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Lean statement

Matrix-multiplication exponent over every field

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.MatrixAllFields.MatrixMultiplication.AllFieldMain.omega_lt_source_constant

Lean statement file: MatrixFields.leanShow on the mapFamily 107. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theoremWho ran the proof-verification programReport 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 omega_lt_source_constant (F : Type u) [Field F] :    Arithmetic.omega F < (2371054886006746 : ℝ) / 10 ^ 15 ∧    (2371054886006746 : ℝ) / 10 ^ 15 < (2371056 : ℝ) / 10 ^ 6 := 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.MatrixAllFields.MatrixMultiplication.Arithmetic.Gate line 15

    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.MatrixAllFields.MatrixMultiplication.Arithmetic.Gate.eval line 26

    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.MatrixAllFields.MatrixMultiplication.Arithmetic.Gate.cost line 34

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

  • inductive type OAI.MatrixAllFields.MatrixMultiplication.Arithmetic.Program line 43

    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.MatrixAllFields.MatrixMultiplication.Arithmetic.MatrixInput line 62

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

  • structure OAI.MatrixAllFields.MatrixMultiplication.Arithmetic.MatrixAlgorithm line 70

    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.MatrixAllFields.MatrixMultiplication.Arithmetic.MatrixAlgorithm.Correct line 84

    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.MatrixAllFields.MatrixMultiplication.Arithmetic.AdmissibleExponent line 92

    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.MatrixAllFields.MatrixMultiplication.Arithmetic.omega line 97

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

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: exponent, field, arithmetic, matrix and omega. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: multiplication.

Statements in Staggered extraction for exact matrix multiplication over every field come first because the Lean file and module names share words with that paper’s title.

Theorem 1.1 of 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
\[
 \omega_\F<2.371054886006746<2.371054887.
\]

Other possible paper matches, closest wording first

  • Theorem 1.1 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: exponents, arithmetic, matrix and omega. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: multiplication.
  • Proposition 6.1 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: real, constant, output and matrix. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: count and finite.
  • 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: arithmetic and omega. Found in the paper’s title, which its main theorem states: exponent 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: field, positive and omega. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: inequalities, finite and bounds.
  • Theorem 3.1 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: step, one, output and positive.
  • 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: field. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: characteristic and bound.

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 10 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 formalized result gives the unconditional bound $\omega(F)<2.371054886006746$ for every field $F$, including finite fields and fields of positive characteristic. The exponent counts additions, subtractions, and multiplications in finite division-free arithmetic programs, with arbitrary positive exponent slack. No numerical inequalities remain as hypotheses.”

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

116 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: 23 to its own declarations, 39 to Mathlib’s documentation. 15 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 OAIopen scoped BigOperators Topology Polynomialnamespace MatrixAllFieldsnoncomputable 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}end MatrixMultiplication.Arithmeticendend MatrixAllFieldsnamespace MatrixAllFields.MatrixMultiplication.AllFieldMainuniverse utheorem omega_lt_source_constant (F : Type u) [Field F] :    Arithmetic.omega F < (2371054886006746 : ℝ) / 10 ^ 15 ∧    (2371054886006746 : ℝ) / 10 ^ 15 < (2371056 : ℝ) / 10 ^ 6 := by  sorryend MatrixAllFields.MatrixMultiplication.AllFieldMainend 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
OpenAI’s Lean catalogue
OpenAI lists this as a main result in its Lean catalogue.
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
14 local vocabulary declarations
Lines of Lean, without comments or blank lines
55
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

Matrix-multiplication exponent over every field

Family 107: Matrix multiplication with exponent at most 9/4

Candidate pairing

Theorem

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

\omega_\F<2.371054886006746<2.371054887.

Staggered extraction for exact matrix multiplication over every field

TeX label thm:main, build/sections/introduction.tex, line 57

The Lean statement says

theorem omega_lt_source_constant (F : Type u) [Field F] :    Arithmetic.omega F < (2371054886006746 : ℝ) / 10 ^ 15 ∧    (2371054886006746 : ℝ) / 10 ^ 15 < (2371056 : ℝ) / 10 ^ 6 := 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.

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

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

More about this statement

  • Follow-up result The release's all-fields Lean statement bounds omega(F) only by about 2.371055 for every field, and its 9/4 statement is over the complex numbers. Zuiddam's derived Lean proof gives omega(F) at most 9/4 over every field, with a non-vacuity theorem and the three standard axioms; Sela Navot's fork reached the all-fields bound a day earlier. Jeroen Zuiddam, Lean proof that the matrix multiplication exponent is at most 9/4 over every field, 2026-10-07 Source

Questions to resolve when comparing this statement with the paper

  • Two independent readers, MathVet and cert-machine, disagree on whether this family's challenges state its headline; a specialist reading would settle it.

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