{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:MCI2DWLAMIW5RLZXVNHEVZW5E7","short_pith_number":"pith:MCI2DWLA","canonical_record":{"source":{"id":"1503.01340","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-03-04T15:23:27Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"e683aabb5d148f4df40cb53836f1ef5863fc7127dc0ef67445be72eb5bd5d9ed","abstract_canon_sha256":"4bbd8576b7e210a2ca6a4ba9d5c04e30c6d2babdcf56a0b2678334c55111e636"},"schema_version":"1.0"},"canonical_sha256":"6091a1d960622dd8af37ab4e4ae6dd27cd68f294b72f88bb92d975d7e99b0dff","source":{"kind":"arxiv","id":"1503.01340","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.01340","created_at":"2026-05-18T02:25:34Z"},{"alias_kind":"arxiv_version","alias_value":"1503.01340v1","created_at":"2026-05-18T02:25:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.01340","created_at":"2026-05-18T02:25:34Z"},{"alias_kind":"pith_short_12","alias_value":"MCI2DWLAMIW5","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"MCI2DWLAMIW5RLZX","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"MCI2DWLA","created_at":"2026-05-18T12:29:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:MCI2DWLAMIW5RLZXVNHEVZW5E7","target":"record","payload":{"canonical_record":{"source":{"id":"1503.01340","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-03-04T15:23:27Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"e683aabb5d148f4df40cb53836f1ef5863fc7127dc0ef67445be72eb5bd5d9ed","abstract_canon_sha256":"4bbd8576b7e210a2ca6a4ba9d5c04e30c6d2babdcf56a0b2678334c55111e636"},"schema_version":"1.0"},"canonical_sha256":"6091a1d960622dd8af37ab4e4ae6dd27cd68f294b72f88bb92d975d7e99b0dff","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:34.758406Z","signature_b64":"bGwJ0A8zA2zE4g5WX/U/RqSgoyGouFuF7IHuCJUOnJD76TlqMus2Jdi9ded3niBasAcAdVepRwyHk5MkRmZlDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6091a1d960622dd8af37ab4e4ae6dd27cd68f294b72f88bb92d975d7e99b0dff","last_reissued_at":"2026-05-18T02:25:34.757940Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:34.757940Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.01340","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:25:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i+giSPBW48jCYvHiQKNJCEYvUOYunsxVIR0IU5zyqh0DG4HCUkCHOZigIN4Ky24yBjhJ2cRUY84fDqUunVX0AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T11:52:59.399165Z"},"content_sha256":"4db2a983dc8cd481bba62d3ae54000fb528d9646a5eceebfb31c11be78e920f6","schema_version":"1.0","event_id":"sha256:4db2a983dc8cd481bba62d3ae54000fb528d9646a5eceebfb31c11be78e920f6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:MCI2DWLAMIW5RLZXVNHEVZW5E7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bounds on Gromov Hyperbolicity Constant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Domingo Pestana, Jose M. Rodriguez, Veronica Hernandez","submitted_at":"2015-03-04T15:23:27Z","abstract_excerpt":"If $X$ is a geodesic metric space and $x_{1},x_{2},x_{3} \\in X$, a geodesic triangle  $T=\\{x_{1},x_{2},x_{3}\\}$ is the union of the three geodesics $[x_{1}x_{2}]$, $[x_{2}x_{3}]$ and $[x_{3}x_{1}]$ in $X$. The space $X$ is $\\delta$-hyperbolic in the Gromov sense if any side of $T$ is contained in a $\\delta$-neighborhood of the union of the two   other sides, for every geodesic triangle $T$ in $X$.\n  If $X$ is hyperbolic, we denote by  $\\delta(X)$ the sharp hyperbolicity constant of $X$, i.e. $\\delta(X) =\\inf \\{ \\delta\\geq 0:{0.3cm}$ X ${0.2cm}$ $\\text{is} {0.2cm} \\delta \\text{-hyperbolic} \\}.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01340","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:25:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LtqBeI1sEshr7jr0yT/zE3OEbkJyozrpZZuIqVum3VQmK7elyL2FtB3LhKZG9A2qbbrf/1pQE2w3VurLzH0LCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T11:52:59.399546Z"},"content_sha256":"830485af41b52d8591df9357f3f646526e0eae4c2ecc7a83de7301d078cd432a","schema_version":"1.0","event_id":"sha256:830485af41b52d8591df9357f3f646526e0eae4c2ecc7a83de7301d078cd432a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MCI2DWLAMIW5RLZXVNHEVZW5E7/bundle.json","state_url":"https://pith.science