A non-mathematician has spent a month and a fortune in tokens asking Claude to pick an open problem, then built a proof of Conway’s 1976 conjecture using Lean. The proof has passed mechanical checks from the Palomar registry, and experts familiar with both Lean and the field say the statement seems correct. The author, who calls themselves Vibed, has posted the full account on their blog.
Conway’s Refinement Conjecture
Conway’s refinement conjecture concerns a property of omnific integers within the surreal number system. The conjecture states that if ab = cd, there are integers e, f, g, h such that a = ef, b = gh, c = eg, d = fh. In other words, any two factorizations of an omnific integer share a common refinement.
John Conway proposed the conjecture in 1976, and it has stood as one of the last unresolved questions about his own number system. Vibed writes that the proof has not been independently verified by mathematicians, but he has decent reasons to believe it is correct and genuinely invites a refutation.
The Surreal Number System
Surreal numbers are Conway’s invention, or discovery, of a previously unknown number system that contains all real numbers, all ordinal numbers, and combinations of both. The system starts from a single rule: take all the numbers you have so far, then “spawn” a new number in every gap between them.
On the first day, the gap is “between nothing and nothing.” Zero is born. On the second day, there are two gaps: “between nothing and zero” and “between zero and nothing.” Two numbers spawn in those gaps, called –1 and 1. On the third day, four gaps appear, and numbers like –2, –1/2, 1/2, and 2 fill them. The process repeats forever.
Jump to the “infinite-th” day, called ω. With an infinite supply of already-born numbers, you find infinitely many new gaps waiting to be filled. These include “between [1, 2, 3, …] and nothing,” “between nothing and […, –3, –2, –1],” “between 0 and [1, 1/2, 1/4, 1/8…],” and “between [positive already born numbers whose squares are below 2] and [positive already born numbers whose squares are above 2].”
By this day, the system contains every real number, every ordinal, and more. The binary tree based on the single spawning rule gives birth to consistently definable arithmetic.
Claude’s Choice
Vibed asked Claude to pick an open problem in the surreal numbers. The request was simple: which unsolved problem pulls you the most?
Claude narrowed the choice to Conway’s arithmetic. Specifically, the question the L’Innocente–Mantova machinery had sharpened to a point: is every irreducible in K((ℝ^≤0)) with infinite support prime? By their reduction, this is exactly equivalent to Conway’s 1976 conjecture.
Claude’s answer was bold. The problem is the last of Conway’s own conjectures about his own numbers still standing, and 2026 is ONAG’s fiftieth birthday. That was enough for Vibed to commit.
The Lean Proof
The proof was built using Lean, a theorem prover. Lean allows for formal verification, meaning the proof can be checked mechanically. The Palomar registry has passed the mechanical checks, and a few people familiar with both Lean and the field have said the statement seems correct.
Vibed admits that Claude’s claim about the problem having been perfectly reduced was wrong. The reduction machinery had not fully resolved the conjecture as Claude initially described. Despite that, the proof stands on its own terms.
What the Author Learned
Vibed’s approach was unconventional. They did not attempt to understand the substance of the problem before solving it. Instead, they leaned on Claude to guide the selection and then worked through the formalization in Lean.
The project took an entire month of free time and a boatload of tokens. The payoff was a Lean proof of a fifty-year-old conjecture.
The Sentimental Reason
The choice of problem was also sentimental. This year is the fiftieth anniversary of ONAG, Conway’s book that introduced the surreal numbers. Vibed chose the problem for that reason, even though they still do not know whether it was indeed Conway’s last standing conjecture about the surreal numbers.
The Verification Gap
The proof has not been independently verified by mathematicians. Vibed acknowledges this openly. They have decent reasons to believe the proof is correct, but they are not claiming finality.
The mechanical checks from the Palomar registry and the expert nods on Lean and the field provide some assurance. Assuming the proof does not rely on a Lean kernel bug, it is likely to be legitimate. But independent verification remains outstanding.
Key Facts Box
| Fact | Detail |
|---|---|
| Conjecture | Conway’s refinement conjecture, 1976 |
| Inventor | John Conway |
| Number system | Surreal numbers |
| Proof tool | Lean |
| Verification | Passed Palomar registry checks |
| Tokens | Boatload, per author |
| Time | One month of free time |
The Comparison Table
| Aspect | Vibed’s Approach |
|---|---|
| Problem selection | Guided by Claude |
| Proof construction | Lean formalization |
| Verification | Mechanical checks |
| Openness | Public blog post |
| Domain expertise | Non-mathematician |
The comparison shows the difference between Vibed’s informal, curiosity-driven path and the traditional academic route. Both approaches have their strengths, but Vibed’s method demonstrates that AI-assisted problem selection can lead to productive work.
The story is a reminder that the tools of mathematics are changing. Lean and similar systems offer a way to formalize proofs and check them mechanically. That is a powerful capability.
Vibed’s journey from casual curiosity to a published proof is a story worth telling. It shows what happens when someone asks Claude to pick a problem and follows the answer. The method worked.
The sentimentality of the choice adds a human dimension to the technical achievement. Choosing a problem tied to the fiftieth anniversary of ONAG gave the work a personal stake. Vibed’s admission that they still do not know whether this was Conway’s last standing conjecture shows the limits of information even when the proof succeeds.
The story is a reminder of the power of curiosity and the tools available today.
See the video the story is built around at overreacted.io.
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