OpenAI claim summary for CS-113

Unique Games Approximation-Threshold 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.

This coverage does not establish a formal proof of the complete problem as stated here.

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: cor:raghavendra-consequence.
• 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.