feat(desktop): add macOS dock badge for chats with unseen activity

Show a count of chats (root sessions) with unseen activity on the macOS
dock icon. The count is computed in the existing tray snapshot (full
cross-project list, not the capped tray view; a subtask's unseen rolls up
to its root only when subtask notifications are enabled) and pushed to the
main process over the existing desktop_tray_update IPC, which calls
app.setBadgeCount (0 clears it). The badge clears as sessions are marked
seen on window focus.

Add a Dock badge toggle in Appearance settings (default on, persisted,
darwin desktop only), localized across all dictionaries, with a matching
settings-search entry whose availability mirrors the render guard exactly.

packages/ui/src/lib/i18n/messages/en.settings.ts

@@ -1606,6 +1606,8 @@ export const settingsDict = {
   'settings.openchamber.visual.actions.reloadingThemes': 'Reloading themes...',
   'settings.openchamber.visual.field.macVibrancy': 'Window transparency',
   'settings.openchamber.visual.field.macVibrancyHint': 'Use the native macOS blur (vibrancy) behind the sidebar. Turn off for fully solid, opaque surfaces.',
+  'settings.openchamber.visual.field.dockBadge': 'Dock badge',
+  'settings.openchamber.visual.field.dockBadgeHint': 'Show a count of chats with unseen activity on the macOS dock icon.',
   'settings.openchamber.visual.actions.saveAndRestart': 'Save & restart',
   'settings.openchamber.visual.actions.restarting': 'Restarting…',
   'settings.openchamber.visual.field.themeImportInfoAria': 'Theme import info',

packages/ui/src/lib/i18n/messages/es.settings.ts

@@ -1572,6 +1572,8 @@ export const settingsDict = {
   "settings.openchamber.visual.actions.reloadThemes": "Recargar temas",
   "settings.openchamber.visual.field.macVibrancy": "Transparencia de la ventana",
   "settings.openchamber.visual.field.macVibrancyHint": "Usa el desenfoque nativo de macOS (vibrancy) detrás de la barra lateral. Desactívalo para superficies totalmente sólidas y opacas.",
+  "settings.openchamber.visual.field.dockBadge": "Insignia en el Dock",
+  "settings.openchamber.visual.field.dockBadgeHint": "Muestra en el icono del Dock de macOS el número de chats con actividad sin ver.",
   "settings.openchamber.visual.actions.saveAndRestart": "Guardar y reiniciar",
   "settings.openchamber.visual.actions.restarting": "Reiniciando…",
   "settings.openchamber.visual.actions.reloadingThemes": "Recargando temas...",

packages/ui/src/lib/i18n/messages/fr.settings.ts

@@ -1762,6 +1762,8 @@ export const settingsDict = {
   'settings.openchamber.visual.option.mobileLayout.new': 'Nouvelle',
   'settings.openchamber.visual.field.macVibrancy': 'Transparence de la fenêtre',
   'settings.openchamber.visual.field.macVibrancyHint': 'Utiliser le flou macOS natif (vibrancy) derrière la barre latérale. Désactivez cette option pour des surfaces entièrement opaques.',
+  'settings.openchamber.visual.field.dockBadge': 'Badge du Dock',
+  'settings.openchamber.visual.field.dockBadgeHint': 'Afficher sur l’icône du Dock de macOS le nombre de discussions avec une activité non vue.',
   'settings.openchamber.visual.actions.saveAndRestart': 'Enregistrer et redémarrer',
   'settings.openchamber.visual.actions.restarting': 'Redémarrage…',
   'settings.openchamber.visual.field.fileEditorKeymap': 'Keymap de l’éditeur de fichiers',

