{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:IK4RHCLPYTVYPJYI7J7M7D3GON","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":"d2f5e848e151c0b7469f8fe104a806060cfbc14ef9e298852d6f46cc6790c8b8","cross_cats_sorted":["math.FA","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2013-09-04T00:24:51Z","title_canon_sha256":"b21594c6f7902f3a7ef3b71c9514b79a8e3a8985a0b42b01fdc788fa7d82e071"},"schema_version":"1.0","source":{"id":"1309.0877","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.0877","created_at":"2026-05-18T02:51:06Z"},{"alias_kind":"arxiv_version","alias_value":"1309.0877v3","created_at":"2026-05-18T02:51:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.0877","created_at":"2026-05-18T02:51:06Z"},{"alias_kind":"pith_short_12","alias_value":"IK4RHCLPYTVY","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"IK4RHCLPYTVYPJYI","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"IK4RHCLP","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:eafeb541c3b110122afc43d41d760d729c530017299686ec05e55f3a0f471f80","target":"graph","created_at":"2026-05-18T02:51:06Z","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":"It is a classical result in complex analysis that the class of functions that arise as the Cauchy transform of probability measures may be characterized entirely in terms of their analytic and asymptotic properties. Such transforms are a main object of study in non-commutative probability theory as the function theory encodes information on the probability measures and the various convolution operations. In extending this theory to operator-valued free probability theory, the analogue of the Cauchy transform is a non-commutative function with domain equal to the non-commutative upper-half plan","authors_text":"John D. Williams","cross_cats":["math.FA","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2013-09-04T00:24:51Z","title":"Analytic Function Theory for Operator-Valued Free Probability"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0877","kind":"arxiv","version":3},"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":"sha256:39135ce7d52381014248a314101d8a2ed094c0732856567152d0d9fce5122ba7","target":"record","created_at":"2026-05-18T02:51:06Z","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":"d2f5e848e151c0b7469f8fe104a806060cfbc14ef9e298852d6f46cc6790c8b8","cross_cats_sorted":["math.FA","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2013-09-04T00:24:51Z","title_canon_sha256":"b21594c6f7902f3a7ef3b71c9514b79a8e3a8985a0b42b01fdc788fa7d82e071"},"schema_version":"1.0","source":{"id":"1309.0877","kind":"arxiv","version":3}},"canonical_sha256":"42b913896fc4eb87a708fa7ecf8f66737d9fde4475f16a32bccbe9bb4b75a54d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"42b913896fc4eb87a708fa7ecf8f66737d9fde4475f16a32bccbe9bb4b75a54d","first_computed_at":"2026-05-18T02:51:06.481864Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:51:06.481864Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yGibIKg769U581R9ezfzHQNAppOX+4piYXOpWjW46jJ+LUrjV0iiWTN6WrhdGlTHiYwOiJqbLUMRX4WVQy+hBA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:51:06.482390Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.0877","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:39135ce7d52381014248a314101d8a2ed094c0732856567152d0d9fce5122ba7","sha256:eafeb541c3b110122afc43d41d760d729c530017299686ec05e55f3a0f471f80"],"state_sha256":"60e13b249f3a96d12d658bb56efb987cb383c16f677e8ede9bb07334db9f5999"}