Differential Algebraic Framework for Hodge Theory: Constructive Hodge Decomposition and Harmonic Forms

This paper establishes a comprehensive differential algebraic framework for constructive Hodge theory on compact K¨ahler manifolds. We define the Hodge closure KHodge, a differentially closed field extension constructed through recursive adjunction processes that incorporate harmonic forms, Hodge decompositions, Dolbeault operators, and spectral data of Laplace operators. Within this closure, we prove that harmonic forms and Hodge decompositions admit unified explicit representations that respect the underlying complex, Riemannian, and algebraic structures. The framework rigorously addresses the interplay between differential geometry, complex analysis, and algebraic topology while preserving graded algebraic structures and compatibility conditions. We provide detailed constructive proofs with complete mathematical rigor, deriving explicit solution formulas with certified error bounds and establishing convergence criteria in appropriate Sobolev and H¨older spaces. Comprehensive algorithms with precise complexity analysis are presented, including stability guarantees and adaptive precision control with certified error bounds via interval arithmetic. The work demonstrates that explicit analytic constructions in Hodge theory exist within appropriately constructed differential algebraic closures, providing new algebraic perspectives on harmonic analysis while maintaining consistency with classical theory.Extensions to Dolbeault cohomology, K¨ahler identities, and connections with the Hodge conjecture establish bridges across mathematical disciplines while maintaining full mathematical rigor.