Theorem: Free causal BRST differential

The differential (eq:s0) preserves the free equations and causal relations. It induces a nilpotent graded derivation of ${\mathcal{A}}_0$.

Proof. The variation of the first metric equation is $AKc=0$. The variation of $Ch-2b$ is $\square c$. The antighost equation varies into $\square b$.

The latter is a consequence of the bosonic equations: the linearized Bianchi identity and

give $\square b=0$. The remaining equations are stable.

For the causal relations, the nontrivial cancellation is the initially zero $h$–$\bar{c}$ relation:

The reversed relation follows by graded symmetry. All other variations use zero cross-blocks or the zero $cc$ and $bb$ blocks. Nilpotence descends from Lemma lem:s0nil.