{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:YMHUPYZSDMAO4FP64RTTNLTZB2","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":"5cabe239fb309f206f147a19dcdaf281604080fb9c71e75ee1934eaffee07dcb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-26T11:47:43Z","title_canon_sha256":"0fa39564c7a8e8707af26d2df56666cc5d81757a833a86a3afbd7031a9fb7a1c"},"schema_version":"1.0","source":{"id":"1205.5876","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.5876","created_at":"2026-05-18T03:28:06Z"},{"alias_kind":"arxiv_version","alias_value":"1205.5876v3","created_at":"2026-05-18T03:28:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.5876","created_at":"2026-05-18T03:28:06Z"},{"alias_kind":"pith_short_12","alias_value":"YMHUPYZSDMAO","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"YMHUPYZSDMAO4FP6","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"YMHUPYZS","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:78d84a08e2176179ff3cf673d6bd98c5db837b3ae27078e8cdc01fe8cf579d86","target":"graph","created_at":"2026-05-18T03:28:06Z","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":"Let $\\D_{v,b,k}$ denote the family of all connected block designs with $v$ treatments and $b$ blocks of size $k$. Let $d\\in\\D_{v,b,k}$. The replication of a treatment is the number of times it appears in the blocks of $d$. The matrix $C(d)=R(d)-\\frac{1}{k}N(d)N(d)^\\top$ is called the information matrix of $d$ where $N(d)$ is the incidence matrix of $d$ and $R(d)$ is a diagonal matrix of the replications. Since $d$ is connected, $C(d)$ has $v-1$ nonzero eigenvalues $\\mu_1(d),...,\\mu_{v-1}(d)$. Let $\\D$ be the class of all binary designs of $\\D_{v,b,k}$. We prove that if there is a design $d^*\\i","authors_text":"E. Ghorbani, G. B. Khosrovshahi, M. R. Faghihi, S. Tat","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-26T11:47:43Z","title":"On optimality of designs with three distinct eigenvalues"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.5876","kind":"arxiv","version":3},"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:eedfd4c8094e883360fb78643386d84a77aa77f6cba1b9d072eb3e60eff3b0ad","target":"record","created_at":"2026-05-18T03:28:06Z","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":"5cabe239fb309f206f147a19dcdaf281604080fb9c71e75ee1934eaffee07dcb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-26T11:47:43Z","title_canon_sha256":"0fa39564c7a8e8707af26d2df56666cc5d81757a833a86a3afbd7031a9fb7a1c"},"schema_version":"1.0","source":{"id":"1205.5876","kind":"arxiv","version":3}},"canonical_sha256":"c30f47e3321b00ee15fee46736ae790e96e51f1c8d48cc3a67ff6f353b12701d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c30f47e3321b00ee15fee46736ae790e96e51f1c8d48cc3a67ff6f353b12701d","first_computed_at":"2026-05-18T03:28:06.064211Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:28:06.064211Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9c4bh1CX5k98ysDRvRFxfD6N0WoR5D4kPYiovkeA/gVWjzaKeQn//o8BQKYCPhyRO1mQdwYN2qZb68HJqZFvCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:28:06.065040Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.5876","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:eedfd4c8094e883360fb78643386d84a77aa77f6cba1b9d072eb3e60eff3b0ad","sha256:78d84a08e2176179ff3cf673d6bd98c5db837b3ae27078e8cdc01fe8cf579d86"],"state_sha256":"e51857d65ab389049bb1f5894b0bd53a73e07a65b99ec1d19f25e54517213238"}