Result 028, Number theory

Uniformly bounded components of Gaussian-prime graphs

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.

Lean formalization Proof

The bigger picture

Why it matters

Can you travel forever through prime numbers in the complex plane using only bounded-length steps? The manuscript reports that you cannot, and claims a stronger limit on how many primes any such journey can reach.

What changes?

Gaussian primes are prime elements among complex numbers whose real and imaginary parts are integers. For any fixed finite distance bound D, join two Gaussian primes when their distance is at most D. The manuscript claims that every connected component, meaning all primes reachable from one another along these links, has at most a number of vertices depending only on D. This includes primes on the coordinate axes. The bound is nonexplicit: no explicit value is supplied.

What does that help mathematicians do?

Ruling out an infinite walk through distinct primes alone would still leave open the possibility of arbitrarily large finite clusters. The stronger claim rules that out too: with D fixed, choosing a different starting prime cannot produce indefinitely larger reachable networks. This gives researchers a uniform restriction on the geometry of Gaussian primes, rather than merely saying that each individual bounded-step exploration eventually exhausts its options.

Are there practical applications?

The immediate value is foundational: the claim describes how Gaussian primes fail to connect across the integer lattice in the complex plane. The reported proof uses a finite, repeating sieve obstruction, geometric sampling and information-theoretic estimates. These are the stated ingredients behind the connectivity barrier, not a demonstrated practical algorithm; the nonexplicit bound supplies no numerical exploration limit.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Bounded-Step Walks on Gaussian Primes

September 26, 2026 29 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026,
  author = {{OpenAI}},
  title = {{Bounded-Step Walks on Gaussian Primes}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/paper.pdf}{OAI:Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/028.md.

Uniformly bounded components of Gaussian-prime graphs

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

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 DD. More strongly, one finite bound depending only on DD 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 DD is supplied.

Comparator links

Result Comparator statement
Uniform bounds for bounded-step Gaussian-prime walks GaussianMoat.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.