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[en] We consider four-dimensional chiral gauge theories defined over a spacetime manifold with topology and periodic boundary conditions over the compact dimension. The effective gauge-field action is calculated for Abelian gauge fields which depend on all four spacetime coordinates (including the coordinate of the compact dimension) and have vanishing components (implying trivial holonomies in the 4-direction). Our calculation shows that the effective gauge-field action contains a local Chern–Simons-like term which violates Lorentz and CPT invariance. This result is established perturbatively with a generalized Pauli–Villars regularization and nonperturbatively with a lattice regularization based on Ginsparg–Wilson fermions.