Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy

No arrangement of points in the plane beats the triangular lattice for any steadily fading repulsion.

4 papers September 23, 2026 Explained by AI

Imagine points spread over a plane, one per unit of area, all pushing each other apart. A Lean formalization proves that the triangular lattice gives the lowest push per point. This holds for every push in a broad, steadily fading family, against every arrangement. The papers also claim to prove the same for charged-particle energies, including the planar Coulomb case. That claim fixes a constant for the best spreading of many points on a sphere.

First step

1953

R. A. Rankin, triangular minimum among lattices

Significance

Major

It settles the planar case of the Cohn–Kumar question and a 2012 Sandier–Serfaty conjecture, if the proofs hold.

Trust

Formal proof listed

Field

Convex and metric geometry

4 papers

The explanation

Spread points over a flat plane, one point per unit of area on average. Let every pair of points repel. The push gets weaker as the pair gets farther apart. Which arrangement makes the total push per point as small as possible? Try two tidy grids. In a square grid, each point has four nearest neighbors, exactly 1 unit away. In a triangular grid, each point has six nearest neighbors, about 1.075 units away.

Score a pair at squared distance t by e^(−πt), a bell-shaped curve called a Gaussian. The four square-grid neighbors score about 0.173 together. The six triangular neighbors score about 0.159. Adding every other point too gives about 0.180 per point for the square grid. The triangular grid gives about 0.160. It has more close neighbors, but each one sits a little farther away. Here that trade wins by about 11 percent.

The papers say this is no accident of one score or two grids. The main theorem covers every completely monotone interactionSource 1. That means a push that never goes negative and keeps getting weaker, smoothly, at every level of detail. Both e^(−πt) and 1/t^p, for any positive power p, qualifySource 2. The competitors are almost any arrangements with one point per unit area. Density is measured in ever larger disks around one center. A competitor need not repeat. It may have points very close together, or crowded patchesSource 3. The theorem compares long-run averages of push per point, and it allows infinite totalsSource 4. A Lean formalization proves that the triangular lattice is never beaten in this sense.

Before this, the comparison was settled only among lattices, the grids made by repeating one tile. Rankin in 1953 and Cassels in 1959 began that work for inverse powers. Ennola and Diananda repaired and simplified it. Montgomery in 1988 handled every Gaussian, still among lattices only. In 2007, Cohn and Kumar asked whether the triangular lattice beats every arrangement. In 2022, Cohn, Kumar, Miller, Radchenko and Viazovska proved the matching statement in dimensions 8 and 24. In the plane, earlier results covered only narrow classes, such as honeycombs or small shifts of the lattice.

The one obstacle is the size of the competition. Checking lattices means tuning a few numbers. Here a rival can be any messy pattern. The papers get around this with a floor. For each Gaussian, they build a helper function that lies below it everywhereSource 5. Its Fourier transform, the recipe of waves that make it up, is never negative. Those two signs force every arrangement's energy above a fixed floor. The helper touches the Gaussian at every lattice point, so the triangular lattice sits exactly on that floor. Building such a helper is the hard part. It must touch at infinitely many points and keep its signs everywhere between them. The papers then write every allowed push as a positive blend of Gaussians, so the floors add upSource 6.

The same tools reach further. Take the push 1/r^s with s between 0 and 2, or the logarithm. Then totals are infinite even on the latticeSource 7. Physicists handle this with a uniform background of opposite charge, called jellium. The second paper claims to prove that the triangular lattice wins this corrected contest tooSource 8. The fourth paper treats the planar Coulomb case directly, by a different methodSource 9. It claims to prove a 2012 conjecture of Sandier and SerfatySource 10. Combined with a 2018 result of Bétermin and Sandier, this settles a related question on a sphereSource 11. Brauchart, Hardin and Saff had predicted a constant in the lowest logarithmic energy of n points there. The predicted constant is about −0.0556. The third paper builds the matching helper for packing equal disks, with density π/(2√3), about 0.907Source 12.

There are limits. The theorem names the lowest value, not every arrangement that reaches itSource 13. It does not build a separate sharp helper for each push, which Cohn and Kumar's conjecture asks forSource 14. It says nothing about s equal to 2Source 15. The sphere result gives a growth rate, not a way to place the pointsSource 16.

How certain is this? The Lean scope note says a Lean formalization proves the universal minimum among these planar arrangements. It also covers the Gaussian helper functions and the disk-packing helper. The jellium results, the Coulomb paper and the sphere constant are outside that check. So is the packing paper's uniqueness statement. No specialist review has been published.

