Theorem: Free differential on the causal algebra

The field rule (eq:s) induces a ghost-number-one nilpotent graded derivation of ${\mathcal{A}}_0$.

Proof. Equation stability is Lemma lem:equations. It remains to check the commutation relations, not merely to postulate a Leibniz rule.

The only nontrivial mixed check is the initially zero relation between $\gamma$ and $\bar{c}$. The ghost block gives

The auxiliary block gives

with its tensor indices understood. Therefore

These are distributional identities tested with compact smearings; no pointwise field multiplication is involved.

All other transformed relations contain only zero cross-blocks, or $cc$ or $bb$ brackets, which vanish. The nonzero $\gamma \gamma$, $\varphi \varphi$, and $c\bar{c}$ brackets have constant right-hand sides and zero transformed left-hand sides. Hence $s_0$ preserves the relation ideal. Nilpotence descends from Lemma lem:nilpotence.