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Lean statement

Squared-logarithmic randomized $k$-server on arbitrary metrics

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.KServer.main_theorem

Lean statement file: KServer.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 110 link it.

See the candidate theoremWho ran the proof-verification programReport a comparison with the paper (GitHub issue)

Problems reported

  • Problem reported Reported as broken in the same way: its cost and policy definitions are listed as holes, so a trivial replacement makes the checked statement vacuous. Alex Meiburg, Several of the comparator challenges in the OpenAI release are broken and hackable, 2026-10-08 Source

The Lean statement

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

theorem main_theorem : MainStatement.{u} := 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.

  • abbreviation OAI.KServer.Configuration line 13

    abbrev Configuration (k : ℕ) (X : Type u) := Fin k → X

  • definition OAI.KServer.LabelDistribution line 15

    def LabelDistribution (k : ℕ) :=  {probability : Fin k → ℝ //    (∀ label, 0 ≤ probability label) ∧ ∑ label, probability label = 1}

  • abbreviation OAI.KServer.Policy line 19

    abbrev Policy (k : ℕ) (X : Type u) :=  List (X × Fin k) → X → LabelDistribution k

  • definition OAI.KServer.serve line 22

    def serve {k : ℕ} {X : Type u} (s : Configuration k X)    (label : Fin k) (request : X) : Configuration k X :=  Function.update s label request

  • definition OAI.KServer.serviceCost line 26

    def serviceCost {k : ℕ} {X : Type u} [MetricSpace X]    (s : Configuration k X) : (requests : List X) → (Fin requests.length → Fin k) → ℝ  | [], _ => 0  | request :: requests, labels =>      dist (s (labels 0)) request +        serviceCost (serve s (labels 0) request) requests (fun index => labels index.succ)

  • definition OAI.KServer.pathProbability line 33

    def pathProbability {k : ℕ} {X : Type u} (A : Policy k X)    (history : List (X × Fin k)) :    (requests : List X) → (Fin requests.length → Fin k) → ℝ  | [], _ => 1  | request :: requests, labels =>      (A history request).val (labels 0) *        pathProbability A (history ++ [(request, labels 0)]) requests          (fun index => labels index.succ)

  • definition OAI.KServer.expectedCost line 42

    def expectedCost {k : ℕ} {X : Type u} [MetricSpace X]    (A : Policy k X) (s : Configuration k X) (requests : List X) : ℝ :=  ∑ labels : Fin requests.length → Fin k,    pathProbability A [] requests labels * serviceCost s requests labels

  • definition OAI.KServer.optimalCost line 47

    def optimalCost {k : ℕ} {X : Type u} [MetricSpace X]    (s : Configuration k X) (requests : List X) : ℝ :=  sInf (Set.range (serviceCost s requests))

  • definition OAI.KServer.MainStatement line 51

    def MainStatement : Prop :=  ∃ C : ℝ, 0 < C ∧ ∀ (k : ℕ), 2 ≤ k →    ∀ (X : Type u) (_ : MetricSpace X),      (∃ embedding : Fin (k + 1) → X, Function.Injective embedding) →      ∀ s : Configuration k X,        ∃ A : Policy k X, ∃ B : ℝ,          0 ≤ B ∧ (Function.Injective s → B = 0) ∧          ∀ requests : List X,            expectedCost A s requests ≤              C * (Real.log (k + 1)) ^ 2 * optimalCost s requests + B

Definitions taken from OpenAI’s library 9

Comparator compares only the name and type of each definition listed here. When a theorem is configured, its proof uses the definition in OpenAI’s solution library. Read that definition as well as the one in this file when comparing the statement with the paper.

  • Comparator checks the name and type of Configuration; its definition is in OpenAI’s library, in the modules that OAI.Combinatorics.KServer.Main imports.
  • Comparator checks the name and type of LabelDistribution; its definition is in OpenAI’s library, in the modules that OAI.Combinatorics.KServer.Main imports.
  • Comparator checks the name and type of Policy; its definition is in OpenAI’s library, in the modules that OAI.Combinatorics.KServer.Main imports.
  • Comparator checks the name and type of serve; its definition is in OpenAI’s library, in the modules that OAI.Combinatorics.KServer.Main imports.
  • Comparator checks the name and type of serviceCost; its definition is in OpenAI’s library, in the modules that OAI.Combinatorics.KServer.Main imports.
  • Comparator checks the name and type of pathProbability; its definition is in OpenAI’s library, in the modules that OAI.Combinatorics.KServer.Main imports.
  • Comparator checks the name and type of expectedCost; its definition is in OpenAI’s library, in the modules that OAI.Combinatorics.KServer.Main imports.
  • Comparator checks the name and type of optimalCost; its definition is in OpenAI’s library, in the modules that OAI.Combinatorics.KServer.Main imports.
  • Comparator checks the name and type of MainStatement; its definition is in OpenAI’s library, in the modules that OAI.Combinatorics.KServer.Main imports.

