{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JVIOPDOBYEEVVXMRYLJ2J3F6SR","short_pith_number":"pith:JVIOPDOB","canonical_record":{"source":{"id":"1506.02006","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-05T18:31:11Z","cross_cats_sorted":[],"title_canon_sha256":"6c4c156d16b7ddf0223490d76b5f0e76ae7a3df1705a158dc5ed3459de0de711","abstract_canon_sha256":"02bb9ed20ca075d8f2caf79b0c9b9e1e2278db1359def425e368eaf3e72d006b"},"schema_version":"1.0"},"canonical_sha256":"4d50e78dc1c1095add91c2d3a4ecbe9470700007ac310710a2cb492cf96fb334","source":{"kind":"arxiv","id":"1506.02006","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.02006","created_at":"2026-05-18T00:50:31Z"},{"alias_kind":"arxiv_version","alias_value":"1506.02006v3","created_at":"2026-05-18T00:50:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.02006","created_at":"2026-05-18T00:50:31Z"},{"alias_kind":"pith_short_12","alias_value":"JVIOPDOBYEEV","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JVIOPDOBYEEVVXMR","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JVIOPDOB","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JVIOPDOBYEEVVXMRYLJ2J3F6SR","target":"record","payload":{"canonical_record":{"source":{"id":"1506.02006","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-05T18:31:11Z","cross_cats_sorted":[],"title_canon_sha256":"6c4c156d16b7ddf0223490d76b5f0e76ae7a3df1705a158dc5ed3459de0de711","abstract_canon_sha256":"02bb9ed20ca075d8f2caf79b0c9b9e1e2278db1359def425e368eaf3e72d006b"},"schema_version":"1.0"},"canonical_sha256":"4d50e78dc1c1095add91c2d3a4ecbe9470700007ac310710a2cb492cf96fb334","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:31.287998Z","signature_b64":"Z557QZlKRlMs8a55ZumhdYoYkJdZQ+8niYiluZ0y4kEoy7fnd+zd4VyVVuhI1mc6G36MbA3Sr8ioRZ4WbZknCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d50e78dc1c1095add91c2d3a4ecbe9470700007ac310710a2cb492cf96fb334","last_reissued_at":"2026-05-18T00:50:31.287447Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:31.287447Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1506.02006","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:50:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2qPKY3dAA7RS1QgNSx927yvfPEYikYIutcvvhI9TwZd4kd3oxjmcYT4YUBS5h7S7119VPwFXAqKWxdsCyhq0AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T09:59:22.663433Z"},"content_sha256":"2d6bd00d20b5dae8e5ea20faa62dea722437840ef0f08fb4708e628ddd4b8abc","schema_version":"1.0","event_id":"sha256:2d6bd00d20b5dae8e5ea20faa62dea722437840ef0f08fb4708e628ddd4b8abc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JVIOPDOBYEEVVXMRYLJ2J3F6SR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Small cocycles, fine torus fibrations, and a ${\\mathbb Z}^2$ subshift with neither","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Alex Clark, Lorenzo Sadun","submitted_at":"2015-06-05T18:31:11Z","abstract_excerpt":"Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam and Skau conjectured that all minimal, free ${\\mathbb Z}^d$ actions on Cantor sets admit \"small cocycles.\" These represent classes in $H^1$ that are mapped to small vectors in ${\\mathbb R}^d$ by the Ruelle-Sullivan (RS) map. We show that there exist ${\\mathbb Z}^d$ actions where no such small cocycles exist, and where the image of $H^1$ under RS is ${\\mathbb Z}^d$. Our methods involve tiling spaces and shape deformations, and along the way we prove a relation between the image of RS and the set of \"virtual eigen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.02006","kind":"arxiv","version":3},"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":"2026-05-18T00:50:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5qmWh+01hhOQmGZw7fqjD8q+MSraXOAaIT/o5t0bEARl4sjduQTGc2Ab8p02hKtHB7E3/LamdeM3Mk3vJDjuDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T09:59:22.663778Z"},"content_sha256":"7ae82b9e557dc153b209c7e2842ae1252337121ff254c3e55bf890e22156dcca","schema_version":"1.0","event_id":"sha256:7ae82b9e557dc153b209c7e2842ae1252337121ff254c3e55bf890e22156dcca"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JVIOPDOBYEEVVXMRYLJ2J3F6SR/bundle.json","state_url":"https://pith.science/pith/