A faster matrix multiplication algorithm through structured optimization

Abstract

We present a new asymptotic matrix multiplication construction yielding $\omega<2.37115924$, and therefore an algorithm that multiplies two $n\times n$ matrices in $O(n^{2.37115924})$ arithmetic operations over the rationals, reals, or complex numbers. The construction improves the bound $2.371177$ of Dupont et al. within the combination-loss framework for the Coppersmith--Winograd tensor. Our approach combines a shared recursive parameter family with a conditioned optimization procedure that addresses two obstacles: weak gradients at low-mass constituents and competing bottlenecks in the retained exponent. A mass- and probability-dependent rescaling supports quasi-Newton optimization in $212\,820$ coordinates, while a six-control correction balances the three extraction roles at each relevant stage. Positive rational entropy-dual factors and an independent ordinary-string verifier turn the resulting construction into an exact certificate. The verified upper-bound expression lies in $[2.3711592385931075066123,\,2.3711592385931075066124]$. The complete witness and an offline verifier using only the Python standard library accompany the paper.