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

Abstract

A finite dynamical system on $\mathbb{F}_p^N$ has a functional graph and combinatorial basins, but nothing on which to run the local analysis a multiplier makes possible. We supply one. Orderings encode configurations as residue classes in $\mathbb{Z}_p$, and an exact rational interpreter over $\mathbb{C}_p$ carries each ball onto its image ball. Rational approximants send, resolution by resolution, every ball onto one of prescribed radius and center. A half-integer pole sphere, with no Archimedean counterpart, settles the expanding case. The prescribed radii are the missing multipliers. They sort balls into contracting, expanding or isometric, giving each configuration a word over three letters, one per scale. Fixed configurations all satisfy $f(a)=a$, yet their words refine them into scale-resolved classes. Contracting balls containing their images trap unique attracting fixed points of the approximant, expanding balls inside their images unique repelling ones. The counts become scores $(\mu_E,\mu_A,\mu_I)$, Haar integrals over $\mathbb{Z}_p$, and minimizing $\mu_E$ over orderings is the finite variational principle. The first score vanishes exactly on Anashin's $p$-adic automata, grading departures by resolution, while the ceiling $(N-1)p^N$ at every ordering isolates the $28$ outer-permutive rules among elementary cellular automata on rings of $N\ge5$ cells. The transition table and ordering determine radii, letters and scores. On a thirteen-gene Boolean network we certify the minimizer over all $13!$ orderings, whose hierarchy separates the ten fixed states by type.