Proposition: The regular product is not globally hyperbolic

For every $\mu \in I$, the regular spacetime $M_{\mu }$ is not globally hyperbolic.

Proof. Fix $r_0>0$, one angle, and $z=0$. Take two events at that spatial point with times $T_{\ast }\pm 2r_0/c$. The events at time $T_{\ast }$ and radii $r_n\downarrow 0$, with the same angle and $z$, belong to their causal diamond: the required radial travel times are less than $r_0/c$. They have no convergent subsequence in $M_{\mu }$. The diamond is noncompact.