{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:SWWISDICPIRN5EVFLHRAXNWNSN","short_pith_number":"pith:SWWISDIC","canonical_record":{"source":{"id":"1311.0738","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-04T15:47:02Z","cross_cats_sorted":["math.GR","math.LO"],"title_canon_sha256":"054e9645112834b299ec2a8758bb0bf85869652ab9cff82ace9b9e1c0faa2456","abstract_canon_sha256":"4971ba4da861cbfc096da5d6027cf8c7154496ecc83f3d8774bef0e82aab0c35"},"schema_version":"1.0"},"canonical_sha256":"95ac890d027a22de92a559e20bb6cd9378917757b6a5fa9c91003599e2fad420","source":{"kind":"arxiv","id":"1311.0738","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.0738","created_at":"2026-05-18T02:48:24Z"},{"alias_kind":"arxiv_version","alias_value":"1311.0738v2","created_at":"2026-05-18T02:48:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.0738","created_at":"2026-05-18T02:48:24Z"},{"alias_kind":"pith_short_12","alias_value":"SWWISDICPIRN","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"SWWISDICPIRN5EVF","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"SWWISDIC","created_at":"2026-05-18T12:27:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:SWWISDICPIRN5EVFLHRAXNWNSN","target":"record","payload":{"canonical_record":{"source":{"id":"1311.0738","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-04T15:47:02Z","cross_cats_sorted":["math.GR","math.LO"],"title_canon_sha256":"054e9645112834b299ec2a8758bb0bf85869652ab9cff82ace9b9e1c0faa2456","abstract_canon_sha256":"4971ba4da861cbfc096da5d6027cf8c7154496ecc83f3d8774bef0e82aab0c35"},"schema_version":"1.0"},"canonical_sha256":"95ac890d027a22de92a559e20bb6cd9378917757b6a5fa9c91003599e2fad420","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:24.631229Z","signature_b64":"WAN662MoFD2Od5DjTxbGzEO/q41T+UyjssXB8Zuq0XXUUE0pp7dU8i/J4CfzeRrJ0dsk51aB8NYQyjLIDxFAAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95ac890d027a22de92a559e20bb6cd9378917757b6a5fa9c91003599e2fad420","last_reissued_at":"2026-05-18T02:48:24.630782Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:24.630782Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1311.0738","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":"2026-05-18T02:48:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Glk7K12Lo+M2y3+8DwELtzilPFhi5V7UUMNI6kMORwes1tRV53P8x+S5Ea5/WC4709V2iJSldMYJfexrvzknDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T21:16:54.030118Z"},"content_sha256":"275080c904f0efe24737b7e95dfc4b37dcf2b1577ae2cdab681c544eadb7b191","schema_version":"1.0","event_id":"sha256:275080c904f0efe24737b7e95dfc4b37dcf2b1577ae2cdab681c544eadb7b191"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:SWWISDICPIRN5EVFLHRAXNWNSN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Every action of a non-amenable group is the factor of a small action","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.LO"],"primary_cat":"math.DS","authors_text":"Brandon Seward","submitted_at":"2013-11-04T15:47:02Z","abstract_excerpt":"It is well known that if $G$ is a countable amenable group and $G \\curvearrowright (Y, \\nu)$ factors onto $G \\curvearrowright (X, \\mu)$, then the entropy of the first action must be greater than or equal to the entropy of the second action. In particular, if $G \\curvearrowright (X, \\mu)$ has infinite entropy, then the action $G \\curvearrowright (Y, \\nu)$ does not admit any finite generating partition. On the other hand, we prove that if $G$ is a countable non-amenable group then there exists a finite integer $n$ with the following property: for every probability-measure-preserving action $G \\c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.0738","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":"2026-05-18T02:48:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iFu4JQGOoyN4K2TIuNkag7GcK2q6Guw+cdHrnygwqlw5DhVDh4bHgCM2dpEgs3VJd3ypxM9Ck1yJJi0TwwVxCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T21:16:54.030857Z"},"content_sha256":"db5748295e3d9aef9a5f89ce7cfaf46459c1b211637fd3c33c83ae77c9ae0642","schema_version":"1.0","event_id":"sha256:db5748295e3d9aef9a5f89ce7cfaf46459c1b211637fd3c33c83ae77c9ae0642"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SWWISDICPIRN5EVFLHRAXNWNSN/bundle.json","state_url":"https://pith.science/pith/