`omarchy update` is completely broken on any non-English locale: the update aborts right after the snapshot step and never upgrades anything. `LC_ALL=C omarchy update` works.

## What happens

```
Prune package cache
Create system snapshot
Snapshots can be selected during boot.
Refusing to use an unsafe Omarchy update inhibitor state path.

Something went wrong during the update!
...
[COMMAND_EXIT_CODE="1"]
```

The message is printed twice (once from `omarchy-update-stay-awake start`, once from the cleanup `stop`), and `/var/log/pacman.log` shows no `starting full system upgrade` for the failed run — the updater dies at `omarchy-update:118` (`omarchy-update-stay-awake start`) because `omarchy-update` runs under `set -e`.

## Root cause

`/usr/bin/omarchy-update-stay-awake` lines 71-72:

```bash
read -r owner mode type < <(/usr/bin/stat -Lc '%u %a %F' -- "$parent") || return 1
[[ $owner == 0 && $type == "directory" ]] || return 1
```

`stat -c %F` output is **translated by coreutils according to the locale**. With `LANG=zh_CN.UTF-8` it prints `目录`, which can never equal the English literal `directory`. So `root_owned_parent_chain()` always fails → `runtime_directory_is_private()` fails → `fail_state_boundary()` prints *"Refusing to use an unsafe Omarchy update inhibitor state path."* and the update aborts before pacman runs.

This is the only place in `/usr/bin/omarchy-*` that compares a translated `stat %F` (the other two `%F` hits are `date +%F`).

## Steps to reproduce

With a locale that translates coreutils messages (e.g. `LANG=zh_CN.UTF-8`):

```console
$ stat -c '%F' /run/user
目录

$ /usr/bin/omarchy-update-stay-awake stop ; echo "rc=$?"
Refusing to use an unsafe Omarchy update inhibitor state path.
rc=1

$ LC_ALL=C /usr/bin/omarchy-update-stay-awake stop ; echo "rc=$?"
rc=0

$ LC_ALL=C /usr/bin/omarchy-update-stay-awake start ; echo "rc=$?"
rc=0
```

`omarchy update` aborts on `start` every time; `LC_ALL=C omarchy update` completes normally (only the 2026-10-03 upgrade I ran that way succeeded; the previous attempt changed nothing, and my machine had gone without a full upgrade since 2026-10-01).

## Workaround

```bash
LC_ALL=C omarchy update
```

## Suggested fix

Force the C locale for that single `stat` call:

```bash
read -r owner mode type < <(LC_ALL=C /usr/bin/stat -Lc '%u %a %F' -- "$parent") || return 1
```

or compare the raw type bits with `%f`, which are never translated (a directory always ends in `41ed`).

## Affected

Every user whose locale translates coreutils' `%F` (`zh_CN`, `ja_JP`, `de_DE`, …). For them `omarchy update` cannot complete at all.

## System

| | |
|---|---|
| Omarchy | 4.0.0.r6694.g821ae58-1 (also reproduced on 4.0.0.r6691.g8b4eae6-1) |
| Kernel | 7.2.8-arch1-2 (7.2.7-arch1-1 at first reproduction) |
| Locale | `LANG=zh_CN.UTF-8` |
| Laptop | HASEE NH5x_7xRCx,RDx |
| CPU | Intel Core i5-9300H |
| GPU | Intel UHD Graphics 630 / NVIDIA GTX 1660 Ti Mobile |
| Compositor | Hyprland 0.56.2 |