Exploration on tExploration on the Correlation Between Modulo 30 Prime Residual Distribution and Real Number Dichotomy Structurehe Correlation Between Modulo 30 Prime Residual Distribution and Real Number Dichotomy Structure

Abstra Abstract: Discrete prime distribution in integer sets and the dichotomous structure of real numbers (algebraic numbers and transcendental numbers) are two fundamental research objects in analytic number theory and real variable theory. Traditional research independently explores the two fields, lacking systematic discussion on their structural homology and correlation. This paper takes the classical modulo 30 reduced residual system as the basic research carrier, based on universally recognized theories including Dirichlet prime theorem, Lebesgue measure theory, finite abelian group theory and fractal set properties. By sorting out the standard structural characteristics of the modulo 30 eighth-order multiplicative group, counting the distribution rules of primes in residual classes, and comparing the topological attributes of discrete integer sets and continuous real number sets, this paper systematically discusses the structural correlation between prime residual segment distribution and real number dichotomy segments. The study verifies that the modulo 30 residual system is the minimal low-order modulus with complete symmetric group structure for prime screening, sorts out the inherent logical consistency between discrete prime-coprime composite dichotomy and continuous algebraic-transcendental number dichotomy in set partition, measure attributes and group isomorphism. On the basis of existing classical theories and verified statistical laws, this paper summarizes the existing research limitations, verifiable correlation characteristics and feasible subsequent research directions, and avoids unproven hypothetical theorems and artificial quantitative models. This work supplements the interdisciplinary comparative analysis paradigm of number theory and real variable topology, and provides reliable empirical and theoretical basis for further research on cross-domain structural correlation.