feat: add CalDAV synchronization with automatic sync

- Add CaldavService with tsdav/ical.js for CalDAV server communication
- Add CaldavController, CaldavRepository, and caldav routes
- Add client-side CaldavConfigService with sync(), config CRUD
- Add CalDAV settings UI with config load/save in settings screen
- Sync on login, auto-login (AuthGuard), periodic timer (calendar), and sync button
- Push single events to CalDAV on server-side create/update/delete
- Push all events to CalDAV after chat event confirmation
- Refactor ChatService to use EventService instead of direct EventRepository
- Rename CalDav/calDav to Caldav/caldav for consistent naming
- Add Radicale Docker setup for local CalDAV testing
- Update PlantUML diagrams and CLAUDE.md with CalDAV architecture

apps/client/src/components/AuthGuard.tsx

@@ -3,6 +3,7 @@ import { View, ActivityIndicator } from "react-native";
 import { Redirect } from "expo-router";
 import { useAuthStore } from "../stores";
 import { useThemeStore } from "../stores/ThemeStore";
+import { CaldavConfigService } from "../services/CaldavConfigService";
 
 type AuthGuardProps = {
   children: ReactNode;
@@ -20,7 +21,16 @@ export const AuthGuard = ({ children }: AuthGuardProps) => {
   const { isAuthenticated, isLoading, loadStoredUser } = useAuthStore();
 
   useEffect(() => {
-    loadStoredUser();
+    const init = async () => {
+      await loadStoredUser();
+      if (!useAuthStore.getState().isAuthenticated) return;
+      try {
+        await CaldavConfigService.sync();
+      } catch {
+        // No CalDAV config or sync failed — not critical
+      }
+    };
+    init();
   }, [loadStoredUser]);
 
   if (isLoading) {