Theorem: IX-C reduction package and release status

Relative to the inherited IX-A and IX-B results and the explicit analytical premise in Assumption ass:calibration, this manuscript establishes:

1. both admitted first-stage products, with locked signs and reverse contraction;

2. actual-functional existence and admission of the three one-edge outputs;

3. the labelled signs, multiplicities, and residual degrees of the three two-edge placements;

4. necessary and sufficient algebraic descent tests for a prescribed labelled coefficient map;

5. necessary and sufficient scalar relation tests for the calibrated assignment on the named partial-output spaces;

6. the empty/singleton/all-real classification of parameters passing those scalar relation tests;

7. the three nontrivial coefficient equations for associativity through order ${\hslash }^3$;

8. conditional placement coherence, scalar reversal, and presentation independence after actual descent.

Neither sharp second multiplication is constructed. No incompatibility excluding every map satisfying Requirement req:interface is proved. The release outcome is therefore an unresolved gate.

Proof. The assertions follow from Theorem thm:first, Proposition prop:oneedges, Proposition prop:placementcount, Theorem thm:descent, Theorem thm:scalarfactor, Proposition prop:parameters, Theorem thm:coherence, Corollary cor:placementcoherence, Proposition prop:scalarreverse, and Theorem thm:presentation.

The two-edge existence and admission requirements in Remark rem:treeboundary have not been discharged. The scalar relation tests have not been evaluated for the actual presentation kernels. Consequently the hypotheses needed to construct the second-stage maps and invoke the conditional coherence results have not been established. No violated relation or other universal incompatibility has been exhibited. The unresolved classification follows.