{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:FQARKJGGOUZESAJA7HQZK5FM4B","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":"398e053994e7044905f67a2fdcf46db91a868c5ee9b12928a45fb82e6b804ed4","cross_cats_sorted":["math.GR"],"license":"","primary_cat":"math.FA","submitted_at":"2005-11-27T16:41:03Z","title_canon_sha256":"703b07252a0ee2a1f2607f04448b282f19b9ee117cc5df6615f89cf0497aa7b7"},"schema_version":"1.0","source":{"id":"math/0511655","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0511655","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"arxiv_version","alias_value":"math/0511655v1","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0511655","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"pith_short_12","alias_value":"FQARKJGGOUZE","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"FQARKJGGOUZESAJA","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"FQARKJGG","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:2852a24950e27a8e7812f229cc3aa8ac48ebe284fcd182c049152e09b6303c1f","target":"graph","created_at":"2026-05-18T01:05:22Z","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":"We prove that the approximation conjecture of Luck holds for all amenable groups in the complex group algebra case. This result was previously proved by\n Dodziuk, Linnell, Mathai, Schick and Yates under the assumption that the group is torsion-free elementary amenable.","authors_text":"Gabor Elek","cross_cats":["math.GR"],"headline":"","license":"","primary_cat":"math.FA","submitted_at":"2005-11-27T16:41:03Z","title":"The Strong Approximation Conjecture holds for amenable groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0511655","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:b58cc2a5326c299e1443a31991ea2d54304108df61772c5439024570bc952a01","target":"record","created_at":"2026-05-18T01:05:22Z","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":"398e053994e7044905f67a2fdcf46db91a868c5ee9b12928a45fb82e6b804ed4","cross_cats_sorted":["math.GR"],"license":"","primary_cat":"math.FA","submitted_at":"2005-11-27T16:41:03Z","title_canon_sha256":"703b07252a0ee2a1f2607f04448b282f19b9ee117cc5df6615f89cf0497aa7b7"},"schema_version":"1.0","source":{"id":"math/0511655","kind":"arxiv","version":1}},"canonical_sha256":"2c011524c67532490120f9e19574ace07087bcf7b24f7a9602f6f34fd10db0a7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2c011524c67532490120f9e19574ace07087bcf7b24f7a9602f6f34fd10db0a7","first_computed_at":"2026-05-18T01:05:22.875950Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:22.875950Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AyVoCY0YwjzChn/qyz+aNqL3EAo6NzMPSbIoYTgRGfruJHA0OdQzU6HnsqgeiWtC7OOOuTI1lKrq5YpWNffLBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:22.876446Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0511655","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b58cc2a5326c299e1443a31991ea2d54304108df61772c5439024570bc952a01","sha256:2852a24950e27a8e7812f229cc3aa8ac48ebe284fcd182c049152e09b6303c1f"],"state_sha256":"e77420470d7dc3b0e62bbbc39b6e6158d0e1b4873c1531a31f554409f17d98b3"}