Lean statement

Homogeneous depth-five bounds for iterated matrix multiplication

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.Problem335.main

Lean statement file: DepthFive.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 135 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 main :    (∃ threshold : ℕ, ∀ size : ℕ, threshold ≤ size →      ∀ circuit : Depth5Circuit ℂ size, circuitValue circuit = imm ℂ size →        (size : ℝ) ^ (Real.sqrt (size : ℝ) / 400) ≤ (circuitSize circuit : ℝ)) ∧    (∃ threshold : ℕ, ∀ (FieldType : Type u) [Field FieldType] [CharZero FieldType],      ∀ size : ℕ, threshold ≤ size → ∀ circuit : Depth5Circuit FieldType size,        circuitValue circuit = imm FieldType size →          (size : ℝ) ^ (Real.sqrt (size : ℝ) / 400) ≤ (circuitSize circuit : ℝ)) ∧    (∀ (FieldType : Type u) [Field FieldType] (size : ℕ), 2 ≤ size →      ∃ circuit : Depth5Circuit FieldType size, circuitValue circuit = imm FieldType size ∧        circuitSize circuit ≤ upperGateBound size ∧        (upperGateBound size : ℝ) ≤ (size : ℝ) ^ (Real.sqrt (size : ℝ) + 4)) := 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.

inductive type OAI.Problem335.D5Leaf line 12

inductive D5Leaf (K : Type u) (n : ℕ) where  | scalar : K → D5Leaf K n  | variable : (Fin n × Fin n × Fin n) → D5Leaf K n

definition OAI.Problem335.d5LeafDegree line 16

def d5LeafDegree {K : Type u} {n : ℕ} : D5Leaf K n → ℕ  | D5Leaf.scalar _ => 0  | D5Leaf.variable _ => 1

structure OAI.Problem335.Depth5Circuit line 20

structure Depth5Circuit (K : Type u) [CommSemiring K] (n : ℕ) where

Show the remaining 28 lines

  leafCount : ℕ  leaves : Fin leafCount → D5Leaf K n  bottomCount : ℕ  bottomInputs : Fin bottomCount → List (K × Fin leafCount)  bottomDegree : Fin bottomCount → ℕ  bottomHomogeneous :    ∀ i entry, entry ∈ bottomInputs i → d5LeafDegree (leaves entry.2) = bottomDegree i  bottomEmpty :    ∀ i, bottomInputs i = [] → bottomDegree i = 0  lowerCount : ℕ  lowerInputs : Fin lowerCount → List (Fin bottomCount)  middleCount : ℕ  middleInputs : Fin middleCount → List (K × Fin lowerCount)  middleDegree : Fin middleCount → ℕ  middleHomogeneous :    ∀ i entry, entry ∈ middleInputs i →      ((lowerInputs entry.2).map bottomDegree).sum = middleDegree i  middleEmpty :    ∀ i, middleInputs i = [] → middleDegree i = 0  upperCount : ℕ  upperInputs : Fin upperCount → List (Fin middleCount)  outputInputs : List (K × Fin upperCount)  outputDegree : ℕ  outputHomogeneous :    ∀ entry, entry ∈ outputInputs →      ((upperInputs entry.2).map middleDegree).sum = outputDegree  outputEmpty :    outputInputs = [] → outputDegree = 0

definition OAI.Problem335.d5LeafValue line 50

def d5LeafValue {K : Type u} [CommSemiring K] {n : ℕ}    (x : D5Leaf K n) : MvPolynomial (Fin n × Fin n × Fin n) K :=  match x with  | D5Leaf.scalar a => MvPolynomial.C a  | D5Leaf.variable i => MvPolynomial.X i

definition OAI.Problem335.bottomValue line 56

def bottomValue {K : Type u} [CommSemiring K] {n : ℕ}    (c : Depth5Circuit K n) (i : Fin c.bottomCount) :    MvPolynomial (Fin n × Fin n × Fin n) K := by  classical  exact ((c.bottomInputs i).map (fun entry =>    MvPolynomial.C entry.1 * d5LeafValue (c.leaves entry.2))).sum

definition OAI.Problem335.lowerValue line 63

def lowerValue {K : Type u} [CommSemiring K] {n : ℕ}    (c : Depth5Circuit K n) (i : Fin c.lowerCount) :    MvPolynomial (Fin n × Fin n × Fin n) K := by  classical  exact ((c.lowerInputs i).map (fun j => bottomValue c j)).prod

