Finite Certificates and Undecidable Gibbs Uniqueness

This preprint studies whether an algorithm can decide uniqueness of equilibrium for a classical lattice model at a fixed positive temperature. It presents an effective construction on the square lattice in which the halting of a Turing machine is equivalent to uniqueness among all physical Gibbs states. Main construction. The interaction is translation invariant and finite range, with fixed support shapes and local energy terms taking only the exact integer values 0, 1 and 20. The finite on-site alphabet depends on the machine and its input. At inverse temperature one, a halting computation gives a unique Gibbs state with exponential clustering of every bounded local physical observable. Without halting, the full Gibbs simplex is affinely isomorphic to the low-temperature, zero-field two-dimensional Ising simplex, with exactly two extremal states. Mechanism. Finite computation histories act as certificates. Character orthogonality cancels invalid selections exactly inside a strictly positive finite-energy model. A surviving accepting certificate produces a positive phase-selection bias. Conditional auxiliary mixing connects the certificate representation to arbitrary physical Gibbs states; an Ising comparison, four independent upper coverings and finite replica exchange then establish uniqueness. The manuscript distinguishes this construction from its applications of established Ising, comparison and disagreement methods. Further results. A distinct soft-certificate table has the same dichotomy at every finite inverse temperature at least 2/5. The selected plus phase has uniform boundary-specific exponential convergence and is computable across all machine inputs. Pressure values and the complete right smooth jet at zero field are uniformly computable, while zero-field differentiability encodes halting and the left derivative is not uniformly computable across the family. On the uniqueness branch, pressure is infinitely differentiable on a witness-dependent field interval, with absolutely summable cumulant formulas. Small finite-range spin-only perturbations preserve uniqueness within a witness-dependent neighbourhood. Exact microscopic symmetrization excludes a positive family-wide radius for arbitrary real perturbations. Scope. The physical alphabet is variable, not fixed. The convergence results retain their stated boundary conventions; they do not assert arbitrary-boundary strong spatial mixing or rapid Glauber dynamics. Smoothness does not assert analyticity. A separate effective compactness argument gives a computable valid all-boundary modulus under a uniqueness promise.