Theorem: Finite-net isotony and spacelike commutation

The assignment $O\mapsto \mathfrak{A}(O)$ is isotone. If $O_1$ and $O_2$ are spacelike separated, then

The net is noncommutative and contains nontrivial local quantum algebras.

Proof. Isotony follows immediately from the inclusion of generating point sets.

Every point of either active branch is causally related to $A$. Every two points on the same branch, including its vertical storage continuation, are causally related because that branch is future timelike. Thus two spacelike-separated regions cannot both contain active points of the same branch. Nor can one contain $A$ and the other an active branch point.

It remains to compare left and right generators. At the common cut immediately after the source gate, the entire left factor set ${\mathcal{H}}_L$ and right factor set ${\mathcal{H}}_R$ are disjoint tensor factors. Their full algebras commute. Conjugating both by the same source unitary preserves commutators:

The premeasurement algebras are subalgebras of these full wing algebras. Every left generator therefore commutes with every right generator. Products, sums, adjoints, and norm closure preserve this commutation, proving (eq:netmicrocausality).

A small region meeting the interior of the left branch contains a represented copy of $\mathcal{B}(Q_{L}W)$, and similarly for the right branch. These are noncommutative matrix algebras.