Proposition: Free causal BRST recovery

The free equation ideal

is $s_0$-stable. The normalized causal relations

are also stable. Consequently $s_0$ induces the nilpotent free causal BRST derivation of X-G-ST.

Proof. Solving $b=(\mathsf{C}r-j)/2$ in the sourced bosonic equations reduces them to

which proves (eq:Grb). The equation variations are $AKc=0$, $\square c$, and $\square b$. The last follows from the divergence of $Ar+{\mathsf{C}}^{\ast }b=0$; the scalar and ghost equations are otherwise inert.

For the brackets, the only cancellation between nonzero blocks is the variation of the initially zero $r$–$\bar{c}$ relation:

The remaining generator pairs use zero cross blocks or ${\Delta }_{bb}=0$. The odd–odd blocks have the required graded symmetry because ${{\delta }_0}^T=-{\delta }_0$. These generator checks prove compatibility.