Theorem: Degree-two intermediate coefficient maps

For each fixed real $c$, the following prescriptions define unique bilinear maps on actual functionals:

Their domains are ${\mathcal{L}}_2\times \mathcal{B}$ and $\mathcal{B}\times {\mathcal{R}}_2$, respectively, and their targets are ${\mathcal{P}}_{\Sigma }$.

Proof. Theorem thm:qinjective gives a unique algebraic presentation tensor for each intermediate argument. The sharp generating map gives a unique test for the other argument. The displayed graph sums are linear in the presentation tensor and in the sharp test, so they descend to bilinear maps. In particular, the coefficient sum, not merely its individual terms, is well defined.

Theorem thm:treeadmission places the positive-order nonconstant outputs in ${\mathcal{P}}_{\Sigma }$. Corollary cor:cdesc handles the scalar coefficient. Classical closure handles order zero.