Unified Analytic Finite Representation Theory in Differential Algebraic Closure and Its Bidirectional Mapping with Differential Equations

This paper establishes a unified framework, within the differential algebraic closure, for explicit analytic representations of solutions to a wide class of differential equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), exterior differential equations (EDEs), and total differential equations (TDEs), as well as their nonlinear counterparts.We provide complete constructive proofs, explicit formulas for the combinatorial coefficients, algorithmic implementations with rigorous error bounds, and extensive numerical validation on a wide range of physical examples (exponential decay, Maxwell–Boltzmann, Fermi–Dirac, Bose–Einstein, Bessel functions, KdV solitons, Maxwell equations, etc.). The framework is shown to be consistent with classical impossibility results and to extend Picard–Vessiot theory, Painlev´e analysis, the Cauchy–Kovalevskaya theorem, exterior differential systems, Lie symmetry methods, and spectral methods. Additionally, we resolve several open problems: (i) the extension of the theory to Itˆo stochastic differential equations, (ii) the existence of a neural network approximant for the high-dimensional combinatorial coefficients with rigorous error bounds, and (iii) the algebraic-geometric interpretation of the differential algebraic closure as the function field of an infinite- dimensional differential scheme.