definition OAI.Problem335.middleValue line 69

def middleValue {K : Type u} [CommSemiring K] {n : ℕ}    (c : Depth5Circuit K n) (i : Fin c.middleCount) :    MvPolynomial (Fin n × Fin n × Fin n) K := by  classical  exact ((c.middleInputs i).map (fun entry =>    MvPolynomial.C entry.1 * lowerValue c entry.2)).sum

definition OAI.Problem335.upperValue line 76

def upperValue {K : Type u} [CommSemiring K] {n : ℕ}    (c : Depth5Circuit K n) (i : Fin c.upperCount) :    MvPolynomial (Fin n × Fin n × Fin n) K := by  classical  exact ((c.upperInputs i).map (fun j => middleValue c j)).prod

definition OAI.Problem335.circuitValue line 82

def circuitValue {K : Type u} [CommSemiring K] {n : ℕ}    (c : Depth5Circuit K n) : MvPolynomial (Fin n × Fin n × Fin n) K := by  classical  exact (c.outputInputs.map (fun entry =>    MvPolynomial.C entry.1 * upperValue c entry.2)).sum

definition OAI.Problem335.circuitSize line 88

def circuitSize {K : Type u} [CommSemiring K] {n : ℕ}    (c : Depth5Circuit K n) : ℕ :=  c.leafCount + c.bottomCount + c.lowerCount + c.middleCount + c.upperCount + 1

definition OAI.Problem335.immLayer line 92

def immLayer (K : Type u) [CommSemiring K] (n : ℕ) (t : Fin n) :    Matrix (Fin n) (Fin n) (MvPolynomial (Fin n × Fin n × Fin n) K) :=  fun i j => MvPolynomial.X (t, i, j)

definition OAI.Problem335.imm line 96

def imm (K : Type u) [CommSemiring K] (n : ℕ) :    MvPolynomial (Fin n × Fin n × Fin n) K := by  classical  by_cases h : 0 < n  · let matrices :        List (Matrix (Fin n) (Fin n)          (MvPolynomial (Fin n × Fin n × Fin n) K)) :=      List.ofFn (fun t : Fin n => immLayer K n t)    exact matrices.prod ⟨0, h⟩ ⟨0, h⟩  · exact 0

definition OAI.Problem335.ceilSqrt line 107

def ceilSqrt (n : ℕ) : ℕ :=  Nat.sqrt n + if (Nat.sqrt n) ^ 2 = n then 0 else 1

definition OAI.Problem335.ceilDiv line 110

def ceilDiv (n t : ℕ) : ℕ :=  (n + t - 1) / t

definition OAI.Problem335.upperGateBound line 113

def upperGateBound (n : ℕ) : ℕ :=  let t := ceilSqrt n  let r := ceilDiv n t  2 * n ^ 3 + r * n ^ 2 + r * n ^ (t + 1) + n ^ (r - 1) + 1

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: computing, homogeneous, least and syntactically.

Theorem 1.1 of Homogeneous depth-five lower bounds for iterated matrix multiplication

Theorem

There exist absolute constants c>0 and n_0 such that, for every integer n\geq n_0, every syntactically homogeneous \Sigma\Pi\Sigma\Pi\Sigma circuit over \C computing \IMM_{n,n} has size at least

n^{c\sqrt n}.

One may take c=1/400.

Section 1, Introduction Read it in the paper TeX source, line 53

Show the TeX

There exist absolute constants $c>0$ and $n_0$ such that, for every integer
$n\geq n_0$, every syntactically homogeneous
$\Sigma\Pi\Sigma\Pi\Sigma$ circuit over $\C$ computing $\IMM_{n,n}$
has size at least
\[
 n^{c\sqrt n}.
\]
One may take $c=1/400$.

Other possible paper matches, closest wording first

Corollary 6.1 of Homogeneous depth-five lower bounds for iterated matrix multiplication. Words of the Lean statement, its names split into words, that also appear in the theorem: bottom, zero, integer, field, homogeneous, circuit and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: n/400, characteristic, fan, support, least, gates and four more.
Proposition 3.1 of Homogeneous depth-five lower bounds for iterated matrix multiplication. Words of the Lean statement, its names split into words, that also appear in the theorem: polynomial, degree, layers, size, matrix, field and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sigma, computed, pi and syntactically.
Proposition 6.2 of Homogeneous depth-five lower bounds for iterated matrix multiplication. Words of the Lean statement, its names split into words, that also appear in the theorem: leaves, field, homogeneous, circuit, imm and sqrt. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: gates, computing, sigma, pi and syntactically.
Proposition 4.1 of Homogeneous depth-five lower bounds for iterated matrix multiplication. Words of the Lean statement, its names split into words, that also appear in the theorem: imm and sqrt. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: large and sufficiently.

