Theorem: The bare insertion is not generally free-closed

For every $m\ge 0$ and every $0<\epsilon <T$, there are admissible nonnegative observation tests $f$ and a smooth free multiplet configuration for which

In particular, this variation is not generally a functional vanishing on all free configurations.

Proof. Work over the Grassmann algebra $\mathbb{C}[\vartheta ]/({\vartheta }^2)$ with $\vartheta$ odd and nonzero. Set

Every equation in (eq:freeshell) holds. The ghost is an affine wave solution, and no ghost initial-data restriction in XI-B excludes it.

Here

so

The spatially constant zero-data wave response is therefore

The spatial components vanish. Ramp symmetry gives

It follows that

where $\cos (mt)=1$ when $m=0$.

Choose $t_{\ast }\in (\epsilon ,T)$ with $\cos (m{t}_{\ast })\ne 0$, and a small time interval on which it has a fixed nonzero sign. Choose $f\ge 0$, nonzero, compactly supported in that interval and in a spatial ball. Since $S_{\epsilon }>0$ there, the ordinary coefficient of $\vartheta$ in (eq:witness) is nonzero.

The switch and plateau may be chosen to contain this observation region while satisfying the frozen support requirements. They do not enter (eq:witness).