SWWISDICPIRN5EVFLHRAXNWNSN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SWWISDICPIRN5EVFLHRAXNWNSN/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":"2026-06-08T21:16:54Z","links":{"resolver":"https://pith.science/pith/SWWISDICPIRN5EVFLHRAXNWNSN","bundle":"https://pith.science/pith/SWWISDICPIRN5EVFLHRAXNWNSN/bundle.json","state":"https://pith.science/pith/SWWISDICPIRN5EVFLHRAXNWNSN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SWWISDICPIRN5EVFLHRAXNWNSN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:SWWISDICPIRN5EVFLHRAXNWNSN","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":"4971ba4da861cbfc096da5d6027cf8c7154496ecc83f3d8774bef0e82aab0c35","cross_cats_sorted":["math.GR","math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-04T15:47:02Z","title_canon_sha256":"054e9645112834b299ec2a8758bb0bf85869652ab9cff82ace9b9e1c0faa2456"},"schema_version":"1.0","source":{"id":"1311.0738","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.0738","created_at":"2026-05-18T02:48:24Z"},{"alias_kind":"arxiv_version","alias_value":"1311.0738v2","created_at":"2026-05-18T02:48:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.0738","created_at":"2026-05-18T02:48:24Z"},{"alias_kind":"pith_short_12","alias_value":"SWWISDICPIRN","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"SWWISDICPIRN5EVF","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"SWWISDIC","created_at":"2026-05-18T12:27:59Z"}],"graph_snapshots":[{"event_id":"sha256:db5748295e3d9aef9a5f89ce7cfaf46459c1b211637fd3c33c83ae77c9ae0642","target":"graph","created_at":"2026-05-18T02:48:24Z","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":"It is well known that if $G$ is a countable amenable group and $G \\curvearrowright (Y, \\nu)$ factors onto $G \\curvearrowright (X, \\mu)$, then the entropy of the first action must be greater than or equal to the entropy of the second action. In particular, if $G \\curvearrowright (X, \\mu)$ has infinite entropy, then the action $G \\curvearrowright (Y, \\nu)$ does not admit any finite generating partition. On the other hand, we prove that if $G$ is a countable non-amenable group then there exists a finite integer $n$ with the following property: for every probability-measure-preserving action $G \\c","authors_text":"Brandon Seward","cross_cats":["math.GR","math.LO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-04T15:47:02Z","title":"Every action of a non-amenable group is the factor of a small action"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.0738","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":"sha256:275080c904f0efe24737b7e95dfc4b37dcf2b1577ae2cdab681c544eadb7b191","target":"record","created_at":"2026-05-18T02:48:24Z","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":"4971ba4da861cbfc096da5d6027cf8c7154496ecc83f3d8774bef0e82aab0c35","cross_cats_sorted":["math.GR","math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-04T15:47:02Z","title_canon_sha256":"054e9645112834b299ec2a8758bb0bf85869652ab9cff82ace9b9e1c0faa2456"},"schema_version":"1.0","source":{"id":"1311.0738","kind":"arxiv","version":2}},"canonical_sha256":"95ac890d027a22de92a559e20bb6cd9378917757b6a5fa9c91003599e2fad420","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"95ac890d027a22de92a559e20bb6cd9378917757b6a5fa9c91003599e2fad420","first_computed_at":"2026-05-18T02:48:24.630782Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:48:24.630782Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WAN662MoFD2Od5DjTxbGzEO/q41T+UyjssXB8Zuq0XXUUE0pp7dU8i/J4CfzeRrJ0dsk51aB8NYQyjLIDxFAAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:48:24.631229Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.0738","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:275080c904f0efe24737b7e95dfc4b37dcf2b1577ae2cdab681c544eadb7b191","sha256:db5748295e3d9aef9a5f89ce7cfaf46459c1b211637fd3c33c83ae77c9ae0642"],"state_sha256":"0b6795515f94763d426f6e811070f3a4459feb98729c905a83f531a931fc074b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xEYsFYsdMTmLIrPnU+YRyuCjySsH/1r1FZ045EUcO/z08y2jgtfESRVgVxWHODO3N1pFIugjafs7FxASFLHPBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T21:16:54.035227Z","bundle_sha256":"7fe5ec3959991cd9be90ec8d4e64142242bf92acd68b3c7168a9df9a57c14950"}}