A Constructive Bidirectional Equivalence Between Classical Analytic Solutions and Differential-Algebraic Closure Series: Rigorous Foundations, Explicit Combinatorics, and Comprehensive Applications to Special Functions and Physical Equations

This paper establishes a rigorous bidirectional equivalence between the classical explicit analytic solutions of differential equations satisfying the Cauchy–Kovalevskaya conditions and the solutions represented by the unified series of the differential-algebraic closure theory. The forward direction proves that every analytic solution can be expanded in a series u =u0+(Φm)1/pmωkmpmψm,m∈I where ψm are basis functions of a linearized operator, Φm are elements of the closure (limits of differential polynomials in the initial data) built from explicit combinatorial coefficients (Stirling numbers for ODEs, multi-index Beta functions for PDEs, sign factors for EDEs), and the series converges uniformly on compact sets. The backward direction shows that any such series satisfies a differential polynomial that is equivalent (up to a constant factor) to the original equation. This equivalence implies that all classical special functions—including Bessel, Legendre, Airy, hypergeometric, elliptic, Painlev´e I–VI, and Lambert W—admit a unified representation. The paper provides complete, self-contained proofs of the equivalence theorem, extensive verification on Einstein field equations (with matter and cosmological constant), KdV, sine-Gordon, nonlinear Schr¨odinger equations, and a comprehensive classification of special functions with explicit tables. All conjectures from the literature are resolved and turned into theorems. Numerical validation, pseudo-code, complexity analysis, and detailed appendices are included.