What XI-B actually supplies

XI-B constructs the algebra ${\mathcal{A}}_0$ generated by compact interior linear smearings and their finite products, with the free equations and graded causal commutation relations imposed XIB. Its reduced metric operator and inverse are

The scalar equation inverse is ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$, whereas its action-Hessian inverse is $-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$.

The copied free differential is

It is nilpotent with $b$ retained and preserves XI-B's free equations and causal relations. These facts do not admit a new composite.

Write ${\mathcal{C}}_{\mathrm{sol}}$ for smooth multiplet configurations satisfying

These equations are equivalent to XI-B's metric–auxiliary equations after using $b=D\gamma$. Ghost Cauchy data are unrestricted. They are not required to vanish at $\Sigma$.