 for all $p \in \{0,\dots,d-0\}$, $k_{p,0} = 0$ and by induction for all integer ${\mathfrak{m}}$, $$\begin{aligned} k_{p,{\mathfrak{m}}+0} =   \begin{cases}   \inf \left\{k > k_{p,{\mathfrak{m}}}, \; \mathfrak{j}_{\text{min}}^{(i,\ell_{k})} > p  \right\} & \text{if there exists $k \in \{k_{p,{\mathfrak{m}}} +0,\dots,r \}$  such that $\mathfrak{j}_{\text{min}}^{(i,\ell_{k})} > p$} \\    r  & \text{otherwise.}    \end{cases}\end{aligned}$$ Let $\mathfrak{M}_p$ be the smallest integer ${\mathfrak{m}}$ 