{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:B6QS35GFDCEDPINRZS6GEA4NRU","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":"5a097e20f5773692bf7d06f23a4d4560a9c6d3013d6d978b6d17dcb05e51a35a","cross_cats_sorted":["math-ph","math.CO","math.MG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-06T23:20:59Z","title_canon_sha256":"61d264bfb82d228028dc015edd28c60ed233e3f02346f722d1696eb4ba64855c"},"schema_version":"1.0","source":{"id":"1907.03203","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.03203","created_at":"2026-07-05T01:38:14Z"},{"alias_kind":"arxiv_version","alias_value":"1907.03203v6","created_at":"2026-07-05T01:38:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.03203","created_at":"2026-07-05T01:38:14Z"},{"alias_kind":"pith_short_12","alias_value":"B6QS35GFDCED","created_at":"2026-07-05T01:38:14Z"},{"alias_kind":"pith_short_16","alias_value":"B6QS35GFDCEDPINR","created_at":"2026-07-05T01:38:14Z"},{"alias_kind":"pith_short_8","alias_value":"B6QS35GF","created_at":"2026-07-05T01:38:14Z"}],"graph_snapshots":[{"event_id":"sha256:2d735a66e773586de02a6c20e01d1a55f591b5bc0bec01e47eb08ebd1f5a878e","target":"graph","created_at":"2026-07-05T01:38:14Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1907.03203/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Leila Sloman, Sourav Chatterjee","cross_cats":["math-ph","math.CO","math.MG","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-06T23:20:59Z","title":"Average Gromov hyperbolicity and the Parisi ansatz"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.03203","kind":"arxiv","version":6},"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":"sha256:99e83d2152eae4beba2ae6f96959b7b688209b47d4492f6687d9f4603a2edcff","target":"record","created_at":"2026-07-05T01:38:14Z","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":"5a097e20f5773692bf7d06f23a4d4560a9c6d3013d6d978b6d17dcb05e51a35a","cross_cats_sorted":["math-ph","math.CO","math.MG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-06T23:20:59Z","title_canon_sha256":"61d264bfb82d228028dc015edd28c60ed233e3f02346f722d1696eb4ba64855c"},"schema_version":"1.0","source":{"id":"1907.03203","kind":"arxiv","version":6}},"canonical_sha256":"0fa12df4c5188837a1b1ccbc62038d8d3babe3a532f2c17388b0c44448aa162b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0fa12df4c5188837a1b1ccbc62038d8d3babe3a532f2c17388b0c44448aa162b","first_computed_at":"2026-07-05T01:38:14.957816Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:38:14.957816Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Kh87FjlhMv4bnm4OWAm3jGlJUaKqJgTkf2oE1kQFq6d7ZfMCZztQGtmWwHtlaIVBy6PB7wVDDT+4bBnFM2ulCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T01:38:14.958225Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.03203","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:99e83d2152eae4beba2ae6f96959b7b688209b47d4492f6687d9f4603a2edcff","sha256:2d735a66e773586de02a6c20e01d1a55f591b5bc0bec01e47eb08ebd1f5a878e"],"state_sha256":"d92c2bbd58ccb3b1d660a4fd2e43fb397a659400a69f87ecc18fce279afd242a"}