OpenAI claim summary for CS-18
computer sciencemathematicsTop Open Problemscontributionliterature_reportClaim: literature report
Summary of a claim published in openai/math; not a claim by this platform.
The claimed hardness gap holds for every fixed pair of positive completeness and soundness errors, and hence establishes the Unique Games conjecture's promise. It also restricts the instances to explicit translation constraints on simple bipartite graphs, which are a subclass of permutation Unique Games.
Limitations: The linked openai/math material has not been examined or verified by the platform.
Manuscript
Contents
The Unique Games Conjecture
OpenAI's claim
OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 102; 2026-09-23), the asserted result is that for any fixed positive completeness and soundness errors, a deterministic polynomial reduction from 3SAT produces fixed-alphabet Unique Games with the requested gap.Mathematical statement of the claimed result
Manuscript 1.
For every fixed $\varepsilon,\delta\in(0,1/2)$, the claim gives $s\ge1$ and a deterministic polynomial-time $3$SAT reduction to explicit unweighted Unique Games over $K=\mathbb F_2^s$, on a simple bipartite graph with translation constraints $a(v_e)=a(u_e)+c_e$, satisfying \[ \varphi\text{ satisfiable}\Rightarrow\operatorname{val}(G_{\varphi})\ge1-\varepsilon, \qquad\varphi\text{ unsatisfiable}\Rightarrow\operatorname{val}(G_{\varphi})\le\delta. \] Alphabet size and polynomial constants depend only on the two fixed errors. For any fixed constraint-language instance $J$ with $A=\operatorname{OPT}(J)<B=\operatorname{SDP}_{R}(J)$, the associated consequence claims NP-hardness of distinguishing value at least $B-\eta$ from value at most $A+\eta$ for any fixed $0<\eta<(B-A)/2$.Outline of the argument
For manuscript 1 (family 102; 2026-09-23), a latent quadratic alphabet and matrix-slice tests supply an outer permutation game. High acceptance decodes agreeing advice strategies, whereas clean-coordinate estimates from the unsatisfiable parity instance force their agreement to decay, giving the soundness contradiction.Formal verification
For manuscript 1 (family 102; 2026-09-23), OpenAI lists Lean statements for this assertion: for any fixed positive completeness and soundness errors, a deterministic polynomial reduction from 3SAT produces fixed-alphabet Unique Games with the requested gap.The selected statement includes the clause used for this problem.
The linked family Comparator checks are DirectedFeedback.lean, MinUncut.lean, OptimalMaxCut.lean, UniqueGamesTheorem.lean, VertexCover.lean.
Their named declarations include OAI.DirectedFeedback.main, OAI.MinUncut.main, OAI.OptimalMaxCut.main, OAI.UniqueGamesTheorem.theorem11, OAI.VertexCover.every_fixed_factor_below_two, OAI.VertexCover.explicit_cover_gap.
This site has not executed the Lean code. Described statement coverage is not an independent proof check.
Status caveat
The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.Sources
- OpenAI, The Unique Games Theorem, 2026-09-23; citation key
OAI:The-Unique-Games-Theorem-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:ugc. - Family 102: Lean scope.
- Comparator DirectedFeedback.json.
- Comparator DirectedFeedback.lean.
- Comparator MinUncut.json.
- Comparator MinUncut.lean.
- Comparator OptimalMaxCut.json.
- Comparator OptimalMaxCut.lean.
- Comparator UniqueGamesTheorem.json.
- Comparator UniqueGamesTheorem.lean.
- Comparator VertexCover.json.
- Comparator VertexCover.lean.
- Family 102: Reasoning summary.
- OpenAI catalogue entry.