Is the generalized Mukai conjecture 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 OpenAI posted a preprint on 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.

In OpenAI's preprint The generalized Mukai conjecture (PDF): https://github.com/openai/math/blob/main/preprints/The-Generalized-Mukai-Conjecture-September-24-2026/article.pdf

Yes, Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More on YouTube (28:48, chapter 48): https://www.youtube.com/watch?v=HqKGp54O-t0&t=979s

Algebraic and complex geometry·OpenAI Math·Result 063

The generalized Mukai conjecture

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 proof of the generalized Mukai conjecture (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 It has not been peer reviewed.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.

Compilation · 28:48 · 14 October 2026 · from 16:19Watch on YouTube →

Subscribe on YouTube

Chapter 48 of 83 in Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · starts at 16:19 · watch from there on YouTube →

What OpenAI claims

Proves the generalized Mukai conjecture: every positive-dimensional smooth complex projective Fano manifold of dimension n, Picard number ρ and pseudoindex ι satisfies ρ(ι-1) ≤ n, with equality exactly for (ℙ^ι-1)^ρ. Here the pseudoindex is the least anticanonical degree of a rational curve.
OpenAI, in its public catalogue of maths results

Paper

The generalized Mukai conjecture
24 September 2026 · Preprint, not peer reviewed · PDF
OpenAI's abstract We prove the generalized Mukai conjecture: every positive-dimensional smooth complex Fano manifold of dimension n, Picard number ρ, and pseudoindex ι satisfies ρ(ι-1) ≤ n. Equality holds precisely for the product of ρ copies of ℙ^ι-1.

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 OpenAI posted a preprint on 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.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 the generalized Mukai conjecture: every positive-dimensional smooth complex projective Fano manifold of dimension n, Picard number ρ and pseudoindex ι satisfies ρ(ι-1) ≤ n, with equality exactly for (ℙ^ι-1)^ρ. Here the pseudoindex is the least anticanonical degree of a rational curve.
Where is the proof?: In the paper, The generalized Mukai conjecture (PDF).
Is there a video?: Yes, on YouTube (28:48, chapter 48).

Also in

Chapter 8 of 10 in Kobayashi's Ampleness Conjecture, Kuznetsov Conjecture, Virasoro Conjecture and 7 More · starts at 2:26 · 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 48 from 16:19 · description, sources and chapters →

Related results in Algebraic and complex geometry

All 36 results in Algebraic and complex geometry →

Keep watching

More from OpenAI Math

All videos →