Real and complex analysis·OpenAI Math·Result 079
Local smoothing in three dimensions
One of the 372 results in OpenAI's public mathematics catalogue, filed under real and complex analysis: 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 ‘Local smoothing in three dimensions’ (real and complex analysis) in a preprint dated 24 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 It has not been peer reviewed.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 82 of 84 in Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More · starts at 28:16 · watch from there on YouTube →
What OpenAI claims
Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions. The estimate holds throughout the full strict range 2 < p < ∞, with Sobolev regularity above max {0,1-3/p}. In particular, the critical L3 estimate holds with every positive Sobolev loss.
Paper
Critical local smoothing for the three-dimensional wave equation
OpenAI's abstract We prove the critical L3 local smoothing estimate for the wave equation in three spatial dimensions, with every positive Sobolev loss. This resolves Sogge's Euclidean local smoothing conjecture in dimension three.
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 OpenAI posted a preprint on 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.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?
- Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions. The estimate holds throughout the full strict range 2 < p < ∞, with Sobolev regularity above max {0,1-3/p}. In particular, the critical L3 estimate holds with every positive Sobolev loss.
- Where is the proof?
- In the paper, Critical local smoothing for the three-dimensional wave equation (PDF).
- Is there a video?
- Yes, on YouTube (29:20, chapter 82).
Also in
- Chapter 7 of 10 in Falconer Distance Conjecture, Bochner–Riesz Conjecture and 8 More · starts at 2:06 · 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 82 from 28:16 · description, sources and chapters →
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