Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures
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Constructs finite-dimensional algebras over a characteristic-two rational-function field that disprove the Auslander–Reiten and Gorenstein-projective conjectures, and Tachikawa’s second conjecture. An associated endomorphism algebra also disproves the classical, generalized and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. The counterexamples persist under every extension of the base field. (OpenAI’s Lean note)
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The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
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.
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
Let A,M be as in Theorem 1.1. 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.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.
Two more possible matches
Theorem 1.1in A counterexample to Tachikawa’s second conjecture, 23 September 2026.
Corollary 1.3(Finitistic dimensions)in A counterexample to Tachikawa’s second conjecture, 23 September 2026.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We disprove the Auslander–Reiten conjecture for Artin algebras. We construct a finite-dimensional algebra Λ over \(k=\mathbb F_2(q,H_1,H_2)\) and a finite-dimensional nonprojective left module Z such that \(\mathop{\mathrm{Ext}}\nolimits ^i_\Lambda(Z,Z)=\mathop{\mathrm{Ext}}\nolimits ^i_\Lambda(Z,\Lambda)=0\) for every i > 0. The module is Gorenstein-projective, so the same example also disproves the Gorenstein-projective conjecture. Both counterexamples persist after every extension of k.
We disprove Tachikawa’s second conjecture by constructing a finite-dimensional symmetric algebra over \(k=\mathbb F_2(q,H_1,H_2)\) with a finite-dimensional nonprojective module whose self-extension groups vanish in every positive degree. The associated endomorphism algebra also gives counterexamples to the classical, generalized, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. These conclusions persist after every extension of k.
Two Lean statements in this family
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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.
The words they share: conclusions, independent, ext, field, integer, dimensional, algebra, projective, finite, modules, gorenstein, nonprojective.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
AuslanderReiten.lean, the release lists it as a main result.
theoremmain : FullStatement.{v} := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
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.
The words they share: symmetric, ext, integer, algebra, finite, module, counterexample, tachikawas, conjecture, nonprojective, dimensional.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
Tachikawa.lean, the release lists it as a main result.
defCounterexample : Prop := ∃ (A : Type) (rA : Ring A),letI := rA ∃ (aA : Algebra k A),letI := aA Module.Finite k A ∧ SymmetricOver k A ∧ ∃ (M : Type) (gM : AddCommGroup M),letI := gM ∃ (mA : Module A M) (mk : Module k M),letI := mAletI := mk IsScalarTower k A M ∧ Module.Finite k M ∧ ¬ Module.Projective A M ∧ ∀ n : ℕ, 0 < n → Subsingleton (CategoryTheory.Abelian.Ext (ModuleCat.of A M) (ModuleCat.of A M) n)⋮theoremmain_theorem : Counterexample := by⋮
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).
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).
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method: 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.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
The gap they name:The endomorphism-algebra counterexamples (three Nakayama conjectures, Auslander–Gorenstein, Wakamatsu tilting) and field-extension persistence of the Tachikawa counterexample are not stated.
“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says so.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The Auslander–Reiten conjecture predicts that a finitely generated module $M$ over an Artin algebra $A$ is projective if $\mathrm{Ext}^i_A(M,M\oplus A)=0$ for every $i>0$. 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.
It also establishes $A/\mathrm{rad}\,A\cong k^8$, $(\mathrm{rad}\,A)^4\ne0$, and preservation of the listed properties after every field extension. The Tachikawa companion is separate.
Tachikawa’s second conjecture predicts that a finite-dimensional module over a finite-dimensional self-injective algebra is projective if all its positive-degree self-Ext groups vanish. The formalized counterexample gives a finite-dimensional symmetric algebra over $\mathbb F_2(q,H_1,H_2)$ and a finite-dimensional nonprojective module $M$ with $\mathrm{Ext}^i(M,M)=0$ for every $i>0$. Symmetric algebras are self-injective, so this contradicts the conjecture.
The later field-extension, endomorphism-algebra, and related homological consequences are not included.
AI-drafted comparisons, with no recorded human review
Auslander–Reiten counterexampleAuslanderReiten
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: over k = F₂(q, H₁, H₂), a finite-dimensional algebra and a nonprojective Gorenstein-projective module with the Auslander–Reiten vanishing, with the stated structural properties.
Tachikawa counterexampleTachikawa
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a symmetric algebra over F₂(q, H₁, H₂) with a nonprojective module whose positive self-Ext groups vanish. The field-extension and homological corollaries are not included.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Auslander–Reiten counterexampleAuslanderReiten
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: the ten-dimensional algebra C, its trivial extension T and trace, the 179-entry cochain (cocycle and boundary identities, evaluation q^3), the resolution of s over C for every i, and the finite identities of the lift — not the Ext vanishing over Lambda (F, Lambda, Z are not published)”
Proposition 4.2 of A counterexample to Tachikawa’s second conjecture, 23 September 2026
Reported by
cert-machine
Method
Exact computation
Their verdict
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: the ten-dimensional algebra C (and its identity with the companion paper’s), T and its symmetrizing form, E = T (x) T symmetric nondegenerate, the corner table of B, the exact sequences of Prop a:dual-dimensions, the resolution of s for every i, the socle element and the finite trace computation — not the Ext vanishing over A (Y, F, Lambda, Z, A, M are not published)”
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
KitaKen1, KitaKen1/tachikawa-characteristic-zero: Tachikawa counterexample over the rationals in Lean 4, 9 October 2026. Source
Counterexample to Tachikawa’s second conjecture over the rationals, Lean.
Hexagon, Hexagon: a repository for research in the mathematical sciences and theoretical computer science, 7 October 2026. Source
hexagon:2610.00143v2 (琪越 唐) extends OpenAI’s counterexample constructions to any field with an element of infinite multiplicative order, covering both Tachikawa conjectures.
References outside the release
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Moritz Firsching, anxkhn, Paul Lezeau, Formal Conjectures: mark problems solved in the OpenAI math release and link formal proofs, 11 October 2026. Source
A Formal Conjectures maintainer, after source inspection, links the counterexample to the Auslander-Reiten conjecture, stated as a negation for Artin algebras.
Questions awaiting review
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Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 199The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Compare the proposed paper and Lean matches
Family 199Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1
In An explicit counterexample to the Auslander-Reiten conjecture, 23 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method
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.
The challenge statements state less than the headline, and OpenAI’s Lean note says so.
Their method
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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 432 s; peak memory 6.5 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: the ten-dimensional algebra C, its trivial extension T and trace, the 179-entry cochain (cocycle and boundary identities, evaluation q^3), the resolution of s over C for every i, and the finite identities of the lift — not the Ext vanishing over Lambda (F, Lambda, Z are not published)”
Proposition 4.2 of A counterexample to Tachikawa’s second conjecture, 23 September 2026
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: the ten-dimensional algebra C (and its identity with the companion paper’s), T and its symmetrizing form, E = T (x) T symmetric nondegenerate, the corner table of B, the exact sequences of Prop a:dual-dimensions, the resolution of s for every i, the socle element and the finite trace computation — not the Ext vanishing over A (Y, F, Lambda, Z, A, M are not published)”
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.