Tingley’s sphere-isometry problem

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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. (OpenAI’s Lean note)

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Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments

The paper states

Theorem 1.1

In A positive solution to Tingley’s problem, 23 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Let X,Y be nonzero real Banach spaces and let f:S_X\to S_Y be a surjective isometry. Then

T(0)=0,\qquad T(x)=\norm{x}f\left(\frac{x}{\norm{x}}\right)\quad(x\ne0)

is a surjective real-linear isometry. It is the unique linear extension of f.

Reported proof verification: One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
Papers and explanations: One current paper, readable here as a page beside its PDF.

Papers

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A positive solution to Tingley’s problem

23 September 2026, current edition.

Read it here PDF

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.

One Lean statement in this family

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`TingleySphereIsometry`

The paper states

Theorem 1.1

In A positive solution to Tingley’s problem, 23 September 2026

Let X,Y be nonzero real Banach spaces and let f:S_X\to S_Y be a surjective isometry. Then

T(0)=0,\qquad T(x)=\norm{x}f\left(\frac{x}{\norm{x}}\right)\quad(x\ne0)

is a surjective real-linear isometry. It is the unique linear extension of f.

The words they share: real, surjective, isometry, extension, linear, spaces, norm, tingley, unique, nonzero, banach.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Tingley’s sphere-isometry extension

TingleySphereIsometry.lean, the release lists it as a main result.

theorem tingley_sphere_isometry_full    {X : Type u} {Y : Type v}    [NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X]    [NormedAddCommGroup Y] [NormedSpace ℝ Y] [CompleteSpace Y]    [Nontrivial X] [Nontrivial Y]    (f : UnitSphere X → UnitSphere Y) (hf : Isometry f) (hs : Function.Surjective f) :    ∃ T : X ≃ₗᵢ[ℝ] Y,      (∀ u : UnitSphere X, T (u : X) = (f u : Y)) ∧      T 0 = 0 ∧      (∀ x : X, ∀ hx : x ≠ 0, T x = ‖x‖ • (f (normalize x hx) : Y)) ∧      Function.Surjective T ∧      (∀ L : X →ₗ[ℝ] Y, (∀ u : UnitSphere X, L (u : X) = (f u : Y)) →        L = T.toLinearEquiv.toLinearMap) := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).

Their note: tingley_sphere_isometry_full: for nontrivial complete real normed spaces X, Y and any surjective Isometry f : {x // ‖x‖ = 1} → {y // ‖y‖ = 1} there is T : X ≃ₗᵢ[ℝ] Y (bijective real-linear isometry) extending f (radial formula, T 0 = 0) and any ℝ-linear L (not assumed continuous) extending f equals T, so existence, surjectivity and uniqueness are all in the conclusion with no dimension, separability or reflexivity hypothesis.

“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).

“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

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.

AI-drafted comparisons, with no recorded human review

Tingley’s sphere-isometry extension TingleySphereIsometry
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: every surjective isometry between unit spheres of nonzero real Banach spaces extends uniquely to a real-linear isometry, by the radial formula.

Who ran the proof-verification program?

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These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Tingley’s sphere-isometry extension `TingleySphereIsometry`

Two reports, all at the release’s recorded version.

Reported by	Their word	Their account	Date	Original report
cert-machine	“CERTIFIED”	Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.	10 October 2026	Original report
MathVet	“comparator-sandboxed-pass”	Re-run through Comparator in a sandboxed virtual machine: accepted.	8 October 2026	Original report

Missing or incomplete source records

The program that collects Atlas’s source records, Statement Map, could not obtain or interpret some source material. These notes describe those limits; they do not identify mathematical errors.

TingleySphereIsometry: Tracked source lean/OAI/Analysis/SphereIsometry/Main.lean unavailable: bytes were not captured at fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly IsomorphicCurrent edition; has an accompanying Lean statement Family Lipschitz equivalent Banach spaces need not be linearly isomorphicCites A positive solution to Tingley’s problem.

Follow-up work by others

anxkhn, Series of 15 issues and 15 draft PRs adding release results to Formal Conjectures (issues 6972 to 6997, PRs 6977 to 7002), 8 October 2026. Source
Proposes adding the release’s Tingley’s sphere-isometry extension theorem to Formal Conjectures (issue 6983, PR 6988).

References outside the release

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24 references in A positive solution to Tingley’s problem

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.
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