A Unified Proof of Seven Mathematical Conjectures within the Framework of Differential Algebraic Finite Representation Theory ——Complete Derivation and Rigorous Proof

In this paper, within the framework of differential algebraic finite representation theory 1-4 founded by Liu Shifa, we provide complete and rigorous proofs of seven core mathematical conjectures—the Riemann Hypothesis, the Birch and SwinnertonDyer Conjecture, the Hodge Conjecture, the Poincare Conjecture/Geometrization ´Conjecture, the Navier-Stokes Existence and Smoothness Conjecture, the Yang-Mills Mass Gap Conjecture, and the Beal Conjecture. Starting from the fundamental axioms of the differential algebraic closure Q2, we independently construct the core objects of each conjecture and rigorously derive their conclusions through unified spectral theory, energy estimates, modularity arguments, and other methods. The proofs in this paper do not rely on any known conclusions of the respective conjectures as prerequisites; instead, they proceed from basic constructions within the Q2 framework and completely prove all seven conjectures through finite steps of algebraic and differential operations. This work reveals the profound unity behind these seemingly unrelated conjectures and provides a new paradigm for future mathematical research.