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: space, length, policy, serves, integer, request, metric, log and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sequence, randomized, points, least, one, initial and finite.

Statements in Squared-logarithmic randomized k-server on arbitrary metrics come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of Squared-logarithmic randomized k-server on arbitrary metrics

Theorem

There is an absolute constant C<\infty such that, for every integer k\ge2, every metric space (X,d) with at least k+1 points, and every s\in X^k, there exist a randomized online policy A and a finite B\ge0 satisfying

\E\,\cost_{A,s}(\sigma) \le C\bigl(\log(k+1)\bigr)^2\OPT_{X,d,k,s}(\sigma)+B

for every finite request sequence \sigma. The constant C is independent of every property of the metric. The number B may depend on (X,d),k,s, but not on the sequence or its length. The same policy serves all finite sequences and horizons. If the initial positions are distinct, one may take B=0.

Section 1, Introduction Read it in the paper TeX source, line 21
Show the TeX
There is an absolute constant \(C<\infty\) such that, for every
integer \(k\ge2\), every metric space \((X,d)\) with at least \(k+1\)
points, and every \(s\in X^k\), there exist a randomized online policy
\(A\) and a finite \(B\ge0\) satisfying
\begin{equation}
 \E\,\cost_{A,s}(\sigma)
 \le C\bigl(\log(k+1)\bigr)^2\OPT_{X,d,k,s}(\sigma)+B
\end{equation}
for every finite request sequence \(\sigma\).
The constant \(C\) is independent of every property of the metric.
The number \(B\) may depend on \((X,d),k,s\), but not on the sequence
or its length. The same policy serves all finite sequences and
horizons. If the initial positions are distinct, one may take \(B=0\).

Other possible paper matches, closest wording first

  • Proposition 9.2 of Squared-logarithmic randomized k-server on arbitrary metrics. Words of the Lean statement, its names split into words, that also appear in the theorem: configurations, expected, integer, request, metric and cost. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: containing and finite.
  • Theorem 2.1 of Uniform computation of the squared-logarithmic k-server bound. Words of the Lean statement, its names split into words, that also appear in the theorem: space, policy, serves, integer, request, metric and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: squared, points, server, randomized, one, least and two more.
  • Proposition 2.2 of Squared-logarithmic randomized k-server on arbitrary metrics. Words of the Lean statement, its names split into words, that also appear in the theorem: probability, policy, metric and cost. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite.
  • Proposition 3.4 of Uniform computation of the squared-logarithmic k-server bound. Words of the Lean statement, its names split into words, that also appear in the theorem: index, length, expected and cost. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unbounded, arbitrary, sequence and finite.
  • Theorem 1.1 of Uniform computation of the squared-logarithmic k-server bound. Words of the Lean statement, its names split into words, that also appear in the theorem: length, serves, history, request, metric, log and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sequence, server, randomized, one, initial and finite.
  • Proposition 8.2 of Squared-logarithmic randomized k-server on arbitrary metrics. Words of the Lean statement, its names split into words, that also appear in the theorem: expected and cost. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

  • Accepted Their term: “accepted”.

    Re-run through Comparator with Lean’s kernel: accepted (the check took 624 s; peak memory 6.4 GiB).

    Reported 11 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.

Definitions compared by name and type only (9): Configuration, LabelDistribution, Policy, serve, serviceCost, pathProbability, expectedCost, optimalCost, MainStatement. Their bodies come from OpenAI’s solution library.

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 110’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: Both the arbitrary-metric O(log^2 (k+1)) bound and the uniform polynomial-time implementation on finite rational metrics (2 <= k < n) are stated; the matching lower bound and any polynomial bound on the additive constant are not (docs: constant not claimed polynomially bounded).

