Lemma: Uniqueness of actual kernels

The symmetric kernels in (eq:polynomial) are uniquely determined by the functional on all smooth off-shell configurations.

Proof. Scaling separates homogeneous degrees. Polarization recovers the symmetric multilinear form in each degree. If that form vanishes on all smooth directions, its kernel annihilates all finite sums of elementary tensor tests. Those sums are dense in the smooth test space on each compact product region. Distributional continuity implies that the kernel is zero.