Lean statement

Optimal Max-Cut hardness gap

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.OptimalMaxCut.main

Lean statement file: OptimalMaxCut.leanShow on the mapFamily 102. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

def MainStatement : Prop :=  ∀ α : ℝ, alphaGW < α → α ≤ 1 → Nonempty (GapReduction α)⋮theorem OptimalMaxCut.main : OptimalMaxCut.MainStatement := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

structure OAI.MaxCutGames.BinaryFormula.Literal line 7

structure Literal where  name : Nat  positive : Bool  deriving DecidableEq

definition OAI.MaxCutGames.BinaryFormula.Literal.eval line 12

def Literal.eval (literal : Literal) (assignment : Nat → Bool) : Bool :=  if literal.positive then assignment literal.name else !(assignment literal.name)

abbreviation OAI.MaxCutGames.BinaryFormula.Clause line 15
```
abbrev Clause := Vector Literal 3
```
structure OAI.MaxCutGames.BinaryFormula.Formula line 22
```
structure Formula where  clauses : List Clause
```

definition OAI.MaxCutGames.BinaryFormula.Formula.Satisfiable line 25

def Formula.Satisfiable (F : Formula) : Prop :=  ∃ assignment : Nat → Bool, ∀ clause ∈ F.clauses, clause.eval assignment = true

definition OAI.MaxCutGames.BinaryEncoding.bitsValue line 34

def bitsValue : List Bool → Nat  | [] => 0  | b :: bits => Nat.bit b (bitsValue bits)

definition OAI.MaxCutGames.BinaryEncoding.frame line 39

def frame : List Bool → List Bool  | [] => [false]  | b :: bits => true :: b :: frame bits

definition OAI.MaxCutGames.BinaryEncoding.parseFrame line 43

def parseFrame : List Bool → Option (List Bool × List Bool)  | false :: rest => some ([], rest)  | true :: b :: rest => do      let (bits, trailing) ← parseFrame rest      return (b :: bits, trailing)  | _ => none

definition OAI.MaxCutGames.BinaryEncoding.nameBits line 50

def nameBits (name : Nat) : List Bool := frame name.bits

definition OAI.MaxCutGames.BinaryEncoding.parseName line 53

def parseName (input : List Bool) : Option (Nat × List Bool) := do  let (digits, rest) ← parseFrame input  let name := bitsValue digits  if digits = name.bits then some (name, rest) else none

definition OAI.MaxCutGames.BinaryEncoding.parseLiteral line 58

def parseLiteral : List Bool → Option (Literal × List Bool)  | sign :: input => do      let (name, rest) ← parseName input      return (⟨name, sign⟩, rest)  | [] => none

definition OAI.MaxCutGames.BinaryEncoding.parseClause line 64

def parseClause (input : List Bool) : Option (Clause × List Bool) := do  let (a, input) ← parseLiteral input  let (b, input) ← parseLiteral input  let (c, rest) ← parseLiteral input  return (#v[a, b, c], rest)

definition OAI.MaxCutGames.BinaryEncoding.parseClauses line 71

def parseClauses : Nat → List Bool → Option (List Clause × List Bool)  | 0, _ => none  | _ + 1, false :: rest => some ([], rest)  | fuel + 1, true :: input => do      let (clause, input) ← parseClause input      let (clauses, rest) ← parseClauses fuel input      return (clause :: clauses, rest)  | _ + 1, [] => none

definition OAI.MaxCutGames.BinaryEncoding.decodeFormula line 80

def decodeFormula (input : List Bool) : Option Formula := do  let (clauses, rest) ← parseClauses (input.length + 1) input  if rest = [] then some ⟨clauses⟩ else none

definition OAI.OptimalMaxCut.alphaGW line 91

noncomputable def alphaGW : ℝ :=  sInf ((fun ρ : ℝ => 2 * Real.arccos ρ / (Real.pi * (1 - ρ))) ''    Set.Ico (-1 : ℝ) 1)

definition OAI.OptimalMaxCut.binary3SAT line 96

def binary3SAT (input : List Bool) : Prop :=  ∃ formula, MaxCutGames.BinaryEncoding.decodeFormula input = some formula ∧    formula.Satisfiable

