According to the blog post dated September 18, 2026, the author spent a month of free time and a large number of tokens working with the AI model Claude to attempt a proof of Conway’s refinement conjecture. The conjecture, originally posed by John Conway in 1976, states that for omnific integers a, b, c, d satisfying ab = cd, there exist omnific integers e, f, g, h such that a = ef, b = gh, c = eg and d = fh. Omnific integers are defined as the integer part of the surreal number tree, which includes all ordinary integers as well as infinite quantities like ω and combinations such as ω/2. The author notes that surreal numbers arise from a simple rule: repeatedly filling gaps between existing numbers, a process that generates all real numbers, all ordinal numbers and further constructions. The author chose to work in the area of surreal numbers after asking Claude to suggest an open problem; Claude pointed to Conway’s arithmetic and, after reduction by the L’Innocente–Mantova machinery, identified the refinement conjecture as equivalent to a statement about irreducibles in K((ℝ^≤0)) with infinite support being prime. The author reports that after initial unsuccessful attempts that produced unfocused output, a more guided interaction with Claude led to a Lean formulation of the conjecture. The resulting Lean proof passed the mechanical checks of the Palomar registry, and a few individuals familiar with both Lean and the field told the author that the statement appeared correct. The author acknowledges that the proof has not been independently verified by mathematicians and invites refutation, while noting that, barring a Lean kernel bug, the proof is likely legitimate. The post also mentions that 2026 marks the fiftieth anniversary of Conway’s book On Numbers and Games (ONAG), which motivated the choice of problem for sentimental reasons. The author outlines the workflow: converting relevant papers to TeX for the model, iterating with Claude, and eventually obtaining a proof that the author believes meets the conjecture’s requirements. No claim is made about the broader significance of the result beyond the personal effort described.
Key facts
- The author used Claude and Lean to attempt a proof of Conway’s refinement conjecture.
- The proof passed mechanical checks from the Palomar registry.
- A few people familiar with Lean and the field said the statement seems correct.
- The proof has not been independently verified by mathematicians.