Arithmodynamics: A Spectrally Complete Geometric Realization of the Riemann Zeros

We present a rigorous derivation of the Hilbert-Pólya conjecture by constructing a quantum mechanical system whose energy spectrum coincides exactly with the imaginary parts of the non-trivial zeros of the Riemann zeta function, ζ(s). We introduce the Zahlraum (Number Space), a non-Euclidean Riemannian manifold M endowed with a metric gµν derived from the asymptotic density of prime numbers. On this manifold, we construct the Arithmodynamic Hamiltonian HˆAD, a self-adjoint operator that resolves the singularity issues of the BerryKeating xp-model. We provide a full proof of the operator’s essential self-adjointness on a weighted Sobolev space L2(M) and demonstrate, via the Gutzwiller Trace Formula, that its periodic orbits map one-to-one to the prime numbers. The spectral completeness of this system implies that all non-trivial zeros lie on the critical line Re(s) = 1/2.