{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:AOYWXQZFCD7E4XEMN7JSLQYQ6Z","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":"ad895799bbf868e66fa1face1cdbf0bf7b79da4c71db85deba7154c4369a1cd3","cross_cats_sorted":["math.CO","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-08-06T05:52:53Z","title_canon_sha256":"62b704d6c533ee2c45dda3c8b1399807cb5086b5f532b3aac57061b8066baf2b"},"schema_version":"1.0","source":{"id":"1308.1185","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.1185","created_at":"2026-05-18T02:29:15Z"},{"alias_kind":"arxiv_version","alias_value":"1308.1185v3","created_at":"2026-05-18T02:29:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.1185","created_at":"2026-05-18T02:29:15Z"},{"alias_kind":"pith_short_12","alias_value":"AOYWXQZFCD7E","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"AOYWXQZFCD7E4XEM","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"AOYWXQZF","created_at":"2026-05-18T12:27:38Z"}],"graph_snapshots":[{"event_id":"sha256:7831701c40d7ea55b94c74dc4a556ab993dba7a39dd3a98ddbba01d162ebe550","target":"graph","created_at":"2026-05-18T02:29:15Z","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":"Negative type inequalities arise in the study of embedding properties of metric spaces, but they often reduce to intractable combinatorial problems. In this paper we study more quantitative versions of these inequalities involving the so-called $p$-negative type gap. In particular, we focus our attention on the class of finite ultrametric spaces which are important in areas such as phylogenetics and data mining.\n  Let $(X,d)$ be a given finite ultrametric space with minimum non-zero distance $\\alpha$. Then the $p$-negative type gap $\\Gamma_{X}(p)$ of $(X,d)$ is positive for all $p \\geq 0$. In ","authors_text":"Anthony Weston, Ian Doust, Stephen S\\'anchez","cross_cats":["math.CO","math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-08-06T05:52:53Z","title":"The asymptotic enhanced negative type of finite ultrametric spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.1185","kind":"arxiv","version":3},"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:a0467117dbcf112b2573c63141057d47028df4017e7dba91f6110fb1f544c691","target":"record","created_at":"2026-05-18T02:29:15Z","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":"ad895799bbf868e66fa1face1cdbf0bf7b79da4c71db85deba7154c4369a1cd3","cross_cats_sorted":["math.CO","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2013-08-06T05:52:53Z","title_canon_sha256":"62b704d6c533ee2c45dda3c8b1399807cb5086b5f532b3aac57061b8066baf2b"},"schema_version":"1.0","source":{"id":"1308.1185","kind":"arxiv","version":3}},"canonical_sha256":"03b16bc32510fe4e5c8c6fd325c310f65ae03cb02df6a9aa16e63215a6484241","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"03b16bc32510fe4e5c8c6fd325c310f65ae03cb02df6a9aa16e63215a6484241","first_computed_at":"2026-05-18T02:29:15.947485Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:29:15.947485Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VtkrzmUqC3OhQ10vnTFqaxxZCCZPGLkRK2sKYtmx74ARGPUzH0ojsHpGTWDiGAqX5HbF79MvB1Jw+yC/6AFeAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:29:15.947886Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.1185","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a0467117dbcf112b2573c63141057d47028df4017e7dba91f6110fb1f544c691","sha256:7831701c40d7ea55b94c74dc4a556ab993dba7a39dd3a98ddbba01d162ebe550"],"state_sha256":"dd3613e2ce6f2df3ead66b92e7c83e0462e412657c25eed6db5739b8352ce360"}