Theorem: Schedule-locality

Let prescribed partitions ${\mathcal{B}}_r$ define layers

Set

Then

If all block diameters are at most $\Delta$, then

Equal input reductions on the ancestor region give equal output reductions on $Y$. For joint output with a reference $E$, equality must hold on the joint ancestor–reference input reduction.

Proof. The adjoint of a trace-preserving map is unital. A block disjoint from current observable support therefore acts on identity and leaves it unchanged. Every other block can enlarge support only within its own interface. Reverse induction gives the inclusion. Each step adds at most one block diameter. Testing all output observables proves the reduced-state statements, with $\mathcal{B}({\mathcal{H}}_E)$ carried unchanged in the referenced proof.