structure OAI.OptimalMaxCut.Graph line 101

structure Graph where  vertices : ℕ  adj : Fin vertices → Fin vertices → Bool  symmetric : ∀ u v, adj u v = adj v u  loopless : ∀ u, adj u u = false

definition OAI.OptimalMaxCut.Graph.cutSize line 108

def Graph.cutSize (G : Graph) (cut : Fin G.vertices → Bool) : ℕ :=  ∑ u : Fin G.vertices, ∑ v : Fin G.vertices,    if u < v ∧ G.adj u v = true ∧ cut u ≠ cut v then 1 else 0

definition OAI.OptimalMaxCut.Graph.maxCut line 113

def Graph.maxCut (G : Graph) : ℕ :=  (Finset.univ.image G.cutSize).sup id

structure OAI.OptimalMaxCut.ScaledGraph line 118

structure ScaledGraph where  graph : Graph  scale : ℕ  scale_pos : 0 < scale

definition OAI.OptimalMaxCut.ScaledGraph.bits line 125

def ScaledGraph.bits (G : ScaledGraph) : List Bool :=  MaxCutGames.BinaryEncoding.nameBits G.graph.vertices ++  MaxCutGames.BinaryEncoding.nameBits G.scale ++  (List.finRange G.graph.vertices).flatMap (fun u =>    (List.finRange G.graph.vertices).map (fun v => G.graph.adj u v))

structure OAI.OptimalMaxCut.GapReduction line 133

structure GapReduction (α : ℝ) where

Show the remaining 13 lines

  yesBound : ℚ  noBound : ℚ  yes_pos : 0 < yesBound  no_nonneg : 0 ≤ noBound  gap : (noBound : ℝ) < α * (yesBound : ℝ)  construct : List Bool → ScaledGraph  computation : Turing.TM2ComputableInPolyTime (id : List Bool → List Bool)    ScaledGraph.bits construct  finiteAlphabet : ∀ k, Finite (computation.tm.Γ k)  completeness : ∀ input, binary3SAT input →    (yesBound : ℝ) ≤ ((construct input).graph.maxCut : ℝ) / (construct input).scale  soundness : ∀ input, ¬ binary3SAT input →    ((construct input).graph.maxCut : ℝ) / (construct input).scale ≤ (noBound : ℝ)

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: gw, cut, max, alpha, simple, unweighted and graphs. Found in the paper’s title, which its main theorem states: optimal.

Statements in A Direct Proof of Optimal Max-Cut Hardness come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of A Direct Proof of Optimal Max-Cut Hardness

Theorem

For every fixed \alpha with \GW<\alpha\le1, it is NP-hard to approximate Max-Cut within a factor of \alpha. This holds for simple unweighted graphs.

Section 1, Introduction Read it in the paper TeX source, line 39

Show the TeX

For every fixed $\alpha$ with $\GW<\alpha\le1$, it is NP-hard to approximate
Max-Cut within a factor of $\alpha$. This holds for simple unweighted graphs.

Other possible paper matches, closest wording first

Theorem 1.1 of The Factor-Two Hardness Threshold for Vertex Cover. Words of the Lean statement, its names split into words, that also appear in the theorem: vertices, edges, formula, output, sat, lists and nine more. Found in the paper’s title, which its main theorem states: two and vertex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: explicit, polynomial and deterministic.
Theorem 1.1 of Constant-factor hardness of Min-UnCut. Words of the Lean statement, its names split into words, that also appear in the theorem: length, formula, binary, encoded, full, bit and ten more. Found in the paper’s title, which its main theorem states: cut. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: explicit, polynomial and deterministic.
Proposition 2.2 of A Direct Proof of Optimal Max-Cut Hardness. Words of the Lean statement, its names split into words, that also appear in the theorem: edge, map, games, pi, output, input and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: polynomial and deterministic.
Theorem 1.2 of A Direct Proof of Optimal Max-Cut Hardness. Words of the Lean statement, its names split into words, that also appear in the theorem: maximum, edges, cut, yes, graphs and rational.
Proposition 6.1 of Constant-factor hardness of Min-UnCut. Words of the Lean statement, its names split into words, that also appear in the theorem: divided, compares, tables, max, constructed, two and ten more.
Theorem 2.2 of Constant-factor hardness of Min-UnCut. Words of the Lean statement, its names split into words, that also appear in the theorem: assignment, none, length, nonempty, full, list and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: explicit, polynomial and deterministic.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

