{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:W4U7XB7X4LEH3S5EYVWJNDOGH5","short_pith_number":"pith:W4U7XB7X","canonical_record":{"source":{"id":"0907.3275","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-07-20T08:33:28Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"bd2f4bed4899a5be0ba7133d195dfff0e8459f85781c0bdc10d4da5858167d96","abstract_canon_sha256":"e024fc511dcb513c2c33581cdb642fb73f4c58c3147bfccc40d3c47a42ef135c"},"schema_version":"1.0"},"canonical_sha256":"b729fb87f7e2c87dcba4c56c968dc63f7c06fed28df75b41aa26035bbb5598d0","source":{"kind":"arxiv","id":"0907.3275","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.3275","created_at":"2026-05-18T04:36:40Z"},{"alias_kind":"arxiv_version","alias_value":"0907.3275v2","created_at":"2026-05-18T04:36:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.3275","created_at":"2026-05-18T04:36:40Z"},{"alias_kind":"pith_short_12","alias_value":"W4U7XB7X4LEH","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"W4U7XB7X4LEH3S5E","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"W4U7XB7X","created_at":"2026-05-18T12:26:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:W4U7XB7X4LEH3S5EYVWJNDOGH5","target":"record","payload":{"canonical_record":{"source":{"id":"0907.3275","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-07-20T08:33:28Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"bd2f4bed4899a5be0ba7133d195dfff0e8459f85781c0bdc10d4da5858167d96","abstract_canon_sha256":"e024fc511dcb513c2c33581cdb642fb73f4c58c3147bfccc40d3c47a42ef135c"},"schema_version":"1.0"},"canonical_sha256":"b729fb87f7e2c87dcba4c56c968dc63f7c06fed28df75b41aa26035bbb5598d0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:36:40.505227Z","signature_b64":"e8YLecasCPTpOPQT82IfnpV0s0DwNK1yKAKFhOqm0CkcKqFjmmhg0EeVSJvpG8yDqq8no2reOWvx+CSzJd4pAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b729fb87f7e2c87dcba4c56c968dc63f7c06fed28df75b41aa26035bbb5598d0","last_reissued_at":"2026-05-18T04:36:40.504754Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:36:40.504754Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0907.3275","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":"2026-05-18T04:36:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0ptE/eLBpRWu6C2nFguynxiYM/iDZf9MIa4tCZl0v5D8RjPtCJm0t1VrzxQfYkhWyOhq9RizGwTUDhGVhWTVBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T16:58:55.824470Z"},"content_sha256":"da2ccd44690362a77e50fbc4cfc9db71b29481d0da35c2036ff231fa45dd2ff1","schema_version":"1.0","event_id":"sha256:da2ccd44690362a77e50fbc4cfc9db71b29481d0da35c2036ff231fa45dd2ff1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:W4U7XB7X4LEH3S5EYVWJNDOGH5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$q$-exchangeability via quasi-invariance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Alexander Gnedin, Grigori Olshanski","submitted_at":"2009-07-20T08:33:28Z","abstract_excerpt":"For positive $q\\neq1$, the $q$-exchangeability of an infinite random word is introduced as quasi-invariance under permutations of letters, with a special cocycle which accounts for inversions in the word. This framework allows us to extend the $q$-analog of de Finetti's theorem for binary sequences---see Gnedin and Olshanski [Electron. J. Combin. 16 (2009) R78]---to general real-valued sequences. In contrast to the classical case of exchangeability ($q=1$), the order on $\\mathbb{R}$ plays a significant role for the $q$-analogs. An explicit construction of ergodic $q$-exchangeable measures invo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.3275","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":"2026-05-18T04:36:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"18k6/t2Gm+lFFxZSDVWxa2FjgH+ZoDyt4QrciJrYdAi+OrnNP5TLuozwo50tF+rOBIjEeSZeRrAZPAw/CfCSCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T16:58:55.825145Z"},"content_sha256":"1e859311eefa5f14e31a7c66fa2d40aef42bfb8e44b23fcac13d4505a40e1c55","schema_version":"1.0","event_id":"sha256:1e859311eefa5f14e31a7c66fa2d40aef42bfb8e44b23fcac13d4505a40e1c55"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W4U7XB7X4LEH3S5EYVWJNDOGH5/bundle.json","state_url":"https://pith.