Proof of the Riemann Hypothesis. Regularization of the Connes Spectral Triple via Weighted Sobolev-Gaussian Topology and the Riemann Hypothesis

In this paper, we construct a rigorous analytical scheme for the regularization ofAlain Connes' noncommutative spectral triple. Connes' original framework (1999)suered from the presence of a continuous spectral component caused by the sharpindicator truncation of the adelic spaces, which led to the loss of compactness of theDirac operator's resolvent and precluded a direct proof of the Riemann Hypothe-sis. We overcome this barrier by deforming the inner product and transitioning toa weighted Sobolev space induced by a smooth Gaussian window on the group ofprime numbers. It is proved that within this new weighted topology, the sequence ofresolvents of the truncated operators converges to the limiting resolvent uniformly inthe operator norm. Using the KatoRellich criterion, we establish the essential self-adjointness of the limiting operator and the complete vanishing of the de Brangesdeciency indices. Based on a modied SelbergConnes trace formula, a strict iden-tity is established between the spectral measure and the discrete measure of thenon-trivial zeros of the Riemann zeta function, completely ruling out the existenceof parasitic states and xing all zeros strictly on the critical line Re(s) = 1/2.