Proposition

Let $S=S_0+\kappa S_1+{\kappa }^2{S}_2$. Under compact variations and total-derivative equivalence, the retained classical master through order ${\kappa }^2$ is

Indexed evaluation gives $(S_0,S_0)$ Constructed, $(S_0,S_1)$ Constructed by inheritance of XVI-N through order $\kappa$, and $2(S_0,S_2)+(S_1,S_1)$ Partial: the scalar sector is Constructed by finite-$\kappa$ invariance of $\int {\mathcal{L}}_q$ plus Taylor expansion, while the gravitational component stays Partial. The full ten-component ${\ell }_4$ generating polynomial, the explicit gravitational expansion at order ${\kappa }^2$, and the complete ghost and antifield order ${\kappa }^2$ on $D[h]$ are Unresolved.