Proposition: Missing sharp second multiplication

The parent assignment $U_c$, the admitted $Q_{f,g}$, and Theorem thm:regularassociation do not define either

where ${\star }_{\Sigma }$ is only a placeholder for a further construction. In particular, the trilinear numerical assignment

is not by itself a binary map on actual intermediate carrier elements.

Proof. The arguments of $U_c$ are observation tests. The arguments of the missing second-stage map would instead include actual functionals such as $Q_{f,g}$ and $B_f{W}_g$. The parent distribution gives no value on arbitrary such intermediate elements or their finite relations.

A binary factorization must prove descent from all test presentations to actual functionals, admission of all nonconstant outputs and their directional descendants, compatible overlap with old products, and the required coherence identities. None of these maps is defined by the scalar pairing (eq:Nc). Equality of two regularized scalar limits therefore cannot supply either missing composition.