Lean statement

Subpolynomial query complexity for log-concave sampling

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Theorem selected for verification: OAI.LogConcaveSampling.exact_source_main

Lean statement file: LogConcaveQuery.leanShow on the mapFamily 139. OpenAI lists this as a main result in its Lean catalogue.

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 exact_source_main :    (∀ ε : ℝ, 0 < ε → ∃ C : ℝ, ∀ d : ℕ, 2 ≤ d →      (queryComplexity d).toENNReal ≤ ENNReal.ofReal (C * (d : ℝ) ^ ε)) ∧    (∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, 2 ≤ d₀ ∧ ∀ d : ℕ, d₀ ≤ d →      ENNReal.ofReal (c * Real.log (d : ℝ)) ≤ (queryComplexity d).toENNReal) ∧    gammaStar = 0 := by  sorry

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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.LogConcaveSampling.Point line 12

abbrev Point (d : ℕ) := EuclideanSpace ℝ (Fin d)

abbreviation OAI.LogConcaveSampling.Reply line 13
```
abbrev Reply (d : ℕ) := ℝ × Point d
```

abbreviation OAI.LogConcaveSampling.Transcript line 14

abbrev Transcript (d q : ℕ) := Fin q → Reply d

structure OAI.LogConcaveSampling.Admissible line 17

structure Admissible {d : ℕ} (V : Point d → ℝ) : Prop where  smooth : ContDiff ℝ 2 V  value_zero : V 0 = 0  gradient_zero : gradient V 0 = 0  hessian_bounds : ∀ x v : Point d,    ‖v‖ ^ 2 ≤ inner ℝ v (fderiv ℝ (gradient V) x v) ∧    inner ℝ v (fderiv ℝ (gradient V) x v) ≤ 2 * ‖v‖ ^ 2

definition OAI.LogConcaveSampling.gibbsDensity line 26

def gibbsDensity {d : ℕ} (V : Point d → ℝ) (x : Point d) : ℝ≥0∞ :=  ENNReal.ofReal (Real.exp (-V x))

definition OAI.LogConcaveSampling.partition line 29

def partition {d : ℕ} (V : Point d → ℝ) : ℝ≥0∞ :=  ∫⁻ x, gibbsDensity V x ∂volume

definition OAI.LogConcaveSampling.gibbs line 33

def gibbs {d : ℕ} (V : Point d → ℝ) : Measure (Point d) :=  (partition V)⁻¹ • volume.withDensity (gibbsDensity V)

definition OAI.LogConcaveSampling.TVAtMost line 36

def TVAtMost {E : Type*} [MeasurableSpace E]    (μ ν : Measure E) (ε : ℝ) : Prop :=  ∀ s : Set E, MeasurableSet s → |μ.real s - ν.real s| ≤ ε

definition OAI.LogConcaveSampling.firstOrderReply line 40

def firstOrderReply {d : ℕ} (V : Point d → ℝ) (x : Point d) : Reply d :=  (V x, gradient V x)

structure OAI.LogConcaveSampling.OracleAlgorithm line 44

structure OracleAlgorithm (Ω : Type*) [MeasurableSpace Ω] (d q : ℕ) where  query : Fin q → Ω × Transcript d q → Point d  query_measurable : ∀ i, Measurable (query i)  output : Ω × Transcript d q → Point d  output_measurable : Measurable output

definition OAI.LogConcaveSampling.OracleAlgorithm.history line 50

def OracleAlgorithm.history {Ω : Type*} [MeasurableSpace Ω] {d q : ℕ}    (A : OracleAlgorithm Ω d q) (V : Point d → ℝ) (ω : Ω) : ℕ → Transcript d q  | 0 => 0  | n + 1 =>      let h := A.history V ω n      if hn : n < q then        Function.update h ⟨n, hn⟩ (firstOrderReply V (A.query ⟨n, hn⟩ (ω, h)))      else h

definition OAI.LogConcaveSampling.OracleAlgorithm.run line 59

def OracleAlgorithm.run {Ω : Type*} [MeasurableSpace Ω] {d q : ℕ}    (A : OracleAlgorithm Ω d q) (V : Point d → ℝ) (ω : Ω) : Point d :=  A.output (ω, A.history V ω q)

definition OAI.LogConcaveSampling.CanSample line 64

def CanSample (d q : ℕ) : Prop :=  ∃ (Ω : Type) (_ : MeasurableSpace Ω) (μ : Measure Ω)    (_ : IsProbabilityMeasure μ) (A : OracleAlgorithm Ω d q),    ∀ V : Point d → ℝ, Admissible V →      Measurable (A.run V) ∧ TVAtMost (μ.map (A.run V)) (gibbs V) (1 / 10)

definition OAI.LogConcaveSampling.queryComplexity line 71

def queryComplexity (d : ℕ) : ℕ∞ :=  sInf ((fun q : ℕ => (q : ℕ∞)) '' {q | CanSample d q})

definition OAI.LogConcaveSampling.ExponentBound line 75

def ExponentBound (γ : ℝ) : Prop :=  0 ≤ γ ∧ ∀ ε : ℝ, 0 < ε → ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d →    (queryComplexity d).toENNReal ≤ ENNReal.ofReal ((d : ℝ) ^ (γ + ε))

definition OAI.LogConcaveSampling.gammaStar line 79

def gammaStar : ℝ := sInf {γ : ℝ | ExponentBound γ}

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: adaptive, algorithms, bound and randomized.

