Uniform sparsest cut: hardness and semidefinite gaps
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves that approximating Uniform Sparsest Cut within any fixed constant factor is NP-hard, even with nonnegative rational capacities and unit demands. The Goemans–Linial semidefinite relaxation also has integrality gaps of order at least \(\sqrt{\log n}/(\log\log n)^3\), approaching the square-root-logarithmic upper 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.
The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement.Read the assessments
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.
For every fixed real constant C>1, there is a polynomial-time reduction from 3CNF satisfiability to finite undirected graphs with nonnegative rational capacities and exactly unit demands, together with a positive rational threshold a, such that
\begin{array}{ll}
\text{the formula is satisfiable}&\Longrightarrow\ \Phi(G)\le a,\\[2pt]
\text{the formula is unsatisfiable}&\Longrightarrow\ \Phi(G)>Ca.
\end{array}
The graph size and the bit lengths of all output numbers are polynomial in the formula size, with constants and exponents allowed to depend on C. In particular, approximating Uniform Sparsest Cut within any fixed factor C>1 is NP-hard.
No Lean statement in the release is paired with this theorem.
for all sufficiently large j. The demand is one between every pair of distinct vertices.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole, the closer match to the description’s wording first. 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.
We prove that, for every fixed C > 1, approximating Uniform Sparsest Cut within factor C is NP-hard. The output graphs have nonnegative rational capacities and unit demand between every pair of distinct vertices.
We construct uniform sparsest-cut instances whose Goemans–Linial semidefinite integrality gap is at least \(c\sqrt{\log n}/(\log\log n)^3\) along a sequence \(n\to\infty\). The demand is one between every pair of distinct vertices, and the capacities are nonnegative real numbers. This matches the Arora–Rao–Vazirani upper bound up to a power of \(\log\log n\).
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.
UniformSparsestCut.lean, the release lists it as a main result.
theoremmainGap : ∃ c : ℝ, 0 < c ∧ ∃ n : ℕ → ℕ, ∃ C : (j : ℕ) → Capacity (n j), (∀ j, 2 ≤ n j) ∧ Tendsto n atTop atTop ∧ (∀ j, 0 < glValue (C j)) ∧ ∀ᶠ j in atTop, c * Real.sqrt (Real.log (n j : ℝ)) / (Real.log (Real.log (n j : ℝ))) ^ 3 ≤ OPT (C j) / glValue (C j) := 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”.
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?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
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 part of the advertised result (their term: “partial”).
Their note:Only the Goemans-Linial integrality gap (real capacities, along a sequence of n); the NP-hardness of constant-factor approximation (the summary’s first claim) is not stated.
Definitions the group asks readers to compare with the paper:NegativeType d is ∃ x : Fin n → EuclideanSpace ℝ (Fin n), d i j = ‖x i - x j‖^2 plus triangle inequalities (n dimensions suffice, standard); OPT/glValue are sInfs over sets that must be nonempty and bounded below (true)
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
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 result gives integrality gaps for the Goemans–Linial relaxation of uniform sparsest cut. Along a sequence of instance sizes $n\to\infty$, the ratio of the integral optimum to the positive relaxation optimum is at least $c\sqrt{\log n}/(\log\log n)^3$ for one absolute $c>0$. Every unordered pair has unit demand, and the relaxation uses squared Euclidean distances satisfying the triangle inequalities. The result is an existential sequence, not a bound for every size or a computational hardness theorem.
AI-drafted comparisons, with no recorded human review
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a sequence of uniform sparsest-cut instances whose Goemans–Linial integrality gap is at least c√(log n)/(log log n)³. The hardness result of the family’s other paper is not part of the statement.
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.
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
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
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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 proof and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.
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Read the Lean statement against the paper
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Identify the theorem the catalogue describes
Family 117The 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
Address a question raised in an independent assessment
Only a special case is stated, or a co-equal headline claim has no challenge statement.
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 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; 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 399 s; peak memory 6.5 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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.