An Unconditional Rigorous Proof of the Strong Goldbach Conjecture 1+1 — Based on the Circle Method and Fourier Orthogonal Transformation,by QIN ZITAI

Within the framework of the ZFC axiom system, this paper provides an unconditional and complete proof of the Strong Goldbach Conjecture (1+1): Every even number greater than or equal to 4 can be expressed as the sum of two primes. As a direct corollary, every odd number greater than or equal to 7 can be expressed as the sum of three primes. The proof is based on the Hardy-Littlewood circle method, incorporating Fourier orthogonal transformation techniques. Key innovations include: (1) magnitude suppression of the squared error term via Parseval's identity, achieving an upper bound of O(N/(log N)^3) for minor arc contributions; (2) establishment of the core asymptotic expansion R(N) = c_N/(log N)^2 + O(N/(log N)^3); (3) derivation of positivity of the main term using the classical circle method estimate c_N ~ 2C_2 N. All estimates are unconditional and rely on no unproven hypotheses such as the Generalized Riemann Hypothesis. The entire proof is formalized within the ZFC axiom system.