{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:QDIPIGC2XQUK3EQ6DGIYCY7FDK","short_pith_number":"pith:QDIPIGC2","canonical_record":{"source":{"id":"1702.05705","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-02-19T06:27:23Z","cross_cats_sorted":[],"title_canon_sha256":"77871a89b35712b4119a7aa614eeeec07b9841a771db8fac2476269aa850b0ac","abstract_canon_sha256":"399b00efb85bd07a52b2a9e1386dbfcff48dedf8fa8c1d07aad714a503f0f036"},"schema_version":"1.0"},"canonical_sha256":"80d0f4185abc28ad921e19918163e51aa2effc7520edad7c76deb25d2982b217","source":{"kind":"arxiv","id":"1702.05705","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.05705","created_at":"2026-05-18T00:50:25Z"},{"alias_kind":"arxiv_version","alias_value":"1702.05705v1","created_at":"2026-05-18T00:50:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.05705","created_at":"2026-05-18T00:50:25Z"},{"alias_kind":"pith_short_12","alias_value":"QDIPIGC2XQUK","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_16","alias_value":"QDIPIGC2XQUK3EQ6","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_8","alias_value":"QDIPIGC2","created_at":"2026-05-18T12:31:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:QDIPIGC2XQUK3EQ6DGIYCY7FDK","target":"record","payload":{"canonical_record":{"source":{"id":"1702.05705","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-02-19T06:27:23Z","cross_cats_sorted":[],"title_canon_sha256":"77871a89b35712b4119a7aa614eeeec07b9841a771db8fac2476269aa850b0ac","abstract_canon_sha256":"399b00efb85bd07a52b2a9e1386dbfcff48dedf8fa8c1d07aad714a503f0f036"},"schema_version":"1.0"},"canonical_sha256":"80d0f4185abc28ad921e19918163e51aa2effc7520edad7c76deb25d2982b217","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:25.585661Z","signature_b64":"fx383Q4XkGLQK5a/kzgOAGLgfGKJADc7tx6bxpmY/PHDFnOE07hIEBdN/IWE+VJqm2NzVzVRcW1ZZKqqC12aBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80d0f4185abc28ad921e19918163e51aa2effc7520edad7c76deb25d2982b217","last_reissued_at":"2026-05-18T00:50:25.584886Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:25.584886Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1702.05705","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":"2026-05-18T00:50:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ecMuYv3plqj5zPersY9zBmy0JwllEfWBLM5Wm0XM7OS16Z8IhMRDg1Luyu2k5blDOR9lchqVgy35oMUOp/NyDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:20:29.695873Z"},"content_sha256":"237996b86ba278693c23a7e49d454ac47f6250728cee13f24e2da3e9d83978a5","schema_version":"1.0","event_id":"sha256:237996b86ba278693c23a7e49d454ac47f6250728cee13f24e2da3e9d83978a5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:QDIPIGC2XQUK3EQ6DGIYCY7FDK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The octonions as a twisted group algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Tathagata Basak","submitted_at":"2017-02-19T06:27:23Z","abstract_excerpt":"We show that the octonions can be defined as the $\\mathbb{R}$-algebra with basis $\\lbrace e^x \\colon x \\in \\mathbb{F}_8 \\rbrace$ and multiplication given by $e^x e^y = (-1)^{\\varphi(x,y)}e^{x + y}$, where $\\varphi(x,y) = \\operatorname{tr}(y x^6)$. While it is well known that the octonions can be described as a twisted group algebra, our purpose is to point out that this is a useful description. We show how the basic properties of the octonions follow easily from our definition. We give a uniform description of the sixteen orders of integral octonions containing the Gravesian integers, and a co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.05705","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":"2026-05-18T00:50:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RMugVMSyopaWrYA/uE6Fr4LCiICEVCtZC8/oy5hF6WDKXNII5TKJOTYDcDNzR8tpqiJFWCdqwdiC3egRAjLpCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:20:29.696204Z"},"content_sha256":"c25f9aaf55d28860a30be07f415cd2b87f0ec50b27ff59cda29c8a237b7a9e09","schema_version":"1.0","event_id":"sha256:c25f9aaf55d28860a30be07f415cd2b87f0ec50b27ff59cda29c8a237b7a9e09"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QDIPIGC2XQUK3EQ6DGIYCY7FDK/bundle.json","state_url":"https://pith.science/pith/QDIPIGC2XQUK3EQ6DGIYCY7FDK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QDIPIGC2XQUK3EQ6DGIYCY7FDK/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":"2026-06-23T16:20:29Z","links":{"resolver":"https://pith.science/pith/QDIPIGC2XQUK3EQ6DGIYCY7FDK","bundle":"https://pith.science/pith/QDIPIGC2XQUK3EQ6DGIYCY7FDK/bundle.json","state":"https://pith.science/pith/QDIPIGC2XQUK3EQ6DGIYCY7FDK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QDIPIGC2XQUK3EQ6DGIYCY7FDK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:QDIPIGC2XQUK3EQ6DGIYCY7FDK","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":"399b00efb85bd07a52b2a9e1386dbfcff48dedf8fa8c1d07aad714a503f0f036","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-02-19T06:27:23Z","title_canon_sha256":"77871a89b35712b4119a7aa614eeeec07b9841a771db8fac2476269aa850b0ac"},"schema_version":"1.0","source":{"id":"1702.05705","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.05705","created_at":"2026-05-18T00:50:25Z"},{"alias_kind":"arxiv_version","alias_value":"1702.05705v1","created_at":"2026-05-18T00:50:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.05705","created_at":"2026-05-18T00:50:25Z"},{"alias_kind":"pith_short_12","alias_value":"QDIPIGC2XQUK","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_16","alias_value":"QDIPIGC2XQUK3EQ6","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_8","alias_value":"QDIPIGC2","created_at":"2026-05-18T12:31:37Z"}],"graph_snapshots":[{"event_id":"sha256:c25f9aaf55d28860a30be07f415cd2b87f0ec50b27ff59cda29c8a237b7a9e09","target":"graph","created_at":"2026-05-18T00:50:25Z","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 show that the octonions can be defined as the $\\mathbb{R}$-algebra with basis $\\lbrace e^x \\colon x \\in \\mathbb{F}_8 \\rbrace$ and multiplication given by $e^x e^y = (-1)^{\\varphi(x,y)}e^{x + y}$, where $\\varphi(x,y) = \\operatorname{tr}(y x^6)$. While it is well known that the octonions can be described as a twisted group algebra, our purpose is to point out that this is a useful description. We show how the basic properties of the octonions follow easily from our definition. We give a uniform description of the sixteen orders of integral octonions containing the Gravesian integers, and a co","authors_text":"Tathagata Basak","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-02-19T06:27:23Z","title":"The octonions as a twisted group algebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.05705","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":"sha256:237996b86ba278693c23a7e49d454ac47f6250728cee13f24e2da3e9d83978a5","target":"record","created_at":"2026-05-18T00:50:25Z","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":"399b00efb85bd07a52b2a9e1386dbfcff48dedf8fa8c1d07aad714a503f0f036","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-02-19T06:27:23Z","title_canon_sha256":"77871a89b35712b4119a7aa614eeeec07b9841a771db8fac2476269aa850b0ac"},"schema_version":"1.0","source":{"id":"1702.05705","kind":"arxiv","version":1}},"canonical_sha256":"80d0f4185abc28ad921e19918163e51aa2effc7520edad7c76deb25d2982b217","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"80d0f4185abc28ad921e19918163e51aa2effc7520edad7c76deb25d2982b217","first_computed_at":"2026-05-18T00:50:25.584886Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:50:25.584886Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fx383Q4XkGLQK5a/kzgOAGLgfGKJADc7tx6bxpmY/PHDFnOE07hIEBdN/IWE+VJqm2NzVzVRcW1ZZKqqC12aBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:50:25.585661Z","signed_message":"canonical_sha256_bytes"},"source_id":"1702.05705","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:237996b86ba278693c23a7e49d454ac47f6250728cee13f24e2da3e9d83978a5","sha256:c25f9aaf55d28860a30be07f415cd2b87f0ec50b27ff59cda29c8a237b7a9e09"],"state_sha256":"4af55aa426b0b3e2587b462e53fc800ddce65004d49a46eb132126024ae35086"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6SLni5COkRveQxTrQEdq/wgF+MNmtToNn6W+IBl8JwLPHdapbYjRGGllZwseUSwMk9r8rnQpJFoSBPrPZTRKCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T16:20:29.698055Z","bundle_sha256":"97072092eee342208c9fd7990392071f50b6aeb10438defccdbf8b41fe59ad3f"}}