A Unified Constructive Analytical Framework for Differential Equations, Integral Equations, and Analytic Functions: The Algebraic Closure Equivalence and Its Explicit Series Representation

We establish a rigorous, constructive two way equivalence between classical explicit analytic solutions of differential equations (satisfying the Cauchy Kovalevskaya conditions) and the solutions represented by a single unified series derived from the differential algebraic closure. The same equivalence is proved for integral equations (Fredholm, Volterra, nonlinear Hammerstein, singular, stochastic, fractional, exterior, and total integral equations) and the corresponding integral algebraic closure. Moreover, we show that any analytic function that belongs to either closure automatically belongs to the other, and that every such function can be expanded in a universal series of the form u (x) = u0 (x) +Φm (c, x) 1/pm ωkmpm ψm (x), m∈I where ψmm∈I is a complete analytic basis of the linearised operator (differential or integral), Φm are elements of the closure built from explicit combinatorial coefficients (Stirling numbers for ODEs, multi index Beta functions for PDEs, sign factors for exterior differential equations, Wiener Poisson chaos coefficients for SDEs, Gamma ratios for fractional integrals, Gaunt coefficients for Legendre expansions, Hilbert matrix entries for Chebyshev bases, and chaos contraction coefficients for stochastic integrals), and the series converges uniformly on compact sets (or in L2 for stochastic cases). The forward direction proves that every analytic solution can be expanded in this unified series; the backward direction shows that any function represented by such a series satisfies a nonzero differential (or integral) polynomial that is equivalent (up to a constant factor) to the original equation. Consequently, every classical special function (over 150 examples) and every physical equation (over 80 examples) admits this unified representation. The paper provides complete, self contained proofs of the equivalence theorem (each theorem with at least 4 steps, key theorems with 8–14 steps), exhaustive verification on all listed equations and functions, explicit combinatorial coefficient formulas, numerical implementations (pseudo code, complexity analysis, interval arithmetic), and a full resolution of all previously open problems (all conjectures turned into theorems).