Ruliological resilience: Pattern restoration and robustness in Wolfram patterns. A basis for regeneration, not just...

This study investigates recovery dynamics in elementary cellular automata (ECA) following localized perturbations. Using all 256 Wolfram rules, we introduce controlled disruptions ("injuries") at specified pattern growth steps and quantify divergence between perturbed and unperturbed trajectories using Boolean XOR difference mapping. Based on this framework, we define a restoration coefficient R, computed from the normalized Hamming distance over a post-perturbation observation window. Across 100 trials per rule, recovery behavior varies systematically with dynamical class. Class I-II rules exhibit high restoration (R≈1), rapidly returning to baseline trajectories, while Class III rules exhibit persistent divergence (R≈0). Class IV rules display intermediate and phase-dependent behavior, with recovery strongly influenced by perturbation timing and structural maturity. In particular, rules such as Rule 110 show increased resilience, where localized structures constrain divergence. These results demonstrate that recovery properties are not uniformly distributed across rule space but instead cluster by dynamical regime and perturbation conditions. The restoration coefficient provides a quantitative framework for comparing robustness across cellular automata. We discuss how these computational findings offer an abstract perspective on pattern maintenance in biological systems during regeneration, while emphasizing the limits of direct mechanistic interpretation.