Single-fold Diophantine representations and undecidability under an at-most-one-solution promise
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Every recursively enumerable set of tuples of natural numbers has a Diophantine representation with exactly one auxiliary solution for each member and none for nonmembers. This proves the single-fold conjecture and hence the finite-fold conjecture. Diophantine solvability over the nonnegative integers remains undecidable even with an at-most-one-solution promise. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
The two independent assessments disagree on how much of the result described above is expressed by the Lean statements.Read the assessments
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
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,
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
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.
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.
Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
Every recursively enumerable set of natural-number tuples has a polynomial Diophantine representation with exactly one complete auxiliary tuple for each member. This proves the single-fold conjecture and, consequently, the finite-fold conjecture.
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
SingleFold
The paper states
Corollary 1.2(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.
The words they share: input, integers, most, polynomial, promise, solution, whether.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
SingleFold.lean, the release lists it as a main result.
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
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”.
Other mathematical statements in OpenAI’s Lean library
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.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
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: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).
“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.
The single-fold Diophantine conjecture asks whether every recursively enumerable set has a polynomial representation with a unique auxiliary witness for each member. 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.
The consequence that solvability remains undecidable under an at-most-one-solution promise is not separately formalized.
AI-drafted comparisons, with no recorded human review
Single-fold Diophantine representationsSingleFold
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: every recursively enumerable set has a single-fold Diophantine representation. The undecidability corollary is not separately formalised.
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.
Single-fold Diophantine representationsSingleFold
Three 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.
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.
Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
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.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
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 proof and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
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
Identify the theorem the catalogue describes
Family 242The 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 242Atlas 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 Single-fold Diophantine representations, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
enumerable, recursively
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
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. 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 with Lean’s kernel: accepted (the check took 321 s; peak memory 6.5 GiB).
Date
10 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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.