Modular Orthogonal Topology of 30/210 Systems and Its Application to Unifying Discrete and Continuous Mathematical Structures

2. Abstract Fundamental mathematics is traditionally divided into four relatively independent branches: number theory, geometry, mathematical analysis, and topology. Long-standing disciplinary separation leads to inconsistent scale criteria and structural incompatibility between discrete integer systems and continuous geometric spaces. Traditional modular research merely takes low-order moduli as elementary screening tools, lacking systematic topological construction and invariant generalization. This study constructs a minimal two-stage 30/210 modular orthogonal topological hierarchical system and establishes six self-consistent fundamental axioms. An improved Mathematical Topology Sequence Projection (MTSP) method is proposed to characterize the statistical correlation between structural modular constraints and prime orbital distribution, correcting conceptual confusions regarding homeomorphism and bijective mapping in previous topological researches. Mathematical derivation verifies that modulo 30 and modulo 210 form complete closed topological bases for all primes greater than 5, possessing orbital orthogonality, operational closure, fractal self-similarity, and global scale conservation properties. This work establishes a series of hierarchical topological invariant formulas, constructing a complete discrete-continuous scale unification framework. The system effectively bridges discrete modular topological rules and continuous Euclidean metric evolution. Based on this self-consistent theoretical system, this paper systematically analyzes the topological mechanism of classic number theory problems, including twin prime asymptotic distribution, Mersenne prime structural constraints, and large integer factorization dimensionality reduction. Furthermore, this study provides unified topological interpretations and systematic theoretical supplements for the core connotations of the seven Millennium Prize Problems. All derivations in this paper are axiom-based, parameter-free, and numerically self-consistent. This research provides a novel unified topological paradigm for analyzing prime distribution laws, constructing modular invariant systems, and promoting the internal logical consistency of fundamental mathematics. 3. Introduction Fundamental mathematics develops through the long-term evolution of number theory, geometry, analysis, and topology. Each branch has formed mature theoretical systems and independent research paradigms. However, inherent logical segmentation persists: discrete number theory focuses on integer congruence and prime iteration but lacks continuous metric description; mathematical analysis accurately describes continuous evolution but does not naturally embed discrete integer constraints; traditional topology separates discrete point-set structures from continuous manifold research. Such separation results in a lack of unified underlying rules for explaining core mathematical phenomena. Prime distribution, modular structural characteristics, and the essential mechanism of many unsolved mathematical problems cannot be uniformly interpreted under a consistent scale benchmark. Existing modular topology studies generally regard modulo 30 and modulo 210 as elementary prime screening tools, ignoring their minimal completeness, orthogonal closure, and hierarchical fractal invariants. Contemporary unified mathematical attempts mostly depend on empirical fitting or hypothetical parameters, lacking pure axiomatic and self-consistent structural systems. To resolve the discrete-continuous scale contradiction in fundamental mathematics, this paper constructs a 30/210 two-stage modular orthogonal topological hierarchical system, derives complete invariant formula groups, establishes discrete-continuous homomorphism rules, and realizes the internal logical unification of multiple mathematical branches. On this basis, this paper analyzes the topological essence of classic number theory problems and Millennium Prize Problems, providing new theoretical perspectives for basic mathematical research.