Remark: Green subtractions are prescription dependent

A chartwise subtraction of the retarded remainder in (eq:defect) changes the specified observable. It does not automatically define a tensorial subtraction on the full cone. Already

for a general compact $\xi$, so the fixed flat inverse does not commute with arbitrary spacetime pullbacks. A global replacement needs its own covariance and initial-data theorem. This is the relevant warning behind the fixed-data subtraction comparison in V-B CheungGradeS.