Lean statement

Perfect-completeness reduction for 2-to-1 games

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.PerfectCompleteness.Theorem11.exists_reduction

Lean statement file: PerfectCompleteness.leanShow on the mapFamily 105. 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.

theorem exists_reduction (δ : ℚ) (positive : 0 < δ) (less_than_one : δ < 1) :    Nonempty (PerfectCompleteness.BinaryGapReduction δ) := 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.

definition OAI.UniqueGamesTheorem.Foundations.Complexity.encodeWord line 7

def encodeWord (n : Nat) : List Bool := List.replicate n true ++ [false]

definition OAI.UniqueGamesTheorem.Foundations.Complexity.encodeWords line 9

def encodeWords : List Nat → List Bool  | [] => []  | n :: ns => encodeWord n ++ encodeWords ns

definition OAI.UniqueGamesTheorem.Foundations.Complexity.MachineFiniteAlphabet.FiniteAlphabet line 19
```
def FiniteAlphabet (M : FinTM2) : Prop := ∀ k, Finite (M.Γ k)
```

structure OAI.PerfectCompleteness.BinaryFormula.Literal line 25

structure Literal where  name : Nat  positive : Bool  deriving DecidableEq

definition OAI.PerfectCompleteness.BinaryFormula.Literal.eval line 30

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

abbreviation OAI.PerfectCompleteness.BinaryFormula.Clause line 33
```
abbrev Clause := Vector Literal 3
```
structure OAI.PerfectCompleteness.BinaryFormula.Formula line 38
```
structure Formula where  clauses : List Clause
```

definition OAI.PerfectCompleteness.BinaryFormula.Formula.Satisfiable line 41

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

definition OAI.PerfectCompleteness.BinaryEncoding.bitsValue line 50

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

definition OAI.PerfectCompleteness.BinaryEncoding.parseFrame line 54

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.PerfectCompleteness.BinaryEncoding.parseName line 61

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.PerfectCompleteness.BinaryEncoding.parseLiteral line 66

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

definition OAI.PerfectCompleteness.BinaryEncoding.parseClause line 72

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.PerfectCompleteness.BinaryEncoding.parseClauses line 78

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.PerfectCompleteness.BinaryEncoding.decodeFormula line 87

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.PerfectCompleteness.BinaryLanguage.language line 95

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

structure OAI.PerfectCompleteness.ProjectionTable line 102

structure ProjectionTable (q : Nat) where  images : Vector (Fin q) (2 * q)  exactlyTwo : ∀ b : Fin q,    ((List.finRange (2 * q)).filter (fun a => decide (images[a] = b))).length = 2

structure OAI.PerfectCompleteness.Edge line 107

structure Edge (leftVertices rightVertices q : Nat) where  left : Fin leftVertices  right : Fin rightVertices  projection : ProjectionTable q

structure OAI.PerfectCompleteness.Instance line 112

structure Instance (q : Nat) where  leftVertices : Nat  rightVertices : Nat  edges : List (Edge leftVertices rightVertices q)  nonempty : edges ≠ []

abbreviation OAI.PerfectCompleteness.Labeling line 118

abbrev Labeling {q : Nat} (G : Instance q) :=  (Fin G.leftVertices → Fin (2 * q)) × (Fin G.rightVertices → Fin q)

definition OAI.PerfectCompleteness.Edge.satisfied line 121

def Edge.satisfied {l r q : Nat} (e : Edge l r q)    (a : Fin l → Fin (2 * q)) (b : Fin r → Fin q) : Bool :=  decide (e.projection.images[a e.left] = b e.right)

definition OAI.PerfectCompleteness.countSatisfied line 125

def countSatisfied {l r q : Nat} (a : Fin l → Fin (2 * q))    (b : Fin r → Fin q) : List (Edge l r q) → Nat  | [] => 0  | e :: es => (if e.satisfied a b then 1 else 0) + countSatisfied a b es

definition OAI.PerfectCompleteness.Instance.maxSatisfied line 134

def maxSatisfied {q : Nat} (G : Instance q) : Nat :=  Finset.univ.sup (fun s : Labeling G => countSatisfied s.1 s.2 G.edges)

definition OAI.PerfectCompleteness.Instance.value line 137

noncomputable def value {q : Nat} (G : Instance q) : ℝ :=  (G.maxSatisfied : ℝ) / G.edges.length

5 more definitions are reached the same way; the full file lists them.

