{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:LK7WFGPJM32D7QGN7UJAQUGUR6","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":"8d24816bb918634bd8b8d9fec18f67798c2a2fd367214305f39729b5bec04fb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-11-05T01:45:56Z","title_canon_sha256":"6b70d2a5e25eec15e441868f89a0710021df9a71e899c457b13af731313b702a"},"schema_version":"1.0","source":{"id":"1811.01477","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.01477","created_at":"2026-05-18T00:01:34Z"},{"alias_kind":"arxiv_version","alias_value":"1811.01477v1","created_at":"2026-05-18T00:01:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.01477","created_at":"2026-05-18T00:01:34Z"},{"alias_kind":"pith_short_12","alias_value":"LK7WFGPJM32D","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"LK7WFGPJM32D7QGN","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"LK7WFGPJ","created_at":"2026-05-18T12:32:37Z"}],"graph_snapshots":[{"event_id":"sha256:12f734797afbc376fee50a05daf350ad2e0ba06a86aa921141eef270bccfe422","target":"graph","created_at":"2026-05-18T00:01:34Z","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 consider Bernoulli bond percolation on the product graph of a regular tree and a line. We show that the triangle condition does not hold at the uniqueness threshold.","authors_text":"Kohei Yamamoto","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-11-05T01:45:56Z","title":"Percolation on the product graph of a regular tree and a line does not satisfy the triangle condition at the uniqueness threshold"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.01477","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:e4ca33e40885072f20e4d715c905f26faa4a98d904626a31d41fb857b8e9a7e2","target":"record","created_at":"2026-05-18T00:01:34Z","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":"8d24816bb918634bd8b8d9fec18f67798c2a2fd367214305f39729b5bec04fb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-11-05T01:45:56Z","title_canon_sha256":"6b70d2a5e25eec15e441868f89a0710021df9a71e899c457b13af731313b702a"},"schema_version":"1.0","source":{"id":"1811.01477","kind":"arxiv","version":1}},"canonical_sha256":"5abf6299e966f43fc0cdfd120850d48f8198dfa72c1e7e81e9bab42219904a96","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5abf6299e966f43fc0cdfd120850d48f8198dfa72c1e7e81e9bab42219904a96","first_computed_at":"2026-05-18T00:01:34.200835Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:01:34.200835Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1305wS0hsKKhLPBYB2bWy7Av90j6w72QtVBnprmxHuRtvA51F2mxLAViN9K3+zTvAU3b/1rZmr3w+0LtwNCXAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:01:34.201349Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.01477","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e4ca33e40885072f20e4d715c905f26faa4a98d904626a31d41fb857b8e9a7e2","sha256:12f734797afbc376fee50a05daf350ad2e0ba06a86aa921141eef270bccfe422"],"state_sha256":"2056510fd538d746fe0e78bed1a7e068d5d2354d3e9be1744b524b125ca605af"}