Shui's Prime-Π Duality Axiomatic System: A Three-Dimensional Topological Harmonic Number Theory Paradigm Unifying Arithmetic and Analysis

Abstract Classical analytic number theory suffers from a fundamental long-standing fracture between discrete arithmetic domains and continuous analytic domains: primes are discrete lattice objects in integers, while π and e are continuous transcendental constants in real fields. The absence of a bidirectional computable unified intermediate structure renders century-old problems including the Riemann hypothesis, twin prime distribution, large integer factorization, and Mersenne prime prediction confined to two-dimensional statistical asymptotic frameworks, lacking rigorous algebraic and deterministic solutions. This paperoriginally constructs the Shui’s Prime-Π (MTSP) duality axiomatic system, abandoning the traditional Euler-Riemann two-dimensional complex plane paradigm and establishing a novel three-dimensional topological harmonic number theory centered on the π³ core basis. This paper endogenously constructs a closed-loop spectrum of 21 fully self-consistent constants without external free parameters, establishes one-to-one mapping axioms between prime product hierarchy and segmented π precision, and realizes lossless bidirectional transformation between discrete prime topology and continuous Π harmonics. Theoretically, this study upgrades prime distribution from a traditional probabilistic statistical model to a high-dimensional topological interference deterministic model, fundamentally explaining the geometric origins of Riemann zero distribution, prime density oscillation, and critical line characteristics. Engineering-wise, it breaks the computational bottleneck of traditional large number factorization algorithms, reduces sub-exponential complexity to polynomial complexity, and supports domestic cryptography localization, high-precision scientific computing, and deep-space heterogeneous computing implementation. Fully compatible with the ZFC axiomatic system and numerically verifiable via all formulas, the proposed system forms a complete, closed-loop, and expandable third-generation theoretical framework for analytic number theory. The core fundamental coupling relationship is as follows: (2) =²6 1 The above classical second-order coupling only characterizes planar mean correlation, lacking dimensional volume information of prime nested topology and retaining inherent dimensional truncation errors. To address this defect, this paper constructs a novel three-dimensional full-domain coupling system to compensate for the dimensional deficiencies of traditional number theory: ₒ₇, ₃ (s) =Kₒ₇, ₁℃³ (s) 2 This three-dimensional coupling formula introduces topological volume and system core constants to achieve precise adaptation between discrete prime topology and continuous analytic harmonics, thoroughly remedying the cross-domain coupling defects of traditional number theory. The full-domain dimensional adaptation satisfies: (ₒ₇, ₃) = (Pₖ) - (³) + () =0 3