Proposition: Local completeness with fixed finite slots

Consider a family on one fixed finite typed skeleton whose support coordinates converge and whose complete readout laws, for a declared input preparation, converge in total variation. Even if its local channels are not Cauchy, a subsequence of its local assignments converges to an admitted circuit with that limiting law.

For matched pairs on the same fixed finite slots, a common subsequence can be chosen so that the two limiting circuits remain matched.

Proof. The finite product of local CPTP-map spaces is compact. Select a convergent subsequence of all local channels and any separately retained finite preparation states. The causal constraints are closed, so the limiting assignment is admitted. Continuity of circuit evaluation gives its readout law. The original laws have a prescribed limit, so the subsequence has that same limit.

For pairs, use compactness of the joint finite product. Equal unchanged channels and equal exterior reductions remain equal under the common subsequence.