Theorem: Fixed-skeleton operational completion

The metric completion of ${{\mathcal{X}}_{\Gamma }}^{\circ }$ is precisely the class ${\mathcal{X}}_{\Gamma }$ with the same fixed local types and the closed causal inequalities

It has the following properties:

1. every compatible local family has a unique limit as labelled local data;

2. its complete circuit channels converge in half-diamond distance to the channel of that limit;

3. admitted finite tensor compositions and acyclic typed gluings are continuous and extend to the completion;

4. every limit object has the matched discrete realization constructed in Section sec:realization.

Proof. For fixed finite types, the space of CPTP maps is closed in the finite-dimensional vector space of linear maps: positivity of every finite Choi block and trace preservation are closed conditions. It is bounded and hence compact, and in particular complete in diamond norm Choi,Watrous. The position space is complete. Closed causal inequalities define a closed subset of their finite product. Thus ${\mathcal{X}}_{\Gamma }$ is complete.

Every closed-causal assignment is approximated by strict-causal ones. Choose a topological rank $r_v$, and replace $t_v$ by $t_v+r_v/k$, leaving positions and instruments unchanged. Every edge gains strictly positive time and the assignments converge. This identifies the completion.

Circuit evaluation is a finite tensor-and-composition expression in its local channels. Lemma lem:telescope proves its continuity and the channel limit. Coordinates specify admissibility and supports; they do not alter the logical channel formula.

A typed gluing forms a finite composite skeleton. When it is acyclic and satisfies the causal constraints, its channel is the corresponding tensor-and-composition expression. The same bound proves continuity. Vertex support coordinates and boundary identifications are retained in that expression. Finally, Theorem thm:realization includes all closed-causal assignments, including boundary connections.