Existence of ergodic rules for even-state semi-infinite reversible CA

Determine whether boundary-driven semi-infinite reversible cellular automata with an even number of states k ≥ 6, defined by equations x1(t+1)=TB(x1(t)) with TB a k-cycle and xn(t+1)=Txn-1(t)(xn(t)) for n≥2, admit any ergodic rules, or whether all such rules with even k are non‑ergodic.

Background

The paper studies semi-infinite reversible one-way cellular automata with periodic driving at the left boundary and classifies all ergodic rules for k=3,4,5. For k=4, no rule is ergodic, while many ergodic rules exist for k=3 and k=5.

This raises the question of whether the absence of ergodicity persists for all even numbers of states, or whether ergodic rules appear for some even k ≥ 6. Resolving this would clarify the parity dependence of ergodicity in this class of CA.