Theorem: Scalar kernel replacement

Let $B=A_{F,\beta }+m^2$, let $U$ be angularly small, let $f\in {C_c}^{\infty }(U;L_{\beta })$, and let $\mathcal{O}\subset U$ be open. Transport $f$ in the local parallel frame to the completion and extend it by zero. For $\sigma \in \{+1,-1\}$, assume

Then

For $E_B$, it suffices to assume

A sufficient common condition for both signs and for $E_B$ is that the time extent of $\mathcal{O}\cup \mathrm{supp}f$ be less than $L_U$. The same replacement holds for the ${\mathsf{E}}_0$ extension of Proposition prop:E0.

Proof. In the parallel chart frame the direct $k=0$ term is the ordinary flat kernel with unit weight. Every other retarded geometric term is supported at or after $T_{\mathrm{arr}}$, and the retarded diffracted term is supported at or after ${\rho }_0$. The global distribution prescription in Theorem thm:closed preserves these support statements at image boundaries. Hence the difference kernel vanishes on the open pair region in (eq:directional-window) for $\sigma =+1$. Time reflection proves the advanced statement with $\sigma =-1$. Integration against $f$ gives (eq:transfer).

The absolute-time condition handles both signs. Lemma lem:window gives the uniform sufficient condition. The ${\mathsf{E}}_0$ correction begins at ${\rho }_0$, so it also vanishes in the same window.