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: rational, reduction, projection, exactly, formula, edge, map, list and eleven more. Found in the paper’s title, which its main theorem states: completeness and perfect. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: preimages, produces, sat, delta, games, deterministic, unweighted, unsatisfiable and three more.

Theorem 1.1 of Perfect completeness for 2-to-1 games

Theorem

For every fixed rational \delta\in(0,1) there are an integer q=q(\delta)\ge2 and a deterministic polynomial-time reduction from 3-SAT to 2-to-1 games with alphabets [2q] and [q] such that:

(i)
a satisfiable formula produces a game of value 1;
(ii)
an unsatisfiable formula produces a game of value at most \delta.

The output contains a nonempty explicit list of unweighted edge occurrences and an explicit table for every projection map. Every table has exactly two preimages for each right answer. The polynomial running-time bound is in the ordinary binary input length; its degree and constants may depend on the fixed \delta.

Section 1, Introduction Read it in the paper TeX source, line 32

Show the TeX

For every fixed rational $\delta\in(0,1)$ there are an integer $q=q(\delta)\ge2$
and a deterministic polynomial-time reduction from $3$-SAT to 2-to-1 games
with alphabets $[2q]$ and $[q]$ such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a satisfiable formula produces a game of value $1$;
\item an unsatisfiable formula produces a game of value at most $\delta$.
\end{enumerate}
The output contains a nonempty explicit list of unweighted edge occurrences
and an explicit table for every projection map.
Every table has exactly two preimages for each right answer.
The polynomial running-time bound is in the ordinary binary input length;
its degree and constants may depend on the fixed $\delta$.

Other possible paper matches, closest wording first

Theorem 2.1 of Perfect completeness for 2-to-1 games. Words of the Lean statement, its names split into words, that also appear in the theorem: gap, encoding, clause, map, listed, formula and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: deterministic, polynomial and unsatisfiable.
Proposition 3.6 of Perfect completeness for 2-to-1 games. Words of the Lean statement, its names split into words, that also appear in the theorem: one, positive, clause, assignment, value, game and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constraint.
Corollary 1.3 of Perfect completeness for 2-to-1 games. Words of the Lean statement, its names split into words, that also appear in the theorem: language, bits, literal, soundness, exactly, one and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constraints and polynomial.
Proposition 8.1 of Perfect completeness for 2-to-1 games. Words of the Lean statement, its names split into words, that also appear in the theorem: less, value, game, two, completeness and perfect. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unsatisfiable.
Theorem 2.6 of Perfect completeness for 2-to-1 games. Words of the Lean statement, its names split into words, that also appear in the theorem: range, vector, table, finite, binary, values and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: depending.
Proposition 5.4 of Perfect completeness for 2-to-1 games. Words of the Lean statement, its names split into words, that also appear in the theorem: returning, decoding, eq, count, positive, alphabet and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: measurable, output and depends.

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 (Lean’s default kernel accepts the solution; the nanoda replay stopped: fatal runtime error: stack overflow, aborting).
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 105’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; NP-hardness is encoded as a polynomial-time Karp gap reduction from binary 3SAT (Mathlib TM2 model, finite stack alphabets).
Definitions the group asks readers to compare with the paper: Same Mathlib TM2ComputableInPolyTime + FiniteAlphabet encoding as family 102; instances are multigraphs (edge list with repeats), outputs written with unary encodeWord
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 rational $0<\delta<1$, a deterministic polynomial-time reduction from binary 3SAT produces a nonempty unweighted game of value exactly $1$ on satisfiable inputs and at most $\delta$ on unsatisfiable inputs. Each constraint projection has exactly two preimages for each output label. The alphabet depends only on $\delta$, and the runtime is measured in the original input bit length.”
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

