Lean statement

Quadratic stabilization of the canonical Foulkes–Howe map

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.Problem346.canonical_foulkes_howe_surjective

Lean statement file: FoulkesHowe.leanShow on the mapFamily 210. 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 canonical_foulkes_howe_surjective :    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 2 ≤ a → a * (a - 1) ≤ b → ∃ μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V μ ∧ Function.Surjective μ ∧ ∀ ν : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V ν → ν = μ := 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.

definition OAI.Problem346.symMonomialRaw line 12

def symMonomialRaw (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]    (v : Fin n → V) : SymmetricAlgebra ℂ V :=  ∏ i : Fin n, SymmetricAlgebra.ι ℂ V (v i)

definition OAI.Problem346.symPowSubmodule line 16

def symPowSubmodule (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :    Submodule ℂ (SymmetricAlgebra ℂ V) :=  Submodule.span ℂ (Set.range (symMonomialRaw n V))

abbreviation OAI.Problem346.SymPow line 20

abbrev SymPow (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :=  ↥(symPowSubmodule n V)

definition OAI.Problem346.symMonomial line 23

def symMonomial (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]    (v : Fin n → V) : SymPow n V :=  ⟨symMonomialRaw n V v, Submodule.subset_span (Set.mem_range_self v)⟩

definition OAI.Problem346.foulkesFormula line 27

def foulkesFormula (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]    (v : Fin b → Fin a → V) : SymPow a (SymPow b V) := by  classical  exact    ((a.factorial : ℂ) ^ b)⁻¹ •      ∑ σ : Fin b → Equiv.Perm (Fin a),        symMonomial a (SymPow b V)          (fun i => symMonomial b V (fun j => v j ((σ j) i)))

definition OAI.Problem346.IsFoulkesMap line 36

def IsFoulkesMap (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]    (μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)) : Prop :=  ∀ v : Fin b → Fin a → V,    μ (symMonomial b (SymPow a V) (fun j => symMonomial a V (v j))) =      foulkesFormula a b V v

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: map, surjective, finite, dimensional, complex and integer. Found in the paper’s title, which its main theorem states: canonical, howe and foulkes. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: space and vector.

Statements in Quadratic stabilization of the canonical Foulkes–Howe map come first because the release lists only that paper for this Lean statement.

Theorem 1 of Quadratic stabilization of the canonical Foulkes–Howe map

Theorem

For every integer a\ge2, every integer b\ge a(a-1), and every finite-dimensional complex vector space V, the map \mu_{a,b,V} in (eq:map) is surjective. Equivalently, A_b=R_b.

Section 1, Introduction Read it in the paper TeX source, line 99

Show the TeX

For every integer $a\ge2$, every integer $b\ge a(a-1)$, and every
finite-dimensional complex vector space $V$, the map
$\mu_{a,b,V}$ in \eqref{eq:map} is surjective. Equivalently, $A_b=R_b$.

Other possible paper matches, closest wording first

Corollary 4 of Quadratic stabilization of the canonical Foulkes–Howe map. Words of the Lean statement, its names split into words, that also appear in the theorem: symmetric, sym, finite, dimensional, complex and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: embedding, sixth, equivariant, power, space and vector.
Proposition 3 of Quadratic stabilization of the canonical Foulkes–Howe map. Words of the Lean statement, its names split into words, that also appear in the theorem: symmetric, sym, finite, dimensional, complex and integers. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: product, space and vectors.
Theorem 1.1 of Foulkes' conjecture for the sixth symmetric power. Words of the Lean statement, its names split into words, that also appear in the theorem: sym, finite, dimensional, complex and integer. Found in the paper’s title, which its main theorem states: foulkes and symmetric. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: equivariant, space and vector.
Corollary 2.1 of Foulkes' conjecture for the sixth symmetric power. Words of the Lean statement, its names split into words, that also appear in the theorem: range, sym, finite, dimensional and complex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: equivariant.
Corollary 2.6 of Foulkes' conjecture for the sixth symmetric power. Words of the Lean statement, its names split into words, that also appear in the theorem: module.
Proposition 3.2 of Foulkes' conjecture for the sixth symmetric power. Words of the Lean statement, its names split into words, that also appear in the theorem: function. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: vector.

