D6: scalar variation, coupling order, and no double count

The field $\varphi$ is allowed to move in the class definition. Its contribution to the full target must retain the scalar-sourced slot. Explicitly,

where the inherited derivative is

This is differentiation of the existing slot, not a new interaction entry. Its zero metric variation does not authorize deleting either term in (eq:scalar-variation).

For a shell-descending member of the frozen class all three terms vanish on the shell because ${Y_{\xi }}^{\varphi }[u]=0$. They were not omitted before applying that theorem.

The candidate cochain (eq:C-definition) has degree two, whereas $\mathcal{O}$ has degree three. It must not be identified with $⟨{{\mathcal{O}}^{\prime }}_{\gamma },r_{\xi }⟩$ merely because $r_{\xi }$ has a plausible retarded support. For instance, a field-linear $r_{\xi }$ makes that slot variation degree three. Homogeneous field scaling distinguishes it from the bilinear target.

The order of a prospective compensating effect here is ${\kappa }^2{\hslash }^0$. No term is silently added at the first coefficient. If a future proposal acts on the lower term $\kappa {\mathcal{A}}_1$, it must print the additional coupling power producing a ${\kappa }^2$ effect and supply its own transformation law.

The filled gravitational slot remains $r_{\mathrm{D}}=-u_{\mathrm{D}}$. It is neither modified nor counted a second time. The admitted parameter-dependent cochain is not inserted into that slot.

Finally, a finite transformation contains $\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$ in its inherited two-jet law. A linear parameter substitution is not a construction of that finite law. The present obstruction already excludes a necessary infinitesimal integrated interface in the class; it therefore requires no assertion about an unconstructed finite action.