185 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: 47 to its own declarations, 139 to Mathlib’s documentation. 7 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 UniqueGamesTheorem.Foundations.Complexitydef encodeWord (n : Nat) : List Bool := List.replicate n true ++ [false]def encodeWords : List Nat → List Bool  | [] => []  | n :: ns => encodeWord n ++ encodeWords nsend UniqueGamesTheorem.Foundations.Complexitynamespace UniqueGamesTheorem.Foundations.Complexity.MachineFiniteAlphabetopen Turingdef FiniteAlphabet (M : FinTM2) : Prop := ∀ k, Finite (M.Γ k)end UniqueGamesTheorem.Foundations.Complexity.MachineFiniteAlphabetnamespace PerfectCompleteness.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 assignmentstructure Formula where  clauses : List Clausedef Formula.Satisfiable (F : Formula) : Prop :=  ∃ assignment : Nat → Bool, ∀ clause ∈ F.clauses, clause.eval assignment = trueend PerfectCompleteness.BinaryFormulanamespace PerfectCompleteness.BinaryEncodingopen PerfectCompleteness.BinaryFormuladef bitsValue : List Bool → Nat  | [] => 0  | b :: bits => Nat.bit b (bitsValue bits)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)  | _ => nonedef 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)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 PerfectCompleteness.BinaryEncodingnamespace PerfectCompleteness.BinaryLanguagedef language (input : List Bool) : Prop :=  ∃ formula, BinaryEncoding.decodeFormula input = some formula ∧ formula.Satisfiableend PerfectCompleteness.BinaryLanguagenamespace PerfectCompletenessstructure ProjectionTable (q : Nat) where  images : Vector (Fin q) (2 * q)  exactlyTwo : ∀ b : Fin q,    ((List.finRange (2 * q)).filter (fun a => decide (images[a] = b))).length = 2structure Edge (leftVertices rightVertices q : Nat) where  left : Fin leftVertices  right : Fin rightVertices  projection : ProjectionTable qstructure Instance (q : Nat) where  leftVertices : Nat  rightVertices : Nat  edges : List (Edge leftVertices rightVertices q)  nonempty : edges ≠ []abbrev Labeling {q : Nat} (G : Instance q) :=  (Fin G.leftVertices → Fin (2 * q)) × (Fin G.rightVertices → Fin q)def Edge.satisfied {l r q : Nat} (e : Edge l r q)    (a : Fin l → Fin (2 * q)) (b : Fin r → Fin q) : Bool :=  decide (e.projection.images[a e.left] = b e.right)def countSatisfied {l r q : Nat} (a : Fin l → Fin (2 * q))    (b : Fin r → Fin q) : List (Edge l r q) → Nat  | [] => 0  | e :: es => (if e.satisfied a b then 1 else 0) + countSatisfied a b esend PerfectCompletenessnamespace PerfectCompleteness.Instancedef maxSatisfied {q : Nat} (G : Instance q) : Nat :=  Finset.univ.sup (fun s : Labeling G => countSatisfied s.1 s.2 G.edges)noncomputable def value {q : Nat} (G : Instance q) : ℝ :=  (G.maxSatisfied : ℝ) / G.edges.lengthend PerfectCompleteness.Instancenamespace PerfectCompleteness.Encodingopen UniqueGamesTheorem.Foundations.Complexitydef tableWords {q : Nat} (table : ProjectionTable q) : List Nat :=  table.images.toList.map Fin.valdef edgeWords {l r q : Nat} (edge : Edge l r q) : List Nat :=  [edge.left.val, edge.right.val] ++ tableWords edge.projectiondef gameWords {q : Nat} (game : Instance q) : List Nat :=  [game.leftVertices, game.rightVertices, q, game.edges.length] ++    game.edges.flatMap edgeWordsdef gameBits {q : Nat} (game : Instance q) : List Bool :=  encodeWords (gameWords game)end PerfectCompleteness.Encodingnamespace PerfectCompletenessopen Turing UniqueGamesTheorem.Foundations.Complexitystructure BinaryGapReduction (δ : ℚ) where  alphabet : Nat  alphabet_ge_two : 2 ≤ alphabet  construct : List Bool → Instance alphabet  computation : TM2ComputableInPolyTime (id : List Bool → List Bool)    Encoding.gameBits construct  finiteAlphabet : MachineFiniteAlphabet.FiniteAlphabet computation.tm  completeness : ∀ input, BinaryLanguage.language input → (construct input).value = 1  soundness : ∀ input, ¬BinaryLanguage.language input → (construct input).value ≤ (δ : ℝ)end PerfectCompletenessnamespace PerfectCompleteness.Theorem11theorem exists_reduction (δ : ℚ) (positive : 0 < δ) (less_than_one : δ < 1) :    Nonempty (PerfectCompleteness.BinaryGapReduction δ) := by  sorryend PerfectCompleteness.Theorem11end 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.

