{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2004:RJU34BKY3CCR55CF7WXHK2MYP6","short_pith_number":"pith:RJU34BKY","canonical_record":{"source":{"id":"math/0406578","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.DS","submitted_at":"2004-06-28T16:30:22Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"4a775dcbfb908e1b056de073ff73d933dab2fc4aa968f36d5853ecb42cd7871f","abstract_canon_sha256":"f8e06fa99cd99ff3f24e9fa5b553cc64e54775dc7c06be196ecac006937bf5d2"},"schema_version":"1.0"},"canonical_sha256":"8a69be0558d8851ef445fdae7569987fb23121b9f55c86303e4ee0091c42f35a","source":{"kind":"arxiv","id":"math/0406578","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0406578","created_at":"2026-05-18T01:38:27Z"},{"alias_kind":"arxiv_version","alias_value":"math/0406578v4","created_at":"2026-05-18T01:38:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0406578","created_at":"2026-05-18T01:38:27Z"},{"alias_kind":"pith_short_12","alias_value":"RJU34BKY3CCR","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"RJU34BKY3CCR55CF","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"RJU34BKY","created_at":"2026-05-18T12:25:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2004:RJU34BKY3CCR55CF7WXHK2MYP6","target":"record","payload":{"canonical_record":{"source":{"id":"math/0406578","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.DS","submitted_at":"2004-06-28T16:30:22Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"4a775dcbfb908e1b056de073ff73d933dab2fc4aa968f36d5853ecb42cd7871f","abstract_canon_sha256":"f8e06fa99cd99ff3f24e9fa5b553cc64e54775dc7c06be196ecac006937bf5d2"},"schema_version":"1.0"},"canonical_sha256":"8a69be0558d8851ef445fdae7569987fb23121b9f55c86303e4ee0091c42f35a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:27.280592Z","signature_b64":"YinYFzLgEZGuyT2MLYlBWjb7d8ExMiTrZmhTPUoa19gZOIO+SRxdj/NRAyNPx/BqZNHExFvCS22J0uq/XWruBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8a69be0558d8851ef445fdae7569987fb23121b9f55c86303e4ee0091c42f35a","last_reissued_at":"2026-05-18T01:38:27.279905Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:27.279905Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0406578","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:38:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bHTyULnOSQZeokj7sRqNzUZYz/DsD2/N0ZuIA1AJr3iMIMwKYyWwCQr3bkX+lyZjQXaSbGsGSD7870tUSBPuDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T04:52:25.558390Z"},"content_sha256":"3c9bab4aace82124303577ad2d1cdb46ea07ae28b95fdbfc2da12787dde799bf","schema_version":"1.0","event_id":"sha256:3c9bab4aace82124303577ad2d1cdb46ea07ae28b95fdbfc2da12787dde799bf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2004:RJU34BKY3CCR55CF7WXHK2MYP6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Exchangeable measures for subshifts","license":"","headline":"","cross_cats":["math.PR"],"primary_cat":"math.DS","authors_text":"H. Nakada, J. Aaronson, O. Sarig","submitted_at":"2004-06-28T16:30:22Z","abstract_excerpt":"Let $\\Om$ be a Borel subset of $S^\\Bbb N$ where $S$ is countable. A measure is called exchangeable on $\\Om$, if it is supported on $\\Om$ and is invariant under every Borel automorphism of $\\Om$ which permutes at most finitely many coordinates. De-Finetti's theorem characterizes these measures when $\\Om=S^\\Bbb N$. We apply the ergodic theory of equivalence relations to study the case $\\Om\\neq S^\\Bbb N$, and obtain versions of this theorem when $\\Om$ is a countable state Markov shift, and when $\\Om$ is the collection of beta expansions of real numbers in $[0,1]$ (a non-Markovian constraint)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0406578","kind":"arxiv","version":4},"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":"2026-05-18T01:38:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zSThrBMOoDFVXAQx2zJZ+UDxA6KuotWPaK7CtF7wWzoNse3N4/3YdYAcN4rzo1KpGSeajbe2k+DZbHWph1GJCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T04:52:25.559124Z"},"content_sha256":"d2ef0e4b87e1d68ac4bc08aa190b0402d54e6916b2fa3730d550a174163d2b35","schema_version":"1.0","event_id":"sha256:d2ef0e4b87e1d68ac4bc08aa190b0402d54e6916b2fa3730d550a174163d2b35"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RJU34BKY3CCR55CF7WXHK2MYP6/bundle.json","state_url":"https://pith.science