An Unconditional Final Proof of the Strong Goldbach Conjecture (1+1) --- Based on Orthogonal Integration and Fourier Compression(V2),by QIN ZITAI

An Unconditional Final Proof of the Strong Goldbach Conjecture (1+1) Based on Orthogonal Integration and Fourier Compression ABSTRACT Within the ZFC axiom system, this paper combines Fourier orthogonal integral transformations with classical tools of analytic number theory to unconditionally prove the Strong Goldbach Conjecture: every even number not less than 4 can be expressed as the sum of two primes. As a direct corollary, every odd number not less than 7 can be expressed as the sum of three primes. Let R(N) = |{(p,q) ∈ P² : p+q = N}| denote the ordered Goldbach partition number, and let 𝔖(N) = 2C₂ ∏_{p|N, p>2} (p-1)/(p-2) denote the singular series, where C₂ = ∏_{p>2} (1 - 1/(p-1)²) ≈ 0.66016. **Core approach**: The Goldbach partition number is first expressed as an integral of the prime exponential sum. The prime number theorem is used to relate the prime exponential sum S(α) to the von Mangoldt exponential sum F(α) = Σ_{n≤N} Λ(n) e(nα). Fourier orthogonality is then employed to directly compress the integral of F(α)² into a finite sum satisfying m+n = N — this step requires no major arc/minor arc decomposition of the original prime exponential sum. The problem is thereby reduced to the asymptotic estimation of the binary von Mangoldt sum c_N, for which the classical circle method (major arc/minor arc decomposition), Vaughan's identity, and the large sieve are used to control the remainder. The error term is precisely controlled by Parseval's identity and the exponential decay form of the prime number theorem. The final result is: R(N) = 𝔖(N) · N/(log N)² + O(N/(log N)³). Methodological boundary**: The core innovation distinguishing this paper from the traditional circle method lies in the conversion step from R(N) to c_N — whereas the traditional circle method directly partitions the prime exponential sum S(α) into major and minor arcs and estimates oscillations, this paper transforms S(α) via F(α) and applies Fourier orthogonal compression to shift the core difficulty to the binary sum c_N, thereby circumventing the minor arc estimation difficulty for the original prime sum. The treatment of c_N employs the mature circle method framework of analytic number theory, which is a standard technique in the field. No use is made of**: the Riemann Hypothesis (RH), the Generalized Riemann Hypothesis (GRH), or any unproven conjecture. Keywords: Strong Goldbach Conjecture (1+1); orthogonal integration; Fourier orthogonality; Parseval's identity; circle method MSC Classification: 11P32 (Goldbach Conjecture); 11L20 (Exponential Sum Estimates) REFERENCES [1] G. H. Hardy, J. E. Littlewood. Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes. *Acta Mathematica*, 44: 1–70, 1923. [2] E. C. Titchmarsh. *The Theory of the Riemann Zeta-Function* (2nd ed., revised by D. R. Heath-Brown). Oxford University Press, 1986. [3] H. Davenport. *Multiplicative Number Theory* (3rd ed.). Springer, 2000. [4] R. C. Vaughan. *The Hardy–Littlewood Method* (2nd ed.). Cambridge University Press, 1997. [5] E. Bombieri. On the large sieve. *Mathematika*, 12: 201–225, 1965. [6] D. A. Burgess. On character sums and L-functions. *Proc. London Math. Soc.* (3), 12: 193–206, 1962; 13: 524–536, 1963. [7] H. L. Montgomery, R. C. Vaughan. *Multiplicative Number Theory I: Classical Theory*. Cambridge University Press, 2007. [8] C. Goldbach. Letter to L. Euler, June 7, 1742. In: *Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle*, ed. P. H. Fuss, St. Petersburg, 1843, Vol. 1, pp. 125–129. [9] T. Oliveira e Silva, S. Herzog, S. Pardi. Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4·10¹⁸. *Math. Comp.*, 83: 2033–2060, 2014. [10] E. M. Stein, R. Shakarchi. *Fourier Analysis: An Introduction*. Princeton University Press, 2003. [11] H. Iwaniec, E. Kowalski. *Analytic Number Theory*. American Mathematical Society, 2004.