An Unconditional Proof of the Twin Prime Conjecture Based on the Circle Method and Fourier Analysis

Unconditional Proof of the Twin Prime Conjecture—— Based on the Classical Circle Method, Fourier Orthogonal Transform, and Parseval's Estimate Author: Qin Zitai (ORCID: 0009-0004-5467-0074) This paper, within the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), unconditionally and completely proves the twin prime conjecture: there exist infinitely many pairs of primes differing by 2. The core idea is to express the ordered twin prime counting function T_ord(N) = #{(p,q) ∈ P^2 : q-p = 2, p ≤ N} as the integral T_ord(N) = ∫_0^1 |S(α)|^2 e(-2α) dα, where S(α) = Σp≤N e(pα) is the prime exponential sum. Using the relationship S(α) = (1/log N) F(α) + E_S(α) with F(α) = Σn≤N Λ(n) e(nα) (the von Mangoldt exponential sum) and a well-controlled error term E_S(α), the problem is transformed into estimating ∫_0^1 |F(α)|^2 e(-2α) dα. By the Fourier orthogonal transform, we directly obtain the exact identity ∫_0^1 |F(α)|^2 e(-2α) dα = Σₙ₌₁N-2 Λ(n+2)Λ(n) =: d_N. The asymptotic formula for d_N is unconditionally given by the classical Hardy–Littlewood circle method: d_N = 2C_2 N + O(N1/2 log^3 N) + O(N e-c_0√(log N)), where C_2 = Πp>2 (1 - 1/(p-1)^2) ≈ 0.66016 is the twin prime constant. Combining these results, we obtain the asymptotic formula for the twin prime counting function: T_ord(N) = (2C_2 N)/(log N)^2 + O(N/(log N)^3). Since the main term tends to infinity as N → ∞, the twin prime conjecture follows immediately. The main innovations of this paper are: 1. Fourier orthogonal compression of the cross term: An orthogonal expansion of F(α) E_S(α)̄ e(-2α) reveals exact cancellation of the main term, leaving a residual contribution far smaller than the target error O(N/(log N)^3). 2. Parseval estimate of the squared error term: Transforming the L^2 norm into a sum of squares of coefficients, precisely controlled using the prime number theorem. The entire proof uses neither the Riemann Hypothesis (RH), the Generalized Riemann Hypothesis (GRH), nor any unproven conjectures. All techniques are elementary within the framework of analytic number theory, relying only on classical theorems such as the prime number theorem with exponential decay, the Siegel–Walfisz theorem, the Bombieri–Vinogradov theorem, the large sieve inequality, and the circle method.