Lean statement

Entropy-rate dimension formula

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.EntropyRateDimension.entropy_rate_dimension

Lean statement file: SelfSimilar.leanShow on the mapFamily 148. 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 entropy_rate_dimension {ι : Type uι} [Fintype ι] [Nonempty ι]    (S : System ι) (μ : Measure ℝ) [IsProbabilityMeasure μ]    (hμ : S.SelfSimilar μ) :    lowerHausdorffDimension μ = ENNReal.ofReal (min 1 (S.entropyRate / S.lyapunov)) := 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.EntropyRateDimension.System line 12

structure System (ι : Type uι) [Fintype ι] where  ratio : ι → ℝ  offset : ι → ℝ  weight : ι → ℝ  ratio_pos : ∀ i, 0 < |ratio i|  ratio_lt_one : ∀ i, |ratio i| < 1  weight_pos : ∀ i, 0 < weight i  weight_sum : ∑ i, weight i = 1

definition OAI.EntropyRateDimension.System.affine line 25

def affine (S : System ι) (i : ι) (x : ℝ) : ℝ := S.ratio i * x + S.offset i

definition OAI.EntropyRateDimension.System.wordAffine line 27

def wordAffine (S : System ι) : List ι → ℝ × ℝ  | [] => (1, 0)  | i :: w => (S.ratio i * (S.wordAffine w).1,      S.ratio i * (S.wordAffine w).2 + S.offset i)

definition OAI.EntropyRateDimension.System.completeMap line 32

def completeMap (S : System ι) {n : ℕ} (w : Fin n → ι) : ℝ × ℝ :=  S.wordAffine (List.ofFn w)

definition OAI.EntropyRateDimension.System.wordWeight line 35

noncomputable def wordWeight (S : System ι) {n : ℕ} (w : Fin n → ι) : ℝ :=  ∏ j, S.weight (w j)

definition OAI.EntropyRateDimension.System.mapSupport line 38

noncomputable def mapSupport (S : System ι) (n : ℕ) : Finset (ℝ × ℝ) := by  classical  exact Finset.univ.image (S.completeMap (n := n))

definition OAI.EntropyRateDimension.System.mapMass line 42

noncomputable def mapMass (S : System ι) (n : ℕ) (g : ℝ × ℝ) : ℝ := by  classical  exact ∑ w ∈ (Finset.univ : Finset (Fin n → ι)).filter (S.completeMap · = g),    S.wordWeight w

definition OAI.EntropyRateDimension.System.walkEntropy line 47

noncomputable def walkEntropy (S : System ι) (n : ℕ) : ℝ :=  (∑ g ∈ S.mapSupport n, Real.negMulLog (S.mapMass n g)) / Real.log 2

definition OAI.EntropyRateDimension.System.entropyRate line 50

noncomputable def entropyRate (S : System ι) : ℝ :=  sInf (Set.range fun n : ℕ => S.walkEntropy (n + 1) / (n + 1 : ℕ))

definition OAI.EntropyRateDimension.System.lyapunov line 53

noncomputable def lyapunov (S : System ι) : ℝ :=  -(∑ i, S.weight i * (Real.log |S.ratio i| / Real.log 2))

definition OAI.EntropyRateDimension.System.SelfSimilar line 56

def SelfSimilar (S : System ι) (μ : Measure ℝ) : Prop :=  μ = ∑ i, ENNReal.ofReal (S.weight i) • μ.map (S.affine i)

definition OAI.EntropyRateDimension.lowerHausdorffDimension line 61

noncomputable def lowerHausdorffDimension (μ : Measure ℝ) : ℝ≥0∞ :=  ⨅ (E : Set ℝ) (_ : MeasurableSet E) (_ : 0 < μ E), dimH E

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: probability, nonempty, similar, measure, self, finite, min and dim. Found in the paper’s title, which its main theorem states: rate, dimension and entropy. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: rw, allowed, chi, positive, mu and overlaps.

Theorem 1.1 of The entropy-rate dimension formula for self-similar measures on the line

Theorem

For every finite nonempty indexed family \varphi_i(x)=r_i x+t_i on \R with 0<|r_i|<1, and every strictly positive probability vector p, its self-similar measure satisfies

\dim_H\mu_{\Phi,p} =\min\left\{1,\frac{h_{\mathrm{RW}}(\Phi,p)}{\chi(\Phi,p)}\right\}.

