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On Sarnak--Strömbergsson conjecture

This paper proves the Sarnak–Strömbergsson conjecture by demonstrating that, among all three-dimensional lattices of unit covolume, the face-centered cubic (FCC) lattice minimizes the theta function for $\alpha > 1$ while the body-centered cubic (BCC) lattice minimizes the Epstein zeta function for $s > 3/2$ .

Original authors: Senping Luo, Juncheng Wei

Published 2026-09-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Senping Luo, Juncheng Wei

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the hidden architecture of the physical world, matter often organizes itself into repeating, crystalline patterns. This tendency, known as crystallization, is driven by a simple principle: atoms and molecules settle into the arrangement that requires the least amount of energy to maintain. For scientists, finding this perfect, lowest-energy state is like solving a complex three-dimensional puzzle where the pieces are not just shapes, but mathematical descriptions of how particles interact across space. The challenge becomes particularly intricate when dealing with three-dimensional space, the dimension we inhabit. While the two-dimensional version of this problem has long been understood, with a single, perfect hexagonal pattern emerging as the winner, the three-dimensional case has remained a stubborn mystery. For decades, researchers have debated which of the two most common crystal structures—the face-centered cubic, where atoms are packed as tightly as possible, or the body-centered cubic, where atoms sit in the center of a cube—truly represents the most efficient, lowest-energy state. The answer depends on the specific nature of the forces between the particles, a factor that changes the rules of the game entirely.

A new study by Senping Luo and Juncheng Wei has finally settled this long-standing debate for a wide range of physical conditions. The researchers tackled the problem by examining two different mathematical models that describe how energy behaves in a lattice, or grid, of points. One model, known as the theta function, describes systems where particles interact through a force that decays very quickly, similar to how a Gaussian curve drops off. The other, the Epstein zeta function, describes systems with longer-range interactions, like gravity or electricity, where the force weakens more slowly. The team's goal was to determine which lattice structure minimizes the energy for every possible strength of these interactions, assuming the volume occupied by the grid remains constant.

The findings reveal a fascinating split in the behavior of these systems. When the interaction force is strong and short-ranged, corresponding to a specific mathematical scale greater than one, the face-centered cubic lattice is the undisputed champion. This structure, which is the most efficient way to pack spheres in three dimensions, consistently provides the lowest energy state. However, as the interaction becomes weaker and longer-ranged, the rules flip. When the scale drops below one, the body-centered cubic lattice takes over as the energy minimizer. At the precise transition point where the scale equals one, both structures are equally efficient, sharing the title of the lowest-energy state. This discovery confirms a major conjecture proposed by mathematicians Peter Sarnak and Arne Strömbergsson, resolving a question that has lingered in the intersection of number theory and solid-state physics.

The situation is even more decisive for the second model involving longer-range forces. In this case, the face-centered cubic lattice is the winner across the entire range of conditions where the math holds together. Unlike the first model, there is no switch to the body-centered structure; the face-centered cubic arrangement remains the most stable configuration regardless of how the interaction strength varies, provided the mathematical series converges. This result is significant because it provides a complete map of the optimal structures for these fundamental energy problems in three dimensions.

To reach this conclusion, the authors did not rely on simulations or approximations that might miss a subtle detail. Instead, they constructed a rigorous mathematical proof that covers every possible shape the lattice could take. They began by eliminating the most extreme and unstable configurations, narrowing the field to a manageable set of shapes. Then, they focused on the two leading candidates, the face-centered and body-centered lattices, proving that any small deviation from these perfect structures would inevitably increase the energy. By combining these local checks with a global analysis that covered the entire remaining space of possibilities, they demonstrated that no other arrangement could ever beat these two. The proof is absolute; it leaves no room for doubt or hidden exceptions.

This work does more than just identify the winners; it clarifies the role of each structure in the physical world. The face-centered cubic lattice, already famous for being the densest way to pack spheres, is confirmed as the ground state for a broad class of long-range interactions. The body-centered cubic lattice, often observed in metals like iron, is shown to be the optimal choice for specific short-range interactions. The study also highlights a unique transition point where the two structures are indistinguishable in terms of energy, a phenomenon that suggests a phase transition where matter could shift from one form to another without a change in energy cost.

The implications of this research extend beyond pure mathematics. By proving exactly which lattice minimizes energy under different conditions, the study provides a solid foundation for understanding why certain materials form the crystals they do. It offers a definitive answer to a question that has puzzled physicists and mathematicians for years, turning a landscape of uncertainty into a clear, mapped territory. The work stands as a testament to the power of rigorous proof in uncovering the fundamental rules that govern the arrangement of matter, showing that even in the complex, three-dimensional world, there is a precise and predictable order waiting to be discovered.

