Theorem: Computational backwards support: the C5 interface

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

with all block maps CPTP. Set

Then

Equal input reductions on $N_{u,t}(Y)$ give equal outputs on $Y$. For joint output with untouched reference $E$, equality is required on the joint input $N_{u,t}(Y)E$, not only on its site marginal.

This set is a computational upper bound, not an identified spacetime causal past.

Proof. A block disjoint from the current observable support acts on identity. Its adjoint is unital and preserves that identity. Every other block can enlarge support only within its own input block. Backward induction proves the first inclusion. Each layer adds at most one block diameter, giving the second.

Output expectations then depend only on the named input reduction. With a reference, the pullback belongs to $\mathcal{A}(N_{u,t}(Y))\otimes \mathcal{B}({\mathcal{H}}_E)$, proving the referenced statement.