Copied ramp premises

For a scalar jet $\phi ={\phi }_0+\kappa {\phi }_1+{\kappa }^2{\phi }_2$, put

The copied expansion is

Its smeared coefficients depend on backgrounds only in neighborhoods of ${\mathcal{D}}_f$, and continuously at fixed $\epsilon$. Setting ${\phi }_1={\phi }_2=0$ defines the bare dressing used here. At $g=0$ this dressing survives: the coordinate prescription is unchanged.

Write

For the formal flow of $\kappa \xi +{\kappa }^2\zeta$, $\xi ,\zeta \in {\mathfrak{g}}_U$,

In particular ${\mathsf{X}}_1[K\xi ]=\xi$ on ${\mathfrak{g}}_U$.

For the excluded edge parameter $\xi =\alpha (t)\beta (\mathbf{x}){\partial }_1$, where $\alpha \ge 0$, $\alpha \not\equiv 0$, $\mathrm{supp}\alpha \subset (0,\epsilon )$, and $\int \beta {\mathrm{d}}^{3}x=1$, X-A proves

The coordinate identity fails in general outside the frozen algebra. This witness is not a defect on ${\mathfrak{g}}_U$.

The first switched scalar vertex is

Its positive Euler coefficient, denoted ${\mathsf{V}}_{1,g}$ in X-A, is here

The second coefficient is

For a free smooth solution $u$, the retarded scalar coefficients are

X-A proves simultaneous-pullback covariance off shell, and background covariance of the on-shell relational observable at fixed free input, through second classical order on ${\mathcal{G}}_U$. This is classical input, not a quantum product theorem.