Improved Almost laws for $SO(3)$

We construct quantitative almost laws for $SO(3)$. More precisely, there exist a constant $c>0$ and non-trivial words $W_n\in F_2$ such that, for every $A,B\in SO(3)$, $$\|W_n(A,B)-I\| \le \exp\!\left(-c |W_n|^{\delta}\right),$$ where $\delta=\log_2(x_0)=0.879146\ldots$ and $x_0>1$ is the real root of $x^3=x^2+x+1$. This improves the exponent $\log_2\varphi$ obtained from Elkasapy's lower-central-series construction. As an application, we show how this result improves the word-length threshold in Kuperberg's Solovay--Kitaev algorithm for single-qubit gates.