About the Riemann Hypothesis. Proof of Compactness of the Dirac Operator's Resolvent in Connes' Adelic Spectral Triple

This paper presents a rigorous functional-analytic proof of the compactness of theresolvent of the modified Dirac operator D = −i(x*d/dx + 1/2)on the space of adelic commensurabilityclasses AQ/Q∗. The justification relies on the compactness of the non-Archimedean topology of the ring of integral adeles bZ, the embedding of weighted Sobolevspaces, Schauder’s finite-rank approximation criterion, and the theory of KMS states onC∗-algebras. The proven compactness implies a purely discrete and real character of theeigenvalue spectrum, which eliminates the existence of de Branges defects and fixes thezeros of the Riemann zeta function on the critical line.