| _version_ | 1866901736109637632 |
|---|---|
| author | Holdway, Craig Edwin |
| author_facet | Holdway, Craig Edwin |
| contents | <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">This record presents TA15 (Continuum Limit of Depth Evolution), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">TA14 established that the effective transport generator G acts as a local adjacency operator on the 320-slot depth graph, inducing weighted averaging over neighbouring depth values rather than deterministic stepping, making depth a continuum-inducing parameter. TA15 performs the continuum limit explicitly.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Taking the elementary depth spacing \( epsilon = 1/320 \) as a small parameter, and noting that after choosing any ordering of the 320 slots compatible with local transport adjacency, the induced transport law reduces locally to nearest-neighbour form up to higher-order corrections, a standard Taylor expansion yields a differential evolution equation in the continuous depth coordinate \( d = n*epsilon \). The result is:</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">\[ partial_t(psi) = c_d * partial_d(psi) + D_d * partial_d^2(psi) + omega * A * psi + O(epsilon^2) \]</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">where \[ c_d * partial_d(psi) \] is the oriented transport term arising from the Mobius/Gray directional bias c_d = v<em>epsilon, D_d * partial_d^2(psi) is the symmetric spreading term from symmetric adjacency averaging (D_d = kappa</em>epsilon^2), omega * A * psi is the local rotational phase term from the reduced generator A = i*sigma_y, and leakage-return corrections from the K†BK block enter at higher order.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Three lemmas support the derivation: the symmetric continuum limit (equal forward and backward rates produce a diffusion equation in depth), the oriented continuum limit (Mobius/Gray directional bias produces a first-order transport equation), and the phase rotation term (the A-block contributes independently of the depth derivative structure). Two regime corollaries identify the transport-dominated case (strong directional bias gives directed propagation along depth) and the diffusion-dominated case (symmetric transport gives depth diffusion). A third corollary identifies the continuum equation as the starting point for TA16 (Smooth-Path / Minimum-Curvature Principle).</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">The theorem does not claim a physical wave equation, Schrodinger equation, gravitational field equation, or any observable identification of the depth coordinate. The continuum limit is a structural result about the 320-slot transport lattice. Explicit coefficient values for c_d, D_d, and omega remain conditional on the full Q5 construction of the leakage generator (open from T126).</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Together, TA14 and TA15 mark a conceptual transition point in the architecture: the depth coordinate moves from a combinatorial indexing label to an emergent coarse coordinate with local generator flow, continuum approximation, and transport/diffusion decomposition. This transition occurs without abandoning the discrete substrate.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">The theorem chain progressively derives the structure of the effective transport generator \[ G_eff = Pi_Y G Pi_Y + K†BK \], from which observable phase, leakage, decoherence, and residual correction emerge as structural consequences of projected transport closure on Q5.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20277176 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Continuum Limit of Depth Evolution 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 record presents TA15 (Continuum Limit of Depth Evolution), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">TA14 established that the effective transport generator G acts as a local adjacency operator on the 320-slot depth graph, inducing weighted averaging over neighbouring depth values rather than deterministic stepping, making depth a continuum-inducing parameter. TA15 performs the continuum limit explicitly.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Taking the elementary depth spacing \( epsilon = 1/320 \) as a small parameter, and noting that after choosing any ordering of the 320 slots compatible with local transport adjacency, the induced transport law reduces locally to nearest-neighbour form up to higher-order corrections, a standard Taylor expansion yields a differential evolution equation in the continuous depth coordinate \( d = n*epsilon \). The result is:</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">\[ partial_t(psi) = c_d * partial_d(psi) + D_d * partial_d^2(psi) + omega * A * psi + O(epsilon^2) \]</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">where \[ c_d * partial_d(psi) \] is the oriented transport term arising from the Mobius/Gray directional bias c_d = v<em>epsilon, D_d * partial_d^2(psi) is the symmetric spreading term from symmetric adjacency averaging (D_d = kappa</em>epsilon^2), omega * A * psi is the local rotational phase term from the reduced generator A = i*sigma_y, and leakage-return corrections from the K†BK block enter at higher order.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Three lemmas support the derivation: the symmetric continuum limit (equal forward and backward rates produce a diffusion equation in depth), the oriented continuum limit (Mobius/Gray directional bias produces a first-order transport equation), and the phase rotation term (the A-block contributes independently of the depth derivative structure). Two regime corollaries identify the transport-dominated case (strong directional bias gives directed propagation along depth) and the diffusion-dominated case (symmetric transport gives depth diffusion). A third corollary identifies the continuum equation as the starting point for TA16 (Smooth-Path / Minimum-Curvature Principle).</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">The theorem does not claim a physical wave equation, Schrodinger equation, gravitational field equation, or any observable identification of the depth coordinate. The continuum limit is a structural result about the 320-slot transport lattice. Explicit coefficient values for c_d, D_d, and omega remain conditional on the full Q5 construction of the leakage generator (open from T126).</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">Together, TA14 and TA15 mark a conceptual transition point in the architecture: the depth coordinate moves from a combinatorial indexing label to an emergent coarse coordinate with local generator flow, continuum approximation, and transport/diffusion decomposition. This transition occurs without abandoning the discrete substrate.</p> <p class="font-claude-response-body break-words whitespace-normal leading-[1.7]">The theorem chain progressively derives the structure of the effective transport generator \[ G_eff = Pi_Y G Pi_Y + K†BK \], from which observable phase, leakage, decoherence, and residual correction emerge as structural consequences of projected transport closure on Q5.</p> |
| title | Continuum Limit of Depth Evolution |
| 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.20277176 |