Proposition: A smooth pure-trace blind sector

Pure-trace tests do not determine an unrestricted compact smooth kernel with two symmetric metric slots and one scalar slot.

Proof. In a local Lorentzian frame choose a nonzero constant symmetric tensor $A^{\mu \nu }$ with $A^{\mu \nu }{\eta }_{\mu \nu }=0$. For nonzero compact smooth functions $b,c$, set

This is a nonzero compact smooth kernel, symmetric in its metric slots. Its contraction with $(\eta {\psi }_1)\otimes (\eta {\psi }_2)\otimes \phi$ vanishes for every test choice. Thus zero and $K$ have identical pure-trace evaluations.