    Definitions the group asks readers to compare with the paper: Hand-rolled bit-operation model for the uniform claim: BitMachine/BitAction/MachineState (read-only input tape, tapes work tapes over Option Bool, emit, yield), run for exactly requestBudget c e L t := c * (L + Nat.clog 2 (t+1) + 1)^e coin flips per request with a deterministic boot of polynomialBudget steps; not Mathlib's TM model. KServer's definition_names list 9 definitions that the solution must reproduce exactly

  • 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 proves an $O((\log(k+1))^2)$ competitive ratio for randomized $k$-server against oblivious finite request sequences. For every $k\ge2$, every metric space containing at least $k+1$ points, and every initial configuration, one policy works for all request sequences, including in infinite and unbounded spaces.”

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

69 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: 19 to its own declarations, 30 to Mathlib’s documentation. 8 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 OAInoncomputable sectionopen scoped BigOperatorsnamespace KServeruniverse uabbrev Configuration (k : ℕ) (X : Type u) := Fin k → Xdef LabelDistribution (k : ℕ) :=  {probability : Fin k → ℝ //    (∀ label, 0 ≤ probability label) ∧ ∑ label, probability label = 1}abbrev Policy (k : ℕ) (X : Type u) :=  List (X × Fin k) → X → LabelDistribution kdef serve {k : ℕ} {X : Type u} (s : Configuration k X)    (label : Fin k) (request : X) : Configuration k X :=  Function.update s label requestdef serviceCost {k : ℕ} {X : Type u} [MetricSpace X]    (s : Configuration k X) : (requests : List X) → (Fin requests.length → Fin k) → ℝ  | [], _ => 0  | request :: requests, labels =>      dist (s (labels 0)) request +        serviceCost (serve s (labels 0) request) requests (fun index => labels index.succ)def pathProbability {k : ℕ} {X : Type u} (A : Policy k X)    (history : List (X × Fin k)) :    (requests : List X) → (Fin requests.length → Fin k) → ℝ  | [], _ => 1  | request :: requests, labels =>      (A history request).val (labels 0) *        pathProbability A (history ++ [(request, labels 0)]) requests          (fun index => labels index.succ)def expectedCost {k : ℕ} {X : Type u} [MetricSpace X]    (A : Policy k X) (s : Configuration k X) (requests : List X) : ℝ :=  ∑ labels : Fin requests.length → Fin k,    pathProbability A [] requests labels * serviceCost s requests labelsdef optimalCost {k : ℕ} {X : Type u} [MetricSpace X]    (s : Configuration k X) (requests : List X) : ℝ :=  sInf (Set.range (serviceCost s requests))def MainStatement : Prop :=  ∃ C : ℝ, 0 < C ∧ ∀ (k : ℕ), 2 ≤ k →    ∀ (X : Type u) (_ : MetricSpace X),      (∃ embedding : Fin (k + 1) → X, Function.Injective embedding) →      ∀ s : Configuration k X,        ∃ A : Policy k X, ∃ B : ℝ,          0 ≤ B ∧ (Function.Injective s → B = 0) ∧          ∀ requests : List X,            expectedCost A s requests ≤              C * (Real.log (k + 1)) ^ 2 * optimalCost s requests + Btheorem main_theorem : MainStatement.{u} := by  sorryend KServerendend 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.

Squared-logarithmic randomized k-server on arbitrary metrics

Theorem

There is an absolute constant C<\infty such that, for every integer k\ge2, every metric space (X,d) with at least k+1 points, and every s\in X^k, there exist a randomized online policy A and a finite B\ge0 satisfying

\E\,\cost_{A,s}(\sigma) \le C\bigl(\log(k+1)\bigr)^2\OPT_{X,d,k,s}(\sigma)+B

for every finite request sequence \sigma. The constant C is independent of every property of the metric. The number B may depend on (X,d),k,s, but not on the sequence or its length. The same policy serves all finite sequences and horizons. If the initial positions are distinct, one may take B=0.

