{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:RQRLU2RCZRGKEZP4YXSED3NWY7","short_pith_number":"pith:RQRLU2RC","canonical_record":{"source":{"id":"1306.1660","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-06-07T09:01:33Z","cross_cats_sorted":[],"title_canon_sha256":"a608ad1e026227d7d10dc1bc9db2e1ba1bd7b019daef96d2bc2a8de5c102a992","abstract_canon_sha256":"fa584fa1fc7fd166233d52295e8f1e1f997ba4c7df4f0604a688bc3a408d2121"},"schema_version":"1.0"},"canonical_sha256":"8c22ba6a22cc4ca265fcc5e441edb6c7d56ba30016c7a936a18dbc5b5a3d5a75","source":{"kind":"arxiv","id":"1306.1660","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.1660","created_at":"2026-05-18T03:21:32Z"},{"alias_kind":"arxiv_version","alias_value":"1306.1660v1","created_at":"2026-05-18T03:21:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.1660","created_at":"2026-05-18T03:21:32Z"},{"alias_kind":"pith_short_12","alias_value":"RQRLU2RCZRGK","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"RQRLU2RCZRGKEZP4","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"RQRLU2RC","created_at":"2026-05-18T12:27:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:RQRLU2RCZRGKEZP4YXSED3NWY7","target":"record","payload":{"canonical_record":{"source":{"id":"1306.1660","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-06-07T09:01:33Z","cross_cats_sorted":[],"title_canon_sha256":"a608ad1e026227d7d10dc1bc9db2e1ba1bd7b019daef96d2bc2a8de5c102a992","abstract_canon_sha256":"fa584fa1fc7fd166233d52295e8f1e1f997ba4c7df4f0604a688bc3a408d2121"},"schema_version":"1.0"},"canonical_sha256":"8c22ba6a22cc4ca265fcc5e441edb6c7d56ba30016c7a936a18dbc5b5a3d5a75","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:32.752947Z","signature_b64":"7cGl/he4YVy5Gm0fFfkGzyN4RcmT4zYPhuSedG5sGPmhxAfvCs5YV4W+vc3RXTG7HMW1yyKQ7wZs6awOxKc3AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c22ba6a22cc4ca265fcc5e441edb6c7d56ba30016c7a936a18dbc5b5a3d5a75","last_reissued_at":"2026-05-18T03:21:32.752158Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:32.752158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1306.1660","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":"2026-05-18T03:21:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"085fsF3snHml9mkL+GQhI+3T50QBIbjiP5BiJF2SuBGPd5ngwCDeo5Eg4lFMsH6T2GnNCNVPcb1l2ys48eUbDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T08:42:44.599558Z"},"content_sha256":"2592afc54a6483ba4cb900cf6bd5e7e82b84996bf41c9cfccff6c1bc5bb72e51","schema_version":"1.0","event_id":"sha256:2592afc54a6483ba4cb900cf6bd5e7e82b84996bf41c9cfccff6c1bc5bb72e51"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:RQRLU2RCZRGKEZP4YXSED3NWY7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Introduction to Clifford's Geometric Algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Eckhard Hitzer","submitted_at":"2013-06-07T09:01:33Z","abstract_excerpt":"Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing, tracking, geographic information systems and neural computing. This tutorial explains the basics of geometric algebra, with concrete examples of the plane, of 3D space, of spacetime, and the popular conformal model. Geometric algebras are ideal to represent geometric transformations in the general framework of Clifford groups (also called versor or Lipschit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.1660","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":"2026-05-18T03:21:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iSFiHLY4EW9Kzzy5GbO+ndQjQWF/59nxwn0P0q7iM5YD6pnNSO1Gs6fqdla2kRolwaRJ+K+JOsdhOKcTqdUyBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T08:42:44.599982Z"},"content_sha256":"849a5487b03075da2a66dafaf05c51138cac322501cd79685429b4905ac44291","schema_version":"1.0","event_id":"sha256:849a5487b03075da2a66dafaf05c51138cac322501cd79685429b4905ac44291"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RQRLU2RCZRGKEZP4YXSED3NWY7/bundle.json","state_url":"https://pith.science