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. We comment on the qualitative difference between 4- and 2-dimensional models.