Densest packings of two translates of a lattice in four dimensions

Let Λ be a lattice in R⁴ and b a vector not in Λ such that distinct points of Λ ∪ (Λ + b) are at distance at least 2. We prove that the covolume of Λ is at least 16, with equality only when Λ ∪ (Λ + b) is congruent to √2 D₄. Thus no packing of unit balls whose centres form two translates of a lattice is denser than the D₄ lattice packing, and every Voronoi cell of such a packing has volume at least 8, the volume of the regular 24-cell.

The archive contains the paper (PDF, LaTeX sources with PNG figures, arXiv tarball), the exact enumeration of the polyhedron of two-periodic packings, the integer certificates, independent checks in C, Julia, Lean 4 and Python, the paper in preprintMusin24CellConjectureTwoTranslates/ and a master zip of the whole repository, and a proof of the vertex and edge table in Lean 4 with Mathlib in which every step is checked by the kernel.

Version 2.0.0 adds Ushashi Bhattacharya as second author (Deep Bhattacharjee is the corresponding author), a two-sentence abstract, a figure of two translates of a lattice, a section of statements and declarations, and files named after the conjecture. It also carries the changes prepared as version 1.2.0, which was not released on its own: a proof of Proposition 5.1 in Lean 4 with Mathlib in which every step is checked by the kernel (verification/lean_kernel), with no native_decide and only the axioms propext, Classical.choice and Quot.sound. Version 1.1.1 added the run logs of crosscheck/ and analysis/, which the earlier versions left out, and moved the comparison with the numerical enumeration of Andreanov and Kallus into a footnote of the paper. Version 1.1.0 added the manuscript in the form submitted to Discrete & Computational Geometry, a copy of the d4-voronoi-cells v2.0.0 code (doi:10.5281/zenodo.23240354) under its own licence, a floating-point test of the statements (G) and (C) of that code on two-periodic packings, and a record of two analytic approaches to (G) and (C) with the points where each stops. The general 24-cell conjecture remains open and is not claimed.