Proposition 4

For the chart of G used to be canonical, it is necessary and sufficient that $\psi_j = 0$ for $j \geq 2$.

This is sufficient by Proposition 3. Suppose that the chart is canonical and that $\psi_i = 0$ for $2 \leq i < n$. Then $nx = x^{[n]} = \sum_{i=0}^n \binom{n}{i} \psi_i(x) = nx + \psi_n(x)$, whence $\psi_n = 0$. Hence $\psi_j = 0$ for $j \geq 2$ by induction on $j$.