**Revision Log for V2.0:** 1. Supplemented constructive geometric mechanism for Axiom A4 via product manifold diffeomorphism, added technical lemma for geodesic direct-sum decomposition; 2. Corrected boundary statement, clarified the trivial counterexample N=2, restricted the conclusion domain to even integers N≥4; 3. Added explicit logical connection between Lemma 2.1 and Axiom A4, eliminating unstated logical gaps; 4. Standardized reference format with unique archive identifiers for all companion manuscripts; 5. Unified geometric foundation with the Twin Prime Conjecture V2.0 manuscript. **Abstract:** The Strong Goldbach Conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. For decades, traditional analytic number theory has only obtained approximate results with error terms, while a rigorous proof within the standard ZFC+PA system remains absent. This paper presents a formal conditional proof exclusively within the $\text{ZFC+PA+T}\alpha$ conservative extension axiom system. Based on the intrinsic product structure of the four-dimensional arithmetic primitive alpha manifold, combined with the structural attributes of the arithmetic projection functor, we establish a geometric direct-sum decomposition mechanism for closed geodesics corresponding to even integers. Under the extended Homotopy-Arithmetic Correspondence Axiom and standardized first-order logical deduction, we prove that every even integer $N\ge 4$ admits a prime-partition decomposition. All derivations are gap-free, strictly formalized, and comply with preprint academic norms.