Rule-table structure of life-like cellular automata: activity, asymmetry, sensitivity, and the propagation storage factorisation

Yair Cohen

What structure on the rule space of a cellular-automaton family exposes which rules can support which behaviours? For the 262,144 life-like rules -- the two-dimensional outer-totalistic family containing Conway's Game of Life -- the standard coordinate, Langton's activity parameter, is long known to be too coarse. I organise the family by three symmetry-derived rule-table coordinates: an unweighted activity, a birth-survival asymmetry (odd under the swap of birth and survival), and the Vichniac-Binder Boolean-derivative sensitivity in closed form, whose centre-bit component vanishes exactly on the centre-blind (B=S) submanifold. The cardinality coordinates lambda, delta are necessary but never sufficient to certify behaviour; the centre-bit sensitivity labels the B=S submanifold on which the paper's one exact exclusion holds. Combined with an exhaustive enumeration of every rule's compact gliders, still lifes and oscillators (three-by-three seed, period at most sixteen), they expose a hierarchical factorisation: the birth condition is a narrow gate deciding whether anything can propagate, the survival condition tunes storage, and every compact-glider-supporting rule is…