The missing second scalar response

The copied positive Euler coefficients are ${\mathsf{L}}_{1,g},{\mathsf{L}}_{2,g}$, and for a smooth free input $u$,

The switched Euler operator ${\mathsf{E}}_g[q]=P_m+\kappa {\mathsf{L}}_{1,g}+{\kappa }^2{\mathsf{L}}_{2,g}$ is covariant under ${\mathcal{G}}_U$. The unswitched curved density operator satisfies naturality under every compactly supported diffeomorphism. These are copied premises from X-A.

X-B-v3 constructs ${\hat{A}}_1=\hat{\varphi }(c_1-{\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f)$ and proves its invariance. Its truncation is

The mixed term has the fixed minus insertion sign:

Its copied finite transformation on ${\mathfrak{g}}_U$ is

This is not an invariance statement.

The classical second coefficient has the additional $\int f{\rho }_{2,g}$. On the complete pure-gauge jet $h=K\xi ,k=\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$, the incomplete coefficient in (eq:old-truncation) is $-\frac{1}{2}\int f{X}_{\xi }^2\hat{\varphi }{\mathrm{d}}^{4}x$. The missing classical response is $\frac{1}{2}{X_{\xi }}^{2}u$. Constructing that response is the present target.