We study generalized gauge theories engineered by taking the low energy limit of the Dp branes wrapping X× Tp - 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 gaugeorigami̲. 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 Ω -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(n→ , k) of crossed and spiked instantons, demonstrated in “BPS/CFT correspondence II: instantons at crossroads, moduli and compactness theorem”.