We show that the mediated three-sector system underlying Theorems 10 and 11 arises as the effective active core of a five-phase bulk-dressed architecture. Integrating out two outer bulk phases via Schur-complement reduction produces dressed detunings that sharpen the detuning-controlled transport asymmetry mechanism without altering its structure. We then introduce an adjacency-structured bulk and derive the leading correction to the dressed detuning difference under asymmetric adjacency coupling. In the symmetric-bulk regime, adjacency enters only at second order, yielding the explicit shift ΔR−ΔL=δ+(EM−B2)(EM−B1)2u2(a′2−a2)+O(u2a4,u4), which provides a compact algebraic relation linking bulk-core coupling, adjacency structure, and left-right asymmetry. A conditional proposition shows that, under a monotonicity assumption, adjacency asymmetry alone can reverse the directional ordering of local recovery failure in the reduced model. We further establish an operator realization of the leading correction on the dressed core as a diagonal side-asymmetry operator PR−PL. While no natural bare-basis global asymmetry operator projects to this form, we construct an explicit lifted realization via a core-bath mixing unitary, and exhibit an exact dynamical implementation within a parameter-tuned regime of the same five-phase Hamiltonian architecture. The question of whether such a lift is structurally forced by the underlying Q5 geometry remains open.
We show that the mediated three-sector system underlying Theorems 10 and 11 arises as the effective active core of a five-phase bulk-dressed architecture. Integrating out two outer bulk phases via Schur-complement reduction produces dressed detunings that sharpen the detuning-controlled transport asymmetry mechanism without altering its structure. We then introduce an adjacency-structured bulk and derive the leading correction to the dressed detuning difference under asymmetric adjacency coupling. In the symmetric-bulk regime, adjacency enters only at second order, yielding the explicit shift ΔR−ΔL=δ+(EM−B2)(EM−B1)2u2(a′2−a2)+O(u2a4,u4), which provides a compact algebraic relation linking bulk-core coupling, adjacency structure, and left-right asymmetry. A conditional proposition shows that, under a monotonicity assumption, adjacency asymmetry alone can reverse the directional ordering of local recovery failure in the reduced model. We further establish an operator realization of the leading correction on the dressed core as a diagonal side-asymmetry operator PR−PL. While no natural bare-basis global asymmetry operator projects to this form, we construct an explicit lifted realization via a core-bath mixing unitary, and exhibit an exact dynamical implementation within a parameter-tuned regime of the same five-phase Hamiltonian architecture. The question of whether such a lift is structurally forced by the underlying Q5 geometry remains open.
We establish two related structural results on the five-dimensional hypercube Q5. First, every symmetric three-sector Hamiltonian of mediated form arises as the unique compression of a Q5 operator family to the complement-symmetric block subspace K = span{|L⟩, |M⟩, |R⟩}, with diagonal energies generated by aI + bB̂ + cΠ̂ and off-diagonal couplings generated by the Gray-weighted transfer operator Aγ. Under this compression, the symmetric coupling condition vL = vR is emergent from complement symmetry rather than imposed, and the bulk asymmetry operator ηadj(PR − PL) is absorbed into the coefficients of I and B̂. Second, in any three-sector Hamiltonian with nonzero mediated couplings vL, vR ≠ 0 and nondegenerate outer energies EL ≠ ER, no eigenstate is localized in a single phase sector: every eigenstate has strictly nonzero amplitude in all three sectors. These results establish that the three-sector Hamiltonian underlying the mediated asymmetry framework is not a primitive model but a compressed object arising from Q5 geometry, and that eigenstates are intrinsically multi-phase distributed objects.
