ON BOND ANISOTROPY FOR THE ISING MODEL THROUGH THE GEOMETRY OF NUMBERS

The paper proves that, for L ∈ X^1_d with couplings J_k(L)=λ_k(L)^{-1}, the random critical temperature Tc(L) has finite p-th moment for p<d when d≥3 and for all p≤2 when d=2, via a mean-field upper bound Tc(J) ≤ Σ_k J_k together with reduction theory/Iwasawa coordinates for d≥3, and Onsager’s exact 2D critical curve plus a Taylor-asymptotic bound for the anisotropic regime for d=2. These ingredients are stated and executed in Section 1.3 (Theorem 1.1 and its proof), including Tc ≤ Σ_k J_k ≤ d/λ_1 in d≥3 and the 2D relation sinh(2J1/Tc) sinh(2J2/Tc)=1 leading to Tc(J) ≪ J1/|log(J1/J2)| and the concluding integral ∫(−a1 log a1)^{-p} a1 da1 that is finite for p≤2 . The candidate solution reaches the same exponents by a different route: Dobrushin uniqueness (yielding Tc ≤ 2 Σ_k J_k), Siegel’s cusp bound to control negative moments of λ_1, and in d=2 a direct inequality on the Onsager curve giving an explicit Lambert-W bound, combined with Minkowski’s second theorem to relate anisotropy to λ_1. Up to constant factors, both arguments coincide on the key scale Tc(L) ≍ λ_1(L)^{-1}/log(1/λ_1(L)) in the 2D cusp, and both deliver the stated integrability. Minor indexing notational issues in the paper’s 2D limit (ratio J1/J2→0 despite J1≥J2 under successive-minima indexing) do not affect the correctness of the proof. Overall, the paper’s proof is correct and complete for the stated result, and the model’s proof is also correct but methodologically distinct.