{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:UUGPOQWNWW2GHI2WYA6J5677KY","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":"60b765b9b181fe368dba73f6681f8b16cf1b6a2b55c73e6d5aed8321fedb2630","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-06-03T08:12:04Z","title_canon_sha256":"6f5d20484f4c7f7f987036b9249799ed97d475fbe1a1e42d91083bea7323a452"},"schema_version":"1.0","source":{"id":"2606.04577","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.04577","created_at":"2026-06-04T01:09:18Z"},{"alias_kind":"arxiv_version","alias_value":"2606.04577v1","created_at":"2026-06-04T01:09:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.04577","created_at":"2026-06-04T01:09:18Z"},{"alias_kind":"pith_short_12","alias_value":"UUGPOQWNWW2G","created_at":"2026-06-04T01:09:18Z"},{"alias_kind":"pith_short_16","alias_value":"UUGPOQWNWW2GHI2W","created_at":"2026-06-04T01:09:18Z"},{"alias_kind":"pith_short_8","alias_value":"UUGPOQWN","created_at":"2026-06-04T01:09:18Z"}],"graph_snapshots":[{"event_id":"sha256:870007e1532c29c5fffa01355e63d121401eb4f1e1494a4a294b873dbe11947c","target":"graph","created_at":"2026-06-04T01:09:18Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.04577/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $A$ be a finite universal algebra. Then the orders of the $n$-generated free algebras $F_n$ in the variety (equational class) generated by $A$ satisfy G. Birkhoff's inequality: $|F_n|\\le |A|^{|A|^n}$ for $n=1,2,\\dots$ It follows that $\\limsup_{n\\to\\infty}\\sqrt[n]{\\log |F_n|}\\le |A|$. When $A$ is a finite group or a finite nonassociative algebra, we obtain a criterion for equality in this estimate; equivalently, a criterion for maximal growth of the sequence $\\{|F_n|\\}_{n=1}^{\\infty}$.","authors_text":"Alexander Olshanskii","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-06-03T08:12:04Z","title":"Finite groups and rings generating varieties with rapid growth"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.04577","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":"sha256:37baa13ea72b86e663e93fe19997d1acd2d763d68c37178d058fae2f892066fb","target":"record","created_at":"2026-06-04T01:09:18Z","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":"60b765b9b181fe368dba73f6681f8b16cf1b6a2b55c73e6d5aed8321fedb2630","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-06-03T08:12:04Z","title_canon_sha256":"6f5d20484f4c7f7f987036b9249799ed97d475fbe1a1e42d91083bea7323a452"},"schema_version":"1.0","source":{"id":"2606.04577","kind":"arxiv","version":1}},"canonical_sha256":"a50cf742cdb5b463a356c03c9efbff5600f1ce3f2850ac83f27f12fd4ee3d285","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a50cf742cdb5b463a356c03c9efbff5600f1ce3f2850ac83f27f12fd4ee3d285","first_computed_at":"2026-06-04T01:09:18.920002Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-04T01:09:18.920002Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4iPXVXTGrsCGg8ZXcO8UGGHGYJIECBXt6h2MDHfORIr4rbpE/6dCZHcc+tkk2JuLNpZgFQSVay/VhoyGA98oDw==","signature_status":"signed_v1","signed_at":"2026-06-04T01:09:18.920726Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.04577","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:37baa13ea72b86e663e93fe19997d1acd2d763d68c37178d058fae2f892066fb","sha256:870007e1532c29c5fffa01355e63d121401eb4f1e1494a4a294b873dbe11947c"],"state_sha256":"fc8058cea23407adae6eb8a564735cac55b9324abf5b78c0a4d250cc4d749303"}