Unified Theory of Hilbert's 7th Problem under the First-Principles Framework of Differential Algebra

This paper constructs a unified mathematical framework from the first principles of differential algebra, providing a comprehensive solution to Hilbert’s 7th problem and its higher-dimensional generalizations. We first rigorously define the exponential-logarithmic differential closure F(n)EL , transforming transcendence problems into problems of differential algebraic solvability. Through recursive construction of differential field extensions, we establish a complete differential algebraic theoretical system. This system not only rigorously reproduces classical results such as the Gelfond–Schneider theorem and the Baker theorem, but also generalizes them to complex scenarios including multivariate linear forms, mixed exponential-logarithmic varieties, and branch choices. We prove an exact closed-form formula for the combinatorial correction coefficients γm,α, provide differential estimates for the radius of convergence,and establish intrinsic connections with traditional p-adic theory, motivic integration, and quantum computation. This paper proposes and proves the Fundamental Theorem of Differential Algebraic Transcendence, fully incorporating Hilbert’s 7th problem into the differential algebraic framework and providing a new research paradigm for transcendental number theory. All proofs start from the axioms of differential fields, avoiding the use of unproven analytical tools, achieving genuine first-principles derivation.