Derivative counting alone is insufficient

These functionals have no field derivatives, are even in $\phi$, and have the indicated engineering dimension when $[\kappa ]=-1$, $[m]=[\phi ]=1$, and $[g]=0$.

The proposition does not claim that all $C_n$ occur as finite renormalizations of a fixed two-insertion construction. Multilinearity and perturbative homogeneity in local coupling insertions may exclude them. Its point is that those restrictions must be stated and used, rather than replaced by the phrase “fixed derivative order.”

Nor does an infinite functional catalogue imply an infinite-dimensional image on a fixed finite sample: every such image is a subspace of a finite-dimensional row space.

Figure fig:VI-catalogue-bounds distinguishes polynomial finiteness from coverage of the allowed response class.