The unconstrained supported extension space

The finite contact dimension should not obscure the larger extension problem.

A translation-invariant scalar difference supported on $\mathcal{W}$ and with transverse scaling degree at most $2$ is $c{\delta }_{\mathcal{W}}$. Without translation invariance its coefficient can be an arbitrary distribution in $t,z$.

For stress, scaling degree at most $4$ permits transverse delta derivatives through order $2$. The almost-homogeneity condition reduces these to order $2$, and the symmetry and Ward equations then give Theorem thm:classification. If almost homogeneity is dropped but the other stated symmetries and conservation are kept, one can additionally add

This is conserved for the reference representative, but it violates the declared massless almost-homogeneous class and, unless accompanied by a changed trace prescription, its trace condition.

For a current, a normal-order bound alone likewise does not select circulation or remove possible longitudinal structures if longitudinal Lorentz covariance is dropped. A statement that there are “only three apex counterterms” without the assumptions of Theorem thm:classification would therefore be false.

These observations are elementary instances of the distinction between a finite normal-jet module and a finite-dimensional space of normalization parameters Dang,CheungXVB. They also explain why no counterterm class from a different companion observable is identified with the class constructed here.