Theorem: All four degree-four prescriptions descend

The assignments

define unique algebraic bilinear maps on ${\mathcal{L}}_4\times \mathcal{B}$ and $\mathcal{B}\times {\mathcal{R}}_4$, with values in ${\mathcal{P}}_{\Sigma }$.

Proof. The two degree-four presentation kernels are zero. All four descent tests in Theorem thm:descent therefore hold. Every value is a previously proved admitted graph or sum of such graphs. No new admission claim is inferred from generator admission alone.