No separation assumption is required; exact overlaps and repeated generators are allowed.

Section 1, Introduction Read it in the paper TeX source, line 78

Show the TeX

For every finite nonempty indexed family
$\varphi_i(x)=r_i x+t_i$ on $\R$ with $0<|r_i|<1$, and every
strictly positive probability vector $p$, its self-similar measure satisfies
\[
 \dim_H\mu_{\Phi,p}
 =\min\left\{1,\frac{h_{\mathrm{RW}}(\Phi,p)}{\chi(\Phi,p)}\right\}.
\]
No separation assumption is required; exact overlaps and repeated
generators are allowed.

Other possible paper matches, closest wording first

Corollary 6.4 of The entropy-rate dimension formula for self-similar measures on the line. Words of the Lean statement, its names split into words, that also appear in the theorem: maps, log, probability, similar, measure, self and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive, mu and overlaps.
Corollary 6.5 of The entropy-rate dimension formula for self-similar measures on the line. Words of the Lean statement, its names split into words, that also appear in the theorem: maps, log, similar, measure, self, min and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: mu and overlaps.
Corollary 6.2 of The entropy-rate dimension formula for self-similar measures on the line. Words of the Lean statement, its names split into words, that also appear in the theorem: real, probability, nonempty, finite, min and dim. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: chi, positive, mu and overlaps.
Proposition 4.2 of The entropy-rate dimension formula for self-similar measures on the line. Words of the Lean statement, its names split into words, that also appear in the theorem: weight, mass, integer and finite.
Corollary 6.3 of The entropy-rate dimension formula for self-similar measures on the line. Words of the Lean statement, its names split into words, that also appear in the theorem: real, nonempty, finite, min and dim. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: overlaps.

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 148’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: entropy_rate_dimension gives lowerHausdorffDimension μ = ofReal (min 1 (entropyRate / lyapunov)) for every System (signed ratios 0<|r_i|<1, positive weights, arbitrary offsets) and every probability measure with μ = Σ w_i f_i μ, with no separation hypothesis, so exact overlaps are allowed. The two corollaries (homogeneous ratio, attractor formula under no exact overlaps) are stated in SelfSimilarCorollaries.
Definitions the group asks readers to compare with the paper: lowerHausdorffDimension μ := ⨅ E measurable with 0 < μ E, dimH E (the inf-dimension, as the docs say; equals dim_H for the exact-dimensional self-similar measure) ; entropyRate S := sInf (range fun n => walkEntropy (n+1)/(n+1)) with walkEntropy the base-2 entropy of the law of the composed affine map (words with equal composition merged), and lyapunov also in base 2: custom but matches the standard h_RW/chi
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 $\dim_H\mu=\min\{1,h_{\mathrm{RW}}/\chi\}$ for the lower Hausdorff dimension of every finite real self-similar probability measure with positive weights and nonzero contraction ratios of absolute value below one. Here $h_{\mathrm{RW}}$ is the entropy rate of random affine compositions and $\chi$ is the corresponding Lyapunov exponent in the same logarithmic base. Ratios may be signed and unequal, and exact overlaps are allowed.”
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

73 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: 12 to its own declarations, 29 to Mathlib’s documentation. 13 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 entropy_rate_dimension 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 12 other declarations of the file; with the theorem itself they span 39 lines. 5 of them are named in the statement itself.

Names in the namespace OAI.EntropyRateDimension.System are listed without it.

structure OAI.EntropyRateDimension.Systemline 12in the statementlibrary
def affineline 25library
def wordAffineline 27library
def completeMapline 32library
def wordWeightline 35library
def mapSupportline 38library
def mapMassline 42library
def walkEntropyline 47library
def entropyRateline 50in the statementlibrary
def lyapunovline 53in the statementlibrary
def SelfSimilarline 56in the statementlibrary
def OAI.EntropyRateDimension.lowerHausdorffDimensionline 61in the statementlibrary
theorem OAI.EntropyRateDimension.entropy_rate_dimensionline 65selected theoremlibrary

