{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:TNAGKQT4WHHGQFZS5KRVN2EYCQ","short_pith_number":"pith:TNAGKQT4","canonical_record":{"source":{"id":"1610.02079","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-10-06T21:44:54Z","cross_cats_sorted":[],"title_canon_sha256":"231e27ee88714ffa9a15bd6d90876395f0258eeb064d38dafc21c5e355014c6d","abstract_canon_sha256":"66fc32f289eceb259a7f6351ae9e4bbb3672a8d03d31e04e1468b4ad8131c807"},"schema_version":"1.0"},"canonical_sha256":"9b4065427cb1ce681732eaa356e898142991267b58295afe5551bd13185ab6cb","source":{"kind":"arxiv","id":"1610.02079","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.02079","created_at":"2026-05-18T00:44:33Z"},{"alias_kind":"arxiv_version","alias_value":"1610.02079v2","created_at":"2026-05-18T00:44:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.02079","created_at":"2026-05-18T00:44:33Z"},{"alias_kind":"pith_short_12","alias_value":"TNAGKQT4WHHG","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"TNAGKQT4WHHGQFZS","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"TNAGKQT4","created_at":"2026-05-18T12:30:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:TNAGKQT4WHHGQFZS5KRVN2EYCQ","target":"record","payload":{"canonical_record":{"source":{"id":"1610.02079","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-10-06T21:44:54Z","cross_cats_sorted":[],"title_canon_sha256":"231e27ee88714ffa9a15bd6d90876395f0258eeb064d38dafc21c5e355014c6d","abstract_canon_sha256":"66fc32f289eceb259a7f6351ae9e4bbb3672a8d03d31e04e1468b4ad8131c807"},"schema_version":"1.0"},"canonical_sha256":"9b4065427cb1ce681732eaa356e898142991267b58295afe5551bd13185ab6cb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:33.695362Z","signature_b64":"rnaaLqwfxbRFwCLDhinqGdvHuaNL2af7FbTJ9/MvolNDaNJ/1kJh1UEQ7G6pqqvA5SUqx0y9OUHlI7GnLXE8DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b4065427cb1ce681732eaa356e898142991267b58295afe5551bd13185ab6cb","last_reissued_at":"2026-05-18T00:44:33.694804Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:33.694804Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.02079","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":"2026-05-18T00:44:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xJ5qx3YayOgHoAWcMRnfaQqspU85miQdswar07PGvZwux4q75XT1vtMvQSj8qccFB8fPLVmXu0Oo8FzhoX6BDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T03:03:26.941050Z"},"content_sha256":"84a69b3645577e1009724aa5084cebaebbad10f4407fffb65d702a86840d8c25","schema_version":"1.0","event_id":"sha256:84a69b3645577e1009724aa5084cebaebbad10f4407fffb65d702a86840d8c25"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:TNAGKQT4WHHGQFZS5KRVN2EYCQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Almost Engel compact groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"E. I. Khukhro, P. Shumyatsky","submitted_at":"2016-10-06T21:44:54Z","abstract_excerpt":"We say that a group $G$ is almost Engel if for every $g\\in G$ there is a finite set ${\\mathscr E}(g)$ such that for every $x\\in G$ all sufficiently long commutators $[...[[x,g],g],\\dots ,g]$ belong to ${\\mathscr E}(g)$, that is, for every $x\\in G$ there is a positive integer $n(x,g)$ such that $[...[[x,g],g],\\dots ,g]\\in {\\mathscr E}(g)$ if $g$ is repeated at least $n(x,g)$ times. (Thus, Engel groups are precisely the almost Engel groups for which we can choose ${\\mathscr E}(g)=\\{ 1\\}$ for all $g\\in G$.)\n  We prove that if a compact (Hausdorff) group $G$ is almost Engel, then $G$ has a finite "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.02079","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":"2026-05-18T00:44:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Dj0qNqXa5vzMJcdVqzt6ThAETzXlrtlkQRnbgnWy8UHe+cQDBmPXrbpvZHN1o2xFcDn6H0LVJay0aKnxXmc1BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T03:03:26.941403Z"},"content_sha256":"f1f83c62f7a341c11c76c095d209c75cf23848fd3ea1530d4d45843479de702c","schema_version":"1.0","event_id":"sha256:f1f83c62f7a341c11c76c095d209c75cf23848fd3ea1530d4d45843479de702c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TNAGKQT4WHHGQFZS5KRVN2EYCQ/bundle.json","state_url":"https://pith.science