Densest sphere packing in dimension 12

Determine the optimal sphere packing density in twelve-dimensional Euclidean space R^12 by identifying a packing that achieves maximal density and establishing its exact density value.

Background

The paper evaluates FlowBoost on finite packings in hypercubes and motivates high-dimensional exploration by referencing the classical infinite-space sphere packing problem. While the optimal sphere packings are known in dimensions 8 and 24 (E8 and the Leech lattice), the case of dimension 12 remains unresolved despite rich lattice structure such as the Coxeter–Todd lattice K12.

Within this context, the authors highlight that the optimal packing density in R12 is still unknown, situating their computational experiments (e.g., N=31 spheres in a 12D hypercube) as exploratory proxies related to this longstanding open question in the continuous setting.

References

Dimension $d = 12$ is particularly interesting as it lies between the solved cases, admits rich lattice structure (e.g., the Coxeter-Todd lattice $K_{12}$), yet the optimal packing density remains unknown, making it a natural testbed for computational exploration.

— Flow-based Extremal Mathematical Structure Discovery  (2601.18005 - Bérczi et al., 25 Jan 2026) in Section 3 (Results), Overview — Sphere Packing