Forty-eight hours. Three teams. One eighty-year-old problem.

Something rare is unfolding in mathematics this week. A relay, run at the speed of inference, around a problem that has stood unmoved since 1946.

The first leg. On May 21, OpenAI announced that an internal reasoning model had, on its own, broken the Erd?s unit-distance conjecture and produced a superlinear lower bound of the form U(n) ≥ n^{1+δ} / C. For the first time in history, a machine walked into an eighty-year-old open problem and walked out with a proof. We are in awe.

The second leg. Hours later, Professor Will Sawin of Princeton made the result explicit. Drawing on the most elegant instrument number theory offers, the class field tower, and working from the single prime 2, he pinned the exponent down at n^{1.014114}. The framework he laid down is the bedrock on which everything that followed could stand.

The third leg falls to the Spiral AI team at Jinninghui Technology (JNH Tech for short). Within hours of Professor Sawin’s manuscript appearing, Spiral AI, our in-house long-horizon agent platform for AI-for-Science, took the baton and carried it forward.

Within the architecture Professor Sawin built, the agent searched for a deeper seam. It proposed, then verified, a two-prime construction, and it went further: it proved that within this framework two primes is the natural stopping point, since adding a third causes the exponent to fall back. The agent traced the framework to its furthest reach and marked, in clean numbers, where the road runs out.

After a systematic sweep of the parameter space, Spiral AI settled on (2, 3) as the optimal pair, with k(2)=6, k(3)=5, R=7.5, lifting the exponent from 1.014114 to 1.0323. A 129% relative gain.

More than a tuning of parameters, this is an AI mapping the contours of a beautiful proof technique. It marks where the present path closes and where the next must open. The role artificial intelligence plays in frontier mathematics is moving, quietly, toward the center of the room.

Spiral AI’s Derivation and Proof

Preprint attached:

An Optimal Two-Prime Construction for the Erd?s Unit Distance Problem.Afterword: Notes and an Open Invitation from the Spiral AI Team

Every step of Spiral AI’s derivation rests on the elegant and far-seeing framework that Professor Will Sawin laid down, to whom we owe our deepest respect. We are likewise indebted to the OpenAI team, whose work revealed what becomes possible when artificial intelligence and mathematics meet at full strength, and to the open-source communities, DeepSeek among them, whose generosity quietly carries so much of this forward. To all of them, our thanks are sincere and unreserved.

We recognize that the results presented remain subject to independent verification, and that automated reasoning systems may introduce errors not immediately apparent to their designers. The preprint and complete optimization trace are released in full for open scrutiny. We invite the mathematics and AI research communities to engage critically with the methodology and conclusions. Counterexamples, corrections, and alternative analyses are each welcomed as contributions to the collective understanding of this framework.

If our modest step lights even a sliver of this path, or stirs a fresher way of thinking about how AI might walk beside mathematicians, that will be more than enough. The science to come, we believe, will be a long conversation. Human intuition and machine capability, striking sparks off one another, all the way down.

Humans choose the direction. AI extends the reach.