{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"fa1c901753202a005135b4004f587a8e9496a306a92b926cfa780fadb02f2c03","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["495dd9d40f4dbb4d4973451583167084e9b39124885ceb58ead1c73288aa9924","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"826fe5a3cd4555b142dc2b84edd482a6c98245b9f0d0df0b92d90e65b2d6ab8d","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":"2","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 $\\lim(6N-c(N))/N^{2/3}$ exists and, if so, its value (expected to be set by the surface energy of the optimal close-packed shape).","problem_ref":"495dd9d40f4dbb4d4973451583167084e9b39124885ceb58ead1c73288aa9924","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 spheres","topic_ref":null},"content_withheld":"false","files":[],"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"ee6ce482673d2863f21d5f6b9b1d371c1d1edad98ca4f96fa6cd117bbe83d160","schema":"pubphys.attested/1"},"id_token":null,"id_token_withheld":"false","envelope":{"attested_hash":"039cedd28619419313919f28d4f6ce302b47fac4745aa0091c1c375340ef9b4b","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"TMuyQbiECz3COb_bitpaZOxw1UWxWbPtxQUXlDoFdQYa-JwiPqgMcNK7nGp7xLkeby64YaLJ6_WJxTwHLkK3Bw"},"schema":"pubphys.envelope/1"},"ots":{"attested":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEIA5zt0oYZQZMTkZ8o1PbOMCtH-sR0WqAJHBw3U0Dvm0vxIAONLdANE97lh0PymhbzwJbhS-lCTwjOFYwMHwNqyD0TCPAgqMEsOwPGAxrKzHkTGxsV4rA3jnCTnheuDNUEu0rYDAEI8CAihZfNeYvqbTaXOkcGCmC_NO7cI8zmNaKReuJxYJkCvwjwIOA4TVB_qhdyins1Fm_YuOk7AJhF3lQC14gtWp4wFNeTCPEgxNJPS4zGdgbtKyVpngmJDSBrvKAvL5SarSvcryXvjhcI8SDZFGw2DlApp2CE79iq7_Zr10S_t4UuAVLnpwZ6Wk8ceQjwIM_uaOPgSae5e7oKY6eaLbC4ERnOug_ROwb2X_EGeV_iCPAghRlAhJmWc0qdN-Mo674D6_tnilkvPnw8TGule_yDRm0I8CBUfn8vND_FHy1q8eURvKLesQojCPxZvV5RtrwTxl9iVwjwIKD8mGRUgbk2Dt1xG-4OxMpCuw1ya5UEUQXsqASURGCtCPAgkPXX-naXR0fRIdS-sEDlzYmtMvSCNH42ESUF0QXb6NUI8CDQo_sIWZiWXaTGRSQBLmU-lHJKskYKqsjO1r02DhB8LAj_8Ajn9YmozHwo9QjwEJ46aM9ZDv3TdtqV3ZbRsNgI8SBOF8pILGf1kodr5MKmQs6Cd93joraGn3g6UPt_-kewPwjwIOG-gmM7bd_eR4AiDMNfluS3xvr7vicLR-sCBr5qtG2cCPAgrVNFrCoeQcCxhlmnE0BzHgR8HQpZAlfqzDDFzfWb8GUI8QRq0WWT8AjZJ2cF_c7k8ACD3-MNLvkMjiwraHR0cHM6Ly9ib2IuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZ_AI9ZMS1zxd7IEI8BDd00WgJZ0RncsC1Fyai_4aCPEgvxMFrnu7F05IW5INiraQ3OVEYMjNx6JfBxw1Q4jPLi4I8QRq0WWS8Ag2zxPAtI5JsP8Ag9_jDS75DI4uLWh0dHBzOi8vYWxpY2UuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZwjxIHAnv4KFFf4uiYEFJvLxiMxSyz8VowJP4roTfDD9h7bICPAgj5tgvbNF7mxSP4jGGE66ezUFRLnI4g1VTkvuV4VtCPAI8CCxlc1HuKORuHgn7DYIYcDhc4oo99OkTWBEvddBypoa7AjwIFcjjaF8kDNDYCZGaNISlRjvX-I7in-QNwsr4JYQbFJvCPAgnl0k4gs7LuY5i1umiHO-J7My3uBThLJUWQoY9LdVSrQI8SA6-Ft11_K_GqMNWYhkVsVqCFvXSYbp4S8JoLsLBnUzvgjxIBC8_mAE7JBP35SNxDBcG7joLJwURpMxdV2vSO64_jEOCPEge04Coj1xSi7CE8UwqiBGNhPyZ5QdD_YZwn-heSdObU4I8SBgdDSgr3VUx1j8m_pK7-OzMwwFgXeLy43ujffCv-hf5AjwIMl-PnAt4xjoBEVU4-nkyN0NKRa9AvX4kEMfllFyIJS9CPFZAQAAAAHz4rV-uwQFg02bVnEMUFgpIdGKjM4tWuID1kTVnrjisAAAAAAA_v___wJURAIAAAAAABYAFBFys2oc0-U-bYrC3VulGLzND5fbAAAAAAAAAAAiaiDwBHLTDgAICPEgMJIBYi46xOIuFDwzJ9nOx14ngW5HbglUf65ORpkoIYcICPAgHgGWEGCDHWgscwT92NOoJ7s-ezbPkyiWr2RnYZDC_dwICPAgtffMyyLu-MscJsDc0dzwGspwSrrmvVAnMNgsgWhHClgICPAgMUl1OfBDUwhp5VZjHwVA0nuz1F5HZTUJCHAnM1L1F3MICPAgStK7UWpnrUueV5EaE7uE0y6BLMp6EANmrbhrTqEBZu8ICPAg5LVW7IHtUL5mNOvPx1hOTfLMmqlQ7t9EFW-KDtrISIQICPAgABgXZ6upFU8qiDQKFtQFpqkjKwl0PsxM4JUjVLlDKbkICPAgEwb96FiOhjlbwI8GFZQ4a5GXwxs4bUdqVOMKrtBWV30ICPEgqBg-wp0c8CAPV9k6J7wIa9_GtQfpWmhpE1zqFZyi2pQICPAg2pZLTVPGJbriiz0UhMytYDVYAXOgQnWDcrMOewK9O8cICPEgxm3QwzbvT4siJzubQGnST67bWv-exYwSPP1KDi7CaUwICPEgm6udUudfjUaM0gUFTylKGDMypj7ptMTG0zmxd_T5yTcICPAgMpOInkjFCVp1pOZhfZnyPAl8hdWGmHzQcrGLgLwSKLgICAAFiJYNc9cZAQPzpjs"],"envelope":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEIfFdiMmCqddR3CiXHAD1TuCWbm5UkWfNVoCkYaXeLz2PxIHxWA2xoSbYei3LELB4fFtV724ovk3XZmjbUQzYvEk-ZCPEgp6roL4qw1P_do5189oYlxBMMhLqbtqiaImnuhNHyw5QI8CAj1pE5UtXsjnABH8MbRIHjG5zOkYeNQ3a98uwfIlQghAjxIJ0WcHuboFxQhKtzYD4VyX0NUFWK_YI7X5jMF6ZmbRC6CPAg7bABo6WZQ2eyXXr99XBQlAA_1Zsqrj616XdoLbfAldoI8SDRVwDUbKiHkLGRwIBP7Zb9IC0P3TOIOPuGjC32wpU2WgjxINa0voVB5dVvA8mG3aK_q_H4RbvesCgQAIqtbZgmnzDbCPAgXx45yhBgitlxXZoPiY801JhflemOgnxVewLKiGj6hiYI8CCCFfNrg2ZeB51PNa84j-z6Sin0AxT6aP9r7HupgUKbMwjxICknnLxa8xs8qmAdhtb0NnANbtwYyrlRTgqcMVLLl8ogCPEgHd9u6OIGCvUV12A1B1xdiOprSJ-4w0cqwfe7CzVX5HsI8CDQo_sIWZiWXaTGRSQBLmU-lHJKskYKqsjO1r02DhB8LAj_8Ajn9YmozHwo9QjwEJ46aM9ZDv3TdtqV3ZbRsNgI8SBOF8pILGf1kodr5MKmQs6Cd93joraGn3g6UPt_-kewPwjwIOG-gmM7bd_eR4AiDMNfluS3xvr7vicLR-sCBr5qtG2cCPAgrVNFrCoeQcCxhlmnE0BzHgR8HQpZAlfqzDDFzfWb8GUI8QRq0WWT8AjZJ2cF_c7k8ACD3-MNLvkMjiwraHR0cHM6Ly9ib2IuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZ_AI9ZMS1zxd7IEI8BDd00WgJZ0RncsC1Fyai_4aCPEgvxMFrnu7F05IW5INiraQ3OVEYMjNx6JfBxw1Q4jPLi4I8QRq0WWS8Ag2zxPAtI5JsP8Ag9_jDS75DI4uLWh0dHBzOi8vYWxpY2UuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZwjxIHAnv4KFFf4uiYEFJvLxiMxSyz8VowJP4roTfDD9h7bICPAgj5tgvbNF7mxSP4jGGE66ezUFRLnI4g1VTkvuV4VtCPAI8CCxlc1HuKORuHgn7DYIYcDhc4oo99OkTWBEvddBypoa