What question did this study set out to answer?

This research aims to unify discrete and continuous structures within number theory through a new axiom system.

July 5, 2026Open Access

π Four-Dimensional Differential Steady- State Field: The Grand Unified Theory of Number Theory and Topological Self- Consistent Proof of the Riemann Hypothesis

Key Points

This research aims to unify discrete and continuous structures within number theory through a new axiom system.
Constructed a π four-dimensional differential steady-state field axiom system.
Established a homomorphism mapping between transcendental numbers and integer structures.
Developed a fourth-order difference curvature operator to analyze various prime types.
Proved the existence of a universal criterion for representing various primes and number structures.
Completed a closed-loop topological proof of the Riemann hypothesis within the axiomatic framework.
Demonstrated a unifying relationship between analytic number theory and differential topology.

Abstract

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.

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