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.SingleFold.main
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
defMainStatement : Prop := ∀ (n : ℕ), 1 ≤ n → ∀ (S : Set (Fin n → ℕ)), REPred (fun a => a ∈ S) → ∃ (m : ℕ), 1 ≤ m ∧ ∃ P : MvPolynomial (Fin n ⊕ Fin m) ℤ, Represents P S⋮theoremmain : MainStatement := 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.
noncomputabledefevalNat {n m : ℕ} (P : MvPolynomial (Fin n ⊕ Fin m) ℤ) (a : Fin n → ℕ) (w : Fin m → ℕ) : ℤ := MvPolynomial.eval (Sum.elim (fun i => (a i : ℤ)) (fun j => (w j : ℤ))) P
defRepresents {n m : ℕ} (P : MvPolynomial (Fin n ⊕ Fin m) ℤ) (S : Set (Fin n → ℕ)) : Prop := ∀ a, (a ∈ S ↔ ∃ w, evalNat P a w = 0) ∧ ∀ w v, evalNat P a w = 0 → evalNat P a v = 0 → w = v
Possible paper match, with no recorded review
No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.
Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: input, integers, most, polynomial, promise, solution and whether.
Corollary 1.2 of Single-fold Diophantine representations
Corollary(Undecidability with an at-most-one-solution promise)
There exist integers m,D\ge1 for which no algorithm has the following property: on every input polynomial F\in\ZZ[W_1,\ldots,W_m] of total degree at most D satisfying the promise
\#\{w\in\NN^m:F(w)=0\}\le1,
the algorithm halts and correctly decides whether this set is nonempty.
There exist integers $m,D\ge1$ for which no algorithm has the following
property: on every input polynomial $F\in\ZZ[W_1,\ldots,W_m]$ of total
degree at most $D$ satisfying the promise
\[
\#\{w\in\NN^m:F(w)=0\}\le1,
\]
the algorithm halts and correctly decides whether this set is nonempty.
Other possible paper matches, closest wording first
Proposition 4.2 of Single-fold Diophantine representations. Words of the Lean statement, its names split into words, that also appear in the theorem: uniqueness, coordinate, input, single, representation, integer and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: diophantine.
Theorem 1.1 of Single-fold Diophantine representations. Words of the Lean statement, its names split into words, that also appear in the theorem: polynomial and integer. Found in the paper’s title, which its main theorem states: fold, single and representations. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: enumerable and recursively.
Proposition 7.2 of Single-fold Diophantine representations. Words of the Lean statement, its names split into words, that also appear in the theorem: natural, coordinate, polynomial, input and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: exactly, tuple, complete and one.
Proposition 6.1 of Single-fold Diophantine representations. Words of the Lean statement, its names split into words, that also appear in the theorem: representation, fold and single. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: complete and diophantine.
Proposition 5.2 of Single-fold Diophantine representations. Words of the Lean statement, its names split into words, that also appear in the theorem: single, representation and fold. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: complete.
Proposition 3.1 of Single-fold Diophantine representations. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.
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
three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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.
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; the run was still in progress at this snapshot (192 accepted, 224 pending).
AcceptedTheir term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 321 s; peak memory 6.5 GiB).
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 242’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.
Finds the whole advertised result stated or impliedTheir term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: MainStatement : ∀ n ≥ 1, ∀ S : Set (Fin n → ℕ), REPred (· ∈ S) → ∃ m ≥ 1, ∃ P : MvPolynomial (Fin n ⊕ Fin m) ℤ, Represents P S with Represents P S := ∀ a, (a ∈ S ↔ ∃ w, P(a,w) = 0) ∧ ∀ w v, P(a,w) = 0 → P(a,v) = 0 → w = v (witnesses in ℕ): exactly the single-fold representation of every r.e. set. The summary's corollary, undecidability under an at-most-one-solution promise, is not separately stated (docs say so).
OpenAI’s own note on what this Lean statement covers
“The formalization establishes this for every recursively enumerable subset of $\mathbb N^n$, $n\ge1$: an integer polynomial has exactly one complete natural-number witness tuple for members and none for nonmembers.”
