{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:XEXOMEUBWDRCOW4WYZ2HMPVUVW","short_pith_number":"pith:XEXOMEUB","canonical_record":{"source":{"id":"1205.0743","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-05-03T15:47:15Z","cross_cats_sorted":["math-ph","math.KT","math.MP","math.OA"],"title_canon_sha256":"e5bc2fe967192eb1bbd41c1cc59b14f264e1544631b7b8c72ec21bfbb191154b","abstract_canon_sha256":"b374b1b0611d763135737c93a3b4ba2efa5634a88c1ab1f1631c21ef93bed091"},"schema_version":"1.0"},"canonical_sha256":"b92ee61281b0e2275b96c674763eb4ad872268ed39c05bf6625fa777332d517b","source":{"kind":"arxiv","id":"1205.0743","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.0743","created_at":"2026-05-18T00:14:29Z"},{"alias_kind":"arxiv_version","alias_value":"1205.0743v1","created_at":"2026-05-18T00:14:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.0743","created_at":"2026-05-18T00:14:29Z"},{"alias_kind":"pith_short_12","alias_value":"XEXOMEUBWDRC","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XEXOMEUBWDRCOW4W","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XEXOMEUB","created_at":"2026-05-18T12:27:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:XEXOMEUBWDRCOW4WYZ2HMPVUVW","target":"record","payload":{"canonical_record":{"source":{"id":"1205.0743","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-05-03T15:47:15Z","cross_cats_sorted":["math-ph","math.KT","math.MP","math.OA"],"title_canon_sha256":"e5bc2fe967192eb1bbd41c1cc59b14f264e1544631b7b8c72ec21bfbb191154b","abstract_canon_sha256":"b374b1b0611d763135737c93a3b4ba2efa5634a88c1ab1f1631c21ef93bed091"},"schema_version":"1.0"},"canonical_sha256":"b92ee61281b0e2275b96c674763eb4ad872268ed39c05bf6625fa777332d517b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:29.473125Z","signature_b64":"RlF8IAyMIol83Bc0jw9JGtPzOo3YIhEtDjb36xym6sZa62qnsHzLw2t3hOWCqZsYq6QmEftB2Bl+Z+XzcHkOCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b92ee61281b0e2275b96c674763eb4ad872268ed39c05bf6625fa777332d517b","last_reissued_at":"2026-05-18T00:14:29.472406Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:29.472406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1205.0743","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":"2026-05-18T00:14:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5Ewi4TZbCwxMb0a8sO8USIYpvqpoDona+5hzGM/PkQ5NwrVAsjvxBggSPHEGP+YKYC4XLYk838yDl2VlyxMgBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-19T23:59:51.059642Z"},"content_sha256":"255869ddb5d68afb2dff67115ee89d2807cdb45ff2bb411672bfb7f3089a0aee","schema_version":"1.0","event_id":"sha256:255869ddb5d68afb2dff67115ee89d2807cdb45ff2bb411672bfb7f3089a0aee"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:XEXOMEUBWDRCOW4WYZ2HMPVUVW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"K-theory of noncommutative Bieberbach manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.KT","math.MP","math.OA"],"primary_cat":"math.QA","authors_text":"Andrzej Sitarz, Piotr Olczykowski","submitted_at":"2012-05-03T15:47:15Z","abstract_excerpt":"We compute $K-theory of noncommutative Bieberbach manifolds, which quotients of a three-dimensional noncommutative torus by a free action of a cyclic group Z_N, N=2,3,4,6."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.0743","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":"2026-05-18T00:14:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G6Px19oNEBjoeAfo7XtNMZhiUAw2f0TbhLBpP9lV0cr49bXRBQuRSvc80XwBjz7vmAMB/hJBj1pmypjbB/+9Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-19T23:59:51.059987Z"},"content_sha256":"a4f86c5cb99de920bbac6d560978a92b9f3c863fae589d9a3fc56cecfb471873","schema_version":"1.0","event_id":"sha256:a4f86c5cb99de920bbac6d560978a92b9f3c863fae589d9a3fc56cecfb471873"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XEXOMEUBWDRCOW4WYZ2HMPVUVW/bundle.json","state_url":"https