{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:IPVNSTXGZ7QTNPKBYXMC4YF7CC","short_pith_number":"pith:IPVNSTXG","canonical_record":{"source":{"id":"1108.0458","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-08-02T02:12:20Z","cross_cats_sorted":[],"title_canon_sha256":"afcf7b6310c88dc607c2422f3d16e4fffbfb64cb898a9f81ec9afe4b0ec3116f","abstract_canon_sha256":"ed9d7662dd8ffaef8d778e17d65051bbcd1d885f81c53239da41c8cdb9e6e8ec"},"schema_version":"1.0"},"canonical_sha256":"43ead94ee6cfe136bd41c5d82e60bf10a66dc69cd2d940e598df0c6f7dd6fe72","source":{"kind":"arxiv","id":"1108.0458","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.0458","created_at":"2026-05-18T03:59:01Z"},{"alias_kind":"arxiv_version","alias_value":"1108.0458v1","created_at":"2026-05-18T03:59:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.0458","created_at":"2026-05-18T03:59:01Z"},{"alias_kind":"pith_short_12","alias_value":"IPVNSTXGZ7QT","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"IPVNSTXGZ7QTNPKB","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"IPVNSTXG","created_at":"2026-05-18T12:26:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:IPVNSTXGZ7QTNPKBYXMC4YF7CC","target":"record","payload":{"canonical_record":{"source":{"id":"1108.0458","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-08-02T02:12:20Z","cross_cats_sorted":[],"title_canon_sha256":"afcf7b6310c88dc607c2422f3d16e4fffbfb64cb898a9f81ec9afe4b0ec3116f","abstract_canon_sha256":"ed9d7662dd8ffaef8d778e17d65051bbcd1d885f81c53239da41c8cdb9e6e8ec"},"schema_version":"1.0"},"canonical_sha256":"43ead94ee6cfe136bd41c5d82e60bf10a66dc69cd2d940e598df0c6f7dd6fe72","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:01.931589Z","signature_b64":"A8+3S4wZ1PwqSyDDi/JuCukkTvtu2b7B2y4+9IMX4p2nBYO1mwIyMqeoRJEc2jIgzNkL7T0NhJWUBS2fbsQVDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"43ead94ee6cfe136bd41c5d82e60bf10a66dc69cd2d940e598df0c6f7dd6fe72","last_reissued_at":"2026-05-18T03:59:01.930921Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:01.930921Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1108.0458","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":"2026-05-18T03:59:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tqvnIgG6NkKz0AX3GnEW3ZMSHaeQSM6nqYH+8wE9kQ4fMEZoJqfFgjKXHSsJF3mLJaISFouSFY8nl3zxWSinDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T06:52:21.246967Z"},"content_sha256":"d9d7db930794b79450fe978a282de8a8c1dcddecb32c1df0ddb7e6ba035c0c64","schema_version":"1.0","event_id":"sha256:d9d7db930794b79450fe978a282de8a8c1dcddecb32c1df0ddb7e6ba035c0c64"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:IPVNSTXGZ7QTNPKBYXMC4YF7CC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The classification of Leonard triples of QRacah type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Hau-wen Huang","submitted_at":"2011-08-02T02:12:20Z","abstract_excerpt":"Let $\\mathbb{K}$ denote an algebraically closed field. Let $V$ denote a vector space over $\\mathbb{K}$ with finite positive dimension. By a Leonard triple on $V$ we mean an ordered triple of linear transformations in ${\\rm End}(V)$ such that for each of these transformations there exists a basis of $V$ with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal. There is a family of Leonard triples said to have QRacah type. This is the most general type of Leonard triple. We classify the L"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0458","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":"2026-05-18T03:59:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EReHkCX7k0v473ji9YvJ3injPVRc5gKjzetuQ2/nT0RelJlHmt73xLKjbxmM7jfkZzzIrvDugNyboE0S7Zt6Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T06:52:21.247335Z"},"content_sha256":"62dfc4aadc71330df848ba4d472d4c5ba9f74d2f008cbe856e767f763fe1c718","schema_version":"1.0","event_id":"sha256:62dfc4aadc71330df848ba4d472d4c5ba9f74d2f008cbe856e767f763fe1c718"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IPVNSTXGZ7QTNPKBYXMC4YF7CC/bundle.json","state_url":"https