{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:ZHOZBTY6YLWPIABZS4GF2TRZWX","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":"b30d12b36e0366e047940597206b25f6b3e5af39dd9ad38c003c0af8897927a3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-05-13T18:29:05Z","title_canon_sha256":"2dc7f4ef010288009bf9dcc807992df2228d34065a82b58c350ee58fe7d934ab"},"schema_version":"1.0","source":{"id":"1105.2799","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.2799","created_at":"2026-05-18T02:37:52Z"},{"alias_kind":"arxiv_version","alias_value":"1105.2799v1","created_at":"2026-05-18T02:37:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.2799","created_at":"2026-05-18T02:37:52Z"},{"alias_kind":"pith_short_12","alias_value":"ZHOZBTY6YLWP","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"ZHOZBTY6YLWPIABZ","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"ZHOZBTY6","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:1c2853c5535d395786bbf97e6fa7cb48e7aa9a55288a0962d68733fc86463179","target":"graph","created_at":"2026-05-18T02:37:52Z","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":"If $G$ is a solvable group, we take $\\Delta (G)$ to be the character degree graph for $G$ with primes as vertices. We prove that if $\\Delta (G)$ is a square, then $G$ must be a direct product.","authors_text":"Mark L. Lewis, Qingyun Meng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-05-13T18:29:05Z","title":"Square character degree graphs yield direct products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.2799","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":"sha256:4e13baf8a48c55af24261e91da6c71b3a14e40012400cc73b74082b6bfac13d7","target":"record","created_at":"2026-05-18T02:37:52Z","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":"b30d12b36e0366e047940597206b25f6b3e5af39dd9ad38c003c0af8897927a3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-05-13T18:29:05Z","title_canon_sha256":"2dc7f4ef010288009bf9dcc807992df2228d34065a82b58c350ee58fe7d934ab"},"schema_version":"1.0","source":{"id":"1105.2799","kind":"arxiv","version":1}},"canonical_sha256":"c9dd90cf1ec2ecf40039970c5d4e39b5f771aed092ba1a8b16f3352136328969","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c9dd90cf1ec2ecf40039970c5d4e39b5f771aed092ba1a8b16f3352136328969","first_computed_at":"2026-05-18T02:37:52.358219Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:37:52.358219Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"X+xN/ogvC+weBsnSm0f1HgM0TvFKkR1s/wvekjo1UMfj/uNO5JBkUCJhjX4Q1bfdfkBn5AKWx4A+iiTOMK06Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:37:52.358849Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.2799","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4e13baf8a48c55af24261e91da6c71b3a14e40012400cc73b74082b6bfac13d7","sha256:1c2853c5535d395786bbf97e6fa7cb48e7aa9a55288a0962d68733fc86463179"],"state_sha256":"d3f4b7de27dac8c53c00edc30435313ef61b1325f84d0562cc2eec8040a587a7"}