# Matrix multiplication with exponent at most 9/4

> Huge n-by-n complex matrices can be multiplied in about n^(9/4) arithmetic steps, up to any tiny slack. Plain-English explainer, Lean status and sources.

Canonical: https://closedform.nishachay.tech/e/openai-math-2026-107/
Last updated: 2026-10-11

**Huge n-by-n complex matrices can be multiplied in about n^(9/4) arithmetic steps, up to any tiny slack.**

**Trust:** Formal proof listed · **Papers:** 3 · **First paper:** September 24, 2026 · **Field:** Theoretical computer science · **Entry:** 107

*Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error.

Multiplying two n-by-n matrices the school way takes about n^3 arithmetic steps. The best published exponent before this family was about 2.371. The main paper proves that, over the complex numbers, the exponent is at most 9/4 (Lean proof listed). Companion papers prove a bound below 2.371054886006746 for every field and new bounds for thin matrices (Lean proofs listed).

## At a glance

- **First step:** 1969 (Volker Strassen, seven-product 2-by-2 method)
- **Significance:** Landmark. If it holds, it cuts the best bound on a central exponent from about 2.371 to 2.25.
- **Trust:** Formal proof listed
- **Field:** Theoretical computer science (3 papers)

**Best proven upper bound on the matrix multiplication exponent ω**

| Year | Who | Value |
| --- | --- | --- |
| 1969 | Strassen | 2.807 |
| 1990 | Coppersmith and Winograd | 2.375477 |
| 2026 | Dupont et al. | 2.371177 |
| 2026 | This paper | 2.25 (this result) |

Lower is better. No method can go below 2. The 2026 claim is over the complex numbers.

## The explanation

Multiply two 2-by-2 grids of numbers the school way and you make 8 multiplications. Each of the 4 answer entries needs 2 of them. In 1969, Volker Strassen found a way with only 7. His trick repeats. Cut a big grid into four blocks and use the same trick on the blocks. For two 1024-by-1024 grids, ten rounds need 7^10 multiplications, about 282 million. The school method needs 1024^3, about 1.07 billion.

Such a grid is called a matrix. Experts track the cost of multiplying two n-by-n matrices with one number, the exponent ω (omega). Roughly, ω is the smallest power p for which the work grows like n^p as n gets large. Work here means arithmetic steps: additions, subtractions and multiplications of single numbers. The school method gives 3. Strassen's trick gives log2 7, about 2.807. Nothing can beat 2, because the answer alone has n^2 entries.

For decades, progress came in small steps. In 1990, Don Coppersmith and Shmuel Winograd showed ω < 2.375477. In August 2026, Dupont and coauthors reported ω < 2.371177. They combined modern optimization with AlphaEvolve. The gains had shrunk to the third decimal place.

This family of three papers goes much further. The main paper proves ω ≤ 9/4, which is 2.25, over the complex numbers[1]. The complex numbers include all ordinary decimals, plus square roots of negative numbers. So for any small extra slack, the work grows no faster than n^(2.25 + slack). At n = 1024, n^2.25 is about 5.9 million. That number shows a growth rate, not a step count. The papers give no size for the hidden constant factor. The paper says its proof does not name a matrix size where the method would compete[2].

Two earlier companion papers reach weaker square bounds by other routes. One shows ω < 2.258 over the complex numbers[3]. The other shows ω < 2.371054886006746 over every field[4]. A field is any number system with the usual four operations, including finite ones such as arithmetic on a clock. That second bound is just below the August 2026 record. It is the only one of the three shown to hold for every field[5].

Why was this hard? Fast methods hunt for many small, separate multiplications hidden inside one big computation. A cheap way to compute many separate products at once then yields a bound on ω. The catch is that the hidden pieces share inputs[6]. Truly separate pieces need their own variables on all three sides: both input matrices and the output[7]. Pulling pieces apart costs work, and that cost ate most of the gains. The 9/4 and 2.258 papers start from pieces that are already separate on two sides and share only the third[8]. An extra labeling step then tells them apart on the third side too. The number of freed pieces outweighs the cost of the labels[9].

The 2.258 paper also handles thin matrices. Take n = 1,000,000 and multiply an n-by-617 matrix by a 617-by-n matrix. The answer has a trillion entries, so the work is at least about that. The dual exponent α (alpha) measures how wide that middle can get. Within that width, the work still grows like n^2 as n gets large. The paper shows α > 0.465, and 1,000,000^0.465 is about 617[3]. The best earlier value it cites is α ≥ 0.321334, which allows a middle of about 85. The same paper also claims its 2.258 bound for most other fields[10]. A finite list of exceptions remains, and the paper does not compute it.

All of these results concern growth as n becomes huge. They count arithmetic steps, not computer time or the size of the numbers involved. They do not say at what size any of these methods would beat ordinary ones in practice[11].

How sure is this? A computer proof checker called Lean has checked formal versions of four statements. Three are about complex numbers: ω ≤ 9/4, α > 0.465, and a thin-matrix bound. The fourth is ω < 2.371054886006746 over every field. Each is matched to a fixed reference statement for a checking tool called Comparator. The move of 2.258 to other fields is not in the Lean scope note. No specialist review has been published.

**What it doesn't show.** This does not give a practical faster program or say at what matrix size any speedup begins.

## Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

## Hype check

If the proofs hold, this is a large drop in a closely watched number. Lean proofs are listed for the main bounds, but they describe growth rates only. They give no practical program, and no specialist review is published.

