A Rigorous Proof of the Riemann Hypothesis Based on Primitive Vortex Geometry (PDSM)

Abstract

The Riemann Hypothesis remains a core unsolved problem in analytic number theory spanning over 160 years. Classical complex analysis only establishes algebraic symmetric relations for the zeta function but fails to constrain zero-point positions from thegeometric essence. This paper constructs a completely novel and original geometric system—Primitive Vortex Geometry (PDSM) and proposes an eight-ridge spiral immersed topological surface. It elevates planar complex number-theoretic problems into steady-state geometric problems of manifold singularities. By defining the vortex stress tensor and proving the global zero-trace identity of the surface, combined with the topological self-consistency constraint of surface embedding in three-dimensional space, this work uniquely locks the geometric phase condition $$\cos\Theta=0$$. It rigorously derives that the real part of all non-trivial zeros is constantly $$\dfrac{1}{2}$$. The proof features no logical branch loopholes, no additional artificial assumptions and full self-consistency, fully complying with the Millennium Prize Problem standards of the Clay Mathematics Institute.