Definition: The homogeneous polynomial class

The admissible class consists of symmetric rank-two tensor fields $S_{\mu \nu }$ satisfying all of the following:

1. $S$ is a finite vacuum-normal-ordered local polynomial in $\varphi$ and its derivatives, on the declared free field domain.

2. Its coefficients are constant, with no new dimensionful parameter or coordinate-dependent coefficient.

3. It is Poincare covariant and parity even; tensor coefficients are built from $\eta$.

4. Every monomial has engineering dimension four, with $\dim \varphi =1$ and $\dim \partial =1$.

5. It is conserved on the free shell.

Expressions are identified by the actual free field equation and normal-product identities, not by their presentation labels.