Proposition: Antifield-independent functionals

Let $F$ be a smooth functional of the fields alone, local or nonlocal, with compactly supported derivative kernels (for instance $\mathcal{O}$, or any element of ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$). Then on ${\mathfrak{F}}_W$

and $s^{2}F=0$ off shell, on all of $\mathfrak{F}$, by kinematic nilpotence alone. This nilpotence does not use the master equation.

Proof. For antifield-independent $F$ only the second term of (eq:antibracket) survives, giving (eq:sF); the pairing is finite because $s{\Phi }^A$ is compactly supported for compact ghosts. Since $s{\Phi }^A$ contains no antifields, $sF$ is again antifield-independent, and $s^{2}F={\sum }_A⟨{F^{\prime }}_A,s^2{\Phi }^A⟩+{\sum }_{A,B}⟨{F^{\prime \prime }}_{AB},s{\Phi }^A\otimes s{\Phi }^B⟩$; the first sum vanishes by XIV-A's kinematic nilpotence lemma, and the second vanishes by graded symmetry of $F^{\prime \prime }$ against the odd directions.