Theorem: Background identity of the all-tree observable

Let $\xi ,\zeta \in {\mathfrak{g}}_U$, and let the classical backgrounds transform by the complete two-jet law

with $\hat{\varphi }$ and $\hat{h}$ held fixed. Then

Term by term: the two-jet part of (eq:all-tree) varies by $-{\kappa }^2{\hat{A}}_1[{\mathcal{L}}_{\xi }\hat{h}]$; ${\mathsf{T}}_{\mathrm{mat}}$ and ${\mathsf{T}}_{\mathrm{self}}$ are invariant; ${\mathsf{T}}_{\mathrm{mix}}$ varies by $+{\kappa }^2{\hat{A}}_1[{\mathcal{L}}_{\xi }\hat{h}]-{\kappa }^2{\hat{A}}_1[{{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{\mathcal{L}}_{\xi }{P}_{\mathrm{gr}}\hat{h}]$, and the last term lies in $\mathfrak{J}$.

Proof. The variation of $\hat{\varphi }(f)+\kappa {\hat{A}}_1[h]+{\kappa }^2(\hat{Q}[h,h]+{\hat{A}}_1[k_{\mathrm{cl}}])$, $h=h_{\mathrm{cl}}+\hat{h}$, is (eq:xe-defect): $-{\kappa }^2{\hat{A}}_1[{\mathcal{L}}_{\xi }\hat{h}]$ modulo $\mathfrak{J}$ and ${\kappa }^3$. ${\mathsf{T}}_{\mathrm{mat}}$ depends on $(f,g,\epsilon ,\hat{\varphi })$ only and ${\mathsf{T}}_{\mathrm{self}}$ on $(f,\epsilon ,\hat{\varphi },\hat{h})$ only; neither depends on $(h_{\mathrm{cl}},k_{\mathrm{cl}})$. ${\mathsf{T}}_{\mathrm{mix}}$ depends on $h_{\mathrm{cl}}$ linearly through ${\mathcal{Q}}^{(1)}[h_{\mathrm{cl}}]$ and not on $k_{\mathrm{cl}}$; since ${\hat{A}}_1$ and ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}$ are linear,

by (eq:jetlaw). For the last term, $d_1[{{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{\mathcal{L}}_{\xi }{P}_{\mathrm{gr}}\hat{h}](x)=\int N_0(x,y)({\mathcal{L}}_{\xi }{P}_{\mathrm{gr}}\hat{h})(y)\mathrm{d}y=-\int {P_{\mathrm{gr}}}_y[{\mathcal{L}}_{\xi }{N}_0(x,\cdot )](y)\hat{h}(y)\mathrm{d}y$ by Lemma lem:DRP, so ${\hat{A}}_1[{{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{\mathcal{L}}_{\xi }{P}_{\mathrm{gr}}\hat{h}]$ is the degree-$(1,1)$ monomial with kernel $(1\otimes P_{\mathrm{gr}})v$, $v:=-(1\otimes {\mathcal{L}}_{\xi })N_0$. The coefficients of ${\mathcal{L}}_{\xi }$ are supported in $\mathrm{supp}\xi \subset U$, so $v$ is compactly supported (in $K_f\times \mathrm{supp}\xi$), and $\mathrm{WF}(v)\subset {\Sigma }_2$ by Lemma lem:admission and Lemma lem:sumzero(b). Lemma lem:gslot gives ${\hat{A}}_1[{{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{\mathcal{L}}_{\xi }{P}_{\mathrm{gr}}\hat{h}]\in \mathfrak{J}$. Summing, the ${\kappa }^2$ variation of ${\hat{\Phi }}^{\mathrm{F}}(f)$ is $-{\hat{A}}_1[{\mathcal{L}}_{\xi }\hat{h}]+{\hat{A}}_1[{\mathcal{L}}_{\xi }\hat{h}]\equiv 0$.