{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:B6QS35GFDCEDPINRZS6GEA4NRU","short_pith_number":"pith:B6QS35GF","schema_version":"1.0","canonical_sha256":"0fa12df4c5188837a1b1ccbc62038d8d3babe3a532f2c17388b0c44448aa162b","source":{"kind":"arxiv","id":"1907.03203","version":6},"attestation_state":"computed","paper":{"title":"Average Gromov hyperbolicity and the Parisi ansatz","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CO","math.MG","math.MP"],"primary_cat":"math.PR","authors_text":"Leila Sloman, Sourav Chatterjee","submitted_at":"2019-07-06T23:20:59Z","abstract_excerpt":"Gromov hyperbolicity of a metric space measures the distance of the space from a perfect tree-like structure. The measure has a \"worst-case\" aspect to it, in the sense that it detects a region in the space which sees the maximum deviation from tree-like structure. In this article we introduce an \"average-case\" version of Gromov hyperbolicity, which detects whether the \"most of the space\", with respect to a given probability measure, looks like a tree. The main result of the paper is that if this average hyperbolicity is small, then the space can be approximately embedded in a tree. The proof u"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1907.03203","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-06T23:20:59Z","cross_cats_sorted":["math-ph","math.CO","math.MG","math.MP"],"title_canon_sha256":"61d264bfb82d228028dc015edd28c60ed233e3f02346f722d1696eb4ba64855c","abstract_canon_sha256":"5a097e20f5773692bf7d06f23a4d4560a9c6d3013d6d978b6d17dcb05e51a35a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:38:14.958225Z","signature_b64":"Kh87FjlhMv4bnm4OWAm3jGlJUaKqJgTkf2oE1kQFq6d7ZfMCZztQGtmWwHtlaIVBy6PB7wVDDT+4bBnFM2ulCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0fa12df4c5188837a1b1ccbc62038d8d3babe3a532f2c17388b0c44448aa162b","last_reissued_at":"2026-07-05T01:38:14.957816Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:38:14.957816Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Average Gromov hyperbolicity and the Parisi ansatz","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CO","math.MG","math.MP"],"primary_cat":"math.PR","authors_text":"Leila Sloman, Sourav Chatterjee","submitted_at":"2019-07-06T23:20:59Z","abstract_excerpt":"Gromov hyperbolicity of a metric space measures the distance of the space from a perfect tree-like structure. The measure has a \"worst-case\" aspect to it, in the sense that it detects a region in the space which sees the maximum deviation from tree-like structure. In this article we introduce an \"average-case\" version of Gromov hyperbolicity, which detects whether the \"most of the space\", with respect to a given probability measure, looks like a tree. The main result of the paper is that if this average hyperbolicity is small, then the space can be approximately embedded in a tree. The proof u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.03203","kind":"arxiv","version":6},"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/1907.03203/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1907.03203","created_at":"2026-07-05T01:38:14.957866+00:00"},{"alias_kind":"arxiv_version","alias_value":"1907.03203v6","created_at":"2026-07-05T01:38:14.957866+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.03203","created_at":"2026-07-05T01:38:14.957866+00:00"},{"alias_kind":"pith_short_12","alias_value":"B6QS35GFDCED","created_at":"2026-07-05T01:38:14.957866+00:00"},{"alias_kind":"pith_short_16","alias_value":"B6QS35GFDCEDPINR","created_at":"2026-07-05T01:38:14.957866+00:00"},{"alias_kind":"pith_short_8","alias_value":"B6QS35GF","created_at":"2026-07-05T01:38:14.957866+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/B6QS35GFDCEDPINRZS6GEA4NRU","json":"https://pith.science/pith/B6QS35GFDCEDPINRZS6GEA4NRU.json","graph_json":"https://pith.science/api/pith-number/B6QS35GFDCEDPINRZS6GEA4NRU/graph.json","events_json":"https://pith.science/api/pith-number/B6QS35GFDCEDPINRZS6GEA4NRU/events.json","paper":"https://pith.science/paper/B6QS35GF"},"agent_actions":{"view_html":"https://pith.science/pith/B6QS35GFDCEDPINRZS6GEA4NRU","download_json":"https://pith.science/pith/B6QS35GFDCEDPINRZS6GEA4NRU.json","view_paper":"https://pith.science/paper/B6QS35GF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1907.03203&json=true","fetch_graph":"https://pith.science/api/pith-number/B6QS35GFDCEDPINRZS6GEA4NRU/graph.json","fetch_events":"https://pith.science/api/pith-number/B6QS35GFDCEDPINRZS6GEA4NRU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/B6QS35GFDCEDPINRZS6GEA4NRU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/B6QS35GFDCEDPINRZS6GEA4NRU/action/storage_attestation","attest_author":"https://pith.science/pith/B6QS35GFDCEDPINRZS6GEA4NRU/action/author_attestation","sign_citation":"https://pith.science/pith/B6QS35GFDCEDPINRZS6GEA4NRU/action/citation_signature","submit_replication":"https://pith.science/pith/B6QS35GFDCEDPINRZS6GEA4NRU/action/replication_record"}},"created_at":"2026-07-05T01:38:14.957866+00:00","updated_at":"2026-07-05T01:38:14.957866+00:00"}