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/hooks/useTraySync.ts

@@ -19,6 +19,7 @@ import { QUOTA_PROVIDERS, formatWindowLabel, formatQuotaValueLabel } from '@/lib
 import { useProjectsStore } from '@/stores/useProjectsStore';
 import { useSessionUIStore } from '@/sync/session-ui-store';
 import { useGitStore } from '@/stores/useGitStore';
+import { useUIStore } from '@/stores/useUIStore';
 import { resolveProjectForSessionDirectory, normalizeProjectPath } from '@/lib/projectResolution';
 import type { ProjectEntry } from '@/lib/api/types';
 import type { WorktreeMetadata } from '@/types/worktree';
@@ -80,6 +81,9 @@ type TraySnapshot = {
   // dropdown (same "configured to show" rule as the header/mobile). Empty
   // groups → the tray omits the Usage submenu entirely.
   usage: TrayUsage;
+  // Number of chats (root sessions) with unseen activity, for the macOS dock
+  // badge. 0 when the user disabled the badge — the main process clears it.
+  dockBadgeCount: number;
 };
 
 // focus-session / new-session are routed natively by the main process through
@@ -389,7 +393,26 @@ const buildSnapshot = (instanceName: string): TraySnapshot => {
 
   const approvals = live.approvals.map((a) => ({ ...a, sessionTitle: titleById.get(a.sessionId) || '' }));
 
-  return { sessions, approvals, instanceName, usage: buildUsage() };
+  // Dock badge: count chats (root sessions) with unseen activity over the FULL
+  // cross-project list — not the MAX_SESSIONS-capped `sessions` above — so the
+  // number is accurate even with many projects. A subtask's unseen rolls up to
+  // its root only when the user opted into subtask notifications, matching the
+  // sidebar's needs-attention rule.
+  const ui = useUIStore.getState();
+  let dockBadgeCount = 0;
+  if (ui.dockBadgeEnabled) {
+    for (const session of allSessions) {
+      if (!session?.id || session.parentID) continue; // roots only
+      let familyUnseen = notif.unseenCount[session.id] ?? 0;
+      if (familyUnseen === 0 && ui.notifyOnSubtasks) {
+        familyUnseen = collectDescendants(session.id)
+          .reduce((sum, id) => sum + (notif.unseenCount[id] ?? 0), 0);
+      }
+      if (familyUnseen > 0) dockBadgeCount += 1;
+    }
+  }
+
+  return { sessions, approvals, instanceName, usage: buildUsage(), dockBadgeCount };
 };
 
 export const useTraySync = (): void => {
@@ -491,6 +514,9 @@ export const useTraySync = (): void => {
     const unsubscribeProjects = useProjectsStore.subscribe(() => scheduleFlush());
     const unsubscribeWorktrees = useSessionUIStore.subscribe(() => scheduleFlush());
     const unsubscribeGit = useGitStore.subscribe(() => scheduleFlush());
+    // The dock-badge toggle and subtask-notification preference live here; a
+    // change must re-push the snapshot so the badge appears/clears immediately.
+    const unsubscribeUI = useUIStore.subscribe(() => scheduleFlush());
     // Cross-project status map: fed live by the sync dispatcher from the global
     // event stream, and seeded/reconciled by the poll below.
     const unsubscribeGlobalStatus = useGlobalSessionStatusStore.subscribe(() => scheduleFlush());
@@ -541,6 +567,7 @@ export const useTraySync = (): void => {
       unsubscribeProjects();
       unsubscribeWorktrees();
       unsubscribeGit();
+      unsubscribeUI();
       unsubscribeGlobalStatus();
       unsubscribeQuota();
       unsubscribeRegistry?.();