/pith/TNAGKQT4WHHGQFZS5KRVN2EYCQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TNAGKQT4WHHGQFZS5KRVN2EYCQ/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":"2026-06-23T03:03:26Z","links":{"resolver":"https://pith.science/pith/TNAGKQT4WHHGQFZS5KRVN2EYCQ","bundle":"https://pith.science/pith/TNAGKQT4WHHGQFZS5KRVN2EYCQ/bundle.json","state":"https://pith.science/pith/TNAGKQT4WHHGQFZS5KRVN2EYCQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TNAGKQT4WHHGQFZS5KRVN2EYCQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:TNAGKQT4WHHGQFZS5KRVN2EYCQ","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":"66fc32f289eceb259a7f6351ae9e4bbb3672a8d03d31e04e1468b4ad8131c807","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-10-06T21:44:54Z","title_canon_sha256":"231e27ee88714ffa9a15bd6d90876395f0258eeb064d38dafc21c5e355014c6d"},"schema_version":"1.0","source":{"id":"1610.02079","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.02079","created_at":"2026-05-18T00:44:33Z"},{"alias_kind":"arxiv_version","alias_value":"1610.02079v2","created_at":"2026-05-18T00:44:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.02079","created_at":"2026-05-18T00:44:33Z"},{"alias_kind":"pith_short_12","alias_value":"TNAGKQT4WHHG","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"TNAGKQT4WHHGQFZS","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"TNAGKQT4","created_at":"2026-05-18T12:30:44Z"}],"graph_snapshots":[{"event_id":"sha256:f1f83c62f7a341c11c76c095d209c75cf23848fd3ea1530d4d45843479de702c","target":"graph","created_at":"2026-05-18T00:44:33Z","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 say that a group $G$ is almost Engel if for every $g\\in G$ there is a finite set ${\\mathscr E}(g)$ such that for every $x\\in G$ all sufficiently long commutators $[...[[x,g],g],\\dots ,g]$ belong to ${\\mathscr E}(g)$, that is, for every $x\\in G$ there is a positive integer $n(x,g)$ such that $[...[[x,g],g],\\dots ,g]\\in {\\mathscr E}(g)$ if $g$ is repeated at least $n(x,g)$ times. (Thus, Engel groups are precisely the almost Engel groups for which we can choose ${\\mathscr E}(g)=\\{ 1\\}$ for all $g\\in G$.)\n  We prove that if a compact (Hausdorff) group $G$ is almost Engel, then $G$ has a finite ","authors_text":"E. I. Khukhro, P. Shumyatsky","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-10-06T21:44:54Z","title":"Almost Engel compact groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.02079","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":"sha256:84a69b3645577e1009724aa5084cebaebbad10f4407fffb65d702a86840d8c25","target":"record","created_at":"2026-05-18T00:44:33Z","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":"66fc32f289eceb259a7f6351ae9e4bbb3672a8d03d31e04e1468b4ad8131c807","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-10-06T21:44:54Z","title_canon_sha256":"231e27ee88714ffa9a15bd6d90876395f0258eeb064d38dafc21c5e355014c6d"},"schema_version":"1.0","source":{"id":"1610.02079","kind":"arxiv","version":2}},"canonical_sha256":"9b4065427cb1ce681732eaa356e898142991267b58295afe5551bd13185ab6cb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9b4065427cb1ce681732eaa356e898142991267b58295afe5551bd13185ab6cb","first_computed_at":"2026-05-18T00:44:33.694804Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:33.694804Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rnaaLqwfxbRFwCLDhinqGdvHuaNL2af7FbTJ9/MvolNDaNJ/1kJh1UEQ7G6pqqvA5SUqx0y9OUHlI7GnLXE8DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:33.695362Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.02079","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:84a69b3645577e1009724aa5084cebaebbad10f4407fffb65d702a86840d8c25","sha256:f1f83c62f7a341c11c76c095d209c75cf23848fd3ea1530d4d45843479de702c"],"state_sha256":"64aa3d38a1bb3d3f8391e043d324eb30b821f6989c5dc5b125c8ca6edffb01ff"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cek7jhZOwaZAi07dGVl2VbGN1yekVSfuDdDRNxFlF9Mm6PXum0nY/2SFq215iaJPeVxHz3edCi2OnfbdmwmXBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T03:03:26.943282Z","bundle_sha256":"08907cc998d84eb9af03199c005203814b878cb714273ba41d9439b0e728f136"}}