{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:NIXMNL7HK76ADYM4LJLLO3C2SA","short_pith_number":"pith:NIXMNL7H","canonical_record":{"source":{"id":"1504.03007","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-04-12T19:11:52Z","cross_cats_sorted":[],"title_canon_sha256":"a44f1ee7bb51602499f5142bbf7f721bfe9535ec11678e194e21e4563e9f3f5c","abstract_canon_sha256":"b3601c83b4e9276388fbb746eae2fe36c875c3796e704456c810a3267261947a"},"schema_version":"1.0"},"canonical_sha256":"6a2ec6afe757fc01e19c5a56b76c5a903213314b2c6fcd358dbbe046a4406442","source":{"kind":"arxiv","id":"1504.03007","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.03007","created_at":"2026-05-18T02:18:03Z"},{"alias_kind":"arxiv_version","alias_value":"1504.03007v2","created_at":"2026-05-18T02:18:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.03007","created_at":"2026-05-18T02:18:03Z"},{"alias_kind":"pith_short_12","alias_value":"NIXMNL7HK76A","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"NIXMNL7HK76ADYM4","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"NIXMNL7H","created_at":"2026-05-18T12:29:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:NIXMNL7HK76ADYM4LJLLO3C2SA","target":"record","payload":{"canonical_record":{"source":{"id":"1504.03007","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-04-12T19:11:52Z","cross_cats_sorted":[],"title_canon_sha256":"a44f1ee7bb51602499f5142bbf7f721bfe9535ec11678e194e21e4563e9f3f5c","abstract_canon_sha256":"b3601c83b4e9276388fbb746eae2fe36c875c3796e704456c810a3267261947a"},"schema_version":"1.0"},"canonical_sha256":"6a2ec6afe757fc01e19c5a56b76c5a903213314b2c6fcd358dbbe046a4406442","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:03.312022Z","signature_b64":"yiNp3W+WWu10O0YCcOMCEMDKCZYgAdnFoYPj6qCUcn3Ue0nDcEQx31ikp1g469dwDgcBFgsGy7OpXrN5AlqmBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a2ec6afe757fc01e19c5a56b76c5a903213314b2c6fcd358dbbe046a4406442","last_reissued_at":"2026-05-18T02:18:03.311187Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:03.311187Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.03007","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":"2026-05-18T02:18:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Jqmg2jOwHQB9ZAuZoc8CwzUrksrzUXtESrWmpI1Jkoe8lvpBfUR/tzHI7Flh0+xCkCjsGT7votKz+X5NKO+qBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T21:28:48.182753Z"},"content_sha256":"91b08080379a00af24b6db9df54932fe85d1417cbab77bedfce9198c96f14904","schema_version":"1.0","event_id":"sha256:91b08080379a00af24b6db9df54932fe85d1417cbab77bedfce9198c96f14904"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:NIXMNL7HK76ADYM4LJLLO3C2SA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Witten Rigidity Theorem for Odd Dimensional Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Fei Han, Jianqing Yu","submitted_at":"2015-04-12T19:11:52Z","abstract_excerpt":"We establish several Witten type rigidity and vanishing theorems for twisted Toeplitz operators on odd dimensional manifolds. We obtain our results by combining the modular method, modular transgression and some careful analysis of odd Chern classes for cocycles in odd $K$-theory. Moreover we discover that in odd dimensions, the fundamental group of manifolds plays an important role in the rigidity."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03007","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":"2026-05-18T02:18:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5cogpARLz+OAq513U7ABplBBp5phRKkMvaGQso4B8pfV03tpsmucOd9rAXyjLokKFUOeRPEBGhhw1Fu5bVi1CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T21:28:48.183401Z"},"content_sha256":"ddb403ad4f5a3e0cf634ab9a8007baee73d0f32696c218789e531c56e64b4eb9","schema_version":"1.0","event_id":"sha256:ddb403ad4f5a3e0cf634ab9a8007baee73d0f32696c218789e531c56e64b4eb9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NIXMNL7HK76ADYM4LJLLO3C2SA/bundle.json","state_url":"https