Dynamic Number Theory: The Deep Structure of Natural Numbers as Evolution Step Counts

Abstract Natural numbers are traditionally regarded as primitive objects—their existence is guaranteed by the Peano axioms or the ZFC axiom of infinity. In this paper, starting from the successor determinacy of Generative Axiom 3, natural numbers are rigorously defined as the evolution step-count trajectory of the successor operator acting on the generative zero element; the phase closure condition of Axiom 4 further endows the step count with a periodic structure. The central discovery is that prime numbers are rotating vectors of the phase coupling equation in the number-theoretic context—their magnitude ln p is rigorously derived from the spectral expansion of ζ'/ζ, and their phase t ln p is the intrinsic angular velocity of the evolution step count. Based on this rigorous construction, this paper proves that the x/ln(x) of the prime number theorem is the inevitable signature of the coherent superposition of all prime rotating vectors on the critical line—it counts not only the density of primes but also the number of independent group generators. The factor 1/ln(x) is the optimal balance between independence and completeness of group generators: a higher density would make generators redundant, while a lower density would leave the group structure incomplete. The Riemann critical line 1/2 is the self-dual fixed point of the rotating vectors under the flip operation—it is simultaneously the position where the phase locking of all group generators attains optimal symmetry under the flip operation. When all prime vectors are superimposed on the complex plane, isoperimetric optimality enforces phase locking, giving rise to the prime spiral—each spiral arm corresponds to an independent U(1) generator. The deep connection between primes and group structure is thereby revealed: the phase periodicity of the rotating vectors intrinsically implies closed paths, and closed paths enforce a group structure. Every prime corresponds to an unavoidable topological constraint on the evolution step-count trajectory—the closed path encircling it constitutes an independent U(1) group generator. The prime number theorem is the counting of groups; the Riemann Hypothesis is the symmetry condition of groups. Three millennium problems are thus unified in the same geometric picture and the same algebraic structure: the prime spiral. Keywords: dynamic number theory; phase coupling equation; evolution step count; prime number theorem; Riemann Hypothesis; rotating vector; prime spiral; group generator; self-dual balance