Hierarchical Non-Archimedean Stability of Finite Discrete Dynamical Systems: A Variational Theory over Coordinate Orderings

Abstract

We develop a non-Archimedean reading of finite discrete dynamical systems in which the order chosen on the coordinates is itself a dynamical observable. For a map $f : \mathbb{F}_p^N \to \mathbb{F}_p^N$, an ordering embeds the phase space into the $p$-adic integers, so that agreement in the first $n$ coordinates means membership in a common ball of radius $p^{-n}$. Realizing $f$ as a compatible family of ball-level maps over $\mathbb{C}_p$, we attach to each fixed point scale-resolved indices of expansion, attraction, and invariance. These indices are computable from the finite data alone, the rational interpreter serving as a theoretical device. The expansion index $μ_E$ is a function on the symmetric group $S_N$, and minimizing it gives a variational principle that selects a coordinate hierarchy intrinsic to $f$. On the Boolean Arabidopsis thaliana floral network ($N=13$, $p=2$) the minimizing ordering recovers the eight documented key regulators with Spearman $ρ=1$, and an exact branch-and-bound search over all $13!$ orderings certifies the global optimum and its four symmetric minimizers. The resulting $A/E/I$ words separate canalized cell fates from transient developmental states, a non-Archimedean analog of Waddington's landscape.