feat(backup): add automatic weekly encrypted backups

A silent safety net against in-app data loss or DB corruption, with zero
friction. Driven off the app lifecycle (startup + when hidden), it writes a
sealed .tanemaki copy into the app's private storage once a week, keeping the
newest 3 (rotating). Reuses the manual backup's encryption via the extracted
ExportImportService.buildSealedBackup(); never throws, so a failed copy can't
crash startup or backgrounding. Not registered on web (no file storage).

Settings shows a quiet reassurance line with the date of the last automatic
copy (localised via intl), hidden where no automatic backup exists.

apps/app_seeds/test/ui/auto_backup_gate_test.dart

@@ -0,0 +1,86 @@
+import 'dart:typed_data';
+
+import 'package:flutter/material.dart';
+import 'package:flutter_test/flutter_test.dart';
+import 'package:tane/db/database.dart';
+import 'package:tane/security/secure_key_store.dart';
+import 'package:tane/services/auto_backup_service.dart';
+import 'package:tane/services/auto_backup_store.dart';
+import 'package:tane/services/export_import_service.dart';
+import 'package:tane/services/file_service.dart';
+import 'package:tane/ui/auto_backup_gate.dart';
+
+import '../support/test_support.dart';
+
+class _NoFiles implements FileService {
+  @override
+  Future<String?> saveFile({
+    required String suggestedName,
+    required Uint8List bytes,
+  }) async => null;
+
+  @override
+  Future<Uint8List?> pickFileBytes({List<String>? allowedExtensions}) async =>
+      null;
+}
+
+/// Counts how many times the lifecycle asked for a backup.
+class _SpyAutoBackup extends AutoBackupService {
+  _SpyAutoBackup(AppDatabase db)
+    : super(
+        exporter: ExportImportService(
+          repository: newTestRepository(db),
+          files: _NoFiles(),
+          keys: SecureKeyStore(store: InMemorySecretStore()),
+        ),
+        store: AutoBackupStore(InMemorySecretStore()),
+        directory: () async => throw UnimplementedError(),
+      );
+
+  int calls = 0;
+
+  @override
+  Future<bool> runIfDue({Duration interval = const Duration(days: 7)}) async {
+    calls++;
+    return false;
+  }
+}
+
+void main() {
+  late AppDatabase db;
+
+  setUp(() => db = newTestDatabase());
+  tearDown(() => db.close());
+
+  testWidgets('runs a backup on startup', (tester) async {
+    final spy = _SpyAutoBackup(db);
+    await tester.pumpWidget(
+      AutoBackupGate(service: spy, child: const SizedBox()),
+    );
+    await tester.pump(); // let the post-frame callback fire
+
+    expect(spy.calls, 1);
+  });
+
+  testWidgets('runs a backup again when the app is hidden', (tester) async {
+    final spy = _SpyAutoBackup(db);
+    await tester.pumpWidget(
+      AutoBackupGate(service: spy, child: const SizedBox()),
+    );
+    await tester.pump(); // startup backup
+
+    tester.binding.handleAppLifecycleStateChanged(AppLifecycleState.inactive);
+    tester.binding.handleAppLifecycleStateChanged(AppLifecycleState.hidden);
+
+    expect(spy.calls, 2);
+  });
+
+  testWidgets('does nothing when no service is provided', (tester) async {
+    await tester.pumpWidget(
+      const AutoBackupGate(child: Text('ok', textDirection: TextDirection.ltr)),
+    );
+    await tester.pump();
+
+    expect(find.text('ok'), findsOneWidget);
+  });
+}