Corollary: An explicit nonempty profile class

Let $\rho \ge 0$, $\int \rho =1$, be compactly supported in

Then $M_2(\mathbf{n})>0$ for every $\mathbf{n}$. Such profiles exist.

For these profiles, any isolated positive-placement neighborhood with $f>0$ at the response point and $g>0$ at the observation point admits the complete-family detecting test. In particular one may impose an output plateau $g=1$.

Proof. In frame coordinates,

Thus the boundary value in $M_2$ is the ordinary positive function $S^{-2}$, and its normalized nonnegative-profile integral is strictly positive. A small nonnegative normalized bump centered at $H_1=0,H_2=(T,0)$, with $T>2s_0$, supplies an example. The test integral is then strictly negative for nonnegative nonzero $\psi$.