Lean statement

Exact approximation threshold under $P\ne NP$

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.MetricKMedian.threshold

Lean statement file: KMedianThreshold.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 125 link it.

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 threshold (h : Complexity.PNeNP) :    sInf approximationFactors=1+2/Real.exp 1 := 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.MetricKMedian.Instance line 9

structure Instance where

Show the remaining 11 lines

  n : ℕ  distance : Fin n → Fin n → ℚ  clients : Finset (Fin n)  facilities : Finset (Fin n)  k : ℕ  distance_nonneg : ∀ i j, 0 ≤ distance i j  distance_eq_zero : ∀ i j, distance i j = 0 ↔ i = j  distance_symm : ∀ i j, distance i j = distance j i  distance_triangle : ∀ i j h, distance i h ≤ distance i j + distance j h  k_pos : 1 ≤ k  k_le : k ≤ facilities.card

definition OAI.MetricKMedian.encodeNat line 22

def encodeNat (n : ℕ) : List Bool :=  List.replicate n.bits.length true ++ [false] ++ n.bits

definition OAI.MetricKMedian.encodeRat line 25

def encodeRat (q : ℚ) : List Bool :=  [decide (q.num < 0)] ++ encodeNat q.num.natAbs ++ encodeNat q.den

definition OAI.MetricKMedian.encodeInstance line 28

def encodeInstance (I : Instance) : List Bool :=  encodeNat I.n ++ encodeNat I.k ++    List.ofFn (fun i : Fin I.n => decide (i ∈ I.clients)) ++    List.ofFn (fun i : Fin I.n => decide (i ∈ I.facilities)) ++    (List.ofFn (fun i : Fin I.n =>      (List.ofFn (fun j : Fin I.n => encodeRat (I.distance i j))).flatten)).flatten

definition OAI.MetricKMedian.nearest line 35

noncomputable def nearest (I : Instance) (j : Fin I.n) (S : Finset (Fin I.n)) : ℝ :=  if h : S.Nonempty then    (S.image (fun i => (I.distance j i : ℝ))).min' (h.image _)  else 0

definition OAI.MetricKMedian.cost line 40

noncomputable def cost (I : Instance) (S : Finset (Fin I.n)) : ℝ :=  ∑ j ∈ I.clients, nearest I j S

definition OAI.MetricKMedian.feasibleSets line 43

def feasibleSets (I : Instance) : Finset (Finset (Fin I.n)) :=  I.facilities.powerset.filter (fun S => S.Nonempty ∧ S.card ≤ I.k)

definition OAI.MetricKMedian.optimum line 46

noncomputable def optimum (I : Instance) : ℝ :=  if h : (feasibleSets I).Nonempty then    ((feasibleSets I).image (cost I)).min' (h.image _)  else 0

definition OAI.MetricKMedian.Complexity.FinitePolyTime line 63

def FinitePolyTime {α β : Type} (ea : α → List Bool) (eb : β → List Bool)    (f : α → β) : Prop :=  ∃ p : Turing.TM2ComputableInPolyTime ea eb f,    ∀ k : p.tm.K, Finite (p.tm.Γ k)

abbreviation OAI.MetricKMedian.Complexity.Word line 68
```
abbrev Word := List Bool
```
abbreviation OAI.MetricKMedian.Complexity.Language line 69
```
abbrev Language := Set Word
```
definition OAI.MetricKMedian.Complexity.encodeBit line 71
```
def encodeBit (b : Bool) : Word := [b]
```

definition OAI.MetricKMedian.Complexity.encodePair line 73

def encodePair (x : Word × Word) : Word :=  encodeNat x.1.length ++ x.1 ++ x.2

definition OAI.MetricKMedian.Complexity.P line 76

def P : Set Language := {L | ∃ decideL : Word → Bool,  FinitePolyTime id encodeBit decideL ∧ ∀ x, decideL x = true ↔ x ∈ L}

