{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:M22GOXKCKX3TTNEMX5ZUQSUXFP","short_pith_number":"pith:M22GOXKC","canonical_record":{"source":{"id":"1501.03367","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-01-14T15:16:53Z","cross_cats_sorted":[],"title_canon_sha256":"2397e782bb1b5ee9611d4dc9387d7195741a79c8ab4b86ef84fe15c598bfc43c","abstract_canon_sha256":"c3efc0a5e6d06de52ebd79893af2b800cc1913df5e61446318e85aec209a6f90"},"schema_version":"1.0"},"canonical_sha256":"66b4675d4255f739b48cbf73484a972bd6439de36d3ca2a42e8324f9f5f5bccc","source":{"kind":"arxiv","id":"1501.03367","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.03367","created_at":"2026-05-17T23:49:18Z"},{"alias_kind":"arxiv_version","alias_value":"1501.03367v4","created_at":"2026-05-17T23:49:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.03367","created_at":"2026-05-17T23:49:18Z"},{"alias_kind":"pith_short_12","alias_value":"M22GOXKCKX3T","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"M22GOXKCKX3TTNEM","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"M22GOXKC","created_at":"2026-05-18T12:29:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:M22GOXKCKX3TTNEMX5ZUQSUXFP","target":"record","payload":{"canonical_record":{"source":{"id":"1501.03367","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-01-14T15:16:53Z","cross_cats_sorted":[],"title_canon_sha256":"2397e782bb1b5ee9611d4dc9387d7195741a79c8ab4b86ef84fe15c598bfc43c","abstract_canon_sha256":"c3efc0a5e6d06de52ebd79893af2b800cc1913df5e61446318e85aec209a6f90"},"schema_version":"1.0"},"canonical_sha256":"66b4675d4255f739b48cbf73484a972bd6439de36d3ca2a42e8324f9f5f5bccc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:18.204864Z","signature_b64":"91QeCmvOtNve7gUzohccEO0MXwfYRGUvtXuFTSS5fzruYBrPtpxLlZIfHEt8uI4B2JGHFUG57EEK/71z5+l2BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"66b4675d4255f739b48cbf73484a972bd6439de36d3ca2a42e8324f9f5f5bccc","last_reissued_at":"2026-05-17T23:49:18.204129Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:18.204129Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.03367","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":"2026-05-17T23:49:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hhsxtZi8NNAHHHATgBwmjR95pdTgMsKhkHPHsNiV3opXq1/TEdgZdTRF7N2RGGwQk6+GMgjNewR5qlzsJGQrBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T17:28:19.992836Z"},"content_sha256":"ea63e2366f7d23a66178225b6e0992cae6d0fcfe27c9973f06c462a90903fc20","schema_version":"1.0","event_id":"sha256:ea63e2366f7d23a66178225b6e0992cae6d0fcfe27c9973f06c462a90903fc20"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:M22GOXKCKX3TTNEMX5ZUQSUXFP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Krieger's finite generator theorem for actions of countable groups II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Brandon Seward","submitted_at":"2015-01-14T15:16:53Z","abstract_excerpt":"We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Ab\\'{e}rt--Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every countable group admits a free ergodic action of positive Rokhlin entropy, we prove that: (i) the Rokhlin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.03367","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":"2026-05-17T23:49:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PiiZwGjH4pHZyv6vk3ViWuEt3mABrcktpESIye7NWQ1q5Yx69ueqIkqUuJzvAhI0buyuWkhjzxRftG/ONV7PBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T17:28:19.993506Z"},"content_sha256":"5620e5cd711d46d5d654262d4a8cbd05defa65f0afca1269c12a2d1da264ef21","schema_version":"1.0","event_id":"sha256:5620e5cd711d46d5d654262d4a8cbd05defa65f0afca1269c12a2d1da264ef21"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/M22GOXKCKX3TTNEMX5ZUQSUXFP/bundle.json","state_url":"https://pith.science/pith/M22GOXKCKX3TTNEMX5ZUQSUXFP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/M22GOXKCKX3TTNEMX5ZUQSUXFP/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":"2026-06-08T17:28:19Z","links":{"resolver":"https://pith.science/pith/M22GOXKCKX3TTNEMX5ZUQSUXFP","bundle":"https://pith.science/pith/M22GOXKCKX3TTNEMX5ZUQSUXFP/bundle.json","state":"https://pith.science/pith/M22GOXKCKX3TTNEMX5ZUQSUXFP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/M22GOXKCKX3TTNEMX5ZUQSUXFP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:M22GOXKCKX3TTNEMX5ZUQSUXFP","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":"c3efc0a5e6d06de52ebd79893af2b800cc1913df5e61446318e85aec209a6f90","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-01-14T15:16:53Z","title_canon_sha256":"2397e782bb1b5ee9611d4dc9387d7195741a79c8ab4b86ef84fe15c598bfc43c"},"schema_version":"1.0","source":{"id":"1501.03367","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.03367","created_at":"2026-05-17T23:49:18Z"},{"alias_kind":"arxiv_version","alias_value":"1501.03367v4","created_at":"2026-05-17T23:49:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.03367","created_at":"2026-05-17T23:49:18Z"},{"alias_kind":"pith_short_12","alias_value":"M22GOXKCKX3T","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"M22GOXKCKX3TTNEM","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"M22GOXKC","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:5620e5cd711d46d5d654262d4a8cbd05defa65f0afca1269c12a2d1da264ef21","target":"graph","created_at":"2026-05-17T23:49:18Z","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 continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Ab\\'{e}rt--Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every countable group admits a free ergodic action of positive Rokhlin entropy, we prove that: (i) the Rokhlin","authors_text":"Brandon Seward","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-01-14T15:16:53Z","title":"Krieger's finite generator theorem for actions of countable groups II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.03367","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:ea63e2366f7d23a66178225b6e0992cae6d0fcfe27c9973f06c462a90903fc20","target":"record","created_at":"2026-05-17T23:49:18Z","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":"c3efc0a5e6d06de52ebd79893af2b800cc1913df5e61446318e85aec209a6f90","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-01-14T15:16:53Z","title_canon_sha256":"2397e782bb1b5ee9611d4dc9387d7195741a79c8ab4b86ef84fe15c598bfc43c"},"schema_version":"1.0","source":{"id":"1501.03367","kind":"arxiv","version":4}},"canonical_sha256":"66b4675d4255f739b48cbf73484a972bd6439de36d3ca2a42e8324f9f5f5bccc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"66b4675d4255f739b48cbf73484a972bd6439de36d3ca2a42e8324f9f5f5bccc","first_computed_at":"2026-05-17T23:49:18.204129Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:49:18.204129Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"91QeCmvOtNve7gUzohccEO0MXwfYRGUvtXuFTSS5fzruYBrPtpxLlZIfHEt8uI4B2JGHFUG57EEK/71z5+l2BQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:49:18.204864Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.03367","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ea63e2366f7d23a66178225b6e0992cae6d0fcfe27c9973f06c462a90903fc20","sha256:5620e5cd711d46d5d654262d4a8cbd05defa65f0afca1269c12a2d1da264ef21"],"state_sha256":"9b2a1802ed280d7df596a4732546602c81a225b1aff7c035aa3a488fd0139eed"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lrKqE3rI7HWveaKnYQ6QYE9v1+fzEqGrqjZbO+ddsWzmkobmAC3dMDxKb05Dm3EfyYF4KbBHAb5JzDM0aE5sBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T17:28:19.997654Z","bundle_sha256":"acc2951d4487483545fe6ed8c688d8dc6e1c40b614e6d53473c09aebb372f9f3"}}