Result 107, Theoretical computer science

Matrix multiplication with exponent at most 9/4

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 n^a 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.

Lean formalization New or sharp bound

The bigger picture

Why it matters

Matrix multiplication asks how much computation is needed to combine two arrays of numbers. These unreviewed manuscripts claim new bounds on that cost, including an exponent of at most 2.25 for square matrices over the complex numbers.

What changes?

The exponent measures how arithmetic-operation counts grow with matrix size. For complex n-by-n matrices, the manuscript reports O_epsilon(n^(9/4 + epsilon)) operations for every positive epsilon, with constants allowed to depend on epsilon. Related manuscripts report a square exponent below 2.258 over every characteristic-zero field, and over fields outside one possible finite set of positive characteristics. A separate bound below 2.371054886006746 applies over every fixed field, meaning every fixed number system supporting the usual field arithmetic.

What does that help mathematicians do?

In characteristic zero, some fixed a greater than 0.465 reportedly permits multiplying n-by-n^a and n^a-by-n matrices in n^(2+o(1)) operations, nearly proportional to the number of output entries. For inner dimension n^0.709, the claimed exponent is below 2.092, also outside the same finite set of positive characteristics. Researchers can use these bounds to identify rectangular computations with low arithmetic costs. They do not establish exponent two for square multiplication.

Are there practical applications?

The immediate value is foundational: these bounds sharpen theoretical cost estimates for computations expressed through matrix multiplication, while specifying which underlying fields are covered. The demonstrated claims concern arithmetic-operation counts, not measured running times. Practical speed would also depend on constants, memory use and implementation details that the supplied abstracts do not establish.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

An Upper Bound of 9/4 for the Matrix Multiplication Exponent

October 2, 2026 13 pages Main result formalized in Lean

We prove that the exponent of matrix multiplication over the complex numbers is at most 9/4.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Matrix-Multiplication-Nine-Fourths-October-2-2026,
  author = {{OpenAI}},
  title = {{An Upper Bound of $9/4$ for the Matrix Multiplication Exponent}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf}{OAI:Matrix-Multiplication-Nine-Fourths-October-2-2026}},
  year = {2026}
}

Complex Matrix Multiplication Below 2.258 and Rectangular Bounds

September 24, 2026 83 pages Main result formalized in Lean 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.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026,
  author = {{OpenAI}},
  title = {{Complex Matrix Multiplication Below $2.258$ and Rectangular Bounds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026.pdf}{OAI:Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026}},
  year = {2026}
}

Staggered extraction for exact matrix multiplication over every field

September 24, 2026 35 pages Main result formalized in Lean

We prove that the arithmetic exponent of square matrix multiplication over every fixed field satisfies ω < 2.371054886006746. This includes every positive characteristic.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026,
  author = {{OpenAI}},
  title = {{Staggered extraction for exact matrix multiplication over every field}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026.pdf}{OAI:Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/107.md.

Matrix multiplication with exponent at most 9/4

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

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.

Comparator links

Result	Comparator statement
Complex square, dual, and rectangular exponent bounds	MatrixMultiplication.lean
Matrix-multiplication exponent over every field	MatrixFields.lean

Posts about this result

Steven Strogatz @stevenstrogatz, Mathematician, Cornell

Many staggering results here. But this is a particularly amazing one: the exponent for matrix multiplication is no more than 2.25. The previous world record had been something like 2.37. This leap in progress is like Bob Beamon's long jump. github.com/openai/math/blob/main/preprints/Ma...

Quoting @OpenAI: We’re releasing a broad range of new mathematical results produced by an internal frontier model. We’ve been consulting with the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute f...

Oct 6, 2026, 6:56 PM ET

Thomas Ahle @thomasahle, Head of AI, Normal Computing

Somebody generalized the OpenAI O(n^{2.25}) algorithm for Fast Matrix Multiplication to general fields (not just C): github.com/selanavot/matrix-multiplication-al.... It seems to check out, including in Lean. Btw, the new FMM paper is surprisingly elegant. It uses a very different approach than previous results. Instead of powering the CW tensor and trimming it, the define a potential function on tensors, and show the whole result by contradiction by studying the simple convolution tensor. Maybe n^{9/4} is actually the right exponent for matrix multiplication...

Oct 7, 2026, 8:06 AM ET

At a glance

Subject: Theoretical computer science
Manuscripts: 3
Pages: 131
Dated: Sep 24 to Oct 2, 2026
Lean formalization: Yes
Reasoning summary: No

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