Lemma: A sufficient speed in the slicing parameter

An actual regular spatial path $x(s),\zeta (s)$ with speed at most

lifts by $(F_s(x(s)),x(s),\zeta (s))$ to a future-causal curve. The bound holds across $s=0$.

Proof. The lower temporal-order estimate and spatial Lipschitz bound give, almost everywhere,

when the full spatial speed is at most $v_{\ast }$. The lift is Lipschitz and has future-causal tangent almost everywhere. The estimate does not require differentiability of the cellulation at zero.