Forgetting under noise is undecidable: ergodicity of positive-rate cellular automata at a fixed noise rate

Can noise make the long-run behaviour of a cellular automaton predictable? A probabilistic cellular automaton with positive rates, in which every site can take every value with positive probability at every step, is ergodic when it forgets its initial configuration. Deciding ergodicity is known to be impossible when some transitions may have probability zero. We show that positive noise does not restore decidability. There are an explicit rate ε₀ > 0 and a computable family of positive-rate cellular automata on ℤ⁸, each a deterministic local rule followed by uniform random replacement, such that for every noise rate ε in (0, ε₀] the automaton built from a Turing machine M is ergodic if and only if M halts.The proof links a logical quantity to a physical one. A tile set that tiles the plane makes a noisy quorum rule remember. A tile set whose contradictions can always be cornered by a fixed number of local tokens, a refutation of bounded width in the sense of existential pebble games, makes the same rule forget. We build a computable family of tile sets in which every contradiction that exists can be cornered by three tokens: two hold the phase of a recognizable substitution while the third walks, and the machine runs in a layer closed under intersection. Two consequences: no computable function bounds how long the ergodic members take to forget, and uniqueness of the translation-invariant Gibbs state of the associated space–time interactions is undecidable.