The Hodge Conjecture: A Spectral Fingerprint Proof & Lean 4 Formal Verification (Repo: https://github.com/merchantmoh-debug/Hodge-Conjecture-Lean-4-Solution-Repository.)
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2026
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| _version_ | 1866901339550777344 |
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| author | Al-Zawahreh, Mohamad |
| author_facet | Al-Zawahreh, Mohamad |
| contents | <p>We prove the Hodge Conjecture: on any non-singular complex projective variety X, every Hodge class is a Q-linear combination of cohomology classes of algebraic subvarieties. Our proof introduces the Spectral Fingerprint Framework (SFF), which exploits the Hodge Laplacian on Kähler manifolds to detect the algebraicity of cohomology classes. The key insight is that every Hodge class admits a spectral expansion in the Laplacian eigenbasis. The coefficients of this expansion—the spectral fingerprint—are algebraic numbers precisely when the class is an algebraic cycle. This is established via the Cattani-Deligne-Kaplan theorem (which proves Hodge loci are algebraic), Deligne's absolute Hodge theory, and codimension induction using the Hard Lefschetz operator. The proof recovers Lefschetz (1,1) for codimension 1 and correctly fails for non-projective Kähler manifolds (Voisin) and integral coefficients (Atiyah-Hirzebruch). Computational validation confirms Hodge symmetry and Hard Lefschetz across multiple varieties. The complete machine-readable proof and Lean 4 repository are publicly available at: https://github.com/merchantmoh-debug/Hodge-Conjecture-Lean-4-Solution-Repository.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18977072 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Hodge Conjecture: A Spectral Fingerprint Proof & Lean 4 Formal Verification (Repo: https://github.com/merchantmoh-debug/Hodge-Conjecture-Lean-4-Solution-Repository.) Al-Zawahreh, Mohamad <p>We prove the Hodge Conjecture: on any non-singular complex projective variety X, every Hodge class is a Q-linear combination of cohomology classes of algebraic subvarieties. Our proof introduces the Spectral Fingerprint Framework (SFF), which exploits the Hodge Laplacian on Kähler manifolds to detect the algebraicity of cohomology classes. The key insight is that every Hodge class admits a spectral expansion in the Laplacian eigenbasis. The coefficients of this expansion—the spectral fingerprint—are algebraic numbers precisely when the class is an algebraic cycle. This is established via the Cattani-Deligne-Kaplan theorem (which proves Hodge loci are algebraic), Deligne's absolute Hodge theory, and codimension induction using the Hard Lefschetz operator. The proof recovers Lefschetz (1,1) for codimension 1 and correctly fails for non-projective Kähler manifolds (Voisin) and integral coefficients (Atiyah-Hirzebruch). Computational validation confirms Hodge symmetry and Hard Lefschetz across multiple varieties. The complete machine-readable proof and Lean 4 repository are publicly available at: https://github.com/merchantmoh-debug/Hodge-Conjecture-Lean-4-Solution-Repository.</p> |
| title | The Hodge Conjecture: A Spectral Fingerprint Proof & Lean 4 Formal Verification (Repo: https://github.com/merchantmoh-debug/Hodge-Conjecture-Lean-4-Solution-Repository.) |
| url | https://doi.org/10.5281/zenodo.18977072 |