What it doesn't show. This does not list every arrangement that reaches the minimum, and it makes no claim at s = 2.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if the proofs hold. Lean covers the universal minimum and the helper functions. The jellium, Coulomb and sphere claims are not machine-checked. No specialist review is published.

Words used

Triangular lattice: The grid in which each point has six equally spaced nearest neighbors, like a honeycomb's cell centers.
Lattice: A grid of points made by repeating one tile in two directions.
Completely monotone: Describes a push that never goes negative and keeps weakening smoothly, at every level of detail.
Gaussian: A bell-shaped score e^(−πt) that fades very fast with squared distance t.
Fourier transform: The recipe of waves that add up to a given function.
Jellium: A model of charged points sitting in a uniform background of opposite charge.
Renormalized energy: An energy with each point's infinite self-energy removed, so arrangements can be compared.
Universal optimality: Being the best arrangement for a whole family of pushes at once.

The original papers (4)

An atomic certificate for triangular-lattice universal optimality (PDF, opens in a new tab)
September 26, 2026
Universal optimality of the triangular lattice (PDF, opens in a new tab)
September 23, 2026
A sharp Fourier certificate for planar circle packing (PDF, opens in a new tab)
September 23, 2026
Triangular minimality for planar Coulomb renormalized energy (PDF, opens in a new tab)
September 23, 2026

Lean notes

All quoted lines (16)

For every smooth nonnegative completely monotone function $g$ on $(0,\infty)$ and every locally finite planar set $\mathcal C$ of centered disk density one, $E_g(\mathcal C)\ge \sum_{a\in A\setminus\{0\}}g(|a|^2)=E_g(A).$
An atomic certificate for triangular-lattice universal optimality Back to text
This includes every Gaussian $g(t)=e^{-\pi\alpha t}$ with $\alpha>0$ and every inverse power $g(t)=t^{-p}$ with $p>0$ .
Universal optimality of the triangular lattice Back to text
The problem goes beyond comparison with other lattices: a competing configuration may be nonperiodic, may contain arbitrarily close pairs, and may have very uneven local point counts.
Universal optimality of the triangular lattice Back to text
The comparison includes infinite energies.
An atomic certificate for triangular-lattice universal optimality Back to text
We seek a real radial Schwartz function $f_\alpha$ satisfying $f_\alpha\le G_\alpha,\qquad \widehat f_\alpha\ge0, \qquad \begin{aligned} f_\alpha(a)&=G_\alpha(a)&& (a\in A\setminus\{0\}),\\ \widehat f_\alpha(w)&=0&& (w\in A^*\setminus\{0\}). \end{aligned}$ The two inequalities give a lower bound for arbitrary competitors. The contacts at the direct and dual lattice points make the bound exact on $A$ .
Universal optimality of the triangular lattice Back to text
Positive mixtures of the resulting energy inequalities, rather than mixtures of the auxiliary functions, give Theorem ….
An atomic certificate for triangular-lattice universal optimality Back to text
For $0<s<2$ , the raw Riesz energy with $g(t)=t^{-s/2}$ is infinite even on $A$ , so the comparison of raw energies does not by itself distinguish configurations in this range.
Universal optimality of the triangular lattice Back to text
Corollary … proves the triangular minimum for every $0\le s<2$ , with interaction $-\log|x|$ at $s=0$ and $|x|^{-s}$ for $0<s<2$ .
Universal optimality of the triangular lattice Back to text
The present proof treats the logarithmic functional directly and uses no universal-optimality result as an input.
Triangular minimality for planar Coulomb renormalized energy Back to text
Their Conjecture 1 asks whether the triangular value is also the minimum over the full admissible class. Theorem … resolves this conjecture positively.
Triangular minimality for planar Coulomb renormalized energy Back to text
This proves the $d=2$ case of the logarithmic-energy conjecture of Brauchart, Hardin, and Saff.
Triangular minimality for planar Coulomb renormalized energy Back to text
We resolve the planar Cohn–Elkies sharpness conjecture: the two-point Fourier bound attains the optimal circle-packing density $\pi/(2\sqrt3)$ .
A sharp Fourier certificate for planar circle packing Back to text
It identifies the minimum value, not all minimizers.
Universal optimality of the triangular lattice Back to text
That stronger per-potential quantifier is not claimed here.
Universal optimality of the triangular lattice Back to text
It asserts neither uniqueness nor exclusion of defects, and makes no claim at $s=2$ or for quantum crystallization.
Universal optimality of the triangular lattice Back to text
The conclusion is an asymptotic value, not a construction of near-optimal point sets or an algorithm for Smale's seventh problem.
Triangular minimality for planar Coulomb renormalized energy Back to text