Lemma: Variation of the gravitational slot

At $\gamma =0$ the slot changes by $s_{\xi }[0]+K\zeta$ with $\zeta =-{\rho }_{\xi }[0]$: the X-A form of the second-order metric law, except that ${\rho }_{\xi }$ is a propagated field supported in $J^{+}(\overline{U})\cap M_T$ and not an element of ${\mathfrak{g}}_U$.

Proof. Multiply XII-D's response law (eq:ledgerD) by $-1$ and use (eq:exi). Linearity of ${\mathcal{A}}_1$ gives the second formula. At $\gamma =0$, $A\gamma =0$ and the last term vanishes.