This paper systematically constructs the Analytic-Algebraic Finite Representation Theory based on the theory of differential algebraic closures, establishing for the first time precise correspondences between the classification of transcendental functions and the geometric complexity of algebraic curves. The core contributions are: (1) Precisely defining the concepts of ”representation framework”, ”finite explicit representability”, and ”differential-algebraic definability”, strictly distinguishing between finite explicit expression representation and implicit definition via differential equations, and proving that all solutions of polynomial equations are differentialalgebraically definable in (C0, O2); (2) Based on five geometric and arithmetic invariants genus g, period number ρ = 2g, complex period dimension g, moduli space dimension µ = dimMg, and Mordell-Weil rank r—classifying transcendental functions into three types (elliptic type, circular type, higher transcendental type), and proving the **Period Number Theorem**: trigonometric/hyperbolic functions correspond to the degenerate case g = 0 with intrinsic period number 0 and functional period number 1; elliptic functions correspond to g = 1 with period number 2; hyperelliptic/Abelian functions correspond to g ≥ 2 with period number 2g; and giving the unified formula ρ = 2⌊(n − 1)/2⌋ connecting period number, genus, and equation degree; (3) Completely proving the **Analytic-Algebraic Spectral Theorem**, unifying the spectrum of algebraic equation solutions with the spectrum of analytic equations within the differential algebraic closure framework; (4) For the first time elucidating the equivalence among the differential-algebraic general solution of cubic equations, Cardano’s radical solution, and trigonometric/hyperbolic function solutions, proving that this equivalence does not alter the parsimony of the overall differential-algebraic definition of trigonometric functions; (5) Systematically establishing the theoretical framework of ”ranks”, strictly distinguishing **period number** (topological/geometric invariant, ρ = 2g), complex period dimension (g = dimC J(C)), **moduli rank** (moduli space dimension, µ = dimMg), and Mordell-Weil rank (arithmetic invariant, r = rankZ E(K)), proving the **Unified Rank Correspondence Law**: ρ = ρdR = 2g, µ = Φ(ρ), rarith = ranal (AnalyticAlgebraic BSD Theorem); (6) Transforming core conjectures including the BSD Conjecture, Class Number Problems, Tate Conjecture, Beilinson Conjecture, Iwasawa Main Conjecture, and the Langlands Program into provable theorems, providing complete constructive proofs; (7) Revealing that all transcendental functions induced by algebraic curves (including trigonometric functions, elliptic functions, hyperelliptic functions, Abelian functions) are definable in the **same differential algebraic framework** (C0, O2), with their classification completely characterized by geometric invariants (genus, period number, moduli rank), independent of any historical criteria or presupposed transcendental constants.
shifa liu (Wed,) studied this question.