Proposition: A local variational test fails for the unchanged law

The source-switched mixed operator $\sigma Q^{(1)}[h]$ is not, for all smooth backgrounds $h$, the Hessian of a local action on unrestricted off-shell fluctuations.

Consequently (eq:response) cannot be identified, on the stated off-shell domain, with the first variational Møller correction of an unspecified switched action merely because its unswitched source was obtained from (eq:fluctuation).

Proof. For fixed $h$, a quadratic-action Hessian is formally self-adjoint under the tensor-density pairing. Proposition prop:Q gives

There is no general equality with $\sigma Q^{(1)}[h]$.

An explicit admissible local example is $h=\eta$. For a positive constant $c$, the four-dimensional Einstein density obeys $\mathcal{E}[cq]=\mathcal{E}[q]$. Hence $D\mathcal{E}[c\eta ]r=c^{-1}{G}_{\mathrm{lin}}[r]$. The background gauge-fixing quadratic action at $c\eta$ also scales as $c^{-1}$: its density contributes $c^2$, the lower metric $c$, and the two divergence factors $c^{-4}$. Therefore

In an open part of the ramp where ${\sigma }^{\prime }\ne 0$,

is a nonzero differential operator. It is detected by compact variations in that open set. Thus $-\sigma {\mathsf{P}}_{\mathrm{gr}}$ is not formally self-adjoint.