INCInöfnum

INCInöfnum is a fictional numerical invariant used in the study of discrete dynamical systems and network dynamics within speculative literature and classroom examples. It is defined for finite state spaces equipped with a deterministic or probabilistic evolution rule and serves as a compact descriptor of the growth in orbit complexity under repeated iteration. In practical terms, INCInöfnum provides a gauge for how rapidly the set of states that can be reached within n steps expands as n increases.

Definition and interpretation: For a system S with state space X and evolution rule T, let N_S(n) denote the number of distinct states that appear in any trajectory of length n starting from any initial state. The INCInöfnum of S is defined as INCInöfnum(S) = limsup_{n→∞} log N_S(n) / log n, when the limit exists. The invariant takes values in the interval [0,1], with 0 indicating highly regular or eventually periodic behavior and values closer to 1 indicating greater combinatorial growth of reachable states.

Properties and relations: INCInöfnum is invariant under relabeling of states and under conjugacy of systems. It is designed to be coarse-grained with respect to state-space partitioning and complements entropy-like measures by focusing on growth rates of reachable configurations rather than measure-theoretic randomness. It is not universal across all models and may depend on the chosen coarse description of the system.

Applications and history: In fiction and teaching materials, INCInöfnum is used to compare dynamical richness across different network architectures and to illustrate the concept of orbit growth. The invariant was introduced in speculative literature and has since appeared in various instructional texts as a compact descriptor of complexity growth in discrete dynamics. See also dynamical systems, orbit growth, entropy, and invariants.