The Matrix Multiplication Exponent

The matrix multiplication exponent $\omega$ is the infimum of all $t$ for which two $n \times n$ matrices can be multiplied in $O(n^t)$ arithmetic operations. Strassen showed in 1969 that $\omega < 3$, and sixty years of work has driven the upper bound down without anyone knowing the true value. Whether $\omega = 2$ is one of the central open questions of algebraic complexity. The current bounds come from the laser method as refined by combination loss analysis. This paper attacks the optimization problem at the core of that refinement, reformulating it so it can be solved in a larger setting, designing a new optimization algorithm for it, and then refining that algorithm with AlphaEvolve. The result is $\omega < 2.371177$, improving the previous best of $2.371339$.

A record, not a resolution, and a small one by design. The bound moves from $2.371339$ to $2.371177$, about $1.6 \times 10^{-4}$, and the authors describe it as a small step. Whether $\omega = 2$ is untouched, and nothing here suggests the laser method can reach it. The interesting claim is methodological rather than numerical. The bottleneck in this line of work is a hard optimization problem, and the paper reports progress by reformulating that problem and then improving the optimizer, with AlphaEvolve doing the final refinement. That is a different kind of contribution from a new mathematical identity, and it is why the entry is filed as computation.