= Solution
For <causal reachability with periodic time>, let $P_t=2\pi T$. The lower-right panel of part <1/c> identifies its top and bottom edges. The vertical curve at fixed $x$ is a <closed timelike curve>: after a positive proper duration $P_t$, it returns to the same <spacetime event>. The quotient is a mathematically well-defined flat <Lorentzian manifold>, but it violates chronology and admits no usual global causal ordering or ordinary global initial-value evolution.
These are physical causality problems, rather than an algebraic contradiction in the metric. Fields and particle histories must satisfy periodic consistency conditions; an arbitrary initial state followed by irreversible evolution will generally fail to return to itself after a temporal lap. The fact that the <causal past> and <causal future> overlap everywhere, established below, explains why the ordinary three-way causal partition degenerates.
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