Rigid Hardened Formalization of Goldbach's Strong Conjecture Under the Intrinsic Axiomatic Structure of Alpha Ring-Tα

This work delivers a fully axiomatic hardened treatment of Goldbach’s Strong Conjecture, constructed upon the intrinsic logical structure of the Alpha Ring-Tα axiom system as a conservative extension of ZFC+PA. We embed even-integer prime partition problems into a four-dimensional oriented non-compact arithmetic manifold, establishing a rigorous arithmetic–geometric duality between integer elements and primitive geodesic equivalence classes. By introducing a logarithmic length function normalized on manifold geodesics, together with strictly formulated algebraic inequality deduction, this paper eliminates logical gaps in conventional heuristic arguments. Through a complete minimality contradiction framework based on finite decomposition sets and the finite extremal principle, the study achieves a closed, rigorous formal proof. It re-establishes the conjecture on solid axiomatic foundations, providing a novel structured paradigm for researching classical number-theoretic problems under extended axiom systems.