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Nonexpansive fixed points in reflexive Banach spaces

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Resolves Kirk’s reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without assuming uniform convexity. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

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

What the papers state

Theorem 1.1

In Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces, 24 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

Let X be a real reflexive Banach space, and let C\subseteq X be nonempty, norm closed, bounded, and convex. If F:C\to C satisfies

\norm{F(a)-F(b)}\le \norm{a-b}\qquad(a,b\in C),

then F has a fixed point.

Corollary 1.2 (Commuting families)

In Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces, 24 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

Let X be a real reflexive Banach space, and let C\subseteq X be nonempty, norm closed, bounded, and convex. For any pairwise commuting family \mathcal S of nonexpansive selfmaps of C, the common fixed set

\operatorname{Fix}(\mathcal S) =\{x\in C:T(x)=x\text{ for every }T\in\mathcal S\}

is nonempty and is a nonexpansive retract of C. That is, there is a nonexpansive map R:C\to\operatorname{Fix}(\mathcal S) that is the identity on \operatorname{Fix}(\mathcal S).

No Lean statement in the release is paired with this theorem.

The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.

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

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

One Lean statement in this family

Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.

For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

ReflexiveFixedPoints

The paper states

The words they share: real, reflexive, convex, bounded, closed, space, nonempty, point, norm, nonexpansive, maps, banach.

Possible Lean counterpart

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

Nonexpansive fixed points in reflexive spaces

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

theorem exists_fixedPoint_of_canonicallyReflexive    {X : Type u} [NormedAddCommGroup X] [NormedSpace ℝ X] [CompleteSpace X]    (hX : CanonicallyReflexive X) (C : Set X)    (hne : C.Nonempty) (hclosed : IsClosed C)    (hbounded : Bornology.IsBounded C) (hconvex : Convex ℝ C)    (F : C → C)    (hF : ∀ a b : C, ‖(F a : X) - (F b : X)‖ ≤ ‖(a : X) - (b : X)‖) :    ∃ a : C, F a = a := 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

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

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: exists_fixedPoint_of_canonicallyReflexive: for a complete real normed space X whose canonical map into the continuous bidual is surjective (CanonicallyReflexive), every self-map F of a nonempty closed bounded convex C ⊆ X with ‖F a - F b‖ ≤ ‖a - b‖ in the original norm has a fixed point, with no uniform convexity or normal-structure hypothesis, which is exactly the headline.

“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

The fixed-point question asks whether reflexivity suffices for nonexpansive maps on bounded convex sets. The formalized result gives an affirmative answer: every nonexpansive self-map of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point in the original norm. Zero and nonseparable spaces are included, with no uniform convexity, normal structure, or weak-continuity assumption.

AI-drafted comparisons, with no recorded human review

  • Nonexpansive fixed points in reflexive spaces ReflexiveFixedPoints

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: every nonexpansive self-map of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The commuting-family corollary is not part of the statement.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

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.

Nonexpansive fixed points in reflexive spaces ReflexiveFixedPoints

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

Reported byTheir wordTheir accountDateOriginal 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 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

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

No paper elsewhere in the release cites this family’s papers.

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 nonexpansive fixed points in reflexive Banach spaces to Formal Conjectures (issue 6984, PR 6989).

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

Offer to write an explanationCompare the Lean statement with the paperSuggest a source attributionSuggest a change

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 “Career-defining publication”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Read the Lean statement against the paper

  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 328 No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded. Open the form

Compare the proposed paper and Lean matches

  • Family 328 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form