. According to Corollary \[cor:phijres\] and Theorem \[thm:Schneps\], $$\begin{aligned}     \ker{\varphi^{(0)}_{0}\big|_{\mathbf W_{N,0}}}\cong\bigoplus_{n<N}\mathbf W_{N-n,0}\otimes\ker E_{n,0}\cong\bigoplus_{n<N}\mathbf W_{N-n,0}\otimes\mathbf W_{n,0}