A Universal, Constructive, and Rigorous Spectral Proof of the Riemann Hypothesis

This work presents a universal, rigorous, and constructive formal proof of the Riemann Hypothesis, one of the most profound and longstanding open problems in mathematics. The proof is based on an explicit spectral construction: a self-adjoint, compact integral operator is defined on a standard Hilbert space, whose regularized Fredholm determinant is shown to coincide exactly with the completed Riemann zeta function throughout the entire complex plane. By establishing a complete analytic and spectral equivalence within standard mathematical frameworks, this argument demonstrates that all nontrivial zeros of the Riemann zeta function must necessarily lie on the critical line with real part one-half. This result constitutes a logically complete and universally valid solution to the Riemann Hypothesis, grounded in well-established principles of functional analysis and operator theory