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We study generalized gauge theories engineered by taking the low energy limitof the $Dp$ branes wrapping $X \times T^{p-3}$, with $X$ a possibly singularsurface in a Calabi-Yau fourfold $Z$. For toric $Z$ and $X$ the partitionfunction can be computed by localization, making it a statistical mechanicalmodel, called the gauge origami. The random variables are the ensembles ofYoung diagrams. The building block of the gauge origami is associated with atetrahedron, whose edges are colored by vector spaces. We show the properlynormalized partition function is an entire function of the Coulomb moduli, forgeneric values of the $\Omega$-background parameters. The orbifold version ofthe theory defines the $qq$-character operators, with and without the surfacedefects. The analytic properties are the consequence of a relative compactnessof the moduli spaces $M({\vec n}, k)$ of crossed and spiked instantons,demonstrated in arXiv:1608.07272.

BPS/CFT Correspondence III: Gauge Origami partition function and qq-characters