Proposition: Failure of the sufficient product test

At a strictly ordered null-collinear triple $z\prec y\prec x$, the ordinary sufficient wavefront product criterion for (eq:triangle) fails. This does not prove nonexistence of an extension, a cancellation in the complete sum, or a normalized retarded observable.

Proof. In slot order $(x,z,y)$, the three covectors can have forms

Their sum is zero precisely when $l=k=-p$. At the stated null triple, the two Feynman factors supply the opposing orientations at the earlier and later endpoints, while the retarded kernel has both signs of its null conormal. The displayed cancellation is therefore allowed. It violates a hypothesis of the usual product theorem Hormander; a failed sufficient hypothesis is not a nonexistence theorem.