{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:DH5I47KF4RSXXIDVQRUUFFG5FB","short_pith_number":"pith:DH5I47KF","canonical_record":{"source":{"id":"1309.6685","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-09-25T22:15:19Z","cross_cats_sorted":[],"title_canon_sha256":"6208ac0c283c0b88ce9c96c18547009b936cd28d3003597de34f9bca533df8f2","abstract_canon_sha256":"c62b4e9fc9327db6438806ca7749d1331e3ef1a2f278f98876da92fecb5ec5b1"},"schema_version":"1.0"},"canonical_sha256":"19fa8e7d45e4657ba07584694294dd28681c57225ad9c48c2c468c6dc1e5e180","source":{"kind":"arxiv","id":"1309.6685","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.6685","created_at":"2026-05-18T03:07:56Z"},{"alias_kind":"arxiv_version","alias_value":"1309.6685v2","created_at":"2026-05-18T03:07:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.6685","created_at":"2026-05-18T03:07:56Z"},{"alias_kind":"pith_short_12","alias_value":"DH5I47KF4RSX","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"DH5I47KF4RSXXIDV","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"DH5I47KF","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:DH5I47KF4RSXXIDVQRUUFFG5FB","target":"record","payload":{"canonical_record":{"source":{"id":"1309.6685","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-09-25T22:15:19Z","cross_cats_sorted":[],"title_canon_sha256":"6208ac0c283c0b88ce9c96c18547009b936cd28d3003597de34f9bca533df8f2","abstract_canon_sha256":"c62b4e9fc9327db6438806ca7749d1331e3ef1a2f278f98876da92fecb5ec5b1"},"schema_version":"1.0"},"canonical_sha256":"19fa8e7d45e4657ba07584694294dd28681c57225ad9c48c2c468c6dc1e5e180","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:56.371946Z","signature_b64":"6KMnI5/xvysnhwsheomeDT79Jf6BiL+8j4oPFvAikRJLyp1cnMHbSC5JyARgLGzG6Or3Y8FkRzy253gA6hJkDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"19fa8e7d45e4657ba07584694294dd28681c57225ad9c48c2c468c6dc1e5e180","last_reissued_at":"2026-05-18T03:07:56.371108Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:56.371108Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.6685","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":"2026-05-18T03:07:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QbR2Sx/bg+YfnnHOLbrqpEaX2JnrfHwpCq5LAaXeigMLLySsLr4eUjCHdtlrrodz9aUbllPR9mBBXC6TTQlECw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T20:48:05.374417Z"},"content_sha256":"4f9e4000bc237dccf5b46a56433cc6c0394f31d2cabb468eabbc6f9ed27524a9","schema_version":"1.0","event_id":"sha256:4f9e4000bc237dccf5b46a56433cc6c0394f31d2cabb468eabbc6f9ed27524a9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:DH5I47KF4RSXXIDVQRUUFFG5FB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fundamental Groups of Spaces with Bakry-Emery Ricci Tensor Bounded Below","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Maree Jaramillo","submitted_at":"2013-09-25T22:15:19Z","abstract_excerpt":"We first extend Cheeger-Colding Almost Splitting Theorem to smooth metric measure spaces. Arguments utilizing this extension of the Almost Splitting Theorem show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abelian. Second, if the smooth metric measure space has Bakry-Emery Ricci curvature bounded from below then the number of generators of the fundamental group is uniformly bounded. These results are extensions of theorems which hold for Riemannian manifolds with Ricci curvature bound"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6685","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":""},"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":"2026-05-18T03:07:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HCBYq2siu8A3mOplroP4MQo/S7UF7iHES1ypNL+FfInqF6doLXJLxFHrBO9aeqY1tU72H4OwGf7+6yestfRZDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T20:48:05.374764Z"},"content_sha256":"8d6a2b08fa9bc1b232c200a4c613e73d3fc13b9b143baf37a28ce7ee7cb0c308","schema_version":"1.0","event_id":"sha256:8d6a2b08fa9bc1b232c200a4c613e73d3fc13b9b143baf37a28ce7ee7cb0c308"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DH5I47KF4RSXXIDVQRUUFFG5FB/bundle.json