{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:REWTMQ5LRH2PG5T4DIXURYZCXB","short_pith_number":"pith:REWTMQ5L","canonical_record":{"source":{"id":"2403.16140","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-03-24T13:24:01Z","cross_cats_sorted":[],"title_canon_sha256":"c48e7ff5358327562d5628e1bdde33b74ec8b40175c091818a91760a9f4bdebe","abstract_canon_sha256":"d9eb6fd4a9104b3c4c5e81611b4564cb13985e0af343c662b3e866c09917bd5d"},"schema_version":"1.0"},"canonical_sha256":"892d3643ab89f4f3767c1a2f48e322b86a7e4968cf73bc879c041e362d706044","source":{"kind":"arxiv","id":"2403.16140","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2403.16140","created_at":"2026-07-05T08:29:24Z"},{"alias_kind":"arxiv_version","alias_value":"2403.16140v2","created_at":"2026-07-05T08:29:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.16140","created_at":"2026-07-05T08:29:24Z"},{"alias_kind":"pith_short_12","alias_value":"REWTMQ5LRH2P","created_at":"2026-07-05T08:29:24Z"},{"alias_kind":"pith_short_16","alias_value":"REWTMQ5LRH2PG5T4","created_at":"2026-07-05T08:29:24Z"},{"alias_kind":"pith_short_8","alias_value":"REWTMQ5L","created_at":"2026-07-05T08:29:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:REWTMQ5LRH2PG5T4DIXURYZCXB","target":"record","payload":{"canonical_record":{"source":{"id":"2403.16140","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-03-24T13:24:01Z","cross_cats_sorted":[],"title_canon_sha256":"c48e7ff5358327562d5628e1bdde33b74ec8b40175c091818a91760a9f4bdebe","abstract_canon_sha256":"d9eb6fd4a9104b3c4c5e81611b4564cb13985e0af343c662b3e866c09917bd5d"},"schema_version":"1.0"},"canonical_sha256":"892d3643ab89f4f3767c1a2f48e322b86a7e4968cf73bc879c041e362d706044","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:29:24.790479Z","signature_b64":"AzWfppAgWDtYv17Fcv8LCyn4LQHNcuQQ9yJPfeKJFWa9mOk+/uaU8Pb8Y9sV5VRXpmZ/UEfOW6Lfv0hD495dBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"892d3643ab89f4f3767c1a2f48e322b86a7e4968cf73bc879c041e362d706044","last_reissued_at":"2026-07-05T08:29:24.790008Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:29:24.790008Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2403.16140","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":"2026-07-05T08:29:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oNwLpyUp3Rt8flsZzEkGpUzBfAi+YqG6Cz162n1S9DSueFNXusrpXJ4cXR8tNhiQoSmV+h/84YVS+SC9Fr4WDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T18:48:26.957636Z"},"content_sha256":"baaf0ac43b0cce8d1c9bd40bdda3b7f18376986a269ddecc354610dfc89d51b9","schema_version":"1.0","event_id":"sha256:baaf0ac43b0cce8d1c9bd40bdda3b7f18376986a269ddecc354610dfc89d51b9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:REWTMQ5LRH2PG5T4DIXURYZCXB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Rearranged Stochastic Heat Equation: Ergodicity and Related Gradient Descent on ${\\mathcal P}({\\mathbb R})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Fran\\c{c}ois Delarue, William R.P. Hammersley","submitted_at":"2024-03-24T13:24:01Z","abstract_excerpt":"This article provides a case study for a recently introduced diffusion in the space of probability measures over the reals, namely rearranged stochastic heat, which solves a stochastic partial differential equation valued in the set of symmetrised quantile functions over the unit circle. This contribution studies probability measure-valued flows perturbed by this noise with a special focus on gradient flows. This is done by introducing a drift to the rearranged stochastic heat equation by means of a vector field from the set of random variables over the unit circle into itself. When the flow i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.16140","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/2403.16140/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":"2026-07-05T08:29:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cY/4pe/K9jyQXFYtAec2OazkufpB1lvrMSfsd+orLAt4YTvPhECap02pkB6tDougXOzBnMt4KW/SXaRfi5u9Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T18:48:26.958191Z"},"content_sha256":"ea446b58d02db637a279d0415256ad086665565d03f2304c62b41580d7def857","schema_version":"1.0","event_id":"sha256:ea446b58d02db637a279d0415256ad086665565d03f2304c62b41580d7def857"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/REWTMQ5LRH2PG5T4DIXURYZCXB/bundle.json","state_url":"https://pith.science