This monograph introduces Configuration Numbers, a new class of objects arising from the dynamics of infinite iteration in hyperoperations, together with the associated concept of k‑DNA. In infinite tetration, the outcome depends not only on the base but on an infinite sequence of logarithmic branch choices made at each iteration step. This sequence, called the k‑DNA, uniquely specifies the iteration process. A Configuration Number is defined as an equivalence class of pairs (B,k) that generate the same asymptotic ensemble of limit points. The monograph proves that arbitrary finite cycles, including generic 2‑cycles, can be realised by appropriate DNA sequences, resolving ambiguities that cannot be captured by finitely many parameters. Configuration Numbers thus extend the classical complex plane by dynamical equivalence rather than algebraic closure. These objects are shown to underlie ensemble and stochastic behaviour observed at higher hyperoperation ranks, with the k‑DNA already present at the tetration level. The purpose of this deposit is to document the framework and conceptual priority of Configuration Numbers as the natural parameters of infinite iteration.
Paweł Łukasz Garycki (Fri,) studied this question.