Proposition: Gather–operate–distribute realization

Consider a gate with input events $p_1,\dots ,p_m$ and output events $q_1,\dots ,q_n$. Suppose there are stations $r_{-}⪯r_{+}$ such that

with specified routes, and the gate has a finite local implementation between $r_{-}$ and $r_{+}$. Then Contract contract:routing gives a Target B realization of that gate.

Proof. Carry the inputs to $r_{-}$, retain them through the local implementation, and carry the outputs from $r_{+}$ to their destinations. Include every ancilla and outcome register in the local implementation. Each wire and elementary gate has causal endpoints and a declared support. The endpoint channel is the intended gate, including with arbitrary references.