{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:XXNJELZQS6BI6TBABQXNDBHHCM","short_pith_number":"pith:XXNJELZQ","schema_version":"1.0","canonical_sha256":"bdda922f3097828f4c200c2ed184e71330727a09b2585356bc42b5c2a164f8f6","source":{"kind":"arxiv","id":"1411.0848","version":3},"attestation_state":"computed","paper":{"title":"Commuting probabilities of finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.GR","authors_text":"Sean Eberhard","submitted_at":"2014-11-04T10:19:37Z","abstract_excerpt":"The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Let P \\subset (0,1] be the set of commuting probabilities of all finite groups. We prove that every point of P is nearly an Egyptian fraction of bounded complexity. As a corollary we deduce two conjectures of Keith Joseph from 1977: all limit points of P are rational, and P is well ordered by >. We also prove analogous theorems for bilinear maps of abelian groups."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1411.0848","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-11-04T10:19:37Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"01e07ef4d4ff7fde54019f4ebde48dd567b31bc09eb53c24e50441746c191be9","abstract_canon_sha256":"91a874db4b49569e62fbc455411c0283f8289ad5fdf1665f36f83ec7d40e3c0c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:57.965961Z","signature_b64":"jcd2hJs4rB8SxUtAZnjP9tqdbOEdCXi0HxAVpKN5X0JHSnJMdQ3prJQ/ZRg+7IdD3Wj0AQzK0JmBCtTnTt/kDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bdda922f3097828f4c200c2ed184e71330727a09b2585356bc42b5c2a164f8f6","last_reissued_at":"2026-05-18T00:50:57.965580Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:57.965580Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Commuting probabilities of finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.GR","authors_text":"Sean Eberhard","submitted_at":"2014-11-04T10:19:37Z","abstract_excerpt":"The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Let P \\subset (0,1] be the set of commuting probabilities of all finite groups. We prove that every point of P is nearly an Egyptian fraction of bounded complexity. As a corollary we deduce two conjectures of Keith Joseph from 1977: all limit points of P are rational, and P is well ordered by >. We also prove analogous theorems for bilinear maps of abelian groups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.0848","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1411.0848","created_at":"2026-05-18T00:50:57.965640+00:00"},{"alias_kind":"arxiv_version","alias_value":"1411.0848v3","created_at":"2026-05-18T00:50:57.965640+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.0848","created_at":"2026-05-18T00:50:57.965640+00:00"},{"alias_kind":"pith_short_12","alias_value":"XXNJELZQS6BI","created_at":"2026-05-18T12:28:57.508820+00:00"},{"alias_kind":"pith_short_16","alias_value":"XXNJELZQS6BI6TBA","created_at":"2026-05-18T12:28:57.508820+00:00"},{"alias_kind":"pith_short_8","alias_value":"XXNJELZQ","created_at":"2026-05-18T12:28:57.508820+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XXNJELZQS6BI6TBABQXNDBHHCM","json":"https://pith.science/pith/XXNJELZQS6BI6TBABQXNDBHHCM.json","graph_json":"https://pith.science/api/pith-number/XXNJELZQS6BI6TBABQXNDBHHCM/graph.json","events_json":"https://pith.science/api/pith-number/XXNJELZQS6BI6TBABQXNDBHHCM/events.json","paper":"https://pith.science/paper/XXNJELZQ"},"agent_actions":{"view_html":"https://pith.science/pith/XXNJELZQS6BI6TBABQXNDBHHCM","download_json":"https://pith.science/pith/XXNJELZQS6BI6TBABQXNDBHHCM.json","view_paper":"https://pith.science/paper/XXNJELZQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1411.0848&json=true","fetch_graph":"https://pith.science/api/pith-number/XXNJELZQS6BI6TBABQXNDBHHCM/graph.json","fetch_events":"https://pith.science/api/pith-number/XXNJELZQS6BI6TBABQXNDBHHCM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XXNJELZQS6BI6TBABQXNDBHHCM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XXNJELZQS6BI6TBABQXNDBHHCM/action/storage_attestation","attest_author":"https://pith.science/pith/XXNJELZQS6BI6TBABQXNDBHHCM/action/author_attestation","sign_citation":"https://pith.science/pith/XXNJELZQS6BI6TBABQXNDBHHCM/action/citation_signature","submit_replication":"https://pith.science/pith/XXNJELZQS6BI6TBABQXNDBHHCM/action/replication_record"}},"created_at":"2026-05-18T00:50:57.965640+00:00","updated_at":"2026-05-18T00:50:57.965640+00:00"}