{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:MYCEXK7X7RISGLWNT73NRL4YTI","short_pith_number":"pith:MYCEXK7X","canonical_record":{"source":{"id":"1801.03401","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2018-01-09T01:19:29Z","cross_cats_sorted":[],"title_canon_sha256":"5fac1e136f5743f28fa747a48a773a4da255ba52c0583d41399cce095de59c58","abstract_canon_sha256":"8635d5a9353ef8152106046a2218788682f5761e125cad102aa0ac8d34b92730"},"schema_version":"1.0"},"canonical_sha256":"66044babf7fc51232ecd9ff6d8af989a19b657ae48fb232d0690f04028ec4495","source":{"kind":"arxiv","id":"1801.03401","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.03401","created_at":"2026-05-18T00:26:17Z"},{"alias_kind":"arxiv_version","alias_value":"1801.03401v1","created_at":"2026-05-18T00:26:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.03401","created_at":"2026-05-18T00:26:17Z"},{"alias_kind":"pith_short_12","alias_value":"MYCEXK7X7RIS","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"MYCEXK7X7RISGLWN","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"MYCEXK7X","created_at":"2026-05-18T12:32:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:MYCEXK7X7RISGLWNT73NRL4YTI","target":"record","payload":{"canonical_record":{"source":{"id":"1801.03401","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2018-01-09T01:19:29Z","cross_cats_sorted":[],"title_canon_sha256":"5fac1e136f5743f28fa747a48a773a4da255ba52c0583d41399cce095de59c58","abstract_canon_sha256":"8635d5a9353ef8152106046a2218788682f5761e125cad102aa0ac8d34b92730"},"schema_version":"1.0"},"canonical_sha256":"66044babf7fc51232ecd9ff6d8af989a19b657ae48fb232d0690f04028ec4495","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:17.177695Z","signature_b64":"rrKG41lnXA7yEjUfNLlj6hxbhaQkxcZa4ipigsSKJeAEVESYVdN6f680BSvQV+k2iWV18eeqcBPfgEPs07x2CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"66044babf7fc51232ecd9ff6d8af989a19b657ae48fb232d0690f04028ec4495","last_reissued_at":"2026-05-18T00:26:17.177091Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:17.177091Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1801.03401","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":"2026-05-18T00:26:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+I2DLWQTR0Qon12kVlTsw9ImwnUILr2mz27QEhk+SOMR4+LNnb6Ft757nzs2qtchDplcsEkLJ0XIXRTQMxDZCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T08:32:39.220037Z"},"content_sha256":"824d6e3e04587f6d137987084a042e7a50ff5c6a1af3a7f549923e4e6492d649","schema_version":"1.0","event_id":"sha256:824d6e3e04587f6d137987084a042e7a50ff5c6a1af3a7f549923e4e6492d649"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:MYCEXK7X7RISGLWNT73NRL4YTI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Free-free-Boolean independence for triples of algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Weihua Liu","submitted_at":"2018-01-09T01:19:29Z","abstract_excerpt":"In this paper, we introduce the notion of free-free-Boolean independence relation for triples of algebras. We define free-free-Boolean cumulants ans show that the vanishing of mixed cumulants is equivalent to free-free-Boolean independence. A free-free -Boolean central limit law is studied."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.03401","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":"2026-05-18T00:26:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W8ExQbMMynXzZQyBQUH2tqGuFI3X9CjGF2ZrtbHtOkETJtX+vecGafs3kkK7CK4W+kFoNgT7yZYckjZalHqBAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T08:32:39.220400Z"},"content_sha256":"2eb9e7a5310eb10eb6a922880c2776b524b1a61482212386a548754f98b8b764","schema_version":"1.0","event_id":"sha256:2eb9e7a5310eb10eb6a922880c2776b524b1a61482212386a548754f98b8b764"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MYCEXK7X7RISGLWNT73NRL4YTI/bundle.json","state_url":"https://pith.science