Remark: The exact quantifier

Failure of the sufficient criterion is not a proof of nonexistence of a distributional extension. Nor does it prove that every specially chosen differential numerator has a nonzero obstruction after all cancellations. No such universal noncancellation claim is needed.

The phrase “no second pair” means no second quadratic pair completing the asserted time-ordered programme by these tools. It does not mean that every other elementary pair, such as a pair of smooth linear fields, is undefined.