= Solution
Let $L/K$ be a finite <Galois extension> of non-Archimedean <local fields>, let $G=\operatorname{Gal}(L/K)$, and normalize the <discrete valuation> by $v_L(L^{\times})=\mathbb Z$. The <lower ramification numbering> is
$$
G_{-1}=G,\qquad
G_i=\{\sigma\in G:v_L(\sigma(x)-x)\geq i+1\text{ for every }x\in\mathcal O_L\},\quad i\geq0.
$$
Here $G_0$ is the <inertia group> and $G_1$ is the <wild inertia group>. For real $s\geq-1$, put $G_s=G_{\lceil s\rceil}$; thus $G_s=G_0$ for $-1<s\leq0$. Define the <Herbrand function> and its inverse by
$$
\varphi(s)=\int_0^s\frac{dt}{[G_0:G_t]},\qquad \psi=\varphi^{-1},
$$
extending $\varphi(s)=s$ on $[-1,0]$. The <upper ramification numbering> is $G^u=G_{\psi(u)}$ for $u\geq-1$.
Let $\alpha^p-\alpha=t^{1-p}$ over $K=\mathbb F_p((t))$, with $t$ its <uniformizer>. The <Artin–Schreier polynomial> has derivative $-1$ in characteristic $p$, so all its roots are distinct. If $\alpha$ is one root and $\beta$ another, then $(\beta-\alpha)^p=\beta-\alpha$, so $\beta-\alpha\in\mathbb F_p$. Conversely every $\alpha+a$, $a\in\mathbb F_p$, is a root. Thus $L=K(\alpha)$ contains all roots and is a <splitting field> of a separable <polynomial>; it is Galois. Every automorphism has the form $\sigma_a(\alpha)=\alpha+a$, and its <Galois group> embeds in the additive group of $\mathbb F_p$.
There is no root in $K$. If $v_K(z)\geq0$, then $z^p-z$ cannot have negative <valuation>. If $v_K(z)<0$, the <ultrametric inequality> gives $v_K(z^p-z)=p\,v_K(z)$, divisible by $p$, whereas $v_K(t^{1-p})=1-p$ is not. The <Galois group> is consequently nontrivial. Its order divides the prime $p$, so
$$
\boxed{[L:K]=p,\qquad G\cong(\mathbb F_p,+).}
$$
Let $e=e(L/K)$ be the <ramification index>. The equation forces $v_L(\alpha)<0$, and hence
$$
p\,v_L(\alpha)=(1-p)e.
$$
As $\gcd(p,p-1)=1$, $p\mid e$. Since $e\leq[L:K]=p$, we get $e=p$, <residue-field degree> one, and $v_L(\alpha)=1-p$. Thus this is a <totally ramified extension>, and
$$
\pi_L=t\alpha\qquad\text{has }v_L(\pi_L)=p+(1-p)=1.
$$
This is a <uniformizer>. The <uniformizer criterion for lower ramification groups> states that for a totally ramified <Galois extension>, $\sigma\in G_i$ exactly when $v_L(\sigma(\pi_L)-\pi_L)\geq i+1$. Here every nonidentity automorphism satisfies
$$
\sigma_a(\pi_L)-\pi_L=a t,\qquad v_L(a t)=p.
$$
Therefore the <ramification break of an Artin–Schreier pole> occurs at $p-1$, and the lower groups are
$$
\boxed{G_i=G\quad(-1\leq i\leq p-1),\qquad G_i=\{1\}\quad(i\geq p).}
$$
This includes $p=2$, where the break is one. There is no off-by-one shift: the condition is $v_L(\sigma\pi_L-\pi_L)\geq i+1$.
For the upper groups, compute the <Herbrand function> explicitly:
$$
\varphi(s)=\begin{cases}s,&0\leq s\leq p-1,\\p-1+\dfrac{s-(p-1)}p,&s\geq p-1.\end{cases}
$$
Hence the upper break is also $p-1$ and, with the stated real-index convention,
$$
\boxed{G^u=G\quad(-1\leq u\leq p-1),\qquad G^u=\{1\}\quad(u>p-1).}
$$
As a consistency check, $\pi_L$ satisfies the <Eisenstein polynomial> $X^p-t^{p-1}X-t$. Its derivative in characteristic $p$ is $-t^{p-1}$, giving different exponent $p(p-1)$, equal to $\sum_{i\geq0}(|G_i|-1)=p(p-1)$ from the <different exponent from ramification groups>.
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