An uncomputable phase diagram at positive temperature for one classical lattice model

Can the phase diagram of one fixed classical lattice model be uncomputable? For quantum spin systems at zero temperature it can: there are families of Hamiltonians whose spectral gap or ground-state phase cannot be computed from the parameter. We prove a classical counterpart at positive temperature. There is one translation-invariant nearest-neighbour interaction on ℤ², with local energies 0, 1 and +∞ (hard constraints) and a fixed universal Turing machine U built in, and there are disjoint intervals J_n with rational endpoints computable from n, such that at every inverse temperature β in J_n the model has exactly one translation-invariant Gibbs measure if U does not halt on input n, and at least two if U halts. So uniqueness of the translation-invariant Gibbs measure at rational inverse temperature is undecidable for this single model. Uniqueness among all Gibbs measures fails at every temperature, because the hard constraints freeze a non-periodic background.The background is a hierarchical tiling that contains arbitrarily large square grids. On each grid a Turing machine reads a label n from the size of the grid and runs U on n. Where U halts, small sensors made of free bits and unit energies couple the corner spins of all grids of that size into an ordinary Ising model, whose coupling is large only when β is close to n log 2. Assuming the exact solution of the two-dimensional Ising model, the inverse temperatures with at least two translation-invariant Gibbs measures form exactly a union of explicit intervals around n log 2 over the halting inputs n; restricted to rational β, this set is Σ⁰₁-complete.