Lemma: Affine slot

$d_1[h]$ is linear in $h$. There is a symmetric bilinear $Q[h,h^{\prime }]$ with

Proof. $A$ and $B$ are linear in $k$ with coefficients $a_1[k]$, $b_1[k]$; so ${\mathsf{L}}_{2,g}[h,k]={\mathsf{L}}_{2,g}[h,0]+{\mathsf{L}}_{1,g}[k]$. By (eq:coordinate-equations) and the zero data, ${\mathsf{X}}_2[h,k]={\mathsf{X}}_2[h,0]+{\mathsf{X}}_1[k]$. In $c_2=-{\partial }_{\mu }(f{B}^{\mu })+\frac{1}{2}{\partial }_{\mu }{\partial }_{\nu }(f{{\mathsf{X}}_1}^{\mu }{{\mathsf{X}}_1}^{\nu })$, $B^{\mu }={{\mathsf{X}}_1}^{\nu }{\partial }_{\nu }{{\mathsf{X}}_1}^{\mu }-{{\mathsf{X}}_2}^{\mu }$, the only $k$-dependence is $+{\partial }_{\mu }(f{{\mathsf{X}}_1}^{\mu }[k])=c_1[k]$. Collecting the $k$-linear terms of (eq:completed) gives $c_1[k]-{\mathsf{L}}_{1,g}[k]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f=d_1[k]$. The remainder is quadratic in $h$; polarize.