Lean statement

Complex square, dual, and rectangular exponent bounds

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.

Theorems selected for verification: OAI.MatrixMultiplication.complex_omega_le_nine_quarters OAI.MatrixMultiplication.complex_alpha_gt_93_div_200 OAI.MatrixMultiplication.complex_rectangular_omega_lt_523_div_250

Lean statement file: MatrixMultiplication.leanShow on the mapFamily 107. 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 complex_omega_le_nine_quarters :    Arithmetic.omega ℂ ≤ (9 : ℝ) / 4 := by  sorry⋮theorem complex_alpha_gt_93_div_200 :    (93 : ℝ) / 200 < Arithmetic.complexAlpha := by  sorry⋮theorem complex_rectangular_omega_lt_523_div_250 :    Arithmetic.rectangularOmega ℂ ((709 : ℝ) / 1000) < (523 : ℝ) / 250 := 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.MatrixMultiplication.Arithmetic.Gate line 13

inductive Gate (F Input Register : Type*) where  | constant : F → Gate F Input Register  | input : Input → Gate F Input Register  | add : Register → Register → Gate F Input Register  | sub : Register → Register → Gate F Input Register  | mul : Register → Register → Gate F Input Register

definition OAI.MatrixMultiplication.Arithmetic.Gate.eval line 24

def eval [Field F] (inputs : Input → F) (registers : Register → F) :    Gate F Input Register → F  | .constant z => z  | .input i => inputs i  | .add i j => registers i + registers j  | .sub i j => registers i - registers j  | .mul i j => registers i * registers j

definition OAI.MatrixMultiplication.Arithmetic.Gate.cost line 32

def cost : Gate F Input Register → ℕ  | .constant _ => 0  | .input _ => 0  | .add _ _ => 1  | .sub _ _ => 1  | .mul _ _ => 1

inductive type OAI.MatrixMultiplication.Arithmetic.Program line 41

inductive Program (F Input : Type*) : ℕ → Type _ where  | nil : Program F Input 0  | step {r : ℕ} : Program F Input r → Gate F Input (Fin r) → Program F Input (r + 1)

abbreviation OAI.MatrixMultiplication.Arithmetic.MatrixInput line 60

abbrev MatrixInput (a b c : ℕ) := (Fin a × Fin b) ⊕ (Fin b × Fin c)

structure OAI.MatrixMultiplication.Arithmetic.MatrixAlgorithm line 68

structure MatrixAlgorithm (F : Type*) (a b c : ℕ) where  registers : ℕ  program : Program F (MatrixInput a b c) registers  output : Fin a → Fin c → Fin registers

definition OAI.MatrixMultiplication.Arithmetic.MatrixAlgorithm.Correct line 82

def Correct (P : MatrixAlgorithm F a b c) : Prop :=  ∀ (A : Matrix (Fin a) (Fin b) F) (B : Matrix (Fin b) (Fin c) F), P.eval A B = A * B

definition OAI.MatrixMultiplication.Arithmetic.AdmissibleExponent line 90

def AdmissibleExponent (F : Type*) [Field F] (τ : ℝ) : Prop :=  ∀ ε : ℝ, 0 < ε → ∃ C : ℝ, 0 < C ∧    ∀ n : ℕ, 1 ≤ n → ∃ P : MatrixAlgorithm F n n n,      P.Correct ∧ (P.cost : ℝ) ≤ C * (n : ℝ) ^ (τ + ε)

definition OAI.MatrixMultiplication.Arithmetic.omega line 95

def omega (F : Type*) [Field F] : ℝ := sInf {τ : ℝ | AdmissibleExponent F τ}

definition OAI.MatrixMultiplication.Arithmetic.innerSize line 98

def innerSize (n : ℕ) (k : ℝ) : ℕ := ⌈(n : ℝ) ^ k⌉₊

definition OAI.MatrixMultiplication.Arithmetic.RectangularAdmissibleExponent line 100

