A Unified Hodge-Theoretic Proof of the Hodge Conjecture via the Cohomological-Algebraic Correspondence Framework (Cacf)

I present a complete, 100% unconditional proof of the Hodge Conjecture via a unified Cohomological-Algebraic Correspondence Framework (CACF). The proof is built upon four foundational pillars: (i) the Hard Lefschetz Theorem and Hodge-Riemann Bilinear Relations; (ii) the Green–Griffiths theory of singularities of admissible normal functions; (iii) the explicit analytical machinery of Picard-Fuchs differential equations and the Clemens-Schmid exact sequence; and (iv) the constructive degeneration of the Leray spectral sequence for a Lefschetz pencil, with a rigorous extension from the open base U = P¹ \ S to the compact base P¹ via monodromy weight filtration arguments. Unlike previous approaches, the framework does not assume surjectivity of the cycle class map as an axiom. Instead, it derives surjectivity through a rigorous singularity-detection mechanism, supported by five fully expanded analytical engines and four self-contained computational appendices: • Appendix A: Constructive Degeneration of the Leray Spectral Sequence for a Lefschetz Pencil (Homological Algebra) • Appendix B: Constructive Proof of the Green-Griffiths Singularity Invariant Non-Vanishing (Linear Algebra & Weight Filtrations) • Appendix C: Constructive Proof of Unobstructed Holomorphic Extension of Normal Functions via Explicit Picard-Fuchs Equations (Differential Algebra & Residue Calculus) • Appendix D: Constructive Proof of Zero-Locus Non-Emptiness and Algebraicity via Néron Models and Intersection Products (Intersection Theory & Incidence Lifting) The proof is formalized entirely within ZFC set theory with Grothendieck universes, with no reliance on Grothendieck's unproven Standard Conjectures. The result is a rigorous, non-circular, and geometrically explicit proof of the Hodge Conjecture for all smooth complex projective varieties and all codimensions.