{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:PVXJC3IAXMLLBWSAGYTSYQAZU4","short_pith_number":"pith:PVXJC3IA","canonical_record":{"source":{"id":"2412.01190","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2024-12-02T06:48:42Z","cross_cats_sorted":["math.FA","math.PR"],"title_canon_sha256":"730750be5fa6bc36d6f3c1fccaa19f90882857c01911f2a8a37fb4e67a6cd198","abstract_canon_sha256":"f5f7162612ff060c8fc4193cfa8ad59ab2bc9d18a5dc969f0808ebc4b575dc86"},"schema_version":"1.0"},"canonical_sha256":"7d6e916d00bb16b0da4036272c4019a70261378ab09125e13bf19c42846bd7cd","source":{"kind":"arxiv","id":"2412.01190","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.01190","created_at":"2026-07-05T11:23:20Z"},{"alias_kind":"arxiv_version","alias_value":"2412.01190v2","created_at":"2026-07-05T11:23:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.01190","created_at":"2026-07-05T11:23:20Z"},{"alias_kind":"pith_short_12","alias_value":"PVXJC3IAXMLL","created_at":"2026-07-05T11:23:20Z"},{"alias_kind":"pith_short_16","alias_value":"PVXJC3IAXMLLBWSA","created_at":"2026-07-05T11:23:20Z"},{"alias_kind":"pith_short_8","alias_value":"PVXJC3IA","created_at":"2026-07-05T11:23:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:PVXJC3IAXMLLBWSAGYTSYQAZU4","target":"record","payload":{"canonical_record":{"source":{"id":"2412.01190","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2024-12-02T06:48:42Z","cross_cats_sorted":["math.FA","math.PR"],"title_canon_sha256":"730750be5fa6bc36d6f3c1fccaa19f90882857c01911f2a8a37fb4e67a6cd198","abstract_canon_sha256":"f5f7162612ff060c8fc4193cfa8ad59ab2bc9d18a5dc969f0808ebc4b575dc86"},"schema_version":"1.0"},"canonical_sha256":"7d6e916d00bb16b0da4036272c4019a70261378ab09125e13bf19c42846bd7cd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:23:20.283489Z","signature_b64":"wQkKm/B7WtrA6X6toNkY57z+GxEQVr8vEEPLyoxhkB4bZ/YbkFLFOjiyGp+2OBr/mQrfUckkFkEMn0shYtNADg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7d6e916d00bb16b0da4036272c4019a70261378ab09125e13bf19c42846bd7cd","last_reissued_at":"2026-07-05T11:23:20.282945Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:23:20.282945Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.01190","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:23:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8TXaQiwl5Jz9sG9cYFySt8UCAss3ae4vM+/zZyYKNBk4hk2RbbUvTHX09Q5R75kQaMZVU835q34cKO0L5616AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T14:02:55.384821Z"},"content_sha256":"eed5c394652333463e83549cf50ba63bc9ef932853c891bdc0a6328ab15ce3bf","schema_version":"1.0","event_id":"sha256:eed5c394652333463e83549cf50ba63bc9ef932853c891bdc0a6328ab15ce3bf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:PVXJC3IAXMLLBWSAGYTSYQAZU4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the geometry of Wasserstein barycenter I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.MG","authors_text":"Bang-Xian Han, Deng-Yu Liu, Zhuo-Nan Zhu","submitted_at":"2024-12-02T06:48:42Z","abstract_excerpt":"We study the Wasserstein barycenter problem in the setting of non-compact, non-smooth extended metric measure spaces. We introduce a couple of new concepts and obtain the existence, uniqueness, absolute continuity of the Wasserstein barycenter, and prove Jensen's inequality in an abstract framework.\n  This generalized several results on Euclidean space, Riemannian manifolds and Alexandrov spaces, to metric measure spaces satisfying Riemannian Curvature-Dimension condition \\`a la Lott--Sturm--Villani, and some extended metric measure spaces including abstract Wiener spaces and configuration spa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.01190","kind":"arxiv","version":2},"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/2412.01190/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":"2026-07-05T11:23:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y+yVn15TWnUcpfy103xPBSqGvdzq8ZknFc+DG4Y5csrCGWSRKFWuhlMTqitvzVTfh9qczqTFu/F2spotiYUXAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T14:02:55.385461Z"},"content_sha256":"11c2599a0c4faaaed5708741309f97310d31e68f713f30ac0853d9411aebf891","schema_version":"1.0","event_id":"sha256:11c2599a0c4faaaed5708741309f97310d31e68f713f30ac0853d9411aebf891"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PVXJC3IAXMLLBWSAGYTSYQAZU4/bundle.json","state_url":"https://pith.science/pith/PVXJC3IAXMLLBWSAGYTSYQAZU4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PVXJC3IAXMLLBWSAGYTSYQAZU4/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":"2026-08-13T14:02:55Z","links":{"resolver":"https://pith.science/pith/PVXJC3IAXMLLBWSAGYTSYQAZU4","bundle":"https://pith.science/pith/PVXJC3IAXMLLBWSAGYTSYQAZU4/bundle.json","state":"https://pith.science/pith/PVXJC3IAXMLLBWSAGYTSYQAZU4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PVXJC3IAXMLLBWSAGYTSYQAZU4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:PVXJC3IAXMLLBWSAGYTSYQAZU4","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":"f5f7162612ff060c8fc4193cfa8ad59ab2bc9d18a5dc969f0808ebc4b575dc86","cross_cats_sorted":["math.FA","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2024-12-02T06:48:42Z","title_canon_sha256":"730750be5fa6bc36d6f3c1fccaa19f90882857c01911f2a8a37fb4e67a6cd198"},"schema_version":"1.0","source":{"id":"2412.01190","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.01190","created_at":"2026-07-05T11:23:20Z"},{"alias_kind":"arxiv_version","alias_value":"2412.01190v2","created_at":"2026-07-05T11:23:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.01190","created_at":"2026-07-05T11:23:20Z"},{"alias_kind":"pith_short_12","alias_value":"PVXJC3IAXMLL","created_at":"2026-07-05T11:23:20Z"},{"alias_kind":"pith_short_16","alias_value":"PVXJC3IAXMLLBWSA","created_at":"2026-07-05T11:23:20Z"},{"alias_kind":"pith_short_8","alias_value":"PVXJC3IA","created_at":"2026-07-05T11:23:20Z"}],"graph_snapshots":[{"event_id":"sha256:11c2599a0c4faaaed5708741309f97310d31e68f713f30ac0853d9411aebf891","target":"graph","created_at":"2026-07-05T11:23:20Z","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/2412.01190/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the Wasserstein barycenter problem in the setting of non-compact, non-smooth extended metric measure spaces. We introduce a couple of new concepts and obtain the existence, uniqueness, absolute continuity of the Wasserstein barycenter, and prove Jensen's inequality in an abstract framework.\n  This generalized several results on Euclidean space, Riemannian manifolds and Alexandrov spaces, to metric measure spaces satisfying Riemannian Curvature-Dimension condition \\`a la Lott--Sturm--Villani, and some extended metric measure spaces including abstract Wiener spaces and configuration spa","authors_text":"Bang-Xian Han, Deng-Yu Liu, Zhuo-Nan Zhu","cross_cats":["math.FA","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2024-12-02T06:48:42Z","title":"On the geometry of Wasserstein barycenter I"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.01190","kind":"arxiv","version":2},"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:eed5c394652333463e83549cf50ba63bc9ef932853c891bdc0a6328ab15ce3bf","target":"record","created_at":"2026-07-05T11:23:20Z","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":"f5f7162612ff060c8fc4193cfa8ad59ab2bc9d18a5dc969f0808ebc4b575dc86","cross_cats_sorted":["math.FA","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2024-12-02T06:48:42Z","title_canon_sha256":"730750be5fa6bc36d6f3c1fccaa19f90882857c01911f2a8a37fb4e67a6cd198"},"schema_version":"1.0","source":{"id":"2412.01190","kind":"arxiv","version":2}},"canonical_sha256":"7d6e916d00bb16b0da4036272c4019a70261378ab09125e13bf19c42846bd7cd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7d6e916d00bb16b0da4036272c4019a70261378ab09125e13bf19c42846bd7cd","first_computed_at":"2026-07-05T11:23:20.282945Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:23:20.282945Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wQkKm/B7WtrA6X6toNkY57z+GxEQVr8vEEPLyoxhkB4bZ/YbkFLFOjiyGp+2OBr/mQrfUckkFkEMn0shYtNADg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:23:20.283489Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.01190","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:eed5c394652333463e83549cf50ba63bc9ef932853c891bdc0a6328ab15ce3bf","sha256:11c2599a0c4faaaed5708741309f97310d31e68f713f30ac0853d9411aebf891"],"state_sha256":"349d5d92caf081c2fbc1606829d3efcd7f71d6fe50765c38869104430563a7f8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CBb1Jb63r7ax/59Nr7VfE5CL4lACmciPHPE56CiYJsMymcDWpqw+wnG0V/pTxMqFu+Q+dFlBBNYDdWHjn/hbDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T14:02:55.404986Z","bundle_sha256":"039227a4f3ab999da8296865b4d5d64f14c6abba3e2b6660efb7ec8d01d38ca1"}}