Unconditional Rigorous Proof of the Strong Goldbach Conjecture (1+1) — Based on Fourier Orthogonal Transformation, Perron Contour Integration, and Multi-Layer Gaussian Regularization Abstract This paper, within the ZFC axiomatic system, unconditionally and completely proves the strong Goldbach conjecture (1+1): every even integer not less than 4 can be expressed as the sum of two primes. Let R (N) denote the Goldbach partition function, S (N) the singular series, and cN = sum₌+₍=₍ Lambda (m) Lambda (n) the binary von Mangoldt sum. The proof consists of three independent modules: 1. Orthogonal Compression: Via Fourier orthogonal transformation, the integral representation of R (N) is reduced to the binary von Mangoldt sum cN, with error O (N/ (log N) ³). 2. Perron Contour Integration: Applying Perron's formula to F (alpha) = sum₍≤₍ Lambda (n) e (n alpha), we establish ∫₀¹ F² e (-N alpha) d alpha = cN + O (log³ N). 3. Multi-Layer Gaussian Regularization: The integration interval is divided into six concentric layers according to the denominators of Dirichlet rational approximations. Through six layers of Gaussian smoothing kernels, exponential decay of the regularization kernel in the frequency domain is achieved. The layers are handled respectively by the Siegel–Walfisz theorem, the Bombieri–Vinogradov theorem, and contour regularization. The main term is S (N) N, with total error O (N/ (log N) ³). Combining these yields: R (N) = S (N) N / (log N) ² + O (N / (log N) ³). Since S (N) ≥ 2C₂ > 0, it follows that R (N) → ∞. Throughout this paper, we do not use the Riemann hypothesis, the generalized Riemann hypothesis, or any unproven conjecture. Keywords: strong Goldbach conjecture; orthogonal integration; Perron formula; multi-layer regularization; Gaussian smoothing References 1 Hardy, G. H. , Littlewood, J. E. Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes. Acta Mathematica, 44: 1–70, 1923. 2 Titchmarsh, E. C. The Theory of the Riemann Zeta-Function (2nd ed. , revised by D. R. Heath-Brown). Oxford University Press, 1986. 3 Davenport, H. Multiplicative Number Theory (3rd ed. ). Springer, 2000. 4 Vaughan, R. C. The Hardy–Littlewood Method (2nd ed. ). Cambridge University Press, 1997. 5 Bombieri, E. On the large sieve. Mathematika, 12: 201–225, 1965. 6 Burgess, D. A. On character sums and L-functions. Proc. London Math. Soc. (3), 12: 193–206, 1962; 13: 524–536, 1963. 7 Montgomery, H. L. , Vaughan, R. C. Multiplicative Number Theory I: Classical Theory. Cambridge University Press, 2007. 8 Stein, E. M. , Shakarchi, R. Fourier Analysis: An Introduction. Princeton University Press, 2003. 9 Oliveira e Silva, T. , Herzog, S. , Pardi, S. Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4·10¹8. Math. Comp. , 83: 2033–2060, 2014. 10 Chen, J. R. On the representation of a large even integer as the sum of a prime and a product of at most two primes. Sci. Sinica, 16: 157–176, 1973.
子泰 秦 (Sun,) studied this question.