Proposition: Pure-trace probes do not authenticate a full kernel

Pure-trace evaluations alone do not determine an unrestricted compact smooth two-metric-slot kernel. They do not, however, create ambiguity between two retarded inverses of the fully specified operator $\mathcal{B}$.

Proof. Choose a nonzero smooth compact trace-free symmetric tensor $a^{\mu \nu }$ in a coordinate patch, and set

It annihilates all pure-trace metric tests and is nonzero. Thus selected trace probes cannot authenticate a full kernel.

On the other hand, Theorem thm:greens proves uniqueness from the complete differential equation and retarded support. The trace-blind perturbation is not asserted to preserve those inverse equations and support conditions.