{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:QE4VDFJKAO7DJWJY35XWTASP26","short_pith_number":"pith:QE4VDFJK","schema_version":"1.0","canonical_sha256":"813951952a03be34d938df6f69824fd7a28f053eec0f669ec7e7ae1803f580fe","source":{"kind":"arxiv","id":"2605.17171","version":1},"attestation_state":"computed","paper":{"title":"Higher Commutativity in Finite Groups, Rigidity, Extremal bounds, and Heisenberg-Type Families","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Robert Shwartz, Vadim E Levit","submitted_at":"2026-05-16T21:59:59Z","abstract_excerpt":"For a finite group $G$ and an integer $r\\ge 2$ let $$ P_r(G):=\\frac{|Hom(\\mathbb Z^r,G)|}{|G|^r}, $$ where $\\Hom(\\mathbb Z^r,G)$ is the set of pairwise commuting $r$-tuples in $G$. This paper studies rigidity and extremal behavior of the hierarchy $\\{P_r(G)\\}_{r\\ge2}$, together with a low-rank representation-theoretic / TQFT counting bridge. The first main direction is cyclic-index rigidity: for groups with an abelian normal subgroup $A$ and cyclic quotient of order $\\omega$, under a natural fixed-subgroup hypothesis we prove the exact all-rank formula $$ P_r(G)=\\frac{1}{\\omega^r}+\\left(1-\\fra"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.17171","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-05-16T21:59:59Z","cross_cats_sorted":[],"title_canon_sha256":"9ce0b22589b188460cb96f1760e57789415b311f1f0fba14538e1159fb5123b4","abstract_canon_sha256":"0b69eed8aa6543ae62345508f0df12edad8cdbff617072f24130756a1a8ec951"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:03:43.146005Z","signature_b64":"CoYCMJ42ZzYGJjr9hQODgGBiQHcndZPPB8I308umydLOxvw1KsbAwd59UDsYUK2e04KvMFmFJhUkd4xoz2PpAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"813951952a03be34d938df6f69824fd7a28f053eec0f669ec7e7ae1803f580fe","last_reissued_at":"2026-05-20T00:03:43.145101Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:03:43.145101Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Higher Commutativity in Finite Groups, Rigidity, Extremal bounds, and Heisenberg-Type Families","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Robert Shwartz, Vadim E Levit","submitted_at":"2026-05-16T21:59:59Z","abstract_excerpt":"For a finite group $G$ and an integer $r\\ge 2$ let $$ P_r(G):=\\frac{|Hom(\\mathbb Z^r,G)|}{|G|^r}, $$ where $\\Hom(\\mathbb Z^r,G)$ is the set of pairwise commuting $r$-tuples in $G$. This paper studies rigidity and extremal behavior of the hierarchy $\\{P_r(G)\\}_{r\\ge2}$, together with a low-rank representation-theoretic / TQFT counting bridge. The first main direction is cyclic-index rigidity: for groups with an abelian normal subgroup $A$ and cyclic quotient of order $\\omega$, under a natural fixed-subgroup hypothesis we prove the exact all-rank formula $$ P_r(G)=\\frac{1}{\\omega^r}+\\left(1-\\fra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.17171","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17171/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T22:33:23.752945Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T22:01:57.978121Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"be5752149eef81b438290a227e277ffa96f36c020b2ed99fb49b695b95130a45"},"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":"2605.17171","created_at":"2026-05-20T00:03:43.145255+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.17171v1","created_at":"2026-05-20T00:03:43.145255+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.17171","created_at":"2026-05-20T00:03:43.145255+00:00"},{"alias_kind":"pith_short_12","alias_value":"QE4VDFJKAO7D","created_at":"2026-05-20T00:03:43.145255+00:00"},{"alias_kind":"pith_short_16","alias_value":"QE4VDFJKAO7DJWJY","created_at":"2026-05-20T00:03:43.145255+00:00"},{"alias_kind":"pith_short_8","alias_value":"QE4VDFJK","created_at":"2026-05-20T00:03:43.145255+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/QE4VDFJKAO7DJWJY35XWTASP26","json":"https://pith.science/pith/QE4VDFJKAO7DJWJY35XWTASP26.json","graph_json":"https://pith.science/api/pith-number/QE4VDFJKAO7DJWJY35XWTASP26/graph.json","events_json":"https://pith.science/api/pith-number/QE4VDFJKAO7DJWJY35XWTASP26/events.json","paper":"https://pith.science/paper/QE4VDFJK"},"agent_actions":{"view_html":"https://pith.science/pith/QE4VDFJKAO7DJWJY35XWTASP26","download_json":"https://pith.science/pith/QE4VDFJKAO7DJWJY35XWTASP26.json","view_paper":"https://pith.science/paper/QE4VDFJK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.17171&json=true","fetch_graph":"https://pith.science/api/pith-number/QE4VDFJKAO7DJWJY35XWTASP26/graph.json","fetch_events":"https://pith.science/api/pith-number/QE4VDFJKAO7DJWJY35XWTASP26/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QE4VDFJKAO7DJWJY35XWTASP26/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QE4VDFJKAO7DJWJY35XWTASP26/action/storage_attestation","attest_author":"https://pith.science/pith/QE4VDFJKAO7DJWJY35XWTASP26/action/author_attestation","sign_citation":"https://pith.science/pith/QE4VDFJKAO7DJWJY35XWTASP26/action/citation_signature","submit_replication":"https://pith.science/pith/QE4VDFJKAO7DJWJY35XWTASP26/action/replication_record"}},"created_at":"2026-05-20T00:03:43.145255+00:00","updated_at":"2026-05-20T00:03:43.145255+00:00"}