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
Xinbao Lu and Kaiwen Yang posted a proof of Banach's 1932 isometric conjecture for every odd n (arXiv 2608.13536, 13 Aug 2026). With Gromov's even-n theorem, this completes the real case: a real Banach space whose n-dimensional subspaces are all isometric must be a Hilbert space. They had reduced the problem to one theorem themselves. The approach to that theorem 'emerged through extensive interactions with ChatGPT 5.5 Pro and ChatGPT 5.6 Pro', and GPT-5.6 Sol drafted the key section. On 18 Aug Acuaviva and Kania extended the method to complex and quaternionic spaces with GPT-5.6 Sol's help.
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- Status: preprints, not peer-reviewed or formalised as of 30 Sep 2026
Science result
- Field
- mathematics / functional analysis / convex geometry
- Problem
- Banach's isometric subspace conjecture (open since 1932)
- Result
- Real case completed (all odd n, with Gromov's even n); complex and quaternionic analogues also proved.
- AI system
- ChatGPT 5.5 Pro, ChatGPT 5.6 Pro, GPT-5.6 Sol
- Human role
- AI-assisted: the humans made the reduction; the key approach came out of dialogue with ChatGPT Pro, and GPT-5.6 Sol drafted proofs that the authors checked and rewrote
- Verification
- Preprints; not peer-reviewed
- Status
- pending
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.
Changelog
- 2026-09-30: created
Related events
- OpenAI broadly releases GPT-5.6 (Sol, Terra, Luna) after government-gated preview ★★★★
- Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help (July–September catalogue) ★★★★
Sources (2)
- paperarXiv 2608.13536: A solution to Banach's isometric conjecture (Lu, Yang)
- paperarXiv 2608.18257: Banach's Isometric Conjecture over the Complex Field (Acuaviva, Kania)
id: 2026-08-13-banach-isometric-conjecture-solved · updated 2026-09-30 · open in the interactive timeline