{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:T6DPRYEPFYWCCTYTLROCGSN4AN","short_pith_number":"pith:T6DPRYEP","canonical_record":{"source":{"id":"1605.06839","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-22T19:40:58Z","cross_cats_sorted":[],"title_canon_sha256":"0f4a6cbaf5143cf52399ddd3087ee0b497ef4aad5db262b7c4472b5fc5ec4835","abstract_canon_sha256":"b11d372fe836f4628a87f928907b8922ad90290cdede7b50099b85d6e71b21b3"},"schema_version":"1.0"},"canonical_sha256":"9f86f8e08f2e2c214f135c5c2349bc03720d70a6d720642c90921dd34a684b39","source":{"kind":"arxiv","id":"1605.06839","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.06839","created_at":"2026-05-18T00:22:29Z"},{"alias_kind":"arxiv_version","alias_value":"1605.06839v3","created_at":"2026-05-18T00:22:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.06839","created_at":"2026-05-18T00:22:29Z"},{"alias_kind":"pith_short_12","alias_value":"T6DPRYEPFYWC","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"T6DPRYEPFYWCCTYT","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"T6DPRYEP","created_at":"2026-05-18T12:30:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:T6DPRYEPFYWCCTYTLROCGSN4AN","target":"record","payload":{"canonical_record":{"source":{"id":"1605.06839","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-22T19:40:58Z","cross_cats_sorted":[],"title_canon_sha256":"0f4a6cbaf5143cf52399ddd3087ee0b497ef4aad5db262b7c4472b5fc5ec4835","abstract_canon_sha256":"b11d372fe836f4628a87f928907b8922ad90290cdede7b50099b85d6e71b21b3"},"schema_version":"1.0"},"canonical_sha256":"9f86f8e08f2e2c214f135c5c2349bc03720d70a6d720642c90921dd34a684b39","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:29.873907Z","signature_b64":"P5hFjodjoswB7StzXZnkS0hNwKVtaIuNi9bjK/zOSq5UKB1IRN9BrWehGCzR7bDKPX+D0aEsENX/Zy3m6XFkDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9f86f8e08f2e2c214f135c5c2349bc03720d70a6d720642c90921dd34a684b39","last_reissued_at":"2026-05-18T00:22:29.873298Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:29.873298Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1605.06839","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:22:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sVyrStR1eSt7OyDZKRgbU5NG26cz5JkCWT8otX1fTVePyre049dbk+763207GPJLHgaECgCU+jXs4HauuIInDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T13:51:23.338554Z"},"content_sha256":"a25fed612f5f7e72f4d89f5225c12340b09d4b55dc85f777b56f26b0eda5a9ca","schema_version":"1.0","event_id":"sha256:a25fed612f5f7e72f4d89f5225c12340b09d4b55dc85f777b56f26b0eda5a9ca"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:T6DPRYEPFYWCCTYTLROCGSN4AN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Geometric inequalities on Heisenberg groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alexandru Krist\\'aly, Kinga Sipos, Zolt\\'an M. Balogh","submitted_at":"2016-05-22T19:40:58Z","abstract_excerpt":"We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group $\\mathbb H^n$. Our results include a natural sub-Riemannian version of the celebrated curvature-dimension condition of Lott-Villani and Sturm and also a geodesic version of the Borell-Brascamp-Lieb inequality akin to the one obtained by Cordero-Erausquin, McCann and Schmuckenschl\\\"ager. The latter statement implies sub-Riemannian versions of the geodesic Pr\\'ekopa-Leindler and Brunn-Minkowski inequalities. The proofs are based on optimal mass transportation and Riemannian approximation of $\\mathbb H^n$ de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06839","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T00:22:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FXNLEHfQcdc07FKgSe1kDvSIrrphI/pBW3dXef2zbsu1dw48TR6a21MJ470SwLYWSeDpbmgdSGtkW3fZFBdxBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T13:51:23.338898Z"},"content_sha256":"72005f3fd645a6803f9df9015ade610d0e10813257dfa34e8d907b15f0eb38cf","schema_version":"1.0","event_id":"sha256:72005f3fd645a6803f9df9015ade610d0e10813257dfa34e8d907b15f0eb38cf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/T6DPRYEPFYWCCTYTLROCGSN4AN/bundle.json","state_url":"https://pith.science/pith/T6DPRYEPFYWCCTYTLROCGSN4AN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/T6DPRYEPFYWCCTYTLROCGSN4AN/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":"2026-06-08T13:51:23Z","links":{"resolver":"https://pith.science/pith/T6DPRYEPFYWCCTYTLROCGSN4AN","bundle":"https://pith.science/pith/T6DPRYEPFYWCCTYTLROCGSN4AN/bundle.json","state":"https://pith.science/pith/T6DPRYEPFYWCCTYTLROCGSN4AN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/T6DPRYEPFYWCCTYTLROCGSN4AN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:T6DPRYEPFYWCCTYTLROCGSN4AN","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":"b11d372fe836f4628a87f928907b8922ad90290cdede7b50099b85d6e71b21b3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-22T19:40:58Z","title_canon_sha256":"0f4a6cbaf5143cf52399ddd3087ee0b497ef4aad5db262b7c4472b5fc5ec4835"},"schema_version":"1.0","source":{"id":"1605.06839","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.06839","created_at":"2026-05-18T00:22:29Z"},{"alias_kind":"arxiv_version","alias_value":"1605.06839v3","created_at":"2026-05-18T00:22:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.06839","created_at":"2026-05-18T00:22:29Z"},{"alias_kind":"pith_short_12","alias_value":"T6DPRYEPFYWC","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"T6DPRYEPFYWCCTYT","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"T6DPRYEP","created_at":"2026-05-18T12:30:44Z"}],"graph_snapshots":[{"event_id":"sha256:72005f3fd645a6803f9df9015ade610d0e10813257dfa34e8d907b15f0eb38cf","target":"graph","created_at":"2026-05-18T00:22:29Z","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"},"paper":{"abstract_excerpt":"We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group $\\mathbb H^n$. Our results include a natural sub-Riemannian version of the celebrated curvature-dimension condition of Lott-Villani and Sturm and also a geodesic version of the Borell-Brascamp-Lieb inequality akin to the one obtained by Cordero-Erausquin, McCann and Schmuckenschl\\\"ager. The latter statement implies sub-Riemannian versions of the geodesic Pr\\'ekopa-Leindler and Brunn-Minkowski inequalities. The proofs are based on optimal mass transportation and Riemannian approximation of $\\mathbb H^n$ de","authors_text":"Alexandru Krist\\'aly, Kinga Sipos, Zolt\\'an M. Balogh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-22T19:40:58Z","title":"Geometric inequalities on Heisenberg groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06839","kind":"arxiv","version":3},"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:a25fed612f5f7e72f4d89f5225c12340b09d4b55dc85f777b56f26b0eda5a9ca","target":"record","created_at":"2026-05-18T00:22:29Z","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":"b11d372fe836f4628a87f928907b8922ad90290cdede7b50099b85d6e71b21b3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-05-22T19:40:58Z","title_canon_sha256":"0f4a6cbaf5143cf52399ddd3087ee0b497ef4aad5db262b7c4472b5fc5ec4835"},"schema_version":"1.0","source":{"id":"1605.06839","kind":"arxiv","version":3}},"canonical_sha256":"9f86f8e08f2e2c214f135c5c2349bc03720d70a6d720642c90921dd34a684b39","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9f86f8e08f2e2c214f135c5c2349bc03720d70a6d720642c90921dd34a684b39","first_computed_at":"2026-05-18T00:22:29.873298Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:29.873298Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"P5hFjodjoswB7StzXZnkS0hNwKVtaIuNi9bjK/zOSq5UKB1IRN9BrWehGCzR7bDKPX+D0aEsENX/Zy3m6XFkDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:29.873907Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.06839","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a25fed612f5f7e72f4d89f5225c12340b09d4b55dc85f777b56f26b0eda5a9ca","sha256:72005f3fd645a6803f9df9015ade610d0e10813257dfa34e8d907b15f0eb38cf"],"state_sha256":"d2b4a82a7770b08b7099e943645729edeea87ddc1e5333a618bfb3635b42b8e4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hf5K+MTqZe26JbBKEb24AHpAagGzxShUWA7YRiSHpuijhb5uMTYbMLrGdwBSwqiE0/cQnSu5h4XUWNmjut/jBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T13:51:23.340904Z","bundle_sha256":"e2bdd2f059827da5614cf8f955e75565d6bc280965c9b8a7039de7b3556a65ba"}}