{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:B6ITDD4ZOAY3YF5N6PJQT6CEKS","short_pith_number":"pith:B6ITDD4Z","canonical_record":{"source":{"id":"1409.0509","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-01T19:06:12Z","cross_cats_sorted":[],"title_canon_sha256":"42fde817f146ddb54b002379eda097288c4efc74e1615bdc0394af12325f987c","abstract_canon_sha256":"915e0627bef0dc1abb34a2b68bb0d1282275496ce9adbd50f047ccd8c330003e"},"schema_version":"1.0"},"canonical_sha256":"0f91318f997031bc17adf3d309f84454be1f2a2370e82fcecf28f0b67b2b8156","source":{"kind":"arxiv","id":"1409.0509","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.0509","created_at":"2026-05-18T02:43:52Z"},{"alias_kind":"arxiv_version","alias_value":"1409.0509v1","created_at":"2026-05-18T02:43:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.0509","created_at":"2026-05-18T02:43:52Z"},{"alias_kind":"pith_short_12","alias_value":"B6ITDD4ZOAY3","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"B6ITDD4ZOAY3YF5N","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"B6ITDD4Z","created_at":"2026-05-18T12:28:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:B6ITDD4ZOAY3YF5N6PJQT6CEKS","target":"record","payload":{"canonical_record":{"source":{"id":"1409.0509","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-01T19:06:12Z","cross_cats_sorted":[],"title_canon_sha256":"42fde817f146ddb54b002379eda097288c4efc74e1615bdc0394af12325f987c","abstract_canon_sha256":"915e0627bef0dc1abb34a2b68bb0d1282275496ce9adbd50f047ccd8c330003e"},"schema_version":"1.0"},"canonical_sha256":"0f91318f997031bc17adf3d309f84454be1f2a2370e82fcecf28f0b67b2b8156","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:52.640690Z","signature_b64":"HWCkvY5LMm/tsG/YSfsoAnc8QSBD46JeZN0lLomrbGL5JB7gOeHP+lgg8XLjzkKOXsL9hpewvD1wjV2KAnaVCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f91318f997031bc17adf3d309f84454be1f2a2370e82fcecf28f0b67b2b8156","last_reissued_at":"2026-05-18T02:43:52.640291Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:52.640291Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1409.0509","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":"2026-05-18T02:43:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cgXPlun4NKKAVYme6T96ZZkbqxzY4yTvsG4vE3ap0cxZBaeiz1UWU8Dwqd5g8xrPV/7cPkM8tP/XIlAVyZ2DAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T19:03:48.923720Z"},"content_sha256":"6699e9d4284d04e2369161bfbbee29ebc1c1f01885bfbd969e967e11192bd66f","schema_version":"1.0","event_id":"sha256:6699e9d4284d04e2369161bfbbee29ebc1c1f01885bfbd969e967e11192bd66f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:B6ITDD4ZOAY3YF5N6PJQT6CEKS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lagrange's Theorem for continued fractions on the Heisenberg group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Joseph Vandehey","submitted_at":"2014-09-01T19:06:12Z","abstract_excerpt":"We prove an analog of Lagrange's Theorem for continued fractions on the Heisenberg group: points with an eventually periodic continued fraction expansion are those that satisfy a particular type of quadratic form, and vice-versa."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0509","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":"2026-05-18T02:43:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mf8Cimw/Yh4OU/IE4Uo3n58DbZf+5/liJelaSYMmMGeAQdkXD/qaAvKiTko0OD/waQ8tjvEifawHdcJpY63sCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T19:03:48.924489Z"},"content_sha256":"2f00626a4547db77a8566bd8342a31a6abb3632676e1d397c34be7164ebf04eb","schema_version":"1.0","event_id":"sha256:2f00626a4547db77a8566bd8342a31a6abb3632676e1d397c34be7164ebf04eb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B6ITDD4ZOAY3YF5N6PJQT6CEKS/bundle.json","state_url":"https://pith.science