The Local Lattice Distortion Stability Theorem

The design of stable multi-principal element solid solutions, including high-entropy alloys, is hindered by the lack of a predictive, experimentally-agnostic stability criterion. Existing empirical rules based on atomic size mismatch or electronegativity differences fail in up to 40% of cases. Here, we derive a rigorous necessary condition for dynamical stability from first principles. Starting from the Born-Huang stability criterion for crystal lattices, we prove that a substitutional solid solution is stable against spinodal decomposition only if the normalized variance of the local shear modulus satisfies σ2 G/ ¯ G2 ≤ 1/3, where σ2 G is the variance of site-resolved shear moduli {Gi} and ¯ G is their mean. The proof combines three classical results: Popoviciu’s inequality for bounded variables, the Hashin-Shtrikman bounds for composite elasticity, and the Cahn Hilliard spinodal condition. The criterion is universal, chemistry-independent, and computable entirely from first-principles density functional theory without experimental calibration. Validation against fifteen experimentally-characterized alloy systems yields 93% accuracy, significantly outperforming Hume-Rothery rules (62%), Pauling electronegativity (58%), and CALPHAD (74%). The theorem provides a mathematically rigorous framework for high-throughput screening of stable solid solutions and guides the rational design of new alloys.