feat: sync device timezone to backend on startup

On shell mount, reads the IANA timezone from the device via
flutter_timezone and POSTs it to PUT /api/settings as user_timezone.
Mirrors the web app's App.vue behaviour so briefing events and the
chat route use the correct local time on Android.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

lib/app.dart

@@ -2,6 +2,8 @@ import 'package:flutter/material.dart';
 import 'package:flutter_riverpod/flutter_riverpod.dart';
 import 'package:go_router/go_router.dart';
 
+import 'package:flutter_timezone/flutter_timezone.dart';
+
 import 'core/constants.dart';
 import 'core/exceptions.dart';
 import 'core/theme.dart';
@@ -159,19 +161,29 @@ class _ShellState extends ConsumerState<_Shell> {
   @override
   void initState() {
     super.initState();
-    // Silent update check — only if we haven't already checked this session.
-    // Skipping when status is not idle/error prevents the dialog from
-    // re-appearing every time the shell re-mounts (e.g. after visiting settings).
     WidgetsBinding.instance.addPostFrameCallback((_) {
+      // Silent update check — only if we haven't already checked this session.
       final repoUrl = ref.read(forgejoRepoUrlProvider);
-      if (repoUrl == null || repoUrl.isEmpty) return;
-      final status = ref.read(updateProvider).status;
-      if (status == UpdateStatus.idle || status == UpdateStatus.error) {
-        ref.read(updateProvider.notifier).check(repoUrl);
+      if (repoUrl != null && repoUrl.isNotEmpty) {
+        final status = ref.read(updateProvider).status;
+        if (status == UpdateStatus.idle || status == UpdateStatus.error) {
+          ref.read(updateProvider.notifier).check(repoUrl);
+        }
       }
+      // Sync device timezone to backend so briefing and chat use local time.
+      _syncTimezone();
     });
   }
 
+  Future<void> _syncTimezone() async {
+    try {
+      final tzInfo = await FlutterTimezone.getLocalTimezone();
+      await ref.read(settingsApiProvider).syncTimezone(tzInfo.identifier);
+    } catch (_) {
+      // Best-effort — failure is non-critical.
+    }
+  }
+
   int _tabIndex(String location) {
     for (var i = 0; i < _tabs.length; i++) {
       if (location.startsWith(_tabs[i])) return i;