Hypothesis: C6: Product-preserving observable realization

For each scale there is a represented observable algebra ${\mathfrak{B}}_{\epsilon }$ containing local subalgebras associated with bounded spacetime operation regions. It has the following properties:

1. Generators assigned to two regions with no scheduled dependency path in either direction commute in the chosen complete observable realization.

2. There are linear comparison maps $j_{\epsilon }:{\mathfrak{B}}_{\epsilon }\to \mathfrak{B}$ into a common $C^{\ast }$-algebra. For the admitted uniformly bounded generator sequences and their finite products, these maps are asymptotically product preserving:

   $$\Vert j_{\epsilon }(a_{\epsilon }{b}_{\epsilon })-j_{\epsilon }(a_{\epsilon })j_{\epsilon }(b_{\epsilon })\Vert ⟶0.$$

   They also preserve the adjoint asymptotically. Admitted generator images converge in norm.

3. The norm limits generate isotone $C^{\ast }$-subalgebras $\mathfrak{A}(O)\subseteq \mathfrak{B}$. The operational laws extend to states on their generated algebra.

The realization must specify how allocation, consumed wires, discarded systems, and dilation ancillas are represented.