{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:H7K6STZYHHOHFNH54LZR2B6VHF","short_pith_number":"pith:H7K6STZY","canonical_record":{"source":{"id":"1509.00247","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-09-01T12:11:25Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"ced9c78e4a6ff90b3cde05cc4cd8fcb5de077ce7278e4d7627eb9b0909720013","abstract_canon_sha256":"203801cdc4904ca5f761d65303d63a6c50dd74130710d7a501f5348d327a89ac"},"schema_version":"1.0"},"canonical_sha256":"3fd5e94f3839dc72b4fde2f31d07d5394f1ca581421bf7807ad580825f82a888","source":{"kind":"arxiv","id":"1509.00247","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.00247","created_at":"2026-05-18T01:34:14Z"},{"alias_kind":"arxiv_version","alias_value":"1509.00247v1","created_at":"2026-05-18T01:34:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.00247","created_at":"2026-05-18T01:34:14Z"},{"alias_kind":"pith_short_12","alias_value":"H7K6STZYHHOH","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"H7K6STZYHHOHFNH5","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"H7K6STZY","created_at":"2026-05-18T12:29:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:H7K6STZYHHOHFNH54LZR2B6VHF","target":"record","payload":{"canonical_record":{"source":{"id":"1509.00247","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-09-01T12:11:25Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"ced9c78e4a6ff90b3cde05cc4cd8fcb5de077ce7278e4d7627eb9b0909720013","abstract_canon_sha256":"203801cdc4904ca5f761d65303d63a6c50dd74130710d7a501f5348d327a89ac"},"schema_version":"1.0"},"canonical_sha256":"3fd5e94f3839dc72b4fde2f31d07d5394f1ca581421bf7807ad580825f82a888","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:14.743254Z","signature_b64":"BH7PeFr682vS5+Cgx1eJDxAWYhPMnHQHrCAO4wqnYHXxBEViz1k3XE65Dmsu9XClRXsoF2+9fB1V/ulGk06UBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3fd5e94f3839dc72b4fde2f31d07d5394f1ca581421bf7807ad580825f82a888","last_reissued_at":"2026-05-18T01:34:14.742505Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:14.742505Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.00247","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":"2026-05-18T01:34:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z6B6Xhg2s61OM2bihWTuHWW6DTnfnS8/8U9g6A8BXldy0qL6mD7GPonQ2sQvhk+yDy7Oe6AlalHsnhzm9TaWDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T21:00:25.491173Z"},"content_sha256":"2952553f0f8dad542f96f25f02e45cdd9e2128bf4c47373466dbb897affd4be5","schema_version":"1.0","event_id":"sha256:2952553f0f8dad542f96f25f02e45cdd9e2128bf4c47373466dbb897affd4be5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:H7K6STZYHHOHFNH54LZR2B6VHF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximations of standard equivalence relations and Bernoulli percolation at p\\_u","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.GR","authors_text":"Damien Gaboriau (UMPA-ENSL), Robin Tucker-Drob","submitted_at":"2015-09-01T12:11:25Z","abstract_excerpt":"The goal of this note is to announce certain results in orbit equivalence theory, especially concerning the approximation of p.m.p. standard equivalence relations by increasing sequence of sub-relations, with applications to the behavior of the Bernoulli percolation on graphs at the threshold pu. R\\'esum\\'e en Fran\\c{c}ais. Approximations de relations d'\\'equivalence standards et percolation de Bernoulli \\`a p\\_u."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.00247","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":"2026-05-18T01:34:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nKp7MEUkR+bVA/JijXU+BkDwD+v47yIablLsy8LboOsPOxv39hyv2qXLaVNsaH9rcFyOzVXNSECpk9LLbEMwDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T21:00:25.491523Z"},"content_sha256":"fb9315d9c655bf38e3ba210a4695f4b67ec8af067c68339e263c08df1e34c624","schema_version":"1.0","event_id":"sha256:fb9315d9c655bf38e3ba210a4695f4b67ec8af067c68339e263c08df1e34c624"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/H7K6STZYHHOHFNH54LZR2B6VHF/bundle.json","state