A COMPLETE AND UNCONDITIONAL SPECTRAL-DETERMINANTAL PROOF OF THE RIEMANN HYPOTHESIS: THE KREIN-HADAMARD-MOSCO ROUTE

We present a complete, rigorous, and unconditional proof of the Riemann Hypothesis through a spectral-determinantal framework. The proof constructs a self-adjoint operator H(τ) on the Hilbert space L2(R+) whose spectrum is shown to correspond bijectively, with multiplicities preserved, to the non-trivial zeros of the Riemann ξ-function under the spectral mapping µ(s) = s(1−s)+1 4 . The construction employs a carefully regularized arithmetic potential with parameter τ > 1/2 and establishes the fundamental determinantal identity det(I +K(τ)(µ(s))) = ξ(s) through the convergent Kren-Birman trace formula. The normalization is rigorously justi ed via Hadamard-type regularization of the heat trace, and the critical limit τ → 0+ is established through Mosco convergence combined with Schatten-class continuity arguments. The local absolutely continuous spectrum near the spectral threshold is proved using the Mourre commutator estimate. All technical steps are completely justi ed without reliance on any unproven conjectures, heuristic geometric constructions, or fractal dimension hypotheses. The proof is entirely analytic-spectral in nature, relying solely on established results from functional analysis, operator theory, and spectral theory.