{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:ROU44SKW55IS3C4DWVU726NDHX","short_pith_number":"pith:ROU44SKW","canonical_record":{"source":{"id":"1408.4470","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-19T20:19:14Z","cross_cats_sorted":[],"title_canon_sha256":"7d1faa3b903a96ca5b5985fe5e4ebe7f749a06435aa92d3cdb26e2c142ac93fb","abstract_canon_sha256":"634ed08bcd071f244bbc4ea37adf1344f862b9db544a0be5615119416f00f6c8"},"schema_version":"1.0"},"canonical_sha256":"8ba9ce4956ef512d8b83b569fd79a33df0d1bb4d82a74abd49813b5cf2f16f5a","source":{"kind":"arxiv","id":"1408.4470","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.4470","created_at":"2026-05-18T02:44:55Z"},{"alias_kind":"arxiv_version","alias_value":"1408.4470v1","created_at":"2026-05-18T02:44:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.4470","created_at":"2026-05-18T02:44:55Z"},{"alias_kind":"pith_short_12","alias_value":"ROU44SKW55IS","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"ROU44SKW55IS3C4D","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"ROU44SKW","created_at":"2026-05-18T12:28:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:ROU44SKW55IS3C4DWVU726NDHX","target":"record","payload":{"canonical_record":{"source":{"id":"1408.4470","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-19T20:19:14Z","cross_cats_sorted":[],"title_canon_sha256":"7d1faa3b903a96ca5b5985fe5e4ebe7f749a06435aa92d3cdb26e2c142ac93fb","abstract_canon_sha256":"634ed08bcd071f244bbc4ea37adf1344f862b9db544a0be5615119416f00f6c8"},"schema_version":"1.0"},"canonical_sha256":"8ba9ce4956ef512d8b83b569fd79a33df0d1bb4d82a74abd49813b5cf2f16f5a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:55.182930Z","signature_b64":"sVIbSRpui9rdW/hvCepf0RqolF8S9VBnYixTzie0RwTvrddUY89m3y8MzU+USrmyeBlJ9HdNAPHb3O8DUxdNCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ba9ce4956ef512d8b83b569fd79a33df0d1bb4d82a74abd49813b5cf2f16f5a","last_reissued_at":"2026-05-18T02:44:55.182515Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:55.182515Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1408.4470","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":"2026-05-18T02:44:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9CDBJGG1CY6vw+EOkR+fOINH6Kjh2AliAtt19qnwpehNCvAQhzz4SDPZjPELZMFpzHVoV7tur6a1JxoEN4OLCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T11:53:27.463853Z"},"content_sha256":"029399597f8d1f7ae233fe671e8ae55856f30e85ce97503a343d85056db3daaa","schema_version":"1.0","event_id":"sha256:029399597f8d1f7ae233fe671e8ae55856f30e85ce97503a343d85056db3daaa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:ROU44SKW55IS3C4DWVU726NDHX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"St\\'ephane Sabourau","submitted_at":"2014-08-19T20:19:14Z","abstract_excerpt":"We establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function whose level set volume is bounded in terms of the volume of the manifold. As a consequence of this sweep-out estimate, there exists an embedded, closed (possibly singular) minimal hypersurface whose volume is bounded in terms of the volume of the manifold."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.4470","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":""},"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":"2026-05-18T02:44:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A0r6ruEIGX27VCahiKka0td5De3JfVEp4nIQq+lawvrF1rlHT2CmA9Dt+VZh+fljaeiye+Zw35GTTOVLDc5eBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T11:53:27.464205Z"},"content_sha256":"95262ba4f2c6be251b53954e9d67cf1fafee189b265f0319eda21f586e39153e","schema_version":"1.0","event_id":"sha256:95262ba4f2c6be251b53954e9d67cf1fafee189b265f0319eda21f586e39153e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ROU44SKW55IS3C4DWVU726NDHX/bundle.json","state_url":"https://pith.science/pith/ROU44SKW55IS3C4DWVU726NDHX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ROU44SKW55IS3C4DWVU726NDHX/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":"2026-07-01T11:53:27Z","links":{"resolver":"https://pith.science/pith/ROU44SKW55IS3C4DWVU726NDHX","bundle":"https://pith.science/pith/ROU44SKW55IS3C4DWVU726NDHX/bundle.json","state":"https://pith.science/pith/ROU44SKW55IS3C4DWVU726NDHX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ROU44SKW55IS3C4DWVU726NDHX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:ROU44SKW55IS3C4DWVU726NDHX","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":"634ed08bcd071f244bbc4ea37adf1344f862b9db544a0be5615119416f00f6c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-19T20:19:14Z","title_canon_sha256":"7d1faa3b903a96ca5b5985fe5e4ebe7f749a06435aa92d3cdb26e2c142ac93fb"},"schema_version":"1.0","source":{"id":"1408.4470","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.4470","created_at":"2026-05-18T02:44:55Z"},{"alias_kind":"arxiv_version","alias_value":"1408.4470v1","created_at":"2026-05-18T02:44:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.4470","created_at":"2026-05-18T02:44:55Z"},{"alias_kind":"pith_short_12","alias_value":"ROU44SKW55IS","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"ROU44SKW55IS3C4D","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"ROU44SKW","created_at":"2026-05-18T12:28:46Z"}],"graph_snapshots":[{"event_id":"sha256:95262ba4f2c6be251b53954e9d67cf1fafee189b265f0319eda21f586e39153e","target":"graph","created_at":"2026-05-18T02:44:55Z","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 a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function whose level set volume is bounded in terms of the volume of the manifold. As a consequence of this sweep-out estimate, there exists an embedded, closed (possibly singular) minimal hypersurface whose volume is bounded in terms of the volume of the manifold.","authors_text":"St\\'ephane Sabourau","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-19T20:19:14Z","title":"Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.4470","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":"sha256:029399597f8d1f7ae233fe671e8ae55856f30e85ce97503a343d85056db3daaa","target":"record","created_at":"2026-05-18T02:44:55Z","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":"634ed08bcd071f244bbc4ea37adf1344f862b9db544a0be5615119416f00f6c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-19T20:19:14Z","title_canon_sha256":"7d1faa3b903a96ca5b5985fe5e4ebe7f749a06435aa92d3cdb26e2c142ac93fb"},"schema_version":"1.0","source":{"id":"1408.4470","kind":"arxiv","version":1}},"canonical_sha256":"8ba9ce4956ef512d8b83b569fd79a33df0d1bb4d82a74abd49813b5cf2f16f5a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8ba9ce4956ef512d8b83b569fd79a33df0d1bb4d82a74abd49813b5cf2f16f5a","first_computed_at":"2026-05-18T02:44:55.182515Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:55.182515Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sVIbSRpui9rdW/hvCepf0RqolF8S9VBnYixTzie0RwTvrddUY89m3y8MzU+USrmyeBlJ9HdNAPHb3O8DUxdNCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:55.182930Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.4470","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:029399597f8d1f7ae233fe671e8ae55856f30e85ce97503a343d85056db3daaa","sha256:95262ba4f2c6be251b53954e9d67cf1fafee189b265f0319eda21f586e39153e"],"state_sha256":"38a6c219da5b374766b7d74946ed8730f574ea9e2658fe6113835f812f3fc138"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U1Olkr6S5uNe9Yfxje40s0Afjvr+6yuUtJw/VmU9uzBBNhux7HxP/su3oDQpAKbEFxg/MfFrPLX7lMJhORadAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T11:53:27.466261Z","bundle_sha256":"2dd9c087012f77aeb8872387e3ed39676a033acd9b91eb7b3251a86a2b0dec89"}}