We study a minimal three-sector mediated system and derive the second-order effective dynamics on the outer subspace. We show that mediated transport is governed by an inverse detuning structure, while asymmetry between transport channels is controlled by detuning imbalance. This produces structural suppression of reconstruction under restricted return operations, while exact reversibility remains preserved under full unitary inversion. The complete proof and unification with the forward dominance theorem is given in the companion paper (Theorem 11b (Unified), DOI: 10.5281/zenodo.19380323).
We study a minimal three-sector mediated system and derive the second-order effective dynamics on the outer subspace. We show that mediated transport is governed by an inverse detuning structure, while asymmetry between transport channels is controlled by detuning imbalance. This produces structural suppression of reconstruction under restricted return operations, while exact reversibility remains preserved under full unitary inversion. The complete proof and unification with the forward dominance theorem is given in the companion paper (Theorem 11b (Unified), DOI: 10.5281/zenodo.19380323).
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 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.
This Theorem identifies the structural form of the effective decoherence rate arising from the irreversible coupling established in T50. The rate is not a free parameter but is determined by three structural inputs from the Q5 transport chain: Γ_eff = γ_loss · G² · [1 + β·A²] · f(d_orbit) · L(δ_detune) Four factors: (1) Base rate γ_loss·G², decay requires irreversible sector coupling; Γ_eff = 0 when γ_loss = 0, regardless of all other factors. (2) Asymmetry term [1 + β·A²], coupling asymmetry A ~ |β_J − β| = |μν'c'b − b'c| from T33; operationally A ~ |g₁ − g₀|/√(g₁² + g₀²) from T8; quadratic dependence consistent with simulation showing decay enhancement scales as (g₁ − g₀)². (3) Orbit depth term f(d_orbit) = 1 − e^{−α(d_orbit − 2)}, saturation form reflecting finite Q5 combinatorial depth; f(2) = 0 at the minimal orbit of T36–T37. (4) Lorentzian detuning suppression L(δ_detune) = 1/(1 + (δ_detune/γ_c)²), detuning suppresses coupling efficiency, not a decay channel. Three structural statements: scrambling is necessary but not sufficient for decoherence; asymmetry amplifies scrambling; irreversible coupling converts scrambling into decay. Three corollaries: zero-loss limit (Γ_eff = 0 when γ_loss = 0); resonance maximum at δ_detune = 0; symmetric coupling limit (A = 0 returns base rate). Status: Asymmetry identification A ~ |μν'c'b − b'c| from T33 solid. f(2) = 0 at minimal orbit solid within T36–T37. Three qualitative boxed statements solid from T50 and factor structure. Overall structural form model-level, functional forms of f and L are natural leading-order models, not uniquely forced. Five structural constants (γ_loss, G, α, β, γ_c) were identified qualitatively; explicit values from the T17 kernel geometry are open. All results inherit T8, T17, T33, T35, T36, T37, T50 conditionality. Dependencies: T8, T11, T29, T33, T35, T36, T37, T50.
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,The resulting reduced evolution operator becomes non-factorizable under local sector decomposition, implying strictly positive entangling power,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.