User blog comment:Mh314159/new YIP notation/@comment-39585023-20190705220451/@comment-35470197-20190707035551

We have \begin{eqnarray*} o(\{0\}_0(x)) & \sim & 2 \omega \\ o(\{0\}_y(x)) & \sim & 2 \omega + 2y \\ o(\{m\}_y(x)) & \sim & 2 \omega \times (1 + m) + 2y \\ [a] & \sim & F_{\omega^2+1}(a). \end{eqnarray*} It is significantly greater than your original single bracket number. I note that \([a]\) just iterate a fixed function \(\{y\}_y\(x)\), and hence the strength of the single bracket number is mainly due to the recursion of \(\{m\}_y(x)\).