Phase–Hamiltonian Correspondence and Mediated Non-Factorization

This paper establishes that the phase coefficient extracted from reduced Q5 transport is identical in structure to the effective mediated coupling of a reduced two-state Hamiltonian system, upgrading the phase from an extracted observable artifact to an effective dynamical coupling parameter. Starting from a five-fibre Hamiltonian H₅ with boundary channels BL, BR, visible channels L, R, and mediator M, two successive Schur eliminations produce the effective visible two-state Hamiltonian \ Hₑff^ (2) = h₀I + hₓσₓ + hᵦσᵦ \ with mediated coupling: \ hᵦ = vL vR / Δ \ In the rotated phase-plane basis α, γ, the mediated bilinear appears as the coefficient of σᵦ rather than σₓ, a consequence of the basis rotation that diagonalizes the symmetric part of the visible operator. On the transport side, TA9 gives \ Q_⊥ = (1/Δ) vvᵀ \ with \ v = (vL, vR) \. The same mediated bilinear structure governs both descriptions, and under the phase normalization, the two are related by \ µ = 2hᵦ + O (δ²) \. The non-factorization result is the strongest structural consequence: \ hᵦ = vL vR / Δ \ vanishes if either channel coupling vanishes, so the phase cannot be assigned to one branch alone. No invariant subsystem supported only on one visible branch can reproduce nonzero hᵦ. The visible two-state system is not primitive, it is the Schur-reduced projection of a higher-dimensional mediated transport system. The correspondence \ Q_⊥ ~ vvᵀ ↔ hᵦ ~ vL vR / Δ \ establishes that phase emergence is equivalent to mediated second-order coupling, not merely analogous to it.