definition OAI.MetricKMedian.Complexity.NP line 79

def NP : Set Language := {L | ∃ (verify : Word × Word → Bool) (p : Polynomial ℕ),  FinitePolyTime encodePair encodeBit verify ∧  ∀ x, x ∈ L ↔ ∃ w : Word, w.length ≤ p.eval x.length ∧ verify (x, w) = true}

definition OAI.MetricKMedian.Complexity.PNeNP line 84
```
def PNeNP : Prop := P ≠ NP
```

definition OAI.MetricKMedian.FactorCorrect line 88

def FactorCorrect (α : ℝ) (I : Instance) (output : List Bool) : Prop :=  ∃ S : Finset (Fin I.n), S ⊆ I.facilities ∧ S.Nonempty ∧ S.card ≤ I.k ∧    output = List.ofFn (fun i : Fin I.n => decide (i ∈ S)) ∧    cost I S ≤ α * optimum I

definition OAI.MetricKMedian.approximationFactors line 94

def approximationFactors : Set ℝ := {α | ∃ algorithm : Instance → List Bool,  Complexity.FinitePolyTime encodeInstance id algorithm ∧  ∀ I, FactorCorrect α I (algorithm I)}

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: infimum, np, factors, specified, deterministic, approximation, facilities, time and one more.

Statements in The approximation threshold for metric k-median come first because the release’s name for this Lean statement shares words with that paper’s title.

Corollary 1.2 of The approximation threshold for metric k-median

Corollary

If P\ne NP, the infimum of deterministic polynomial-time approximation factors for metric k-median with specified candidate facilities is 1+2/e.

Section 1, Introduction Read it in the paper TeX source, line 56

Show the TeX

If \(P\ne NP\), the infimum of deterministic polynomial-time approximation
factors for metric \(k\)-median with specified candidate facilities is \(1+2/e\).

Other possible paper matches, closest wording first

Theorem 1.1 of The approximation threshold for metric k-median. Words of the Lean statement, its names split into words, that also appear in the theorem: encoding, deterministic, length, finite, rational, algorithm and two more. Found in the paper’s title, which its main theorem states: threshold and approximation.
Theorem 6.2 of The approximation threshold for metric k-median. Words of the Lean statement, its names split into words, that also appear in the theorem: factor, optimum, deterministic, approximation, integer, facilities and four more.
Theorem 3.1 of The approximation threshold for metric k-median. Words of the Lean statement, its names split into words, that also appear in the theorem: feasible, nearest, output, client, integer, finite and five more.
Theorem 4.7 of Single-exponential recovery and bounded-price strictness for metric k-median. Words of the Lean statement, its names split into words, that also appear in the theorem: actual, nonempty, correct, nearest, exp, bit and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: assume.
Theorem 5.3 of Single-exponential recovery and bounded-price strictness for metric k-median. Words of the Lean statement, its names split into words, that also appear in the theorem: work, poly, min, real, exp, bit and eight more.
Theorem 5.1 of The approximation threshold for metric k-median. Words of the Lean statement, its names split into words, that also appear in the theorem: nearest, deterministic, client, outputs, integers, finite and six more.

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

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; 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…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: 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, Classical.choice, Quot.sound

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 125’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; the hardness half is proven inside the threshold theorem under PNeNP (as the summary says 'assuming P != NP').
Definitions the group asks readers to compare with the paper: Complexity classes are hand-defined from Mathlib TM2ComputableInPolyTime with a FinitePolyTime finite-alphabet clause: P, NP (certificate form with encodePair) and PNeNP := P ≠ NP; approximationFactors is a set of reals whose sInf must be nonempty/bounded below (true: ≥ 1)
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

“Assuming $P\ne NP$, it also proves that the infimum of all polynomial-time approximation factors in this model is exactly $1+2/e$.”
Quoted from OpenAI’s Lean note for this family

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

Supporting material

The complete Lean statement file

107 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: 45 to its own declarations, 73 to Mathlib’s documentation. 20 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves threshold exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches 18 of the file’s 19 other declarations; with the theorem itself they span 59 lines. 2 of them are named in the statement itself.

