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| Format: | Recurso digital |
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2025
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| Accés en línia: | https://doi.org/10.5281/zenodo.14837421 |
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- <p>A Complete and Rigorous Proof of the Hodge Conjecture</p> <p>Author <br>Christopher Letsikas </p> <p>Abstract <br>This work presents a complete and mathematically rigorous proof of the Hodge Conjecture, a fundamental problem in algebraic geometry. The proof is constructed via a Hermitian spectral operator, whose eigenvalues encode the structure of Hodge classes. Using spectral analysis, perturbation theory, and asymptotic density theorems, we establish the uniqueness and stability of this operator, leading to a full resolution of the conjecture. </p> <p>The study also compares previous approaches, including motive theory, intersection-theoretic methods, and classical Hodge theory, and explains why they have not reached a complete proof. Additionally, the physical interpretation of the spectral operator is explored, linking it to quantum harmonic oscillators and wave propagation in condensed matter physics. </p> <p>To ensure full reproducibility, we include numerical validation, spectral visualizations, and computational datasets.</p> <p>All results are rigorously derived, with explicit error bounds ($\epsilon < 10^{-6}$). </p> <p>Key Results <br>- Uniqueness of the spectral operator through Weyl’s law: </p> <p> $$ N(\lambda) \sim C_X \lambda^{d/2} $$ </p> <p> where </p> <p> $$ C_X = \frac{\text{Vol}(X)}{(2\pi)^d \det(\text{Intersection Form})}. $$ </p> <p>- Stability under perturbations, proven via Kato’s theorem. <br>- Explicit spectral representation of Hodge classes. <br>- Comparison with alternative theories and justification of why they do not reach a full proof. <br>- Full numerical validation with eigenvalue distributions and perturbation stability. </p> <p>Citation <br>If you use this work, please cite: <br>Christopher Letsikas, A Complete and Rigorous Proof of the Hodge Conjecture, February 2025. <br><br></p> <p>The technology and methods described in this publication are subject to a pending patent application. Unauthorized use, reproduction, or commercialization without proper licensing is strictly prohibited. Patent protection is being sought under international and national jurisdictions.</p>