Matrix multiplication with exponent at most 9/4

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 numbersSource 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 competeSource 2.

Two earlier companion papers reach weaker square bounds by other routes. One shows ω < 2.258 over the complex numbersSource 3. The other shows ω < 2.371054886006746 over every fieldSource 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 fieldSource 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 inputsSource 6. Truly separate pieces need their own variables on all three sides: both input matrices and the outputSource 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 thirdSource 8. An extra labeling step then tells them apart on the third side too. The number of freed pieces outweighs the cost of the labelsSource 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 617Source 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 fieldsSource 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 practiceSource 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.