Why the triangular lattice minimizes so many planar energies

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Family 090 · Convex and metric geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-14

1. The question

Place repelling particles in the plane at a fixed average density. Their preferred arrangement depends on how the repulsion changes with distance. Could the same arrangement minimize energy for an entire class of interactions, even when its competitors are not periodic?

The triangular lattice is the natural candidate. Each point has six equally distant nearest neighbors, and the associated Voronoi cells are regular hexagons. Its circle-packing optimality is classical. Universal energy optimality is a stronger and different question: it compares interactions at every distance, rather than merely requiring a minimum separation.

The four manuscripts in this family connect three settings. M1 and M2 claim a universal energy minimum. M3 constructs a sharp Fourier certificate for the already-known optimal packing density. M2 and M4 address neutralized long-range energies, and M4 deduces an exact next-order coefficient for logarithmic energy on the sphere.

2. Background and essential definitions

Normalize the triangular lattice $A$ to have one point per unit area. A locally finite configuration $C$ has centered disk density one if

$$\frac{N_R}{\pi R^2}\longrightarrow1, \qquad N_R=\#(C\cap B_R),$$

where $B_R$ is the disk of radius $R$ centered at the origin. Local finiteness means every bounded region contains finitely many points. This density assumption does not give uniform control over counts in every translated small disk.

For a nonnegative interaction $g$ of squared distance, define the lower energy per particle by

$$E_g(C)=\liminf_{R\to\infty}\frac1{N_R} \sum_{\substack{x,y\in C\cap B_R\\x\ne y}}g(|x-y|^2).$$

The sum counts ordered pairs. A completely monotone function satisfies $(-1)^j g^{(j)}(t)\geq0$ for every nonnegative integer $j$. Examples include Gaussians $g(t)=e^{-\pi\alpha t}$, $\alpha>0$, and inverse powers $g(t)=t^{-p}$, $p>0$. Such functions admit positive-mixture representations by exponential kernels.

Long-range interactions need another definition. In two dimensions, $|x|^{-s}$ with $0<s<2$ has a divergent raw lattice sum, and the Coulomb interaction is $-\log|x|$. A jellium energy includes an oppositely charged uniform background. A renormalized energy also subtracts the singular self-energy near each point. These operations are part of the energy’s definition, not a consequence of comparing two infinite raw sums.

3. Development history and previous partial results

Classical work by Rankin, Cassels, Ennola and Diananda established the triangular minimum for convergent inverse-power sums among lattices. Montgomery’s Minimal theta functions (1988) proved the Gaussian minimum among planar lattices of a fixed density. These results compare lattices with lattices; they do not control an arbitrary nonperiodic arrangement.

Cohn and Kumar developed the universal-optimality program and singled out the triangular, $E_8$ and Leech lattices. Cohn, Kumar, Miller, Radchenko and Viazovska later established the eight- and twenty-four-dimensional cases through sharp Fourier auxiliary functions and interpolation. Planar partial results treated restricted periodic competitors or sufficiently small bounded displacements of the triangular lattice.

The Fourier packing method began with Cohn and Elkies. Viazovska’s dimension-eight construction and its dimension-twenty-four counterpart attained the sharp packing bounds there. The plane already had Thue’s geometric density theorem; the unresolved analytic question was whether the two-point Fourier bound itself could attain that value. M3 addresses this sharp-certificate question, rather than claiming the first planar packing theorem.

For Coulomb energy, Sandier and Serfaty’s vortex-lattice work established the infinite-system framework, existence of a minimum, periodic approximation and triangular optimality among lattices. Their conjecture asked for the unrestricted minimum. Petrache and Serfaty’s 2020 paper connected universal optimality to Coulomb and Riesz crystallization.

Bétermin and Sandier’s spherical-energy analysis had already shown that the unknown planar minimum determines the linear term of optimal spherical logarithmic energy. M4 uses that established bridge after claiming the planar minimum directly.

4. What the new manuscripts claim

Universal energy comparison

M1 and M2 both state that every locally finite density-one configuration satisfies

$$E_g(C)\geq\sum_{a\in A\setminus\{0\}}g(|a|^2)=E_g(A)$$

for every smooth nonnegative completely monotone $g$. The comparison includes infinite values. No minimum separation, periodicity or bound on local occupancy is required.

