Theorem: Both selected associations

For every fixed real common parent $c$, both selected associations are admissible in Definition def:product and satisfy

Proof. The first-stage output has degree-four component $B_f{W}_g$ and $\hslash$ times degree-two component $Q_{f,g}$. Their tensors with $B_{f^{\prime }}$ lie in the declared left second-stage domains. The reverse assertion is identical.

Insert the coefficient table and collect powers of $\hslash$. The order-one sum contains exactly the three one-edge outputs. The order-two sum contains exactly the three two-edge outputs. The scalar term comes from $\hslash Q$ followed by its order-two scalar coefficient, hence has order ${\hslash }^3$, not ${\hslash }^2$. Both sides use the same $U_c$. All maps involved have already descended and glued.