Modular Ackermann maps and hierarchical structures

Abstract

We introduce and study modular truncations of the Ackermann function, formalized as discrete dynamical trajectories on the set of least non-negative residues. These maps form a hierarchy of rapidly increasing compositional complexity indexed by recursion depth. We investigate their structural properties, sensitivity to depth variation, and the induced distributions modulo powers of two. While such hierarchical constructions are superficially motivated by hash-type mixing functions, we analyze how powers of two interact with the recursive structure modulo $2^k$, leading to strong saturation effects in the depth $m=3$ case. Instead of the expected asymptotic equidistribution, an absorption phenomenon occurs where the uniform measure concentrates onto a localized subset of residues, driving the total variation distance from the uniform distribution to 1.

Version 2 completely revised. Following an analysis of the modular recurrence modulo powers of two, a universal absorption phenomenon was established, disproving the initial equidistribution conjecture. The title, abstract, and core sections have been updated to focus on this structural collapse