Answers Jacobsthal’s quadratic-bound question: every interval of \(Ck^2\) consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute constant C. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss. (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
It bears on Erdős problem #970, as listed by Thomas Bloom, erdosproblems.com.
In A quadratic bound for Jacobsthal’s function
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.
There is an absolute constant C>0 such that
In A quadratic bound for Jacobsthal’s function
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
There are absolute constants c>0 and z_0 with the following property. For every real z\ge z_0 and every choice of a residue class a_p\pmod p for each prime p\le z,
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.
Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.
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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.
JacobsthalThe paper states
There is an absolute constant C>0 such that
The words they share: integer, quadratic, jacobsthals, bound.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
Jacobsthal.lean, the release lists it as a main result.
def JacobsthalQuadratic : Prop := ∃ C : ℝ, 0 < C ∧ ∀ k : ℕ, 0 < k → ∃ m : ℕ, IsJacobsthalBound k m ∧ (m : ℝ) ≤ C * (k : ℝ) ^ 2theorem erdos_970_quadratic : NumberTheoryLean.Targets.JacobsthalQuadratic := 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.
One independent group reports that Comparator accepted a run for this Lean statement file: cert-machine. Who ran the proof-verification program
JacobsthalImprovedThe paper states
There is an absolute constant C>0 such that
The words they share: integer, log, jacobsthals, bound, absolute.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
JacobsthalImproved.lean, the release lists it as a supporting statement.
def JacobsthalIteratedLog : Prop := ∃ C : ℝ, 0 < C ∧ ∀ k : ℕ, 0 < k → ∃ m : ℕ, IsJacobsthalBound k m ∧ (m : ℝ) ≤ C * (k : ℝ) ^ 2 / (Real.log (Real.log (3 * (k : ℝ)))) ^ 2 theorem erdos_970_iterated_log : NumberTheoryLean.Targets.JacobsthalIteratedLog := 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.
One independent group reports that Comparator accepted a run for this Lean statement file: cert-machine. 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.
IsJacobsthalBound Model.lean, line 12JacobsthalQuadratic Model.lean, line 16MathVet 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; both the quadratic and the (log log 3k)^-2 improved bound are stated with elementary definitions.
“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
Let $h(k)$ be the least interval length that guarantees an integer coprime to any prescribed positive modulus with at most $k$ distinct prime factors. The formalization proves the paper’s strengthened Jacobsthal bound $h(k)\le Ck^2/(\log\log(3k))^2$ for one absolute $C>0$ and every $k\ge1$. Intervals may begin at any signed integer. The earlier quadratic bound is also retained as a separate statement; the optimal order of growth is not determined.
Quadratic Jacobsthal bound Jacobsthal
Unreviewed AI draft: states a weaker form of a paper’s theorem. States the earlier quadratic bound h(k) ≤ Ck², which the main theorem improves by the factor (log log 3k)²; the note keeps it as a separate statement.
Jacobsthal bound with an iterated-logarithm saving JacobsthalImproved
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem h(k) ≤ Ck²/(log log 3k)² for every k ≥ 1, with intervals allowed to start at any integer.
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.
JacobsthalOne report, 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 |
JacobsthalImprovedOne report, 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 |
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
mo271, GreensOpenProblems/46: green_46.improve_upper is answered by the improved Jacobsthal bound (issue 6908), 7 October 2026. Source
The quadratic Jacobsthal bound h(k) << k^2/(log log 3k)^2 answers the open upper-bound variant.
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.
This family’s papers cite no paper elsewhere in the release.
No paper elsewhere in the release cites this family’s papers.
Luka Carroll, Murray Elder, Testing epimorphism onto the bicyclic monoid is $\mathsf{NP}$-complete, 6 October 2026. Source
Carroll and Elder remark that substituting OpenAI’s quadratic bound for Jacobsthal’s function would improve the exponent in their Theorem C, but they leave their weaker bound in place.
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
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.
Offer to write an explanationCompare one Lean statement with its paper: Jacobsthal, JacobsthalImprovedSuggest a source attributionSuggest a change
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.
Author not stated. Source
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 Jacobsthal bounds to Erdos problem 970 and adds an iterated-log variant.
eigendash, eigendash/openai-math-visualised: numerical probes of seven result families, 8 October 2026. Source
For the primorials P_k with k up to 10, the ratio of Jacobsthal’s function to k^2/(log log 3k)^2 rises to about 0.71 and then stays flat, consistent with the quadratic bound over that small range.
Thomas Bloom, Erdős problems in the OpenAI proofs, 8 October 2026. Source
Bloom lists Erdős 970 (a quadratic-type bound for Jacobsthal’s function), names Iwaniec’s earlier bound, and links the formalisation.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Why this passage is shown
Theorem 1.1
In A quadratic bound for Jacobsthal’s function
Why this passage is shown
Theorem 1.2
In A quadratic bound for Jacobsthal’s function
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, Jacobsthal
Proof-verification report
cert-machine, JacobsthalImproved
Lean statement
Jacobsthal
Lean statement
JacobsthalImproved