Proposition: The gauge-fixed second metric coefficient

Let $\gamma ={\gamma }_0+\kappa {\gamma }_1+O({\kappa }^2)$ and $\varphi ={\varphi }_0+O(\kappa )$ solve the gauge-fixed equations of $S_{\mathrm{D},\mathrm{gf}}$ through order $\kappa$ with $P_m{\varphi }_0=0$, and let ${\gamma }_1$ have zero initial data. Then $L{\gamma }_0=0$ and

Proof. By XII-C, $e_{\mathrm{D}}=A\gamma +\kappa ({\mathcal{J}}_{\mathrm{D}}[\gamma ]-\frac{1}{2}g{T}_0[\varphi ])+O({\kappa }^2)$ and $e_{\mathrm{gf}}=2D^{\ast }D\gamma$. Insert the expansions; since ${\mathcal{J}}_{\mathrm{D}}$ is quadratic, ${\gamma }_1$ enters the order-$\kappa$ coefficient only through $(A+2D^{\ast }D){\gamma }_1=L{\gamma }_1$. The coefficients are $L{\gamma }_0=0$ and $L{\gamma }_1=-{\mathcal{J}}_{\mathrm{D}}[{\gamma }_0]+\frac{1}{2}g{T}_0[{\varphi }_0]$. The zero-data solution is ${G_{\mathrm{gr}}}^{\mathrm{R}}$ of the right side, with the local cutoff construction of XII-D for the nonlocal part. The scalar-sourced term is XI-E's $r_g$ XIE.