{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:X7M7B4NCJBYKQRPGOEISQ64XHH","short_pith_number":"pith:X7M7B4NC","canonical_record":{"source":{"id":"1305.5444","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-05-23T15:02:31Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"583f8aa39b3472c9b941d5303c3667ecfa969639667277eb237fe8d20d707875","abstract_canon_sha256":"bf5a355bd90f4bbd86cb25ab7be43d743818dc3c4e60fe5b0f83515457768e1b"},"schema_version":"1.0"},"canonical_sha256":"bfd9f0f1a24870a845e67111287b9739c730450146261e578d4aea485666e788","source":{"kind":"arxiv","id":"1305.5444","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.5444","created_at":"2026-05-18T01:35:16Z"},{"alias_kind":"arxiv_version","alias_value":"1305.5444v2","created_at":"2026-05-18T01:35:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.5444","created_at":"2026-05-18T01:35:16Z"},{"alias_kind":"pith_short_12","alias_value":"X7M7B4NCJBYK","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"X7M7B4NCJBYKQRPG","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"X7M7B4NC","created_at":"2026-05-18T12:28:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:X7M7B4NCJBYKQRPGOEISQ64XHH","target":"record","payload":{"canonical_record":{"source":{"id":"1305.5444","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-05-23T15:02:31Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"583f8aa39b3472c9b941d5303c3667ecfa969639667277eb237fe8d20d707875","abstract_canon_sha256":"bf5a355bd90f4bbd86cb25ab7be43d743818dc3c4e60fe5b0f83515457768e1b"},"schema_version":"1.0"},"canonical_sha256":"bfd9f0f1a24870a845e67111287b9739c730450146261e578d4aea485666e788","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:16.836211Z","signature_b64":"dmkvB2aYqRMTA9e1mrsKLdtQl3gnKMSasIyjD6p11JxsN03gApaJJzsfGB3q6PxAJTHXmbOq5LOtmi6DP7g5BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bfd9f0f1a24870a845e67111287b9739c730450146261e578d4aea485666e788","last_reissued_at":"2026-05-18T01:35:16.835722Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:16.835722Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1305.5444","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":"2026-05-18T01:35:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X/80oejeWU9TN88YsduPseuubAm6WagZ6EnC3hGG2u8maut94XyscnVymA5fYZNOQXU2W9ylxDqOX33YXmSEDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T13:29:28.727155Z"},"content_sha256":"982f0f4f27ed79c27f5e40496494e38055cef4bc39345c4ae67a950e7d5e0aef","schema_version":"1.0","event_id":"sha256:982f0f4f27ed79c27f5e40496494e38055cef4bc39345c4ae67a950e7d5e0aef"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:X7M7B4NCJBYKQRPGOEISQ64XHH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The time of bootstrap percolation in two dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"B\\'ela Bollob\\'as, Paul Balister, Paul Smith","submitted_at":"2013-05-23T15:02:31Z","abstract_excerpt":"We study the distribution of the percolation time $T$ of two-neighbour bootstrap percolation on $[n]^2$ with initial set $A\\sim\\mathrm{Bin}([n]^2,p)$. We determine $T$ with high probability up to a constant factor for all $p$ above the critical probability for percolation, and to within a $1+o(1)$ factor for a large range of $p$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.5444","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":"2026-05-18T01:35:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dd2ebopo80KCPZPGeXmKmhdy+q5seKPIooVZluGoFIDO+d13B6b2+qCR7HD1HIaoT43z9VMvvq63Ydm3s1BjDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T13:29:28.727503Z"},"content_sha256":"26174e599dfdf83c621b7eee636995df53433854161e3522578196b95f8227a2","schema_version":"1.0","event_id":"sha256:26174e599dfdf83c621b7eee636995df53433854161e3522578196b95f8227a2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X7M7B4NCJBYKQRPGOEISQ64XHH/bundle.json","state_url":"https://pith.science