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Mediated Asymmetry and Forward Dominance in Three-Sector Quantum Systems

Craig Edwin Holdway

Published Mar 31, 2026• 0 citations
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Abstract

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.

DOI
10.5281/zenodo.19380323

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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.
We exhibit an exact three-mode packet realization of a mediated three-sector system in which forward transport strictly dominates complementary reconstruction on a nontrivial coherence window. The coefficient hierarchy forces the forward amplitude to remain close to the interior coherence kernel while the return amplitude is uniformly suppressed at order 1/318. Whenever the coherence kernel exceeds 1/158, forward dominance holds exactly. Since the kernel equals 1 at t=0, a nontrivial forward-dominance interval is guaranteed to exist. The result is structural and occurs in a closed system without environmental coupling or irreversibility.
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 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 present three theorems that complete the structural account of coherence and decoherence in the reduced transport sector of the Q5 framework. This mathematical system derives operator structures from the combinatorial geometry of the five-dimensional hypercube and its Gray code Hamiltonian cycle. Theorem 50 identifies coherence as the survival of the commutator-driven transverse mode under transport, and establishes a precise structural distinction between reversible phase scrambling and irreversible decoherence. No new operators are introduced; all objects are inherited from the prior theorem chain (T1–T49). Theorem 51 derives the structural form of the effective decoherence rate as a product of four independently motivated factors: a base loss rate, a quadratic coupling asymmetry term, a saturating orbit depth function, and a Lorentzian detuning suppression factor. Each factor is grounded in a specific prior result in the Q5 transport chain. Theorem 52 establishes that the coherence decay rate is the unique globally attracting fixed point of a self-consistent kernel-induced dynamical map. Existence and uniqueness are proved via the intermediate value theorem applied to a strictly decreasing map; global convergence follows from even/odd subsequence monotonicity. The exponential saturation law is derived as the asymptotic reduction of the fixed-point structure in the strong-suppression regime, rather than postulated. The primary remaining open step is the explicit derivation of the coupling parameter β from the commutator [M_n, D_n]Y geometry of Theorem 29. All results are conditional on the admissible linked-pair reduction of Theorem 17 and the retained-pairing assumption of Theorem 36. Epistemic classifications are provided for each result.
D T58 develops a conditional entangling-power framework arising from irreducible rotational structure in the reduced Q5 transport sector. The theorem analyzes the antisymmetric rotational invariant generated by phase-asymmetric transport amplitudes and studies its consequences under a bipartite reduced-sector decomposition. When the reduced interaction generator contains a non-removable rotational interaction component,Aint≠0,\mathcal A_{\mathrm{int}}\neq0,Aint​=0,The resulting reduced evolution operator becomes non-factorizable under local sector decomposition, implying strictly positive entangling power,ep(U)>0.e_p(U)>0.ep​(U)>0.The theorem, therefore, identifies a structural pathway from irreducible mediated rotational transport to nonlocal reduced-sector coupling behaviour. T58 does not claim that Q5 derives quantum entanglement from first principles, nor that this mechanism is unique. Rather, it establishes that under the stated decomposition and interaction assumptions, the mediated rotational residue generated by asymmetric transport produces nonfactorizable reduced evolution with positive entangling capability. The theorem further clarifies that vanishing rotational structure disables this specific entangling channel without excluding alternative entanglement mechanisms. Status: solid for the nonfactorizable-unitary and positive-entangling-power conclusions under the stated operator assumptions; conditional on the bipartite tensor decomposition and the mapping from reduced rotational structure to interaction generators; speculative for broader physical interpretation or equivalence with quantum entanglement phenomenology.
This document is not a new theorem, but a structural map of the Q_5 theorem series, isolating the minimal dependency chain and the forced architecture that emerges from it. We present a structural compression showing that the combinatorial geometry of the 5-dimensional hypercube Q_5, equipped with parity grading, admissible transport, and Gray-constrained traversal, forces a uniquely determined operator algebra and state-space structure on C[Q5]. Within this forced architecture, a minimal defect-supported carrier emerges canonically, yielding a three-dimensional Lie algebra C_X isomorphic to su(2) under its natural commutator relations. Order-sensitive transport along Gray-constrained paths generates a nonfactorizable phase defect that survives projection through a sequence of reduction operators and is extracted via a canonical linear functional L. This phase is not imposed, but forced by the underlying combinatorial constraints. We further show that the embedded defect coefficient governing this phase is generically nontrivial and cannot vanish identically. This nontriviality arises from an inversion asymmetry inherent in the transport architecture. Under a compatibility condition between defect extraction and the retained transport sector, the phase reduces to a universal constant value. The resulting structure exhibits a clear correspondence with key features of quantum mechanical formalism, including graded state spaces, noncommutative operator algebras, and phase-carrying transport mechanisms. While no claim of full equivalence to quantum mechanics is made, this synthesis isolates a minimal set of combinatorial conditions under which such structures are forced, suggesting a potential geometric origin for phase coherence and its persistence under constrained transport.