Mediated Phase Emergence, Transport Reduction, and Non-Factorization

This Theorem establishes a complete transport-to-visible reduction theorem for a five-fibre system governed by Gray-structured transport on Q5. The cyclic reflected Gray code produces a unique surviving antisymmetric ordered source, the (5,1) closure class with ordered-pair count C₅₁ = 16. This reduces under the canonical projection–pairing–retention pipeline to a nearest-neighbour antisymmetric commutator operator K on a three-dimensional carrier space. Eliminating the mediator channel via Schur reduction forces a rank-one quadratic visible operator Q⊥ ~ vvᵀ, whose phase-bearing component is extracted by μ(Q) = ½Tr(QR). Path-class decomposition gives the single-sheet coefficient law μ = 2κ_adj + κ_nonadj; cross-sheet doubling gives the observable coefficient law: μ_obs = 4κ_adj + 2κ_nonadj Independently, Schur reduction of the five-fibre Hamiltonian gives h_z = v_L·v_R/Δ, bilinear in left and right couplings. Under the transport–Hamiltonian matching hypothesis in the near-symmetric regime, μ = 2h_z + O(δ²_eff). The bilinearity is irreducible (no subsystem supported on a single visible branch can reproduce a nonzero coefficient), establishing that the observable two-state dynamics are intrinsically non-factorizable. A geometric holonomy lemma establishes that μ_obs is a loop invariant, nontrivial despite vertex-level closure, vanishing on commutative transport classes. A Berry Phase Correspondence Criterion identifies the conditions under which γ_Berry = μ_obs + O(δ²), without asserting those conditions are satisfied. Status: Gray ordered-pair counts, uniqueness of (5,1) closure source, zero-pattern of K, rank-one quadratic emergence, phase extraction, cross-sheet doubling, bilinearity, and no-one-sided-support result all solid. Conditional on phase dressing assumption (full derivation from T17 kernel generators open. Conditional on path-class multiplicity), a 2:1 ratio requires explicit enumeration from T17 kernel geometry. Conditional on transport–Hamiltonian matching, bridge requires a near-symmetric regime hypothesis and second-order leg matching justification.