OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves \(\omega\le9/4\) over ℂ, giving \(O_\varepsilon(n^{9/4+\varepsilon})\) arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension na with a > 0.465 permits \(n^{2+o(1)}\) rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field. (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
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.
The other possible matches are Theorem 1.1 in Complex Matrix Multiplication Below 2.258 and Rectangular Bounds () and Theorem 1.1 in An Upper Bound of 9/4 for the Matrix Multiplication Exponent ().
Over every fixed field \F, the arithmetic exponent of square matrix multiplication satisfies
\omega_\F<2.371054886006746<2.371054887.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
— secondary writeup Over every field of characteristic zero, we prove that the square matrix-multiplication exponent satisfies ω < 2.258, the dual exponent satisfies α > 0.465, and \(\omega(1,0.709,1)\lt 2.092\). The strict square and k = 0.709 rectangular bounds also hold over every field except possibly in one finite set of positive characteristics, in the arithmetic-operation model.
We prove that the arithmetic exponent of square matrix multiplication over every fixed field satisfies ω < 2.371054886006746. This includes every positive characteristic.
Two Lean statements in this family
Choose one to see it beside the paper theorem it is paired with.
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.
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.
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.
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:ω(C) <= 9/4, α > 0.465 and ω(F) < 2.371054886006746 for every field are stated; 'ω < 2.258 outside finitely many positive characteristics’ is not. Exponents are sInf of uniform-constant, non-uniform straight-line program families (standard).
Definitions the group asks readers to compare with the paper:omega F := sInf {τ | AdmissibleExponent F τ} is only meaningful if the set is nonempty and bounded below (true: τ = 3 admissible, and cost ≥ n² forces τ ≥ 2; junk value 0 otherwise); complexAlpha := sSup {k ∈ [0,1] | rectangularOmega ℂ k = 2}; the first-conjunct-only reading of omega_lt_source_constant is padded with a pure numeric inequality 2.371054886006746 < 2.371056
“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.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).
The gap they name:'ω < 2.258 outside finitely many positive characteristics’ is not stated for any field of positive characteristic; the characteristic-zero rectangular clause is stated over C only. Both would follow from the C results by standard characteristic-transfer arguments (Schönhage-type), but no challenge states them.
“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.
The formalized results bound the complex matrix-multiplication exponent by $\omega(\mathbb C)\le9/4$, the dual exponent by $\alpha>0.465$, and the rectangular exponent at aspect ratio $0.709$ by $\omega(\mathbb C;1,0.709,1)<2.092$. The dual exponent is the supremum of rectangular aspect ratios attainable with exponent $2$. The arithmetic model counts additions, subtractions, and multiplications in finite division-free programs, with arbitrary positive exponent slack. The square bound implies the paper’s weaker $2.258$ headline bound.
The formalized result gives the unconditional bound $\omega(F)<2.371054886006746$ for every field $F$, including finite fields and fields of positive characteristic. The exponent counts additions, subtractions, and multiplications in finite division-free arithmetic programs, with arbitrary positive exponent slack. No numerical inequalities remain as hypotheses. The result concerns arithmetic complexity, rather than bit complexity or practical crossover sizes.
AI-drafted comparisons, with no recorded human review
Matrix-multiplication exponent over every fieldMatrixFields
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the staggered-extraction paper: ω_F < 2.371054886006746 over every field F, with no numerical hypotheses left.
Complex square, dual, and rectangular exponent boundsMatrixMultiplication
Unreviewed AI draft: states a paper’s theorem as the paper does. States ω(C) ≤ 9/4, α > 0.465 and the rectangular bound at 0.709. Since 9/4 < 2.258, this implies all three bounds of the main theorem; the 9/4 bound itself is the subject of a later family paper that the note does not link.
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.
Matrix-multiplication exponent over every fieldMatrixFields
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original 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.
Other people’s accounts of running proof-verification software, with links to their reports.
GasStationManager, GasStationManager/SafeVerifyAgent: audit log including a corpus look and a read-through of family 107, 8 October 2026. Source
Headline omega(C) <= 9/4 EXACT against a reference statement; builds on three standard axioms; export accepted by four independent checkers.
