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/screens/settings_screen.dart

@@ -1,3 +1,4 @@
+import 'package:file_picker/file_picker.dart';
 import 'package:flutter/material.dart';
 import 'package:flutter_colorpicker/flutter_colorpicker.dart';
 import 'package:flutter_riverpod/flutter_riverpod.dart';
@@ -6,6 +7,7 @@ import '../l10n/app_localizations.dart';
 import '../models/pen_tool.dart';
 import '../models/pressure_curve.dart';
 import '../providers/settings_provider.dart';
+import '../services/vault_service.dart';
 import '../utils/stroke_stabilizer.dart';
 
 /// Material 3 settings screen for BadNote.
@@ -282,6 +284,12 @@ class SettingsScreen extends ConsumerWidget {
             ),
           ),
           const Divider(),
+          _SectionHeader(title: 'Vault', icon: Icons.folder_special),
+          const Padding(
+            padding: EdgeInsets.symmetric(horizontal: 16, vertical: 8),
+            child: _VaultSettings(),
+          ),
+          const Divider(),
           _SectionHeader(title: 'About', icon: Icons.info),
           Padding(
             padding: const EdgeInsets.symmetric(horizontal: 16, vertical: 8),
@@ -320,6 +328,93 @@ class SettingsScreen extends ConsumerWidget {
   }
 }
 
+/// Shows the current vault folder path and lets the user re-pick it. Re-uses
+/// the same `getDirectoryPath` flow as the first-run [VaultSetupScreen],
+/// persisting the choice through [VaultService.setVaultRoot].
+class _VaultSettings extends StatefulWidget {
+  const _VaultSettings();
+
+  @override
+  State<_VaultSettings> createState() => _VaultSettingsState();
+}
+
+class _VaultSettingsState extends State<_VaultSettings> {
+  VaultService? _vault;
+  String? _path;
+  bool _busy = false;
+
+  @override
+  void initState() {
+    super.initState();
+    _load();
+  }
+
+  Future<void> _load() async {
+    final vault = await VaultService.getInstance();
+    if (!mounted) return;
+    setState(() {
+      _vault = vault;
+      _path = vault.vaultRoot;
+    });
+  }
+
+  Future<void> _changeFolder() async {
+    final vault = _vault;
+    if (vault == null) return;
+    final l = AppLocalizations.of(context);
+    final messenger = ScaffoldMessenger.of(context);
+    setState(() => _busy = true);
+    try {
+      final path = await FilePicker.platform.getDirectoryPath(
+        dialogTitle: l.vaultSetupTitle,
+        lockParentWindow: true,
+      );
+      if (path != null) {
+        await vault.setVaultRoot(path);
+        if (!mounted) return;
+        setState(() => _path = path);
+        messenger.showSnackBar(SnackBar(content: Text(l.vaultUpdated)));
+      }
+    } catch (e) {
+      if (!mounted) return;
+      messenger.showSnackBar(
+        SnackBar(content: Text(l.vaultPickFailed(e.toString()))),
+      );
+    } finally {
+      if (mounted) setState(() => _busy = false);
+    }
+  }
+
+  @override
+  Widget build(BuildContext context) {
+    final l = AppLocalizations.of(context);
+    final colorScheme = Theme.of(context).colorScheme;
+    return Column(
+      crossAxisAlignment: CrossAxisAlignment.start,
+      children: [
+        Text(
+          l.vaultFolderLabel,
+          style: const TextStyle(fontWeight: FontWeight.w500),
+        ),
+        const SizedBox(height: 4),
+        Text(
+          (_path == null || _path!.isEmpty) ? l.vaultNoneSelected : _path!,
+          style: TextStyle(
+            fontSize: 13,
+            color: colorScheme.onSurfaceVariant,
+          ),
+        ),
+        const SizedBox(height: 12),
+        OutlinedButton.icon(
+          onPressed: _busy ? null : _changeFolder,
+          icon: const Icon(Icons.drive_folder_upload),
+          label: Text(l.vaultChangeFolder),
+        ),
+      ],
+    );
+  }
+}
+
 class _SectionHeader extends StatelessWidget {
   final String title;
   final IconData icon;