{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:YZC5ISFWXYDQ4RTJVYCYSSMLYM","short_pith_number":"pith:YZC5ISFW","canonical_record":{"source":{"id":"1609.07820","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2016-09-26T00:29:16Z","cross_cats_sorted":[],"title_canon_sha256":"30dc6df4bcb4924c43e90c2453752776f7f03506365743a2456ce28242a41099","abstract_canon_sha256":"2247fcc901e04bac2facbe7ba3ab55f026945a12d6a05a13fd2733f443d53a4e"},"schema_version":"1.0"},"canonical_sha256":"c645d448b6be070e4669ae0589498bc338cb164656d7e0f61ba8dd38435c07fd","source":{"kind":"arxiv","id":"1609.07820","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.07820","created_at":"2026-05-17T23:54:37Z"},{"alias_kind":"arxiv_version","alias_value":"1609.07820v1","created_at":"2026-05-17T23:54:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.07820","created_at":"2026-05-17T23:54:37Z"},{"alias_kind":"pith_short_12","alias_value":"YZC5ISFWXYDQ","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"YZC5ISFWXYDQ4RTJ","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"YZC5ISFW","created_at":"2026-05-18T12:30:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:YZC5ISFWXYDQ4RTJVYCYSSMLYM","target":"record","payload":{"canonical_record":{"source":{"id":"1609.07820","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2016-09-26T00:29:16Z","cross_cats_sorted":[],"title_canon_sha256":"30dc6df4bcb4924c43e90c2453752776f7f03506365743a2456ce28242a41099","abstract_canon_sha256":"2247fcc901e04bac2facbe7ba3ab55f026945a12d6a05a13fd2733f443d53a4e"},"schema_version":"1.0"},"canonical_sha256":"c645d448b6be070e4669ae0589498bc338cb164656d7e0f61ba8dd38435c07fd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:37.002274Z","signature_b64":"AbdsdczhcjO8HlCqk7NRvZZDRRONU78m6QS7sPecPw4UEUSb3J6TCGcoGk7sZeB0mRAGEXjg80yuYjK5V5qlAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c645d448b6be070e4669ae0589498bc338cb164656d7e0f61ba8dd38435c07fd","last_reissued_at":"2026-05-17T23:54:37.001733Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:37.001733Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1609.07820","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":"2026-05-17T23:54:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1eFzEJmJ2I/s+lA9E74pVdWH+QIAaJrQ0F4yxJJjuo8L7xFwVlvEwNAo3HKkcHBEXzzlub0hOB6pIwnILE5oDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T14:59:48.306423Z"},"content_sha256":"7f4b80004c7cc03ab278e20a6b46f9d7255c7f7f49b4717516c8bc7fdfb0deb5","schema_version":"1.0","event_id":"sha256:7f4b80004c7cc03ab278e20a6b46f9d7255c7f7f49b4717516c8bc7fdfb0deb5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:YZC5ISFWXYDQ4RTJVYCYSSMLYM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Conditionally Bi-Free Independence with Amalgamation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Paul Skoufranis, Yinzheng Gu","submitted_at":"2016-09-26T00:29:16Z","abstract_excerpt":"In this paper, we introduce the notion of conditionally bi-free independence in an amalgamated setting. We define operator-valued conditionally bi-multiplicative pairs of functions and construct operator-valued conditionally bi-free moment and cumulant functions. It is demonstrated that conditionally bi-free independence with amalgamation is equivalent to the vanishing of mixed operator-valued bi-free and conditionally bi-free cumulants. Furthermore, an operator-valued conditionally bi-free partial $\\mathcal{R}$-transform is constructed and various operator-valued conditionally bi-free limit t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.07820","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":"2026-05-17T23:54:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H4lwCbymY6r/PIuTxyO7zQwbqvgKlLTVDqid92YOqXzIh3t9f9rxHkO7P8ZArPOU6dF9OFJ6c5osjdTQSbI3Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T14:59:48.306770Z"},"content_sha256":"98c37a31b99cbe010b34755fb8c241f0ab9678de8cea7b15c0c009c63da2b747","schema_version":"1.0","event_id":"sha256:98c37a31b99cbe010b34755fb8c241f0ab9678de8cea7b15c0c009c63da2b747"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YZC5ISFWXYDQ4RTJVYCYSSMLYM/bundle.json","state_url