Abstract

We formulate the binary coherence problem for the ordered word $B_f,W_g,B_{f^{\prime }}$ on the actual-functional carrier of IX-A. Both admitted first-stage products are fixed, including the reverse contraction and its sign. We identify the three one-edge outputs and give the labelled expressions for the three two-edge outputs, without assuming that a labelled expression defines a distributional product. We prove exact algebraic descent criteria for second-stage coefficient maps and for the calibrated scalar assignment ${u_{\mu }}^{\rho }+cr$. The scalar relation tests restrict the real parent parameter to an empty set, a singleton, or all of $\mathbb{R}$. We also prove the coefficientwise equations required for associativity through order ${\hslash }^3$, and distinguish observation compatibility, involution, presentation independence, and bulk-relative agreement.

The release outcome is an unresolved gate. Neither sharp second multiplication is constructed, and no incompatibility excluding all such multiplications is proved. The completed results are the admitted partial stages and the factorization and coherence reductions. The numerical triangle calibration remains conditional on the explicit VII-D analytical premise restated in IX-B-v2.