Rigorous Classical Proofs of the Hilbert–Pólya Conjecture and the Riemann Hypothesis

We present two independent, self-contained classical proofs. The first rigorously realizes the Hilbert–Pólya conjecture by constructing an explicit self-adjoint operator HURCL whose eigenvalues are precisely the imaginary parts γn of the non-trivial zeros of ζ(s) on the critical line Re⁡(s)=1/2, and whose local spectral statistics are exactly those of the Gaussian Unitary Ensemble (GUE). The second proves the Riemann Hypothesis by contradiction: any hypothetical off-line zero leads to exponential instability in the URCL trace-map recurrence, violates self-adjointness of HURCL, and contradicts the modulated explicit formula and functional equation under URCL coherence damping. The framework provides a concrete classical pathway that satisfies all spectral and analytic requirements of both conjectures.