{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:7HZARSAVDQA2EETONUZIF5JNVI","short_pith_number":"pith:7HZARSAV","schema_version":"1.0","canonical_sha256":"f9f208c8151c01a2126e6d3282f52daa0efa7c7c4cf5be63ac316b2d4051a199","source":{"kind":"arxiv","id":"2605.20976","version":1},"attestation_state":"computed","paper":{"title":"A compensation theorem for the Sylow-integral invariant and counterexamples to an \\texorpdfstring{$A_5$}{A5}-characterization conjecture","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Yaoran Yang, Yutong Zhang","submitted_at":"2026-05-20T10:06:19Z","abstract_excerpt":"Let \\(\\nu_p(G)\\) be the number of Sylow \\(p\\)-subgroups of a finite group \\(G\\), let \\(\\sigma_p(G)\\) be their common order, and set \\[\n  \\gamma(G)=\\int_0^1\\sum_{p\\in\\pi(G)}\\nu_p(G)x^{\\sigma_p(G)}\\,dx\n  =\\sum_{p\\in\\pi(G)}\\frac{\\nu_p(G)}{\\sigma_p(G)+1}. \\] A recent conjectural extension of the simple-group theorem for this invariant asserted that a nonsolvable finite group has \\(\\gamma(G)=9/2\\) precisely when \\(G\\cong A_5\\). We disprove this assertion by a direct and verifiable construction. More generally, we prove an exact direct-product compensation formula for \\(A_5\\) with an arbitrary nilpo"},"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":"2605.20976","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.GR","submitted_at":"2026-05-20T10:06:19Z","cross_cats_sorted":[],"title_canon_sha256":"5cb2ddb76baed0b8f15fef323d7060fdc77f0e1f16ae4bf199e6c30dae543718","abstract_canon_sha256":"0d310ded6f7f7a3915a2157990038a81898ed9cd0fec81f2a81cd49feedc0c50"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-21T01:05:30.706176Z","signature_b64":"1ceU0HTRaIeCvYsFkQZsOcdHr8ac2KieVgmUMTLRY5X1LSs8EKuhRmpY57+KhHCXd/BnJe/B6fs3IZy8F4ALCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f9f208c8151c01a2126e6d3282f52daa0efa7c7c4cf5be63ac316b2d4051a199","last_reissued_at":"2026-05-21T01:05:30.705411Z","signature_status":"signed_v1","first_computed_at":"2026-05-21T01:05:30.705411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A compensation theorem for the Sylow-integral invariant and counterexamples to an \\texorpdfstring{$A_5$}{A5}-characterization conjecture","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Yaoran Yang, Yutong Zhang","submitted_at":"2026-05-20T10:06:19Z","abstract_excerpt":"Let \\(\\nu_p(G)\\) be the number of Sylow \\(p\\)-subgroups of a finite group \\(G\\), let \\(\\sigma_p(G)\\) be their common order, and set \\[\n  \\gamma(G)=\\int_0^1\\sum_{p\\in\\pi(G)}\\nu_p(G)x^{\\sigma_p(G)}\\,dx\n  =\\sum_{p\\in\\pi(G)}\\frac{\\nu_p(G)}{\\sigma_p(G)+1}. \\] A recent conjectural extension of the simple-group theorem for this invariant asserted that a nonsolvable finite group has \\(\\gamma(G)=9/2\\) precisely when \\(G\\cong A_5\\). We disprove this assertion by a direct and verifiable construction. More generally, we prove an exact direct-product compensation formula for \\(A_5\\) with an arbitrary nilpo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20976","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/2605.20976/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2605.20976","created_at":"2026-05-21T01:05:30.705528+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.20976v1","created_at":"2026-05-21T01:05:30.705528+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.20976","created_at":"2026-05-21T01:05:30.705528+00:00"},{"alias_kind":"pith_short_12","alias_value":"7HZARSAVDQA2","created_at":"2026-05-21T01:05:30.705528+00:00"},{"alias_kind":"pith_short_16","alias_value":"7HZARSAVDQA2EETO","created_at":"2026-05-21T01:05:30.705528+00:00"},{"alias_kind":"pith_short_8","alias_value":"7HZARSAV","created_at":"2026-05-21T01:05:30.705528+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/7HZARSAVDQA2EETONUZIF5JNVI","json":"https://pith.science/pith/7HZARSAVDQA2EETONUZIF5JNVI.json","graph_json":"https://pith.science/api/pith-number/7HZARSAVDQA2EETONUZIF5JNVI/graph.json","events_json":"https://pith.science/api/pith-number/7HZARSAVDQA2EETONUZIF5JNVI/events.json","paper":"https://pith.science/paper/7HZARSAV"},"agent_actions":{"view_html":"https://pith.science/pith/7HZARSAVDQA2EETONUZIF5JNVI","download_json":"https://pith.science/pith/7HZARSAVDQA2EETONUZIF5JNVI.json","view_paper":"https://pith.science/paper/7HZARSAV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.20976&json=true","fetch_graph":"https://pith.science/api/pith-number/7HZARSAVDQA2EETONUZIF5JNVI/graph.json","fetch_events":"https://pith.science/api/pith-number/7HZARSAVDQA2EETONUZIF5JNVI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7HZARSAVDQA2EETONUZIF5JNVI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7HZARSAVDQA2EETONUZIF5JNVI/action/storage_attestation","attest_author":"https://pith.science/pith/7HZARSAVDQA2EETONUZIF5JNVI/action/author_attestation","sign_citation":"https://pith.science/pith/7HZARSAVDQA2EETONUZIF5JNVI/action/citation_signature","submit_replication":"https://pith.science/pith/7HZARSAVDQA2EETONUZIF5JNVI/action/replication_record"}},"created_at":"2026-05-21T01:05:30.705528+00:00","updated_at":"2026-05-21T01:05:30.705528+00:00"}