/pith/MCI2DWLAMIW5RLZXVNHEVZW5E7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MCI2DWLAMIW5RLZXVNHEVZW5E7/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-22T11:52:59Z","links":{"resolver":"https://pith.science/pith/MCI2DWLAMIW5RLZXVNHEVZW5E7","bundle":"https://pith.science/pith/MCI2DWLAMIW5RLZXVNHEVZW5E7/bundle.json","state":"https://pith.science/pith/MCI2DWLAMIW5RLZXVNHEVZW5E7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MCI2DWLAMIW5RLZXVNHEVZW5E7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:MCI2DWLAMIW5RLZXVNHEVZW5E7","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":"4bbd8576b7e210a2ca6a4ba9d5c04e30c6d2babdcf56a0b2678334c55111e636","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-03-04T15:23:27Z","title_canon_sha256":"e683aabb5d148f4df40cb53836f1ef5863fc7127dc0ef67445be72eb5bd5d9ed"},"schema_version":"1.0","source":{"id":"1503.01340","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.01340","created_at":"2026-05-18T02:25:34Z"},{"alias_kind":"arxiv_version","alias_value":"1503.01340v1","created_at":"2026-05-18T02:25:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.01340","created_at":"2026-05-18T02:25:34Z"},{"alias_kind":"pith_short_12","alias_value":"MCI2DWLAMIW5","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"MCI2DWLAMIW5RLZX","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"MCI2DWLA","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:830485af41b52d8591df9357f3f646526e0eae4c2ecc7a83de7301d078cd432a","target":"graph","created_at":"2026-05-18T02:25:34Z","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":"If $X$ is a geodesic metric space and $x_{1},x_{2},x_{3} \\in X$, a geodesic triangle  $T=\\{x_{1},x_{2},x_{3}\\}$ is the union of the three geodesics $[x_{1}x_{2}]$, $[x_{2}x_{3}]$ and $[x_{3}x_{1}]$ in $X$. The space $X$ is $\\delta$-hyperbolic in the Gromov sense if any side of $T$ is contained in a $\\delta$-neighborhood of the union of the two   other sides, for every geodesic triangle $T$ in $X$.\n  If $X$ is hyperbolic, we denote by  $\\delta(X)$ the sharp hyperbolicity constant of $X$, i.e. $\\delta(X) =\\inf \\{ \\delta\\geq 0:{0.3cm}$ X ${0.2cm}$ $\\text{is} {0.2cm} \\delta \\text{-hyperbolic} \\}.$","authors_text":"Domingo Pestana, Jose M. Rodriguez, Veronica Hernandez","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-03-04T15:23:27Z","title":"Bounds on Gromov Hyperbolicity Constant"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01340","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":"sha256:4db2a983dc8cd481bba62d3ae54000fb528d9646a5eceebfb31c11be78e920f6","target":"record","created_at":"2026-05-18T02:25:34Z","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":"4bbd8576b7e210a2ca6a4ba9d5c04e30c6d2babdcf56a0b2678334c55111e636","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-03-04T15:23:27Z","title_canon_sha256":"e683aabb5d148f4df40cb53836f1ef5863fc7127dc0ef67445be72eb5bd5d9ed"},"schema_version":"1.0","source":{"id":"1503.01340","kind":"arxiv","version":1}},"canonical_sha256":"6091a1d960622dd8af37ab4e4ae6dd27cd68f294b72f88bb92d975d7e99b0dff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6091a1d960622dd8af37ab4e4ae6dd27cd68f294b72f88bb92d975d7e99b0dff","first_computed_at":"2026-05-18T02:25:34.757940Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:25:34.757940Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bGwJ0A8zA2zE4g5WX/U/RqSgoyGouFuF7IHuCJUOnJD76TlqMus2Jdi9ded3niBasAcAdVepRwyHk5MkRmZlDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:25:34.758406Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.01340","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4db2a983dc8cd481bba62d3ae54000fb528d9646a5eceebfb31c11be78e920f6","sha256:830485af41b52d8591df9357f3f646526e0eae4c2ecc7a83de7301d078cd432a"],"state_sha256":"d99201460b90a6b2306275ef8346e5d92f8ab89465f033641ced4caf6313332d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IoVADj5JRSPMD8RTZo527E5cOvUe9YG2MCUa8njca6w6WQk22nDseYOEUNn28PvVPlZc2fVYmMzYTyoEzbBMDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T11:52:59.401683Z","bundle_sha256":"17ba8447c4cfb4374e079469722d34e8849238ad156d8cb593b424d5f3e87cc1"}}