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Provided the constraint density satisfies $\\Delta \\gg 1$, such an instance is unsatisfiable with high probability. The \\emph{refutation} problem is to efficiently find a proof of unsatisfiability.\n  We show that whenever the predicate $P$ supports a $t$-\\emph{wise uniform} probability distribution on its satisfying assignments, the sum of squares "},"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":"1701.04521","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2017-01-17T03:36:34Z","cross_cats_sorted":[],"title_canon_sha256":"dc3a93b38cc37db6e796f61a303c7f4def7af999cdfea359d39a777febe2fe44","abstract_canon_sha256":"02e088a4d52cd382056a172d9c62d26149884669ef453ebc7c98a4852ef793e3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:43.442611Z","signature_b64":"GBytp6tDgRT49lg9d9I/j+4jNz3WChbEQPlpNmnbxjOE+5xQwvTZic6fOt6g38GIDgyW/D++aeuRmDYb7oxTAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"04811e4e730d890feb2bc44c1799f7046c77efdcc4fc7f907c3ab53d997ce2ca","last_reissued_at":"2026-05-18T00:52:43.442125Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:43.442125Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sum of squares lower bounds for refuting any CSP","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"David Witmer, Pravesh K. Kothari, Ryan O'Donnell, Ryuhei Mori","submitted_at":"2017-01-17T03:36:34Z","abstract_excerpt":"Let $P:\\{0,1\\}^k \\to \\{0,1\\}$ be a nontrivial $k$-ary predicate. Consider a random instance of the constraint satisfaction problem $\\mathrm{CSP}(P)$ on $n$ variables with $\\Delta n$ constraints, each being $P$ applied to $k$ randomly chosen literals. Provided the constraint density satisfies $\\Delta \\gg 1$, such an instance is unsatisfiable with high probability. The \\emph{refutation} problem is to efficiently find a proof of unsatisfiability.\n  We show that whenever the predicate $P$ supports a $t$-\\emph{wise uniform} probability distribution on its satisfying assignments, the sum of squares "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.04521","kind":"arxiv","version":1},"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":"1701.04521","created_at":"2026-05-18T00:52:43.442206+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.04521v1","created_at":"2026-05-18T00:52:43.442206+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.04521","created_at":"2026-05-18T00:52:43.442206+00:00"},{"alias_kind":"pith_short_12","alias_value":"ASAR4TTTBWEQ","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_16","alias_value":"ASAR4TTTBWEQ72ZL","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_8","alias_value":"ASAR4TTT","created_at":"2026-05-18T12:31:08.081275+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1908.06130","citing_title":"Average-Case Lower Bounds for Learning Sparse Mixtures, Robust Estimation and Semirandom Adversaries","ref_index":19,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ASAR4TTTBWEQ72ZLYRGBPGPXAR","json":"https://pith.science/pith/ASAR4TTTBWEQ72ZLYRGBPGPXAR.json","graph_json":"https://pith.science/api/pith-number/ASAR4TTTBWEQ72ZLYRGBPGPXAR/graph.json","events_json":"https://pith.science/api/pith-number/ASAR4TTTBWEQ72ZLYRGBPGPXAR/events.json","paper":"https://pith.science/paper/ASAR4TTT"},"agent_actions":{"view_html":"https://pith.science/pith/ASAR4TTTBWEQ72ZLYRGBPGPXAR","download_json":"https://pith.science/pith/ASAR4TTTBWEQ72ZLYRGBPGPXAR.json","view_paper":"https://pith.science/paper/ASAR4TTT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.04521&json=true","fetch_graph":"https://pith.science/api/pith-number/ASAR4TTTBWEQ72ZLYRGBPGPXAR/graph.json","fetch_events":"https://pith.science/api/pith-number/ASAR4TTTBWEQ72ZLYRGBPGPXAR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ASAR4TTTBWEQ72ZLYRGBPGPXAR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ASAR4TTTBWEQ72ZLYRGBPGPXAR/action/storage_attestation","attest_author":"https://pith.science/pith/ASAR4TTTBWEQ72ZLYRGBPGPXAR/action/author_attestation","sign_citation":"https://pith.science/pith/ASAR4TTTBWEQ72ZLYRGBPGPXAR/action/citation_signature","submit_replication":"https://pith.science/pith/ASAR4TTTBWEQ72ZLYRGBPGPXAR/action/replication_record"}},"created_at":"2026-05-18T00:52:43.442206+00:00","updated_at":"2026-05-18T00:52:43.442206+00:00"}