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