Research on String Theory Vacuum Configuration Screening and Cosmic Unified Model Based on Prime Topology Axioms

Traditional string theory and high-dimensional geometric research have long relied on artificially preset topological structures and empirical parameter fitting, which generally suffer from non-unique geometric bases, artificial constraint systems, massive redundant vacuum solutions, and fragmented multi-scale mathematical frameworks. The essential defect lies in the lack of a native number-theoretic skeleton adaptable to high-dimensional topological spaces and capable of covering multi-scale degrees of freedom hierarchically. To break through the long-standing research paradigm of "physical assumptions precede mathematical constraints" in high-dimensional geometric physics, this paper pioneers a novel topological number theory system dominated by nested three-level modular bases (30-mod, 210-mod, 30030-mod) based on coprime prime sets at the frontier of fundamental number theory. This study redefines the topological mathematical connotations of primes and composites and establishes a dual number-theoretic structure: primes correspond to orthogonal perturbation-free steady-state topology, while composites correspond to coupled residual distorted topology. By constructing a three-level prime modular sequence, a standardized topological framework covering full low-, medium-, and high-dimensional scales is established to replace traditional artificial geometric construction methods. Based on the orthogonal uncoupled properties of modular reduced residue systems, a rigorous mathematical screening axiom for topological configurations is proposed to eliminate massive redundant solutions in high-dimensional spaces and uniquely lock steady-state topological structures. Relying on six-dimensional coprime prime orthogonal bases, a mathematical criterion for rigid locking of high-dimensional manifold parameters is constructed to solve the long-standing problem of free-floating high-dimensional geometric parameters. This paper establishes a complete set of quantitative topological indicators including topological residual operators, spacetime steady-state purity, and topological distortion degree, forming a quantifiable, iterative, and extendable novel topological evaluation system. Through global dual-invariant mathematical construction, self-consistent unification of multi-scale topological systems is realized, avoiding numerical divergence and scale fracture in traditional geometric modeling. Numerical verification confirms that the proposed three-level prime modular system possesses strong mathematical convergence, rigorous hierarchical structure, complete orthogonality, and excellent scale adaptability. Breaking the limitations of discrete analysis in classical number theory, this research elevates prime combinatorial structures to the underlying mathematical skeleton of high-dimensional topology, initiating a new research direction of prime modular topological number theory. It provides an original mathematical paradigm and standardized number-theoretic tool for high-dimensional differential geometry, topological field theory, multi-scale mathematical modeling, and constrained topological optimization.