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.ArExplicit.Statement.main
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
theoremmain : FullStatement.{v} := bysorry
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.
defConclusions (F R Z : Type u) [Field F] [Ring R] [Algebra F R] [AddCommGroup Z] [Module R Z] [Module F Z] [IsScalarTower F R Z] : Prop :=
Show the remaining 10 lines
FiniteDimensional F R ∧ FiniteDimensional F Z ∧ ¬ Module.Projective R Z ∧ TotallyAcyclicWitness R Z ∧ (∀ i : ℕ, 0 < i → Subsingleton (Abelian.Ext (ModuleCat.of R Z) (ModuleCat.of R Z) i)) ∧ (∀ i : ℕ, 0 < i → Subsingleton (Abelian.Ext (ModuleCat.of R Z) (ModuleCat.of R R) i)) ∧ Nonempty ((R ⧸ Ring.jacobson R) ≃ₐ[F] (Fin 8 → F)) ∧ (Ring.jacobson R)^4 ≠ ⊥
AlgebraicIndependent (ZMod 2) parameters ∧ ∃ s : System, Conclusions K s.algebraCarrier s.moduleCarrier ∧ ∀ (E : Type v) [Field E] [Algebra K E], ∃ action : Module (E ⊗[K] s.algebraCarrier) (E ⊗[K] s.moduleCarrier),letI := action ∃ tower : IsScalarTower E (E ⊗[K] s.algebraCarrier) (E ⊗[K] s.moduleCarrier),letI := tower (∀ (a b : E) (r : s.algebraCarrier) (z : s.moduleCarrier), (a ⊗ₜ[K] r) • (b ⊗ₜ[K] z) = (a*b) ⊗ₜ[K] (r • z)) ∧ Conclusions E (E ⊗[K] s.algebraCarrier) (E ⊗[K] s.moduleCarrier)
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: conclusions, independent, ext, field, integer, dimensional, algebra, projective and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: gorenstein and nonprojective.
Statements in An explicit counterexample to the Auslander-Reiten conjecture come first because the release’s name for this Lean statement shares words with that paper’s title.
Theorem 1.1 of An explicit counterexample to the Auslander-Reiten conjecture
Theorem
Let \kk=\mathbb F_2(q,H_1,H_2), where q,H_1,H_2 are algebraically independent. There exist a finite-dimensional \kk-algebra \Lambda and a finite-dimensional nonprojective left \Lambda-module Z such that
\Ext^i_\Lambda(Z,Z)=0=\Ext^i_\Lambda(Z,\Lambda)
\qquad\text{for every integer }i\geq1.
The module Z is Gorenstein-projective. The semisimple quotient is \Lambda/\rad\Lambda\simeq\kk^8, so the algebra is split with eight isomorphism classes of simple modules; also \rad(\Lambda)^4\ne0. These conclusions persist after every field extension of \kk.
Let $\kk=\mathbb F_2(q,H_1,H_2)$, where $q,H_1,H_2$ are algebraically
independent. There exist a finite-dimensional $\kk$-algebra $\Lambda$
and a finite-dimensional nonprojective left $\Lambda$-module $Z$ such that
\[
\Ext^i_\Lambda(Z,Z)=0=\Ext^i_\Lambda(Z,\Lambda)
\qquad\text{for every integer }i\geq1.
\]
The module $Z$ is Gorenstein-projective. The semisimple quotient is
$\Lambda/\rad\Lambda\simeq\kk^8$, so the algebra is split with eight
isomorphism classes of simple modules; also $\rad(\Lambda)^4\ne0$.
These conclusions persist after every field extension of $\kk$.
Other possible paper matches, closest wording first
Proposition 3.2 of A counterexample to Tachikawa’s second conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: totally, acyclic, complex, hom, ext, dimensional and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonprojective.
Proposition 9.2 of An explicit counterexample to the Auslander-Reiten conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: conclusions, field, algebra and module.
Proposition 3.1 of An explicit counterexample to the Auslander-Reiten conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: map, ext, hom, field, integer, dimensional and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: gorenstein and nonprojective.
