For every fixed C > 0, a positive proportion of consecutive prime gaps exceed \(C\log p_n\), throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where \(p_n/n\) increases have positive lower density, answering Erdős and Prachar. (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 #968, as listed by Thomas Bloom, erdosproblems.com.
In Positive lower density of large prime gaps, 25 September 2026
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
For every fixed real C>0, there are constants c(C)>0 and N_0(C) such that, for every integer N\ge N_0(C),
In Positive lower density of large prime gaps, 25 September 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
The set
has positive lower asymptotic density.
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.
PrimeGapsThe paper states
It is paired with Theorem 1.1 and Corollary 1.2, each shown below.
In Positive lower density of large prime gaps, 25 September 2026
For every fixed real C>0, there are constants c(C)>0 and N_0(C) such that, for every integer N\ge N_0(C),
In Positive lower density of large prime gaps, 25 September 2026
The set
has positive lower asymptotic density.
The words it shares with Theorem 1.1: log, real, integer, large, gaps, prime, density, lower; with Corollary 1.2: asymptotic, density, lower, positive.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
PrimeGaps.lean, the release lists it as a supporting statement.
theorem large_gaps_and_ratio_density : (∀ C : ℝ, 0 < C → ∃ c : ℝ, 0 < c ∧ ∃ N₀ : ℕ, ∀ N : ℕ, N₀ ≤ N → c * (N : ℝ) ≤ ((LargePrimeGaps.largeGapIndices C N).card : ℝ)) ∧ 0 < lowerAsymptoticDensity ratioIncreaseIndices := 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
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; challenge states the large-gap theorem and the ratio corollary. Lower density is a custom sSup-defined liminf (checked).
Definitions the group asks readers to compare with the paper: def lowerAsymptoticDensity (A : Set ℕ) : ℝ := sSup {d : ℝ | ∃ N0 : ℕ, ∀ N : ℕ, N0 ≤ N → d ≤ (initialCount A N : ℝ) / (N : ℝ)} (custom liminf-style density; correct since the set is nonempty and bounded by 1)
“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 formalized supplement proves that the indices $n$ for which $p_n/n<p_{n+1}/(n+1)$ have positive lower asymptotic density, where $p_n$ is the $n$th prime. Thus the normalized prime sequence has a positive-density set of increases. This is the prime-ratio corollary associated with the paper’s theorem on a positive lower density of large prime gaps.
Positive lower density of prime-ratio increases PrimeGaps
Unreviewed AI draft: states a paper’s theorem as the paper does. The statement’s first conjunct is the main theorem (for every C > 0, a positive proportion of n ≤ N have p_{n+1} − p_n > C log p_n) and its second is the prime-ratio corollary. The scope note describes only the corollary.
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.
PrimeGapsTwo 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.
This family’s papers cite no paper elsewhere in the release.
No paper elsewhere in the release cites this family’s papers.
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
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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 proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. 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 large prime gaps result to Erdos problem 968.
Thomas Bloom, Erdős problems in the OpenAI proofs, 8 October 2026. Source
Bloom lists Erdős 968 (a prime gap density question), notes it follows from a stronger large gaps result using sieve tools of Goldston, Pintz and Yildirim and Maynard and Tao, and links the formalisation.
eigendash, eigendash/openai-math-visualised: numerical probes of seven result families, 8 October 2026. Source
For primes up to 2 times 10^8, the proportion of gaps exceeding C log p is about 0.60, 0.34 and 0.11 at C = 0.5, 1 and 2, positive as the theorem predicts for every fixed C.
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 Positive lower density of large prime gaps, 25 September 2026
Why this passage is shown
Corollary 1.2
In Positive lower density of large prime gaps, 25 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, PrimeGaps
Proof-verification report
MathVet, PrimeGaps
Lean statement
PrimeGaps