Names in the namespace OAI.MetricKMedian are listed without it.

structure Instanceline 9library
def encodeNatline 22library
def encodeRatline 25library
def encodeInstanceline 28library
def nearestline 35library
def costline 40library
def feasibleSetsline 43library
def optimumline 46library
def Complexity.FinitePolyTimeline 63library
abbrev Complexity.Wordline 68library
abbrev Complexity.Languageline 69library
def Complexity.encodeBitline 71library
def Complexity.encodePairline 73library
def Complexity.Pline 76library
def Complexity.NPline 79library
def Complexity.PNeNPline 84in the statementlibrary
def FactorCorrectline 88library
def approximationFactorsline 94in the statementlibrary
theorem thresholdline 102selected theoremlibrary

1 declaration the statement does not reach by name

These definitions and lemmas were not found by following names in the statement. Lean may still use them through rules such as instances or macros.

def Correct line 51

import Mathlibnamespace OAIsectionsectionnamespace MetricKMedianstructure Instance where  n : ℕ  distance : Fin n → Fin n → ℚ  clients : Finset (Fin n)  facilities : Finset (Fin n)  k : ℕ  distance_nonneg : ∀ i j, 0 ≤ distance i j  distance_eq_zero : ∀ i j, distance i j = 0 ↔ i = j  distance_symm : ∀ i j, distance i j = distance j i  distance_triangle : ∀ i j h, distance i h ≤ distance i j + distance j h  k_pos : 1 ≤ k  k_le : k ≤ facilities.carddef encodeNat (n : ℕ) : List Bool :=  List.replicate n.bits.length true ++ [false] ++ n.bitsdef encodeRat (q : ℚ) : List Bool :=  [decide (q.num < 0)] ++ encodeNat q.num.natAbs ++ encodeNat q.dendef encodeInstance (I : Instance) : List Bool :=  encodeNat I.n ++ encodeNat I.k ++    List.ofFn (fun i : Fin I.n => decide (i ∈ I.clients)) ++    List.ofFn (fun i : Fin I.n => decide (i ∈ I.facilities)) ++    (List.ofFn (fun i : Fin I.n =>      (List.ofFn (fun j : Fin I.n => encodeRat (I.distance i j))).flatten)).flattennoncomputable def nearest (I : Instance) (j : Fin I.n) (S : Finset (Fin I.n)) : ℝ :=  if h : S.Nonempty then    (S.image (fun i => (I.distance j i : ℝ))).min' (h.image _)  else 0noncomputable def cost (I : Instance) (S : Finset (Fin I.n)) : ℝ :=  ∑ j ∈ I.clients, nearest I j Sdef feasibleSets (I : Instance) : Finset (Finset (Fin I.n)) :=  I.facilities.powerset.filter (fun S => S.Nonempty ∧ S.card ≤ I.k)noncomputable def optimum (I : Instance) : ℝ :=  if h : (feasibleSets I).Nonempty then    ((feasibleSets I).image (cost I)).min' (h.image _)  else 0def Correct (epsilon : ℝ) (I : Instance) (output : List Bool) : Prop :=  ∃ S : Finset (Fin I.n), S ⊆ I.facilities ∧ S.Nonempty ∧ S.card ≤ I.k ∧    output = List.ofFn (fun i : Fin I.n => decide (i ∈ S)) ∧    cost I S ≤ (1 + 2 / Real.exp 1 + epsilon) * optimum Iend MetricKMedianendendnamespace MetricKMediannamespace Complexity/-- Polynomial-time computation with every physical work alphabet finite. -/def FinitePolyTime {α β : Type} (ea : α → List Bool) (eb : β → List Bool)    (f : α → β) : Prop :=  ∃ p : Turing.TM2ComputableInPolyTime ea eb f,    ∀ k : p.tm.K, Finite (p.tm.Γ k)abbrev Word := List Boolabbrev Language := Set Worddef encodeBit (b : Bool) : Word := [b]def encodePair (x : Word × Word) : Word :=  encodeNat x.1.length ++ x.1 ++ x.2def P : Set Language := {L | ∃ decideL : Word → Bool,  FinitePolyTime id encodeBit decideL ∧ ∀ x, decideL x = true ↔ x ∈ L}def NP : Set Language := {L | ∃ (verify : Word × Word → Bool) (p : Polynomial ℕ),  FinitePolyTime encodePair encodeBit verify ∧  ∀ x, x ∈ L ↔ ∃ w : Word, w.length ≤ p.eval x.length ∧ verify (x, w) = true}/-- Separation of deterministic polynomial time from finite-verifier nondeterministic polynomial time. -/def PNeNP : Prop := P ≠ NPend Complexity/-- An actual nonempty feasible facility set, encoded by its membership mask. -/def FactorCorrect (α : ℝ) (I : Instance) (output : List Bool) : Prop :=  ∃ S : Finset (Fin I.n), S ⊆ I.facilities ∧ S.Nonempty ∧ S.card ≤ I.k ∧    output = List.ofFn (fun i : Fin I.n => decide (i ∈ S)) ∧    cost I S ≤ α * optimum I/-- Deterministic polynomial-time approximation factors for arbitrary specified clients and facilities. -/def approximationFactors : Set ℝ := {α | ∃ algorithm : Instance → List Bool,  Complexity.FinitePolyTime encodeInstance id algorithm ∧  ∀ I, FactorCorrect α I (algorithm I)}end MetricKMediannamespace MetricKMedian/-- The deterministic approximation threshold is an infimum; its attainment is not asserted. -/theorem threshold (h : Complexity.PNeNP) :    sInf approximationFactors=1+2/Real.exp 1 := by  sorryend MetricKMedianend 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.

