{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:QQO4PKVCL5KEY7MYNPCGJZI3Q2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c619b9c7f6714829976ee2f2500f7163001163e379bb9d48f9f5ff96d03a703d","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-12-26T16:51:54Z","title_canon_sha256":"c968b2fd19e18ad1ae46f6281bf8d8b46532219abf5335506b3c778df9bf42ca"},"schema_version":"1.0","source":{"id":"1212.6211","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.6211","created_at":"2026-05-18T03:37:42Z"},{"alias_kind":"arxiv_version","alias_value":"1212.6211v1","created_at":"2026-05-18T03:37:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.6211","created_at":"2026-05-18T03:37:42Z"},{"alias_kind":"pith_short_12","alias_value":"QQO4PKVCL5KE","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"QQO4PKVCL5KEY7MY","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"QQO4PKVC","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:29e2260fd40cbfcc7c8f344ed8b17bd22d6a41fc6e7c02e376f4c67d17183e2a","target":"graph","created_at":"2026-05-18T03:37:42Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"For $N$-point best-packing configurations $\\omega_N$ on a compact metric space $(A,\\rho)$, we obtain estimates for the mesh-separation ratio $\\gamma(\\omega_N,A)$, which is the quotient of the covering radius of $\\omega_N$ relative to $A$ and the minimum pairwise distance between points in $\\omega_N$. For best-packing configurations $\\omega_N$ that arise as limits of minimal Riesz $s$-energy configurations as $s\\to \\infty$, we prove that $\\gamma(\\omega_N,A)\\le 1$ and this bound can be attained even for the sphere. In the particular case when N=5 on $S^2$ with $\\rho$ the Euclidean metric, we pro","authors_text":"A. V. Bondarenko, D. P. Hardin, E. B. Saff","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-12-26T16:51:54Z","title":"Mesh ratios for best-packing and limits of minimal energy configurations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.6211","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:16c7b8399c5ae6d85cdd0348f4122d42799ccb56159c9a6e7f380d887916735e","target":"record","created_at":"2026-05-18T03:37:42Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c619b9c7f6714829976ee2f2500f7163001163e379bb9d48f9f5ff96d03a703d","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-12-26T16:51:54Z","title_canon_sha256":"c968b2fd19e18ad1ae46f6281bf8d8b46532219abf5335506b3c778df9bf42ca"},"schema_version":"1.0","source":{"id":"1212.6211","kind":"arxiv","version":1}},"canonical_sha256":"841dc7aaa25f544c7d986bc464e51b869471e6e9fa72f0bf3f586f4f510be3e4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"841dc7aaa25f544c7d986bc464e51b869471e6e9fa72f0bf3f586f4f510be3e4","first_computed_at":"2026-05-18T03:37:42.391957Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:37:42.391957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3ZJeZjOE1ynBAy4KF7EzQUSobxgl/ikq6vkIZmdMfYgmwyo17KS1rcbswf8I+c9k9A0iLVdBI1RxwqgSghhJAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:37:42.392739Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.6211","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:16c7b8399c5ae6d85cdd0348f4122d42799ccb56159c9a6e7f380d887916735e","sha256:29e2260fd40cbfcc7c8f344ed8b17bd22d6a41fc6e7c02e376f4c67d17183e2a"],"state_sha256":"7e38c36dacb8e74829193686ab514d17a4850bd1b3cfbddd9053801f5d65917d"}