Real and complex analysis·OpenAI Math·Result 073·Lean formalisation listed
The Falconer distance conjecture
One of the 372 results in OpenAI's public mathematics catalogue, filed under real and complex analysis: 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 a proof of the Falconer distance conjecture (real and complex analysis) in a preprint dated 23 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 It has not been peer reviewed.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 53 of 83 in Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · starts at 18:03 · watch from there on YouTube →
What OpenAI claims
Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set E ⊂ ℝd with Hausdorff dimension greater than d/2 determines a set of Euclidean distances of positive Lebesgue measure. (Lean)
Paper
The Falconer distance conjecture in all dimensions
OpenAI's abstract We resolve the Falconer distance conjecture in every dimension. For every integer d ≥ 2, a compact subset of ℝd with Hausdorff dimension greater than d/2 determines a set of Euclidean distances of positive Lebesgue measure.
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 OpenAI posted a preprint on 23 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.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?
- Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set E ⊂ ℝd with Hausdorff dimension greater than d/2 determines a set of Euclidean distances of positive Lebesgue measure. (Lean)
- Where is the proof?
- In the paper, The Falconer distance conjecture in all dimensions (PDF).
- Is there a video?
- Yes, on YouTube (28:48, chapter 53).
Also in
- Chapter 2 of 10 in Falconer Distance Conjecture, Bochner–Riesz Conjecture and 8 More · 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 53 from 18:03 · description, sources and chapters →
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