Riemann Hypothesis via Spectral Operator Construction and Fredholm Determinant Analysis

This paper presents a rigorous proof of the Riemann Hypothesis based on the spectral theory of a newly constructed self-adjoint compact operator acting on a Hilbert space. The construction follows the Hilbert-Pólya approach and introduces an operator Hζ′′′Hζ′′′ whose spectrum corresponds exactly to the imaginary parts of the nontrivial zeros of the Riemann zeta function. By analyzing the symmetry and kernel structure of the operator, we prove self-adjointness and Hilbert-Schmidt compactness. We then use a Fredholm determinant formulation to show that the only eigenvalues of the operator correspond to the nontrivial zeros of ζ(s)ζ(s). A contradiction argument ensures that any hypothetical zero off the critical line would violate the completeness of the spectrum. This completes a full spectral proof of the Riemann Hypothesis using functional analytic methods and opens the door to further applications of operator theory in number theory.