Universal optimality of the triangular lattice

A complete proof that the triangular lattice is universally optimal in the plane, following OpenAI's proof (2026): among all point configurations of density one it has the least energy for every completely monotone interaction of the squared distance. The linear programming bound for configurations of density one; completely monotone interactions as positive mixtures of exponentials; the reduction to one normalized statement about functions of one variable; the construction of these functions from a sine product, cardinal interpolation, Gaussian damping and the Fourier transform; the solution of the interpolation system; and the proof of the two sign conditions. Three steps are finite sets of inequalities between explicitly defined numbers, established by computations with rigorous error bounds; the programs are printed in the lessons. A last lesson derives that the triangular lattice has the least periodic energy for the interactions |x|^-s, 0 < s < 2, and -log|x| among all periodic configurations.

Thirteen lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Energy, density and the linear programming bound
2. Completely monotone interactions
3. Shells and Gaussian duality
4. The sine product and cardinal functions
5. Gaussian damping and Fourier pairs
6. The transformed function at distant nodes
7. A positive quadrature rule
8. The reference lists
9. Exact interpolation
10. Quotients on the half-gaps
11. Positivity on the half-gaps
12. The tail and the theorem
13. Periodic Riesz and logarithmic energies

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Finite Radon representation on locally compact spaces (Harmonic analysis on locally compact abelian groups)
• Real-variable calculations for the Fourier examples (Harmonic analysis on locally compact abelian groups)
• The Fourier inversion theorem and the dual Haar measure (Harmonic analysis on locally compact abelian groups)
• The Poisson summation formula (Harmonic analysis on locally compact abelian groups)
• The Pontryagin duality theorem (Harmonic analysis on locally compact abelian groups)
• Lebesgue integration (core course Measure and Integration) and the Weierstrass approximation theorem (core course Real Analysis II); the lessons prove the other facts they use.