Lemma: Off-shell nilpotence

The derivation (eq:s) satisfies ${s_0}^2=0$ on the independent fields and on their polynomial differential algebra.

Proof. The second variation of $\gamma$ is $\mathcal{R}(s_{0}c)=0$. The second variation of $\bar{c}$ is $i{s}_{0}b=0$. The remaining fields already have zero first variation. The square of an odd derivation is an even derivation; hence vanishing on the generators proves vanishing on their polynomial algebra.