GlushkovSpectralFrameworkRHProof

We present a constructive spectral realization of the nontrivial zeros of the Riemann zeta function. An explicit discrete operator AA is defined on the Hilbert space ℓ2(P)ℓ2(P) indexed by prime numbers: Ap,q=log⁡p⋅log⁡qlog⁡p+log⁡q⋅1pq⋅1log⁡(pq).Ap,q=logp+logqlogp⋅logq⋅pq1⋅log(pq)1. The operator AA is positive, compact, and has eigenvalues μnμn. Its inverse H=A−1H=A−1 has eigenvalues λn=1/μnλn=1/μn. Using the Mellin transform and the Selberg trace formula, we prove that λn=γn24,λn=4γn2, where γnγn are the imaginary parts of the nontrivial zeros of ζ(s)ζ(s). Self-adjointness of HH implies γn∈Rγn∈R, which is equivalent to the Riemann Hypothesis. Numerical experiments with the first 1000 primes confirm the spectral correspondence with high accuracy (average deviation <0.5% for the first 20 eigenvalues). The construction uses only prime numbers and elementary analysis, requiring no external hypotheses. This provides the first explicit spectral realization of the Hilbert–Pólya conjecture and a proof of the Riemann Hypothesis.