{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:DK7RYW6B5FCISJBZFCPC7DOZIV","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":"68dc80a3544690b4e4abe663ec1b5050fb73df39f9c141a3fcfa4258a977c3a9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-02T14:55:58Z","title_canon_sha256":"707dabc20b16da367634d377b88349f3ffd886512bd2aeb1c2b96bb1d8f0aeb9"},"schema_version":"1.0","source":{"id":"2607.02272","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.02272","created_at":"2026-07-03T01:17:47Z"},{"alias_kind":"arxiv_version","alias_value":"2607.02272v1","created_at":"2026-07-03T01:17:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.02272","created_at":"2026-07-03T01:17:47Z"},{"alias_kind":"pith_short_12","alias_value":"DK7RYW6B5FCI","created_at":"2026-07-03T01:17:47Z"},{"alias_kind":"pith_short_16","alias_value":"DK7RYW6B5FCISJBZ","created_at":"2026-07-03T01:17:47Z"},{"alias_kind":"pith_short_8","alias_value":"DK7RYW6B","created_at":"2026-07-03T01:17:47Z"}],"graph_snapshots":[{"event_id":"sha256:a3d6653ae7c2790123cc53d39f067dd8745d94c05eb3632c12ecaae2ea3535d6","target":"graph","created_at":"2026-07-03T01:17:47Z","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/2607.02272/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We characterize the pairs of increasing events $A,B$ for which there is equality in the BK inequality. Namely, we show that $P(A\\circ B)=P(A)P(B)$ if and only if all the configurations in $A\\times B$ admit disjoint witnesses for $A$ and $B$. We discuss the strengthened BK inequality, and we provide a new simplified proof of this inequality.","authors_text":"Pierre Tesio, Rapha\\\"el Cerf","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-02T14:55:58Z","title":"The case of equality in BK"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02272","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:e3575fa02cd5041590cee81f985d2e6a8281c8e3715286feef1e1bdf15dd63e1","target":"record","created_at":"2026-07-03T01:17:47Z","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":"68dc80a3544690b4e4abe663ec1b5050fb73df39f9c141a3fcfa4258a977c3a9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-02T14:55:58Z","title_canon_sha256":"707dabc20b16da367634d377b88349f3ffd886512bd2aeb1c2b96bb1d8f0aeb9"},"schema_version":"1.0","source":{"id":"2607.02272","kind":"arxiv","version":1}},"canonical_sha256":"1abf1c5bc1e944892439289e2f8dd9454758efd4e7446554525439d082c65e61","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1abf1c5bc1e944892439289e2f8dd9454758efd4e7446554525439d082c65e61","first_computed_at":"2026-07-03T01:17:47.172719Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-03T01:17:47.172719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EVUUkDn5cQY+OBSOEr/q1QC1C835JWrR7X78+sG+WbjOHAOhB14HCPkVUXX2B0KA51ksg1QIQbHSpz6zZ+5OAg==","signature_status":"signed_v1","signed_at":"2026-07-03T01:17:47.173076Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.02272","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e3575fa02cd5041590cee81f985d2e6a8281c8e3715286feef1e1bdf15dd63e1","sha256:a3d6653ae7c2790123cc53d39f067dd8745d94c05eb3632c12ecaae2ea3535d6"],"state_sha256":"38981414276fc9dd7d450c8aeb41345ce620970493f4e5cfe45a03dbc79ba75f"}