{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:VFV7QCGBXPO4DSSEC5BESV2MTW","short_pith_number":"pith:VFV7QCGB","canonical_record":{"source":{"id":"2607.22329","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-07-24T14:07:55Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"dcfd4ec60b082fd74f012a336408da1f04ed875b8819968a0ac18dc9112e944f","abstract_canon_sha256":"185c85ec177e127a72bb866bfa2989f83c190483f11d08a3548f3fe6a33fb9e0"},"schema_version":"1.0"},"canonical_sha256":"a96bf808c1bbddc1ca44174249574c9da7070a78e2385b187881f9aeeefe0286","source":{"kind":"arxiv","id":"2607.22329","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.22329","created_at":"2026-07-27T01:21:03Z"},{"alias_kind":"arxiv_version","alias_value":"2607.22329v1","created_at":"2026-07-27T01:21:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.22329","created_at":"2026-07-27T01:21:03Z"},{"alias_kind":"pith_short_12","alias_value":"VFV7QCGBXPO4","created_at":"2026-07-27T01:21:03Z"},{"alias_kind":"pith_short_16","alias_value":"VFV7QCGBXPO4DSSE","created_at":"2026-07-27T01:21:03Z"},{"alias_kind":"pith_short_8","alias_value":"VFV7QCGB","created_at":"2026-07-27T01:21:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:VFV7QCGBXPO4DSSEC5BESV2MTW","target":"record","payload":{"canonical_record":{"source":{"id":"2607.22329","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-07-24T14:07:55Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"dcfd4ec60b082fd74f012a336408da1f04ed875b8819968a0ac18dc9112e944f","abstract_canon_sha256":"185c85ec177e127a72bb866bfa2989f83c190483f11d08a3548f3fe6a33fb9e0"},"schema_version":"1.0"},"canonical_sha256":"a96bf808c1bbddc1ca44174249574c9da7070a78e2385b187881f9aeeefe0286","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-27T01:21:03.132350Z","signature_b64":"QYMONuN9w15ukpA88sDD8zoVM0l5xmLRnc/M1NA8cX0J0jJSEivVfAQvQMA6xbG5tUtXyqkf1RRzOcqZc1w+Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a96bf808c1bbddc1ca44174249574c9da7070a78e2385b187881f9aeeefe0286","last_reissued_at":"2026-07-27T01:21:03.131492Z","signature_status":"signed_v1","first_computed_at":"2026-07-27T01:21:03.131492Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.22329","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":"2026-07-27T01:21:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZC3B/KmqkqHUUG7lPuuEToBCrkX69Nr5RT+aUKvWD66bNABdkrwd/NUkQlqwQTg99SPNtRIvjfoTW8mDo83cCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T20:46:17.783825Z"},"content_sha256":"271503e6e613d7286a4876a7f9c0770f8b571235d0d637f37089aa4caf43fdaa","schema_version":"1.0","event_id":"sha256:271503e6e613d7286a4876a7f9c0770f8b571235d0d637f37089aa4caf43fdaa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:VFV7QCGBXPO4DSSEC5BESV2MTW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Homogeneous Landau Equation with Regularised Thermal Noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"Manh Hong Duong, Zhengyan Wu, Zihui He","submitted_at":"2026-07-24T14:07:55Z","abstract_excerpt":"We introduce and analyze a fluctuating homogeneous Landau equation with regularised thermal noise. The model is motivated by the nonlocal gradient flow structure of the deterministic Landau equation, the fluctuation--dissipation principle, and the covariance of the martingale fluctuations of a Kac-like conservative Landau particle system. The noise is written in Landau-divergence form, is antisymmetric in the pair of velocities, and is interpreted in the Stratonovich sense after introducing a velocity correlation. To handle the vacuum singularity of the square-root mobility and the nonlocal St"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.22329","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.22329/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":"2026-07-27T01:21:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9mXfGwI614t5oo+3LrPprYrNBhylct+zbTCtgbuuEWn5st7J1RX2uni3q8WZ/JCJiVQIsBOyCElTLnSU9mA4BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T20:46:17.784326Z"},"content_sha256":"3dc6cf6e8fa715fabbb5cd2d58b922b635fae5f0c7b190f76cac9fe081e42403","schema_version":"1.0","event_id":"sha256:3dc6cf6e8fa715fabbb5cd2d58b922b635fae5f0c7b190f76cac9fe081e42403"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VFV7QCGBXPO4DSSEC5BESV2MTW/bundle.json","state_url":"https://