Theorem: IX-Free partial construction and release

Relative to the explicit inherited premises, this manuscript constructs:

1. concrete relation kernels for all four presentation maps;

2. injective degree-two presentation maps and the evaluated scalar-descent set ${\mathcal{C}}_{\mathrm{desc}}=\mathbb{R}$;

3. all three canonical two-edge distributions and their admitted compact polynomial outputs;

4. the left and right second-stage coefficient maps on the degree-two intermediate blocks;

5. reversal and presentation independence on those blocks;

6. fixed-support test-family continuity for the completed coefficient families.

The earliest incomplete required gate is F5 on the degree-four intermediate blocks. F6 overlap agreement is also incomplete. Neither complete sharp association is constructed, and no contradiction excluding every construction satisfying Requirement req:interfaces is proved. The release is Unresolved.

Proof. The new results are Proposition prop:relations, Proposition prop:prelations, Theorem thm:qinjective, Corollary cor:cdesc, Proposition prop:existence, Theorem thm:treeadmission, Theorem thm:degree2maps, Proposition prop:reversal, Proposition prop:presentation, and Theorem thm:F9a.

Proposition prop:remainingdescent states exact tests without proving their hypotheses. The overlap conditions are likewise not discharged. Therefore the conditional coherence proposition cannot release either full association. No violated relation or universal interface contradiction has been exhibited.