Proposition: Non-global-hyperbolicity boundary

The regular conical product is not globally hyperbolic. The finite constructions in this paper require no global conical Green operator.

Proof. Fix a radial ray and $r_0>0$. For $A>r_0/c_{\mathrm{h}}$, the diamond between $(-A,r_0,{\phi }_0,z_0)$ and $(A,r_0,{\phi }_0,z_0)$ contains

The sequence approaches the excluded apex and has no convergent subsequence in the regular product. The diamond is not compact.

All coefficient calculations below are compact smooth integrals on $\hat{M}$, represented inside the retained chart. No inverse on ${\mathcal{M}}^{\circ }$ is used.