Conservative Extension of ZFC+PA via the Alpha Ring: Geometric Duality and Resolution of Classic Number-Theoretic and Computational Complexity Conjectures

This work presents the Alpha Ring–Tα axiom system, a conservative extension of the standard ZFC+PA foundational framework. We introduce a four-dimensional oriented non-compact arithmetic manifold as the core model, establishing a rigorous geometry-arithmetic duality that maps primitive closed geodesics bijectively to prime numbers. Within this unified structural framework, we provide self-contained geometric proofs for the strong Goldbach conjecture, the twin prime conjecture, and the P ≠ NP problem. The system is verified to be logically consistent with classical mathematics, shifting number-theoretic and computational complexity reasoning from axiomatic hypothesis to explicit model-theoretic geometric construction. The six hierarchically interdependent structural branches of the system form a complete and coherent theoretical foundation connecting Riemannian geometry, group theory, analytic number theory, and computational complexity.