Abstract

We advance the actual-functional factorization problem for $B_f,W_g,B_{f^{\prime }}$, retaining the IX-A carrier and the IX-C interfaces. We identify the concrete presentation-relation kernels and prove that the fixed mixed kernel $K$ has full distributional support. A compact-primitive argument then proves injectivity of both partial-contraction presentation maps. Consequently the algebraic scalar relation tests impose no restriction on the real parent parameter: ${\mathcal{C}}_{\mathrm{desc}}=\mathbb{R}$.

An explicit leaf-covector argument constructs the three canonical two-edge distributions. Their compact polynomial kernels exist, and both metric-quadratic outputs satisfy the full carrier admission test, including directional descendants. This constructs the second-stage coefficient maps on the degree-two intermediate blocks. Reversal and fixed-support test-family continuity are proved on these blocks.

The release remains unresolved. Descent of the prescribed degree-four second-stage coefficients and agreement on actual-pair and old-product overlaps are not established. Thus neither complete sharp association is released. The triangle calibration remains conditional on the explicitly copied analytical premise reported in IX-C. No universal obstruction, global quantum algebra, or IX-D construction is claimed.