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arXiv · 2610.28627

Matrix Multiplication from Polynomial Packings: A Constructive and Elementary Proof of $ω\le9/4$

Rafael Pass
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Abstract

We give a self-contained constructive proof that the matrix multiplication exponent satisfies $ω\le9/4$. This bound was recently claimed by OpenAI~\cite{original} through non-constructive arguments in Strassen's asymptotic spectral framework, counting exact scalar arithmetic operations over the complex numbers. We explicitly construct a Strassen-style matrix multiplication identity and obtain the bound by ordinary block recursion. For every $γ>0$, the resulting algorithm multiplies two $n\times n$ integer matrices using $O(n^{9/4+γ})$ integer arithmetic operations.

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Rafael Pass. 2026-10-20. Matrix Multiplication from Polynomial Packings: A Constructive and Elementary Proof of $ω\le9/4$. https://arxiv.org/abs/2610.28627

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