Result 322, Functional analysis

Tingley’s sphere-isometry problem

Resolves Tingley's problem: every surjective isometry between the unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the whole spaces. No dimension restriction is imposed, so the metric geometry of the unit sphere determines the Banach space up to linear isometry.

Lean formalization Proof

The bigger picture

Why it matters

Can the geometry of a space's unit sphere reveal its entire linear structure? The manuscript claims it can for real Banach spaces, even in infinite dimensions, linking distances on the sphere to the whole space.

What changes?

A real Banach space is a complete real vector space with a norm measuring length; its unit sphere consists of vectors of length one. The unreviewed manuscript reports that every distance-preserving map from one such sphere onto another extends uniquely to an onto map of the whole spaces preserving distances, addition and real scalar multiplication. This claims to resolve Tingley's problem for all nonzero real Banach spaces, with no dimension restriction or additional assumptions.

What does that help mathematicians do?

The claimed extension makes reconstruction explicit: fix zero, and send each nonzero vector to its length times the image of its unit-length direction. The substantive claim is that this construction must preserve both distances and linear operations. Consequently, researchers could rule out sphere isometries between real Banach spaces known not to be linearly isometric, without separately analyzing all possible distance-preserving maps between their spheres.

Are there practical applications?

Its immediate value is foundational: it would let functional analysts translate questions about distance-preserving symmetries of unit spheres into questions about linear isometries of the entire spaces. This connects metric and linear approaches to classifying Banach spaces. The supplied material describes a theoretical argument using fixed points and convexity, not a demonstrated computational or technological application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A positive solution to Tingley’s problem

September 23, 2026 12 pages Main result formalized in Lean

Every surjective isometry between the unit spheres of real Banach spaces extends uniquely to a surjective real-linear isometry, giving an affirmative solution to Tingley's problem. If distances between different radii are not preserved, we realize the positive maximal defect in a possibly enlarged pair of Banach spaces. We then align extremal chords using common supports and Darbo's fixed-point theorem and obtain a contradiction from support and convexity estimates.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-positive-solution-to-Tingleys-problem-September-23-2026,
  author = {{OpenAI}},
  title = {{A positive solution to Tingley's problem}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/paper.pdf}{OAI:A-positive-solution-to-Tingleys-problem-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/322.md.

Tingley’s sphere-isometry problem

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Tingley's problem asks whether a surjective isometry between the unit spheres of Banach spaces extends to a linear isometry of the spaces. The formalized result gives a unique surjective real-linear isometric extension for arbitrary nonzero real Banach spaces, with no finite-dimensionality, separability, reflexivity, or convexity assumptions. For complex spaces viewed as real spaces, the conclusion is real linearity rather than complex linearity.

Comparator links

Result Comparator statement
Tingley's sphere-isometry extension TingleySphereIsometry.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 10 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.