Theorem: Three-relation separation

Consider a finite acyclic network of completely positive trace-preserving operations, with complete dependency bookkeeping and intervention-independent initial preparation. Then

for distinct events, and the inclusion can be strict.

Neither relation is automatically contained in the pullback of $R_{\sup }$ along an independently chosen location map. If every dependency edge is certified to respect $R_{\sup }$, then scheduled ancestry, and hence operational influence, is contained in that pullback.

These statements express the separation adopted from VII-J; they do not identify the three relations.

Proof. Pull an output observable backwards through the circuit. Operations outside its dependency ancestors act on factors carrying the identity observable and disappear because the adjoint of a trace-preserving map preserves the identity. Thus an intervention outside the ancestor set cannot change the output channel. The argument remains valid after tensoring with an untouched reference. This is the finite-channel content of VII-J's scheduled-support and no-signalling statements CheungVIIJ (thm:schedule, thm:nosignal).

For strictness, let a scheduled wire from $p$ enter an operation at $q$ whose observed output is the replacement channel

Every trace-preserving intervention upstream gives ${\mathcal{R}}_{\sigma }\circ \mathcal{I}={\mathcal{R}}_{\sigma }$. There is scheduled ancestry but no influence on that output.

An independent location assignment need not respect the schedule. For example, assign the endpoints of a scheduled identity transmission the locations $(1,x)$ and $(0,x)$, in that order. Its operational transmission is unchanged, while (eq:supplied) fails. Conversely, certified causal containment of every edge propagates along a path by transitivity of the supplied relation. This is the separate embedding hypothesis of VII-J CheungVIIJ (thm:physical).