","state_url":"https://pith.science/pith/DH5I47KF4RSXXIDVQRUUFFG5FB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DH5I47KF4RSXXIDVQRUUFFG5FB/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":"2026-06-24T20:48:05Z","links":{"resolver":"https://pith.science/pith/DH5I47KF4RSXXIDVQRUUFFG5FB","bundle":"https://pith.science/pith/DH5I47KF4RSXXIDVQRUUFFG5FB/bundle.json","state":"https://pith.science/pith/DH5I47KF4RSXXIDVQRUUFFG5FB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DH5I47KF4RSXXIDVQRUUFFG5FB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:DH5I47KF4RSXXIDVQRUUFFG5FB","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":"c62b4e9fc9327db6438806ca7749d1331e3ef1a2f278f98876da92fecb5ec5b1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-09-25T22:15:19Z","title_canon_sha256":"6208ac0c283c0b88ce9c96c18547009b936cd28d3003597de34f9bca533df8f2"},"schema_version":"1.0","source":{"id":"1309.6685","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.6685","created_at":"2026-05-18T03:07:56Z"},{"alias_kind":"arxiv_version","alias_value":"1309.6685v2","created_at":"2026-05-18T03:07:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.6685","created_at":"2026-05-18T03:07:56Z"},{"alias_kind":"pith_short_12","alias_value":"DH5I47KF4RSX","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"DH5I47KF4RSXXIDV","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"DH5I47KF","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:8d6a2b08fa9bc1b232c200a4c613e73d3fc13b9b143baf37a28ce7ee7cb0c308","target":"graph","created_at":"2026-05-18T03:07:56Z","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 first extend Cheeger-Colding Almost Splitting Theorem to smooth metric measure spaces. Arguments utilizing this extension of the Almost Splitting Theorem show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abelian. Second, if the smooth metric measure space has Bakry-Emery Ricci curvature bounded from below then the number of generators of the fundamental group is uniformly bounded. These results are extensions of theorems which hold for Riemannian manifolds with Ricci curvature bound","authors_text":"Maree Jaramillo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-09-25T22:15:19Z","title":"Fundamental Groups of Spaces with Bakry-Emery Ricci Tensor Bounded Below"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6685","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":"sha256:4f9e4000bc237dccf5b46a56433cc6c0394f31d2cabb468eabbc6f9ed27524a9","target":"record","created_at":"2026-05-18T03:07:56Z","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":"c62b4e9fc9327db6438806ca7749d1331e3ef1a2f278f98876da92fecb5ec5b1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-09-25T22:15:19Z","title_canon_sha256":"6208ac0c283c0b88ce9c96c18547009b936cd28d3003597de34f9bca533df8f2"},"schema_version":"1.0","source":{"id":"1309.6685","kind":"arxiv","version":2}},"canonical_sha256":"19fa8e7d45e4657ba07584694294dd28681c57225ad9c48c2c468c6dc1e5e180","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"19fa8e7d45e4657ba07584694294dd28681c57225ad9c48c2c468c6dc1e5e180","first_computed_at":"2026-05-18T03:07:56.371108Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:56.371108Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6KMnI5/xvysnhwsheomeDT79Jf6BiL+8j4oPFvAikRJLyp1cnMHbSC5JyARgLGzG6Or3Y8FkRzy253gA6hJkDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:56.371946Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.6685","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4f9e4000bc237dccf5b46a56433cc6c0394f31d2cabb468eabbc6f9ed27524a9","sha256:8d6a2b08fa9bc1b232c200a4c613e73d3fc13b9b143baf37a28ce7ee7cb0c308"],"state_sha256":"fae33f3c9a9b7d3f96d3e3c9b5fef683eadd8202d3bd18fc528cd73ddd5ea115"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lqiHwVjEatIE/MrmYJMpWjglEO3uIvjIfbucsbzyNIWzjT+8fS9vXqwi9S43KB1efK7Vdcmi6aZ+c/wsL3hpDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T20:48:05.376675Z","bundle_sha256":"e512eb097841e6ac802fe417a6d180d9bcab23d46829b1bd09cc5364266caf7b"}}