{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:4FGBNTC77UXXNZZI6NR3LJRJQ4","short_pith_number":"pith:4FGBNTC7","canonical_record":{"source":{"id":"1704.02865","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2017-03-25T12:49:27Z","cross_cats_sorted":[],"title_canon_sha256":"21ec2ef82d0be39c0a251bdf1da1b2f2b271c63e191469291b31347684c1ea28","abstract_canon_sha256":"5be1da1e7d56816a022782251cfbf2bb89f9af96dcc690cfc4817472c469bc11"},"schema_version":"1.0"},"canonical_sha256":"e14c16cc5ffd2f76e728f363b5a629870e87fbba910cdf71006deab170b7cfc0","source":{"kind":"arxiv","id":"1704.02865","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.02865","created_at":"2026-05-18T00:19:04Z"},{"alias_kind":"arxiv_version","alias_value":"1704.02865v1","created_at":"2026-05-18T00:19:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.02865","created_at":"2026-05-18T00:19:04Z"},{"alias_kind":"pith_short_12","alias_value":"4FGBNTC77UXX","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"4FGBNTC77UXXNZZI","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"4FGBNTC7","created_at":"2026-05-18T12:30:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:4FGBNTC77UXXNZZI6NR3LJRJQ4","target":"record","payload":{"canonical_record":{"source":{"id":"1704.02865","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2017-03-25T12:49:27Z","cross_cats_sorted":[],"title_canon_sha256":"21ec2ef82d0be39c0a251bdf1da1b2f2b271c63e191469291b31347684c1ea28","abstract_canon_sha256":"5be1da1e7d56816a022782251cfbf2bb89f9af96dcc690cfc4817472c469bc11"},"schema_version":"1.0"},"canonical_sha256":"e14c16cc5ffd2f76e728f363b5a629870e87fbba910cdf71006deab170b7cfc0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:04.172793Z","signature_b64":"OIiUwQqXevbJF3w7H/tUPvrKlTGjs0Ehhc80tOsVqkBvdsjeQldTG5XLzGWPElAyl8HvOV16tbNeMqhxzNX/DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e14c16cc5ffd2f76e728f363b5a629870e87fbba910cdf71006deab170b7cfc0","last_reissued_at":"2026-05-18T00:19:04.172130Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:04.172130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1704.02865","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":"2026-05-18T00:19:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YGGrSXGsKztskuhbEA/GbFqTRuOnBBXQyFV3RGb7Yvyd72LxkWg0k3naJAH1nHisms7QuWEiDqi08iDdriVjBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T08:55:10.143590Z"},"content_sha256":"c20dad02dc197efd778f28e7d44fbd019f665e7434a1d6742c535a41a6c24f85","schema_version":"1.0","event_id":"sha256:c20dad02dc197efd778f28e7d44fbd019f665e7434a1d6742c535a41a6c24f85"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:4FGBNTC77UXXNZZI6NR3LJRJQ4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Algebraic properties of bi$-$periodic dual Fibonacci quaternions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Fatma Ate\\c{s}, Ismail G\\\"ok, Nejat Ekmekci","submitted_at":"2017-03-25T12:49:27Z","abstract_excerpt":"The purpose of the paper is to construct a new representation of dual quaternions called bi$-$periodic dual Fibonacci quaternions. These quaternions are originated as a generalization of the known quaternions in literature such as dual Fibonacci quaternions, dual Pell quaternions and dual $k-$Fibonacci quaternions. Furthermore, some of them have not been introduced until this time. Then, we give generating function, Binet formula and Catalan's identity in terms of these quaternions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.02865","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":"2026-05-18T00:19:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2wSSPWP91Ij7z51964aIE765oFAClvCGHIrxazakl3Hca6gvjdJ7GfcRo/pifAScVhjWgmXmP5/R+MwUYJmXCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T08:55:10.143936Z"},"content_sha256":"f5a212014c1fb1f231f8b4302b8b761c5a503bca36f3686af2019930eea5464c","schema_version":"1.0","event_id":"sha256:f5a212014c1fb1f231f8b4302b8b761c5a503bca36f3686af2019930eea5464c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4FGBNTC77UXXNZZI6NR3LJRJQ4/bundle.json","state_url":"https://pith.science/pith/4FGBNTC77UXXNZZI6NR3LJRJQ4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4FGBNTC77UXXNZZI6NR3LJRJQ4/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":"2026-06-30T08:55:10Z","links":{"resolver":"https://pith.science/pith/4FGBNTC77UXXNZZI6NR3LJRJQ4","bundle":"https://pith.science/pith/4FGBNTC77UXXNZZI6NR3LJRJQ4/bundle.json","state":"https://pith.science/pith/4FGBNTC77UXXNZZI6NR3LJRJQ4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4FGBNTC77UXXNZZI6NR3LJRJQ4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:4FGBNTC77UXXNZZI6NR3LJRJQ4","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":"5be1da1e7d56816a022782251cfbf2bb89f9af96dcc690cfc4817472c469bc11","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2017-03-25T12:49:27Z","title_canon_sha256":"21ec2ef82d0be39c0a251bdf1da1b2f2b271c63e191469291b31347684c1ea28"},"schema_version":"1.0","source":{"id":"1704.02865","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.02865","created_at":"2026-05-18T00:19:04Z"},{"alias_kind":"arxiv_version","alias_value":"1704.02865v1","created_at":"2026-05-18T00:19:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.02865","created_at":"2026-05-18T00:19:04Z"},{"alias_kind":"pith_short_12","alias_value":"4FGBNTC77UXX","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"4FGBNTC77UXXNZZI","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"4FGBNTC7","created_at":"2026-05-18T12:30:58Z"}],"graph_snapshots":[{"event_id":"sha256:f5a212014c1fb1f231f8b4302b8b761c5a503bca36f3686af2019930eea5464c","target":"graph","created_at":"2026-05-18T00:19:04Z","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":"The purpose of the paper is to construct a new representation of dual quaternions called bi$-$periodic dual Fibonacci quaternions. These quaternions are originated as a generalization of the known quaternions in literature such as dual Fibonacci quaternions, dual Pell quaternions and dual $k-$Fibonacci quaternions. Furthermore, some of them have not been introduced until this time. Then, we give generating function, Binet formula and Catalan's identity in terms of these quaternions.","authors_text":"Fatma Ate\\c{s}, Ismail G\\\"ok, Nejat Ekmekci","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2017-03-25T12:49:27Z","title":"Algebraic properties of bi$-$periodic dual Fibonacci quaternions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.02865","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:c20dad02dc197efd778f28e7d44fbd019f665e7434a1d6742c535a41a6c24f85","target":"record","created_at":"2026-05-18T00:19:04Z","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":"5be1da1e7d56816a022782251cfbf2bb89f9af96dcc690cfc4817472c469bc11","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2017-03-25T12:49:27Z","title_canon_sha256":"21ec2ef82d0be39c0a251bdf1da1b2f2b271c63e191469291b31347684c1ea28"},"schema_version":"1.0","source":{"id":"1704.02865","kind":"arxiv","version":1}},"canonical_sha256":"e14c16cc5ffd2f76e728f363b5a629870e87fbba910cdf71006deab170b7cfc0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e14c16cc5ffd2f76e728f363b5a629870e87fbba910cdf71006deab170b7cfc0","first_computed_at":"2026-05-18T00:19:04.172130Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:19:04.172130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OIiUwQqXevbJF3w7H/tUPvrKlTGjs0Ehhc80tOsVqkBvdsjeQldTG5XLzGWPElAyl8HvOV16tbNeMqhxzNX/DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:19:04.172793Z","signed_message":"canonical_sha256_bytes"},"source_id":"1704.02865","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c20dad02dc197efd778f28e7d44fbd019f665e7434a1d6742c535a41a6c24f85","sha256:f5a212014c1fb1f231f8b4302b8b761c5a503bca36f3686af2019930eea5464c"],"state_sha256":"a90b1276c0e46b26def8848d4ee6182fe987c9cb8dc732bb7b737033326fd7d1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oRyq1J2FimoozqKNnnB4240VACvSUPwT/jYjKbTU0Zf8mUb20zX8Wri8eEr8lpULaQdY+DdnmWijkORvgaviCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T08:55:10.147050Z","bundle_sha256":"121a54ab1896f829e66ad23de37faec361e0b17c7dc7822cfdfcb53e852c0a0e"}}