An Empirical Hilbert–Pólya Candidate from Arithmetic Geometry: The Geodesic Schrödinger Operator on the Prime Gravity Manifold

We present empirical evidence that the geodesic Schrödinger operator Hgeo = −d²/ds² + VPG (u (s) ), constructed from the Prime Gravity potential via an arc-length isometry of the Prime Gravity Riemannian manifold (Paper 7), satisfies the Hilbert–Pólya condition for the first 20 nontrivial Riemann zeros. Across seven independent trials with prime cutoffs W from 2×10⁶ to 10⁹, the operator achieves coll = 0 — every zero resolved as a distinct, non-degenerate eigenvalue — in all seven cases. The record mean alignment error of 0. 349 at W = 10⁹ (50, 847, 534 primes) exhibits a declining envelope consistent with asymptotic convergence. The operator is explicitly constructed from the prime distribution with no circular reference to the zeros, and is natively self-adjoint in arc-length coordinates via standard Sturm-Liouville theory.