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.

翻译：人工生命中的开放进化（OEE）通常由不可解释的黑箱神经网络复杂度指标驱动，导致类生命系统与复杂度的物理理论脱节。我们提出MSPD（多尺度路径散度，记作DP）——一种受重整化群启发的标量，用于量化局部转换规律异质性的时间多尺度组织。MSPD在群体层面定义为实现轨迹的泛函，并通过窗口化有限分辨率估计量计算，两者一致性以命题形式阐述。该指标具有显式公式且扮演双重角色：作为无梯度适应度函数，以及作为揭示局部转换规律的任意仿真事后分析透镜。实验表明，MSPD优化参数产生的留出复杂度得分优于同一基底匹配随机参数。高H_Δt态对应对外部干预更不稳定的状态，因此该指标追踪底层动力学的生物学特性而非噪声。更高MSPD对应更强尺度依赖性阻挫：高复杂度系统在不同空间尺度展现的动力学差异更大，这使得MSPD直接关联Vanchurin等人[23]提出的生物学复杂度阻挫准则。相同协议可迁移至Flow-Lenia基底之外的类生命元胞自动机与Particle Life++系统，其中C1、C2和C5均成立。因此，单个显式公式既能引导开放进化，又能提供黑箱驱动者无法实现的与复杂度物理学的原理性桥梁。