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 B_L, B_R, visible channels L, R, and mediator M, two successive Schur eliminations produce the effective visible two-state Hamiltonian \[ H_eff^(2) = h₀I + h_xσ_x + h_zσ_z \] with mediated coupling: \[ h_z = v_L v_R / Δ \] In the rotated phase-plane basis {α, γ}, the mediated bilinear appears as the coefficient of σ_z rather than σ_x, 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 \[ v = (v_L, v_R) \]. The same mediated bilinear structure governs both descriptions, and under the phase normalization, the two are related by \[ µ = 2h_z + O(δ²) \]. The non-factorization result is the strongest structural consequence: \[ h_z = v_L v_R / Δ \] 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 h_z. The visible two-state system is not primitive, it is the Schur-reduced projection of a higher-dimensional mediated transport system. The correspondence \[ Q_⊥ ~ vvᵀ ↔ h_z ~ v_L v_R / Δ \] establishes that phase emergence is equivalent to mediated second-order coupling, not merely analogous to it.