(a) Derived normalisations

For $Y=K\cdot F(h,h)$ the components of $\sigma Y=(S_0,Y)$ are produced by the same operators that build the rows ($X_{hhh}$ by the $\delta S_0/\delta h$ contraction, $X_{Khc}$ by $h\to s_{0}h(c)$). These operators commute, so ${\sigma }^2=0$ fixes the relative sign ${\lambda }_F=-1$; likewise ${\lambda }_P=-1$ for $Y=\bar{c}\cdot \Phi$, and ${\lambda }_G=+1$ for $Y=c^{\ast }\cdot G$. Check: at ${\lambda }_F=-1$, 139 of 204 generators lie in the kernel. All 65 failures carry a $K$- or $h$-trace monomial, which points to the $\eta$-trace convention in the probe contraction (open). The Q08 $G$ and $P$ constructions do not close at the derived values (0/33, 0/134); they were assembled by hand and are open. The Q08 fitted counts are superseded.