{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:CQEM26RPGOOXBBLU7FGL4EE4B4","short_pith_number":"pith:CQEM26RP","canonical_record":{"source":{"id":"0901.0580","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2009-01-05T23:51:56Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"e52bc8a43de4febffe19f3caf8e5d83b4624421a524c27bd053999dbdd17f285","abstract_canon_sha256":"dea0557a1ce90f7854b00f3159e57c7c3a1acc4f98dfeb598d96a1dbe629dbeb"},"schema_version":"1.0"},"canonical_sha256":"1408cd7a2f339d708574f94cbe109c0f12553fa0b15b950b7f7bde6dedcb46a9","source":{"kind":"arxiv","id":"0901.0580","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0901.0580","created_at":"2026-05-18T03:18:06Z"},{"alias_kind":"arxiv_version","alias_value":"0901.0580v3","created_at":"2026-05-18T03:18:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.0580","created_at":"2026-05-18T03:18:06Z"},{"alias_kind":"pith_short_12","alias_value":"CQEM26RPGOOX","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"CQEM26RPGOOXBBLU","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"CQEM26RP","created_at":"2026-05-18T12:25:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:CQEM26RPGOOXBBLU7FGL4EE4B4","target":"record","payload":{"canonical_record":{"source":{"id":"0901.0580","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2009-01-05T23:51:56Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"e52bc8a43de4febffe19f3caf8e5d83b4624421a524c27bd053999dbdd17f285","abstract_canon_sha256":"dea0557a1ce90f7854b00f3159e57c7c3a1acc4f98dfeb598d96a1dbe629dbeb"},"schema_version":"1.0"},"canonical_sha256":"1408cd7a2f339d708574f94cbe109c0f12553fa0b15b950b7f7bde6dedcb46a9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:06.661750Z","signature_b64":"3Jv2d0w3KNyG319kH1so0BLeSZEYKx0JYUIuBMroX1x0f8z+F2cwDpzpqsXwsI/lMhuo28zxf3yWtq5u9zURDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1408cd7a2f339d708574f94cbe109c0f12553fa0b15b950b7f7bde6dedcb46a9","last_reissued_at":"2026-05-18T03:18:06.661084Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:06.661084Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0901.0580","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:18:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JoTF4Atn2EOYD7rLIj9gKQFo86TOXvNxQ5QR1t2TsVtbLjUMQATe+G8iw/8K6iN3xzMHKIwD2pL3SQY34197Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T20:28:48.790125Z"},"content_sha256":"3d188908835ceb8cfc82e1d3145e36345fd66580155fb697b35ac4b188176ac2","schema_version":"1.0","event_id":"sha256:3d188908835ceb8cfc82e1d3145e36345fd66580155fb697b35ac4b188176ac2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:CQEM26RPGOOXBBLU7FGL4EE4B4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bilinear Forms and Fierz Identities for Real Spin Representations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"gr-qc","authors_text":"Eric O. Korman, George Sparling","submitted_at":"2009-01-05T23:51:56Z","abstract_excerpt":"Given a real representation of the Clifford algebra corresponding to $R^{p+q}$ with metric of signature $(p,q)$, we demonstrate the existence of two natural bilinear forms on the space of spinors. With the Clifford action of $k$-forms on spinors, the bilinear forms allow us to relate two spinors with elements of the exterior algebra. From manipulations of a rank four spinorial tensor, we are able to find a general class of identities which, upon specializing from four spinors to two spinors and one spinor in signatures (1,3) and (10,1), yield some well-known Fierz identities. We will see, surp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.0580","kind":"arxiv","version":3},"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":"2026-05-18T03:18:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nwA/T0jt57V+n67UZ86sH93TpPwxBUJ2BxoBSAtgq+tSQ7MQzcQPRQsIDX0UChfD6YReYiAtIpnjtNWOY58MAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T20:28:48.790499Z"},"content_sha256":"df9733e879140a5819fe27f5b7520a68fce077cab1adfe7dacebea7cac7ca02d","schema_version":"1.0","event_id":"sha256:df9733e879140a5819fe27f5b7520a68fce077cab1adfe7dacebea7cac7ca02d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CQEM26RPGOOXBBLU7FGL4EE4B4/bundle.json","state_url":"https