Why the one-line proof stops

There is now one uncontracted field slot in each entry and two contracted slots. In a proposed forbidden product cancellation, the uncontracted covectors may vanish while the two contracted covectors in an entry are opposite and nonzero. Their sum is zero, so the $Z_3$ bounds do not force them individually to vanish.

This is the exact place where Lemma lem:one-line no longer applies. It is not evidence that the distribution cannot exist; it identifies a hypothesis needing additional analysis or extension.

For a nonlocal composite, the contracted observable variables need not coincide with the observation point. Consequently extension is not automatically reduced to one diagonal between two local densities. Standard local time-ordered constructions specify maps on local entries and their subsequently defined multilocal domain; they do not supply the present joint nonlocal extension merely from star-product admission FR (Sections 6–7).