{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:HRWIPV5CW5CG53FWEODQQ76NHD","short_pith_number":"pith:HRWIPV5C","canonical_record":{"source":{"id":"2506.01279","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2025-06-02T03:21:36Z","cross_cats_sorted":[],"title_canon_sha256":"f0da2f4f0213e843b52e328151625f6489261c5c63cc1c5dfb62aee28b22488f","abstract_canon_sha256":"f23592b8e35b6a1c681baaff30207866ca2fb70723c0961a58a51750a9dfa701"},"schema_version":"1.0"},"canonical_sha256":"3c6c87d7a2b7446eecb62387087fcd38dc5777424f7e029ed5ca896e4db0fff9","source":{"kind":"arxiv","id":"2506.01279","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.01279","created_at":"2026-07-05T11:25:45Z"},{"alias_kind":"arxiv_version","alias_value":"2506.01279v2","created_at":"2026-07-05T11:25:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.01279","created_at":"2026-07-05T11:25:45Z"},{"alias_kind":"pith_short_12","alias_value":"HRWIPV5CW5CG","created_at":"2026-07-05T11:25:45Z"},{"alias_kind":"pith_short_16","alias_value":"HRWIPV5CW5CG53FW","created_at":"2026-07-05T11:25:45Z"},{"alias_kind":"pith_short_8","alias_value":"HRWIPV5C","created_at":"2026-07-05T11:25:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:HRWIPV5CW5CG53FWEODQQ76NHD","target":"record","payload":{"canonical_record":{"source":{"id":"2506.01279","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2025-06-02T03:21:36Z","cross_cats_sorted":[],"title_canon_sha256":"f0da2f4f0213e843b52e328151625f6489261c5c63cc1c5dfb62aee28b22488f","abstract_canon_sha256":"f23592b8e35b6a1c681baaff30207866ca2fb70723c0961a58a51750a9dfa701"},"schema_version":"1.0"},"canonical_sha256":"3c6c87d7a2b7446eecb62387087fcd38dc5777424f7e029ed5ca896e4db0fff9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:25:45.406901Z","signature_b64":"wdqYrpYZlDCMODihDRX5PE/Ce0S8WTa2vtiGUF//SLSungsJG7GahOF9UFw9iqDALkOnB5GoZVBD9IooNP9OCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3c6c87d7a2b7446eecb62387087fcd38dc5777424f7e029ed5ca896e4db0fff9","last_reissued_at":"2026-07-05T11:25:45.406388Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:25:45.406388Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.01279","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":"2026-07-05T11:25:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"weJgXZn3+Srsv0mJZinQpzlB+jMhb/udsVg7cqZXUan1Wbx3r94DrcfW/8+Hs+2FmhN0CYE05aunARIfKoiZBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T04:07:49.957197Z"},"content_sha256":"d40f633cd5470052a15c8a446c7290ccae3b0ed3b28e86c5fe96b638dc41394c","schema_version":"1.0","event_id":"sha256:d40f633cd5470052a15c8a446c7290ccae3b0ed3b28e86c5fe96b638dc41394c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:HRWIPV5CW5CG53FWEODQQ76NHD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$W$-entropy formulas and Langevin deformation on the $L^q$-Wasserstein space over Riemannian manifolds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Rong Lei, Xiang-Dong Li, Yu-Zhao Wang","submitted_at":"2025-06-02T03:21:36Z","abstract_excerpt":"We first prove the $W$-entropy formula and rigidity theorem for the geodesic flow on the $L^q$-Wasserstein space over a complete Riemannian manifold with bounded geometry condition. Then we introduce the Langevin deformation on the $L^q$-Wasserstein space over a complete Riemannian manifold, which interpolates between the $p$-Laplacian heat equation and the geodesic flow on the $L^q$-Wasserstein space, where ${1\\over p}+{1\\over q}=1$, $1< p, q<\\infty$. The local existence, uniqueness and regularity of the Langevin deformation on the $L^q$-Wasserstein space over the Euclidean space and a compac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.01279","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/2506.01279/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":"2026-07-05T11:25:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r9k5fmVhdp9rIvbqwrm13Z7ATpZ5yoa6cniheymOvNLgOi/FBflbmGHQned4KzScPda545CuaiMyX0xgNi8zAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T04:07:49.957714Z"},"content_sha256":"da4efc78a2a923e41d2cfa46a4ec0da890c567c0defbeaf55b09e695c14e27c2","schema_version":"1.0","event_id":"sha256:da4efc78a2a923e41d2cfa46a4ec0da890c567c0defbeaf55b09e695c14e27c2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HRWIPV5CW5CG53FWEODQQ76NHD/bundle.json","state_url":"https://pith