Spectral Rigidity and Algebraicity: A Unified Framework for the Hodge Conjecture

Abstract

This paper presents a novel symbolic analytic framework to address the Hodge Conjecture, utilizing a refined invariant called the Hermitian spectral fingerprint. We modify the fingerprint functional to specifically exclude $(k,k)$ components, demonstrating its vanishing for rational classes of type $(k,k)$. Critically, we develop a comprehensive proof strategy to establish the converse: the vanishing of this refined fingerprint across all realization functors (de Rham and $\ell$adic) implies the class is absolute Hodge. By fundamental theorems in arithmetic algebraic geometry, absolute Hodge classes of type $(k,k)$ are equivalent to algebraic cycles. This framework offers a new, robust criterion for detecting algebraic cycles, reformulating the conjecture into a problem of establishing the exhaustive spanning properties of GaussManin derivatives and Galois actions within their respective cohomology spaces. While building upon established deep results, this approach provides a fresh perspective and a pathway towards a complete resolution.