Corollary: Per-SWAP half-diamond allowance

Assume exact encoding, exact output reduction, exact waits, and exact spectator storage. Suppose each implemented SWAP is a channel on the same two-site interface and differs from the ideal SWAP channel by at most ${\eta }_j$ in half-diamond distance. Then

Thus $j{\eta }_j\to 0$ is sufficient for uniform convergence on bounded-distance null targets.

Proof. Tensoring a local gate difference with identity channels on the other registers does not enlarge its diamond norm. All wait ticks have zero error under the stated assumptions. Apply Lemma lem:telescoping to the $m_j$ nontrivial gates.