Banach's isometric conjecture settled in the remaining odd dimensions
A real Banach space whose n-dimensional subspaces are all isometric to one another must be a Hilbert space for every odd n, which together with Gromov's even-dimensional theorem closes a question Banach asked in 1932.
- Lab
- Independent
- Model
- ChatGPT 5.5 Pro, ChatGPT 5.6 Pro and GPT-5.6 Sol
- Field
- Mathematics
- Date
- 2026-08-13
- Human collaborators
- Xinbao Lu, Kaiwen Yang
- Problem posed
- 1932 · open 94 yrs
What was found
Banach asked in 1932 whether a real Banach space X all of whose n-dimensional subspaces are isometric, for some fixed n strictly between 1 and the dimension of X, has to be a Hilbert space. Gromov settled every even n, and later work closed several odd cases, leaving a gap. This 21-page paper proves the remaining odd n, combining bundle topology with Brouwer degree theory, which completes the conjecture in the real case. The complex case is not addressed.
Novelty check
The conjecture is a named 1932 problem of Banach with a documented partial-results ladder: Gromov's theorem covers even n, and subsequent work settled particular odd dimensions, which is the prior state of the art the paper states and cites. Searched arXiv and MathSciNet for a proof covering all odd n; none appears before this preprint, and the surveys of the problem still list the general odd case as open. The result is a new proof rather than a retrieval.
Caveats and known objections
A preprint, unrefereed and not formalized, and only days old at entry. Autonomy graded ai-assisted rather than collaborative: the declaration states that the authors had already reduced the main problem to proving Theorem 3.10 before using generative AI, that an approach to that theorem then emerged through extensive interactions with ChatGPT 5.5 Pro and 5.6 Pro, that GPT-5.6 Sol generated the initial draft of Section 3 and corresponding material in Section 2, and that the authors checked and rewrote it and take full responsibility. The reduction, which is the framing step, is human, and the machine contribution is one theorem inside it. The weaker defensible reading applies.
The open problem
MathDBBanach's isometric conjecture
383555
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Also recorded at
vibemathedbanach-s-isometric-conjecture
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
Nobody outside the lab has checked this yet.
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Entry history (1 event)
- AddedEntered the registry graded Author verified and AI-assisted.
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Graded author verified for verification and ai-assisted for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Banach's isometric conjecture settled in the remaining odd dimensions. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-13-banach-isometricBibTeX
@misc{whataifound-independent-2026-isometric,
title = {Banach's isometric conjecture settled in the remaining odd dimensions},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: AI-assisted.},
url = {https://whataifound.org/finding/2026-08-13-banach-isometric}
}