{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2005:FXWDGK6LPJ4UFKFFJ4377PYKVB","short_pith_number":"pith:FXWDGK6L","canonical_record":{"source":{"id":"math/0508245","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2005-08-14T19:15:17Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"732d89e9184f7193013e21cd5ca42cf0ff40a974d878245cc0a55a05af10fcc4","abstract_canon_sha256":"7735fb08ecb396e711b598b40c146228c7a13d73fd1bca2552f85d1813bb4012"},"schema_version":"1.0"},"canonical_sha256":"2dec332bcb7a7942a8a54f37ffbf0aa879f97fb1d1fbad1436bba4440c3152c9","source":{"kind":"arxiv","id":"math/0508245","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0508245","created_at":"2026-05-18T04:39:29Z"},{"alias_kind":"arxiv_version","alias_value":"math/0508245v2","created_at":"2026-05-18T04:39:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0508245","created_at":"2026-05-18T04:39:29Z"},{"alias_kind":"pith_short_12","alias_value":"FXWDGK6LPJ4U","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"FXWDGK6LPJ4UFKFF","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"FXWDGK6L","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2005:FXWDGK6LPJ4UFKFFJ4377PYKVB","target":"record","payload":{"canonical_record":{"source":{"id":"math/0508245","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2005-08-14T19:15:17Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"732d89e9184f7193013e21cd5ca42cf0ff40a974d878245cc0a55a05af10fcc4","abstract_canon_sha256":"7735fb08ecb396e711b598b40c146228c7a13d73fd1bca2552f85d1813bb4012"},"schema_version":"1.0"},"canonical_sha256":"2dec332bcb7a7942a8a54f37ffbf0aa879f97fb1d1fbad1436bba4440c3152c9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:29.100477Z","signature_b64":"rmgiWW1JgqdlwO1Un9DU/fjr34gv7DGSWJ30ABUfYikOMzPiS6xbN8YD44O02TYpJe7ZKuAf5guGVpyHrMB5Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2dec332bcb7a7942a8a54f37ffbf0aa879f97fb1d1fbad1436bba4440c3152c9","last_reissued_at":"2026-05-18T04:39:29.099826Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:29.099826Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0508245","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":"2026-05-18T04:39:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+4T9bp+/JNZD3+6fxEwbwr1dfGuh5ss4zGV9oTiXXa/wuAI6HPwG3VJ4u7tCAPwx+vwdEF4Y6vq06QLttZJGCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T04:48:52.892113Z"},"content_sha256":"14856a54af256baa2315c25cf01566fc0d83ae7b97145c27be1c69296c6d377a","schema_version":"1.0","event_id":"sha256:14856a54af256baa2315c25cf01566fc0d83ae7b97145c27be1c69296c6d377a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2005:FXWDGK6LPJ4UFKFFJ4377PYKVB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Arithmetic of Distributions in Free Probability Theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.OA","authors_text":"F. G\\\"otze, G. Chistyakov","submitted_at":"2005-08-14T19:15:17Z","abstract_excerpt":"We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and of Schur functions. We consider the set of probability distributions as a semigroup $\\bold M$ equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of $\\bold M$ contains either indecomposable (\"prime\") factors or it belongs to a class, say $I_0$, of distributions without indecomposable factors. In contrast to the classical convolution semigroup in the free ad"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0508245","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":"2026-05-18T04:39:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HFzuFuuhylM9XHzPhJAuImshzUb6yY5R3/ngIAHyh3cAuThv7Od0EBZVr4NtksXPtIecT/Unil2Cd5IxLWG+AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T04:48:52.892474Z"},"content_sha256":"29c5f7662796903e4d3392cc852dc48b825e7ef4459ce5af5b00a9cc7b901e95","schema_version":"1.0","event_id":"sha256:29c5f7662796903e4d3392cc852dc48b825e7ef4459ce5af5b00a9cc7b901e95"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FXWDGK6LPJ4UFKFFJ4377PYKVB/bundle.json","state_url":"https://