import Mathlibnamespace OAIuniverse uιopen scoped BigOperators ENNRealopen MeasureTheorynamespace EntropyRateDimensionstructure System (ι : Type uι) [Fintype ι] where  ratio : ι → ℝ  offset : ι → ℝ  weight : ι → ℝ  ratio_pos : ∀ i, 0 < |ratio i|  ratio_lt_one : ∀ i, |ratio i| < 1  weight_pos : ∀ i, 0 < weight i  weight_sum : ∑ i, weight i = 1namespace Systemvariable {ι : Type uι} [Fintype ι]def affine (S : System ι) (i : ι) (x : ℝ) : ℝ := S.ratio i * x + S.offset idef wordAffine (S : System ι) : List ι → ℝ × ℝ  | [] => (1, 0)  | i :: w => (S.ratio i * (S.wordAffine w).1,      S.ratio i * (S.wordAffine w).2 + S.offset i)def completeMap (S : System ι) {n : ℕ} (w : Fin n → ι) : ℝ × ℝ :=  S.wordAffine (List.ofFn w)noncomputable def wordWeight (S : System ι) {n : ℕ} (w : Fin n → ι) : ℝ :=  ∏ j, S.weight (w j)noncomputable def mapSupport (S : System ι) (n : ℕ) : Finset (ℝ × ℝ) := by  classical  exact Finset.univ.image (S.completeMap (n := n))noncomputable def mapMass (S : System ι) (n : ℕ) (g : ℝ × ℝ) : ℝ := by  classical  exact ∑ w ∈ (Finset.univ : Finset (Fin n → ι)).filter (S.completeMap · = g),    S.wordWeight wnoncomputable def walkEntropy (S : System ι) (n : ℕ) : ℝ :=  (∑ g ∈ S.mapSupport n, Real.negMulLog (S.mapMass n g)) / Real.log 2noncomputable def entropyRate (S : System ι) : ℝ :=  sInf (Set.range fun n : ℕ => S.walkEntropy (n + 1) / (n + 1 : ℕ))noncomputable def lyapunov (S : System ι) : ℝ :=  -(∑ i, S.weight i * (Real.log |S.ratio i| / Real.log 2))def SelfSimilar (S : System ι) (μ : Measure ℝ) : Prop :=  μ = ∑ i, ENNReal.ofReal (S.weight i) • μ.map (S.affine i)end Systemnoncomputable def lowerHausdorffDimension (μ : Measure ℝ) : ℝ≥0∞ :=  ⨅ (E : Set ℝ) (_ : MeasurableSet E) (_ : 0 < μ E), dimH Eopen MeasureTheorytheorem entropy_rate_dimension {ι : Type uι} [Fintype ι] [Nonempty ι]    (S : System ι) (μ : Measure ℝ) [IsProbabilityMeasure μ]    (hμ : S.SelfSimilar μ) :    lowerHausdorffDimension μ = ENNReal.ofReal (min 1 (S.entropyRate / S.lyapunov)) := by  sorryend EntropyRateDimensionend 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.

The entropy-rate dimension formula for self-similar measures on the line

Theorem

For every finite nonempty indexed family \varphi_i(x)=r_i x+t_i on \R with 0<|r_i|<1, and every strictly positive probability vector p, its self-similar measure satisfies

\dim_H\mu_{\Phi,p} =\min\left\{1,\frac{h_{\mathrm{RW}}(\Phi,p)}{\chi(\Phi,p)}\right\}.

No separation assumption is required; exact overlaps and repeated generators are allowed.

Section 1, Introduction Read it in the paper TeX source, line 78

Show the TeX

For every finite nonempty indexed family
$\varphi_i(x)=r_i x+t_i$ on $\R$ with $0<|r_i|<1$, and every
strictly positive probability vector $p$, its self-similar measure satisfies
\[
 \dim_H\mu_{\Phi,p}
 =\min\left\{1,\frac{h_{\mathrm{RW}}(\Phi,p)}{\chi(\Phi,p)}\right\}.
\]
No separation assumption is required; exact overlaps and repeated
generators are allowed.

Configuration and file facts

File: lean/ComparatorChallenges/SelfSimilar.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/MeasureTheory/SelfSimilar/Main.lean
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: 12 local vocabulary declarations
Lines of Lean, without comments or blank lines: 39
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