{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:VI4E5GGVDNXBETATTKNZQPWV2M","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":"1a256d34af21aec9c664c52a5fad777e3989eaaf65d67881c813d2319a02a1df","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-11T16:38:17Z","title_canon_sha256":"07e7172c015c03857710ec8da8fe1a92c306636c611b565bbe5419c5993a6285"},"schema_version":"1.0","source":{"id":"1807.04238","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.04238","created_at":"2026-05-17T23:56:29Z"},{"alias_kind":"arxiv_version","alias_value":"1807.04238v2","created_at":"2026-05-17T23:56:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.04238","created_at":"2026-05-17T23:56:29Z"},{"alias_kind":"pith_short_12","alias_value":"VI4E5GGVDNXB","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"VI4E5GGVDNXBETAT","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"VI4E5GGV","created_at":"2026-05-18T12:32:59Z"}],"graph_snapshots":[{"event_id":"sha256:406a96addabb48d49b5bd1eca0a2a2c3a9711ec7865b29a7d98da45ceb5690d5","target":"graph","created_at":"2026-05-17T23:56:29Z","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":"A Butson Hadamard matrix $H$ has entries in the kth roots of unity, and satisfies the matrix equation $HH^{\\ast} = nI_{n}$. We write $\\mathrm{BH}(n, k)$ for the set of such matrices. A complete morphism of Butson matrices is a map $\\mathrm{BH}(n, k) \\rightarrow \\mathrm{BH}(m, \\ell)$. In this paper, we develop a technique for controlling the spectra of certain Hadamard matrices. For each integer $t$, we construct a real Hadamard matrix $H_{t}$ of order $n_{t} = 2^{2^{t-1}-1}$ such that the minimal polynomial of $\\frac{1}{\\sqrt{n_{t}}}H_{t}$ is the cyclotomic polynomial $\\Phi_{2^{t+1}}(x)$. Such","authors_text":"Eric Swartz, Padraig O Cathain, Ronan Egan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-11T16:38:17Z","title":"Spectra of Hadamard matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.04238","kind":"arxiv","version":2},"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:a155cb10810222d4d27f10036e34065f2b6c07b5e9c92aa3e6d25802b2df2f8c","target":"record","created_at":"2026-05-17T23:56:29Z","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":"1a256d34af21aec9c664c52a5fad777e3989eaaf65d67881c813d2319a02a1df","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-07-11T16:38:17Z","title_canon_sha256":"07e7172c015c03857710ec8da8fe1a92c306636c611b565bbe5419c5993a6285"},"schema_version":"1.0","source":{"id":"1807.04238","kind":"arxiv","version":2}},"canonical_sha256":"aa384e98d51b6e124c139a9b983ed5d33ac43695b2f1a8ea7f4ad7f8ee516891","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aa384e98d51b6e124c139a9b983ed5d33ac43695b2f1a8ea7f4ad7f8ee516891","first_computed_at":"2026-05-17T23:56:29.656907Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:56:29.656907Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MzJ+z3LNdDTQEMeeeRBl1uyjBmdpROB3SkRn7Tmu9ZuI5qtfmOcYBvLOn8wlHpb4llSpFW7lnCIupFs5/RerCQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:56:29.657263Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.04238","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a155cb10810222d4d27f10036e34065f2b6c07b5e9c92aa3e6d25802b2df2f8c","sha256:406a96addabb48d49b5bd1eca0a2a2c3a9711ec7865b29a7d98da45ceb5690d5"],"state_sha256":"4a89419226be458fb76e5060b5351f9409381d9e171b96778fd442e01547f06b"}