Impossibility of spontaneous vector flavor symmetry breaking on the lattice

I show that spontaneous breaking of vector flavor symmetry on the lattice is impossible in gauge theories with a positive functional-integral measure, for discretized Dirac operators linear quark masses, if corresponding propagator and its commutator generators can be bounded norm independently configuration uniformly volume. Under these assumptions, any order parameter vanishes symmetric limit fermions equal masses. assumptions are satisfied by staggered, minimally doubled Ginsparg-Wilson fermion mass, value spacing, so continuum this exists. They instead not Wilson fermions, which known to take place Aoki phase. The existence regularizations unaffected residual doubling cannot break spontaneously establishes rigorously (at physicist's level) impossibility number flavors.