Proposition: Bounded perturbation allowance

Suppose two Hamiltonians preserve the declared bounded-particle domain, and their difference on that domain has norm at most $\nu (s)$ at time $s$. Then their prepared complete channels at time $t$ have half-diamond distance at most

Proof. Duhamel's formula on the invariant domain gives a unitary difference bounded by ${{\int }_0}^t\nu (s)\mathrm{d}s$. Restrict to the prepared isometry and apply Lemma lem:isometry.