Proposition: A normalized negative witness

For every $m\ge 0$, an admitted packet exists with $f\ge 0$, $\int f=1$, and a compact larger-support generator $\xi$, constant near $\mathrm{supp}g$, such that

but

The gap is independent of the baseline $h$.

Proof. Choose an interior event $(t_{\ast },0)$ and free scalar data at $t=t_{\ast }$ equal to $x^1$ near zero, with zero time derivative, and smoothly compactly supported in space. The resulting free solution satisfies ${\partial }_1{\varphi }_0(t_{\ast },0)=1$. Choose a small plateau neighborhood $U$ where ${\partial }_1{\varphi }_0\ge 1/2$, and a unit-integral nonnegative test $f$ there. Choose $g=1$ near $\overline{U}$.

A compact smooth ${\chi }_{\xi }$, supported after $0$, can be chosen equal to one near the entire $\mathrm{supp}g$. Set $\xi ={\chi }_{\xi }{\partial }_1$. All derivatives of $\xi$ vanish near $\mathrm{supp}g$, so $gK\xi =0$. The scalar response has zero variation, while the coordinate response changes by $\xi$. Therefore

The cutoff of $\xi$ occurs outside the plateau containing the observation. Its support is not contained in $U$.