://pith.science/pith/CQEM26RPGOOXBBLU7FGL4EE4B4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CQEM26RPGOOXBBLU7FGL4EE4B4/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":"2026-06-25T20:28:48Z","links":{"resolver":"https://pith.science/pith/CQEM26RPGOOXBBLU7FGL4EE4B4","bundle":"https://pith.science/pith/CQEM26RPGOOXBBLU7FGL4EE4B4/bundle.json","state":"https://pith.science/pith/CQEM26RPGOOXBBLU7FGL4EE4B4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CQEM26RPGOOXBBLU7FGL4EE4B4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:CQEM26RPGOOXBBLU7FGL4EE4B4","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":"dea0557a1ce90f7854b00f3159e57c7c3a1acc4f98dfeb598d96a1dbe629dbeb","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2009-01-05T23:51:56Z","title_canon_sha256":"e52bc8a43de4febffe19f3caf8e5d83b4624421a524c27bd053999dbdd17f285"},"schema_version":"1.0","source":{"id":"0901.0580","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0901.0580","created_at":"2026-05-18T03:18:06Z"},{"alias_kind":"arxiv_version","alias_value":"0901.0580v3","created_at":"2026-05-18T03:18:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.0580","created_at":"2026-05-18T03:18:06Z"},{"alias_kind":"pith_short_12","alias_value":"CQEM26RPGOOX","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"CQEM26RPGOOXBBLU","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"CQEM26RP","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:df9733e879140a5819fe27f5b7520a68fce077cab1adfe7dacebea7cac7ca02d","target":"graph","created_at":"2026-05-18T03:18:06Z","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 a real representation of the Clifford algebra corresponding to $R^{p+q}$ with metric of signature $(p,q)$, we demonstrate the existence of two natural bilinear forms on the space of spinors. With the Clifford action of $k$-forms on spinors, the bilinear forms allow us to relate two spinors with elements of the exterior algebra. From manipulations of a rank four spinorial tensor, we are able to find a general class of identities which, upon specializing from four spinors to two spinors and one spinor in signatures (1,3) and (10,1), yield some well-known Fierz identities. We will see, surp","authors_text":"Eric O. Korman, George Sparling","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2009-01-05T23:51:56Z","title":"Bilinear Forms and Fierz Identities for Real Spin Representations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.0580","kind":"arxiv","version":3},"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":"sha256:3d188908835ceb8cfc82e1d3145e36345fd66580155fb697b35ac4b188176ac2","target":"record","created_at":"2026-05-18T03:18:06Z","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":"dea0557a1ce90f7854b00f3159e57c7c3a1acc4f98dfeb598d96a1dbe629dbeb","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2009-01-05T23:51:56Z","title_canon_sha256":"e52bc8a43de4febffe19f3caf8e5d83b4624421a524c27bd053999dbdd17f285"},"schema_version":"1.0","source":{"id":"0901.0580","kind":"arxiv","version":3}},"canonical_sha256":"1408cd7a2f339d708574f94cbe109c0f12553fa0b15b950b7f7bde6dedcb46a9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1408cd7a2f339d708574f94cbe109c0f12553fa0b15b950b7f7bde6dedcb46a9","first_computed_at":"2026-05-18T03:18:06.661084Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:18:06.661084Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3Jv2d0w3KNyG319kH1so0BLeSZEYKx0JYUIuBMroX1x0f8z+F2cwDpzpqsXwsI/lMhuo28zxf3yWtq5u9zURDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:18:06.661750Z","signed_message":"canonical_sha256_bytes"},"source_id":"0901.0580","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3d188908835ceb8cfc82e1d3145e36345fd66580155fb697b35ac4b188176ac2","sha256:df9733e879140a5819fe27f5b7520a68fce077cab1adfe7dacebea7cac7ca02d"],"state_sha256":"edf8ad00e86d7f07d908d8c1c3d5e9019b28374e8ce5ca0fa2c610b506d6fd50"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SBOOGeCwOC9oiFZvl9UmRap8Fjd4Op8BJN32XcUXKNOmFZ+Emxh/EY3QVke4JzZD/QRq2U94B4C4rdqFoZMWCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T20:28:48.792444Z","bundle_sha256":"f71617e93b1cb4d2caf259f0e97699bd224d81d3426d67b1ff588864a81c6ea1"}}