OpenAI claim summary for CS-144
computer sciencemathematicsTop Open Problemscontributionliterature_reportClaim: literature report
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
Constant-factor hardness of Min-UnCut
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 Min-UnCut admits no fixed constant-factor approximation unless the stated NP-hard distinction can be decided.Mathematical statement of the claimed result
Manuscript 1.
Define $\operatorname{OPT}_{\rm uncut}(G)=\min_{S\subseteq V}|E(S)|+|E(V\setminus S)|$ for a finite simple undirected graph. For every fixed integer $K\ge2$, the claimed reduction transforms $3$SAT instances in deterministic polynomial time into $(G,k)$, with $G$ unweighted and binary integer $k\ge1$, such that \[ \varphi\text{ satisfiable}\Rightarrow\operatorname{OPT}_{\rm uncut}(G)\le k, \qquad\varphi\text{ unsatisfiable}\Rightarrow\operatorname{OPT}_{\rm uncut}(G)>Kk. \] For fixed $K$, all construction and bit costs are polynomial in formula length, without advice or an oracle.Outline of the argument
For manuscript 1 (family 102; 2026-09-23), folded bit comparisons turn face correlation into a large odd Fourier coefficient. Alphabet-independent subbox decoding converts that coefficient into outer-game lists, and rational Gaussian tables are realized by an exact simple unweighted graph with a strict gap.Formal verification
For manuscript 1 (family 102; 2026-09-23), OpenAI lists Lean statements for this assertion: Min-UnCut admits no fixed constant-factor approximation unless the stated NP-hard distinction can be decided.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, Constant-factor hardness of Min-UnCut, 2026-09-23; citation key
OAI:Constant-factor-hardness-of-Min-UnCut-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.