://pith.science/pith/XEXOMEUBWDRCOW4WYZ2HMPVUVW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XEXOMEUBWDRCOW4WYZ2HMPVUVW/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":"2026-05-19T23:59:51Z","links":{"resolver":"https://pith.science/pith/XEXOMEUBWDRCOW4WYZ2HMPVUVW","bundle":"https://pith.science/pith/XEXOMEUBWDRCOW4WYZ2HMPVUVW/bundle.json","state":"https://pith.science/pith/XEXOMEUBWDRCOW4WYZ2HMPVUVW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XEXOMEUBWDRCOW4WYZ2HMPVUVW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:XEXOMEUBWDRCOW4WYZ2HMPVUVW","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":"b374b1b0611d763135737c93a3b4ba2efa5634a88c1ab1f1631c21ef93bed091","cross_cats_sorted":["math-ph","math.KT","math.MP","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-05-03T15:47:15Z","title_canon_sha256":"e5bc2fe967192eb1bbd41c1cc59b14f264e1544631b7b8c72ec21bfbb191154b"},"schema_version":"1.0","source":{"id":"1205.0743","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.0743","created_at":"2026-05-18T00:14:29Z"},{"alias_kind":"arxiv_version","alias_value":"1205.0743v1","created_at":"2026-05-18T00:14:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.0743","created_at":"2026-05-18T00:14:29Z"},{"alias_kind":"pith_short_12","alias_value":"XEXOMEUBWDRC","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XEXOMEUBWDRCOW4W","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XEXOMEUB","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:a4f86c5cb99de920bbac6d560978a92b9f3c863fae589d9a3fc56cecfb471873","target":"graph","created_at":"2026-05-18T00:14:29Z","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 compute $K-theory of noncommutative Bieberbach manifolds, which quotients of a three-dimensional noncommutative torus by a free action of a cyclic group Z_N, N=2,3,4,6.","authors_text":"Andrzej Sitarz, Piotr Olczykowski","cross_cats":["math-ph","math.KT","math.MP","math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-05-03T15:47:15Z","title":"K-theory of noncommutative Bieberbach manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.0743","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":"sha256:255869ddb5d68afb2dff67115ee89d2807cdb45ff2bb411672bfb7f3089a0aee","target":"record","created_at":"2026-05-18T00:14:29Z","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":"b374b1b0611d763135737c93a3b4ba2efa5634a88c1ab1f1631c21ef93bed091","cross_cats_sorted":["math-ph","math.KT","math.MP","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-05-03T15:47:15Z","title_canon_sha256":"e5bc2fe967192eb1bbd41c1cc59b14f264e1544631b7b8c72ec21bfbb191154b"},"schema_version":"1.0","source":{"id":"1205.0743","kind":"arxiv","version":1}},"canonical_sha256":"b92ee61281b0e2275b96c674763eb4ad872268ed39c05bf6625fa777332d517b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b92ee61281b0e2275b96c674763eb4ad872268ed39c05bf6625fa777332d517b","first_computed_at":"2026-05-18T00:14:29.472406Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:29.472406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RlF8IAyMIol83Bc0jw9JGtPzOo3YIhEtDjb36xym6sZa62qnsHzLw2t3hOWCqZsYq6QmEftB2Bl+Z+XzcHkOCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:29.473125Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.0743","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:255869ddb5d68afb2dff67115ee89d2807cdb45ff2bb411672bfb7f3089a0aee","sha256:a4f86c5cb99de920bbac6d560978a92b9f3c863fae589d9a3fc56cecfb471873"],"state_sha256":"770e2e49e3404cc59e6622496a5c93e147408502c39c034a9061aefc9f3f5946"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tvTiFAZg3KUiOax/qTxplBSKv6/GuDFWJMrBUvFuxWpt/hzonxt9llxe/z6oo9ZTt8+ghARuOnZPUdBTgkS2BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-19T23:59:51.062330Z","bundle_sha256":"5b36f6215be06e9ace20ead72056871d6d44e76e058872e290dc326560ba33ba"}}