= Causal reachability with periodic time
{title2=$J^+(p)=J^-(p)=\mathcal M$}
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.
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