7AjwIFcjjaF8kDNDYCZGaNISlRjvX-I7in-QNwsr4JYQbFJvCPAgnl0k4gs7LuY5i1umiHO-J7My3uBThLJUWQoY9LdVSrQI8SA6-Ft11_K_GqMNWYhkVsVqCFvXSYbp4S8JoLsLBnUzvgjxIBC8_mAE7JBP35SNxDBcG7joLJwURpMxdV2vSO64_jEOCPEge04Coj1xSi7CE8UwqiBGNhPyZ5QdD_YZwn-heSdObU4I8SBgdDSgr3VUx1j8m_pK7-OzMwwFgXeLy43ujffCv-hf5AjwIMl-PnAt4xjoBEVU4-nkyN0NKRa9AvX4kEMfllFyIJS9CPFZAQAAAAHz4rV-uwQFg02bVnEMUFgpIdGKjM4tWuID1kTVnrjisAAAAAAA_v___wJURAIAAAAAABYAFBFys2oc0-U-bYrC3VulGLzND5fbAAAAAAAAAAAiaiDwBHLTDgAICPEgMJIBYi46xOIuFDwzJ9nOx14ngW5HbglUf65ORpkoIYcICPAgHgGWEGCDHWgscwT92NOoJ7s-ezbPkyiWr2RnYZDC_dwICPAgtffMyyLu-MscJsDc0dzwGspwSrrmvVAnMNgsgWhHClgICPAgMUl1OfBDUwhp5VZjHwVA0nuz1F5HZTUJCHAnM1L1F3MICPAgStK7UWpnrUueV5EaE7uE0y6BLMp6EANmrbhrTqEBZu8ICPAg5LVW7IHtUL5mNOvPx1hOTfLMmqlQ7t9EFW-KDtrISIQICPAgABgXZ6upFU8qiDQKFtQFpqkjKwl0PsxM4JUjVLlDKbkICPAgEwb96FiOhjlbwI8GFZQ4a5GXwxs4bUdqVOMKrtBWV30ICPEgqBg-wp0c8CAPV9k6J7wIa9_GtQfpWmhpE1zqFZyi2pQICPAg2pZLTVPGJbriiz0UhMytYDVYAXOgQnWDcrMOewK9O8cICPEgxm3QwzbvT4siJzubQGnST67bWv-exYwSPP1KDi7CaUwICPEgm6udUudfjUaM0gUFTylKGDMypj7ptMTG0zmxd_T5yTcICPAgMpOInkjFCVp1pOZhfZnyPAl8hdWGmHzQcrGLgLwSKLgICAAFiJYNc9cZAQPzpjs"]},"log":{"leaf_index":"3410","tree_size":"3804","proof":["96d2f4d1ccf044d13bfb5e435f3b6fc04f30398b1d3b00cf454b4b5b0ec4e171","c2495d9fdde1a8c4f659517e40f1e4700504ce59220941c8d1c20451458f7781","99d9f106c889d1f3689478f00d453622acb5a584128d4d8d86c5f573c4186389","86f049497ab2c15e6252cdd1dd520df946a9aa1b29e33c8c57a935cbf6b23b31","0981ca5bafd71738c7ae26eebac8a066b68f089b6240a352ce4ab7019e21ffaf","9d431ce7742af24a57989023456a175d95e430366c461878574a00aa046e43b8","217966f7b365dc70af6d616a83ae450e616d3fef50ead5372fb26577cf221836","f543c2acee32c00eed64ca6a002c4cdf20cd84f300a60f431863f4556659c7bc","5bc8f1bbb7a36ffa2afc301ec767424419006f77d79dfc7ac3adb0dab9eb36e8","98b5a14b42b3a202b6ef825f66833fa505601404c4c0f14575f3ec922607a858","af34d80d0065c0ab502282775e865492e2cb49c9493f4594d63ba5577922854a","3c578dccc15502c5f3efe228cadc99fee76ab536ed01f66b14d316fdcc157429"],"checkpoint":"pubphys.com/log/v1\n3804\nAK6t4qgu96+M9wScTrIlsJ10d/zhKVEtYguyM3aeJvo=\n\n— pubphys.com/log/v1 JFPwIPP8EtoxgcsQg7V7CwIdL4UKPqJtpv+eB8sFNJjyQHIdyA1VabmBsPyLqq2bLxJPn57rA6fQi8y0QDDjRXHvlQg=\n","promise":"pubphys.com/log/v1 promise\nc9d9229922c681df0fea60edfd204e50171c0fb49eee6b05053438d0ed0df111\n3410\n2026-10-15T23:43:57Z\n2026-10-16T01:43:57Z\n\n— pubphys.com/log/v1 JFPwIBO2a+SNK1r487ZT1XJsJ9iGmw30MOsIz0P+bl4ShGAGacq/gVN0iUWp+eEgbkP1PCWGBpw4g9ELAGlscygW6QE=\n"},"orcid_key_evidence":[],"attesting":null}