Algebraic and complex geometry·OpenAI Math·Result 037
The ordinary-double-point volume gap
One of the 372 results in OpenAI's public mathematics catalogue, filed under algebraic and complex geometry: 2 preprints, the latest dated 24 September 2026. This page quotes OpenAI's own summary of the claim, links the papers and abstracts, and embeds the animation.
In one line: OpenAI reports a result titled ‘The ordinary-double-point volume gap’ (algebraic and complex geometry) in 2 preprints, the latest 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 It has not been peer reviewed.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 32 of 83 in Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · starts at 10:48 · watch from there on YouTube →
What OpenAI claims
Proves the ordinary-double-point volume-gap conjecture: every singular complex algebraic klt germ of dimension n ≥ 2, with zero boundary, has normalized volume at most 2(n-1)n. Equality holds precisely for an analytic ordinary double point.
Papers (2)
The ordinary-double-point gap in every dimension
OpenAI's abstract We prove the ordinary-double-point gap conjecture for boundary-zero complex algebraic klt germs in every dimension: a singular n-dimensional germ has normalized volume at most 2(n-1)n, with equality precisely at an analytic ordinary double point.
The normalized-volume gap in dimension four
OpenAI's abstract We prove that every singular complex algebraic klt fourfold germ with zero boundary has normalized volume at most 162, with equality precisely for an analytic ordinary double point. This resolves the ordinary-double-point volume-gap conjecture in dimension four.
Status
Preprint. The papers have 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 OpenAI posted 2 preprints, the latest dated 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.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 ordinary-double-point volume-gap conjecture: every singular complex algebraic klt germ of dimension n ≥ 2, with zero boundary, has normalized volume at most 2(n-1)n. Equality holds precisely for an analytic ordinary double point.
- Where is the proof?
- In the paper, The ordinary-double-point gap in every dimension (PDF).
- Is there a video?
- Yes, on YouTube (28:48, chapter 32).
Also in
- Chapter 2 of 10 in Nagata's Conjecture, Bloch's Conjecture and 8 More in Algebraic and Complex Geometry · starts at 0:21 · 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 32 from 10:48 · description, sources and chapters →
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