Proposition: Classical master equation of S inv through O(κ 2)

Under the declared conventions every component of $(S,S)$ through $O({\kappa }^2)$ is a total divergence, in invariant and in $\Psi$-transformed variables, with all mixed cancellations retained: required_master_residuals=[\,].

Proof. Engine parts: exact Euler-operator divergence test. Gravitational af0: exact polynomial identity $E_{c^{\mu }}(W_n)\equiv 0$ on the exports. Structurally, $\sqrt{-g}R$ is a scalar density, so ${\delta }_{\xi }(\sqrt{-g}R)={\partial }_{\mu }({\xi }^{\mu }\sqrt{-g}R)$; $L_{\Gamma \Gamma }=\sqrt{-g}R+{\partial }_{\mu }{B}^{\mu }$ (C3, corrected orientation), and $s{\partial }_{\mu }{B}^{\mu }={\partial }_{\mu }s{B}^{\mu }$. af1 rows are $-2\sum {\Phi }^{\ast }{s}^2\Phi$ and vanish by the derived nilpotency.