Proof of the Riemann Hypothesis version 2

Abstract

This paper presents a proof of the Riemann Hypothesis by analyzing the topological phase synchronization of non-trivial zeros. Using a finite-N analogue ofthe Riemann zeta function derived via the Euler-Maclaurin formula, we demonstrate that the phase synchronization identity is an unavoidable algebraic necessity directly required by the convergence of the real and imaginary parts at anyzero. We formulate these trigonometric arguments as continuous phase flowslifted to the universal covering space of the circle, establishing that the phasetrajectories are strictly locked within a punctured neighborhood of the zeros.Crucially, by rigorously bounding the remainder term, we geometrically provethat the phase perturbation for a sufficiently large finite N is strictly restrictedto ∆θ < π/2. This mathematical constraint guarantees the topological stabilityof the phase branch, ensuring that residual error terms cannot induce branchjumps prior to taking the asymptotic limit. Based on this topological rigidity,we apply a proof by contradiction, assuming the existence of a zero off the critical line (a ̸=1/2). We show that disparities in the real parts strictly violate thephase synchronization condition as the continuous phase flow approaches thezero, leading to a mathematical contradiction. Consequently, we conclude thatall non-trivial zeros must lie exactly on the critical line a = 1/2.