Section 1, Introduction Read it in the paper TeX source, line 21
Show the TeX
There is an absolute constant \(C<\infty\) such that, for every
integer \(k\ge2\), every metric space \((X,d)\) with at least \(k+1\)
points, and every \(s\in X^k\), there exist a randomized online policy
\(A\) and a finite \(B\ge0\) satisfying
\begin{equation}
 \E\,\cost_{A,s}(\sigma)
 \le C\bigl(\log(k+1)\bigr)^2\OPT_{X,d,k,s}(\sigma)+B
\end{equation}
for every finite request sequence \(\sigma\).
The constant \(C\) is independent of every property of the metric.
The number \(B\) may depend on \((X,d),k,s\), but not on the sequence
or its length. The same policy serves all finite sequences and
horizons. If the initial positions are distinct, one may take \(B=0\).

Uniform computation of the squared-logarithmic k-server bound

Theorem

There are an absolute constant C and a polynomial p with the following property. Let n\ge3, 2\le k<n, let d be a rational metric on X=\{1,\ldots,n\}, and let s\in X^k be an initial labeled tuple, with repetitions allowed. Write L for the total binary encoding length of n,k,d,s. A single uniform randomized online algorithm preprocesses these data in at most p(L) bit operations and serves request r_t, for every reachable history and internal state, in at most

p\!\left(L+\left\lceil\log_2(t+1)\right\rceil\right)

additional bit operations. It uses only unbiased random bits and moves exactly one chosen server to each request, permitting a zero-length move. For every fixed finite request sequence \sigma independent of those bits,

\E\cost_{A,s}(\sigma) \le C\bigl(\log(k+1)\bigr)^2\OPT_s(\sigma)+B_{d,k,s},

where B_{d,k,s} is finite and independent of \sigma and its length. The algorithm does not need to know that length in advance.

Section 1, Introduction Read it in the paper TeX source, line 21
Show the TeX
There are an absolute constant \(C\) and a polynomial \(p\) with the following
property. Let \(n\ge3\), \(2\le k<n\), let \(d\) be a rational metric on
\(X=\{1,\ldots,n\}\), and let \(s\in X^k\) be an initial labeled tuple,
with repetitions allowed. Write \(L\) for the total binary encoding length
of \(n,k,d,s\). A single uniform randomized online algorithm preprocesses
these data in at most \(p(L)\) bit operations and serves request \(r_t\),
for every reachable history and internal state, in at most
\[
 p\!\left(L+\left\lceil\log_2(t+1)\right\rceil\right)
\]
additional bit operations. It uses only unbiased random bits and moves
exactly one chosen server to each request, permitting a zero-length move.
For every fixed finite request sequence \(\sigma\) independent of those bits,
\begin{equation}
 \E\cost_{A,s}(\sigma)
 \le C\bigl(\log(k+1)\bigr)^2\OPT_s(\sigma)+B_{d,k,s},
\end{equation}
where \(B_{d,k,s}\) is finite and independent of \(\sigma\) and its length.
The algorithm does not need to know that length in advance.
Configuration and file facts
OpenAI’s Lean catalogue
OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 110 link it.
Solution module
OAI.Combinatorics.KServer.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.

Statement comparison

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

Plectis Atlas

Squared-logarithmic randomized $k$-server on arbitrary metrics

Family 110: Optimal-order randomized k-server on arbitrary metrics

Candidate pairing

Theorem

There is an absolute constant C<\infty such that, for every integer k\ge2, every metric space (X,d) with at least k+1 points, and every s\in X^k, there exist a randomized online policy A and a finite B\ge0 satisfying

\E\,\cost_{A,s}(\sigma) \le C\bigl(\log(k+1)\bigr)^2\OPT_{X,d,k,s}(\sigma)+B

for every finite request sequence \sigma. The constant C is independent of every property of the metric. The number B may depend on (X,d),k,s, but not on the sequence or its length. The same policy serves all finite sequences and horizons. If the initial positions are distinct, one may take B=0.

Squared-logarithmic randomized k-server on arbitrary metrics

TeX label thm:main, build/source/latex/sections/01_introduction.tex, line 21

The Lean statement says

theorem main_theorem : MainStatement.{u} := 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.

Comparator checks only the name and type of 9 definitions, among them Configuration; their bodies come from the solution.

Atlas selected this possible match because the two statements share words. No review of whether they mean the same thing is recorded.three independent groups report that Comparator accepted runs for this file at the release version shown here.Statement review: none recorded

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

Questions to resolve when comparing this statement with the paper

  • 9 definition holes: Comparator compares these by name and type only, so a reader checks what each solution body means.

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