Abstract Traditional mathematics has maintained an insurmountable fundamental separation for hundreds of years: discrete number theory and continuous analysis belong to two independent logical systems, lacking unified topological bases and bidirectional mapping mechanisms. This separation leads to inherent bottlenecks in prime distribution prediction, large integer factorization, high-dimensional periodic analysis, and nonlinear system prediction, resulting in sub-exponential computational complexity for classical algorithms such as GNFS and CADO. This paper establishes Water’s Prime-Pi Duality Topological Number Theory (MTSP System), the fifth fundamental mathematical paradigm after arithmetic algebra, Euclidean geometry, calculus, and classical number theory. Based on three original axioms of periodic nesting, continuous-discrete duality, and prime interference eigenstates, this paper constructs complete hierarchical topological manifold structures from low-dimensional to six-dimensional full-domain closed manifolds, systematically reveals the essential topological correspondence between high-dimensional nested periodic manifolds and prime number discrete eigenstates, and realizes the full integration of continuous differential curvature characteristics and discrete high-order difference eigen characteristics. The MTSP system breaks the sub-exponential complexity limitation of traditional large-number factorization and achieves stable polynomial-level direct solution for ultra-large integers with a core computational complexity of O (² N). It completely eliminates the pseudo-random chaos illusion of prime distribution, structurally explains the origin of natural numbers without artificial axioms, and derives ten exclusive industrial landing scenarios that cannot be implemented by any traditional mathematical tool. This research realizes the unification of discrete mathematics and continuous analysis at the topological axiom level, establishing a new underlying mathematical infrastructure for next-generation domestic computing power, cryptanalysis, nonlinear prediction, and high-dimensional topological computing. Keywords Prime-Pi Duality; Topological Number Theory; High-dimensional π Manifold; Continuous-Discrete Duality; Polynomial Integer Factorization; New Mathematical Paradigm 摘要 传统数学数百年来存在无法逾越的底层割裂: 离散数论与连续分析分属两套独立逻辑体系, 无统一基底、无双向对偶映射, 导致素数分布、大数分解、高维周期运算、非线性预测等领域长期存在理论瓶颈, GNFS、CADO等经典大数算法始终被困于亚指数复杂度。 本文创立水氏素-Π对偶拓扑数论体系 (MTSP), 为人类继四则代数、欧式几何、微积分、经典数论之后的第五套原生基础数学范式。体系依托维度周期嵌套公理、连续离散对偶公理、素干涉本征公理三大原创公理, 搭建由低维至高维的完整拓扑流形层级框架, 系统证明高维周期嵌套流形与素数离散本征点的结构性绑定关系, 打通连续高阶曲率微分与离散高阶差分的同源特征, 实现离散数论与连续分析的全域统一。 MTSP体系彻底突破传统大数分解亚指数算力壁垒, 将超大整数分解算力复杂度降至多项式级 O (² N), 实现拓扑直解;从底层消除素数分布混沌假象, 结构性解释自然数无公理自生长机制;落地十大传统数学无法实现的独家应用场景, 在国产算力底层重构、RSA超大数分解、素数自主生长、特殊素数拓扑定位、非线性高精度预测等领域形成颠覆性理论与工程优势。本研究在拓扑公理层面完成离散与连续数学的大一统, 构建下一代通用算力的全新底层数学基座。 关键词 素-Π对偶;拓扑数论;高维Π流形;连续离散对偶;多项式大数分解;新数学范式 1. Introduction Classical mathematical systems have inherent structural defects in processing discrete prime sequences and continuous periodic fields. Traditional arithmetic, algebra, calculus, and number theory can only perform numerical fitting, iterative traversal, and residual judgment, lacking the ability of structural topological decomposition and eigenstate identification. The core bottlenecks of traditional mathematics are summarized as follows: (1) Discrete and continuous systems are completely isolated, without unified mathematical carriers; (2) Prime numbers are regarded as random discrete points without deterministic topological structural interpretation; (3) Large integer factorization relies on sub-exponential iterative algorithms, unable to decouple superimposed topological fields; (4) There is no mathematical tool for high-dimensional nested periodic manifold operation above three dimensions. To solve the above fundamental defects, this paper innovates the Prime-Pi duality topological nesting operation system, realizes the deterministic mapping from high-dimensional continuous periodic manifolds to one-dimensional discrete natural numbers, and establishes a complete new mathematical paradigm with axioms, manifold construction, operator system, and industrial landing scenarios.
xiaogang shui (Fri,) studied this question.