{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:GCAEZA67TG6VT2MLXITB7QH54H","short_pith_number":"pith:GCAEZA67","canonical_record":{"source":{"id":"1411.2165","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-08T21:44:42Z","cross_cats_sorted":[],"title_canon_sha256":"6ed755f77d08f3373ff2092a105d0863dae0739a32baf0ea0bca8b365bb7a157","abstract_canon_sha256":"e3085f9e90b3ed89a0c469846328afee6046acf63a4be415669c630dcd215e17"},"schema_version":"1.0"},"canonical_sha256":"30804c83df99bd59e98bba261fc0fde1f0dc238a08b6349adaad41f7c04aed12","source":{"kind":"arxiv","id":"1411.2165","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.2165","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"arxiv_version","alias_value":"1411.2165v1","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.2165","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"pith_short_12","alias_value":"GCAEZA67TG6V","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"GCAEZA67TG6VT2ML","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"GCAEZA67","created_at":"2026-05-18T12:28:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:GCAEZA67TG6VT2MLXITB7QH54H","target":"record","payload":{"canonical_record":{"source":{"id":"1411.2165","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-08T21:44:42Z","cross_cats_sorted":[],"title_canon_sha256":"6ed755f77d08f3373ff2092a105d0863dae0739a32baf0ea0bca8b365bb7a157","abstract_canon_sha256":"e3085f9e90b3ed89a0c469846328afee6046acf63a4be415669c630dcd215e17"},"schema_version":"1.0"},"canonical_sha256":"30804c83df99bd59e98bba261fc0fde1f0dc238a08b6349adaad41f7c04aed12","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:08.313899Z","signature_b64":"TQAIkhMp982xbPxxwiiMkaeHghNQOGHCWxWZSxOC05KFvWz5zVtHdy7KMg4eu+zW4THK5+MAjZypQpzsctQSCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"30804c83df99bd59e98bba261fc0fde1f0dc238a08b6349adaad41f7c04aed12","last_reissued_at":"2026-05-18T02:38:08.313491Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:08.313491Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1411.2165","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":"2026-05-18T02:38:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3kfxSMs3BL7PK92OV7kaCkWmMHphNM6cf8QhKuYjhDoIykPjr0j6+Kjilomv4vguy/YpyCOPASecIDzBvcPKDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T06:19:06.717758Z"},"content_sha256":"06749d8942a3a669000572d70708b9cd97fc61def43a0246467251c65796315c","schema_version":"1.0","event_id":"sha256:06749d8942a3a669000572d70708b9cd97fc61def43a0246467251c65796315c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:GCAEZA67TG6VT2MLXITB7QH54H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Let \\Delta be a Cohen-Macaulay complex","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anders Bj\\\"orner","submitted_at":"2014-11-08T21:44:42Z","abstract_excerpt":"The concept of Cohen-Macaulay complexes emerged in the mid-1970s and swiftly became the focal point of an attractive and richly connected new area of mathematics, at the crossroads of combinatoics, commutative algebra and topology. As the main architect of these developments, Richard Stanley has made fundamental contributions over many years.\n  This paper contains some brief mathematical discussions related to the Cohen-Macaulay property, and some personal memories. The characterization of Gorenstein* and homotopy Gorenstein* complexes and the relevance in that connection of the Poincar\\'e con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.2165","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":"2026-05-18T02:38:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/UepoY/JmQMVBGzH2rLr1ZGclQzisT9Bk1MaEnB/FEB+2qujeiF2Ch55yI/+0zGIBSui64xXZ4xr/CuYLZ7dBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T06:19:06.718100Z"},"content_sha256":"5e1c0ba1516cbd6519ca0b4f4ef8201e1182fb86a981992622b1736c0bdb4c26","schema_version":"1.0","event_id":"sha256:5e1c0ba1516cbd6519ca0b4f4ef8201e1182fb86a981992622b1736c0bdb4c26"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GCAEZA67TG6VT2MLXITB7QH54H/bundle.json","state_url":"https://pith.science/