $q$-Painlev\'e equation of type $A_7^{(1)'}$ as a Riemann-Hilbert problem on a circle. Then, we consider the Fredholm determinant built from the jump of this Riemann-Hilbert problem and prove that it satisfies bilinear relations equivalent to $P(A_7^{(1)'})$. We also find the minor expansion of this Fredholm determinant in explicit factorized form and prove that it coincides with the Fourier series in $q$-deformed conformal blocks, or partition functions of the pure $5d$ $\mathcal{N}=1$ $SU(2)$ gauge theory, including the cases with the Chern-Simons term. Finally, we solve the connection problem for these isomonodromic tau functions, finding in this way their global behavior.