[en] We consider four-dimensional chiral gauge theories defined over a spacetime manifold with topology ${\mathbb{R}}^3\times S^1$ and periodic boundary conditions over the compact dimension. The effective gauge-field action is calculated for Abelian $U(1)$ gauge fields $A_{\mu }(x)$ which depend on all four spacetime coordinates (including the coordinate $x^4\in S^1$ of the compact dimension) and have vanishing components $A_4(x)$ (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.