Both papers construct sharp auxiliary functions for each Gaussian, then mix the energy inequalities. They do not claim that this mixing constructs a sharp auxiliary function for every mixed potential in the stronger per-potential formulation of the Cohn–Kumar conjecture. Nor do they classify all configurations attaining the energy minimum.

A sharp certificate for packing

With Fourier transform $\widehat f(\xi)=\int f(x)e^{-2\pi i x\cdot\xi}\,dx$, M3 claims a smooth rapidly decreasing radial function satisfying

$$\widehat f(0)=1,\quad f(0)=\frac2{\sqrt3},\quad \widehat f\geq0,\quad f(x)\leq0\ \text{when }|x|\geq1.$$

The Cohn–Elkies bound then gives the optimal density $\pi/(2\sqrt3)$ for disks of radius $1/2$, including nonperiodic packings. The certificate also recovers triangular uniqueness among periodic equality cases. Its extra zeros mean that it does not satisfy a separate, stronger conjectured requirement prescribing exactly the lattice-shell zero set.

Long-range energies and the sphere

M2 claims triangular minimality for its planar logarithmic and Riesz renormalized energies, with $0<s<2$ in the Riesz case, and the corresponding thermodynamic jellium minima. Its exact periodic minimizers are the $n^2$ triangular lattice classes on compatible tori $\mathbb R^2/(nA)$. This is not an assertion of equality on arbitrary square tori.

M4 gives an independent direct logarithmic result. Its admissible fields satisfy

$$\operatorname{div}E=2\pi\left(\sum_{p\in\Lambda}\delta_p-1\right), \qquad \operatorname{curl}E=0,$$

where $\Lambda$ is simple and locally finite, and its centered disk counts divided by area are uniformly bounded for radii greater than one. Here $\delta_p$ is a unit point mass and the constant one is the background density.

M4 first removes small disks of radius $\eta$ around charges from the half-Dirichlet energy $\frac12\int\chi_R|E|^2$, adds the self-energy correction $\pi\log\eta\sum_p\chi_R(p)$, and takes $\eta\downarrow0$. It then divides by the area of $[-R,R]^2$ and takes a large-square limsup, using cutoffs with fixed-width boundary layers. The claimed minimum is the triangular field’s value. On a square torus of area $n$, every configuration of $n$ distinct points has total renormalized energy at least $n$ times that value; equality on such a torus is not asserted.

For $n$ points on the unit sphere, using ordered pairs and Euclidean chord distance, M4 consequently states

$$\min\sum_{i\ne j}-\log|y_i-y_j| =\left(\frac12-\log2\right)n^2 -\frac n2\log n+C_{\mathrm{BHS}}n+o(n),$$

where

$$C_{\mathrm{BHS}}=2\log2+\frac12\log\frac23 +3\log\frac{\sqrt\pi}{\Gamma(1/3)}.$$

The gamma function extends factorials and is defined for positive $t$ by $\Gamma(t)=\int_0^\infty x^{t-1}e^{-x}\,dx$. The remainder $o(n)$, divided by $n$, tends to zero.

5. Why the advance matters

Importance: 5/5. Universal optimality of the triangular lattice is a central crystallization problem. The claimed comparison would allow fully nonperiodic competitors, including strongly uneven local arrangements, rather than only other lattices or small perturbations of one.

The long-range claims add content that raw universal energy cannot provide when all relevant sums diverge. Neutralization and self-energy subtraction create a finite comparison capable of distinguishing configurations for Coulomb and slowly decaying Riesz interactions.

The sphere consequence is also more than the leading uniform-distribution term. Its linear coefficient records microscopic organization after the bulk and logarithmic terms are removed. Identifying that coefficient would settle the stated two-dimensional logarithmic-energy prediction of Brauchart, Hardin and Saff.

At the same time, an energy minimum is not a complete crystallization classification. Local defects or compact modifications can leave an infinite-volume energy per unit area unchanged. The manuscripts do not infer that every minimizer must literally be a perfect lattice everywhere.

