feat(vault): pick a notebook vault folder on first run

Phase 0 of the file-based storage plan (docs/plans/
2026-06-24-file-based-storage.md). Foundation only — no editor or
DB change yet.

- VaultService (SharedPreferences): stores the vault root path,
  vaultRootValid() = path set AND directory exists.
- VaultSetupScreen: first-run folder picker (file_picker, Windows).
- main.dart gates HomeScreen behind a valid vault, re-prompting if
  the saved folder is gone.
- Settings: a Vault section to change the folder.

Editors still use SQLite; later phases move annotations into
per-file sidecars under the vault. analyze clean, tests green.

lib/main.dart

@@ -10,7 +10,9 @@ import 'editor/pdf/pen_capture_region.dart';
 import 'l10n/app_localizations.dart';
 import 'providers/settings_provider.dart';
 import 'screens/home_screen.dart';
+import 'screens/vault_setup_screen.dart';
 import 'services/database_service.dart';
+import 'services/vault_service.dart';
 
 Future<void> main() async {
   // Kind-aware binding (extends WidgetsFlutterBinding) must be the active
@@ -64,7 +66,7 @@ class BadNoteApp extends ConsumerWidget {
             GlobalCupertinoLocalizations.delegate,
           ],
           supportedLocales: AppLocalizations.supportedLocales,
-          home: const HomeScreen(),
+          home: const VaultGate(),
         );
       },
     );
@@ -78,3 +80,56 @@ class BadNoteApp extends ConsumerWidget {
         ),
       );
 }
+
+/// Startup gate: shows [HomeScreen] only once a valid vault root folder has been
+/// chosen. If none is set — or the saved folder no longer exists — it shows
+/// [VaultSetupScreen] first (re-prompting on a missing folder rather than
+/// silently scattering data elsewhere). Phase 0: records the vault path only;
+/// editors still use SQLite.
+class VaultGate extends StatefulWidget {
+  const VaultGate({super.key});
+
+  @override
+  State<VaultGate> createState() => _VaultGateState();
+}
+
+class _VaultGateState extends State<VaultGate> {
+  VaultService? _vault;
+  bool _valid = false;
+  bool _hadStoredPath = false;
+  bool _loading = true;
+
+  @override
+  void initState() {
+    super.initState();
+    _check();
+  }
+
+  Future<void> _check() async {
+    final vault = await VaultService.getInstance();
+    final valid = await vault.vaultRootValid();
+    if (!mounted) return;
+    setState(() {
+      _vault = vault;
+      _valid = valid;
+      // A stored-but-invalid path means the chosen folder went missing.
+      _hadStoredPath = (vault.vaultRoot?.isNotEmpty ?? false);
+      _loading = false;
+    });
+  }
+
+  @override
+  Widget build(BuildContext context) {
+    if (_loading) {
+      return const Scaffold(
+        body: Center(child: CircularProgressIndicator()),
+      );
+    }
+    if (_valid) return const HomeScreen();
+    return VaultSetupScreen(
+      vaultService: _vault!,
+      missing: _hadStoredPath,
+      onVaultReady: () => setState(() => _valid = true),
+    );
+  }
+}