Lipschitz equivalent Banach spaces need not be linearly isomorphic
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Constructs separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic, resolving the separable Lipschitz-isomorphism problem negatively. Thus even the complete metric structure up to bi-Lipschitz equivalence does not determine a separable Banach space’s linear isomorphism class. (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
The space X contains a linearly isometric copy of c_0(\ell_2), while Y contains no closed subspace linearly isomorphic to c_0(\ell_2). In particular, X and Y are not linearly isomorphic.
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
There is a separable real Banach space Z containing no closed linear subspace isomorphic to c_0, and an onto bi-Lipschitz map
F:Z\oplus_\infty c_0\longrightarrow Z.
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
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
There are separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic. This gives a negative answer to the separable Banach-space Lipschitz-isomorphism problem.
We construct a separable real Banach space Z that contains no linear copy of c0, yet is bi-Lipschitz equivalent to \(Z\oplus_\infty c_0\). The same space contains a bi-Lipschitz image of every separable metric space.
Two Lean statements in this family
Choose one to see it beside the paper theorem it is paired with.
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.
C0Absorption
The paper states
No paper theorem is paired with this Lean statement.
no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
Statements it shares words with:
Corollary 8.1in Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0, 26 September 2026.
Theorem 1.1in Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0, 26 September 2026.
Proposition 4.3(Onto-ness of the nonlinear shift)in Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0, 26 September 2026.
Proposition 7.5(Selection of compatible detectors)in Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0, 26 September 2026.
Proposition 7.6in Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0, 26 September 2026.
Proposition 3.1(Weighted graph criterion)in Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic, 24 September 2026.
Theorem 1.1in Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic, 24 September 2026.
Theorem 2.1(Compact reduction for a weakly null sequence)in Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic, 24 September 2026.
C0Absorption.lean, the release lists it as a main result.
defMainConclusion : Prop := ∃ (Z : Type) (_ : NormedAddCommGroup Z) (_ : NormedSpace ℝ Z), CompleteSpace Z ∧ TopologicalSpace.SeparableSpace Z ∧ NoLinearC0 Z ∧ (∃ F : Z × C0 → Z, Function.Surjective F ∧ BiLip F) ∧ (∃ f : C0 → Z, BiLip f) ∧ MetricUniversal Z ∧ ¬ Nonempty ((Z × C0) ≃L[ℝ] Z)⋮theoremmain_result : MainConclusion := bysorry
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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
The space X contains a linearly isometric copy of c_0(\ell_2), while Y contains no closed subspace linearly isomorphic to c_0(\ell_2). In particular, X and Y are not linearly isomorphic.
The words they share: 76/25, copy, contains, separable, real, banach, spaces, isometric, ell, bijection, linearly, isomorphic.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
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.
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:LipschitzEquivalence.main: ∃ separable real Banach spaces X, Y and a bijection Ψ with (4/21)‖s-t‖ ≤ ‖Ψ s - Ψ t‖ ≤ (76/25)‖s-t‖, IsEmpty (X ≃L[ℝ] Y), X ⊇ isometric c₀(ℓ₂), Y has no bounded-below linear image of c₀(ℓ₂): exactly the summary’s bi-Lipschitz-equivalent but not linearly isomorphic pair. C0Absorption adds the second paper’s single space Z (no linear c₀, Z × c₀ bi-Lipschitz onto Z, metrically universal, ¬ Nonempty ((Z × c₀) ≃L Z)).
“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.
The formalized counterexample gives separable real Banach spaces $X,Y$ that are bi-Lipschitz equivalent but not linearly isomorphic. The bijection has lower Lipschitz bound $4/21$ and upper bound $76/25$. The linear obstruction is explicit: $X$ contains a linear isometric copy of $c_0(\ell_2)$, whereas $Y$ contains no bounded linear copy of that space.
The formalized result constructs one separable real Banach space $X$ that is bi-Lipschitz equivalent to $X\times c_0$ but contains no closed linear subspace isomorphic to $c_0$. The same space contains a bi-Lipschitz copy of $c_0$, is bi-Lipschitz universal for separable metric spaces, and is not linearly isomorphic to $X\times c_0$. Thus nonlinear absorption of $c_0$ does not force a linear copy of it.
AI-drafted comparisons, with no recorded human review
Bi-Lipschitz absorption of $c_0$C0Absorption
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the absorption paper: a separable Banach space bi-Lipschitz equivalent to its sum with c₀ that contains no linear copy of c₀.
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: separable Banach spaces X, Y with a bi-Lipschitz bijection (constants 4/21 and 76/25) that are not linearly isomorphic.
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.
Bi-Lipschitz absorption of $c_0$C0Absorption
Three 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.
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 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 bi-Lipschitz equivalent nonisomorphic separable Banach spaces to Formal Conjectures (issue 6994, PR 6999).
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.
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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 counterexample and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
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
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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 324No 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 324Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1
In Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Atlas records no review confirming that the two statements mean the same thing.
Why this passage is shown
Theorem 2.1 (Compact reduction for a weakly null sequence)
In Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic, 24 September 2026
How it was found
Its words were compared with the words of the Lean statement C0Absorption.
Words they share
dist, complete, metric, space
Review
Atlas records no review confirming that the two statements mean the same thing.
Independent assessment
MathVet
Their term
“full”
What their term means
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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
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; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 309 s; peak memory 6.3 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
No paper theorem is paired with this Lean statement.
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.