{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:HGN7HFHAWFVMG3CJ7ZXPNRCYWH","short_pith_number":"pith:HGN7HFHA","canonical_record":{"source":{"id":"1910.06637","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-10-15T10:22:55Z","cross_cats_sorted":["math.DG","math.MG"],"title_canon_sha256":"76238b0ffb01bfaecab799a41882fcf5d8cef57c7369394fd6945dda429b39d4","abstract_canon_sha256":"baad0c8e91fdab95e83b6781bda75e512e303227c491feefacf3cc985b9375ed"},"schema_version":"1.0"},"canonical_sha256":"399bf394e0b16ac36c49fe6ef6c458b1f3ca141dd442e65ec1192ab558fa7b94","source":{"kind":"arxiv","id":"1910.06637","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1910.06637","created_at":"2026-07-05T06:45:46Z"},{"alias_kind":"arxiv_version","alias_value":"1910.06637v3","created_at":"2026-07-05T06:45:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1910.06637","created_at":"2026-07-05T06:45:46Z"},{"alias_kind":"pith_short_12","alias_value":"HGN7HFHAWFVM","created_at":"2026-07-05T06:45:46Z"},{"alias_kind":"pith_short_16","alias_value":"HGN7HFHAWFVMG3CJ","created_at":"2026-07-05T06:45:46Z"},{"alias_kind":"pith_short_8","alias_value":"HGN7HFHA","created_at":"2026-07-05T06:45:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:HGN7HFHAWFVMG3CJ7ZXPNRCYWH","target":"record","payload":{"canonical_record":{"source":{"id":"1910.06637","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-10-15T10:22:55Z","cross_cats_sorted":["math.DG","math.MG"],"title_canon_sha256":"76238b0ffb01bfaecab799a41882fcf5d8cef57c7369394fd6945dda429b39d4","abstract_canon_sha256":"baad0c8e91fdab95e83b6781bda75e512e303227c491feefacf3cc985b9375ed"},"schema_version":"1.0"},"canonical_sha256":"399bf394e0b16ac36c49fe6ef6c458b1f3ca141dd442e65ec1192ab558fa7b94","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:45:46.660157Z","signature_b64":"gTsE6fNaUMI0z5GKla3YvsYsG1eW6wpTTfpmqvk34kHJ3OVQT/XvJuM++hN47o9dd2TlajUfLj4SRLYVLjnYAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"399bf394e0b16ac36c49fe6ef6c458b1f3ca141dd442e65ec1192ab558fa7b94","last_reissued_at":"2026-07-05T06:45:46.659736Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:45:46.659736Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1910.06637","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:45:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ulA/ujrFtE9d3O/WI68HYHObghSzPZV0+umfqjidw4BzhnGYVLwgxxHn1Ufid5KxqLxxjo2ZlfuA8pB+FcpFDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T11:44:31.603153Z"},"content_sha256":"e8dea16876d142db30ea277bb46a21f4e98a07c024e66b8615b6c681bc7e738b","schema_version":"1.0","event_id":"sha256:e8dea16876d142db30ea277bb46a21f4e98a07c024e66b8615b6c681bc7e738b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:HGN7HFHAWFVMG3CJ7ZXPNRCYWH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Quantitative Obata's Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.MG"],"primary_cat":"math.FA","authors_text":"Andrea Mondino, Daniele Semola, Fabio Cavalletti","submitted_at":"2019-10-15T10:22:55Z","abstract_excerpt":"We prove a quantitative version of Obata's Theorem involving the shape of functions with null mean value when compared with the cosine of distance functions from single points. The deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise. These results are obtained in the general framework of (possibly non-smooth) metric measure spaces with curvature-dimension conditions through a quantitative analysis of the transport-rays decompositions obtained by the localization method."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.06637","kind":"arxiv","version":3},"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/1910.06637/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:45:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"68fJ/z2LkhOfU6ybzUk+iSGyxXCbWG/+AV76RaeVTdK6nfNeXG2kCjc72e/ziErXumSyPpT4Awia9cTdCoLTAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T11:44:31.603549Z"},"content_sha256":"925d3f48bf9c2f176f7994a7f422d9e6fadc4f8e55083ea4fe1e87736cc4bcdd","schema_version":"1.0","event_id":"sha256:925d3f48bf9c2f176f7994a7f422d9e6fadc4f8e55083ea4fe1e87736cc4bcdd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HGN7HFHAWFVMG3CJ7ZXPNRCYWH/bundle.json","state_url":"https://pith.science