{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"2d37fa2c71c8ec5495758987a4e49910846ae0dfab17a7cba636345e4fecbd03","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"9ad83123549be84c8f2f7c8c4246bbba273066fd20f9e2d5cf97cb33f59c5ce8","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.crystallization.sticky-spheres-3d","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"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.","posed_since":"","precise":"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).","problem_ref":null,"references":"","settled_by":"A structure theorem for maximal-contact configurations of N spheres together with the sharp surface constant.","status_note":"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.","title":"Close-packed ground states of sticky hard 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