feat: Integrate Expo mobile application into monorepo (#16)

- Add mobile/ directory with Expo React Native project
- Create API client with JWT auth and token refresh using AsyncStorage
- Implement AuthContext for login/register/logout flow
- Add screens: Login, Register, Library, Search, Settings
- Set up React Navigation with AuthStack and MainTabs
- Create packages/shared/ with shared types and utilities
- Add shared validation utilities (email, password strength)
- Update root package.json workspaces to include mobile + shared
- Add spec document docs/backend/009-expo-integration.md

mobile/src/navigation/MainTabs.tsx

@@ -0,0 +1,55 @@
+import { type ReactNode } from "react";
+import { createBottomTabNavigator } from "@react-navigation/bottom-tabs";
+import { Text } from "react-native";
+import LibraryScreen from "../screens/LibraryScreen";
+import SearchScreen from "../screens/SearchScreen";
+import SettingsScreen from "../screens/SettingsScreen";
+
+export type MainTabParamList = {
+  Library: undefined;
+  Search: undefined;
+  Settings: undefined;
+};
+
+const Tab = createBottomTabNavigator<MainTabParamList>();
+
+function TabIcon({ label, focused }: { label: string; focused: boolean }) {
+  return (
+    <Text style={{ fontSize: 22, opacity: focused ? 1 : 0.5 }}>
+      {label === "Library" ? "📚" : label === "Search" ? "🔍" : "⚙️"}
+    </Text>
+  );
+}
+
+export default function MainTabs(): ReactNode {
+  return (
+    <Tab.Navigator
+      screenOptions={({ route }) => ({
+        tabBarIcon: ({ focused }: { focused: boolean }) => (
+          <TabIcon label={route.name} focused={focused} />
+        ),
+        tabBarActiveTintColor: "#4f8ef7",
+        tabBarInactiveTintColor: "#888",
+        headerStyle: { backgroundColor: "#1a1a2e" },
+        headerTintColor: "#fff",
+        tabBarStyle: { backgroundColor: "#1a1a2e", borderTopColor: "#333" },
+      })}
+    >
+      <Tab.Screen
+        name="Library"
+        component={LibraryScreen}
+        options={{ title: "My Library" }}
+      />
+      <Tab.Screen
+        name="Search"
+        component={SearchScreen}
+        options={{ title: "Search" }}
+      />
+      <Tab.Screen
+        name="Settings"
+        component={SettingsScreen}
+        options={{ title: "Settings" }}
+      />
+    </Tab.Navigator>
+  );
+}