.science/pith/HRWIPV5CW5CG53FWEODQQ76NHD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HRWIPV5CW5CG53FWEODQQ76NHD/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":"2026-08-08T04:07:49Z","links":{"resolver":"https://pith.science/pith/HRWIPV5CW5CG53FWEODQQ76NHD","bundle":"https://pith.science/pith/HRWIPV5CW5CG53FWEODQQ76NHD/bundle.json","state":"https://pith.science/pith/HRWIPV5CW5CG53FWEODQQ76NHD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HRWIPV5CW5CG53FWEODQQ76NHD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:HRWIPV5CW5CG53FWEODQQ76NHD","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":"f23592b8e35b6a1c681baaff30207866ca2fb70723c0961a58a51750a9dfa701","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2025-06-02T03:21:36Z","title_canon_sha256":"f0da2f4f0213e843b52e328151625f6489261c5c63cc1c5dfb62aee28b22488f"},"schema_version":"1.0","source":{"id":"2506.01279","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.01279","created_at":"2026-07-05T11:25:45Z"},{"alias_kind":"arxiv_version","alias_value":"2506.01279v2","created_at":"2026-07-05T11:25:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.01279","created_at":"2026-07-05T11:25:45Z"},{"alias_kind":"pith_short_12","alias_value":"HRWIPV5CW5CG","created_at":"2026-07-05T11:25:45Z"},{"alias_kind":"pith_short_16","alias_value":"HRWIPV5CW5CG53FW","created_at":"2026-07-05T11:25:45Z"},{"alias_kind":"pith_short_8","alias_value":"HRWIPV5C","created_at":"2026-07-05T11:25:45Z"}],"graph_snapshots":[{"event_id":"sha256:da4efc78a2a923e41d2cfa46a4ec0da890c567c0defbeaf55b09e695c14e27c2","target":"graph","created_at":"2026-07-05T11:25:45Z","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/2506.01279/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We first prove the $W$-entropy formula and rigidity theorem for the geodesic flow on the $L^q$-Wasserstein space over a complete Riemannian manifold with bounded geometry condition. Then we introduce the Langevin deformation on the $L^q$-Wasserstein space over a complete Riemannian manifold, which interpolates between the $p$-Laplacian heat equation and the geodesic flow on the $L^q$-Wasserstein space, where ${1\\over p}+{1\\over q}=1$, $1< p, q<\\infty$. The local existence, uniqueness and regularity of the Langevin deformation on the $L^q$-Wasserstein space over the Euclidean space and a compac","authors_text":"Rong Lei, Xiang-Dong Li, Yu-Zhao Wang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2025-06-02T03:21:36Z","title":"$W$-entropy formulas and Langevin deformation on the $L^q$-Wasserstein space over Riemannian manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.01279","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:d40f633cd5470052a15c8a446c7290ccae3b0ed3b28e86c5fe96b638dc41394c","target":"record","created_at":"2026-07-05T11:25:45Z","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":"f23592b8e35b6a1c681baaff30207866ca2fb70723c0961a58a51750a9dfa701","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2025-06-02T03:21:36Z","title_canon_sha256":"f0da2f4f0213e843b52e328151625f6489261c5c63cc1c5dfb62aee28b22488f"},"schema_version":"1.0","source":{"id":"2506.01279","kind":"arxiv","version":2}},"canonical_sha256":"3c6c87d7a2b7446eecb62387087fcd38dc5777424f7e029ed5ca896e4db0fff9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3c6c87d7a2b7446eecb62387087fcd38dc5777424f7e029ed5ca896e4db0fff9","first_computed_at":"2026-07-05T11:25:45.406388Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:25:45.406388Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wdqYrpYZlDCMODihDRX5PE/Ce0S8WTa2vtiGUF//SLSungsJG7GahOF9UFw9iqDALkOnB5GoZVBD9IooNP9OCA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:25:45.406901Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.01279","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d40f633cd5470052a15c8a446c7290ccae3b0ed3b28e86c5fe96b638dc41394c","sha256:da4efc78a2a923e41d2cfa46a4ec0da890c567c0defbeaf55b09e695c14e27c2"],"state_sha256":"f5f3495bb6728d039e41898c1df9e96df9bd20ddc10fd0b2be2ab7b841e2e4e4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sSg1Zrrs3A/yj9yLtjO4u/mzM77YDMnaNJFk/fI+oPCiZuZQgzrRyRfuwMmhyfmoBIQqzoaJ7lhP7ys7Pq+hAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T04:07:49.961332Z","bundle_sha256":"902f2c7c72e3566db15b8a7e62aac9ebe3eae8508dabfe6056d07478a959b46c"}}