{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:U5VXJDRHNUNWAC77GXM2352DM3","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":"6a5d2f9bdf49c06603ca12218cb9b6adebb972a33a747150be2681a0ac66d97d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-24T00:47:02Z","title_canon_sha256":"5ad37ec5e7c32ed11070a98efa639bd75bdd3623c8493496fed4ff2ee3fa9d65"},"schema_version":"1.0","source":{"id":"1202.5351","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.5351","created_at":"2026-05-18T02:28:54Z"},{"alias_kind":"arxiv_version","alias_value":"1202.5351v5","created_at":"2026-05-18T02:28:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.5351","created_at":"2026-05-18T02:28:54Z"},{"alias_kind":"pith_short_12","alias_value":"U5VXJDRHNUNW","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"U5VXJDRHNUNWAC77","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"U5VXJDRH","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:29c52f0d807995e194b4aa098853c7f9390ae4c44ce8a4d52e3d8f8aaf374a16","target":"graph","created_at":"2026-05-18T02:28:54Z","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 Hamming torus of dimension $d$ is the graph with vertices $\\{1,\\dots,n\\}^d$ and an edge between any two vertices that differ in a single coordinate. Bootstrap percolation with threshold $\\theta$ starts with a random set of open vertices, to which every vertex belongs independently with probability $p$, and at each time step the open set grows by adjoining every vertex with at least $\\theta$ open neighbors. We assume that $n$ is large and that $p$ scales as $n^{-\\alpha}$ for some $\\alpha>1$, and study the probability that an $i$-dimensional subgraph ever becomes open. For large $\\theta$, we","authors_text":"Christopher Hoffman, David Sivakoff, James Pfeiffer, Janko Gravner","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-24T00:47:02Z","title":"Bootstrap percolation on the Hamming torus"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.5351","kind":"arxiv","version":5},"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:debb34f7213e0f088b163758843766f281d50154146653a9c862d9eab7f5e2fb","target":"record","created_at":"2026-05-18T02:28:54Z","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":"6a5d2f9bdf49c06603ca12218cb9b6adebb972a33a747150be2681a0ac66d97d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-02-24T00:47:02Z","title_canon_sha256":"5ad37ec5e7c32ed11070a98efa639bd75bdd3623c8493496fed4ff2ee3fa9d65"},"schema_version":"1.0","source":{"id":"1202.5351","kind":"arxiv","version":5}},"canonical_sha256":"a76b748e276d1b600bff35d9adf74366cec9529dc0e6a42b2368a87a232b2f4c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a76b748e276d1b600bff35d9adf74366cec9529dc0e6a42b2368a87a232b2f4c","first_computed_at":"2026-05-18T02:28:54.273152Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:28:54.273152Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lpqIDuN3NoAf/W6rZEuAdpf84S9p8PiBvi5mhDPq+gwSOCUozyB68dISPa0LnH7qOFFMQ+LlqRG6nhFLb1ikDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:28:54.273486Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.5351","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:debb34f7213e0f088b163758843766f281d50154146653a9c862d9eab7f5e2fb","sha256:29c52f0d807995e194b4aa098853c7f9390ae4c44ce8a4d52e3d8f8aaf374a16"],"state_sha256":"bca45f26604fd1569e450409eb658d140743f17b24ad9a72d9c8c502574c90db"}