Theorem 1.1 of Subpolynomial query complexity for well-conditioned log-concave sampling

Theorem

In the oracle model above, for every \varepsilon>0 there is C_\varepsilon<\infty such that

Q(d)\le C_\varepsilon d^\varepsilon \qquad(d\ge2).

There are also absolute constants c>0 and d_0 such that

Q(d)\ge c\log d\qquad(d\ge d_0).

The lower bound applies to arbitrary randomized adaptive first-order algorithms. In particular, \gamma_*=0.

Section 1, Introduction Read it in the paper TeX source, line 51

Show the TeX

In the oracle model above, for every \(\varepsilon>0\)
there is \(C_\varepsilon<\infty\) such that
\[
                  Q(d)\le C_\varepsilon d^\varepsilon
                  \qquad(d\ge2).
\]
There are also absolute constants \(c>0\) and \(d_0\)
such that
\[
                  Q(d)\ge c\log d\qquad(d\ge d_0).
\]
The lower bound applies to arbitrary randomized adaptive
first-order algorithms. In particular, \(\gamma_*=0\).

Other possible paper matches, closest wording first

Theorem 8.5 of Subpolynomial query complexity for well-conditioned log-concave sampling. Words of the Lean statement, its names split into words, that also appear in the theorem: 1/10, gradient, upper, measurable, value, execution and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: varepsilon and subpolynomial.
Theorem 9.1 of Subpolynomial query complexity for well-conditioned log-concave sampling. Words of the Lean statement, its names split into words, that also appear in the theorem: private, randomness, potentials, execution, algorithm, query and two more.
Proposition 8.2 of Subpolynomial query complexity for well-conditioned log-concave sampling. Words of the Lean statement, its names split into words, that also appear in the theorem: run, admissible, total, variation, zero, output and one more.
Theorem 2.1 of Subpolynomial query complexity for well-conditioned log-concave sampling. Words of the Lean statement, its names split into words, that also appear in the theorem: measurable, one, tv, dimension, execution, uniform and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nabla.
Corollary 6.6 of Subpolynomial query complexity for well-conditioned log-concave sampling. Words of the Lean statement, its names split into words, that also appear in the theorem: functions, deterministic, zero, one, dimension, relative and five more.
Theorem 6.7 of Subpolynomial query complexity for well-conditioned log-concave sampling. Words of the Lean statement, its names split into words, that also appear in the theorem: positive, maps, value, deterministic, one, sample and seven more.

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Reported Comparator runs

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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

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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

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Do family 139’s Lean statements express the advertised result?

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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 claim is about the number of exact value-and-gradient queries only (no time/bit-cost statement), as in the summary.
Definitions the group asks readers to compare with the paper: Hand-rolled oracle model (OracleAlgorithm, OracleAlgorithm.history/run, TVAtMost := ∀ s measurable, |μ s - ν s| ≤ ε, queryComplexity := sInf over feasible integer budgets in ℕ∞); non-uniform in d (one algorithm per (d, q))
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

“For potentials with $V(0)=0$, $\nabla V(0)=0$, and $I\le\nabla^2V\le2I$, the least worst-case query budget achieving total-variation error at most $1/10$ is at most $C_\varepsilon d^\varepsilon$ for every $\varepsilon>0$, and at least $c\log d$ eventually. Hence the infimal polynomial exponent is zero.”
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

93 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: 37 to its own declarations, 38 to Mathlib’s documentation. 17 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 exact_source_main 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 16 other declarations of the file; with the theorem itself they span 55 lines. 2 of them are named in the statement itself.

