Kaplansky’s quasitrace conjecture reported disproved by an unreleased OpenAI model

In 1951 Irving Kaplansky introduced AW*-algebras, an algebraic generalization of von Neumann algebras. Two questions about them stayed open for decades. The first asks whether the quasitrace of a finite AW*-factor must be a trace, which is equivalent to asking whether every 2-quasitrace on a unital C*-algebra is a trace. The second asks whether every AW*-algebra is monotone complete. Haagerup proved the first for exact C*-algebras in 2014.

On 19 Oct 2026 Jananan Arulseelan and James E. Hanson posted a counterexample to the monotone completeness conjecture for finite AW*-algebras. They start from “the quasitrace manuscript”, an unpublished September 2026 manuscript in which an unreleased OpenAI model produced a claimed counterexample to the quasitrace conjecture. They reproduce that disproof by a modified construction and combine it with model-theoretic results (quasitracial ultraproducts, countable saturation, separable reflection). In their AI disclosure they say GPT 6 Astra completed a preliminary write-up of their own proof idea, which they then “significantly reworked and fixed”.

OpenAI has not published the quasitrace manuscript, and it does not appear among the files we can check. Everything here comes from the Arulseelan–Hanson paper.

If it holds, an AI system has settled one of the oldest problems in operator algebras, and specialists rebuilt the result within weeks and pushed it to the companion conjecture. It also shows that results from unreleased OpenAI models are reaching mathematicians privately before any public announcement.