{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:EKC7G4HLN3ERK7DH2BZUDR76AK","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":"1f8902f54e20b2c0709199116f257f8c7d4c8ab3cfc8a925615c820d1d77fd13","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-06-22T14:29:20Z","title_canon_sha256":"2333c668cf5434f8eefa905c7222af2686ca6045bba30ce135c8c68862a21689"},"schema_version":"1.0","source":{"id":"1706.07338","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.07338","created_at":"2026-05-18T00:41:51Z"},{"alias_kind":"arxiv_version","alias_value":"1706.07338v1","created_at":"2026-05-18T00:41:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.07338","created_at":"2026-05-18T00:41:51Z"},{"alias_kind":"pith_short_12","alias_value":"EKC7G4HLN3ER","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_16","alias_value":"EKC7G4HLN3ERK7DH","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_8","alias_value":"EKC7G4HL","created_at":"2026-05-18T12:31:12Z"}],"graph_snapshots":[{"event_id":"sha256:33cee16a016d6a047f7130beba89c4aeebaf9d71d986a679573190c11247de8b","target":"graph","created_at":"2026-05-18T00:41:51Z","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":"In the polluted bootstrap percolation model, vertices of the cubic lattice $\\mathbb{Z}^3$ are independently declared initially occupied with probability $p$ or closed with probability $q$. Under the standard (respectively, modified) bootstrap rule, a vertex becomes occupied at a subsequent step if it is not closed and it has at least $3$ occupied neighbors (respectively, an occupied neighbor in each coordinate). We study the final density of occupied vertices as $p,q\\to 0$. We show that this density converges to $1$ if $q \\ll p^3(\\log p^{-1})^{-3}$ for both standard and modified rules. Our pri","authors_text":"Alexander E. Holroyd, David Sivakoff, Janko Gravner","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-06-22T14:29:20Z","title":"Polluted Bootstrap Percolation in Three Dimensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.07338","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":"sha256:4494ed5a3f270056d8a4d6c59b651117ca78c044690bdc5b4fbfb3f96bcf28d0","target":"record","created_at":"2026-05-18T00:41:51Z","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":"1f8902f54e20b2c0709199116f257f8c7d4c8ab3cfc8a925615c820d1d77fd13","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-06-22T14:29:20Z","title_canon_sha256":"2333c668cf5434f8eefa905c7222af2686ca6045bba30ce135c8c68862a21689"},"schema_version":"1.0","source":{"id":"1706.07338","kind":"arxiv","version":1}},"canonical_sha256":"2285f370eb6ec9157c67d07341c7fe02a3465cb16791c6cee0971ac15a3d04d7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2285f370eb6ec9157c67d07341c7fe02a3465cb16791c6cee0971ac15a3d04d7","first_computed_at":"2026-05-18T00:41:51.949604Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:41:51.949604Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UcNJrASzzxBr9rn+HI3qucTyziKI+qTh6KRBQRPzKFxtLTFF6j2/xOxp9RJhpbiXs6WeKmrMyJhhNfVEoOdkCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:41:51.950083Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.07338","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4494ed5a3f270056d8a4d6c59b651117ca78c044690bdc5b4fbfb3f96bcf28d0","sha256:33cee16a016d6a047f7130beba89c4aeebaf9d71d986a679573190c11247de8b"],"state_sha256":"78359f53634b8715b1e5dd9dfa99fec7e75fab77bffe9548157c7a2e73c95566"}