{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:M4A7J3DKERSEUSIWGHGXWKSXVD","short_pith_number":"pith:M4A7J3DK","canonical_record":{"source":{"id":"2206.12335","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2022-06-24T15:13:41Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"7d30e9401cdb9de9cddeb8d5dff9f8faa7feff879074d052921507c0a7edfec3","abstract_canon_sha256":"0dbd7f16be59302388f1971e5ce3004a2a26229bc0df11b214fb2af3fc0f66e9"},"schema_version":"1.0"},"canonical_sha256":"6701f4ec6a24644a491631cd7b2a57a8f145f6c6bd5f1dc51562464e8b58e3a3","source":{"kind":"arxiv","id":"2206.12335","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2206.12335","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"arxiv_version","alias_value":"2206.12335v2","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2206.12335","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"pith_short_12","alias_value":"M4A7J3DKERSE","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"pith_short_16","alias_value":"M4A7J3DKERSEUSIW","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"pith_short_8","alias_value":"M4A7J3DK","created_at":"2026-07-05T11:25:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:M4A7J3DKERSEUSIWGHGXWKSXVD","target":"record","payload":{"canonical_record":{"source":{"id":"2206.12335","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2022-06-24T15:13:41Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"7d30e9401cdb9de9cddeb8d5dff9f8faa7feff879074d052921507c0a7edfec3","abstract_canon_sha256":"0dbd7f16be59302388f1971e5ce3004a2a26229bc0df11b214fb2af3fc0f66e9"},"schema_version":"1.0"},"canonical_sha256":"6701f4ec6a24644a491631cd7b2a57a8f145f6c6bd5f1dc51562464e8b58e3a3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:25:36.213547Z","signature_b64":"lCJreniQnuv/GRzJJeReTi53vHyvl46/ID6PGt9HKQWMnrtegJP+85JVt2nHVSh+bdOjTCqubD46ERle59zpAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6701f4ec6a24644a491631cd7b2a57a8f145f6c6bd5f1dc51562464e8b58e3a3","last_reissued_at":"2026-07-05T11:25:36.213047Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:25:36.213047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2206.12335","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":"2026-07-05T11:25:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W7hnx/GpqRMGA00jPY7/h6Z9ne8sdKY2YFEJ+RrUE9QB+fn9iaMLOOB7S0dgK8JCT6GPtQqHVG3hRnzHcLAxAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T07:41:08.889911Z"},"content_sha256":"a317225a36658858253be921b0cef09eb77420da6b48a1f5323cab5c340942e5","schema_version":"1.0","event_id":"sha256:a317225a36658858253be921b0cef09eb77420da6b48a1f5323cab5c340942e5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:M4A7J3DKERSEUSIWGHGXWKSXVD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Improved bounds for 1-independent percolation on $\\mathbb{Z}^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Alex Scott, Michael Savery, Paul Balister, Tom Johnston","submitted_at":"2022-06-24T15:13:41Z","abstract_excerpt":"A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$, the states of the edges in $S_1$ are independent of the states of the edges in $S_2$. Such a model is said to percolate if the random subgraph has an infinite component with positive probability. In 2012 the first author and Bollob\\'as defined $p_{\\max}(G)$ to be the supremum of those $p$ for which there exists a 1-independent bond percolation model on $G$ in which each edge is present in the random subgraph with p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.12335","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2206.12335/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T11:25:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E4HY/HcaiKxAJ8r/xkgE76WU+UuqifCobvMQWEIS1/wIORotGEAXEEQm0WSfQVbH43kg2iOv4tAIeLNqqdv1DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T07:41:08.890463Z"},"content_sha256":"43a6ff439aa867721cdf11c1e3e105db45370ac3b11aa4d9f4502932aa5962fc","schema_version":"1.0","event_id":"sha256:43a6ff439aa867721cdf11c1e3e105db45370ac3b11aa4d9f4502932aa5962fc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/M4A7J3DKERSEUSIWGHGXWKSXVD/bundle.json","state