Construction

The gauge fixing is the canonical shift by $\Psi =\bar{c}\cdot F(h)$ with $B$ integrated out. So the antifield of $h$ enters only through $\tilde{K}=K+\bar{c}\cdot F^{\prime }(h)$, and the Q08 ghost equation ($\rho =-2$) is this statement on the amplitudes. A local ghost-number $-1$ functional must therefore be built from $\tilde{K}$: $Y_F=\tilde{K}\cdot F(h,h)$ has the $\bar{c}$ partner $\rho T(K\cdot F)$, and the Q08 $P$ family is absorbed into it. $Y_G=c^{\ast }\cdot G(c,h)$ gives $X_{Khc}$ ($c^{\ast }\to D^T\tilde{K}$), its $\bar{c}$ partner, and the $c^{\ast }cc$ piece ($h\to s_{0}h$, antisymmetrised in the two ghosts). Q08 lacked that last piece. $\sigma =(S_0,\cdot )$ is applied with the same channel operators that build the rows, with the antibracket sign $-1$ between antifield-type and field-type pieces. Every generator gets one test: rows, Bose and ghost equation. No per-generator fit is made.