Hodge Conjecture: A Unified Reduction Framework via Cohomological Stratification, Birational Classification and Formal Verification

This paper presents a structured derivation of the Hodge conjecture within theframework of compact K¨ahler manifolds. The study restricts to integral (p, p)-typediagonal Hodge classes, leveraging the Hodge decomposition and the minimal modelprogram to reduce the problem to three canonical classes: Fano, Calabi–Yau, andgeneral-type minimal varieties. The analysis introduces the Quillen spectral sequencein algebraic K-theory to constrain the period map from Chow groups to Hodge cohomology, establishing conditions for surjectivity. Arithmetic reduction to characteristicp, Witt-vector cohomology lifting, and p-adic Hodge theory are employed to avoid analytic ambiguities. Nonabelian Hodge correspondence is invoked to complete higherorder Hodge structures. All definitions, stratifications, spectral-sequence arguments,dimension inductions, and lifting steps are formalized in Lean4, yielding a closed,machine-verifiable derivation framework.