Proposition: The regular product supplies no global Cauchy inverse

The regular conical product is not globally hyperbolic. No global Green operator on that product is used here.

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

This sequence approaches the excluded worldsheet and has no convergent subsequence in ${\mathcal{M}}^{\circ }$. The diamond is not compact.

The Green operators constructed below are instead the specified Minkowski operators on $\hat{M}$.