  We consider a QED$_{d+1}$, $d=1,3$ lattice model with emergent Lorentz or
chiral symmetry, both when the interaction is irrelevant or marginal. While the
correlations present symmetry breaking corrections, we prove that the
Adler-Bardeen (AB) non-renormalization property holds at a non-perturbative
level even at finite lattice: all radiative corrections to the anomaly are
vanishing. The analysis uses a new technique based on the combination of
non-perturbative regularity properties obtained by exact renormalization Group
methods and Ward Identities. The AB property, essential for the
renormalizability of the standard model, is therefore a robust feature imposing
no constraints on possible symmetry breaking terms, at least in the class of
lattice models considered.
