A Fractal–Spectral Approach to the Poincar´e Conjecture: An Alternative Geometric Proof Strategy
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| Format: | Recurso digital |
| Language: | English |
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2025
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| author | Maréchal, Thierry |
| author_facet | Maréchal, Thierry |
| contents | <p>We propose a fractal–spectral operator-based method as an alternative proof strategy for the Poincar´e Conjecture in dimension three. Specifically, we construct a self-adjoint, multi-scale “fractal” operator H3D on a closed 3-manifold M. By enforcing spectral stability (no large gaps or spurious clusters) under a fractal variant of Weyl’s law, we argue that only the simply connected manifold topologically equivalent to S3 passes the test.</p> <p><strong>Key points:</strong></p> <ul> <li>Bounded fractal potential: We embed local topological data (e.g. fundamental group signatures) into a bounded fractal potential, ensuring H3D is self-adjoint.</li> <li>Fractal Weyl counting in 3D: A log-corrected Weyl law reveals “spectral anomalies” for any non-spherical or non-simply connected manifold, leading to contradictions in the O(logE) margin.</li> <li>Conclusion: A closed, simply connected M3 that remains spectrally stable must be homeomorphic to S3.</li> </ul> <p>While not a fully rigorous replacement for Perelman’s Ricci-flow solution, this fractal–spectral framework provides a new lens bridging multi-scale PDE methods, prime-problem analogies, and geometric classification in 3D topology.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_16852189 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Fractal–Spectral Approach to the Poincar´e Conjecture: An Alternative Geometric Proof Strategy Maréchal, Thierry Mathematic <p>We propose a fractal–spectral operator-based method as an alternative proof strategy for the Poincar´e Conjecture in dimension three. Specifically, we construct a self-adjoint, multi-scale “fractal” operator H3D on a closed 3-manifold M. By enforcing spectral stability (no large gaps or spurious clusters) under a fractal variant of Weyl’s law, we argue that only the simply connected manifold topologically equivalent to S3 passes the test.</p> <p><strong>Key points:</strong></p> <ul> <li>Bounded fractal potential: We embed local topological data (e.g. fundamental group signatures) into a bounded fractal potential, ensuring H3D is self-adjoint.</li> <li>Fractal Weyl counting in 3D: A log-corrected Weyl law reveals “spectral anomalies” for any non-spherical or non-simply connected manifold, leading to contradictions in the O(logE) margin.</li> <li>Conclusion: A closed, simply connected M3 that remains spectrally stable must be homeomorphic to S3.</li> </ul> <p>While not a fully rigorous replacement for Perelman’s Ricci-flow solution, this fractal–spectral framework provides a new lens bridging multi-scale PDE methods, prime-problem analogies, and geometric classification in 3D topology.</p> |
| title | A Fractal–Spectral Approach to the Poincar´e Conjecture: An Alternative Geometric Proof Strategy |
| topic | Mathematic |
| url | https://doi.org/10.5281/zenodo.16852189 |