Actual tensor relations and an algebraic criterion

Double-ordering tests are necessary but not sufficient. Every actual linear relation among prescribed tensors must also be respected. This obligation is distinct from the already closed degree-four descent.

Write $S(d)={\star }_c(d)$ on ${\mathcal{D}}_2$ and define

The inherited classical coefficient of $A$ is multiplication. Equation (eq:free-star) implies

Let $\{(d_i,t_i){\}}_{i\in I}$ be the collection of all prescribed values: $t_i=S(d_i)$ for every ordered pair, together with $t_i={{\mathsf{T}}_2}^{\mathrm{old}}(d_i)$ for every tensor on a supplied old overlap. The index set need not be finite. Assume these values and their required descendants are admitted, and their classical coefficients equal multiplication. Put

Let

Conjugation induces an antilinear involution on $Q$.