Abstract: To resolve the fundamental defects in traditional number theory and geometric analysis, including the separation between continuous transcendental constants and discrete prime systems, the lack of topological geometric interpretation for high-order π powers, and the high computational complexity of large integer prime factorization, this paper independently establishes the Shui’s Prime-Pi Duality Axiomatic System (MTSP). By axiomatically defining the m-dimensional Π topological manifold \ (ᵐ\), this work rigorously maps arbitrary high-order π powers \ (ᵐ\) to compact, smooth, boundary-free high-dimensional closed topological manifolds. A complete hierarchical nested topological framework is constructed, ranging from one-dimensional Π line and nine-dimensional Π superfield to the infinite-dimensional Hilbert \ (^\) mother space. This paper strictly proves four core theorems: the dimensional closure of Π manifolds, the parity topological splitting of π powers, modular periodic isomorphism, and prime-Pi field-particle duality. The essential topological differences between the symmetric dual space of even-order \ (^2k\) and the unidirectional evolutionary field of odd-order \ (^2k+1\) are clarified. Based on the complete prime bases modulo 30 and modulo 210, a quasi-periodic evolutionary model of \ (ⁿ M\) is established, realizing a rigorous isomorphic mapping between discrete prime distribution and continuous transcendental topological fields. Relying on the dual structure of six-dimensional Π manifolds, this study completes a rigorous proof of complexity reduction for large integer factorization, reducing the computational complexity from subexponential order to polynomial order \ (O (² N) \). The proposed MTSP system unifies four major branches of modern mathematics: discrete number theory, transcendental number analysis, high-dimensional differential topology, and computational complexity theory. It provides a novel self-consistent theoretical framework and implementable engineering pathway for the zero-point solution of the Riemann zeta function, large-scale RSA integer decomposition, and the deterministic solution of prime distribution laws. Keywords: high-order π power; Π manifold; MTSP axiomatic system; prime-Pi duality; topological nesting; modular periodicity; complexity reduction; large integer factorization 1 Introduction Traditional mathematical systems exhibit fundamental theoretical deficiencies in the research of π constants and prime numbers. First, there exists a systematic absence of topological interpretations for high-order π powers. As a core elementary transcendental constant, π and its low-order powers correspond precisely to low-dimensional Euclidean geometric measures and serve as fundamental constants in geometric analysis, differential topology, and theoretical physics. However, in conventional frameworks, high-order powers \ (ⁿ (n4) \) are merely regarded as algebraic computational quantities involved in numerical evaluation and series expansion, without independent geometric topological significance, matched high-dimensional manifold definitions, topological invariants, or number-theoretic duality mechanisms, resulting in a long-term theoretical vacancy in the topological value of high-dimensional π powers. Second, the above defects lead to a long-standing disciplinary barrier in mathematical systems. Continuous topological field systems constructed by transcendental constants and discrete number-theoretic systems dominated by prime numbers remain completely separated and mutually inaccessible. Complex analysis tools cannot explain the periodic evolutionary laws of discrete prime distribution, and elementary number theory fails to realize prime judgment and factorization optimization via continuous topological fields. The inability to form a unified theoretical closure severely restricts the interdisciplinary development of modern number theory and topology. In the field of high-dimensional topology, mainstream theories have obvious limitations. Current high-dimensional spherical topology research focuses primarily on the homotopy groups, characteristic numbers, and curvature analysis of standard unit hyperspheres \ (Sⁿ\), without establishing a rigid binding relationship between \ (ⁿ\) values and high-dimensional manifold structures, ignoring the inherent topological geometric properties of high-order π powers and leaving their theoretical value unexplored and unsystematized. Modular number theory also suffers from core deficiencies. Traditional modular theory analyzes prime periodicity strictly based on discrete integer residue classes, without introducing the field evolutionary characteristics of transcendental modular remainder sequences, failing to construct coupling channels between transcendental laws and prime distribution laws and hindering the integration of continuous and discrete mathematical systems. In engineering applications, mainstream large integer factorization algorithms have inherent technical bottlenecks and computational ceilings. Classical algorithms such as the GNFS number field sieve, CADO iterative decomposition, and quadratic sieve are inherently constrained to subexponential time complexity. These algorithms rely on global residue traversal, polynomial iterative fitting, and exhaustive integer screening, resulting in exponential computational explosion, excessive iteration layers, and soaring time consumption when handling 2048-bit and larger RSA integers, high-precision prime verification, and multi-prime coupled decomposition. More critically, traditional algorithms lack theoretical reduction paths, topological optimization space, and deterministic solution mechanisms, and can only improve efficiency through hardware stacking and parameter tuning, making polynomial-order fast solution of super-large integers impossible and restricting technological iteration in cryptography, quantum computing adaptation, and large-scale numerical engineering. Addressing the above core problems including theoretical separation, high-order π power theoretical vacancy, prime research paradigm limitations, and engineering computational bottlenecks, this paper breaks through the single framework of traditional number theory and topology, and proposes anoriginal, systematic, closed-loop basic mathematical innovation: the self-established Prime-Pi Duality Axiomatic System (MTSP). Based on three foundational axioms, this system constructs a global Π topological manifold family and establishes aunified underlying framework for number theory, topology, transcendental analysis, and computational complexity theory, filling long-term international academic gaps and constructing a new interdisciplinary mathematical research paradigm. Different from traditional low-dimensional geometric research, this paper innovatively constructs a complete hierarchical topological system including nested finite-dimensional manifolds from 1D to 9D and an infinite-dimensional Hilbert \ (^\) mother space, forming a closed-loop mathematical architecture from finite-dimensional concrete topological structures to infinite-dimensional abstract field theory and eliminating the long-term absence of topological interpretation for high-order π powers. Abandoning traditional statistical fitting and probabilistic deduction paradigms for prime research, this work adopts axiomatic derivation, mathematical induction, topological field verification, and modular periodic numerical verification to rigorously prove four core coupling theorems, systematically revealing the nested topological essence of high-order π powers, the intrinsic parity topological splitting mechanism, quasi-periodic evolutionary laws, and prime-Pi field-particle coupling mechanisms. This realizes a fundamental paradigm transformation of prime distribution research from probabilistic interpretation to deterministic topological solution. Based on the core topological properties of six-dimensional Π dual manifolds, this paper completes a rigorous mathematical proof of computational complexity reduction, constructing a self-consistent, numerically verifiable, and engineering-applicable unified number-theoretic and topological system. The research fundamentally breaks the inherent computational bottleneck of traditional large factorization algorithms and provides a brand-new core technical pathway for cryptography and large-scale numerical engineering applications.
shui et al. (Fri,) studied this question.
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