CATALOGUE 322
Tingley's sphere-isometry problem
This catalogue contains one manuscript claiming that a surjective isometry between unit spheres of real Banach spaces extends uniquely to a real-linear isometry of the spaces. The overview follows its proof by maximizing the distance defect between different radii, aligning chords, and eliminating the defect through convexity.
Paper in this catalogue
These summaries give the principal scope; consult the cited original results for the complete hypotheses and statements.
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| # | Manuscript and overview | Principal claim | Role and available overview |
|---|---|---|---|
| 01 | A positive solution to Tingley’s problem →Short name: A positive solution to Tingley’s problemSeptember 23, 2026 · 12 pagesOriginal paper (GitHub) ↗ | Every surjective isometry between unit spheres of nonzero real Banach spaces extends uniquely to an onto real-linear isometry, without dimension or separability restrictions. Theorem 1.1 · p. 1 ↗ | Attain a maximal distance defect, align chords through common supports and Darbo’s theorem, then eliminate the defect by support and convexity arguments. Read the proof overview → |
Guide to the article
The overview of A positive solution to Tingley’s problem first presents the overall proof of Theorem 1.1. It then explains the invariant closed convex set and decreasing noncompactness in Proposition 4.2, followed by the support and convexity contradiction in Proposition 5.3. Darbo’s fixed-point theorem and the Mazur–Ulam reflection argument are classical inputs; their application conditions are distinguished from the constructions supplied by the manuscript.
Sources and checking scope
The official subject is Functional analysis, covering catalogue numbers 322–332. The catalogue entry and paper order use the fixed official revision checked on 20 October 2026. The manuscript is dated 23 September 2026 and has 12 pages.
The key OpenAI26Cat denotes OpenAI’s OpenAI Research Catalog, fixed at commit 1581d4cbdd69a0faf34b2089614fe2a814316a6a.
[OpenAI26Cat, Catalogue 322; Functional analysis, p. 35]
Theorem 1.1; §§3–6 · pp. 1, 3–9
The main statement, central reductions, and statements and application conditions of the direct classical inputs were compared with the sources. This does not assert independent verification of the complete proof or expert peer review. Recursive checking of external proofs and a complete audit of prior work are outside scope; the article records the detailed scope and version notes.