Plectis Atlas
Search

↑ and ↓ move, Enter opens, Esc closes.

Lean statement

Randomized metric $k$-median approximation below two

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.global_application

Lean statement file: KMedianRecovery.leanShow on the mapFamily 125. OpenAI lists this as a main result in its Lean catalogue.

See the closest candidatesWho ran the proof-verification programReport a comparison with the paper (GitHub issue)

The Lean statement

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

def globalApplicationClaim : Prop :=  ∃ σ : ℝ, 0 < σ ∧ ∀ a : ℝ, 0 < a → ∃ A : BinaryRandomizedAlgorithm,    (∀ (I : RationalMetricInput) (s : A.Seed I), I.Feasible (A.output I s)) ∧    (∀ I : RationalMetricInput,      1 - Real.rpow ((I.size : ℝ) + 2) (-a) ≤        A.successProbability I ((2 - σ) * (I.optimum : ℝ))) ∧    (∀ I : RationalMetricInput, A.expectedCost I ≤ (2 - σ) * (I.optimum : ℝ))theorem global_application : globalApplicationClaim := 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.comparisonSets line 31

    def comparisonSets (I : RationalMetricInput) : Finset (Finset (Fin I.pointCount)) :=  I.facilities.powerset.filter (fun S => S.Nonempty ∧ S.card ≤ I.budget)

  • definition OAI.MetricKMedianRecovery.RationalMetricInput.optimum line 41

    def optimum (I : RationalMetricInput) : ℚ :=  I.comparisonSets.inf' I.comparisonSets_nonempty I.cost

  • definition OAI.MetricKMedianRecovery.RationalMetricInput.natBits line 44

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

  • definition OAI.MetricKMedianRecovery.RationalMetricInput.rationalBits line 47

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

  • definition OAI.MetricKMedianRecovery.RationalMetricInput.bits line 50

    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 57

    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 62

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

  • structure OAI.MetricKMedianRecovery.BinaryRandomizedAlgorithm line 65

    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)

  • definition OAI.MetricKMedianRecovery.BinaryRandomizedAlgorithm.bitCount line 73

    def bitCount (A : BinaryRandomizedAlgorithm) (I : RationalMetricInput) : ℕ :=  A.randomBits.eval I.bits.length

  • abbreviation OAI.MetricKMedianRecovery.BinaryRandomizedAlgorithm.Seed line 76

    abbrev Seed (A : BinaryRandomizedAlgorithm) (I : RationalMetricInput) :=  Fin (A.bitCount I) → Bool

  • definition OAI.MetricKMedianRecovery.BinaryRandomizedAlgorithm.output line 79

    def output (A : BinaryRandomizedAlgorithm) (I : RationalMetricInput) (s : A.Seed I) :    Finset (Fin I.pointCount) :=  I.decodeOutput (A.run (I.bits, List.ofFn s))

  • definition OAI.MetricKMedianRecovery.BinaryRandomizedAlgorithm.successProbability line 83

    noncomputable def successProbability (A : BinaryRandomizedAlgorithm) (I : RationalMetricInput)    (bound : ℝ) : ℝ := by  classical  exact ((Finset.univ.filter (fun s : A.Seed I => (I.cost (A.output I s) : ℝ) ≤ bound)).card : ℝ) /    2 ^ (A.bitCount I)

  • definition OAI.MetricKMedianRecovery.BinaryRandomizedAlgorithm.expectedCost line 89

    noncomputable def expectedCost (A : BinaryRandomizedAlgorithm) (I : RationalMetricInput) : ℝ :=  (∑ s : A.Seed I, (I.cost (A.output I s) : ℝ)) / 2 ^ (A.bitCount I)

No confident match with a paper statement

No paper theorem is paired with this Lean statement.

Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

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.

Other possible paper matches, closest wording first

  • 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: application, global, bit, input, length, randomized and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sigma, absolute, always, opt, returns and median.
  • 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, output, expected, inputs, client, integer and nine more.
  • 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: binary, length, finite, rational, metric, algorithm and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: opt, returns and median.
  • 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: expected, client, outputs, integers, randomized, finite and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: resulting and solution.
  • 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, metric, cost, algorithm and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: factor, approximation, opt and median.

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 linked final approximation result applies to arbitrary finite rational metrics. For one absolute $\sigma>0$, a polynomial-time fair-bit algorithm always returns a feasible solution, has expected cost at most $(2-\sigma)\mathrm{OPT}$, and attains the same factor with arbitrarily high polynomial confidence.”

