What was proved, and what you are looking at

The question

A Gaussian integer is a point a + bi of the plane with whole-number coordinates. Some of them cannot be factored into smaller ones: those are the Gaussian primes. Plot them and you get the eight-fold starfield on this page.

In 1962 Basil Gordon asked: starting near the centre, can you walk off to infinity stepping only on Gaussian primes, if no step may be longer than some fixed length? The gaps you would have to jump are called moats.

What was proved

For every step length there is a finite number that bounds the size of every connected cluster of Gaussian primes. So no walk on distinct primes with bounded steps goes on forever, wherever it starts.

That is Theorem 1.1 of “Bounded-Step Walks on Gaussian Primes”, paraphrased. It is stronger than the original question: it covers every starting prime, not only the centre, and gives one bound per step length for all of them.

The paper is credited to OpenAI and dated 26 September 2026. It was released on 6 October 2026 in github.com/openai/math, a collection of 722 manuscripts produced by an unreleased internal model. The repository includes a formalisation in Lean, a language in which a computer checks each step of a proof: the formal statement is short and worth reading.

What was not proved

Where the moats are. The bound is non-explicit: the proof shows that a number exists for each step length and gives no way to compute it. For steps longer than 6, nobody knows how far the central island reaches. The film ends in the dark for that reason.

This is a new result. It is a preprint with a formal proof attached, not yet a paper that other mathematicians have refereed. The formal statement says what is claimed above.

The picture is not the proof

Everything drawn here is computed. Nothing is illustrative. But a computation can only ever find one moat at a time, for one step length, and there are infinitely many step lengths. The pictures were available for decades and settled nothing.

The proof is a different kind of object. It shows that sieving the grid by a finite, well-chosen set of primes already leaves a repeating pattern that no infinite walk can thread, using an entropy argument about what a long walk would have to know about its own position. None of that is visible on this page.

What is computed here

Live rows are computed in your browser when the page opens: an exact sieve finds every Gaussian prime within 10,760 of the centre, and a breadth-first search walks them. Counts are for the whole plane, all eight symmetric copies included.

Ahead of time: the same algorithm, run once ahead of time on every prime out to 230,000, and stored in this page as a map whose cells are 32 to 256 units wide. At that scale a pixel holds thousands of primes; it lights when the walk first enters it, and its brightness is the density of reached primes.

Published rows are from Nobuyuki Tsuchimura’s 2004 computation and are drawn only as labelled rings. They were not recomputed. The first seven rows agree with his table and with Gethner, Wagon and Wick (1998).

Colour shows order of arrival within each step length, dark first and gold last, so the bright rim of each region is where that walk died. The faint bands are contours of equal step count. The discs you see up close have a diameter of one step: two primes are within a step of each other exactly when their discs touch, so an island is a cluster of touching discs and a moat is the water around it.