Proposition: Variational reciprocity is a separate test

If $J$ is the variational derivative of a twice differentiable classical functional on the declared even variation domain, then

for every admitted pair $u,v$. Neither this identity nor its failure is a ghost-number-one master equation.

Proof. The two terms are the mixed second variations of the same functional. Their equality is symmetry of the second derivative on even variations.

The ghost-number and parity assignments follow from the arguments in (eq:reciprocity); they differ from those of ${\mathcal{R}}_{\mathrm{cl}}$.