Phase–Hamiltonian Correspondence and Mediated Non-Factorization
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| Natura: | Recurso digital |
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Zenodo
2026
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| _version_ | 1866901732456398848 |
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| author | Holdway, Craig Edwin |
| author_facet | Holdway, Craig Edwin |
| contents | <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">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.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">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:</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">\[ h_z = v_L v_R / Δ \]</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">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 \[ µ = 2h_z + O(δ²) \].</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">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.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">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.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20263841 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Phase–Hamiltonian Correspondence and Mediated Non-Factorization Holdway, Craig Edwin Q5 hypercube penteract Gray code Hamiltonian cycle hypercube graph discrete geometry combinatorial topology defect bond holonomy transport operator phase defect linked-pair reduction ordered residue kernel structure fibre reduction Lie algebra representation theory spinor commutator projection operator orientation invariant grading phase transport barrier crossing paired transport continuous limit discrete-to-continuous unitary evolution phase accumulation Born rule interference entanglement decoherence quantum measurement reduced sector observable interferometry quadrature cycling first-harmonic bias falsifiable prediction fringe visibility ZPH independent research preprint <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">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.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">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:</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">\[ h_z = v_L v_R / Δ \]</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">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 \[ µ = 2h_z + O(δ²) \].</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">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.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">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.</p> |
| title | Phase–Hamiltonian Correspondence and Mediated Non-Factorization |
| topic | Q5 hypercube penteract Gray code Hamiltonian cycle hypercube graph discrete geometry combinatorial topology defect bond holonomy transport operator phase defect linked-pair reduction ordered residue kernel structure fibre reduction Lie algebra representation theory spinor commutator projection operator orientation invariant grading phase transport barrier crossing paired transport continuous limit discrete-to-continuous unitary evolution phase accumulation Born rule interference entanglement decoherence quantum measurement reduced sector observable interferometry quadrature cycling first-harmonic bias falsifiable prediction fringe visibility ZPH independent research preprint |
| url | https://doi.org/10.5281/zenodo.20263841 |