Proposition: Scheduled support and conditional physical containment

For ${\Phi }_{s,t}={\Phi }_t\circ \cdots \circ {\Phi }_{s+1}$,

Equal input reductions on $N_{s,t}(Y)$ therefore give equal output reductions on $Y$. The statement is stable under an untouched reference when the joint ancestor–reference reductions agree.

Physical causal containment additionally follows if every dependency edge, including control and guard inputs, has a location in a separately specified transitive physical causal relation.

Proof. For one layer, an observable is the identity on every disjoint block. The corresponding adjoints preserve that identity. An intersecting block can enlarge support only to its own input block. Iterate backward to obtain the inclusion.

Pairing with input states proves snapshot determinacy. Tensoring the linear maps with a reference identity gives the joint referenced statement.

Under the additional embedding hypothesis, each dependency path maps to a chain in the physical causal relation. Transitivity places its initial event in the physical past of its final event.