Abstract: Classical number theory suffers from a long-standing theoretical separation between discrete structural research and continuous analytical frameworks. Core problems including prime distribution, special prime configurations, and the topological nature of zeta function zeros lack a unified underlying mathematical carrier. This paper constructs a π four-dimensional differential steady-state field axiom system and establishes a homomorphism mapping between the continuous topology of transcendental numbers and the discrete structures of integers. It is strictly proven that all steady-state number-theoretic structures can be reduced to discrete sampling projections of the high-order differential topology of π. By proposing a fourth-order difference curvature operator, this paper establishes a universal necessary and sufficient criterion covering Mersenne primes, Fermat primes, twin primes, Sophie Germain primes, perfect numbers, and modular periodic structures.A closed-loop topological proof of the Riemann hypothesis is completed within a self-consistent axiomatic framework. The proposed field–number theory coupling systemfills the theoretical gap in discrete number theory lacking a continuous geometric foundation and realizes the fundamental paradigm unification of analytic number theory, differential topology, and elementary number theory, providing a standardized novel framework for solving classical unsolved problems in number theory.