{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:TDLYVZ47GV7NRUNTHNTZG77QAX","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":"c3d35ecfca3bd65ccaea0293e9b18fd7698c5cacf4665d5892bef1ecbb013ed6","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-10-24T13:09:16Z","title_canon_sha256":"e049309e34f0ab6ac893d4d2b0417b6225851d2afb6337a0fcb7df1078c0c19e"},"schema_version":"1.0","source":{"id":"1510.07143","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.07143","created_at":"2026-05-18T01:25:57Z"},{"alias_kind":"arxiv_version","alias_value":"1510.07143v1","created_at":"2026-05-18T01:25:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.07143","created_at":"2026-05-18T01:25:57Z"},{"alias_kind":"pith_short_12","alias_value":"TDLYVZ47GV7N","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"TDLYVZ47GV7NRUNT","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"TDLYVZ47","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:a63ad025399b96496626ed0bbfd4a38c3129dc4e7e57212ad4566d8815fdd285","target":"graph","created_at":"2026-05-18T01:25:57Z","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":"In his seminal paper, half a century ago, Hyman Bass established a commutator formula in the setting of (stable) general linear group which was the key step in defining the K_1 group. Namely, he proved that for an associative ring A with identity, E(A)=[E(A),E(A)]=[GL(A),GL(A)] where GL(A) is the stable general linear group and E(A) is its elementary subgroup. Since then, various commutator formulas have been studied in stable and non-stable settings, and for a range of classical and algebraic like-groups, mostly in relation to subnormal subgroups of these groups. The major classical theorems ","authors_text":"N. Vavilov, R. Hazrat, Z. Zhang","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-10-24T13:09:16Z","title":"The commutators of classical groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07143","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":"sha256:42c852203eddbc8996946eb1d1e4e95ba94165a3a3ea95286bcb1f5731c7229d","target":"record","created_at":"2026-05-18T01:25:57Z","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":"c3d35ecfca3bd65ccaea0293e9b18fd7698c5cacf4665d5892bef1ecbb013ed6","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-10-24T13:09:16Z","title_canon_sha256":"e049309e34f0ab6ac893d4d2b0417b6225851d2afb6337a0fcb7df1078c0c19e"},"schema_version":"1.0","source":{"id":"1510.07143","kind":"arxiv","version":1}},"canonical_sha256":"98d78ae79f357ed8d1b33b67937ff005c18ce581bfb0a3e36afdf3b40e3bb0a8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"98d78ae79f357ed8d1b33b67937ff005c18ce581bfb0a3e36afdf3b40e3bb0a8","first_computed_at":"2026-05-18T01:25:57.018907Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:57.018907Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hYJ9WIbql0Cb3Y8GMGRi2ISOXjdlvfAPdT7D5cOVXufUs19aatXhMurD+iY9fADK2ZyN6GoOBwoPDajDN2KMCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:57.019527Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.07143","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:42c852203eddbc8996946eb1d1e4e95ba94165a3a3ea95286bcb1f5731c7229d","sha256:a63ad025399b96496626ed0bbfd4a38c3129dc4e7e57212ad4566d8815fdd285"],"state_sha256":"a3760860af466e162b046f8d6d7137c86ad96718c2f016671d790660da817867"}