A Spectral Determinant Proof of the Riemann Hypothesis

We present a spectral determinant framework for the Riemann Hypothesis based on a self-adjoint operator construction and Krein spectral shift theory. Starting from the completed Riemann xi-function, we define a family of regularized Schrödinger operators as bounded perturbations of the harmonic oscillator, for which trace-class resolvent differences and perturbation determinants are rigorously available. Using spectral zeta regularization, we relate the logarithmic derivative of the perturbation determinant to a spectral shift function associated with the pair . The central result is a non-circular identification of this spectral shift function with the boundary phase of the xi-function, established via explicit trace and Fourier analysis in the regularized setting and normalized by high-energy asymptotics. This yields an exact determinant identity between the operator-theoretic perturbation determinant and the xi-function, up to a nonvanishing entire factor. Since zeros of the determinant correspond to eigenvalues of a self-adjoint operator, all zeros of the xi-function are shown to be real. The regularization is then removed by controlled trace-norm convergence. Together with the Supplementary Information, the argument provides a complete operator-theoretic proof of the Riemann Hypothesis, obtained via a spectral determinant and spectral shift framework.