Mechanism behind ash–seed density correspondence in lifelike cellular automata

Determine the causal explanation for the observed close correspondence between ash density (the mean population density after evolving a 50% random “soup” for 100 generations) and seed density (the mean population density after evolving all 2^16 configurations of a 4×4 central patch for 100 generations) across 64 of the 100 sampled lifelike cellular automata rulesets, and account for why Conway’s Game of Life exhibits a substantially larger discrepancy (~7%) between these two densities.

Background

In the paper’s sampling and results, the authors compare two aggregate population measures for each ruleset: the ‘ash’ density obtained by evolving high-entropy soups and the ‘seed’ density obtained by evolving all possible 4×4 initial configurations. Surprisingly, for many lifelike cellular automata (LCA) rulesets, these two averaged densities nearly coincide.

Conway’s Game of Life (CGOL) deviates from this pattern, showing a notably larger difference between ash and seed densities, raising the question of what structural or dynamical properties of these rulesets lead to this correspondence and why CGOL differs.