Lean statement

Deterministic metric $k$-median approximation at the threshold

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.main_finite_work

Lean statement file: MetricKMedian.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 main_finite_work : ∀ ε : ℝ, 0 < ε →    ∃ (algorithm : Instance → List Bool) (P : Turing.TM2ComputableInPolyTime encodeInstance id algorithm),      (∀ k : P.tm.K, Finite (P.tm.Γ k)) ∧ ∀ I, Correct ε I (algorithm I) := 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 12

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 25

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

definition OAI.MetricKMedian.encodeRat line 28

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

definition OAI.MetricKMedian.encodeInstance line 31

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 38

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 43

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

definition OAI.MetricKMedian.feasibleSets line 46

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 49

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.Correct line 54

def 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 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: encoding, length, finite, rational, algorithm and time. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: varepsilon, deterministic, median, metric and polynomial.

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.

Theorem 1.1 of 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 \(\varepsilon\).

Other possible paper matches, closest wording first

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 four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: metric and polynomial.
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: optimum, integer, facilities, cost, algorithm and time. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: varepsilon, deterministic, approximation, median, metric and polynomial.
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, client, outputs, integers, finite, rational and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: gives, deterministic, metric and polynomial.
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 seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: polynomial.
Corollary 1.2 of The approximation threshold for metric k-median. Words of the Lean statement, its names split into words, that also appear in the theorem: facilities and time. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: candidate, specified, deterministic, approximation, median, metric and one more.
Proposition 9.1 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: compute, real, facility, cost, algorithm and time. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: polynomial.

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, 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 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

“The formalization gives, for every fixed $\varepsilon>0$, a deterministic polynomial-time $(1+2/e+\varepsilon)$-approximation for metric $k$-median on finite rational metrics with specified candidate facilities. Every output is a nonempty subset of the candidate facilities and opens at most $k$ of them.”
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

68 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: 20 to its own declarations, 47 to Mathlib’s documentation. 10 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 main_finite_work 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 all 9 other declarations of the file; with the theorem itself they span 42 lines. 3 of them are named in the statement itself.

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

structure Instanceline 12in the statementlibrary
def encodeNatline 25library
def encodeRatline 28library
def encodeInstanceline 31in the statementlibrary
def nearestline 38library
def costline 43library
def feasibleSetsline 46library
def optimumline 49library
def Correctline 54in the statementlibrary
theorem main_finite_workline 59selected theoremlibrary

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 Itheorem main_finite_work : ∀ ε : ℝ, 0 < ε →    ∃ (algorithm : Instance → List Bool) (P : Turing.TM2ComputableInPolyTime encodeInstance id algorithm),      (∀ k : P.tm.K, Finite (P.tm.Γ k)) ∧ ∀ I, Correct ε I (algorithm I) := by  sorryend MetricKMedianendendend 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/MetricKMedian.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.Main
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: 9 local vocabulary declarations
Lines of Lean, without comments or blank lines: 42
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