The symmetric mass generation (SMG) approach to the construction of lattice chiral gauge theories
attempts to use interactions to render mirror fermions massive without symmetry breaking, to
obtain the desired chiral massless spectrum (before the gauge field is turned on). If the zeros that
often replace the mirror poles of fermion two-point functions in an SMG phase are “kinematical”
singularities, general constraints can be formulated on the existence of a chiral fermion spectrum
which are valid in the presence of (non-gauge) interactions of arbitrary strength, including in any
SMG phase. Constructing a one-particle lattice hamiltonian describing the fermion spectrum, we
discuss the conditions for the applicability of the Nielsen-Ninomiya theorem to this hamiltonian.
If these conditions are satisfied, the massless fermion spectrum must be vector-like.