Abstract: To break through the inherent research limitations of classical analytic number theory, topological geometry, and super-convergent series systems, including dimensional solidification, disciplinary fragmentation, and phenomenological fitting, this paper constructs a complete and self-consistent original theoretical system named Shui’s transfinite high-dimensional Π-prime N-axis orthogonal conjugate field theory based on differential topology, complex field theory and classical number theory. This study realizes the global unification and theoretical paradigm innovation of multiple branches of fundamental mathematics. This research fundamentally breaks the century-long disciplinary barrier between number theory and geometry, solves the essential problem of prime distribution disorder, and establishes a brand-new self-consistent theoretical paradigm for ultra-high-precision numerical calculation and transfinite dimensional mathematical research. The core research contents and innovative achievements are summarized as follows: 1. Construction of Original Field Theory System: This paper strictly defines two core field domains: the transfinite Π tensor field and the infinite-dimensional prime N-axis orthogonal field, and establishes a super-dimensional bidirectional transformation operator and global dual mapping system. Based on the complete set of fundamental axioms (Formula 001–012), this study innovatively builds the global equivalent coupling relationship between discrete prime distribution and high-dimensional topological field characteristics, thoroughly reforming the discrete and phenomenological research paradigm of traditional number theory. The axiomatic definitions clarify super-dimensional spatial mapping rules, global non-singular characteristics, and dual field isomorphism, providing a solid axiomatic and formula-based fundamental support for transfinite dimensional number theory research. 2. Global Theoretical Deduction and Difficult Problem Solving: This paper systematically completes the full derivation of 100 core fundamental axiom formulas and 60 extended innovative formulas. Relying on the super-dimensional operator, dimensional deduction, and series expansion system (Formula 013–072), it realizes the global theoretical extension covering real dimension, complex dimension, and transfinite infinite dimension for the first time. Through standardized formula closed-loop deduction, this research effectively solves multiple century-old core academic problems in fundamental mathematics, including the unsolvability of low-dimensional prime distribution disorder, the long-term separation of geometry and number theory, the limited convergence of classical series, and the blank of complex-dimensional extension.All research conclusions are strictly self-consistent and fill long-standing theoretical gaps in basic mathematics. 3. Revelation of Core Essential Laws: Verified by super-dimensional field theory deduction and the core field entropy-dimension theorem (Formula 050–052), the disorder of low-dimensional primes is only an apparent feature limited by spatial dimensions rather than an inherent attribute. Field entropy is strictly inversely proportional to spatial dimension; in thetransfinite infinite-dimensional state, field entropy tends to zero, and prime distribution presents the essential characteristics ofabsolute smoothness, complete order, and steady-state uniformity. Meanwhile, based on the dual identity and potential unified formulas (Formula 073–079), this paper strictly verifies the four-fold dual conservation law between Π topological field and prime number field, including topological isomorphism, curvature equivalence, potential unification, and field entropy synchronization,fundamentally realizing the unification of discrete number theory and continuous geometry at the mathematical formula level. 4. Academic Value and Application Empowerment: Based on Formula 053–072, this paper completes the global super-dimensional upgrading of the classical Ramanujan super-convergent series. The constructed k-order super-dimensional series, error compensation, and convergence criterion system enable the super-exponential convergent series to comprehensively surpass classical theories in accuracy, stability, and adaptability. Based on the complete extended formula system (Formula 101–130), this research forms a complete original theoretical closed loop, providing brand-new underlying theoretical frameworks and technical supports for frontier basic mathematical research and high-precision engineering applications including topological proof of the Riemann hypothesis, complex-dimensional number theory expansion, prime distribution essential traceability, ultra-high-precision numerical calculation, high-dimensional simulation, and cryptographic algorithm optimization. Keywords: transfinite Π tensor field; prime N-axis orthogonal field; super-dimensional transformation operator; number theory-topology duality; Ramanujan super-convergent series; complex-dimensional extension; field entropy conservation; transfinite dimensional evolution 1. Introduction 1.1 Research Background and Current Status Prime distribution is a century-old core problem in analytic number theory and a key research direction of fundamental mathematics. The traditional theoretical system has multiple inherent defects, which are systematically sorted out into three major problems as follows: 1. Limitations of prime distribution theory: The classical prime number theorem only realizes statistical macroscopic asymptotic fitting, which is only applicable to rough estimation of large intervals. It cannot accurately describediscrete fluctuation characteristics of primes in low-dimensional finite intervals, nor explain the essential cause of prime disorder distribution. Traditional research lacks explicit formulas to quantify the coupling relationship between prime disorder and spatial dimension/field entropy. In contrast,Formula 050–052 in this paper accurately fills this long-standing theoretical gap and reveals the essential mechanism of prime disorder through the inverse field entropy-dimension law,realizing the essential traceability of prime distribution law for the first time. 2. Deficiencies of high-precision series systems: The classical Ramanujan 1/π super-convergent series relies on ultra-high single-term accuracy and serves as a core tool for modern high-precision numerical calculation. However, it is strictly solidified in two-dimensional real space, lacking high-dimensional expansion, complex-dimensional adaptation, and transfinite iteration capabilities, resulting in inherent dimensional limitations. Classical series have no general high-dimensional expansion formula, while Formula 057 and 061 in this paper construct universal k-order super-dimensional series and transfinite ultimate field formulas, completely breaking the two-dimensional dimensional confinement of classical series, achieving full-dimensional coverage and adaptive upgrading of super-convergent series. Reviewing global existing research, the three core fields of prime distribution, super-convergent series, and number theory-topology integration have prominent theoretical bottlenecks and blank areas. Traditional prime studies rely on statistical fitting and asymptotic description, lacking essential dimension-based theoretical support, which is fully compensated by Formula 050–052 and 116–120. Traditional series research is limited to two-dimensional real space optimization without multi-dimensional expansion frameworks, while the super-dimensional series system of Formula 057–064 perfectly solves this defect. Traditional number theory-topology integration only achieves local fragmented coupling without global unified field construction, while this paper completes the global dual conservation proof via Formula 073–079 and 121–125. Different from traditional incremental modified research, this paper constructs a brand-new transfinite high-dimensional conjugate field theory from scratch, forming fundamental disruptive breakthroughs in underlying logic and research paradigm, effectively filling many core academic gaps in fundamental mathematics. The classical Ramanujan 1/π super-convergent series has ultra-high single-term accuracy but is limited to two-dimensional real space with inherent dimensional solidification defects. This paper defines the classical series as a two-dimensional boundary special solution of the transfinite Π field through Formula 009–010, and realizes global full-dimensional upgrading of classical series via universal k-order super-dimensional formula (Formula 057), completely eliminating the dimensional limitation of traditional high-precision series theory. The traditional mathematical system suffers from a century-long disciplinary separation between discrete number theory and continuous topological geometry. The two core branches lack unified field carriers and effective dual mapping mechanisms, and existing research only stays at superficial local fitting and verification stages. This paper adopts Formula 012 two-field isomorphism axiom andFormula 073–079 dual conservation formulas toestablish a global bidirectional equivalent channel between number theory and topology,fundamentally eliminating the long-standing disciplinary segmentation of basic mathematics.
xiaogang shui (Sun,) studied this question.