Corollary: Approximate payload implementation

Suppose the exact payload channel at vertex $v$ is replaced by a channel with half-diamond error at most $e_{v,n}$, including all its classical controls and output records. Then

Proof. Replace payloads one at a time. The maps before and after a replacement are channels, including identity extensions by the other live factors. Composition with channels cannot increase the diamond norm. The triangle inequality gives (eq:telescoping).