Proposition: Finite-circuit operational convergence

Suppose the logical implementation of vertex $v$ has error $e_{v,n}$, and the complete logical implementation of connection $e$ has identity error $r_{e,n}$. Then

If the right-hand side tends to zero, all complete finite readout laws converge in total variation, uniformly over logical inputs and untouched references.

Proof. Evaluate the finite DAG in a topological order, tensoring each step with identities on the other live logical factors. Apply Lemma lem:telescope. A complete terminal readout is CPTP and cannot increase the error. Its classical trace distance is total variation.