The cochain stays outside

The admitted cochain ${\mathcal{C}}_{\xi }=-{\mathfrak{B}}_{\xi }[D\gamma ,\varphi ]$ cancels the pairing algebraically on ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$ and is not an integrated completion in ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$ (Input statement input:XIIIAD). It is not inserted into $\mathcal{O}$: that would change the observable and void the reuse of $u_{\ast }$ (Section sec:B0). It is not inserted into $S_{\mathrm{ext}}$: the only ghost-dependent form it could take is ${\mathcal{C}}_{c/2}$, which is odd of ghost number one and cannot be a summand of an even, ghost-number-zero action; a parameter-indexed family is not a term of one functional. B4 ends Proved obstructed for both lifts.