A Differential-Algebraic Proof of the Hodge Conjecture: A Constructive Framework Based on Finite Representations and Period Closures

This paper, based on the differential-algebraic closure theory and the method of finite representations of transcendental functions developed in “On Differential Algebraic Closure Solutions, Finite Representations of Transcendental Functions, and a Framework for Applications in Number Theory” (referred to as the Framework), presents the first constructive proof of the Hodge Conjecture. The core contributions are: (1) The introduction of Differential-Algebraic Hodge Theory, which completely incorporates the Hodge structures and period mappings of complex projective varieties into the differential-algebraic closure framework, proving that Hodge classes admit finite representations in extensions generated by differential-algebraic solutions of polynomial equations; (2) The construction of an explicit differential-algebraic model for Hodge cycles, where for any rational (p, p)-Hodge class η on a smooth complex projective variety X, we construct a differentialalgebraic cycle Zη whose defining equations are explicitly given by the differential-algebraic period data of X, and prove that it represents η up to rational equivalence; (3) The establishment of a rigid comparison theorem for period closures, proving that the period domain generated by the differential-algebraic closure is isomorphic to the classical period domain,thereby ensuring that the constructed cycle is both algebraic and precisely matches the given Hodge class; (4) A solution for singular varieties and compactifications of moduli spaces, extending the proof to arbitrary complex projective varieties through the introduction of quasi-differential-algebraic closures and boundary Hodge theory. This paper not only proves the Hodge Conjecture but also provides an operational differential-algebraic algorithm for explicit computation and algebraic realization of Hodge classes, elevating Hodge theory from an existential statement to a constructive theory.