Dense sphere packings on the integer lattice $\mathbb{Z}^3$ for arbitrary exclusion distance: the lattice analogue of Kepler's conjecture

The hard-core lattice gas on the unit cubic lattice $\mathbb{Z}^3$ with Euclidean exclusion distance $D$ has, at fugacity $u>1$, ground states given by the dense-packings: admissible lattice configurations of maximal particle density, i.e. packings of equal spheres with centers on $\mathbb{Z}^3$. Mazel, Stuhl and Suhov classified these packings only for $D^2\in\{2,3,4,5,6,8,9,10,11,12\}$ and $D^2=2\ell^2$ (the latter via Hales' proof of Kepler's conjecture), and posed as open the lattice version of Kepler's conjecture: for every attainable $D$, does a periodic dense-packing exist; is one of them a sublattice of $\mathbb{Z}^3$; can the dense-packing sublattices be described number-theoretically as in the solved two-dimensional case; and which $D^2$ exhibit sliding? Beyond the solved families the three-dimensional case remains open, while the two-dimensional prototype is completely settled. The problem is open.

Background

The hard-core (HC) model is a lattice gas in which particles occupy sites of a lattice and an exclusion rule forbids particles from sitting too close; each particle contributes a fugacity factor $u>0$ to the statistical weight, so for $u>1$ configurations with more particles dominate. Mazel, Stuhl and Suhov study this model on the unit cubic lattice $\mathbb{Z}^3$ with a Euclidean exclusion distance $D>0$: an admissible configuration is a set of lattice points whose pairwise Euclidean distances are all at least $D$. Only attainable values of $D$ matter, namely those with $D^2=m^2+n^2+k^2$ for integers $m,n,k$, since these are exactly the squared distances realized between lattice sites; a non-attainable $D$ is equivalent to the nearest larger attainable value. A ground state is an admissible configuration whose particle density cannot be increased by any modification localized in a finite region; for $u>1$ the ground states are precisely the dense-packings, i.e. admissible configurations of maximal particle density (also called perfect configurations), and a periodic one (invariant under a full-rank group of lattice translations) is a periodic ground state.

Identifying dense-packings is the discrete analogue of the classical sphere-packing problem with centers constrained to $\mathbb{Z}^3$. In $\mathbb{R}^3$, by Hales' proof of Kepler's conjecture, the densest packings are the face-centered cubic (FCC) lattice $A_3$ and its layered stackings; the $\mathbb{Z}^3$ problem inherits this structure whenever the scaled lattice $D\cdot A_3$ embeds in $\mathbb{Z}^3$, which happens exactly for $D^2=2\ell^2$, $\ell\in\mathbb{N}$. Mazel–Stuhl–Suhov solved the packing problem on $\mathbb{Z}^3$ for $D^2=2,3,4,5,6,8,9,10,11,12$ ($D^2=7$ is unattainable, $D^2=1$ trivial) and for $D^2=2\ell^2$, the latter family using Hales' star decomposition and score functions (the machinery of the Kepler proof, which partitions space around each sphere center into star-shaped regions carrying an additive score that bounds the local density) in place of Voronoi cells. Depending on the arithmetic of $D^2$, three behaviors occur: finitely many periodic ground states (dense-packing sublattices of deformed FCC or body-centered-cubic (BCC) type); countably many layered periodic ground states ($D^2=5,6$, and $D^2=2\ell^2$ with $3\mid\ell$), where parallel layers may be shifted against each other; and sliding ($D^2=4,11$), where a continuum of non-layered perfect configurations exists and perturbations of arbitrarily large diameter cost only a bounded excess energy — the mechanism that makes the two-dimensional hard-square model at half filling critical. Using the Pirogov–Sinai theory (a general method constructing periodic pure phases of a lattice model at low temperature or high fugacity as gases of localized low-energy perturbations around ground states, valid when a Peierls-type bound controls perturbation costs), they further identified the periodic pure phases at large fugacity for most of the solved values; for $D^2=11$ the analysis is computer-assisted.

The two-dimensional prototype is completely understood: the same authors solved the disk-packing problem on $\mathbb{Z}^2$ for all $D$ by classifying the minimal-area triangles inscribed in $\mathbb{Z}^2$ with side lengths at least $D$, via norm equations in the ring $\mathbb{Z}[\sqrt[6]{-1}]$ (a cyclotomic integer ring in which squared lattice distances factor), and identified the complete finite list of sliding values, independently confirmed by Krachun; triangular and honeycomb lattices were treated similarly. Physics simulations of $k$-th-nearest-neighbor exclusion models on the cubic lattice (Vigneshwar–Mandal–Damle–Dhar–Rajesh) motivated the rigorous program but contain no proofs. In three dimensions no counterpart of the planar classification exists beyond the solved families: for an attainable value such as $D^2=13,14,16,17,\dots$ not even the maximal packing density, nor a single periodic maximizer, is currently known.

