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The cotype–cotype conjecture under the approximation property

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

Resolves the cotype–cotype conjecture for real Banach spaces with the approximation property. Such a nonzero space is K-convex if and only if both it and its dual have finite Rademacher cotype, with possibly different exponents. Equivalently, these cotype assumptions force nontrivial Rademacher type. (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.

The two independent assessments disagree on how much of the result described above is expressed by the Lean statements. Read the assessments

What the papers state

Theorem 1.1

In The cotype–cotype conjecture under the approximation property, 23 September 2026

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

Let X be a nonzero real Banach space with the ordinary approximation property. Then X is K-convex if and only if

C_q(X)<\infty \quad\hbox{and}\quad C_r(X^*)<\infty \qquad\text{for some }q,r\in[2,\infty).

The exponents q and r may differ. The forward implication holds for every real Banach space, without an approximation hypothesis.

Corollary 1.2 (Uniform cotype bounds and entropy duality)

In The cotype–cotype conjecture under the approximation property, 23 September 2026

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

Fix q,r\in[2,\infty) and C,D\in[1,\infty). There is a finite constant \kappa=\kappa(q,r,C,D) such that every nonzero real Banach space X with AP and

C_q(X)\le C,\qquad C_r(X^*)\le D

satisfies K(X)\le\kappa. Consequently there are a,b\ge1, depending only on q,r,C,D, such that for every n\ge1, all origin-symmetric convex bodies K,L\subset\R^n, and every t>0,

\frac1b\log N(L^\circ,atK^\circ) \le \log N(K,tL) \le b\log N(L^\circ,a^{-1}tK^\circ),

provided either X_K or X_L satisfies the displayed primal and dual cotype bounds.

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.

Cotype

The paper states

Theorem 1.1

In The cotype–cotype conjecture under the approximation property, 23 September 2026

Let X be a nonzero real Banach space with the ordinary approximation property. Then X is K-convex if and only if

C_q(X)<\infty \quad\hbox{and}\quad C_r(X^*)<\infty \qquad\text{for some }q,r\in[2,\infty).

The exponents q and r may differ. The forward implication holds for every real Banach space, without an approximation hypothesis.

The words they share: property, real, convex, space, approximation, cotype, nonzero, banach.

Possible Lean counterpart

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

Cotype–cotype equivalence with approximation property

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

theorem mainTarget_proved [Nontrivial X] : MainTarget X := 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 disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).

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: mainTarget_proved : [CompleteSpace X] [Nontrivial X], ApproximationProperty X → (KConvex X ↔ FiniteCotype X ∧ FiniteCotype (X →L[ℝ] ℝ)) is the headline for every nonzero real Banach space with AP; the summary’s ‘equivalently, these cotype assumptions force nontrivial type’ is the classical K-convex ⇔ nontrivial-type reformulation (Pisier), which is not stated in Lean.

Definitions the group asks readers to compare with the paper: KConvex E := ∃ K ≥ 0, ∀ n (f : Cube n → E), cubeL2 (radProjection f) ≤ K * cubeL2 f (uniform L²-boundedness of the Rademacher projection, the standard characterization) and HasCotype q := 2 ≤ q ∧ ∃ C ≥ 0, ∀ n (x : Fin n → E), (∑ ‖x i‖^q)^(1/q) ≤ C * cubeL2 (rademacherSum x): hand-built, standard (L² average in place of the L^q average, equivalent by Kahane)

“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 a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).

The gap they name: Only the ‘Equivalently … nontrivial Rademacher type’ sentence, a reformulation of the stated equivalence through Pisier’s K-convexity theorem; no notion of type appears in the challenge.

“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.

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 cotype–cotype problem asks whether finite cotype of a Banach space and its dual characterizes $K$-convexity. The formalized result establishes this equivalence for every nonzero real Banach space with the approximation property: $X$ is $K$-convex exactly when both $X$ and $X^*$ have finite cotype.

AI-drafted comparisons, with no recorded human review

  • Cotype–cotype equivalence with approximation property Cotype

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a nonzero real Banach space with the approximation property is K-convex exactly when it and its dual have finite cotype.

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.

Cotype–cotype equivalence with approximation property Cotype

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.

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.

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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 “Solid publication, could lead to tenure”. 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 326 No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded. Open the form

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 326 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form