Generating functions for Higgs/Coulomb branch operators from 1d-3d cohomological equivalence

We provide a proof for the conjectured equality of the generating function of integrated Higgs and Coulomb branch topological operators in 3d $\mathcal{N}\ge 4$ gauge theories and the three sphere partition function deformed by mass or FI parameters. The equality is the result of cohomological equivalence and applies to all theories in this class, including ABJM and other generalized Gaiotto-Witten models, and those without an explicit supersymmetric Lagrangian.