The composite entry

Neither (eq:sV-shell) nor (eq:sO) is set to zero. The auxiliary remainder ${\mathfrak{B}}_{\xi }$ is even and has ghost number zero; it is not a term in these ghost-number-one formulas. A compact finite shift $K\xi$ is not $s_0$, and the finite two-jet term $\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$ is not an infinitesimal ghost variation.

We do not apply the differential to the binary datum and do not form the binary gauge-consistency defect. That separate question remains closed, including in the region where the tree datum is constructed.