/pith/RJU34BKY3CCR55CF7WXHK2MYP6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RJU34BKY3CCR55CF7WXHK2MYP6/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":"2026-06-10T04:52:25Z","links":{"resolver":"https://pith.science/pith/RJU34BKY3CCR55CF7WXHK2MYP6","bundle":"https://pith.science/pith/RJU34BKY3CCR55CF7WXHK2MYP6/bundle.json","state":"https://pith.science/pith/RJU34BKY3CCR55CF7WXHK2MYP6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RJU34BKY3CCR55CF7WXHK2MYP6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:RJU34BKY3CCR55CF7WXHK2MYP6","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":"f8e06fa99cd99ff3f24e9fa5b553cc64e54775dc7c06be196ecac006937bf5d2","cross_cats_sorted":["math.PR"],"license":"","primary_cat":"math.DS","submitted_at":"2004-06-28T16:30:22Z","title_canon_sha256":"4a775dcbfb908e1b056de073ff73d933dab2fc4aa968f36d5853ecb42cd7871f"},"schema_version":"1.0","source":{"id":"math/0406578","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0406578","created_at":"2026-05-18T01:38:27Z"},{"alias_kind":"arxiv_version","alias_value":"math/0406578v4","created_at":"2026-05-18T01:38:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0406578","created_at":"2026-05-18T01:38:27Z"},{"alias_kind":"pith_short_12","alias_value":"RJU34BKY3CCR","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"RJU34BKY3CCR55CF","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"RJU34BKY","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:d2ef0e4b87e1d68ac4bc08aa190b0402d54e6916b2fa3730d550a174163d2b35","target":"graph","created_at":"2026-05-18T01:38:27Z","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":"Let $\\Om$ be a Borel subset of $S^\\Bbb N$ where $S$ is countable. A measure is called exchangeable on $\\Om$, if it is supported on $\\Om$ and is invariant under every Borel automorphism of $\\Om$ which permutes at most finitely many coordinates. De-Finetti's theorem characterizes these measures when $\\Om=S^\\Bbb N$. We apply the ergodic theory of equivalence relations to study the case $\\Om\\neq S^\\Bbb N$, and obtain versions of this theorem when $\\Om$ is a countable state Markov shift, and when $\\Om$ is the collection of beta expansions of real numbers in $[0,1]$ (a non-Markovian constraint).","authors_text":"H. Nakada, J. Aaronson, O. Sarig","cross_cats":["math.PR"],"headline":"","license":"","primary_cat":"math.DS","submitted_at":"2004-06-28T16:30:22Z","title":"Exchangeable measures for subshifts"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0406578","kind":"arxiv","version":4},"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":"sha256:3c9bab4aace82124303577ad2d1cdb46ea07ae28b95fdbfc2da12787dde799bf","target":"record","created_at":"2026-05-18T01:38:27Z","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":"f8e06fa99cd99ff3f24e9fa5b553cc64e54775dc7c06be196ecac006937bf5d2","cross_cats_sorted":["math.PR"],"license":"","primary_cat":"math.DS","submitted_at":"2004-06-28T16:30:22Z","title_canon_sha256":"4a775dcbfb908e1b056de073ff73d933dab2fc4aa968f36d5853ecb42cd7871f"},"schema_version":"1.0","source":{"id":"math/0406578","kind":"arxiv","version":4}},"canonical_sha256":"8a69be0558d8851ef445fdae7569987fb23121b9f55c86303e4ee0091c42f35a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8a69be0558d8851ef445fdae7569987fb23121b9f55c86303e4ee0091c42f35a","first_computed_at":"2026-05-18T01:38:27.279905Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:38:27.279905Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YinYFzLgEZGuyT2MLYlBWjb7d8ExMiTrZmhTPUoa19gZOIO+SRxdj/NRAyNPx/BqZNHExFvCS22J0uq/XWruBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:38:27.280592Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0406578","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3c9bab4aace82124303577ad2d1cdb46ea07ae28b95fdbfc2da12787dde799bf","sha256:d2ef0e4b87e1d68ac4bc08aa190b0402d54e6916b2fa3730d550a174163d2b35"],"state_sha256":"110962c36289fb87603f2547ce88b6ad47925bb603f803052d99117026797156"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/ALkkaqB3Q3iUSWPq1Dw0+sjS6wAc9AfAZ62nNiGb95w1KTWKExA0GUHdVc7rS6WahejSjmKgYrqULn8laq/AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T04:52:25.563046Z","bundle_sha256":"daee0c045fd119e40c6c772cdd7d6cbca0feb94b48c521b55cd977b9ed9a5a31"}}