[proofplan] We compute the upper ramification filtration of $K^{\mathrm{ab}}/K$ by taking inverse limits of the upper ramification filtrations of finite sub-extensions $K_m L_n / K$. The key ingredients are: (1) the upper ramification of the Lubin--Tate extensions $L_n/K$ has integer breaks (by the [Upper Ramification Groups of $L_n/K$](/theorems/???)); (2) adjoining an unramified extension does not change the upper ramification above $t = -1$ (since unramified extensions contribute only at $t = -1$); (3) the [Generalized Local Kronecker--Weber Theorem](/theorems/???) gives $K^{\mathrm{ab}} = K^{\mathrm{ur}} L_\infty = \varinjlim K_m L_n$. We pass to the limit and use the Artin map to translate the result into norm group language. [/proofplan] [step:Compute $G^t(K_m L_n / K)$ for finite composita] Let $K_m / K$ be the unramified extension of degree $m$ and $L_n / K$ the $n$-th Lubin--Tate extension. Their compositum $K_m L_n / K$ is a finite abelian extension with $\operatorname{Gal}(K_m L_n / K) \cong \operatorname{Gal}(K_m / K) \times \operatorname{Gal}(L_n / K)$ (since $K_m \cap L_n = K$: $K_m/K$ is unramified while $L_n/K$ is totally ramified, so their intersection is $K$). For $t > -1$, the upper ramification of a compositum $LM/K$ where $L/K$ is unramified satisfies $G^t(LM/K) \cong G^t(M/K)$. This is because the unramified part contributes only to $G^{-1}/G^0$ (the Galois group modulo inertia), and for $t > -1$ we are within the inertia subgroup and beyond. Explicitly: the inertia group of $K_m L_n / K$ is $\operatorname{Gal}(K_m L_n / K_m) \cong \operatorname{Gal}(L_n / K)$ (since $K_m L_n / K_m$ is totally ramified), and the upper ramification groups for $t \geq 0$ live inside the inertia group. By the [Compatibility of Upper Numbering with Quotients](/theorems/???), restriction to $L_n$ gives \begin{align*} G^t(K_m L_n / K) = G^t(L_n / K) \quad \text{for } t > -1, \end{align*} where we identify $G^t(L_n/K)$ with its image in $\operatorname{Gal}(K_m L_n / K)$ via inflation. More precisely, viewing $G^t(K_m L_n / K)$ as a subgroup of $\operatorname{Gal}(K_m L_n / K)$: \begin{align*} G^t(K_m L_n / K) = \operatorname{Gal}(K_m L_n / K_m L_{\lceil t \rceil}) \quad \text{for } 0 \leq t \leq n - 1, \end{align*} and $G^t(K_m L_n / K) = \{1\}$ for $t > n - 1$. This follows from the [Upper Ramification Groups of $L_n/K$](/theorems/???), which gives $G^t(L_n/K) = \operatorname{Gal}(L_n / L_{\lceil t \rceil})$. [/step] [step:Pass to the inverse limit to obtain $G^t(K^{\mathrm{ab}}/K)$] By the [Generalized Local Kronecker--Weber Theorem](/theorems/???), $K^{\mathrm{ab}} = K^{\mathrm{ur}} L_\infty = \varinjlim_{m,n} K_m L_n$. The Galois group is the inverse limit \begin{align*} \operatorname{Gal}(K^{\mathrm{ab}}/K) = \varprojlim_{m,n} \operatorname{Gal}(K_m L_n / K). \end{align*} The upper ramification filtration is compatible with inverse limits: for a tower of abelian extensions $M_\alpha / K$ with $\bigcup_\alpha M_\alpha = M$, we have \begin{align*} G^t(M/K) = \varprojlim_\alpha G^t(M_\alpha / K). \end{align*} Applying this to $M = K^{\mathrm{ab}}$ and the directed system $(K_m L_n)_{m, n}$: \begin{align*} G^t(K^{\mathrm{ab}}/K) &= \varprojlim_{m,n} G^t(K_m L_n / K) = \varprojlim_{m, n \geq \lceil t \rceil} \operatorname{Gal}(K_m L_n / K_m L_{\lceil t \rceil}). \end{align*} Taking the union of the fixed fields: the fixed field of $G^t(K^{\mathrm{ab}}/K)$ is \begin{align*} (K^{\mathrm{ab}})^{G^t} = \bigcup_{m, n \geq \lceil t \rceil} K_m L_{\lceil t \rceil} = K^{\mathrm{ur}} L_{\lceil t \rceil}. \end{align*} Therefore \begin{align*} G^t(K^{\mathrm{ab}}/K) = \operatorname{Gal}(K^{\mathrm{ab}} / K^{\mathrm{ur}} L_{\lceil t \rceil}) \quad \text{for } t > -1. \end{align*} [/step] [step:Compute the Artin preimage $\operatorname{Art}_K^{-1}(G^t(K^{\mathrm{ab}}/K))$] We need to identify which elements of $K^\times$ map to $G^t(K^{\mathrm{ab}}/K) = \operatorname{Gal}(K^{\mathrm{ab}} / K^{\mathrm{ur}} L_{\lceil t \rceil})$ under the Artin map. For finite levels: $G^t(K_m L_n / K) = \operatorname{Gal}(K_m L_n / K_m L_{\lceil t \rceil})$, and \begin{align*} \operatorname{Art}_K^{-1}(\operatorname{Gal}(K_m L_n / K_m L_{\lceil t \rceil})) = N(K_m L_{\lceil t \rceil} / K) = \langle \pi^m \rangle \cdot U_K^{(\lceil t \rceil)}. \end{align*} This uses the norm group computation from the proof of the [Generalized Local Kronecker--Weber Theorem](/theorems/???): $N(K_m L_j / K) = \langle \pi^m \rangle \cdot U_K^{(j)}$ for any $m, j$. Taking the intersection over all $m$ and $n \geq \lceil t \rceil$: \begin{align*} \operatorname{Art}_K^{-1}(G^t(K^{\mathrm{ab}}/K)) &= \bigcap_{m \geq 1} \langle \pi^m \rangle \cdot U_K^{(\lceil t \rceil)}. \end{align*} An element $\pi^a u$ (with $u \in U_K$) lies in $\langle \pi^m \rangle \cdot U_K^{(\lceil t \rceil)}$ for all $m$ if and only if $u \in U_K^{(\lceil t \rceil)}$ and $m \mid a$ for all $m \geq 1$. The condition $m \mid a$ for all $m$ forces $a = 0$ (the only integer divisible by every positive integer is $0$). Therefore \begin{align*} \operatorname{Art}_K^{-1}(G^t(K^{\mathrm{ab}}/K)) = U_K^{(\lceil t \rceil)}. \end{align*} [guided] The computation of the Artin preimage proceeds in two stages. At finite level, the Artin map identifies $\operatorname{Gal}(K_m L_n / K_m L_{\lceil t \rceil})$ with $K^\times / N(K_m L_{\lceil t \rceil}/K)$, so the preimage of this Galois subgroup under $\operatorname{Art}_K$ is the norm group $N(K_m L_{\lceil t \rceil}/K) = \langle \pi^m \rangle \cdot U_K^{(\lceil t \rceil)}$. Passing to the inverse limit means intersecting over all $m$ and $n$. The intersection $\bigcap_{m \geq 1} \langle \pi^m \rangle \cdot U_K^{(\lceil t \rceil)}$ eliminates the powers of $\pi$: an element $\pi^a u$ with $u \in U_K^{(\lceil t \rceil)}$ lies in $\langle \pi^m \rangle \cdot U_K^{(\lceil t \rceil)}$ if and only if $m \mid a$ (since $\pi^a u = \pi^{mk} \cdot (\pi^{a - mk} u')$ requires $a - mk \in m\mathbb{Z}$... more carefully: $\pi^a u \in \langle \pi^m \rangle \cdot U_K^{(\lceil t \rceil)}$ means $\pi^a u = \pi^{mk} v$ for some $k \in \mathbb{Z}$ and $v \in U_K^{(\lceil t \rceil)}$, which requires $a = mk$ and $u = v \in U_K^{(\lceil t \rceil)}$). For this to hold for all $m \geq 1$, we need $m \mid a$ for all $m$, forcing $a = 0$. So the intersection is $\{u \in U_K : u \in U_K^{(\lceil t \rceil)}\} = U_K^{(\lceil t \rceil)}$. This gives the clean result: $\operatorname{Art}_K^{-1}(G^t(K^{\mathrm{ab}}/K)) = U_K^{(\lceil t \rceil)}$. The upper ramification filtration on $\operatorname{Gal}(K^{\mathrm{ab}}/K)$ corresponds, via the Artin map, to the filtration by higher unit groups $U_K^{(j)}$, $j = 0, 1, 2, \ldots$ This is the explicit link between the arithmetic of $K$ (the unit group filtration) and the Galois theory of $K^{\mathrm{ab}}$ (the ramification filtration). [/guided] [/step]