{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:MOCISQGKWXGENQJZPVZKEI4QZU","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":"661173f3a15bb699d563484993a9b0910648da27176b9b69ad1c53764372c6e5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-08-03T19:30:44Z","title_canon_sha256":"c32a9a9bf46a4e15caa4755d320aed9742af7330a78bb316a1dcae4b1e0ddd61"},"schema_version":"1.0","source":{"id":"1308.0733","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.0733","created_at":"2026-05-18T03:16:48Z"},{"alias_kind":"arxiv_version","alias_value":"1308.0733v1","created_at":"2026-05-18T03:16:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.0733","created_at":"2026-05-18T03:16:48Z"},{"alias_kind":"pith_short_12","alias_value":"MOCISQGKWXGE","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_16","alias_value":"MOCISQGKWXGENQJZ","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_8","alias_value":"MOCISQGK","created_at":"2026-05-18T12:27:52Z"}],"graph_snapshots":[{"event_id":"sha256:310265699bb9041f4dcdeabb63da77bcb0a29d451ef340fd1363994003d4b203","target":"graph","created_at":"2026-05-18T03:16:48Z","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 note we report the solution of the problem of defining the $S$-transform in Free Probability Theory in arbitrary dimensions. This is achieved by generalising the theory and embedding it into an algebraic-geometric framework. Finally, we classify the groups arising as distributions of s-tuples of non-commutative random variables.","authors_text":"John McKay, Roland M Friedrich","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-08-03T19:30:44Z","title":"The $S$-transform in arbitrary dimensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.0733","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":"sha256:31048f79f5706960f53a548b0ae1612d596aa92b6fba4c67b5ac2047d1a247f7","target":"record","created_at":"2026-05-18T03:16:48Z","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":"661173f3a15bb699d563484993a9b0910648da27176b9b69ad1c53764372c6e5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-08-03T19:30:44Z","title_canon_sha256":"c32a9a9bf46a4e15caa4755d320aed9742af7330a78bb316a1dcae4b1e0ddd61"},"schema_version":"1.0","source":{"id":"1308.0733","kind":"arxiv","version":1}},"canonical_sha256":"63848940cab5cc46c1397d72a22390cd0d5d94f4405756eb910f2ed4b81d6caa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"63848940cab5cc46c1397d72a22390cd0d5d94f4405756eb910f2ed4b81d6caa","first_computed_at":"2026-05-18T03:16:48.352086Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:16:48.352086Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xQguD8TfnpgYJcgzotG1kDeFYfKIqxNUZ0TtjTQ3cVz99jiLaRAmh2eq6juNh96sTnq41G7jUAr2Uy/CTZTaCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:16:48.352608Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.0733","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:31048f79f5706960f53a548b0ae1612d596aa92b6fba4c67b5ac2047d1a247f7","sha256:310265699bb9041f4dcdeabb63da77bcb0a29d451ef340fd1363994003d4b203"],"state_sha256":"0e1e3013b74afa5b51e1e5e702e8da54cd54504b86ba4600b5b11e771aa94207"}