pith.science/pith/FXWDGK6LPJ4UFKFFJ4377PYKVB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FXWDGK6LPJ4UFKFFJ4377PYKVB/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":"2026-06-05T04:48:52Z","links":{"resolver":"https://pith.science/pith/FXWDGK6LPJ4UFKFFJ4377PYKVB","bundle":"https://pith.science/pith/FXWDGK6LPJ4UFKFFJ4377PYKVB/bundle.json","state":"https://pith.science/pith/FXWDGK6LPJ4UFKFFJ4377PYKVB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FXWDGK6LPJ4UFKFFJ4377PYKVB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:FXWDGK6LPJ4UFKFFJ4377PYKVB","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":"7735fb08ecb396e711b598b40c146228c7a13d73fd1bca2552f85d1813bb4012","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2005-08-14T19:15:17Z","title_canon_sha256":"732d89e9184f7193013e21cd5ca42cf0ff40a974d878245cc0a55a05af10fcc4"},"schema_version":"1.0","source":{"id":"math/0508245","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0508245","created_at":"2026-05-18T04:39:29Z"},{"alias_kind":"arxiv_version","alias_value":"math/0508245v2","created_at":"2026-05-18T04:39:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0508245","created_at":"2026-05-18T04:39:29Z"},{"alias_kind":"pith_short_12","alias_value":"FXWDGK6LPJ4U","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"FXWDGK6LPJ4UFKFF","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"FXWDGK6L","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:29c5f7662796903e4d3392cc852dc48b825e7ef4459ce5af5b00a9cc7b901e95","target":"graph","created_at":"2026-05-18T04:39:29Z","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":"We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and of Schur functions. We consider the set of probability distributions as a semigroup $\\bold M$ equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of $\\bold M$ contains either indecomposable (\"prime\") factors or it belongs to a class, say $I_0$, of distributions without indecomposable factors. In contrast to the classical convolution semigroup in the free ad","authors_text":"F. G\\\"otze, G. Chistyakov","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2005-08-14T19:15:17Z","title":"The Arithmetic of Distributions in Free Probability Theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0508245","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":"sha256:14856a54af256baa2315c25cf01566fc0d83ae7b97145c27be1c69296c6d377a","target":"record","created_at":"2026-05-18T04:39:29Z","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":"7735fb08ecb396e711b598b40c146228c7a13d73fd1bca2552f85d1813bb4012","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2005-08-14T19:15:17Z","title_canon_sha256":"732d89e9184f7193013e21cd5ca42cf0ff40a974d878245cc0a55a05af10fcc4"},"schema_version":"1.0","source":{"id":"math/0508245","kind":"arxiv","version":2}},"canonical_sha256":"2dec332bcb7a7942a8a54f37ffbf0aa879f97fb1d1fbad1436bba4440c3152c9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2dec332bcb7a7942a8a54f37ffbf0aa879f97fb1d1fbad1436bba4440c3152c9","first_computed_at":"2026-05-18T04:39:29.099826Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:39:29.099826Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rmgiWW1JgqdlwO1Un9DU/fjr34gv7DGSWJ30ABUfYikOMzPiS6xbN8YD44O02TYpJe7ZKuAf5guGVpyHrMB5Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:39:29.100477Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0508245","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:14856a54af256baa2315c25cf01566fc0d83ae7b97145c27be1c69296c6d377a","sha256:29c5f7662796903e4d3392cc852dc48b825e7ef4459ce5af5b00a9cc7b901e95"],"state_sha256":"773a8be1bc620bd30f826bc8a63cf9def9547461e381178aad026de20135b4ad"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jTVSxJ8ve/+jhfl6j/ZUZT7AOr7SAlxbS/xUvUjtnEzPKNy/9DMqh+CUVPe/hmcX144hZptFd9AiUrTrUjQfCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T04:48:52.894554Z","bundle_sha256":"9f4ffb70067879548ebb047328394f1e46273f975073e48cde272cf418d0fe81"}}