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 135’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
Definitions the group asks readers to compare with the paper: Depth5Circuit is a bespoke structure (leaf/bottom/lower/middle/upper/output arrays with bottomHomogeneous, middleHomogeneous, outputHomogeneous degree-equality fields) and circuitSize counts all nodes; homogeneity is syntactic and the model charges nodes, not wires
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 that, over every characteristic-zero field, syntactically homogeneous depth-five $\Sigma\Pi\Sigma\Pi\Sigma$ circuits computing this polynomial require at least $n^{\sqrt n/400}$ gates for all sufficiently large $n$, with a threshold independent of the field. Arbitrary bottom support, finite fan-in and fan-out, and sharing 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

136 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: 38 to its own declarations, 94 to Mathlib’s documentation. 16 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 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 15 other declarations of the file; with the theorem itself they span 104 lines. 5 of them are named in the statement itself.

Names in the namespace OAI.Problem335 are listed without it.

inductive D5Leafline 12library
def d5LeafDegreeline 16library
structure Depth5Circuitline 20in the statementlibrary
def d5LeafValueline 50library
def bottomValueline 56library
def lowerValueline 63library
def middleValueline 69library
def upperValueline 76library
def circuitValueline 82in the statementlibrary
def circuitSizeline 88in the statementlibrary
def immLayerline 92library
def immline 96in the statementlibrary
def ceilSqrtline 107library
def ceilDivline 110library
def upperGateBoundline 113in the statementlibrary
theorem mainline 118selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped BigOperatorsuniverse unamespace Problem335inductive D5Leaf (K : Type u) (n : ℕ) where  | scalar : K → D5Leaf K n  | variable : (Fin n × Fin n × Fin n) → D5Leaf K ndef d5LeafDegree {K : Type u} {n : ℕ} : D5Leaf K n → ℕ  | D5Leaf.scalar _ => 0  | D5Leaf.variable _ => 1structure Depth5Circuit (K : Type u) [CommSemiring K] (n : ℕ) where  leafCount : ℕ  leaves : Fin leafCount → D5Leaf K n  bottomCount : ℕ  bottomInputs : Fin bottomCount → List (K × Fin leafCount)  bottomDegree : Fin bottomCount → ℕ  bottomHomogeneous :    ∀ i entry, entry ∈ bottomInputs i → d5LeafDegree (leaves entry.2) = bottomDegree i  bottomEmpty :    ∀ i, bottomInputs i = [] → bottomDegree i = 0  lowerCount : ℕ  lowerInputs : Fin lowerCount → List (Fin bottomCount)  middleCount : ℕ  middleInputs : Fin middleCount → List (K × Fin lowerCount)  middleDegree : Fin middleCount → ℕ  middleHomogeneous :    ∀ i entry, entry ∈ middleInputs i →      ((lowerInputs entry.2).map bottomDegree).sum = middleDegree i  middleEmpty :    ∀ i, middleInputs i = [] → middleDegree i = 0  upperCount : ℕ  upperInputs : Fin upperCount → List (Fin middleCount)  outputInputs : List (K × Fin upperCount)  outputDegree : ℕ  outputHomogeneous :    ∀ entry, entry ∈ outputInputs →      ((upperInputs entry.2).map middleDegree).sum = outputDegree  outputEmpty :    outputInputs = [] → outputDegree = 0def d5LeafValue {K : Type u} [CommSemiring K] {n : ℕ}    (x : D5Leaf K n) : MvPolynomial (Fin n × Fin n × Fin n) K :=  match x with  | D5Leaf.scalar a => MvPolynomial.C a  | D5Leaf.variable i => MvPolynomial.X idef bottomValue {K : Type u} [CommSemiring K] {n : ℕ}    (c : Depth5Circuit K n) (i : Fin c.bottomCount) :    MvPolynomial (Fin n × Fin n × Fin n) K := by  classical  exact ((c.bottomInputs i).map (fun entry =>    MvPolynomial.C entry.1 * d5LeafValue (c.leaves entry.2))).sumdef lowerValue {K : Type