Proposition: Flat-gauge residual

Replace Definition def:bgf by X-E's flat gauge ${S^{(0)}}_{\mathrm{gf}}[u]$ at every background, so that ${\mathcal{F}}^{(1)}\equiv 0$, ${{\mathcal{Q}}^{(1)}}_{\mathrm{flat}}[h_{\mathrm{cl}}]={\mathcal{G}}_2(h_{\mathrm{cl}},\cdot )$, and define ${{\mathsf{T}}_{\mathrm{mix}}}^{\mathrm{flat}}$, ${\hat{\Phi }}_{\mathrm{flat}}^{\mathrm{F}}$ by (eq:trees), (eq:all-tree) with this ${\mathcal{Q}}^{(1)}$. Under the frozen transformation of Theorem thm:main,

Proof. As in Theorem thm:main, with Proposition prop:key's flat-gauge statement: $-{{\mathcal{Q}}^{(1)}}_{\mathrm{flat}}[K\xi ]\hat{h}=-[{\mathcal{L}}_{\xi },P_{\mathrm{gr}}]\hat{h}+[{\mathcal{L}}_{\xi },{\mathcal{F}}^{(0)}]\hat{h}$. The first term gives ${\mathcal{L}}_{\xi }\hat{h}$ modulo $\mathfrak{J}$ after ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}$ and ${\hat{A}}_1$, cancelling (eq:xe-defect); the second is the residual. It is admitted: $[{\mathcal{L}}_{\xi },{\mathcal{F}}^{(0)}]\hat{h}$ is compactly supported in $\mathrm{supp}\xi$, and the kernel is $-(1\otimes [{\mathcal{L}}_{\xi },{\mathcal{F}}^{(0)}{]}^{†})N_0$, in ${\Sigma }_2$.