Two questions and five distinct objects

The first question is classical. Fix a free scalar solution $\varphi$, a Cauchy surface, and a scalar smearing $f$. The metric-to-scalar response is linear in the metric perturbation. When does that functional equal a pairing with a smooth compact conserved tensor on a declared space of vacuum metric solutions?

The second question is quantum and perturbative. Start from a specified compactly switched vertex, not from a presumed observable. What is its first retarded product with one scalar insertion, and on which domain is the resulting expression represented?

The two questions have different inputs and different conclusions:

A result along one arrow is not a premise completing the other.