Signed free multiplet

For a symmetric tensor $r$, put

XI-B supplies

Here ${{\mathsf{E}}_0}^{\mathrm{R}}$ is the zero-initial-data inverse of $\square$. Its advanced counterpart has zero final data. Their causal support and smooth Cauchy properties are those used in XI-B XIB,BGP.

The coefficient of $\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$ is $-1$, not $-1/2$. The scalar equation inverse is ${{\mathsf{E}}_{\mathsf{m}}}^{\pm }$; its action-Hessian inverse is $-{{\mathsf{E}}_{\mathsf{m}}}^{\pm }$. Thus

The auxiliary field is retained, with

The classical XI-A parameter $K\xi$ corresponds to $c=2\xi$. Eliminating $b$ gives only on-shell nilpotence of the reduced antighost rule. Ghosts and antifield source slots are spectators in the contraction below.

XI-B also gives

Consequently the plateau algebra is not a nontrivial residual action on free de Donder solutions. Its classical background action and the free graded differential are different objects.