Theorem: Scheduled ancestry is an influence upper bound

Let $(X,\mathrm{dist})$ be a supplied finite metric space. At tick $j$, let the channel be a product over partition blocks of diameter at most $\Delta$:

Define backward sets

Then

The result includes initially entangled states. Unreported operations outside this ancestry cannot change the terminal reduced output.

Proof. A block disjoint from the current observable support acts on identity, which a channel adjoint preserves. Backward induction gives the support inclusion. Each backward step adds at most one block diameter, so the triangle inequality gives the metric envelope. Proposition prop:snapshot supplies reduced-channel dependence. The proof uses no assumption of initial product states.