OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Multiplies two n-bit integers exactly at every input length in deterministic worst-case time \(O(n(\log n)^{1-\kappa})\), with \(\kappa=2^{-182}\), on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen \(n\log n\) optimality conjecture in the ordinary multitape bit model.
The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.
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 deterministic Turing machine A, with a fixed finite alphabet and a fixed finite number of one-dimensional tapes, that computes this exact product for every n\ge1 and every pair of n-bit inputs. Its worst-case running time satisfies
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
There are deterministic machines with fixed finite alphabets and fixed finite numbers of one-dimensional tapes performing the following tasks in O(n(\lg n)^{1-\kappa}) worst-case time, for every n\ge1. For x,y\in\{0,1\}^n, put a=\operatorname{val}(x) and b=\operatorname{val}(y). If b>0, the division machine takes x\#y and returns \operatorname{bin}_n(q)\#\operatorname{bin}_n(r), where q=\lfloor a/b\rfloor and r=a-bq. The square-root machine takes x and returns \operatorname{bin}_n(\lfloor\sqrt a\rfloor). Leading zeroes are allowed, and \kappa is as in Theorem 1.
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
The release has no Lean statement for this family. The papers are available to read and assess.
We give a deterministic algorithm that multiplies two n-bit integers in \(O(n(\lg n)^{1-\kappa})\) worst-case time, with \(\kappa=2^{-182}\), on one fixed finite-alphabet Turing machine with a fixed finite number of one-dimensional tapes. The algorithm is exact for every input length and disproves the \(n\log n\) optimality conjecture of Schönhage and Strassen in this model.
Lean statements
The release has no Lean statement for this family. Its papers remain available to read, and contributors can propose a formal statement of a result. Suggest a change
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.
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
Prove2Me, Campaign: Integer Multiplication Below n log n, 11 October 2026. Source
Community exponents tracked as open Lean missions: kappa of 5.10e-5, 6.40e-5 and 1.26e-4, all far above the release’s 2^-182; none formalised yet.
Douglas Colkitt, integer-mult-bounds, 10 October 2026. Source
A community effort has pushed the conditional exponent for integer multiplication well beyond the manuscript’s, with merged pull requests, exact certificates and partial Lean transfer; the claims remain conditional and are not a complete Lean proof.
Douglas Colkitt, integer-mult-bounds, 10 October 2026. Source
The maintainer-reviewed selection on 10 October gives T(n) = O(n (log n)^(1 - kappa)) with kappa = 90446448750933/125000000000000000, about 0.000724 or 2^-10.43, against the manuscript’s 2^-182; it remains conditional on the original framework’s interfaces and is checked by finite replay and written review, not a complete Lean proof.
Swapnil Jain, Swapnil-jain/integer-mult-kappa, 9 October 2026. Source
A second team posted a series of conditional exponent improvements for the integer multiplication result, reaching about 2 to the power minus 10.5 in eleven rounds on 8 and 9 October, with Lean files for finite certificates and analytic premises assumed.
Zvi Mowshowitz, New Math from OpenAI, 9 October 2026. Source
Doug Colkitt published an update tightening the integer-multiplication saving by about a factor of 48 million (witness 8.3e-11) within two days.
erayyap, erayyap/integer-multiplication-kappa: improving kappa from 2^-182 to 2^-59.335, 7 October 2026. Source
kappa improved from 2^-182 to 2^-59.335 (7 Oct).
dysfunctor, dysfunctor/network-coding-counterexample: counterexample to the network coding conjecture from the XOR swap network, 7 October 2026. Source
Network coding conjecture counterexample derived from the family 109 construction (unreviewed).
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 “Career-defining publication”. The page says that nothing on it is verified.
tparker, Philip Weiss, Will Sawin, Nik Weaver, Pace Nielsen, mathworker21, Joshua P. Swanson, Timothy Chow, Daniel Soltesz, MathOverflow: how many of the released results went against the conventional wisdom?, 8 October 2026. Source
An answerer says a deterministic integer multiplication algorithm faster than n log n overturns what experts, including the authors of the best earlier algorithm, expected to be optimal.
tparker with nine answerers, How many of the results released by OpenAI went against the conventional mathematical wisdom?, 8 October 2026. Source
Integer multiplication faster than n log n overturns a conjectured optimum; Harvey and van der Hoeven had described it as the expected complexity.
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.
Identify the theorem the catalogue describes
Family 109The 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
State the result in Lean
Family 109The release has no Lean statement for this family.Open the form
Why this passage is shown
Theorem 1
In Integer multiplication below n log n, 23 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.