Based on the existing theory of differential algebraic closure, this paper introduces the difference operator and systematically constructs a differential-difference algebraic closure, providing a finite representation framework for classical transcendental functions such as the gamma function. By defining the gamma function as an abstract element satisfying the algebraic difference equation (z+1) =z (z) and a finite number of key point values (e. g. , (1) =1, (1/2) =), we prove its existence and uniqueness in the differential-difference closure. Using Newton series expansion, we derive an explicit analytic expression (z) =₍=₀^z-1/2n (-1) ^n (2n) !2^{2n (n!) ²} from the special values, and prove its convergence and consistency with the classical gamma function. On this basis, a large number of transcendental constants (, e, , , etc. ) are reduced to values of basic transcendental functions at algebraic points, greatly reducing the number of required initial constants. Furthermore, we establish a rigorous axiomatic classification system for functions based on genus, number of periods, and differential/difference/reflection orders, revealing the intrinsic relationships among these invariants and proving a DDR-Riemann-Roch type index theorem. We develop a comprehensive categorical, Galois-theoretic, cohomological, and geometric foundation for the theory, proving the independence of the DDR-axioms, establishing the model theory of DDR-fields, and constructing deformation theories for DDR-schemes. We introduce the reflection operator into the closure, prove its consistency and commutation relations, and provide a rigorous characterization of the Riemann zeta function within this framework, proving its functional equation and its connection to the reflection operator. The possibility of multiplicative operators and generalized closures is explored, proving the non-existence of finite-order multiplicative algebraic difference equations for the zeta function using Baker's theorem. We establish deep connections with modular forms (proving the Eichler-Shimura isomorphism for DDR-modular forms), L-functions (deriving the functional equation from the DDR-structure), transcendental number theory (proving a DDR-six exponentials theorem), quantum groups (deriving the quantum DDR-Yang-Baxter equation), and motives (constructing explicit DDR-motives for elliptic curves and the zeta function). This culminates in a Tannakian duality theorem that unifies the entire framework and provides a reformulation of the Riemann Hypothesis as a problem in spectral theory within the DDR-closure. Finally, we formulate the Grand Unified DDR-Langlands Conjecture, which unifies the classical Langlands correspondence, the geometric Langlands correspondence, the theory of special values of L-functions, the Riemann Hypothesis, and the Birch-Swinnerton-Dyer conjecture into a single framework.
shifa liu (Wed,) studied this question.