A Unified Differential-Algebraic Theory of Hilbert's 19th Problem: Variational Theory of Differential-Algebraic Closures

Based on differential-algebraic methods, this paper establishes a unified theoretical framework for variational problems. By constructing a categorified differential algebraic closure functor K : Vark → DiffFieldsk, we incorporate various classes of variational problems into a single algebraic system. We prove the existence and uniqueness of this closure and derive a complete theory for combinatorial correction coefficients Γ(d)m,α, including recurrence formulas, branching selections, and singularity classifications. Traditional variational theories (elliptic, parabolic, hyperbolic, stochastic) all become special cases of this framework and can be expressed through unified formulas. For high-dimensional non-elliptic problems, we prove partial regularity theorems; for stochastic evolution equations, we provide deterministic representations; in geometric deep learning, we establish geometric flow descriptions of training dynamics.This paper also provides a complete algorithmic implementation framework and formal verification scheme, ensuring the computability and reliability of the theory.