Lemma: Finiteness under explicit polynomial bounds

Suppose a local counterterm class is generated by a finite set of field, ghost, switch, and background components and their derivatives up to a fixed order. Suppose also that:

1. the total degree in these generators is bounded;

2. the allowed mass and scale dependence lies in a fixed finite-dimensional coefficient space.

Then the resulting vector space of local polynomial densities is finite-dimensional. Imposing linear symmetry conditions and quotienting by linear relations preserves finite-dimensionality.

Proof. The component set and derivative bound provide finitely many jet variables, say $z_1,\dots ,z_n$. A total polynomial degree bound $p$ allows only finitely many monomials ${z_1}^{a_1}\cdots {z_n}^{a_n}$, with ${\sum }_i{a}_i\le p$. Tensor contractions select a subspace of this finite-dimensional space. Multiplying by a finite basis of the allowed mass and scale dependence still gives a finite list.

Linear covariance, parity, or Ward constraints define a subspace. Linear identities and integrations by parts define a quotient. Subspaces and quotients of finite-dimensional spaces are finite-dimensional.