Phase Coefficient Structure from Reduced Transport

This paper establishes the structural origin of the observable phase coefficient \[ µ_obs = 4κ_adj + 2κ_nonadj \] from first principles, completing the transport chain: Gray ordered source → reduced commutator → triplet carrier → phase plane → observable coefficient The derivation proceeds in six parts. First, the cyclic reflected Gray code on Q₅ is shown to have a unique maximal ordered bias \[ C₅₁ = 16 \], with all lower-lower ordered pairs cancelling exactly. This identifies X₅₁ as the unique dominant ordered transport source. Second, under the reduction pipeline \[ Π_col ∘ Π_pair ∘ Π_ret \] applied to the commutator [Mₙ, Dₙ], the full ordered-pair structure reduces to X₅₁, and the retained triplet operator K on \[ span{u₂, u₃, u₄} \] is antisymmetric with nearest-neighbour zero pattern: no primitive u₂ ↔ u₄ transition exists. Third, Schur elimination of the internal mode u₃ produces a forced rank-one quadratic phase-plane operator \[ Q_⊥ = (1/Δ)vvᵀ\] , establishing that visible coupling is mediated, not primitive. Fourth, phase extraction via \[ µ(Q) = ½Tr(QR) \] yields \[ µ = (ab/Δ)sin θ \], which vanishes if either adjacent coupling is absent. Fifth, path-class multiplicity analysis of X₅₁ gives 2 adjacent classes and 1 nonadjacent class, producing \[ µ = 2κ_adj + κ_nonadj \] on one sheet. Sixth, the sheet-exchange involution doubles the observable coefficient to \[ µ_obs = 2µ \], giving the main result. The coefficient ratio 4:2 arises from 2:1 path-class multiplicity, followed by cross-sheet doubling, not from raw Gray counts directly.