A product-preserving comparison, not arbitrary compression

Because the algebras are finite dimensional, the algebra generated by any collection of point algebras is already generated by a finite subcollection: successive strict increases of subalgebra dimension can occur only finitely often. Thus every observable of $\mathfrak{A}(O)$ can be treated by the finite-generator localization statement when its selected generators lie positively inside $O$.

This logical-identification construction is the finite analogue of the comparison discipline used in En05-C CheungEn05C. It avoids identifying a moving physical register address with one fixed observable in an ambient infinite tensor product.