Same shape, not the same?
Two AI preprints claim separable Banach spaces that match distances up to fixed factors but are not linearly the same
Covers result 324 in OpenAI’s catalog: all 2 papers. See its connections to other results and the work it cites.
3:51. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Two Banach spaces are linearly isomorphic when a linear map matches them, stretching or shrinking distances by at most fixed factors. Drop linearity and keep only the distance bounds, and you get a bi-Lipschitz bijection. In the plane this changes nothing, since any two norms there are linearly isomorphic anyway. From the square norm to the round norm the best linear map distorts by the √2, and every nonlinear map our computer tried did worse.
In 1978 Aharoni and Lindenstrauss found two non-separable spaces that are bi-Lipschitz equivalent but not linearly isomorphic, and wrote that it is of course of interest to find also separable examples. Kalton listed the separable question as Problem 3 in his 2008 survey, and a tool of Godefroy and Kalton showed why the old approach cannot work for separable spaces. Papers still called it open as late as October 5, 2026.
Result 324 is two OpenAI papers, of 23 and 25 pages, dated September 24 and 26 and made public October 6, 2026. The first claims separable real Banach spaces X and Y and a bijection between them that keeps every distance between 4/21 and 76/25 of the original, yet X and Y are not linearly isomorphic. X contains a copy of c₀ of Hilbert spaces, a sequence of whole Hilbert spaces, with no distortion at all, and Y contains none, not even a distorted one.
What it doesn’t claim
The second paper gives an independent example: a space with no linear copy of c₀ that is bi-Lipschitz equivalent to itself plus c₀, which also answers Kalton’s Problem 2. Not claimed: anything about the classical spaces ℓ₁, L₁ or C[0,1], a reflexive example, or complex scalars. And the first paper’s Y may still contain ordinary c₀.
Where it stands
- Posted as preprints
- OpenAI, 2 papers, 23 and 25 pages, dated September 24 and 26, 2026
- Peer reviewed
- Not yet
- Formalized in Lean
- Both main theorems (docs)
Last checked October 19, 2026. Both papers are unrefereed and unchanged since release. Unusually, both main theorems are formalized in Lean with statements matching the papers, including the first paper’s exact constants, in about 19,000 lines, not run by us. Two outside projects re-ran the checks and they were accepted. No specialist has published an assessment, and we found no reported error or competing claim. OSVerify has not reviewed it.
How the proof goes
- A Hilbert hotel with slots. The key map is a Hilbert hotel. Split Hilbert space into finite blocks and mark one slot in each. Move block m’s slot coefficient to block 2m, and put the m-th newcomer into block 2m − 1. Frozen, this fits Hilbert space plus an extra copy of itself onto Hilbert space exactly; in a small finite version, our computer confirms it preserves lengths. Then the slots are allowed to turn, depending on the point, but only extremely slowly, keeping distances within 4%. The blocks grow fast: the eighth already has over 7 billion dimensions.
- Telling X and Y apart. A triangular change of variables with an explicit inverse turns this into the bi-Lipschitz bijection between X and Y. To tell them apart, the proof shows a key linear operator is completely continuous. Together with a known completeness property of the space it acts on, this means any copy of c₀ of Hilbert spaces in Y would collapse. The difference is a growth pattern: in c₀, 1,024 basis vectors sum to length 1; in a Hilbert space, 1,024 perpendicular unit vectors sum to 32.
The numbers in the video
- Ours, sampled: from the square norm to the round norm, the best linear map distorts by √2 ≈ 1.41; the nonlinear maps we tried gave 2.51, 3.49 and 21.4.
- The paper’s constants: 4/21 and 76/25; slots keep distances within 24/25 to 26/25; block 8 has 7,151,611,609 dimensions (our arithmetic from the paper’s formula).
- Ours: in c₀, N basis vectors sum to length 1; in a Hilbert space, N perpendicular unit vectors sum to √N (32 for N = 1,024).
- Lean: 60 + 34 files and 18,725 lines in the two import closures (counted by us; not run by us).
Read the source
- Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic (PDF), 23 pages.
- Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0 (PDF), 25 pages.
- The Lean formalization notes.
- The collection’s catalog, where this is result 324.
The 2 papers in this result
OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.
Transcript
Read the full script
0:00The question
Two Banach spaces are linearly isomorphic when a linear map matches them, stretching or shrinking distances by at most fixed factors. Drop linearity and keep only the distance bounds, and you get a bi-Lipschitz bijection. In the plane this changes nothing, since any two norms there are linearly isomorphic anyway. From the square norm to the round norm the best linear map distorts by the √2, and every nonlinear map our computer tried did worse. In 1978 Aharoni and Lindenstrauss found two non-separable spaces that are bi-Lipschitz equivalent but not linearly isomorphic, and wrote that it is of course of interest to find also separable examples. Kalton listed the separable question as Problem 3 in his 2008 survey, and a tool of Godefroy and Kalton showed why the old approach cannot work for separable spaces. Papers still called it open as late as October 5, 2026.
0:59The claim
Result 324 is two OpenAI papers, of 23 and 25 pages, dated September 24 and 26 and made public October 6, 2026. The first claims separable real Banach spaces X and Y and a bijection between them that keeps every distance between 4/21 and 76/25 of the original, yet X and Y are not linearly isomorphic. X contains a copy of c₀ of Hilbert spaces, a sequence of whole Hilbert spaces, with no distortion at all, and Y contains none, not even a distorted one. The second paper gives an independent example: a space with no linear copy of c₀ that is bi-Lipschitz equivalent to itself plus c₀, which also answers Kalton’s Problem 2. Not claimed: anything about the classical spaces ℓ₁, L₁ or C[0,1], a reflexive example, or complex scalars. And the first paper’s Y may still contain ordinary c₀.
2:02How the proof goes
The key map is a Hilbert hotel. Split Hilbert space into finite blocks and mark one slot in each. Move block m’s slot coefficient to block 2m, and put the m-th newcomer into block 2m − 1. Frozen, this fits Hilbert space plus an extra copy of itself onto Hilbert space exactly; in a small finite version, our computer confirms it preserves lengths. Then the slots are allowed to turn, depending on the point, but only extremely slowly, keeping distances within 4%. The blocks grow fast: the eighth already has over 7 billion dimensions. A triangular change of variables with an explicit inverse turns this into the bi-Lipschitz bijection between X and Y. To tell them apart, the proof shows a key linear operator is completely continuous. Together with a known completeness property of the space it acts on, this means any copy of c₀ of Hilbert spaces in Y would collapse. The difference is a growth pattern: in c₀, 1,024 basis vectors sum to length 1; in a Hilbert space, 1,024 perpendicular unit vectors sum to 32.
3:14Where it stands
Both papers are unrefereed and unchanged since release. Unusually, both main theorems are formalized in Lean with statements matching the papers, including the first paper’s exact constants, in about 19,000 lines, not run by us. Two outside projects re-ran the checks and they were accepted. No specialist has published an assessment, and we found no reported error or competing claim. If it holds, even the full metric shape of a separable real Banach space, up to bounded distortion, does not determine its linear structure.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.