{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:VVS3SWADWSTDMLMBY5Y5SVNLBI","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":"3b2b208c4b613d41fc2fe89751596d70a9a2ae2c53fe0abebd89c055666b6d9f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-10-11T00:14:15Z","title_canon_sha256":"97eb134f2f1bb23063e68fde6ae145ad812f707652f838b9b65f2bf16c2696e0"},"schema_version":"1.0","source":{"id":"1610.03136","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.03136","created_at":"2026-05-18T00:13:48Z"},{"alias_kind":"arxiv_version","alias_value":"1610.03136v3","created_at":"2026-05-18T00:13:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.03136","created_at":"2026-05-18T00:13:48Z"},{"alias_kind":"pith_short_12","alias_value":"VVS3SWADWSTD","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VVS3SWADWSTDMLMB","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VVS3SWAD","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:7e1e58fbe87d6f19fc417c10de48d7c7f17d48cfadded1f99c75d2b6f23b34de","target":"graph","created_at":"2026-05-18T00:13:48Z","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":"Let $F$ be a field of characteristic $\\ne 2,3$ and let $A$ be a unital associative $F$-algebra. Define a left-normed commutator $[a_1, a_2, \\dots , a_n]$ $(a_i \\in A)$ recursively by $[a_1, a_2] = a_1 a_2 - a_2 a_1$, $[a_1, \\dots , a_{n-1}, a_n] = [[a_1, \\dots , a_{n-1}], a_n]$ $(n \\ge 3)$. For $n \\ge 2$, let $T^{(n)} (A)$ be the two-sided ideal in $A$ generated by all commutators $[a_1, a_2, \\dots , a_n]$ ($a_i \\in A )$. Define $T^{(1)} (A) = A$.\n  Let $k, \\ell$ be integers such that $k > 0$, $0 \\le \\ell \\le k$. Let $m_1, \\dots , m_k$ be positive integers such that $\\ell$ of them are odd and ","authors_text":"Alexei Krasilnikov, Galina Deryabina","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-10-11T00:14:15Z","title":"Products of several commutators in a Lie nilpotent associative algebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.03136","kind":"arxiv","version":3},"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":"sha256:97bf108a82e03f0663a59df6dcc09281513201dca184afbd8f39fa25e56aa163","target":"record","created_at":"2026-05-18T00:13:48Z","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":"3b2b208c4b613d41fc2fe89751596d70a9a2ae2c53fe0abebd89c055666b6d9f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-10-11T00:14:15Z","title_canon_sha256":"97eb134f2f1bb23063e68fde6ae145ad812f707652f838b9b65f2bf16c2696e0"},"schema_version":"1.0","source":{"id":"1610.03136","kind":"arxiv","version":3}},"canonical_sha256":"ad65b95803b4a6362d81c771d955ab0a3d1af0401265e90045af253846a8c62e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ad65b95803b4a6362d81c771d955ab0a3d1af0401265e90045af253846a8c62e","first_computed_at":"2026-05-18T00:13:48.342426Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:13:48.342426Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MvfV+REjUAkJ6NWEmLZCU9AqGtaGvbuSUPZgibOxqkOvhZ+rk4spto0Iljoze8cgRWXlin4D+0R1mOkp7wbnAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:13:48.343019Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.03136","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:97bf108a82e03f0663a59df6dcc09281513201dca184afbd8f39fa25e56aa163","sha256:7e1e58fbe87d6f19fc417c10de48d7c7f17d48cfadded1f99c75d2b6f23b34de"],"state_sha256":"caab4c28b0b35394402de77613812accb0afceef3674e3c95269719d21fb7017"}