This paper establishes the structural origin of the observable phase coefficient \ µₒbs = 4κₐdj + 2κₙonadj \ 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 \ Πcol ∘ Πₚair ∘ Πᵣet \ applied to the commutator Mₙ, Dₙ, the full ordered-pair structure reduces to X₅₁, and the retained triplet operator K on \ spanu₂, 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 = 2κₐdj + κₙonadj \ on one sheet. Sixth, the sheet-exchange involution doubles the observable coefficient to \ µₒbs = 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.
Craig Edwin Holdway (Sat,) studied this question.