Theorem: Realized-route cone certificate

A spatial route parametrized over $s\in [a,b]$, with metric speed at most

lifts under $s\mapsto (F_s(x(s)),x(s))$ to a future-causal curve. In particular, if an actual regular route of length $D$ joins $x$ to $y$, then

is sufficient for a causal route between its corresponding slice ports. The statement holds across $s=0$.

Proof. Lemma lem:slices gives, for an increment $\delta s>0$,

The lift is Lipschitz. At almost every parameter its time derivative is at least $1-\lambda |\dot{x}|$. When $|\dot{x}|\le v_{\ast }$, this is at least $|\dot{x}|$, so the lifted tangent is future causal. For the final assertion, traverse the supplied spatial route at constant arc-length speed $D/(b-a)$, allowing waits.