2026-09-04
Ω is zero for one attractor and for pure noise; it rises only when a trajectory revisits multiple phenotypes. On N=100 RBNs, PBN switching and modal gating keep the signal.
Open-ended evolution (OEE) still lacks a portable diagnostic. Evolutionary Activity Statistics need a pre-chosen set of "components," and those components can change function. MODES is richer, but some of its scores only make sense on platforms such as AVIDA. Counting new states is worse: a process that jumps at random looks endlessly novel and never settles.
Binghamton, Cambridge, Salamanca, and Simón Bolívar want a finite-horizon test that does not depend on the substrate. The question is whether a trajectory repeatedly enters several long-lived cyclic phenotypes, rather than collapsing to one attractor or wandering as unstructured noise.
They define a score Ω. On a trajectory of length T, a detector finds recurrence episodes: a repeated state starts a cycle of length k (the gap between visits); dwell time d counts how long the trajectory stays in that already-explored region; a never-seen state counts as an escape and allows a new episode. In the deterministic limit, Ω is (1/T²) times the sum of d×k over episodes.
The T² normalization is the point. A single attractor has d growing with T and bounded k, so Ω goes to zero. Pure noise treated as T point attractors with k=d=1 also sends Ω to zero. Ω rises only when the trajectory keeps entering multiple persistent cyclic phenotypes.
The testbed is Random Boolean Networks with N=100 nodes, 1,000 networks per mean connectivity K, T=10⁶ steps, K from 1.1 to 4.5. Two architectures: homogeneous (Poisson degree, synchronous update) and heterogeneous (Exponential degree, CUBEWALKERS random asynchronous update). Five state-dependent mechanisms sit on the same wiring: probabilistic Boolean networks (PBN) that switch whole rule tables, annealed rule mutation (ARM) that flips lookup entries on read, paraconsistent local contradictions, modal necessary/possible gating, and a quantum-inspired mix of superposition collapse plus paired-state coupling. Each family is represented by the parameter set that maximizes area under the Ω-K curve.
On homogeneous synchronous nets, PBN wins at low-to-medium K: piecewise-deterministic motion inside a context, punctuated by rare switches, concatenates long residences from different attractor families. At high K, modal gating takes over. "Necessary" constraints cut chaotic overshoot; "possible" gates open alternatives when the local context allows.
Heterogeneous asynchronous nets invert the ranking. PBN's Ω collapses toward zero across K: asynchronous updates break long cycles, so switches hop among a few short attractors and the d×k sum cannot keep up with T². Modal logic still spikes at high K. Paraconsistent logic has a broad shoulder near K≈2.3. Quantum-inspired rules do best at very low K. The classical deterministic curve sinks everywhere. ARM is consistently poor: indiscriminate LUT flips erase attractor geometry. They add plasticity, and they do not add the structured, state-dependent plasticity Ω rewards.
External checks: elementary cellular automata with fixed-point rules give Ω(T)≈1/T; Rule 54 has the highest average, and additive Rules 90 and 150 also score high. Curated Boolean gene-regulatory networks sit near the finite-horizon floor. Ω is not unique-state growth, node entropy, or compressibility. High noise can pair with low Ω; structured recurrence can score higher at moderate novelty.
The main text does not quote absolute Ω values. Comparisons live in the figures.
For anyone running discrete regulatory models or designing synthetic circuits, Ω is a drop-in diagnostic: no component ontology, no platform lock-in. The design rules are concrete. Homogeneous low-K nets want rare context switching. Homogeneous high-K nets want modal gating. Heterogeneous asynchronous nets want modal logic at high K and paraconsistent or quantum-inspired rules when wiring is sparse. High mutation rates are the wrong lever if the goal is sustained novelty.
For open-ended AI, the paper turns a formal claim (strong OEE needs undecidability) into a computable finite-horizon proxy. What it measures is recurrence-weighted novelty, not fitness, meaning, or unbounded complexity.
The authors are explicit: Ω is not a full OEE verdict. The node set and phenotype map are fixed in advance, so evolution that changes which variables exist is out of scope. Hidden variables can fake recurrence or hide it. The non-classical operators are mechanistic motifs, not a claim that real gene regulation implements those logics. The main experiments are random ensembles; curated GRNs are a scale check. The continuous/hybrid extension is a formula, not an experiment.