{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:5I2NJLHUHOFIVY6UFT6SFXYSIW","short_pith_number":"pith:5I2NJLHU","canonical_record":{"source":{"id":"1509.07718","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-09-25T13:56:40Z","cross_cats_sorted":[],"title_canon_sha256":"109f00e5855000b7586f1c9faa2fb740bb397b228018da90d6900540e7c8e056","abstract_canon_sha256":"f90c85efba1b02dac4a6769b5eb29eaaa93dbf149bf9804e59da9cb3a14a919b"},"schema_version":"1.0"},"canonical_sha256":"ea34d4acf43b8a8ae3d42cfd22df1245b49ef686381b305eb70825ed76d024fd","source":{"kind":"arxiv","id":"1509.07718","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.07718","created_at":"2026-05-18T01:32:02Z"},{"alias_kind":"arxiv_version","alias_value":"1509.07718v1","created_at":"2026-05-18T01:32:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07718","created_at":"2026-05-18T01:32:02Z"},{"alias_kind":"pith_short_12","alias_value":"5I2NJLHUHOFI","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"5I2NJLHUHOFIVY6U","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"5I2NJLHU","created_at":"2026-05-18T12:29:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:5I2NJLHUHOFIVY6UFT6SFXYSIW","target":"record","payload":{"canonical_record":{"source":{"id":"1509.07718","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-09-25T13:56:40Z","cross_cats_sorted":[],"title_canon_sha256":"109f00e5855000b7586f1c9faa2fb740bb397b228018da90d6900540e7c8e056","abstract_canon_sha256":"f90c85efba1b02dac4a6769b5eb29eaaa93dbf149bf9804e59da9cb3a14a919b"},"schema_version":"1.0"},"canonical_sha256":"ea34d4acf43b8a8ae3d42cfd22df1245b49ef686381b305eb70825ed76d024fd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:02.092884Z","signature_b64":"g21ZYGv2iHVnZLurk554hXbhr0ifJ99Zo6X5oK2osYaBFBPI5IjX2DLNc3k6FNYByd1sysCmRhQFcLgwQx6CBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea34d4acf43b8a8ae3d42cfd22df1245b49ef686381b305eb70825ed76d024fd","last_reissued_at":"2026-05-18T01:32:02.092471Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:02.092471Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.07718","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":"2026-05-18T01:32:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IoKyRzjbQUmsB+Ecjf9WNdpwcLq4uYoO52GO93hOHlzMPEIGdRqQ/XiP5fUgzspxyTwyXQnzDgeaJCgyu5kzAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T05:03:05.643263Z"},"content_sha256":"fe04ae1a7f2634541960e0d8c5c9649b36efc84b8a505ce1b787e33a2f5201f2","schema_version":"1.0","event_id":"sha256:fe04ae1a7f2634541960e0d8c5c9649b36efc84b8a505ce1b787e33a2f5201f2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:5I2NJLHUHOFIVY6UFT6SFXYSIW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Octonion associators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Stephen J. Sangwine","submitted_at":"2015-09-25T13:56:40Z","abstract_excerpt":"The algebra of octonions is non-associative (as well as non-commutative). This makes it very difficult to derive algebraic results, and to perform computation with octonions. Given a product of more than two octonions, in general, the order of evaluation of the product (placement of parentheses) affects the result. Inspired by the concept of the commutator $[x,y]= x^{-1}y^{-1}xy$ we show that an associator can be defined that multiplies the result from one evaluation order to give the result from a different evaluation order. For example, for the case of three arbitrary octonions $x$, $y$ and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07718","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":"2026-05-18T01:32:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a9w55GW4w7D6JnbkQalm5kK99TwYdUbJBYbK1ym3SNQApsri4NIWCfOooNEEs0E9mQDpH61IBEaEZdQ761W1CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T05:03:05.643623Z"},"content_sha256":"1a109aa37fac6dc8cf087942aa81ef6b6432fb5c8bb0b1950e973ea232ecaa08","schema_version":"1.0","event_id":"sha256:1a109aa37fac6dc8cf087942aa81ef6b6432fb5c8bb0b1950e973ea232ecaa08"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5I2NJLHUHOFIVY6UFT6SFXYSIW/bundle.json","state_url":"https://pith