A Pseudo-Riemannian Metric on the Critical Strip of the Riemann Zeta Function: Schwarzschild Correspondence, Geodesic Structure, and a Geometric Reformulation of the Riemann Hypothesis

We dene a pseudo-Riemannian (Lorentzian) metric tensor gij = ∂i∂jΦζ on the critical strip of the Riemann zeta function ζ(s), where Φζ (s) = − log |ζ(s)|. By the harmonicity of Φζa direct consequence of ζ being holomorphicthe metric satises gσσ + gtt = 0 identically, forcing Lorentzian signature (+, −). We establish ve main results: (1) gij has Lorentzian signature forced by analyticity; (2) the pole at σ = 1 is a coordinate singularity (curvature K nite), in exact analogy with the Schwarzschild horizon; (3) the critical line Re(s) = 1 2 is a geodesic of gij via Γ σ tt|σ=1/2 = 0, proved from the functional equation; (4) the geodesic has variable causal character with a null point at t ≈ 5.561; (5) non-trivial zeros of ζ are curvature singularities with K ∼ cn/|s − ρn| 2 . The Riemann Hypothesis is equivalent to: all curvature singularities lie on the geodesic Re(s) = 1 2 . All results are veried numerically at 25-digit precision using mpmath over a 40 × 60 grid. Keywords: Riemann zeta function, pseudo-Riemannian metric, Lorentzian signature, Schwarzschild metric, geodesic, curvature singularity, FisherRao metric, Riemann Hypothesis. MSC 2020: 11M06, 53B30, 81T20, 53C50.