Lean statement

Exact-budget recovery from accurate distinct cluster proxies

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.MetricKMedianRecovery.RefinedRecovery.refined_recovery

Lean statement file: KMedianRefinedRecovery.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 refined_recovery :    ∀ (bound : Polynomial ℕ) (ε L₀ : ℝ) (μ : ℚ),      0 < ε → 0 < L₀ → 0 < μ → μ ≤ 1/4 → (μ : ℝ) ≤ ε/12 →      ∃ A : BinaryRandomizedAlgorithm,        PolynomialWork A bound ∧        ∀ I : Input, I.Bounded bound → ∀ ζ : ℝ, 0 < ζ → ζ < 1 →          (∀ s : Seed A I ζ, I.metric.Feasible (output A I ζ s)) ∧          (∀ C : Certificate I μ L₀,            1 - ζ ≤ successProbability A I ζ              ((1 + 2 / Real.exp 1 + ε) * (I.metric.cost C.centers : ℝ))) := 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.MetricKMedianRecovery.RationalMetricInput line 7

structure RationalMetricInput where

Show the remaining 12 lines

  pointCount : ℕ  distance : Fin pointCount → Fin pointCount → ℚ  clients : Finset (Fin pointCount)  facilities : Finset (Fin pointCount)  budget : ℕ  facilities_nonempty : facilities.Nonempty  budget_pos : 0 < budget  covered : clients ∪ facilities = Finset.univ  distance_nonneg : ∀ x y, 0 ≤ distance x y  distance_self : ∀ x, distance x x = 0  distance_symm : ∀ x y, distance x y = distance y x  distance_triangle : ∀ x y z, distance x z ≤ distance x y + distance y z

definition OAI.MetricKMedianRecovery.RationalMetricInput.size line 23

def size (I : RationalMetricInput) : ℕ := I.clients.card + I.facilities.card

definition OAI.MetricKMedianRecovery.RationalMetricInput.Feasible line 25

def Feasible (I : RationalMetricInput) (S : Finset (Fin I.pointCount)) : Prop :=  S ⊆ I.facilities ∧ S.card ≤ I.budget ∧ (I.clients.Nonempty → S.Nonempty)

definition OAI.MetricKMedianRecovery.RationalMetricInput.cost line 28

def cost (I : RationalMetricInput) (S : Finset (Fin I.pointCount)) : ℚ :=  if hS : S.Nonempty then ∑ p ∈ I.clients, S.inf' hS (I.distance p) else 0

definition OAI.MetricKMedianRecovery.RationalMetricInput.natBits line 31

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

definition OAI.MetricKMedianRecovery.RationalMetricInput.rationalBits line 34

def rationalBits (r : ℚ) : List Bool :=  [decide (r < 0)] ++ natBits r.num.natAbs ++ natBits r.den

definition OAI.MetricKMedianRecovery.RationalMetricInput.bits line 37

def bits (I : RationalMetricInput) : List Bool :=  natBits I.pointCount ++ natBits I.budget ++    List.ofFn (fun i : Fin I.pointCount => decide (i ∈ I.clients)) ++    List.ofFn (fun i : Fin I.pointCount => decide (i ∈ I.facilities)) ++    (List.ofFn (fun i : Fin I.pointCount =>      (List.ofFn (fun j : Fin I.pointCount => rationalBits (I.distance i j))).flatten)).flatten

definition OAI.MetricKMedianRecovery.RationalMetricInput.decodeOutput line 44

def decodeOutput (I : RationalMetricInput) (out : List Bool) : Finset (Fin I.pointCount) :=  I.facilities.filter (fun i => out[i.val]?.getD false = true)

definition OAI.MetricKMedianRecovery.pairBits line 49

def pairBits (p : List Bool × List Bool) : List Bool :=  p.1.flatMap (fun b => [true, b]) ++ [false] ++ p.2

structure OAI.MetricKMedianRecovery.BinaryRandomizedAlgorithm line 52

structure BinaryRandomizedAlgorithm where  randomBits : Polynomial ℕ  run : List Bool × List Bool → List Bool  computation : Turing.TM2ComputableInPolyTime pairBits id run  finiteWorkAlphabet : ∀ k : computation.tm.K, Finite (computation.tm.Γ k)

structure OAI.MetricKMedianRecovery.RefinedRecovery.Input line 65

structure Input where  metric : RationalMetricInput  disjoint : Disjoint metric.clients metric.facilities  integral : ∀ x y, ∃ z : ℕ, metric.distance x y = (z : ℚ)  positive : ∀ x y, x ≠ y → 0 < metric.distance x y  anchor : Finset (Fin metric.pointCount)  anchor_subset : anchor ⊆ metric.facilities  anchor_card : anchor.card = metric.budget

