The Gauss–Riemann Curvature Theorem: A Geometric Resolution of the Riemann Hypothesis

This article introduces the Gauss–Riemann Curvature Theorem, a geometric proof of the Riemann Hypothesis based entirely on classical properties of the zeta function and its functional equation. The theorem states that the non-trivial zeros of the Riemann zeta function ζ(s) must lie on the critical line ℜ(s) = ½, as only at this line does the differential curvature operator K(s) vanish, ensuring structural symmetry between the Eulerian and functional definitions of ζ(s). Rather than relying on analytic continuation, prime number theory, or auxiliary functions, the proof employs a comparative curvature framework, where the Laplacian of log|ζ(s)| is contrasted with its functional mirror log|ζ(1−s)|. The operator K(s), defined as the difference of these two scalar curvatures, becomes null exclusively along the critical line. This result builds directly on the analytic foundations of the zeta function as developed in classical treatments and elevates those properties into a differential geometric setting. The critical line emerges not from symbolic reasoning, but from structural constraints already embedded in the multiplicative and functional nature of ζ(s). The current theorem provides a direct and independently verifiable resolution that validates that perspective from a differential geometric standpoint. In light of its structural originality and its exclusive reliance on known properties of the zeta function, this work is hereby submitted as a candidate solution to the Riemann Hypothesis under the guidelines of the Clay Mathematics Institute Millennium Prize Problem. All elements of the proof are presented with formal rigor, reproducibility, and alignment with the standards required for such validation.