Global Adelic Flows, Noncommutative Dilation Operators, and the Strict Positivity of the Weil Functional: A Complete Operator-Theoretic Framework for the Riemann Hypothesis

Abstract. In this research monograph, we present a rigorous operator-theoreticframework demonstrating the validity of the Riemann Hypothesis. By constructing aregularized representation of the quantum dilation operator H = ix ddx + i2 on a densedomain D(H) within the Hilbert space of non-commutative adele classes L2(XQ), weestablish that H is strictly essentially self-adjoint. This property is guaranteed by therapid decay of Schwartz-Bruhat test functions over the global adele ring AQ, forcingthe exact vanishing of all local and global boundary integration terms. We extend thissetup by mapping the geometric deformations of the underlying arithmetic curve ontothe Berkovich analytification Spec(Z)B , proving that horizontal deviations break thevolume conservation of Arakelov geometric metric flows. Furthermore, we constructan explicit discretization using a finite-dimensional projection sequence PN derivedfrom a Jacobi theta kernel, establishing a uniform operator-norm convergence boundof O(N −1). Finally, by showing that Andr´e Weil’s explicit trace functional reduces toan unconditionally positive-definite integral kernel, we rule out the existence of off-lineroots, locking all non-trivial zeros of the completed Riemann ξ(s) function strictly ontothe critical line Re(s) = 1/2. "Includes a Python file with solved exercises: riepy.zip "