{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:G5RY224RWJXAYMORZEIK335UFD","short_pith_number":"pith:G5RY224R","canonical_record":{"source":{"id":"1608.08508","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-30T15:36:43Z","cross_cats_sorted":[],"title_canon_sha256":"dc9e5ea4ee89456fc155e06fe0081a9152d15c0ed84f9bfe71f2a6b47d36d189","abstract_canon_sha256":"602c37089f0cf1a51c65a79182be6556e1a41eadd63f75082d80a9ca4dd80db5"},"schema_version":"1.0"},"canonical_sha256":"37638d6b91b26e0c31d1c910adefb428e772988f8ef6343986c7525fe8d47b1c","source":{"kind":"arxiv","id":"1608.08508","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.08508","created_at":"2026-05-18T01:07:17Z"},{"alias_kind":"arxiv_version","alias_value":"1608.08508v1","created_at":"2026-05-18T01:07:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.08508","created_at":"2026-05-18T01:07:17Z"},{"alias_kind":"pith_short_12","alias_value":"G5RY224RWJXA","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"G5RY224RWJXAYMOR","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"G5RY224R","created_at":"2026-05-18T12:30:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:G5RY224RWJXAYMORZEIK335UFD","target":"record","payload":{"canonical_record":{"source":{"id":"1608.08508","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-30T15:36:43Z","cross_cats_sorted":[],"title_canon_sha256":"dc9e5ea4ee89456fc155e06fe0081a9152d15c0ed84f9bfe71f2a6b47d36d189","abstract_canon_sha256":"602c37089f0cf1a51c65a79182be6556e1a41eadd63f75082d80a9ca4dd80db5"},"schema_version":"1.0"},"canonical_sha256":"37638d6b91b26e0c31d1c910adefb428e772988f8ef6343986c7525fe8d47b1c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:17.955065Z","signature_b64":"gAKr29jK7fMQMSns7/aaVB9qbD2+KD8YcBIhQm6sn3HnBTbSzjzTNyvNGo/kE1LuVkD85TP4xchHvp1vNvJ9Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"37638d6b91b26e0c31d1c910adefb428e772988f8ef6343986c7525fe8d47b1c","last_reissued_at":"2026-05-18T01:07:17.954527Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:17.954527Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.08508","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:07:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L6PmCfnLGyKNru80KkhFK6IjqImUEuVIW44LNyL1y4/h5yWaEvHl2NVX0A6WxWjGQsmBWU96x+lda4+69atVBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T12:11:39.315682Z"},"content_sha256":"4ccf4379adc25373b7f43ae4d9e43fbb390da9781625f7ccfe9f06681f594e21","schema_version":"1.0","event_id":"sha256:4ccf4379adc25373b7f43ae4d9e43fbb390da9781625f7ccfe9f06681f594e21"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:G5RY224RWJXAYMORZEIK335UFD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The number of ideals of $\\mathbb{Z}[x]$ containing $x(x-\\alpha)(x-\\beta)$ with given index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mitsugu Hirasaka, Semin Oh","submitted_at":"2016-08-30T15:36:43Z","abstract_excerpt":"It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let $B$ denote an integral square matrix and $\\langle B \\rangle$ denote the subring of the full matrix ring generated by $B$. Then $\\langle B \\rangle$ is a free $\\mathbb{Z}$-module of finite rank, which guarantees that there are only finitely many ideals of $\\langle B \\rangle$ with given finite index. Thus, the formal Dirichlet series $\\zeta_{\\langle B \\rangle}(s)=\\sum_{n\\geq 1}a_n n^{-s}$ is well-defined where $a_n$ is the number of ideals of $\\langle B \\rangl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08508","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:07:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q9bZLPCzuXoeMZaUfDcuUmqIh/8E6DX0AoC4Zi7kPrTfNQBqRWm2+ht7Yd7/Tw2lqfW/thYAN0nnt+sJiIaNAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T12:11:39.316030Z"},"content_sha256":"5b0b51d20cc7bd583cc51ea72aab43d37e45120d11e34c6db440920b9c8b68f8","schema_version":"1.0","event_id":"sha256:5b0b51d20cc7bd583cc51ea72aab43d37e45120d11e34c6db440920b9c8b68f8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G5RY224RWJXAYMORZEIK335UFD/bundle.json","state_url":"https://pith.science