Hierarchical p-Adic Framework for Gene Regulatory Networks: Theory and Stability Analysis

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 $\mathbb{Z}_p$ -- 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 $\mathbb{Z}_p$ and working over the complete, algebraically closed field $\mathbb{C}_p$, 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 $\mu$ -- 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 $\mu$ and the classification, although their definition and dynamical meaning require the analytical tools of $\mathbb{C}_p$, are fully determined by the discrete data. Minimizing $\mu$ over all $N!$ gene orderings defines an optimal regulatory hierarchy; for the Arabidopsis thaliana floral development network ($N=13$, $p=2$), a $\mu$-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.