Integration by parts can introduce a second switch derivative

The absence of ${\sigma }^{\prime \prime }$ in (eq:Dsymbol) is compatible with its appearance in (eq:ibp). The latter derivative arose from transferring a derivative in an integral, not from changing the source symbol.

This calculation assumes a smooth scalar amplitude. It does not prove that an amplitude obtained from an unidentified singular retarded kernel is smooth, and it does not evaluate the completed gravitational insertion.