A Conditional Proof of the Hodge Conjecture

We outline a conditional approach toward a proof of the Hodge Conjecture, employing Deligne cohomology, the theory of normal functions, the structure of the Hodge locus in a variation of Hodge structure, and the spread of cycles over algebraic families. The strategy shows that rational Hodge classes on smooth projective complex varieties must arise from algebraic cycles under natural assumptions supported by recent advances in Hodge theory. This manuscript is part of an ongoing effort to resolve the Hodge Conjecture, one of the Millennium Prize Problems. The approach combines theoretical insights from algebraic geometry and Hodge theory, and reflects computational explorations conducted in collaboration with AI reasoning systems.