  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.
