OpenAI claim summary for CS-140
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Summary of a claim published in openai/math; not a claim by this platform.
The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 102; 2026-09-23).
Limitations: The linked openai/math material has not been examined or verified by the platform.
Manuscript
Contents
Optimal Max-Cut approximation hardness
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 approximating Max-Cut above the Goemans--Williamson ratio is NP-hard on simple unweighted graphs.Mathematical statement of the claimed result
Manuscript 1.
Let $\operatorname{MaxCut}(G)$ denote the largest weight crossing a vertex bipartition, and set \[ \alpha_{\rm GW}=\min_{-1\le\rho<1}\frac{2\arccos\rho}{\pi(1-\rho)}. \] For each fixed $\alpha\in(\alpha_{\rm GW},1]$, the claim is NP-hardness of returning a cut of weight at least $\alpha\operatorname{MaxCut}(G)$, even on simple unweighted graphs. Its more explicit weighted gap uses any rational $0<t<1$, $B(t)=2\arcsin(t)/\pi$, and $0<\varepsilon<(t-B(t))/4$: distinguishing the two promises \[ \operatorname{MaxCut}(G)\ge(1+t)/2-\varepsilon \quad\hbox{and}\quad \operatorname{MaxCut}(G)\le(1+B(t))/2+\varepsilon \] is NP-hard when the rational edge weights sum to one.Outline of the argument
For manuscript 1 (family 102; 2026-09-23), an affine game is composed directly with folded Gaussian sign tables. Excess stability reveals a hidden coordinate through restriction Fourier mass and vector-valued hashing, yielding endpoint labelers that contradict the outer game's low value.Formal verification
For manuscript 1 (family 102; 2026-09-23), OpenAI lists Lean statements for this assertion: approximating Max-Cut above the Goemans--Williamson ratio is NP-hard on simple unweighted graphs.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, A Direct Proof of Optimal Max-Cut Hardness, 2026-09-23; citation key
OAI:A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - 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.