":"https://pith.science/pith/YZC5ISFWXYDQ4RTJVYCYSSMLYM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YZC5ISFWXYDQ4RTJVYCYSSMLYM/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":"2026-06-08T14:59:48Z","links":{"resolver":"https://pith.science/pith/YZC5ISFWXYDQ4RTJVYCYSSMLYM","bundle":"https://pith.science/pith/YZC5ISFWXYDQ4RTJVYCYSSMLYM/bundle.json","state":"https://pith.science/pith/YZC5ISFWXYDQ4RTJVYCYSSMLYM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YZC5ISFWXYDQ4RTJVYCYSSMLYM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:YZC5ISFWXYDQ4RTJVYCYSSMLYM","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":"2247fcc901e04bac2facbe7ba3ab55f026945a12d6a05a13fd2733f443d53a4e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2016-09-26T00:29:16Z","title_canon_sha256":"30dc6df4bcb4924c43e90c2453752776f7f03506365743a2456ce28242a41099"},"schema_version":"1.0","source":{"id":"1609.07820","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.07820","created_at":"2026-05-17T23:54:37Z"},{"alias_kind":"arxiv_version","alias_value":"1609.07820v1","created_at":"2026-05-17T23:54:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.07820","created_at":"2026-05-17T23:54:37Z"},{"alias_kind":"pith_short_12","alias_value":"YZC5ISFWXYDQ","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"YZC5ISFWXYDQ4RTJ","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"YZC5ISFW","created_at":"2026-05-18T12:30:53Z"}],"graph_snapshots":[{"event_id":"sha256:98c37a31b99cbe010b34755fb8c241f0ab9678de8cea7b15c0c009c63da2b747","target":"graph","created_at":"2026-05-17T23:54:37Z","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 conditionally bi-free independence in an amalgamated setting. We define operator-valued conditionally bi-multiplicative pairs of functions and construct operator-valued conditionally bi-free moment and cumulant functions. It is demonstrated that conditionally bi-free independence with amalgamation is equivalent to the vanishing of mixed operator-valued bi-free and conditionally bi-free cumulants. Furthermore, an operator-valued conditionally bi-free partial $\\mathcal{R}$-transform is constructed and various operator-valued conditionally bi-free limit t","authors_text":"Paul Skoufranis, Yinzheng Gu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2016-09-26T00:29:16Z","title":"Conditionally Bi-Free Independence with Amalgamation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.07820","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":"sha256:7f4b80004c7cc03ab278e20a6b46f9d7255c7f7f49b4717516c8bc7fdfb0deb5","target":"record","created_at":"2026-05-17T23:54:37Z","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":"2247fcc901e04bac2facbe7ba3ab55f026945a12d6a05a13fd2733f443d53a4e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2016-09-26T00:29:16Z","title_canon_sha256":"30dc6df4bcb4924c43e90c2453752776f7f03506365743a2456ce28242a41099"},"schema_version":"1.0","source":{"id":"1609.07820","kind":"arxiv","version":1}},"canonical_sha256":"c645d448b6be070e4669ae0589498bc338cb164656d7e0f61ba8dd38435c07fd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c645d448b6be070e4669ae0589498bc338cb164656d7e0f61ba8dd38435c07fd","first_computed_at":"2026-05-17T23:54:37.001733Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:54:37.001733Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AbdsdczhcjO8HlCqk7NRvZZDRRONU78m6QS7sPecPw4UEUSb3J6TCGcoGk7sZeB0mRAGEXjg80yuYjK5V5qlAg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:54:37.002274Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.07820","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7f4b80004c7cc03ab278e20a6b46f9d7255c7f7f49b4717516c8bc7fdfb0deb5","sha256:98c37a31b99cbe010b34755fb8c241f0ab9678de8cea7b15c0c009c63da2b747"],"state_sha256":"d5107702579d4f7916774bd44c6212a6456bf4a529955e080beda444ff485a47"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jNc+RWoGoH89vqSzY2hslG9+cZEGTiLEXsRaiFbTgeYDJ1d20ima7LAJzc8+8cOfHP2CcdatEju/Og/TZduwBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T14:59:48.308631Z","bundle_sha256":"2a48a57a1aa3d9ffd23e3390ba5a6433f4251dc3f003f756e8e5d62bc20445a5"}}