Proposition: Pure-gauge jet through second order

Let $\xi \in {\mathfrak{g}}_U$, $\zeta =0$, and take the pure-gauge jet of the flat configuration,

Then

and ${\mathsf{X}}_1,{\mathsf{X}}_2$ are independent of $\epsilon$ for every $\epsilon$ with $\overline{U}\subset \{t>\epsilon \}$.

Proof. The jet is $F_{\kappa }^{\ast }(\eta ,{\varphi }_0)$ with $\zeta =0$. Since ${\mathsf{X}}_{\eta }=x$, Theorem thm:coordinate gives ${\mathsf{X}}_q=F_{\kappa }^{\ast }x=x+\kappa \xi +{\kappa }^2\frac{1}{2}{\xi }^{\nu }{\partial }_{\nu }\xi$. The first coordinate equation was verified directly in Proposition prop:preflight. For the second, the identities (eq:Vid)–(eq:Wid) at $q=\eta$ read ${\mathcal{V}}_{\epsilon }[K\xi ]=D\square -\square X$ and ${\mathcal{W}}_{\epsilon }[K\xi ,\frac{1}{2}{\mathcal{L}}_{\xi }K\xi ]+{\mathcal{V}}_{\epsilon }[K\xi ]X+\frac{1}{2}\square X^2=\frac{1}{2}{D}^2\square$, so that

with zero data. Then ${\Phi }^{(1)}=X_{\xi }{\varphi }_0-{\xi }^{\mu }{\partial }_{\mu }{\varphi }_0=0$, and

because ${X_{\xi }}^2{\varphi }_0={\xi }^{\nu }{\partial }_{\nu }{\xi }^{\mu }{\partial }_{\mu }{\varphi }_0+{\xi }^{\mu }{\xi }^{\nu }{\partial }_{\mu }{\partial }_{\nu }{\varphi }_0$. Regulator independence holds because ${\sigma }_{\epsilon }=1$ and ${\sigma }_{\epsilon }^{\prime }=0$ on the supports of all sources.