Input statement: X-A: scalar completion and its covariance

For the formal metric $q=\eta +\kappa H+{\kappa }^{2}k$ with smooth symmetric $H,k$, a free scalar input $P_m{\varphi }_0=0$, and the regulated coordinates and switched scalar jet of X-A, the smeared relational scalar has second coefficient ${{\mathcal{A}}_2}^{\mathrm{sc}}[f,g](H,k;{\varphi }_0)$, displayed in (eq:Asc) below. For every $\xi \in {\mathfrak{g}}_U$ (X-A's group element $\exp (\kappa {\mathcal{L}}_{\xi })$),

Also ${\mathcal{A}}_1[f,g](K\zeta ;{\varphi }_0)=0$ for $\zeta \in {\mathfrak{g}}_U$, and the pure-gauge two-jet $(K\xi ,\frac{1}{2}{\mathcal{L}}_{\xi }K\xi )$ of the flat configuration has vanishing second coefficient.