Proposition: Direct decomposition of the new domain

The new domain is the algebraic direct sum

The prescribed coefficient maps therefore glue uniquely on $D_{\mathrm{new}}$.

Proof. The zero intersections involving the degree-four blocks are the inherited field-degree results of IX-Free. The locked first-stage domains have species degrees distinct from all second-stage domains and from each other.

All relations can be projected onto the distinct homogeneous argument-species components. The only component containing two listed domains is $((1,1);(1,1))$. Its two domains have zero intersection by Theorem thm:separation. Thus that sum is direct as well.

This proves a direct decomposition of the entire sum, not merely pairwise agreement. Each coefficient restriction is an actual linear map by the inherited first stages, inherited degree-two descent, and Theorem thm:degree4. Their sum is consequently a unique linear map on $D_{\mathrm{new}}$.