We study the complement-paired bulk-mediated extension of the fibre transport architecture arising from the closure-parity and fibrewise effective dynamics framework on the five-dimensional hypercube Q5Q_5Q5. Starting from the single-fibre defect generator, we pass to a parity-adapted basis in which the dynamics separates into independent transport and asymmetry phase planes. We prove that in a complement-paired reduction with shared bulk mediation, crossing compatibility forces a strict separation between the transport and asymmetry sectors. On the crossing-even transport plane, we show that nontrivial bulk-mediated coupling together with preservation of an exact inherited transport rotor forces the effective correction to be nonzero, singular, symmetric, and rank one. A direct matrix argument then establishes that the inherited rotor condition is equivalent to annihilation of the endpoint line, uniquely forcing the correction to take the form A=ρ ∣M+⟩⟨M+∣A = \rho\,|M_+\rangle\langle M_+|A=ρ∣M+⟩⟨M+∣. We further show that this structure is perturbatively rigid: the middle-projector branch is the unique infinitesimal deformation preserving the inherited transport channel. The paired transport holonomy then decomposes into a protected channel with exact inherited phase e±iηpe^{\pm i\eta p}e±iηp and a bulk-dressed complementary channel. The resulting spectral mismatch defines an exact relative phase defect Δφ(τ)=4βτ2(p−p2−4a2),\Delta\varphi(\tau) = 4\beta\tau^2\bigl(p - \sqrt{p^2 - 4a^2}\bigr),Δφ(τ)=4βτ2(p−p2−4a2), which is invariant under fibre exchange but irreducible to independent single-fibre contributions. We identify this structure as nonfactorizable complementary holonomy: a shared phase constraint that arises only at the paired level. The bulk mediator does not act as a transport channel, but as a constraint-enforcing mechanism. All results are established within a minimal complement-paired bulk-mediated model and are explicitly classified by epistemic status and are structural and algebraic within a minimal complement-paired bulk-mediated model. This work develops Theorem 15 in the Q5 framework. See associated Zenodo records for Theorems 7, 8, 12, and 14.
This paper establishes a structural and exact result for mediated three-sector systems, demonstrating asymmetry between transport and reconstruction channels in a closed finite-dimensional setting. A minimal three-sector model is introduced in which outer states are coupled only through a mediated intermediate state. Using second-order reduction, it is shown that outer-sector transport is governed by inverse detuning structure, while asymmetry between channel-resolved amplitudes is controlled by detuning imbalance. This produces suppression of reconstruction under restricted return operations, while exact reversibility remains preserved under full unitary evolution. An exact realization of this mechanism is then provided through a three-mode packet model derived from the Q₅ framework. In this setting, forward and return amplitudes are computed explicitly, yielding a coefficient hierarchy in which forward contributions are order-one while return contributions are suppressed by approximately three orders of magnitude. This leads to a rigorously defined coherence window in which forward amplitude strictly dominates return amplitude. The results establish that forward dominance is not an isolated numerical artifact, but arises from a structural asymmetry inherent in mediated multi-channel systems. Together, the structural theorem and exact realization provide a unified account of reconstruction suppression and transport dominance in a closed-system setting. This work forms part of a broader program investigating operator structure, grading behavior, and sector-resolved dynamics on the five-dimensional hypercube (Q₅), with related results available in accompanying white papers and operator decomposition studies.
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.
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 Heff(2)=h0I+hxσx+hzσz with mediated coupling: hz=vLvR/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=(vL,vR). The same mediated bilinear structure governs both descriptions, and under the phase normalization, the two are related by µ=2hz+O(δ2). The non-factorization result is the strongest structural consequence: hz=vLvR/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↔hzvLvR/Δ establishes that phase emergence is equivalent to mediated second-order coupling, not merely analogous to it.
What happens at the precise threshold where two coupled quantum channels are about to split, and what symmetry structure governs that moment? This paper answers that question within the five-dimensional hypercube Q5 framework. Working from the complement-paired bulk-mediated architecture established in Theorems 12 and 15, we study the linked interface between two fibres and prove three things. First, the threshold behaviour is controlled by a single basis-independent discriminant D = |κ|² − ((ω_A − ω_B)/2)², and this discriminant defines a symmetry class that is invariant under the full architecture-preserving subgroup G_AB of the B5 symmetry group. The threshold is a structural invariant, not a coordinate artifact. Second, eliminating the dressed sector via Schur complement generates no protected-channel transport bridge between the fibres at any order. The interface carries a threshold class without carrying a transport operator: a clean structural separation. Third, at the generic nontrivial threshold, the dressed helicity block undergoes non-semisimple coalescence. An explicit Jordan chain is constructed, showing that the two eigenvalues collide in a way that cannot be diagonalized: a signature of genuine structural degeneracy. Together, these results characterize the linked complement-reversed interface as a mathematically precise object: a G_AB-invariant threshold class realized by Jordan coalescence, in the absence of any protected transport channel. The open boundary (deriving the admissible coupling form from Q5 first principles) is clearly stated and does not affect the theorem. Dependencies: Theorems 7, 12, 14, 15.