A nilpotent cellular automaton that remembers under noise

Does every positive zero-range perturbation of a nilpotent cellular automaton have a unique attracting stationary law? We construct a counterexample on ℤ³, answering a question of Marcovici, Sablik and Taati. Each site carries a bit and a bounded countdown. The deterministic rule reaches the all-zero configuration after a fixed number of steps, while independent uniform replacements renew the countdowns and allow a horizontal bit plane to retain its initial value. With stationary countdowns, the plane is exactly a north-east-centre majority process with independent gates and independent replacement faults. We calculate the gate law and apply Toom's stability theorem in the quantitative version of Gács to obtain two translation-invariant stationary laws. For the same fixed automaton, a dependency-path estimate proves uniform exponential ergodicity at sufficiently small and sufficiently large noise rates. We also construct, for every prescribed separation, a nilpotent automaton, a set of replacement events at pairwise distance exceeding that separation, and two initial configurations whose bits differ on a whole plane at all times under those common events.The countdown also lifts other rules. Applied to the computable family of positive-rate automata on ℤ⁸ whose ergodicity at a fixed noise rate is undecidable (Douglas and Mghirbi, Forgetting under noise is undecidable), it gives a computable family of radius-one cellular automata on ℤ⁹, all nilpotent within the same number of steps, with alphabets that grow with the machine. At every rate in [ε₀/2, ε₀], where ε₀ = 490000^(−99), uniform replacement noise makes the member built from a Turing machine M ergodic if and only if M halts. For every n, we also construct one nilpotent automaton on ℤ³ with n disjoint intervals of noise rates that carry two translation-invariant stationary laws, alternating with intervals of uniform exponential ergodicity. Whether a single nilpotent rule can have infinitely many such windows is open.Erasure in two steps already suffices, while erasure in one step forces ergodicity under every zero-range noise. We give a four-symbol cellular automaton F on ℤ³ with F∘F constant whose uniform replacement perturbation has two translation-invariant stationary laws at every rate in [2^(−44), 2^(−42)] and is uniformly exponentially ergodic at sufficiently small and sufficiently large rates. Each site reads a seed, the word of 2^51 auxiliary symbols above it, which every deterministic update resets to zero. Exact rational seed partitions give a compiler: for fixed rationals 0 < ε < 2ρ, it maps every attractive binary probabilistic cellular automaton on ℤ^d with rational transition probabilities in [ρ, 1 − ρ] computably to a cellular automaton on ℤ^(d+1), nilpotent in two steps, whose uniform replacement perturbation at rate ε is ergodic, uniformly exponentially ergodic or has two translation-invariant stationary laws exactly when the original automaton has that property.