Sixteenfold Classification of Many-Body Multipoles from Rotation-Compatible Canonical Symmetries

The conventional multipole framework provides a standard symmetry-based language for one-body electronic degrees of freedom, but fails to distinguish physically distinct sectors in many-body operator space. We show that canonical symmetries compatible with rotations provide this structure through two additional 2 labels associated with the particle-number gauge transformation Gπ/2 and the particle-hole transformation CA. These labels separate operators with different body numbers and particle-number changes, leading to a sixteenfold classification that organizes selection rules for nonzero expectation values, induced multipoles, and symmetry-allowed couplings in many-body multipole space.