.science/pith/5I2NJLHUHOFIVY6UFT6SFXYSIW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5I2NJLHUHOFIVY6UFT6SFXYSIW/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":"2026-06-03T05:03:05Z","links":{"resolver":"https://pith.science/pith/5I2NJLHUHOFIVY6UFT6SFXYSIW","bundle":"https://pith.science/pith/5I2NJLHUHOFIVY6UFT6SFXYSIW/bundle.json","state":"https://pith.science/pith/5I2NJLHUHOFIVY6UFT6SFXYSIW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5I2NJLHUHOFIVY6UFT6SFXYSIW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:5I2NJLHUHOFIVY6UFT6SFXYSIW","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":"f90c85efba1b02dac4a6769b5eb29eaaa93dbf149bf9804e59da9cb3a14a919b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-09-25T13:56:40Z","title_canon_sha256":"109f00e5855000b7586f1c9faa2fb740bb397b228018da90d6900540e7c8e056"},"schema_version":"1.0","source":{"id":"1509.07718","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.07718","created_at":"2026-05-18T01:32:02Z"},{"alias_kind":"arxiv_version","alias_value":"1509.07718v1","created_at":"2026-05-18T01:32:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07718","created_at":"2026-05-18T01:32:02Z"},{"alias_kind":"pith_short_12","alias_value":"5I2NJLHUHOFI","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"5I2NJLHUHOFIVY6U","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"5I2NJLHU","created_at":"2026-05-18T12:29:05Z"}],"graph_snapshots":[{"event_id":"sha256:1a109aa37fac6dc8cf087942aa81ef6b6432fb5c8bb0b1950e973ea232ecaa08","target":"graph","created_at":"2026-05-18T01:32:02Z","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 algebra of octonions is non-associative (as well as non-commutative). This makes it very difficult to derive algebraic results, and to perform computation with octonions. Given a product of more than two octonions, in general, the order of evaluation of the product (placement of parentheses) affects the result. Inspired by the concept of the commutator $[x,y]= x^{-1}y^{-1}xy$ we show that an associator can be defined that multiplies the result from one evaluation order to give the result from a different evaluation order. For example, for the case of three arbitrary octonions $x$, $y$ and ","authors_text":"Stephen J. Sangwine","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-09-25T13:56:40Z","title":"Octonion associators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07718","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:fe04ae1a7f2634541960e0d8c5c9649b36efc84b8a505ce1b787e33a2f5201f2","target":"record","created_at":"2026-05-18T01:32:02Z","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":"f90c85efba1b02dac4a6769b5eb29eaaa93dbf149bf9804e59da9cb3a14a919b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-09-25T13:56:40Z","title_canon_sha256":"109f00e5855000b7586f1c9faa2fb740bb397b228018da90d6900540e7c8e056"},"schema_version":"1.0","source":{"id":"1509.07718","kind":"arxiv","version":1}},"canonical_sha256":"ea34d4acf43b8a8ae3d42cfd22df1245b49ef686381b305eb70825ed76d024fd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ea34d4acf43b8a8ae3d42cfd22df1245b49ef686381b305eb70825ed76d024fd","first_computed_at":"2026-05-18T01:32:02.092471Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:32:02.092471Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g21ZYGv2iHVnZLurk554hXbhr0ifJ99Zo6X5oK2osYaBFBPI5IjX2DLNc3k6FNYByd1sysCmRhQFcLgwQx6CBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:32:02.092884Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.07718","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fe04ae1a7f2634541960e0d8c5c9649b36efc84b8a505ce1b787e33a2f5201f2","sha256:1a109aa37fac6dc8cf087942aa81ef6b6432fb5c8bb0b1950e973ea232ecaa08"],"state_sha256":"0dc8592fb1d05aaae70a51c64d6c070b249e1d74a5cbce50ccae544bc5e02932"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rV4DPvJVXsCAZTpMdEuluGOpacp2kvPcpkmrSnjq/0zN3o2K/ALk2rFf7vNxxnBhWEzbXqahIgoThGRWFNbyAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T05:03:05.645531Z","bundle_sha256":"33a3a2704fd341101668b2569eb49ee83832345285921c45345e37c42d450720"}}