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The Euclidean Steinitz–Bergström bound

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

Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within \(C\sqrt d\), independently of length. Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums. A matching lower bound gives the optimal order \(S_2(d)=\Theta(\sqrt d)\). (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 (Prescribed-order signed prefixes)

In The Euclidean Steinitz–Bergström theorem, 24 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

There is an absolute constant C such that, for all integers d,N\ge1 and every sequence v_1,\ldots,v_N\in\R^d with \lVert v_i\rVert_2\le1, there are signs \varepsilon_1,\ldots,\varepsilon_N\in\{-1,1\} for which

\max_{0\le k\le N} \left\lVert\sum_{i=1}^k\varepsilon_i v_i\right\rVert_2\le C\sqrt d.

The same signs control all prefixes, and C is independent of d, N and the sequence.

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

Theorem 1.2 (Euclidean Steinitz–Bergström bound)

In The Euclidean Steinitz–Bergström theorem, 24 September 2026

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

There is an absolute constant C such that, for all integers d,N\ge1 and every indexed family v_1,\ldots,v_N\in\R^d satisfying \lVert v_i\rVert_2\le1 and \sum_{i=1}^N v_i=0, some permutation \pi\in\mathfrak S_N satisfies

\max_{0\le k\le N}\left\lVert\sum_{i=1}^k v_{\pi(i)}\right\rVert_2 \le C\sqrt d.

The constant in Theorem 1.1 also suffices here.

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

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.

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.

The Euclidean Steinitz–Bergström theorem

24 September 2026, current edition.

Read it herePDF

Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝd has one signing for which every signed prefix has norm at most \(C\sqrt d\), with C absolute and independent of the sequence length. Consequently, every indexed zero-sum family of unit-ball vectors admits an ordering with the same bound for its unsigned partial sums. This determines the Euclidean Steinitz constant up to absolute factors, \(S_2(d)=\Theta(\sqrt d)\), and resolves the Euclidean Steinitz–Bergström conjecture.

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.

SteinitzBergstrom

The paper states

Corollary 6.1 (Finite-dimensional \ell_p bounds)

In The Euclidean Steinitz–Bergström theorem, 24 September 2026

Let 1\le p\le\infty, with 1/\infty=0, and let d,N\ge1 be integers. Every sequence v_1,\ldots,v_N\in\R^d with \lVert v_i\rVert_p\le1 admits signs \varepsilon_i\in\{-1,1\} such that

\max_{0\le k\le N} \left\lVert\sum_{i=1}^k\varepsilon_i v_i\right\rVert_p \le C d^{\max\{1/p,\,1-1/p\}}.

If additionally \sum_{i=1}^N v_i=0, there is a permutation \pi\in\mathfrak S_N such that

\max_{0\le k\le N} \left\lVert\sum_{i=1}^k v_{\pi(i)}\right\rVert_p \le C d^{\max\{1/p,\,1-1/p\}}.

Here C is the same absolute constant as in Theorems 1.1 and 1.2, independent of p,d,N. For 1\le p\le2, both bounds have sharp dimension order d^{1/p}: for every d there is a unit-ball sequence for which every signing has a prefix of norm at least d^{1/p}, and a finite indexed zero-sum unit-ball family for which every permutation has a prefix of norm at least \tfrac13d^{1/p}.

The words they share: order, pi, prefix, signs, integers, bounds, unit, finite, zero, ball, norm, family, permutation.

Possible Lean counterpart

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

Euclidean signed and reordered prefix bounds

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

theorem main : ∃ C : ℝ, SignedPrefixBound C ∧ OrderingPrefixBound C := 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 — 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: Signed-prefix and zero-sum reordering bounds with an absolute constant are stated; the matching Omega(sqrt d) lower bound (optimal order) is not.

“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: the matching lower bound S_2(d) = Ω(sqrt d), hence the Θ statement

“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 formalized result gives the Euclidean Steinitz–Bergström bound with one absolute constant $C$. For every finite family of vectors in the unit ball of $\mathbb R^d$, signs can be chosen so that every prefix in the prescribed order has norm at most $C\sqrt d$. When the vector sum is zero, a permutation makes every unsigned prefix satisfy the same bound. Repeated and zero vectors are included. The result is existential and does not supply an online algorithm.

AI-drafted comparisons, with no recorded human review

  • Euclidean signed and reordered prefix bounds SteinitzBergstrom

    Unreviewed AI draft: states a paper’s theorem as the paper does. States both main theorems: signed prefixes of norm at most C√d in a prescribed order, and, for zero-sum families, a reordering with every prefix of norm at most C√d.

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.

Euclidean signed and reordered prefix bounds SteinitzBergstrom

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 232 s; peak memory 6.6 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

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 Euclidean Steinitz-Bergstrom prefix bounds to Formal Conjectures (issue 6985, PR 6990).

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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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 097 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

Compare the proposed paper and Lean matches

  • Family 097 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form