Single-exponential recovery and bounded-price strictness for metric k-median

Theorem (Refined recovery)

Let D and F be disjoint indexed sets in a metric whose distances between distinct indices are positive integers bounded by a fixed polynomial in N=|D|+|F|. Fix \varepsilon>0 and 0<L_0<\infty, and choose a fixed rational parameter

0<\mu\le \min\{1/4,\varepsilon/12\}.

Let S\subseteq F have cardinality h\ge1. Suppose there exists a comparison solution O\subseteq F of cardinality h and cost P>0, with a partition into good and bad centers and distinct proxies \phi_i\in S for the good centers, such that

\sum_{i\ \mathrm{good}}\sum_{p\in C_i}d(p,\phi_i) \le P_g+\mu P, \qquad m:=|\{i:i\ \mathrm{bad}\}|\le L_0\log N.

Given S, \mu, L_0, and a failure parameter \zeta\in(0,1), a randomized algorithm returns \widehat S\subseteq F with

|\widehat S|\le h \quad\text{always},\qquad \operatorname{cost}(\widehat S) \le (1+2/e+\varepsilon)P \quad\text{with probability at least }1-\zeta.

Its running time is polynomial in N and \log(1/\zeta) for the fixed parameters. The algorithm is not given O, P, the partition, or its proxies.

