Breakthrough Tracker record
Two preprints claim to settle Banach’s isometric conjecture over real, complex and quaternionic spaces
Xinbao Lu and Kaiwen Yang claim the remaining real odd-dimensional cases, while Antonio Acuaviva and Tomasz Kania claim the remaining complex cases and a quaternionic counterpart. Together with earlier results, the two first-version preprints would settle the classical finite-dimensional real and complex isometric-subspace conjectures.
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- Stable ID
math-banach-isometric-conjecture-real-case-2026- Revision
math-banach-isometric-conjecture-real-case-2026.v2- Field
- Mathematics · Functional analysis, convex geometry and topology
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional claimed proof
- Last checked
AI role
Lu and Yang report extensive interactions with ChatGPT 5.5 Pro and 5.6 Pro after reducing the problem to a key theorem. Acuaviva and Kania report using ChatGPT 5.6 Sol for technical details and extensions. Both groups state that the human authors checked and own the results.
Record details
- Problem or result
- Banach’s 1932 isometric-subspace conjecture over real and complex scalars, plus a quaternionic analogue
- Authors
- Xinbao Lu and Kaiwen Yang; Antonio Acuaviva and Tomasz Kania
- Institutions
- Tongji University; Lancaster University; Czech Academy of Sciences and Jagiellonian University
- Result date
- Real-case preprint submitted August 13 and revised August 31; complex/quaternionic preprint submitted August 18, 2026
Why it matters
If both arguments hold, they close the remaining finite-dimensional cases of a 94-year-old problem connecting normed spaces, ellipsoids, bundle topology and Brouwer degree theory, and extend the mechanism to quaternionic spaces.
Limits
The real-case manuscript is now version 2, with a substantially expanded and restructured proof but an unchanged theorem claim. Both manuscripts remain non-peer-reviewed preprints. No independent specialist correctness assessment or formal proof was located, and the complex/quaternionic paper adapts the real-case mechanism rather than independently confirming it.
Sources
- Primary: Xinbao Lu and Kaiwen Yang, A solution to Banach’s isometric conjecture
- Primary: Antonio Acuaviva and Tomasz Kania, Banach’s Isometric Conjecture over the Complex Field
Correction and revision history
- 2026-08-22 — Added after primary-source, scope, status, AI-role and limitation review.
- 2026-09-01 — Rechecked real-case version 2, which expands and restructures the proof while retaining the same theorem claim; provisional status retained.
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