{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:VCOGKBH4DEOWAJDGGT2QVEOBXZ","short_pith_number":"pith:VCOGKBH4","canonical_record":{"source":{"id":"1511.02177","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-11-06T18:03:51Z","cross_cats_sorted":["math.CA","math.MP","math.QA"],"title_canon_sha256":"19196fcfca3709e114ecd5798aa8d6073ee85c2350af96baa11d9b0c75c78eaf","abstract_canon_sha256":"563d6257516cbebf433c41808a0f0154f977600164e0d3c40999883ce789117b"},"schema_version":"1.0"},"canonical_sha256":"a89c6504fc191d60246634f50a91c1be760674ab8e92e65f56a5a40c405d1a3a","source":{"kind":"arxiv","id":"1511.02177","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.02177","created_at":"2026-05-18T00:50:52Z"},{"alias_kind":"arxiv_version","alias_value":"1511.02177v2","created_at":"2026-05-18T00:50:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.02177","created_at":"2026-05-18T00:50:52Z"},{"alias_kind":"pith_short_12","alias_value":"VCOGKBH4DEOW","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"VCOGKBH4DEOWAJDG","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"VCOGKBH4","created_at":"2026-05-18T12:29:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:VCOGKBH4DEOWAJDGGT2QVEOBXZ","target":"record","payload":{"canonical_record":{"source":{"id":"1511.02177","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-11-06T18:03:51Z","cross_cats_sorted":["math.CA","math.MP","math.QA"],"title_canon_sha256":"19196fcfca3709e114ecd5798aa8d6073ee85c2350af96baa11d9b0c75c78eaf","abstract_canon_sha256":"563d6257516cbebf433c41808a0f0154f977600164e0d3c40999883ce789117b"},"schema_version":"1.0"},"canonical_sha256":"a89c6504fc191d60246634f50a91c1be760674ab8e92e65f56a5a40c405d1a3a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:52.421088Z","signature_b64":"HKVZNcNzNK7St8v9XxfOLd1GZraS4wHvjJNr0RJe47lroE4uvVuA8Rw+Er5jS2JObuMLBjdQkfYDHvsMprDwDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a89c6504fc191d60246634f50a91c1be760674ab8e92e65f56a5a40c405d1a3a","last_reissued_at":"2026-05-18T00:50:52.420638Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:52.420638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.02177","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:50:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dcPV/qA6T1xJRTQnhB8+9XHKZzBE5Ts/m/7NyD96cTcLNdRG1wUMWkSiYJk4NAaloMnI5DNjWhipzCk28VeKDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T01:11:38.574120Z"},"content_sha256":"c15e625f52f3ce492b64ad8da034fc0fedc69029ea0853611ca49daaaa34533c","schema_version":"1.0","event_id":"sha256:c15e625f52f3ce492b64ad8da034fc0fedc69029ea0853611ca49daaaa34533c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:VCOGKBH4DEOWAJDGGT2QVEOBXZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The $\\mathbb{Z}_2^n$ Dirac-Dunkl operator and a higher rank Bannai-Ito algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.MP","math.QA"],"primary_cat":"math-ph","authors_text":"Hendrik De Bie, Luc Vinet, Vincent X. Genest","submitted_at":"2015-11-06T18:03:51Z","abstract_excerpt":"The kernel of the $\\mathbb{Z}_2^{n}$ Dirac-Dunkl operator is examined. The symmetry algebra $\\mathcal{A}_{n}$ of the associated Dirac-Dunkl equation on $\\mathbb{S}^{n-1}$ is determined and is seen to correspond to a higher rank generalization of the Bannai-Ito algebra. A basis for the polynomial null-solutions of the Dirac-Dunkl operator is constructed. The basis elements are joint eigenfunctions of a maximal commutative subalgebra of $\\mathcal{A}_{n}$ and are given explicitly in terms of Jacobi polynomials. The symmetry algebra is shown to act irreducibly on this basis via raising/lowering op"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.02177","kind":"arxiv","version":2},"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":"2026-05-18T00:50:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+ZmGNyC1okdCNZ8qJ+Bvy1/6m0c5LnzdZhc7PGOFmT8eFL8whSJMQkasj5l+y7rDuSGq2t9NWdhkxaV4lMVtAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T01:11:38.574492Z"},"content_sha256":"46035f154046c7388ca918d0d59d68dd1fae6fd10fa63d2ebeec6d0746637d2f","schema_version":"1.0","event_id":"sha256:46035f154046c7388ca918d0d59d68dd1fae6fd10fa63d2ebeec6d0746637d2f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VCOGKBH4DEOWAJDGGT2QVEOBXZ/bundle.json","state_url":"https://pith.science