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

abbrev Pointline 12library
abbrev Replyline 13library
abbrev Transcriptline 14library
structure Admissibleline 17library
def gibbsDensityline 26library
def partitionline 29library
def gibbsline 33library
def TVAtMostline 36library
def firstOrderReplyline 40library
structure OracleAlgorithmline 44library
def OracleAlgorithm.historyline 50library
def OracleAlgorithm.runline 59library
def CanSampleline 64library
def queryComplexityline 71in the statementlibrary
def ExponentBoundline 75library
def gammaStarline 79in the statementlibrary
theorem exact_source_mainline 81selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen MeasureTheory Filteropen scoped ENNReal NNReal Topologynamespace LogConcaveSamplingabbrev Point (d : ℕ) := EuclideanSpace ℝ (Fin d)abbrev Reply (d : ℕ) := ℝ × Point dabbrev Transcript (d q : ℕ) := Fin q → Reply d/-- Normalized twice continuously differentiable potentials with Hessian between the identity and twice the identity. -/structure Admissible {d : ℕ} (V : Point d → ℝ) : Prop where  smooth : ContDiff ℝ 2 V  value_zero : V 0 = 0  gradient_zero : gradient V 0 = 0  hessian_bounds : ∀ x v : Point d,    ‖v‖ ^ 2 ≤ inner ℝ v (fderiv ℝ (gradient V) x v) ∧    inner ℝ v (fderiv ℝ (gradient V) x v) ≤ 2 * ‖v‖ ^ 2/-- The unnormalized Gibbs density relative to Euclidean volume. -/def gibbsDensity {d : ℕ} (V : Point d → ℝ) (x : Point d) : ℝ≥0∞ :=  ENNReal.ofReal (Real.exp (-V x))def partition {d : ℕ} (V : Point d → ℝ) : ℝ≥0∞ :=  ∫⁻ x, gibbsDensity V x ∂volume/-- The Gibbs probability law, normalized by its partition function. -/def gibbs {d : ℕ} (V : Point d → ℝ) : Measure (Point d) :=  (partition V)⁻¹ • volume.withDensity (gibbsDensity V)def TVAtMost {E : Type*} [MeasurableSpace E]    (μ ν : Measure E) (ε : ℝ) : Prop :=  ∀ s : Set E, MeasurableSet s → |μ.real s - ν.real s| ≤ εdef firstOrderReply {d : ℕ} (V : Point d → ℝ) (x : Point d) : Reply d :=  (V x, gradient V x)/-- Measurable adaptive queries and output, parametrized by private randomness. -/structure OracleAlgorithm (Ω : Type*) [MeasurableSpace Ω] (d q : ℕ) where  query : Fin q → Ω × Transcript d q → Point d  query_measurable : ∀ i, Measurable (query i)  output : Ω × Transcript d q → Point d  output_measurable : Measurable outputdef OracleAlgorithm.history {Ω : Type*} [MeasurableSpace Ω] {d q : ℕ}    (A : OracleAlgorithm Ω d q) (V : Point d → ℝ) (ω : Ω) : ℕ → Transcript d q  | 0 => 0  | n + 1 =>      let h := A.history V ω n      if hn : n < q then        Function.update h ⟨n, hn⟩ (firstOrderReply V (A.query ⟨n, hn⟩ (ω, h)))      else hdef OracleAlgorithm.run {Ω : Type*} [MeasurableSpace Ω] {d q : ℕ}    (A : OracleAlgorithm Ω d q) (V : Point d → ℝ) (ω : Ω) : Point d :=  A.output (ω, A.history V ω q)/-- A uniform sampler with a deterministic query budget on every execution and total-variation error at most one tenth. -/def CanSample (d q : ℕ) : Prop :=  ∃ (Ω : Type) (_ : MeasurableSpace Ω) (μ : Measure Ω)    (_ : IsProbabilityMeasure μ) (A : OracleAlgorithm Ω d q),    ∀ V : Point d → ℝ, Admissible V →      Measurable (A.run V) ∧ TVAtMost (μ.map (A.run V)) (gibbs V) (1 / 10)/-- The smallest feasible integer query budget, or infinity when no budget is feasible. -/def queryComplexity (d : ℕ) : ℕ∞ :=  sInf ((fun q : ℕ => (q : ℕ∞)) '' {q | CanSample d q})/-- The quantified dimension-exponent upper bound with every positive exponent slack. -/def ExponentBound (γ : ℝ) : Prop :=  0 ≤ γ ∧ ∀ ε : ℝ, 0 < ε → ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d →    (queryComplexity d).toENNReal ≤ ENNReal.ofReal ((d : ℝ) ^ (γ + ε))def gammaStar : ℝ := sInf {γ : ℝ | ExponentBound γ}theorem exact_source_main :    (∀ ε : ℝ, 0 < ε → ∃ C : ℝ, ∀ d : ℕ, 2 ≤ d →      (queryComplexity d).toENNReal ≤ ENNReal.ofReal (C * (d : ℝ) ^ ε)) ∧    (∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, 2 ≤ d₀ ∧ ∀ d : ℕ, d₀ ≤ d →      ENNReal.ofReal (c * Real.log (d : ℝ)) ≤ (queryComplexity d).toENNReal) ∧    gammaStar = 0 := by  sorryend LogConcaveSamplingendend 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.

Subpolynomial query complexity for well-conditioned log-concave sampling