Characterizing Open-Ended Evolution Through Undecidability Mechanisms in Random Boolean Networks
Discrete dynamical models underpin systems biology, but we still lack substrate-agnostic diagnostics for identifying finite-horizon dynamical signatures that may be relevant to open-ended evolution (OEE), such as the recurrent production of novel phenotypic states rather than rapid settling or unstructured noise. We introduce a simple, model-independent metric, {\Omega}, that summarizes the residence-time-weighted contribution of attractor cycle lengths across the sequence of recurrent episodes realized within a finite observation window. {\Omega} is zero for single-attractor dynamics and also vanishes for pure novelty without recurrence, while increasing when trajectories repeatedly enter multiple persistent cyclic phenotypes. Using Random Boolean Networks (RBNs) as a controlled testbed, we compare classical Boolean dynamics with biologically motivated non-classical mechanisms (probabilistic context switching, annealed rule mutation, paraconsistent logic, modal necessary/possible gating, and quantum-inspired superposition/paired-state coupling) under homogeneous and heterogeneous updating schemes. Our results support the view that undecidability-adjacent, state-dependent mechanisms -- implemented as probabilistic context switching, modal necessity/possibility gating, paraconsistent logic, or quantum-inspired correlated branching -- are enabling conditions for sustained novelty. At the end of our manuscript we outline a practical extension of {\Omega} to continuous/hybrid state spaces, positioning {\Omega} as a portable proxy for OEE in biological modeling and a guide for engineering evolvable synthetic circuits.