OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves that no finite modification of the primes can be written as \(A+B\) with \(A,B\subseteq\mathbb Z_{\ge0}\) each containing at least two elements. This resolves Ostmann’s inverse Goldbach conjecture on additive indecomposability. (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.
Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
If A,B\subseteq\N_0 satisfy |A|,|B|\ge2, then (A+B)\mathbin{\triangle}\mathcal P is infinite. Equivalently, every finite modification of \mathcal P is additively indecomposable into two sets with at least two elements each.
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 prove Ostmann’s inverse Goldbach conjecture: no set differing from the primes by finitely many elements can be written as \(A+B\), where A and B are sets of nonnegative integers with at least two elements each.
Two Lean statements in this family
Choose one to see it beside the paper theorem it is paired with.
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.
OstmannComplete.lean, the release lists it as a supporting statement.
defInverseGoldbach : Prop := ∀ A B : Set ℕ, A.Nontrivial → B.Nontrivial → Set.Infinite (@symmDiff (Set ℕ) (@SemilatticeSup.toMax (Set ℕ) (@Lattice.toSemilatticeSup (Set ℕ) (@CompleteLattice.toLattice (Set ℕ) (@CompleteBooleanAlgebra.toCompleteLattice (Set ℕ) (@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra (Set ℕ) (@Set.instCompleteAtomicBooleanAlgebra ℕ)))))) Set.instSDiff (sumset A B) primes)⋮theoreminverseGoldbach : InverseGoldbach := 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.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
If A,B\subseteq\N_0 satisfy |A|,|B|\ge2, then (A+B)\mathbin{\triangle}\mathcal P is infinite. Equivalently, every finite modification of \mathcal P is additively indecomposable into two sets with at least two elements each.
The words they share: infinite, primes, elements, least, finite, indecomposable, two.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
OstmannPrimes.lean, the release lists it as a supporting statement.
theoremmain : ∀ A B : Set ℕ, A.Nontrivial → B.Nontrivial → Set.Infinite ((A + B) ∆ {n : ℕ | Nat.Prime n}) := 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.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet 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:None; standard Mathlib objects only.
“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 formalization proves Ostmann’s inverse Goldbach conjecture. For any two sets $A,B$ of nonnegative integers, each containing at least two elements, the symmetric difference between their sumset $A+B$ and the set of primes is infinite. Thus no set differing from the primes by only finitely many elements can be decomposed as such a sumset. The linked statements include the full result and its two-infinite-summands case.
AI-drafted comparisons, with no recorded human review
Ostmann’s inverse Goldbach theorem and the infinite-summands caseOstmannComplete
Unreviewed AI draft: states a paper’s theorem as the paper does. States Ostmann’s inverse Goldbach conjecture as in the paper’s main theorem, together with the case of two infinite summands.
Additive indecomposability of the primes up to finite changesOstmannPrimes
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem directly: for sets A, B of natural numbers with at least two elements each, (A + B) and the primes have infinite symmetric difference.
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.
Ostmann’s inverse Goldbach theorem and the infinite-summands caseOstmannComplete
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.
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
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
Follow-up work by others
The Justin Sun Prize, The Justin Sun Prize: problem bank and award records, 11 October 2026. Source
JSP-000353 lists the release’s claim as Solved with independent mathematical review pending; Lean proof column: reported complete, independent verification pending.
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.
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 Ostmann’s inverse Goldbach result to Erdos problem 431.
grwtsk, [AUDIT.00] Audit the formal trust surface of the OpenAI math release (issue 2), 9 October 2026. Source
Some Ostmann inverse-Goldbach theorems carry explicit Published* hypotheses and are conditional until a provider theorem discharges them; a child issue plans the discharge graph.
Thomas Bloom, Erdős problems in the OpenAI proofs, 8 October 2026. Source
Bloom lists Erdős 431 (primes are not a sumset, Ostmann’s conjecture) in an 80-page PDF with the formalisation linked.
Questions awaiting review
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
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 013The 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 013Atlas 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 (Ostmann’s conjecture)
In The additive indecomposability of the primes, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
elements, finite, least, modification, ostmann
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 the headline as OpenAI wrote it.
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 with Lean’s kernel: accepted (the check took 307 s; peak memory 7.6 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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 2045 s; peak memory 8.8 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.