Spectral Stability and Continuum Limit of the Sobolev-Ozok Lattice (SOL) Model
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| Natura: | Recurso digital |
| Lingua: | inglese |
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2025
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| _version_ | 1866901516444499968 |
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| author | Ozok, Ozcan |
| author_facet | Ozok, Ozcan |
| contents | <p>This work investigates the mathematical well-posedness of the Sobolev–Ozok Lattice (SOL) framework by analyzing the spectral properties of a discrete third-order Sobolev curvature operator defined on a Planck-scale lattice. The primary objective is to determine whether the model admits resolution-independent observables and a meaningful continuum limit, which are necessary prerequisites for physical consistency in any discrete spacetime formulation.</p> <p>We demonstrate numerically that the eigenvalue spectrum of the discrete operator stabilizes under lattice refinement, indicating spectral convergence as the number of lattice cells increases. Based on this result, we formulate a precise mode-tracking procedure that identifies physical states with stable eigenvalue branches rather than fixed eigenvalue indices, thereby resolving the ambiguity associated with index drift in discretized systems.</p> <p>Physical observables are then defined as continuous functions of these stable curvature eigenvalues. Under this formulation, invariance of the observable quantities in the continuum limit follows directly from spectral stability and does not depend on lattice labeling conventions. As a consistency check, we show that a single global mass–curvature mapping reproduces the characteristic mass scales of the charged leptons, while the electron is treated explicitly as a geometric boundary case.</p> <p>An interpretive appendix discusses a possible coherence-closure mechanism at the third Sobolev order that may constrain the number of stable charged lepton configurations, while remaining clearly separated from the formally established results of the main text.</p> <p>Overall, this paper establishes that the SOL lattice admits well-defined, resolution-independent spectral observables and provides a mathematically disciplined foundation for further investigations of geometry-based physical interpretations within the SOL framework.</p> <div dir="auto"><strong>Decleration of Tools Used:</strong></div> <div dir="auto">This paper was prepared and formatted using Overleaf (LaTeX editor). Text refinement and language polishing were assisted by Overleaf AI Editor. all scientific content, derivations, and conclusions are original and autored by the undersigned.</div> <div dir="auto"> </div> <div dir="auto"> <p>This paper is part of the Sobolev–Ozok Lattice (SOL) research program.</p> <p><strong>Project webpage (papers, figures, updates):</strong></p> <p><a href="https://ozokozcasol.github.io/Sobolev-Ozok-Lattice/" target="_blank" rel="noopener">https://ozokozcasol.github.io/Sobolev-Ozok-Lattice/</a></p> </div> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17924486 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Spectral Stability and Continuum Limit of the Sobolev-Ozok Lattice (SOL) Model Ozok, Ozcan Discrete spacetime Spectral stability Continuum limit Sobolev operators Lattice models Eigenvalue convergence Resolution independence Mode tracking Discrete curvature Mathematical physics Lattice regularization Quantum gravity Discrete differential geometry Operator spectra Planck-scale lattice Geometric mass interpretation <p>This work investigates the mathematical well-posedness of the Sobolev–Ozok Lattice (SOL) framework by analyzing the spectral properties of a discrete third-order Sobolev curvature operator defined on a Planck-scale lattice. The primary objective is to determine whether the model admits resolution-independent observables and a meaningful continuum limit, which are necessary prerequisites for physical consistency in any discrete spacetime formulation.</p> <p>We demonstrate numerically that the eigenvalue spectrum of the discrete operator stabilizes under lattice refinement, indicating spectral convergence as the number of lattice cells increases. Based on this result, we formulate a precise mode-tracking procedure that identifies physical states with stable eigenvalue branches rather than fixed eigenvalue indices, thereby resolving the ambiguity associated with index drift in discretized systems.</p> <p>Physical observables are then defined as continuous functions of these stable curvature eigenvalues. Under this formulation, invariance of the observable quantities in the continuum limit follows directly from spectral stability and does not depend on lattice labeling conventions. As a consistency check, we show that a single global mass–curvature mapping reproduces the characteristic mass scales of the charged leptons, while the electron is treated explicitly as a geometric boundary case.</p> <p>An interpretive appendix discusses a possible coherence-closure mechanism at the third Sobolev order that may constrain the number of stable charged lepton configurations, while remaining clearly separated from the formally established results of the main text.</p> <p>Overall, this paper establishes that the SOL lattice admits well-defined, resolution-independent spectral observables and provides a mathematically disciplined foundation for further investigations of geometry-based physical interpretations within the SOL framework.</p> <div dir="auto"><strong>Decleration of Tools Used:</strong></div> <div dir="auto">This paper was prepared and formatted using Overleaf (LaTeX editor). Text refinement and language polishing were assisted by Overleaf AI Editor. all scientific content, derivations, and conclusions are original and autored by the undersigned.</div> <div dir="auto"> </div> <div dir="auto"> <p>This paper is part of the Sobolev–Ozok Lattice (SOL) research program.</p> <p><strong>Project webpage (papers, figures, updates):</strong></p> <p><a href="https://ozokozcasol.github.io/Sobolev-Ozok-Lattice/" target="_blank" rel="noopener">https://ozokozcasol.github.io/Sobolev-Ozok-Lattice/</a></p> </div> |
| title | Spectral Stability and Continuum Limit of the Sobolev-Ozok Lattice (SOL) Model |
| topic | Discrete spacetime Spectral stability Continuum limit Sobolev operators Lattice models Eigenvalue convergence Resolution independence Mode tracking Discrete curvature Mathematical physics Lattice regularization Quantum gravity Discrete differential geometry Operator spectra Planck-scale lattice Geometric mass interpretation |
| url | https://doi.org/10.5281/zenodo.17924486 |