One independent group reports that Comparator accepted a run for this Lean statement file: MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

No verdict reached Their term: “REFUSED”.
Re-run through Comparator did not reach a verdict (the job ended (failure) in the step "comparator, Lean’s kernel and nanoda" without uploading a result: the runner terminated it (GitHub: Process completed with exit code 143.)).
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-pass”.
Re-run through Comparator: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 102’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: None found; all five hardness statements are formalized as polynomial-time (Mathlib TM2, finite stack alphabets) Karp gap reductions from binary 3SAT (or an arbitrary NP verifier language), which is the standard NP-hardness encoding. Cone is ~190k lines and uses 'Std'/'Lean' (core) only; the OptimalMaxCut axiom_files flag is a comment-only false positive.
Definitions the group asks readers to compare with the paper: Complexity notions are Mathlib's Turing.TM2ComputableInPolyTime (note: Mathlib's FinTM2 only forces the input stack alphabet to be finite, so every challenge adds finiteAlphabet : ∀ k, Finite (computation.tm.Γ k)), outputs are measured in explicit unary/binary encodings (gameBits writes every number in unary); alphaGW is defined as an sInf over ρ ∈ [-1,1) (equals the GW constant, checked); InNP is the certificate definition via NPVerifier with a polynomial witnessBound
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“For every fixed $\alpha_{\mathrm{GW}}<\alpha\le1$, it gives a deterministic polynomial-time reduction from binary 3SAT to a strictly separated Max-Cut gap on finite simple unweighted graphs. The reduction includes the explicit positive integer scaling used in the gap statement.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

