Research on Spatial Mapping Model of Eight-Ridge Spiral Prime Conch and π Harmonic Ripple under the Primitive Vortex Framework

Traditional prime distribution theories are centered on the complex plane analytic continuation of the Riemann ζ-function, which constructs prime distribution laws based on a two-dimensional complex plane topology and suffers from inherent limitations including dimensional constraints, static topology, and the inability to explain the endogenous mechanisms of prime clustering and gaps. Based on the Primitive Vortex Theory (PDSM), this paper establishes a rigorous spatial mapping system of the three-dimensional eight-ridge spiral prime conch topology and the π harmonic ripple field. A unified cylindrical coordinate system is adopted to construct the discrete vortex skeleton equations of primes and the continuous π harmonic field equations. A vortex projection mapping operator is defined to realize triple bindings of coaxial constraint, scale mapping, and phase coupling between prime singularities and π ripple fields. Breaking through the traditional planar ripple assumption, this paper proposes a coaxial double-layer spiral symbiotic model, enabling the π ripple to ascend axially synchronously with the prime spiral. Four core topological effects are derived, including longitudinal vortex waveguide, eight-fold symmetric interference ring layers, constant scale difference evolution, and high-order critical decoupling. Quantitative closed-loop theoretical relationships between geometric topology and observable physical quantities are established via prime density-harmonic amplitude coupling equations and phase periodic synchronization constraints. Numerical simulations verify that the linear radial expansion of the prime conch and the square-root scale growth of π ripples generate inherent scale differences in high-dimensional intervals. Prime vortices gradually decouple from the π harmonic field at elevated axial heights, perfectly characterizing the distribution features of ultra-large prime intervals. This model constructs a brand-new topological paradigm for prime distribution that differs from the traditional complex analysis system, providing novel theoretical support for large integer factorization, periodic prime distribution, and the construction of vortex number theory systems.