Proposition 5.4 of A counterexample to Tachikawa’s second conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: action, scalar, groups and dimensional.
Theorem 1.1 of A counterexample to Tachikawa’s second conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: independent, ext, integer, dimensional, algebra, finite and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonprojective.
Proposition 5.2 of A counterexample to Tachikawa’s second conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: polynomial, groups, maps, integer and module. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonprojective.
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.
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…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
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.
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 199’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.
Finds part of it stated, or a narrower or weaker statementTheir term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: Stated: ArExplicit.Statement.main (FullStatement: over K = Frac(F₂[X₁,X₂,X₃]) a finite-dimensional algebra and a non-projective Gorenstein-projective module Z (TotallyAcyclicWitness) with Ext^i(Z,Z) and Ext^i(Z,A) subsingleton for i > 0, A/rad ≅ K^8, rad^4 ≠ 0, all conclusions persisting under every field extension E ⊗_K −) and Tachikawa.main_theorem (symmetric finite-dimensional algebra over k with a non-projective f.d. module with vanishing self-Ext). Not stated: the 'associated endomorphism algebra' consequences (classical/generalized/strong Nakayama, Auslander-Gorenstein, Wakamatsu tilting) named in the summary's second sentence (docs: 'The later field-extension, endomorphism-algebra, and related homological consequences are not included'; the docs also contradict themselves on field-extension persistence, which the Lean does state).
Finds part of it stated, or a narrower or weaker statementTheir term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: The endomorphism-algebra counterexamples (three Nakayama conjectures, Auslander–Gorenstein, Wakamatsu tilting) and field-extension persistence of the Tachikawa counterexample are not stated.
OpenAI’s own note on what this Lean statement covers
“The formalized counterexample is a finite-dimensional algebra over $k=\mathbb F_2(q,H_1,H_2)$ and a finite-dimensional nonprojective Gorenstein-projective module with this vanishing.”
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
77 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: 28 to its own declarations, 58 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.
Let \kk=\mathbb F_2(q,H_1,H_2), where q,H_1,H_2 are algebraically independent. There exist a finite-dimensional \kk-algebra \Lambda and a finite-dimensional nonprojective left \Lambda-module Z such that
\Ext^i_\Lambda(Z,Z)=0=\Ext^i_\Lambda(Z,\Lambda)
\qquad\text{for every integer }i\geq1.
The module Z is Gorenstein-projective. The semisimple quotient is \Lambda/\rad\Lambda\simeq\kk^8, so the algebra is split with eight isomorphism classes of simple modules; also \rad(\Lambda)^4\ne0. These conclusions persist after every field extension of \kk.
Let $\kk=\mathbb F_2(q,H_1,H_2)$, where $q,H_1,H_2$ are algebraically
independent. There exist a finite-dimensional $\kk$-algebra $\Lambda$
and a finite-dimensional nonprojective left $\Lambda$-module $Z$ such that
\[
\Ext^i_\Lambda(Z,Z)=0=\Ext^i_\Lambda(Z,\Lambda)
\qquad\text{for every integer }i\geq1.
\]
The module $Z$ is Gorenstein-projective. The semisimple quotient is
$\Lambda/\rad\Lambda\simeq\kk^8$, so the algebra is split with eight
isomorphism classes of simple modules; also $\rad(\Lambda)^4\ne0$.
These conclusions persist after every field extension of $\kk$.
where q,H_1,H_2 are algebraically independent. There exist a finite-dimensional associative unital symmetric k-algebra A and a finite-dimensional unital left A-module M such that M is nonprojective and
\Ext_A^i(M,M)=0\qquad\text{for every integer }i>0.
In particular, Tachikawa’s second conjecture is false.
Let
\[
k=\mathbb F_2(q,H_1,H_2),
\]
where $q,H_1,H_2$ are algebraically independent.
There exist a finite-dimensional associative unital symmetric
$k$-algebra $A$ and a finite-dimensional unital left $A$-module $M$
such that $M$ is nonprojective and
\[
\Ext_A^i(M,M)=0\qquad\text{for every integer }i>0.
