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Subpolynomial dimension reduction in Lp

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

For every fixed \(1\lt p\lt \infty\) and distortion D > 1, every n-point subset of real Lp embeds into \(\ell_p^d\) with distortion at most D and dimension \(d=n^{o(1)}\), answering Naor’s sublinear-dimension question for p ≠ 2. In contrast, exact embeddings require worst-case dimension \(\Theta(n^2)\) when p ≠ 2. (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

The paper states

Theorem 1.1

In Subpolynomial dimension reduction in Lp, 23 September 2026

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

Fix 1<p<\infty, p\ne2, and put

\gamma(p)= \begin{cases} 2-p,&1<p<2,\\ 1-2/p,&2<p<\infty. \end{cases}

For every D>1 there is a constant C_{p,D}<\infty such that, for every integer n\ge2,

\frac{\log n}{\log(1+2D)} \le d_p(n,D) \le \exp\!\bigl(C_{p,D}(\log n)^{\gamma(p)}\bigr).

For exact embeddings,

\left\lfloor\frac{n-1}{4}\right\rfloor^2 \le d_p(n,1)\le\binom n2\qquad(n\ge9).

Consequently,

\lim_{n\to\infty}\frac{\log d_p(n,D)}{\log n} =\begin{cases}0,&D>1,\\2,&D=1.\end{cases}

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

The release does not say which paper theorem its catalogue describes. The statement above shares 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; three 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.

Subpolynomial dimension reduction in Lp

23 September 2026, current edition.

Read it herePDF

For every fixed \(1\lt p\lt \infty\) and D > 1, every n-point subset of a real Lp space embeds into \(\ell_p^d\) with distortion at most D and subpolynomial dimension \(d=n^{o(1)}\). The target has the same exponent p, and the embedding need not be linear.

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.

SubpolynomialLp

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:

  • Theorem 1.1 in Subpolynomial dimension reduction in Lp, 23 September 2026.

The Lean statement

No paper theorem is paired with it.

Dimension reduction in $L_p$

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

theorem source_main (p : ℝ) (hp : 1 < p) (hp2 : p ≠ 2) :    (∀ D : ℝ, 1 < D → ∃ C : ℝ, ∀ n : ℕ, 2 ≤ n →      GoodDimension.{u} p n D (dimension.{u} p n D) ∧      Real.log (n : ℝ) / Real.log (1 + 2 * D) ≤ (dimension.{u} p n D : ℝ) ∧      (dimension.{u} p n D : ℝ) ≤ Real.exp (C * Real.log (n : ℝ) ^ gamma p)) ∧    (∀ n : ℕ, 9 ≤ n →      GoodDimension.{u} p n 1 (dimension.{u} p n 1) ∧      ((n - 1) / 4) ^ 2 ≤ dimension.{u} p n 1 ∧      dimension.{u} p n 1 ≤ n.choose 2) ∧    (∀ D : ℝ, 1 ≤ D →      Tendsto (fun n : ℕ => Real.log (dimension.{u} p n D : ℝ) / Real.log (n : ℝ))        atTop (𝓝 (if 1 < D then 0 else 2))) := 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.

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”.

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 — DECLARED”).

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: None; both the subpolynomial upper bound and the quadratic exact-embedding bounds are stated for every p > 1, p != 2.

Definitions the group asks readers to compare with the paper: dimension p n D := sInf {d | GoodDimension p n D d} is a natural-number infimum (junk 0 if empty); the statement itself asserts GoodDimension... (dimension...), ruling that out

“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 — DECLARED”).

The gap they name: The headline’s first sentence quantifies over every 1 < p < ∞; the Lean theorem assumes p ≠ 2, so the (classical, Johnson–Lindenstrauss) p = 2 case of the subpolynomial-dimension clause is not stated.

“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says 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

For $p>1$, $p\ne2$, the formalization bounds the least dimension needed to embed every $n$-point subset of real $L_p$ with distortion $D$. For each fixed $D>1$, there is a constant $C=C(p,D)$ such that this dimension lies between $\log n/\log(1+2D)$ and $\exp(C(\log n)^{\gamma(p)})$ for every $n\ge2$, where $\gamma(p)=2-p$ for $p<2$ and $1-2/p$ for $p>2$. At distortion $1$ and $n\ge9$, it lies between $\lfloor(n-1)/4\rfloor^2$ and $\binom n2$. The least dimension is attained, and its logarithm divided by $\log n$ tends to $0$ for $D>1$ and to $2$ for $D=1$.

AI-drafted comparisons, with no recorded human review

  • Dimension reduction in $L_p$ SubpolynomialLp

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for p > 1, p ≠ 2 and fixed distortion D > 1, the least embedding dimension for n-point subsets of L_p is at most exp(C(log n)^{γ(p)}), with the stated lower bound and the distortion-one case.

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.

Dimension reduction in $L_p$ SubpolynomialLp

Three 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
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 561 s; peak memory 6.5 GiB).11 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

Papers elsewhere in the release that cite 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 094 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 094 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form