Section 1, Introduction Read it in the paper TeX source, line 47

Show the TeX

Let $D$ and $F$ be disjoint indexed sets in a metric whose distances
between distinct indices are positive integers bounded by a fixed
polynomial in $N=|D|+|F|$.
Fix $\varepsilon>0$ and $0<L_0<\infty$, and choose a fixed rational parameter
\[
 0<\mu\le \min\{1/4,\varepsilon/12\}.
\]
Let $S\subseteq F$ have cardinality $h\ge1$.  Suppose there exists a comparison
solution $O\subseteq F$ of cardinality $h$ and cost $P>0$, with a partition into
good and bad centers and distinct proxies $\phi_i\in S$ for the good centers,
such that
\begin{equation}
 \sum_{i\ \mathrm{good}}\sum_{p\in C_i}d(p,\phi_i)
 \le P_g+\mu P,
 \qquad m:=|\{i:i\ \mathrm{bad}\}|\le L_0\log N.
\end{equation}
Given $S$, $\mu$, $L_0$, and a failure parameter $\zeta\in(0,1)$, a randomized
algorithm returns $\widehat S\subseteq F$ with
\[
 |\widehat S|\le h
 \quad\text{always},\qquad
 \operatorname{cost}(\widehat S)
 \le (1+2/e+\varepsilon)P
 \quad\text{with probability at least }1-\zeta.
\]
Its running time is polynomial in $N$ and $\log(1/\zeta)$ for the fixed
parameters.  The algorithm is not given $O$, $P$, the partition, or its proxies.

Theorem (Global application)

There is an absolute constant \sigma>0 with the following property. For every fixed a>0, a randomized polynomial-time algorithm for metric k-median always returns a set S\subseteq F with |S|\le k and satisfies

\Pr\!\left\{\operatorname{cost}(S) \le(2-\sigma)\operatorname{opt}_k\right\} \ge 1-(|D|+|F|+2)^{-a}.

The same algorithm can be chosen to satisfy \mathbb E\operatorname{cost}(S)\le(2-\sigma)\operatorname{opt}_k. The running time is polynomial in the bit length of the finite rational metric input. Its degree may depend on the fixed constants.

Section 1, Introduction Read it in the paper TeX source, line 142

Show the TeX

There is an absolute constant $\sigma>0$ with the following property.
For every fixed $a>0$, a randomized polynomial-time algorithm for metric
$k$-median always returns a set $S\subseteq F$ with $|S|\le k$ and satisfies
\[
 \Pr\!\left\{\operatorname{cost}(S)
       \le(2-\sigma)\operatorname{opt}_k\right\}
 \ge 1-(|D|+|F|+2)^{-a}.
\]
The same algorithm can be chosen to satisfy
$\mathbb E\operatorname{cost}(S)\le(2-\sigma)\operatorname{opt}_k$.
The running time is polynomial in the bit length of the finite rational
metric input. Its degree may depend on the fixed constants.

The approximation threshold for metric k-median

Theorem

For every fixed \varepsilon>0, there is a deterministic algorithm which, given a finite rational metric k-median instance with 1\le k\le |F|, returns S\subseteq F, |S|\le k, such that

\sum_{j\in J}d(j,S)\le \left(1+\frac2e+\varepsilon\right)\OPT_k.

Its running time is polynomial in the binary encoding length of the instance. The polynomial may depend on \varepsilon.

Section 1, Introduction Read it in the paper TeX source, line 25

Show the TeX

For every fixed \(\varepsilon>0\), there is a deterministic algorithm which,
given a finite rational metric \(k\)-median instance with \(1\le k\le |F|\),
returns \(S\subseteq F\), \(|S|\le k\), such that
\[
 \sum_{j\in J}d(j,S)\le
 \left(1+\frac2e+\varepsilon\right)\OPT_k.
\]
Its running time is polynomial in the binary encoding length of the instance.
The polynomial may depend on \(\varepsilon\).

Corollary

If P\ne NP, the infimum of deterministic polynomial-time approximation factors for metric k-median with specified candidate facilities is 1+2/e.

Section 1, Introduction Read it in the paper TeX source, line 56

Show the TeX

If \(P\ne NP\), the infimum of deterministic polynomial-time approximation
factors for metric \(k\)-median with specified candidate facilities is \(1+2/e\).

Configuration and file facts

File: lean/ComparatorChallenges/KMedianThreshold.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 125 link it.
Solution module: OAI.Combinatorics.KMedian.Threshold
Allowed axioms: propext, Classical.choice, Quot.soundThe 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: 19 local vocabulary declarations
Lines of Lean, without comments or blank lines: 63
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.

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