Causal reachability with periodic time ($J^{+}(p)=J^{-}(p)=\mathcal{M}$)

In the flat quotient $t\equiv t+P_t$ with noncompact space, choose any sufficiently large positive or negative temporal winding to make a lifted displacement timelike. Every event is in both the causal future and causal past of every other event. Constant-position curves are closed timelike curves. The quotient is a valid Lorentzian manifold, but it violates chronology and cannot support arbitrary globally consistent initial-value evolution.