Constructs a countable uniformly discrete metric space whose real Lipschitz-free Banach space has the approximation property but not the bounded approximation property, answering Kalton’s question negatively. Finite-rank operators approximate the identity on every compact set, but their norms cannot share a finite bound. (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
In A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces, 26 September 2026
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
There is a countable, unbounded, nonproper pointed metric space (M,d,o) such that d(x,y)\ge1 for distinct points and \F(M) has AP but fails \lambda-BAP for every finite \lambda\ge1.
In A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces, 26 September 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
For each integer p\ge1, there are a bounded countably infinite pointed metric space (M_p,d_p,o) and a finite set A_p\subseteq M_p containing o such that
and every bounded finite-rank linear operator T_0:\F(M_p)\to\F(M_p) with \norm{T_0}\le p satisfies
No Lean statement in the release is paired with this theorem.
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DiscreteLipschitzFreeThe paper states
In A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces, 26 September 2026
There is a countable, unbounded, nonproper pointed metric space (M,d,o) such that d(x,y)\ge1 for distinct points and \F(M) has AP but fails \lambda-BAP for every finite \lambda\ge1.
The words they share: ap, finite, countable, metric, bap, space, free, bounded, unbounded, distinct, fails, pointed.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
DiscreteLipschitzFree.lean, the release lists it as a supporting statement.
theorem source_endpoints : BlockStatement ∧ MainStatement := by sorryThis 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
RealL1RenormingThe paper states
In A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces, 26 September 2026
For each integer p\ge1, real \ell_1 has an equivalent norm \norm{\cdot}_p such that
and (\ell_1,\norm{\cdot}_p) has 2B_p-BAP but fails p-BAP. For p=1, this norm fails MAP.
The words they share: real, map, integer, 1/2, norm, bap, equivalent, fails, ell.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
RealL1Renorming.lean, the release lists it as a supporting statement.
def RealL1RenormingStatement : Prop := ∀ p : ℕ, 1 ≤ p → ∃ N : AddGroupNorm RealL1, (∀ (r : ℝ) (c : RealL1), N (r • c) = |r| * N c) ∧ (∀ c : RealL1, (1 / 2 : ℝ) * ‖c‖ ≤ N c ∧ N c ≤ blockBound p * ‖c‖) ∧ (let m : Module ℝ RealL1 := inferInstance letI : NormedAddCommGroup RealL1 := N.toNormedAddCommGroup ∃ s : NormedSpace ℝ RealL1, s.toModule = m ∧ letI := s CompleteSpace RealL1 ∧ HasBAP RealL1 (2 * blockBound p) ∧ ¬ HasBAP RealL1 (p : ℝ) ∧ (p = 1 → ¬ HasBAP RealL1 1))theorem real_l1_quantitative_renorming : RealL1RenormingStatement := by sorryThis 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
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.
HasAP Model.lean, line 191BlockStatement Model.lean, line 196MainStatement Model.lean, line 205MathVet 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: MainStatement (inside source_endpoints) gives a countable metric space M with all distinct distances ≥ 1, unbounded and not proper, such that the real Lipschitz-free space FreeSpace o has HasAP but ¬ HasBAP for every Λ ≥ 1, exactly the summary; BlockStatement (finite quantitative blocks) and the ℓ¹ renorming with 2B_p-BAP but not p-BAP are the retained earlier results.
Definitions the group asks readers to compare with the paper: FreeSpace o := closure of span {evaluation o x} ⊆ (Lip0 o →L[ℝ] ℝ) with Lip0 o normed by the slope map into ℓ^∞(M × M): hand-built Lipschitz-free space, standard; HasAP/HasBAP state approximation of the identity on every compact set by finite-rank operators (with ‖T‖ ≤ Λ for BAP)
“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.
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.
OpenAI’s own note on what its Lean covers
Kalton’s question asks whether uniform discreteness of a metric space forces its Lipschitz-free space to have the bounded approximation property. The formalization constructs a countable metric space with all distinct points at distance at least one whose real Lipschitz-free space has the approximation property but fails every bounded approximation bound. The metric space is unbounded and not proper.
The earlier quantitative renorming result for real $\ell_1$ is retained: for each integer $p\ge1$, an equivalent complete norm has the $2B_p$-bounded approximation property but fails the $p$-bounded approximation property, where $B_p=300p^2+150p+5$.
Uniformly discrete Lipschitz-free space with AP but no BAP DiscreteLipschitzFree
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a countable uniformly discrete metric space whose Lipschitz-free space has the approximation property but no bounded approximation property.
Quantitative renorming of real $\ell_1$ RealL1Renorming
Unreviewed AI draft: states a consequence of a paper’s theorem. States the paper’s renorming corollary for real ℓ₁ (an equivalent norm with the 2B_p-bounded but not the p-bounded approximation property), which the note calls the earlier quantitative result.
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.
DiscreteLipschitzFreeThree 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 |
| Neal McBurnett, openai-math-lean-tooling | “accepted” | Re-run through Comparator with Lean’s kernel: accepted (the check took 86 s; peak memory 6.2 GiB). | 11 October 2026 | Original report |
RealL1RenormingThree 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 |
| Neal McBurnett, openai-math-lean-tooling | “accepted” | Re-run through Comparator with Lean’s kernel: accepted (the check took 81 s; peak memory 6.2 GiB). | 11 October 2026 | Original report |
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.
This family’s papers cite no paper elsewhere in the release.
No paper elsewhere in the release cites this family’s papers.
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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 “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
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Why this passage is shown
Theorem 1.1
In A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces, 26 September 2026
Why this passage is shown
Theorem 1.2
In A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces, 26 September 2026
Why this passage is shown
Corollary 1.3
In A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces, 26 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, DiscreteLipschitzFree
Proof-verification report
MathVet, DiscreteLipschitzFree
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, DiscreteLipschitzFree
Proof-verification report
cert-machine, RealL1Renorming
Proof-verification report
MathVet, RealL1Renorming
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, RealL1Renorming
Lean statement
DiscreteLipschitzFree
Lean statement
RealL1Renorming