Proposition: No global cone Cauchy claim

The regular conical product is not globally hyperbolic.

Proof. Choose a fixed radial ray, $r_0>0$, $z_0$, and $A>r_0/c_{\mathrm{h}}$. Between the events with coordinates $(-A,r_0,{\phi }_0,z_0)$ and $(A,r_0,{\phi }_0,z_0)$, the causal diamond contains

Their only completion limit lies at the excluded apex. They have no convergent subsequence in the regular product, so that diamond is not compact.