{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:6T7VUQYDVGDKWEHLJUIFCFVUPC","short_pith_number":"pith:6T7VUQYD","schema_version":"1.0","canonical_sha256":"f4ff5a4303a986ab10eb4d105116b478af2409e8586a5f82f709399dc5fd59ac","source":{"kind":"arxiv","id":"1509.00327","version":2},"attestation_state":"computed","paper":{"title":"On the critical group of the missing Moore graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Joshua E. Ducey","submitted_at":"2015-09-01T14:44:05Z","abstract_excerpt":"We consider the critical group of a hypothetical Moore graph of diameter $2$ and valency $57$. Determining this group is equivalent to finding the Smith normal form of the Laplacian matrix of such a graph. We show that all of the Sylow $p$-subgroups of the critical group must be elementary abelian with the exception of $p = 5$. We prove that the $5$-rank of the Laplacian matrix determines the critical group up to two possibilities."},"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":"1509.00327","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-01T14:44:05Z","cross_cats_sorted":[],"title_canon_sha256":"b9ab03256c4ec0a1c2027a1f2ede4c8fbe9c8a36b18cb9cf508897ec6ba5a386","abstract_canon_sha256":"a439a250f609cb3c8f67f0ad4a10ad0b38a8ea4b52865a0b2785fd244281c41c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:06.432270Z","signature_b64":"L9xKVeoZn4O2nOK8Z4QBcWU9znNNZ8wOoiVG3HpoXkCs0nvH5v3OC60UtCBxI2bFZm5xP8D+CBz5Cfl71QCACg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f4ff5a4303a986ab10eb4d105116b478af2409e8586a5f82f709399dc5fd59ac","last_reissued_at":"2026-05-18T01:03:06.431610Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:06.431610Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the critical group of the missing Moore graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Joshua E. Ducey","submitted_at":"2015-09-01T14:44:05Z","abstract_excerpt":"We consider the critical group of a hypothetical Moore graph of diameter $2$ and valency $57$. Determining this group is equivalent to finding the Smith normal form of the Laplacian matrix of such a graph. We show that all of the Sylow $p$-subgroups of the critical group must be elementary abelian with the exception of $p = 5$. We prove that the $5$-rank of the Laplacian matrix determines the critical group up to two possibilities."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.00327","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1509.00327","created_at":"2026-05-18T01:03:06.431712+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.00327v2","created_at":"2026-05-18T01:03:06.431712+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.00327","created_at":"2026-05-18T01:03:06.431712+00:00"},{"alias_kind":"pith_short_12","alias_value":"6T7VUQYDVGDK","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_16","alias_value":"6T7VUQYDVGDKWEHL","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_8","alias_value":"6T7VUQYD","created_at":"2026-05-18T12:29:07.941421+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/6T7VUQYDVGDKWEHLJUIFCFVUPC","json":"https://pith.science/pith/6T7VUQYDVGDKWEHLJUIFCFVUPC.json","graph_json":"https://pith.science/api/pith-number/6T7VUQYDVGDKWEHLJUIFCFVUPC/graph.json","events_json":"https://pith.science/api/pith-number/6T7VUQYDVGDKWEHLJUIFCFVUPC/events.json","paper":"https://pith.science/paper/6T7VUQYD"},"agent_actions":{"view_html":"https://pith.science/pith/6T7VUQYDVGDKWEHLJUIFCFVUPC","download_json":"https://pith.science/pith/6T7VUQYDVGDKWEHLJUIFCFVUPC.json","view_paper":"https://pith.science/paper/6T7VUQYD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.00327&json=true","fetch_graph":"https://pith.science/api/pith-number/6T7VUQYDVGDKWEHLJUIFCFVUPC/graph.json","fetch_events":"https://pith.science/api/pith-number/6T7VUQYDVGDKWEHLJUIFCFVUPC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6T7VUQYDVGDKWEHLJUIFCFVUPC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6T7VUQYDVGDKWEHLJUIFCFVUPC/action/storage_attestation","attest_author":"https://pith.science/pith/6T7VUQYDVGDKWEHLJUIFCFVUPC/action/author_attestation","sign_citation":"https://pith.science/pith/6T7VUQYDVGDKWEHLJUIFCFVUPC/action/citation_signature","submit_replication":"https://pith.science/pith/6T7VUQYDVGDKWEHLJUIFCFVUPC/action/replication_record"}},"created_at":"2026-05-18T01:03:06.431712+00:00","updated_at":"2026-05-18T01:03:06.431712+00:00"}