://pith.science/pith/IPVNSTXGZ7QTNPKBYXMC4YF7CC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IPVNSTXGZ7QTNPKBYXMC4YF7CC/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":"2026-06-24T06:52:21Z","links":{"resolver":"https://pith.science/pith/IPVNSTXGZ7QTNPKBYXMC4YF7CC","bundle":"https://pith.science/pith/IPVNSTXGZ7QTNPKBYXMC4YF7CC/bundle.json","state":"https://pith.science/pith/IPVNSTXGZ7QTNPKBYXMC4YF7CC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IPVNSTXGZ7QTNPKBYXMC4YF7CC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:IPVNSTXGZ7QTNPKBYXMC4YF7CC","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":"ed9d7662dd8ffaef8d778e17d65051bbcd1d885f81c53239da41c8cdb9e6e8ec","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-08-02T02:12:20Z","title_canon_sha256":"afcf7b6310c88dc607c2422f3d16e4fffbfb64cb898a9f81ec9afe4b0ec3116f"},"schema_version":"1.0","source":{"id":"1108.0458","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.0458","created_at":"2026-05-18T03:59:01Z"},{"alias_kind":"arxiv_version","alias_value":"1108.0458v1","created_at":"2026-05-18T03:59:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.0458","created_at":"2026-05-18T03:59:01Z"},{"alias_kind":"pith_short_12","alias_value":"IPVNSTXGZ7QT","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"IPVNSTXGZ7QTNPKB","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"IPVNSTXG","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:62dfc4aadc71330df848ba4d472d4c5ba9f74d2f008cbe856e767f763fe1c718","target":"graph","created_at":"2026-05-18T03:59:01Z","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":"Let $\\mathbb{K}$ denote an algebraically closed field. Let $V$ denote a vector space over $\\mathbb{K}$ with finite positive dimension. By a Leonard triple on $V$ we mean an ordered triple of linear transformations in ${\\rm End}(V)$ such that for each of these transformations there exists a basis of $V$ with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal. There is a family of Leonard triples said to have QRacah type. This is the most general type of Leonard triple. We classify the L","authors_text":"Hau-wen Huang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-08-02T02:12:20Z","title":"The classification of Leonard triples of QRacah type"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0458","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:d9d7db930794b79450fe978a282de8a8c1dcddecb32c1df0ddb7e6ba035c0c64","target":"record","created_at":"2026-05-18T03:59:01Z","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":"ed9d7662dd8ffaef8d778e17d65051bbcd1d885f81c53239da41c8cdb9e6e8ec","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-08-02T02:12:20Z","title_canon_sha256":"afcf7b6310c88dc607c2422f3d16e4fffbfb64cb898a9f81ec9afe4b0ec3116f"},"schema_version":"1.0","source":{"id":"1108.0458","kind":"arxiv","version":1}},"canonical_sha256":"43ead94ee6cfe136bd41c5d82e60bf10a66dc69cd2d940e598df0c6f7dd6fe72","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"43ead94ee6cfe136bd41c5d82e60bf10a66dc69cd2d940e598df0c6f7dd6fe72","first_computed_at":"2026-05-18T03:59:01.930921Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:59:01.930921Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A8+3S4wZ1PwqSyDDi/JuCukkTvtu2b7B2y4+9IMX4p2nBYO1mwIyMqeoRJEc2jIgzNkL7T0NhJWUBS2fbsQVDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:59:01.931589Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.0458","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d9d7db930794b79450fe978a282de8a8c1dcddecb32c1df0ddb7e6ba035c0c64","sha256:62dfc4aadc71330df848ba4d472d4c5ba9f74d2f008cbe856e767f763fe1c718"],"state_sha256":"6a27fbc1c867e5a37a58c554fda64463a046cb7d45ed3ef61db0866f91fa62c1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fl/EVkWXDXB6jU88IEYIgUtlWBKP5EN/F4T5yYbMQignqeMnWijkHYNEWxFCJHxBW1TXYWpE3Y+sEBx5swzBDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T06:52:21.249272Z","bundle_sha256":"d9d9886f2a015942cf25c58e460e4acf276533082cb075e625d529794f5c12ee"}}