Proposition: Preparation memory despite a common plateau

There are two admitted switches sharing an observation plateau, the same metric coefficients, the same free scalar, and the same harmonic prescription, for which $C_1[g_1]\ne C_1[g_2]$. The example can be chosen in the massless coupling-jet class.

Proof. Take $m=0$, ${\varphi }_0=t$, and $h=2\eta$. Then $A(h)=2\eta$, $\mathcal{C}[h]=0$, and $w=0$. Choose a compact spatial cutoff equal to one near the spatial projection of ${\mathcal{D}}_f$. On that causal region let $g_i(t,x)=q_i(t)$, with $q_i$ compactly supported after $0$ and equal to one near all observation times.

Since

the zero-data solution is

Choose $q_2=q_1+\epsilon p$, where $p\ge 0$ is a nonzero smooth pulse strictly before the observation plateau and supported where $q_1=0$. Choose $\epsilon >0$ small enough to keep $0\le q_2\le 1$. For $\int f=1$,

To impose the metric equations, choose $k$ from (eq:sourced-example) for ${\varphi }_0=t$. The constant $h$ has vanishing linear and quadratic Einstein coefficients, so the same $k$ works for both switches.