Proposition: Timed scheduled support

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

Consequently, equal reductions on $N_{s,t}(Y)$ give equal output reductions on $Y$. A trace-preserving intervention outside $N_{s,t}(Y)$ cannot signal to $Y$, including jointly with an untouched entangled reference.

Proof. Apply Lemma lem:onelayer successively in reverse tick order. This proves (eq:support); Proposition prop:comparison then proves snapshot determinacy.

An intervention on the complement preserves the reduction to the ancestor set, because its adjoint preserves identity. With a reference $Q$, it also preserves the joint ancestor–$Q$ reduction when it does not act on $Q$. Apply the reference factorization in Proposition prop:comparison.