{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:7XRGZASYZL2EPWLATL5SZDGSRE","short_pith_number":"pith:7XRGZASY","canonical_record":{"source":{"id":"1108.6232","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-08-31T13:59:15Z","cross_cats_sorted":["math.GR","math.KT"],"title_canon_sha256":"68d5a277dab3502d42eec68c589c4822368f241c6f167dd9c5a9f524c803c504","abstract_canon_sha256":"b1d4bf1ca1dbb3cf9a6aef2d135bd4abcd02aec39d95d13f567d6781fed1d9dd"},"schema_version":"1.0"},"canonical_sha256":"fde26c8258caf447d9609afb2c8cd2891f14386ec07f03abb483ed55cd1b569b","source":{"kind":"arxiv","id":"1108.6232","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.6232","created_at":"2026-05-18T02:41:43Z"},{"alias_kind":"arxiv_version","alias_value":"1108.6232v1","created_at":"2026-05-18T02:41:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.6232","created_at":"2026-05-18T02:41:43Z"},{"alias_kind":"pith_short_12","alias_value":"7XRGZASYZL2E","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7XRGZASYZL2EPWLA","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7XRGZASY","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:7XRGZASYZL2EPWLATL5SZDGSRE","target":"record","payload":{"canonical_record":{"source":{"id":"1108.6232","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-08-31T13:59:15Z","cross_cats_sorted":["math.GR","math.KT"],"title_canon_sha256":"68d5a277dab3502d42eec68c589c4822368f241c6f167dd9c5a9f524c803c504","abstract_canon_sha256":"b1d4bf1ca1dbb3cf9a6aef2d135bd4abcd02aec39d95d13f567d6781fed1d9dd"},"schema_version":"1.0"},"canonical_sha256":"fde26c8258caf447d9609afb2c8cd2891f14386ec07f03abb483ed55cd1b569b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:43.694404Z","signature_b64":"tFXqFrXjxjV1uSKRmJvmYRNRLw6xRRqXYA6nqv99kP/wYOUhB/zkg+rxsDiymxc0l+l9OmP3t2nF/epE8a/ZDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fde26c8258caf447d9609afb2c8cd2891f14386ec07f03abb483ed55cd1b569b","last_reissued_at":"2026-05-18T02:41:43.693656Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:43.693656Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1108.6232","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:41:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5Ga6lda2UMGRWwvQNACTWcjRZuWrtddj1CSA7VTV6+cjaIUdDLAya9O/HlkI+UzYLYVbF+ZrjS94duDAPY4wCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:55:32.976137Z"},"content_sha256":"8466467a8ceb1e140873c27685fd1f8f410d64cbbac09a1282d89e26209254d5","schema_version":"1.0","event_id":"sha256:8466467a8ceb1e140873c27685fd1f8f410d64cbbac09a1282d89e26209254d5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:7XRGZASYZL2EPWLATL5SZDGSRE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Expanders and Property A","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.KT"],"primary_cat":"math.GT","authors_text":"A. Khukhro, N. J. Wright","submitted_at":"2011-08-31T13:59:15Z","abstract_excerpt":"We give a cohomological characterisation of expander graphs, and use it to give a direct proof that expander graphs do not have Yu's property A."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.6232","kind":"arxiv","version":1},"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":"2026-05-18T02:41:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sX5RZoQ4eI3QHi+GhPpfRB3hrrcIVfmLDEUrukyWMNxtloYNTta8qTdqrGnBY2+OcUr1AY2dTNwkv+cTGy/6Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:55:32.976805Z"},"content_sha256":"d6735593748291c02115595cb83270592ed2113a51097d81a938d2fe0a38d3e0","schema_version":"1.0","event_id":"sha256:d6735593748291c02115595cb83270592ed2113a51097d81a938d2fe0a38d3e0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7XRGZASYZL2EPWLATL5SZDGSRE/bundle.json","state_url":"https://pith.science/pith/7XRGZASYZL2EPWLATL5SZDGSRE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7XRGZASYZL2EPWLATL5SZDGSRE/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":"2026-05-27T14:55:32Z","links":{"resolver":"https://pith.science/pith/7XRGZASYZL2EPWLATL5SZDGSRE","bundle":"https://pith.science/pith/7XRGZASYZL2EPWLATL5SZDGSRE/bundle.json","state":"https://pith.science/pith/7XRGZASYZL2EPWLATL5SZDGSRE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7XRGZASYZL2EPWLATL5SZDGSRE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:7XRGZASYZL2EPWLATL5SZDGSRE","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":"b1d4bf1ca1dbb3cf9a6aef2d135bd4abcd02aec39d95d13f567d6781fed1d9dd","cross_cats_sorted":["math.GR","math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-08-31T13:59:15Z","title_canon_sha256":"68d5a277dab3502d42eec68c589c4822368f241c6f167dd9c5a9f524c803c504"},"schema_version":"1.0","source":{"id":"1108.6232","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.6232","created_at":"2026-05-18T02:41:43Z"},{"alias_kind":"arxiv_version","alias_value":"1108.6232v1","created_at":"2026-05-18T02:41:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.6232","created_at":"2026-05-18T02:41:43Z"},{"alias_kind":"pith_short_12","alias_value":"7XRGZASYZL2E","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7XRGZASYZL2EPWLA","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7XRGZASY","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:d6735593748291c02115595cb83270592ed2113a51097d81a938d2fe0a38d3e0","target":"graph","created_at":"2026-05-18T02:41:43Z","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 give a cohomological characterisation of expander graphs, and use it to give a direct proof that expander graphs do not have Yu's property A.","authors_text":"A. Khukhro, N. J. Wright","cross_cats":["math.GR","math.KT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-08-31T13:59:15Z","title":"Expanders and Property A"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.6232","kind":"arxiv","version":1},"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":"sha256:8466467a8ceb1e140873c27685fd1f8f410d64cbbac09a1282d89e26209254d5","target":"record","created_at":"2026-05-18T02:41:43Z","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":"b1d4bf1ca1dbb3cf9a6aef2d135bd4abcd02aec39d95d13f567d6781fed1d9dd","cross_cats_sorted":["math.GR","math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-08-31T13:59:15Z","title_canon_sha256":"68d5a277dab3502d42eec68c589c4822368f241c6f167dd9c5a9f524c803c504"},"schema_version":"1.0","source":{"id":"1108.6232","kind":"arxiv","version":1}},"canonical_sha256":"fde26c8258caf447d9609afb2c8cd2891f14386ec07f03abb483ed55cd1b569b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fde26c8258caf447d9609afb2c8cd2891f14386ec07f03abb483ed55cd1b569b","first_computed_at":"2026-05-18T02:41:43.693656Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:43.693656Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tFXqFrXjxjV1uSKRmJvmYRNRLw6xRRqXYA6nqv99kP/wYOUhB/zkg+rxsDiymxc0l+l9OmP3t2nF/epE8a/ZDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:43.694404Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.6232","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8466467a8ceb1e140873c27685fd1f8f410d64cbbac09a1282d89e26209254d5","sha256:d6735593748291c02115595cb83270592ed2113a51097d81a938d2fe0a38d3e0"],"state_sha256":"262426114f6c56251180d3a0fe769468afd01517c4278dfd54cafb1a3fd491f6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"US9Ri+TV+e1GZmv1pSEtJR0K9G6u6PhChgr3W90mTHtuqepcsXTGbC7rF6RPl9U49VH1B4qwTBjdDcKiSjiKCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T14:55:32.980227Z","bundle_sha256":"0daa47021e043797a92b6f56d3a4795b079f30ae7ffafe90fe1d97db18cd55ca"}}