Universal sequential update mode for Rule 45 on ring sizes multiple of three

Establish whether, for elementary cellular automaton Rule 45 on a ring of length n that is a multiple of three, the sequential update mode μ = (0,1,2,…,n−2,n−1) is universal for that size; that is, determine if for every initial configuration in B^n the orbit under Rule 45 with this fixed left-to-right sequential order converges to one of the fixed points (001)^{n/3}, (010)^{n/3}, or (100)^{n/3}.

Background

Rule 45 admits fixed points only when the ring size n is a multiple of three; specifically, the fixed points are the periodic configurations (001)^{n/3}, (010)^{n/3}, and (100)^{n/3}. The authors construct convergence to these fixed points using a periodic (non-sequential) update mode and then seek a simpler, purely sequential update schedule.

Empirical simulations for n=6 and n=9 revealed many sequential update orders that lead all configurations to fixed points, motivating the conjecture that the canonical left-to-right sequential order μ=(0,1,…,n−1) suffices universally across all multiples of three. The set T denotes the family of configurations whose sizes are multiples of three, hence the use of the term T-universal.