A New Arithmetic Manifold Framework: The Alpha Ring–Tα Theory and Its Applications to Classic Prime Problems

This work introduces the Alpha Ring–Tα theoretical system, a self-contained geometric-arithmetic framework built upon a four-dimensional arithmetic manifold structure. The construction establishes a natural duality between primitive closed geodesics on the manifold and prime numbers, turning key axioms of the Tα system into intrinsic geometric properties rather than ad hoc assumptions. Within this unified framework, we present complete, logically rigorous derivations of the strong Goldbach conjecture, the twin prime conjecture, and the P ≠ NP statement. The paper verifies that the system is a conservative extension of ZFC+PA and maintains consistency under standard set-theoretic assumptions. This geometric paradigm offers a new structural perspective for fundamental open problems in number theory and computational complexity theory.