Lemma: Pure-trace evaluations do not determine a three-slot kernel

The family of evaluations on all pure-trace tensor pairs does not determine a compact smooth three-slot kernel with symmetric tensor slots.

Proof. Choose a nonzero constant symmetric contravariant tensor $A$ such that $A^{\mu \nu }{\eta }_{+,\mu \nu }=0$, for example $A^{12}=A^{21}=1$ and all other components zero. Choose nonzero real compact smooth functions $b,c$ and set

This is a nonzero compact smooth kernel symmetric under exchange of its tensor slots. Each contraction with ${\eta }_{+}{\psi }_1\otimes {\eta }_{+}{\psi }_2\otimes \varphi$ vanishes because $A:{\eta }_{+}=0$. Hence $K$ and zero have identical data on the entire pure-trace family.