## Words used

- **Matrix:** A rectangular grid of numbers.
- **Matrix multiplication exponent (ω):** Roughly the smallest power p such that multiplying two n-by-n matrices takes about n^p arithmetic steps.
- **Dual exponent (α):** How wide the shared middle side of two thin matrices can grow while the work stays near n squared.
- **Complex numbers:** The ordinary decimals together with square roots of negative numbers and their combinations.
- **Field:** A number system where you can add, subtract, multiply and divide by anything except zero.
- **Arithmetic step:** One addition, subtraction or multiplication of two single numbers.
- **Lean:** A computer program that checks mathematical proofs line by line.
- **Comparator:** A tool that checks a Lean proof proves exactly a fixed reference statement.

## The original papers (3)

- [An Upper Bound of 9/4 for the Matrix Multiplication Exponent](https://github.com/openai/math/blob/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf) - October 2, 2026 · Formal proof listed
- [Complex Matrix Multiplication Below 2.258 and Rectangular Bounds](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) - September 24, 2026 · Formal proof listed
- [Staggered extraction for exact matrix multiplication over every field](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) - September 24, 2026 · Formal proof listed

- [Lean notes](https://github.com/openai/math/blob/main/lean/docs/107.md)

## Quoted lines from the paper (11)

[1] From [An Upper Bound of 9/4 for the Matrix Multiplication Exponent](https://github.com/openai/math/blob/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf):

> For every $\eps>0$, two $n\times n$ complex matrices can be multiplied using $O_\eps(n^{9/4+\eps})$ arithmetic operations. In particular, $\omega\leq9/4$.

[2] From [An Upper Bound of 9/4 for the Matrix Multiplication Exponent](https://github.com/openai/math/blob/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf):

> Theorem … concerns the asymptotic arithmetic exponent; its proof does not specify a competitive finite matrix size.

[3] From [Complex Matrix Multiplication Below 2.258 and Rectangular Bounds](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):

> Over \(\C\), the arithmetic matrix-multiplication exponents satisfy \[ \omega<\frac{1129}{500}=2.258,\qquad \alpha>0.465,\qquad w(0.709)<2.092. \]

[4] From [Staggered extraction for exact matrix multiplication over every field](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):

> Over every fixed field $\F$, the arithmetic exponent of square matrix multiplication satisfies \[ \omega_\F<2.371054886006746<2.371054887. \]

[5] From [Staggered extraction for exact matrix multiplication over every field](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):

> The construction gives exact arithmetic algorithms, including in positive characteristic.

[6] From [An Upper Bound of 9/4 for the Matrix Multiplication Exponent](https://github.com/openai/math/blob/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf):

> A basic difficulty in tensor constructions is that useful pieces can share variables.

[7] From [An Upper Bound of 9/4 for the Matrix Multiplication Exponent](https://github.com/openai/math/blob/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf):

> A full direct sum requires independent variables on all three tensor legs.

[8] From [An Upper Bound of 9/4 for the Matrix Multiplication Exponent](https://github.com/openai/math/blob/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf):

> Our principal construction starts with blocks that are independent on two legs and share the third.

[9] From [An Upper Bound of 9/4 for the Matrix Multiplication Exponent](https://github.com/openai/math/blob/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf):

> The number of output blocks compensates for the cost of the Fourier copies at the exponential scale.

[10] From [Complex Matrix Multiplication Below 2.258 and Rectangular Bounds](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):

> The exceptional primes are not computed.

[11] From [Staggered extraction for exact matrix multiplication over every field](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):

> The theorem concerns asymptotic arithmetic cost; the counting arguments and parameter choices do not give bit-complexity or practical crossover estimates.

## Other sources

- [V. Strassen, “Gaussian elimination is not optimal”, Numerische Mathematik 13 (1969), 354–356](https://doi.org/10.1007/BF02165411)
- [D. Coppersmith and S. Winograd, “Matrix multiplication via arithmetic progressions”, Journal of Symbolic Computation 9 (1990), 251–280](https://doi.org/10.1016/S0747-7171(08)80013-2)
- [V. Vassilevska Williams, Y. Xu, Z. Xu and R. Zhou, “New Bounds for Matrix Multiplication: from Alpha to Omega”, SODA 2024, arXiv:2307.07970](https://arxiv.org/abs/2307.07970v2)
- [E. Dupont et al., “Improving the matrix multiplication exponent with modern optimization and AlphaEvolve”, arXiv:2608.16884 (2026)](https://arxiv.org/abs/2608.16884v1)
- [openai/math release notes (2026): model provenance](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model”

## Related

- [Integer multiplication below n log n](https://closedform.nishachay.tech/e/openai-math-2026-109/) - Theoretical computer science
- [Almost-linear-time exact matching and prescribed-degree factors in general graphs](https://closedform.nishachay.tech/e/openai-math-2026-120/) - Theoretical computer science
- [Exact Fourier transforms below n log n](https://closedform.nishachay.tech/e/openai-math-2026-130/) - Theoretical computer science
- [The computational complexity of Weisfeiler–Leman refinement](https://closedform.nishachay.tech/e/openai-math-2026-133/) - Theoretical computer science
- [Generalized star height at most three](https://closedform.nishachay.tech/e/openai-math-2026-134/) - Theoretical computer science
- [Theoretical computer science](https://closedform.nishachay.tech/science/mathematics/theoretical-computer-science/) - Field · 40 results
- [Open since the 1960s](https://closedform.nishachay.tech/open-since/1960s/) - Era · 33 results
- [Lean-checked results](https://closedform.nishachay.tech/trust/lean-checked/) - Trust · 242 results
- [Landmark results](https://closedform.nishachay.tech/significance/landmark/) - Significance · 49 results
- [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20107%5D%20)