{"paper":{"title":"Unlocking the Hodge Conjecture: A Spectral Fingerprint Approach via Gauss-Manin Derivatives","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Bita Hajebi, Pooya Hajebi","submitted_at":"2025-07-17T12:29:44Z","abstract_excerpt":"We present a symbolic analytic framework for addressing the Hodge Conjecture, based on a refined invariant called the Hermitian spectral fingerprint. By projecting out $(k,k)$ components from holomorphic forms and their Gauss Manin derivatives, we define a fingerprint functional that vanishes identically for any rational cohomology class of type $(k,k)$. We prove unconditionally that the projected derivatives span the entire orthogonal complement of $H^{k,k}(X)$ in $H^{2k}(X,\\mathbb{C})$, implying structural vanishing. This vanishing criterion across realizations leads to absolute Hodge behavi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.13064","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.13064/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}