Directing Open-Ended Evolution in Artificial Life via Multi-Scale Path Divergence

• URL: http://arxiv.org/abs/2606.17091v1
• Date: Fri, 12 Jun 2026 22:24:12 GMT
• Title: Directing Open-Ended Evolution in Artificial Life via Multi-Scale Path Divergence
• Abstract summary: MSPD (Multi-Scale Path Divergence) is a renormalization-group-inspired scalar.<n>It quantifies the temporal multiscale organization of heterogeneity in local transition laws.<n>MSPD is an explicit formula and plays a dual role: as a gradient-free fitness function and as a post-hoc analytical lens.
• Score: 0.24214594180459362
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Open-ended evolution (OEE) in artificial life is typically driven by uninterpretable, black-box neural-network complexity metrics, leaving life-like systems disconnected from physical theories of complexity. We introduce MSPD (Multi-Scale Path Divergence, denoted DP ), a renormalization-group-inspired scalar that quantifies the temporal multiscale organization of heterogeneity in local transition laws. MSPD is defined at the population level as a functional of the realised trajectory and is computed as a windowed finite-resolution estimator, with consistency between the two stated as a proposition. The metric is an explicit formula and plays a dual role: as a gradient-free fitness function and as a post-hoc analytical lens on any simulation that exposes local transition laws. Empirically, MSPD-optimized parameters produce higher held-out complexity scores than matched random parameters from the same substrate. High-$H_{Delta_t}$ states correspond to states with higher instability to external interventions, so the metric tracks the biology of the underlying dynamics rather than noise. Higher MSPD corresponds to stronger scale-dependent frustration: high-complexity systems exhibit larger differences between the dynamics expressed at different spatial extents, linking MSPD directly to the frustration criterion of biological complexity in the sense of Vanchurin et al. [ 23 ]. The same protocol transfers beyond the primary Flow-Lenia substrate to Life-like cellular automata and Particle Life++, where C1, C2 and C5 all hold. A single explicit formula thus both directs open-ended evolution and provides a principled bridge to the physics of complexity that black-box drivers do not.

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