pith.science/pith/VFV7QCGBXPO4DSSEC5BESV2MTW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VFV7QCGBXPO4DSSEC5BESV2MTW/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":"2026-08-08T20:46:17Z","links":{"resolver":"https://pith.science/pith/VFV7QCGBXPO4DSSEC5BESV2MTW","bundle":"https://pith.science/pith/VFV7QCGBXPO4DSSEC5BESV2MTW/bundle.json","state":"https://pith.science/pith/VFV7QCGBXPO4DSSEC5BESV2MTW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VFV7QCGBXPO4DSSEC5BESV2MTW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VFV7QCGBXPO4DSSEC5BESV2MTW","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":"185c85ec177e127a72bb866bfa2989f83c190483f11d08a3548f3fe6a33fb9e0","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-07-24T14:07:55Z","title_canon_sha256":"dcfd4ec60b082fd74f012a336408da1f04ed875b8819968a0ac18dc9112e944f"},"schema_version":"1.0","source":{"id":"2607.22329","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.22329","created_at":"2026-07-27T01:21:03Z"},{"alias_kind":"arxiv_version","alias_value":"2607.22329v1","created_at":"2026-07-27T01:21:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.22329","created_at":"2026-07-27T01:21:03Z"},{"alias_kind":"pith_short_12","alias_value":"VFV7QCGBXPO4","created_at":"2026-07-27T01:21:03Z"},{"alias_kind":"pith_short_16","alias_value":"VFV7QCGBXPO4DSSE","created_at":"2026-07-27T01:21:03Z"},{"alias_kind":"pith_short_8","alias_value":"VFV7QCGB","created_at":"2026-07-27T01:21:03Z"}],"graph_snapshots":[{"event_id":"sha256:3dc6cf6e8fa715fabbb5cd2d58b922b635fae5f0c7b190f76cac9fe081e42403","target":"graph","created_at":"2026-07-27T01:21:03Z","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/2607.22329/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce and analyze a fluctuating homogeneous Landau equation with regularised thermal noise. The model is motivated by the nonlocal gradient flow structure of the deterministic Landau equation, the fluctuation--dissipation principle, and the covariance of the martingale fluctuations of a Kac-like conservative Landau particle system. The noise is written in Landau-divergence form, is antisymmetric in the pair of velocities, and is interpreted in the Stratonovich sense after introducing a velocity correlation. To handle the vacuum singularity of the square-root mobility and the nonlocal St","authors_text":"Manh Hong Duong, Zhengyan Wu, Zihui He","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-07-24T14:07:55Z","title":"The Homogeneous Landau Equation with Regularised Thermal Noise"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.22329","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":"sha256:271503e6e613d7286a4876a7f9c0770f8b571235d0d637f37089aa4caf43fdaa","target":"record","created_at":"2026-07-27T01:21:03Z","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":"185c85ec177e127a72bb866bfa2989f83c190483f11d08a3548f3fe6a33fb9e0","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-07-24T14:07:55Z","title_canon_sha256":"dcfd4ec60b082fd74f012a336408da1f04ed875b8819968a0ac18dc9112e944f"},"schema_version":"1.0","source":{"id":"2607.22329","kind":"arxiv","version":1}},"canonical_sha256":"a96bf808c1bbddc1ca44174249574c9da7070a78e2385b187881f9aeeefe0286","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a96bf808c1bbddc1ca44174249574c9da7070a78e2385b187881f9aeeefe0286","first_computed_at":"2026-07-27T01:21:03.131492Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-27T01:21:03.131492Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QYMONuN9w15ukpA88sDD8zoVM0l5xmLRnc/M1NA8cX0J0jJSEivVfAQvQMA6xbG5tUtXyqkf1RRzOcqZc1w+Cg==","signature_status":"signed_v1","signed_at":"2026-07-27T01:21:03.132350Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.22329","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:271503e6e613d7286a4876a7f9c0770f8b571235d0d637f37089aa4caf43fdaa","sha256:3dc6cf6e8fa715fabbb5cd2d58b922b635fae5f0c7b190f76cac9fe081e42403"],"state_sha256":"2813dcf471000b9f24b3de8b46b6204a54d70070f5fb0af9ec050d428bc38c6f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Fv+S38ek8MWXAWVRYKo877Hrfd0uhINvBJiZ5UjP636mm26jeBG52XaTYBtz/Wq0D2eqJpiS65JGsmhhYDcTBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T20:46:17.788155Z","bundle_sha256":"ce0fdb8db3569c328393b7ad6dddff0faff426e927628caf013628ec8369fa76"}}