Abstract Traditional analytic number theory, sieve theory and probabilistic number theory have long relied on statistical fitting, residual period induction and stochastic probability hypotheses to interpret prime distribution laws, and possess four essential inherent defects: inability to explain the inherent energy-level stratification differences of special primes, inability to establish homologous distribution mechanisms for various special primes, inability to construct dual evolution systems for prime and composite substrates, and inability to eliminate the stochastic cognition of prime distribution from geometric essence. To completely break the fragmented, empirical and probabilistic research shackles of traditional number theory, this paper completely abandons the classical analytical framework and stochastic probability presupposition, and constructs the world’s first unified topological field theory of primes based on the minimal indivisible self-closed Ω=2×3 topological lattice as the underlying geometric substrate and π³~π¹⁸ multi-order higher-dimensional curvature manifolds as the evolution space. This paper achieves subversive paradigm-level original breakthroughs in the field of analytic number theory: it pioneers a unified closed classification system for structured special primes and independently defines the exclusive Shui-type prime system, which classifies traditionally isolated and scattered twin primes, cousin primes and Mersenne primes as structured subclasses with different coupling orders and energy-level gradients under a single high-dimensional topological field. Relying on the fourth-order differential topological mutation operator, modular topological dilution conservation model, interval boundary topological entropy increase defect mechanism, high-dimensional orbital symmetry breaking rules, and π-dimensional curvature field evolution equations, this paper provides a complete, self-consistent, zero-exception closed-loop geometric explanation for all core number theory anomalies, including the global sparse evolution of primes, fixed energy-level differences of special primes, orbital polarization of Mersenne primes, invalidation of finite interval boundaries, and continuous expansion of composite substrates. It realizes the historic paradigm innovation of number theory research: transforming disciplinary logic from numerical statistics to geometric topology, from probabilistic randomness to field-theoretic necessity, and from discrete empirical fitting to continuous high-dimensional field projection. This paper constructs five original topological axioms, six core topological theorems, the innovative π-prime dimensional homology theory, and the exclusive three-class closed classification system for Shui-type primes, together with a complete set of independently derived analytical closed-form solutions and differential evolution equations. It comprehensively covers all evolution laws of structured Shui-type primes, ordinary discrete primes, and composite gap duality, forming a contradiction-free, omission-free, exception-free and self-consistent closed-loop prime distribution system. This research completely breaks the centuries-old academic barrier separating discrete prime research and continuous π geometric system research, establishes the fundamental homologous relationship between high-dimensional irrational curvature geometry and discrete prime number sets for the first time, and provides a brand-new original theoretical cornerstone for special prime mechanism research, high-precision primality modeling, novel prime sieve algorithm development and high-dimensional number theory system expansion. Keywords: Prime Distribution; Ω Topological Lattice; Fourth-Order Differential Operator; π Higher-Dimensional Manifold; Shui-type Prime; Topological Energy Level; Modular Dilution; Symmetry Breaking; π-Prime Homology Theory; Unified Field Theory 1. Introduction The prime distribution problem is the core fundamental problem of analytic number theory. Traditional research has adopted multiple technical approaches including sieve optimization, asymptotic formula derivation, probabilistic statistical modeling and residual period analysis, yet has long been trapped in the inherent research dilemma of "phenomenon fitting, mechanism absence, and probability fallback". Existing theories can only approximately describe the macroscopic statistical laws of primes and fail to provide necessary geometric explanations for numerous deterministic topological anomalies. The core unsolved contradictions can be systematically summarized into six categories, forming an insurmountable systematic bottleneck for traditional classical number theory. First, the continuous prime topological upper bound anomaly. In large number domains, there exist no triple consecutive prime structures in natural numbers, and the maximum length of continuous prime marks is strictly 2, with only small-value exceptions. Traditional theories only provide superficial explanations through the sieve of multiples of 2 and 3, without essential prohibition proofs from the perspectives of dimensional topology, lattice structure and active site arrangement, failing to illustrate the inevitability of this phenomenon. Second, the finite interval boundary defect anomaly. The head and tail boundary regions of any finite natural number segment inherently lose the generation capability of high-energy prime pairs and cannot produce high-order structured Shui-type primes such as twin primes and cousin primes; only the complete lattice middle segment of the interval possesses high-energy prime generation conditions. Traditional number theory lacks relevant mechanisms for boundary topological incompleteness, entropy attenuation and resonance failure, and cannot explain the differential primality distribution characteristics of intervals. Third, the special prime energy-level stratification anomaly. Three classical types of special primes (twin primes, cousin primes, Mersenne primes) exhibit permanently fixed and immutable hierarchical differences in distribution density, field response intensity, structural stability and topological energy level, which are scale-independent and universally constant. The existing academic system lacks unified energy-level sorting criteria, quantitative difference formulas and underlying topological mechanism support, and always regards the three types of primes as isolated special cases. Fourth, the global field evolution asymmetry anomaly. The forward primality topological pulse decays logarithmically with the expansion of numerical scale, leading to the continuous collapse and extinction of structured high-energy primes; the negative composite vacuum substrate field deepens steadily with continuous expansion of gaps. The evolution rate, trend and logic of the dual field domains are completely asymmetric. Traditional analytical systems cannot distinguish the essential differences between binary field domains nor explain the geometric origin of asymmetric evolution. Fifth, the modular topological dilution reverse anomaly. High-order modular expansion greatly increases the number of prime candidate active orbits, which should statistically improve the generation probability of special primes. However, in actual topological field responses, the signal intensity of high-order Shui-type primes such as twin primes continues to decay and energy levels decrease, presenting a reverse evolution law that contradicts traditional probabilistic cognition, which cannot be self-consistently explained by any classical theory. Sixth, the special prime orbital polarization anomaly. The two active orbits of the Ω lattice are geometrically equivalent but completely inequivalent in energy-level bearing and structural adaptability. Mersenne primes are only stably distributed on the 6k+1 forward orbit with no effective Mersenne prime distribution on the 6k+5 negative orbit, indicating inherent high-dimensional topological symmetry breaking in the prime system, which lacks dimensional mechanism modeling of orbital polarization and symmetry breaking in traditional number theory. Targeting all the above systematic defects and unsolved problems, this paper completely abandons the traditional probabilistic statistical thinking and empirical numerical fitting research paradigm that has dominated number theory for thousands of years, takes the Ω=2×3 minimal self-closed indivisible topological lattice as the sole underlying geometric substrate for natural number primality distribution, and constructs a global full-dimensional topological unified field theory covering one-dimensional discrete values to eighteen-dimensional π curvature manifolds. Through a strictly self-consistent system of original topological axioms, core topological theorems, high-dimensional field evolution equations, and exclusive Shui-type prime classification system, this paper realizes the geometric inevitability, field-theoretic unification, mechanism closed-loop and quantitative numerical interpretation of all prime distribution anomalies for the first time in history, fills the long-standing underlying theoretical gaps in basic number theory, and completes the fundamental paradigm upgrade and disciplinary reconstruction of prime distribution theory.
xiaogang shui (Fri,) studied this question.
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