Directing Open-Ended Evolution in Artificial Life via Multi-Scale Path Divergence
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 $D_P$), a renormalization-group-inspired scalar that quantifies how the heterogeneity of a system's local transition laws is organized across temporal and spatial scales. 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. On a Flow-Lenia substrate we establish three claims. (C1) Under fixed-context replay of the exact pathwise objective, MSPD-optimized parameters score higher than matched random parameters. (C2) Along optimized trajectories, states whose local transition laws are more heterogeneous yield larger future divergence than matched, less-heterogeneous states under exact-state stochastic continuations, so the metric tracks intrinsic dynamics rather than injected noise. (C5) Higher MSPD corresponds to stronger scale-dependent frustration --- larger differences between the dynamics expressed at different spatial extents --- linking MSPD to the frustration criterion of biological complexity in the sense of Vanchurin et al. All three transfer to Life-like cellular automata and Particle Life++, indicating that MSPD is not specific to a single substrate. A single explicit formula thus both \emph{directs} open-ended evolution and provides a principled bridge to the physics of complexity that black-box drivers do not.