An Extremal Property of the Square Lattice

The paper’s Theorem 1 asserts, for each fixed ring Ar at radius r around the deep hole p in Z^2, a lower bound ∑_{δ∈Cr} ||p−δ|| − ∑_{z∈Ar} ||p−z|| ≥ r |Ar| d(Δ,Z^2)^2, and an analogous bound for convex φ, with d defined by d(Δ,Z^2)=sqrt(||v−v'||^2+||w−w'||^2) (equivalent up to constants for other norms) . For the unimodular shear v'=v, w'=w+sv (det=1), taking the first nonempty ring r=√(1/2) with Ar of size 4, one computes explicitly that the LHS equals r s^2 + o(s^2) as s→0, while the RHS equals 4 r s^2, contradicting the claimed inequality for all sufficiently small s≠0. Since φ(x)=x is permitted, the convex-φ statement reduces to the same false claim. The definitions of Ar and Cr, and the setup, match the paper’s notation . Therefore, the model’s counterexample is valid and Theorem 1 (linear-distance and convex cases) is false as stated.