/pith/RQRLU2RCZRGKEZP4YXSED3NWY7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RQRLU2RCZRGKEZP4YXSED3NWY7/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":"2026-06-21T08:42:44Z","links":{"resolver":"https://pith.science/pith/RQRLU2RCZRGKEZP4YXSED3NWY7","bundle":"https://pith.science/pith/RQRLU2RCZRGKEZP4YXSED3NWY7/bundle.json","state":"https://pith.science/pith/RQRLU2RCZRGKEZP4YXSED3NWY7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RQRLU2RCZRGKEZP4YXSED3NWY7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:RQRLU2RCZRGKEZP4YXSED3NWY7","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":"fa584fa1fc7fd166233d52295e8f1e1f997ba4c7df4f0604a688bc3a408d2121","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-06-07T09:01:33Z","title_canon_sha256":"a608ad1e026227d7d10dc1bc9db2e1ba1bd7b019daef96d2bc2a8de5c102a992"},"schema_version":"1.0","source":{"id":"1306.1660","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.1660","created_at":"2026-05-18T03:21:32Z"},{"alias_kind":"arxiv_version","alias_value":"1306.1660v1","created_at":"2026-05-18T03:21:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.1660","created_at":"2026-05-18T03:21:32Z"},{"alias_kind":"pith_short_12","alias_value":"RQRLU2RCZRGK","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"RQRLU2RCZRGKEZP4","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"RQRLU2RC","created_at":"2026-05-18T12:27:59Z"}],"graph_snapshots":[{"event_id":"sha256:849a5487b03075da2a66dafaf05c51138cac322501cd79685429b4905ac44291","target":"graph","created_at":"2026-05-18T03:21:32Z","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":"Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing, tracking, geographic information systems and neural computing. This tutorial explains the basics of geometric algebra, with concrete examples of the plane, of 3D space, of spacetime, and the popular conformal model. Geometric algebras are ideal to represent geometric transformations in the general framework of Clifford groups (also called versor or Lipschit","authors_text":"Eckhard Hitzer","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-06-07T09:01:33Z","title":"Introduction to Clifford's Geometric Algebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.1660","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":"sha256:2592afc54a6483ba4cb900cf6bd5e7e82b84996bf41c9cfccff6c1bc5bb72e51","target":"record","created_at":"2026-05-18T03:21:32Z","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":"fa584fa1fc7fd166233d52295e8f1e1f997ba4c7df4f0604a688bc3a408d2121","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-06-07T09:01:33Z","title_canon_sha256":"a608ad1e026227d7d10dc1bc9db2e1ba1bd7b019daef96d2bc2a8de5c102a992"},"schema_version":"1.0","source":{"id":"1306.1660","kind":"arxiv","version":1}},"canonical_sha256":"8c22ba6a22cc4ca265fcc5e441edb6c7d56ba30016c7a936a18dbc5b5a3d5a75","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8c22ba6a22cc4ca265fcc5e441edb6c7d56ba30016c7a936a18dbc5b5a3d5a75","first_computed_at":"2026-05-18T03:21:32.752158Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:21:32.752158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7cGl/he4YVy5Gm0fFfkGzyN4RcmT4zYPhuSedG5sGPmhxAfvCs5YV4W+vc3RXTG7HMW1yyKQ7wZs6awOxKc3AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:21:32.752947Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.1660","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2592afc54a6483ba4cb900cf6bd5e7e82b84996bf41c9cfccff6c1bc5bb72e51","sha256:849a5487b03075da2a66dafaf05c51138cac322501cd79685429b4905ac44291"],"state_sha256":"bd18b9540131d33849213113cf0c4a585d05c82e4778747587d48c78b159e88b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6tF4HuVOmoEqOGzaW1Kgeae0lLlHX6/PfCA5o2YnR8PBOFr2bugcegoHUUF3l9vvgntdj6HiTjxaWHFPKzg+AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T08:42:44.601862Z","bundle_sha256":"56aecc7195d26f6822a4fda8d9cd9d0b98d2a51aa2829f49c3fcc7a07a39ec85"}}