Theorem: Explicit convolution-covariance defect

There are admissible plateau parameters and free massless scalar data for which the first-order variation of ${{\mathcal{O}}_{f,ϵ}}^{\mathrm{conv}}$, under the unchanged pullback action on the original fields, is strictly positive.

In a suitable compact dependence neighbourhood, one can choose

with a compact realization of $\xi \in {\mathfrak{g}}_U$, and obtain

Here $f\ge 0$, $f\not\equiv 0$, and the sampled region is above the initial surface.

Proof. Choose the compact parameter support strictly above $\Sigma$, and $ϵ$ small enough that its smearing still vanishes near $\Sigma$. Constant-coefficient differentiation commutes with convolution. Since $CK\xi =\square \xi$,

The first-order scalar pullback is $\delta \varphi =\kappa {\xi }^{\mu }{\partial }_{\mu }\varphi$. The variation therefore is

Take a compact cutoff equal to one on a neighbourhood of the $ϵ$-sampling region of $\mathrm{supp}f$, and set $\xi =t{\partial }_t$ there. The printed scalar is globally smooth and satisfies $\square \varphi =0$. Evenness and normalization give, on that neighbourhood,

Substitution proves (eq:convdefect). A smooth nonnegative radial mollifier of integral one has $m_{00}>0$.