{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:SJ2O4RGFK2MRB2PPPEYLSJMIUV","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":"ca9ae8f7f91cddd0fad031b046419e957e7d0d63a4aa017e5c3d7192fb8eddd5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-06-26T18:53:41Z","title_canon_sha256":"97d62e5a6a248f34298af0528bb97ce34f101cd498944dcd32c65963f072d089"},"schema_version":"1.0","source":{"id":"1006.5160","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.5160","created_at":"2026-05-18T04:31:42Z"},{"alias_kind":"arxiv_version","alias_value":"1006.5160v2","created_at":"2026-05-18T04:31:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.5160","created_at":"2026-05-18T04:31:42Z"},{"alias_kind":"pith_short_12","alias_value":"SJ2O4RGFK2MR","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"SJ2O4RGFK2MRB2PP","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"SJ2O4RGF","created_at":"2026-05-18T12:26:13Z"}],"graph_snapshots":[{"event_id":"sha256:9a36b21cbe42c31664a4e5a2327d8e8854076b339c80259517627fb8d54add68","target":"graph","created_at":"2026-05-18T04:31:42Z","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":"By adapting the classical proof of Jordan's theorem on finite subgroups of linear groups, we show that every approximate subgroup of the unitary group U_n(C) is almost abelian.","authors_text":"Ben Green, Emmanuel Breuillard","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-06-26T18:53:41Z","title":"Approximate groups, III: the unitary case"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.5160","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":"sha256:4398b0eeb0dfc5895c8ec3f497df5a5f697938897a7957dbef3d62b46989dbff","target":"record","created_at":"2026-05-18T04:31:42Z","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":"ca9ae8f7f91cddd0fad031b046419e957e7d0d63a4aa017e5c3d7192fb8eddd5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-06-26T18:53:41Z","title_canon_sha256":"97d62e5a6a248f34298af0528bb97ce34f101cd498944dcd32c65963f072d089"},"schema_version":"1.0","source":{"id":"1006.5160","kind":"arxiv","version":2}},"canonical_sha256":"9274ee44c5569910e9ef7930b92588a564b896c31e6e0faeae8e97f3c7802889","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9274ee44c5569910e9ef7930b92588a564b896c31e6e0faeae8e97f3c7802889","first_computed_at":"2026-05-18T04:31:42.181974Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:31:42.181974Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ddAA2JL3+b2y+PeJcyBydcqP/CeoMIhVaMA7LlVvt921b3u8U1xfGUDYZwGiiR9/DX37NJSDxxzZ2nq31WTVAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:31:42.182418Z","signed_message":"canonical_sha256_bytes"},"source_id":"1006.5160","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4398b0eeb0dfc5895c8ec3f497df5a5f697938897a7957dbef3d62b46989dbff","sha256:9a36b21cbe42c31664a4e5a2327d8e8854076b339c80259517627fb8d54add68"],"state_sha256":"1259b916d8aef779ca5b1c7a0d73a8db0c1917321750c408b85970c8ec5cc126"}