Characterize nonlinear FCA with reversibility period greater than one

Establish necessary and sufficient conditions that identify nonlinear one-dimensional finite cellular automata whose reversibility sequence has period greater than 1 under common boundary conditions (e.g., null or periodic), thereby enabling precise classification of such rules.

Background

The paper introduces the reversibility sequence—a binary sequence indexed by cell count n indicating reversibility—and demonstrates that many nonlinear finite CA exhibit a period of 1 (eventual complete irreversibility), while a minority have larger periods (e.g., rule 11010010).

The authors explicitly ask for a necessary and sufficient condition to identify the latter class, indicating a structural characterization problem for nonlinear rules whose reversibility alternates periodically with n.