Lemma: Kinematic nilpotence

$s^2=0$ on the independent fields and on their graded polynomial differential algebra.

Proof. Put $w^{\mu }=c^{\nu }{\partial }_{\nu }{c}^{\mu }$. The graded identity ${\mathcal{L}}_c{\mathcal{L}}_c={\mathcal{L}}_w$ holds: for a scalar $u$, $c^{\nu }{\partial }_{\nu }(c^{\mu }{\partial }_{\mu }u)=w^{\mu }{\partial }_{\mu }u+c^{\nu }{c}^{\mu }{\partial }_{\nu }{\partial }_{\mu }u$ and the last term vanishes by anticommutativity; for covectors one extra term is treated identically, and the identity extends to tensors by the graded Leibniz rule (X-G's kinematic lemma, normalization independent). Then, since $s$ is an odd derivation and ${\mathcal{L}}_{c}q$ is linear in $c$,

hence $s^2\gamma =0$. Likewise $s^2\varphi =\frac{{\kappa }^2}{4}(w^{\mu }{\partial }_{\mu }\varphi -{\mathcal{L}}_c{\mathcal{L}}_c\varphi )=0$. For the ghost,

the first two terms cancelling after renaming and the third vanishing by antisymmetry. Finally $s^2\bar{c}=isb=0$ and $s^{2}b=0$. The square of an odd derivation is an even derivation, so vanishing on generators suffices.