/pith/MYCEXK7X7RISGLWNT73NRL4YTI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MYCEXK7X7RISGLWNT73NRL4YTI/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":"2026-06-04T08:32:39Z","links":{"resolver":"https://pith.science/pith/MYCEXK7X7RISGLWNT73NRL4YTI","bundle":"https://pith.science/pith/MYCEXK7X7RISGLWNT73NRL4YTI/bundle.json","state":"https://pith.science/pith/MYCEXK7X7RISGLWNT73NRL4YTI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MYCEXK7X7RISGLWNT73NRL4YTI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:MYCEXK7X7RISGLWNT73NRL4YTI","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":"8635d5a9353ef8152106046a2218788682f5761e125cad102aa0ac8d34b92730","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2018-01-09T01:19:29Z","title_canon_sha256":"5fac1e136f5743f28fa747a48a773a4da255ba52c0583d41399cce095de59c58"},"schema_version":"1.0","source":{"id":"1801.03401","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.03401","created_at":"2026-05-18T00:26:17Z"},{"alias_kind":"arxiv_version","alias_value":"1801.03401v1","created_at":"2026-05-18T00:26:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.03401","created_at":"2026-05-18T00:26:17Z"},{"alias_kind":"pith_short_12","alias_value":"MYCEXK7X7RIS","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"MYCEXK7X7RISGLWN","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"MYCEXK7X","created_at":"2026-05-18T12:32:40Z"}],"graph_snapshots":[{"event_id":"sha256:2eb9e7a5310eb10eb6a922880c2776b524b1a61482212386a548754f98b8b764","target":"graph","created_at":"2026-05-18T00:26:17Z","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":"In this paper, we introduce the notion of free-free-Boolean independence relation for triples of algebras. We define free-free-Boolean cumulants ans show that the vanishing of mixed cumulants is equivalent to free-free-Boolean independence. A free-free -Boolean central limit law is studied.","authors_text":"Weihua Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2018-01-09T01:19:29Z","title":"Free-free-Boolean independence for triples of algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.03401","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":"sha256:824d6e3e04587f6d137987084a042e7a50ff5c6a1af3a7f549923e4e6492d649","target":"record","created_at":"2026-05-18T00:26:17Z","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":"8635d5a9353ef8152106046a2218788682f5761e125cad102aa0ac8d34b92730","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2018-01-09T01:19:29Z","title_canon_sha256":"5fac1e136f5743f28fa747a48a773a4da255ba52c0583d41399cce095de59c58"},"schema_version":"1.0","source":{"id":"1801.03401","kind":"arxiv","version":1}},"canonical_sha256":"66044babf7fc51232ecd9ff6d8af989a19b657ae48fb232d0690f04028ec4495","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"66044babf7fc51232ecd9ff6d8af989a19b657ae48fb232d0690f04028ec4495","first_computed_at":"2026-05-18T00:26:17.177091Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:26:17.177091Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rrKG41lnXA7yEjUfNLlj6hxbhaQkxcZa4ipigsSKJeAEVESYVdN6f680BSvQV+k2iWV18eeqcBPfgEPs07x2CA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:26:17.177695Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.03401","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:824d6e3e04587f6d137987084a042e7a50ff5c6a1af3a7f549923e4e6492d649","sha256:2eb9e7a5310eb10eb6a922880c2776b524b1a61482212386a548754f98b8b764"],"state_sha256":"978e0884aeae50677f48e11b31745637c38c672b1821ad6411fabb26462d0875"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WmLjwrtq8FcdzoTKHz24I0JFXOECHopuEnBqYRrYcO9WfyZgztYhaIjsVf44PPCuDDSD6DePd3GpnMqbucklBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T08:32:39.222421Z","bundle_sha256":"dae7324b8f424966941fceb568751f4c228da3b265907cb4a5784f4a0ba60b84"}}