Gauge origami and quiver W-algebras IV: Pandharipande--Thomas $qq$-characters

We develop a contour integral formalism for computing the K-theoretic equivariant 3-vertex. Within the Jeffrey–Kirwan (JK) residue framework, we show that, by an appropriate choice of the reference vector, both the equivariant Donaldson–Thomas (DT) and Pandharipande–Thomas (PT) 3-vertices can be extracted from the same integrand. We analyze three distinct limits of the PT 3-vertex, recovering the unrefined topological vertex, the refined topological vertex, and the Macdonald refined topological vertex. Higher-rank extensions of PT counting and the DT/PT correspondence are also explored. From a quantum algebraic perspective, we construct an operator version of the equivariant PT 3-vertex and term it the Pandharipande–Thomas $qq$-character. We then discuss its connection with the quantum toroidal ${\mathfrak{g}\mathfrak{l}}_1$.