feat(pwa): app-icon badge with the highlight count via navigator.setAppBadge (#154)

Same number as the tab title (leaf-room highlights), cleared at zero.
Skipped under Tauri where the native set_badge_count owns the badge, and
silently absent where the Badging API is not available.

Closes #154

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01PPmy3tPq869XDW4njjVaKA

LOTUS_FEATURES.md

@@ -1179,6 +1179,10 @@ Links you paste, send, edit, or merely _see_ lose ad/analytics identifiers — `
 
 The syncable subset of Lotus settings (theme, composer toolbar order, notification/quiet-hour preferences, call keys, privacy toggles, …) is mirrored to the `io.lotus.settings` account-data event on the user's own homeserver and applied on every other device. Device-bound keys stay local (`DEVICE_LOCAL_KEYS` in `src/app/utils/settingsSync.ts`: page zoom, media auto-load, animation pause, glassmorphism, noise-suppression tier/model, bitrates, volumes, notification permission, developer tools, PTT mode, camera-on-join, drawer state). Conflicts are last-write-wins on an `updatedAt` stamp forced monotonic per device; a per-account `lastSyncedAt` marker in localStorage stops a device from echoing a snapshot it just applied. **Settings → General → Sync** has the toggle (itself device-local), **Push now** (make this device win everywhere) and **Clear synced copy**. Hook: `src/app/hooks/useSettingsSync.ts`, mounted from `ClientNonUIFeatures`.
 
+### PWA App-Icon Badge (Gitea #154)
+
+When Lotus Chat is installed as a PWA (Android Chrome, desktop Chrome/Edge), the app icon carries a numeric badge via the Badging API (`navigator.setAppBadge`). The number is the same highlight count (mentions/DMs, leaf rooms only) that the tab title shows, so the two can never disagree; it clears when the count reaches zero. Lives in `FaviconUpdater` (`ClientNonUIFeatures.tsx`) next to the title/favicon logic. No-op in a plain browser tab (the API is absent) and under Tauri, where the native `set_badge_count` already owns the badge.
+
 ### Media Gallery
 
 `MediaGallery.tsx` — a right-side drawer for browsing room media.