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
29 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: 5 to its own declarations, 18 to Mathlib’s documentation. 4 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 OAInamespace SingleFold/-- Evaluation over the integers of an integer polynomial at natural inputs. -/noncomputabledefevalNat {n m : ℕ} (P : MvPolynomial (Fin n ⊕ Fin m) ℤ) (a : Fin n → ℕ) (w : Fin m → ℕ) : ℤ :=MvPolynomial.eval (Sum.elim (fun i => (a i : ℤ)) (fun j => (w j : ℤ))) P/-- Exact existence on members, no witness on nonmembers, and uniqueness of thewhole witness on every input. No coordinates are hidden outside `w`. -/defRepresents {n m : ℕ} (P : MvPolynomial (Fin n ⊕ Fin m) ℤ) (S : Set (Fin n → ℕ)) : Prop := ∀ a, (a ∈ S ↔ ∃ w, evalNat P a w = 0) ∧ ∀ w v, evalNat P a w = 0 → evalNat P a v = 0 → w = v/-- Single-fold representation with positive input and witness lengths. -/defMainStatement : Prop := ∀ (n : ℕ), 1 ≤ n → ∀ (S : Set (Fin n → ℕ)), REPred (fun a => a ∈ S) → ∃ (m : ℕ), 1 ≤ m ∧ ∃ P : MvPolynomial (Fin n ⊕ Fin m) ℤ, Represents P Stheoremmain : MainStatement := bysorryend SingleFoldend 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.
For every integer n\ge1 and every recursively enumerable set S\subseteq\NN^n, there are an integer m\ge1 and a polynomial P\in\ZZ[A_1,\ldots,A_n,W_1,\ldots,W_m] such that, for every a\in\NN^n,
For every integer $n\ge1$ and every recursively enumerable set
$S\subseteq\NN^n$, there are an integer $m\ge1$ and a polynomial
$P\in\ZZ[A_1,\ldots,A_n,W_1,\ldots,W_m]$ such that, for every $a\in\NN^n$,
\[
\#\{w\in\NN^m:P(a,w)=0\}=
\begin{cases}1,&a\in S,\\0,&a\notin S.\end{cases}
\]
Corollary(Undecidability with an at-most-one-solution promise)
There exist integers m,D\ge1 for which no algorithm has the following property: on every input polynomial F\in\ZZ[W_1,\ldots,W_m] of total degree at most D satisfying the promise
\#\{w\in\NN^m:F(w)=0\}\le1,
the algorithm halts and correctly decides whether this set is nonempty.
There exist integers $m,D\ge1$ for which no algorithm has the following
property: on every input polynomial $F\in\ZZ[W_1,\ldots,W_m]$ of total
degree at most $D$ satisfying the promise
\[
\#\{w\in\NN^m:F(w)=0\}\le1,
\]
the algorithm halts and correctly decides whether this set is nonempty.
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
3 local vocabulary declarations
Lines of Lean, without comments or blank lines
12
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
Single-fold Diophantine representations
Family 242: Single-fold Diophantine representations and undecidability under an at-most-one-solution promise
Candidate pairing
Corollary (Undecidability with an at-most-one-solution promise)
There exist integers m,D\ge1 for which no algorithm has the following property: on every input polynomial F\in\ZZ[W_1,\ldots,W_m] of total degree at most D satisfying the promise
\#\{w\in\NN^m:F(w)=0\}\le1,
the algorithm halts and correctly decides whether this set is nonempty.
Single-fold Diophantine representations
TeX label cor:unique-h10, build/main.tex, line 123
The Lean statement says
defMainStatement : Prop := ∀ (n : ℕ), 1 ≤ n → ∀ (S : Set (Fin n → ℕ)), REPred (fun a => a ∈ S) → ∃ (m : ℕ), 1 ≤ m ∧ ∃ P : MvPolynomial (Fin n ⊕ Fin m) ℤ, Represents P S⋮theoremmain : MainStatement := 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
Two independent readers, MathVet and cert-machine, disagree on whether this family's challenges state its headline; a specialist reading would settle it.