Research on the Deep Correlation Between Prime Gaps and π Based on the Superposition of Eight-Ridge and Π-Ridge Topology

Abstract

Abstract: Based on the PDSM-NT primitive vortex number theory system, this paper constructs an eight-ridge topological model of primes under the CY₃ Calabi–Yau manifold and introduces the core axial structure, the Π-ridge. Breaking the limitation of traditional number theory that studies prime gaps only from the perspective of discrete arithmetic, this paper defines prime gaps as the topological geodesic distance of vortex singularities on the eight-ridge belt. It proves that the distribution, resonant small gaps, mismatched large gaps, and scale evolution of prime gaps are all jointly determined by the coupling, resonance, and mismatch effects of the mod-120 discrete periodic constraint of the eight-ridge and the 2π continuous spiral phase field of the Π-ridge. This paper establishes the Shui stress tensor as the core quantitative operator to uniformly describe manifold topological deformation and prime gap fluctuation, correlates the topological essence of non-trivial zeros of the Riemann zeta function, provides a novel topological interpretation for the twin prime conjecture and Cramér’s gap conjecture, and constructs a new theoretical system for the intercommunication of geometric topology and prime arithmetic statistics.