/pith/HGN7HFHAWFVMG3CJ7ZXPNRCYWH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HGN7HFHAWFVMG3CJ7ZXPNRCYWH/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":"2026-07-06T11:44:31Z","links":{"resolver":"https://pith.science/pith/HGN7HFHAWFVMG3CJ7ZXPNRCYWH","bundle":"https://pith.science/pith/HGN7HFHAWFVMG3CJ7ZXPNRCYWH/bundle.json","state":"https://pith.science/pith/HGN7HFHAWFVMG3CJ7ZXPNRCYWH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HGN7HFHAWFVMG3CJ7ZXPNRCYWH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:HGN7HFHAWFVMG3CJ7ZXPNRCYWH","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":"baad0c8e91fdab95e83b6781bda75e512e303227c491feefacf3cc985b9375ed","cross_cats_sorted":["math.DG","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-10-15T10:22:55Z","title_canon_sha256":"76238b0ffb01bfaecab799a41882fcf5d8cef57c7369394fd6945dda429b39d4"},"schema_version":"1.0","source":{"id":"1910.06637","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1910.06637","created_at":"2026-07-05T06:45:46Z"},{"alias_kind":"arxiv_version","alias_value":"1910.06637v3","created_at":"2026-07-05T06:45:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1910.06637","created_at":"2026-07-05T06:45:46Z"},{"alias_kind":"pith_short_12","alias_value":"HGN7HFHAWFVM","created_at":"2026-07-05T06:45:46Z"},{"alias_kind":"pith_short_16","alias_value":"HGN7HFHAWFVMG3CJ","created_at":"2026-07-05T06:45:46Z"},{"alias_kind":"pith_short_8","alias_value":"HGN7HFHA","created_at":"2026-07-05T06:45:46Z"}],"graph_snapshots":[{"event_id":"sha256:925d3f48bf9c2f176f7994a7f422d9e6fadc4f8e55083ea4fe1e87736cc4bcdd","target":"graph","created_at":"2026-07-05T06:45:46Z","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/1910.06637/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove a quantitative version of Obata's Theorem involving the shape of functions with null mean value when compared with the cosine of distance functions from single points. The deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise. These results are obtained in the general framework of (possibly non-smooth) metric measure spaces with curvature-dimension conditions through a quantitative analysis of the transport-rays decompositions obtained by the localization method.","authors_text":"Andrea Mondino, Daniele Semola, Fabio Cavalletti","cross_cats":["math.DG","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-10-15T10:22:55Z","title":"Quantitative Obata's Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.06637","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:e8dea16876d142db30ea277bb46a21f4e98a07c024e66b8615b6c681bc7e738b","target":"record","created_at":"2026-07-05T06:45:46Z","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":"baad0c8e91fdab95e83b6781bda75e512e303227c491feefacf3cc985b9375ed","cross_cats_sorted":["math.DG","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-10-15T10:22:55Z","title_canon_sha256":"76238b0ffb01bfaecab799a41882fcf5d8cef57c7369394fd6945dda429b39d4"},"schema_version":"1.0","source":{"id":"1910.06637","kind":"arxiv","version":3}},"canonical_sha256":"399bf394e0b16ac36c49fe6ef6c458b1f3ca141dd442e65ec1192ab558fa7b94","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"399bf394e0b16ac36c49fe6ef6c458b1f3ca141dd442e65ec1192ab558fa7b94","first_computed_at":"2026-07-05T06:45:46.659736Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:45:46.659736Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gTsE6fNaUMI0z5GKla3YvsYsG1eW6wpTTfpmqvk34kHJ3OVQT/XvJuM++hN47o9dd2TlajUfLj4SRLYVLjnYAw==","signature_status":"signed_v1","signed_at":"2026-07-05T06:45:46.660157Z","signed_message":"canonical_sha256_bytes"},"source_id":"1910.06637","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e8dea16876d142db30ea277bb46a21f4e98a07c024e66b8615b6c681bc7e738b","sha256:925d3f48bf9c2f176f7994a7f422d9e6fadc4f8e55083ea4fe1e87736cc4bcdd"],"state_sha256":"3cecbdb26f0bf321110f6e442fc5f86b422b761b11b050129eb64002939150fb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YkJ1VF80tRb7t3R0HtPkrKfwjXACHOABnoy8VYrf5FAVIN7R4nB39n6RcojpKGpuTaHsAKsxtXH17PgxE0GjDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T11:44:31.605538Z","bundle_sha256":"a5e762b44b08155a2d9da9c404d8b07fbfcea50e112de337c1b1f213ada711aa"}}