Functional analysis·OpenAI Math·Result 329·Lean formalisation listed
A counterexample to metric-entropy duality
One of the 372 results in OpenAI's public mathematics catalogue, filed under functional 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 counterexample to metric-entropy duality (functional 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 It has not been peer reviewed.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 28 of 84 in Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More · starts at 9:27 · watch from there on YouTube →
What OpenAI claims
Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and those of the polar bodies, even when the covering body is a cube. (Lean)
Paper
Counterexamples to the duality conjecture for metric entropy
OpenAI's abstract We disprove Pietsch's dimension-free duality conjecture for metric entropy. For every proposed pair of universal constants, we construct origin-symmetric convex bodies that violate the corresponding covering-entropy inequality, already when the covering body is a cube.
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 OpenAI posted a preprint on 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.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?
- Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and those of the polar bodies, even when the covering body is a cube. (Lean)
- Where is the proof?
- In the paper, Counterexamples to the duality conjecture for metric entropy (PDF).
- Is there a video?
- Yes, on YouTube (29:20, chapter 28).
Also in
- Chapter 4 of 10 in Tingley's Problem, Separable Quotient Problem and 8 More in Functional Analysis · starts at 1:03 · 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 28 from 9:27 · description, sources and chapters →
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