/pith/B6ITDD4ZOAY3YF5N6PJQT6CEKS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B6ITDD4ZOAY3YF5N6PJQT6CEKS/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":"2026-05-31T19:03:48Z","links":{"resolver":"https://pith.science/pith/B6ITDD4ZOAY3YF5N6PJQT6CEKS","bundle":"https://pith.science/pith/B6ITDD4ZOAY3YF5N6PJQT6CEKS/bundle.json","state":"https://pith.science/pith/B6ITDD4ZOAY3YF5N6PJQT6CEKS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B6ITDD4ZOAY3YF5N6PJQT6CEKS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:B6ITDD4ZOAY3YF5N6PJQT6CEKS","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":"915e0627bef0dc1abb34a2b68bb0d1282275496ce9adbd50f047ccd8c330003e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-01T19:06:12Z","title_canon_sha256":"42fde817f146ddb54b002379eda097288c4efc74e1615bdc0394af12325f987c"},"schema_version":"1.0","source":{"id":"1409.0509","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.0509","created_at":"2026-05-18T02:43:52Z"},{"alias_kind":"arxiv_version","alias_value":"1409.0509v1","created_at":"2026-05-18T02:43:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.0509","created_at":"2026-05-18T02:43:52Z"},{"alias_kind":"pith_short_12","alias_value":"B6ITDD4ZOAY3","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"B6ITDD4ZOAY3YF5N","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"B6ITDD4Z","created_at":"2026-05-18T12:28:22Z"}],"graph_snapshots":[{"event_id":"sha256:2f00626a4547db77a8566bd8342a31a6abb3632676e1d397c34be7164ebf04eb","target":"graph","created_at":"2026-05-18T02:43:52Z","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":"We prove an analog of Lagrange's Theorem for continued fractions on the Heisenberg group: points with an eventually periodic continued fraction expansion are those that satisfy a particular type of quadratic form, and vice-versa.","authors_text":"Joseph Vandehey","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-01T19:06:12Z","title":"Lagrange's Theorem for continued fractions on the Heisenberg group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0509","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":"sha256:6699e9d4284d04e2369161bfbbee29ebc1c1f01885bfbd969e967e11192bd66f","target":"record","created_at":"2026-05-18T02:43:52Z","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":"915e0627bef0dc1abb34a2b68bb0d1282275496ce9adbd50f047ccd8c330003e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-01T19:06:12Z","title_canon_sha256":"42fde817f146ddb54b002379eda097288c4efc74e1615bdc0394af12325f987c"},"schema_version":"1.0","source":{"id":"1409.0509","kind":"arxiv","version":1}},"canonical_sha256":"0f91318f997031bc17adf3d309f84454be1f2a2370e82fcecf28f0b67b2b8156","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0f91318f997031bc17adf3d309f84454be1f2a2370e82fcecf28f0b67b2b8156","first_computed_at":"2026-05-18T02:43:52.640291Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:43:52.640291Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HWCkvY5LMm/tsG/YSfsoAnc8QSBD46JeZN0lLomrbGL5JB7gOeHP+lgg8XLjzkKOXsL9hpewvD1wjV2KAnaVCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:43:52.640690Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.0509","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6699e9d4284d04e2369161bfbbee29ebc1c1f01885bfbd969e967e11192bd66f","sha256:2f00626a4547db77a8566bd8342a31a6abb3632676e1d397c34be7164ebf04eb"],"state_sha256":"1dfa9170e83e1ba8ede4b191c7d8a812c7c59c2f25852847fabc0b832d74967a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"81/kfprINEHzTQQrjJWs+7rWqGe2K6M0anMuQh0PwjZ4mMh9bZTHjxXgACvgtq26d+Ess5l51bLKjQtVed4+Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T19:03:48.928374Z","bundle_sha256":"c14a47f25dec99c7b4b6c0ed6f05ad17eeffc2688032f1fe702c758c421c05b9"}}