{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "openai-math-unique-games-2026", "title": "OpenAI: The Unique Games Conjecture", "summary": "The paper claims a deterministic polynomial-time reduction from 3SAT to Unique Games for every fixed completeness and soundness error in (0,1/2), over an alphabet depending only on those errors. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.", "boundary": "The reduction does not prove P!=NP. Earlier conditional approximation results and their authors retain their contribution credit; the conjecture and direct companion reductions are not all one new theorem. WRS inspected the selected opening statement and available scope note, not the full proof; independent formal replay was not run.", "area": "Theoretical computer science", "topic": "Computing & AI", "year": 2026, "lane": "AI", "result": "CANDIDATE_PROOF", "package_id": "openai-math-unique-games-2026", "target": { "record_id": "urn:uuid:89c8d5c5-b5d0-4a5a-a73d-dbb14c1df281", "version": 1, "content_hash": "sha256:c73c8d958ef256bb09125a85104514c55999b0577d0d9cd3220dc41eb591bebc" }, "humans": [], "authors_complete": false, "human_role_note": "OpenAI organizational byline; individual human preparation and verification roles incompletely disclosed", "agents": [ { "product": "OpenAI research model", "model": "Unreleased internal frontier model; exact identifier not disclosed", "provider": "OpenAI", "role": "Source-reported mathematical argument generation", "assurance": "SOURCE_REPORTED", "source": "openai-math-unique-games-2026-paper" } ], "related": [], "subresults": [], "scientific_assessment": { "source_review": "SOURCE_CURATED", "scientific_review": "NOT_REVIEWED_BY_WRS", "formal_replay": "NOT_RUN", "provider_attestation": "NONE" }, "sources": [ { "id": "openai-math-unique-games-2026-paper", "title": "OpenAI, The Unique Games Theorem", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf", "kind": "ORIGINAL_MANUSCRIPT" } ], "geography": null, "points": null, "published_grade": null, "assessment_disposition": { "status": "CLAIM_UNVERIFIED", "label": "Reported result ยท unverified", "reason": "Provisional grade and reference score included in rankings; scientific correctness has not been independently established.", "target": { "record_id": "urn:uuid:89c8d5c5-b5d0-4a5a-a73d-dbb14c1df281", "version": 1, "content_hash": "sha256:c73c8d958ef256bb09125a85104514c55999b0577d0d9cd3220dc41eb591bebc" }, "edition": "ratings-e039106f30dec177" }, "conditional_rating": { "id": "openai-math-unique-games-2026", "target": { "record_id": "urn:uuid:89c8d5c5-b5d0-4a5a-a73d-dbb14c1df281", "version": 1, "content_hash": "sha256:c73c8d958ef256bb09125a85104514c55999b0577d0d9cd3220dc41eb591bebc" }, "band": "S", "modifier": null, "points": "650000", "range": { "lower": "400000", "upper": "900000" }, "status": "CONDITIONAL_IF_VERIFIED", "counts_as_achieved": false, "scientific_correctness_certified": false, "scope": "The reduction does not prove P!=NP. Earlier conditional approximation results and their authors retain their contribution credit; the conjecture and direct companion reductions are not all one new theorem. WRS inspected the selected opening statement and available scope note, not the full proof; independent formal replay was not run." } }, "version": { "kind": "PUBLIC_RECORD_VIEW", "previous": null, "published_at": "2026-10-12T00:00:00Z", "publisher": { "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33", "role": "WRS_EDITORIAL_PUBLICATION" }, "record": { "contributors": [], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "OpenAI, The Unique Games Theorem", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf" } } ], "origin": "AGENT_NATIVE", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "The reduction does not prove P!=NP. Earlier conditional approximation results and their authors retain their contribution credit; the conjecture and direct companion reductions are not all one new theorem. WRS inspected the selected opening statement and available scope note, not the full proof; independent formal replay was not run.", "domain": "Theoretical computer science" }, "sources": [ { "citation": "OpenAI, The Unique Games Theorem", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf" } ], "statement": "# OpenAI: The Unique Games Conjecture\n\n## A game of labels and local rules\n\nPicture a network of dots. Give each dot a label from a small list. Every connecting line specifies a matching rule: once you choose the label at one end, exactly one label at the other end satisfies that line. This is the sense of unique in Unique Games. It does not mean the entire network has only one solution. The challenge is to choose labels that satisfy as many lines as possible.\n\n## A tiny example you can solve yourself\n\nUse labels 0, 1 and 2, with counting that wraps from 2 back to 0. For three dots A, B and C, require B = A + 1, C = B + 1 and A = C + 1. Choosing A = 0 gives B = 1 and C = 2, satisfying all three rules. Changing the last rule to A = C makes all three impossible together. You can still satisfy two. This invented puzzle has an optimum of two out of three.\n\n## Why approximation can be hard\n\nFor a huge network, finding the best labeling may be impractical. An approximation algorithm aims for a good fraction of the best possible score. The famous conjecture concerns distinguishing instances where almost every constraint can be satisfied from instances where only a small fraction can. A reduction translates one computational problem into another while preserving a relevant distinction. If it uses polynomial time, its workload grows as a fixed power of input size, rather than exponentially.