6. The main idea

A lower bound that becomes exact on a lattice

For a Gaussian interaction, seek a function $f$ lying below it with nonnegative Fourier transform. Fourier inversion turns a finite pair sum into an integral involving

$$\widehat f(\xi)\left|\sum_{x\in C\cap B_R}e^{2\pi i x\cdot\xi}\right|^2.$$

Positivity permits a lower bound from frequencies near zero. After removing diagonal terms, the bound involves $\widehat f(0)-f(0)$. Contact with the Gaussian at nonzero triangular lattice points, together with zeros on the dual lattice, makes Poisson summation identify this bound with the triangular energy.

The technical construction must control a function and its Fourier transform simultaneously, at infinitely many radii. Trigonometric factors encode a periodic superset of the triangular shell radii. Value-and-derivative interpolation imposes exact contacts; Gaussian damping supplies rapid decay.

M1 uses two finite interpolation blocks of size twenty and an infinite summable correction. M2 uses a different, larger finite construction. In both, solving the interpolation equations is separate from establishing signs between nodes and on the unbounded tail. The manuscripts describe interval arithmetic, polynomial bounds and analytic tail estimates for those tasks. These are parts of their claimed arguments, not calculations performed by this digest.

Positive-mixture representations then pass from Gaussians to completely monotone interactions at the level of nonnegative energy sums. Lower-limit arguments retain singular potentials and infinite energies without requiring a well-behaved mixed auxiliary function.

Neutralization and a direct cell comparison

M2 uses heat-kernel comparisons to connect Gaussian inequalities to neutralized periodic and field energies. Periodic approximation then transfers the minimum to the infinite system, with the source’s order of cutoffs and limits retained.

M4 follows a separate geometric route for the logarithmic case. It reduces the full-plane minimum to square-periodic fields and first studies a minimizing configuration on each torus. Separation is obtained for those finite minimizers, not imposed on every competitor.

Voronoi cells divide the torus into regions closest to individual charges. The manuscript constructs a single trial potential whose formulas match across cell boundaries and whose logarithmic singularities match the true potential. An exact energy identity leaves a sum of sector contributions plus a nonnegative square measuring the difference of gradients.

A sharp sector bound is calibrated on the regular hexagon. Areas and angles sum exactly, while Euler’s relation limits the average number of sectors to six per point. These facts give the triangular lower bound. Earlier periodic approximation and the Bétermin–Sandier asymptotic formula then deliver the full-plane and spherical consequences.

7. Scope, limitations, and open questions

The raw energy uses centered disks, ordered pairs and a liminf. M4’s field energy removes the puncture cutoff first and then takes a square limsup. M2 uses its own field normalization and order of limits. Their constants should not be exchanged without the conversion specified in the sources.

The Riesz range is strictly $0<s<2$; no endpoint theorem at $s=2$ or quantum crystallization statement is asserted. Infinite raw energies remain legitimate in the universal comparison but do not replace the renormalized theorem.

The packing certificate settles a sharpness question for an analytic bound whose numerical optimum was already known geometrically. It does not satisfy every stronger zero-set conjecture. M3 also notes a related 2026 thesis announcement whose precise relationship it does not determine from the available abstract; the digest makes no priority judgment beyond the cited accounts.

Neither the infinite-volume minimum nor the spherical asymptotic supplies an algorithm producing optimal point sets. In particular, the spherical formula is not presented as an algorithmic solution of Smale’s seventh problem.

8. Sources and further reading

M1 and M2 give distinct Gaussian interpolation constructions with the same universal-energy endpoint. M3 treats packing signs and exact normalization. M4 treats Coulomb energy directly and fixes the spherical ordered-pair convention.

The linked Sandier–Serfaty, Petrache–Serfaty and Bétermin–Sandier papers explain the long-range framework and its established implications. Brauchart, Hardin and Saff’s next-order conjectures give the spherical context. The manuscripts’ historical sections distinguish earlier lattice-only comparisons from unrestricted configuration claims.

Source manuscripts

• M1. OpenAI, An atomic certificate for triangular-lattice universal optimality (September 26, 2026).
• M2. OpenAI, Universal optimality of the triangular lattice (September 23, 2026).
• M3. OpenAI, A sharp Fourier certificate for planar circle packing (September 23, 2026).
• M4. OpenAI, Triangular minimality for planar Coulomb renormalized energy (September 23, 2026).

Sources consulted on 14 October 2026 at repository revision 1581d4c. Download the manuscript citations.