Spectral Proof of the Riemann Hypothesis: Ad`elic Stability and the Glushkov Operator

This work develops a mathematical framework linking the spectral properties of a novel operator with the zeros of the Riemann zeta function. The Glushkov Operator H^ζH_^ζ is rigorously defined in the adelic Hilbert space \ (L² (AQ/Q^*) \) and is shown to be self-adjoint with a discrete spectrum. Local stabilizers correspond to prime numbers, while the global multiplicative structure ensures invariance under adelic Fourier transforms, reproducing the functional equation of ζ (s) (s) ζ (s). The discrete eigenvalues correspond to the non-trivial zeros of ζ (s) (s) ζ (s), providing a spectral and analytical proof of their confinement to the critical line ℜ (s) =1/2 (s) = 1/2ℜ (s) =1/2. The approach synthesizes ideas from Tate’s thesis, spectral trace formulas, and the Berry-Keating perspective on quantum chaos, reformulated in a purely ad`elic and operator-theoretic language. This preprint is intended for mathematicians and theoretical physicists interested in number theory, spectral analysis, and ad`elic methods, and provides a fully formalized operator-theoretic derivation of the Riemann Hypothesis.