Jeroen Zuiddam, Lean proof that the matrix multiplication exponent is at most 9/4 over every field, 7 October 2026. Source
Recorded Slurm run on 7 Oct: lake build, axiom check and Comparator on the all-fields challenge all succeeded; GitHub build workflow green (3 of 3 runs).
erenciracioglu-dotcom, erenciracioglu-dotcom/openai-math-9-4-lean-check: Windows build and statement check for family 107, 7 October 2026. Source
Lean build succeeds and statements match the challenge after pp.all elaboration; Comparator not run.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The review reports that Comparator accepted OpenAI’s solution for this challenge, run end to end on Lean 4.34.1 with the default kernel and only the three standard axioms.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The review reports that Comparator accepted OpenAI’s solution for this challenge, run end to end on Lean 4.34.1 with the default kernel and only the three standard axioms.
Other proofs and exact calculations
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
“REFUSED” The exact code did not decide it; cert-machine records this as refused, which is no evidence against the claim.
What it covers
“finite components only: the three rectangular witnesses (alpha > 0.465, w(0.709) < 2.092, family omega < 2.267) as finite computations; for omega < 2.258 the barrier, both charts (the 18 MB chart verified as the recursion’s output), the residual eps, the chart comparisons and the W-seed polynomial. NOT the interior lemma (cost), NOT the boundary lemma / W-seed library (records not published), hence NOT Theorem thm:certificate; no theory”
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: the parameter certificate — the five integer arrays, decoded by the printed rules and contracted by the printed finite formulas, give every printed table value and enclosure and 3(8 log 7 - H_0 - C_*)/S_* < 2.371054886006746 with every feasibility condition; NOT the extraction theory that turns this ratio into a bound on omega”
Comment by Alex Meiburg: the result is O(n^(9/4+epsilon)) for every epsilon, not O(n^(9/4)); Aaronson fixed the post.
Explanations by others
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source
Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.
Jeroen Zuiddam, Lean proof that the matrix multiplication exponent is at most 9/4 over every field, 7 October 2026. Source
Only one step of the complex 9/4 proof uses more than algebraic closure: Proposition 3.1 (Fourier separation) needs a primitive 5M-th root of unity, which fails when the characteristic divides 5M; a separation period of 3M-2 or 3M-1 keeps the downstream bound unchanged.
Missing or incomplete source records
The program that collects Atlas’s source records, Statement Map, could not obtain or interpret some source material. These notes describe those limits; they do not identify mathematical errors.
MatrixFields: Tracked source lean/OAI/LinearAlgebra/MatrixFields/Main.lean unavailable: bytes were not captured at fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
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.
PALOMAR-2026-10-11-000007 (Sela Navot) extends the 9/4 bound to every associative unital ring.
Sela Navot, Matrix multiplication: omega at most 9/4 over every ring (PALOMAR-2026-10-11-000007), 11 October 2026. Source
Registered 2026-10-11T06:55:37Z; adapts both the OpenAI paper and the author’s own all-fields repository.
Sela Navot, selanavot/omega_le_nine_quarters_all_rings: omega at most 9/4 over every associative unital ring, 9 October 2026. Source
omega_le_nine_quarters_all_rings: the 9/4 bound holds over every ring in the same arithmetic model.
acprk, acprk/openai-math-followups: follow-up results on three OpenAI families, 9 October 2026. Source
Independently of THM-4568 (credited as first), an explicit extremal profile satisfies all five properties the 9/4 proof uses with 1/t = 4/3, so they cannot yield an exponent below 9/4, and the diagonal-growth lemma is exactly optimal; the note extends the barrier to further valid inequalities and extractions.
Palomar registry, Palomar: a registry of Lean-verified mathematics, 8 October 2026. Source
Sela Navot’s Lean extension of omega <= 9/4 to arbitrary fields (PALOMAR-2026-10-08-000005) and a further extension to every ring (PALOMAR-2026-10-11-000007).
Sela Navot, Matrix multiplication exponent at most 9/4 over arbitrary fields (PALOMAR-2026-10-08-000005), 8 October 2026. Source
Prove2Me, Campaign: matrix multiplication exponent (omega), 7 October 2026. Source
Mission ‘Matrix multiplication exponent over every field’ completed 7 October 2026 with bound 2.25.
