Topology·OpenAI Math·Result 315
The four-dimensional Singer conjecture
One of the 372 results in OpenAI's public mathematics catalogue, filed under topology: 1 preprint, the latest dated 25 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 a proof of the four-dimensional Singer conjecture (topology) in a preprint dated 25 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 It has not been peer reviewed.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 21 of 84 in Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More · starts at 7:00 · watch from there on YouTube →
What OpenAI claims
Proves that the L2-Betti numbers of the universal cover of every closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same conclusion holds for every finite connected aspherical integral Poincaré complex of formal dimension four, proving the four-dimensional Singer conjecture in this wider class.
Paper
The Singer conjecture in dimension four
OpenAI's abstract We prove the four-dimensional Singer conjecture: the L2-Betti numbers of the universal cover of a closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same vanishing holds for finite connected aspherical integral Poincaré complexes of formal dimension four, including nonorientable ones.
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 OpenAI posted a preprint on 25 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.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 the L2-Betti numbers of the universal cover of every closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same conclusion holds for every finite connected aspherical integral Poincaré complex of formal dimension four, proving the four-dimensional Singer conjecture in this wider class.
- Where is the proof?
- In the paper, The Singer conjecture in dimension four (PDF).
- Is there a video?
- Yes, on YouTube (29:20, chapter 21).
Also in
- Chapter 10 of 10 in Kervaire Invariant Problem at Prime 3, The Grothendieck Homotopy Hypothesis and 8 More · starts at 3:09 · 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 21 from 7:00 · description, sources and chapters →
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