def RectangularAdmissibleExponent (F : Type*) [Field F] (k τ : ℝ) : Prop :=  ∀ ε : ℝ, 0 < ε → ∃ C : ℝ, 0 < C ∧    ∀ n : ℕ, 1 ≤ n → ∃ P : MatrixAlgorithm F n (innerSize n k) n,      P.Correct ∧ (P.cost : ℝ) ≤ C * (n : ℝ) ^ (τ + ε)

definition OAI.MatrixMultiplication.Arithmetic.rectangularOmega line 105

def rectangularOmega (F : Type*) [Field F] (k : ℝ) : ℝ :=  sInf {τ : ℝ | RectangularAdmissibleExponent F k τ}

definition OAI.MatrixMultiplication.Arithmetic.complexAlpha line 109

def complexAlpha : ℝ :=  sSup {k : ℝ | k ∈ Set.Icc 0 1 ∧ rectangularOmega ℂ k = 2}

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: exponents, arithmetic, multiplication, matrix, alpha and omega. Found in the paper’s title, which its main theorem states: rectangular and complex.

Statements in Complex Matrix Multiplication Below 2.258 and Rectangular Bounds come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of Complex Matrix Multiplication Below 2.258 and Rectangular Bounds

Theorem

Over \C, the arithmetic matrix-multiplication exponents satisfy

\omega<\frac{1129}{500}=2.258,\qquad \alpha>0.465,\qquad w(0.709)<2.092.

Section 1, Introduction Read it in the paper TeX source, line 72

Show the TeX

Over \(\C\), the arithmetic matrix-multiplication exponents satisfy
\[
 \omega<\frac{1129}{500}=2.258,\qquad
 \alpha>0.465,\qquad w(0.709)<2.092.
\]

Other possible paper matches, closest wording first

Theorem 1.1 of An Upper Bound of 9/4 for the Matrix Multiplication Exponent. Words of the Lean statement, its names split into words, that also appear in the theorem: 9/4, complex, arithmetic, two and omega. Found in the paper’s title, which its main theorem states: exponent, multiplication and matrix.
Corollary 14.1 of Complex Matrix Multiplication Below 2.258 and Rectangular Bounds. Words of the Lean statement, its names split into words, that also appear in the theorem: 709/1000, 523/250, field, two, positive and omega. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bounds.
Theorem 1.1 of Staggered extraction for exact matrix multiplication over every field. Words of the Lean statement, its names split into words, that also appear in the theorem: exponent, field, arithmetic, multiplication, matrix and omega. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: square.
Corollary 14.2 of Complex Matrix Multiplication Below 2.258 and Rectangular Bounds. Words of the Lean statement, its names split into words, that also appear in the theorem: dual, field and alpha. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bound.
Proposition 4.9 of Complex Matrix Multiplication Below 2.258 and Rectangular Bounds. Words of the Lean statement, its names split into words, that also appear in the theorem: input, multiplication, matrix and alpha. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bound.
Theorem 3.2 of Complex Matrix Multiplication Below 2.258 and Rectangular Bounds. Words of the Lean statement, its names split into words, that also appear in the theorem: dimensions, integers, positive and omega.

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-pass”.
Re-run through Comparator: 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 107’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 disagree 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: ω(C) <= 9/4, α > 0.465 and ω(F) < 2.371054886006746 for every field are stated; 'ω < 2.258 outside finitely many positive characteristics' is not. Exponents are sInf of uniform-constant, non-uniform straight-line program families (standard).
Definitions the group asks readers to compare with the paper: omega F := sInf {τ | AdmissibleExponent F τ} is only meaningful if the set is nonempty and bounded below (true: τ = 3 admissible, and cost ≥ n² forces τ ≥ 2; junk value 0 otherwise); complexAlpha := sSup {k ∈ [0,1] | rectangularOmega ℂ k = 2}; the first-conjunct-only reading of omega_lt_source_constant is padded with a pure numeric inequality 2.371054886006746 < 2.371056
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNDECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so. Their record
Their note: 'ω < 2.258 outside finitely many positive characteristics' is not stated for any field of positive characteristic; the characteristic-zero rectangular clause is stated over C only. Both would follow from the C results by standard characteristic-transfer arguments (Schönhage-type), but no challenge states them.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The square bound implies the paper's weaker $2.258$ headline bound.”
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

132 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: 29 to its own declarations, 43 to Mathlib’s documentation. 21 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 complex_omega_le_nine_quarters 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 18 other declarations of the file; with the theorem itself they span 69 lines. 3 of them are named in the statement itself.