Perfect completeness for 2-to-1 games

Theorem

For every fixed rational \delta\in(0,1) there are an integer q=q(\delta)\ge2 and a deterministic polynomial-time reduction from 3-SAT to 2-to-1 games with alphabets [2q] and [q] such that:

(i)
a satisfiable formula produces a game of value 1;
(ii)
an unsatisfiable formula produces a game of value at most \delta.

Section 1, Introduction Read it in the paper TeX source, line 32

Show the TeX

For every fixed rational $\delta\in(0,1)$ there are an integer $q=q(\delta)\ge2$
and a deterministic polynomial-time reduction from $3$-SAT to 2-to-1 games
with alphabets $[2q]$ and $[q]$ such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a satisfiable formula produces a game of value $1$;
\item an unsatisfiable formula produces a game of value at most $\delta$.
\end{enumerate}
The output contains a nonempty explicit list of unweighted edge occurrences
and an explicit table for every projection map.
Every table has exactly two preimages for each right answer.
The polynomial running-time bound is in the ordinary binary input length;
its degree and constants may depend on the fixed $\delta$.

Corollary (Maximum k-colorable subgraph)

There are absolute constants C>0 and an integer k_0\ge3 such that, for every fixed integer k\ge k_0, it is NP-hard to distinguish the following cases for a finite unweighted graph G=(W,F) with F\ne\varnothing:

(i)
G is k-colorable;
(ii)
every coloring \chi:W\to[k] satisfies
\frac{|\{\{u,v\}\in F:\chi(u)\ne\chi(v)\}|}{|F|} \le 1-\frac1k+C\frac{\log k}{k^2}.

The fraction on the left is the proportion of edges properly colored by \chi, or equivalently cut by its partition into k color classes.

Section 1, Introduction Read it in the paper TeX source, line 189

Show the TeX

There are absolute constants $C>0$ and an integer $k_0\ge3$ such that, for
every fixed integer $k\ge k_0$, it is NP-hard to distinguish the following
cases for a finite unweighted graph $G=(W,F)$ with $F\ne\varnothing$:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $G$ is $k$-colorable;
\item every coloring $\chi:W\to[k]$ satisfies
\[
 \frac{|\{\{u,v\}\in F:\chi(u)\ne\chi(v)\}|}{|F|}
 \le 1-\frac1k+C\frac{\log k}{k^2}.
\]
\end{enumerate}
The fraction on the left is the proportion of edges properly colored by
$\chi$, or equivalently cut by its partition into $k$ color classes.

Corollary (Satisfiable Not-Two and three-query PCPs)

Let \operatorname{NTW}(a,b,c) be the Boolean predicate accepting exactly zero, one, or three true inputs. For every fixed 0<\eps<3/8, it is NP-hard to distinguish a satisfiable instance with constraints \operatorname{NTW}(\ell_1,\ell_2,\ell_3) from one in which every assignment satisfies at most a 5/8+\eps fraction of the constraints; each literal \ell_i may be a variable or its negation. Consequently, every language in NP has a polynomial-time verifier using O(\log n) random bits and three nonadaptive proof-bit queries, with perfect completeness and soundness at most 5/8+\eps, where n is the input length.

Section 1, Introduction Read it in the paper TeX source, line 222

Show the TeX

Let $\operatorname{NTW}(a,b,c)$ be the Boolean predicate accepting exactly
zero, one, or three true inputs. For every fixed $0<\eps<3/8$, it is NP-hard
to distinguish a satisfiable instance with constraints
$\operatorname{NTW}(\ell_1,\ell_2,\ell_3)$ from one in which every assignment
satisfies at most a $5/8+\eps$ fraction of the constraints; each literal
$\ell_i$ may be a variable or its negation.
Consequently, every language in NP has a polynomial-time verifier using
$O(\log n)$ random bits and three nonadaptive proof-bit queries, with perfect
completeness and soundness at most $5/8+\eps$, where $n$ is the input length.

Configuration and file facts

File: lean/ComparatorChallenges/PerfectCompleteness.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Computability/PerfectCompleteness/Theorem11.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: 30 local vocabulary declarations
Lines of Lean, without comments or blank lines: 101
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