Theorem: Instrument–observable refinement

For each admitted logical observable $A\in \mathcal{B}({\mathcal{H}}_0)$, the sequence $A_n={\iota }_n(A)$ has the exact comparison image

Finite products and adjoints converge through the same maps exactly, and the same statement holds for bounded logical sequences whose comparison images converge in norm.

For an observable generated by finitely many point-labelled generators inside an open region $O$, its physical representative is localized in $O$ for every sufficiently large $n$, provided those locations have positive distance from $O^c$. For positively spacelike-separated compact localization sets, the corresponding admitted representatives commute at all sufficiently fine scales, and their limit algebras commute.

Proof. The exact assertions follow from Lemma lem:obscomparison. If $j_n(A_n)\to A$ and $j_n(B_n)\to B$ in norm, uniform boundedness and

give norm convergence to $AB$. Adjoints are continuous.

For localization, take the minimum positive distance from the finitely many declared generator locations to $O^c$. The geometric error tends to zero, so all representatives eventually lie in $O$. The physical routed dilation factors and their local operations provide the asserted representatives.

For two compact localization sets with a positive spacelike margin, the same geometric convergence preserves spacelike separation for all sufficiently fine scales. The logical representatives commute by Theorem thm:microcausality, and their physical pullbacks commute by the $\ast$-homomorphism ${\iota }_n$. Their products therefore retain the same zero commutator under $j_n$.

No equality of full regional algebras at regions tangent to localization boundaries is asserted.