156 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 36 to its own declarations, 106 to Mathlib’s documentation. 6 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlibnamespace OAInamespace MaxCutGames.BinaryFormulastructure Literal where  name : Nat  positive : Bool  deriving DecidableEqdef Literal.eval (literal : Literal) (assignment : Nat → Bool) : Bool :=  if literal.positive then assignment literal.name else !(assignment literal.name)abbrev Clause := Vector Literal 3def Clause.eval (clause : Clause) (assignment : Nat → Bool) : Bool :=  (clause[0].eval assignment || clause[1].eval assignment) || clause[2].eval assignment/-- An ordinary conjunction of three-slot clauses with sparse variable names.Repeated names and repeated literals are permitted in every clause. -/structure Formula where  clauses : List Clausedef Formula.Satisfiable (F : Formula) : Prop :=  ∃ assignment : Nat → Bool, ∀ clause ∈ F.clauses, clause.eval assignment = trueend MaxCutGames.BinaryFormulanamespace MaxCutGames.BinaryEncodingopen BinaryFormuladef bitsValue : List Bool → Nat  | [] => 0  | b :: bits => Nat.bit b (bitsValue bits)/-- Prefix-free framing on the two-symbol input alphabet. -/def frame : List Bool → List Bool  | [] => [false]  | b :: bits => true :: b :: frame bitsdef parseFrame : List Bool → Option (List Bool × List Bool)  | false :: rest => some ([], rest)  | true :: b :: rest => do      let (bits, trailing) ← parseFrame rest      return (b :: bits, trailing)  | _ => nonedef nameBits (name : Nat) : List Bool := frame name.bits/-- Equality with the canonical digits rejects alternate padded encodings. -/def parseName (input : List Bool) : Option (Nat × List Bool) := do  let (digits, rest) ← parseFrame input  let name := bitsValue digits  if digits = name.bits then some (name, rest) else nonedef parseLiteral : List Bool → Option (Literal × List Bool)  | sign :: input => do      let (name, rest) ← parseName input      return (⟨name, sign⟩, rest)  | [] => nonedef parseClause (input : List Bool) : Option (Clause × List Bool) := do  let (a, input) ← parseLiteral input  let (b, input) ← parseLiteral input  let (c, rest) ← parseLiteral input  return (#v[a, b, c], rest)/-- Fuel bounds the number of clause iterations, not the size of any name. -/def parseClauses : Nat → List Bool → Option (List Clause × List Bool)  | 0, _ => none  | _ + 1, false :: rest => some ([], rest)  | fuel + 1, true :: input => do      let (clause, input) ← parseClause input      let (clauses, rest) ← parseClauses fuel input      return (clause :: clauses, rest)  | _ + 1, [] => nonedef decodeFormula (input : List Bool) : Option Formula := do  let (clauses, rest) ← parseClauses (input.length + 1) input  if rest = [] then some ⟨clauses⟩ else noneend MaxCutGames.BinaryEncodingnamespace OptimalMaxCutopen scoped BigOperators/-- The exact Goemans–Williamson ratio, as an infimum over -1 ≤ ρ < 1. -/noncomputable def alphaGW : ℝ :=  sInf ((fun ρ : ℝ => 2 * Real.arccos ρ / (Real.pi * (1 - ρ))) ''    Set.Ico (-1 : ℝ) 1)/-- The ordinary binary 3SAT language, including rejection of malformed words. -/def binary3SAT (input : List Bool) : Prop :=  ∃ formula, MaxCutGames.BinaryEncoding.decodeFormula input = some formula ∧    formula.Satisfiable/-- A finite simple unweighted graph with its full Boolean adjacency table. -/structure Graph where  vertices : ℕ  adj : Fin vertices → Fin vertices → Bool  symmetric : ∀ u v, adj u v = adj v u  loopless : ∀ u, adj u u = false/-- Each unordered distinct edge crossing the cut is counted exactly once. -/def Graph.cutSize (G : Graph) (cut : Fin G.vertices → Bool) : ℕ :=  ∑ u : Fin G.vertices, ∑ v : Fin G.vertices,    if u < v ∧ G.adj u v = true ∧ cut u ≠ cut v then 1 else 0/-- The maximum over all Boolean cuts of the finite vertex set. -/def Graph.maxCut (G : Graph) : ℕ :=  (Finset.univ.image G.cutSize).sup id/-- The positive integer scale Q is part of the output. It need not equal thenumber of graph edges. -/structure ScaledGraph where  graph : Graph  scale : ℕ  scale_pos : 0 < scale/-- Canonical delimited binary vertex count and scale, followed by the fulladjacency table in lexicographic order. -/def ScaledGraph.bits (G : ScaledGraph) : List Bool :=  MaxCutGames.BinaryEncoding.nameBits G.graph.vertices ++  MaxCutGames.BinaryEncoding.nameBits G.scale ++  (List.finRange G.graph.vertices).flatMap (fun u =>    (List.finRange G.graph.vertices).map (fun v => G.graph.adj u v))/-- The rational gap bounds and the finite machine are fixed before the input.The bounds compare the maximum cut divided by the positive output scale. -/structure GapReduction (α : ℝ) where  yesBound : ℚ  noBound : ℚ  yes_pos : 0 < yesBound  no_nonneg : 0 ≤ noBound  gap : (noBound : ℝ) < α * (yesBound : ℝ)  construct : List Bool → ScaledGraph  computation : Turing.TM2ComputableInPolyTime (id : List Bool → List Bool)    ScaledGraph.bits construct  finiteAlphabet : ∀ k, Finite (computation.tm.Γ k)  completeness : ∀ input, binary3SAT input →    (yesBound : ℝ) ≤ ((construct input).graph.maxCut : ℝ) / (construct input).scale  soundness : ∀ input, ¬ binary3SAT input →    ((construct input).graph.maxCut : ℝ) / (construct input).scale ≤ (noBound : ℝ)def MainStatement : Prop :=  ∀ α : ℝ, alphaGW < α → α ≤ 1 → Nonempty (GapReduction α)end OptimalMaxCuttheorem OptimalMaxCut.main : OptimalMaxCut.MainStatement := by  sorryend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

The Unique Games Theorem

Theorem (Unique Games Theorem)

For every fixed \varepsilon,\delta\in(0,1/2), there are an integer s\ge1 and a deterministic polynomial-time reduction from \mathrm{3SAT} to explicit unweighted Unique Games instances G_\varphi over K=\F_2^s such that

\begin{aligned} \varphi\text{ satisfiable} &\quad\Longrightarrow\quad \val(G_\varphi)\ge1-\varepsilon,\\ \varphi\text{ unsatisfiable} &\quad\Longrightarrow\quad \val(G_\varphi)\le\delta. \end{aligned}

The graph is simple and bipartite, and every constraint is a translation a(v_e)=a(u_e)+c_e of K. The alphabet, the degree of the running-time polynomial, and its constants depend only on \varepsilon,\delta.

