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.

test/vault_service_test.dart

@@ -0,0 +1,81 @@
+// Proves the Phase-0 vault root recording: the path persists through
+// SharedPreferences, vaultRootValid() is true ONLY when a path is set AND its
+// directory exists, and clearVaultRoot() forgets it. Uses a real temp dir for
+// existence checks and a mock SharedPreferences for persistence.
+
+import 'dart:io';
+
+import 'package:flutter_test/flutter_test.dart';
+import 'package:shared_preferences/shared_preferences.dart';
+
+import 'package:badnote/services/vault_service.dart';
+
+void main() {
+  TestWidgetsFlutterBinding.ensureInitialized();
+
+  late Directory tempDir;
+
+  setUp(() async {
+    SharedPreferences.setMockInitialValues({});
+    tempDir = await Directory.systemTemp.createTemp('vault_service_test');
+  });
+
+  tearDown(() async {
+    if (await tempDir.exists()) {
+      await tempDir.delete(recursive: true);
+    }
+  });
+
+  Future<VaultService> makeService() async {
+    final prefs = await SharedPreferences.getInstance();
+    return VaultService.forTest(prefs);
+  }
+
+  test('starts with no vault root and is invalid', () async {
+    final vault = await makeService();
+    expect(vault.vaultRoot, isNull);
+    expect(await vault.vaultRootValid(), isFalse);
+  });
+
+  test('setVaultRoot persists the path', () async {
+    final vault = await makeService();
+    await vault.setVaultRoot(tempDir.path);
+    expect(vault.vaultRoot, tempDir.path);
+
+    // Re-read through a fresh service over the SAME prefs store: persisted.
+    final prefs = await SharedPreferences.getInstance();
+    expect(prefs.getString(VaultService.vaultRootKey), tempDir.path);
+    final reopened = VaultService.forTest(prefs);
+    expect(reopened.vaultRoot, tempDir.path);
+  });
+
+  test('vaultRootValid is true only when set AND the dir exists', () async {
+    final vault = await makeService();
+
+    // Set to an existing dir → valid.
+    await vault.setVaultRoot(tempDir.path);
+    expect(await vault.vaultRootValid(), isTrue);
+
+    // Point at a path that does not exist → invalid (folder went missing).
+    final missing = '${tempDir.path}${Platform.pathSeparator}gone';
+    await vault.setVaultRoot(missing);
+    expect(vault.vaultRoot, missing);
+    expect(await vault.vaultRootValid(), isFalse);
+  });
+
+  test('clearVaultRoot forgets the path and becomes invalid', () async {
+    final vault = await makeService();
+    await vault.setVaultRoot(tempDir.path);
+    expect(await vault.vaultRootValid(), isTrue);
+
+    await vault.clearVaultRoot();
+    expect(vault.vaultRoot, isNull);
+    expect(await vault.vaultRootValid(), isFalse);
+  });
+
+  test('empty stored path is treated as invalid', () async {
+    final vault = await makeService();
+    await vault.setVaultRoot('');
+    expect(await vault.vaultRootValid(), isFalse);
+  });
+}