{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ALQIQ7Y7IIQETUHX6TH23MCCGC","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":"df326febf47945758a37e0c4c28756aed5f69425a635e5c423af17fe9716941a","cross_cats_sorted":["math.SP"],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-23T04:01:11Z","title_canon_sha256":"61866e63d30e63ba86e0f2cd85c8dc6494eb620eb53de8dacd1aa60ef1a8d193"},"schema_version":"1.0","source":{"id":"2607.20907","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.20907","created_at":"2026-07-24T00:23:41Z"},{"alias_kind":"arxiv_version","alias_value":"2607.20907v1","created_at":"2026-07-24T00:23:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.20907","created_at":"2026-07-24T00:23:41Z"},{"alias_kind":"pith_short_12","alias_value":"ALQIQ7Y7IIQE","created_at":"2026-07-24T00:23:41Z"},{"alias_kind":"pith_short_16","alias_value":"ALQIQ7Y7IIQETUHX","created_at":"2026-07-24T00:23:41Z"},{"alias_kind":"pith_short_8","alias_value":"ALQIQ7Y7","created_at":"2026-07-24T00:23:41Z"}],"graph_snapshots":[{"event_id":"sha256:ad593caef67898247b3f1e3abba36e834db87ea195131ffbe764f5ace79e32df","target":"graph","created_at":"2026-07-24T00:23:41Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.20907/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For a multiset $S$ on the cyclic group $\\mathbb{Z}/N\\mathbb{Z}$, we study finite sums of cosine functions of rational angles associated to $S$ by translating them as evaluations of elements in the group ring $\\mathbb{Z}[\\mathbb{Z}/N\\mathbb{Z}]$. Using vanishing sums of roots of unity, especially the Lam-Leung theory, we obtain criteria for the vanishing of the cosine sums under some conditions, and prove a small-weight Fourier rigidity. We then apply these algebraic results to cyclic Cayley graphs, deriving the zero-eigenvalue criteria, multiplicity bounds for nonzero eigenvalues in the small-","authors_text":"Qin Xue","cross_cats":["math.SP"],"headline":"","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-23T04:01:11Z","title":"On structured cosine sums and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.20907","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":"sha256:89fd608a3ee94ad928ef27a32446152ed664a35535c6430b3adf12643c6c1a9a","target":"record","created_at":"2026-07-24T00:23:41Z","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":"df326febf47945758a37e0c4c28756aed5f69425a635e5c423af17fe9716941a","cross_cats_sorted":["math.SP"],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-23T04:01:11Z","title_canon_sha256":"61866e63d30e63ba86e0f2cd85c8dc6494eb620eb53de8dacd1aa60ef1a8d193"},"schema_version":"1.0","source":{"id":"2607.20907","kind":"arxiv","version":1}},"canonical_sha256":"02e0887f1f422049d0f7f4cfadb0423082be39208c36beb3ea79c6670e3bfd91","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"02e0887f1f422049d0f7f4cfadb0423082be39208c36beb3ea79c6670e3bfd91","first_computed_at":"2026-07-24T00:23:41.166057Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-24T00:23:41.166057Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1YDnamAXor281B8rNKy0T4erXDGB/KJOeFwGVSYW8Nz3qNdHDkKb1ciJpnOriTdmvbh7UcqSYAsKPxV/8OIkAg==","signature_status":"signed_v1","signed_at":"2026-07-24T00:23:41.166989Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.20907","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89fd608a3ee94ad928ef27a32446152ed664a35535c6430b3adf12643c6c1a9a","sha256:ad593caef67898247b3f1e3abba36e834db87ea195131ffbe764f5ace79e32df"],"state_sha256":"378acdb5be1b7d1c9342c0cc2d4d98275442ee130b2c2e80318bd1f134ce2299"}