Uniformly bounded components of Gaussian-prime graphs
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have uniformly bounded steps. More strongly, for every distance bound D, the graph joining Gaussian primes at distance at most D has uniformly bounded finite component sizes, depending only on D, including primes on the coordinate axes. (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.
For every finite real D there is a finite B_D such that every connected component of G_D has at most B_D vertices. The bound is independent of the starting prime and includes primes on the coordinate axes. Consequently every sequence of distinct Gaussian primes whose successive distances are at most D has at most B_D terms.
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every D\ge1 there is a finite set \mathcal P_D of rational primes congruent to 1 modulo 4 such that \mathcal A(\mathcal P_D) contains no infinite sequence of distinct points with successive distances at most D. The set \mathcal P_D depends only on D.
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.
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.
We prove the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have bounded steps. More strongly, for each fixed finite step bound, the connected components of the Gaussian-prime graph have uniformly bounded size. This bound applies to every starting prime, including primes on the coordinate axes, and is nonexplicit. The proof constructs a finite periodic sieve obstruction using geometric sampling and information-theoretic estimates.
One Lean statement in this family
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For every finite real D there is a finite B_D such that every connected component of G_D has at most B_D vertices. The bound is independent of the starting prime and includes primes on the coordinate axes. Consequently every sequence of distinct Gaussian primes whose successive distances are at most D has at most B_D terms.
The words they share: real, component, gaussian, finite, uniform, prime, connected, bound.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
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
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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; both the no-infinite-walk statement and the uniform finite component/walk-length bound are stated for every real D.
“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 Gaussian moat problem asks whether an infinite walk through distinct Gaussian primes can have bounded step lengths. The formalized result gives a negative answer for every real step bound $D$. More strongly, one finite bound depending only on $D$ limits the size of every connected component of the bounded-step graph and the length of every injective bounded-step walk. Axis primes and all associates are included. No explicit function of $D$ is supplied.
AI-drafted comparisons, with no recorded human review
Uniform bounds for bounded-step Gaussian-prime walksGaussianMoat
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for every step bound D, the connected components of the bounded-step Gaussian-prime graph have uniformly bounded size, so no infinite bounded-step walk exists.
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.
Uniform bounds for bounded-step Gaussian-prime walksGaussianMoat
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.
These reports examine claims in the papers using HOL Light, a different proof assistant, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
A finite nonempty set in the plane is not invariant under a nonzero translation
HOL Light card 028
Reported by
Ben Knill, openai-math-hol-cards
Method
Proof of a selected step in HOL Light
Their verdict
“PASS” The report records a proof of a supporting fact.
What it covers
Cross-check of a Lean-covered result; on-screen value
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.
Follow-up work by others
The Justin Sun Prize, The Justin Sun Prize: problem bank and award records, 11 October 2026. Source
JSP-000792 lists the release’s claim as Solved with independent mathematical review pending; Lean proof column: reported complete, independent verification pending.
Kenta Kitamura, Add three variants of Erdős Problem 952 (issue 7091), 10 October 2026. Source
Three Erdős 952 variants with Lean proofs proposed, the first being the release’s uniform Gaussian moat theorem.
KitaKen1, KitaKen1/erdos-952-binary-form-moat-classification: Lean proofs generalising the Gaussian moat results to binary forms, 10 October 2026. Source
Generalised moat and binary-form classification stated beyond OpenAI’s uniform Gaussian moat, with Lean proofs.
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 the Gaussian-prime moat result to Erdos problem 952, recording a norm against distance difference.
Thomas Bloom, Erdős problems in the OpenAI proofs, 8 October 2026. Source
Bloom lists Erdős 952 (bounded step walks on Gaussian primes), notes the problem likely began with Gordon and Motzkin, and links the formalisation.
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
Identify the theorem the catalogue describes
Family 028The 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 028Atlas 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 (Uniform component bound)
In Bounded-Step Walks on Gaussian Primes, 26 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
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 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 180 s; peak memory 6.6 GiB).
Date
11 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.
Uniform bounds for bounded-step Gaussian-prime walks
Role
the release lists it as a main result
Paired with
Theorem 1.1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.