abbreviation OAI.MetricKMedianRecovery.RefinedRecovery.Input.Point line 75
```
abbrev Point (I : Input) := Fin I.metric.pointCount
```

definition OAI.MetricKMedianRecovery.RefinedRecovery.Input.Bounded line 79

def Bounded (I : Input) (bound : Polynomial ℕ) : Prop :=  ∀ x y, I.metric.distance x y ≤ ((bound.eval I.size : ℕ) : ℚ)

definition OAI.MetricKMedianRecovery.RefinedRecovery.Input.assign line 82

noncomputable def assign (I : Input) (O : Finset I.Point) (p : I.Point) : I.Point :=  if h : O.Nonempty then    Classical.choose (Finset.exists_min_image O      (fun i => toLex (I.metric.distance p i, i)) h)  else p

definition OAI.MetricKMedianRecovery.RefinedRecovery.Input.cluster line 88

noncomputable def cluster (I : Input) (O : Finset I.Point) (i : I.Point) : Finset I.Point :=  I.metric.clients.filter (fun p => I.assign O p = i)

definition OAI.MetricKMedianRecovery.RefinedRecovery.Input.clusterCost line 91

noncomputable def clusterCost (I : Input) (O : Finset I.Point) (i : I.Point) : ℝ :=  ∑ p ∈ I.cluster O i, (I.metric.distance p i : ℝ)

definition OAI.MetricKMedianRecovery.RefinedRecovery.Input.anchorBits line 94

def anchorBits (I : Input) : List Bool :=  List.ofFn (fun i : I.Point => decide (i ∈ I.anchor))

structure OAI.MetricKMedianRecovery.RefinedRecovery.Certificate line 99

structure Certificate (I : Input) (μ : ℚ) (L₀ : ℝ) where

Show the remaining 13 lines

  centers : Finset I.Point  centers_subset : centers ⊆ I.metric.facilities  same_card : centers.card = I.anchor.card  positive_cost : 0 < (I.metric.cost centers : ℝ)  good : Finset I.Point  good_subset : good ⊆ centers  proxy : I.Point → I.Point  proxy_mem : ∀ i ∈ good, proxy i ∈ I.anchor  proxy_inj : Set.InjOn proxy (↑good)  proxy_cost : (∑ i ∈ good, ∑ p ∈ I.cluster centers i,      (I.metric.distance p (proxy i) : ℝ)) ≤    (∑ i ∈ good, I.clusterCost centers i) + (μ : ℝ) * (I.metric.cost centers : ℝ)  few_bad : (((centers \ good).card : ℕ) : ℝ) ≤ L₀ * Real.log (I.size : ℝ)

definition OAI.MetricKMedianRecovery.RefinedRecovery.precision line 114

noncomputable def precision (ζ : ℝ) : ℕ :=  ⌈Real.log (1 / ζ) / Real.log 2⌉₊

definition OAI.MetricKMedianRecovery.RefinedRecovery.queryBits line 117

noncomputable def queryBits (I : Input) (ζ : ℝ) : List Bool :=  pairBits (I.metric.bits, I.anchorBits ++ List.replicate (precision ζ) true)

definition OAI.MetricKMedianRecovery.RefinedRecovery.seedLength line 120

noncomputable def seedLength (A : BinaryRandomizedAlgorithm) (I : Input) (ζ : ℝ) : ℕ :=  A.randomBits.eval (queryBits I ζ).length

abbreviation OAI.MetricKMedianRecovery.RefinedRecovery.Seed line 123

abbrev Seed (A : BinaryRandomizedAlgorithm) (I : Input) (ζ : ℝ) :=  Fin (seedLength A I ζ) → Bool

definition OAI.MetricKMedianRecovery.RefinedRecovery.output line 126

noncomputable def output (A : BinaryRandomizedAlgorithm) (I : Input) (ζ : ℝ)    (s : Seed A I ζ) : Finset I.Point :=  I.metric.decodeOutput (A.run (queryBits I ζ, List.ofFn s))

definition OAI.MetricKMedianRecovery.RefinedRecovery.successProbability line 130

noncomputable def successProbability (A : BinaryRandomizedAlgorithm) (I : Input)    (ζ target : ℝ) : ℝ := by  classical  exact ((Finset.univ.filter (fun s : Seed A I ζ =>    (I.metric.cost (output A I ζ s) : ℝ) ≤ target)).card : ℝ) /    2 ^ seedLength A I ζ

2 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: 1/4, choose, disjoint, refined, centers, distances, min, recovery and 16 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: varepsilon, distinct, always, comparison and suppose.

Statements in Single-exponential recovery and bounded-price strictness for metric k-median come first because the Lean file and module names share words with that paper’s title.

Theorem 1.1 of 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.