\]
In particular, Tachikawa's second conjecture is false.
Corollary(Homological conjectures)
Let A,M be as in Theorem thm:main. For every field extension K/k, put A_K=K\otimes_k A, M_K=K\otimes_k M, and
\Gamma_K=\End_{A_K}(A_K\oplus M_K)^{\mathrm{op}}.
Then \Gamma_K is a finite-dimensional K-algebra with the following properties, in the category of finite left \Gamma_K-modules.
1.
Every term of the minimal injective resolution of \Gamma_K is projective, but \idim_{\Gamma_K}\Gamma_K=\infty.
2.
There is a nonzero simple module S such that \Ext_{\Gamma_K}^i(S,\Gamma_K)=0 for every i\geq0.
3.
There is a projective-injective Wakamatsu tilting module W which is not tilting.
Consequently the classical, generalized, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture are false for finite-dimensional algebras over K.
Let $A,M$ be as in Theorem~\ref{thm:main}. For every field extension
$K/k$, put $A_K=K\otimes_k A$, $M_K=K\otimes_k M$, and
\[
\Gamma_K=\End_{A_K}(A_K\oplus M_K)^{\mathrm{op}}.
\]
Then $\Gamma_K$ is a finite-dimensional $K$-algebra with the following
properties, in the category of finite left $\Gamma_K$-modules.
\begin{enumerate}
\item Every term of the minimal injective resolution of $\Gamma_K$
is projective, but $\idim_{\Gamma_K}\Gamma_K=\infty$.
\item There is a nonzero simple module $S$ such that
$\Ext_{\Gamma_K}^i(S,\Gamma_K)=0$ for every $i\geq0$.
\item There is a projective-injective Wakamatsu tilting module $W$
which is not tilting.
\end{enumerate}
Consequently the classical, generalized, and strong Nakayama conjectures,
the Auslander--Gorenstein conjecture, and the Wakamatsu tilting conjecture
are false for finite-dimensional algebras over $K$.
Corollary(Finitistic dimensions)
Let k be the field in Theorem thm:main, let K/k be any field extension, and let \Gamma_K be the algebra in Corollary cor:homological. For every integer n\geq1, there is a finite-dimensional left \Gamma_K-module C_n such that
\pd_{\Gamma_K}C_n=n.
Consequently \Gamma_K has infinite little left finitistic dimension and infinite big left finitistic dimension.
Let $k$ be the field in Theorem~\ref{thm:main}, let $K/k$ be any field
extension, and let $\Gamma_K$ be the algebra in
Corollary~\ref{cor:homological}. For every integer $n\geq1$, there is a
finite-dimensional left $\Gamma_K$-module $C_n$ such that
\[
\pd_{\Gamma_K}C_n=n.
\]
Consequently $\Gamma_K$ has infinite little left finitistic dimension
and infinite big left finitistic dimension.
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
6 local vocabulary declarations
Lines of Lean, without comments or blank lines
49
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
Auslander–Reiten counterexample
Family 199: Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures
Candidate pairing
Theorem
Let \kk=\mathbb F_2(q,H_1,H_2), where q,H_1,H_2 are algebraically independent. There exist a finite-dimensional \kk-algebra \Lambda and a finite-dimensional nonprojective left \Lambda-module Z such that
\Ext^i_\Lambda(Z,Z)=0=\Ext^i_\Lambda(Z,\Lambda)
\qquad\text{for every integer }i\geq1.
The module Z is Gorenstein-projective. The semisimple quotient is \Lambda/\rad\Lambda\simeq\kk^8, so the algebra is split with eight isomorphism classes of simple modules; also \rad(\Lambda)^4\ne0. These conclusions persist after every field extension of \kk.
An explicit counterexample to the Auslander-Reiten conjecture
TeX label thm:main, build/01-introduction.tex, line 26
The Lean statement says
theoremmain : FullStatement.{v} := bysorry
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.
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.
Questions to resolve when comparing this statement with the paper
Both independent readers find this family's challenges state less than its headline, while the atlas's AI draft classes this file as stating a paper's theorem as the paper does; check which part of the headline it covers.