Two actual separating probes

Choose real compact $f,h$ such that

Such choices are explicit in the following sense. Take nonzero nonnegative compact bumps whose Fourier transforms are nonzero near zero. Then

For $K$, the pair-creation kernel is $-p\cdot q\hat{h}(p+q)$, and

There are arbitrarily small noncollinear positive-energy momenta for which both factors are nonzero. Both vectors lie in ${\mathcal{D}}_E$.

For $S=\lambda T+cI$, define the translation residual

where $q^S$ uses $S^{0\nu }{X}_{\nu }$, and its collar contains $\mathrm{supp}f$. Define

These are genuine field matrix elements, not assumed abstract coordinate functionals.

By Theorem thm:classification,