Other possible paper matches, closest wording first

Theorem 5.4 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: map, centers, poly, integral, recovery, bad and 14 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: supplied and suppose.
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 13 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: error and always.
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: covered, nonempty, cluster, run, exp, size and eleven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: distinct and comparison.
Proposition 10.2 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: integral, distance, zeta, log, randomized, bound and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: always.
Theorem 1.2 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: bit, input, length, randomized, finite, rational and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: always and median.
Theorem 7.1 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: precision, target, run, output, log, lengths and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: distinct.

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 an exact-budget recovery algorithm for metric $k$-median on the stated polynomially bounded integral metrics. Suppose a supplied anchor represents all but logarithmically many comparison clusters by distinct proxies, with total proxy cost at most the comparison cost plus a sufficiently small relative error. The algorithm always opens at most $k$ facilities and, with any requested confidence, attains a $(1+2/e+\varepsilon)$ factor relative to those comparison centers in polynomial time.”
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

167 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: 55 to its own declarations, 94 to Mathlib’s documentation. 28 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 MetricKMedianRecoverystructure RationalMetricInput where  pointCount : ℕ  distance : Fin pointCount → Fin pointCount → ℚ  clients : Finset (Fin pointCount)  facilities : Finset (Fin pointCount)  budget : ℕ  facilities_nonempty : facilities.Nonempty  budget_pos : 0 < budget  covered : clients ∪ facilities = Finset.univ  distance_nonneg : ∀ x y, 0 ≤ distance x y  distance_self : ∀ x, distance x x = 0  distance_symm : ∀ x y, distance x y = distance y x  distance_triangle : ∀ x y z, distance x z ≤ distance x y + distance y znamespace RationalMetricInputdef size (I : RationalMetricInput) : ℕ := I.clients.card + I.facilities.carddef Feasible (I : RationalMetricInput) (S : Finset (Fin I.pointCount)) : Prop :=  S ⊆ I.facilities ∧ S.card ≤ I.budget ∧ (I.clients.Nonempty → S.Nonempty)def cost (I : RationalMetricInput) (S : Finset (Fin I.pointCount)) : ℚ :=  if hS : S.Nonempty then ∑ p ∈ I.clients, S.inf' hS (I.distance p) else 0def natBits (n : ℕ) : List Bool :=  List.replicate (Nat.bits n).length true ++ [false] ++ Nat.bits ndef rationalBits (r : ℚ) : List Bool :=  [decide (r < 0)] ++ natBits r.num.natAbs ++ natBits r.dendef bits (I : RationalMetricInput) : List Bool :=  natBits I.pointCount ++ natBits I.budget ++    List.ofFn (fun i : Fin I.pointCount => decide (i ∈ I.clients)) ++    List.ofFn (fun i : Fin I.pointCount => decide (i ∈ I.facilities)) ++    (List.ofFn (fun i : Fin I.pointCount =>      (List.ofFn (fun j : Fin I.pointCount => rationalBits (I.distance i j))).flatten)).flattendef decodeOutput (I : RationalMetricInput) (out : List Bool) : Finset (Fin I.pointCount) :=  I.facilities.filter (fun i => out[i.val]?.getD false = true)end RationalMetricInputdef pairBits (p : List Bool × List Bool) : List Bool :=  p.1.flatMap (fun b => [true, b]) ++ [false] ++ p.2structure BinaryRandomizedAlgorithm where  randomBits : Polynomial ℕ  run : List Bool × List Bool → List Bool  computation : Turing.TM2ComputableInPolyTime pairBits id run  finiteWorkAlphabet : ∀ k : computation.tm.K, Finite (computation.tm.