Close-packed ground states of sticky hard spheres
In plain words
Model atoms as hard balls that gain energy only when they touch. Prove that the best arrangement of many such balls stacks flat triangular layers, as in fcc or hcp crystals.
Precise statement
For $V(r)=+\infty \text{ for } r<1, -1 \text{ for } r=1, 0 \text{ for } r>1 \text{ in } R^3$, minimizing $E_N$ is maximizing the contact number $c(N)$ of $N$ non-overlapping unit-diameter spheres. Prove that for large N every minimizer coincides, except for $o(N)$ spheres, with a subset of a close packing (a stacking of triangular layers), and determine whether $\operatorname{lim}(6N-c(N))/N^{2/3}$ exists and, if so, its value (expected to be set by the surface energy of the optimal close-packed shape).
What would settle it
A structure theorem for maximal-contact configurations of N spheres together with the sharp surface constant.
Status in the literature
The 2D sticky-disk problem is solved (Heitmann, Radin 1980); in 3D only bounds on $c(N)$ of the form $6N - C N^{2/3}$ are known.