Operator algebras·OpenAI Math·Result 296·Lean formalisation listed
The generator problem for finite factors
One of the 372 results in OpenAI's public mathematics catalogue, filed under operator algebras: 1 preprint, the latest dated 23 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports an answer to the generator problem for finite factors (operator algebras) in a preprint dated 23 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More, is on YouTube.
Chapter 6 of 84 in Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More · starts at 1:46 · watch from there on YouTube →
What OpenAI claims
Proves that every type II1 factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion P ⊂ M of such factors, the unitaries u with M=W*(P,u) form a dense Gδ subset in the trace 2-norm topology. (Lean)
Paper
Relative generation and the generator problem for finite factors
OpenAI's abstract We prove that every type II1 factor with separable predual is generated by one operator, equivalently by two self-adjoint operators. Combined with the established direct-integral reduction to this case, this gives an affirmative solution of the generator problem for von Neumann algebras with separable predual.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 23 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- Proves that every type II1 factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion P ⊂ M of such factors, the unitaries u with M=W*(P,u) form a dense Gδ subset in the trace 2-norm topology. (Lean)
- Where is the proof?
- In the paper, Relative generation and the generator problem for finite factors (PDF).
- Is there a video?
- Yes, on YouTube (29:20, chapter 6).
Also in
- Chapter 5 of 9 in Kaplansky's Quasitrace Conjecture, Naimark's Problem and 7 More in Operator Algebras · starts at 1:24 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More · Compilation · 29:20 · chapter 6 from 1:46 · description, sources and chapters →
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