= Solution
For any representative event $(t,x)$, choose an integer $n$ large enough that $c(t+nP_t)>|x|$. A future-directed timelike segment from the origin to that lift projects to a <causal curve> on the quotient. Therefore
$$
\boxed{J^+(0)=\text{the entire spacetime}.}
$$
Even points with large $|x|$ can be reached by allowing sufficiently many temporal windings. The purple lower-right panel in part <1/c> denotes simultaneous membership in both the <causal future> and <causal past>.
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