Proposition: Two different source comparisons

Assume the imported layer property $\mathrm{supp}{\mathcal{D}}_{\sigma }[\gamma ]\subset {\Lambda }_{\epsilon }$, with the correction vanishing at the plateaus as in the supplied XII-C/XII-D preparation. Then, for smooth $\gamma$,

Comparison with the unramped source gives

It need not be supported in ${\Lambda }_{\epsilon }$.

Define the comparison function

Then the locally integrable tensor

has distributional support in ${\Lambda }_{\epsilon }$.

Proof. For $t\le 0$, $\sigma =0$ and the imported correction vanishes. For $t\ge \epsilon$, $\sigma =1$ and the correction vanishes. This proves (eq:pastzero) and (eq:plateau-source), and substitution proves (eq:unramped-comparison).

For the witness pair, the unramped $00$ source equals $-2{\lambda }^2{e}^{2\mathrm{i}\lambda t}$ even at negative times. Thus (eq:unramped-comparison) is genuinely not a layer-supported comparison on this family.

The coefficient $\sigma -H$ vanishes for $t<0$ and $t>\epsilon$. Since the evaluated $J_3$ is smooth, $(\sigma -H)J_3$ is locally integrable and supported in the closed layer. Adding the layer-supported correction proves (eq:sharp-comparison).