packages/ui/src/lib/i18n/messages/ja.settings.ts

@@ -1606,6 +1606,8 @@ export const settingsDict = {
   'settings.openchamber.visual.actions.reloadingThemes': 'テーマを再読み込み中...',
   'settings.openchamber.visual.field.macVibrancy': 'ウィンドウ透過',
   'settings.openchamber.visual.field.macVibrancyHint': 'サイドバーの背後に macOS ネイティブのぼかし（Vibrancy）を使用します。完全に不透明なサーフェスにする場合はオフにします。',
+  'settings.openchamber.visual.field.dockBadge': 'Dock バッジ',
+  'settings.openchamber.visual.field.dockBadgeHint': '未読のアクティビティがあるチャットの数を macOS の Dock アイコンに表示します。',
   'settings.openchamber.visual.actions.saveAndRestart': '保存して再起動',
   'settings.openchamber.visual.actions.restarting': '再起動中...',
   'settings.openchamber.visual.field.themeImportInfoAria': 'テーマインポート情報',

packages/ui/src/lib/i18n/messages/ko.settings.ts

@@ -1572,6 +1572,8 @@ export const settingsDict = {
   'settings.openchamber.visual.actions.reloadThemes': '테마 다시 로드',
   'settings.openchamber.visual.field.macVibrancy': '창 투명도',
   'settings.openchamber.visual.field.macVibrancyHint': '사이드바 뒤에 macOS 기본 블러(vibrancy)를 사용합니다. 완전히 불투명한 표면을 원하면 끄세요.',
+  'settings.openchamber.visual.field.dockBadge': 'Dock 배지',
+  'settings.openchamber.visual.field.dockBadgeHint': '확인하지 않은 활동이 있는 채팅 수를 macOS Dock 아이콘에 표시합니다.',
   'settings.openchamber.visual.actions.saveAndRestart': '저장 후 재시작',
   'settings.openchamber.visual.actions.restarting': '재시작 중…',
   'settings.openchamber.visual.actions.reloadingThemes': '테마 다시 로드 중...',

packages/ui/src/lib/i18n/messages/pl.settings.ts

@@ -907,6 +907,8 @@ export const settingsDict = {
   'settings.openchamber.visual.actions.reloadThemes': 'Przeładuj motywy',
   'settings.openchamber.visual.field.macVibrancy': 'Przezroczystość okna',
   'settings.openchamber.visual.field.macVibrancyHint': 'Używaj natywnego rozmycia macOS (vibrancy) za panelem bocznym. Wyłącz, aby uzyskać w pełni nieprzezroczyste powierzchnie.',
+  'settings.openchamber.visual.field.dockBadge': 'Plakietka w Docku',
+  'settings.openchamber.visual.field.dockBadgeHint': 'Pokazuj na ikonie w Docku macOS liczbę czatów z nieprzeczytaną aktywnością.',
   'settings.openchamber.visual.actions.saveAndRestart': 'Zapisz i uruchom ponownie',
   'settings.openchamber.visual.actions.restarting': 'Ponowne uruchamianie…',
   'settings.openchamber.visual.actions.reloadingThemes': 'Przeładowywanie motywów...',

packages/ui/src/lib/i18n/messages/pt-BR.settings.ts

@@ -1572,6 +1572,8 @@ export const settingsDict = {
   "settings.openchamber.visual.actions.reloadThemes": "Recarregar temas",
   "settings.openchamber.visual.field.macVibrancy": "Transparência da janela",
   "settings.openchamber.visual.field.macVibrancyHint": "Usa o desfoque nativo do macOS (vibrancy) atrás da barra lateral. Desative para superfícies totalmente sólidas e opacas.",
+  "settings.openchamber.visual.field.dockBadge": "Selo no Dock",
+  "settings.openchamber.visual.field.dockBadgeHint": "Mostra no ícone do Dock do macOS o número de conversas com atividade não vista.",
   "settings.openchamber.visual.actions.saveAndRestart": "Salvar e reiniciar",
   "settings.openchamber.visual.actions.restarting": "Reiniciando…",
   "settings.openchamber.visual.actions.reloadingThemes": "Recarregando temas...",

