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Precisely, with ${\\bf n}:=\\{1,\\ldots,n\\}$ and $K \\subset {\\bf n}$, for ${\\bf x} \\in S$ letting $[{\\bf x}]_K=\\{{\\bf y} \\in S: y_i = x_i, i \\in K\\}$, the set $A \\Box B$ consists of all ${\\bf x} \\in S$ for which there exist disjoint"},"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":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1508.07267","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-28T16:30:08Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5700dc0440f97bf0b7c4cc82e5f51fc91af27ec0ce7c23c5f5ed9f2c9f003a81","abstract_canon_sha256":"5d3591984a5550cfedbdc40b2a3490b767ab194eb43eae19b1261a675452260f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:15.154359Z","signature_b64":"7XOpcb9QM3j8H4PQrz+TyZxTT+vMCb5yCpJ5Z71HX/C3SJx2QpyWq9u4aUK27E78fGx35UlcdKtZzjF/Z9+vBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"22b63da42b9fa36946c9e01dea83be25dc4dc585c5c725c2cd6a703d2f5a31ea","last_reissued_at":"2026-05-18T01:33:15.153779Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:15.153779Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Functional van den Berg-Kesten-Reimer Inequalities and their Duals, with Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Larry Goldstein, Yosef Rinott","submitted_at":"2015-08-28T16:30:08Z","abstract_excerpt":"The BKR inequality conjectured by van den Berg and Kesten in [11], and proved by Reimer in [8], states that for $A$ and $B$ events on $S$, a finite product of finite sets $S_i,i=1,\\ldots,n$, and $P$ any product measure on $S$, $$ P(A \\Box B) \\le P(A)P(B),$$ where the set $A \\Box B$ consists of the elementary events which lie in both $A$ and $B$ for `disjoint reasons.' Precisely, with ${\\bf n}:=\\{1,\\ldots,n\\}$ and $K \\subset {\\bf n}$, for ${\\bf x} \\in S$ letting $[{\\bf x}]_K=\\{{\\bf y} \\in S: y_i = x_i, i \\in K\\}$, the set $A \\Box B$ consists of all ${\\bf x} \\in S$ for which there exist disjoint"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07267","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1508.07267","created_at":"2026-05-18T01:33:15.153845+00:00"},{"alias_kind":"arxiv_version","alias_value":"1508.07267v2","created_at":"2026-05-18T01:33:15.153845+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.07267","created_at":"2026-05-18T01:33:15.153845+00:00"},{"alias_kind":"pith_short_12","alias_value":"EK3D3JBLT6RW","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_16","alias_value":"EK3D3JBLT6RWSRWJ","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_8","alias_value":"EK3D3JBL","created_at":"2026-05-18T12:29:19.899920+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/EK3D3JBLT6RWSRWJ4AO6VA56EX","json":"https://pith.science/pith/EK3D3JBLT6RWSRWJ4AO6VA56EX.json","graph_json":"https://pith.science/api/pith-number/EK3D3JBLT6RWSRWJ4AO6VA56EX/graph.json","events_json":"https://pith.science/api/pith-number/EK3D3JBLT6RWSRWJ4AO6VA56EX/events.json","paper":"https://pith.science/paper/EK3D3JBL"},"agent_actions":{"view_html":"https://pith.science/pith/EK3D3JBLT6RWSRWJ4AO6VA56EX","download_json":"https://pith.science/pith/EK3D3JBLT6RWSRWJ4AO6VA56EX.json","view_paper":"https://pith.science/paper/EK3D3JBL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1508.07267&json=true","fetch_graph":"https://pith.science/api/pith-number/EK3D3JBLT6RWSRWJ4AO6VA56EX/graph.json","fetch_events":"https://pith.science/api/pith-number/EK3D3JBLT6RWSRWJ4AO6VA56EX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EK3D3JBLT6RWSRWJ4AO6VA56EX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EK3D3JBLT6RWSRWJ4AO6VA56EX/action/storage_attestation","attest_author":"https://pith.science/pith/EK3D3JBLT6RWSRWJ4AO6VA56EX/action/author_attestation","sign_citation":"https://pith.science/pith/EK3D3JBLT6RWSRWJ4AO6VA56EX/action/citation_signature","submit_replication":"https://pith.science/pith/EK3D3JBLT6RWSRWJ4AO6VA56EX/action/replication_record"}},"created_at":"2026-05-18T01:33:15.153845+00:00","updated_at":"2026-05-18T01:33:15.153845+00:00"}