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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)
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Title A Complete Spectral–Adelic Proof of Goldbach’s Conjecture via GRH and Operator-Theoretic Methods with Explicit Minor Arc Bounds and the Identity Dχ(s) ≡ Ξ(s, χ)
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)
Work type Unclassified
Tags χ), fredholm-type determinant dχ(s, a complete spectral–adelic proof of goldbach’s co
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Identifier 2510083263691
Entry date Oct 8, 2025, 6:10 PM UTC
License All rights reserved
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Author 100.00 %. Holder JOSE MANUEL MOTA BURRUEZO. Date Oct 8, 2025.
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