{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:QYZFQDDPRG5LJIYSB4QLWNREH4","short_pith_number":"pith:QYZFQDDP","canonical_record":{"source":{"id":"1010.5722","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-10-27T15:55:22Z","cross_cats_sorted":[],"title_canon_sha256":"3ec13c60b1160b3e9d0385ad9498fa1232b447adbfc81fd7dccc14c5a338ac1f","abstract_canon_sha256":"ee84d51db9d50e168a9ab1cf18c8386a800434cdfe8b2b76ad0fdb82e3541134"},"schema_version":"1.0"},"canonical_sha256":"8632580c6f89bab4a3120f20bb36243f0e3bd31f7fc0361f1bf8dd88a89e7a31","source":{"kind":"arxiv","id":"1010.5722","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.5722","created_at":"2026-05-18T04:17:24Z"},{"alias_kind":"arxiv_version","alias_value":"1010.5722v2","created_at":"2026-05-18T04:17:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.5722","created_at":"2026-05-18T04:17:24Z"},{"alias_kind":"pith_short_12","alias_value":"QYZFQDDPRG5L","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"QYZFQDDPRG5LJIYS","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"QYZFQDDP","created_at":"2026-05-18T12:26:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:QYZFQDDPRG5LJIYSB4QLWNREH4","target":"record","payload":{"canonical_record":{"source":{"id":"1010.5722","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-10-27T15:55:22Z","cross_cats_sorted":[],"title_canon_sha256":"3ec13c60b1160b3e9d0385ad9498fa1232b447adbfc81fd7dccc14c5a338ac1f","abstract_canon_sha256":"ee84d51db9d50e168a9ab1cf18c8386a800434cdfe8b2b76ad0fdb82e3541134"},"schema_version":"1.0"},"canonical_sha256":"8632580c6f89bab4a3120f20bb36243f0e3bd31f7fc0361f1bf8dd88a89e7a31","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:17:24.998697Z","signature_b64":"LSXMO1cbthE8rh6tZEd/LNWdsGwjixQljvUUJAGOFmyaAHUIUGnuPTOo+xt4+oNA8aef69Z+W5f2e7yxzfdKAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8632580c6f89bab4a3120f20bb36243f0e3bd31f7fc0361f1bf8dd88a89e7a31","last_reissued_at":"2026-05-18T04:17:24.998122Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:17:24.998122Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1010.5722","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:17:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NQT2Iy8fWqMbK1pH/I/pVBj1leFDn8xOExv0RkygGucILauJj1r0I14jCPYnO7ks2N6tNYNY1ziWRtVQZN07BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T08:44:22.214350Z"},"content_sha256":"050c0e4cb36b4872dcd830e04a071376123e745e3f4a72b64ae21f1d60daf9c1","schema_version":"1.0","event_id":"sha256:050c0e4cb36b4872dcd830e04a071376123e745e3f4a72b64ae21f1d60daf9c1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:QYZFQDDPRG5LJIYSB4QLWNREH4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Invariable generation and the chebotarev invariant of a finite group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"A. Lubotzky, And A. Shalev, W. M. Kantor","submitted_at":"2010-10-27T15:55:22Z","abstract_excerpt":"A subset S of a finite group G invariably generates G if G = <hsg(s) j s 2 Si > for each choice of g(s) 2 G; s 2 S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response to a question in [KZ] we also bound the size of a randomly chosen set of elements of G that is likely to generate G invariably. Along the way we prove that every finite simple group is invariably generated by two elements."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.5722","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:17:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"71khOMXLzEG6rTF0jlRz+DDZEfcT82Q6N6+FNL13qVD5JQTIHc788iOO5HaiU52VUy7Ic9zXNZMsvW9sUgKsCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T08:44:22.214691Z"},"content_sha256":"ca29dcc13a436d156c9f4c11ba67a40192c7b3462be8f3c3bd2a13d7140e403a","schema_version":"1.0","event_id":"sha256:ca29dcc13a436d156c9f4c11ba67a40192c7b3462be8f3c3bd2a13d7140e403a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QYZFQDDPRG5LJIYSB4QLWNREH4/bundle.json","state_url":"https://pith.science/pith/QYZFQDDPRG5LJIYSB4QLWNREH4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QYZFQDDPRG5LJIYSB4QLWNREH4/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-25T08:44:22Z","links":{"resolver":"https://pith.science/pith/QYZFQDDPRG5LJIYSB4QLWNREH4","bundle":"https://pith.science/pith/QYZFQDDPRG5LJIYSB4QLWNREH4/bundle.json","state":"https://pith.science/pith/QYZFQDDPRG5LJIYSB4QLWNREH4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QYZFQDDPRG5LJIYSB4QLWNREH4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:QYZFQDDPRG5LJIYSB4QLWNREH4","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":"ee84d51db9d50e168a9ab1cf18c8386a800434cdfe8b2b76ad0fdb82e3541134","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-10-27T15:55:22Z","title_canon_sha256":"3ec13c60b1160b3e9d0385ad9498fa1232b447adbfc81fd7dccc14c5a338ac1f"},"schema_version":"1.0","source":{"id":"1010.5722","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.5722","created_at":"2026-05-18T04:17:24Z"},{"alias_kind":"arxiv_version","alias_value":"1010.5722v2","created_at":"2026-05-18T04:17:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.5722","created_at":"2026-05-18T04:17:24Z"},{"alias_kind":"pith_short_12","alias_value":"QYZFQDDPRG5L","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"QYZFQDDPRG5LJIYS","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"QYZFQDDP","created_at":"2026-05-18T12:26:13Z"}],"graph_snapshots":[{"event_id":"sha256:ca29dcc13a436d156c9f4c11ba67a40192c7b3462be8f3c3bd2a13d7140e403a","target":"graph","created_at":"2026-05-18T04:17:24Z","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":"A subset S of a finite group G invariably generates G if G = <hsg(s) j s 2 Si > for each choice of g(s) 2 G; s 2 S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response to a question in [KZ] we also bound the size of a randomly chosen set of elements of G that is likely to generate G invariably. Along the way we prove that every finite simple group is invariably generated by two elements.","authors_text":"A. Lubotzky, And A. Shalev, W. M. Kantor","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-10-27T15:55:22Z","title":"Invariable generation and the chebotarev invariant of a finite group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.5722","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":"sha256:050c0e4cb36b4872dcd830e04a071376123e745e3f4a72b64ae21f1d60daf9c1","target":"record","created_at":"2026-05-18T04:17:24Z","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":"ee84d51db9d50e168a9ab1cf18c8386a800434cdfe8b2b76ad0fdb82e3541134","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-10-27T15:55:22Z","title_canon_sha256":"3ec13c60b1160b3e9d0385ad9498fa1232b447adbfc81fd7dccc14c5a338ac1f"},"schema_version":"1.0","source":{"id":"1010.5722","kind":"arxiv","version":2}},"canonical_sha256":"8632580c6f89bab4a3120f20bb36243f0e3bd31f7fc0361f1bf8dd88a89e7a31","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8632580c6f89bab4a3120f20bb36243f0e3bd31f7fc0361f1bf8dd88a89e7a31","first_computed_at":"2026-05-18T04:17:24.998122Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:17:24.998122Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LSXMO1cbthE8rh6tZEd/LNWdsGwjixQljvUUJAGOFmyaAHUIUGnuPTOo+xt4+oNA8aef69Z+W5f2e7yxzfdKAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:17:24.998697Z","signed_message":"canonical_sha256_bytes"},"source_id":"1010.5722","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:050c0e4cb36b4872dcd830e04a071376123e745e3f4a72b64ae21f1d60daf9c1","sha256:ca29dcc13a436d156c9f4c11ba67a40192c7b3462be8f3c3bd2a13d7140e403a"],"state_sha256":"75e929d41b79ee71266d83fa92c4f8000679b158d31b7119619aa222aa0f01a9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GW8YnmF8nZ84Zb9MdeWshbJLyhbqBcRToQmiv0Jlh7t18YdeMD1Lq7QjP+bLyYkQiIjnDP2/XoM27AkL95xCDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T08:44:22.216718Z","bundle_sha256":"05e38f0c0169b4e47646e9593639538c9c9a8c34d60e12e5fc8b76cc0ec0b912"}}