fix(ui): preserve mobile home route and confirm Android exit

ui/flutter/lib/shared/navigation/app_exit_confirmation_controller.dart

@@ -1,3 +1,5 @@
+import 'dart:async';
+
 import 'package:flutter/services.dart';
 import 'package:flutter/widgets.dart';
 
@@ -7,18 +9,37 @@ import '../widgets/app_toast.dart';
 class AppExitConfirmationController {
   static const confirmationWindow = Duration(seconds: 2);
 
-  DateTime? _lastBackAt;
+  OverlayEntry? _toast;
+  Timer? _timer;
 
   Future<void> handleBack(BuildContext context) async {
-    final now = DateTime.now();
-    final lastBackAt = _lastBackAt;
-    if (lastBackAt != null && now.difference(lastBackAt) <= confirmationWindow) {
-      _lastBackAt = null;
+    if (_toast != null) {
+      dispose();
       await SystemNavigator.pop();
       return;
     }
 
-    _lastBackAt = now;
-    showAppToast(context, context.l10n.pressBackAgainToExit);
+    final message = context.l10n.pressBackAgainToExit;
+    final themes = InheritedTheme.capture(from: context, to: Overlay.of(context, rootOverlay: true).context);
+    _toast = OverlayEntry(
+      builder: (_) => IgnorePointer(
+        child: Center(
+          child: Padding(
+            padding: const EdgeInsets.all(24),
+            child: themes.wrap(AppToastContent(message: message)),
+          ),
+        ),
+      ),
+    );
+    Overlay.of(context, rootOverlay: true).insert(_toast!);
+    _timer = Timer(confirmationWindow, dispose);
+  }
+
+  void dispose() {
+    _timer?.cancel();
+    _timer = null;
+    _toast?.remove();
+    _toast?.dispose();
+    _toast = null;
   }
 }

ui/flutter/lib/shared/widgets/app_toast.dart

@@ -11,11 +11,6 @@ enum AppToastType { info, success, error }
 /// short messages stay compact while long messages wrap before becoming wider
 /// than the configured maximum. Errors and normal notices use the same layout.
 void showAppToast(BuildContext context, String message, {AppToastType type = AppToastType.info}) {
-  final palette = AppPalette.of(context);
-  final borderColor = switch (type) {
-    AppToastType.error => palette.error.withValues(alpha: 0.55),
-    AppToastType.success || AppToastType.info => palette.border,
-  };
   shad.showToast(
     context: context,
     location: shad.ToastLocation.topCenter,
@@ -23,16 +18,34 @@ void showAppToast(BuildContext context, String message, {AppToastType type = App
       alignment: Alignment.topCenter,
       widthFactor: 1,
       heightFactor: 1,
-      child: Container(
-        key: const ValueKey('app-toast-content'),
-        padding: const EdgeInsets.symmetric(horizontal: 14, vertical: 12),
-        decoration: BoxDecoration(
-          color: palette.cardBg,
-          borderRadius: BorderRadius.circular(8),
-          border: Border.all(color: borderColor),
-        ),
-        child: Text(message, style: TextStyle(color: palette.textPrimary, fontSize: 13, height: 1.2)),
-      ),
+      child: AppToastContent(message: message, type: type),
     ),
   );
 }
+
+/// Shared toast surface for edge notifications and centered exit prompts.
+class AppToastContent extends StatelessWidget {
+  const AppToastContent({super.key, required this.message, this.type = AppToastType.info});
+
+  final String message;
+  final AppToastType type;
+
+  @override
+  Widget build(BuildContext context) {
+    final palette = AppPalette.of(context);
+    final borderColor = switch (type) {
+      AppToastType.error => palette.error.withValues(alpha: 0.55),
+      AppToastType.success || AppToastType.info => palette.border,
+    };
+    return Container(
+      key: const ValueKey('app-toast-content'),
+      padding: const EdgeInsets.symmetric(horizontal: 14, vertical: 12),
+      decoration: BoxDecoration(
+        color: palette.cardBg,
+        borderRadius: BorderRadius.circular(8),
+        border: Border.all(color: borderColor),
+      ),
+      child: Text(message, style: TextStyle(color: palette.textPrimary, fontSize: 13, height: 1.2)),
+    );
+  }
+}