_url":"https://pith.science/pith/M4A7J3DKERSEUSIWGHGXWKSXVD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/M4A7J3DKERSEUSIWGHGXWKSXVD/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":"2026-08-22T07:41:08Z","links":{"resolver":"https://pith.science/pith/M4A7J3DKERSEUSIWGHGXWKSXVD","bundle":"https://pith.science/pith/M4A7J3DKERSEUSIWGHGXWKSXVD/bundle.json","state":"https://pith.science/pith/M4A7J3DKERSEUSIWGHGXWKSXVD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/M4A7J3DKERSEUSIWGHGXWKSXVD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:M4A7J3DKERSEUSIWGHGXWKSXVD","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":"0dbd7f16be59302388f1971e5ce3004a2a26229bc0df11b214fb2af3fc0f66e9","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2022-06-24T15:13:41Z","title_canon_sha256":"7d30e9401cdb9de9cddeb8d5dff9f8faa7feff879074d052921507c0a7edfec3"},"schema_version":"1.0","source":{"id":"2206.12335","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2206.12335","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"arxiv_version","alias_value":"2206.12335v2","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2206.12335","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"pith_short_12","alias_value":"M4A7J3DKERSE","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"pith_short_16","alias_value":"M4A7J3DKERSEUSIW","created_at":"2026-07-05T11:25:36Z"},{"alias_kind":"pith_short_8","alias_value":"M4A7J3DK","created_at":"2026-07-05T11:25:36Z"}],"graph_snapshots":[{"event_id":"sha256:43a6ff439aa867721cdf11c1e3e105db45370ac3b11aa4d9f4502932aa5962fc","target":"graph","created_at":"2026-07-05T11:25:36Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2206.12335/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$, the states of the edges in $S_1$ are independent of the states of the edges in $S_2$. Such a model is said to percolate if the random subgraph has an infinite component with positive probability. In 2012 the first author and Bollob\\'as defined $p_{\\max}(G)$ to be the supremum of those $p$ for which there exists a 1-independent bond percolation model on $G$ in which each edge is present in the random subgraph with p","authors_text":"Alex Scott, Michael Savery, Paul Balister, Tom Johnston","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2022-06-24T15:13:41Z","title":"Improved bounds for 1-independent percolation on $\\mathbb{Z}^n$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.12335","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:a317225a36658858253be921b0cef09eb77420da6b48a1f5323cab5c340942e5","target":"record","created_at":"2026-07-05T11:25:36Z","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":"0dbd7f16be59302388f1971e5ce3004a2a26229bc0df11b214fb2af3fc0f66e9","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2022-06-24T15:13:41Z","title_canon_sha256":"7d30e9401cdb9de9cddeb8d5dff9f8faa7feff879074d052921507c0a7edfec3"},"schema_version":"1.0","source":{"id":"2206.12335","kind":"arxiv","version":2}},"canonical_sha256":"6701f4ec6a24644a491631cd7b2a57a8f145f6c6bd5f1dc51562464e8b58e3a3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6701f4ec6a24644a491631cd7b2a57a8f145f6c6bd5f1dc51562464e8b58e3a3","first_computed_at":"2026-07-05T11:25:36.213047Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:25:36.213047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lCJreniQnuv/GRzJJeReTi53vHyvl46/ID6PGt9HKQWMnrtegJP+85JVt2nHVSh+bdOjTCqubD46ERle59zpAg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:25:36.213547Z","signed_message":"canonical_sha256_bytes"},"source_id":"2206.12335","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a317225a36658858253be921b0cef09eb77420da6b48a1f5323cab5c340942e5","sha256:43a6ff439aa867721cdf11c1e3e105db45370ac3b11aa4d9f4502932aa5962fc"],"state_sha256":"5899eae6b8fed0a814d926120032bfed584a0aff8c3b178c30a5c5664ff9e5ae"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mraeo1yx3K29y29zja1oibPJBrko7Vdqsl+JyGEjwCGByje5ws/RFVYI3g2YtKlf8k7SyyhCIuqunt0DF+fjCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-22T07:41:08.894892Z","bundle_sha256":"37674ca3f2cf0a4205b9fbab33c7eb26d46dc38965d6be006f20f18a60302457"}}