Corollary: Exact agreement on a confinement event

On the event that no path of cost at most $t$ reaches $\partial {\Lambda }_L$, the finite and infinite infected sets agree up to time $t$:

The probability of disagreement is at most the bound in Lemma lem:escape.

Proof. Restricted passage times are no smaller than unrestricted ones. If an unrestricted path of cost at most $u\le t$ leaves the box, its initial segment reaches the boundary with cost at most $t$. On the stated event every relevant path remains inside.