/pith/X7M7B4NCJBYKQRPGOEISQ64XHH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X7M7B4NCJBYKQRPGOEISQ64XHH/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":"2026-06-02T13:29:28Z","links":{"resolver":"https://pith.science/pith/X7M7B4NCJBYKQRPGOEISQ64XHH","bundle":"https://pith.science/pith/X7M7B4NCJBYKQRPGOEISQ64XHH/bundle.json","state":"https://pith.science/pith/X7M7B4NCJBYKQRPGOEISQ64XHH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X7M7B4NCJBYKQRPGOEISQ64XHH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:X7M7B4NCJBYKQRPGOEISQ64XHH","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":"bf5a355bd90f4bbd86cb25ab7be43d743818dc3c4e60fe5b0f83515457768e1b","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-05-23T15:02:31Z","title_canon_sha256":"583f8aa39b3472c9b941d5303c3667ecfa969639667277eb237fe8d20d707875"},"schema_version":"1.0","source":{"id":"1305.5444","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.5444","created_at":"2026-05-18T01:35:16Z"},{"alias_kind":"arxiv_version","alias_value":"1305.5444v2","created_at":"2026-05-18T01:35:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.5444","created_at":"2026-05-18T01:35:16Z"},{"alias_kind":"pith_short_12","alias_value":"X7M7B4NCJBYK","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"X7M7B4NCJBYKQRPG","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"X7M7B4NC","created_at":"2026-05-18T12:28:06Z"}],"graph_snapshots":[{"event_id":"sha256:26174e599dfdf83c621b7eee636995df53433854161e3522578196b95f8227a2","target":"graph","created_at":"2026-05-18T01:35:16Z","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 study the distribution of the percolation time $T$ of two-neighbour bootstrap percolation on $[n]^2$ with initial set $A\\sim\\mathrm{Bin}([n]^2,p)$. We determine $T$ with high probability up to a constant factor for all $p$ above the critical probability for percolation, and to within a $1+o(1)$ factor for a large range of $p$.","authors_text":"B\\'ela Bollob\\'as, Paul Balister, Paul Smith","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-05-23T15:02:31Z","title":"The time of bootstrap percolation in two dimensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.5444","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":"sha256:982f0f4f27ed79c27f5e40496494e38055cef4bc39345c4ae67a950e7d5e0aef","target":"record","created_at":"2026-05-18T01:35:16Z","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":"bf5a355bd90f4bbd86cb25ab7be43d743818dc3c4e60fe5b0f83515457768e1b","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-05-23T15:02:31Z","title_canon_sha256":"583f8aa39b3472c9b941d5303c3667ecfa969639667277eb237fe8d20d707875"},"schema_version":"1.0","source":{"id":"1305.5444","kind":"arxiv","version":2}},"canonical_sha256":"bfd9f0f1a24870a845e67111287b9739c730450146261e578d4aea485666e788","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bfd9f0f1a24870a845e67111287b9739c730450146261e578d4aea485666e788","first_computed_at":"2026-05-18T01:35:16.835722Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:35:16.835722Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dmkvB2aYqRMTA9e1mrsKLdtQl3gnKMSasIyjD6p11JxsN03gApaJJzsfGB3q6PxAJTHXmbOq5LOtmi6DP7g5BA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:35:16.836211Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.5444","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:982f0f4f27ed79c27f5e40496494e38055cef4bc39345c4ae67a950e7d5e0aef","sha256:26174e599dfdf83c621b7eee636995df53433854161e3522578196b95f8227a2"],"state_sha256":"6cb375ef3accd241d561c7c6ed043ec387b86a9fb50063fb8e4af24f5346fbd9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4ZXu06pVJLSZ6eDqezHe6iXS71d/UGGbNBUzFpvQmkIFtAJjDsRaCnPoOdlfCCEa4aWp/0cEk5AsohonZgE6Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T13:29:28.729275Z","bundle_sha256":"97657433fe20ab197666543b19112b49dcea93f2a1273b2f40cff8c0ef7453b1"}}