No interacting scalar is reconstructed

In the free model we have constructed $L$ by a normal product and local quadratic approximation. In an interacting theory, the formal expression

would not establish locality, mixed products, domain compatibility, or affiliation. This paper supplies no such interacting construction.

Likewise, deleting $Q$ from a list of labels does not delete ${\partial }_{\mu }Q$, or prove that $Q$ is absent from a represented net. A reduced algebra has to be analyzed on its actual generators and closures. Renaming its list is not a membership argument.

Four-dimensional scale-to-conformal arguments and their generalized-free-field loopholes are discussed in DKST,DFKLP. Local renormalization-group identities with background sources provide a different framework Baume. Those results are comparisons; our theorem is a free-model calculation with its own declared field algebra.