Unlocking the Hodge Conjecture: A Spectral Fingerprint Approach via Gauss-Manin Derivatives

Abstract

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 behavior and, by deep results in arithmetic geometry, confirms algebraicity. Thus, the Hodge Conjecture is resolved within this framework.