Assumption: Additional supplied microlocal statements

The supplied XII-D admission excerpt applies to the kernel in Input input:kernel, with the same insertion, response normalization, preparation data, scalar slot, and source polarization. It gives

The root statement reproduced in the supplied XVI-I gives

The root represents a compact smooth root map:

in the distributional kernel sense.

For the separate trisolution consequence only, the inherited XI-G transport lemma is used in its stated compact-kernel quantifier: it transports component covectors in Minkowski frames, may move their base points, and restricts each surviving entry to a null or zero covector.