Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture. (OpenAI’s Lean note)
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Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments
In An algebra of infinite little finitistic dimension, 23 September 2026
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
There is a finite-dimensional unital complex algebra A such that, for every integer m\ge1, there is a finite-dimensional left A-module N_m with
In particular, \findim A=\infty.
No Lean statement in the release is paired with this theorem.
In An algebra of infinite little finitistic dimension, 23 September 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
There is a finite-dimensional unital complex algebra \Lambda such that
All four dimensions here use left modules. Thus the little and big finitistic dimensions of \Lambda are infinite on the left and zero on the right. Injective left \Lambda-modules do not generate the unbounded derived category of all left \Lambda-modules, whereas injective right \Lambda-modules generate the unbounded derived category of all right \Lambda-modules.
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FinitisticAsymmetryThe paper states
In An algebra of infinite little finitistic dimension, 23 September 2026
There is a finite-dimensional unital complex algebra \Lambda such that
All four dimensions here use left modules. Thus the little and big finitistic dimensions of \Lambda are infinite on the left and zero on the right. Injective left \Lambda-modules do not generate the unbounded derived category of all left \Lambda-modules, whereas injective right \Lambda-modules generate the unbounded derived category of all right \Lambda-modules.
The words they share: injective, asymmetry, extreme, finitistic, generate, little, derived, category, dimensions, complex, algebra, dimensional, modules, finite, unbounded, zero, infinite.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
FinitisticAsymmetry.lean, the release lists it as a main result.
theorem extreme_left_right_asymmetry : ∃ (Λ : Type) (_ : Ring Λ) (_ : Algebra ℂ Λ), FiniteDimensional ℂ Λ ∧ littleFinitisticDimension Λ = ⊤ ∧ bigFinitisticDimension Λ = ⊤ ∧ littleFinitisticDimension Λᵐᵒᵖ = 0 ∧ bigFinitisticDimension Λᵐᵒᵖ = 0 ∧ ¬ InjectivesGenerate Λ ∧ InjectivesGenerate Λᵐᵒᵖ := by sorryThis 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.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program
LittleFinitisticThe paper states
In An algebra of infinite little finitistic dimension, 23 September 2026
There is a finite-dimensional unital complex algebra \Lambda such that
All four dimensions here use left modules. Thus the little and big finitistic dimensions of \Lambda are infinite on the left and zero on the right. Injective left \Lambda-modules do not generate the unbounded derived category of all left \Lambda-modules, whereas injective right \Lambda-modules generate the unbounded derived category of all right \Lambda-modules.
The words they share: algebra, category, complex, derived, dimensional, dimensions, finite, finitistic, generate, infinite, left, little, modules, right, unbounded, zero.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
LittleFinitistic.lean, the release lists it as a main result.
theorem exists_counterexample : ∃ (A : Type) (_ : Ring A) (_ : Algebra ℂ A), FiniteDimensional ℂ A ∧ littleFinitisticDimension A = ⊤ ∧ ∀ m : ℕ, 0 < m → ∃ N : ModuleCat.{0} A, Module.Finite A N ∧ ((2*m-2 : ℕ) : WithBot ℕ∞) ≤ projectiveDimension N ∧ projectiveDimension N < ⊤ := by sorryThis 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.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
InjectivesGenerate Model.lean, line 15MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
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 or imply the whole advertised result (their term: “full”).
Their note: exists_counterexample : ∃ A (_ : Ring A) (_ : Algebra ℂ A), FiniteDimensional ℂ A ∧ littleFinitisticDimension A = ⊤ ∧ ∀ m > 0, ∃ N, Module.Finite A N ∧ (2m-2 : WithBot ℕ∞) ≤ projectiveDimension N ∧ projectiveDimension N < ⊤, where littleFinitisticDimension A := ⨆ (M) (_ : Module.Finite A M) (_ : projectiveDimension M < ⊤), projectiveDimension M (Mathlib projectiveDimension): unbounded finite projective dimensions over a finite-dimensional complex algebra, as in the summary. FinitisticAsymmetry is an extra secondary result.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
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 or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
OpenAI’s own note on what its Lean covers
The little finitistic-dimension conjecture predicts that finitely generated modules of finite projective dimension over a fixed finite-dimensional algebra have bounded projective dimensions. The formalized counterexample is a finite-dimensional complex algebra with a finitely generated module of projective dimension at least $2m-2$ and still finite for every $m\ge1$.
A separate formalized result gives a finite-dimensional complex algebra whose little and big finitistic dimensions are infinite on the left and zero on the right. Injectives fail to generate its unbounded derived category on the left and generate it on the right.
Left/right finitistic-dimension asymmetry FinitisticAsymmetry
Unreviewed AI draft: states a consequence of a paper’s theorem. States the paper’s corollary: an algebra whose finitistic dimensions are infinite on the left and zero on the right, with the corresponding behaviour of injective generation.
Infinite little finitistic dimension LittleFinitistic
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a finite-dimensional complex algebra with finitely generated modules of finite projective dimension at least 2m − 2 for every m.
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.
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FinitisticAsymmetryTwo 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. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
LittleFinitisticTwo 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. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
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.
Hexagon, Hexagon: a repository for research in the mathematical sciences and theoretical computer science, 7 October 2026. Source
hexagon:2610.00140v4 (琪越 唐) simplifies OpenAI’s algebra of infinite little finitistic dimension and extends it to every field, with a Lean formalisation added on 10 October.
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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.
Author not stated. Source
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Why this passage is shown
Theorem 1.1
In An algebra of infinite little finitistic dimension, 23 September 2026
Why this passage is shown
Corollary 1.2 (Extreme left–right asymmetry)
In An algebra of infinite little finitistic dimension, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, FinitisticAsymmetry
Proof-verification report
MathVet, FinitisticAsymmetry
Proof-verification report
cert-machine, LittleFinitistic
Proof-verification report
MathVet, LittleFinitistic
Lean statement
FinitisticAsymmetry
Lean statement
LittleFinitistic