u} [CommSemiring K] {n : ℕ}    (c : Depth5Circuit K n) (i : Fin c.lowerCount) :    MvPolynomial (Fin n × Fin n × Fin n) K := by  classical  exact ((c.lowerInputs i).map (fun j => bottomValue c j)).proddef middleValue {K : Type u} [CommSemiring K] {n : ℕ}    (c : Depth5Circuit K n) (i : Fin c.middleCount) :    MvPolynomial (Fin n × Fin n × Fin n) K := by  classical  exact ((c.middleInputs i).map (fun entry =>    MvPolynomial.C entry.1 * lowerValue c entry.2)).sumdef upperValue {K : Type u} [CommSemiring K] {n : ℕ}    (c : Depth5Circuit K n) (i : Fin c.upperCount) :    MvPolynomial (Fin n × Fin n × Fin n) K := by  classical  exact ((c.upperInputs i).map (fun j => middleValue c j)).proddef circuitValue {K : Type u} [CommSemiring K] {n : ℕ}    (c : Depth5Circuit K n) : MvPolynomial (Fin n × Fin n × Fin n) K := by  classical  exact (c.outputInputs.map (fun entry =>    MvPolynomial.C entry.1 * upperValue c entry.2)).sumdef circuitSize {K : Type u} [CommSemiring K] {n : ℕ}    (c : Depth5Circuit K n) : ℕ :=  c.leafCount + c.bottomCount + c.lowerCount + c.middleCount + c.upperCount + 1def immLayer (K : Type u) [CommSemiring K] (n : ℕ) (t : Fin n) :    Matrix (Fin n) (Fin n) (MvPolynomial (Fin n × Fin n × Fin n) K) :=  fun i j => MvPolynomial.X (t, i, j)def imm (K : Type u) [CommSemiring K] (n : ℕ) :    MvPolynomial (Fin n × Fin n × Fin n) K := by  classical  by_cases h : 0 < n  · let matrices :        List (Matrix (Fin n) (Fin n)          (MvPolynomial (Fin n × Fin n × Fin n) K)) :=      List.ofFn (fun t : Fin n => immLayer K n t)    exact matrices.prod ⟨0, h⟩ ⟨0, h⟩  · exact 0def ceilSqrt (n : ℕ) : ℕ :=  Nat.sqrt n + if (Nat.sqrt n) ^ 2 = n then 0 else 1def ceilDiv (n t : ℕ) : ℕ :=  (n + t - 1) / tdef upperGateBound (n : ℕ) : ℕ :=  let t := ceilSqrt n  let r := ceilDiv n t  2 * n ^ 3 + r * n ^ 2 + r * n ^ (t + 1) + n ^ (r - 1) + 1theorem main :    (∃ threshold : ℕ, ∀ size : ℕ, threshold ≤ size →      ∀ circuit : Depth5Circuit ℂ size, circuitValue circuit = imm ℂ size →        (size : ℝ) ^ (Real.sqrt (size : ℝ) / 400) ≤ (circuitSize circuit : ℝ)) ∧    (∃ threshold : ℕ, ∀ (FieldType : Type u) [Field FieldType] [CharZero FieldType],      ∀ size : ℕ, threshold ≤ size → ∀ circuit : Depth5Circuit FieldType size,        circuitValue circuit = imm FieldType size →          (size : ℝ) ^ (Real.sqrt (size : ℝ) / 400) ≤ (circuitSize circuit : ℝ)) ∧    (∀ (FieldType : Type u) [Field FieldType] (size : ℕ), 2 ≤ size →      ∃ circuit : Depth5Circuit FieldType size, circuitValue circuit = imm FieldType size ∧        circuitSize circuit ≤ upperGateBound size ∧        (upperGateBound size : ℝ) ≤ (size : ℝ) ^ (Real.sqrt (size : ℝ) + 4)) := by  sorryend Problem335endend 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.

Homogeneous depth-five lower bounds for iterated matrix multiplication

Theorem

There exist absolute constants c>0 and n_0 such that, for every integer n\geq n_0, every syntactically homogeneous \Sigma\Pi\Sigma\Pi\Sigma circuit over \C computing \IMM_{n,n} has size at least

n^{c\sqrt n}.

One may take c=1/400.

Section 1, Introduction Read it in the paper TeX source, line 53

Show the TeX

There exist absolute constants $c>0$ and $n_0$ such that, for every integer
$n\geq n_0$, every syntactically homogeneous
$\Sigma\Pi\Sigma\Pi\Sigma$ circuit over $\C$ computing $\IMM_{n,n}$
has size at least
\[
 n^{c\sqrt n}.
\]
One may take $c=1/400$.

Configuration and file facts

File: lean/ComparatorChallenges/DepthFive.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 135 link it.
Solution module: OAI.Algebra.DepthFive.Bounds
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: 15 local vocabulary declarations
Lines of Lean, without comments or blank lines: 104
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