Theorem: Backwards support for prescribed block channels

For ticks $r=1,\dots ,T$, let ${\mathcal{B}}_r$ be prescribed partitions of $X$, with $\mathrm{diam}(B)\le \Delta$, and let

where every ${\Phi }_{r,B}$ is CPTP on its fixed block interface.

For $0\le u\le t\le T$, define

Then

and

Equal input reductions on $N_{u,t}(Y)$ give equal output reductions on $Y$. With an untouched finite-dimensional reference $R$, equal joint input reductions on $N_{u,t}(Y)\cup R$ give equal joint output reductions on $Y\cup R$.

The scheduled ancestor set is an upper bound on dependence. It need not be the minimal influence region and is not identified with a spacetime causal past.

Proof. At one backwards tick, a block disjoint from the current support acts on an identity factor. Its adjoint is unital and preserves that identity. All remaining support can enlarge only within the union of blocks meeting the current support. Backwards induction gives eq:support.

A point added at one step is within $\Delta$ of a point in the previous backwards set. The triangle inequality proves eq:metric-support. For $Y=\varnothing$, all backwards sets are empty and the conclusion is trivial.

The reduced-state conclusions follow from lem:regional. In the referenced version, block adjoints act as identities on $R$, so the pullback lies in $\mathcal{A}(N_{u,t}(Y))\otimes L({\mathcal{H}}_R)$. Equality of the joint input reduction is the required hypothesis; equality on the ancestor sites alone is insufficient for a joint referenced conclusion.