{"paper":{"title":"Intrinsic Geometry of Hard Disk Clusters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Fabi\\'an Henry Vilaxa, Jos\\'e Ayala Hoffmann","submitted_at":"2026-08-06T18:52:10Z","abstract_excerpt":"Put \\(n\\) identical coins on a table with no two overlapping. Which arrangement makes the perimeter of the convex hull of the cluster as small as possible? Despite its elementary statement, the solution of this problem is known only up to four disks.\n  We produce a calculus for hard disk clusters of arbitrary finite size, providing class criticality conditions, first order descent tests, and second order spectral criteria for perimeter minimisation. A central difficulty is that the perimeter formula changes with the hull combinatorics, while the admissible first order geometry changes with the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.06513","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.06513/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}