User blog comment:Mh314159/new YIP notation/@comment-39585023-20190714235606/@comment-35470197-20190718141730

Here is the analysis for the second version, i.e. the one with proposed new Universal Relationships: \begin{eqnarray*} f_a(b) & \sim & F_{2a}(b) \\ f_x(x) & \sim & F_{2 \omega}(x) \\ O(\{\{0\}0\}_0(x)) & \sim & 2 \omega \\ O(\{\{m\}n\}_{1+y}(x)) & \sim & O(\{\{m\}n\}_0(x)) + 2(1+y) \\ O(\{\{1+m\}n\}_0(x)) & \sim & O(\{\{m\}n\}_0(x)) + 2 \omega + 1 \\ O(\{\{0\}(1+n)\}_0(x)) & \sim & O(\{\{0\}n\}_0(x)) + (2 \omega + 1) \times \omega \\ O(\{\{x\}x\}_x(x)) & \sim & 2 \omega + ((2 \omega + 1) \times \omega) \times \omega \\ [a] & \sim & F_{2 \omega + ((2 \omega + 1) \times \omega) \times \omega}^a(4) \sim F_{\omega^3 + 1}(a) \end{eqnarray*} So the limit of this notation is \(\omega^3 + 1\) in FGH, if I am correct. Unlike the first version, you only used the single candidate of the base function \(\{\{0\}0\}_0(x)\).