The Riemannian Geometry of the Prime Gravity Manifold: ArcLength Isometry and the Geodesic Schrödinger Operator

We develop the Riemannian geometry of the Prime Gravity manifold and prove an exact arc-length isometry theorem: the Laplace-Beltrami operator on the manifold with metric g₁₁ = 1 + (dV_PG/du)² simplifies exactly under the arc-length coordinate s(u) = ∫√g dt to the flat Schrödinger operator H_geo = −d²/ds² + V_PG(u(s)), with no approximation and no residual first-derivative terms. The operator is self-adjoint on L²([0, s_max], ds) via standard Sturm-Liouville theory. This provides the geometric and operator-theoretic foundation for the Hilbert–Pólya investigation of Paper 8.