An Absolute Resolution of the Hodge Conjecture via Cohomological Rigidity and Algebraic Lifting

This work presents a claimed complete proof of the Hodge Conjecture, asserting that every rational cohomology class of type (p,p) on a smooth projective complex variety is algebraic. The paper develops a proof architecture based on spread methodology, correspondence functoriality, variation of Hodge structure, and Deligne-style rigidity principles for horizontal sections, organized through deformation closure and algebraic lifting. The manuscript positions its main contribution as a global closure argument: if the Hodge Conjecture holds at one fiber in a connected smooth projective family, then it holds throughout the family. The text synthesizes known verified cases, spread constructions, functorial transfer, rigidity arguments, and a formal verification skeleton in Coq and Lean. Subtitle: Deformation Closure, Spread Functoriality, and the Deligne Principle. Important note: The manuscript presents a claimed affirmative resolution of the Hodge Conjecture, a Millennium Prize Problem. This metadata describes the document as submitted and does not imply independent validation of the claim.