JVIOPDOBYEEVVXMRYLJ2J3F6SR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JVIOPDOBYEEVVXMRYLJ2J3F6SR/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":"2026-05-31T09:59:22Z","links":{"resolver":"https://pith.science/pith/JVIOPDOBYEEVVXMRYLJ2J3F6SR","bundle":"https://pith.science/pith/JVIOPDOBYEEVVXMRYLJ2J3F6SR/bundle.json","state":"https://pith.science/pith/JVIOPDOBYEEVVXMRYLJ2J3F6SR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JVIOPDOBYEEVVXMRYLJ2J3F6SR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JVIOPDOBYEEVVXMRYLJ2J3F6SR","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":"02bb9ed20ca075d8f2caf79b0c9b9e1e2278db1359def425e368eaf3e72d006b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-05T18:31:11Z","title_canon_sha256":"6c4c156d16b7ddf0223490d76b5f0e76ae7a3df1705a158dc5ed3459de0de711"},"schema_version":"1.0","source":{"id":"1506.02006","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.02006","created_at":"2026-05-18T00:50:31Z"},{"alias_kind":"arxiv_version","alias_value":"1506.02006v3","created_at":"2026-05-18T00:50:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.02006","created_at":"2026-05-18T00:50:31Z"},{"alias_kind":"pith_short_12","alias_value":"JVIOPDOBYEEV","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JVIOPDOBYEEVVXMR","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JVIOPDOB","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:7ae82b9e557dc153b209c7e2842ae1252337121ff254c3e55bf890e22156dcca","target":"graph","created_at":"2026-05-18T00:50:31Z","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":"Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam and Skau conjectured that all minimal, free ${\\mathbb Z}^d$ actions on Cantor sets admit \"small cocycles.\" These represent classes in $H^1$ that are mapped to small vectors in ${\\mathbb R}^d$ by the Ruelle-Sullivan (RS) map. We show that there exist ${\\mathbb Z}^d$ actions where no such small cocycles exist, and where the image of $H^1$ under RS is ${\\mathbb Z}^d$. Our methods involve tiling spaces and shape deformations, and along the way we prove a relation between the image of RS and the set of \"virtual eigen","authors_text":"Alex Clark, Lorenzo Sadun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-05T18:31:11Z","title":"Small cocycles, fine torus fibrations, and a ${\\mathbb Z}^2$ subshift with neither"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.02006","kind":"arxiv","version":3},"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":"sha256:2d6bd00d20b5dae8e5ea20faa62dea722437840ef0f08fb4708e628ddd4b8abc","target":"record","created_at":"2026-05-18T00:50:31Z","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":"02bb9ed20ca075d8f2caf79b0c9b9e1e2278db1359def425e368eaf3e72d006b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-05T18:31:11Z","title_canon_sha256":"6c4c156d16b7ddf0223490d76b5f0e76ae7a3df1705a158dc5ed3459de0de711"},"schema_version":"1.0","source":{"id":"1506.02006","kind":"arxiv","version":3}},"canonical_sha256":"4d50e78dc1c1095add91c2d3a4ecbe9470700007ac310710a2cb492cf96fb334","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4d50e78dc1c1095add91c2d3a4ecbe9470700007ac310710a2cb492cf96fb334","first_computed_at":"2026-05-18T00:50:31.287447Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:50:31.287447Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Z557QZlKRlMs8a55ZumhdYoYkJdZQ+8niYiluZ0y4kEoy7fnd+zd4VyVVuhI1mc6G36MbA3Sr8ioRZ4WbZknCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:50:31.287998Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.02006","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2d6bd00d20b5dae8e5ea20faa62dea722437840ef0f08fb4708e628ddd4b8abc","sha256:7ae82b9e557dc153b209c7e2842ae1252337121ff254c3e55bf890e22156dcca"],"state_sha256":"7dc17c30f0fb269c89ab90ebb76590842f54b1d8047da09b7942e4bf6b677942"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I5B9qv6oCOaImXnCxr+E/JKKMvJtW4LRpd89m7IEborHwmd9aQRpWhziMX1YN/R6AI/oak0e+G2tkKnpkS10AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T09:59:22.665998Z","bundle_sha256":"c14401e03ed3998a2ed3c4b2f9cd2a5bc816d83b16307bd8767a041a51c0b450"}}