Prime-Pi Dual-State Mathematics: Prime Conch Topology and Bidirectional Vortex Resonance Singularity Cluster Research

Abstract

Based on the original theoretical framework of Prime-Pi Dual-State Mathematics, this paper proposes the intrinsic free-angle mapping rule for natural numbers and constructs an eight-ridge helical geometric structure constrained by prime residue classes modulo 30. The model defines primes as discrete particle states and π as the global wave state, and establishes a three-dimensional spherical harmonic interference field system. It rigorously proves that the eight-ridge helix naturally differentiates into 4 left-handed and 4 right-handed symmetric vortex structures. The intersection region of opposite-direction vortices generates coherent interference resonance and spontaneously forms stable discrete singularity clusters. Through long-range numerical simulation with N_max=500, the global stability of conch topology is verified. Based on the point cloud dataset of singularity clusters, the implicit analytical equation of the prime conch topological envelope is obtained via the least squares method. This paper establishes a complete theoretical system containing six core axioms and one hundred graded formulas, and for the first time reveals that prime distribution possesses strictly self-consistent three-dimensional smooth conch topological geometric characteristics, breaking the cross-domain correlation between discrete number theory, continuous differential geometry and wave field theory. Keywords: Prime-Pi Dual-State Mathematics; Prime Conch Topology; Modulo-30 Eight-Ridge Structure; Bidirectional Vortex; Interference Singularity Cluster; Number Theory Geometry; Spherical Harmonics; Least Squares Fitting