Theorem: Rank-zero direct graded projection

For ten representatives $u_a\in {\mathcal{P}}_3$, the matrix of $e{P}_2$ is

The original minor is empty with determinant one. The zero map descends through every relation subspace.

In a nonnegative formal $\kappa$-series, an order-${\kappa }^3$ addition cannot change the order-${\kappa }^2$ coefficient under operations that do not lower the filtration.

Proof. Homogeneity gives $P_2{u}_a=0$. The full left kernel and empty minor certify rank zero. A zero map annihilates all relations. For the formal-order assertion, products of nonnegative powers cannot create a term below the order of the addition.