://pith.science/pith/NIXMNL7HK76ADYM4LJLLO3C2SA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NIXMNL7HK76ADYM4LJLLO3C2SA/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":"2026-06-05T21:28:48Z","links":{"resolver":"https://pith.science/pith/NIXMNL7HK76ADYM4LJLLO3C2SA","bundle":"https://pith.science/pith/NIXMNL7HK76ADYM4LJLLO3C2SA/bundle.json","state":"https://pith.science/pith/NIXMNL7HK76ADYM4LJLLO3C2SA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NIXMNL7HK76ADYM4LJLLO3C2SA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:NIXMNL7HK76ADYM4LJLLO3C2SA","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":"b3601c83b4e9276388fbb746eae2fe36c875c3796e704456c810a3267261947a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-04-12T19:11:52Z","title_canon_sha256":"a44f1ee7bb51602499f5142bbf7f721bfe9535ec11678e194e21e4563e9f3f5c"},"schema_version":"1.0","source":{"id":"1504.03007","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.03007","created_at":"2026-05-18T02:18:03Z"},{"alias_kind":"arxiv_version","alias_value":"1504.03007v2","created_at":"2026-05-18T02:18:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.03007","created_at":"2026-05-18T02:18:03Z"},{"alias_kind":"pith_short_12","alias_value":"NIXMNL7HK76A","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"NIXMNL7HK76ADYM4","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"NIXMNL7H","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:ddb403ad4f5a3e0cf634ab9a8007baee73d0f32696c218789e531c56e64b4eb9","target":"graph","created_at":"2026-05-18T02:18:03Z","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 several Witten type rigidity and vanishing theorems for twisted Toeplitz operators on odd dimensional manifolds. We obtain our results by combining the modular method, modular transgression and some careful analysis of odd Chern classes for cocycles in odd $K$-theory. Moreover we discover that in odd dimensions, the fundamental group of manifolds plays an important role in the rigidity.","authors_text":"Fei Han, Jianqing Yu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-04-12T19:11:52Z","title":"On the Witten Rigidity Theorem for Odd Dimensional Manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03007","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":"sha256:91b08080379a00af24b6db9df54932fe85d1417cbab77bedfce9198c96f14904","target":"record","created_at":"2026-05-18T02:18:03Z","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":"b3601c83b4e9276388fbb746eae2fe36c875c3796e704456c810a3267261947a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-04-12T19:11:52Z","title_canon_sha256":"a44f1ee7bb51602499f5142bbf7f721bfe9535ec11678e194e21e4563e9f3f5c"},"schema_version":"1.0","source":{"id":"1504.03007","kind":"arxiv","version":2}},"canonical_sha256":"6a2ec6afe757fc01e19c5a56b76c5a903213314b2c6fcd358dbbe046a4406442","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6a2ec6afe757fc01e19c5a56b76c5a903213314b2c6fcd358dbbe046a4406442","first_computed_at":"2026-05-18T02:18:03.311187Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:18:03.311187Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yiNp3W+WWu10O0YCcOMCEMDKCZYgAdnFoYPj6qCUcn3Ue0nDcEQx31ikp1g469dwDgcBFgsGy7OpXrN5AlqmBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:18:03.312022Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.03007","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:91b08080379a00af24b6db9df54932fe85d1417cbab77bedfce9198c96f14904","sha256:ddb403ad4f5a3e0cf634ab9a8007baee73d0f32696c218789e531c56e64b4eb9"],"state_sha256":"9b954f0a94ca8ab0b3a599aae21516f4680fb2c834baf9befb3b7f669c50ca79"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vXHffZ98a908B3Nrl19i1f+kBUdgN149Il0LhoueYiRe931P3D2jF9AMwcaB4wJHlFxSXTJKn+HCr4ZZyokWDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T21:28:48.188725Z","bundle_sha256":"ec1847908c109a141f67833ffe660a258a3846e524c94bd1d938f2a5d4b871bb"}}