Commutator-Driven Phase Extraction with Irreducible Transport

This Theorem identifies Mₙ, Dₙ as the unique algebraic source of order-sensitive contribution to the scalar phase coefficient μ. The ordered branch difference UA − UB = XMₙ, DₙY for common transport factors X and Y. The null classes of the T28 extraction functional L are classified explicitly (branch-null, collapse-null, axis-null, non-rotational transverse), and under the hypothesis that the remainder Ωᵣem is null-class decomposable, μ reduces to: μ = L̃ (Πₛub Πₚair Πᵣet (Mₙ, DₙY) ) The diagonal transport factor X is absorbed into the extraction pipeline under explicit structural hypotheses. The weighted adjacency factor Y is shown to be not retained-support invariant (non-commuting with Πᵣet) and acts irreducibly on the connected support graph: it cannot be reduced further and remains as essential transport weighting. A corollary evaluates μ explicitly on the Gray code kernel support g (14), g (15), g (16) = 9, 8, 24 with cycle-edge adjacency matrix and diagonal transport X = −2I, giving μ = −8βh₁, confirming a nonzero generically present phase coefficient whenever β ≠ 0 and h₁ ≠ 0. Status: Conditional. The main theorem requires three explicit structural hypotheses: (a) remainder null-class decomposability, (b) diagonal transport absorption, (c) connected support graph. Corollary, conditional on T17 kernel support identification, cycle-edge weight assignment, diagonal transport evaluation, and fibre-to-vertex correspondence from T14. Dependencies: T17, T20, T26, T27, T28.