This paper establishes the Analytic Mathematics Finite Representation Theory, inaugurating the Constructive Unification Program for Analytic Mathematics. We prove that all equations—algebraic equations of arbitrary degree, differential equations, integral equations, difference equations, and variational equations—admit explicit analytic solutions and finite representations in the language LDA(Q), where coefficients lie in the algebraic numbers Q and definitions involve only algebraic operations and differentiation. Building on this foundational result, we develop a complete classification of transcendental functions defined by algebraic curves under two orthogonal spectra: elementary closed form existence (E) and differential-algebraic definability (D). The Dual Spectrum Theorem reveals that all such functions share D = Q (definability without transcendental constants), while E distinguishes trigonometric functions (E = Q(π, e)) from elliptic and abelian functions (E = ∞). The Degeneration Limit Theorem unifies these classes by showing trigonometric functions arise as limits of elliptic functions within the LDA(Q) framework. The theory extends to number theory: we prove that BSD coefficients, class numbers, and modular form coefficients are definable in LDA(Q), providing a differential-algebraic formulation of the Langlands program where functoriality becomes algebraic transformation between representations. We introduce the Analytic Mathematical Closure eQ math, a universal domain containing all objects constructible from Q by finite algebraic, analytic, number-theoretic, geometric, topological, and categorical operations. We prove that eQ math is the initial object in the category of such closures, establishing its universal property. Furthermore, we show that eQ math carries a natural Tannakian structure, and the Tannakian Galois group G unifies all Galoistype groups in mathematics—absolute Galois groups, differential Galois groups, monodromy groups, and Mumford-Tate groups appear as quotients of G. This yields a representationtheoretic description of all mathematics: eQ math ≃ RepQ (G). Finally, we develop a complete system of fourteen characteristic invariants that classify objects in eQ math, proving the fundamental relations among them and establishing that these invariants determine objects up to isomorphism. This provides a quantifiable, computable foundation for the entire unification program.
shifa liu (Wed,) studied this question.