Machine-learning interatomic potentials face a fundamental conflict: rotational symmetry (SO(3) equivariance) and long-range electrostatic interactions cannot coexist in real space. Local message-passing architectures achieve equivariance by construction but cannot represent the slowly decaying, anisotropic multipolar correlations that govern real materials. Existing long-range extensions either break the symmetry or sacrifice energy-force consistency.

EquiEwald resolves this by moving to reciprocal space. The Ewald summation — a classical technique that splits Coulomb interactions into short-range (real-space) and long-range (Fourier-space) components — is embedded within an equivariant neural architecture. Equivariant message passing in reciprocal space through learned k-space filters captures the tensorial long-range correlations while maintaining SO(3) equivariance automatically. The inverse transform returns to real space with the physics intact.

The structural insight is about choosing the right representation space for the problem. In real space, equivariance and long-range anisotropy fight each other — maintaining rotational symmetry while representing slowly decaying directional interactions requires compromises that degrade either symmetry or range. In reciprocal space, the same physics becomes tractable: long-range real-space correlations become short-range in k-space, and equivariant operations on the Fourier coefficients are natural. The conflict was not fundamental — it was an artifact of working in the wrong coordinate system.

(arXiv:2603.18389)