Definition of matrix-multiplication exponent $\omega$
The matrix multiplication exponent being $\omega$ does not guarantee that there is an algorithm that runs in time $O(n^\omega)$, but only that for each $\epsilon &gt; 0$, there is an algorithm that runs in $O(n^{\omega+\epsilon})$. Indeed if you can find an algorithm that runs in time $O(n^2 \mathrm{polylog}(n))$, then this shows that $\omega = 2$.

You can find the formal definition in the book Algebraic Complexity Theory by Peter Bürgisser, Michael Clausen, Amin Shokrollahi.