The Arc Threshold of the Two-Row Ising Torus: Exact Factorization, an Algebraic Critical Coupling, Uniformity in the Circumference, and the Rate of Convergence

Abstract

The Lee-Yang circle theorem places every zero of a ferromagnetic Ising partition polynomial on the unit circle. The arc condition, that every zero lie in the closed left half-plane, turns the circle theorem into a product formula for the characteristic function of the occupation count and makes it usable for local limit theorems. For the two-row periodic Ising torus we give the exact reflection factorization of the row transfer matrix, reduce the crossing at zeta = i to a real cubic, and identify the critical coupling a_c* = 0.8637850566870... as the unique root in (0,1) of an explicit octic. We prove that every a > a_c* satisfies the arc condition simultaneously for all circumferences, that no smaller coupling does, and that the finite thresholds approach a_c* at the rate C/n^2 with an explicit algebraic constant C. The factorization and the 2x2 threshold are formalized in Lean 4 against Mathlib; the zeta = i reduction, uniformity and rate are analysis-level.