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This 3SUM conjecture implies $\\Omega(n^2)$ lower bounds on numerous problems in computational geometry and a variant of the conjecture implies strong lower bounds on triangle enumeration, dynamic graph algorithms, and string matching data structures.\n  In this paper we refute the 3SUM conjecture. We prove that the decision tree complexity of 3SUM is $O("},"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":"1404.0799","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2014-04-03T08:30:03Z","cross_cats_sorted":["cs.CC","cs.CG"],"title_canon_sha256":"bd937b5d7cf8f37c0dd775d32e4611efc29f2fd524d9bd8f82d68b99edbb9537","abstract_canon_sha256":"afd100ad4715b3713e6df60a1de4a7a934b26e4645794eb65d9568cf9f8f8677"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:50.160216Z","signature_b64":"fl/7pBGjU2RuKuqguGImEz6UK+blK90gfffyME9k+CrU9XgaRl9kdp6sg74wL0aw+no9tco8w62TENrJDYdUAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d2e8a5af703c8cbb926c660a68c5003ac8df0edc616dbae0bfc760ec3a2cc0a","last_reissued_at":"2026-05-18T02:50:50.159688Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:50.159688Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Threesomes, Degenerates, and Love Triangles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.CG"],"primary_cat":"cs.DS","authors_text":"Allan Gr{\\o}nlund, Seth Pettie","submitted_at":"2014-04-03T08:30:03Z","abstract_excerpt":"The 3SUM problem is to decide, given a set of $n$ real numbers, whether any three sum to zero. 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