Add background sync runner and rejected-sync UX

test/features/sync/sync_auto_trigger_controller_test.dart

@@ -0,0 +1,65 @@
+import 'dart:async';
+
+import 'package:flutter_test/flutter_test.dart';
+import 'package:relationship_saver/features/sync/sync_auto_trigger_controller.dart';
+import 'package:relationship_saver/features/sync/sync_coordinator.dart';
+
+void main() {
+  test('triggers sync once and respects cooldown interval', () async {
+    int callCount = 0;
+    DateTime now = DateTime(2026, 2, 15, 12);
+
+    final SyncAutoTriggerController controller = SyncAutoTriggerController(
+      syncNow: () async {
+        callCount += 1;
+        return _result(success: true);
+      },
+      now: () => now,
+      minimumInterval: const Duration(seconds: 60),
+    );
+
+    final bool first = await controller.trigger();
+    final bool second = await controller.trigger();
+    now = now.add(const Duration(seconds: 61));
+    final bool third = await controller.trigger();
+
+    expect(first, isTrue);
+    expect(second, isFalse);
+    expect(third, isTrue);
+    expect(callCount, 2);
+    expect(controller.lastTriggeredAt, now);
+  });
+
+  test('does not run a second sync while a run is still in progress', () async {
+    int callCount = 0;
+    final Completer<void> completer = Completer<void>();
+
+    final SyncAutoTriggerController controller = SyncAutoTriggerController(
+      syncNow: () async {
+        callCount += 1;
+        await completer.future;
+        return _result(success: true);
+      },
+      now: DateTime.now,
+      minimumInterval: Duration.zero,
+    );
+
+    final Future<bool> first = controller.trigger();
+    final bool second = await controller.trigger();
+    completer.complete();
+    final bool firstCompleted = await first;
+
+    expect(firstCompleted, isTrue);
+    expect(second, isFalse);
+    expect(callCount, 1);
+  });
+}
+
+SyncRunResult _result({required bool success}) {
+  return SyncRunResult(
+    success: success,
+    operation: 'sync',
+    message: success ? 'ok' : 'fail',
+    pendingAfter: 0,
+  );
+}