packages/ui/src/lib/i18n/messages/uk.settings.ts

@@ -1572,6 +1572,8 @@ export const settingsDict = {
   "settings.openchamber.visual.actions.reloadThemes": "Перезавантажити теми",
   "settings.openchamber.visual.field.macVibrancy": "Прозорість вікна",
   "settings.openchamber.visual.field.macVibrancyHint": "Використовувати нативне розмиття macOS (vibrancy) під сайдбаром. Вимкніть для повністю непрозорих поверхонь.",
+  "settings.openchamber.visual.field.dockBadge": "Лічильник у доку",
+  "settings.openchamber.visual.field.dockBadgeHint": "Показувати на іконці в доку кількість чатів із непрочитаною активністю.",
   "settings.openchamber.visual.actions.saveAndRestart": "Зберегти та перезапустити",
   "settings.openchamber.visual.actions.restarting": "Перезапуск…",
   "settings.openchamber.visual.actions.reloadingThemes": "Перезавантаження тем...",

packages/ui/src/lib/i18n/messages/zh-CN.settings.ts

@@ -1572,6 +1572,8 @@ export const settingsDict = {
   'settings.openchamber.visual.actions.reloadThemes': '重新加载主题',
   'settings.openchamber.visual.field.macVibrancy': '窗口透明度',
   'settings.openchamber.visual.field.macVibrancyHint': '在侧边栏后使用 macOS 原生模糊（vibrancy）。关闭以获得完全不透明的界面。',
+  'settings.openchamber.visual.field.dockBadge': 'Dock 角标',
+  'settings.openchamber.visual.field.dockBadgeHint': '在 macOS Dock 图标上显示有未读动态的会话数量。',
   'settings.openchamber.visual.actions.saveAndRestart': '保存并重启',
   'settings.openchamber.visual.actions.restarting': '正在重启…',
   'settings.openchamber.visual.actions.reloadingThemes': '正在重新加载主题...',

packages/ui/src/lib/i18n/messages/zh-TW.settings.ts

@@ -1491,6 +1491,8 @@
   'settings.openchamber.visual.actions.reloadThemes': '重新載入主題',
   'settings.openchamber.visual.field.macVibrancy': '視窗透明度',
   'settings.openchamber.visual.field.macVibrancyHint': '在側邊欄後使用 macOS 原生模糊（vibrancy）。關閉以獲得完全不透明的介面。',
+  'settings.openchamber.visual.field.dockBadge': 'Dock 標記',
+  'settings.openchamber.visual.field.dockBadgeHint': '在 macOS Dock 圖示上顯示有未讀動態的對話數量。',
   'settings.openchamber.visual.actions.saveAndRestart': '儲存並重新啟動',
   'settings.openchamber.visual.actions.restarting': '正在重新啟動…',
   'settings.openchamber.visual.actions.reloadingThemes': '正在重新載入主題...',

packages/ui/src/lib/settings/search.ts

@@ -20,6 +20,8 @@ export interface SettingsSearchResult extends SettingsSearchItem {
 interface SettingsSearchAvailabilityContext extends SettingsRuntimeContext {
   isMobile: boolean;
   isDesktopLocalOrigin: boolean;
+  // macOS desktop shell — for controls that only render on darwin (e.g. dock badge).
+  isMac: boolean;
 }
 
 const SETTINGS_SEARCH_ITEMS: readonly SettingsSearchItem[] = [
@@ -65,6 +67,16 @@ const SETTINGS_SEARCH_ITEMS: readonly SettingsSearchItem[] = [
     keywords: ['transparent', 'transparency', 'vibrancy', 'blur', 'macos', 'opaque'],
     isAvailable: (ctx) => ctx.isDesktopLocalOrigin,
   },
+  {
+    id: 'appearance.dock-badge',
+    page: 'appearance',
+    titleKey: 'settings.openchamber.visual.field.dockBadge',
+    descriptionKey: 'settings.openchamber.visual.field.dockBadgeHint',
+    keywords: ['dock', 'badge', 'unread', 'unseen', 'counter', 'count', 'notification', 'macos'],
+    // Exactly matches the render guard in OpenChamberVisualSettings: any darwin
+    // Electron shell (isMac already implies isDesktopShell), local or remote host.
+    isAvailable: (ctx) => ctx.isMac,
+  },
   {
     id: 'appearance.pwa-install-name',
     page: 'appearance',