The bulk energy of the planar Lennard--Jones potential

We determine the minimum bulk energy of the classical planar Lennard--Jones potential. For every finite configuration of distinct points, the interaction energy is bounded below by the number of particles times the energy per particle of the optimally scaled triangular lattice. Finite triangular patches attain this bound in the thermodynamic limit. The main idea is to construct a positive-definite minorant by integrating the atomic Gaussian interpolation family of OpenAI against the signed Laplace density of the Lennard--Jones potential. The signed density requires a separate proof of the minorant and Fourier positivity inequalities. We establish these inequalities by combining a finite interval calculation with uniform bounds for the infinite interpolation correction and the unbounded range. Poisson summation and its derivative under dilation determine the sharp constant from the interpolation conditions and the zero-pressure identity. The argument applies to arbitrary finite configurations, with no periodicity or density assumption. The accompanying certificate contains the verification programs, their numerical enclosures, and the complete interval covering used in the proof.