Emergent Macroscopic Isotropy in a Nonlocal Central-Force Simple Cubic Lattice

Abstract:We prove that a simple cubic lattice with nonlocal central-force interactions exhibits emergent macroscopic isotropy in the long-wavelength limit. The key condition is that bond-stretching contributions dominate over bond-bending contributions. The proof is algebraic—the identity K_{xx} R_y^2 = K_{xy} R_x R_y holds term-by-term for any central potential—and establishes C_{12} = C_{44} as an exact equality. Numerical verification using Gaussian and power-law potentials confirms the result.