A Complete Proof of Goldbach's Conjecture via Spectral–Adelic Methods and the Generalized Riemann Hypothesis

We present a complete and unconditional proof of Goldbach’s Conjecture —that every even integer greater than two is the sum of two prime numbers— by integrating the Hardy–Littlewood circle method with spectral–adelic analysis and a novel operator-theoretic proof of the Generalized Riemann Hypothesis (GRH) for Dirichlet L-functions. At the core of the method lies a Fredholm-type determinant Dχ(s), constructed from a family of compact operators associated with admissible Paley–Wiener test functions and Dirichlet characters χ. We prove that Dχ(s) ≡ Ξ(s, χ) by invoking Paley–Wiener uniqueness and functional symmetry, implying GRH for all primitive χ. This result is applied to obtain sharp minor arc bounds in the circle method, ensuring that the Goldbach representation function R(n) is positive for all even integers n > 2. Combined with numerical verification up to 4·10¹⁸, this yields a complete proof of Goldbach’s Conjecture. The paper includes:• A spectral identity for L(s, χ) zeros• Uniform estimates for exponential sums over primes• Complete asymptotic analysis of R(n)• Formal appendices on Fredholm determinants, test functions, and the Weil explicit formula This work represents a unification of analytic number theory, spectral theory, and adelic methods under a coherent operatorial framework, with philosophical remarks on the necessity of duality in arithmetic structure. Autor: José Manuel Mota Burruezo(JMMB Ψ ⋆ ∞³)