F5: the flat-gauge residual

The class of (eq:flat-residual) modulo $\mathfrak{J}$ is not decided on this sheet. What is known:

1. $[{\mathcal{L}}_{\xi },{\mathcal{F}}^{(0)}]\hat{h}=-\frac{1}{4}{\mathcal{L}}_{\xi }(\mathcal{P}KC[\hat{h}])+\frac{1}{4}\mathcal{P}KC[{\mathcal{L}}_{\xi }\hat{h}]$. On a configuration with $C[h_0]=0$ the first term vanishes and ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}$ of the second is $-K{{\mathsf{E}}_0}^{\mathrm{R}}C[{\mathcal{L}}_{\xi }{h}_0]$, by ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}\mathcal{P}Kv=-4{{\mathsf{E}}_0}^{\mathrm{R}}Kv=-4K{{\mathsf{E}}_0}^{\mathrm{R}}v$: a pure-gauge configuration supported in $J^{+}(\mathrm{supp}\xi )$, not compactly supported. X-D's representative $d_1[K\xi ]=P_m(\xi \cdot \partial {{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f)$ is proved for $\xi \in {\mathfrak{g}}_U$ and does not apply to it without an extension off the plateau, which is not frozen.

2. On a configuration with $C[h_0]\ne 0$ the first term contributes a constraint term. X-E's witness pair $h=x^1{\delta }^2\otimes {\delta }^2$ has $C^{\mu }[h]=-\frac{1}{2}{{\delta }^{\mu }}_1$ and evaluates ${\hat{A}}_1[{\mathcal{L}}_{\xi }\hat{h}]$, a different functional; it does not decide (eq:flat-residual).

3. If the class is nonzero, the ${\kappa }^2$ background identity is gauge-fixing dependent at this order in a non-covariant gauge, which is the expected behaviour of a background-field identity and the reason the background gauge is frozen. If it is zero, the flat gauge would also reach ending (1). Either way Theorem thm:main is unaffected.