Abstract: Classical number theory suffers from a long-standing theoretical separation betweendiscrete structural research and continuous analytical frameworks. Core problems includingprime distribution, special prime configurations, and the topological nature of zeta functionzeros lack a unified underlying mathematical carrier. This paper constructs a π four-dimensional differential steady-state field axiom system and establishes a homomorphismmapping between the continuous topology of transcendental numbers and the discretestructures of integers. Itisstrictlyproventhatallsteady-state number-theoreticstructurescanbe reducedtodiscretesamplingprojectionsofthe high-orderdifferentialtopology ofπ. By proposing a fourth-order difference curvature operator, this paper establishes a universalnecessary and sufficient criterion covering Mersenne primes, Fermat primes, twin primes,Sophie Germain primes, perfect numbers, and modular periodic structures.Aclosed-looptopologicalproofofthe Riemann hypothesis iscompletedwithinaself-consistentaxiomatic framework. The proposed field‒number theory coupling systemfillsthetheoreticalgap indiscrete numbertheory lackingacontinuousgeometric foundationandrealizesthefundamentalparadigmunificationofanalytic numbertheory, differentialtopology, andelementary numbertheory, providing a standardized novel framework for solving classicalunsolved problems in number theory.
xiaogang shui (Fri,) studied this question.