{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:JJERQKVQ3KL2HIISHKSX6G6E6A","short_pith_number":"pith:JJERQKVQ","canonical_record":{"source":{"id":"2404.15205","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-04-23T16:38:49Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"f34f9f04e7d3d90140fef5663ad590bad8c7e43ef44e158fc9159619b36eca2b","abstract_canon_sha256":"db1d8f782c3b9f3695e09ba1cea0de72b2bee65bb354efec29d734bc1f68d8fe"},"schema_version":"1.0"},"canonical_sha256":"4a49182ab0da97a3a1123aa57f1bc4f003a49418802c3d31a2f0cabbcba76f3f","source":{"kind":"arxiv","id":"2404.15205","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.15205","created_at":"2026-07-05T08:16:09Z"},{"alias_kind":"arxiv_version","alias_value":"2404.15205v2","created_at":"2026-07-05T08:16:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.15205","created_at":"2026-07-05T08:16:09Z"},{"alias_kind":"pith_short_12","alias_value":"JJERQKVQ3KL2","created_at":"2026-07-05T08:16:09Z"},{"alias_kind":"pith_short_16","alias_value":"JJERQKVQ3KL2HIIS","created_at":"2026-07-05T08:16:09Z"},{"alias_kind":"pith_short_8","alias_value":"JJERQKVQ","created_at":"2026-07-05T08:16:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:JJERQKVQ3KL2HIISHKSX6G6E6A","target":"record","payload":{"canonical_record":{"source":{"id":"2404.15205","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-04-23T16:38:49Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"f34f9f04e7d3d90140fef5663ad590bad8c7e43ef44e158fc9159619b36eca2b","abstract_canon_sha256":"db1d8f782c3b9f3695e09ba1cea0de72b2bee65bb354efec29d734bc1f68d8fe"},"schema_version":"1.0"},"canonical_sha256":"4a49182ab0da97a3a1123aa57f1bc4f003a49418802c3d31a2f0cabbcba76f3f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:16:09.115042Z","signature_b64":"QVvzg/yBdCe/uaH2LqpZKrMESiTqx3+4UrbYeHqIohNExrgQSLozw1LnnLxWmFkUoOJ+0Jiy8/6nkvqq/3aqBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a49182ab0da97a3a1123aa57f1bc4f003a49418802c3d31a2f0cabbcba76f3f","last_reissued_at":"2026-07-05T08:16:09.114587Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:16:09.114587Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2404.15205","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":"2026-07-05T08:16:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mIikOCZ0DJ/spd95oBltGQI3Q0BpSjVgRjCPP6BT0+FvBYYvzxdtSkLKEI10NClEZ3y/+jiviqUShGruEqKKAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T20:55:41.322086Z"},"content_sha256":"317dedd4af754737b56714a30275c8f1de4be341efab5786c0b2ced5bc6c6b67","schema_version":"1.0","event_id":"sha256:317dedd4af754737b56714a30275c8f1de4be341efab5786c0b2ced5bc6c6b67"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:JJERQKVQ3KL2HIISHKSX6G6E6A","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Heat flow, log-concavity, and Lipschitz transport maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"Francesco Pedrotti, Giovanni Brigati","submitted_at":"2024-04-23T16:38:49Z","abstract_excerpt":"In this paper we derive estimates for the Hessian of the logarithm (log-Hessian) for solutions to the heat equation. For initial data in the form of log-Lipschitz perturbation of strongly log-concave measures, the log-Hessian admits an explicit, uniform (in space) lower bound. This yields a new estimate for the Lipschitz constant of a transport map pushing forward the standard Gaussian to a measure in this class. Further connections are discussed with score-based diffusion models and improved Gaussian logarithmic Sobolev inequalities. Finally, we show that assuming only fast decay of the tails"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.15205","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/2404.15205/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":"2026-07-05T08:16:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hpdi/0iAMrdRvkMaVSiEGk0UNH5fjD6i1u4XKoonmy32F3SV40LSNG/IW/nhvT80rVb87yHySPeIE+InHZ4vAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T20:55:41.322453Z"},"content_sha256":"0276afc2b14176b7f0d6323fb0fa22793cde4bfdb4a9588b6c870ec58da9ee4c","schema_version":"1.0","event_id":"sha256:0276afc2b14176b7f0d6323fb0fa22793cde4bfdb4a9588b6c870ec58da9ee4c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JJERQKVQ3KL2HIISHKSX6G6E6A/bundle.json","state_url":"https://pith.science