Section 1, Introduction Read it in the paper TeX source, line 22

Show the TeX

For every fixed \(\varepsilon,\delta\in(0,1/2)\), there are an integer
\(s\ge1\) and a deterministic polynomial-time reduction from
\(\mathrm{3SAT}\) to explicit unweighted Unique Games instances
\(G_\varphi\) over \(K=\F_2^s\) such that
\[
\begin{aligned}
 \varphi\text{ satisfiable}
 &\quad\Longrightarrow\quad \val(G_\varphi)\ge1-\varepsilon,\\
 \varphi\text{ unsatisfiable}
 &\quad\Longrightarrow\quad \val(G_\varphi)\le\delta.
\end{aligned}
\]
The graph is simple and bipartite, and every constraint is a translation
\(a(v_e)=a(u_e)+c_e\) of \(K\). The alphabet, the degree of the
running-time polynomial, and its constants depend only on
\(\varepsilon,\delta\).

A Direct Proof of Optimal Max-Cut Hardness

Theorem

For every fixed \alpha with \GW<\alpha\le1, it is NP-hard to approximate Max-Cut within a factor of \alpha. This holds for simple unweighted graphs.

Section 1, Introduction Read it in the paper TeX source, line 39

Show the TeX

For every fixed $\alpha$ with $\GW<\alpha\le1$, it is NP-hard to approximate
Max-Cut within a factor of $\alpha$. This holds for simple unweighted graphs.

Theorem

Fix a rational t\in(0,1) and a constant 0<\eps<(t-B(t))/4. For graphs equipped with a rational probability distribution on edges, it is NP-hard to distinguish

\begin{aligned}\textnormal{YES:}\quad&\Val\ge\frac{1+t}{2}-\eps,\\ \textnormal{NO:}\quad&\Val\le\frac{1+B(t)}{2}+\eps,\end{aligned}

where \Val is the maximum cut weight.

Section 1, Introduction Read it in the paper TeX source, line 52

Show the TeX

Fix a rational $t\in(0,1)$ and a constant
$0<\eps<(t-B(t))/4$. For graphs equipped with a rational probability
distribution on edges, it is NP-hard to distinguish
\begin{align*}
 \textnormal{YES:}\quad&\Val\ge\frac{1+t}{2}-\eps,\\
 \textnormal{NO:}\quad&\Val\le\frac{1+B(t)}{2}+\eps,
\end{align*}
where $\Val$ is the maximum cut weight.

The Factor-Two Hardness Threshold for Vertex Cover

Theorem (Explicit cover gap)

For every fixed integer m\geq4, there is a deterministic polynomial-time reduction from \mathrm{3SAT} to finite simple undirected unweighted graphs with the following guarantees:

\begin{array}{ll} \text{the formula is satisfiable} &\Longrightarrow\quad \tau(G)<\left(\dfrac12+\dfrac1m\right)|V(G)|,\\[6pt] \text{the formula is unsatisfiable} &\Longrightarrow\quad \tau(G)>\left(1-\dfrac1m\right)|V(G)|. \end{array}

The output lists all vertices and edges explicitly.

Section 1, Introduction Read it in the paper TeX source, line 15

Show the TeX

For every fixed integer \(m\geq4\), there is a deterministic
polynomial-time reduction from \(\mathrm{3SAT}\) to finite simple
undirected unweighted graphs with the following guarantees:
\[
\begin{array}{ll}
\text{the formula is satisfiable}
 &\Longrightarrow\quad
 \tau(G)<\left(\dfrac12+\dfrac1m\right)|V(G)|,\\[6pt]
\text{the formula is unsatisfiable}
 &\Longrightarrow\quad
 \tau(G)>\left(1-\dfrac1m\right)|V(G)|.
\end{array}
\]
The output lists all vertices and edges explicitly.

