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| Main Author: | |
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| Format: | Recurso digital |
| Language: | English |
| Published: |
Zenodo
2025
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| Subjects: | |
| Online Access: | https://doi.org/10.5281/zenodo.16852189 |
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Table of 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>