D2: support of the constructed free outputs

For a smooth compact source $s$, the defining retarded Green property gives

this is the usual Green-operator support convention FR. A specified smooth comparison

is therefore legitimate when its source is smooth and compact and its chosen inverse has the stated Cauchy or retarded property. It remains a smooth classical composition.

Zero-Cauchy inversion agrees with the corresponding retarded solution only under the appropriate zero past-data hypothesis. It is not a Feynman inverse, and arbitrary homogeneous data cannot be discarded.