Proposition: Gather–operate–distribute

Suppose all gate inputs have supplied causal routes to one station event $r_{-}$, all outputs have supplied routes from a later station event $r_{+}$, and a finite local implementation fits between $r_{-}$ and $r_{+}$. Then Assumption ass:routing realizes that gate, with its records, ancillary factors, and references, as Target B.

Proof. Carry the input factors to the station, execute the stipulated local channel, and distribute its outputs. Identity carriage and the local channel give the intended complete composition on all operators. Every elementary dependency has an explicit causal route.