Theorem: Reference-preserving scheduled support

For every endpoint observable $A\in \mathcal{A}(Y)$,

Consequently, the endpoint channel factors through the input reduction on $N_{u,t}(Y)$. If two input states, possibly with a spectator reference, have equal reductions on $N_{u,t}(Y)$ together with that reference, their endpoint-reference outputs are equal.

In particular, a local trace-preserving intervention supported outside $N_{u,t}(Y)$ cannot influence this endpoint.

Proof. A block disjoint from the current support acts on an identity factor. Its adjoint is unital and preserves that identity. A block meeting the support can enlarge it only within the same block. Iterating proves the support assertion.

All endpoint observables therefore pull back to the ancestor algebra. Equal input reductions give equal expectations of every pullback, hence equal endpoint states. The argument tensors with the identity on a reference.

For explicit factorization, choose any fixed state on the nonancestor sites, insert it after the input reduction, and then apply the endpoint channel. The equality of reductions proves that this constructed channel agrees with the original one on every input. A local trace-preserving intervention outside the ancestor set preserves its reduction, including the reference.