Proposition: The Koszul–Tate sector resolves 's shell, not

At antifield degree zero, $s_{\Psi }{\Phi }_A^{\ast }={\delta }^{\mathrm{R}}{S}_{\Psi }/\delta {\Phi }^A{|}_{{\Phi }^{\ast }=0}$ is the Euler derivative of the gauge-fixed interacting action (eq:Sgf). Its ${\kappa }^0$ coefficient is the system

which is XII-A's complete shell ${\mathcal{C}}_{\mathrm{sol}}$. Its ${\kappa }^1$ coefficient is not zero on ${\mathcal{C}}_{\mathrm{sol}}$: at the configuration $\gamma =0$, $b=0$, $\varphi ={\varphi }_m$, $c=\bar{c}=0$, which lies in ${\mathcal{C}}_{\mathrm{sol}}$,

Hence the Koszul–Tate differential of the relative complex does not vanish on ${\mathcal{C}}_{\mathrm{sol}}$; it resolves the equations of $S_{\Psi }$, which differ from the free shell at order $\kappa$. The ghost equation of $S_{\Psi }$, $\square c+\kappa D({\mathcal{L}}_c\gamma )=0$, is likewise not $\square c=0$.

Proof. The ${\kappa }^0$ system is read from (eq:Sgf): the cubic-only modification does not change the free Hessian (XII-C), the auxiliary and ghost terms are XI-B's, and XI-B proves $A\gamma +2D^{\ast }b=0$, $b=D\gamma$ is equivalent to $-\square \mathcal{P}\gamma =0$, $b=D\gamma$. For the witness, XII-C's exact metric Euler derivative is $e_{\mathrm{D}}=\frac{2}{\kappa }\sqrt{-q}G-\frac{\kappa }{2}g\sqrt{-q}{T}_q+\kappa {Z_{\sigma }}_{\epsilon }$; at $\gamma =0$ the Einstein tensor and ${Z_{\sigma }}_{\epsilon }$ vanish and $T_q=T_0$, so $e_{\mathrm{D}}=-\frac{\kappa }{2}g{T}_0[{\varphi }_m]$ exactly, and ${T_0}^{00}[{\varphi }_m]=\frac{1}{2}$. The gauge-fixing contribution $2D^{\ast }b$ vanishes at $b=0$ and the ghost contribution at $c=\bar{c}=0$.