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2026
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| Online Access: | https://doi.org/10.5281/zenodo.19580746 |
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| author | Trees, Nala |
| author_facet | Trees, Nala |
| contents | <p class="MsoNormal">This paper establishes the arithmetic Aharonov-Bohm package for the SGOC programme in the bounded-domain setting.</p> <p class="MsoNormal"> </p> <p class="MsoNormal">Given a primitive Dirichlet character chi of conductor q and a compact swallowed window O with a character-seeded placed screen complex, the residual period vector encodes a U(1)-valued arithmetic holonomy class on the cycle basis of H_1(O; Z). Conditional on the continuum flat U(1) realisation of the residual period class and cut-neighbourhood trivialisation, a nodal-free stationary state acquires the Aharonov-Bohm phase Delta_phi = Arg(Per_{chi,O}(Gamma)) around every cycle class Gamma. Once the residual periods are fixed on basis cycles, the phase on any remaining cycle is predicted exactly.</p> <p class="MsoNormal"> </p> <p class="MsoNormal">The paper makes the theorem concrete with the first worked example: the primitive character modulo 5 with chi(2) = i. For a one-cycle bouquet this gives the quarter-flux ring ladder E_n(pi/2) = (hbar^2 / 2mR^2)(n - 1/4)^2, and for a two-cycle screen it yields the phase predictions theta_1 = pi/2, theta_2 = -pi/2 with trivial and half-flux classes on the sum and difference cycles.</p> <p class="MsoNormal"> </p> <p class="MsoNormal">The bouquet-class continuum realisation theorem is proved unconditionally: on a planar bouquet screen every arithmetic period homomorphism is realised by a flat U(1) connection on a trivial line bundle. The arithmetic scale is fixed by coherence matching rather than by spectral fitting.</p> <p class="MsoNormal"> </p> <p class="MsoNormal"><strong>Keywords: </strong>Aharonov-Bohm effect, Dirichlet characters, arithmetic topology, holonomy, residual periods, SGOC, ring spectrum, flux quantisation</p> <p class="MsoNormal"> </p> <p class="MsoNormal"><strong>Related Programme: </strong>Spectral Geometry of Coherence (SGOC)</p> <p class="MsoNormal"> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19580746 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Arithmetic Aharonov--Bohm Phase Shift, Held-Out Cycle Prediction, Ring Benchmarks, and Multi-Cycle Holonomy Classes Trees, Nala <p class="MsoNormal">This paper establishes the arithmetic Aharonov-Bohm package for the SGOC programme in the bounded-domain setting.</p> <p class="MsoNormal"> </p> <p class="MsoNormal">Given a primitive Dirichlet character chi of conductor q and a compact swallowed window O with a character-seeded placed screen complex, the residual period vector encodes a U(1)-valued arithmetic holonomy class on the cycle basis of H_1(O; Z). Conditional on the continuum flat U(1) realisation of the residual period class and cut-neighbourhood trivialisation, a nodal-free stationary state acquires the Aharonov-Bohm phase Delta_phi = Arg(Per_{chi,O}(Gamma)) around every cycle class Gamma. Once the residual periods are fixed on basis cycles, the phase on any remaining cycle is predicted exactly.</p> <p class="MsoNormal"> </p> <p class="MsoNormal">The paper makes the theorem concrete with the first worked example: the primitive character modulo 5 with chi(2) = i. For a one-cycle bouquet this gives the quarter-flux ring ladder E_n(pi/2) = (hbar^2 / 2mR^2)(n - 1/4)^2, and for a two-cycle screen it yields the phase predictions theta_1 = pi/2, theta_2 = -pi/2 with trivial and half-flux classes on the sum and difference cycles.</p> <p class="MsoNormal"> </p> <p class="MsoNormal">The bouquet-class continuum realisation theorem is proved unconditionally: on a planar bouquet screen every arithmetic period homomorphism is realised by a flat U(1) connection on a trivial line bundle. The arithmetic scale is fixed by coherence matching rather than by spectral fitting.</p> <p class="MsoNormal"> </p> <p class="MsoNormal"><strong>Keywords: </strong>Aharonov-Bohm effect, Dirichlet characters, arithmetic topology, holonomy, residual periods, SGOC, ring spectrum, flux quantisation</p> <p class="MsoNormal"> </p> <p class="MsoNormal"><strong>Related Programme: </strong>Spectral Geometry of Coherence (SGOC)</p> <p class="MsoNormal"> </p> |
| title | Arithmetic Aharonov--Bohm Phase Shift, Held-Out Cycle Prediction, Ring Benchmarks, and Multi-Cycle Holonomy Classes |
| url | https://doi.org/10.5281/zenodo.19580746 |