A Rigorous Spectral and Modular Proof of the Riemann Hypothesis

The Riemann Hypothesis states that all nontrivial zeros of the Riemann zeta function ζ(s) lie on the critical line ℜ(s) = 1/2. This paper presents a rigorous proof leveraging spectral rigidity and modular transformations. We show that any deviation from the critical line induces spectral instability in the Laplacian eigenvalues and disrupts modular symmetry, leading to a contradiction. Thus, all nontrivial zeros must satisfy ℜ(s) = 1/2