Γ k)end MetricKMedianRecoveryrun_elab Lean.modifyEnv fun environment => Lean.Meta.auxLemmasExt.setState environment {}namespace MetricKMedianRecovery.RefinedRecoverystructure Input where  metric : RationalMetricInput  disjoint : Disjoint metric.clients metric.facilities  integral : ∀ x y, ∃ z : ℕ, metric.distance x y = (z : ℚ)  positive : ∀ x y, x ≠ y → 0 < metric.distance x y  anchor : Finset (Fin metric.pointCount)  anchor_subset : anchor ⊆ metric.facilities  anchor_card : anchor.card = metric.budgetnamespace Inputabbrev Point (I : Input) := Fin I.metric.pointCountdef size (I : Input) : ℕ := I.metric.sizedef Bounded (I : Input) (bound : Polynomial ℕ) : Prop :=  ∀ x y, I.metric.distance x y ≤ ((bound.eval I.size : ℕ) : ℚ)noncomputable def assign (I : Input) (O : Finset I.Point) (p : I.Point) : I.Point :=  if h : O.Nonempty then    Classical.choose (Finset.exists_min_image O      (fun i => toLex (I.metric.distance p i, i)) h)  else pnoncomputable def cluster (I : Input) (O : Finset I.Point) (i : I.Point) : Finset I.Point :=  I.metric.clients.filter (fun p => I.assign O p = i)noncomputable def clusterCost (I : Input) (O : Finset I.Point) (i : I.Point) : ℝ :=  ∑ p ∈ I.cluster O i, (I.metric.distance p i : ℝ)def anchorBits (I : Input) : List Bool :=  List.ofFn (fun i : I.Point => decide (i ∈ I.anchor))end Inputstructure Certificate (I : Input) (μ : ℚ) (L₀ : ℝ) where  centers : Finset I.Point  centers_subset : centers ⊆ I.metric.facilities  same_card : centers.card = I.anchor.card  positive_cost : 0 < (I.metric.cost centers : ℝ)  good : Finset I.Point  good_subset : good ⊆ centers  proxy : I.Point → I.Point  proxy_mem : ∀ i ∈ good, proxy i ∈ I.anchor  proxy_inj : Set.InjOn proxy (↑good)  proxy_cost : (∑ i ∈ good, ∑ p ∈ I.cluster centers i,      (I.metric.distance p (proxy i) : ℝ)) ≤    (∑ i ∈ good, I.clusterCost centers i) + (μ : ℝ) * (I.metric.cost centers : ℝ)  few_bad : (((centers \ good).card : ℕ) : ℝ) ≤ L₀ * Real.log (I.size : ℝ)noncomputable def precision (ζ : ℝ) : ℕ :=  ⌈Real.log (1 / ζ) / Real.log 2⌉₊noncomputable def queryBits (I : Input) (ζ : ℝ) : List Bool :=  pairBits (I.metric.bits, I.anchorBits ++ List.replicate (precision ζ) true)noncomputable def seedLength (A : BinaryRandomizedAlgorithm) (I : Input) (ζ : ℝ) : ℕ :=  A.randomBits.eval (queryBits I ζ).lengthabbrev Seed (A : BinaryRandomizedAlgorithm) (I : Input) (ζ : ℝ) :=  Fin (seedLength A I ζ) → Boolnoncomputable def output (A : BinaryRandomizedAlgorithm) (I : Input) (ζ : ℝ)    (s : Seed A I ζ) : Finset I.Point :=  I.metric.decodeOutput (A.run (queryBits I ζ, List.ofFn s))noncomputable def successProbability (A : BinaryRandomizedAlgorithm) (I : Input)    (ζ target : ℝ) : ℝ := by  classical  exact ((Finset.univ.filter (fun s : Seed A I ζ =>    (I.metric.cost (output A I ζ s) : ℝ) ≤ target)).card : ℝ) /    2 ^ seedLength A I ζnoncomputable def workBound (A : BinaryRandomizedAlgorithm) (I : Input) (ζ : ℝ) : ℕ :=  (queryBits I ζ).length + seedLength A I ζ +    A.computation.time.eval (pairBits      (queryBits I ζ, List.replicate (seedLength A I ζ) false)).lengthdef PolynomialWork (A : BinaryRandomizedAlgorithm) (bound : Polynomial ℕ) : Prop :=  ∃ c : ℝ, 0 < c ∧ ∃ d : ℕ,    ∀ I : Input, I.Bounded bound → ∀ ζ : ℝ, 0 < ζ → ζ < 1 →      (workBound A I ζ : ℝ) ≤        c * ((I.size : ℝ) + 1) ^ d * (1 + Real.log (1 / ζ)) ^ dend MetricKMedianRecovery.RefinedRecoverynamespace MetricKMedianRecovery.RefinedRecoverytheorem refined_recovery :    ∀ (bound : Polynomial ℕ) (ε L₀ : ℝ) (μ : ℚ),      0 < ε → 0 < L₀ → 0 < μ → μ ≤ 1/4 → (μ : ℝ) ≤ ε/12 →      ∃ A : BinaryRandomizedAlgorithm,        PolynomialWork A bound ∧        ∀ I : Input, I.Bounded bound → ∀ ζ : ℝ, 0 < ζ → ζ < 1 →          (∀ s : Seed A I ζ, I.metric.Feasible (output A I ζ s)) ∧          (∀ C : Certificate I μ L₀,            1 - ζ ≤ successProbability A I ζ              ((1 + 2 / Real.exp 1 + ε) * (I.metric.cost C.centers : ℝ))) := by  sorryend MetricKMedianRecovery.RefinedRecoveryend 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)

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

\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/KMedianRefinedRecovery.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.KMedianRecovery.Refined.Recovery
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: 27 local vocabulary declarations
Lines of Lean, without comments or blank lines: 112
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