Lemma: Endpoint times and strict causal slack

The four matter-port times are

Both cross-port dependencies are causal. Their conservative slicing-time margin obeys

Proof. On a ray, $K=0$ and $H=2r_{\ast }/3$. At negative $s$, the $H$ branch is active; at positive $h_{\ast }>2r_{\ast }/3$, the $s$ branch is active. The apex time is $s$ before and $2s$ after the event. Since $d_C(o,e)=r_{\ast }$, the causal claims follow from Theorem thm:conegates. Finally, ${\lambda }_0<1$ gives