Lemma: Signed cotangent lift

One has $(H_s,{\Phi }^A)=R^A$ for every field. Moreover,

as a local variational class.

Proof. Put ${ϵ}_A=|{\Phi }^A|$. The biderivation rule and the canonical coordinate bracket give

The prefactor in (eq:Hs) therefore gives $(H_s,{\Phi }^A)=R^A$, for even and odd species. In particular the minus signs in the ghost and antighost terms of (eq:Hsexplicit) are required.

The bracket square of an antifield-linear Hamiltonian is antifield linear. Its action on every field is the square of the induced evolutionary vector field, by the graded Jacobi identity. That square is zero by Theorem thm:kinematic. Consequently the bracket square is zero modulo a divergence: a variationally trivial antifield-linear functional has no nonzero antifield-independent constant.