A Spectral-Geometric Proof of the Goldbach Conjecture

Abstract

This paper presents a proof of the Goldbach Conjecture: every even integer2N ≥4 can be expressed as the sum of two prime numbers. The argument is builtwithin spectral geometry. We embed the prime-pair decomposition condition intothe zero locus of a singular spectral potential defined on a compact one-dimensionalclosed manifold obtained by one-point compactification of a line segment. The maintheorem establishes that for every integer N ≥ 2, the spectral potential possesses atleast one lattice point corresponding to a pair of primes (p,q) satisfying p+q = 2N.The proof is non-constructive and finite; it does not rely on any unproven numbertheoretic hypotheses, prime density asymptotics, or conditional analytic conjectures.All geometric objects are constructed explicitly, and the spectral correspondence isestablished via mollification of singular distributions, essential self-adjointness ofDirac-type operators, spectral zeta regularization, and the Selberg trace formula oncompact Riemannian 1-manifolds