Primitive Vortex Mathematics (PDSM): Prime Singularities, Residues and Geometric Theory of the Riemann ζ-Function on Eight-Ridge Spiral Manifolds

Based on Primitive Vortex Mathematics (PDSM) and Vortex Dynamic Differential Geometry (V-DG), this paper constructs an eight-ridge spiral induced manifold as the geometric carrier of prime number distribution. Covariant vortex equations and intrinsic manifold covariant residues are defined on the manifold. Relying on the eight-fold rotationally symmetric phase attraction dynamics, stable prime singularities are screened, and the scaling relation between singularity residues and prime numbers is established. Conformal inversion duality enables topological mapping between the original singularity cluster and the dual singularity cluster at infinity, realizing the geometric reconstruction of the Euler product of the Riemann ζ-function. This paper systematically presents the complete topological, analytical, and dynamical properties of singular residues on the manifold, geometrically interprets the intrinsic mechanism of the ζ-function functional equation, and proposes the Primitive Vortex Geometry Conjecture to construct an intuitive geometric image of the non-trivial zeros of the Riemann ζ-function. This work provides a novel geometric research framework for prime distribution and core problems in analytic number theory.