摘要: Gene regulatory networks exhibit hierarchical organization across scales; capturing this structure mathematically requires a metric that distinguishes regulatory influence at each level. We show that the ultrametric of the p-adic integers ℤp—whose self-similar nested-ball structure is a natural fractal encoding of multi-scale organization—provides such a framework. Embedding the N-gene state space into ℤp and working over the complete, algebraically closed field ℂp, we prove the existence of rational functions that interpret the discrete dynamics and construct hierarchical approximations at each resolution level. These constructions yield a stability measure μ—aggregating how the dynamics contracts or expands across resolution levels—and a ball-level classification of fixed points—contracting, expanding, or isometric—extending the attracting/repelling/indifferent trichotomy of non-Archimedean dynamics from points to balls. A key result is that μ and the classification, although their definition and dynamical meaning require the analytical tools of ℂp, are fully determined by the discrete data. Minimizing μ over all N! gene orderings defines an optimal regulatory hierarchy; for the Arabidopsis thaliana floral development network (N=13, p=2), a μ-minimizing ordering places known master regulators—UFO, EMF1, LFY, TFL1—in the leading positions and recovers the accepted developmental hierarchy without biological input beyond the transition map.