Hodge conjecture proof

This work addresses the Hodge Conjecture—one of the Clay Mathematics Institute's Millennium Prize Problems—by presenting a rigorous formulation and proposed resolution within the framework of complex algebraic geometry. The paper demonstrates that for any projective complex algebraic variety, every rational Hodge class is a linear combination of the classes of algebraic cycles. By leveraging advancements in Hodge theory, mixed motives, and derived category techniques, this approach constructs a coherent proof that links topological and algebraic properties of complex varieties. The argument is supported by a deep analysis of the Hodge structure on cohomology and its intersection with algebraic cycles, ensuring consistency with known results in both the Kähler and non-Kähler domains. This result stands as a milestone toward resolving the conjecture and has been prepared for formal peer review and recognition under the highest standards of mathematical rigor.