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 210’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.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: Quadratic stabilization is stated: canonical_foulkes_howe_surjective : ∀ V f.d., ∀ a b, 2 ≤ a → a * (a - 1) ≤ b → ∃ μ, IsFoulkesMap a b V μ ∧ Function.Surjective μ ∧ (unique). The headline sixth case of Foulkes' conjecture for all b >= 6 is only reached for b >= 30 = 6*5 (docs: 'The sixth-power specialization is covered for b ≥ 30; the companion's full range b ≥ 6 is not included'); the equivariant embedding is only implicit (dual of surjectivity).
Definitions the group asks readers to compare with the paper: foulkesFormula a b V v := (a!)^(-b) • Σ_{σ : Fin b → Perm (Fin a)} sym_a(fun i => sym_b(fun j => v j (σ j i))) and IsFoulkesMap define the 'canonical' map by hand on SymPow n V := span of degree-n products in SymmetricAlgebra ℂ V; the theorem also asserts uniqueness, which pins the definition.
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: The sixth Foulkes case for 6 ≤ b < 30 is not stated (only b ≥ 30 follows), and the equivariant embedding itself is not stated even for b ≥ 30.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalized result proves surjectivity of the canonical averaged Foulkes–Howe map $\mathrm{Sym}^b(\mathrm{Sym}^a\,V)\to\mathrm{Sym}^a(\mathrm{Sym}^b\,V)$ for every finite-dimensional complex vector space, $a\ge2$, and $b\ge a(a-1)$. It also covers the bijective $a=1$ case, the vanishing-on-products consequence, and the corresponding equivariant embedding in the reverse direction. The sixth-power specialization is covered for $b\ge30$; the companion's full range $b\ge6$ is not included.”
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

50 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: 27 to its own declarations, 34 to Mathlib’s documentation. 7 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 canonical_foulkes_howe_surjective 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 6 other declarations of the file; with the theorem itself they span 27 lines. 2 of them are named in the statement itself.

Names in the namespace OAI.Problem346 are listed without it.

def symMonomialRawline 12library
def symPowSubmoduleline 16library
abbrev SymPowline 20in the statementlibrary
def symMonomialline 23library
def foulkesFormulaline 27library
def IsFoulkesMapline 36in the statementlibrary
theorem canonical_foulkes_howe_surjectiveline 42selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped BigOperatorsuniverse unamespace Problem346def symMonomialRaw (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]    (v : Fin n → V) : SymmetricAlgebra ℂ V :=  ∏ i : Fin n, SymmetricAlgebra.ι ℂ V (v i)def symPowSubmodule (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :    Submodule ℂ (SymmetricAlgebra ℂ V) :=  Submodule.span ℂ (Set.range (symMonomialRaw n V))abbrev SymPow (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V] :=  ↥(symPowSubmodule n V)def symMonomial (n : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]    (v : Fin n → V) : SymPow n V :=  ⟨symMonomialRaw n V v, Submodule.subset_span (Set.mem_range_self v)⟩def foulkesFormula (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]    (v : Fin b → Fin a → V) : SymPow a (SymPow b V) := by  classical  exact    ((a.factorial : ℂ) ^ b)⁻¹ •      ∑ σ : Fin b → Equiv.Perm (Fin a),        symMonomial a (SymPow b V)          (fun i => symMonomial b V (fun j => v j ((σ j) i)))def IsFoulkesMap (a b : ℕ) (V : Type u) [AddCommGroup V] [Module ℂ V]    (μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V)) : Prop :=  ∀ v : Fin b → Fin a → V,    μ (symMonomial b (SymPow a V) (fun j => symMonomial a V (v j))) =      foulkesFormula a b V vtheorem canonical_foulkes_howe_surjective :    ∀ (V : Type u) [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] (a b : ℕ), 2 ≤ a → a * (a - 1) ≤ b → ∃ μ : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V μ ∧ Function.Surjective μ ∧ ∀ ν : SymPow b (SymPow a V) →ₗ[ℂ] SymPow a (SymPow b V), IsFoulkesMap a b V ν → ν = μ := by  sorryend Problem346endend 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.