Theorem: Exact encoded causal containment: Target A

Suppose:

1. the entire encoded–computed–decoded experiment is a complete finite network;

2. every dependency edge has a physical endpoint certificate;

3. input and output localizations use the same independent causal relation.

For a joint output $B$, let ${\mathsf{Phys}}^{-}(B)$ be the input factors whose assigned events can causally reach at least one output event in $B$. Then

Inputs outside ${\mathsf{Phys}}^{-}(B)$ cannot signal to $B$, even with an untouched entangled reference.

Proof. Theorem thm:schedule gives the first inclusion. Every scheduled path is a chain of certified causal endpoint relations. Transitivity gives a causal relation between its first and last events. Hence every scheduled ancestor belongs to the physical past. The signalling conclusion follows from Theorem thm:nosignal.