Proposition: Failure of a sufficient product test

Suppose a wavefront condition $P$ is sufficient for the existence of a specified ordinary product. Failure of $P$ alone does not prove nonexistence of every product or generalized composition.

Proof. Sufficiency has the form $P\Rightarrow E$, where $E$ is the existence conclusion under the theorem's hypotheses. It does not imply $\neg P\Rightarrow \neg E$. A converse exclusion theorem would be an additional result.