A conditional cocycle test

Fix the ordinary vector-field bracket convention

For a proposed action on functionals, choose the representation convention

If another pullback convention is used, its corresponding sign must be changed consistently.

A necessary test for a correction $F$ satisfying ${\delta }_{\xi }F=-{\mathfrak{B}}_{\xi }$ is

Indeed, under (eq:representation), the difference of the first two terms is $-[{\delta }_{\xi },{\delta }_{\zeta }]F=-{\delta }_{[\xi ,\zeta ]}F$.

This test is well posed on the shell quotient only after the candidate $\delta$ preserves the equation ideal. That action has not been supplied here. In particular, the constant off-shell metric-shift derivatives

commute with each other. They do not furnish the nonabelian representation (eq:representation), and they generally do not preserve ${\mathcal{N}}_{\mathrm{sol}}$. They therefore cannot be inserted into (eq:cocycle) as if they were a declared action of the vector-field Lie algebra on the shell.

For a specified action and allowed correction class, failure of (eq:cocycle) excludes a correction of the stated affine type. Success supplies only a necessary compatibility condition, not solvability. Without the action, interpreting a formal failure as a universal completion obstruction would be invalid.