Corollary

For every fixed real \alpha with 1\leq\alpha<2, approximating Vertex Cover within factor \alpha is NP-hard. Thus, unless \mathrm P=\mathrm{NP}, no deterministic polynomial-time algorithm achieves factor 2-\varepsilon for any fixed \varepsilon\in(0,1].

Section 1, Introduction Read it in the paper TeX source, line 33

Show the TeX

For every fixed real \(\alpha\) with \(1\leq\alpha<2\),
approximating Vertex Cover within factor \(\alpha\) is NP-hard.
Thus, unless \(\mathrm P=\mathrm{NP}\), no deterministic
polynomial-time algorithm achieves factor \(2-\varepsilon\)
for any fixed \(\varepsilon\in(0,1]\).

Constant-factor hardness of Min-UnCut

Theorem

For every fixed integer K\ge2, there is a deterministic polynomial-time many-one reduction mapping a 3SAT formula \varphi to a pair (G,k), where G is an explicit finite simple undirected unweighted graph and k\ge1 is a binary-encoded integer, such that

\begin{aligned} \varphi\text{ satisfiable}&\quad\Longrightarrow\quad \OPT_{\rm uncut}(G)\le k,\\ \varphi\text{ unsatisfiable}&\quad\Longrightarrow\quad \OPT_{\rm uncut}(G)>Kk. \end{aligned}

For each fixed K, the graph size, full construction work, and bit complexity are polynomial in the bit length of \varphi. All constants are effective; no advice or oracle is supplied to the reduction.

Section 1, Introduction Read it in the paper TeX source, line 50

Show the TeX

For every fixed integer $K\ge2$, there is a deterministic polynomial-time
many-one reduction mapping a $3$SAT formula $\varphi$ to a pair $(G,k)$, where
$G$ is an explicit finite simple undirected unweighted graph and $k\ge1$ is
a binary-encoded integer, such that
\[
 \begin{aligned}
 \varphi\text{ satisfiable}&\quad\Longrightarrow\quad
       \OPT_{\rm uncut}(G)\le k,\\
 \varphi\text{ unsatisfiable}&\quad\Longrightarrow\quad
       \OPT_{\rm uncut}(G)>Kk.
 \end{aligned}
\]
For each fixed $K$, the graph size, full construction work, and bit complexity
are polynomial in the bit length of $\varphi$. All constants are effective;
no advice or oracle is supplied to the reduction.

Corollary

For every fixed real C>1, minimum 2CNF clause deletion is NP-hard to approximate within factor C. In particular, a deterministic polynomial-time C-approximation would imply \mathsf P=\NP.

Section 1, Introduction Read it in the paper TeX source, line 86

Show the TeX

For every fixed real $C>1$, minimum $2$CNF clause deletion is NP-hard to
approximate within factor $C$. In particular, a deterministic polynomial-time
$C$-approximation would imply $\mathsf P=\NP$.

Constant-factor hardness of directed feedback vertex set

Theorem

For every fixed real constant A\ge1, it is NP-hard to approximate minimum directed feedback vertex set within factor A on unweighted digraphs. More precisely, there is a deterministic polynomial-time reduction from an NP-hard promise problem to instances (G,k) with k\in\N_{>0}. The polynomial and its constants may depend on A. The reduction has the following alternatives:

\textnormal{YES: }\DFVS(G)\le k, \qquad \textnormal{NO: }\DFVS(G)>Ak.

Section 1, Introduction Read it in the paper TeX source, line 50

Show the TeX

For every fixed real constant $A\ge1$, it is NP-hard to approximate
minimum directed feedback vertex set within factor $A$ on unweighted
digraphs. More precisely, there is a deterministic polynomial-time
reduction from an NP-hard promise problem to instances $(G,k)$ with
$k\in\N_{>0}$. The polynomial and its constants may depend on $A$.
The reduction has the following alternatives:
\[
  \textnormal{YES: }\DFVS(G)\le k,
  \qquad
  \textnormal{NO: }\DFVS(G)>Ak.
\]

Configuration and file facts

File: lean/ComparatorChallenges/OptimalMaxCut.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Computability/MaxCut/Main.lean
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 24 local vocabulary declarations
Lines of Lean, without comments or blank lines: 94
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas