Theorem: Actual degree-two second-stage maps

For every fixed $c\in \mathbb{R}$, there are unique algebraic bilinear maps on ${\mathcal{L}}_2\times \mathcal{B}$ and $\mathcal{B}\times {\mathcal{R}}_2$ with

All values are in ${\mathcal{P}}_{\Sigma }$.

Proof. Theorem thm:qinjective provides the unique algebraic presentation tensor of the intermediate argument. The other argument has a unique sharp test. The displayed values are linear in those presentations. Their target membership follows from classical closure, Theorem thm:treeadmission, and Corollary cor:cdesc.