Jeroen Zuiddam, Lean proof that the matrix multiplication exponent is at most 9/4 over every field, 7 October 2026. Source
All-fields Lean theorem omega_le_nine_quarters_all_fields built from 143 modules of the release library.
Jeroen Zuiddam, Lean proof that the matrix multiplication exponent is at most 9/4 over every field, 7 October 2026. Source
The release’s all-fields Lean statement bounds omega(F) only by about 2.371055 for every field, and its 9/4 statement is over the complex numbers. Zuiddam’s derived Lean proof gives omega(F) at most 9/4 over every field, with a non-vacuity theorem and the three standard axioms; Sela Navot’s fork reached the all-fields bound a day earlier.
selanavot, matrix-multiplication-all-fields, 7 October 2026. Source
Using an AI coding agent, the author extended the 9/4 exponent bound from the complex numbers to every field in Lean, and reports that Lean and Comparator checks pass.
Przemyslaw Uznanski, Rectangular matrix multiplication from shared-leg entropy, 7 October 2026. Source
Uznanski extends OpenAI’s square matrix-multiplication analysis to rectangular products, giving omega(1,k,1) <= 2 for k up to 1/2 and an n^2.5 bound for Zwick’s directed APSP algorithm.
Eliott Cassidy, eliottcassidy2000/math: THM-4568, the 9/4 growth lemma is exactly sharp, 7 October 2026. Source
The symmetric, separately concave profiles satisfying the tripling inequality used in the 9/4 proof have an explicit least element, built from Gamma(m+1/3)/(Gamma(4/3)Gamma(m)), that meets the rank bound exactly at t = 3/4: 9/4 is the ceiling of this argument, and the paper’s shift a - 1 is the smallest valid one. Checked in exact rationals for small cases.
From github.com/selanavot/matrix-multiplication-all-fields, 6 October 2026. Source
Sela Navot’s fork proves omega(F) <= 9/4 for every field F in Lean, replacing the complex-only hypothesis; Lean and Comparator pass per the author.
selanavot reports that Codex extended the matrix multiplication bound omega <= 2.25 to every field with a Lean proof passing Comparator (repository selanavot/matrix-multiplication-all-fields); it was later registered on Palomar.
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.
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 “Would win a Fields Medal”. The page says that nothing on it is verified.
Scientific American (byline not recorded), The most exciting claims from OpenAI’s heap of new proofs, 8 October 2026. Source
Virginia Vassilevska Williams (MIT) notes the 9/4 exponent is far below the 2.37 reached by Vassilevska Williams and colleagues and describes it, with some irony, as language models speeding themselves up.
Raemon, Are OpenAI’s math results “creative” in an important way?, 7 October 2026. Source
Thomas Ahle calls the 9/4 matrix multiplication paper surprisingly elegant, using a potential function on tensors and a contradiction argument on the convolution tensor.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The Lean statement gives the exponent bound 9/4 over the complex numbers, which is stronger than the 2.258 bound in the paper the documentation links; the nine-fourths manuscript it actually follows is not linked.
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
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 107The 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
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. 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 does not say 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…
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.
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.
Exact calculations for finite claims, cert-machine
Their verdict
“REFUSED”
Claim tested or proved
The exact code did not decide it; cert-machine records this as refused, which is no evidence against the claim.
What it covers
“finite components only: the three rectangular witnesses (alpha > 0.465, w(0.709) < 2.092, family omega < 2.267) as finite computations; for omega < 2.258 the barrier, both charts (the 18 MB chart verified as the recursion’s output), the residual eps, the chart comparisons and the W-seed polynomial. NOT the interior lemma (cost), NOT the boundary lemma / W-seed library (records not published), hence NOT Theorem thm:certificate; no theory”
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: the parameter certificate — the five integer arrays, decoded by the printed rules and contracted by the printed finite formulas, give every printed table value and enclosure and 3(8 log 7 - H_0 - C_*)/S_* < 2.371054886006746 with every feasibility condition; NOT the extraction theory that turns this ratio into a bound on omega”