Theorem: Descended degree-four coefficient maps

The four prescriptions (eq:lambdaL)–(eq:lambdaR) define unique algebraic bilinear maps

with the prescribed values.

Proof. Theorem thm:pinjective makes both relation kernels zero. Thus all four annihilation tests (eq:annihilation) hold.

More explicitly, an element of ${\mathcal{L}}_4$ has a unique presentation tensor in ${\mathcal{E}}_L$, and an element of $\mathcal{B}$ has a unique sharp test. The graph prescriptions are linear in that tensor and in the additional test. They therefore define unique bilinear maps on the actual images. The same argument applies on the right.

Their values are finite sums of inherited admitted graphs, so belong to ${\mathcal{P}}_{\Sigma }$, including the required admission descendants. Classical multiplication defines $L_0,R_0$.