{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2005:EJTPN5COMAG766GRZKAPKFNSKZ","short_pith_number":"pith:EJTPN5CO","canonical_record":{"source":{"id":"math/0502438","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.AC","submitted_at":"2005-02-21T15:27:37Z","cross_cats_sorted":["math.CO","math.GR"],"title_canon_sha256":"eef95378ad176aca8e7eff97e864da42e970f2735efe13354e157b289be7ceb7","abstract_canon_sha256":"108b79f9784f13c37e8373e0ce114a1ec8d60fc9dd34a95092291084cc29c252"},"schema_version":"1.0"},"canonical_sha256":"2266f6f44e600dff78d1ca80f515b2565bcb7f04b96033e08a68d663290951e9","source":{"kind":"arxiv","id":"math/0502438","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0502438","created_at":"2026-05-18T04:38:49Z"},{"alias_kind":"arxiv_version","alias_value":"math/0502438v2","created_at":"2026-05-18T04:38:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0502438","created_at":"2026-05-18T04:38:49Z"},{"alias_kind":"pith_short_12","alias_value":"EJTPN5COMAG7","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"EJTPN5COMAG766GR","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"EJTPN5CO","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2005:EJTPN5COMAG766GRZKAPKFNSKZ","target":"record","payload":{"canonical_record":{"source":{"id":"math/0502438","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.AC","submitted_at":"2005-02-21T15:27:37Z","cross_cats_sorted":["math.CO","math.GR"],"title_canon_sha256":"eef95378ad176aca8e7eff97e864da42e970f2735efe13354e157b289be7ceb7","abstract_canon_sha256":"108b79f9784f13c37e8373e0ce114a1ec8d60fc9dd34a95092291084cc29c252"},"schema_version":"1.0"},"canonical_sha256":"2266f6f44e600dff78d1ca80f515b2565bcb7f04b96033e08a68d663290951e9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:38:49.084657Z","signature_b64":"HPgPPec2uoMYHvjm8fzjcU6L1s8h6zyZl3B2spkSRiwHu88dnwBoQcLojv+XSzuXoxYJj+H3FceB3M84LlE+BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2266f6f44e600dff78d1ca80f515b2565bcb7f04b96033e08a68d663290951e9","last_reissued_at":"2026-05-18T04:38:49.084190Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:38:49.084190Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0502438","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":"2026-05-18T04:38:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A6PccpnP3U6W06uGavVB/6YKL8JmQ0NmVfFcmhKHdmLQGSJtA7KVmqzp48nQ0Q0thttzlKjRgAB1npvMGIsxBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T03:27:20.681833Z"},"content_sha256":"28ed5d1a9ef0686c33b59e1cd9a99add739744810efedd3d15a6f2f5496d6e31","schema_version":"1.0","event_id":"sha256:28ed5d1a9ef0686c33b59e1cd9a99add739744810efedd3d15a6f2f5496d6e31"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2005:EJTPN5COMAG766GRZKAPKFNSKZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence","license":"","headline":"","cross_cats":["math.CO","math.GR"],"primary_cat":"math.AC","authors_text":"Alexander I. Suciu, Henry K. Schenck","submitted_at":"2005-02-21T15:27:37Z","abstract_excerpt":"If \\A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G=\\pi_1(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A=H^*(X,\\k), viewed as a module over the exterior algebra E on \\A: \\theta_k(G) = \\dim_\\k Tor^E_{k-1}(A,\\k)_k, where \\k is a field of characteristic 0, and k\\ge 2. The Chen ranks conjecture asserts that, for k sufficiently large, \\theta_k(G) =(k-1) \\sum_{r\\ge 1} h_r \\binom{r+k-1}{k}, where h_r is the number of r-dimensional components of the projective resonance variety R^1(\\A). Our"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0502438","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":"2026-05-18T04:38:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a08NjamwqA+u/Zm6ag1NJhHEq1JIjRKxUgjK1nAbjFS/VPi7DItGxqN1yjvjNj/jj66lSKyx9nWj7oU3fE2gDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T03:27:20.682459Z"},"content_sha256":"5cd0e5f80c6cf20e4034db25bec89eca0fe3233e6f4221c3c952acc8f1c17455","schema_version":"1.0","event_id":"sha256:5cd0e5f80c6cf20e4034db25bec89eca0fe3233e6f4221c3c952acc8f1c17455"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EJTPN5COMAG766GRZKAPKFNSKZ/bundle.json","state_url":"https://pith.science