Multiply n x n matrices in n^{2.371338} time — beat omega < 2.371339 — or break the Ambainis–Filmus–Le Gall 2.3725 laser-method barrier some other way

Record

omega < 2.371339

Record held by

Alman, Duan, Vassilevska Williams, Xu, Xu & Zhou, SODA 2025 (arXiv:2404.16349, \"More Asymmetry Yields Faster Matrix Multiplication\")

Hardness

no superquadratic lower bound known; laser-method analyses of powers of the Coppersmith–Winograd tensor cannot prove omega < 2.3725 (and broad generalizations cannot go below ~2.3078)

Hardness from

Ambainis, Filmus & Le Gall, STOC 2015

Endgame

none known

Problems

Matrix Multiplication

Statement

Let omega be the matrix multiplication exponent: the infimum of c such that two n x n matrices can be multiplied in O(n^c) arithmetic operations. Trivially omega <= 3 and omega >= 2, and the running conjecture is omega = 2. The problem: publish any bound omega < 2.371339. An equally valid closing move is structural: prove that some concrete new technique escapes the Ambainis–Filmus–Le Gall barrier, which shows the standard "laser method" applied to powers of the Coppersmith–Winograd tensor can never prove omega < 2.3725.

Current record

The record is omega < 2.371339, by Alman, Duan, Vassilevska Williams, Xu, Xu and Zhou (SODA 2025), the latest step in a line that revived progress after a decade of stasis at 2.3728: Duan–Wu–Zhou (FOCS 2023) found and fixed a "combination loss" in prior laser-method analyses, Vassilevska Williams–Xu–Xu–Zhou (SODA 2024) reached 2.371552, and the 2025 paper squeezed further via asymmetric analyses of the Coppersmith–Winograd tensor's powers. All of these analyze the same underlying tensor; Ambainis, Filmus and Le Gall (STOC 2015) proved this family of analyses hits a wall at 2.3725 for the classical laser method, and that even substantially generalized monomial-degeneration approaches on that tensor cannot reach below roughly 2.3078. So the remaining slack inside the current framework is thin, and known barriers block the road to 2.

Why it matters

Omega is a load-bearing constant across the entire reduction web: transitive closure, context-free parsing, all-pairs shortest paths subroutines, dense linear algebra, and the polynomial-method algorithms cited by several other problems all inherit their exponents from it. Any improvement propagates automatically through dozens of published reductions. Conversely, a clean new barrier theorem prunes the search space for everyone.

Attack surface

Inside the framework: the recent gains came from bookkeeping refinements (asymmetry, combination-loss repair) that were sitting unnoticed for years — systematic, machine-assisted search over laser-method parameter spaces (hash-function moduli, distribution splits) is genuinely plausible territory for optimization tooling. Outside it: candidate replacement tensors (e.g., analyzing tensors other than Coppersmith–Winograd), the group-theoretic approach of Cohn–Umans, or degeneration notions that evade the known barrier definitions. First cheap experiment: rerun the 2025 asymmetric optimization on the next unanalyzed tensor power and see whether the numerical program still improves.

Verification notes

Checked against the literature in August 2026: no published bound below 2.371339 was found; sources through late 2025 still cite omega < 2.371339 (arXiv:2404.16349) as the record. Adjacent 2026 activity (e.g., asymptotic-rank speedup theorems, arXiv:2605.21738) does not claim a new omega bound.