Operational–algebraic compatibility

Equation (eq:selectivepairing) is stronger than agreement of the terminal outcome law: it identifies every selective retained-output expectation. Conditional normalized states are obtained only on positive-probability branches.

Theorem thm:pairing, Proposition prop:nosignalling, and Proposition prop:Belllaw establish simultaneously:

1. spacelike commutation of local observable algebras;

2. unconditioned independence from the remote setting;

3. nonclassical joint Bell correlations;

4. an exact complete-instrument refinement, not just a law limit.