Prime Conch Geometry and Riemann Zeta Zero Analysis Based on Modulo-210 Periodic Vortex Topology (PDSM Framework)

Abstract

Abstract: 1. This paper constructs the Prime Vortex Dynamic System (PDSM) and establishes an eight-ridge prime conch topological structure constrained by dual modulo-30 and modulo-210 periodic boundaries, breaking the coarse-grained limitations of traditional prime geometric models. 2. Based on the refined subdivision of the classical eight-main-ridge topology via modulo-210 iteration, a 48-subridge subdivided vortex structure is generated. A novel strong coupling modulation mechanism between subridges and the transcendental number π is established, enabling dynamic and precise regulation of vortex stress tensors and geometric annihilation points. 3. The core number-theoretic von Mangoldt function Λ(n) is geometrized as the weight of the vortex field, forming the Mangoldt conch vortex field. Strict geometric correspondences are built between the conch surface integrals and the prime-counting function π(x) and Chebyshev ψ function. 4. By adopting the complex analytic continuation of the vortex field, a pure geometric integral expression for the Riemann zeta function is derived, constructing an isomorphic mapping system between conch-space annihilation points and non-trivial zeta zeros. 5. To address the widespread pseudo-zero interference problem in numerical simulation, the eigenvalue equation of the vortex stress tensor is deduced, and a numerically implementable quantitative screening criterion for valid zeros is established to accurately distinguish true zeta zeros from pseudo-annihilation points induced by π noise and subridge interference. 6. A complete polar coordinate discrete simulation scheme and error propagation system is constructed, and the theoretical precision upper bound of zero mapping is derived to ensure the reliability and rigor of numerical simulation results. 7. This study ultimately establishes a novel self-consistent research system of “number-theoretic periodicity—geometric topology—vortex field evolution—zero analyticity”, providing a brand-new theoretical paradigm and numerical method for the geometric research of prime distribution laws and the Riemann hypothesis.