Copied conventions

The ramp ${\sigma }_{\epsilon }$, switch $g\in {C_c}^{\infty }((0,T)\times {\mathbb{R}}^3;[0,1])$, label test $f$, plateau $P_g$, region $U$, and algebra ${\mathfrak{g}}_U={C_c}^{\infty }(U;TM)$ are those of X-B. Backgrounds $(h,k)$ range over $\mathcal{B}=C^{\infty }(M;S^2{T}^{\ast }M{)}^2$. With $a_1[h]$, $b_1[h]$ as in X-B,

a formally self-adjoint operator with smooth coefficients supported in $\mathrm{supp}g$. The vertex is $V_{1,g}[h,\phi ]=-\frac{1}{2}⟨\phi ,{\mathsf{L}}_{1,g}[h]\phi ⟩$, so that $V_{1,g}^{\prime }=-{\mathsf{L}}_{1,g}[h]\phi$ and $V_{1,g}^{\prime \prime }=-{\mathsf{L}}_{1,g}[h]$. For smooth $u$, ${\rho }_{1,g}[h;u]=-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{L}}_{1,g}[h]u$. The single normalized map is

The frozen insertion tests $c_0=f$, $c_1[h]$, $c_2[h,k]$ are smooth, supported in $K=\mathrm{supp}f$, and polynomial and continuous in $(h,k)$ at fixed $\epsilon$ (X-B, Proposition prop:linear). The completed coefficient is ${\hat{A}}_1[h]=\hat{\varphi }(c_1[h]-{\mathsf{L}}_{1,g}[h]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f)$ and the retained mixed term is $\mathcal{M}[h]=R_1(V_{1,g}[h];F_{c_1[h]})$. The general insertion pair of X-B is

1. $V=-\frac{1}{2}⟨\phi ,\mathsf{L}\phi ⟩$ with $\mathsf{L}$ formally self-adjoint with smooth compactly supported coefficients;

2. $F_c$ with $c\in \mathcal{D}$.