/pith/JJERQKVQ3KL2HIISHKSX6G6E6A/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JJERQKVQ3KL2HIISHKSX6G6E6A/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":"2026-08-22T20:55:41Z","links":{"resolver":"https://pith.science/pith/JJERQKVQ3KL2HIISHKSX6G6E6A","bundle":"https://pith.science/pith/JJERQKVQ3KL2HIISHKSX6G6E6A/bundle.json","state":"https://pith.science/pith/JJERQKVQ3KL2HIISHKSX6G6E6A/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JJERQKVQ3KL2HIISHKSX6G6E6A/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:JJERQKVQ3KL2HIISHKSX6G6E6A","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":"db1d8f782c3b9f3695e09ba1cea0de72b2bee65bb354efec29d734bc1f68d8fe","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-04-23T16:38:49Z","title_canon_sha256":"f34f9f04e7d3d90140fef5663ad590bad8c7e43ef44e158fc9159619b36eca2b"},"schema_version":"1.0","source":{"id":"2404.15205","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.15205","created_at":"2026-07-05T08:16:09Z"},{"alias_kind":"arxiv_version","alias_value":"2404.15205v2","created_at":"2026-07-05T08:16:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.15205","created_at":"2026-07-05T08:16:09Z"},{"alias_kind":"pith_short_12","alias_value":"JJERQKVQ3KL2","created_at":"2026-07-05T08:16:09Z"},{"alias_kind":"pith_short_16","alias_value":"JJERQKVQ3KL2HIIS","created_at":"2026-07-05T08:16:09Z"},{"alias_kind":"pith_short_8","alias_value":"JJERQKVQ","created_at":"2026-07-05T08:16:09Z"}],"graph_snapshots":[{"event_id":"sha256:0276afc2b14176b7f0d6323fb0fa22793cde4bfdb4a9588b6c870ec58da9ee4c","target":"graph","created_at":"2026-07-05T08:16:09Z","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/2404.15205/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we derive estimates for the Hessian of the logarithm (log-Hessian) for solutions to the heat equation. For initial data in the form of log-Lipschitz perturbation of strongly log-concave measures, the log-Hessian admits an explicit, uniform (in space) lower bound. This yields a new estimate for the Lipschitz constant of a transport map pushing forward the standard Gaussian to a measure in this class. Further connections are discussed with score-based diffusion models and improved Gaussian logarithmic Sobolev inequalities. Finally, we show that assuming only fast decay of the tails","authors_text":"Francesco Pedrotti, Giovanni Brigati","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-04-23T16:38:49Z","title":"Heat flow, log-concavity, and Lipschitz transport maps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.15205","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":"sha256:317dedd4af754737b56714a30275c8f1de4be341efab5786c0b2ced5bc6c6b67","target":"record","created_at":"2026-07-05T08:16:09Z","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":"db1d8f782c3b9f3695e09ba1cea0de72b2bee65bb354efec29d734bc1f68d8fe","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-04-23T16:38:49Z","title_canon_sha256":"f34f9f04e7d3d90140fef5663ad590bad8c7e43ef44e158fc9159619b36eca2b"},"schema_version":"1.0","source":{"id":"2404.15205","kind":"arxiv","version":2}},"canonical_sha256":"4a49182ab0da97a3a1123aa57f1bc4f003a49418802c3d31a2f0cabbcba76f3f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4a49182ab0da97a3a1123aa57f1bc4f003a49418802c3d31a2f0cabbcba76f3f","first_computed_at":"2026-07-05T08:16:09.114587Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:16:09.114587Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QVvzg/yBdCe/uaH2LqpZKrMESiTqx3+4UrbYeHqIohNExrgQSLozw1LnnLxWmFkUoOJ+0Jiy8/6nkvqq/3aqBA==","signature_status":"signed_v1","signed_at":"2026-07-05T08:16:09.115042Z","signed_message":"canonical_sha256_bytes"},"source_id":"2404.15205","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:317dedd4af754737b56714a30275c8f1de4be341efab5786c0b2ced5bc6c6b67","sha256:0276afc2b14176b7f0d6323fb0fa22793cde4bfdb4a9588b6c870ec58da9ee4c"],"state_sha256":"e596308b310a2bc85444f842da7f5ecc1aa7f8b80443ff2b0d703c42c7167705"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7hE494ObZCIbw4uklBw7jSK3O6KX9JTFAeX8o/b8kr+WB1Ne48a4WOO79Cy8e6rCRX2zct6HFg/6mFnNqW6lCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-22T20:55:41.324992Z","bundle_sha256":"0b40f05b27ebfcd39250e5ed69b1f2b9b5541d48ea69119407f4b602a26a076f"}}