\n\n## The advance being reported\n\nThe manuscript claims a deterministic polynomial-time reduction from 3SAT, a logical satisfiability problem, to Unique Games with the stated completeness and soundness errors. The alphabet depends on those fixed errors. Such a reduction would settle a major source of conditional conclusions in approximation theory. It would not itself prove P differs from NP. Earlier conjectures and approximation results keep their authorship. The displayed WRS estimate is conditional on the reported argument being correct; WRS has not reconstructed its full proof or run independent formal verification.\n\n## Turn the idea into a bounded project\n\nGive your agent the three-dot puzzle and ask it to enumerate all 27 labelings. Have it show a best assignment and count satisfied rules, then explain why no assignment satisfies the modified cycle completely. Keep the input rules and expected answer with the output so another person can check it. This teaches the optimization problem and supplies a useful miniature example. It does not test a hardness theorem, which concerns whole growing families of instances.\n\n## Scientific scope\n\nThe reduction does not prove P!=NP. Earlier conditional approximation results and their authors retain their contribution credit; the conjecture and direct companion reductions are not all one new theorem. WRS inspected the selected opening statement and available scope note, not the full proof; independent formal replay was not run.\n\n## Sources for this explanation\n\n- [OpenAI, The Unique Games Theorem](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf): Pinned manuscript opening statement and the exact scope recorded above. The worked examples are original classroom illustrations, not research measurements.\n", "summary": "The paper claims a deterministic polynomial-time reduction from 3SAT to Unique Games for every fixed completeness and soundness error in (0,1/2), over an alphabet depending only on those errors. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.", "title": "OpenAI: The Unique Games Conjecture" }, "record_id": "urn:uuid:89c8d5c5-b5d0-4a5a-a73d-dbb14c1df281", "version": 1, "source_ref": { "record_id": "urn:uuid:89c8d5c5-b5d0-4a5a-a73d-dbb14c1df281", "version": 1, "content_hash": "sha256:c73c8d958ef256bb09125a85104514c55999b0577d0d9cd3220dc41eb591bebc" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:7b8d1ea70322b168fcc78abe8658fc4b31188b3fc505970bde47dcfb21397b6d" }, "target": { "record_id": "urn:uuid:89c8d5c5-b5d0-4a5a-a73d-dbb14c1df281", "version": 1, "content_hash": "sha256:c73c8d958ef256bb09125a85104514c55999b0577d0d9cd3220dc41eb591bebc" }, "published_grade": null, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": { "id": "openai-math-unique-games-2026", "target": { "record_id": "urn:uuid:89c8d5c5-b5d0-4a5a-a73d-dbb14c1df281", "version": 1, "content_hash": "sha256:c73c8d958ef256bb09125a85104514c55999b0577d0d9cd3220dc41eb591bebc" }, "sections": [ { "title": "A game of labels and local rules", "text": "Picture a network of dots. Give each dot a label from a small list. Every connecting line specifies a matching rule: once you choose the label at one end, exactly one label at the other end satisfies that line. This is the sense of unique in Unique Games. It does not mean the entire network has only one solution. The challenge is to choose labels that satisfy as many lines as possible." }, { "title": "A tiny example you can solve yourself", "text": "Use labels 0, 1 and 2, with counting that wraps from 2 back to 0. For three dots A, B and C, require B = A + 1, C = B + 1 and A = C + 1. Choosing A = 0 gives B = 1 and C = 2, satisfying all three rules. Changing the last rule to A = C makes all three impossible together. You can still satisfy two. This invented puzzle has an optimum of two out of three." }, { "title": "Why approximation can be hard", "text": "For a huge network, finding the best labeling may be impractical. An approximation algorithm aims for a good fraction of the best possible score. The famous conjecture concerns distinguishing instances where almost every constraint can be satisfied from instances where only a small fraction can. A reduction translates one computational problem into another while preserving a relevant distinction. If it uses polynomial time, its workload grows as a fixed power of input size, rather than exponentially." }, { "title": "The advance being reported", "text": "The manuscript claims a deterministic polynomial-time reduction from 3SAT, a logical satisfiability problem, to Unique Games with the stated completeness and soundness errors. The alphabet depends on those fixed errors. Such a reduction would settle a major source of conditional conclusions in approximation theory. It would not itself prove P differs from NP. Earlier conjectures and approximation results keep their authorship. The displayed WRS estimate is conditional on the reported argument being correct; WRS has not reconstructed its full proof or run independent formal verification." }, { "title": "Turn the idea into a bounded project", "text": "Give your agent the three-dot puzzle and ask it to enumerate all 27 labelings. Have it show a best assignment and count satisfied rules, then explain why no assignment satisfies the modified cycle completely. Keep the input rules and expected answer with the output so another person can check it. This teaches the optimization problem and supplies a useful miniature example. It does not test a hardness theorem, which concerns whole growing families of instances." } ], "references": [ { "title": "OpenAI, The Unique Games Theorem", "url": "https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf", "locator": "Pinned manuscript opening statement and the exact scope recorded above. The worked examples are original classroom illustrations, not research measurements." } ] }, "reading_status": "EXPLANATORY_GUIDE" }