Banach’s isometric conjecture (1932) completed in the real case with key steps from ChatGPT 5.5/5.6 Pro; complex and quaternionic cases follow five days later
Awaiting review
Importance: major (4 of 5)The takeaway
Xinbao Lu and Kaiwen Yang posted a proof of Banach’s 1932 isometric conjecture for every odd n (arXiv 2608.13536, 13 Aug 2026).
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 30 September 2026
Your AI and this story
- GPT-6 Astra105 days after its cutoff
- Claude Opus 5.544 days after its cutoff
- Gemini 4 Hydrogen135 days after its cutoff
- Grok 4.774 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 44 days before it.
Key facts
- Banach (1932): if all n-dimensional subspaces of a real Banach space X (fixed 1 < n < dim X) are linearly isometric, is X a Hilbert space? Gromov proved it for even n; several odd cases were settled later; Lu–Yang cover every odd n
- Method: principal bundle theory combined with a Brouwer degree argument (arXiv 2608.13536, 22 pages)
- AI declaration (Lu–Yang): ‘The authors had reduced the main problem to proving Theorem 3.10 before using generative AI tools. An approach to that theorem subsequently emerged through extensive interactions with ChatGPT 5.5 Pro and ChatGPT 5.6 Pro. The initial draft of Section 3 and the corresponding parts in Section 2 were generated by GPT 5.6 Sol … and subsequently checked and rewritten by the authors’
- Complex and quaternionic case: Acuaviva & Kania (arXiv 2608.18257, 18 Aug 2026) adapt ‘the bundle-degree mechanism introduced by Lu and Yang’; ‘ChatGPT 5.6 Sol assisted with certain technical details needed to carry out these extensions’
- Status: preprints, not peer-reviewed or formalised as of 30 Sep 2026
What happened
Lu and Yang closed the odd-dimensional cases of Banach’s question about isometric subspaces, using ChatGPT Pro to find the approach to the decisive theorem. The complex and quaternionic versions followed from another team within a week, also with AI help.
Why it matters
Banach’s conjecture is one of the oldest questions in the geometry of normed spaces, and Gromov’s even-dimensional solution had left the remaining odd cases open. The human–AI division of labour is unusually well documented: human reduction, AI-found approach, AI-drafted proof, human verification.
What is disputed or not yet verified
| Verification | Preprints; not peer-reviewed |
|---|
Sources
2 sources from 1 site. Numbers match the chips in the text.
2 sources: 2 primary
Primary
Changes
- Filed