Translative kissing number of the four-dimensional cross-polytope

Prove that the translative kissing number of the four-dimensional cross-polytope C₄ is κ(C₄) = 40, and characterize all configurations attaining this value as the lattice generated by (2,0,0,0), (1,1,0,0), (1,0,1,0), and (1/2,1/2,1/2,1/2), up to coordinate permutations and coordinate sign changes.

Background

The paper proves the lattice kissing number κ*(C₄) = 40, but the translative kissing number κ(C₄) allows arbitrary discrete packing arrangements rather than restricting the translates to a lattice. The authors provide a lattice configuration with 40 contacts, so κ(C₄) is at least 40, but they do not establish the matching upper bound for all translative packings.

The conjecture further asserts uniqueness of the extremal configuration, modulo coordinate permutations and sign changes.

References

Conjecture 4.2. In $\mathbb{E}4$, we have $$\kappa(C_4)=40,$$ and it can be attained if and only if by the lattice $\Lambda$ which generated by ${(2,0,0,0),$ $(1,1,0,0), (1,0,1,0), (\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})}$, up to permutation of coordinates and signs of coordinates.

— On Generalized Kissing Numbers of Convex Bodies (II)  (2501.06792 - Li et al., 12 Jan 2025) in Section 4, immediately after Conjecture 4.1