Sorting topological stabilizer models in three dimensions

The S -matrix invariant is known to be complete for translation invariant topological stabilizer models in two spatial dimensions, as such models are phase equivalent to some number of copies of toric code. In three dimensions, much less is understood about translation invariant topological stabilizer models due to the existence of fracton topological order. Here we introduce bulk commutation quantities inspired by the 2D S -matrix invariant that can be employed to coarsely sort 3D topological stabilizer models into qualitatively distinct types of phases: topological quantum field theories, foliated or fractal type-I models with rigid string operators, or type-II models with no string operators.