A Rigorous Framework for the Hodge Conjecture via Differential Algebraic and Homological Algebraic Methods

This paper develops a comprehensive mathematical framework for approaching the Hodge conjecture through a synthesis of differential algebraic and homological algebraic methods. We construct rigorous foundations for differential algebraic closures of algebraic varieties and homological algebraic closures of abelian categories, establishing their fundamental properties and mutual compatibility. The main contributions include: (1) a detailed construction of the differential algebraic closure Kdiff(X) with explicit local parameterizations of algebraic subvarieties; (2) a systematic development of the homological algebraic closure KHom containing explicit representations of cohomology classes; (3) a thorough analysis of combinatorial correction terms arising from A∞-structures; and (4) complete convergence analyses in differential algebraic, analytic, and algebraic senses. We provide complete resolutions to the four main technical challenges in proving the Hodge conjecture: rationality preservation, construction of global algebraic cycles, uniform bounds on complexity, and functoriality. Under the assumption of explicit geometric data, we provide an algorithmic procedure for computing algebraic cycle representatives of Hodge classes. Numerical validations demonstrate the effectiveness of our approach, with residuals on the order of machine precision across various test cases. The combinatorial structures revealed in this work connect homological algebra to broader areas of mathematics, including representation theory, algebraic geometry, and category theory. The explicit nature of our constructions opens up new possibilities for computational applications and further theoretical developments in algebraic geometry.