science/pith/W4U7XB7X4LEH3S5EYVWJNDOGH5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W4U7XB7X4LEH3S5EYVWJNDOGH5/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":"2026-06-06T16:58:55Z","links":{"resolver":"https://pith.science/pith/W4U7XB7X4LEH3S5EYVWJNDOGH5","bundle":"https://pith.science/pith/W4U7XB7X4LEH3S5EYVWJNDOGH5/bundle.json","state":"https://pith.science/pith/W4U7XB7X4LEH3S5EYVWJNDOGH5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W4U7XB7X4LEH3S5EYVWJNDOGH5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:W4U7XB7X4LEH3S5EYVWJNDOGH5","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":"e024fc511dcb513c2c33581cdb642fb73f4c58c3147bfccc40d3c47a42ef135c","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-07-20T08:33:28Z","title_canon_sha256":"bd2f4bed4899a5be0ba7133d195dfff0e8459f85781c0bdc10d4da5858167d96"},"schema_version":"1.0","source":{"id":"0907.3275","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.3275","created_at":"2026-05-18T04:36:40Z"},{"alias_kind":"arxiv_version","alias_value":"0907.3275v2","created_at":"2026-05-18T04:36:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.3275","created_at":"2026-05-18T04:36:40Z"},{"alias_kind":"pith_short_12","alias_value":"W4U7XB7X4LEH","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"W4U7XB7X4LEH3S5E","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"W4U7XB7X","created_at":"2026-05-18T12:26:02Z"}],"graph_snapshots":[{"event_id":"sha256:1e859311eefa5f14e31a7c66fa2d40aef42bfb8e44b23fcac13d4505a40e1c55","target":"graph","created_at":"2026-05-18T04:36:40Z","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":"For positive $q\\neq1$, the $q$-exchangeability of an infinite random word is introduced as quasi-invariance under permutations of letters, with a special cocycle which accounts for inversions in the word. This framework allows us to extend the $q$-analog of de Finetti's theorem for binary sequences---see Gnedin and Olshanski [Electron. J. Combin. 16 (2009) R78]---to general real-valued sequences. In contrast to the classical case of exchangeability ($q=1$), the order on $\\mathbb{R}$ plays a significant role for the $q$-analogs. An explicit construction of ergodic $q$-exchangeable measures invo","authors_text":"Alexander Gnedin, Grigori Olshanski","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-07-20T08:33:28Z","title":"$q$-exchangeability via quasi-invariance"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.3275","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:da2ccd44690362a77e50fbc4cfc9db71b29481d0da35c2036ff231fa45dd2ff1","target":"record","created_at":"2026-05-18T04:36:40Z","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":"e024fc511dcb513c2c33581cdb642fb73f4c58c3147bfccc40d3c47a42ef135c","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-07-20T08:33:28Z","title_canon_sha256":"bd2f4bed4899a5be0ba7133d195dfff0e8459f85781c0bdc10d4da5858167d96"},"schema_version":"1.0","source":{"id":"0907.3275","kind":"arxiv","version":2}},"canonical_sha256":"b729fb87f7e2c87dcba4c56c968dc63f7c06fed28df75b41aa26035bbb5598d0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b729fb87f7e2c87dcba4c56c968dc63f7c06fed28df75b41aa26035bbb5598d0","first_computed_at":"2026-05-18T04:36:40.504754Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:36:40.504754Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"e8YLecasCPTpOPQT82IfnpV0s0DwNK1yKAKFhOqm0CkcKqFjmmhg0EeVSJvpG8yDqq8no2reOWvx+CSzJd4pAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:36:40.505227Z","signed_message":"canonical_sha256_bytes"},"source_id":"0907.3275","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:da2ccd44690362a77e50fbc4cfc9db71b29481d0da35c2036ff231fa45dd2ff1","sha256:1e859311eefa5f14e31a7c66fa2d40aef42bfb8e44b23fcac13d4505a40e1c55"],"state_sha256":"fce342a7dba44d99253f7ff1fc75be4f69979091db608853186d9d6885170536"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eYFh1r5uOO1kzoFdNTM4Q2Y4Z8YV8uY/mcTuMKw5ni3wpFENw2KmUt4dqxOo1l00DgaGbJ3u1Tfb7YHarM+eAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T16:58:55.828987Z","bundle_sha256":"19972842f7933a93cbec44ec2fd84e81e7b8c9a56962c700b3eeb3383e6cc13c"}}