Theorem: A negative plateau gap

An admitted supported packet exists with real compact $f_w,g,\xi$ and a real compact-data free scalar ${\varphi }_w$ such that

while

Both metric representatives satisfy $G^{(1)}(h)=0$.

Proof. Choose $t_{\ast }\in (0,b)$. On the auxiliary flat Cauchy slice $t=t_{\ast }$, prescribe

where ${\chi }_0$ is compact smooth and equals one near zero. Klein–Gordon evolution gives compact smooth initial data at $0$, and ${\partial }_1{\varphi }_w(t_{\ast },0)=1$. Choose a small relatively compact neighborhood $O$ of $(t_{\ast },0)$ on which ${\partial }_1{\varphi }_w\ge 1/2$. Let $f_w\ge 0$ have unit integral and support in $O$.

Take a compact smooth $g=1$ near $\mathrm{supp}{f}_w$. Choose ${\chi }_{\xi }$ compactly supported after $0$, equal to one near the entire $\mathrm{supp}g$, and put $\xi ={\chi }_{\xi }{\partial }_1$. All supports can be chosen inside ${{\mathcal{D}}_{\mathcal{R},b}}^{\circ }$ by enlarging the laboratory first if necessary.

Every derivative of $\xi$ vanishes near $\mathrm{supp}g$. Thus $gK\xi =0$, and both switched responses in (eq:gap) are zero. But $w[K\xi ]=\xi$, and $\xi ={\partial }_1$ on $\mathrm{supp}{f}_w$. This proves (eq:gap). Finally, $G^{(1)}K=0$.