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It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds.\n  For a communication problem $f$, let $f \\wedge f$ denote the function that evaluates $f$ on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem $f$ with $\\mathbf{UPP}(f)= O(\\log n)$, and $\\mathbf{UPP}(f \\wedge"},"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":"1903.00544","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2019-03-01T21:29:13Z","cross_cats_sorted":[],"title_canon_sha256":"1d7151a5f3887fbc93331fbb5e7ddc32f87a8ab0b17a8afd61315b5b9d8f4bb6","abstract_canon_sha256":"17566b3cb58ed64c3ca2c2a2ce75307727b93600758d446dcf598b817409fd5b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:52:14.196250Z","signature_b64":"TPROYta83/CKIa25RE7zF/xAL5y8ZJfQb19ZTWsTVl5Ql0vcudpstVL516uzMiTCvXBXT6cSLaTF9SbtlozkBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b53caafe3ea00fa8f3af14242277886328acce5a07e3c80df9def652524d6441","last_reissued_at":"2026-05-17T23:52:14.195608Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:52:14.195608Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sign-Rank Can Increase Under Intersection","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Justin Thaler, Mark Bun, Nikhil S. 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