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Almost-linear approximation of edit distance

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

For every fixed rational \(\varepsilon\in(0,1)\), gives a randomized \((1+\varepsilon)\) approximation to unit-cost edit distance in worst-case expected time \(N^{1+o(1)}\), with success probability at least 2/3. The strings have total length N and polynomially bounded integer symbols. This is an asymptotic guarantee at fixed accuracy. (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

The paper states

Theorem 1.1

In An Almost-Linear Approximation Scheme for Edit Distance, 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 one uniform randomized algorithm that, given x,y and a rational \varepsilon\in(0,1), returns a nonnegative integer \widehat D such that

\Pr\bigl[\ED(x,y)\le \widehat D\le(1+\varepsilon)\ED(x,y)\bigr] \ge \frac23.

For every fixed rational \varepsilon, its worst-case expected running time on strings of total length N is N^{1+o(1)} as N\to\infty on the logarithmic-word RAM. The expectation is over internal randomness only, and the bound includes input processing, preprocessing, numerical work, random sampling and all data-structure operations. Equal strings return zero deterministically.

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

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An Almost-Linear Approximation Scheme for Edit Distance

24 September 2026, current edition.

Read it herePDF

We give a uniform randomized approximation scheme for unit-cost edit distance. For every fixed rational \(\varepsilon\in(0,1)\), it estimates the distance between arbitrary explicitly stored strings of total length N within a factor \(1+\varepsilon\) with probability at least 2/3, in worst-case expected time \(N^{1+o(1)}\) on a logarithmic-word RAM. The algorithm supports polynomially bounded integer alphabets and returns zero deterministically on equal strings.

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.

EditApproximation

The paper states

Theorem 1.1

In An Almost-Linear Approximation Scheme for Edit Distance, 24 September 2026

There is one uniform randomized algorithm that, given x,y and a rational \varepsilon\in(0,1), returns a nonnegative integer \widehat D such that

\Pr\bigl[\ED(x,y)\le \widehat D\le(1+\varepsilon)\ED(x,y)\bigr] \ge \frac23.

For every fixed rational \varepsilon, its worst-case expected running time on strings of total length N is N^{1+o(1)} as N\to\infty on the logarithmic-word RAM. The expectation is over internal randomness only, and the bound includes input processing, preprocessing, numerical work, random sampling and all data-structure operations. Equal strings return zero deterministically.

The words they share: equal, expected, fixed, integer, length, randomized, rational, return, strings, total, work, zero.

Possible Lean counterpart

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

Almost-linear randomized approximation of edit distance

EditApproximation.lean, the release lists it as a supporting statement.

theorem almostLinearEditDistanceRawBand (epsilon : OAI.EditApproximation.BinaryFraction)    (hepsilon : 0 < epsilon.value) (_hepsilonOne : epsilon.value < 1)    :    (∀ (source target : List ℤ) (N : ℕ), source.length + target.length ≤ N →      (2 / 3 : ℝ) ≤ (OAI.EditApproximation.computedFullLaw (source.map OAI.EditApproximation.integerSymbolCode) (target.map OAI.EditApproximation.integerSymbolCode) N epsilon).real        {outcome | EditDistortion.edit source target ≤ OAI.EditApproximation.integerBinaryAnswerRawBand source target N epsilon outcome ∧          (OAI.EditApproximation.integerBinaryAnswerRawBand source target N epsilon outcome : ℝ) ≤            (1 + (epsilon.value : ℝ)) * OAI.EditDistortion.edit source target}) ∧    (∀ (source target : List ℤ) (N : ℕ), source = target → ∀ outcome,      OAI.EditApproximation.integerBinaryAnswerRawBand source target N epsilon outcome = 0) ∧    (∀ (C : ℕ) (eta : ℝ), 0 < eta → ∀ᶠ N : ℕ in atTop,      ∀ source target : List ℤ, source.length + target.length ≤ N →        (∀ z ∈ source ++ target, z.natAbs ≤ (N + 2) ^ C) →        (∫ outcome, OAI.EditApproximation.integerBinaryWorkRawBand source target N epsilon outcome          ∂OAI.EditApproximation.computedFullLaw (source.map OAI.EditApproximation.integerSymbolCode) (target.map OAI.EditApproximation.integerSymbolCode) N epsilon) ≤            (N + 2 : ℝ) ^ (1 + eta)) ∧

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-pass”.

Read the complete Lean statement

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

MathVet 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: Statement matches the summary (2/3 success, (1+eps) estimate, expected work (N+2)^(1+eta), poly space), but the ‘expected time’ is a hand-instrumented work counter in a 19k-line bespoke model with no link to a RAM/TM semantics; it should be treated as unaudited for the complexity claim (the correctness/probability part is meaningful). Not audited line by line.

Definitions the group asks readers to compare with the paper: Bespoke cost model: integerBinaryWorkRawBand := computedFullOutputWork... built from ...WithWork functions with hard-coded charges (e.g. def trimBitWordWithWork : List Bool → List Bool × ℕ | bit :: bits =>... (…, tail.2 + 3), queryInsertWithWork... (…, test.2 + 2)) and inductive BitQuery | read (key) (next) | charge (work : ℕ) (next); ‘random’ is a Measure (computedFullLaw, rejectionRuntimeArrayLaw), not random bits; the integer-symbol coding integerSymbolCode and BinaryFraction for ε are custom; the file also carries attribute [local instance] plumbing and ~1,200 declarations that the theorem statement silently depends on

“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 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.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The formalization gives a randomized $(1+\varepsilon)$-approximation for unit-cost edit distance between explicitly stored integer strings, for every fixed rational $0<\varepsilon<1$. Its estimate is at least the true distance and at most $(1+\varepsilon)$ times it with probability at least $2/3$; equal strings return zero on every execution.

For total length at most $N$ and integer symbols bounded by a fixed polynomial in $N$, expected work is at most $(N+2)^{1+\eta}$ for every fixed $\eta>0$ and all sufficiently large $N$. Space is polynomially bounded on every execution.

AI-drafted comparisons, with no recorded human review

  • Almost-linear randomized approximation of edit distance EditApproximation

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem in an explicit machine model: a randomised (1 + ε)-approximation of edit distance with expected work (N + 2)^{1+η} for every η > 0, for integer symbols bounded by a polynomial in N, which corresponds to the paper’s logarithmic-word RAM.

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.

Almost-linear randomized approximation of edit distance EditApproximation

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-pass”Re-run through Comparator: accepted.10 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

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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 algorithm and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.

    Author not stated. Source

  • Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source

    The review marks this statement as possibly weaker: it bounds a work counter built from hand-written costs rather than a machine model, with no undercount found but not every cost audited.

Questions awaiting review

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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 121 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

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 121 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form