Non-uniform number-conserving elementary cellular automata on the infinite grid: A tale of the unexpected

In this paper, we study non-uniform elementary cellular automata on the infinite grid in the context of number conservation. These automata operate in a one-dimensional setting, where individual cells can employ distinct Wolfram rules for updating their states. The result is an exhaustive characterization of such number-conserving cellular automata. Until now, such a characterization was known only for finite grids, for which research hypotheses could be derived on the basis of computer experiments. It turns out that when considering number conservation for non-uniform cellular automata, the infinite grid cannot be treated as a limiting case of finite grids, i.e., there are number-conserving non-uniform cellular automata on the infinite grid that have no analogous counterpart on finite grids.