{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:MV7ZNOEOE6LYMNHASW7AVM2CLT","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":"469724d099fc08266c520b6c818edf441de3efd8d19e80945d34d3b1fc37eaca","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-01T11:48:17Z","title_canon_sha256":"3c2db2429b8b922351eef98080dc9f8e1c66ac84f4448b8a402e7ae6df40601a"},"schema_version":"1.0","source":{"id":"2607.00825","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.00825","created_at":"2026-07-02T01:17:56Z"},{"alias_kind":"arxiv_version","alias_value":"2607.00825v1","created_at":"2026-07-02T01:17:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.00825","created_at":"2026-07-02T01:17:56Z"},{"alias_kind":"pith_short_12","alias_value":"MV7ZNOEOE6LY","created_at":"2026-07-02T01:17:56Z"},{"alias_kind":"pith_short_16","alias_value":"MV7ZNOEOE6LYMNHA","created_at":"2026-07-02T01:17:56Z"},{"alias_kind":"pith_short_8","alias_value":"MV7ZNOEO","created_at":"2026-07-02T01:17:56Z"}],"graph_snapshots":[{"event_id":"sha256:1a8f68172d8accecf5d9d407c631ae797f34290a3a83479cd495484db93d5c99","target":"graph","created_at":"2026-07-02T01:17:56Z","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/2607.00825/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We determine the minimal absolute value of a non-vanishing sum of $n$ fifth roots of unity chosen with repetition, and characterize the corresponding sums. As a function of $n$, the minimal absolute value is monotone non-increasing over congruence classes of $n$ modulo $5$ and its only jumps occur when $n=5F_m$, $n=L_m$, or $n=2L_m$, where $F_m$ and $L_m$ denote the $m$-th Fibonacci and Lucas numbers respectively. To prove our results we reduce the problem to a series of inequalities involving rational approximations of the golden ratio $\\varphi=(1+\\sqrt{5})/2$, the solutions of which can be c","authors_text":"Akihiro Munemasa, Guillermo N\\'u\\~nez Ponasso","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-01T11:48:17Z","title":"The Minimal Absolute Value of Sums of Fifth Roots of Unity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.00825","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:a94b2669d728527c4ae3c75eaf7e9813f0edb0ff3d3dea27a8c511b2b469411c","target":"record","created_at":"2026-07-02T01:17:56Z","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":"469724d099fc08266c520b6c818edf441de3efd8d19e80945d34d3b1fc37eaca","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-01T11:48:17Z","title_canon_sha256":"3c2db2429b8b922351eef98080dc9f8e1c66ac84f4448b8a402e7ae6df40601a"},"schema_version":"1.0","source":{"id":"2607.00825","kind":"arxiv","version":1}},"canonical_sha256":"657f96b88e27978634e095be0ab3425cfdb1ca16e91c04ad28adec98780a324c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"657f96b88e27978634e095be0ab3425cfdb1ca16e91c04ad28adec98780a324c","first_computed_at":"2026-07-02T01:17:56.377540Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-02T01:17:56.377540Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"58Eb2zgGMXVAcAYv8yjmO+0hjcwnpP8Bki9Rht8huoV+mwgbM+aH3/+3BvzveZ1c3GMiTGyRo05TQYJm1OG2Dw==","signature_status":"signed_v1","signed_at":"2026-07-02T01:17:56.377905Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.00825","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a94b2669d728527c4ae3c75eaf7e9813f0edb0ff3d3dea27a8c511b2b469411c","sha256:1a8f68172d8accecf5d9d407c631ae797f34290a3a83479cd495484db93d5c99"],"state_sha256":"6597c2a39762367d1429532d577fc9f396013e129b470c0adf3aaf1812d8811c"}