Dynamics in a Low-Rank Separable Field Cellular Automaton

Complex collective dynamics in cellular automata are usually associated with local-neighborhood combinatorics, yet it remains unclear whether long-lived dynamical organization requires such explicit local interaction structure. Here, we introduce a Separable-Field Cellular Automaton (SFCA), a normalized-field cellular automaton in which local neighbor counting is replaced by a rank-one-like row-column field. Each cell is updated according to a normalized field, with survival and birth governed by two threshold intervals. Systematic scans over interval widths and positions revealed four outcome classes: extinction, fixed points, cycles, and long transients. The outcome phase diagram was organized by the relative geometry of the survival and birth intervals: fixed points dominated when born interval was contained in survival interval, whereas long transients concentrated near the boundary between partial overlap and no overlap. A fine scan along this transition showed that the long-transient region forms a narrow but persistent ridge separating two qualitatively distinct cycle-dominated regimes. One side produced dense, high-change-rate cycles approximating global period-2 alternation, whereas the other produced sparse, low-change-rate, stripe-like cycles. Damage-spreading further supported a basin-competition interpretation, in which the long-transient ridge reflects delayed selection between two cyclic attractor families rather than random nonconvergence, while finite-size analysis shows that the long-transient ridge remains robust across tested grid sizes. These results show that structured long-transient dynamics can arise under compressed separable field coupling, suggesting that nontrivial collective organization does not necessarily require full local-neighborhood combinatorics.