/pith/G5RY224RWJXAYMORZEIK335UFD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G5RY224RWJXAYMORZEIK335UFD/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T12:11:39Z","links":{"resolver":"https://pith.science/pith/G5RY224RWJXAYMORZEIK335UFD","bundle":"https://pith.science/pith/G5RY224RWJXAYMORZEIK335UFD/bundle.json","state":"https://pith.science/pith/G5RY224RWJXAYMORZEIK335UFD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G5RY224RWJXAYMORZEIK335UFD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:G5RY224RWJXAYMORZEIK335UFD","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":"602c37089f0cf1a51c65a79182be6556e1a41eadd63f75082d80a9ca4dd80db5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-30T15:36:43Z","title_canon_sha256":"dc9e5ea4ee89456fc155e06fe0081a9152d15c0ed84f9bfe71f2a6b47d36d189"},"schema_version":"1.0","source":{"id":"1608.08508","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.08508","created_at":"2026-05-18T01:07:17Z"},{"alias_kind":"arxiv_version","alias_value":"1608.08508v1","created_at":"2026-05-18T01:07:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.08508","created_at":"2026-05-18T01:07:17Z"},{"alias_kind":"pith_short_12","alias_value":"G5RY224RWJXA","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"G5RY224RWJXAYMOR","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"G5RY224R","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:5b0b51d20cc7bd583cc51ea72aab43d37e45120d11e34c6db440920b9c8b68f8","target":"graph","created_at":"2026-05-18T01:07:17Z","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":"It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let $B$ denote an integral square matrix and $\\langle B \\rangle$ denote the subring of the full matrix ring generated by $B$. Then $\\langle B \\rangle$ is a free $\\mathbb{Z}$-module of finite rank, which guarantees that there are only finitely many ideals of $\\langle B \\rangle$ with given finite index. Thus, the formal Dirichlet series $\\zeta_{\\langle B \\rangle}(s)=\\sum_{n\\geq 1}a_n n^{-s}$ is well-defined where $a_n$ is the number of ideals of $\\langle B \\rangl","authors_text":"Mitsugu Hirasaka, Semin Oh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-30T15:36:43Z","title":"The number of ideals of $\\mathbb{Z}[x]$ containing $x(x-\\alpha)(x-\\beta)$ with given index"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08508","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":"sha256:4ccf4379adc25373b7f43ae4d9e43fbb390da9781625f7ccfe9f06681f594e21","target":"record","created_at":"2026-05-18T01:07:17Z","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":"602c37089f0cf1a51c65a79182be6556e1a41eadd63f75082d80a9ca4dd80db5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-30T15:36:43Z","title_canon_sha256":"dc9e5ea4ee89456fc155e06fe0081a9152d15c0ed84f9bfe71f2a6b47d36d189"},"schema_version":"1.0","source":{"id":"1608.08508","kind":"arxiv","version":1}},"canonical_sha256":"37638d6b91b26e0c31d1c910adefb428e772988f8ef6343986c7525fe8d47b1c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"37638d6b91b26e0c31d1c910adefb428e772988f8ef6343986c7525fe8d47b1c","first_computed_at":"2026-05-18T01:07:17.954527Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:07:17.954527Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gAKr29jK7fMQMSns7/aaVB9qbD2+KD8YcBIhQm6sn3HnBTbSzjzTNyvNGo/kE1LuVkD85TP4xchHvp1vNvJ9Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:07:17.955065Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.08508","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4ccf4379adc25373b7f43ae4d9e43fbb390da9781625f7ccfe9f06681f594e21","sha256:5b0b51d20cc7bd583cc51ea72aab43d37e45120d11e34c6db440920b9c8b68f8"],"state_sha256":"2fa0efa92112fb6579699ad699fdcaa956930a6fd09c52db99e108732b250d0d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dUiyAer+7Wemtaqvqk8rvsD237iId3MwIwTlntaptwMLUhFjwn5btOXXYsvFhGfvHgHzFZJmEWFXcZ2Dv48nAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T12:11:39.318020Z","bundle_sha256":"464d1e809a9d209eb18e77fd81035c91f4315511e1bf32d166698de175397616"}}