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