Deep logarithm subgroup of a formal group ($m>\frac{e}{p-1},F({\pi }^m{\mathcal{O}}_K)\cong ({\mathcal{O}}_K,+)$)

Let $F$ be a formal group law over the valuation ring of a finite extension $K/{\mathbb{Q}}_p$, with $v_K(\pi )=1$ and $e=v_K(p)$. If $m>e/(p-1)$, its formal logarithm is an isomorphism $F({\pi }^m{\mathcal{O}}_K)\to ({\pi }^m{\mathcal{O}}_K,+)$. Indeed, its scaled series is $U+\pi R(U)$ with $R$ integral and convergent on ${\mathcal{O}}_K$; solving by contraction produces the formal group exponential. The subgroup has index $q^{m-1}$ in $F(\pi {\mathcal{O}}_K)$, where $q$ is the residue-field order.