Functional analysis·OpenAI Math·Result 326·Lean formalisation listed
The cotype–cotype conjecture under the approximation property
One of the 372 results in OpenAI's public mathematics catalogue, filed under functional 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 cotype–cotype conjecture under the approximation property (functional analysis) in a preprint dated 23 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 It has not been peer reviewed.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 27 of 84 in Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More · starts at 9:05 · watch from there on YouTube →
What OpenAI claims
Resolves the cotype–cotype conjecture for real Banach spaces with the approximation property. Such a nonzero space is K-convex if and only if both it and its dual have finite Rademacher cotype, with possibly different exponents. Equivalently, these cotype assumptions force nontrivial Rademacher type. (Lean)
Paper
The cotype–cotype conjecture under the approximation property
OpenAI's abstract We prove the cotype–cotype conjecture under the ordinary approximation property. A nonzero real Banach space with this property is K-convex if and only if both the space and its dual have finite Rademacher cotype, possibly with different exponents.
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 OpenAI posted a preprint on 23 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.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 cotype–cotype conjecture for real Banach spaces with the approximation property. Such a nonzero space is K-convex if and only if both it and its dual have finite Rademacher cotype, with possibly different exponents. Equivalently, these cotype assumptions force nontrivial Rademacher type. (Lean)
- Where is the proof?
- In the paper, The cotype–cotype conjecture under the approximation property (PDF).
- Is there a video?
- Yes, on YouTube (29:20, chapter 27).
Also in
- Chapter 3 of 10 in Tingley's Problem, Separable Quotient Problem and 8 More in Functional Analysis · starts at 0:41 · 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 27 from 9:05 · description, sources and chapters →
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