arXiv2026
Using the Weil--Gel'fand--Zak transform of Faddeev's quantum dilogarithm, we construct a compact state integral for positively shaped triangulations of oriented pseudo three-manifolds. The integral is labelled by a circle-valued first cohomology class of the vertex complement (a ``sector'') and takes boundary values in a line bundle over compatible boundary cocycles. We prove absolute convergence, smoothness, invariance under shape gauge transformations and under compatible shaped $2$--$3$ moves, branching independence, and a single-valued global meromorphic continuation in all shape and boundary parameters for each fixed quantum parameter. Gluing is expressed by a conditional Haar integral which retains the final cohomology class; the ordinary trace of integrated cut amplitudes computes an average of the sectors over a torus of classes determined by the cut, and is a sewing formula for every cut of a closed oriented three-manifold, but not for arbitrary pseudo-manifolds. For closed admissible triangulations, the original Teichmüller partition function is the normalized average of the sectors; this yields its global meromorphic continuation. A translation law relates sectors differing by twice an extending class, with phases given by linking pairings; for closed three-manifolds the sectors are supported in a distinguished coset of twice the torsion group and are determined by a single amplitude, which a nonvanishing twisted quadratic Gauss sum recovers from the original partition function.