{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:BW6QCJUFFDVJWETQ7DWTENE72F","short_pith_number":"pith:BW6QCJUF","canonical_record":{"source":{"id":"1504.06652","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-04-24T22:09:01Z","cross_cats_sorted":[],"title_canon_sha256":"d6400ef2339bc46adbc37ebabfa510e8c52aaed08eceb15699d33389db23518a","abstract_canon_sha256":"8bf892ed21399ca8d3b8306dc09dd55caba565800e9853638ca622e83ee8e11b"},"schema_version":"1.0"},"canonical_sha256":"0dbd01268528ea9b1270f8ed32349fd1432c64a89785d2d61da98c920c2db1ba","source":{"kind":"arxiv","id":"1504.06652","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.06652","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"arxiv_version","alias_value":"1504.06652v1","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.06652","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"pith_short_12","alias_value":"BW6QCJUFFDVJ","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"BW6QCJUFFDVJWETQ","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"BW6QCJUF","created_at":"2026-05-18T12:29:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:BW6QCJUFFDVJWETQ7DWTENE72F","target":"record","payload":{"canonical_record":{"source":{"id":"1504.06652","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-04-24T22:09:01Z","cross_cats_sorted":[],"title_canon_sha256":"d6400ef2339bc46adbc37ebabfa510e8c52aaed08eceb15699d33389db23518a","abstract_canon_sha256":"8bf892ed21399ca8d3b8306dc09dd55caba565800e9853638ca622e83ee8e11b"},"schema_version":"1.0"},"canonical_sha256":"0dbd01268528ea9b1270f8ed32349fd1432c64a89785d2d61da98c920c2db1ba","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:17:52.868711Z","signature_b64":"21MstU5KTNBM/fUuvvMpHhAy6AlQYz0JhIH3li7IPYi480Ill9o72D+fae7w0EiLo+pgwZs0ohwaW/9k68fPBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0dbd01268528ea9b1270f8ed32349fd1432c64a89785d2d61da98c920c2db1ba","last_reissued_at":"2026-05-18T02:17:52.868219Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:17:52.868219Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.06652","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":"2026-05-18T02:17:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YR6oyvBcPg6SZALTVqlQtL2a7frUbQ0EUuUgki6vYVB3SA5wSYfYzFoIADBzB5Z7rJGBUIUQkCSNsdwPe+X9CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T02:07:44.975898Z"},"content_sha256":"b10f7f7a3dac0a3d5d4e471cef424b653c104c930bb8e953a0e4cda822c76541","schema_version":"1.0","event_id":"sha256:b10f7f7a3dac0a3d5d4e471cef424b653c104c930bb8e953a0e4cda822c76541"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:BW6QCJUFFDVJWETQ7DWTENE72F","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Linear independence of trigonometric numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Arno Berger","submitted_at":"2015-04-24T22:09:01Z","abstract_excerpt":"Given any two rational numbers $r_1$ and $r_2$, a necessary and sufficient condition is established for the three numbers $1$, $\\cos (\\pi r_1)$, and $\\cos (\\pi r_2)$ to be rationally independent. Extending a classical fact sometimes attributed to I. Niven, the result even yields linear independence over larger number fields. The tools employed in the proof are applicable also in the case of more than two trigonometric numbers. As an application, a complete classification is given of all planar triangles with rational angles and side lengths each containing at most one square root. Such a class"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.06652","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":"2026-05-18T02:17:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fHkfuBbZqDQ5nzR0UU+JGWSWgP/cNs0IorwJZZyfhR0FGSNBtCfmKHwOso0KuBI5/bl3Edjyt97R0NZAtwz3Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T02:07:44.976368Z"},"content_sha256":"52f36ddf39fa9a7da237992f4cad119ca14a5fd5531e4050a4a2cdd6d3b437df","schema_version":"1.0","event_id":"sha256:52f36ddf39fa9a7da237992f4cad119ca14a5fd5531e4050a4a2cdd6d3b437df"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BW6QCJUFFDVJWETQ7DWTENE72F/bundle.json","state_url":"https://pith.science/pith/BW6QCJUFFDVJWETQ7DWTENE72F/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BW6QCJUFFDVJWETQ7DWTENE72F/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":"2026-06-09T02:07:44Z","links":{"resolver":"https://pith.science/pith/BW6QCJUFFDVJWETQ7DWTENE72F","bundle":"https://pith.science/pith/BW6QCJUFFDVJWETQ7DWTENE72F/bundle.json","state":"https://pith.science/pith/BW6QCJUFFDVJWETQ7DWTENE72F/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BW6QCJUFFDVJWETQ7DWTENE72F/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:BW6QCJUFFDVJWETQ7DWTENE72F","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":"8bf892ed21399ca8d3b8306dc09dd55caba565800e9853638ca622e83ee8e11b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-04-24T22:09:01Z","title_canon_sha256":"d6400ef2339bc46adbc37ebabfa510e8c52aaed08eceb15699d33389db23518a"},"schema_version":"1.0","source":{"id":"1504.06652","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.06652","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"arxiv_version","alias_value":"1504.06652v1","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.06652","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"pith_short_12","alias_value":"BW6QCJUFFDVJ","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"BW6QCJUFFDVJWETQ","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"BW6QCJUF","created_at":"2026-05-18T12:29:14Z"}],"graph_snapshots":[{"event_id":"sha256:52f36ddf39fa9a7da237992f4cad119ca14a5fd5531e4050a4a2cdd6d3b437df","target":"graph","created_at":"2026-05-18T02:17:52Z","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":"Given any two rational numbers $r_1$ and $r_2$, a necessary and sufficient condition is established for the three numbers $1$, $\\cos (\\pi r_1)$, and $\\cos (\\pi r_2)$ to be rationally independent. Extending a classical fact sometimes attributed to I. Niven, the result even yields linear independence over larger number fields. The tools employed in the proof are applicable also in the case of more than two trigonometric numbers. As an application, a complete classification is given of all planar triangles with rational angles and side lengths each containing at most one square root. Such a class","authors_text":"Arno Berger","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-04-24T22:09:01Z","title":"Linear independence of trigonometric numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.06652","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:b10f7f7a3dac0a3d5d4e471cef424b653c104c930bb8e953a0e4cda822c76541","target":"record","created_at":"2026-05-18T02:17:52Z","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":"8bf892ed21399ca8d3b8306dc09dd55caba565800e9853638ca622e83ee8e11b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-04-24T22:09:01Z","title_canon_sha256":"d6400ef2339bc46adbc37ebabfa510e8c52aaed08eceb15699d33389db23518a"},"schema_version":"1.0","source":{"id":"1504.06652","kind":"arxiv","version":1}},"canonical_sha256":"0dbd01268528ea9b1270f8ed32349fd1432c64a89785d2d61da98c920c2db1ba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0dbd01268528ea9b1270f8ed32349fd1432c64a89785d2d61da98c920c2db1ba","first_computed_at":"2026-05-18T02:17:52.868219Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:17:52.868219Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"21MstU5KTNBM/fUuvvMpHhAy6AlQYz0JhIH3li7IPYi480Ill9o72D+fae7w0EiLo+pgwZs0ohwaW/9k68fPBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:17:52.868711Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.06652","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b10f7f7a3dac0a3d5d4e471cef424b653c104c930bb8e953a0e4cda822c76541","sha256:52f36ddf39fa9a7da237992f4cad119ca14a5fd5531e4050a4a2cdd6d3b437df"],"state_sha256":"5d587f16fd3f2f83142fad0ade8404267e2488c3bdcd03190e27548dfccf2b41"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GMYydI9HW+12rdwSR0HQGgHUD26g9HUbmwmcIKvqcZa0st1Mx+C4Lis3vkMJTrkOVF5vl4TNnpYwV0SnyVqNCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T02:07:44.980409Z","bundle_sha256":"740c1caad4bf1288a74179ef8d5d73bc86d7c8a6615d9aee101fde765955a3fd"}}