/pith/EJTPN5COMAG766GRZKAPKFNSKZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EJTPN5COMAG766GRZKAPKFNSKZ/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":"2026-06-20T03:27:20Z","links":{"resolver":"https://pith.science/pith/EJTPN5COMAG766GRZKAPKFNSKZ","bundle":"https://pith.science/pith/EJTPN5COMAG766GRZKAPKFNSKZ/bundle.json","state":"https://pith.science/pith/EJTPN5COMAG766GRZKAPKFNSKZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EJTPN5COMAG766GRZKAPKFNSKZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:EJTPN5COMAG766GRZKAPKFNSKZ","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":"108b79f9784f13c37e8373e0ce114a1ec8d60fc9dd34a95092291084cc29c252","cross_cats_sorted":["math.CO","math.GR"],"license":"","primary_cat":"math.AC","submitted_at":"2005-02-21T15:27:37Z","title_canon_sha256":"eef95378ad176aca8e7eff97e864da42e970f2735efe13354e157b289be7ceb7"},"schema_version":"1.0","source":{"id":"math/0502438","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0502438","created_at":"2026-05-18T04:38:49Z"},{"alias_kind":"arxiv_version","alias_value":"math/0502438v2","created_at":"2026-05-18T04:38:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0502438","created_at":"2026-05-18T04:38:49Z"},{"alias_kind":"pith_short_12","alias_value":"EJTPN5COMAG7","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"EJTPN5COMAG766GR","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"EJTPN5CO","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:5cd0e5f80c6cf20e4034db25bec89eca0fe3233e6f4221c3c952acc8f1c17455","target":"graph","created_at":"2026-05-18T04:38:49Z","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":"If \\A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G=\\pi_1(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A=H^*(X,\\k), viewed as a module over the exterior algebra E on \\A: \\theta_k(G) = \\dim_\\k Tor^E_{k-1}(A,\\k)_k, where \\k is a field of characteristic 0, and k\\ge 2. The Chen ranks conjecture asserts that, for k sufficiently large, \\theta_k(G) =(k-1) \\sum_{r\\ge 1} h_r \\binom{r+k-1}{k}, where h_r is the number of r-dimensional components of the projective resonance variety R^1(\\A). Our","authors_text":"Alexander I. Suciu, Henry K. Schenck","cross_cats":["math.CO","math.GR"],"headline":"","license":"","primary_cat":"math.AC","submitted_at":"2005-02-21T15:27:37Z","title":"Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0502438","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":"sha256:28ed5d1a9ef0686c33b59e1cd9a99add739744810efedd3d15a6f2f5496d6e31","target":"record","created_at":"2026-05-18T04:38:49Z","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":"108b79f9784f13c37e8373e0ce114a1ec8d60fc9dd34a95092291084cc29c252","cross_cats_sorted":["math.CO","math.GR"],"license":"","primary_cat":"math.AC","submitted_at":"2005-02-21T15:27:37Z","title_canon_sha256":"eef95378ad176aca8e7eff97e864da42e970f2735efe13354e157b289be7ceb7"},"schema_version":"1.0","source":{"id":"math/0502438","kind":"arxiv","version":2}},"canonical_sha256":"2266f6f44e600dff78d1ca80f515b2565bcb7f04b96033e08a68d663290951e9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2266f6f44e600dff78d1ca80f515b2565bcb7f04b96033e08a68d663290951e9","first_computed_at":"2026-05-18T04:38:49.084190Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:38:49.084190Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HPgPPec2uoMYHvjm8fzjcU6L1s8h6zyZl3B2spkSRiwHu88dnwBoQcLojv+XSzuXoxYJj+H3FceB3M84LlE+BA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:38:49.084657Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0502438","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:28ed5d1a9ef0686c33b59e1cd9a99add739744810efedd3d15a6f2f5496d6e31","sha256:5cd0e5f80c6cf20e4034db25bec89eca0fe3233e6f4221c3c952acc8f1c17455"],"state_sha256":"d6542bccd1581bd6951efde804c5dfe98efe2f6f9a0a1169d849b0403337a921"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a7mCZvI1ZLmVK1E726SJ/XBxhACb+buiodYuJ1uhHuH10NBxv52ABoWbC7YvZy0zDb2qmcYmh/Ygsu/addPkDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T03:27:20.684691Z","bundle_sha256":"6eb3e155bcc5d7fb703582da9d2f4633ffa92e78f59e89455241332c571393d0"}}