Abstract: Traditional number theory generally adopts the simplified assumption of equal-weight and homogeneous primes. Although effective for macroscopic statistical analysis, thisparadigm fails to explain refined characteristics of prime distribution, local numericalfluctuation and spectral evolution. Consequently, core problems including the symmetry ofRiemann zeta zeros, twin-prime distribution, structural features of π, and high-efficiency large-integer factorization lack a unified geometric interpretation framework. This paper proposes aself-consistent double-layer topological model for primes. Based on structural contributiondifferences in integer modular spaces, primes are classified into inner primitive structuralprimes and outer high-order harmonic primes. Relying on low-level primitive primes, threehierarchical periodic bases (30, 210, 30030) are constructed, forming a dual-layer topologicalanalysis framework together with phase-spectrum decomposition methods. The modelsuggests that small inner primes dominate the fundamental modular structure and symmetry ofnatural number spaces, while large outer primes mainly introduce local spectral perturbationswithout altering the global topological framework. This framework provides new geometricinterpretations for prime orbital constraints, zeta zero symmetry, twin-prime aggregation andhierarchical characteristics of π. Meanwhile, structural stratification effectively optimizestraditional traversal algorithms and improves the efficiency of large-integer primality testingand factorization. As a supplementary, explanatory and verifiable geometric number theoryframework, the proposed model enriches structural analysis tools for discrete number theoryand complements empirical paradigms in classical arithmetic research.
xiaogang shui (Sat,) studied this question.