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We show that there exist invertible $W, W', W'' $ in ${\\rm End}(V)$ such that (i) $A$ commutes with $W$ and $W^{-1}BW-C$; (ii) $B$ commutes with $W'$ and $(W')^{-1}CW'-A$; (iii) $C$ commutes with $W''$ and $(W'')^{-1}AW''-B$. Moreover each of $W,W', W''$ is unique up to multiplication by a nonzero scalar in $\\mathbb F$. We show that the three elements $W"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1609.05488","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2016-09-18T14:03:25Z","cross_cats_sorted":[],"title_canon_sha256":"040802ac3b32cb8fdcf2849a7194199647c08bad71cfe2160852c13061e8e86d","abstract_canon_sha256":"5065bf4b1ae235be1157e146ed6fb48486ab070e319f09b7f33bc81012ca1086"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:26.274762Z","signature_b64":"FqcLQL0HGjInf+BIcNHKReFvrLYtSsKuvCBLlM7o4vCg4B/Nrf/AIfEKWI5h+vGtkNFgUbHDTmOqtm0k0uZvCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a26c5ba46f096f3f331de4c16e5d1469f3e44b537a0c4335e286be68fced0c2f","last_reissued_at":"2026-05-18T01:04:26.274034Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:26.274034Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Leonard triples of $q$-Racah type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Paul Terwilliger","submitted_at":"2016-09-18T14:03:25Z","abstract_excerpt":"Let $\\mathbb F$ denote a field, and let $V$ denote a vector space over $\\mathbb F$ with finite positive dimension. Pick a nonzero $q \\in \\mathbb F$ such that $q^4 \\not=1$, and let $A,B,C$ denote a Leonard triple on $V$ that has $q$-Racah type. We show that there exist invertible $W, W', W'' $ in ${\\rm End}(V)$ such that (i) $A$ commutes with $W$ and $W^{-1}BW-C$; (ii) $B$ commutes with $W'$ and $(W')^{-1}CW'-A$; (iii) $C$ commutes with $W''$ and $(W'')^{-1}AW''-B$. Moreover each of $W,W', W''$ is unique up to multiplication by a nonzero scalar in $\\mathbb F$. We show that the three elements $W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05488","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1609.05488","created_at":"2026-05-18T01:04:26.274145+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.05488v1","created_at":"2026-05-18T01:04:26.274145+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.05488","created_at":"2026-05-18T01:04:26.274145+00:00"},{"alias_kind":"pith_short_12","alias_value":"UJWFXJDPBFXT","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_16","alias_value":"UJWFXJDPBFXT6MY5","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_8","alias_value":"UJWFXJDP","created_at":"2026-05-18T12:30:46.583412+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UJWFXJDPBFXT6MY54TAW4XIUNH","json":"https://pith.science/pith/UJWFXJDPBFXT6MY54TAW4XIUNH.json","graph_json":"https://pith.science/api/pith-number/UJWFXJDPBFXT6MY54TAW4XIUNH/graph.json","events_json":"https://pith.science/api/pith-number/UJWFXJDPBFXT6MY54TAW4XIUNH/events.json","paper":"https://pith.science/paper/UJWFXJDP"},"agent_actions":{"view_html":"https://pith.science/pith/UJWFXJDPBFXT6MY54TAW4XIUNH","download_json":"https://pith.science/pith/UJWFXJDPBFXT6MY54TAW4XIUNH.json","view_paper":"https://pith.science/paper/UJWFXJDP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.05488&json=true","fetch_graph":"https://pith.science/api/pith-number/UJWFXJDPBFXT6MY54TAW4XIUNH/graph.json","fetch_events":"https://pith.science/api/pith-number/UJWFXJDPBFXT6MY54TAW4XIUNH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UJWFXJDPBFXT6MY54TAW4XIUNH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UJWFXJDPBFXT6MY54TAW4XIUNH/action/storage_attestation","attest_author":"https://pith.science/pith/UJWFXJDPBFXT6MY54TAW4XIUNH/action/author_attestation","sign_citation":"https://pith.science/pith/UJWFXJDPBFXT6MY54TAW4XIUNH/action/citation_signature","submit_replication":"https://pith.science/pith/UJWFXJDPBFXT6MY54TAW4XIUNH/action/replication_record"}},"created_at":"2026-05-18T01:04:26.274145+00:00","updated_at":"2026-05-18T01:04:26.274145+00:00"}