/pith/VCOGKBH4DEOWAJDGGT2QVEOBXZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VCOGKBH4DEOWAJDGGT2QVEOBXZ/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":"2026-05-28T01:11:38Z","links":{"resolver":"https://pith.science/pith/VCOGKBH4DEOWAJDGGT2QVEOBXZ","bundle":"https://pith.science/pith/VCOGKBH4DEOWAJDGGT2QVEOBXZ/bundle.json","state":"https://pith.science/pith/VCOGKBH4DEOWAJDGGT2QVEOBXZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VCOGKBH4DEOWAJDGGT2QVEOBXZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:VCOGKBH4DEOWAJDGGT2QVEOBXZ","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":"563d6257516cbebf433c41808a0f0154f977600164e0d3c40999883ce789117b","cross_cats_sorted":["math.CA","math.MP","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-11-06T18:03:51Z","title_canon_sha256":"19196fcfca3709e114ecd5798aa8d6073ee85c2350af96baa11d9b0c75c78eaf"},"schema_version":"1.0","source":{"id":"1511.02177","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.02177","created_at":"2026-05-18T00:50:52Z"},{"alias_kind":"arxiv_version","alias_value":"1511.02177v2","created_at":"2026-05-18T00:50:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.02177","created_at":"2026-05-18T00:50:52Z"},{"alias_kind":"pith_short_12","alias_value":"VCOGKBH4DEOW","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"VCOGKBH4DEOWAJDG","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"VCOGKBH4","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:46035f154046c7388ca918d0d59d68dd1fae6fd10fa63d2ebeec6d0746637d2f","target":"graph","created_at":"2026-05-18T00:50:52Z","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 kernel of the $\\mathbb{Z}_2^{n}$ Dirac-Dunkl operator is examined. The symmetry algebra $\\mathcal{A}_{n}$ of the associated Dirac-Dunkl equation on $\\mathbb{S}^{n-1}$ is determined and is seen to correspond to a higher rank generalization of the Bannai-Ito algebra. A basis for the polynomial null-solutions of the Dirac-Dunkl operator is constructed. The basis elements are joint eigenfunctions of a maximal commutative subalgebra of $\\mathcal{A}_{n}$ and are given explicitly in terms of Jacobi polynomials. The symmetry algebra is shown to act irreducibly on this basis via raising/lowering op","authors_text":"Hendrik De Bie, Luc Vinet, Vincent X. Genest","cross_cats":["math.CA","math.MP","math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-11-06T18:03:51Z","title":"The $\\mathbb{Z}_2^n$ Dirac-Dunkl operator and a higher rank Bannai-Ito algebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.02177","kind":"arxiv","version":2},"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:c15e625f52f3ce492b64ad8da034fc0fedc69029ea0853611ca49daaaa34533c","target":"record","created_at":"2026-05-18T00:50:52Z","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":"563d6257516cbebf433c41808a0f0154f977600164e0d3c40999883ce789117b","cross_cats_sorted":["math.CA","math.MP","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-11-06T18:03:51Z","title_canon_sha256":"19196fcfca3709e114ecd5798aa8d6073ee85c2350af96baa11d9b0c75c78eaf"},"schema_version":"1.0","source":{"id":"1511.02177","kind":"arxiv","version":2}},"canonical_sha256":"a89c6504fc191d60246634f50a91c1be760674ab8e92e65f56a5a40c405d1a3a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a89c6504fc191d60246634f50a91c1be760674ab8e92e65f56a5a40c405d1a3a","first_computed_at":"2026-05-18T00:50:52.420638Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:50:52.420638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HKVZNcNzNK7St8v9XxfOLd1GZraS4wHvjJNr0RJe47lroE4uvVuA8Rw+Er5jS2JObuMLBjdQkfYDHvsMprDwDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:50:52.421088Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.02177","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c15e625f52f3ce492b64ad8da034fc0fedc69029ea0853611ca49daaaa34533c","sha256:46035f154046c7388ca918d0d59d68dd1fae6fd10fa63d2ebeec6d0746637d2f"],"state_sha256":"e29fa7d56d997110eeb918cce15181535e2365398c95c12a6828e564391282d7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c+v97Ysb74MVSlcAARarxylKjhwJgt/H1VeP1teZAQdRrPyalkJDGtLWHtIRrsM1pZuxBggB6efbIv9bQXHYBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T01:11:38.576517Z","bundle_sha256":"3ee33d8dfa73ba46624b09966cbe6932f63a8384b48f465037e3043494795a91"}}