{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:WZGA24H2SSME5A2HR47HBUTBSV","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":"8024dce23d7cc09b2339370295d1f8fb1c037961821d3c77674941c956c837c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-27T21:32:56Z","title_canon_sha256":"64c9a77b11bd1c6a0a0ecdc2e95d70c2f26a0f2443a2097b3d6fefdc9420bfc2"},"schema_version":"1.0","source":{"id":"1408.6573","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.6573","created_at":"2026-05-18T02:44:01Z"},{"alias_kind":"arxiv_version","alias_value":"1408.6573v1","created_at":"2026-05-18T02:44:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.6573","created_at":"2026-05-18T02:44:01Z"},{"alias_kind":"pith_short_12","alias_value":"WZGA24H2SSME","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_16","alias_value":"WZGA24H2SSME5A2H","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_8","alias_value":"WZGA24H2","created_at":"2026-05-18T12:28:54Z"}],"graph_snapshots":[{"event_id":"sha256:7855f469c346c159f903163c6dc62929f1b83e9c4585450805edc033bd1803e6","target":"graph","created_at":"2026-05-18T02:44:01Z","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":"There are various results connecting ranks of incidence matrices of graphs and hypergraphs with their combinatorial structure. Here, we consider the generalized incidence matrix $N_2$ (defined by inclusion of pairs in edges) for one natural class of hypergraphs: the triple systems with index three. Such systems with nonsingular $N_2$ (over the rationals) appear to be quite rare, yet they can be constructed with PBD closure. In fact, a range of ranks near $\\binom{v}{2}$ is obtained for large orders $v$.","authors_text":"Kseniya Garaschuk, Peter J. Dukes","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-27T21:32:56Z","title":"Threefold triple systems with nonsingular $N_2$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.6573","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:c7562a710ed78550ac97bdd67d6a826417cd6c9162f22e9025068d297212400b","target":"record","created_at":"2026-05-18T02:44:01Z","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":"8024dce23d7cc09b2339370295d1f8fb1c037961821d3c77674941c956c837c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-27T21:32:56Z","title_canon_sha256":"64c9a77b11bd1c6a0a0ecdc2e95d70c2f26a0f2443a2097b3d6fefdc9420bfc2"},"schema_version":"1.0","source":{"id":"1408.6573","kind":"arxiv","version":1}},"canonical_sha256":"b64c0d70fa94984e83478f3e70d261955b5326bfc51f258bf84257a78625b92a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b64c0d70fa94984e83478f3e70d261955b5326bfc51f258bf84257a78625b92a","first_computed_at":"2026-05-18T02:44:01.300688Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:01.300688Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CJa/PJBG4HuAQAtStSBwnzEVAh/gwdP9uHB14HGjpTznksYu8hNxvrJm9MnOG8CPOV2fJyNgGI2BKt+KKxM8CA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:01.301094Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.6573","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c7562a710ed78550ac97bdd67d6a826417cd6c9162f22e9025068d297212400b","sha256:7855f469c346c159f903163c6dc62929f1b83e9c4585450805edc033bd1803e6"],"state_sha256":"ed7960190b1f1b1de59d285f6ae5354805ce96347b9fbed47a5f703e75e681d0"}