Proposition: A smooth blind sector

Pure-trace metric evaluations do not determine a general quadratic insertion kernel, even within the smooth compact class.

Proof. Choose a nonzero smooth compact symmetric contravariant tensor density $A^{\mu \nu }$ with

For example, in the background coordinates take only compactly supported $12$ and $21$ entries.

The nonzero linear functional

vanishes for every $h_{\mu \nu }={\bar{g}}_{\mu \nu }\rho$. Thus the smooth compact quadratic kernel

vanishes on every pair of pure-trace metric inputs but is not zero. Adding $B_A$ to a kernel leaves all such probe values unchanged.