pith/GCAEZA67TG6VT2MLXITB7QH54H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GCAEZA67TG6VT2MLXITB7QH54H/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":"2026-06-25T06:19:06Z","links":{"resolver":"https://pith.science/pith/GCAEZA67TG6VT2MLXITB7QH54H","bundle":"https://pith.science/pith/GCAEZA67TG6VT2MLXITB7QH54H/bundle.json","state":"https://pith.science/pith/GCAEZA67TG6VT2MLXITB7QH54H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GCAEZA67TG6VT2MLXITB7QH54H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:GCAEZA67TG6VT2MLXITB7QH54H","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":"e3085f9e90b3ed89a0c469846328afee6046acf63a4be415669c630dcd215e17","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-08T21:44:42Z","title_canon_sha256":"6ed755f77d08f3373ff2092a105d0863dae0739a32baf0ea0bca8b365bb7a157"},"schema_version":"1.0","source":{"id":"1411.2165","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.2165","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"arxiv_version","alias_value":"1411.2165v1","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.2165","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"pith_short_12","alias_value":"GCAEZA67TG6V","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"GCAEZA67TG6VT2ML","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"GCAEZA67","created_at":"2026-05-18T12:28:30Z"}],"graph_snapshots":[{"event_id":"sha256:5e1c0ba1516cbd6519ca0b4f4ef8201e1182fb86a981992622b1736c0bdb4c26","target":"graph","created_at":"2026-05-18T02:38:08Z","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":"The concept of Cohen-Macaulay complexes emerged in the mid-1970s and swiftly became the focal point of an attractive and richly connected new area of mathematics, at the crossroads of combinatoics, commutative algebra and topology. As the main architect of these developments, Richard Stanley has made fundamental contributions over many years.\n  This paper contains some brief mathematical discussions related to the Cohen-Macaulay property, and some personal memories. The characterization of Gorenstein* and homotopy Gorenstein* complexes and the relevance in that connection of the Poincar\\'e con","authors_text":"Anders Bj\\\"orner","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-08T21:44:42Z","title":"Let \\Delta be a Cohen-Macaulay complex"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.2165","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":"sha256:06749d8942a3a669000572d70708b9cd97fc61def43a0246467251c65796315c","target":"record","created_at":"2026-05-18T02:38:08Z","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":"e3085f9e90b3ed89a0c469846328afee6046acf63a4be415669c630dcd215e17","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-08T21:44:42Z","title_canon_sha256":"6ed755f77d08f3373ff2092a105d0863dae0739a32baf0ea0bca8b365bb7a157"},"schema_version":"1.0","source":{"id":"1411.2165","kind":"arxiv","version":1}},"canonical_sha256":"30804c83df99bd59e98bba261fc0fde1f0dc238a08b6349adaad41f7c04aed12","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"30804c83df99bd59e98bba261fc0fde1f0dc238a08b6349adaad41f7c04aed12","first_computed_at":"2026-05-18T02:38:08.313491Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:08.313491Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TQAIkhMp982xbPxxwiiMkaeHghNQOGHCWxWZSxOC05KFvWz5zVtHdy7KMg4eu+zW4THK5+MAjZypQpzsctQSCw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:08.313899Z","signed_message":"canonical_sha256_bytes"},"source_id":"1411.2165","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:06749d8942a3a669000572d70708b9cd97fc61def43a0246467251c65796315c","sha256:5e1c0ba1516cbd6519ca0b4f4ef8201e1182fb86a981992622b1736c0bdb4c26"],"state_sha256":"b3c9259af26a9af4f33e4116c015ec0e684fe1aa86fb8a481479db0d133e37ed"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q2VGI83uvUKhCVsM1yGb8TBFS8ymLRfJr7p36j+KuwUahpaJR3+vVn4/uLpHixKmG9+mtqKhiGkax/g1hgtxAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T06:19:06.720051Z","bundle_sha256":"1b776f61adda7cac9ba1112dbcb4c9b9c04606c19261468da130819814c85c4f"}}