Retained action and background gauge

Keep the curvature convention and local action of X-E:

Euler derivatives mean compact variations. Neither global action integrability nor decay is assumed. The Palatini variation and transverse-traceless sign are retained from the stated action, not selected anew XE,XEST.

For $\bar{q}=\eta +\kappa h$, retain the background de Donder vector

and the fluctuation gauge-fixing density

Only the fluctuation is gauge fixed.

Let $\mathcal{E}[q]=J_q{G}^{\mu \nu }[q]$, with its Euler density type, and define

This fixes $Q_w$ without an unspecified vertex. It is a local formally symmetric differential operator, linear in the two-jet of $w$, with $\mathrm{supp}(Q_{w}r)\subset \mathrm{supp}w$.

Naturality of both terms in (eq:Q) gives

Indeed, differentiate $Q[F_s^{\ast }\eta ](F_s^{\ast }r)=F_s^{\ast }(Q[\eta ]r)$. This verifies the typed commutator used later; it does not rebuild the three response kernels.

The frozen mixed source map is

For general smooth inputs the source is prepared by its smooth onset and interpreted in the past-compact Green domain. For the compact backgrounds used in the obstruction, it is a compact smooth source.