Theorem: No exact source-preserving variational upgrade

There is no upgrade satisfying Definition def:upgrade on the unrestricted smooth background class of the frozen X-F source law.

The contradiction persists if off-shell representatives are changed by terms that vanish on every free configuration. It is not removed by algebraic auxiliary elimination, ghost loops, or counterterms beginning at positive formal powers of $\hslash$.

Proof. Linearize the claimed reduced interaction at zero fluctuation, zero scalar, and zero ghosts in the compact background $w$ of (eq:witnessdata). Its mixed Hessian must satisfy Lemma lem:reciprocity. Proposition prop:Dsigned contradicts that necessary condition.

A change annihilating every free configuration does not change $N_{w}u,N_{w}v$ on the detecting pair, so it does not change their reciprocity defect.

Eliminating a nondegenerate algebraic auxiliary field from an action gives another action. At quadratic order its reduced Hessian is the symmetric Schur complement

It still satisfies the reciprocity condition. The explicit witness therefore survives auxiliary elimination.

Finally, at zero ghosts the classical bosonic Hessian used in the contradiction is unchanged by ghost loops or positive-quantum-order terms. For arbitrary coefficients $a_n$,

because its constant coefficient is strictly positive.