One noisy cellular automaton with an uncomputable noise diagram

Can varying only the noise rate of one cellular automaton encode the halting problem? We construct one computably specified cellular automaton on Z^6, with a finite alphabet and a finite neighbourhood, whose ergodicity under uniform replacement noise does so. There are pairwise disjoint open intervals I_n with rational endpoints computable from n such that, at every rate in I_n, the perturbed automaton is ergodic exactly when a fixed universal machine halts on input n. In the nonhalting case it has at least two translation-invariant stationary laws; in the halting case it has one stationary law attracting every initial law.Finite rulers in a hierarchical tiling turn the noise rate into a cutoff in a sequence of local flags. Computation boards read that cutoff and simulate the selected input. A threshold checker preserves a global bit when the computation does not halt. When it halts, interval-valued sensors erase every surviving chart, and a common-noise coupling removes dependence on every initial state. The guaranteed intervals accumulate at zero and lie at astronomically small rates. The space-time Gibbs correspondence gives a finite-energy interaction on Z^7 with undecidable translation-invariant Gibbs uniqueness. Thus one fixed local rule has an uncomputable noise diagram.