The next product obligation

A subsequent factorization theorem must construct binary maps on named actual-functional domains, verify every resulting coefficient and directional descendant in ${\mathcal{P}}_{\Sigma }$, prove descent through all presentation relations and agreement on old-product overlaps, and establish the coefficientwise coherence equations for the same fixed parent $U_c$. Naming ${\mathcal{N}}_c$ is not that theorem.

Even a separately constructed partial free product would not automatically give time-ordered products, retarded products, a switched interacting differential, or a quantum Ward identity FR,CheungSuccessors. This sheet makes none of those transfers.