{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:5GQU5AGWJCEKXDJW5XHSOKRVHB","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":"cb43bd7cab35007e51d86d8c3fe73a1209e5b9eea6f6595962f031dc8f109f52","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-12-26T18:43:26Z","title_canon_sha256":"b6891d2bad3894cf7d6384e308dc3bde5eaf63105eb03b8c2d4a249c4c46aff3"},"schema_version":"1.0","source":{"id":"1112.5879","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.5879","created_at":"2026-05-18T04:05:37Z"},{"alias_kind":"arxiv_version","alias_value":"1112.5879v1","created_at":"2026-05-18T04:05:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.5879","created_at":"2026-05-18T04:05:37Z"},{"alias_kind":"pith_short_12","alias_value":"5GQU5AGWJCEK","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"5GQU5AGWJCEKXDJW","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"5GQU5AGW","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:f392405f378d6c0dd7ad18fa7f587863a9f09d2ce1f7c1940dde6090418e66de","target":"graph","created_at":"2026-05-18T04:05:37Z","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":"For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite exponent $e$ whose union contains all $\\gamma_k$-values in $G$, it is shown that $\\gamma_k(G)$ has finite $(e,k,s)$-bounded exponent. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite rank $r$ whose uni","authors_text":"Cristina Acciarri, Pavel Shumyatsky","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-12-26T18:43:26Z","title":"On profinite groups in which commutators are covered by finitely many subgroups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.5879","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":"sha256:3b6c1a603123ab81a586bd555835cd1e7bec8b0c71f28dfa8c332db3b69f97d7","target":"record","created_at":"2026-05-18T04:05:37Z","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":"cb43bd7cab35007e51d86d8c3fe73a1209e5b9eea6f6595962f031dc8f109f52","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-12-26T18:43:26Z","title_canon_sha256":"b6891d2bad3894cf7d6384e308dc3bde5eaf63105eb03b8c2d4a249c4c46aff3"},"schema_version":"1.0","source":{"id":"1112.5879","kind":"arxiv","version":1}},"canonical_sha256":"e9a14e80d64888ab8d36edcf272a35386302ccc329265493f24349b9f0dc1718","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e9a14e80d64888ab8d36edcf272a35386302ccc329265493f24349b9f0dc1718","first_computed_at":"2026-05-18T04:05:37.375458Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:05:37.375458Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"k/HP6VYwGssC5isyFdim213u8Ab+toUFeFz2r/89NjS+kAirX+gyYdWeyMsQ9IyAFpaWiV6sHcQO7Avjx/gPDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:05:37.376248Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.5879","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3b6c1a603123ab81a586bd555835cd1e7bec8b0c71f28dfa8c332db3b69f97d7","sha256:f392405f378d6c0dd7ad18fa7f587863a9f09d2ce1f7c1940dde6090418e66de"],"state_sha256":"97f1a42bb70097c2f72628ab1993261f4105be254cb2d425e3b538ab38c77300"}