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

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: 30 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.

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 comparisonSets (I : RationalMetricInput) : Finset (Finset (Fin I.pointCount)) :=  I.facilities.powerset.filter (fun S => S.Nonempty ∧ S.card ≤ I.budget)lemma comparisonSets_nonempty (I : RationalMetricInput) : I.comparisonSets.Nonempty := by  obtain ⟨i, hi⟩ := I.facilities_nonempty  refine ⟨{i}, ?_⟩  simp only [comparisonSets, Finset.mem_filter, Finset.mem_powerset,    Finset.singleton_subset_iff, Finset.singleton_nonempty, Finset.card_singleton, true_and]  exact ⟨hi, I.budget_pos⟩def optimum (I : RationalMetricInput) : ℚ :=  I.comparisonSets.inf' I.comparisonSets_nonempty I.costdef 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)namespace BinaryRandomizedAlgorithmdef bitCount (A : BinaryRandomizedAlgorithm) (I : RationalMetricInput) : ℕ :=  A.randomBits.eval I.bits.lengthabbrev Seed (A : BinaryRandomizedAlgorithm) (I : RationalMetricInput) :=  Fin (A.bitCount I) → Booldef output (A : BinaryRandomizedAlgorithm) (I : RationalMetricInput) (s : A.Seed I) :    Finset (Fin I.pointCount) :=  I.decodeOutput (A.run (I.bits, List.ofFn s))noncomputable def successProbability (A : BinaryRandomizedAlgorithm) (I : RationalMetricInput)    (bound : ℝ) : ℝ := by  classical  exact ((Finset.univ.filter (fun s : A.Seed I => (I.cost (A.output I s) : ℝ) ≤ bound)).card : ℝ) /    2 ^ (A.bitCount I)noncomputable def expectedCost (A : BinaryRandomizedAlgorithm) (I : RationalMetricInput) : ℝ :=  (∑ s : A.Seed I, (I.cost (A.output I s) : ℝ)) / 2 ^ (A.bitCount I)end BinaryRandomizedAlgorithmdef globalApplicationClaim : Prop :=  ∃ σ : ℝ, 0 < σ ∧ ∀ a : ℝ, 0 < a → ∃ A : BinaryRandomizedAlgorithm,    (∀ (I : RationalMetricInput) (s : A.Seed I), I.Feasible (A.output I s)) ∧    (∀ I : RationalMetricInput,      1 - Real.rpow ((I.size : ℝ) + 2) (-a) ≤        A.successProbability I ((2 - σ) * (I.optimum : ℝ))) ∧    (∀ I : RationalMetricInput, A.expectedCost I ≤ (2 - σ) * (I.optimum : ℝ))theorem global_application : globalApplicationClaim := by  sorryend MetricKMedianRecoveryend 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
OpenAI’s Lean catalogue
OpenAI lists this as a main result in its Lean catalogue.
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
18 local vocabulary declarations
Lines of Lean, without comments or blank lines
70
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

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

Plectis Atlas

Randomized metric $k$-median approximation below two

Family 125: The metric k-median approximation threshold and recovery

No paper statement selected

no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

The Lean statement says

def globalApplicationClaim : Prop :=  ∃ σ : ℝ, 0 < σ ∧ ∀ a : ℝ, 0 < a → ∃ A : BinaryRandomizedAlgorithm,    (∀ (I : RationalMetricInput) (s : A.Seed I), I.Feasible (A.output I s)) ∧    (∀ I : RationalMetricInput,      1 - Real.rpow ((I.size : ℝ) + 2) (-a) ≤        A.successProbability I ((2 - σ) * (I.optimum : ℝ))) ∧    (∀ I : RationalMetricInput, A.expectedCost I ≤ (2 - σ) * (I.optimum : ℝ))theorem global_application : globalApplicationClaim := 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.

No paper theorem is paired with this Lean statement.two independent groups report that Comparator accepted runs for this file at the release version shown here.Statement review: none recorded

18 local vocabulary declarationsAllowed axioms: propext, Quot.sound, Classical.choiceopenai/math at fd4aeebwcook04.github.io/plectis-atlas/statements/KMedianRecovery.html

Recorded comparisons with the paper

Atlas records no contributed comparison of this statement with the paper.

Contributed Comparator run reports

Atlas records no contributed report of running Comparator for this statement file.

Compare this statement with the paperShare a Comparator run reportSuggest a change