Technical Summary: On the Sarnak–Strömbergsson Conjecture in Dimension Three

Problem Statement
This paper addresses the minimization of two fundamental lattice energy functions in three dimensions ( $\mathbb{R}^3$ ) under a fixed covolume constraint (set to 1). The two functions are:

The Theta Function: $\Theta(\alpha, L) = \sum_{v \in L} e^{-\pi \alpha |v|^2}$ for $\alpha > 0$ , representing the energy of a lattice under a Gaussian potential.
The Epstein Zeta Function: $E(L, s) = \sum_{v \in L \setminus \{0\}} |v|^{-2s}$ for $s > 3/2$ , representing the energy under a Riesz potential.

The central problem is to identify the lattice structures that minimize these energies. While the two-dimensional case is well-understood (the hexagonal lattice is optimal for all parameters), the three-dimensional case has remained largely open. Specifically, the paper aims to resolve the Sarnak–Strömbergsson conjecture [41], which posits specific minimizers for these functions across different parameter regimes.

Methodology
The authors employ a rigorous "local-global" strategy, combining analytic estimates with computer-assisted verification. The methodology proceeds through several distinct stages:

Geometric Reduction and Cusp Exclusion:
The space of lattice shapes is reduced using a modified Grenier fundamental domain. The authors first exclude "cusps" (lattices with extremely short vectors) by showing that such lattices have strictly higher energy than the reference lattices (Face-Centered Cubic, FCC, and Body-Centered Cubic, BCC). This reduces the minimization problem to a compact set of lattice shapes.
Local Analysis (Taylor Expansions):
The authors perform high-order local expansions of the energy functions around the FCC and BCC reference lattices.
- Theta Function: They analyze the Hessian and higher-order terms near FCC and BCC. A key difficulty is the transition at the self-dual scale ( $\alpha=1$ ), where the minimizer switches from FCC to BCC. They utilize Poisson summation to relate the behavior at $\alpha$ to $1/\alpha$ , allowing them to control the BCC neighborhood via duality.
- Epstein Zeta Function: They utilize the Ewald representation and a positive kernel $G(a, x)$ to express the energy. They compute the Hessian and third/fourth-order derivatives, retaining signed cubic and quartic terms to enlarge the radius of local stability. This allows them to prove local minimality for FCC across the entire range $s > 3/2$ .
Global Comparison (Exterior Estimates):
For lattice shapes outside the local neighborhoods of FCC and BCC, the authors use finite sum approximations and convexity arguments.
- For the Theta function, they use a "convex continuation" lemma. By establishing a positive derivative and a value comparison at a specific scale ( $\alpha=5$ ), they extend the inequality to all $\alpha \ge 5$ .
- For the Epstein function, they use "upper-curvature interpolation" and convex power-sum comparisons. They bound the energy difference on parameter boxes using finite sums of competitor vectors and rigorous bounds on the reference tails.
Computer-Assisted Verification:
The proofs rely heavily on interval arithmetic and rational bounds to verify inequalities on finite sets of parameters (exponents and shape coordinates). The authors provide rigorous enclosures for kernel bounds, shell multiplicities, and derivative estimates, ensuring that all numerical approximations are mathematically sound.

Key Contributions and Results
The paper provides a complete solution to the minimization problem in dimension three, confirming the Sarnak–Strömbergsson conjecture.

Theta Function Minimization: The minimizer depends on the Gaussian scale $\alpha$ :
- If $\alpha > 1$ , the FCC lattice is the unique global minimizer.
- If $\alpha < 1$ , the BCC lattice is the unique global minimizer.
- If $\alpha = 1$ , both FCC and BCC are global minimizers (they have equal energy).
  This result highlights a phase transition at the self-dual scale, contrasting with the two-dimensional case where the hexagonal lattice is optimal for all scales.
Epstein Zeta Function Minimization:
- For all $s > 3/2$ , the FCC lattice is the unique global minimizer.
- This result holds uniformly across the entire convergence range, including the critical finite part at $s = 3/2$ (related to the spectral height).

Significance
The paper claims to provide the first comprehensive, rigorous proof of global optimality for these lattice energy problems in three dimensions.

Resolution of a Major Conjecture: It confirms the Sarnak–Strömbergsson conjecture, which was highlighted by Blanc and Lewin as a significant open problem in both analytic number theory and solid-state physics.
Clarification of Dimensional Differences: It rigorously establishes the sharp contrast between the 2D case (where the hexagonal lattice is universally optimal) and the 3D case (where the optimal structure depends on the interaction range/scale).
Physical Relevance: The results validate the observation that FCC is the ground state for many short-range interactions (consistent with Kepler's conjecture and atomistic crystallization), while BCC emerges as the minimizer for specific long-range interaction potentials or specific Gaussian scales.
Methodological Advancement: The work extends the methods of Sarnak and Strömbergsson from the critical exponent $s=3/2$ to the full range $s > 3/2$ and provides a robust framework for handling the transition between FCC and BCC in the theta function problem.

The authors emphasize that their results are derived from rigorous analytic estimates and computer-assisted proofs, avoiding heuristic arguments, and that the situation in three dimensions, previously described as "almost completely open," is now comprehensively resolved for these specific energy functionals.

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