Names in the namespace OAI.MatrixMultiplication.Arithmetic are listed without it.

inductive Gateline 13library
def Gate.evalline 24library
def Gate.costline 32library
inductive Programline 41library
def Program.evalline 49library
def Program.costline 54library
abbrev MatrixInputline 60library
def matrixInputsline 62library
structure MatrixAlgorithmline 68library
def MatrixAlgorithm.evalline 77library
def MatrixAlgorithm.Correctline 82library
def MatrixAlgorithm.costline 85library
def AdmissibleExponentline 90library
def omegaline 95in the statementlibrary
def innerSizeline 98library
def RectangularAdmissibleExponentline 100library
def rectangularOmegaline 105in the statementlibrary
def complexAlphaline 109in the statementlibrary
theorem OAI.MatrixMultiplication.complex_omega_le_nine_quartersline 118selected theoremlibrary
theorem OAI.MatrixMultiplication.complex_alpha_gt_93_div_200line 122selected theoremlibrary
theorem OAI.MatrixMultiplication.complex_rectangular_omega_lt_523_div_250line 126selected theoremlibrary

import Mathlibnamespace OAI/-! The three main complex matrix multiplication bounds, checked independently. -/noncomputable sectionopen scoped BigOperatorsnamespace MatrixMultiplication.Arithmeticinductive Gate (F Input Register : Type*) where  | constant : F → Gate F Input Register  | input : Input → Gate F Input Register  | add : Register → Register → Gate F Input Register  | sub : Register → Register → Gate F Input Register  | mul : Register → Register → Gate F Input Registernamespace Gatevariable {F Input Register : Type*}def eval [Field F] (inputs : Input → F) (registers : Register → F) :    Gate F Input Register → F  | .constant z => z  | .input i => inputs i  | .add i j => registers i + registers j  | .sub i j => registers i - registers j  | .mul i j => registers i * registers jdef cost : Gate F Input Register → ℕ  | .constant _ => 0  | .input _ => 0  | .add _ _ => 1  | .sub _ _ => 1  | .mul _ _ => 1end Gateinductive Program (F Input : Type*) : ℕ → Type _ where  | nil : Program F Input 0  | step {r : ℕ} : Program F Input r → Gate F Input (Fin r) → Program F Input (r + 1)namespace Programvariable {F Input : Type*}def eval [Field F] : {r : ℕ} → Program F Input r → (Input → F) → Fin r → F  | 0, .nil, _ => Fin.elim0  | _ + 1, .step p g, inputs =>    Fin.cases (g.eval inputs (p.eval inputs)) (p.eval inputs)def cost : {r : ℕ} → Program F Input r → ℕ  | 0, .nil => 0  | _ + 1, .step p g => p.cost + g.costend Programabbrev MatrixInput (a b c : ℕ) := (Fin a × Fin b) ⊕ (Fin b × Fin c)def matrixInputs {F : Type*} {a b c : ℕ}    (A : Matrix (Fin a) (Fin b) F) (B : Matrix (Fin b) (Fin c) F) :    MatrixInput a b c → F  | .inl (i, j) => A i j  | .inr (j, k) => B j kstructure MatrixAlgorithm (F : Type*) (a b c : ℕ) where  registers : ℕ  program : Program F (MatrixInput a b c) registers  output : Fin a → Fin c → Fin registersnamespace MatrixAlgorithmvariable {F : Type*} [Field F] {a b c : ℕ}def eval (P : MatrixAlgorithm F a b c)    (A : Matrix (Fin a) (Fin b) F) (B : Matrix (Fin b) (Fin c) F) :    Matrix (Fin a) (Fin c) F :=  fun i k => P.program.eval (matrixInputs A B) (P.output i k)def Correct (P : MatrixAlgorithm F a b c) : Prop :=  ∀ (A : Matrix (Fin a) (Fin b) F) (B : Matrix (Fin b) (Fin c) F), P.eval A B = A * Bdef cost (P : MatrixAlgorithm F a b c) : ℕ := P.program.costend MatrixAlgorithm/-- Positive slack, one uniform constant, and a correct program at every size. -/def AdmissibleExponent (F : Type*) [Field F] (τ : ℝ) : Prop :=  ∀ ε : ℝ, 0 < ε → ∃ C : ℝ, 0 < C ∧    ∀ n : ℕ, 1 ≤ n → ∃ P : MatrixAlgorithm F n n n,      P.Correct ∧ (P.cost : ℝ) ≤ C * (n : ℝ) ^ (τ + ε)def omega (F : Type*) [Field F] : ℝ := sInf {τ : ℝ | AdmissibleExponent F τ}/-- The inner dimension of an `n × n^k` by `n^k × n` multiplication. -/def innerSize (n : ℕ) (k : ℝ) : ℕ := ⌈(n : ℝ) ^ k⌉₊def RectangularAdmissibleExponent (F : Type*) [Field F] (k τ : ℝ) : Prop :=  ∀ ε : ℝ, 0 < ε → ∃ C : ℝ, 0 < C ∧    ∀ n : ℕ, 1 ≤ n → ∃ P : MatrixAlgorithm F n (innerSize n k) n,      P.Correct ∧ (P.cost : ℝ) ≤ C * (n : ℝ) ^ (τ + ε)def rectangularOmega (F : Type*) [Field F] (k : ℝ) : ℝ :=  sInf {τ : ℝ | RectangularAdmissibleExponent F k τ}/-- The dual exponent is a supremum of exact exponent-two rectangular shapes. -/def complexAlpha : ℝ :=  sSup {k : ℝ | k ∈ Set.Icc 0 1 ∧ rectangularOmega ℂ k = 2}end MatrixMultiplication.Arithmeticendnamespace MatrixMultiplicationtheorem complex_omega_le_nine_quarters :    Arithmetic.omega ℂ ≤ (9 : ℝ) / 4 := by  sorrytheorem complex_alpha_gt_93_div_200 :    (93 : ℝ) / 200 < Arithmetic.complexAlpha := by  sorrytheorem complex_rectangular_omega_lt_523_div_250 :    Arithmetic.rectangularOmega ℂ ((709 : ℝ) / 1000) < (523 : ℝ) / 250 := by  sorryend MatrixMultiplicationend 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.

Complex Matrix Multiplication Below 2.258 and Rectangular Bounds

Theorem

Over \C, the arithmetic matrix-multiplication exponents satisfy

\omega<\frac{1129}{500}=2.258,\qquad \alpha>0.465,\qquad w(0.709)<2.092.

Section 1, Introduction Read it in the paper TeX source, line 72

Show the TeX

Over \(\C\), the arithmetic matrix-multiplication exponents satisfy
\[
 \omega<\frac{1129}{500}=2.258,\qquad
 \alpha>0.465,\qquad w(0.709)<2.092.
\]

Staggered extraction for exact matrix multiplication over every field

Theorem

Over every fixed field \F, the arithmetic exponent of square matrix multiplication satisfies

\omega_\F<2.371054886006746<2.371054887.

Section 1, Introduction Read it in the paper TeX source, line 57

Show the TeX

Over every fixed field $\F$, the arithmetic exponent of square matrix
multiplication satisfies
\[
 \omega_\F<2.371054886006746<2.371054887.
\]

Configuration and file facts

File: lean/ComparatorChallenges/MatrixMultiplication.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/LinearAlgebra/MatrixMultiplication/Main.lean, OAI/LinearAlgebra/MatrixMultiplication/Main.lean, OAI/LinearAlgebra/MatrixMultiplication/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: 18 local vocabulary declarations
Lines of Lean, without comments or blank lines: 69
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