Critical Line Convexity: The Zeta-Gravity Metric and Geometric Structure of Zeta Zero Loci 临界线凸性与 ζ 引力度规:黎曼 ζ 函数的算术几何场理论

Abstract

Abstract This paper puts forward a novel geometric framework centered on the Zeta-Gravity Metric to systematically investigate the distribution of non-trivial zeros of the Riemann zeta function. Two core discoveries are established: first, the Critical Line Convexity (CLC) property. For any non-trivial zero , the horizontal profile attains its unique global minimum at , strictly decreasing on and strictly increasing on . Second, the Zeta-Gravity Metric defined on the phase space. Along the critical line, this metric is positive-definite, diagonally symmetric and fully decoupled, while its induced Gaussian curvature flips sign across , forming a geometric phase transition boundary. We perform high-precision numerical verification of the CLC property for the first 100 non-trivial zeros, and adopt the Lipschitz propagation method to deliver rigorous interval enclosures for the first 50 zeros. Numerical results demonstrate that the effective potential generates a restoring force pointing toward for all tested zeros; geodesic dynamics under the Zeta-Gravity Metric also exhibit universal convergence toward the critical line. Preliminary computations on primitive Dirichlet -functions indicate that the CLC property is universal across the entire -function family. This paper objectively identifies four core analytical gaps that prevent the current numerical and geometric evidence from constituting a complete rigorous proof of the Riemann Hypothesis. Even without a full proof of RH, the CLC property and Zeta-Gravity Metric represent original, independent mathematical results within zeta function theory.