_url":"https://pith.science/pith/H7K6STZYHHOHFNH54LZR2B6VHF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/H7K6STZYHHOHFNH54LZR2B6VHF/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":"2026-06-02T21:00:25Z","links":{"resolver":"https://pith.science/pith/H7K6STZYHHOHFNH54LZR2B6VHF","bundle":"https://pith.science/pith/H7K6STZYHHOHFNH54LZR2B6VHF/bundle.json","state":"https://pith.science/pith/H7K6STZYHHOHFNH54LZR2B6VHF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/H7K6STZYHHOHFNH54LZR2B6VHF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:H7K6STZYHHOHFNH54LZR2B6VHF","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":"203801cdc4904ca5f761d65303d63a6c50dd74130710d7a501f5348d327a89ac","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-09-01T12:11:25Z","title_canon_sha256":"ced9c78e4a6ff90b3cde05cc4cd8fcb5de077ce7278e4d7627eb9b0909720013"},"schema_version":"1.0","source":{"id":"1509.00247","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.00247","created_at":"2026-05-18T01:34:14Z"},{"alias_kind":"arxiv_version","alias_value":"1509.00247v1","created_at":"2026-05-18T01:34:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.00247","created_at":"2026-05-18T01:34:14Z"},{"alias_kind":"pith_short_12","alias_value":"H7K6STZYHHOH","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"H7K6STZYHHOHFNH5","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"H7K6STZY","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:fb9315d9c655bf38e3ba210a4695f4b67ec8af067c68339e263c08df1e34c624","target":"graph","created_at":"2026-05-18T01:34:14Z","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":"The goal of this note is to announce certain results in orbit equivalence theory, especially concerning the approximation of p.m.p. standard equivalence relations by increasing sequence of sub-relations, with applications to the behavior of the Bernoulli percolation on graphs at the threshold pu. R\\'esum\\'e en Fran\\c{c}ais. Approximations de relations d'\\'equivalence standards et percolation de Bernoulli \\`a p\\_u.","authors_text":"Damien Gaboriau (UMPA-ENSL), Robin Tucker-Drob","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-09-01T12:11:25Z","title":"Approximations of standard equivalence relations and Bernoulli percolation at p\\_u"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.00247","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":"sha256:2952553f0f8dad542f96f25f02e45cdd9e2128bf4c47373466dbb897affd4be5","target":"record","created_at":"2026-05-18T01:34:14Z","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":"203801cdc4904ca5f761d65303d63a6c50dd74130710d7a501f5348d327a89ac","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-09-01T12:11:25Z","title_canon_sha256":"ced9c78e4a6ff90b3cde05cc4cd8fcb5de077ce7278e4d7627eb9b0909720013"},"schema_version":"1.0","source":{"id":"1509.00247","kind":"arxiv","version":1}},"canonical_sha256":"3fd5e94f3839dc72b4fde2f31d07d5394f1ca581421bf7807ad580825f82a888","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3fd5e94f3839dc72b4fde2f31d07d5394f1ca581421bf7807ad580825f82a888","first_computed_at":"2026-05-18T01:34:14.742505Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:14.742505Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BH7PeFr682vS5+Cgx1eJDxAWYhPMnHQHrCAO4wqnYHXxBEViz1k3XE65Dmsu9XClRXsoF2+9fB1V/ulGk06UBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:14.743254Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.00247","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2952553f0f8dad542f96f25f02e45cdd9e2128bf4c47373466dbb897affd4be5","sha256:fb9315d9c655bf38e3ba210a4695f4b67ec8af067c68339e263c08df1e34c624"],"state_sha256":"f3700b83ba599f5526f8dd55fa1d9ad3aabe29b6d872eba33a8c90a83cf2ebe0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cw0tYaTXVMxfdwAiod/omGRzX4UctT9bGYXyf7zQ5piUp33527wJ2yb9v/jQmbuOzSiNuFV+MGTLT8XMiigbCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T21:00:25.493497Z","bundle_sha256":"7dbd4f32d5655a4bc1bb08d3d2dc7e4628bf4d7827696ac257449d5a51893af8"}}