Algebraic and complex geometry·OpenAI Math·Result 043
P = W for fixed-determinant SLn moduli spaces
One of the 372 results in OpenAI's public mathematics catalogue, filed under algebraic and complex geometry: 1 preprint, the latest dated 24 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 result titled ‘P = W for fixed-determinant SLn moduli spaces’ (algebraic and complex geometry) in a preprint dated 24 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More, is on YouTube.
Chapter 37 of 83 in Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · starts at 12:31 · watch from there on YouTube →
What OpenAI claims
Proves Pk=W2k=W2k+1 on the full rational cohomology of smooth coprime fixed-determinant, trace-free Higgs moduli spaces and their character varieties for composite ranks over smooth projective complex curves of genus at least two. This includes variant cohomology and, together with the known prime-rank theorems, establishes the fixed-determinant P = W conjecture in every coprime rank.
Paper
P=W in composite rank for fixed determinant
OpenAI's abstract We prove the P = W conjecture on the full rational cohomology of fixed-determinant, trace-free Higgs moduli spaces in composite rank and coprime degree, for smooth projective complex curves of genus at least two. Together with the established prime-rank cases, this gives the equality in every coprime rank.
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 OpenAI posted a preprint on 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.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 Pk=W2k=W2k+1 on the full rational cohomology of smooth coprime fixed-determinant, trace-free Higgs moduli spaces and their character varieties for composite ranks over smooth projective complex curves of genus at least two. This includes variant cohomology and, together with the known prime-rank theorems, establishes the fixed-determinant P = W conjecture in every coprime rank.
- Where is the proof?
- In the paper, P=W in composite rank for fixed determinant (PDF).
- Is there a video?
- Yes, on YouTube (28:48, chapter 37).
Also in
- Chapter 7 of 10 in Nagata's Conjecture, Bloch's Conjecture and 8 More in Algebraic and Complex Geometry · starts at 2:04 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · Compilation · 28:48 · chapter 37 from 12:31 · description, sources and chapters →
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