Vanishing reachability ratio for non–peripherally-linear elementary cellular automata

Prove that for every elementary cellular automaton whose local rule is not peripherally-linear, the reachability ratio ρ(n)—defined as the fraction of pairs of n-cell configurations (x,y) such that y is reachable from x in some finite time using only the two boundary cells as controls at each time step—tends to zero as the number of cells n tends to infinity.

Background

The paper studies regional controllability in one-dimensional elementary cellular automata (ECA) where control is exerted by setting the two boundary cells over time. It defines the reachability ratio ρ(n) as the fraction of pairs of n-bit configurations (x,y) for which y can be reached from x under boundary control.

Empirical results via a SAT-based method show that peripherally-linear rules are fully controllable (ρ(n)=1), while for other rules the estimated reachability ratio decreases rapidly as n grows. Motivated by these observations, the authors explicitly formulate a conjecture asserting that ρ(n)→0 as n→∞ for all non–peripherally-linear ECA.