/pith/REWTMQ5LRH2PG5T4DIXURYZCXB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/REWTMQ5LRH2PG5T4DIXURYZCXB/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":"2026-08-12T18:48:26Z","links":{"resolver":"https://pith.science/pith/REWTMQ5LRH2PG5T4DIXURYZCXB","bundle":"https://pith.science/pith/REWTMQ5LRH2PG5T4DIXURYZCXB/bundle.json","state":"https://pith.science/pith/REWTMQ5LRH2PG5T4DIXURYZCXB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/REWTMQ5LRH2PG5T4DIXURYZCXB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:REWTMQ5LRH2PG5T4DIXURYZCXB","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":"d9eb6fd4a9104b3c4c5e81611b4564cb13985e0af343c662b3e866c09917bd5d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-03-24T13:24:01Z","title_canon_sha256":"c48e7ff5358327562d5628e1bdde33b74ec8b40175c091818a91760a9f4bdebe"},"schema_version":"1.0","source":{"id":"2403.16140","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2403.16140","created_at":"2026-07-05T08:29:24Z"},{"alias_kind":"arxiv_version","alias_value":"2403.16140v2","created_at":"2026-07-05T08:29:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.16140","created_at":"2026-07-05T08:29:24Z"},{"alias_kind":"pith_short_12","alias_value":"REWTMQ5LRH2P","created_at":"2026-07-05T08:29:24Z"},{"alias_kind":"pith_short_16","alias_value":"REWTMQ5LRH2PG5T4","created_at":"2026-07-05T08:29:24Z"},{"alias_kind":"pith_short_8","alias_value":"REWTMQ5L","created_at":"2026-07-05T08:29:24Z"}],"graph_snapshots":[{"event_id":"sha256:ea446b58d02db637a279d0415256ad086665565d03f2304c62b41580d7def857","target":"graph","created_at":"2026-07-05T08:29:24Z","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/2403.16140/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This article provides a case study for a recently introduced diffusion in the space of probability measures over the reals, namely rearranged stochastic heat, which solves a stochastic partial differential equation valued in the set of symmetrised quantile functions over the unit circle. This contribution studies probability measure-valued flows perturbed by this noise with a special focus on gradient flows. This is done by introducing a drift to the rearranged stochastic heat equation by means of a vector field from the set of random variables over the unit circle into itself. When the flow i","authors_text":"Fran\\c{c}ois Delarue, William R.P. Hammersley","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-03-24T13:24:01Z","title":"Rearranged Stochastic Heat Equation: Ergodicity and Related Gradient Descent on ${\\mathcal P}({\\mathbb R})$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.16140","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:baaf0ac43b0cce8d1c9bd40bdda3b7f18376986a269ddecc354610dfc89d51b9","target":"record","created_at":"2026-07-05T08:29:24Z","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":"d9eb6fd4a9104b3c4c5e81611b4564cb13985e0af343c662b3e866c09917bd5d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-03-24T13:24:01Z","title_canon_sha256":"c48e7ff5358327562d5628e1bdde33b74ec8b40175c091818a91760a9f4bdebe"},"schema_version":"1.0","source":{"id":"2403.16140","kind":"arxiv","version":2}},"canonical_sha256":"892d3643ab89f4f3767c1a2f48e322b86a7e4968cf73bc879c041e362d706044","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"892d3643ab89f4f3767c1a2f48e322b86a7e4968cf73bc879c041e362d706044","first_computed_at":"2026-07-05T08:29:24.790008Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:29:24.790008Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AzWfppAgWDtYv17Fcv8LCyn4LQHNcuQQ9yJPfeKJFWa9mOk+/uaU8Pb8Y9sV5VRXpmZ/UEfOW6Lfv0hD495dBw==","signature_status":"signed_v1","signed_at":"2026-07-05T08:29:24.790479Z","signed_message":"canonical_sha256_bytes"},"source_id":"2403.16140","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:baaf0ac43b0cce8d1c9bd40bdda3b7f18376986a269ddecc354610dfc89d51b9","sha256:ea446b58d02db637a279d0415256ad086665565d03f2304c62b41580d7def857"],"state_sha256":"2b390146307bfec6850439bbb2b1a61b8ee3f6fa8abaf6e1fd073e1c052086a9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zfPvZjyqI/ac+0y9aC2l3+QhGWX7Dqk0AV1VGkDqRFLzxBHys9QvkvpJeBicBXJm/81RocnyhnyWUGZROKEVAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T18:48:26.964737Z","bundle_sha256":"7df2ce6638d38adda32c48f5b356535004dcbcae028b213204e4d959aacdb78f"}}