{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:576XC73FFTOSK4EXQXEFQNHCLL","short_pith_number":"pith:576XC73F","schema_version":"1.0","canonical_sha256":"effd717f652cdd25709785c85834e25ad6da6f7d926ee6c1cf4cc2ab25bea2d4","source":{"kind":"arxiv","id":"2605.15066","version":1},"attestation_state":"computed","paper":{"title":"The critical activation density in graph bootstrap percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For every graph H the critical H-percolation threshold p_c(n,H) is located in terms of the limiting density ρ(H) of the graphs that activate an edge most efficiently.","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Brett Kolesnik, Maksim Zhukovskii, Pawe{\\l} Pra{\\l}at, Rajko Nenadov, Tam\\'as Makai, Xavier P\\'erez-Gim\\'enez","submitted_at":"2026-05-14T16:56:35Z","abstract_excerpt":"In graph bootstrap percolation, edges of an Erd\\H{o}s-R\\'enyi random graph ${\\mathcal G}_{n,p}$ are initially active. Activation spreads to other edges of the complete graph $K_n$ by an iterative process governed by a fixed graph $H$, whereby an edge becomes active whenever it is the only inactive edge in a copy of $H$. If all edges of $K_n$ are eventually activated, we say the process $H$-percolates. The case $H=K_3$ corresponds to the classical sharp threshold for connectivity in ${\\mathcal G}_{n,p}$. When $H=K_4$, there are close connections with $2$-neighbor bootstrap percolation from stat"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.15066","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-05-14T16:56:35Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"372081e84101975713fd6fd29401cf8a59eb10c946fbf4918c1621f8733569d7","abstract_canon_sha256":"02af2e2dc014bd743712f2bbe6b1cd8d60f293039b69f508eaf8a90b0a61084b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:38:54.229963Z","signature_b64":"ktURxUft9wJf+y0Rgd8eAfZDgA9EVlB/5LZUV4/U4912TmU7eG/ApJUgXcozWS7WAiuAN3HTmgcDwR6aYgQ2CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"effd717f652cdd25709785c85834e25ad6da6f7d926ee6c1cf4cc2ab25bea2d4","last_reissued_at":"2026-05-17T23:38:54.229160Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:38:54.229160Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The critical activation density in graph bootstrap percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For every graph H the critical H-percolation threshold p_c(n,H) is located in terms of the limiting density ρ(H) of the graphs that activate an edge most efficiently.","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Brett Kolesnik, Maksim Zhukovskii, Pawe{\\l} Pra{\\l}at, Rajko Nenadov, Tam\\'as Makai, Xavier P\\'erez-Gim\\'enez","submitted_at":"2026-05-14T16:56:35Z","abstract_excerpt":"In graph bootstrap percolation, edges of an Erd\\H{o}s-R\\'enyi random graph ${\\mathcal G}_{n,p}$ are initially active. Activation spreads to other edges of the complete graph $K_n$ by an iterative process governed by a fixed graph $H$, whereby an edge becomes active whenever it is the only inactive edge in a copy of $H$. If all edges of $K_n$ are eventually activated, we say the process $H$-percolates. The case $H=K_3$ corresponds to the classical sharp threshold for connectivity in ${\\mathcal G}_{n,p}$. When $H=K_4$, there are close connections with $2$-neighbor bootstrap percolation from stat"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"for every graph H, we locate the critical H-percolation threshold p_c(n,H)","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the critical limiting density ρ(H) of graphs that most efficiently activate a given edge is well-defined and finite for every H, and that the threshold is determined by this quantity.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For any fixed graph H the critical percolation threshold p_c(n,H) equals the value determined by the minimal activation density ρ(H) of graphs that efficiently activate an edge.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For every graph H the critical H-percolation threshold p_c(n,H) is located in terms of the limiting density ρ(H) of the graphs that activate an edge most efficiently.