Problem Statement

Settle, for the hard-core model on $\mathbb{Z}^3$ with Euclidean exclusion distance $D$ and every attainable $D$ (i.e. $D^2=m^2+n^2+k^2$ with $m,n,k\in\mathbb{Z}$), the structure of its dense-packings — admissible configurations (sets of lattice points at pairwise Euclidean distance at least $D$) of maximal particle density. This is the lattice version of Kepler's conjecture posed explicitly by Mazel–Stuhl–Suhov. Each part below has independent standing and may be settled separately; a solution of (b) subsumes (a).

(a) Prove or disprove: for every attainable $D$ there exists at least one periodic (invariant under a full-rank group of translations of $\mathbb{Z}^3$) dense-packing configuration in $\mathbb{Z}^3$ (equivalently, a periodic ground state of the model at fugacity $u>1$).

(b) Prove or disprove the stronger statement: for every attainable $D$ there exists a dense-packing which is a sublattice of $\mathbb{Z}^3$, i.e. a sublattice that is itself an admissible configuration of maximal density. (This assertion is strictly stronger than (a), and not equivalent to it.)

(c) In the affirmative case of (b), develop a number-theoretical description of the dense-packing sublattices of $\mathbb{Z}^3$ and of the maximal density as a function of $D$, similarly to the two-dimensional solution via norm equations in $\mathbb{Z}[\sqrt[6]{-1}]$; a complete answer should classify all dense-packing sublattices and give the maximal density for every attainable $D$ in this case.

(d) Determine the complete list of attainable $D^2$ that exhibit sliding — values for which a continuum of non-layered perfect configurations (perfect configuration = dense-packing) exists, with perturbations of arbitrarily large diameter whose excess energy stays bounded. Here a dense-packing is layered if it is a union of parallel copies of a two-dimensional dense-packing, possibly with relative translations, and the excess energy of a perturbation is the amount by which its energy exceeds the ground-state (dense-packing) energy at activity $u>1$. In particular, decide whether this list is finite, as it is in two dimensions.

The known classifications on the solved families $D^2\in\{2,3,4,5,6,8,9,10,11,12\}$ and $D^2=2\ell^2$, $\ell\in\mathbb{N}$, provide a consistency check for any general result.

References

1. Mazel A., Stuhl I., Suhov Y., The hard-core model on $\mathbb{Z}^3$ and Kepler's conjecture, arXiv:2304.08642 (2023), DOI 10.48550/arXiv.2304.08642, https://arxiv.org/abs/2304.08642
2. Mazel A., Stuhl I., Suhov Y., Kepler's conjecture and phase transitions in the high-density hard-core model on $\mathbb{Z}^3$, arXiv:2112.14250 (2021; v2 2023), https://arxiv.org/abs/2112.14250
3. Mazel A., Stuhl I., Suhov Y., High-density hard-core model on $\mathbb{Z}^2$ and norm equations in ring $\mathbb{Z}[\sqrt[6]{-1}]$, arXiv:1909.11648 (2019), https://arxiv.org/abs/1909.11648
4. Krachun D., Triangles on $\mathbb{Z}^2$ and sliding phenomenon, arXiv:1912.07566 (2019), https://arxiv.org/abs/1912.07566
5. Mazel A., Stuhl I., Suhov Y., The hard-core model on planar lattices: the disk-packing problem and high-density phases, arXiv:2011.14156 (2020), https://arxiv.org/abs/2011.14156
6. Hales T., A proof of the Kepler conjecture, Annals of Mathematics 162(3) (2005) 1065–1185, DOI 10.4007/annals.2005.162.1065, https://doi.org/10.4007/annals.2005.162.1065
7. Vigneshwar N., Mandal D., Damle K., Dhar D., Rajesh R., Phase diagram of a system of hard cubes on the cubic lattice, Physical Review E 99 (2019) 052129, DOI 10.1103/PhysRevE.99.052129, https://doi.org/10.1103/PhysRevE.99.052129
8. Jauslin I., Lebowitz J., High-Fugacity Expansion, Lee–Yang Zeros, and Order–Disorder Transitions in Hard-Core Lattice Systems, Communications in Mathematical Physics 364 (2018) 655–682, DOI 10.1007/s00220-018-3269-7, https://doi.org/10.1007/s00220-018-3269-7
9. He Q., Jauslin I., High-Fugacity Expansion and Crystallization in Non-sliding Hard-Core Lattice Particle Models Without a Tiling Constraint, Journal of Statistical Physics 191 (2024), article 135, DOI 10.1007/s10955-024-03349-x, arXiv:2402.02615, https://arxiv.org/abs/2402.02615
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11. Mazel A., Stuhl I., Suhov Y., The Pirogov–Sinai Theory for Infinite Interactions, Journal of Statistical Physics 191 (2024), article 158, DOI 10.1007/s10955-024-03370-0, arXiv:2409.02328, https://arxiv.org/abs/2409.02328
12. Mazel A., Stuhl I., Suhov Y., A Nearest-Neighbor Hard-Core Model on a Penrose Graph, arXiv:2604.21086 (2026), https://arxiv.org/abs/2604.21086
13. Mazel A., Stuhl I., Suhov Y., A Reverse Hard-Core Model in $\mathbb{A}_2$: an Application of the Pirogov–Sinai Theory, arXiv:2606.14903 (2026), https://arxiv.org/abs/2606.14903