Algorithms·matmul:omega
Exponent of matrix multiplication
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Fidelity F0: Absent. No formal statement is attached to the result.
The smallest ω such that two n×n matrices can be multiplied in about n^ω operations. The schoolbook method gives 3, and ω is at least 2; the record is the best proven upper bound.
Record history
15 steps·exponent (upper bound), lower is betterStrassen's 1969 algorithm first brought the exponent below 3. By 1990 the methods of Coppersmith, Winograd and Strassen had taken it to 2.3755, and refinements since 2012 have moved it in the third and fourth decimal places. The 2026 step came from a new optimisation algorithm for the combination-loss analysis, refined with AlphaEvolve.
A reformulated optimisation problem and a new optimisation algorithm, refined with AlphaEvolve.
More asymmetry yields faster matrix multiplication.
From alpha to omega.
Asymmetric hashing.
A refined laser method.
Powers of tensors and fast matrix multiplication.
Multiplying matrices faster than Coppersmith–Winograd.
Matrix multiplication via arithmetic progressions; the record for two decades.
The laser method.
On the asymptotic complexity of matrix multiplication.
Disjoint sums of tensors.
Partial and total matrix multiplication, and the asymptotic sum inequality.
Approximate algorithms and border rank.
Trilinear aggregating.
Gaussian elimination is not optimal: seven multiplications for 2×2, applied recursively.
AI activity
How grades workAn upper bound of 2.371177 on the exponent of matrix multiplication, improving 2.371339.
Reasoning and sources
Autonomy
The authors reformulated the optimisation problem and designed a new algorithm for it; AlphaEvolve played a supporting role, refining that algorithm, as the abstract says.
Fidelity
How fidelity is gradedF0 no formal statement. Absent. No formal statement is attached to the result.
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qed.bot, “Exponent of matrix multiplication”, https://qed.bot/t/matmul-omega, as of 30 Sep 2026.