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"de4d5078fc9a83ec4ba0afa448f4e8bef3c934ded18058d468407166489f8e52"},"source":{"id":"2605.15066","kind":"arxiv","version":1},"verdict":{"id":"ac963031-7fbb-4e57-a814-21e4cf41cf3d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:57:30.338857Z","strongest_claim":"for every graph H, we locate the critical H-percolation threshold p_c(n,H)","one_line_summary":"For any fixed graph H the critical percolation threshold p_c(n,H) equals the value determined by the minimal activation density ρ(H) of graphs that efficiently activate an edge.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the critical limiting density ρ(H) of graphs that most efficiently activate a given edge is well-defined and finite for every H, and that the threshold is determined by this quantity.","pith_extraction_headline":"For every graph H the critical H-percolation threshold p_c(n,H) is located in terms of the limiting density ρ(H) of the graphs that activate an edge most efficiently."},"references":{"count":30,"sample":[{"doi":"","year":1988,"title":"M. Aizenman and J. L. Lebowitz,Metastability effects in bootstrap percolation, J. Phys. A21(1988), no. 19, 3801–3813","work_id":"32a08250-75c2-4560-b60c-880d47df233d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1985,"title":"N. Alon,An extremal problem for sets with applications to graph theory, J. Combin. Theory Ser. A40(1985), no. 1, 82–89","work_id":"0e5f2e1d-37cf-4e60-9169-d2eaaaa69502","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"N. Alon and J. H. Spencer,The probabilistic method, fourth ed., Wiley Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, 2016","work_id":"4499de7c-30a3-4c53-87d6-15694724ad69","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"O. Angel and B. Kolesnik,Sharp thresholds for contagious sets in random graphs, Ann. Appl. Probab.28(2018), no. 2, 1052–1098","work_id":"41cb610d-ae93-4915-8d84-8e2d75c86b6b","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":",Large deviations for subcritical bootstrap percolation on the Erd˝ os–Rényi graph, J. Stat. Phys.185(2021), no. 2, Paper No. 8, 16","work_id":"dc0e7b4d-8ff3-4b1d-8ac6-666e7cfc27e3","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":30,"snapshot_sha256":"ded220e1a6d40e552d9364e3a039c3f426434a349ce47a064514e90b340e29ee","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.15066","created_at":"2026-05-17T23:38:54.229297+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.15066v1","created_at":"2026-05-17T23:38:54.229297+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.15066","created_at":"2026-05-17T23:38:54.229297+00:00"},{"alias_kind":"pith_short_12","alias_value":"576XC73FFTOS","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_16","alias_value":"576XC73FFTOSK4EX","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_8","alias_value":"576XC73F","created_at":"2026-05-18T12:33:37.589309+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/576XC73FFTOSK4EXQXEFQNHCLL","json":"https://pith.science/pith/576XC73FFTOSK4EXQXEFQNHCLL.json","graph_json":"https://pith.science/api/pith-number/576XC73FFTOSK4EXQXEFQNHCLL/graph.json","events_json":"https://pith.science/api/pith-number/576XC73FFTOSK4EXQXEFQNHCLL/events.json","paper":"https://pith.science/paper/576XC73F"},"agent_actions":{"view_html":"https://pith.science/pith/576XC73FFTOSK4EXQXEFQNHCLL","download_json":"https://pith.science/pith/576XC73FFTOSK4EXQXEFQNHCLL.json","view_paper":"https://pith.science/paper/576XC73F","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.15066&json=true","fetch_graph":"https://pith.science/api/pith-number/576XC73FFTOSK4EXQXEFQNHCLL/graph.json","fetch_events":"https://pith.science/api/pith-number/576XC73FFTOSK4EXQXEFQNHCLL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/576XC73FFTOSK4EXQXEFQNHCLL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/576XC73FFTOSK4EXQXEFQNHCLL/action/storage_attestation","attest_author":"https://pith.science/pith/576XC73FFTOSK4EXQXEFQNHCLL/action/author_attestation","sign_citation":"https://pith.science/pith/576XC73FFTOSK4EXQXEFQNHCLL/action/citation_signature","submit_replication":"https://pith.science/pith/576XC73FFTOSK4EXQXEFQNHCLL/action/replication_record"}},"created_at":"2026-05-17T23:38:54.229297+00:00","updated_at":"2026-05-17T23:38:54.229297+00:00"}