{"paper":{"title":"On Arrangements of Six, Seven, and Eight Spheres: Maximal Bonding of Monatomic Ionic Compounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Samuel Reid","submitted_at":"2016-03-27T10:48:36Z","abstract_excerpt":"Let $C(n)$ be the solution to the contact number problem, i.e., the maximum number of touching pairs among any packing of $n$ congruent spheres in $\\mathbb{R}^3$. We prove the long conjectured values of $C(6)=12, C(7)=15$, and $C(8)=18$. The proof strategy generalizes under an extensive case analysis to $C(9)=21, C(10) = 25, C(11) = 29, C(12) = 33$, and $C(13) = 36$. These results have great importance for condensed matter physics, materials science, crystallography, organic and physical chemistry of interfaces."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.08201","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}