Lemma: A strongly separated comparison class

Suppose $f,g$ have times in $[0,T]$ and spatial supports in $B({\mathbf{x}}_0,r)$ and $B({\mathbf{y}}_0,s_0)$. If

and all retained shadows fit in the laboratory, then both actual orderings hold.

Proof. A point of $P_F\cup F$ has time in $[0,T]$ and spatial distance at most $T+r$ from ${\mathbf{x}}_0$. Its distance from any point of $G$ is therefore greater than $T$, while their time difference is at most $T$. The upper support sets are strictly spacelike, and actual supports are their subsets.