From Dobrushin Comparison to a Quasi-Local C*-State: A Lean-Checked Construction for the Two-Dimensional Ising Model

We present a Lean 4 formalization of a thermodynamic-limit construction for the anisotropic nearest-neighbour Ising model on the two-dimensional integer lattice. A telescoping comparison argument and the classical Dobrushin resolvent produce a volume-uniform expectation bound. Exact restriction and reindexing maps between finite Gibbs measures yield convergence of the complete free-boundary sequence, independence of auxiliary envelopes and cofinal samplings, stability under receding boundary perturbations, and equality of free and periodic limits.Finite-support cylinder presentations are quotiented by equality of their represented functions on the full spin space. The resulting local algebra carries a genuine lattice-translation action, and the limiting functional is positive, normalized, real-linear, and invariant under every integer translation. We equip this algebra with its intrinsic uniform norm, construct its complex star-algebra representation, take the corresponding commutative C*-closure, and extend the limiting functional to a positive complex-linear functional of norm one. We also formalize normalized Gibbs conditional kernels for arbitrary finite conditioning sets and prove positivity, exterior locality, idempotence, and the exact finite-volume Gibbs tower identity.All results remain within the classical anisotropic Dobrushin region. The infinite-volume DLR fixed-point equation for the completed state is not claimed.